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The nuclear piston engine and pulsed gaseous core reactor power systems

Material Information

Title:
The nuclear piston engine and pulsed gaseous core reactor power systems
Creator:
Dugan, Edward Thomas, 1946-
Publisher:
Edward Dugan
Publication Date:
Language:
English
Physical Description:
xxxviii, 509 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Engines ( jstor )
Gas pressure ( jstor )
Gas temperature ( jstor )
Gas turbines ( jstor )
Neutrons ( jstor )
Photoneutrons ( jstor )
Piston engines ( jstor )
Pistons ( jstor )
Reflectors ( jstor )
Turbines ( jstor )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF
Engines.
Nuclear Engineering Sciences thesis Ph. D.
Nuclear fuels.
Pistons.

Notes

General Note:
Typescript.
General Note:
Vita.
General Note:
Bibliography: leaves 501-507.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
000171116 ( aleph )
02948820 ( oclc )
AAT7538 ( notis )

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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 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. Mockel, 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. Mockel. 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.

Iii

%










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 Engineering. 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 opera-

tional 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 availa-

bility 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 carry-

ing on more extensive investigations of the nuclear piston

engine concept. It is recognized that a complete system

analysis and optimization will not only be difficult but

also expensive. A demonstration '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 organi-

zation of this 'dissertation. First, most of the equations

and derivations used in the neutronics and energetic analy-

sis of the hUclear 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 the-appropriate equation(s) and corresponding

appendix where necessary. It is felt that this approach

renders a more convenient and ordered presentation and facil-

itates 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 pre-

sented in Chapter IV. The line of reasoning was to examine

these simpler engines first, in some detail, before pro-

ceeding 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 promis-

ing venture, work was begun on the more intricate four-

stroke configurations. Results from this phase of the

research are presented .in Chapter V.















TABLE OF CONTENTS


Paqe
ACKNOWLEDGMENTS . . . . ii

PREFACE . . . . . iv

LIST OF TABLES . . . . . xii

LIST OF FIGURES . . . . xviii

LIST OF SYMBOLS AND ABBREVIATIONS . . xxvii

ABSTRACT. . . . . . xxxvi

CHAPTER

I INTRODUCTION . . . . 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 Pre-
vious 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 Moderating-Reflector
Studies . ... . . 54
Remarks Concerning the Algorithms Used in
the NUCPISTN Code . . . 57
Parametric Studies with NUCPISTN for D20-
Reflected 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 Helium-
to-U235 Mass Ratio. . . 93
Effects of Variations in the Neutron
Source Strength . .. . 93
Performance Analysis of Two D20-Reflected
Piston Engines. . . . 98
NUCPISTN Results-Compared with Higher Order
Steady-State Neutronic Calculations . 107
Some Remarks Regarding the. D20 Tempera-
tures . . . . .. 116
Exhaust Gas Temperature Calculations for
the Two-Stroke Engines . . .117
Mass Flow Rates for the Two-Stroke Engines. 120
Thermodynamic Studies for 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


v'i i 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 Tempera-
ture 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 Tech-
niques 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, Effec-
tive B'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









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 Tempera-
ture 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 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 GENERAL POINT REACTOR KINETICS EQUATIONS 434

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

E THE ENERGETIC 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 Moderating-
Reflector Materials at 2900K . . 55

8 Beryllium-D20 Composite Reflector Study at
2900K . . . . 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 Piston-
Gas Turbine-Steam Turbine System Which Uses
Piston Engine #2 . . . 126.


xii









LIST OF TABLES (continued)


TABLE Page

12 Summary of Thermodynamic .Results for the Piston-
Steam Turbine System Which Uses Piston Engine
.# . . .. . . 1 2 9

13 Summary of Thermodynamic Results for the Piston-
Cascaded 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 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#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 3.0M/W Lattice . 175

31 Burnup Calculations for a System Using 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









LIST OF TABLES (continued)


TABLE Page

40 Six Factor .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 UF6 Using the Engine #8 Configura-
tion at the TDC Position . . . 257

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

46 Fuel Temperature Coefficient of Reactivity for
80% Enriched UF6 Using the Engine #8 Configura-
tion at the TDC Position . . . 259

47 D20 Moderating-Reflector Density or Void Co-
efficient 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. Compensa-
tion 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 Con-
stants from NUCPISTN for Engine #I0 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 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









LIST OF TABLES (continued)

TABLE Page

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 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 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


xvii















LIST OF FIGURES

FIGURE Page

1 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 Temoerature 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 Graphite-
Reflected Engine. .. . . . 31

