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 oneyear
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, quasisteadystate
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 theappropriate 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 twostroke (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 fourstroke systems. Later,
after it became apparent from the twostroke 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 TWOSTROKE ENGINE STUDIES . 36
Introduction . . . .. 36
Comparison of Initial Results with Pre
vious Nuclear Piston Engine Analyses. 40
GraphiteReflected Systems. . .. 42
GraphiteReflected Systems Compared with
Systems Using Other ModeratingReflector
Materials . . . .. 47
ModeratingReflector Power .' 51.
Some Composite ModeratingReflector
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 StepReflector Thicknesses. ... 83
Effects of Variations in the Cycle
Fraction Position for StepReflector
Addition and StepReflector Removal 86
Effects of Variations in the Helium
toU235 Mass Ratio. . . 93
Effects of Variations in the Neutron
Source Strength . .. . 93
Performance Analysis of Two D20Reflected
Piston Engines. . . . 98
NUCPISTN ResultsCompared with Higher Order
SteadyState Neutronic Calculations . 107
Some Remarks Regarding the. D20 Tempera
tures . . . . .. 116
Exhaust Gas Temperature Calculations for
the TwoStroke Engines . . .117
Mass Flow Rates for the TwoStroke Engines. 120
Thermodynamic Studies for Three Nuclear
Piston Engine Power Generating Systems. 121
Nuclear PistonGas TurbineSteam
Turbine System. . . .. 122
PistonSteam Turbine System . 123
PistonCascaded 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 FOURSTROKE 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 ModeratingReflector Tempera
ture Coefficients of Reactivity . 254
D20 ModeratingReflector 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 FourStroke 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
SteadyState Neutronic Analysis . 373
ModeratingReflector Studies. .. 375
Fuel Studies . . . 376
Neutron Cross Section Libraries . 377
ModeratingReflector 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
HetoU Mass Ratio Studies. .. . 387
StepReflector Addition and Removal 388
Parametric Studies. . . .. 389
ModeratingReflector 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
HeUF6toHe 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 TWOGROUP, TWOREGION, ONEDIMENSIONAL
DIFFUSION THEORY EQUATIONS USED IN THE
NUCPISTN CODE WHEN PHOTONEUTRONS ARE IGNORED 405
B TWOGROUP, TWOREGION, ONEDIMENSIONAL
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
GraphiteReflected Piston Engines Analyzed by
Kylstra et al . . . 27
2 Comparison of a Typical GraphiteReflected, UF6
Piston Engine with the Nordberg Diesel . 34
3 Atom Densities and Temperatures for Large
GraphiteReflected Engines at the TDC
Position . . . . 43
4 Neutron Multiplication Factors for the Large
GraphiteReflected Engines at the TDC
Position . . . . 44
5 Neutron Multiplication Factors for Small
Engines with Various ModeratingReflector
Materials at the TDC Position. . .. 48
6 Moderator Neutron Temperatures and Neutron
Lifetimes for Small Engines with Various
ModeratingReflector Materials at the TDC
Position . . . . .. 52
7 ModeratingReflector Power for Some Moderating
Reflector Materials at 2900K . . 55
8 BerylliumD20 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 TurbineSteam 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 ModeratingReflector 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 ModeratingReflector 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 #9Like 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 BRTl 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 TurbineSteam 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 TurbineSteam 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 StepReflector 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
GraphiteReflected 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 SteoReflector 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 TwoStroke Engine 69
xviii
LIST OF FIGURES (continued)
FIGURE Page
12 Mechanical Efficiency and Peak Gas Temperature
Versus Compression Ratio for a TwoStroke
Engine . . . . . 70
13 Mechanical Power and Peak Gas Pressure Versus
Clearance Volume for a TwoStroke Engine. .. 71
14 Mechanical Efficiency and Peak Gas Temperature
Versus Clearance Volume for a TwoStroke Engine ..72
15 Mechanical Power and Peak Gas Temperature Versus
Initial Gas Pressure for a TwoStroke Engine. 74
16 Mechanical Efficiency and Peak Gas Temperature
Versus Initial Gas Pressure for a TwoStroke
Engine . . . . .. . 75
17 Mechanical Power and Pe.ak Gas Pressure Versus
Initial Gas Temperature for a TwoStroke Engine 76
18 Mechanical Efficiency and Peak Gas Temperature
Versus Initial Gas Temperature for a TwoStroke
Engine . . . . . 77
19 Mechanical Power and Peak Gas Pressure Versus
Engine Speed for a TwoStroke Engine. . 79
20 Mechanical Efficiency and Peak Gas Temperature
Versus Engine Speed for a TwoStroke Engine 80
21 Mechanical Power and Peak Gas Pressure Versus
