Neutronics of a coupled multiple chamber gaseous core reactor powersystem

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Material Information

Title:
Neutronics of a coupled multiple chamber gaseous core reactor powersystem
Physical Description:
xxi, 364 leaves : ill. ; 28 cm.
Language:
English
Creator:
Panicker, Mathew M., 1945-
Publication Date:

Subjects

Subjects / Keywords:
Pulsed reactors   ( lcsh )
Nuclear reactors   ( lcsh )
Nuclear Engineering Sciences thesis Ph. D
Dissertations, Academic -- Nuclear Engineering Sciences -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references (leaves 359-362)
Statement of Responsibility:
by Mathew M. Panicker.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001483556
oclc - 21094823
notis - AGZ5602
System ID:
AA00002131:00001

Full Text












NEUTRONICS
GASEOUS


OF A
CORE


COUPLED
REACTOR


MULTIPLE CHAMBER
POWER SYSTEM


MATHEW


PANICKER


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY


UNIVERSITY


OF FLORIDA



































To my parents.















ACKNOWLEDGEMENTS


The


author


wishes


express


his


sincere


thanks


and


appreciation


to the


members


of his


supervisory


committee,


Alan


Jacobs,


Nils


Diaz,


Calvin


Oliver,


and


Robert


Hanrahan


their


willingness


ass


preparation


of this


dissertation.


The


author


particularly


thankful


to Dr.


Edward


Dugan,


his


supervisory


committee


chairman,


his


guidance,


patience,


author


and


understanding


grateful


having


throughout


studied


this


under


research.


the


The


direction


such


splendid


personality


as Dr.


Edward


Dugan


whose


knowledge


gaseous


core


neutronics


has


been


truly


invaluable


to the


preparation


this


dissertation.


This


research


was


supported


funds


from


the


Frederick


Hauck


Fund,


the


University


Florida,


and


the


Air


Force


Wright


Aeronautical


Laboratories


(AFWAL).


The


AFWAL


work


was


performed


the


Innovative


Science


and


Technology


Directorate


the


Strategic


Defense


Initiative


Organization


within


the


Innovative


Nuclear


Space


Power


Institute


(INSPI)


the


University


of Florida.


The


author


wishes


express


-- S


a a


. .


-- a


_









this


thanks


research.


and


The


appreciation


author

to Dr.


also

Nils


wishes


Diaz


express


Director


special

INSPI,


affording


him


the


opportunity


to work


INSPI


as a


research


assistant.


Computations


this


research


work


were


done


University


of Florida


facilities


(Northeast


Regional


Data


Center


and


the


CAD/CAM


center)


and


the


San


Diego


Supercomputer


Center.


The


author


grateful


the


College


Engineering


the


local


computer


support


and


the


National


Science


Foundation


computer


time


the


San


Diego


Supercomputer


Center.


The


author


thankful


to fellow


graduate


students


with


whom


he had


many


fruitful


discussions


on various


aspects


this


research.


Special


thanks


to Mrs.


Mary


Cole


for


her


assistance


the


final


preparation


this


dissertation.


Finally,


an indescribable


degree


appreciation


goes


the


author


s wife,


Annamma,


and


son,


Dinesh,


their


patience,


support,


and


understanding


throughout


the


author


stay


the


University of


Florida.


















TABLE


OF CONTENTS


ACKNOWLEDGEMENTS................ .... . ....


v111
*iii

* *
XIlI


LIST

LIST


ABSTRACT...... . ****** ** .

CHAPTERS


....... 1


INTRODUCTION..................


Brief Description of the Multiple Cavity
Gaseous Core Reactor Power System.....
Aim and Scope of the Dissertation.......
Organization of the Dissertation........


Pulsed
0.*..0...
......O..
........0


STUDIES


Introduction


Choice of Weighting
Selection of Phase
Treatment of Coupli
Coupling Coeffici
Delay Time Effect
Comments...........


ON COUPLED

.....**. *


Funct
Space
ng Eff
ents..


CORE


ions...
Regions
ects...


NEUTRONICS......

*.e a .* ** a .

* e S S S S *
.. e .. ****
.................
.................


PREVIOUS
AT THE


STUDIES ON
UNIVERSITY


PULSED GASEOUS
OF FLORIDA....


CORE


REACTOR


Introduction.....
UF6 Piston Engine


Neutronic Ana
Energy Model.
Results......
Pulsed Nuclear
Gas Generator
Pulaed Nuelea


of Kylstra
sis........


Piston and Pulse
Systems........
r Piston System.


.... ....** **
al...............
* .*.****.. * .
.....oo...........
*.*.*.......*****..
*...*....* **


. .. ..* ........ ..


PREVIOUS


OF TABLES................ ........-. .- .....-.

OF FIGURES.............................. ...-.... .


i










STATIC NEUTRONIC
CAVITY GASEOUS


STUDIES ON COUPLED MULTIPLE
CORE REACTOR SYSTEM.............


Introduction...
The Bimodal Gas
Steady State Ne
Steady State Ca


I


Core
utron
icula


.....*. ... .. *..
Reactor System
ic Analysis....
tions Procedure


ription


scussioS
Effect a
Beryl
the B:
Effect
Varia
Effect
Beryl
Core
Effect
Beryl
BPGCR
Effect
the B


n of Resu
of PGCR C
lium Thic
PGCR Core
of Outer
tion.....
of BPGCR
lium Thic
Fuel Load
of PGCR C
lium Thic
Core Fue
of Uneaua


PGCR


Core


Its
ore
kne
Is
Ber

Cor
kne
ing
ore
kne


1
1


Loadi:
ss Var
Unfue
yllium

e Fuel
ss Var
Is Fi
Fuel
ss Var


Loading
PGCR Co


ng a
iati
led.
Thi


nd
ons


Inner
When


ckness


Loading and
nations When


xed....
Loading
nations
Is Fix
re Fuel


Is Unfueled....


