UFDC Home  myUFDC Home  Help 



Full Text  
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 X5 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 SquareLattice Honeycomb System ..............................................6... Background on an Evaluated Nuclear Data File............... .....................................7... 2 CR O SSSECTION PR O CE SSIN G ......................................................... ...............9... CrossSections Processing Codes .......................................................9...... N JOY N nuclear D ata Processing System ........................................... ...............9... TRANSX 2001 a Code for Interfacing MATXS CrossSection 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 CrossSections 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 CrossSections 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 SQUARELATTICE 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 CrossSections Library Generation ....................................49 Choosing the Correct W eight Function.................................................... 51 G odiva Calculations .................................................................. ........... ...............5.. 51 SquareLattice 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 CROSSSECTION L IB R A R IE S .............................................................................................................. 10 1 B ENERGY GROUP STRUCTURES FOR SQUARELATTICE HONEYCOMB .. 106 C AUTOMATION PROCESS FOR PENDF, ACER, AND DTFR MODULES ........ 109 D DETAILED CALCULATION OF NUMBER DENSITIES IN FUEL REGIONS OF SQUARELATTICE HONEYCOM B ....................... .................... ..................... 125 E DETAILED CALCULATION OF THE LOCATIONS OF EACH RING MATERIALS IN THE SQUARELATTICE 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 41. Comparison calculations of the "true" SquareLattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region h om og en ization ........................................................................................................ 32 42. Number densities of the isotopes in the fuel region...............................................37 43. Properties of nonfuel elements in the SquareLattice Honeycomb heterogeneous m o d el ..................................................................................................... ......... .. 3 8 44. Properties of fuel elements in the SquareLattice Honeycomb heterogeneous m o d el ...................................................................................................... ........ .. 3 9 45. Properties of the SquareLattice Honeycomb reactor .........................................42 46. Properties of nonfuel materials in the SquareLattice Honeycomb homogeneous m o d el ...................................................................................................... ........ .. 4 3 47. The properties of axial region 1 fuel materials in the SquareLattice Honeycomb hom ogeneous m odel ... ................................................................... .............. 44 48. The properties of axial region 2 fuel materials in the SquareLattice Honeycomb hom ogeneous m odel ... ................................................................... .............. 45 49. The properties of axial region 3 fuel materials in the SquareLattice Honeycomb hom ogeneous m odel ... ................................................................... .............. 46 410. The properties of axial region 4 fuel materials in the SquareLattice Honeycomb hom ogeneous m odel ... ................................................................... .............. 47 411. The properties of axial region 5 fuel materials in the SquareLattice Honeycomb hom ogeneous m odel ... ................................................................... .............. 48 51. Comparison of PARTISN calculations utilizing correct and incorrect multigroup neutron crosssections for four surrounding shells materials with the radius of 235U Godiva is 6.7 cm .. ................................................................................. 66 52. Comparison of PARTISN calculations utilizing "true" and "false" multigroup neutron crosssections for the SquareLattice Honeycomb................................67 53. The intermediate temperature conditions and the operating temperature conditions for the SquareLattice Honeycomb ................................................... 68 A1. List of essential isotopes in the RGOUW crosssection libraries........................... 102 A2. List of thermal scattering data in the RGOUW crosssection libraries .................. 105 B1. The 45energy group structures...... ............ .......... ...................... 107 B2. The LANL187 energy group structures........................................108 LIST OF FIGURES Figure page 41. SquareLattice Honeycomb nuclear reactor geometry description.......................29 42. The SquareLattice Honeycomb fuel wafers fabrication into fuel element ............30 43. The fabrication of the SquareLattice Honeycomb fuel elements into fuel a sse m b ly ............................................................................................................ .. 3 1 44. Energy spectra of "true" SquareLattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region homogenization. ...........33 45. Energy spectra of "true" SquareLattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region homogenization in the therm al energy range (less than leV ).................................................. ................ 34 46. Geometry description of the SquareLattice Honeycomb heterogeneous model.....35 47. Here is the geometry description of the SquareLattice Honeycomb homogeneous m o d el ...................................................................................................... ........ .. 3 6 51. Energy spectrum comparison between the SLHC heterogeneous and SLHC hom ogeneous m odels at 293.6 K ........................................................ ................ 61 52. Energy spectrum comparison between the SLHC heterogeneous and SLHC homogeneous models in thermal energy range at 293.6 K .................................62 53. Energy spectrum comparison between heterogeneous and homogeneous models of the SquareLattice Honeycomb in fast energy range at 293.6 K ......................63 54. Geometry description of the Godiva sphere surrounded by hydrogen gas ............64 55. Energy spectrum comparison between heterogeneous model of the SquareLattice Honeycomb and 23 5U Godiva surrounded by H2 gas at 293.6 K obtained from M C N P ................................................................................................................ .. 