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Steady-state and time-dependent behavior of fusion-fission hybrid systems

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Title:
Steady-state and time-dependent behavior of fusion-fission hybrid systems
Creator:
Vernetson, William G ( William Gerard ), 1945-
Publication Date:
Language:
English
Physical Description:
xx, 423 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Blankets ( jstor )
Flux density ( jstor )
Fusion reactors ( jstor )
Modeling ( jstor )
Neutrons ( jstor )
Plasma temperature ( jstor )
Plasmas ( jstor )
Temperature control ( jstor )
Transfer functions ( jstor )
Tritium ( jstor )
Nuclear reactors -- Models -- Testing ( lcsh )
City of Gainesville ( local )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 411-422).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by William G. Vernetson.

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
023062343 ( ALEPH )
AAK4663 ( NOTIS )
05715688 ( OCLC )

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Full Text







STEADY-STATE AND TIME-DEPENDENT BEHAVIOR OF
FUSION-FISSION HYBRID SYSTEMS






By


WILLIAM G.


VERNETSON


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


























Dedicated to

Theresa

without whom this work

would have been impossible.













ACKNOWLEDGMENTS



The author would like to express his appreciation to his graduate


committee for their assistance during the course of thi


research.


Special


thanks are extended to Dr. H. D. Campbell, chairman of the author's super-

visory committee for providing guidance and encouragement throughout the


course of this work.


Dr. Campbell


many helpful comments and suggestions


have greatly aided the completion of this work


Thanks are also extended


to Dr. E. E


Carroll, Dr. R


. T. Schneider, and Dr.


L. Bailey who have


also served on the author's supervisory committee.

Special thanks are extended to Dr. M. J. Ohanian for the research and


teaching


assistantship opportunities presented which enabled the author


to pursue the doctorate.

The author's studies at the University of Florida have been supported,


in part, by a National


Science Foundation Traineeship and also by a one-


year Fellowship from the University of Florida and this support is grate-

fully acknowledged.

A large portion of the funds for the computer analysis were furnished

by the Northeast Regional Data Center on the University of Florida campus


through the College of Engineering.


difficult to obtain


Special thank


This help, though at times meager and


, is also acknowledged.

are due to Dr. N. J. Diaz without whose efforts and







Special thanks are also due to Dr. E.


T. Dugan whose knowledge of


computer analysis and nuclear reactor physics was of great


assistance


during much of thi


work.


In addition, thanks are extended to Mr


Maya for his aid with some of the plasma calculations and their impli


cations


of helpful


Thanks are also extended to Mr. B. G. Schnitzler for a number


consultations.


Finally, the author would like to extend his deepest appreciation to

his wife whose support and encouragement made it possible to complete this

work.














TABLE OF CONTENTS


?agfi


ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES


ABSTRACT.


* a S S S S S S S S S S S S S S


S~~~ ~~ ~~~~~ 0 5 55 0 S S S S S S


CHAPTER


INTRODUCTION


Preliminary Concepts for Fusion-Fi
Review of Fusion Blanket Studies
Critical Review of Hybrid Blanket
Review of Controlled Thermonuclear
Stability Analyses .
Motivation for the Research. .
Summary of the Research. .


ssion Reactors


Studies.
Reactor


Thermal


* a a a a a
* S S S S S S
* S S S S S S S S S


THE PLASMA MODEL


Intro
The P
The L
Trans
Stabi


duction to
oint-Model
in ari 7PrH


fer F
lity


the Plasma
Plasma
P1 a~ ma Mnorl


Model


unction Representation of Plasma Character
Analysis of the Linearized Plasma Model.


istic


A HYBRID REACTOR ANALYTICAL MOD

Development of the Hybrid Model
Thp I inpAri pd Mvrid id Mndel


. 116


S S S S a S S S S 5 116


Incorporation of Feedback Effects into the
Nonlinear and Linearized Hybrid Model Summ
Transfer Function Representation of the Hy
Stability Criteria for the Hybrid System

HYBRID PLASMA OPERATIONAL CONSIDERATIONS


. .
Hybr
ary.
brid


S 1 31
id Model. 138
* 143
S 147


* S S 5 5 5 1 60


* S S S S S S164.


T ~ ~ ~ L -l,, 4-, .4.,~, 3 4. 2 .I4. II 4 rl *'2f,... r-..


Irn







Diie


Uncontrolled Plasma Response to Perturbations.
Predicted Stability Versus Point-Model Response
Short-Term Plasma Transient Response .
Plasma Response With Feedback. .

HYBRID BLANKET ANALYSIS. .. .


introduction .
lanket Calculations Using
homogeneous Diffusion Th
inetic Parameters
transport Theory Calculati
homogeneous Transport Th
ime-Dependent Blanket Con


or
ec
s-


* S S 181
.* S 196
.* S S 202
219


S258


25
Diffusion Theory 26
3ry Calculations. 30

ns. 32
ory Calculations. 34
iderations. 34


CONCLUDING COMMENTS 356

Discussion and Conclusions 356
Suggestions for Further Work 361


APPENDICES


GLOBAL


BLANKET


ENERGY MULTIPLICATION


HYBRID SYSTEM PHYSICAL CHARACTERISTIC


BURNUP AND
PLASMA


SENSITIVITY CONSIDERATIONS FOR THE HYBRID
S S S S S S S S 388


COMPUTER CODE DESCRIPTIONS


REFERENCES.


* 0 0 a a ..396


S S S S S S S S S S S S P S S S a S S S 4-1 1


BIOGRAPHICAL SKETCH


S 0 5 S S S P 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4. 3


ec













LIST OF TABLES


Table


Paae


Fusion Reaction Parameters


1-II


, -III


Dependence of Tritium Breeding Ratios and Energy


Deposition Rates for Lee'


Summary Descriptions of ORN
Blanket Designs .


Fusion Blankets

L Ootimistic an


d Conservative


,-IV


Summary of


teiner's Tritium Breeding Calculations Per


Incident 14 MeV Neutron


Neutron Economy of Lidsky's Hybrid Blanket.


1-VI

1-VII


Lidsky'


Lee'


Hybrid Reactor Parameters.


Neutron Balance in Infinite Met


a S S


1-VIII


Subcritical Fast Fission Blanket Component


tudied by


Fast Fission Hybrid Neutron Economy Per 14 MeV Neutron
Calculated by Lee .


Neutron Economy for Thorium-Fueled Blankets

Neutron Economy for Uranium-Fueled Blankets


1-XII


* S a *

* S a a a


Comparison of Best Natural Uranium-Fueled and Extrapo-
lated Thori um-Fueled Blankets .


Earl


-XIV


y PNL Hybrid Neutron Balance.


y PNL Hybrid Specifications


PNL Hybrid Blanket Analysis


1-XVI


'-XVII


Critical Temperatures for D-T Fusion Reactors


Predicted Blanket Global Response per 14 MeV Neutron.


1-IX






Table

4-I


4-II


Page


Selected Spectrum of Equilibrium Operating Conditions
for the Hybrid Plasma With Constant Confinement. 173

Hybrid Plasma Equilibrium Operating Conditions for TEo
= 1.7 sec to Meet Required Power Production. ... 175


4-III


4-IV


4-V


Equilibrium Plasma Conditions Selected for Transient
Analysis With R = 2 and qPo = 1.41 x 1011 nts/cm3-se


c.


Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a +5% Perturbation in the Temperature .

Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a -5% Perturbation in the Temperature. .


. 178


.187


. 188


4-VI


Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Increase in the Steady-State


Source Feedrate.


. S 0189


4-VII



4-VIII


4-IX


4-X



4-XI



4-XII


4-XIII


Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Decrease in the Steady-State
Source Feedrate .

Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a +5% Perturbation in the Ion Density. .

Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a -5% Perturbation in the Ion Density.

Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Increase in the Steady-State
Injection Energy .

Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Decrease in the Steady-State
Injection Energy .


Summary of Predicted Stabilization Requirements for
Instantaneous Temperature Feedback on the Feedrate

Comparison of Confinement Time Effects on Plasma
Temperature at 10 sec Following +5% Perturbation in
Feedrate versus +5% Perturbation in Temperature for
Six Hypothetical Hybrid Equilibrium States .


.190


. 191


. 192



. 193



. 194


. 197




. .218


Boundaries for Four-Group Critical i ty Calculation.


5-11


. 263


BRT-1 Cell-Smeared Thermal Constants for 1.35%
Fnrih Ped FIP. 264






Table


5-IV


Flux Dep
Lattices


ress


ion Factors for the 1.35% Enriched


Average Cros
Column. .


Sections for


23U and 238U in th


5-VI


Space Point Placement for BRT-1 Calculation Over Inner


Half of th


5-ViI


Hybrid Blanket.


Space Point Placement for BRT-1 Calculation Over Outer


Half of th
Refl sector


ssion Lattice and


nto the Graphit


VIII


Summary of PHROG Calculations by


Region


Resonance Region Scattering Cross
Nuclides. .


Sections


Four-Group, 13 Region Constants for 1
at 570K from BRT-1 and PHROG .


.35%


for Blanke


Enrichment


5-XI


PHROG-Generated Macroscopic Downscattering Cross


tions for 1.35% Enrichment, 5700K, and 1


Regions.


5-XII


Four-Group,


13-Region Constants for 1


.35%


Enrichment


at 9700K from BRT-1 and PHROG


5-XIII


PHROG-Generated Macroscopic Downscattering Cross


Sections for 1.35% Enrichment, 970K, and


5-XIV


Regions.


Results of Diffusion Theory Criticality Calculations.


Summary of Inhomogeneous CORA Calculations for
Variations in Enrichment and Temperature .


5-XVI


Yield Fractions for Si


Delayed Neutron Precursor


Groups.


5-XVII


Delayed Neutron Energy Spectrum Yield Fractions for
4-Group CORA Calculations .


5-XVIII


Blanket Kineti


Parameters


5-XIX


5-XX


Source Weighting Factors in Four Groups and Ten
Regions


Effectiveness of Uniform Volume Sources for Design
Power Level


Page






Table


5-XXIII


Effective Moderator Scattering Cros


Sections Per


Absorber Atom.


S S S S S S SS 5 327


5-XXIV


5-XXV


XXVI


Isotopi
NITAWL


Resonance Integra


Values Obtained from


SS SS S S S S S S S S SS S S 5 323


Hybrid Blanket Anal


S4 Quadrature Constants.


XSDRNPM 43-Group Energy Boundaries


330


S S S S S S S S 333


5-XXV II


XSDRNPM


Group and 11-Group Energy Boundari


. 335


XVIII


5-XXI


XSDRNPM 6-Group Cross Section Energy Boundari


XSDRNPM kef Results for a
edition at the Vacuum Wall.


* 337


Zero-Flux Boundary Con-
337


5-XXX


Transmission Ratio for 14 MeV Neutrons Through the


Hybrid Blanket


S S S S S S S S S S S S S S 34l6


Hybrid Blanket Equivalent Unit Cell Geometry


Fuel Column Spherical Micropartici


B -III


. 375


Design Parameters.


377


Temperature-Independent Fuel-Pin-Averaged Nuclide


Number Density Variation with Enrichment


S. S 378


B-IV


Hybrid Blanket Shield Composition.


a S S S* S S 5 330


Helium and Natural Lithium Number Density Variation


with Temperature


B-VI


S S S S S S S S S S S S S S 381


Effects of Vacuum Wall Radius on Blanket Power


Requirements and Power Density


S. 5 S S S S S 384


Point-Model Comparison of Confinement Times and
Related Plasma Parameters in UWMAK-III and the Hybrid


Plasma


S S S S S S 5 5 5 5 S S S S S S 5 5 390


Page












LIST OF FIGURES


Figure


Page


The essential components of a Tokamak fusion reactor


Comparison of spatially-dependent heating rates for vacuum


wall regions in two designs .

Early PNL fusion-fission hybrid
configuration. .


ubcritical blanket


Comparison of Lawson breakeven and plasma equilibrium


regions


Time variation of point-model plasma temperature and
density for constant confinement and charged particle
heating


Typical Lawson breakeven curve for a 50-50 D-T plasma and
33% overall efficiency showing relative position of hybrid
sys terns. .


Predicted variation of blanket fusion neutron energy
multiplication with blanket effective neutron multi-
plication factor .


Transfer function formulation for a point-model fusioning
plasma


Block diagram for the point-model plasma system.


Partially-reduced block diagram for the point-model
plasma system. .


Alternate block diagram for the point-model plasma system.

Partially-reduced block diagram for the alternate point-
model plasma system formulation. .

Reduced open-loop block diagram for the point-model plasma

Routh array for open-loop point model fusioning plasma


with burnuD


a a a a a a a -






Figure


Page


Variation of F(T)


= nr with temperature


. .* .* 105


Block diagram for the point-model plasma with temperature


feedback to the feedrate.


.a.*.. 5 ** 110


Block diagram schematic for point-model blanket kineti
retaining both source and reactivity perturbations .


cs
. 152


Block diagram of the
hybrid reactor model


inearized global fusion-fi


ssion


Partially-reduced hybrid block diagram with no artificial


feedback.


S S S S a S S S S S S S S S 5 156


Simplified reduced hybrid
artificial feedback


stem block diagram with no


Closed-loop block diagram for the linearized point-model


plasma with temperature feedback to the feedrate.


. 159


Routh array for the cubic denominator for blanket effect


in the


overall hybrid transfer function


. 161


Equilibrium curves for various equilibrium plasma


conditions.


S S S U S S S S U S U S S SS U 174


Mills steady-state curves


including burnup for R


- 2.


. 180


Illustration of the feedforward effectiveness of the


feedrate and the injection energy on plasma
conditions and transient behavior .


Arbitrary equilibrium curve with a hypothetical


source


equilibrium
S 184


stable


hybrid state at point A plus a possible equilibrium curve


containing a perturbed unstable


state at point B.


Variation of plasma temperature following a


step increase


in the temperature of the six hypothetical hybrid equilib-
ri um states .


Variation of plasma volumetric neutron production rate


following a


5% step increase in the temperature of the six


hypothetical hybrid equilibrium states.

Variation of plasma temperature following a


in the temperature of the si


rlum


step decrease


hypothetical hybrid equilib-


states


Variation of olasma volumetri


neutron production rate


S f f 1541


200


o






Figure


Page


Variation of the heating rate in the first wall region of
the UMAK-III design. .


Variation of plasma temperature following a


step


crease


in the feedrates for the si


hypothetical hybrid


equilibrium states


Variation of plasma volumetri


following a


neutron production rate


step increase in the feedrates of the


hypothetical hybrid equilibrium states .

Variation of plasma temperature following a


decrease in the feedrates for the si


step


hypothetical hybrid


equilibrium states


Variation of plasma volumetri


following a


neutron production rate


step decrease in the feedrates for the six


hypothetical hybrid equilibrium states


Variation of plasma temperature following a


incr


ease


in the feedrate of an equilibrium


5% step
state at TE
-o


sec with delayed shutoff times


Variation of plasma volumetri


following a
equil ibrium
times.


neutron production rate


step increase in the feedrate of an


state at TEo
-o


sec with delayed shutoff


Variation of plasma temperature following a
increase in the feedrate of an equilibrium
1.7 sec with delayed shutoff times .


Variation of plasma volumetri


following a


5% step
state at TE
-o


neutron production rate


step increase in the feedrate of an


equilibrium state at TEo
ti mes


=117


sec with delayed shutoff


Variation of plasma temperature following a


crease


car (A'


step


in the feedrate of an equilibrium state at TE
444 rA/l c hnI nc Cki+n-FF + 0


3 C L l I LI I UC I UJ cU


.JU VI IULAJI U1111f


Variation of plasma volumetri


following a


neutron production rate


step increase in the feedrate of an


equilibrium state at Tr


sec with delayed shutoff


ti umes U

Variation of plasma temperature following a
decrease in the feedrate of an equilibrium
1 5 npr with dplavpd hutnff timp


5% step
state at TE
0


I I _







Figure

45. Variation of plasma temperature following a


decrease


in,' Ia


step


in the feedrate of an equilibrium state at TE
44h tin1- d l dr rk u4-nC 4-h-I f i


ace w e aye s uto f t mes


Variation of plasma volumetric neutron production rate


step decrease .in the feedrate of an


equilibrium state at rE_


- 1.7


sec with delayed shutoff


times. u

Variation of plasma temperature following a


decrease


CcaC I,!


5% step


in the feedrate of an equilibrium state at XE
S+fh aHol n aId ckh nff +4-mc o


.J'..t.. III LoE At. I UJ C 4 311U LAJ I II.IIC


Variation of plasma volumetric neutron production rate


fol lowina a
equilibri um
times.


step decrease in the feedrate of an


state at rEo
o


sec with delayed shutoff


Variation of plasma temperature with timperature feedback


following a 5%
1 c rr nt r'


step increase in the feedrate of the TE
i t l d l di;/-v/ h r' t4l rff f Ue o


sec equ r um s ate p us e aye s u o o


Variation of plasma temperature with temperature feedback


following a 5
1.7 sec equil


ih


step increase in the feedrate of the TE
v im cf+ nfine l aun chit+nt-Ff nC &R 0


P i uTi UL lui U3 U UU C 3IU3 IUV 4~IttI I 'U


Variation of plasma temperature with temperature feedback
following a 5% steo increase in the feedrate of the TE =
2.0 sec equilibrium state plus delayed shutoff of 6S .

Variation of plasma temperature with temperature feedback


following a 5
1.5 sec equil


ib


step decrease in the feedrate of the rt
rium state olus delayed shutoff of 6S


0


Variation of plasma temperature with temperature feedback


following a 5%
1 7 1 L


step decrease in the feedrate of the rE


1.7 sec equilibrium state plus delayed shu f


Variation of plasma temperature with temperature feedback


following a 5%


steo decrease in the feedrate of the TE
,1-,, -- .je o


2.0 sec equilibrium state plus delayed s f


Variation of plasma temperature with temperature feedback


following a 5%
1 -, *4 1 -1.


step decrease in the temperature of the Eo
*f a^ ***i r~* ^ ^'/ -


t~UL I et juI` I I u I .111 lLclLU


Variation of plasma temperature with temperature feedback


following a 5%
93


step increase in the temperature of the TEo


1 7 car annilibrium efate


following a


Page


r- ---J






Figure


Page


Variation of plasma temperature with temperature feedback


following a 5%
fl fl ^ ^. 1


step increase in the temperature of the TEo


2.0 sec equilibrium state.


Variation of plasma temperature with temperature feedback


following a 5%
r 1 L


step decrease in the temperature of the TE
U -o


c es e q u i l i b r i um s t a te.


Variation of plasma temperature with temperature feedback


following a
1.7 sec equ


step decrease in the temperature of the TEo
*o


m uirbili s t a t e.


Variation of plasma temperature with temperature feedback


following a
2.0 sec equ


5% step decrease in the temperature of the TE


bili r i u m s ta te.


Variation of plasma volumetric neutron production rate with


temperature feedback following a


step increase in


temperature of the TEo


= 1.5


sec equilibrium state


Variation of plasma volumetric neutron production rate with


temperature feedback following a


temperature of the XEo
0o


= 1.7


step increase in


sec equilibrium state


Variation of plasma volumetric neutron production rate with


temperature feedback following a


temperature of the rEo


step increase in


= 2.0 sec equilibrium state


Variation of plasma volumetric neutron production rate with


temperature feedback following a


step decrease in


temperature of the TEo


-'.5


sec equilibrium state


Variation of plasma volumetric neutron production rate with


temperature feedback following a


temperature of the TEo


step decrease in


= 1.7 sec equilibrium state


Variation of plasma volumetric neutron production rate with


temperature feedback following a


step decrease in


temperature of the TEo


= 2.0 sec equilibrium


state


BRT-1 thermal flux profiles across the equivalent unit cell


for 1


.35% enrichment at 290K, 5700K, and 9700K.


Typical paths for an unscattered neutron in an equivalent
unit cell and an actual unit cell of a nuclear reactor


BRT-1 thermal flux profiles across the inner half of the
hybrid blanket for 1.35% enrichment at 2900K, 570K, and
9700K with zero-flux vacuum wall boundary condition .


- "1 L-


*






Figure


Page


BRT-1 thermal flux profiles across the outer half of the
fission lattice out to 12 cm of graphite reflector for
1.35% enrichment and 2900K, 5700K, and 9700K .

Thermal flux profiles from BRT-1 calculations across the
outer 18 cm of graphite reflector and 30 cm of shield for
2900K, 5700K, and 9700K.


Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 2900K with zero-current vacuum
wall boundary condition. .


Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 570K with zero-current vacuum
wall boundary condition. .

Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 9700K with zero-current vacuum
wall boundary condition. .

Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 9700K with zero-flux facuum wall


boundary condition


Variation of blanket effective neutron multiplication
factor with temperature for the 1.35% enrichment using
four-group diffusion theory. .


Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.35% enrichment at 5700K. .

Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.35% enrichment at 9700K. .

Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.50% enrichment at 570K. .

Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.50% enrichment at 9700K. .


Blanket power density variation for 6500 MWth for
enrichment at 5700K and 970K. .


.35%


Blanket power density variation for 6500 MWth for 1.50
onrirhmpnnt at r700k ind Q700






Figure


Paqe
-.,,- .- -*..


Six-


group fundamental mode flux profile


from XSDRNPM for


1.35% enrichment at 9000K with zero-flux vacuum wall


boundary condition


Six-group flux profiles for a
10' nts/cmL-sec in group 1 t
enrichment and 900K .


surface source


of 1.336


o generate 6500 MWth at 1


x
.35%


Fractional transmit


on of 14 MeV neutrons through th


hybrid blanket


Power transient in th


hybrid blanket following a 5


step


increase in the neutron source for a forced-critical


sten


Hybrid blanket power transient derived for a subcritical


Conceptual Tokamak fusion-fission hybrid reactor


stem.


Overall hybrid blanket slab geometry used in neutronics
al culations


Selected PNL hybrid blanket module geometry for Tokamak
fusion-fission hybrid. .


Hybrid thermal fission lattice unit cell


Geometri


arrangement of the inner convertor with inner


breeder and outer breeder. .

Reactivity and sensitivity variation with temperature for
the D-T fusion reaction. .








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


STEADY-STATE AND TIME-DEPENDENT BEHAVIOR OF
FUSION-FISSION HYBRID SYSTEMS

By


William G


. Vernetson


June 1979


Chairman:


Huah D. Campbell


Major Department:


Nuclear Engineering Sciences


study examined stability analysis of point-model systems repre-


sending pure fusioning plasmas as well as coupled fusion-fission


stems.


The stability criteria for these systems were derived for constant plasma

confinement conditions based on engineering perturbations of the system


feedrate.


The result


of linearized point-model plasma stability anal


ysis


of the thermal instability were shown to be applicable to hybrid plasmas

and to be attainable from considerations of engineering-related per-


turbations in the extrinsi


plasma feedrate variabi


A Tokamak fusion-fi


ssion hybrid design was


ected for further,


more specific analy


sis.


The modeled hybrid system in linearized form was


found to be stable provided certain hybrid plasma temperature and con-


finement time limits are met.

absolute stability is not suf


However, for realistic installations,


'ficient; nor is it guaranteed by linearized


anal


ysis.


Therefore, hybrid plasma behavior was examined under transient


and overpower conditions.


Time-dependent analysis


of a low reactivity hybrid plasma (8 keV






with perturbations to pure fusion plasmas with high plasma reactivity.


In addition


, the predictions of plasma stability ranges were verified


for various confinement times.


transients following -


The slowly developing hybrid plasma


temperature or feedrate perturbations were


found to be significant for the control of the power-producing hybrid.


Neutrons and their


associated energy are multiplied in hybrid


blank


ets;


therefore, the global equation in use to relate the blanket


energy deposition per fusion neutron to the blanket effect


multiplication factor was investigated.


indicate the oloba


neutron


Results were obtained which


approach supplies a poor estimate of blanket energy


multiplication for a fusion neutron source and an even poorer


estimate


for fission energy neutrons.


Although results


showed the blanket energy deposition per fusion


neutron to be some


below point-model predictions, the


selected blanket


is still a significant multiplier, by a factor of


25 or more, of th


neutron energy entering the blanket via fusion neutrons.


The documenta-


tion of the reduced worth of fusion neutrons, entering the blanket through

a convertor region, may be a significant factor in redesigning vacuum


wall


of hybrid reactors despite the advantages of reduced 14 MeV wall


loadings.


Diffusion theory and di


screte ordinates transport theory anal


ysis


were both applied to establish the relative importance of the inner con-


vertor region for power generation.


The results of the


calculation were used to determine the source


size


transport


required by volume


equivalence with the Tokamak geometry to produce 6500 MWth in the blanket.

The source value was used to establish the steady-state requirements on






Tokamak hybrid plasma volume involved.


In addition, the


calculation


was used to show that only about 6% of the 14 MeV fusion neutrons reach


the thermal fission latti


without a collision


These transmission


results indicate graphically why the blanket is less effective at energy


multiplication than

Finally, space


expected from previous reports.


-time kinetics calculations were performed on the


blanket to demonstrate the fast response of the blanket in keeping with


its millisecond prompt neutron lifetimes and subcriticality


Al though


no time-dependent feedback effects were examined, the speed of response

of the system was determined for typical transients and some character-

istics for hybrid operational controllability were established.












CHAPTER 1

INTRODUCTION


Preliminary Concepts for Fusion-Fission Reactors


The fusion-fission hybrid reactor concept is a combination of a


sub-Lawson


fusion reactor and a subcritical


ssion


reactor in a single


power-producing system.


Fiss


ion reactors are


"power rich" but


"neutron


poor," while anticipated D-T


fusion reactors will


"neutron rich" but


"power poor.


these


Hence,


two systems to u


the essential

se excess fus


hybrid feature


the combination of


;ion neutrons to breed fissil


fuel


while simultaneous


y sustaining and driving the


sys ten


for useful


power er


using fission energy multiplication of the fusion neutron source


~.ne rg:


Limited studies


, concentrating on blanket neutronics


have been


done on hybrid system


in parallel with pure fusion blan


t work


how-


ever


, no system dynamics or stability investigations


have been reported


for hybrids


Some research effort ha


been devoted to global


stability


anal


of th


plasma


in pure fusion device


present research


extends such pure fusion time-dependent studies


into the area of hybrid


systems.


continued development of the hybrid in parallel


with the


fast breeder reactor is supported by the hybrid's potential a


alternate and attainable energy and fuel


producing concept.


In fact,




-2-


Much research effort and capital investment have been committed to

the realization of a mixed burner-breeder nuclear reactor economy planned


for the end of this century.


This effort is justified by


expected con-


tinued growth of energy needs


sumption of fossil fuel


within the past few deca


des.


and by a marked shift from direct con-


secondary consumption of electrical energy


With the growth in nuclear generating capacity


limited fissil


fuel


reserves have caused the thrust of research and deve


opment in the


nuclear industry to shift to the fast breeder reactor


LHFBR)


Even with


the projected impact of the commercial LMFBR sometime after 1


siderabl


0, con-


additional enriching capacity and capital investment will be


required for fueling burner reactors.

Current emphasis on the safety and the environmental impact of


nuclear generating faci liti


as well


as certain technological and


political objection


make it increasingly unlikely that high gain breeder


reactors will make


a significant


rmpact prior to the mid-1990


s or later.


Even if the breeder i


introduced sooner, the relatively lono doubling


times under consideration (1


years or more


may not be adequate for


generation of sufficient additional fuel to support an


reactor economy


existing burner


IWith so much effort and capital investment committed


to the realization of the mixed burner-breeder economy planned for the

1990's, the availability of an effective alternate concept to produce


fissil


fuel


could be important.


One candidate for producing fissile fuel is the controlled thermo-


nuclear reactor utilizing the D-T cycl


C1


e. Deuterium resources are virtu-
7







fusion neutrons can be used to breed fissile material.


By diverting


neutrons from tritium production, the tritium supply can be maintained


reasonable


averted to fissi


evel while fertile

reactor fuel.


materials (238U and Th) are con-


Unfortunately the realization of pure


fusion power is too far removed and uncertain to be counted upon to pro-


duce fissil


fuel in the near term.


The alternative concept currently receiving renewed attention is the


coupled fusion-fission hybrid system combining a less than


self-sustaining


(energy) fusion reactor with a subcritical but power producing fission


reactor.


Although achievement of pure fusion power i


not yet possible,


recent advances indicate the plasma requirements for hybrids will be


reached while the fission power component of the


still increasing.


electrical economy i


9
Then, as an alternative to the LMFBR for fi


and power production, the hybrid can be very useful

The hybrid concept has many potential advantages over the LMFBR for


providing power and fissile fuel in the latter part of thi


century.


First, the hybrid reactor


possesses


great potential as a breeder of


fissil

be abl


fuel.


With its abundant supply of neutrons, the hybrid should


to produce fissil


material more rapidly than any of the current


breeder reactor concepts to keep pace with power requirements.


Second, the hybrid makes an alternative fuel cycle available for


existing burner reactors.


Reliance on the


U-239Pu fuel cycle with its


weapons grade plutonium can be reduced in favor of the


233U fue


cycle.


Third


, hybrid development allows early introduction of fusion


232Th_




-4-


hybrid system, current advanced reactor technology would require only

modest extensions to produce a hybrid system as a natural link in the


development leading from pure fission to pure fusion power.


Finally, the hybrid concept using


from a safety standpoint


subcritical blankets is attractive


since it would diminish the need for critical


nuclear reactors.


4.10


The current concern over reactor safety and


core meltdown could b


essentially eliminated.11


Past

hybrid ana


studies of the hybrid concept have been restrictive.


1


yses


Typical


limited to steady-state evaluation of the technical


characteristics of a concept with emphasis on the


neutron economy of the


conceptual blanket.


-3,12-14


Important features in such hybrid studies


parallel ordinary fusion reactor blanket


studies and include:


Tritium conversion ratio and doubling time.

Fissile breeding ratio and doubling time.

Energy production and multiplication in the blanket


Constraint


on the fusion plasma due to neutronics.


5. Vacuum wall loading and neutron energy transport.

The neutron economy and energy multiplication of the hybrid blanket


have been of primary interest in these initial studio


by fission events


both are enhanced


Little consideration has been given the fusioning


plasma in these hybrid designs beyond setting plasma characteristics


necessary to achieve the assumed blanket power performance.


Basic fusion


reactor blanket studies and hybrid blanket work to date are reviewed in


the next


section; the similarity of the two


remarkable despite the


increased importance of energy production in hybrid blankets.








factor, keff, less than unity)


as well


blanket, no time-dependent analysis


as the heat generation rates in the


has been considered; dynami


behavior


and associated safety of the hybrid fusion-fission system have been


ignored.


The effects of perturbations on the coupled system have also


been ignored.


Some studies on safety and control analysis


of pure fusion reactors


have been reported.1


5-24


Mills'


described the stability requirements


on a


state


teady-state,

(equilibrium


point model, fusioning plasma, and found the steady-


plasma unstable against various parameter fluctua-


tions below a critical


ion temperature.


The effects of artificial feed-


back were


simulated at lower temperatures to control this thermal


instability and maintain equilibrium operation below the critical


temperature.


The work of Mill


is a benchmark work in fusioning plasma


global dynamics and control


work on stability by


Ohta et al


18
is one of the most complete


thermal stability


studi


of point model thermonuclear plasmas.


Stability


criteria were established using linear analysis


energy balance plasma equations.


of coupled particle and


The thermal instability was evaluated


and suitable feedback control was implemented to allow stable operation

below the critical plasma temperature set by the stability criteria.


Stacey


as well as Usher and Campbell23'24 have reported extensions of


this work to more


sophisticated plasma model


Yamato et al


19,20 have
have


extended such stability studies to simpi


comparable e


i homogeneous plasmas with


results.


Since such time-dependent analysis was neglected in previous hybrid







of fusion energy blanket multiplication not previously considered.


much larger hybrid blanket energy multiplication demands a coupled time-


dependent analysis


establishment of specific safety and operating


character ri


stics


for a


coupled hybrid system is


necessary


for the con-


tinued development of the


concept into a viable energy alternative.


The effect of thermal instabilities in the fusioning plasma on the

fissioning blanket are analyzed in this work to establish hybrid system


interactions,

deficiency in


safety, and ea


existing studies


of control

of hybrid


This work


systems


eliminates a major


so that a decision can


be made on its pi


ace


in the power industry of this country in the last


decades of this century.



Review of Fusion Blanket Studies


The Fusion Process


Since hybrids depend on fusion neutrons to breed fissile fuel, at

least two fusion reactions have potential for use in a hybrid reactor.


These


are the deuterium-tritium and the deuterium-deuterium reactions


which have the following balances:


2 3D+
D + ,T
~1 1


He (3.52) + n (14
He (3.52) + n (14.06)
2 0


e (0.82) +
2He (0.82) + n (2.45)


(1 .01


+ p (3.03)


where the two D-D branches have nearly equal probabilities at energies


2 3
D D
1







The properties of the D-T fusion reaction are far superior to those


of the D-D reaction.


For energies below 200 keV the D-T reaction cross


section with its broad resonance at 110 keV i


nearly two orders of


magnitude above the D-D


cross


sections.


The probabili


for a fusion


reaction occurring i


characterized by the reactivity or rate coefficient,





, which i


an average of the product of the cross section


for the


fusion reaction in question and the relative speed


, v, of the reactants.


The reactivity can usually be approximated using a Maxwellian distribu-


tion of particle speeds.


With a broad resonance around 65 keV


the D-T


reaction rate coefficient i


much greater than the D-D reaction rate


coefficient below 100 keV


Finally


, the energy released per fusion reaction, QF


, is signifi-


cantly higher for D-T fusion events.


These comparative values are


summarized in Tabl


e -I25


and indicate why near term fusion reactors and


hence hybrids are limited to the D-T fuel cycle.



Table 1-I

Fusion Reaction Parameters


o (barns) (cm3/sec) Q (MeV
Reaction at 100 keV at 65 keV F '


0.46


x 10-16

x 10-15


3.65


17.6


As noted in Eq


1), the 17.6 MeV per D-T fusion reaction is divided




-8-


plasma, but the 14.06 MeV of neutron energy must be recovered in surround-

ing blanket regions.



Fusion Reactor Blanket Studies


Since only limited quantities of tritium occur in nature, sufficient

tritium must be generated through nuclear reactions to refuel operating


fusion devi


ces.


The 14.06 MeV fusion neutrons are used for this purpose


in two lithium reactions:


6
Li +
3


1
n (slow)


SHe + T3 + 4.8 MeV
2 1T


3Li + On (fast)


4He+?T+


- 2.82


where natural lithium has the composition:


7.56%


6Li and


92.44%


The exothermic reaction has a


thermi


2.9 b resonance at 0.25 MeV while


reaction, with its threshold at


the endo-


has a 450 mb resonance


at 8


.0 MleV.


For the usual toroidal fusion reactor using

for the magnetic confinement, the position of thj


superconducting coils


blanket used for heat


recovery and tritium generation is illustrated in Fig. 1.


This con-


figuration conforms to the Tokamak designs most often considered for


economic, power-producing fusion machines.


27-33


Refractory metals such


as vanadium, molybdenum, and niobium are usually postulated


as the vacuum


and structural material due to the hiqh heat and stress load as well


as the need for (n,2n) reactions to enhance tritium breeding


raphi te





-9-











































C-,

a)


LI




w Ad


o o c C


o Co -
(3 0
I-


0

4-,
a





a
E
0


C-)



a)
LI




E-



I-

1,

a)
I.-




-10-


tritium breeding are confined to the inner reflector/moderator regions

of the overall blanket, the outer regions shield the low temperature

superconducting magnets from the deposition of energy by high energy


particle


generated within the fusioning plasma and inner blanket region.


A typical thickness for the


total heat recovery and


shielding regions of


the blan


about two (2) meters with actual heat recovery and tritium


production confined to the first meter.


Many early studies were conducted to evaluate tritium breeding and


heat generation in idealized fusion blankets.


These initial studies in-


dictated that adequate tritium generation was possible but with severe


heat transfer requirements on the vacuum wall


This problem was partly


due to the fact that only the exothermic lithium reaction was known and


used in the earliest studi


Myers et a


>11


used diffusion theory to examine homogeneous cylin-


drica


blan


ts of varying thicknes


from 9 to 96 cm


Material


tested


included a lithium beryllium-fluoride

natural lithium metal and 6Li metal.


salt (LiF + BeF2) called "flibe,"

All but 6Li provided adequate


tritium breeding ratios above 1.45; the value of only 0.976 for "L

demonstrated the potential significance of 7Li breeding reactions.


Impink


and Homeyer


also examined the effects of blanket composi-


tion on tritium breeding and on spatial heating rates


, respectively


Graphite was used


the neutron moderator with molybdenum


as the vacuum


wall material because of its neutronic and refractory characteristic


The flibe coolant and tritium generation medium was

electromagnetic resistance to coolant circulation.


ected to avoid


For variations in





-11-


Since nuclear heating rate calculations showed extreme peaking near


the first wall based on 14 MieV neutron energy flux of only


MW/m


on the


vacuum wall, Homeyer concluded that cooling of the vacuum wall would be


most


severe heat removal problem in the blanket.


blanket energy was


calculated to be


The recoverable


17.4 MeV oer entering 14-MeV neutron.


used multigroup transport calculations to analyze an infinite


annular blanket and concluded that pure lithium is an attractive breeding

material but requires a thicker blanket than one containing beryllium.


Unfortunately beryllium is probably too


expensive to justify its large


volume usage in systems of the si


of power-producing fusion devi


ces.


Realistic blanket designs required more detailed neutron


studies


to consider structural and heat generation requirements as well as the


tradeoff between tritium breeding and energy generation


as shown in more


S8,27,38-43
recent, detailed calculations. '


used Monte Carlo theory to calculate neutronics results for


a three zone


spherical annular blanket with outer radii of


302 cm for a 100 cm radius plasma.


Structural effects were simulated


by homogeneous volume fractions of niobium chosen for its refractory,


fabricating, and welding characteristic


exce


llent results were obtained


for a structureless lithium blanket.


simulated by making Zone


in Zones


More realistic blankets were


(1 cm) all niobium and diluting the lithium


and 4 with increasing volume fractions of niobium structure.


Lee's results are


summarized in Table 1-II where the increase in energy


generation per fusion event i


due to Nb(n,y) reactions


Since


enrichment was found to be ineffective and only


5 to 6% niobium is




-12-


Electromagnetic resistance to lithium flow may


excessive


near the


vacuum wall where high coolant velocity


are needed.


Induced


currents


in the lithium act to retard lithium flow across magnetic field lines;


but such resistance


great


reduced in the outer blanket regions where


heating rates and hence flow rates are reduced.


Tabl


1 -Il


Dependence of Tritium Breeding Ratios and Energy Deposition


Rates for Lee'


Fusion Blank


Nb (Volume Per Cent)


QB (MeV)


1 .38
1.16
1.00


19.60
20.20
20.50


Steiner8'39


analyzed the neutronic behavior of two designs based on


the ORNL standard blanket configuration containing niobium structure,


coolant, and graphite reflector.


optimistic (D


These two blankets reflected an


esign 1) and a conservative (Design 2) outlook on the problem


cooling the


vacuum wall


Design 1 contained lithium throughout the


blanket


Design


assumed that flibe must be used to cool th


vacuum wall


with lithium elsewhere.


Steiner rejected flibe coolant throughout the


blanket since


it produced an inadequate


= 0.95) tritium breeding


ratio.


Neutron activation problems were also first revealed by Steiner.


Niobium was selected over molybdenum as the vacuum wall and struc-




-13-


lower sputtering ratio despite molybdenum's demonstrated


superiority for tritium breeding.

reflector in both designs. Summa


with


typical


Graphite was employed as the moderator/


ry descriptions of these two blankets


niobium structure are presented in Table 1-III to indicate


blanket model


Table


1-I''


Summary Descriptions of ORNL Optimistic (1) and Conservative (2)
Blanket Designs


Region
Number


Description
of Region


Thickness
by Region


Volume Composition by Region


Design 1


Design


Coolant


94% Li


94% Flibe


Structure


6% Nb


Second wall


Coolant


94% Li


94% Li


60.0


Structure


6% Nb


Moderator-
reflector


30.0


Gra white


Graphite


Coolant


94% Li


94% Li


Structure


6% Nb


The basic 100 cm Design 1 blanket with first wall at 200 cm radius
2A' n n + O rl a c *h n c+anA hl n n\ a* mn^al 3-a +ka MIa tI rn nar r c Zncc in c F


as well







44
National Laboratory (ORNL) in June 1971. This blanket has been frequently

used to check neutronics calculations.

Transport theory-was applied in slab geometry to obtain the tritium


breeding results listed in Tab


1-IV where the breeding ratio of 1.35 in


Design


is some 10% above the 1


value for Design


Slab geometry is


adequa


to th


large plasma radii


meters) for steady-state


fusion reactors.


33,45


Tabl


e 1-IV


Summary of Steiner's Tritium Breeding Calculations per Incident
14 MeV Neutron


Design T/n Neutron Leakage


0.023

0.020


If hypothesized low levels of tritium holdup


46,47


are realized, then


breeding rati


only slightly


above unity


.01) will be sufficient for


seven year doubling times.


Therefore, Steiner


s relatively low 1


breeding ratio i


sufficient to obtain the one month doubling time to


establish initial tritium inventories.


Steiner'


results for spatially dependent, nuclear-heating rates


were


based on a standard first wall energy transport of 10 MW/mn


due to


the 14 MeV neutron flux


Extreme peaking of nuclear-heating rates was


I,,,, :, 1,, ,, S


-n 4: n 4


-J


f


n,,:,, 1





-15-


---- DESIGN 1 (STEINER)


DESIGN


(STEINER)


170-

160 -

150 -

140-

1 30 -

1 20-


110-1



90-

80-

70-

60-

50

40-

30-


FIRST WALL


SECOND WALL


COOLANT & STRUCTURE


-- -


Distance from Vacuum iWal (cm


Figure


Comparison of


spatially-dependent heating rates for vacuum wall


*


200 -




-16-

heating rate peak at the vacuum wall will be 5-10% more extreme than in-


dicated


These extreme heating rates (power densities) near the first wall


along with the


excess


ve fusion neutron wall loading represent a major


technological problem for all Tokamak fusion power reactors.30,33,47


Steiner'


work


supported previous work


indicating that blankets employ-


ing lithium


the only coolant are superior to those employing flibe since:


Design 1 has a 10


higher tritium breeding ratio.


Design
since


has a 50


lower heat load in the niobium vacuum walls


high gamma cross section of flibe has been removed.


Neutron irradiation effects within the vacuum wall are essen-


tially the same in both designs along with


rates nea


excess


ive heating


r the first wall.


Blow et al.40 used Monte Carlo calculations in cylindrical geometry

with first wall at 150 cm to examine Steiner's two basic 100 cm thick


blanket model


with varying (


2-8%)


niobium structural content.


Good


breeding rati


cl usive


1.15-1.54) were reported for all


use of flibe


cases


except the


coolant in the entire blanket where T/n


= 1.027.


Blow reported additional good breeding results (T/n


= 1.58) for blankets


of Design


where niobium was replaced with


molybdenum.


Examination


of molybdenum was justified because the


Mo) has the neutronic characteristic


characteristics


alloy TZM (0


of pure molybdenum but welding


similar to niobium.


A modular blanket design using heat pipes has been proposed by


Werner et al


in which neutron


behavior was examined in a 100 cm


thick cylindrical annulus with 200 cm inside diameter.


n relocating the


"standard" vacuum wall of a thermonuclear reactor beyond the neutron-


moderatina. enerov-convertina blanket (at 320 cm). the entire moderator




-17-


to eliminate the neutronic losses and structural buckling problems of

previous designs.

The interlocking modular blanket units incorporated heat pipes which

remove radiant energy from the inner module surface and flatten the power


distribution in the blanket by moving


excess


energy outward to power-


deficient zones


WJerner's blanket model contained beryllium for neutron multiolica-


tion


, lithium for tritium breeding, sodium for energy generation


niobium for structural strength.


The 100 cm moderator section of the


blanket was divided into two zones


Zone 1 contained


Li and


wh i e


Zone


contained varying volume percentages of Be, Na, and Li


Both


zones contained


% 20%


volume for heat pipe voids.