.....0..
and Inn
When th
ed......
Loading
a.....a..


Inner
the PGCR


er
e
e


When


Suimmary.........


DYNAMIC
CAVITY


NEUTRONIC
GASEOUS


Introduction..
Method of Soluw
Reactor Kine
COUPKIN....
Discussion of '
Integral Par<
Neutron Gel
Effect oi
beryl:
BPGCR
Effect o
variai
Effect o
inner
when
Effect o
inner
when
fuel
Effect o
when


tion
tics


STUDIES ON COUPLED
CORE REACTOR SYSTEM


of Coupled
Equations


Results.
meters:
neration


PGCR
ium th
chamber
outer
ions..
BPGCR
beryll


Couplin
Time, an
ore fuel
ckness va
is unfue
beryllium


core fuel
ium thickn


MULTIPLE


...*Core Point*
core Point


Co
Re
oad
iat


ed......
thicknes


oading


S


PGCR gas loading i
f PGCR core fuel 1
beryllium thickne
the BPGCR core has


load
f un
the


ing..
equal
BPGCR


PGC
cor


R core
e is un


var
ixel
ing
var
nom


- -


1


lient
ity.
nd i
when


* ...
nner
the
.O...


and
nations

and
nations


inal


1 load
led...


ing


Di


A --


!










Fueled PGCR cores
BPGCR core....


Summary.....


nominally

* ***


fueled


* . 4 0**
* ********


CONCLUSION
OF FOURTH

Introducti
Summary of
Variatio
and In
Unfuel
Effect o
(20-70
Variatio
and In
PGCR F
Variatio
Inner
Chambe
Unequa
Static and
Bimodal
Steady
Integr
Coup


Gas


RECOMMENDATIONS
RESEARCH.......


on. . ..
I~tl ni ,a a


n
ne
ed
f
c


n PGC
Be T
BPGCR
uter
S). ..


n in BPGCR
ner Be Thi
uel Loadin
n in PGCR
Be Thickne
r Loading
1 PGCR Cor
Dynamic N
Gaseous Co
State Neu
al Kinetic


led


Neutron K
Feedbacks
Moderator
Fuel Stud
Flow/Heat


Gaseous
Summary....


Core
neti
due
Refl
es..
Tran


Core
a ....


s.
CS
to
ec


Rea


R Fuel Loading
hickness (10-40
Chamber.......
Be Thickness Va


F
<*


CK
g
Css
09
(2
e
eu
re
tr
S


ue
ne
(1
re


.....
Load
s (10
atm)
Loadi


(20-30
5 atm)..
Loading.
tronic M
Reactor
onic Ana
Paramete


ing
-40

ng
cm)


ode
sy
lys
rs


Calculation
Temperature
tor Studies.


r Studie
tA--


. JL o
at a a


AND


AREAS


* a S
................

................


(20-6
cm) ,


** *
atm)


riation


a....
(25-
cm) ,

(10-5
, Fix


* ..
Is f
stem
is..
for


Chang....
Chang


for


* O O
100 atm)
Fixed

0 atm) and
ed BPGCR


. .


247

247
248


249

249


250


. ..


the


Individual


.... ...
* ...


*4*****
Bimodal


* a a a0 a S 4 *e "
* O a ...a." a a
....********************


APPENDICES


MCNP-A GENERAL MONTE
PHOTON TRANSPORT..


CARLO


CODE


FOR


NEUTRON


AND


COUPLED CORE POINT
AND THE SOLUTION


BENCHMARK
CODES ON
GAS CORE


REACTOR KINETICS
PROCEDURE.......


CALCULATIONS WITH XSDRNPM
REFERENCE SPHERICAL AND
REACTOR SYSTEMS........


EQUATIONS


AND MCNP
CYLINDRICAL


COMPARISON OF


RESULTS


FROM


COUPKIN


AND


.


!



b
L


_ -


I
1
I


Concus.LUa
















LIST


OF TABLES


Tables


Summary


Kinetics


of Results


Models


from


for


the


the


Different


PNP


Neutron


Configuration.


...... 59


Typical Equilibrium
Generator System


Cycle


Pulsed


Operating


Gas


Conditions...........


Typical Equilibrium
System Operating


Cycle


Piston


-Driven


Conditions..


Core


Physics


Gaseous


Core


Parameters


Reactor


the


Bimodal


System..


............. 82


System


and
of P
Core


Neutron
Neutron


GCR


Gas


Reactor


Multiplication


Removal
Loading
System


Lifetime


for
with


the


Factor (keff)
(R) as a Function


Bimodal


Inner


Gas


Beryllium


Thickness


cm..


........................... 83


System


and
of P
TIBE


Neutro
Neutron


GCR


Gas


of 20


n


Multiplication


Removal
Loading
cm......


Lifetime


the


Factc
(B)
BGCR


>r (keff)
as a Function


S


system


with


System
and


Neutron
Neutron


of PGCR


Gas


Multiplication


Removal
Loading


Lifetime


the


Factc
(G)
BGCR


>r (keff)
as a Function


System


with


TIBE


of 25


cm......