6 5 56. This figure shows the system's energy spectrum of the SquareLattice Honeycomb Heterogeneous model at room, intermediate, and operating temperatures..............69 57. This figure presents the fuel region's energy spectrum of the SquareLattice Honeycomb Heterogeneous model at room, intermediate, and operating tem p eratu re s. ............................................................................................................ 7 0 58. This plot presents the moderator region's energy spectrum of the SquareLattice Honeycomb Heterogeneous model at room, intermediate, and operating tem p eratu re s. ............................................................................................................ 7 1 59. This plot shows the system's energy spectrum of the SquareLattice Honeycomb Homogeneous model at room, intermediate, and operating temperatures ............72 510. The plot presents the total crosssection data of several important isotopes in the SquareL attice H oneycom b ...................................... ....................... ................ 73 511. The plot presents the axial power distribution of the SquareLattice Honeycomb in the first and second radial fuel regions................................................ ................ 74 512. The figure presents the radial power distribution of the SquareLattice Honeycomb in the second, third and fourth axial fuel regions ................................ ................ 75 513. The figure shows the radial power distribution of the SquareLattice Honeycomb in the first and fifth axial fuel regions ..................................................... ................ 76 514. The plot illustrates the fast neutron energy (> 65 keV) axial flux profiles of the SquareLattice Honeycomb in the two radial fuel regions..................................77 515. The epithermal neutron energy (2.5 eV 65 keV) axial flux profiles of the Square Lattice Honeycomb in the two radial fuel regions. .............................................78 516. The figure illustrates the thermal neutron energy (< 2.5 eV) axial flux profiles of the SquareLattice Honeycomb in the two radial fuel regions...............................79 517. The figure presents the fast neutron energy (> 65 keV) radial flux profiles of the SquareLattice Honeycomb in the five axial fuel regions...................................80 518. The epithermal neutron energy (2.5 eV 65 keV) radial flux profiles of the Square Lattice Honeycomb in the five axial fuel regions. ..............................................81 519. The figure presents the thermal neutron energy (< 2.5 eV) radial flux profiles of the SquareLattice Honeycomb in the five axial fuel regions...................................82 520. The plot shows the plot of fuel temperature coefficient of the SquareLattice H oneycom b during startup ....................................... ....................... ................ 83 521. The plot of fuel temperature coefficient of the SquareLattice Honeycomb at the interim ediate tem perature ranges ......................................................... ................ 84 522. The plot of fuel temperature coefficient of the SquareLattice Honeycomb at the operating tem perature ranges ..................................... ...................... ................ 85 523. The plot of moderator temperature coefficient of the SquareLattice Honeycomb at the operating tem perature ranges ........................................................ ................ 86 524. The plot presents the system temperature coefficient of the SquareLattice H o n ey co m b ............................................................................................................ 8 7 525. The plot illustrates the thermal scattering cross section of 1H and Zr in zirconium h y d rid e .................................................................................................... ........ .. 8 8 526. Plot of the 1H neutron absorption cross section..................................................89 527. Critical configurations of the control drums in the SquareLattice Honeycomb at 2 9 3 .6 K ................................................................................................................. ... 9 0 528. The "fullyin" configurations of the control drums in the SquareLattice Honeycomb, which has keff of 0.89858 + 0.00005 at 293.6 K..............................91 529. The "fullyout" configurations of the control drums in the SquareLattice Honeycomb, which has keff of 1.05961 + 0.00006 at 293.6 K..............................92 530. The reactivity worth plot of the control drums from "fullyin" to "fullyout" positions at 293.6 K .. .................................................................................... 93 531. Three different configurations and their keff values of two control drums jammed at the "fullyout" position ........................................ ........................ ................ 94 532. Three different configurations and their keff values with three control drums jam m ed at the "fullyout" position ...................................................... ................ 95 533. Three different configurations and their keff values with four control drums jam m ed at the "fullyout" position ...................................................... ................ 96 534. Configuration of the SquareLattice Honeycomb reactor for water submersion a c c id e n t.................................................................................................................. ... 9 7 535. Modified configuration of the SquareLattice Honeycomb reactor for water subm version accident. ............................... .. ....................... .......................... 98 E1. The locations of fuel ring material in the SquareLattice Honeycomb homogeneous m odel ............. . ............................................................. .......... 12 9 E2. The locations of fuel material in the SquareLattice Honeycomb heterogeneous m o d el ............. . ............................................................. ....... .. 13 0 E3. Locations of graphite coating in the SquareLattice 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 crosssections. 