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THE NUCLEAR PISTON ENGINE AND PULSED GASEOUS CORE REACTOR POWER SYSTEMS












By

EDWARD T. DUGAN


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





UNIVERSITY OF FLORIDA


1976














ACKNOWLEDGMENTS


The author would like to express his appreciation to

his graduate committee for their assistance during the

course of this research. Special thanks are due to Dr. N. J.

Diaz, chairman of the author's supervisory committee for providing practical and theoretical guidance and patient encouragement throughout the course of this work.

Thanks are also due to Dr. M. J. Ohanian whose endeavors, along with those of Dr. Diaz, were responsible for securing most of the funds for the computer analysis phase of this research. The dedication, knowledge, and sources of information which were provided by both of these individuals helped

make this work possible.

The author feels fortunate to have studied and worked

with Dr. A. J. Mockel, now at Combustion Engineering. His excellent scientific knowledge of computer analysis and nuclear reactor physics was of great assistance during the initial phases of this work. The author recognizes that

much of his own knowledge in these fields was assimulated

during his years of association with Dr. Mockel. Also to be

recognized is Dr. H. D. Campbell whose criticisms and suggestions were a stimulus for some of the calculations which

appear in this work.
I-ii










The author's studies at the University of Florida have been supported, in part, by a United States Atomic Energy Commission Special Fellowship and also by a one-year Fellowship from the University of Florida and this support is gratefully acknowledged.

A large portion of the funds for the computer analysis were furnished by the University of Florida Computing Center through the College of Engineerinq. This help, though at times meager and difficult to obtain, is also acknowledged.

Thanks are also due to.those fellow students whose

comments, criticisms, and suggestions have also been a source of inspiration,.

Finally, thanks are due to the author's parents for their patient understanding and support which has been a constant source of encouragement.














PREFACE


The fundamental objective of this work has been to gain an insight into the basic power producing and operational characteristics of the nuclear piston engine, a concept which involves a type of pulsed, quasi-steady-state gaseous core nuclear reactor. The studies have consisted primarily of neutronic and energetic analyses supplemented by some reasonably detailed thermodynamic studies and also by some heat transfer and fluid mechanics calculations.

This work is not to be construed as beinq a complete expos of the nuclear piston engine's complex neutronic and energetic behavior. Nor are the proposed power producing systems to be interpreted as being the ultimate or optimum conditions or configurations. This dissertation is rather a beginning or a foundation for future pulsed, gaseous core reactor studies.

Despite being hampered by a rather limited availability of computer funds, it is believed that the models and results presented in this work are indeed indicative of the type of performance which can be anticipated from nuclear piston engine power generating systems and that










they form valuable tools and guidelines for future research on pulsed, gaseous core reactor systems. Indeed, part of this work has been the basis -for major research proposals which have been submitted by the University of Florida's Department of Nuclear Engineering for the purpose of carrying on more extensive investigations of the nuclear piston engine concept. It is recbgniz6d that a complete system analysis and optimization will not only be difficult but also expensive. A demonstratioA of technical feasibility will require the cooperation of not only other departments from within the university but also contributions from other institutions and agencies.

A few remarks should be made concerning the organization of this dissertation. First, most of the equations and derivations used in the neutronics and energetics analysis of the hcldar piston dhgine have been ordered or grouped into appendices. Very few equations appear in the text of the dissertation itself. References are made from the text to theappropriate equation(s) and corresponding appendix where necessary. It is felt that this approach renders a more convenient and ordered presentation and facilitates reading of the text.

The research conducted on the nuclear piston engine has consisted of two reasonably distinct segments. The first phase focused on simple two-stroke (compression and










power stroke) engines. Results from these studies are presented in Chapter IV. The line of reasoning was to examine these simpler engines first, in some detail, before proceeding to the more complex four-stroke systems. Later, after it became apparent from the two-stroke engine studies that the nuclear piston engine concept was indeed a promising venture, work was begun on the more intricate fourstroke configurations. Results from this phase of the research are presented in Chapter V.















TABLE OF CONTENTS


Paae

ACKNOWLEDGMENTS . . .ii

PREFACE . . . iv

LIST OF TABLES . . . . .xii LIST OF FIGURES . . .xviii

LIST OF SYMBOLS AND ABBREVIATIONS . . xxvii

ABSTRACT. . . .xxxvi

CHAPTER

I INTRODUCTION I. .1

Description of Engine Operation . . 1
Applications and Highlights of the Nuclear
Piston Engine Concept . . 3
Dissertation Organization . . 6

II PREVIOUS STUDIES ON GASEOUS CORE CONCEPTS. 8

Gaseous Cores Analytical Studies . . 8
Neutronic Calculations for Gaseous Core
Nuclear Rockets . . 12
Comparison of Theoretical Predictions with
Experimental Results . . 14
Comments . . . 17

III PREVIOUS GASEOUS CORE NUCLEAR PISTON ENGINE
STUDIES AT THE UNIVERSITY OF FLORIDA 19

Introduction . . . 19
Neutronic Model . 23
Energy Model . . . 25
Analytical Results . . 28


vii










TABLE OF CONTENTS (continued)


CHAPTER Page
IV RESULTS FROM TWO-STROKE ENGINE STUDIES . . . . 36

Introduction . . .36
Comparison of Initial Results with Previous Nuclear Piston Engine Analyses. . . 40 Graphite-Reflected Systems . .42
Graphite-Reflected Systems Compared with Systems Using Other Moderating-Reflector
Materials . 47 Moderating-Reflector Power . . . . 51
Some Composite Moderatinq-Reflector
Studies . . . . . . . . 54
Remarks Concerning the Algorithms Used in
the NUCPISTN Code . 57
Parametric Studies with NUCPISTN for D20Reflected Systems . . . 62
Effects of Compression Ratio and
Clearance Volume Variations . .68
Effects of Initial Pressure and Initial
Temperature Variations . 73
Effects of Engine Speed and Neutron
Lifetime Variations . . 78
Effects of Variations in the Initial
and Step-Reflector Thicknesses. .83
Effects of Variations in the Cycle
Fraction Position for Step-Reflector
Addition and Step-Reflector Removal. 86
Effects of Variations in the Heliumto-U235 Mass Ratio . . .93
Effects of Variations in the Neutron
Source Strength .93
Performance Analysis of Two D 20-Reflected
Piston Engines . . . . . . . . . . . . . . . 98
NUCPISTN Results-Compared with Higher Order
Steady-State Neutronic Calculations . . . 107
Some, Remarks Regarding the D20 Temperatures . . . . . 116
Exhaust Gas Temperature Calculations for
the Two-Stroke Engines . . 117 Mass Flow Rates for the Two-Stroke Engines, 120
Thermodynamic Studies f.or Three Nuclear
Piston Engine Power Generating Systems. 121
Nuclear Piston-Gas Turbine-Steam
Turbine System . . .122 Piston-Steam Turbine System . 123 Piston-Cascaded Gas Turbine System. .123


vii i









TABLE OF CONTENTS (continued)


CHAPTER Page

IV Preliminary Heat Exchanger Analysis . . . 132
(cont.) Comparison of the Nuclear Piston Engine
Power Generating Systems . .134
Timestep Size Selection for the Neutron
Kinetics Equations . . .142 Delayed Neutron Effects . . 147
Engine Startup: Approach to Equilibrium in
the Presence of Delayed Neutrons . .148 Nuclear Piston Engine Blanket Studies . . 164 Neutron Lifetime Results . . .177 Summary . . .184

V RESULTS FROM FOUR-STROKE ENGINE STUDIES. . . . 192

Introduction . 192
Engine Startup: Approach to Equilibrium in
the Presence of Delayed and Photoneutrons 197
An Examination of Reactor Physics Parameters
as They Vary During the Piston Cycle . . 214 Flux Shape Changes During the Piston Cycle. 223
Fuel and D20 Moderating-Reflector Temperature Coefficients of Reactivity . . 254
D20 Moderating-Reflector Density or Void
Coefficient of Reactivity . . .262
"Cycle Fractions" for the Engines of This
Chapter . . .264
Effect of Uranium Enrichment on Engine
Performance . 265
Effects of Delayed and Photoneutrons on
Engine Performance . 266
Timestep Size Selection for the Energetics
Equations . . . . 273
Effects of C Formula Selection and of
Neutron Kinetics Equations Numerical Techniques on Engine Behavior . . .276 Blanket Studies for Four-Stroke Engines . . 278
Material Densities and Group Constants for
Nuclear Piston Engine #10 . 281
Neutron Lifetimes, Generation Times, Effective f3s, keffectives, and Inhomogeneous
Source Weighting Functions for Engine #10
from Different Computational Schemes. . . . 286
A Further Comparison of keffs for Engine
#10 from Various Computational Schemes. . . 295 NUCPISTN Cycle Results for Engine #10 . . . 299
Power Transients for Engine #10 Induced by
Loop Circulation Time Variations . .306









TABLE OF CONTENTS (continued)


CHAPTER Page

V Thermodynamic Studies for Nuclear Piston
(cont.) Engine Power Generating Systems Utilizing
the Engine #10 Configuration . . 311
"Gas Generator" Nuclear Piston Engines. . 319
NUCPISTN Cycle Results for a "Gas Generator"
Engine . . 325
Thermodynamic Studies for Nuclear Piston
Engine Power Generating Systems Utilizing
"Gas Generator" Engines . 332
Summary . . .343

VI RELATED RESEARCH AND DEVELOPMENTS . .355

Introduction . . . . . 355
Related Research and Developments at the
University of Florida . . . 356
Other Related Research and Developments in
Progress . . . 356

VII CONCLUSIONS; REFINEMENTS AND AREAS FOR FURTHER
RESEARCH . . . 364

Introduction . . .364 Applications . . .368
Analytical Model for Piston Neutronics and
Energetics . . .373
Steady-State Neutronic Analysis . . . 373
Moderating-Reflector Studies . .375 Fuel Studies . . .376 Neutron Cross Section Libraries . . . . 377
Moderating-Reflector and Fuel Temperature Coefficients of Reactivity . . . 378
Neutron Kinetics Calculations . .379
Neutronic Coupling Between Piston Engine
Cores in an Engine Block . .383 Equation of State for the HeUF6 Gas . 385
Fluid Flow in the Piston Engine . . . . 385
Temperature Distribution and Piston
Engine Heat Transfer Studies . 386 He-to-U Mass Ratio Studies . . . . 387 Step-Reflector Addition and Removal 388
Parametric Studies . . .389
Moderating-Reflector Density or Void
Coefficients of Reactivity . .390 Blanket Studies and Breeding Prospects. 391
Comments . . . .391










TABLE OF CONTENTS (continued)


CHAPTER Page

VII Analytical Models for Systems External to
(cont.) the Piston'Engine . .394
Thermodynamic Cycles for the Turbines 394
Turbine Loop Energetics, Heat Transfer,
and Fluid Mechanics Studies . .395 HeUF6-to-He Exchanger Studies . .396 Comments . . .397
Economic Model for the Nuclear Piston Enqine
Power Generating System . . .397
Fixed Charges (Capital and Cost Related
Charges) . . .397 Fuel Cycle Costs . 398
Power Production Costs . .399 Comments . 400
Safety Analysis and Methods of Control. . 401

APPENDICES

A TWO-GROUP, TWO-REGION, ONE-DIMENSIONAL
DIFFUSION THEORY EQUATIONS USED IN THE
NUCPISTN CODE WHEN PHOTONEUTRONS ARE IGNORED 405

B TWO-GROUP, TWO-REGION, ONE-DIMENSIONAL
DIFFUSION THEORY EQUATIONS USED IN THE
NUCPISTN CODE WHEN PHOTONEUTRONS ARE INCLUDED. 422

C GENERAL POINT REACTOR KINETICS EQUATIONS . . . 434

D THE POINT REACTOR KINETICS EQUATIONS USED IN
THE NUCPISTN CODE . . .456

E THE ENERGETICS EQUATIONS USED IN THE NUCPISTN
CODE . . . 474

F GROUP STRUCTURES AND VARIOUS REACTOR PHYSICS
CONSTANTS USED IN THE NUCLEAR PISTON ENGINE
COMPUTATIONS . . .487

G LISTING OF THE TASKS PERFORMED BY THE NUCPISTN
SUBROUTINES AND A FLOW DIAGRAM FOR THE
NUCPISTN CODE . . .492

Listing of Tasks Performed by the NUCPISTN
Subroutines . . 492
NUCPISTN Code Flow Diagram . .495

LIST OF REFERENCES . . . 501

BIOGRAPHICAL SKETCH . . . 508














LIST OF TABLES


TABLE Page

1 Values of Primary Independent Parameters for
Graphite-Reflected Piston Engines Analyzed by
Kylstra et al . . . 27

2 Comparison of a Typical Graphite-Reflected, UF6
Piston Engine with the Nordberg Diesel . . . 34

3 Atom Densities and Temperatures for Large
Graphite-Reflected Engines at the TDC
Position . . 43

4 Neutron Multiplication Factors for the Large
Graphite-Reflected Engines at the TDC
Position . . . 44

5 Neutron Multiplication Factors for Small
Engines with Various Moderating-Reflector
Materials at the TDC Position . . 48

6 Moderator Neutron Temperatures and Neutron
Lifetimes for Small Engines with Various
Moderating-Reflector Materials at the TDC
Position . . . 52

7 Moderating-Reflector Power for Some ModeratingReflector Materials at 290'K . 55

8 Beryllium-D20 Composite Reflector Study at
290�0.K . . .56

9 Operating Characteristics for Engines #1 and
#2 . . . 63

10 Cycle Results from NUCPISTN for Engines #1 and
#2 . . . 99

11 Summary of Thermodynamic Results for the PistonGas Turbine-Steam Turbine System Which Uses
Piston Engine #2 . . . 126


xii









LIST OF TABLES (continued)


TABLE Page

12 Summary of Thermodynamic Results for the PistonSteam Turbine System Which Uses Piston Engine
#I . . . . 129

13 Summary of Thermodynamic Results for the PistonCascaded Gas Turbine System Which Uses Piston
Engine #2 . . 133

14 A Comparison of Thermodynamic Results for the
Three Nuclear Piston Engine Power Generating
Systems Which Use Piston Engines #1 and #2 . , , 135

15 Reactor Volume per Unit Power for Three Operational Nuclear Reactor Power Systems and for
the Three Nuclear Piston Engine Power Generating
Systems Which Use Piston Engines #1 and #2 138

16 Heat Rate and Fuel Cost Estimates for the Three
Nuclear Piston Engine Power Generating Systems
Which Use Piston Engines#1 and #2 . .141

17 Operating Characteristics for Engine #3 . .143

18 Cycle Results from NUCPISTN for Engine #3 . 144

19 Effects of Neutron Kinetics Equations Timestep
Size Variation on Engine #3 Performance . 146

20 Operating Characteristics for Engines #4 and #5. 150

21 Startup Procedure for Engine #4 in the Presence
of Delayed Neutrons . . .151

22 Startup Procedure for Engine #5 in the Presence
of Delayed Neutrons . . .152

23 Equilibrium Cycle Results from NUCPISTN for
Engines #4 and #5 . . .162

24 Operating Characteristics for Engine #6 . 165

25 Equilibrium Cycle Results from NUCPISTN for
Engine #6 . . .166


xiii









LIST OF TABLES (continued)

TABLE Page

26 Equivalent Cylindrical Cell Data and Pure
Blanket Material Densities . . . 168

27 Homogenized Densities for a Blanket Using a
1.5M/W Lattice . . . 171

28 Burnup Calculations for a System Using Engine
#6, an 80cm D20 Reflector Region, and a
Blanket Region with a 1.5M/W Lattice .172

29 Burnup Calculations for a System Using Engine
#6, a 70cm D20 Reflector Region, and a
Blanket Region with a 1.5M/W Lattice . . . . . 173

30 Burnup Calculations for a System Using Engine
#6, an 80cm D20 Reflector Region, and a
Blanket Region with a 3.OM/W Lattice . .175

31 Burnup Calculations for a System Usinq Engine
#6, a 70cm D20 Reflector Region, and a Blanket
Region with a 3.OM/W Lattice . . . 176

32 Operating Characteristics for Engine #7. . . . 178

33 Equilibrium Cycle Results from NUCPISTN for
Engine #7 . . . 179

34 Neutron Multiplication Factors, Neutron Lifetimes and Neutron Generation Times at Various
Cycle Positions for Engine #7 as Obtained from
CORA 6nd NUCPISTN . . . 181

35 Operating Conditions for Engine #8 . . 198

36 Startup Procedure for Engine #8 in the Presence
of Delayed and Photoneutrons . . . . . 201

37 Equilibrium Cycle Results from NUCPISTN for
Engine #8 . . . 213

38 Core Radii and Neutron Multiplication Factors
as Obtained from CORA and NUCPISTN for Engine
#8 at Various Cycle Positions . . 215

39. Flux Ratios as Obtained from CORA and NUCPISTN
for Engine #8 at Various Cycle Positions . . . 217


xiv










LIST OF TABLES (continued)


TABLE Page40 Six Factor Formula Parameters as Obtained from
CORA for Engine #8 at Various Cycle Positions. 219

41 Neutron Lifetimes and Generation Times as Obtained from CORA for Engine #8 at Selected Cycle
Positions . . .221

42 U235 Enrichment Effects on the Neutron Multiplication Factor for Engine #8 at the TDC Position. 221

43 D20 Moderating-Reflector Temperature Coefficients
of Reactivity Using the Engine #8 Configuration
at the TDC Position . . .256

44 Fuel Temperature Coefficients of Reactivity for
100% Enriched UF6 Using the Engine #8 Configuration at the TDC Position . . .257

45 Fuel Temperature Coefficient of Reactivity for
93% Enriched UF6 Using the Engine #8 Configuration at the TDC Position . . .258

46 Fuel Temperature Coefficient of Reactivity for
80% Enriched UF6 Using the Engine #8 Configuration at the TDC Position . . .259
47 D20 Moderating-Reflector Density or Void Coefficient of Reactivity Using the Engine #8 263
Configuration at the TDC Position .

48 Operating Characteristics for Engine #9 . .266

49 Equilibrium Cycle Results from NUCPISTN for
Engine #9 . . .267

50 Effect of Uranium Enrichment on Required Fuel
Loading for Engine #9 . . . .269

51 Compensating for the Absence of Delayed and/or
Photoneutrons by Increased Fuel Loading for
Engine #9 . . . . . . . . . 271

52 Engine #9 Behavior in the Absence of Delayed
and/or Photoneutrons When There Is No. Compensation by Increased Fuel Loading . . .272









LIST OF TABLES (continued)


TABLE Page

53 Effects of Energetics Equations Timestep Size
Variation on Engine #9 Performance . . .274

54 Effects of Specific Heat Formula and of Neutron
Kinetics Equations Numerical Techniques on
Engine #9 Performance . . .277

55 Burnup Calculations for a System Using an
Engine #9-Like Configuration, a 70cm D20
Reflector Region, and a Blanket Region with a
3.0M/W Lattice . . 280

56 Operating Characteristics for Engine #10 . . . 282

57 Equilibrium Cycle Results from NUCPISTN for
Engine #10 . . .283

58 Material Densities and Core Thermal Group Constants from NUCPISTN for Engine #IO at the TDC
Position . . .285

59 Fast and Thermal Collapsed Group Constants from
PHROG and BRT-l for Engine #10 at the TDC
Position . . .287

60 Reactor Physics Parameters for Engine #10 at the
TDC Position from Various Computational Schemes. 288

61 Reactor Physics Parameters for Engine #10 at the
0.056 Cycle Fraction from Various Computational
Schemes . . . .290

62 Neutron Multiplication Factors for Engine #10
from Various Computational Schemes for the TDC
and 0.056 Cycle Fraction Positions . . .296

63 Summary of Thermodynamic Results for the PistonGas Turbine-Steam Turbine System Which Uses the
Modified #10 Piston Engine . . . . . . . 313

64 Summary of Thermodynamic Results for the PistonSteam Turbine System Which Uses Piston Engine
#10 . . .314


xvi









LIST OF TABLES (continued)

TABLE Page

65 Summary of Thermodynamic Results for the PistonCascaded Gas Turbine System Which Uses the
Modified #10 Piston Engine . . . 316
66 A Comparison of Thermodynamic Results for the
Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engine #10 and the
Modified #10 Piston Engine . . .318

67 Operating Characteristics for Engine #11 . . . . 321

68 Equilibrium Cycle Results from NUCPISTN for
Engine #11. . . . .323

69 Some Operating Characteristics and NUCPISTN
Equilibrium Cycle Results for "Gas Generator"
Piston Engines . . .324

70 Summary of Thermodynamic Results for the PistonGas Turbine-Steam Turbine System Which Uses the
Modified #11 Piston Engine . . .333

71 Summary of Thermodynamic Results for the PistonSteam Turbine System Which Uses Piston Engine
#II . . .335

72 Summary of Thermodynamic Results for the PistonCascaded Gas Turbine System Which Uses the
Modified #11 Piston Engine . . .336


xvii















LIST OF FIGURES

FIGURE Page

I Simple Schematic of a UF6 Nuclear Piston
Engine . . . . 20

2 Illustration of Step-Reflector Addition and
Removal . . . 22

3 UF6 Phase Diaqram . . .24

4 Neutron Multiplication Factor Versus UF6 Partial
Pressure for an Infinite Graphite Reflector . . 29

5 Gas Pressure and Temnerature as a Function of
Percent Travel Through the Piston Cycle for a
Graphite-Reflected Engine . . 30

6 Average Core Thermal Neutron Flux and Neutron
Multiplication Factor as a Function of Percent Travel Through the Piston Cycle for a GraphiteReflected Engine . . . 31

7 UF6 Nuclear Piston Engine Performance for the
Case of Graphite Step-Reflector Addition at the
0.1 Cycle Fraction . . -. 33

8 Neutron Multiplication Factor Versus U235 Atom
Density for Systems Which Have No Helium Gas
Present in the Core and a 100cm Thick D20
Reflector at 2906K .64

9 Neutron Multiplication Factor Versus U235 Atom
Density for Systems Which Have Helium Gas Present in the Core and a 100cm Thick D20
Reflector at 2900K. .65

10 Neutron Multiplication Factor Versus U235 Atom
Density for Systems Which Have Helium Gas Present in the Core and a 100cm Thick D20
Reflector at 5700K . . . 66

11 Mechanical Power and Peak Gas Pressure Versus
Compression Ratio for a Two-Stroke Engine . . . 69


xviii










LIST OF FIGURES (continued)

FIGURE Page

12 Mechanical Efficiency and Peak Gas Temperature
Versus Compression Ratio for a Two-Stroke
Engine . . . . . . 70

13 Mechanical Power and Peak Gas Pressure Versus
Clearance Volume for a Two-Stroke Engine . 71

14 Mechanical Efficiency and Peak Gas Temperature
Versus Clearance Volume for a Two-Stroke Engine 72

15 Mechanical Power and Peak Gas Temperature Versus
Initial Gas Pressure for a Two-Stroke Engine. 74

16 Mechani-cal Efficiency and Peak Gas Temperature
Versus Initial Gas Pressure for a Two-Stroke
Engine . . . . .75

17 Mechanical Power and Peak Gas Pressure Versus
Initial Gas Temperature for a Two-Stroke Engine . 76

18 Mechanical Efficiency and Peak Gas Temperature
Versus Initial Gas Temperature for a Two-Stroke
Engine. . . 77

19 Mechanical Power and Peak Gas Pressure Versus
Engine Speed for a Two-Stroke Engine . .79

20 Mechanical Efficiency and Peak Gas Temperature
Versus Engine Speed for a Two-Stroke Engine 80

21 Mechanical Power and Peak Gas Pressure Versus
Neutron Lifetime for a Two-Stroke Engine . 81

22 Mechanical Efficiency and Peak Gas Temperature
Versus Neutron Lifetime for a Two-Stroke Engine . 82

23 Mechanical Power and Peak Gas Pressure Versus
Initial D20 Reflector Thickness for a Two-Stroke
Engine. . . 84

24 Mechanical Efficiency and Peak Gas Temperature
Versus Initial D20 Reflector Thickness for a
Two-Stroke Engine . . .85


xix










LIST OF FIGURES (continued)


FIGURE Page

25 Mechanical Power and Peak Gas Pressure Versus
D20 Step-Reflector Thickness for a Two-Stroke
Engine . . . . 87

26 Mechanical Efficiency and Peak Gas Temperature
Versus D20 Step-Reflector Thickness for a TwoStroke Engine . . . 88

27 Mechanical Power and Peak Gas Pressure Versus
the Cycle Fraction for Step-Reflector Addition
for a Two-Stroke Engine . . . . 89

28 Mechanical Efficiency and Peak Gas Temperature
Versus the Cycle Fraction for Step-Reflector
Addition for a Two-Stroke Engine . . 90

29 Mechanical Power and Peak Gas Pressure Versus
the Cycle Fraction for Step-Reflector Removal
for a Two-Stroke Engine . . 91

30 Mechanical Efficiency and Peak Gas Temperature
Versus the Cycle Fraction for Step-Reflector
Removal for a Two-Stroke Engine . . 92

31 Mechaniff Power and Peak Gas Pressure Versus
He-to-U Mass Ratio for a Two-Stroke Engine 94

32 Mechanical Effj cency and Peak Gas Temperature
Versus He-to-ULi Mass Ratio for a Two-Stroke
Engine . . 95

33 Mechanical P.ower and Peak Gas Pressure Versus
Neutron Source Strength for a Two-Stroke
Engine . . . 96

34 Mechanical Efficiency and Peak Gas Temperature
Versus Neutron Source Strength for a Two-Stroke
Engine . . . 97

35 D20 Reflector Thickness Versus Cycle Fraction
for Engine 1I . . 100

36 Gas Temperature Versus Cycle Fraction for
Engine #1 . . 101

37' Gas Pressure Versus Cycle Fraction for
Engine #I . . 102










LIST OF FIGURES (continued)


FIGURE Paqe

38 Average Core Thermal Neutron Flux Versus
Cycle Fraction for Engine #1. . . I.103

39 Neutron Multiplication Factor Versus Cycle
Fraction for Engine WI1 . . . 104
40 Neutron Multiplication Factor Versus D20
Reflector Thickness as Obtained from Two-Group
NUCPISTN Calculations . . . 112

41 Neutron Multiplication Factor Versus D20
Reflector Thickness as Obtained from FourGroup CORA Calculations . . . 113

42 Fast and Thermal Neutron Flux Versus Radius
for a D20-Reflected Core as Obtained from
Two-Group CORA Calculations . . . 114

43 Fast and Thermal Adjoint Neutron Flux Versus
Radius for a D20-Reflected Core as Obtained
from Two-Group CORA Calculations . . 115

44 Piston-Gas Turbine-Steam Turbine Schematic
for the Power System Which Uses Piston
Engine #2 . . . 124

45 Steam and Gas Turbine Temperature-Entropy
Diagrams for the Piston-Gas Turbine-Steam
Turbine System Which Uses Piston Engine #2. 125

46 Piston-Steam Turbine Schematic for the System
Which Uses Piston Engine #1. . .127

47 Steam Turbine Temperature-Entropy Diagram for
the Piston-Steam Turbine System Which Uses
Piston Engine #1. . . . . . . . . 128

48 Piston-Cascaded Gas Turbine Schematic for the
System Which Uses Piston Engine #2 . . 130

49 Gas Turbine Temperature-Entropy Diaqram for
the Piston-Cascaded Gas Turbine System Which
Uses Piston Engine #2 . . . 131

50 Diagram of a D20-Reflected, 3-to-l Compression
Ratio Nuclear Piston Engine at the TDC
Position . . . . . 139


xxi









LIST OF FIGURES (continued)


FIGURE

51 Sketch of an 8-Cylinder Nuclear Piston Engine
Block for 40-50 Mw(e) Power Generating
Systems - . . . . . . . . . 140

52 Delayed Neutron Precursor Concentration Buildup During Startup for Engine #4 . . 153

53 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #4 155

54 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #4
(continued) . . . 156

55 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #4
(continued) . . . 157

56 Delayed Neutron Precursor Concentration Buildup During Startup for Engine #5 . . 158

57 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #5 . . 160

58 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #5
(continued) . . . . . . . . . 161

59 Typical Blanket Region Unit Cell Diagram. 167

60 Delayed Neutron and Photoneutron Precursor
Concentration Buildup During Startup for
Engine #8 . . . 204

61 Delayed Neutron and Photoneutron Precursor
Concentration Buildup During Startup for
Engine #8 (continued) . . . 205

62 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #8 207

63 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for EnQine #8
(continued) . . . 208


xxii









LIST OF FIGURES (continued)


FIGURE Paqe

64 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #8
(continued) . . . 209

65 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #8
(continued) . . . 210

66 Fast and Thermal Neutron Flux Versus Radial
Distance for Enqine #8 at Timestep Number
351 . . . . 225

67 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
351 When Inhomogeneous Photoneutron Sources
Are Ignored . . . 226

68 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
501 . . . . 227

69 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
701 . . . 228

.70 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
901 . . . 229

71 Fast'and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
901 When Inhomogeneous Photoneutron Sources
Are Ignored . . . 230

72 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
1551 . . . 231

73 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
1801 . . 232

74 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
2026 . . . 233


xxiii










LIST OF FIGURES (continued)


FIGURE Page

75 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
2326 . . .234

76 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
3226 . . .236

77 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
3226 When Inhomogeneous Photoneutron Sources
Are Ignored . . . .237

78 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
3726. . . . . 238

79 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
3976 . . 239

80 Fast and Thermal Neutron Flux Versus Radial
Distance for Engine #8 at Timestep Number
4426 . . . 240

81 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 351 . . .241

82 .East and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 501 . . . . 242

83 Fa'st and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 701 . . . . .243

84 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 901 . . 244

85 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 1551 . . . 245

86 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 1801 . . .246


xxi v










LIST OF FIGURES (continued)


FIGURE Page

87 Fast and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep
Number 2026 . . 247

88 Fast and Thermal Adjoint'Neutron Flux Versus
Radial .Distance for Engine #8 at Timestep
Number. 2326 . . 248

89 Fast and Thermal Adjoint Neutron Flux Versus
Radial'Distance for Engine #8 at Timestep
Number 3226 . .249

90 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 3726 . . .250

91 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 3976 . . . 251

92 Fast and Thermal Adjoint Neutron Flux Versus
Radial Distance for Engine #8 at Timestep
Number 4426 . .252

93 D20 Reflector Thickness Versus Piston Cycle
Time for Engine #10. . . . 300

94 Gas Temperature Versus Piston Cycle Time for
Engine #10 . . . 301

95 Gas Pressure Versus Piston Cycle Time for
Engine #10 . . 302

96 Average Core Thermal Neutron Flux Versus
Piston Cycle Time for Engine #10 . .303

97 Neutron Multiplication Factor Versus Piston
Cycle Time for Engine #10. . 304

98 Slow Power Transients for Engine #10 Induced
by Changes in th.e Loop Circulation Time . . 307

99 Intermediate Level Power Transients for
Engine #10 Induced by Changes in the Loop
Circulation Time . . .308


xxv










LIST OF FIGURES (continued)

FIGURE Paqe

100 Rapid Power Transients for Engine #10 Induced by Changes in the Loop Circulation Time . . . 309 101 D20 Reflector Thickness Versus Piston Cycle Time for Engine #11 . . . . . . . 326 102 Gas Temperature Versus Piston Cycle Time for Engine #11. . . 327 103 Gas Pressure Versus Piston Cycle Time for Engine #11 . . .328 104 Average Core Thermal Neutron Flux Versus Piston Cycle Time for Engine #11 . .329 105 Neutron Multiplication Factor Versus Piston Cycle Time for Engine #11. . . .330 106 Schematic of NASA's NERNUR--A Large Power Generating System Utilizing a UF 6 Gas Core
Nuclear Reactor . . .362


xxvi















LIST OF SYMBOLS AND ABBREVIATIONS


A constant appearing in the steady-state, two-group, tworegion diffusion theory flux expressions
2
A cross-sectional area of exhaust valve (m2)
e
A. cross-sectional area of intake valve (m )
1

C D fraction of fast neutrons which leave the core and
return as thermal neutrons

' fraction of fast neutrons from th.e fast. inhomogeneous source in the core which leave the core and return as
thermal neutrons.
P PPP
a parameter which is the product of fP times y times 5

B constant appearing in the steady-state, two-group, tworegion diffusion theory flux expressions
D
f3. yield fraction for delayed neutron precursor group j

D "effective" yield fraction for delayed-neutron precursor group j

P
yield fraction for photoneutron precursor group j

BR breeding ratio

BWR boiling water reactor

c subscript indicating the core region

C constant appearing in the steady-state, two-group, tworegion diffusion theory flux expressions; also, an abbreviation which is used to designate the compression
stroke

cD, effective" delayed neutron precursor concentration J. for delayed group j

C. delayed neutron precursor concentration for delayed group j


xxvii









C photoneutron precursor concentration for photoneutron
precursor group j

Ce flow coefficient for the exhaust valve C. flow coefficient for the intake valve gc
c gas specific heat at constant volume
v
Cp gas sPecific heat at constant pressure

X the normalized energ spectruW for fission neutron
emission; X = [X (l- ) + Exi. ]
p

Xj the normalized energy spectrum for delayed fission neutron group j
the normalized energy spectrum for prompt fission neutrons

CR conversion ratio D diffusion coefficient

DB2 average product of the-neutron diffusion coefficient
and the buckling squared

V symbol indicating the gradient operation

6 parameter appearing in the numerical form of the neutron
kinetics equations; for the two-point finite difference relations, 6 = 1.0 while for the three-point integration
formulas, 6 = 3/2
P
6 the ratio of the average thermal neutron density in the
core to the average fast neutron density in the moderatingreflector

e, subscript indicating the exhaust phase of the cycle E symbol for the quantity or variable of energy; also, an
abbreviation which is used to designate the exhaust
stroke

Ef energy released per fission C fast fission factor; also, the overall efficiency

m mechanical efficiency


xxviii










cycle fraction for step-reflector addition �2 cycle fraction for step-reflector removal n neutron production factor, i.e.,'the average number of
neutrons produced in thermal fission over the total
thermal absorption in the fuel

TIT turbine efficiency TIC compressor efficiency f thermal utilization; as a superscript it indicates the
fast group; as a subscript it indicates either fission
or forced flow

production operator or volume integral of the adjoint
weighted fission source
d
f fraction of the removal cross section which is downscattering
fP fraction of the gamma rays emitted by the photoneutron
precursors which penetrate from the core to the
moderating-reflector with energy above the (y, n)
threshold
fv geometry factor equal to the core volume over the
moderating-reflector volume

yg geometry factor given by ( 3 - R3)
P
'Y fraction of those gammas reaching the moderatingreflector region with energy above the (y, n) threshold
which actually induce photoneutrons

h enthalpy (B/lb .); subscript indicating hydraulic
m
AH increase in enthalpy of a system (BTU's) he energy or enthalpy of the mass leaving the system (B/Ib ) hi energy of enthalpy of the mass entering the system (B/lbM) HPT high pressure turbine HTGR high temperature gas-cooled reactor i subscript indicating the intake phase of the cycle


xxix










abbreviation.which is used to designate the intake stroke

J neutron current or vector flux
i D number of delayed neutron precursor groups

JP number of photoneutron precursor groups

K1. K coefficients which are convenient groupings of varius reactor physics parameters used in solving the 2group, 2-region neutron diffusion theory equations

k, keff, k-effective the effective (static) neutron multiplication factor for a reactor or system

kd the (effective) dynamic neutron multiplication factor
for a reactor of system

ko the infinite medium neutron multiplication factor, i.e.,
the neutron multiplication factor in the absence of
leakage

[k]c convenient grouping of reactor physics parameters defined
as [vz fIc/[ E aIc

Kf the inverse square root of the age to thermal of fast m neutrons in the moderating-reflector when f0 is unity

t
Km the inverse thermal diffusion length in the moderatingreflector
Kc the inverse thermal diffusion length in the core when is unity

neutron lifetime

the infinite medium neutron lifetime, i.e,, the neutron
lifetime in the absence of leakage

[. ]c convenient grouping of reactor physics parameters defined as (l/V)c/[cEalc

A neutron generation time usually defined as 1R,/k]
D
A delayed neutron precursor decay constant for delayed
group j
P
X photoneutron precursor decay constant for photoneutron
precursor group j

LPT low pressure turbine


xxx









mass flow rate out of the cylinder M. mass flow rate into the cylinder m subscript indicating the moderating reflector; symbol
denoting mass, usually the HeUF 6 mass mw coolant water flow rate M gas mass which enters or leaves the system during the
m time At

neutron density (neutrons/cm3); symbol indicating neutrons

N neutron population

235
N uranium-235 atom density (atoms/barn-cm)
NU uranium atom density (atoms/barn-cm) ns one of two components of the shape function; the units
are arbitrary, depending upon the normalization applied
to the amplitude function

V average number of neutrons released per fission

V, the average number of neutrons released per fission
required for criticality w engine speed in rpm's

symbol- for the vector variable indicating direction or
angle

p resonance escape probability; pressure

P total power output; also, abbreviation which is used
to designate the power stroke P m mechanical power output P(t) amplitude factor or amplitude function PWR pressurized water reactor
t
PNL thermal non-leakage probability

scalar neutron flux


xxxi










S average thermal neutron flux in the core due only to the inhomogenous fast neutron source in the core
+ scalar adjoint flux for a time independent critical
reference system

shape factor or shape function D angular neutron flux

D + angular adjoint neutron flux for a time independent o critical reference system Q net amount of heat added to a system from the surroundings

Qf fission heat release AqR heat of reaction; the fission heat from a nuclear reacti on

Qf the rate of fission-heat release Q(t) weighted source term~appearing in the point reactor
neutron kinetics equations MAX
QEA ratio of the maximum fission heat released in any
energetics equation timestep to the total fission heat
released during the piston cycle

Q MAX ratio of the maximum fission heat released in any neutron
kinetics equation timestep to the total fission heat
released during the piston cycle

total heat transfer rate

%out rate of heat rejection r symbol for the vector variable indicating position

rpm revolutions per minute R core radius or position at the core-reflector interface;
gas. constant equal to the universal gas constant divided
by the gas molecular weight

extrapolated reactor radius or position at the outer
(extrapolated) edge of the moderating-reflector region r symbol for the variable indicating position


xxxii









p reactivity; density; parameter appearing in the numerical form of the neutron kinetics equations which has a
value of zero for the two-point finite difference relations and a value of one-half for the three-point
integration relations

s entropy (B/IbmOR)

So inhomogeneous fast neutron source strength in the core
(neutrons/sec)
f
5 1 inhomogeneous fast neutron source term for the moderatingreflector region due to photoneutron production 1P
number of thermal neutrons generated per unit time and c per unit volume in the core as a result of photoneutron production in the moderating-reflector
t
c5 average thermal neutron density per unit time in the c. core due solely to the inhomogeneous fast neutron source in the core
f
c fast neutron source strength per unit volume in the core; equal to So divided by the core volume for the
gas cores of concern

S(.,t)neutron source strength distribution per unit volume
(neutrons/cm sec)

S(t) average neutron sou ce strength (in a region) per unit
volume (neutrons/cm sec)
macroscopic neutron absorption cross section
a
macroscopic neutron scattering cross section

Er macroscopic neutron removal cross section Ef macroscopic neutron fission cross section

out microscopic cross section for neutron transfer out of a group by scatter

ft macroscopic cross section for neutron scatter from the fast to the thermal group
isotropic component of the macroscopie neutron elastic
0 transfer cross section

linearly anisotropic, component of the macroscopic neuS tron elastic transfer cross section


xxxiii










Etr macroscopic neutron transport cross section .of microscopic neutron fission cross section T temperature

7 average gas temperature

Te exhaust gas temperature Tf gas temperature at the end of the compression-power cycle Ti initial gas temperature in the inlet line or at the
beginning Of the compression-power cycle Tn neutron temperature t symbol for the variable indicating time; as a superscript, it indicates the thermal group

T Fermi age of fission neutrons to thermal energy

Td delay time between the generation of a fast photoneutron
inthe moderating-reflector region and its appearance as
a thermal neutron in the core

TD delayed neutron precursor mean lifetime for delayed
group j

T photoneutron precursor mean lifetime for photoneutron
precursor groups

TDC top dead center position for the piston TZN alloy of niobium,-zirconium, and titanium AU increase in stored or internal energy of a system V volume

v velocity

W mechanical work output; net amount of work done by a
system on the surroundings

Wc weighting function or weighting term for the core inhomogeneous source(s)

Wm weighting function or weighting term for the moderatingreflector inhomogeneous source(s)


xxxiv









Wf net flow work performed by the system on the gas as it
passes across the system (piston engine) boundaries

average logarithmic energy decrement per collision for
neutrons or the average increase in lethargy per collision

Y net expansion factor for compressible flow through
e the exhaust valve

Y. net expansion factor for compressible flow through the
1 intake valve

core fast absorption factor


xxxv









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

THE NUCLEAR PISTON ENGINE AND PULSED
GASEOUS CORE REACTOR POWER SYSTEMS By

Edward T. Dugan

March, 1976

Chairman: Dr. N. J. Diaz
Maior Department: Nuclear Enqineering Sciences

Nuclear piston engines operating on gaseous fissionable fuel should be capable of providing economically and energetically attractive power generating units.

A fissionable gas-fueled engine has many of the advantages associated with solid-fueled nuclear reactors but fewer safety and economical limitations. The capital cost per unit power installed (dollars/kwe) should not spiral for small gas-fueled plants to the extent that it does for solidfueled plants. The fuel fabrication (fuel and cladding, spacer grids, etc.) is essentially eliminated; the engineering safeguards and emergency core cooling requirements are reduced significantly.

The investigated nuclear piston engines consist of a pulsed; gaseous core reactor enclosed by a moderatingreflecting cylinder and piston assembly and operate on a thermodynamic cycle similar to the internal combustion engine. The primary working fluid is a mixture of uranium


xxxvi










hexafluoride, UF6, and helium, He, gases. Highly enriched UF6 gas is the reactor fuel. The helium is added to enhance the thermodynamic and heat transfer characteristics of the primary working fluid and also to provide a neutron flux flattening effect in the cylindrical core.

Two-and four-stroke engines have been studied in which a neutron source is the counterpart of the sparkplug in the internal combustion engine. The piston motions which have been investigated include oure simple harmonic, simple harmonic with dwell periods, and simple harmonic in combination with non-simple harmonic motion.

Neutronically, the core goes from the subcritical

state, through criticality and to the suDercritical state during the (intake and) compression stroke(s). Supercriticality is reached before the piston reaches top dead center (TDC), so that the neutron flux can build up to an adequate level to release the required energy as the piston passes

TDC.

The energy released by the fissioning gas can be extracted both as mechanical power and as heat from the circulating gas. External equipment is used to remove fission products, cool the gas, and recycle it back to the pistpn engine. Mechanical power can be directly taken by
0
means of a conventional crankshaft operating at low speeds.

For the purpose of evaluating the nuclear piston engine cycle behavior, a computer code was developed which couples the necessary energetics and neutronics equations.


xxxvii









The code, which has been named NUCPISTN, solves for the neutron flux, delayed and photoneutron precursor concentrations, core volume, gas temperature, qas pressure,. fission heat release, and mechanical (pV) work throughout the piston cycle.

As a circulating fuel reactor, the nuclear piston engine's quasi-steady-state power level is capable of being controlled not only by variations in the neutron multiplication factor but also by changes in the loop circulation time. It is shown that such adjustments affect the delayed and photoneutron feedback into the reactor and hence provide an efficient means for controlling the reactor power level

The results of the conducted investigations indicate good performance potential for the nuclear piston engine with overall efficiencies of as high as 50% for nuclear piston engine power generating units of from 10 to 50 Mw(e) capacity. Larger plants can be conceptually designed by increasing the number of pistons, with the mechanical complexity and physical size as the probable limiting factors.

The primary uses for such power systems would be for small mobile and fixed ground-based power generation (especially for peaking units for electrical utilities) and also for nautical propulsion and ship power.


xxxviii
















CHAPTER I

INTRODUCTION


Description of Engine Operation

The investigated nuclear piston engines consist of a pulsed, gaseous core reactor enclosed by a moderatingreflecting cylinder and piston assembly, and operate on a thermodynamic cycle similar to the internal combustion engine. The primary working fluid is a mixture of uranium hexafluoride, UF6, and helium, He, gases. [ighly enriched UF6 gas is the reactor fuel. The He is added to enhance the thermodynamic and heat transfer characteristics of the primary working fluid and also to provide a flux flattening effect in the cylindrical core. ,:

Both two-and four-stroke engines have been studied in which a neutron source is the counterpart of the sparkplug in an internal combustion engine. The piston motions which have been investigated include pure simple harmonic, simple harmonic with dwell periods, and simple harmonic in combination with non-simple harmonic motion.

Neutronically, the core goes from the subcritical state through criticality and to the supercritical state









during the (intake and) compression stroke(s). Supercriticality is reached before the piston reaches top dead center (TDC) so that the neutron flux can build up to an adequate level to release the required energy as the piston passes TDC.
The energy released by the fissioning gas can be extracted both as mechanical power and as heat from the circulating gas. External equipment is used to remove fission products, cool the gas, and recycle it back to the piston engine. Mechanical power can be directly taken by means of a conventional crankshaft operating at low speeds.

To utilize the significant amount of available energy in the hot gas, an external heat removal loop can be designed. The high temperature (%1200 to 1600'K) HeUF
6
exhaust gas can be cooled in an HeUF6-to-He heat exchanger. The heated He (%IO00K to 1400'K) is then passed either directly through gas turbines or is used in a steam generator to produce steam to drive a turbine.

The total mechanical plus turbine power per nuclear piston or per cylinder ranges from around 3 to 7 Mw(e) depending on the selected piston engine operating characteristics and the external turbine equipment arrangement. Thus, power generating units of from 10 to 50 Mw(e) capacities would consist of a cluster of 4 to 8 pistons in a nuclear piston engine block. Larger power plants can be conceptually









designed by increasing the number of pistons with the mechan-. ical complexity and physical size as the probable limiting factors. Overall efficiencies are as high as 50% implying heating rates of around 6800 BTU/kr-hr. Fuel costs are presently estimated as being below $0.20 per million BTU*or around 1.4 mills/kwe-hr.


Applications and Highlights of the Nuclear Piston Engine Concept

Some of the primary uses for nuclear piston engine power generating systems would be for peaking units for electrical utilities, for small mobile and fixed groundbased power systems, for nautical propulsion and ship electrical power and for process heat. Further possible applications will be discussed in Chapter VII.

Most current peaking units operate on conventional fossil fuels. These units are, in general, expensive, wasteful, and inefficient. Fuel costs range from $0.50 to $1.30 per million BTUs,* heat rates are as high as 15,000 to 21,000 BTU/kw-hr, and efficiencies are not much greater than 20%.

A conversion from wasteful, conventionally fueled peaking units to efficient, nuclear-fueled peaking units would yield significant savings in fossil fuels. The fuel thus saved could be reallocated for more critical applications. This consideration alone should be incentive enough to

*Based on fiscal year 1974 costs.









investigate any promising, nuclear-fueled peaki'ng unit concept--even if the nuclear-fueled unit's power production costs should be estimated to be as high as for the conventionally fueled units. The fact that preliminary estimates indicate that a nuclear piston engine peaking unit should be more economical than any of the fossil-fueled units now employed makes this concept that much more attractive.

Already-developed nuclear reactor concepts like pressurized water reactors (PWRs), boiling water reactors (BWRs), and high temperature gas-cooled reactors (HTGRs) can be economically competitive only when they are incorporated into large capacity power systems. Given the fuel cycle costs and operation and maintenance costs for these reactor concepts, it is their high capital costs which economically prevent them from being used on a scaled-down basis for 20-50-100 Mw(e) units. The cost per unit power installed (dollars/kwe) for scaled-down units operating on these already-developed solid-fueled core concepts would be extremely high.

A nuclear piston engine power plant, however, will not require the sophisticated and costly engineered safeguards and auxiliary systems associated with the solidfueled cores of current large capacity nuclear power plants. The inherent safety of an expanding gaseous fuel can b.e engineered to take the place of many of the functions of the









safeguards systems. Hence,. while gaseous core, nuclear piston engine power plants would possess relatively high costs per unit power installed as compared to comparably sized fossil-fueled units, their capital costs per unit power installed would be considerably less than for any scaled-down nuclear units operating on current solidfueled core concepts.

In addition to decreased capital costs, the nuclear piston engine should possess fuel cycle costs which are about half the fuel cycle costs of most present large capacity nuclear plants. Fuel fabrication costs, transportation costs to and from the fabricator, and transportation costs to and from the reprocessor will all be eliminated. These costs typically comprise from 40 to 50% of the current nuclear fuel cycle costs.*
Thus, it would appear as if power production costs for a nuclear piston engine will not only be less than those of conventionally fueled peaking units, but that they should also approach the power production costs of large-scale fossil and large-scale nuclear-fueled plants.

With regard to power generation for nautical applications, the nuclear units utilized by ships are more expensive than conventionally fueled units. The major advantage of current nuclear-fueled vessels is their tremendous range between refuelings as compared to conventionally powered vessels. It is for this strategic reason rather than for economic

*Based on 1974 fiscal year costs.









reasons that the U.S. Navy maintains nuclear-powered vessels. On the other hand, the economic disadvantage is the primary reason why nuclear-powered vessels have not been able to replace conventionally powered commercial" vessels. Ships powered by nuclear piston engine gas core systems, however, should be able to compete economically with conventionally powered vessels while still retaining the, advantage of long ranges between refuelings. The extensive use of such nuclear power units by ships would, of course, also lead to significant fossil fuel savings.


Dissertation Organization

In the chapter which follows, a summary is presented of some of the more important nuclear studies which have been performed on gaseous core, externally moderated reactors. It presents the models employed to analyze the neutronics of gaseous cores, calculations performed, experi'ments conducted and appropriate comparisons between analytical and experimental results.

This is followed by a chapter describing the previous work which was done on the gaseous core nuclear piston engine concept by other authors here at the University of Florida where the idea originated.

Results from work which has since been performed by

the author on two-stroke nuclear piston engines is presented









in Chapter IV. Since these investigations indicated good performance potential for the nuclear piston engine concept, more sophisticated, four-stroke engines were studied. The results of these studies are presented in Chapter V.

Chapter VI discusses other ongoing and related research in the field of gaseous core reactors. The results of some of these other studies will certainly have an impact on the future of further research on the nuclear piston engine concept.

The last chapter presents conclusions. Suggestions are made for refinements in the neutronics and energetics equations used in the nuclear piston engine analysis. Also discussed are areas where further studies are needed before the technical feasibility of the nuclear piston engine concept can be firmly established.

Finally, all neutronics and energetics equations used in the nuclear piston engine analysis have been placed in appendices. References are made from Chapters IV and V to the appropriate appendix for equation development and presentation.















CHAPTER II

PREVIOUS STUDIES ON GASEOUS CORE CONCEPTS



Gaseous Cores Analytical Studies

The first report on analytical studies of a gas core nuclear reactor was due to George Bell of Los Alamos in 1955 [1]. Age and diffusion theories were used to analyze the neutronics of a spherical gaseous cavity surrounded by a moderating-reflector. Age theory was used to describe neutron slowing down in the moderating-reflector (no slowing down or fast neutron absorption in the fuel was permitted) and diffusion theory was used to describe thermal neutron diffusion into the cavity and the resulting fissions in the fuel. The reactors considered were strictly thermal with UF6 gas cores and D 0, Be and graphite reflectors.
6 2'
In 1958, a report of a study on externally moderated reactors was published by Safonov [2]. The study was based on the prime assumption of complete external moderation. Fissile material was contained in a central or "interior" region while the moderating-reflector material surrounding the fuel comprised the "exterior" region. The analysis included, but was not limited to gaseous cores. Low density, liquid-metal-fueled, externally moderated reactors









were also considered. Fermi age and diffusion theory were used to describe the neutronics of the exterior moderatingreflector while diffusion theory was used for the interior cores with Es >>E and transport theory was applied to the interior if << a*

Safonov investigated U233, U235 and Pu 239-fueled
systems with D20, Be and graphite moderating-reflectors. Since the critical particle densities of fissionable atoms correspond to molecular densities of gases of less than atmospheric pressure, the term "cavity reactor" was applied to these low interior density systems.

Breeding prospects for externally moderated systems

were also looked at by Safonov. A U233-fueled gaseous core with a non-infinite, D20 moderating-reflector was surrounded with an external thorium blanket. For a 1 meter diameter core with a 50cm thick D20 moderating-reflector, a potential breeding ratio of 1.23 was calculated; for a 100cm thick D20 region, the potential breeding ratio was still 1.03.

Both Bell and Safonov's models were restricted to thermal, spherically symmetric systems of low interior greyness (interior greyness being defined as the ratio of the thermal neutron current. into the interior to the thermal neutron flux at the interior boundary). Small radii systems or large radii systems of high gas density are normally too grey to thermal neutrons to permit analysis by these models.









In 1961 Ragsdale and Hyland [3] looked at cylindrical and spherical, D20 reflector-moderated, U235-fueled, gaseous core reactors. A parametric study was made with variations in moderator thickness, fuel region radius for a given cavity radius, the effect of the cavity liner, and the moderator temperature. Six-group, one-dimensional diffusion theory was used for the spherical configurations while four-group, two-dimensional diffusion theory was used for the cylindrical systems. The reflector temperature was assumed to govern the neutron energy and fast absorption and slowing down in the fuel region was disregarded. The criteria as established by Safonov for the validity of diffusion theory were used in this analysis and were a function of the "cavity greyness." Thermal cross sections were obtained in the analyses for the moderator temperature, regardless of the region.

In 1963, Ragsdale, Hyland, and Gunn [4] extended

their work. They considered in this work only cylindrical geometry using four-group, two-dimensional diffusion theory.
239 235
The fuels considered were Pu and U, while D20 (at 300'K) and graphite (at 300'F and at 32000F) were studied ,as moderating-reflector materials. They looked at the effect on critical mass of a variable fuel region radius in a fixed size cavity. The moderating-reflector was 100cm in thickness since earlier work has shown this would be optimum for reducing critical mass without incurring an excessive weight penalty. Assumptions were made for flow









rates, pressure, temperatures, etc., and within these constraints, cavity radii of 40cm and 150cm were investigated. Thermal cross sections were all computed at the moderator temperature, the effect of Doppler broadening due to the elevated fuel temperature was considered and a Maxwellian flux shape was used in determining mean average thermal cross sections.

Kaufman et al. [5] performed an extensive parametric survey on gaseous core reactors in cylindrical geometry in 1965. One-dimensional Sn transport theory studies were used initially to evaluate the effectiveness of various moderating-reflectors (Be, D20 and graphite) for a variety of cavity dimensions and moderator temperatures. Also studied were composite reflectors and pressure vessel and liner effects on the critical mass. The effects of geometric variations, such as radius-to-height ratios, were looked at using two-dimensional S transport theory.
n
Comparisons made between calculations using 24, 15, 13 and

3 collapsed broad groups showed that a good set of collapsed three-group constants was adequate to yield critical masses, fluxes, absorptions, and leakages.

Herwig and Latham [6], after studying a hot gaseous core containing hydrogen, concluded in 1967 that upscattering of neutrons returning to the core by the hot hydrogen is an important effect and that for such cores, a multithermal group approach is essential. One-dimensional diffusion theory with 6 fast and 12 thermal groups was used









to evaluate reactor characteristics for changes in moderatingreflector parameters such as temperature, slowing down power, (n,2n) production, thermal scattering, and thermal absorption. D 20, Be and graphite-moderated reactors were studi ed.

Neutronic Calculations for Gaseous Core Nuclear Rockets

Nuclear studies of gaseous core nuclear rocket engines were carried out by Plunkett [7] in 1967. S8 transport theory calculations using 14, 15 and 16 groups were performed and compared with multi-group diffusion theory results. Thermal cross sections were obtained by averaging over a Maxwellian neutron distribution at the moderator temperature, and Doppler broadening was considered. Diffusion theory critical loadings and fluxes were in good agreement with the transport theory results.

Latham conducted a series of extensive calculations (1966-1969) [8-12] for a nuclear light bulb model (closed system) gaseous core nuclear rocket engine. BeO and graphite moderating-reflectors with U 233, U235 and Pu239 fuels were investigated. A series of 24-group, one-dimensional, S4 transport theory calculations were performed with ANISN [13]. Fourteen of the 24 groups were in the 0 to 29eV range because of the large temperature differences in adjacent regions and the consequent importance of upscattering by the hot hydrogen and neon in the 1200 to 7000'K core.









Fast neutron cross sections were obtained from GAM-I [14] with the slowing down spectrum in the core being assumed to be that of the moderating-reflector region. TEMPEST [15] was used for the thermal absorption probabilities and SOPHIST [16] for thermal up- and down-scattering probabilities. Flux and volume weighted four-group cross sections were obtained from the 24-group, one-dimensional transport calculations for use in two-dimensional calculations. Two-dimensional transport theory calculations were then performed with DOT [17] while two-dimensional diffusion theory calculations were obtained from the EXTERMINATOR-II code [18]. The objectives of Latham's works were to (a) evaluate effects of variations in engine design on U233 critical mass, (b) compare U233 U235, and Pu239 critical mass requirements, and (c) to evaluate various factors affecting engine dynamics. In addition to critical masses, material worths, pressure, temperature and reactivity coefficients, and neutron lifetimes were determined.

Critical masses and the effects of variations in

cavity size, fuel-to-cavity radius and reflector thickness for U233- and U 235-fueled, open, gas core nuclear rocket engines were looked at by Iyland in 1971 [19]. A composite, D20-BeO-D20 moderating-reflector was used. Fuel temperatures reached 44,000�C and moderating-reflector temperatures varied from 93'C to 560'C. The cavity wall temperature









was 1115'C. A total of 19 groups (7 thermal) were used with fast group constants obtained from GAM-II [20] and thermal group constants from GATHER [21]. One-dimensional, S4 transport calculations in spherical geometry were performed with the TDSN code [22]. Hyland found that a large amount of a light gas, like hydrogen, in the core increases the absorptions and upscattering caused by the hydrogen in the cavity between the uranium and the moderators. This means more fuel absorption in the higher energy levels which are less productive (fewer fissions per absorbed neutron) and hence an increase in the critical mass requirement. Other observations were (a) U233 has a lower

critical mass than U235, (b) for U233 there is less change in the critical mass requirement from the startup temperature than for U235, (c) the critical mass increases with increasing cavity diameters while the critical fuel density decreases, (d) above a total reflector thickness of more than one meter little reduction in critical mass is obtained, and (e) the critical mass increases rapidly with decreasing fuel to cavity radius.


Comparison of Theoretical Predictions with Experimental Results
In a report on reflector-moderated reactors, Mills [23] in 1962 compared theoretical predictions using Sn transport theory with experimental data for gaseous uranium core reactors and attained fairly good agreement. Mills also









performed parametric studies of gas-filled, reflectormoderated reactors to establish minimum critical loadings. Large spherical and cylindrical cavities reflected by D20$ 235
Be, and graphite were investigated; gaseous U was the only fuel studied. Some of the conclusions reached by 235
Mills were (a) systems are sensitive to the U content,

(b) the systems are not sensitive to core diameter changes for a constant amount of U 235, (c) systems are sensitive to absorption either in the liner or in the moderatingreflector, and (d) the critical mass increases approximately as the radius squared (this is in contrast to internally moderated cores where the critical mass increases approximately as radius cubed).

In 1965, Jarvis and Beyers [24] of Los Alamos made

a comparison between diffusion theory predictions and experimental results for a D20-reflected cavity reactor. The maximum discrepancy between calculated and experimental results for this system was 3% in reactivity.

A series of critical experiments (1967-1969) [25-32] were performed by Pincock, Kunze et al. to test the ability of various calculational procedures to evaluate criticality and other reactor parameters in a configuration closely resembling a coaxial-flow (open system) gaseous core nuclear rocket engine. In some configurations, fuel was dispersed as small foils in various patterns representing fuel distributions in a gas core reactor. Other configurations









contained UF6 gas. D20 reflectors were employed which for some configurations had Be slabs (heat shields) in the D20 spaced 5 to 10cm from the cavity wall. The theoretical calculations included one-dimensional, 19-group diffusion theory, one-dimensional, S4 and S8 transport theory, and four-and seven-group, two-dimensional diffusion theory results. Comparison of these results showed little difference between S4 and S8 calculations and reasonably good agreement between the multigroup diffusion theory and the transport theory calculations. The transport theory calculations themselves were in good agreement with experimental results.

A benchmark critical experiment with sphericalsymmetry was conducted on the gas core nuclear reactor concept in 1972 by Kunze, Lofthouse, and Cooper [33]. Nonspherical perturbations were experimentally determined and found to be small. The reactor consisted of a low density, central uranium hexafluoride gaseous core, surrounded by an annulus of either void or low density hydrocarbon which in turn was surrounded by a 97cm thick D20 moderatingreflector. One configuration looked at also contained a

0.076cm thick stainless steel liner located on the inside of the cavity wall. Critical experiments to measure reactivity, power and flux distributions and material worths were performed. Theoretical predictions were made using 19 groups (7 thermal) in an S4 transport calculation with the SCAMP code [34]. Fast group cross sections were obtained from









PHROG [35] and thermal group cross sections from INCITE [36]. The predicted eigenvalues were in good agreement with experiment as was the reactivity penalty for the stainless steel liner. Fuel worths and the reactivity penalty for the hydrocarbon however were substantially underpredicted.



Comments

While the above is not a complete listing of all the nuclear studies which have been performed on gaseous core, externally moderated reactors, it is a representative sampling of the types of models employed, calculations performed and experiments conducted on gaseous core reactors.

The models of Bell and Safonov are reasonable only for thermal, spherically symmetric systems which are not very grey. For systems which possess a significant degree of greyness., the general conclusion to be drawn from the above investigations is that in "most cases," multigroup diffusion theory is adequate provided "good" fast and thermal group constants are used and provided there is a multithermal-group approach which allows for full UDscattering and downscattering.

Because of the wide range of geometries, temperatures, temperature differences, pressures, densities and materials which can be employed in gaseous core reactors, each new gaseous-core reactor concept demands individual scrutiny. The applicability of diffusion theory to the processes of neutron birth in the core, thermalization in the moderating-










reflector, and the diffusion of the thermal neutrons back into the core should be checked by performing some transport calculations for at least a few reference configurations which are typical of the particular gaseous core reactor design being investigated. Attempts to apply or extend conclusions from previous gas core analyses to new gaseous core concepts cannot be justified by presently established theoretical and/or experimental evidence. In particular, the gas core "cavity" type reactors are not directly comparable to the dynamic situation in a gaseous core-piston engine. Significant differences exist in the modus operandi of each concept. In the steady-state condition, however, valuable analogies can be drawn between these sytems.

Before concluding this section, credit should be given to J.D. Clement and J.R. Williams for their report on gascore reactor technology [37]. Besides outlining the important work which had been done in the field of gas core reactor neutronics calculations up until mid-1970, this report contains an extensive listing of helpful references.















CHAPTER III

PREVIOUS GASEOUS CORE NUCLEAR PISTON ENGINE STUDIES
AT THE UNIVERSITY OF FLORIDA



Introduction

A thermodynamic cycle, similar to the internal combustion engine using a gaseous fissionable fuel,was first proposed by Schneider and Ohanian [38]. Preliminary feasibility studies by Kylstra et al. [39] showed that a UF6fueled, Otto-type fission engine has a very good performance potential. Power in the MW/cylinder range and thermodynamic efficiencies of up to 50% seemed feasible with a low fuel cycle cost making the process economically attractive.

The nuclear piston engine studied by Kylstra and

associates (see Figure 1) was a pulsed gaseous core reactor enclosed by a neutron-reflecting cylinder and piston. The fuel was 100% enriched UF6 and the ignition process was triggered by an "external" neutron source. The engine was a two-stroke model with intake and compression occurring on the first stroke and expansion and exhaust on the following stroke. Neutronically, it was desired that the core go from the subcritical state, through criticality, and to the supercritical state during the compression stroke.







20








EAT L

HANGER









4 MODERATING-REFLECTOR







PISTON MODERATING-REFLECTOR


FIGURE 1. Simple Schematic of A UF6 Nuclear Piston Engine


I I









Supercriticality was to be reached before the piston reached top dead center so that the neutron flux could build up to an adequate level to release the required power as the piston passed TDC. To avoid releasing fission heat after the piston was already well into the power stroke, it was then required that the reactor be rapidly shut down. To attain the desired time sequence of subcritical to super-. critical to subcritical behavior for the reactor, the reflector thickness was varied throughout the cycle. At the compression stroke start, the reflector was a thin reflector, increasing in thickness slowly, then stepping to a thick reflector at some cycle fraction, cl, in the compression stroke and then continuing to increase slowly until TDC. The step-reflector was then removed, going back to a thin reflector at some cycle fraction, �2' usually at TDC (see Figure 2).

The moderating-reflector surrounding the cylindrical core and the piston itself were made of graphite, with a nickel liner being used for protection of the graphite from the UF6 [40]. External equipment was to be employed to remove the fission products, cool the gas, and recycle it back to the engine.

A HeUF6 mixture rather than pure UF 6 gas was used for the engine's primary working fluid. The addition of helium improved the working fluid's thermodynamic and heat transfer properties while also leading to a flattening of the neutron flux in the core. For the systems studied, the UF6 partial









CORE
MODE RAT IN G -REFLECTOR








LOW LEAKAGE,
SUPERCRITICAL
ARRANGEMENT





AFTER STEP-REFLECTOR ADDITION AT





CORE

MODERATING-REFLECTOR





HIGH LEAKAGE, SUBCRITICAL
ARRANGEMENT




AFTER STEP-REFLECTOR REMOVAL AT c OR BEFORE STEP-REFLECTOR ADDITION AT cI


FIGURE 2. ILLUSTRATION OF STEP-REFLECTOR ADDITION AND REMOVAL









pressures and temperatures throughout the cycle were such that the UF6 remained in the vapor phase (see Figure 3) and underwent no dissociation due to thermal kinetic energy [41].

The engine was to be operated at high graphite temperatures (1000-1200�F) so as to minimize the convective and conductive heat losses from the core region. The compression ratio for the engine was l0-to-I with a clearance volume of 0.24m3 and an engine shaft speed of 100rpm.


Neutronic Model

For the steady-state solution of the neutron balance, a two-group, two-region diffusion theory approximation in spherical geometry was used with the following assumptions:

1) no interactions for fast neutrons in the core.
Thus, the fast core equation was replaced by a boundary condition for the net neutron current
into the moderator.

2) no absorption in the moderator.

3) no delayed neutrons (the delayed neutron precursors are swept out of the cylindrical core with
the exhaust gas before they exert any influence).

4) no time dependence.

5) no angular dependence.

6) for the sake of simplifying the analysis, the
cylindrical two-region piston was represented by
a two-region spherical model. The spherical core
volume and reflector thickness were then varied
to simulate the motion in the corresponding
cylindrical piston.














100


10 L


SOLID


LIQUID


TRIPLE POINT


VAPOR


0.1 I-


FIGURE 3. UF6 Phase


0 300 350 400 450 5uu 55u
T--MPVRATURE (@K)


Di a g rarn









The set of equations resulting from the above assumptions and approximations was solved for both the neutron multiplication factor, k-effective, and for the average steady-state thermal neutron flux in the core, as a function of piston position.

As the neutron multiplication factor approached and

exceeded one, the average core thermal neutron flux at each time step was calculated from a single-group, point reactor kinetics equation rather than from the expression derived from the two-group, steady-state analysis. The justification offered for the uncoupling of time and space (implied in the point model treatment) was the difference in time scale between the speed of the piston (or rate of change of geometry) and the diffusion speed or cycle time of the neutrons. The use of the diffusion equations was justified only by their simplicity compared to Sn or P approximan n
tions to the neutron transport equation.

A knowledge of the average core thermal flux throughout the compression and power strokes permitted the calculation of the fission heat as a function of time for use as the heat source term in the energy balance equation.



Energy Model
The conservation of energy equation for a non-flow, closed system was used. It was assumed that the HeUF6 mixture was an ideal gas of constant composition and heat









loss to the walls was neglected. The energy equation balanced the rate of increase of the internal energy of the HeUF6 gas against the rate of performance of the pV mechanical work by the gas on the piston. The initial pressure, temperature, volume and piston position as a function of time were input parameters. A numerical form of the energy equation was then used to determine the gas temperature, T(t+At), in terms of T(t). Since the piston position at t+At is known, so is the cylinder volume V(t+At) and the ideal gas equation was then solved to obtain the gas pressure, p(t+At). This process was continued over the entire compression-power stroke cycle to yield not only the gas pressure and temperature variations throughout the cycle but to also permit the determination of the total pV work and total fission heat released during the piston cycle. The neutron flux, neutron multiplication factor and atom density were also monitored as a function of piston position throughout the cycle.

Primary independent variables of the engines studied by Kylstra et al. and their range of values are shown in Table 1. The clearance volume is the core volume with the piston at TDC.














TABLE 1

Values of Primary Independent Parameters for
Graphite-Reflected Piston Engines
Analyzed by Kylstra et al.[39]


Gas Mixture Initial Temperature (OK) Gas Mixture Initial Pressure (atm) U235 Loading (kg)


Engine Speed (rpm) Clearance Volume m 3) Compression Ratio


Cycle Fraction for Step-Reflector Addition, el Cycle Fraction for Step-Reflector Removal, �2 Neutron Source Strength (neutrons/sec)


= 400 = 1-4 = 1.7-3.1 = 100 = 0.24 = 10 -to-1 = 0.1-0.3 = 0.5 = Ix109









Analytical Results

Shown in Figure 4 is the neutron multiplication

factor for an engine with an infinite graphite reflector as a function of the UF6 partial pressure at a temperature of 400'1K. At this temperature, for UF6 partial pressures greater than about 1 atmosphere, the core becomes so black to neutrons that additional uranium is ineffective.

Figures 5 and 6 show the total gas pressure, gas temperature, average core thermal neutron flux and neutron multiplication factor over a complete compression-power stroke cycle for a typical set of independent parameters. The stepreflector addition was at a cycle fraction cI = 0.10 while the step-reflector removal and reactor shutdown occurred at a c2 =0.50 cycle fraction. The maximum pressure for this system was 31 atmospheres and the maximum temperature was 1230�K (1754�F). Both. the temperature and pressure peak at TDC, which is the point at which the step-reflector is removed and the reactor is shut down. Since the cylinder walls were to be maintained at 1O00-1200�F, a peak temperature of 1754�F did not represent an excessive thermal pulse.

The neutron multiplication factor in Figure 6 increases to a value greater than 1 upon step-reflector addition at

1 = 0.1 and then gradually decreases as the core decreases insize. The increased neutron leakage as the piston moves towards TDC and the decreased U235 cross section with increased temperature are thus more important than the















































0.3 1.0 3.0 10


UF6 PRESSURE (atm)

(Gas Temperature = 400K)


FIGURE 4.


Neutron Multiplication Factor Versus UF6 Partial Pressure for an Infinite Graphite Reflector [39]


1.25 1.0 1.75




0.5


























1250 1000
0




750 500


25 50 75


PERCENT TRAVEL THROUGH CYCLE






Initial Gas Pressure = 1 atm Initial Gas Temp.rature 400K U-235 Mass = 2.15 k


FIGURE 5.


Gas Pressure anJ Temperature as a Function of Percent Travel Through the Piston Cycle for a Graphite-P.fI1 LrLed Engine [39]











































L I I 10.4
0 25 50 75 10


PERCENT TRAVEL THROUGH CYCLE





Initial Gas Pressure 1 atm
Initial Gas Temperature = 400'K
,-235 Mass = 2.15 kg


FIGURE 6.


Average Core Thermal Neutron Flux and Neutron Multiplication Factor as a Function of Percent Travel Through the Piston Cycle for a GraphiteReflected Engi,. [39]


10 15









increase in U 235 density. The multiplication factor then drops rapidly upon removal of the stop-reflector at E2 = 0.5.

Figure 7 shows typical performance results obtained by Kylstra et al. for the UF6 piston engine. The data for Figure 7 is for an engine with the step-reflector imposed at the 10% cycle position, the same as Figures 5 and 6; A maximum neutron multiplication factor of from 1.07 to 1.10 was reached for these systems with k-effective dropping to 0.99 to 1.01 as TDC was approached. This behavior of k-effective greatly increases the control safety since the power doubling time is large at high power. Increasing' the U235
U 2 loading leads to larger k-effectives but it also reduces the helium content in the gas mixture for the same initial pressure. Thus, the efficiency and power curves of Figure 7 are concave downward to reflect the higher specific heat and hence poorer thermodynamic properties of the gas mixture as more UF6 is added at the expense of He.

Kylstra et al. compared one of their piston engines with a large stationary diesel power plant [42]. The results of this comparison are shown in Table 2. The Nordberg Diesel which was used in the comparison has 6 to 12 cylinders, a 29-inch bore, and a 40-inch stroke. The fuel cost estimates were obtained by assuming a 20�/gal price for diesel fuel, and a $12/gm U charge for the UF6 fuel.















2 2.10 2.15 2.20 2.25 2.30 2.35
10


EFFIC [IENCY
60

40 ci


I-4
INITIAL. PRESSURE - 20 ,
I = I atm- 0 .
I0if[ 2 ata II
I11 = 4 ar t.i

U
e- E: = 0.[ 1l
:.�n : = 0.5

U2

M3


II)




-Cl
1 0 10


',.C.

--,l
10 J 4I Ii 1
U \ ENSUREE




C i02t10
10J



Cc Fat.








2. 10 2 .15 2 .2 0 2 .25 2 . 02 1
10- !! I !I .3 101
235

Compression Ratio = 1O-to-1
(Ch,,ramc,: Vo o (3) = 0.24 EII, ille SpOAe (ripm 's) = 1O0

FIG.URE 7.- UF 6 Nuclear Piston Enqine Performan ce for the Case
of Graphite Step-Reflector Addition at the 0.1
Cycle Fraction















TABLE 2

Comparison of a Typical Graphite-Reflected, UF
Piston Engine (as analyzed by Kylstra et al. 6
with the Nordberg Diesel [39]


Characteristic Diesel UF6 Engine


Clearance Volume (m ) 0.0394 0.24

Compression Ratio 12-to-i l0-to-I

Displacement Volume (m ) 0.434 2.16


Shaft Speed (rpm) 200 100

Type of Cycle (# of strokes) 2 2

Net work per cycle (Mw-sec) 0.224 0.86

Power (Mw) 0.746 1.436
3
Power Density (w/cm ) 1.72 0.665

Efficiency (%) 20-30 42

Fuel Cost ($/106 BTU) 1.39 0.21

Relative Fuel Cost 6.6 1









One of the results of the study by Kylstra et al. was the observation that the time of application of both the step-reflector addition and removal was rather critical. Analytically, the application of this step function in the reflector thickness is easy to attain; practically, it can only be approximated. Recognizing this, Kylstra and his associates were led to conclude that rather than simple harmonic motion, a better pattern would involve the use of dwell periods by means of 4-bar linkage systems.

Since the preliminary feasibility studies conducted by Kylstra indicated such good performance potential for a nuclear piston engine, further work on the gaseous core nuclear piston engine was warranted in order to better judge the technical feasibility of the concept. The additional work which has been conducted to date is discussed in detail in the next two chapters,. As will be seen, these studies have led to piston engine performances and designs which differ significantly from those of Kylstra and his associates.















CHAPTER IV

RESULTS FROM TWO-STROKE ENGINE STUDIES



Introduction
Many of the studies on gaseous core reactor concepts which were discussed in Chapter II have proven to be valuable guides for the gaseous core, nuclear piston engine neutronics calculations. Large differences exist, however, in sizes, pressures, temperatures, densities, and materials between the cores analyzed in these previous studies and the nuclear piston engine. These differences preclude any extrapolation of predicted behavior to the nuclear piston engine. A complete, thorough investigation of the nuclear characteristics, including extensive parametric surveys, is therefore an essential step if the nuclear piston engine's technical feasibility is to be demonstrated.

Initially, only two-stroke engines were analyzed. The compression-power stroke cycles for these engines were examined with the intention of eventually proceeding to more complex four-stroke engines if the results from the two-stroke engine proved encouraging.

The first two-stroke engines which were looked at

neglected delayed and photoneutron effects. Later two-stroke









engines considered first the influence of delayed neutrons and then the influence of both delayed and photoneutrons on the nuclear piston engine's performance.

The effects of variations in the nuclear piston

engine operating conditions were analyzed by means of a reasonably simple, analytical model, incorporated into a computer code which has been named NUCPISTN. The code is similar in function to the code used by Kylstra et al. [39]. The NUCPISTN code, however, is much more sophististed than the original code and has been modified and improved in several important aspects. Two-group, two-region diffusion theory equations in spherical geometry were still used for the steady-state spatial flux dependence and for the neutron multiplication factor throughout the piston cycle. Five of the six assumptions which were used in the solution of the steady-state neutron balance by Kylstra et al. (see Chapter III) were initially maintained. Only the second assumption was altered in that neutron absorption in the moderating-reflector was no longer neglected. A complete development of the two-group, two-region, steady-state diffusion theory equations used in the NUCPISTN code in the absence of photoneutrons is given in Appendix A.

Initially, a single point reactor kinetics equation was again used for the time dependence of the neutron flux. The point kinetics equation(s) used in the NUCPISTN code are presented in Appendix D. However, Appendix C,






38


which presents the general point reactor kinetics equations, should be examined first both for familiarization with the notation as well as for a development of some expressions which are used in Appendix D. The actual one-speed (thermal) point reactor kinetics equation used by NUCPISTN when delayed and photoneutrons are ignored is given by equation (D23).

The same non-flow, closed system conservation of

energy equation as used by Kylstra et al. is again used in the NUCPISTN code for the two-stroke engines. A development of this equation is set forth in the first section of Appendix E.

The NiUCPISTN code then couples the neutronics and

energetics equations and solves for the neutron multiplication factor, neutron flux, core volume, gas temperature,. gas pressure, pV work, and fission heat release over the piston compression-power cycle.

As already mentioned, the NUCPISTN code is much more flexible than its predecessor. It is able to accommodate a wider variety of initial conditions and possible piston motions including dwell periods in the piston cycle and nonsimple harmonic motion. The piston behavior during the cycle is closely monitored and the final cycle output information is much more extensivethan in the prior code. Time steps are chosen during the cycle on the basis of the current k-effective of the engine and also on the current rate










of energy release in an efficient and systematic manner. Non-I/v variations in the uranium microscopic cross sections with temperature are now accounted for by means of Wescott factors and thermal absorption in the reflector is considered. Corrected and improved thermodynamic constants for the HeUF6 gas are used [41, 43].
Neutron multiplication factors and cross sections output by the NUCPISTN code at various cycle positions for different piston engine models have been compared with more elaborate calculations. The first comparisons were made with results obtained from two- and four-group, one-dimensional diffusion theory calculations performed with CORA [44] in spherical geometry and with corresponding two-dimensional diffusion theory calculations performed by EXTERMINATOR-I1 [18] in cylindrical geometry. The collapsed fast group constants for CORA and EXTERMINATOR-II were obtained from a standard 68-group, PHROG B-I calculation [35]. The thermal group constants used in CORA and EXTERMINATOR-II were obtained from a 30-group BRT-l [45] calculation. BRT-l is the Battelle-revised version of the industry benchmark computer program, THERMOS. The PHROG, BRT-l, and NUCPISTN group constants were then compared with collapsed group constants obtained from 123-group, one-dimensionalSn transport theory calculations which were performed with the powerful transport scheme of XSDRN [46] in spherical geometry. S4 and

S6 quadratures were used.









