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NEUTRONICS GASEOUS OF A CORE COUPLED REACTOR MULTIPLE CHAMBER POWER SYSTEM MATHEW PANICKER 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 To my parents. ACKNOWLEDGEMENTS The author wishes express his sincere thanks and appreciation to the members of his supervisory committee, Alan Jacobs, Nils Diaz, Calvin Oliver, and Robert Hanrahan their willingness ass preparation of this dissertation. The author particularly thankful to Dr. Edward Dugan, his supervisory committee chairman, his guidance, patience, author and understanding grateful having throughout studied this under research. the The direction such splendid personality as Dr. Edward Dugan whose knowledge gaseous core neutronics has been truly invaluable to the preparation this dissertation. This research was supported funds from the Frederick Hauck Fund, the University Florida, and the Air Force Wright Aeronautical Laboratories (AFWAL). The AFWAL work was performed the Innovative Science and Technology Directorate the Strategic Defense Initiative Organization within the Innovative Nuclear Space Power Institute (INSPI) the University of Florida. The author wishes express  S a a . .  a _ this thanks research. and The appreciation author to Dr. also Nils wishes Diaz express Director special INSPI, affording him the opportunity to work INSPI as a research assistant. Computations this research work were done University of Florida facilities (Northeast Regional Data Center and the CAD/CAM center) and the San Diego Supercomputer Center. The author grateful the College Engineering the local computer support and the National Science Foundation computer time the San Diego Supercomputer Center. The author thankful to fellow graduate students with whom he had many fruitful discussions on various aspects this research. Special thanks to Mrs. Mary Cole for her assistance the final preparation this dissertation. Finally, an indescribable degree appreciation goes the author s wife, Annamma, and son, Dinesh, their patience, support, and understanding throughout the author stay the University of Florida. TABLE OF CONTENTS ACKNOWLEDGEMENTS................ .... . .... v111 *iii * * XIlI LIST LIST ABSTRACT...... . ****** ** . CHAPTERS ....... 1 INTRODUCTION.................. Brief Description of the Multiple Cavity Gaseous Core Reactor Power System..... Aim and Scope of the Dissertation....... Organization of the Dissertation........ Pulsed 0.*..0... ......O.. ........0 STUDIES Introduction Choice of Weighting Selection of Phase Treatment of Coupli Coupling Coeffici Delay Time Effect Comments........... ON COUPLED .....**. * Funct Space ng Eff ents.. CORE ions... Regions ects... NEUTRONICS...... *.e a .* ** a . * e S S S S * .. e .. **** ................. ................. PREVIOUS AT THE STUDIES ON UNIVERSITY PULSED GASEOUS OF FLORIDA.... CORE REACTOR Introduction..... UF6 Piston Engine Neutronic Ana Energy Model. Results...... Pulsed Nuclear Gas Generator Pulaed Nuelea of Kylstra sis........ Piston and Pulse Systems........ r Piston System. .... ....** ** al............... * .*.****.. * . .....oo........... *.*.*.......*****.. *...*....* ** . .. ..* ........ .. PREVIOUS OF TABLES................ ......... . ...... OF FIGURES.............................. ....... . i STATIC NEUTRONIC CAVITY GASEOUS STUDIES ON COUPLED MULTIPLE CORE REACTOR SYSTEM............. Introduction... The Bimodal Gas Steady State Ne Steady State Ca I Core utron icula .....*. ... .. *.. Reactor System ic Analysis.... tions Procedure ription scussioS Effect a Beryl the B: Effect Varia Effect Beryl Core Effect Beryl BPGCR Effect the B n of Resu of PGCR C lium Thic PGCR Core of Outer tion..... of BPGCR lium Thic Fuel Load of PGCR C lium Thic Core Fue of Uneaua PGCR Core Its ore kne Is Ber Cor kne ing ore kne 1 1 Loadi: ss Var Unfue yllium e Fuel ss Var Is Fi Fuel ss Var Loading PGCR Co ng a iati led. Thi nd ons Inner When ckness Loading and nations When xed.... Loading nations Is Fix re Fuel Is Unfueled.... .....0.. and Inn When th ed...... Loading a.....a.. Inner the PGCR er e e When Suimmary......... DYNAMIC CAVITY NEUTRONIC GASEOUS Introduction.. Method of Soluw Reactor Kine COUPKIN.... Discussion of ' Integral Par< Neutron Gel Effect oi beryl: BPGCR Effect o variai Effect o inner when Effect o inner when fuel Effect o when tion tics STUDIES ON COUPLED CORE REACTOR SYSTEM of Coupled Equations Results. meters: neration PGCR ium th chamber outer ions.. BPGCR beryll Couplin Time, an ore fuel ckness va is unfue beryllium core fuel ium thickn MULTIPLE ...*Core Point* core Point Co Re oad iat ed...... thicknes oading S PGCR gas loading i f PGCR core fuel 1 beryllium thickne the BPGCR core has load f un the ing.. equal BPGCR PGC cor R core e is un var ixel ing var nom   1 lient ity. nd i when * ... nner the .O... and nations and nations inal 1 load led... ing Di A  ! Fueled PGCR cores BPGCR core.... Summary..... nominally * *** fueled * . 4 0** * ******** CONCLUSION OF FOURTH Introducti Summary of Variatio and In Unfuel Effect o (2070 Variatio and In PGCR F Variatio Inner Chambe Unequa Static and Bimodal Steady Integr Coup Gas RECOMMENDATIONS RESEARCH....... on. . .. I~tl ni ,a a n ne ed f c n PGC Be T BPGCR uter S). .. n in BPGCR ner Be Thi uel Loadin n in PGCR Be Thickne r Loading 1 PGCR Cor Dynamic N Gaseous Co State Neu al Kinetic led Neutron K Feedbacks Moderator Fuel Stud Flow/Heat Gaseous Summary.... Core neti due Refl es.. Tran Core a .... s. CS to ec Rea R Fuel Loading hickness (1040 Chamber....... Be Thickness Va F <* CK g Css 09 (2 e eu re tr S ue ne (1 re ..... Load s (10 atm) Loadi (2030 5 atm).. Loading. tronic M Reactor onic Ana Paramete ing 40 ng cm) ode sy lys rs Calculation Temperature tor Studies. r Studie tA . JL o at a a AND AREAS * a S ................ ................ (206 cm) , ** * atm) riation a.... (25 cm) , (105 , Fix * .. Is f stem is.. for Chang.... Chang for * O O 100 atm) Fixed 0 atm) and ed BPGCR . . 247 247 248 249 249 250 . .. the Individual .... ... * ... *4***** Bimodal * a a a0 a S 4 *e " * O a ...a." a a ....******************** APPENDICES MCNPA GENERAL MONTE PHOTON TRANSPORT.. CARLO CODE FOR NEUTRON AND COUPLED CORE POINT AND THE SOLUTION BENCHMARK CODES ON GAS CORE REACTOR KINETICS PROCEDURE....... CALCULATIONS WITH XSDRNPM REFERENCE SPHERICAL AND REACTOR SYSTEMS........ EQUATIONS AND MCNP CYLINDRICAL COMPARISON OF RESULTS FROM COUPKIN AND . ! b L _  I 1 I Concus.LUa LIST OF TABLES Tables Summary Kinetics of Results Models from for the the Different PNP Neutron Configuration. ...... 59 Typical Equilibrium Generator System Cycle Pulsed Operating Gas Conditions........... Typical Equilibrium System Operating Cycle Piston Driven Conditions.. Core Physics Gaseous Core Parameters Reactor the Bimodal System.. ............. 82 System and of P Core Neutron Neutron GCR Gas Reactor Multiplication Removal Loading System Lifetime for with the Factor (keff) (R) as a Function Bimodal Inner Gas Beryllium Thickness cm.. ........................... 83 System and of P TIBE Neutro Neutron GCR Gas of 20 n Multiplication Removal Loading cm...... Lifetime the Factc (B) BGCR >r (keff) as a Function S system with System and Neutron Neutron of PGCR Gas Multiplication Removal Loading Lifetime the Factc (G) BGCR >r (keff) as a Function System with TIBE of 25 cm...... . . . . . . 85 System and of Neutron Neutron PGCR Gas Multiplication Removal Loading Lifetime the Factc (e) BGCR >r (keff) as a Function S system with TIBE of 30 m ...... ............................. 86 System and Neutron Neutron Multiplication Removal Lifetime Factc (e) ,r (keff) as a Function Pages System from Neutron MCNP Thickness Multiplication as a Function (TIBE) Various Factor Inner B PGCR (k ef) ery lium Core Gas Loadings... System from Neutron XSDRNPM Thickness Multiplication as a Function (TIBE) Various Factor (keff) f Inner Beryllium PGCR Core Gas Loadings. Total PGCR Lifetime Beryllium Gas at Loading and Criticality Thicknesses Neutron for the Removal Various BGCR S Inner ystem........ 410 Neutron Inner B PGCRGas removal erylliu Loadin Lifetime Thickness as Predic as a Function for ted Various MCNP............ 411 System and Neutron Neutron of Outer TIBE of 20 Multiplication Removal Beryllium Lifetime Thickness Factor (keff) (1) as a Function (TOBE) for cm......... 412 System and Neutron Neutron of Outer Multiplication Removal Beryllium Lifetime Thickness Factor (keff) (R) as a Function (TOBE) for TIBE of 30 cm. .... 413 System and Neutron Neutron BPGCR Gas Multiplication Removal Loading Lifetime Factor (keff) (1) as a Function TIBE 414 System and Neutron Neutron of BPGCR Gas Multiplication Removal Loading Lifetime for TIB Factc (E) E of or (keff) as a Function 2 cm. . .... 415 System and Neutron Neutron of BPGCR Gas Multiplication Removal Lifetime Loading TIBE Factor (keff) (1) as a Function cm. 416 System and Neutron Neutron of BPGCR Gas Multiplication Removal Loading Lifetime TIBE Factor as a of 40 f) unction cm. 417 System from Neutron MCNP Multiplication as a Function Factor Inner B (ke f) ery lum cm.......... _ v 418 System Neutron Multiplication Factor (keff) from XSDRNPM as a Function of Inner Beryllium Thickness (TIBE) for the Spherical Mockup of BGCR System for Various BPGCR Gas Loadings... 419 420 421 51 System Neutron Removal Lifetime (i) as Function of Inner Beryllium Thickness for the BGCR System at Various BPGCR GasFuel Loadings.................... System Neutron Mu from XSDRNPM as Thickness (TIBE of BGCR System Loadings...... (TIBE) Itiplication Factor (keff) a Function of Inner Beryllium ) for the Spherical Mockup for Various BPGCR Gas System Neutron Multiplication Factor (k ff) and System Neutron Removal Lifetime () of the BGCR System as Predicted by MCNP for Various Unequal PGCR Gas Loadings......... Integral Kinetics Parameters as PGCR Core FuelGas Loading for with Inner Beryllium Thickness a Function of the BGCR System of 10 cm....... Integral Kinetics Parameters as PGCR Core FuelGas Loading for with Inner Beryllium Thickness Integral Kinetics Parameters as PGCR Core FuelGas Loading for with Inner Beryllium Thickness Integral Kinetics Parameters as PGCR Core FuelGas Loading for with Inner Beryllium Thickness Integral Kinetics Parameters as PGCR Core FuelGas Loading for with Inner Beryllium Thickness a Function of the BGCR System of 20 cm....... a Function of the BGCR System of 25 cm......... a Function of the BGCR System of 30 cm......... a Function of the BGCR System of 40 cm....... Integral Outer System Integral Outer Kinetics Beryllium with TIBE Kinetics Beryllium Parameters Thickness of 20 cm. Parameters Thickness a aa as a (TOBE) as a (TOBE) Function for the Function for the of BGCR of BGCR ........... I BGCR System with TIBE Integral BPGCR Kinetics Core Parameters FuelGas Loading as a Function the BGCR of System with TIBE cm. 510 Integral Core TIBE Kinetics FuelGas of 30 cm. Parameters Loading as a Function the . .. .........* BGCR stem BPGCR with ..*..*..*..* ..** **S** S* ** 511 Integral Core TIBE Kinetics FuelGas of 40 Parameters Loading as a Function the BGCR stem BPGCR with cm. 512 Integral Core TIBE Kinetics FuelGas Parameters Loading as a Function the cm. BGCR stem PGCR with S. . " 513 Integral Core TIBE Kinetics FuelGas of 20 Parameters Loading as a Function the BGCR stem of PGCR with cm... Average Cores of When the Coupling the PGCR Coefficients BGCR Core System w Loading Among ith the a TIBE Pattern PGCR of 2 [60(4) 20(4) , 10(4)].... 515 Average Cores When Coupling for the the PGCR Coeffic BGCR Core ients System Loading Among with the PGCR a TIBE Pattern 50(4), 30(4) , 20(4)]. . a. .. .