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Nuclear Design Methodology for Analyzing an Ultra High Temperature Highly Compact Ternary Carbide Reactor


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NUCLEAR DESIGN METHODOLOGY FOR ANALYZING ULTRA HIGH TEMPERATURE HIGHLY COMPAC T TERNARY CARBIDE REACTOR By REZA RAYMOND GOUW A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Reza Raymond Gouw

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ii ACKNOWLEDGMENTS This research would not be complete without the contributions of several people. I would like to thank Dr. Samim Anghaie fo r his support and encouragement throughout the entire project. His guidance and wisdom ha ve allowed me to complete this work in a timely fashion. I also would like to expre ss my thanks to Dr. Edward Dugan for his unfailing support throughout my undergraduate and graduate time at the University of Florida. His guidance both with the research and with my curriculu m was invaluable. I would like to thank Dr. Robert Little and the entire X-5 Data Team for their support and guidance during my stay at the Los Alamos National Laboratory. I would like to send special thanks to Dr. Morgan White for his s upport and guidance as a mentor as well as a friend. I also would like to thank Dr. Bob McFarlane of Los Alamos National Laboratory for his help and guidance in learning NJOY nuclear codes. I would like to thank Dr. Travis Knight for his help throughout this study. Last but not least, I would also like to thank Dr. Darryl Butt for agreeing to take th e time and effort to be on my supervisory committee. This study would not be complete without assistance from the Department of Energy in the form of DOE Nuclear Engi neering/Health Physics Fellowship. Furthermore, the Los Alamos National Laborat ory provided its facili ty and its staff to help the completion of this study. I would also like to express my thanks to the faculty, my colleagues and friends in the Nuclear and Radiological Engineering department. Specifi cally, I would like to thank Beth Bruce for her help in the academic re gistration. Their support, suggestions and

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iii encouragement helped me to complete this re search. Finally, I woul d like to thank my wife, Virginia Pangalila, for her support a nd sacrifices during th e completion of this study.

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iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS..................................................................................................ii LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 Background on Nuclear Thermal Rockets....................................................................1 Nuclear vs. Chemical Propulsion..........................................................................1 The KIWI and Tory Programs...............................................................................2 Rover and NERVA................................................................................................3 Background on Square-Lattice Honeycomb System....................................................6 Background on an Evaluated Nuclear Data File...........................................................7 2 CROSS-SECTION PROCESSING..............................................................................9 Cross-Sections Processing Codes.................................................................................9 NJOY Nuclear Data Processing System................................................................9 TRANSX 2001 a Code for Interfacing MATXS Cross-Section Libraries to Nuclear Transport Codes.................................................................................10 Generating PENDF library.........................................................................................11 RECONR Module...............................................................................................11 BROADR Module...............................................................................................12 UNRESR Module................................................................................................13 HEATR Module..................................................................................................14 THERMR Module...............................................................................................14 Automation of Processes.....................................................................................16 Generating Cross-Sections for Monte Carlo Code.....................................................16 PURR Module.....................................................................................................16 GASPR Module...................................................................................................17 ACER Module.....................................................................................................17 Automation of Processes.....................................................................................18 Generating Cross-Sections for Deterministic Code....................................................18 GROUPR Module...............................................................................................19 DTFR Module.....................................................................................................19

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v MATXSR Module...............................................................................................20 TRANSX Calculation..........................................................................................20 3 NUCLEAR TRANSPORT CODES DESCRIPTIONS..............................................21 MCNP Version 5........................................................................................................21 PARTISN Version 3.56..............................................................................................22 4 SQUARE-LATTICE HONEYCOMB (SLHC) NUCLEAR ROCKET ENGINE DESCRIPTION..........................................................................................................24 Geometry Description.................................................................................................24 MCNP5 Geometry Representation of SLHC......................................................25 PARTISN 3.51 Geometry Representation of SLHC...........................................27 Materials Description..................................................................................................27 Materials in Heterogeneous SLHC......................................................................27 Materials in Homogeneous SLHC.......................................................................28 5 METHODOLOGY.....................................................................................................49 Monte Carlo Neutron Cross-S ections Library Generation.........................................49 Choosing the Correct Weight Function......................................................................51 Godiva Calculations............................................................................................51 Square-Lattice Honeycomb Calculations............................................................53 Neutronics Analysis....................................................................................................54 Energy Spectrums Characterization....................................................................54 Power Distributions and Flux Profiles Analyses.................................................55 Temperature Coefficient Analyses......................................................................57 Control Drums Analyses.....................................................................................58 Water Submersion Accident Analysis.................................................................60 6 CONCLUSIONS........................................................................................................99 APPENDIX A LISTS OF NUCLIDES PROCESSED IN THE RGOUW CROSS-SECTION LIBRARIES..............................................................................................................101 B ENERGY GROUP STRUCTURES FOR SQUARE-LATTICE HONEYCOMB..106 C AUTOMATION PROCESS FOR PENDF ACER, AND DTFR MODULES........109 D DETAILED CALCULATION OF NUMBER DENSITIES IN FUEL REGIONS OF SQUARE-LATTICE HONEYCOMB......................................................................125

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vi E DETAILED CALCULATION OF THE LOCATIONS OF EACH RING MATERIALS IN THE SQUARE-LA TTICE HONEYCOMB HOMOGENEOUS MODEL....................................................................................................................129 LIST OF REFERENCES.................................................................................................134 BIOGRAPHICAL SKETCH...........................................................................................135

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vii LIST OF TABLES Table page 4-1. Comparison calculations of the tru e Square-Lattice Honeycomb (SLHC) heterogeneous model and SLHC hete rogeneous model with fuel region homogenization........................................................................................................32 4-2. Number densities of the is otopes in the fuel region.................................................37 4-3. Properties of non-fuel elements in th e Square-Lattice Honeycomb heterogeneous model........................................................................................................................38 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous model........................................................................................................................39 4-5. Properties of the Square-Lattice Honeycomb reactor..............................................42 4-6. Properties of non-fuel materials in the Square-Lattice H oneycomb homogeneous model........................................................................................................................43 4-7. The properties of axial region 1 fuel materials in the Square-Lattice Honeycomb homogeneous model.................................................................................................44 4-8. The properties of axial region 2 fuel materials in the Square-Lattice Honeycomb homogeneous model.................................................................................................45 4-9. The properties of axial region 3 fuel materials in the Square-Lattice Honeycomb homogeneous model.................................................................................................46 4-10. The properties of axial region 4 fuel ma terials in the Square-Lattice Honeycomb homogeneous model.................................................................................................47 4-11. The properties of axial region 5 fuel materials in the Square-Lattice Honeycomb homogeneous model.................................................................................................48 5-1. Comparison of PARTISN calculations u tilizing correct and incorrect multigroup neutron cross-sections for f our surrounding shells material s with the radius of 235U Godiva is 6.7 cm...............................................................................................66 5-2. Comparison of PARTISN calculations utilizing true and false multigroup neutron cross-sections for th e Square-Lattice Honeycomb......................................67

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viii 5-3. The intermediate temperature conditions and the operating temperature conditions for the Square-Lattice Honeycomb..........................................................................68 A-1. List of essential isotopes in the RGOUW cross-section libraries...........................102 A-2. List of thermal scattering data in the RGOUW cross-section libraries..................105 B-1. The 45-energy group structures..............................................................................107 B-2. The LANL-187 energy group structures.................................................................108

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ix LIST OF FIGURES Figure page 4-1. Square-Lattice Honeycomb nuclear reactor geometry description..........................29 4-2. The Square-Lattice Honeycomb fuel wa fers fabrication into fuel element.............30 4-3. The fabrication of the Square-Lattice Honeycomb fuel elements into fuel assembly. .................................................................................................................31 4-4. Energy spectra of tru e Square-Lattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region homogenization.............33 4-5. Energy spectra of tru e Square-Lattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model w ith fuel region homogenization in the thermal energy range (less than 1eV).......................................................................34 4-6. Geometry description of the Square -Lattice Honeycomb heterogeneous model.....35 4-7. Here is the geometry description of the Square-Lattice Honeycomb homogeneous model........................................................................................................................36 5-1. Energy spectrum comparison betwee n the SLHC heterogeneous and SLHC homogeneous models at 293.6 K.............................................................................61 5-2. Energy spectrum comparison betwee n the SLHC heterogeneous and SLHC homogeneous models in ther mal energy range at 293.6 K......................................62 5-3. Energy spectrum comparison between hete rogeneous and homogeneous models of the Square-Lattice Honeycomb in fast energy range at 293.6 K.........................63 5-4. Geometry description of the G odiva sphere surrounded by hydrogen gas..............64 5-5. Energy spectrum comparison between he terogeneous model of the Square-Lattice Honeycomb and 235U Godiva surrounded by H2 gas at 293.6 K obtained from MCNP.......................................................................................................................65 5-6. This figure shows the systems energy spectrum of the Square-Lattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures..............69

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x 5-7. This figure presents the fuel region s energy spectrum of the Square-Lattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures.............................................................................................................70 5-8. This plot presents the moderator regi ons energy spectrum of the Square-Lattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures.............................................................................................................71 5-9. This plot shows the systems energy spectrum of the Square-Lattice Honeycomb Homogeneous model at room, intermediate, and operating temperatures...............72 5-10. The plot presents the tota l cross-section data of seve ral important isotopes in the Square-Lattice Honeycomb......................................................................................73 5-11. The plot presents the axial power dist ribution of the Square-Lattice Honeycomb in the first and second radial fuel regions.....................................................................74 5-12. The figure presents the radial power di stribution of the Square-Lattice Honeycomb in the second, third and f ourth axial fuel regions.....................................................75 5-13. The figure shows the radial power distri bution of the Square-Lattice Honeycomb in the first and fifth axial fuel regions..........................................................................76 5-14. The plot illustrates the fast neutron energy (> 65 keV) axial flux profiles of the Square-Lattice Honeycomb in the two radial fuel regions.......................................77 5-15. The epithermal neutron energy (2.5 eV 65 keV) axial flux prof iles of the SquareLattice Honeycomb in the two radial fuel regions...................................................78 5-16. The figure illustrates the thermal neutr on energy (< 2.5 eV) axial flux profiles of the Square-Lattice Honeycomb in the two radial fuel regions.................................79 5-17. The figure presents the fast neutron ener gy (> 65 keV) radial flux profiles of the Square-Lattice Honeycomb in th e five axial fuel regions........................................80 5-18. The epithermal neutron energy (2.5 eV 65 keV) radial flux profiles of the SquareLattice Honeycomb in the five axial fuel regions....................................................81 5-19. The figure presents the thermal neutron en ergy (< 2.5 eV) radial flux profiles of the Square-Lattice Honeycomb in th e five axial fuel regions........................................82 5-20. The plot shows the plot of fuel temp erature coefficient of the Square-Lattice Honeycomb during startup.......................................................................................83

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xi 5-21. The plot of fuel temperature coeffici ent of the Square-Lattice Honeycomb at the intermediate temperature ranges..............................................................................84 5-22. The plot of fuel temperature coeffici ent of the Square-Lattice Honeycomb at the operating temperature ranges...................................................................................85 5-23. The plot of moderator temperature coe fficient of the Square-Lattice Honeycomb at the operating temperature ranges.............................................................................86 5-24. The plot presents the system temper ature coefficient of the Square-Lattice Honeyc omb. .............................................................................................................87 5-25. The plot illustrates the thermal scatteri ng cross section of 1H and Zr in zirconium hydride......................................................................................................................88 5-26. Plot of the 1H neutr on absorption cross section.......................................................89 5-27. Critical configurations of the contro l drums in the Square-Lattice Honeycomb at 293.6 K.....................................................................................................................90 5-28. The fully-in configurations of th e control drums in the Square-Lattice Honeycomb, which has k-e ff of 0.89858 + 0.00005 at 293.6 K..............................91 5-29. The fully-out configurations of th e control drums in the Square-Lattice Honeycomb, which has k-e ff of 1.05961 + 0.00006 at 293.6 K..............................92 5-30. The reactivity worth plot of the control drums from fully-in to fully-out positions at 293.6 K..................................................................................................93 5-31. Three different configura tions and their k-eff values of two control drums jammed at the fully-out position.........................................................................................94 5-32. Three different configura tions and their k-eff values with three control drums jammed at the fully-out position...........................................................................95 5-33. Three different configura tions and their k-eff values with four control drums jammed at the fully-out position...........................................................................96 5-34. Configuration of the Square-Lattice Honeycomb r eactor for water submersion accident.....................................................................................................................97 5-35. Modified configuration of the Squa re-Lattice Honeycomb reactor for water submersion accident.................................................................................................98

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xii E-1. The locations of fuel ring material in the Square-Lattice Honeycomb homogeneous model......................................................................................................................129 E-2. The locations of fuel material in th e Square-Lattice Honeycomb heterogeneous model......................................................................................................................130 E-3. Locations of graphite coating in th e Square-Lattice Honeycomb homogeneous model......................................................................................................................132

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xiii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NUCLEAR DESIGN METHODOLOGY FOR ANALYZING ULTRA HIGH TEMPERATURE HIGHLY COMPAC T TERNARY CARBIDE REACTOR By Reza Raymond Gouw December 2006 Chair: Samim Anghaie Major: Nuclear Engineering Sciences Recent studies at the Innovative Nuclear Space Power and Propulsion Institute (INSPI) have demonstrated the feasibility of fa bricating solid solutions of ternary carbide fuels such as (U,Zr,Nb)C, (U,Zr,Ta)C, (U,Z r,Hf)C and (U,Zr,W)C. The necessity for accurate nuclear design analysis of these te rnary carbides in highly compact nuclear space systems prompted the development of nuclear design methodology for analyzing these systems. This study will present the improvement made in the high temperature nuclear cross-sections. It will show the re lation between Monte Carlo and Deterministic calculations. It will prove the significant role of the energy spectrum in the multigroup nuclear cross-sections generation in the high ly-thermalized-nuclear system. The nuclear design methodology will address several issu es in the homogenization of a nuclear system, such as energy spectrum comparison between a heterogeneous system and homogeneous system. It will also addre ss several key points in the continuous and multigroup nuclear cross-sections generation. The study will present the methodology of

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xiv selecting broad energy group structures. Fi nally, a comparison betw een the Monte Carlo and Deterministic methods will be performe d for the Square-Lattice Honeycomb Nuclear Space Reactor. In the comparison calculations, it will include the system characterization calculations, such as energy spectrum compar ison, 2-D power distributions, temperature coefficient analysis, and water submersion accident analysis.

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1 CHAPTER 1 INTRODUCTION Background on Nuclear Thermal Rockets Nuclear propulsion has been around for ove r 40 years. In 1955 the Atomic Energy Commission (AEC) initiated research and deve lopment work, and it was directed toward the utilization of a nuclear thermal rocket (N TR) concept to propel si ngle-stage, ballistic missiles over intercontinental trajectories. At the same time, the Los Alamos Scientific Laboratory (LASL), now Los Alamos Nationa l Laboratory and the Lawrence Livermore Laboratories, had begun explor atory propulsion research pr ograms. At Los Alamos Scientific Laboratory the program was known as the KIWI program, and the Tory program was at the Lawrence Livermore Laborat ories. However, in 1970 the work came to a halt because of the lack of interest and funds (Gunn, 2001). Nuclear vs. Chemical Propulsion All liquid rocket propulsion systems rely on the creation of a continuous supply of high-pressure, high-temperature gas, and the expansion of that gas through a suitable supersonic nozzle to the systems low-pressure environment. In conventional, chemical system, the combustion process defines th e temperature and the average molecular weight of the propellants, thereby determini ng the performance (specific impulse) of the propulsion system. A nuclear thermal rocket, on the other hand, depends on a temperature source of thermal energy, typically a high-temperature, solid-core reactor, to heat a single

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2 propellant to as high a temper ature as possible, and then to expand the resulting hightemperature gas to the exhaust pressure. In NTR engine, th e propellant is pumped from its supply tank to the reactor-nozzle subsystem, where it is first used to cool the nozzle and the reactor pressure shell, the neutron re flector, and the core s upport structure. Next, the propellant passes through th e reactor core where most of the fission energy being released by uranium atoms is absorbed by the propellant. Finally, the super-heated propellant is expanded out of the supersonic nozzle. The KIWI and Tory Programs The choice of systems propellant was infl uenced by concerns over the potential for chemical interactions between the fuel el ements and the propellant, as well as by the availability of suitable feed systems equipmen t to deliver the propellant. The choice of propellant in the KIWI reactors was ammonia, and nitrogen was selected for the Tory reactors. Graphite was chosen for the re quisite fuel elements because of its hightemperature capabilities and its ability to serve both as a container of uranium atoms and as an effective high-energy, fissi on-produced neutron moderator. In 1957, the AEC reviewed the two nuclear rocket reactor programs and finally chose to focus their resources on the KIWI r eactor, part of the Rover program. LASL, in turn, reviewed its development plans and d ecided to switch the propellant from ammonia to hydrogen gas. A model liquid hydrogen pump developed by Rocketdyne Division of North American Aviation initiated the propellant switch because it was capable of delivering 38 liters per minute at 105 atm. Initially, the KIWI test program was concentrated on relatively modest power-density reactor. The power of KIWI A, A, and A3 were rated at 100 MW. The three KIWI r eactors were tested at the nuclear rocket development site in Nevada in July 1959 a nd in July and October 1960. The results of

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3 these tests met their objectives and clearly demonstrated that coated graphite-fuel elements could be used to heat hydrog en to temperature in excess of 2000 K. With the successful test results, LASL focu sed its attention on the development of the KIWI-B reactor series. These reactors generated heat in excess of 1,000 MW. A total of five KIWI-B reactors were tested at the Nevada test site between December 7, 1961, and September 10, 1964. The final and comple te verification of the capabilities of the KIWI reactor occurred on July 28, 1964, when the KIWI B-4E reactor was operated at full power and temperature for 8 minutes. Then, on September 10, 1964, the same reactor was restarted and operated at full power and temperature for an additional 2.5 minutes. The test was judged as a complete success. The formation of NASA in July 1958 produ ced a major change in the thrust and subsequent scope of the USAs nuclear rocket program. Possible interest areas shifted from the Air Forces missile application to manned lunar missions and later to planetary missions. A single point was designated the Office for Space Nuclear Propulsion (SNPO), which assumed the responsibility for the Rover program. This office was staffed and jointly financed by the AEC and NASA (Pelaccio and El-Genk, 1994). Rover and NERVA The first major program decision made by the SNPO was the creation of the Nuclear Engine for Rocket Vehicle App lication (NERVA) program in 1961. NERVA focused on utilizing and integrating the KIWI B reactor design into a flight-packaged nuclear rocket engine. Following the KIWI B-4E full-power test, LASL turned its attention to the experimental development of more powerful reactors. Accordingly, LASL adopted a new Rover program goal of a 5,000 MW propulsion reactor, thereby establishing the Phoebus r eactor program (Robbins, 1991).

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4 Phoebus LASL used the proven KIWI B-4E fuel el ements as a core building block for the Phoebus 1A and 1B. Phoebus 1A and 1B also incorporated improved fuel elements to facilitate a 50% up rating in test power, chamber pressure, and equivalent thrust. Phoebus 1A and 1B were tested on June 25, 1965, and February 23, 1967. Phoebus 1A was operated at 245 kN of thrust for 10.5 mi nutes, while Phoebus 1B was operated at 325 kN of thrust for 30 minutes. The Phoebus 1B test result proved to be a significant milestone because it demonstrated a practical power, thrust, and pe rformance level that correspond with the view of the projected NTR propulsion require ments for planetary exploration. The Phoebus 2A reactor, test ed for 12 minutes on June 26, 1968, achieved a power level of 4,200 MW, which is significantly greater than any commercial nuclear power plant operating today (Gunn, 2001). Advanced fuel elements Attention was then directed towards th e development of more advanced fuel elements capable of higher operating temper atures and longer opera ting life. To reduce the lead time and the expense associated with high-power reactor tests, LASL decided to build a much smaller reactor, designated Pe wee1, to evaluate such advanced fuel elements. It was designed to generate 500 MW of power and reach a hydrogen temperature of 2300 K. Pewee 1 was tested at full power and the average core-exit gas temperatures of 2300 K were held for 20 minutes in December, 1968. The final phase of advanced fuel element development was directed towards an evaluation of composite (a mixture of gra phite and carbides) fuel elements and allcarbides fuel elements. The composite offe red the potential of roughly 2700 K core-exit temperature, while the all-carbides fuel elements gave promise of core-exit gas

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5 temperature of around 3200 K. As a further step in reducing the power level (and cost) of the test reactor, and in uti lizing a test system that would provide for complete containment and subsequent disposal of th e radioactive fission products, LASL designed and constructed a 50-MW test reactor designated the nuclear furnace (NF-1). Four high-power NF-1 tests were conduc ted between June 29 and July 27, 1972. The power level developed was 54 MW, the core-exit gas temperature was 2400 K and the total run duration was 108 minutes. Postmo rtem examination of the fuel elements indicated that the composite fuel elements were generally in good condition, and that any subsequent test could be ta rgeted for a core-exit gas te mperature between 2600 K and 2800 K. With regard to the all-carbide fuel elements there was extensive cracking, but this was as expected because of their brittle ness. However, the fuel elements condition had no meaningful effect on the ability to heat the hydrogen gas to its measured exit temperature (Gunn, 2001). Testing NERVA By the spring of 1966 the NERVA reactor development program had completed all but two of its planned reactor tests, and wa s in the process of conducting the first NRXEngine System Test (NRX-EST) series. The pr imary objectives of this test series were to: (1) demonstrate the feasibility of a hot gas bleed-turbine drive cycle; (2) demonstrate the capability of boot-strap st art-ups; and (3) map the engi ne operating envelope over a wide range of design flow and chamber pr essures. These test objectives were accomplished in a series of three low-power tests conducted in March 1966. A total of 24 minutes at full power (110 MW) was accumulated, and resulted in a clear demonstration of the feasibility of nuclear rocket engines.

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6 Next, the NRX A-5 reactor was tested in June 1966 with back-to-back full power runs of 15 minutes each. The r eactor used fuel elements very similar to those used in Phoebus 1A; postmortem examination of thes e fuel elements revealed a generally satisfactory condition. The fi nal NERVA reactor test (NRX A-6) was conducted on December 13, 1967. The test operations were a complete successthe reactor was brought to the targeted 2300 K core-exit gas temp erature, and then held at that operating condition for 62 minutes. The stage was now set for the final, flight-packaged NERVA engine system (XE-Prime) tests. The XE-Prime engine featured a close-c oupled, flight-type conf iguration, but was designed for ground-test development. For the XE -Prime tests, a new engine test facility, designated engine test facility 1 (ETF-1), was constructed. The test stand provided for vertical downward firing of the engine in a simulated flight-stage structure. Nuclear powered tests were initiated on March 20, 1969, and were concluded on September 11, 1969. The objectives of the powered test se ries included investigation of start-up characteristics under different control modes, determination of engine and component performance parameters and investigations of engine shutdown and pulse-cooling characteristics. During this test period, 24 start-ups were accomplished, as well as a fullpower test at 1,100 MW and a core-ex it gas temperature of 2300 K (Gunn, 2001). Background on Square-Lattice Honeycomb System The technical accomplishments of the Rover and NERVA programs were remarkable. In a period of some 15 years, a totally new concept for rocket propulsion was developed to the point where the experimental development had been accomplished. There is now a renewed interest in examini ng nuclear propulsion in the context of Mars and other planetary exploration. From a t echnical and programmatic point of view it

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7 would seem that the nuclear thermal rocket engine could offer significant space transfer advantages to mission planners. For this reason, the Square-Lattice Honeycomb (SLHC) Space Nuclear Rocket Engine is being expl ored. The Square-Lattice Honeycomb (SLHC) Space Nuclear Rocket Engine is a NERVA De rivative Reactor core with a new nuclear fuel design. It is an attemp t to reduce the weight of th e nuclear rocket engine and simplify the core design while increasing the thrust level. Background on an Evaluated Nuclear Data File To perform the analysis of SLHC, nuclear cross section lib raries must be created. These nuclear cross-sections libraries will consist of high temperature nuclear cross sections. These nuclear cross-sections librari es are generated from a standard data file consisting of nuclear data. This nuclear data file is known as an Evaluated Nuclear Data File or ENDF. The ENDF system was develo ped to provide a unified format that could be used to store and retrieve evaluated sets of neutron cross-sections It was designed to allow easy exchange of cross-section inform ation between various national laboratories. The initial system contained format speci fications for neutron cross-sections and other related nuclear constants. During the la ter stages of development, the formats were expanded to include photon in teraction cross-sections, photon production data (photon produced by neutron interacti ons) and nuclear structure data The basic data formats developed for the library are versatile enough to allow accurate description of the crosssections considered for a wide range of incident neutron energies (10-5 eV to 20 MeV). The ENDF formats are flexible in the sense th at almost any type of neutron interaction mechanism can be accurately described. Th ey are restrictive in that only a limited number of different representations ar e allowed for any given neutron reaction mechanism.

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8 There are two different types of evaluate d data libraries are maintained at the National Nuclear Data Section Center (NNDC). The ENDF/A library contains either complete or partial data sets (partial in the sense that the data set may be, for example, an evaluation of the fissi on cross-section for 235U in the energy range 100 keV to 15 MeV only). This library may also contain several di fferent evaluations of the cross-sections for a particular nuclide. Th e ENDF/B library, on the other hand, contains only one evaluation of the cross-sections for each materi al in the library, but each material contains cross-sections for all signifi cant reactions. The data set selected for the ENDF/B library is the set recommended by the Cross Secti on Evaluation Working Group (CSEWG). The ENDF/B library contains reference data se ts with which other information may be compared, as opposed to data sets that are revised often on the basis of new information so as to constitute current standard data sets. ENDF/B is primarily intended as the main input to a cross-section pr ocessing program (Kinsey, 1975).

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9 CHAPTER 2 CROSS-SECTION PROCESSING Cross-Sections Processing Codes A high temperature neutron cross-sections li brary needs to be generated before the Square-Lattice Honeycomb analysis can begi n. These neutron cross-sections will provide information needed by the computer co des to perform the neutronics analysis of SLHC. As discussed in the pr evious chapter, ENDF/B contai ns evaluated cross-sections data sets in a form that can be used in vari ous neutronics calculati ons. However, if the existing neutronics-codes require data librari es that are quite diffe rent from the ENDF library, a code or series of code have been wr itten that read the ENDF library as input and generate a secondary cross-sections librar y. One of these codes is known as the NJOY Nuclear Data Processing System. NJOY Nuclear Data Processing System The NJOY Nuclear Data Processing System is a comprehensive computer code package for producing pointwise and multigr oup nuclear cross-sections and related quantities from evaluated nuclear in the E NDF format. The NJOY code purpose is to take the basic data from the nuclear data libr ary and convert it into the forms needed for applications. The NJOY code consists of a set of modules, each performing a welldefined-processing task (M acFarlane and Muir, 1994). Several considerations were made for ge nerating the nuclear cross-section library used in the neutronics analysis. One of considerations is the type of a librarya

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10 pointwise or a multigroup nuclear cross-sec tion library. The point wise nuclear crosssection library is used in the Monte Carlo code, and the multigroup nuclear cross-section library is used in the deterministic code. This report required pr oducing both a pointwise and a multigroup cross-section library. The nuc lides and the number of temperatures to be processed need to be considered. There are 10 different temperatures in the nuclear cross-section library generated fo r this report. The list of nuc lides in the library is shown in Appendix A. The other consideration was th e type of nuclear data to be included in the libraryneutron data only, photon data on ly, or neutron and photon data. For this analysis, the libraries generated will only consist of neutron data. For multigroup nuclear cross-sections, the number of broad and fine energy groups also needs to be specified. The multigroup cr oss-sections libraries have one fine energygroup structure and one broad energy-group stru cture. These energy-group structures are shown in Appendix B. A weight function is needed to generate this multigroup nuclear cross-sections library. A wei ght function is the neutron ener gy spectrum of the system or generalized-system weight functions. TRANSX 2001 a Code for Interfacing MATX S Cross-Section Libraries to Nuclear Transport Codes Discrete-ordinates (SN) transport codes, which solve the Boltzmann equation for the distribution of neutrons and photons in nuc lear systems, have reached high level of development. The early one-dimensional code s are very widely used. The development of effective acceleration methods as well as increasing computer speed and capacity has made detailed transport calculations more economical; as a result, codes such as PARTISN are seeing increasing use. The DI F-3D diffusion code a nd Monte-Carlo codes with multigroup capability like MCNP are also used frequently.

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11 However, many of the users of transport codes have the same complaint: it is hard to get good, up-to-date, documented cross-sect ion data and prepare them for input into these codes. TRANSX works together with a generalized crosssection library called MATXS (for material cross-sec tion library) to give the tran sport code user easier access to appropriate nuclear data and some capabi lities difficult or impossible to get with any other system. The code can be used to construct data for fusion reactors, fast fission reactors, thermal fission reactors, and sh ielding problems. Its main weakness is in computing resonance effects in thermal reactors. TRAN SX was originally developed in the late seventies to handle cross sections for fissi on, fusion, and shielding applications at Los Alamos National Laboratory. In the early ei ghties, extensions to handle heterogeneous self-shielding problems for fast re actors were added (MacFarlane, 1992). Generating PENDF Library There is a common processing path fo r producing a pointwise and a multigroup library. This processing path produces an intermediate library f ile that is commonly called pointwise-ENDF (PENDF). In this process, the NJOY code utilized the RECONR, BROADR, UNRESR, HEAT R, and THERMR modules. RECONR Module The NJOY processing sequences start with RECONR, which serves two roles. First, it goes through all the re actions included on the ENDF library and chooses a union grid that allows all cross sections to be represented using linear interpolation to a specified accuracy. This step remove s any nonlinear interpolation ranges ( e.g., log-log, linear-log). It also makes it possible for a ll summation reactions to be reconstructed as the sum of their parts ( e.g., total, total inelastic, total fission). Second, for resonance

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12 materials, it reconstructs th e resonance cross sections (ela stic, fission, capture) on a union grid that allows them all to be represented within certain accuracy criteria, and then combines the resonance data with the othe r linearized and unioni zed cross sections. RECONR has the following features: Efficient use of dynamic storage allocation and a special stack st ructure allow large problems to be run on small machines The unionized grid improves the accuracy, usefulness, and ENDF/B compatibility of the output. All summation cross sec tions are preserved on the union grid. Approximate and Doppler broadening functions may be used in some cases to speed up reconstructions for narrow-resonance materials. A resonance-integral criterion is added to the normal linearization criterion in order to reduce the number of points added to th e tabulation to represent unimportant resonances. All ENDF-6 formats are handled except Generalized R-Matrix parameters, energydependent scattering radius and the calculation of a ngular distributions from resonance parameters. BROADR Module This module generates Doppler-broadened cr oss sections. The input cross-sections can be from RECONR or a previous BROADR run. The me thod utilized in BROADR is based on Cullens SIGMA1 method. The method is often called kernel broadening because it uses a detailed integration of the integral equation defining the effective cross section. BROADR has the following features: An alternate calculation is used for low ener gies and high temperat ures that corrects a numerical problem of the original SIGMA1 Variable dimensioning is used, which allows the code to be r un on large or small machines with full use of whatever storage is made available All low-threshold reactions are broadened in parallel on a union grid. This makes the code run faster than the original SIGMA1

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13 The union grid is constructed adaptively to give a linearized re presentation of the broadened cross section with tolerances c onsistent with those used in RECONR. Energy points may be added to or removed fr om the input grid as required for the best possible representation. Binary input and output can be used. This roughly halves the time required for a typical run on some computer systems, and it allows the full accuracy of the machine to be used. The summation cross sections total, non-elasti c, and sometimes fission or (n, 2n) are reconstructed to equa l the sum of their parts. UNRESR Module This module is used to produce effective se lf-shielded cross sections for resonance reactions in the unresolved energy range. In ENDF-format evaluations, the unresolved range begins at energy where it is difficult to measure individual resonances and extends to energy where the effect of fluctuations in the resonance cross-sections becomes unimportant for practical calculations. The resonance information for this energy range is given as average values for resonance wi dths and spacings togeth er with distribution functions for the widths and spacings. This representation can be converted into effective cross-sections suitable for codes that use the background cross-section method, often called the Bondarenko method, using a met hod originally developed for the MC2 code and extended for the ETOX code. UNRESR has the following features: Flux-weighted cross sections are produced for the total, elastic, fission, and capture cross sections, including competition with inelastic scattering A current-weighted total cro ss section is produced for cal culating the effective selfshielded transport cross section Up to 10 values of temperature and 10 values of 0 are allowed The energy grid used is consistent with the grid used by RECONR The computed effective cross sections are written on the PENDF tape in a specially defined section (MF2, MT152) for use by other modules

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14 The accurate quadrature scheme from the MC2-2 code is used for computing averages over the ENDF stat istical distribution functions. HEATR Module This module generates pointwise heat production cross sectio ns and radiation damage energy production for specified reacti ons and adds them to an existing PENDF library. Heating is an important parameter of any nuclear system. It may represent the product being soldas in a pow er reactoror it may affect the design of peripheral systems such as shields, and structural com ponents. Nuclear heati ng can be divided into neutron heating and photon heating. The neutr on heating at given lo cation is proportional to the local neutron flux and arises from th e kinetic energy of the charged products of a neutron-induced reaction. Similarly, the photon heating is proportional to the flux of secondary photons transported fr om the site of previous neut ron reactions. It is also traceable to the kinetic energy of char ged particles. HEATR has the following advantages: Heating and damage are computed in a consistent way All ENDF/B neutron and photon data are used ENDF-6 charged-particle distribut ions are used when available Kinematics checks are available to improve future evaluations Both energy-balance and kinematics KE RMA (Kinetic Energy Releasing in Materials) factors can be produced. THERMR Module At thermal energies, that is up to about 0.5 eV for temperatures around room temperature and maybe up to as high as 4 eV for hotter materials, the energy transferred by the scattering of a neutron is similar to the kinetic energies of motion of the atoms in liquids and to the energies of excitations in molecules and crystalline lattices. Therefore,

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15 you cannot picture the target atoms as being init ially stationary and recoiling freely as is normally done for higher neutron energies. The motion of the target atoms and their binding in liquids and solids affects both cro ss sections and the dist ribution in energy and angle of the scattered neutrons. The TH ERMR module of NJOY is used to compute these effects. For free-gas scattering, where only the therma l motion of the targets is taken into account, not internal modes of excitation, THER MR can generate the cross sections and scattering distributions using analytic formul as. For real bound scattering, it uses an input scattering function and other parameters from an ENDF-format thermal evaluation in File 7 format. THERMR has the following features: The energy grid for coherent elastic scat tering is produced adaptively so as to represent the cross section between the sh arp Bragg edges to a specified tolerance using linear interpolation The secondary energy grid for inelastic in coherent scattering is produced adaptively so as to represent all structure with linear interpolation Incoherent cross sections are computed by integrating the incoherent matrix for consistency Free-atom incoherent scattering is norma lized to the Doppler broadened elastic scattering cross section in order to pr ovide an approximate representation of resonance scattering to preserve the correct total cross section Discrete angle representations are used to avoid the limitations of Legendre expansions Hard-to-find parameters for the ENDF /B-III evaluations are included in the THERMR code for the users convenience ENDF-6 format files can be processed. This gives the evaluator more control over the final results, because all parameters n eeded to compute the cross sections are contained in the file.