Two of the piston engines studied were next selected for incorporation into several nuclear piston engine power generating systems. These systems included the nuclear piston engines, a HeUF6-to-He heat exchanger, and gas or steam turbines, along with associated auxiliaries including pumps, compressors, condensers, regenerators, etc.

Thermodynamics analyses were performed and fuel cost

estimates were made for these piston engine power generating systems. In addition, a preliminary analysis for the HeUF6to-He heat exchanger was carried out for one of the piston engine power generating systems. As a result of this analysis, it was possible to estimate the HeUF6 circulation time,
235
and hence the U inventory in the primary loop for this
particular nuclear piston engine power generating system.


Comparison of Initial Results with
Previous Nuclear Piston Engine Analyses

For a given set of initial conditions, the NUCPISTN code has yielded piston engine performances which are significantly different from the results obtained by Kylstra et al., even when thermal absorption in the reflector has been neglected.

Some of these differences are due to the improved calculaLvjndl scheme utilized in NUCPISTN and also to the improved thermodynamic constants. Most of the differences, however, are a result of the improved group constants used









in the NUCPISTN code. The crude approximations to the cross sections and their temperature dependence employed in the code used by Kylstra and his associates were extremely inaccurate.

Upon inclusion of thermal absorption in the reflector, the performance differences between the previous and current studies became even greater. A wide variety of configurations and loading schemes were consequently investigated. From these investigations, it became apparent that the thermal absorption correction in the graphite reflector was of major importance. In fact, this correction so severely limited the graphite-reflected engine's performance, for the core sizes and operating conditions of interest, that these engines had to be discarded. As will be seen, heavy waterreflected cores eventually became the basic component for the nuclear piston engine studies of this work.

Referring to Table 1, it will be noted that the engines analyzed by Kylstra et al. typically had compression ratios of around lO-to-I and clearance volumes of 0.24m 3. The resulting strokes were therefore excessively large--around 19 to 20 feet. The heavy water-reflected nuclear piston engines have been restricted to more reasonable stroke sizes of from arounid,3 to 5 feet. Neutronic considerations require that the clearance volume be large compared to conventional internal combustion engines in order to achieve criticality.









Hence, the compression ratios are therefore limited to around

3 or 4-to-l for 3-to 5-foot strokes.


Graphite-Reflected Systems

Presented in Table 3 are the uranium-235, fluorine, graphite, and helium atom densities (in atoms per barn-cm) for five graphite-reflected engines at the TDC position. Also presented are the core and reflector region average physical temperatures and the engine dimensions at the TDC position.

Shown in Table 4 are the neutron multiplication factors for these engines at TDC as obtained from various computational schemes. The NUCPISTN results were from twogroup, two-region diffusion theory equations in which the fast core equation was replaced by a boundary condition. Hence, in this scheme, no fast interactions in the core are permitted (see Appendix A).

The XSDRN results are from 123-group, two-region Sn

transport theory calculations in which S4 and S6 quadratures were used. The CORA results are from two-group, two-region diffusion theory calculations in which the thermal group constants were obtained from BRT-l and the fast group constants from PHROG. Fast interactions in the core were included in this scheme.

All of the schemes utilized one-dimensional spherical geometry. The two-region, two-dimensional cylinder was












TABLE 3


Atom Densities and Temperatures for Large GraphiteReflected Engines at the TDC Position


Characteristic/Engine G-1 G-2 G-3 G-4 G-5

Core Atom Densities
-5 -5
U-235 (atoms/barn-cm) 7.192xi0 2.24xi0 2.67x0-5 2.694x0-5 2.687x0-5
44-4 -4 -4
F-19 (atoms/barn-cm) 4.315x0- 1.344x10 1.602xlO4 1.616xO-4 1.612xOHe-4 (atoms/barn-cm) 0.0 1.60xlO " 1.56xlO4 3.39xl04 5.24xl04

Reflector Atom Density

C-12 (atoms/barn-cm) 8.08x10-2 8.27xl0-2 8.08xlO2 3.13xlO-2 8.18xlO-2

Core Temperature ('K) 2576 1000 2400 1800 1530

Reflector Temperature (�K) 2000 820 2000 1500 1270


Core Radius = 54.8cm
Core Height = 100cm
Graphite Reflector Thickness = 100cm Core Volume = 0.915m3












TABLE 4


Neutron Multiplication Factors for the Large GraphiteReflected Engines at the TDC Position

Computational 2-group 2-group 123-group 123-group
Engine Scheme NUCPISTN CORA XSDRN XSDRN
Ief * **f k **k *
'eff. keff. eff. eff.


G-l 1.179 1.221

G-2 1.005 1.052 0.934 0.934

G-3 1.025 1.067 0.954 0.954

G-4 1.032 1.080 0.962 0.962

G-5 1.037 1.086 0.969 0.969

*no fast interactions in core
**viith fast interactions in core
***vjith fast interactions in core; S4 quadrature
****with fast interactions in core; S6 quadrature









replaced by a two-region spherical configuration which possessed an equivalent volume core and an equivalent thickness reflector. The CORA and XSDRN calculations were restricted to two regions in order to be compatible with NUCPISTN which can handle only two-region systems. The rationale for this restriction as well as justifications for using "equivalent" spherical systems will be elaborated on in a later section in this chapter and also in Appendix A. More detailed accounts of the procedures used in generating the PHROG and BRT-l constants will also be given in a later section in this chapter.

In examining the results of Table 4, it will be noted that the two-group NUCPISTN keffectives are all around 4.to 5% lower than the corresponding two-group CORA results. Some of this discrepancy is due to differences which exist between the thermal group constants generated by N UCPISTN and the BRT-l thermal group constants which were used in CORA. The major portion of the discrepancy however is due to the fact that CORA includes fast core interactions and NUCPISTN does not; hence, the higher keffectives for the CORA results.

The 123-group XSDR14 keffectives are all around 11 to 12% lower than the two-group CORA keffectives The thermal group cutoff in BRT-l is 0.683eV and full upscattering and downscattering below this energy are accounted for. The use of the BRT-l and P[IROG constants in CORA means that any










.upscattering to above 0.683eV is neglected. Th'e 123-group XSDRN calculation on the other hand allows for complete upscattering and downscattering and an examination of the XSDRN results reveals that there is some upscattering to above*0.683eV.

The two-group CORA keffectives are thus higher than the XSDRN keffectives for two reasons. First, the two-group analyses do not give proper emphasis to the non-thermal groups which are less productive than the thermal group. The two-group CORA calculations hence tend to overpredict keffective' Second, the two-group CORA results neglect the upsca-ttering which occurs to above 0.683eV and this also causes k effective to be overpredicted. Of the two effects, the former is the more significant and this will be more clearly illustrated in the next chapter.

It will be noted that all of the above graphitereflected engines at the TDC position are rather large. For any reasonable compression ratio, the resultant stroke would therefore also be large--too large in fact for serious con-.

sideration for the nuclear piston'engine. The component mechanical stresses for such an engine would be so great that the engine lifetime would indeed be short.










Graphite-Reflected Systems Compared with Systems
Using Other Moderating-Reflector Materials

Smaller sized engines with other moderatinq-reflector materials were therefore investigated. Neutron multiplication factors for some of these systems at the TDC position are tabulated in Table 5 for different computational schemes. The uranium-235, fluorine, and helium atom densities for these particular systems are the same as for engine G-5 in Table 3. The core radius has been reduced from 54.8 to 34.55cm, the core height at TDC from 100 to 64cm, and the
3 3
core volume at TDC from 0.915m to 0.240m

The NUCPISTN and CORA results in Table 5 are again for one-dimensional "equivalent" spheres. The EXTERMINATOR-II results are for two-dimensional cylinders. Both CORA and EXTERMINATOR-II make use of the PHROG and BRT-l fast and thermal group constants.

It will be noted that for these smaller sized engines, the graphite-reflected configurations are far-subcritical. These same engines were analyzed by Kylstra et al. and their results indicated that these systems would be supercritical. Their thermal group constants however were in considerable error and they also neglected thermal absorptions in the reflector (see Chapter III). It should be pointed out that the graphite-reflected systems in Table 5 could have their 'keffectives increased somewhat by increasing the uranium loading. However, these small systems are already rather











TABLE 5

Neutron Multiplication Factors for Small Engines with Various 'oderating-Reflector ilaterials at the TDC Position

iloderating- Reflector Two-group Two-group Four-group Two-group Four-group Two-group
Reflector Physical. NUCPISTN* CORA* CORA* CORA** CORA** EXTERMI,Material Temperature k k NATOR**
(OK)eff eff keff effeff


D20
D 2O0


290

290 370
570

290 570 970
1270


0.965

1.183 1.148
1.087

0.845 0.817 0. 794 0.776


0.963 1.183 1.151 1.092

.846


0.978
1.132 1.102 1.044 0.796


0.970 1.200 1.180
1.120 0.864


1.012 1.178 1.154 1.097

0.838


1.210 1.201
1.143


Core radius = 34.55cm Core height = 64cm Reflector thickness = 100cm Core volume = 0.24m3


Core atom densities are the same as for Engine G-5 in Table 3
*No fast interactions in core
**With fast interactions in core









black to thermal neutrons so that large increases in uranium loading yield but small increases in the system
k
effective.

For essentially infinitely thick reflector regions (from a neutronics standpoint), the D20-reflected systems possess the highest neutron multiplication factor for a given geometry and core loading. Most of the engines which are examined in this work have heavy water-reflected cores as their basic component. Some composite material moderatingreflector studies have been done, and it is anticipated that future piston engine designs will most probably make use of such composite reflectors.

In returning to Table .5, it will be noted that there is very little disagreement between the two-group NUCPISTN and the two-group CORA results in which fast core interactions have been neglected. This table clearly illustrates the statement made regarding the results in Table 4. That is, that the differences between the thermal group NUCPISTN constants and the thermal group BRT-l constants used in CORA are not of great significance. The inclusion or omission of fast core interactions is a much more significant factor. The inclusion of fast core interactions for the twogroup computations in Table 5 leads to keffectives which are 2 to 3% higher than for the corresponding cases which neglect these interactions. For the four-group computations, the inclusion or omission of fast core interactions









leads to differences in k effective which are as high

as 5%.

In comparing the two-dimensional, cylindrical geometry EXTERMINATOR-II results with the one-dimensional, spherical geometry CORA results, one observes that the latter possess ke which are 1 to 2% lower than the former. The
kffeti yes
difference is due to the fact that the "equivalent" spheres experience less fast leakage to the moderating-reflector region (where neutrons must undergo slowing down before they can efficiently produce fissions in the core) than do the actual cylinders. Hence, the CORA keffectives are consistently lower than the EXTERMINATOR-II results.

When comparing the four-group results for the D20reflected systems with the corresponding two-group results, the latter have keffectives which are always higher than the corresponding four-group results. This is because the twogroup computations do not give proper emphasis to the nonthermal interactions which are less productive than the thermal reactions. The two-group problems hence tend to overpredict the neutron multiplication factors for these systems.

In contrast, the two-group keffectives for the Bereflected systems tend to be lower than the four-group results. The reason is that the two-group results do not properly account for the (n, 2n) production which occurs in










the beryllium at high energies. A listing of the group structure utilized in the four-group calculations is to be found in Appendix F.

Presented in Table 6 are physical temperatures for

various reflector materials and their corresponding moderator neutron temperatures. The neutron temperatures were obtained from BRT-l calculations in which the reflector thickness was 100cm. The core composition was observed to have very little effect on these neutron temperatures and the results presented are in fact for engines with the geometry of Table 5 and with the core composition of engine G-5 of Table 3. Also presented are neutron lifetime results obtained from two-group, two-dimensional EXTERMINATOR-Il perturbation calculations. The reactor geometry was again that of Table 5 and the core composition that of engine G-5 in Table 3. The large size of the neutron lifetime in the moderating-reflector region relative to the core region lifetime is to be noted.



Moderating-Reflector Power

In speaking of moderator characteristics, one frequently encounters the terms "slowing down power" and"moderating ratio." The slowing down power is defined as s where & is the average logarithmic energy decrement per collision or the average increase in lethargy per collision. If one











TABLE 6


Moderator Neutron Temperatures and Neutron Lifetimes for Small Engines with
Various Moderating-Reflector Materials at the TDC Position


Moderator Moderator Neutron Neutron Total Neutron
Physical Neutron Lifetime Lifetime Lifetime
Moderator Temperature Temperature In Core in Reflector in System
Ma teri al ( - K) (0 K) (msec) (msec) (msec)


D 0 D 20 D 20


D2O D 20 D20


290

290 320
370 420 470 490 520 570

290 570 970 1270


396

384 424 490 558 624 660 691
750

439 790 1295 1560


0.210 0.207




0.199


2.028

1 .798 1 .488


2.238 2. 005 1.687


Engine geometry same as for engines of Table 5 Core atom densities same as for Engine G-5 in Table 3 Moderator neutron temperatures from BRT-l calculations Neutron lifetimes from two-group, two-dimensional EXTERMINATOR-II calculations in which fast core interactions are included.










considers the moderating materials of Be, BeO, C, D20, and H20 and orders them from best to worst according to slowing down power, the order is H20, D20, Be, BeO and then C. The moderating ratio is defined as ( s/Za )and the ordering of the above moderators from best to worst according to the moderating ratio is D20, C, BeO, Be, and then H20.

Various reports on externally moderated, gaseous core reactors have attempted to order moderating-reflector materials according to neutronic efficiency by using various lumpings of reactor physics parameters. Some have used the moderating ratio, others have used the Fermi age or the square root of the Fermi age in combination with the thermal neutron mean free path. While some of these groupings give the correct ordering for two or three of these materials, none give the correct ordering for all five materials.

It is argued that a more reasonable grouping of

parameters is (T m tDt)I or (fd/T E t Dit) which has been
m a ni M m a m
m m
given the name "moderating-reflector power." If one considers a two-group, externally moderated gaseous core reactor, it is desired that Dt and E a for the moderating in a
m f
reflector be small. It is also desired that Df for the
m
f
moderator be small and that E , the removal cross section
m
from the fast group, be large (assuming that most of the removal cross section is downscattering to the thermal group). Since the Fermi age, Tm' for this region can be defined as









f f t t
(Df/E ) it is hence desirable that the combination T D E
rm r
II mt I l m in a m

be small or that (TmDm) be large. The term f is the

fraction of the removal cross section which is downscattering. Since it is desirable that a large fraction of the removal cross section from the fast group be downscattering rather than absorption the term (fo/TmDt m i.e., the
m
"moderating-reflector power" should be larger, the better moderating-reflector material. This is a rather simple grouping of constants. When one is considering the desirability of a moderating-reflector material from a neutronics standpoint, this combination takes into account most of the important effects. It does not, of course, account for all effects. For example, (n-2-) production in beryllium is
n
ignored by this grouping. Table 7 lists moderating-reflector powers for the above five materials at 290'K. For externally moderated, gaseous core reactors possessing an essentially infinite (from a neutronics standpoint) moderating-reflector, this simple combination of constants properly orders the above materials.


Some Composite Moderating-Reflector Studies

Appearing in Table 8 are some results for a system

possessing a moderating-reflector region of varying composition. The core composition and geometry is fixed and the total moderating-reflector thickness is also fixed at 70cm. The 020 and Be thicknesses are allowed to vary from 70 to 0












TABLE 7

Moderating-Reflector Power for Some
Moderating-Reflector Materials at 2900K

Moderating-Reflector Power
Mater al d tt
z at Dt-1d _[ E t
m D) (fIIIm ma m
mIl


D 20


BeO


Be


C


H2 0


303.0 35.4


18.9


13.5 11 .7


301 .6 33.5 17.7


13.5 11 .6


Above results for an essentially infinite moderatingreflector region

t t
a and Dm obtained from BRT-'l calcuations


and fd obtained from PHROG calculations
III II









TABLE 8

Beryllium-D20 Composite Reflector Study at 2900K


Inner Reflector Region (Be) Thickness
(c111)


0
5

10

15 20 25 30 35 40 45 50 55 60 65 70


Outer Reflector Region (D 0) Thickn es
(cm)


70

65 60

55 50

45 40

35 30

25 20

15 10

5 0


Four-Group
CORA keff


1 .100 1 .046 1 .027 1 .017

1 .009 1 .002 0.997

0.994 0.003

0.993 0.993 0.992 0.991 0.990 0.989


Engine geometry same as for reflector thickness is 70cm

Core atom densities are the Table 3.


engines of Table 5 except


engines of Table 5 except rather than 100cm. same as for Engine G-5 in


'Moderating-reflector physical temperature = 290'K. Fast interactions in core included.










and from 0 to 70cm respectively. The beryllium region is next to the core, when present, and the moderator physical temperature is 290'K. The penalty in the neutron multiplication factor decrease for going from a pure D20 reflector to a pure Be reflector for this particular configuration is but around 11%.

The nuclear piston engine studies which will be presented in the remainder of this chapter and in the following chapter will have pure D20 moderating-reflectors in order to maintain two-region systems for the neutronic calculations. Future investigations will probably utilize a composite reflector in which the inner 10 or 20cm consist of either Be or BeO and the remaining 60 to 80cm consist of D20. The Be or BeO will allow for structural integrity and separate the liquid D20 from the gaseous core. The Be or BeO will probably be lined with nickel for low temperature-cores or with a niobium alloy (e.g., TZN) for high temperature engines to protect the inner moderating-reflector region from the corrosive UF6 gas.


Remarks Concerning the Algorithms Used in the NUCPISTN Code

The restriction of the steady-state calculations in NUCPISTN to two groups and two regions in which the fast core equation is replaced by a boundary condition allows one to obtain fairly short computer execution times. As is






58


explained in detail at the end of Appendix A, about five or


six thousand timesteps are required for equations for each piston cycle. Solut


solving the neutronlcs


ion of complete two-


group, two-region or three-region problems by standard


diffusion theory codes would involve IBM


70 computer


execution times of about 20 minutes for each piston cycle


anal


By using the simpler equations developed in Appendix


A, the 370 computer execution time for each piston cycl


analysis is reduced to around 0.3 of a minute.


As has been


demonstrated., replacement of the fast core equation by a boundary condition and neglecting the fast interactions in the core leads to errors in effective which are of the


order of


The restriction at this point to two region


is justified


since this work is an attempt,to gain insight into the basic power producing and operational characteristics of the nuclear


piston engine concept.


Although some of the higher order-


neutronic calculations which were performed could easily have
been extended to three or more regions, they were -generally restricted to two regions so that the results could be compared directly with the two-region NUCPISTN results.


The thermal group core constants used in the NUCPISTN code










and BRT-l) and treated as constant during the piston cycle. Although the moderating-reflector constants depend to an extent on the core composition, treating them as constant during the cycle is a very good approximation if the reflector dimensions do not change. The two-stroke engines discussed in Chapter III and in this chapter however utilize step-reflector additions and removals to obtain the desired subcritical to supercritical to subcritical behavior. For these engines the moderating-reflector group constants of the thick reflector system are input into NUCPISTH. During those portions of the cycle when the thin reflector is applied, the thick moderating-reflector group constants are generally in error by about 7 or 8%,o as compared to the actual thin moderating-reflector group constants. Theerror in the system keffective however is usually only 1 or 2%. It is to be noted that these portions of the cycle are relatively unimportant anyway since the system is far-subcritical. These errors will not therefore noticeably affect the engine's behavior and the use of the thick moderating-reflector qrout constants over the entire cycle is, even for these systems, a very qood approximation.

As mentioned in Chapter III, the step function in the reflector thickness is easy to attain analytically. Practically, it can only be approximated. One method of simulating this behavior would involve using a sheath whose motion would be synchronized to alternately expose and










shield the bulk of the reflector region from the core region. The sheath could be-made of a mild neutron absorber material such as stainless steel. Another method, not involving any moving components, would depend on the moderating-reflector region being constructed so that its thickness varies in the proper manner along the length of the cylinder. Another approach would involve the use of. a few poison or control rods rather than of changes in the reflector thickness. The rods would be inserted into the moderating-reflector region and their motion could be timed so as to attain the desired subcritical to supercritical to subcritical behavior during the piston cycle. The NUCPISTN code can accommodatL either poison additions and removals in the reflector region or step additions and removals in the reflector thickness.

It is recognized that the use of moving sheaths or control rods complicates the piston engine design and, in fact, "gas generator" engines are covered in the next chapter which require no variations in reflector thickness or control rod motions during their normal cycling operation.

The UF6 gas specific heat formula utilized by Kylstra

et al. was


-3 42
C = 32.43 + (7.936xlo- )T- (32.068xlO )/T2 (cal/mole�K) (1)
p









where T is in degrees Kelvin [47, 48]. This formula however is valid only at temperatures around 4000K and is rather inaccurate at elevated temperatures. A formula which better fits the existing UF6 data [41, 43] is given by


C = 37.43 + (0.15x163 )T - (.6450xlO6)T2 (cal/mole�K) (2)



where T is again in degrees Kelvin. This formula agrees quite well with the compiled UF6 data over the temperature range from 400 to 2400'K. The NUCPISTN code allows the user the option of selecting either one of the above formulas.

As discussed in Chapter III, both helium and UF6 are treated as ideal gases and comments on this approximation will be made in Chapter VII. A discussion of the numerical methods used for solving the NUCPISTN energetics and neutronics equations, of the procedures used for timestep selection for the energetics equations, of the effects of fuel enrichment variations on engine performance, and detailed comparisons of the influence of delayed and photoneutrons on engine behavior will all be presented in the next chapter. The remainder of this chapter will focus on the operating characteristics of the simple two-stroke, D20-reflected nuclear piston engine and on the qualities of power generating systems which have these engines as their basic component.









Parametric Studies with NUCPISTN
for D20-Reflected Systems

Initial operating conditions for two such D20-reflected engines appear in Table 9. Engine #1 differs slightly from Engine #2 in that its initial gas pressure is 14.6 atmospheres rather than 14.5 Also, Engine #1 has the step-reflector applied at the l = 0.050 cycle fraction and removed at the C2 = 0.650 cycle fraction; Engine #2 has the step-reflector applied at the �I = 0.100 cycle fraction and removed at the �2 = 0.700 cycle fraction.

A series of pertinent results obtained from NUCPISTN calculations are shown in Figures 8 through 10. Figure 8 shows the neutron multiplication factor for an essentially infinite D20 reflector as a function of U235 atom density for various piston engine volumes. The reflector temperature is 290'K and the helium-to-U 23 mass ratio is zero, i.e., the core is 100% UF6 gas. For U235 densities greater
20 3
than around 10 atoms/cm , the core is so black to neutrons that additional uranium has no effect on the system multiplication factor. Figure 9 contains the same information as 235
Figure 8 except the helium-to-U mass ratio is 0.322. The helium at this concentration (the gas mixture helium mole fraction is 0.95) has no detectable effect on the neutron multiplication factor. For a given U235 density, the neutron multiplication factor is the same in Figure 9 as for the corresponding atom density in Figure 8. Figure 10 shows

PAGE 1

THE NUCLEAR PISTON ENGINE AND PULSED GASEOUS CORE REACTOR POWER SYSTEMS By EDWARD T. DUGAN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1976

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ACKNOWLEDGMENTS The aut.hor would like to express his appreciation to his graduate committee for their assistance during the course of this research. Special thanks are due to Dr~ N. J. Diaz, chairman of the author's supervisory committee for pro viding practical and theoretical guidance and patient en couragement throughout the course of this work. Thanks are also due to Dr. M. J. Ohanian whose endeav ors, along with those of Dr. Diaz, were responsible for secur ing most of the funds for the computer analysis phase of this. research .. The dedication, knowledge, and sources of informa tion which were provided by both of these individuals helped make this work possible. The author feels fortunate to have studied and worked with Dr. A. J. Mackel, now at Combustion Engineering. His excellent scientific knowledge of computer analysis and nu clear reactor physics was of great assistance during the initial phases of this work. The author recognizes that much of his own knowledge in these fields was assimulated during his years of association with Dr. Mackel. Also to be recognized is Dr. H. D. Campbell whose criticisms and sug gestions were a stimulus for some of the calculations which appear in this work. i i

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The authorts studies at the University of Florida have been supported, in part, by a United States Atomic Energy Commission Special Fellowship and also by a one-year Fellowship from the University of Florida and this support is gratefully ackhowledged . . A large portion of the funds for the computer analysi~ were furnished by the Unive~sity of Florida Crimputing Center I thtough t~e College of Engineeri~g. This help, though at times meager and difficult to o~tain, is also acknowledged. Thanks are also due to . those fellow students whose comments, criticisms, and suggestions have also been a source of inspiration~ Finally, thanks ar~ du~ to the author's parents for their patient understanding and support which has been a constant source of _ encouragement. i i i

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PREFACE The fundamental objective of this work has b~en to gain an insight into the basi~ 'power producing and opera tional characteristics of the nuclear piston engine, a concept which.involves a type of ' 1 pulsed, quasi-steady-state gaseouscore nuclear reactor. The stJdies have consisted primarily of neutronic and energetic analyses supplement~d by some reason~bly detailed thermodynamic studies and also by some heat transfer arid fluid mechanics calculations. This work is not to be construed as beinq a complete expos~ of the nuclear piston engine's complex neutronic and energetic behavior. Nor are the proposed power producing systems to be interpreted as being the ultimate or optimum conditions or configurations. This dissertation is rather a beginning or a foundation for future pulsed, gaseous core reactor studies. Despite being hampered by a rather limited availa bility of computer funds, it is believed that the models and results presented in this wo~k are indeed indic~tive of the type of ~erformanci which can be anticipated from nuclear piston engine power generating systems and that iv

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they furm valuable tools and guidelines for future research on pulsed, gaseous core reactor systems. Indeed, part of this work has been the basis for major research proposals which have been submitted by the University of Florida's Department of Nuclear Engineering for the purpose of carry ing on more extensive investigations of the nuclear piston engine concept. It is recrigniz~d that a co~plete system. analysis and optimization will not only be difficult but also expensive. A demonstration :of technical feasibility ~ill require the cooperation of not only other departments from within the university but also contributions from other institutions and agencies. A few remarks should be made concerning the organi zation of this 'dissertation.' Fi"rst, most of the equations and derivations used •in the neutronics and' energetics analy sis of the iiLiclear piston engine have been ordered or , :,, grouped into appendices. Very few equations appear in the text of the dissertation itself. Refer~nces are made from the text to the-appropriate equation(s) and corresponding appendix where necessary. It is felt that this approach ', renders a more convenient and ordered presentation and facilitates reading of the text. The reseaich conducted on the nuclear piston engine has consisted of two reasonably distinct segments. The first phase focused on simple two-stroke (compression and V

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power stro ke) engines. Results from these studies are presented in Chapter IV. '. ' The line of reasoni~g was to examine these simpler engines first, in some detail, before pro ceeding to th~ more complex four-stroke systems; Later, after it . became apparent from the two-stroke engine stu~ies that the nuclear piston engine concept was indeed a promis~ ing venture, work was begun on the more intricate four stroke configurations. Results ffom this phase ~f the resear~h are presented . in Chapter V. vi

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TABLE OF CONTENTS Paqe ACKNOWLEDGMENTS i' i PREFACE . . iv LIST OF TABLES. Xi i LIST OF FI GU RES XV i i i LIST OF SYMBOLS AND ABBREVIATIONS xxvii ABSTRACT. . . . . . . . . . . xxxvi c'HAPTER I I I I I I INTRODUCTION ... ! Description of Engine Operation . . . . l Applications and Highlights of the Nuclear Piston Engine Concept . . . . . . . . . 3 Dissertation Organization . . . . . . . 6 PREVIOUS STUDIES ON GASEOUS CORE CONCEPTS. 8 Gaseous Cores Analytical Studies. . . . 8 Neutronic Calculations for Gaseous Core Nuclear Rockets . . . . . . . . . . . 12 Comparison of Theoretical Predictions with Experimental Results. . . . . . . 14 Comments. . . . . . . . . . . . . 17 PREVIOUS GASEOUS CORE NUCLEAR PISTON ENGINE STUDIES AT THE UNIVERSITY OF FLORIDA Introduction .... Neutronic Model .. Energy Model .... Analytical Results. vii l 9 l 9 23 25 28

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CHAPTER IV TABLE OF CONTENTS (continue~) RESULTS FROM TWO-STROKE ENGINE STUDIES .. In tr rid u ct ion. . . . . . . . . . . . . . Comparison of Initial Results with Pre vious Nuclear Piston Engine Analyses .. Graphite-Reflected Systems ...... . Graphite-Reflected Systems Compared with Systems U~inq Other Moderating~Reflector Materials ............ . Moderating-Reflector Power ..... .'. Some Composite Moderatinq-Refl~ctor Studies ...... , . ... ; ... . Remarks Concerning the Algorithms Used in the NUCP I STN Co de . . . . . . . . . . . . Parametric Studies with NUCPISTN for Reflected Systems ........... . Effects of Comptession Ratio and Clearance Volume Variations ..... Effects of Initial Pre~s~re and Initial Temperature Variations ....... . Effects of Engine Speed and Neutron Lifetime Variations ........ . Effects of Variations in the Initial and Step-Reflector Thicknesses .... Effects of Variations in the Cycle Fraction Position for Steo-Reflector Addition and Step-R~flect6r Removal.• Effect~ of Variations in the Heliumto-u235 Mass Ratio ......... . Effects of Variations ih the Neutron Source Strenqth .......... . Performance Anal;sis of Two D 2 0-Reflected Pis to . n Engines. . . . . . . . . . . . . . NUCPISTN Results Compared with Higher Order Steady-State Neutronic Calculations ... Some . Remarks Regarding the . o 2 o Temperatu res , . . . . . . . . . . . . . . . . . . Exha~st Gas Temperature Calculations for thi Two-Stroke Engines ......... i Mass Flow Rates for the Two-Stroke Enqines~ Thermodynamic Studies for Three Nuclear ri~ton Engine Power Generatinq Systems. Nuclear Piston~Gas Turbine-Steam Turbine System .......... . Piston-Steam Turbine System ... . ~iston-Cascaded Gas Turbine System, Page 36 . 36 40 42 .4 7 51 . 54 57 62 68 73 78 83 86 93 93 98 1 07 11 6 . 117 120 121 122 123 . l 2 3

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CHAPTER IV (c.ont.) V TABLE OF CONTENTS (continued) Prelimi.nary Heat. Exchanger Analysis .. Comparison of the Nuclear Piston Enqine Power Generating Systems. . . ... Timestep Size Selection for the Neutron Kinetics Equations. . . . . . . . . . . Delayed Neutron Effects .. ....... . Engine Startup: Approach to Equilibrium in the Presence of Delayed Neutrons ... Nuclear Piston Engine Blanket Studies Neutron Lifetime Results ...... . Summary ............... . RESULTS FROM FOUR-STROKE ENGINE STUDIES. l 3 2 1 34 142 147 148 1 64 . 177 184 192 Introduction. . . . . . . . . . . . . 192 Engine Startup: Approach to E~uilibrium in the Presence of Delayed and Photoneutrons . 197 An Examination of Reactor Physics Parameters as They Vary During the Pisto~ Cycle .... 214 Flux Shape Changes During the Piston Cycle. 223 Fuel and D20 Moderating-Reflector Temperature Coefficients of Reactivity . . . . . 254 D20 Moderating-Reflector Density or Void Coefficient of Reactivity. . . . . . . . 262 11 Cy c 1 e Fr act i on s II for the Eng i n es of Thi s Chapter . . . . . . . . . . . . . . . . 264 Effect of Uranium Enrichment on Engine Performance . . . . . . . . . . . . . . . 265 Effects of Delayed and Photoneutrons on Engine Performance. . . . . . . . . . . . 26d Timestep Size Selection for the Energetics Equations . . . . . . . . . . . . . . . . . 273 Effects of C Formula Selection and of Neutron Kine~ics Equations Numerical Techniques on Engine Behavior ......... 276 Blanket Studies for Four-Stroke Engines .. 278 Material Densities and Group Constants for Nuclear Piston Engine #10 ......... 281 Neutron Lif~times, Generation Times, Effective B 1 s, keffectives• and Inhomogeneous Source Weighting Functions for Engine #10 from Different Computational Schemes. . . 286 A Further Comparison of keffs for Engine #10 from Various Computational Schemes. 295 NUCPISTN Cycle Results for Engine #10 . 299 Power Transients for Engine #10 Induced by Loop Circulation Time Variations. . . . 306 ix

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CHAPTER V (cont.) VI TABLE OF CONTENTS (continued) Thermodynamic Studies for Nuclear Piston Engine Power Generating Systems Utilizing the Engine #10 Configuration. . . . . . . 311 11 Gas Gener at or 11 Nu cl ear Pi st on Eng i n es . . 31 9 NUCPISTN Cycle Results for a "Gas Generator" Engine. . . . . . . . . . . . . . . . . . 325 Thermodyriamic Studies for Nuclear Piston Engine Power Generating Systems Utilizinq "Gas Generator" Engines . . . . 332 Summary . . . . . . . . . . 343 RELATED RESEARCH AND DEVELOPMENTS. 355 Introduction. . . . . . . . 355 Related Research and Developments at the University of Florida . . . . . . 356 Other Related Research and Developments in Prag res s. . . . . . . . . . . 3 56 VII CONCLUSIONS; REFINEMENTS AND AREAS FOR FURTHER RESEARCH . . . . 164 Introduction. 364 Applications. . . 368 Analytical Model for Piston Neutronics and E ne rg et i cs. . . . . . . . . . . . . 3 7 3 Steady-State Neutronic Analysis 373 Moderating-Reflector Studies ...... 375 Fuel Studies. . . . . . . . . . . . . . 37 6 Neutron Cross Section Libraries .... 377 Moderating-Reflector and Fuel Temperature Coefficients of Reactivity . 378 Neutron Kinetics Calculations ..... 379 Neutronic Coupling B~tween Piston Engine Cores in an Engine Block. . . . . . 383 Equation of State for the HeUF6 Gas 385 Fluid Flow in the Piston Engine . . 385 Temperature Distribution and Piston Engine Heat Transfer Studies. . . . 386 He-to-U Mass Ratio Studies. . . . . 387 Step-Reflector Addition and Removal 388 Parametric Studies. . . . . . . . . 389 Moderating-Reflector Density or Void Coefficients of Reactivity. . . . . . 390 Blanket Studie~ and Breeding Prospects. 391 Comments. . . . . . . . . . . . . . . . 391 X

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CHAPTER VII (cont.) TABLE OF CONTENTS (continued} Analytical Models for Systems External to the PistonEngine . . . . . . . . . . . . 394 Thermodynamic Cycles for the Turbines . 394 Turbine Loop Energetics, Heat Transfer, and Fluid Mechanics Studies . . . 395 HeUF5-to-He Exchanger Studies ..... 396 Comments ........... . . . . . 397 Economic Model for the Nuclear Piston Engine Power Generating System ........... 397 Fixed Charges (Capital and Cost Related Charges). . . . . . . . 397 Fuel Cycle Costs. . . . . . . . 398 Power Production Costs. . . . . 399 Comments. . . . . . .. . . . 400 Safety Analysis and Methods of Control. 401 APPENDICES A TWO-GROUP, TWO~REGION, ONE-DIMENSIONAL DIFFUSION THEORY EQUATIONS USED IN THE NUCPISTN CODE WHEN PHOTONEUTRONS ARE IGNORED 405 B TWO-GROUP; TWO-REGION, ONE-DIMENSIONAL DIFFUSION THEORY EQUATIONS USED IN THE NUCPISTN CODE WHEN PHOTONEUTRONS ARE INCLUDED. 422 C GENE.RAL POINT REACTOR KINETICS EQUATIONS . . 434 D THE POINT REACTOR KINETICS EQUATIONS USED IN THE NUCPISTN CODE. . . . . . . . . . . . 456 E THE ENERGETICS EQUATIONS USED IN THE NUCPISTN CODE . . . . . . . . . . . . . . . . . . . . 474 F GROUP STRUCTURES AND VARIOUS REACTOR PHYSICS CONSTANTS USED IN THE NUCLEAR PISTON ENGINE COMPUTATIONS . . . . . . . . . . . . . 487 G LISTING Of THE TASKS PERFORMED BY THE NUCPISTN SUBROUTINES AND A FLOW DIAGRAM FOR THE NUC PI STN CODE. . . . . . . . . . . . . . 492 Listing of Tasks Performed by the NUCPISTN Subroutines . . . . . . . . 492 NUCPISTN Code Flow Diagram. 495 LIST OF REFERENCES. 501 BIOGRAPHICAL SKETCH 508 xi

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TABLE l 2 3 4 5 6 7 8 9 l O l l LIST OF TABLES Values of Primary Independent Parameters for Graphite-Reflected Piston Engines Anal.yzed by Kylstra et al ................ . Comparison of a Typical Graphite-Reflected, UF 6 Piston Engine with the Nordberg Diesel .... Atom Densities and Temperatures for Large Graphite-Reflected Engines at the TDC Position . . . . . . ... .• . Neutron Multiplication Factors for the Large Graphite-Reflected Engines at the TDC Position .............. . Neutron Multiplication Factors for Small Engines with Various Moderating-Reflector Materials at the TDC Position ...... . Moderator Neutron Temperatures and Neutron Lifetimes for Small Engines with Various Moderating-Reflector Materials at the TDC Position ................ . Moderating-Reflector Power for Some ModeratingReflector Materials at 290K ...... . Beryllium-D 2 o Composite Reflector Study at 290K .................. . Operating Characteristics for Engines #1 and #2 . . . . . . . . . . . . . . . . . . . . Cycle Results from NUCPISTN for Engines #1 and # 2 . . . . . . . . . . . . . . . . . . . Summary of Thermodynamic Results for the Piston Gas Turbine-Steam Turbine System Which Uses Piston Engine #2 . . . . . . . . . . . . . . . xii 27 34 43 44 48 52 55 56 63 99 l 2 6

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LIST OF TABLES (continued) TABLE Page 12 Summary of Thermodynamic .Results for the Piston Steam Turbine System Which Uses Piston Engine .#1 ..... .................. 129 13 Summary of Thermodynamic Results for the Piston Cascaded Gas Turbine System Which Uses Piston Engine # 2.. . . . . . . . . . . . . . . . . . . . . l 3 3 14 A Comparison of Thermodynamic Results for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engines #1 and #2 ... 135 15 Reactor Volume per Unit Power for Three Opera tional Nuclear Reactor Power Systems and for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engines #1 and #2 ... 138 16 Heat Rate and Fuel Cost Estimates for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engines#l and #2. . 141 17 Operating Characteristics for Engine #3. 143 18 Cycle Results from NUCPISTN for Engine #3. 144 19 Effects of Neutron Kinetics Equations Timestep Size Variation on Engine #3 Performance. 146 20 Operating Characteristics for Engines #4 and #5. 150 21 Startup Procedure for Engine #4 in the Presence of Delayed Neutrons. . . . . . .... 151 22 Startup Procedure for Engine #5 in the Presence of Delayed Neutrons. . . . . 152 23 Equilibrium Cycle Results from NUCPISTN for Engines #4 and #5... . . . . . . . . 162 24 Operating Characteristics for Engine #6. 165 25 Equilibrium Cycle Results from NUCPISTN for Engine #6. . . . . . . . . . . . . . . . . . 1 66 X i i i

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LIST OF TABLES (continued) TABLE 26 Equivalent Cylindrical Cell Data and Pure Blanket Material Densiti~s . . . . . . . . 168 27 Homogenized Densities for a Blanket Using a l 5 M / vJ L a t t i c e . . . . . . 1 7 l 28 Burnup Calculations for a System Using Engine #6, an 80cm D20 Reflector Region, and a B l a n k e t R e g i o n w i t h a 1. 5 M / vi L a t t i c e . . . . . .1 7 2 29 Burnup Calculations for a System Usinq Engine #6, a 70cm D20 Reflector Region, and a Blanket Region with a l.5M/W Lattice . . . . . 173 30 Burnup Calculations for a System Using Engine #6, an 80cm D20 Reflector Region, and a Blanket Region with a 3.0M/W Lattice . . . . . 175 31 Burnup Calculations for a System Usinq Engine #6, a 70cm D20 Reflector Region, and a Blanket Region with a 3.0M/W Lattice . . . . 176 32 Operating Characteristics for Engine #7. . . 178 33 Equilibrium Cycle Results from NUCPISTN .for Engine #7.................. 179 34 Neutron Multiplication Factors, Neutron Life times and Neutron Generation Times at Various . Cycle Positions for Engine #7 as Obtained from CORA and NUCPISTN. . . . . . . 181 35 Operating Conditions for Engine #8 198 36 Startup Procedure for Engine #8 in the Presence of Delayed and Photoneutrons . . . . . 201 37 Equilibrium Cycle Results from NUCPISTN for Engine #8. . . . . . . . . . . . . . . . . . 213 38 Core Radii and Neutron Multiplication Factors as Obtained from CORA and NUCPISTN for Engine #8 at Various Cycle Positions. . . . . . . . . 215 39. Flux Ratios as Obtained from CORA and NUCPISTN for Engine #8 at Various Cycle Positions . . . 217 xiv

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TABLE 40 LIST OF TABLES (continued) Six Factof formula Parameters as Obtained from CORA for Engine #8 at Various Cycle Positions .. 219 41 Neutron Lifetimes and Generation Times as Ob tained from CORA for Engine #8 at Selected Cycle Positions .................... 221 42 U235 Enrichment Effects on the Neutron Multipli cation Factor for Engine #8 at the TDC Position. 221 43 D20 Moderating-Reflector Temperature Coefficients of Reactivity Using the Engine #8 Configuration at the TDC Position ............... 256 44 Fuel Temperature Coefficients of Reactivity for 100% Enriched UF5 Using the Engine #8 Configuration at the TDC Position ............ 257 45 Fuel Temperature Coefficient of Reactivity for 93% Enriched UF5 Using the Engine #8 Configuration at the TDC Position ............ 258 46 Fuel Temperature Coefficient of Reactivity for 80% Enriched UF5 Using the Engine #8 Configuration at the TDC Position . . . . . . . . 259 47. 020 Moderating-Reflector Density or Void Coefficient of Reactivity Using the Engine #8 263 Configuration at the TDC Position .. 48 Operating Characteristics for Engine #9. 266 49 Equilibrium Cycle Results from NUCPISTN for Engine #9.......... . . . . . 267 50 Effect of Uranium Enrichment on Required Fuel Loading for Engine #9 .. ........... 269 51 Compensating for the Absence of Delayed and/or Photoneutrons by Increased Fuel Loading for Eng i.ne # 9. . . . . . . . . . . . . . . . . . . 27 l 52 Engine #9 Behavior in the Absence of Delayed and/or Photoneutrons When There Is Nn Compensation by Increased Fuel Loading ......... 272 xv

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LIST OF TABLES (continued) TABLE 53 Effects of Energetics Equations Timestep Size Variation on Engine #9 Performance . . . . . . 274 54 Effects of Specific Heat Formula and of Neutron Kinetics Equations Numerical Techniques on Engine #9 Performance. . . . . . . . . . . 277 55 Burnup Calculations for a'system Using an Engine #9-Like Configuration,a 70cm 020 Reflector Region, and a Blanket Region with a 3.0M/W Lattice . . . . . . . . . . . 280 56 Operating Characteristics for Engine #10 282 57 Equilibrium Cycle Results from NUCPISTN for Engine #10 . . . . . . . . . . . . . . . . . 283 58 Material Densities and Core Thermal Group Con stants from NUCPISTN for Engine #10 at the TDC Position . . . . . . . . . . . . . . . . . 285 59 Fast and Thermal Collapsed Group Constants from PHROG and BRT-1 for Engine #10 at the TDC Position . . . . . . . . . . . . . . . . . 287 60 Reactor Physics Parameters for Engine #10 at the TDC Position from Various Computational Schemes. 288 61 Reactor Physics Parameters for Engine #10 at the 0.056 Cycle Fraction from Various Computational Sc hen1e s. . . . . . . . . . . . . . . . . . . . 290 62 Neutron Multiplication Factors for Erigine #10 from Various Computational Schemes for the TDC and 0.056 Cycle Fraction Positions 296 63 Summary of Ther~odynamic Results for the Piston Gas Turbine-Steam Turbine System Which Uses the Modified #10 Piston Engine ........... 313 64 Summary of Thermodynamic Results for the Piston Steam Turbine System Which Uses Piston Engine #10 ... ................... 314 xvi

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LIST OF TABLES (continued) TABLE 65 Summary of Thermodynamic Results for the Piston Cascaded Gas Turbine System Which Uses the Modified #10 Piston Engine . . . . . . . . . . 316 66 A Comparison of Thermodynamic ~esults for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engine #10 and the Modified #10 Piston Engine . . . . . 318 67 Operating Characteristics for Engine #11 . . 321 68 Equilibrium Cycle Results from NUCPISTN for Engine # l 1 . . . . . . . . . . . . . 323 69 Some Operating Characteristics and NUCPISTN Equilibrium Cycle Results for 11 Gas Generator" Piston Engines . . . . . . . . . 324 70 Summary of Thermodynamic Results for the Piston Gas Turbine-Steam Turbine System Which Uses the Modified #11 Piston Engine ........... 333 71 Summary of Thermodynamic Results for the Piston Steam Turbine System Which Uses Piston Engine #11 ....................... 335 72 Summary of Thermodynamic Results for the Piston Cascaded Gas Turbine System Which Uses the Modified #11 Piston Engine ........... 336 XV ii

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LIST OF FIGURES FIGURE 1 Simple Schematic of a UF 6 Nuclear Piston Engine. . . . . . . . . . . . . . 20 2 Illustration of Step-Reflector Addition and Removal . 22 3 UF 6 Phase Diagram 24 4 Neutron Multiplication Factor Versus UF5 Partial Pressure for an Infinite Graphite Reflector . 29 5 Gas Pressure and Temoerature as a Function of Percent Travel Through the Piston Cycle for a Graphite-Reflected Engine . . . . . . . . . . 30 6 Avera~e Core Thermal .Neutron Flux and Neutron Multiplication Factor as a Function of Percent Travel Through the Piston Cycle for a Graphite Reflected Engine. . .. . . . . . . . . . . . . . 31 7 UF5 Nuclear Piston Engine Performance for the Case of Graphite Stec-Reflector Addition at the 0 . l Cy c 1 e F r a c t i o n . . . . . . . . . . . . . . . 3 3 8 Neutron Multiplication Factor Versus u 235 Atom Density for Systems Which Have No Helium Gas Present in the Core and a 100cm Thick o 2 o Reflector at 290K. . . . . . . . . . . . 64 9 10 1 l Neutron Multiplication Factor Versus u 235 Density for Systems Which Have Helium Gas Present in the C~re and a 100cm Thick o 2 o Reflector at 290 K ... ........ . Neutron Mult1olication Factor Versus u 235 . . . Density for Systems Which Have Helium Gas Present irt the Core and a 100cm Thick o 2 o Reflector at 570K. . ...... . Atom Atom Mechanical Power and Pe~k Gas Pressure Versus Compression Ratio for a Two-Stroke Enqine ... XV iii 65 66 69

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LIST OF FIGURES (continued) FIGURE 12 Mechanical Efficiency and Peak Gas T~mperature Versus Compression Ratio for a Two-Stroke Engine. . . . . . . . . . . . . . . . . . . . 70 13 Mechanical Power and Peak Gas Pressure Versus Clearance Volume for a Two-Stroke Engine. . . . 71 14 Mechanical Efficiency and Peak Gas Temoerature Vers~s Clearance Volume for a Two-Stroke Engine 72 15 Mechanicil Power and Peak Gas Temperature Versus Initial Gas Pressure for a Two~Stroke Engine .. 74 16 Mechanical Efficiency and Peak Gas Temperature Versus Initial Gas Pressure for a Two-Stroke Engine ........ : ............ 75 17 Mechanical Power and P~ak Gas Pressure Versus Initial Gas Temperature for a Two~Stroke Engine . 76 18 Mechanical Efficiency and Peak Gas Temperature Versus Initial Gas T~mperature for a Two-Stroke Engine. . . . . . . . . . . . . . . . . . . . . 77 19 Mechanical Power and Peak Gas Pressure Versus Engine Speed for a Two~Stroke Engine. . . . . 79 20 Mechanical Efficiency and Peak Gas Temperature Versus Engine Speed for a Two-Stroke Engine . 80 21 Mechanical Power and Peak Gas Pressure Versus Neutron Lifetime for a Two-Stroke Engine. . . 81 22 Mechanic~l Efficiency and Peak Gas Temperature Versus Neutron Lifetime for a Two-Stroke Engine 82 23 Mechanical Power and Peak Gas Pressure Versus Initial o 2 o Reflector Thickness for a Two-Stroke Engine. . . . . . . . . . . . . . . . . . . . . 84 24 Mechanical Efficiency and Peak Gas Temperature Versus Initial D20 Reflector Thickness for a Two-Stroke Engine . . . . . . . . . . . . . . . 85 xix

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LIST OF FIGURES (continued) FIGURE 25 Mechanical Power anrl Peak Gas Pressure Versus Step-Reflector Thickness for a Two-Stroke Engine. . . . . . . . . . . . . . . . . . . . . 87 26 Mechanical Efficiency and Peak Gas Temperature Versus D20 Step-Reflector Thickness for a Two~ Stroke Engine . . . . . . . . . . . . . . . . 88 27 Mechanical Power and Peak Gas Pressure Versus the Cycle Fraction for Step-Reflector Addition for a Two-Stroke Engine . . . . . . . . . . . . 89 28 Mechanical Efficiency and Peak Gas Te~peratufe Versus the Cycle FractiQn for Step-Reflector Addition for a Two-Stroke Engine. : .... : 90 29 Mechanical Power and Peak Gas Pressure Versus th~ Cycle Fraction for Step-Reflector Removal for a Two-Stroke Engine ........... 91 30 Mechanical Efficiency and Peak Gas Te~perature Versus the Cycle Fraction for Step-Reflector Removal for a Two-Stroke Engine . . . . . . . 92 31 Mechani~~! Power and Peak Gas Press~re Versus He-to-U Mass Ratio for a Two-Stroke Enqine 94 32 Mechanical Effiiency and Peak Gas Temperature Versus He-to-U~ Mass Ratio for a Two-Stroke Engine. . . . . . . . . . . . . . . . . . 9 5 33 Mechanical ~ower and Peak Gas Pressure Versus Neutron Source Strength for a Two-Stroke Engine .. _ . . . . . . . . . . . . . . . . . . 96 34 Mechanical Efficiency and Peak Gas Temperature Versus Neutron Source Strength for a Two-Stroke Engine. . . . . . . . . . . . . . . . . . . 97 35 Reflector Thickness Versus Cycle Fraction for Engine #1 . . . . . . . . . . . . . 100 36 Gas Temperature Versus Cycle Fraction for Engine #1 . . . . . . . . . . 101 37 Ga~ Pressure Versus Cycle Fraction for Engine #1 : . . . 102 xx

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LIST OF FIGURES (continued) FIGURE 38 Average Core Thermal Neutron Flux Vers~s Cy~le Fraction for Engine #1. . . . . 103 39 Neutron Multiplication Factor Versus Cycle Fraction for Engine #1. . . . . . . . . . . 104 40 Neutron Multiplication Factor Versus D20 Reflector Thickness as Obtained from Two-Group NUCPISTN Calculations . . . . . . . . . . . 112 41 Neutron Multiplication Factor Versus 020 Reflector Thickness as Obtained from FourGroup CORA Calculations . . . . . . . . . . 113 42 Fast and Thermal Neutron Flux Versus Radius for a Core as Obtained from Two-Group CORA Calculations . . . . . . . . 114 43 Fast and Thermal Adjoint Neutro n Flux Versus Radius for a 020-Reflected Core as Obtained from Two-Group CORA Calc~lations. . . . . . 115 44 Piston-Gas Turbine-Steam Turbine Schematic for the Power System Which Uses Piston Engine #2................ 124 45 Steam and Gas Turbine Temperature-Entropy Diagrams for the Piston-Gas Turbine-Steam Turbine System Which Uses Piston Engine #2. 125 46 Piston-Steam Turbine Schematic for the System Which Uses Piston Engine #1 . . . . . . . . . 127 47 Steam Turbirie Temperature-Entropy Diagram for t e P i s t o n : S t e a m T r b i rl S f,~ t e m ~J h i c h U s e s Piston Engine #1 .... ..... 128 48 Piston-Cascaded Gas Turbine Schematic for the System Which Uses Piston Engine #2. . 130 49 Gas Turbine Temperature-Entropy Diagram for the Piston-Cascaded Gas Turbine System Which Uses Piston Engine #2 . . . . . . . . . . 131 50 Diagram of a D20~Reflected, 3-to-l Compression Ratio Nuclear Piston Engine at the TDC Position. . . . . . . . . . . . . . . . . 139 xxi

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LIST OF FIGURES (continued) FIGURE 51 Sketch of an 8-Cylinder Nuclear Piston Engine Block for 40-50 Mw(e) Power Generating Sys terns . . . . . . . . . . . . . . . . 140 52 Delayed Neutron Precursor Concentration Buildup During Startup for Engine #4 . . . . 153 53 Peak Gas Temperature and Mechanical Power put Behavior During Startup for Engine #4 54 Peak Gas Temperature and Mechanical Power put Behavior During Startup for Engine #4 (continued) .............. . 55 Peak Gas Temperature and Mechanical Power put Behavior During Startup for Engine #4 (continued) ............. . Out155 Out156 Out157 56 Delayed Neutron Precursor Concentration Buildup During Startup for Engine #5 . . . . . 158 57 Peak Gas Temperature and Mechanical Power Output Behavior During Startup for Engine #5 160 58 Peak Gas Temperature and Mechanical Power Out. put Behavior During Startup for Engine #5 (continued) ....... ......... 161 59 Typical Blanket Region Unit Cell Diagram. 167 60 Delayed Neutron and Photoneutron Precursor Concentration Buildup During Startup for Engine #8 . . . . . . . . . . . . . . . . . 204 . ' 61 Delayed Neutron and Photoneutron Precursor Concentration Buildup During Startup for Engine #8 (continued) . . . . . . . . . . 205 62 Peak Gas Temperature anrl Mechanical Power Output Behavior During Startup for Engine #8 207 63 Peak Gas Temperature and Mechanical Power Out put Behavior During Startup for Enoine #8 (continued) . . . . . . . . . . . . . . . 208 xx ii

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LIST OF FIGURES (continued) FIGURE 64 Peak Gas Temperature and Mechanical Power Out put Behavior During Startup for Engine #8 (continued) . . _. . . . . . . . . . . . . 2 09 65 Peak Gas Temperature and Mechanical Power Out put Behavior During Startup for Engine #8 (continued) . . . . . . . . . . . . . . . 210 66 Fast and Thermal Neutron. Flux Versus Radial Distance for Enqine #8 at Timestep Number 3 51 . . . . . . . . . . . . . . . . . . . . 2 2 5 67 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 351 When Inhomogeneous Photoneutron Sources Are Ignored . . . . . . . . . . . . . . . . 226 68 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 501 .. . . . . . . . . . . . . . . . . . . 227 69 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 701 . . . . . . . . . . . . . . . . . 228 70 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 901 . . . . . . . . . . . . . . . . . . . . 229 71 Fast'and Thermal Neutron ~lux Versus Radial Distance for Engine #8 at Timestep Number 901 When Inhomogeneous Photoneutron Sources Are Ignored . . . . . . . . . . . . . . . . 2 30 72 Fast and Therma1 Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 1551 ................... . 23 73 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 ~t Timestep Number 1801 ...... .............. 232 74 Fast and Thermal Neutron Flux Versus Radial Di~tance for Engine #8 at Timestep Number 2026. . . . . . . . . . . . . . . . . . . . 233 XX iii

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FIGURE 75 76 77 LIST OF FIGURES (continued) Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 2326. . . . . . . . . . . . . . . . . . . . Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 3226. . . . . . . . . . . . . . . . . . . . Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 3226 When Inhomogeneous Photoneutron Sourtes . Are I gn ored . . . . . . . . . . . . . . . . Paqe 234 236 237 78 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 3726 ...... . ... . ........ .. .. 238 79 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 3976 ......... .. . .......... 239 80 Fast and Thermal Neutron Flux Veisus Radial Distance for Engine #8 at Timestep Number 44 26. . . . . . . . . . . . . . . . . . . . . . 240 81 Fast and Thermal Adjoint Neutron Flux Versus ~adial Distance for Engine #8 at Ti . mest~p Number 351 .............. . ..... 241 82 . fast and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 501. . . . . . . . . . . . . . . . . . ; 242 83 Fa~t and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 701 ............ . . . . . . 243 84 Fast and Th~rmal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Nu1nber 901 . . . . . . . . . . . . . . . . . . 85 Fast and Thermal Adjoint Neutron Flux Versus 244 Radial Distance for Engine #8 ~t Timestep Number l551 .................. 245 86 Fast and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Nurnber 1801 . . . . . . . . . . . . . . . .. 246 XX i V

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LIST OF tIGURES ( _ continued) FIGURE 87 Fast and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 2026 ...... . . . . . . . . . . . . 247 88 Fast and Thermal Adjoint Neutron Flux Versus Radial . Distance for Engine #8 at Timestep Nu nt be r . 2 3 2 6 . . . . . . . . . . . . . . . . . . 2 4 8 89 Fast ahd Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 3 226 . . . . . . . . . . . . . . . . . 24 9 90 Fast and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 3726 ... . ... . . . . . . . . . . . 250 91 Fast and Thermal Adjoint Neutron Flux Versus Radial Distanc~ for Engine #8 at Timestep Number 3976 . . . . . . . . . . . . . . . . . 251 92 Fast and Thermal Adjoint Neutron Flux Versus Radial Distance for Engine #8 at Timestep Number 4426 ..... . . . . . . . . 252 93 020 Reflector Thickness Versus Piston Cycle Time for Engine #10 . . 300 94 . Gas Temperature Versus Piston Cycle Time for Engine #10. . . . . . . . 301 95 Gas Pressure Versus Piston Cycle Time for Engine #10. . . . . . . . . . . . 302 96 Average Core Thermal Neutron Flux Versus Piston Cycle Time for Engine #10. . . . . 303 97 Neutron Multipl~cation Factor Versus Piston Cycle Time for Engine #10 . . . . . . . . . 304 98 Slow Power Tranii~nts for Engine #10 Induced by C h a n g e s i n t h . e L o o p . C i r c u l a t i o n T i 111 e . 3 0 7 99 Intermediate Level Power T~ansients for Engine #10 Induced by Changes in the Loop Ci . rculation Time. . . . . . . . 308 xxv

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LIST OF FIGURES (continued) FIGURE Paqe 100 . Rapid Power Transients for Engine #10 Induced by Changes in the Loop Circulation Time . . 309 101 020 Reflector Thickness Versus Piston Cycle Time for Engine #11 .. . . . . . . . 326 102 Gas Temperature Versus Piston Cycle Time for Engine #ll. . . . . . . . . . . . . . 327 103 Gas Pre~sure Versus Piston Cycle Time for Engine . #11. . . . . . . . . . . . 328 104 Average ~ore Thermal Neutron Flux Versus Piston Cycle Time for Engine #11 . . . . . . . . . 329 105 Neutron Multiplication Factor Versus Piston Cycle Time for Engine #11 . . . . . . . . . 330 106 Schematic of NASA's NERNUR--A Large Power Generating System Utilizing a UF 6 Gas Core Nuclear Reactor . . . . . . . . .... . .. 362 xxvi

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LIST OF SYMBOLS AND ABBREVIATIONS A constant appearing in the steady-state, two-group, two region diffusion theory flux expressions Ae cross-sectional area of exhaust valve (m 2 ) A. 1 p a B p f3 . J BR BHR C C D . c. J . c~ J cross-sectional area of intake valve (m 2 ) fra.ction of fast neutrons which leave the core and return as thermal neutrons fraction of fast neutrons from t~e fast. inhomogeneous source in the core which leave the core and return as thermal neutrons. a parameter which is the product of fp times yp times 8P constant appearing in the steady-state, two-group, two region diffusion theory flux expressions yield fraction for delayed neutron precursor group j 11 effective 11 yield fraction for delayed -neutron precur sor group j yield fraction for photoneutron precursor group j breeding ratio boiling watet reactor subscript indicating the core region constant appearing in the steady-state, two-group, two region diffusion theory flux expressions; also, an ab breviation which is used to designate the compression stroke "effective" delayed neutron precursor concentration for delayed group j delayed neutron precursor concentration for delayed gro11p j xxvii

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. CV X J CR 'i/ photoneutron precursor concentration for photoneutron precursor group j flow coefficient for the exhaust valve flow coefficient for the intake valve gas specific heat at constant volume gas specific heat at constant pressure the normalized erier~~ spectr~~ for fission neutron emission; x = [x (l-B ) + Ex-Sj] p j J the normalized energy spectrum for delayed fission neu tron group j the normalized energy spectrum for prompt fission neu trons conversion ratio diffusion coefficient average product of the neutron diffusion coefficient and the buckling squared symbol indicating the gradient operation o parameter appearing in the numerical form of the neutron kinetics equations; for the two-point finite difference relations, c = 1 .0 while for the three-point integration formulas, o = 3/2 p o the ratio of the average thermal neutron density in the core to the average fast neutron density in the moderating reflector e, subscript indicating the exhaust phase of the cycle E symbol for the quantity or variable of energy; also, an abbreviation which is used to designate the exhaust stroke Ef energy released per fission E fast fission factor; also, the overall efficiency Em mechanical efficiency xxviii

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F h L'.H he . h. 1 HPT HTGR i cycle fraction for step-reflector addition cycle fraction for step-reflector removal neutron production factor, i.e., 'the average number of neutrons produced in thermal fission over the total thermal absorption in the fuel turbine effici~ncy compressor efficiency thermal utilization; as a superscript it indicates the fast group; as a subscript it indicates either fission or forced flow production operator or volume integral nf the adjoint weighted fission source fraction of the removal cross section which is down scattering fraction of the gamma rays emitted by the photoneutron precursors which penetrate from the core to the moderating-reflector with energy above the (y, n) . threshold geometry factor equal to the core volume over the moderating-reflector volume. geometry factor given by (R 3 R 3 ) fraction of those gammas reaching the moderating reflector region with energy above the (y, n) threshold which actually induce photoneutrons enthalpy ( B / l b .) ; m subscript indicating hydraulic increase in enthalpy of a system (BTU's) energy or enthalpy of the mass leaving the system (B/lb ) m energy of enthalpy of the mass entering the system (B/lbm) high pressure turbine high temperature gas-cooled reactor subscript indicating the intake phase of the cycle xxix

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I abbrevtation . which is used to designate the . intake stroke J neutron current or vector flux JD number of delayed neutron precursor groups JP number of photoneutron precursor groups K 1 ... K 12 coefficients which are convenient groupings of vari ous reactor physics parameters used in solving the 2group, 2-region neutron diffusion theory equations k, keff• k-effective the effective (static) neutron multipli cation factor for a reactor or system k 00 the (effective) dynamic neutron multiplication factor for a reactor of system the infinite medium neutron multiplication factor, i.e., the neutron multiplication factor in the absence of leakage [k 00 ]c convenient grouping of reactor physics parameters defined as [vrf]c/[ ra]c . the inverse square root of the age to therm~l of fast neutrohs in the moderating-reflector when fm is unity the inverse thermal diffusion length in the moderating reflector the inverse thermal diffusion l~ngth in the core when is unity neutron lifetime the infinite medium neutron lifetime, i .e,, the neutron lifetime in the absence of leakage [2 00 ]c convenient grouping of reactor physics parameters de fined as (l/v)c/[cra]c LPT neutron generation time usually defined as [i/k] delayed neutron precursor decay constant for delayed group j photoneutron precursor decay constant for photoneutr.on precursor group j low pressure turbine XXX

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m 8 m n \) \) ,.. I., W. p p PWR mass flow rate out of the cylinder mass flow rate into the cylinder subscript indicating the moderating reflector; symbol denoting mass, usually the HeUF 6 mass coolant water flow rate gas mass which enters or leaves the system during the time At neutron density (neutrons/cm 3 ); symbol indicating neu trons neutron population uranium-235 atom density (atoms/barn-cm) uranium atom density (atoms/barn-cm) one of two components of the shape function; the units are arbitrary, depending upon the normalization applied to the amplitude function average number of neutrons released per fission the averag~ number of neutrons released per fission required for criticality engine speed in rpm's symbolfor the vector variable indicating direction or angle resonance escape probability; pressure total power output; also, abbreviation which is used to designate the power stroke mechanical power output ampli.tude factor or amplitude function pressurized water reactor thermal non-leakage probability scalar neutron flux xxxi

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Q QMAX N r rpm R r average thermal neutron flux in the core due only to the inhomogenous fast neutron source in the core scalar a~joint flux for a time independent critical reference system shape factor or shape function angular neutron flux angular adjoint neutron flux for a time independent critical reference system net amount of heat added to a system from the surround ings fission h~at release heat of r~action; the fission heat from a nuclear reac tion the rate of fissi~n heat release weighted source term,appearing in the point reactor neutron kinetics equati _ ons ratio of the maximum fission heat released in any energetics equation timestep to the total fission heat released during the piston cycle ratio of the maximum fission he~t released in any neutron . kinetics equation timestep to the total fission heat released during the piston cycle total heat transfer rate rate of heat rejection symbol for the vector variable indicating position revolutions per minute core radius or position at the core-reflector interface; gas constant equal to the universal gas constant divided by the gas molecular weight extrapolated reactor radius or position at the outer (extrapolated) edge of the moderating-reflector region symbol for the variable indicating position xxxii

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p s ~t C ~f C reactivity; density; parameter appearing in the numeri cal form of the neutron kinetics equations which has a value of zero for the two-point finite difference rela tions and a value of one-half for the three-ooint integration relations _entropy (B/lbm 0 R) inhomogeneous fast neutron source strength in the core (neutrons/sec) inhomogeneous. fast neutron source term for the moderating reflector region due to photoneutron production number of thermal neutrons generated per unit time and per unit volume in the core as a result of photoneutron production in the moderating-reflector average thermal neutron density per unit time in the core due solely to the inhomogeneous fast neutron source in the core fast neutron source strength per unit volume in the core; equal to S 0 divided by the core volume for the gas cores of concern + S(r,t)neutron soure strength distribution per unit volume (neutrons/cm sec) f+t E. average neutron sou 3 ce strength (in a region) per unit volume (neutrons/cm sec) macroscopic neutron absorption cross section macroscopic neutron scattering cross section macroscopic neutron re~oval cross section macroscopic neutron fission cross section macroscopic cross section for neutron transfer out of a group by scatter macroscopic cross section for neutron scatter from the fast to the thermal group isot~opic component of the macroscopie neutron elastic. transfer cross section linearly anisotropic component of the .macroscopic neu tron elastic transfer cross section xxxiii

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1 D 1, J p 1. J TDC TZN V V w macroscopic neutron transport cross section microscopic neutron fission cross section temperature average gas temperature exhaust gas temperature gas temperature at the end of the compression-power cycle initial gas temperature in the inlet line or at the beginning rif the compression-power cycle neutron temperature symbol for the variable indicating time; as a super script, it indicates the thermal group . Fermi age of fission neutrons to thermal energy delay time between the generation of a fast photoneutron inthe moderating~reflector region and its appearince as a thermal neutron in the core delayed neutron precursor mean lifetime for delayed group j photoneutron precursor mean lifetime for photoneutron precursor group j top dead center position for the piston alloy of niobium, zirconium, and titanium increase in stored or internal energy_ of a system volume velocity mechanical work output; net amount of work done by a system on the surroundings weighting function or weighting term for the core inhomogeneous source(s) weighting function or weighting term for the moderating reflector inhomogeneous source(s) xxxiv

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y e y . 1 net flow work performed by the system on the gas as it passes across the system (piston engine) boundaries average logarithmic energy decrement per collision for neutrons or the average increase in lethargy per col lision net expansion factor for compressible flow through the exhaust valve net expansion factor for compressible flow through the intake valve core fast absorption factor XXXV

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE NUCLEAR PISTON ENGINE AND PULSED GASEOUS CORE REACTOR POWER SYSTEMS By Edward T. Dugan March, 1976 Chairman: Dr. N. J. Diaz Major Department: Nuclear Enqineering Sciences Nuclear piston engines operating on qaseous fission able fuel should be capable of providinq economically and enerqetically attractive power 9.enerating units. A fissionable gas-fueled enaine has many of the ad vantages associated with solid-fueled nuclear reactors but fewer safety and economical limitations. The capital cost per unit power installed (dollars/kwe) should no~ spiral for small gas-fueled plants to the extent that it does for solid fueled plants. The fuel fabrication (fuel and claddinq, spacer grids, etc.) is essentially eliminated; the engineer~ inq safeguards and emergency core coolinq requirements are reduced significantly. The investiqated nuclear piston engines consist of a pulsed; gaseous core reactor enclosed by a moderatinq reflectino cylinder and piston assembly and operate on a thermodynamic cycle similar to the internal combustion engine. The primary working fluid is a mixture of uranium xxxvi

PAGE 37

hexafluoride, UF 6 , and helium, He, gases. Hiqhly enriched UF 6 gas is the reactor fuel. The helium is added to enhance the thermodynamic and heat transfer characteristics of the primary working fluid and also to provide a neutron flux flatt~ning effect in the cylindrjcal core. Two-and four-stroke engines have been studied in which a neutron source is the counterpart of the sparkolug in the internal tombustion engine. The piston motions which have been investigated include oure simple harmonic, simple harmonic with dwell pefiods, and simple harmonic in combi~ nation with non-simple harmonic motion. Neut~onically, the core qoes from the subcritical state, through criticality and to the suoercritical state during the (intake and) compression stro~e(s). Supercriti cality is reached before the piston reaches top dead center (TDC), s6 that the neutron flux can build up to an adequate level to release the required energy as the piston passes TDC. The energy released by the fissioning gas can be extracted both as mechanical power and as heat from the circulating gas. External equipment is used to remove fis sion products, cool the gas, and recycle it back to the pistpn engine. Mechanical power can be directly taken by 0 means of a conventional crankshaft operating a~ low speeds. For the purpose of evaluating the nuclear piston en gine cycle behavior, a computer code was developed which couples the necessary energetics and neutronics equations. xxxvii

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The code, which has been named NUCPISTN, solves for the neutron flux, delayed and photoneutron precursor co~centra tions, core volume, gas tempetature, qas pressure,. fission heat release, and mechanical {pV) work throughout the piston cycle. As a circulatinq fuel reactor, the nuclear piston en gine's quasi-steady-state power level is capable of beinq controlled not only by variations in the neutron multipli cation factor but also by changes in the loop circulation time. It is shown that such adjustments affect the delayed and photoneutron feedback into the reactor and hence pro vide an efficient means for controlling the reactor power level. The results of the conducted investiqations indicate good performance potential for the nuclear piston engine with overall ~fficiencies of as high as 50% for nuclear piston engine power qenerating units of from 10 to 50 M~(e) capacity. Larger plants can be conceptually designed by increasing the number of pistons, with the mechanical com plexity and physical size as the probable limiting factors. The primary uses for such power systems would be for small mobile and fixed ground-based power generation (es pecially for peaking units for electrical utilities) and also for nautical propulsion and ship power. xxxviii

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CHAPTER I INTRODUCTION Description of Engine Operation The investigated nuclear piston engines consist of a pulsed, gaseous core reactor enclosed by a moderatirig reflecting cylinder and piston assembly, and operate on a thermodynamic cycle similar to the internal combustion engine. The primary working fluid is a mixture of uranium hexafluoride, UF 6 , and helium, He, gases. Highly enriched UF 6 gas is the reactor fuel. The He is added to enhance the thermodyna~ic and heat transfer characteristics of the primary working fluid and also to provid~ a fl~x flattening effect in the cylindrical core. Both two-and four-stroke engines have been studied in which a neutron source is the counterpart of the sparkplug in an internal combustion engine. The piston motions which have been investigated include p~re simp1e harmonic, simple harmonic with dwell periods, and simple harmonic in combina tion with n6n-simple harmonic motion. Neutronically, the core goes from the subcritical stite through criticality and to the supercritical state l

PAGE 40

during the (intake and) compression stroke(s). Super criticality is reached before the piston reaches top dead center (TDC) so that the neutron flux can build up to an adequate level to release the required energy as the piston passes TDC. The energy released by the fissioning gas can be extracted both as mechanical power and as heat from the circulating gas. External equipment is used to remove fission products, cool the gas, and recycle it back to the piston engine. Mechanical power can be directly taken by means of a conventional crankshaft operating at low speeds. 2 To utilize the significant amount of available energy in the hot gas, an external .heat removal loop can be de signed. The high temperature (~1200 to 1600K) HeUF 6 exhaust gas can be cooled in an HeUF 6 -to-He heat exchanger. The heated He (~1000K to 1400K) is then.passed either directly through gas turbines or is used in a steam genera tor to produce steam to drive a turbine. The total mechanical plus turbine power per nuclear piston or per cylinder ranges from around 3 to 7 Mw(e) depending on the selected piston engine operating character istics and the external turbine equipment arrangement. Thus, power generating units of from 10 to 50 Mw(e) capacities would consist of a cluster of 4 to 8 pistons in a nuclear piston engine block, Larger power plants can be conceptually

PAGE 41

3 designed by increasing the number of pistons with the mechan-. ical complexity and physical size as the probable limiting factors; Overall efficiencies are as high as 50% implying heating rates of around 6800 BTU/kr-hr. Fuel costs are presently estimated as being below $0.20 per million BTU*or around 1.4 mills/kwe-hr. Applications and Highlights of the Nuclear Piston Engine Concept Some of the primary uses for nuclear piston engine . power generating systems would be for peaking units for electrical utilities, for small mobile and fixed ground based power systems, for nautical propulsion and ship elec trical power and for process heat. Further possible appli cations will be discussed in Chapter VII. Most current peaking units operate on conventional fossil fuels. These units are, in general, expensive, wastef~l, and inefficient. Fuel costs range from $0.50 to $1 .30 per million BTUs,* heat rates are as high as 15,000 to 21,000 BTU/kw-hr, and efficiencies are not much greater than 20%. A conversion from wasteful, conventionally fueled peak ing units to efficient, nuclear-fueled peaking units would yield significant savings in fossil fuels. The fuel thus sa~ed could be reallocated for more critical applications. This consideration alone should be incentive enough to *Based on fiscal year 1974 costs.

PAGE 42

investigate any promising, nuclear-fueled peakfng unit concept--even if the nuclear-fueled unit's power production costs should be ~stimated to be as high as for the con ventionally fueled units. The fact that preliminary esti mates indicate that a nuclear piston engine peaking unit should be more economical than any of the fossil-fueled units now employed makes this concept that much more attractive. 4 Already-developed nuclear reactor concepts like pres surized water reactors (PWRs), boiling water reactors (BWRs), and high temperature gas-cooled reactors (HTGRs) can be economically competitive only when they are incorporated into large capacity power systems. Given the fuel cycle costs and operation and maintenance costs for these reactor concepts, it is their high capital costs which economically prevent them from being used on a scaled-down basis for 20-50-100 Mw(e) units. The cost per unit power installed (dollars/kwe) for scaled-down units operating on these already-developed solid-fueled core concepts would be extremely high. A nuclear piston engine power plant, however, will not require the sophisticated and costly engineered safe guards and auxiliary systems associated with the solid fueled cores of current large capacity nuclear power plants. The inherent safety of an expanding gaseous fuel can b~ engineered to take the place of many of the functions of the

PAGE 43

safeguards systems. Hencei while gaseous core, nuclear piston engine power plants would pos~ess relatively high costs per unit power installed as compared to comparably sized fossil-fueled units, their capital costs per unit power installed would be considerably less than for any scaled-down nuclear units operating on current solid fueled core concepts. In addition to decreased capital costs, the nuclear piston engine should possess fuel cycle costs which are 5 -about half the fuel cycle costs of most present large capa city nuclear plants. Fuel fabrication costs, transportation costs to ~nd from the fabricator, and transportation costs to and from the _reprocessor will all be eliminated. These costs typically comprise from 40 to 50% of the current nu clear fuel cycle costs.* Thus, it would appear as if power production cos.ts for a nuclear piston engine will not only be less than those of conventionally fueled peaking units, but that they should a)so approach the power production costs of large-scale fossil and large-scale nuclear-fueled plants. With regara to power generation for nautical applica tions, the nuclear units utilized by ship~ are more expensive than conventionally fueled units. The major advantage of cur rent nuclear-fueled vessels is their tremendous rang~ between refuelings as compared to conventionally powered vessels. It is for this strategic reason rather than for economic ------*Based on 1974 fiscal year costs.

PAGE 44

6 ' reasons that the U.S. Navy maintains nuclear-powered vessels. On the other hand, the economic disadvantage is the primary reason why nuclear-powered vessels have not been able to replace conventionally powered commercial vessels. Ships powered by nuclear piston engine gas core sy~tems,_ however; should be able to compete economically wi.th conventionally powered vessels while still retaining the_ advantage of long ranges ~etwe~n refuelin~s. The extensive use of such nuclear power units by ships would, of course, also lead to significant fossil fuel savings. Dissertation Organization In the chapter which follows, a summary is presented of some of the more impqrtant nuclear studies which have been performed on gaseous core, externally moderated reac tors. It presents the models employed to analyze the neu tronics of gaseous cores, calculations performed, experiments conducted and appropriate comparisons between analyti cal arid expe~imental results. This is followed by a chapter describing the previous work which was done on the gaseous core nuclear piston engine concept by other authors here at the University of Florida where the idea originated. Results from work which has since been performed by the author on two-stroke nuclear piston engines is presented

PAGE 45

7 in Chapter IV. Since these investigations indic~te~ good performance potential for the nuclear piston engine concept, more sophisticaied, four-stroke engines were studied. The results of these studies ~re presented in Chapter V. Chapter VI discusses other ongoing and related research in the field of gaseous core reactors. The results of some of these other studies will certainly have an impact on the future of further research on the nuclear piston engine concept. The last chapter ~resents conclusions. Suggestions are made for refinements in the neutronics and energetics equa tibns .used in the nuclear piston engihe analysis. Also discu~sed are areas where further studies are needed before the technical feasibility of the nuclear piston engine con cept can be firmly established, Finally, all neutronics and energetics equations used in the nuclear piston engine analysis have been placed in appendices. References are made from Chapters IV and V to the appropriate appendix for equation development and presentation.