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 crosssections generation in the highlythermalizednuclear 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 crosssections 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 SquareLattice Honeycomb Nuclear Space Reactor. In the comparison calculations, it will include the system characterization calculations, such as energy spectrum comparison, 2D 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 singlestage, 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 highpressure, hightemperature gas, and the expansion of that gas through a suitable supersonic nozzle to the system's lowpressure 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 hightemperature, solidcore 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 reactornozzle 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 superheated 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 highenergy, fissionproduced 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 powerdensity 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 graphitefuel 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 KIWIB reactor series. These reactors generated heat in excess of 1,000 MW. A total of five KIWIB 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 B4E 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 ElGenk, 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 flightpackaged nuclear rocket engine. Following the KIWI B4E fullpower 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 B4E 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 highpower 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 coreexit 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 coreexit temperature, while the allcarbides fuel elements gave promise of coreexit 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 50MW test reactor designated the nuclear furnace (NF1). Four highpower NF1 tests were conducted between June 29 and July 27, 1972. The power level developed was 54 MW, the coreexit 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 coreexit gas temperature between 2600 K and 2800 K. With regard to the allcarbide 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 (NRXEST) series. The primary objectives of this test series were to: (1) demonstrate the feasibility of a hot gas bleedturbine drive cycle; (2) demonstrate the capability of 'bootstrap' startups; 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 lowpower 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 A5 reactor was tested in June 1966 with backtoback 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 A6) was conducted on December 13, 1967. The test operations were a complete successthe reactor was brought to the targeted 2300 K coreexit gas temperature, and then held at that operating condition for 62 minutes. The stage was now set for the final, flightpackaged NERVA engine system (XEPrime) tests. The XEPrime engine featured a closecoupled, flighttype configuration, but was designed for groundtest development. For the XEPrime tests, a new engine test facility, designated engine test facility 1 (ETF1), was constructed. The test stand provided for vertical downward firing of the engine in a simulated flightstage 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 startup characteristics under different control modes, determination of engine and component performance parameters and investigations of engine shutdown and pulsecooling characteristics. During this test period, 24 startups were accomplished, as well as a full power test at 1,100 MW and a coreexit gas temperature of 2300 K (Gunn, 2001). Background on SquareLattice 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 SquareLattice Honeycomb (SLHC) Space Nuclear Rocket Engine is being explored. The SquareLattice 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 crosssections libraries will consist of high temperature nuclear cross sections. These nuclear crosssections 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 crosssections. It was designed to allow easy exchange of crosssection information between various national laboratories. The initial system contained format specifications for neutron crosssections and other related nuclear constants. During the later stages of development, the formats were expanded to include photon interaction crosssections, 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 crosssection for 235U in the energy range 100 keV to 15 MeV only). This library may also contain several different evaluations of the crosssections for a particular nuclide. The ENDF/B library, on the other hand, contains only one evaluation of the crosssections for each material in the library, but each material contains crosssections 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 crosssection processing program (Kinsey, 1975). CHAPTER 2 CROSSSECTION PROCESSING CrossSections Processing Codes A high temperature neutron crosssections library needs to be generated before the SquareLattice Honeycomb analysis can begin. These neutron crosssections 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 crosssections data sets in a form that can be used in various neutronics calculations. However, if the existing neutronicscodes 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 crosssections 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 crosssections 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 definedprocessing task (MacFarlane and Muir, 1994). Several considerations were made for generating the nuclear crosssection library used in the neutronics analysis. One of considerations is the type of a librarya pointwise or a multigroup nuclear crosssection library. The pointwise nuclear cross section library is used in the Monte Carlo code, and the multigroup nuclear crosssection library is used in the deterministic code. This report required producing both a pointwise and a multigroup crosssection library. The nuclides and the number of temperatures to be processed need to be considered. There are 10 different temperatures in the nuclear crosssection 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 libraryneutron 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 crosssections, the number of broad and fine energy groups also needs to be specified. The multigroup crosssections libraries have one fine energy group structure and one broad energygroup structure. These energygroup structures are shown in Appendix B. A weight function is needed to generate this multigroup nuclear crosssections library. A weight function is the neutron energy spectrum of the system or generalizedsystem weight functions. TRANSX 2001 a Code for Interfacing MATXS CrossSection Libraries to Nuclear Transport Codes Discreteordinates (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 onedimensional 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 DIF3D diffusion code and MonteCarlo 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, uptodate, documented crosssection data and prepare them for input into these codes. TRANSX works together with a generalized crosssection library called MATXS (for material crosssection 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 selfshielding 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 "pointwiseENDF" (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., loglog, linearlog). 