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Fissioning uranium plasma for rocket propulsion

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Title:
Fissioning uranium plasma for rocket propulsion
Creator:
Atwater, Henry Foust, 1937-
Publisher:
University of Florida
Publication Date:
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English
Physical Description:
xiii, 206 leaves : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Coolants ( jstor )
Graphite ( jstor )
Hydrogen ( jstor )
Neutrons ( jstor )
Plasma composition ( jstor )
Plasma temperature ( jstor )
Plasmas ( jstor )
Temperature distribution ( jstor )
Uranium ( jstor )
Uranium plasmas ( jstor )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF ( lcsh )
Nuclear Engineering Sciences thesis, Ph.D. ( lcsh )
Plasma rockets ( lcsh )
Uranium ( lcsh )

Notes

Bibliography:
Bibliography: leaves 200-205.
General Note:
Manuscript copy.
General Note:
Thesis - University of Florida.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright Henry Foust Atwater. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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021597777 ( ALEPH )
13166470 ( OCLC )
ACX0023 ( NOTIS )

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University of Florida Theses & Dissertations
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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. 1. 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 acknowledged. 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 . iii 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 80,000�K . 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 nucleonic 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 analysis 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 advanced because of the magnitude of the complex physical phenomena which must be understood and the many technological 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 reactors (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,* I s, 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 desirable. 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 considered 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 propellant 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 (9,13,15,1.6). The vortex flow reactor illustrated 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 associated 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 tangentially 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


ssure shell Propellant inlet



/Y
11v






Rotating olasma fuel



Figure 1. Vortex flow plasma reactor.






Neutron reflector
Pressure shell Propellant inlet











Current-carrying Magnetic field
coils lines
Plasma fuel


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


lector






- 8 -


Propellant exhaust


Propella nt_/,' 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






- 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 o2 these methods.

The effects of the high fuel and coolant temperatures 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 hydrogenland uranium mixing is developed. In Chapter VIII the calculated plasma compositions and the simulated flow distributions are used to determine the temperature distributions 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 integrodifferential 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 .(, , t
(r,v t) + � + Et(r'v) (r'v''t)



x(v) f dv, f df ' cv'Zfc ,v') ( ,' , ' T 1



+ f dv' f d sE rv'-v, ' w'r'v"'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 -


angular 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'1, ' into a speed and direction range dv, dQ at v,


The individual terms of (2.1) describe the following reactions:

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


�- V =


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


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


Xfdv'fdQ'V lf =




fdv'fd-,'Z Sp =


number of neutrons of direction Q2 and speed v gained per unit time in dr from fissions due to neutrons with all speeds v and directions

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


ES (~


(rVA t)






- 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 500'K to 6001K 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 30001K (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 distributions of nuclei and thermal neutrons can be described by


-- 14






- 15 -


modified Maxwell-Boltzmann distributions having a characteristic 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 therma2ization theory are used to treat the effects of nuclear moLion 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 100,0001K 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 -


vn v n neutron velocity VN = VN N = nuclear velocity v = relative velocity


The relative speed v, is then defined as


v = n- VN, where v = V2n m (2.2)
i n ->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(rv, ,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

* The expressions for these collision processes become:


Neutrons removed by absorption and scattering =

f dVN .Nd NN(-r,VNN a)t(v)n f J ')(>,vn, nt)v
VNN


Neutrons gained by fission =


X(v) f dVrN N V dv'n f , d (v'n)N(' ,VN,r N,t)af(v)
N N n n

x� ~ n v n',t v v n





- 18 -


Neutrons gained by "scattering in" =
dVf d f v ,Nrvr *,,n ]n
VN N dN v n f ' N f N 'N t)Gs(, vv, JVN dv' n n



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


Tn(r, n,~,t) + v n Vn(rv n n t)