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CHAPTER II PREVIOUS STUDIES ON GASEOUS CORE CONCEPTS Gaseous Cores Analytical Studies The first report on analytical studies of a gas core nuclear reactor was due to George Bell of Los Alamos in 1955 [1]. Age and diffusion theories were used to analyze the neutronics of a spherical gaseous cavity surrounded by a moderating-reflector. Age theory was used t6 describe neutron slowing down in the moderating-reflector (no slowing down or fast neutron absorption in the fuel was permitted) and diffusion theory was used to describe thermal neutron diffusion into the cavity and the resulting fissicins in the fuel. The reactors considered were strictly thermal with UF 6 gas cores and Be and graphite reflectois. In 1953, a report of a study on externally moderated reactors was published by Safonov [2]. The study was based on the prime assumption of complete external moderation. F i s s i l e mater i a 1 v-1 as cont a i n e d i n a cent r a 1 or 11 i n t er i or 11 region while the rnoder~ting-reflector material surrounding the fuel comprised the 11 exterior 11 region. The analysis included, but was not limited to gaseous cores. Low density, liquid-metal-fueled, externally moderated reactors 8

PAGE 47

were also considered. Fermi age and diffusion theory were used to describe the neutronics of the exterior moderating reflector while diffusion theory was used for the interior cores with L >>L and transport theory was applied to the s a interior if Ls<
PAGE 48

l 0 In 1~61 Ragsdale and Hyland [3] looked at cylindrical and spherical, o 2 o reflector-moderated, u 235 -fueled, gaseous core reactors. A parametric study was made with variations in moderator thickness, fuel region radius for a given ca~ity radius, the effect of the cavity liner, and the moderator temperature. _Six-group, one-dimensional diffusion theory was used for the spherical configurations while four-group, two-dimensional diffusion theory was used for the cylindrical systems. The reflector temperature was assumed to govern the neutron energy and fast absorption and slowing down in the fuel region was disregarded. The criteria as established by Safonov for the validity of diffusion theory were used in this analysis and were a function of the 11 cavity greyness. 11 Thermal cross sections were obtained inthe analyses for the moderator tempera ture, regardless of the region. In 1963, Ragsdale, Hyland, and Gunn [4] extended their work. They considered in this work only cylindrical geometry using four-group, two-dimensional diffusion theory. The fuels considered were Pu 239 and u 235 while o 2 o (at 300K) and graphite (at 300F and at 3200F) were studied as moderating-reflector materials. They looked at the effect on critical mass of a variable fuel reiion radius in a fixed size cavity. The moderating-reflector was 100cm in thickness since earlier work has shown this would be optimum for reducing critical mass without incurring an excessive weight penalty. Assumptions were made for flow

PAGE 49

rates, pressure, temperatures, etc., and within these constraints, cavity radii of 40cm and 150cm were investi gated .. Thermal cross sections.were all computed at the moderator temperature, the effect of Doppler broadening due to the elevated fuel temperature was considered and a Maxwellian flux shape was used. in determining mean average thermal cro~s sections. Kaufman et al. [5] performed an extensive par~metric survey on gaseous core reactors in cylindrical geometry in 1965. One-dimensional S transport theory studies were . n . used initially to evaluate the effectiveness of various moderating-reflectors (Be, D 2 0 and graphite) for a variety of cavity dimensions and moderator temperatures. Also l l studied were composite reflectors and pressure vessel and liner effects on the critical mass. The effects of geomet ric variations, such as radius-to-height ratios, were looked at using two-dimensional S transport theory. n . . Compari~ons made between.calculations using 24, 15, 13 and 3 collapsed broad groups showed that a good set of collapsed three-group constants was adequate to yield critical masses, fluxes, absorptions, and leakages. Herwig and Latham [6], after studying a hot gaseous core containing hydrogen, concluded in 1967 that upscat tering of neutrons returning to th~ core by the hot hydrog~n is ijn important effect and that for such cores, a multi thermal group approach is essen~ial. One-dimensional diffusion theory with 6 fast and 12 thermal groups was used

PAGE 50

l 2 to evaluate reactor characteristics for changes in moderating reflector parameters such as temperature, slowing down power, (n,2n) production, thermal scattering, and thermal absorption. o 2 o, Be and graphite-moderated reactors were studied. Neutronic Calculations for Gaseous Core Nuclear Rockets Nuclear studies of gaseous core nuclear rocket engines were carried out by Plunkett [7] in 1967. s 8 transport theory calculations u~ing 14, 15 and 16 groups were per formed and compared with multi~group diffusion theory results. Thermal cross sections were obtained by averaging over a Maxwellian neutron distribution at the moderator temperature, and Doppler broadening wa~ considered. Dif fusion theory critical loadings and fluxes were in good agreement with the transport theory resul.ts. Latham conducted a series of extensive calculations (1966-1969) [8-12] for a nuclear light bulb model (closed system) ga~eous core nuclear rocket engine. BeO and graphite moderating-reflectors with u 233 , u 235 and Pu 239 fuels were investigated. A series of 24-group, one-dimensional, s 4 transport theory calculations were performed with ANISN [13]. Fourteen of the 24 groups were in the Oto 29eV range because of the large temperature differences in adja cent regions and the consequent imp~rtance of upscattering .by the hot hydrogen and neon in the 1200 to 7000K core.

PAGE 51

Fast neutron cross sections were obtained from GAM-1 [14] with the slowing down spectrum in the core being assumed to be that of the moderating-reflector region. TEMPEST [15] was used for the thermal absorption probabili ties and SOPHIST [16] for thermal upand down-scattering probabilities. Flux and volume weighted four-group cross sections were obtained from the 24-group, one-dimensional transport calculations for use in two-dimensional calcula tions. Two-dimensional transport theory calculations were then performed with DOT [17] while two-dimensional diffusion theory calculations were obtained from the EXTERMINATOR-II code [18]. The objectives of Latham's works were to (a) evaluate effects of variations in . . d . u233 l (b) u233 engine es1gn on cr1 ,ca mass, compare u 235 , and Pu 239 critical mass requirements, and (ci to evaluate various factors affecting engine dynamics. In addition to critical masses, material worths, pressure, temperature and reactivity coefficients, and neutron lifetimes were determined. Critical masses and the effects of variations in cavity size, fuel-to-cavity radius and reflector thickness 233 235 for U and U -fueled, open, gas core nuclear rocket l 3 engines were looked at by ltyland in 1971 _[19]. A composite, moderating-reflector was used. Fuel tempera tures reached 44,OOOC and moderating-reflector temperatures varied from 93C to 56OC. The cavity wall temperature

PAGE 52

was 1115C. A total of 19 groups {7 thermal) were used wfth fast group constants obtained from GAM-II [20] and thermal group constants from GATHER [21]. One-dimensional, s 4 transport calculations in spherical geometry were per formed with the TDSN code [22]. Hyland found that a large amount of a light gas, like hydrogen, in the core increases the absorptions and upscattering caused by the hydrogen in the cavity between the uranium and the moderators. This means more fuel absorption in the higher energy levels which are less productive (fewer fissions per absorbed neutron) and henc~ an increase in the critical mass requirement. Other observations were (a) u 233 has a lower critical mass than u 235 , (b) for u 233 there is less change in the critical mass_ requirement from the startup tempera235 ture than for U , (c) the critical mass increases with increasing cavity diameteri while the critical fuel density decreases, (d) above a total reflector thickness of more 1 4 than one meter little reduction in critical mass is obtained, and (e) the critical mass increases rapidly with decreasing fuel to cavity radius. Comparison of Theoretical Predictions with ExBerimental Results In a report on reflector-moderated reactors, Mills [23] in 1962 compared theoretical predictions using Sn transport theory with experimental data for gaseous uranium core reactors and attained fairly good agreement. Mills also

PAGE 53

performed parametric studies of gas-filled, reflector moderated reactors to establish minimum critical loadings. Large spherical and cylindrical cavities reflected by o 2 o, Be, and graphite were investigated; gaseous u 235 was the only fuel studied. Some of the conclusions reached by Mills were (a) systems are sensitive to.the u 235 content, {b) the systems are not sensitive to core diameter changes 235 for a constant amount of U , (c) systems are sensitive l 5 to absorption either in the liner or in the moderating reflector, and (d) the critical mass increases approximately as the radius squared (this is in contrast to internally moderated cores where the critical mass increases approxi mately as radius cubed). In 1965, Jarvis and Beyers [24] of Los Alamos made a comparison between diffusion theory predictions and experi mental results for a o 2 o-reflected cavity reactor. The maximum discrepancy between calculated and experimental results for t~is system was 3% in reactivity. A series of critical experiments (1967-1969) [25-32] were performed by Pincock, Kunze et al. to test the ability of various calculational procedures to evaluate criticality and other reactor parameters in a configuration closely resembling a coaxial-flow (open system) gaseous core nuclear rocket engine. In some configurations, fuel was dispersed as small foils in various patterns representing fuel distri butions in a gas core reactor. Other configurations

PAGE 54

l 6 contained UF 6 gas. o 2 o reflectors were employed which for some configurations had Be slabs (heat shields) in the o 2 o spaced 5 to 10cm from the cavity wall. The theoretical calculations included one-dimensional, 19-grotip diffusion theory, one-dimensional, s 4 and s 8 transport theory, and fou~and seven-group, two-dimensional diffusion theory results. Comparison of these results showed little difference between s 4 and S calculations and reasonably good agreement 8 . between the multigroup diffusion theoiy and the transport theory calculations. The transport theory calculations themselves were in good agreement with ~xperimental results. A benchmark critical experiment with sphericalsymmetry was conducted on the gas core nuclear reactor concept in 1972 by Kunze, Lofthouse, and Cooper [33]. Nonspherical perturbations were experimentally determined and found to be small. The reactor consisted of a low density, central urani~m hexafluoride gaseous core, surrounded by an annulus of either void or low density hydrocarbon which in turn was ,surrounded by a 97cm thick o 2 o moderating reflector. One configuration looked at also contained a 0.076cm thick stainless steel liner located on the inside of the cavity wall. Critical experiments to measure reactivity, power and flux distributions and material worths were per formed. Theoretical predictions were made using 19 groups (7 thermal) in an s 4 transpo,t calculation with the SCAMP code [34]. Fast group cross sections were obtained from

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l 7 PHROG [35] and thermal group cross sections from INCITE [36]. The predicted eigenvalues were in good agreement with experiment as was the reactivity penalty for the stainless steel liner. Fuel worths and the reactivity penalty for the hydrocarbon however were substantially underpredicted. Comments While the above is not a complete listing of all the nuclear studies which have been performed on gaseous core, externally moderated reactors, it is a representative sampling of the types of models employed, calculations per formed and experiments conducted on gaseous core reactors. The models of Bell and Safonov are reasonable only for thermal, spherically symmetric systems which are not very grey. For systems which possess a significant degree of greyness., the general conclusion to be drawn from the above investigations is that in "most cases," multigroup diffusion theory is adequate provided 11 good 11 fast and thermal group constants are used and provided the~e is a multi thermal-group approach which allows for full uoscatterinq and downscattering. Because of the wide range of geometries, temperatures, temperature differences, pressures, densities and materials which can b employed in gaseous core reactors, each new gaseous core reactor concept demands individual scrutiny. The applicability of diffusion theory to the processes of neutron birth in the core, thermalization in the moderating

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l 8 reflector, and the diffusion of the thermal neutrons back into the core should be checked by performing some transport calculations for at least a few reference configurations which are typical of the particular gaseous core reactor design being investigated. Attempts to apply or extend conclusions from previous gas core analyses to new gaseous core concepts cannot be justified by presently established theoretical and/or experimental evidence. In particular, the gas core 11 cavity 11 type reactors are not directly compara ble to the dynamic situation in a gaseous core-piston engine. Significant differences exist in the modus operandi of each concept. In the steady-state condition, however, valuable analogies can be drawn between these sytems. Before concluding this section, credit should be given to J.D. Clement and J.R. Williams for their report on gas core reactor technology [37]. Besides outlining the important work which had been done in the field of gas core reactor neutronics calculations up until mid-1970, this report contains an extensive listing of helpful references.

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CHAPTER III PREVIOUS GASEOUS CORE NUCLEAR PISTON ENGINE STUDIES AT THE UNIVERSITY OF FLORIDA Introduction A thermodynamic cycle, similar to the internal combus tion engine using a gaseous fissionable fuel,was first proposed by Schneid~r and Ohanian [38]. Preliminary feasi bility studies by Kylstra et al. [39] showed that a ~F 6 fueled, Otto-type fission engine has a very good performance potential. Power in the MW/cylinder range and thermodynamic efficiencies of up to 50% ~eemed f~asible with a low fuel cycle cost making the process economically attractive. The nuclear piston engine studied by Kylstra and associates (see Figure 1) was a pulsed gaseous core reactor enclosed by a neutron-reflecting cylinder and piston. The fuel was 100% enriched UF 6 and the ignition process was triggered by an 11 external 11 neutron source. The engine was a two-stroke model with intake and compression occur ring on the first stroke and expansion and exhaust on the following stroke. Neutronically, it was desired that the core go from the subcritical state, through criticality, and to the supercritical state during the compression stroke. 1 9

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REPROCESSING PLANT HEAT EXCHANGER 20 MODERATING-REFLECTOR ""PISTON MODERATING-REFLECTOR FIGURE l. Simple Schematic of A UF 6 Nutlear Piston Engine

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Supercriticality was to be reached before the piston reached top dead center so that the neutron flux could build up to an adequate level to release the required power as the piston passed TDC. To avoid releasing fission heat after the piston was already well into the power stroke, it was 21 then required that the reactor be rapidly shut down. To attain the desi.red time sequence of subcritical to super critical to subcritical behavior for the reactor, the reflec tor thickness was varied throughout the cycle. At the compression stroke start, the reflector was a thin reflector, increasing in thickness slowly, then stepping to a thick •reflector at some cycle fraction, El, in the compression stroke and then continuing to increase slowly until TDC. The step-reflector was then removed, going back to a thin reflector at some cycle fraction, 2 , usually at TDC (see Figure 2) . . The moderating-reflector surrounding the cylindrical core and the piston itself were made of g~aphite, with a nickel liner being used for protection of the graphite from the UF 6 [40]. External equipment was to be employed to rem~ve the fission products, cool the gas, and recycle it back to the engine. A HeUF 6 ~ixture rather than pure UF 6 gas was used for the engine's primary working fluid. The addition of helium improved the working fluid's thermodynamic and heat transfer properties while also leading to a flattening of the neutron flux in the core. For the systems studied, the UF 6 partial

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CORE MODERATING-REFLECTOR LOW LEAKAGE, SUPERCRITICAL ARRANGEMENT AFTER STEP-REFLECTOR ADDITION AT El CORE MODERATING-REFLECTOR HIGH LEAKAGE, SUBCRITICAL ARRANGEMENT AFTER STEP-REFLECTOR REMOVAL AT E 2 OR BEFORE STEP~REFLECTOR ADDITION AT El 22 FIGURE 2. ILLUSTRATION OF STEP~REFLECTOR ADDITION AND REMOVAL

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pressures and temperatures throughout the cycle were such that the UF 6 remained in the vapor phase (see Figure 3) and underwent no dissociation due to thermal kinetic energy [41]. 23 The engine was to be operated at high graphite tempera tures (1000-1200F) so as to minimize the convective and conductive heat losses from the core region. The compres~ sion ratio for the engine was 10-to-l with a clearance volume of 0.24m 3 and an engine shaft sp~ed of lOOrpm. Neutronic Model For the steadystate sol u ti on of the neutron b al an c.e , a two-group, two~region diffusion theory approximation in spherical geomet~y was used with the following assumptions: 1) no interactions for fast neutrons in the core. Thus, the fast core equation was replaced by a boundary condition for the net neutron current into the moderator. 2) no absorption in the moderator. 3) no delayed neutrons (the delayed neutron precur sors are swept out of the cylindrical core with the exhaust gas before they exert any influence). 4) no time dependenc~. 5) no angular dependence. 6) for the sake of simplifying the analysis, the cylindrical two-region piston was represented by a two-region spherical model. The spherical core volume and reflector thickness were then varied to simulate the motion in the corresponding cylindrical piston.

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100 ,-. 8 ... <"l .._, ::, 1.0 Vl Vl ll< 0.1 250 300 350 !, (J() T~PERATURE (K) 450 FIGURE 3 . . UF 6 Phase Diagram 5lJU .'.:l5U

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The set of equations resulting from the above assump tions and approximations was solved for both the neutron multipiication factor, k-effective, ahd for the average steady-state thermal neutron flux in the core, as a function of piston position. 25 As the neutron multiplication factor approached and exceeded one, the average core thermal neutron flux at each time step was calculated from a single-group, point reactor kinetics equation rather than from the expression derived from the two-group, steady-state analysis. The justification offered for the uncoupling of time and space (implied in the point model treatment) was the difference in time scale between the speed of the piston (or rate of change of geometry) and the diffusion speed or cycle time of the neutrons. The use of the diffusion equations was justified only by their simplicity compared to S or P approximan n tions to the neutron transport equation. A knowledge of the average core thermal flux throughout the compression and power strokes permitted the calculation of the fission heat as a function of time for use as the heat source term in the energy balance equation. Energy Model The conservation of energy equation for a non-flow, clo~ed system was used. It was assumed that the HeUF 6 mixture was an ideal gas of constant composition and heat

PAGE 64

loss to the walls was neglected. The energy equation balanced the rate of increase of the internal energy of the HeUF 6 gas ~gainst the rate of performance of the pV mechanical work by the gas on the piston. The initial pressure, temperature, volume and piston position as a function of time were input parameters. A numerical. form of the energy equation was then used to determine the gas temperature, T(t+6t), in terms of T(t). Since the piston position at t+6t is known, so is the cylinder volume 26 V(t+6t) and the ideal gas equation was then solved to obtain the gas pressure, p(t+6t). This process was continued over the entire compression-power stroke cycle to yield not only the gas pressure and temperature variations through out the cycle but to also permit the determination of the total pV work and total fission heat released during the piston cycle. The neutron flux, neutron multiplication factor and atom density were also monitored as a function of piston position throughout the cycle. Primary independent variables of the engines studied by Kylstra et al. and their range of values are shown in Table l. The clearance volume is the core volume with the piston at TDC.

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TABLE l Values of Primary Iridependent Parameters for Graphite-Reflected Piston Engines Analyzed by Kylstra ~~.[39] Gas Mixture Initial Temperature (K) Gas Mixture Initial Pressure (atm) U 235 L d' oa 1ng (kg) Engine Speed (rpm) . 3 Clearance Volume (m) Compression Ratio Cycle Fraction for Step-Reflector Addition, 1 Cycle Fraction for Step~Reflector Removal, 2 Neutron Source Strength (neutrons/sec) = = = = = = = = = 27 400 1 -4 l.7-3.l 100 0.24 10-to-1 0.1-0;3 0. 5 lxl0 9

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Analytical Results Shown in Figure 4 is the neutron multiplication factor for an engine with an infinite graphite reflector as a function of the UF 6 partial pressure at a temperature of 400K. At this temperature, for UF 6 partial pressures greater than about 1 atmosphere, the core becomes so black to neutrons that additional uranium is ineffective. 28 Figures 5 and 6 show the total gas pressure, gas tem perature, average core thermal neutron flux and neutron mul ti_plication factor over a complete compression-power stroke cycle for a typical set of independent parameters. The step reflector addition was at a cycle fraction El = 0.10 while t~e step-reflector removal and reactor shutdown occurred at a E 2 = 0.50 cycle fraction. The maximum pressure for this system was 31 atmospheres and the maximum temperature was 1230K (1754F). Both.the temperature and pressure peak at TDC, which is the point at which the step-reflector is removed and the reactor is shut down. Since the cylinder walls were to be maintained at 1000~200F, a peak tempera ture of l 754F did not represent an excessive thermal pulse. The, neutron multiplication factor in Figure 6 increases to a value greater than l upon step-reflector addition at El = 0.1 and then gradually decreases as the core decreases i n s i z e . T h e i n c re a s e d n e u t r o n l e a k a g e a s t"h e p i s to n mo v e s towards TDC and the decreased u 235 cross section with increased temperature ar.e thus more important than the

PAGE 67

0 E--< u < , ~"' o H E--< < u 1-1 .~ p, H H ,-:i .-:" ( z 0 E--< . ::.:, w 7. 1.25 1.0 1. 75 0.5 0.1 0.3 1.0 3.0 UF 6 PRESSURE (atm) (Gas Temperature= 400K) . 29 10 F I GU I{ F /\ . Neu tr on Mu l . t i p I i cat i on Factor V c~ r s u s U F 6 Pa rt i al Pressure fur ,111 Infinite Grt1pl : ite Reflector [39]

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,,...... E u ('j 1250 30 1000 20 750 10 500 --~----_._ ____ _,__ ____ _J, ____ _, 0 25 50 75 PERCENT TRAVEL THROUGH CYCLE Initial Gas Pressure= 1 atm Initial Cus Temperature= ~00K U-2.JS M:1s,.; = 2 .15 kL~ l 00 30 FfGURE 5. Gas Pressure nnJ Temperature as a Function of Perce 11 t Tr ,1 v c l { h .o ugh the P i ton Cy c 1 e for a Graph i t e P. ,) f ,~ 1: c J Eng i n e [ 3 9]

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.,...._ 0 (l) Ul I N 0 s:: ,-.l .., ,....1 < Q i:.:. ~E:--< . 0 u .1 0 < 1015 10 10 105 10 1 0 25 so 75 PERCENT TRAVEL THROUGH CYCLE Initial Gas Pressure~ 1 atm lnit ial Gas Ter.!p~rature = l100K U-23'1 :,lass = 2. 15 kg 31 1.1 0 E:--< u 1.0 < .,.. z 0 0.9 H E:--< < u H 0.8 •. 1 i::.. 1--1 1~ ,-.l 0.7 c-, •'. 7. 0 0.6 p,: E:--< >4 ~. 0. 5 0.4 100 FIGURE 6. Average Core Thormal Neutron Flux and Neutron Multiplicaticin factor as a Function of Percent T r a v e l T h r o u q Ii l I; e P i s t o n Cy c l e f o r a G r a p h i t e R e f l e c t e d E 11 ~Ji I k [ 3 9 J

PAGE 70

increase in u 235 density. The multiplication factor then drops rapidly upon removal of the stop-reflector at E:2 = Q .• 5. Figure 7 shows typical performance results obtained by Kylstra et al. for the UF 6 piston engine. The data for Figure 7 is for an engine with the step-reflector imposed at the 10% cycle position, the same as Figures 5 and 6: A maximum neutron multiplication factor of from 1.07 to 1.10 was reached for these systems with k-effective dropping 32 to 0.99 to 1.01 as TDC was approached. This behavi~r of k-effective greatly incre~ses the control safety since the pow e r d o u b 1 i n g t i me i s 1 a r g e a t h _i g h p ow e r . I n c re a s i n g t 11 e u 235 loading leads to larger k-effectives but it also reduces the helium content in the gas mixture for the same initial pressure. Thus, the efficiency and power curves of Figure 7 are concave downward to reflect the higher specific heat and hence poorer thermodynamic properties of the gas mix ture as more UF 6 is added at the expense of He. Ky1stra et al. compared one of their piston engines with a large stationary diesel power plant [42]. The re sults of this comparison are shown in Table 2. The Nordberg Diesel which was used in the comparison has 6 to 12 cylinders, a 29-inch bore, and a 40-inch stroke. The fuel cost esti mates were obtained by assuming a 20/gal price for diesel fuel, and a $12/gm U charge for the UF 6 fuel.

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33 2.10 2. 15 2.20 2.25 2.10 2,]5 'J. Jo..-----,------,------.------..-------.------. r...< C.:] ~, EFFICCE:-:CY DlfTL\L PRESSU~E I 1 atm lf 2 :llm III /1 atm c o ;c.. 10 -l 0 ~-~. ;;: u 5~ 2. l 0 2. l 5 Compression Ratio Cl,•:1r;1n<:,: \'ul11m,• (m1) Engine SpQ• u w H u 20 H i:,. i.,. 0 el •: u H ., .~ :i~ u :! ..-. 2. 2 5 2. 10 2.]'l 101 FIGURE 7. UF 6 Nuclear Piston Engine Performance for the Case of Graphite Step-Reflector Addition at the 0. l Cycle Fraction

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TABLE 2 Comparison of a Typical Graphite-Reflected~ UF 6 Piston Engine (as analyzed by Kylstra et al.) with the Nordberg Diesel [39] Characteristic 3 Clearance Volume (m ) Compression Ratio Displacement Volume (m 3 ) Shaft Speed (rpm) Type of Cycle (# of strokes) Net wo~k per cycle (Mw-sec) Power (Mw) P~wer Density (w/cm 3 ) Efficiency (%) 6 Fuel Cost ($/10 BTU) Relative Fuel Cost Diesel 0.0394 l 2 -to -1 0.434 200 2 0.224 0.746 l. 7 2 'v20-30 l. 39 6.6 UF 6 Engine 0.24 10-to-l 2. 16 100 2 0.86 l . 43 6 0.665 42 0. 21 1 34

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One of the results of the study by Kylstra et al. was the observation that the time of application of both the step-reflector addition and removal was rather criticaJ. Analytically, the application of this step function in the reflector thickness is easy to attain; practically, it can only be approximated. Recognizing this, Kylstra and his associates were led to conclude that rather than simple ha~monic motion, a better pattern would involve the use of dwell periods by means of 4-bar linkage systems. Since the preliminary feasibility studies conducted by Kylstra indicated such good performance potential for a nuclear piston engine, further work on the gaseous ccire nuclear piston engine was warranted in order to better judge the technical feasibility of the concept. The addi tional work which has been conducted to date is discussed in detail in the next two chapters. As will be seen, these studies have led to piston engine performances and designs which differ significantly from those of Kylstra and his associates. 35

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CHAPTER IV RESULTS FROM TWO-STROKE ENGINE STUDIES Introduction Many of the studies on gaseous core reactor concepts which were discussed in Chapter II have prove~ to be valua ble guides for the gaseous core, nuclear piston engine neutronics calculations. Large differences exist, however, in siz~s, pressures, temperatures, densities, and materials between the cores analyzed in these previous studies and the nuclear piston engine. These differences preclude any_ extrapolation of predicted behavior to the nuclear piston engine. A complete, thorough investigation of the nuclear characteristics, including extensive parametric surveys, is therefore an essential step if the nuclear piston engine's technical feasibility is to be demonstrated. Initially, only two-stroke engines were analyzed. The compression-power stroke cycles for these engines were examined with the intention of eventually proceeding to more complex four-stroke engines if the results from the twd-stroke engine pr6ved encouraging. The first two-stroke engines which were looked at neglected delayed and photoneutron effects. Later two-stroke 3.6

PAGE 75

engines considered first the influence of delayed neutrons and then the influence of both delayed and photoneutrons on the nuclear piston engine's performance. The effects of variations in the nuclear piston engine operating conditions were analyzed by means of a reasonably simple, analytical model, incorporated into a computer code which has been named NUCPISTN. The code is. similar in function to the code used by Kylstra et al. [39]. The NUCPISTN code, ho0ever, is much more sophististe1 than the original code and has been modified and improved in several important aspects. Two-group, two-region diffusion theory equations in spherical geometry were still used for the steady-state spatial flux dependence and for the neutron multiplication factor throughout the piston cycle. Five of the six assumptions which were used in the solution of the steady-state neutron balance by Kylstra et al. (see Chapter III) were initially maintained. Only the second assumption was ~ltered in that neutron absorption in the moderating-reflector was no longer neglected. A complete development of the two-group, two-region, steady-state diffusion theory equations used in the NUCPISTN code in the absence of photoneutrons is given in Appendi~ A. Initially, a single point reactor kinetics equation was again used for th~ time dependence of the neutron flux. The point kinetics equation(s) used in the NUCPISTN code are presented in Appendix D. However, Appendix C, 37

PAGE 76

38 which presents the general point reactor kinetics equations, should be ~xamined first both fo~ familiarization with the notation as well as for a development of some expressibns which are used in Appendix D. The actual one-spe~d (thermal) point reactor kinetics equation used by NUCPISTN when de layed and photoneutrons are ignored is given by equation (D2 3) . The same non-flow, closed system conservation of energy equation as used by Kylstra et al. is again used in the NUCPISTN code for the two-str6ke engines. A develop ment of this equation is set forth in the first section of Appendix E. The liU C P IS TN code then coup l es the neutron i cs and energetics equations and solves for the neutron multiplica tion factor, neutron flux, core volume, gas temperature,. gas pressure, pV work, and fission heat release over the piston compreision-power cycle. As already mentioned, the NUCPISTN code is much more flexible than its predecessor. It is able to accommodate a wider variety of initial conditions and possible piston motions including dwell periods in the piston cycle and non simple harmonic motion .. The piston behavior during the cycle ~s closely monitored and the final cycle output infor mation is much more extensive than in the prior code. Time steps are chosen during the cycle on the basis of the cur rent k-effective of the engine and also on the current rate

PAGE 77

39 of energy release in an efficient and systematic manner. Non-1/v variations in the uranium microscopic cross sections with temperature are now account~d for by means of Wescott factors and thermal absorption in the reflector is considered. Corrected and _ improved thermodynamic constants for the HeUF 6 gas are used [41, 43]. Neutron multiplication factors and cross sections out~ put by the NUCPISTN code at various cycle positions for different piston engine models have been compared with more elaborate calculatibns. The first comparisons were made with results obtained from twoand four-group, one-dimensional diffusion theory calculations performed ~ith CORA [44] in spherical geometry and with . corresponding two-dimensional diffusion theory calculations performed by EXTERMINATOR-II [18] in cylindrical geometry. The collapsed fast group con~ stantsfor CORA and EXTERMINATOR-II were obtained from a standard 68-group, PHROG B-1 calculation [35]. The thermal group constants used in CORA and EXTERMINATOR-II were obtained from a 3q~group BRT-1 [45] calculation. BRT-1 is the Battelle-revised version of ihe industry benchmark computer program, THERMOS. The PHROG, BRT-1, and NUCPISTN group constants were then . compai~d with collapsed group _ constants obtained from 123-group, one-dimensional,Sn transport theorj calculations which were performed with the powerful trans port scheme of XSDRN [46] in spherical geometry. s 4 and s 6 quadratures were used.

PAGE 78

Two of the piston engines studied were next selected for incorporation into several nuclear piston engine power generating systems. These systems included the nuclear piston engines, a HeUF 6 -to-He heat exchanger, and gas or steam turbines, along with associated auxiliaries including pumps, compressors, condens crs, regenerators, etc. 40 Thermodynamics analyses were performed and fuel cost estimates were made for these piston engine power generating systems . In addition, a preliminary analysis for the HeUF 6 to-He heat exchanger was carried out for one of the piston ' engine power generating systems. As a result .of this analysis, it was possible to estimate the HeUF 6 circulation time, d th u 235 . t . th . l f h. an ,1ence . e 1nven ory 1n e primary oop or t 1s particular nuclear piston engine power generating system. Comparison of Initial Results with Previous Nuclear Piston Engine Analyses For a given set of initial conditions, the NUCPISTN code has yielded piston engine performaric~s which are sig nificantly different from the results obtained by Kylstra et al. i even when thermal absorption in the reflector has been neglected. Some of these differences are due to the improved cal culatiunal scheme utilized in ~WCPISTN and also to the im proved therm6dynamic cbnstants. Most of the differences, ho~ever, are a result of the improved group constants used

PAGE 79

41 in the NUCPISTN code. The crude approximations to the cross sections and their temperature dependence employed in the code used by Kylstra and his associates were extremely inaccurate. Upon inclusion of thermal absorption in the reflector, the performance differences between the previous and current studies became even greater. A wide variety of configura tions and loading schemes were consequently investigated. From these investigations, it be~ame apparent that the thermal absorption correction in the graphite reflector was of major importance. In fact, this correction so severely limited the graphite-reflected engine's performance, for the core sizes and operatirig conditions of interest, that these engines had to be discarded. As will be seen, heavy water reflected cores eventually became the basic component for the nuclear piston engine studies of this work. Referring to Table l, it will be noted that the engines analyzed by Kylstra et al. typically had compression ratios of around 10-to-l and clearance volumes of 0.24m 3 . The re sulting strokes were therefore excessively large--around 19 to 20 feet. The heavy water-reflected nuclear piston engines have been restricted to more reasonable stroke sizes of from arou~d ,3 to 5 feet. Neutronic c-0nsiderations require that the clearance volume be large compared to conventional in ternal combustion engines in order to achieve criticality.

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42 Hence, the compression ratios are therefore limited to around 3 oi 4-to-l for 3-to 5-foot strokes. Graphite-Reflected Systems Presented in Table 3 are the uranium-235, fluorine, graphite, and helium atom densities (in atoms per barn-cm) for five graphite-reflected engines at the TDC position. Also presented are the core and reflector region average physical temperatures and the engine dimensioos at the TDC position. Shown in Table 4 ar~ the neutron multiplication fac tors for these engines at TDC as obtained from various com putational schemes. The NUCPISTN results were from two group, two-regiorr diffusion theory equations in which the fast core equati-0n was replaced by a boundary condition. Hence, in this scheme, no fast interactions in the core are permitted (see Appendix A). The XSDRN results are from 123-group, two-region Sn transport theory calculations in which s 4 and s 6 quadratures were used. The CORA results are from two-group, two-region diffusion theory calculations in which the thermal group constants were obtained from BRT-1 and the fast group con stants from PHROG. Fast interactions in the core were in cluded in this scheme. All of the schemes utilized one-dimensional spherical geometry. The two-region, two-dimensional cylinder was

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TABLE 3 Atom Densities and Temperatures Reflected ~ngines at the Characteristic/Engine G-1 . G-2 Core Atom Jensities U-235 (atoms/barn-cm) 7. l 92x 10 -5 . 2.24xl0 -5 F-19 (atorns/barn-cm) H e4 (atoms/barn-cm) Reflector Atom Uensity C-12 (atoms/barn-cm) Core Temperature (OK) Reflector Temperature Core Radius= 54.8cm Core Height= 100cm ( o K) 4.315 x l0 -4 0.0 . -2 8.08 x l0 . 2576 2000 Graphite Reflector Thic k ness= 100cm Core Volume= 0.915m3 . l.344xl0 -4 -4 . l . 60 x l O 8.27 x l0 -2 1000 820 for L a rge GraphiteTDC Position G-3 G-4 G-5 2.67 x l0 -5 2.694 x 10 -5 2.687xl0 -5 . -4 l.602 x l0 l.616 x l0 -4 . -4 l.612 x l0 . -4 l. 56 x 10 3.39 x l0 -4 5.24xl0 -4 8. 08 x l0 -2 3. l3 x l0 -2 8. l8xl0 -2 2400 1800 1530 2000 1500 1270

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TABLE 4 Neutron M~ltiplication Factors for the Large Graphite Reflected Engines at the TDC Position Computational Engine Scheme G-1 G-2 G-3 G-4 G-5 2-group rwc p I STN k * eff. l . l 7 9 l . 005 l.025 l . 032 1. 037 *no fast interactions in core **with fast interactions in core ***with fast interactions in core; ****with fast interacti6ns in core; 2-group CORA k ** eff. l . 2 21 l . 052 1 . 06 7 l. 080 l . 086 S quadrature s 4 quadrature . 6 123-group XSDRN k . *** eff. 0.934 0.954 0.962 0.969 123-group XSDRN k **** eff. 0.934 0.954 0.962 0.969

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45 replaced by a two-region spherical configuration which pos sessed an equivalent volume core and an equivalent thick ness reflector. The CORA and XS0RN calculations were restricted to two regions in order to be compatible with NUCPISTN which can handle only two-region systems. The rationale for this restriction as well as justifications for using 11 equivalentil spherical systems will be -elaborated on in a later section in this chapter and also in Appendix A. More detailed accounts of the procedures used in generating the PHROG and BRT-1 constants will also be given in a later section in this chapter. In exami~ing the results of Table 4, it will be noted that the two-group NUCPISTN. keffectives are all around 4 .to 5% lower than the corresponding two-group CORA results. Some of this discrepancy is due to differences which exist between the thermal group constants generated by NUCPISTN and the BRT-1 thermal group constants which were used in CORA. The major portion of the discrepancy however is due to the fact that CORA includes fast core interactions and NUCPISTN does not; hence, the higher keffectives for the CORA results. The 123-group XSDRN k ff t . are all around 11 to e ec 1ves 12% lower than the two-group CORA k ff t The thermal e ec 1ves. group cutoff in BRT-1 is O.683eV and full upscattering and downscattering be)ow this energy are accounted for. The use o f t h e B RT 1 a nd P H ROG c o n s ta n t s i n C O RA me a n s t h a t a n y

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. upscattering to above 0.683eV is neglected. Th e 123-group XSORN calculation on the other hand allows for complete upscattering and downscattering and an examination of the XSORN results reveals that there is some upscattering to above 0.683eV. 46 The two-group CORA keffectives are thus highet than the XSORN keffectives for two reasons . First, the two-group analyses do not give proper emphasis to the non-thermal groups which a~e less productive than the thermal group. The two-group CORA calculations hen~e tend to overpredict k Second, the two-group CORA results neglect the effective upsca-ttering which occurs to above 0.681eV and this also causes k ff t to be overpredicted. Of the two effects, e ec 1ve the former is the more significant and this will be more cle~rly illustrated in the next chapter . . It will be noted that all of the above gr~phite r~flected engines at the TDC position are rather large. For any reasonable compression ratio, the resultant stroke would therefore also be large -too large in fact for serious con. sideration for the nucleqr piston engine. The component mechanical itresses for such an engine would be so great that the engine lifetime would indeed be short.

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Graphite-Reflected Systems Compared with Systems Using Other Moderating-Reflector Materials 47 Smaller sized engines with other moderatinq-reflector materials were therefore investigated. Neutron multiplica tion factors for some of these systems at the TDC position are tabulated in Table 5 for different computational schemes. The uranium-235, fluorin~, and helium atom densities for these particular systems are the same as for engine G-5 in Table 3. The core radius has be~n reduced from 54.8 to 34.55cm, the core height at TDC from 100 to 64cm, and the 3 3 core volume at TDC from 0.915m to 0.240m . The NUCPISTN and CORA results in Table 5 are aqain for one-dimensional llequivalent 11 spheres. The EXTERMINATOR-II results are for two-dimensional ~ylinders. Both CORA and EXTERMINATOR-II make use of the 'PHROG and BRT-1 fast and thermal group constants. It will be noted that for these smaller sized engines, the gra.phite-reflected configurations are far-subcritical. These same engines were analyzed by Kylstra et al. and their resulti indicated that these systems would be supercritical. Their thermal group constants however were in considerable error and they also ne9lected thermal absorptions in the reflector (see Chapter III). It should be pointed out that the graphite-reflected syst~ms in Table 5 could have their 'keffectives increased somewhat by increasing the uranium loading. However, these small systems are already rather

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. TABLE 5 Neutron : u l t i p l i c a t i on Factors for Small Engines vii th Various Moderating-Reflector Haterials at the TDC Position r1oderatingReflector Tv-m-group Two-group Four-group Two-group Four-group . Reflector Physical . i ' la teri a 1 Temperature (OK) Be 290 020 290 i) 0 370 o 2 o 570 2 C 290 C 570 C 970 C 1270 Core radius= 34.55cm Core height= 64cm Reflector thickness 3 = 100cm Core volume= 0.24m NUCPISTN* CORA* CORA* keff keff keff 0. 965 0.963 0.978 1. 183 1. 183 l. 132 1. 148 l . 15 l l. l 02 1. 087 1. 092 1. 044 0.845 .846 . 0. 796 0.817 0. 794 0. 776 Core atom densities are the same as for Engine G-5 in Table 3 *No fast interactions in core **Hith fast interactions in core CO RA** CORA** keff keff 0.970 1. 012 l. 200 l. 178 1. l 80 l. 154 1. 120 1. 097 0.864 0.338 Two-group EXTER~1IW!,TQR** k eff l. 210 1. 201 1. 143

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black to thermal neutrons so that large increases in uranium loading yield but small increases in the system keffective. For essentially infinitely thick reflector regions 49 (from a neutronics standpoint), the o 2 O-reflected systems possess the highest neutron multiplication factor for a given geometry and core loading. Most of the engines which are examined in this work have heavy water-reflected cores as their basic component. Some composite material moderating reflector studies have been done, and it is anticipated that future piston engine designs will most probably make us~ of such composite reflectors. In returning to Table .5, it will be noted that there is very little disagreement between the two-group NUCPISTN and the two-group CORA results in which fast core interac tions have been rieglected. This table clearly illustrates the statement made regarding the results in Table 4. That is, that the differences between the thermal group NUCPISTN constants and the .thermal group BRT-1 constants used in CORA are not of great significance. The inclusion or omis sion of fast core interactions is a much more significant factor. The inclusion of fast core interactions for the twogroup computations in Table 5 leads to k ff t which e cc 1ves are 2 to 3% higher than for the corresponding cases which neglect these interactions. For the four-group computations, the inclusion or omission of fast core interactions

PAGE 88

leads to differences in keffective which are as high as 5%. 50 In comparing the two-dimensional, cylindrical geometry EXTERMINATOR-II res~lts with the one-dimensional, spherical geometry CORA results, one observes that the latter possess keff~ctives which are 1 to 2% lower than the former. The difference is due to the fact that the 11 equivalent 11 spheres experience less fast leakage to the moderating-reflector region (where neutrons must undergo slowing down before they can efficiently produce fissions in the. core) than do the actual cylinders. Hence, the CORA keffectives are consis tently lower than the EXTERMINATOR-II results. When comparing the four-group results for the o 2 reflected systems with the corresponding two-.group results, the latter have k ff t which are always higher than the e ec 1ves corresponding four-group results. This is because the twogroup computations do not give proper emphasis to the non thermal interactions which are less productive than the thermal reactions. The two-group problems hence tend to . overpredict the ~eutron multiplication factors for these systems. In contrast, the two-group k ff t for the Bee ec 1ves reflected systems tend to be lower than the four-group results. The reason is that the two-group results do not properly account for the {n, 2n) production which occurs in

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the berylli~m at high energies. A listing of the group structure utilized in the four-group calculations is to be found in Appendix F. 51 Presented in Table 6.are physical temperatures for various reflector materials and their corresponding modera tor neutron temperatures. The neutron temperatures were obtained from BRT-1 calculations in which the reflector thickness was 100cm. The core composition was observed to have very little effect on these neutron temperatures and the results presented are in fact for engines with the geometry of Table 5 and with the core composition of engine G-5 of Table 3. Also presented are neutron lifetime results obtained from two-group, two-dimensional EXTERMINATOR-II perturbation calculations. The •reactor geometry was again that of Table 5 and the core composition that of engine G-5 in Table 3. The large size of the neutron lifetime in the modera~ing-reflector region relative to the ~ore region 1;fetime is to be noted. Moderating-Reflector Power In speaking of moderator characteristics, one frequently e n c o u n t e r s t h e t e rm s II s 1 ow i n g d own p owe r II a n d 11 m o d e r a t i n g r a t i o . 11 T h e s 1 ow i n g d o \'In p owe r i s d e f i n e d a s t: s w he re i s the average logarithmic energy decrement per collision or the average increase in lethargy per collision. If one

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TABLE 6 Moderator fleutron Temperatures arid Neutron Lifetimes for Small Engines with Various Moderating-Reflector Materials at the TDC Position Moderator Physical Mocierator Temperature :1aterial ( "K) Be 290 l) 0 290 [)20 320 ;lo 370 420 o 2 o 470 o 2 o 490 D 2 0 520 t/0 57 0 2 C 290 C 570 C 970 C 1270 i-1 o d e r a t o r Neutron Temperature ( o K) 396 384 424 490 558 624 660 691 439 790 1295 1560 Neutron Lifetime In'Core (msec) 0. 21 0 0.207 0. 199 Engine geometry same as for engines of Table 5 Core atom densities same as for Engine G-5 in Table 3 Moderator neutron temperatures from BRT-1 calculations fJ eu t ro n Lifetime in Reflector (msec) 2.028 l . 7 98 l. 488 Total Neutron Lifetime in System (msec) 2.238 2.005 l . 68 7 Neutron lifetimes from two-group, two-dimensional EXTERMINATOR-II calculations in which fast core interactions are included. <.n N

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53 considers the moderating materials of Be, BeO, C, o 2 o, and H 2 0 and orders them from best to worst according to slow ing dovrn po1t1er, the order is H 2 0, o 2 o, Be, BeO and then C. The moderating ratio is. defined as (~rs/ra) and the ordering of the above moderators from best to worst according to the moderating ratio is o 2 o, C, BeO, Be, and then H 2 0. Various reports on externally moderated, gaseous core reactors have attempted to order mode~ating-reflector ma terials according to neutronic efficiency by using various. lumpings of reactor physics parameters. Some have used the moderating ratio, others have used the Fermi age or the square root of the Fermi age in combination with the th~rmal neutron mean free path. While some of these gro~pings give the correct ordering for two or three of these materials, none give the correct ordering for all five materials. It is argued that a more reasonable grouping of P arameters is (TE tDt)-l o (fd/T E tot) which has been m a m r m m a m m m given the name "moderating-reflector power." If one considers a two-group, externally moderated gaseous core reactor, it is desired that ot and rt for m am the moderating reflector be small. It is also desired that f D for the m . f moderator be small and that L'. , the removal cross section rm from the fast group, be large (assuming that most of the removal cross section is downscattering to the thermal group). Since the Fermi age, T, for this region can be defined as m

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54 (Of/If), it is hence desirable Ill r that the combination T DtI t m t t -1 be small or that (1mDmia ) be large. The 111 fraction of the removal cross section which mm a 111 . d term f is the m i s d o ~rn s c a t t e r ing. Since it is desirable that a large fraction of the removal cross section from the fast group be downscattering rather than absorption the term {fi/1mD~Iat), i.e., the m 11 moderating-reflector power 11 should be larger, the better moderating-reflector material. This is a rather simple grouping of constants. When one is considering the desira bility of a moderating-reflector material from a neutronics standpoint, this combination takes into account most of the important effects. It does not, of course, account for all effects. For example, (n-2 -) production in beryllium is n ignored by this grouping. Table 7 lists moderating-reflector powers for the above five materials at 290K. For externally moderated, gaseous core reactors possessing an essentially infinite (from a neutronics standpoint) moderating-reflector, this simple combination of constants properly orders the above materials. Some Composite Moderating-Reflector Studies Appearing in Table 8 are some results for a ~ystem possessing a moderating-reflector region of varying composi ti~n. The core composition and geometry is fi-xed and the total moderating-reflector thickness is also fixed at 70cm. The o 2 o and Be thicknesses are allowed to vary from 70 to 0

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Material D 0 2 Be0 Be C H 2 0 TABLE 7 Moderating-Reflector P6wer for Some Moderating-Reflector Materials at 290K Moderating-Reflector Power 303.0 35.4 18. 9 13.5 11. 7 3 01 . 6 33. 5 l 7. 7 l 3. 5 11. 6 Above results for an essentially infinite moderating refl~ctor region t:a t and D~ obtained from BRTcalcuations m Tm and f~ obtained from PHR0G calculations 55

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56 TABLE 8 Beryllium-D 2 0 Composite Reflector Study at 290K Inner Reflector Region (Be) Thickness Outer Reflector Region (D?O) Thi c kne s 5 (cm) Four-Group CORA ( C Ill) keff 0 70 l . l 00 5 65 l . 04 6 l 0 60 l . 027 l 5 55 l . 01 7 20 50 l. 009 25 45 l . 002 30 40 0.997 35 35 0.994 40 30 0.003 45 25 0.993 50 20 0.993 55 l 5 0.992 60 l 0 0.991 65 5 0.990 70 0 0.989 Engine geometry same as for engines of Table 5 except reflector thickness is 70cm rather than 100cm. Core atom densities are the same as for Engine G-5 in Table 3. Moderating-reflector physical temperature= 290K. Fast interactions in core included.

PAGE 95

and from Oto 70cm respectiv~ly. The beryllium region is next to the core, when present, and the moderator physical 57 temperature is 290K. The penalty in the neutron multipli cation factor decrease for going from a pure o 2 o reflector to a pure Be reflector for this particular configuration is but around 11%. The nuclear pist6n engine studies which will be pre sented in the remainder of this chapter and in the following chapter will have pure o 2 o moderating-reflectors in order to maintain two-region systems for the neutronic calculations. Future investigations will probably utilize a composite reflector in which the inner 10 or 20cm consist of either Be or BeO and the remaining 60 to 80cm consist of o 2 o. The Be or BeO will allow for structural integrity and separate the liquid o 2 o from the gaseous core. The Be or BeO will probably be lined with nickel for low temperature cores or with a niobium illloy (e.g., TZN) for high temperature engines to protect the inner woderating-reflector region from the corrosive UF 6 gas. Remarks Concerning the Algorithms Used in the NUCPISTN Code The restriction of the steady-state calculations in NUCPISTN to two groups and twd regions in which the fast core equation is replaced by a boundary condition allows one to obtain fairly short computer execution times. As is

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58 explained in detail at the end of Appendix A, about five or six thousand timesteps are required for solving the neutronics equations for each piston cycle. Solution of complete two group, two-region or three-region problems by standard diffusion theory codes would involve IBM 370 computer execution times of about 20 minut~s for each piiton cycle analysis. By using the simpler equations developed in Appendix A, the 370 computer execution time for each piston cycle analysis is reduced to around 0.3 of a minute. As has been demonstrated, replacement of the fast core equation by a boundary condition and neglecting the fast interactions iri the core leads to errors ink which are of the effective order of 2 to 5 % The restriction at this point to two regions is justified since this work is an attempt . to ga~n insight irito the basi~ power producing and operational characteristics of the nuclear piston engine concept. Although some of the higher ordef neutronic calculations which were perform~d could easily have been extended to three or more regions, they were generally restricted to two regions so that the results could be compared directly with the two-region NUCPISTN results. The thermal grou . p core consta~ts used in the NUCPISTN code are generated internally as a function of the gas density throug~out the piston cycle. The fast and thermal group moderating-reflector constants are read into the code. They are obtained by independent means (e.g., from PHROG

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59 and BRT-1) and treated as constant during the piston cycle. Although the mode~ating-reflect6r constants depend to an extent on the ccire comaosition, treating them as constant during the cycle is a very good approximation if the reflec tor dimensions do not chan~e. The two-stroke engines dis cussed in Chapter III and in this chapter however utilize step-reflector additions and removals to obtain the desired subcritical to supercritical to subcritical behavior. For these engines the moderating-reflector qroup constants of the thick reflector system are input into NUCPISTN. During those portions of the cycle when the thin reflector is applied, the thick moderating-reflector group constants are generally in error by about 7 or 8% as compared to the actual thin moderating-reflector group constants. The error in the system k ff t however is usually onl . y l or 2 % . rt is to be noted e ec 1ve that these portions of the cycle are relatively unimportant anyway since the system is far-subcritical. These errors will not therefore noticeably affect the enqine 1 s behavior and the use of the thick moderatinq-reflector qrouo constants over the entire cycle is, even for these systems, a very good approximation. As mentioned in Chapter III, the step function in the reflector thickness is easy to attain analytically. Prac tically, it can only be approximated. One method of simu lating this behavior would involve usinq a sheath whose motion would b~ synchronized to alternately expose and

PAGE 98

shield the bulk of the reflecJor region from the core region. The sheath could be made of a mild neutron ab sorber material such as stainless steel. Another method, not involving any moving components, would depend on the moderating-reflector region being constructed so that its thickness varies in the proper manner along the length of the cylinder. An~ther approach would involve the use of . 60 a few poison or control rods rather than of changes in the reflector thickness. The rqds would be inserted into the moderating-refl~ctor region and their motion could be timed so as to attain the desired subcritical to supercritical to subcritical behavior during the piston cycle. The NUCPISTN code can accommodat(; P.ither poison additions and removals in the reflector region or step additions and removals in th~ reflector thickness: It is recognized that the use of moving sheaths or control rods complicates the piston engine design and, in fact, "gas generator" engines are covered in the next chapter ~hich require no variations in reflector thickness or control rod motions during their normal cycling operation. The UF 6 gas specific heat formula utilized by Kylstra etal. was

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61 where Tis i~ degrees Kelvin [47, 48]~ This formula however is valid only at temperatures around 400K and is rather. inaccurate at elevated temperatures. A formula which better fits the existing UF 6 data [41, 43] is given by C = 37.43 + (0.15xlO 3 )T (.6450xlo 6 )T 2 (cal/moleK) (2) p . where Tis again in degre~s Kelvin. This formula agrees quite well with the compiled UF 6 data over the temperature range from 400 to 2400K. The NUCPISTN code allows the user the option of selecting either one of the above formulas. As discussed in Chapter III, both helium and UF 6 are treated as ideal gases and comments on this approximation will be made in Chapter VII. A discussion of the numerical methods used for solving the NUCPISTN energetics and neu tronics equations, of the procedufes used for tim~tep selection for the energetics equations, of the effects of fuel enrichment variations on engine performance, and de tailed comparisons of the influence of delayed and photo neutrons on engine behavior will all be presented in the next chapter. The remainder of this chapter will focus on the operating characteristics of the simple two-stroke, piston engine and on the qualities of power generating systems which have these engines as their basic component.

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Param~tric Studies with NUCPISTN for Systems 62 Initial operating conditions for two such o 2 0-reflected engines appear in Table 9. Engine #1 differs slightJy from Engine #2 in thatits initial gas pressure is 14.6 atmospheres rather than 14.5 Also, Engine #1 has the step-reflector applied at the El = 0.050 cycle fraction and removed at the E 2 = 0.650 cycle fraction; Engine #2 has the step-reflector applied at the El = 0.100 cycle fraction and removed at the E 2 = 0.700 cycle fr~ction. A series of pertinent results obtained from NUCPISTN ~alculations are shown in Figures 8 through 10. Figure 8 shows the neutron multiplication factor for an essentially infinite o 2 o ieflector as afunction of u 235 atom density for various piston engine volumes. The reflector tempera ture is 290K and the helium-to-u 235 mass ratio is zero, i.e., the core is 100% UF 6 gas. For u 235 densities greater 20 3 than around 10 atoms/cm , the core is so black to neutrons that additional uranium has no effect on the system multi plication factor. Figure 9 contains the same information as Figure 8 except th~ helium-to-u 235 mass ratio is 0.322. The helium at this concentration (the gas mixture heli~m mole fra~tion is 0.95} has no detectable effect on the neutron lt . l. t f t F . u 235 d t th t mu 1p ,ca 10n ac or. or a given ens, y, e neuron multiplication factor is the same in Figure 9 as for the correspon~ing atom density in Figure 8. Figure 10 shows

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T/\BLE 9 Operating Characteristics for Engines #l and #2 Ch a r act er i st i c Number of strokes Compression ratio . Cle~rance volume (m 3 ) S troke (ft) Initial gas pressure (atm) Initial gas temperature (K) Cycle f~action for step-reflector addition removal Neutron source strength (n/sec) Engine speed (rpm) He-to-U mass ratio % U235 enrichment Photone~trons cdnsidered . Delayed neutrons considered Pure simple harmonic motion . Initial D20 reflector thickness (cm) D20 reflector thickness after step addition (cm) Neutron lifeti~e (msec) Piston cycle time (sec) u mass in cylinder (kg) He mass in cylinder (kg) Average DO reflector physical temperature {K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) Engine #1 2 3.0 to l 0. 18 0 3.8 14.6 400 0 : 05 0.65 1.ox10 9 100 0 . 332 100 No No Yes 30 100 l . 68 7 0.60 2.751 0.913 570 5.0 57.94 3 i . 32 Engine # 2 2 3.0 to l 0. 180 3.8 14.5 400 0. l 0 0.70 1.ox109 . 100 0.332 l 00 No No Yes 30 l 00 l . 68 7 0.60 2.727 0.903 570 5. 0 57.94 31 . 3 2 0) w

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FIGURE 8. 1.50 l. 4C 0,: l. 20 C iu < :,.. z C 1.10 .... f-1 < u j i::.. f-: :.oc = c::: ,,_. 5 0.9() z 0.80 0. 7C lB 10 Results from Two-Group ":\~}C?ISTX" C;)iculations 019 1 , \' 0.48 3 0.36 !r.J V m \' 0.24 3 :n V 0.18 3 rn Reflector Neutron Tem?crnt~re = 3S~K Reflector Physical Temperature• 290K D?O.Reflcctor T~ickncss = 100 cm H;-to-u 235 Mass Ra~io = 0,0 He-Ur 6 G?.s Te:nperature = 400K Li5 j L' ,\70: ! J::::--iSITY (.;tcr::s/cr.. ) , ()21 ..:. 1 Neutron Multiplication Factor Versus u 235 Atom Density for Syste~s Which Have No Helium Gas Present in the Core and a 100cm Thick o 2 o Reflector at 290K

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FIGURE 9. C ,... < u . .... 1. so . 1.40 _ 1.20 1.10 1.00 0,90 0 "r ov Results from T~o-Group ":,:JCPIST:-." CalcuL1tions 0.10)018 :ol9 V O.L.8 3 m 0.36 3 \' m 0.24 3 V m V 0.19 3 m Reflector Neutron Tcrnpcracure 3S~K Refl e ctor Physical Te~perature = 2~0K DO Reflector Thickness= !0 0 cm 2 235 Ec-t o-U Ma ss R~t io = 0.332 He-UF 6 Gas Tcmpcr~ture = 400K Neutron Multiolication Factor Versus u 2 35 Atom Density for Systems Which Have Helium G~s Present in the Core and a 100cm Thick o 2 o Reflector at 290K ' u,

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FIGURE 10. .... c... ,... ;-, I l. 40 , ., " ..,,. .Jv 1. 20 l. 10. 1.00 0.90 0.80 0.70 Results fro~ Two-Group ":-;L'CPIST~" C11lcul.:1tions , r,20 i' V 0.1,8 3 r., V 0.36 3 r., V = 0.24 3 m 0.18 3 V r., Rcflcctor-~cutron Tempcr~turc = 750K Reflector Physic.:11 ~cmper.:1turc 570K D 2 o Reflector ~hickncss 100 cm 235 Hc-to-U X.:1ss Ratio= 0.332 1021 1 f!235 , .... V . "." H"T""' . 3 .; n,O .. D •. ~.h.Y (nt0rr.s/cm) Neutron Multiplication Factor Versus u 235 Atom Oe~sity for Systems Which Have Helium.Gas Present in the Core and a 100cm Thick Reflector at 570K CJ) CJ)

PAGE 105

the multiplication factor for an infinite o 2 o reflector as a function of u 235 density for various piston volumes, at a reflector temperature of 570K. The helium-to-u 235 mass ratio i~ 0.332. Relative to the curves of Figures 8 and 9, the res11lts shown in Figure 10 indicate a noticeably lower k-effective for a given u 235 density due to the elevated reflector temperature. 67 Parametric studies to determine the effect which vari ations in the nuclear piston engine initial operating con ditions would have.on engine performance were carried out with the NUCPISTN code. The compression ratio~ initial gas pressure, clearance volume, engine speed, neutron lifetime, initial reflector thickness, step-reflector thickness, .h t u 235 t e 1umomass ra 10, cycle fraction for step-reflector addition, cycle fraction for step-reflector removal, initial gas temperature and neutron source strength were each inde pendently varied while holding all the other initial condi tions constant. Results for such a parametric study for Engine #1 are presented below. The behavior of the cycle power, peak temperature, peak pressure, and mechanical efficiency (net mechanical work p~rformed divided by the total fission heat released) as a function of the varied initial conditions are discussed.

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Effects of Compression Ratio and Clearance Volume Variations In Figures 11 and 12, the mechanica . l power, peak tem perature, peak pressure and mechanical efficiency increase with the compression ratio, The higher compression ratio system have increased displacement or working volumes. This allows for a greater amount of pV work. Also, for fixed clearance volumes, the . higher compression ratio systems have larger reactor volumes throughout the cycle ex~ept, of c-0urse, at the TDC position. This allows for a greater amount of fission energy release. For fixed initial pressures and temperatures, the larger working volumes and greater fission energy releases of the higher compression ratio systems mean larger peak pressures and largfar peak temperatures. 68 The clearance volume is the core volume when the piston is at TDC. Increased clearance volumes for a fixed compres sion ratio and for a fixed initial temper~ture and press~re mean increased initial volumes with a corresponding increased initial gas loading. This effectively means a larger reacto~ throughout the cycle and hence the . increase in mechanical power, peak temperature, and peak pressure with the clearance volume in Figuresl3 and 14 . Despite the increase in mechani. cal power or pV work, the m~chanical efficiency is seen to decrease because the fission heat released increases at a faster rate than does the pV work. This implies that an

PAGE 107

10.0 ;-: ...., 4,0 2.G All Initial Conditions Except the Cocprcssion Rntio Are These of Engine lll ~'hich Appear in Table 9 _._ --2.88 r 7..90 ! 2.92 2 2.94 2.98 3.00 ' ,,, , ,, ,, / ,,, 3.02 I , I I I I I 3.04 I I I I I I I I 3.06 280 240 160 120 -: ea 40 3.05 Mechanical Power and Peak Gas Press~re Versus Compression qatio for a Two-Stroke Engine O"I I.O

PAGE 108

2~.9 28.8 ,... 28. 7 >u 7:,.; u . G l:. : ,.J < u ... z < 28.S u 28.4 All Initial Conditions EY.cept the Co~prcs3ion ~tio Aie These of Er.gine /fl \' ~h Appear in Table 9 ----,I / / , , I I I I I I I I I I / . I. . 4000 3500 JOCO Q . . ._, l!) 2000 1500 L,J p.. 1000 . ------J.02 J.04 J.06 J.08 2.86 FIGURE 12; 2.88 2.'JO 2.92 2.94 2.96 2.98 CO~!PP.F.SSIO~: R,\TIO 11echanical Power and Peak Gas Temperature Versus Compression Ratio for a Two-Stroke Engine '-I 0

PAGE 109

. ,-J ...: u .... 14 .c I 12.0 10.0 S 6.0 u 2 4.0 2.0 ,Ul Initial ConJ itions Except the Clearance Volu~c Are Those of Engine fll h'hich Appear in Table 9 ------------., ... ., ... ,,,, ,,,, ,,,, . ,,,, ,, / ,, / / / / 240 / 200 160 120 80 40 0.182 0.183 6 ...., "' ..., t,l 5 .C/l C/l p.. t.:) ~w p.. 'J.172 0.173 0.174 0.175 0,176 0.177 C. 178 0.1/9 ' 0.180 C.181 FIGURE 13. ' ~echanical Power and Peak Gas Pressure Versus Clearance Volume for a Two-Stroke Engine '-J _,

PAGE 110

29.2 29.l 28.~ 28.8 28.7 28.6 t.11 Initial Ccndi~ion~ Except the Clearance Volume lire Those o f Engine /1 1 \:" .• :!. :;h Appear in T.iblc 9 -----,I ,, ,I ,, ,, ,, 0.179 0.180 0.181 ,I ,I ,I ',1 ,I / I I 0.182 0.183 4000 3500 ,... :,.: . 3000 '-" i:! ....i 25 00 I"' < L? 2000 p.. . 1500 1000 . 0.172 0.173 0.174 0.175 0.176 0.177 0.178 FIGURE 14. Mechanical Efficiency and P~ak Gas Temperature Versus Clearance Velum ~ for a Two-Stroke Engine -....J N

PAGE 111

increasing fraction of the fission heat released is going towards increasing the gas mixture's internal energy and a decreased fraction towards pV work. Effects of Initial Pressure and Initial Temperature Variations In Figures15 and 16, peak pressure, peak temperature, and mechanical power increase with increased initial pres sure, because of the greater initial u 235 loading and hence greater fission energy release. At an initial pressure of 73 14. 6 a t m o s p h er e s , the u235 atom density is but l.3xlo 19 atoms/ 3 for this engine. Referring back to Figure 1 0 , it i s cm seen that this is well below the region where the core is so black to neutrons that additional u 235 is ineffective. Here again, the fission heat released increases faster than the pV work. An increasing fraction of the fission heat released contributes to an increase in internal energy and the mechan cal efficiency decreases with increased initial pressures. The effect -of the initial gas temperature on the engine behavior is shown in Figuresl7 and 18. From the ideal gas equation of state, it is seen that increased gas temperatures mean decreased gas loadings if the other parameters are fixed. Hence, the decrease in engine mechanical power, peak temperature and peak pressure with increased initial temperature. Despite the decrease in pV work, the mechani cal efficiency increases since an increasing fraction of the

PAGE 112

< u ;;,: < :c u s:! 12.0 6.0 4,0 2.0 All !n!tial Cnnditiona Except the Initial Gas i:='.c ssure Ar,~ Those of Engine /11 1,,'hich Appear in Table 9 ... ------------------,.,.. .... , , , , , I , I I , I I I I , , 240 200 160 120 80 40 e .. "' ,_.. !,l rt! ::, . V) V) -Q. t., !,l ""' _,,__--r,-:-;;---;-:--:--13.8 13 ., 4.0 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14 . 8 14.9 FIGURE 15. r:;ITIAL GAS PRESSL'RE (ntm) ~echanical Power and Peak Gas Te~oerature Versus Initial Gas Pressure for a Two-Stroke Engine

PAGE 113

29.J :! :, . l U 29.0 < :!3.9 '28.7 13.8 FIGURE 16. ... ... -----13.9 14.0 14.1 All Initial Con~itions Except the Initial Gas Pressure Arc Those of Engine lll ~"hich Appear in Table 9 .,. .,. .,. 14.2 14.3 14.4 14.5 INITIAL GAS !'!'..ESS!.T .. F. (at::i) / ,I ,I , I I 14.6 I I I I I I I I I 14.7 I I I I I I 14.8 14.9 3200 2800 ,..,. 2400 ::.: 0 t,J ex: i:;:. 2000 w 0... .... w .... U') ..,; t., 1600 t,.l 0... 1200 800 r,:echanical Efficiency and Peak Gas Temperature Versus Initial Gas Press::re fer a Two-Stroke Engine

PAGE 114

..l < . :.) ' 8.C 6.() ' ' \ ' \ ' ' ' ' ' ' ' ' ' ' All Ir.1t1al Concitions Except the Initial Gas Temperature Are Those of Engine 01 ~hich A?pear in Table 9 .... .... -i:--:-240 200 160 e ... Ill ...., t..l c;,: ::, Vl Vl ::::l P.. .. I.!> 120 80 "--l p.. 4.0 ltr-:~ Ic,il ---------. Pc:.ri:;I? 2. C 408 410 . 392 394 396 398 400 402 404 406 FIGURE 17. INITIAL GAS Tf:TERAT!:RE (K) Mechanical Power a _ nd Peak Gas Pressure Versus Initial for a Two-Stroke Engine 40 412 4 Gas Temoerature '• --..J 0)

PAGE 115

:,.. u 29.3 29.2 29.1 t2 29.0 ..... ~ .... '--t: --:: 28.9 u .... 28.8 26.7 392 FIGURE 18. \ '\ \ ' '\ \ \. ' ' \. . ' 396 398 All Initial Concitions Except the Initial Gas Te~pcrature A rc Those of Engine Ul which A?pcar in :able 9 ... ......... .... ...... ... ........... .... ... ... _ 400 402 404 406 INITIAL G.',S :E~:J'!::lA:URE ( K) 4000 3500 JOC9 2500 2000 1500 -1000 ---408 410 4 2 414 M echanical tfficiency and , Peak Gas Temperature Versus Initial Gas Temperature for a Two-Str-0ke Engine ,-.. :,.:: 0 ~ "-l ,/ p... ti t-< c.:> :,.:: < w "" ......., .......,

PAGE 116

78 fission energy released goes towards the pV work rather than towards increasing the gas mixture's internal energy. Effects of Engine Speed and Neutron Lifetime Variations Figuresl9 and 20 show that for a given neutron life time, there is definite range of desirable engine speeds for practical engine performance. At high engine speeds there is not sufficient time to allow for significant neu tron flux buildup. Therefore, the fission heat release, the mechanical power, the peak temperature, and the peak pressure are all small. At slow speeds, there is too much time for flux buildup and the peak pressure and peak tem perature rapi~ly increase beyond physically acceptable limits. A comparable effect is shown in Figures21 and 22, where for a fixed engine speed, a definite range of desirable neutron lifetimes for practical engine performance exists. For long neutron lifetimes, the engine speed is too rapid ' to allow a significant flux buildup and fission heat release. Therefore, the mechanical power, peak temperature, and peak pressure are all small. At small neutron lifetimes, signifi cant flux buildup at the given engine speed is possib.le. The peak pressure and peak temperature begin therefore to increase rapidly with decreasing neutron lifetime. Factors which determine the n~utron lifetime are the core-fuel

PAGE 117

--. :r :l: ._, ::,: ..l -,; u ... :,:: u !;l ... FIGURI: 19. \ \ . 12 .o 10.0 8.0 6.0 4.0 z.o 94 96 \ \ \ \ \ \ 98 100 102 All In!tial Conditions Except the Engine Speed Are Those of Engine Dl . \..'hich A;ipca.r in Table 9 ........ ... ----------104 106 108 110 E~CICE SPEED (r~~•a) L 200 160 120 80 40 ,... G ... "' ._, Mechanical Power and Peak Gas Pressure Versus Engine Speed for a Two Stroke Engine -...J I..C

PAGE 118

,.... .... FIGURE 20. 29.2 29.1 29.0 28.9 28.8 28.7 28.6 \ \ 9 \ \ \ 9p All Initial Conditions Except the Engine Speec l1rc These of Engine fJl ~'hich Appear in Table 9 ' ' \ 1 0 ' ' ' ' ... ... .... ...... 192 , 4 E~GINE sr:ED (rpo's) .... .... .... ... ... 10 4000 3500 ,.... 3000 . ._, 2500 w t.!) 2000 p.. 1500 1000 ... ---1 8 11 112 ~echanical Efficiency and Peak Gas Ternpefature Versus Engine Soeed for a Two-Stroke Engine

PAGE 119

14 .0 \ \ \ 12.0 10.0 ,.0 4.0 2.0 l. 62 FIGURE 21. \ ' \ \ . , . \ \ ' 1.64 \ \ ' ' ' 1.66 , ' ' 1.68 , .... 1.70 l 1. 72 . r All Initial Conditions Except the Neutron Lifeti~e Axe Those of Enr,ine Ill \..'hich Appear in Table 9 1. 74 r .......... ... ---l.7S I 1.78 t --1.80 l !,!::UTRO!{ LIFETl}!.E ( m sec) 1. 82 l r 240 200 160 120 80 40 ,... 5 .. ro ._, ~echilnical Power and Peak Gas Pressur~ Versus Neutron Lifetime for a Two-Stroke Engine co _:,

PAGE 120

,.... "" ._,, >u :.--: .... '-'• .... J ::;; u .... w 29.2 29.l 29.0 28.;; 28.8 28.7 2. (, \ \ \ \ \ \ \ ' ' ' ' \ ' ' ' All Initial Conditions Except the Neutron Lifetime Arc Those of Engine (/1 Which Appear in Table 9 ---..... -. ----4000 :!500 3000,.... 0 ._,, gJ 2500 w S: w t-< .,, 2000 i3 1500 1000 1.62 1.64 1.66 1.68 1.70 1. 72 1.74 1.76 1.78 1.80 1.82 1.84 FIGURE 22. ' NEt:TRO!: LIF!:TH'.!: (Msec) Mechanical Efficiency and Peak Gas Temperature Versus Neutron LifetimG for a Two-Stroke Engine co N

PAGE 121

83 characteristics, the moderator-reflector material, the core and moderator-reflector temperatures, as well as the system geometry and region dimensions~ Effects of Variations in the Initial ~..!:!.S!---2.~-Reflector Thicknesses* Figures23 and 24 show the effects of variations in the initial reflector thickness. The step-reflector addi tion which is made to the initial reflector thickness at the 1 = 0.05 cycle fraction position is such that the total reflector thickness after 1 is 100cm. For small initial reflector thicknesses, the system during the t i me be f o r e 1 i s fa r s u b c r i t i. c a 1 . T he c y c 1 e po r t i o n b e for e the step-reflector addition is therefore of little importance and this is reflected in the flatness of the mechanical power, peak pressure, peak temperature and efficiency curves in this region. As the initial o 2 o reflector thickness approaches 55cm in thickness the system, during the time before 1 , is approaching critical. The cycle fraction before the step-reflector addition is now important and this is reflected in the rapidly increasing mechanical power, peak temperature and peak pressure with increasing initial. *The reader is referred to the Introduction in the previous chapter for a discussion of the reason for applying a thick step-reflector to an initially thin reflector at some piston engine cycle fraction before TDC.

PAGE 122

,,-: :,.J :, 0 p.. ,..J < u .... u :.: 14 .c 12.0 10.0 8.0 6.0 4.0 2.0 0 FIGURE 23. All Initial Conditions Except the Initial DO and n 2 o Str-~•Reflector Thicknesses Are Those of Engine 01 'Wh::..ch Appear in Table 9 (the step-reflector addition is varied in such a 1,:ay that the initial plus the step-reflector thickness is always 100 cm) G;,..s pRr.ssuRl'.. -?l?.f>R -: -----------------10 1 20 2 3 280 240 160 120 80 40 40 45 5 55 Mechanical Power and Peak Gai Pressure Versus Initial o 2 o Reflector Thickn~ss for a Two-Stroke Engine ,.... I= ..., "' ....,

PAGE 123

,.... N -29.0 27.0 26.0 :!J. (I 24.0 FIGURE 24. 0 All In.itial Conditions Except the !.nitial o 2 o and o 2 o Step-Reflector Thicknesses kc Those . of Engine lll ~hich Appear in Table 9 (the step-reflector addition is varied in such a w~y that the initial plus the step-reflector thi~kncss is always 100 cm) -----------5 10 20 25 30 35 I!HTIAL D 2 0 REFLECTOR T!iICK!:ES5 (cm) / / 40 45 / / I I I I I . I I I 50 I I I I I \ 55 4000 3500 Q 3000 i:: t.l 25:JO (..; V) < c.., 20Ci0 t,J "" 1500 1000 Mechanical Efficiency and Peak Gas Temperature Versus Initial o 2 o Refl~ctor Thickness for a Two-Stroke Engine co (;)'1

PAGE 124

86 The effects of varying the thickness of the o 2 o step reflector which is added tothe initial reflector thickness at the El = 0.05 cycle position are shown in Figures 25 and 26. The initial o 2 o reflector thickness is 30cm. The mechanical power, peak temperature and peak pressure in crease with ir1creased o 2 o step-reflector thickness. This is due to tl1e system becomirig more and more supercritical upon application of increasingly thick step-reflectors. Effects of Variations iri the Cycle Fraction Position for Step-Reflector Addition and Step-Reflector Removal Figures 27 and 28 show that there is a range of de sirable times or positions in the cycle for application of the step-reflector in order to obtain attractive engine per formances. If the step-reflector is imposed late, an in sufficient fraction of the cycle is left for neutron flux and power buildup and the peak pressure, peak temperature and mechanical power output are all small. As the step-reflector is applied at earlier and earlier times or positions, the neutron flux buildup and fission heat release increase rapidly. Hence the rapid increase in mechanical power, peak pressure and peak temperature. The effectsof variati6ns in the time or position for removal of the step-reflector are shown in Figures 29 and 30. Asthe step-reflector is removed at later and later times or positions, the neutron flux buildup and fission heat

PAGE 125

...... ._, ....: < u .... 14.0 12. 0 . ..1 . .. _ 8.0 4.0 2.0 62 . All Initial Conditions Except the n 2 o Step-Reflector Thickness he Those of En;;ine 111 Which Appear in Table 9 -.. ------------------61 67 ' .... ..... 63 ' 69 I D20 STEP-REFLECTOR TH! CK::r:ss (ctn) 70 I I I I 71 , I I I I I I I I 72 I 73 280 240 200 160 a ., d ...., _ ti) < <.:) 120 co 40 p.. FIGURE 25. Mechanical Power and Peak Gas Pressure Versus o 2 o Step-Reflector Thic ~ ~ ~ ss for a Two-Stroke Engine 00 -..J

PAGE 126

,-,. "' ._,, >t u :a! w .... u .... . t: 29,3 29.1 29.0 2s. s 28.8 28.7 FIGURE 26. (,2 All Initial Conditions Except the D20 Step-Reflector Thickness he Those of Engine Ill t..'hich Appear in Table 9 6:l 64----,.., .... -65 66 67 68 69 D 2 0 STEP-REFLECTOR TI!!C!c-ESS (cm) / / / / 70 I / I I I I 71 I I I I I I I I I 72 73 4000 3500 . 3000 "' !:'= < c,:; w E-, 2500 Ci r V, < l.: 2000 < w ;l. 1500 1000 Mechanical Efficiency and Peak Gas Temperature Versus Step-ReflectJr Thickness for a Two-Stroke Engine co co

PAGE 127

14.0 12.0 ,.... 10.0 2 ct: w :< C 0.. s.o .J 0 :I: u 6.0 4.0 2.0 o.oo FIGURE 27. ' ' ' \ \ \ \ \ \ ., 0.02 ' ' ' ' ' ' 0.03 0.04 All Initial Conditions Except the Cycle Fraction for Step-Reflector Adc!ition Are Those of Engine lll ~hich Appear in Table 9 ..... ... .. ... ... ... .. ... ---0.05 0.06 0 07 0.08 0.09 I CYCL" FRACTIO!< FOR STEP-R!:FLECTOR /IDJITION 350 300 250 200 150 100 --50 0.10 0.11 ,.... 8 .., r.: ,_, Mechanic~l Power and Peak Gas Pressure Versus the Cycle Fraction for SL2D Reflector Addition for a Two-Stroke Engine co I..O

PAGE 128

29.0 28.9 2!!. E 2tl. 7 28.6 FIGURE 28. \ \ ' \ \ \ \ \ \ ' \. ' \ \. ' 0.02 ' ' All Initial Conditions Except the Cycle Fraction for Step-~eflcctor Addition Are Those of Engine Ill l..'hich t.ppear in Table 9 . ' ' ' ' ' .. . 0,03 . 0.04 o. 05 0.06 0.07 CYCLE FRACTIO~ FOR STEP-R!:FLECTOR /,:lDITION f 5600 480 0 40 00 ,-.. :,.. 0 _ ....., 32 00 < ix: ;.,J .J 24 00 t'.;;! l: :-,: < w p.. 1600 ---~--.. -800 o _ . oa 0.09 0.10 0.11 M echanical Efficiency and Peak Gas Temperature Versus the Cycle Fract icn for Step-Reflector Addition for a Two-Stroke En9ine I..O 0

PAGE 129

ll, .0 12 .() :0. (, ,..,. & ex: w i! 8.0 ""' ,-l < u z G 6.0 w :i: I.,.(; 2.G 0.58 FIGURE 29. All Initial Condftions Except the Cycle Fraction for Step-Reflector Removal .i're Those of Engine Ul Vhich Appears in Table 9 ----------0.61 0.62 0.63 0.64 0.65 CYCLE FMCTIOX FOR STEP-R!:FLI:CTOR RE!•:OVAL I I / 0.66 I I I I I 0.67 I I I I I I I I 0.68 0.6' 310 270 230 110 70 ,..,. e .... "' ...... ''.echanical Pov,er and Peak Gus Pressure Versus the ~ycle Fraction for S:p ~cflector Removul fdr il Two-Stroke Engine <..O .....