7 UF6 Nuclear Piston Engine Performance for the
Case of Graphite Steo-Reflector Addition at the
0.1 Cycle Fraction . . . 33
235
8 -Neutron Multiplication Factor Versus U Atom
Density for Systems Which Have No Helium Gas
Present in the Core and a 100cm Thick D20
Reflector at 290K. . . . 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 Mechanical Efficiency and Peak Gas Temperature
Versus Initial Gas Pressure for a Two-Stroke
Engine . . . . .. . 75

17 Mechanical Power and Pe.ak 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 020 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 Temperature
Versus the Cycle Fractio.n 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 Mechanial Power and Peak Gas Pressure Versus
He-to-U Mass Ratio for a Two-Stroke Engine 94

32 Mechanical Eff icency and Peak Gas Temperature
Versus He-to-U 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 #1 . . . . 100

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

37' Gas Pressure Versus Cycle Fraction for
Engine #1 . . . . 102










LIST OF FIGURES (continued)


FIGURE Page

38 Average Core Thermal Neutron Flux Versus
Cycle 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 D20
Reflector Thickness as Obtained from Four-
Group 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 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
I4


xxi









LIST OF FIGURES (continued)


FIGURE Page

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

52 Delayed Neutron Precursor Concentration Build-
up During Startup for Engine #4 . 153

53 Peak Gas Temperature and Mechanical Power Out-
put Behavior During Startup for Engine #4 155

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

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

56 Delayed Neutron Precursor Concentration Build-
up During Startup for Engine #5 . 158

57 Peak Gas Temperature and Mechanical Power Out-
put 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 and Mechanical Power Out-
put Behavior During Startup for Engine #8 207

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


xxii









LIST OF FIGURES (continued)


FIGURE Page

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

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
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 Paqe

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 Fast 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


xxiv










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


100. Rapid Power Transients for Engine #10 Induced
by Changes in the Loop Circulation Time .


Page


. 309


D20 Reflector Thickness Versus Piston Cycle
Time for Engine #11 ... . . 326

Gas Temperature Versus Piston Cycle Time for
Engine #11. . . . . .. 327

Gas Pressure Versus Piston Cycle Time for
Engine #11. . . . . 328

Average Core Thermal Neutron Flux Versus Piston
Cycle Time for Engine #11 . . 329

Neutron Multiplication Factor Versus Piston
Cycle Time for Engine #11 . . .... 330

Schematic of NASA's NERNUR--A Large Power
Generating System Utilizing a UF6 Gas Core
Nuclear Reactor . . . .. . 362


xxvi


101


102


103


104


105


106















LIST OF SYMBOLS AND ABBREVIATIONS


A constant appearing in the steady-state, two-group, two-
region diffusion theory flux expressions

2
A cross-sectional area of exhaust valve (m )

A. cross-sectional area of intake valve (m )

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

aD fraction of fast neutrons from th.e fast. inhomogeneous
source in the core which leave the core and return as
thermal neutrons.

a a parameter which is the product of f times y times 6

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

B. "effective" yield fraction for delayed-neutron precur-
sor group j

P
yield fraction for photoneutron precursor group j
.j
BR breeding ratio

BWR boiling water reactor

c subscript indicating the core region

C 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

c. "effective" delayed neutron precursor concentration
J. for delayed group j

C. delayed neutron precursor concentration for delayed
group j


xxvii









P
C photoneutron precursor concentration for photoneutron
precursor group j

Ce flow coefficient for the exhaust valve

C. flow coefficient for the intake valve

c gas specific heat at constant volume

c gas specific heat at constant pressure

X the normalized energy spectrum for fission neutron
emission; x = [Xp i- ) + zXjB. ]

X. the normalized energy spectrum for delayed fission neu-
tron group j

XP the normalized energy spectrum for prompt fission neu-
trons

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

6 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

m mechanical efficiency


xxviii










cl cycle fraction for step-reflector addition

e2 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

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

F production operator or volume integral of the adjoint
weighted fission source

f fraction of the removal cross section which is down-
scattering
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

f geometry factor equal to the core volume over the
moderating-reflector volume

Y geometry factor given by (R3 R3)

'Y fraction of those gammas reaching the moderating-
reflector region with energy above the (y, n) threshold
which actually induce photoneutrons

h enthalpy (B/lb m.); subscript indicating hydraulic

AH increase in enthalpy of a system (BTU's)

he energy or enthalpy of the mass leaving the system (B/lb )

hi energy of enthalpy.of the mass entering the system (B/lb )