. . . . . . 85


System


and
of


Neutron
Neutron


PGCR


Gas


Multiplication


Removal
Loading


Lifetime


the


Factc
(e)
BGCR


>r (keff)
as a Function


S


system


with


TIBE


of 30


m ......


............................. 86


System
and


Neutron
Neutron


Multiplication


Removal


Lifetime


Factc
(e)


,r (keff)
as a Function


Pages










System
from


Neutron


MCNP


Thickness


Multiplication


as a Function


(TIBE)


Various


Factor
Inner B


PGCR


(k ef)
ery lium
Core


Gas


Loadings...


System
from


Neutron
XSDRNPM


Thickness


Multiplication
as a Function


(TIBE)


Various


Factor (keff)
f Inner Beryllium
PGCR Core


Gas


Loadings.


Total


PGCR


Lifetime
Beryllium


Gas
at


Loading


and


Criticality


Thicknesses


Neutron


for
the


Removal


Various
BGCR S


Inner


ystem........


4-10


Neutron


Inner B
PGCRGas


removal
erylliu
Loadin


Lifetime


Thickness
as Predic


as a Function


for
ted


Various


MCNP............


4-11


System
and


Neutron
Neutron


of Outer


TIBE


of 20


Multiplication


Removal


Beryllium


Lifetime
Thickness


Factor (keff)
(1) as a Function


(TOBE)


for


cm.........


4-12


System
and


Neutron
Neutron


of Outer


Multiplication


Removal


Beryllium


Lifetime
Thickness


Factor (keff)
(R) as a Function
(TOBE) for


TIBE


of 30


cm. ....


4-13


System
and


Neutron
Neutron


BPGCR


Gas


Multiplication


Removal


Loading


Lifetime


Factor (keff)
(1) as a Function


TIBE


4-14


System
and


Neutron
Neutron


of BPGCR


Gas


Multiplication


Removal
Loading


Lifetime


for


TIB


Factc
(E)
E of


or (keff)
as a Function


2


cm. . ....


4-15


System


and


Neutron
Neutron


of BPGCR


Gas


Multiplication


Removal


Lifetime


Loading


TIBE


Factor (keff)
(1) as a Function


cm.


4-16


System


and


Neutron
Neutron


of BPGCR


Gas


Multiplication


Removal


Loading


Lifetime


TIBE


Factor


as a


of 40


f)
unction


cm.


4-17


System


from


Neutron


MCNP


Multiplication


as a Function


Factor
Inner B


(ke f)
ery lum


cm..........


_


v









4-18


System Neutron Multiplication Factor (keff)
from XSDRNPM as a Function of Inner Beryllium
Thickness (TIBE) for the Spherical Mockup of
BGCR System for Various BPGCR Gas Loadings...


4-19


4-20





4-21




5-1


System Neutron Removal Lifetime (i) as
Function of Inner Beryllium Thickness
for the BGCR System at Various BPGCR
Gas-Fuel Loadings....................


System Neutron Mu
from XSDRNPM as
Thickness (TIBE
of BGCR System
Loadings......


(TIBE)


Itiplication Factor (keff)
a Function of Inner Beryllium
) for the Spherical Mockup
for Various BPGCR Gas


System Neutron Multiplication Factor (k ff)
and System Neutron Removal Lifetime () of
the BGCR System as Predicted by MCNP for
Various Unequal PGCR Gas Loadings.........


Integral Kinetics Parameters as
PGCR Core Fuel-Gas Loading for
with Inner Beryllium Thickness


a Function of
the BGCR System
of 10 cm.......


Integral Kinetics Parameters as
PGCR Core Fuel-Gas Loading for
with Inner Beryllium Thickness

Integral Kinetics Parameters as
PGCR Core Fuel-Gas Loading for
with Inner Beryllium Thickness


Integral Kinetics Parameters as
PGCR Core Fuel-Gas Loading for
with Inner Beryllium Thickness


Integral Kinetics Parameters as
PGCR Core Fuel-Gas Loading for
with Inner Beryllium Thickness


a Function of
the BGCR System
of 20 cm.......


a Function of
the BGCR System
of 25 cm.........


a Function of
the BGCR System
of 30 cm.........


a Function of
the BGCR System
of 40 cm.......


Integral
Outer
System


Integral
Outer


Kinetics
Beryllium
with TIBE


Kinetics
Beryllium


Parameters
Thickness
of 20 cm.


Parameters
Thickness
a aa


as a
(TOBE)


as a
(TOBE)


Function
for the


Function
for the


of
BGCR


of
BGCR


...........


I









BGCR


System


with


TIBE


Integral
BPGCR


Kinetics


Core


Parameters


Fuel-Gas


Loading


as a Function


the


BGCR


of
System


with


TIBE


cm.


5-10


Integral


Core
TIBE


Kinetics


Fuel-Gas


of 30


cm.


Parameters


Loading


as a Function


the


. .. .........*


BGCR


stem


BPGCR


with


..*..*..*..* ..** **S** S* **


5-11


Integral


Core
TIBE


Kinetics


Fuel-Gas


of 40


Parameters


Loading


as a Function


the


BGCR


stem


BPGCR


with


cm.


5-12


Integral


Core
TIBE


Kinetics


Fuel-Gas


Parameters


Loading


as a Function


the


cm.


BGCR


stem


PGCR


with


S. . "


5-13


Integral


Core
TIBE


Kinetics


Fuel-Gas


of 20


Parameters


Loading


as a Function


the


BGCR


stem


of PGCR
with


cm...