PAGE 130

34.0 32. C _ 30.0 R ....... 28.0 2&. < J 24. 0 22. () All Initial Conditions Except the Cycle Fraction for Step-Reflector Removal Arc Those of Engine lll \./hich Appear in Table 9 ----_ _ _ ,_.,. --0.65 0.66 I I I 0.67 I I I I I I I I I 0.63 0.69 5500 4800 4000 ..-.. ::.: 3-200 L,l 1,-< Ill 2400 t3 1(,00 800 L,l p.. Q .58 0 59 0.60 0.61 0.62 0.63 . 0. 64 FIGURE 30. CYCLE FRACTION FO R STEPR EFLECTOR RE~!OVAL Mechanical Efficiency and Peak Gas Temperature Ve r sus the Cycle Fracti o n for Step-Reflector Removal for a Two-Stroke Engin e I.O N

PAGE 131

release increase rapidly. The mechanical power, peak temperature, and peak pressure therefore also increase rapidly. Early step-reflector removals cut into the flux buildup and. fission heat release and therefore cause de creased mechanical power outputs, peak temperatures, and peak pressures. Effects of Variations in the Helium-to-u235 Nass Ratio 93. 23 Effects of variations in the helium-to-U 5 mass ratio on engine performance are shown in Figures 31 and 32. For a fixed initial pressure, an increased helium-to-u 235 mass ratio means a decreased u 235 loading and fission heat re lease during the cycle. The peak temperature, pressure, and power thus drop off as the helium-to-u 235 mass ratio increases. The increase in this ratio means a lower a~erage specific heat for the gas mixture which means that a decreas ing fraction of the fission heat released goes towaids in creasing the gas mixtures internal energy. It is for this reason that the mechanical efficiency increases with increas235 ing He-to-U mass ratio. Effects of Variations in the ~eutron Source Strength Figures 33 ~nd 34 show that a range of desirable neu tron source strengths exist for practical engine performances. At low source strengths, the initial flux, resultant flux

PAGE 132

\ \ \ 14.0 \ \ \ \ \ \ 12.0 \ \ \ ' 3 10.0 :.: ,_, tr! w c . 3.C ',...l -; L) .... 7. :5 u 6.0 w :.: 4.0 2. (I 0.324 0.326 0.328 0.330 0.332 ?35 All Initial Conditions Except the He-to-u l-'~1ss R.a tio ge Those of Engine f/1 l.'hich A?pear in Table . 9 ... .... .... ... .... ... .... --0.334 0.336 0.338 0.340 0.342 !le-TO-U2J5 }!ASS RATIO 300 260 220 180 140 100 . 60 .346 ,,...._ E ... r., ...... W tr! ;::, t/l t/l w r.,: p.. t/l ..: w p.. FIGURE 31. . 23 5 Mechanical Power and Peak Gas Pressure Versus He-to-U Mass Ratio fer a Two-Stroke Engine

PAGE 133

29.4 28.8 28.6 23.L . ' ' , . ' ' ' ' ' ' ' ' 0.324 0.326 All Initial Conditions Except the He-to-v235 Y..:iss Ratio Are Those of Engine ll l ~'hich Appear in Table 9 .... 0.328 0,330 0.332 0,334 I 0.336 He~To-u 2 3 5 }~SS PJ\TIO -----0.338 0.340 0.342 0.344 0,346 15600 4800 4000 ,,.... :,,: . ._, L? 2400 1600 800 '-'l p.. FIGURE 32. Mechanical Efficiency and Peak Gas Temperature Versus He-to-u 235 Mass ~~tio for a Two-Stroke Engine <.D l11

PAGE 134

14. Q 12.0 10.0 C 1 ..,J ... , 6.0 4.0 . 2. 0 All In1t1nl Conditions Except . the Neutron Source Strength kc Those of Engine 01 . l..'hich Appear in Table 9 I / / I I 1 X 10 8 1 x 10Y llEUTRON SOURCE STRE'.;GTII (nct:trono/scc) I I I I I I I I I I I 290 240 200 . e ... 160 ::, ti) ti) ;,.. ti) .,: 120 (.!) til p.. 80 40 FIGURE 33. Mechanical Power and Peak Gas Pressure Versus Neut~on Source Strength for a Two-Stroke Engine '-0 '

PAGE 135

29. l 29.(; 28 . H 28. '/ 2e. FIGURE 34. A!l Initial Conditions Except the Ncu~ron Source Strenr,th l\:r:c Those of E:igin<' Ill ~hi::h ,\ppenr in Table 9 ----I i I I 1 X 10 8 1 X 10 NEUTRO!l SOt:RCE STRE!:GTH (neutrons/sec) I I I I I / I I I I I I I I I I 4000 3500 3000 ,... i.: . ...., l>l 2500 5 i-;';'? i... . p. :r: i... i-2000 <.:> ;j i:.. 1500 . 1000 1 e c h c1 n i c a 1 E ff i c i ency a n d Pe a k Ga s Tempe r a tu re V e rs u s N e u tr o n Sou r c e Strength for a Two-Stroke En9ine I..O ......,

PAGE 136

98 buildup, and fission heat release are all very small .. As the source strength increases, the initial flux, resultant flux buildup and fission heat release all increase rapidly . . The mechanical power, peak temperature, and peak pressure therefore also increase rapidly. Performance Analysis of Two ~ 2 0-Reflected Piston Engines NUCPISTN compression-power cycle results for Engines #1 and #2 are presented in Table 10 .. The mechanical power output for Engine #2 is slightly less than for Engine #1 while its peak and exit temperatures are higher. Piston Engines #1 and #2 have been incorporated into model nuclear piston engine power generating systems which will be elab orated on late.r in this chapter. NUCPISTN results for Engine #1 are illustrated in . Figures 35 through 39. Figure 35 shows the variation bf the 020 reflector thickness during the compression-power cycle. The average gas temperature, average gas pressure, average core thermal neutron flux, and neutron multiplication factor or k ff t during the compression-power cycle are shown e ec 1ve in figures 36, 37, 38, and 39 respectively. The system becomes supercritical upon application of the step-reflecto~ at the 0.050 cycle fraction and becomes subcritical again at the 0.650 cycle fraction upon removal of the step reflector. The discontinuities in the average core thermal

PAGE 137

TABLE 10 Cycle Results from NUCPISTN for Engines #1 and #2 characteristic Maximum gas temperature (K) Maximum gas pressure (atm) j,1 a x i mum k e f f Exhaust gas temperature (K)* Average core therm~l neutron flux during the cycle . (n/cm 2 sec) Average gas temperature during the cycle (K) Average gas pressure during the cycle (atm) Average keff during the cycle Mechanical power output (Mw) Fission hea t released (Mw) Mechanical efficiency (%) 3 Photoneutron ~recursor concentration (#/cm ) 3 Delayed neutron precursor concentration (#/cm ) Thermal (1/v) in core (sec/cm) Fast (1/v) in reflector (sec/cm) Maximum co~e thermal neutron flux during cycle (n/cm sec) Average mass flow rate (lbm/hr) Gas Temperature at Cycle Finish ( K)* Gas Pressure at Cycle Finish (atm) Engine #1 2090 l 4 9. 6 1 . l 0 9 l 2 51 2. 17xlo 14 993 5 9. l 0.926 3.75 13.00 28.8 ----6 2.54xl0 7 l.123x106.lO x lo15 6.6lxl04 l 609 58. 7 Engine #2 2144 l 2 8. l l . l O 7 1368 2.30xlo 14 974 55.7 0.924 3.21 13.70 23.4 ----6 2.54xl0 1.123x10-7 6.3lxlo 15 . 6.56xl04 1774 64.3 *The exhaust gas temperature is the temperature of the gas emerging from the piston; the gas temperature at the cycle finish is the gas temperature at the end of the compression power cycle before it is released from the cylinder. \D \D

PAGE 138

~ , . , j 4 0: .. J 0'\ ' ) o. :. 1 :. o o9 o~ ... 70 -50 40 i) 20 ... 10 0.0 . . Co:-:1prcssion RD.tic = 3.0 235 i'bSs Ratio 0.332 He-to-U = . Initial Gas Temperature = 400K I I ' I Initial Gas Pressure = 14.6 .::itm .. Step Reflector Addition = 0.050 S tep Reflector Removal = 0.650 Cle.::irance Volume = 0.180 3 . m . . . . .. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 . ' I I I I I I I ' CYCLE FR. \ CTIO}! FIGURE 35. n 2 o Reflector Thickness Versus Cycle Fraction for Engin~ #1

PAGE 139

,...... 28G8_ 2400 2000 1200 2JO Compression Ratio= 3.0 235 '1 R . 0 332 He-to-lJ :ass . :1t1.o = . Initial Gas Temperature= 400K Initial Gas Pressure= 14.6 atm Step Reflector Addition= 0.050 Step Reflector Removal= 0.650 3 Clearance Volume= 0.180 m 400 L---------0.0 0.1 0.2 0.3 0.4 C .,5 ' I CYCLE :-:-:!,C'I'ION FIGURE 36. Gas Tem.perature Versus 0.6 0.7 0.8 0.9 1.0 ! __, 0 __, Cycle Fra.ction for Enqine #1

PAGE 140

120 100 ,-... .,.J .. .._, :::::: 80 ;:J ::.r. er.. t.:J ::::::: ::i., ::.r. < ~; 60 40 20 Compression Ratio= 3,0 u 235 '1 R . 0 332 He-to~ass atio = . I~itial Gas Temperature= 400K Initial Gas Pressure= 14.6 atm Step Reflector Addition= 0.050 Step Reflector Removal= 0.650 Clearance Volume= 0,180 m 3 1.-0_. o ___ o.:...1 ___ 0_ .!..2 ____ 0J..._J ___ o.1... 4 ___ 0_. __ 5 ____ 0.1... 6 ___ o_ .J.. Y ____ o...:.._8 ___ 0-1.. 9 ___ 1j FIGURE 37. Gas Pressure Versus Cycle Fraction for Engine #1 0 N

PAGE 141

10 14 ,....._ tJ '.r. N E :., 1012 = '-' ~< -. :.:.., . :?. lOlO ':::::i t.:..: z ...J ::5 10 8 c2 ,., r . ' =, u 10 6 ::J v < w > < 104 0.0 0.1 0.2 I FIGURE 38. Average 0.3 Core Thermal Compression Ratio= 3,0 Hc-to-u 235 ~f.:iss Ratio = 0. 332 Initial Gas Temperature= 400K Initial Gas Pres~ure = 14.6 atm Step Reflector Additi on= 0,050 Step ' Reflector Removal= 0.650 3 Clearance Volume= 0,180 m 0.4 0.5 . 0. 6. CYCLE FR ,\ C:TION 0.7 0.8 Neu t ron Flux Versus c'yc le Fraction I 0.9 1.0 ' . C .:.,.; for Enqine #l

PAGE 142

1.0 ::r: 0.9 ~ z 0 ,-I 0. 8 ..... < u H ~ -< 2 0.7 .: . . ,. . .. ,. ,.... :::i 70.6 0.5 0.0 FIGURE 3 9. 0.1 U 235 " R . i O 332 He-to~ ass ,at o = . Initial Gas Tc~perature = 400 K Initial Gas Pressure= 14.6 atm Step Refl ec tor Addition= 0.050 Step Reflec tor Removal= 0.650 Clearance Volume= . 0.180 m 3 0.2 0.3 0.4 0.5 CYCLE F?...-\C:':o:-: Neutron Multiplication Factor Versus 0.6 0.7 0.8 0.9 1.0 __, 0 -r:::. Cycle Fraction for Enoine # 1

PAGE 143

neutron flux at the 0.050 and 0.650 cycle fractions are a result of the system passin~ from the subcritical to the supercritical and. from the supercritical to the subcriti cal states, respectively. 105 Referring to Figures36 and 37, it will be noted that for Engine #1, the gas pressure and temperature peak at around the 0.675 cycle fraction position and then decrease steadily ~ntil the end of the cycle. The mechanical power output from Engine #1 could be increased by shifting these pressure and temp~rature curves s6 that they peak at earlier cycle times. It should be apparent that such a shift will result in cooler exhaust gases from the engine. The mechani cal power will continue to increase,-as the curves are shifted, until the pressure and temperature peak near the 0.500 cycle fraction (TDC). Further shifting of peak pres sure and temperature to earlier cycle times will then lead to decreases in mechanical power as well as cooler exhaust gas temperatures. The gas pressure and temperature for Engine #2 (with the lower mechanical power and higher exhaust gas tempera ture) as a function of cycle position are not shown. How ever, they yield curves quite similar to those of Engine #1 except that_ they peak later in the cycle--at around the 0.725 cycle fraction position. This is expected in view of the above discussion. The hotter exhaust gases for Engine#2

PAGE 144

l 06 mean that it should yield higher turbine power outputs than Engine #1. A tradeoff is thus seen to exist between mechani cal power from the piston and turbine power extracted from the hot exhiust gases. One can be i~creased at the expense of the other, or alternately, the configuration which yields the maximum combined mechanical plus turbine power can be sought. Aside from physically applying and removing the step reflector earlier or later in the cycle, there is another way to effect a shift in the cycle position for the peaking of the gas pressure and temperature. Shifts can also be ob tained by varying the u 235 loading (and hence the k-effective). B . th u 235 1 d. k th t y 1ncreas1ng e oa 1ng, one can ma e e sys em reach a given maximum pressure and temperature sooner in the cycle. It will be noted that a higher u 235 loading and the different step-reflector application both contribute to the earlier pressure and temperature peaking for Engine #1. It should be brought out at this time that the maximum permissible gas temperature is limited by the increased dis sociation of UF 6 at elevated temperatures. The temperature at which dissociation sets in increases as the UF 6 partial pressure increases. For Engines #1 and #2, this temperature limit is around 2200K. Higher peak gas temperatures would mean higher mechanical powers and/or exit gas temperatures. These, however, could be attained only by going to higher UF 6

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partial pressures and higher overall gas pressures. The maximum permissible overall pressure is, of course, limited t:iy physical considerations. The maximum UF 6 11 partial 11 pressure (u 235 atom density) is limited by a practical consideration already discussed. Beyond a certain density, the core is so black to neutrons that additional u 235 has no effect and is essentially wasted. NUCPISTN Results Compared with Higher Order Steady-State Neutronic Calculations 107 Some comparisons bf NUCPISTN results with higher order steady-state calculations have already been made in the be ginning of this chapter. Neutron multiplication factors from NUCPISTN were compared ~ith results from CORA, XSDRN, and EXTERt1INATOR-Il. A fe\-1 remarks wi 11 no11 be made con cerning the systematic comparisons which have been made be tween the various computational schemes for all of the reference engines presented in this work. For each of the reference engines, the keffective and the core thermal ~roup cross sections output by the NUCPISTN code at various cycle positions were singled out for com parison with more elaborate calculations. As previously discussed, the static calculations in NUCPISTN were per formed using t~o-group, two-region, one-dimensional diffus sion theory equations in spherical geometry. These were c-0mpared with results from twoand four-group, one dimensional, diffusion theory-calculations performed with

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l 08 the CORA calculational scheme in spherical geometry and with corresponding two-dimensional diffusion theory calcu lations performed vlith the EXTERMINATOR-II code in cylindri cal geometry .. In addition to normal forward eigenvalue calculations which yielded neutr6n multiplication factors and flux distribut~ons, adjoint and perturbation calculations were also performed with CORA and EXTERMINATOR-II. The latter calculations permitted a determination of the first order perturbation approximation to the prompt neutron life time for thesys1ans and a cal.culation of effective delayed neutron fractions. The fast group constants (l or 3 collapsed fast groups) for these calculations were .obtained from 68-group PHROG calculations. For the reflector, a B-1 approximation was used to obtain the neutron flux and current spectra. The computed energy spectrum was then used to collapse the reflector constants into the desired broad-group structure. The core fast group constants were obtained from a cell calculation in which the reflector fluxes and currents were input and the core broad-group cross sections were averaged over this spectrum [2]. The thermal group constants (1 collapsed thermal group) were obtained from 30-group BRT~l calculations. BRT-1 com putes the scalar thermal neutron spectrum as a function of . position by numerically solving the integral transport equation. Thirty space points and anisotropic scattering co~rections were applied in the BRT-1 calculations.

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109 The.two-group k from the NUCPISTN code were e ff e ct iv es consistently lower than the values from the two-group EXTERMINATOR-II and CORA calculations, the average difference generally being around 3 or 4%. As already discussed, this difference is due primarily to the fact that NUCPISTN neglects fast core interactions. For the investigated pis ton positions, for all configurations, the maximum observed difference between these sets of k ff t was a little e ec ,ves over 5%. Except for the cor~ thermal diffusion coefficient (or core thermal transport cross section), the core thermal group cross sections from NUCPISTN were within 2.5% of the core thermal group constants from BRT-1. The differences in the core thermal diffusion coefficients were as great as 18%. This result was certainly not unexpected. The core current-weighted thermal diffusion coefficients obtained from the BRT-1 transport calculation were repeatedly smaller than the core thermal diffusion coefficients from NUCPISTN. The NUCPISTN constants were obtained from a very simple algorithm in which ( 3 ) The quantity ( 4 )

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was approximated by ( 5 ) Group constants and neutron multiplication factors were also obtained from 123-group, one-dimensional, Sn transport theory calculations in spherical geometry using the XSDRN code. s 4 and s 6 quadratures were used. The differences in the keffectives and group constants for the 11 0 s 4 and s 6 calculations were negli~ible indicating the ade quacy of an s 4 level computation. Collapsed group constants obtained from XSDRN were generally within 2 to 3% of the corresponding PHROG and BRT~l constants used in CORA and EXTERMINATOR-II. The XSDRN thermal core constants were also in reasonably good agreement with the NUCPISTN constants. Illustrations of the agreement between these various sets of group constants will be presented for one of the four stroke engines in the next chapter. As already mentioned, the XSDRN results have shown that there is some upscattering to above 0.683eV in the piston engine systems which have been examined. The neglect of this upscattering results in errors in keffective which are of the order of l to 2%. This will also be clearly demonstrated in the following chapter. In view of this sma error and in view of the tremendous savings in computer costs, the use of the PHROG and BRT-1 moderating-reflector

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constants in the NUCPISTN code rather than the use of XSDRN moderating-reflector constants is easily justified. l l l Tl1e neutron multiplication factor as a function of the o 2 o reflector thickness is shown in Figures 40 and 41 for a reflector temperature of 570K and a core volume of 0.24m 3 . The results displayed in Figure 40 are from two group NUCPISTN calculations while those in Figure 41 are from four-group CORA calculations. The 100cm thick o 2 o reflector is seen to yield k-effectives which are ~90% of the k-effectives obtained with_ an infinitely thick o 2 o reflector. Fast and thermal group fluxes from a two-group CORA calculation for a D 2 0-reflected core are shown in Figure 42 as a function of reactor radial position. The D 2 0 reflector is at 570K and the core volume is 0.24m 3 . Fast and thermal adjoint fluxes from a two-group CORA calculation for the same configuration are shown as a functiori of radial position in Figure 43. Attempts to account for the perpendicular neutrori leakages by means' of bucklings, and attempts to simulate two-dimensional calculations by means of CORA buckling iteration problems in one-dimension have encountered signifi cant difficulties. Theoretic~lly, there is no justification for the use of bucklings to account for perpendicular neu tron leakages in the general case. The approach, however, is a fairly common one which has historically met with reasonably

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1.2 1.0 ::,: 0 .... ,.; ~ t,.. z 0.8 0 .... ... < u .... ...i ;... .... ... ...i -, 0.6 ;:: 7. 0 "' I-< :=, w 0.4 10 FIGURE 40. 20 30 40 50 60 7C Fast Interactions !n Core Neglected Reflector Neutron Te~perature 750K Reflector Physical Te::nperature 570K Voiume o[ He-UF 6 Gas• 0.240 m 3 l!e-to-u 235 Hass Ratio . 0.332 235 U ! lv,s fo Enuine 2.516 kg !',• _ -UF 6 GusTc>m!)c>rnture l(,0()K 235 19 3 U Atom De~nity 2.~e7 x 10 (atom9/c~) eo 90 lC0 110 120 130 140 150 1~0 170 160 190 2C0 n 2 c R!.:FLECTC,l T'.t!CK:,F.SS (cm) ~eutron Multiplication Factor Versus o 2 o Reflector Thicknes~ as Obtained from Two-Group N UCPISTN CalculJtions __, ...., N

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!..2 LO
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1.0 0.9 0.8 0.7 I>( ::, 0.6 "t,J :> 0.5 ..... i::i :.J 0.4 /01. 0.3 0.2 0.1 FAST FLUX CORI! TI!!::R.'1AL FLUX REFLECTOR ,, Fas~ Interactions in Core !ncluded Reflector Neutron Tc~pera:ure 750K Reflector Physic~l Te~perature 570K Voluce of HeCF 6 Gan• 0.24 reJ lle-ta-u 235 Mass Ratio• 0.332 215 . U Mass in Engine• 2.516 kg HeUF 6 G~s Ten?erature 1600"K D20 Reflector 7hickness 100 c~ . . 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 lj5 120 RADIAL DISV.:;cc ~cin) FIGURE 42. Fast and Thermal Neut~on . Flux Versus Radius far a Core as Obtained from Two-Group CORA Calculations __,

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l.O 0.9 r;: I-< 0.8 :z .... 0 0.7 ; < C.6 .... f... < ..J 0.5 0.4 0.3 0.2 0.1 0 FIGURE 43. 'rHER.V.AL ADJOINT FLUX FAST ADJOINT FLUX CORE , , I REnECTOR Fist Incera~tions in Core Included Reflector Neutron Ter.iperature 750"K Reflector _ Phy~ieal Temperature• 570K Volume of lleUF 6 Gas •.0.24m 3 He-to-u 235 Mass Ratio• o : 332 u 235 Hass in Engine• 2.516 ki lleUF 6 Gas Temperature• 1600K o 2 o Reflector Thickness . 100 cm ,. ' 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 _105 110 115 120 RADIAL D!ST/~;cE (cr.i) Fast and Thermal Adjoint Neutron Flex Versus Radius for a Co~e as Obtained from Two-Group CORA Calculations ..... __, u,

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11 6 good success in treating solid-fuel reactors. For the gaseous cores considered, the flux is generally so flat that there is essentially no flux -curvature (in the core) and the diffusion coefficients are so large that the use of 2 OB terms to account for perpendicular neutron leakages generally yields very poor results. Hence, explicit two dimensional calculations will be required to accurately . account for neutron leakage and flux distiibutions in both the radial and height directions for the cylindrical cores. Some Remarks Regarding the o 2 o Temperatures The average temperatur~ of the o 2 o in the moderating reflector for the reference engines in this chapter was generally taken to be 570K. The pressure required to pre vent boiling of the o 2 o at this temperature would be about 1180 psia. This would be difficult to achieve in a large tank and going to individual pressure tubes in the moderating reflector region would overly complicate the system. There are several justifications offered for using this high temperature in the initial nuclear piston engine in vestigations. Fir~t, with respect to the moderating reflector's neutronics properties, this condition represents a 'w o r s t c a s e 11 c o n d i t i on . T ha t i s , i f t h e s y s t em c a n a t ta i n the necessary keffectives for a given fuel loading at this temperature, there will be no difficulty in attaining the

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required k ff t. at lower moderator tem~eratures. e ec 1ves . Second, the systems analyzed in this chapter used 100% enriched UF 6 . Actual systems will probably operate with 90 to 93% enriched UF 6 . The use of 100% enriched fuel tends to overestimate the actual k which will be effectives 11 7 obtained. The use of S70K as the moderator temperature on the other hand tends to underestimate the keffectives which will be obtained and these two effects partially cancel one another. In the next chapter, systems with less than 100% enrichment and with o 2 o average temperatures of 490K are examined. The pressure required to prevent boiling in the D 2 o at this temperature is but about 315 psi a. Finally, it should be recognized that the actual moderating-reflector region will probably be a composite reflector with Be or BeO in the inner portion and the n 2 o in the outer segment. For this arrangement, if the average overall moderating-reflector temperature is around 600K, the o 2 o portion would have an average temperature which is considerably lower than this. Exhaust Gas Temperature Calculations for the Two-Stroke Engines As will be discussed in the neit chapter, the average exhaust gas temperature for the four-stroke engines is obtained by averaging the temperature of the gas as it passes through the exhaust valve during the exhaust stroke.

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11 8 During the exhaust stroke, an open system, non-steady flow energy equation ii used. For the two-stroke engines, a somewhat different method must be used to obtain the exhaust gas temperature .. Only compression and power strokes are considered for these ~ngines and the energy eq~ation is the non-flow, closed system equation ( 6 ) where the terms in the above equation are defined in Appendix E. The nuclear piston engine is a reciprocating machine, however, and if the properties at various points within the system vary cyclically, the flow through the engine can be treated as steady. The open system steady flow equation is given by (7) The net amount of heat added to the system from the sur round i n gs , Q , i s zero i n the above e qua ti o ns . The heat of the nuclear reaction, qR, and the net amount of work done by the system on the surroundings, vi, have the same values in the above equation as in equation (6). The total heat of the nuclear reaction, the total pV work and the temperature it the finish of the compression power cycle, Tf, are obtained by the NUCPISTN code as it

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solves equation (6) during the piston cycle. By imposing ' ( the balance given in equation (7) on this quasi-steady state system we have for the HeUF 6 ideal gas mixture ( 8) In the above equation, cp and cv are the average specific heats at constant pressure and volume respectively, Ti is the initial gas temperature and Te is the gas temperature after exhaust from the engine. The quantity 6Uc is the change in internal energy for the closed, non-flow system while 6H 0 is the change in enthalpy for the open, steady flow system. 11 9 In the above treatment, any friction losses or fric tional pressure drops outside the engine in the external loop are ignored. If-the total mass in the closed system is exhausted from the engine we have the following relation (Te-T.)c l p ( 9) The above equation .can .then be solved for the exhaust gas temperature, Te. The temperature drop which the gas under• goes when it is exhausted fro~ the cylinder is thus (Tf-Te). For this case, the change in internal energy for the open, steady flow system in not 6Uc but rather 6U 0 where 6U 0 = m(T -T.)c e 1 V ( l O)

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The difference between 6H 0 and 6U 0 is the net flow work performed by the system on the gas as it passes across the system (eng1ne) boundaries. Mass Flow Rates for the Two-Stroke Engines For the four-stroke engines which will b~ considered 120 in the next chapter, the quasi-steady-state overall mass flow rates depend upon the input mass, the piston motion and pos sibly the back pressure. If the engine executes simple harmonic motion, the residual gas mass left in the cylinder after the exhaust stroke depends on the engine clearance vcilume and back pressure. Once this residual mass is known, the overall quasi-steady-state mass flow rate can be dete~ mined. Design considerations for a nuclear piston engine are quite different from those of a conventional internal com bustion engine. For the nuclear piston engine it is extremely desirable that essentially all of the gas be ex hausted from the cylinder on each-stroke. If this were not the case, the engine performance would change in time e~en if the charging gas properties and other engine characteris tics were maintained as constant. The change in performance would be brought on by the change in time of the nuclear properties of any residual gas remaining in the cylinder. This would mean that control of the engine could become extrernely complicated. Thus, the engines of the next chapter

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utilize a combination of simple and non-simple harmonic motion. Most of the cycle experiences simple harmonic l 21 motion. A portion of the intake and exhaust strokes however undergo non-simple harmonic motion and essentially all of the gas in the cylinder is physically forced from the cylin der during the last portion of the exhaust stroke. The overall mass flow rate in this case depends only on the input mass and the piston cycle time . . For the two-stroke engines which consider only the com pression and power strokes, the overall mass flow rate can be determined by pursuing one of two assumptions. Either it can be assumed that all of the gas is somehow forced from the cylinder or it can be assumed that a portion of the gas remains in the cylinder and its mass is determined by the clearance volume and the back pressure. For compatibility with the four-stroke engine analyses of the next chapter, the former assumption was used when estimating the overall mass flow rates for the two-stroke engines. Thermodynamic Studies for Three Nuclear Piston Engine Power Generating Systems Engines #1 and #2 were incorporated into several nuclear piston engine power generating systems. These systems con sist .of the nuclear piston engine, a HeUF 6 -to-He heat exchanger, and gas or steam turbines along with associated

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auxiliaries .including pumps, compressors, condensers, regenerators, etc. This arrangement differs significantly from the one studied by Kylstra et al. in which only piston mechanical power was to be extracted. Their results indicated that 122 exit temperatures would be around 650K {~710F) and since the cycled gas was to feed back into the piston at 400K, they properly suggested just coolihg the gas _without attempt ing to extract any more work from the He-UF 6 . The present calculations, however, indicate that exit temperatures for .theie same configurations will be around 1300K (~1900F). Just cooling the He-UF 6 gas from 1300K to 400K would represent a tremendous waste of available energy. The hot He-UF 6 can be us~d to ~enerate hot helium by means of a heat exchanger. The hot helium can then be used to provide power directly by means of a g~s turbine or it can be used to produce steam to provide power indirectly by means of a steam turbine. Nuclear Piston-Gas Turbine Steam Turbine System The first power generating system investigated included both gas and steam turbines. The gas turbine makes effi cient use of the He-UF 6 high temperature range while th~ steam turbine makes efficient use of the He-UF 6 low tempera tuie range. The nuclear piston-gas turbine-steam turbine

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123 schematic is shown in. Figure 44. Temperature-entropy (TS) diagrams for the steam turbine and gas turbine cycles are shown in Figure 45. The turbine and compressor efficiencies and regenerator effectiveness for the gas turbine loop com~ ponents are representative of the state of current helium gas turbine technology, as developed by Gulf General Atomic [49]. Some results from the thermodynamic analysis of the piston~steam turbine-gas turbine power generating system ar~ presented in Table 11. Piston Engine #2 was the basic com pon~nt for this power system. Piston-Steam Turbine System The second power generating system investigated had only a steam turbine loop. A schematic for this system is shown in Figure 46 while Figure 47 contains a Ts diagram for the steam turbine cycle. Some results from the thermo dynamic analysis of the piston-steam turbine power generating system are presented in Table 12. Piston Engine #1 was the basic component for this power system. Piston-Cascaded Gas Turbine System The final power generating system investigated had only gas turbine loops. Figure 48 shows the system schematic while Figure 49 contains the Ts diagram for the gas turbine cytles. Some results from the thermodynamic analysis of the

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FIGURE 44. YASTZ $!OR.AC! PISTON r:;::;JHE BLDCX .'I Fl6T0SS) %13 ,.i. 2io•, XAr..!-t.rP CA!! ~--"• __ _, Piston-Gas \! h i c h U s e s Turbine-Steam Piston. Engine ] 11,•, 4H p,u T 11 l" bin e i/2 II• Schematic for the System

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T 630F 496.6F n 90;; C nT a 85:l: Steam Exit Quality= 91.2% 443F 420F 102.1F 102F 101.7F 1 psia 5 7 81 214 psia cle ficiency J l. 9:, ------------------------------s 16so•r n 90% C nT 90:7. STEAM TURBINE CYCLE Regenerator Effectiveness= 87% 3' 1115F. ____________ __ 105s•r 9':16F ------------------. -:;'>-I} d' saa•, 435•; 316f __ 292f -cl' sorb' r.) \.0 l' Cycle Efficiency 50.4 5' 6a' 125 .._ _____________________ ...;... _________ 5 FIGURE 45. GAS TURB I 11E CYCLE Steam and Gas Turbine Temperature-Entropy Diagr&ms for the Piston-GJ~ Turbine-Steam T u r b i n e Sy_ s t e 111 W h i c h U s e s P i s t o n E n ~I i n e /! 2

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126 TJ\CLE 11 Summary of Thermodynamic Results for the Piston-Gas Turbine-Steam Turbine System Which Uses Piston Engine #2 Energy I11put to Fluids Per Piston Energy Output from Fluids Per Piston Fission Heat Rate Compressor Po\.'1er Pump Power 13. 69 M\v Gas Turbine Power l . 68 M1v Pis ton Mee hani cal Power 0.01 ~w Steam Turbine Power Qout {gas turbine loop) 15. 38 ~l\ Qout (condenser) Qout (steam generator Output Power Per Piston Piston Mechanical Power Net Gas Turbine Power Net Steam Turbine Power (Net Turbine Power 3. 21 Mv, 2. 12 Mw l. 42 M\.'I 6. 75 Mw Power Breakdown for 54 Mw(e) Unit 8 pistons at 6.75 Mw(e) per piston for a 54 Mw(e) unit exit) Piston Mechanical Power Output 25.68 Mw Net Gas Turbine Power Output 16.96 Mw Net Steam Turbine Power Output 11 .36 Mw 54.0 Mw (Total Turbine Power 28.32 Mw) 3.80 M\.'1 3.21 Mw l. 43 ~hv 2. l O Mw 3. 02 M~t 1.82 rhv 15. 38 M\.'I

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!..'ASTE SIO!(.\GE 17C;0F 262 psia PISTON c:;cn:r. BLOCK (8 PI~TONS) 260F liUT cu:.-.::-m M.1\K[-UP ~---~ G'S 214 psia " •,.:ASTE s:o;tAGE COLD cu:tS-L'P 260F 214 psia He 80nF J50F He He lie COOLER STEA."'! 1050F 5()0 psia 1050F !!PT :----T 2400 p~i.t:~/1 t:l 2.5 in Hg :as.7F 2400 p!lia 110.JF I-< . V, [V-_____. V"' 1Cii.7~F Pi.,':lP 2 .5 in Hg 0 N :,:: 0 N ::; FIGURE 46. Piston-Steam Turbine Schematic for the System Which Uses Pistcn Enqine #1 __, N ---.J

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1050F 662F 640' . 110.3F 109.S"F 108.7F T 11 904 C llT 907. Ste a m E x it Quality= 92.67. Cicle Efficiency= 41.07. 2.5 in Hg 128 5 7 81 ~-----..;_ __________ .;._ ____________ s FIGURE 47. Steam Turbine ~ Temperature-Entropy Diagram for the Piston-Stet1111 Turbine SystP.rn \-lhich Uses P i s t o n E n i n t ! '! l .

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129 TABLE 12 Summary of Thermodynamic Results for the Piston-Steam Turbine System Which Uses Piston Engine #1 Energy Input to Fluids per Piston Energy Output from Fluids per Piston Fission Heat Rate Pump Po\'1er 13.00 Mw Steam Turbine Power 0.04 Mw Piston Mechanical Power . 3 .08 Mv1 3.75 ~\'/ 13.04 Mw Qout (condenser) 4.35 Mw . Qout (steam generator exit) 1.86 Mw output Power per Piston Piston Mechanical Power 3.75 Mw Net Steam Turbine Power 3.04 Mw 6.79 Mw Power Breakdown for 54 Mw(e) Unit 8 pistons at 6.79 Mw(e) per piston for a 54 MvJ(e) unit Piston Mechanical Power Output 30.00 Mw Net Steam Turbine Power Output 24.32 Mw 54.32 Mw 13. 04 Mv1

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FIGUllE 2015F Hor CLE,\JI-UP 282 psla COLll HSTON E:iG !:,;; !!LOCK (8 PISTONS) Cl.L\:1lleUF 6 UP !1.\KE-UP GAS LOil Tn!PERATL1lE 80' He 435"t He I6sor 667 p!>la He 1364r 445 psia 1J64 F 667 psia He L Re _ __,, ~---I He 4 33F R 2 o He 0 ..... ;?, 1120 w c.:, 105(f F lOOO psb He w .., 0 0 u o: w i-. :-: .... 130 .\ 74 f' 250 P"ia' l096F l[J!Xlpsl.1 He t..J 80't ...: /1 0 'IU, 1000 psla He CWJ.rn '50psla 269F JL0F He u 2 o 632•r 445 p9la He Pi s ton Ca s c .:i cl(• d Ga s Tur bi n e Schein a t i c Sy s t e rn vi h i c h U s e s P i s t o n E n ~I i n e N 2 for 'the

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FIGURE 49. 1 31 T Cascaded Turbine Overall Cycle Efflclency 35.5%. n 90% C l'JT 90% Regenerator Efiectlveness 80% 1650F 5 .1364 f 1332'F 1096F lOSO"F 632F 477"F 433'F 269F 250F so•r l b' 6a 4 --1068F High Tecp. Turbine . Cycl~ Efficiency• 13.07. l' 1oso•r _ __ 3' I 474'.! ---}--' I . 4i' 310Fe' Lou T•cp. Turblne Cycle Efficiency• 32.31 4'a " Gas Turbine Temperature-Entropy Diagram for the Piston-Cascaded Gas Turbine System Which Uses P i s t o n E n ~J i n c # 2

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piston-cascaded gas turbine power generating system are presented in Table 13. The cascaded gas turbine arrange ment which is used for this system has a single reheat stage in the high temperature gas turbine cycle and single stage intercooling in the low temperature gas turbine 132 cycle. The heat rejected from the high temperature gas turbine cycle (6a-l) is used for heating by the low tempera~ t u r e g a s tu r b i n e c y c l e (-d ' 3 1 ) T he n e t t u r b i n e po \'t e r o u t p u t from the cascaded gas turbine arrangement is seen to be about 2.75 MW(e) per piston. In contrast, a single gas turbine cycle extending over the entire He-UF 6 temperature range could provide, at most, about 2.25 MW(e) per piston. The basic component for this power system was again Engine #2. Preliminary Heat Exchanger Analysis A preliminary analysis for the HeUF 6 -to-He heat ex changer has been performed for the 54 MW(e) piston-steam turbine power generating system. The necessary property values for the UF 6 gas such as specific heats, thermal conductivities, viscosities, etc. were gathered from tables and formulas found in references 41, 43, 47, and 48. The total heat t~ansf~r rate for the heat exchanger is 2.52xlo 8 BTU/hr. The total surface area for heat trans fer is 6340 ft in a once-through counterflow arrangement. Three hundred and twenty nickel-coated coaxial tubes located

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133 TABLE 13 Summary of Thermodynamic Results for the Piston-Cascaded Gas Turbine Syste1n Which Uses Piston Engine #2 Energy Input to Fluids per Piston Energy Output for Fluids per Piston Fission Heat Rate Compressor Power 13.69 Mw Gas Turbine Power 6.39 Mw Piston Mechanical Power . Qout {gas turbine loop) 20.08 Mw . Qout (He buffer-loop} Output Power per Piston Piston Mechanical Power Net Gas Turbine Power 3.21 Mw 2.76 Mw 5.97 Mw Power Breakdown for 48 Mw(e) Unit 8 pistons at 5.97 Mw(e) per piston for a 48 Mw(e) unit Piston Mechanical Power Output 25.68 Mw Net Gas Turbine Power Output 22.08 Mw 47.76 Mw 9.15 Mw 3.21 Mw 4.86 2.86 20.08 Mw

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134 on approximately 4.5 in centers form the heat transfer surface area. The inner tubes carrying the helium are of 1.25 in inner diameter and of 1.50 in outer diameter. The outer tubes enclosing the He-UF 6 are of 1.95 in inner diame ter and 2.25 in outer diameter. The helium is heated from 80F to 1420F while the He~UF 6 is cooled from 1790F to -260F. The overall heat exchanger dimensions are 6 feet.in diameter by 60 feet in length. The He-UF 6 recirculation time in the piston loop is five seconds leading t~ a u 235 loop i n v en to r y of 1 8 5 kg f o r t hi s 5 4 M \•,( e ) sys t em . Th i s i n v en tor y estimate does not include any UF 6 that would be in cleanup or in stdrage_ for make-up gas purposes. Comparison of the Nuclear Piston m_i n e p O \'/er G en e r a t i n 9 sys t e Ill$_ Presented in Table 14 is a summary of the thermody namic analysis for the three nuclear piston engine power generating systems. The piston-gas turbine-steam turbine setup yields the highest turbine power per piston of the three arrangements. This is a consequence of the gas tur bine being able to make efficient use of the He-UF 6 high temperature range and of the steam turbine making use of the He-UF 6 lower temperature range. However, because of the high HeUF 6 exhaust gas temperatures (from the #2 Piston Engine ~sed for this system), the mechanical power outputs are lower than for the engines used in the piston-steam

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System Pistoncascaded gas turbine Pistonsteam turbine Pistongas turbine:.. steam turbine TABLE 14 A Comparison of Thermodynamic Results for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engines # 1 and # 2 Tota 1 Fission Heat Rate of Heat lO verall Mech anical lurbine P0\ 1er! Heat Transfer Cooling ~.lat er Po1 , 1er Heat Release Rejection Efficiency Po\'1er per per Piston Rate per Un it Fl m v Rate per Output Rate per per Piston (%) Piston (Mw) Power Unit Pmver per Piston U1w) ' (Hw) Efficiency (i3TU/hr 1 1\ i) (gal/min rlt1) Pis ton (f iw) Efficiency ( % ) ( i;hv) ( % ) 3.21 2.76 7 5.97 13 . 69 7.72 43.6 1 . 33x 10 295 23.4 20.2 3.75 3.04 1.15xl0 7 6.79 13.00 6.21 52.2 209 (46.8)* 28.8 23 . 4 3.21 3.54 1 . 23xl0 7 6. 75 13.69 6.94 49.2 235 23.4 25.8 * Overall efficiency when the 11 mechanical 11 efficiency is the same as for the piston-cascaded gas turbine and piston-gas turbine-steam turbine systems~ _, w t11

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136 turbine setup. (Recall the discussion on page 10~) Thus, despite the high turbine power, the total turbine plus mechanical power and overall efficiency for this system is less than foi the piston-steam .turbine system which utilizes Piston Engine #1. One other point to note on the piston gas turbine-steam turbine setup is that this system is the most complex in terms of the number of required components. The piston-steam turbine setup is not able to make as efficient use of the high temperature He-UF 6 as is the piston-cascaded gas turbine setup. However, it is able to make use of the He-UF 6 in the 735F to 260F range which the cascaded gas turbine system cannot do. Consequently the piston-steam turbine arrangement has a higher turbine po0er per piston than the piston-gas turbine arrangement. Because it utilizes the #1 Piston Engine with the cooler exhaust gas, it also has a higher mechanical power output than the piston gas turbine setup. The piston-steam turbine setup will be more costly than the piston-gas turbin~ arrangement, however, because of the existence of the steam generator between the HeUF 6 -to~He heat exchanger and the steam turbines. For the piston-gas turbine system, the helium is used directly in the gas turbines and no component comparable to a steam generator exists between the HeUF 6 -to-He heat exchanger and the turbines. Economic studies will be required to determine wh~ther or not the increased costs of the more complex systems override the better efficiencies and power outputs

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137 of these systems. Also, the desirability of turbine versus mechanical power will have to be assessed for each individual application. Table 15 gives the reactor volume per unit power for the three nuclear piston engine power generating systems. Also included are corresponding numbers for three operational nuclear reactor power systems. The reactor volumes per unit power for tne nuclear piston engine systems are seen to be much smaller than for the large CO 2 reactor, and quite comparable to the moderately sized Russian, graphite moderated BWR. The reactor volumes per unit power for the relatively compact HTGR reactor are about one-half the values for the nuclear piston engine systems. Figure 50 is a diagram of the core and moderating reflector arrangement used for Piston Engines #1 and #2, which were the basis for the three nuclear piston engine power generating systems. A sketch of an 8-cylinder nuclear piston engine block alon~ with overall dimensions appears in Figure 51. From the three power generating systems examined, it can be seen that such 8-cylinder piston engine blocks will be the basis for power generating units in the 44 to 52 MW(e) capacity range. Heat rates and fuel cost estimates for the three nuclear piston engine power generating systems appear in Table 16. Th~ fuel cost estimates are based on the use of 90% enriched UF 6 at $11 ,000/kg U.

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TAGLE 15 Reactor Volume per Unit Power for Three Operational Nuclear Reactor Power Systems and for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engines #1 and #2 Hinkley Point A CO 2 Reactor Russian Graphite i-1od era ted B\~R Piston-Gas Turbine Unit Piston-Steam Turbine Unit Piston-Gas Turbine Steam Turbine Unit Peach Bottom HTGR Reactor Volume p e r U n i t P o 1 e r ( m 3 I : \'I e ) 8.92 2.43 2. 41 2. 1 2 2. 1 3 1 . 06 Reactor Volume per Un it Po 1ve r (ft3/Mwe) 314 85.9 84.9 74.5 75.0 39.8 T o t a 1 P o \•/ e r (;hie) 500 100 47.8 54.3 54.0 40.0 w co

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2.65 m -------e ti ' ,..; rl l a ..., e u -.:, N J_ 139 ~-100 cm ,_ 65 .cm -•-I•-100 cm I FIGURE 50. Diagram of a n 2 0-Reflected 3-to-l Compression Ra t i o Nu c l e a P i s to n Eng i n e il t t he T O C Po s i t i on

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FIGURE 51. I I 11 ,, ,, ,, 1 1 11 ,, " rLi.. 1.., I I : I 1 I I I I I I I I I I I L,-.,J I I : I 7.6:11 ----------------y , -----.1 . .,,--..... .,,,---, ,,,,---, .,,,--, ,"o o 1 ," o o) { o o./ "o o I ..._._,,,.,, ,_., , __ .,,." •! 11 ,, 11 11 ,1 ,, , . ,1 11 ,1 r!..i.. .. t, I I I I I I I I I I : I . r l l I -r-r-' I I I ,,-, ,'o o.,/ ... _ ... I I II 11 ,, I I II li :: r._, _ _.1, I I I I I I I I I I I i I I I I I I I '-,--,1 I I ! ~LOCK VOLU~.E 115 ,.J 11 11 ,, I ,_,1 X' r I I I I I I I I I I I I I I I I / I I / ~( I I Sketch of an 3-Cylinder Nutleilr Piston Engine Glock for 40-50 Mw(e) Power G~nerating Systems __,

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TABLE 16 Heat Rat~ and Fuel Cost Estimates for the Three Nuclear Piston . Engine Power Generating Systems Which Use Piston Engines#l and #2 ilet Powe x Output Overall for an 8-Piston Efficiency Heat Rate Fuel Cost* Unit = P/Q % ) f (BTU/kwe-:hr) (mills/b-1e-hr) Piston-Gas Tu rb i'ne Piston-Steam Turbine Piston-Gas Turbine-Steam Turbine (Mw) . 47: 8 43. 6 54.3 52. 2 54.0 49.5 *Based on the use . of . 90% enriched UF 6 at Sll ,000 per kgU 7830 l. 57 6S50 l. 32 G890 1 . 38

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Timestep Size Selection for the Neutron Kinetics Equations 142 The timestep sizes used in solving the energetics equations are on the average 8 to 10 times larger than the timesteps used in solving the neutron kinetics equations. During those portion of the cycle when the pressure and temperature are increasing rapidly and when the rate of fission heat release is large, NUCPISTN will automatic~lly select smaller timesteps for the solution of the energetics equations. The actual size selection depends on the rate of fission heat release and the rate of temperature change. A restriction on the energy equation timestep size is that it can never become smaller than the timestep size used in solv ing the neutron kinetics equations. More will be said about the timestep size selection for the energy equations in the next chapter. In Table 17 are the operating characteristics for reference Engine #3 and the NUCPISTN cycle re~ults for. this engine are in Table 18. Effects of variations in the timestep size used in solving the neutron kinetics equation were investigated for thi~ ~ngine. The user has the option of specifying timezones in the NUCPISTN code and different stepsizes can be specified for each of the ti~ezones. For this particular engine, three timezones were used. Time zone #1 started at the beginning of the cycle and ended at the 0.05 cycle fraction. Timezone #2 extended from the 0.05 to the 0.650 cycle fraction while timezone #3 extended

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143 TABLE 17 Operating Characteristics for Engine #3 Characteri sti.c Number of strokes Compression ratio Clearance volume (m 3 ) Stroke (ft) Initial gas pressure (atm) Initial gas temperature (K) Cycle fraction for step-reflector addition removal Neutron source strength (n/sec) Engine speed (rpm) He-to-U mass ratio % U235 enrichment Photoneutrons considered Delayed neutrons considered Pure simple harmonic motion Initial o 2 o reflector thickness (cm) o 2 o re~lector thickness after step addition (cm) Neutron lifetime (msec) Piston cycle time (sec) U mass in cylinder (kg) He mass in cylinder (kg) Average o 2 o reflector physical temperature (K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) Engine #3 2 3.0 to 1 0. 180 3.8 14.6 0.050 0.650 1.0 X 10 9 100 0.332 100 no no yes 30 100 1 . 68 7 0.600 2.744 0. 91 l 570 s. o 57. 94 31 . 3 2

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144 TABLE 18 Cycle Results from NUCPISTN for Engine #3 Characteristic Maximum gas temperature (K) Maximum gas pressure (atm) t-1 a x i mum k e ff Exhaust gas temperature (K) Average core thermal neutron flux during the cycle (n/cm 2 sec) Average gas temperature during the cycle (K) Average gas pres~ure during the cycle (atm) Average keff during the cycle Mechanical power output (Mw) Fission heat released (lh-J) Mechanical efficiency (%) Photoneutron precursor concentration (#/cm 3 ) Delayed neutron precursor concentration (#/cm 3 ) Thermal (l/v) in core (sec/cm) Fast (Tlv) in reflector (sec/cm) Maximum core thermal neutron flux during cycle (n/cm2sec) Average mass flow rate (lb /hr) m Engine #3 l 9 20 l 3 9. l l . 1 09 11 5 4 l.93 X 10 14 943 56.6 0.926 3.36 11 . 5 5 2 9. 1 2.54 X 106 l.l23xl07 5.44x 10 15 6.59 X 10 4

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145 from the 0.650 cycle fraction to the end of the cycle. For the results shown in Table 18, the stepsize used in all three timezones was 0. 12 msec; the average neutron lifetime for this system ts 1.687 msec and the piston cycle time is 0.6 seconds. During timezones #1 and #3, the engine is sub critical .and changes in the size of the timestep used for solving the neutron kinetics equation have very little effect. Effects of variations in the timestep size used in solving the neutron kinetics equations in timezone #2 are shown in Table 19 for Engine #3. It will be recalled that the timesteps used in solving the energetics equations can be no smaller than the time steps used in solving the neutronics equation. In examining Table 19, it will be no~ed that for large timesteps, a rather large percentage of the total fission heat release occurs during one individual timestep. For these coarse timestep sizes, the error in 6U o~er the cycle is also high. As the timesteps decrease in size, the error in 6U over the cycle decreases and the maximum amount of fission heat released in any neutron kinetics equation timestep becomes more reason able. For timestep sizes of about 0.36 msec to about 0.04 msec, there is also little variation in the mechanical power output and in the maximum gas temperature. For smaller or larger timesteps, these quantities fluctuate noticeably. As the timesteps continue to decrease in size the maximum fission heat released in a neutron kinetics equation

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146 TJ\BLE 19 Effects of N~utron Kinetics Equations Timestep Size Variation on Engine #3 Perfo~mance Stepsize (111sec) Time Zone #2 3.6 1.8 l. 2 o. 72 0.36 0.18 0.12 0.06 0.04 0.03 0.01 0.005 Humber of Timesteps per Neutron Lifetime 0.47 0.94 1.40 2.34 4.69 9.37 14. l 28. l 42 .. 2 56.2 168. 7 337.4 Mechanical Pav,er. Output (Mw) 2.83 3.08 3. 17 3.25 3.32 3.36 3.36 3.~7 3.38 3.41 3.67 3.92 :~axir.ium Gas Temperature ( o K) 1701 1802 1842 1875 1902 1921 1920 1924 1928 1939 2049 2226 nmax 'IN (%) 15. 06 7.95 5.40 3. 29 l. 66 0.83 0.56 0.28 0.28 0.28 0. 28. 0.28 Error in 6U over Cycle (%) 0.430 0.229 0. 140 0.098 0.050 0.017 0.0002 0.043 0.077 0. l 30 0.420 0 . .797 Stepsizes in Timezones ill and #2 are fixed at 0.12 msec Q max(%) =[Maximum Fission Heat Released in Any Neutron Kinetics Equation N Timeste Total Fission Heat Released Over the Piston Cycle Percent Error in 6U Over the Cycle= [ 6 UcV 6 LJT)Jxl00 6UT . where 6U cV In the above, Qf is the total fission heat released and Wis the net mechanical v1ork performed during the cycle. 100

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147 timestep reaches a constant minimum and the error in U over the cycle begins to grow again. In this region the time step sizes have become too small and computer roundoff errors mount rapidly. It will be observed from the table that the most desirable neutron kinetics equation timestep sizes occur when the number. of timesteps per neutron life time is from around 4 to 40. This result has been found to also hold when the kinetics equations include delayed and photoneutron effects. Delayed Neutron Effects The current version of NUCPISTN can handle up to six groups of delayed neutrons. The default delayed neutron parameters used by the code are listed in Appendix F (Table 'F-5). Delayed neutron parameters (e.g., effective delayed neutron fractions) may also be input by the user and any input values automatically override the cod~s default values. The reader is referred to Appendix C for a presen tation of the general point reactor kinetics equations. This appendix should be examined for familiarization with notation as well as for a development of ~xpressions which are used in Appendix D. A complete derivation of the point reactor kinetics equations which are solved by the NUCPISTN cod,e will be found in Appendix D. The particular point reactor kinetics equations which are solved by NUCPISTN when only delayed neutrons are

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considered are equations (D-44) and (D-45) with the in homogeneous source term due to photoneutron production in the reflector,~ , set to zero. The numerical forms of . m 148 these two equations are given by equations (D-57) and with Wm set to zero. The uset has the option of selecting a two-point finite different or a three-point integration scheme for solving these equations. Higher order numerical techniques were tested and were found to yield no signifi cant gain over the three-point integration technique. More will be said on the_se numerical methods in the conclusions which appear in Chapter VII. Enqine Startup: Apptoach to Equilibrium in the Presence of Delayed Neutrons The nuclear piston engine is a circulating fuel, pulsed, quasi-steady-state reactor when operating at some fixed power level. When delayed neutrons are included in the analysis, the time required for the engine to go from the shutdown condition to some given fixed power level is determined by the time required for the delayed neutron precursors to come into equilibrium at this power level. During the core or piston engtne residence time, delayed neutron precursors are produced and a fraction also decay. The gas is then exhausted from the engine and passes thraugh the external loop where fission product removal and cooling occurs. Uuring this period another fraction of the pre cursors undergo d~cay. When the fuel finally re-enters the

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engi.ne, a fraction of the remaining delayed neutron pre cursors decay and act as a type of external or extraneous neutron source. 149 The procedure -used to attain full power equilibrium from thi shutdown position for the two-stroke engines in volved varying the initial gas pressure (and hence the fuel loading) and/or varying the loop circulation time. In Table 20 are the engine operating conditions for Enginei #4 and #5. These characteristics are identical for the two engines. The final equilibrium cycle results as obtained from NUCPISTN for these two engines are, however, different. The different equilibrium power and precursor levels for these two engines are a result of the different startup paths which were followed. The startup path for Engine #4 is outlined in Table 21. The changes in the loop circulation time and initial pres sure which were made to attain equilibrium are to be noted. The loop c~cle number is to be distinguished from the piston cycle number. For a loop circulation time of 5.4 seconds and a core residence time of 0.6 seconds the total circuit .time is 6.0 seconds. During this interval the piston under goes 10 cycles. The startup path for Engine #5 appears in Table 22. The two startup paths are identical up through the forty-first loop cycle after which time the paths vary. The total delayed neutron precursor population is shown as a function of time for Engine #4 in Figure 52.

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TABLE 20 Operating Characteristics for Engi~es #4 and #5 Characteristic Number of strokes Compression ratio Clearahce volume (m 3 ) Stroke (ft) Initi~l gas pressure (atm) Initial gas temperatures (K) Cycle fraction for step-reflector addition removal Neutron so~rce strength (n/sec) Engine speed (rpm) He-to-U mass ratio % U235 enrichment Photoneutrons considered Delayed neutrons considered Pure simple harmonic motion Initial reflector thickness (cm) reflector thickness after step addition (cm) Neutron lifetime (msec) Piston cycle time (sec) U mass in cylinder (kg) He mass in cylinder (kg) Average 8 2 0 reflector physical temperature (K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) Engine # 4 2 3.0 to l 0, 180 3.8 11. 7 400 0.050 0.650 l .Oxl0 9 100 0.332 100 no yes yes 30 l 00 1. 687 0.600 2.208 0.733 570 7. 9 57 .95 31 . 32 Eng,ne #5 2 3.0 to l 0. l 80 3.8 11. 7 400 0.050 . 0. 650 1.ox109 1 00 0.332 100 no yes yes 30 l 00 1 . 687 0.600 2.203 0.733 570 7.9 57.94 31. 32
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TABLE 21 Startup Procedure for Engine #4 i n the Presence of Delayed fleutrons Initial Maximum De 1 ayed Total Loop Cycle Loop Circ. Gas Mechanical Gas ~leutron Elapsed f1umber Time Pressure Pov!er Temperature Precursor Time (sec) (atm) ( r lvJ) ( o K) Concentration (sec) ( .!1 I 3 . .. cm 1 5.4 11. 75 'v 0 645 3.79 X 1 o 4 6.0 10 5.4 11. 75 'v0 645 2.69 X 106 60.0 20 5.4 11. 75 0.03 647 6.21 :x 10 7 120.0 25 5.4 11. 75 0. 16 . 649 2.91 X 108 150.0 30 5.4 11. 75 0.78 900 9 180.0 l. 36 x l 09 32 5.4 11. 75 1.44 1221 2.52 X 10 9 192.0 "/I 5.4 11. 75 2.66 1795 4.67 x l0 9 204.0 .J...35 5.4 11. 75 3.61 2246 6.35 X 10 9 210.0 36 8.0 . 11. 70 4 .11 2480 6.80 X 10 218.6 ... f . 109 41 8.0 11. 70 2.87 1899 7.02 X 261. 6 42 8.0 11. 75 3.05 1980 7. 15 X 10 9 270.2 + + 10 9 50 8.0 11 . 75 3.56 2221 8.37 X 339.0 51 7.9 11. 70 3.38 2136 8.40 X 10 9 347.5 + + 109 100 7.9 11. 70 3.53 2208 9. 10 X 764.0 + f 109 124 7.9 11. zo 3.52 2205 9. 10 X 968.0 (JI

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TABLE 22 Startup Procedure for Engine # . 5 in the Initial Loop Cycle Loop . Cir. Gas Mechanical iJumber Time Pressure Pm,er . (sec.) ( a tm) . (Mr;) l S.4 11 . 75 '\,o 10 5.4 11. 75 'v0 20 5.4 11. 75 0.03 25 5.4 11. 75 0. 16 30 5.4 11. 75 0. 78 32 5.4 11. 75 l.44 34 5 . 4 11. 75 2.66 35 5.4 l l. 75 3.61 36 8.0 11. 70 4. 11 + + 41 8.0 11. 70 2.87 42 7.9 11.70 2.84 + + 50 7.9 11. 70 2.82 + + 80 7.9 11. 70 2.89 . + + 114 7.9 11. 70 2.~o Presence of Delayed Neutrons :,1aximum Delayed Gas Neutron Teriperature Precursor ( o K) Concentration ( l /cm 3 ) 645 3.79 X 104 645 2.69 X 10 6 647 . 6. 21 . X l o 7 649 8 2.~l x 10 9 910 l . .:>6 x 10 9 1221 2.52 x 109 1795 4.67 x 109 2246 6.35 X 10 9 2480 6.80 X 10 9 1899 7.02 x 10 9 . 1881 7.03 X 10 1872 7. 15 X 10 9 l 908 7.44 X 10 9 1914 7 . 48 X 1 o 9 Total Elapsed Time (sec) 6.0 60.0 120. 0 . 1 so. b 180.0 192. 0 204.0 210.0 218.6 261 .6 270. l 338. 1 593. l 882. l (.11 N

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0 ::..:> :r. :.:::: G c2 .... , ,,11 tl = t2 = 5,5 min I I 3.65mi~ I I b-j I I I I at tl concentration at t2 concentration at t concentration 00 3 10 9 (7 5% equilibriu::i) Uf I cm ) 6.8 X of 3 10 9 (!I I cm ) = 8.2 X (90% of equilibriurr:) C' I 3 ) " cm = 9~1 X 10 9 ( cc, u il i b r i um v;:ilue) c .~o 7 :::,: '""" :::, :.,;J z L:.00 500 :T•!E ( SC!C) 0 100 200 300 600 700 800 900 F I G U R E 5 2 u e 1 u y_e d N e u t r o n P r e c _u r s o r C o n c e n t r a t i o n B Li i-1 d u p O u r i n g S t a r t u p f o r Engine !14 ...... u, w

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154 The s~arp break at t 1 {about 220 seconds) occurs as the initial pressure is dropped from 11.75 to 11.70 atmo~pheres while the loop circulation time is increased from 5.4 to 8.0 seconds. At t 1 the total delayed neutron precursor level is 75% of its equilibrium value. The time required to reach 90% of the equilibrium value is about 330 seconds (t 2 ). In Figures 53 through 55 the mechanical power and maximum gas temperature are shown for Engine #4 as a function of time. The power and temperature giowth are abruptly halted by the above described changes at t 1 . At about 260 seconds the power and temperature decline are turned around by increasing the initial pressure from 11.7 to 11.75 atmosphetes. ~t about 340 seconds, the mild growth in power level and gas temperature is damped out by dropping the initial gai pressure back to 11.70 atmospheres while decreas ing the loop circulation time from 8.0 to 7.9 seconds. The system then settles into equilibrium with a maximum gas temperature of 2205K and a mechanical power output of 3.52 MW. The equilibrium total delayed neutron precursor concentration is 9. lxl0 9 /cm 3 . The total delayed neutron population as a function of time for Engine #5 is shown in Figure 56. The same sharp break as for Engine #4 occurs at t 1 when the 1.oop circulation -time is increased from 5.4 to 8.0 seconds and the initial ga~ pressure drops from 11.75 to 11.70 atmospheres. At t 1 the total delayed neutron precursor level for this engine is

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.; t, 2800 I 6 I TI.MPERATURE 5 2400 ,....., ,....._ :,.:: 0 .,_, ;;:; ......., :::,-: """' ,...._ -, . .'.2000 4 i::... ,..; :::> ;,0 u ::::: :,.:.1 , . 2: '3 ~ ~.::::, lGOO ,-; ,-.l < < u ,..; c,:: :...J z ; ;..;.1 ,..; 1200 2 u V: < , ' ,..., < . ' 1 800 325 100 125 150 175 200 225 250 275 300 TI~ff. (sec) FIGU1E 53. Peak Gas Temperature and ~echanical Power Output Behavior During Startup for Engine #4 u, u,

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:::: ,-,, =-. I f2 6 ~<•r,n ,._ --.J'.J I I 5 2400 I TI:fPERATURE .,,...._ 2 ..... .._., 4 2000 ;:, p... f-< :::> 0 PO'.{ER 3 .. .,. .! r: 'JO I 0 I ...:: I nt t1 tcmper.::iturc (OK) = 2480 < :...> I ,.... at t2 tc1::crnture ("K) = 2202 1200 I 2 :r: at t tcr.ipcrnture (OK) = 2205 u I 00 .... I I 1 800 I I I 300 325 350 375 400 425 450 475 500 525 TH1E (sec) FIGURE 54 . . Peak Gas Temperature and Mechanical Paver Output Behavior During Startup for Engine #4 (continued) u, O'\

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.::soo'""' 2400 0 '-' ~., ;:::: :.:.o: ,..... .-.: 2000 ;.,-, , . ~-:.:, 160.0 < :.,J ::.. ;::: w !:--< U'.) .:..200 < -:.:; '~ ::.... 800500 FIGL!Rc 55. 5 TEI>PERATURE r-. ,._, '-' 4 ,::.::, ::.. !-, ::.::, 0 ~POWER •-:. ... J 5 p.. at tl power (Mw) a 4 .11 ..:: < u at t2 power Gw) = 3.50 H z .:it t pciwer (Mw) = 3.52 _2 $ (XJ u w "' -1 I I I I I I I 550 660 (,50 700 75U 800 850 ':JOO 950 TI~!E (sec) Peak Gas Tempervture and Mech~nical Power Output Behavior Ouring Startup for Engine #4 (contin~ed) u, --.J

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0 C 0 ::r. 0:: ;::J u :..:.J ::::'. z 0 :.:: --;,-. < ..J w 0 -7 10 FIGURE ----tz = 5.5 min 7 0 -,.. :;, l). I I I I nt tl at tz at t I I I (X) I I I I I I I I I I 100 200 JOO 400 concentration concentration concentration 500 T l~-:E ( s cc) 3 (II I cm ) 3 (fl/ cm ) 3 (II I cm ) 600 = 6.8 x 10 9 (90.5% of equilibrium) 7.13 x 10 9 (95% of equilibrium) = 7 . 5 x l O 9 ( c q u i l i b r i mn v ;J : l! e ) 700 800 900 Delayed Neutron Precursor Conce~tration Buildup During Startup for Engine f!5 (.Tl a,

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159 90.5% of its equilibrium value. The time required to reach 95% of the equilibrium value is about 330 seconds {t 2 ). The mechanical power and maximum gas temperature are shown as a function of time for Engine #5 in Figures 57 and 58. The power and temperature growth are again abruptly halted at t 1 . At abciut 260 seconds the temperature and power decline are leveled out by dropping the loop circula tion time from 8.0 to 7.9 seconds and the system settles into equilibrium. The equilibrium mechanical power and maxi mum gas temperature for this engine are 2.90 MW and 1914K respectively. The equilibrium total delayed neutron precursor concentration is 7.5xl0 9 /cm 3 . Equilibrium cycle re sults from NUCPISTN for Engines #4 and #5 are pres~nted in Table 23. It is thus seen that t~e two-stroke nuclear piston engine is capable of rather rapid startups. The time re quired to go from the shutdown condition to an esseritially equilibrium condition is of the order of only 5 to 10 minutes. Safety considerations should not lengthen this time significantly since the engine is capable of rapid shutdown. Pressure relief valves in the core and depressurization of the o 2 o reflector by conventional relief valvi quenching tank systems can bring about essentially instantaneous shut down should the gas temperature or pressure in the core exceed some critical limit. Boiling and voiding of the

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:::::: z c--' 2000 2400 2000 1200 lGO 125 2.50 175 275 300 6 5 4 3 2 1 325 :::i ::..F!GURE 57. Peak Gas Temperature and ~echan{cal Power Output Behavior Juring Startup for Engine #5 O"I 0

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2sooL ,,..._ :.a::: .._, 240 C . .. C ..., . ...: ::,2000 , _ _ ,, .;. 6()0 B b-' :r. < 1200 :./. <; -':.... S CIO 300 t 2 I 6 I . I 8t tl pc w er ( ~ h v ) = 4.11 I nt t2 power c :w) = 2.82 5 I nt t power (!'1w) = 2.90 00 2 .,;.. 1 '-' I ...., I 4 TE >~ ? E MTURE b-' ...., 0 p~ .. , C 3 -:_ >-" < . ' ._, PQl :ER ,-; ~= :5 2 8 at t1 ter.1percture (OK) = 2480 at tz te m per n cu re (OK) 1873 at t te:-:-:pcr2ture (OK) = 1914 00 1 I I J .SLJ 400 450 500 550 600 650 700 750 s oo s so P~ak Ga~ Temperature and Mechanical Power Output Be~aiior ~ur : ~; Startup for E~gine #5 (co~tinued)

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TABLE 23 Equilibrium Cysle Results from NUCPISTN for Engines # 4 and #5 Characteristic Ma x imum gas temperature (K) ~a x i m um gas pressure (atm) :-1 ax i m um k e ff Exhaust gas temperature {K) Average co~e thermal neutron flu x during the cycle (n/cm sec) . Average gas te m perature durin9 the cycle (K) Average gas pressure during the cycle (atm) Average keff during the cycle M echanical power output (~w) Fission heat released (Mw) Mechanical efficiency ( % ) Photoneutron precursor concentration ( # /cm 3 ) 3 Jelayed neutron precursor concentration (#/cm) ThermaJ_j_l/v) in core (sec/cm) Fast (1/v) in reflector (sec/cm) ~a x imum co~e thermal neutron flux during cycle (n/cm sec) . Average mass flow rate (lb /hr) m Engine # 4 2205 1 31 . 5 l . 061 1290 2.36 x lo 14 1078 52.8 . 0. 871 3.52 11 .30 31. 1 --9 9. l x lO _ 6 2.54xl0 _ 7 l. 123 x l0 l 5 3.66 x l0 4 5.31 x 10 Engine # 5 1914 11 4. 3 l . 061 1134 1.94xlo 14 977 48.3 0.871 2. 9 . 0 9 . 33 3 l. 1 ---9 7.48xlo_ 6 2.54xl0 _ 7 l.123xl0 3.02 x lo 15 5.3lxlo 4 0) N

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163 pressurized _o 2 o and valve adjustments in the external loop to increase the loop circulation time are additional effec tive, though less rapid, means of achieving shutdown. Shut down and safety will be discussed in more detail in Chapter V I I . In the next chapter, startup times in the presence of both delayed and photoneutrons will be examined for the four stroke engines. Equilibrium is attained by varying the loop circulation time, the intake line gas pressure, the intake line initial mass flow rate and the cycle fraction at which the step-reflector is removed. Chanqes in the cycle fraction for shutting the intake valve and variations in the 9as temperature in the intake line can also be used as variables in the approach to equilibrium procedure for these four stroke engines. It should be mentioned that the above described pro cedures used in going from shutdown to equilibrium can also be used in going from one equilibrium level (or power level) to another. Examples of such changes will be illustrated in Chapter V. The effects of the delayed neutrons on the engine per formance can be seen by comparing the results of Table 23 with those of either Table 10 or Table 18 which are for engines which did not include delayed neutron effects. The maximum k~ffective has dropped from about 1.109 to 1.061

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164 and the average keffective over the cycle has dropped from 0.926 to 0.871. Also, the uranium loading has decreased from 2.74 kg to 2.21 kg. Nucl~ar Piston Engine Blanket Studies Appearing in Tables 24 and 25 are the nuclear piston engine operating characteristic. and equilibrium cycle re~ sults from NUCPISTN for reference Engine #6. This engine and a very similar one with a larger loading and thinner reflector were the basis for some blanket studies done for the nuclear piston engine. Since the nuclear piston engine's fuel is continually recycled and cleaned up and since make up gas is used to keep the gas composition essentially con stant under steady~state conditions, burnup effects in ~e core were neglected in the nuclear piston engine blanket studies which follow. The investi.gated blankets consisted of rods of uranium and thorium dioxide arranged in a square lattice. The rods in this blanket region were surrounded by o 2 o. A typical unit cell for the blanket region is shown in Figure 59. Two different lattice structures were used: al .5 metal to-water ratio lattice and a 3.0 metal-to-water ratio lat tice. Radii for the equivalent cylindrical cells as well as volume fractions and region thicknesses appear in Table 26. Pure blanket material densities also appear in Table 26 for

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TABLE 24 Operating Characteristics for Engine #6 Characteristic Number of strokes Compression ratio Clearance volume (m 3 ) Stroke (ft) Initial gas pressure (atm) Initial gas temperature (K) Cycle fraction for step-reflector addition removal Neutron source strength (n/sec) Engine speed (rpm) He-to-U mass ratio % U235 enri.chment rhotoneutrons considered Delayed neutrons considered Pure simple harmonic motion Initial reflector thickness (cm) Reflector thickness after step addition (cm) Neutron lifetime (msec) Piston cycle time (sec) U mass in cylinder (kg) He mass in cylinder (kg) Average reflector physical temperature (K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) 165 Engine #6 2 3.0 to l 0. 180 380 11 . 8 5 400 0.050 0.650 . (l 1.0 X 10" 100 0.332 100 no yes yes 30 100 l . 68 7 0.6Uu 2.240 0.744 570 8.60 57. 94 31 . 3 2

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166 TABLE 25 Equilibrium Cycle Results from NUCPISTN for Engine #6 Characteristic Engine #6 ------------------------------Maximum gas temperature (K) Maximum gas pressure (atm) [ a x i mum k e ff Exhaust gas temperature (K) Average core th 2 rma1 neutron flux during the cycle (n/cm sec). Average gas temperature during the cycle (K) Average gas pressure during the cycle (atm) Average keff during the cycle Mechanical power output (Mw) Fission heat released (MvJ) 11 e ch an i ca e ff i c i ency ( % ) Photoneutron precursor concentration (#/cm3) De1ayed neutron precursor concentration (#/cm3) T h e rm a l ( TTv ) i n c o re ( s e c / cm ) F a s t ( 1. / v ) i n r e f l e c t o r ( s e c / c m ) Maximum core th 2 rma1 neutron flux during c y c l e ( n / c ni s e c ) Average mass flow rate (lb /hr) m 2167 128.8 1 . 060 1270 2.3 X 10 14 l 06 7 52.2 0.869 3.43 l l . 01 31 . 2 6.35 X 10 9 2.54 X 106 1. 123 X 107 3.51 X 10 15 5.37 X 10 4

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( \ / Broken Cylinder l{epresents Equivall,nt Cylindrical Cell. --RegLon 4 Region 1 Region 1 uo or Th0 2 2 Region 2 Void or Gap lkgion 3 Zircaloy Clad r~ .. ~g i.\)n !i ... D:l \ ) 167 FIGURE 59. Ty p i c a l G l ,1 n k e t R e 9 i o n U n i t C e l l D i a 9 r a m

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. TJl.BLE 26 ( Equivalent Cyl i ndri cal Cell uata and Pure Blanket Material (at the beginning of the blanket lifetime) Equivalent Ct l i n d r i c a l Ce 11 Data Case Pitch Region Material Radius (cm) ( C r.1) . l. 5M/W l. 3543 l uo 2 or ThO 2 .5334 Lattice 2 Gap .5410 3 Cl ad .5918 4 o 2 o .7638 3. or 1 Iv! 1 . 2 09 0 l uo or Th0 2 .5334 ' Lattice 2 . 2 Gap .5410 3 Clad .5918 4 D 0 2 .6822 Densities Volume Fraction .4877 .0140 .0986 .3996 . 61 l 5 .0176 . l 2 37 .2472 Thickness (cm) .5334 .0076 .0508 . l 7 21 .5334 .0076 .0508 .0904 ...... O"l co

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REgion "Fuel 11 Zircaloy Cladding D 2 0 Moderator Table 26 (continued) Pure Blanket Material Densities Comments For the case where 1% of the rods are 93% enriched U02 and 99% of the rods are Th0 2 Average o 2 o Physical Temperature is 370K Material U-238 U-235 Th-232 0-16 Ni Sn Fe Cr Zr H-2 0-16 Densi ty (atoms/barn-cm) l.575 x 105 2.092 X 104 2.l0xlo2 4.499 X 102 4.6 X 105 5.55 X 104 1.02 X 104 7.5 X 105 4.215 X 102 -2 6.382 X '10 3.191 X 102 ...., O"I I.O

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170 t~e case when 99% of the rods are Th0 2 and 1% of the rods are 93% enriched uo 2 . Homogenized blanket material densi ties for this case are Shown in Table 27 for the l .5 metal to-water ratio lattice. Shown below Table 27 is a one dimensional sketch of the core-reflector-blanket arrangement for the #6 type Nuclear Piston Engines. As already menti9ned, NUCPISTN is capable of handling only two regions. Hence the PHROG and BRT-1 constants for the reflector, blanket and liner regions were flux and volume-weighted so as to obtain a single set of 11 moderating r e fl e c tor II reg i on constants for i n put i n to NU C P IS Tr~ The blanket study calculations were then performed by coupling the one-dimensional CORA diffusion theory code with a zero-dimensional, four-group fuel depletion and burnup code named BURNUP which was written at the University of Florida. The co~e composition and fluxes were taken as being constant during the burnup calculations for reasons which 0ere discussed at the beginning of this section. The blanket region composition and fluxes, however, varied during the burnup calculation. Shown in Tables 28 and 29 are results from burnup c a l c u l a t i o n s f o r t h e l . 5 m e t a l t o 'iJ a t e r r a t i o l a t t i c e s . The results of Table 28 ar~ for a system with an 80cm thick o 2 o reflector region while the Table 29 results are for. a system with a 70cm thick o 2 o reflector region. In both cases, the blanket thickness is 20cm. It will be noted

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l 71 TABLE 27 Homogenized Densities for a Blanket Using a l .5M/W Lattice (at the beginning of the blanket lifetime) Mate.rial U-238 UTh-232 0-16 Ni. Sn Fe Cr Zr H-2 Note: Blanket Region Homogenized Density (atoms/barn cm) 7.68 X 106 1.023 X 104 -2 1.0242 X 10 3.465 X 102 4.54 X 106 r. 5.47 X l 0;_i l. 006 X ,o-5 7. 4 X ,o-6 -3 4. l 56 x l 0 2.55 X 102 The above table is for the case.where 1% of the rods are 93% enriched U02 and 99% of the rods are Th02, i.e., the pure material densities are those of Table 26. Core-Reflector-Blanket Region Sketch CORE -t----3 5 C Ill•---+ I D20 REFLECTOR ti70 or 80cm -t BLANKET ------2 0 C 1111--TZN liner, 0.03cm .TZtlliner, 0. 03c111 Average gas temperature in core= 1067K Average o 2 o moderating reflector temperature= 570K Average temperature in blanket region= 370K

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TABLE 28 Burnu~ Calculations for a System U~ing Engirie # 6, an 80cm DO Reflector Region, and a Blanket Region with a l .5M/W Latti~e t of Blanket Rod s lJ h i c h Are uo 2 l. 2 l . 0 0.5 0.0 o/~ of Blanket Rods ~!hich Are uo 2 l ; 2 l . 0 0. 5 0.0 Cr . or BR 0;444 0.478 0 : 592 0.772 Initial U235 Mass ( k g ) 115.19 95.94 47.97 0.0 Fin al U235 Mass (kg ) 101.57 84.595 42.30 0.0 Blanket volume= 2.41m 020 reflector thickness = Blanket thickness= 20cm 80cm 6 2 10 n/cm 2 sec) time and Blanket Thermal r ( 8.70 7.33 3.91 0.49 Doubling Time (yrs) Fraction of Fuel I n V e n t O r y vJ h i C h Is in Core 0. 153 0. 153 0. 153 0. 153 Initial U233 Mass . (kg) Final U233 Mass (kg) I n i t i a l Th232 '.'-1ass (kg) Final Th232 Mass (kg) 0.0 0.0 0.0 0.0 13.065 13.093 13.167 13.219 9487.3 9507.7 9561.6 9598.7 Blanket is ThO?, U02 and D20 U02 is 93% enrtched uranium Burnup time is 1200 days 9473.0 9493.4 9 54 7. l 9584.2 during = 2.56 X l. 9 5 x 6.22 X 3.65 X 10~ n/cm 2 sec) 10 12 n/cm 2 sec) 10 n/cm sec) volume averaged blanket fluxes Total . U235 consumed in core 1200 days= 18.lkg -core 14 2 4 = 2.3 x 10 n/cm sec Qcore = f

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TABLE 29 Burnup Calculations foi a System Using Engine #6, a 70cm DzO Reflector Region, and a a1anket Region with a l .5M/W Lattice ~ ; of Blanket Rods v!h i Ch Are uo 2 l. 2 1.0 0. 5 0. 0 . ~ ; of Blanket Rods iJhich Are uo 2 l . 2 l . 0 0.5 0.0 Cr or BR 0.491 0.533 0.675 0.916 Initial U235 Mass (kg) 115.19 95.94 47.97 , 0. 0 Blanket volume= 2.41m Fin al U235 Mass (kg) 98.97 82.43 41. 22 0.0 Blanket The rm al Power ( M\'J) 10.48 8.85 4.78 0.71 Initial U233 Mass {kg) o.o 0.0 0.0 0.0 Doubling Time (yrs) Fraction of Fuel Inventory Hhich Is in Core 0. l 5"3 0. l 5 3 0. l 53 0. l 53 Final U233 Mass (kg) Initial Th232 Mass {kg) Final Th232 Mass ( k g ) 15.591 15.625 15.713 15.774 9487.3 9507.7 9 5 61 :6 9598.7 9470.0 9490.4 9544_1 9581.2 D20 reflector thickne~s: 70c~ Blanket thickness= 20cm Blanket is Th0z, U0z and U0z is 93 % enriched uranium Burnup time is 1200 days 1. 99 = ..,..1 1. 12 2.2 = S::.3 = 2.36 ct 4 = 4.40 7 2 X 10 9 n/cm 2 sec) X . l o 1 0 n/cm 2 _ sec) X lo 12 n/cm 2 sec) X 10 n/c, sec) tirrie and volume averaged blanket fluxes Total U235 consumed in core during 1200 days= 18.1kg i~ore = 2.3 x 10 14 n/cm 2 sec Q core f =