+ J dV N f d NN(',VN' N't)rt(v)n(rvn n't)V'n

SX(v) dVN d dv I d V(vf)N rVM


+ ~ N df N~ nv f nN ,NNta(rV 'Vt
x n( ~ n n t~v'n
Nr NJnnnn
+ d f d Nf dvI d I N(r, VN1 N't)as(-r ,v' v, n' 4 V v n)n

x n(r,v n, n t~v' (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 surrounded 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 Nc = 0.08 X i024/cc and an epithermal microscopic scattering cross section of ac 4.8 barns. The corresponding scattering mean free path is

Xs =1I/Nca = 2.6cm
graphite 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 Nu = i019/cc. With an average epithermal scattering cross section of 10 barns, the scattering mean free path in uranium is

Xs =1NUa S =14 C
=1/Nua = i0~ cm
U Uu


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
a
zero power plasma reactor is about Xu = 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 arguments, 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 demonstrated that multigroup diffusion theory is adequate for calculations of this type. In the present study, the effects of elevated temperatures and the resulting exaggerated 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
- DiV2i (r) + ti i() = xiS(r) + ,. i (r)
1 j=l
j 7i
(2.4)

where

N
E a~ + D.iB. 2 +IE i-i
ti=Ei i j=lsij
1 4


S( ) X (Vxf~ ii(r
.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 t.= total removal from i-th group


E .= absorption cross section for the i-th
a.
1 group
B.2= 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 absorption cross section, Za*, will include the effects of the motion of uranium nuclei in the treatment of the Doppler broadening of the uranium absorption resonances. The scattering cross section, E si--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


fEui Ea (E)r dE

ELi

a. E
I Eui J (E4,E)dE

L. (2.5)


where Eui and EL. are the upper and lower boundaries, 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

rr
a () wXgjny
aE =r - 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
1 2
E = j Vn2 (2.7)
r 2 n

where v is the neutron speed in the laboratory coordinate system, and i 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 ry and r are the radiative capture and neutron widths (27) which define a total width F as


S= Y + Fn


- 25 -


(2.9)






- 26 -


The reduced neutron wavelength is given by


Xh
27r 27 (2pE r) 1/2 (2.10)


The statistical spin factor gj (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, o0, occurs at E = E0 and has the value

F
4 4.g7r 2 n (2.11)


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

r1
a (Er) = 0 1 2 (2.12)
1 + , r/2



If the absorbing nuclei were stationary, Ea. would be evaluated from (2.5), with a (E r) 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 100,0001K 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


E 1 - (2.13)
r- n )

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 +rn - E (2.14)
r m n M N m + M

where the following relations have been used:


En - mv2 = kinetic energy of the neutron
n 2 n

E 1
En 2 MVN2 = kinetic energy of the nucleus


Pn mvn = momentum of the neutron PN = MVN = momentum of the nucleus

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,


- 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 terra is approximately 0.001. Therefore, the second term is considered 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 N)
- M +v nvV z (2.15)
m+M m+M n


Using (2.15) and the expression for the relative kinetic
1 2eqain214
energy for stationary nuclei, E . - pvn , equation (2.14) is simplified as


Er -- p v - vVn E - v-E Vz (2.16)


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 witi velocity V z 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) f J�(Er)P(Vz)dVz (2.18)


Using equation (2.16), the effective resonance absorption cross section is

EE dEr
�a (E) = Ja(Er)P( d 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
(E) -Y e(Ox) (2.20)
a o0 r

where

1 O 2 (X
e r.a exp[- e(x - y),] T(e,x) -] 1 + - dy (2.21)
2 - 1 wy






- 31 -


and


x = 2(E - E0)/r, y 2 (Er - E]}/r, e = FA, A = (41EOkT/MN)1/2 (2.22)

Doppler broadened resonance absorption cross section in solid fuel reactors are computed using (2.20), with T(e,x) evaluated from tabulated values based on the resonance parameters F 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 100,0000K 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 reactor criticality and stability.