Neutron Lifetime for a TwoStroke Engine. . 81
22 Mechanical Efficiency and Peak Gas Temperature
Versus Neutron Lifetime for a TwoStroke Engine 82
23 Mechanical Power and Peak Gas Pressure Versus
Initial D20 Reflector Thickness for a TwoStroke
Engine . . . .. . 84
24 Mechanical Efficiency and Peak Gas Temperature
Versus Initial D20 Reflector Thickness for a
TwoStroke Engine . . . . 85
xix
LIST OF FIGURES (continued)
FIGURE Page
25 Mechanical Power and Peak Gas Pressure Versus
D20 StepReflector Thickness for a TwoStroke
Engine . . .. . . 87
26 Mechanical Efficiency and Peak Gas Temperature
Versus 020 StepReflector Thickness for a Two
Stroke Engine . . . . 88
27 Mechanical Power and Peak Gas Pressure Versus
the Cycle Fraction for StepReflector Addition
for a TwoStroke Engine . . . 89
28 Mechanical Efficiency and Peak Gas Temperature
Versus the Cycle Fractio.n for StepReflector
Addition for a TwoStroke Engine. . .. 90
29 Mechanical Power and Peak Gas Pressure Versus
the Cycle Fraction for StepReflector Removal
for a TwoStroke Engine . . .. 91
30 Mechanical Efficiency and Peak Gas Temperature
Versus the Cycle Fraction for StepReflector
Removal for a TwoStroke Engine . . 92
31 Mechanial Power and Peak Gas Pressure Versus
HetoU Mass Ratio for a TwoStroke Engine 94
32 Mechanical Eff icency and Peak Gas Temperature
Versus HetoU Mass Ratio for a TwoStroke
Engine . . . . . 95
33 Mechanical P.ower and Peak Gas Pressure Versus
Neutron Source Strength for a TwoStroke
Engine. . . . . .. 96
34 Mechanical Efficiency and Peak Gas Temperature
Versus Neutron Source Strength for a TwoStroke
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 TwoGroup
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 D20Reflected Core as Obtained from
TwoGroup CORA Calculations . . 114
43 Fast and Thermal Adjoint Neutron Flux Versus
Radius for a D20Reflected Core as Obtained
from TwoGroup CORA Calculations. . .. 115
44 PistonGas TurbineSteam Turbine Schematic
for the Power System Which Uses Piston
Engine #2 .. . . . . 124
45 Steam and Gas Turbine TemperatureEntropy
Diagrams for the PistonGas TurbineSteam
Turbine System Which Uses Piston Engine #2. 125
46 PistonSteam Turbine Schematic for the System
Which Uses Piston Engine #1 . . 127
47 Steam Turbine TemperatureEntropy Diagram for
the PistonSteam Turbine System Which Uses
Piston Engine #1. . .. . 128
48 PistonCascaded Gas Turbine Schematic for the
System Which Uses Piston Engine #2. .. 130
49 Gas Turbine TemperatureEntropy Diagram for
the PistonCascaded Gas Turbine System Which
Uses Piston Engine #2 . . . 131
50 Diagram of a D20Reflected, 3tol Compression
Ratio Nuclear Piston Engine at the TDC
Position . . . . .. 139
I4
xxi
LIST OF FIGURES (continued)
FIGURE Page
51 Sketch of an 8Cylinder Nuclear Piston Engine
Block for 4050 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 NERNURA 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 steadystate, twogroup, two
region diffusion theory flux expressions
2
A crosssectional area of exhaust valve (m )
A. crosssectional 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 steadystate, twogroup, two
region diffusion theory flux expressions
D
D. yield fraction for delayed neutron precursor group j
J
B. "effective" yield fraction for delayedneutron 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 steadystate, twogroup, 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 twopoint finite difference
relations, 6 = 1.0 while for the threepoint 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 stepreflector addition
e2 cycle fraction for stepreflector 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
moderatingreflector with energy above the (y, n)
threshold
f geometry factor equal to the core volume over the
moderatingreflector 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 gascooled 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, 2region neutron diffusion theory equations
k, k eff, keffective 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 moderatingreflector 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 uranium235 atom density (atoms/barncm)
N uranium atom density (atoms/barncm)
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 nonleakage 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 corereflector 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 moderatingreflector 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 twopoint finite difference rela
tions and a value of onehalf for the threepoint
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 moderatingreflector
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 compressionpower cycle
Ti initial gas temperature in the inlet line or at the
beginning of the compressionpower 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 moderatingreflector 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 gasfueled engine has many of the ad
vantages associated with solidfueled nuclear reactors but
fewer safety and economical limitations. The capital cost
per unit power installed (dollars/kwe) should not spiral for
small gasfueled 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.