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174 that as the percentage of the blanket rods which are uo 2 is decreased from 1.2% to 0%, the conversion ratio increases while the therma1 power output from the blanket decreases. For the system which has only a 70cm o 2 o reflector region, the time and volume-averaged fluxes in the blanket are higher than for the system with the 80cm thick reflector . . The corresponding conversion ratios and blanket thermal power outputs are also higher for this system. Also appearing in Tables 28 and 29 are the initial and final masses of U235, U233, and Th232 in the blanket . f b t f 1200 d The Pu 240 , Pu 241 , region or a urnup ,me o ays. u 238 , u 234 , and Pa 233 mass changes are all small or zero and are therefore not listed. None of the arrangem~nts appearing in Tables 28 and 29, with the l .5 metal-to-water ratio blanken lattices, are able to breed. Results for 3.0 metal-to-water ratio blanket lattices are shown in Tables 30 and 31. The Table 30 results are for a system with an 80cm thick o 2 o reflector region while the results of Table 31 are for a system with a 70 cm thick o 2 o region. In both cases, the blanket region thickness is again 20cm. The system with the 70cm thick o 2 o reflector region has higher time and volume-averaged fluxes than does the system with the 80cm thick o 2 o reflector region. It also possesses higher conversion ratios and blanket thermal po~er outputs. It will be observed that the last configura tions in Table 31, in which all of the blanket rods are Th0 2 ,

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TABLE 30 Burnup Calculations for a System Using Engine #6, an 80cm Reflector Region, and a B-lanket Region with a 3.0M/W Lattice 5{ of Blanket Rods \.J hi ch Are uo 2 1. 2 1. 0 0.5 0.0 ~{ of Blanket R o d s lJ h i c h Are uo 2 1. 2 1. 0 0. 5 0.0 Cr or BR 0.500 0.543 0.694 0.956 Initial U235 Mass ( kg ) 143.87 119.89 59.94 0.0 Bla~ket volume= 2.4lm 3 Blanket Thermal Power (Mw) Final U235 Mass (kg ) 126.86 105.71 52.86 0.0 10.87 9. l 6 4.89 0.62 Initial U233 _Mass (kg) 0.0 0.0 0.0 0.0 uoubling Time (yrs) Final U233 Mass (kg) 16.351 16.363 16.453 16.529 Fraction of Fuel I n v e n t o r y vi h i c h Is in Core Initial Th232 Mass (kg) 11873.0 1 l 88 2 . 3 11854.4 12003.0 0. 153 0. l 53 0. l 53 0. 153 Fin al Th232 Mass (kg) 11855.2 11864.4 11929.3 11984.9 D20 reflector thickness= 80cm Blanket thickness= 20cm Blanket is Th02, U02 and J20 U02 is 93% enriched uranium Burnup time is 1200 days (;) = 2.56 X -1 l . 9 5 4;2 = X 93 = 6.22 X Q4 = 3.65 X 10~ n/cm~sec) and time 10 9 n/cm 2 sec) volume averaged l o 12 n/cm 2 sec) blanket fluxes 10 n/cm sec) Total U235 consumed in core during 1200 days= 18.1kg ~ore= 2.3 x 10 14 n/cm 2 sec 0 core = f -....J (JI

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o /c, TABLE 31 Burnup Calculations for a System Using Engine #6, a 70cm 020 Reflection Region, and a B1anket Region with a 3.0M/W Lattice of Blanke•t CR Blanket Doubling Fraction of Fuel Rods vl hi ch or The rm al Time Inventory Which Are uo 2 BR Pov;e r (Mw) (yrs) Is in Core l . 2 0.548 13.09 0. l 53 l . 0 0.599 11 . 06 0. l 53 0.5 0.785 5.98 0. l 53 0.0 l . l 2 9 0.89 20.0 0. l 53 0.0 l . l 2 9 0.89 l O. 2 0.300 0.0 1 . l 2 9 0. 8 9 6. l 2 0.500 0/ of Blanket Initial Final Initial Final Initial Final /0 Rods Hhich U235 Mass U235 Mass U233 Mass U233 Mass Th232 r1a s s Th232 Mass Are uo 2 (kg) ( kg ) l 2 143.87 123.61 1 . 0 119.89 103.01 0.5 59.94 51 . 50 0.0 0.0 0.0 Blanket volume= 241m 3 DO reflector thickness= 70cm Blanket thickness= 20cm 7 2 ( kg) (kg) ( kg) 0.0 19.512 11873.l 0.0 19.527 11882.3 0.0 l 9. 6 34 11947.3 0.0 19.725 12003.0 Blanket is Th02, U02 and D20 U02 is 93% enriched uranium Burnup time is 1200 days (kg) 11851.5 11860.7 11925.5 11981.2 l. 99 .2.1 = X l. 12 .!-2 X 10 9 n/cm 2 sec) 10 10 n/cm 2 sec) time and Total U235 mass consumed in core during 1200. days = 18.1kg = .2.3 = 2.36 X 10 12 n/cm 2 sec) q,4 = 4.40 X 10 n/cm sec) volume averaged blanket fluxes -core 14 2 ~ 4 = 2.3 x 10 n/cm sec 0 core = f l L 01 Mwth -....J C)

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l 7 7have a breeding ratio of 1.129. The system doubling time for the case where the fraction of the fuel inventory which is in the core is 0.153 is 20 years. Some studies which have been done on large, steady state, circulating fuel gas core reactors have indicated that the fraction of the fuel inventory which is in the core will be around 0.5. In view of the cleanup and continuous feed processes required,this number is felt to be a bit high and has led to overly optimistic doubling times. The frac tion of O. 153 used here is a 11 worst case estimate 11 and hence the 20-year doubling time is felt not to be unreasonable. The fraction of the fuel in the loop could conceivably be r~duced enough so that the fraction of the fuel inventory in the core would go to perhaps 0.3 which would lead to doubling times of around 10 years. Additional blanket study results will be presented in the next chapter for a four stroke nuclear piston engine. Neutron Lifetime Results Appearing in Tables 32 and 33 are the engine operating conditions ~nd equilibrium cycle results from NUCPISTN for reference Engine #7. The engine differs from the previously examined engines in two significant respects. First, photoneutrons are considered in addition to delayed neutrons .. Setond, control rod or poison removal and addition in the

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TABLE 32 Operating Characteristics for Engine #7 Characteristic Number of strokes Compression ratio 3 Clearance volume (m ) Stroke (ft) Initial gas pressure (atm) Initial gas temperature (K) Cycle fraction for control rod or poison removal poison addition Neutron source strength (n/sec) Engine speed (rpm) He-to-U mass ratio % U235 enrichment Photoneutrons considered Delayed neutrons considered Pure simple harmonic motion Initial o 2 o reflector thickness (cm) reflector thickness after step addition (cm) Neutro~ lifetime (msec) Piston cycle time (sec) U mass in cylinder (kg} He mass in cylinder (kg) Average o 2 o reflector physical temperature (K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) 178 Engine #7 2 3.0 to l 0.180 3.80 11 . 00 400 0.050 0.600 1.0 X 10g l 00 0.332 1 00 yes yes yes 100 NA l. 687 0.600 2.075 0.689 570 8.00 57.94 31 . 3 2

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l 7 9 TABLE 33 Equilibrium Cycle Results from NUCPISTN for Engine #7 Characteristic Maximum gas temperature (K) Maximum gas pressure (atm) Maximum keff Exhaust gas temperature (K) Average core therma1 neutron flux during the cycle (n/cm sec) . Average gas temperature durihg the cycle ( o K) Average gas pressure during the cycle (atm) Average keff during the cycle Mechanical power output (Mw) Fission heat released (Mw) Mechanical efficiency (%) Photoneutron precursor concentration (#/cm 3 ) Delayed neutron precursor concentration . (#/cm3) Thermal (l;v) in core (sec/cm) Fast (l/v) in reflector (sec/cm) Maximum core thermal neutron flux during cycle (n/cm2sec) Average mass flow rate (lb /hr) m Engine #7 2160 145.6 1 .. 04 6 1188 2. 1 7 X 1014 1129 55.4 0.824 3.33 9.81 33.9 4.90 X 1 0 8 7.30 X 1 o 9 2.54 X 1 o6 l.123 X 10 -7 2.73 X 1 0 l 5 4.99 X l o 4

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180 moderating-reflector region is used in place of step reflector addition and removal to achieve the desired sub critical_ to supercritical to subcritical behavior. The effects of the photoneutrons on the engine per formance can be observed by comparing the results of Table 33 with those of Engines #4 and #5 which appear in Table 23 and which considered only delayed neutron effects. The maximum k has dropped from 1.061 to 1.046 while the average effective keff~ctive during the cycle has dropped from 0.871 to 0.824. Also, the uranium loading has decreased from 2.208 kg to 2.075 kg. More will be said about the photoneutron effects in the next chapter when approaches to equilibrium in the presence of both delayed and photoneutrons are treated. One-dimensional, two-group forward eigenvalue calcu lations in spherical geometry were done with the CORA code at various cycle po~itions for Engine #7. Calculations were done for the case when fast core interactions are neglected (as is the case with NUCPISTtl) and for the case when fast core interactions are included. The results are listed in Table 34. Also listed are neutron generation times, A, and neutron lifetimes, i , at these various posi tions. The generation time and lifetime results 'were ob tained by performing auxiliary adjoint and perturbation cal culations with CORA at each of the selected positions. The two-group lifetime results are for the case when. fast

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TABLE 34 Neutron Multiplication Factors, Neutron Lifetimes and Neutron Generation Times at Various Cycle Position~ for Engine # 7 as Obtained from CORA arid NUCPISTN Time Step Number l 01 *** 302 602 l 002 2202 3002*** Elapsed Time into Cycle (sec) .015 .045 .090 . 150 .330 .450 Core Radius (cm) 50.45 49.89 48.08 44.12 35.59 44.14 Two-group rJUCP1STN* keff 0.500 0. 989 0.999 1. 018 1. 045 0.548 TVJo-group CORA* keff 0.494 0.988 0.999 l. 017 1.044 0 . 547 Tv,o-group CORA** keff 0.528 1. 005 ' 1. 015 1. 037 1.071 0.580 Tvm-group CORJ\** 9(msec) 0.866 2.030 1. 935 1. 732 l. 362 0.739 T w o-group CORA** :l(msec) 1. 641 2.020 1. 906 l. 672 1 . 271 1. 273 One-group CORA x. (msec) 0.506 1. 550 1. 478 1. 306 1. 000 0.425 Inhomogeneous sources in the moderating-reflector due to photoneutrons are included. *Fast core interactions neglected. **Fast core interactions included ***Poison or control rods are in~erted into the moderating-reflector region One-group . CORA / 1. (msec) 0.%9 1. 540 l. 454 l.262 0.933 0.734 __, 00 __,

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182 core. interactions are included. One-group (thermal) life time results as obtained from CORA at each of the selected positions are also presented in Table 34. Some neutron lifetime results were described in the beginning of this chapter in connection with Table 6. It will be recalled that the major contribution to the total neutron lifetime is from the moderating-reflector region: Hence, changes in the core size do not geometrically, directly influence the neutron lifetime to a significant extent. It is the resultant changes rather in keffective and in the flux shapes which lead to variations in the neutron lifetime throughout the cycle. Considering Table 34 and the timesteps at which the system is supercritical, the neutron lifetime and generation time become shorter as the system becomes more and more supercritical. For those timesteps where the system is subcritical, the lifetimes are short relative to the supercritical life times. The reason for this is that the subcritical con figurations here have highly absorbing control rods inserted into the moderating-reflector region and this tends to sig nificantly shorten the neutron lifetimes in this region. It will be seen in the next chapter that when step-reflector addition and removal is used in place of control rods or poisons, the neutron lifetimes of the subcriti.cal con figurations are long relative to, the supercritical

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configuration lifetimes. This is the type of behavior w h i c h wo u l d n o rm a 11 y b e e x p e c t e d o n a n i n t u i t i v e b a s i s . 183 It should at this time be pointed out t~at the neutron lifetimes which appear in the tables listing the "engine operating characteristics 11 in this work are all cycle averaged neutron lifetimes. They were obtained by perform ing.adjoint and perturbation calculations at selected time steps or positions during the piston cycle. A least squares fit for the neutron lifetime was then made to the selected timestep or position lifetimes. It was then possible to obtain cycle-averag~d neutron 'lifetimes for the engine of concern. The reason for performing lifetime calculations at only a selected nmber of timesteps (1\,25) during the piston cycle and. then least squares fitting is a purely economical one. Adjoint and perturbation calculations performed at each of the five or six thousand timesteps or at ev~n of the order of 100 timesteps for each analyzed piston cycle would have been prohibitively expensive. Finally, the neutron lifetime and generation times from.the 11 one-group 11 CORA calculations are in significant error. They are generally from 25 to 40% lower than the two-group results. Results from two-group calculations will be matched with four-group neutron lifetimes in the next chapter.

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184 Summary The work on the nuclear piston engine in this chapter has centered on the two-stroke engine in which only the com pression and power strokes were explicity considered. This work has involved three major phases and five lesser phases. The first major phase invol~ed d~velopment and use of the NUCPISTN code for parametric studies. The second major phase included comparisons of NUCPISTN nucleonic results with those obtained from more sophisticated steady-state nuclear computations. The third major phase centered on thermo dynamic and performance studies for nuclear piston engine power generating systems. The five lesser phases have covered (1) timestep size selection for solving the neutron kinetics equations, (2) engine startup behavior in the presence of delayed neutrons, (3) effects of delayed and photoneutrons on engine behavior, (4) blanket studies for the nuclear piston engine, and finally (5) some observations on the neutron lifetime be havior during the piston cycle. Some of the more important results obtained which delineate main identifiable characteristics of the nuclear piston engines are highlighted int he following discussion. Graphite has been found to be unacceptable as a moderating-reflector material for the nuclear piston engine concept. The core sizes and resultant strokes for these systems are considered to be too large to be practical ..

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o 2 o is the best moderating-reflector from a neutron economy standpoint ~nd most of the studies performed in this work cent~r on o 2 o-reflected eng_ines. Some composite reflector material studies have been carried out and it would appear as if the most desirable moderating-reflector designs will involve such reflectors with the inner portion consisting of Be or BeO and the o~ter region containing D 2 0. It has been found that because of the relatively large clearance volumes required ~y nuclear piston engines, they will probably be restricted to small compression ratios (~3-to-l ). Larger compression ratio engines would, of course, yield greater mechanical power outputs, but the required stroke lengths become too large to be practical. Given a gas temperature in the piston engine core, the u 235 atom density should not exceed a certain limit since the core beco~es so black to neutrons that further u 235 is i n e ff e c t i v e . F o r e x a m p l e , a t a D 2 0 t em p e r a t u r e of 2 9 O: K t h e u 235 atom density should not exceed about l .Oxlo 20 atorns/cm 3 , while at 570K this density should not be more than l .6xlo 20 . 3 atoms/cm . For all practical He-to-u 235 mass ratios and engine temperatures, the helium has been found to exert no notice able effect on the piston engine's neutron multiplication factor. However, it has led to a noticeable flattening of the thermal flux in the piston engine core. The helium

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186 also tends to enhance the thermodynamic and heat transfer properties of the primary working fluid. HeUF 6 mixtures with higher helium contents yield higher mechanical effi ciencies for the piston engine, as well as more compact HeUF 6 -to-He heat exchangers. A tr~deoff is seen to exist between mechanical power from the piston and turbine power extracted from the piston 1 s hot exhaustg~ses. exhaust gases are obtained by having the temperature (and pressure) peak late in the piston cycle. Shifting these peaks towards the TDC position results in cooler exhaust gases but increased mechanical power. The maximum permissible gas temperature is limited by the tendency for UF 6 gas to dissociate at elevated tempera tures. The higher the UF 6 partial pressure, the higher the temperature limit before dissociation. For the engines pre sented in this chapter, this limit was around 22OOK. The average difference between the simple NUCPISTN calculations and the more sophisticated CORA and EXTERMINATOR II results for the neutron multiplication .factor was 3 to 4%; the maximum difference was a little over 5%. Except for the thermal diffusion coefficients, the thermal group cross sec tions for the core regions obtained from NUCPISTN were within about 2.5% of the corresponding BRT-1 group tonstants. Con sidering the elementary nature of the. algorithms employed in•NUCPISTN for the neutronic calculations, the attained agreement was better than anticipated.

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187 The BRT and PHROG constants used in CORA and EXTERMINATOR-II and the NUCP1STN constants were then com pared with collapsed group constant.sand keffectives from S transport _theory calculations. 123-group XSDRN cal cun . 1 a t i on s i n s M e r i c a l g e o me t r y \'i e re p e r f o rm e d f o r g r o u p c o n st ants using :both S 4 and S 6 quadratures . The res u l ts i n di cated that aD s 4 quadrature was adequate. Both the keffectives and the collapsed group constants obtained from XSDRN were in good agreement with the PHROG, BRT-1, and NUCPISTN con stants. Fo r pi s ~on engines w i th es sent i a 11 y 11 i n fin i t e II re fl e c tors~ the position at which the thermal flux peaks in the reflector was found to be relatively independent of the en gine operating conditions. For the 100cm thick engines of this chapter, this distance was found to be about 15cm from the core-reflector interface. Of the three nuclear piston engine power generating systems examined, the piston-steam turbine arrangement had the highest overall efficiency and the highest total power. The piston-gas turbine-steam turbine system had the highest turbine power but was the most complex system. The piston gas turbine system was the least complex but it also had the lowest total power and overall efficienci. Economic studies will be required to determine whether or not the in~reased costs of the more complex systems override the better efficiencies and power outputs of these systems.

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188 The attractiveness of turbine versus mechanical power will also have to be considered for each application. Overall efficiencies for the three nuclear piston engine power generating systems ranged from ~44 to 52%. The corresponding heating rates ranged from 6550 to 7750 BTU/kwe-hr. In contrast, overall efficiencies for most fossil-fueled pe~king units lie in the 16 to 23% range for heating rates of from 15,000 to 21,000 BTU/kwe-hr. The fuel costs associated with the nuclear piston engines of this ch~pter are estimated as being below $0.20 per million BTU's or around 1.4 mills/kwe-hr.* Fuel costs for the present fossil-fueled peaking units on the other hand range from $0.50 to $1.30 per million BTU.** It should be pointed out that reliability of the fuel supply will be of prime im portance in establishing the industry preference regarding electric ~ower production, as well as conservation of our irreplaceable fossil fuels. The reactor volume requirements found, by the simpli fied model described, for the nuclear piston engine are not at all excessive. This quantity has compared favorably with *These estimates do not include recent increases in the cost of yellowcake (mostly u 3 oR) which has gone from $8 to $10 per pound to $20 to $25 per pound. **These costs do not consider the recent series of large increases in the cost of oil which will severely effect many of the_enefgy production costs.

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189 the corresponding numbers for large operational nuclear power systems. Other significant characte~istics, presently established for the nuclear piston engine, are the lo~d carrying flexibility, the short shutdown-to-on-line time requirement and the low fuel cycle cost. All above factors considered, this power system seems to offer a most reasonable combination of power vs. physical size for the low power range studied. The fuel inventory requirement for the primary loop appears to be reasonable. Estimates for the 54 MW(e) piston steam turbine system indicate a U requirement of around 185 kg. Regarding the timestep sizes to be used when solving the point reacto~ kinetics equations, it has been found that the best results occur when the number of timesteps per ne~tron lifetime is from around 4 to 40. Larger timesteps are too coarse and the resultant energy released in a single timestep becomes excessively arge. The output piston engine behavior for such timesteps is in significant error. Ftir smaller timesteps, computer roundoff errors accumulate and these also lead to inaccurate res~lts or predictions of engine behavior. The two-stroke nuclear_ piston engine is capable of rapid startups in the presence of delayed neutrons. The ti~e required to go from the shutdown condition to an essen tially equilibrium condition is of the order of 5 to 10 minutes. The startup procedures used in this chapter

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1 90 utilized variations in the initial gas pressure and in the loop circulation time. Variations in the initial gas tempera' ture co~ld also be used. Changes in .the loop circulation time provide ari efficient and effective means of qoinq from one equilibrium condition or power level to another and this will be more clearly illustrated in the next chapter. The effect which delayed neutrons have on the equi librium engine performance is that they lower the required fuel loading or keffective In the examples cited in this chapter the maximum keffective dropped from 1.109 to 1.061 and the average k ff t durinq the_cycle fell from 0.926 e ec 1ve to 0.871 when delayed neutron effects were included. The required uranium~235 loading decreased from 2.74kg to 2.21kg. The effect which photoneutrons had on the equilibrium performance was similar but of a lesser magnitude. When photoneutrons were included the maximum k fell from effectives 1 .061 to 1.046 while the averane k ff t durinn the cycle ';:t e ec 1ve ';:t dropped from 0.871 to 0.824. The uranium loading decreased from 2.208kg to 2.075kg . . Finally, the neutron lifetime was found to vary sig nificantly over the piston cycle because of changes in the .flux shapes and k-effectives. The neutron lifetime in the moderating-reflector region is large relative to the neu tron lifetime in the core. Neutron lifetime calculations

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l 91 were performed by doing adjoint and perturbation calcula tions with both CORA and EXTERMINATOR-II. Results from two group calculations were presented in this chapter. The two-group results will be compared with four-group neutron lifetime calculations in the next chapter.

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CHAPTER V RESULTS FROM FOUR-STROKE ENr,INE STUDIES Introduction A significant amount of insight was gained into the power producing and operational characteristics of the nu clear pi~ton engine ~oncept from the two~stroke engines which were analyzed in the previous chapter. Since these results proved encouraging, it was decided to undertake the study of four-stroke sys\ems. That is, in addition to the compression and power strokes, the intake and exhaust strokes were to be explicitly examined . . All the engines examined in this chapter include the effects of both delayed and photoneutrons. It will be re called that in the last chapter delayed neutrons were ini tially ignored. Later, delayed neutron and then photoneutron effects were included in~e engine studies. Both delayed and photoneutrons were observed to exert a non-negligible effect on the engine behavior. Also, varying the loop cir culation time, and hence the precursor population entering the engine, was found to be an efficient and effective way of achieving fast engine startup and of going from one equ1librium power level to another. 192

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Consideration of the intake and exhaust strokes means that the energetics equations have to be extended to include the open system, non-steady flow equations. The develop ment of these equations, for both the intake and exhaust pro cesses, will be found in Appendix E. The exhaust gas properties can now be explicitly calculated from the gas behavior as it leaves the cylinder arid crosses the•system boundary. In contrast, for the two-stroke engine studies of the last chapter, the exhaust gas properties were obtained by imposing a thermodynamic balance after making some simpli fying assumptions. The mass flow rat~ calculations are also now less ambiguous with the inclusion of the intake and exhaust process in the engine analysis. In the previous chap ter it will be recalled that the assumption was made that the gas mass was completely exhausted from the cylinder after each compression-power stroke cycle. An alternative logical assumption would have involved maintaining a residual amount of gas in the cylinder whose mass would be determined by the clearance volume .and final gas pressure and temperature. The reasons. for not choosing the latter assumption were expounded upon in the previous chapter. It will be seen that the attainable characteristics (e.g., mechanical power, overall efficiencies, gas tempera tures, etc.) for the four-stroke engines are comparable to the performance levels ~chieved with the two-stroke

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194 engines. The operating characteristics for these engines, however, are somewhat different. For example, in progress ing from the two-stroke engines to the four-stroke engines, the most desirable engine speeds go from 100 rpm 1 s to around 170 or 180 rpm 1 s for the o 2 o-reflected systems. The four stroke engine clearance volumes are smaller and the compres sion ratios larger than for the two-stroke configurations. Also, the four-stroke systems have larger fuel loadings. Some parametric studies for the four-stroke engines were carried out and the curves generated from these in vestigations are similar to those obtained for the two-stroke engines (see Figures 11 through 34). The only significant difference in the results was that the four-stroke engines tended to be somewhat less sensitive to parametric variations than were the two-stroke engines. Hence, the correspondin9 curves for the four-stroke systems need not ~e presented. Effects of variations in parameters which are unique to the four-stroke engine (e.g., the cycle fraction for closing the _intake valves) will however be illustrated in tables in this chapter. It was mentioned in the last chapter that it is de sirable for the nuclear piston engine to exhaust essentially all of the gas at the end of e~ch pist~n cycle. For the purpose of attaining this result the four-stroke engines utilize a combination of motions. Over most of the cycle,

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195 the piston undergoes pure simple harmonic motion but during the iriitial portion of the intake stroke and the final por tion of the exhaust stroke, the motion is non-simple har monic. Thus the instantaneous mass flow rate during most of the exhaust stroke is determined by the instantaneous properties of the gas in the cylinder, the valve characteris tics and the back pressure. When the gas pressure in the cylinder drops to the level of the back pressure, the in~ stantaneous mass flow rate depends on the piston motion (see Appendix E). During the last portion of the exhaust stroke, any gas remaining in the cyli_nder is physically forced from the cylinder by the piston as it executes its non-simple harmonic motion. The desired overall piston motion can be achieved by means of a four-bar linkage system [50]. For the four-stroke engines, the helium-to-uranium mass ratio has been decreased from 0.332 to 0.300 or to o.iso. This results in somewh~t poorer thermodynamic pr~per ties for the HeUF 6 gas mixture. However, the increase in the UF 6 partial pressure means that the temperatures at which dissociation effects set in are also increased [41]. It was for the purpose of decreasing any possible dissoci ation effects (i.e., for an added safety factor against such problems) that the change in the helium-to-uranium mass ratio. was made. The NUCPISTN results were again compared with higher order neutronic calculations by using the procedures and

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196 codes outlined in the previous chapter. Multigroup, one dimensional diffusion theory results obtained with MONA [51] represent the only addition to the previously described methods~ Lifetime calculations as a function of cycle position are again presented and plots of forward and ad joint flux distributions as a function of cycle position are also included. Moderator temperature coefficients of reactivity, fuel temperature coefficients of reactivity, and moderat~r void coefficients of reactivity for a engine are tabulated and discussed. The effects of using enrichments of less than 100%, the effects of delayed and photoneutrons, effects of spe cific heat formula selection, of numeral methods choice, and of timestep size selection forthe energy equations are all examined. Engirie startup in fue presence of both delayed and photoneutrons, power transients, and blanket studies are also included in this chapter. Finally thermodynamic analyses, similar to those of the last chapter, are presented for two of the four-stroke reference engines.

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Engine Startuo; Approach to Equilibrium in the Presence of Delayed and Photoneutrons 197 Engine startup from the shutdown condition to an equilibrium power level was examined in the last chapter for the case when delayed neutrons are pre~ent. The startup procedures used variations in the loop circulation time and in the initial qas pressure. Engine startup in the presence of both delayed and photoneutrons will now be investiqated for the four-stroke engine whose operating characteristics are shown in Table 35. The startup procedure used for this engine made use of changes in the following parameters: l) intake line gas pressure, 2) initial mass flow rate through the intake valves, 3) cycle fraction for the step-reflector removal, and 4) the loop circulation time. The NUCPISTN code in its present 1orm can. handle up to eight groups of photoneutron precursors. The default photo neutron precursor parameters are listed in Appendix F (Table_ F-6). The user can input values for any of these parameters to override the default values. The detault photoneutron precursor fractions are for. saturation fission product activity for u 235 fissions in . p p The parameters f and y (see Appendix B) are used to account for the fact that not all gammas released by the photoneutron precursors in the core yield photoneu trons in the o 2 o reflector. fp is the fraction of the gamma rays from the photoneutron precursors which penetrate froni the core to the moderating-reflector region. yp is the

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TABLE 35 Operating Conditions for Engine #8 Characteristic Number of str6kes Compression ratio Clearance volume (m 3 ) Stroke (ft) l~take line gas ~ressure (atm) Intake line gas temperature (K) Cycle fraction for closing intake valves Cycle fraction for opening exhaust valves Neutron source strength (n/sec) Engine speed (rpm) Cycle fraction for step-reflector removal He-to-U mass ratio % U235 enrichment Photoneutrons considered Delayed neutrons considered Pure simple harmonic motion o 2 o reflector thickness (cm) before step removal after step removal Neutron lifetime (msec) Piston cycle time (sec) U mass in cylinder (kg) He mass in cylinder (kg) Average o 2 o reflector physical temperature (K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) l 98 Engine #8 4 4.0 to l o. 135 5. 18 2 0. l 400 0. 150 0.750 1.0 X 10 9 170 0.570 .o. 300 l 00 yes yes no l 00 30 l . 600 0.8235 2.260 0.678 570 8.55 52.64 28.46

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199 fraction of the energetic gamma rays (with an energy of at least 2.3MeV) reaching the o 2 o which actually induce photo neutrons through the (y, n) reaction. For the engines examined in this chapter, the product of these parameters was 0.30. That is, 30% of the gammas released by the photoneu tron precursors in the core eventually produced photoneutrons in the moderating-reflector thfough th~ (y, n) reaction .. fpyp products of as low as 0.05 were investiqated and were observed, even at this level, to have a noticeable influence on the engine performance. In contrast, the more or less generally accepted con clusion for iolid-fueled, heavy-water-moderated heterogeneous core reactors is that the photoneutrons are of little impor tance. The nuclear piston engine, however, differs signifi cantly from these solid core systems not only in structure but also in its modus operandi. The gaseous core and thin niobium alloy or nickel liner do not present nearly as great an ob stacle to the photoneutron precursor gamma rays as do the solid fuel elements and their cladding. For the circulating fuel, pulsed nuclear piston engine the photoneutron precur sors, upon entering the engine on th~ intake stroke, act as a delayed extraneous or inhomogeneous fast neutron source in the moderating-reflector region. It should be mentioned that although there is general agreement regarding the lack of importance o~ the photoneu trons for the solid-fueled, o 2 0-moderated, heterogeneous

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200 core reactors, there is reasonable disagreement with regard to the photoneutron precursor parameters. Experimental re sults for these parameters from various sources are in conflict with one another, especially for the yield fraction value~ [52, 53, 54]. The photoneutron parameters used inthe nuclear pistori engine studies are what were thought to be the best of the available information. However, until the source of conflict in the experimental results can be de termined or until experiments can be conducted on a mockup of th~ nuclear piston erigine itself, the precise degree of the effect which photoneutrons will have on the system behavior will be unresolved. The poi n t k i net i cs e qua ti on s sol v e d by the t! UC P I S Trl code are to be found in Appendix D. In particular, int he presence of delayed and photoneutrons, the equations of in terest are (D-44) through (D-46). The numerical forms of these equations are given by (D-57) through (0-59). The startup procedure utilized for Engine #B is out lined in Table 36. As discussed in the previous chapter, the loop cycle number is to be distinguished from the piston cycle number. The total delayed and photoneutron precursor popula tions as a function of time are shown in Figures 60 and 61. The sharp break at about 80 seconds (t 1 ) occurs when the . . initial mass flow rate into the engine on the intake stroke

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TABLE 36 Startup Procedure for Engine #8 in the Presence of Delayed and Photoneutrons Loop Loop Intake Intake Line i-1ech. rlaximum Delayed P!1otoneutron . Total Cycle Circ. Line Initial Mass P01er Gas r~eutron Precursor Elapsed [lumber Time Gas Fl m<1 Rate ( r~v,) Temp. Precursor Conc:ntr~tion Time (sec) Pressure (kg/sec) ( o K) Cone. 3 (.c I cm ) (sec) (atm =/cm 1 8.0 18.0 26.0 0.99 600 , 4 .292 X 10: 8.3 9.02 X 10 5 2 3.0 18.0 26.0 0.99 600 3.51 X 10 6 .118 X 10 5 17.7 3 8.0 18.0 26.0 0.99 600 l. 05 x l 06 .362 X 10 6 26.5 ,.. 3.0 18.0 26.0 1.00 601 44. 1 ::, 7.83 X 10 7 .276 X 10 7 7 8.0 18.0 26.0 1.05 603 6.34 x 109 . 223 x l 08 61.8 9 8.0 18.0 26.0 2.38 1282 l . 42 x l 09 . 475 x l 03 79.4 10 8.0 18.0 23.5 l.86 l 071 l. 80 x l 09 .676 X 103 88.2 11 8.0 18.0 23.5 1.84 . l 057 2. l O x l 09 . 866 x 109 97. l 12 8.0 18.0 23.5 l.96 1137 2.46 x 109 . l 08 x 109 105.9 13 8.0 18.0 23.2 1.33 l 069 2.61 X 10 . 124 X 10 114. 7 -1-1-1109 l o 9 17 8.0 18.0 23.2 2.38 1426 3.37 X .205 X 150.8 18 8.0 18.0 22.8 2.23 1361 3.98 X 109 .227 X 10 9 158.8 + + + 10 9 109 21 8.0 18.0 22.8 3.00 1873 5.45 X .Jl9 X 150.8 22 8.0 18.0 22.3 2.69 1717 5.71 X 10 9 .347 X 109 194. l -1+ + 0 1 o 9 25 8.0 18.0 22.3 2.92 1875 6.54 X l 0~ .433 X 220.6 26 8.0 1s. o23.2 2.44 1743 6.72 X lOJ .459 X 109 229.4 + + + 9 9 34 8.0 18.4 23.2 3.03 2165 8.79 x 109 .696 X 10 9 300.0 35 8.0 18.4 23. 1 2.80 2018 8.81 X 10 . 720 X 10 308.8 + + + N 0 __,

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T,ABL E 36 (continued) Loop Loop Intake Intake Line ~1ech. Maximum Delayed Photoneutron Total Cycle r'-' 1rc. Line I n it i al '. a s s PO\-:er Gas Neutron Precursor Elapsed r~umber Time Gas FlmJ Rate (Mvl) Temp. Precursor Concentration Time (sec) Pressure (kg/sec) ( o K) Co~c. 3 (#/cm3) (sec) ( atm) :i/Cm 43 8.0 18.4 23. l 2.83 2032 9.05 X 10 9 .890 X 10 9 379.4 44 3.2 18.4 23. l 2.35 2051 9.06 X 10 9 . 911 x l o 9 . 388.4 + + + 9 10 lO 52 8.2 18.4 23. l 2.90 2083 9.24 X 109 . l 06 x 460.6 53 8.4 18.4 23.l 2. 91 2094 9.21 X 10 . l 03 x l O 10 469.8 + + + 9 10 60 8.4 18.4 23.l 2.88 2069 9.09 X 10 9 .118 X lQlO 534.4 61 8.45 18.4 23.l 2.88 2073 9.07 X 10 . 1_20 x l 0 543.7 + + + 9 10 65 8.45 18.4 23. l 2.88 2072 9.05 x 109 . 126 x l 0 1 O 580.8 66 8.50 18.4 23.l 2.88 2070 9.03 X 10 . 127 x l 0 590. l + + + 9 10 69 8.50 18.4 23. l 2.87 2064 8.98 x 109 .131 X 10 10 618. l 70 8.35 20.0 24.3 2.63 2031 8.96 X 10 . 132 x l 0 627.2 +. + + 10 9 10 76 8.35 20.0 24.3 2.67 2060 8.98 X . 139 X 10, 0 682.3 77 8.40 20.0 24.3 2.67 2063 8.99 X 10 9 . 141 X 10 691. 5 + + + 109 l O l 0 86 8.40 20.0 24.3 2.69 2074 9.02 X . 151 X 774.5 87 8.45 20.0 24.3 2.69 2076 9.01 X 109 . 152 X l O l 0 783.8 + + + 10 9 10 99 8.45 20.0 24.3 2.69 2078 8.98 X . 163 x l Ol O 895.1 100 3.00 18.0 22.6 2.66 2069 8.93 X 10 9 . 163 x l 0 903.9 + + + . N 0 N

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TABLE 36 (continued) Loop Loop Intake Intake Line Hech. f.laximum Delayed Cycle Circ. Line Initial :~ass Power GJS l~eutron rJurnber Time Gas Fl ov1 Rate (r1\v) Temp. Precursor (sec) Pressure (kg/sec) ( o K) Cone. (atm) (ff I cm 3 ) 111 8.00 18.0 22.6 2.66 2075 8.77 X 10 9 112 7.95 18 .. o 22.6 2.66 2073 8.76 X 109 + + + l o 9 138 7.95 18.0 22.6 2.66 2070 8.75 X 139 8.50 20. l 25.,5 2.98 2079 8.82 X 109 + + + 10 9 149 8.50 20. l 25.5 2.98 2078 8.98 X 150 8.55 20. 1 25.5 2.98 2079 9.00 X 1 o 9 + + + l o 9 154 8.55 20. l 25.5 3.00 2080 9.00 X + + + 10 9 175 8.55 20.1 25.5 3.00 2082 9:00 X Loop cycles l-25: step-reflector re~oval at the 0.585 cycle fraction Loop cycles 26-69:step-reflector removal at the 0.575 cycle fraction Loop cycles 7~175:step-reflector removal at the _0.570 cycle fraction Photoneutron Precursor Concentration (#/cm3) . 169 X lO 10 . 170 X lO 10 1010 .185 X . 185 X lO l 0 lO l 0 .194 X . 195 X 1010 . 198 X l O l 0 .209 X lO l 0 Tota 1 Elapsed Tir.ie (sec) l 001. O l 009. 7 1237.8 1247.2 1340. 4 1349. 8 1387. 3 1585.2' N 0 w

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........ M E u ::... ll lO L I t 2 =Li. 75r:1in I I I I t 1 =SOsccr-=1010 I Q 10 s ' " .i '10 6 105 0 I I I I I I I 100 200 I \lb I I 300 DI:LAYED NEUTRON PRECURSOR co:\C[NTRATION ./!lOTONEL'IlWI( PJZICURSOR COI-:CENTlu\TION at t 1 PX concentration (C/crn 3 ) = 5.15 x 7 at t 2 PN concentration (#/cm 3 ) = 6.45 x 10 8 at 900 sec PN concentration (#/crn 3 ) = 1.63 x 10 9 at 1800 sec PN concentration (#/crn 3 ) = 2.30 x 10 9 400 500 600 700 300 TI.:! L ( s c C ) 900 FIGURE 60. Delayed Neutron and Photoneutron Precursor Concentration Buildup During Startup for Engine #8

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P'. C U) :::> u i:..:.J 1010 . DELAYED ;;E'CTRot~ PRECURSOR cm:c:c~;TRATION F-----------------------------------------107 900 FIGURE 61. PHOTO~Et:TlWN PRECCRSOR CONCENTRATION at tl DN concentr.:ition Ui/ cm 3 -) = 1.44 X 10 9 (16% of equilibriu:n) at t2 DN concentration UJ /cm 3 ) = 8.13 X 10 9 (90% of equilibrb!!l) at t Dr{ concentr.:ition (f,!/cm 3 ) = 9.00 X 10 9 (equilibrium v2lue) 0:, .1000 1100 1200 1300 l!;QQ 1500 1600 1700 1800 TH1E (sec) Delayed Neutron and Photoneutrnn Precursor Concentration Buildup During Startup for Engine #8 (continu~c) N. 0 Ul

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206 is dropped from 26 to 23.5kg/sec. At t 1 , the total delayed neutron precursor population is 16% of its equilibrium value. The time required for the delayed neutron precursor population to reach 90% of its equilibrium value is about 285 sec (t 2 ). It will be noted in Figure 61, that while the delayed neutron precursor population has attained complete equilibrium, the photoneutron precursor population is still rising at a very slow rate. For all practical purposes, the photoneutron precursors can be considered to be in equilibrium. Their very slow growth rate in this region can easily be compensated for. In Figures 62 through 65 the mechanical power output and maximum gas temperature.are shown as a function of time. At t 1 , the power and temperature growth are halted by the above described change in the mass flow rate. Eventually the power and maximum gas temperature slowly increase again and at about 106 seconds, the intake line initial mass flow rate is further dropped from 23.5 to 23.2kg/sec. After a small dip, the power and maximum gas temperature start to rise again until at about 158 seconds, the intake line initial mass flow rate is once more decreased from 23.2 to 22.8kg/sec. The maximum gas temperature and power first decrease and then increase again until the mass flow rate is cut from 22.8 to 22.3kg/sec at a time of about 185 sec. A drop and then rise in power and maximum gas temperature

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t 24.QCI 11 I I 6 I ,--._ I :.<: 0 I '--' 5 ....... 2000 I ..,_, w "'-" '"' H ::::> --;::.., ;.H u 4 :::i 0 1600. -~ ,,_ oc:: .. , •a,., ~.2 :.::) 3 ,-l < l~OC u 1-1 z w H ::...) :....l o;.r. < 2 ::z: C 80 < :.:.l ::,.., 1 4r,r v ... 0 25 50 75 100 125 150 175 200 225 rr:-:E (sec) N 0 "'-,J FIGURE 62. Peak Gas Temperature and :-1 e c h a n i c a l P 01e r Output Gehavior During Startup for Engine #8

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6 2400 ,....._ 0 -...., 2000 5 ,-... "-• C E-< z :::i ...... >u 1600 4 E-< 0 ~'.3 0 _, 3 , < ,., 0 "" .... vv u ....; ti pm,:ER [-< u Cl) :::l .,_. < -;_;; 800 2 :.,: < C. 1 400 225 250 275 JOO 325 350 375 400 425 450 TI~\E (sec) FIGURE 63. Peak Gas Te~perature and Mechanical Power Output Behavior During Start.up for Engine #8 (continued) l'\J 0 o::>

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24 0 0 ,,...... .... (> .__, 0:::: 2000 , ., z ....: ;>l600 z ...... .. , ~.,, ? ,< 1200 co: ::.. ~:. Cl) < (.:) 8 . 00 0 ;:... 400 400 FIGURE 64. at tl at t2 at t C'O 450 500 TE:!PI:'.\.ATURE POWER temperature (OK) = 1304 te:r.peracure (OK) = 2050 temperature (OK) 2075 550 650 TI?-!E (sec) 6 5 4 3 2 1 700 750 800 850 Peak Gas Temp~rJture and Mechanical Power Output Behavior Ourinn Startup for Engine tB (continued) ,...._ ;;: '-" ,.... ::, t'. c t.:..: ,-J < u ..... z '~ ;:c: u !.:.J ;:.: N 0 \0

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vV ,,...._ :,::: 200 0 '--' fj ,...._ z ..... ...l ::,..; u z ...., ,., ,., =' !-t.l p.. ~w E--< u: 8() < c.,:; :,::: j ;::.. 40 850 FIGURE G5. 6 TD!PERATl:RE 5 4 POWER 3 nt tl power (~1w) = 2.52 2 at t2 power (Mw) = 2.88 at t power (}lw) = 3.00 00 1 950 1050 1150 1250' 1350 1450 1550 1650 1750 TINE (sec) Peak Gas Temoerature and Mechanical Power Outout Behavior Durinq Startup for Enqine fl8 (continued) ,-., "' ........, F-< ;:::i i::... f-, :::, 0 .,_.; ~,. 0 P,_ < u ..... z < ::i:: u 1-:.l "' N 0

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211 ensue until an elapsed time of about 220 seconds is reached. At this point the mass flow rate is inc~eased from 22.3 t-0 23.2kg/~ec and the intake line gas pressure is also iri creased from 18.0 to 18.4 atmospheres. The cycle fraction for step-reflector removal is altered from 0.585 to 0.575 and hence the drop in power and maximum gas temperature after this point. The next significant change is at about 300 seconds when the power and maximum gas temperature increase are damped out by dropping the intake line initial mass flow rate from . 23.2 to 23.lkg/sec. Several other adjustments are made later at about 380 seconds, at 460 seconds, at 534 seconds, and at 580 seconds. The effects of these changes however are almost undetectable in Figures 63 and 64. At about 620 seconds, the loop circulation time is cut from 8.5 to 8.35 seconds, the intake line gas pressure is increased from 18.4 to 20.0 atmospheres and the intake line initial mass flow rate i~ increased from 23.1 to 24.3kg/sec. The cycle frac tion for step-reflector removal is adjusted from 0.575 to 0.570 and thus a small drop . in power level and maximum gas pressure is observed. Several more a~justments are . made at 682 seconds, at 775 seconds, at 895 ~econds, and at 1001 seconds. The effects of these alterations are again undetectable in the figures. At 1238 seconds, the loop circulation time is

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212 raised from 7.95 to 8.5 seconds and the intake line gas pressure is boosted from 18.0 to 20.l atmospheres while the intake line mass flow rate is increased from 22.6 to 25.5 kg/sec. The result is a small jump in the mechanical power output lev~l which is readily seen in Figure 65. At 1340 seconds the loop c'irculation time is increased from 8.5 to 8.55 seconds and this is the final adjustment made for Engine #3. The equilibrium cycle results for this engine are listed in Table 37. It is thus seen that the four-stroke nuclear piston engine, in the presence of delayed and photoneutrons, is capable of rather rapid startups. The time requi~ed to go from the shutdown condition to an essentially equilibriu,n condition (photoneutron precursor concentration varying very slowly) is of the order of about 8 to 12 minutes. Other parameters which could be easily adjusted to assist in the approach to equilibrium procedure are the intake line gas temperature and the cycle fraction at which the intake valves are closed. Power transients which take off from an e~uilibrium condition and which are induced solely by variations in the loop circ~lation time will be examined later in this chapter.

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21 3 TABLE 37 Equilibrium Cycle Results from NUCPISTrl for Engine #8 Characteristic Maximum gas temperature (K) Maximum gas pressure (atm) Maximum keff Exhaust gas temperature (K) Exhaust gas pressure (atm) Average core thermal neutron flux during the cycle (n/cm2sec) Average gas temperature during the cycle (K) Average gas pressufe during the cycle (atm) Average keff during the cycle Mechanical power output (Mw) fission heat released (Mw) Mechanical efficiency (%) Photoneutron precursor concentration (#/cm 3 ) Delayed neutron precursor concentration (#/cm 3 ) Thernial (1/v) in core (sec/cm) Fast (1/v) in reflector (sec/cm) Maximum core thermal neutron flux during cycle (n/cm2)sec) Average mass flow rate (lb /hr) m Engine #8 2082 138.5 1 . 06 6 l 21 7 20.9 1.63 X 10 14 872 33.0 0.713 3.00 7.94 37.8 2.09 X 10 9 9.00 X 10 9 2.54 X 106 1.123xl07 4 . l 7 x 1 0 l 5 3.89 X 10 4

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An Examination of Reactor Physics Parameters as They Vary During the Piston Cycle 214 Nuclear piston Engine #8 has a piston cycle time of 0.8235 seconds and an averag~ neutron lifetime of 1 .600 seconds. A total of 5250 timesteps were used in solving the point reactor neutron kinetics equations. Twenty-four distributed cycle positions were selected for a detailed examination of how various reactor physics parameters varied during the piston cycle. The 24 positions are identified by the timestep number at which th~y occurred. These timesteps are listed in Table 38 along with the real or total elapsed time into the piston cycle, the core radius for the 11 equivalent 11 sphere and the system keffective a~ obtained from three different computational schemes. The two-group NUcrrsTN and two-group CORA results for the case when fast interactions in the core are neglected are in excellent agreement. The inclusion of fast core interactions leads to two-nrouo k ff t which afe gene ec 1ves erally 2 to 3% higher than for the case when these interactions are neglected. As the HeUF 6 mixture is drawn in during the intake _stroke, the keffective for the system increases. At the in stant when the intake valves are closed, i.e., at an elapsed time of 0.165 seconds, the system keffective is 1 .035. The in t a k e 1 s tr o k e II however has not been comp l et e d and a s th i s stroke

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TABLE 38 Core Radii and Neutron Multiplication Factors as Obtained from CO:"
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216 continues, the core volume increases and the gas density and k decrease. When the elapsed time is 0.235 seconds effective the intake stroke ends and the compression stroke.begins; at this point, the system keffective isl .013. During the compression process,'the system k ff t increases to a e ec 1ve maximum value of 1.066 at the TDC position. The power stroke follows, with the core volume increasing while the gas density and k decrease. When the total elapsed effective time in the cycle is 0.461 seconds, the step-reflector is removed and the system k drops from l .063 to effective 0;677. The k continues to drop during the remainder effective of the power stroke. At an elapsed time of 0.588 seconds, the exhaust valves are opened and the exhaust stroke begins. The keffective continues to drop during the exhaust stroke as the HeUF 6 exits from the system. The production of neutrons by the {y, n) reaction in the moderating-reflector region gives rise to a fast inhomogene ous source term for this ~egion (see Appendix B). The effects of this source term on the flux distribution are illustrated in Table 39. of the_ average thermal flux in the moderating-reflector to ( -t -f) the average thermal flux in the core while / is the C m ratio of the average thermal flux in the core to fhe average fast flux in the moderating-reflector. These quantities are listed in Table 39 for different computational schemes at the 24 selected positions during the piston cycle for Engine #8.

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TABLE 39 -Flux Ratios as Obtained from CORA and NUCPISTN for Engine #8 at Vario~s Cycle Positions Stroke Designation I I I I I I I I C C C C C p p p p p Time Step .u 1T 151 351 501 701 901 1051 1201 1351 1551 1801 2026 2326 2626 2926 2976 3226 3476 3726 Two-group NUCPISTN* 6 t fcbt m 'f'c .407 .397 .389 .378 .364 Two-group Two-group CORA** CORA* "7t 1 -t Tt /cbt ~m
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218 The two-group NUCPISTN and CORA results which include the inhomogeneous source terms are in good agreement over the whole cycle. The two-group CORA calculations which neglect this inhomogeneous source term however substantially underpredict (lt,it) and overpredict (~t,~f) when the system m c c m is far-subcritical. As the system approaches critical and goes supercritical, thi~ source term becomes of negligible inportance as far as determining flux distributions is concerned. For all three calculation schemes used in Table 39, fast interactions inthe core were neglected. In Table 40, the parameters appearing in the ~ix factor formula are tabulated at the 24 selected positions during the piston cycle for.Engine #8. The results are from two-group CORA calculations which include fast inte~actions in the core. During the time up until when the intake valves are th f closed, f, E, PNL' and PNL all increase as the UF 6 fuel is drawn into the cylinder; the resonance escape probability, p, on the other hand decreases during this period. The parameter undergoing the largest change in this span is P~t while f undergoes the next largest variation. Afterwards,. up until the time when the step-reflector is removed, the changes which occur in the six-factor formula parameters can be directly related to the gas density-changes which occur.

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TABLE 40 Six-Factor Formula Parameters as Obtained from CORA for Engine #8 at Various Cycle Positions Stroke Time f pth Designation Step f n E p k PNL k . # 00 NL I l 51 0.878 2.06 l. 042 0.981 l. 848 0.9994 0.317 0.585 I 351 0.915 2.06 l. 044 0.974 1. 918 0.9995 0.417 0.799 I 501 0.926 2.06 1. 048 0.970 1.939 0.9995 0.453 0.877 I 701 0.933 2.06 1. 048 0.969 l. 952 0.9995 0.483 0.942 I 901 0.938 2.06 l. 048 0. 967 l. 958 0.9995 0.515 l. 008 I N T A K E V A L V E S C L O S E D I l 051 0.942 2.06 1. 048 0. 965 1. 962 0.9996 0.536 l. 051 I 1801 0.940 2.06 .044 0. 968 l. 957 0.9996 0.531 1. 038 I 1351 0.939 2.06 l. 041 0.970 1. 953 0.9996 0.527 l. 029 C l 5_51 0.938 2.06 l. 041 0.970 l. 952 0.9996 0.526 l. 027 C 1801 0.940 2.06 1. 044 0.968 1. 956 0.9996 0. 537 1. 038 C 2026 0.944 _ 2. 06 1.051 0. 963 1. 968 0.9996 0.538 1. 059 C 2326 0.948 2.06 1. 070 0. 951 l. 987 0.9995 0.547 l. 087 C 2626 0. 951 2.06 l. 088 0.940 2.003 0.9995 0.547 l. 095 p 2926 0.949 2.06 l. 070 0.951 l. 989 0.9995 0.547 l. 087 p 2976 0.989 2.06 l. 103 0.946 2. 127 0.837 0.399 0.710 p 3226 0.988 2.06 1. 082 0.958 2. 110 0.841 0.375 0.666 p 3476 0.986 2.06 1.072 0.964 2.099 0.843 0.357 0.632 p 3726 0.986 2.06 1. 069 0.966 2.097 0.844 0.350 0.619 E X H A U S T V A L V E S 0 P E N E D E 3776 0.986 2.06 1. 066 0.968 2.096 0.844 0.340 0. 601 E 3976 0.980 2.06 l. 054 0. 977 2.079 0.842 0.271 0.475 E 4176 0.973 2.06 1. 049 0.982 2.065 0.839 0.219 0.379 E 4426 0. 965 2.06 l. 046 0.986 2.050 0.834 0. 170 0. 291 E 4676 0.955 2.06 l. 046 0.988 2.033 0.827 0. 130 0.219 E 4976 0.940 2.06 1.048 0.990 2.009 0.819 0. l 01 0. 166 All results are from two-group CORA calculations in which fast core interactionswere included. N __, \.0

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220 For the remainder of the intake stroke, the fuel or gas d~nsity decreases and f, E, and P~t also decrease while p increases. During th~ compression stroke the gas density increases and f, E, and also increase while p decreases. During the power stroke the gas density decreases again yielding a decrease inf, E, and P~t and an increase in p. The qua n t i t y , n , d u r i n g t h i s w h o l e p e r i o d i .s e s sen t i a 11 y . c~nstant while PN[ has also hardly changed. Shortly after .the power stroke starts (at a total elapsed time of 0.461 seconds} the step-reflector is removed and five of the six-factor parameters (n excepted) are seen to take sudden jum~s in their values. The decrease 1n the reflector size from 1.00 to 30cm. has caused an increase inthe fast and thermal leakages from the system and hence P fast d pth d . 'f• tl Th . NL an . NL rop s1gn1 ,can y. e resonance escape probability also drops while f and Eincrease as a result -Of the step-reflector removal. For the remainder of the power ~troke, these parameters agaih behave in the normal manner, i.e., f, E, and decrease while p increases as the gas density decreases. The exhaust val.ves are_ then opened and the exhaust stroke commences. As the UF 6 -fuel is removed th f from the cylinder, f, E, PNL' and PNL decrease; only p undergoes an increase during this stroke. Presented in Table 41 are some prompt neutron life ti~e ~nd generation times at a few of the selected positions

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221 TABLE 41 Neutron Lifetimes and Generation Times as Obtained from CORA for Engine #8 at Selected Cycle Positions Stroke Time Core t A Designation Step k eff Radius (msec) (msec) No. (cm) I 1 51 0.570 23.44 2.86 5.03 I 9 O l 0.992 42.29 2.01 2.03 I l 3 51 1 . 01 6 49.97 l. 94 1. 91 C 1976 l . 03 7 45. 17 l. 7 3 l.67 C 2526 1 . 066 32.42 l . 4 l 1. 32 Results are from two-group CORA calculations in which fast interactions in the core are neglected. The inhomogeneous source term in the reflector due to photoneutrons has been included. TABLE 42 U235 Enrichment Effects on the Neutron Multiplication Factor for Engine #8 at tile TDC Position U235 Enrichment % l 00 93 80 hrn group* T\'rn group** CORA keff CORA keff l . 06 5 l . 0 51 l. 050 l . 09 5 l . 07 5 l. 07 2 Four-group* CORI\ keff l . 0 21 l . 007 l . 006 *Fast interactions in the core are neglected. **Fast interactions in the core are included. Four-group** CORA keff l . 03 6 l . 020 l . 0 l 8

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for Engine #8. The results were obtained from CORA two-group adjoint and perturbatio~ calculations in which fast core interactions were neglected. 222 As the keffective increases from the subcriiical to far-supercritical, the neutron lifetime and gener~tion time decrease continuously. This is somewhat different from the results of the last chapter where for Engine #7 the far-subcritical systems were observed to have shorter generation times. The far-subcritical configurations for Engine #7, however, were obtained by inserting highly absorbing poison or control rods into the moderating reflector region. These poisons greatly shortened the neutron lifetime in the moderating-reflector and hence the system neutron lifetime was also shortened. The neglect of fast interactions in the core results in very small errors in the neutron lifetime and generation time val~es since the bulk of the neutron lifetime is in fact determined by the time which the neutrons spend in the moderating-reflector region. The neglect of the inhomogeneous sources for the near-critical and super critical systems also has a negligible effect on the neutron lifetime calculations. Howeveri for the far-subcritical configurations, the neglect of these inhomogeneous sources leads to significant errors in the neutron lifetime results.

PAGE 261

223 This is to be expected in view of the just completed. discussion on the effects which these sources have on the neutron flux distributions. Pres~nted in Table 42 are results which illustrate how the system keffective is influenced by going to less than 100% enriched uranium hexafluoride fuels. Actual engines will probably operate on highly (~90%) enriched UF 6 . The results of Table 42 indicate that the penalty or the keffective drop in going from 100% to 93% enriched UF 6 is but about 1.5%. In fact, the nuclear piston engine designs presented in this study could readily operate with enrichments of as low as 80%. The influence of enrich ment variations will be further examined later in this chapter. The possibility of using U-233 fuels will be covered in Chapter VII. Flux Shape Changes During the Piston Cycle Shown in Figures 66 through 92 are plots of the fast and thermal forward fluxes and the fast and thermal adjoint fluxes for Engine #8 at various selected cycle positions. The results are all from two-group CORA for ward and adjoint cilculations in which fast interactions in the core were neglected. In examining these plots, it should be remembered that not only is the system keff cha~ging throughout the piston cycle but also that the core geometry is continuously changing.

PAGE 262

224 The. changes in the forward flux shapes during the piston cycle follow.a pattern which parallels the behavior exhibited by the parameters of the six-facto~ formula as they varied during the cycle. In Figures 66, 68, 69 and 70 as the syst~m goes from far-subcritical towards critical and as the HeUF 6 enters the cylinder the fast flux in the core is observed to grow relative to the core thermal fl~x. The intake valves are then closed and as previously described, the engine then proceeds to complete the intake ~troke. During this period the gas density and keffective decrease and the thermal core flux increases relative to the core fast flux (figures illustrating this effect are not shown). The engine then begins the compression stroke (Figure 72) and the gas density and k ff• t increase. e ec 1ve As the system continues towards TDC and becomes more and more supercritical, the fast core flux grows relative to the core thermal flux (Figures 73, 74 and 75). Next, the power stroke starts and the gas density and keffective decrease. The core thermal flux then increases relative to the core fast flux. Shortly after the power stroke commences, however, the step-reflector removal occurs and the core thermal flux takes a sudden slight rise relative to the core fast flux (not shown in the figures). The power stroke then continues with the gas density and keffective decreasing and the core thermal flux increasing

PAGE 263

X _, , ~. alo.s THEP~'...i\L FLUX FAST FLUX Two-group CORA calcul2tion no fast interactions in core with inhomogeneous photone~tron so~rces timcstep ncr:1ber 351/intake stroke neutron multiplication factor= 0.782 ;.:... C', _cl---------------1 ;..J :::: 0.4 o. :' 0 FIGURE 66. 10 REFLECTOR CORE 20 30 40 50 GO 70 80 90 100 RA".HAL DISTA::CE (c:1) Fas t and Therm a 1 Ne Lit:-on Fl u x Ver s u s Rad i a l D i stance for Eng i n e ff 8 a t Ti !:! es t e p Number 351 N N CJ1

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X 0.8 CORE I I. I I 0.4 I I I I I I 0.2 I I I I_ I 0 10 20 30 REfLI:CTOR 40 so RADIAL DISTA::cr:: T\,'o-group CORA calculation no fast interactions in core no inho~ogcneous photoncutron sources timestcp number 351/intake stroke neutron multiplication factor= 0.782 60 70 80 90 ( c:11) 100 FI"GURE 67. Fast Jnd Thermal Ncutro.n Flux Versus Rc1dial Distance for Engine t/D at TimGstcp Nu~ber 351 When Inhomogeneous Photoneutron Sources Are Igndred N N '

PAGE 265

>:: -c. ,. ;_;., :..-~ :.-l .l.O THEm:.AL FLUX 0.8 C..--------------C'.G 0.4 0.2 0 10 FIGURE 68. FAST FLUX . CORE 20 30 Fast and Thermal l;um be r 5 0 l I I I 40 r~eu t ron REFLI:CTOR 50 Two-group CORA calcul2.tion no fast interactions in core with inhomo8eneous ?hotone~tron sources timestep number 501/~ntake stroke neutron multiplication factor= 0.859 60 70 80 9U 100 RADIAL D:!:s-.rx:cE (cm) Flux Versus Radial Distance for Engine #8 at Timestep N N --.J

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0.8 X 3 ,..:... '.). 6 0.4 0.2 0 FIGURE 69. CORE 20 30 Two-group CORA calc:.:lation nc fast interactions in sore with inhomogeneous phctoncutron sources timestep number 701/intake stroke neutron multiplication factor= 0.924 REFLECTOR 40 50 60 70 so 90 100 R/1DIAL DISL"t'.:CE (cm) Fast ~nd Thermal Neutron F1ux Versus Radial Distance for [ngine #8 at Ti~estcp 1 u m be r 7 O 1 N N CX)

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l.J Two-group CORA calculation no fast interactions in core with inho~ogcncous photoncutron sources timestep number 901/ictake stroke neutron multiplic&tion.factor = 0.991 0.8 J---------==========-TH[~-1.i\L FLUX :J. C 0.4 0.2 0 lU FIGURE 70. FAST FLUX CORE REFLECTOR 20 40 50 60 70 80 YO 100 RADIAL :ns:-:,::f..:;[ (cm) Fast and Thermal Neutron Flux Versus Radia~ Distance for Engine #8 at Timestep iJumber 901 N N I.O

PAGE 268

'1 .l.. v Two-group CORA calculation no fast intcracticns in core 0.8 f==========-----==::::::::::::..., no inhomogeneous photoneut~on sources timestep nu:nber 901/intake stroke neutron IT.ultiplication factor= 0.991 _, ' ,..:.. :c..: o. r_; r ;.>f-, < ..J :...:i 0.2 0 r:GU?.E 71. 10 CORE THERNAL FLUX REFLECTOR 20 30 40 50 60 70 80 YO 100 RADIAL DISL\::CE (cm) Fast and Thermal Neutron Flux Ve~sus Radial Distance for Engine# at Ti~cstep Number 901 .When Inhomogeneous Photoneutron Sources Are Ignored N w 0

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. (' .: .• J THE~!AL FLUX Two-group CO:-L-\ cc!lculation no fast intcrJctions in core with inho~ogeneous photoneutron sources tirnestep number 1551/co~pression strpke multiplic~tion f~ctor = 1.013 0.8 6..---------~--------:,; :....: (' ,. r 0.4 0.2 0 10 FAST FLUX CORE 20 30 I I I I I I I I I I I I ,, 40 50 60 RADIAL DISTAXC: (c::1) REFLECTOR ,_ 70 80 90 100 Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Ti~cstep I'll: m b e r l 5 5 1 N w __,

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l.C Two-group CORA calculation no fast interactions in core with inhocogcneous photoneutron sources tirnestep nu~bcr 1801/compression stroke neutron nultiplication fact~r 1.023 0.81::------------------------r ~-'. 0. 6 ..... !-, < ....J :..:.:: ~-0.4 0.2 0 lU FIGURE 73. THEH~L\L CORE REFLECTOR 2U JU 4U !:>U uU ,u 80 iUU RADIAL DISTA~C[ (c:n) Fast and Therma1 fleutron F1ux Versus Radia1 Distance for Engine #8 at Tir:;estep Number 1801 N w N

PAGE 271

" Al. ' ,, ....... ...., I 0.8 i=---~--------------X REFLECTOR . CORE C.4 0.2 0 10 20 30 40 50 60 RADIAL DISTA:~CE (cr.1) FIGURE 74, Fast an9 Thermal Neutron Flux Versus Radial Humber 2026 Two-group CORA calcuL::ion no fGst intcrDctions in core with inhomogeneous photoneutron sour~es tincstep number 2026/conpression stroke neutron multiplic~tion f2ctor = ~.041 TEER,'-1AL 'FLUX 70 . 80 90 100 Distance for Engine #8 at Timestep N w w

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>< ....., I .: . or----------.:....:.' o.s Two-group CORA calcul3tion no fast inter~ctions in core with inhomogeneous photoncutron sources timestep number 2326/comprcssion stroke neutron multiplication factor= 1.064 -.~. :. ~. CORE 0.4 o. 2 I I I FAST FLUX REFLECTOR I I L_ __ j_ __ J_ __ j__..:___1_ __ +~-_J_ __ _I====:=t:======~l.,._ O 10 LU JU 40 50 60 70 8U ':IU lUii RADIAL D!STA~CS (c~) FIG~~E 75. Fast and Thermal tleutron Flux Versus. Radi&l i~ um be r 2 3 2 6 ~istance for Engtne 18 at Timcstcp

PAGE 273

235 relative to the core fast flux. This trend then continues during the exhaust stroke (Figures 76, 78, 79 and 80) as the HeUF 6 is purged from the cylinder and the system be comes more and more subcritical. The effect on the flux distributions of neglecting the fast inhomogeneous source term due to photoneutrons is illustrated in Figures 67, 71 and 77. Comparing Figure 67 with Figure 66, Figure _71 with Figure 70 ~nd Figure 77 with Figure 76, it ~ill be noted that the neglect of the fast inhomogeneous source term in the reflector causes the thermal core flux to be depressed relative to the fast core flux. The reason .for this is that the inhomogen eous photoneutron production in the moderating-reflector acts as an additional thermal neutron source for the core since essentially all of the slowing down and hence essentially all of the core thermal neutrons are generated in the moderating-reflector. It will be observed in these figure~ that the influence of the photoneutron production on the flux distribution becomes less and less pronounced . as the system approaches critical . . Figures 81 through 92 show adjoint fluxes for the same positions at whicl1 the Engine #8 forward fluxes were examined. The changes in the adjoint flux distributions which take place during the cycle can be described in excictly the same manner as were the changes in the forward flux distributions. In fact, if one substitutes the words

PAGE 274

FAST FLVX ' __ , ____________________ I 0.8 Tl:r.ltl/lJ... FLUX r , > c-' < ,-l u 0-::: CORE 0.4 0.2 0 10 20 JO FIGURE 76. Fast and Thermal Number 3226 40 Rlwii\L Neutron Flux I 50 DIST11:-;cr: Versus Two-group CORA calculation no fast interactions in core with inhomogeneous photoncutron sources timestep nu~ber 3226/power stroke neutron nultiplication factor= 0,.635 60 70 80 90 100 (cm) Radial Jistance for Engine #8 at Timestep N w a-,

PAGE 275

FAST FLUX l.O r=----------------....:..---. Two-gr
PAGE 276

:..i.. f-, < --l :.,.:J c,: l.C FAST FLUX ,-----------------------0.8 Ti-!ER!-1Al FLUX CORE 0.4 0.2 0 10 20 30 40 50 RADIAL DISTA::CE FIGURE 78. Fast and Thermal I~ e u tr on Flux Versus Number 3726 60 (cm) Radial Two-g!'oup CORA calculation no fast inteiactions in core with inhomogeneous photoneutron sources timestep number 3726/power stroke neutron multiplic~tion factor= 0.594 70 80 90 100 Distance for Engine #8 at Tir.iestep N w co

PAGE 277

0.8 X > .-1 f--1 < j ::,::: 0.4 0.2 0 TH.Sm1AL FLUX F.I\ST FLUX CORE 10 20 30 40 50 Jli\DIJ\L DISTA'.\CE Two-group CORA ~alculation no fast interactions in core with inhomogeneous photo!1<2utron sources timestep number 3976/exh,rnst stroke neutron multiplication factor= 0.458 RI:FLECTOR 70 80 90 100 FIGURE 79. Fast and Thermal Neutron Flux Versus Radial Distance for Engine #8 at Timestep Numbe-r 3976 N w I..O

PAGE 278

n:rn~-U,L FLUX :.~ r ------~------------r 0.8 CORE :_c: OJ r------------------...!.. < ..J :..:.J cc 0.4 0.2 0 FIGURE 10 8 0. FAST FLUX 20 30 Fast and Thermal Number 4426 I I 40 Neutron F..ADIAL Flux 50 T~o-group CORA calculation no fast interactions in core ,dth inhomogeneous photoheutron sources timestep number 4426/exhaust stroke neutron multiplication factor= 0.280 60 70 80 90 DIS:'A\'CI (cm) Versus Radial Distance for Engine ii8 at N 100 -~ 0 Timestep

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..;.. THER.:AL ADJOINT ~.or--------------0.8 0.2 FAST ADJOINT CORE I I REFLECTOR Two-group CORA calculation no fast interactions .in core with inhomogeneous photoneutron sources timcstep nu~ber 351/intake stro!
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I 1.0 ,---T_E_Er_~_lA_L_A_::>_J_o_r_N_T ____ :1 l\•.o-group C0:11\ calculation no fast interactions in core_ 0.8 0.4 C.2 0 FIGURE 82. with inho~ogcneous photoncutron sources ti~estep number 501/intake s~roke neutron multiplication= 0.859 F,\ST ,\DJOINT CORE 10 20 Fast and Timestep Rr:FLECTOR 30 40 50 60 70 80 90 lUU I'-J\DIAL DISTA::CE (er.-:) Thermal Adjoint Neutron Flux Vers~s Radial Distance for Engine #8 at r•; um b e r 5 o l

PAGE 281

THE~l.AL ADJO r~n ~-0 r----------------..... 0.8 FAST ADJOGT .....J 0.4 CORE REFLECTOR 0.2 0 10 20 30 40 50 RADIAL DIST!•.XCE FIGURE 83. Fa.st and Thermal Adjoint Neutron Flux Timestep riurnber 7 0 l Two-group CORA calculation no fast interactions in ccre \,ith inhomogeneous photoneutron sources timestep number 701/intake stroke neutron multiplication factor= 0.924 !10 70 80 90 (cm) Versus Radial Disttince for Engine N 100 ..,. w Jl 0 1TLJ at

PAGE 282

0.8 0.4 0 FIGURE 84. THET-1.i\L ADJOINT f AST ADJO I!:.'T CORE 10 20 30 Fast and Thermal Adjoint Timestep Number 901 40 P..ADL\L Two-group CORA calculation no fast interactions in core with inhomogeneous photoncutron sources timcstcp number 901/intake stroke neutron multiplication factor= 0.991 REFL!::C'IOR so 60 70 oU ~u DlSL\~CI: (cm) Neutron F1 ux Versus Radial Distance for Engine N ..;:,. lUU -~ #8 at

PAGE 283

:.o THERNAL ADJOINT 0.8 FAST ADJOINT >< -, -; :...... ... ;:....:;, . (, r -5 ,., -~ 0.4 CORE 0 ., .... 0 10 20 30 ~rSURE 85. Fast and Thermal Adjoint Timestep Nur.1ber 1551 40 RADIAL Neutron so DIST;\>:cr: Two-group CORA cnlculation no fast interactions in core with inhomogeneous phcitoneutron sources timestep number 1551/comprcssion stroke neutron multiplication facto~= 1.013 ~EFLECTOR 60 70 1..:0 90 100 (cm) Flux Versus Radial Distance for Engine t? 8 N Ul at

PAGE 284

r:'EEPJ-!.AL ADJOINT I I 1 1-----------------------1 0.8 FAST ADJOINT :..:... CORE 0.4 0.2 I D 10 20 30 40 RADIAL FIGt.:RE [3 6 . Fast and Thermal Adjoint r; e u tr on Titnestep Number 18.01 I I I SQ Two-group COR./1. calcul2 tion no f2st core interactions with inhomogeneous photoneutron sources timestep nuDber 1801/com?ression stroke multiplication factor= 1.023 REfL[CTOR 60 70 80 90 DIST,\:{CE (cm) Flux Versus Radial Distance for Engine #8 N -l:s 100 0, at

PAGE 285

T!IER?-!AL :\DJOEH l.Or------------~------! 0.8 FAST ADJOINT 0.4 CORE 0.2 0 10 20 JO 40 RADIAL Rr:ru::CTOR 50 uO :JI STA'.•:Cr: (cm) Two-group CORA calculation no fast interactions in core with inhomogeneous photone~tron sources timestep number 2026/co~pression stroke neutron multiplicatioc fQctor = 1.041 70 80 90 100 FIGURE 87. Fc:.st and Thermal /\djoint f!eutron Flux Versus Radial Distance for Engine .'JO ;;u Timestep i!umber 2026 N .J::> -....j.,at

PAGE 286

n:EP~1AL ADJOINT "'',-------~ 0.8 FAST ADJOnT :;.. 0.4 CORZ 0.2 0 10 20 30 FIGURE 88. Fast and Thermal Adjoint Tinest~p Number 2326 T\.;o-group CORA calculation no fnst interactions in core with inhomogeneous photcncutron sources ti:::es tep nu:::ber 2326/ compression s tro::cr: (cm) Heutron Flux Ve rs.us Radial Distnnce for Engine 100 ff 8 at N .p. 00

PAGE 287

TEEr-~-1/\L ADJODlT , r~----.:__ ___ .:___ J O,Si=-------F_A_s_T_AD_J_o_r_1rr ________ --1:___ X ......, •. I\ F ..... -..; -'0.4 u 10 CORE 20 JO 40 I I r RADIAL 50 DIST,\;:Cl.: FIGURE 89. Fast and Thermal Adjoint !ieutron Flux Tir.-:ester Number 3226 Two-group CORA calculation no fast interactions in core with inhomogeneous photoneutron source tirncstep nur:1ber 3226/pm,,cr stcoke neutron multiplication factor= 0.635 GO 70 (:o 90 (rn) 'Jersus Radial Ji stance for Engine lUU t,!8 at N ..:::,. I.O

PAGE 288

. TE[?21AL ADJOIXT r------__:...:__~---FAST ADJOU:T 0.8 1/. -; 0.4 CORE 0.2 II 0 10 20 30 40 50 60 P,ADIAL DIST/,:~CE (c~) FI Gu RE 90. Fast and Thermal Adjoint Neutron Flux Versus Timestep Number 3726 T"o-group COPv\ calcuL1tion no fnst interactions in core with inhomo~cneous photoneutron sources timestep number 3726/power stroke neutron multiplication f~ctor = 0.594 70 30 90 lOG Radial Distcnce for Engine If 8 at N u, 0

PAGE 289

FAST ADJOI~T l.Or----------------__; _______ _ Tl!Efilf.i\L J\DJOI~T 0.8 CORE 0.4 n ; ,.; . 0 10 :w 30 FIGURE 91. Fast and Thermal Adjoint Ti~cstep ~umber 3976 40 50 MDIAL DIST.\:~cr: rieutron Flux I 60 (cm) Versus Two-group CORA calculation no fast intcrnctions in core with inhomogeneous photoncutron sources timestep nu~ber 3976/cxhaust stroke neutron multiplication factor= 0.458 70 80 90 100 R a d i a 1 Distance for Engine #3 at N <.n __,

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0.8 0.4 0 F i G Li R E 9 2 .• THE!vl.AL ADJ O rrn CORE lU 20 JO 40 SU 60 RADIAL :JISTA:~cr: (cm) Two-group car.A calculnt~on no fast interactions in ccre with inho:'iogc:1cous phctoncutron so1:.,.-:-~s ti1:iestep num_bcr L:426/cxhaust stroke neutron multi?lic3tlon factor= 0.280 REFLECTOR 70 oU YO .iUU Fast and Thermal Adjoint ~eutron Flux Versus Radial Jistance for Timestep Number 4426 Engine #3 at N C.7l N

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11 c ore fa s t ad j o i n t fl u x " for 11 c ore therm a 1 fl u x " a n d "core thermal adjoint fli.Jx 11 for 11 core fast flux 11 , the above description can be applied directly to the adjoint flux curves. From Table 39, Table 41 and Figures 66 through 92 253 it is apparent that significant changes . occur in the neutron flux distributions and neutron lifeti~es during the piston cycle. Consequently, the adequacy of using a pure, simple po i n t r e a c t o r k i n e t i c s mod e 1 f o r h a n d l i n g t h e s y s t em 1 s neutron kinetics is questionable. This is true even if cjcle~averaged values for parameters such as the ne~tron . lifetime, the effective delayed fractions, and the inhomo geneous source weighting functions are employed (see Appendix D). Thus, the NUCPISTN code was modified to allow for time-vaiying effective delayed neutron fractions, ti"me-varying neutron l _ ifetimes, and time-varying inhomogen eous source weighting functions . for the neutron kinetics equations. The effect of this improvement is an extension from the simple point reactor kinetics model to an adiabatic method for treating the sysfe~s neutron kinetic~. The effects of time-varying source weighting functions w i l l , b e o f i m p o r t a n c e o n l y \~ h e n t h e s y s t em i s f a r s u b c r i t i cal. The effettive ~elayed neutron fractions are generally of the order of 0.0067 to 0.0068 and time-varying effects for . this parameter will not be nearly as important as

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2 5'4 effects from a time-varying neutron generation time. This topic will be discussed further in Chapter VII. Some of the derivations in Appendix B dep~nded on the assumptions that the relative flux distribu~ion~ and rela tive neutron populations do not change significantly over a neutron lifetime. Referring to Table 39, the elapsed time between timestep #701 and timestep .#901 is 0.031 seconds. During this interval, 200 timesteps were used in solving the neutron kinetics equations. The ratios ~/t and t/f change from 0.378 to 0.364 and from 37.2 m C C Ill to 38.7 respectively during this period. While these changes are occurring, approximately 19 neutron lifetimes elapse. Hence, the assumption of sma.11 changes for these ratios over a single neutron lifetime is quiti good. Fuel and DO Moderating-Reflector Temperature cefficients of Reactivity Moderator temperature coefficients of reactivity for the configuration of Engine #8 at the TDC position were calculated for average moderator temperaWres of from 320 to 570K. Results obtained from two-group CORA calcu lations in which fast interactions in the core were first neglected and then included are shown in Table 43. The o 2 o moderator density was held constant for these calcula tions in order to isolate temperature effects. D 2 O moderating-reflector density changes and voiding effects will be cohsidered separately.

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255 In examining Table 43, it will be observed that over the entire chosen temperature range, the D 2 o moderatingre fl e c to r t em p e r a tu re c o e f f i c i en t o f r ea c t i v i t y i s l a r g. e and negative. This is obviously a favorable situation from a safety and control standpoint for the nuclear piston engine concept. Fuel temperature coefficients of reactivity for the configuration of Engine #8 at the TDC position were calcu lated for average fuel temperatures of from 300 to 2100K. Results obtained from two-and four-group CORA calculations in which fast interactions in the core were included are presented in Table 44 for the case of 100% enriched UF 6 . Results for 93% and 80% enriched UF 6 fuel are shown in Tables 45 and 46 respectively. For the case of 100% enriched Uf 6 , the fuel tempera ture coefficient is extremely small and positive over the range of examined temperatures. An increase in fuel temperature influences the system keff for this case primarily as a result of changes in the epithermal u 235 absorption and fission cross sections. The u 235 has epithermal resonances at about 0.3eV and at l.15~V. With the increase in fuel temperature, the epithermal absorption and fission cross sections both increase because of Doppler broadening of these resonances. However, fa~ the Engine #8 core, the epithermal group average energy at 300K is 0.50eV while at 2100K, the epithermal group

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TABLE 4.3 o 2 o Moderatin~-Reflector Temperature Coefficients of Reactivity Using the Engine #8 Configuration at the TDC Position Average o 2 o Two-group Two-group Temperature CORA* l dk * CORA** 1 dk ** ( o K) keff k dT keff k ctT C 320 1. 1428 1.1676 345 -3.46lx10 -4 -3.195xl0 370 1.1232 1.1491 395 -2.361x10 -4 -2. 177x10 420 1. 1100 1.1366 445 -2.845xl0 -4 -2.620x10 470 1.0943 1.1218 495 . -4 -2.685xl0 -2.47lxl0 520 1 .0797 1.1080 256 -4 -4 -4 -4 545 -2.62lxl0 -4 -2.409xl04 .. 570 1.0657 1 . 0948 320-570 -2.793xl0 -4 .*Fast interactions in the core are neglected. **Fast interactions in the core are included. Average gas temperature in the core. is 1500K. -2.574x10 -4

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257 TABLE 44 . Fuel Temperature Coefficients of Reactivity for 100% Enriched UF 6 Using the Engine #8 Configuration at the TDC Position Average Fuel Two-group Four-group Temperature CORA 1 dk CORA 1 dk ( o K) ketf k dT keff k df 300 1.092290 1. 035602 450 2.227xl0 -6 8.586xlo7 600 l. 093021 1.035869 750 -6 -7 1.728xl0 6.818xl0 900 1. 093587 1.036081 1050 -6 ' ' ' -7 l.377xl0 5.524xl0 1200 l. 094039 1. 036253 1350 -7 . 7 8.222xl0 3.107xl0 1500 1.094309 l. 036349 1650 -6 -7 l .047xl0 4.19lx10 1800 1. 094653 1.036480 1950 -7 -7 9.606x10 3.997x10 2100 1.094968 1.036604 300900 -6 -7 l.978x10 7.702xl0 900-1500 -6 ' 7 l.099xl0 4.315xl0 1500-2100 ' -6 ' 7 ' l.004x10 4 . . 094xl0 300-2100 . -6 -7 l.360xl0 5.37lxl0 Fast interactions in the co~e are included for all cases. The average o 2 o moderating-feflecto~ temperature ~s 570K.