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

JD number of delayed neutron precursor groups

J number of photoneutron precursor groups

K ...K2 coefficients which are convenient groupings of vari-
1'6us reactor physics parameters used in solving the 2-
group, 2-region neutron diffusion theory equations

k, k eff, k-effective the effective (static) neutron multipli-
cation factor for a reactor or system

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

k 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 [vZflc/[ za c

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

K I the inverse thermal diffusion length in the moderating-
reflector

Kc the inverse thermal diffusion length in the core when
is unity

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

[z ] convenient grouping of reactor physics parameters de-
.c fined as (1/v) c/[zac

A neutron generation time usually defined as [9,/k]
D
X delayed neutron precursor decay constant for delayed
group j

X photoneutron precursor decay constant for photoneutron
precursor group j

LPT low pressure turbine


XXX









tge mass flow rate out of the cylinder

Mi mass flow rate into the cylinder

m subscript indicating the moderating reflector; symbol
denoting mass, usually the HeUF6 mass

mw coolant water flow rate
MS gas mass which enters or leaves the system during the
time At

n neutron density (neutrons/cm3); symbol indicating neu-
trons

N neutron population

235
N uranium-235 atom density (atoms/barn-cm)
N 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

vc 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

Pm mechanical power output

P(t) amplitude factor or amplitude function

PWR pressurized water reactor

PNL thermal non-leakage probability

S scalar neutron flux


xxxi










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

S shape factor or shape function

D angular neutron flux

+ angular adjoint neutron flux for a time independent
0 critical reference system

Q net amount of heat added to a system from the surround-
ings

Qf fission heat release
AqR heat of reaction; the fission heat from a nuclear reac-
tion

Qf the rate of fission heat release

Q(t) weighted source term.appearing in the point reactor
neutron kinetics equations
MAX
QE ratio of the maximum fission heat released in any
energetic equation timestep to the total fission heat
released during the piston cycle
MAX
QN ratio of the maximum fission heat released in any neutron
kinetics equation timestep to the total fission heat
released during the piston cycle

q total heat transfer rate

0out 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

R 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 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-point
integration relations

s entropy (B/1bm R)

S inhomogeneous fast neutron source strength in the core
(neutrons/sec)
f
5 f 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
t
5 average thermal neutron density per unit time in the
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(r,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)

Sa macroscopic neutron absorption cross section

Es macroscopic neutron scattering cross section

Er macroscopic neutron removal cross section

f. macroscopic neutron fission cross section

Eout macroscopic cross section for neutron transfer out of
a group by scatter
E ft macroscopic cross section for neutron scatter from the
fast to the thermal group

E.s isotropic component of the macroscopie neutron elastic
0 transfer cross section

Es linearly anisotropic component of the macroscopic neu-
1 tron elastic transfer cross section


xxxiii










Etr macroscopic neutron transport cross section

Of microscopic neutron fission cross section

T temperature

T 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 super-
script, 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

T. delayed neutron precursor mean lifetime for delayed
group j
P
T. photoneutron precursor mean lifetime for photoneutron
precursor group j

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 sources)

Wm weighting function or weighting term for the moderating-
reflector inhomogeneous sources)


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 col-
lision

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

Y. net expansion factor for compressible flow through the
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
Major Department: Nuclear Enqineering Sciences

Nuclear piston engines operating on gaseous fission-

able fuel should be capable of providing economically and

energetically attractive power generating units.

A fissionable gas-fueled engine 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 not spiral for
small gas-fueled plants to the extent that it does for solid-

fueled plants. The fuel fabrication (fuel and cladding,

spacer grids, etc.) is essentially eliminated; the engineer-

ing safeguards and emergency core cooling requirements are

reduced significantly.

The investigated nuclear piston engines consist of a

pulsed, gaseous core reactor enclosed by a moderating-

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


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 combi-

nation 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 strokess. Supercriti-

cality 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 fis-

sion products, cool the gas, and recycle it back to the

pistpn engine. Mechanical power can be directly taken by

means of a conventional crankshaft operating at low speeds.

For the purpose of evaluating the nuclear piston en-

gine cycle behavior, a computer code was developed which

couples the necessary energetic and neutronics equations.


xxxvii









The code, which has been named NUCPISTN, solves for the

neutron flux, delayed and photoneutron precursor concentra-

tions, 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 en-

gine's quasi-steady-state power level is capable of being

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 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 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
















CHAPTER I

INTRODUCTION


Description of Engine Operation

The investigated nuclear piston engines consist of a

pulsed, gaseous core reactor enclosed by a moderating-

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, UF6, and helium, He, gases. Highly 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. A'"

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 combina-

tion 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 strokess. 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.