Zone 1 was used to


buffer the energy density in the fluid so that all nuclear and radiant

heating energy could be removed by convective heat-transfer through the


heat pi

densiti

Thi


resulting in power flattening and increased average power


tradeoff between tritium breeding and energy multiplication


through use of beryllium or

tions in a 90 cm thick Zone


neutron up to


odium was examined for varying volume frac-


Increased energy generation per fusion


.0 MeV for beryllium and 26.05 MeV for sodium was obtained


but with a reduction in the tritium breeding ratio.


Unless maximum energy


is very important, Werner recommended maintenance of tritium breeding--

probably because of beryllium costs and sodium activation.


Struve and Tsoulfanidi


used Monte Carlo methods to calculate


tritium breeding ratios and heating rates for two proposed blanket designs





-18-


The two blanket configurations included a basic Steiner-type

the vacuum wall surrounds the plasma and a Werner-type where the


where


vacuum


wall surrounds the blanket.


To avoi


the problem of


coolant flow,


Struve proposed a heat transfer fluid such


as helium which would be un-


affected by magnetic field lines and transparent to neutron


It was


simulated by


volume void in the lithium.


Breeding ratios


above


were obtained and agreed


areas


onably


well with previous blanket


studio


using niobium structure.


8,40,42


The use of helium


as a fusion blanket


coolant has


been


investigated by Hopkins and Mlelese-d'Hosoital and


others at Genera


patiall


Atomic Company.31


dependent nuclear-heating rates for the two blankets


showed high vacuum wall heating and agreed with previous results.


Steiner's


generally


higher calculated heating rates


were


caused by niobium blanket


structure.

These detailed neutronic studies of fusion blankets indicate ample


tritium breeding


possible in realistic blankets.


The inability to


breed tritium is not a problem in fusion designs


The real problems in-


clude providing adequate heat removal for the first wall and protecting

and designing the vacuum wall to withstand the required 15 MeV neutron


fluxes.


These fusion reactor blanket


scoping studies have formed the basis


for a number of design studies for Tokamak fusion power reactors of


either full commercial scale or demonstration


size


28-32


various


oure fusion Tokamak blankets use either flibe, natural lithium


as coolant and flibe, natural lithium, or


or hel ium


ome lithium-bearing medium




-19-


of flibe.


All blankets are on the order of 100 cm thick and some


20-25


MeV are deposited in the blanket per 14 MeV neutron entering the blanket


with extreme peaking of heating rates near the first wall


are not expected then to be


The blankets


significantly energy multiplying.


In genera


the tendency i


toward more compact fusioning plasmas with


an associated reduction in the first wall neutron flux to well below 10


NiH~/r


of 14 lMeV neutron energy transport.


28-33


The basis for such reduc-


tions is the extreme technological problems of designing a first wall


which will function for at least two years or more.


If such cannot be


accomplished, then fusion power plants that are viable in other respects


are likely to be


too limited in outage maintenance


time to compete


economically


with other


electrical power sources.


33,49


Critical Review of Hybrid Blanket Studies


Overview of Hybrid Blanket Studies


Fusion blanket designs attempt to maximize energy generation while


maintaining the tritium breeding ratio.


The inclusion of fissionable


materials in the blanket is an obvious possibility for achieving signifi-


cant power and neutron multiplication.


Such a hybrid blanket must still


meet the basic fusion blanket requirements of adequate tritium breeding,

heat transfer, and magnet shielding as well as produce energy multipli-


cation and/or fissile material


As with pure fusion systems,


previous


evaluations of hybrid concepts have been based primarily on the cal-

culated neutronic behavior of the conceptual blanket as reflected in the




-20-



Fusion plasma characteristics.

Neutron first wall loading.


The tritium breeding ratio must be


sufficient to refuel operating


hybrid


systems


and fuel new on


As for pure fusion


systems, adequate


values are in the range T/n


- 1.1


and are relatively


y easy


obtain.


Simultaneously, a hybrid may also be required to produce or


even breed significant amounts of fissile fuel


,3,6


Energy deposition in the blanket per fusion event i


a very important


hybrid criterion.


Usually


D-T fusion


systems assume a blanket energy


deposition, QB

neutron and th


of about 20 MeV per fusion to account for the 14.1 MeV


MeV per


6)3T reaction.
Li(n, a) T reaction.


Fusion blanket studies


show thi

position


energy deposition is relatively insensitive to design or com-


with calculated values per fusion neutron ranging from a


maximum of


26 MeV for Werner's41 best design down to 18.


MeV evaluated


by Leonard


for the ORNL standard design.


Although fusion blankets are limited in their energy multiplication


capabil iti


this is not the


case


for hybrids which are evaluated for


significantly increased blanket energy deposition per fusion event


through fission energy multiplication.


Interest in subsystem interactions


and dynamics studies of such a coupled hybrid system is certainly justi-

fied when the potential for energy generation through energy multiplication


in the


subcritical blanket i


considered.


The third area of technical assessment of hybrids involve


fusion plasma characteristics


required to achieve the assumed blanket


Th' ic fnr nm a ic cc cm n cmn+t da c iro-l ma i-n f-ho hin.nL'ot onnn nv/


nn rf:n mi ~ n r a




-21-


to reach overall breakeven in energy production or scientific breakeven.

The breakeven nT-value varies inversely with the total energy generated


per fusion event.


Therefore


, the potential value of a hybrid system is


characterized by its ability to relax the Lawson condition through effec-


fission increase of energy released per fusion event.


Finally


, the required transport of neutron energy through the fi


vacuum wall


an important figure of merit.


Previous projections of 10


MW/m


impose


stringent material problems so more recent designs attempt


to achi


eve


val 1 loadings in the range 0.25 to


MW/m


1,3,13


hybrid relaxation of first wall


pure fusion


oadings is a technical advantage over


systems


Such potential for breeding fissile fuel with fission energy multi-


plication of the fusion neutron source


strength to sustain and drive the


coupled


stem h


been examined by many researchers.


Early concepts


were summarized adequately by Leonard and have little more than historical

significance.1



Lontai Attenuator Model


The first detailed calculations on the neutron economy of hybrid


blankets


were


performed by Lontai in 1965.10


He assumed a


steady-state,


D-T clyindrical plasma with a 5.0 MW/m


energy transport of 14 MeV


neutrons but performed the neutron balance calculations for an infinite


slab source geometry.


Lontai's results were based on blanket configura-


tions using flibe coolant channelled in a graphite matri


Neutron


iI *





-22-


ratios


Such a


scope of study and results reported set the stage for


most of the hybrid studi


Lontai


which followed.


s best results were reported for a blanket concept consisting


of a


cm molybdenum vacuum wall


, 1.5 cm coolant (flibe) region, and 49 cm


attenuator region containing 21


graphite by volume with 70


salt bearing


uranium (LiF


- BeF2


- UF4)


natural lithium


case


had insufficient


tritium breeding.


Adequate tritium breeding was calculated only by using


lithium


salt enriched to 50% Li and varying composition.


The fi


ssion


energy multiplication increased by nearly a factor of two over non-fissile


blankets with better heat transfer characteristics.


Similar calculations


for 90% enriched 6Li resulted in much lower fissile fuel production with no


increase in energy multiplication.


Plasma requirements are not relaxed much


by such small amounts of fission energy deposition; however, Lontai opti-


mistically labeled th


6Li attenuator practical because of possible


reduced plant


Lontai'


capital


costs


hybrid feasibility


study currently has little more than


historical


significance because of inherent deficiencies:


Faiure to consider values of plutonium production.


Failure to consider cost of maintaining high


6Li enrichment.


Failure to


consider


U present in depleted uranium.


Use of obsolete computer


Lidsky


codes and poor cross section data.


Symbiosis Concept


A novel approach to the fusion-fission hybrid concept was proposed


-* .a I- *tJ- b j. S




-23-


feature of this symbioti


tritium and fissile nuclei


device such


scheme was a fusion system breeding sufficient

to fuel itself and a power-producing fission


as an MSCR.


A cylindrical


m radius torus of D-T plasma


was used in the


symbiosi


The basi


duplex blanket configuration contained a thorium-


bearing blanket fl i be


salt composed of LiF


:SeF2:


ThF4 in the ratio


71:0O


2:27


and lithium depleted in


Th


neutron


properties of pure


molybdenum with


ts large Mo(n,2n) cross section, were utilized in the


TZM structural alloy


Since Lidsky'


s fusion reactor was designed for


, not power production, a graphi


moderating region was used to


prevent thorium fission products from poisoning the blanket during opera-


tion.


only


p055


ibie at initial operation until fissile


233 is
U ls


produced which impli


which Lidsky ignored.


frequent refueling and possible cost penalties


Lidsky used SN transport theory to evaluate the


neutron


this


economy of th


hybrid blanket configuration


as well as variations in the base design are


results for


shown i n Tabi


,-V.


Since


simultaneous production of fissile nuclei and tritium


be attainable


system can be


over a range of production ratios


optimi


found to


each component of the


for power or fuel production to utilize the strong


points of both fusion reactors (neutron rich) and fission reactors (power

rich).


The reactors in the symbiosi


were coupled by the production of fuel


for th


ssion reactor by the fusion reactors


Lidsky6


also analyzed


equations for the time dependence of the fuel inventori


of the two


rPacrtnrR in thp fuiinn-nficinn




-24-


Tabi


Neutron Economy of Lidsky' s Hybrid Blanket


Events per 14-neV Source Neutron


Calcul ated


Range


Tritium production


Thori u
Total


captu


covers


1.126
0.325
1.451


0.05-0.50


2 1.40


Lidsky


results demonstrated that the fuel doubling time of such a


balanced hybrid


system


determined entirely by the neutron-rich fusion


reactor component.


Lidsky


power production analysis


indicated further


that the net


power production in such a dual system i


determined pri-


marily by the fission reactor component since the fusion power reactor


is onl


y a small perturbation on the net power


r of real


systems.


Thus each


symbiosis can theoretically be optimized for its


respective primary purpose of fuel or power production. T

important point to remember with respect to hybrid reactor


Lidsky


his i


system design.


selected a CTR-MSCR power plant with 1500 MWe output and a


10 year doubling time for symbiosis study


MSCR was rated at 4450


14JMWth with a fuel conversion ratio of 0.96 operating on the


cle.


Lidsky calculated a 10 year fissile doubling time with a tritium


linear fuel doubling time of 0.113


years.


For a 40% thermodynanmi


efficiency the fusion reactor would be a net consumer of


the overall


MWe while


stem was calculated to be able to provide 1690 fWe net


subsystem in


233U_23~Th





-25-


Required plasma characteristics were encouraging since the vacuum

wall loading due to 14 IMeV neutrons was only 1.00 MW/m --well below that


necessary to assure technological feasibility in pure fusion plants


In addition, there was no energy multiplication in the fusion reactor

blanket of the symbiotic scheme; this assumption was clearly not accurate


as soon


as some fissile fuel breeding has occurred.


Plasma parameters


are near Lawson conditions as indicated by the hybrid parameters summary


in Table 1-VI and the fact that only


MWth was required to support


the fuel-producing fusion system.



Table 1-VI


Lidsky


Hybrid Reactor Parameters


= 1014 ions/cm


= 0.625


sec


0 keV


Wall loading
233
U production


=1 MIW/r


= 1.1 kg/day


The symbiosis has a number of advantages.


simplify


First, this scheme


construction of power plants capable of breeding and


processing all requisite fuel in situ.


Second


, the lessening of fuel


cost constraints makes the modifications of existing reactors possible

to avoid thermal pollution. Finally, by developing this concept, the




-26-


In addition to the symbiotic hybrid concept and the usual power-

producing hybrid concept, Lidsky has also formalized consideration of a


third hybrid concept called the augean concept.


concept involves


using the hybrid blanket to burn the actinide waste from fission


reactors.


augean concept is of little interest for dynamic


consideration.



Lee's Fast Fission Hybrid Concept


eliminated Lidsky'


separate fusion and fission reactors in


favor of the so-called subcritical fast fission blanket.


Monte Carlo


Transport theory was used to perform neutron balance calculations in


infinite media of pure thorium, pure

the breeding potential of hybrid blan

1-VII are in good agreement with expe


Weal


238U, and natural uranium to verify

kets. The results shown in Table


measurements done by


51
et al


Table 1-VII

Lee's Neutron Balance in Infinite Media


Blanket


QB (MeV)


Breeding Reactions
per 14-MeV Neutron


Thorium


2.7 [232Th(n,y)]

4.4 [23U(n,y)]

5.0 [238U(n,y)]


Natural Uranium




-27-


sperhical annulus having an inner radius of 200 cm and an outer radius of


300 cm with composition as listed in Table 1-VIII.


For constant blanket


qeometry and material volume fractions, the following optimum results

were obtained for depleted lithium (4% Li) and depleted uranium (0.04%


U) per 14 MeV neutron:


= 103 MeV


= 0.986;


U(n,y) reactions


- 1.


Because of the 1.68 239Pu breeding reactions per D-T fusion


event, Lee chose


Pu as the fissile fuel.


Tabl

Subcritical Fast Fi


e 1-VIII


on Blanket Components


Studied by Lee


Element


Volume Fraction


Zone 1
(30 cm thick)


0.95
0.05


Zone


cm thick)


Nb
Heavy Element


0.30
0.05
0.65


Lee also studied the neutronics effects of changes in the thickness


of Zone 1 and material volume fractions in Zone


for the composition


shown in Table 1-VIII results were reported for the following heavy

element material variations:


Depleted uranium versus


U content.


Metallic and oxide mixtures of plutonium and uranium versus 239p




-28-


Best energy generation with


sufficient breeding was reported for


the metallic uranium blanket with 4% plutonium.


poisoned with


case


fission products are summarized in Table


and one

-IX.


Tabl


e 1-IX


Fast Fission Hybrid Neutron Economy per 14 MeV
Neutron Calculated by Lee


Plutonium
Tritium Conversion B
Material Production Ratio (MeV) eff

4% Pu-U 1.38 3.14 431 0.84

4% Pu-U + 8% FP 1.18 3.03 306


The usefulness of a hybrid concept is contingent upon a short


ssile fuel doubling time


. Lee estimated a very high 14 MeV neutron


wall loading of 12.


MW/m


to obtain a


year plutonium doubling time


for the


Leonard la


FP blanket but reports no fusion plasma characteristics.

ter claimed that the 306 MeV blanket energy release per fusion


neutron


n Lee's


8% FP model would lead to a three-fourths reduction of


the usual Lawson breakeven condition.


However, current engineering con-


siderations indicate that such first wall power loadings will almost

certainly make fusion power unrealistic due to the need for frequent


first wal


Since hi


times over non-fissil


replacement


results indicated energy production increases of 10 to 20


blankets with simultaneous adequate tritium and




-29-


advantage over other concepts except as a fuel producer.


Considerabi


additional research has been reported on blankets and hybrid systems


using the fast fission concept.


All have emphasized fuel production versus power production and


have


worked with reduced first wall neutron loadings of 1-5 MW/m


advantages of using fusion neutrons for fast fission as well as breeding


fuel in situ


are probably only applicable in the true symbiotic conceptS


where the hybrid is not a system energy producer but a fuel producer,


since blan


fission.


multiplication


Hence, th


lowered for low enrichments with fast


fast fission hybrid is of little interest in this


current study.



Texas Fast Fission Hybrid


Parish and Draoer


presented extensive hybrid neutronics results


for their no


del which


was also a fast fission design.


They investigated


the potential of 14 MleV fusion neutrons to fission fertile material


(232


Th and


U) while


maintaining adequate fusion blanket performance.


Parish and Draper based the a

tive abundance of such fertile


attractiveness of this concept on the rela-


fuels and the elimination of dependence


on breeding fissil


fuel for hybrid usage.


The large fission energy


multiplications obtained in other studi


1,3 were not paralleled in thi


hybrid; however, the potential of both thorium and natural uranium-fueled

fast fission blankets to produce both fission power and fissile material

was demonstrated.
.. /L .. .




-30-


various calculational methods.


To verify methods of analysis


Parish and


Draper calculated the neutronic and photonic characteristic


standard fusion blanket model using ENDF/B-III cross


of the


section data in the


ANISN5


code for a P3-S4 transport approximation.


The resultant standard


blanket neutron


economy compared wel


with Steiner


s latest results on


the same standard.59


Good agreement was obtained for breeding


1.445


versus T/n


= 1.452) and (n,2n) reactions as well as neutron leakage


despite Steiner's use of pre-ENDF/B-III cross section data


hybrid was one of the first hybrid studies to account for (n,3n


s Texas


reactions


which become very important in such poorly multiplying blankets.


Since high energy neutrons are needed to fission fertile fuels


fission material regions in this concept were placed as


as possibi


to the vacuum walls.

offset by (n,2n) and


Low energy neutron absorption was only


n,3n


partially


reactions.


The volume fractions of fuel, clad


niobium)


and coolant (lithium)


in the model


tively


were


maintained constant at 0.45, 0.15, and 0.40,


, to approximate fuel regions


n a LMFBR


respec-


The tritium breeding


fissil

region


breeding, fission power, and spatial heat deposition by


were presented in the T


blanket


exas study for various blanket fuel thick-


nesses.


blank


The results of these calculations for two thorium-fueled


and four uranium-fueled blankets are presented in Tabi


and 1-XI.


The calculation of


spatial heat deposition rates in the standard


and fertile fueled blankets in this work emphasized the problems with


ow mul tipli


cation hybrid blankets.




-31-


Table 1-X

Neutron Economy for Thorium-Fueled Blankets


Thorium Fue


Reactions/Fusion Event


Region Thickness


2Th(n,f)


232Th(n,)
Th(n, )


6 cm


.3012


.0310


.1326


13 cm


1.0964


.0472


.3118


Table


U "LXI


Neutron Economy for Uranium-Fueled Blankets


Natural Uranium


Reactions/Fusion Event


Region Thickness


U(n,f)


235U(n,f)


Total
Fission


238U(ny)


10 cm


1.3252


.133


.0133


.1463


.2487


1.2694


20 cm


1.0865


.0161


.0259


.2024

.2096


.3818

.5320


0.9614


.1986


.0315


.2301


.6654


For the large 10 MW/m


first wall neutron loading limit, the two


thorium-fueled blankets showed peak power densities of 200 W/cm


3
For


the 1


wall


of 20


cm natural uranium case


ranged from 510 to 364 W/cm


to 146 W/cm


, the power density between the niobium


the related thorium case had a ranqe


Fuel was eliminated in the 3 cm region between


----1


L~.......-ll -11


-1




-32-


density, Parish and Draper have claimed these hybrid blanket power den-


siti


are acceptable.


This is doubtful because of the


ow power den-


siti


at blanket positions removed from the vacuum wall and the resultant


unit cost of


electrical and fusion power produced.


The superiority of natural uranium to thorium as a fast fission


hybrid blanket fuel because of its larger fast fission cros


illustrated in Parish


presented in Tabl


section is


s comparison of the best case for each fuel

II.


Table 1-XII


Comparison of Best Natural Uranium-Fueled and
Extrapolated Thorium-Fueled Blankets


Uranium


Thorium


Tritium Breeding Ratio
Fusion Blanket Energy Multiplication
Fissile Nuclei Produced per Fusion Event
Peak Power Density at Nb First Wall


1.09
, 20


'"" 3
409 W/cm3


1.15
0.5
0.31
S200 W/cm


However, the low return of the fissioning blanket renders this concept

uneconomical versus other concepts relying on better fissile blankets.

Increasing fuel costs could make this concept more attractive at some

future date but others seem more appropriate.



Light Water Hybrid Reactors


The feasibility of fusion-fission hybrid reactors based on breeding




-33-


Princeton Plasma Physics Laboratory.60 Emphasi


was placed on fuel


-self-


sufficient (FSS) hybrid power reactors fueled with natural uranium.


Light Water Hybrid Reactors (LWHR) considered included FSS-LWHR'


Other


fueled


with spent fuel from Light Water Reactors (LWR's), and LWHR's to sup-

plement LWR's by providing a tandem LWR-LWHR power economy that would be


fuel self-sufficient


similar to Lidsky


symbiotic concept


Nuclear


power economies based on any of these LWHRs were found to be free from


the need for uranium enrichment and for the separation of plutonium.


They


offer a high utilization of uranium resources (including depleted uranium)

and have no doubling-time limitations.


study investigated the property


atti


of subcritical thermal


for hybrid applications and concluded that light water is the


best moderator for FSS hybrid reactors for power generation.


latti


Several


geometries and compositions of particular promise for LWHR'swere


identified with thicknesses up to 250 cm.


The performance of several


conceptual LWHR blankets was investigated and optimal blanket designs


were identified for natural uranium-fueled lattices.


The effect of


blanket conversion efficiency and the feasibility of separating the

functions of tritium breeding and of power generation to different


blankets were investigated.


Optimal iron-water shields for LWHR


were


also determined.

The evolution of the blanket properties with burnup was evaluated


along with fuel management schemes.


The feasibility of using the lithium


system of the blanket to control the blanket power amplitude and


was also investigated


shape


A parametric study of the energy balance of LWHR




-34-


with critical systems and delineated the advantages of such hybrids in

alleviating nuclear technology problems relating to resource utilization,


prol iferation


and safety issues.


In general


, this study reported the


same types of


information as previous studio


but for a different blanket


design.



PNL--Thermal Fission Hybrid


Pacific Northwest Laboratories (PNL)1'61 initially studied a hybrid


fusion reactor utilizing a subcritical thermal fission latti


around the


usual


cylindrical D-T plasma


The four distinct regions of the hybrid


blanket configuration are illustrated in Fig.


cm thick neutron convertor region was filled with niobium-clad


pins of both depleted uranium carbide and natural lithium.


Niobium


structure


walls are used along with helium coolant.


The 150 cm thick


thermal fission lattice, consisting of a graphite-moderated, natural


uranium-fueled, helium-cooled matrix,


was designed for fi


ssion power


generation.


The last 50 cm of blanket thickness are filled with graphite


reflector and natural lithium absorber


, respectively


The ENDF/B III cross section data were used in the HRG362


Revised-Thermos (BRT-1)63


and Battelle-


cross section codes to obtain fast and thermal


broad group data

trained using a P


, respectively


The final neutron balance results ob


transport calculation in ANISN58


are summarized in


column


of Table 1


-XII I.


Neutronic effects from slight enrichment of


the uranium in the fi


ssion lattice are also shown in the neutron balances


I-~~ -5- II tt-


r I





-35-


S'- 4
A 4


~C~L~ll*ll~:
t
I~ ,&




-36-


Table 1-XIII

Early PNL Hybrid Neutron Balance


Events per Source 14-MeV Neutron


235
U Atom Percent Enrichment


0.7196


0.80


0.90


Tritium Production


0.956

0.019


1.188

0.019


1.763

0.020


Total Tritium Production


0.975


1.207


.783


Fissions


0.234


1.936


U Captures


1.121


0.251


.776


0.988


.292


4.863

0.853


U Absorptions


Estimated kff
eff


0.84


0.884


0.928


1050


Based on their composite behavior with fi


enrichment, an


enrichment was predicted


0.77 atom


) for which both the tritium and


fissile conversion ratios could be optimized to


excee


d unity.


The cal-


culated energy deposition in the blanket for the best case was calculated


to be about 500 MeV per


pl ication of about


significantly 0.7


source neutron corresponding to an energy multi-


This PNL optimum hybrid is attract


nce


enriched uranium can be produced than the higher




-37-


reactor capabilities which is very low.


This power density was used to


determine the plasma and blanket specifications shown in Table I-XIV where

the plasma requirements are substantially less than for a nonmultiplying


blanket and the vacuum wall loading i


Tabl


very low.


e 1-XIV


Early PNL Hybrid Specifications


Blanket


Plasma


Specific power
Thermal power
Vacuum wall loading


keV)


0.75 W/cm3
20 MW/m
0.05 MW/m2


nT (steady


tate) (sec/cm3)


x 1013
x 1013


Since a non-negligible fraction of the thermal energy produced in the

blanket must be used to sustain such a plasma, the need for investigation


of controls i


tion i


justified, especially since the fission energy multiplica-


predicted to be so high.


This preliminary PNL hybrid design was faced with drawbacks such


as large


(2 m thick blanket) and low power density


0.75 W/cm3).


However,


it was favored with low wall loading and plasma conditions re-


duced to


S1/6 Lawson Criterion value.


Since the hybrid objective


energy multiplication with adequate breeding of tritium and fissile fuel

are attainable, the PNL concept appeared to be a promising competitor for


the LMFBR program.


Much additional work has been performed including




-38-


studies have identified and delineated the merits of the helium-cooled,

thermal fission hybrid fueled with natural or slightly enriched uranium


moderated with graphite, and cooled with helium.


In addition, the


optimal use of lithium for breeding has been delineated.


This PNL concept of a fusion-fission


system has been developed to a


considerable


degree


as reported by many studio


64-67


The most complete


results on blanket parameters were reported by the combined efforts of


Lawrence Livermore Laboratory and Pacific Northwest Laboratories.


Although thi


hybrid blanket design was intended for use in the spherical


geometry of Livermore'


mirror (Yin-Yang) fusion reactor


concept, the


basic blanket geometry is very similar to that shown in Fig.


modul


Blanket


of varying composition were analyzed using a fuel pin lattice


geometry similar to that used in High-Temperature Gas-Cooled Reactors.


Results reported for th


hybrid blanket analysis are included in Table


1-XV showing seven (7) different


cases


analyzed, all of approximately


200 cm thickness.


The inner convertor region was closest to the plasma


and contained helium coolant and stainl


steel structure as well as


depleted uranium to enhance neutron multiplication.


The inner thin


breeder contained lithium for fast neutron tritium breeder while the

thicker outer lithium breeder contained lithium for thermal neutron


breeding of tritium.


The reflector, where used


was composed of graphite


and the thermal fission latti


was composed of hexagonal unit cell


slightly enriched (as noted) fuel pins in a helium-cooled graphite matrix.


The fuel pin geometry and


cell pitch were optimized using


transport


calculations.




-39-


Tabi


1-XV


PNL Hybrid Blanket Analysis


Tritium Fissile Blanket
Case Blanket Breeding Breeding Fusion Energy
Arrangement Ratio Ratio Multiplication

1 8.5 cm convertor
1.5 cm breeder
150 cm lattice (1.0%)* 0.766 1.59 18.9
20 cm reflector
15 cm breeder


2 10 cm convertor-breeder mix
150 cm lattice (1.0%) 0.725 1.57 19.8
20 cm reflector
15 cm breeder


3 10 cm convertor-breeder mix
180 cm lattice (1.0 %) 0.365 1.62 25.2
10 cm breeder


4 8.5 cm convertor
1.5 cm breeder
0.737 1.55 20.0
180 cm lattice
10 cm breeder


5 8.5 cm convertor
1.5 cm breeder
1e 0.893 1.22 31.8
180 cm lattice (1.25%)
10 cm breeder


6 8.5 cm convertor
1.5 cm breeder
15cbed 41.26 0.984 59.6
180 cm lattice (1.50%)
10 cm breeder


7 8.5 cm converter
1.5 cm breeder
n 00 1. 11 39 .8
180 cm lattice (1.35%)
10 cm breeder




-40-


unity


In addition, the energy multiplication of the fusion power was


found to be very large for thi


best case (MB


= 39.8).


This energy multiplication was claimed to be related to the effec-


neutron multiplication of the


blanket and the neutrons produced per


fission in the blanket by the following global parameter equation:


200 MeV 1)
14 MeV


eff


- kff
eff


where 200 and 14 represent the energy deposited due to fission reactions


and fusion neutrons, respectively,


v is the number of neutrons released


per fission and


tion factor.


keff i


usual blanket effective neutron multiplica-


this equation related global parameters and


since


4 MeV source i


introduced inhomogeneously, the current work was


partially directed at determining if this equation might be inadequate

despite its frequent use in describing and analyzing results from cal-

culations performed on hybrid blankets.



Review of Controlled Thermonuclear Reactor


Thermal Stability Ana


yses


Fusioning Plasma Operational Criteria


The first determinations of operational criteria for thermonuclear

reactors were performed using global or point-model reactor parameters.


Rigorous descriptions of comply


plasma dynamics with attendant spatial


variations were usually beyond the scope of such criteria development.

The first attempt to soecifv fusion reactor operational criteria




-41-


uniform temperature, T, and confined for a time, r, after which cooling


was allowed.


Conduction losses were entirely neglected.


This initial


work established values of temperature and the product of ion density


and confinement time, nr, for a zero-power but

nuclear system. A system energy balance was u


self-sustaining thermo-


sed in which the energy to


heat the plasma, E and the energy to overcome bremsstrahlung radiation


losses

energy


, Eg, were


upplied


supplied to produce fusion reaction energy


as well


as the fusion reaction energy, was assumed to be


recoverable


and converted to useful output energy at


some efficiency, n.


The minima


condition for breakeven is simply defined as follows:


[EF + E + Ep]n


EB + Ep


where n is the overall system energy conversion efficiency


For a D-T fusion system as described above, the so-called Lawson

Criterion for breakeven becomes simply:


( )p(l


- p)QFDT


where


= fuel ion density (ions/cm3 )

= plasma temperature (keV)


DT


= reactivity of D-T plasma (cm


3/sec


n = overall


system energy conversion efficiency


= proportionality constant for bremsstrahlung radiation

= tritium fraction of ion density
r.. ./i.. \


- bT1/2




-42-


The Lawson Criterion for the pure D-T fuel cycle is represented


by a series of parametri


spectrum of curves in Fig.


curves in the efficiency as shown in the lower


Points on such parametric curves represent


minimum nr and T values for breakeven fusion energy production


no net


fusion energy i


produced.


If the energy per fusion event can be aug-


mented by fission reactions in the hybrid blanket, then the requirements

on the plasma can be significantly relaxed.

Cyclotron or impurity radiation losses are not considered in Lawson-


type anal


yses.


No stability i


considered


since the conditions quoted


from such analyses refer to minimum requirements for overall breakeven.


Another


early study of the reactor energy balance was done by


Jensen et al


Again the


D-T reaction was of primary concern though


subsidiary fusion reactions were also treated.


Jensen reported on the


effects of finite energy transfer rates and found self-sustaining

reactors were possible over an increased parameter range, although all


ion speci


were treated at a uniform temperature.


The major


shortcoming


of Jensen


energy balances was its failure to consider particle confine-


ment times of diffusion losses.


Additional energy balance


considerations


were reported by Woods.


Horton and Kammash


have also considered energy balances and


operating conditions for the D-T fusion cycle.


since alpha particles are


a significant plasma heating mechanism, energy and particle conservation

equations were introduced for the alphas created in D-T fusion reactions.


Both bremsstrahlung and synchrotron radiation 1


losses


were treated along


with the effects of cold and energetic fuel injection.


This work was




-43-


temperature and density, some of which were applied in the later stability


work of Mill


15-17


and Ohta et al


A similar but more realistic condition than the Lawson Criterion

for minimal operation has been developed by Mills for a system using only


the D-T reaction.


15,16


This model i


based upon continuous injection of


cold fuel where fusion temperatures are assumed to be supported by alpha


heating.


Mills used particle


and energy conservation equations for the


ion density


follows:


dt


- n/T


d 3
dt 2


+ Te)]


= p(1l


- p)n 2 DTCQ


S(T. + T )
1 e'


where


= fuel ion density (ions/cm")


= fuel ion injection feedrate


nuclei/cm3-sec)


= tritium fraction of ion density

= confinement time against all plasma losses including
fusion (sec)

= alpha particle energy from D-T fusion events (3520 keV)

= fraction of alpha energy retained in the plasma for
heating


i,e


= temperature for ion and electron


species respectively (keV)


DT


= D-T fusion reaction reactivity or rate coefficient
(cm3/sec)


For steady-state operation with this model


Mills found that the


following equilibrium condition must be maintained if operating charac-




-44-


2p(1


+ T )


- p)DTcQ


This result is similar to the Lawson condition but more conservative


since only a fraction of th


alpha particle energy i


retained to sustain


the plasma while none of the neutron kinetic energy is retained.


addition, the Mills


condition i


a steady-state condition based only on


the plasma while the Lawson Criterion attempts to account for all in-


fluences on


losses


tem efficiency.


The constant, c, accounts for energy


due to bremsstrahlung and synchrotron radiation.


feature of thi


work i


An important


the temperature difference allowed between the


ion and electron speci


in general, Mill


s found that the electron


temperature is elevated due to preferential alpha heating


Figure 4


illustrates the Lawson breakeven region for


to 45% efficiency


com-


pared to the Mill


' equilibrium region


= 0.8, p


= 0.25 and 0.50)


Since Mills' model i


concerned only with alpha heating and radia-


tion 1


losses


within the plasma, energy release to neutrons was not con-


sidered.


Though actual power generation capabilities were not considered


by Mills, comparison with the zero power condition developed in Lawson

model does indicate net overall power production as expected for

equilibrium operation.


Fusion devices producing values above Mills


equilibrium region in


Fig.


4 can be operated only in the pulsed mode.


similarly, devices pro-


viding ni-values below the Lawson region can never operate


as power-


producing reactors, while those falling between the two criteria will


4tn, l nnvn 4n4r4n,,n C; .; n a n 1, n n + I I


r.. c.,ll ..,.+., cn inhn ; $





-45-


10"15


PLASMA EQUILIBRIUM


REGION (c


= 0.8)


p = 0.25


- 0.50


n = 45%
n = 40%
n = 35%


LAWSON BREAKEVEN REGION


20 40 60


Ion Temperature (keV)


I,


t~


14
10


13
10




-46-


Plasma Thermal Stability Considerations


Plasma global thermal stability studies were initiated by Mills


based on the operational equilibrium


studies.15-17
studies


Mills demonstrated


that the equilibrium condition is equivalent to requiring the constancy


of a function


follows:


= STr2p(


- p)S


where the so-called stability function,


cm2/keV-sec)


varies with ion


temperature Ti


as S


" cT~ which exhibits a broad resonance
%CT1


peak around


In the


first approximation Mills treated the alpha energy re-


tention fraction


as a constant.


For stable equilibrium, the


ogarithmic


variation of i(S,T,p,T.) must vanish.


Therefore, Mills found that the


operational equilibrium i


unstable


against fluctuations in the fuel


feedrate


th


confinement time, the


fuel mixture (unl


), and


the ion temperature except when the


exponent in Ta falls to zero above
1


28 keV.

Although the exact behavior of the confinement time with ion tem-

perature was not known (nor is it known today), the p-function formaliza-


tion showed that if t i


nuclear reactor below


s independent of T.


keV is impossible without


stable operation of a thermo-


some form of control.


Below


, departures from equilibrium are supported due to the posi-


tive


slope of the


stability function.


It is not until the negative


slope


region of the


stability function i


reached above


keV that the in-


herent instability against fluctuations in T. is controlled and the
1
+ maarnnr s -I I r t i -u h3Pfn4 annII 1i h nr I~ +n ar,4 km




-47-


In fact, most fusion reactor


system design studies currently


operating temperatures below 20 keV


28-32


But at temperatures below


keV, Mills showed that control i


extreme departures from equilibrium.


necessary to avoid the predicted

This control can be implemented


via the feedrate, the fuel mixture, the confinement time, or radiation


losses


dependent on injection of impurities.


Initially, Mill


favored


control via the confinement time


5,17 but later work has emphasized


feedrate control


More recent studio


by Ohta et al.


18 have confirmed


the use of feedrate as a viable method by which to control stability.


If the confinement time i


temperature-dependent


, then it may be


useful for inherent control by introducing temperature dependency into


the p-function.


Mills hypothesized Bohm-type diffusion (r


ST-1) as a
CLT) as a


possible inherent control to allow stable operation below the 28 keV


cutoff indicated for constant confinement operating conditions.


fixed feedrate and fuel mixture, Mill


used the p-function variationa


method to demonstrate inherent stabilization of plasma equilibria for


this Bohm-type diffusion for temperatures in the 7 to


keV range.


analyzing the dynamic behavior of thermonuclear plasmas, Mills

lished the self-stabilizing influence of Bohm diffusion below


estab-


temperatures


as the perturbed plasma temperatures (ion and


electron)


and ion density were shown to approach equilibrium with time.


Mill


In this


justified operation near the 12 keV temperature to take ad-


vantage of the optimal D-T reaction rate73 without the necessity of

introducing artificial control.


Mills17


also presented details on calculations to evaluate the time




-48-


exchange between ion species


as an instantaneous process.


Results were


reported only for plasma time behavior for attempted initial equilibrium

operation about a temperature of 11 keV with 50% deuterium and 50% tritium


fuel injection


leading to ion densiti


x 10 ions/cm


stability of plasma operating conditions in thi


region was verified for


constant confinement and shown to result in rapid plasma runaway in less


than three seconds


The plasma temperatures (T. and T ) were shown to
1 e


runaway above or below ignition depending to extreme acc


uracy on whether


or not the constant plasma confinement time was too


ong or too


short


artificial control was found to be essential below


Mills17


also investigated feedback control via the fuel mixture


using the monitored plasma electron temperature.


When the


ectron tem-


perature was set below a preselected control temperature, the injected


fuel mixture was maintained at the original 50


D, 50


T; when the tem-


perature exceeded the control temperature, tritium injection was replaced


with pure deuterium.


effect of stopping tritium injection was to


reduce fusion events and lower temperature


the stabilizing effect of


this mixture control feedback was achieved by making the time average of


p(l-p) low enough to compensate for


excess


ive confinement time.


Control


to a temperature that was too low to provide the nr-equilibrium condition


was found to result in the reacting plasma


extinguishing itself.


Mill


also noted that


excess


ive confinement time will result in severe initial


temperature overshoot.


These investigations by Mill


constituted the first efforts to


study the dynamics and control of thermonuclear reactor plasmas. The




-49-


incomplete stability criteria development in Mill


work is its most


significant deficiency.

The same stability problems of point model D-T plasmas have been


investigated in more detail by Ohta et al


18 but using the following


global nonlinear balance equations for plasma density and temperature

(energy):


- n/T


n (


d(nT) n2f)
dt


- nT+ ST


(13)
s


where


DT


f(T)


- 1.12


-1015 1/2
x 10 T


= plasma ion density (ions/cm3)

= uniform plasma temperature (keV)

= particle and energy confinement times (1/sec)


= fuel injection feedrate (ions/cm


DT


-sec


= fuel ion inject energy (keV)

= alpha particle energy from D-T fusion events (3520 keV
3
D-T fusion reaction rate coefficient (cm /sec).


Ohta addressed only the D-T reaction; the fusion reaction was not

considered an important loss mechanism in the particle conservation equa-


tion in essential agreement with Mills.


and the bremsstrahlung energy 1


Both the fusion energy source


terms were included in f(T) but


dn
dt




-50-


No temperature difference was allowed between the electron and ion


species which is a limitation in contrast to Mill


attempt to treat


differing temperatures


The advantages of Ohta'


model include accounting


for energy diffusion with particles and energetic ion inj


section as well


as including an explicit expression for bremsstrahlung radiation.


obtained the following form of th


state subscriptt o)


Ohta


Mills equilibrium condition for steady-


evaluation of the balance equations:


oEE


T
E
_o


0
(14)


f(T


which indicates the reduction in required nT-values by the inclusion of

Ohta's injection heating option.

Efforts by Ohta to examine steady-state plasma stability can be

categorized into two areas:


Linear analysis establishing temperature-dependent stability
criteria in possible operating regions for future fusion
plasmas, and


Nonlinear dynamic


subject to


simulation of the plasma balance equations


small perturbations with and without feedback


effects to verify agreement with linear stability analysis


and control possibility


in unstable operating regimes


Linearized analysis


will usually predict stability regimes.


If a


system is not


stable, linearized analysis will not predict true con-


sequences of the


unstabi


situation--hence


the need for dynamic simula-


tions.


ability criteria to predict whether small plasma perturbations


will grow or diminish with time were developed by Ohta from linearized


forms of the density and temperature balance


Pouations.


The elimination




-51-


and small temperature perturbations, 6T(t), which Ohta assumed to vary

exponentially with time.

Stability is assured provided the real part of the growth rate is


negative.


Ohta obtained general stability criteria by solving for the


growth rate after substituting the density and temperature variations into

the linearized density and temperature equations.


To proceed beyond such general stability criteria


, the functional


dependence of both the particle and energy confinement times were re-


Because the exact density and temperature dependence of confine-


ment time was uncertain, Ohta based the analy


functional dependence of confinement time on density and temperature


T n Tm

this genera


upon the following


It is the derivation of stability criteria on the basi


diffusion model that represents the major contribution of


Ohta


s stability analysis.


To obtain useful stability criteria, Ohta


used three diffusion models to get specific values for


and m:


Constant confinement


T % constant (z


= 0, m


= 0).


Bohm confinement:


T T-l (e


Classical confinement:


= O, m


T % n-'T1/2 (A


= -1)


= 1/2).


The minimum temperature satisfying the stability criteria for each


confinement model is known as the critical temperature, T

temperature above which operating conditions are predicted


that is, the


d to be stable


as described by Mill


s' work.


Representative temperature results pre-


dicted by these stability criteria are listed in Table 1-XVI for both

charged particle and injection heating for all three diffusion models.

Ohta also dynamically simulated the balance equations to check the


quired.




-52-


densities, however, were perturbed a small amount above and below


equilibrium and th

and temperature ca


effect on the temporal behavior of the plasma density


culated as presented in Fig. 5.


Table 1-XVI


Critica


Confinement Model


Temperatures for D-T Fusion Reactors


c (keY)
C


Charged particle Heating


Injection Heating*


(*n E/


= 10)


(n /TE


- 1)


T = constant

T T-


T fl


*Ion Injection Energy:


For the c

which the


ase


= 150 keV.


of constant confinement and charged particle heating for


critical temperature T


keV was found also by Mills.


Ohta's results are depicted in Fig. 5 for three initial equilibrium tem-


peratures of 10 keV


30 keV


and 50 keV.


For equal magnitude density


perturbations, equilibrium density is always approached with time which

indicates plasma stability under isolated density perturbations.


Similarly


, temperature transients resulting from the density perturbations


out for ca


30 keV to 50 keV) where T


However, for the


subcritical 10 keV initial temperature, the time evolution of temperature

is unstable as shown in Fig. 5 and predicted in Table 1-XVI.




-53-


From Ohta et al


T = 10 keV
ST = 30 keV
T = 50 keV


0 2 4 6 8 10


Time


sec)


= 10 keV
= 30 keV


T = 50 keV


0 2 4 6 8 10


Time (sec)


Figure


Time variation of point-model plasma temperature and density
for constant confinement and charged particle heating.




-54-


general, the quick plasma response on the order of a few seconds was


found for all these anal


yses


of unstable plasma variations in pure fusion


plasmas.


Ohta'


s behavior agreed with previous fusion plasma analyses.

results demonstrate the need for stabilizing control to allow


fusion reactors to operate below the critical


temperatures


as planned by


current fusion reactor design studies.


stabilization for the


Ohta et al


balance equation


The case of feedback


constant confinement model was also examined by


Stability criteria were again derived from linearized

ns. Density feedback control was introduced by adding the


term,


6n(t)


, but was not able to stabilize the system because the


balance equations are stable for isolated density perturbations.


since


temperature instabilities can grow independently, various types of tem-

perature feedback were introduced by adding the stabilizing feedback term,


6T(t)
a T
T
0
tions.


, to either one or both of the perturbed linearized balance equa-

New stability criteria were derived dependent on the value of the


feedback coefficient,


in implementing control

ferred by Ohta et al. i


. Although many parameters are possible for use

, feedback via the injection feedrate was pre-


n agreement with Mills.


Ohta demonstrated control of the temperature instability through

dual temperature and density feedback which was introduced through the

injection rate and its "small" variation about equilibrium as follows:


an


S(t)


+65S


T


- At)


where


a is the feedback coefficient and At is the delay time between a




-55-


applicable to realistic control situations.


The effectiveness of feedback


stabilization was found to be dependent on both feedback parameters:


and At.


For the applicable plasma model Ohta found a stabilized region


in the aAt-plane from the linear stability analysis


of thi


s feedback


effect.


In general larger negative feedbacks and


shorter delay times


were found to yield more effective stabilization.


At or small


subcritical (T


For sufficiently large


feedback stabilization was found to be ineffective in all


cases.


As expected, Ohta found the unrealistic case of


zero delay time to be


the most effective


feedback.


However, when the delay time and feedback


coefficient were within the stability region predicted by linear analysis,

an equilibrium temperature was always approached; however, the amplitude

of oscillations was found to increase with delay time as the limits of


the stability regime were neared.


