• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Application of diffusion theory...
 Nucleonic analysis of externally...
 Pressure-temperature analysis of...
 Pressure-temperature analysis of...
 Pressure temperature analysis of...
 Simulation of hydrogen-uranium...
 Radiant energy transfer and temperature...
 Conclusions and recommendation...
 Appendix
 Uplaz-2 program
 Hplaz program
 Huplaz-2 program
 Reference
 Biographical sketch
 Copyright














Title: Fissioning uranium plasma for rocket propulsion
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00084186/00001
 Material Information
Title: Fissioning uranium plasma for rocket propulsion
Physical Description: xiii, 206 leaves : illus. ; 28 cm.
Language: English
Creator: Atwater, Henry Foust, 1937-
Publisher: University of Florida
Place of Publication: Gainesville
Publication Date: 1968
 Subjects
Subject: Plasma rockets   ( lcsh )
Uranium   ( lcsh )
Nuclear Engineering Sciences thesis Ph. D   ( lcsh )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Bibliography: leaves 200-205.
General Note: Manuscript copy.
General Note: Thesis - University of Florida.
General Note: Vita.
 Record Information
Bibliographic ID: UF00084186
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000546065
oclc - 13166470
notis - ACX0023

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
    Abstract
        Page xi
        Page xii
        Page xiii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Application of diffusion theory to neutrons in a fissioning uranium plasma
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
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        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Nucleonic analysis of externally moderated uranium plasma
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Pressure-temperature analysis of uranium plasma fuel region
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
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        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
    Pressure-temperature analysis of hydrogen plasma coolant region
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
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        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
    Pressure temperature analysis of hydrogen-uranium mixing region
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
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        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
    Simulation of hydrogen-uranium mixing in steady flow
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
    Radiant energy transfer and temperature distribution
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
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    Conclusions and recommendations
        Page 169
        Page 170
        Page 171
        Page 172
        Page 173
        Page 174
        Page 175
        Page 176
    Appendix
        Page 177
    Uplaz-2 program
        Page 178
        Page 179
        Page 180
        Page 181
        Page 182
        Page 183
        Page 184
        Page 185
        Page 186
        Page 187
        Page 188
        Page 189
        Page 190
    Hplaz program
        Page 191
        Page 192
        Page 193
        Page 194
        Page 195
        Page 196
        Page 197
    Huplaz-2 program
        Page 198
        Page 199
    Reference
        Page 200
        Page 201
        Page 202
        Page 203
        Page 204
        Page 205
    Biographical sketch
        Page 206
        Page 207
    Copyright
        Copyright
Full Text














FISSIONING URANIUM PLASMA
FOR ROCKET PROPULSION












By
HENRY FOUST ATWATER













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
1968


































To my Father and Mother,

for their continued, encouragement.














ACKNOWLEDGEMENTS


The author expresses his sincere appreciation to

faculty members and fellow students for their interest and

assistance during the preparation of this dissertation.

Special appreciation is extended to Dr. R. B. Perez, who

served as chairman of the author's graduate committee,

until leaving to accept another position. The author

appreciates the guidance of Dr. M. J. Ohanian, who assumed

the duties of chairman following Dr. Perez's departure.

The author is extremely grateful to Dr. H. E. Wilhelm for

many explanations and suggestions which were essential to

the entire study. The participation of Dr. W. H. Ellis is

appreciated. Several helpful suggestions from Dr. R. T.

Schneider are acknowledged. The author also thanks

Dr. R. G. Blake and Dr. R. A. Blue for their participation.

Special appreciation is extended to Mr. W. G.

Wolfer for patiently explaining the notation of molecular

and atomic spectroscopy. The author appreciates the

invaluable assistance so generously provided by Mrs. Jennie

Grossman of the University of Florida Computing Center.

The financial assistance received from two Atomic

Energy Commission Special Fellowships and from the National


iii











Aeronautics and Space Administration is gratefully ac-

knowledged. This study was performed under NASA Grant

NGR 10-005-068.

It is a pleasure to thank my wife, Vann, for

typing the first draft.














TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS ..................................... ii

LIST OF TABLES ........................................ vii

LIST OF FIGURES .................................... viii

ABSTRACT ................ ............................ xi


Chapter

I. INTRODUCTION .................................. 1

II. APPLICATION OF DIFFUSION THEORY TO
NEUTRONS IN A FISSIONING URANIUM
PLASMA ....................................... 11

III. NUCLEONIC ANALYSIS OF EXTERNALLY
MODERATED URANIUM PLASMA .................. 41

IV. PRESSURE-TEMPERATURE ANALYSIS OF
URANIUM PLASMA FUEL REGION ................ 50

V. PRESSURE-TEMPERATURE ANALYSIS OF
HYDROGEN PLASMA COOLANT REGION ............ 82

VI. PRESSURE-TEMPERATURE ANALYSIS OF
HYDROGEN-URANIUM MIXING REGION ........... 108

VII. SIMULATION OF HYDROGEN-URANIUM
MIXING IN STEADY FLOW ..................... 127

VIII. RADIANT ENERGY TRANSFER AND
TEMPERATURE DISTRIBUTION ......... ........ 134

IX. CONCLUSIONS AND RECOMMENDATIONS ............ 169


Appendices

A. UPLAZ-2 PROGRAM ............................... 178













TABLE OF CONTENTS--Continued


Appendices Page

B. HPLAZ PROGRAM ................ ................. 191

C. HUPLAZ-2 PROGRAM .............................. 198


LIST OF REFERENCES .................................. 200

BIOGRAPHICAL SKETCH .................................. 206


~















LIST OF TABLES


Table Page

1. THEORETICAL SPECIFIC IMPULSE ................ 4

2. THEORETICAL IONIZATION POTENTIALS
OF URANIUM .................................. 179

3. DATA USED IN HYDROGEN PLASMA
COMPOSITION CALCULATIONS .................. 197


vii













LIST OF FIGURES


Figure Page

1. Vortex flow plasma reactor ................... 7

2. Axial flow plasma reactor with magnetic
containment ........................... ....... 7

3. Wheel flow plasma reactor .................... 8

4. Axial flow plasma reactor ................... 8

5. Neutron spectrum in graphite ................. 43

6. Neutron spectrum in graphite moderated
uranium plasma ............................... 46

7. Thermal neutron flux in graphite
reflected uranium plasma reactor ............. 48

8. Uranium plasma composition at pressure
of 100 atmospheres ........................... 72

9. Uranium plasma composition at pressure
of 500 atmospheres ........................... 73

10. Uranium plasma composition at pressure
of 1000 atmospheres .......................... 74

11. Effect of temperature on total uranium
density ...................................... 77

12. Pressure trajectories for constant
uranium density ........... ... ................ 80

13. Hydrogen plasma composition at pressure
of 100 atmospheres ........................... 105

14. Effect of temperature on total hydrogen
density ...................................... 106

15. Composition of hydrogen-uranium plasma
at pressure of 500 atmospheres ............... 126


viii












LIST OF FIGURES--Continued


Figure Page

16. Radial density profiles of hydrogen
and uranium ................................ 129

17. Rosseland opacity of uranium ................ 153

18. Rosseland opacity of hydrogen .............. 154

19. Effect of coolant seeding on graphite
temperature ................................. 159

20. Plasma temperature distribution at
100 atmospheres ................ ............ 160

21. Temperature distributions for centerline
temperatures from 20,0000K to 80,000K ...... 161

22. Pressure required forcritical plasma
reactor ...................................... 163

23. Net power and fission fragment power as
a function of centerline temperature ........ 167

24. Energy of hydrogen coolant nuclei due
to flow and thermal motion ................ 174

25. Uranium plasma composition at pressure
of 100 atmospheres .......................... 181

26. Uranium plasma composition at pressure
of 200 atmospheres .......................... 182

27. Uranium plasma composition at pressure
of 300 atmospheres .......................... 183

28. Uranium plasma composition at pressure
of 400 atmospheres .......................... 184

29. Uranium plasma composition at pressure
of 500 atmospheres .......................... 185

30. Uranium plasma composition at pressure
of 600 atmospheres .......................... 186











LIST OF FIGURES--Continued


Figure Page

31. Uranium plasma composition at pressure
of 700 atmospheres .......................... 187

32. Uranium plasma composition at pressure
of 800 atmospheres .......................... 188

33. Uranium plasma composition at pressure
of 900 atmospheres ................. ........ 189

34. Uranium plasma composition at pressure
of 1000 atmospheres ......................... 190

35. Hydrogen plasma composition at pressure
of 100 atmospheres .......................... 192

36. Hydrogen plasma composition at pressure
of 200 atmospheres .......................... 193

37. Hydrogen plasma composition at pressure
of 300 atmospheres .......................... 194

38. Hydrogen plasma composition at pressure
of 400 atmospheres .......................... 195

39. Hydrogen plasma composition at pressure
of 500 atmospheres .......................... 196












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


FISSIONING URANIUM PLASMA
FOR ROCKET PROPULSION


By

Henry Foust Atwater


March, 1968


Chairman: Dr. M. J. Ohanian
Major Department: Nuclear Engineering Sciences


A fissioning uranium plasma for use as a high

temperature self-sustaining energy conversion device is

studied theoretically. The study is oriented with par-

ticular emphasis on the use of the fissioning plasma as

a heat source for rocket propulsion systems, although the

ideas and methods which are developed can also be applied

to a closed cycle heating system which utilizes the

fissioning plasma principle.

The effects of coupling mechanisms between nucle-

onic and plasma properties of the fissioning plasma are

identified and analyzed for a reference model. The model

is based on a central fuel region of fully enriched

uranium surrounded by a hydrogen coolant region, contained

within a cylindrical graphite chamber. The uranium region











serves as a heat source through liberation of radiant energy

from fissioning uranium nuclei. The radiant heat transfer

from the uranium plasma to the hydrogen coolant plasma is

accomplished by seeding the coolant with a material which

is sufficiently opaque to reduce the coolant-wall interface

temperature below the melting point of graphite. The

radial mixing of flowing uranium and hydrogen is simulated

by use of analytic expressions. Numerical techniques are

developed for the engineering analysis of the principle

characteristics of the fissioning plasma reactor, under

conditions of local thermodynamic equilibrium.

The reference model study leads to several basic

requirements which must be satisfied in order to develop

an operable fissioning plasma reactor. The nuclear analy-

sis of such problems as critical mass, reactivity balance,

and kinetic behavior must treat both the fuel and coolant

as plasma regions rather than ordinary gases. In order to

sustain a critical uranium mass at the desired high

temperatures, pressures as high as 1000 atmospheres will

be required. In order to heat the relatively transparent

hydrogen coolant and also to prevent melting of the

graphite containment, the coolant will have to be seeded.