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 1600'K) HeUF6
6
exhaust gas can be cooled in an HeUF6-to-He heat exchanger.

The heated He (,0000K 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









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,

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 peak-

ing 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 peaking unit

concept--even if the nuclear-fueled unit's power production

costs should be estimated 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.

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.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 solid-

fueled 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 capa-

city 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 nu-

clear 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 applica-

tions, the nuclear units utilized by ships are more expensive

than conventionally fueled units. The major advantage of cur-

rent nuclear-fu.eled 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 reac-

tors. It presents the models employed to analyze the neu-

tronics of gaseous cores, calculations performed, experi-

'ments conducted and appropriate comparisons between analyti-

cal 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 energetic equa-

tions 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 con-

cept can be firmly established.

Finally, all neutronics and energetic 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 slow-

ing 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 D00, Be and graphite reflectors.

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 moderating-

reflector while diffusion theory was used for the interior

cores with E s and transport theory was applied to the

interior if E << a*.
s 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 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 Pu239 and U235 while D20 (at

3000K) and graphite (at 3000F and at 32000F) were studied

Sas 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 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 cross sections.

Kaufman et al. [5] performed an extensive parametric

survey on gaseous core reactors in cylindrical geometry

in 1965. One-dimensional S 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 geomet-

ric variations, such as radius-to-height ratios, were

looked at using two-dimensional S transport theory.

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 upscat-

tering of neutrons returning to the core by the hot hydrogen

is an important effect and that for such cores, a multi-

thermal group approach is essential. One-dimensional

diffusion theory with 6 fast and 12 thermal groups was used








to evaluate reactor characteristics for changes in moderating-

reflector parameters such as temperature, slowing down

power, (n,2n) production, thermal scattering, and thermal

absorption. D 20, Be and graphite-moderated reactors were

studio 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 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 was considered. Dif-

fusion 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 adja-

cent regions and the consequent importance of upscattering

by the hot hydrogen and neon in the 1200 to 70000K core.









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 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 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
233 233
engine design on U critical mass, (b) compare U ,

U 235, 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 life-

times 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 Hyland in 1971 [19]. A composite,

D20-BeO-D20 moderating-reflector was used. Fuel tempera-

tures reached 44,000C and moderating-reflector temperatures

varied from 93C to 5600C. The cavity wall temperature









was 11150C. 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 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 hence an increase in the critical mass re-

quirement. 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 tempera-
235
ture than for U (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 S

transport theory with experimental data for gaseous uranium

core reactors and attained fairly good agreement. Mills also









performed parametric studies of gas-filled, reflector-

moderated 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

Mills were (a) systems are sensitive to the U235 content,

'(b) the systems are not sensitive to core diameter changes
235
for a constant amount of U2, (c) systems are sensitive

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 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 distri-

butions 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 spherical-

symmetry 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 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 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 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 "good" fast and thermal

group constants are used and provided there is a multi-

thermal-group approach which allows for full upscattering

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 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 systems.

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.















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 Schneider and Ohanian [38]. Preliminary feasi-

bility studies by Kylstra et al. [39] showed that a UF6-

fueled, 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 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.








20










EAT
HANGER










4- MODERATING-REFLECTOR








PISTON MODERATING-REFLECTOR


FIGURE 1. Simple Schematic of A UF6 Nuclear Piston Engine


I I


- ^------.--''3-: ..^LI--r -_--.__.-(*._L_









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 reflec-

tor thickness was varied throughout the cycle. At the com-

pression stroke start, the reflector was a thin reflector,

increasing in thickness slowly, then stepping to a thick

reflector at some cycle fraction, Ec, 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 UF6 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
MODERATING-REFLECTOR








LOW LEAKAGE,
SUPERCRITICAL
ARRANGEMENT





AFTER STEP-REFLECTOR
ADDITION AT I





CORE

MODERATING-REFLECTOR





HIGH LEAKAGE, SUBCRITICAL
ARRANGEMENT



AFTER STEP-REFLECTOR REMOVAL
AT c OR BEFORE STEP-REFLECTOR
ADDI ION AT eI


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 tempera-

tures (1000-1200'F) so as to minimize the convective and

conductive heat losses from the core region. The compres-

sion ratio for the engine was 10-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 precur-
sors 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