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258 TABLE 45 Fuel Temperature Coefficient of Reactivity for 93% Enriched UF 6 Using the Engine #8 Configuration at the TDC Position Average Fuel Two-group Four-group Temperature CORA 1 dk CORA 1 dk ( o K) keff k dT keff I TT 300 1.074388 1.019753 450 1. 629xl o6 5.17lxl0 -7 600 1.074913 l.019912 750 -6 -7 l.355xl0 4.735xl0 900 l. 075350 1.020056 1050 -6 -7 l.082xl0 4.xl0 . 1200 1. 075699 1.020183 1350 -7 -7 6.934xl0 2.045xl0 1500 l. 075923 1.020245 1650 -7 -7 9. 142xl0 3.486xl0 1800 1.076219 l. 020352 19,50 8.315xlo7 3.306xlo7 2100 1.076487 1. 020453 300900 l.492xl0 -6 4.953xlo7 900-1500 -7 -7 8.877xl0 3.088xl0 1500-2100 -7 -7 8.73lxl0 3.396xl0 300-.2100 l.084xl0 -6 3.812xlo7 Fast interactions in the core are included for all cases. The average moderating-reflector temperature is 570K.

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259 TABLE 46 Fuel Temperature toefficient of Reactivity for 80% Enriched UF 6 Usinq the Engine #8 Configuration at Average Fuel Temperature ( o K) 300 450 600 750 900 l 050 1200 1350 1500 1650 1800 1950 2100 300900 900-1500 l_ 5 0021 00 300-2100 Two-group_ CORA keff 1.0712885. l .0713789 1.0714987 l.0716177 1.0716206 1.0717394 1.0718664 l dk k dT . -7 2.813xl0 3.727xl0 -7 3.702xl07 . -9 9.02lxl0 2.714xl0 -7 3.950xl0 -7 3.270xl07 ' -7 l .896xl0 -7 3.822xl0 -7 2.996xl0 the TDC Position Four-qroup CORA 1 d k keff k dT l. 0177896 . -7 -3.557x10 1.0176810 -l.572xlo7 l. 0176330 -7.632xl0 -8 1.0176097 -2.038cl0 -7 1.0175475 4.586xl0 -9 1.0175489 3.964xl0 -8 1.0175610 -2.565xl0 -7 -l.400xl0 -7 2.2llxl0 -8 -l .248xlo7 Fast interactions in the core are included for all cases. __ Theaverage o 2 o moderating-reflector temperature is 570K

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260 average energy is 0.57eV. For. neutron energies in this vicinity, a, the capture-to-fission ratio decreases with increasing neutron energy. The fission cross secti~n in crease therefore outweighs the absorption cross section increase and hence the positive fuel temperature coeffi cient of reactivity for Engine #8 for the case of 100% enrichment. At increasing fuel temperatures, the rate of decreas~ in a decreases and this is reflected in Table 44 in the decreasing magnitude of the positive fuel tempera ture coefficient of reactivity with increasing fuel temperature. For the case of 93% enrichment, the effect of the u 238 is to decrease the magnitude of the small positive fuel temperatu~e coefficient of reactivity by about 30% for the four-group results and by about 20% for the two1 Th t . u 238 t "b t . d grou~ resu ts. e nega 1ve con r1 u 10n 1s ue primarily to increased neutron absorption as a consequence of Do~pler broadening of the 6.67eV resonance with increased fuel temperatures. At the 93% enrichment, the amount of u 238 present is not yet sufficient to override the above described u 235 effect and bring about a negative coeffi cient of reactivity for the fuel mixture. At 80% enrichment, the four-group results indicate 238 that the amount of U present is just baiely sufficient to tause the fuel temperature coefficient of reactivity to

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be negative. The coefficient is negative up until about 1350 6 K at which point the continuing decrease in a is 261 able to override the.u 238 effect. Thus at higher tempera tures, although the coefficient is positive, it is extremely small. At 1650K it is almost 100 times smaller than for the 100% enriched case while at 1950K it is approxi mately ten times smaller than for the corresponding 100% enric~ed case. The two-group results for the 80% enriched case indi cate a positive tempefature coefficient of ~eactivity over the entire temperature range. Even at those tempera tures where the two-group and four-group coefficients agree in sign, however, a relatively large difference exists between the magnitudes of the two coefficients. The two group calculations are in fact too coarse to accurately predict the very small fuel temperature coefficient of reactivity. In particular, it has been pointed out that the epithermal energy region effects are of major signifi cance for this coefficient. This coefficient is thus not properly determined from a calculation which makes use of a single fast broad group. This is true not only for the 80% enrichment case but also for the 93 and 100% enrichment cases. In contrast, relatively good agreement was obtained between the two-group and four~group calcu lations for the larger moderator temperature coefficient of reactivity.

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~20 Moderating-Reflector Density or Void Coefficient of Reactivity 262 Presented in Table 47 are the results of studies for which the average density of the o 2 o moderating-reflector was varied from 2.397 x 102 (density of the saturated -3 liquid phase of o 2 o at 570K) to _l .441 x 10 atoms/barn-cm (density of the saturated vapor phase of o 2 o at 570K). The configuration used for these studies was Engine #8 at t~e TDC position. Two-group CORA calculations which neglected fast core interactions were used. Table 47 shows that boiling or voiding in the D 2 0 is an effective means for obtaining engine shutdown. When the average D 2 0 density drops to the point where the aver age amount of o 2 o which is tn the vapor state is but 1% by weight, the system keffective goes from 1 .066 to just subcritic~l. When the average amount of o 2 o which is in the vapor state is 4% by weight, the system is already far-subcritical. During normal engine operation, the D 2 0 in the moderating-reflector will be pressurized. Should the gas temperature and/or pressure in the core exceed some critical limit, the o 2 o moderating-reflector-can be depressurized by a relatively conventional relief-valve quenching tank system. Upon depressurization, the o 2 o will undergo boilinq and void ing. As can be seen from the results of Table 47, these processes

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263 TABLE 47 o 2 o Moderating-Reflector Density or Yoid Coefficient of Reactivity Using the Engine #8 Configuration Average o 2 o Density (atoms/barn-cm) -2 2.397xl0 2~200xl02 -2 2. OOOxlO . -2 l.800xl0 . -2 l. 600xl0 -2 l.400x10 -2 l.200xl0 -2 l.OOOxlO -3 7.500xl0 -3 5.00 xlO 4.00 xl03 -3 3.00 xlO . -3 2.50 xlO -3 l.441xl0 at the TDC Position Average D20 Specific Vol~me (ft /1bm) 0 . 0201 0.0219 0.02409 0.02677 0.03012 0.3442 0. 04015 . 0.04818 0.06425 0.09637 0. 12046 . 0.16061 0.24092 0.33438 Averaqe Weight % of D20 \-Jhich is in the Liquid Phase 100.00 99.43 98.73 97.88 96.81 . 95.44 93.62 91.06 85.95 75.73 68 . 07 55.29 45.07 0.0 From 100% Liquid to 100 % Vapor Two group CORA keff 1.0659 1. 0153 0 . 9576 0.8927 0.8191 0.7353 0.6397 ()53. 12 0. 3778 0.2158 0.1552 0.10.17 0.0786 0.0393 The average .D 2 0 moderating-refle . ctor temperature is 570K. The average gas temperature in the core is 900K. F~st interactions in the core were neglected for all cases. l dk k d( '.0 void) . -2 -8.545xl0 . -2 -8.346xl0 -8.257x102 -8.039x10 -2 -7.869xl0 -2 -7.636xl0 -2 . -2 -7.243x10 -6. 604xl02 -5. 342x102 -4.262x10'" 2 -3. 259x102 -2.507x102 . -2 -1. 927x10 . -2 -1.858xl0

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.264 seriously disrupt the neutron thermalization process in the controlling external reflector and bring about rapid and effective shutdown. In addition, one could utilize either core burst diaphrams set torupture or core emergency relief valves set to open should the core pressure exceed some preset critical limit. The HeUF 6 gas would, under the over pressure condition, be vented to a large tank. The rapid decrease in gas density and keff would bring about an essentially instantaneous shutdown. Even without depres surization of the reflector's o 2 o or without venting of the HeUF 6 gas, an over-temperature condition in the core would eventually be felt in the moderating-reflector and it would take very little boiling and voiding to then cause reactor shutdown. The latter mechanism is of course less rapid due to the time lag for heat transfer from the core to the moderating-reflector. 11 Cy c 1 e F r a c t i o n s 11 f o r t he Engines of This Chapter As has already been discussed, the four-stroke engines of this chapter undergo a combination of simple harmonic and non-simple harmonic motion. The bulk of the cycle exper iences simple harmonic motion with non-simple harmonic motion occurring at the beginning of the intake stroke and at the end of the exhaust stroke. The latter motion

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265 occurs in a manner so as to effect essentially complete removal of the fuel load from the cylinder at the end of each piston cycle. When 11 cycle fractions 11 are referred to in listing the operational charact~ristic for these engines, the fractions are for the simple harmonic motion only. To illustrate, the piston cycle time for Engine #8 is 0.8235 seconds. The time period for which the piston undergoes pure simple harmonic motion is 0.7059 secohds. It also undergoes non-simple harmonic motion during the first .0588 seconds of the intake stroke and during the last .0588 seconds of the exhaust stroke for the total cycle time of 0.8235 seconds. If the intake valves are shut at the 0. 150 cycle fraction, the time into the cycle at which this event occurs is [0.150 (.7059) + .0588] = 0.1647 seconds. If the step-reflector is removed at the 0.570 cycle fraction~ the time into the cycle at which this event occurs is [0.570 (.7059) + .0588] = 0.4612 seconds. Effect of Uranium Enrichment on Engine Performance Shown in Table 48 are the operating characteristics for Engine #9. This engine differs from Engine #8 pri marily in four characteristics. The clearance volume has been increased from 0.135m 3 to 0.160m 3 , the compression ratio has been reduced from 4-to-l to 3.8-to-l, the average o 2 o moderating-reflector temperature has been lowered from 570K to 490K, and the helium-to-uranium mass ratio has been decreased from 0.30 to 0.25.

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TABLE 48 Operating Characteristics for Engine #9 Characteristic Number of strokes Compression ratio Clearance volume (m 3 ) Stroke (ft) Intake line gas pressure (atm) Intake line gas temperature (K) Cycle fraction for closing intake valves Cycle fraction for opening exhaust valves Neutrbn source strength (n/sec) Engine speed (rpm) Cycle fraction for step-reflector removal He-to-U mass ratio % U235 enrichment Photo~eutrons considered Delayed neutrons considered Pure simple harmonic motion . o 2 o Re~l_ector thic.kness (cm) before step removal after step removal Neutron lifetime (msec) Piston cycle time (sec) U mass in cylinder {kg) He mass in cylinder {kg) Average o 2 o reflector physical temperature (K) Loop circulation time (s~c) Core height at TDC (cm) Core radius at TDC (cm) 266 Engine #9 4 3.8 to l 0. 1 60 5. l 2 19.4 400 0. l 50 0.750 1.0 X 10 9 170 0.560 0.250 100 yes yes no 100 30 1. 600 0.8320 3. 14 1 0.785 490 8.45 5 5. 91 30.21

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267 TABLE 49 Equilibrium Cycle Results from NUCPISTN for Engin~ #9 Characteristic Maximum gas temperature (K) Maximum gas pressure (atm) Maximum keff Exhaust gas temperature (K) Exhaust gas pressure (atm) Average core thermal neutron flux during the cycle (n/cmisec) Average gas temperature during the cycle (K) Average gas pressure during the cycle (atm) Average keff during the cycle i~ e c h a n i c a l p o we r o u t p u t ( M vi) Fission heat releas~d (Mw) Mechanical efficiency (%) Photoneutron precursor concentration (#/cm 3 ) 0elayed neutron precursor concentration (#/cm3) Thermal (1/v) in core (sec/cm) Fast (1/v) in reflector (sec/cm) Maximum core thermal neutron flux during cycle (n/cmlsec) Av e r a g e ma s s fl o \
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268 Th t t 1 . d . u 235 1 ct e o a wranium an uranium oa 1ngs as a function of enrichment are shown in Table 50 for Engine #9: The loadings wer~ determined by requiring that the maximum keffective during the piston cycle be the same at all enrichments. The results in Table 50 indicate that the different systems also had the same average keffective over the cycle. In addition, the total mechanical plus turbine power obtained from the systems was also almost identical. This latter result is not as obvious as it might at first seem. The lower enrichment systems had higher mass flow rates and slightly higher mechanical power outputs. The exhaust gas temperatures for these systems, however, were lower than for the high enrichment systems. Thus, the penalty in going from 100% to 90% enriched UF 6 . 11 Th . d u 235 1 ct . b b t 1% 1s sma . e require oa 1ng increases ya ou a while the total uranium loading increases by about 12%. Effects of Delayed and Photoneutrons on Engine Performance In the previous chapter, it was shown that the re quired fuel loading for a given engine performance was decreased when delayed neutron effects were included. It was also shown that this loading was even further decreased when photoneutron effects were included.

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TACLE 50 Effect of Uranium Enrichment on Required Fuel Loading Enrichment 1 oo~; 99% 95% Characteristic To ta 1 Uranium Loading (kg) 3. 141 3.175 3. 31 9 U235 Loading (kg) 3. 14 1 3. 14 3 3. 1 50 :-1ax ir.ium k effective 1 . 0 5 6 1 . 056 l . 05 6 Average k eff O"ve r cycle 0.710 0.71-0 0.710 It TDC 1 . f at (cm ) 0. 1568 0.1569 0. 1575 core . L. t TDC l 0. l8q0 0. 1861 0. 1868 at (cm ) acore All results, including cross sections, are from the NUCPISTN code. for Engine 93% 3.399 3. 1 58 1 . 056 0.710 0. 1580 0.1374 #9 90% 3.524 3. 169 l . 056 0. 710 0 .. l 584 0.1379 N 0) I.O

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270 The effect which these neutrons have on four-stroke engine performance is illustrated in Ta~les 51 and 52. The results are for Engine #9. In Table 51, the required fuel loadings, maximum k ff t and cycle-averaged k ff t are compared for e ec 1ves e ec 1ves three different cases. The basis for comparis~n was that . . the total o~tput mechanical plus turbine power be the same under all conditions. In the first case, effects of delayed and photoneutrons were included. In the second case, the photoneutron effects were ignored and the required uranium loading increased from 3.141kg to 3.240kg while the maximum keff rose from 1.056 to 1.061. The third case neqlected the effects of both delayed _ and photoneutrons. It is appar ent that the delayed neutrons exert a stronger influence on engine behavior than do the photoneutrons. The required fuel loading relative to the case where only photoneutrons were neglected has risen from 3.240kg to 3.717kg while the maximum keff has increased from 1.061 to 1.091. The effects of delayed and photoneutrons are shown in a slightly different fashion in Table 52. Here, no compen. . sation by way of increased fuel loadings is made for the absence of delayed and/or photoneutrons. The degradation in engine performance is considerable. For example, when delayed neutrons are neglected, the mechanical power output

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TABLE 51 Compensatin9 for the Absence of Delayed and/or Photoneutrons by Increased Fuel Loading for Engine #9 Condition Characteristic Uranium loading (kg) Maximum keff Average keff over cycle Uranium enrichment is 100% Hith Delayed and Photoneutrons 3. l 41 l . 056 0. 711 vJithout Photoneutrons 3.240 l. 061 0. 71 6 v! i t h o u t D e l aye d or Photoneutrons 3.717 l . 090 0.740 N -....J

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TABLE 52 Engine #9 Behavior in the Absence of Delayed and/or Photoneutrons When There Is No Compensation by Increased Fuel Loading Condition Characteristic Uranium loading Maximum k eff (kg) Maxi mum gas temperature ( o k) Maximum gas pre.s sure (atm) Mechanical power output (Mw) Uranium enrichment is 100% With Delayed and Photoneutrons 3. 1 41 1 . 056 2079 155.9 3. l 7 Without Delayed Neutrons 3. 1 41 l . 056 1030 86.6 l. 3 7 vlithout Delayed or Photoneutrons 3. l 41 l . 056 590 63.4 0.413 N ......., N'

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273 drops by about 57%; when photoneutrons are also neglected, the mechanical power output drops by about 87%. Timestep Size Selection for the Energetics Equations The optimum timestep sizes for solution of the ener getics equations are quite different from the optimum timestep sizes for solution of the neutronics equations. In the NUCPISTN code, the user can choose different timestep sizes for solution of these different equations. The user can also establish timezones and can select different timesteps from timezone to timezone. In addi tion, th~re is a routine in NUCPISTN which automatically chops the energetics equation timestep size in regions of rapid neutron flux changes so that pressure and tempera ture changes over an energetics equation timestep are within predefined limits. The energetics equation timesteps, however, cannot be chopped or cut to values which are less than the timestep size being used to solve the neutron kinetics equations. Studies were performed in which a different number of timesteps (i.e., different timestep sizes) were used in solving the energetics equations while holding the timestep size used in solving the kinetics equations fixed. The results are shown in Table 53 for Engine #9.

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TABLE 53 Effects of Enerqetics Equations Timestep Size Variation on Enqine #9 Performance # of Timesteps # of Timesteps Maximum Gas Exhaust Gas ~MAX for Energetics for Neutron Temperature Temperature Equations Kinetics (OK) ( o K) E E uations ( ~$) 5304 5304 2095 1265 0.38 1778 5304 2084 1252 l. 12 896 5304 2079 1245 2.22 604 5304 2079 1241 3.22 457 5304 2081 1244 4.26 348 5304 2083 1247 5.73 282 5304 2088 1255 7.08 241 5304 2100 1260 7.82 209 53C4 2195 1269 8.86 184 5304 2291 1291 10.18 = Maximum Fission Heat Released in any Energetics Equation Timestep Q~AX(%) Total Fission Heat Released Over the Piston Cycle Percent error in pV work over the cycle= [WpV WT] x lOO vJT Error in pV Work Over Cycle (10 x l 00 . 1513 .0492 .0247 . 0151 . 0092 . 0129 . 0461 .0746 . 1255 .2048 In the above, Qf is the total fission heat released, DU is the total chanqe in internal energy and Wflow is the net flow work performed tiy the system on the gas as it passes across the system (engine) boundaries.

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275 For this particular example when the number of time steps used in the energetics equations was between 350 .and -1000, the observed engine behavior was fairly constant. For example, the maximum gas temperature and exhaust gas temperature varied only by a few degrees. When the number of timesteps used decreased beyond the above limit, the observed behavior changed noticeably. The energy equation timesteps in this region are too coarse and the error in the work over the cycle and the maximum fission heat released in any one energetics equation timestep rise rapidly. When the number of timesteps is increased beyond the above limit, the observed engine behavior again undergoes large changes. While the fission ene~gy released in one energetics equation timestep is small, the error in the pV work over the cycle rises once again .. The tirnestep sizes in this region are too small and computer roundoff errors begin to cause problems. In the above example, the optimum energetics equation timesteps were, on the average, 5 to 15 times larger than the timesteps used in solving the neutron kinetics equations. It will be recalled from the last chapter that the most desirable neutron kinetics equation timestep sizes occur when the number of neutron kinetics equation timesteps per neutron lifetime is somewhere between 4 and 40. Both of the~e rules have been found to be generally applicable for all the examined nuclea1 piston engine configurations.

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fffects of CP Formula Selection and of Neutron Kinetics Equations Numerical Techniques on Engine Behavior The information compiled in Table 54 was obtained 276 by using the Engine #9 configuration. A comparison of the results of column th_ree with the results of column o~e shows the effect of going from a two-point finite difference scheme to a three-point integration technique for solution of the neutron kinetics equations. The observable dif ferences in engine performance (e.g., mechanical power output, fission heat release rate, maximum gas temperature, etc.) for the two numerical techniques are not very great. The reduction in the error in the pV work over the cycle is from .0169% to .0151%. In the initial studies which were performed with the two-stroke engines, even higher order numerical techniques (fourth order Runge-Kutta and five point integration) were tested. The gain in accuracy as compared with the three-point integration technique was minimal and therefore not worth the added computational costs. The current version of NUCPISTN a11ows the user a choice between the two-point finite difference and the three-point integration method for solution of the neutron kinetics equations. A comparison of column two of Table 54 with ~olumn one reveals the effects which C formula selection has on p engine performance. The old C formula is equation p

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TABLE 54 Effects of Specific Heat Formula and of Neutron Kinetics Equations Numerical Techniques on Enqine #9 Performance Condition 3 pt integration and 3 pt integration and 2 pt finite difference and Characteristic new CP formula old CP formula new C 0 formula Maximum gas temperature (K) Mechanical power output (Mw) Rate of fission heat release (Mw) Mechanical efficiency(%) 2079 3. 17 9.30 34.2 Average core thermal neutron 14 flux over cycle (n/cm2sec) l.28xl0 Error in pV work over cycle (%) .0151 t'1IXTURE over cycle (B/lb F) .2672 P m ~IXTU~E over cycle (B/lbmF) . 1911 ~F6 over cycle (B/lbmF) .1017 2035 2062 3. 13 3. 14 9.55 9.21 32.8 34.2 l.32xl0 14 l.27xl0 14 0. 154 . 0169 . .2749 .2672 . 1983 . 1911 . 1107 . 1017 ~F6 over cycle (B/lbmF) .0962 . 1046 .0961 Old specific heat (CD) formula is given b_y equation (1) in Chapter IV. Nei,.1 soecific heat (C ) formula is given by equation (2J in Chapter IV. P Percent error in pV work over cycle= (WpV WT) x 100 where WpV = J pdV and WT= (Qf 6U + Wflow) WT cycle _ In the above, Qf is the total fission heat released, tU is the total change in internal energy, and Wflow is the net flow •.-1ork performed b_y the system on the gas as it passes across the system (enqine) boundaries. N -...J -...J

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278 (l) of Chapter IV. This is the equation used originally by Kylstra et al. and is really only v~lid for temperatures which are in the vicinity of 400K. The new Cp formula is equation (2) of Chapter IV and it fits the experimental and theoretical data reasonably well_ over the temperature range from 400 to 2400K. The effect which C formula p selection has on engine performance is not enormous but it is significant; The changes in engine performance are of course due to the influence which the formulas have on the specific heat values, which are computed throughout the piston cycle as a function of temperature. For example, the cycle-averaged UF 6 specific heats at constant volume and constant pressure as obtained from the two formulas for this engine differ by about 9%. Blanket Studies for Four-Stroke Engines Blanket studies were carried out for the four-stroke engine~ in a manner similar to those which were conducted for the two-stroke engines of Chapter IV. Reference is made to Figure and Tables 26 and 27 for the unit cell structure and material densities used in the blanket region. The four-stroke system analyses were performed for an engine similar to #9 using the four-group, zero-dimensional BURNUP depletion code in combination with the one-dimensional C0R~ diffusion theory code as described in the previous chapter.

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279 The metal-to-water ratio in the blanket regions for these studies was 3.0 and some results are shown in Table 55. Because of the low flux levels which exist during the intake and exhaust strokes, the average core fluxes over the piston cycle are lower for the four-stroke engines than for the two-stroke engines which were examined in the last chapter. Consequently, the fluxes seen in the blanket region are also lower. As a result, the conversion and/or breeding ratios are lower and the doubling times longer for these four stroke systems. For the case where 1.2% of the blanket rods are uo 2 , the conversion ratio is low, 0.53, while the blanket thermal power is high, 10.3Mw. When all of the rods in the blanket are Th0 2 ,_ the system breeds. The breeding ratio of 1 .07 should be compared with the two-stroke engine result of Table 31 where for the same blanket configuration, the breeding ratio was 1.13. The corresponding doubling time of 63.5 yeats when the fraction of the fuel inventory which is in the core is 0.15 is to be compared with the two-stroke system doubling time of 20.0 years. It was mentioned in the last chapter that this low fraction of fuel in the core represents a "worst case"condition. Reductions in the out-of-core fuel inventory could conceivably be made so as to i n crease the fr act i on of the f u e 1 i n vent or y, which i s inthe core to a value of around 0.3 and this would reduce the doubling time to around 32 years.

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280 TABLE 55 Burnup Calculations for a System Using an Enqine #9-Like Configuration, a 70cm D20 Reflector Region, and a Blanket Region with a 3.0M/W Lattice % of Blanket CR Blanket Doubling Fraction of Fuel Rods Which or Thermal Time Inventory Hhich Are uo 2 BR Power (yrs) Is in Core ( M11) 1.2 0.529 l 0. 29 0. 150 1.0 0.578 8.67 0. 150 0.5 0.752 4. 61 0.150 0.0 l. 070 0.55 63.5 0. 150 0.0 l. 070 0.55 31.8 0.300 o .. o l. 070 0.55 19. 0 0.500 % of Blanket Final Initial Initial Final Initial Final Rods Which U235 U235 U234 U235 Th232 Th232 Are uo 2 Mass Mass Mass Mass Mass Mass ( kq) (kq) (kg) (kq) (kg) (kq) l. 2 143.87 127.78 0.0 15.482 11873. l 11856. 1 ,l.O 119. 89 106.48 0.0 15. 4 94 11882. 3 ll8n5.4 0.5 59.94 53.24 0.0 15.579 11947.3 11930.3 0.0 0.0 0.0 0.0 15.652 12003.0 11986.0 Blanket volume 241m 3 Blanket is Th0 2 , U0 2 and D 0 D O reflector thickness = 70cm U02 is 93% enriched uraniu~ Blanket thickness= 20cm Burnup time is 1200 days t 1 = l:56xlO~n/cm~sec ) time and volume .cr. 2 = 8.-78xl0 n/cm ~ec ) averaged blanket ~ 3 = l.85xlolOn/cm sec) fluxes 4 = 3.44xlol2n/cm2sec) Total U235 consumed in core during 1200 days= 15.4kq -:-eore 14 2 ~ 4 = l.28xl0 n/cm sec Q core = f . 9. 30Mwth

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281 As has been discussed, some studies which have been performed on large, fixed geometry, steady-state circu lating fuel gaseous core reactors have used values for the fraction of the fuel inventory which is in the core of as high as 0.5. However, it is felt that this value is a bit high and leads to over-optimistic predictions for the system doubling time. For the four-stroke engine, a value of 0.5 for this fraction yields a system doubling time of 19 years. For the two-stroke engine of Table 31, the system doubling time would be reduced to just 6 years. if the fraction of the core fuel inventory is given a value of 0.5. It should be pointed out that the breeding ratios and doubling times appearing in many of the gas core reports, especially tho-se from NASA subcontractors, are not based on fuel depletion and burnup calculations. The breeding ratios quoted are actually "potential" breeding ratios and are grossly misrepresentative of physically attainable breeding ratios. The doubling time~ in these reports are therefore much too optimistic and very misleading. Material Densities and Group Constants for Nuclear Piston Engine #10 Listed in Tables 56 and 57 are the operating conditions and e~uilibrium cycle results from NUCPISTN for Engine #10. The differences between this engine and Engine #9 are slight. The intake line gas pressure has been increased from 19.4

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TABLE 56 Operating Characteristics for Engine #10 Characteristic Number of strokes Compression ratio Clearance volume (m 3 ) S t ro k e ( f t ) Intake line gas pressure (atm) Intake lin.e gas temperature (K) Cycle fracti.on for closing intake valves Cycle fraction for opening exhaust valves Neutron source strength (n/sec) Engine speed (rpm) Cycle fraction for step-reflector removal He-to-U mass ratio % U235 enrichment Photoneutrons considered Delayed neutrons considered Pure simpl~ harmonic motion u 2 o reflector thickness (cm) before step removal after step removal Neutron lifetime (msec) Piston cycle time (sec) U mas~ in cylinder (kg) He mass in cylinder (kg) Average o 2 o reflector physical temperature ( o K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) 282 Engine #10 4 3.8 to l 0. l 60 5. l 2 l 9. 5 400 0. 155 0.750 1.0 X 10 9 170 0.565 0.250 100 yes yes no 100 30 1. 600 0.8320 3.219 0.805 490 8.40 5 5. 91 30.21

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283 .. TABLE 57 Equilibrium Cycle Results from NUCPISTN for Engine #10 Characteristic Maximum gas temperature (K) Maximum-gas pressure (atm) Maximum keff Exhaust gas temperature (K) Exhaust gas pressure (atm) Average core therma1 neutron fl~x during the cycle (n/cm sec) Average gas temperature during the cycle ( o K) Average gas pressure during the cycle (atm) Average keff during the cycle Me c ha n i ca 1 power output ( M \v) Fission heat released (Mw) Mechanical efficiency (%) Photoneut 3 on precursor concentration (#/cm ) Delayed neutron precursor concentration (#/cm3) _ T he rm a 1 ( l/v) i n c o re ( s e c / cm ) Fast (1/v) in reflector (sec/cm) Maximum core th~rmal neutron flux during cycle (n/cm sec) Average mass flow rate (lbm/hr) Engine #10 2072 1 52. 81 . 054 1252 23.9 1.30 X 10 14 911 37. 2 0.712 3. l 9 9.59 33.3 2.25 X 10 9 1 . 03 X 1 0 l O 3.03 X 106 l . 1 23 X 1 07 2.59 X 10 15 5.33 X 10 4

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284 atmospheres to 19.5 atmospheres, the cycle fraction for closing the intake valves has been shifted from 0.150 to 0. 155 and the cycle fraction for shutdown by step-reflector removal has been changed from 0.560 to 0.565. Listed in Table 58 are the material densities used in analyzing Engine #10. The average o 2 o moderating reflector temperature is 490K and the core densities listed are for the TDC position (core volume= 0.160m 3 ). Also . shown in this table are the core thermal group constants at TDC as output by the NUCPISTN code. Shown in Table 59 are the two-group and four-group constants which were used in CORA and EXTERMINATOR-II in analyzing Engine #10 at the TDC position. The fast group constants (1 or 3 collapsed fast groups) were obtained from 68-group PHROG calculations. For the o 2 o reflector, a B-1 calculation was used to obtain the neutron flux and current spectra which were then used in collapsing the reflector consiants into the desired braid group structure. The core fast group constants were obtained from a cell calculation in which the reflector fluxes and currents were input and the core broad group cross sections were averaged o v e r t he s e s p e c t r a . T h e therm a 1 g r o u p c o n s t a n t s ( 1 c o l lapsed thermal group)were obtained from 30-group BRT-1 calculations. Thirty space points and anisotropic scattering corrections were applied in solving for the core and moderating reflector thermal group const~nts. A listing of the four-group energy structure will be found in Table F-4 of Appendix F.

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285 TABLE 58 Material Densities and Core Thermal Group Constants from HUCPISTN for Engine #10 at the TDC Position Material Densities at TDC Reaion Core Reflector Material U-235 F-19 He-4 H-2 0-16 Core he~ght is 55.91cm Core radius is 30.21 cm Reflector thickness is 100cm Density (atoms/barn-cm) 5.156 x 105 3 . 0 9 4 x l 4 7.57 x 104 5.603 X 10~ 2 2.801 x 102 Core Thermal Group Constants from NUCPISTN at TDC Lt (.cm1 ) = a l . 856 X l 02 C t (cm1 ) \JLf = 3.800 X ,o-2 C Lt ( cm l ) = l . 564 X l 2 f C

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286 In comparing the core thermal group constants from BRT-1 and from NUCPISTN, good agreement iS observed for all but the di_ffusion coefficient constants. The BRT-1 valu~s in Table 59 are current-weighted diffusion coefficients . . The BRT-1 flux-weighted diffusion coefficient for this configuration is 16.250 which is reasonably close to the NUCPISTN value. Neutron Lifetimes, Generation Times, Effective B 1 s, ~effectives' and_Inhomo3eneo~s Source Weighting Functions for Engine #10 from Different Computat1onal Schemes Listed in Table 60 are several nuclear parameters as calculated from different computational schemes for Engine #10 at the TDC position. The two-and four-group CORA and EXTERMINATOR-II calculations used the group constants appearing in Table 59. The two-group NUCPISTN code generates its own core thermal group constants. The fast and thermal moderating-reflector constants for NUCPISTN were input to the code as discussed in the previous chapter. Among the parameters listed in Table 60 are prompt neutron lifetimes, neutron generation times, effective delayed neutron fractions, k ff t , weighting functions e ec ives for fast inhomogeneous sources in the core, Wc, and weighting_ functions for fast inhomogeneous sources in the mo~erating reflector region, W. The latter two quantities are defined m and more fully explained in Appendix D.

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Region Core Reflector TABLE 59 Fast and Thermal Collapsed Group Constants from PHROG and BRT-1 for Engine #10 at the TDC Position Constant D(cm) -1 I:a(cm ) I: 2(cm-1) I: . 3 ( cm1 ) 1 of 3 1. 043x10 2 7.609x105 1. 76lx104 6.550x105 6. 153x104 6. 530x109 o:o 2. 195 -4 3.249x10 -2 6.192xl0 0.0 2 of 3 1. 404xl0 2 -4 l.705x10 -4 3.092x10 -4 1.269xl0 7. 913xl05 0.0 l. 442 0.0 -2 .3.0l3xl0 3 of 3 1 8.597x10 -3 l.784xl0 -3 2.800xl0 -3 l.152xl0 4. 63xl05 l. 478 -7 1.465xl0 0.0 0.0 1.762x102 1 of 1 1 . 2 l. 008x 0 -3 l.039x10 -3 1.65lxl0 -4 6.782xl0 . -5 2.532xl0 l. 563 -5 4.39lx10 -3 9.618x10 Thermal 13.579 l.85lx102 . -2 3.810xl0 -2 l.568x10 1. 1083 -5 2.236xl0 l of 3, 2 of 3, and 3 of 3 are the three collapsed fast group constants from PHROG. 1 of l are the single collapsed fast group constants from PHROG. Thermal are the collapsed thermal 9roup constants from BRT-1. For the transfer cross sections (i.i., and under the l of 3 column the x has a value of l; under the 2 of 3 column the x has a value of 2; under the 3 of 3 column the x has a value of 3~ and under the l of l column the x again has a value 1. N 00 ......,

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TABLE 60 Reactor Physics Parameters for Engin~ #10 at the rot Position from Various Computational Schemes Parameter Two-group Four-group Two-group Four-qroup CORA CORA EXTERMINATOR EXTERMINATOR keffective l. 080 l. 012 .l.115 l. 052 9., core (msec) 0. 110 0. 113 0. 112 0.134 9, refl (msec) l. 157 l. 359 l. 291 l. 493 .Q,total(msec) l. 267 l. 472 l. 463 l. 627 l\ (msec) l. 172 l. 453 l. 260 l. 544 BD . 0.006502 0.006800 0.006502 0.006771 WC .0584 .0463 Wm . 725 .626 (k/t)(,;,, /k )~ .0584 .0470 . 0545 .0444 (X) (X) Piston cycle time is 0.8320 seconds Elapsed time at TDC is 0.4160 seconds Note: The two-group NUCPISTN calculation neglects fast interactions in the core. Also, times and vim values are "cycle-averaged" values \'/hich are input into NUCPISTN and over the cycle. Two-qroup NUCPISTN l. 045 0. 132 l. 468 l. 600 1.530 0.006502 .0510 .700 .0510 the neutron lifeheld constant N 00 00

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289 Listed in Table 61 are parameters as calculated from the different computational schemes for Engine #10 at the 0.056 cycle fraction. The cycle time for this enqine is 0.8320 seconds and the 0.056 cycle fraction occurs at an elapsed time of 0. 1026 seconds into the cycle. The group constants used in CORA and EXTERMINATOR-II at this position are not listed but the method of computation with the PHROG and BRT-1 codes is the same as described above for obtain ing the group constants at the TDC position . . In reviewing the results of Tables 60 and 61, the observed trends in the keff behavior are the same as for previously described results. The two-group NUCPISTN keffs are again 2 to 3% lower than the two-group CORA keffs be cause of the omission of fast core interactions by the former. The four-group keffs are 5 to 6% lower than the corresponding two-group keffs from the same computational scheme. Finally, the two-dimensional results (EXTERMINATOR11) are 3 to 4% higher than the corresponding one-dimensional (CORA) results. It will be recalled that the 11 equivalent 11 volume sphere experiences less fast neutron leakage to the controlling moderator-reflector region, where the fission neutrons must slow down before they can efficiently produce fissions in the core, than does the actual cylinder. In examining the lifetime and generation time results, the four-group values (with the lower keffectives) are larger than the corresponding two-group values (with the higher

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Parameter keffective (msec) core .Q,refl (msec) .Q,tota 1 (msec) A (msec) SD WC ~1m ( k/ 9,) ( 00 / k,J TABLE 61 Re actor Phys i cs Par am et er s for Eng i n e # 1 0 at the O . 0 5 6 Cy c 1 e Fr a c ti on from Various Computational Schemes Tv10-group CORA 0.922 0.240 1. 685 1. 925 2.089 0.006502 .0823 .889 .0823 Four-group CORA o.871 0.242 1. 920 2. 162 2.494 0. 006774 .0682 .785 .0690 T1110-group Four-group EXTERMINATOR EXTERMINATOR 0.944 0.889 0.236 0.238 1. 731 1. 941 1. 967 2. 179 2. 081 2.450 0.006502 0.006738 .0825 . 0702 Piston cycl~ time is 0.8320 seconds. Elapsed time at the 0.056 cycle fraction is 0.1026 seconds. Two-qroup NUCPISTN 0.904 0. 132 1. 468 1. 600 1.770 0.006502 .0759 0.700 .0759 Note: The two-group NUCPISTN calculation neglect~ fast interactions in the core. Also, the neutron life times and W values are "cycle-averaged" values v,hich are input into NUCPISTN and held constant over the cy~le. N \D 0

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2 91 k ff t~ ) for a given computational scheme. This is the e ec 1ves behavior which would be expected physically, on an intuitive basis. The neutron lifetimes from the one-dimensional computations are smaller than the co_rresponding two dimensional results because of geometric effects. The equivalent-in-core-volume sphere has the same thickness of reflector (100cm) as the actual cylinder. The reflector volume for the sphere is thus significantly smaller (by 25 to 30%) than for the actual cylinder. Since the bulk of the total neutron lifetime is spent in the reflector, it is not, therefore, surprising that the lifetimes from the t \•JO d i mens i, on a 1 cal cul a ti on s are 1 a r g er than for the 11 equivalent 11 spherical system. As shown in Tables 60 and 61, the neutron lifetime in the core is approximately an order of magnitude smaller than the neutron lifetime in the mod~rating-reflector region. In comparing the effective delayed neutron yield fractions from the two-dimensional calculations with the one-dimensional results, there is seen to be very little disagreement. The values are within 0.5% of each other. It will be observed that the relative effectiveness factors for delayed neutrons (~ 0 ;s 0 ) are small for the nuclear piston engine with the values generally being from 1.04 to .os.

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292 The weighting function, Wc, for the fast inho~ogeneo~s source term in the core is derived in Appendix D where it. is proven that the two-group expression for the gaseous cores of concern is ( 11 ) The parameters (1 )t and (~)tare defined in Appendix 00 C 00 C D where it is asserted that the above relation is also a good approximation for four-group or even for multigroup calculations for the gaseous core nuclear piston engine. The results of Tables 60 and 61 show that Wc and (k/1) [(t )t/(k )t] are indeed equal for the_ two-group calculaoo c ooc tions and very close for the four-group computations. The weighting function, W , for the fast inhomogeneous moderatingm reflector region sources is also derived in Appendix D. Equation_ (D-52) is the algorithm used in obtaining the Wm values for Tables 60 and 61. The EXTERMINATOR-II code in its present form requires that an adjoint calculation be immediately preceded by the corresponding forward problem and thai the forward problem not have any inhomogeneous sources. Hence, no values for Wc and Wm appear under the EXTERMINATOR-II results in Tables 60 and 61.

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293 The k ff t and the weighting function for fast e ec 1ve inhomogeneous sources_ in the core are calculated internally_ by the NUCPISTN code at each new timestep. The neutron lifetime, effective delayed neutron fraction, and weighting function for fast inhomogeneous sources in the moderating reflector were obtained from independ~nt calc~lations and input into NUCPISTN. The input values were constant cycle averaged quantities. They v1ere obtained by selecting of the order of 24 positions during the piston cycle, performing the independent calculations for each of the 24 configura tions, and then least squaresfitting the parameters over the piston cycle. The appropriate averages could then be obtained in a straightforward manner. The number of positions or points was limited to around 24 because it was found that reasonably good least squares fits could be obtained for this number of points. More points would of course provide better fits but the independent calculations: are expensive and were therefore kept to a minimum whenever possible. The NUCPIStN code has been updated to allow for tim~ varying effective delayed neutron fractions, neutron life times, and weighting functions for fas_t inhomogeneous sources in the moderating-reflector region. That is, the least squarffi fitted curves for these parameters can be input to the code rather than just the constant cycle-' averaged values obtained from the curves.

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294 Since the delayed neutrons have a relative effective ness factor which is fairly imall (1.04 to 1.05), the effect of variations in this parameter over the cycle wt11 be small. The fast inhomogeneous sources in the moderating-_ reflector have a significant influence on the engine behavior primarily when the system is subcritical. Hence, time v a r i a t i o n s i n t h e w e i g h t i n g f lJ n c t i o n s f o r t hes e s o u r c es w i l l have only a mild effect on the engine behavior. The inclu sion of a time-varying neutron lifetime will have an influence over the entire piston cycle and thus represents a major improvement over the use of a constant, averaged rteutron lifetime during the cycle. As previously discussed, accounting for time variations in the above parameters means a progression from the point kinetics ~odel to an adiabatic model for handling the system~ neutron kinetics. This improvement, however, will also represent a considerable increase in computational costs s1nce an iterative procedure between the NUCPISTN code and the independent means of obtaining the parameters of concern will be required. This will be further elaborated on in Chapter VII. For the present, let it suffice to say that the attainable engine performances as presented in this. chapter will not be degraded at all by this calculational refinement. The required fuel loading or engine speed might be somewhat alfered as a result of the refinement but the attainable performance levels will remain essentially unchanged.

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A Further Comparison of keffs for Engine #10 from Various Computat1onal Schemes 295 A further comparison of keff results from various computational schemes for Engine #10 at the TDC and 0.056 cycle fraction appears in Table 62. The table duplicates some of the results appearing in Tables 60 and 61. Table 62, howeier, also includes results from one-dimensional, s 4 computations performed with XSDRN using 123, 21 and 4 groups. Also included are results from a 21-grour, two dimensional EXTERMINATOR-II calculation and from a 21-group, one-dimensional MONA calculation. MONA is a diffusion theory code which is the multigroup version of the few group CORA diffusion theory code. In the previous secti6n, the effect on keff of first neglecting and then including fast core interactions was discussed for two-group computations. Results illustrating this effect are included in Table 62 for both two-group and four-group calculations. It will be observed that the inclusion of fast core interactions leads to an increase in keff of from 1 to 3% over the corresponding case where these interactions are neglected. It was also mentioned that the four-group keffs are 5 to 6% lower than the corresponding two-group keffs An additional observation which can be made from Table 62 is that the 21-group results are 2 to 3% lower than the cor responding four-group results. In comparing the 123-group

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296 TABLE 62 Neutron Multiplication Factors for Engine #10 from Various Compuiational Schemes for the TDC and 0.056 Cycle Fraction Positions Computational Scheme Geometry Number of keff at keff at 0.056 cycle Dimensions TDC fraction Two-group NUCPISTN* SPH 1 l. 045 0.904 Two-group CORA* SPH 1 1. 045 0.901 Two-group CORA SPH 1 1.080 0.922 Two-group EXTERMINATOR II CYL 2 1.115 0.944 Four-group CORA* SPH l 0.994 0.860 Four-group CORA SPH l l. 012 0. 871 Four-group EXTERMINATOR II CYL 2 l. 052 0.889 Four~group XSDRN s 4 SPH 1 0.984 21-group MONA SPH l 0.984 21-group EXTERMINATOR II CYL 2 l. 025 21-group XSDRN s 4 SPH l 0.966 123-group XSDRN s 4 SPH 1 0.960 I *Fast interactions in the core are neglected.

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297 XSDRN results with the 21-group XSDRN results, one can observe that the former have a k ff t which is about 1% e ec 1ve lower than the latter. The two-dimensional EXTERMINATOR-II results.have been found to be 3 to 4% higher than the one-dimensional CORA results for both the two-group and four-group cases. The same trend is observed in Table 62 for the 21-group case_ when the two-dimensional EXTERMINATOR-II results are compared with the one-dimensional MONA results. In comparing the one-dimensional XSDRN s 4 results with the corresponding one-dimensional diffusion theory results, the XSDRN k ff are observed to be 2 to 3% lower e s than the diffusion theory results. These differences can be traced to two factors. First, the diffusion theory results used thermal group cross sections from BRT and fast group cross sections from PHROG. The cross section libraries for PHROG and BRT-1 are not identical with the XSJRN cross section library. Second, the upper energy cutoff point in BRT-1 is 0.683eV. Hence, the diffusion theory results cannot account for any upscattering above this point. The XSORN results, which include complete upscattering and downscattering, indicated a presence of some upscattering.to above 0.683eV, The neglect of up scatterin9 accounts for over half of the above cited errors, with the library cross section differences accounting for the remainder.

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298 The computer code library which has been assembled at the University of Florida does not presently include a two~ dimensional Sn theory code. The expense involved in running such a code (e.g., DOT [17], TDSN [22], or TWOTRAN2 [55]) was felt to be too high to be justified for the current feasi bility studies. However, one would anticipate that the twodimensional s 4 k ff t would be of the order of 3 to 4% e ec 1ves higher than the corresponding one-dimensional s 4 keffectives This would imply that a 21-group, two-dimensional s 4 calculation w~uld yield a keffective of approximately l .000. That this is a reasonable estimate can be substantiated in an alternative fashion. Consider, fdr example, the four group CORA keffective calcu_lation. Relative to the 21-qroup calculations, the four-group calculations were said to over predict keffective by 2 to 3%. The use of one-dimensional rather than two-dimensional geometry causes an underpredic tion of keff by about 3 to 4%, while neglecting upscattering to above 0.683eV causes an overprediction in keff by about 2%. Hence, relative to a 21-qroup, two-dimensional s 4 cal culation one would expect the four-group, one-dimensional CORA code to overpredict keff by about 1%. Through this line of reasoning, one arrives at a keff for a 21-qroup, s 4 two-dimensional calculation of about 1.002. Alternatively, the 21-group EXTERMINATOR-II calculation would be expected to yield a keff of about l .000 if the neglected upscattering were included. Relative to this value, the two-group NUCPISTN code is seen to overpredict k ff t by between 4 and 5%. e ec 1ve

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299 Thus, it can be concluded that excellent keffectives can be obtained from two-dimensional multigroup (i.~., around 21 groups) diffusion theory if 11 good 11 group constants or cross sections are employed. Four-group, two-dimensional. diffusion theory calculations utilizing 11 good 11 cross sec tions would be expected to overpredict keffective by about 3% for the nuclear piston engine systems. The two~group NUCPISTN k calculations were initially justified effective by their simplicity and the consequent large savings in computer time which they afforded when analyzing the nuclear piston engine systems. The fact that thes~ calculations are in error by only about 4% is fortuitous, due to a pa rt i a 1 c an c e 1 l i n g e ff e c t by c om pen s a t i n g e r r or s . NUCPISTN Cycle Results for Engine #10 Figure 93 shows the variation of the o 2 o reflector thickness during the Engine #10 cycle. The average gas temperature, average gas pressure, average core thermal neutron flux and neutron multipli cation factor during the piston cycle are shown in Figures 94, 95, 96 and 97 respectively. The step-reflector is removed at the 0.565 cycle fraction or at 0.462 seconds into the cycle. The system passes from the supercritical (keff 1.05) to the subcritical state (keff 0.63) at this point and the gas temperature, gas pressure and the thermal neutron flux, which had all been growing, undergo

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1 40 12 0 ,....._ E :.J ._,, Cf) 1 0 0 '.fl :.:.J ....,. ;..::: :..) ,..., 80 ,...,. : ~ :.) :...J 6 0 , . :..:.J cG C N Ci 40 '.2 0 1 I 'l'DC I I I I ~JT / dZ E _J I I I VALV ES I I I OPE~J I I . C O:t-!PRE S S IO l\ I rS T""' Q''E ---l I I .L ~\. .L\.. I I I I I P OW ER --i I I I I S TROKE I I I I I I I ' EXlli\.U ST I I 1 r= VALV:C S I I OPEN ' ~ I I I I I I. I L'iTAK E I I . I I I S T:Z O KE I r--1 I I I I I I I I I I I I I I I I I I I I I I I I I I l I 0.0 0 . 0 63 0.13 4 0 .204 0. 275 0.34 5 0.4i6 0 . 1 1 8 7 0 .5 57 0.6213 0. 6 9 8 0. 769 0.832 ':'1~ 1E ( sec ) FIG U RE 93. D 2 0 R ef l ecto r Thic k ness Vers us Pis t on C ycle Ti m e fo r Eng ine #1 0 w 0 0

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2100 ,-... :L 1800 0 ._,, :,: :;) lSOO :2 c... "' .c. :..2 0 0 < e. r . 900 C; :~ ..... ,....l G 600 300 ! :\TAK E VALV: CS OPEN I I -1 . I I I I I 1 I I I , I I I TDC: I . I I I I I I I . ~cm~ I'RE SSim~_J STROKE . I I I 1 I I I I I I I I I I I I I I I I POWER STROKE I I I E X:!.A[S T V,\ L\'ES OPEN l I I I . I I I I I I I o.o 0.063 o.134 0.204 0.215 o.345 o.416 o.~s7 o.557 o.628 o.698 o .7 69 o.632 F I GURf 94. Gas Temperature Versus Piston Cycle Time for Engine # 10 w 0

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140 ,...,, 120 E ..... 1 I I I I I I I 0.832 w 0 N

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,...._ i C""" w :::::: 0 u 1016 1015 1014 1013 1012 10 11 I:,TAKE VALVES OPE1~ I I --i I I I I I I I I I I I .I I I f--SO:1PRESSI0;'; STROKE --9j I I I I I I I I I I I I I I I I I I I ' I P01!ER STROKE I I -1 [X!1.\:..:ST '.7Al.,V}:.S DPE:'.'J I I o.o O.G63 0.134 0.204 0.275 0.345 0.416 0.487 0.557 0.628 0.693 0.769 0.832 TI~![ (sec) FIGURE 9G. Average Core Thermal :leutron Flux Versus Piston Cycle Time for Engine filO w 0 w

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. 1. 2 c::: 0 [... --;:: 0.8 0 ...... .... < :_.) ...... 0.6 :-< ....l 2 -'0.2 I'.JTAKE V,\LVES OPE?\ TDC I I I I Co ,,pn~s"TO"l I ,~ l\L .J l. !\ . I S:'ROKE 4 I I I I I I -~ I rrnmR STROKE I P-1 I I I I I E~-:!I!~L'ST \',\L\'ES' Ol't1\ I I I r,,.I I I I I I I I o.o 0.063 0.134 0.204 0.275 0.345 0.416 0.487 0.557 0.628 0.698 0.769 0.8]2 TD:E (sec) FIGURE 97. Neutron Multiplication Factor Versus Piston Cycle Time for Engine #10

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305 an initial rapid drop after the step-reflector removal. During the remainder of the power stroke the system keffective decreases slowly (to about 0.575) as the HeUF 6 gas density decreases. The gas temperature, gas pressure and average core thermal neutron flux are also seen to undergo a less rapid decrease during this period. At th~ 0.750 cycle fraction (0.5920 seconds into the cycle) the exhaust valves are opened and the HeUF 6 begins to undergo removal from the cylinder. The neutron multiplication factor, gas temperature, gas pressure and average thermal neutron flux in the co~e all begin to experience a more rapid decrease at this point in the cycle. An examination of Figures 94 and 95 reveals a small decrease in the gas pressure and temperature at about the 0.250 cycle fraction (0.240 seconds into the cycle). Up until this point, the amount of fission heat which has been released is too small to significantly affect the gas temperature and pressure. They, therefore, undergo i cosine-like behavior, mirroring the piston's motion. The gas pressure and temperature in this region vary primarily as a result of changes in the cylinder volume. An interesting type of behavior can be obse.rved in Figure 97 for the neutron multiplication factor. Once the intake valves are closed at the 0.155 cycle fraction (0:172 seconds into the cycle) the keff is seen to undergo

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306 a cosine-like behavior, first decreasing during the remainder of the intake stroke and then increasing during the beginning portipn of the compression stroke. If the keff variations were due only to density changes, the keff should continue to increase throughout the compression stroke up until TDC. However, at about 0.345 seconds into the cycle, the keff is seen to decrease. At this point the UF 6 gas. is quite black to thermal neutrons and further increases in gas density lead to very little increased fission neutron production. On the other hand, the system leakage is now increasing rapidly enough so that the effective neutron multiplication factor actually begins to decrease and hence the dip in keff around the TDC position in Figure 97. Power Transients for Engine #10 Induced by Loop Circulation Time Variations Shown in Figures 98, 99 and 100 are power transients for Engine #10. The transients were induced solely by variations in the loop circulation time. The starting point was th~ equilibrium condition for Engine #10, which is listed in Table 57. The loop circulation time at this eq~ilbriu~ steady-state power level is 8.4 seconds. The power rise seen in Figure 98 was produced by changing the loop circulation time from 8.4 to 8.2 seconds; the power drop was produced by altering the loop circulation time from 8.4 to 8.55 seconds. The transients in this

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,--.. :.<: ::..:i f< u: < _, ::.,:: < :.z.:i ;::..., 2800 2000 l.600 1200. 300 0 FIGURE 98. --50 Equilibrium loop circul~tion time ls 8.4 s0conds . TP ERATl'RE -TE;-iPERATVRE --------.-----. Loop circulation Loop ci-::-culation 100 150 200 time is 8.20 time is 8.55 250 TE!E (sec) .... seconds seconds 300 ------for pow~r rise for power drop 350 400 5 4 3 2 1 450 Slow Power Transients for Engine #10 Induced by Changes in the Loop Circulation Time ,-, :::,:; 0 p._, w C) .......,

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2800 2400 w Cl z H ....l >u ~::; 2000 ,-; ::.. f.; VJ < c., 1200 < :.,..:i 0.. 800 0 FIGURE 99. 5 4 -3 -25 ------POh'ER -------Equilibril;1m loop circulation time is 8.4 seconds 2 Loop circulation ti~c is 8.0 seconds for powerrise Loop circulation time is 8.8 seconds for power drop 1 50 75 100 125 . 150 TrnE (sec) 175 200 225 250 275 I n t e rm e d i a t e L c v c l P o \'J e r T r u n s i e n t s f o r E n g i n e } 1 0 I n d u c e d by C h c:1 n g e s in the Loop Circulation Ti~e w 0 00

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Ec,uili~riu~ loop circulation ti~e is 8.4 seconds 3600 c:::: 3200 i.i.J ,...., :z; -< ,..J 2800 z ..... 5 2400 "" ,., 2000 1600 1200 ' ...... ..... ...... 0 25 FIGURE 100. ...... ..... .... so Loop circulation time is 7.5 seconds for power rise Loop circul.::ition time is 10. 0 seconds for power clrop ------------..... ----75 100 125 150 175 200 225 TI~!E (sec) -Rapid Power Transients Loop Circulation Ti~e for Engine ii~ n ,. l V Induced by Chilnges 8 7 ,....._ 6 ,.:.. 5 4 3 2 '--' ~l c p... ,..J < u ..... in the w 0 I.O

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310 figure are fairly slow with the mechanical power rise being but about 2.5% per minute while the mechanical power drop is even smaller, about 1.75% per minute. In Figure 99, the power rise was induced by changing the loop circulation time from 8.4 to 8.0 seconds while the power drbp ~as induced by changing the loop circulation time, from 8.4 to 8.8 seconds. The mechanical power rises at a rate of about 6.5% per minute while the mechanical power drop is about 4.2% per minute. Finally, the Figure 100 power rise was brought on by decreasing the loop circulat,on time from 8.4 to 7.5 seconds while the power drop occurred as a result of in creasing the loop circulation time from 8.4 to 10.0 seconds. The mechanical power rise is now quite large, about a 24% increase per minute in mechanical power. For the power drop, the rate of decline in mechanical power is about 10% per minute. Control of the loop circulation time is an effective means of controlling the engine performance because the delayed and photoneutron precursors entering the cylinder on the intake stroke act as an extraneous or "external" neutron source .. Re-examination of Figures 33 and 34 in the previous chapter shows that the 11 external 11 neutron source strength has a strong influence on the engine behavior.

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Power transients induced by variations in the loop circulation time can be damped out by further variations 311 .in the loop ciriulation time. That is, when the trarisient approaches the new desired operating level, the system can be made to stabilize at the desired level, solely by changes in the loop circulation time. If desired, other parameters may be varied to speed up the stabilization process. For example, the cycle fraction for closing the intake valves, the intake line gas pressure or the intake line gas tempera ture could be varied in combination with the l~op circulation. time. The more rapid transients would of course be more difficult to damp out and if they are toorapid it is possible that they could lead to an over-pressure or over-temperature tondition in the core. This sh6uld pose no serious safety hazard, however, because of the ability to rapidly shut down the engine by one of the several mechanisms already dis cussed in the section on moderating-reflector void coefficients of reactivity. Safety will receive further consideration in Chapter VII. Thermodynamic Studies for Nuclear Piston Engine Power Generating Systems Utilizing the Engine #10 Configuration The four-stroke #10 Engine and a modified four-stroke. #10 Engine were incorporated into three nuclear power genera ting systems. The studies are similar to those done in the previous chapter for the two-st~oke engines.

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3l2 The results from the nuclea~ piston-gas turbinesteam turbine system study are shown in Table 63. A modifi~d #10 Engine was used for this system. The modification. involved changing the intake line gas pressure from 19.5 to 19.6 atmospheres. This adjustment yielded an increase in the exhau~t gas temperature of from 1252 to 1370K and an increase in the exhaust gas pressure of from 23.9 to 26.0 atmospheres. Mechanical power output for the modified #10 Engine is .2.78Mw as compared to 3.19 Mw for the standard #10 Engine. The piston-gas turbine-steam turbine schematic is essentially identical with Figure 44. HeUF 6 gas exiting from the piston engine block_ is now at 2006F and 382.2 psi a rather than at 2015F and 282 psi a while the HeUF 6 gas entering the block is at 288.l psia instead of 213 psia. T h e. re i s n o c h a n g e i n t h e. i n 1 e t He U F 6 g a s temp er a tu r e from the 260F shown i~ Fi~ure 44 .. Ts dia•grams for the s t e am a n d g a s t u r b i n e c y c 1 e s a re a s g i v e n i n F i g 11 r e 4 5 . The results for the piston-steam turbine arrangement are shown in Table 64. The standard #10 Engine was used for this system and the schematic for this system is essent ially indentical with the Figure 56 schematic. HeUF 6 now exits from the piston engine block at 1793F rather than at 1790F while the exit pressure is up from 262 psia to 351 psia. The inlet temperature to the engine block is still 260F. Gas pressure in the intake line is 286.6 psia as

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313 TABLE 63 Summary of Thermodynamic Results for the Piston-Gas Turbine . Steam Turbine System Which Uses the Modified #10 Piston Engine Enerqy__!_!,put to Fluids per Piston Energy Output from Fluids per Piston Fission Heat Rate . Compression Power Pump Power 10.02 . Mw 1.16 Mw 0.005 Mw 11. 185 Mw Gas Turbine Power Piston Mechanical Power Steam Turbine Power . . Qout (gas tufbine loop) . Qout (conden~er) 2.62 Mw 2. 78 Mw 0.985 Mw l. 45 Mw 2.08 Mw 0out (steam generator exit)l.27 Mw ll .185 Mw Power Source Description Power/Pis ton (Mw) Efficiency ( t ) Net Steam Turbine Power 0 . 98 Net Gas Turbine Power 1.46 Net Steam arid Gas Turbine Power 2.44 Pis ton Mechani ca 1 Power 2.78 Net Steam Turbine and Mechanical 3.76 Power 4.24 Net Gas Turbine and Mechanical Power 5.22 Steam and Gas Turbine and Mechanical Power Pov-1er Breakdown . for 42 Mw(e) Unit ' (8 pistons at 5.22 Mw(e) per piston) Piston Mechanical Power Output 22.24 Mw Net Gas Turbine Power Output 11:68 Mw Net Steam Turbine Power Output 7.84 Mw 41. 76 M11 9.8 14.6 24.3 27. 7 37.5 42.3 52 , 0

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314 TABLE 64 Summary of Thermodynamic Results for the Piston-Sf~am' Turbine System Which Uses Piston Engine #10 Energy Input to Fluids per Piston Energy Output from Fluids per Piston Fission Heat Rate PU!TIP Power 9.60 Mw Steam Turbine Power 0.025 Mw Piston Mechanical Power 2 .115 Mw 3.19 Mw . 9 _ 625 Mw Qout (condenser) 3.03 Mw Q (steam generator exit)l.29 Mw out 9.625 Mw Power Source Description Power/Piston {M1v) Efficiency (%) Net Steam Turbine Power 2.09 Piston Mechanical Power 3. 19 flet Steam Turbine and Piston 5.28 Mechanical Power Power Breakdown for 42 Mw(e) Unit (8 pistons at 5.28 Mw(e) per piston) Piston Mechanical Power Output Net Steam Turbine Power Output 25.52 Mw 16. 72 M1v 42. 24 M1v 21.8 33.2 55.0

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31 5 compared to the 214 psia shown in Figure 46. Figure 47 gives the steam turbtne cycle Ts diagram for this pow~r system. The results for the piston-cascaded gas turbine sys tem are shown in Table 65. The modified #10 Engine was used for this system and the system schematic is essentially that of Figure 48. Exit temperature from the piston engine block for the HeUF 6 is now 2006F rather than 20l5F while the exit pressure is 382.2 psia instead of 282 psia. The inlet gas temperature remains at 260F while the inlet pressure is raised from the 213 psia shown in Figure 48 to 288.1 psi a. Figure 49 gives the cascaded gas turbine cycle Ts diagrams. The four-stroke engine systems have a higher mechani cal efficiency but a lower mechanical power output than the two-stroke engine systems of the previous chapter. They also possess lower mass flow rates and smaller He-to~uranium mass ratios and hence yield lower turbine power outputs than. the two-stroke systems. The overall efficiencies, however, are slightly better than for the two-stroke engine systems. The total power per piston for the four-stroke engine units is around 5Mw(e) as compared to the 6 to 7Mw(e) for the two-stroke engine systems. Nuclear piston engine power generating systems using 8-cylinders and the two-stroke engines were found to yield 50Mw(e) units with overall efficiencies of between 40 and50%.

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316 TABLE 65 Summary of Thermodynamic Results for the Piston-Cascaded Gas Turbine System Which Uses the Modified #10 Piston Enqine Energy Input to Fluids Energy Output from Fluids Fission Heat Rate Compressor Power 10.02 Mw 4.41 Mw 14.43 Mw Power Source Description Net Gas Turbine Power Piston Mechanical Power Net Gas Turbine and Mechanical Power Gas Turbine Power Piston Mechanical Power Qout (gas turbine loop) Qout (He buffer loop) Power/Piston (Mw) 1. 91 2.78 4.69 Power Breakdown for 37.5 Mw(e) Unit (8 pistons at 4.69 Mw(e) per piston) Piston Mechanical Power Output 22.24 Mw Net Gas Turbine Power Output 15.28 Mw 37.52 Mw 6.32 Mw 2. 78 Mw 3.35 Mw 1. 9,8 Mw 14.43 Mw Efficiency (%) 19. l 27.7 46.7

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The nuclear piston engine power systems using 8-cylinders and the four-stroke engines will yield units of around 40Mw(e) capacity but with efficiencies which are slightly higher than for the two-stroke units. Presented in Table 66 is a summary of the thermo3 l 7 dynamic analyses for the three nuclear piston engine power generating systems. The piston-gas turbine-steam turbine setup yields the highest turbine power per piston of the three arrangements. This is a consequence of the gas turbine being able to make efficient use of the HeUF 6 high temperature range and of the steam turbine making efficient use of the 11eUF 6 lower temperature range. However, because of the higher HeUF 6 exhaust gas temperature from the modified #10 Engine, the niechanical power output is lower than for the standard #10 Engine. Thus, despite the high turbine power, the total turbine plus mechanical power and overall efficiency for this system is less than for the piston-steam turbine system which utilizes the standard #10 Engine. Another point to .note regarding the piston-gas turbine-steam turbine setup is that this system is the most complex in terms of the number of required components. The piston-steam turbine setu~ is not able to make as efficient use .of the high temperature HeUF 6 as is the piston-cascaded gas turbine setup. It is, however, able to make use of the HeUF 6 in the 735F to 260F range ~hich

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TABLE 66 A Comparison of Thermodynamic Results for the Three Nuclear Piston Engine Power Generating Systems Which Use Piston Engine #10 and the Modified e10 Piston Engine Total Fission Rate of Mechanical Turbine Power Heat Cooling via ter System Power Heat Heat Power per per Piston Transfer Flow Rate per Output Release Rejection Overall Piston (Mw) (Mw) Rate per Unit Power per Rate per Efficiency Unit Power (gal/min Mw) Piston per Piston (%) Efficiency Efficiency (BTU/_hr Mi1) (Mw) Piston (Mw) (%) (%) (Mw) Piston2.78 l. 91 cascaded --7 Gas Turbine 4.69 . 02 5.33 46.7 27.7 19. 0 1.16 X 10 258 Piston3. 19 2.09 Steam 55.0 -107 Turbine 5.28 9.60 4.32 (49.5)* 33.2 21.8 l. 02 X 186 PistonGas Turbine2.78 2.44 Steam --7 Turbine 5.22 l 0. 02 4.80 52.0 27.7 24.3 l. 095 x l 0 210 *Overall efficiency \vhen the "mechanical" efficiency is the same as for the piston-cascaded qas turbine and piston-gas turbine-steam turbine systems. w 00

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319 the cascaded gas turbine system cannot do. Consequently, the piston-steam turbine arrangement has a higher turbine power per piston than the piston-c~scaded gas turbine system. Because it utilizes the standard #10 Engine with the cooler exhaust gas, it also has a higher mechanical power than th~ piston-cascaded gas turbine setup. The piston-steam turbine arrangement, however, will be more costly than the piston cascaded gas turbine system because of the existence of the steam generator between the HeUF 6 -to-He heat exchanger and the steam turbines. For the piston-cascaded gas turbine s y s t em s. , t he H e i s u s e d d i r e c t l y i n t h e g a s t u r b i n e s a n d n o component comparable to the steam generator exists between the HeUF 6 -to-He heat exchanger and the turbines. Economic studies will be required to determine whether or not the increased costs of the more comple~ systems override the better efficiencies and power outputs of these systems. Also, the desirability of turbine versus mechanical power will have to be assessed for each individual application. 11 Gas Gener at or 11 Nu cl e,a r Pi st on Eng i n es It was discussed in the previous chapter that shifting the temperature and pressure curves so that they peak later in the cycle results in hotter exhaust gases but at the expense of a decrease in mechanical power output. These shifts were effected by removing the step-reflector at later and

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later times in the piston cycle. In this section, 11 gas generator 11 nuclear piston engines will be examined in which there is no step-reflector removal (or addition). The temperatureand pressure peak very late in the cycle and the mechanical power output is quite small. Such 320 engines have the advantage of being much simpler than the previously analyzed engines. The difficulties involved with having a step-reflector removal or poison addition (to the moderator region) at some cycle fraction to attain shutdown are eliminated. These engines have a smaller UF 6 loading and hence a lower peak keffective than the previously investigated engines and the fission heat release during the cycle is therefor~ more .gradual. Despite the slightly faster engine speeds (180 rpms versus 170 rpms) the mass flow rates for these systems are slightly less than for the previous four-stroke engines because of the smaller loading. Due to the higher exhaust gas temperatures, the output tur bine power for these systems is slightly increased relative to the previously analyzed engines. The total power output for the gas generator systems will be seen to be in the 3 to 4Mw(e) ~ange. This is in contrast to the just studied four-stroke engines which yielded power outputs of about 5Mw(e) per piston. Listed in Table 67 are the operating characteristics for Engine #11. The equilibrium cycle NUCPISTN results for

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TABLE 67 Operating Characteristics for Engine #11 Characteristic Number of strokes Compression ratio . ( 3 Clearance volume m) Stroke (ft) Intake line gas pressure (atm) Intake line gas temperature (K) Cycle fraction for closing intake valves Cycle fraction for opening exhaust valves Neutron source strength (n/sec) Engine speed (rpm) Cycle fraction for step-reflector removal He-to-U mass ratio % U235 enrichment Photoneutrons considered Delayed neutrons considered Pure simple harmonic motion D 2 0 reflector thickness (cm) before step removal after step removal Neutron lifetime (msec) Piston cycle time (sec) U mass in cylinder (kg) He mass in cylinder (kg) Average D 2 o reflector physical temperature (K) Loop circulation time (sec) Core height at TDC (cm) Core radius at TDC (cm) 321 Engine #11 4 3.8 to l 0. 160 5. 1 2 19.3 400 0. 150 0.750 1 . 0 X 1 o 9 180 NA 0.250 100 yes yes no 100 NA 1 . 600 0.7857 2. 9 31 0.783 490 8.45 55.91 30. 21

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322 th i s 11 gas generator 11 are shown i n Table 6 8 . Presented in Table 69 are some operating characteristics and NUCPISTN res u 1 ts for sever a 1 11 gas generator II en g i n es . Except for t h e i n t a k e 1 i n e i n i t i a 1 m a s s f 1 o w r a t e a n d t h e i n t a k e 1 i n e gas pressure, the operating characteristics are identical with those tabulated in Table 67. Appearing among the Table 69 entries are the Engine #11 configuration itself as well as a modified #11 Engine both of which are used later in thermodynamic studies for nuclear piston engine power generating systems. Examination of Table 69 shows that exhaust gas temperatures of from 1200 to 1400K will involve peak gas temperatures of from around 1300 to 1500K. In con trast, the previously studied engines would require peak gas temperatures of from 2000 to 2200K for the production of 1200 to 1400K exhaust gas temperatures .. This is an important consideration if the data concerning UF 6 dis sociation sho~ld be in error. The available data indicate that the enginei studied in the past sections will not. encounter serious problems with dissociation. For a given UF 6 partial pressure, the gas temperatures for these engines were not permitted to exceed levels where significant dis sociation sets in. However, if new data should indicate that dissociation a~ises at much lower temperatures, the 11 g a .s genera to r II en g i n e s s ho u 1 d s t i 11 be a bl e to p e r form satisfactorily.

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323 TABLE 68 Equilib~ium Cycle Results from NUCPISTN for Engine #11 Characteristic Maximum gas temperature (K) Maximum gas pressure (atm) Maximum keff Exhaust gas temperature (K) Exhuast gas pressure (atm) Averaqe core thermal neutron flux during the cycle (n/cm2sec) Average gas temperature during the cycle (K) Average keff during the cycle Mechanical power output (Mw) Fission heat released (Mw) Mechanical effic~ency (%) Photoneutron precursor concentration (#/cm 3 ) Delayed nutron precursor ~oncentration (#/cm) Thermal (1/v) in core (sec/cm) Fast (T7v). in reflector (sec/cm( Maximum core thermal neutron flux during cycle (n/cm2sec) Avera~e mass flow rate (lbm/hr) Engine #11 1404 66.6 l. 04 0 1299 21 . 6 l.10 X 10 14 726 22.4 0.60 7. 1 2 8.4 2.24 X 10 9 7.45 X 10 9 3.097xl06 1. 123xl0-? 7.11 X 10 14 5.14 X 10 4

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TABLE 69 Some Operating Ch a r a ct er i st i cs and NUCPISTN Equilibrium Cycle .Results for "Gas Gener at or." Piston Engines Intake Line Intake Line Ma>.
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NUCPISTN Cycle Results for a 11 Ga s G e n e r a t o r II E n g i n e_ 325 Fi g u re l O l shows that the D 2 0 ref l e c tor th i ck n es s du ing the Engine #11 cycle is constant at 100cm. The average gas temperature, average gas pressure, averaqe core thermal neutron flux and neutron multi~lication factor during the piston cycle are shown in Figures 102, 103, 104, and 105 respectively. During the portion of the intake stroke when the valves are open, the HeUF 6 is drawn into the cylinder and the neutron multiplication factor rises steadily (Figure 105). The intake valves are shut at the 0.150 cycl'e frac tion or at 0 . . 160 seconds into the cycle. From this point, up until around TDC, the amount of fission heat released is too small to signific an~ly affect the gas temperature and gas pressure. Th~y, therefore, undergo a cosine-like be havior, mirroting the piston's motion. The gas pressure and temperature in this region vary primarily as a result of changes in the cylinder volume. After TDC, the amount of fission heat released becomes large enough to start affecting the gas behavior. During the power stroke, the gas temperature begins to deviate from the cosine behavior and eventually undergoes a rapid growth as the flux and fission heat release rate climb. During the initial portion of the powef strike, the gas temperature remains at a fairly constant level and the . gas pressure drops rapidly as the cylinder volume increases.

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I TDC I I I I I I :_t,O I r:-:TAKE I I VAI..V:CS -.; I op::::::; f1I:Xl-l.i\l.!ST ,,l 120 I V.\LV:C:S ,,...., I OPEl: IE I r.., ._, I er., 100 I :.r:, :.:.J I z ::L: u I I COXPRESSIO); t,,J I so ':!' I STROKE I I .._, I I ;..:..) '"'60 I I rnT.t\KE er:: 0 sT:wrrn I 117-1 N I PO\\'ER 0 I j-c:;40 STROKE I I I I 20 I I I I I I I I o.o O.UGO U.126 0.193 0.260 0.326 0 "()'"• .. -'., .) 0. /.,60 0.526 U.593 0.660 0.726 0.786 Tl'.-! (sec) FIGURE l O 1. 020 Reflector Thickness Versus P1ston Cycle Time.for Engine Ill l w N 0)

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,-., 1200 C '-' :.:.l ,_; :...... 1000 < :..:..J ;: .. , S00 .,.., tL ,..... 600 ,-.; ....:l ;,-. u 400 20tJ I~TAKE \'ALVES OPE!J I I I I I TDC I __ co:-1P:1Essio:--: i r-S"'...,O"E l. 1\. ,I.'-. I I I I I I I . I I I I POWER STROKE _ EXH,\L'ST V,\LVES o~o 0.126 o.193 0.260 o.326 o.393 o.4so o.526 o.593 o.660 o.726 o.786 Tl'.E (sec) FIGU~E 102. Gas Tor:iperv.ture '/crsus Piston Cycle Tirne for Engine t-11. w N ......,

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,-... E .u ....., ::::: tr:. v .. , ..-; .r. < C ...... , ;..-. u I I 70 I I I 60 I!,TAI~ VALVES --Pf OP[N I 50 I I 40 I 30 20 10 I TDC I I I cmTRESSIO:J I r-STR0~
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......... J; ... ::< 5 ' ...,.. ;:: :--< 3 :--; ,.... !:.i.l ca:: 0 u ,., l.? __. _, ;_.) ;::..< I , 6 I 10~ I~:T1\I~E VALVES OPEX I , 10~.) I I 14 10 10 13 1012 11 10 101Cl 'l"DC I I I I CO;-!PRESSIO:{ I STROKE -e, I . I I I I I . I l I I I I L>O:-'ER STROKE I I I EXHAL'ST V/,LVES OPEN 0.U U.UbU 0.126 U.193 U.26U U.326 O.J93 0.460 U.526 0.593 0.660 0.726 0.786 TL'![ (sec) FIGURE 104. Average Core Thermal Neutron Flux Versus Piston Cycle Time for Engine #11 ,. w N <.O

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oc ::_: < ,. ,,-.. C ,...., c... < :..:, ,.... . i...:i ::.i :;;: -,,, f1 i::.> -,,, 1. 2 1.0 0.8 0.6 0.4 0.2 I ,. I TDC I I I I I . EXllt\l' S'l' .,, EITJ\.KE ..... _ ->i I V1\LVIS VALVES OPEH I I I I I I Cm!PRESSION I STROKE I I I I I I I I I I I I I I I I I k.POh'ER I I I STROKE I I I I I I I I I I I I I I I I I I I I I I I o.o 0.060 0.126 0.193 0.160 0.326 0.39J 0,460 0.526 0.593 0.660 0.726 0,786 TI~.fE (sec) FIGU~E 105. :!cutron i1ultiplication Fuctor Versus Piston Cycle Tirr!c for Engineill w w C)

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Later in the power stroke the gas temperature rises and when the rate of increase in the temperature exceeds 331 the rat~ of increase in the cylinder volume, the gas pres sure also begins to increase. At the 0.750 cycle fraction (or at 0.560 seconds into the pi st on . c y c 1 e ) the exhaust val ve s are opened and the gas pressure, density and keffective all drop rapidly. The neutron flux also declines but the rate of fisiion heat release at this point is large enough to sustain a further gas temperature increase during the initial portion of the exhaust stroke. Examination of Figure 105 shows a small dip in keffective around the TDC position. It is similar to the dip observed in Figure 97 for Engine #10 and the reason for the existence of the dip i s again due to the fact that the UF 6 gas at this point is quite black to thermal neutrons and increases in gas density lead to very little increased fission neutron production. At the same time, the system leakage is increasing rapidly so that the effec tive neutron multiplication factor acutally shows a decrease in the TDC vicinity.