Average


Cores of
When the


Coupling


the
PGCR


Coefficients


BGCR
Core


System w
Loading


Among


ith


the


a TIBE


Pattern


PGCR


of 2


[60(4)


20(4) ,


10(4)]....


5-15


Average
Cores
When


Coupling


for
the


the
PGCR


Coeffic


BGCR
Core


ients


System
Loading


Among


with


the


PGCR


a TIBE


Pattern


50(4),


30(4)


, 20(4)].


. a. .. .* *


5-16 Average
Cores
When


Coupling


for
the


the
PGCR


Coeffic


BGCR
Core


ients


System
Loading


Among


with


the


PGCR


a TIBE


Pattern


[70(4),


30(4),


20(4)].


......*.* ........""****


5-17 Average
Cores
When


Coupling


for
the


the
PGCR


Coeffic


BGCR
Core


ients


System
Loading


Among


with


the


a TIBE


Pattern


PGCR
of 20


[80(4),


40(4),


30(4)].


... ..S .........S***


5-18


Average
Cores
When


Coupling


for
the


the
PGCR


Coefficients


BGCR
Core


System
Loading


Among


with


the


a TIBE


Pattern


PGCR
of 20


[70(6),


. .. .* *


flnes.CP4


an 4' e


4bIa


a t rI 1 I u S t *= I r_ | -J1X L-nat *


'lnnrnl 4 ni


am -7 -a


PGCR


cm..............* ...


10(6)]........


K-,1


Stta iraT









A-i


MCNP Surface Cards........ ........**************.


Basic


MCNP


Tallies..


. .. .* * *


Input Cards for the
Reactor System...


Bimodal


Gaseous


Core


MCNP


Input


a Single


Core


System k from XSDRNPM Benchmark Calculations
for th eeference Gaseous Core Reactor-Cylindrical
Geometry.....................*.*** ****. ***-.**-


System k from XSDRNPM Benchmark Calculations
for the Reference Gaseous Core Reactor-Spheric
Geometry Mockup........................... ...


System keff from MCNP Benchmark
for the Reference Gaseous Core


Calculation
Reactor System...


XSDRNPM

XSDRNPM


Collapsed

Collapsed


Relative Neut
of Time as
a System of
1.0058....


26-Group

4-Group


ron Level
Predicted
Neutron


Structures.............

Structure...............


(N(t)/N(O))
by ANCON and
Multiplicatio


as a Function
COUPKIN for
n Factor


.a . .. . a.. a. a a a


Relative Neutron Leve
of Time as Predicte
a System of Neutron
0.9280..............


1 (N(t)/N(O)) a
d by ANCON and
Multiplication


a Function
OUPKIN for
Factor


. .. .a a a* a a


Relative Neu
of Time as
a System o
1.0421....


tron Level
Predicted
f Neutron


(N(t)/N(O))
by ANCON and
Multiplicatio


as a Function
COUPKIN for
n Factor


- .


Relative Neutron Level (N(t)/N(O)) as a Function
of Time as Predicted by ANCON and COUPKIN for
a System of Neutron Multiplication Factor
1.1171..... ............... -.. -..-.-.-.-.-.-...-


.... . o. ****************.


PGCR... ......... ... .


__















LIST


OF FIGURES


Bimodal


Gas


Core


Reactor


Power


System


Schematic....


Top


and


Side


View


of Bimodal


Gas


Core


Reactor..


Schematic Representation
Power Breeder; (a) Top


of Coupled Fast-Thermal
View (b) Side View.....


Schematic


Representation


of UF6


Piston


Engine......


Neutron Multiplication Factor Versus
Pressure for an Infinite Graphite


UF Partial
Reflector...


Relative Thermal
Reflector for


Flux in
the UF6


Core
Piston


and Graphite
Engine..............


Average Core Thermal Neutron
Multiplication Factor as a
Travel Through the Piston
Reflected Engine..........


Flux and
Function
Cycle for


Neutron
of Percent
a Graphite-


UFg Piston
Imposed


Engine Performance
at 10 Percent Cycle


for Thick
Position.


Refl
. .


ector


General Schematic
Gas Core Nuclear


for a
Power


Piston-Driven
System......


Pulsed


Delayed Neutron and Photoneutron Precursor
Concentration Buildup During Startup from
for Pulsed Nuclear Piston Engine.........


Peak Gas Temperature and Mechanical
Time During Startup from Shutdown
Nuclear Piston Engine............


Shutdown


Power Versus
for the Pulsed
. e....


Average Thermal
Cycle Time for
Kinetics Model


-I- n


Neutron Flux in
the PNP Engine
Is used.......


tvarrna 'T'harmn 1 Ui


the
When


W inv i n thh


Core
the


mnrs


Versus
Adiabatic


Vsrs lls


0... 5


Mbiit rhn









Average
Cycle


Kinetics


Thermal


Time


Model


Neutron


the


PNP


Is Used


Flux


Engine
with 1:


the
When


Core
the


Versus
Point


msec.


-12 Pulsed


Gas


Generator


Schematic.......


3-13


Average
Time


Cor
for


Nuclear


e


Thermal


a Typical


Neutron


Gas


Flux


Generator


Versus
Pulsed


Cycle
Gas


Core


System....


3-14


Average
Time


Cor
for


e


Thermal


a Typical


Neutron


Flux


Piston-Driven


Versus
Pulse


Cycle
d Gas


Core


Nuclear


System..