Additional theoretical and experimental studies of plasma

instabilities associated with the coaxial flow of uranium

and hydrogen are needed to analyze such problems as system


xii











stability and fuel loss rate. Finally, a substantial high

pressure, high temperature experimental program is needed

to determine statistical weights and energy levels of the

various uranium ions.


xiii













CHAPTER I

INTRODUCTION


This study presents the identification and analysis

of some of the prominent physical effects which must be

examined in order to develop a coherent understanding of

a fissioning uranium plasma. For the benefit of those who

(like this writer) have had more experience in the area of

nuclear engineering than in plasma physics, it will be

helpful to have a basic definition of a fissioning uranium

plasma. In the most general sense, a fissioning uranium

plasma is a mass of ionized uranium gas which is undergoing

nuclear fission. From this definition, it is immediately

seen that the analysis of such a system must be built upon

a combination of two areas of physics which are ordinarily

treated separately. The first is nuclear physics, which

forms the basis for the disciplines of reactor physics and

nuclear engineering. The second is plasma physics, which

treats the static and dynamic behavior of high temperature

ionized gases.

In terms of reactor technology, a fissioning uranium

plasma can be considered as another member of the large

collection of reactor types that have been conceived over

the past two decades as successors to present-day operating


- 1 -





- 2 -


power reactors. Even so, the plasma core reactor must

certainly be designated as one of the most advanced

concepts in view of its position in the evolution from

solid to liquid to plasma fuel reactors. It is more

important however, to recognize that the concept is ad-

vanced because of the magnitude of the complex physical

phenomena which must be understood and the many techno-

logical problems associated with the high temperature,

high pressure plasma experiments which must be performed,

before such a system is built and operated.

A common characteristic of the solid, liquid and

plasma core reactors is the energy production mechanism.

In each case, energy is released when the fuel nuclei

undergo neutron induced fission. After this initial

energy production, the plasma reactor concept differs

greatly from either the solid or liquid core reactor. In

the solid and liquid core reactors, the energy from fission

heats some working fluid or coolant by conductive and

convective heat transfer. In the plasma core reactor, the

fuel plasma is heated by collisional energy transfer from

the fission fragments. The coolant plasma is then heated

by radiant energy transfer from the uranium plasma.

Radiant energy transfer is the characteristic which most

clearly distinguishes a plasma core reactor from a solid

or liquid core reactor. Since radiation has not been






- 3 -


utilized as the primary energy transfer mechanism in a

nuclear reactor, it is expected that radiant energy transfer

will also produce some of the most difficult but interesting

theoretical and engineering problems associated with plasma

core concept.

The usual impetus for studying a new reactor concept

is derived from the continuing efforts of the nuclear

industry to develop more efficient and economical re-

actors (1,2,3). The plasma core reactor is an exception.

The initial interest in the gaseous or plasma reactor

concept (4-11) has resulted in research directed toward

its use as a high performance space propulsion device.

In order to compare the propulsion potential of the plasma

core reactor with other concepts, the specific impulse,*

Isp, is used as a relative measure of performance. Since

specific impulse is related to propellant mass (M) and

temperature (T) through the relation Isp /T79 (12), a

high temperature, low molecular weight propellant is desira-

ble. The maximum propellant temperature is limited by the

maximum allowable temperature of the reactor fuel. The

temperature limit in a solid core reactor is the fuel

melting point, while the temperature of a liquid core


*Specific impulse is defined as engine thrust per
unit weight flow rate of propellant.






- 4 -


reactor is limited by the fuel boiling point. The plasma

core reactor can theoretically operate at arbitrarily high

temperatures. In practice the temperature limit will be

determined by the maximum fuel temperature for which the

plasma containment chamber can be cooled below its melting

point. The following are typical values of specific

impulse (13,14) indicating the high performance that can

be expected from a plasma reactor propulsion system.


TABLE 1

THEORETICAL SPECIFIC IMPULSE, I (seconds)
sp

Chemical Rocket 400
Solid Core Reactor 1000
Liquid Core Reactor 2500
Plasma Core Reactor 6500


Although the present interest in plasma core

reactors seems to be entirely concerned with rocket

propulsion applications, the concept should not be con-

sidered as a single purpose device. Without specifying

its application, a fissioning plasma embedded in a coolant

plasma can be considered as a general energy conversion

device. The high temperature coolant might be used in

numerous open or closed cycle systems. Central station

power plants operating with gas heat exchangers or directly

driven gas turbines could utilize the plasma reactor as

the energy source. When considering the maximum cycle






5 -




efficiency as given by the Rankine cycle, the possibility

of extremely high coolant temperatures in a plasma reactor

are particularly attractive.

A brief survey of recent plasma reactor* research

will be indicative of the wide range of problems which are

being studied. The basic plasma core concept is simple.

A heavy, low velocity gaseous fuel and a light, high

velocity gaseous propellant are injected into a reaction

chamber having walls composed of some suitable neutron

reflecting material. Fissions in the fuel heat the pro-

pellant which is then exhausted through a nozzle to give

thrust. The gaseous state of the fuel gives the previously

mentioned advantage of high temperatures, but also causes

a serious problem of fuel separation and containment.

Some of the fuel will mix with the propellant and be

exhausted from the reactor. In order to maintain a

critical fuel mass, additional fuel must be continually

injected into the reaction chamber. This is an inefficient

use of expensive nuclear fuel which also results in a

reduction of specific impulse, due to the increase in the

mean mass of the propellant. Several approaches to the

fuel containment problem have been proposed. Four of the

concepts which have received attention are the vortex flow,


The plasma core reactor is also referred to as a
gaseous core or cavity core reactor.











axial flow with magnetic containment, wheel flow, and

coaxial flow (9,13,15,16). The vortex flow reactor illus-

trated in Figure 1 is designed to achieve fuel-propellant

separation through use of centrifugal forces. Hydrogen

propellant diffuses radially inward through the gaseous

uranium vortex, with the centrifugal forces associated

with the heavier uranium counteracting the diffusion drag

of the hydrogen. The disadvantage of this concept is due

to the hydrogen drag, which is great enough to cause large

uranium losses except at very low hydrogen flow rates.

Figure 2 illustrates the magnetic containment scheme.

Magnetic fields of force confine the uranium plasma while

the propellant flows axially around the fuel region.

This concept involves the technical difficulties associ-

ated with magnetic containment, in addition to propellant

drag problems similar to those of the vortex flow. Another

variation is the wheel flow concept shown in Figure 3.

Hydrogen propellant enters and leaves the reactor tan-

gentially along the outer periphery after circulating

around the inner cylindrical uranium region. In this

concept the main difficulty still seems to be excessive

fuel loss due to hydrogen-uranium mixing. The coaxial

flow reactor is shown in Figure 4. This concept is based

on the coaxial flow of a central cylindrical uranium mass

and a surrounding annulus of hydrogen. No vortex motion


- 6 -










Neutron


assure shell Propellant inlet




__"__[_ _..-___ V e




Rotating plasma fuel



Figure 1. Vortex flow plasma reactor.





Neutron reflector
Pressure shell Propellant inlet


e






Current-carrying Magnetic field
coils lines
Plasma fuel


Figure 2. Axial flow plasma reactor with
magnetic containment.


lector






- 8 -


Propellant
exhaust


Propellant/,'
inlet



Plasma fuel -


Propellant
inlet


Figure 3. Wheel flow plasma reactor.






Neutron reflector


Pressure shell


Fuel inlet


Plasma fuel


Figure 4. Axial flow plasma reactor.






- 9 -


is involved. It is hoped that the hydrogen-uranium mixing

can be controlled by proper selection of initial velocity

profiles and upstream geometry.

Studies related to the plasma reactor concept can

be divided into three areas. These are nuclear reactor

physics, radiant energy transfer, and turbulent plasma

flow. A very comprehensive summary of NASA/Lewis supported

research in these areas has recently been published by

Ragsdale and Rom (17). The past studies have generally

approached the three problem areas by treating each area

separately, without including the coupling mechanisms by

which nuclear, plasma and radiation effects are related

physically. The present study will consider only the

coaxial flow system, which appears to be the simplest

concept. This system presents a sufficient number of

problems, without considering the added complexities of

vortex flow or magnetic containment. The purpose of the

present study is to identify specific problems in the

above three areas, develop methods for relating these

problems, and analyze them in a consistent manner. The

high temperature uranium is properly treated as a plasma

rather than an ordinary gas, and the effects of the plasma

characteristics on nuclear criticality are determined.

The effects of fuel and coolant inhomogeneities on radiant

energy transfer are included. Temperature distributions


1






- 10 -


determined by radiant energy transfer are then included

in the calculation of the critical uranium mass. The

emphasis is on the development and application of general

methods of analysis, rather than a conceptual design of

an entire plasma reactor system. In order that methods

of this study may be extended to actual design problems,

numerical techniques are developed for several of these

methods.

The effects of the high fuel and coolant tempera-

tures on neutron reactions are treated in Chapters II and

III, within the framework of the diffusion approximation

to the Boltzmann equation. The composition of the

uranium plasma region and the hydrogen plasma region,

and the influence of plasma effects on reactor criticality

are analyzed in Chapters IV and V. The composition of

the hydrogen-uranium mixing region is analyzed in

Chapter VI. In Chapter VII an analytic simulation of

hydrogen and uranium mixing is developed. In Chapter VIII

the calculated plasma compositions and the simulated flow

distributions are used to determine the temperature distri-

butions resulting from radiant energy transfer. The

temperature distributions are then related to the critical

uranium mass. Recommended theoretical and experimental

extensions of this study are presented in Chapter IX.













CHAPTER II

APPLICATION OF DIFFUSION THEORY TO NEUTRONS
IN A FISSIONING URANIUM PLASMA


A General Boltzmann Equation for Neutrons
in a Fissioning Uranium Plasma


A fundamental macroscopic description of the

neutron population in a solid fuel reactor is given by

the Boltzmann neutron transport equation. This integro-

differential equation specifies the neutron distribution

in space, velocity, time and direction of motion. Using

notation similar to that of Weinberg and Wigner (18), the

transport equation is


1 a 4 ,
v at(r,vQt) + V(r,v,,t) + t (r,v) (r,v, ,t)


= X(v) f dv' f d (v')Z (r,v')((rv' ',t)
V' t '


+ J dv' f dS 's C v' v ( 'rv','t) (2.1)



External neutron sources and delayed neutrons are not

included in (2.1).