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16 Automation of Processes For processing a large number of nuclides, it is nice to automate the process. To automate the generation of the PENDF library, shell-scripts were uti lized. These scripts automatically obtained the appropriate ENDF library from the High Performance Storage System (HPSS), and then create an NJOY i nput file to perform a calculation in the RECONR, BROADR, UNRESR, HEATR, and TH ERMR modules, and finally store the PENDF library back into HPSS. Example of these scripts can be found in Appendix C. Generating Cross-Sections for Monte Carlo Code Once the PENDF library of all nuclides need ed for the analysis is produce, codespecific cross-sections libraries are generate d. The Monte Carlo code that was used to perform the analysis in this report was M onte Carlo N-Particle (MCNP) version 5. A detailed description and uses of the MCNP5 code is discussed in the next chapter of this report. The ACER module of the NJOY code can produce a library to be used by the MCNP5 code. However, additional calculatio ns must be performed before the ACER calculation. PURR Module The unresolved self-shielding data gene rated by UNRESR is suitable for use in multigroup methods after processing by the GROUPR Module, discussed later in this section, but the so-called Bondarenko method is not very useful for continuous-energy Monte Carlo codes like MCNP5. The natural approach for treating unresolved-resonance self-shielding for Monte Carlo codes is th e Probability Table method. This module produces probability tables to trea t unresolved-resonance self-shielding.

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17 GASPR Module In many practical applica tions, it is important to know the total production of protons (hydrogen), alphas (helium), and othe r light charged particle s resulting from the neutron flux. Therefore, it is convenient to have a set of special "gas production" or "charged-particle production" cros s-sections that can be used in application codes. The ENDF format provides a set of MT numb ers for these quantities, but only a few evaluators have added them to their files: MT=203 -total proton production MT=204 -total deuteron production MT=205 -total triton production MT=206 -total He-3 production MT=207 -total alpha production The GASPR module goes through all of the reactions given in an ENDF-format evaluation, determines which charged partic les would be produced by the reaction, and it adds up the particle yield times the reacti on cross-section to produce the desired gas production cross-sections. They are then ava ilable for plotting, multigroup averaging, or reformatting for the MCNP code. ACER Module After the calculations of the two prev ious modules were completed, the ACER module was utilized to create a MCNP nuclear cross-section library. The ACER module prepares libraries in ACE format (A Co mpact ENDF) for the MCNP continuous-energy neutron-photon Monte Carlo code MCNP requires that all th e cross-sections be given on a single union energy grid suitable for linear interpolation. Although the energy grid and cross-section data on an NJOY PENDF libra ry are basically consistent with the

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18 requirements of MCNP, there is still one problem. Many ENDF evaluations produce energy grids with very large numbers of poi nts. The ACER modul e provides thinning algorithms to control the accuracy loss and balance it against the computer memory requirements. The ACE library files come in three different types in order to allow for efficiency, portability, and history. Type 1 is a simple formatted file suitable for exchanging ACE libraries between different computers Type 2 is a FORTRAN-77 dire ct-access binary file for efficient use during actual MCNP runs Type 3 is a word-addressable direct-access binary file. It uses nonstandard read and write call, and it is normally used only at Los Alamos National Laboratory. Type 3 is only used for the fast library not for dosimetry, thermal, or photoatomic data. Automation of Processes Like the generation of the PENDF librar y, the generation of the MCNP5 library utilizes shell-scripts. Th ese scripts automatically obtai n the appropriate ENDF and PENDF library from the High Performance Stor age System (HPSS), and then create the NJOY input file to perform a calculation in the PURR, GASPR, and ACER modules, and finally store the ACER library back into HPSS. Example of these scripts can be found in Appendix D. Generating Cross-Sections for Deterministic Code This report also performed analysis using a deterministic code. The deterministic code used to perform the analysis was PA RTISN version 3.51. A detailed description and uses of PARTISN version 3.51 is discussed in the next chapter of this report. The MATXSR module in NJOY was used to gene rate a multigroup nuclear cross-section library. Additional calculations need to be performed to prepare the library for MATXSR

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19 module. Finally, an additional cross-section processing code was needed to organize the MATXSR multigroup library into a working cr oss-section library for PARTISN. The TRANSX code version 2001 performed these ta sks. The other path of organizing the multigroup cross-section is through the DTFR module. GROUPR Module This module produces self-shielded multig roup cross sections, anisotropic groupto-group scattering matrices, and anisotropic photon production matrices for neutrons from ENDF/B-IV, V, or VI evaluated nucle ar data. The Bondarenko narrow-resonance weighting scheme is usually used. Ne utron data and photonproduction data are processed in a parallel manne r using the same weight f unction and quadrature scheme. Two-body scattering is computed with a cen ter-of-mass (CM) Gaussian quadrature, which gives accurate results even for sma ll Legendre components of the group-to-group matrix. Output is written to an output groupwise-ENDF (GENDF) file for further processing by a formatting m odule (DTFR, CCCCR, MATXSR). DTFR Module This module is used to prepare libraries fo r discrete-ordinates transport codes that accept the DTF format. Transport tables in DTF format are organized to mirror the structure of data inside a di screte-ordinates transport code. These codes start with the highest energy group and work downward. The ba sic table consists of the three standard edits, namely, particle balance absorption, fission neutron producti on cross section, and total cross section. These st andard edits are followed by the group-to-group scattering cross sections.

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20 MATXSR Module This module formats the GENDF tape in to a generalized CCCC-type interface format for neutron, photon, and charged-partic le data, including cro ss sections, group-togroup matrices, temperature variations, se lf-shielding, and time-dependence. MATXS libraries can be used with the TRANSX code to produce effective cross sections for a wide variety of application codes, such as PARTISN. TRANSX Calculation As discussed earlier, the MATXS modul e in NJOY produced a generalized CCCCtype interface format for neutron, photon, a nd charged-particle data. However, this format is not very useful in the PARTISN code. Additional processing is needed to create the PARTISN library. This is where the TRANSX code comes into play. The TRANSX code performed additi onal cross-section processing, such as homogenization of materials, self-shielding, and Dancoff corre ction. TRANSX also organized the crosssections into a library format useful for the PARTISN code.

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21 CHAPTER 3 NUCLEAR TRANSPORT CO DES DESCRIPTIONS This chapter discusses the nuclear transpor t codes are used to perform the analysis of SLHC. MCNP Version 5 MCNP is a general-purpose Monte Carlo NParticle code that can be used for neutron, photon, electron, or coupled neutron/photon/electr on transport, including the capability to calculate eigenvalu es for critical systems. Pointwise cross-section data are used. For neutrons, all reactions given in a particular cross-section evaluation (such as ENDF/B-VI) are accounted for. Thermal neut rons are described by both the free gas and S( ) models. For photons, the code takes account of incoherent and coherent scattering, the possibility of fluorescent emission after photoe lectric absorption, absorption in pair production with local emission of annihila tion radiation, and bremsstrahlung. A continuous slowing down mode l is used for electron transport that includes positrons, k x-rays, and bremsstrahlung but does not include external or selfinduced fields. Important standard features that make MCNP very versatile and easy to use include a powerful general source, cr iticality source, and surface source; both geometry and output tally plotter; a rich collection of variance reduction techniques; a flexible tally structure; and extensive collections of cross-section data (Hendricks, 1997).

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22 PARTISN Version 3.56 The PARTISN and DANTSYS code package ar e essentially the same code with the difference that PARTISN is a more modern in the sense that it us es Fortran 90 language and it also support parallel processors e nvironment. The PARTISN code package includes the following transport code s: ONEDANT, TWODANT, TWODANT/GQ, TWOHEX, AND THREEDANT. This code p ackage is a modular computer program package designed to solve time-independent, multigroup discrete ordinates form of the Boltzmann transport equation in several differe nt geometries. The modular construction of the package separates the input processing, the transport equation solving, and the post processing (or edit) functions into distinct code modules: the Input Module, one or more Solver Modules, and the Edit Module, respec tively. The Input and Edit Modules are very general in nature and are common to all the Solver Modules. The ONEDANT Solver Module contains a one-dimensional (slab, cy linder, and sphere), time-independent transport equation solver using the st andard diamond-differencing method for space/angle discretization. Al so included in the package are Solver Modules named TWODANT, TWODANT/GQ, THREEDAN T, AND TWOHEX. The TWODANT Solver Module solves the time-independent two dimensional transport equation using the diamond-differencing method for space/angle discretization. An adaptive weighed diamond differencing (AWDD) method for spa tial and angular discretization is also introduced in TWODANT as an option. The TWOHEX Solver Module solves the timeindependent two-dimensional transport equati on on a equilateral triangle spatial mesh. The THREEDANT Solver Module solves th e time-independent, three-dimensional transport equation for XYZ and RZ symmetries using both diamond differencing with

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23 set-to-zero fixup and the AWDD method. The TWODANT/GQ Solver Module solves the two-dimensional time-independent tran sport equation in XY and RZ symmetries using a spatial mesh of arbitrary quadrilatera ls. The spatial differe ncing method is based upon the diamond differencing method with se t-to-zero fix up with changes to accommodate the generalized special meshing (Alcouffe et al., 2002).

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24 CHAPTER 4 SQUARE-LATTICE HONEYCOMB (SLHC) NUCLEAR ROCKET ENGINE DESCRIPTION This chapter discusses the general de scription of Square -Lattice Honeycomb Nuclear Rocket Engine. It describes the ge ometry as well as the materials in SLHC. Geometry Description The Square-Lattice Honeycomb nuclear reac tor geometry descri ption is shown in Figure 4-1. The overall diameter and height of the SLHC reactor core are 31.0 cm and 45.0 cm, respectively. Beryllium reflectors in the radial and top ax ial directions surround the reactor. The thickness of the radial beryllium reflector is 20.0 cm, and the thickness of the axial beryllium reflector is 20.0 cm. The core is fueled with a solid solution of 93% enriched (U,Zr,Nb)C, which is one of several ternary uranium car bides that are under consideration for this concept. The fuel is to be fabricated as 1 mm grooved (U,Zr,Nb)C wafers. The fuel wafers are used to form square-lattice honeycomb fuel elements, containing 30% cross-sectional flow area, shown in Figure 4-2 (Furman, 1999). The fu el elements dimensions are 4.0 cm in diameter and 1.5 cm in height. The fuel s ub assembly consists of six fuel elements stacked axially, shown in Figure 4-3 (Furman, 1999). Each fuel sub assembly has a 0.5cm thick graphite coating and a 0.5-cm thick zirconium oxide coating. Five fuel sub assemblies are stacked axially to form one fuel assembly. Finally, the assemblies are then arranged in the circ ular pattern inside a zirconium hydride matrix.

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25 The control system of the SLHC is in the form of control drums inside the radial reflector region. The control drums have an inner diameter, an outer diameter and a height of 13.6 cm, 18.0 cm, and 45.0 cm, resp ectively. The materials of the control drums will be discussed in the next section of this chapter. MCNP5 Geometry Representation of SLHC The SLHC is modeled as a heterogeneous model and a homogenized representation of the heterogeneous model. Heterogeneous representation of SLHC The SLHC heterogeneous model models th e regions of the SLHC exactly except for the fuel regions, which they are homoge nized. The reason for the homogenization in the fuel is for saving computation time. The homogenization in the fu el region is selected to simply the geometry description in the problem. The fuel re gions are homogenized using straight-forward hom ogenization method. This met hod utilizes the straight conversion of heterogeneous number densit y into homogeneous number density through the use of volume fractions. Table 4-1 pres ents the result of calculation of the true SLHC heterogeneous model and the heteroge neous model with homogenization in fuel region. Figure 4-4 presents the fuel re gion energy spectra comparison of true heterogeneous SLHC model and SLHC he terogeneous model with fuel region homogenization. Figure 4-5 presents the fuel region en ergy spectra of true SLHC heterogeneous model and SLHC heterogeneous model with fuel region homogenization in the thermal energy range (less than 1eV). Based on the result in Table 4-1, Figure 4-4 and Figure 4-5, the SLHC heterogeneous m odel with fuel region homogenization is a valid representation of the true SLHC hete rogeneous model because in the both models show similar results in k-eff and the energy spectrum. However, the time to obtain the

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26 similar statistical error on the results is al most 3 times less in the SLHC heterogeneous model with fuel homogenization. Utilizing 128 parallel-processors, it takes about 85000 minutes of computation time to complete th e analysis of the true SLHC heterogeneous model. However, the analysis can be co mpleted in only 30000 minutes of computation time if the fuel homogenization is utilized. The SLHC heterogeneous model is shown in Figure 4-6. The SLHC heterogeneous m odel is used to generate the SLHC heterogeneous models energy spectrum. The SLHC heterogeneous model is utilized for finding the minimum numbers of control drum s required to shutdown the Square-Lattice Honeycomb reactor. Finally, the water s ubmersion accident analysis is also being performed using the SLHC heterogeneous model. Homogeneous representation of SLHC The homogenized model of the SLHC in MCNP will be used for a comparison between the Monte Carlo method and the de terministic method. Figure 4-7 shows the geometry description of the SLHC. The contro l system is modeled as an absorber ring instead of control drums. In the homogenized model, each of the SLHC regions is represented as a ring of materials. The important constraint when developing this homogeneous model is that mass is conserved; therefore area is conserved. The locations of each ring material are based on the center location of the fuel materials. Please refer to Appendix E for the detailed calculation of the locations of the fuel rings in the SquareLattice Honeycomb homogeneous model. With the utilization of rings of materials, the homogenization is unlike th e conventional homogenization method. This method rearranges the regions in the SLHC into rings of materials which have the same composition as the heterogeneous model. This method eliminates the necessity of

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27 defining a unit cell representation. This me thod also eliminates the requirement for calculating the homogenized number densities. PARTISN 3.51 Geometry Representation of SLHC In PARTISN, the geometry of the SLHC has to be represented as a homogenized model. The homogenized model of the SLHC in PARTISN is exactly the same as the homogenized model in MCNP because the resu lts from the PARTISN calculation and the MCNP calculation will be compared. Materials Description The SLHC nuclear reactor is fueled by a (U,Zr,Nb)C fuel. The propellant of the reactor is hydrogen gas. The reflector materi al is beryllium, and the moderator material is zirconium hydride. Materials in the Heterogeneous SLHC In the SLHC heterogeneous model, the fuel region consists of a homogenization of (U,Zr,Nb)C and hydrogen gas. The fuel regi on has 70% solid volume fraction. The core is divided into five axial temperature regions Each axial temperature region has height of 9 cm. For each temperature region, the de nsity of the (U,Zr,Nb)C is varied according to Table 4-2. Please refer to Appendix D for detailed calculation of the number densities in the SLHC heterogeneous model. A 0.5-cm thick graphite coating follow ed by a 0.5-cm zirconium oxide coating surrounds the fuel region. The purpose of these coatings is to act as an insulator between the hot zone (fuel region) and the moderato r region. As described earlier, the fuel assemblies were placed in the circular patt ern inside the zirconium hydride moderator. Finally, axial and radial beryl lium reflectors enclose the reacto r core. The control system of the SLHC is placed inside the radial beryllium reflector in the form of control drums.

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28 These control drums consist of neutron-abso rber and neutron-reflecting material. The neutron-absorber material is boron carbide, and the beryllium is the neutron-reflecting material. Table 4-3 presents properties of all the non-fuel elements in the SLHC heterogeneous model and includes the densi ties, isotopes, volume, and mass of each region. Table 4-4 presents the properties of all fuel elements in the SLHC heterogeneous model and includes the densities, isotopes, volume, and mass of each region. Finally, Table 4-5 presents some properties of the Square-Lattice Honeycomb reactor. Materials in the Homogeneous SLHC Table 4-6 presents properties for all the non-fuel regions in the homogenized model of the SLHC with their isotopes, volume fraction, heterogeneous number density, and homogeneous number density. Both the heterogeneous and homogeneous number densities are presented in these tables to illu strate that there is no need to calculate the homogeneous number densities for the ring me thod utilization. Table 4-7 presents the first axial region fuel materials in the SLHC homogeneous model and includes the number density, isotopes, and volume fractions of each region. Table 4-8 presents the second axial region fuel materials in th e SLHC homogeneous model and includes the number density, isotopes, and volume fractions of each region. Table 4-9 presents the thir d axial region fuel material s in the SLHC homogeneous model and includes the number density, isot opes, and volume fractions of each region. Table 4-10 presents the fourth axial region fuel materials in the SLHC homogeneous model and includes the number density, isot opes, and volume fractions of each region. Table 4-11 presents the fifth axial region fuel materials in the SLHC homogeneous model and includes the number density, isotopes, and volume fractions of each region.

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29 65 cm 71 cm Figure 4-1. Square-Lattice Honeycomb nuc lear reactor geometry description.

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30 Figure 4-2. The Square-Lattice Honeycomb fuel wafers fabrication into fuel element.

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31 9 cm 4 cm 1.5 cm 9 cm 4 cm 9 cm 4 cm 4 cm 4 cm 1.5 cm Figure 4-3. The fabrication of the Square-L attice Honeycomb fuel elements into fuel assembly.

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32 Table 4-1. Comparison calculations of the true Square-Lattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region homogenization. Properties True SLHC Heterogeneous Model SLHC Heterogeneous Model with fuel region homogenization % Difference k-eff 0.99913 + 0.00005 0.99997 + 0.00005 0.08 Number of cycles 25000 20000 -20 Computation time (minutes) 85000 30000 -65 Real time (hours) 34 12 -65

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33 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalized Flux SLHC Heterogeneous with fuel homogenization "True" SLHC Heterogeneous Figure 4-4. Energy spectra of true Square-Lattice Honeyc omb (SLHC) heterogeneous model and SLHC heterogeneous mode l with fuel region homogenization.

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34 0.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05 1.00E+06 1.20E+06 1.40E+061.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01Energy (eV)Normalized Flux SLHC Heterogeneous with fuel homogenization "True" SLHC Heterogeneous Figure 4-5. Energy spectra of true Square-Lattice Honeyc omb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region homogenization in the thermal energy range (less than 1eV).

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35 31 cm 71 cm Outer radius of fuel = 2.00 cm Outer radius of graphite coating = 2.50 cm Outer radius of zirconium oxide coating = 3.00 cm Inner radius of center hole = 2.25 cm Outer radius of center hole = 2.50 cm Inner radius of control drum = 6.80 cm Outer radius of control drum = 9.00 cm 31 cm 71 cm Outer radius of fuel = 2.00 cm Outer radius of graphite coating = 2.50 cm Outer radius of zirconium oxide coating = 3.00 cm Inner radius of center hole = 2.25 cm Outer radius of center hole = 2.50 cm Inner radius of control drum = 6.80 cm Outer radius of control drum = 9.00 cm Figure 4-6. Geometry description of the Square-Lattice Honeycomb heterogeneous model.

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36 Hydrogen hole (Outer radius = 2.25 cm) Zirconium tube (Outer radius = 2.50 cm) First zirconium hydride (Outer radius = 4.16 cm) First zirconium oxide coating (Outer radius = 4.81 cm) First graphite coating (Outer radius = 5.35 cm) First fuel region (Outer radius = 7.25 cm) Second graphite coating (Outer radius = 7.79 cm) Second zirconium oxide coating (Outer radius = 8.44 cm) Second zirconium hydride (Outer radius = 9.95 cm) Third zirconium coating (Outer radius = 10.63 cm) Third graphite coating (Outer radius = 11.18 cm) Second fuel region (Outer radius = 13.16 cm) Fourth graphite coating (Outer radius = 13.71 cm) Fourth zirconium oxide coating (Outer radius = 14.39 cm) Third zirconium hydride (Outer radius = 15.50 cm) First beryllium reflector (Outer radius = 16.25 cm) First beryllium and hydrogen gas (Outer radius = 29.20 cm) Boron carbide (Outer radius = 30.93 cm) Second beryllium and hydrogen gas (Outer radius = 34.75 cm) Second beryllium reflector (Outer radius = 35.50 cm) Figure 4-7. Here is the geometry descri ption of the Square-Lattice Honeycomb homogeneous model.

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37 Table 4-2. Number densities of th e isotopes in the fuel region. Axial Region Temperature ( K ) Uranium density ( g/cm3 ) 1 600 0.7 2 1000 0.9 3 1200 1.2 4 2000 1.2 5 2500 1.2

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38 Table 4-3. Properties of non-fuel elemen ts in the Square-Lattice Honeycomb heterogeneous model. Region Isotope Number Density (Atoms/bcm) Volume (cm3) Density (g/cm3) Mass (g) Hydrogen hole 1H 8.6000E-03 7.1569E+02 1.4393E-02 1.0301E+01 Top Hydrogen hole 1H 1.7200E-02 3.7937E+03 2.8787E-02 1.0921E+02 Zirconium tube 90Zr 2.2114E-02 1.6788E+ 02 6.5110E+00 1.0931E+03 91Zr 4.8226E-03 92Zr 7.3715E-03 94Zr 7.4703E-03 96Zr 1.2035E-03 90Zr 2.2231E-02 9.7264E+ 03 6.6900E+00 6.5069E+04 Zirconium Hydride Region 91Zr 4.8481E-03 92Zr 7.4104E-03 94Zr 7.5097E-03 96Zr 1.2099E-03 1H 8.6418E-02 Bottom Graphite Coating 12C 1.4139E-01 2.2619E+02 2.8200E+00 6.3787E+02 Bottom ZrO2 Coating 90Zr 1.4810E-02 2.2619E+ 02 5.8900E+00 1.3323E+03 91Zr 3.2297E-03 92Zr 4.9367E-03 94Zr 5.0029E-03 96Zr 8.0600E-04 16O 5.7571E-02 6 Inner Cylinder of Control Drums 9Be 1.2362E-01 3.9222E+ 04 1.8500E+00 7.2561E+04 6 Outer Cylinder of Control Drums (Beryllium) 9Be 1.2362E-01 1.4742E+ 04 1.8500E+00 2.7273E+04 10B 9.1506E-02 1.4742E+04 2.5100E+01 3.7003E+04 6 Outer Cylinder of Control Drums (B4C) 11B 2.2876E-02 12C 2.8596E-02 Beryllium reflector 9Be 1.2362E-01 8.6205E+ 04 1.8500E+00 1.5948E+05 9Be 1.1991E-01 6.6468E+ 04 1.7962E+00 1.1618+05 Beryllium reflector and Hydrogen gas 1H 1.0320E-03

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39 Table 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous model. Region Isotope Number Density (Atoms/bcm) Volume (cm3) Density (g/cm3) Mass (g) 1H 1.0320E-02 2.0358E+03 5.0873E+00 1.0356E+04 18 First Axial Fuel Region 12C 2.7880E-02 90Zr 1.0482E-02 91Zr 2.2858E-03 92Zr 3.4939E-03 94Zr 3.5407E-03 96Zr 5.7043E-04 93Nb 6.7908E-03 235U 6.6658E-04 238U 5.0173E-05 18 First Axial Graphite Coating 12C 1.4139E-01 1.1451E+03 2.8200E+00 3.2292E+03 90Zr 1.4810E-02 1.3996E+ 03 5.8900E+00 8.2435E+03 18 First Axial ZrO2 Coating 91Zr 3.2297E-03 92Zr 4.9367E-03 94Zr 5.0029E-03 96Zr 8.0600E-04 16O 5.7571E-02 1H 1.0320E-02 2.0358E+03 5.1577E+00 1.0500E+04 18 Second Axial Fuel Region 12C 2.7805E-02 90Zr 1.0315E-02 91Zr 2.2494E-03 92Zr 3.4382E-03 94Zr 3.4843E-03 96Zr 5.6134E-04 93Nb 6.6826E-03 235U 9.9987E-04 238U 7.5259E-05 18 Second Axial Graphite Coating 12C 1.4139E-01 1.1451E+03 2.8200E+00 3.2292E+03 90Zr 1.4810E-02 1.3996E+ 03 5.8900E+00 8.2435E+03 18 Second Axial ZrO2 Coating 91Zr 3.2297E-03 92Zr 4.9367E-03 94Zr 5.0029E-03 96Zr 8.0600E-04 16O 5.7571E-02

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40 Table 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous model. (continued) Region Isotope Number Density (Atoms/bcm) Volume (cm3) Density (g/cm3) Mass (g) 1H 1.0320E-02 2.0358E+03 5.2644E+00 1.07174E+04 18 Third Axial Fuel Region 12C 2.7656E-02 90Zr 9.9802E-03 91Zr 2.1764E-03 92Zr 3.3267E-03 94Zr 3.3714E-03 96Zr 5.4314E-04 93Nb 6.4660E-03 235U 1.6665E-03 238U 1.2543E-04 18 Third Axial Graphite Coating 12C 1.4139E-01 1.1451E+03 2.8200E+00 3.2292E+03 90Zr 1.4810E-02 1.3996E+ 03 5.8900E+00 8.2435E+03 18 Third Axial ZrO2 Coating 91Zr 3.2297E-03 92Zr 4.9367E-03 94Zr 5.0029E-03 96Zr 8.0600E-04 16O 5.7571E-02 1H 1.0320E-02 2.0358E+03 5.2633E+00 1.0715E+04 18 Fourth Axial Fuel Region 12C 2.7656E-02 90Zr 9.9802E-03 91Zr 2.1764E-03 92Zr 3.3267E-03 94Zr 3.3714E-03 96Zr 5.4314E-04 93Nb 6.4660E-03 235U 1.6665E-03 238U 1.2543E-04 18 Fourth Axial Graphite Coating 12C 1.4139E-01 1.1451E+03 2.8200E+00 3.2292E+03 90Zr 1.4810E-02 1.3996E+ 03 5.8900E+00 8.2435E+03 18 Fourth Axial ZrO2 Coating 91Zr 3.2297E-03 92Zr 4.9367E-03 94Zr 5.0029E-03 96Zr 8.0600E-04 16O 5.7571E-02

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41 Table 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous model. (continued) Region Isotope Number Density (Atoms/bcm) Volume (cm3) Density (g/cm3) Mass (g) 1H 1.0320E-02 2.0358E+03 5.2622E+00 1.0712E+04 18 Fifth Axial Fuel Region 12C 2.7656E-02 90Zr 9.9802E-03 91Zr 2.1764E-03 92Zr 3.3267E-03 94Zr 3.3714E-03 96Zr 5.4314E-04 93Nb 6.4660E-03 235U 1.6665E-03 238U 1.2543E-04 18 Fifth Axial Graphite Coating 12C 1.4139E-01 1.1451E+03 2.8200E+00 3.2292E+03 90Zr 1.4810E-02 1.3996E+ 03 5.8900E+00 8.2435E+03 18 Fifth Axial ZrO2 Coating 91Zr 3.2297E-03 92Zr 4.9367E-03 94Zr 5.0029E-03 96Zr 8.0600E-04 16O 5.7571E-02

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42 Table 4-5. Properties of the Square-Lattice Honeycomb reactor. Properties Value Reactor diameter (cm) 31.0 Reactor height (cm) 45.0 Radial reflector thickness (cm) 20.0 Top axial reflector thickness (cm) 20.0 Fuel element height (cm) 9.0 Fuel type Solid solution of (U,Zr,Nb)C Fuel enrichment (%) 93 Uranium densities (g/cm3) 0.7 1.2 Uranium mass (g) 10600 Reflector material Beryllium Absorber material Boron Carbide

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43 Table 4-6. Properties of non-fuel materi als in the Square-Lattice Honeycomb homogeneous model. Region Isotope Volume Fraction Heterogeneous Number Density (atoms/bcm) Homogeneous Number Density (atoms/bcm) 1 1H 1.0000 8.6000E-03 8.6000E-03 2 90Zr 1.0000 2.2114E-02 2.2114E-02 91Zr 1.0000 4.8226E-03 4.8226E-03 92Zr 1.0000 7.3715E-03 7.3715E-03 94Zr 1.0000 7.4703E-03 7.4703E-03 96Zr 1.0000 1.2035E-03 1.2035E-03 13 90Zr 1.0000 8.6418E-02 8.6418E-02 91Zr 1.0000 2.2231E-02 2.2231E-02 92Zr 1.0000 4.8481E-03 4.8481E-03 94Zr 1.0000 7.4104E-03 7.4104E-03 96Zr 1.0000 7.5097E-03 7.5097E-03 1H 1.0000 1.2099E-03 1.2099E-03 14 12C 1.0000 1.4139E-01 1.4139E-01 15 90Zr 1.0000 5.7571E-02 5.7571E-02 91Zr 1.0000 1.4810E-02 1.4810E-02 92Zr 1.0000 3.2297E-03 3.2297E-03 94Zr 1.0000 4.9367E-03 4.9367E-03 96Zr 1.0000 5.0029E-03 5.0029E-03 16O 1.0000 8.0600E-04 8.0600E-04 16 9Be 1.0000 1.2362E-01 1.2362E-01 17 9Be 0.9835 1.1991E-01 1.1991E-01 1H 0.0165 5.1600E-04 5.1600E-04 18 10B 1.0000 9.1506E-02 9.1506E-02 11B 1.0000 2.2876E-02 2.2876E-02 12C 1.0000 2.8596E-02 2.8596E-02

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44 Table 4-7. The properties of axial region 1 fuel materials in the Square-Lattice Honeycomb homogeneous model. Region Isotope Volume Fraction Heterogeneous Number Density (atoms/bcm) Homogeneous Number Density (atoms/bcm) 3 1-1 235U 0.3111 1.1665E-03 1.1665E-03 238U 0.3111 8.7803E-05 8.7803E-05 90Zr-Fuel 0.3111 1.0231E-02 1.0231E-02 91Zr-Fuel 0.3111 2.2311E-03 2.2311E-03 92Zr-Fuel 0.3111 3.4103E-03 3.4103E-03 94Zr-Fuel 0.3111 3.4561E-03 3.4561E-03 96Zr-Fuel 0.3111 5.5679E-04 5.5679E-04 93Nb 0.3111 6.6284E-03 6.6284E-03 12C-Fuel 0.3111 2.7768E-02 5.1600E-03 1H-Fuel 0.1333 5.1600E-03 2.7768E-02 4 1-2 235U 0.3111 1.1665E-03 1.1665E-03 238U 0.3111 8.7803E-05 8.7803E-05 90Zr-Fuel 0.3111 1.0231E-02 1.0231E-02 91Zr-Fuel 0.3111 2.2311E-03 2.2311E-03 92Zr-Fuel 0.3111 3.4103E-03 3.4103E-03 94Zr-Fuel 0.3111 3.4561E-03 3.4561E-03 96Zr-Fuel 0.3111 5.5679E-04 5.5679E-04 93Nb 0.3111 6.6284E-03 6.6284E-03 12C-Fuel 0.3111 2.7768E-02 5.1600E-03 1H-Fuel 0.1333 5.1600E-03 2.7768E-02