Cross-Sectional


Core
BGCR


View


Reactor


XSDRNPM


of Cylindrical


MCNP


and


Spherical


Bimodal


Gas


Mock-up


Calculation..


ke
ke fr
for

k
eff
for


and


TIBE


and


as a Function


of 10


cm..


as a Function


of PGCR
...of PGCR...

of PGCR


Gas

* Gas
Gas


Loading


Loading


TIBE


of 20


cm.....


keff
for


and


TIBE


as a Function


of 25


PGCR


cm..


Gas
. .


Loading


ke for


and


as a Function


PGCR


Gas


Loading


TIBE


cm..


keff


and i
TIBE


as a Function


cm.....


of PGCR
. 0..


Gas
....


Loading


System


Neutron


XS DRNPM


Multiplication


as a Function


of PGC


Factor
R Gas


from


Loading.


System
from


Neutron


MCNP


Multiplication


as a Function


Factor, (k if)
Inner Beryltlum


Thickness


System


Neutron


XSDRNPM


Multiplication


as a Function


Factor


Inner


from


Beryllium


Thickness...


4-10


Neutron


Removal


Function


Lifetime


of Inner


Vanri an


Beryllium


Gasf


from


MCNP


Thickness


Tadina.. -


as a
(TIBE)


.- -. 105


PCnpR









System Neutron Multiplication
and Neutron Removal Lifetime
of Outer Beryllium Thickness
TIBE of 20 cm...............


System Neutron Multiplication
and Neutron Removal Lifetime
of Outer Beryllium Thickness
of 30 cm....................


Factor
(1) as
(TOBE)


Factor
(t) as
(TOBE)
.......


(keff)
a Function
for


(keff) .
a Function
for TIBE


4-13


System keff and I as a
Loading for TIBE of


Function
10 cm....


BPGCR


Gas


4-14


System
Gas


keIf and t
LoaIdng for


as a
TIBE


Function
of 20 cm


BPGCR


4-15


4-16


System ke f and I as a
Loading for TIBE of


System ket and I as a
Loading or TIBE of


Function
30 cm....


Function
40 cm....


BPGCR


of BPGCR


Gas


Gas


4-17


System Neutron Multiplication Factor (k ) from
MCNP as a Function of Inner Berylliume sickness
for Various BPGCR Chamber Fuel-Gas Loadings....


4-18


System k ^ from XSDRNPM as
Inner Beryllium Thickness
Chamber Gas-Fuel Loadings


a Function
for Various


of
BPGCR


4-19


System Neutron Removal Lifetime (I) as a Function
of Inner Beryllium Thickness as Predicted by
MCNP for BPGCR Chamber Gas Loadings............


4-20


System keff and
Loading with
BPGCR Chamber


I as a Function
5 atmosphere of
for TIBE of 20


of PGCR Gas
Fuel-Gas in
cm..........


4-21


4-22


System keff and
Loading with
BPGCR Chamber


System keff
Thickness
PGCR Gas


I as a Function of PGCR Gas
5 atmospheres of Fuel-Gas in
for TIBE of 30 cm...........


as a Function of Inner Beryllium
Predicted by XSDRNPM for Various
Loadings.........................


4-11


4-12










4-24


Relative Radial Thermal Neutron
in the BGCR System with BPGCR
Atmospheres and PGCR Cores at
Fuel-Gas Loading.............


Flux Distribution
Chamber at 100
10 Atmospheres


Average Coupling Coefficients (aZ2k)
of PGCR Gas Loading for a BGCR with
10 cm... ..... ... ......... ...........


Average Coupling Coefficients (a2,k)
of PGCR Gas Loading for a BGCR with
20 cm...............


Average Coupling Coefficients (a2k)
of PGCR Gas Loading for a BGCR with
25 cm..............................


Average Coupling Coefficients (a2uk)
of PGCR Gas Loading for a BGCR with
30 cm..............................


as a
TIBE


as a
TIBE


as a
TIBE


as a
TIBE


Average Coupling Coefficients ( a2 ) as
of PGCR Gas Loading for a BGCR with T
40 cm................................

Average Coupling Coefficients (a 3) as
of PGCR Gas Loading for Various Inner
Thickness.................. ...... ...


Average Coupling Coefficients
of Inner Beryllium Thickness
Gas Loadings................


a
IBE


Function
of


Function
of


Function
of


Function
of


Function
of


a Function
Beryllium


(a2 ~) as a Function
for Various PGCR
....................I


Reactivity
Core Gas
Thickness


of BGCR
Loading
....e...


System as a
for Various
*...... ...


Function of PGCR
Inner Beryllium


Neutron Generation Time as
Beryllium Thickness for
Chamber Gas Loading of 2


a Function
BGCR with
atm Helium


f Inner
BPGCR
Gas.........


5-10


Average Coupling Coefficients
of Outer Beryllium Thickness
and 30 cm...................


(a2 ) as
for TIBE


Function
20 cm


F -11


Ctra om Doar+- iA tr nae


a nunr +'i nn


nf f irt-or









5-12


Average Coupling Coefficients (a. ) due to
Neutronic Contribution from PG Cores to BPGCR
Chamber as a Function of BPGCR Core Fuel-Gas
Loading........................................ .


5-13


5-14


Average Coupling Coefficients (a. ) due to
Neutronic Contribution from PG Cores to
Chamber as a Function of Inner Beryllium
Thickness...............................


Average Coupling
PGCR Cores (a2
Beryllium Thic
Loading of 10


BPGCR


Coefficients Between Adjacent
-3 13) as a Function of Inner
kn6ess for a BGCR with a PGCR Gas
atmospheres UF6-He Mixture......