Since delay times of


to 3


seconds


are outside the


linear stability regime predicted for this case, extreme


amplitude of oscillation for these delays


was found as


expected


Usher and Campbell23'24 extended point-model thermal stability


analyses to other fuel cycle


and other plasma diffusion models with


similar results and


speeds of response.


In addition burnup was treated


in this extension of Ohta


s analysis with essentially similar results for


the D-T fuel cycle.


Stacey'


point model plasma


stability analysis of the D-T fuel cycle


extended point model plasma


stability analysis of the D-T fuel cycle to


include more detailed plasma behavior including four balance equations


to represent the following plasma parameters:




-56-


Alpha particle density.

Electron energy density.


Again the temperature instability was found in certain regimes.


Effective


stabilization to control operation about an unstable equilibrium point


through use of controlled ion injection rate as well


as controlled D-T


fuel mixture was demonstrated


temperature instability has


also been examined for radially


homogeneous D-T fusion plasmas by Yamato, Ohta


and Mori


, using partici


and energy balance equations.


19-21


The results of thi


s inhomogeneous


anal


ysis


support the validity of decoupling


excursions in the overall


particle densities and temperatures from excursions in the spatial density


and temperature distributions.


When the injection of fuel is uniform


the temperature instability can develop only in the zero order mode.

Stability criteria were developed similar to those for the uniform plasma


with similar results


, including feedback stabilization through temperature


to allow operation below the critical temperature.

There have been no investigations of hybrid plasmas to examine the


temperature instability discussed in thi


review.


an area that


requires study because large hybrid blanket energy multiplication values

coupled with large plasma transients and neutron release could have con-


as wel


as safety significance.


Motivation for the Research


As is evident from the preceding critical review of hybrid studies,


- hn rn vmn m v n \F r r P vnn 4n 4 nncn n# i An ct M c n h r


hlar c f I I rl; o c n n h ~rh ri rl c r ;I n







the dynamic interaction of the two components of the hybrid system.


investigations have not been reported in the literature to date.


Such


Thus,


the objective was not to devi


a new system but to take the


somewhat


arbitrary approach of selecting a previously establi


shed hybrid concept


with necessary adjustments.


Many different types


of hybrid machines have been proposed with many


different methods of application.


Power-producing


Tokamak hybrids are of


most interest for


control


and dynamic


considerations


and so such a model


was selected for thi


work.


entially thi


hybrid design is compatible


with various hybrid advantages


Laboratory


delineated in the recent Princeton Plasma


tens study of Tokamak fusion-fission hybrid


reactors which concluded that the most economical mix of power- and


fuel-producing hybrids should emphasis


power production


An optimized


hybrid machine should be a substantial


power producer with a by-product


of fissionable fuel,


the optimum ratio of fuel


production to power pro-


duction being determined by economics.

An early demonstration of hybrids could allow a very reassuring


program for future development of the utility industry.


A guarantee of


future reasonable fuel costs could promote the accelerated installation


of current LWR plants


straints on all


to fill


sectors of th


near-term power needs while


United States energy economy


loosening con-


subsequent


commercial


development of hybrids could supplement LWR's,


provide them


with fue

eventual


1, and take up the load of retired power stations followed by

introduction of the pure fusion reactor sometime in the coming


century.




-58-


enrichment operations run for nuclear power plants and defense purposes.


The hybrid may be a better way to burn


U reserves with possible elimina-


tion of some enrichment requirements and perhaps elimination of plutonium


separation if bred plutonium i


burned in situ.


scenario i


especially important in


eight of the continuing breeder controversy


the recent Three Mil


Island accident ,74


which will undoubtedly delay


introduction of the breeder still longer due to safety considerations.

Since the hybrid represents an alternate concept for power production

and orderly progression to long-range utility application of pure fusion,


its characteristics require analysis

central station power production.


scribe hybrid


prior to its being approved for


One parameter frequently used to de-


characteristics is the global relationship for the blanket


neutron energy deposition per fusion neutron


QB, derived in Appendi


f
=--- [-
v
-Li


kff
eff


]+E


- keff


where


= blanket energy deposition per entering fusion neutron


= fission energy deposited in the blanket per fission event
(192.9 MeV)75

= average number of fission neutrons produced per fission
event


keff


= effective blanket neutron multiplication factor

= energy of the fusion neutron (14.06 PMeV)


= addition
due, for


energy generated and deposited in the blanket


example, to exothermi


neutron absorption


reactions


+ 6E




-59-


Several forms of the global relationship of Eq. (16) have been used


tensively to describe hybrid blankets.


1,64,76,


However, no results have


been reported on its validity.


If the parameter is to be used as a


figure of merit characterizing the multiplicative capabilities of hybrid

blankets, then its applicability must be verified and its limitations

established.

Hybrid blankets are expected to have substantial energy deposition

per fusion event so it becomes imperative that safety studies be undertaken


to examine the implications of this characteristic.


For non-multiplying


pure fusion blankets, the energy deposition per fusion event is


to be about 20 MeV.


expected


For hybrid blankets, even extremely modest ones


with keff = 0

fusion event.


.8 are predicted by Eq


. (16) to have 316 MeV deposited per


energy deposition and fusion energy multiplication


predicted by Eq. (16) for possible blanket keff values are listed in

Table 1-XVII.


Table 1-XVII

Predicted Blanket Global Response per 14 MeV Neutron


Effective Blanket Blanket Energy Blanket Fusion Neutron
Neutron Multiplication Deposition* Energy Multiplication
keff QB (MeV) MB


0.80
0.85
0.90


0.94
0.95
0.98
0.99


439
687
872
1181
1429
3654
7364


ex-




-60-


The accepted variation of blanket fusion neutron energy multiplica-


tion with blanket values of keff i


depicted graphically in Fig. 6 to


demonstrate the hybrid capability for high energy multiplication with in-


creasing but still far subcritical blanket systems.


Despite the impossi-


ability of reaching a critical fi


ssion reactor state


in such


stems,


variations in the plasma operating conditions could cause blanket energy


production rates beyond the technical limitations or the technical


fications of the design.


speci-


Even with no danger of supercritical behavior,


large uncontrolled thermal instabilities in the plasma could


ead to


cess


ive energy deposition


in the power-producing hybrid blanket.


In addi-


tion, there is the possibility of criticality at low temperatures prior to


power startup.


If plasma


startup is very quick, then the plasma neutron


production may drive the blanket to large overpower ratings before the

temperature defect can reduce the effective blanket neutron multiplication

factor, keff


Although relatively small quantities of therma


energy are contained


in the plasma a full-scale hybrid system generating 6500 MWth of steady-


state thermal power will require large numbers of 14 MeV neutrons.


a far-subcritical blanket (keff


tion of the fusion neutron


fission neutrons in the blanket.


Even


Z 0.9) can cause considerable multiplica-


available as an external source for providing


The component interactions as well a


the control and stability of such power-producing hybrid systems must be

well-understood.

The Lawson Criterion for hybrid reactors is modified as follows to

account for fusion and fission sources of thermal power with zero energy




-61-


x 1014


1012


LAWSON BREAKEVEN CURVE


Temperature (keV)


Fioure 6.


Tvoical Lawson breakeven curve for a 50-50 D-T plasma and


14
10


13
10




-62-


12 T


l-fl DT(Q


- 4bT1/2


where QB is the blanket energy deposition per fusion neutron and n i


usual overall


system efficiency defined for the Lawson Criterion.


50.69


Obviously


f significant energy is produced in the fissile blanket, the


requisite hybrid plasma parameters can be relaxed to allow earlier utili-


zation of fusion power in


combination with a subcritical fission reactor


to take full advantage of inherent hybrid safety features.

The typical effect of hybrid operation with blanket energy multipli-


cation is a reduction in the required


n product is depicted in Fig. 7.


The production of fission energy effectively reduces the need for fusion-


produced energy.


The hybrid-revised Lawson Criterion of Eq. (18) is


greatly relaxed because QB is on the order of hundreds of l1eV versus the


usual QB used for pure fusion systems which i


s limited to about


0 [IeV


including exothermic blanket reactions.


As noted, this interactive multi-


plication demonstrates the need to examine the dynamics and controllability

of hybrid systems.

Previous studies have been restricted to steady-state neutron


balance calculations and associated technological limitations.


There


has been no analysis of the time-dependent behavior associated with

hybrids, when subjected to reasonable perturbations in the characterizing


parameters


In addition, there have been no reports of analysis of hybrid


plasmas in the reduced reactivity regions where plasmas are not self-


sustaining.


The development of a model to describe the dynamic


S Sl -I I I .4 I r I


nT-


+ Q,)


F





-63-


PREDICTED: MB vs. k ff


0.80


0.85


0.90


0.95


Effective Neutron Multiplication Factor


Figure 7.


Predicted variation of blanket fusion neutron energy
multiplication with blanket effective neutron multiplication




-64-


hybrid system must be established when subjected to effects


due to the thermal instability analyzed by Mills15-17


such as those


and by Ohta et al


for pure fusion plasmas.


The desired result was a hybrid


system model whose analysis


would


yield useful operational characteristic


then enabi


of hybrid machines which could


the hybrid to make a contribution to power production before


the turn of the century


These


various investigations will only be possible


if both the plasma and blanket components are modeled and coupled to allow

dynamics and stability analysis to be performed.



Summary of the Research


The research reported here began with the Ohta plasma model18


burnup effects included after Campbell and Usher


with


and developed plasma


stability criteria based upon source feedrate perturbations and other

engineering considerations for plasma changes affecting the output neutron


production rate.


Essentially, an effort was made to develop an analytical


model for pure fusion plasma


stability and control based on a global


parameter treatment of a linearized fusioning plasma model using concepts


of classical control theory and transfer functions.


Feedback effects


were also incorporated into the model which was kept independent of


specific


design concepts.


The analytical model and its stability pre-


dictions were compared with Ohta 's results to develop an engineering-


oriented mode


which could have broad application to more sophisticated


plasma models in the future.


Perturbations causing plasma transients


r' II .1 ... I I J -




-55-


With the completion of this plasma stability and transfer function


analysis, the effort was extended to develop a


simplified hybrid model


from which general stability criteria were developed for the interacting


components of a hybrid system.


Again, the model was kept independent of


specific hybrid concepts except that the plasma confinement time was


assumed to be a constant


The model was


, independent of plasma temperature and density.


specifically developed and related to engineering con-


iderations of hybrid system perturbations


as well as dynamic


imulation


and control


Inherent as well as artificial feedback effects were in-


corporate where appropriate


The entire effort was directed to develop-


ment of a


simple, linearized


closed-loop model in transfer function


format which could be used for future extensions of this work on dynamic


and stability characteristic


of hybrids.


Of course the nonlinear form


was retained for dynamic


simulations.


The hybrid analytical model was then used to examine the properties


of a particular hybrid system.


The various augean and symbiotic concepts


and variations proposed by Lidsky2and analyzed parametrically in the


Princeton Study


were rejected for this research since they are not


primarily intended for power production.


s left essentially two


choices:

the possib


a fast fission blanket or a thermal fission blanket.


1


need for


To avoid


significant enrichments and to take advantage of


expected higher multiplication factors, a thermal fission concept was


selected.


The most advanced and promising design was reported by PNL


and Livermore workers under Wolkenhauer


This PNL blanket design was based primarily on existing technology




-66-


severely power-limited, the only substantive change for this research wa


the conversion to a Tokamak-driven hybrid versus the mirror-device hybrid

to promote larger power output and allow consideration of thermal in-

stability effects.

Since the physical arrangement of the hybrid blanket selected cor-


responded to the


reported PNL concept as nearly as possible, the results


of oreviousl


performed parametric anal


yses


of optimized region width


ordering of


zones, and region material constituents were used as the basis


for extending steady-state neutronic analysis


Tokamak-driven blanket design used is described


Detailed neutroni


of the blanket.


n Appendi


claculations were performed on the blanket for


selected design whose thermal lattice unit cell enrichment and global


temperature were the only varied parameters.


The cell enrichment was


varied from natural uranium up to 1 .50% enriched while the temperature


was varied from 2900K up to 970K


BRT-16


This work was performed using the


(one thermal group) and PHROG79 (three fast groups) codes to get


4-group constants.


The 4-group CORA diffusion theory code80 was then


used for criticality calculations and acquisition of fundamental flux


shapes.


The doppler defect was also calculated as a function of the


blanket operating temperature.


Only the more promising blankets with


keff


0.90 at elevated temperatures were considered


n detail.


This


limitation minimized blanket dependence on the fusion component of the

hybrid system.

Adjoint and perturbation calculations were performed on the system


to provide parameters to characterize the kinetic property


of the




-67-


source weighting factors,


of the hybrid blanket were calculated using


diffusion theory


Additional


inhomogeneous calculations for blankets driven by planar


sources of group


mate the fi


fast neutrons (10 MeV


ssion energy source size


- 0.821 MeV) were used to approxi-


required to produce a nominal design


power of 6500 MWth.


Volume source calculations were also run to investi-


gate the difference in the worth of the diffusion theory group 1 source

neutron power production depending on the point of introduction into the


blanket


This investigation was accomplished to analyze the validity of


global parameter relationship for the blanket energy deposition per


fusion neutron presented in Eq. (16).


relationship was expected to


yield reasonable


agreement with diffusion theory


simulations since the


source neutrons were introduced at nearly fission spectrum energy


The series of diffusion theory results were used essentially as


scoping


calculations to


select the best enrichment for further, more


detailed and exact


transport theory analysis using the AMPX code


package


available from ORNL


The blanket neutronic analysis


performed


with the XSDRNPM


code82 from AMPX was the first reported application of


the ORNL-developed AMPX package to such hybrid studies.


using the AMPX package


In P2-S4 analysis


, the fusion neutron source energy was treated more


nearly


as a true 14 MeV source.


The required


source strength for pro-


during the 650 MW


design power was determined for the toroidal


to establish finally the applicable degree of validity


expected in cal


culating or predicting the blanket energy deposition per entering fusion


neutron using Eq


(16).


The flux


hapes were also investigated again but




-68-


On the basis of the XSDRNPM-predicted fusion neutron source strength

required for a 6500 MWth hybrid plant, the required plasma conditions were


estimated.


The corresponding plasma temperature, density, constant con-


finement time, source feedrate, and injection energy characteristics were

then parametrically varied to establish reasonable hybrid plasma operating


conditions.


Perturbations in various parameters with emphasis on plasma


feedrate and temperature were then


simulated to investigate the thermal


instability of the hybrid plasma and the results compared with stability


predictions and expected dynamic behavior under transient conditions.


way the plasma component of the hybrid plant was examined with


respect to the thermal instability to establish operational characteristics

necessary for planning proper deployment of hybrid power plants.


Finally


transient phenomena,


time variations

driving the bla


since hybrid plasmas are expected to be subjected to various


especially thermal instability-driven transients,


n the design magnitude of the 14 MeV neutron source


nket were considered on the basis of those transients


resulting from the perturbed behavior of the hybrid plasma examined pre-


viously.


Kinetics calculations representing the effects of plasma-caused


perturbations on the fusion neutron source driving the blanket were run


and changes in power level were examined for one


patial dimension and


six delayed neutron groups


These kineti


calculations were performed


using the space-time kinetics code GAKIN II83


with si


neutron groups


whos


group constants were obtained from the previous XSDRNPM, P2-S4


calculations.


Although no time-dependent feedback effects were examined,


the speed of response of the system was determined for typical transients













CHAPTER


THE PLASMA MODEL



Introduction to the Plasma Model


First generation fusion power plants are expected to utilize the

basic deuterium-tritium (D-T) fuel cycle


2 3
D + T


S-


4He (3.52 MeV) + 1
2He (3.52 PMeV) + On (14.06 MeV


(19)


Because of its large cross section and reactivity, its minimized plasma


temperature requirements and its relatively large energy rel


ease


reaction, no other fuel cycle is given serious consideration for use in


early pure fusion power reactors.

and demonstration fusion power syst


Certainly the near term experimental

ems are expected to use D-T fuel 29,32,84


The United States Department of Energy effort toward implementation of

central station fusion power plants has clearly recognized the superiority


of this fuel cycle in the overall development programs


85-87


Even the utility industry has recognized the need for future choices


in types of power generating


systems and is supporting the effort to


develop fusion reactors using the D-T fuel cyc1


The major magneti


confinement efforts to produce fusion power in other countries have also


been directed toward the D-T fuel cycle.


89,90


Even so, D-T fueled fusion




-70-


The complexity and difficulty involved in achieving fusion power is

amply demonstrated in full scale commercial fusion power plant design


studies.


28-30


Because economic fusion power is such a large


made the


technological challenge, no factor can be dismissed which wil


development proceed more easily.


designs


The one common factor in different


for fusion power plants in a closed, steady-state mode of opera-


tion (Tokamak) has


been the universal selection of the D-T fuel


Hence, although the D-T fuel cycle has the drawback of producing high


energy, penetrating neutrons


serious choi


Mill


, its other advantages make it the only


for fusion fuel for many years.


demonstrated that the fusion reaction rate and fusion power


production are maximized for thermonuclear plasmas which have a 50%


deuterium-50% tritium fuel ion composition.


the most favorable fuel cycle


This 50-50 D-T mixing ratio


for the production of fusion energy.


With thi


cycle, not only i


the demonstration of scientific breakeven


in a


self-sustaining fusioning plasma more easily accomplished but the


steady-state production of net energy in a fusion power plant can be


accomplished at minimized


evel


of plasma particle density, temperature,


and confinement time.


These inherent advantages


of the D-T fuel


cycle in reducing plasma


requirement


have been


illustrated in various analyses of equilibrium


requirements and conditions including those of Lawson in which the


criterion for energy breakeven was first presented.50


of the 50-50 D-T fuel


The superiority


cycle for reaching and maintaining thermonuclear


power-producing conditions has been uniformly demonstrated in extensive




-71-


Because fusion-fission hybrids are expected to serve


as an inter-


mediate energy-producing stepping block between current LWR plants and

the eventual development of pure fusion power, the usual 50-50 D-T fuel


cycle


was logically sel


ected for this hybrid analysis.


This choice was


aimed at optimizing the time


scal


for th


implementation of the hybrid


power-producing concept.


The Point Model Plasma


In this work


, time-dependent point model balance equations were first


established for the plasma ion density, n(t), the plasma energy density,


3n(t)T(t), and the volumetri


plasma neutron production rate, qp(t)


these three balance equations for the plasma ion (particle) density,

temperature, and neutron production rate state variables are presented

as follows:


Plasma Ion (Particle) Density:


dn(t
= S(t)


n(t)


t)DT


Plasma Energy Density:


d[3n(t)T(t)]


n2 (t)DT Q+
=---D + T(t)


3n(t)T(t)
t) -


- bn(t)T1/2(t)


Plasma Volumetric Neutron Production Rate:


n (t)DT


t4-"




-72-


Conventional definitions for symbol


equations are listed below:


n(t)

T(t)


used in these nonlinear point model


-3
= plasma ion density for 50-50 D-T plasma (ions/cm3 )

= plasma temperature (keV)


= external fuel volumetri


injection rate


ions/cm


sec)


= volumetri


Ts(t
7


fusion neutron production rate (#/cm


sec)


= external particle (ion) injection energy (keV)

= particle confinement time (sec)


= energy confinement time


DT


sec)


= completely plasma confined fusion-produced alpha particle
energy (3520 keV)

= D-T fusion reaction rate coefficient (cm3/sec)


= proportionality coefficient for plasma energy loss rate via
Bremsstrahlung radiation (3.36 x 10-15 cm3 keV1/2/sec)18


the plasma in this analysis was treated


as a global system,


only a


single average plasma temperature was


considered


that i


distinction was made between ionic species or between ion and electron


temperatures.


The inclusion of the burnup term in the plasma ion density


equation i


an improvement to the model used by Ohta


corporate by others


23,24,95


18 that has been in-


In stability studies on pure fusion devices


this burnup term and its effects have frequently been neglected because


burnup causes small


changes in the


stable


e temperature operating regimes of


D-T fusion systems.


This analysis was intended for application to a


hybrid system where most of the energy would be produced in the blanket


so burnup predictions were


even lower than in pure fusion devices


that


, plasma temperature and plasma density are both expected to be lower




-73-


burnup increases due to temperature increases


will be directly respon-


sible for lowering neutron yields which are proportional to the square


of the ion density


All of the alpha particle energy produced in fusion


was assumed to be deposited within the


plasma to help heat the system.


Others have


assumed fractional deposition,


but there is no loss of


applicability in assuming full alpha energy deposition.


For this initial analysis


of point model kineti


the plasma volume,


, was treated as a constant; for linearized stability analysis, thi


adequate because only small plasma system perturbations were considered.

For time-dependent, nonlinear analysis energy and neutron production are

overpredicted by the assumption of constant volume since both are pro-


portional to the square of the ion density.


More detailed anal


yses


the future will incorporate temperature-dependent as well as magnetic


and other dynamic


conditions that can affect the volume occupied by the


plasma independent of whether the plasma density has changed.


liminary global analy


Some pre-


of such plasma volume variations have been


19-21,96
reported for pure fusion models


and additional work is under-


97,98The analysis was directed ultimately to the kinetic behavior


of the hybrid so the inclusion of the added complication of a variable

plasma volume in this initial treatment of the plasma neutron source

driving a power-producing blanket was not justified.


The inherent behavior and characteristic


of the point model fusion-


ing plasma used for analysis


in thi


study i


completely described by


Eq. (20), Eq. (21), and Eq.


In fact


, the plasma response to any


input perturbation


as well as its equilibrium characteristic




-74-


with the driven nature of the hybrid subcritical blanket, the third

equation for the specific neutron production rate was also necessary;

without neutrons produced in and hence output from the plasma, no inter-


action i


possib


between the two component halves of the hybrid system.


Note that these neutrons are produced in the plasma and inherently drive


the blanket; however, there i


plasma is affected by the neutrons themsel


no inherent reverse effect whereby the


or by the blanket itself.


The neutrons and their effects are strictly feedforward in nature.


The volumetric neutron production rate


q (t), is an intrinsic


variable--characteristic of the condition of the plasma represented by


the state variabi


of ion density and temperature only.


The volumetric


neutron production rate was multiplied by the effective plasma volume,


V to obtain the


total plasma neutron production rate, Q (t


follows:


Qp(t)


= q (t)
'p


p (23)


where the total neutron production rate is an extrinsic variable charac-


teristic of a


specific plasma with constant effect


volume, V


other words, Q (t) is


characteristic not of all plasmas in a state de-


scribed by an ion density and a temperature


but only of those specific


plasmas whose


volumes satisfy Eq. (23)


This extrinsic variable could


useful for relating specific


blanket; however, for this genera


plasmas to the corresponding hybrid


development, the volumetric neutron


production rate was more useful


since it is the intrinsic variabi


from


which any specific pure fusion or hybrid plasma can be analyzed.


Indeed,




-75-


constant volume will be simply multiplicative--the larger the plasma,

the greater the system power production.

The density equation was rewritten in the following simplified form:


dn(t)= S(t)
=S(t)


- n(t) f2
nf (T)n (t
n


where the temperature-dependent coefficient, fl(T), was defined as follows


to simplify the burnup


term:


f,(T)


Similarly, Eq


DT
2


1) for the plasma energy density was also simplified


preparatory to linearization by rewriting it in the following form after


Ohta:


d[n(t)T(t)]
dt


f,(T)n


n(t)T(t) +
fE


TsS(t)


where the temperature-dependent coefficient, f2(T), was used to account

for charged alpha particle heating and bremmstrahlung radiation,


respectively


CclV>D T~c


fo(T)


(27)


bT32
3


Although it was not so complicated, the equation for the volumetric

neutron production rate was also redefined as follows:


qn(t)


= g(T)n2(t)


(28)


v




-76-


It is noteworthy that g(T) in the neutron production equation and


fl(T) in the burnup term of the particle


a factor of two (2)


density equation differ only by


follows:


= 2fl(T)


which simply means that two (2) ions must undergo fusion burnup for each

neutron produced.


The Linearized Plasma Mode


The global plasma equations were linearlized in order to facilitate


analysis of stability regimes in the frequency domain.


At this point


contrary to previous work


3,24,95


specific perturbations were intro-


duced into the point model plasma equations


Since the feedrate,


is the only external influence appearing in both the density and tem-

perature point model equations, the feedrate was chosen as the typical


source perturbation for the


choice was logical


examination of global plasma stability


the driving force for the entire fusioning energy


producer is ultimately supplied by


the plasma feedrate.


The same depen-


dence on feedrate is applicable for the hybrid


stem, since the hybrid


will be entirely dependent for energy production on the plasma-produced


neutrons because of the blanket subcriticality.

neutrons is ultimately governed by the state of


But the production of


' the plasma (ion density


and temperature) which itself i


driven and sustained by the


feedrate of


energetic fuel ions.


Therefore, examination of the hybrid system response


.. -~ L. -.1
fl 5" J*% .n S in L.. 5 S La S -.
j**. ri .,. S. -. n .. -d 4- 1 -


S(t),




-77-


For inherent stability and control analyses, the system response to


small external or internal perturbations was a primary concern.


Depen-


dent variable perturbations about steady-state values were used to generate


a dynamic variation in the point model equations.


the following necessarily small variabi


For linear analysis


perturbations were used:


n(t)


T(t)


+ 6n(t)


(31a)


+ 6T(t)


(31b)


+ 6S(t)


(3)


qp(t)


= qPo


+ 6qp(t)


(31d)


where the subscript "o" was used to designate a system variable at an


initial


steady-state equilibrium value about which a small perturbation


in the variable, represented by


6-terms was


introduced so the system could


subsequently examined for stability in linearized form.


In other words,


the time-dependent arbitrary perturbations in ion density, 6n(t), plasma


temperature, 6T(t)


source feedrate


(t), and volumetric neutron pro-


duction rate


6qp,


were required to be small to validate the lineariza-


tion of the point model equations.


These perturbed variable


were sub-


stituted into the point model dynamics equations along with first order

linear expansions for all the density- and temperature-dependent coeffi-


clients in these equations.


The objective was to obtain linearized


perturbed equations from the three original nonlinear plasma dynamics

equations for the plasma ion density, temperature, and volumetric neutron




-78-


The inverse confinement time coefficients were examined using pro-


cedures from previous analyses of global plasma behavior.


15,16,23,24


Both the particle and energy confinement times were assumed to depend


exclusively


y on the plasma ion density and temperature state variabi


= F(n,T)


Therefore


, fo


owing the


example of Ohta


both inverse confinement times


were


expanded about an initial steady-state in first order Taylor series


in the dependent density and temperature variables as follows:


Tn(t)


5T(t)


6n(t) 1 6T(t)
n T, T


(33a)


1~


1
-~ -


The constants 1/1,nl


1

6n(t) + E
0


6T(t)


6n(t) + 1 6T(t)
n E,, T


ITT,


, 1/nl, ,and


(33b)


/rT1 were used to reduce


complexity of analytical manipulations.


to denote quantiti


The subscript "o" was used again


in an initial steady-state condition about which the


system was somehow to be perturbed and subsequently examined for stability


- +


*1


1
T




-79-


Each of the other three temperature-dependent coefficients (no


density dependence) was also


From the particle


expanded


expanded in a simple linear Taylor series.


equation the temperature-dependent coefficient was


as follows:


6T(t)


The temperature-dependent bremsstrahlung and alpha heating coefficient


in the energy density equation was expanded


af2(T)


fjTi


Finally


similarly:


6T(t)


, the temperature-dependent coefficient in the equation for the


volumetric neutron production rate becomes:


g(T)


= g(To) + aT)
yv 4


6T(t)


(36)


A linearized system of plasma equations can now be produced by substitu-

tion of the perturbed variables from Eq. (33) and the various coefficient

expansions into the plasma model equations.

Substitution of the first order variable and coefficient expansions


into the plasma particle


equation yields the following equation:


+ 6S(


- [+
1
C +
n
0


,..r\


n( t) 1 i 6T(t)
n -- ---]no


+ an(t)]


dan(T)
dt


fl


f2(T~)+




-80-


The steady-state condition in the expanded particle equation was eliminated

using the steady-state equilibrium condition.

Two additional coefficients were defined from the effect of including


burnup via fl (T) in thi


model


specifically, the


effective confinement


time for particle


burnup effects.


due to any 1


Certainly


mechanisms is reduced by including


, fusion of particles is a loss mechanism.


subscript "b" was used in defining inverse confinement time terms to

account for the increased loss of plasma ions by fusion as follows:


S= 2n f
T. o0 1


af, (T


= n T
o o


(39)


By eliminating the steady-state solution and neglecting all terms above

the first order in the perturbed variables and coefficients, the following

linearized equation for the perturbed plasma ion density was obtained:


d6n(t)
dt


6S(t) 1_ +
n T


+ 1 n(t) 1 6T(t)
b, o T1 b, o


.(40)


The inclusion of burnup results in an additional burnup-dependent

inverse confinement time term as well as the usual density- and


temperature-perturbed terms. This dual effect fc

the dependence of burnup on both state variables.


allows directly from


Note that the addition


of inverse confinement time terms results in lowered overall particle




Full Text

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INGEST IEID EHQTFQPW2_EADT6B INGEST_TIME 2011-08-09T16:44:35Z PACKAGE AA00002202_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES



STEADY-STATE AND TIME-DEPENDENT BEHAVIOR OF
FUSION-FISSION HYBRID SYSTEMS
By
HILL IAM G. VERNETSON
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

Dedicated to
Theresa
without whom this work
would have been impossible.

ACKNOWLEDGMENTS
The author would like to express his appreciation to his graduate
committee for their assistance during the course of this research. Special
thanks are extended to Dr. H. D. Campbell, chairman of the author's super¬
visory committee for providing guidance and encouragement throughout the
course of this work. Dr. Campbell's many helpful comments and suggestions
have greatly aided the completion of this work. Thanks are also extended
to Dr. E. E. Carroll, Dr. R. T. Schneider, and Dr. T. L. Bailey who have
also served on the author's supervisory committee.
Special thanks are extended to Dr. M. J. Ohanian for the research and
teaching assistantship opportunities presented which enabled the author
to pursue the doctorate.
The author's studies at the University of Florida have been supported,
in part, by a National Science Foundation Traineeship and also by a one-
year Fellowship from the University of Florida and this support is grate¬
fully acknowledged.
A large portion of the funds for the computer analysis were furnished
by the Northeast Regional Data Center on the University of Florida campus
through the College of Engineering. This help, though at times meager and
difficult to obtain, is also acknowledged.
Special thanks are due to Dr. N. J. Diaz without whose efforts and
encouragement this study might never have been completed.

Special thanks are also due to Dr. E. T. Dugan whose knowledge of
computer analysis and nuclear reactor physics was of great assistance
during much of this work. In addition, thanks are extended to Mr. I.
Maya for his aid with some of the plasma calculations and their impli¬
cations. Thanks are also extended to Mr. B. G. Schnitzler for a number
of helpful consultations.
Finally, the author would like to extend his deepest appreciation to
his wife whose support and encouragement made it possible to complete this
work.

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES xi
ABSTRACT xviii
CHAPTER
1 INTRODUCTION 1
Preliminary Concepts for Fusion-Fission Reactors 1
Review of Fusion Blanket Studies 5
Critical Review of Hybrid Blanket Studies 19
Review of Controlled Thermonuclear Reactor Thermal
Stability Analyses 40
Motivation for the Research 56
Summary of the Research 64
2 THE PLASMA MODEL 69
Introduction to the Plasma Model 69
The Point-Model Plasma 71
The Linearized Plasma Model 76
Transfer Function Representation of Plasma Characteristics . 84
Stability Analysis of the Linearized Plasma Model 95
3 A HYBRID REACTOR ANALYTICAL MODEL 116
Development of the Hybrid Model 116
The Linearized Hybrid Model 131
Incorporation of Feedback Effects into the Hybrid Model. . . 138
Nonlinear and Linearized Hybrid Model Summary 143
Transfer Function Representation of the Hybrid 147
Stability Criteria for the Hybrid System 160
4 HYBRID PLASMA OPERATIONAL CONSIDERATIONS 164
Introduction to Hybrid Plasma Time-Dependent Behavior. . . . 164
Selecting a Spectrum of Hybrid Plasma Equilibrium
Conditions 170
v

Page
Uncontrolled Plasma Response to Perturbations 181
Predicted Stability Versus Point-Model Response 196
Short-Term Plasma Transient Response 202
Plasma Response With Feedback 219
5 HYBRID BLANKET ANALYSIS 258
Introduction 258
Blanket Calculations Using Diffusion Theory 261
Inhomogeneous Diffusion Theory Calculations 309
Kinetic Parameters 313
Transport Theory Calculations 322
Inhomogeneous Transport Theory Calculations 342
Time-Dependent Blanket Considerations 348
6 CONCLUDING COMMENTS 356
Discussion and Conclusions 356
Suggestions for Further Work 361
APPENDICES
A GLOBAL BLANKET ENERGY MULTIPLICATION 366
B HYBRID SYSTEM PHYSICAL CHARACTERISTICS 369
C BURNUP AND SENSITIVITY CONSIDERATIONS FOR THE HYBRID
PLASMA 388
D COMPUTER CODE DESCRIPTIONS 396
REFERENCES 411
BIOGRAPHICAL SKETCH 423
VI

LIST OF TABLES
Table Page
1-1 Fusion Reaction Parameters 7
I -11 Dependence of Tritium Breeding Ratios and Energy
Deposition Rates for Lee's Fusion Blankets 12
1-111 Summary Descriptions of ORNL Optimistic and Conservative
Blanket Designs 13
1 -1V Summary of Steiner's Tritium Breeding Calculations Per
Incident 14 MeV Neutron 14
1-V Neutron Economy of Lidsky's Hybrid Blanket 24
1 -VI Lidsky's Hybrid Reactor Parameters 25
1 -V11 Lee's Neutron Balance in Infinite Media 26
1-VIII Subcritical Fast Fission Blanket Components Studied by
Lee 27
1 -1X Fast Fission Hybrid Neutron Economy Per 14 MeV Neutron
Calculated by Lee 28
1-X Neutron Economy for Thorium-Fueled Blankets 31
1 -XI Neutron Economy for Uranium-Fueled Blankets 31
1 - X11 Comparison of Best Natural Uranium-Fueled and Extrapo¬
lated Thorium-Fueled Blankets 32
1-XIII Early PNL Hybrid Neutron Balance 36
1-XIV Early PNL Hybrid Specifications 37
1 -XV PNL Hybrid Blanket Analysis 39
1-XVI Critical Temperatures for D-T Fusion Reactors 52
1 - XV11 Predicted Blanket Global Response per 14 MeV Neutron. . . 59
2-1 Stability Criteria for a D-T Fusion Reactor 106

Tabl e Page
4-1 Selected Spectrum of Equilibrium Operating Conditions
for the Hybrid Plasma With Constant Confinement 173
4-II Hybrid Plasma Equilibrium Operating Conditions for x¡r
= 1.7 sec to Meet Required Power Production °. . 175
4-111 Equilibrium Plasma Conditions Selected for Transient
Analysis With R = 2 and qpQ = 1.41 x 10^ nts/cm3-sec. . 178
4-IV Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a +5% Perturbation in the Temperature 187
4-V Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a -5% Perturbation in the Temperature 183
4-VI Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Increase in the Steady-State
Source Feedrate 189
4-V11 Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Decrease in the Steady-State
Source Feedrate 190
4-V III Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a +5% Perturbation in the Ion Density 191
4-1X Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a -5% Perturbation in the Ion Density 192
4-X Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Increase in the Steady-State
Injection Energy 193
4-XI Final Uncontrolled Hybrid Plasma Equilibrium Conditions
Following a 5% Step Decrease in the Steady-State
Injection Energy 194
4-X11 Summary of Predicted Stabilization Requirements for
Instantaneous Temperature Feedback on the Feedrate . . . 197
4-XIII Comparison of Confinement Time Effects on Plasma
Temperature at 10 sec Following +5% Perturbation in
Feedrate versus +5% Perturbation in Temperature for
Six Hypothetical Hybrid Equilibrium States 218
5-1 Boundaries for Four-Group Criticality Calculation. . . . 263
5-II BRT-1 Cell-Smeared Thermal Constants for 1.35%
Enriched Fuel 264
5-111 Graphite Moderator Region Scattering Properties 268
v i i i

Table Page
5-IV Flux Depression Factors for the 1.35% Enriched
Lattices 268
5-V Average Cross Sections for and in the Fuel
Column 269
5-VI Space Point Placement for BRT-1 Calculation Over Inner
Half of the Hybrid Blanket 273
5-V11 Space Point Placement for BRT-1 Calculation Over Outer
Half of the Fission Lattice and Into the Graphite
Reflector 274
5-VIII Summary of PHROG Calculations by Region 280
5-1X Resonance Region Scattering Cross Sections for Blanket
Nuclides 281
5-X Four-Group, 13 Region Constants for 1.35% Enrichment
at 570°K from BRT-1 and PHROG 283
5-XI PHROG-Generated Macroscopic Downscattering Cross
Sections for 1.35% Enrichment, 570°K, and 13 Regions. . 286
5-XII Four-Group, 13-Region Constants for 1.35% Enrichment
at 970°K from BRT-1 and PHROG 287
5-XI11 PHROG-Generated Macroscopic Downscattering Cross
Sections for 1.35% Enrichment, 970°K, and 13 Regions. . 290
5-XIV Results of Diffusion Theory Criticality Calculations. . 292
5-XV Summary of Inhomogeneous CORA Calculations for
Variations in Enrichment and Temperature 305
5-XVI Yield Fractions for Six Delayed Neutron Precursor
Groups 315
5-XV11 Delayed Neutron Energy Spectrum Yield Fractions for
4-Group CORA Calculations 315
5-XVIII Blanket Kinetic Parameters 316
5-XIX Source Weighting Factors in Four Groups and Ten
Regions 318
5-XX Effectiveness of Uniform Volume Sources for Design
Power Level 321
5-XXI AMPX Master Library 123-Group Energy Boundaries .... 324
5-XX 11 Nuclides Selected from the AMPX Library 325

Table Page
5-XXIII Effective Moderator Scattering Cross Sections Per
Absorber Atom 327
5-XXIV Isotopic Resonance Integral Values Obtained from
NITAWL 328
5-XXV Hybrid Blanket Analysis S^ Quadrature Constants 330
5-XXVI XSDRNPM 43-Group Energy Boundaries 333
5-XXV11 XSDRNPM 26-Group and 11-Group Energy Boundaries 335
5-XXVIII XSDRNPM 6-Group Cross Section Energy Boundaries 337
5-XXIX XSDRNPM keff Results for a Zero-Flux Boundary Con¬
dition at tne Vacuum Wall 337
5-XXX Transmission Ratio for 14 MeV Neutrons Through the
Hybrid Blanket 346
B-I Hybrid Blanket Equivalent Unit Cell Geometry 375
B-II Fuel Column Spherical Microparticle Design Parameters. . 377
B-III Temperature-Independent Fuel-Pin-Averaged Nuclide
Number Density Variation with Enrichment 378
B-IV Hybrid Blanket Shield Composition 380
B-V Helium and Natural Lithium Number Density Variation
with Temperature 381
B-VI Effects of Vacuum Wall Radius on Blanket Power
Requirements and Power Density 384
C-I Point-Model Comparison of Confinement Times and
Related Plasma Parameters in UWMAK-III and the Hybrid
Plasma 390
x

LIST OF FIGURES
Figure Page
1. The essential components of a Tokamak fusion reactor .... 9
2. Comparison of spatially-dependent heating rates for vacuum
wall regions in two designs 15
3. Early PNL fusion-fission hybrid subcritical blanket
configuration 35
4. Comparison of Lawson breakeven and plasma equilibrium
regions 45
5. Time variation of point-model plasma temperature and
density for constant confinement and charged particle
heating 53
6. Typical Lawson breakeven curve for a 50-50 D-T plasma and
33% overall efficiency showing relative position of hybrid
systems 61
7. Predicted variation of blanket fusion neutron energy
multiplication with blanket effective neutron multi¬
plication factor 63
3. Transfer function formulation for a point-model fusioning
plasma 85
9. Block diagram for the point-model plasma system 87
10. Partially-reduced block diagram for the point-model
plasma system 91
11. Alternate block diagram for the point-model plasma system. . 92
12. Partially-reduced block diagram for the alternate point-
model plasma system formulation 93
13. Reduced open-loop block diagram for the point-model plasma . 94
14. Routh array for open-loop point model fusioning plasma
with burnup 97
15. Routh array: Open-loop plasma model with constant
confinement 100
XI

Figure Page
16. Variation of F(T) = m with temperature 105
17. Block diagram for the point-model plasma with temperature
feedback to the feedrate 110
18. Block diagram schematic for point-model blanket kinetics
retaining both source and reactivity perturbations 152
19. Block diagram of the linearized global fusion-fission
hybrid reactor model 154
20. Parti ally-reduced hybrid block diagram with no artificial
feedback 156
21. Simplified reduced hybrid system block diagram with no
artificial feedback 157
22. Closed-loop block diagram for the linearized point-model
plasma with temperature feedback to the feedrate 159
23. Routh array for the cubic denominator for blanket effects
in the overall hybrid transfer function 161
24. Equilibrium curves for various equilibrium plasma
conditions 174
25. Mills steady-state curves including burnup for R = 2 180
26. Illustration of the feedforward effectiveness of the source
feedrate and the injection energy on plasma equilibrium
conditions and transient behavior 184
27. Arbitrary equilibrium curve with a hypothetical stable
hybrid state at point A plus a possible equilibrium curve
containing a perturbed unstable state at point B 200
28. Variation of plasma temperature following a 5% step increase
in the temperature of the six hypothetical hybrid equilib¬
rium states 204
29. Variation of plasma volumetric neutron production rate
following a 5% step increase in the temperature of the six
hypothetical hybrid equilibrium states 205
30. Variation of plasma temperature following a 5% step decrease
in the temperature of the six hypothetical hybrid equilib¬
rium states 205
31. Variation of plasma volumetric neutron production rate
following a 5% step decrease in the temperature of the six
hypothetical hybrid equilibrium states 207

Figure Page
32. Variation of the heating rate in the first wall region of
the UMAK-III design 211
33. Variation of plasma temperature following a 5% step
increase in the feedrates for the six hypothetical hybrid
equilibrium states 214
34. Variation of plasma volumetric neutron production rate
following a 5% step increase in the feedrates of the six
hypothetical hybrid equilibrium states 215
35. Variation of plasma temperature following a 5% step
decrease in the feedrates for the six hypothetical hybrid
equilibrium states 216
36. Variation of plasma volumetric neutron production rate
following a 5% step decrease in the feedrates for the six
hypothetical hybrid equilibrium states 217
37. Variation of plasma temperature following a 5% step
increase in the feedrate of an equilibrium state at t£ =
1.5 sec with delayed shutoff times °. . . 220
38. Variation of plasma volumetric neutron production rate
following a 5% step increase in the feedrate of an
equilibrium state at xr =1.5 sec with delayed shutoff
times 9 221
39. Variation of plasma temperature following a 5% step
increase in the feedrate of an equilibrium state at x^ =
1.7 sec with delayed shutoff times °. . . 222
40. Variation of plasma volumetric neutron production rate
following a 5% step increase in the feedrate of an
equilibrium state at xF =1.7 sec with delayed shutoff
times 9 223
41. Variation of plasma temperature following a 5% step
increase in the feedrate of an equilibrium state at x^ =
2.0 sec with delayed shutoff times °. . . 224
42. Variation of plasma volumetric neutron production rate
following a 5% step increase in the feedrate of an
equilibrium state at x£ =2.0 sec with delayed shutoff
times 9 225
43. Variation of plasma temperature following a 5% step
decrease in the feedrate of an equilibrium state at x£ =
1.5 sec with delayed shutoff times °. . . 226
44. Variation of plasma volumetric neutron production rate
following a 5% step decrease in the feedrate of an
equilibrium state at x¡r =1.5 sec with delayed shutoff
xi i i
227