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 narrowresonance materials. * A resonanceintegral 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 ENDF6 formats are handled except Generalized RMatrix parameters, energy dependent scattering radius, and the calculation of angular distributions from resonance parameters. BROADER Module This module generates Dopplerbroadened cross sections. The input crosssections 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 lowthreshold 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, nonelastic, and sometimes fission or (n, 2n) are reconstructed to equal the sum of their parts. UNRESR Module This module is used to produce effective selfshielded cross sections for resonance reactions in the unresolved energy range. In ENDFformat 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 crosssections 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 crosssections suitable for codes that use the background crosssection 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: * Fluxweighted cross sections are produced for the total, elastic, fission, and capture cross sections, including competition with inelastic scattering * A currentweighted 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 MC22 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 soldas in a power reactoror 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 neutroninduced 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 * ENDF6 chargedparticle distributions are used when available * Kinematics checks are available to improve future evaluations * Both energybalance 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 freegas 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 ENDFformat 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 * Freeatom 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 * Hardtofind parameters for the ENDF/BIII evaluations are included in the THERMR code for the user's convenience * ENDF6 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, shellscripts 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 CrossSections for Monte Carlo Code Once the PENDF library of all nuclides needed for the analysis is produce, code specific crosssections libraries are generated. The Monte Carlo code that was used to perform the analysis in this report was Monte Carlo NParticle (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 selfshielding data generated by UNRESR is suitable for use in multigroup methods after processing by the GROUPR Module, discussed later in this section, but the socalled Bondarenko method is not very useful for continuousenergy Monte Carlo codes like MCNP5. The natural approach for treating unresolvedresonance selfshielding for Monte Carlo codes is the "Probability Table" method. This module produces probability tables to treat unresolvedresonance selfshielding. 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 "chargedparticle production" crosssections 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 He3 production * MT=207  total alpha production The GASPR module goes through all of the reactions given in an ENDFformat evaluation, determines which charged particles would be produced by the reaction, and it adds up the particle yield times the reaction crosssection to produce the desired gas production crosssections. 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 crosssection library. The ACER module prepares libraries in ACE format (A Compact ENDF) for the MCNP continuousenergy neutronphoton Monte Carlo code. MCNP requires that all the crosssections be given on a single union energy grid suitable for linear interpolation. Although the energy grid and crosssection 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 FORTRAN77 directaccess binary file for efficient use during actual MCNP runs * Type 3 is a wordaddressable directaccess 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 shellscripts. 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 CrossSections 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 crosssection library. Additional calculations need to be performed to prepare the library for MATXSR module. Finally, an additional crosssection processing code was needed to organize the MATXSR multigroup library into a working crosssection library for PARTISN. The TRANSX code version 2001 performed these tasks. The other path of organizing the multigroup crosssection is through the DTFR module. GROUP Module This module produces selfshielded multigroup cross sections, anisotropic group togroup scattering matrices, and anisotropic photon production matrices for neutrons from ENDF/BIV, V, or VI evaluated nuclear data. The Bondarenko narrowresonance weighting scheme is usually used. Neutron data and photonproduction data are processed in a parallel manner using the same weight function and quadrature scheme. Twobody scattering is computed with a centerofmass (CM) Gaussian quadrature, which gives accurate results even for small Legendre components of the grouptogroup matrix. Output is written to an output "groupwiseENDF" (GENDF) file for further processing by a formatting module (DTFR, CCCCR, MATXSR). DTFR Module This module is used to prepare libraries for discreteordinates transport codes that accept the DTF format. Transport tables in DTF format are organized to mirror the structure of data inside a discreteordinates 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 grouptogroup scattering cross sections. MATXSR Module This module formats the GENDF tape into a generalized CCCCtype interface format for neutron, photon, and chargedparticle data, including cross sections, groupto group matrices, temperature variations, selfshielding, and timedependence. 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 chargedparticle 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 crosssection processing, such as homogenization of materials, selfshielding, 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 generalpurpose Monte Carlo NParticle 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 crosssection data are used. For neutrons, all reactions given in a particular crosssection evaluation (such as ENDF/BVI) 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 xrays, 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 crosssection 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 timeindependent, 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 onedimensional (slab, cylinder, and sphere), timeindependent transport equation solver using the standard diamonddifferencing 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 timeindependent two dimensional transport equation using the diamonddifferencing 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 twodimensional transport equation on a equilateral triangle spatial mesh. The THREEDANT Solver Module solves the timeindependent, threedimensional transport equation for XYZ and RZ symmetries using both diamond differencing with 23 settozero fixup and the AWDD method. The TWODANT/GQ Solver Module solves the twodimensional timeindependent 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 settozero fix up with changes to accommodate the generalized special meshing (Alcouffe et al., 2002). CHAPTER 4 SQUARELATTICE HONEYCOMB (SLHC) NUCLEAR ROCKET ENGINE DESCRIPTION This chapter discusses the general description of SquareLattice Honeycomb Nuclear Rocket Engine. It describes the geometry as well as the materials in SLHC. Geometry Description The SquareLattice Honeycomb nuclear reactor geometry description is shown in Figure 41. 