5-15


Neutr
Inn
wit
UF6


on Generation T
er Beryllium Th
h PGCR Core Gas
-He Mixture....


ime (A)
ickness
Loading


as a Function of
for a BGCR System
of 10 Atmospheres


5-16


5-17


Neutron
BPGCR
Inner


Generation Time (A) as
Gas Loading for a BGCR
Beryllium Thickness of


System Reactivity
Loading. ......


as a Function


a Function
System for
10 cm and


of BPGCR


cm.....


Gas


5-18


5-19


Average Coupling Coefficients due to
Contribution from BPGCR Chamber to
as a Function of PGCR Gas Loading.


Average coupling Coefficients
Contribution from PGCR Cores
of the BGCR as a Function of
Loading.....................


Neutronic
PGCR Cores


due to Neutronic
to BPGCR Chamber
PGCR Core Gas


5-20


Average
Cores
BGCR


Coupling Coefficients
as a Function of PGCR
for both Unfueled and


Between Adjacent PGCR
Gas Loading for the
Fueled BPGCR Chamber..


5-21


Neutron Generation
Loading for the
BPGCR Chamber...


Time
BGCR


as
with


a Function
Fueled and


of PGCR Core
Unfueled


Gas

200


5-22


System Reactivity as a Funct
Gas Loading for the BGCR w


ion
ith


of PGCR
Unfueled


Core
and









5-24 Relative P
Function


GCR


Neutron


of Time


5-25 Relative PGCR
Function of


Neutron


Time


for


Level
TIBE


Level
TIBE


and
of


and
of


keff
25 cm


keff
30 cm


as a


S......... 214


as a


......... 215


5-26 Relative PGCR
Function of


Neutron


Time


Level


and


a TIBE


kf
40


as a


cm.


......... 216


5-27 Relative


for
with


PGCR


a Cycle
Inner


and


Neutron


Time


Beryllium
40 cm with


Level


as a Function


seconds


Thickness


ejk=0
)k0


for
of 20


During


BGCR
cm,


Exhaust


Time


System


cm,
...... 218


5-28 Relative PGCR
Function of


Neutron


Time


Level


an


a TIBE


d kf
of


as a


cm and


Cycle


Time of

5-29 Relative P
Function
During E


0.3

GCR
of


seconds

Neutron


Time


xhaust...........


with

Level


During


ajk=0nd
and


a TIBE


keff
40


Exhaust


... 221


as a


cm with


ajk


.. 222


5-30 Relative


Time


Various


PGCR


Neutron


a BGCR
Values


Level


System


with


Coupling


as a Function


TIBE


Coeffi


clients


of Cycle
cm for


(ejk)


.... 224


5-31 Relative


PGCR


a BGCR


Values


Neutron


with


TIBE


Coupling


Level


of 40


as a Function


cm at


Coefficients


Time


Various


. 225


5-32 Relative


Time


Various


PGCR


Neutron


a BGCR
Valies


with


Level


a TIBE


as a Function


of 30


Core-to-Core


of Cycle


cm for


Delay


Times


(rjk)............................................ 226


5-33 Relative


PGCR


Neutron


Level


as a Function


Cycle


Time


Various
(Tjk)--


a BGCR
Values


with


a TIBE


Core-to-Core


cm for


Delay


Times


5-34 Relative


Time


PGCR


the
cm,


Neutron


BGCR
and


Level


System


as a Function


with


cm with


TIBE


Core-to-Core


of Cycle


cm,
Delay


Time,


sec


ajk=0


During


Exhaust


...... 229


5-35


Nj(t)/N.(O)
for aJ BGCR


a


nd C (t)/C (0)
with TIBE f 4


Two


cm with


Cycles
the


r jk=0


4


I









5-36


5-37


5-38


N. (t)/N.(0) and C (t)/C (0) for
for a BGCR with TIBE Af 40 cm
Coupling Reduced to 0.6 Times
Values......................


N.(t)/N.(0) and C (t)/C (0) for
a BGCR with TIBE of 46 cm with
Coupling Coefficients Reduced
the Actual Values.............


N.(t)/N (0) and C (t)/C (0) for
a BGC with TIBE of 3 cm with
Coupling Reduced to 0.50 times


Two Cycles
with Core-to-Core
the Actual


Five Cycles for
Core-to-Core
to 0.55 times


Four Cycles for
Core-to-Core
the Actual Values.


5-39




5-40


Relative PGCR and BPGCR Neutron Levels as a
Function of Cycle Time for a BGCR with a
TIBE of 30 cm with Coupled PGCR and BPGCR
Cores....................................


PGCR Core Neutron Multiplication Factor as
Function of Cycle Time for a BGCR System
with TIBE of 30 cm......................


5-41




5-42


5-43


Relative PGCR
Function of
TIBE of 30
from BPGCR


and BPGCR Neutron
Cycle Time for a
cm with PGCR Cores
Core..............


Levels as
BGCR with
Uncoupled


Relative PGCR and BPGCR Neutron Levels as
Function of Cycle Time for a BGCR with
TIBE of 20 cm with Coupled PGCR and
BPGCR Cores............................


PGCR Core Neutron Multiplication Factor as
Function of Cycle Time for a BGCR System
with TIBE of 20 cm......................


5-44




A-i


Relative PGCR
Function of
TIBE of 20
from BPGGCR


and BPGCR Neutron
Cycle Time for a
cm with PGCR Cores
Core.............


Geometry Splitting/Russian
in MCNP.................


Levels as
BGCR with
Uncoupled


Roulette
.0. 0.0..


Technique


Detail


of the


Weight


Window....