The symbols are defined in the usual manner:


- 11 -






- 12 -


Singular neutron flux (number of neutrons
per unit area per unit time per unit
solid angle per unit speed in. the volume
-4-
element dr about r, traveling within the
solid angle d in the direction and
having a speed v in dv at time t)


Zt(r,v) = macroscopic collision cross section for
neutrons at r with speed v

X(v) = spectrum of fission neutrons

v(v) = mean numbers of neutrons resulting from
a fission caused by a neutron with
speed v


= macroscopic scattering cross section at
r for changing the speed and direction
v', S' into a speed and direction range
dv, dQ at v,


The individual terms of (2.1) describe the following

reactions:

1 9 = change per unit time in the net number
v t of neutrons of direction and speed v
in dr


V4 =


net loss per unit time from dr of neutrons
of direction in dn by leakage


St = number of neutrons of direction and
speed v which are removed per unit time
from dr by absorption and scattering
collisions


XJdv'fdQ'vZf =




jdv'fdri'Z E =
s


number of neutrons of direction 2 and
4-
speed v gained per unit time in dr from
fissions due to neutrons with all speeds
v and directions

number of neutrons of direction 3 and
speed v gained per unit time in dr from
scattering collisions which scatter
neutrons from all speeds v' and directions
-I


E (S ,v'+v,6'+ )
s


S(r,v,At)







- 1.3 -


For a steady state neutron population, the first

term of (2.1) is zero. This gives the steady state

Boltzmann equation for which there are many analytical

and numerical methods for obtaining solutions. Exact and

approximate solutions which have been obtained for solid

core reactors are numerous and will not be presented here.

A qualitative description of the usual treatment of the

steady state Boltzmann equation will demonstrate that

modifications are necessary, in order to properly describe

neutrons in a plasma core reactor. When applying the

Boltzmann equation to neutrons in a solid core reactor, the

usual procedure is to classify neutrons as either epithermal

or thermal, according to their energy. The high energy

epithermal group includes those neutrons which are emitted

at fission energies (average fission energy = 2 MeV) and

slowed down to some thermal energy boundary (usually taken

as 1 eV to 2 eV). Thermal neutrons are those which have

slowed down and are in approximate thermal equilibrium at

some characteristic temperature, T, of the medium. Thus

the thermal energy range extends from zero to a few eV.

In the third, fourth, and fifth terms of (2.1), the

neutron speed v (or v'), appearing in the macroscopic cross

sections is the relative speed between neutron and nucleus.

Since solid core reactors typically operate with coolant

temperatures in the 5000K to 600K range (corresponding to












an energy range kT = 0.04 eV to 0.05 eV), these scattering

nuclei are considered to be unbound and also stationary

with respect to epithermal neutrons. In this case, the

relative speed is the absolute speed of the neutron. As

long as neutron speeds are above a few electron volts,

nuclear motion does not affect the neutron slowing down

process. All scattering collisions between epithermal

neutrons and stationary nuclei will result in a decrease

in neutron energy. The absorption of epithermal neutrons

by nuclei having large resonance cross sections cannot be

treated by assuming that the absorbing nuclei are at rest.

At the maximum solid core fuel temperatures of about 30000K

(corresponding to an energy kT = 0.26 eV), the translational

motion of the nuclei is large enough to strongly influence

resonance absorption through the Doppler effect. Resonance

absorption will be discussed later in this chapter.

When neutrons are slowed down from the epithermal

energy range, into the thermal energy range, the neutron

speeds are comparable to nuclear speeds and the nuclear

motion cannot be ignored. The effects of nuclear motion

are important in two aspects of neutron interactions. The

first involves the treatment of thermal neutron scattering

reactions and the second deals with the specifications of

low energy absorption cross sections. The energy distri-

butions of nuclei and thermal neutrons can be described by


- 14 -






- 15 -


modified Maxwell-Boltzmann distributions having a character-

istic energy kT. Because of the distribution of energies

of the scattering nuclei, a scattering collision does not

always result in a decrease in neutron energy. Collisions

in which the neutron gains energy become possible and the

probability for collisions with an energy loss become

smaller. Thermal neutron scattering is further complicated

by chemical binding effects associated with the lattice

structure of solids and the molecular structure of liquids.

The effects of nuclear motion and chemical binding on

scattering collisions are included in the theory of neutron

thermalization (19), and will not be treated here. A

concise description of neutron thermalization problems is

given by Beckurts and Wirtz (20).

Microscopic thermal neutron absorption cross

sections are functions of the relative neutron speed. In

general, this speed dependent cross section is not the

same as the experimentally measured average cross section.

In order to use measured absorption cross sections in

reactor calculations, the experimentally determined values

must be properly weighted to include the effects of relative

speed (21).

The preceding discussion of the Boltzmann transport

equation has considered only neutrons in a solid core

reactor. To summarize this discussion, the application






- IG -


of the Boltzmann equation to solid core reactors can be

characterized by two features: (1) the assumption that

epithermal neutrons are slowed down by collisions with

stationary nuclei and (2) the methods of thermalization

theory are used to treat the effects of nuclear motion

and chemical binding on thermal neutron collisions.

In order to properly apply the Boltzmann equation

to neutrons in plasma core reactor, several modifications

are necessary. The need for these modifications is due to

the extremely high temperatures of the plasma core reactor,

as compared to solid core reactor temperatures. Tempera-

tures on the order of 100,0000K are anticipated in the

plasma core reactor. Scattering and absorbing nuclei at

this elevated temperature have an energy kT = 8.6 eV.

Recalling that scattering and absorbing nuclei in a solid

core reactor have a maximum energy of about 0.05 eV and

0.26 eV respectively, it is clear that nuclear motion

becomes a more important consideration in a plasma core

reactor. To formally account for the possible effects of

this increased nuclear motion, the usual form of the

Boltzmann equation (2.1) is modified to include the

dependence of collision processes on the relative speed.

The following symbols for the several velocities are

used:






- 17 -


v v = neutron velocity
n n n

S= VNN = nuclear velocity

v = relative velocity

The relative speed v, is then defined as

v = | = i NI, where v = V2E m (2.2)
in -N n n

Collision processes are related to the relative speed

through the neutron flux < = n(r,v,,t)v, and the energy

dependent microscopic cross sections, a(E), where n(r,v,,t)

is the neutron density. For a neutron of speed vn and a

nucleus of speed VN, the collision processes of (2.1) must

now be integrated over all nuclear speeds VN and directions

N. The expressions for these collision processes become:

Neutrons removed by absorption and scattering =


SdVN dN(- ,VN Nt)t(v)n(,vn" nt)
N N

Neutrons gained by fission =


X(v) f dVN N dv'n dv(v')N(r,V NN,t)f(v)
N N n n

x n(r,v'n ,t)v'n





- 18 -


Neutrons gained by "scattering in" =

SdN dN dv'n d 'nN( ,VN' Nt)o s ,v'v,'n
VN v n
x n(r,v' n t)v' n

The general Boltzmann equation for neutrons in a fissioning
plasma can now be written as


-~(,vnr,,t) + vn Vn(,vn nt)

+ f dVN d NN(N,VN',N,t)at(v)n(r,vn nt)V'n

= X(v) dV d dv' dI (n )N(r,VN ,t)f(v)

x n(r,vnn tvn n

+ I dV N dN f dv' f di'nN(,VN'N,t)as(rv'v' n n)

x n(r,vnn,t)v'n (2.3)

Equation (2.3) shows the means by which the relative
velocity enters into the individual collision processes.
This form of the Boltzmann equation is still quite general.
As such, it does not immediately suggest obvious methods
of reduction to simpler forms which are more amenable to






- 19 -


analytical or numerical solutions. In order to retain the

significant effects that may result from the inclusion of

exaggerated nuclear motion, some characteristics of a

typical plasma core reactor will now be considered. The

introduction of a model of such a reactor allows physical

arguments to be used in making simplifying assumptions.

For purposes of describing the nucleonic characteristics,

the reactor is basically a central region of uranium plasma

surrounded by a cylindrical annulus of hydrogen coolant

plasma. To provide the desired neutron reflection while

operating at high temperatures, the reaction chamber is

assumed to be graphite. The determination of the neutron

energy spectrum for this model is of primary importance,

since all nuclear characteristics of the system will be

directly affected by the neutron spectrum. The simplest

case to consider is the zero power plasma core reactor

which consists of a critical mass of uranium plasma sur-

rounded by graphite. No hydrogen coolant is included

since there is no significant heat production at zero

power. For an adequately reflected zero power reactor,

leakage is assumed to be negligible. For a high power

rocket reactor, leakage due to neutron streaming through

the nozzle would have to be included. The neutron spectrum

in the zero power reactor will be determined by scattering

collisions with graphite and uranium, and by absorption






- 20 -


collisions with uranium. The relative importance of

scattering collisions in graphite and uranium can be

determined by comparing the scattering mean free paths

of neutron in the two materials. Graphite has an atom

density of N = 0.08 x 1024/cc and an epithermal micro-

scopic scattering cross section of ac = 4.8 barns. The

corresponding scattering mean free path is


Xs i = 1/N a s = 2.6 cm
graphite c c

Nuclear criticality calculations (which are discussed in

the following chapter) indicate that a typical critical

uranium-235 mass can be achieved with an atom density of

approximately N = 10'9/cc. With an average epithermal

scattering cross section of 10 barns, the scattering mean

free path in uranium is


us = 1/Nu a = 104 cm


These mean free path values show that a neutron slowing

down from fission energy to thermal energy will have about

4000 scattering collisions in graphite for each scattering

collision in uranium. Thus the neutron spectrum will be

determined primarily by collisions in graphite. This

conclusion greatly simplifies the computation of effective

absorption cross sections in the fuel region since the






- 21 -


neutron spectrum in graphite can be used to obtain averaged

uranium cross sections, even though the uranium contains

no graphite. The neutron spectrum of graphite will

actually be slightly modified in the fuel region due to

epithermal and thermal absorptions in uranium. The

temperature of the graphite will essentially determine

the distribution of thermal neutrons. The influence of

uranium absorptions on the thermal neutron spectrum is

through second order effects, since the absorption mean

free path for thermal neutrons in the fuel region of a

zero power plasma reactor is about Xua = 1700 cm. For

this value, a 78 meter thickness of uranium plasma would

be required to attenuate the thermal neutron intensity

by one per cent. The epithermal neutron spectrum of

graphite will be modified at energies coinciding with large

uranium resonance cross sections. From the above argu-

ments, the zero power plasma reactor is seen to be well

moderated, with an epithermal neutron spectrum determined

by slowing down collisions in graphite, and a thermal

spectrum determined by the graphite temperature.

Extensive one and two dimensional transport theory

calculations have been used to study the effects of

materials, temperature and dimensions on the criticality

of gaseous core reactors (22). For the purpose of

calculating detailed design parameters, it will probably

be necessary to use the more exact computational methods






- 22 -


such as these. However, since the reactor is very thermal,

it is reasonable to expect that diffusion theory methods

will suffice for survey and comparison calculations. In a

parametric study of gaseous core reactors using both

transport and diffusion theories, Plunkett (23) has demon-

strated that multigroup diffusion theory is adequate for

calculations of this type. In the present study, the

effects of elevated temperatures and the resulting ex-

aggerated nuclear motion will be treated within the

framework of multigroup diffusion theory. This approach

leads to the development of calculational methods which

are designed specifically for application to diffusion

theory codes. Such methods may be directly applicable to

transport theory codes. If not, it will be necessary to

include the effects of nuclear motion by reduction of

equation (2.3) to a form which can be solved numerically.