Figure Page
45. Variation of plasma temperature following a 5% step
decrease in the feedrate of an equilibrium state at xE =
1.7sec with delayed shutoff times °. . . 228
46. Variation of plasma volumetric neutron production rate
following a 51 step decrease in the feedrate of an
equilibrium state at te =1.7 sec with delayed shutoff
times 9 229
47.Variation of plasma temperature following a 5% step
decrease in the feedrate of an equilibrium state at x^
2.0 sec with delayed shutoff times °.
230
48.Variation of plasma volumetric neutron production rate
following a 5% step decrease in the feedrate of an
equilibrium state at = 2.0 sec with delayed shutoff
times 9
231
49. Variation of plasma temperature with timoerature feedback
following a 5% step increase in the feedrate of the x¡r =
1.5 sec equilibrium state plus delayed shutoff of 6S .°. . . 234
50. Variation of plasma temperature with temperature feedback
following a 5% step increase in the feedrate of the x-r =
1.7sec equilibrium state plus delayed shutoff of 6S .°. . . 235
51. Variation of plasma temperature with temperature feedback
following a 5l step increase in the feedrate of the x£ =
2.0 sec equilibrium state plus delayed shutoff of 6S .°. . . 236
52. Variation of plasma temperature with temperature feedback
following a 5% step decrease in the feedrate of the xc =
1.5 sec equilibrium state plus delayed shutoff of 6S .°. . . 237
53. Variation of plasma temperature with temperature feedback
following a 5% step decrease in the feedrate of the x^ =
1.7sec equilibrium state plus delayed shutoff of 6S .°. . . 238
54. Variation of plasma temperature with temperature feedback
following a 5% step decrease in the feedrate of the x^ =
2.0 sec equilibrium state plus delayed shutoff of 6S .°. .
55. Variation of plasma temperature with temperature feedback
56.
following a 5% step
1.5 sec equilibrium
decrease
state. .
in the temperature of the x^
Variation of plasma temperature with temperature feedback
following a 5% step increase in the temperature of the xj:
1.7sec equilibrium state. ....
239
242
243
XT V

Figure Page
57. Variation of plasma temperature with temperature feedback
following a 5% step increase in the temperature of the x¡r =
2.0 sec equilibrium state °. . 244
58. Variation of plasma temperature with temperature feedback
following a 5% step decrease in the temperature of the x¡r =
1.5 sec equilibrium state °. . 245
59. Variation of plasma temperature with temperature feedback
following a 5% step decrease in the temperature of the x£ =
1.7 sec equilibrium state ? . 246
60. Variation of plasma temperature with temperature feedback
following a 5" step decrease in the temperature of the x¡r =
2.0 sec equilibrium state ? . 247
61. Variation of plasma volumetric neutron production rate with
temperature feedback following a 5% step increase in
temperature of the x¡rQ = 1.5 sec equilibrium state 252
62. Variation of plasma volumetric neutron production rate with
temperature feedback following a 5% step increase in
temperature of the xF =1.7 sec equilibrium state 253
Lo
63. Variation of plasma volumetric neutron production rate with
temperature feedback following a 5% step increase in
temperature of the xr =2.0 sec equilibrium state 254
o
64. Variation of plasma volumetric neutron production rate with
temperature feedback following a 5% step decrease in
temperature of the xF =1.5 sec equilibrium state 255
o
65. Variation of plasma volumetric neutron production rate with
temperature feedback following a 5% step decrease in
temperature of the x£q =1.7 sec equilibrium state 256
66. Variation of plasma volumetric neutron production rate with
temperature feedback following a 5% step decrease in
temperature of the xr = 2.0 sec equilibrium state 257
o
67. BRT-1 thermal flux profiles across the equivalent unit cell
for 1.35% enrichment at 290°K, 570°K, and 970°K 265
68. Typical paths for an unscattered neutron in an equivalent
unit cell and an actual unit cell of a nuclear reactor . . . 267
69. BRT-1 thermal flux profiles across the inner half of the
hybrid blanket for 1.35% enrichment at 290°K, 570°K, and
970°K with zero-flux vacuum wall boundary condition 272
xv

Ü3.ure Page
70. BRT-1 thermal flux profiles across the outer half of the
fission lattice out to 12 cm of graphite reflector for
1.35% enrichment and 290°K, 570°K, and 970°K 275
71. Thermal flux profiles from BRT-1 calculations across the
outer 18 cm of graphite reflector and 30 cm of shield for
290°K, 570°K, and 970°K 277
72. Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 290°K with zero-current vacuum
wall boundary condition 293
73. Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 570°K with zero-current vacuum
wall boundary condition 294
74. Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 970°K with zero-current vacuum
wall boundary condition 295
75. Four-group fundamental mode flux profiles from CORA for
the 1.35% enrichment at 970°K with zero-flux facuum wall
boundary condition 296
76. Variation of blanket effective neutron multiplication
factor with temperature for the 1.35% enrichment using
four-group diffusion theory 299
77. Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.35% enrichment at 570°K 307
78. Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.35% enrichment at 970°K 308
79. Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.50% enrichment at 570°K 309
80. Four-group flux profiles from inhomogeneous CORA run
using group 1 surface source to generate 6500 MWth for
1.50% enrichment at 970°K 310
81. Blanket power density variation for 6500 MWth for 1.35%
enrichment at 570°K and 970°K 311
82. Blanket power density variation for 6500 MWth for 1.50%
enrichment at 570°K and 970°K 312
83. Six-group fundamental mode flux profiles from XSDRNPM for
1.35% enrichment at 900°K with zero-flux vacuum wall
boundary condition 339
xvi

Page
Figure
84. Six-group fundamental mode flux profiles from XSDRNPM for
1.35% enrichment at 900°K with zero-flux vacuum wall
boundary condition 341
85. Six-group flux profiles for a surface source of 1.336 x
10'3 nts/cm -sec in group 1 to generate 6500 MWth at 1.35%
enrichment and 900°K 345
86. Fractional transmission of 14 MeV neutrons through the
hybrid blanket 347
87. Power transient in the hybrid blanket following a 5% step
increase in the neutron source for a forced-criti cal
system 350
88. Hybrid blanket power transient derived for a subcritical
system 352
B1. Conceptual Tokamak fusion-fission hybrid reactor system. . . 370
B2. Overall hybrid blanket slab geometry used in neutronics
calculations 371
B3. Selected PNL hybrid blanket module geometry for Tokamak
fusion-fission hybrid 374
B4. Hybrid thermal fission lattice unit cell 376
B5. Geometric arrangement of the inner convertor with inner
breeder and outer breeder 379
Cl. Reactivity and sensitivity variation with temperature for
the D-T fusion reaction 395
xvi i

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
STEADY-STATE AND TIME-DEPENDENT BEHAVIOR OF
FUSION-FISSION HYBRID SYSTEMS
By
William G. Vernetson
June 1979
Chairman: Hugh D. Campbell
Major Department: Nuclear Engineering Sciences
This study examined stability analysis of point-model systems repre¬
senting pure fusioning plasmas as well as coupled fusion-fission systems.
The stability criteria for these systems viere derived for constant plasma
confinement conditions based on engineering perturbations of the system
feedrate. The results of linearized point-model plasma stability analysis
of the thermal instability were shown to be applicable to hybrid plasmas
and to be attainable from considerations of engineering-related per¬
turbations in the extrinsic plasma feedrate variable.
A Tokamak fusion-fission hybrid design was selected for further,
more specific analysis. The modeled hybrid system in linearized form was
found to be stable provided certain hybrid plasma temperature and con¬
finement time limits are met. However, for realistic installations,
absolute stability is not sufficient; nor is it guaranteed by linearized
analysis. Therefore, hybrid plasma behavior was examined under transient
and overpower conditions.
Time-dependent analysis of a low reactivity hybrid plasma (8 keV,
13 3
9.55 x 10 ions/cm ) subjected to various parameter perturbations showed
the resultant transients to be less quickly developing than those associated
xvi 11

with perturbations to pure fusion plasmas with high plasma reactivity.
In addition, the predictions of plasma stability ranges were verified
for various confinement times. The slowly developing hybrid plasma
transients following - 5% temperature or feedrate perturbations were
found to be significant for the control of the power-producing hybrid.
Neutrons and their associated energy are multiplied in hybrid
blankets; therefore, the global equation in use to relate the blanket
energy deposition per fusion neutron to the blanket effective neutron
multiplication factor was investigated. Results were obtained which
indicate the global approach supplies a poor estimate of blanket energy
multiplication for a fusion neutron source and an even poorer estimate
for fission energy neutrons.
Although results showed the blanket energy deposition per fusion
neutron to be some 60% below point-model predictions, the selected blanket
is still a significant multiplier, by a factor of 25 or more, of the
neutron energy entering the blanket via fusion neutrons. The documenta¬
tion of the reduced worth of fusion neutrons, entering the blanket through
a convertor region, may be a significant factor in redesigning vacuum
walls of hybrid reactors despite the advantages of reduced 14 MeV wall
1oadings.
Diffusion theory and discrete ordinates transport theory analysis
were both applied to establish the relative importance of the inner con¬
vertor region for power generation. The results of the Sn transport
calculation were used to determine the source size required by volume
equivalence with the Tokamak geometry to produce 6500 MWth in the blanket.
The source value was used to establish the steady-state requirements on
plasma temperature and density from geometric considerations of the
xix

Tokamak hybrid plasma volume involved. In addition, the Sn calculation
was used to show that only about 61 of the 14 MeV fusion neutrons reach
the thermal fission lattice without a collision. These transmission
results indicate graphically why the blanket is less effective at energy
multiplication than expected from previous reports.
Finally, space-time kinetics calculations were performed on the
blanket to demonstrate the fast response of the blanket in keeping with
its millisecond prompt neutron lifetimes and subcriticality. Although
no time-dependent feedback effects were examined, the speed of response
of the system was determined for typical transients and some character¬
istics for hybrid operational controllability were established.
xx

CHAPTER 1
INTRODUCTION
Preliminary Concepts for Fusion-Fission Reactors
The fusion-fission hybrid reactor concept is a combination of a
sub-Lawson fusion reactor and a subcritical fission reactor in a single
power-producing system. Fission reactors are "power rich" but "neutron
poor," while anticipated D-T fusion reactors will be "neutron rich" but
"power poor." Hence, the essential hybrid feature is the combination of
these two systems to use excess fusion neutrons to breed fissile fuel
while simultaneously sustaining and driving the system for useful power
using fission energy multiplication of the fusion neutron source
1
e ne rgv.
Limited studies, concentrating on blanket neutronics, have been
done on hybrid systems in parallel with pure fusion blanket work; how¬
ever, no system dynamics or stability investigations have been reported
for hybrids. Some research effort has been devoted to global stability
analysis of the plasma in pure fusion devices. The present research
extends such pure fusion time-dependent studies into the area of hybrid
systems. This continued development of the hybrid in parallel with the
fast breeder reactor is supported by the hybrid's potential as an
2
alternate and attainable energy and fuel producing concept. In fact,
some researchers suggest replacing the breeder reactor with the hybrid
+ 3
concept.
-1-

-2-
Much research effort and capital investment have been committed to
the realization of a mixed burner-breeder nuclear reactor economy planned
for the end of this century. This effort is justified by expected con¬
tinued growth of energy needs, and by a marked shift from direct con¬
sumption of fossil fuels to secondary consumption of electrical energy
4
within the past few decades.
With the growth in nuclear generating capacity, limited fissile fuel
reserves have caused the thrust of research and development in the
nuclear industry to shift to the fast breeder reactor (LI1FBR). Even with
the projected impact of the commercial LMFBR sometime after 1990, con¬
siderable additional enriching capacity and capital investment will be
required for fueling burner reactors.
Current emphasis on the safety and the environmental impact of
nuclear generating facilities as well as certain technological and
5
political objections make it increasingly unlikely that high gain breeder
reactors will make a significant impact prior to the mid-1990's or later.
Even if the breeder is introduced sooner, the relatively long doubling
times under consideration (15 years or more) may not be adequate for
generation of sufficient additional fuel to support an existing burner
r
reactor economy.0 With so much effort and capital investment committed
to the realization of the mixed burner-breeder economy planned for the
1990's, the availability of an effective alternate concept to produce
fissile fuel could be important.
One candidate for producing fissile fuel is the controlled thermo¬
nuclear reactor utilizing the D-T cycle. Deuterium resources are virtu-
9 7
ally unlimited (enough for 10J years).7 Since tritium can be bred from
7 8
lithium at rates resulting in less than one month doubling times, ’

-3-
fusio.n neutrons can be used to breed fissile material. By diverting
neutrons from tritium production, the tritium supply can be maintained
at a reasonable level while fertile materials ( U and Th) are con¬
verted to fissile reactor fuel. Unfortunately the realization of pure
fusion power is too far removed and uncertain to be counted upon to pro¬
duce fissile fuel in the near term.
The alternative concept currently receiving renewed attention is the
coupled fusion-fission hybrid system combining a less than self-sustaining
(energy) fusion reactor with a subcritical but power producing fission
reactor. Although achievement of pure fusion power is not yet possible,
recent advances indicate the plasma requirements for hybrids will be
reached while the fission power component of the electrical economy is
g
still increasing. Then, as an alternative to the LMFBR for fissile fuel
and power production, the hybrid can be very useful.
The hybrid concept has many potential advantages over the LMFBR for
providing power and fissi 1 e fuel in the latter part of this century.
First, the hybrid reactor possesses great potential as a breeder of
fissile fuel. With its abundant supply of neutrons, the hybrid should
be able to produce fissile material more rapidly than any of the current
2 3
breeder reactor concepts to keep pace with power requirements. ’
Second, the hybrid makes an alternative fuel cycle available for
238 239
existing burner reactors. Reliance on the U- Pu fuel cycle with its
232 233
weapons grade plutonium can be reduced in favor of the Th- U fuel
cycle.
Third, hybrid development allows early introduction of fusion
reactors while protecting the large capital investment represented in
operating thermal reactors. By using fission blanket power to drive the

-4-
hybrid system, current advanced reactor technology would require only
modest extensions to produce a hybrid system as a natural link in the
1 3 4
development leading from pure fission to pure fusion power. ’ ’
Finally, the hybrid concept using subcritical blankets is attractive
from a safety standpoint since it would diminish the need for critical
nuclear reactors J >3>4,10 curreny COncern over reactor safety and
core meltdown could be essentially eliminated.^
Past studies of the hybrid concept have been restrictive. Typical
hybrid analyses are limited to steady-state evaluation of the technical
characteristics of a concept with emphasis on the neutron economy of the
1-3 12-14
conceptual blanket. ’ Important features in such hybrid studies
parallel ordinary fusion reactor blanket studies and include:
1. Tritium conversion ratio and doubling time.
2. Fissile breeding ratio and doubling time.
3. Energy production and multiplication in the blanket
4. Constraints on the fusion plasma due to neutronics.
5. Vacuum wall loading and neutron energy transport.
The neutron economy and energy multiplication of the hybrid blanket
have been of primary interest in these initial studies; both are enhanced
by fission events. Little consideration has been given the fusioning
plasma in these hybrid designs beyond setting plasma characteristics
necessary to achieve the assumed blanket power performance. Basic fusion
reactor blanket studies and hybrid blanket work to date are reviewed in
the next section; the similarity of the two is remarkable despite the
increased importance of energy production in hybrid blankets.
Although some hybrid safety considerations have been made based on
the subcriticality of the blanket (effective neutron multiplication

-5-
factor, k less than unity) as well as the heat generation rates in the
blanket, no time-dependent analysis has been considered; dynamic behavior
and associated safety of the hybrid fusion-fission system have been
ignored. The effects of perturbations on the coupled system have also
been ignored.
Some studies on safety and control analysis of pure fusion reactors
15-24 i5_i7
have been reported. Mills " described the stability requirements
on a steady-state, point model, fusioning plasma, and found the steady-
state (equilibrium) plasma unstable against various parameter fluctua¬
tions below a critical ion temperature. The effects of artificial feed¬
back were simulated at 1ower temperatures to control this thermal
instability and maintain equilibrium operation below the critical
temperature. The work of Mills is a benchmark work in fusioning plasma
global dynamics and control.
18
The work on stability by Ohta et al. is one of the most complete
thermal stability studies of point model thermonuclear plasmas. Stability
criteria were established using linear analysis of coupled particle and
energy balance plasma equations. The thermal instability was evaluated
and suitable feedback control was implemented to allow stable operation
below the critical plasma temperature set by the stability criteria.
22 23 24
Stacey as well as Usher and Campbell 5 have reported extensions of
19 20
this work to more sophisticated plasma models. Yamato et al. ’ have
extended such stability studies to simple inhomogeneous plasmas with
comparable results.
Since such time-dependent analysis was neglected in previous hybrid
studies, this research analyzes some dynamic characteristies of a fusion-
fission coupled system along with certain steady-state characteristics

-6-
of fusion energy blanket multiplication not previously considered. The
much larger hybrid blanket energy multiplication demands a coupled time-
dependent analysis. The establishment of specific safety and operating
characteristics for a coupled hybrid system is necessary for the con¬
tinued development of the concept into a viable energy alternative.
The effect of thermal instabilities in the fusioning plasma on the
fissioning blanket are analyzed in this work to establish hybrid system
interactions, safety, and ease of control. This work eliminates a major
deficiency in existing studies of hybrid systems so that a decision can
be made on its place in the power industry of this country in the last
decades of this century.
Review of Fusion Blanket Studies
The Fusion Process
Since hybrids depend on fusion neutrons to breed fissile fuel, at
least two fusion reactions have potential for use in a hybrid reactor.
These are the deuterium-tritiurn and the deuterium-deuterium reactions
which have the following balances:
25
+ ^T —► ^He (3.52) + Jn (14.06)
(1)
gHe (0.82) + Jn (2.45)
(2a)
?T (1 .01) + ]p (3.03)
(2b)
where the two D-D branches have nearly equal probabilities at energies
of interest. The numbers listed in parentheses are the energies (MeV)
with which the fusion reaction products emerge.

-7-
The properties of the D-T fusion reaction are far superior to those
of the D-D reaction. For energies below 200 keV the D-T reaction cross
section with its broad resonance at 110 keV is nearly two orders of
magnitude above the D-D cross sections. The probability for a fusion
reaction occurring is characterized by the reactivity or rate coefficient,
, which is an average of the product of the cross section, o, for the
fusion reaction in question and the relative speed, v, of the reactants.
The reactivity can usually be approximated using a Maxwellian distribu¬
tion of particle speeds. With a broad resonance around 65 keV, the D-T
reaction rate coefficient is also much greater than the D-D reaction rate
coefficient below 100 keV.
Finally, the energy released per fusion reaction, Qp, is signifi¬
cantly higher for D-T fusion events. These comparative values are
25
summarized in Table 1-1 and indicate why near term fusion reactors and
hence hybrids are limited to the D-T fuel cycle.
Table 1-1
Fusion Reaction Parameters
a (barns)
(cm /sec)
Qf (MeV)
Reaction
at 100 keV
at 65 keV
D-D
0.46
8 x 10"16
3.65
D-T
5.0
9 x 10"lD
17.6
As noted in Eq. (1), the 17.6 MeV per D-T fusion reaction is divided
between the resultant neutron and alpha particle. The alpha particle
ultimately is expected to impart most of its energy to the fusioning

-8-
plasnia, but the 14.06 MeV of neutron energy must be recovered in surround¬
ing blanket regions.
Fusion Reactor Blanket Studies
Since only limited quantities of tritium occur in nature, sufficient
tritium must be generated through nuclear reactions to refuel operating
fusion devices. The 14.06 MeV fusion neutrons are used for this purpose
7 8
in two lithium reactions: ’
3LÍ + Jn (slow) > 2He + ]J + 4-8 MeV (3)
^Li + Jn (fast) * 2He + ]3T + Jn - 2.82 MeV (4)
where natural lithium has the composition: 7.56% ^Li and 92.44% \i.
The exothermic reaction has a 2.9 b resonance at 0.25 MeV while the endo¬
thermic reaction, with its threshold at 2.8 MeV, has a 450 mb resonance
26
at 8.0 MeV.
For the usual toroidal fusion reactor using superconducting coils
for the magnetic confinement, the position of the blanket used for heat
recovery and tritium generation is illustrated in Fig. 1. This con¬
figuration conforms to the Tokamak designs most often considered for
27-33
economic, power-producing fusion machines. Refractory metals such
as vanadium, molybdenum, and niobium are usually postulated as the vacuum
wall and structural material due to the high heat and stress load as well
g
as the need for (n,2n) reactions to enhance tritium breeding. Graphite
is the usual outer moderator/reflector material and the shield region is
8 31 33
usually of lead and steel composition. ’ ’
While heat recovery and

BLANKET
Figure 1. The essential components of a Tokamak fusion reactor.

-10-
tritium breeding are confined to the inner reflector/moderator regions
of the overall blanket, the outer regions shield the low temperature
superconducting magnets from the deposition of energy by high energy
particles generated within the fusioning plasma and inner blanket region.
A typical thickness for the total heat recovery and shielding regions of
the blanket is about two (2) meters with actual heat recovery and tritium
production confined to the first meter.J
Many early studies were conducted to evaluate tritium breeding and
heat generation in idealized fusion blankets. These initial studies in¬
dicated that adequate tritium generation was possible but with severe
heat transfer requirements on the vacuum wall. This problem was oartly
due to the fact that only the exothermic lithium reaction was known and
O
used in the earliest studies.
34
Myers et al. ‘ used diffusion theory to examine homogeneous cylin¬
drical blankets of varying thicknesses from 9 to 96 cm. Materials tested
included a lithium beryl 1ium-fluoride salt (LiF + BeF^) called "flibe,"
6 6
natural lithium metal and Li metal. All but Li provided adequate
5
tritium breeding ratios above 1.45; the value of only 0.976 for Li
demonstrated the potential significance of \i breeding reactions.
35 36
ImpinkJ0 and Homeyer also examined the effects of blanket composi¬
tion on tritium breeding and on spatial heating rates, respectively.
Graphite was used as the neutron moderator with molybdenum as the vacuum
wall material because of its neutronic and refractory characteristics.
The flibe coolant and tritium generation medium was selected to avoid
electromagnetic resistance to coolant circulation. For variations in
C
vacuum wall thickness, Li enrichment and flibe-Be composition, Impink
obtained tritium breeding ratios, T/n, as high as 1.55.

-n-
Since nuclear heating rate calculations showed extreme peaking near
2
the first wall based on 14 MeV neutron energy flux of only 1 MW/m on the
vacuum wall, Homeyer concluded that cooling of the vacuum wall would be
the most severe heat removal problem in the blanket. The recoverable
blanket energy was calculated to be 17.4 MeV per entering 14-MeV neutron.
37
Bell used multigroup transport calculations to analyze an infinite
annular blanket and concluded that pure lithium is an attractive breeding
material but requires a thicker blanket than one containing beryllium.
Unfortunately beryllium is probably too expensive to justify its large
volume usage in systems of the size of power-producing fusion devices.
Realistic blanket designs required more detailed neutronics studies
to consider structural and heat generation requirements as well as the
tradeoff between tritium breeding and energy generation as shown in more
recent, detailed calculations.^>27,38-43
38
Lee used Monte Carlo theory to calculate neutronics results for
a three zone spherical annular blanket with outer radii of 101, 202, and
302 cm for a 100 cm radius plasma. Structural effects were simulated
by homogeneous volume fractions of niobium chosen for its refractory,
fabricating, and welding characteristics; excellent results were obtained
for a structure!ess lithium blanket. More realistic blankets were
simulated by making Zone 2 (1 cm) all niobium and diluting the lithium
in Zones 3 and 4 with increasing volume fractions of niobium structure.
Lee's results are summarized in Table 1 -11 where the increase in energy
generation per fusion event is due to Nb(n,y) reactions. Since Li
enrichment was found to be ineffective and only 5 to 6% niobium is
necessary for real blanket structure, Lee concluded the simplest blanket
containing lithium and structure can meet tritium breeding requirements.

-12-
Electromagnetic resistance to lithium flow nay be excessive near the
vacuum wall where high coolant velocities are needed. Induced currents
in the lithium act to retard lithium flow across magnetic field lines;
but such resistance is greatly reduced in the outer blanket regions where
heating rates and hence flow rates are reduced.
Table 1-II
Dependence of Tritium Breeding Ratios and Energy Deposition
Rates for Lee's Fusion Blankets
Nb (Volume Per Cent)
T/n
Qb (MeV)
0
2.10
17.14
5
1.77
18.63
10
1 .38
19.60
15
1 .16
20.20
20
1 .00
20.50
8 39
Steiner ’ analyzed the neutronic behavior of two designs based on
the ORNL standard blanket configuration containing niobium structure,
coolant, and graphite reflector. These two blankets reflected an
optimistic (Design 1) and a conservative (Design 2) outlook on the problem
of cooling the vacuum wall. Design 1 contained lithium throughout the
blanket; Design 2 assumed that f1ibe must be used to cool the vacuum wall
with lithium elsewhere. Steiner rejected flibe coolant throughout the
blanket since it produced an inadequate (T/n = 0.95) tritium breeding
g
ratio. Neutron activation problems were also first revealed by Steiner.
Niobium was selected over molybdenum as the vacuum wall and struc¬
tural material because of superior fabrication and welding characteristics

-13-
as well as lower sputtering ratio despite molybdenum's demonstrated
g
superiority for tritium breeding. Graphite was employed as the moderator/
reflector in both designs. Summary descriptions of these two blankets
with 6% niobium structure are presented in Table 1 -111 to indicate
typical blanket models.
Table 1 -111
Summary Descriptions of ORNL Optimistic (1) and Conservative (2)
Blanket Designs
Region
Number
Description
of Region
Thickness
by Region
(cm)
Volume Composi
tion by Region
Design 1
Design 2
1
First wal1
0.5
Nb
Nb
Cool ant
94% Li
94% FIibe
2
+
3.0
Structure
6% Nb
6% Nb
3
Second wall
0.5
Nb
Nb
Coolant
94% Li
94% Li
4
+
60.0
Structure
6% Nb
6% Nb
5
Moderator-
ref! ector
30.0
Graph ite
Graphite
Cool ant
94% Li
94% Li
6
+
6.0
Structure
6% Nb
6% Nb
The
basic 100 crn Design
1 blanket
with first wall at
200 cm radius
was adopted as the standard blanket model at the Neutronics Session of
the Working Sessions on Fusion Reactor Technology held at Oak Ridge

-14-
National Laboratory (ORNL) in June 1971. This blanket lias been frequently
used to check neutronics calculations.
Transport theory was applied in slab geometry to obtain the tritium
breeding results listed in Table 1 -IV where the breeding ratio of 1.35 in
Design 1 is some 10% above the 1.23 value for Design 2. Slab geometry is
adequate due to the large plasma radii (1-5 meters) for steady-state
33 45
fusion reactors. ’
Table 1-IV
Summary of Steiner's Tritium Breeding Calculations per Incident
14 MeV Neutron
Design
T/n
Neutron Leakage
1
1 .35
0.023
2
1 .22
0.020
46 47
If hypothesized low levels of tritium holdup ’ are realized, then
breeding ratios only slightly above unity 1.01) will be sufficient for
seven year doubling times. Therefore, Steiner's relatively low 1.3
breeding ratio is sufficient to obtain the one month doubling time to
establish initial tritium inventories.
Steiner's results for spatially dependent, nuclear-heating rates
2
were based on a standard first wall energy transport of 10 MW/m due to
the 14 MeV neutron flux. Extreme peaking of nuclear-heating rates was
found in the first wall regions as shown in Fig. 2; Design 1 yielded
3
heating rates up to 180 Vi/cm while Design 2 with f 1 ibe first wall coolant
3
was less but still over 120 W/cm . Due to plasma radiation, the usual

Heating Rate (W/cm )
-15-
Figure 2. Comparison of spatially-dependent heating rates for vacuum wall
regions in two designs.

-16-
heating rate peak at the vacuum wall will be 5-10% more extreme than in¬
dicated. These extreme heating rates (power densities) near the first wall
along with the excessive fusion neutron wall loading represent a major
30 33 47
technological problem for all Tokamak fusion power reactors. ’ ’
Steiner's work supported previous work indicating that blankets employ
ing lithium as the only coolant are superior to those employing flibe since
1. Design 1 has a 10% higher tritium breeding ratio.
2. Design 1 has a 50% lower heat load in the niobium vacuum walls
since the high gamma cross section of flibe has been removed.
3. Neutron irradiation effects within the vacuum wall are essen¬
tially the same in both designs along with excessive heating
rates near the first wall.
40
Blow et al. used Monte Carlo calculations in cylindrical geometry
with first wall at 150 cm to examine Steiner's two basic 100 cm thick
blanket models with varying (2-8%) niobium structural content. Good
breeding ratios (1.15-1.54) were reported for all cases except the ex¬
clusive use of flibe coolant in the entire blanket where T/n = 1.027.
Blow reported additional good breeding results (T/n = 1.58) for blankets
of Design 1 where niobium was replaced with 2% molybdenum. Examination
of molybdenum was justified because the alloy TZM (0.5% Ti, 0.1% Zr,
99.4% Mo) has the neutronic characteristics of pure molybdenum but welding
characteristics similar to niobium.
A modular blanket design using heat pipes has been proposed by
41 42
Werner et al. ’ in which neutronic behavior was examined in a 100 cm
thick cylindrical annulus with 200 cm inside diameter. In relocating the
"standard" vacuum wall of a thermonuclear reactor beyond the neutron¬
moderating, energy-converting blanket (at 320 cm), the entire moderator
was placed in a cylindrical vacuum envelope in clear view of the plasma

-17-
to eliminate the neutronic losses and structural buckling problems of
previous designs.
The interlocking modular blanket units incorporated heat pipes which
remove radiant energy from the inner module surface and flatten the power
distribution in the blanket by moving excess energy outward to power-
42
deficient zones.
Werner's blanket model contained beryllium for neutron multiplica¬
tion, lithium for tritium breeding, sodium for energy generation, and
niobium for structural strength. The 100 cm moderator section of the
blanket was divided into two zones; Zone 1 contained 75% Li and 5% Nb
while Zone 2 contained varying volume percentages of Be, Na, and Li. Both
zones contained % 20% volume for heat pipe voids. Zone 1 was used to
buffer the energy density in the fluid so that all nuclear and radiant
heating energy could be removed by convective heat-transfer through the
heat pipes resulting in power flattening and increased average power
densities.
The tradeoff between tritium breeding and energy multipi cation
through use of beryllium or sodium was examined for varying volume frac-
41
tions in a 90 cm thick Zone 2. Increased energy generation per fusion
neutron up to 23.0 MeV for beryllium and 26.05 MeV for sodium was obtained
but with a reduction in the tritium breeding ratio. Unless maximum energy
is very important, Werner recommended maintenance of tritium breeding--
probably because of beryllium costs and sodium activation.
43
Struve and Tsoulfanidis used Monte Carlo methods to calculate
tritium breeding ratios and heating rates for two proposed blanket designs
utilizing vanadium as the structural material in lithium. Vanadium was
used for its reduced activation and after-heat advantages, although it
has the disadvantage of a low operating temperature.

-18-
O
The two blanket configurations included a basic Steiner-type where
the vacuum wall surrounds the plasma and a Werner-type where the vacuum
42
wall surrounds the blanket. To avoid the problem of coolant flow,
Struve proposed a heat transfer fluid such as helium which would be un¬
affected by magnetic field lines and transparent to neutrons. It was
simulated by 20% volume void in the lithium. Breeding ratios above 1.3
were obtained and agreed reasonably well with previous blanket studies
8 40 42
using niobium structure. ’ ’ The use of helium as a fusion blanket
fin
coolant has been investigated by Hopkins and fleiese-d'Hospital ' and by
0*1
others at General Atomic Company.J
The spatially dependent nuclear-heating rates for the two blankets
showed high vacuum wall heating and agreed with previous results. Steiner's
generally higher calculated heating rates were caused by niobium blanket
structure.
These detailed neutronic studies of fusion blankets indicate ample
tritium breeding is possible in realistic blankets. The inability to
breed tritium is not a problem in fusion designs. The real problems in¬
clude providing adequate heat removal for the first wall and protecting
and designing the vacuum wall to withstand the required 15 Me'/ neutron
„ 33
fluxes.
These fusion reactor blanket scoping studies have formed the basis
for a number of design studies for Tokamak fusion power reactors of
28-32
either full commercial scale or demonstration size. ° These various
pure fusion Tokamak blankets use either flibe, natural lithium, or helium
as coolant and flibe, natural lithium, or some lithium-bearing medium as
breeding material with tritium breeding ratios from 1.15 to 1.3. Most
pure fusion design studies use lithium or helium as the coolant, instead

-19-
33
of f1ibe. All blankets are on the order of 100 cm thick and some 20-25
MeV are deposited in the blanket per 14 MeV neutron entering the blanket
with extreme peaking of heating rates near the first wall. The blankets
are not expected then to be significantly energy multiplying.
In general the tendency is toward more compact fusioning plasmas with
an associated reduction in the first wall neutron flux to well below 10
2 28-3 3
MW/m of 14 F1eV neutron energy transport. The basis for such reduc¬
tions is the extreme technological problems of designing a first wall
which v/ill function for at least two years or more. If such cannot be
accomplished, then fusion power plants that are viable in other respects
are likely to be too limited in outage maintenance time to compete
33 49
economically with other electrical power sources. ’
Critical Review of Hybrid Blanket Studies
Overview of Hybrid Blanket Studies
Fusion blanket designs attempt to maximize energy generation while
maintaining the tritium breeding ratio. The inclusion of fissionable
materials in the blanket is an obvious possibility for achieving signifi¬
cant power and neutron multiplication. Such a hybrid blanket must still
meet the basic fusion blanket requirements of adequate tritium breeding,
heat transfer, and magnet shielding as well as produce energy multipli¬
cation and/or fissile material. As with pure fusion systems, previous
evaluations of hybrid concepts have been based primarily on the cal¬
culated neutronic behavior of the conceptual blanket as reflected in the
following parameters:
1. Tritium breeding ratio and fissile fuel production.
2. Energy production in the blanket per fusion event.

-20-
3. Fusion plasma characteristics.
4. Neutron first wall loading.
The tritium breeding ratio must be sufficient to refuel operating
hybrid systems and fuel new ones. As for pure fusion systems, adequate
values are in the range T/n = 1.15 - 1.3 and are relatively easy to
1 3
obtain. ’° Simultaneously, a hybrid nay also be required to produce or
2 3 6
even breed significant amounts of fissile fuel. ’ ’
Energy deposition in the blanket per fusion event is a very important
hybrid criterion. Usually D-T fusion systems assume a blanket energy
deposition, 0^, of about 20 MeV per fusion to account for the 14.1 MeV
6 3
neutron and the 4.8 MeV per Li(n,a)JT reaction. Fusion blanket studies
show this energy deposition is relatively insensitive to design or com-
33
position with calculated values per fusion neutron ranging from a
41
maximum of 26 MeV for Werner's' best design down to 18.3 MeV evaluated
by Leonard^ for the ORNL standard design.
Although fusion blankets are limited in their energy multiplication
capabilities, this is not the case for hybrids which are evaluated for
significantly increased blanket energy deposition per fusion event
through fission energy multiplication. Interest in subsystem interactions
and dynamics studies of such a coupled hybrid system is certainly justi¬
fied when the potential for energy generation through energy multiplication
in the subcritical blanket is considered.
The third area of technical assessment of hybrids involves the
fusion plasma characteristics required to achieve the assumed blanket
performance. This assessment is directly related to the blanket energy
deposition, Q^, per fusion neutron. The Lawson Criterion sets the plasma
values of density, n, confinement time, t, and temperature, T, required

-21-
50
to reach overall breakeven in energy production or scientific breakeven.
The breakeven m-value varies inversely with the total energy generated
per fusion event. Therefore, the potential value of a hybrid system is
characterized by its ability to relax the Lawson condition through effec¬
tive fission increase of energy released per fusion event.
Finally, the required transport of neutron energy through the first
vacuum wall is an important figure of merit. Previous projections of 10
2
MW/m impose stringent material problems so more recent designs attempt
to achieve wall loadings in the range 0.25 to 3.5 Any
hybrid relaxation of first wall loadings is a technical advantage over
pure fusion systems.
Such potential for breeding fissile fuel with fission energy multi¬
plication of the fusion neutron source strength to sustain and drive the
coupled system has been examined by many researchers. Early concepts
were summarized adequately by Leonard and have little more than historical
. ... 1
significance.
Lontai Attenuator Model
The first detailed calculations on the neutron economy of hybrid
blankets were performed by Lontai in 1965. ^ He assumed a steady-state,
2
D-T clyindrical plasma with a 5.0 MW/m energy transport of 14 MeV
neutrons but performed the neutron balance calculations for an infinite
slab source geometry. Lontai's results were based on blanket configura¬
tions using flibe coolant channelled in a graphite matrix. Neutron
balance ranges were reported for various molybdenum wall thicknesses,
UF^ concentrations and ^Li enrichments to increase poor tritium breeding

-22-
ratios. Such a scope of study and results reported set the stage for
most of the hybrid studies which followed.
Lontai's best results were reported for a blanket concept consisting
of a 1 cm molybdenum vacuum wall, 1.5 cm coolant (flibe) region, and 49 cm
attenuator region containing 21% graphite by volume with 70% salt bearing
uranium (LiF - BeF2 - UF^). The natural lithium case had insufficient
tritium breeding. Adequate tritium breeding was calculated only by using
lithium salt enriched to 50% ®Li and varying composition. The fission
energy multiplication increased by nearly a factor of two over non-fissile
blankets with better heat transfer characteristics. Similar calculations
for 90% enriched Li resulted in much lower fissile fuel production with no
increase in energy multiplication. Plasma requirements are not relaxed much
by such small amounts of fission energy deposition; however, Lontai opti¬
mistically labeled the 50% ^Li attenuator practical because of possible
reduced plant capital costs.
Lontai's hybrid feasibility study currently has little more than
historical significance because of inherent deficiencies:
1. Faiure to consider values of plutonium production.
2. Failure to consider cost of maintaining high ^Li enrichment.
235
3. Failure to consider U present in depleted uranium.
4. Use of obsolete computer codes and poor cross section data.
Lidsky1s Symbiosis Concept
A novel approach to the fusion-fission hybrid concept was proposed
2 6
by L.M. Lidsky, ’ who analyzed the characteristics of a hybrid fusion
reactor which, when coupled with a Molten Salt Converter Reactor (MSCR),
would constitute a viable central station power plant.^ The essential

-23-
feature of this symbiotic scheme was a fusion system breeding sufficient
tritium and fissile nuclei to fuel itself and a power-producing fission
device such as an MSCR.
A cylindrical, 1.25 m radius torus of D-T plasma was used in the
symbiosis. The basic duplex blanket configuration contained a thorium¬
bearing blanket flibe salt composed of LiFrBeF^:ThF^ in the ratio
r
71:02:27 and lithium depleted in ^Li. The neutronic properties of pure
molybdenum with its large Mo(n,2n) cross section, were utilized in the
TZii structural alloy. Since Lidsky's fusion reactor was designed for
fuel, not power production, a graphite moderating region was used to
prevent thorium fission products from poisoning the blanket during opera-
233
tion. This is only possible at initial operation until fissile U is
produced which implies frequent refueling and possible cost penalties
which Lidsky ignored. Lidsky used S». transport theorv to evaluate the
neutron economy of the hybrid blanket configuration. The results for
this as well as variations in the base design are shown in Table 1-V.
Since simultaneous production of fissile nuclei and tritium was found to
be attainable over a range of production ratios, each component of the
system can be optimized for power or fuel production to utilize the strong
points of both fusion reactors (neutron rich) and fission reactors (power
rich).
The reactors in the symbiosis were coupled by the production of fuel
for the fission reactor by the fusion reactors. Lidsky6 also analyzed
equations for the time dependence of the fuel inventories of the two
reactors in the fusion-fission symbiosis combination.

-24-
Table 1-V
Neutron Economy of Lidsky's Hybrid Blanket
Events per 14-MeV
Source Neutron
Calculated
Range
Tritium production
1 .1 26
Thorium captures
0.325
0.05-0.50
Total conversion
1 .451
¿ 1.40
Lidsky's results demonstrated that the fuel doubling time of such a
balanced hybrid system is determined entirely by the neutron-rich fusion
reactor component. Lidsky's power production analysis indicated further
that the net power production in such a dual system is determined pri¬
marily by the fission reactor component since the fusion power reactor
is only a small perturbation on the net power of real systems. Thus each
subsystem in the symbiosis can theoretically be optimized for its
respective primary purpose of fuel or power production. This is an
important point to remember with respect to hybrid reactor system design.
Lidsky selected a CTR-MSCR power plant with 1500 MWe output and a
10 year doubling time for symbiosis study. The MSCR was rated at 4450
233 232
IT,1th with a fuel conversion ratio of 0.96 operating on the U- Th
cycle. Lidsky calculated a 10 year fissile doubling time with a tritium
linear fuel doubling time of 0.113 years. For a 40% thermodynamic
efficiency the fusion reactor would be a net consumer of 89 MWe while
the overall system was calculated to be able to provide 1690 MWe net
power with 35.6% plant efficienty. Lidsky predicted net plant efficien¬
cies above 40% for near-classical plasma confinement times.

-25-
Required plasma characteristics were encouraging since the vacuum
2
wall loading due to 14 MeV neutrons was only 1.00 MW/m --well below that
33
necessary to assure technological feasibility in pure fusion plants.
In addition, there was no energy multiplication in the fusion reactor
blanket of the symbiotic scheme; this assumption was clearly not accurate
as soon as some fissile fuel breeding has occurred. Plasma parameters
are near Lawson conditions as indicated by the hybrid parameters summary
in Table 1 -VI and the fact that only 295 f'Wth was required to support
the fuel-producing fusion system.
Table 1-VI
Lidsky1s Hybrid Reactor Parameters
n
T
T
Wall loading
233
U production
in14 . . 3
= 10 ions/cm
= 0.625 sec
= 20 keV
= 1 MW/m2
=1.1 kg/day
The symbiosis has a number of advantages. First, this scheme
simplifies the construction of power plants capable of breeding and
processing all requisite fuel in situ. Second, the lessening of fuel
cost constraints makes the modifications of existing reactors possible
to avoid thermal pollution. Finally, by developing this concept, the
system under construction at any time could take full advantage of the
existing state of development of both fusion and fission technology
without final commitment to either.

-26-
In addition to the symbiotic hybrid concept and the usual power-
producing hybrid concept, Lidsky has also formalized consideration of a
third hybrid concept called the augean concept. This concept involves
using the hybrid blanket to burn the actinide waste from fission
2 1 3
reactors. ’ The augean concept is of little interest for dynamic
consideration.
Lee's Fast Fission Hybrid Concept
45
Lee eliminated Lidsky s separate fusion and fission reactors in
favor of the so-called subcritical fast fission blanket. Monte Carlo
Transport theory was used to perform neutron balance calculations in
238
infinite media of pure thorium, pure U, and natural uranium to verify
the breeding potential of hybrid blankets. The results shown in Table
1 -V11 are in good agreement with experimental measurements done by
Weale et al.~^
Table 1 -VII
Lee's Neutron Balance in Infinite Media
B1anket
Qb (MeV)
Breeding Reactions
per 14-MeV Neutron
Thorium
64
2.7 [232Th(n,Y)]
238u
233
4.4 [238U(n,y)]
Natural Uranium
309
5.0 [238U(n,Y) ]
Lee examined blankets containing varying concentrations of niobium,
lithium, and a fertile nuclide. The blanket gemoetry was a two-zone,

-27-
sperhical annul us having an inner radius of 200 cm and an outer radius of
300 cm with composition as listed in Table 1 -VIII. For constant blanket
geometry and material volume fractions, the following optimum results
6
were obtained for depleted lithium (4% Li) and depleted uranium (0.04%
^U) per 14 MeV neutron: Qg = 103 MeV; T/n = 0.986; ^U(n,y) reactions
239
= 1.58. Because of the 1.68 Pu breeding reactions per D-T fusion
239
event, Lee chose Pu as the fissile fuel.
Table 1 -VI11
Subcritical Fast Fission Blanket Components
Studied by Lee
Element
Volume Fraction
Zone 1
(30 cm thick)
Li
0.95
Nb
0.05
Zone 2
(70 cm thick)
Li
0.30
Nb
0.05
Heavy Element
0.65
Lee also studied the neutronics effects of changes in the thickness
of Zone 1 and material volume fractions in Zone 2; for the composition
shown in Table 1 -V111 results were reported for the following heavy
element material variations:
235
1. Depleted uranium versus U content.
239
2. Metallic and oxide mixtures of plutonium and uranium versus Pu
content.
233
3. Thorium versus U content.