* * 516 Average Cores When Coupling for the the PGCR Coeffic BGCR Core ients System Loading Among with the PGCR a TIBE Pattern [70(4), 30(4), 20(4)]. ......*.* ........""**** 517 Average Cores When Coupling for the the PGCR Coeffic BGCR Core ients System Loading Among with the a TIBE Pattern PGCR of 20 [80(4), 40(4), 30(4)]. ... ..S .........S*** 518 Average Cores When Coupling for the the PGCR Coefficients BGCR Core System Loading Among with the a TIBE Pattern PGCR of 20 [70(6), . .. .* * flnes.CP4 an 4' e 4bIa a t rI 1 I u S t *= I r_  J1X Lnat * 'lnnrnl 4 ni am 7 a PGCR cm..............* ... 10(6)]........ K,1 Stta iraT Ai MCNP Surface Cards........ ........**************. Basic MCNP Tallies.. . .. .* * * Input Cards for the Reactor System... Bimodal Gaseous Core MCNP Input a Single Core System k from XSDRNPM Benchmark Calculations for th eeference Gaseous Core ReactorCylindrical Geometry.....................*.*** ****. ***.** System k from XSDRNPM Benchmark Calculations for the Reference Gaseous Core ReactorSpheric Geometry Mockup........................... ... System keff from MCNP Benchmark for the Reference Gaseous Core Calculation Reactor System... XSDRNPM XSDRNPM Collapsed Collapsed Relative Neut of Time as a System of 1.0058.... 26Group 4Group ron Level Predicted Neutron Structures............. Structure............... (N(t)/N(O)) by ANCON and Multiplicatio as a Function COUPKIN for n Factor .a . .. . a.. a. a a a Relative Neutron Leve of Time as Predicte a System of Neutron 0.9280.............. 1 (N(t)/N(O)) a d by ANCON and Multiplication a Function OUPKIN for Factor . .. .a a a* a a Relative Neu of Time as a System o 1.0421.... tron Level Predicted f Neutron (N(t)/N(O)) by ANCON and Multiplicatio as a Function COUPKIN for n Factor  . Relative Neutron Level (N(t)/N(O)) as a Function of Time as Predicted by ANCON and COUPKIN for a System of Neutron Multiplication Factor 1.1171..... ............... .. .......... .... . o. ****************. PGCR... ......... ... . __ LIST OF FIGURES Bimodal Gas Core Reactor Power System Schematic.... Top and Side View of Bimodal Gas Core Reactor.. Schematic Representation Power Breeder; (a) Top of Coupled FastThermal View (b) Side View..... Schematic Representation of UF6 Piston Engine...... Neutron Multiplication Factor Versus Pressure for an Infinite Graphite UF Partial Reflector... Relative Thermal Reflector for Flux in the UF6 Core Piston and Graphite Engine.............. Average Core Thermal Neutron Multiplication Factor as a Travel Through the Piston Reflected Engine.......... Flux and Function Cycle for Neutron of Percent a Graphite UFg Piston Imposed Engine Performance at 10 Percent Cycle for Thick Position. Refl . . ector General Schematic Gas Core Nuclear for a Power PistonDriven System...... Pulsed Delayed Neutron and Photoneutron Precursor Concentration Buildup During Startup from for Pulsed Nuclear Piston Engine......... Peak Gas Temperature and Mechanical Time During Startup from Shutdown Nuclear Piston Engine............ Shutdown Power Versus for the Pulsed . e.... Average Thermal Cycle Time for Kinetics Model I n Neutron Flux in the PNP Engine Is used....... tvarrna 'T'harmn 1 Ui the When W inv i n thh Core the mnrs Versus Adiabatic Vsrs lls 0... 5 Mbiit rhn Average Cycle Kinetics Thermal Time Model Neutron the PNP Is Used Flux Engine with 1: the When Core the Versus Point msec. 12 Pulsed Gas Generator Schematic....... 313 Average Time Cor for Nuclear e Thermal a Typical Neutron Gas Flux Generator Versus Pulsed Cycle Gas Core System.... 314 Average Time Cor for e Thermal a Typical Neutron Flux PistonDriven Versus Pulse Cycle d Gas Core Nuclear System.. CrossSectional Core BGCR View Reactor XSDRNPM of Cylindrical MCNP and Spherical Bimodal Gas Mockup Calculation.. ke ke fr for k eff for and TIBE and as a Function of 10 cm.. as a Function of PGCR ...of PGCR... of PGCR Gas * Gas Gas Loading Loading TIBE of 20 cm..... keff for and TIBE as a Function of 25 PGCR cm.. Gas . . Loading ke for and as a Function PGCR Gas Loading TIBE cm.. keff and i TIBE as a Function cm..... of PGCR . 0.. Gas .... Loading System Neutron XS DRNPM Multiplication as a Function of PGC Factor R Gas from Loading. System from Neutron MCNP Multiplication as a Function Factor, (k if) Inner Beryltlum Thickness System Neutron XSDRNPM Multiplication as a Function Factor Inner from Beryllium Thickness... 410 Neutron Removal Function Lifetime of Inner Vanri an Beryllium Gasf from MCNP Thickness Tadina..  as a (TIBE) . . 105 PCnpR System Neutron Multiplication and Neutron Removal Lifetime of Outer Beryllium Thickness TIBE of 20 cm............... System Neutron Multiplication and Neutron Removal Lifetime of Outer Beryllium Thickness of 30 cm.................... Factor (1) as (TOBE) Factor (t) as (TOBE) ....... (keff) a Function for (keff) . a Function for TIBE 413 System keff and I as a Loading for TIBE of Function 10 cm.... BPGCR Gas 414 System Gas keIf and t LoaIdng for as a TIBE Function of 20 cm BPGCR 415 416 System ke f and I as a Loading for TIBE of System ket and I as a Loading or TIBE of Function 30 cm.... Function 40 cm.... BPGCR of BPGCR Gas Gas 417 System Neutron Multiplication Factor (k ) from MCNP as a Function of Inner Berylliume sickness for Various BPGCR Chamber FuelGas Loadings.... 418 System k ^ from XSDRNPM as Inner Beryllium Thickness Chamber GasFuel Loadings a Function for Various of BPGCR 419 System Neutron Removal Lifetime (I) as a Function of Inner Beryllium Thickness as Predicted by MCNP for BPGCR Chamber Gas Loadings............ 420 System keff and Loading with BPGCR Chamber I as a Function 5 atmosphere of for TIBE of 20 of PGCR Gas FuelGas in cm.......... 421 422 System keff and Loading with BPGCR Chamber System keff Thickness PGCR Gas I as a Function of PGCR Gas 5 atmospheres of FuelGas in for TIBE of 30 cm........... as a Function of Inner Beryllium Predicted by XSDRNPM for Various Loadings......................... 411 412 424 Relative Radial Thermal Neutron in the BGCR System with BPGCR Atmospheres and PGCR Cores at FuelGas Loading............. Flux Distribution Chamber at 100 10 Atmospheres Average Coupling Coefficients (aZ2k) of PGCR Gas Loading for a BGCR with 10 cm... ..... ... ......... ........... Average Coupling Coefficients (a2,k) of PGCR Gas Loading for a BGCR with 20 cm............... Average Coupling Coefficients (a2k) of PGCR Gas Loading for a BGCR with 25 cm.............................. Average Coupling Coefficients (a2uk) of PGCR Gas Loading for a BGCR with 30 cm.............................. as a TIBE as a TIBE as a TIBE as a TIBE Average Coupling Coefficients ( a2 ) as of PGCR Gas Loading for a BGCR with T 40 cm................................ Average Coupling Coefficients (a 3) as of PGCR Gas Loading for Various Inner Thickness.................. ...... ... Average Coupling Coefficients of Inner Beryllium Thickness Gas Loadings................ a IBE Function of Function of Function of Function of Function of a Function Beryllium (a2 ~) as a Function for Various PGCR ....................I Reactivity Core Gas Thickness of BGCR Loading ....e... System as a for Various *...... ... Function of PGCR Inner Beryllium Neutron Generation Time as Beryllium Thickness for Chamber Gas Loading of 2 a Function BGCR with atm Helium f Inner BPGCR Gas......... 510 Average Coupling Coefficients of Outer Beryllium Thickness and 30 cm................... (a2 ) as for TIBE Function 20 cm F 11 Ctra om Doar+ iA tr nae a nunr +'i nn nf f irtor 512 Average Coupling Coefficients (a. ) due to Neutronic Contribution from PG Cores to BPGCR Chamber as a Function of BPGCR Core FuelGas Loading........................................ . 513 514 Average Coupling Coefficients (a. ) due to Neutronic Contribution from PG Cores to Chamber as a Function of Inner Beryllium Thickness............................... Average Coupling PGCR Cores (a2 Beryllium Thic Loading of 10 BPGCR Coefficients Between Adjacent 3 13) as a Function of Inner kn6ess for a BGCR with a PGCR Gas atmospheres UF6He Mixture...... 515 Neutr Inn wit UF6 on Generation T er Beryllium Th h PGCR Core Gas He Mixture.... ime (A) ickness Loading as a Function of for a BGCR System of 10 Atmospheres 516 517 Neutron BPGCR Inner Generation Time (A) as Gas Loading for a BGCR Beryllium Thickness of System Reactivity Loading. ...... as a Function a Function System for 10 cm and of BPGCR cm..... Gas 518 519 Average Coupling Coefficients due to Contribution from BPGCR Chamber to as a Function of PGCR Gas Loading. Average coupling Coefficients Contribution from PGCR Cores of the BGCR as a Function of Loading..................... Neutronic PGCR Cores due to Neutronic to BPGCR Chamber PGCR Core Gas 520 Average Cores BGCR Coupling Coefficients as a Function of PGCR for both Unfueled and Between Adjacent PGCR Gas Loading for the Fueled BPGCR Chamber.. 521 Neutron Generation Loading for the BPGCR Chamber... Time BGCR as with a Function Fueled and of PGCR Core Unfueled Gas 200 522 System Reactivity as a Funct Gas Loading for the BGCR w ion ith of PGCR Unfueled Core and 524 Relative P Function GCR Neutron of Time 525 Relative PGCR Function of Neutron Time for Level TIBE Level TIBE and of and of keff 25 cm keff 30 cm as a S......... 214 as a ......... 215 526 Relative PGCR Function of Neutron Time Level and a TIBE kf 40 as a cm. ......... 216 527 Relative for with PGCR a Cycle Inner and Neutron Time Beryllium 40 cm with Level as a Function seconds Thickness ejk=0 )k0 for of 20 During BGCR cm, Exhaust Time System cm, ...... 218 528 Relative PGCR Function of Neutron Time Level an a TIBE d kf of as a cm and Cycle Time of 529 Relative P Function During E 0.3 GCR of seconds Neutron Time xhaust........... with Level During ajk=0nd and a TIBE keff 40 Exhaust ... 221 as a cm with ajk .. 222 530 Relative Time Various PGCR Neutron a BGCR Values Level System with Coupling as a Function TIBE Coeffi clients of Cycle cm for (ejk) .... 224 531 Relative PGCR a BGCR Values Neutron with TIBE Coupling Level of 40 as a Function cm at Coefficients Time Various . 225 532 Relative Time Various PGCR Neutron a BGCR Valies with Level a TIBE as a Function of 30 CoretoCore of Cycle cm for Delay Times (rjk)............................................ 226 533 Relative PGCR Neutron Level as a Function Cycle Time Various (Tjk) a BGCR Values with a TIBE CoretoCore cm for Delay Times 534 Relative Time PGCR the cm, Neutron BGCR and Level System as a Function with cm with TIBE CoretoCore of Cycle cm, Delay Time, sec ajk=0 During Exhaust ...... 229 535 Nj(t)/N.(O) for aJ BGCR a nd C (t)/C (0) with TIBE f 4 Two cm with Cycles the r jk=0 4 I 536 537 538 N. (t)/N.(0) and C (t)/C (0) for for a BGCR with TIBE Af 40 cm Coupling Reduced to 0.6 Times Values...................... N.(t)/N.(0) and C (t)/C (0) for a BGCR with TIBE of 46 cm with Coupling Coefficients Reduced the Actual Values............. N.(t)/N (0) and C (t)/C (0) for a BGC with TIBE of 3 cm with Coupling Reduced to 0.50 times Two Cycles with CoretoCore the Actual Five Cycles for CoretoCore to 0.55 times Four Cycles for CoretoCore the Actual Values. 539 540 Relative PGCR and BPGCR Neutron Levels as a Function of Cycle Time for a BGCR with a TIBE of 30 cm with Coupled PGCR and BPGCR Cores.................................... PGCR Core Neutron Multiplication Factor as Function of Cycle Time for a BGCR System with TIBE of 30 cm...................... 541 542 543 Relative PGCR Function of TIBE of 30 from BPGCR and BPGCR Neutron Cycle Time for a cm with PGCR Cores Core.............. Levels as BGCR with Uncoupled Relative PGCR and BPGCR Neutron Levels as Function of Cycle Time for a BGCR with TIBE of 20 cm with Coupled PGCR and BPGCR Cores............................ PGCR Core Neutron Multiplication Factor as Function of Cycle Time for a BGCR System with TIBE of 20 cm...................... 544 Ai Relative PGCR Function of TIBE of 20 from BPGGCR and BPGCR Neutron Cycle Time for a cm with PGCR Cores Core............. Geometry Splitting/Russian in MCNP................. Levels as BGCR with Uncoupled Roulette .0. 0.0.. Technique Detail of the Weight Window.... . ... a a a aaala t a T  ,E n,,,LU~ II(L1~ ~1 nAUA Abstract of the of Di ssertation :he University Requirements J Presented Florida the Degree the Partial I of Doctor Graduate School fulfillment Philosophy NEUTRONICS GASEOUS OF A COUPLED CORE REACTOR MULTIPLE POWER CHAMBER SYSTEM MATHEW May . PANICKER , 1989 Chairman Major De : Dr. Edward apartment Static . Dugan : Nuclear dynamic n Engineering eutronics Sciences an innovative, coupled, multiple chamber beryllium moderated bimodal gaseous core reactor (BGCR) , capable generating both and high power space applications are inves tigated. found that the coreto core neutronic coupling effects contribute significantly the power behavior any chamber during both power and high power operation. a consequence the strong coupling, steady state operation even power increases can be achieved with the cores operating a subcritical condition. The core power levels can be controlled the core fuel loadings as well as the neutronic coupling effects among the cores The BGCR system consists of a larqe , central w  w chamber and a surrounding annular ring of cylindrical power pulsed gaseous core reactor (PGCR) chambers embedded beryllium moderator. The BPGCR operates as an open flow stem For and PGCR employs s the an MHD gaseous generator fuel mixture energy conversion. cyclically injected into PGCR chambers during the intake phase, heated during the power phase, and then discharged energy conversion system. With multiple PGCR chambers and proper timing of their operation, the conducted neutronics analysis indicates that the power system should able to provide a relatively continuous source of hot pressurized the power conversion system. The effects of PGCR and BPGCR core loadings and inner and outer beryllium thickne sses on the system neutron multiplication factor, system neutron removal lifetime, the core tocore neutronic coupling coefficients have been examined static (steady state) neutronic analy the BGCR. The dynamic behavior the BGCR evaluated the program, COUPKIN, which was developed as part this work. COUPKIN solves the coupled core point reactor kineti equations. integral kinetics parameters the COUPKIN program are obtained from the static analyst the system. The n taken eutronic into coupling account effect among coupled the core cores kineti explicitly equations CHAPTER INTRODUCTION Pulsed cyclic gaseous core nuclear reactor systems have undergone extensive theoretical and experimental investigations at the University Florida. Neutroni CS energetic analyses pulsed gas core reactor systems have lead a basic scientific understanding the behavior devices. that promi associated The with results the pulsed sing nuclear gaseous r energy the these core conceptual research reactor concept that operation efforts these indicate a versatile has and attractive features space power generation as compared conventional solid fueled nuclear reactor systems(1 These features include low critical mass , high fuel utilization, adaptability to different energy conversion systems, high operating temperature and efficiency, and good control and safety characteristic CS. They have additional advantages of compactness, relatively high power density, response, and wide good operating operational ranges excellent flexibility . The dynamic high temperature the coolant/working fluid the core, heat rejection space. The gaseous nature the fuel gives the added advantage rapid startup capabilities and the simple geometry the core structure helps minimize the thermal shock and thermal stress especially during the rapid startup. Pulsed gaseous core reactors (PGCR) are energy intensive, cyclic fission driven power systems that have uniquely attractive neutronic and energetic characteristics. the power chamber or core a single chamber PGCR, fissionable circulating gas is pulsed from a predominantly subcritical state a short lived, power rich nearcritical, critical, or supercritical state and then the hot gas exhausted. The energy released the fissioning gas is extracted any suitable power conversion system. Previous analyses pulsed, cyclic gaseous core reactor power systems have been performed single neutronically isolated chambers. These previous studies have not included the effects on system performance the coretocore neutronic coupling which would exist array pulsed gas core reactor chambers. The pulsed gaseous core reactor system space power generation investigated this dissertation PGCR s are expected to provide long term stationkeeping surveillance power a space based system. If a pulsed gaseous core reactor system with just one or two chambers used pulsation thi effects purpose, will flow make discontinuities difficult due to extract continuous conversion supply system hot . The exhaust multiple gas chamber any power design the power pulsed system will be capable as the analysis Chapters 4 and 5 will show, delivering a relatively continuous source high temperature, pressurized gas power conversion system associated with the reactor uninterrupted power generation . Thi can be achieved properly timing the operation the individual chambers between the intake, power, and exhaust phases. Also the multiple cavity design should have a favorable impact on the system reliability since power can generated even a few the individual chambers fail. Brief Description Gaseous Core The Multiple Reactor Power Cavity System Pulsed The investigated gas core reactor power system consists (Figures and 12) a large , central burst power oscillating gas core reactor (BPGCR) chamber  .. a ~. n, e~cr~n~l 1 ,.~I 4,%A.4 n at a 1 J &T.T AC ~m~ll IYYIIY~ L~ ~ I. I~~(1~Y L 4 i e I  'nq' w Se S gi I 11 a a____ a__ _..a,,.' *  MU frt * I I I S II a mixture uranium hexafluoride (UF6) and helium gas. Highly enriched UFg the reactor fuel. The helium gas added to enhance the thermodynamic, transport, heat transfer characteristics of the fluid. The identical upper and lower halves the reactor system are separated a common moderator slab the mid plane. This moderator slab separates the magnetohydrodynamic (MHD) disk generator regions the top and bottom central BPGCR chambers. The annular ring PGCR chambers designed provide low power station keeping/surveillance purposes. In the PGCR chambers the fissile gaseous fuel (UF6 mixture) cyclically injected into the power chambers, which are individually pulsed from far subcritical to nearcritical, critical, or a supercritical state and the heated gas then discharged energy conversion; each chamber designed produce a few MW(e) power. With the multiple PGCR chamber design and proper timing of operation the individual chambers among their intake, power, and exhaust (discharge) phases, the system expected to provide a continuous source high temperature pressurized gas any suitable power fail. During power operation the system, the central BPGCR chamber will either be voided or have constant, pressure nonfuel gas flow. The transition from station keeping the high burst power mode of operation can achieved circulating a gaseous uranium fuel through the central BPGCR cavity. The neutronic coupling effects the PGCR chambers on the gaseous fuel bearing central chambers provide the necessary criticality/heating condition generate a partially ionized plasma 20004000 The high pressure fuel gas mixture driven into the disk MHD generator through a supersonic nozzle. The energy conversion occurs the disk MHD generator. The fuel gas then passes through a diffuser and a radiator heat exchanger before being circulated back into the central BPGCR chamber a compressor. more detailed description the bimodal gaseous core reactor power system and basic operational details are given Chapter Aim and Scone The Dissertation The primary aim this dissertation to perform static (steady state) and dynamic neutronic esign neutronic coupling among the cavities separated the moderator/reflector; effects variation inner and outer moderator thickness; and the effects variation core fuel gas loadings on the core neutron level the system various cycle times operation. The parameters required the dynamic neutronic analysis the system such as reactivity, neutron generation time, neutronic coupling coefficients among the various from cores, static coretocore neutronic delay analysis times the s etc., system. are obtained Steady state neutronic analysis the system includes examining the effects of variation PGCR gas loading and inner and outer moderator/reflector thickness on the system neutron multiplication factor (keff), system neutron removal lifetime neutronic neutron coupling generation coefficients time (ajk) (A), among and the the various chambers. Calculations have been performed both equal and unequal gas loading among the PGCR chambers. Organization of The Dissertation Chapter the dissertation includes a brief survey of coupled core neutronics analysis methods that have been used in previous studies various coupled core nuclear ( ), analysis which leads to a better understanding of the neutronic behavior of coupled core systems. Chapter discusses the highlights previous gas core research efforts that have been performed the University of Florida . This chapter includes a brief summary methods used the analyses and presents key results obtained from these previous studies. A basic understanding provided the single chamber pulsed gaseous core reactor system concept. Chapters 4 and 5 deal with the static and dynamic neutronic studies, respectively, the multiple cavity gaseous includes core the reactor. results The from static the neutronic general analysis purpose, mainly three dimensional Monte Carlo transport code, MCNP (3), and the onedimensional discrete ordinates transport code, XSDRNPM (4) . The static neutronic analysis includes the effects variation inner and outer moderator thickness and the PGCR and BPGCR gas loading on the system neutron multiplication factor (keff) , the neutron removal lifetime neutronic (). Chapter analysis the which system contains includes the the dynamic effects variation inner and outer moderator thickness and gas loading on the core coupling coefficients, system Chapter 6 draws conclusions from the results obtained this study. This chapter also provides some recommendations various areas future research to be undertaken towards further development multiple gaseous core nuclear reactor power systems. The appendices that are given at the end this dissertation include sections on the derivation and method of solution of coupled core point reactor kinetics equations as employed research, a discussion the basic features and capabilities the MCNP Monte Carlo code, benchmark neutronic calculations with XSDRNPM MCNP codes on reference spherical and cylindrical gaseous core reactor systems, and comparative evaluation results from the coupled core kinetics program, COUPKIN, a single core the point reactor kinetics code, ANCON (5), from the Alamos National Laboratory. CHAPTER PREVIOUS STUDIES ON COUPLED CORE NEUTRONICS Introduction The spatial and spectral coupling effects reactor kinetics were treated many authors (624) during the 1960s and 1970s. The coupled nuclear reactor systems