Expanding equation (2.13) as before and retaining all terms, the relative energy is



1 2
E r = E + f PV N2 - 2E V z (2.23)



where E P v2 again denotes the relative kinetic energy

of the neutron stationary nuclei. The relative energy is now a function of all three velocity components (Vx, Vy, V z), 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, V z) is proportional to the three dimensional Maxwellian distribution of nuclei P(VN), where



P(V d'V 3(2 e 3/2 2/2kT 4 2dV (2.24)
N 'N 2 M7TkT eVN MN


where V = Vx2 + Vy2 + Vz2. The effective resonance absorption cross section for a neutron with kinetic energy E is






- 33 -


CO CO CO

a(E) = a f (Er)P(VNfdVN = I (a(Er)P(VN)dVxdV dz
VN - 0 _ 00

(2.25)


where Oa(Er) is given by equation (2.12), E r 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, V zp) = (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)dE - 2 e-EN/kT (2.26)
N) N (7kT) 3/2 N


Since all directions of motion of the absorbing nuclei are equally probable, EX= E E where Ei MV2 E
x y z i 2 1 EN;
i = x, y, z. The velocity component of the nucleus parallel to the direction of travel of the neutron is






- 34 --


Vz = /2Tz7H = "f (2.27)


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


Er E + EN - 2 IN, 3M (2.28.)


Using (2.12), (2.26) and (2.28), the computational form of the effective resonance cross section is

r
a aE) = a0 T H(a, f, Y) (2.29)


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



H( , Y) = 2 e e-+
� 0 1 + Y (a+ )


(2.30)

where


oa(E, T) = (E - E0)/kT

(E, T) = 2V-1iTE/3M

y(F, 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(, , 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 criticality 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 investigated. The effective cross section is defined by considering 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 --


NHs nr = s nVr = 'snVn (2.32)


where the effective macroscopic scattering cross section for hydrogen is given by


s V r VIn (2.33)

The symbols are


V = average relative velocity between hydrogen
and neutrons

Vn = average neutron speed = (8kT n/Tmn) 1/2 VH = average hydrogen speed = (8kTH/TrmH) 1/2

a = microscopic scattering cross section of
hydrogen

The effective multigroup hydrogen cross sections are calculated by the following relation:


N H a % - Vi
S.= 1 = N.ij = Es Zij
1 Vi 1 j

(2.34)

where

VH - Vi = 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
1

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


The GAM-l (32) and SOPHIST-I (33) codes are used to calculate 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 E 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 diffusion 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 PLASMA


The multigroup diffusion approximation to the

Boltzmann equation is used to make a parametric investigation of some of the essential nuclear characteristics of an externally moderated uranium plasma. These calculations 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 calculations 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 calculated using the GAM-I 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 thermalepithermal 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 reflector. Since the reflector contains no uranium, the spectrum is determined entirely by scattering and absorption 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
P
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 i(E) has a maximum at
E = kT, while the thermal lethargy flux (u) has a maximum at a lethargy corresponding to E = 2kT.

























4--






-~ 44


superposition of an epithermal graphite spectrum and a thermal uranium spectrum. Now consider the heterogeneous system consisting of a central UJ-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 n'eutron 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 = 1018/cc, the epithermal spectrum in the uranium plasma is equal to the spectrum in graphite, since resonance absorptions in uranium are negligible for such dilute quantities of U-235. As the uranium density is increased to N235 = 1019/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






- 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 = 1019/cc was used in these two calculations.