LIQUID


10
SOLID






1.0 TRIPLE POINT




VAPOR


0.1








250 300 350 400 450 5UU 55U
TEMPERATURE (K)



FIGURE 3.. UF6 Phase Diagram
.6









The set of equations resulting from the above assump-

tions 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 neu-

trons. The use of the diffusion equations was justified

only by their simplicity compared to S or P approxima-
n 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 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 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, eI


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 4000K. 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 tem-

perature, average core thermal neutron flux and neutron mul-

tiplication factor over a complete compression-power stroke

cycle for a typical set of independent parameters. The step-

reflector addition was at a cycle fraction E1 = 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

12300K (17540F). 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 1900-1200F, a peak tempera-

ture of 1754F 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

1I = 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





























1.25




1.0




1.75


0.1 0.3 1.0 3.0 10

UF6 PRESSURE (atm)

(Gas Temperature = 400K)










FIGURI. 4. Neutron Multiplication Factor Versus UF6 Partial
Pressure for ai Infinite Graphlite Reflector [39]


























1250





1000 2
0




750 t-




500


25 50 75


PERCENT TRAVEL THROUGH CYCLE





Initial Gas Pressure = 1 atm
Initial Cas Tompaerature = 400K
U-235 Mass = 2.15 kg


FIGURE 5.


Gas Pressure anJ Temperature as a Function of
Percent Travel Through the Piston Cycle for a
Graphi te-PR.fi r Led Engine [39]













































L I I 10.4
0 25 50 75 100


PERCENT TRAVEL THROUGH CYCLE





Initial Gas Pressure = 1 atm
Initial Cas Temperature = 400K
11-235 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 Graphite-
Reflected Engiina, [39]


1015









235
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
235
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 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 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 UF6 fuel.
6














2 2.10 2.15 2.20 2.25 2.30 2.35
10 -----------

EFFIC [ ENCY
60 0

40

I-4
INITrTAL PRESSURE ~ 20 0
I = t1 ;tmr- 0 .
if = 2 aun T
10iI =1 4 atni TOI

~ E. = 0.1 -
.1 ,: = 0.5







S100 10













~ .






2.10 2.15 2.20 2.25 2.30 .
10 I I I 101
U oL\ASS (kg)
Comiiipres;sion Ratio = 10-to-I
Cl ar;i ti Volin (in3) = 0.24
Engilme Spe-d rpm'ss) = 100

FIGURE 7. UF6 Nuclear Piston Engine Performance 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 3) 0.0394 0.24

Compression Ratio 12-to-i 10-to-1

Displacement Volume (m 3) 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

Power Density (w/cm 3) 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 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.















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 valua-

ble 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 de-

layed and photoneutrons are ignored is given by equation (D-

23).

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 develop-

ment of this equation is set forth in the first section of

Appendix E.

The NUCPISTN code then couples the neutronics and

energetic 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 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 non-

simple harmonic motion.. The piston behavior during the

cycle is 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










of energy release in an efficient and systematic manner.

Non-1/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 out-

put 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-II

[18] in cylindrical geometry. The collapsed fast group con-

stants for 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 ob-

tained from a 30-group BRT-1 [45] calculation. BRT-l is the

Battelle-revised version of the industry benchmark computer

program, THERMOS. The PHROG, BRT-1, and NUCPISTN group

constants were then compared with collapsed group constants

obtained from 123-group, one-dimensional,Sn transport theory

calculations which were performed with the powerful trans-

port scheme of XSDRN [46] in spherical geometry. S4 and

S6 quadratures were used.






40


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 HeUF6-

to-He heat exchanger was carried out for one of the piston

engine power generating systems. As a result of this analy-

sis, 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 sig-

nificantly 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 cal-

culaLui' dal scheme utilized in NUCPISTN and also to the im-

proved 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 configura-

tions 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 water-

reflected 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 10-to-I and clearance volumes of 0.24m3. 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

around ,3 to 5 feet. Neutronic considerations require that

the clearance volume be large compared to conventional in-

ternal 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 fac-

tors for these engines at TDC as obtained from various com-

putational schemes. The NUCPISTN results were from two-

group, 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 S

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 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












TABLE 3


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


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

Core Atom Densities

U-235 (atoms/barn-cm) 7.192x10-5 2.24x10-5 2.67x10-5 2.694x10-5 2.687x10-5

F-19 (atoms/barn-cm) 4.315x10-4 1.344x10-4 1.602x10-4 1.616x10-4 1.612x10-4

He-4 (atoms/barn-cm) 0.0 1.60x10-4 1.56x10-4 3.39x10-4 5.24x10-4

Reflector Atom Density

C-12 (atoms/barn-cm) 8.08xi0-2 8.27x10-2 8.08x10-2 8.13x10-2 8.18x10"2

Core Temperature (K) 2576 1000 2400 1800 1530

Reflector Temperature (oK) 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 Graphite-
Reflected Engines at the TDC Position

Computational 2-group 2-group 123-group 123-group
Engine Scheme NUCPISTN CORA XSDRN XSDRN
ik ** k *** k ****
eff. eff. eff. eff.