-28-
Best energy generation with sufficient breeding was reported for
the metallic uranium blanket with 4% plutonium. This case and one
poisoned with 8% fission products are summarized in Table 1 -1X.
Table 1 - IX
Fast Fission Hybrid Neutron Economy per 14 MeV
Neutron Calculated by Lee
Material
Tri ti urn
Production
PIutoniurn
Conversion
Ratio
%
(MeV)
k
eff
4% Pu-U
1 .38
3.14
431
0.84
4% Pu-U + 8% FP
1.18
3.93
306
The usefulness of a hybrid concept is contingent upon a short
fissile fuel doubling time. Lee estimated a very high 14 MeV neutron
2
wall loading of 12.8 MW/m to obtain a 5 year plutonium doubling time
for the 8% FP blanket but reports no fusion plasma characteristics.
Leonard^ later claimed that the 306 MeV blanket energy release per fusion
neutron in Lee's 8% FP model would lead to a three-fourths reduction of
the usual Lawson breakeven condition. However, current engineering con¬
siderations indicate that such first wall power loadings will almost
certainly make fusion power unrealistic due to the need for frequent
33
first wall replacement.
Since his results indicated energy production increases of 10 to 20
times over non-fissile blankets with simultaneous adequate tritium and
fissile fuel breeding, Lee concluded fast fission hybrid blankets were
feasible. However, the fast fission blanket has no clear neutronic

-29-
advantage over other concepts except as a fuel producer. Considerable
additional research has been reported on blankets and hybrid systems
52-57
using the fast fission concept.
All have emphasized fuel production versus power production and
O
have worked with reduced first wall neutron loadings of 1-5 MW/m . The
advantages of using fusion neutrons for fast fission as well as breeding
5
fuel in situ are probably only applicable in the true symbiotic concept
where the hybrid is not a system energy producer but a fuel producer,
since blanket multiplication is lowered for low enrichments with fast
fission. Hence, the fast fission hybrid is of little interest in this
current study.
Texas Fast Fission Hybrid
12
Parish and Draper presented extensive hybrid neutronics results
for their model which was also a fast fission design. They investigated
the potential of 14 MeV fusion neutrons to fission fertile materials
(" Th and U) while maintaining adequate fusion blanket performance.
Parish and Draper based the attractiveness of this concept on the rela¬
tive abundance of such fertile fuels and the elimination of dependence
on breeding fissile fuel for hybrid usage. The large fission energy
multiplications obtained in other studies1’" were not paralleled in this
hybrid; however, the potential of both thorium and natural uranium-fueled
fast fission blankets to produce both fission power and fissile material
was demonstrated.
44
The use of the standard ORNL fusion blanket with natural lithium
coolant and niobium structure provided the model for comparison between

-30-
various calculationai methods. To verify methods of analysis Parish and
Draper calculated the neutronic and photonic characteristics of the
standard fusion blanket model using ENDF/B-III cross section data in the
ANISN^ code for a P^-S^ transport approximation. The resultant standard
blanket neutron economy compared well with Steiner's latest results on
59
the same standard. Good agreement was obtained for breeding (T/n =
1.445 versus T/n = 1.452) and (n,2n) reactions as well as neutron leakage
despite Steiner's use of pre-ENDF/B-111 cross section data. This Texas
hybrid was one of the first hybrid studies to account for (n,3n) reactions
which become very important in such poorly multiplying blankets.
Since high energy neutrons are needed to fission fertile fuels, the
fission material regions in this concept were placed as close as possible
to the vacuum walls. Low energy neutron absorption was only partially
offset by (n,2n) and (n,3n) reactions.
The volume fractions of fuel, clad (niobium), and coolant (lithium)
in the model were maintained constant at 0.45, 0.15, and 0.40, respec¬
tively, to approximate fuel regions in a LMFBR. The tritium breeding,
fissile breeding, fission power, and spatial heat deposition by blanket
region were presented in the Texas study for various blanket fuel thick¬
nesses. The results of these calculations for two thorium-fueled
blankets and four uranium-fueled blankets are presented in Tables 1-X
and 1-XI.
The calculation of spatial heat deposition rates in the standard
and fertile fueled blankets in this work emphasized the problems with
low multiplication hybrid blankets.

-31-
Table 1-X
Neutron Economy for Thorium-Fueled Blankets
Thorium Fuel
Reactions/Fusion Event
Region Thickness
T/n
232t, , ,,
Th(n,f)
232t. , r~
Th(n,y)
6 cm
1.3012
.0310
.1326
1 3 cm
1.0964
.0472
.3118
Table 1-XI
Neutron Economy for Uranium-hueled
Blankets
Natural Uranium
Fuel
Region Thickness
Reactions/Fusion
Event
T/n
238U(n,f) 235U(n,f)
Total
Fission
Z38U(n,y)
10 cm
1.3252
.133 .0133
.1463
.2487
1 3 cm
1.2694
.1863 .0161
.2024
.3818
20 cm
1.0865
.1837 .0259
.2096
.5320
26 cm
0.9614
.1986 .0315
.2301
.6654
For the large
10 MW/m2
first wall neutron loading limit,
the two
3
thorium-fueled blankets showed peak power densities of 200 W/cm . For
the 13 cm natural uranium case, the power density between the niobium
3
walls ranged from 510 to 364 W/cm ; the related thorium case had a range
3
of 203 to 145 W/cm . Fuel was eliminated in the 3 cm region between
niobium walls for all other uranium cases to prevent vacuum wall cooling
3
problems. Since the LMFBR is designed for 500 W/cm average power

-32-
density, Parish and Draper have claimed these hybrid blanket power den¬
sities are acceptable. This is doubtful because of the low power den¬
sities at blanket positions removed from the vacuum wall and the resultant
unit cost of electrical and fusion power produced.
The superiority of natural uranium to thorium as a fast fission
hybrid blanket fuel because of its larger fast fission cross section is
illustrated in Parish's comparison of the best case for each fuel
presented in Table 1 -X11.
Table 1-XII
Comparison of Best Natural Uranium-Fueled and
Extrapolated Thorium-Fueled Blankets
Uranium
Thoriurn
Tritium Breeding Ratio
Fusion Blanket Energy Multiplication
Fissile Nuclei Produced per Fusion Event
Peak Power Density at Nb First Wall
1.09
a, 20
0.53 -
409 W/cnr
1 .15
0.5
0.31 .
% 200 W/cn
However, the low return of the fissioning blanket renders this concept
uneconomical versus other concepts relying on better fissile blankets.
Increasing fuel costs could make this concept more attractive at some
future date but others seem more appropriate.
Light Water Hybrid Reactors
The feasibility of fusion-fission hybrid reactors based on breeding
light water thermal fission systems has recently been investigated at the

-33-
Princeton Plasma Physics Laboratory Emphasis was placed on fuel-self-
sufficient (FSS) hybrid power reactors fueled with natural uranium. Other
Light Water Hybrid Reactors (LWHR) considered included FSS-LWHR's fueled
with spent fuel from Light Water Reactors (LWR's), and LWHR's to sup¬
plement LWR's by providing a tandem LWR-LWHR power economy that would be
fuel self-sufficient similar to Lidsky's symbiotic concept. Nuclear
power economies based on any of these LWHRs were found to be free from
the need for uranium enrichment and for the separation of plutonium. They
offer a high utilization of uranium resources (including depleted uranium)
and have no doubling-time limitations.
This study investigated the properties of subcritical thermal
lattices for hybrid applications and concluded that light water is the
best moderator for FSS hybrid reactors for power generation. Several
lattice geometries and compositions of particular promise for LWHR'swere
identified with thicknesses up to 250 cm. The performance of several
conceptual LWHR blankets was investigated and optimal blanket designs
were identified for natural uranium-fueled lattices. The effect of
blanket conversion efficiency and the feasibility of separating the
functions of tritium breeding and of power generation to different
blankets were investigated. Optimal iron-water shields for LWHR's were
also determined.
The evolution of the blanket properties with burnup was evaluated
along with fuel management schemes. The feasibility of using the lithium
system of the blanket to control the blanket power amplitude and shape
was also investigated. A parametric study of the energy balance of LWHR
power plants was carried out, and performance parameters expected from
LWHR's were estimated. This investigation of LWHRs also compared LWHR's

-34-
with critical systems and delineated the advantages of such hybrids in
alleviating nuclear technology problems relating to resource utilization,
prol iteration, and safety issues. In general, this study reported the
same types of information as previous studies but for a different blanket
design.
PNL--Thermal Fission Hybrid
Pacific Northwest Laboratories (PNL)^’^ initially studied a hybrid
fusion reactor utilizing a subcritical thermal fission lattice around the
usual cylindrical D-T plasma. The four distinct regions of the hybrid
blanket configuration are illustrated in Fig. 3.
The 8 cm thick neutron convertor region was filled with niobium-clad
pins of both depleted uranium carbide and natural lithium. Niobium
structural walls are used along with helium coolant. The 150 cm thick
thermal fission lattice, consisting of a graphite-moderated, natural
uranium-fueled, helium-cooled matrix, was designed for fission power
generation. The last 50 cm of blanket thickness are filled with graphite
reflector and natural lithium absorber, respectively.
O
The ENDF/B III cross section data were used in the HRG3 and Battelle-
63
Revised-Thermos (BRT-1) cross section codes to obtain fast and thermal
broad group data, respectively. The final neutron balance results ob¬
tained using a P^-Sg transport calculation in ANISN^ are summarized in
column 1 of Table 1 -X111. Neutronic effects from slight enrichment of
the uranium in the fission lattice are also shown in the neutron balances
of columns 2 and 3 in Table 1-XIII.

\
\
\
\
1
\
1
PLASMA I
I
I
I
I
I
I
/
/
/
160 cm
RADIUS
NEUTRON
CONVERTOR
THERMAL FISSION LATTICE
GRAPHITE
© ®
MODERATOR LITHIUM
REFLECTOR ABSORBER
© ©
© ®
® ©
® ®
® ©
® ®
I
Co
c_n
i
Figure 3. Early PNL fusion-fission hybrid subcritical blanket configuration.

-36-
Table 1-XI11
Early PNL Hybrid Neutron Balance
Events per
Source 14-
-MeV Neutron
235
U Atom Percent Enrichment
0.7196
0.80
0.90
Tritium Production
6U
0.956
1.188
1 .763
7Li
0.019
0.019
0.020
Total Tritium Production
0.975
1 .207
1.783
Fissions
23Su
0.234
0.251
0.292
235y
1 .936
2.776
4.863
238.. r .
U Captures
235U Absorptions
1 .121
0.988
0.853
Estimated k ,,
eff
0.847
0.884
0.928
Qb
455
625
1050
Based on their composite behavior with fissile enrichment, an
enrichment was predicted (0.77 atom %) for which both the tritium and
fissile conversion ratios could be optimized to exceed unity. The cal¬
culated energy deposition in the blanket for the best case was calculated
to be about 500 MeV per source neutron corresponding to an energy multi¬
plication of about 25. This PNL optimum hybrid is attractive since
significantly 0.77% enriched uranium can be produced than the higher
percentages required for fast or thermal fission breeder concepts.
This early PNL concept assumed a thermal power generation rate of
3
0.75 W/cm corresponding to advanced, gas-cooled, uranium metal graphite

-37-
reactor capabilities which is very low. This power density was used to
determine the plasma and blanket specifications shown in Table 1-XIV where
the plasma requirements are substantially less than for a nonmultiplying
blanket and the vacuum wall loading is very low.
Table 1-XIV
Early PNL Hybrid Specifications
B1anket
Specific power
Thermal power
Vacuum wal1 1oading
Plasma
T (keV)
10
20
0.75 W/cm3
20 MW/m
0.05 MW/m2
ni (steady state) (sec/cm2)
3.5 x 1013
1.8 x 1013
Since a non-negligible fraction of the thermal energy produced in the
blanket must be used to sustain such a plasma, the need for investigation
of controls is justified, especially since the fission energy multiplica¬
tion is predicted to be so high.
This preliminary PNL hybrid design was faced with drawbacks such
as large size (2 m thick blanket) and low power density (0.75 W/cm ).
However, it was favored with low wall loading and plasma conditions re¬
duced to ^ 1/6 Lawson Criterion value. Since the hybrid objectives of
energy multiplication with adequate breeding of tritium and fissile fuel
are attainable, the PNL concept appeared to be a promising competitor for
the LMFBR program. Much additional work has been performed including
further scoping calculations to determine the best design of the PNL
helium-cooled hybrid blanket design. These detailed parametric analytical

-38-
studies have identified and delineated the merits of the helium-cooled,
thermal fission hybrid fueled with natural or slightly enriched uranium,
moderated with graphite, and cooled with helium. In addition, the
optimal use of lithium for breeding has been delineated.
This PNL concept of a fusion-fission system has been developed to a
considerable degree as reported by many studies.^4-67 mosy complete
results on blanket parameters were reported by the combined efforts of
64
Lawrence Livermore Laboratory and Pacific Northwest Laboratories.
Although this hybrid blanket design was intended for use in the spherical
geometry of Livermore's mirror (Yin-Yang) fusion reactor concept, the
basic blanket geometry is very similar to that shown in Fig. 3. Blanket
modules of varying composition were analyzed using a fuel pin lattice
geometry similar to that used in High-Temperature Gas-Cooled Reactors.
Results reported for the hybrid blanket analysis are included in Table
1 -XV showing seven (7) different cases analyzed, all of approximately
200 cm thickness. The inner convertor region was closest to the plasma
and contained helium coolant and stainless steel structure as well as
depleted uranium to enhance neutron multiplication. The inner thin
breeder contained lithium for fast neutron tritium breeder while the
thicker outer lithium breeder contained lithium for thermal neutron
breeding of tritium. The reflector, where used, was composed of graphite
and the thermal fission lattice was composed of hexagonal unit cells of
slightly enriched (as noted) fuel pins in a helium-cooled graphite matrix.
The fuel pin geometry and cell pitch were optimized using Sn transport
calculations.
The tritium breeding and fissile breeding ratios were very en¬
couraging particularly for Case 7 where both were reported to exceed

-39-
Table 1 -XV
PNL Hybrid Blanket Analysis
Tritium
Fissile
Blanket
Case
B1anket
Breeding
Breeding
Fusion Energy
Arrangement
Ratio
Ratio
Multipiication
1
3.5
cm convertor
1 .5
cm breeder
150
cm lattice (1.0%)*
0.766
1.59
18.9
20
cm reflector
15
cm breeder
2
10
cm convertor-breeder mix
150
20
cm lattice (1.0%)
cm reflector
0.725
1 .57
19.8
15
cm breeder
3
10
cm convertor-breeder mix
180
cm lattice (1.0 %)
0.365
1 .62
25.2
10
cm breeder
4
8.5
cm convertor
1 .5
180
cm breeder
cm lattice
0.737
1 .55
20.0
10
cm breeder
5
8.5
cm convertor
1.5
180
cm breeder
cm lattice (1.25%)
0.893
1 .22
31 .8
10
cm breeder
6
8.5
cm convertor
1 .5
180
cm breeder
cm lattice (1.50%)
1 .26
0.984
59.6
10
cm breeder
7
8.5
cm converter
1.5
180
cm breeder
cm lattice (1.35%)
1 .00
1 .11
39.8
10
cm breeder
*235
U enrichments denoted in parentheses.

-40-
unity. In addition, the energy multiplication of the fusion power was
found to be very large for this best case (MD = 39.8).
D
This energy multiplication was claimed to be related to the effec¬
tive neutron multiplication of the blanket and the neutrons produced per
fission in the blanket by the following global parameter equation:
M
B
1200 MeVwl w keff n
1 14 MeVMvMl - k
eff
(5)
where 200 and 14 represent the energy deposited due to fission reactions
and fusion neutrons, respectively, v is the number of neutrons released
per fission and k ^ is the usual blanket effective neutron multiplica¬
tion factor. Since this equation related global parameters and since
the 14 MeV source is introduced inhomogeneously, the current work was
partially directed at determining if this equation might be inadequate
despite its frequent use in describing and analyzing results from cal¬
culations performed on hybrid blankets.
Review of Controlled Thermonuclear Reactor
Thermal Stability Analyses
Fusioning Plasma Operational Criteria
The first determinations of operational criteria for thermonuclear
reactors were performed using global or point-model reactor parameters.
Rigorous descriptions of complex plasma dynamics with attendant spatial
variations were usually beyond the scope of such criteria development.
The first attempt to specify fusion reactor operational criteria
50 69
was undertaken by Lawson and refined by Ribe et al. This model
assumed instantaneous heating of the plasma at an ion density, n, to a

-41-
uniform temperature, T, and confined for a time, t, after which cooling
was allowed. Conduction losses were entirely neglected. This initial
work established values of temperature and the product of ion density
and confinement time, nx, for a zero-power but self-sustaining thermo¬
nuclear system. A system energy balance was used in which the energy to
heat the plasma, Ep, and the energy to overcome bremsstrahlung radiation
losses, Eg, were supplied to produce fusion reaction energy, Ep. The
energy supplied as well as the fusion reaction energy, was assumed to be
recoverable and converted to useful output energy at some efficiency, n.
The minimal condition for breakeven is simply defined as follows:
(6)
where n is the overall system energy conversion efficiency.
For a D-T fusion system as described above, the so-called Lawson
Criterion for breakeven becomes simply:
3T
(7)
f^)p(l - p)Qp<°v>DT - bT1/2
where
3
n = fuel ion density (ions/cm )
T = plasma temperature (keV)
3
p-|. = reactivity of D-T plasma (cm /sec)
n = overall system energy conversion efficiency
b = proportionality constant for bremsstrahlung radiation
p = tritium fraction of ion density
Qp = energy release per fusion reaction (keV).

-42-
The Lawson Criterion for the pure D-T fuel cycle is represented
by a series of parametric curves in the efficiency as shown in the lower
spectrum of curves in Fig. 4. Points on such parametric curves represent
minimum nx and T values for breakeven fusion energy production; no net
fusion energy is produced. If the energy per fusion event can be aug¬
mented by fission reactions in the hybrid blanket, then the requirements
on the plasma can be significantly relaxed.
Cyclotron or impurity radiation losses are not considered in Lawson-
type analyses. No stability is considered since the conditions quoted
from such analyses refer to minimum requirements for overall breakeven.
Another early study of the reactor energy balance was done by
Jensen et al.^ Again the D-T reaction was of primary concern though
subsidiary fusion reactions were also treated. Jensen reported on the
effects of finite energy transfer rates and found self-sustaining
reactors were possible over an increased parameter range, although all
ion species were treated at a uniform temperature. The major shortcoming
of Jensen's energy balances was its failure to consider particle confine¬
ment times of diffusion losses. Additional energy balance considerations
were reported by Woods.^
72
Horton and Kammash have also considered energy balances and
operating conditions for the D-T fusion cycle. Since alpha particles are
a significant plasma heating mechanism, energy and particle conservation
equations were introduced for the alphas created in D-T fusion reactions.
Both bremsstrahlung and synchrotron radiation losses were treated along
with the effects of cold and energetic fuel injection. This work was
distinguished by its treatment of several different models for diffusion
and hence several different types of confinement time variation with

-43-
temperature and density, some of which were applied in the later stability
work of Mills15"17 and Ohta et al.18
A similar but more realistic condition than the Lawson Criterion
for minimal operation has been developed by Mills for a system using only
the D-T reaction.15,15 This model is based upon continuous injection of
cold fuel where fusion temperatures are assumed to be supported by alpha
heating. Mills used particle and energy conservation equations for the
ion density as follows:
dn
dt
= S - n/x
(8)
n 5t [l (Ti + Te)] = p(' " P)n20TcQ« " I S(Ti + V (9)
where
T.
i ,e

n = fuel ion density (ions/cm )
3
S = fuel ion injection feedrate (nuclei/cm -sec)
p = tritium fraction of ion density
i = confinement time against all plasma losses including
fusion (sec)
Qa = alpha particle energy from D-T fusion events (3520 keV)
c = fraction of alpha energy retained in the plasma for
heating
= temperature for ion and electron species respectively (keV)
= D-T fusion reaction reactivity or rate coefficient
(cm^/sec) .
DT
For steady-state operation with this model, Mills found that the
following equilibrium condition must be maintained if operating charac¬
teristics are to remain unchanged:

This result is similar to the Lawson condition but more conservative
since only a fraction of the alpha particle energy is retained to sustain
the plasma while none of the neutron kinetic energy is retained. In
addition, the Mills condition is a steady-state condition based only on
the plasma while the Lawson Criterion attempts to account for all in¬
fluences on system efficiency. The constant, c, accounts for energy
losses due to bremsstrahlung and synchrotron radiation. An important
feature of this work is the temperature difference allowed between the
ion and electron species; in general, Mills found that the electron
temperature is elevated due to preferential alpha heating. Figure 4
illustrates the Lawson breakeven region for 35 to 45% efficiencies com¬
pared to the Mills1 equilibrium region (c = 0.8, p = 0.25 and 0.50).^
Since Mills' model is concerned only with alpha heating and radia¬
tion losses within the plasma, energy release to neutrons was not con¬
sidered. Though actual power generation capabilities were not considered
by Mills, comparison with the zero power condition developed in Lawson's
model does indicate net overall power production as expected for
equilibrium operation.
Fusion devices producing values above Mills' equilibrium region in
Fig. 4 can be operated only in the pulsed mode. Similarly, devices pro¬
viding nx-values below the Lawson region can never operate as power-
producing reactors, while those falling between the two criteria will
require energetic injection of fuel. Controlling a reactor to keep it
exactly at the equilibrium condition is the preferable mode of operation
which leads to stability considerations.

nr-Product (sec/ciri )
-45-
1 i i â–  r i r
0 20 40 60 80 100
Ion Temperature (keV)
Figure 4. Comparison of Lawson breakeven and plasma equilibrium regions.

-46-
Plasma Thermal Stability Considerations
Plasma global thermal stability studies were initiated by Mills
15-1 7
based on the operational equilibrium studies. Mills demonstrated
that the equilibrium condition is equivalent to requiring the constancy
of a function, ip, as follows:
il» = Sx2p(l - p)S (11 )
2
where the so-called stability function, S (cm /keV-sec), varies with ion
temperature T. as S cT^ which exhibits a broad resonance peak around
28 keV. In the first approximation Mills treated the alpha energy re¬
tention fraction as a constant. For stable equilibrium, the logarithmic
variation of iKS,x,p,T.j) must vanish. Therefore, Mills found that the
operational equilibrium is unstable against fluctuations in the fuel
feedrate, the confinement time, the fuel mixture (unless p = 1/2), and
the ion temperature except when the exponent in T'j falls to zero above
28 keV.
Although the exact behavior of the confinement time with ion tem¬
perature was not known (nor is it known today), the ^-function formaliza¬
tion showed that if x is independent of T., stable operation of a thermo¬
nuclear reactor below 28 keV is impossible without some form of control.
Below 28 keV, departures from equilibrium are supported due to the posi¬
tive slope of the stability function. It is not until the negative slope
region of the stability function is reached above 28 keV that the in¬
herent instability against fluctuations in T. is controlled and the
temperature driven back to equilibrium.
15
Mills acknowledged a preference for operating at lower temperatures,
73
perhaps near the 12 keV temperature for the optimal D-T reaction rate.

-47-
In fact, most fusion reactor system design studies currently select
operating temperatures below 20 keV. ~ But at temperatures below
28 keV, Mills showed that control is necessary to avoid the predicted
extreme departures from equilibrium. This control can be implemented
via the feedrate, the fuel mixture, the confinement time, or radiation
losses dependent on injection of impurities. Initially, Mills favored
15 17
control via the confinement time ’ but later work has emphasized
feedrate control.^ More recent studies by Ohta et al have confirmed
the use of feedrate as a viable method by which to control stability.
If the confinement time is temperature-dependent, then it may be
useful for inherent control by introducing temperature dependency into
the ^-function. Mills hypothesized Bohm-type diffusion (t ^ T as a
possible inherent control to allow stable operation below the 28 keV
1 5
cutoff indicated for constant confinement operating conditions. For
fixed feedrate and fuel mixture, Mills used the ^-function variational
method to demonstrate inherent stabilization of plasma equilibria for
this Bohm-type diffusion for temperatures in the 7 to 28 keV range. By
analyzing the dynamic behavior of thermonuclear plasmas, Mills estab¬
lished the self-stabilizing influence of Bohm diffusion below 28 keV
temperatures as the perturbed plasma temperatures (ion and electron)
15
and ion density were shown to approach equilibrium with time. In this
way, Mills justified operation near the 12 keV temperature to take ad-
73
vantage of the optimal D-T reaction rate without the necessity of
introducing artificial control.
Mills'*^ also presented detals on calculations to evaluate the time
evolution of the parameters in a fusioning plasma. The calculations
accounted for bremsstrahlung and synchrotron radiation by treating energy

-48-
exchange between ion species as an instantaneous process. Results were
reported only for plasma time behavior for attempted initial equilibrium
operation about a temperature of 11 keV with 50% deuterium and 50% tritium
14 3
fuel injection leading to ion densities of ^ 5 x 10 ions/cm . The in¬
stability of plasma operating conditions in this region was verified for
constant confinement and shown to result in rapid plasma runaway in less
than three seconds. The plasma temperatures (T.. and T ) were shown to
runaway above or below ignition depending to extreme accuracy on whether
or not the constant plasma confinement time was too long or too short so
artificial control was found to be essential below 28 keV.
Mills^7 also investigated feedback control via the fuel mixture
using the monitored plasma electron temperature. When the electron tem¬
perature was set below a preselected control temperature, the injected
fuel mixture was maintained at the original 50% D, 50% T; when the tem¬
perature exceeded the control temperature, tritium injection was replaced
with pure deuterium. The effect of stopping tritium injection was to
reduce fusion events and lower temperature; the stabilizing effect of
this mixture control feedback was achieved by making the time average of
p(l-p) low enough to compensate for excessive confinement time. Control
to a temperature that was too low to provide the nx-equilibrium condition
was found to result in the reacting plasma extinguishing itself. Mills
also noted that excessive confinement time will result in severe initial
temperature overshoot.
These investigations by Mills constituted the first efforts to
study the dynamics and control of thermonuclear reactor plasmas. The
omission of other than D-T fusion reactions and the incomplete treatment
of the synchrotron radiation represent deficiencies in Mills' work. The

-49-
incomplete stability criteria development in Mills' work is its most
significant deficiency.
The same stability problems of point model D-T plasmas have been
18
investigated in more detail by Ohta et al. but using the following
global nonlinear balance equations for plasma density and temperature
(energy):
4j- = S - n/x (12
dt n v
= n2f(T) - — + ST
dt K ' xr s
03)
where
f(T) =
QgDT
12
1 .12 x 10"15 T1/2
and
n
T
Tn,E
S
3T
s
Q
a

DT
3
plasma ion density (ions/cm )
uniform plasma temperature (keV)
particle and energy confinement times (1/sec)
fuel injection feedrate (ions/cm -sec)
fuel ion inject energy (keV)
alpha particle energy from D-T fusion events (3520 keV)
3
D-T fusion reaction rate coefficient (cm /sec) .
Ohta addressed only the D-T reaction; the fusion reaction was not
considered an important loss mechanism in the particle conservation equa¬
tion in essential agreement with Mills. Both the fusion energy source
and the bremsstrahlung energy loss terms were included in f(T) but
synchrotron radiation losses were neglected. Injection energy was
specifically included in the energy equations as 3T$ for convenience of
notation.

-50-
No temperature difference was allowed between the electron and ion
species which is a limitation in contrast to Mills' attempt to treat
differing temperatures. The advantages of Ohta's model include accounting
for energy diffusion with particles and energetic ion injection as well
as including an explicit expression for bremsstrahlung radiation. Ohta
obtained the following form of the Mills equilibrium condition for steady-
state (subscript o) evaluation of the balance equations:
n T r-
o E
E T
1 T
Tn To °
o J
“W
04)
which indicates the reduction in required m-values by the inclusion of
Ohta's injection heating option.
Efforts by Ohta to examine steady-state plasma stability can be
categorized into two areas:
1. Linear analysis establishing temperature-dependent stability
criteria in possible operating regions for future fusion
plasmas, and
2. Nonlinear dynamic simulation of the plasma balance equations
subject to small perturbations with and without feedback
effects to verify agreement with linear stability analysis
and control possibilities in unstable operating regimes.
Linearized analysis will usually predict stability regimes. If a
system is not stable, linearized analysis will not predict true con¬
sequences of the unstable situation--hence the need for dynamic simula¬
tions. Stability criteria to predict whether small plasma perturbations
will grow or diminish with time were developed by Ohta from linearized
forms of the density and temperature balance equations. The elimination
of nonlinear terms is valid only for small density perturbations, ón(t),

-51-
and small temperature perturbations, 6T(t), which Ohta assumed to vary
exponentially with time.
Stability is assured provided the real part of the growth rate is
negative. Ohta obtained general stability criteria by solving for the
growth rate after substituting the density and temperature variations into
the linearized density and temperature equations.
To proceed beyond such general stability criteria, the functional
dependences of both the particle and energy confinement times were re¬
quired. Because the exact density and temperature dependence of confine¬
ment time was uncertain, Ohta based the analysis upon the following
functional dependence of confinement time on density and temperature:
p|i
t ^ n T . It is the derivation of stability criteria on the basis of
this general diffusion model that represents the major contribution of
Ohta's stability analysis. To obtain useful stability criteria, Ohta
used three diffusion models to get specific values for £ and m:
1. Constant confinement: x ^ constant (£ = 0, m = 0).
2. Bohm confinement: x ^ T ^ (i, = 0, m = -1).
-1 1/2
3. Classical confinement: x^n 1 ' (£=-l,m= 1/2).
The minimum temperature satisfying the stability criteria for each
confinement model is known as the critical temperature, Tc; that is, the
temperature above which operating conditions are predicted to be stable
as described by Mills' work. Representative temperature results pre¬
dicted by these stability criteria are listed in Table 1 -XVI for both
charged particle and injection heating for all three diffusion models.
Ohta also dynamically simulated the balance equations to check the
stability predictions of the linear analysis. For these numerical simula¬
tions, initial equilibrium temperatures were assumed; the initial

-52-
densities, however, were perturbed a small amount above and below
equilibrium and the effect on the temporal behavior of the plasma density
and temperature calculated as presented in Fig. 5.
Table 1-XVI
Critical Temperatures for D-T Fusion Reactors
Confinement Model
T (keV)
c
Charged particle Heating
Injection Heating^
(Tn/TE=1)
t = constant
28
21
T-1
i ^ T
14
5
-1 T1 / 2
i ^ n T
42
33
*Ion Injection Energy: 3Ts = 150 keV.
For the case of constant confinement and charged particle heating for
which the critical temperature Tc is 28 keV was found also by Mills.
Ohta's results are depicted in Fig. 5 for three initial equilibrium tem¬
peratures of 10 keV, 30 keV, and 50 keV. For equal magnitude density
perturbations, equilibrium density is always approached with time which
indicates plasma stability under isolated density perturbations.
Similarly, temperature transients resulting from the density perturbations
die out for cases (30 keV to 50 keV) where T > T . However, for the
subcritical 10 keV initial temperature, the time evolution of temperature
is unstable as shown in Fig. 5 and predicted in Table 1 -XVI.
Ohta's dynamic simulations supported the linear analysis stability
criteria for both methods of heating and all three diffusion models. In

-53-
ro
Figure 5. Tine variation of point-model plasma temperature and density
for constant confinement and charged particle heating.

-54-
general, the quick plasma response on the order of a few seconds was
found for all these analyses of unstable plasma variations in pure fusion
plasmas. This behavior agreed with previous fusion plasma analyses.
Ohta's results demonstrate the need for stabilizing control to allow
fusion reactors to operate below the critical temperatures as planned by
28-32
current fusion reactor design studies. The case of feedback
stabilization for the constant confinement model was also examined by
I O
Ohta et al. Stability criteria were again derived from linearized
balance equations. Density feedback control was introduced by adding the
term, a but was not able to stabilize the system because the
o
balance equations are stable for isolated density perturbations. Since
temperature instabilities can grow independently, various types of tem¬
perature feedback were introduced by adding the stabilizing feedback term,
a -j---, to either one or both of the perturbed linearized balance equa-
o
tions. New stability criteria were derived dependent on the value of the
feedback coefficient, a. Although many parameters are possible for use
in implementing control, feedback via the injection feedrate was pre-
1 C
ferred by Ohta et al. in agreement with Mills.
Ohta demonstrated control of the temperature instability through
dual temperature and density feedback which was introduced through the
injection rate and its "small" variation about equilibrium as follows:
an
S(t) - Sq + 6S(t) = S + ^ o
where a is the feedback coefficient and At is the delay time between a
temperature variation, <5T(t), and the corresponding feedback effect on
the injection rate. Inclusion of feedback delay time makes Ohta's results

-55-
applicable to realistic control situations. The effectiveness of feedback
stabilization was found to be dependent on both feedback parameters: a
and At. For the applicable plasma model Ohta found a stabilized region
in the ctAt-plane from the linear stability analysis of this feedback
effect. In general larger negative feedbacks and shorter delay times
were found to yield more effective stabilization. For sufficiently large
At or small a, feedback stabilization was found to be ineffective in all
subcritical (T < T ) cases.
c
As expected, Ohta found the unrealistic case of zero delay time to be
the most effective feedback. However, when the delay time and feedback
coefficient were within the stability region predicted by linear analysis,
an equilibrium temperature was always approached; however, the amplitude
of oscillations was found to increase with delay time as the limits of
the stability regime were neared. Since delay times of 2 to 3 seconds
are outside the linear stability regime predicted for this case, extreme
amplitude of oscillation for these delays was found as expected.
23 24
Usher and Campbell ’ extended point-model thermal stability
analyses to other fuel cycles and other plasma diffusion models with
similar results and speeds of response. In addition burnup was treated
in this extension of Ohta's analysis with essentially similar results for
the D-T fuel cycle.
Stacey's point model plasma stability analysis of the D-T fuel cycle
extended point model plasma stability analysis of the D-T fuel cycle to
include more detailed plasma behavior including four balance equations
22
to represent the following plasma parameters:
1. Ion particle density.
2. Ion energy density.

-56-
3. Alpha particle density.
4. Electron energy density.
Again the temperature instability was found in certain regimes. Effective
stabilization to control operation about an unstable equilibrium point
through use of controlled ion injection rate as well as controlled D-T
fuel mixture was demonstrated.
The temperature instability has also been examined for radially in¬
homogeneous D-T fusion plasmas by Yamato, Ohta, and Mori, using particle
19-21
and energy balance equations. The results of this inhomogeneous
analysis support the validity of decoupling excursions in the overall
particle densities and temperatures from excursions in the spatial density
and temperature distributions. When the injection of fuel is uniform,
the temperature instability can develop only in the zero order mode.
Stability criteria were developed similar to those for the uniform plasma
with similar results, including feedback stabilization through temperature
to allow operation below the critical temperature.
There have been no investigations of hybrid plasmas to examine the
temperature instability discussed in this review. This is an area that
requires study because large hybrid blanket energy multiplication values
coupled with large plasma transients and neutron release could have con¬
trol as well as safety significance.
Motivation for the Research
As is evident from the preceding critical review of hybrid studies,
there are many different versions of hybrids. New studies on hybrids can
either design new blanket models or make use of existing designs with
appropriate changes. The primary objective of this work was to analyze

-57-
the dynamic interaction of the two components of the hybrid system. Such
investigations have not been reported in the literature to date. Thus,
the objective was not to devise a new system but to take the somewhat
arbitrary approach of selecting a previously established hybrid concept
with necessary adjustments.
Many different types of hybrid machines have been proposed with many
different methods of application. Power-producing Tokamak hybrids are of
most interest for control and dynamics considerations and so such a model
was selected for this work. Essentially this hybrid design is compatible
with various hybrid advantages delineated in the recent Princeton Plasma
Physics Laboratory systems study of Tokamak fusion-fission hybrid
reactors which concluded that the most economical mix of power- and
3
fuel-producing hybrids should emphasize power production. An optimized
hybrid machine should be a substantial power producer with a by-product
of fissionable fuel, the optimum ratio of fuel production to power pro¬
duction being determined by economics.
An early demonstration of hybrids could allow a very reassuring
program for future development of the utility industry. A guarantee of
future reasonable fuel costs could promote the accelerated installation
of current LWR plants to fill near-term power needs while loosening con¬
straints on all sectors of the United States energy economy. Subsequent
commercial development of hybrids could supplement LWR's, provide them
with fuel, and take up the load of retired power stations followed by
eventual introduction of the pure fusion reactor sometime in the coming
century.
The hybrid could also prove to be inherently superior to the fast
3
breeder reactor for using the depleted uranium reserves built up from

-58-
enrichment operations run for nuclear power plants and defense purposes.
238
The hybrid may be a better way to burn 1) reserves with possible elimina¬
tion of some enrichment requirements and perhaps elimination of plutonium
separation if bred plutonium is burned in situ. This scenario is
5
especially important in light of the continuing breeder controversy and
11 74
the recent Three Mile Island accident ’ which will undoubtedly delay
introduction of the breeder still longer due to safety considerations.
Since the hybrid represents an alternate concept for power production
and orderly progression to long-range utility application of pure fusion,
its characteristics require analysis prior to its being approved for
central station power production. One parameter frequently used to de¬
scribe hybrid characteristics is the global relationship for the blanket
neutron energy deposition per fusion neutron, QR, derived in Appendix A:
Gf
% â–  = W
v
'eff
- k
eff
-] + E + 6.
(16)
where
Qg = blanket energy deposition per entering fusion neutron
Gf = fission energy deposited in the blanket per fission event
T (192.9 MeV)75
v = average number of fission neutrons produced per fission
event
k = effective blanket neutron multiplication factor
= energy of the fusion neutron (14.06 MeV)
= additional energy generated and deposited in the blanket
due, for example, to exothermic neutron absorption
reactions.
The associated blanket fusion neutron energy multiplication, Mg, is then:
07)

-59-
Several forms of the global relationship of Eq. (16) have been used ex¬
tensively to describe hybrid blankets J >64,76,77 |-|oweverj no results have
been reported on its validity. If the parameter is to be used as a
figure of merit characterizing the multiplicative capabilities of hybrid
blankets, then its applicability must be verified and its limitations
established.
Hybrid blankets are expected to have substantial energy deposition
per fusion event so it becomes imperative that safety studies be undertaken
to examine the implications of this characteristic. For non-multiplying
pure fusion blankets, the energy deposition per fusion event is expected
to be about 20 MeV. For hybrid blankets, even extremely modest ones
with k^^ = 0.8 are predicted by Eq. (16) to have 316 MeV deposited per
fusion event. The energy deposition and fusion energy multiplication
predicted by Eq. (16) for possible blanket k ^ values are listed in
Table 1 -XVII.
Table 1-XVII
Predicted Blanket Global Response per 14 MeV Neutron
Effective Blanket
Neutron Multiplication
k ,,
eff
Blanket Energy
Deposition*
Qb (MeV)
Blanket Fusion Neutron
Energy Multiplication
M
B
0.80
316
22.5
0.85
439
31 .2
0.90
687
48.8
0.92
872
62.0
0.94
1181
84.0
0.95
1429
102
0.98
3654
260
0.99
7364
524
*Based on Gf = 192.9 MeV, v = 2.6 nts/fission, E = 14.06 MeV, and
6E = 4.84 MeV n

-60-
The accepted variation of blanket fusion neutron energy multiplica¬
tion with blanket values of k ^ is depicted graphically in Fig. 6 to
demonstrate the hybrid capability for high energy multiplication with in¬
creasing but still far subcritical blanket systems. Despite the impossi¬
bility of reaching a critical fission reactor state in such systems,
variations in the plasma operating conditions could cause blanket energy
production rates beyond the technical limitations or the technical speci¬
fications of the design. Even with no danger of supercritical behavior,
large uncontrolled thermal instabilities in the plasma could lead to ex¬
cessive energy deposition in the power-producing hybrid blanket. In addi¬
tion, there is the possibility of criticality at low temperatures prior to
power startup. If plasma startup is very quick, then the plasma neutron
production may drive the blanket to large overpower ratings before the
temperature defect can reduce the effective blanket neutron multiplication
factor, k^f.
Although relatively small quantities of thermal energy are contained
in the plasma a full-scale hybrid system generating 6500 MWth of steady-
state thermal power will require large numbers of 14 MeV neutrons. Even
a far-subcritical blanket (k ^ ~ 0.9) can cause considerable multiplica-
eff
tion of the fusion neutrons available as an external source for providing
fission neutrons in the blanket. The component interactions as well as
the control and stability of such power-producing hybrid systems must be
wel1-understood.
The Lawson Criterion for hybrid reactors is modified as follows to
account for fusion and fission sources of thermal power with zero energy
fuel injection into the plasma:

Lawson n-r---Product (sec/cm )
-61-
5 x 10
10
10
Figure 6
4 _
n i i I " i r~~ ~~r
2 5 10 20 50 100 200
Temperature (keV)
Typical Lawson breakeven curve for a 50-50 D-T plasma and 33%
overall efficiency showing relative position of hybrid
systems.

-62-
12 T
(18)
1-n
n
where Qg is the blanket energy deposition per fusion neutron and n is the
50 69
usual overall system efficiency defined for the Lawson Criterion. ’
Obviously, if significant energy is produced in the fissile blanket, the
requisite hybrid plasma parameters can be relaxed to allow earlier utili¬
zation of fusion power in combination with a subcritical fission reactor
to take full advantage of inherent hybrid safety features.
The typical effect of hybrid operation with blanket energy multipli¬
cation is a reduction in the required nx-product is depicted in Fig. 7.
The production of fission energy effectively reduces the need for fusion-
produced energy. The hybrid-revised Lawson Criterion of Eq. (18) is
greatly relaxed because Qg is on the order of hundreds of 1-leV versus the
usual Qg used for pure fusion systems which is limited to about 20 MeV
including exothermic blanket reactions. As noted, this interactive multi¬
plication demonstrates the need to examine the dynamics and controllability
of hybrid systems.
Previous studies have been restricted to steady-state neutron
balance calculations and associated technological limitations. There
has been no analysis of the time-dependent behavior associated with
hybrids, when subjected to reasonable perturbations in the characterizing
parameters. In addition, there have been no reports of analysis of hybrid
plasmas in the reduced reactivity regions where plasmas are not self-
sustaining. The development of a model to describe the dynamics of
fusion-fission coupled systems was one of the primary objectives of this
research. The basis for operation, stability, and control of a coupled

Blanket Fusion Neutron Energy Multiplication
-63-
Effective Neutron Multiplication Factor
Figure 7. Predicted variation of blanket fusion neutron energy
multiplication with blanket effective neutron multiplication
factor.

-64-
hybrid system must be established when subjected to effects such as those
due to the thermal instability analyzed by Mills15-^ and by Ohta et al.1^
for pure fusion plasmas.
The desired result was a hybrid system model whose analysis would
yield useful operational characteristics of hybrid machines which could
then enable the hybrid to make a contribution to power production before
the turn of the century. These various investigations will only be possible
if both the plasma and blanket components are modeled and coupled to allow
dynamics and stability analysis to be performed.
Summary of the Research
18
The research reported here began with the Ohta plasma model with
23 24
burnup effects included after Campbell and Usher ’ and developed plasma
stability criteria based upon source feedrate perturbations and other
engineering considerations for plasma changes affecting the output neutron
production rate. Essentially, an effort was made to develop an analytical
model for pure fusion plasma stability and control based on a global
parameter treatment of a linearized fusioning plasma model using concepts
of classical control theory and transfer functions. Feedback effects
were also incorporated into the model which was kept independent of
specific design concepts. The analytical model and its stability pre¬
dictions were compared with Ohta's results to develop an engineering-
oriented model which could have broad application to more sophisticated
plasma models in the future. Perturbations causing plasma transients
were specifically related to engineering expectations instead of
theoretical hypotheses.