Based on the four group diffusion theory calculations, 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 i019/cc to 1.6 x I019/cc will be required to achieve a critical plasma reactor at the desired









- 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 availability 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 100,0001K 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 temperatures 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 ionization 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 0 U + + e U1+ : U2+ + e U2 + U3+ + e






i+ + e


U11+ U,2 + e


(4.1)


where


0
U = neutral uranium 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 Nu. 2 mu k 3/2
Ni+l Ne 2 i+1 e 3/2 -E/kT 4.2)
2- ( TY 9- T!e-E4./kT
N.U


where

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

N. = 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+1 = partition function of (i+l)-times ionized
uranium

ui = 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 i=O


where


Zi = charge of i-th uranium ion


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 = NTOT kT
1iO


NTOT kT =


12
NikT = NekT + I NikT
1 i=O 1


12
N TOT N e+ I N.
NTOT= Ne 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


(T) U*~ 2 mek 3/2
u. 3 / T3/2 e-Ei/kT


The i-th Saha equation is then written as


(4.5)


(4.3)






- 54-


N. e- K. (T) (4.6)
N.


Combining equations (4.3), (4.4), and (4.6) gives the complete set of equations describing the plasma composition.


N 2 N
NI-= K1 (4.7-1)
N11



N N
N =K 2 (4.7-2)








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


N 13 N
-- K12 N12


13
Ne= Z. N
e i=


(4.7-12)


(4.7-13)


(4.7-14)


13
N =N + [ N NTOT Ne
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:


Density of U0 in partic les/cc

N = density of U +in particles/cc







N 13 = 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 < 1. Dividing each term in equations (4.7-1) through (4.7-14) by NTOT gives the following set of scaled equations.

C C
- -= K1 (4.8-1)
1


C3 Ce
2 = K2, (4.8-2)
C2






- 56 -


C i+ Ce
C. K
1


C C 13 e
c12


13
c =
e i=l


- K *
12


zi Ci


(4.8-i)


(4.8-12)


(4.8-13) (4.8-14)


]3
1= Ce + [ Ci
1


Ci = Ni/NTo, Ce = Ne/N TOT, Ki* = Ki/NToT (4.9)


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


C1
C2 - C Kl *
e
C2 C1 C1
C3 K2* Kl* K - K'2
ee C 2 C 2


where





- 57 -


C3 CI CI
* K 1 * K * 1
C4 eK3 3 1 2 3 3
e e






In general,


_CiI * C.--K.
C1 Ce i-ie


C1 * * *
1K K2 . . K

C 2


C
K
Ce1 ie


i
l *
K. = I .
1 j=l ]


From equations (4.8-13) and (4.10)


13
c e Z ici
Si=l 1


13
i=z


C
K.
Ci-i
e


13
=c
i=l


z .
Ce
e
(4.12)


From equations (4.8-14) and (4.10)

13 13
C + I c =c e+C1 + I c,
i=l i=2


]3 K'
1=C + C1(l + Ae 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
e ____ e1- C, -= (4.14)
13 K'.-1
1 +
i e 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(PTC e 1 + K3eK' = 0 (4.15)
ei-i3 i-i
i i-l I=2C l
i=2 C i
e e

For a given temperature and pressure, F(P,T,C e) F(C e)

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

expressed as

G (Ce
F(Ce) = e (4.16)
e (CT(416
e

where


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



13 K'. 13 K'.
H(C e) = (1 + 1( -7424.) (4.18)
i=2 C i=2 C
e e

Since all C , K', and Z. are positive, H(C e) is always
i' 1 C






- 59 -


positive and the zeros of F(C e) are given by the zeros of G(C e). Thus, the electron fraction, C e, is given by the solution of


G(Ce) = 0 (4.19)


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


13 2-l1
G(C) = Ce + 1 1 + Z-i)Ce Z. C e
i=2 1 e


x K' i-1 = 0 (4.20)


Equation (4.20) is a 13th degree polynomial in Ce with one real positive root. From physical intuition there should be only one real root, since for a given temperature 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 NewtonRhapson recurrence formula for the solution to (4.20) is

G (Ce(n))
Ce (n + 1) = C (n) - (Ce(n)) (4.21)






-- 60-


where


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

Ce (n) = value of Ce after n iterations

C (i) = initial estimate of C e e

dGC
e , e C = C (n)
e e

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


13 ~1i-~
G'(Ce) = 1 + I (2-i) (l+Zi)C - (1-i)Z. C K'
e' i=2 L i


(4.22)