Twoand fourstroke 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 nonsimple 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 quasisteadystate 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 groundbased 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 twoand fourstroke 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 nonsimple 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 HeUF6toHe 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/krhr. Fuel costs are
presently estimated as being below $0,20 per million BTU*or
around 1.4 mills/kwehr.
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/kwhr, and efficiencies are not much greater
than 20%.
A conversion from wasteful, conventionally fueled peak
ing units to efficient, nuclearfueled 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, nuclearfueled peaking unit
concepteven if the nuclearfueled 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 fossilfueled
units now employed makes this concept that much more
attractive.
Alreadydeveloped nuclear reactor concepts like pres
surized water reactors (PWRs), boiling water reactors (BWRs),
and high temperature gascooled 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 scaleddown basis for
2050100 Mw(e) units. The cost per unit power installed
(dollars/kwe) for scaleddown units operating on these
alreadydeveloped solidfueled 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 fossilfueled units, their capital costs per unit
power installed would be considerably less than for any
scaleddown 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 largescale
fossil and largescale nuclearfueled 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 nuclearfu.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 nuclearpowered
vessels. On the other hand, the economic disadvantage is
the primary reason why nuclearpowered 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 twostroke nuclear piston engines is presented
in Chapter IV. Since these investigations indicated good
performance potential for the nuclear piston engine concept,
more sophisticated, fourstroke 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 moderatingreflector. Age theory was used to describe
neutron slowing down in the moderatingreflector (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 moderatingreflector material surrounding
the fuel comprised the "exterior" region. The analysis
included, but was not limited to gaseous cores. Low
density, liquidmetalfueled, 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 239fueled
systems with D20, Be and graphite moderatingreflectors.
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 U233fueled gaseous core
with a noninfinite, D20 moderatingreflector was surrounded
with an external thorium blanket. For a 1 meter diameter
core with a 50cm thick D20 moderatingreflector, 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 reflectormoderated, U235fueled, 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. Sixgroup, onedimensional diffusion
theory was used for the spherical configurations while
fourgroup, twodimensional 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 fourgroup, twodimensional diffusion theory.
The fuels considered were Pu239 and U235 while D20 (at
3000K) and graphite (at 3000F and at 32000F) were studied
Sas moderatingreflector materials. They looked at the
effect on critical mass of a variable fuel region radius
in a fixed size cavity. The moderatingreflector 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. Onedimensional S transport theory studies were
used initially to evaluate the effectiveness of various
moderatingreflectors (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 radiustoheight ratios, were
looked at using twodimensional S transport theory.
Comparisons made between calculations using 24, 15, 13 and
3 collapsed broad groups showed that a good set of collapsed
threegroup 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. Onedimensional
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 graphitemoderated 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 multigroup 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
(19661969) [812] for a nuclear light bulb model (closed
system) gaseous core nuclear rocket engine. BeO and graphite
moderatingreflectors with U 233, U235 and Pu239 fuels were
investigated. A series of 24group, onedimensional, 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 GAM1
[14] with the slowing down spectrum in the core being
assumed to be that of the moderatingreflector region.
TEMPEST [15] was used for the thermal absorption probabili
ties and SOPHIST [16] for thermal up and downscattering
probabilities. Flux and volume weighted fourgroup cross
sections were obtained from the 24group, onedimensional
transport calculations for use in twodimensional calcula
tions. Twodimensional transport theory calculations
were then performed with DOT [17] while twodimensional
diffusion theory calculations were obtained from the
EXTERMINATORII 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, fueltocavity radius and reflector thickness
for U233 and U 235fueled, open, gas core nuclear rocket
engines were looked at by Hyland in 1971 [19]. A composite,
D20BeOD20 moderatingreflector was used. Fuel tempera
tures reached 44,000C and moderatingreflector 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 GAMII [20] and
thermal group constants from GATHER [21]. Onedimensional,
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 reflectormoderated 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 gasfilled, 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 D20reflected cavity reactor. The
maximum discrepancy between calculated and experimental
results for this system was 3% in reactivity.