-55-
With the completion of this plasma stability and transfer function
analysis, the effort was extended to develop a simplified hybrid model
from which general stability criteria were developed for the interacting
components of a hybrid system. Again, the model was kept independent of
specific hybrid concepts except that the plasma confinement time was
assumed to be a constant, independent of plasma temperature and density.
The model was specifically developed and related to engineering con¬
siderations of hybrid system perturbations as well as dynamic simulation
and control. Inherent as well as artificial feedback effects were in¬
corporated where appropriate. The entire effort was directed to develop¬
ment of a simple, linearized, closed-loop model in transfer function
format which could be used for future extensions of this work on dynamic
and stability characteristics of hybrids. Of course the nonlinear form
was retained for dynamic simulations.
The hybrid analytical model was then used to examine the properties
of a particular hybrid system. The various augean and symbiotic concepts
2
and variations proposed by Lidsky and analyzed parametrically in the
3
Princeton Study were rejected for this research since they are not
primarily intended for power production. This left essentially two
choices: a fast fission blanket or a thermal fission blanket. To avoid
the possible need for significant enrichments and to take advantage of
expected higher multiplication factors, a thermal fission concept was
selected. The most advanced and promising design was reported by PNL
64
and Livermore workers under Wolkenhauer.
This PNL blanket design was based primarily on existing technology
and intended for use in spherical geometry with a Yin-Yang spherical
78
mirror device for a plasma core. Since this spherical concept is

-66-
severely power-1imi ted, the only substantive change for this research was
the conversion to a Tokamak-driven hybrid versus the mirror-device hybrid
to promote larger power output and allow consideration of thermal in-
stabi1ity effects.
Since the physical arrangement of the hybrid blanket selected cor¬
responded to the reported PNL concept as nearly as possible, the results
of previously performed parametric analyses of optimized region widths,
ordering of zones, and region material constituents were used as the basis
for extending steady-state neutronic analysis of the blanket. The
Tokamak-driven blanket design used is described in Appendix B.
Detailed neutronic claculations were performed on the blanket for
the selected design whose thermal lattice unit cell enrichment and global
temperature were the only varied parameters. The cell enrichment was
varied from natural uranium up to 1.50% enriched while the temperature
was varied from 290°K up to 970°K. This work was performed using the
BRT-1^ (one thermal group) and PHROG^ (three fast groups) codes to get
80
4-group constants. The 4-group CORA diffusion theory code was then
used for criticality calculations and acquisition of fundamental flux
shapes. The doppler defect was also calculated as a function of the
blanket operating temperature. Only the more promising blankets with
keff £ 0.90 at elevated temperatures were considered in detail. This
limitation minimized blanket dependence on the fusion component of the
hybrid system.
Adjoint and perturbation calculations were performed on the system
to provide parameters to characterize the kinetic properties of the
system. Specifically, the average global delayed neutron fraction,
the average prompt neutron lifetime, a , and the so-called neutron

-67-
source weighting factors, c , of the hybrid blanket were calculated using
diffusion theory.
Additional inhomogeneous calculations for blankets driven by planar
sources of group 1 fast neutrons (10 MeV - 0.821 MeV) were used to approxi¬
mate the fission energy source size required to produce a nominal design
power of 6500 MWth. Volume source calculations were also run to investi¬
gate the difference in the worth of the diffusion theory group 1 source
neutron power production depending on the point of introduction into the
blanket. This investigation was accomplished to analyze the validity of
the global parameter relationship for the blanket energy deposition per
fusion neutron presented in Eq. (16). This relationship was expected to
yield reasonable agreement with diffusion theory simulations since the
source neutrons were introduced at nearly fission spectrum energies.
The series of diffusion theory results were used essentially as
scoping calculations to select the best enrichment for further, more
detailed and exact S transport theory analysis using the AMPX code
81
package available from ORNL. The blanket neutronic analysis performed
82
with the XSDRNPM code from AMPX was the first reported application of
the ORNL-developed AMPX package to such hybrid studies. In P-^-S^ analysis
using the AMPX package, the fusion neutron source energy was treated more
nearly as a true 14 MeV source. The required source strength for pro¬
ducing the 650 MW design power was determined for the toroidal system
to establish finally the applicable degree of validity expected in cal¬
culating or predicting the blanket energy deposition per entering fusion
neutron using Eq. (16). The flux shapes were also investigated again but
for a six energy group structure to more exactly model blanket effects
caused by energetic fusion neutrons.

-68-
On the basis of the XSDRNPM-predicted fusion neutron source strength
required for a 6500 MWth hybrid plant, the required plasma conditions were
estimated. The corresponding plasma temperature, density, constant con¬
finement time, source feedrate, and injection energy characteristics were
then parametrically varied to establish reasonable hybrid plasma operating
conditions. Perturbations in various parameters with emphasis on plasma
feedrate and temperature were then simulated to investigate the thermal
instability of the hybrid plasma and the results compared with stability
predictions and expected dynamic behavior under transient conditions. In
this way the plasma component of the hybrid plant was examined with
respect to the thermal instability to establish operational characteristics
necessary for planning proper deployment of hybrid power plants.
Finally, since hybrid plasmas are expected to be subjected to various
transient phenomena, especially thermal instability-driven transients,
time variations in the design magnitude of the 14 MeV neutron source
driving the blanket were considered on the basis of those transients
resulting from the perturbed behavior of the hybrid plasma examined pre¬
viously. Kinetics calculations representing the effects of plasma-caused
perturbations on the fusion neutron source driving the blanket were run
and changes in power level were examined for one spatial dimension and
six delayed neutron groups. These kinetics calculations were performed
oo
using the space-time kinetics code GAKIN II with six neutron groups
whose group constants were obtained from the previous XSDRNPM, P^-S^
calculations. Although no time-dependent feedback effects were examined,
the speed of response of the system was determined for typical transients
and some characterisecs for hybrid operational controllability were
established.

CHAPTER 2
THE PLASMA MODEL
Introduction to the Plasma Model
First generation fusion power plants are expected to utilize the
basic deuterium-tritiurn (D-T) fuel cycle
^D + ^T v 2He (3-52 MeV) + Jn (14.06 MeV) . (19)
Because of its large cross section and reactivity, its minimized plasma
temperature requirements and its relatively large energy release per
reaction, no other fuel cycle is given serious consideration for use in
early pure fusion power reactors. Certainly the near term experimental
29 32 84
and demonstration fusion power systems are expected to use D-T fuel. ’ ’
The United States Department of Energy effort toward implementation of
central station fusion power plants has clearly recognized the superiority
of this fuel cycle in the overall development programs.^“^7
Even the utility industry has recognized the need for future choices
in types of power generating systems and is supporting the effort to
88
develop fusion reactors using the D-T fuel cycle. The major magnetic
confinement efforts to produce fusion power in other countries have also
89 90
been directed toward the D-T fuel cycle. ’ Even so, D-T fueled fusion
power plants are not expected to have significant impact on the utility
91
industry until well into the next century.
-69-

-70-
The complexity and difficulty involved in achieving fusion power is
amply demonstrated in full scale commercial fusion power plant design
28-30 92 93
studies. ’ ’ Because economic fusion power is such a large
technological challenge, no factor can be dismissed which will made the
development proceed more easily. The one common factor in different
designs for fusion power plants in a closed, steady-state mode of opera¬
tion (Tokamak) has been the universal selection of the D-T fuel cycle.
Hence, although the D-T fuel cycle has the drawback of producing high
energy, penetrating neutrons, its other advantages make it the only
serious choice for fusion fuel for many years.
1 5
Mills demonstrated that the fusion reaction rate and fusion power
production are maximized for thermonuclear plasmas which have a 50°l
deuterium-50% tritium fuel ion composition. This 50-50 D-T mixing ratio
is the most favorable fuel cycle for the production of fusion energy.
With this cycle, not only is the demonstration of scientific breakeven
in a self-sustaining fusioning plasma more easily accomplished but the
steady-state production of net energy in a fusion power plant can be
accomplished at minimized levels of plasma particle density, temperature,
and confinement time.
These inherent advantages of the D-T fuel cycle in reducing plasma
requirements have been illustrated in various analyses of equilibrium
requirements and conditions including those of Lawson in which the nx-
50
criterion for energy breakeven was first presented. The superiority
of the 50-50 D-T fuel cycle for reaching and maintaining thermonuclear
power-producing conditions has been uniformly demonstrated in extensive
analyses of thermonuclear, steady-state, net energy-producing systems as
94 71 18
reported by Mills, Woods, Ohta et al., as well as Usher and
Campbell.23,24

-71-
Because fusion-fission hybrids are expected to serve as an inter¬
mediate energy-producing stepping block between current LWR plants and
the eventual development of pure fusion power, the usual 50-50 D-T fuel
cycle was logically selected for this hybrid analysis. This choice was
aimed at optimizing the time scale for the implementation of the hybrid
power-producing concept.
The Point Model Plasma
In this work, time-dependent point model balance equations were first
established for the plasma ion density, n(t), the plasma energy density,
3n(t)T(t), and the volumetric plasma neutron production rate, 0p(t);
these three balance equations for the plasma ion (particle) density,
temperature, and neutron production rate state variables are presented
as follows:
Plasma Ion (Particle) Density:
dn(t) = sft\ n(t) n (t)-qv-DT
dt t 2
(20)
Plasma Energy Density:
d[3n(t)T(t)]
dt
\i n (t)nTQ
DPa
+ Ts(t)S(t)
bn(t)T1/2(t)
(21)
Plasma Volumetric Neutron Production Rate:
qp(t) =
n¿(t) DT
4
(22)

-72-
Conventional definitions for symbols used in these nonlinear point model
equations are listed below:
O
n(t) = plasma ion density for 50-50 D-T plasma (ions/cm )
T(t) = plasma temperature (keV)
S(t) = external fuel volumetric injection rate (ions/cm sec)
q (t) = volumetric fusion neutron production rate (#/cm sec)
Ts(t) = external particle (ion) injection energy (keV)
= particle confinement time (sec)
t£ = energy confinement time (sec)
Q = completely plasma confined fusion-produced alpha particle
energy (3520 keV)
3
pT = D-T fusion reaction rate coefficient (cm /sec)
b = proportionality coefficient for plasma energy loss rate via
Bremsstrahlung radiation (3.36 x 10~^5 cm3 keV'/2/sec)18
Since the plasma in this analysis was treated as a global system,
only a single average plasma temperature was considered; that is, no
distinction was made between ionic species or between ion and electron
temperatures. The inclusion of the burnup term in the plasma ion density
18
equation is an improvement to the model used by Ohta that has been in-
23 24 95
corporated by others. ’ ’ In stability studies on pure fusion devices
this burnup term and its effects have frequently been neglected because
burnup causes small changes in the stable temperature operating regimes of
D-T fusion systems. This analysis was intended for application to a
hybrid system where most of the energy would be produced in the blanket
so burnup predictions were even lower than in pure fusion devices; that
is, plasma temperature and plasma density are both expected to be lower
for hybrid plasmas than for pure fusion plasmas. However, large tempera¬
ture variations in perturbed nonlinear systems can occur at which point

-73-
burnup increases due to temperature increases will be directly respon¬
sible for lowering neutron yields which are proportional to the square
of the ion density. All of the alpha particle energy produced in fusion
was assumed to be deposited within the plasma to help heat the system.
15
Others have assumed fractional deposition, but there is no loss of
applicability in assuming full alpha energy deposition.
For this initial analysis of point model kinetics, the plasma volume,
V , was treated as a constant; for linearized stability analysis, this is
adequate because only small plasma system perturbations were considered.
For time-dependent, nonlinear analysis energy and neutron production are
overpredicted by the assumption of constant volume since both are pro¬
portional to the square of the ion density. More detailed analyses in
the future will incorporate temperature-dependent as well as magnetic
and other dynamic conditions that can affect the volume occupied by the
plasma independent of whether the plasma density has changed. Some pre¬
liminary global analyses of such plasma volume variations have been
19-21 96
reported for pure fusion models ’ and additional work is under-
27 97 98
way. ’ ’ The analysis was directed ultimately to the kinetic behavior
of the hybrid so the inclusion of the added complication of a variable
plasma volume in this initial treatment of the plasma neutron source
driving a power-producing blanket was not justified.
The inherent behavior and characteristics of the point model fusion¬
ing plasma used for analysis in this study is completely described by
Eq. (20), Eq. (21), and Eq. (22). In fact, the plasma response to any
input perturbation, as well as its equilibrium characteristics are
determined by only the first two equations relating ion and energy
density. However, since this analysis of hybrid reactors was concerned

-74-
with the driven nature of the hybrid subcritical blanket, the third
equation for the specific neutron production rate was also necessary;
without neutrons produced in and hence output from the plasma, no inter¬
action is possible between the two component halves of the hybrid system.
Note that these neutrons are produced in the plasma and inherently drive
the blanket; however, there is no inherent reverse effect whereby the
plasma is affected by the neutrons themselves or by the blanket itself.
The neutrons and their effects are strictly feedforward in nature.
The volumetric neutron production rate, q (t), is an intrinsic
P
variable--characteristic of the condition of the plasma represented by
the state variables of ion density and temperature only. The volumetric
neutron production rate was multiplied by the effective plasma volume,
Vp, to obtain the total plasma neutron production rate, Q (t), as
follows:
Qp(t) - qp(t) • Vp (23)
where the total neutron production rate is an extrinsic variable charac¬
teristic of a specific plasma with constant effective volume, V . In
other words, Qp(t) is characteristic not of all plasmas in a state de¬
scribed by an ion density and a temperature, but only of those specific
plasmas whose volumes satisfy Eq. (23). This extrinsic variable could
be useful for relating specific size plasmas to the corresponding hybrid
blanket; however, for this general development, the volumetric neutron
production rate was more useful since it is the intrinsic variable from
which any specific pure fusion or hybrid plasma can be analyzed. Indeed,
if pure global analysis is used throughout a fusion plasma study or even
a hybrid study, then any effect on total power production of such a

-75-
constant volume will be simply multipiicative--the larger the plasma,
the greater the system power production.
The density equation was rewritten in the following simplified form:
= S(t) - - f,(T)n2(t) (24)
Tn
where the temperature-dependent coefficient, f^(T), was defined as follows
to simplify the burnup loss term:
<0V>nT
fi(T) = —• (25)
Similarly, Eq. (21) for the plasma energy density was also simplified
preparatory to linearization by rewriting it in the following form after
Ohta:18
dMtmtii = f2(T)n2(t). "ííiiíti + Vfi {26)
where the temperature-dependent coefficient, f2(T), was used to account
for charged alpha particle heating and bremmstrahlung radiation,
respectively:
f2(T)
nTQ
D not
12
bT1/2
3
(27)
Although it was not so complicated, the equation for the volumetric
neutron production rate was also redefined as follows:
qp(t) = g(T)n2(t) (28)
where the temperature-dependent coefficient, g(T), was defined as follows:
g(T) =
4
(29)

-76-
It is noteworthy that g(T) in the neutron production equation and
f 1 (T) in the burnup term of the particle density equation differ only by
a factor of two (2) as follows:
g(T) = 2 fi(T) (30)
which simply means that two (2) ions must undergo fusion burnup for each
neutron produced.
The Linearized Plasma Model
The global plasma equations were 1 inearlized in order to facilitate
analysis of stability regimes in the frequency domain. At this point,
18 23 24 95
contrary to previous work, ’ ’ ’ specific perturbations were intro¬
duced into the point model plasma equations. Since the feedrate, S(t),
is the only external influence appearing in both the density and tem¬
perature point model equations, the feedrate was chosen as the typical
source perturbation for the examination of global plasma stability. The
choice was logical since the driving force for the entire fusioning energy
producer is ultimately supplied by the plasma feedrate. The same depen¬
dence on feedrate is applicable for the hybrid system, since the hybrid
will be entirely dependent for energy production on the plasma-produced
neutrons because of the blanket subcriticality. But the production of
neutrons is ultimately governed by the state of the plasma (ion density
and temperature) which itself is driven and sustained by the feedrate of
energetic fuel ions. Therefore, examination of the hybrid system response
to perturbations in the plasma feedrate is logical for such a global
plasma model.

-77-
For inherent stability and control analyses, the system response to
small external or internal perturbations was a primary concern. Depen¬
dent variable perturbations about steady-state values were used to generate
a dynamic variation in the point model equations. For linear analysis
the following necessarily small variable perturbations were used:
n(t) = nQ + 6n(t) ,
(31a)
T(t) = Tq + fiT(t) ,
(31b)
S(t) = SQ + 6S(t) ,
(31c)
yo * % + %(t>
(31 d)
where the subscript "o" was used to designate a system variable at an
initial steady-state equilibrium value about which a small perturbation
in the variable, represented by 6-terms was introduced so the system could
be subsequently examined for stability in linearized form. In other words,
the time-dependent arbitrary perturbations in ion density, Sn(t), plasma
temperature, 6T(t), source feedrate, 6S(t), and volumetric neutron pro¬
duction rate, óq (t), were required to be small to validate the lineariza¬
tion of the point model equations. These perturbed variables were sub¬
stituted into the point model dynamics equations along with first order
linear expansions for all the density- and temperature-dependent coeffi¬
cients in these equations. The objective was to obtain linearized
perturbed equations from the three original nonlinear plasma dynamics
equations for the plasma ion density, temperature, and volumetric neutron
production rate.

-78-
The inverse confinement time coefficients were examined using pro¬
cedures from previous analyses of global plasma behavior16,23,24
Both the particle and energy confinement times were assumed to depend
exclusively on the plasma ion density and temperature state variables:
Tn?E = F(n,T) . (32)
1 g
Therefore, following the example of Ohta, both inverse confinement times
were expanded about an initial steady-state in first order Taylor series
in the dependent density and temperature variables as follows:
1_
T
n
8(7-)
n
9n
3(t")
<$n(t) +
9T
0
6T(t)
and
1 _ 1 + 1 6n(t) + 1 6T(t)
T T T -I n Tti T
n n nl 0 T1 0
0
1_
tE
9n
6n(t) + y— 6T(t)
0 0
(33a)
1 _ 1 + 1 ¿n(t) + 1 T r Tr £ , n £xn T
E EQ nl 0 T1 0
(33b)
The constants 1/x^^, 1/i.p, 1/e^, and l/r-^ were used to reduce the
complexity of analytical manipulations. The subscript "0" was used again
to denote quantities in an initial steady-state condition about which the
system was somehow to be perturbed and subsequently examined for stability
in linearized form.

-79-
Each of the other three temperature-dependent coefficients (no
density dependence) was also expanded in a simple linear Taylor series.
From the particle equation the temperature-dependent coefficient was
expanded as follows:
3MTl
fl(T) = fl(To} + 3T~ 6T(t) • (34)
0
The temperature-dependent bremsstrahlung and alpha heating coefficient
in the energy density equation was expanded similarly:
f2m = f2d0) +
3f2(T)
9T~
6T(t)
0
(35)
Finally, the temperature-dependent coefficient in the equation for the
volumetric neutron production rate becomes:
g(T) - g(T0) +
<5T(t)
0
(36)
A linearized system of plasma equations can now be produced by substitu¬
tion of the perturbed variables from Eq. (33) and the various coefficient
expansions into the plasma model equations.
Substitution of the first order variable and coefficient expansions
into the plasma particle equation yields the following equation:
d dt
S + 6S(t) - [—
o v Ll
1 _ 6n(_t). 1 (ST(t)
nl
iTLLi][nn + cSn(t)]
1 Ti I U
TI o
^(To^ +
3f1(T)
8T
<5T(t)][nQ + 6n(t)]2
0
(37)

-80-
The steady-state condition in the expanded particle equation ivas eliminated
using the steady-state equilibrium condition.
Two additional coefficients were defined from the effect of including
burnup via f-j(T) in this model; specifically, the effective confinement
time for particles due to any loss mechanisms is reduced by including
burnup effects. Certainly, fusion of particles is a loss mechanism. The
subscript "b" was used in defining inverse confinement time terms to
account for the increased loss of plasma ions by fusion as follows:
and
= 2n f,(T )
o 1 o'
(38)
9f-,(T)
noTo 3T
0
(39)
3y eliminating the steady-state solution and neglecting all terms above
the first order in the perturbed variables and coefficients, the following
1inearized equation for the perturbed plasma ion density was obtained:
1 dón(t) _ ¿S(t)
n dt n
o o
rJ_ + JL + _!_] «HU). r_L + _L] ¿T(t)
Tn Tnl Tbn no tT1 X To
o 1 ¿
.(40)
The inclusion of burnup results in an additional burnup-dependent
inverse confinement time term as well as the usual density- and
temperature-perturbed terms. This dual effect follows directly from
the dependence of burnup on both state variables. Note that the addition
of inverse confinement time terms results in lowered overall particle
confinement time as expected.

-81-
Next the variable and coefficient expansions were substituted into
the energy density equation to obtain:
gt i[nQ + 6n(t)][T0 + 6T(t)]} = [nQ + 6n(t)]2[f2(T) + 6T(t)]
0
i T S(t)
- [nQ + <5n(t)][Tq + <5T( t) ] [-■— + enl 6n(t) + e-^ 6T(t)] + —^
ro
(41)
where the injection energy, T , is constant; allowance for variation in
this injection energy term is another possibility for future analysis of
fusioning plasma stability and dynamic response.
The steady-state solution was eliminated from Eq. (41) as usual.
All terms above first order in perturbations were also eliminated since
products of perturbations are negligible with respect to first order
terms. In parallel with Ohta's work, the following two coefficients
l/x-j and 1 /t2 were defined to simplify the analysis:
and
, , T
— = f- [2n f,(T ) - —
t T L o 2V o e ,â– 
oo nl
1_
t2
i-[n2
n o
o
3f2(T)
~TT
o
n
]
T1
(42a)
(42b)
After rearrangement of terms, the following linearized equation was ob¬
tained for the perturbed plasma energy density equation:
1_ d*n(t) , 1_ d6T(t) = /]_
n dt T dt i,
o o
6n(t) + (1
4T(t) +
T
o
Ts 6S(t)
3T n.
o 1
(43)

-82-
The equation for the volumetric neutron production rate was also
linearized; substitution of the first order variable and coefficient
expansions yielded:
+ iq„(t) = [g(T ) +
3T
(44)
0
Since all terms above first order in the perturbed variations are negli¬
gible with respect to first order perturbation terms, these higher order
terms were neglected. The usual steady-state condition was also eliminated
to obtain the linearized perturbed fusion neutron source equation:
<5q (t) = 2n g(T )6n(t) + n^ -^-1—
Mpv oav o o sT
<5T(t)
(45)
0
The three linearized point model plasma equations which account for plasma
input as well as output are summarized below:
Plasma Ion Density:
1_ dón(t) + , 1 + _L + J__) 4n( t) + / 1 + 1 i ¿T(t) = ¿S(t)
n dt t , i, ' n vxT1 t, ' T n
o n ni b, o TI b0 o o
o I 2
(46)
Plasma Energy Density:
1 d (Sn (t) 1__
n dt T
o o
dt
= /]_ 1 \ ¿n(t) /I 1 , fiT(t) s 6S(t)
t-i " tc ‘ n W " tc ; T 3T n
1 o 2 E o oo
o o
Plasma Volumetric Neutron Production Rate:
6q (t) = 2n g(T )ón(t) + n^
Mp' os o o 3T
6T(t)
(48)

-83-
The stability of these equations was examined in the Laplace or
frequency domain using the standard applicable methods of classical con-
99
trol theory. By Laplace transforming these three linearized differential
equations and rearranging terms, the following set of three linear
algebraic equations was obtained in the frequency domain:
An (s)
[s +
J_ + J_j + Ms}
Tnl Tb, To
r_l_ + 1 ~i _ AS (S )
ltTi t, j n
TI b^ o
(49)
An(s)
[s +
T1
AT(s)
T
[s +
-k1’
3T.
AS(S)
n
o
(50)
AOp(s) -
2n g(T )An(s) + n'
o o c
MU
9f
AT(s)
(51)
This set of three algebraic equations can now be analyzed to determine
stability information about the plasma which is modeled by these equa¬
tions. This stability analysis was performed in the frequency domain
not only because algebraic equations are now involved instead of dif¬
ferential equations, but also because the methods of stability analysis
of linearized systems of equations are easily applied in the Laplace, s,
domain. Since this simplified system of three algebraic equations con¬
tains four (4) unknown transformed variables in the frequency domain, the
ratio of any pair of dependent perturbations is easily determined. All
other parameters were assumed to remain constant, although future analyses
may allow the injection energy as well as the plasma volume to be time-
dependent variables.

-84-
As noted previously, the driving input perturbation to this plasma
system was represented as a variation, 6S(t), in the feedrate. The cor¬
responding output response is a perturbation or variation, óq^t), in the
volumetric neutron production rate. There are other internal system
changes in response to changes in feedrate such as variations in density
or temperature, but the final plasma output response to a change in feed-
rate is a change in the neutron production rate. The changed neutron
production rate is then effective in altering parameters such as energy
production and temperature in the subcritical blanket surrounding the
plasma in the basic hybrid design. Previous anslysis by Ohta has effec¬
tively considered the temperature change as the output; however, for
hybrid analysis, the neutron production rate change is a more important
engineering system output, for which the temperature is only a partial
system indicator. Of course, for the hybrid power producing blanket, the
plasma neutron production rate is the important system parameter
ultimately.
Transfer Function Representation of Plasma Characteristics
The linearized system stability characteristics, which govern the
output response of the system to an arbitrary input perturbation, are
contained in the transfer function, t(s), for the system. This transfer
function, with its characteristic system stability information, is
defined as the ratio of the output to the input variable perturbations
in the s-domain for this plasma system:
T(s)
Aq (S)
Asirr
(52)

-85-
So, the transfer function for any system or element within the
system is defined as the transformed output of that system or element
divided by the corresponding input in the frequency domain; that is,
ignoring all initial conditions, the transfer function, T(s), of the
plasma system is defined as that factor which when multiplied with the
Laplace-transformed input, AS(s), yields the Laplace-transformed output,
Aqp(s). This linearized plasma model is represented by the open-loop
block diagram in Fig. 8 for which the output, Aq^s) is obtained by the
simple algebraic multiplication of the system transfer function and the
input perturbation, AS(s).
The stability of a time-invariant linear system is determined from
the characteriStic equation. The denominator of the system transfer
function set equal to zero is the characteristic equation. Conse¬
quently, if all the roots of the denominator have negative real parts,
the system is demonstrated, within the constraints of the linearization,
to be stable. Therefore, the system or element within an overall system
characteristic information concerning stability is contained in the
transfer function, t(s), for the overall system or an elemental part
of the system.
Basic block diagram and transfer function methodology in the
frequency domain were used to show how the input perturbation, 6S(t),
in the plasma feedrate can cause a corresponding output perturbation,
qp(t), in the volumetric neutron production rate. This effect is shown
in Fig. 9. The block flow diagram in Fig. 9 is based upon the re¬
arrangement of the relation for the linearized volumetric neutron
production rate of Eq. (51) into the following form:

PLASMA SYSTEM CHARACTERISTICS
Figure 8. Transfer function formulation for a point-model fusioning plasma.
-86-

AS(s)
Figure 9. Block diagram for the point-model plasma system.
-87-

-88-
(53)
O
The representation of transformed equations such as Eq. (53) in block
diagrams such as shown in Fig. 9 is straightforward because of the
algebraic method of operation of transfer functions in the Laplace
domain.
A more easily analyzed equivalent block diagram of Fig. 9 is
shown in Fig. 10 after application of the appropriate block diagram
transformation. The transfer function for pairs of dependent variables
denoted as ratios in the block diagram were determined from the system
of algebraic equation. The transfer functions for the plasma system
with no feedback as they are presented here are all ratios of poly¬
nomials. The following two polynomial ratios required to analyze the
frequency response of the system in the block diagrams of Fig. 9 and
Fig. 10 were obtained:
+ a£S + a^
(54)
(55)

-89-
where the coefficients a^, a^, a^, and a^ are given by
T
1 1
5 (-1- + r1-) >
t2 3To tT1 Tb
(56a)
a„ = — — + —+ 1 1
2 T I- X2 Tr
Tnl Tb] tT1 Tb2
(56b)
a, = (-L + -J- + -L)(-L - ]—) + (J- + -J-)(^ !-) , (56c)
3 Tn Tnl Tb, tE t2 tT1 V T1 tE
o lo 2 o
a4 1 ' 3T
(56d)
5 T r-
t
1 1
T
T
'0
i 3T ’T T , Tl
on nib,
o
(56e)
The elemental transfer functions An(s)/AS(s) and AT(s)/An(s) given in
Eqs. (54) and (55) as well as the simpler factors,2nQg(To) and n3 |^- ,
operate in the transfer function methodology via algebraic multiplication
of transform inputs to produce the various transform outputs shown in the
block diagram in Fig. 9. A third elemental transfer function is required
to complete the block diagram representation of the linearized set of
three trnasformed equations applicable for this plasma model; this trans¬
fer function is presented as the only remaining ratio of perturbed,
frequency-dependent variables as shown in Eq. (57):
AT(s) =
AS(s )
T T
- ir-cd - 5
3T + a4^
+ a2s + a^
(57)

-90-
The verification of the identity presented in Eq. (57) was used to check
not only the algebraic multiplication of these elemental transfer func¬
tions but also the correctness of the transfer function relations in
Eqs. (54), (55), and (57):
An(s) _ aT(s) _ An(s)
aS(s) aS(s) ' AT(s)
(58)
where ^y|is simply the inverse of | j) j from Eq. (55).
The transfer function, is presented for completeness because
the linearized neutron source Eq. (51) can be rearranged as follows in
contrast to the form in Eq. (53):
qp(s) = 2nQg(To)An(s) +
2 12.
o 3T
AT(s)
An (s)
An(s)
(59)
The point model plasma system using this form of the linearized
volumetric neutron production rate is represented by the somewhat dif¬
ferent block diagram shown in Fig. 11. The requirement for the transfer
function element, |~j|y, is explicitly indicated. The block diagram of
Fig.. 11 is reduceable to the simplified form shown in Fig. 12 which again
is different from its equivalent counterpart in Fig. 10. The algebraic
nature of the equations allows formulation for optimum system coordina¬
tion.
The two reduced block diagram representations of Fig. 10 and Fig. 12
are completely equivalent since each reduces to the overall plasma model
shown in Fig. 13 where the input is a transform perturbation in the
feedrate and the output is a corresponding transform perturbation in
the volumetric neutron production rate in the plasma.

2n g(T ) + n^
oa 0 o 3T
. AT(s)
0
Figure 10. Partially-reduced block diagram for the point-model plasma system.
-91-

aS(s)
Figure 11. Alternate block diagram for the point-model plasma system.
-92-

2 32
O 3T
2n g(T )
oa o
An(s)
atTTT
Aq (s)
>
Figure 12. Partially-reduced block diagram for the alternate point-model plasma
system formulation.
-93-

Figure 13. Reduced open-loop block diagram for the point-model plasma.
-94-

-95-
The overall open loop transfer function for the point-model plasma
system is presented in Eq. (60) as the perturbed neutron source output
divided by the initiating perturbation in the plasma feedrate as follows:
%
stir
o aT
0
aT(s)
AS(s )
2n g(T )
o3 o
An(s)
AS (S )
(60)
The elemental transfer functions, An(s)/AS(s) and aT(s)/aS(s), and Eqs.
(54) and (57) were used to evaluate the overall open-loop plasma transfer
function. After substitution and simplifying rearrangement of terms,
the overall transfer function for the point model plasma without feedback
becomes
qqp(s)
aS(s)
s[2n g(T ) - n T ^
o3V o o o 3T
a„] + 2n g(T )a, -
4J o3 o 1
+ a2S + a3
n T
o o
iS.
3T
• (61)
Equation (61) is the overall open-loop plasma source feedrate transfer
function; this result is similar in application and meaning to the source
transfer function derived for fissile subcritical assemblies where a
neutron source perturbation causes a changed flux or neutron level in
99
the system.
Stability Analysis of the Linearized Plasma Model
Individual elements in block diagrams used to model mechanical,
electrical, or other physical systems are usually required to be stable--
that is, to have stable impulse responses. In contrast, the individual
elements in the open-loop plasma block diagrams in Fig. 9 and Fig. 11 do
not necessarily have stable impulse responses. For example, the elemental

-96-
transfer function, An(s)/AT(s), may well be unstable when considered
independently since it has the same number of zeroes and poles. This
situation occurs because of the effort to compartmentalize the plasmas
system and its describing linearized equations in the Laplace domain in
order to analyze the overall system. No such independent element exists
in the plasma; both the density and the temperature state variables are
interdependent and this separation has no clear physical meaning. How¬
ever, this overall transfer function was still analyzed to determine the
conditions for system stability which are contained in the characteristic
equation of the overall system transfer function.
The system characteristic equation was obtained from the system
transfer function by setting the polynomial in its denominator equal to
zero as follows:
0 = s^ + a„s + a_ = s^ + s(—— - —
2 3 Tr T,
nl
T1
+ (— + +
: -i T.
nl bn
-) + (: 1
T1
(62)
This characteristic equation was used to determine the open-loop stability
of the modeled plasma; that is, if all the roots of the denominator have
negative real parts, the system is inherently stable to small disturbances
in the feedrate. The Routh Stability Criterion was selected from the
several methods of stability analysis available from classical control
theory. The Routh array in Fig. 14 was constructed from the coefficients
of the character!'stic equation.

-97-
1
a
3
a2
0
a3 0
0
Figure 14. Routh array for open-loop point model fusioning plasma
with burnup.
Within the limits of the assumptions of small perturbations to
obtain linearized equations, the requirements of the Routh criterion for
absolute stability are that there be no sign changes in column 1 of the
array. Therefore, for the plasma system model represented by the open-
loop black diagram of Fig. 9, the inequality requirements for absolute
stability are twofold:
a2 xr
— + — + — + 1
1 1
[ , T, IT1
nl b.
> 0
(63a)
and
3 T T n T, Tr To Txn
n nl b, E 2 T1
o 1 o
1 â– )(!-- JL) >o
T, ’ ' Tn Tr
b2 1 E(
(63b)
18
These stability criteria are identical to those of Ohta but were ob¬
tained differently using a system-related phys8cal model of the plasma
behavior and its sources of perturbations via the feedrate. The two
conditions of Eqs. (63a) and (63b) represent general stability criteria

-98-
for the modeled plasma. Unless some assumptions are made concerning
diffusion properties in the plasma, this type of analysis cannot proceed
any further.
Because understanding and treatment of diffusion in fusioning
plasmas are clouded by anomalous behavior and the lack of data on diffusion
in full scale thermonuclear plasmas, plasma diffusion behavior has been
represented by a wide range of different analytical models. The three
100
most frequently used analytical diffusion models are
1. Constant confinement where t r = constant.
n,E
-1 1/2
2. Classical confinement where t c n T .
n,E
3. Bohm confinement where x r <* T ^ .
n,E
These three postulated types of diffusion can be used to model a wide
range of expected possibilities for diffusion in future fusioning plasmas
when they are produced for steady-state operation. The constant con¬
finement case has frequently been chosen not only for simplicity but also
because its predictions of diffusion losses fall between the most en¬
couraging model (classical) and the least advantageous model (Bohm) for
attaining fusion reactor conditions.
18
Following Ohta et al. the confinement time was conveniently
represented in all these diffusion models by a general dependence on
density and temperature:
i * nV1 (64)
where (a) z = m = 0 for constant confinement, (b) z = -1, m = 1/2 for
classical confinement, and (c) Z = 0, m = -1 for Bohm confinement.
The assumption of constant confinement time was made to simplify
the open-loop transfer function of Eq. (61) by the elimination of all

-99-
terms containing the inverse confinement time expansion terms 1/t, and
1/tj-j. The following resultant open-loop transfer function was obtained:
Vs>
iSisT
s[2n g(T ) - n T
o3 o o o 3T
a4] + 2n0g(T0)ab - ff
0
s + a8s + ag
(65)
where the reduced coefficients are given by
a6 T
11 si
E T2 3To Tbo
o 2
(66 a)
ay - -L- - (-1- + -±-) ,
' Tr TI O1,
'0
H J o Tn Tb,
o
(66b)
a = J_ . 1_ + J_ + _L J_
8 tE t2 Tn Tb, Tb„
o o 1 2
(66c)
.9-(-L + -L)(-L-i-)t-L(i-.-L,
9 T T. Tr To Tl Tt Tr
n b, E„ 2 b0 1 E
o 1 o 2 o
(66d)
The linearized response of the open-loop plasma volumetric neutron pro¬
duction rate to a small perturbation in the plasma feedrate is contained
in this transfer function. The characteristic equation to be examined
for stability was reduced to the simplified expression in Eq. (67) which
was obtained by setting the denominator of the transfer function of
Eq. (65) equal to zero:
° = s2 + a8s + a9 =
s2 +s(——
T r To T
2 n„
o o
rL) + (-L + TL)(
n 'b,
o 1
—-—) +_1_(1_ __L
tE t2 Tb, T1 tE.
■) •
"o
(67)

-100-
The further simplified Routh array for Eq. (67) is presented in Fig.
1 5.
a8 ^
ag 0
0
Figure 15. Routh array: Open-loop plasma model with constant con-
f i nement.
The dual stability requirements become simply:
and
a8 t r
> 0
(68a)
= (:
—h^-
-i ♦ J- (L
-) > 0
(68b)
When the effects of burnup were also neglected, then the two inverse
confinement times accounting for burnup disappeared so that the stability
requirements become simply:
t2
> 0
(69a)
and

-101-
Tz‘
(69b)
18
which are the same stability results as those obtained by Ohta. Here,
however, the results were obtained in a manner related directly to
reasonably expected perturbations of the plasma feedrate from the
engineering viewpoint of affecting the resultant volumetric neutron
production rate emitted from the plasma to drive the blanket for energy
production. Further, this is a completely general approach which can be
applied whenever such plasma systems are analyzed.
Typical steady-state fusioning plasma equilibrium conditions were
used to demonstrate that the burnup-related terms, 1/t. and 1/t, , are
o 1 d2
negligible with respect to the equilibrium inverse particle confinement
time, 1/t , as well as the equilibrium inverse energy confinement time,
1 o
1/t^ . The negligible effect of burnup for both pure fusion designs such
0 30
as UWMAK-III and hybrid designs such as that developed in this study
was easily demonstrated using equilibrium operating conditions for the
two types of systems: pure fusion and hybrid fusion-fission. This
comparison was simply based on comparing particle confinement times ex¬
cluding burnup effects to particle confinement times including burnup
effects. In other words, the characteristic fusion time, ip, is defined
as follows:
= 2
tF n(t)DT
(70)
The effect of this confinement time on the temperature and density of the
plasma is shown to be negligible in Appendix C, in comparison to similar
effects of the ordinary particle confinement time.

-102-
A1though the comparisons of Appendix C are presented between
equilibrium parameters for the two systems, the small perturbations
applicable in the linearized analysis maintain the validity of the com¬
parison. In addition, the types of nonlinear transients of most interest
in this analysis involve relatively small, t 5l perturbations in some
operating conditions. This study of hybrid reactors was concerned with
the transient behavior over time intervals on the order of a few seconds.
The total resultant nonlinear temperature transients were not expected to
be large within the five to fifteen seconds of interest for such opera¬
tional perturbations. As noted in Appendix C, if very large nonlinear,
short time transients are considered or even if slowly changing transients
are allowed to grow for long periods of time, then assumptions in the
plasma model used here will be invalid. For example, if large temperature
variations are obtained by nonlinear time variations in the parameters
of the plasma model, then the results will be invalid because they
violate the basic assumption of constant plasma volume.
As expected, stability analysis of such full scale point model D-T
fusioning plasmas has demonstrated that the stability criteria as well
as time-dependent transient development due to departures of plasma
temperature from equilibrium are nearly identical for the two pure
23 24 95
fusion cases. ’ ’ In other words, whether burnup is included or
removed from consideration makes little difference in stability criteria
and transient plasma development for which the current plasma model is
applicable. The calculations of Appendix C support such results.
When the general confinement model of Eq. (64) was used to obtain
expressions for 1/t^, I/t-j-j, 1/t-j, and 1/t^ from their original defini¬
tions in Eqs. (33a), (33b), (42a), and (42b), respectively, the two

-103-
general stability requirements in Eqs. (69a) and (69b) were expanded
to yield:
Stability Condition I:
3f,(T)
J-- [n
o 3T
T
3(V
o 3T
0
a(f)
] + -L + n "
n n
0 o
o 3n
- T
3(1-)
n
o 3T
> 0
(71)
Stability Condition II:
1
»(*->
t no 3n
n
o
n
! 3f?(T)
^ - "o -ir-
0 o
+ T
o 3T
+ T
3()-)
n
o 3T
,-fvVlL. n ’‘V
T
o 3n
-] > 0 .
(72)
0 o
Several definitions were then introduced in parallel with Ohta's
treatment to simplify the application of these stability criteria. First,
the ratio of particle to energy confinement time in the steady-state was
designated by the constant, R:
R = — (73)
which is predicted to exceed unity for the full-scale, power-producing
fusioning plasmas considered in large conceptual design studies.30,32,101,102
This predicted behavior of the R-ratio is consistent with theoretical
results indicating that particles of higher energy diffuse out of plasma
devices more quickly than lower energy particles do. In addition, R-values
above unity account for other anomalous energy losses

-104-
The second identity is related to Lawson's nr^-criterion; that is,
at steady-state, the Lawson relation using the more meaningful energy
confinement time is defined by
(74)
o
which is a more restrictive condition used by Ohta in contrast to Lawson's
original criterion using the particle confinement time, t . The variation
of this relation with temperature is presented in Fig. 16 for both
charged particle heating and injection heating cases in general agreement
with results presented in Ohta's paper. The derived stability criteria
are dependent on the behavior of the F(T) function.
Finally, to simplify notation, a combination of recurring parameters was
defined by the following constant:
L =1-1-1
^2 R 3T
(75)
o
The three parameters defined in Eqs. (73), (74), and (75) together with
Eq. (64) for the general confinement model were used to simplify the two
expanded versions of the stability conditions given in Eqs. (71) and (72).
After appropriate substitution as well as combination and rearrange¬
ment of terms, the two stability criteria originally determined by
18
Ohta ° were obtained as follows:
Criterion I:
3F
9T
0
(76)

Equilibrium F(T)-Product (sec/cm
-105-
Figure 16. Variation of F(T) = m with temperature.

-106-
Criterion II:
3F
9T
0
>
2mF
o
T
o
(77)
where stability is shown to be strongly dependent on the Lawson n -
o
product and its derivative at the existing equilibrium plasma condition.
For the three previously defined diffusion models, these two
stability criteria are summarized in Table 2-1. The confinement time
effects are contained only in the Fo-term via and to a less effective
o
degree, in the via the R-ratio.
Table 2-1
Stability Criteria for a D-T Fusion Reactor
Confinement
Model
Stability
Criteria
Critical Temperature
(Tc) (keV)
CPH* IH**
x^ = constant
_o 3F
Fo 5T
> 0
28
21
tE .t
-1
(Bohm)
0
3F
F
0
3T
T
0
3F
F
0
3T
T
0
3F
F
0
3T
T
0
3F
F
3T
> -2
-i/ec
14
T1 /2 -1
x^ ^ T n
(Classical)
> 1/2
Zf
(1 - 5/R)
’2
42
33
★
★ *
Charged particle heating (Ts =0) R = 1, = 1.
Injection heating (Ts / 0) Ts = 150 keV, R = 10.

-107-
These results are identical to those presented by Ohta; the fusioning
plasma is stable and self-controlling only if its operating steady-state
temperature exceeds the critical temperature limits defined by the two
stability criteria for each model.
Since most full scale fusion power plant designs are predicted to
utilize plasmas at steady-state temperatures in the range of 15-20 keV,
these plasmas are expected to require feedback control due to the thermal
30 92 93 102
instability problem. ’ ’ ’ A similar but significantly lower range
of plasma operating temperatures is anticipated for the core of hybrid
reactors. The hybrid plasma discussed in Chapter 4 will operate in the
steady-state equilibrium condition at a temperature of only 8.0 keV.
Therefore, feedback control was selected to ensure plasma stability for
closed loop pure fusion plasma operation at low temperatures.
In accordance with Ohta's efforts to demonstrate global plasma
stability, an artificial feedback mechanism was incorporated to effect
changes in the plasma sustaining feedrate via corresponding changes in
the plasma temperature as follows:
6S(t) = a, Y2- 6T(t - t , ) .
1 o al
(78)
The delay time, t , , was incoroorated to account for the possible lag time
al
between the occurrence of a change in the steady-state plasma temperature,
fiT(t), and the corresponding application of a change in the feedrate,
6S(t). The inclusion of this delay time, at least in the hypothetical
model, is necessitated by the limitations on engineering speed of response.
The factor, n /T , was used to normalize the feedback relationship from

-108-
energy to particle density while the factor, , was designated as the
temperature (plasma) feedback coefficient with units of inverse seconds.
To examine stability for this feedback case, this plasma source
feedrate perturbation was transformed to the Laplace domain to obtain the
following exponential equation:
-t , s
n d1
AS(s) = y2- e aT(s) . (79)
o
Due to the exponential delay term, this feedback equation is not purely
algebraic and linear in the transform domain. Since the delay time was
assumed small, the equation was linearized by expanding the exponential
term in a first order Taylor series to obtain the desired algebraic form
for the transformed feedback equation:
n
AS(s) = cu (1 - t, s) aT(s) . (80)
o al
The transform feedback relation of Eq. (80) is in the polynomial form
suited to the Laplace domain stability analysis methods which were
applied to the open-loop system. Note that, if delay time is neglected
in the application of feedback, then this equation can be simplified to
the following form:
n
AS(S ) = a-, J2- AT( S ) . (81 )
0
This form was actually applied to the hybrid plasma model developed in
Chapter 3.
The four linearized equations which were developed in the Laplace
domain--one each for plasma ion density, plasma energy density or

-109-
temperature, and plasma volumetric neutron production rate plus a fourth
linearized equation for plasma feedrate dependent on plasma tempera ture--
were rewritten for ease of handling as follows:
Plasma Ion (Particle) Density:
(s +
1 + 1 ' An(s) _ AS(s)
x t -i n n
n ni o o
o
1 AT(s)
rT1 T
o
(82)
Plasma Energy Density:
(S - - '-) Mil + (s + j-
' E T1
1.) AT(s)
t2
T
n T
o o
AS(s) (83)
Plasma Volumetric Neutron Production Rate:
Aqp(s) - g(T0) An(s) + nQ
AT(s)
(84)
Plasma Temperature Feedback:
AS(S) = a, y2 (1 - td s) AT(S) .
0 al
(85)
The block diagram applicable to this neutron-producing plasma system with
feedback is presented in Fig. 17 where the only change from the reduced
schematic in Fig. 12 is the stabilizing feedback loop. The open-loop
elemental transfer function, aT(s)/aS(s), was presented in Eq. (57). By
applying block diagram reduction methods, the transfer function for this
closed loop system was determined from Fig. 17 to be as shown in Eq.
(84).