G-1 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
**with fast interactions in core
***with fast interactions in core; S4 quadrature
****with fast interactions in core; S6 quadrature









replaced by a two-region spherical configuration which pos-

sessed an equivalent volume core and an equivalent thick-

ness 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 NUCPISTN

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 XSDRN 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 PHIROG constants in CORA means that any










.upscattering to above 0.683eV is neglected. The 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

effective. 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 graphite-

reflected 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 multiplica-

tion 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 0.240m3

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 gra.phite-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 Moderating-Reflector Materials at the TDC Position


Moderating- Reflector Two-group Two-group -Four-group Two-group Four-group Two-group
Reflector Physical NUCPISTN* CORA* CORA* CORA** CORA** EXTERMI-
Material Temperature kff keff keff k ef kf NATOR**
(oK) ke
eff


D20
D 20


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 moderating-

reflector 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 interac-

tions 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-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 two-

group 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 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

k fftiv which are 1 to 2% lower than the former. The
effectives
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 consis-

tently lower than the EXTERMINATOR-II results.

When comparing the four-group results for the D20-

reflected 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 two-

group 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 neutron multiplication factors for these

systems.

In contrast, the two-group keffectives for the Be-

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










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 modera-

tor 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-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

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
Material ( K) (K) (msec) (msec) (msec)


D 0
D 0
D 0
D O
D 2 O
D' 0O
D 0


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 slow-

ing down power,the order is H20, D20, Be, BeO and then C.

The moderating ratio is defined as (E /Z) 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 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 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 (TE D ) or (fd /T E D ) which has been
m a iM m a m
m m
given the name "moderating-reflector power." If one con-

siders a two-group, externally moderated gaseous core reac-

tor, it is desired that Dt and E t for the moderating
m f
reflector be small. It is also desired that D for the
m
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










(D /Z ), it is hence desirable that the combination T D tE
Sr m m a
Sm t t -1 d m
be small or that (TmDmza ) be large. The term f is the

fraction of the removal cross section which is downscatter-

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 (fd/T D E ), i.e., the
m mIm a
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 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 2900K. 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 composi-

tion. 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 290K

Moderating-Reflector Power
Material t t1 d tt
(T DI) (f /Tma m
mam m mI m
m m


D 20


BeO


Be


C


H20


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 moderating-
reflector region
t t
Eam and Dm obtained from BRT-'I calculations
a o m

T and fd obtained from PHROG calculations









TABLE 8

Beryllium-D20 Composite Reflector Study at 2900K


Inner Reflector
Region (Be)
Thickness
(cm)


0

5

10

15

20

25

30

35

40

45

50

55

60

65

70


Outer Reflector
Region (D 0)
Thickn ess
(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 = 2900K.

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 2900K. The penalty in the neutron multipli-

cation 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 pre-

sented 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










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 minutes for each piston 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 in

the core leads to errors in effective which are of the

order of 2 to 5%

The restriction at this point to two regions 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

are generated internally as a function of the gas density

throughout 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










and BRT-1) and treated as constant during the piston cycle.

Although the moderating-reflector constants depend to an

extent on the core comDosition, treating them as constant

during the cycle is a very good approximation if the reflec-

tor dimensions do not change. 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 group 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

effective 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 grout 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 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 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. 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 accommodate 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


C = 32.43 + (7.936xO1-3)T (32.068x104)/T2 (cal/moleK) (1)
P









where T is in degrees Kelvin [47, 48], This formula how-

ever 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/moleOK) (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 24000K. 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 energetic and neu-

tronics equations, of the procedures used for timestep

selection for the energetic 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,

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 El = 0.050 cycle fraction and removed at the

E2 = 0.650 cycle fraction; Engine #2 has the step-reflector'

applied at the cI = 0.100 cycle fraction and removed at the

e2 = 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 tempera-
235
ture is 2900K and the helium-to-U 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 multi-

plication 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




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