Diffusion Approximation to Boltzmann Equation

The previous discussion of the plasma core reactor

has considered the zero power reactor which contains no

hydrogen. A power producing plasma reactor will operate

with a high velocity hydrogen coolant plasma surrounding

the uranium plasma. In this case the exaggerated motions

of both absorbing fuel nuclei and scattering coolant nuclei

become important. The inclusion of these effects will






- 23 -


require special treatment of several terms in the multigroup

diffusion equation. Using notation similar to that of

Baller (24), the multigroup diffusion equation is


N
D.V2.() + i (r) = X.S(r) + E S .j- (r)
Di 1i t. 1 1 s ,j
1 j=l
j i
(2.4)

where

N
t = i + DiB. j=l


N
S(-r) = (vxf)i(Wr
i=l


for 1 < i < N, N = number of groups.

The symbols are defined as


.i(r) = neutron flux in i-th group at position r

D. = diffusion coefficient for the i-th group
1

E. = total removal from i-th group
1

E = absorption cross section for the i-th
a group

B2 = transverse buckling for the i-th group


s,i+j = scattering from group i to group j

Xi = integral of the fission spectrum over the
energy range represented by group i






- 24 -


S(r) = fission source distribution

(vEf)i = average number of neutron produced by
fission in the i-th group times the
fission cross section in the i-th group

X = criticality eigenvalue


The effects of nuclear motion of absorbing nuclei and

scattering nuclei will be treated separately. The ab-

sorption cross section, Eai, will include the effects of
1
the motion of uranium nuclei in the treatment of the

Doppler broadening of the uranium absorption resonances.

The scattering cross section, incle
s,1-j, will include the

effects of the motion of hydrogen nuclei.


Effect of Thermal Motion of Absorbing Nuclei on Epithermal
Neutrons

The macroscopic absorption cross section for the

i-th group is the flux weighted average cross section

which is defined as


J E



a. E
S JE i(r,E)dE
E
i (2.5)


where Eui and EL. are the upper and lower boundaries,
respectively, of the i-th group. In the epithermal energy
respectively, of the i-th group. In the epithermal energy











range, the uranium-235 microscopic absorption cross

section, a (E), exhibits a large number of resonances

(25). The usual procedure in evaluating the integral in

the numerator of (2.5) is to assume that each well defined

resonance is accurately represented by the single level

Breit-Wigner formula (26). This formula expresses the

energy dependent absorption cross section as

r r
o0(E) = 2- gj n y
a (Er E0)2 + (r/2)2 (2.6)


The symbol E0 denotes the energy of the resonance and Er

is the kinetic energy associated with the relative velocity

between neutron and nucleus. In the case of a stationary

nuclei, Er is given by


E = 1 2~ (2.7)
r 2 n

where v is the neutron speed in the laboratory coordinate

system, and p is the reduced mass as defined by

mM (2.8)
m + M

with m and M representing the mass of the incident neutron

and the target nucleus, respectively. The symbols Fy and

r are the radiative capture and neutron widths (27) which
n
define a total width r as


F = F + F
"Y n


- 25 -


(2.9)






- 26 -


The reduced neutron wavelength is given by


1 h
27r 27T(2pE ) 1/2 (2.10)


The statistical spin factor g (28) is equal to (2J + 1)/

2(21 + 1), where J is the spin quantum number of the

compound nucleus formed by the target nucleus and the

neutron, and I is the spin quantum number of the target

nucleus. The peak value of the total (absorption plus

scattering) cross section, a0, occurs at E = EO and has

the value

r
S= 4Xt2gj n (2.11)


Using (2.11) and (2.6), the absorption cross section

becomes

r1
a (Er) = 0 o 2 (2.12)
E E
T/2


If the absorbing nuclei were stationary, Eai would

be evaluated from (2.5), with a (Er ) given by (2.12) and

Er given by (2.7). In reality, all absorbing nuclei have

a kinetic energy at least as great as that corresponding

to the temperature of the surrounding medium. The thermal

motion associated with the kinetic energy effectively






- 27 -


shifts the energy at which neutrons are absorbed to higher

or lower values, depending on the direction of the target

nuclei with respect to that of the incident neutron. This

energy shift is referred to as Doppler broadening (18) of

the resonance cross section, since the neutron "sees" a

cross section which is spread over an increasingly wider

energy range as the nuclear motion increases. Doppler

broadening strongly affects neutron absorption since the

neutron density is a rapidly varying function in the

neighborhood of a resonance. At the comparatively low

temperatures of solid core reactors, the Doppler effect

is very important in determining reactor criticality and

stability. The kinetic energy of an absorbing nuclei in

a plasma core reactor will correspond to temperatures in

the 50,0000K to 100,0000K range, as compared to a maximum

of about 30000K for a solid core reactor. At such high

fuel temperatures, the effects of Doppler broadening in a

plasma core reactor are expected to be much larger than

the corresponding effects in a solid core reactor. In

order to compare the Doppler broadening equations in the

plasma and solid core reactors, the approximations used

in treating the Doppler effect in a solid core reactor

will be examined briefly.

In a solid core reactor operating at any appreciable

power level, the fuel nuclei will have sufficient thermal












energy to require an accurate determination of the relative

energy appearing in the resonance cross section as given in

equation (2.12). The relative kinetic energy of the

neutron-nucleus collision is



r I n VN
E = n. (2.- 13)


where vn is the velocity of the incident neutron, VN the

velocity of the nucleus, and I the reduced mass as defined

by (2.8). Expanding (2.13), the relative energy can be

written as

pn P
E E + E n N (2.14)
r m n M N m + M14

where the following relations have been used:


E mv 2 = kinetic energy of the neutron
n 2 n
S1
En 2 MVN2 = kinetic energy of the nucleus


P = mn = momentum of the neutron
n n

N = MV = momentum of the nucleus
"N N

Note that, if the nuclei were stationary, the second and

third terms of (2.14) would be zero and the expression

reduces to (2.6). The kinetic energy of nuclei at the

maximum solid core fuel temperature of 30000K is 0.26 eV,


_IL~i


- 28 -






- 29 -


while a neutron at the lowest uranium-235 resonance has an

energy of 1.0 eV. From these values, the second term of

(2.14) is seen to be much smaller than the first, since

the largest ratio of the second to the first term is

approximately 0.001. Therefore, the second term is con-

sidered negligible in dealing with solid fuel resonance

absorption. The third term of (2.14) is proportional to

the component of momentum of the target nucleus parallel

to the direction of motion of the incident neutron. Due

to its large mass as compared to that of the neutron, the

nucleus has a large enough momentum to cause the third

term to have a significant effect on the relative energy.

Denoting by Vz the component of the velocity of the nucleus

which is parallel to that of the neutron, the third term

can be written as

Pn PN Nm)

S N n)v V (2.15)
m+M m+M nz

Using (2.15) and the expression for the relative kinetic
i 2 equation (2 14)
energy for stationary nuclei, E =- 2v equation (2.14)

is simplified as


E = E v V = E 7E" V (2.16)
r m n nz z

Since only the nuclear motion parallel to the

neutron direction is considered to be significant, the






- 30 -


probability of a neutron collision with a nucleus with

velocity Vz is proportional to the number of nuclei with

this velocity. The number of such nuclei is given by the

one dimensional Maxwellian distribution


MN 1/2
P(Vz) = (2 e-MNVz2/2kT (2.17)

Hence the effective cross section for a neutron with kinetic

energy E is


O(E) = ((Er)P(Vz)dVz (2.18)


Using equation (2.16), the effective resonance absorption

cross section is

E E dE
a (E) = o a(Er)P( r (2.19)


This cross section is evaluated by substituting (2.12) and

(2.17) into (2.19). Using the notation of Dresner (29),

this substitution yields

r
ca(E) = C0 -Y Y(e,x) (2.20)
a 0 r

where


SrC. exp[- e2(x y)2]
Y(e,x) 2/ _- 1 +- dy (2.21)
2 VT -2


L






- 31 -


and


x = 2(E EO)/r, y 2(Er EO)/, 0 = FA, A = (41pEkT/) 1/2


(2.22)

Doppler broadened resonance absorption cross section

in solid fuel reactors are computed using (2.20), with

Y(9,x) evaluated from tabulated values based on the reso-

nance parameters r and E0, and the fuel temperature T.

Doppler broadening in a plasma core reactor is

expected to differ quantitatively from the solid core

reactor, due to the larger kinetic energy of the plasma

core fuel nuclei. The following treatment of the Doppler

effect in a plasma reactor is based on the assumption that

the single level Breit-Wigner formula (2.12) accurately

describes resonance absorption cross sections. The relative

kinetic energy of the neutron-nucleus collision is again

given by equation (2.14). From the same arguments used in

the investigation of the Doppler effect in solid core

reactors, the third term of (2.14) will have a significant

effect on the relative energy in the plasma fuel reactor

also. For fuel nuclei at a temperature of 100,000K and

a neutron at the energy of the first uranium resonance,

the ratio of the second to the first term is 0.04, or forty

times larger than the corresponding ratio for the solid

core reactor. Thus, the second term of (2.14) can result in






- 32 -


a significant correction to the relative energy, and hence

to the amount of resonance absorption. At the extremely

high temperatures which are being considered in the plasma

core reactor, this correction may strongly influence re-

actor criticality and stability.

Expanding equation (2.13) as before and retaining

all terms, the relative energy is


E E + V 2 /2iE V (2.23)
r 2 N z


where E = v2 again denotes the relative kinetic energy

of the neutron stationary nuclei. The relative energy is

now a function of all three velocity components (V, Vy, V ),

rather than the single component Vz as in the case of solid

fuel nuclei. Hence, the probability of a neutron collision

with a nucleus with velocity VN = (Vx, Vy Vz) is proportional

to the three dimensional Maxwellian distribution of nuclei

P(VN), where



P(V )dV = ( /2 e MVN2/2kT 4V2dV (2.24)
N N =27 kT e4VN N


where V 2 V + V 2 + V2. The effective resonance
N x y z
absorption cross section for a neutron with kinetic energy

E is






- 33 -


CO CO CO

a(E) a(Er)P(VNdV N = a(Er}P(VN) ddV dVz
N -CO -C 0co

(2.25)


where a (Er) is given by equation (2.12), Er by (2.23)

and P(VN) by (2.24).

Equation (2.25) is the plasma fuel equivalent of

the solid fuel cross section given by (2.18). An analytical

evaluation of (2.25) was investigated by expanding the

integrand in a three-variable Taylor's series about the

most probable velocities (Vxp, Vyp, Vzp ) = (kT, kT, kT).