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 squarelattice honeycomb fuel elements, containing 30% crosssectional flow area, shown in Figure 42 (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 43 (Furman, 1999). Each fuel sub assembly has a 0.5 cm thick graphite coating and a 0.5cm 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 straightforward homogenization method. This method utilizes the straight conversion of heterogeneous number density into homogeneous number density through the use of volume fractions. Table 41 presents the result of calculation of the "true" SLHC heterogeneous model and the heterogeneous model with homogenization in fuel region. Figure 44 presents the fuel region energy spectra comparison of "true" heterogeneous SLHC model and SLHC heterogeneous model with fuel region homogenization. Figure 45 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 41, Figure 44 and Figure 45, 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 keff 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 parallelprocessors, 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 46. 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 SquareLattice 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 47 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 42. Please refer to Appendix D for detailed calculation of the number densities in the SLHC heterogeneous model. A 0.5cm thick graphite coating followed by a 0.5cm 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 neutronabsorber and neutronreflecting material. The neutronabsorber material is boron carbide, and the beryllium is the neutronreflecting material. Table 43 presents properties of all the nonfuel elements in the SLHC heterogeneous model and includes the densities, isotopes, volume, and mass of each region. Table 44 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 45 presents some properties of the SquareLattice Honeycomb reactor. Materials in the Homogeneous SLHC Table 46 presents properties for all the nonfuel 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 47 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 48 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 49 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 410 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 411 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 41. SquareLattice Honeycomb nuclear reactor geometry description. Figure 42. The SquareLattice Honeycomb fuel wafers fabrication into fuel element. E~ 4 cm  1.5 cm I 9 cm Figure 43. The fabrication of the SquareLattice Honeycomb fuel elements into fuel assembly. Table 41. Comparison calculations of the "true" SquareLattice 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 keff 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.OOE01 Energy (eV) Figure 44. Energy spectra of "true" SquareLattice 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 45. Energy spectra of "true" SquareLattice Honeycomb (SLHC) heterogeneous model and SLHC heterogeneous model with fuel region homogenization in the thermal energy range (less than 1eV). Figure 46. Geometry description of the SquareLattice 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 47. Here is the geometry description of the SquareLattice Honeycomb homogeneous model. Table 42. 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 43. Properties of nonfuel elements in the SquareLattice Honeycomb heterogeneous model. Number Density Volume Density Mass Region Isotope (Atoms/bcm) (cm3) (g/cm3) (g) Hydrogen hole 8.6000E03 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 1.7200E02 2.2114E02 4.8226E03 7.3715E03 7.4703E03 1.2035E03 2.2231E02 4.8481E03 7.4104E03 7.5097E03 1.2099E03 8.6418E02 1.4139E01 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 5.7571E02 1.2362E01 1.2362E01 9.1506E02 2.2876E02 2.8596E02 1.2362E01 1.1991E01 1.0320E03 1H 1.4393E02 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 2.8787E02 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 44. Properties of fuel elements in the SquareLattice 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 1.0320E02 2.7880E02 1.0482E02 2.2858E03 3.4939E03 3.5407E03 5.7043E04 6.7908E03 6.6658E04 5.0173E05 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 1.4139E01 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 5.7571E02 1.0320E02 2.7805E02 1.0315E02 2.2494E03 3.4382E03 3.4843E03 5.6134E04 6.6826E03 9.9987E04 7.5259E05 1.4139E01 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 5.7571E02 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 44. Properties of fuel elements in the SquareLattice 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 1.0320E02 2.7656E02 9.9802E03 2.1764E03 3.3267E03 3.3714E03 5.4314E04 6.4660E03 1.6665E03 1.2543E04 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 1.4139E01 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 5.7571E02 1.0320E02 2.7656E02 9.9802E03 2.1764E03 3.3267E03 3.3714E03 5.4314E04 6.4660E03 1.6665E03 1.2543E04 1.4139E01 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 5.7571E02 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 44. Properties of fuel elements in the SquareLattice 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 1.0320E02 2.7656E02 9.9802E03 2.1764E03 3.3267E03 3.3714E03 5.4314E04 6.4660E03 1.6665E03 1.2543E04 2.0358E+03 5.2622E+00 1.0712E+04 18 Fifth Axial Graphite Coating 18 Fifth Axial ZrO2 Coating 1.4139E01 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 5.7571E02 1.1451E+03 1.3996E+03 2.8200E+00 3.2292E+03 5.8900E+00 8.2435E+03 42 Table 45. Properties of the SquareLattice 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 46. Properties of nonfuel materials in the SquareLattice Honeycomb homogeneous model. Heterogeneous Homogeneous Volume Number Density Number Density Region Isotope Fraction (atoms/bcm) (atoms/bcm) 1 1H 1.0000 8.6000E03 8.6000E03 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 