-. ... a a- a -aaala


t a T -


,E


n,,,LU~


II(L1~ ~1


nAUA















Abstract


of
the


of Di


ssertation


:he University
Requirements J


Presented


Florida


the


Degree


the


Partial I
of Doctor


Graduate


School


fulfillment


Philosophy


NEUTRONICS


GASEOUS


OF A COUPLED


CORE


REACTOR


MULTIPLE


POWER


CHAMBER


SYSTEM


MATHEW


May


. PANICKER

, 1989


Chairman
Major De


: Dr.


Edward


apartment


Static


. Dugan


: Nuclear

dynamic n


Engineering


eutronics


Sciences

an innovative,


coupled,


multiple


chamber


beryllium


moderated


bimodal


gaseous


core


reactor


(BGCR)


, capable


generating


both


and


high


power


space


applications


are


inves


tigated.


found


that


the


core-to-


core


neutronic


coupling


effects


contribute


significantly


the


power


behavior


any


chamber


during


both


power


and


high


power


operation.


a consequence


the


strong


coupling,


steady


state


operation


even


power


increases


can


be achieved


with


the


cores


operating


a subcritical


condition.


The


core


power


levels


can


be controlled


the


core


fuel


loadings


as well


as the neutronic


coupling


effects


among


the


cores


The


BGCR


system


consists


of a larqe


, central


w -- w









chamber


and


a surrounding


annular


ring


of cylindrical


power


pulsed


gaseous


core


reactor


(PGCR)


chambers


embedded


beryllium


moderator.


The


BPGCR


operates


as an open


flow


stem


For


and


PGCR


employs


s the


an MHD


gaseous


generator


fuel


mixture


energy


conversion.


cyclically


injected


into


PGCR


chambers


during


the


intake


phase,


heated


during


the


power


phase,


and


then


discharged


energy


conversion


system.


With


multiple


PGCR


chambers


and


proper


timing


of their


operation,


the


conducted


neutronics


analysis


indicates


that


the


power


system


should


able


to provide


a relatively


continuous


source


of hot


pressurized


the


power


conversion


system.


The


effects


of PGCR


and


BPGCR


core


loadings


and


inner


and


outer


beryllium


thickne


sses


on the


system


neutron


multiplication


factor,


system


neutron


removal


lifetime,


the


core


-to-core


neutronic


coupling


coefficients


have


been


examined


static


(steady


state)


neutronic


analy


the


BGCR.


The


dynamic


behavior


the


BGCR


evaluated


the


program,


COUPKIN,


which


was


developed


as part


this


work.


COUPKIN


solves


the


coupled


core


point


reactor


kineti


equations.


integral


kinetics


parameters


the


COUPKIN


program


are


obtained


from


the


static


analyst


the


system.


The n

taken


eutronic


into


coupling


account


effect


among


coupled


the


core


cores

kineti


explicitly


equations















CHAPTER


INTRODUCTION


Pulsed


cyclic


gaseous


core


nuclear


reactor


systems


have


undergone


extensive


theoretical


and


experimental


investigations


at the


University


Florida.


Neutroni


CS-


energetic


analyses


pulsed


gas


core


reactor


systems


have


lead


a basic


scientific


understanding


the


behavior


devices.


that

promi


associated


The


with


results


the pulsed

sing nuclear


gaseous

r energy


the


these


core


conceptual


research


reactor


concept


that


operation


efforts


these


indicate


a versatile


has


and


attractive


features


space


power


generation


as compared


conventional


solid


fueled


nuclear


reactor


systems(1


These


features


include


low


critical


mass


, high


fuel


utilization,


adaptability


to different


energy


conversion


systems,


high


operating


temperature


and


efficiency,


and


good


control


and


safety


characteristic


CS.


They


have


additional


advantages


of compactness,


relatively


high


power


density,


response,


and


wide

good


operating

operational


ranges


excellent


flexibility


. The


dynamic

high


temperature


the


coolant/working


fluid


the


core,










heat


rejection


space.


The


gaseous


nature


the


fuel


gives


the


added


advantage


rapid


startup


capabilities


and


the


simple


geometry


the


core


structure


helps


minimize


the


thermal


shock


and


thermal


stress


especially


during


the


rapid


startup.


Pulsed


gaseous


core


reactors


(PGCR)


are


energy


intensive,


cyclic


fission


driven


power


systems


that


have


uniquely


attractive


neutronic


and


energetic


characteristics.


the


power


chamber


or core


a single


chamber


PGCR,


fissionable


circulating


gas


is pulsed


from


a predominantly


-subcritical


state


a short


lived,


power


rich


near-critical,


critical,


or supercritical


state


and


then


the


hot


gas


exhausted.


The


energy


released


the


fissioning


gas


is extracted


any


suitable


power


conversion


system.


Previous


analyses


pulsed,


cyclic


gaseous


core


reactor


power


systems


have


been


performed


single


neutronically


isolated


chambers.


These


previous


studies


have


not


included


the


effects


on system


performance


the


core-to-core


neutronic


coupling


which


would


exist


array


pulsed


gas


core


reactor


chambers.


The


pulsed


gaseous


core


reactor


system


space


power


generation


investigated


this


dissertation











PGCR


s are


expected


to provide


long


term


stationkeeping


surveillance


power


a space


based


system.