The advantages of this approximate analytic approach do

not justify the difficulty involved in obtaining sufficient

accuracy. A more direct method of computing the effective

cross section is based on the use of the energy dependent

form of the Maxwellian distribution. Expressed in terms

of the kinetic energy, EN, of the nucleus, the distribution

is



P(EN)dEN 27w /EN e-E/kT (2.26)
1N4) N (7kT)3/2 N


Since all directions of motion of the absorbing nuclei are
1 2 1
=E = E where E. 1 MV 1
equally probable, E = E z i = 2 MV = EN;

i = x, y, z. The velocity component of the nucleus parallel

to the direction of travel of the neutron is






- 34 -


V = /2E~7M = /2E 73M (2.27)


From (2.23) and (2.27), the relative energy is

Er E + EN 2/"EEN 3M (2.28.)


Using (2.12), (2.26) and (2.28), the computational form

of the effective resonance cross section is

r
oa(E) = a0 4 H(a, 8, y) (2.29)

with the Doppler broadening function H(a, 3, y,) defined as



H(a, 8, y) = de-
/ 0 1 + ya / + E)

(2.30)

where


a(E, T) = (E EO)/kT

g(E, T) = 2/VifkTE/3M

y(r, T) = 4kT/r2 (2.31)


and the reduced energy variable is e = EN/kT.


The Doppler broadened resonance absorption cross

section in a plasma core reactor is given by (2.29), with

H(a, 6, y) evaluated from tabulated values based on the

resonance parameters F and E0, and the fuel temperature T.






- 35 -


Effect of Thermal Motion of Hydrogen Nuclei

In the previously discussed zero power plasma

reactor with a graphite reflector, the thermal neutron

spectrum was described by a Maxwellian distribution

corresponding to the graphite temperature. When the

reactor operates at significantly high power levels, the

hydrogen coolant required for heat removal will exist at

temperatures which result in a large increase in the thermal

motion of these scattering nuclei. When neutrons in thermal

equilibrium with graphite have a scattering collision with

hydrogen nuclei at a higher temperature, the neutrons will

gain kinetic energy. If the collision rate with hydrogen

is comparable to that with graphite, the characteristic

temperature of the neutron spectrum will be shifted to a

higher value. This shift, or "hardening," of the spectrum

will influence reactor criticality by reducing the neutron

density at energies where the uranium absorption cross

section is large.

Based on two dimensional criticality calculations

of gaseous core reactors containing hydrogen, Hyland et

al. (30) have concluded that neutron scattering collisions

with hydrogen have a negligible effect on the critical mass.

This conclusion does not appear to be justified since the

"upscattering" effects of hydrogen are not included. The

calculations are based on the assumption that the thermal






- 36 -


neutron spectrum is determined by the graphite temperature,

and do not include the spectral hardening due to hydrogen

at a higher temperature. While it may be true that hydrogen

scattering collisions are negligible in certain restricted

cases, this is not expected to be true in most cases of

interest. The extent to which hydrogen influences criti-

cality will be determined by the mass of hydrogen in the

core, and by the spatial distribution of the hydrogen

density and temperature. Thus, in order to properly

evaluate the effects of hydrogen, the heat transfer and

fluid flow characteristics of hydrogen must be considered

simultaneously with the neutron scattering problem.

A more rigorous treatment of hydrogen scattering

has been used by Herwig and Latham (31) in their multigroup

diffusion theory calculations of spherical gaseous core

reactors containing hot hydrogen. By defining an effective

macroscopic hydrogen scattering cross section, the effects

of high temperature hydrogen on critical mass are investi-

gated. The effective cross section is defined by con-

sidering the scattering collision rate of a Maxwellian

distribution of neutrons at temperature Tn with a

Maxwellian distribution of hydrogen atoms at T For a

neutron density n and a hydrogen density NH, the collision

rate is






- 37 -


NH a nV = E nV = Z' nV (2.32)
Hs r s r s n

where the effective macroscopic scattering cross section

for hydrogen is given by


ss Vr/ n (2.33)

The symbols are


Vr = average relative velocity between hydrogen
and neutrons

V = average neutron speed = (8kT /nm ) 1/2

H = average hydrogen speed = (8kTH/TmH) 1/2

o = microscopic scattering cross section of
hydrogen

The effective multigroup hydrogen cross sections

are calculated by the following relation:


NHo V V)
N s a H 1
= = NCs.C.ij = zs.-1i
1 V 13 1

(2.34)

where


VH V. = average relative speed between hydrogen
at a given temperature and neutrons in
group i

as. = microscopic scattering cross section for
I group i






- 38 -


V = average neutron velocity in group i

pij= neutron energy transfer coefficient from
group i to group j


The GAM-1 (32) and SOPHIST-1 (33) codes are used to calcu-

late the transfer coefficients. Using the effective

hydrogen scattering cross section with fuel and reflector

cross sections computed in the usual manner, a one

dimensional multigroup diffusion theory was used to make

a parametric study based on various hydrogen temperatures

and pressures, fuel nuclei, reflector materials, and fuel

distributions. The details of these calculations and the

assumptions involved are presented in the original article

and will not be discussed here. However, it is worthwhile

to discuss the results of this study, which are significant

for two reasons. First, the group-averaged cross sections

include the spectral hardening effects of high temperature

hydrogen and second, the fact that high temperature hydrogen

strongly influences the critical mass through a complex

interaction of multiple processes is clearly demonstrated.

Within the framework of multigroup diffusion theory, the

technique of using an effective hydrogen scattering cross

section is a practical approach to the treatment of thermal

neutron scattering by high temperature hydrogen. Before

adopting this technique as a standard method of calculating






- 39 -


thermal group cross sections, a more rigorous and possibly

more accurate method should be considered. The fundamental

approach to the treatment of thermal neutron scattering is

through application of appropriate scattering kernels

within the framework of thermalization theory. In the

hydrogen cooled plasma core reactor, the thermal group

averaged cross section of hydrogen, Es, is determined by

a mixed neutron spectrum. The spectrum results from

scattering collisions with both the high temperature

hydrogen and the relatively low temperature reflector

material. The scattering cross section Es in (2.34) was

obtained using the TEMPEST code (34). This code averages

microscopic cross sections over a thermal neutron spectrum

based upon one of the following approximations:


1. The Wigner-Wilkins light moderator equation

2. The Wilkins heavy moderator equation

3. The Maxwellian distribution

Since each of these approximations is characterized by a

single moderator temperature, it appears that Es in (2.34)

is determined from a spectrum based upon a temperature and

scattering kernel corresponding to either hydrogen or the

reflector, rather than from a mixed spectrum. A more

rigorous calculation of Es would require the determination

of the mixed spectrum. Such a spectrum could be computed






- 40 -


from an appropriate weighting of a hydrogen spectrum based

upon a free gas scattering kernel and a reflector spectrum

based upon the proper scattering kernel (such as the Parks

kernel for graphite) at the reflector temperature. The

spectral weighting would be determined by the spatial

distribution of the hydrogen density and temperature. The

decision to use either the mixed spectrum method or the

effective hydrogen cross section method should be based on

the particular reactor being studied. The effective cross

section method may be adequate for most parameter studies,

while the mixed spectrum method might be required for

detailed core design studies. If a very elaborate mixed

spectrum calculation is necessary to obtain multigroup

cross sections, then the accuracy of the multigroup dif-

fusion approximation may be less than that involved in the

spectrum calculation. In this case, the accuracy of the

cross sections would provide no additional accuracy in the

multigroup results. If additional accuracy is required,

one and two dimensional transport theory codes should be

used.














CHAPTER III

NUCLEONIC ANALYSIS OF EXTERNALLY MODERATED
URANIUM PL:F.'.'


The multigroup diffusion approximation to the

Boltzmann equation is used to make a parametric investi-

gation of some of the essential nuclear characteristics

of an externally moderated uranium plasma. These calcu-

lations provide a quantitative check on the assumption

that the neutron energy spectrum in the uranium plasma is

primarily determined by the moderating characteristics of

the surrounding reflector. The results of these calcu-

lations are used to define a reference model of the plasma

core reactor. This model is used to study the coupling

between the plasma and nucleonic characteristics of a

fissioning uranium plasma.

The neutron energy spectrum in graphite is calcu-

lated using the GAM-1 code (32) for the epithermal portion

of the spectrum and the TEMPEST code (34) for the thermal

portion. A value of 1.125 eV is used for the thermal-

epithermal energy boundary (maximum graphite temperature

corresponds to 0.35 eV), and the U-235 fission spectrum

is used as the neutron source. The calculated spectrum

represents the energy distribution of neutrons which

originate from fissions in a central U-235 plasma and are


- 41 -






- 42 -


slowed down by collisions in a surrounding graphite re-

flector. Since the reflector contains no uranium, the

spectrum is determined entirely by scattering and ab-

sorption in graphite. The calculated neutron spectra

corresponding to several reflector temperatures are shown

in Figure 5. Changing the graphite temperature affects

the thermal portion of the spectrum by shifting the most

probable energy, E = kT, at which the thermal neutron

flux is a maximum.* Once the graphite temperature is

specified, the calculation of the neutron energy spectrum

and multigroup cross sections in the reflector is

straightforward.

The neutron spectrum and cross sections in the

central uranium plasma are calculated using the following

argument. Consider a homogeneous mixture of graphite and

a small amount of U-235. Most of the epithermal neutron

scattering will be due to graphite, while thermal neutron

absorption will be almost entirely due to uranium. For a

sufficiently small amount of U-235, the epithermal

scattering effects of uranium and the thermal absorption

effects of graphite are negligible. In this case, the

total neutron spectrum can be considered to be the



The thermal energy flux c(E) has a maximum at
E = kT, while the thermal lethargy flux (u) has a maximum
at a lethargy corresponding to E = 2kT.










bK-i

Ii
10- /







6 -

Graphite Temperature
r -- K2000K -1
-1 J 3-20000K I

30000K

Sj -40000K _
2
I \ -j



Lethargy 20 15 I 1 5
Energy .02eV ley 10eV lkeV IkeV i0keV 1MeV MeV


Figure 5. Neutron spectrum in graphite.






- 44 -


superposition of an epithermal graphite spectrum and a

thermal uranium spectrum. Now consider the heterogeneous

system consisting of a central U-235 plasma surrounded by

a graphite reflector. Due to the low uranium density, the

epithermal macroscopic scattering cross section of uranium

is negligible compared to that of graphite and the uranium

plasma is essentially transparent to high energy neutrons.

Hence, the higher energy epithermal neutrons in the core

region have a spectrum determined by the reflector and

experience a negligible attenuation in the uranium plasma.