â– +- t +
Figure 17. Block diagram for the point-model plasma with temperature feedback to
the feedrate.
-no-

-111-
Aqp(s)
AS(s)
t
0
n
o
[n
a
2
o 9T
10
s + a
(a4s + a7) -
11 ' “l ^ " td
2n^q(T )a
o
T
o
s)(a.s+
1
6
(86)
where the reduced coefficients, a^ and a^, apply for negligible burnup
and are given by
'10
nl
1
tT1
(87a)
*11 = (7
1-)(rL
nl
—) + — (— -
t2 tT1 t1
(87b)
The assumption of constant confinement time was employed not only to
simplify analysis of this transfer function but also because the constant
confinement model is an effective average for the spectrum of possible
diffusion behavior in fusion systems. Diffusion losses are probably
overpredicted by the unrealistic Bohm diffusion model (t %T "* ) because
of the so-called collisionless regime at relatively high plasma tempera¬
tures and possibly underpredicted by the classical diffusion model
(t ^ n ^j^/2).100 por ^-¡g hybrid analysis the constant confinement
model was used exclusively as an effective average diffusion behavior
because this study was concerned primarily with typical parameters and
behavior rather than actual design base calculations.
By combining terms into polynomials in the s-domain, the overall
closed-loop plasma transfer function was reduced to
flVs)
AS( s )
[g(T ) - t
J 0 0 3T
a4]s + 9(To)(7
0
1 1 \ J 3g
p ~ T2 ~ 0
0.
12
a13S + al4S + al5
0
(88)

-112-
where the reduced coefficients, a]2_a15’ aPP^ f°r negligible burnup with
constant confinement and are given by
a
12
1 1
0 no
(89a)
al 3 1 + al td-,
(89b)
14
T- + J
x2
n
" ala4 + alV
'12
(89c)
'15
t2> " “lal2
(89 d)
Application of the Routh criterion to the characteristic equation for this
transfer function yielded the following three stability conditions
(criteria).
Stability Criterion
I:
h 3 =
Stabi1ity
Criterion
II:
1
1
t2
0
Stabi1ity
Criterion
III:
1 dn
T
s
3T
o
, T alf=2 td1
+ 7~ ' nl(1 ' " 0
o
o
0
(90a)
(90b)
1,1 1
12
15 xn xE t2
0 0
(— - —) + > 0
(90c)

-113-
These criteria are presented dependent on Criterion I being positive which
is the case only when the injection energy, T , exceeds the thermal
o
temperature, T , by a factor of three or more since Criterion I implies
T 0
s -1
al > ^d (yr— • Should the reverse be true for Criterion I, then
1 o
the direction of the inequalities in Eqs. (90b) and (90c) must be re¬
versed; that is, all three coefficients of the characteristic polynomial
are required to be the same sign, regardless of whether it is positive or
negative. Of course, from an engineering point of view, the injection
energy makes no sense unless it is much larger than the equilibrium plasma
temperature, Tq; otherwise, the capital investment used to inject
energetic particles would be wasted. In fact, pure fusion machines are
expected to operate with injection energies of hundreds of keV versus
29 30 92
plasma temperatures of tens of keV. ’ ’ The recent successful in¬
jection heating experiments on the Princeton Large Torus (PLT) Tokamak
system support this heating method. Temperatures above 5 keV and possibly
as high as 7 keV were reached for the first time in a large Tokamak
, • 9,103
device.
For the case when feedback is applied, Criteria II and III remain
very similar to those which were developed without feedback. Criterion I
is completely new since it is introduced essentially because of the
feedback. No confinement times are involved in Criterion I which depends
only on the feedback coefficient and its delay time in addition to the
ratio of injected energy to plasma temperature. Although the open loop
(no feedback) plasma model is found to be unstable with respect to
feedrate perturbations for certain temperature ranges, the closed loop
system can be made stable to small perturbations over much of the pre¬
viously unstable temperature range.

-114-
Again, these results for plasma stability agree with those of Ohta;
but in this development, the system concept of transfer functions is used
with more reliance on the physics of the plasma system to justify ex¬
amining perturbations in feedrate as the governing plasma parameter. The
present analysis has much broader application to more sophisticated
systems as needed. Note that it is the feedrate alone--assuming as is
15 18 19 21-24
usually done, ’ ’ 5 that the injection energy is constant--that
affects the plasma content and behavior, as an external influence in these
global models. All other influences on the plasma depend on the model
used and parameters within the system itself. Only the feedrate, S(t),
or the feed energy, Ts(t), affects the plasma from an external source.
Some studies have attempted to look at other external influences on
the plasma. Specifically, the primary effect considered by these other
studies has been the effect of a changing magnetic field and concurrently
(causally) a changing plasma volume. ’ To include such effects in
this model would have needlessly increased the complexity and number of
equations needed to represent the fusioning plasma and its dynamic be¬
havior. Such increased complexity was not justified because the
fusioning plasma is only the driver of the hybrid, dual fusion-fission
system which is the primary subject of this analysis. Therefore, the
basic plasma model as utilized here to drive a hybrid blanket is
complete.
The very important result was noted that these three stability
criteria are the same as those obtained by Ohta et al.^ Although the
plasma neutron production rate is now included, this additional compli¬
cation of the overall model does not affect stability requirements which
was expected. Essentially, the previously applied point-model plasma and

-115-
its related stability were given more general applicability by the
development of the transfer function basis in this work. This approach
also allows more general model development. It is also important to
realize that linearized analysis usually predicts correctly whether or
not a system or a model is stable. However, if the system is unstable,
linearized analysis is not capable of predicting the true consequences
resulting from input perturbations.

CHAPTER 3
A HYBRID REACTOR ANALYTICAL MODEL
Development of the Hybrid Model
Consideration was next given to an analytical model which could be
used to describe the dynamic behavior of the fusion-fission hybrid system.
This same model, in linearized form, was used to establish stability
criteria for the hybrid system. The plasma model for a power-producing
fusion reactor has already been presented in Chapter 2 and was assumed
to apply directly to the hybrid reactor plasma core. Although the plasma
will not be self-sustaining for the hybrid case, the plasma core must
still be in steady-state equilibrium for the neutron production rate
required to meet the power rating of the hybrid system. Since the hybrid
plasma is not expected to be self-sustaining, more energy will be re¬
quired to confine and maintain it than is produced by fusion reactions.
The same three equations for plasma particle density, n(t), plasma
energy density,3n(t)T(t), and plasma volumetric neutron production rate,
qp(t), were used to describe the global dynamic behavior of the hybrid
reactor plasma core as follows:
Plasma Ion Density:
dnM = S(t) - ^ - f.(T)n2(t) (91)
dt xn 1
-116-

-117-
Plasma Energy Density:
Plasma Volumetric Neutron Production Rate:
qp(t) = g(T)n2(t) . (93)
In the absence of any feedback effects, the global dynamic behavior of
the plasma core of the hybrid is completely determined by these three
equations where temperature-dependent coefficients are defined as before:
m
fl(T) = —’
(94a)
f 2 (T)
nyQ 1/9
= - bT1/2(t) ,
(94b)
m <0V>DT
g(J) = 4
(94c)
Only ion density and plasma temperature are needed to describe completely
the state of the plasma in this model; that is, only the density and
energy equations are actually needed for stability analysis of the plasma
in isolation. However, to analyze the hybrid system, the specific
neutron production rate was explicitly included in the model as noted in
Chapter 2. In addition, since the neutrons produced by the plasma are
its only means of effecting changes in the hybrid blanket flux dis¬
tributions and power production in this model, the plasma volumetric
neutron source equation was required for physical interpretation and

-118-
modeling of the functioning hybrid. These three equations completely
determine the nonlinear dynamic behavior of the hybrid plasma under the
assumption that no feedback is applied.
To determine the global stability criteria for the hybrid system,
the same procedure was applied as for the pure fusion system. The plasma
model consists of essentially the same equations although the plasma
itself is considerably less reactive as shown in Table C-I of Appendix C.
The linearized perturbed form of the plasma equations was obtained and
linear stability of the hybrid was determined for small perturbations in
the steady-state equilibrium operation of the hybrid plasma core which
is not self-sustaining. Burnup in the fusioning plasma was neglected
from the beginning in the formulation of the hybrid linearized model
because the inclusion of burnup effects was demonstrated in Appendix C
as well as in comparison with previous analyses of pure fusion
23 24 95
plasmas 5 ’ to be nearly negligible in its effect on the less
reactive hybrid plasmas of interest in this analysis. Therefore, such
small burnup-induced effects on total particle confinement time were
completely eliminated in the simplified linearized plasma model equa¬
tions. The linearized plasma equations in the perturbed variables are
repeated without burnup from Eqs. (46), (47), and (48) in Chapter 2 as
fol1ows:
Plasma Ion Density:
1 d iS n (t)
n dt
o
o
) MO
' n
o
<51(0
T
0
+ MO
n
(95)

-119-
Plasma Energy Density:
1_ d¿n(t) , 1_ dóT(t) = ,1
n dt T dt 'x
o o
1 \ ¿n(t)
s <5S(t)
(96)
Plasma Volumetric Neutron Production Rate:
óqp(t) = 2nQg(To) 6n(t) + np — ST(t) .
(97)
0
Except for feedback effects, the linearized stability analysis of the
hybrid plasma core was completely determined by these three linearized
plasma equations just as for the pure fusion system.
The fissile hybrid blanket model was developed using the point
reactor kinetics equations which were used to describe the kinetic be¬
havior of the neutron population in the blanket. Typically, seven point
reactor kinetics equations are applicable. One equation gives the neutron
density, N(t), which can be related to neutron flux or power by a multi¬
plicative factor; one additional equation applies each precursor con¬
centration, C^(t), corresponding to one of six possible delayed neutron
groups. This model is summarized in the following well-known fission
104
reactor formulation of global reactor kinetics:
Mil s [£(t)^- B] H(t) + l x.c.(t)+qR (t)
(98)
and
dC.(t) Bi
= N(t) - A.C.(t)
dt
i = 1,2,...6
(99)

-120-
where the symbols used in these equations were defined as follows:
N(t) = neutron density in the blanket (#/cm )
C.j (t) = effective delayed neutron precursor concentration for ith
delayed neutron group in the blanket (#/cm3)
p(t) = blanket reactivity
= effective neutron source strength in the blanket
e (#/cm3-sec)
3 = total effective delayed neutron fraction in the fissile
blanket
6. = effective delayed neutron fraction of the ith group of
precursors in the blanket
A = neutron generation time in the blanket (sec)
A. = decay constant for the ith group of precursors (1/sec).
The effective delayed neutron fractions and the neutron generation
time in the fissile blanket were assumed time-independent for the hybrid
model.
The effective neutron source strength was defined by the following
equation:
qR (t) = Jf J f qR (r,E,t)/(r,E)drdE (100)
Beff Af ; E BACT 0
where angular dependence has been neglected and,
+(r,E) = the adjoint flux for the critical reference system
as a function of position, r, and energy, E
q„ (r,E,t) = the true or actual source strength as a function of
JACT position, energy, and time
f = the production operator.

-121-
Standard definitions for the production operator, f, as well as
C.j(t), 6, B.¡, p(t), and A, can be found in the usual references on reactor
kinetics.
105,106
The effective neutron source strength, qR (t), is a weighted
eff
quantity which was used to account, in lumped parameter fashion, for the
neutron source in the fissile hybrid blanket due to the fusion neutrons
produced in the plasma. This fusion neutron source represents a unique
characteristic of fissile hybrid blankets versus the usual fission reac¬
tor kinetic analysis on subcritical fissile systems; for example, neutron
sources are used to produce an easily monitored neutron flux level in a
subcritical fuel assembly prior to startup in a power reactor.^^,108
The blanket neutron energy deposition per fusion event, Qg, is a
global parameter which consists of three terms derived in Appendix A due
to fission energy deposition, fusion neutron energy deposition, and
exothermic reaction energy deposition as follows:
'eff
-1 - k
eff
-] + E +6.
J n V
(101)
where
Gf = fission energy deposited in the blanket per fission event
v = average number of fission neutrons produced per fission
event
kg^jT = effective blanket neutron multiplication factor
E = energy associated with a source neutron entering the
n blanket
6^ = any additional energy deposited in the blanket due to
exothermic neutron absorption reactions for example.
The resultant blanket neutron energy multiplication factor, Mg, is simply
the ratio of energy deposited in the blanket (QR) to energy input to the

-122-
blanket as follows:
(102)
which is also strictly global; despite its extensive use in previous
1 ,64,76,77
analyses of hybrid blankets and their neutronics,
results ob¬
tained in this work and reported in Chapter 5 indicate that Eq. (101) is
inadequate for a spatially-dependent analysis of multiplying hybrid
blanket properties. Estimation of the theoretical global power involves
only the use of a relationship such as
(103)
blanket power production, PR , incorporates the shortcomings associated
DrcT
EST
with the blanket neutron energy deposition factor so it is of no use for
scaling as shown in Appendix A.
The hybrid blanket is driven to power production by a surface in¬
homogeneous source leaving the plasma. For the hybrid model the surface
source strength, q (t), from the plasma was expressed as follows:
Qn(t)
where A$ is the inner blanket surface area facing the plasma and Q (t)
is the total plasma neutron production rate. The total plasma neutron
production rate was derived as follows:
(105)

-1 23-
where Vp is the constant plasma volume and qp(t) is the lumped plasma
volumetric neutron production rate.
For use in a point kinetics calculation, this actual surface source
was converted to an effective surface source using Eq. (100). This is
the usual adjoint-weighted source term used in the point kinetics
equations. Note that the integration involved in this definition would
have to be transformed from a volume to a surface integral in this case.
A1ternatively, a surface conversion coefficient, ^, can be deter¬
mined by comparing the results of lumped parameter or point-model blanket
power (neutron density) calculations using the blanket neutron energy
multiplication in Eq. (103) with the results of space- and energy-
dependent power (neutron density or flux) calculations performed using
diffusion theory, transport theory, or some other space-dependent
neutronic model for determining actual neutron density distributions and
power levels in a fissile medium. So the surface conversion coefficient
for use in the hybrid model was defined as follows:
K,
1
PR
kact
PR
best
(106)
where PD is the actual or true system power calculated from the sur-
BACT
face source in a space- and energy-dependent calculation, and PR is
bEST
the estimated power generation. Calculations described in Chapter 5 found
the actual power was less than the estimated power so that the conversion
coefficient, , is less than unity--at least for the thermal fission
hybrid system considered in Chapter 5. Since the neutron source enters
the blanket at a surface and not within the main blanket volume, the

-124-
surface conversion coefficient was expected to be less than one although
the increased energy of fusion neutrons over fission spectrum neutrons
was expected to compensate for most, if not all, of the decreased neutron
worth. The surface source conversion coefficient was used to relate the
geometrically "equivalent" surface source, q (t), to the effective sur¬
face source for inclusion in the point kinetics equations as follows:
007)
and
V
(108)
where the effective surface source is related directly to the plasma
neutron volumetric production rate. Note that such an effective area
source requires the point kinetics equations to be modified to get a
dimensionally consistent set.
Each of the calculations performed to obtain the actual blanket
power generation involved an inhomogeneous, multigroup, spatially-
dependent, steady-state calculation over the blanket while the corres¬
ponding theoretical global power calculation involved only the use of a
relationship such as Eq. (103). For a given total neutron source pro¬
duction rate or source strength in the plasma, Q (t), the estimation of
blanket power generation,PD , was performed using qlobal blanket cal-
D rcT
EST
culations based on the blanket energy deposition factor, QD. Some energy
D
dependence was included by basing the average neutrons per fission on
the neutron spectrum involved in the inhomogeneous calculation. Implicit
in such global calculations is the assumption that the planar source

-125-
of neutrons can be treated in a manner that is not spatial vary¬
ing.
Depending on the available computational tools, it may be desirable
or necessary to work with an "equivalent" volumetric source rather than
a surface source. In fact, this is usually the case so that there is
no need to correct dimensions in the point kinetics equations. It is
more convenient to work with volume sources in the kinetic equations
because the other terms are related to volumes. The correspondíng volu¬
metric neutron source strength in the hybrid blanket can be written as
follows:
qB(t) = vj qs(t)
(109)
where Vg is the effective hybrid fissile blanket volume. This equation is
again based on purely geometric equivalence or conservation of neutrons
and is also written as follows:
0 (t)
(no)
B
Note that this volumetric source, qB(t), conserves neutrons but not their
effectiveness. It is simply an "equivalent" volumetric source obtained
from geometrical consideration of the actual surface source just as
qs(t) was for the surface case. When this neutron source strength,
qg(t), is properly weighted, it provides a simple conceptual input for
the source term which is used to drive the subcritical fissileblanket.
This weighted or effective inhomogeneous source term, along with the
other kinetic parameters, was used in the point model fission kinetic
equations to obtain the variation of neutron density (or power or flux)

-126-
with time following source insertion or alteration as well as any other
type of neutronic perturbation in the blanket system. However, the
underlying assumption for use of the kinetics equations in general, and
the source definition in Eq. (109) in particular, is that a global or
point model treatment of the blanket is valid. This "equivalent" volu¬
metric source, qg(t), is not sufficient for use in the point-model
equations but must first be converted to an effective source in the same
way that the surface source was converted, except the actual calculation
involves a distributed volume source, qg(t), instead of a surface source,
qs(t). The source term, qg(t), is presented only as a volume-normalized
source in Eq. (110); however, the fusion neutron source driving the
blanket is actually a nearly planar (large radius toroidal surface for
the actual blanket model presented in Appendix B) source of 14.06 MeV
neutrons entering the hybrid blanket through the inner surface containing
the plasma. Such a geometry is not conducive inherently to a global
treatment--at least, not without some further assumptions. Of course,
the neutrons are also not introduced with a fission spectrum which is
another reason why they are not amenable to a simple global treatment.
Equation (100) can be directly used to perform this conversion of
the "equivalent" volume source to an effective volume source in order to
obtain the adjoint-weighted volumetric neutron source for inclusion in
the point kinetics equations. Alternatively, a second conversion coef¬
ficient, can be defined. This volume conversion coefficient was
obtained from the same equation as the surface conversion coefficient:
c2
BACT
best
(HI)

-127-
except that the actual power, PR , was obtained from a space-dependent
bACT
and energy-dependent neutronics calculation using a distributed volume
source instead of a surface source. The estimated power was obtained
from the same global energy multiplication/deposition relation in Eq.
(103). So the volume conversion coefficient relates the geometrically
"equivalent" volume source to the effective volume source for inclusion
in the point kinetics equations as follows:
or
and
qR (t) = qR(t)
Beff 2 B
A
VJ‘> = c2 VTqs(t)
eff ’B
V
qB (t) = ?2 V2 qD(t)
Beff 2 VB p
(112)
(113)
(114)
where the conversion coefficient, ^ relates the effective neutron
source, qR , required (as by flux distribution calculations using an
eff
actual surface driving source) to the calculated source strength, q^,
obtained by simple global analysis using only geometric effects as in
Eqs. (109) and (110). Essentially, the inclusion of either conversion
coefficient, ^ or t;0, in this model recognizes the inhomogeneous nature
of surface fusion neutron sources and the resultant inadequacy of a
simple global representation of the source itself.
Analysis of neutron density (or power) transients using the point-
model kinetics equations is limited to amplitude changes; the point-model
formulation cannot account for changes in the neutron flux shape during

-128-
a transient. As the name indicates, the point model is applicable to a
global reactor or blanket treatment (a zero-dimensional system) during
a transient. Of course, the plasma model is also zero-dimensional since
Eqs. (91-93) cannot account for spatial variations in any of the plasma
parameters such as ion density, temperature, or volumetric neutron pro¬
duction rate.
In more sophisticated transient analyses using the adiabatic or
quasistatic methods, the slow variation or the shape function is used to
justify point-model calculations of transient neutron density changes
over relatively long intervals with companion recalculations of the
applicable changed shape function at widely spaced intervals. In this
way relatively long transients or transients involving significant spatial
changes in the neutron population can be analyzed. 105,106 ^ not
the case for pure point-model analysis.
A generalized neutron density variable, N(r,E,s),t), can be written
as a function of position, energy, direction of neutron movement, and
time, respectively. Since the neutron energy, E, and direction of
motion, ft, variables are of secondary concern in this dynamic analysis,
the neutron density can be reduced to a dependence on only position and
time as N(r,t).
The point reactor concept is based on a reassignment of the spatial
and temporal dependence of the neutron density as the product of an
amplitude function, N(t), and a shape function, ijj(r,t), as follows:
N(r,t) = N(t) • ip(r,t) . (115)
All spatial variations in the neutron density are contained in the nor¬
malized shape function, ^(r,t), which is assumed in this global model to

-129-
be slowly varying in time. Such a global reduction to consideration of
only temporal variations is a corcmon method of analysis if transient
shapes are not expected to vary appreciably from steady-state equilibrium
.. , .. 105
di stn butions.
When the point reactor kinetics equations are applied to a reactor
such as the hybrid blanket, only the magnitude of the neutron density (or
equivalently the power or flux) is taken into account during any hypo¬
thetical transient. Temporal changes in the neutron density shape
function cannot be analyzed with these equations; only temporal changes
in the magnitude of the neutron density are contained in the amplitude
function, N(t). Therefore, the application of the point model equations
yields the magnitude of the neutron density during a transient assuming
the final shape, ^(r,t), at time, t, is unchanged from the initial shape,
large, asymmetrically introduced, or allowed to run long enough so the
flux shape changes significantly then results based on the point model
will be inadequate to describe the exact dynamic behavior of the neutron
population. Although inaccurate, such results can still be used to in¬
dicate trends and, for small transients, yield approximate values of
maximum power densities and possible destruction or excessive power
density zones. Such point-model results cannot be used for final design
analysis for which the full space-time analysis of transients is required.
Nevertheless, the point-model dynamics results can be used for studying
stability and transients in reactor systems. Therefore, the basic
assumption implied in using these point-model equations to model the
kinetic behavior of a subcritical hybrid blanket is that the shape func¬
tion for the spatial dependence of the neutron density does not change

-1 30-
significantly over the time scale during which these equations are used
to describe transient behavior.
This assumption is very adequate when the sizes of typical "realis¬
tic" hybrid plasma transients are considered. Calculations for typical
hybrid plasma perturbations in Chapter 4 demonstrate that relatively
small transients in plasma neutron production result from typical (Í 5%)
perturbations in different hybrid plasma equilibrium conditions such as
plasma temperature, plasma ion density, source feedrate, and injection
energy.
Because the hybrid blanket will be subcritical, the effects of de¬
layed neutrons are expected to be less important in the subcritical
blanket of the hybrid system than in a critical power-producing reactor.
The delayed neutrons will not be needed to maintain power-producing
capability. In fact, since proposed operating parameters for the hybrid
blanket indicate subcritica!ity at room temperature, the typical tem¬
perature defect in core reactivity due to heat up and ultimate buildup
of blanket poisons indicates that the blanket will be far subcritical
during actual hybrid power operation. This large negative reactivity
109
will make delayed neutrons even less important. Therefore, the six
delayed neutron groups were combined into one effective group with no
loss in applicability of the model to typical hybrid systems. The seven
point reactor kinetics equations were reduced to the usual pair of equa¬
tions for the neutron density, N(t), and the total precursor concentra¬
tion, C(t):
_dNiti „ [P(t) - 8] N(t) + ,C(t) + qB (t)
ef f
(116)
and

-131-
(117)
where C(t) is the effective precursor concentration for all six delayed
groups; q^ (t) is the effective blanket neutron source whether an
eff
effective surface source or an effective volume source; and A is the
average decay constant for six groups of delayed neutron precursors:
(118)
The Linearized Hybrid Model
Equation (112) for the neutron density is nonlinear due to the
reactivity-density product term, p(t)N(t)/A; therefore, these equations
were linearized prior to stability analysis. These equations are, how¬
ever, far less nonlinear than the plasma model equations developed in
Chapter 2. The following usual small perturbation expansions of the
dependent variables about an initial equi1ibriurn were used:
N(t) = Nq + 6N(t) ,
(119a)
C(t) = Cq + SC(t) ,
(119b)
p(t) = pQ + 5p(t) ,
(119c)
(119d)
where the variations in the reactivity and the inhomogeneous volume source
in the blanket were both retained. This simultaneous retention of the

-132-
reactivity as well as the blanket source perturbation is unique to hybrid
blanket system analysis where both perturbations are required in deter¬
mining how the neutron density and power changes in the symbiotic blanket
system are dependent on the plasma feedrate. No other work has con¬
sidered or even commented upon this set of unique conditions in a hybrid
model.
After substitution of the perturbed variables of Eqs. (119a)-(119d)
into the point-model reactor kinetics equations, the following pair of
expanded kinetics formulations were obtained:
(t) fiP(t)6N(t)
+ aCq+A6C(t)
(120)
d-6^ = 7 N + f 6N(t) - AC - A6C( t)
dt A 0 A v 0
(121)
Because of the assumption of small perturbations, the second order, non¬
linear term, 6p(t)6N(t)/A, was neglected as usual. The two initial
equilibrium conditions were also eliminated so that linearized perturbed
point reactor kinetics equations of the following form were produced:
Perturbed Blanket Neutron Density:
eff
(122)
Perturbed Precursor Concentration:
dsC(t) _ 3_
dt A
= f 6N(t) - A6C(t)
(123)

-133-
These linearized perturbed point kinetics equations are in the same form
encountered in fission reactor kinetics with one exception--both the
inhomogeneous effective volumetric neutron source strength perturbation,
6qR (t), and the reactivity perturbation, 6p(t), of the blanket are
eff
incorporated in this model simultaneously. This unique feature of hybrid
reactor kinetics analysis introduces considerable interplay of variable
dependences in the hybrid model.
In the linearized kinetics analysis of ordinary fission reactors,
one or the other of these two variations of transient-producing terms is
completely negligible. In the usual analysis of subcritical systems,
the reactivity perturbation due to a source variation for a zero-power
system is expected to be zero while the source variation is itself
retained; similarly, in the kinetic analysis of critical or supercritical
power-producing reactors, the source perturbation in the presence of a
reactivity variation is assumed to be zero since inhomogeneous source
variations have a negligible effect on a critical power-producing
. no
system.
For example, in a subcritical assembly--perhaps a research system
or a LWR during fuel loading--interest is directed toward the so-called
source perturbation and the associated-source transfer function,
AN(s)/AqR (s), for stability analysis. It is this quantity in ordinary
eff
subcritical reactor kinetics which is likened to the volumetric source
feedrate transfer function, Aq (s)/AS(s), derived in Chapter 2 for
characterization of plasma stability. Both transfer functions represent
output neutron densities (or power) due to input source perturbations in
either the ion feedrate for the plasma or the neutron source for the
subcritical assembly. On the other hand, in a critical power-producing

-134-
system such as a large PWR or even a research reactor at power, the
changes in reactivity, 6p(t), and the corresponding reactivity transfer
function, AN(s)/Ap(s), are the quantities of interest. Any perturbations
due to an inhomogeneous source are negligible in a transient analysis.
In the hybrid blanket, the effective volumetric neutron source,
qR (t), will be so large that its perturbations were included alonq
eff
with perturbations in the reactivity because both the fusioning plasma
with its volumetric driving source of neutrons, qp(t), and the fissioning
blanket with its driven reactivity, p(t), are required to sustain the
overall system in a power-producing mode. Since both the effective
neutron source and the reactivity are needed, perturbations in both
variables were included in the linearized hybrid kinetics equations.
Only for the hybrid case, where the inhomogeneous source is so large and
the blanket is subcritical, is there a need to account for the effects
of both perturbed variables simultaneously in determining system stability
and dynamic behavior.
In addition, Eq. (114) for the volumetric neutron source conversion
coefficient was used to relate the effective volumetric neutron source
in the blanket to the volumetric neutron production rate in the fusion¬
ing plasma as follows:
eff
(t) =
q (t)
B
(124)
The surface neutron conversion coefficient could have been used but volumes
are more applicable. As usual perturbations in both volumetric sources
about the steady-state [qn (t)
eff
qR ,, + <$qR (t) and q (t) = q
Beffo Beff P Po
+ 6q (t)] were used to obtain the following perturbed and linearized

-1 35-
volumetric neutron conversion coefficient equation:
<5qB (t)
^2 Vp %(t>
eff
V
B
(125)
where the steady-state condition has been eliminated.
Because the hybrid is a power-producing system, a power scaling
equation was required to relate neutron density to power density as
follows:
p(t) = Gf Ef v N(t) (126)
where
3
p(t) = power density in the fissile hybrid blanket (W/cm )
Gf = average energy per fission (ergs/fission)
v = average neutron velocity in the fissile blanket (cm/sec)
= average macroscopic fission cross section of the fissile
blanket (1/cm).
The three multiplicative coefficients (G^,v,E^) were assumed con¬
stant as in typical transient analyses. Certainly for the short-term
blanket transients caused by plasma perturbations considered here, these
coefficients will not change significantly. Although already in linear
form, this scaling equation was converted into perturbed form to be
compatible with the global plasma and blanket kinetics equations which
were perturbed for linear stability analysis. The blanket power density
was represented as the sum of a steady-state, equilibrium power level
plus the usual small perturbation, p(t) = pQ + perturbed form of the linear power density scaling equation was obtained
as follows:

-136-
6p(t) = Gf Ef v 6N(t) , (127)
since only the neutron density was allowed to vary with time.
At this point some form of heat transfer relationship was needed to
provide the overall hybrid model with a means of removing energy from the
system. Based on the typical heterogeneous unit cell arrangement con¬
sisting of helium coolant, graphite moderator, and fuel material, a lumped
parameter model using three temperature nodes could have been used to
represent the heat transfer processes in the blanket. No generality
would be lost by such an assignment of constituents and nodes in the
unit cell since other fuels, other moderators, and other coolants could
be treated in the same way. They were not included here to prevent un¬
necessary complication in the basic hybrid model.
For this initial linearized analysis of system stability and kinetic
behavior, a single temperature node was selected for the fissile blanket.
The blanket temperature was treated as a global parameter just as the
neutron density was treated in the neutron dynamics model of Eqs. (98)
and (99). By utilizing a single average blanket temperature, Tg, the
simplicity of the model was retained along with the interaction of plasma
and blanket effects.
This global blanket temperature model can be employed for linear
stability studies as well as for the dynamics analysis of the hybrid
system. Although informative dynamics results can be obtained, more
detailed multinodal temperature analysis of the blanket will be required
prior to selection of a hybrid design for final analysis and installation.
The present study was not so concerned with the temperature-dependence in
the blanket but only its general behavior through development of the
overall hybrid model.

-137-
To proceed with development of the hybrid analytical model, the
global blanket temperature was related to power. A single, effective
helium coolant temperature, T^, was assumed along with Newton's law of
cooling to obtain an overall fissile blanket energy balance as follows:
dTR(t)
CpBmB (Tt~ = p(t)VB - hAc[TB(t) - Tc] (128)
where
c = fissile blanket average specific heat (ergs/gm°K)
PB
mB = fissile blanket mass (gm)
Tg(t) = average fissile blanket temperature (°K)
p(t) = average fissile blanket power density W/cm )
h = effective average blanket conductance between graphite
modulator and helium coolant (W/cm2-°K)
Aj. = total effective coolant channel area across which heat
flows from the graphite moderator to the helium coolant
(cm^)
= average bulk helium coolant temperature (°K)
Vg = effective fissile blanket volume (cm ).
The steady-state condition was represented by
0 ' PoVB - hflC(TB - V <129)
0
where the coolant temperature was essentially assumed constant and was
o
used to represent the average blanket temperature for the initial equilib¬
rium state. Certainly, variations in the fuel and moderator temperatures
would be expected to be much larger than variation in the temperature of
the coolant which can have more or less heat removed in a heat exchanger
to keep it constant. Since coolant temperatures do not vary as much as

-138-
fuel temperatures anyway, the helium coolant temperature was eliminated
from Eqs. (128) and (129) to obtain the following result:
dTR(t)
CppmB dt = VBP^ " PoVB " hAc^TB(t) - Tg ] . (130)
B o
This equation has frequently been employed in linear stability studies. ^
To make this equation compatible with the previously linearized hybrid
equations, small variations about the equilibrium state were again
utilized:
(t) = Tb + ÓT (t)
(131a)
0
p(t) = PQ+ 6p(t) •
(131b)
These perturbed variables of Eqs. (131a) and (131b) were substituted into
Eq. (130) to obtain the following linearized equation in the perturbations
d CpBmB Ht = VB<5p(t) - hVTB(t) • (132)
With the development of this blanket temperature equation, the feedforward
part of the linearized hybrid system model is complete; however, feedback
effects are also necessary to complete a general hybrid analytical model.
Incorporation of Feedback Effects into the Hybrid Model
At this point the analytical hybrid model was sufficiently developed
to trace the dynamic effects due to an initial perturbation in the plasma
feedrate from effects on the plasma density and temperature through

-139-
effects on the blanket neutron density and power production and finally
to changes in the global blanket temperature. The global equations
presented can be used to analyze the dynamic behavior and, in linearized
form, to predict the stability limits of the point-model system in the
absence of controlling feedback effects. However, several feedback terms
were required to complete the overall analytical model of global fusion-
fission reactor dynamics. Although the linearized hybrid dynamics model
is valid only for small perturbations, the influence of feedback effects
on stability criteria cannot be ignored. This is especially true since
uncontrolled pure fusion plasmas are predicted unstable to temperature
perturbations for proposed operating temperatures and expected confinement
18 23 24
characteristies. ’ ’ Indeed, much the same condition was found to
hold for hybrid plasmas in Chapter 4 depending on the equilibrium operating
conditions selected.
First, the neutron-producing plasma subsystem may be unstable to
temperature and other perturbations at expected thermonuclear operating
temperatures in the hybrid core below 28 keV for the constant confinement
model (x ^constant). Therefore, the artificial temperature feedback
previously applied for pure fusion systems was incorporated into the
hybrid model as follows:
n
S(t) = S + a, t9- [T(t) - T ]u(t - t, ) (133)
o 0 i
which was used to show that the feedback due to a plasma temperature
variation from a steady-state temperature, T , is applied to the feed-
rate after a delay time, t^ , via the delayed unit step function,

-140-
u(t - t, ). The factor, n /T , was used to normalize the temperature
Q -j 0 0
variation to the feedrate so that the feedback coefficient, , has units
of inverse seconds. The usual perturbed formulation, 6S(t), about
equilibrium was introduced to obtain the following feedback equation:
n
6S(t) = ou <5T(t - t , )
1 o dl
(134)
A second, less effective source of artificial feedrate feedback
was incorporated through variations in the global blanket temperature.
In this model blanket power fluctuations can cause changes in the
effective blanket temperature. This feedback was applied to the plasma
feedrate in the following form similar to that for the plasma temperature
feedback:
n
s(t) = SQ + a2 -5- [TR(t) - T]u(t - td ) (135)
where the normalization factor, nQ/TB , was used so this externally con-
0
trolled or artificial power feedback coefficient, c^, would have the
same units of inverse seconds as the artificial plasma temperature feed¬
back coefficient. The usual perturbed formulation of this feedback
equation was obtained:
i5S( t) =
l2 T° 6TB^t " W
o
(136)
These two delayed, artificial feedback effects in Eqs. (132) and (134)
were combined into a single analytical expression relating perturbations
in plasma temperature and blanket temperature to the driving plasma

-141-
feedrate via delayed artificial feedback effects:
n
n
6S(t) - a, 1 To d1 2 Tg B d2
o
(137)
The remaining feedback effect is a natural result of temperature
changes within the blanket. The reactivity of the hybrid blanket was
treated as a sum
P(t) = po + Pp(t) (138)
where pq is the reactivity (negative by design) of the subcritical blanket
in the initial, steady-state, equilibrium condition and Pp(t) is the
additional reactivity of either sign added due to feedback effects re¬
sulting from departures from the initial equilibrium.
The blanket temperature was assumed to be the only parameter whose
change would inherently affect the fissile blanket reactivity. The usual
linear truncation of the Taylor series expansion representing the general
nonlinear relation between reactivity and blanket temperature was included
as follows:
p(t) - p0 + ot-j-CTg(t) - Tg ]
Do
(139)
where
ay = the overall effective prompt temperature coefficient of
reactivity
Tg(t) = the average blanket temperature
T = the reference initial steady-state temperature for the
Bo equilibrium operation of the blanket system.

-142-
This natural temperature feedback was primarily based upon the doppler
effect within the fuel for which there is little delay. Therefore, the
assumption of instantaneous feedback on the blanket reactivity is
reasonable. The model is simplistic, however, since the feedback effects
are based on criticality evaluations of the fissile blanket for different
equilibrium operating temperatures. Therefore, the net reactivity feed¬
back effect was applied with no delay in this model in Eq. (139) as if
the blanket temperature feedback were due exclusively to effects within
the fuel such as doppler broadening.
Since primary interest is in the stability of the linearized hybrid
model as well as short-term response to small perturbations in the plasma
feedrate, only prompt reactivity feedback effects were included to main¬
tain a simplified model upon which later analyses can expand to include
the many delayed effects which must be considered prior to final design
of the hybrid blanket.
Feedback directly to the reactivity, p(t), via control rod assemblies
may also be present but would be relatively long-acting. Such control
rods would allow end-of-life operation in the event that optimum plasma
operation were to be reached with no reserve capability remaining to
increase the fusion source sufficiently to overcome reduced blanket
multiplication. Hence, to maintain blanket power level given by
PR (t) - Q (t) • Q (140)
“ACT p b
in the presence of a decreasing reactivity, the energy production per
fusion neutron must be increased by removing control rods and increasing
the effective neutron multiplication factor of the blanket.