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45 Table 4-8. The properties of axial region 2 fuel materials in the Square-Lattice Honeycomb homogeneous model. Region Isotope Volume Fraction Heterogeneous Number Density (atoms/bcm) Homogeneous Number Density (atoms/bcm) 5 2-1 235U 0.3111 1.4998E-03 1.4998E-03 238U 0.3111 1.1289E-04 1.1289E-04 90Zr-Fuel 0.3111 1.0064E-02 1.0064E-02 91Zr-Fuel 0.3111 2.1947E-03 2.1947E-03 92Zr-Fuel 0.3111 3.3546E-03 3.3546E-03 94Zr-Fuel 0.3111 3.3996E-03 3.3996E-03 96Zr-Fuel 0.3111 5.4769E-04 5.4769E-04 93Nb 0.3111 6.5201E-03 6.5201E-03 12C-Fuel 0.3111 2.7693E-02 3.8700E-03 1H-Fuel 0.1333 3.8700E-03 2.7693E-02 6 2-2 235U 0.3111 1.4998E-03 1.4998E-03 238U 0.3111 1.1289E-04 1.1289E-04 90Zr-Fuel 0.3111 1.0064E-02 1.0064E-02 91Zr-Fuel 0.3111 2.1947E-03 2.1947E-03 92Zr-Fuel 0.3111 3.3546E-03 3.3546E-03 94Zr-Fuel 0.3111 3.3996E-03 3.3996E-03 96Zr-Fuel 0.3111 5.4769E-04 5.4769E-04 93Nb 0.3111 6.5201E-03 6.5201E-03 12C-Fuel 0.3111 2.7693E-02 3.8700E-03 1H-Fuel 0.1333 3.8700E-03 2.7693E-02

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46 Table 4-9. The properties of axial region 3 fuel materials in the Square-Lattice Honeycomb homogeneous model. Region Isotope Volume Fraction Heterogeneous Number Density (atoms/bcm) Homogeneous Number Density (atoms/bcm) 7 3-1 235U 0.3111 1.9997E-03 1.9997E-03 238U 0.3111 1.5052E-04 1.5052E-04 90Zr-Fuel 0.3111 9.8130E-03 9.8130E-03 91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03 92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03 94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03 96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04 93Nb 0.3111 6.3576E-03 6.3576E-03 12C-Fuel 0.3111 2.7581E-02 2.5800E-03 1H-Fuel 0.1333 2.5800E-03 2.7581E-02 8 3-2 235U 0.3111 1.9997E-03 1.9997E-03 238U 0.3111 1.5052E-04 1.5052E-04 90Zr-Fuel 0.3111 9.8130E-03 9.8130E-03 91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03 92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03 94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03 96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04 93Nb 0.3111 6.3576E-03 6.3576E-03 12C-Fuel 0.3111 2.7581E-02 2.5800E-03 1H-Fuel 0.1333 2.5800E-03 2.7581E-02

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47 Table 4-10. The properties of axial region 4 fuel materials in the Square-Lattice Honeycomb homogeneous model. Region Isotope Volume Fraction Heterogeneous Number Density (atoms/bcm) Homogeneous Number Density (atoms/bcm) 9 4-1 235U 0.3111 1.9997E-03 1.9997E-03 238U 0.3111 1.5052E-04 1.5052E-04 90Zr-Fuel 0.3111 9.8130E-03 9.8130E-03 91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03 92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03 94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03 96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04 93Nb 0.3111 6.3576E-03 6.3576E-03 12C-Fuel 0.3111 2.7581E-02 1.9350E-03 1H-Fuel 0.1333 1.9350E-03 2.7581E-02 10 4-2 235U 0.3111 1.9997E-03 1.9997E-03 238U 0.3111 1.5052E-04 1.5052E-04 90Zr-Fuel 0.3111 9.8130E-03 9.8130E-03 91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03 92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03 94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03 96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04 93Nb 0.3111 6.3576E-03 6.3576E-03 12C-Fuel 0.3111 2.7581E-02 1.9350E-03 1H-Fuel 0.1333 1.9350E-03 2.7581E-02

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48 Table 4-11. The properties of axial region 5 fuel materials in the Square-Lattice Honeycomb homogeneous model. Region Isotope Volume Fraction Heterogeneous Number Density (atoms/bcm) Homogeneous Number Density (atoms/bcm) 11 5-1 235U 0.3111 1.9997E-03 1.9997E-03 238U 0.3111 1.5052E-04 1.5052E-04 90Zr-Fuel 0.3111 9.8130E-03 9.8130E-03 91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03 92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03 94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03 96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04 93Nb 0.3111 6.3576E-03 6.3576E-03 12C-Fuel 0.3111 2.7581E-02 1.2384E-03 1H-Fuel 0.1333 1.2384E-03 2.7581E-02 12 5-2 235U 0.3111 1.9997E-03 1.9997E-03 238U 0.3111 1.5052E-04 1.5052E-04 90Zr-Fuel 0.3111 9.8130E-03 9.8130E-03 91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03 92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03 94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03 96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04 93Nb 0.3111 6.3576E-03 6.3576E-03 12C-Fuel 0.3111 2.7581E-02 1.2384E-03 1H-Fuel 0.1333 1.2384E-03 2.7581E-02

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49 CHAPTER 5 METHODOLOGY Monte Carlo Neutron Cross-Sections Library Generation The starting point of this re search is the generation of a Monte Carlo neutron crosssection library. These neutron cross-sections will be processed from the ENDF/B VI data. As described in Chapter 2, the ENDF /B VI data are processed using the NJOY nuclear code through several modules. A set of ten temperatures are selected, ranging from 293.6 K to 3000 K. The essential isotope to be processed are also selected; they are 1H, 9Be, 10B, 11B, 12C, 16O, 90Zr, 91Zr, 92Zr, 94Zr, 96Zr, 93Nb, 235U, and 238U. In addition, essential thermal scattering kernels are also processed. These thermal-scattering-kernels or S( ) kernels are 1H in water, 1H in zirconium hydride, 9Be in beryllium metal, 9Be in beryllium oxide, 12C in graphite, and zircon ium in zirconium hydride. MCNP5 utilizes the newly created Monte Carlo neutron cross-sections library to generate energy spectra of the Square-Lat tice Honeycomb. The Monte Carlo analysis involves generating energy spectra for bot h the SLHC heterogeneous and SLHC homogeneous models. These energy spectra are generated using MCNP with 620-energy bins. The energy spectra generated have a re lative error less than 0.1 in each of the 620energy bins. The analyses utilize 100,000 pa rticles per cycle and 20000 cycles. Each analysis utilizes 128-pa rallel processors at the Los Alam os National Laboratory. With 128 processors, each analysis requires 30,000 mi nutes of computation time. Figure 5-1 shows the energy spectra of both the SLHC heterogeneous and SLHC homogeneous

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50 model of the Square-Lattice Honeycomb at 293.6 K. Figure 5-1, it shows higher spectrum in the thermal energy region system and lower energy spectrum in the fast energy region for the heterogeneous model. Figure 5-2 and Figure 53 show a closer look at the thermal and fast energy regions. Figur e 5-2 shows slightly higher values for the flux in the lower end of the thermal region for the SLHC heterogeneous model. It is difficult to see, but in Figure 5-3, there are ve ry slightly lower valu es for the flux in the epithermal and fast regions of the spectrum for the SLHC heterogeneous model. The magnitude of the SLHC heterogeneous energy spectrums peak is 4.5 106 at 0.04 eV. For comparison, the magnitude of the SLHC homogeneous energy spectrums peak is 4.2 106 also at 0.04 eV. The magnitude of th e SLHC homogeneous energy spectrums peak is 6.7% lower than the magnitude of the SLHC heterogeneous energy spectrums peak. The average value of the SLHC heterogeneous energy spectrum is 4.04 105, while the average value of the SLHC homogeneous energy spectrum is 3.81 105. The average value of the SLHC homogeneous energy spectrum is 5.6% lower than the average value of the SLHC heterogeneous energy spectrum. These differences will contribute to the accuracy level of the weight function utilized for the multigroup neutron cross-sections generation. Other important features in the Figure 51, which are clearly pr esented in Figure 52, are the four small peaks at the energy belo w 0.01 eV. The first peaks size that are positioned at 0.009 eV is 3.0 106, at 0.0033 eV is 2.1 106, at 0.0016 eV is 1.4 106, and at 0.00046 eV is 4.7 105. These peaks are the results of the thermal scattering treatments in ACE files. The thermal scatteri ng kernel was divided into discrete angles. The numbers of bin affect the accuracy of th e thermal spectrum. With a larger bin, the

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51 peaks should not be observed on the plot. Du e to time constraint and limited access to high performance computer, this affect co uld not be demonstrated in this study. Choosing the Correct Weight Function In the multigroup neutron cross-sections gene ration process, it is essential to select the correct weight function. A significant ca lculation error can be observed when an incorrect weight function is utilized to analyze a highly compact hydrogen-rich nuclear system, such as the Square-Lattice Honeycomb. Godiva Calculations To analyze the importance of choosing the correct weight function when analyzing a highly-compact hydrogen-rich nuclear system, additional calculations were performed. One of the calculations is to compare the k-eff results obtained from MCNP and PARTISN calculations. The MCNP result is assumed to be the t rue result, and it is used as a benchmark for the other calculati ons. Before the analys is can begin, a test model is created. Figure 5-4 shows the 235U Godiva sphere surrounded by a shell of zirconium hydride. First the test model is analyzed using MCNP, and an energy spectrum of the system is also generated. Next, two sets of the multigroup neutron crosssections are generatedcorr ect and incorrect cross-sections sets. These crosssections consist of a 187-fine-energy-gr oup structure with 55 thermal-energy groups. The difference between the two multigroup neutr on cross-sections is in the way they are generated. The correct multigroup neutron cro ss-section set is generated utilizing the Godiva energy spectrum as its weight func tion. However, the incorrect multigroup neutron cross-section is gene rated utilizing the Square-Latti ce Honeycomb heterogeneous model energy spectrum as its weight functi on. A PARTISN nuclear code is used to perform the analysis. The an alysis performed is a 1-D calculation utilizing 187 energy-

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52 groups, 55 thermal-groups, P3 order, and S16 order. Figure 5-5 presents a comparison between the two energy spectra for the Godiva and the SLHC heterogeneous model that are obtained from MCNP. The magnitude of the SLHC heterogeneous energy spectrums peak is 4.5 106 at 0.04 eV. For comparison, the ma gnitude of the Godiva surrounded by zirconium hydride energy sp ectrums peak is 4.2 106 also at 0.04 eV. The magnitude of the SLHC heterogeneous model energy spectrums peak is 6.7% lower than the magnitude of the Godiva surrounded by zirconium hydride energy spectrums peak. The average value of the SLHC heter ogeneous model energy spectrum is 4.04 105, while the average value of the Godiva surrounded by zircon ium hydride energy spectrum is 3.81 105. The average value of the SLHC heterogeneous model energy spectrum is 5.6% lower than the average value of the Godiva surrounded by zirconium hydride energy spectrum. The k-eff results are presen ted in Table 5-1. As shown in Table 5-1, the incorrect cross-section data produces a k-eff which is 3% lower than the k-eff produced by using the corre ct cross-section data. In Table 5-1, the values of percent error that are presented below the k-eff values are related to the relative error of each k-eff value to its corresponding k-eff value obtained from MCNP, while the values in th e percent difference column are the relative difference between both values of k-eff from PARTISN data. However, this error is amplified in the highly compact hydrogen ri ch nuclear system. Additional tests are performed by replacing zirconium hydride wi th beryllium metal, beryllium oxide, and graphite. The results are presented also in Table 5-1. Although there are differences in the k-eff for each case, the differences were not as significant as in the hydrogen gas case. For the Be, BeO and graphite case, the average neutron energy is at eV, while the average

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53 neutron energy of zirconium hydrid e case is at eV. Neutrons are manage to slow down to a lower energy in the zirconium hydride case compare to the other moderator material because the presence of hydrogen. The di fference in the average neutron energy contributes to the error in the k-eff value between zirconium hydrid e case and the other moderator cases. In the zirconium hydride case, the system is more sensitive to the weight function used because there are large numbers of thermal neutron compare to the other moderator cases. These thermal neutrons need to be correctly model in the weight function. The cross-section goes as 1/E, theref ore as the average neut ron energy is lower, the cross-section value is increasing. In th e zirconium hydride case, there is a significant difference of energy spectra in the thermal en ergy range. This diffe rence contributes to the large error in k-eff values. Square-Lattice Honeycomb Calculations As discussed above, the correct weight f unction plays a significant role in the multigroup neutron cross-section generation process. As shown in Figure 5-1 to 5-3, there are differences in the heterogene ous and homogeneous energy spectra. The question is how these differences affect the accuracy of the neutron cross-section data generated. To answer this que stion, two nuclear cross-sectio n sets are generated to be used in analyzing the homogeneous model of the Square-Lattice Honeycomb. The true multigroup nuclear cross-section set is ge nerated utilizing the homogeneous energy spectrum of the Square-Latti ce Honeycomb, while the fal se multigroup nuclear crosssection set is generated usi ng the heterogeneous energy spectrum. As in the previous experiment, an MCNP calculation is used as a benchmark for this analysis. The homogeneous model of the Square-Latti ce Honeycomb is analyzed by MCNP5 to determine the true k-eff for the homogeneous configuration.

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54 A two dimensional analysis is performed by PARTISN using each multigroup nuclear cross-section set. PARTISN utili zes 187 energy-groups with 55 thermal-energygroups, P3 order of scattering, S16 quadrature order, 155-mesh intervals in the r-direction, and 256-mesh intervals in the z-direction. Th e results of the PARTISN calculations are also shown in Table 5-2. Thus, the answer to the question can be observed in Table 5-2. Neutronics Analysis To perform the neutronics analysis of the Square-Lattice Honeycomb, Monte Carlo and Deterministic methods are used. The neutronics analysis includes the following analysis: Energy spectra characterizati on at startup, intermediate and operating temperatures. Power Distribution analysis at operating temperature Temperature coefficient analysis at startup, intermediate and operating temperatures. Control drums analysis at startup, inte rmediate and operating temperatures. Water submersion accident analysis Energy Spectra Characterization Energy spectra characterizat ion utilizes the Monte Carlo method. The method is used to characterize both the heterogeneous and homogeneous models. As discussed in the beginning of this chapter, the room te mperature energy spectra are generated for both heterogeneous and homogeneous models of the Square-Lattice Honeycomb. In addition, energy spectra characterization at intermedia te and operating temperatures are also performed. To analysis these two additional temperatures, condition for intermediate and operating temperatures has to be defined. Tabl e 5-3 defines the intermediate temperature conditions and the operating temperature cond itions. Figure 5-6 through Figure 5-9 show the graphs of these energy spectra.

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55 Figure 5-6, Figure 5-7, and Figure 5-8 show the S quare-Lattice Honeycomb Heterogeneous models energy spectrum of th e system, fuel region, and moderator region at 293.6 K and 1200 K and 2500K, while Figure 5-9 describes the Square-Lattice Honeycomb Homogeneous models energy spectrum. Power Distributions and Flux Profiles Analyses While the earlier analysis is performed utilizing the 187-fine-energy-group crosssections library, this next section utilizes the 45-broadenergy-group cross-sections library. This 45-broad-energy-group library has 13 thermal-energy groups. A method is needed to be developed in selecting thes e energy groups. Figure 5-10 shows the total cross-section plot of severa l important isotopes in SLHC Based on this figure, a 45broad-energy group is created. The 45-energy group is also based on the combination of the LASER-THERMOS 35-Group Structure and the LANL 30-Group Structure (MacFarlane and Muir, 1994). The therma l energy range is resembled the LASERTHERMOS 35-Group Structure, while the ep ithermal and fast energy ranges are resembled the LANL 30-Group Structure. Ho wever, the thermal energy below 0.01 eV is chosen to be represented into a single group because it is found to help the k-eff accuracy. Additional energy groups are added in the vicinity of the resonance region. The goal is to create a broad-energy-group st ructure that will best model these crosssections data. A 45-energy structure is found to have the best representation of the crosssection data. The selection of energy stru cture is based on the cross-section data presented in Figure 5-10. Based in Figure 5-10, the resonance regions have the energy between 1eV to 100 keV. This energy region needs to be represen ted well in the energy group selection. Once this group-structure is developed, a new crosssection library is generated based on it for all temperatures.

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56 Another important characterization of the system is the power density distribution and flux profile analyses. These analyses ar e performed at the operating temperature. The analyses of power density and flux distri bution at operating temperature are scaled to 100-MW-thermal power. The deterministic method is used to perform these analyses. Figure 5-11 presents the axial power distributions in the mi ddle of the first and second radial fuel regions. Figure 5-12 shows the ra dial power distribution in the middle of the second, third, and fourth axial fuel regions, while Figure 5-12 shows the radial power distribution in the middle of th e first and fifth axial fuel regions. As shown in these figures, power is only generated in the fuel regions. Figure 5-13 shows discontinuities in the power density plot; these discontinuities represent the three different axial fuel regions in the core. In these axial regions, ur anium density is varied as shown in Table 42. Figure 5-14 and Figure 5-15 show the fa st (energy greater than 65 keV) and epithermal (energy between 2.5 eV to 65 keV) energy neutron axial flux profiles in the two radial fuel regions, respectively. Figur e 5-16 presents the thermal energy neutron (energy less than 2.5 eV) axial flux profiles in the two radial fuel regions. Figure 5-17 shows the fast energy neutron ra dial flux profiles of the Squa re-Lattice Honeycomb in the two radial fuel regions. Fi gure 5-18 and Figure 5-19 presen t the epithermal and thermal energy neutron radial flux profiles of the Squa re-Lattice Honeycomb in the five axial fuel regions. In Figure 5-17 the peak s in the fast energy flux are f ound to be in the two radial fuel regions, while in Figure 5-19, the dips are fo und to be in the two radial fuel regions. The fast neutrons are generated in the fuel region, then they are moderated in the moderator and reflector region, and finally they returns to fuel region to be absorbed by uranium isotopes.

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57 Temperature Coefficient of Reactivity Analyses The fuel temperature coefficient of the Square-Lattice Honeycomb can be calculated by varying the temperature in the fuel region while keepi ng the temperature of the other region constant. The fuel temperat ure coefficient analyses consist of three different ranges of temperatures: startup te mperature range, intermediate temperature range, and operating temperature range. Figur e 5-20 presents the startup fuel temperature coefficient analysis. Figure 5-21 shows the intermediate fuel temperature coefficient analysis. Finally, Figure 5-22 describes the operating fuel temperature coefficient analysis. In the operating temperature range the moderator temperature coefficient is analyzed by varying the moderator temperature while keeping the temperature in the fuel and other regions constant. The plot of this analysis is presented in Figure 5-23. The combine temperature coefficient is called the sy stem overall temperature coefficient. To obtain the system overall temperature coeffici ent, the temperature profiles in Table 5-3 are utilized. Fuel temperatures are used to identify the points in the plot. For room temperature analysis, all regions are represente d to be at the room temperature. Figure 524 presents the system overall temperature co efficient. Based on the figures, the fuel temperature coefficients are f ound to be negative. However, the main concern is in the moderator temperature coefficient, which turned out to be positive. The positive moderator temperature coefficient is as results of the scattering cross-section increase in zirconium hydride, as shown in Figure 5-25, and the absorption cro ss-section decrease in hydrogen isotope, as shown in Figure 5-26. As the temperature increases, the average neutron energy increases resu lting in lower absorption cr oss-section of hydrogen. A lower absorption and fission in 235U will also occur as a result of the increase in average neutron energy. However, in this analysis, the fuel temperature remains constant, so

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58 neutron average energy in the fuel rema ins constant throughout the moderator temperature coefficient analysis. Therefore, the only change in the cross-section is found in the moderator region. Since the averag e neutron energy changes only affect the moderator region, the absorption cross-secti on of hydrogen plays a dominant role in the k-eff. Since higher average neutron ener gy means lower absorption in hydrogen, k-eff will increase as the temperature in the moderator increases. This reactor is behaving differently from the TRIGA reactor that al so uses zirconium hydride because in the TRIGA reactor, zirconium hydride is integrated with the fuel in the form of UraniumZirconium-Hydride fuel. The TRIGA reactor al so uses light water in the core. The behavior of light water in high temperature is that its density decreases, reducing the number density of hydrogen, which in turned reduces the moderation of the neutron. The TRIGA reactor incorporates the zirconium hydr ide with uranium; hence, the temperature effect is simultaneous. As the fuel temp erature increases, the average neutron energy increases which makes the absorption of in uranium (also fission) and hydrogen decreases. Since there is more uranium than hydrogen, the temperature coefficient will be negative. Control Drums Analyses The Square-Lattice Honeycomb reactor is c ontrolled by six control drums that are located in the reflector region. These cont rol drums have both absorber and reflecting materials in part of their regi ons. Figure 5-27 presents the critical configuration of the control drums of the Square-Lattice Honeycomb. This position places the reactor at the critical condition at zero power There are two extreme posit ions of the control drums: fully-in and fully-out. The fully-in position is when the c ontrol drums have the most absorbing property or when the absorber region is closest to the reactor core. At

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59 this position the reactor is subcritical, and the k-eff at zero power is 0.89858 + 0.00005. Figure 5-28 shows this configuration. The fully-out position is the opposite of the fully-in position; it is when the absorber re gion is farthest from the reactor core. At this position the reactor is supercritica l, and the k-eff at zero power is 1.05961 + 0.00006. Figure 5-29 presents this configuration. These three positions are examined to obt ain the reactivity worth of the control drums at startup, intermediate and operating temperature ranges. Figure 5-30 presents the reactivity worth of the control drums at room temperature. This figure shows that the control drums have their highest differential reactivity worth when their position is near the half-way position. The half-way posi tion is the position wh en the control drums are 90 from the fully-in positions. How many control drums are needed to s hutdown the reactor? To answer this question, we need to perform several analyses to find the fewest number of the control drums to shutdown the reactor. These analys es are performed under the assumption that there are independent controls for each cont rol drum. Figure 5-31 shows three different configurations and their k-eff values with two control drums jammed in the fully-out position. Next, Figure 5-32 shows three different configurations and their k-eff values with three control drums jammed in the fu lly-out position. Fi nally, Figure 5-33 shows three different configurations and their k-eff values with four control drums jammed in the fully-out positi on. Based on these analyses a nd the assumption of independent control of each control drum, the minimum numb er of control drums needed to shutdown the reactor is three control drums.

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60 Water Submersion Accident Analysis The final analysis of th e Square-Lattice Honeycomb is the water submersion accident analysis. This analysis predicts th e performance of the r eactor if it accidentally drops into a large body of water, such as an o cean or lake. To perform this analysis, the heterogeneous model of the Square-Lattice H oneycomb is utilized. The configuration of the control drums for this analysis will be at the fully-in position. Water will surround the reactor from all sides, and water w ill also replace all the empty spaces (hydrogen holes) in the reactor, as shown in Figure 5-34. The ideal performance of the reactor during such an accident is to stay in the subcritical condition. Based on the analysis performed utilizing Monte Carlo method, the k-eff of the reactor is found to be 0.95824 + 0.00007. Although, the reactor is indeed s ubcritical during the water submersion accident, the shutdown margin (i.e., the fractional k/k value below critical of 0.04 k/k is too low. Therefore, additional absorber mate rials are required to increase the margin of safety. A boron carbide absorber is placed at the center hydrogen hole, as shown in Figure 5-35. The reactors k-eff after this modification is 0.83376 + 0.00006, which yield a shutdown margin of almost 0.17 k/k.

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61 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalized Flux SLHC Heterogeneous Spectrum SLHC Homogeneous Spectrum Figure 5-1. Energy spectrum comparison betw een the SLHC heterogeneous and SLHC homogeneous models at 293.6 K

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62 0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06 3.00E+06 3.50E+06 4.00E+06 4.50E+06 5.00E+061.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01Energy (eV)Normalized Flux SLHC Heterogeneous Spectrum SLHC Homogeneous Spectrum Figure 5-2. Energy spectrum comparison betw een the SLHC heterogeneous and SLHC homogeneous models in th ermal energy range at 293.6 K

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63 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+031.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalized Flux SLHC Heterogeneous Spectrum SLHC Homogeneous Spectrum Figure 5-3. Energy spectrum comparison be tween heterogeneous and homogeneous models of the Square-Lattice Honeyc omb in fast energy range at 293.6 K

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64 Figure 5-4. Geometry description of the Godiva sphere surrounded by hydrogen gas.

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65 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (MeV)Normalized Flux SLHC Heterogeneous Spectrum Zirconium Hydride Godiva Spectrum Figure 5-5. Energy spectrum comparison betw een heterogeneous model of the SquareLattice Honeycomb and 235U Godiva surrounded by H2 gas at 293.6 K obtained from MCNP.

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66 Table 5-1. Comparison of PARTISN calcula tions utilizing correct and incorrect multigroup neutron cross-sections for f our surrounding shells materials with the radius of 235U Godiva is 6.7 cm Type of surrounding shell Shell Outer Radius (cm) MCNP Result PARTISN result with correct cross-section PARTISN result with incorrect cross-section % difference H2 32 0.99367 + 0.00002 1.00032 0.67% 0.97015 -2.37% -3.02 Be metal 32 1.00334 + 0.00001 1.00356 0.02% 1.00366 0.03% 0.01 BeO 32 1.00295 + 0.00001 1.00366 0.07% 1.00397 0.10% 0.03 Graphite 32 0.99667 + 0.00001 0.99587 -0.08% 0.99307 -0.36% -0.28

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67 Table 5-2. Comparison of PARTISN calculati ons utilizing true and false multigroup neutron cross-sections for the Square-Lattice Honeycomb Type of Calculations Results % Different % Difference from Overall MCNP Heterogeneous (Benchmark Overall) 0.99997 + 0.00005 MCNP Homogeneous (Benchmark for PARTISN) 0.99522 + 0.00002 0.48 PARTISN Homogeneous with Homogeneous Cross Sections 0.99001 -0.52 -1.00 PARTISN Homogeneous with Heterogeneous Cross Sections 0.95569 -3.97 -4.43

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68 Table 5-3. The intermediate temperature conditions and the operating temperature conditions for the Square-Lattice Honeycomb Region Intermediate Temperature (K) Operating Temperature (K) Hydrogen hole 1000 2000 Top Hydrogen hole 400 600 Zirconium tube 800 1200 Zirconium hydride 600 1000 Bottom Graphite Coating 1000 2000 Bottom ZrO2 Coating 800 1200 6 Inner Cylinder of Control Drums 400 600 6 Outer Cylinder of Control Drums (Beryllium) 400 600 6 Outer Cylinder of Control Drums (B4C) 400 600 Beryllium reflector 400 600 Beryllium reflector and Hydrogen gas 400 600 18 First Axial Fuel Region 400 800 18 First Axial Graphite Coating 400 600 18 First Axial ZrO2 Coating 400 600 18 Second Axial Fuel Region 600 1200 18 Second Axial Graphite Coating 400 1000 18 Second Axial ZrO2 Coating 400 800 18 Third Axial Fuel Region 800 1600 18 Third Axial Graphite Coating 600 1200 18 Third Axial ZrO2 Coating 400 1000 18 Fourth Axial Fuel Region 1000 2000 18 Fourth Axial Graphite Coating 800 1600 18 Fourth Axial ZrO2 Coating 600 1000 18 Fifth Axial Fuel Region 1200 2500 18 Fifth Axial Graphite Coating 1000 2000 18 Fifth Axial ZrO2 Coating 800 1200

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69 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalize Flux 293 K 1200 K 2500 K Figure 5-6. This figure shows the systems energy spectrum of the Square-Lattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures.

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70 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalized Flux 293 K 1200 K 2500 K Figure 5-7. This figure presents the fuel regions energy sp ectrum of the Square-Lattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures.

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71 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalized Flux 293 K 1200 K 2500 K Figure 5-8. This plot presents the modera tor regions energy spect rum of the SquareLattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures.

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72 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+071.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Energy (eV)Normalize Flux 293 K 1200 K 2500 K Figure 5-9. This plot shows the systems energy spectrum of the Square-Lattice Honeycomb Homogeneous model at room, intermediate, and operating temperatures.

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73 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+061.00E-11 1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03Energy (MeV)Total cross section (b) Be-9 B-10 B-11 Zr-90 Zr-91 Zr-92 Zr-94 Zr-96 Nb-93 U-235 U-238 Figure 5-10. The plot presents th e total cross-section data of several important isotopes in the Square-Lattice Honeycomb.

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74 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 1.40E+00 1.60E+00 0.010.020.030.040.050.060.070.0 Height (cm)Power density (MW/cm2) First Radial Fuel at 6.30 cm Second Radial Fuel at 12.49 cm Figure 5-11. The plot presents the axial power distribution of the Square-Lattice Honeycomb in the first and second radial fuel regions.

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75 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 1.40E+00 0.05.010.015.020.025.030.035.040.0 Radius (cm)Power density (MW/cm2) Second Axial Fuel at 31.5 cm Third Axial Fuel at 22.5 cm Fourth Axial fuel at 13.5 cm Core boundar y Inside boundary of B4C Outside boundary of B4C Reactor boundar y Figure 5-12. The figure presen ts the radial power distri bution of the Square-Lattice Honeycomb in the second, third and fourth axial fuel regions.

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76 0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 0.05.010.015.020.025.030.035.040.0 Radius (cm)Power density (MW/cm2) First Axial Fuel at 40.5 cm Fifth Axial Fuel at 5.5 cm Core boundar y Inside boundary of B4C Outside boundary of B4C Reactor boundar y Figure 5-13. The figure shows the radial power distribution of the Square-Lattice Honeycomb in the first and fifth axial fuel regions.

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77 0.00E+00 2.00E+20 4.00E+20 6.00E+20 8.00E+20 1.00E+21 1.20E+21 1.40E+21 1.60E+21 1.80E+21 0.010.020.030.040.050.060.070.0 Height (cm)Fluxes (neutron/cm2.s) Fast Flux in First Radial Fuel at 6.3 cm Fast Flux in Second Radial Fuel at 12.49 cm Figure 5-14. The plot illustrates the fast ne utron energy (> 65 keV) axial flux profiles of the Square-Lattice Honeycomb in the two radial fuel regions.

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78 0.00E+00 1.00E+18 2.00E+18 3.00E+18 4.00E+18 5.00E+18 6.00E+18 7.00E+18 8.00E+18 9.00E+18 1.00E+190.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0Height (cm)Fluxes (neutron/cm2.s) Epithermal Flux in First Radial Fuel Region at 6.3 cm Epithermal Flux in Second Radial Fuel Region at 12.49 cm Top core boundar y Top reactor boundar y Figure 5-15. The epithermal neutron energy ( 2.5 eV 65 keV) axial flux profiles of the Square-Lattice Honeycomb in th e two radial fuel regions.

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79 0.00E+00 2.00E+13 4.00E+13 6.00E+13 8.00E+13 1.00E+14 1.20E+14 0.010.020.030.040.050.060.070.0 Height (cm)Fluxes (neutron/cm2.s) Thermal Flux in First Radial Fuel Region at 6.3 cm Thermal Flux in Second Radial Fuel Region at 12.49 cm Top core boundar y Top reactor boundar y Figure 5-16. The figure illust rates the thermal neutron energy (< 2.5 eV) axial flux profiles of the Square-Lattice Honeycomb in the two radial fuel regions.

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80 0.00E+00 2.00E+20 4.00E+20 6.00E+20 8.00E+20 1.00E+21 1.20E+21 1.40E+21 1.60E+21 1.80E+21 0.05.010.015.020.025.030.035.040.0 Radius (cm)Fluxes (neutron/cm2.s) Fast Flux in First Axial Fuel Region at 40.5 cm Fast Flux in Second Axial Fuel Region at 31.5 cm Fast Flux in the Third Axial Fuel Region at 22.5 cm Fast Flux in the Fourth Axial Fuel Region at 13.5 cm Fast Flux in the Fifth Axial Fuel Region at 5.5 cm Core boundar y Inside boundary of B4C Outside boundary of B4C Reactor boundar y Figure 5-17. The figure presents the fast neutron energy (> 65 keV) radial flux profiles of the Square-Lattice Honeycomb in the five axial fuel regions.

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81 0.00E+00 1.00E+18 2.00E+18 3.00E+18 4.00E+18 5.00E+18 6.00E+18 7.00E+18 8.00E+18 9.00E+18 1.00E+19 0.05.010.015.020.025.030.035.040.0 Radius (cm)Fluxes (neutron/cm2.s) Epithermal Flux in First Axial Fuel Region at 40.5 cm Epithermal Flux in Second Axial Fuel Region at 31.5 cm Epithermal Flux in Third Axial Fuel Region at 22.5 cm Epithermal Flux in Fourth Axial Fuel Region at 13.5 cm Epithermal Flux in Fifth Axial Fuel Region at 5.5 cm Core boundar y Inside boundary of B4C Outside boundary of B4 C Reactor boundar y Figure 5-18. The epithermal neutron energy (2.5 eV 65 keV) radial flux profiles of the Square-Lattice Honeycomb in th e five axial fuel regions.