2.2114E02 4.8226E03 7.3715E03 7.4703E03 1.2035E03 8.6418E02 2.2231E02 4.8481E03 7.4104E03 7.5097E03 1.2099E03 1.4139E01 5.7571E02 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 1.2362E01 1.1991E01 5.1600E04 9.1506E02 2.2876E02 2.8596E02 2.2114E02 4.8226E03 7.3715E03 7.4703E03 1.2035E03 8.6418E02 2.2231E02 4.8481E03 7.4104E03 7.5097E03 1.2099E03 1.4139E01 5.7571E02 1.4810E02 3.2297E03 4.9367E03 5.0029E03 8.0600E04 1.2362E01 1.1991E01 5.1600E04 9.1506E02 2.2876E02 2.8596E02 44 Table 47. The properties of axial region 1 fuel materials in the SquareLattice Honeycomb homogeneous model. Heterogeneous Homogeneous Volume Number Density Number Density Region Isotope Fraction (atoms/bcm) (atoms/bcm) 3 11 235U 0.3111 1.1665E03 1.1665E03 238U 0.3111 8.7803E05 8.7803E05 "oZrFuel 0.3111 1.0231E02 1.0231E02 91ZrFuel 0.3111 2.2311E03 2.2311E03 92ZrFuel 0.3111 3.4103E03 3.4103E03 94ZrFuel 0.3111 3.4561E03 3.4561E03 "9ZrFuel 0.3111 5.5679E04 5.5679E04 93Nb 0.3111 6.6284E03 6.6284E03 12CFuel 0.3111 2.7768E02 5.1600E03 1HFuel 0.1333 5.1600E03 2.7768E02 4 12 235U 0.3111 1.1665E03 1.1665E03 238U 0.3111 8.7803E05 8.7803E05 9ZrFuel 0.3111 1.0231E02 1.0231E02 91ZrFuel 0.3111 2.2311E03 2.2311E03 92ZrFuel 0.3111 3.4103E03 3.4103E03 94ZrFuel 0.3111 3.4561E03 3.4561E03 96ZrFuel 0.3111 5.5679E04 5.5679E04 93Nb 0.3111 6.6284E03 6.6284E03 12CFuel 0.3111 2.7768E02 5.1600E03 1HFuel 0.1333 5.1600E03 2.7768E02 Table 48. The properties of axial region 2 fuel materials in the SquareLattice Honeycomb homogeneous model. Heterogeneous Homogeneous Volume Number Density Number Density Region Isotope Fraction (atoms/bcm) (atoms/bcm) 5 21 235U 0.3111 1.4998E03 1.4998E03 238U 0.3111 1.1289E04 1.1289E04 "9ZrFuel 0.3111 1.0064E02 1.0064E02 91ZrFuel 0.3111 2.1947E03 2.1947E03 92ZrFuel 0.3111 3.3546E03 3.3546E03 94ZrFuel 0.3111 3.3996E03 3.3996E03 "9ZrFuel 0.3111 5.4769E04 5.4769E04 93Nb 0.3111 6.5201E03 6.5201E03 12CFuel 0.3111 2.7693E02 3.8700E03 1HFuel 0.1333 3.8700E03 2.7693E02 6 22 235U 0.3111 1.4998E03 1.4998E03 238U 0.3111 1.1289E04 1.1289E04 9ZrFuel 0.3111 1.0064E02 1.0064E02 91ZrFuel 0.3111 2.1947E03 2.1947E03 92ZrFuel 0.3111 3.3546E03 3.3546E03 94ZrFuel 0.3111 3.3996E03 3.3996E03 96ZrFuel 0.3111 5.4769E04 5.4769E04 93Nb 0.3111 6.5201E03 6.5201E03 12CFuel 0.3111 2.7693E02 3.8700E03 1HFuel 0.1333 3.8700E03 2.7693E02 46 Table 49. The properties of axial region 3 fuel materials in the SquareLattice Honeycomb homogeneous model. Heterogeneous Homogeneous Volume Number Density Number Density Region Isotope Fraction (atoms/bcm) (atoms/bcm) 7 31 235U 0.3111 1.9997E03 1.9997E03 238U 0.3111 1.5052E04 1.5052E04 9ZrFuel 0.3111 9.8130E03 9.8130E03 91ZrFuel 0.3111 2.1400E03 2.1400E03 92ZrFuel 0.3111 3.2710E03 3.2710E03 94ZrFuel 0.3111 3.3149E03 3.3149E03 96ZrFuel 0.3111 5.3404E04 5.3404E04 93Nb 0.3111 6.3576E03 6.3576E03 12CFuel 0.3111 2.7581E02 2.5800E03 1HFuel 0.1333 2.5800E03 2.7581E02 8 32 235U 0.3111 1.9997E03 1.9997E03 238U 0.3111 1.5052E04 1.5052E04 9ZrFuel 0.3111 9.8130E03 9.8130E03 91ZrFuel 0.3111 2.1400E03 2.1400E03 92ZrFuel 0.3111 3.2710E03 3.2710E03 94ZrFuel 0.3111 3.3149E03 3.3149E03 96ZrFuel 0.3111 5.3404E04 5.3404E04 93Nb 0.3111 6.3576E03 6.3576E03 12CFuel 0.3111 2.7581E02 2.5800E03 1HFuel 0.1333 2.5800E03 2.7581E02 47 Table 410. The properties of axial region 4 fuel materials in the SquareLattice Honeycomb homogeneous model. Heterogeneous Volume Number Density Isotope Fraction (atoms/bcm) 235U 0.3111 1.9997E03 238U 0.3111 1.5052E04 "9ZrFuel 0.3111 9.8130E03 91ZrFuel 0.3111 2.1400E03 92ZrFuel 0.3111 3.2710E03 94ZrFuel 0.3111 3.3149E03 "9ZrFuel 0.3111 5.3404E04 93Nb 0.3111 6.3576E03 12CFuel 0.3111 2.7581E02 1HFuel 0.1333 1.9350E03 90ZrFuel 91ZrFuel 92 ZrFuel 94 ZrFuel 96ZrFuel 12CFuel 1HFuel 0.3111 0.3111 0.3111 0.3111 0.3111 0.3111 0.3111 0.3111 0.3111 0.1333 1.9997E03 1.5052E04 9.8130E03 2.1400E03 3.2710E03 3.3149E03 5.3404E04 6.3576E03 2.7581E02 1.9350E03 Homogeneous Number Density (atoms/bcm) 1.9997E03 1.5052E04 9.8130E03 2.1400E03 3.2710E03 3.3149E03 5.3404E04 6.3576E03 1.9350E03 2.7581E02 1.9997E03 1.5052E04 9.8130E03 2.1400E03 3.2710E03 3.3149E03 5.3404E04 6.3576E03 1.9350E03 2.7581E02 Region 9 41 10 42 Table 411. The properties of axial region 5 fuel materials in the SquareLattice Honeycomb homogeneous model. Heterogeneous Homogeneous Volume Number Density Number Density Region Isotope Fraction (atoms/bcm) (atoms/bcm) 11 51 235U 0.3111 1.9997E03 1.9997E03 238U 0.3111 1.5052E04 1.5052E04 "9ZrFuel 0.3111 9.8130E03 9.8130E03 91ZrFuel 0.3111 2.1400E03 2.1400E03 92ZrFuel 0.3111 3.2710E03 3.2710E03 94ZrFuel 0.3111 3.3149E03 3.3149E03 "9ZrFuel 0.3111 5.3404E04 5.3404E04 93Nb 0.3111 6.3576E03 6.3576E03 12CFuel 0.3111 2.7581E02 1.2384E03 1HFuel 0.1333 1.2384E03 2.7581E02 12 52 235U 0.3111 1.9997E03 1.9997E03 238U 0.3111 1.5052E04 1.5052E04 9ZrFuel 0.3111 9.8130E03 9.8130E03 91ZrFuel 0.3111 2.1400E03 2.1400E03 92ZrFuel 0.3111 3.2710E03 3.2710E03 94ZrFuel 0.3111 3.3149E03 3.3149E03 96ZrFuel 0.3111 5.3404E04 5.3404E04 93Nb 0.3111 6.3576E03 6.3576E03 12CFuel 0.3111 2.7581E02 1.2384E03 1HFuel 0.1333 1.2384E03 2.7581E02 CHAPTER 5 METHODOLOGY Monte Carlo Neutron CrossSections Library Generation The starting point of this research is the generation of a Monte Carlo neutron cross section library. These neutron crosssections 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 thermalscatteringkernels 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 crosssections library to generate energy spectra of the SquareLattice 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 620energy 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 128parallel processors at the Los Alamos National Laboratory. With 128 processors, each analysis requires 30,000 minutes of computation time. Figure 51 shows the energy spectra of both the SLHC heterogeneous and SLHC homogeneous model of the SquareLattice Honeycomb at 293.6 K. Figure 51, it shows higher spectrum in the thermal energy region system and lower energy spectrum in the fast energy region for the heterogeneous model. Figure 52 and Figure 53 show a closer look at the thermal and fast energy regions. Figure 52 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 53, 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 crosssections generation. Other important features in the Figure 51, 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 crosssections 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 hydrogenrich nuclear system, such as the SquareLattice Honeycomb. Godiva Calculations To analyze the importance of choosing the correct weight function when analyzing a highlycompact hydrogenrich nuclear system, additional calculations were performed. One of the calculations is to compare the keff 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 54 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" crosssections