A series of critical experiments (19671969) [2532]
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 coaxialflow (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 onedimensional, 19group diffusion
theory, onedimensional, S4 and S8 transport theory, and
fourand sevengroup, twodimensional 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
thermalgroup 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 corepiston
engine. Significant differences exist in the modus operandi
of each concept. In the steadystate 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 mid1970, 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, Ottotype 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 neutronreflecting cylinder and piston. The
fuel was 100% enriched UF6 and the ignition process was
triggered by an "external" neutron source. The engine
was a twostroke 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 MODERATINGREFLECTOR
PISTON MODERATINGREFLECTOR
FIGURE 1. Simple Schematic of A UF6 Nuclear Piston Engine
I I
 ^.''3: ..^LIr _.__.(*._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 stepreflector was then removed, going back to a thin
reflector at some cycle fraction, 2', usually at TDC (see
Figure 2).
The moderatingreflector 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
MODERATINGREFLECTOR
LOW LEAKAGE,
SUPERCRITICAL
ARRANGEMENT
AFTER STEPREFLECTOR
ADDITION AT I
CORE
MODERATINGREFLECTOR
HIGH LEAKAGE, SUBCRITICAL
ARRANGEMENT
AFTER STEPREFLECTOR REMOVAL
AT c OR BEFORE STEPREFLECTOR
ADDI ION AT eI
FIGURE 2. ILLUSTRATION OF STEPREFLECTOR 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 (10001200'F) so as to minimize the convective and
conductive heat losses from the core region. The compres
sion ratio for the engine was 10toI with a clearance
volume of 0.24m3 and an engine shaft speed of 100rpm.
Neutronic Model
For the steadystate solution of the neutron balance,
a twogroup, tworegion 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 tworegion piston was represented by
a tworegion 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, keffective, and for the average
steadystate 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 singlegroup, point reactor
kinetics equation rather than from the expression derived
from the twogroup, steadystate 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 nonflow,
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 compressionpower 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
GraphiteReflected 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 StepReflector Addition, eI
Cycle Fraction for StepReflector Removal, 2.
Neutron Source Strength (neutrons/sec)
= 400
= 14
= 1.73.1
= 100
= 0.24
= 10to1
= 0.10.3
= 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 compressionpower stroke
cycle for a typical set of independent parameters. The step
reflector addition was at a cycle fraction E1 = 0.10 while
the stepreflector 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 stepreflector is
removed and the reactor is shut down. Since the cylinder
walls were to be maintained at 19001200F, 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 stepreflector 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
U235 Mass = 2.15 kg
FIGURE 5.
Gas Pressure anJ Temperature as a Function of
Percent Travel Through the Piston Cycle for a
Graphi tePR.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
11235 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 stopreflector 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 stepreflector 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 keffective dropping
to 0.99 to 1.01 as TDC was approached. This behavior of
keffective greatly increases the control safety since the
power doubling time is large at high power. Increasing' the
235
U 235 loading leads to larger keffectives 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 29inch bore, and a 40inch stroke. The fuel cost esti
mates were obtained by assuming a 20/gal price for diesel
fuel, and a $12/gm U charge for the UF6 fuel.
6
2 2.10 2.15 2.20 2.25 2.30 2.35
10 
EFFIC [ ENCY
60 0
40
I4
INITrTAL PRESSURE ~ 20 0
I = t1 ;tmr 0 .
if = 2 aun T
10iI =1 4 atni TOI
~ E. = 0.1 
.1 ,: = 0.5
S100 10
~ .
2.10 2.15 2.20 2.25 2.30 .
10 I I I 101
U oL\ASS (kg)
Comiiipres;sion Ratio = 10toI
Cl ar;i ti Volin (in3) = 0.24
Engilme Sped rpm'ss) = 100
FIGURE 7. UF6 Nuclear Piston Engine Performance for the Case
of Graphite StepReflector Addition at the 0.1
Cycle Fraction
TABLE 2
Comparison of a Typical GraphiteReflected, UF
Piston Engine (as analyzed by Kylstra et al.)6
with the Nordberg Diesel [39]
Characteristic Diesel UF6 Engine
Clearance Volume (m 3) 0.0394 0.24
Compression Ratio 12toi 10to1
Displacement Volume (m 3) 0.434 2.16
Shaft Speed (rpm) 200 100
Type of Cycle (# of strokes) 2 2
Net work per cycle (Mwsec) 0.224 0.86
Power (Mw) 0.746 1.436
Power Density (w/cm 3) 1.72 0.665
Efficiency (%) \2030 42
Fuel Cost ($/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
stepreflector 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 4bar 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 TWOSTROKE 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 twostroke engines were analyzed.