Using equations (4.20) and (4.22) with (4.21), the electron fraction is then given by C (n+l) = Ce (n)


Ca(n) + 3 2l+i)C (n) - Z (n) Kii=2
13
1 + n) I 2-i) (l+zi)Ci(n) - iZ C e-i(n)K K'
i= i(4.23)

Beginning with an assumed average electron concentration of Ce (1) = 0.5 as the initial estimate, expression (4.23)






-61 -


is evaluated repeatedly until the desired degree of accuracy is obtained. The electron density is then given by


ee N
e TOT

From equation (4.12),


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

i=2 CiFrom equation (4.7-i), Ni+1 = K N e/Ni 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 10,0001K, the iterative method for solving equation (4.14) does not readily converge. Although uranium is not highly ionized at temperatures below 10,0001K, 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 temperatures.






- 62 -


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

the only species that are present in significant quantities are U1, U1 and electrons. A single reaction accounts for the equilibrium ionization and recombination of these three species.


U0 t U'+ + e (4.24)

The corresponding Saha equation is


N+ e 2 7m kT 3/2 u+ _E
2 (e e/kT (4.25)
No h2 uo


The total particle density is NP
NTOT - - = No + N + e (4.26)


The charge neutrality is given by N+ =e (4.27)

Using N, = N0, N2 = N+, N = e, and


27rm kT 3/2 U E0kT
K = 2 e +) --e/k
h2 U 0

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


N2 Ne K (4.28)
N,






- 63 -


NTOT = NI + N2 + Ne


(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* N K
NTOT


The solutions of equation (4.31) are


Ce = K* (- 1 � V/-+ T7Since K* > 0, Vi + 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


Ne NTOT K* (-a. + V1 + l/K3*)


N, = NTOT


- 2N


N2 = N
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)


u = giCe-Xij/kT



-= , + 1 -X/kT + e-X2/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 = (2Lij 2S + 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 Li'j and Si,j 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 gi,j values are calculated from (4.36), using the Li'j and Si,j values of the neutral atom having i orbital electrons. The Xi'j 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 microfields 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 ionization potential, Ei eff is defined as


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






- 66 --


for the reaction Ui+ -U(i+l)+ where

E. = uncorrected ionization potential AE. = lowering of ionization potential

Griem (44) has used the Debye-fHuckel 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(Tr/kT)l/2(Ne + 7 g2Ni)-1/2 i=2
(4.38)

where

Zi = 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
00

u.= g g e- Xij/kT (435)
3=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 = , the partition function diverges since Xi,j approaches a constant value (the ionization energy), and gi,j 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, nc, 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 (439)
ui =j0gi,j e-(.9 j=O

The cut-off quantum number is a function of the effective ionization potential since nc (i) is the largest integer eff Frteit rnu
value for j for which X. < E. . For the i-th uranium ion, the cut-off number is expressed as


nc(i) = 3 Xij < Eief (4.40)



The relation between principal quantum number and energy level for a hydrogen-like atom is (45)






- 68 -


2
E =f (4.41)
n 2h2n2

where I 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 aZ. (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 = 1 (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 calculations.

For a given pressure and temperature, the UPLAZ-2 iteration is carred 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.