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Thermodynamit Studies for Nuclear Piston Engine Power Generating Systems U ti 1 i z i n g 11 Gas Gener at or II Eng in es 332 The fotjr-stroke #11 Engine and the modified #11 Engine (see Table 69) were incorporated into three nuclear power generating systems. The studies are analogous to those just discussed for Engine #10. The results from the nuclear piston-gas turbine steam turbine system study are shown in Table 70. The modified #11 Engine was used for this system. For a schematic of the piston-gas turbine-steam turbine system, reference is made to the Figure 44 diagram. While this figure gives the proper component layout, it should be realized that most of the temperatures and pressures are now different from the Figure 44 yalues. They have been altered to accommodate the hotter exhaust gas from the "gas generator" engine. For instance, the HeUF 6 gas exiting from the piston engine block is at 2358F and at 380 psia instead of the 2015F and 282 psia which are shown in Figure 44. The HeUF 6 gas entering the _block is at 288.l psia rather than at 213 psia while inlet gas temperature remains unchanged at 260F .. Ts diagrams for the steam and gas turbine cycles are similar to but not identical with the Figure 45 diagrams. The helium gas turbine inlet temperature is now 1950F

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333 . TABLE 70 Summary of Thermodynamic Results for the Piston-Gas Tt.irbine ~ Steam Turbine System Which Uses the Modified #11 Piston Engine Eriergy Input to Fluids per Piston Energy Output from Fluids per Piston Fission Heat Rate Compressor Powe~ Pumpt Power 9.060 Mw 0.007 Mw 10.417 Mw Gas Turbine Power Piston Mechanical Power Steam Turbine Power . Qout {gas turbine loop) . Qout (condenser) 3.390 Mw 0.580 Mw 1.140 Mw 1. 712 Mw 2.320 Mw Oout (steam generator exit)l.275 Mw Power Source Description Power/Piston (Mw) Piston Mechanical Power Net Steam Turbine Power Net Steam Turbine and Mechanical Power Net Gas Turbine Power Net Gas Turbine and Mechanical Power Net Steam and Gas Turbine Power Steam and Gas Turbine and Mechanical Power 0.580 1. 133 1. 713 2.040 2.620 3.173 3.753 Power Breakdown for 37.5 Mw(e) Unit (10 pistons at 3. 753 M\v(e) per Piston) Piston Mechanical Power Net Steam Turbine Power Net Gas Turbine Power 5.800 Mw 11. 330 Mw 20;400 Mw 37.53 Mw 10.417 Mw Efficiency( % ) 6.4 12.5 18.9 22.6 29.0 . 35.0 41. 5

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instead of 1650F and the gas turbine cycle efficiency is up from 50.4% to 54.5%., Steam turbine inlet conditi6ns 334 are now 720F and 800 psia rather than 680F and 660 psia. The steam turbine cycle efficiency is 32.9% as compared to the prevtous value of 31 ,9%. Single stage reheating and singl_e stage intercooling are still used in the gas turbine cycle while the steam turbine cycle again has a single reheat stage. The 1950F gas turbine inlet temperature is not considered excessive since inlet temperatures of 1700F are now common for industrial open cycle gas turbines and 2000F is being approached [56]. The results for the piston-steam turbine arrangement are shown in Table 71. A standard #11 Engine was used fot this system. The piston-steam turbine schematic is essen tially identical with the schematic of Figure 46. Exit temperature from the engine block for the HeUF 6 is 1878F rather than 1790F while the exit pressure is 318 psia ' . instead of 262 psia. The intake pressure is 284 psi~ as compared to the value of 214 psia which is shown in Figure 46. Inlet temperature for the HeUF 6 remains unchanged at 260F. Figure 47 gives the Ts diagram for the ste~m turbine cycle. The results for the piston-cascaded gas turbine system are shown in Table 72. The modified #ll Engine was used for this system. Reference is made to Figure 48 for a schematic of the piston-cascaded gas turbine system.

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335 TABLE 71 Summary of-Thermodynamic Results for the Piston-Steam Turbine Syste~ Which Uses Piston Engine #11 Energy Input to Fluids per.Piston Fission Heat Rate Pump Power 7. 110 Mw 0.026 Mw 7 .136 Mw Energy Output from Fluids Steam Turbine Power Piston Mechanical Power 0 out (condenser) oout (steam generator) per Piston 2.176 Mw 0.60 Mw 3.09 Mw 1. 27 Mw 7.136 Mw Power Source Description Power/Piston (M11) Efficiency (%) Piston Mechanical Power 0.60 Net. Net Steam Turbine Power 2. 15 Steam Turbine and Mechanical 2.75 Pov1er Power Breakdown for 27.5 Mw(e) Unit (10 pistons at 2.75 Mw(e) per piston) Piston Mechanical Power Output 6.00 Mw Net Steam Turbine Power Output 21.50 Mw 27.5 Mw 8.45 30.25 38.7

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336 TABLE 72 Summary of Thermodyna~ic Results for the Piston-Cascaded Gas~ Turbine System Which Uses the Modified #11 Piston Engine _!nergy Input to Fluids Fission Heat Rate 9.06 Mw Compressor Power 4.04 Mw 13. l O Mw Power Source Description Piston Mechanical Power Net Gas Turbine Power Mechanical and Gas Turbine Power Energy Output from Fluids Gas Turbine Power 6.50 r1w Piston Mechanical Power 0.58 Mw . Qout (gas turbine loop) 4.12 Mw . Qout (Hebuffer loop) 1.90 Mw 13. l O Mw Power/Piston (Mw) Efficiency 0.58 6.4 2.46 27.2 3.04 33.6 Power Breakdown for 30 Mw(e) Unit Piston Mechanical Power Output 5.80 Mw Net Gas Turbine Power Output 24.60 Mw 30.40 Mw (t)

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337 However, once again, many of the temperatures and pressures have been altered to accommodate the hot exhaust g a s from the II g as genera tor '' en g i n e . T he He U F 6 ex i t t em perature from the block is 2358F rather than 2015F and the exit pressure is 380 psia instead of 282 psia. Inlet gas tem~erature remains unchanged at 260F while the inlet gas pressure is up from 213 psia to 288.1 psia. The Ts diagrams for the cascaded gas turbine cycles are somewhat different from the Figure 49 diagrams. Turbine and compressor efficiencies remain at 90%. The high temperature gas turbine cycle regenerator effective ness is 30% v1hile the low temperature gas turbine cycle regenerator effectiveness is 40%. Previously, only the low temperature gas turbine cycle had a regenerator and its effectiveness was 80%. Single stage intercooling has been added to the high temperature gas turbine cycle while single stage reheating has been added to the low temperature gas turbine cycle. The helium inlet temperature for the high temperature gas turbine is now l950F rather than 1650F. Helium temperature at this turbine outlet is 1558F instead of the previous 1364F. The high te~perature turbine inlet pressure is up from 1000 psia to 1150 psia. Pressures after eipansion in the high temperature gas turbines are 697 and_423 psia instead of 667 and 445 psia. Aside from the addition of single stage reheating, the low temperature gas turbine cycle is the same as in Figure 49.

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The high temperature gas turbine cycle efficiency, the low temperature gas turbine cycle efficiency, and the cascaded gas turbine overall. cycle efficiency are 19.8%, 33.6% and 42.0%.respectively. The corresponding Figure 49 values are 13.0%, 32.3% and 35.5%. In Table 73 is a summary of the thermodynamic analyses for the three nuclear piston engine power gene rating systems. The mechanical power output for these 3,38 11 gas generator 11 systems is seen to be small, about 0.6Mw per piston as compared to about 3Mw per piston for the pre viously analyzed engines. The mechanical efficiencies are also very low for these gas generator engines, being around 6 to 9%. Despite the slightly lower mass flow rates for these 11 gas generator 11 engines, the turbine power outputs and efficiencies for the turbine portions of the systems are higher than for the previously analyzed systems (see Table 66). This is -because of the higher exhaust gas temperatures pro duced by the 11 gas generator" engines._ Due to the low mechani cal power and low mechanical efficiency, however, the total power output per piston and the overall efficiencies for these power units are less than for the previous systems. The total power per piston is in the 3 to 4Mw range and hence power units consisting of 10 cylinders would have 30 to 40Mw(e) capacities with overall efficiencies of about 35 to 40%.

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TABLE 73 A Comparison of Thermodynamic Results for the Three Nuclear ~iston Engine Power Generating Systems Which Us _ e Piston Entjine #11 and the Modified #11 Piston Engine 1 Total / t-1ss1on Rate of Mechanical Turbine Heat Transfer Cool i ng via ter Power Heat Heat Power per Power per Rate per . Flow Rate per _ System Output , Rel ease Rejection Overall Piston (Mw) Piston (~1w) Unit Power Unit Power . per Rate per per Efficiency (BTU/hr Mw) (qal/min Mw) Piston Piston Piston ( % ) Efficiency Efficiency (Mw) (Mw) ( MvJ) ( % ) ( % ) Piston0.58 2,46 . . cascaded --x l o 7 Gas Turbine 3.04 9.06 6.02 33.6 6.4 27.2 2. 14 448 . . Piston0.60 2. 15 38.7 Stearn --X 10 7 Turbine 2.75 7. 11 4.36 (36.6)* 8.5 30.2 2. 01 364 Piston' Gas Turbine0.58 3.173 Stearn -. 7 Turbine 3. 7531 9.06 5;307 41. 5 6.4 35.0 l.75 X 10 322 *Overall efficiency when the mechanical efficiency is the same as for the piston-cascaded qas turbine and piston-gas turbine-steam turbine systems. w w I.O

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340 The piston-gas turbine-steam turbine system is the most costly and ~omplex in terms of the number of required , components. It yields a higher total power and higher overall efficiency than either of the other two systems. This. is a consequence of the gas turbine being able to make efficient use of the HeUF 6 high temperature range and of the steam turbine making efficient use of the HeUF 6 low temperature range. The piston-steam turbine system has a lower power output than the cascaded gas turbine system but a slightly higher overall efficiency. This is possible because the modified #11 Engine which is used with the cascaderl gas turbine system has a greater amount of available energy in its exhaust gas. The HeUF 6 exit temperature from this engine is 2358F as opposed to the 1878F exhaust tempera ture from the standard #11 Engine. The steam turbine is efficient over its entire temperature range. The cascaded gas turbine unit operates over a much larger temperature range and while it is very efficient at the high temperatures, it is unable to make use of the HeUF 6 in the 700 to 260F range. Thus its overall efficiency is slightly less than for the piston-steam turbine syste~ despite.the fact that it yields almost 15% more power. As previously mentioned, because of the presence of :the steam generator component, the piston-steam turbine configuration will be more expensive than the cascaded gas turbine system.

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An asset of the ugas generator" piston engines is that they are able to take full advantage of the latest 341 gas turbine technological advances which allow inlet temperatures approaching 2000F. The previously analyzed piston engines which yield the high mechanical powers are capable of providing He gas at temperatures up to about 1700K. To produce hotter He gas temperatures with these engines would require the HeUF 6 temperature to reach levels during the cycle where significant dissociation of the UF 6 occurs. With the "gas generator" engines, He at 2000F can be provided without having the HeUF 6 anywhere near the dissociation temperatures. For the power generating systems utilizing gas tur bines, an attractive arrangement, especially for the low mechanical-power "gas generator" engines, might involve making use of free pistons and compound bounce cylinders containing helium. The helium in the bounce ~ylinders would be compressed during the intake and power strokes of the opposed large pistons which move in the cylinder con taining the HeUF 6 . The expansion of the He in some of the bounce cylinders would be sufficient to move the large pistons towards each other for compacting the HeUF 6 during the compression stroke and for expelling the HeUF 6 during the exhaust stroke. The compressed helium inthe remaining bounce cylinders could be fed to the gas turbines. Thus, instead of providing mechanical power through a crankshaft,

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342 the nuclear piston engines could be used to assist in the compression of the helium gas for the gas turbine cycl~. ~ Economic studies will again be required to determine . whether or not the increased costs of the more complex systems override the better efficiencies and power outputs of these systems. Also, the desirability of turbine versus mechanical power will have to be assessed for each indi vidual application. Such considerations will determine whether one chooses a low-mechanical-power "gas generator" nuclear piston engine, a high-mechanical-power nuclear piston engine, or some other nuclear piston engine con figuration like the free piston system described above. Economics and the particular. application for the engine will also determine whether one chooses a design which makes use of a blanket region. The blanket region itself could be designed either for breeding or for high conversion with low thermal power output or for low conversion with high thermal power output. The blanket thermal power could then be used, for example, for process heating. It should be emphasized that the thermodynamic studies and cycles used in this and the previrius chapter do not represent the ultimate or opti~um conditions. Careful consideration, however, went into the selection of cycles~ temperatures, pressures, flow rates, temperature drops, and component efficiencies. The listed nuclear piston engine power generating system characteristics are therefore felt

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343 to be representative and notoverly optimistic of the type of performance which can be anticipated from these system~;-~ Indeed, detailed thermodynamic optimization and contingent economic studies should lead to at least small improvements over the performance r~sults listed in this.study. An example of an area requiring such study is the HeUF 6 -toHe heat exchanger. The temperature drops across this unit must be fairly large if one is to have a reasonable size and moderate cost. Large temperature drops, however, yield . large losses in available energy and further studies on this component could easily lead to some improvements in the overall system performance. Summary The attainable engine performance for the four-stroke engines was found to be comp~rable to t~e performances achieved with the two-stroke engines. Operating charac teristics for the four-stroke engines, however, are somewhat different than for the two-stroke engines. The most de sirable engine speeds are around 170 or 180 rpm's rather than 100 rpm's. Clearance volumes are reduced from around 0. 18m 3 to 0. 16m 3 while the compression ratios are increased from 3-to-l to around 4-to-l. The fuel loadings for the four-stroke engines are somewhat larger than for the two stroke configurations and parametric studies showed that the four-stroke designs were less sensitive to parametric

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344 variations than were the two-stroke engines. Piston motion for the four-stroke engines is a combination of simple and-~ non-simple harmonic motion instead of pure simple harmonic motion as used with the two-stroke systems of the previous chapter. The helium-to-uranium mass ratio for the four stroke engine studies was reduced fro~ 0.332 to 0.30 or 0.25 so as to increase the UF 6 partial pressure and provide an added safety factor against UF 6 dissociation. Engine startup studies for the four-stroke systems in the presence of delayed and photoneutrons showed that the time required to go from shutdown to an essentially equilibrium condition is of the order of 8 to 12 minutes. The equilibrium condition was attained by making adjustments . . inthe intake line gas pressure, the intake line initial mass flow rate, and the loop circulation time. Other vari ables which could be readily altered to assist in the approach to equilibrium are the intake line gas temperature and the cycle fraction at which the intake valves are closed. The behavior of neutronic parameters during the piston cycle were examined in detail. Parameters included in this study were keff' the neutron lifetime, the neutron genera tion time, flux ratios, flux shapes and each of the individual factors of the six-factor formula. The observed variations in all of these parameters over thepiston cycle were explained in reasonable detail. It was found that the inhomogeneous sources in the

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345 moderating-reflector resulting from photoneutron production had an important effect on fl u x shapes and fl u x rat i o s w h en the system was far-subcritical. As the system approached critical, the effect of these inhomogeneous source terms diminished rapidly. The neglect of fast core interactions was found to have almost no effect on neutron lifetimes and flux shapes. On the other ha~d, the neutron multiplication factor was found to be underestimated by about 2 to 3% when fast core interactions were neglected. Plots of forward and adjoint flux distributions as a function of cycle position were presented and reasons for their observed variations during the piston cycle discussed . . in detail. The flux shapes .and neutron lifetime were found. to vary significantly over the piston cycle and the use of a pure point kinetics model for describing the engine's neutron kirtetics behavior is questionable. Thus, the NUCPISTN code was updated to allow not only for time varying neutron lifetimes but also for time-varying effec tive delayed neutron fractions and time-vaiying inhomogeneous source weighting functions. The result of these improve ments is an extension from the simple point kinetics model to an adiabatic method for treating the system~s neutron kinetics. o 2 o moderating-reflector -temperature coefficients of rea~tivity were calculated over the temperature range from 320 to 570K. The coefficients were found to be large and

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346 negative (~-2.5 x 6k/k per K) and this is obviously a favorable situation from a s~fety and control standpoint. Fuel temperature coefficients of reactivity were calculated over the temperature range from 300 to 2100K. For 100% enriched UF 6 the coefficients were small and positive (~5 x l07 ). The 93% enriched UF 6 coefficients were slightly smaller (~3.8 ~' but still positive. _ The coefficients for 80% enriched UF 6 were small and nega tive (~-1.2 x The positive fuel temperature coeffi cients of reactivity for this gas core reactor represent no real safety or control hazard. The gaseous core system is totally different from solid and liquid core systems with regard to potential accident behavior. As has been discussed on several occasions in this chapter, effective means for attaining essentially instantaneous reactor shut down are available and they do not rely in any way on the fuel temperature coefficient of reactivity. Safety and control will be further discussed in Chapter VII. The o 2 o moderating-reflector density or voiding coefficients of reactivity were found to be large and nega tive. For the examined system, it was found that conversion of as little as 1% by mass of the o 2 o to the vapor state is enough to cause reactor shutdown. The effect of using less than 100% enriched UF 6 on en~ine performance was examined and found to be small. For example, the penalty in going from 100 to 90% enriched UF 6

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347 was but a 1% increase in U235 loading for equivalent engine behavior. The effects of delayed and photoneutrons on engine behavior were examined by two different procedures. The first method involved compensating for the absence of de layed and/or photoneutrons by increased fuel loadings. For the examined system, the maximum keff when delayed and photoneutron effects were included was 1 .055. When photo neutron were neglected, in,order to maintain the engine 1 s performance level, the maximum keffective had to be in creas~d to 1.061. When both delayed and photoneutrons were neglected, the maximum keffective had to be increased to 1.09 in order to maintain the engine performance level. The second method examined the engine performance level when no compensation was made for the absence of delayed and/or photoneutrons. When delayed neutrons were neglected, the mechanical power output dropped by about 57%. When photo neutrons were also neglected, the mechanical power output dropped by about 87%. Thus, the delayed and photoneutrons both exert a noticeable influence on engine behavior with the delayed neutron influence being the stronger of the two. Tirnestep selection for solution of the energetics eq~ations was examined. It was found that the optimum energetics equation timesteps were on the average 5 to 15 ti~es larger than the timesteps used in solving the neutron kinetics equations. It will be recalled from the last

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chapter that the most desirable neutron kinetics equation timestep sizes occur when the number of neutron kinetics equation timesteps per neutron lifetime is somewhere be tween 4 and 40. 348 For solution of the neutron kinetics equations, it was found that a simple three-point finite_ integration techni qu~ yielded good accuracy. The qain in acc~racy from higher order numerical techniques was minimal and not worth the added computational costs. The effect of C formula selection on overall engine p performance is not tremendous but it is certainly significant. The formula used in this work is given by equation (2) and is roughly valid over the 400 to 2400K temperature range. Equati-0n (l) was used by Kylstra et al. and is only valid for temperatures around 400K. Blanket studies were carried out for the four-stroke engines in a manner similar to those conducted for the two stroke engines of the previous chapter. Because of the lower cycle-averaged fluxes for the four-stroke engines, the conversion ratios and breeding ratios are lower than for the two-stroke systems. The doubling times are corre spondingly longer for the four-stroke engines. Values for this parameter could be as long as 60 to 65 years. However, it is felt that values of 30 to 35 years should be readily attainable by reducing the out of core fuel inventory and by more detailed blanket optimization studies.

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Group constants as output by BRT-1, PHROG and NUCPI.STN were given for a selected nuclear piston engine configuration. The core thermal group constants output by NUCPISTN and BRT-1 were seen to be in good agreement with one another. 349 The two-group NUCPISTN keffectives which neglect fast core interactions are 2 to 3% lower than the two-group CORA calculations which incJude fast core interactions. Fourgroup k ff t values were found to be consistently lower e ec 1ve than the two-group keffectives from the corresponding computational scheme by 5 to 6%. Twenty-one group calculations keffectives were observed to be 2 to 3% lower than the four group keffectives from the correspondirig computational scheme. The difference between the 21-group and 123-group keffectives was very small, less than 1%. It should be re iterated that all of these results are for nuclear _piston engines which possess a o 2 o moderating-reflector. Two-dimensional k ff t from EXTERMINATOR-II e e c .1 v es were consistently 3 to 4% higher than the corresponding onedimensional results from CORA or MONA. The "equivalent" volume spheres used in CORA and MONA do not properly repre sent the fast leakage from the cylindrical core to the controlling moderator-reflector region. When k ff t from one-dimensional XSDRN s 4 cale ec 1ves culations were compared with corresponding one-dimensional diffusion theory results, tile latter ~,ere found to be 2 to

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3% higher than the former. Less than half of this differ ence could be traced to disparities in the cross section libraries used in generating the group constants used, in XSDRN and in CORA (or MONA). The remainder of the differ ences is due to the fact that some upscattering to above 0.683eV was neglected in the diffu~ion theory calculations. The neutron lifetimes obtained from four-~roup cal.culations (with the lower keffectives) are larger than the neutron lifetimes obtained from two-group calclations (with the higher keffectives) for a given computational schem~. This is the behavior which would be expected on an intuitive basis. 350 The one~dimensional computations yielded neutron life times which were smaller than the neutron lifetimes obtained from two-dimensional calculations. The difference is due to geometric effects as the equivalent-in-core volume sphere has a reflector volume which is significantly smaller (by 25 to 30%) than for the actual cylinder. For the systems which do not use poison insertions and removals in the moderating-reflector region, the neu tron lifetime and generation time decrease continuously as keffective increases from far-subcritical to supercritical. Fo~ the systems which use poison additions in the moderating reflector, the neutron lifetimes in the far-subcritical configurations are shorter because of the presence of the highly absorbing poison. The bulk of the neutron lifetime

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in the D 2 0-reflected piston engine systems is spent in the moderating-reflector region; the lifetime in this region is usually about an order of magnitude greater than the core lifetime. 351 The conclusion reached regarding the neutronic cal culational schemes was that excellent values for the system k and neutron lifetime can be obtained from twoeffective dimensional, multigroup (i.e., around 21 !1roups) diffusion theory provided 11 good 11 group constants or cross sections are employed. Because of partial cancelling by compensating errors, the NUCPISTN code yields keffectives which are in error by only about 4 or 5%. The elementary two-group com putational scheme used in NUCPISTN for obtaining the neu tron multiplication factor was initially justified by its simplicity and consequent large savings in computer time. Power transients or power level changes can be initi ated solely by varying the loop circulation time for the nuclear piston engines. These transients can then be damped out by further variations in the loop circulation time. If desired, the system can be made to settle at a new power level by varying additional parameters such as the intake line gas pressure, the intake line gas tempera ture, the intake line initial mass flow rate, or the cycle fraction for closing the intake valves. Thus, variation of the loop circulation time is seen to be an effective and efficient means not only for attaining equilibrium during

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352 engine startup but also for proceeding from one equilibrium power level to another. Thermodynamic studies were carried out for four-stroke nuclear piston engine power generating systems in a manner similar to the studies which were performed in the previous chapter for the two-stroke engines. The four-stroke systems were found to have higher mechanical efficiencies but lower mechanical power outputs, than the two-stroke engines. They also possess lower mass flow rates and smaller He-to uranium mass ratios and hence yield lower turbine outputs than for the two-stroke sjstems. The total power per piston for the four-stroke systems is around 5Mw(e) as compared to the 6 to 7Mw(e) for the two-stroke engine systems. The overall efficiency (~50%), however, is slightly bettir for the four-stroke configurations than for the two-stroke power systems. For the three nuclear piston engine power generating systems examined, the piston-steam turbine system again had the highest overall efficiency and highest total power. The piston-gas turbine-steam turbine system had the highest turbine power but was the most complex arrangement. The piston-gas turbine system was the least complex but it also had the lowest total power and overall efficiency. "Gas generator" nuclear piston engines were also examined in which there is no step-reflector addition or removal or no poison addition or removal during the piston

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353 cycle. The gas pressure and temperature peak very late in the cycle for these engines and the mech~nical power outpu~ and the mechanical efficiency are quite small. The turbine power outputs and turbine power efficiencies for these ''gas generators, 11 however, are slightly better than for the pre viously analyzed systems. Thermodynamic studies for power generating systems based on the 11 gas generator" nuclear piston engine showed that the total power output per piston is about 3 to 4Mw(e) while.overall efficiencies range from 34 to 42%. The piston-gas turbine-steam turbine system was the most cdmplex but it also yielded the highest turbine power, the highest total power and the highest overall efficiency. The piston-gas turbine power system was the least complex but it also had the lowest overall efficiency. The piston steam turbine system had the lowest turbine power and low est total p6wer although its overall efficiency is slightly better than for the piston-gas turbine arrangement. The 11 gas generator 11 nuclear piston engines would ap pear to be most compatible with power generating systems employing gas turbines. This is because the "gas generator 11 piston engines are able to take full advantage of the latest gas turbine technological advances, which allow inlet tem peratures approaching 2000F. The 11 gas generator" nuclear p'iston engines can provide helium gas at such temperatures without encountering any UF 6 dissociation difficulties. In

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354 contrast, the high mechanical power nuclear piston engines can supply helium for gas turbines at maximum t~wp~raturel of from 1600 to 1700F. To provide higher temperature helium would force these engines to exceed temperatures where a significant degree of UF 6 dissociation occurs._

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CHAPTER VI RELATED RESEARCH AND DEVELOPMENTS Introduction Many i~portant pr~ctical and materials problems will have to be solved before technical feasibility o1 the nuclear piston engine concept can be demonstiated. Among these are lubrication of the piston and piston rings and selection of su~table liner materials for protection of susceptible m~ terials from tl1e corrosive UF 6 at high temperatur~s. Also, the degree of UF 6 decomposition induced by fission frag ments, the effects of neutron i~radiation damage and of th~rmal-mechanical stresses on fue piston components over extended operation periods, ~swell as the details or methods for cleaning up the UF 6 gas will ~11 have to be resolved. Many of these practical and materials problems, of which only a few are listed above, are already the subject of extensive investi~ation by other gaseous co~e nuclear reactor research programs. The nuclear piston engine con cept will of course benefit greatly from this related re sea~ch which is the subject of this chapter. Problems ~hich are more or less unique to the nuclear piston . engine 355

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concept and which require further, separate research pro grams will be covered in the next chapter. Related Research and Developments at the University of Florida 356 An on-going and related project being conducted here at the University of Florida and sponsored by NASA is re searching the thermodynamics of UF 6 and UF 6 -He mixtures [57]. The specific heat of UF 6 and UF 6 -He mixtures and the viscous coupling between UF 6 and He have been measured to l300F using a ballistic piston to create the conditions required for the measurements [43]. The piston is to eventually be introduced into the thermal column of the University of Florida Training Reactor to determine the thermodynamic properties of the gas in the presence of the fission pro cess. These ballistic piston studies should provide some of the necessary thermodynamics information to properly descri~e feedback mechanisms for the nuclear piston engine neutron kinetic studies; Other Related Research and Developments in Progress An occurrence which should prove very advantageous to the development of a nuclear piston engine power generating system is the commitment by NASA to a comprehensive re search and test program [58, 59] of critical gaseous fuel nuclear reactor experimenti. The experiments ~re planned

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357 to pro~eed with stepwise increases in power and the overall test program offers the expectation of multiple technologi... cal uses at various performance levels. The first pha~e of the NASA experiments will be basic criticality tests and general reactor physics research at low power for a core with a static filling of 93% enriched UF 6 . Gas temperatures are expected to be from 300 to 500K while power outputs will be but 0.1 to l.0kw. The second phase of the program includes the capabil ity of flowing the UF 6 through the reactor cavity. Gas temperatures will again be in the .300 to 500K range while power levels will be from 1 to l0kw. Various reactor physics and UF 6 handling problems are expected to be re solved in this experiment. UF 6 chemistry in a nuclear environment is another research area which will be explored. This includes the chemical interactinns of the UF 6 gas with the confining surfaces as well as the chemical sta bility of the gas itself. Also, the effect of fission power d~iven pressure fluctuations in the gas will be studied. The third phase of the program will incorporate the separation of the gaseous UF 6 fuel cloud from the surfaces by a confining flow of buff~r gas. Gas temperatures will be at about 500K v1hile power levels \'-/ill be from l to l0kw. Reactivity feedback mechanisms such as fuel tempera ture, fuel mass, fuel .volume, and buffer gas temperature and density effects will be experimentally measured and

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compared with theoretical predictions. Another reactivity feedback mechanism which will 'be studied is the effect which results from the removal of a part of the delayed neutrons in the UF 6 flow. The next phase of the program will involve a bench358 mark experiment to demonstrate the technical feasibility of a new type of power reactor~~the gas core nuclear reactor. Ga~ temperatures will be around 1500K and power levels will range from 10 to lOOkw. The final phase in the program is expected to be a plasma core reactor test in which gas temperatures will reach 6000K and power levels will be around 5Mw. The above outlined research program is anticipated by NASA to extend over a six-year period. The completion of the program is expected to yield the physics principles and experimental know~how.as a base for gaseous UF 6 and plasma core reactor technology development. Feasib1lity studies of potential applications will be conducted along with the reactor research. The NASA program of reactor ex periments is being supplemented by a strong program of basic research in atomic and molecular physics and chemistry. NASA's interest in uranium plasmas and gas core reac tors has in ~e past focused on the ad~anced propulsion capa~ bilities of these systems for future space missions. In recent years, consideration has also been given to the broad spectrum of possible applicationsfor these systems

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359 which exist outside of the range of space propulsion. The w i d e r a n g e o f a pp 1 i cat i on s res u l t s from t he w i d e ran g es of -~ power, temperature, and pressure at which gaseous core reactors can be made to operate. The features of the gaseous core concepts allow these s y s te m s to o p e r a t e r e ad i l y n o t o n l y a s bu r n e r o r c o n v e r t er power reactors but also as breeders. The gas core reactors can be readily coupled with various types of power conver sion systems (MHD conversion, thermionic conversion, etc.). They can find use ror industrial processing by thermochemical and photochemical means and can also be used for direct con version of fission energy into laser radiation. The latter ability will exist if-the el~ctromagnetic radiation emitted by a fissioning gas or plasma can be tailored in its spectral properties so as to be capable of transforming a useful percentage of the fission energy into narrow band radiation. Experimental investigations are not yet complete, but the evidence thus far suggests that such tailoring will indeed be possible for a fissioning gas or plasma. Another important feature of gas core reactors is their ability to _be used for burning or for transmuting nuclear wastes. Because of their high neutron fluxes, relatively small critical mass, and the ease of processing and recycling because of the gaseous state of the fuel, gas cor~ reactors lend t~emselves much more favorably to the transmutation task .than do other reactor concepts. The

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360 nuclear piston engine as a pulsed, quasi-steady-state gaseous core reactor possesses the characteristics nec~ssa~~ to operate in any of the above described modes. Because of its pulsed nature, it would be particularly well suited to provide nuclear pumping for lasers. Aside from their wide range of applications, NASA cites other features whith make gas core reactors attrac tive. These include 1) low critical mass of fissile fuel 2) small accident h~zard since the gas core does not contain excess fuel and fission products (the latter can be continuously removed during normal operation) 3) low fuel cycle costs because of the absence of fuel fabrication and fuel handling costs 4) high efficiency and low thermal pollution 5) complete tolerance for loss of coolant accidents 6) low capital and operating costs and 7) major prospects for growth in technology. All of these features are possessed by the nuclear piston engine gaseous core reactor and have already been touched upon in the preceding chapters. More will be said about some of these qualities in the last chapter. Another related NASA project involves the investiga tion into the possible development of a large power, steady state, gaseous core nuclear reactor [60]. The reactor, which has been named NERNUR, will be fueled by hig~ly

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3 61 enriched UF 6 gas. The core will probably contain a He-UF 6 gas mixture and will have a heavy water moderating-reflecfcir (see Figure 106). In ~ont~ast to the nuclear piston engine, the tIASA system will be a static rather than a pulsed or quasi-steady-state system. Because the essentially motionless He-UF 6 in the NERNUR has much poorer heat transfer 1 properties than the flowing He-UF 6 in the nuclear piston engine system, large heat trans fer surface areas will be required. Since the available heat transfer surface area is determined by the core size, the NERNUR is necessarily restricted to much larger power ratings than the nuclear piston engine systems. The nuclear analyses of the two systems should comple ment each other since there are overlaying areas of interest in the proposed systems. Other studies which are performed for one system will likely be directly applicable to the other system. For example, chemical studies aimed at de veloping methods to reprocess or clean-up the poisoned UF 6 gas will be usable by both the NERNUR and the nuclear piston engine systems. Metallurgical studies ~eeking to find ma terials suitable to withstand the hot, corrosive UF 6 will also benefit both reactor concepts. Also, while methods of control during normal reactor operation might differ somewhat for the two systems, emergency control or shutdown mechanisms which are useful for the NERNUR should also work quite well for the piston ~ngine system. Finally, safety

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FIGURE 106. Dt:~J> TA~K BCRO!<' i::1:;RGE~CY FLOOD I~,:; SYSTE!-! •--BO!tO'/ SOLUTION P!'.ESSl!!llZER 0 0 0 0 0 0 0 0 lo o' ..._ _:,,..rJ 0 0 0 .n 0 0 0 0 0 Di v ~:OD. 0 i I 0 1F:.!:P!\O PROC!:SS RF'..AT I coc: ... rnc PU!IP Schematic of NASA'S NERNLlR-~A Larqe Power Generat;ng System Utilizing a UF 6 Gas Core ~uclear keactor w 0) N

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363 and containment problems, with regard to the UF 6 gas, both i n the co re an d i n t he 1 o op o u ts i de t he c ore , s ho u 1 d be -'" quite similar for the two. systems~ A major difference here will be that the UF 6 gas for the piston engine will 1 ikely be at a higher mean pressure than the UF 6 gas for the lffRNUR.

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CHAPTER VII CONCLUSIONS; REFINEMENTS AND AREAS FOR FURTHER RESEARCH Introduction Because of the wide range of temperatures and pres sures and because of the generation of thermal, radiative and mech.anical power, the nuclear piston engine is a very versati}e concept. The safety and economic aspects of the nuclear piston power engine appear at this point to be extremely attractive. The contribution which such an engine could make to our technological societ especially in a time of dwindling, expensive, irreplaceable fossil fuel supplies, is obvious. The nuclear piston engine concept, as a result of the.work presented in this paper, has evolved to a point at which a major research effort is required to establish its technical feasibility. A significant amount of insight has already been . gained into the power producing and operational character istics of the nuclear ~iston engine from the studies which were presented in Chapters IV and V. The nuclear piston engine is a pulsed, quasi-steady-state gaseous core reactor. A complete and accurate ~escription of the complex neutronic 364

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365 and energetic behavior and complete system optimization will be not only difficult but expensive. Because of the pre-~ liminary nature of this investigation and because of the lirnit~d budget, relatively ele~entary models were employed for some of the studies presented in this work. While the shortcomings of ~ome of these models are recognized (e.g., the use of point reactor neutron kinetics equations), it is felt that they have been extremely valuable-tools for this initial investigation. The series of parametric studies, ther~odynamic and nuclear system analyses and preliminary fuel cost estimates performed in this work have established a quantitative framework from which valuable characteri~ tics were found and from whi-ch comparisons could be made .. It is not claimed that any of these investigations repre sent optimum or ultimate conditions. It is recognized that the thermodynamic studies, blanket s~udies and even th~ piston cycle itself should be the.subject of more sophisti_cated and detaile~ investigations, both theoretical and experimental. Since the results of this study have indicated such good performance potential for the nuclear piston engine, major research proposals have been submitted to various ~gen cies by the University of Florida's Department of Nuclear Engineering. The ~roposed research program aims to estab liih on better grounds the probable operating characteris tics, power output and economics of the nuclear piston

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366 engine power systems. Improved and necessartly more com plex analytical models will be. established for use in the engine experimental and design states. The models will use, as much as possible, already existing computerized compu tational schemes, but will have to be incorporated into a 11 workhorse 11 computational system. The difficulty in accurately describing, analytically, the behavior of the space and time changing neutron popula tion has been mentioned on several occasions already, The development of more sophisticated neutron kinetics models should precede any further development of the power system. During the task of assembling analytical neutronic models, the relative gain in accuracy obtained from very intricate calculations will have to be weighed against the expediency (and economic attractiveness from the computer expense standpoint) of less complex schemes. The neutronic model will set the mechanical power and working g~s conditions for the power system. When the neu tronic model is firmly established, a more sophisticated thermodynamic analysis of the turbine power system will be . coupled with it. A complete 11 workhorse 11 calculational scheme will then be used for the sys~em's analysis and per formance optimization studies. In preparation for this proposed research, a very complete and up-to-date library of tomputer codes has been assembled and made operational at the University of Florida Computer Center. All of the

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367 nuclear reactor computational schemes deemed necessary at this point are ready to be utilized in the proposed re~ea~ch as soon as funding becomes available. The research will b~ performed in order toestablish a firm analytical base on which a technical feasibility program for the nuclear piston engine can be founded. Whil~ this analytical phase iS being conducted, plans for the experimental phase can be developed to fully assess the nuclear piston engine concept capabilities. The proposed research program which is based upon the results of this work has the following objectives: l) to produce more complete and detailed analytical models for the neutronic and energetic description of the nuclear piston engine; 2) to establish analytical heat transfer, thermodynamic, and fluid mechanics modelsfor the segments of the power system external to the nuclear piston engine and to inte grate these models with the above models; 3) to optimize, for some given engine application, the power system cyclesby matching th~ neutronics, energetics, heat transfer, thermodynamic and fluid mechanics behavior; 4) to establish an economics model that will roughly predict the capital costs, fuel cycle costs, and operation and maintenance costs of the nuclear piston engine power systems;

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5) to perform preliminary safety analysis on the nuclear piston engine; 368 6) to perform blanket,,optimization and more detailed breeding potential studies; and 7) to make recommendations on further steps which Would be necessary to achieve the final objective of a technically feasible, safe nuclear piston engine power generating unit within engineering, economic and energetic constraints based on the results obtained from the proposed research and from the results presented in this work. In the sections which follow, detailed accounts will be given of areas requiring further research and of possible refinements to be made to the models employed in this work. First, however, a section will be presented which briefly outlines some of the many potential uses and applications for nuclear piston engine power units. Applications Some of the primary uses as listed in Chapter I for nuclear piston engine power generating systems were for peaking units for electrical utilities, for small ground based power systems, for mobile power units, for process .heat, for nautical propulsion and ship electric power. Be cause of the high efficiencies and low specific mass of th~se engines, other attrac.tive uses would be for space

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station and space base power or as propulsion units for space shuttle vehicles. 369 It has been mentioned that the versatility of these units stems from (1) the wide range of pressures and tem peratures at w~ich these engines can operate and (2) from the fact that they generate radiative, thermal, and mechani cal energy. ~alculations presented in this work have con sidered the use and/or conversion of the latter two forms of energy. No quantitative calculations regarding the use of radiative energy from the fissioning UF 6 are given in this paper. While the utilization of the radiative energy coul.d eventually have significant applications, the bulk of the power will be thermal and mechanical and they should properly be of more immediate concern. Only a short quali tative discussion will therefore be given on the possible utilization of radiative energy from the nuclear piston engine. The tradeoff between mechanical power from the piston. and thermal power from the engine~ exhaust gases has been covered several times. The mechanical ~ower can be used directly for example for nautical propulsion, or it can be converted by means of generators and alternators into elec tric power, or by means of free pistons and bounce cylinders it can be used to assist in compressing helium for use in gas turbines. The thermal power would be extracted primarily

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370 by means of heat engines (e.g., gas and steam turbines) and then converted into electricity by means of generators. Thermoelectric devices and thermionic diodes could provide possible "topping cycles" for further utilization of the thermal power. The thermal power from the gas core could also be used to provide process heat if so desired. A blanket region could be designed so as to yield a low con version ratio but a high thermal power level which tould also be a source of process heat .. It has been demonstrated in this work that gas core power reactors can operate as burners, converters, or as breeders. From the above and previous discussions, it is also apparent that they can readily be coupled with various types of conversion systems. An additional example of this would involve using part of the nuclear piston engine's output energy for industrial processing by thermochemical or photochemical means. In particular photolysis of media such as the dissociation of water for hydrogen production could be an important appli cation. The fact that these engines can operate at a wide range of temperatures and pressures can be seen by recon~ sidering the parametric survey (Figures 11 through 34). Or, for the "gas generator 11 en g i n es , one need only exam i n e Table 69 to get an idea of the relative ease with which a wide range of temperatures can be reached.

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371 While investigations are not yet complete, experimen tal evidence to date indicates that the electromagneti~ _ radiation emitted by a fissioning gas or plasma can be tailored in its spectral properties and may thus be capable of transforming a us~ful percentage of the fission energy into narrow band radiation .. One could thus have a direct conv~rsion of fission energy into laser radiation or the photon energies of the radiant flux could be matched to those (energies) of .desired chemical reactions (industrial processing by photochemical means). The use of radiative energy from the nuclear piston engine for such applications could have important consequences and would constitute a t y p e_ o f II t o p p i n g c y c l e II to t ho s e c y c l e s u s i n g t h e p i s to n engine's thermal and mechanical power. It should be men tioned that the nuclear piston engine, because of its pulsed nature, would be particularly well suited to provide nuclear pumping for lasers. Another potential use for the nuclear piston engine is for burn i n g or trans mu ti _n g nu cl ear wastes . (3 e cause of the high thermal neutron flux, relatively low critical mass and the ease of processing and recycling because of the gaseous state of the fuel, the nuclear piston engine and gas core reactors ingeneral have a distinct advantage over other reactor types for this applicati6n. Studies have been carried out in which the fission product and actinide wastes

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372 gene~ated by a gas core reactor were recycled and burned [ 6 l ] . T h e s e i n v e s t i g a 't i o 'n s s h o ~, e d t h a t t h e i n c r e a s e d f u e f .... loading penalty due to the presenc~ of these recycled wastes in both the core and' the moderating-reflector region was but a few percent. These studies also revealed that the rate of burnup and production of certain actinides reached an equilibrium state after some time of operation. No net increase of these wastes in the system can then occur, re gardl~ss of the time period of operation. Actinides and fissiori products which cannot be transmuted must be processed and stored. Studies on the use of gas core reactors to burn certain waste products from other power reactors, ineluding iodine 129, have yielded favorable results and further research i n this area i s currently being carried but by NASA [58]. Before closing this section, mention should be made of the many possibilities which exist "' i th regard to the us~ of binary, trifluid, or multifluid cycles for utiliza tion of the thermal power ffom the nuclear piston engine. The existence of these multipl~ possibilities is due of course to the wide range of temperatures which can be r~ached with the nuclear piston engine. In this work, studiss were done only for binary HeUF 6 -He and trifluid HeUF 6 -He-H 2 0 cycles. The binary .fluid cycles made use of He gas turbines while the trifluid cycles used either He gas turbines in combination with steam turbines or just steam

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373 turbines. For the combination gas a_nd steam turbine system, the .gas turbine cycles tapped the thermal energy of the high temperature HeUF 6 while the steam turbine made use of the thermal energy of the.:HeUF 6 at lower temperatures. Another variation of this combination would have the steam turbine cycle maki~g use of some of the heat rejected by the hi-gh t~mperature gas turbines. Other reasonable tri~ fluid and multifluid cycles would be HeUF 6 -He-Hg, HeUF 6 -He Hg7H20, and HeUF 6 -He-Na-Hg-H 2 0. The inert helium serves either as a buffer between the hot, radioactive HeUF 6 and the other working fluids or it can serve the dual purpose of both a buffer and a working fluid. While the more com plex systems are attractive _from a thermodynamics and effi ciency standpoint, economic considerations may render many ' ' of the more elaborate possibilities impractical . . Analytical Model for Piston Neutronics and Energetics Steady-State N~utronic Analysis As has been discussed, attempts to account for neutron leakages in the perpendicular direction(s) by means of bucklings for gas core systems leads to difficulties. Two or three-dimensional si.mulation in a one-dimensional calcu lation by means of bucklings to account for perpendicular neutron leakages is a procedure which has been used quite ' i '.,'

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374 successfully with solid-f~eled reactors. In spite of the common usage, however, there is no theoretical justifica... tion for the method in the general case. In particular, for most gaseous core re~ctbrs, the flux curvature in the core is so small and the diffusion coefficients so large, that attempts to account for perpendicular neutron leakages by means of DB 2 terms have genera 11 y l e d to prob l ems . This has been the experience with the gaseous core nuclear piston engine. While the use of "equivalent-in-core:.. volume" spheres to simulate the two-dimensional piston has met with better success, it is st1ll only an approxima tion. The approximation consistently underestimates keff and also underpredicts the neutron lifetime. The approxima tion itself is best near the TDC position when one has a 11 square or "right" cylinder and poorest at those positions where the height-to-radius ratio is largest. For the ini tial studies, the spherical approximation in NUCPISTN is certainly justified because of the tremendous reduction in computer costs. Further studies should concentrate more on two-dimensional calculations for a proper description of neutron flux distributions, neutron leakages, lifetimes, and multiplication factors. It should be realized that conver gence for two-dimensional diffusion and transport theory calculations for gaseous core reactors is. comparatively slow and such calculations are consequently quite expen5ive.

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375 The conclusions reached regarding the nuclear piston engine steady-state neutronic analysis is that excellerit results can be obtained from two-dimensional, multigroup (around 21 groups) diffusion theory provided 11 good 11 group constants are used. Also, good results can be obtained from few-group (around four groups), two-dimensional diffusion theory prov i de d II good 11 group constants are used ( i . e . , up . scattering is completely accounted for). The four-group results for keffective will generally differ from the multi~ group results by but 2 to 3%. Moderating-Reflector Studies Graphite, o 2 o, H 2 0 and. beryllium have been considered in the moderating-reflector materials studies. Investiga tions for these materials have been made at a few representa tive temperatures and at a few selected thicknesses (see Table 5,e.g. ). As previously discussed, H 2 0 and graphite are no longer being considered as possible primary moderating reflector materials. Their thermal absorption cross sections have proven to be too high for practical piston engine con figurations. Both Be (with its n-2n reaction) and o 2 o (with its small thermal absorption) are good moderating reflector materials from a neutronics standpoint. BeO should also be an excellent material for this region. Some composite moderating-reflector region studies have been performed for Be-o 2 o combinations at 290K (see Table 8).

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376 A desirable composite reflector will 1 -probably consist of an i n n er Be or Be O reg i on o f a r bu n d a 1 O to 2 O cm th i c kn e s s -~ 1 followed by an outer o 2 o region of around a 60 to 80cm thickness. : ' 1 The IJ e or Be O a 1 'l ow s for s tr u c tu r a l i n t e g r i t y and separates the liquid o 2 o.-from the gaseous core. An addi tional consideration will be liner materials for protecting the moderating-reflect~r fro~ the corrosive UF 6 . The moderating-reflector region studies conducted thus far have been limited by the existing cross section libraries. If funding is obtained from the submitted re search proposals, these libraries will be made more exten sive. The reflector region -calculations should be conducted at ~any more temperatures and tl1icknesses and for addi tional composite material combinations. The optimization of this region's design will have to consider not only neu tronic characteristics but heat transfer and structural p 1 roperties as \>1ell as cost. Fuel Studies Thus far; most of the calculations have centered on the u s e of l O O % en r i c he d U 2 3 5 f u e l . Stu d'i es , ho \'I ever , ha v e been conducted for uranium-235 fuels of less than 100% enrich ment. Fuels of from 90 to 93% enrichment are more.realistic and the penalties in increased U235 loading requirements '\

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incurred by goihg to these lower enrichments are small (see Table 50). 1 377 Studies need yet to be conducted to determine how low a U235 enri~hment can be reasonably tolerated. Also, the extent of core poisoning by fission products has yet to be determined. The extent of this poisoning will help to 'dic tate the rate of the gas feed-and-bleed process. Finally, studies should also be extended to include u 233 and Pu fuels. From 1 a ~eutronics standpoint, these fuels are more desirable than u 235 in that the critical mass for the thermal nuclear piston engine system will be reduced, especially for the u 233 fuels. ~eutron Cross Section Libraries Neutron cross section l.ibraries at higher temper~tures will have to be made more extensive than the presently ~Vat1abl~ libraries. Th~~standard BRT-1 [45] .and XSDRN [46] libraries for exampl~ do not contain BeO and cross sections for many materials have been compiled at only a few 11 representative 11 temperatures above 400K. The genera tion of the necessary and more extensive nuclide cross sec~ t i o n s a t h i g h e r t em p e r. a t u r e s a n d t h e a d d i t i o n o f B e O t r o 111 the ENDF/B [62] data to the above mentioned libraries will be carried out here at the University of Florida when funds ' ' are obtained from the research proposals which have been

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378 submitted. The generation bf these data will be expensive but necessary if an accurate prediction of the piston neutronics is to be obtained over the entire expected range of piston engine operating temper~tures. ~oderating-Reflector and Fuel Temperature Coefficients of Reactivity Thus far, effects of ~oderator temperatur~ changes.on the neutron spectrum and reactivity have been examined and averaged over only widely spaced temperatures for o 2 o (see Table 43). With the more complete cross section li braries it will be possible to look at the spectrum and re~ctivity variations brought on by tempetature changes in much finer detail, over a given temperature range, not only for o 2 o but also for other moderating-reflector materials like Be and BeO. Also, the effects of fuel temperature changes on reactivity through Doppler broadening will be further in vestigated. For the externally moderated, gaseous core nuclear pfston engine, moderator temperature changes will have a much more pronounced effect on the neutron spectrum and reactivity than will any highly enriched-fuel tempera ture variations. However, the fuel temperature effects cannot be neglected. A few studies have been performed for 100%, 93%, and 80% enriched U235 fuels for a particu lar reference engine configuration (see Tables 44, 45, and 46). However, these need to be sup~lemented by more

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379 extensive calculations not only.for U235/U238 fuels but also for Th 232 ;u 233 and for Pu fuels. Neutr6n Kinetics Calculations Time-dependent studjes of the nuclear piston engine gaseous core are complicated by the changing reactor geometry during the piston cycle. Two levels of time de~ pendence are present: a) a time dependence intrinsic to the piston motion and changing core geometry and b) a space-time dependence intrinsic to the kinetics of the neutron population, driven not only by the core cl1anges with. piston position but also by the excess or deficiency of _neutrons (in the overall neutron balance) and their spatial distribution. For every piston position, a set of steady-state nuclear parameters (and a corresponding neutron distribu tion) can be attributed to the geometry and core conditions under considerat.ion. These parameters and associated core constants can then be construed to change with time, in a well-described manner, between each step of the piston cycle. The parameters established for this sequence of. engine states can then be used in a complete space-time dependent neutron and power distribution calculation for theengine under consideration.

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380 The initial neutron kinetic analysis performed with the tWCPISTt~ code made use of but a single point reactor_--.. kinetics equation. The code was later changed to accommo date up to six delayed neutron precursor groups and up to eight photoneutron precursor groups. As has been described, the delayed neutrons have been observed to have a signifi cant effect on~e nuclear piston engine behavior. The effect of photoneutron production in ~he o 2 o moderating reflector was also significant though less important than _the delayed neutron effect. This is in contrast to studies on solid core systems where the photrrneutron effect has generally been regarded to be essentially unimportant. For the studies in this work, the photoneutron precursor frac tions used for saturation fission product activity for U235 fissions in were taken from reference [52]. The product p p f y , the fraction of the photoneutron gamma rays which penetrate from the gas core to the moderating-reflector region and eventually induce photoneutrons, was taken to be 0.3. It should be mentioned that there is reasonable disagreement with regard t-0 the photoneutron parameters. Experimental results for these parameters from various sources are in conflict with one another, especially for the yield fractions [52, 53, 54]. The precise degree of the effect which photoneutrons will have on the nuclear piston en~ine behavior will therefore be unresolved until the

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381 reasons for these experimental discrepancies are determined or until experiments can be conducted on a mock-up of the nuclear piston engine itself. Flux shape changes, neutron lifetime changes, and source term weighting factor variations during the piston cycle have been examined in detail. The changes have been found to be significant and the adequacy of the point reac tor model for handling the neutron kinetics is questionable, even when cycle-averaged values for these parameters are used. NUCPISTN was therefore updated to allow for time varying neutron lifetimes, source term weighting functions, and delayed neutron effective yield fractions. The method involves an extension from~ point kinetics model to an adiabatic method for treating the system's neutron kinetics. This improvement results in a considerable increase in com putational costs since an iterative procedure is required between NUCPISTN and an independent code (e.g., CORA or EXTERMINATOR-II) which is used for obtaining the parameters of concern at various selected points throughout the cycle. Studies have been started to analyze the effects of this improvement but will require much additional funding before they can be completed. The flux shape changes, neutron lifetime variations, and source weighting factor variations which occur throughout th~ cycle have been examined thus far primarily with the

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382 one-dimensional CORA code. Some two-dimensional studies have been done but they have been restricted because of a shortage of computer funds. Future studies should be directed towards the detailed examination of these changes with two-dimensional codes (e.g., EXTERMINATOR-II) .. The use of one-dimensional space-time dependent dif fusion theory codes (e.g., WIGL2 [63] or GAKIN [64]) in .. spherical geometry or of two-dimensional space-time dependent diffusion theory codes (e.g~, TWIGL [65]) for nuclear piston engine kinetic studies will be a difficult task. Because the reactor geometry is continually changing in time, these codes would have to undergo major alterations before they could be applied to_ the neutronic analysis of the nuclear piston engine cycle. It is felt that it would be more reasonable to work first with the above proposed adiabatic scheme. Extensive analysis should be done first _with this model and the results of these investigations might then indicate whether a complete space-time dependent analysis throughout the piston cycle would be a necessary and worth while endeavor. The above described adiabatic scheme, as used in NUCPISTN, ~ill eventually be ext~nded to include the following feedback effects: a) fuel temperature feedback (Doppler broadening and gas density effects), b) moderator temperature feedback (spectral shift and density effects for no change of phase of the o 2 o),

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383 c) void feedback due to boiling of D 2 0 moderator ( d en s i t y e ff e c ts res u 1 ti n g from c h an g e of p ha s e ) , --.... and d) mass motion feedback (resulting from variations from normal bulk fluid motion in the cylinder). Neutronic Coupling Between Piston ~ngine Cores in an Engine Block The extent of the neutronic coupling between the piston cores in an engine block will have to be determined. If each of the cores in the block is surrounded by an essen tially infinite reflector, then the cores would be neutroni cally isolated. The engine block for such an arrangement would be extremely large. By reducing the total separation between the cores to the thickness of an essentially infinite reflector (~100cm for D 2 0) or to a lesser thickness, the block volume can be reduced considerably. Under such an arrangement, however, the cores will exhibit a neutronic coupling, with the level of the coupling increasing as the separating reflector thickness decreases. A rel~tively weak neutronic coupling could quite conceivably be an asset. Through proper piston timing and reflector separation dis tance, the residual flux in the block could be maintained at a level sufficient to 11 ignite 11 the cores at the start of each piston cycle. This situation would decrease the need for having a strong neutron source in each piston core

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384 to i~uce firing during each cycle of operation. A neutron source unde~ the coupled condition would then be required essentially for engine startup only. The full implications of neutronic coupling during engine operation and the degree of coupling which can be tolerated will have to be determined. Inhomogeneous problems with external boundary sources will be solved in order to evalu ate the effects of the neutronic coupling. EXTERMINATOR-II . can readily treat these types of problems. The static-flux and importante distributions, within each core, from the EX T E RM Ir l ATOR I I _c a l c u l a t i on s , c a n t h e n b e u s e d to c a l c u l a t e both coupling coefficients or coupling parameters and also time-delay terms for neutro~s that originate in one core and arrive in another core. The coupling coefficients a~d time-delay terms will then be used with the appropriate kinetics equations to arrive at a system of coupled core equations which can be incorporated into the NUCPISTN code. For example, for weak neutronic coupling, a possible formalism would involve the use of a system of point reactor equations, one for each core. The coupling could then be accounted for by intro. ducing a source term into the point-kinetics equation for each core to represent the interaction with other cores.

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385 Equation of State for the HeUF 6 ~ T h e H e U F 6 g a s m i x t u re \'Ja s a s s u med to b e a II p e r f e c t .. gas 11 for all calculations performed thus far and the ideal gas equatioh of state was utilized for all HeUF 6 calcula tions. Under the partial pressures and temperatures con sidered, the assumption is quite good for the UF 6 gas com pon~nt. The helium, however, does not behave as an ideal gas for the range of its partial pressures and temperatures. Corrections for deviations from ideal gas behavior will thus have to be made for the helium gas. Since the helium is not in the vicinity of the critical point under any fore seeable conditions, the departure from ideal behavior can be accounted for either by means of a generalized compressi bility factor utilizing pseudoreduced pressures and tempera tures or by means of the van der Waals equation of state. Fluid Flow in the Piston Engine Approximate models for bulk fluid motion in the cylin der of the engine need to be developed. Estimates ar~ necessary in order to determine effects of bulk energy transport for heat transfer calculations. Also, flow in the intake and exhaust ports should be considered to deter mine pressure drops that may be encountered. Based on these preliminary studies, a flow model for future experimental tests would then be proposed.

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Temperature Distribution and Piston Engine Heat Transfer Studies 386 The desired temperature distribution for the moderating reflector has to be established. The required heat removal rate from this region will then be determined not only by the rate of energy deposition by heat transfer from the hot core but also by the rate of energy deposition resulting from the attenuation of high energy neutrons and gamma rays. The reflector temperature exerts a major influence on the neu tronic behavior of the. core during engine operation which in turn affects the engine energetics and power output. The engine energetics and neutronics establish the energy depo sition rates in the reflector, and hence determine the cool ant rates which will be required if the moderating-reflector temperature is to be maintained at some desired level. Given the coolant rates in the reflector and the neu tronic and energetic behavior of the core, the moderating reflector and core temperature distributions could then be calculated from heat transfer studies which would proceed in the following manner: a) transport property data for heat transfer in the piston engine will be assembled. The relative importance of radiative and conductive heat transfer as well as convection of energy due to bulk fluid motion will be determined;

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387 b) the temperature distribution in the moderatingreflector will be calculated using an iterati~e approach. The required information for internal energy input-distributed volume heat sources will be estimated from previous nuclear calculations. A spatial periodic { 11 steady-state 11 assumed) tem perature solution will be sought using ippropriate boundary conditions for the conceptual design. The penetration depth and thermal phase lag as well as the amplitude excursions of cyclic thermal .. variations will be deter~ined for use in subse quent nuclear calculations. Improved nuclear calculations will then lead to improved temperature predictions. Thermal calculations for the moderating-reflector will of course have to be interfaced with those for the fluid in the core~ He-to-U Mass Ratio Studies Further investigations on the detailed effects of vari ations in the He-to-U mass ratio need to be conducted. The addition of helium means improved thermodynamic and heat transfer properti~s for the primary working fluid. It also leads to an overall flux flattening effect i~ the core which is a highly favorable situation. Since there is a minimum amount of UF 6 required for the piston engine to attain the desired fission heat release,

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388 increased He-to-U mass ratios for a fixed U mass mean increased operating pressures throughout the piston cycle. This condition results in both increased me~hanical power outputs and increased meihanical efficiencies. It is the increased pressures rather than any diluting effect on the fissionable UF 6 gas which eventu~lly limit the amouht of helium which can be added for a fixed U mass. ....... On the other hand, if the initial pressure is held fixed, increased He-to-U mass ratios mean increased mechanical efficiencies but decreased mechanical power outputs (see Figures 31 and 32). The precise degree to which the improved thermodynamic properties of the primary working fluid, under such a conditi~n, can be made to compensate for the loss of mechanical power by increased turbine power will have to be established. Finally, lower He-to-U mass ratios mean increased UF 6 partial pressures and hence higher permissible peak gas temperatures before dissociation of the UF 6 becomes a problem. ~~Reflector Addition and Removal For the four-stroke engine it was found that there was no longer a need for step-refl~ctor addition (or poison removal) at some critical point during the compression (or intake) portion of the cycle. At most, it is required that there be a step-reflector removal (or poison addition) at

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389 some desired position for shutdown during the power stroke o f t he c y c l e . I f t h e II g a s g e n er a to r II a r r a n g em e Ii t v, i t h t h e .. lower uranium loading is used, the step-reflector require ment is eliminated completely. The gas temperature and pressure for this engine peak near the end of the cycle. Piston mechanical power is low and power is extracted pri marily from the hot exhaust gases. The engine desi~n for the "gas generator" is much simpler than for the high mechanical power systems. Methods of obtaining the required subcritical-to-supercritical~to-subcritical behavior for the latter type engines were suggested at the beginning of Chapter IV. These included the use of mildly absorbing neutron sheaths or strongly absorbing poison and control rods whose motion in the moderating-reflector region could then be properly synchronized with the piston motion. These and other methods require further consideration and study to determine not only their .practicality but also their expected reliability, serviceability, and safety during extended periods of engine operation. Parametric Studies Some parametric studies were performed for the four stroke engines and the engine operating characteristics used for the reference engines in this work were selected after careful consideration of the parametric survey results. No claim is made, however, that these are the best possible

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390 operating conditions and further parametric studies need to be performed. Also, optimization studies need to be carried~~ out for these same parameters which take into account not only the piston engine behavior itself but also the charac teristics of any heat engine cycles (e.g., gas or steam turbine cycle~) which will be used in conjunction with the nuclear piston engine. Engine startup and equilibrium power level transition curves need to be generated. Some startup and power transi. ent calculations were carried out in this work. However, there are an essentially infinite number of paths which can be followed in going from shutdown to some equilibrium power level or in going from one equilibrium power level to another. The curves in this work need to be supplemented in order to better know what kind of behavior can be a~ticipated during these transients and also to establish which are the most desirable procedures or paths to follow. Moderaiing~Reflector Density or Void Coefficients of Reactivi!_y The need for more extensive moder&tinq-reflector temperature coefficient of reactivity and fuel temperature coefficient of reactivity studies was mentioned. The void coefficient of reactivity studies which were presented in this work. were for a pure o 2 o moderating-reflector region at an averaqe o 2 o temperature of 570K. More void coefficient of reactivity calculations need to be performed at

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391 other temperatures for the o 2 o and for other moderatingreflector materials as well as for composite material ~, reflector regions. Blanket Studies and Breeding Prospects Blanket studies were conducted in this work for both two-stroke and four-stroke engines to investigate potential breeding and/or power production from such regions. The blankets studied consisted of a lattice arrangement contain ing varying percentages of Th0 2 and 93% enriched uo 2 rods in o 2 o. The zero-dimensional fuel depletion code BURNUP was coupled with one-dimensional, four-group diffusion theory CORA computations for the blanket analysis. As already m~ntioned, the presented results were the best of the -in vestigated configuration. No extensive blanket optimization studies were carried out and it is ariticipated that such studies should lead to improvements over the presented de signs. Also, future studies should, if possible, make use of a two-dimensional fuel depletion and burnup code like PDQ5 [66]. Comments As already stated, the pressure, temperatures, source. strengths, dimensions and other operating conditions which haie been presented for the reference engines represent the more favorable of the investigated conditions which are

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392 within practical engineering limits. They are not, by any me a n s , t h e o p t i mu m or u 1 t i ma t e e n g i n e o p e r a t i n g c o n d i t -, o n .. The attainment of optimized conditions is one of the objects of proposed future research on the nuclear piston engine. When the piston engine operating characteristics optimization is per~ormed, full account will have to be taken of the characteristics of the portions of the power generating systems which are external to the piston e~gine itself. Gas inlet line temperature and pressure, inlet mass flow rate and loop circulation time are parameters which can easily be adjusted to effect engine control during normal operation. The neutron lifetime is determined by the ~ngine core and reflector materials, by the gas tempera ture and pressure, by the reflector temperature, and by the region dimensions. The neutron 1 ifetime in turn determines the desirable range of engine operating speeds (see Figure 1 9) . Mention should be made of the apparent conflict which exists in the dissociation data for UF 6 gas. Theoretical studies done in this country [41] and in the U.S.S.R. [67] are in reasonable agreement. Experimental investigations have been conducted thus far only at relatively low tempera tures and the experimental results in this country are not only in disagreement with theoretical data concerning UF 6 dissociation but they also are in disagreement with each other. Part of the problem may be due to the presence of

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393 impurities in the UF 6 gas since these can have a marked effect not only on the UF 6 dissociation but also on its measured physical properties. While experiments in this country generally indicate a somewhat higher degree of dis sociation than.would be anticipated from the theoretical predictions, Russian experiments [68, 69, 70] have not yet encountered any dissociation difficulties .. The Russians. have added fluorinating agents like BrF 3 and ClF 3 to the UF 6 and tl1ey claim that these additions should permit operation at even higher temperatures than the theoretical predictions would allow. Ho~efully, the gas core research program being conducted by NASA [58, 59] will provide more insight into the UF 6 dissociation problem. In the studies done to date, no account has been taken for the spatial distribution of the inhomogeneous photoneu tron sources in the moderating-reflector. An even spatial distribution has been assumed for these sources throughout the moderating-reflector region. This assumption will probably be maintained in future research until experimental mock-ups of the engine provide not only data on the spatial distribution of these sources but also more information on the general photoneutron precursor parameters. Finally, a new discovery in the gas core reactor area (presently under recommendation for a patent by the University of Florida [71]) could have significant influence

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394 on the development of the nuclear ~iston engine. Gas-fueled, h~terogeneous core reactors are currently being examin~d at the University of Florida. They offer promise for a I reduction of both size and mate~ials problems for th~ nuclear piston engine as well as promise for improved thermodynamic performance. Analytical Models for Systems External to the Piston Eng_!!l~ Thermodynamic Cyciles for the Turbines The thermodynamic cycles used for the gas and/or steam turbine units for the nuclear piston engine reference ~yst e m s pr e s e n t e d i n t h i s \'JO r k -we r e t h e b e s t o f t he i n v e s t i g a t e d cycles for each of the corresponding configurations. As with the piston engine operating conditions for the refer ence engines, the cycles presented are by no means optimized. They are, hopefully, models which give reasonable estimates for the turbine powers to be obtained from the portions of the nuc1~ar piston engine system which are external to the piston itself; Detailed studies need to be performed for the purpose of obtaining optimized thermodyn~mic cycles for the turbines with full consideration for the piston engin~ operating characteristics and HeUF 6 ex~aust gas conditiohs. The optimization of these cycles in conjunction with the piiton engine operating conditions is therefore also a part of' future proposed research.

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Turbine Loop Energetics, Heat Transfer, and Fluid Mechanics Studies 395 The various turbine, coolant, and fuel-makeup and purification loops of the system and. their associated com ponents are coupled with the neutron equations through the temperature and pressure responses to the neutron level changes. These loops are in turn coupled to each other through the heat exchanger (and/or steam generator, if a~y) and system pressure and temperature variations. Complete energetics, heat transfer, and fluid mechanics studies for these loops and of the coupling between them hav~ not yet been carried out and are a part of future proposed research. It should be pointed out that the analysis of any steam turbine loops and of their associated He-to-H 2 o steam generators will benefit greatly from the technology -developed by Gulf General Atomic for the HTGR. The possible turbine cycles are not limited to those presented in this ~rnrk. As suggested under the 11 Applications 11 section of this chapter many other possible binary, trifluid, or multifluid cycles are possible. The work to be done on the turbine cycles will involve studying these possible cycles by optimizing or balancing, for example, steam versus gas turbine power so as to obtain the maximum possible power per piston from the external loop without incurring counter balancing economic penalties for going to loop arrangements. which are too elaborate.

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396 The use of the HeUF 6 exhaust gas directly in a gas turbine would lead to much higher system efficiencies be cause available energy losses due to temperaturedrops across the intermediate heat exchangerswould be avoided. Although some reports have given serious consideration to the direct use of HeUF 6 or of UF 6 in gas turbines, the author feels that the UF 6 is too corrosive and too hot (radioac-. tively) and that turbine tolerances are too fine for this to ever be practical. It is difficult to conceive of a UF 6 gas turbine design which could give safe and reliable per formance over an extended period of operation. HeUE._ 6 -to-He Heat Exchanger Studies The HeUF 6 -to-He heat exchanger is an engineering problem w h i c 11 w i l l re q u i re e x t e n s i v e h ea t t r a n s f e r a n d fl u i d me c h a n i c s studi~s, as well as economic studies. A preliminary analysis for a HeUF 6 -to-He heat exchanger was performed for the 54MWe piston-steam turbine system of Chapter IV. This analysis was performed primarily for the purpose of demonstrating .pr a c t i ca l i t y , i. e . , to s how th a t the He U F 6 proper t i e s are such that a HeUF 6 -to-He heat exchanger would not be a pro hibitively large and costly component. It is expected that reductions in size from this preliminary analysis can be readily achieved from more thorough investigations.

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397 Comments The overall optimization of a nuclear piston engine power generating system will invol.ve simultaneous adjust ments in three interdependent areas. The nuclear piston engine operating conditions will hav~ to be adjusted dif ferently according to the external loop power system. On the oth~r hand, any gas and/or steam turbine thermodynamic cycles must be optimized so as to take best advantage of the fundamental nuclear piston engine operating character istics. The adjustments .which are made in both these areas must at the same time be made in a manner so that resultant cost increases do not overshadow any engine performance im ~rovements. Thus, the final optimization will involve not only neutronics and energetics, but also economic considera tions. An economic model, as outlined in the research pro posals which have been submitted for future nuclear piston engine studies, is discussed below. Economic Model for the Nuclear Piston Engine Power Generating Syste~ Fixed Charges (Capital and Cost Related Charges) Fixed charges can be determined by means of the ORCOST [72] and CONCEPT [73] computer codes. ORCOST is a quick, simple program for estimating direct and indirect capital costs of nuclear and fossil-fueled steam-electric

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398 power plants. The CONCEPT computer package also provides capital cost estimates for riuclear and fossil-fueled plani;: It is a much more elaborate code than ORCOST both in tetms of required input and printed output. Cost estimates can be made as afunction of plant type, size, location, and date of operation. A detailed breakdown of the cost estimate into direct and indirect costs is given according to the USAEC accounting system. CONCEPT makes use of extensive files containing both cost model data for the various types of plants and historical data for labor and material costs for various locations which can be continuously updated and modified. The detailed capital cost estimates include costs for land and land rights, structure and facility costs, reactor plant equipment costs, turbine plant equipment costs, electric plant equipment costs, costs for miscellaneous plant equipment and special materials, construction facilities equipment and service costs, engineering and construction management service costs, interest during construction costs, and miscellaneous other costs. Fuel Cycle Costs Engineering economy predictions of the fuel cycle costs can be made with the CINCAS [74] nuclear fuel cycle cost code. The fuel cycle cost will be reduced and the cal cul~tion of such costs simplified by the lack of fuel

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399 fabrication costs and the lack of transportation costs both to a n d f r om t he fa b r i c a to r a n d t o a n d f r om t he re p r o c es s o r-: ... Fuel cycle costs incurred are normally assigned to ' cost categories or accounts. The CINCAS code uses six direct or expense cost categories and four inventory cost categories: (a), uranium expense, {b) plutonium credit, (c) fabrication e~pense, (d) shipping expense, (e) repro~ cessing or clean-up expense, (f) reconversion expense, (g) uranium inventory cost, (h) plutonium inventory cost, (i) fabrication inventory cost, and {j) post-irradiation inven~ tory cost. The fabrication expense, reconversion expense, fabrication inventory cost,and post-irradiation inventory cost for the gaseous core nuclear piston engine will all be eliminated. The shipping expense cost will be greatly re-. duced since the only transportation co~ts will be for ship ment rif the UF 6 from the gaseous diffusion plant and for ship ment of wastes for disposal. Finally, plutonium related costs for any highly u 235 -enriched fuel should be but of rela tively minor importance. Power Production Costs An overall ,economic analysis for the nuclear piston engine power generating system will have to provide total power production costs including the fixed charges (capital and cost related charges), fuel cycle costs, and operation and maintenance costs. The POWERCO [75] computer code for

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calculating the cost of electricity produced by nuclear power stations can be employed for this purpose. Comments 400 Th~ lack of fuel cladding and the continuous removal of fission products will mean extremely high fuel burnups for the gaseous core nuclear piston engine, High burnups, lack of fuel fabrication costs, and lack of transportation costs to and from the fabricator and to and from the re processor will all contribute towards low fuel cycle costs. Fuel cost estimates for a nuclear piston engine power gen erating syste~ are presently around 1 .4 rnills/kwe-hr. Most large capacity nuclear plants have fuel costs around 2.5 mills/kwe-hr while fuel costs for large fossil fuel plants are generally between 3.5 and 5.0 mills/kwe/hr. Fuel costs for fossil-fueled peaking units range anywhere from around 7 to as high as 27 mills/kwe-hr.* Increased overall efficiency and ~ower per piston will eventually be obtained only by going to more costly and elaborate system setups. The energetic and neutronic opti mization can provide only limit~d increases in efficiencies and power outputs at fixed system costs. Once this point is reached, a balance will have to be struck beh1een efficiency *Again, these costs do not consider the recent large in~ creases inthe cost of oil, v1hich will severely affect many of the energy production costs. Also, these costs do not consider the recent large increases in the cost of u 3 o 8 .

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4 01 and power output gains versus the increased costs of more complex system setups. Safety Analysis and Methods of Control . .... Important and essential phases of the nuclear piston engine system will center around methods of control and safety analysis. A preliminary safety analysis as well as methods of load following, control and shutdown will be integrated into the technical feasibility demonstration program. A complete ~afety analysis would naturally follow an actual demonstration of technical feasibility and precede the construction of a nuclear piston engine test facility. Providing means for adequate cooling for the piston reflector and shielding and for containment of the HeUF 6 gas under all normal or abnormal conditions will be two of the intermediate and primary safety concerns. Emergency 11 c o r e II c o o l i n g f a c i l i t i e s w i l l , o f c o u r s e• , b e u n n e c e s s a r y for the gaseous core piston engines. Pfoblems of radiation containment will be restricted to the engines themselves and the HeUF 6 working fluid loop. Any steam turbine loops will be isolated from the HeUF 6 loop by a He buffer loop; For the gas turbine i oops, the inert helium with essentially .zero absorption cross section for neutron~ is i-tself a sufficient buffer between the radioac tive HeUF 6 gas and any gas turbine power generatinq com ponents. Another safety-related interest will be in the

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methods of handling and disposing of the wastes from the HeUF 6 clean-up system. 402 Since the HeUF 6 piston inlet temperature and pressure are rather sensitive mechanisms for control, load following for the nuclear piston engine system should be readily achieved through the HeUF 6 -to-He heat exchanger. Other methods of obtaining control under normal operation would involve varying the inlet mass flow rate, the loop circula tion time, the He-to-UF 6 ratio in the HeUF 6 gas mixture or varying the fission product poisoning concentration in the gas by adjusting the rate of the gas feed-and-bleed process. Th f k f . u 235 . h ld e use o ma e-up gas o varying. . enr,c ments cou provide another means for long-term control. Rather large negative moderator temperature coeffi cients of reactivity (see Table 43) have been calculated for the reflector-moderated engines (under a no change of phase condition). This is obviously a favorable situation from a safety and control standpoint. Both normal and emergency control can be achieved by the use of neutron absorbing material in the important moderating-reflector region. Total control system worths exceeding 10% 6k/k have been easily attained in this manner for experimental gaseous core reactors [76]. The primary emergency shutdown mechanisms will probably depend on boiling and voiding in the o 2 o reflector and on pressure relief valves in the core.

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403 The o 2 o in the reflector will be pressurized during n or ma 1 o p e r a t i on . \.I h e n t he g a s p r e s s u r e a n d / o r t em per a t u re ... in the core exceed some critical limit, the o 2 o reflector will be depressurized by means of a relatively conventional relief valve-quenching tank system, Upon depressurization the o 2 o will undergo boiling and voiding. These processes seriously disrupt the neutron therma1ization process in the controlling external reflector and bring about rapid and effective shutdown. Core pressure relief valves would also be triggered if the core gas pressure exceeded some predetermined 1eve1. The rapid escape of the gas from the core and resulting d u 235 d t 1d 1t . . 11 . rop ,n ens, y wou resu 1n an essent1a y 1nstantaneous shutdown. An accidental locking or freezing of the piston could also be designed to trigger a reactor protection system action to safely shutdown the core. Such a protection system mi~ht include any or all of the above described mechanisms. Such action would also. be initiated under conditions of normal piston movement should the flux exceed certain preset limits. These protection system actio~s would be assisted by the large negative moderator temperature coefficient in attaining rapid shutdown.

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APPENDICES

PAGE 443

APPENDIX A TWO-GROUP, TWO-REGION, ONE-DIMENSIONAL DIFFUSION THEORY EQUATIONS USED IN THE NUCPISTN CODE WHEN PHOTONEUTRONS ARE IGNORED The general two-group, two-region, time independent, one-dimensional diffusion the~ry equations may be written as + fd I f qi f = 0 C r C C If
PAGE 444

The macroscopic fast removal cross section in the core _region is given by {J\-5) 406 while the fraction of the fast removal cross section for the core which is due to downscattering is given by Combining equations (A-5) and (A-6) yields fd E f = C r C (A-6) (A-7) The macroscopic fast removal cross section for the moderating-reflector region, Er:' and the fraction of the fast r~moval cross section for the moderating-reflector region which is due to downscattering, f:, are defined in an analogous manner. The boundary conditions to be applied for solution of equations (A-1) through (A-4) are (A-8) ( A9)

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407 q/ ( R) = q/ ( R) (A-10) m C ..... cp~(R) = cp ( R) ( A11 ) cp f ( 0 ) i s finite (A-12) C cpt(O) is finite (A-13) C [Dtvcpt] = [Dtvcpt] (A-14) c c r=R m m r=R [Df f] = [Dfvcpf] (A-15) cvcpc r=R m m r=R where R is the core radius or position at the core-reflector '\, interface and R is the extrapolated reactor radius or the position at the outer (extrapolated) edge of the moderator reflector region. For the nuclear piston engine gas cores, there are relatively few fast group interactions in the core. Replac ing the f~st core equation, (A-1 ), with a boundary condition is therefore a reasonable approximation and the resulting set of equations requiresa significantly shorter time for solu tion than the above system of equations. The boundary con dition which replaces (A-1) is. t R t ( E ) f (r)dV V f C 't'C 0

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408 and this condition implies no fast interactions at all in the core (i.e., no absorptions, no fissions, and no slowin~'~ down}. Note that if we can obtain by independent means a fast fission factor, E, defined as E = total productions = thermal productions (A-17} then the fast fissions in the core can be accounted for. Also, if we can obtain by independent means a core fast absorption factor, s, defined as s = total core absorptions thermal core absorptions = (A-18} then we can approximately account for the (few) fast ab so r pt i on i n t er act i on s w h i c h o--c c u r i n th e core . i~ h en the E and s factors are greater than unity, condition (A-1~} still simpl ies and E f = E f_ re ac E ands will be no slowing down in the core, i.e., fd = 0 C Procedures used for obtaining values for discussed at the end of this appendix. The two-group, two-region, ti~e independent one dimensional diffusion theory equations are now given by

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409 (A-J9) ..... (A-20) (A-21) while the boundary conditions to be applied for solution of equations (A-19) through (A-21) are f I\, 0 cp m ( R) = cpt(R) = 0 m cpt(R) = cpt(R) . m c. . t R t (vEf)cfo cpc(r)dV 4nR 2 (A-22) (A-23) (A-24) (A-25) (A-26) (A-27)

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410 The above system of equations will now be solved for. the case where one has spherical geometry. The solution fo~ t(r) is given by C (A-28) where use has been made of boundary condition (A-26), C is a constant to be determined, and The solution for ~f( ) is 't'm r f f '\, A -K rQ 2K (r-R}-J = -e m l e m r (A-29) (A-30) where use has been made of boundary condition (A-22), A is a constant to be determined, and f/Df r Ill m t The expression for m(r) is t ( r) m (A-31) t '\, r, 2K (r-R)J L e m (A-32)

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411 where use has been -made of boundary condition (A-23), Bis a constant to be determined, and (A-33) t t . In the limit, as I and K go to zero, for the case of zero a m m or negligible thermal absorption in the moderator-reflector, it can be shown that equation (A-32) becomes (A-34) Using the solutions (A-28), (A-30), and (A-32) for ~(r), f(r), and 1 (r) respectively along with the boundary crinm Ill ditions (A-24), (A-25), and (A-27) leads to three homogeneous e q u a t i o n s i n v o l v i n g t he u n k n own c o n s t a n t s A , B , a n d C . (A-35) (A-36) (A-37) The coefficients K 1 through K 9 in the above equations are defined as follows:

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K = 0 8 . e m 2Kf (R-R)] 412 (A-39) (A-40) (A-42) (A-43) ( A4 4 )(A-45) (A-46)

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413 For the case of zero or negligible thermal absorption in t~e moderator-reflector K 1 , K 2 , K 4 , and K 5 become (A-47) l l K = (-R ~) 2 R (A-4g) (. f~ 2Kf (R-RJ l [. 2Kf (R-RJ ) K l+e m + 1-e m m R . (A-4g) (A-50) The remaining K coefficients, i.e., K 3 , K 6 , K 7 , Kg, and Kg, are unchanged. In order for equations (A-35) through (A-37) to have a non-trivial solution, it is required, that the determinant of the K coefficients be zero. The 11 c r i t i ca l " determinant is thus given by Kl K2 K3 K4 K5 K6 = 0 (A-51) K7 Kg Kg

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Solving for the value of the number of neutrons per fission required for criticality yields The effective static neutron multiplication factor for the system can then be obtained from (A-53) or using equation (A-52) gives (A-54) where v in the above equation has a value of 2.43. 414 For the case of zero or negligib.le thermal absorptio~ in the moderator-reflector, equation (A-54) can be shown to be equivalent to

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415 When keff < l, the equilibrium or steady-state flux magnitudes are determined by the inhomogeneous source(i). -~ A fast inhomogeneous neutron source of constant strength is assumed to ejist in the core and emit S neutrons per 0 second. For the gas cores of concern, the source strength per unit volume can be approximated throughout the core region as (S 0 /rrR 3 ). This is true even \'Jhen S 0 is. a single point source or.an unsymmetrical distribution of point sources in the core because of the relatively small number of fast group interactions which take place in the core. Consequently, adding the core fast source term as a uniformly distributed source at the core-reflector inter face is not an unreasonable approach for the particular type of gas cores being considered. Such a source distri bution greatly facilitates solution of the inhomogeneous system of equations. Thus, if boundary condition (A-27) is replaced by 11/ Vcp f -1 Lm m . r=R (A-56) then the inhomogeneous problem includes , 0 quations (A-19) through (A-21) and boundary ~onditions (A-22) through (A-26) as -well as boundary condition (A-56). For the inhomogeneous problem, the equations corresponding to (A-35) through (A-37) are

PAGE 454

416 K 1 A + K 2 B + K 3 C = 0 (A-57) K 4 A + K 5 B + K 6 C = 0 (A-58) K 9 C 2 (A-59) K 7 A + K 8 B + = (S /4TTR ). 0 These equations can then be solved for the constants A, B, and C to yield the following results: (A-60) (/\-61) (A-62) .... , The average steady-state thermal flux in the core is obtained from (/\-63) Substit~tion of (A-28) into this express{6n yields

PAGE 455

417
PAGE 456

418 . The average steady-state thermal flux in the moderator reflector is obtained by solving -t

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The average steady-state fast flux in the moderator reflector is obtained from '\, 419 cii 1 ~ = f RR qi~ (r ) d V (A-71) -ny 3 g Substitution of (A-30) into the above gives (A-72) .....