sets. These cross sections consist of a 187fineenergygroup structure with 55 thermalenergy groups. The difference between the two multigroup neutron crosssections is in the way they are generated. The "correct" multigroup neutron crosssection set is generated utilizing the Godiva energy spectrum as its weight function. However, the "incorrect" multigroup neutron crosssection is generated utilizing the SquareLattice Honeycomb heterogeneous model energy spectrum as its weight function. A PARTISN nuclear code is used to perform the analysis. The analysis performed is a 1D calculation utilizing 187 energy groups, 55 thermalgroups, P3 order, and S16 order. Figure 55 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 keff results are presented in Table 51. As shown in Table 51, the "incorrect" crosssection data produces a keff which is 3% lower than the keff produced by using the correct crosssection data. In Table 51, the values of percent error that are presented below the keff values are related to the relative error of each keff value to its corresponding keff value obtained from MCNP, while the values in the percent difference column are the relative difference between both values of keff 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 51. Although there are differences in the keff 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 keff 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 crosssection goes as 1/E, therefore as the average neutron energy is lower, the crosssection 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 keff values. SquareLattice Honeycomb Calculations As discussed above, the correct weight function plays a significant role in the multigroup neutron crosssection generation process. As shown in Figure 51 to 53, there are differences in the heterogeneous and homogeneous energy spectra. The question is how these differences affect the accuracy of the neutron crosssection data generated. To answer this question, two nuclear crosssection sets are generated to be used in analyzing the homogeneous model of the SquareLattice Honeycomb. The "true" multigroup nuclear crosssection set is generated utilizing the homogeneous energy spectrum of the SquareLattice 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 SquareLattice Honeycomb is analyzed by MCNP5 to determine the "true" keff for the homogeneous configuration. A two dimensional analysis is performed by PARTISN using each multigroup nuclear crosssection set. PARTISN utilizes 187 energygroups with 55 thermalenergy groups, P3 order of scattering, S16 quadrature order, 155mesh intervals in the rdirection, and 256mesh intervals in the zdirection. The results of the PARTISN calculations are also shown in Table 52. Thus, the answer to the question can be observed in Table 52. Neutronics Analysis To perform the neutronics analysis of the SquareLattice 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 SquareLattice 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 53 defines the intermediate temperature conditions and the operating temperature conditions. Figure 56 through Figure 59 show the graphs of these energy spectra. Figure 56, Figure 57, and Figure 58 show the SquareLattice 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 59 describes the SquareLattice Honeycomb Homogeneous model's energy spectrum. Power Distributions and Flux Profiles Analyses While the earlier analysis is performed utilizing the 187fineenergygroup cross sections library, this next section utilizes the 45broadenergygroup crosssections library. This 45broadenergygroup library has 13 thermalenergy groups. A method is needed to be developed in selecting these energy groups. Figure 510 shows the total crosssection plot of several important isotopes in SLHC. Based on this figure, a 45 broadenergy group is created. The 45energy group is also based on the combination of the LASERTHERMOS 35Group Structure and the LANL 30Group Structure (MacFarlane and Muir, 1994). The thermal energy range is resembled the LASER THERMOS 35Group Structure, while the epithermal and fast energy ranges are resembled the LANL 30Group 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 keff accuracy. Additional energy groups are added in the vicinity of the resonance region. The goal is to create a broadenergygroup structure that will best model these cross sections data. A 45energy structure is found to have the best representation of the cross section data. The selection of energy structure is based on the crosssection data presented in Figure 510. Based in Figure 510, 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 groupstructure is developed, a new crosssection 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 100MWthermal power. The deterministic method is used to perform these analyses. Figure 511 presents the axial power distributions in the middle of the first and second radial fuel regions. Figure 512 shows the radial power distribution in the middle of the second, third, and fourth axial fuel regions, while Figure 512 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 513 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 514 and Figure 515 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 516 presents the thermal energy neutron (energy less than 2.5 eV) axial flux profiles in the two radial fuel regions. Figure 517 shows the fast energy neutron radial flux profiles of the SquareLattice Honeycomb in the two radial fuel regions. Figure 518 and Figure 519 present the epithermal and thermal energy neutron radial flux profiles of the SquareLattice Honeycomb in the five axial fuel regions. In Figure 517 the peaks in the fast energy flux are found to be in the two radial fuel regions, while in Figure 519, 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 SquareLattice 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 520 presents the startup fuel temperature coefficient analysis. Figure 521 shows the intermediate fuel temperature coefficient analysis. Finally, Figure 522 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 523. The combine temperature coefficient is called the system overall temperature coefficient. To obtain the system overall temperature coefficient, the temperature profiles in Table 53 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 crosssection increase in zirconium hydride, as shown in Figure 525, and the absorption crosssection decrease in hydrogen isotope, as shown in Figure 526. As the temperature increases, the average neutron energy increases resulting in lower absorption crosssection 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 crosssection is found in the moderator region. Since the average neutron energy changes only affect the moderator region, the absorption crosssection of hydrogen plays