subjected kinetics studies included fastthermal reactors , modular cores large thermal power reactors, clustered rocket reactors, and Argonaut type reactors. From the point view of coupling, that the amount reactivity supplied the mutual neutronic interactions among subsystems, the above reactors range from very tightly coupled systems instance, the fastthermal reactors) very loosely coupled systems (like the coupled rocket reactors). Except the coupled core nuclear rocket engines, the other works on coupled core kinetics referenced this chapter are directed towards reactors with solid fuel cores In view the existence many apparently analysis, different seems formulations appropriate coupled compare core various neutronic models br nriaoraBanni4 in , tho nouiitrnnni r hph ~i nr rnimli Ar rr kalln~ Coupled core reactor kinetics was initiated 1958 Robert Avery (6,7) with his investigation of dynamic characteristics the coupled fastthermal breeder reactors. The investigated fastthermal reactor system that couples a fast and thermal assembly was designed obtain relatively long prompt neutron lifetime that characteristic of a thermal assembly and a high breeding gain that characteristic fast assembly (Figure The coupled system considered Avery consisted of a Pu239 fueled cylindrical fast core surrounded an annular blanket of natural uranium, coolant and structural material followed an annular beryllium moderator surrounded outer blanket consisting primarily depleted uranium. The inner blanket serves as the core the thermal system, a barrier the low energy neutrons between the moderator and fast core, and as a "reflector" for the fast core. In Avery' formulation, each core or system divided into nodes the point model kinetics equations are constructed coupling effects each node, among the taking nodes. into The account integral the and neutronic coupling parameters the coupled core point kineti formulation are obtained from the steady state analysis the corresponding system cores or subsystem. Shield Outer Blanket Inner Blanket Inner Blanket Moderator Outer Blanket Shield Figure 21. Schemati Representation of Coupled Fast Thermal Power Breeder; moderator/reflector medium Masaoki Komato (8) . Komato developed two models; one multinode kinetic theory model with time dependent coupling parameters and a time dependent multicore parameters, diffusion such theory the model diffusion which nuclear coefficient, microscopic cross section etc., are invariable with respect space each region but can vary with time. Modifications and alterations the methods developed Avery and Komato have been achieved many authors and are referenced this chapter (624) Except Avery' approach modifications others, other formulations make use the assumption factorized into that a time neutron dependent angular neutron flux can angular flux an amplitude (power level) function *(r,E,n,t) = N(t) (r,E,n,t) (21) with the restriction that fd3rldEfd2f w(r n,E) t(r,E,f,t) (22) where w(r,n,E) an arbitrary weighting function. The major differences the various coupled core neutronic analyses that are examined this work can classified into three categories, namely, the choice a a   *1e n r~ 44'. a ' r. a  S*1 fl 'I 1.e., ,,,: c, klh .~CIL: ~Y C%A nk a ~h, AC various cores that include the coupling coefficients coretocore delay times. Choice of Weiahtina Functions Three procedures, namely , straight averaging the neutron flux over individual cores or reactor regions, importance weighting, and weighting an average fission density can be distinguished. In the formulation straight averaging of spacetime reactor technique, kinetics as by the . Kaplan based on a onegroup diffusion theory approximation, the averaging done over the core volume fd3r *(r,t) (23) where, 9(r,t) the spacetime dependent flux, and the core volume used as a basis the averaging process leading the nodal equations. The transport theory development of BellaniMorante (10) introduces a spaceangle averaged density, as a basic variable Mj(t) fd3r fd2n F(x,n,t) (24) where F(x,n,t) the neutron angular flux. 4j(t) choice of importance function depends on the nature coupled core array i.e., whether the cores are loosely tightly coupled, whether or not a buffer zone exists between the regions, and whether neutron exchange primarily unidirectional or multidirectional. Another criterion the selection importance function whether the function describing the importance distribution should valid over entire reactor or the importance function calculated separately each core. the reactor system can be analyzed a single importance onedimensional function geometry entire the adoption reactor most suitable isolated (as, the other example hand, , the the KIWI cores cores) are nearly weighting the neutron distribution with an importance function associated with each core is much more reasonable ,17). terms the degree coupling among cores, a single importance function the entire reactor suitable a tightly coupled system, whereas a loosely coupled system the use individual core weighting functions is required. comparison Cockrell and Perez and Hansen (11) approaches illustrates the above criteria regarding the selection treatment the the importance University function. Florida Cockrell Training his Reactor the entire reactor. On the other hand , Hansen s treatment (11) the loosely coupled, clustered KIWI cores employed an importance function associated with each the individual cores. Although each case must be considered separately, these examples from two extreme cases bear out the general guidelines suggested above. Avery his formulation of coupled core kinetics introduced function In the a fission the application source determination Avery density the s method as the integr Kom weighting al parameters. lato, a time dependent ssion source density was used as the weighting function. Gyftopaulos (18) and Gage et al. (13) their formulation of coupled core kinetics employed, addition to the fundamental mode adjoint function, the fission source density fission Sf(r,E,t) density as the weighting as a weighting function. function means The that use the dependent variables reflect the exact growth and decay tendencies the average power individual cores. Because this approach exact treatment of neutron density within the boundaries of a particular core, the only assumption required are therefore the source term that provides the coupling effects from the other cores the array. based on the assumption that only spatial effects were significant the coupling between spatial regions or nodes (cores) the reactor system considered. the case KIWI cores with a leakage spectrum characterized intermediate energy range or the Argonaut reactor with largely kinetics parameter thermal equations are neutron are properly interaction, considered weighted. energy sufficient cases independent provided such large coupled core reactors a better accounting the spectral effects of leakage neutrons would be desirable, especially, the study of short time transients because the spectrum any the cores composed of two parts; one, which a distribution character stic fast, the intermediate, material and thermal composition the core, and two, a distribution characteristic the coupling or leakage neutrons from the other cores. the dynamics a single core, a common time dependency neutron energy groups a good approximation since the neutron spectrum adjusts any nuclear, mechanical, thermal changes the core within a few neutron generations after the changes are introduced. But coupled cores, assumption a common time dependency neutron groups a bad approach because of the spectrum contribution the spectrum due to leakage neutrons may require a longer period to adjust. An example coupled thermal cores where the prompt neutron lifetime may be of the order of 10 second. For a short transient which appreciable changes occur the range milliseconds, assumption a common time dependency neutron groups would ignore the important spectral coupling effects. Cockrell Perez (12), their formulation which they applied on the Argonaut type UFTR, accounted spectral effects the leakage neutrons explicitly. In it the which angular flux, originated k^(r,4n, E) in region \ refers the energy neutron flux group Multiplication the transport equation the adjoint angular flux I (r,n,E) integration over jth spatial region and over the ath energy group leads an ordinary differential Njk aB equation d2n in terms the d3r variable k1 (r,f,IE, t). 25) n E The quantity Njk(t) represents the neutron number density the jth spatial region and ath energy group due neutrons born kth spatial region and fith energy (E) *(r,n,E)v example an m group n region problem, this formulation requires (mn) 2 basic equations, while the other approaches require a much smaller number equations. This formulation may be classified as overly descriptive system; i.e. provides more equations than are required describe the neutronic behavior a reasonable degree accuracy. Treatment Couolinq Effects The reactivity contribution due to neutronic coupling effects among the cores introduced a manner consistent with the treatment of the time dependent variables selection phase space regions,and the selection weighting functions. A particular technique treating the coupling effect may more compatible with the one formulation than with another. Couolina Coefficients Methods introducing coupling effects can classified as falling under either the "reactivity" approach or the "effective source" technique. For comparison various treatments of coupling effects that are examined this chapter, we can classify the methods of Cockrell and Perez (12), Hansen (11), Kaplan 1. (9), Kohler and Plaza Hansen (21), Gage (13), and BellaniMorante (10) the "effective source" technique. In the generalized reactivity reactivity approach, expression assumed a given that core the contains implicitly the contribution due coupling from the other cores. In this approach, account the coupling interaction with the transport theory formalism and importance weighting, the leakage term VI(r fn,E,t) transformed surface into coupling a surface accounts integral. the This neutronic concept interaction between the cores. The derivation the coupled core kinetics equations analogous to that of the generalized point reactor kinetics equations as developed Henry (22) or Bell and Glasstone (23); the adjoint flux an arbitrary stationary state the unperturbed interacting reactor used as a weighting function. Cockrell s coupled core kinetics formulation introduces both spatial and spectral coupling and arrives at generalized coupled point reactor kineti equations region for and energy (t) energy c which Iroup the tim number e t tha of neutrons t originated region and energy group and Cik(t) the precursor a aI both the spatial and spectral effects. Cockrell' (12) formulation assumes that cores have common interfaces with one another and that the coupling neutrons travel only through interfaces. Cockrell assumptions are too restrictive, and exclude many realistic physical situations which coupling between isolated regions which not share common interfaces important. A second di calculation fficulty the with source Cockrell' integral s approach the that the formulation but the simplest geometrical configurations may impractical. dependent Third, coefficient the only partial under lifetime certain O * a time circumstances. Finally from a practical standpoint, the large number equations significant introduced Cockrell disadvantage; s formulation a mgroup, nregion presents problem this formulation, as already indicated, requires (mn) 2 