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420 Comments -1 t should be recognized that the effective neutron ... mult1plication factor for the system is changing throughout the piston cycle primarily as the result of two simultaneous processes .. First, the core densities and hence the macroscopic neutron cross sections for the core are c.on tinually changing during the piston cycle. Second, the . reactor geometry and h~nce t~e neutron leakages are also constantly varying. For the NUCPISTN point reactor kinetics calculations, the relation p = [(keff-l)fkeff] isused where keff is taken to be the effective, static neutron multiplication factor obtained from two-group, two-region, time independent, one-dimensional diffusion theory calculations. For a dis cuss i on on t_h e a ppr op r i ate n es s of the use of a II static 11 neutron multiplication factor, of the possible use of a 11 d y n am i c II n eu tr o n mu l t i p l i ca t i o n fa c t o r a n d o t he r p e rt i n e n t refinements, the reader is referred to Appendix C. The evaluation of the point reactor kinetics equations in the NUCPISTN code generally involves on the order of five or six thousand time steps for each individual piston cycle. Solution of the complete two-group, two-r~gion diffusion theory equations at each time step for the effective neutron multiplication factor (and/or for the 11 steady-state 11 flux diitributions) requires a large amount of computer time. Although standard diffusion theory codes like CORA [44]

PAGE 459

421 were found to consume less than 0.01 of a minute of IBM 370 computer time for such a calculation, the repetition of this process, at each time step, would run the 370 computer time for the analysis of each piston cycle to around 20 minutes. By replacing the fast core equation with a boundary condition, the diffusion theory equations are simplified significantly arid the time required for the complete analysis of a piston cycle by the NUCPISTN code is reduced to around 0.30 to a minute on the IBM 370 computer~ The 11 independent 11 calculations referred to at the be ginning of this appendix for obtaining the fast fission and fast absorption factors were two-group, two-region, time independent diffusion theory calculations in which the tore fast group equation was not replaced with a boundary condi tion. Of the order of 10 to 15 positions during the piston cycle were selected for analysis by the CORA few-group dif fusion theory code. The quantities and c as given by (A-17) and (A-18) were calculated from the CORA results for each of the s~lected positions. The corrections entailed by these parameters were small and the ~ararneters were found to change very little over the piston cycle. Hence, the quantities E and c were taken as being constant over the piston .cycle and the values input into NUCPISTN were average values obtained from the 10 or 15 evaluated positions.

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APPENDIX B TWO-GROUP, TWO-REGION, ONE~DIMENSIONAL DIFFUSION THEORY EQUATIONS USED IN THE NUCPISTN CODE WHEN PHOTONEUTRONS ARE INCLUDED When the external moderator-reflector region contains. large amounts of beryllium or heavy water, photoneutron production from the (y, n) reaction can occur in the Be or D 2 0 to a significant degree. The (y, n) threshold in heavy water is 2.3MeV and 1.63 MeV for beryllium. Certain fission products (photoneutron precursors) decay in the core with the emission of energetic y-rays. A fraction, fP, of these gamma rays penetrate from the core to the moderator-reflector with energy above the threshold level. A fraction, yP, of these energetic gamma rays which reach the moderator-reflector then induce photoneutrons through the (y, n) reaction. This photoneutron production in the moderator-reflector analytically necessitates the introduction of an inhomogeneous, fast neutron source term for this region into the neutron balance equations. The two-group, two-region, time independent, one-dimensional diffusioo theory equations may thus be written as 422

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423 ot,;lctit I tcpt i;; = 0 C C ac c ( B1 ) of ~lctif I fcpf + sf = 0 m m rm m m (B-2) 0 t 9 2
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424 The presence of the -inhomogeneous, fast neutron source term, S~, in the reflector does not affect the neutron multiplication factor, keff' which is still obtained from (A-54) or (A-55). The relations for the steady-state flux ..... distributions are, however, changed by the presence of this source term. The above system of equations will be solved for the case where one has spherical geometry. The solution for ~(r) is given by (B-10) where use has been made of boundary condition (B-9), C is a constant to be determined, and ( 8-11) It will be noted that the solution for tis of the same C form as for the case when photoneutrons were neglected. The constant C, however, is no longer given by (A-60). As will presently be seen, the expression for evaluating the constant, The -f C, now contains terms dependent upon S . Ill . f f . expression or m 1s (B-12)

PAGE 463

425 where use has been made of boundary condition (B~5), A is a constant to be determined, and The expression for ~t is m -Ktr . t '\, . Be m 2K (r-R) ] + --l e m r -........ (B-13) (B-14) where use has been made of boundary condition (B-6), Bis a constant to be determined and ( 8-15)

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426 When keff < l, the equilibrium or steady-state flux magnitudes are determined by the inhomogeneous sources -which now include the fast neutron source in the core, S , . 0 and the fast photoneutron source in the moderator-reflector, ~f. Using the solutions (B-10), (B-12), and (B-14) for m ~, ~, and~ respectively along with the boundary conditions (B-4), (B-7), and (B-8) leads to three equations in volving the unknown constants A, B, and C. (B-16) (B-17) (B-18) The coefficients K 1 through K 9 are given by (A-38) through (A-46). The remaining coefficients are given by the following relations: ( B-19)

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427 f '\, ) t 2 K (R-R) f l ( K ) e 111 . ( K -) m rn R (B-20) (B-21) Solving equations (B-16) through (B-18) for the constants A, B, and C yields C = ( K 2 K 4 -K 1 K 5 )(Kl)( S/4nR 2 ) + K 2 K 7 (K 4 K 10 -K 1 K 11 ) + ( K 2 K 4 .-K 1 K 5 ) ( Kl K 12 -K 7 K 10 ) {K1K9-K3K7)(K2K4-K1K5) + K2K7(K3K4-K1K5) B = C(Kl1<6-K3K4) + (K4KlO-KlKll) (K2K4-K1K5) A= C(K2K6-K3K5) + {K5Kl0-K2Kll) ( Kl K5-K4 K2) (B-22) (B-23) (B-24) The average steady-state thermal flux in the core is obtained by substitution of (B-10) into (A-63). The resulting expres sion is

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428 (B-25) where C i•s given by (B-22). The average steady-state thermal flux in the reflector is obtained by substitution. of (B-14) into (A-67). The re sult of this substitution yields the relation ( 'v. Kf (i<-R) ) 2R em ? . m -KtR 3Be 111 Yg ~ 'v t 'v ] 2 R K {R-R) em + Kt m . l'tR t 'v -,m ( 2K (R-RJ + 3Be R Kt l+e m + (Kt)2 m _ Yg m l 2Kt (R-R'.) J ) 1-e m R f 'v . [ f [ 'v K (R-R)) K R-Re m I n J

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429 [ Kf(R-R))J[Kt 1 2 1-e m I m J lf . K m t I\., [i t[ I\., K (R-R) 1 + K -R+Re m I m J t I\., ( ~l(R-Rh+ l 1-e J 2 Kf m Kt Ill (B-26) where Band A ire given by (B-23) and (B-24) respectively and the geometry factor, Yg, is given by (A-69). The average ,steady-state fast flux int he reflector is obtained by substitution of (B-12) into (A-71). The equation resulting from this substitution is given by -KfR r f I\., 3Ae 111 R ( f cKm (R-RJ + f 2 K l+e ( K ) m Ill yg l r, 2Kf (R-RJl ) .+ Rll-e m . J (B-27) ......... Thus far, no expression has been given for the fast, inhomogeneous ph~toneutron source term, ~f, in the p Ill moderator-reflector. If [.(t) is the average photoneutron J

PAGE 468

430 .precursor concentration in the core at time t for precursor group j and if A~ is the corresponding photoneutron preJ cursor decay constant, then the photoneutron precursor decay rate per unit volume in the core at time t is . p where J is the number of photoneutron precursor groups. This is also equal to the rate of emission of energetic gamma rays per unit volume by the decaying precursors in the core. The gamma ray emission rate in the core from the decaying photoneutron precursors at time t is then A~ C~ ( t ) J V ( t ) J J C where V (t) is the core volume at time t. C The number of these gamma rays penetrating from the core to the reflector per unit time and with energy above the (y, n) threshold at time tis The number of fast photoneutrons produced in the reflector per unit time is therefore

PAGE 469

431 [ JP J p -P L'. ),_. C.(t) j = l J J Division of the above quantity by the reflector vol ume, Vm' gives the average number of fast photoneutroris produced per unit volume per unit time in the reflector at time t. This is in fact the definition of the source term, ~f(t), which is thus given by m . (B-28) \'Jhere the factor \(t) is given by (B-29) Note that in the above derivatio~ it has been assumed that the photoneutron production is uniform throughout the moderating-reflector region. It has also been assumed p p that f and y are tim~ independent factors. It has been seen that ~f(t) is the number of fast m photoneutrons produced per unit time per unit volume in the reflector at time t. Let n~(t) be the average thermal ne 1 utrbn density in the core at time t and let ~f(t) be the Ill average fast neutron density in the moderating-reflector at time t. If the ratio [nt(t)/nf(t) does not change sigc m nificantly over a neutron 1 ifetime~ then the number of

PAGE 470

432 thermal neutrons being generated per unit time and per unit volume in the core as a result of photoneutron release in-~ the moderating-reflector is (B-30) Td is the delay time which exists between the genera tion of a fast photoneutron in the moderator-reflector and its appearance as a thermal neutron in the core. To a good approximation, this delay time is equal to the neutron life time of the system so that (B-31) . If the parameter op(t) is defined as

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433 a~d if the parameter ap(t) is defined as (B-33) and if equation (B-28) is used in equation (B-31 ), it can be -P . shown that the sourc~ term S (t+l) is given by C p a (t)f (t). V (B-34) Note that no acco~nt has been taken in the above develop ment for the time required f6r they-rays in the core to travel to the moderating-reflector and to induce photoneu trons. The process may be considered to be instantaneous since the time steps used in the point kinetics equations in NUCPISTN are generally no smaller than a tenth of a neutron lifetime. The neutron lifetime itself is of the order of a millisecond.

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APPENDIX C GENERAL POINT REACTOR KINETICS EQUATIONS Complete derivations of the general equations fnr the time dependent behavior of a (point) reactor can readily be found in the literature and will not be reproduced here. The final point reactor kinetics equations, however, will be list~d along with expressions for the various parameters which appear in these equations. Some of these expressions are subsequently modified by taking into account properties of the gas core reactors of ~oncern and these results are then used in the next appendix. The notation utilized throughout this appendix generally follows that of Bell and G1asstone [77]. The equations describing the kinetic behavior of a (point) reactor may be written as dP (t) dt D . de. ( t) J -dtD J D D P(t) + E A.c.(t) + Q(t) j=l J J . (C-1) D 1 , 2, . . . .J (C-2) 434

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435 -D -D D( The parameters p(t), Sj(t), a (t), A(t), cj t), and Q(t) are defi~ed below. JD is the nOmber of delayed n~u-~ tron precursor groups and A~ is the delayed neutron pre cursor decay constant for group j. The quantity p(t) is given by + ++ + + -;tx ljJ(r, Q', E', t) cp 0 (r, O, E) dVdQdEd1l'dE' (C-3) where the zero subscript indicates values in the time independent (critical~ reference state and the summation overxtf is for all non-fission interactions from which neutrons emerge. The 6 1 s represent differences between the respective quantities, Ef and E, in the time varying stati and in the time-independent (critical) reference state, e.g., 6E=(E-E 0 ). The remaining parameters in (C~l) and (C-2) are given by -:--D l D + +~ a. ( t) = -F I . .. I X. ( E) a . v~f ( r' EI ' t) 1/J ( r' II I ' EI ' t) J ' J J ( C-4)

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436 (C-5) (C-6) 1 J D+ ++ + + = AF ffx.(E) C.(r, t) (r, n, E)dVdndE ii J J 0 ( C-7) Q(t) 1 + + ++ + + = AF J fl S ( r , r2 , E, t ) O ( r , r2 , E ) d V dnd E {C-8) The factor Fin the preceding definitions is usually defined as the volume integral of the adjoint weighted fission .source, i.e., F(t) + x dO'dE' (C-9) and can .thus be interpreted as a production operator. It should be realized that the set of definitions for the above parameters is to some extent arbitrary. The vari ous parameters are, however, related so that they must be defined consistently. Once a consistent choice has been ma d e, t h e i n d i v id u a 1 pa ram e t e rs a r e q u i t e d e f i n i t e a 1 t h o u g h a simple physical interpretation for these parameters may not always be possible.

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437 Thus far, no mention has been made of the term, P(t), appearing in equations (C-1) and (C-2). P(t) is genera11;-, termed the amplitude factor or the amplitude function and as will be seen, it can be normalized independently in any convenient manner. It is related to, the angular neutron flux by + + cp(r, n, E, t) + = P(t). i(r, n, E, t) (C-10) + \there the quantity i(r, n, E, t) is termed the shape factor or shape function. For the case when 1jJ is independent of time, then equations (C-1) and (C-2) can be properly re ferred to as 11 point 11 reactor kinetics equations. The shape function itself is often written as the product of two components, i.e., {C-11) The angular neutron flux can thus be written as + + + + + f(r, E, n, t) = v n(r, E, n, t) = P{t)[~(r, n, E, t) + + where. n(r, E, n, t) is the angular neutron density.

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438 The total neutron population is given by . N(t) = ff jn(1, E, n, t)dVdEdri .. {C-13) By means of equation {C-12), the total neutron populatiDn can be expressed as 1,J(t) = fff[~] jJr, E, ~, t)dVdDdE r11 -i= P ( t ) ff f l v J 1/J ( r , E , Q , t ) d V d Qd E = P ( t )f ff n s ( r , r, , E , t ) d V d Qd E. ( C l 4 ) In writing the angular neutron flux inthe form of {C-10), the intent isthat the amplitude fact~r, P(t), should describe most of the time dependence whereas the shape factor, 1/J, should change very little with time. It has already been stated that the amplitude factor may be normalized in any convenient manner. The shape factor is normalized so that (C-15) The purpose of the above normalization is to satisfy the requirement that

PAGE 477

439 _ 3P(t) l + + + + + .+ at [JJJ(-) (r, ~, E)~(r, a, E, t)dVJndE]. V 0 (C-16) If the amplitude fa~tor is normalized to be the neu tron population at time t 0 , then (C-17) The use of (C-17) in equation (C-14) then yields + + + l + + JJJ n (r, a, E, t }dVd~dE = JJJ(-}~(r, a, E, t }dV~ldE = l. S O V 0 (C-18) This relation fixes the normalization of the shape function, ~' at time t=t 0 and from (C-15), the normalization on~ is then determined at all other times t. It will be noted that with the above normalization,~ has units of (l/cm 2 -sec) while n 5 has units of (l/cm 3 ). In addition, the amplitude function, P(t), will approximately represent the neutron population at any other time t provided the shape function does not change significantly. If the amplitude function is normalized to be the average neutron density of the system at time t 0 , tl1en (C-19) The use of (C-19) in equation (C-14) then yields

PAGE 478

440 V(t ). 0 The above relation fixes the normalization of the shape function, l/J, at time t=t and from (C-15), the normaliza o tion ori l/J is then determined at all other times. With the above normalization, l/J now has units of (cm/sec) while ns is dimensionless. The amplitude function, P(t),will ap proximately represent the average neutron density of the system at any other time t provided the flux shape does not change significantly. If the amplit~de f~nction is normalized to be the average neutron density in .the core of a two-region reac tor at time t 0 , then (C-21) Use of (C-21) in equation (C-14) then gives the expression + + 7 JJJ ns(r, E, n, t )dVd11dE system 0 (C-22) ...... If the integration is carried out just over the core region, it follows that

PAGE 479

441 (C-23) Noting that t (t )V (t ) + n (t )V (t 0 )] c o c o m o m = n ( t ) . C O (C-24) the result that 1 ff!. n (r, E, n, t )dVd;~E = ff! (-) (r, a, E, to)dVdndE mod/refl s O mod/refl v (C-25) can be obtained from (C-22) if use is made of (C-23). Thus, either relation (C-22) or relations (C-23) and (C-25) fix the normalization of the shape function~ at time t=t 0 and from (C-15), the normalization on~ is then determined at all other times. The amplitude function,

PAGE 480

442 P(t), will approximately represent the average neutron density in core at any other time t, provided the shape function for the system does not ~hange significantly. Or alternatively, it can be said that the amplitude func tion, P(t), will approximately represent the average neu...... tron density in the core at any other time t, provided the shape function in the core does not change significantly and provided the ratio (nm/nc) does not change significantly. For the case of a one-speed, two-region ~ystem in which the angular dependence is ignored, equation (C-22) becomes f n Cr, t )dV = systems 0 f (l) (r,.to)dV = system v [N(t )/n (t 0 )] 0 C (C-26) and equations (C-23) and (C-25) become f n (r, t )dV cores 0 (C-27) J n (1, t 0 )dV mod/refl s = 1 (1) c;. t )dv = mod/refl v 0 [n (t )/n (t )Jv (t ) . m o c o m o (C-28) Some particular forms of the parameters (C-3) through (C-9) for 11 one-speed systems" in which the angular de pendence is ignored and in which the amplitude function is normalized to be the average neutron density in the core will now be considered.

PAGE 481

443 For a one-region reactor, (~-9) becomes F ( t) -+ + -+ = [vEf(t)J / \/J(r, t) 0 (r)dV system (C-29) while for a two-region reactor, (C-9) is F(t) (C-30) For a one-region reactor, (C-6) becomes with the help of (C-29) A ( t) = ( l / V) (-+ ) + (-+) d ( l / V) l\ . F ( t) / 1jJ r, t
PAGE 482

444 in equation (C-33) then gives (C-35) From the definition of the neutron generation time, however, J\(t)' = [dt)/k(t)J , (C-36) Combining (C-35) and {C-36) gives for a one-speed, one region system, the following relation for the neutron lifetime Q, ( t) (C-37) For a two-region reactor; (C-6) becomes with the help of (C-30) J\ ( t) = J l / V ) . f ljJ (r ,' t ) + (r ) d V F system 0 ((38)

PAGE 483

445 If [k J and [1 J are defined in the following 00 00 C C manner, (C-39) [1 00 ] (1/v)/[tra] C C (C-40) th~n the following expression follows directly from these definitions [1 00 ] /[k 00 ] = (1/v)/[vrf] . C C C (C-41) Utilizing this in {C-38) yields A(t) = [1 (t)J I 1/JCr, t) 4 /(r)dv 00 c system 0 (C-42) [k (t)] 00 C Rewriting the above then gives J\(t) = [1 (t)] 00 C [k (t)J 00 C l + + + + f ljJ(r, t) (r)dV mod/refl 0 + + + f ~,(r, t) (r)dV core 0 {C-43) ..... Making use of the definition of the neutron generation time, i.e., equation (C-36), gives for the neutron lifetime of a bne-speed, two-region system

PAGE 484

446 k(t) = [k (t)] 00 C + + + ( J ijJ(r, t) (r)dV ) [iaJt)J 1 + mod/refl+ +: . c J ljJ(r, t)cp (r)dV core 0 From (C-4), the effective delayed neutron fraction becomes for a one-region system, with the help of (C-29), -D l D + + + D rl-.(t) = F[\!t:f(t)] J S.1/J(r, t) (r)dV = s .. J system J O J (C-45) While from (C-4), the effective delayed neutron fraction becomes for a two-region system, with the help of (C-30), s~(t) J D B .. J (C-46) The source term, given by (C-8)~ becomes for a one-region system + + + Q( t) = f S(r, t) (r)dV system 0 AF + ++ J S(r, t) (r)dV = system 0 I\[\! f ( t ) ] f ;p (; , . system + + t) (r)dV 0 where use has been made of (C-29). (C-47)

PAGE 485

447 By utilizing the one-speed equivalent of (C-11) in which the angular dependence is neglected, i.e., --+ ljJ(r, t) = n (r, t) V s the above can be put in the form Q(t) = jsys tern S (;;, t ): (r)dV A[vEf(t)] J n (r, t)cp+(f)ctV -(1/v) systems 0 Making use of (C-31) gives Q(t) -+ + -+ J ~(r, 0 (r)dV = system . J n (~, t)cp+(r)dV systems 0 (C-48) (C-49) (C-50) For the one-speed, one-region gaseous core _systemi cif in+ -+ terest, cp 0 (r) c6nstant and hence J S(r, t)dV -S-(t)V (t) Q ( t) ~sy~s_t_em ____ _ J n Cr, t)dV systems = system J n (r, t)dV systems (C-51) Using (C-20) in the above then yields ...... (C-52)

PAGE 486

448 The source term, (C-8) for a two-region system is Q(t) +. + + + J S (r, t) (r)dV + J S (r, C O m = core mod/refl J\F + + + + J S (r, t) (r)dV + f . S (r, = core c O mod/refl m J\ [v~f(t)J f ~(;, t)+(r)dV (C-53) c core 0 where use has been made of (C-30). Sc(r, t) is the core source distribution, and Sm(1, t) is the moderator reflector source distribution. Introducing (C-48) into the above and using (C-38) leads to a form for the two region system which is analogous to (C-50), i.e., 1 s (r, t)q/(r)dv + 1 s er, Q{t) = core c O mod/refl m -------------'-------! n (r, t)~+(r)dV systems 0 (C-54) For the one-speed, two-region gaseous core systems of intetest, [:(r)~core = :(R) = constant (see,for example,Figures 81 through 92) with R being the position at the core-reflector interface. Equation (c..:-53_) can thus be written as

PAGE 487

449 Q(t) J S (;, t)dV core c ---------A[vEf(t)] J ~(;, t)dV 1 s (r, t)+(r)dv + _m_o_d-'--/ r_e_f_l _m_+ ___ 0 __ +___ .' ( C--s 5 ) A[Vtf(t)J o(R)J ~(r, t)dV c core c core From equation (C-23) if follows that (~)J ~(r,, t)dV = core v ( t) . . c (C-56) Using the above along with (C-36) and (C-41) in e qua ti on (.C 5 5) g i v es Q(t) S (t)[k(t)/(t)] [ (t)J /[k (t)J . C oo . oo C C 1 s (r, t)+(r)dv m o + [k(t)/(t)]{ [ (t)] /[k (t)J }-mo_d ____ /r_e_fl ____ _ OO C OO C +(R)V C (C-57) If there are no sources in the moderator-reflector region, then (C-57) becomes Q(t) SC(t)[k(t)/(t)]{[ 00 (t)J /[k 00 (t)]} . (C-58) C C Considering next the delayed neutron precursors, for a one-region system, (C-7) becomes

PAGE 488

= i) + J C.(r, t)cp (r)dV system J 0 + A[vif(t)J f ~(r, t)cp (r)dV system 0 where use has been made of (C-29). in the above yields D J C.(r, system J + t)cp (r)dV 0 f n (r, system s + . t)cp (r)dV 0 450 (C-59). Using (C-48) and (C-31) (C-60) For the one-speed, one-region gaseous core systems of inter est, :(r) = constant and hence -d) c~(t): C.(t) V (t) J J system J n (r, t)dV system 5 = C.(t). J (C-61) For a two-region system, (C-7) becomes . D + c~(t) = f C.(r, t)

PAGE 489

4.51 D + J C.(r, t)cp (r)dV = core J 0 (C-:.a.62}.... where use has been made of (C-30). Introducing (C-48) into the above and using (C-38) leads to a form for the two r e g i o n s y s t e m \I h i c h i s a n a l o g o u s t o ( C 6 0 ) , i . e . , c~(t) J J C~(r, t)cp+(r)dV = core J 0 J n (r, t)cp (r)dV systems O (C-63) For the one-speed,two-region gaseous core systems of interest., [cp:(r)J = constant and (C-62) becomes core D D J C.(r, t)dV c.(t)= coreJ J J\[vc'.f(t)\ J tjJ(r, t)dV core (C-64) Using {C-56), (C-36) and (C-41) in the above equation yields. the result that c~(t) ~(t)[k(t)/(t)]{{ (t)J /[k (t)]} J J . 00 00 C C (C-65) Finally, considering the parameter which is usually designated as the reactivity, for a one-region system, (C-3) is

PAGE 490

' . 7 6[I(t)]f ~(r, 0 (r)dV . + + + J -system _ Using (C-29) yields p(t) = trvrf(t)J 6[Ia(t)] vI:f(t) 452 (C-66) (C-67) Letting the subscript zero stand for the parameters of a critical reference system, (C-68) From the definition given by (C-32) it follows that. (C-69) k 00

PAGE 491

453 Making use of (C-37) and noting that [k(t)J = 1 yields 0 the expression '[k 00 (t)J = [1 00 (t)/1(t)J . (C-70) 0 0 Equation (C-69) can thus be written as p(t) = (k 00 -l) [La(t)J /[La(tll{[1 00 (t)/1(t)] 1} 0 0 (C-71) Recalling that the above relations are strictly valid only if the 6 1 s are small, it follows that and [La(t)] /[La(t)J l 0 [1 (t)/1(t)J. 00 Equation (C-69) can then be rewritten as p ( t) k 00 (C-72) (C-73) The results of (C-72) and (C-73) are actually more general than for a one-speed, one-region system. It can

PAGE 492

454 be shown that these relations are valid for other systems p r o v i d e d t h a t t h e s h a p e f u n c t i on , iJi , h a s t h e s am e d e p end ~.... -+ -+ e n c e o n r , n , .a n d E a s the f u n d am e n t a l a n g u l a r f l u x eigenfunction, ~ 0 (r, ~' E), for the reference critical system. In the more general case the shape function does not vary with r, ~. and E in the same way as does the eigenfunction of k in the critical state and under these circumstances, pis not simply related to the static multiplication constant, k. Under this condition, (C-3) represents a generalization to dynamics problems and it may be used. to define a dynamic multiplication factor, kd' through the relation (C-74) and kd is then only indirectly related to k for a static problem. When~ can be closely approximated by the eigen function of k, theh kd=k and the interpretation of pas reactivity is reasonable. However, if iJi cannot be closely approximated by the fundamental eigenfunction, the computation of the more fundamental quantities (p/A) and (S/A) is more appropri ate. The relationships (C-36) and (C-74) need not be used (except perhaps for computing a kd). The computation of the static ~ultiplication factor from the diffusion theory equations (see Appendices A and B) is no longer necessary

PAGE 493

455 but perturbat•ion calculations for the quantities (p/J\) and (B/J\) must now be performed during the cycle. If these calculations are performed at each of the 5 to 6 thousand time steps of the piston cycle, the process becomes cessively expensive. Even if the quantities (p/J\) and (S/J\) are calculated at relatively few time steps (say of the order of 20 to 50) and a least squares fit performed for these quantities during the piston cycle, the IBM/370 computer time required would be of the order of 25 to 30 minutes for each piston cycle computation. It should be noted that the above approach would in volve at least an adiabatic approximation for the neutron kinetics analysis since the shape factor used at each of the 2_0 to 50 selected time steps would be obtained from the k-eigenval~e calculation for that step. A more sophis ticated treatment would involve going to a quasistatic approximation which would in turn mean a further increase in required computer time. Finally, another possible improvement would involve the use of a-eigenvalue rather than k-eigenvalue calcula tions for determining the shape factor when the system is around or above prompt critical.

PAGE 494

APPENDIX D THE POINT REACTOR KINETICS EQUATIONS USED Ili THE NUCPISTN CODE .... . .. Before proceeding to a development of the point kinetics equations Used in {he NUCPISTN code, some useful expressions will be derived from the two-group, two-region diffusion theory equations of Appendices A and B. Recalling that there is essentially no slowing down . -t in the core, if we define Sc{t) to be the average thermal neutron density per unit fim~ in the c-0re due solely to the inhomogeneous fas~ neutron source in the core then (D-l) The* indicates that the core thermal neutron flux which results solely from the inhomogeneous fast neutron source in the core is to be used. From (A-28), t where C* is obtained by setting (vEf}c or Kg equal to zerd in (A-60), i.e., 456

PAGE 495

457 C* = Kl (K 4 K 2 -K 1 K 5 )(S/4TTR 2 ). K2K7 ( K4 K3-Kl K6) K3K7 ( K4K2-K1 K5). (D-3) Using (D-2) 1n equation (D-1) gives (D-4) If the expression for C*, i.e., (D-3), is used in the above -t equation, the following relation is obtained for Sc (D-5) Comparing the above with (A-65) shows that S~ can be re written as (D-6) If the definitions for Kt and [t lt C coC ' i . e . , ( 07) (D-8)

PAGE 496

458 are used in combination with (D-6), it can be shown that (D-9) or = ,I, I [i ]t 'I'S 00 C (b-10) If a 0 is defined to be the fraction of all fast neu trons which leave the core and eventually return as thermal neutrons, then {D-11) where use has bee~ made of (A-56). U s i n g ( A 2 8 ) f o r cp t ( r ) a n d ( A 6 0 ) f o r C t h e n y i e 1 d s C , t 4 3-:--t [E(Vtf) + S /-;:rr3 R ] C O C (D-12) where -t c is given by (A-64). If we let a; be the fraction of fast neutrons from the fast inhomogeneous source in the core which leave the core and return as thermal neutrons then

PAGE 497

459 ( 0-J 3) t* Using (D-2) for c (r) and (D-3) for C* enables a 0 to be expressed as (D-14) where s is given by (A-65). For the case where there is no photoneutron production in the moderating reflector, a 0 , so from (D-12) and (D-14) it follows that t t V [E(VEf) + S C C 0 (D-15) It is to be noted that a 0 could also have been developed from the definition of S~, i.e., 1 = thermal neutrons in core due to fast inhomogeneous source = ao fast inhomogeneous source strength (D-16)

PAGE 498

460 Making. use of (D-6) in the above equation then gives 1. ao = as before. If one now considers the point reactor kinetics equa tion for the case where delayed and photoneutron precursors are ignored, wi1ere the amplitude function is normalized to be the average neutron density in the core, and where only the thermal group is considered, from (C-1) it follows that (D-17) 'Using (C-36) and (C-73), the above can be rewritten as di_t(t) t(t)-~ t . [Q (t.)Ja~ C = (t) + _C __ cit i(t) C (1/V)t C (D-18) where ~D has been set to zero. The source term can be expressed by means of (C-58) as = Sc(t)[k(t)/i(t)]{[t (t)Jt / [k (t)]t} 00 C 00 C (D-19)

PAGE 499

where [1 00 (t)]~ is given by (D-81 and The neutron source strength in the core,~ (t), is C . 461 (D-20) Making use of (D-16) in the above equation leads to the result that -t where the expression for S is given by (D-9). Substituting C (u-21) into (D-19) yields or a~ QC (t) = (1/v)~ (D-22) T h u s , t h e o n e s p e e d ( t he rm a l ) , t\-10 r e g i o n g a s e o u s core reactor -point kinetics equation when delayed and photoneutrons are neglected is

PAGE 500

462 k(t) = [k (t)Jt oo C The expression for the inhomogeneous source term given by either equation (C-58) or (D-19) which has been incor porated into the above equation is more general than for the case of a one-speed (thermal) two-region gaseous core system. It can be shown that the weighting function, (k/l)[l ]t/[k ]tc, is also valid for the two-group, twooo c oo region gaseous core reactors of concern. Hence, the point reactor kinetics equation for a two-speed, two-region gaseous core reactor when delayed and photoneutrons are neglected and when the amplitude function is hormalized to the average thermal neutron density in the core is also given by (D;..23). A somewhat different and perhaps intuitively more satisfying approach to the development of the point reactor kinetics equation with reg~rd to the source tirm follows. It involves the use of a spatially dependent, inhomogeneous neutron source term, S~(r), defined as ( D24)

PAGE 501

The* indicates the thermal neutron current at the core-reflector interface which results solely from the inhomogeneous fast neutron souice, S 0 , in the core is to 463 be used. R is the position at the core-reflector interface. Equation (D-24) implies that the source is distributed uni formly at the core-reflector interface which is the same distributi6n implied in the diffusion theory equations by means of conditions (A-56) or (B-4). Noting that (D-25) and using (D-2) and (D-3) in combihation with the above two equations yields (D-26) Inserting this equation in (C-53) gives Q (t) = JR qi [I ]t{R/3)o(r-R)+(r) 4nr 2 dr C 0 S a C 0 (D-27) J\F[a 0 J or (D-28)

PAGE 502

464 From (C-30) and (C-36) it follows that for a onespeed (thermal)-, two-region gaseous core system .. Making use of the approxi~ation constant and also utilizing (C-56) in (D-29) yields (D-30) From (D-8) and (D-20) one can obtain the relation (D-31) Equation (D-30) then becomes (D-32) 4 3 Noting that Ve= 1nR it is obvious that the use of (D-32} in (D-28) will lead to the expression (D-33)

PAGE 503

465 or equivalently, (D-34) which agrees with the previous result obtained in e~uation (D-22). If the point reactor kinetics equation given by (0~23) is to agree in the steady-state limit with the steady-state d i f f u s i o n t he o r y e q u a t i o n s , t he s o u r c e t e rm mu s t b e [ / t ] , i . e . , ( D -2 3 ) mu s t be re p 1 a c e d by (D-35) Solving the above yields (D-36) Assuming that l 5 (t) constant (or slowly varying compared to i~(t), it is easy to show that upon carrying out the integration in (D-36) one obtains r k-1) ((t) = T C Ae l (D-37)

PAGE 504

466 If ~(o) = 0, then A= ~s/(k-1) and (D-38) t . For k
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467 Relative to the source term (~sit), the point reactor kinetics source term is seen to be multiplied by the f~cto~ ~ (k/[k ]t) . . co C t ' From the definition of [kco]c' i.e., (D-20), from the definition of n, and from the four-factor formula, it can be shown that for the gas cores of concern [k ]t f c:: p = k co C . co (D-41) where c:: is th~ fast fission factor, p is the resonance escape probability, and f is th~ system thermal utilization factor. Similarly, it can be shown that (D-42) t ' f where PNL is the thermal non-leakage probability and PNL is the fast non-leakage probability. For a one-speed (ther~al) system, p = E: = P fast= . NL 1. 0 and equation (0-42) becomes [k Jt f p th= k co c NL (D-43)

PAGE 506

468 It is thus observed that for a one-speed system the point reactor kinetics source term is (
PAGE 507

469 where there are JD delayed neutron precursor groups and JP photoneutron precursor groups. It has been shown that the weighting fun~tion for the core source term for the gaseous cores of concern is (D-47) and that {D-48) where. (D-49) Thus, the desired quantity, a~[Qc(t)], is given by (D-21) or (D-33). The source term Qm{t) is due to the production of photoneutrons in the moderating-reflector from the {y, n) reaction. From (B-28) it follows that (D-50}

PAGE 508

4 70 . By analogy with (D-48) one can write a relation for Qm(t) as (D-51) where the \'1eighting function, Wm' can be obtained from (C-53) and is simply l + f f cfi dV. J\F mod/refl 0 (D-52) Making use of (B-32) and (B-33) in combination with (D-50) and (D-51) yields the result that (D-53) I t i s now poss i bl e to re \'i r i t e e qua t i on s ( D 4 4 ) t hr-o ugh (D-46) in the foll6wing manner + a_P_(t_)f_v_(t_) [ i\tc~ (t~l H (t) ( l /v) j = l J J m (D-54)

PAGE 509

471 (D-56) where use ha~ been made of the fact that fo.r the gaseous -0 D d -0 D [ ( ) ] cores of concern, s. s. an S S see C-46 . J J The general expressions for the two-point finite difference or three-point integration formulas which correspond to the point reactor kinetics equations for the case where delayed and photoneutrons ar~ included can be shown to be

PAGE 510

472 (D-57) + p(>,.c.(t-1)6t] P _ .. J [(l/v) (t-1)] (t-1) D (k(t-l)S~ 6t t t ) J J i{ t-1) C C (D-58) p,,_.p (k(t-1 )B~ 6t ) + o[>,.C.(t-1) 6t] P J [(l./v)tc(t-l)J "tc(t-l) . (D-59) J J i(t-1)

PAGE 511

In the above relations, 8=1.0 and p=O for the two-point finite difference equations and 8=3/2 and p=l/2 for the three-point integration formulas. Equations (D-57) 473 through (D-59) are the numerical forms of the kinetics equations used in the NUCPISTN code when delayed and photo neutrons are present. For definitions of ap and f , refer V to equations (B-33) and (B-29) of Apperidix B. Values for Wm{t) as obtained from equation (D-52) are arrived at by independent p~rturbation theory calculations performed by either the CORA or EXTERMINATOR-II codes. If no value for Wm(t) is input into the code, the code internally sets Wm(t)=O. l.

PAGE 512

APPENDIX E THE ENERGETICS EQUATIONS USED IN THE N U C P I ST [ 1 l C OD E The energy equation used for the HeUF 6 mixture from the time at which the intake valves are closed until the time at which the exhaust val v2s are opened is the non flow, closed system equation ( El ) Wis th~ net amount of work done by the system on the sur ioundings, Q is the net amount ofheat added to the system from the surroundings, 6U is the increase in stored or internal energy of the system, and qR is the heat of reaction. The (nuclear) reaction is the fission process and thus, ( E 2 ) where Ef is the energy per fission and N~ 35 is the U235 atom density in the core. 474 ......

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If the HeUF 6 mixture is considered to behave as an ideal gas then (E-3) 475 where mis the total HeUF 6 mass, cv is the mixture average specific heat at c~nstant volume and f is the mixture average temperature. The work term, when friction losses are neglected, is W = J pdV core = mR f. l s!_I{_ core V (E-4) where use has been madeof the ideal gas equation of state pV = mRT. {E-5) The gas constant, R, is equal to the universal gas constant divided by the molecular weight of the gas. The term Q includes heat loss terms from the core (cylinder) to the walls and reflector. Maintaining the moderating reflector at anelevated temperature will minimize this loss and no heat transfer relations are included in the present version of NUCPISTN to explicitly account for this

PAGE 514

476 .heat loss. The term Q was therefore set to zero while the energy per •fission, Ef, was take_n tobe 180 MeV. The energy equation in NUCPISTN for the closed por tion of the cycle is thus J t 235 -t TdV -Ef[af cf N (t) (t)Vc(t)dt mR f -V= m c 6(T). c c . core v (E-6) 235 Noting that N (t)V (t) = constant for the closed system C C and using the following approximations ( E-7) and mRJ T ~v mR ~(t+6t1 + T(ttls . ~I core L J core V (E-8)

PAGE 515

477 gives the following form for the energy equation for the closed portion of the cycle The above equation is solved to determine T(t+tt). Since the piston position is known as a function of time, V{t+6t) is always known. The ideal gas equation is then solved to determine p(t+tt). The energy equation used for the ~eUF 6 mixture before the intake valve is closed is the open system, nori-steady flow equation fore. Q W + h-om. = 6U 6qR. l l {E-10) Q, W, 6U, and tqR all have the same meaning as be om. is the gas mass which enters the system during l the time 6t, i.e., om.= dm = M.{t)tt l l (E-11)

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478 where ~-(t) is the mass flow rate of gas into the cylinder l at time t. Thus, m ( t + L\t ) = [ m ( t ) + M i ( t ) L\ t J . (E-12) his the energy of the mass entering the system and is l given by h. = C T .. l p . l l T. is the gas average temperature in the intake line l and c is the mixture average specific heat at constant pi pressure in the intake line. Wis again given by (E-4), L\qR by (E-2), Q is again zero, and L\U is (E-14) It can thus be shown that equation (E-10) can b~ written as E [cr ]t J N 235 (t) t(t)V (t)dt R J T dV f fc C C. C m V core + Jc T. dm = fc f dm + f m c dT. p. l V V l (E-15)

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479 Using (E-8) along with the following approximations E [ Jt N 235 (t) ~t(t) V (t)dt f 0 f C C ~c C 2 Jc T.dm = c T.J [1.(t)dt c fJ1.(t)6t P; l pi 1 l P; l l (E-17) (E-18) (E-19) yields the following form for the energy equation for the portion of the cycle before intake valve closure l+ C T.M.(t)M mR ff(t+M)+T(tjl n tc(t+6tj . . L 2 Vc{t) P; 1 1

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480 The above equation is solved to determine T(t+~t). Since the piston position is known as a function of time, V(t+6t) is always kno~n, m(t+6t) is also known and hence p(t+6t) is solved from the ideal gas equation of state. The mass flow rate during the intake portion of the cycle is allowed to vary according to t _ dm M . ( t) = Y . ( t )C . A. p. p ( t) . d t . _ 1 1 1 1 1 1 (E-21) In the above expression, p. is the pressure in th~ intake 1 line, P is the density in the intake line obtained from 1 p. = p./RT., 1 1 1 (E-22) p(t) is the gas pressure in the cylinder, Yi is the net expansion factor for compressible f}ow through the intake valve, Ci is the flow coefficieni for the intake valve, and A. is the cross sectional area of the intaie valve. 1 The paramet~r v 1 is a function of the specific heat ratio, the.ratio of the cylinder diameter to the valve diameter

PAGE 519

481 and the ratio of the cylinder to valve pressure. Yi can be read from standard tables as a function of these other-parameters [78]. of the Reynolds valve diameter. The flow coefficient, C., is a function l number and the ratio of the cylinder to It also can be read from standard tables as a fun~tion of the pertinent parameters. For anticipated intake flow rates, Ci is essentially constant over the cor responding range of Reynolds numbers. If pis in atmos pheres, A is in m 2 , pin {kg/m 3 ) and Yi and Ci dimension less, then r\ (t)[kg/sec] = (105T. l l l l 1 {_E2 3) The procedure used for computing ~.(t) during the intake l portion of the cycle is then as follows. An initial value for ~-{t) is specified, Y.(t) is read from tables and from l l the value of p(t), pi and pi the product AiCi, a constant over the intake portion of the cycle, is determined. Mi(t) is then assumed constant over a time step and T(t+6t) and p(t+6t) are computed as described previously. A value of Yi(t+6t) is then read from the tables and since the product A.C. is known, ~.(t+6t) can be computed from (E-23). l . 1 l The input of gas through the intake valve is stopped if (a) the cycle fraction becomes equal to the requested value for shutoff or (b) if p becomes equal top., the gas pres, sure in the intake line.

PAGE 520

482 The energy equation used for the HeUF 6 mixture after the exhaust valves are opened is the op~n system, non~ steady flow equation Q W h om = dU 6qR. e e (E-24) Q, W, 6U, and 6qR have already been described and ome is the gas mass which leaves the system during the time 6t, (E-25) where ~e(t) is the mass flow rate of gas out of the system at time t. Thus, (E-26) he is the energy of the mass leaving the system and is given by (E-27) T is the average temperature of the HeUF 6 mixture in the cylinder at the time of concern. Wis given by (E-4), 6qR by (E-2), Q is zero, 6U is given by (E-14) and thus (E-24) becomes

PAGE 521

= f c T dm + J m c df V V (E-28) Using (E-8), (E-16), and (E-19) along with the following approximations = -cp f(t+!it)+f(t) t~ (t) L'lt 2 e (E-29) Jc f dm -c r-(t+L'lt)+f{tLl M (t)L'lt v v[ 2 _I e (E-30) 483 yields the following form for the energy equation for the portion of the cycle after the exhaust valves are opened

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484 The above equation is solved to determine T(t+tt)and the ideal gas equation of state is then solved for p(t+tt). The mass flow rate during the exhaust portion of the cycle is allowed to vary according to [ M ( t ) J = Ye ( t ) C A ( p ( t ) p ) p ( t ) = ddmt_ e h e e e (E-32) Pe is the back pressure in the exhaust line, pis the gas density in the cylinder computed from p = p/Rl, (E-33) p(t) is the gas pressure in the cylinder, Ye is the net ex pansion factor for compressible flow through the exhaust valve,C is the flow coefficient for the exhaust valve and e I Ae is the cross sectional area of the exhaust valve. The behavior of the parameters Ye and Ce is completely analogous to the beh~vior of Y. and C. described for the l l intake phase of the cycle with C being essentially con e stant over the range of Reynolds numbers for anticipated exhaust flow rates. Thus,

PAGE 523

(E-34} when the parameters have the same units as described in connection with (E-23}. 485 The procedure for computing [~e(t)]h is then analogous to the procedure used for ~.(t). An initial ~alue for 1 [~e(t)Jh is specified, Ye(t) is read from tables and from the values for p(t), Pe and p(t), the product AeCe, a con stant over the exhaust portion of the cycle, is determined. [~e(t)Jh is then assumed constant over a time step and T(t+6t) and p(t+6t) are computed as described abo~e. A value of Ye(t+6t) is then read from tables and since the product AeCe is known, [~e(t+6t)]h is obtained from (E-34). Note that for engines in which all the gas is ex hausted or forced from the cylinder, a forced flow rate, [~e(t)Jf, is computed in addition to the hydraulic flow rate, [~ 2 (t)]h, from the expression dm(t) [M (t)J = e f dt p ( t )_ [ d V c { t )/ d t] ~-----R T{t) {E-35) The forced flow rate is due solely to the physical motion of the piston and d\{:/dt is the rate of change of the cylin der volume. If [~e]f > [~e]h or for times during the exhaust phase of the cycle during which the piston is undergoing

PAGE 524

the 11 drift 11 or non-simple harmonic motion, (E-36) When [~e]h > [~e]f and if the piston is undergoing simple harmonic motion then (E-37) 486

PAGE 525

APPENDIX f GROUP STRUCTURES AND VARIOUS REACTOR PHYSICS CONSTANTS USED IN THE NUCLEAR PISTON ENGINE COMPUTATIONS Table F-1 XSDRN Collapsed 21-Group Structure and Prompt Fission Neutron Chi Is Group E (lower) E (upper) x~ 3.0lMeV 15MeV 2.067 X l 0l 2 2.23MeV 3.0lMeV l.352 X l O l 3 1.65MeV 2.23MeV 1. 423 X l 0l 4 l. 35MeV . 65MeV !3.914 X 1 02 . 5 821 keV. l.35MeV l. 805 x l O l 6 498keV 821keV l. l 48 x ,0-1 7 3 3.4 k e V 498keV 5.485 X 1 0-:2 8 l83keV 334keV 4.398 X ,0-2 9 52.5keV 183keV 2.796 X l 02 l O ll.7keV 52.5keV 4.545 x l o3 1 l 5.53keV ll.7keV l 2 3.36keV 5.53keV l 3 96leV 3.36keV l 4 275eV 96leV 1 5 78.9eV 275eV 1 6 2.26eB 78.9eV l 7 6.48eV 22.6eV l 8 l .86eV 6.48eV l 9 l.13eV l. 86eV 20 0.655eV l. l 3eV 21 0.0047eV 0.655eV 487

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488 Table F--2 .... .PHROG-BRT-1 Collapsed 21-Group Structure and Prompt Fission Neutron Chi I s Group E (lower) E (upper) X 1 2.87MeV lOMeV 2.276 X 1 0l 2 2.23MeV 2.87MeV 1 . 1 4 l x l 0l 3 l.74MeV 2.23MeV 1.189 X 1 01 4 l.35MeV l.74MeV 1. 124 X 1 O l 5 82lkeV 1. 35MeV 1. 805 X l O l 6 498keV 82lkeV l . 148 x 10-l 7 302keV 498keV 6.479 X 1 02 8 183keV 302keV 3.404 X ,0-2 9 52.5keV 183keV 2.796 X 10-2 1 0 ll.7keV 52.5keV 4.806 X ,o-3 1 1 5.53keV 11.7keV 1 2 3.35keV 5.53keV 1 3 961eV 3.35keV 14 275eV 96leV 1 5 78.9eV 275eV 1 6 22.6eV 78.9eV 1 7 6 .48eV 22.6eV 18 l.86eV 6.48eV 1 9 l.13eV 1 .86eV 20 0.683eV l.13eV 21 'v0e V 0.683eV

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Group l 2 3 4 Group l 2 3 4 Table F-3 XSDRN Collapsed Four-Group Structure and Prompt Fission Neutron Chi's E (lower) 498keV 3.36keV 0.655eV 0.0047eV E (upper) l 5MeV 498keV 3.36keV 0.655eV Table F-4 X 0.8637 0.1313 0.0 0.0 PHR0G-BRT-1 Collaised Four-Group Structure and Prompt Fission Neutron Chi I s E ( l o \\le r) 498keV 3.36keV 0.683eV 'u0eV E (upper) l 0MeV 498keV 3.36keV 0.683eV X 0.8684 0.1316 0.0 0.0 489

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BO Delayed Group j J l 0.000215 2 0.001424 3 0.00)274 4 0.002568 5 0.000748 6 0.000273 0.006502 Table F-5 Delayed Neutron Parameters Used in the Nuclear Piston Engine Computations >. D 1* 2* T J J X X l J J (sec ) (sec) 0.0124 80.65 0.330 0.670 0.0305 32.74 0.640 0.360 0. 111 9.01 0.620 0.380 0. 301 3.32 0.690 0.310 1 . 1 4 0.877 0.590 0.410 3.01 0.332 0.590 0.410 The above data i s for thermal fission of .U 23 5. *The superscripts 1 ' 2 ' 3 ' and 4 apply to broad groups 1 ' 2 ' 3 ' and as given in Table F-4. 3* 4* xj X J 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4 respectively

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491 Table F-6 Photoneutron Precursor Parameters Used in ..... the Nuclear Piston Engine Computations Photoneutron A. p p r::* (3~xl0 5 Precursor J 1. (3.xlO::, l J rou~ j (sec ) J ' J l 6.27xl0 -7 18. 5 days 0.042 0.050 2 ' 6 3.63xl0 3.l9days 0. l 06 0. l 03 3 4.37xl0 -5 6.36hrs 0.335 0.323 4 1.17xl0 -4 2.37hrs 2. 41 2.34 5 4.28xl0 -4 38.9 min 2. 13 2.07 6 1. 50xl0 -3 11. 1 min 3.47 3.36 7 4.8lxl0 -3 3.47min 7.22 7.00 8 -2 59.2 21. 1 20.4 l.69xl0 sec 9 1 3.6 67.0 6 5. 1 2.77xl0 sec 103.81 100.75 The above f3~ 1 s are for saturation fission product activity for u235 fi~sions in 020. The parameters fP and yP (see Appendices Band D) are used to account for the fact that not ~ll ~ammas released by the photoneutron precursors yield photoneutrons in the o 2 o. *These values are from reference [54]. The other parameters in the table are from reference [52] and are the.NUCPISTN c o d e d e f a u l t v a l u e s . T h e N IJ C P I ST r~ c o d e i n i t s p re s e n t f o rm haridles eight photoneutron groups with group l in the above table being neglected.

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APPENDIX G LISTING OF THE TASKS PERFORMED BY THE NUCPISTN SUBROUTINES.AND A FLOW DIAGRAM FOR THE NUCPISTN CODE Listing of Tasks Performed by the NUCPISTN Subroutines 1. MAIN This is the overall master or controlling program for the NU CPI ST N code . Al l i n put data a re read by this routine. It also monitors and prints out data during the compression and power strokes. 2. OUTl This is the ~irst of three subroutines used for outputting most of the NUCPISTN cycle results. Quantities output by this subroutine include re sults from steady-state neutronic calculations (e.g., steady-state fluxes and flux ratios and values for the inhomogeneous source in the moderating-reflector due to photoneutrons). 3. OUT2 This is the second of three subroutines used for outputting most of the NUCPISTN cycle results. Quantities output by this subroutine include cycle averaged parameters, cycle peak values for various parameters, code-computed thermal group constants for the core, neutron and piston cycle time char acteristics, total fission heat release, mechani cal power output and other miscellaneous cycle results. 4. OUT3 This is the third of three subroutines used for outputting most of the NUCPISTN cycle results. Quantities output by this subroutine include mass flow rate results, gas specific heat data and delayed and photoneutron precursor concentrations at thi cycle start, cycle finish, and after passing through the loop. 5. XSECTThis subroutine computes number densities, core thermal group cross sections, and reactor dimen sions throughout the piston cycle. 492

PAGE 531

493 6. KEFF This subroutine computes the static neutron mul tiplication factor, keff, and various lumped reactor physics parameters for use by otheT 'S~b~ routines throughout the piston cycle. 7. HEATC This is a subroutine which calculates specific heat values for the HeUF 6 mixture as a function of temperature throughout the piston cycle. 8. FLUX This is a subroutine which computes the steady state fast -and thermal reflector-fluxes and steady-state thermal core flux throughout the piston cycle when photoneutrons are neglected. 9. FLUXP This is a subroutine which computes the steadystate fast and thermal reflector fluxes and , steady-state thermal core flux throughout the piston cycle when photoneutrons are included. 10. PTKIN This subroutine solves the point reactor kinetics equations for the instantaneous value of the average core thermal neutron flux, the delayed neutron precursor concentrations and the photo neutron precursor concentrations. 11. WESCOTThis subroutine computes Wescott non-1/v factors for use in the core thermal group cross section calculations. 12. INlT This is a subfoutine which initializes parameters at the beginning of each piston cycle. 13. INTAKEThis is the controlling or master subroutine during the intake stroke. It also monitors and prints out data during the intake stroke. 14. SUM This is a subroutine which sums quantities at each time step. Some of these sums are then used in obtaining cycle-averaged values for various parameters. 15. TIMESETThis subroutine computes the total elapsed time into the piston cycle and the cycle fraction at each timestep during the piston cycle. 16. SUM4 This is another s~broutine which sums quantities at each timestep. Some of these sums are then used in obtaining cycle-averaged values for various parameters.

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494 17. EXHAUS This is the master or controlling subroutine during the exhaust stroke. It also monitors a n d pr i n ts o u t d a ta dur i n g the ex ha u s t s t r o k e<; 18. RESET After each timestep, this subroutine s~ifts or resets the values for various parameters which are used in solving the kinetics and energetics numerical equations. 19. FIT This is a subroutine used to account for time varying neutron lifetimes, effective delayed neutron fractions and inhomogeneous source \,1eighting functions during the piston cycle. This subroutine allows the treatment of the neutron kinetics to be extended f~om a pure point reactor model to an adiabatic model. 20. XFLOW When the intake valves are open, this subrou~ tine obtains quantities at each timestep which are used in calculating the mass flow rate through the intake valve. 21. YFLOW When the exhaust valves are open, this sub routine obtains quantities at each timestep which are used in caltulating the mass flow r a t e. t h r o u g h th e e x h a u s t v a l v e . 22. ENERGY This subroutine solves the energetics equa tions for the gas temperature and then the ideal gas equation for the gas pressure. The subroutine then calculates the fission heat release and mechanical power out~ut as well as other thermodynamic quantities for the time step of interest. 23. TSTEPE This is~ subroutine which determines the size of the timesteps which are used in solving the energetics equations. The timestep size selec tion depends on the rate of fission heat re lease and rate of temperature change of the gas. This timestep can never be smaller than the timestep used in solving the neutron kinetics equations.

PAGE 533

CAI.L EXIT ST.\RT A R~ad in !cput p~r1~~ters e~d c~t d~fault values for para::ictcr~ not rt!Jd in. PROGR,\.'I p_;r:, Obtain ~escott f~ctors for the ccutr0n tecp~r~tur~ of inttrest. SUBROl:T /NE l.c.SCOT Q taln spec!flc l:edt d. :,c,1 r the H~UF6 gJs at th~ itial sas"tccp~rature. SUBFOUTl'.iE ll:.ATC lnitializ" .v.,r lous par .. i:nttters ir. prepJration for a ncY plsto~ cycl~ cJ\c~\Jtluu. SUBROUTil::C: I~IT Neutron ldn~tics model V CO 10 C 495 Obtain initial values for the ti~~-varying quantltle, 1, I\ , t eff w.,, and (1/v)c• flt'l'!()II

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Compute r,lr.Dbt-"r d~nsit !es, core theroal grol!p const~nts, ~nJ reactor d irr,en~ !.ens. 5'.'BPOlJTI1;E XSEC Compute keff a~d various luruped reJctor physics para:nett'rs. SUBROU.:It:E HFF CO TOE 496 Obtain 1nitl3l pJram~trrs (e.g., initial ~a~s flow ra,e) rcqulred for intak~ stroke at1alysis. SU6~0UTI~E I~TAK Coa.pute steaJy-state fast ar.J th~rn.,l r~flectur fluxe" anJ Liie stcady-!>tate tht."rmal core f L,x SUBROUTl~c FLUXP CO TO F

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E' Comput~ st~ddy-st.J.te fagt and tht'rrial reflector fluxes a~d the ste•dy-st~te thcr~3l core flux. SUB'
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I Comp:.ite th~ lk'JF ga•i speclflc heat v.1~ut:':::1 at tl1~ n~w gJs SUP,ROUII:n: m:,\lC G' Sun mis.:~lL.mcous quantitit!~ and s.:ive for u:;e in obt.1inin~ cyclc-cJ.v~r.,g!::J p.1r.iLit!tcr$. S~ EROUTI '.IE Sl:i Sma otht!r miscellaneous qu.:1.1t lties and save for u_s.~ in obt.1 ini~1g, cycle-av~r.:ig•:d par,tmetC'rs. SU meet I :CE Sl11~ Shift or rcsut valu~s of varioua pJr~ttCr!; '-'hi...:h ..!.re llSt-:d in oalvlnt the en~rgetlcs and n~utron kln~tl~s equations. CALL EXIT SUBROUTINI:: RESH Compute ti,~ total elapsed ti~~ into the pbton cycle .rnd the cycle fraction. strr 0 ounr,;,: T1~1::sn GO to K 498 Readjust the tl::J.,step.slu u~ed in ~v\vln~ the eueq~~tlcs equ:idcns. SUBP.O~'T I!:E TSTEPE CO TO J Obtain n~w value~ for th~ ti~~-varyln~ quantities 1, ll , II , and (1 / v) l j eff DI C SUBRUUTINE FlT

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Output pbton cycle results. SU BF.OU ru:~ OUTL Output piston cycle results. SUBROUTl'.H, OliT2 Output pl,ton cycl~ result::J. SUBRUUTlllE oi;n V CO TO 11 Obtain qu3ntit1es us~d for calculati:ig th~ t1~1s::1 flow t3te thruu~h th~ 1 ntake valves. SUBROUTI!lE XFLOI/ CO 'IO I> 499 co 1'0 :l

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Are th~re ai [ ~ !::'::,?;:::: ~, ... , -&• CO 10 B c;n TO A CALL EXIT lt •. used icr c:as~ fly,,,1 the exh l~E YFLO~ust ~alv l GO TO p 500

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l. 2 . 3 . 4. 5. 6. 7. 8. 9. LIST OF REFERENCES G.I. Bell, "Calculation of the Critical. Mass of UF 6 as a Gaseous Core, with Reflectors of D 0, Be and c, 11 USAEC Report LA-1874, Los Alamos Scient~fic Laboratory (February 1955). G.Safonov, "Externally ModPrr1ted Reactors," Proceed inqs of the Second United Nations International Con ference on the Peaceful Uses of Atomic Eneray, Geneva, Vol . 1 2, P. nrs--n 9 58 . R.G. Rags2~ 5 e and R.E. Hyland, "Some Nuclear Calcula tions of U-D O Gase_ous-Core Cavity Reactors," Report NASA-TN-475, NaEional Aeronautics and Soace Administra t1on, Lewis Research Center (October 19~1 ). R.E. Hyland, R.G. Ragsdale, and E.J. Gunn, "Two-Dimensional Criticality Calculations of Gaseous-Core Cylindrical Cavity Reactors," Report-NASA-TN-0-1575, National Aero nautics and Space Administration, Lewis Research Center (March 1963). R.M. Kaufman, W.F. Osborn, Jr., J.R. Simmons, and E.B. Roth, "Reactor Physics Calculations for the Gaseous Core Cavity Reactor," Allison Res. Eng., 8:11-18'(1965). L.O. Hen-ligand T.W. Latham, "Nuclear Characteristics of Large Refl~ctor-Moderated Gaseous-Fueled Cavity Reactors Containing Hot Hydrogen," AIAA (Amer. Inst. Aeronaut. Astronaut.) J., 5(5):930 (Mayl967). T . F . P l u n' k et t , 11 Nu c l ea r An a l y s i s of Ga s e o u s Core Nu c l ea r Rockets," Nucl. Appl., 3:178 (March 1967). T.L. Latham, "Nuclear Criticality Study of a Specific Vortex~ St ab i 1 i zed Gaseous Nu c 1 ear Rocket Eng i n e , 11 Report E-910375-1, United Aircraft Corporation, Research Labora tories (October 1966). T.S. Latham, "Nuclear Criticality Study of a Specific Light Bulb and Open-Cycle Gaseous Nuclear Rocket System," Report F-910375-2, United Aircraft Corporation (September '1967). , 501

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502 10. T.S. Latham, "Criticality Studies of a Nuclear Light BulbEngine, 11 Fourth AIAA Propulsion Joint Soecialist C o n f e r e n c e , C l e v e l a n d , 0 h i o , P a p e r N o . 6 8 5 7 l ( J u n e l 9 6.8.J . 11. T.S. Latham, "Nuclear Studies of the Nuclear Light Bulb Rocket Engine," Report G-910375-3, United Aircraft Cor-. poration (September 1968). 12. T.S. Latham, H.E. Bauer, and R.J. Rod~ers, "Studies of Nuclear Light Bulb Startup Conditions and Engine Dy namics," Report il-910375-4, United Aircraft Corporation (September 1969). 13. W.W. Engle, "A Users Manual for ANISN, A One-Dimensional Discrete Ordinates Transport Code with Anisotropic Scattering," USAEC Report K-1693, Union Carbide Corpora tion, Nuclear Division (March 30, 1967). 14. G.C. Joanou and ,1.s. Dudek, "GAM-1, A Consistent P-1 Multigroup Code for the Calculation of Fast Neutron Spectra andMultigroup Constants," USAEC Report GA-1850, Gulf General Atomic Incorporated (June 1961 ). 15. R.H. Shudde and J.Dyer, "TEMPEST-11, A Neutron Thermali zation Code," USAEC, Report TID-18284, Atomics International (June 1962). 16. E.H. Canfield, R.N. Stuart, R.P. Freis, and W.H. Collins, 11 SOPHIST-1, An IBM 709/7090 Code 14hich Calculates Multi group Transfer Coefficients for Gaseous Moderators, 11 USAEC Report UCRL-5956, University of California, Lawrence Radiation Laboratory (October 1961 ). 17. R.F, Mynatt, 11 A Users Manual for DOT, A Two-Dimensional Discrete Ordinates Transport Code with Anisotropic Scattering," USAEC Report K-1964, Union Carbide Cor poration, Nuclear Division (1964); 18. T.B. Fowler, M.L. Tobias, and D.R. Vandy, 11 EXTERMINATOR II: A Fortran IV Code for Solving Multigroup Neutron Diffusion Equations in Two Dimensions," USAEC Report ORNL-4078, Oak Ridge National Laboratory (April 1967). 19. R.E. Hyland, 11 Evaluation of Critical Mass for Open Cycle Gas-Core Rocket Reactors," Nucl. Tech., 12(2): 152 (October 1971). 20. G.D. Joanou and J.S. Dudek, 11 GAM-II, A 8-3 Code for the .Calculation of Slowing Down Spectrum and Associated Multigroup Constants," GA-4265, General Dynamics Cor poration (1963).

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21. H.A. Vieweg, G.D. Joanou, and C.V. Smith, 11 GATHER-II, an IBM-7090 FORTRAN-II Program for the Computation __ of 503 Thermal Neutron Spectra and Associated Multiqroup. Cross Sect i on s , 11 GA 4 l 3 2 , Gener al Dyna mi cs Corpora ti on (1963). 22. C.E. Barber, 11 A FORTRAN-IV, Two-Dimensional Discrete Angular Segmentation Program, 11 TN-D-3573, NASA (1966). 23. C.B. Mills, 11 Reflector Moderated Reactors, 11 Nucl. Sci. Eng., 13(4):301 (August 1962). 24. G.A. Jarvis and C. G. Beyers, 11 Critical Mass Measure-. ments for Various Fuel Config~rations in the LASL D 0 Reflected Cavity Reactor, 11 AIAA Propulsion Joint Sp~cialist Conference, Colorado Springs, Colorado, Paper No. 65-555. (june 1965). 2 5 . G .. D . P i n c o c k a n d J . F . Ku n z e , 11 C a v i t y Re a c to r C r i t i c a l Experiment, 11 Vol. 1, USAEC Report NASA-CR-72234, General Electric Company, Nuclear Materials and Propulsion Operation (September 1967). 26. G.D. Pincock, J.F. Kunze, R.E. Hood and R.E. Hyland, 11 Cavity Reactor Engineering Mockup Critical Experiment, 11 Trans. Amer. Nucl. Soc., 11:29 (1968); also Reoort NASACR 1 2 4 1 5 ( J u n e l 9 6 8 ) . . 27. G.D. Pincock and J.F. Kunze, 11 Cavity Reactor Critical Exp er i rn en t , 11 Vol . I I , Report NASA CR7 2 4 l 5 (May l 9 6 8 ) 28. G.D. Pincock and J.F. Kunze, 11 Cavity Reactor Critical Experiments," Vol. III, Report NASA-CR-72384 (November 1968). 29. W.B. Henderson and J.F. Kunze, 11 Analysis of Cavity Reactor Experiinents, 11 Report NASA-CR-72484 (January 1969). 30. J.F. Kunze, G.D. Pincock and R.E. Hyland, 11 Cavity Reactor Critical Experirnents, 11 Nucl. Appl., 6(February 1969). 31. G.D. Pincock and J.F. Kunze, 11 Cavity Reactor Critical Experiment, 11 Vol. IV, Report, NASA-CR-72550 (October 1969). 32. G.D. Pincock and P.L. Chase, 11 Cavity Reactor Critical Experiment, 11 Vol. V, Report NASA-CR-72577 (November 1969). 33. J.F. Kunze, J.H. Lofthouse and C.G. Cooper, 11 Benchmark Gas Core Critical Experiment, 11 Nucl. Sci. Enq., 47(1 ):59 (January 1972).

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34. 504 C.L. Beck and G.E. Putnam, 11 SCAMP--An S Code for Analysis of Multigroup Problems," Letter Cornmun~cation, Idaho Nuclear Corporation (1970). 35. R.L. Curtis, G.L. Singer, F.J. Wheeler, and R.A. Grimesey, 11 PHROG-A FORTRAN-IV Program to Generate Fast Neutron Spectra and Average Multigroup Constants, 11 IN-1435 36. 37. 38. 39. 40. 4 l. 42. 43. 44. (April 1971). R.L. Curtis and R.A. Grimesey, "INCITE-A FORTRAN-IV Program to Generate Thermal Neutron Spectra and Multi group Constants Using Arbitrary Scattering Kernels," . IN-1062 (November 1967). J.D. Clement and J.R. Williams, "Gas-Core Reactor Tech nology,11 Reactor Technoloqy, 13(3):233 (Summer 1970). R.T. Schneider and M.J. Ohanian, patent disclosure to NASA (1970). C.D. Kylstra, J.L. Cooper and B: E. Miller, 11 UF Plasma Engine," Second Sm osium on UraniumPlasmas: ~esearch and Appl i cations , A I AA , New York l 9 71 ... C.F. Hale et al., 11 High Temperature Corrosion of Some Metals and Ceramics in Fluorinating .l\tmospheres, 11 R-1459, Union Carbide (September 1960). H A . H a s s a n a n d J . E . De e s e , 11 T he rm o d y n am i c P r o p e r t i e s o f UF 6 at High Temperatures," Report, North Carolina State UnTversity, Raleigh, N.C. (1973). Mechanical Engineers Handbook, sixth edition, McGraw-Hill (l9p4). D.E. Sterritt, G.T. Lalos and R.T. Schneider, 11 Thermodyn am i c Proper t i es of U F 6 M.e a sured w i th a Ba l l i st i c P i s ton Compressor," NASA Contract NGL 10-005-089 (March 1973). E.C. Anderson and G.E. Putnam, 11 CORA: A Few Group Dif fusion Theory Code for One-Dimensional Reactor Analysis, 11 ID0-17199 (August 1970). 4 5 . C . L . B e n n e t t a n d l1 . L . P u r c e l l , 11 B R T l : B a t t e l l e R e v i s e d THERMOS , 11 B NW L l 4 3 4 (June l 9 7 0) . 4 6. N . M. Gree n_e and C . W . Craven , Jr. , 11 XS DR N : A Di s c re t e 0. r d i n a t e s S p e c t r a l A v e r a g i n g C o d e , 11 0 R N L T M 2 5 0 0 ( J u l y .1969).

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505 . 47. J.J. Katz and E. Rabinowitch, The Chemistry of Uranium, th~ Elements, Its Binary and Related Com ounds, Dover 48. 49. 50. 51. 52. Publications, Inc., New York 1961 . R. DeW~tt, Uranium Hexafluoride: A Survey of the Physito~ Chemical Proe~rties, Goodyear Atomic Corporation, Portsmouth, Ohio (August 1960). P. Fortescue, "Dry Cooling of Power Plants and the HTGR Gas Turbine System," GA-Al2026 (March 1972). D. Tesar, "The Desiqn of Linkage Dwell Systems," NSF Grant # GK-44O6, University of Florida (1969). G.E. Putnam, IIMONA-A Multigroup One-Dimensional Neutronics Analysis Code," ANCR 1051, TID4500 (March 1972). G.R. Keepin, Phfsics of Nuclear Kinetics, Addison-Wesley, Reading, Mass. 1965). . 53 . . N.P. Baumann, R.L. Currie, and 2 3 '5. Pellarin, "Production of Delayed Photoneutrons from U in Heavy Water Moderator , 11 T r _ a n s . Amer . Nu c l . Soc . , l 7 : 4 9 9 ( l 9 7 3 ) . 54. ANL-5800, Reactor Physics Constants, Second Edition, pp. 20 and 475, U.S.A.E.C. Division ofTechnical Information (July 1963). . . 55. K.D. Lathrop and F.W. Brinkley, 11 TWOTRAN2, A Two-Dimensional Mu1tigroup Transport Theory Code," LA-4848-MS (1973). 56. T.W. Schoene, "The HTGR Gas Turbine Plant with Dry Air Cooling," Nuclear Engineering and Design, 26:170 (1974). 57. R.T : Schneider, "Experimental Investigation of a Uranium Plasma: Pertinent to a Self-Sustaining Plasma Source," NASA Contract NGL-10-005-089 (March 1973). 58. K. Thom and F.C. Schwenk, "Gaseous Fuel Nuclear Reactor Research," Frontiers of Power Technology Conference, Oklahoma State University (October 9-10, 1974). 59. K. Thom, R.T. Schneider, and F.C. Schwenk, "Physics and Potentials of Fissioning Plasmas for Space Power and Propulsion," International Astronautical Federation X X V t h C O n g r e s s, Am s t e r d a m ( S e p t em b e r 3 0 0 c t o b e r -~ l 9 7 4 ) . 60. R.T. Schneider, private communication (1974).

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61. 506 R. Paternoster, M.J. Ohanian, R.T. Schneider, and K. Thom, 11 Nuclear Waste Disposal Utilizing a Gaseous Core Reactor," Trans. Amer. Nucl. Soc., 17(2) (1974). 62. H.C. Honek, ENDF/8--Specifications for an Evaluated Nuclear Data File for Reactor Applications, 11 BNL 50066 (May 1966). . 63. A.F. Henry and A.V. Vota, 11 WIGL2: A Program for the Solution of the One-Dimensional, Two-Group, Space-Time Diffusion Equations Accounting for Temperature, Xenon, a n ct C o n t r o l F e e d b a c k , 11 \•J A P D TM 5 3 2 ( l 9 6 5 ) . 64. K.R. Hansen an d S.R. Johnson, 11 GAKIN: A One-Dimensional, Multigroup Kinetics Code," GA-7543 (1967). 6 5 . J . B . Ya s i n s k y , M . N a tel so n a n d L . A . Ga gem a n , 11 T\H G L : A Program to Sol~e the Two-Dimensional, Two-Group, Spacet i me N e u t r o n D i f f u s i o n w i t h T em p e r a t u r e F e e d b a c k , 11 ~, A P D TM-743 (1968). 66. L.A. Hageman and C.J. Pfeifer, "The Utilization of the Two-Dimensional Neutron Diffusion-Depletion Prdgram PDQ5 , 11 lJ APT D TM 3 9 5 ( J a nu a r y l 9 6 5 ) . 67. V.A. Dmitrievskii and L.A. zaklyaz'minskii, 11 Induction Magnetohydrodynamic Generator with a Hollow Nuclear Reactor, 11 Teplofizika Vysokileh Temperatur, Vol. 9, No. 2, pp. 405-412 (March-April 1971). . 68. I.K. Kikoin, V.A. Dmitrievskii et al., "Experimental Reactor vJith Gaseous Fiss ionable Substance (UF ), 11 Proceedings of the 2nd Int~rnational Conferenc~ on the P_~ a C e f u l u s e s O f ' A t Om i C E n e r g y , V O l . 2 , p . 2 3 2 , MO s C O \'/ 11959). ' 69 . . V.A. Dmitrievskii, E.M. Voinow and S.D. Tetel 'baum, "Use of Uranium Hexafluoride in Nuclear Power Plants, 11 Atomnaya EnergiB_, Vol. 29, No. 4, _ pp. 251-255 (October 1959). 70. V. A. Dmitrievskii et al., "Simulator of Uranium Hexa fluoride Steam Turbine Power Plant, 11 Atom~aya Enerqiya, Vol. 32, No. 5, pp. 446-447 (May 1972). . . 71. N.J. Diaz and E.T. Dugan, "Gas Fueled Heteroqeneous Core Reactors, 11 Disclosure of Invention, University of Florida (October 15, 1974). 72. L.C. Fuller, C.A. Sweet and H.I. Bowers, 11 0RCOST--A Computer Code for Summary Capital Cost Estimates of Steam Electric Power Plants," ORNL-TM-3743 (September 1972).

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507 73. H.J. Bowers, L.D. Reynolds, R.C. Delozier and B.E. Srite, 11 CONCEPT--Computerized Conceptual Cost Estimates for Ste . am-Electri . c Power Plants," ORNL-4809 (April _1973). . . . --~"74. "CINCAS--A Nuclear Fuel Cycle Engineering Economy and Accounting Forecasting Code,'' Argonne Code Center Abstract Reference #354 (November 1968). 75. R. Salmon, "A Procedure and a Computer Code (POWERCO) for Calculating the Cost of Electricity Produced by Nuclear Power Stations," ORNL-3944 (June 1966). 76. J.H. Lofthouse and J.F. Kunze, "Spheri~al Gas Core Reactor Criticai Experiment," Report NASA-C-67747-A (February 1971 ). 77. G.J. Bell and S. Glasstone, Nuclear Reactor Theory, pp. 463-482, Van Nostrand Reinhold Company, New York (1970). 78. "Flow of Fluids Through Valves, Fittings, and Pipe," Ninth Printing, Crane Co., Technical Paper No. 410, Chicago (1965).

PAGE 546

BIOGRAPHICAL SKETCH Edward T. Dugan was born in Fountain Hill, Pennsylvania, pn June 10, 1946. He graduated as ~ale~ictorian of his class from Notre Dame High School, Green Pond, Pennsylvania, in June, 1964. In June, 1968, he was graduated magna cum laude from the University of Notre Dame, -Notre Dame, Indiana, with a degree in mechanical engineering. In September, 1968, he started graduate school at the University of Florida. His.studies were interrupted by military service which began in January, 1969. On August 17, 1970, he received an honorable discharge from the United States Army after serving 13 months in Vietnam. His military a~ards included the National Defense Ribbon, the Good Conduct Medal, the Army Commendation Medal with Two Oak Leaf Clusters, the Vietnam Service Medal and the Vietnam Campaign Ribbon. He returned to the University of Florida to contin~e his graduate st~dies inSepte~ber, 1970. He completed the requirements for an M.S.E. iri Nuclear Engineering Sciences in March, 1972, with his studies being supported by a U.S. Atomic Energy Commission Special Fellowship, by a Graduate School Fellowship from the University of Florida, and by a 508

PAGE 547

509 University of Florida Graduate Assistantship. He is a member of Pi Tau Sigma and of the American Nuclear Soc:'i~tS,......

PAGE 548

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presehtation and is fully adequate,in scope and quality, as a dissertation for the degree of Doctor of Philosophy. < Nils J. rman Associa r of Nuclear Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly• presentation and is fully adequate, in scope and quality, as a dissertation for the degree of.,,..Q..oJ t-or of Philosophy. Enqi neeri nq I certify that i have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the 1~gree of Doctor of Philosophy. /uJt~ /JI{~ Richard T. Schneider Professor of Nuclear Enqineering ,. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the deqree of Doctor of Philosophy . . C "~ c:::: . c2{?A&" Cavin C .7' iv.er Professor of Mechanical Engineering

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I certify that I have read this study and that.in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associate Professor of Nuclear Chemistry This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partiijl fulfillment of the requirements for the degree of Occtor of Philosophy. March, 1976 Dean, Graduate School