The compressionpower stroke cycles for these engines
were examined with the intention of eventually proceeding
to more complex fourstroke engines if the results from
the twostroke engine proved encouraging.
The first twostroke engines which were looked at
neglected delayed and photoneutron effects. Later twostroke
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. Twogroup, tworegion diffusion
theory equations in spherical geometry were still used for
the steadystate 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 steadystate neutron balance by Kylstra et al. (see
Chapter III) were initially maintained. Only the second
assumption was altered in that neutron absorption in the
moderatingreflector was no longer neglected. A complete
development of the twogroup, tworegion, steadystate
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 onespeed (thermal)
point reactor kinetics equation used by NUCPISTN when de
layed and photoneutrons are ignored is given by equation (D
23).
The same nonflow, closed system conservation of
energy equation as used by Kylstra et al. is again used in
the NUCPISTN code for the twostroke 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 compressionpower 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 keffective of the engine and also on the current rate
of energy release in an efficient and systematic manner.
Non1/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 fourgroup, onedimensional
diffusion theory calculations performed with CORA [44] in
spherical geometry and with .corresponding twodimensional
diffusion theory calculations performed by EXTERMINATORII
[18] in cylindrical geometry. The collapsed fast group con
stants for CORA and EXTERMINATORII were obtained from a
standard 68group, PHROG B1 calculation [35]. The thermal
group constants used in CORA and EXTERMINATORII were ob
tained from a 30group BRT1 [45] calculation. BRTl is the
Battellerevised version of the industry benchmark computer
program, THERMOS. The PHROG, BRT1, and NUCPISTN group
constants were then compared with collapsed group constants
obtained from 123group, onedimensional,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 HeUF6toHe 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
toHe 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 graphitereflected 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 10toI and clearance volumes of 0.24m3. The re
sulting strokes were therefore excessively largearound 19
to 20 feet. The heavy waterreflected 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 4tol for 3to 5foot strokes.
GraphiteReflected Systems
Presented in Table 3 are the uranium235, fluorine,
graphite, and helium atom densities (in atoms per barncm)
for five graphitereflected 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, tworegion 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 123group, tworegion S
transport theory calculations in which S4 and S6 quadratures
were used. The CORA results are from twogroup, tworegion
diffusion theory calculations in which the thermal group
constants were obtained from BRTl and the fast group con
stants from PHROG. Fast interactions in the core were in
cluded in this scheme.
All of the schemes utilized onedimensional spherical
geometry. The tworegion, twodimensional cylinder was
TABLE 3
Atom Densities and Temperatures for Large Graphite
Reflected Engines at the TDC Position
Characteristic/Engine G1 G2 G3 G4 G5
Core Atom Densities
U235 (atoms/barncm) 7.192x105 2.24x105 2.67x105 2.694x105 2.687x105
F19 (atoms/barncm) 4.315x104 1.344x104 1.602x104 1.616x104 1.612x104
He4 (atoms/barncm) 0.0 1.60x104 1.56x104 3.39x104 5.24x104
Reflector Atom Density
C12 (atoms/barncm) 8.08xi02 8.27x102 8.08x102 8.13x102 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 2group 2group 123group 123group
Engine Scheme NUCPISTN CORA XSDRN XSDRN
ik ** k *** k ****
eff. eff. eff. eff.
G1 1.179 1.221
G2 1.005 1.052 0.934 0.934
G3 1.025 1.067 0.954 0.954
G4 1.032 1.080 0.962 0.962
G5 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 tworegion 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 tworegion 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 BRTl 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 twogroup NUCPISTN keffectives are all around 4.to
5% lower than the corresponding twogroup CORA results.
Some of this discrepancy is due to differences which exist
between the thermal group constants generated by NUCPISTN
and the BRTl 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 123group XSDRN keffectives are all around 11 to
12% lower than the twogroup CORA keffectives. The thermal
group cutoff in BRTl is 0..683eV and full upscattering and
downscattering below this energy are accounted for. The use
of the BRTl and PHIROG constants in CORA means that any
.upscattering to above 0.683eV is neglected. The 123group
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 twogroup CORA keffectives are thus higher than the
XSDRN keffectives for two reasons. First, the twogroup .
analyses do not give proper emphasis to the nonthermal
groups which are less productive than the thermal group.
The twogroup CORA calculations hence tend to overpredict
effective. Second, the twogroup 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 largetoo 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.