basic equations, while other approaches require a much smaller number equations. Plaza and Kohler (19), their coupled core formulation consider a group of reactors or cores interacting with each other through a nonmultiplying coupling medium. The point reactor kinetics equations contribution using the surface coupling technique (19). Hansen importance (11) weighting negroup consists transport theory of dividing the approach total using flux (Fs) the core into two parts: which refers neutrons time born t and the which core refers and to all remaining other there neutrons until the core identified with neutron leakage into core Consequently he developed his theory starting from two transport equations and derived two sets kinetics equations and N' . Coupled core formulations Baldwin (20), Seale and Hansen (21), and Gage et al. (12) include the contribution from coupling effects as an effective source; the fact that the reactivity coupling requirement recognized but not each core explicitly reduced taken the into account. The (tacit) assumption inherent these derivations that calculating the reactivity associated with the core, the contributions due "effective sources In the on the Baldwin surface (20) that formulation core for must the be set two zero. core configuration the Argonaut reactor, which two slightly subcritical slabs about two feet apart are immersed large graphite reflector, the coupling effect introduced which accounts the interaction between the two cores. In this expression the additional source component, ' is the delay time due the exchange neutrons between two cores and characteristic the graphite reflector, represents the neutron flux core k and the interaction probability between the two cores. The interaction component the source core e12(tr) and that core i.e., the interaction component proportional the average flux the other core. Seale and Hansen (21) uses Baldwin (20) model, generalizing to M cores but retaining a source component the above form. Gage et al. (13), their coupled core analyst introduced a source stribution function Sj (r,E,t) that composed a series of terms representing the power coupling with the other cores the reactor. Since there a finite time lapse from the birth of a neutron one core capture another core causing fission, coupling power growth assigned a retarded argument the form Sj (r,E,t) = E ek(r,E,t) k=lk (26) aI nr,1n 1 4 4. 1 Ja * 6201(tT), *pj(r,E,t)Nk(tTjk) r ir F +> r.rk hra C~~C ~L Al i C ZIIArr energy the core. The term ,E,t) represents the new shape function that accounts the power distribution arising from the effective surface source of coupling neutrons. In introducing such effective source term, the contributions the reactivity expression each core due to the coupling neutrons from other cores must be set zero. Avery formulation of coupled cores treats the system terms integral parameters which explicitly characterize the individual reactors and the coupling between them. The total fission source in the core set equal to the fission source contribution arising that core and the other (Ml) cores (27) The coupling coefficient defined Sjk (28) kjk where kjk the expectation value or probability that fission neutron core (reactor) k gives rise to a next generation of fi ssion neutrons The equations describing the kinetic behavior the n Al' Wi A mw a M 7W = 2 nrr Ejk(r,E,t)pj(r ^ i ^rAn ~rrrrCArn dCki and (210 AiCki. In Equations (29) and (210) and are the effective delayed neutron fraction and decay constant, respectively, the group delayed neutron precursors, the total effective delayed neutron fraction, Cki is a properly weighted measure delayed neutron precursors the group core ND is number of delayed neutron groups, M is the number cores the reactor, kjk the expectation value or probability that a fi ssion neutron core k gives rise to a next generation fission neutrons core and the prompt neutron lifetime the process. The term a measure kjk the cross coupling from core k to core The Equations (29) and (210) can be rewritten terms the time rate change of Njk(=ljkSjk) neutrons equal the difference between the production (including delayed as well as prompt neutrons) and the loss these neutrons (1 ) M Nkm m=l 1km (211) Nj 1jk ljk + E i=1 dCki and =i m=l Ckm. (212) ikm dN jk dt = i j+k = kjk Nkm produce the next generation of neutrons The term dNjk/dt represents the difference in production and loss rate Njk type neutrons The first term on the right hand side Equation (211) , the kjk(l3) Sk, production rate the Njk type of neutrons prompt emission. The term DiSk represents the precursor type, so that this term represents Cki. The the term production XiCki rate represents term the Equation loss (212) rate The total number delayed source neutrons in core k is Cki the so that the production neutron last rate emission. term of Njk Finally equation type Njk/ljk (211) of neutrons represents kjk SXiCki delayed the loss rate type neutrons equation (211). The power level each core the reactor system proportional to the sum the partial fission neutron sources: a S k=l Sjk' (213) a S Nijk k=l In applying this theory the coupled fastthermal breeder rector experiments, a spectral definition the interacting thermal and fast cores terms thermal and fast sources was chosen. The definitions these sources are given Nj k fma 9() system (rv) system vc d3r (215) where the thermal cutoff speed and Vmax the speed at the maximum neutron energy considered. Since the dependent variables Avery's kinetics equations are partial neutron densities (Njk) , this formulation requires equations where M i the number cores Delay the Time system. Effects In Avery's formulation, the delay times associated with the transfer neutrons between cores are not introduced explicitly. When considering pulsed experiments with coupled cores or other aspects the behavior of loosely coupled systems, desirable, sometimes necessary introduce the delay times explicitly into the coupled core kinetics formalism. One possible representation the delay times can the probability jk(tt that neutron born the a th energy group in the k th core time ' enters the core within dt at a time This probability should be chosen so as to represent the physical and nuclear characteristics the system being considered. In Baldwin s formulation (20) two loosely coupled core loadings the Argonaut type, the interaction term the interaction term the delay time the neutron transfer process between the cores. Chezem model and to analyze Helmick the (24) developed Alamos Coupled a general core theoretical experiment which two cores were separated beryllium reflector. pulse of neutrons was introduced into the core time and the neutron response each the core was obtained from neutron The general detectors model located developed the Chezem center of each Helmick core. consists a single neutron balance equation terms the neutron population Ni(t) the i th core of a loosely coupled array cores given dt dt P(T)dT + A6il(t) ai4JN0(tT) 0 (216) where when isolated the excess from the prompt reactivity environment and the the core neutron lifetime. The source term A6il(t) represents a delta pulse of neutrons core 1 and time t=0. The term aijNj (tT)P(T)dT source contribution core from fissions coupling core coefficient times tT core from Quantity fission neutrons the the Ni(t)+ where an "effective interaction time" neutrons entering core from core This expression P(t) will lead to a simple form transfer function and consequently a simple analytical form the time dependent flux. The interaction time which may also called delay time does really provide the function P(T) represent the minimum time delay corresponding the minimum neutron transit time. For example , in the case an internally pulsed "driver" core coupled to a "passive" core, time the response the external the pulse passive core no matter would how close start or far the apart the two cores are. This simple model would lead a simple exponential behavior the driver core while there are indications that a quasioscillatory response possible and may have occurred the KIWITransient Nuclear Test (KIWITNT) experiments (25,26). For these reasons desirable introduce slightly more complicated representation P(T) that will include a finite delay time. Hansen (11) his neutronics studies a cluster of reactors, gives the source term the i th core due interaction the other cores = ij N. (  (tT) (218) Pij (r) C, effectiveness fission neutron that moves from core core i in inducing fission and eventual neutron population and Pij(r) the time distribution function this transfer. the cluster array of coupled cores, Hansen (11) assumed that = 6 (tTij) (219) where the effective mean drift time of neutrons between that the the sum boundaries first three moment mean cores the times; and distribution the time to be noted function fission Pij () i neutron to leak out core from birth, the drift time from the boundary of j the boundary , and the time from entrance absorption In Avery and Komato formulations of coupled core through neutronics a partial coretocore lifetime, ljk(t) delay , the time mean introduced lifetime neutrons the process (j,k) , .e., of neutrons born core k which diffuse the system ultimately cause fission core the formulation Gage et al. (13) nonlinear stability of a coupled core reactor system, the coupling term that accounts the contribution from remaining cores to a specific core introduced with retarded time Pij (v) diffuse from core k to core through any intervening medium which separates the cores. Comments Salient features the major formulations coupled reactor kinetics are presented this chapter. The various major formulations coretocore differ coupling mainly effects that their include treatment the the coupling coefficients approaches h and delay handling the times. couplin Although ca effects the can various broadly classified into "effective source" and "reactivity" techniques, the various approaches differ their actual implementation during the derivation the coupled core kinetics these ap equations. preaches, Based three on the distinct actual im categories plementation can identified the introduction the coupling effects the coupled core kinetics equations: introduction these additional sources into the basic neutron balance equation (14,15,20); (ii) introduction coupling sources during the derivation the coupled point reactor kinetics equations from neutron balance equations replace the contribution from the surface integral (11,19,24); and (iii) the partitioning the neutron flux and hence the neutron source into partial sources that contain the contribution The method introduction time delay also differs among the various formulations. Delay time can introduced as a probability distribution function time finite time a retarded time argument the power levels the remaining cores. The delay times can neglected a situation where the delay times are very short compared to the prompt neutron generation time. The expected influence delay times coupled core kinetics a decrease the systems stability (13). However, the maximum delay time, 'max so short that N(t)N(trmax) << N(t) or N(t)N (t rmax) times of practical interest, then the delay times can be expected to have very little or no practical effect on the time behavior the system. Most the coupled core reactor kinetics formulations that have been previewed this chapter are applied solid systems onedimensional plane geometry with very restrictive conditions. the system has a complex geometry, the case the bimodal gaseous core reactor system, the calculation of precise coupling coefficients delay times becomes complicated and hence should be obtained from Monte Carlo methods. CHAPTER PREVIOUS REACTOR STUDIES AT THE ON PULSED UNIVERSITY GASEOUS CORE OF FLORIDA Introduction Pulsed cyclic gaseous core reactors have undergone extensive theoretical and experimental investigation the University Florida One the earliest conc epts that was proposed Kylstra et al consists a pulsed plasma core reactor enclosed a cylinder and a piston and analogous to a gasoline internal combustion engine (Figure 31) The fuelworking fluid a UF6 mixture. The reactor made critical during the compare ssion stroke and the neutron flux allowed to build to a significant power level The neutronic and energy calculations have shown that significant power (MWs) per cylinder with high efficiencies (40% 50%) can be achieved Kylstra feasibility studi on pulsed gas core reactors have indicated such good performance potential that Dugan . (28) were encouraged to undertake extens theoretical and experimental neutronics and energetic studies on a variety gas core reactor concepts. These a~I~I a mi l.. a 4t h1&n Itt, ~e'e 'DUnun\ T.dli t nl I*C 4 A Thtln A A.