-_G 9-






-- 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 = 2, 3, . ., 13
c1 1

(4.43)

Step 2.--Returning to equation (4.23), the particle densities are again calculated, this time including n c(i) (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 (2 i) = [nc (i)] (4.44)



where [nc (i)] is the largest integer value not exceeding j, for which


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

where


E. = uncorrected ionization potential

AEi = current value of ionization potential lowering






- 71 -


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

Step 2 is repeated r times until nc(r) (i) - nc (r-) (i) 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 50001K to 120,0001K. 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. Additional 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 -


10


Elcto
U 3+ U 4+

U 5+ U 6+




FH
Cl)


C)






0
E-l
z




w 7
P4
U -+







01
rZ


-0l0







o.~~ 1 _L ,__10




0 40,000 80,000 120,000

Temperature, 'K


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






- 73 -


100


10



E-_
-H
rf) U7+
z
W

P




[r-1

P








(4 80008 ,0 2 ,0
F:
H

H


0


0

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


10
H




UU
H







0

0
EI







z1
U














0.1
0 40,000 80,000 120,000

Temperature, 0K


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)
e i=O


where

Pe = N ekT, partial pressure of electrons

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 ionization 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 (=) , 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 temperatures greater than 40,000�K. 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 i019/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 -


1021 ] --- -IT F'-i-F-T














O 1020


ir

H


H





1019 00 IT r e





10 1





0 40,000

Temperature, 'K


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 temperature 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-temperature 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(MuT) (4.47)

where

Pc = 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 120,000K, 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 i019/cc. Calculated 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 i019/cc to 0.5 x 1019/cc, the pressure required to operate the reactor at 100,0000K is reduced from 950 atm to 480 atm.





- 80 -


1000
800 600 400 300


200- U-235 density(/cc)





100



















10
0 40,000 80,000 120,000

Temperature, OK 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 ionization 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 ; H+ + e H + e 2 H + +
2 +e2 + e

+ H
2 4 -I' H + H


H + H + H2+ + H

+ 4 4+ H
H + H + H 2

H + H+ H +
2 3

H2 + H2 + H + + H H + H + H + H2 + H2 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 H + H.--The

equilibrium relation between molecular and atomic hydrogen involved in molecular dissociation at absolute temperature T is (39)



NH N 2H2 mm3k T 3 UH UH EH 2/kT
H - H 1 ___ T3/2 ~ e 2(5.1)
NH2 mH h2 UH2


where

mH = atomic weight of H

mH = atomic weight of H2
2
uH = partition function of H UH2 = partition function of H2


E' = dissociation energy of H2
H2

k = Boltzmann's constant

h = Planck's constant

The partition function of atomic hydrogen is


uH = g. j Xj/kT (5.2)
J

where

gj = statistical weight of j-th term Xj = excitation energy of j-th term






- 85 -


The n-th state of atomic hydrogen is 2n2 times degenerate X0
. ,0 where n is the principal
and has an energy Xn =0 n
n
quantum number and X0 is the ground state energy level. On this basis, the partition function may be written explicitly as

n c (1 - 1X0/kT
uH = I 2n2 en- (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



u = 2e - n n2 e /n (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 a". (48) can be applied to the partition function. Replacing the nearcontinuum portion of (5.4) by an integral,



UH = 2e -a FL ne e/n2 + c n2e a/n2d (5.5)
F orn


For


/ 2 <<






- 86-


n n
UH 2eca C n2 en/n2 + c
- -+ n 2d



UH = 2e-a


x Lea + 4nc+ +.+ n2 ea,'nL + 3 3 LJ


(5.6)


The number of terms to be retained in the series (5.6) is determined by the requirement that nc 3 > 3n L This requirement also provides that nc3 >> n L so that the partition function may now be written in a form which is convenient for computation:


U 2e-a n2 e a/n2+1n2(57
uH = + n (5.7)
n1



The quantum number, nc, at which the electron becomes free due to preionization of hydrogen, is a function of the charge density of the plasma. IvanovKholodnyi et aZ. have derived a relation between nc and the electron density Ne as follows:


log Ne = 21.65 - 6 log (nc + 1)


(5.8)






- 87 -


This expression is consistent with the integral approximation 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 uv (5.9)


where

uE = electronic partition function u R = rotational partition function

uv = vibrational partition function The electronic partition function is



uE = gj e-Xj/kT (5.10)
J


The statistical weight is given by g. = 2J(j) + 1 (5.11)
j H2


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

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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