a dominant role in the keff Since higher average neutron energy means lower absorption in hydrogen, keff 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 ZirconiumHydride 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 SquareLattice 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 527 presents the critical configuration of the control drums of the SquareLattice Honeycomb. This position places the reactor at the critical condition at zero power. There are two extreme positions of the control drums: "fullyin" and "fullyout". The "fullyin" 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 keff at zero power is 0.89858 + 0.00005. Figure 528 shows this configuration. The "fullyout" position is the opposite of the "fullyin" position; it is when the absorber region is farthest from the reactor core. At this position the reactor is supercritical, and the keff at zero power is 1.05961 + 0.00006. Figure 529 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 530 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 "halfway" position. The "halfway" position is the position when the control drums are 90 from the "fullyin" 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 531 shows three different configurations and their keff values with two control drums jammed in the "fullyout" position. Next, Figure 532 shows three different configurations and their keff values with three control drums jammed in the "fullyout" position. Finally, Figure 533 shows three different configurations and their keff values with four control drums jammed in the "fullyout" 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 SquareLattice 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 SquareLattice Honeycomb is utilized. The configuration of the control drums for this analysis will be at the "fullyin" 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 534. 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 keff 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 535. The reactor's keff 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.OOE01  S1.00E02 1.OOE03 1.OOE04 1.OOE05 1.OOE06 1.OOE07 0 0 0 0 0 0 0 0 0 0 0 0 0 Energy (eV) Figure 51. 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 52. 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.OOE02 ^ ^ ^ 5 1.00E03 1 .00E03  1.00E04 1.00E05 1.00E06 1.00E07 .. 0 0 0 0 0 0 0 Energy (eV) Figure 53. Energy spectrum comparison between heterogeneous and homogeneous models of the SquareLattice Honeycomb in fast energy range at 293.6 K  13.4 cm 32.0 cm Figure 54. 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 ^ ^ 1.00E02 1.00E03  1.00E04 1.00E05 1.00E06  1.00E07 Energy (MeV) Figure 55. 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 51. Comparison of PARTISN calculations utilizing correct and incorrect multigroup neutron crosssections 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 crosssection crosssection 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 52. Comparison of PARTISN calculations utilizing "true" and "false" multigroup neutron crosssections for the SquareLattice 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 53. The intermediate temperature conditions and the operating temperature conditions for the SquareLattice 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 OOE01 1 OOE02 1 OOE03 1 OOE04 1 OOE05 1 OOE06 1 OOE07 ..... Energy (eV) Figure 56. This figure shows the system's energy spectrum of the SquareLattice 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 2500K 1.00E+03 1.OOE+02 1.OOE+01 1.OOE+00 S1.OOE01 1.OOE02 1.OOE03 1.OOE04 1.OOE05 1.OOE06 1.OOE07 Energy (eV) Figure 57. This figure presents the fuel region's energy spectrum of the SquareLattice 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.OOE01 1.OOE02 1.OOE03 1.OOE04 1.OOE05 1.OOE06 1.OOE07 Energy (eV) Figure 58. 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 OOE01 1 OOE02 1 OOE03 1 OOE04 1 OOE05 1 OOE06 1 OOE07 ..... 2 Ct C 0 rl Ct tf Q r Energy (eV) Figure 59. This plot shows the system's energy spectrum of the SquareLattice Honeycomb Homogeneous model at room, intermediate, and operating temperatures. 73 1.OOE+06 Be9 B10 1.OOE+05 B11 Zr90 Zr91 Zr92 1.OOE+04  Zr94 Zr96 1.OOE+03  Nb93 U235 O U238 1.00E+00 1.OOE+00 1.OOE01 1.OOE02 1.OOE03 .... o o o o o o o o o o o Energy (MeV) Figure 510. The plot presents the total crosssection data of several important isotopes in the SquareLattice Honeycomb. 1.60E+00 1.40E+00 1.20E+00 1.00E+00 8.00E01 6.00E01 4.00E01 2.00E01 0.00E+00 0.0 10.0 20.0 30.0 40.0 50.0 60.0 Height (cm) Figure 511. The plot presents the axial power distribution of the SquareLattice 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 512. The figure presents the radial power distribution of the SquareLattice Honeycomb in the second, third and fourth axial fuel regions. 1.40E+00 1.20E+00 1.OOE+00 8.00E01 ^ 6.00E01 4.00E01 2.00E01 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 513. The figure shows the radial power distribution of the SquareLattice Honeycomb in the first and fifth axial fuel regions. 6.00E01 5.00E01 4.00E01  3.00E01 I 2.00E01 1.OOE01 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 514. The plot illustrates the fast neutron energy (> 65 keV) axial flux profiles of the SquareLattice 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 515. The epithermal neutron energy (2.5 eV 65 keV) axial flux profiles of the SquareLattice 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 516. The figure illustrates the thermal neutron energy (< 2.5 eV) axial flux profiles of the SquareLattice 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 517. The figure presents the fast neutron energy (> 65 keV) radial flux profiles of the SquareLattice 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 518. The epithermal neutron energy (2.5 eV 65 keV) radial flux profiles of the SquareLattice 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 1T0 0.0 5.0 10.0 15.0 20.0 Radius (cm) 0 0 0 25.0 30.0 35.0 40.0 Figure 519. The figure presents the thermal neutron energy (< 2.5 eV) radial flux profiles of the SquareLattice 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=1.2323E04x+ 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 520. The plot shows the plot of fuel temperature coefficient of the SquareLattice Honeycomb during startup. 0.957 0.956 0.955 0.954 0.953 y 1.5778E05x + 9.7299E01 0.952  0.951 1000 1050 1100 1150 1200 1250 1300 1350 1400 Temperature (K) Figure 521. The plot of fuel temperature coefficient of the SquareLattice Honeycomb at the intermediate temperature ranges. 0.9424 0.9422 0.9420 Sy=5.0472E06x + 9.5406E01 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 522. The plot of fuel temperature coefficient of the SquareLattice Honeycomb at the operating temperature ranges. 