The low energy portion of the epithermal neutron spectrum

will be determined by graphite scattering plus uranium

absorption due to the U-235 resonances in the 1 eV to 100

eV range. Epithermal neutrons which are slowed down by

scattering collisions in the reflector become the source

of thermal neutrons in the core region. The thermal

neutron spectrum emerging from the reflector is then

modified by absorptions in the uranium plasma. The flux

weighted thermal and epithermal multigroup cross sections

of the uranium plasma region are calculated using the

neutron spectrum of graphite containing a dilute quantity

of homogeneously mixed U-235. This method of flux weighting

includes the spectral effects of both graphite scattering

and uranium absorption and thereby provides a more accurate






- 45 -


description of the physical system than does the method of

weighting uranium cross sections by the unmodified neutron

spectrum in graphite.

Several calculated neutron spectra in a graphite

moderated uranium plasma are shown in Figure 6. These

spectra were calculated by the methods described in the

preceding paragraph. When the uranium density is less

than N235 = 101/cc, the epithermal spectrum in the uranium

plasma is equal to the spectrum in graphite, since reso-

nance absorptions in uranium are negligible for such

dilute quantities of U-235. As the uranium density is

increased to N235 = 101/cc, resonance absorptions begin

to slightly depress the epithermal spectrum in the 1 eV

to 100 eV range. At a density of N235 = 1020/cc, the

neutron spectrum is significantly modified by thermal and

epithermal absorptions in the uranium plasma.

The spectra shown in Figures 5 and 6 were used to

calculate multigroup cross sections for the reflector and

uranium regions, respectively. Using the AIM-6 multigroup

diffusion theory code (35), a parameter study was made to

determine the range of values of uranium density, core

size, flux shapes, and reflector thickness for which a

critical U-235 plasma reactor can be realized. The purpose

of these calculations is to determine a reasonable

reference model without attempting to optimize all the





i I i I I I I I I I i I i 1 1 i -- .


1o I
i- /


f \ Graphite Temp = 40000K / \\

I








i_ /U-235 Plasma Density _

_I I_ -
1-



\2 /
S1019 --i
/ 1019/cc
S10 2o /cc -

Lethargy 2 15 5 1
Energy .02eV leV LeV IOkeV lMeV i MsH


Figure 6. Neutron spectrum in graphite moderated uranium plasma.






- 47 -


nuclear characteristics of the system. Typical thermal

flux profiles of critical reactor configurations are shown

in Figure 7. The thermal flux attenuation is seen to be

very insensitive to changes in the uranium density profile,

in the case of low fuel density. An average uranium

density of N235 = 101/cc was used in these two calcu-

lations.

Based on the four group diffusion theory calcu-

lations, the uranium plasma reactor reference model is

defined as follows:

Core radius 4 ft

Core height 8 ft

Graphite reflector thickness 2 ft

Graphite temperature at core
interface 40000K

Uranium composition 100% U-235

The critical mass of U-235 will depend on the uranium

density distribution, the plasma temperature distribution,

and pressure, and the hydrogen coolant density and

temperature distribution.

These parameters will have to be specified in

order to compute the critical mass. The diffusion theory

calculations indicate that average U-235 densities on the

order of 0.5 x 1019/cc to 1.6 x 1019/cc will be required

to achieve a critical plasma reactor at the desired




w4W


10

oe-r

x 8




6



r1 4
2







Fuel Graphite reflector
Fuel Graphite reflctorI

0 40 80 120 160 190

Radius, cm

Figure 7. Thermal neutron flux in graphite reflected uranium plasma reactor.


MW






- 49 -


high temperatures. These densities correspond to U-235

masses of about 25 kg to 75 kg. The choice of U-235 as

the fuel for the reference model is based on the availa-

bility of both plasma and nuclear data for uranium. Either

Pu-239 or U-233 appears to be a better plasma reactor fuel

when only the nucleonic criticality effects are considered

(23,31). However, since the plasma characteristics of the

fuel also have an important effect on the critical mass,

the per cent ionization, as a function of pressure and

temperature, must be considered when comparing the relative

merits of the three fuels.













CHAPTER IV

PRESSURE-TEMPERATURE ANALYSIS OF URANIUM
PLASMA FUEL REGION


Introduction

In order for the uranium fuel region to function

efficiently as a source of energy in the plasma core

reactor, the uranium should exist at temperatures in the

50,0000K to 100,0000K range. Under these conditions, heat

transfer is accomplished primarily by radiation. At these

temperatures, the uranium fuel is highly ionized and must

therefore be treated as a plasma rather than an ordinary

gas. Since uranium becomes appreciably ionized at tempera-

tures above a few thousand degrees Kelvin, the plasma

effects are important at all temperatures of practical

interest. In performing a nuclear analysis or radiative

heat transfer analysis of an externally moderated high

temperature reactor, these plasma effects must be considered.

In order to account for these effects, it is necessary to

determine the composition of the uranium plasma fuel region

as a function of plasma temperature and pressure.


Basic Equations for Arbitrary Temperatures

An uranium plasma in local thermodynamic equilibrium

is characterized by the following set of equilibrium ion-

ization and recombination reactions. The uranium atom and


- 50 -






- 51 -


all ions through twelve-times ionized uranium are included.

The more highly ionized species are negligible for the

temperatures and pressures being considered in this study.


U O Ui+ + e

U1+ : U2+ + e

U2+ + U3+ + e






i+ (i+l)+
U +U + e


U11+ t U2+ + e


(4.1)


where


U = neutral uranium

i+
U = i-times ionized uranium

e = electron


The relation between the number of ions and

electrons in an equilibrium ionization-recombination

reaction is described by the following Saha equation (36):






- 52 -


N N u. 2im k 3/2
Ni+le = 2 1+1 e T3/2e-Ei/kT L4.2)
i i h

where

Ni1 .= density of (i+1)-times ionized uranium

Ni = density of i-times ionized uranium

Ne = density of electrons

m = electron mass
e

k = Boltzmann's constant

h = Planck's constant

T = plasma temperature

ui+ = partition function of (i+l)-times ionized
uranium

u. = partition function of i-times ionized
uranium

E. = ionization potential for the reaction


Ui + U(i+l)+ + e


Due to the electrical neutrality of the plasma,

the net charge is zero. This balance is expressed as:


Total negative charge of electrons = Total positive charge
of ions






- 53 -


12
N = I Z. N.
e iO=


where


Z. = charge of i-th uranium ion
1


The temperature, pressure and total particle density

are related by the equation of state for a perfect gas.

The total pressure is the sum of the partial pressures of

the individual species as given by



P = Pi = NOT kT
1


N kT =
TOT


12
. N.kT = NekT + N.kT
1 i=O


12
N TOT N + N.
NTOT Ne i=0 Ni
i=0


(4.4)


where


NTOT = total particle density


The right side of the Saha equation (4.2) is

dependent on the temperature as given by


u 2nm k 3/2
u. 3/2 T3/2eEi/kT
Ki(T) E 2 i+l ( T3/2 e-/k
u1 i h

The i-th Saha equation is then written as


(4.5)


(4.3)







- 54 -


Ni+ N
N. Ki(T) (4.6)



Combining equations (4.3), (4.4), and (4.6) gives the

complete set of equations describing the plasma composition.


N, N
e K, (4.7-1)
1



N N
Se = K, (4.7-2)
2








N. N
+ e K. (4.7-i)
N. 1
1


N 13 N
N 1 2 K12
N12


13
N = Z. N
e i= 1


(4.7-12)


(4.7-13)


(4.7-14)


13
N = N + + i
TOT e 1
i=l







- 55 -


Note that each uranium index (i) has been increased by

one, so that within the formulation of the numerical

expressions used in the computer program, the following

definitions are used:


N = density of U in particles/cc

N = density of U+ in particles/cc







N13 = density of U12+ in particles/cc


Since the electrons and various uranium species will have

densities with numerical values in the approximate range

of 0 to 1020 particles/cc, it is convenient to scale

equations (4.7-1) through (4.7-14) so that all particle

densities can be expressed as dimensionless concentrations

Ci, where 0 < C. < 1. Dividing each term in equations

(4.7-1) through (4.7-14) by NTOT gives the following set

of scaled equations.



C, Ce *
c- K (4.8-1)
C 1
1


C3 C
2 e K2* (4.8-2)
2






- 56 -


C C
i+1 Ce
C. K
1


C C
13 e
12


13
C =e
i=1


= K *
12


Zi Ci
1 1


(4.8-i)


(4.8-12)


(4.8-13)





(4.8-14)


1 = C + c.
i=l


Ci = Ni/NTT Ce = Ne/N Ki* = Ki/NTT


(4.9)


From equations (4.8-1)through (4.8-12),


C2 = -- K*
e

C2 C1 C1
C3 K2* K* K2* K2
e C C2
e e


where





- 57 -


C3 C C
K K K
C, = K3 1 K2* K3 K'3
e C C
e e






In general,


C.
C ---K =
Ci C i-i
e


C
1-1 K K2. Ki-1
C


C
K
-e1 i-1
e


i
K = H K.
1 j=l 3


From equations (4.8-13) and (4.10)


]3
C = Z iC
i=l


]3
= 1 Zi
i=l


C
S i-i
C
e


13
i=l


Z ,
K.
C
e
(4.12)


From equations (4.8-14) and (4.10)

13 ]3
1 = C + C. = C + C1 + C.
i=l1 i=2


]3 K'
1 = C + Cl + i-1
e i=2 C


(4.13)


Solving for C, from equations (4.12) and (4.13) and equating


the results,


where


(4.10)


(4.11)






- 58 -


C 1 C
13 K- C 13 K' (4.14)
-1-1 i-1
Zil i 1 +
i=1 C i=2 C
e e

The set of equations (4.8-1) through (4.8-14) has thus

been reduced to a fundamental expression relating the

electron concentration to the pressure and temperature.

Equation (4.14) can be written as

C C -1
F(P,T,C ) e + e-- 3 = 0 (4.15)
13 K' 13 K1
i-l i-1
i 1-1 1 + 1-1
i=2 C i=2 C
e e

For a given temperature and pressure, F(P,T,Ce) = F(Ce)

becomes a polynomial function of Ce, and (4.15) can be

expressed as


G(C )

e
F(Ce) -= (4.16)


where


13 K' 13 K'
G(C) = C(l + ) + (Ce 1) Z. --i (4.17)
i=2 C i=2 C
e e


13 K'. 13 K'.
H(Ce) = (1 + I -- )( Z -i-i) (4.18)
i=2 C i=2 C
e e

Since all C K'., and Z. are positive, H(C ) is always
'*-- ~ J -L






- 59 -


positive and the zeros of F(C ) are given by the zeros of

G(C ). Thus, the electron fraction, Ce, is given by the

solution of


G(C ) = 0 (4.19)

Simplifying equation (4.17), the polynomial equation (4.19)

is

13
G(C) = Ce + 1 + Zi)Ce Z. C
i=2 e

x K' i = 0 (4.20)
i-i

Equation (4.20) is a 13th degree polynomial in C with

one real positive root. From physical intuition there

should be only one real root, since for a given tempera-

ture and pressure, the number of electrons in a plasma is

uniquely determined. This can be shown rigorously by

application of Descartes' rule of signs (37) to equation

(4.20). Solutions to equation (4.20) are obtained using

the iterative Newton-Rhapson method (38). The Newton-

Rhapson recurrence formula for the solution to (4.20) is


G(Ce(n))
Ce(n + 1) = C (n) G- (4.21)
e e G7~(n) )






- 60 -


where


Ce (n+l) = value of Ce after n + 1 iterations

C (n) = value of C after n iterations
e e
C (1) = initial estimate of C
e e

dG
G' (C (n)) = d-
e ,e Ce = C (n)


G(Ce(n)) is obtained directly from (4.20). G' (Ce(n)) is

obtained by differentiation of (4.20).