^ It^^I A 1 k nrrcrCI hnr Figure 31. Schematic Representation Piston Engine. piston assembly which operates on a thermodynamic cycle similar the internal combustion engine and the Pulsed Gas Generator (PGG) that employs a core of fixed dimensions. These compact , cyclic, fission driven systems have been found to have unique power producing characteristics are capable of generating Piston 10 to 100 Enaine MW(e) Kvlstra power. et al. The piston engine developed Kylstra et al. (27) consists of a cylinder a pi ston surrounded graphite moderator with a nickel lining protection from the fuel working fluid, UF6 During the intake stroke, the UF6 fuelworking fluid drawn into the cylinder and during exhaust stroke the gas and the fission products are ejected from the cylinder. The reactor made critical during compares sion stroke before the piston reaches the top dead center (TDC). To minimize the fission heat release after piston has already passed the power stroke, the reactor required to shut down rapidly. The reflector surrounding the cylinder piston graphite. The graphite has an inner nickel lining because corrosion resistant properties a UF6 atmosphere. The graphite chosen as the reflector low neutron UF6 thermodynamic properties the working fluid. External equipment to be used remove the fission products, cool gas, and recycle back to the cylinder. The reflector thickness varied during the operation of the engine to simulate the desired time sequence the subcriticalsupercriticalsubcritical state the reactor. the compression stroke starts, the reflector starts as a thin reflector, then increases thickness slowly until a step increase reflector removed going back a thin reflector; reflector continues to decrease thickness until the piston bottoms. The engine designed to operate at a high graphite temperature approximately 1000 1200" so as to minimize heat loss from core. The compression ratio the engine with a clearance volume of 0.24 m3 The engine shaft speed rpm and the UF6 100% enriched. Neutronic Analysis For steady state, twogroup, two region diffusion theory analysis, the following assumptions were made. No fast neutron interaction the core. The fast neutron core equation replaced a boundary condition the fast neutron current into the moderator. iii) No delayed neutrons. assumed that the delayed neutron precursors are swept out the cylinder before they exert any influence. No time No angular dependence. dependence. simplicity, the cylindrical geometry was replaced the analysis a spherical geometry with the core volume and reflector thickness conserved times; but they change time simulating the motion the piston within the cylinder. The set of equations based on the above assumptions are solved average thermal 1 core and moderator flux and the neutron multiplication factor (keff) as a function the position the piston the cylinder. the multiplication factor approaches and exceeds unity, a one group, point reactor kinetics equation used to solve the time dependent thermal flux the core. This time dependent solution method adopted because the slow variation system configuration (due to piston motion) compared to the diffusion speed or neutron cycle time. Enerav Model The conservation energy equation a nonflow system is used the energy model the system. fission fragments negligible and that the heat transferred to the walls also negligible. The energy equation balances the rate of variation the internal energy the working fluid (UF6+He) against the rate fission heat release net work done the gas the piston. This equation solved the bulk temperature the core. Results Figure (32) shows the multiplication factor, keff a system with an infinite graphite reflector as a function UF6 pressure partial pressure. of greater than seen atmosphere, that the a UF6 core partial becomes black the neutrons that additional uranium ineffective. Figure (33) shows the relative thermal neutron flux distribution a UF6 partial pressure 0.67 atmosphere and a core volume of 0.44 These two figures imply that low UF6 loading required minimize the nonuniform flux core so that the fission heat deposited a thin shell next to the reflector. The average core thermal flux and neutron multiplication through the factor cycle (keff) Figure as a function shows that of percent the time keff increases to a value greater than when the thick reflector 1.25 1.0 0.75 0.5 0.3 1.0 3.0 10 Pressure (Gas Temperature (atm) = 400K) Figure 32. Neutron Multiplication Factor Vers Pn Tnf nHio lrrnh nb i' us UF6 Par of I or. rr 'tial DyoC cnrro 1.5 u 0 N E U rl.0 tol E 0) C 20 40 60 80 Radius, Figure Relative Reflector Thermal Flux the UF6 in Core Piston and Graphite Engine T 1 J' 1015 Cu N I rz 110 0 X: 4C 'a 4I 00 C 0.9 5 0 *) 4 r' a HI 0.8 S 0.7 c 0.6 Z 0.5 0.4 < 5 0 25 50 75 Percent Initial Initial U235 Travel Through Pressure Mass Temperature = 2.15 ka Cycle = 400K Figure 34. Average Core Thermal Neutron Flux before the thick reflector removed slightly subcritical, causes majority of heat to be added during the last i.e., the compression stroke. While this long cycle time permits easier control the reactor, a little at this larger point; uranium this loading results yields the a keff majority greater the than heat being added a 1525 ms period as the piston passing top dead center (TDC). Figure (35) shows efficiency, power, and pressure versus U235 loading an engine with the thick reflector imposed at the cycle position and removed the position (40% cycle available buildup neutron flux) A maximum keff of 1 to 1 reached these as the TDC systems with the approached. keff This dropping behavior 0.99 keff from 1.01 greatly increases the control safety since the time cons tant larger at higher pressure Increasing U235 loading leads to larger keff but also reduces the helium the mixture, the same initial pressure. Thus, efficiency and the power curves are concave downward reflect the lower average specific heat and hence the poorer thermodynamic properties the gas mixture as more UF6 added the expense of helium. 2.10 2.20 2.25 2.30 2.35 r4 e >4 l0 a) 0 00 &4 "o n 2.15 2.30 1 101 2.35 U235 Mass, KG researchers at the University of Florida to embark on more refined and comprehensive studies on gaseous core reactor systems with and without movable pistons. Pulsed Nuclear Piston and Pulsed Gas Generator Systems Two basic alternative pulsed gaseous core reactor (PGCR) systems that have undergone extensive theoretical and experimental investigations since Kylstra s work are the Pulsed Nuclear Piston (PNP) and Pulsed Gas Generator (PGG) systems concept (29). except These that two the reactors PNP are consists based on a similar of a pulsed gaseous core reactor enclosed a moderating reflector and a piston assembly while the employs a core fixed dimensions. These studies have establi shed a basic scientific understanding of the conceptual operation physical of pulsed phenomena gaseous assoc core s iated with systems and the have resulted favorable response from the scientific and engineering Results community detailed on their theoretical use and space experimental applications. neutronic analyses descriptions processes used the selection of optimum configuration the PNP and can be found 2831). This section presents a brief description the 4ileonic a ranrirtT" nnncnrm't  PNP and Pa.a and ynr 1 c~rl t.th IrnrLS Pulsed Nuclear Piston System The pulsed Nuclear Piston (PNP) system consists pulsed gaseous core reactor enclosed a moderating reflector cylinder a piston assembly (Figure 36) which operates on a thermodynamic cycle similar to the internal combustion engine. The primary working fluid a mixture uranium hexafluoride (UF6) and helium; highly enriched UF6 the reactor fuel and helium gas enhances the thermodynamic, transport, and heat transfer characteristics of the working fluid mixture. The energy released the fissioning gas can extracted both as mechanical power and as heat from the circulating gas which recycles back the core the pressure differential established the power chamber. Mechanical power can be derived from the engine means a conventional crankshaft operating at low speeds. addition the mechanical power, a significant amount energy from the hot gas can removed an external heat removal exhaust loop. gas can The high be cooled temperature (1000 an UF6/He o1300 toHe heat UF6/He exchanger and the heated helium gas can be used a suitable power conversion cycle drive a turbine. Summary PNP Results BeO Moderating Reflector Region DO 20" Moderating Reflector Region He/UF6 BeO Piston research at the University Florida. Twostroke engine analysis considered only the compression and power strokes while the fourstroke engine analysis explicitly examined the intake and exhaust phases operation addition the compression and power strokes. While the earlier studies neglected subsequent studies delayed include neutron d first, and photoneutro the influence effects, of delayed neutrons and then, the influence both the delayed and photoneutrons on the PGCR s performance. Graphite was found to be unacceptable as the moderating reflector since the dimen sions the system with graphite as a moderator were found to be too large to be practical. was determined that BeO) and D20 were the most desirable moderatingreflector materials from the standpoint of neutron neutron li economy fetimes and for size the b the eryllium system. The moderated prompt system were found be half as great as for the D20 reflected system. Although D20 has a significantly lower thermal absorption cross section than beryllium , the Be reflected system has smaller critical mass. This due to the large (n,2n) production that occurs in beryllium. It has been estimated that the (n,2n) production leads an effective increase the average number neutrons per fission (30). found to be large negative .5x10 Ak/k per whereas the moderator temperature coefficient of reactivity Be reflected systems was found to be small and positive 1x10 Ak/k per Although this made or BeO undesirable as a moderator material was suggested that composite, with inner or BeO region surrounded would ideal moderator arrangement with an overall moderator temperature coefficient of reactivity large negative 2x10 Ak/k per Moreover with the (n,2n) production beryllium, this configuration will lead reduced critical mass. Fuel temperature coefficients of reactivity 100% enriched UF6 were found to be small and positive 5x10 Ak/k per the fuel enrichment was reduced the fuel temperature coefficient