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82 0.00E+00 5.00E+13 1.00E+14 1.50E+14 2.00E+14 2.50E+14 3.00E+14 0.05.010.015.020.025.030.035.040.0 Radius (cm)Fluxes (neutron/cm2.s) Thermal Flux in First Axial Fuel Region at 40.5 cm Thermal Flux in Second Axial Fuel Region at 31.5 cm Thermal Flux in Third Axial Fuel Region at 22.5 cm Thermal Flux in Fourth Axial Fuel Region at 13.5 cm Thermal Flux in Fifth Axial Fuel Region at 5.5 cm Core boundar y Inside boundary of B4C Outside boundary of B4C Reactor boundar y Figure 5-19. The figure presents the ther mal neutron energy (< 2.5 eV) radial flux profiles of the Square-Lattice Honeycomb in the five axial fuel regions.

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83 y = -1.2323E-04x + 1.0241E+00 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995 250300350400450500550 Temperature (K)K-eff Figure 5-20. The plot shows the plot of fuel temperature coefficient of the Square-Lattice Honeycomb during startup.

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84 y = -1.5778E-05x + 9.7299E-01 0.951 0.952 0.953 0.954 0.955 0.956 0.957 100010501100115012001250130013501400 Temperature (K)K-eff Figure 5-21. The plot of fuel temperature co efficient of the Square-Lattice Honeycomb at the intermediate temperature ranges.

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85 y = -5.0472E-06x + 9.5406E-01 0.9406 0.9408 0.9410 0.9412 0.9414 0.9416 0.9418 0.9420 0.9422 0.9424 230023502400245025002550260026502700 Temperature (K)K-eff Figure 5-22. The plot of fuel temperature co efficient of the Square-Lattice Honeycomb at the operating temperature ranges.

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86 y = 5.9774E-06x + 9.3147E-01 0.935 0.935 0.936 0.936 0.937 0.937 0.938 0.938 50060070080090010001100 Temperature (K)K-eff Figure 5-23. The plot of m oderator temperature coeffici ent of the Square-Lattice Honeycomb at the operating temperature ranges.

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87 y = -2.1156E-05x + 9.8995E-01 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 050010001500200025003000 Temperature (K)K-eff Figure 5-24. The plot presents the system te mperature coefficient of the Square-Lattice Honeycomb.

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88 Figure 5-25. The plot illustrates the thermal scattering cross section of 1H and Zr in zirconium hydride.

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89 Figure 5-26. Plot of the 1H ne utron absorption cross section

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90 Boron Carbide region Beryllium region Boron Carbide region Beryllium region Figure 5-27. Critical configurations of the control drums in the Square-Lattice Honeycomb at 293.6 K.

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91 Figure 5-28. The fully-in configurations of the control drums in the Square-Lattice Honeycomb, which has k-e ff of 0.89858 + 0.00005 at 293.6 K.

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92 Figure 5-29. The fully-out configurations of the control drums in the Square-Lattice Honeycomb, which has k-e ff of 1.05961 + 0.00006 at 293.6 K.

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93 Figure 5-30. The reactivity worth plot of the c ontrol drums from fully -in to fully-out positions at 293.6 K 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 020406080100120140160180 Angle positions (deg)K-eff

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94 k eff = 0.95646 +/-0.00006 k eff = 0.94793 +/-0.00006 k eff = 0.94720 +/-0.00006 k eff = 0.95646 +/-0.00006 k eff = 0.94793 +/-0.00006 k eff = 0.94720 +/-0.00006 Figure 5-31. Three different c onfigurations and their k-eff values of two control drums jammed at the fully-out position.

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95 k eff = 0.95646 +/-0.00006 k eff = 0.94793 +/-0.00006 k eff = 0.94720 +/-0.00006 k eff = 0.95646 +/-0.00006 k eff = 0.94793 +/-0.00006 k eff = 0.94720 +/-0.00006 Figure 5-32. Three different c onfigurations and their k-eff values with three control drums jammed at the fully-out position.

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96 k eff = 1.00997 + 0.00006 k eff = 1.00087 + 0.00006 k eff = 1.00263 + 0.00006 k eff = 1.00997 + 0.00006 k eff = 1.00087 + 0.00006 k eff = 1.00263 + 0.00006 Figure 5-33. Three different conf igurations and their k-eff valu es with four control drums jammed at the fully-out position.

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97 31 cm 71 cm 131 cm Water Surrounding water 31 cm 71 cm 131 cm Water Surrounding water Figure 5-34. Configuration of the Squa re-Lattice Honeycomb reactor for water submersion accident.

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98 31 cm 71 cm 131 cm Absorber material (B4C) in the hydrogen hole Surrounding water 31 cm 71 cm 131 cm Absorber material (B4C) in the hydrogen hole Surrounding water Figure 5-35. Modified configuration of the Square-Lattice Honeycomb reactor for water submersion accident.

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99 CHAPTER 6 CONCLUSIONS After observing the results discussed in Chapter 5, the Square-Lattice Honeycomb design has several advantages. The SLHC has a desirable performance in the water submersion accident by maintaining subcritical ity. The SLHC significantly reduces the amount of 235U needed to make the system critical. The amount of 93% enriched uranium required is 10.6 kg. This therma l spectrum characteristic is due to the moderation provided by zirconiu m hydride. The SLHC is a compact nuclear system. The SLHC has negative fuel temperature coeffi cient of reactivity. This feature is very desirable for the system because as the temper ature of the fuel increases, the power level decreases. However, the moderator temperat ure coefficient of reactivity is positive. However, the temperature coefficient of reactiv ity of the overall system is negative. It becomes less negative as the temperature incr eases. The important discovery from this study is the importance of the system spectru m characteristic in the multigroup nuclear cross-section generation for a highly moderate d system. As discussed in Chapter 5, the multigroup nuclear cross-sections are closely related to the wei ght function that is used to generate them. To obtain an accurate repr esentation of the multigroup model of a space nuclear reactor, one needs to know the spectral ch aracteristic of the sp ace nuclear system. On the criticality comparison between probabi listic and deterministic methods using the same cross-section library, ther e is a difference of 1 %. However, this difference is

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100 amplified if one does not ge nerate the multigroup nuclear cross-sections based on the systems that are being compared. Further analysis is required for bot h the SLHC heterogeneous and SLHC homogeneous models. A coupled neutroni c-thermal-hydraulic an alysis has to be performed to determine a more accurate model of the system behavior. Additional nuclear cross-sections are needed to be generated to accommodate these new temperatures, especially the thermal scattering data. The generation of cross sections at these new temperatures will help to obtain an accurate analysis of the system. Further analysis of the temperature coefficient of react ivity is needed, to include the effect of the thermal expansion of materials in the reactor.

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101 APPENDIX A LISTS OF NUCLIDES PROCESSED IN THE RGOUW CROSS-SECTION LIBRARIES Appendix A lists all of the nuclides in the RGOUW MCNP cross-section library and RGOUW multigroup cross-secti on library. The lists are se parated into two different setsessential isotopes, and thermal scatteri ng data. Table A-1 presents the essential isotopes in the RGOUW MCNP cross-sec tion library and RGOUW multigroup crosssection library. These crosssection libraries are processe d from the ENDF/B VI data. Next, table A-2 presents the thermal s cattering data in the RGOUW MCNP crosssection library and RGOUW mu ltigroup cross-section library.

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102 Table A-1. List of essential isotopes in the RGOUW cross-section libraries Isotope ENDF Release Number Temperature (K) MCNP Cross-Section Library 187-energy group Cross-Section Library 45-energy group Cross-Section Library 8 293.6 1001.00c h1-00 h1-00 400 1001.01c h1-01 h1-01 600 1001.02c h1-02 h1-02 800 1001.03c h1-03 h1-03 1000 1001.04c h1-04 h1-04 1200 1001.05c h1-05 h1-05 1600 1001.06c h1-06 h1-06 2000 1001.07c h1-07 h1-07 2500 1001.08c h1-08 h1-08 H1 1 3000 1001.09c h1-09 h1-09 8 293.6 4009.00c be9-00 be9-00 400 4009.01c be9-01 be9-01 600 4009.02c be9-02 be9-02 800 4009.03c be9-03 be9-03 1000 4009.04c be9-04 be9-04 1200 4009.05c be9-05 be9-05 1600 4009.06c be9-06 be9-06 2000 4009.07c be9-07 be9-07 2500 4009.08c be9-08 be9-08 Be4 9 3000 4009.09c be9-09 be9-09 1 293.6 5010.00c b10-00 b10-00 400 5010.01c b10-01 b10-01 600 5010.02c b10-02 b10-02 800 5010.03c b10-03 b10-03 1000 5010.04c b10-04 b10-04 1200 5010.05c b10-05 b10-05 1600 5010.06c b10-06 b10-06 2000 5010.07c b10-07 b10-07 2500 5010.08c b10-08 b10-08 B5 10 3000 5010.09c b10-09 b10-09 8 293.6 5011.00c b11-00 b11-00 400 5011.01c b11-01 b11-01 600 5011.02c b11-02 b11-02 800 5011.03c b11-03 b11-03 1000 5011.04c b11-04 b11-04 1200 5011.05c b11-05 b11-05 1600 5011.06c b11-06 b11-06 2000 5011.07c b11-07 b11-07 2500 5011.08c b11-08 b11-08 B5 11 3000 5011.09c b11-09 b11-09 6 293.6 6000.00c cnat00 cnat00 400 6000.01c cnat01 cnat01 600 6000.02c cnat02 cnat02 800 6000.03c cnat03 cnat03 1000 6000.04c cnat04 cnat04 1200 6000.05c cnat05 cnat05 1600 6000.06c cnat06 cnat06 2000 6000.07c cnat07 cnat07 2500 6000.08c cnat08 cnat08 C6 12 3000 6000.09c cnat09 cnat09

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103 Table A-1. List of essentia l isotopes in the RGOUW cross-section libraries (continued) Isotope ENDF Release Number Temperature (K) MCNP Cross-Section Library 187-energy group Cross-Section Library 45-energy group Cross-Section Library 8 293.6 8016.00c o16-00 o16-00 400 8016.01c o16-01 o16-01 600 8016.02c o16-02 o16-02 800 8016.03c o16-03 o16-03 1000 8016.04c o16-04 o16-04 1200 8016.05c o16-05 o16-05 1600 8016.06c o16-06 o16-06 2000 8016.07c o16-07 o16-07 2500 8016.08c o16-08 o16-08 O8 16 3000 8016.09c o16-09 o16-09 8 293.6 40090.00c zr9000 zr9000 400 40090.01c zr9001 zr9001 600 40090.02c zr9002 zr9002 800 40090.03c zr9003 zr9003 1000 40090.04c zr9004 zr9004 1200 40090.05c zr9005 zr9005 1600 40090.06c zr9006 zr9006 2000 40090.07c zr9007 zr9007 2500 40090.08c zr9008 zr9008 Zr40 90 3000 40090.09c zr9009 zr9009 8 293.6 40091.00c zr9100 zr9100 400 40091.01c zr9101 zr9101 600 40091.02c zr9102 zr9102 800 40091.03c zr9103 zr9103 1000 40091.04c zr9104 zr9104 1200 40091.05c zr9105 zr9105 1600 40091.06c zr9106 zr9106 2000 40091.07c zr9107 zr9107 2500 40091.08c zr9108 zr9108 Zr40 91 3000 40091.09c zr9109 zr9109 8 293.6 40092.00c zr9200 zr9200 400 40092.01c zr9201 zr9201 600 40092.02c zr9202 zr9202 800 40092.03c zr9203 zr9203 1000 40092.04c zr9204 zr9204 1200 40092.05c zr9205 zr9205 1600 40092.06c zr9206 zr9206 2000 40092.07c zr9207 zr9207 2500 40092.08c zr9208 zr9208 Zr40 92 3000 40092.09c zr9209 zr9209 8 293.6 40094.00c zr9400 zr9400 400 40094.01c zr9401 zr9401 600 40094.02c zr9402 zr9402 800 40094.03c zr9403 zr9403 1000 40094.04c zr9404 zr9404 1200 40094.05c zr9405 zr9405 1600 40094.06c zr9406 zr9406 2000 40094.07c zr9407 zr9407 2500 40094.08c zr9408 zr9408 Zr40 94 3000 40094.09c zr9409 zr9409

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104 Table A-1. List of essentia l isotopes in the RGOUW cross-section libraries (continued) Isotope ENDF Release Number Temperature (K) MCNP Cross-Section Library 187-energy group Cross-Section Library 45-energy group Cross-Section Library 8 293.6 40096.00c zr9600 zr9600 400 40096.01c zr9601 zr9601 600 40096.02c zr9602 zr9602 800 40096.03c zr9603 zr9603 1000 40096.04c zr9604 zr9604 1200 40096.05c zr9605 zr9605 1600 40096.06c zr9606 zr9606 2000 40096.07c zr9607 zr9607 2500 40096.08c zr9608 zr9608 Zr40 96 3000 40096.09c zr9609 zr9609 6 293.6 41093.00c nb9300 nb9300 400 41093.01c nb9301 nb9301 600 41093.02c nb9302 nb9302 800 41093.03c nb9303 nb9303 1000 41093.04c nb9304 nb9304 1200 41093.05c nb9305 nb9305 1600 41093.06c nb9306 nb9306 2000 41093.07c nb9307 nb9307 2500 41093.08c nb9308 nb9308 Nb41 93 3000 41093.09c nb9309 nb9309 5 293.6 92235.00c u23500 u23500 400 92235.01c u23501 u23501 600 92235.02c u23502 u23502 800 92235.03c u23503 u23503 1000 92235.04c u23504 u23504 1200 92235.05c u23505 u23505 1600 92235.06c u23506 u23506 2000 92235.07c u23507 u23507 2500 92235.08c u23508 u23508 U92 235 3000 92235.09c u23509 u23509 5 293.6 92238.00c u23800 u23800 400 92238.01c u23801 u23801 600 92238.02c u23802 u23802 800 92238.03c u23803 u23803 1000 92238.04c u23804 u23804 1200 92238.05c u23805 u23805 1600 92238.06c u23806 u23806 2000 92238.07c u23807 u23807 2500 92238.08c u23808 u23808 U92 238 3000 92238.09c u23809 u23809

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105 Table A-2. List of thermal scattering da ta in the RGOUW cross-section libraries Isotope ENDF Release Number Temperature (K) MCNP Cross-Section Library 187-energy group Cross-Section Library 45-energy group Cross-Section Library 3 293.6 hh2o.00t hh2o00 hh2o00 400 hh2o.01t hh2o01 hh2o01 600 hh2o.02t hh2o02 hh2o02 800 hh2o.03t hh2o03 hh2o03 H1 1 in water 1000 hh2o.04t hh2o04 hh2o04 3 293.6 hzrh.00t hzrh00 hzrh00 400 hzrh.01t hzrh01 hzrh01 600 hzrh.02t hzrh02 hzrh02 800 hzrh.03t hzrh03 hzrh03 1000 hzrh.04t hzrh04 hzrh04 H1 1 in ZrH2 1200 hzrh.05t hzrh05 hzrh05 3 293.6 bem.00c bem-00 bem-00 400 bem.01c bem-01 bem-01 600 bem.02c bem-02 bem-02 800 bem.03c bem-03 bem-03 1000 bem.04c bem-04 bem-04 Be4 9 in Be-metal 1200 bem.05c bem-05 bem-05 3 293.6 beo.00c beo-00 beo-00 400 beo.01c beo-01 beo-01 600 beo.02c beo-02 beo-02 800 beo.03c beo-03 beo-03 1000 beo.04c beo-04 beo-04 Be4 9 in BeO 1200 beo.05c beo-05 beo-05 3 293.6 graph.00c grph00 grph00 400 graph.01c grph01 grph01 600 graph.02c grph02 grph02 800 graph.03c grph03 grph03 1000 graph.04c grph04 grph04 C6 12 in Graphite 1200 graph.05c grph05 grph05 1600 graph.06c grph06 grph06 2000 graph.07c grph07 grph07 3 293.6 zrzrh.00c zrh-00 zrh-00 400 zrzrh.01c zrh-01 zrh-01 600 zrzrh.02c zrh-02 zrh-02 800 zrzrh.03c zrh-03 zrh-03 1000 zrzrh.04c zrh-04 zrh-04 Zrnat 40 in ZrH2 1200 zrzrh.05c zrh-05 zrh-05

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106 APPENDIX B ENERGY GROUP STRUCTURES FO R SQUARE-LATTICE HONEYCOMB Appendix B describes two energy group st ructures: a fine energy group structure and a broad energy group structure. The broad energy group consists of a 45-energygroup structure that is developed exclus ively for analyzing the Square-Lattice Honeycomb reactor and is shown in Table B1. The fine energy group structure is based on the LANL-187 energy group structures which is shown in Table B-2.

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107 Table B-1. The 45-energy group structures Group Index Group Lower Energy Boundaries (eV) N EL (N+0) EL (N+1) EL (N+2) EL (N+3) EL (N+4) 1 1.0000E-05 1.2397E-02 3.0613E -02 5.6925E-02 1.1159E-01 6 2.2770E-01 3.0113E-01 4.1704E -01 5.0326E-01 6.2493E-01 11 9.5070E-01 1.0137E+00 1.4575E +00 1.8550E+00 2.3824E+00 16 3.0590E+00 3.9279E+00 5.0435E +00 6.4760E+00 8.3153E+00 21 2.2603E+01 6.1442E+01 1.6702E +02 4.5400E+02 7.4852E+02 26 1.2341E+03 3.3546E+03 9.1188E +03 2.4788E+04 6.7379E+04 31 1.8316E+05 3.0197E+05 3.8774E +05 4.9787E+05 8.2085E+05 36 1.3534E+06 1.7377E+06 2.2313E +06 2.8650E+06 3.6788E+06 41 6.0653E+06 7.7880E+06 1.0000E +07 1.2214E+07 1.7000E+07 46 2.0000E+07

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108 Table B-2. The LANL-187 energy group structures Group Index Group Lower Energy Boundaries (eV) N EL (N+0) EL (N+1) EL (N+2) EL (N+3) EL (N+4) 1 1.0000E-05 2.5399E-04 7.6022E -04 2.2769E-03 6.3247E-03 6 1.2396E-02 2.0492E-02 2.5500E -02 3.0612E-02 3.5500E-02 11 4.2755E-02 5.0000E-02 5.6922E -02 6.7000E-02 8.1968E-02 16 1.1157E-01 1.4572E-01 1.5230E -01 1.8443E-01 2.2769E-01 21 2.5103E-01 2.7052E-01 2.9074E -01 3.0112E-01 3.2063E-01 26 3.5767E-01 4.1499E-01 5.0323E -01 6.2506E-01 7.8208E-01 31 8.3368E-01 8.7642E-01 9.1000E -01 9.5065E-01 9.7100E-01 36 9.9200E-01 1.0137E+00 1.0427E +00 1.0525E+00 1.0623E+00 41 1.0722E+00 1.0987E+00 1.1254E +00 1.1664E+00 1.3079E+00 46 1.4574E+00 1.5949E+00 1.7261E +00 1.8554E+00 2.1024E+00 51 2.3824E+00 2.6996E+00 3.0590E +00 3.4663E+00 3.9279E+00 56 4.4509E+00 5.0435E+00 5.7150E +00 6.4760E+00 6.8680E+00 61 7.3382E+00 8.3153E+00 9.4225E +00 1.0677E+01 1.2099E+01 66 1.3710E+01 1.5535E+01 1.7604E +01 1.9947E+01 2.2603E+01 71 2.5613E+01 2.9023E+01 3.2888E +01 3.7267E+01 4.2229E+01 76 4.7851E+01 5.4223E+01 6.1442E +01 6.9623E+01 7.8893E+01 81 8.9398E+01 1.0130E+02 1.1479E +02 1.3007E+02 1.4739E+02 86 1.6702E+02 1.8926E+02 2.1445E +02 2.4301E+02 2.7536E+02 91 3.1203E+02 3.5358E+02 4.0065E +02 4.5400E+02 5.1445E+02 96 5.8295E+02 6.6057E+02 7.4852E +02 8.4818E+02 9.6112E+02 101 1.0891E+03 1.2341E+03 1.3984E +03 1.5846E+03 1.7956E+03 106 2.0347E+03 2.3056E+03 2.6126E +03 2.9605E+03 3.3546E+03 111 3.8013E+03 4.3074E+03 4.8810E +03 5.5308E+03 6.2673E+03 116 7.1017E+03 8.0473E+03 9.1188E +03 1.0333E+04 1.1709E+04 121 1.3268E+04 1.5034E+04 1.7036E +04 1.9305E+04 2.1875E+04 126 2.4788E+04 2.6058E+04 2.8088E +04 3.1828E+04 3.6066E+04 131 4.0868E+04 4.6309E+04 5.2475E +04 5.9462E+04 6.7380E+04 136 7.6351E+04 8.6517E+04 9.8037E +04 1.1109E+05 1.2588E+05 141 1.4264E+05 1.6164E+05 1.8316E +05 2.0754E+05 2.3518E+05 146 2.6649E+05 3.0197E+05 3.4218E +05 3.8774E+05 4.3937E+05 151 4.9787E+05 5.6416E+05 6.3928E +05 7.2440E+05 8.2085E+05 156 9.3015E+05 1.0540E+06 1.1943E +06 1.3534E+06 1.5336E+06 161 1.7377E+06 1.9691E+06 2.2313E +06 2.5284E+06 2.8651E+06 166 3.2465E+06 3.6788E+06 4.1686E +06 4.7237E+06 5.3526E+06 171 6.0653E+06 6.8729E+06 7.7880E +06 8.8250E+06 1.0000E+07 176 1.1000E+07 1.2000E+07 1.3000E +07 1.3500E+07 1.3750E+07 181 1.3940E+07 1.4200E+07 1.4420E +07 1.4640E+07 1.5000E+07 186 1.6000E+07 1.7000E+07 2.0000E+07

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109 APPENDIX C AUTOMATION PROCESS FOR PENDF, ACER, AND DTFR MODULES This appendix presents some examples of the script files used to perform NJOY calculations. The first example is th e script for the PENDF generation of 1H. PENDF generation script 1H psi get tape20:/hpss/rgouw/endfvib/neutron/h/1h psi get tape30:/hpss/rgouw/endfvib/thermal/hh2oc psi get tape33:/hpss/rgouw/endfvib/thermal/hzrhc cat>input <
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110 1 125 16 5 4 0 2 222 0/ 293.6 400 600 800 1000 .005 4.5/ thermr 33 -24 -25/ 7 125 16 6 4 12 1 225 0/ 293.6 400 600 800 1000 1200 .005 4.5/ gaspr -21 -25 -27/ moder -27 28 stop EOF xnjoyinput <
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111 rm output rm input rm tape25 rm tape27 The third example is the script for the GENDF generation of 1H at 293.6 K, which is a group-average cross-section. GENDF generation script 1H psi get tape20:endfvib/neutron/h/1h psi get tape30:isotope/001-h/pendf/1h cat>input <
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112 6.7379E+04 1.8316E+05 3.0197E+05 3.8774E+05 4.9787E+05 8.2085E+05 1.3534E+06 1.7377E+06 2.2313E+06 2.8650E+06 3.6788E+06 6.0653E+06 7.7880E+06 1.0000E+07 1.2214E+07 1.7000E+07 2.0000E+07/ 0. 0. 0 0 1 622 622 5 1.00000E-05 1.00000E-10 1.02500E-04 1.11392E+05 1.07500E-04 1.17078E+04 1.12500E-04 1.14934E+04 1.17500E-04 1.26800E+04 1.23750E-04 1.30923E+04 1.31250E-04 1.34318E+04 1.38750E-04 1.42703E+04 1.46250E-04 1.58507E+04 1.55000E-04 1.63414E+04 1.65000E-04 1.82058E+04 1.75000E-04 1.73768E+04 1.85000E-04 1.99047E+04 1.95000E-04 2.03242E+04 2.05000E-04 2.14583E+04 2.15000E-04 2.28833E+04 2.25000E-04 2.35536E+04 2.35000E-04 2.45236E+04 2.47500E-04 2.67204E+04 2.62500E-04 2.80131E+04 2.75000E-04 2.94375E+04 2.90000E-04 3.06289E+04 3.10000E-04 3.44205E+04 3.30000E-04 4.39002E+04 3.50000E-04 5.48456E+04 3.70000E-04 5.74250E+04 3.90000E-04 5.97437E+04 4.12500E-04 2.37094E+05 4.37500E-04 3.94987E+05 4.62500E-04 4.13698E+05 4.87500E-04 2.37432E+05 5.12500E-04 1.27756E+05 5.37500E-04 1.19987E+05 5.62500E-04 1.29028E+05 5.87500E-04 1.31017E+05 6.15000E-04 1.36522E+05 6.45000E-04 1.46427E+05 6.75000E-04 1.47829E+05 7.05000E-04 1.52335E+05

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113 7.40000E-04 1.57270E+05 7.80000E-04 1.59279E+05 8.20000E-04 1.69389E+05 8.60000E-04 1.77644E+05 9.00000E-04 1.82363E+05 9.40000E-04 1.86206E+05 9.80000E-04 1.91534E+05 1.02500E-03 1.94651E+05 1.07500E-03 2.02277E+05 1.12500E-03 1.96166E+05 1.17500E-03 2.22202E+05 1.23750E-03 2.34042E+05 1.31250E-03 2.42536E+05 1.38750E-03 2.89091E+05 1.46250E-03 8.96979E+05 1.55000E-03 1.23997E+06 1.65000E-03 3.69749E+05 1.75000E-03 3.79239E+05 1.85000E-03 3.94793E+05 1.95000E-03 3.86523E+05 2.05000E-03 4.17001E+05 2.15000E-03 4.22666E+05 2.25000E-03 4.49554E+05 2.35000E-03 4.53076E+05 2.47500E-03 4.67133E+05 2.62500E-03 4.80566E+05 2.75000E-03 4.93421E+05 2.90000E-03 6.50854E+05 3.10000E-03 6.41867E+05 3.30000E-03 1.86963E+06 3.50000E-03 1.82744E+06 3.70000E-03 9.32410E+05 3.90000E-03 8.16610E+05 4.12500E-03 8.46620E+05 4.37500E-03 9.41475E+05 4.62500E-03 1.03928E+06 4.87500E-03 1.19626E+06 5.12500E-03 1.41696E+06 5.37500E-03 1.54108E+06 5.62500E-03 1.54855E+06 5.87500E-03 1.61868E+06 6.15000E-03 1.69974E+06 6.45000E-03 1.85409E+06 6.75000E-03 1.79930E+06 7.05000E-03 1.93360E+06 7.40000E-03 2.20656E+06 7.80000E-03 2.38016E+06 8.20000E-03 2.38413E+06 8.60000E-03 2.63303E+06 9.00000E-03 2.70718E+06 9.40000E-03 2.53813E+06 9.80000E-03 2.64016E+06 1.02500E-02 2.50710E+06 1.07500E-02 2.66570E+06 1.12500E-02 2.68812E+06 1.17500E-02 2.88462E+06 1.23750E-02 2.88435E+06

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114 1.31250E-02 2.97052E+06 1.38750E-02 3.01688E+06 1.46250E-02 3.13765E+06 1.55000E-02 3.15415E+06 1.65000E-02 3.35913E+06 1.75000E-02 3.40535E+06 1.85000E-02 3.46876E+06 1.95000E-02 3.56306E+06 2.05000E-02 3.57929E+06 2.15000E-02 3.77911E+06 2.25000E-02 3.82247E+06 2.35000E-02 3.96044E+06 2.47500E-02 3.97508E+06 2.62500E-02 4.06084E+06 2.75000E-02 4.02143E+06 2.90000E-02 4.05598E+06 3.10000E-02 4.13888E+06 3.30000E-02 4.12935E+06 3.50000E-02 4.17327E+06 3.70000E-02 4.08533E+06 3.90000E-02 4.20377E+06 4.12500E-02 4.06855E+06 4.37500E-02 3.98461E+06 4.62500E-02 3.84963E+06 4.87500E-02 3.65546E+06 5.12500E-02 3.62225E+06 5.37500E-02 3.47186E+06 5.62500E-02 3.36277E+06 5.87500E-02 3.29152E+06 6.15000E-02 3.13327E+06 6.45000E-02 3.02761E+06 6.75000E-02 2.87461E+06 7.05000E-02 2.79251E+06 7.40000E-02 2.62614E+06 7.80000E-02 2.43235E+06 8.20000E-02 2.26002E+06 8.60000E-02 2.10355E+06 9.00000E-02 1.94470E+06 9.40000E-02 1.76296E+06 9.80000E-02 1.60178E+06 1.02500E-01 1.37091E+06 1.07500E-01 1.20795E+06 1.12500E-01 1.06356E+06 1.17500E-01 9.34709E+05 1.23750E-01 7.97453E+05 1.31250E-01 6.45185E+05 1.38750E-01 5.21301E+05 1.46250E-01 4.29124E+05 1.55000E-01 3.65265E+05 1.65000E-01 3.00617E+05 1.75000E-01 2.50810E+05 1.85000E-01 2.22226E+05 1.95000E-01 2.03960E+05 2.05000E-01 1.94698E+05 2.15000E-01 1.70843E+05 2.25000E-01 1.56413E+05 2.35000E-01 1.39063E+05

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115 2.47500E-01 1.23933E+05 2.62500E-01 1.07513E+05 2.75000E-01 9.44459E+04 2.90000E-01 8.38652E+04 3.10000E-01 7.79995E+04 3.30000E-01 7.69131E+04 3.50000E-01 7.78976E+04 3.70000E-01 7.13618E+04 3.90000E-01 6.25304E+04 4.12500E-01 5.74419E+04 4.37500E-01 5.41677E+04 4.62500E-01 5.37839E+04 4.87500E-01 5.27516E+04 5.12500E-01 4.83931E+04 5.37500E-01 4.40871E+04 5.62500E-01 4.14834E+04 5.87500E-01 4.04210E+04 6.15000E-01 4.02728E+04 6.45000E-01 3.75293E+04 6.75000E-01 3.47404E+04 7.05000E-01 3.29599E+04 7.40000E-01 3.26086E+04 7.80000E-01 3.08642E+04 8.20000E-01 2.82453E+04 8.60000E-01 2.72272E+04 9.00000E-01 2.68425E+04 9.40000E-01 2.47227E+04 9.80000E-01 2.35090E+04 1.02500E+00 2.28553E+04 1.07500E+00 2.14013E+04 1.12500E+00 2.02889E+04 1.17500E+00 1.98370E+04 1.23750E+00 1.89241E+04 1.31250E+00 1.80412E+04 1.38750E+00 1.70493E+04 1.46250E+00 1.62187E+04 1.55000E+00 1.52852E+04 1.65000E+00 1.43153E+04 1.75000E+00 1.34840E+04 1.85000E+00 1.27385E+04 1.95000E+00 1.20301E+04 2.05000E+00 1.13302E+04 2.15000E+00 1.09432E+04 2.25000E+00 1.04680E+04 2.35000E+00 1.00116E+04 2.47500E+00 9.49607E+03 2.62500E+00 8.93245E+03 2.75000E+00 8.50770E+03 2.90000E+00 8.04132E+03 3.10000E+00 7.48494E+03 3.30000E+00 7.10110E+03 3.50000E+00 6.64629E+03 3.70000E+00 6.31124E+03 3.90000E+00 6.07504E+03 4.12500E+00 5.73795E+03 4.37500E+00 5.37180E+03 4.62500E+00 5.02473E+03

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116 4.87500E+00 4.68734E+03 5.12500E+00 4.54694E+03 5.37500E+00 4.32900E+03 5.62500E+00 4.13483E+03 5.87500E+00 3.94967E+03 6.15000E+00 3.70241E+03 6.45000E+00 3.41485E+03 6.75000E+00 3.34873E+03 7.05000E+00 3.26840E+03 7.40000E+00 3.19410E+03 7.80000E+00 3.03976E+03 8.20000E+00 2.87112E+03 8.60000E+00 2.60650E+03 9.00000E+00 2.51726E+03 9.40000E+00 2.48136E+03 9.80000E+00 2.41557E+03 1.02500E+01 2.30748E+03 1.07500E+01 2.20551E+03 1.12500E+01 2.09494E+03 1.17500E+01 1.95049E+03 1.23750E+01 1.85348E+03 1.31250E+01 1.81473E+03 1.38750E+01 1.69967E+03 1.46250E+01 1.62874E+03 1.55000E+01 1.53606E+03 1.65000E+01 1.43939E+03 1.75000E+01 1.37265E+03 1.85000E+01 1.28287E+03 1.95000E+01 1.19011E+03 2.05000E+01 1.15110E+03 2.15000E+01 1.11625E+03 2.25000E+01 1.07491E+03 2.35000E+01 1.01084E+03 2.47500E+01 9.74785E+02 2.62500E+01 9.23510E+02 2.75000E+01 8.82623E+02 2.90000E+01 8.45904E+02 3.10000E+01 7.86770E+02 3.30000E+01 7.31076E+02 3.50000E+01 6.76713E+02 3.70000E+01 6.63233E+02 3.90000E+01 6.28552E+02 4.12500E+01 5.96331E+02 4.37500E+01 5.65185E+02 4.62500E+01 5.37678E+02 4.87500E+01 5.07791E+02 5.12500E+01 4.79598E+02 5.37500E+01 4.65498E+02 5.62500E+01 4.37618E+02 5.87500E+01 4.27300E+02 6.15000E+01 4.12057E+02 6.45000E+01 3.93173E+02 6.75000E+01 3.76012E+02 7.05000E+01 3.57708E+02 7.40000E+01 3.41777E+02 7.80000E+01 3.27351E+02 8.20000E+01 3.10020E+02