GraphiteReflected Systems Compared with Systems
Using Other ModeratingReflector Materials
Smaller sized engines with other moderatinqreflector
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 uranium235, fluorine, and helium atom densities for
these particular systems are the same as for engine G5 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
onedimensional "equivalent" spheres. The EXTERMINATORII
results are for twodimensional cylinders. Both CORA and
EXTERMINATORII make use of the PHROG and BRTl fast and
thermal group constants.
It will be noted that for these smaller sized engines,
the gra.phitereflected configurations are farsubcritical.
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 graphitereflected 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 ModeratingReflector Materials at the TDC Position
Moderating Reflector Twogroup Twogroup Fourgroup Twogroup Fourgroup Twogroup
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 G5 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 D20reflected 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 waterreflected 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 twogroup NUCPISTN
and the twogroup 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 BRT1 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 fourgroup computations,
the inclusion or omission of fast core interactions
leads to differences in effective which are as high
as 5%.
In comparing the twodimensional, cylindrical geometry
EXTERMINATORII results with the onedimensional, 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 moderatingreflector
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 EXTERMINATORII results.
When comparing the fourgroup results for the D20
reflected systems with the corresponding twogroup results,
the latter have keffectives which are always higher than the
corresponding fourgroup 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 twogroup problems hence tend to
overpredict the neutron multiplication factors for these
systems.
In contrast, the twogroup keffectives for the Be
reflected systems tend to be lower than the fourgroup
results. The reason is that the twogroup 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 fourgroup 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 BRTl 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 G5
of Table 3. Also presented are neutron lifetime results
obtained from twogroup, twodimensional EXTERMINATORII
perturbation calculations. The reactor geometry was again
that of Table 5 and the core composition that of engine G5
in Table 3. The large size of the neutron lifetime in the
moderatingreflector region relative to the core region
lifetime is to be noted.
ModeratingReflector 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 ModeratingReflector 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 G5 in Table 3
Moderator neutron temperatures from BRTl calculations
Neutron lifetimes from twogroup, twodimensional EXTERMINATORII 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 moderatingreflector 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 "moderatingreflector power." If one con
siders a twogroup, 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
"moderatingreflector power" should be larger, the better
moderatingreflector material. This is a rather simple
grouping of constants. When one is considering the desira
bility of a moderatingreflector 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, (n2 ) production in beryllium is
n
ignored by this grouping. Table 7 lists moderatingreflector
powers for the above five materials at 2900K. For externally
moderated, gaseous core reactors possessing an essentially
infinite (from a neutronics standpoint) moderatingreflector,
this simple combination of constants properly orders the
above materials.
Some Composite ModeratingReflector Studies
Appearing in Table 8 are some results for a system
possessing a moderatingreflector region of varying composi
tion. The core composition and geometry is fixed and the
total moderatingreflector thickness is also fixed at 70cm.
The 020 and Be thicknesses are allowed to vary from 70 to 0
TABLE 7
ModeratingReflector Power for Some
ModeratingReflector Materials at 290K
ModeratingReflector 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
BerylliumD20 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
FourGroup
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 G5 in
Moderatingreflector 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 moderatingreflectors in order to
maintain tworegion 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 moderatingreflector region
from the corrosive UF6 gas.
Remarks Concerning the Algorithms
Used in the NUCPISTN Code
The restriction of the steadystate 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, tworegion or threeregion 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 tworegion 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
moderatingreflector constants are read into the code.
They are obtained by independent means (e.g., from PHROG
and BRT1) and treated as constant during the piston cycle.
Although the moderatingreflector 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 twostroke engines dis
cussed in Chapter III and in this chapter however utilize
stepreflector additions and removals to obtain the desired
subcritical to supercritical to subcritical behavior. For
these engines the moderatingreflector 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 moderatingreflector group constants are generally in
error by about 7 or 8% as compared to the actual thin
moderatingreflector 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 farsubcritical. These errors
will not therefore noticeably affect the engine's behavior
and the use of the thick moderatingreflector 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
moderatingreflector 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
moderatingreflector 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 + (7.936xO13)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 twostroke,
D20reflected nuclear piston engine and on the qualities of
power generating systems which have these engines as their
basic component.
Parametric Studies with NUCPISTN
for D20Reflected Systems
Initial operating conditions for two such D20reflected
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 stepreflector
applied at the El = 0.050 cycle fraction and removed at the
E2 = 0.650 cycle fraction; Engine #2 has the stepreflector'
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 heliumtoU 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 heliumtoU 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