reactivity was found to decrease and at 80% enrichment the coefficient was found be small and negative 1.2x10 Ak/k per The effects delayed neutrons photoneutrons the system behavior were examined different combinations of procedures. One procedure involved increasing the fuel loading to compensate the absence of delayed neutrons and/or p neutrons hotoneutrons. n.. *U nhotoneutrons example, were when nealecte both . the delayed maximum U were present) to 1.091 order to maintain the engine performance level. When photoneutrons were neglected, order to maintain the system s performance level, the maximum neutron multiplication factor had to be increased 1.061. When the delayed neutrons and photoneutrons were neglected the analysis, the power output dropped relative to the case where only photoneutrons were ignored. Thus, the delayed neutrons and photoneutrons both exert noticeable influence on system behavior with the delayed neutron The influence pulsed being gaseous the core stronger system of the two. capable of rapid startups the presence delayed neutrons The startup procedures used the referenced work (30) include variation circulation inlet time gas Changes pressure the and loop the loop circulation time change the fraction of delayed neutrons which undergo decay while the external circulation loop. Those precursors which do not decay the loop , but decay upon reentering the core after pas sing through the loop, provide an extra neutron source and hence can be used to adjust the power level. Figure gives the delayed neutron and photoneutron precursor densities as a function time during the PNP 1011 21010 0 100 200 300 400 500 600 700 800 Time (sec) Figure 37. Delayed Neutron and Photoneutron Precursor Concentration Buildup During Startup from  r I  I I   I 1200 900 600 0 100 200 Time (sec) Figure 38. Peak Gas Temperature and Mechanical Power Versus Time During Startup from Shutdown Pulsed Neutron Piston Engine. 2000 1750 1500 W1250 (0 (U 'CU 0 1.8 'O 1.6 u 1.4 300 400 500 600 700 800 Time sec Figure . Peak Temperature and Mechanical Power (continued) Versus Time During Engine. Startup from Shutdown with a loop circulation time 11.3 seconds was found to be S13 minutes. Dugan included (30) both the a simple PNP point PGG reactor neutron kinetics kinetics model analysis as well as a more sophisticated adiabatic kinetics model. Changes during the engine cycle, the flux shape, and hence the neutron lifetime and source weighting functions were thoroughly examined and found to be significant. The point reactor kinetics solution procedure was found inadequate handle the significant variations these parameters. correct the inadequacy a simple point reactor kinetics model, a more complex adiabatic model was implemented. This method involves an iterative procedure between the point reactor kinetics program and other independent static neutronic analysis codes to obtain the integral parameters various selected points throughout the cycle of operation the system. Figures 310, which give the average core thermal neutron flux as a function of cycle time from the adiabatic model and the point reactor kinetics model, respectively, illustrate the inadequacy the point reactor kinetics model the neutronic analysis. Results from the point reactor kinetics model were 1015 1014 1013 1012 10 1011 1010 0.00 0 .15 0.45 0.60 0.75 0.90 1.05 1.20 Cycle Time (sec) Figure 39. Average Thermal Cycle Time for Kinetics Model Neutron Flux the PNP Engin Ti Used. in the Core e When Versus the Adiabatic 1015 2 o14 be Cu C: 0 O 4 13 110 0) z 0 La 1 010 U 0f 4 adO10 O u 1 0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 Cycle Time (Sec) Figure 310. Average Thermal Neutron Flux the Core Versus cycle Time Point Kinetics Model P Engine used with when .557 msec. adiabatic analysis were used. However was found to be possible to obtain an adjusted "equivalent" prompt neutron lifetime to be used the point reactor kinetics calculations. The results from such a calculation were found to be very close to those obtained from the more complex adiabatic calculations. The variation the average core thermal neutron flux with cycle time obtained from a point reactor kinetics calculation with "equivalent" prompt neutron lifetime of 1. milliseconds (ms) shown Figure 311. These results are much closer the results from the adiabatic kinetics model (Figure 39) than those obtained from the point reactor kinetics model with a prompt neutron lifetime 2.557 msec (Figure 310) which the cycleaveraged prompt neutron lifetime value obtained from the adiabatic analy sis. Results from these three different models a specific configuration are given Table 31. Generator Pulsed Gas Core Nuclear System The pulsed gas generator (PGG) gas core nuclear concept similar to the pulsed nuclear piston concept except that the gas generator employs a core of fixed dimensions (Figure 312). The gas generator system simpler operation design than the nuclear piston concept and offers a more x O sr 11 ~10D 10 (U Z rid 10 o 1$0 1 Eio12 0) XC ES o( C) 0lO' to0 0 0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 Cycle Time (Sec) Figure 311. Average Cycle Thermal Neutron Time Kinetics Model Is Used Flux Engine with Core When the =1.718 Versus Point msec. rfl CM w r r *r 0 C o C k S00 r0 Z a s E 0 'C *c 90 () 0 H rl r6 k O ,C 0 U *rlW0 c e . % * U O M C co 0 0 ri uQI X 0) #I a e w Inner Moderating Reflector Region Outer Moderating Reflector Region Rotating Absorber He/UF ~6 Fuel Gas Core Region Rotating Absorber I pulsing to supercriticality achieved means of a pile oscillator with a rotating absorber the reflector surrounding the core. The system has the advantage mechanical simplicity over the PNP concept while maintaining the advantages of a pulsed system mentioned elsewhere this dissertation. These include the capacity attaining high peak gas temperature 2000 and high neutron fluxes neutrons/cm2 sec while maintaining relatively average gas temperatures (800900 degree Tables and compare typical operating conditions the PGG and PNP systems, respectively, and reveal that gas generator capable operating at lower enrichments than the nuclear piston system. Figures 313 314 which illustrate the variation core average thermal neutron flux with cycle time the PGG and PNP systems, respectively, reveal that the neutron flux gas generator peaks late cycle relative the piston driven system. The system can provide average exhaust gas temperatures that are closer the peak gas temperature than can the PNP system. UF6 dissociation considerations limit the peak gas temperature to around 2000 then the piston driven system where gas temperature peaks earlier cycle can supply helium gas turbines temperatures of 1015 Table 32. Typical Equilibrium Cycle Generator System Operating Pulsed Gas Conditions. Core Intak Intak Heate Syste Vol e L :e d G Tm C Loop Cir Average Core Hei Core Rad Uranium Helium M Helium M Uranium Maximum Maximum Maximum Flux (n/ Average Average Cycl Cycle Av Cycle Av Flux ( Fission ume ( mine G Line as Av ycle culat Promp ght ( ius ( Mass as ;ol Enr Gas Gas Cor cm Exh Exhau e Ave erage erage n/cm Energy m3) as Pressu Gas Tempe erage Mas Time (mse ion Time t Neutron cm) cm) in Core ( n Core (k reaction i hment (wt emperatur pressure ( Thermal N c) st Gas Te st Gas Pr rage Gas Gas Pres Core The sec) Release Chamber (Mw) Net Turbine Power Chamber (Mw) Efficiency (%) Output (atm) ture (K) Flow Rate ) sec) Lifetime kg) g) n Gas %) e (K) atm) eutron (kg/sec) (msec) Mixture mperature ( essure (atm) Temperature sure (atm) rmal Neutron Rate K) (K) Per Per X 1 14 31 74 20 X 10 16.4 The is The is inner 30 cm outer 70 cm BeO thick D20 thick region and at region and at of the moderatingreflector an average temperature of of the moderatingreflector an average temperature of regions 620K. region 490K. Table 33. Typical Equilibrium Cycle Operating Conditions. PistonDriven System Compression Ratio Clearance Volume ( Intake Line Gas Pr Intake Line Gas Te Engine Speed (rpm) Piston Cycle Time Loop circulation T Average Prompt Neu Core Height at TDC Core Radius at TDC Uranium Mass in Co Helium Mass in Cor Uranium Enrichment Maximum Gas Temper Maximum Gas Pressu Maximum Core Therm Flux (n/cm sec) Average Exhaust Ga Average Exhause Ga Cycle Average Gas Cycle Average Gas Cycle Average Core Flux (n/cm sec) Fission Energy Rel Chamber (Mw) Mechanical (Shaft) Per Chamber (Mw) Net Turbine Power Chamber (Mw) Total Power Output Overall Efficiency m3) essure (atm) mperature (' K (sec) ime (sec) tron Lifetime (cm) (cm) re (kg) e (kg) (wt%) ature (K) re (atm) [al Neutron (msec) s Temperature (K) s Pressure (atm) Temperature (K) Pressure (atm) Thermal Neutron ease Power Output Per (%) Rate Per Output Per Chamber (Mw) toi .180 9.5 50 00 .2 3.3 .90 8.1 1.4 .503 .876 3 011 29.9 1.065 15 X 10 X 1014 5.222 1.670 1.446 3.116 59.7 Adiabatic Neutron Kientics Simple Harmonic Motion for D20 ModeratorReflector at Model Employed Connecting Rod 490K and 100 cm thick 1015 114 C 10 1 r*l o a) z " 13 S10 0 0) z SC .l011 C) 4 (U 0.11 810' 0 I3 Ool o] 0.00 0.09 0.18 0.27 0.36 0.45 0.54 0.63 Cycle Time 0.72 (Sec) Figure 313. Average Cycle Core Time Pulsed Gas Thermal for Core Neutron a Typical Nuclear Flux Versus Gas Generator System. x 14 S101 0 1 S013 10 210 l 11 o 0) C < 10 0 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 Cycle Time (seconds) Figure 314. Average Core Cycle Time P sied Gas Thermal for Core Neutron a Typical Nuclear Flux Versus stonDriven System. helium, would require to exceed temperatures where significant degree of UF6 dissociation occurs. Comparison between the PGG and PNP systems from Table and indicates that the PGG concept has the disadvantage of lower overall efficiency relative the PNP concept due to the fact that only turbine power and mechanical or shaft power extracted from the system. Comments The promising results from the gaseous core reactor research efforts undertaken at the University of Florida Dugan and others (28) have encouraged the engineering research community the University Florida to continue extensive research on various advanced gas core reactor concepts and different power conversion systems especially space applications. The gas core reactor neutronics energetic research have indicated that cyclic PGCR' are capable attaining high peak gas temperatures 20002500 and high thermal neutron fluxes neutrons/cm sec) while maintaining relatively low cycle averaged gas temperatures 1000 K to 1200 The pulsed gaseous core systems are, thus, able to utilize very high temperature and pressure working fluid while the structural stress and thermal shock are kept to a minimum. 