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117 8.60000E+01 2.96310E+02 9.00000E+01 2.81527E+02 9.40000E+01 2.71001E+02 9.80000E+01 2.62460E+02 1.02500E+02 2.49756E+02 1.07500E+02 2.38263E+02 1.12500E+02 2.29706E+02 1.17500E+02 2.15895E+02 1.23750E+02 2.08350E+02 1.31250E+02 1.96853E+02 1.38750E+02 1.86736E+02 1.46250E+02 1.77788E+02 1.55000E+02 1.68005E+02 1.65000E+02 1.57999E+02 1.75000E+02 1.48808E+02 1.85000E+02 1.40065E+02 1.95000E+02 1.32228E+02 2.05000E+02 1.28678E+02 2.15000E+02 1.22949E+02 2.25000E+02 1.16982E+02 2.35000E+02 1.12146E+02 2.47500E+02 1.06776E+02 2.62500E+02 1.01285E+02 2.75000E+02 9.88564E+01 2.90000E+02 9.05281E+01 3.10000E+02 8.40352E+01 3.30000E+02 8.12130E+01 3.50000E+02 7.73429E+01 3.70000E+02 7.26886E+01 3.90000E+02 6.97050E+01 4.12500E+02 6.62174E+01 4.37500E+02 6.23773E+01 4.62500E+02 5.91608E+01 4.87500E+02 5.62886E+01 5.12500E+02 5.35397E+01 5.37500E+02 5.12097E+01 5.62500E+02 4.90359E+01 5.87500E+02 4.69523E+01 6.15000E+02 4.50036E+01 6.45000E+02 4.36525E+01 6.75000E+02 4.01648E+01 7.05000E+02 3.94226E+01 7.40000E+02 3.73955E+01 7.80000E+02 3.58432E+01 8.20000E+02 3.42065E+01 8.60000E+02 3.26461E+01 9.00000E+02 3.13037E+01 9.40000E+02 2.96581E+01 9.80000E+02 2.89464E+01 1.02500E+03 2.72184E+01 1.07500E+03 2.64064E+01 1.12500E+03 2.52277E+01 1.17500E+03 2.40533E+01 1.23750E+03 2.30500E+01 1.31250E+03 2.18300E+01 1.38750E+03 2.07924E+01 1.46250E+03 2.08161E+01

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118 1.55000E+03 1.76014E+01 1.65000E+03 1.74955E+01 1.75000E+03 1.66352E+01 1.85000E+03 1.57081E+01 1.95000E+03 1.50039E+01 2.05000E+03 1.41243E+01 2.15000E+03 1.37841E+01 2.25000E+03 1.30844E+01 2.35000E+03 1.25811E+01 2.47500E+03 1.20231E+01 2.62500E+03 1.20745E+01 2.75000E+03 9.33325E+00 2.90000E+03 1.01686E+01 3.10000E+03 9.62168E+00 3.30000E+03 9.08781E+00 3.50000E+03 8.61297E+00 3.70000E+03 8.84983E+00 3.90000E+03 7.28551E+00 4.12500E+03 7.17218E+00 4.37500E+03 7.08166E+00 4.62500E+03 6.51075E+00 4.87500E+03 6.30747E+00 5.12500E+03 5.99315E+00 5.37500E+03 5.88637E+00 5.62500E+03 5.63845E+00 5.87500E+03 5.00148E+00 6.15000E+03 5.05743E+00 6.45000E+03 5.17337E+00 6.75000E+03 4.67550E+00 7.05000E+03 4.25091E+00 7.40000E+03 4.17998E+00 7.80000E+03 4.00553E+00 8.20000E+03 3.89706E+00 8.60000E+03 3.80123E+00 9.00000E+03 3.46863E+00 9.40000E+03 3.43835E+00 9.80000E+03 3.29948E+00 1.02500E+04 3.16057E+00 1.07500E+04 3.03167E+00 1.12500E+04 2.91143E+00 1.17500E+04 2.83028E+00 1.23750E+04 2.66317E+00 1.31250E+04 2.62427E+00 1.38750E+04 2.36993E+00 1.46250E+04 2.29341E+00 1.55000E+04 2.16959E+00 1.65000E+04 2.31644E+00 1.75000E+04 1.78987E+00 1.85000E+04 1.92499E+00 1.95000E+04 1.72663E+00 2.05000E+04 1.68523E+00 2.15000E+04 1.65458E+00 2.25000E+04 1.63681E+00 2.35000E+04 1.48643E+00 2.47500E+04 1.45667E+00 2.62500E+04 1.39253E+00 2.75000E+04 1.34175E+00

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119 2.90000E+04 1.28224E+00 3.10000E+04 1.21440E+00 3.30000E+04 1.14166E+00 3.50000E+04 1.09411E+00 3.70000E+04 1.04148E+00 3.90000E+04 1.04817E+00 4.12500E+04 9.52045E-01 4.37500E+04 8.86440E-01 4.62500E+04 8.60768E-01 4.87500E+04 8.21000E-01 5.12500E+04 8.16519E-01 5.37500E+04 7.78143E-01 5.62500E+04 7.57025E-01 5.87500E+04 7.12712E-01 6.15000E+04 7.05326E-01 6.45000E+04 6.70146E-01 6.75000E+04 6.47471E-01 7.05000E+04 6.49592E-01 7.40000E+04 5.94036E-01 7.80000E+04 5.97701E-01 8.20000E+04 5.59031E-01 8.60000E+04 5.53158E-01 9.00000E+04 5.23244E-01 9.40000E+04 5.13646E-01 9.80000E+04 4.99129E-01 1.02500E+05 4.84361E-01 1.07500E+05 4.69020E-01 1.12500E+05 4.55019E-01 1.17500E+05 4.42498E-01 1.23750E+05 4.28230E-01 1.31250E+05 4.12943E-01 1.38750E+05 3.99563E-01 1.46250E+05 3.86109E-01 1.55000E+05 3.70919E-01 1.65000E+05 3.57720E-01 1.75000E+05 3.44193E-01 1.85000E+05 3.33287E-01 1.95000E+05 3.24873E-01 2.05000E+05 3.18704E-01 2.15000E+05 3.11395E-01 2.25000E+05 3.02932E-01 2.35000E+05 2.94863E-01 2.47500E+05 2.87225E-01 2.62500E+05 2.81422E-01 2.75000E+05 2.74762E-01 2.90000E+05 2.63875E-01 3.10000E+05 2.57601E-01 3.30000E+05 2.51576E-01 3.50000E+05 2.36205E-01 3.70000E+05 2.22911E-01 3.90000E+05 2.17551E-01 4.12500E+05 2.03821E-01 4.37500E+05 1.90210E-01 4.62500E+05 1.91767E-01 4.87500E+05 1.92974E-01 5.12500E+05 1.89199E-01 5.37500E+05 1.79901E-01

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120 5.62500E+05 1.74840E-01 5.87500E+05 1.65975E-01 6.15000E+05 1.45840E-01 6.45000E+05 1.51314E-01 6.75000E+05 1.54678E-01 7.05000E+05 1.51678E-01 7.40000E+05 1.49008E-01 7.80000E+05 1.47575E-01 8.20000E+05 1.38031E-01 8.60000E+05 1.30317E-01 9.00000E+05 1.21610E-01 9.40000E+05 1.14556E-01 9.80000E+05 1.06877E-01 1.05000E+06 1.02365E-01 1.15000E+06 9.93513E-02 1.25000E+06 9.31924E-02 1.35000E+06 8.55025E-02 1.45000E+06 8.34146E-02 1.55000E+06 7.77918E-02 1.65000E+06 7.17097E-02 1.75000E+06 6.70446E-02 1.85000E+06 6.00620E-02 1.95000E+06 5.53431E-02 2.05000E+06 5.00383E-02 2.15000E+06 4.70291E-02 2.25000E+06 4.29705E-02 2.35000E+06 4.05840E-02 2.45000E+06 3.71654E-02 2.55000E+06 3.42405E-02 2.65000E+06 3.08154E-02 2.75000E+06 2.68251E-02 2.85000E+06 2.32806E-02 2.95000E+06 2.09033E-02 3.05000E+06 2.23718E-02 3.15000E+06 1.95975E-02 3.25000E+06 1.72013E-02 3.35000E+06 1.55411E-02 3.45000E+06 1.42750E-02 3.55000E+06 1.33494E-02 3.65000E+06 1.26129E-02 3.75000E+06 1.19895E-02 3.85000E+06 1.17057E-02 3.95000E+06 1.13024E-02 4.05000E+06 1.10453E-02 4.15000E+06 1.04078E-02 4.25000E+06 9.29036E-03 4.35000E+06 8.43375E-03 4.45000E+06 8.21338E-03 4.55000E+06 7.85495E-03 4.65000E+06 7.48915E-03 4.75000E+06 7.13397E-03 4.85000E+06 6.54038E-03 4.95000E+06 6.09976E-03 5.05000E+06 5.71734E-03 5.15000E+06 5.23266E-03 5.25000E+06 5.01462E-03 5.35000E+06 4.50778E-03

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121 5.45000E+06 4.32385E-03 5.55000E+06 4.04844E-03 5.65000E+06 3.73203E-03 5.75000E+06 3.50559E-03 5.85000E+06 3.27342E-03 5.95000E+06 2.99521E-03 6.05000E+06 2.79417E-03 6.15000E+06 2.62603E-03 6.25000E+06 2.29804E-03 6.35000E+06 2.13045E-03 6.45000E+06 2.14103E-03 6.55000E+06 2.05380E-03 6.65000E+06 1.91286E-03 6.75000E+06 1.77415E-03 6.85000E+06 1.65621E-03 6.95000E+06 1.58032E-03 7.05000E+06 1.47963E-03 7.15000E+06 1.37417E-03 7.25000E+06 1.23015E-03 7.35000E+06 1.04327E-03 7.45000E+06 9.58872E-04 7.55000E+06 9.06643E-04 7.65000E+06 8.50671E-04 7.75000E+06 7.39195E-04 7.85000E+06 6.97696E-04 7.95000E+06 6.65709E-04 8.05000E+06 6.21309E-04 8.15000E+06 5.75894E-04 8.25000E+06 5.53358E-04 8.35000E+06 5.22186E-04 8.45000E+06 4.88980E-04 8.55000E+06 4.57344E-04 8.65000E+06 4.21226E-04 8.75000E+06 3.86891E-04 8.85000E+06 3.59791E-04 8.95000E+06 3.33857E-04 9.05000E+06 3.04194E-04 9.15000E+06 2.79452E-04 9.25000E+06 2.61515E-04 9.35000E+06 2.42883E-04 9.45000E+06 2.28576E-04 9.55000E+06 2.12582E-04 9.65000E+06 1.96471E-04 9.75000E+06 1.82368E-04 9.85000E+06 1.70061E-04 9.95000E+06 1.58978E-04 1.00500E+07 1.49620E-04 1.01500E+07 1.42300E-04 1.02500E+07 1.33991E-04 1.03500E+07 1.25919E-04 1.04500E+07 1.18060E-04 1.05500E+07 1.09277E-04 1.06500E+07 1.01455E-04 1.07500E+07 9.22656E-05 1.08500E+07 8.37989E-05 1.09500E+07 7.52020E-05 1.10500E+07 7.01292E-05

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122 1.11500E+07 6.53782E-05 1.12500E+07 6.11156E-05 1.13500E+07 5.72758E-05 1.14500E+07 5.37162E-05 1.15500E+07 5.05691E-05 1.16500E+07 4.64913E-05 1.17500E+07 4.22156E-05 1.18500E+07 3.86214E-05 1.19500E+07 3.48774E-05 1.20500E+07 3.20858E-05 1.21500E+07 3.07121E-05 1.22500E+07 2.92626E-05 1.23500E+07 2.70863E-05 1.24500E+07 2.56979E-05 1.25500E+07 2.38945E-05 1.26500E+07 2.16938E-05 1.27500E+07 2.05949E-05 1.28500E+07 1.86273E-05 1.29500E+07 1.64469E-05 1.30500E+07 1.52802E-05 1.31500E+07 1.46858E-05 1.32500E+07 1.36387E-05 1.33500E+07 1.30023E-05 1.34500E+07 1.22519E-05 1.35500E+07 1.12322E-05 1.36500E+07 9.95144E-06 1.37500E+07 9.36110E-06 1.38500E+07 8.80181E-06 1.39500E+07 7.94005E-06 1.40500E+07 7.17461E-06 1.41500E+07 6.86953E-06 1.42500E+07 6.65197E-06 1.43500E+07 6.09464E-06 1.44500E+07 5.51679E-06 1.45500E+07 5.09217E-06 1.46500E+07 4.79646E-06 1.47500E+07 4.37883E-06 1.48500E+07 3.84796E-06 1.49500E+07 3.63492E-06 1.50500E+07 3.36755E-06 1.51500E+07 3.14592E-06 1.52500E+07 3.02779E-06 1.53500E+07 2.81348E-06 1.54500E+07 2.44568E-06 1.55500E+07 2.47534E-06 1.56500E+07 2.04586E-06 1.57500E+07 1.96005E-06 1.58500E+07 1.88672E-06 1.59500E+07 1.64431E-06 1.60500E+07 1.52578E-06 1.61500E+07 1.40991E-06 1.62500E+07 1.39023E-06 1.63500E+07 1.29623E-06 1.64500E+07 1.13336E-06 1.65500E+07 1.11841E-06 1.66500E+07 1.00389E-06 1.67500E+07 9.97824E-07

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123 1.68500E+07 8.68072E-07 1.69500E+07 7.80811E-07 1.70500E+07 7.33071E-07 1.71500E+07 6.99449E-07 1.72500E+07 5.81615E-07 1.73500E+07 5.66391E-07 1.74500E+07 5.78709E-07 1.75500E+07 5.23847E-07 1.76500E+07 4.28170E-07 1.77500E+07 4.49953E-07 1.78500E+07 3.89436E-07 1.79500E+07 3.15445E-07 2.00000E+07 1.00000E-10/ 3/ 3 221 'free'/ 3 222 'h in water thermal scattering'/ 3 225 'h in zrh inelastic thermal scattering'/ 3 226 'h in zrh elastic thermal scattering'/ 3 251 'mubar 293.6'/ 3 252 'xi'/ 3 259 '1/v'/ 6/ 6 221 'free'/ 6 222 'h in water thermal scattering'/ 6 225 'h in zrh inelastic thermal scattering'/ 6 226 'h in zrh elastic thermal scattering'/ 0/ 0/ stop EOF xnjoyinput <
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124 221 0 6/ 0 0/ h1-00 125 1 293.6/ / stop EOF xnjoy
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125 APPENDIX D DETAILED CALCULATION OF NUMBER DENSITIES IN FUEL REGIONS OF SQUARE-LATTICE HONEYCOMB The ratios of uranium carbide, zirconi um carbide, and niobium carbide are calculated based on their lattice parameters. Th e detailed calculation of number densities of isotopes in the fuel element is presented. 31000 4T U U U U Uwa R N Av M N (D-1) where: U is uranium density (g/cm3) UN is uranium number density (atoms/b cm) UM is uranium atomic weight UR is uranium carbide atom ratio Av is Avogadros number (0.6022 atoms/b cm) Twa is total weighted lattice para meters of UC, ZrC, and NbC. The total weighted lattice parameters of UC, ZrC, and NbC can be defined as the sum of each weighted lattice parameter. NbC ZrC UC Taw aw aw aw (D-2)

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126 NbC Nb NbC ZrC Zr ZrC UC U UCa R aw a R aw a R aw (D-3) where: UCwa is the UC weighted lattice parameter ZrCwa is the ZrC weighted lattice parameter NbCwa is the NbC weighted lattice parameter UCa is the UC lattice parameter (0.4961 10-9 m) ZrCa is the ZrC lattice parameter (0.4697 10-9 m) NbCa is the NbC lattice parameter (0.4469 10-9 m) ZrR is the ZrC atom ratio NbR is the NbC atom ratio Assuming the constant RZr:RNb = 3:1, the ZrC ratio and Nb C ratios can be related to the UC ratio by: U Nb U ZrR R R R 1 25 0 1 75 0 (D-4) Substituting Equation D-4 into Equation D-2, NbC U ZrC U UC U Ta R a R a R aw 1 25 0 1 75 0 NbC ZrC NbC ZrC UC U U NbC NbC U ZrC ZrC UC Ua a a a a R R a a R a a a R25 0 75 0 25 0 75 0 25 0 25 0 75 0 75 0 Let NbC ZrC NbC ZrC UCa a C a a a C25 0 75 0 25 0 75 02 1

PAGE 142

127 And the equation becomes: 2 1C C R waU T (D-5) Substitute D-5 into D-1, U U U U U U U U U U U U U U U U U u T UAv M R C C C R C C R C R M R C C C R C C R C R Av M R C C R Av M R wa Av 250 3 3 3 3 250 250 2503 2 2 2 1 2 2 1 2 3 1 3 3 2 2 2 1 2 2 1 2 3 1 3 3 2 1 3 0 250 3 33 2 2 2 1 2 2 1 2 3 1 3 C Av M C C R C C R C RU U U U U (D-6) Divide D-6 by C1 3 to get: 0 250 3 33 1 3 2 3 1 2 2 1 3 1 2 2 1 2 3 C C C Av M C C R C C C R RU U U U U 00 1 2 2 3 A A R A R RU U U (D-7) where: 3 1 3 2 0C C A 3 1 2 2 1 1250 3 C Av M C C AU U 1 2 23 C C A

PAGE 143

128 Equation D-7 can be solved using the Cubic Formula or the Cardano Method, and one of its roots will be the atom ratio of uranium carbide. Finally, the atom ratios of zirconium carbide and niobium carbide can also be determined. Us ing these ratios, the number density of all isotopes in th e fuel element can be calculated.

PAGE 144

129 APPENDIX E DETAILED CALCULATION OF THE LOCATIONS OF EACH RING OF MATERIAL IN THE SQUARE-LATTICE HONEYCOMB HOMOGENEOUS MODEL The locations of each ring of materi al in the Square-Lattice Honeycomb homogeneous model, shown in Figure E-1, are calculated based on th e center locations, RCF, of fuel materials in the heterogene ous model, shown in Figure E-2. These calculations are based on the c onservation of mass in the model. Furthermore, they can be simplified as the conservation of area of each material in the model. Based on Figure E-1, each of the radii can be represented as follows: ROF1RCF1ROG1 ROF2RCF2ROG3 x2x1 ROF1RCF1ROG1 ROF2RCF2ROG3 x2x1 Figure E-1. The locations of fuel ring ma terial in the Square-Lattice Honeycomb homogeneous model.

PAGE 145

130 RCF1RCF2 RCF1RCF2 Figure E-2. The locations of fuel materi al in the Square-Lattice Honeycomb heterogeneous model. 2 2 2 2 2 3 1 1 1 1 1 1x R R x R R x R R x R RCF OF CF OG CF OF CF OG (E-1) where: RCF1 is center of first radial fuel location RCF2 is center of second radial fuel location ROG1 is the outer radius of first graphite coating ROF1 is the outer radius of first fuel material ROG3 is the outer radius of third graphite coating ROF2 is the outer radius of second fuel material Based on Figure E-2, the area of the fuel, graphite coating, a nd zirconium oxide coating for each fuel element are calculated as follows: 2 2 2 2 2 ZRO G ZRO G F G F FR R a R R a R a (E-2) where: aF is the area of fuel material aG is the area of graphite coating aZRO is the area of zirconium oxide coating

PAGE 146

131 Furthermore, since there are six fuel elemen ts in the first radial location, the total area of fuel, graphite coating, and zirconium oxi de coating of the first radial location can be calculated as follows: 2 2 1 2 2 1 2 16 6 6 6 6 6ZRO G ZRO ZRO G F G G F F FR R a A R R a A R a A (E-3) where: AF1 is the total area of fuel material in the first radial location AG1 is the total area of graphite coating in the first radial location AZRO1 is the total area of zirconium oxid e coating in the first radial location The total areas of fuel, graphite, and zircon ium oxide of 12 fuel elements in the second radial location are calculated as follows: 2 2 1 2 2 1 2 112 12 12 12 12 12ZRO G ZRO ZRO G F G G F F FR R a A R R a A R a A (E-4) where: AF2 is the total area of fuel material in the second radial location AG2 is the total area of graphite co ating in the second radial location AZRO2 is the total area of zirconium oxide coating in the second radial location Finally, the ring locations in Figure E-1 can be calculated as follows: 1 2 1 1 1 2 2 1 1 1 2 1 2 1 1 1 2 1 2 2 1 1 2 1 1 22 3 2 3 2 2 6 6CF F CF F CF CF CF CF F CF CF FR R x x R R x x R R x x R R R x R x R R 1 2 1 1 1 2 1 12 3 2 3CF F CF OF CF F CF OGR R R R R R R R (E-5)

PAGE 147

132 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 23 3 2 2 12 12CF F CF F CF CF CF CF F CF CF FR R x x R R x x R R x x R R R x R x R R 2 2 2 2 2 2 2 23 3CF F CF OF CF F CF OGR R R R R R R R (E-6) ROG2RG1RZRO1 ROG4ROG3RZRO3 ROF2ROF1 y1y2 ROG2RG1RZRO1 ROG4ROG3RZRO3 ROF2ROF1 y1y2 Figure E-3. Locations of graphite coa ting in the Square -Lattice Honeycomb homogeneous model. The next calculation shows the determina tion of the locations of the graphite coating in the first and second radial locations as shown in Figure E-3. 2 2 4 2 3 3 1 1 2 1 1 1y R R y R R y R R y R ROF OG OG OZRO OF OG OG OZRO (E-7) The detail of the calculation of the graphite coating ri ng location follows:

PAGE 148

133 1 1 2 2 1 1 1 1 1 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 1 2 1 2 2 2 1 1 2 1 2 1 2 1 1 2 23 3 2 2 6 6OG OF G F OG OF G F OG OG OG OF OF OF G F OG OG OF OF G FR R R R y y R y R R R y y R R R R y y R R R R y R R R y R R R 1 1 2 2 1 1 1 1 2 2 1 23 3OG OF G F OG OZRO OG OF G F OF OGR R R R R R R R R R R R (E-8) 3 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 2 2 2 2 2 2 26 6 2 2 12 12OG OF G F OG OF G F OG OG OG OF OF OF G F OG OG OF OF G FR R R R y y R y R R R y y R R R R y y R R R R y R R R y R R R 3 2 2 2 3 3 3 2 2 2 2 43 3OG OF G F OG OZRO OG OF G F OF OGR R R R R R R R R R R R (E-9) The rest of the locations can be calculated using the same method.

PAGE 149

134 LIST OF REFERENCES Alcouffe, R.E., Baker, R.S., Dahl, J.A., and Turner, S.A. PARTISN 2.99, MultiDimensional, Time-Independent or Ti me-Dependent, Multigroup, Discrete Ordinates Transport Code System Los Alamos, NM: Los Alamos National Laboratories, 2002. Furman, E. Thermal Hydraulic Design Anal ysis of Ternary Carbide Fueled SquareLattice Honeycomb Nuclear Rocket Engine, 16th Symposium on Space Nuclear Power and Propulsion edited by M.S. El-Genk, New York: AIP Conference Proceedings, 1999. Gunn, S. Nuclear propulsiona hist orical perspective, Space Policy Volume: 17, Issue: 4, pp. 291-298, November 4, 2001. Hendricks, J.S. MCNP4B, Monte Carlo N-Part icle Transport Code System Manual. RSICC Computer Code Collection Los Alamos, NM: Los Alamos National Laboratories, March 19, 1997. Kinsey, R. ENDF-102 Data Formats and Proced ures for the Evaluated Nuclear Data File ENDF Upton, NY: Brookhaven Nationa l Laboratory, October 1975. MacFarlane, R.E., and Muir, D.W. The NJ OY Nuclear Data Proce ssing System Version 91 Los Alamos, NM: Los Alamos National Laboratories, 1994. MacFarlane, R.E. TRANSX 2: A Code for Interfacing MATXS Cross-Section Libraries to Nuclear Transport Codes Los Alamos, NM: Los Alamos National Laboratories, 1992. Pelaccio, D.G., and El-Genk, M.S. A Re view of Nuclear Thermal Propulsion Carbide Fuel Corrosion and Key Issues, Final Re port, Albuquerque, NM: University of New Mexico, November 1994. Robbins, W. H. An Historical Perspectiv e of the NERVA Nuclear Rocket Engine Technology Program, Brook Park, OH: Le wis Research Center, July 1991.

PAGE 150

135 BIOGRAPHICAL SKETCH Reza R. Gouw was born on March 03, 1972, under the name of Reza Widargo in Malang, East Java, Indonesia. He lived in Indonesia for 18 years before his family immigrated to the United States in 1991. In January 1998, he was sworn in to become a United States citizen, and in March 1998, he changed his name to Reza Raymond Gouw. He completed his elementary education at the Trinitas Elementary School in Jakarta, Indonesia. He continued his e ducation at the Karangturi High School in Semarang, Central Java. When he was in 11t h grade, his family moved to Melbourne, Florida. He then finished his high school at the Florida Air Academy. He graduated from the Florida Air Academy on May 31, 1992, with excellencies in physics and math. After his graduation from the Florida Air Acad emy, he enrolled at the University of Florida, Gainesville, Florida, where he studi ed nuclear engineering. He graduated with honors and received his Bachelor of Scien ce degree in nuclear engineering from University of Florida in August 1997. In August 1997, he decided to continue his education by enrolling in graduate program at University of Florida. He has worked for Innovative Nuclear Space Power and Propulsion Institute (IN SPI) from 1997 to present. He was also a recipient of Department of Energy (DOE) Nuclear Engineering/Health Physics Fellowship. He received his Ma ster of Engineering degree in nuclear engineering science from Univer sity of Florida in May 2000. Also in 2000, he did his practicum at the Los Alamos National Laborat ory, when he was hired as a graduate

PAGE 151

136 research assistant in 2003 to 2004. His res ponsibilities and research interests are in the area of space nuclear power and propulsion.


Permanent Link: http://ufdc.ufl.edu/UFE0009660/00001

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Title: Nuclear Design Methodology for Analyzing an Ultra High Temperature Highly Compact Ternary Carbide Reactor
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Copyright Date: 2008

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Material Information

Title: Nuclear Design Methodology for Analyzing an Ultra High Temperature Highly Compact Ternary Carbide Reactor
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
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NUCLEAR DESIGN METHODOLOGY FOR ANALYZING ULTRA HIGH
TEMPERATURE HIGHLY COMPACT TERNARY CARBIDE REACTOR

















By

REZA RAYMOND GOUW


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


2006

































Copyright 2006

by

Reza Raymond Gouw















ACKNOWLEDGMENTS

This research would not be complete without the contributions of several people. I

would like to thank Dr. Samim Anghaie for his support and encouragement throughout

the entire project. His guidance and wisdom have allowed me to complete this work in a

timely fashion. I also would like to express my thanks to Dr. Edward Dugan for his

unfailing support throughout my undergraduate and graduate time at the University of

Florida. His guidance both with the research and with my curriculum was invaluable. I

would like to thank Dr. Robert Little and the entire X-5 Data Team for their support and

guidance during my stay at the Los Alamos National Laboratory. I would like to send

special thanks to Dr. Morgan White for his support and guidance as a mentor as well as a

friend. I also would like to thank Dr. Bob McFarlane of Los Alamos National Laboratory

for his help and guidance in learning NJOY nuclear codes. I would like to thank Dr.

Travis Knight for his help throughout this study. Last but not least, I would also like to

thank Dr. Darryl Butt for agreeing to take the time and effort to be on my supervisory

committee. This study would not be complete without assistance from the Department of

Energy in the form of DOE Nuclear Engineering/Health Physics Fellowship.

Furthermore, the Los Alamos National Laboratory provided its facility and its staff to

help the completion of this study.

I would also like to express my thanks to the faculty, my colleagues and friends in

the Nuclear and Radiological Engineering department. Specifically, I would like to thank

Beth Bruce for her help in the academic registration. Their support, suggestions and









encouragement helped me to complete this research. Finally, I would like to thank my

wife, Virginia Pangalila, for her support and sacrifices during the completion of this

study.















TABLE OF CONTENTS
Page

A C K N O W L E D G M E N T S .................................................................................................. ii

LIST OF TABLES ............................................. ............................ vii

LIST OF FIGURES .................................................. ............................ ix

ABSTRACT ............................................ ............. ................. xiii

CHAPTER

1 IN TR O D U C T IO N ........ .. ......................................... ..........................................1.

Background on Nuclear Thermal Rockets.................................. ..................
N nuclear vs. Chem ical Propulsion ....................... ............................................. 1
The K IW I and Tory Program s.................. ....................................................2...
R over and N ER V A ........................................................... .. ...... .................... .3
Background on Square-Lattice Honeycomb System ..............................................6...
Background on an Evaluated Nuclear Data File............... .....................................7...