-143-
The entire question of control rod feedback was ignored in developing
this linearized model. Only short-term feedback effects were considered
to ensure stability for small perturbations where flux shapes in the
fission blanket remain nearly constant.
Nonlinear and Linearized Hybrid Model Summary
The basic set of nonlinear global model equations derived to de¬
scribe the dynamic behavior of a point-model, fusion-fission hybrid
system is summarized in the following set of ten (10) equations:
Plasma Ion Density:
dn(t) _ c-m rut)_ DT ,
dt [ 1 t " 2 1
n
Plasma Energy Density:
d[n(t)T(t)]
dt
TsS(t)
(141b)
Plasma Volumetric Neutron Production Rate:
(141c)
Blanket Neutron Density:
(141d)

-144-
Blanket Delayed Neutron Precursor Concentration:
= | N(t) - AC(t) (141 e)
Plasma to Blanket Neutron Conversion Coefficient:
V
qR (t) = ??/qn(t) (141 f)
eff ¿ VB p
Blanket Power Density:
p(t) = 6f Yf v N(t) (141g)
Blanket Temperature and Heat Removal:
dTR(t)
CpBmB It— = p(t)VB - hAC[TB(t) - V (141h)
Blanket Temperature (Doppler) Feedback:
p(t) = P0 + aT[TB(t) - Tg ] (1411)
0
Plasma Feedrate Feedback (Blanket and Plasma Temperature):
5(t) = S + tpl [T(t) . T ]u(t - t ) + ^2 [IB(t) - Tb ]u(t - t ) .
o 1 Bq o 2
(141 j )
At this point the entire perturbed and linearized set of coupled
dynamics equations for the hybrid system is also summarized. First, two

-145-
g 1 obal equations for plasma particle and energy density were applied in
the plasma:
]_ d¿n(t) _ _,J_
n dt 't
o n„
¿n(t) _ 1
n
o
tT1
â– ST(t) ,
T
o
6S(t)
(142)
1 d5n(t) 1 d4T(t) ,1 1
ñ dt T dt V " xr
o o 1 E0
In addition, two point-model reactor kinetics equations for neutron
density and precursor concentration were applied in the blanket with one
effective group of delayed neutrons:
6n(t)
+ (r-
t2
JL\ ¿T(t) , 6S(t)
Tr 1 T 3T n
Eo ° oo
—= (P° A B) 6N(t) + -f- 6p(t) + A6C(t)
(144)
= A ¿N^ ~ A6C(t) • (145)
These two sets of basic describing equations for plasma and blanket
dynamics were coupled in the feedforward direction by the linearized
fusion neutron source equation for the perturbed plasma output:
fiqp(t) =
2n g(T )6n(t) + n'
o o c
At
ST
6T(t)
(146)
and by the relationship for the surface conversion coefficient, ^, or
the volume conversion coefficient, > giving the neutron ivorth for the
perturbed blanket input as follows:

-146-
V
óqB (t) = h * óqn(t) (147a)
eff s p
V
6q (t) = c yfi- • <5q (t) . (147b)
eff ¿ VB p
For the effective volumetric source applied in this work, Eq. (147b) was
used. Since this analysis was concerned with the stability of the system
for producing power and keeping within design limitations, two other
linearized, perturbed equations were required--the scaling equation
relating blanket power to neutron density and the blanket heat removal
equation relating the average blanket temperature to the blanket power
1evel:
6p(t) = Gf Ef v 6N(t) (148)
and
d«Tn(t)
cp mB dt " 6p^ ’ ,llAc,5TB^t^ '
D
The feedback equations were also included to complete a closed-loop
model for this hybrid system. First, the blanket temperature feecback
affecting the reactivity was included as follows:
6p(t) = i|<5Tg(t) (150)
where only prompt temperature effects such as those associated with
doppler feedback were considered.
Two additional artificial feedback effects were incorporated into
a single equation to affect the plasma feedrate via plasma temperature

-147-
variations with delay,
delay, t, :
a2
n
6S(t) = a ^ 0
and via blanket temperature variations with
- td ) +
al
'2 T° 6TB(t ' td2)
bo
(151)
Each of these feedback effects, natural as well as artificial, is
activated only when there is departure from the initial equilibrium state
of the power-producing hybrid system. Because the blanket temperature
feedback effect on the plasma feedrate is a duplicate of the plasma
temperature feedback on the feedrate, this effect is included in a dotted
box as unnecessary for basic stability analysis. It was included in this
hybrid model to close the overall loop but it would be included in a
realistic model only as a backup for the feedback based on the plasma
temperature variation unless allowance were made for some mechanism by
which the coolant channels could be blocked or the coolant lost so that
the blanket temperature increased independent of the driving fusion
source. This addition, however, would no longer satisfy the hybrid model
established here.
Transfer Function Representation of the Hybrid
The entire set of ten (10) linearized perturbed hybrid model equa¬
tions was transformed to the Laplace domain to simplify the linearized
stability analysis of the hybrid model; the transformed equations were
rearranged into the following formulation:
An(s)
n
o
[s +
1
1 -| AT(s) 1 AS(s)
T T TT " n
ni o TI o
+
0 52)

-148-
An(s)
n
o
[s +
0
[s +
TS AS(S)
3T n
o o
(153)
AN( s)
[s + = f Ap(.s) + AAC(s) + AqB (S)
Def f
(154)
AC(s)[s + A] = J aN(s)
V
Aqg (s) = • Aq (s)
eff
Aq (s) = -^¡S-
Mp o 8T
aT(s) + 2nQg(To)An(s)
Ap(s) = v AM(s)
hA V
aTb(s)[s + 7r~] = ^r- Ap(s)
c nip c mD
PB B PB B
(155)
(156)
(157)
(158)
(159)
Ap(s) = aTATB(s) (160)
n ~std, n "std?
AS(s) = a, e 'aT(s) + a2 =2. e ATB(s) . (161)
o o
This system of equations is completely linearized in the Laplace domain
except for Eq. (161) where the delayed feedback effects on the plasma
feedrate in the time domain result in exponential terms in the frequency
domain. The previous assumption of short delay times was used to justify
expanding the exponential terms in Eq. (161) in first order Taylor series

-149-
to yield truncated first order polynomials as follows:
-t, s
d-,
e
- 1 - t , s
d.
(162a)
e
(162b)
These expansions allow valid smal1-transient predictions of stability
while maintaining the ease of treatment associated with transfer func¬
tions whose numerators and denominators are polynomials in the frequency
variable, s.
Dynamic analysis of large transients is not possible with this model
and was not the concern of this stability analysis. Of course, the basic
assumption of negligible burnup would also no longer be valid for very
large transients. The following linearized approximation for Eq. (161)
was obtained:
n
o
AS(t) - a-j j— (1 - t^ s)aT(s) + a
o 1
At this point the procedure followed for the point-model plasma
equations in Chapter 2 was repeated to produce a block diagram for the
overall fusion-fission hybrid system. Some simplification of this
system of algebraic equations was needed. First, since it effectively
spans the predictions of several theories on diffusion in fusioning
plasmas and also greatly simplified the stability analysis, the constant
confinement model was utilized for plasma diffusion. As a result, both

-150-
expansion terms, 1/t ^ and 1 /x-p-j , were set to zero in Eq. (152). This
assumption is consistent with the feedback controlled plasma system
described in Chapter 2.
Second, the two blanket reactor kinetics equations were combined as
usual to eliminate the precursor concentration variation as follows:
AN(s)[s
— — N
] = ~~ Ap (S ) + Aq (s)
A(s + A) A eff
(164)
Note that neither the blanket source transfer function,
AN(s)/AqD (s), nor the blanket reactivity transfer function,
eff
AN(s)/Ap(s), can be isolated in this hybrid model; the symbiotic rela¬
tionship is absolute--both the blanket source perturbation, AqR (s),
eff
and the blanket reactivity perturbation, Ap(s), were retained for this
analytical model as explained following Eq. (123). This is a significant
difference in comparison with traditional fission reactor kinetics
studies. One or the other of the two perturbations is always neglected
99
for ordinary stability analysis of fissile systems.
The transfer function formulation in Eq. (165) was obtained upon
rearrangement of Eq. (164):
AN(s) = [—
2
s
S + A
+ ('
" Pr
+ a)s
N
] [7- Ap(s) + AqR (s)]
Apo A eff
A
(165)
where the first factor, involving a first order polynomial in s divided
by a quadratic in s, is the usual factor derived for the source or
reactivity transfer function of fission reactors. This factor was
identified as the blanket reactor transfer function, B(s), after the
similar form used in ordinary fission reactor kinetics:

-151-
B(s)
s + X
9 3 - P _ Ap
> +(-r-a+ »)s-nr
(166)
The plasma block diagram presented in Fig. 9 of Chapter 2 is com¬
pletely applicable for the plasma subsystem of the hybrid. The only
difference in the hybrid plasma case is the less reactive nature of the
plasma which is no longer self-sustaining. To account for the presence
of both source and reactivity perturbations in the blanket kinetics
equation, the summation method presented in Fig. 18 was utilized in the
overall hybrid system diagram for the linearized, perturbed hybrid model
and its associated overall transfer function.
In addition, the power scaling factor relating neutron density to
power density in Eq. (158) was used in the form of a transfer function
as follows:
Ap(s) _ r
MTJ ~ Gf
V E,
(167)
Finally, the transfer function relating blanket temperature changes to
blanket power density changes was obtained from the perturbed blanket
heat transfer relation in Eq. (159) as follows:
Tb(s) VCpsmB
p(s) hA^ ~
(168)
These various subsystems including the plasma diagram of Fig. 9 in
Chapter 2, and the blanket kinetics diagram of Fig. 13, together with the

AqR
Beff
DRIVER
Figure IS. Block diagram schematic for point-model blanket kinetics retaining
both source and reactivity perturbations.
-152-

-153-
connecting relations and feedback components were combined to yield the
overall fusion-fission hybrid block diagram presented schematically in
Fig. 19. This overall block diagram relating transfer functions for the
various components of the linearized hybrid model was entirely derived
from the linearized, Laplace-domain model presented in Eqs. (152)-(l60)
plus Eq. (163). Figure 19 graphically illustrates the linear dependence
of the perturbed hybrid system with the various interactions of blanket
and plasma effects.
The linearlized and perturbed equations of the model are based on
the entire set of ten original model Eqs. (141 a)-(141j ) of which five
are nonlinear. Therefore, although the block diagram in Fig. 19 is in¬
formative and useful for stability analysis, it is not applicable in
transient analysis involving the temporal evolution of perturbed
systems unless the perturbations are small and the time interval to be
examined is short.
The usual rules of block diagram control systems were applied to
obtain the overall transfer function for the hybrid system, ATg(s)/AS(s).
This ratio is a natural result; however, the ratio of blanket power to
plasma feedrate might be more directly useful for this model because of
the basic simplicity of the blanket temperature description given in
Eq. (130).
First, the overall hybrid transfer function, aT^(s)/aS(s), was
obtained without any artificial feedback. The artificial feedback
coefficients (a^ and a^) were set to zero to simplify the open-loop,
plasma-related portion of the hybrid system so that the open-loop plasma
source transfer function of Eq. (169) was obtained in agreement with
Eq. (61) of Chapter 2:

THE BASIC HYBRID MODEL*
Figure 19. Block diagram of the linearized global fusion-fission hybrid reactor model.
-154-

-155-
¿qp(s)
AS (S )
open
1 oop
s[2n q(T ) - n T
o3 o o o 3T
a.] + 2n g(T )(—
4 o3 O X.
1 % t 3q
—) - n T —3-
i2 o o 3T
12
2 , 7 1
s + s(-
tE t2
o
1_ + _L) + JL (J_ . J_)
O 0
(169)
where the terms, l/x^ and 1/x-j, were set to zero for constant confine¬
ment and the terms, 1/x, and 1/x, , were assumed negligible for the
D1 d2
low-burnup hybrid plasma. The coefficient, a^, is defined in Eq. (56d)
and the coefficient, a^» is defined in Eq. (89a).
Therefore, the open-loop plasma source transfer function,
Aq (s)/aS(s), can be incorporated as shown in Fig. 13 of Chapter 2 to
simplify the plasma subsystem. Alternatively, this same result for
Aqp(s)/AS(s) could be obtained directly using the linearized hybrid
model by substituting Eqs. (152) and (153) into Eq. (157) and solving
for the open-loop plasma source transfer function presented in Eq. (169).
The other feedforward elements of the blanket system including the
blanket reactor transfer function, the power scaling factor, and the
transfer function relating blanket temperature changes to blanket power
changes were combined to produce the reduced hybrid block diagram
depicted in Fig. 20. After reduction of the blanket reactivity feed¬
back loop, the simplified hybrid system block diagram of Fig. 21 was
obtained where the blanket temperature feedback effect has been combined
into the overall forward loop using a single blanket transfer function,
Tb(s), given by:
Vs) =
Gf V Ef Vg(s + A)/Cp
B
hAr
(170)
[s + ——]Ls2 + ( - „ P° + A)s - —£] + an,(s + A)
L cmDJL ' A J 16v '
PB B

A$(s )
DRIVER
Figure 20
Partially-reduced hybrid block diagram with no artificial feedback.

- as(s) ®—*
DRIVER
PERTURBATION
DUE TO DRIVER
*Open-loop plasma has no feedback.
Figure 21. Simplified reduced hybrid system block diagram with no artificial feedback.
-zsi-

-158-
where the reducing coefficient, a-jg, is given by
No aT Gf V ^f VB
a16 A
(171)
From Fig. 21 which is representative of the modeled hybrid system with
inherent blanket temperature feedback but without artificial feedback,
the overall, open-loop transfer function was obtained as follows:
ATB(s)
AS(s)
loop loop L c A A a 16
PB
(172)
Aqp(s)
AS(s)
V Co
P 2
f v Lf B/ pR B
hA,
*p.
where the plasma elemental source transfer function, Aq (s)/AS(s), is
given in Eq. (169) and does not include the effect of artificial tem¬
perature feedback on the feedrate.
When the artificial plasma temperature feedback on the feedrate was
included, the closed-loop transfer function for the plasma resembled that
of Eq. (86) in Chapter 2 without 1/t -j and l/iji due to using the constant
confinement model and without 1/x^ and l/i^ which are negligible for
the less reactive hybrid plasmas. The pertinent plasma portion of the
overall hybrid diagram of Fig. 19 is repeated in Fig. 22 for which the
closed-loop (now including artificial feedback) source transfer function
is given in Eq. (173):
Aqp(s)
AS(s)
rn Í3.
n L o 3T
o
-r9(To,(s+xr'i;
0 0 Eo 2
>]
s2 + s(- 7-+ 7-) +7— (7^— 1—) - «lO - td s) tE t2 Tn n tE t2 1 al 4 u
closed
loop
(173)

Figure 22. Closed-loop block diagram for the linearized point-model plasma with temperature feed¬
back to the feedrate.
-159-

-160-
The overall, closed-loop, hybrid system transfer function, ATD(s)/aS(s),
D
presented in Eq. (174) remains in the same form as Eq. (172) except that
the closed-loop plasma source transfer function given in Eq. (173)
replaces the open-loop one used in Eq. (172).
ATB(s)
AS(s)
m As)
closed
loop
ASTsT
closed
1 oop
Vi
VB
Gf v 'f VB/cpBmB
s + a1?s + a18s + a]g
(174)
where the polynomial coefficients, a^, a^g, and a^g are given by
hAr
~ Pr
17 c mD A
PR B
+ A
(175a)
hA„ 3 - p _ .
ai8= (r"sr)(—í + ' —
PB B
Ap N a, G. V Er Vn
off f B
(175b)
-N aT G,. V Er VD A Ap
Off f B yo
'19
hAr
izr-^r) â– 
A vc m
PB B
(175c)
Stability Criteria for the Hybrid System
The Routh Criterion was applied to this overall transfer function
which includes both inherent (doppler) feedback and artificial plasma
temperature feedback. Since the poles of the denominator in Eq. (174)
decide the stability of the system and since the plasma and the blanket
are only interacting in the feedforward mode via the source conversion
coefficient, > the Routh Criterion was applied individually to each
multiplicative segment of the overall hybrid transfer function. Stability

-161-
criteria for the closed-loop plasma source transfer function, Aq (s)/aS(s),
were obtained using the Routh Criterion which supplied the three
stability criteria of Eqs. (90a)-(90c) in Chapter 2. Similarly the
Routh array of Fig. 23 for the second polynomial in the denominator of
Eq. (174) was used to obtain three stability criteria based on blanket
parameters.
1
18
0
al 7
al 9
0
a
18
0
0
a
19
0
0
0
Figure 23. Routh array for the cubic denominator for blanket effects
in the overall hybrid transfer function.
The results obtained by applying the Routh Stability Criterion con¬
sist of six stability criteria in all which are presented in Eqs.
(176a)-(176f).
S tab i 1ity Criterion I:
1 + “iV1 ' 3r}
o
(176a)

-162-
Stability Criterion II:
”l r,2td.
Stability Criterion III:
Stability Criterion IV:
- + ——
Tn
0
â–  - a
III:
; <
n
0
i
te
0
IV:
? -
o|
Q.
lv ‘ 3T
> 0
alc2
> 0
hA,
+ X +
c mD
Pr B
(176b)
(176c)
(176d)
Stability Criterion V:
-Xp hAr 3 - p _ _ _ N
[ r^- + ———— ( 7 ~ + X) - G, V £, Vp aT -“]
LA cm' A ' f fBTAJ
PB B
Xp hAr N
[~¡C + ^ >0
S - p _ hA
C a ~ + A + —r]
A
Cn
PB B
(176e)
Stability Criterion VI:
-Xp hAr N
[——— ( ) - G, V Er- VD A aT ■“] > 0
L A vc m ' f f B T A J
PB B
(176f)

-163-
The first three stability criteria were obtained from the closed-
loop plasma source transfer function and are repeated from Eqs. (90a)-
(90c) in Chapter 2; the second three criteria were obtained from the
closed-loop blanket transfer function using the Routh array of Fig. 23
where all first column coefficients were required to be positive for a
stable hybrid system. All six of these criteria must be satisfied
simultaneously; when appropriate hybrid system values are included,
all six conditions can be met provided feedback is included. Indeed,
the three blanket conditions are met automatically for realistic systems,
regardless of the specific system because of blanket subcriticality
(p0< 0).
The secondary artificial feedback on the plasma feedrate (with delay
time, t , ) is important only for backup control or for treating blocked
a2
coolant channels or failure to remove sufficient blanket heat while in a
steady-state condition. This artificial feedback is not intended for
primary plasma transient control. The plasma source feedback based on
plasma temperature variations is more reliable and faster acting to con¬
trol plasma perturbations directly. In addition, this work was not
really concerned with blanket heat transfer considerations except as
needed to complete the simplified hybrid model developed in this analysis.
Therefore, aside from including the second artificial feedback term in
the overall hybrid nonlinear model as well as in the linearized block
diagram, it was given no further consideration. Such an artificial
feedback term from the overall transformed response, ATg(s), back to the
original transformed perturbation, AS(s), does show the possibility of a
completely closed loop model. Therefore, the completed hybrid model
for this study is complete although somewhat restrictive. The model does
allow consideration of many types of perturbations and resulting transients.

CHAPTER 4
HYBRID PLASMA OPERATIONAL CONSIDERATIONS
Introduction to Hybrid Plasma Time-Dependent Behavior
The blanket neutronic analysis described in Chapter 5 was used to
determine the required volumetric neutron generation rate in the plasma
to produce 6500 MWth in the hybrid system. Calculations were performed
to determine the specific hybrid plasma core conditions required to drive
the hybrid blanket described in Appendix B. Additional calculations
were also performed to establish the stability and the transient behavior
of the hybrid plasma when subjected to small perturbations. Before
specific transient hybrid plasma phenomena were examined, scoping studies
were performed to investigate and characterize various hybrid plasma
equilibrium states and establish ranges of the plasma equilibrium tem¬
perature, Tq, and ion density, n , required to produce the proper neutron
source to drive the power-producing system.
112
The CLASSIC2 Code was used for this hybrid plasma analysis
since it employs a point-model system of equations similar to those
presented in Chapter 2. The CLASSIC2 Code was used to describe both
equilibrium and time-dependent plasma conditions which could affect the
surrounding fissile hybrid blanket. The point-model CLASSIC2 Code is
described in Appendix D along with input requirements for using the code
to examine specific equilibrium and time-dependent plasma conditions.
-164-

-165-
The basic point-model plasma equations applied in CLASSIC2 are
repeated here:
MiL-sm HÜ1 DTn2(t)
dt *[Z> ' T 2
n
077)
d[n(t)T(t) ]
dt
nxQ n2(t) S(t)T (t)
UI a , S
12 ' 3
n(t)T(t)
tE
bT1/2(t)n2(t)
3
(173)
where the only difference when compared with the model presented in
Chapter 2 is the explicit allowance for time-dependent variations in the
injection energy, T (t). An important parameter which must also be
specified in the input to the CLASSIC2 Code is the R-ratio defined as
the ratio of particle and energy confinement times, t /t^, as in Chap¬
ter 2.
In all the hybrid analyses both confinement times were assumed to
vary identically with temperature and density. In agreement with the
generality of this hybrid model and the uncertainty connected with any
actual selection of a confinement model, the constant confinement model
was chosen for use with the CLASSIC2 Code.
The volumetric fusion neutron production rate, q (t), was com¬
puted in CLASSIC2 using Eq. (141c) repeated as follows from Chapter 3:
q (t) =
ip\
DTn (t)
(179)
where the data for the fusioning plasma reactivity, py, was obtained
11 3
from the standard work by Greene. For reference purposes, Greene's
reactivity data obtained using a Maxwellian distribution of plasma ion

-166-
speeds to weight the cross section is presented in Fig. Cl of Appendix
C. These data were used as standard input for CLASSIC2.
1 O O] 1C
As suggested by Ohta et al. ’as well as Mills, two standard
feedback options are available with CLASSIC2. One option is plasma
temperature feedback on the plasma source feedrate; the other option is
plasma temperature feedback on the plasma injection energy. Many other
feedback choices on such parameters as confinement time variation and
impurity concentration to affect bremsstrahlung have been hypothesized,
but these two choices have been proposed as most easily implemented.
In agreement with the closed-loop hybrid plasma model presented in
Chapter 3, the plasma temperature feedback on the source feedrate option
was utilized in all the time-dependent hybrid analysis using CLASSIC2.
The feedrate feedback used with CLASSIC2 is of the form given in Eq.
(130) which is similar to Eq. (141j):
S(t) = Sq + K$[To - T(t)] (180)
where the total feedback coefficient, K , corresponds to the following
formulation in the hybrid model of Chapter 3:
Ks = "alno/To
(181)
The total feedback coefficient employed in the code is not normalized and
2
has units of ions/cm -sec-keV compared to the normalized feedback coef¬
ficient, a-|, which has units of inverse seconds.
One other simplification is incorporated in CLASSIC2 which sets the
feedback to react instantaneously to a change in plasma temperature.
Very short response times have been utilized in magneto-hydrodynamic

-167-
plasma feedback studies as well as in plasma position control studies to
effect control over gross plasma breakdown and escape where response
times in the range of a millisecond down to hundreds of microseconds
are considered possible for engineering implementation of the feed-
97 98 na
back. ’ ’ ' Based on these short times, the assumption of instantaneous
feedback effectiveness is not a great restriction on the model. This was
found to be particularly true in the dynamic response of the hybrid
plasma following various 5% parameter perturbations. The time constants
such as the energy and particle confinement times in the basic point-
model plasmas examined using Eqs. (177) and (178) were on the order of
seconds. The application of feedback within a few tenths of a second was
not expected to be very different from instantaneous application in its
effectiveness; that is, the effectiveness of any feedback on the feed-
rate was not expected to depend strongly on the speed of response in the
possible millisecond to tenths of seconds range.
In particular, increasing values of the total feedback coefficient
can be postulated to overcome any small delays necessitated by engineer¬
ing problems in applying the feedback. Although full-scale fusioning
plasmas are expected to be limited in the size of allowable feedback,
previous work has shown and the results of this work support the fact
that the magnitude of the feedback coefficient in a transient simulation
can be made very large to overcome delayed implementation for low-
reactivity, highly-driven plasmas of the type used in hybrids.
Hence, a large range of feedback coefficients was used to simulate or
account for the possibility of a wide range of feedback effects varying
from very large feedback effects applied with some delay time down to
relatively small feedback effects implemented instantaneously.

-163-
Independent of feedback considerations of CLASSIC2 Code utilized
six basic input parameters for the point-model plasma represented by
Eqs. (177) and (178). Initially, the R-ratio was selected and then any
three of the remaining five variables of ion density, n, temperature, T,
evergy confinement time, t^, source feedrate, S, and injection energy,
T , were specified. From this basic selection corresponding equilibrium
conditions for the remaining two variables were established and, as
desired, time-dependent transients were examined by perturbing any one
of four equilibrium variables of ion density, temperature, feedrate,
or injection energy.
The basic assumption in CLASSIC2 and the entire global plasma
analysis is that the plasma occupies a constant volume. Ordinarily, an
increase in volume is expected to accompany an increase in temperature
since the plasma pressure is given approximately by the following equa¬
tion of state:
P(t) = nT(t)T(t) (182)
where
T(t) = plasma temperature (keV)
nT(t) = plasma total particle (ions and electrons) density
(ft/cm3)
3
P(t) = plasma pressure (keV/cm ).
Obviously, if temperature increases, then pressure is expected to
respond. Nevertheless, this deficiency in failing to account for pressure
and volume changes was disregarded as unimportant in this work for the
relatively small transients of primary concern. So the work of estab¬
lishing some base calculations for hybrid plasma transients was expected

-169-
to overestimate neutron production rates since the neutron production
rate would decrease if the plasma were allowed to expand against the
confining magnetic field. Therefore, the predicted neutron production
rates were conservative but sufficiently accurate for reasonably small
transients in operational hybrid plasmas. Accounting for this effect
would represent additional complexity without supplying additional
fundamental information on plasma transient behavior as it affects the
overall hybrid system.
The consideration of small transients or short time intervals was
most important for analysis of the linearized model on which stability
predictions were based. When transients were considered for large time
intervals, it was only for comparison purposes since the model is not
strictly applicable for large temperature transients (either above or
below the original equilibrium state).
Although relatively small temperature transients were of primary
concern, the resultant volumetric fusion neutron source transients were
not expected to be quite so small. For a plasma with a Maxwellian
Distribution of ion energies, plasma reactivity shows extremely rapid,
nonlinear variation with temperature as shown in Appendix C. A sensi¬
tivity analysis of the volumetric neutron production rate for the 8 keV
temperature selected for the hybrid plasma core showed that small frac¬
tional changes in temperature can yield over three times that same
fractional increase in the neutron production rate; that is, the neutron
production rate is very sensitive to the plasma ion temperature. This
dependence is explained in Appendix C where the following formulation is
presented for the sensitivity, Spy(T):
if2* sdt(t> ' ^ • (,83>

-170-
The dimensionless sensitivity factor, SgT(T), varies smoothly with
temperature; therefore, for a temperature change, ^y, the neutron pro¬
duction rate was expected to increase fractionally by a factor,
Spy • ^y. Variation of the sensitivity factor in the temperature range
of interest for hybrid work (5-20 keV) is shown in Figure Cl. The
sensitivity factor is largest 3.2) at low energies and decreases as
temperature increases until it reaches unity at the 60 keV peak in
reactivity.
Selecting a Spectrum of Hybrid Plasma Equilibrium Conditions
To begin consideration of the hybrid, the plasma particle and
energy relations of Eqs. (177) and (173) were solved for a spectrum of
equilibrium conditions from which reasonable selections were made for
more specific operating equilibria. Further transient analysis was per¬
formed only on those cases selected as most interesting for an actual
hybrid system.
Equilibrium particle densities, n , and temperatures, T , were
predetermined by the neutron production rate required to drive the hybrid
blanket for proper design power levels. These predetermined values of
13 3
ion density (n = 9.56 x 10 ions/cm ) and temperature (Tq = 8.0 keV)
were needed to quarantee proper blanket power production. The proper
design value for the neutron production rate was dependent only on the
density and temperature since the plasma volume was fixed. From these
two preset values, an entire range of energy confinement times was
selected for investigation for a possible set of operational equilibrium
conditions. The range investigated included energy confinement times up

-171-
to 5 seconds in increments of 0.1 seconds for successively varied R-
ratios of 1.0, 1.5, 2.0, 2.5, 3.0, and 10.0.
The confinement time cannot actually be fixed at a value but is
rather characteristic of or dependent upon the plasma operating condi¬
tions which are represented by the particle density and the temperature
in the plasma. The R-ratio characterizes the plasma; that is, the plasma
conditions such as collision frequency, contained magnetic field as well
as temperature and density determine the R-ratio from detailed plasma
dynamics effects. However, the selection of such a large range of values
of energy confinement time did certainly quarantee inclusion of those
values which a functioning hybrid plasma could be expected to achieve
operationally with some modification of constraining magnetic fields.
The feedrates and injection energies required to satisfy plasma
equilibrium operation for the given ranges of the R-ratio and energy
confinement time were initially examined over a wide spectrum of possible
sets of plasma operational equilibrium conditions. To reduce the
parameter variation and hence the number of different sets of plasma
conditions to be examined, the range of confinement times was narrowed
to the more interesting and tractable range of 1.5 sec < < 2.0 sec.
In addition, only integer values of the R-ratio (1, 2, 3, and 10) were
examined in detail since this variation still encompasses predicted
30 101 102
plasma system R-ratios. ’ ’ Operational values of the R-ratio are
currently predicted to be much lower than the unrealistic R-ratios in the
range of 10-50 for which many previous fusion control studies have been
, , 17,18
performed.
Scoping calculations within the indicated limited variable ranges
yielded the data on equilibrium operating conditions presented in

-172-
Table 4-1. After the equilibrium particle density, nQ, and t'ne tempera¬
ture, Tq, were set, for each value of energy confinement time, t^. , a
o
unique correspondí'ng set of values of the equilibrium injection energy,
Tr , and source feedrate, S , was determined using CLASSIC2. Therefore,
o 0
the information contained in Table 4-1 consists of six unique sets of
possible equilibrium operating conditions for n , T , xr , S , and T
r 3 r 3 o o E o o
presented for each of four (4) values of the R-ratio.
The various equilibrium conditions set forth in Table 4-1 were used
to determine where these plasma equilibrium conditions place the hybrid
plasma on a Hills-like equilibrium curve.^ For a given R-ratio and
13 3
assumed density and temperature conditions (nQ = 9.56 x 10 ions/cm
and Tq = 8.0 keV) to give the required volumetric neutron production
rate, the correspondí'ng feedrate, injection energy, and energy confinement
time were determined as presented in Table 4-1 or similar presentations.
Based on these three quantities (S , T<~ , and ) plus the R-ratio, the
o o
corresDonding variation of n xr with plasma temperature was determined
0 Eo
by evaluating new equilibrium conditions for various temperatures; that
is, for the specified values of R and nQ in Table 4-1, the required
corresponding values of were determined as a function of Tq. A
o
specific set of such curves corresponding to the equilibrium cases listed
in Table 4-II are depicted in Fig. 24. All of the curves in Fig. 24
contain the required density and temperature points for proper neutron
production for the case of = 1.7 sec.

-173-
Table 4-1
Selected Spectrum of Equilibrium Operating Conditions for the
Hybrid Plasma With Constant Confinement
(sec) n t£ (sec/cm )
o
SQ (ions/cm -sec)
(keV)
1
1.5
1 .434
X
10^ 4
o14
u14
10 4
10 4
1° 4
10 4
6.401
X
10^ 3
o13
J1 3
10 3
10 3
10 3
10IJ
17.50
1.6
1 .530
X
6.003
X
17.07
1 .7
1 .625
X
5.652
X
16.64
1 .8
1 .721
X
5.339
X
16.21
1 .9
1 .816
X
5.060
X
15.78
2.0
1 .912
X
4.808
X
15.35
2
1.5
1 .434
X
1014
o14
u14
10 4
10 4
10 4
1014
3.215
X
10^ 3
U1 3
10 3
10 3
1° 3
10 3
10 J
34.85
1 .6
1.530
X
3.016
X
33.98
1.7
1 .625
X
2.840
X
33.11
1 .8
1.721
X
2.684
X
32.24
1 .9
1 .816
X
2.544
X
31.38
2.0
1.912
X
2.418
X
30.51
3
1.5
1 .434
X
10^4
u14
10 4
1° 4
10 4
10 4
10
2.153
X
10^ 3
o13
U1 3
10 3
1° 3
10 3
10U
52.04
1.6
1.530
X
2.020
X
50.73
1.7
1 .625
X
1 .903
X
49.42
1 .8
1 .721
X
1.799
X
48.11
1.9
1 .816
X
1 .705
X
46.81
2.0
1 .912
X
1 .622
X
45.50
10
1 .5
1 .434
X
1014
o14
14
10 4
10 4
10 4
10 4
6.655
X
1012
o12
U1 2
1° 2
1° 2
1° 2
10
168.3
1 .6
1.530
X
6.257
X
163.8
1.7
1 .625
X
5.905
X
159.2
1 .8
1 .721
X
5.593
X
154.7
1 .9
1 .816
X
5.313
X
150.2
2.0
1 .912
X
5.062
X
145.8
*Power production conditions were the same in all cases: T0 = 8.0 keV,
nn = 9.56 x lO^3 ions/cm^, and qn =1.41 x 1011 nts/cm3-sec.
u h0

-174-
O
CD
in
*3-
i
o
X
(J
Z3
TD
O
S-
Q_
E
ZJ
S-
-Q
13
cr
10.0
7.0 “
5.0 "
2.0 "
1.0 -
0.7 -
0.5
0.4
These equilibrium curves apply for no
injection energy (Ts = 0) as well as for
injection energy and source feedrate
corresponding to equilibrium conditions
for nQ = 9.56 x 10^3 ions/cm3, TQ = 8.0
keV, and if = 1.7 sec for the R-ratios
indicated.
Ts = 0 (all R-ratios)
i
10
20
T
30
r
40
T
50
Plasma Temperature (keV)
Figure 24. Equilibrium curves for various equilibrium plasma conditions.

-175-
Table 4- 11
Hybrid Plasma
Equilibrium Operating Conditions for x^ =
to Meet Required Power Production o
1 .7 sec
R
3
n (ions/cm )
0
T0 (keV)
x£ (sec)
0
3
Sq (ions/cm -sec)
T (keV)
0
1
9.56 x 1013
8.00
1 .7
5.652 x 1013
16.64
2
9.56 x 1013
8.00
1.7
2.840 x 1013
33.11
3
9.56 x 1013
8.00
1.7
1 .903 x 101 3
49.42
10
9.56 x 1013
8.00
1 .7
5.905 x 1012
159.2
After S and T were selected as equilibrium values for the hybrid
° so
steady-state, CLASSIC2 was rerun for each R-ratio with the S and T
0 so
values for each equilibrium as temperature was allowed to vary from 4 to
50 keV. Equilibrium conditions were determined for each temperature
yielding products of equilibrium density and energy confinement time as
a function of temperature. The resultant variation of the equilibrium
n tg. -product is shown in Fig. 24. The corresponding equation for the
o
equilibrium n x^ -product at steady-state is simply:
o
n
DTQa . dtTs
4 2
(184)
where burnup is included.
For comparison purposes the same plasma simulations were repeated for
a hypothetical zero injection energy. The resultant equilibrium n -
o
curves were no longer dependent on feedrate or R-ratios since the
equilibrium Mills condition, again including burnup, is given by

-176-
3T
n ir = n
o E^ nTQ 1/0
0 DT a _ ^-j-l/2
(135)
which only varies with plasma temperature and not with burnup. The
resultant equilibrium curve for the case of zero injection energy is
also presented in Fig. 24 where the curve is located above the curves
obtained with injection energy and exhibits the characteristic parabolic
shape.
Essentially, the curves in Fig. 24 indicate that the hybrid plasma
is low in reactivity and will require significant blanket conversion of
energy to sustain the system. Although many more curves were generated,
this Mills-like set of curves including the initial equilibrium condition
of i £ =1.7 sec and R = 2 is representative of reasonable operating
o
conditions.
Based on the equilibrium operating data presented in Table 4-1,
certain operating ranges were eliminated. For example, the case of R = 10
was eliminated from further consideration for several reasons. First,
the injection energies required (140-170 keV) represent significant
technological problems, especially at the injection rates required for
the low temperature hybrid system. There is no need to make the hybrid
more complex than necessary. If the hybrid is to be used as an inter¬
mediate step in the development of pure fusion, it must be based on
technology available in the near term.
Second, the presence of large beam energies with associated large
feedrates requires different methods of analysis to evaluate the particle
and energy equations. Specifically, the beam-plasma reactivity must be
accounted for in such cases as shown in recent workJ^ ^ ^ Effects of

-1 77-
non-Maxwellian fusion reactions of the injected beam with the thermal
plasma and the resultant finite slowing down time of the injected beam
particles cannot be neglected in a Two-Component Torus.
To reduce the number of sets of operating equilibrium conditions
to be examined still further, the R = 1 and R = 3 cases were also re¬
moved from consideration. The R = 1 case was removed from further con¬
sideration because the required source feedrates are nearly as large
as the plasma particle density. In addition, the ts/3Tq ratio is less
than unity which is not a condition of interest for hybrids. Such a
condition also presents unique problems in determining equilibrum con¬
ditions. If plasmas can be self-ignited and sustained, then a Tokamak
fusion-fission hybrid will have no place in the power industry except as
2
a possible breeder of fissile fuel in an Augean production system. In
addition, the required injection energy was found to be so low (15-17
keV) that it probably would not justify the system complication required
to implement it. Such energies are simply not under consideration for
presently planned pure fusion or hybrid systems. Of the remaining two
cases, the R = 3 case was eliminated in favor of R = 2 simply because
30
some very significant pure fusion design studies such as UWMAK-III and
29 101
others ’ have been based on low R-ratios in the vicinity of R = 2.
After R = 2 was selected as reasonably representative of possible
fusion devices, the remaining six cases showing a range for the energy
confinement time were each examined to determine operational properties
for the hybrid system as well as to determine the effects that increasing
confinement efficiency has on the capability of the hybrid plasma to be
controlled. Since the sensitivity of the neutron production rate to
temperature changes was calculated to be about 2.7 at plasma temperatures

-1 78-
of 8 keV, even small temperature transients were expected to result in
significant increases or decreases in the size of the neutron source
driving the blanket.
If hybrid blankets are to be optimized for first wall region power
density as carefully as pure fusion blankets, then such surges in neutrons
and power production will be very important design considerations. They
may severely limit overpower ratings as well as average blanket operating
power densities.
The complete equilibrium conditions selected for further analysis
are presented in Table 4-III. Sample analyses of transients and equilib¬
rium conditions at R = 1 and R = 3 indicated little difference from the
R = 2 case in the plasma conditions and responses to perturbations.
Essentially the speed of transient development and difficulty of control
were found to increase with the R-ratio because the R-ratio is an indi¬
cator of the efficiency with which particles are confined and able to
compensate the driving source to promote transient development.
Table 4-1II
Equilibrium Plasma Conditions Selected for Transient Analysis
With R = 2 and q =1.41 x 10'1 nts/cm^-sec
Ho
n (ions/cm^) T (keV) tf (sec) S (ions/cm3 sec) T (keV)
o o o o o
9.56
X
101 J
8.0
1 .5
3.215
X
1013
34.85
9.56
X
1013
8.0
1 .6
3.016
X
1013
33.98
9.56
X
1013
8.0
1.7
2.840
X
1013
33.11
9.56
X
1013
8.0
1 .8
2.684
X
1013
32.24
9.56
X
1013
8.0
1 .9
2.544
X
1013
31 .38
9.56
X
1013
8.0
2.0
2.418
X
1013
30.51

-1 79-
For preset density and temperature values, the equilibrium feedrate
and injection energy values were determined for the preselected R = 2
value and equilibrium energy confinement times ranging from 1.5 to
2.0 sec. Then the Mills-Condition curves were generated for each con¬
finement time value. The effects of burnup were included using CLASSIC2.
The resultant values for the six equilibria summarized in Table 4-111 are
presented graphically in Fig. 25. The results are presented graphically
in Fig. 25 to illustrate the n xr -curves on which each of the six sets
0 Eo
of equilibrium operating hybrid plasma conditions fall. The case of
generating these curves for zero injection energy is also included for
comparison. Figure 25 showing the hybrid plasma position demonstrates
the driven nature of hybrid plasma which means significant blanket
fission energy generation will be needed to sustain the plasma and pro¬
duce net energy for the utility grid.
For more efficient, larger confinement times, lower values of feed-
rate and injection energy are required for equilibrium as summarized in
Table 4-111. Since the driving feedrate and injection energy tend to
impede plasma transient development, it is not surprising that plasmas
with higher confinement times were found to be less controllable when
subjected to perturbations displacing the plasma from the hypothetical
equilibrium states presented in Table 4-111.
Each of the six sets of equilibrium operating conditions in Table
4-111 was examined for its response to a variety of different perturba¬
tions. Since the confinement time was chosen to be time-invariant, and
since a spectrum of possible values was examined, there was no point in
perturbing the confinement times. The perturbations examined included
instantaneous positive and negative 5% step changes in each of four

-180-
o.o H \ 1 1 1 r
0 10 20 30 40 50
Plasma Temperature (keV)
Figure 25. Mills steady-state curves including burnup for R = 2.

-181-
equilibrium variables (n, T, S, T ). Variations in the confinement
time were not considered because it is actually dependent on the state
of the system and cannot realistically be changed independently, but only
in response to some other system parameter change such as density or
temperature.
Uncontrolled Plasma Response to Perturbations
Although a standard 5% perturbation of the equilibrium parameter was
selected for examining the time-dependent response of the hybrid plasma,
some other larger and smaller perturbations were also examined. But the
spectrum of system equilibria examined subject to 5% perturbations is
adequate since larger perturbations become increasingly non-perturbing
and more like large-scale disruptions or accident conditions. Responses
to several smaller perturbations in the range of 0.5-1.0% of an equilib¬
rium parameter were examined and showed very sluggish system response
indicative of the non-interesting nature of exceedingly small perturba¬
tions in the simplified hybrid plasma model. This sluggish behavior
was especially evident for lower R-ratios and low energy confinement
times because the hybrid plasma is not self-sustaining but maintained by
the feedrate. Because its state is retarded from movement by the driving
feedrate, the hybrid was found to react slowly even for 5% perturbations.
Each of the eight system perturbations (- 5% nQ, t 5% T , i 5% SQ,
and ± 5% T ) was introduced into the six different equilibrium hybrid
o
plasmas represented by the R = 2 sets of equilibrium conditions pre¬
sented in Table 4-111. These sets of conditions are referred to as
hypothetical hybrid equilibrium states.

-182-
To explain the general transient results of these perturbations,
the basic global plasma equations relating particle density and tempera¬
ture are repeated here in slightly different form from Eqs. (177) and
(178) as follows:
d[n(t)T(t)]
dt
dn(t) n(t) DTn (t) _ , ,
dt ' T " 2
n
DTQrtn2(t) , n(t)T(t) , bT1/2(t)n2(t)
12 te 3
S(t)Ts(t)
3
(186)
(187)
The source feedrate, S(t), and the injection energy, Ts(t), are so-
called extrinsic variables; that is, S(t) and T$(t) act as inhomogeneous
source terms similar to inhomogeneous neutron source terms in the point-
model reactor kinetics equations. For low-reactivity plasmas such as
those used to drive subcritical fission lattices, the analogy is very
appropriate and useful. Such extrinsic variables are characteristic
only of some predetermined, externally applied conditions. Without the
source feedrate to drive the fusioning plasma, the plasma subsystem can¬
not survive; the same is essentially true for the flux distribution in a
subcritical assembly such as the hybrid blanket. Without an inhomogeneous
neutron source, there can be no power-producing, neutron flux distri¬
bution in the subcritical blanket assembly despite the presence of
fissionable fuel .
The extrinsic nature of the source feedrate and the injection energy
variables is best illustrated by noting that they affect the global
plasma behavior by an external driving force which can be removed or
retained based on engineered actions taken external to the global plasma.
Neither of these variables is intrinsically or inherently affected by

-183-
the transient behavior or time development of the plasma itself. No
plasma conditions are inherently effecting changes in the feedrate or
injection energy. This feedforward, irreversible nature of the driving
variables, S and T , is illustrated in Fig. 26.
Perturbations in the source feedrate or in the injection energy are
really step changes externally introduced into the plasma. CLASSIC2
modeled the changes in these variables as permanent until the initial
alteration was removed or changed with time by some external action
represented as artificial or engineered feedback. The time scale for
such removal may range from instantaneous up to many seconds or even
minutes depending on the physical nature of the perturbation and the
corresponding feedback engineered into the system. However, there is
nothing in the system response that can inherently remove or affect
changes in an external driving force represented by the perturbed feed-
rate or the perturbed injection energy. This type of perturbation re¬
mains effective and unchangeable until removed by some external action
which contrasts directly with perturbations in the temperature or the
particle density which disappear with time as the plasma undergoes
transient development.
Physically, these changes in S or Tg are not internal perturbations
but rather correspond to external system malfunctions where either the
injection rate or the injection energy might be suddenly set to a new
value different from its equilibrium value. This may be a temporary or
even permanent (as far as one fusioning plasma duty cycle is concerned)
value different from the design conditions for these driving variables.
Therefore, when step changes in the extrinsic variables were introduced
into the system, the plasma simply underwent a transient response to

Figure 26. Illustration of the feedforward effectiveness of the source feedrate and the
injection energy on plasma equilibrium conditions and transient behavior.
-184-

-185-
reach a new equilibrium where the feedrate or the injection energy
assumed equilibrium values corresponding to the new value of the per¬
turbed feedrate or injection energy.
In contrast, the plasma density and the plasma temperature are
intrinsic variables, just as the neutron density and the precursor con¬
centration are in a fissioning assembly. In contrast to the inhomo¬
geneous source terms, changes in other plasma variables or operating
conditions result in changes in such intrinsic variables: There variables
are character!'Stic of the state of the system described. If the state is
changed, then the describing variables must change. For example, if the
density is changed, then the temperature is expected to change auto¬
matically depending on how the density is changed; the essential compen¬
sation of intrinsic variables is inherent to describe the evolution state
of the system. However, the externally-fixed source feedrate and in¬
jection energy will not change unless some external effect is intro¬
duced.
This conceptually simple dichotomy of variables is important to the
proper understanding of the dynamic response of the hybrid plasma to the
different types of perturbations for which it was examined. The
transient growth or decay of the plasma from an equilibrium state can be
characterized according to the general category of the perturbed vari¬
able. Thus the transient development of the hybrid plasma was qualita¬
tively categorized prior to examination of the actual perturbations
because the plasma was expected to behave differently for perturbations
in temperature and ion density versus perturbations in the inhomogeneous
feedrate and injection energy.

-186-
Perturbations in the plasma temperature or the particle density,
about an equilibrium condition were found to result in the plasma re¬
turning to the original system temperature, density, and neutron pro¬
duction rate or evolving to some other operating regime depending on the
stability of the global plasma system at the state coordinates in effect
as a result of the perturbation. Depending on the proximity to a stable
operating regime, the system was found to react more or less quickly to
adjust its state coordinates to the parameter perturbations; for the
intrinsic variables, the perturbations were lost within the evolving
system response.
The final equilibrium plasma states (density, np, temperature, Tp,
and volumetric neutron production rate, q ) resulting from ± 5% step
PF
changes in each of the four plasma variables are presented in Tables
4-1V through 4-XI. Each table contains entries for each of the six
initial hypothetical hybrid equilibrium conditions presented in Table
4-111. These results in Tables 4-IV through 4-XI were obtained by per¬
turbing each plasma state and following the resultant nonlinear tempera¬
ture development until the plasma temperature (and density) reached
steady state. The plasma conditions were found to return to the initial
state involved or some other different final state characteristic of the
R = 2, xr , S , and T conditions established at the outset for the
’ E o s
o o
particular perturbation run in question.
No further consideration was given to the density perturbations.
Previous work has shown that density feedback cannot be used to control
an inherently unstable plasma. Here, the insensitivity of the density
to the plasma state involved was demonstrated by the relatively small
fractional changes exhibited by the final densities recorded in Tables

-187-
Table 4- IV
Final Uncontrolled Hybrid Plasma Equilibrium Conditions Following
a +%5 Perturbation in the Temperature
R*
1
2
3
10
(sec)
npT^ (sec/cm)
(ions/cm ) Tc (keV)
(nts/crri -sec)
1
.5
1.434
X
10^
1 A
9.
.560
X
io 3
8.
.00
1
.410
X
10
1
.6
1 .455
X
10 4
9,
.093
X
10 1
38.
.06
1
.601
X
10
1
.7
1 .539
X
10 4
9,
.051
X
1° 3
40.
.90
1
.629
X
10
1
.8
1 .622
X
10 1
9.
.011
X
1° 3
43.
.58
1
.665
X
10
1
.9
1 .705
X
10 4
8,
.974
X
10 3
46,
.13
1
.682
X
10
2
.0
1 .788
X
1014
8,
.939
X
10 3
48,
.55
1
.693
X
10
1
.5
1 .434
X
,014
9.
.560
X
1°¡3
8.
.00
1
.410
X
10
1
.6
1 .397
X
1° 4
8.
.730
X
10 3
36,
.24
1
.438
X