1015 mode of operation. The detailed description the system and static and dynamic neutronic characteristics are given Chapters 4 and STATIC NEUTRONIC CAVITY CHAPTER STUDIES GASEOUS CORE ON COUPLED REACTOR SY MULTIPLE STEM Introduction This chapter examines the bimodal gaseous core reactor system detail predicts the results from the steady state reactor neutronic system. analysis Steady state the bimodal neutronic multiple analysis cavity the system includes examining the effects variations chamber dimen sions, inner and outer beryllium moderator/reflector thicknesses, the BPGCR gas loading and of equal and unequal gas loading among the PGCR chambers on the neutron system removal neutron multiplication lifetime system factor neutron (keff) system generation time (A), and coreto core neutronic coupling coefficients (ajk) The power requirements space based systems under consideration range from s of KW(e) to about 10's of MW(e) the station keeping or surveillance mode operation and 100' of MW(e) few GW(e) the burst ( ), I low power pulsed gas core reactor system were employ only one or two small chambers, compressor ( would needed to provide the necessary driving pressure not only during the system startup, but also during the steady state operation. With the multiple chamber design poss ible arrange the chambers the system so as to "optimize neutronic coupling and reduce the system critical mass requirement. Therefore, proposed employ a PGCR system high with power multiple chamber; chambers multiple surrounding chamber the large design central the low power PGCR system would eliminate the need compressor( , except during system startup, and would reduce the fluid pulsation effects or flow discontinuiti the external loop compared to single chamber design) With the multiple cavities , the system can be designed assure that any given time, one or more chambers are the exhaust phase provide a relatively continuous source of pressurized gas for power generation. The multiple cavity design should have a favorable impact on system reliability since power can be generated even a few reactor chambers fail. The Bimodal Gas Core Reactor System Description annular ring power pulsed gaseous core reactor (PGCR) chambers (Figures and 12) . The central cylindrical chamber and the annular ring of PGCR s are separated inner beryllium moderating/reflecting material. The ring PGCR surrounded an outer beryllium moderator/ reflector region. The identical upper and lower halves the BPGCR are separated a common moderator slab the midplane and the MHD disk generator regions the top bottom central, high power chambers. The bimodal reactor system also provided with top and bottom moderator/ reflectors of appropriate thickness. The PGCR chambers are intended to provide low power station keeping/surveillance purposes. The primary working fluid a mixture of highly enriched uranium hexafluoride (UF6) added which the to enhance reactor the fuel and thermodynamic, helium gas transport, which and heat transfer characteristics the primary working fluid. The fissile gaseous fuel (UF6 He) mixture cyclically injected into PGCR chambers, which are individually pulsed nearcritical, critical, or supercritical state and the heated gas then discharged energy conversion; each chamber capable of providing a few MW(e) power. Neutronically the PGCR cores from farsubcritical state core will have achieved an adequate neutron flux level sufficient duration to generate a hot gas the energy conversion system. The fissionable gaseous fuel whose nuclei undergo fission primarily thermal neutrons neither moderates nor absorbs fast epithermal neutrons any significant degree. Fast chamber, slow moderator, and neutrons down released to thermal diffuse fission energies as thermal neutrons leave the the beryllium thereafter until they leak or are absorbed either the chambers or in the reflector. With multiple PGCR chambers and proper timing operation the individual chambers among their intake, power, and exhaust (discharge) phases the system expected to provide a relatively continuous source of pressurized gas any suitable power conversion system. During power PGCR operation voided, have system, a constant the nonfuel central gas cavity flow, will or have either a low density fuel gas mixture flow. The transition from station keeping burst power operation can be achieved circulating a uranium bearing fuel gas through the central BPGCR cavity. The neutronic coupling effects of the PGCR chambers on the gas fuel bearing central cavity provide the necessary initial power provided the central BPGCR chamber; the surrounding PGCR's continue to provide low or station keeping power. The energy conversion the BPGCR chamber occurs the disk MHD generator. The fuel gas then passes through a radiator heat exchanger before being circulated back into the central chamber a compressor. In addition to providing station keeping power , the PGCR's support burst power operation the central chamber the following reasons: The neutronically coupled PGCR chambers surrounding central chamber are nearcritical, critical, supercritical during a significant portion their power for cycle. the They central relax BPGCR. the This reactivity provides requirements greater flexibility selecting the BPGCR/MHD system working fluid compo sition. The PGCR system s provide and maintain a base the level system neutron "hot" flux so that the a more rapid transition to burst power mode possible than the PGCR s were absent. The pulsed nature of the system greatly minimizes the thermal stresses as compared to those typically experienced comparable performance steady state systems. Also the Steady State Neutronic Analysis Steady state neutronic analysis the multiple cavity bimodal gaseous core reactor system has been performed primarily using the Monte Carlo Neutron Photon (MCNP) transport code and discrete ordinates (SN) transport code, XSDRNPM The complex geometry of the bimodal system and the tenuous nature of the fuelgas mixture core require multidimensional or Monte Carlo transport type calculations the static neutronic analysis. One dimensional static neutronic analysis of the system performed using an "equivalent" sphere which the volume of the core and thickness the moderator are preserved. the later results show, this can lead considerable errors in the values parameters such as the system neutron multiplication factor, neutron removal lifetime, etc. Even though a twodimensional transport calculation can be expected to yield good results system neutron multiplication factor and neutron removal lifetime, cannot be expected to provide an accurate estimate themselves the and coupling between coefficients central among chamber the and PGCR the cores PGCR cores. The complex geometry the bimodal system requires  51   lur a a a a 1 at'I 4. LA a a 4 . Ir 1 1 C1 probabilities interaction can be simulated the MCNP code provides the capability estimate the integral kinetics parameters, especially the coupling coefficients, with a high degree of confidence. MCNP a general purpose, continuous energy, generalized threedimensional geometry, timedependent, coupled neutronphoton Monte Carlo transport code which has the capability calculating the keff eigenvalues fissile systems. has an elaborate tally structure and user interface that allow a user to calculate almost anything conceivable and has elaborate and complete cross section data. has a rich collection variance reduction techniques. The tally structure MCNP capable of producing cellaveraged particle flux, surface neutron current and flux, and neutron, photon and fission heating a cell. The cell flagging and surface flagging capabilities of MCNP allow one to determine the tally contribution of particle tracks from flagged cell surface system. any This other specified capability cell of MCNP or surface used the to determine the neutronic coupling coefficients which are a measure the degree influence the kth core neutron flux on the jth core neutron flux. Both free gas thermal and S(a,$) (ajk) implicitly included the MCNP treatment considering total the average number neutrons generated per fission. more detailed description the MCNP code, associated nuclear cross section data library, capabilities and features, and sample inputs a single chamber PGCR and multiple chamber bimodal system are given Appendix Neutronic studies the bimodal reactor system have utilized the generalized three dimensional geometry capability the MCNP code. However, because the gaseous nature the cores, the convergence the source from initial guess to a distribution fluctuating around the eigenmode solution has been found to be too slow Therefore, order to get a reasonably converged value the eigenvalue (keff) and the neutron removal lifetime has been necessary run MCNP many more cycles generations with a larger source distribution per cycle than would have been required with a denser, solid fueled core. XSDRNPM a onedimensional, time independent, screte ordinates transport code that capable calculating neutron multiplication factor and angular and spatial neutron flux (4). XS DRNPM used the "equivalent" one dimen sional spherical mockups the three four region (Figure 41) spherical configuration with equivalent The volume spherical core mockup and has equivalent a central thickness spherical moderator. chamber (BPGCR) surrounded an annular ring beryllium, followed an annular finally ring an outer gaseous moderator fuel ring. (PGCR The and 's) , keff then values from XSDRNPM guesses have for mainly the been respect iv used e MCN as the P runs required order initial to have keff the MCNP eigenvalue and neutron flux converge faster. XSDRNPM keff values have also been used to explain trends the neutronic behavior different configurations the BGCR system. Steady State Calculations Procedure A seri preliminary calculations were made with MCNP code various configurations the bimodal gas core reactor system (Figure 12) . These initial steady state calculations were done system configurations with inner beryllium thicknesses (TIBE) of 10 and cm during which the outer beryllium thickness (TOBE) was maintained at 50 cm. For each value the inner beryllium thickness the PGCR gas fuel (UF6 He) mixture pressure was varied from 20 atmospheres to 60 atmospheres. PGCR Cylindrical Geometry MCNP Calculation PGCR I mixture and the central BPGCR cavity was maintained pressure of 20 atmospheres helium gas. For each fuel gas loading and system geometry configuration the MCNP code provides the the system system neutron neutron removal multiplication lifetime factor system (keff), neutron generation time (A), and the track length averaged neutron flux each the PGCR chambers and cells the system. The cell flagging capability the MCNP code which the particle c individual to compute contribution cell the from or a group neutronic c a flagged of cells oupling cell can any tallied coefficients other used the among PGCR cells. To study the variation the system neutron multiplication factor with the outer beryllium moderator thickness, calculations were done with the MCNP code which the outer beryllium thickness was changed from cm for two cases inner beryllium thicknesses of 20 cm and cm. The gaseous fuel mixture was maintained constant pressure 50 atmospheres each the PGCR chambers. MCNP calculations were then repeated for two configurations the system with inner beryllium thicknesses cm and 30 Th a cm but with unequal fueleras ( ), situation which a fraction the total number PGCR chambers will always intake, power, or exhaust (discharge) phases operation so that the reactor will capable delivering a continuous supply pressurized to the PGCR power conversion system. Steady state calculations were also done with central central burst chamber power chamber fuelgas fueled. mixture this pressure was case the varied from atmospheres atmospheres. This series of steady state calculations examined system configurations with inner beryllium thicknesses of 10 and cm. The outer beryllium thickness was maintained at 50 cm. The fuelgas of the pressure individual was PGCR maintained chambers, 10 atmospheres this each series calculations. For these calculations, addition the system neutron multiplication factors, neutron removal lifetimes, and the fluxes various BPGCR fuelgas mixture pressures, the neutronic coupling effect the central chamber on the PGCR chambers, the neutronic coupling effects the individual PGCR chambers on the central BPGCR chamber, and the coupling effects among the PGCR chambers themselves were calculated separately. Static neutronic calculations were also performed 