2 CR O SS-SECTION PR O CE SSIN G ......................................................... ...............9...

Cross-Sections Processing Codes .......................................................9......
N JOY N nuclear D ata Processing System ........................................... ...............9...
TRANSX 2001 a Code for Interfacing MATXS Cross-Section Libraries to
N nuclear Transport Codes ......................................................... 10
G enerating PE N D F library ......................................... .......................................... 11
R E C O N R M odule .............. .................. ................................................. .. 11
B R O A D R M odule .............. .................. ............................................... 12
U N R E SR M odule .. .................................................................... ... ............ 13
H E A TR M odule ............... ................ ............................................... 14
TH ER M R M odule .............. .................. ............................................... 14
A utom ation of P rocesses ................................................................ ............... 16
Generating Cross-Sections for M onte Carlo Code ................................ ................ 16
PURR Module ............................... ......... ....................... 16
GA SPR M odule ..... .. ................................ ........................................ 17
ACER Module ........................ ........... ............................... 17
A utom ation of P rocesses ................................................................ ............... 18
Generating Cross-Sections for Deterministic Code .............................................. 18
G R O U PR M odule .............. .................. ............................................... 19
D T F R M o du le ..................................................................................................... 19









M A T X SR M o du le ............................................................................................... 2 0
T R A N SX C alcu lation ..........................................................................................2 0

3 NUCLEAR TRANSPORT CODES DESCRIPTIONS.........................................21

M C N P V e rsio n 5 ........................................................... ............................................. 2 1
P A R T ISN V version 3.56 ..................................................................... ................ 22

4 SQUARE-LATTICE HONEYCOMB (SLHC) NUCLEAR ROCKET ENGINE
D E SC R IP T IO N .......................................................................................................... 24

G eom etry D description ............................................................................ ................ 24
MCNP5 Geometry Representation of SLHC .................................................25
PARTISN 3.51 Geometry Representation of SLHC......................................27
Materials Description............................. ........ .....................27
M materials in Heterogeneous SLHC..................................................... 27
M materials in H om ogeneous SLH C .................................................. ................ 28

5 M E T H O D O L O G Y ................................................... ............................................ 49

Monte Carlo Neutron Cross-Sections Library Generation ....................................49
Choosing the Correct W eight Function.................................................... 51
G odiva Calculations .................................................................. ........... ...............5.. 51
Square-Lattice H oneycom b Calculations ....................................... ................ 53
N eutronics A naly sis ................................. ........ ................ .... .. .. ...... ........... 54
Energy Spectrum s Characterization ............................................... ................ 54
Power Distributions and Flux Profiles Analyses............................................55
Tem perature Coefficient A nalyses ................................................. ................ 57
C control D rum s A nalyses .............. .................................................. ................ 58
W ater Subm version A accident A nalysis............................................ ................ 60

6 C O N C L U SIO N S .................................................. .............................................. 99

APPENDIX

A LISTS OF NUCLIDES PROCESSED IN THE RGOUW CROSS-SECTION
L IB R A R IE S .............................................................................................................. 10 1

B ENERGY GROUP STRUCTURES FOR SQUARE-LATTICE HONEYCOMB .. 106

C AUTOMATION PROCESS FOR PENDF, ACER, AND DTFR MODULES ........ 109

D DETAILED CALCULATION OF NUMBER DENSITIES IN FUEL REGIONS OF
SQUARE-LATTICE HONEYCOM B ....................... .................... ..................... 125









E DETAILED CALCULATION OF THE LOCATIONS OF EACH RING
MATERIALS IN THE SQUARE-LATTICE HONEYCOMB HOMOGENEOUS
M O D E L ................................................................................................................. ... 12 9

LIST O F R EFEREN CE S .. .................................................................... ............... 134

BIOGRAPH ICAL SKETCH .................. .............................................................. 135















LIST OF TABLES


Table page

4-1. Comparison calculations of the "true" Square-Lattice Honeycomb (SLHC)
heterogeneous model and SLHC heterogeneous model with fuel region
h om og en ization ........................................................................................................ 32

4-2. Number densities of the isotopes in the fuel region...............................................37

4-3. Properties of non-fuel elements in the Square-Lattice Honeycomb heterogeneous
m o d el ..................................................................................................... ......... .. 3 8

4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous
m o d el ...................................................................................................... ........ .. 3 9

4-5. Properties of the Square-Lattice Honeycomb reactor .........................................42

4-6. Properties of non-fuel materials in the Square-Lattice Honeycomb homogeneous
m o d el ...................................................................................................... ........ .. 4 3

4-7. The properties of axial region 1 fuel materials in the Square-Lattice Honeycomb
hom ogeneous m odel ... ................................................................... .............. 44

4-8. The properties of axial region 2 fuel materials in the Square-Lattice Honeycomb
hom ogeneous m odel ... ................................................................... .............. 45

4-9. The properties of axial region 3 fuel materials in the Square-Lattice Honeycomb
hom ogeneous m odel ... ................................................................... .............. 46

4-10. The properties of axial region 4 fuel materials in the Square-Lattice Honeycomb
hom ogeneous m odel ... ................................................................... .............. 47

4-11. The properties of axial region 5 fuel materials in the Square-Lattice Honeycomb
hom ogeneous m odel ... ................................................................... .............. 48

5-1. Comparison of PARTISN calculations utilizing correct and incorrect multigroup
neutron cross-sections for four surrounding shells materials with the radius of
235U Godiva is 6.7 cm .. ................................................................................. 66

5-2. Comparison of PARTISN calculations utilizing "true" and "false" multigroup
neutron cross-sections for the Square-Lattice Honeycomb................................67









5-3. The intermediate temperature conditions and the operating temperature conditions
for the Square-Lattice Honeycomb ................................................... 68

A-1. List of essential isotopes in the RGOUW cross-section libraries........................... 102

A-2. List of thermal scattering data in the RGOUW cross-section libraries .................. 105

B-1. The 45-energy group structures...... ............ .......... ...................... 107

B-2. The LANL-187 energy group structures........................................108















LIST OF FIGURES


Figure page

4-1. Square-Lattice Honeycomb nuclear reactor geometry description.......................29

4-2. The Square-Lattice Honeycomb fuel wafers fabrication into fuel element ............30

4-3. The fabrication of the Square-Lattice Honeycomb fuel elements into fuel
a sse m b ly ............................................................................................................ .. 3 1

4-4. Energy spectra of "true" Square-Lattice Honeycomb (SLHC) heterogeneous
model and SLHC heterogeneous model with fuel region homogenization. ...........33

4-5. Energy spectra of "true" Square-Lattice Honeycomb (SLHC) heterogeneous
model and SLHC heterogeneous model with fuel region homogenization in the
therm al energy range (less than leV ).................................................. ................ 34

4-6. Geometry description of the Square-Lattice Honeycomb heterogeneous model.....35

4-7. Here is the geometry description of the Square-Lattice Honeycomb homogeneous
m o d el ...................................................................................................... ........ .. 3 6

5-1. Energy spectrum comparison between the SLHC heterogeneous and SLHC
hom ogeneous m odels at 293.6 K ........................................................ ................ 61

5-2. Energy spectrum comparison between the SLHC heterogeneous and SLHC
homogeneous models in thermal energy range at 293.6 K .................................62

5-3. Energy spectrum comparison between heterogeneous and homogeneous models
of the Square-Lattice Honeycomb in fast energy range at 293.6 K ......................63

5-4. Geometry description of the Godiva sphere surrounded by hydrogen gas ............64

5-5. Energy spectrum comparison between heterogeneous model of the Square-Lattice
Honeycomb and 23 5U Godiva surrounded by H2 gas at 293.6 K obtained from
M C N P ................................................................................................................ .. 6 5

5-6. This figure shows the system's energy spectrum of the Square-Lattice Honeycomb
Heterogeneous model at room, intermediate, and operating temperatures..............69









5-7. This figure presents the fuel region's energy spectrum of the Square-Lattice
Honeycomb Heterogeneous model at room, intermediate, and operating
tem p eratu re s. ............................................................................................................ 7 0

5-8. This plot presents the moderator region's energy spectrum of the Square-Lattice
Honeycomb Heterogeneous model at room, intermediate, and operating
tem p eratu re s. ............................................................................................................ 7 1

5-9. This plot shows the system's energy spectrum of the Square-Lattice Honeycomb
Homogeneous model at room, intermediate, and operating temperatures ............72

5-10. The plot presents the total cross-section data of several important isotopes in the
Square-L attice H oneycom b ...................................... ....................... ................ 73

5-11. The plot presents the axial power distribution of the Square-Lattice Honeycomb in
the first and second radial fuel regions................................................ ................ 74

5-12. The figure presents the radial power distribution of the Square-Lattice Honeycomb
in the second, third and fourth axial fuel regions ................................ ................ 75

5-13. The figure shows the radial power distribution of the Square-Lattice Honeycomb in
the first and fifth axial fuel regions ..................................................... ................ 76

5-14. The plot illustrates the fast neutron energy (> 65 keV) axial flux profiles of the
Square-Lattice Honeycomb in the two radial fuel regions..................................77

5-15. The epithermal neutron energy (2.5 eV 65 keV) axial flux profiles of the Square-
Lattice Honeycomb in the two radial fuel regions. .............................................78

5-16. The figure illustrates the thermal neutron energy (< 2.5 eV) axial flux profiles of
the Square-Lattice Honeycomb in the two radial fuel regions...............................79

5-17. The figure presents the fast neutron energy (> 65 keV) radial flux profiles of the
Square-Lattice Honeycomb in the five axial fuel regions...................................80

5-18. The epithermal neutron energy (2.5 eV 65 keV) radial flux profiles of the Square-
Lattice Honeycomb in the five axial fuel regions. ..............................................81

5-19. The figure presents the thermal neutron energy (< 2.5 eV) radial flux profiles of the
Square-Lattice Honeycomb in the five axial fuel regions...................................82

5-20. The plot shows the plot of fuel temperature coefficient of the Square-Lattice
H oneycom b during startup ....................................... ....................... ................ 83









5-21. The plot of fuel temperature coefficient of the Square-Lattice Honeycomb at the
interim ediate tem perature ranges ......................................................... ................ 84

5-22. The plot of fuel temperature coefficient of the Square-Lattice Honeycomb at the
operating tem perature ranges ..................................... ...................... ................ 85

5-23. The plot of moderator temperature coefficient of the Square-Lattice Honeycomb at
the operating tem perature ranges ........................................................ ................ 86

5-24. The plot presents the system temperature coefficient of the Square-Lattice
H o n ey co m b ............................................................................................................ 8 7

5-25. The plot illustrates the thermal scattering cross section of 1H and Zr in zirconium
h y d rid e .................................................................................................... ........ .. 8 8

5-26. Plot of the 1H neutron absorption cross section..................................................89

5-27. Critical configurations of the control drums in the Square-Lattice Honeycomb at
2 9 3 .6 K ................................................................................................................. ... 9 0

5-28. The "fully-in" configurations of the control drums in the Square-Lattice
Honeycomb, which has k-eff of 0.89858 + 0.00005 at 293.6 K..............................91

5-29. The "fully-out" configurations of the control drums in the Square-Lattice
Honeycomb, which has k-eff of 1.05961 + 0.00006 at 293.6 K..............................92

5-30. The reactivity worth plot of the control drums from "fully-in" to "fully-out"
positions at 293.6 K .. .................................................................................... 93

5-31. Three different configurations and their k-eff values of two control drums jammed
at the "fully-out" position ........................................ ........................ ................ 94

5-32. Three different configurations and their k-eff values with three control drums
jam m ed at the "fully-out" position ...................................................... ................ 95

5-33. Three different configurations and their k-eff values with four control drums
jam m ed at the "fully-out" position ...................................................... ................ 96

5-34. Configuration of the Square-Lattice Honeycomb reactor for water submersion
a c c id e n t.................................................................................................................. ... 9 7

5-35. Modified configuration of the Square-Lattice Honeycomb reactor for water
subm version accident. ............................... .. ....................... .......................... 98









E-1. The locations of fuel ring material in the Square-Lattice Honeycomb homogeneous
m odel ............. . ............................................................. .......... 12 9

E-2. The locations of fuel material in the Square-Lattice Honeycomb heterogeneous
m o d el ............. . ............................................................. ....... .. 13 0

E-3. Locations of graphite coating in the Square-Lattice Honeycomb homogeneous
m o d el ............. . ............................................................. .......... 132















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

NUCLEAR DESIGN METHODOLOGY FOR ANALYZING ULTRA HIGH
TEMPERATURE HIGHLY COMPACT TERNARY CARBIDE REACTOR

By

Reza Raymond Gouw

December 2006

Chair: Samim Anghaie
Major: Nuclear Engineering Sciences

Recent studies at the Innovative Nuclear Space Power and Propulsion Institute

(INSPI) have demonstrated the feasibility of fabricating solid solutions of ternary carbide

fuels such as (U,Zr,Nb)C, (U,Zr,Ta)C, (U,Zr,Hf)C and (U,Zr,W)C. The necessity for

accurate nuclear design analysis of these ternary carbides in highly compact nuclear

space systems prompted the development of nuclear design methodology for analyzing

these systems. This study will present the improvement made in the high temperature

nuclear cross-sections. It will show the relation between Monte Carlo and Deterministic

calculations. It will prove the significant role of the energy spectrum in the multigroup

nuclear cross-sections generation in the highly-thermalized-nuclear system. The nuclear

design methodology will address several issues in the homogenization of a nuclear

system, such as energy spectrum comparison between a heterogeneous system and

homogeneous system. It will also address several key points in the continuous and

multigroup nuclear cross-sections generation. The study will present the methodology of









selecting broad energy group structures. Finally, a comparison between the Monte Carlo

and Deterministic methods will be performed for the Square-Lattice Honeycomb Nuclear

Space Reactor. In the comparison calculations, it will include the system characterization

calculations, such as energy spectrum comparison, 2-D power distributions, temperature

coefficient analysis, and water submersion accident analysis.















CHAPTER 1
INTRODUCTION

Background on Nuclear Thermal Rockets

Nuclear propulsion has been around for over 40 years. In 1955 the Atomic Energy

Commission (AEC) initiated research and development work, and it was directed toward

the utilization of a nuclear thermal rocket (NTR) concept to propel single-stage, ballistic

missiles over intercontinental trajectories. At the same time, the Los Alamos Scientific

Laboratory (LASL), now Los Alamos National Laboratory and the Lawrence Livermore

Laboratories, had begun exploratory propulsion research programs. At Los Alamos

Scientific Laboratory the program was known as the KIWI program, and the Tory

program was at the Lawrence Livermore Laboratories. However, in 1970 the work came

to a halt because of the lack of interest and funds (Gunn, 2001).

Nuclear vs. Chemical Propulsion

All liquid rocket propulsion systems rely on the creation of a continuous supply of

high-pressure, high-temperature gas, and the expansion of that gas through a suitable

supersonic nozzle to the system's low-pressure environment. In conventional, chemical

system, the combustion process defines the temperature and the average molecular

weight of the propellants, thereby determining the performance (specific impulse) of the

propulsion system.

A nuclear thermal rocket, on the other hand, depends on a temperature source of

thermal energy, typically a high-temperature, solid-core reactor, to heat a single









propellant to as high a temperature as possible, and then to expand the resulting high-

temperature gas to the exhaust pressure. In NTR engine, the propellant is pumped from

its supply tank to the reactor-nozzle subsystem, where it is first used to cool the nozzle

and the reactor pressure shell, the neutron reflector, and the core support structure. Next,

the propellant passes through the reactor core where most of the fission energy being

released by uranium atoms is absorbed by the propellant. Finally, the super-heated

propellant is expanded out of the supersonic nozzle.

The KIWI and Tory Programs

The choice of system's propellant was influenced by concerns over the potential for

chemical interactions between the fuel elements and the propellant, as well as by the

availability of suitable feed systems equipment to deliver the propellant. The choice of

propellant in the KIWI reactors was ammonia, and nitrogen was selected for the Tory

reactors. Graphite was chosen for the requisite fuel elements because of its high-

temperature capabilities and its ability to serve both as a container of uranium atoms and

as an effective high-energy, fission-produced neutron moderator.

In 1957, the AEC reviewed the two nuclear rocket reactor programs and finally

chose to focus their resources on the KIWI reactor, part of the Rover program. LASL, in

turn, reviewed its development plans and decided to switch the propellant from ammonia

to hydrogen gas. A model liquid hydrogen pump developed by Rocketdyne Division of

North American Aviation initiated the propellant switch because it was capable of

delivering 38 liters per minute at 105 atm. Initially, the KIWI test program was

concentrated on relatively modest power-density reactor. The power of KIWI A, A', and

A3 were rated at 100 MW. The three KIWI reactors were tested at the nuclear rocket

development site in Nevada in July 1959 and in July and October 1960. The results of









these tests met their objectives and clearly demonstrated that coated graphite-fuel

elements could be used to heat hydrogen to temperature in excess of 2000 K.

With the successful test results, LASL focused its attention on the development of

the KIWI-B reactor series. These reactors generated heat in excess of 1,000 MW. A

total of five KIWI-B reactors were tested at the Nevada test site between December 7,

1961, and September 10, 1964. The final and complete verification of the capabilities of

the KIWI reactor occurred on July 28, 1964, when the KIWI B-4E reactor was operated

at full power and temperature for 8 minutes. Then, on September 10, 1964, the same

reactor was restarted and operated at full power and temperature for an additional 2.5

minutes. The test was judged as a complete success.

The formation of NASA in July 1958 produced a major change in the thrust and

subsequent scope of the USA's nuclear rocket program. Possible interest areas shifted

from the Air Force's missile application to manned lunar missions and later to planetary

missions. A single point was designated the Office for Space Nuclear Propulsion

(SNPO), which assumed the responsibility for the Rover program. This office was

staffed and jointly financed by the AEC and NASA (Pelaccio and El-Genk, 1994).

Rover and NERVA

The first major program decision made by the SNPO was the creation of the

Nuclear Engine for Rocket Vehicle Application (NERVA) program in 1961. NERVA

focused on utilizing and integrating the KIWI B reactor design into a flight-packaged

nuclear rocket engine. Following the KIWI B-4E full-power test, LASL turned its

attention to the experimental development of more powerful reactors. Accordingly,

LASL adopted a new Rover program goal of a 5,000 MW propulsion reactor, thereby

establishing the Phoebus reactor program (Robbins, 1991).









Phoebus

LASL used the proven KIWI B-4E fuel elements as a core building block for the

Phoebus 1A and IB. Phoebus 1A and IB also incorporated improved fuel elements to

facilitate a 50% up rating in test power, chamber pressure, and equivalent thrust.

Phoebus 1A and IB were tested on June 25, 1965, and February 23, 1967. Phoebus 1A

was operated at 245 kN of thrust for 10.5 minutes, while Phoebus IB was operated at 325

kN of thrust for 30 minutes. The Phoebus IB test result proved to be a significant

milestone because it demonstrated a practical power, thrust, and performance level that

correspond with the view of the projected NTR propulsion requirements for planetary

exploration. The Phoebus 2A reactor, tested for 12 minutes on June 26, 1968, achieved a

power level of 4,200 MW, which is significantly greater than any commercial nuclear

power plant operating today (Gunn, 2001).

Advanced fuel elements

Attention was then directed towards the development of more advanced fuel

elements capable of higher operating temperatures and longer operating life. To reduce

the lead time and the expense associated with high-power reactor tests, LASL decided to

build a much smaller reactor, designated Peweel, to evaluate such advanced fuel

elements. It was designed to generate 500 MW of power and reach a hydrogen

temperature of 2300 K. Pewee 1 was tested at full power and the average core-exit gas

temperatures of 2300 K were held for 20 minutes in December, 1968.

The final phase of advanced fuel element development was directed towards an

evaluation of composite (a mixture of graphite and carbides) fuel elements and all-

carbides fuel elements. The composite offered the potential of roughly 2700 K core-exit

temperature, while the all-carbides fuel elements gave promise of core-exit gas









temperature of around 3200 K. As a further step in reducing the power level (and cost) of

the test reactor, and in utilizing a test system that would provide for complete

containment and subsequent disposal of the radioactive fission products, LASL designed

and constructed a 50-MW test reactor designated the nuclear furnace (NF-1).

Four high-power NF-1 tests were conducted between June 29 and July 27, 1972.

The power level developed was 54 MW, the core-exit gas temperature was 2400 K and

the total run duration was 108 minutes. Postmortem examination of the fuel elements

indicated that the composite fuel elements were generally in good condition, and that any

subsequent test could be targeted for a core-exit gas temperature between 2600 K and

2800 K. With regard to the all-carbide fuel elements there was extensive cracking, but

this was as expected because of their brittleness. However, the fuel elements condition

had no meaningful effect on the ability to heat the hydrogen gas to its measured exit

temperature (Gunn, 2001).

Testing NERVA

By the spring of 1966 the NERVA reactor development program had completed all

but two of its planned reactor tests, and was in the process of conducting the first NRX-

Engine System Test (NRX-EST) series. The primary objectives of this test series were

to: (1) demonstrate the feasibility of a hot gas bleed-turbine drive cycle; (2) demonstrate

the capability of 'boot-strap' start-ups; and (3) map the engine operating envelope over a

wide range of design flow and chamber pressures. These test objectives were

accomplished in a series of three low-power tests conducted in March 1966. A total of

24 minutes at full power (110 MW) was accumulated, and resulted in a clear

demonstration of the feasibility of nuclear rocket engines.









Next, the NRX A-5 reactor was tested in June 1966 with back-to-back full power

runs of 15 minutes each. The reactor used fuel elements very similar to those used in

Phoebus lA; postmortem examination of these fuel elements revealed a generally

satisfactory condition. The final NERVA reactor test (NRX A-6) was conducted on

December 13, 1967. The test operations were a complete success-the reactor was

brought to the targeted 2300 K core-exit gas temperature, and then held at that operating

condition for 62 minutes. The stage was now set for the final, flight-packaged NERVA

engine system (XE-Prime) tests.

The XE-Prime engine featured a close-coupled, flight-type configuration, but was

designed for ground-test development. For the XE-Prime tests, a new engine test facility,

designated engine test facility 1 (ETF-1), was constructed. The test stand provided for

vertical downward firing of the engine in a simulated flight-stage structure. Nuclear

powered tests were initiated on March 20, 1969, and were concluded on September 11,

1969. The objectives of the powered test series included investigation of start-up

characteristics under different control modes, determination of engine and component

performance parameters and investigations of engine shutdown and pulse-cooling

characteristics. During this test period, 24 start-ups were accomplished, as well as a full-

power test at 1,100 MW and a core-exit gas temperature of 2300 K (Gunn, 2001).

Background on Square-Lattice Honeycomb System

The technical accomplishments of the Rover and NERVA programs were

remarkable. In a period of some 15 years, a totally new concept for rocket propulsion

was developed to the point where the experimental development had been accomplished.

There is now a renewed interest in examining nuclear propulsion in the context of Mars

and other planetary exploration. From a technical and programmatic point of view it









would seem that the nuclear thermal rocket engine could offer significant space transfer

advantages to mission planners. For this reason, the Square-Lattice Honeycomb (SLHC)

Space Nuclear Rocket Engine is being explored. The Square-Lattice Honeycomb (SLHC)

Space Nuclear Rocket Engine is a NERVA Derivative Reactor core with a new nuclear

fuel design. It is an attempt to reduce the weight of the nuclear rocket engine and

simplify the core design while increasing the thrust level.

Background on an Evaluated Nuclear Data File

To perform the analysis of SLHC, nuclear cross section libraries must be created.

These nuclear cross-sections libraries will consist of high temperature nuclear cross

sections. These nuclear cross-sections libraries are generated from a standard data file

consisting of nuclear data. This nuclear data file is known as an Evaluated Nuclear Data

File or ENDF. The ENDF system was developed to provide a unified format that could

be used to store and retrieve evaluated sets of neutron cross-sections. It was designed to

allow easy exchange of cross-section information between various national laboratories.

The initial system contained format specifications for neutron cross-sections and

other related nuclear constants. During the later stages of development, the formats were

expanded to include photon interaction cross-sections, photon production data (photon

produced by neutron interactions) and nuclear structure data. The basic data formats

developed for the library are versatile enough to allow accurate description of the cross-

sections considered for a wide range of incident neutron energies (10.5 eV to 20 MeV).

The ENDF formats are flexible in the sense that almost any type of neutron interaction

mechanism can be accurately described. They are restrictive in that only a limited

number of different representations are allowed for any given neutron reaction

mechanism.









There are two different types of evaluated data libraries are maintained at the

National Nuclear Data Section Center (NNDC). The ENDF/A library contains either

complete or partial data sets (partial in the sense that the data set may be, for example, an

evaluation of the fission cross-section for 235U in the energy range 100 keV to 15 MeV

only). This library may also contain several different evaluations of the cross-sections for

a particular nuclide. The ENDF/B library, on the other hand, contains only one

evaluation of the cross-sections for each material in the library, but each material contains

cross-sections for all significant reactions. The data set selected for the ENDF/B library

is the set recommended by the Cross Section Evaluation Working Group (CSEWG). The

ENDF/B library contains reference data sets with which other information may be

compared, as opposed to data sets that are revised often on the basis of new information

so as to constitute current standard data sets. ENDF/B is primarily intended as the main

input to a cross-section processing program (Kinsey, 1975).















CHAPTER 2
CROSS-SECTION PROCESSING

Cross-Sections Processing Codes

A high temperature neutron cross-sections library needs to be generated before the

Square-Lattice Honeycomb analysis can begin. These neutron cross-sections will

provide information needed by the computer codes to perform the neutronics analysis of

SLHC. As discussed in the previous chapter, ENDF/B contains evaluated cross-sections

data sets in a form that can be used in various neutronics calculations. However, if the

existing neutronics-codes require data libraries that are quite different from the ENDF

library, a code or series of code have been written that read the ENDF library as input and

generate a secondary cross-sections library. One of these codes is known as the NJOY

Nuclear Data Processing System.

NJOY Nuclear Data Processing System

The NJOY Nuclear Data Processing System is a comprehensive computer code

package for producing pointwise and multigroup nuclear cross-sections and related

quantities from evaluated nuclear in the ENDF format. The NJOY code purpose is to

take the basic data from the nuclear data library and convert it into the forms needed for

applications. The NJOY code consists of a set of modules, each performing a well-

defined-processing task (MacFarlane and Muir, 1994).

Several considerations were made for generating the nuclear cross-section library

used in the neutronics analysis. One of considerations is the type of a library-a









pointwise or a multigroup nuclear cross-section library. The pointwise nuclear cross-

section library is used in the Monte Carlo code, and the multigroup nuclear cross-section

library is used in the deterministic code. This report required producing both a pointwise

and a multigroup cross-section library. The nuclides and the number of temperatures to

be processed need to be considered. There are 10 different temperatures in the nuclear

cross-section library generated for this report. The list of nuclides in the library is shown

in Appendix A. The other consideration was the type of nuclear data to be included in

the library-neutron data only, photon data only, or neutron and photon data. For this

analysis, the libraries generated will only consist of neutron data.

For multigroup nuclear cross-sections, the number of broad and fine energy groups

also needs to be specified. The multigroup cross-sections libraries have one fine energy-

group structure and one broad energy-group structure. These energy-group structures are

shown in Appendix B. A weight function is needed to generate this multigroup nuclear

cross-sections library. A weight function is the neutron energy spectrum of the system or

generalized-system weight functions.

TRANSX 2001 a Code for Interfacing MATXS Cross-Section Libraries to Nuclear
Transport Codes

Discrete-ordinates (SN) transport codes, which solve the Boltzmann equation for

the distribution of neutrons and photons in nuclear systems, have reached high level of

development. The early one-dimensional codes are very widely used. The development

of effective acceleration methods as well as increasing computer speed and capacity has

made detailed transport calculations more economical; as a result, codes such as

PARTISN are seeing increasing use. The DIF-3D diffusion code and Monte-Carlo codes

with multigroup capability like MCNP are also used frequently.









However, many of the users of transport codes have the same complaint: it is hard

to get good, up-to-date, documented cross-section data and prepare them for input into

these codes. TRANSX works together with a generalized cross-section library called

MATXS (for material cross-section library) to give the transport code user easier access

to appropriate nuclear data and some capabilities difficult or impossible to get with any

other system.

The code can be used to construct data for fusion reactors, fast fission reactors,

thermal fission reactors, and shielding problems. Its main weakness is in computing

resonance effects in thermal reactors. TRANSX was originally developed in the late

seventies to handle cross sections for fission, fusion, and shielding applications at Los

Alamos National Laboratory. In the early eighties, extensions to handle heterogeneous

self-shielding problems for fast reactors were added (MacFarlane, 1992).

Generating PENDF Library

There is a common processing path for producing a pointwise and a multigroup

library. This processing path produces an intermediate library file that is commonly

called "pointwise-ENDF" (PENDF). In this process, the NJOY code utilized the

RECONR, BROADER, UNRESR, HEATR, and THERMR modules.

RECONR Module

The NJOY processing sequences start with RECONR, which serves two roles.

First, it goes through all the reactions included on the ENDF library and chooses a union

grid that allows all cross sections to be represented using linear interpolation to a

specified accuracy. This step removes any nonlinear interpolation ranges (e.g., log-log,

linear-log). It also makes it possible for all summation reactions to be reconstructed as

the sum of their parts (e.g., total, total inelastic, total fission). Second, for resonance









materials, it reconstructs the resonance cross sections (elastic, fission, capture) on a union

grid that allows them all to be represented within certain accuracy criteria, and then

combines the resonance data with the other linearized and unionized cross sections.

RECONR has the following features:

* Efficient use of dynamic storage allocation and a special stack structure allow large
problems to be run on small machines

* The unionized grid improves the accuracy, usefulness, and ENDF/B compatibility
of the output. All summation cross sections are preserved on the union grid.

* Approximate x and X Doppler broadening functions may be used in some cases to
speed up reconstructions for narrow-resonance materials.

* A resonance-integral criterion is added to the normal linearization criterion in order
to reduce the number of points added to the tabulation to represent "unimportant"
resonances.

* All ENDF-6 formats are handled except Generalized R-Matrix parameters, energy-
dependent scattering radius, and the calculation of angular distributions from
resonance parameters.

BROADER Module

This module generates Doppler-broadened cross sections. The input cross-sections

can be from RECONR or a previous BROADR run. The method utilized in BROADR is

based on Cullen's SIGMAl method. The method is often called "kernel broadening"

because it uses a detailed integration of the integral equation defining the effective cross

section. BROADR has the following features:

* An alternate calculation is used for low energies and high temperatures that corrects
a numerical problem of the original SIGMAl

* Variable dimensioning is used, which allows the code to be run on large or small
machines with full use of whatever storage is made available

* All low-threshold reactions are broadened in parallel on a union grid. This makes
the code run faster than the original SIGMA1









* The union grid is constructed adaptively to give a linearized representation of the
broadened cross section with tolerances consistent with those used in RECONR.
Energy points may be added to or removed from the input grid as required for the
best possible representation.

* Binary input and output can be used. This roughly halves the time required for a
typical run on some computer systems, and it allows the full accuracy of the
machine to be used.

* The summation cross sections total, non-elastic, and sometimes fission or (n, 2n)
are reconstructed to equal the sum of their parts.

UNRESR Module

This module is used to produce effective self-shielded cross sections for resonance

reactions in the unresolved energy range. In ENDF-format evaluations, the unresolved

range begins at energy where it is difficult to measure individual resonances and extends

to energy where the effect of fluctuations in the resonance cross-sections becomes

unimportant for practical calculations. The resonance information for this energy range

is given as average values for resonance widths and spacings together with distribution

functions for the widths and spacings. This representation can be converted into effective

cross-sections suitable for codes that use the background cross-section method, often

called the Bondarenko method, using a method originally developed for the MC2 code

and extended for the ETOX code. UNRESR has the following features:

* Flux-weighted cross sections are produced for the total, elastic, fission, and capture
cross sections, including competition with inelastic scattering

* A current-weighted total cross section is produced for calculating the effective self-
shielded transport cross section

* Up to 10 values of temperature and 10 values of oo are allowed

* The energy grid used is consistent with the grid used by RECONR

* The computed effective cross sections are written on the PENDF tape in a specially
defined section (MF2, MT 152) for use by other modules









* The accurate quadrature scheme from the MC2-2 code is used for computing
averages over the ENDF statistical distribution functions.

HEATR Module

This module generates pointwise heat production cross sections and radiation

damage energy production for specified reactions and adds them to an existing PENDF

library. Heating is an important parameter of any nuclear system. It may represent the

product being sold-as in a power reactor-or it may affect the design of peripheral

systems such as shields, and structural components. Nuclear heating can be divided into

neutron heating and photon heating. The neutron heating at given location is proportional

to the local neutron flux and arises from the kinetic energy of the charged products of a

neutron-induced reaction. Similarly, the photon heating is proportional to the flux of

secondary photons transported from the site of previous neutron reactions. It is also

traceable to the kinetic energy of charged particles. HEATR has the following

advantages:

* Heating and damage are computed in a consistent way

* All ENDF/B neutron and photon data are used

* ENDF-6 charged-particle distributions are used when available

* Kinematics checks are available to improve future evaluations

* Both energy-balance and kinematics KERMA (Kinetic Energy Releasing in
Materials) factors can be produced.

THERMR Module

At thermal energies, that is up to about 0.5 eV for temperatures around room

temperature and maybe up to as high as 4 eV for hotter materials, the energy transferred

by the scattering of a neutron is similar to the kinetic energies of motion of the atoms in

liquids and to the energies of excitations in molecules and crystalline lattices. Therefore,









you cannot picture the target atoms as being initially stationary and recoiling freely as is

normally done for higher neutron energies. The motion of the target atoms and their

binding in liquids and solids affects both cross sections and the distribution in energy and

angle of the scattered neutrons. The THERMR module of NJOY is used to compute

these effects.

For free-gas scattering, where only the thermal motion of the targets is taken into

account, not internal modes of excitation, THERMR can generate the cross sections and

scattering distributions using analytic formulas. For real bound scattering, it uses an

input scattering function and other parameters from an ENDF-format thermal evaluation

in File 7 format. THERMR has the following features:

* The energy grid for coherent elastic scattering is produced adaptively so as to
represent the cross section between the sharp Bragg edges to a specified tolerance
using linear interpolation

* The secondary energy grid for inelastic incoherent scattering is produced adaptively
so as to represent all structure with linear interpolation

* Incoherent cross sections are computed by integrating the incoherent matrix for
consistency

* Free-atom incoherent scattering is normalized to the Doppler broadened elastic
scattering cross section in order to provide an approximate representation of
resonance scattering to preserve the correct total cross section

* Discrete angle representations are used to avoid the limitations of Legendre
expansions

* Hard-to-find parameters for the ENDF/B-III evaluations are included in the
THERMR code for the user's convenience

* ENDF-6 format files can be processed. This gives the evaluator more control over
the final results, because all parameters needed to compute the cross sections are
contained in the file.









Automation of Processes

For processing a large number of nuclides, it is nice to automate the process. To

automate the generation of the PENDF library, shell-scripts were utilized. These scripts

automatically obtained the appropriate ENDF library from the High Performance Storage

System (HPSS), and then create an NJOY input file to perform a calculation in the

RECONR, BROADR, UNRESR, HEATR, and THERMR modules, and finally store the

PENDF library back into HPSS. Example of these scripts can be found in Appendix C.

Generating Cross-Sections for Monte Carlo Code

Once the PENDF library of all nuclides needed for the analysis is produce, code-

specific cross-sections libraries are generated. The Monte Carlo code that was used to

perform the analysis in this report was Monte Carlo N-Particle (MCNP) version 5. A

detailed description and uses of the MCNP5 code is discussed in the next chapter of this

report. The ACER module of the NJOY code can produce a library to be used by the

MCNP5 code. However, additional calculations must be performed before the ACER

calculation.

PURR Module

The unresolved self-shielding data generated by UNRESR is suitable for use in

multigroup methods after processing by the GROUPR Module, discussed later in this

section, but the so-called Bondarenko method is not very useful for continuous-energy

Monte Carlo codes like MCNP5. The natural approach for treating unresolved-resonance

self-shielding for Monte Carlo codes is the "Probability Table" method. This module

produces probability tables to treat unresolved-resonance self-shielding.