13
G'(C ) = 1 + I (2-i) (+Z )C (l-i)Z. C K'
i=2 1

(4.22)


Using equations (4.20) and (4.22) with (4.21), the electron

fraction is then given by


C (n+l) = C (n)


Ce(n) + [(l+Zi)C (n) Zi 1(n) K'
i=2

1 + I. (2-i)(l+zi)C 1-i(n) (-i)z. C (n Ki'.
i=2 e e -1

(4.23)


Beginning with an assumed average electron concentration

of C (1) = 0.5 as the initial estimate, expression (4.23)






- 61 -


is evaluated repeatedly until the desired degree of accu-

racy is obtained. The electron density is then given by


N e C N
e =e TOT

From equation (4.12),


C
C 13e and N = C NTOT
S 13 K'. 1 1 TOT
1-1
zi i-1
i=2 C


From equation (4.7-i), N.i+ = K. N /N for i = 1, 2, .,

12, gives the particle densities N2, N3, ., N13. The

particle densities Ne N1, N2, ., N13 then completely

describe the uranium plasma composition.


Low Temperature Equations

For temperatures less than about 10,0000K, the

iterative method for solving equation (4.14) does not

readily converge. Although uranium is not highly ionized

at temperatures below 10,0000K, a knowledge of the low

temperature uranium plasma composition is necessary in

order to describe initial or startup conditions in a

gaseous-core reactor. A different mathematical formulation

is used to calculate the plasma composition at low tempera-

tures.






- 62 -


In the temperature range from 10000K to 10,0000K,

the only species that are present in significant quantities

are U, U and electrons. A single reaction accounts for

the equilibrium ionization and recombination of these three

species.


U t U1+ + e (4.24)

The corresponding Saha equation is


N+ e 2r7m kT 3/2 u
N 2 ( e -- e kT (4.25)
No h2 o


The total particle density is

P
NOT -= T = No + N + e (4.26)

The charge neutrality is given by


N = e (4.27)

Using N, = No, N2 = N+, Ne = e, and


27rm kT 3/2 u -/
K = 2 e) e-Eo/kT
h2 uo

equations (4.25), (4.26) and (4.27) are simplified as


N2 N
N2e K (4.28)
N,






- 63 -


NTOT = N + N + Ne
TOT 2 e


(4.29)


(4.30)


N2 = N


Dividing each term in equations (4.28), (4.29), and(4.30)

by NTOT and combining these equations gives


C 2 + 2K*C K* = 0
e e


where


(4.31)


K* = K
TOT

The solutions of equation (4.31) are


Ce = K*(- 1 /i + T7K-)


Since K* > 0, /i + 1/K* > 1, and the positive radical must

be used to give a positive value of the electron concentration.

The low temperature plasma composition is then given by


N = NTO K* (-1 + V1 + 1/K*)


N = NTOT


- 2Ne


N2 = N
2 e


(4.32)


(4.33)


(4.34)


Uranium Partition Functions

The partition functions appearing in the general

Saha equation (4.2) are functions of the plasma temperature






-64 -


and the electron configuration of the particular uranium

atom or ion. The electronic partition function of i-times

ionized uranium is given by (39)


C -Xi j/kT
ui = gij eij/kT


Si0 + i,l e-X/kT + i,2 e-X/kT + (4.35)


where

gi = statistical weight of the j-th term of
'J i-times ionized uranium

X. = exitation energy level of the j-th term
of i-times ionized uranium

The j-th statistical weight for the i-times ionized uranium

is (40)


9i,j = (2Lj + 1)2Si + 1) (4.36)

where

L. = orbital angular momentum

S. = spin angular momentum


The statistical weights of the uranium atom are

calculated from the spectroscopic data of Steinhaus, Blaise,

and Diringer (41). Since experimental uranium ion data

from which the necessary Lij and Sij values can be

determined are not presently available, the iso-electronic






- 65 -


approximation (42) is used to calculate the statistical

weights and excitation energies of the uranium ions. This

approximation is based on the observation that atoms or

ions having the same number of orbital electrons have

similar electron configurations. The gij values are

calculated from (4.36), using the Li-j and Sij values of

the neutral atom having i orbital electrons. The Xij are

obtained from spectroscopic measurements (43) of energy

levels of the neutral atoms.


Lowering of Ionization Potentials

The ionization potential Ei in equation (4.2)

represents the energy required for the ionization of an

isolated atom or ion. At high pressures the plasma

density increases until the atoms and ions can no longer

be considered as isolated. Each particle then moves in

an electric field associated with adjacent charged particles.

The ionization reactions are influenced by plasma micro-

fields associated with electrostatic polarization (Debye

effect). The resulting effect is a lowering of the energy

required for ionization of an atom or ion, the effect

increasing with increasing charge density. To account for

the lowering of the ionization potentials, an effective
eff
ionization potential, E. is defined as


E.eff = E. AE. (4.37)
1 1 1






- 66 --


for the reaction Ui U(i+l)+ + e

where

E. = uncorrected ionization potential

AE. = lowering of ionization potential

Griem (44) has used the Debye-Huckel theory (36) for

ionized gases to obtain an analytical expression for the

amount of potential lowering. For an uranium plasma with

a maximum of twelve degrees of ionization, the potential

lowering is given by

13
AEi = 2(Zi + l)e3(r/kT)/2(Ne + Zi2Ni)1/2
i=2
(4.38)

where

Z. = charge of i-th ion

e = electrostatic charge of the electron

N = electron density

N. = i-th ion density


Cut-off Quantum Number as a Function of Effective Ionization
Potential

The electronic partition function of uranium was

given by


u = e-Xi,j/kT (4.35)
j=0






- 67 -


The index j is the principal quantum number of the j-th

energy level of i-times ionized uranium. The partition

function of a hypothetical isolated particle cannot be

evaluated exactly since (4.35) involves an infinite number

of terms. When the summation extends to j = m, the

partition function diverges since Xij approaches a

constant value (the ionization energy), and gij is not

monotonically decreasing as j -- However, due to the

effect of the Debye field, the summation can be terminated

at a finite cut-off quantum number, n corresponding to

the highest energy level at which the electron is still

bound. The calculational form of the partition function

based on bound electron states is


n (i)
c Xi,j/kT (4.39)
S j=0

The cut-off quantum number is a function of the effective

ionization potential since nc (i) is the largest integer
eff
value for j for which X. < E. For the i-th uranium
1,3 1
ion, the cut-off number is expressed as


nc(i) = 3 Xij < E ef (4.40)



The relation between principal quantum number and

energy level for a hydrogen-like atom is (45)






- 68 -


pZ2 e4
eff
E = (4.41)
n 2h2n2

where p is the reduced mass, Z the nuclear charge, e the

electrostatic electron charge, h the reduced Planck's

constant, and n the principal quantum number. For the

multi-electron atoms and ions in the uranium plasma, the

energy levels and principal quantum numbers are not

explicitly related by expressions such as (4.41). However,

it can be argued that a highly excited electron in a

multi-electron atom is nearly hydrogenic, since the atom

can be regarded as a nucleus having an electron moving

about it in a large orbit. Based on this reasoning,

Drellishak et al. (46) have empirically treated the

lowering of the ionization potential by assuming that the

amount of potential lowering should vary inversely with

the square of the cut-off quantum number, according to

E.
AE = (4.42)
n 2(i)
c

where the symbols are defined as before. This relation

will be used in the following section to obtain initial

estimates of the cut-off quantum numbers of uranium atoms

and ions.











Computation of Uranium Plasma Composition

The computation of the uranium plasma composition

requires the determination of a consistent relation

between (1) the electron fraction (4.23) (or (4.31) for

low temperatures), (2) the lowering of the ionization

potential (4.38), and (3) the cut-off quantum number (4.40).

These expressions are seen to be completely coupled since

the plasma composition is dependent on the partition

functions, the partition functions are dependent on the

effective ionization potentials, and effective ionization

potentials are dependent on the plasma composition. The

uranium plasma composition is calculated using an iterative

technique which searches for the consistent solution of

equations (4.23), (4.38), and (4.40). A program called

UPLAZ-2* was written to perform these iterative calcu-

lations.

For a given pressure and temperature, the UPLAZ-2

iteration is carried out as follows:

Step 1.--Starting with equation (4.23), the electron,

atom, and ion densities are computed using only the first

term (ground state) of (4.39) in the partition functions.

These particle densities are used to compute the ionization


*Details of UPLAZ-2 and other programs written for
this study are given in the Appendices.


- 69 -






- 70 -


potential lowering, AEi, according to (4.38). From

equation (4.42), the initial estimate of the quantum

cut-off numbers are calculated as


N ()(i) = /E~.E i = 2, 3, ., 13
c 1 1

(4.43)

Step 2.--Returning to equation (4.23), the particle

densities are again calculated, this time including

n () (i) terms in the partition function series (4.39).

Using the new number densities, the next value of the

ionization potential lowering is calculated from (4.38).

The second estimate of the cut-off quantum numbers is

then determined from



c (i) = [nc(i)] (4.44)


where [nc(i)] is the largest integer value not exceeding

j, for which


X < E. AE. (4.45)
1, 1 1

where


E. = uncorrected ionization potential

AEi = current value of ionization potential lowering






- 71 -


Step 3.--If n (2) (i) n1) (i) > 1 for any i,

Step 2 is repeated r times until n(r) (i) nc (i) < 1

for all i. When this condition is satisfied, the plasma.

composition is determined consistent with the ionization

potential lowering and cut-off quantum numbers.


Results of Uranium Plasma Composition Calculations

The UPLAZ-2 program was used to calculate the

uranium plasma composition for pressures from 1 atm to

1000 atm and temperatures from 50000K to 120,0000K.

Typical results are shown in Figures 8, 9, and 10, for

pressures of 100, 500, and 1000 atmospheres. The electron

density and the densities of individual uranium species

are given in per cent of total particle density. Ad-

ditional calculated uranium plasma compositions for

pressures from 100 atm to 1000 atm are shown in Figures 25

through 34, Appendix A.