GASPR Module

In many practical applications, it is important to know the total production of

protons (hydrogen), alphas (helium), and other light charged particles resulting from the

neutron flux. Therefore, it is convenient to have a set of special "gas production" or

"charged-particle production" cross-sections that can be used in application codes. The

ENDF format provides a set of MT numbers for these quantities, but only a few

evaluators have added them to their files:

* MT=203 -- total proton production

* MT=204 -- total deuteron production

* MT=205 -- total triton production

* MT=206 -- total He-3 production

* MT=207 -- total alpha production

The GASPR module goes through all of the reactions given in an ENDF-format

evaluation, determines which charged particles would be produced by the reaction, and it

adds up the particle yield times the reaction cross-section to produce the desired gas

production cross-sections. They are then available for plotting, multigroup averaging, or

reformatting for the MCNP code.

ACER Module

After the calculations of the two previous modules were completed, the ACER

module was utilized to create a MCNP nuclear cross-section library. The ACER module

prepares libraries in ACE format (A Compact ENDF) for the MCNP continuous-energy

neutron-photon Monte Carlo code. MCNP requires that all the cross-sections be given on

a single union energy grid suitable for linear interpolation. Although the energy grid and

cross-section data on an NJOY PENDF library are basically consistent with the









requirements of MCNP, there is still one problem. Many ENDF evaluations produce

energy grids with very large numbers of points. The ACER module provides thinning

algorithms to control the accuracy loss and balance it against the computer memory

requirements. The ACE library files come in three different types in order to allow for

efficiency, portability, and history.

* Type 1 is a simple formatted file suitable for exchanging ACE libraries between
different computers

* Type 2 is a FORTRAN-77 direct-access binary file for efficient use during actual
MCNP runs

* Type 3 is a word-addressable direct-access binary file. It uses nonstandard read
and write call, and it is normally used only at Los Alamos National Laboratory.
Type 3 is only used for the fast library not for dosimetry, thermal, or photoatomic
data.

Automation of Processes

Like the generation of the PENDF library, the generation of the MCNP5 library

utilizes shell-scripts. These scripts automatically obtain the appropriate ENDF and

PENDF library from the High Performance Storage System (HPSS), and then create the

NJOY input file to perform a calculation in the PURR, GASPR, and ACER modules, and

finally store the ACER library back into HPSS. Example of these scripts can be found in

Appendix D.

Generating Cross-Sections for Deterministic Code

This report also performed analysis using a deterministic code. The deterministic

code used to perform the analysis was PARTISN version 3.51. A detailed description

and uses of PARTISN version 3.51 is discussed in the next chapter of this report. The

MATXSR module in NJOY was used to generate a multigroup nuclear cross-section

library. Additional calculations need to be performed to prepare the library for MATXSR









module. Finally, an additional cross-section processing code was needed to organize the

MATXSR multigroup library into a working cross-section library for PARTISN. The

TRANSX code version 2001 performed these tasks. The other path of organizing the

multigroup cross-section is through the DTFR module.

GROUP Module

This module produces self-shielded multigroup cross sections, anisotropic group-

to-group scattering matrices, and anisotropic photon production matrices for neutrons

from ENDF/B-IV, V, or VI evaluated nuclear data. The Bondarenko narrow-resonance

weighting scheme is usually used. Neutron data and photon-production data are

processed in a parallel manner using the same weight function and quadrature scheme.

Two-body scattering is computed with a center-of-mass (CM) Gaussian quadrature,

which gives accurate results even for small Legendre components of the group-to-group

matrix. Output is written to an output "groupwise-ENDF" (GENDF) file for further

processing by a formatting module (DTFR, CCCCR, MATXSR).

DTFR Module

This module is used to prepare libraries for discrete-ordinates transport codes that

accept the DTF format. Transport tables in DTF format are organized to mirror the

structure of data inside a discrete-ordinates transport code. These codes start with the

highest energy group and work downward. The basic table consists of the three standard

edits, namely, particle balance absorption, fission neutron production cross section, and

total cross section. These standard edits are followed by the group-to-group scattering

cross sections.









MATXSR Module

This module formats the GENDF tape into a generalized CCCC-type interface

format for neutron, photon, and charged-particle data, including cross sections, group-to-

group matrices, temperature variations, self-shielding, and time-dependence. MATXS

libraries can be used with the TRANSX code to produce effective cross sections for a

wide variety of application codes, such as PARTISN.

TRANSX Calculation

As discussed earlier, the MATXS module in NJOY produced a generalized CCCC-

type interface format for neutron, photon, and charged-particle data. However, this

format is not very useful in the PARTISN code. Additional processing is needed to

create the PARTISN library. This is where the TRANSX code comes into play. The

TRANSX code performed additional cross-section processing, such as homogenization of

materials, self-shielding, and Dancoff correction. TRANSX also organized the cross-

sections into a library format useful for the PARTISN code.















CHAPTER 3
NUCLEAR TRANSPORT CODES DESCRIPTIONS

This chapter discusses the nuclear transport codes are used to perform the analysis

of SLHC.

MCNP Version 5

MCNP is a general-purpose Monte Carlo N-Particle code that can be used for

neutron, photon, electron, or coupled neutron/photon/electron transport, including the

capability to calculate eigenvalues for critical systems. Pointwise cross-section data are

used. For neutrons, all reactions given in a particular cross-section evaluation (such as

ENDF/B-VI) are accounted for. Thermal neutrons are described by both the free gas and

S(cL, P) models. For photons, the code takes account of incoherent and coherent

scattering, the possibility of fluorescent emission after photoelectric absorption,

absorption in pair production with local emission of annihilation radiation, and

bremsstrahlung. A continuous slowing down model is used for electron transport that

includes positrons, k x-rays, and bremsstrahlung but does not include external or self-

induced fields.

Important standard features that make MCNP very versatile and easy to use include

a powerful general source, criticality source, and surface source; both geometry and

output tally plotter; a rich collection of variance reduction techniques; a flexible tally

structure; and extensive collections of cross-section data (Hendricks, 1997).









PARTISN Version 3.56

The PARTISN and DANTSYS code package are essentially the same code with the

difference that PARTISN is a more modem in the sense that it uses Fortran 90 language

and it also support parallel processors environment. The PARTISN code package

includes the following transport codes: ONEDANT, TWODANT, TWODANT/GQ,

TWOHEX, AND THREEDANT. This code package is a modular computer program

package designed to solve time-independent, multigroup discrete ordinates form of the

Boltzmann transport equation in several different geometries. The modular construction

of the package separates the input processing, the transport equation solving, and the post

processing (or edit) functions into distinct code modules: the Input Module, one or more

Solver Modules, and the Edit Module, respectively. The Input and Edit Modules are very

general in nature and are common to all the Solver Modules. The ONEDANT Solver

Module contains a one-dimensional (slab, cylinder, and sphere), time-independent

transport equation solver using the standard diamond-differencing method for

space/angle discretization. Also included in the package are Solver Modules named

TWODANT, TWODANT/GQ, THREEDANT, AND TWOHEX. The TWODANT

Solver Module solves the time-independent two dimensional transport equation using the

diamond-differencing method for space/angle discretization. An adaptive weighed

diamond differencing (AWDD) method for spatial and angular discretization is also

introduced in TWODANT as an option. The TWOHEX Solver Module solves the time-

independent two-dimensional transport equation on a equilateral triangle spatial mesh.

The THREEDANT Solver Module solves the time-independent, three-dimensional

transport equation for XYZ and RZ symmetries using both diamond differencing with






23


set-to-zero fixup and the AWDD method. The TWODANT/GQ Solver Module solves

the two-dimensional time-independent transport equation in XY and RZ symmetries

using a spatial mesh of arbitrary quadrilaterals. The spatial differencing method is based

upon the diamond differencing method with set-to-zero fix up with changes to

accommodate the generalized special meshing (Alcouffe et al., 2002).















CHAPTER 4
SQUARE-LATTICE HONEYCOMB (SLHC) NUCLEAR ROCKET ENGINE
DESCRIPTION

This chapter discusses the general description of Square-Lattice Honeycomb

Nuclear Rocket Engine. It describes the geometry as well as the materials in SLHC.

Geometry Description

The Square-Lattice Honeycomb nuclear reactor geometry description is shown in

Figure 4-1. The overall diameter and height of the SLHC reactor core are 31.0 cm and

45.0 cm, respectively. Beryllium reflectors in the radial and top axial directions surround

the reactor. The thickness of the radial beryllium reflector is 20.0 cm, and the thickness

of the axial beryllium reflector is 20.0 cm.

The core is fueled with a solid solution of 93% enriched (U,Zr,Nb)C, which is one

of several ternary uranium carbides that are under consideration for this concept. The

fuel is to be fabricated as 1 mm grooved (U,Zr,Nb)C wafers. The fuel wafers are used to

form square-lattice honeycomb fuel elements, containing 30% cross-sectional flow area,

shown in Figure 4-2 (Furman, 1999). The fuel element's dimensions are 4.0 cm in

diameter and 1.5 cm in height. The fuel sub assembly consists of six fuel elements

stacked axially, shown in Figure 4-3 (Furman, 1999). Each fuel sub assembly has a 0.5-

cm thick graphite coating and a 0.5-cm thick zirconium oxide coating. Five fuel sub

assemblies are stacked axially to form one fuel assembly. Finally, the assemblies are

then arranged in the circular pattern inside a zirconium hydride matrix.









The control system of the SLHC is in the form of control drums inside the radial

reflector region. The control drums have an inner diameter, an outer diameter and a

height of 13.6 cm, 18.0 cm, and 45.0 cm, respectively. The materials of the control

drums will be discussed in the next section of this chapter.

MCNP5 Geometry Representation of SLHC

The SLHC is modeled as a heterogeneous model and a homogenized representation

of the heterogeneous model.

Heterogeneous representation of SLHC

The SLHC heterogeneous model models the regions of the SLHC exactly except

for the fuel regions, which they are homogenized. The reason for the homogenization in

the fuel is for saving computation time. The homogenization in the fuel region is selected

to simply the geometry description in the problem. The fuel regions are homogenized

using straight-forward homogenization method. This method utilizes the straight

conversion of heterogeneous number density into homogeneous number density through

the use of volume fractions. Table 4-1 presents the result of calculation of the "true"

SLHC heterogeneous model and the heterogeneous model with homogenization in fuel

region. Figure 4-4 presents the fuel region energy spectra comparison of "true"

heterogeneous SLHC model and SLHC heterogeneous model with fuel region

homogenization. Figure 4-5 presents the fuel region energy spectra of "true" SLHC

heterogeneous model and SLHC heterogeneous model with fuel region homogenization

in the thermal energy range (less than leV). Based on the result in Table 4-1, Figure 4-4

and Figure 4-5, the SLHC heterogeneous model with fuel region homogenization is a

valid representation of the "true" SLHC heterogeneous model because in the both models

show similar results in k-eff and the energy spectrum. However, the time to obtain the









similar statistical error on the results is almost 3 times less in the SLHC heterogeneous

model with fuel homogenization. Utilizing 128 parallel-processors, it takes about 85000

minutes of computation time to complete the analysis of the "true" SLHC heterogeneous

model. However, the analysis can be completed in only 30000 minutes of computation

time if the fuel homogenization is utilized. The SLHC heterogeneous model is shown in

Figure 4-6. The SLHC heterogeneous model is used to generate the SLHC

heterogeneous model's energy spectrum. The SLHC heterogeneous model is utilized for

finding the minimum numbers of control drums required to shutdown the Square-Lattice

Honeycomb reactor. Finally, the water submersion accident analysis is also being

performed using the SLHC heterogeneous model.

Homogeneous representation of SLHC

The homogenized model of the SLHC in MCNP will be used for a comparison

between the Monte Carlo method and the deterministic method. Figure 4-7 shows the

geometry description of the SLHC. The control system is modeled as an absorber ring

instead of control drums. In the homogenized model, each of the SLHC regions is

represented as a ring of materials. The important constraint when developing this

homogeneous model is that mass is conserved; therefore area is conserved. The locations

of each ring material are based on the center location of the fuel materials. Please refer to

Appendix E for the detailed calculation of the locations of the fuel rings in the Square-

Lattice Honeycomb homogeneous model. With the utilization of rings of materials, the

homogenization is unlike the conventional homogenization method. This method

rearranges the regions in the SLHC into rings of materials which have the same

composition as the heterogeneous model. This method eliminates the necessity of









defining a unit cell representation. This method also eliminates the requirement for

calculating the homogenized number densities.

PARTISN 3.51 Geometry Representation of SLHC

In PARTISN, the geometry of the SLHC has to be represented as a homogenized

model. The homogenized model of the SLHC in PARTISN is exactly the same as the

homogenized model in MCNP because the results from the PARTISN calculation and the

MCNP calculation will be compared.

Materials Description

The SLHC nuclear reactor is fueled by a (U,Zr,Nb)C fuel. The propellant of the

reactor is hydrogen gas. The reflector material is beryllium, and the moderator material

is zirconium hydride.

Materials in the Heterogeneous SLHC

In the SLHC heterogeneous model, the fuel region consists of a homogenization of

(U,Zr,Nb)C and hydrogen gas. The fuel region has 70% solid volume fraction. The core

is divided into five axial temperature regions. Each axial temperature region has height

of 9 cm. For each temperature region, the density of the (U,Zr,Nb)C is varied according

to Table 4-2. Please refer to Appendix D for detailed calculation of the number densities

in the SLHC heterogeneous model.

A 0.5-cm thick graphite coating followed by a 0.5-cm zirconium oxide coating

surrounds the fuel region. The purpose of these coatings is to act as an insulator between

the hot zone (fuel region) and the moderator region. As described earlier, the fuel

assemblies were placed in the circular pattern inside the zirconium hydride moderator.

Finally, axial and radial beryllium reflectors enclose the reactor core. The control system

of the SLHC is placed inside the radial beryllium reflector in the form of control drums.









These control drums consist of neutron-absorber and neutron-reflecting material. The

neutron-absorber material is boron carbide, and the beryllium is the neutron-reflecting

material. Table 4-3 presents properties of all the non-fuel elements in the SLHC

heterogeneous model and includes the densities, isotopes, volume, and mass of each

region. Table 4-4 presents the properties of all fuel elements in the SLHC heterogeneous

model and includes the densities, isotopes, volume, and mass of each region. Finally,

Table 4-5 presents some properties of the Square-Lattice Honeycomb reactor.

Materials in the Homogeneous SLHC

Table 4-6 presents properties for all the non-fuel regions in the homogenized model

of the SLHC with their isotopes, volume fraction, heterogeneous number density, and

homogeneous number density. Both the heterogeneous and homogeneous number

densities are presented in these tables to illustrate that there is no need to calculate the

homogeneous number densities for the ring method utilization. Table 4-7 presents the

first axial region fuel materials in the SLHC homogeneous model and includes the

number density, isotopes, and volume fractions of each region. Table 4-8 presents the

second axial region fuel materials in the SLHC homogeneous model and includes the

number density, isotopes, and volume fractions of each region.

Table 4-9 presents the third axial region fuel materials in the SLHC homogeneous

model and includes the number density, isotopes, and volume fractions of each region.

Table 4-10 presents the fourth axial region fuel materials in the SLHC homogeneous

model and includes the number density, isotopes, and volume fractions of each region.

Table 4-11 presents the fifth axial region fuel materials in the SLHC homogeneous model

and includes the number density, isotopes, and volume fractions of each region.














I< 71 cm >i


65 cm












Figure 4-1. Square-Lattice Honeycomb nuclear reactor geometry description.









































Figure 4-2. The Square-Lattice Honeycomb fuel wafers fabrication into fuel element.
























-E~--- 4 cm -


1.5 cm I


9 cm


Figure 4-3. The fabrication of the Square-Lattice Honeycomb fuel elements into fuel
assembly.
































Table 4-1. Comparison calculations of the "true" Square-Lattice Honeycomb (SLHC)
heterogeneous model and SLHC heterogeneous model with fuel region homogenization.
Properties "True" SLHC SLHC Heterogeneous Model %
Heterogeneous Model with fuel region homogenization Difference
k-eff 0.99913 + 0.00005 0.99997 + 0.00005 0.08
Number of cycles 25000 20000 -20
Computation time (minutes) 85000 30000 -65
Real time (hours) 34 12 -65


































1.OOE+07

1.OOE+06

1.OOE+05

1.OOE+04

1.OOE+03

1.00E+02

1.OOE+01

. 1.OOE+00

S1.OOE-01


Energy (eV)



Figure 4-4. Energy spectra of "true" Square-Lattice Honeycomb (SLHC) heterogeneous
model and SLHC heterogeneous model with fuel region homogenization.






























1.40E+06
--SLHC Heterogeneous with fuel
homogenization
1.20E+06
-- "True" SLHC Heterogeneous

1.00E 06


S8.00E+05


6.00E+05


4.00E+05


2.00E+05


O.OOE+00 -




Energy (eV)



Figure 4-5. Energy spectra of "true" Square-Lattice Honeycomb (SLHC) heterogeneous
model and SLHC heterogeneous model with fuel region homogenization in
the thermal energy range (less than 1eV).








































Figure 4-6. Geometry description of the Square-Lattice Honeycomb heterogeneous
model.


























I Hydrogen hole (Outer radius = 2 25 cm)
Zirconium tube (Outer radius 2 50 cm)
First zirconium hydride (Outer radius = 4 16 cm)
First zirconium oxide coating (Outer radius= 4 81 cm)
First graphite coating (Outer radius = 5 35 cm)
First fuel region (Outer radius = 7 25 cm)
Second graphite coating (Outer radius = 7 79 cm)
Second zirconium oxide coating (Outer radius = 8 44 cm)
7 Second zirconium hydride (Outer radius = 9 95 cm)
Third zirconium coating (Outer radius = 10 63 cm)
Third graphite coating (Outer radius = 11 18 cm)
Second fuel region (Outer radius = 13 16 cm)
Fourth graphite coating (Outer radius = 13 71 cm)
Fourth zirconium oxide coating (Outer radius = 14 39 cm)
Third zirconium hydride (Outer radius = 15 50 cm)
First beryllium reflector (Outer radius = 16 25 cm)
First beryllium and hydrogen gas (Outer radius = 29 20 cm)
Boron carbide (Outer radius = 30 93 cm)
I Second beryllium and hydrogen gas (Outer radius = 34 75 cm)
Second beryllium reflector (Outer radius = 35 50 cm)



Figure 4-7. Here is the geometry description of the Square-Lattice Honeycomb
homogeneous model.
























Table 4-2. Number densities of the isotopes in the fuel region.
Axial Region Temperature Uranium density
(K) (g/cm3)
1 600 0.7
2 1000 0.9
3 1200 1.2
4 2000 1.2
5 2500 1.2











Table 4-3. Properties of non-fuel elements in the Square-Lattice Honeycomb
heterogeneous model.
Number Density Volume Density Mass
Region Isotope (Atoms/bcm) (cm3) (g/cm3) (g)


Hydrogen hole


8.6000E-03


Top Hydrogen hole

Zirconium tube






Zirconium Hydride
Region






Bottom Graphite
Coating

Bottom ZrO2 Coating







6 Inner Cylinder of
Control Drums

6 Outer Cylinder of
Control Drums
(Beryllium)

6 Outer Cylinder of
Control Drums (B4C)


Beryllium reflector


Beryllium reflector
and Hydrogen gas


1.7200E-02

2.2114E-02
4.8226E-03
7.3715E-03
7.4703E-03
1.2035E-03

2.2231E-02
4.8481E-03
7.4104E-03
7.5097E-03
1.2099E-03
8.6418E-02


1.4139E-01

1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04
5.7571E-02


1.2362E-01



1.2362E-01

9.1506E-02
2.2876E-02
2.8596E-02

1.2362E-01

1.1991E-01

1.0320E-03


1H


1.4393E-02


1.0301E+01


7.1569E+02

3.7937E+03

1.6788E+02






9.7264E+03








2.2619E+02

2.2619E+02








3.9222E+04



1.4742E+04

1.4742E+04




8.6205E+04

6.6468E+04


2.8787E-02 1.0921E+02

6.5110E+00 1.0931E+03






6.6900E+00 6.5069E+04








2.8200E+00 6.3787E+02

5.8900E+00 1.3323E+03








1.8500E+00 7.2561E+04



1.8500E+00 2.7273E+04

2.5100E+01 3.7003E+04




1.8500E+00 1.5948E+05

1.7962E+00 1.1618+05











Table 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous
model.
Number Density Volume Density Mass
Region Isotope (Atoms/bcm) (cm3) (g/cm3) (g)


18 First Axial Fuel
Region


1H
12C
90Zr
91Zr
92Zr
94Zr
96Zr
93Nb
235U
2381


1.0320E-02
2.7880E-02
1.0482E-02
2.2858E-03
3.4939E-03
3.5407E-03
5.7043E-04
6.7908E-03
6.6658E-04
5.0173E-05


2.0358E+03


5.0873E+00


1.0356E+04


18 First Axial
Graphite Coating

18 First Axial ZrO2
Coating







18 Second Axial Fuel
Region











18 Second Axial
Graphite Coating

18 Second Axial
ZrO2 Coating


1.4139E-01

1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04
5.7571E-02

1.0320E-02
2.7805E-02
1.0315E-02
2.2494E-03
3.4382E-03
3.4843E-03
5.6134E-04
6.6826E-03
9.9987E-04
7.5259E-05


1.4139E-01

1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04
5.7571E-02


1.1451E+03

1.3996E+03








2.0358E+03














1.1451E+03

1.3996E+03


2.8200E+00 3.2292E+03

5.8900E+00 8.2435E+03








5.1577E+00 1.0500E+04














2.8200E+00 3.2292E+03

5.8900E+00 8.2435E+03







40



Table 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous
model. (continued)
Number Density Volume Density Mass
Region Isotope (Atoms/bcm) (cm3) (g/cm3) (g)


18 Third Axial Fuel
Region


12c
90Zr
91Zr
92 Zr
94Zr
96 Zr
93mb


1.0320E-02
2.7656E-02
9.9802E-03
2.1764E-03
3.3267E-03
3.3714E-03
5.4314E-04
6.4660E-03
1.6665E-03
1.2543E-04


2.0358E+03


5.2644E+00


1.07174E+04


18 Third Axial
Graphite Coating

18 Third Axial ZrO2
Coating







18 Fourth Axial Fuel
Region











18 Fourth Axial
Graphite Coating

18 Fourth Axial ZrO2
Coating


1.4139E-01

1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04
5.7571E-02

1.0320E-02
2.7656E-02
9.9802E-03
2.1764E-03
3.3267E-03
3.3714E-03
5.4314E-04
6.4660E-03
1.6665E-03
1.2543E-04


1.4139E-01

1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04
5.7571E-02


1.1451E+03

1.3996E+03








2.0358E+03














1.1451E+03

1.3996E+03


2.8200E+00 3.2292E+03

5.8900E+00 8.2435E+03








5.2633E+00 1.0715E+04














2.8200E+00 3.2292E+03

5.8900E+00 8.2435E+03







41



Table 4-4. Properties of fuel elements in the Square-Lattice Honeycomb heterogeneous
model. (continued)
Number Density Volume Density Mass
Region Isotope (Atoms/bcm) (cm3) (g/cm3) (g)


18 Fifth Axial Fuel
Region


12c
90Zr
91Zr
92 Zr
94Zr
96 Zr
93mb


1.0320E-02
2.7656E-02
9.9802E-03
2.1764E-03
3.3267E-03
3.3714E-03
5.4314E-04
6.4660E-03
1.6665E-03
1.2543E-04


2.0358E+03


5.2622E+00


1.0712E+04


18 Fifth Axial
Graphite Coating

18 Fifth Axial ZrO2
Coating


1.4139E-01

1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04
5.7571E-02


1.1451E+03

1.3996E+03


2.8200E+00 3.2292E+03

5.8900E+00 8.2435E+03







42






















Table 4-5. Properties of the Square-Lattice Honeycomb reactor.
Properties Value
Reactor diameter (cm) 31.0
Reactor height (cm) 45.0
Radial reflector thickness (cm) 20.0
Top axial reflector thickness (cm) 20.0
Fuel element height (cm) 9.0
Fuel type Solid solution of (U,Zr,Nb)C
Fuel enrichment (%) 93
Uranium densities (g/cm3) 0.7 1.2
Uranium mass (g) 10600
Reflector material Beryllium
Absorber material Boron Carbide







43






Table 4-6. Properties of non-fuel materials in the Square-Lattice Honeycomb
homogeneous model.
Heterogeneous Homogeneous
Volume Number Density Number Density
Region Isotope Fraction (atoms/bcm) (atoms/bcm)
1 1H 1.0000 8.6000E-03 8.6000E-03


1.0000
1.0000
1.0000
1.0000
1.0000

1.0000
1.0000
1.0000
1.0000
1.0000
1.0000

1.0000

1.0000
1.0000
1.0000
1.0000
1.0000
1.0000

1.0000

0.9835
0.0165

1.0000
1.0000
1.0000


2.2114E-02
4.8226E-03
7.3715E-03
7.4703E-03
1.2035E-03

8.6418E-02
2.2231E-02
4.8481E-03
7.4104E-03
7.5097E-03
1.2099E-03

1.4139E-01

5.7571E-02
1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04

1.2362E-01

1.1991E-01
5.1600E-04

9.1506E-02
2.2876E-02
2.8596E-02


2.2114E-02
4.8226E-03
7.3715E-03
7.4703E-03
1.2035E-03

8.6418E-02
2.2231E-02
4.8481E-03
7.4104E-03
7.5097E-03
1.2099E-03

1.4139E-01

5.7571E-02
1.4810E-02
3.2297E-03
4.9367E-03
5.0029E-03
8.0600E-04

1.2362E-01

1.1991E-01
5.1600E-04

9.1506E-02
2.2876E-02
2.8596E-02







44











Table 4-7. The properties of axial region 1 fuel materials in the Square-Lattice
Honeycomb homogeneous model.
Heterogeneous Homogeneous
Volume Number Density Number Density
Region Isotope Fraction (atoms/bcm) (atoms/bcm)
3 1-1 235U 0.3111 1.1665E-03 1.1665E-03
238U 0.3111 8.7803E-05 8.7803E-05
"oZr-Fuel 0.3111 1.0231E-02 1.0231E-02
91Zr-Fuel 0.3111 2.2311E-03 2.2311E-03
92Zr-Fuel 0.3111 3.4103E-03 3.4103E-03
94Zr-Fuel 0.3111 3.4561E-03 3.4561E-03
"9Zr-Fuel 0.3111 5.5679E-04 5.5679E-04
93Nb 0.3111 6.6284E-03 6.6284E-03
12C-Fuel 0.3111 2.7768E-02 5.1600E-03
1H-Fuel 0.1333 5.1600E-03 2.7768E-02

4 1-2 235U 0.3111 1.1665E-03 1.1665E-03
238U 0.3111 8.7803E-05 8.7803E-05
9Zr-Fuel 0.3111 1.0231E-02 1.0231E-02
91Zr-Fuel 0.3111 2.2311E-03 2.2311E-03
92Zr-Fuel 0.3111 3.4103E-03 3.4103E-03
94Zr-Fuel 0.3111 3.4561E-03 3.4561E-03
96Zr-Fuel 0.3111 5.5679E-04 5.5679E-04
93Nb 0.3111 6.6284E-03 6.6284E-03
12C-Fuel 0.3111 2.7768E-02 5.1600E-03
1H-Fuel 0.1333 5.1600E-03 2.7768E-02






















Table 4-8. The properties of axial region 2 fuel materials in the Square-Lattice
Honeycomb homogeneous model.
Heterogeneous Homogeneous
Volume Number Density Number Density
Region Isotope Fraction (atoms/bcm) (atoms/bcm)
5 2-1 235U 0.3111 1.4998E-03 1.4998E-03
238U 0.3111 1.1289E-04 1.1289E-04
"9Zr-Fuel 0.3111 1.0064E-02 1.0064E-02
91Zr-Fuel 0.3111 2.1947E-03 2.1947E-03
92Zr-Fuel 0.3111 3.3546E-03 3.3546E-03
94Zr-Fuel 0.3111 3.3996E-03 3.3996E-03
"9Zr-Fuel 0.3111 5.4769E-04 5.4769E-04
93Nb 0.3111 6.5201E-03 6.5201E-03
12C-Fuel 0.3111 2.7693E-02 3.8700E-03
1H-Fuel 0.1333 3.8700E-03 2.7693E-02

6 2-2 235U 0.3111 1.4998E-03 1.4998E-03
238U 0.3111 1.1289E-04 1.1289E-04
9Zr-Fuel 0.3111 1.0064E-02 1.0064E-02
91Zr-Fuel 0.3111 2.1947E-03 2.1947E-03
92Zr-Fuel 0.3111 3.3546E-03 3.3546E-03
94Zr-Fuel 0.3111 3.3996E-03 3.3996E-03
96Zr-Fuel 0.3111 5.4769E-04 5.4769E-04
93Nb 0.3111 6.5201E-03 6.5201E-03
12C-Fuel 0.3111 2.7693E-02 3.8700E-03
1H-Fuel 0.1333 3.8700E-03 2.7693E-02







46











Table 4-9. The properties of axial region 3 fuel materials in the Square-Lattice
Honeycomb homogeneous model.
Heterogeneous Homogeneous
Volume Number Density Number Density
Region Isotope Fraction (atoms/bcm) (atoms/bcm)
7 3-1 235U 0.3111 1.9997E-03 1.9997E-03
238U 0.3111 1.5052E-04 1.5052E-04
9Zr-Fuel 0.3111 9.8130E-03 9.8130E-03
91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03
92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03
94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03
96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04
93Nb 0.3111 6.3576E-03 6.3576E-03
12C-Fuel 0.3111 2.7581E-02 2.5800E-03
1H-Fuel 0.1333 2.5800E-03 2.7581E-02

8 3-2 235U 0.3111 1.9997E-03 1.9997E-03
238U 0.3111 1.5052E-04 1.5052E-04
9Zr-Fuel 0.3111 9.8130E-03 9.8130E-03
91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03
92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03
94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03
96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04
93Nb 0.3111 6.3576E-03 6.3576E-03
12C-Fuel 0.3111 2.7581E-02 2.5800E-03
1H-Fuel 0.1333 2.5800E-03 2.7581E-02







47














Table 4-10. The properties of axial region 4 fuel materials in the Square-Lattice


Honeycomb homogeneous model.
Heterogeneous
Volume Number Density
Isotope Fraction (atoms/bcm)
235U 0.3111 1.9997E-03
238U 0.3111 1.5052E-04
"9Zr-Fuel 0.3111 9.8130E-03
91Zr-Fuel 0.3111 2.1400E-03
92Zr-Fuel 0.3111 3.2710E-03
94Zr-Fuel 0.3111 3.3149E-03
"9Zr-Fuel 0.3111 5.3404E-04
93Nb 0.3111 6.3576E-03
12C-Fuel 0.3111 2.7581E-02
1H-Fuel 0.1333 1.9350E-03


90Zr-Fuel
91Zr-Fuel
92 Zr-Fuel
94 Zr-Fuel
96Zr-Fuel

12C-Fuel
1H-Fuel


0.3111
0.3111
0.3111
0.3111
0.3111
0.3111
0.3111
0.3111
0.3111
0.1333


1.9997E-03
1.5052E-04
9.8130E-03
2.1400E-03
3.2710E-03
3.3149E-03
5.3404E-04
6.3576E-03
2.7581E-02
1.9350E-03


Homogeneous
Number Density
(atoms/bcm)
1.9997E-03
1.5052E-04
9.8130E-03
2.1400E-03
3.2710E-03
3.3149E-03
5.3404E-04
6.3576E-03
1.9350E-03
2.7581E-02

1.9997E-03
1.5052E-04
9.8130E-03
2.1400E-03
3.2710E-03
3.3149E-03
5.3404E-04
6.3576E-03
1.9350E-03
2.7581E-02


Region
9 4-1













10 4-2





















Table 4-11. The properties of axial region 5 fuel materials in the Square-Lattice
Honeycomb homogeneous model.
Heterogeneous Homogeneous
Volume Number Density Number Density
Region Isotope Fraction (atoms/bcm) (atoms/bcm)
11 5-1 235U 0.3111 1.9997E-03 1.9997E-03
238U 0.3111 1.5052E-04 1.5052E-04
"9Zr-Fuel 0.3111 9.8130E-03 9.8130E-03
91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03
92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03
94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03
"9Zr-Fuel 0.3111 5.3404E-04 5.3404E-04
93Nb 0.3111 6.3576E-03 6.3576E-03
12C-Fuel 0.3111 2.7581E-02 1.2384E-03
1H-Fuel 0.1333 1.2384E-03 2.7581E-02

12 5-2 235U 0.3111 1.9997E-03 1.9997E-03
238U 0.3111 1.5052E-04 1.5052E-04
9Zr-Fuel 0.3111 9.8130E-03 9.8130E-03
91Zr-Fuel 0.3111 2.1400E-03 2.1400E-03
92Zr-Fuel 0.3111 3.2710E-03 3.2710E-03
94Zr-Fuel 0.3111 3.3149E-03 3.3149E-03
96Zr-Fuel 0.3111 5.3404E-04 5.3404E-04
93Nb 0.3111 6.3576E-03 6.3576E-03
12C-Fuel 0.3111 2.7581E-02 1.2384E-03
1H-Fuel 0.1333 1.2384E-03 2.7581E-02















CHAPTER 5
METHODOLOGY

Monte Carlo Neutron Cross-Sections Library Generation

The starting point of this research is the generation of a Monte Carlo neutron cross-

section library. These neutron cross-sections will be processed from the ENDF/B VI

data. As described in Chapter 2, the ENDF/B VI data are processed using the NJOY

nuclear code through several modules. A set of ten temperatures are selected, ranging

from 293.6 K to 3000 K. The essential isotope to be processed are also selected; they are

H, 9Be, 10B, B, 12C, 160, 90Zr, 91Zr, 92Zr, 94Zr, 96Zr, 93Nb, 235U, and 238U. In addition,

essential thermal scattering kernels are also processed. These thermal-scattering-kernels

or S(ca,p) kernels are 1H in water, 1H in zirconium hydride, 9Be in beryllium metal, 9Be in

beryllium oxide, 12C in graphite, and zirconium in zirconium hydride.