The fact that the fuel region of the "gaseous"

core reactor is a plasma, rather than a ordinary gas, is

clearly demonstrated in Figures 8, 9, and 10. At a

temperature of 20,0000K, the total number of particles in

the fuel region consists of about 50% uranium and 50%

electrons. Above 40,0000K, the fuel region contains

approximately 80% electrons and only 20% uranium. In






- 72 -


100 71













U5+ U6+

10


H

w 7+
ClU


z8+





O 9+






010
0
















0 40,000 80,000 120,000

Temperature, oK


Figure 8. Uranium plasma composition at pressure of 100 atmospheres.





- 73 -


100


10



Z U7+









E-1
U
H





H











0.1
H


z
U











0.1

0 40,000 80,000 120,000

Temperature, oK


Figure 9. Uranium plasma composition at pressure of 500 atmospheres.





- 74 -


100


H- UO' U'
10
H




U U
H






0+


S1
U














0.1
0 40,000 80,000 120,000

Temperature, oK


Figure 10. Uranium plasma composition at pressure of 1000 atmospheres.






- 75 -


order to determine the radiative emission and absorption

characteristics of the uranium plasma fuel region, the

particle densities of the various uranium species must

be considered, because each ion has different radiative

properties. In order to determine the nuclear criticality

characteristics of the reactor, only the total uranium

density is considered, since neutron induced fission is

a nuclear effect and is not influenced by ionization of

uranium atoms. This is not meant to imply that plasma

effects have no influence on nuclear criticality, since

the plasma characteristics of uranium actually determine

the critical uranium mass. This static coupling between

plasma and nuclear effects is shown by the equation of

state (4.3), which can be written as


12
P = P + Y P. (4.46)
i=0

where

P = N kT, partial pressure of electrons
e e

P. = N.kT, partial pressure of the uranium species
1 1
Ui+


This expression shows that in thermodynamic equilibrium,

one free electron creates the same partial pressure as one

uranium atom or ion. Thus, for a given constant reactor






- 76 -


pressure P, the formation of each free electron by ioni-

zation is balanced by the removal of one uranium atom or

ion. From Figures 8, 9, and 10, the electron fraction is

seen to increase monotonically as the temperature increases,

which means that any temperature increase at constant

pressure requires a reduction in the amount of uranium

in the fuel region. The quantitative effects of temperature

on the uranium density are shown in Figure 11. The slope
dN
of the curve (i-) gives the rate of uranium reduction
P
with increasing temperature at constant pressure. The

uranium reduction rate decreases at higher temperatures,

since the electron fraction changes very slowly for tempera-

tures greater than 40,0000K. From Figure 10, a basic

characteristic of the plasma core reactor is seen to be

strong dependence on the uranium density on the plasma

temperature. For a given constant pressure, there is

only one temperature which will sustain the critical

uranium mass. Suppose that the reactor is to be operated

at a constant pressure of 400 atm and that the critical

mass corresponds to a total uranium density of l019/cc.

From Figure 11, it is seen that a temperature of 59,0000K

would be required to sustain the desired uranium mass. An

increase in the temperature would cause the reactor to

become subcritical and similarly, a temperature decrease






- 77 -


C.)
1021
















S1028 II


















0 40,000

Temperature, 'K
H




H
z








I 1 "













0 40,000


Temperature, 0K


Pressure, atmospheres


1000


Effect of temperature on total uranium density.


Figure 11.






- 78 -


would result in a supercritical reactor. This dependence

of critical mass on plasma temperature indicates that a

control system which automatically compensates for tempera-

ture variations would be a necessary component of the

plasma reactor system. In an operating rocket reactor, a

change in the thrust requirement, accomplished by varying

the power output, will involve temperature changes in the

uranium plasma. Since a large temperature change at

constant pressure can significantly affect the criticality

and hence the reactor stability, the pressure must be

allowed to vary in order to maintain a constant uranium

mass as the temperature changes. Thus, the criticality

problem becomes a question of finding a pressure-tempera-

ture relation which sustains the desired uranium mass.

This relation can be expressed mathematically as a problem

of finding temperature dependent pressure trajectories

along which the uranium mass has the critical value. The

general pressure trajectory is expressed as


Pc = f(MU,T) (4.47)

where

P = pressure trajectory

MU = critical uranium mass


T = plasma temperature






- 79 -


f = functional relation between Pc, MU, and T

The functional relations which define pressure

trajectories are determined numerically using the UPLAZ-2

program. For the uranium density corresponding to a given

critical mass and a given temperature, the UPLAZ-2 equations

are solved for the pressure P. By solving the equations

for temperatures from 50000K to 120,0000K, the resulting

pressures define trajectories along which the uranium mass

is constant. The nuclear calculations discussed in

Chapter III indicate that a critical reactor would require

uranium densities in the neighborhood of 1019/cc. Calcu-

lated pressure trajectories for several uranium densities

are shown in Figure 12. The most important trend shown

by these curves is that as the critical uranium density

increases, extremely high pressures are required in order

to operate the reactor at high temperatures. Therefore,

every effort should be made to reduce the critical mass

by optimizing the reactor geometry, the reflector geometry

and materials, and the fluid flow characteristics. From

Figure 11, it is seen that if the critical uranium density

can be reduced from 1.0 x 1019/cc to 0.5 x 101'/cc, the

pressure required to operate the reactor at 100,0000K is

reduced from 950 atm to 480 atm.





- 80 -


1000
800
0.5xl0
600


400

300


200 U-235 density(/cc)

m O.1xlO10 19



100



















10
0 40,000 80,000 120,000

Temperature, K


Figure 12. Pressure trajectories for constant uranium density.






- 81 -


The pressure trajectories for an actual hydrogen

cooled plasma core reactor will differ quantitatively

from those in Figure 12. This is to be expected since

the critical mass will not remain constant over wide

temperature variations, due to the temperatures dependence

of hydrogen scattering and uranium resonance absorption

as discussed in Chapter II. However, the application of

pressure trajectories in determining the critical mass

will still be valid when these two effects are included,

and the general form of the trajectories in Figure 12 can

be expected to be similar to those in the hydrogen cooled

uranium plasma.














CHAPTER V

PRESSURE-TEMPERATURE ANALYSIS OF HYDROGEN
PLASMA COOLANT REGION


Introduction

The high temperature plasma core reactor will be

cooled by hydrogen which is injected concentrically around

the central uranium region. Cooling will be accomplished

by radiant energy transfer from the high temperature

uranium plasma to the lower temperature hydrogen. When

injected at low temperatures, the hydrogen will be in a

diatomic molecular gaseous state. As the hydrogen coolant

flows coaxially around the uranium fuel, the hydrogen

temperature will increase due to absorption of radiant

energy from the fissioning uranium plasma. At high

temperatures hydrogen becomes dissociated and ionized so

that the exhausted coolant will consist of numerous hydrogen

species resulting from the various dissociation and ioni-

zation reactions. In order to determine the neutron

scattering effects and radiant energy absorption properties

of the hydrogen coolant, it is necessary to calculate the

composition of a hydrogen plasma as a function of plasma

temperature and pressure.


- 82 -







- 83 -


Equilibrium Hydrogen Reactions

For the range of pressures and temperatures

expected in the plasma core reactor, the hydrogen coolant

may undergo any of the following equilibrium reactions:


1. Molecular dissociation

2. Atomic ionization

3. Atomic electron
attachment

4. Molecular ionization

5. Dissociative recombination

6. Dissociation of molecular
ions

7., 8., 9. Two-body
combination





10., 11. Three-body
recombination


H2 Z H + H




H + e < H
H +

+ +
H2 +e H + H

+ H+
H2 + + H


H + H2 H,

H2 + H+ H3


H + H + H2 + H


H + H + H H2 + H


From an examination of the reaction energies, absorption

properties, and expected coolant temperatures, the hydrogen

coolant is considered to be adequately described by the

first four reactions. Reaction 9 can indirectly enhance

the absorption properties of hydrogen at higher pressures

(47), but will not be included in the present analysis.






- 84 -


Hydrogen Reaction Equations

1. Molecular dissociation H2 f H + H.--The

equilibrium relation between molecular and atomic hydrogen

involved in molecular dissociation at absolute temperature

T is (39)



N NH 27mHmk 3/2 UH H -E'H /kT
H H _____ T3/2 e 2 (5.1)
NH mH h2 H,2


where

mH = atomic weight of H

mH = atomic weight of H

uH = partition function of H

UH2 = partition function of H


E'H_ = dissociation energy of H2
H2

k = Boltzmann's constant

h = Planck's constant

The partition function of atomic hydrogen is



uH = gj -Xj/kT (5.2)


where

gj = statistical weight of j-th term

X = excitation energy of j-th term






- 85 -


The n-th state of atomic hydrogen is 2n2 times degenerate

X0
and has an energy X, = X0 --, where n is the principal
n
quantum number and XO is the ground state energy level.

On this basis, the partition function may be written

explicitly as

nc (1 XO/kT
UH = 2n2 e n- (5.3)
n=l

where nc is the principal quantum number of the last bound

state. Defining a reduced energy variable a t X0/kT,

(5.3) is simplified as

n
H = 2e n n2 en2 (5.4)
n=l


For a sufficiently large quantum number, nL, the energy

levels approach a continuum. For n > nL the integral

approximation of Ivanov-Kholodnyi et at. (48) can be

applied to the partition function. Replacing the near-

continuum portion of (5.4) by an integral,


nL n/(5
UH = 2e-a n2 + n e dn (5.5)
n=l nL


For


a/n 2 < 1






- 86-


n n
uH 2e- n2 e + ndn
n= nL

-ij

UH = 2e-


x e + 4e/4 + 9e/9 + n eC/nL2 + 1(nc n3


(5.6)


The number of terms to be retained in the series (5.6) is

determined by the requirement that n 3 3 e /nL

This requirement also provides that n3 >> nL 3 so that

the partition function may now be written in a form which

is convenient for computation:


nL
u = 2e-a n2 e + 1 n (5.7)
n=1


The quantum number, n at which the electron

becomes free due to preionization of hydrogen, is a

function of the charge density of the plasma. Ivanov-

Kholodnyi et aZ. have derived a relation between nc and

the electron density Ne as follows:


log Ne = 21.65 6 log (nc + 1)
e C


(5.8)






- 87 -


This expression is consistent with the integral approxi-

mation used in (5.5) and will be used to calculate the

quantum number appearing in the partition function (5.7).

The usual form of the partition function of

molecular hydrogen is


uH E uR u (5.9)


where

uE = electronic partition function

uR = rotational partition function

u = vibrational partition function

The electronic partition function is



u = g ej/kT (5.10)


The statistical weight is given by


g. = 2J(j) + 1 (5.11)
j H2


where J is the total angular momentum of the j-th state
H2
of H,.

When the diatomic hydrogen molecule is pictured as

a rigid dumbbell with moment of inertia I, rotating in

three dimensions, the energy and degeneracy of the J-th




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