MCNP5 utilizes the newly created Monte Carlo neutron cross-sections library to

generate energy spectra of the Square-Lattice Honeycomb. The Monte Carlo analysis

involves generating energy spectra for both the SLHC heterogeneous and SLHC

homogeneous models. These energy spectra are generated using MCNP with 620-energy

bins. The energy spectra generated have a relative error less than 0.1 in each of the 620-

energy bins. The analyses utilize 100,000 particles per cycle and 20000 cycles. Each

analysis utilizes 128-parallel processors at the Los Alamos National Laboratory. With

128 processors, each analysis requires 30,000 minutes of computation time. Figure 5-1

shows the energy spectra of both the SLHC heterogeneous and SLHC homogeneous









model of the Square-Lattice Honeycomb at 293.6 K. Figure 5-1, it shows higher

spectrum in the thermal energy region system and lower energy spectrum in the fast

energy region for the heterogeneous model. Figure 5-2 and Figure 5-3 show a closer look

at the thermal and fast energy regions. Figure 5-2 shows slightly higher values for the

flux in the lower end of the thermal region for the SLHC heterogeneous model. It is

difficult to see, but in Figure 5-3, there are very slightly lower values for the flux in the

epithermal and fast regions of the spectrum for the SLHC heterogeneous model. The

magnitude of the SLHC heterogeneous energy spectrum's peak is 4.5 x 106 at 0.04 eV.

For comparison, the magnitude of the SLHC homogeneous energy spectrum's peak is 4.2

x 106 also at 0.04 eV. The magnitude of the SLHC homogeneous energy spectrum's

peak is 6.7% lower than the magnitude of the SLHC heterogeneous energy spectrum's

peak. The average value of the SLHC heterogeneous energy spectrum is 4.04 x 105,

while the average value of the SLHC homogeneous energy spectrum is 3.81 x 105. The

average value of the SLHC homogeneous energy spectrum is 5.6% lower than the

average value of the SLHC heterogeneous energy spectrum. These differences will

contribute to the accuracy level of the weight function utilized for the multigroup neutron

cross-sections generation.

Other important features in the Figure 5-1, which are clearly presented in Figure 5-

2, are the four small peaks at the energy below 0.01 eV. The first peak's size that are

positioned at 0.009 eV is 3.0 x 106, at 0.0033 eV is 2.1 x 106, at 0.0016 eV is 1.4 x 106,

and at 0.00046 eV is 4.7 x 105. These peaks are the results of the thermal scattering

treatments in ACE files. The thermal scattering kernel was divided into discrete angles.

The numbers of bin affect the accuracy of the thermal spectrum. With a larger bin, the









peaks should not be observed on the plot. Due to time constraint and limited access to

high performance computer, this affect could not be demonstrated in this study.

Choosing the Correct Weight Function

In the multigroup neutron cross-sections generation process, it is essential to select

the correct weight function. A significant calculation error can be observed when an

incorrect weight function is utilized to analyze a highly compact hydrogen-rich nuclear

system, such as the Square-Lattice Honeycomb.

Godiva Calculations

To analyze the importance of choosing the correct weight function when analyzing

a highly-compact hydrogen-rich nuclear system, additional calculations were performed.

One of the calculations is to compare the k-eff results obtained from MCNP and

PARTISN calculations. The MCNP result is assumed to be the "true" result, and it is

used as a benchmark for the other calculations. Before the analysis can begin, a test

model is created. Figure 5-4 shows the 235U Godiva sphere surrounded by a shell of

zirconium hydride. First the test model is analyzed using MCNP, and an energy

spectrum of the system is also generated. Next, two sets of the multigroup neutron cross-

sections are generated-"correct" and "incorrect" cross-sections sets. These cross-

sections consist of a 187-fine-energy-group structure with 55 thermal-energy groups.

The difference between the two multigroup neutron cross-sections is in the way they are

generated. The "correct" multigroup neutron cross-section set is generated utilizing the

Godiva energy spectrum as its weight function. However, the "incorrect" multigroup

neutron cross-section is generated utilizing the Square-Lattice Honeycomb heterogeneous

model energy spectrum as its weight function. A PARTISN nuclear code is used to

perform the analysis. The analysis performed is a 1-D calculation utilizing 187 energy-









groups, 55 thermal-groups, P3 order, and S16 order. Figure 5-5 presents a comparison

between the two energy spectra for the Godiva and the SLHC heterogeneous model that

are obtained from MCNP. The magnitude of the SLHC heterogeneous energy spectrum's

peak is 4.5 x 106 at 0.04 eV. For comparison, the magnitude of the Godiva surrounded

by zirconium hydride energy spectrum's peak is 4.2 x 106 also at 0.04 eV. The

magnitude of the SLHC heterogeneous model energy spectrum's peak is 6.7% lower than

the magnitude of the Godiva surrounded by zirconium hydride energy spectrum's peak.

The average value of the SLHC heterogeneous model energy spectrum is 4.04 x 105,

while the average value of the Godiva surrounded by zirconium hydride energy spectrum

is 3.81 x 105. The average value of the SLHC heterogeneous model energy spectrum is

5.6% lower than the average value of the Godiva surrounded by zirconium hydride

energy spectrum. The k-eff results are presented in Table 5-1. As shown in Table 5-1,

the "incorrect" cross-section data produces a k-eff which is 3% lower than the k-eff

produced by using the correct cross-section data.

In Table 5-1, the values of percent error that are presented below the k-eff values

are related to the relative error of each k-eff value to its corresponding k-eff value

obtained from MCNP, while the values in the percent difference column are the relative

difference between both values of k-eff from PARTISN data. However, this error is

amplified in the highly compact hydrogen rich nuclear system. Additional tests are

performed by replacing zirconium hydride with beryllium metal, beryllium oxide, and

graphite. The results are presented also in Table 5-1. Although there are differences in

the k-eff for each case, the differences were not as significant as in the hydrogen gas case.

For the Be, BeO and graphite case, the average neutron energy is at eV, while the average









neutron energy of zirconium hydride case is at eV. Neutrons are manage to slow down to

a lower energy in the zirconium hydride case compare to the other moderator material

because the presence of hydrogen. The difference in the average neutron energy

contributes to the error in the k-eff value between zirconium hydride case and the other

moderator cases. In the zirconium hydride case, the system is more sensitive to the

weight function used because there are large numbers of thermal neutron compare to the

other moderator cases. These thermal neutrons need to be correctly model in the weight

function. The cross-section goes as 1/E, therefore as the average neutron energy is lower,

the cross-section value is increasing. In the zirconium hydride case, there is a significant

difference of energy spectra in the thermal energy range. This difference contributes to

the large error in k-eff values.

Square-Lattice Honeycomb Calculations

As discussed above, the correct weight function plays a significant role in the

multigroup neutron cross-section generation process. As shown in Figure 5-1 to 5-3,

there are differences in the heterogeneous and homogeneous energy spectra. The

question is how these differences affect the accuracy of the neutron cross-section data

generated. To answer this question, two nuclear cross-section sets are generated to be

used in analyzing the homogeneous model of the Square-Lattice Honeycomb. The "true"

multigroup nuclear cross-section set is generated utilizing the homogeneous energy

spectrum of the Square-Lattice Honeycomb, while the "false" multigroup nuclear cross-

section set is generated using the heterogeneous energy spectrum. As in the previous

experiment, an MCNP calculation is used as a benchmark for this analysis. The

homogeneous model of the Square-Lattice Honeycomb is analyzed by MCNP5 to

determine the "true" k-eff for the homogeneous configuration.









A two dimensional analysis is performed by PARTISN using each multigroup

nuclear cross-section set. PARTISN utilizes 187 energy-groups with 55 thermal-energy-

groups, P3 order of scattering, S16 quadrature order, 155-mesh intervals in the r-direction,

and 256-mesh intervals in the z-direction. The results of the PARTISN calculations are

also shown in Table 5-2. Thus, the answer to the question can be observed in Table 5-2.

Neutronics Analysis

To perform the neutronics analysis of the Square-Lattice Honeycomb, Monte Carlo

and Deterministic methods are used. The neutronics analysis includes the following

analysis:

* Energy spectra characterization at startup, intermediate and operating temperatures.
* Power Distribution analysis at operating temperature
* Temperature coefficient analysis at startup, intermediate and operating
temperatures.
* Control drums analysis at startup, intermediate and operating temperatures.
* Water submersion accident analysis

Energy Spectra Characterization

Energy spectra characterization utilizes the Monte Carlo method. The method is

used to characterize both the heterogeneous and homogeneous models. As discussed in

the beginning of this chapter, the room temperature energy spectra are generated for both

heterogeneous and homogeneous models of the Square-Lattice Honeycomb. In addition,

energy spectra characterization at intermediate and operating temperatures are also

performed. To analysis these two additional temperatures, condition for intermediate and

operating temperatures has to be defined. Table 5-3 defines the intermediate temperature

conditions and the operating temperature conditions. Figure 5-6 through Figure 5-9 show

the graphs of these energy spectra.









Figure 5-6, Figure 5-7, and Figure 5-8 show the Square-Lattice Honeycomb

Heterogeneous model's energy spectrum of the system, fuel region, and moderator region

at 293.6 K and 1200 K and 2500K, while Figure 5-9 describes the Square-Lattice

Honeycomb Homogeneous model's energy spectrum.

Power Distributions and Flux Profiles Analyses

While the earlier analysis is performed utilizing the 187-fine-energy-group cross-

sections library, this next section utilizes the 45-broad-energy-group cross-sections

library. This 45-broad-energy-group library has 13 thermal-energy groups. A method is

needed to be developed in selecting these energy groups. Figure 5-10 shows the total

cross-section plot of several important isotopes in SLHC. Based on this figure, a 45-

broad-energy group is created. The 45-energy group is also based on the combination of

the LASER-THERMOS 35-Group Structure and the LANL 30-Group Structure

(MacFarlane and Muir, 1994). The thermal energy range is resembled the LASER-

THERMOS 35-Group Structure, while the epithermal and fast energy ranges are

resembled the LANL 30-Group Structure. However, the thermal energy below 0.01 eV is

chosen to be represented into a single group because it is found to help the k-eff

accuracy. Additional energy groups are added in the vicinity of the resonance region.

The goal is to create a broad-energy-group structure that will best model these cross-

sections data. A 45-energy structure is found to have the best representation of the cross-

section data. The selection of energy structure is based on the cross-section data

presented in Figure 5-10. Based in Figure 5-10, the resonance regions have the energy

between leV to 100 keV. This energy region needs to be represented well in the energy

group selection. Once this group-structure is developed, a new cross-section library is

generated based on it for all temperatures.









Another important characterization of the system is the power density distribution

and flux profile analyses. These analyses are performed at the operating temperature.

The analyses of power density and flux distribution at operating temperature are scaled to

100-MW-thermal power. The deterministic method is used to perform these analyses.

Figure 5-11 presents the axial power distributions in the middle of the first and second

radial fuel regions. Figure 5-12 shows the radial power distribution in the middle of the

second, third, and fourth axial fuel regions, while Figure 5-12 shows the radial power

distribution in the middle of the first and fifth axial fuel regions. As shown in these

figures, power is only generated in the fuel regions. Figure 5-13 shows discontinuities in

the power density plot; these discontinuities represent the three different axial fuel

regions in the core. In these axial regions, uranium density is varied as shown in Table 4-

2. Figure 5-14 and Figure 5-15 show the fast (energy greater than 65 keV) and

epithermal (energy between 2.5 eV to 65 keV) energy neutron axial flux profiles in the

two radial fuel regions, respectively. Figure 5-16 presents the thermal energy neutron

(energy less than 2.5 eV) axial flux profiles in the two radial fuel regions. Figure 5-17

shows the fast energy neutron radial flux profiles of the Square-Lattice Honeycomb in the

two radial fuel regions. Figure 5-18 and Figure 5-19 present the epithermal and thermal

energy neutron radial flux profiles of the Square-Lattice Honeycomb in the five axial fuel

regions. In Figure 5-17 the peaks in the fast energy flux are found to be in the two radial

fuel regions, while in Figure 5-19, the dips are found to be in the two radial fuel regions.

The fast neutrons are generated in the fuel region, then they are moderated in the

moderator and reflector region, and finally they returns to fuel region to be absorbed by

uranium isotopes.









Temperature Coefficient of Reactivity Analyses

The fuel temperature coefficient of the Square-Lattice Honeycomb can be

calculated by varying the temperature in the fuel region while keeping the temperature of

the other region constant. The fuel temperature coefficient analyses consist of three

different ranges of temperatures: startup temperature range, intermediate temperature

range, and operating temperature range. Figure 5-20 presents the startup fuel temperature

coefficient analysis. Figure 5-21 shows the intermediate fuel temperature coefficient

analysis. Finally, Figure 5-22 describes the operating fuel temperature coefficient

analysis. In the operating temperature range, the moderator temperature coefficient is

analyzed by varying the moderator temperature while keeping the temperature in the fuel

and other regions constant. The plot of this analysis is presented in Figure 5-23. The

combine temperature coefficient is called the system overall temperature coefficient. To

obtain the system overall temperature coefficient, the temperature profiles in Table 5-3

are utilized. Fuel temperatures are used to identify the points in the plot. For room

temperature analysis, all regions are represented to be at the room temperature. Figure 5-

24 presents the system overall temperature coefficient. Based on the figures, the fuel

temperature coefficients are found to be negative. However, the main concern is in the

moderator temperature coefficient, which turned out to be positive. The positive

moderator temperature coefficient is as results of the scattering cross-section increase in

zirconium hydride, as shown in Figure 5-25, and the absorption cross-section decrease in

hydrogen isotope, as shown in Figure 5-26. As the temperature increases, the average

neutron energy increases resulting in lower absorption cross-section of hydrogen. A

lower absorption and fission in 235U will also occur as a result of the increase in average

neutron energy. However, in this analysis, the fuel temperature remains constant, so









neutron average energy in the fuel remains constant throughout the moderator

temperature coefficient analysis. Therefore, the only change in the cross-section is found

in the moderator region. Since the average neutron energy changes only affect the

moderator region, the absorption cross-section of hydrogen plays a dominant role in the

k-eff Since higher average neutron energy means lower absorption in hydrogen, k-eff

will increase as the temperature in the moderator increases. This reactor is behaving

differently from the TRIGA reactor that also uses zirconium hydride because in the

TRIGA reactor, zirconium hydride is integrated with the fuel in the form of Uranium-

Zirconium-Hydride fuel. The TRIGA reactor also uses light water in the core. The

behavior of light water in high temperature is that its density decreases, reducing the

number density of hydrogen, which in turned reduces the moderation of the neutron. The

TRIGA reactor incorporates the zirconium hydride with uranium; hence, the temperature

effect is simultaneous. As the fuel temperature increases, the average neutron energy

increases which makes the absorption of in uranium (also fission) and hydrogen

decreases. Since there is more uranium than hydrogen, the temperature coefficient will

be negative.

Control Drums Analyses

The Square-Lattice Honeycomb reactor is controlled by six control drums that are

located in the reflector region. These control drums have both absorber and reflecting

materials in part of their regions. Figure 5-27 presents the critical configuration of the

control drums of the Square-Lattice Honeycomb. This position places the reactor at the

critical condition at zero power. There are two extreme positions of the control drums:

"fully-in" and "fully-out". The "fully-in" position is when the control drums have the

most absorbing property or when the absorber region is closest to the reactor core. At









this position the reactor is subcritical, and the k-eff at zero power is 0.89858 + 0.00005.

Figure 5-28 shows this configuration. The "fully-out" position is the opposite of the

"fully-in" position; it is when the absorber region is farthest from the reactor core. At

this position the reactor is supercritical, and the k-eff at zero power is 1.05961 + 0.00006.

Figure 5-29 presents this configuration.

These three positions are examined to obtain the reactivity worth of the control

drums at startup, intermediate and operating temperature ranges. Figure 5-30 presents the

reactivity worth of the control drums at room temperature. This figure shows that the

control drums have their highest differential reactivity worth when their position is near

the "half-way" position. The "half-way" position is the position when the control drums

are 90 from the "fully-in" positions.

How many control drums are needed to shutdown the reactor? To answer this

question, we need to perform several analyses to find the fewest number of the control

drums to shutdown the reactor. These analyses are performed under the assumption that

there are independent controls for each control drum. Figure 5-31 shows three different

configurations and their k-eff values with two control drums jammed in the "fully-out"

position. Next, Figure 5-32 shows three different configurations and their k-eff values

with three control drums jammed in the "fully-out" position. Finally, Figure 5-33 shows

three different configurations and their k-eff values with four control drums jammed in

the "fully-out" position. Based on these analyses and the assumption of independent

control of each control drum, the minimum number of control drums needed to shutdown

the reactor is three control drums.









Water Submersion Accident Analysis

The final analysis of the Square-Lattice Honeycomb is the water submersion

accident analysis. This analysis predicts the performance of the reactor if it accidentally

drops into a large body of water, such as an ocean or lake. To perform this analysis, the

heterogeneous model of the Square-Lattice Honeycomb is utilized. The configuration of

the control drums for this analysis will be at the "fully-in" position. Water will surround

the reactor from all sides, and water will also replace all the empty spaces (hydrogen

holes) in the reactor, as shown in Figure 5-34. The ideal performance of the reactor

during such an accident is to stay in the subcritical condition. Based on the analysis

performed utilizing Monte Carlo method, the k-eff of the reactor is found to be 0.95824 +

0.00007. Although, the reactor is indeed subcritical during the water submersion

accident, the shutdown margin (i.e., the fractional Ak/k value below critical of 0.04 Ak/k

is too low. Therefore, additional absorber materials are required to increase the margin of

safety. A boron carbide absorber is placed at the center hydrogen hole, as shown in

Figure 5-35. The reactor's k-eff after this modification is 0.83376 + 0.00006, which yield

a shutdown margin of almost 0.17 Ak/k.








61





















1.OOE +07
1.00E+06 SLHC Heterogeneous Spectrum
1.00E+05 SLHC Homogeneous Spectrum
1.00E+04
1.00E+03
1.00E+02
1.00E+01
4 1 .00EO -00
1.OOE-01 -
S1.00E-02
1.OOE-03
1.OOE-04
1.OOE-05
1.OOE-06
1.OOE-07

0 0 0 0 0 0 0 0 0 0 0 0 0


Energy (eV)

Figure 5-1. Energy spectrum comparison between the SLHC heterogeneous and SLHC
homogeneous models at 293.6 K























5.00E+06
4.50E+06 SLHC Heterogeneous Spectrum
E\ SLHC Homogeneous Spectrum
4.00E 06
3.50E+06

3.00E+06
S2.50E+06
S2.00E+06
1.50E+ 06









Energy (eV)
Figure 5-2. Energy spectrum comparison between the SLHC heterogeneous and SLHC
homogeneous models in thermal energy range at 293.6 K
5.00E+05 --- n --- --------------------


0.00E+00 ....----------------^----





homogeneous models in thermal energy range at 293.6 K







63




















1.OOE+03

1.00E+02 SLHC Heterogeneous Spectrum
m e oteqrLt t- SLHC Homogeneous Spectrum


1.00E+00
l.OOE -01 ---------^ ^-------------

43 l.OOE-02 ----------------^ ^ ^-------

5 1.00E-03
1 .00E-03 ----------------------------


1.00E-04

1.00E-05

1.00E-06

1.00E-07 ..

0 0 0 0 0 0 0

Energy (eV)

Figure 5-3. Energy spectrum comparison between heterogeneous and homogeneous
models of the Square-Lattice Honeycomb in fast energy range at 293.6 K

































- 13.4 cm


32.0 cm
Figure 5-4. Geometry description of the Godiva sphere surrounded by hydrogen gas.






























1.00E+07

1.OOE+06 -- SLHC Heterogeneous Spectrum

1.00E+05 Zirconium Hydride Godiva Spectrum

1.00E+04

1.00E+03 -

1.00E+02 -

& 1.00E+01

1.00E+00

1 .0 0 E -0 1 ---------------------------^ ^----
1.00E-02



1.00E-03 -

1.00E-04

1.00E-05

1.00E-06 -

1.00E-07




Energy (MeV)

Figure 5-5. Energy spectrum comparison between heterogeneous model of the Square-
Lattice Honeycomb and 235U Godiva surrounded by H2 gas at 293.6 K
obtained from MCNP.































Table 5-1. Comparison of PARTISN calculations utilizing correct and incorrect
multigroup neutron cross-sections for four surrounding shells materials with
the radius of 235U Godiva is 6.7 cm
Type of Shell Outer PARTISN result PARTISN result
surrounding Radius with "correct" with "incorrect" %
shell (cm) MCNP Result cross-section cross-section difference
H2 32 0.99367 + 0.00002 1.00032 0.97015 -3.02
0.67% -2.37%
Be metal 32 1.00334 + 0.00001 1.00356 1.00366 0.01
0.02% 0.03%
BeO 32 1.00295 + 0.00001 1.00366 1.00397 0.03
0.07% 0.10%
Graphite 32 0.99667 + 0.00001 0.99587 0.99307 -0.28
-0.08% -0.36%






























Table 5-2. Comparison of PARTISN calculations utilizing "true" and "false" multigroup
neutron cross-sections for the Square-Lattice Honeycomb
Type of Calculations Results % Different % Difference from Overall
MCNP Heterogeneous (Benchmark 0.99997 + 0.00005
Overall)
MCNP Homogeneous (Benchmark for 0.99522 + 0.00002 0.48
PARTISAN)
PARTISN Homogeneous with 0.99001 -0.52 -1.00
Homogeneous Cross Sections
PARTISN Homogeneous with 0.95569 -3.97 -4.43
Heterogeneous Cross Sections
















Table 5-3. The intermediate temperature conditions and the operating temperature
conditions for the Square-Lattice Honeycomb
Intermediate Operating
Temperature Temperature
Region (K) (K)
Hydrogen hole 1000 2000
Top Hydrogen hole 400 600
Zirconium tube 800 1200
Zirconium hydride 600 1000
Bottom Graphite Coating 1000 2000
Bottom ZrO2 Coating 800 1200
6 Inner Cylinder of Control Drums 400 600
6 Outer Cylinder of Control Drums (Beryllium) 400 600
6 Outer Cylinder of Control Drums (B4C) 400 600
Beryllium reflector 400 600
Beryllium reflector and Hydrogen gas 400 600
18 First Axial Fuel Region 400 800
18 First Axial Graphite Coating 400 600
18 First Axial ZrO2 Coating 400 600
18 Second Axial Fuel Region 600 1200
18 Second Axial Graphite Coating 400 1000
18 Second Axial ZrO2 Coating 400 800
18 Third Axial Fuel Region 800 1600
18 Third Axial Graphite Coating 600 1200
18 Third Axial ZrO2 Coating 400 1000
18 Fourth Axial Fuel Region 1000 2000
18 Fourth Axial Graphite Coating 800 1600
18 Fourth Axial ZrO2 Coating 600 1000
18 Fifth Axial Fuel Region 1200 2500
18 Fifth Axial Graphite Coating 1000 2000
18 Fifth Axial ZrO, Coating 800 1200








69






















1 OOE+07

1 OOE+06 V

1 OOE+05

1 OOE+04 --293 K
-- 1200 K
1 OOE+03 --2500 K

1 OOE+02

1 OOE+01

1 OOE+00

1 OOE-01
1 OOE-02

1 OOE-03

1 OOE-04

1 OOE-05

1 OOE-06

1 OOE-07 .....




Energy (eV)


Figure 5-6. This figure shows the system's energy spectrum of the Square-Lattice
Honeycomb Heterogeneous model at room, intermediate, and operating
temperatures.








70



















1.OOE+07
293 K
1.00E +06 ..- 1200 K


1.OOE+04
1.00E+053 -------------2500----------------K--

1.00E+03

1.OOE+02

1.OOE+01

1.OOE+00

S1.OOE-01

1.OOE-02

1.OOE-03

1.OOE-04

1.OOE-05

1.OOE-06

1.OOE-07



Energy (eV)

Figure 5-7. This figure presents the fuel region's energy spectrum of the Square-Lattice
Honeycomb Heterogeneous model at room, intermediate, and operating
temperatures.








71



















1.OOE+07

1.00E 06 293K

1.OOE+05 -1200 K

1.OOE+04 2500 K -

1.OOE+03

1.OOE+02

E 1.OOE+01

1.OOE+ 00

1.OOE-01

1.OOE-02

1.OOE-03

1.OOE-04

1.OOE-05

1.OOE-06

1.OOE-07




Energy (eV)


Figure 5-8. This plot presents the moderator region's energy spectrum of the Square-
Lattice Honeycomb Heterogeneous model at room, intermediate, and
operating temperatures.








72
























1 OOE+07

1 OOE+06

1 OOE+05

1 OOE+04 --293 K
-- 1200 K
1 00E+03 --2500 K

1 OOE+02

1 OOE+01

t 1 OOE+00

1 OOE-01

1 OOE-02

1 OOE-03

1 OOE-04

1 OOE-05

1 OOE-06

1 OOE-07 .....
2 Ct C 0 rl Ct tf Q r-



Energy (eV)


Figure 5-9. This plot shows the system's energy spectrum of the Square-Lattice
Honeycomb Homogeneous model at room, intermediate, and operating

temperatures.








73























1.OOE+06
-Be-9 B-10

1.OOE+05 B-11 -Zr-90
Zr-91 Zr-92
1.OOE+04 ---|---
Zr-94 Zr-96
1.OOE+03 -- Nb-93 U-235

O -U-238



1.00E+00
1.OOE+00

1.OOE-01

1.OOE-02

1.OOE-03 ....

o o o o o o o o o o o

Energy (MeV)

Figure 5-10. The plot presents the total cross-section data of several important isotopes in
the Square-Lattice Honeycomb.



























1.60E+00

1.40E+00


1.20E+00

1.00E+00

8.00E-01

6.00E-01

4.00E-01


2.00E-01

0.00E+00


0.0 10.0 20.0 30.0 40.0 50.0 60.0
Height (cm)

Figure 5-11. The plot presents the axial power distribution of the Square-Lattice
Honeycomb in the first and second radial fuel regions.


First Radial Fuel at 6.30 cm
Second Radial Fuel at 12.49 cm

Z_7

Z

































- Second Axial Fuel at 31.5 cm
- Third Axial Fuel at 22.5 cm
- Fourth Axial fuel at 13.5 cm









00

U


0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
Radius (cm)

Figure 5-12. The figure presents the radial power distribution of the Square-Lattice
Honeycomb in the second, third and fourth axial fuel regions.


1.40E+00


1.20E+00


1.OOE+00


8.00E-01


^ 6.00E-01


4.00E-01


2.00E-01


0.00E+00


































First Axial Fuel at 40.5 cm
Fifth Axial Fuel at 5.5 cm







,,,
7U

o, i'
Ui


2 -U ',
C: -
"--------- 1 1 -- C5
oD u'
2 ^ c
C a


0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
Radius (cm)

Figure 5-13. The figure shows the radial power distribution of the Square-Lattice
Honeycomb in the first and fifth axial fuel regions.


6.00E-01



5.00E-01



4.00E-01



- 3.00E-01



I 2.00E-01



1.OOE-01



0 OF0+00







77
















1.80E+21

1.60E+21

1.40E+21

1.20E+21

1.00E+21

S 8.00E+20

6.00E+20

4.00E+20 Fast Flux in First Radial Fuel at 6.3 cm

2.00E20 Fast Flux in Second Radial Fuel at 12.49 cm


0.00E+00
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0
Height (cm)

Figure 5-14. The plot illustrates the fast neutron energy (> 65 keV) axial flux profiles of
the Square-Lattice Honeycomb in the two radial fuel regions.







78




















1.OOE+19

9.00E+18

8.00E+18

7.00E+18 2 o_

6.00E+18

S 5.00E+18

4.00E+18

3.00E+18

2.00E+18
Epithermal Flux in First Radial Fuel Region at 6.3 cm
1.OOE+18 /
Epithermal Flux in Second Radial Fuel Region at 12.49 cm
0.00E 00 ..


Height (cm)

Figure 5-15. The epithermal neutron energy (2.5 eV 65 keV) axial flux profiles of the
Square-Lattice Honeycomb in the two radial fuel regions.


























1.20E+14



1.OOE+14
I \

'" 8.00E 13



S 6.00E+13



4.00E+13



2.00E+13 Thermal Flux in First Radial Fuel Region at 6.3 cm

Thermal Flux in Second Radial Fuel Region at 12.49 cm :

0.OOE+00
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0
Height (cm)

Figure 5-16. The figure illustrates the thermal neutron energy (< 2.5 eV) axial flux
profiles of the Square-Lattice Honeycomb in the two radial fuel regions.







80



















1.80E+21
Fast Flux in First Axial Fuel Region at 40.5 cm
1.60E+21
Fast Flux in Second Axial Fuel Region at 31.5 cm
1.40E+21 Fast Flux in the Third Axial Fuel Region at 22.5 cm

1.20E+21 Fast Flux in the Fourth Axial Fuel Region at 13.5 cm

Fast Flux in the Fifth Axial Fuel Region at 5.5 cm
S1.OOE+21 _______________________________

S 8.00E+20 -n- -
0/ \,/ *o
S6.00E+20 : '

4.00E+20 --- ---

2.00E+20 --

O .OOE + 00 ...
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
Radius (cm)

Figure 5-17. The figure presents the fast neutron energy (> 65 keV) radial flux profiles of
the Square-Lattice Honeycomb in the five axial fuel regions.































I _____


Epithermal Flux in First Axial Fuel Region at 40.5 cm
Epithermal Flux in Second Axial Fuel Region at 31.5 cm
Epithermal Flux in Third Axial Fuel Region at 22.5 cm
Epithermal Flux in Fourth Axial Fuel Region at 13.5 cm
Epithermal Flux in Fifth Axial Fuel Region at 5.5 cm


'N


.72


0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0


Radius (cm)

Figure 5-18. The epithermal neutron energy (2.5 eV 65 keV) radial flux profiles of the
Square-Lattice Honeycomb in the five axial fuel regions.


1.OOE+19

9.00E+18


8.00E+18

7.00E+18

6.00E+18

i 5.00E+18

4.00E+18

3.00E+18


2.00E+18


1.OOE+18

0.OOE+00


>*,
rt
3


J


8





























































1-T0


0.0 5.0 10.0 15.0 20.0

Radius (cm)


0

0
0


25.0 30.0 35.0 40.0


Figure 5-19. The figure presents the thermal neutron energy (< 2.5 eV) radial flux

profiles of the Square-Lattice Honeycomb in the five axial fuel regions.


- Thermal Flux in First Axial Fuel Region at 40.5 cm

- Thermal Flux in Second Axial Fuel Region at 31.5 cm

Thermal Flux in Third Axial Fuel Region at 22.5 cm

Thermal Flux in Fourth Axial Fuel Region at 13.5 cm

-Thermal Flux in Fifth Axial Fuel Region at 5.5 cm


3.00E+14




2.50E+14




' 2.00E+14




1.50E+14




1.OOE+14




5.00E+13




0.00E+00


-




-




_


-







83
















0.995


0.990 *

8 y-=-1.2323E-04x+ 1.0241E+00
0 .9 8 5 ------S---------------------



S0.9809


0.975


0.970


0.965


0.960
250 300 350 400 450 500 550
Temperature (K)

Figure 5-20. The plot shows the plot of fuel temperature coefficient of the Square-Lattice
Honeycomb during startup.
































0.957



0.956



0.955



0.954



0.953
y -1.5778E-05x + 9.7299E-01


0.952 -



0.951
1000 1050 1100 1150 1200 1250 1300 1350 1400
Temperature (K)

Figure 5-21. The plot of fuel temperature coefficient of the Square-Lattice Honeycomb at
the intermediate temperature ranges.





























0.9424


0.9422


0.9420
Sy=--5.0472E-06x + 9.5406E-01
0.9418

0.9416


0.9414

0.9412


0.9410


0.9408

0.9406
2300 2350 2400 2450 2500 2550 2600 2650 2700
Temperature (K)

Figure 5-22. The plot of fuel temperature coefficient of the Square-Lattice Honeycomb at
the operating temperature ranges.