If a pulsed


gaseous


core


reactor


system


with


just


one


or two


chambers


used


pulsation


thi


effects


purpose,


will


flow


make


discontinuities


difficult


due


to extract


continuous


conversion


supply


system


hot


. The


exhaust


multiple


gas


chamber


any


power


design


the


power


pulsed


system


will


be capable


as the


analysis


Chapters


4 and


5 will


show,


delivering


a relatively


continuous


source


high


temperature,


pressurized


gas


power


conversion


system


associated


with


the


reactor


uninterrupted


power


generation


. Thi


can


be achieved


properly


timing


the


operation


the


individual


chambers


between


the


intake,


power,


and


exhaust


phases.


Also


the


multiple


cavity


design


should


have


a favorable


impact


on the


system


reliability


since


power


can


generated


even


a few


the


individual


chambers


fail.


Brief


Description


Gaseous


Core


The


Multiple


Reactor


Power


Cavity
System


Pulsed


The


investigated


gas


core


reactor


power


system


consists


(Figures


and


1-2)


a large


, central


burst


power


oscillating


gas


core


reactor


(BPGCR)


chamber


- .. a ~. n, -e~cr~n~l 1 ,-.~I 4,%A.4 n at


a


1 J &T.T


AC ~m~ll


IYYIIY~ L~


~ I. I~~(1~Y


L 4











































i-- e --I -

'nq-' w Se S


gi----------
I

-11
a a____ a__ _..a,,.'
* ---------


























































MU frt


* I I I


S |II











a mixture


uranium


hexafluoride


(UF6)


and


helium


gas.


Highly


enriched


UFg


the


reactor


fuel.


The


helium


gas


added


to enhance


the


thermodynamic,


transport,


heat


transfer


characteristics


of the


fluid.


The


identical


upper


and


lower


halves


the


reactor


system


are


separated


a common


moderator


slab


the


mid


plane.


This


moderator


slab


separates


the


magnetohydrodynamic


(MHD)


disk


generator


regions


the


top


and


bottom


central


BPGCR


chambers.


The


annular


ring


PGCR


chambers


designed


provide


low


power


station


keeping/surveillance


purposes.


In the


PGCR


chambers


the


fissile


gaseous


fuel


(UF6


mixture)


cyclically


injected


into


the


power


chambers,


which


are


individually


pulsed


from


far-


subcritical


to near-critical,


critical,


or a supercritical


state


and


the


heated


gas


then


discharged


energy


conversion;


each


chamber


designed


produce


a few


MW(e)


power.


With


the


multiple


PGCR


chamber


design


and


proper


timing


of operation


the


individual


chambers


among


their


intake,


power,


and


exhaust


(discharge)


phases,


the


system


expected


to provide


a continuous


source


high


temperature


pressurized


gas


any


suitable


power











fail.


During


power


operation


the


system,


the


central


BPGCR


chamber


will


either


be voided


or have


constant,


pressure


non-fuel


gas


flow.


The


transition


from


station


keeping


the


high


burst


power


mode


of operation


can


achieved


circulating


a gaseous


uranium


fuel


through


the


central


BPGCR


cavity.


The


neutronic


coupling


effects


the


PGCR


chambers


on the


gaseous


fuel


bearing


central


chambers


provide


the


necessary


criticality/heating


condition


generate


a partially


ionized


plasma


2000-4000


The


high


pressure


fuel


gas


mixture


driven


into


the


disk


MHD


generator


through


a supersonic


nozzle.


The


energy


conversion


occurs


the


disk


MHD


generator.


The


fuel


gas


then


passes


through


a diffuser


and


a radiator


heat


exchanger


before


being


circulated


back


into


the


central


BPGCR


chamber


a compressor.


more


detailed


description


the


bimodal


gaseous


core


reactor


power


system


and


basic


operational


details


are


given


Chapter


Aim


and


Scone


The


Dissertation


The


primary


aim


this


dissertation


to perform


static


(steady


state)


and


dynamic


neutronic


esign











neutronic


coupling


among


the


cavities


separated


the


moderator/reflector;


effects


variation


inner


and


outer


moderator


thickness;


and


the


effects


variation


core


fuel


gas


loadings


on the


core


neutron


level


the


system


various


cycle


times


operation.


The


parameters


required


the


dynamic


neutronic


analysis


the


system


such


as reactivity,


neutron


generation


time,


neutronic


coupling


coefficients


among


the


various


from


cores,


static


core-to-core


neutronic


delay


analysis


times

the s


etc.,


system.


are


obtained


Steady


state


neutronic


analysis


the


system


includes


examining


the


effects


of variation


PGCR


gas


loading


and


inner


and


outer


moderator/reflector


thickness


on the


system


neutron


multiplication


factor


(keff),


system


neutron


removal


lifetime


neutronic


neutron


coupling


generation


coefficients


time


(ajk)


(A),


among


and


the


the


various


chambers.


Calculations


have


been


performed


both


equal


and


unequal


gas


loading


among


the


PGCR


chambers.


Organization


of The


Dissertation


Chapter


the


dissertation


includes


a brief


survey


of coupled


core


neutronics


analysis


methods


that


have


been


used in


previous


studies


various


coupled


core


nuclear


( ),











analysis


which


leads


to a better


understanding


of the


neutronic


behavior


of coupled


core


systems.


Chapter


discusses


the


highlights


previous


gas


core


research


efforts


that


have


been


performed


the


University


of Florida


. This


chapter


includes


a brief


summary


methods


used


the


analyses


and


presents


key


results


obtained


from


these


previous


studies.


A basic


understanding


provided


the


single


chamber


pulsed


gaseous


core


reactor


system


concept.


Chapters


4 and


5 deal


with


the


static


and


dynamic


neutronic


studies, respectively,


the


multiple


cavity


gaseous

includes


core

the


reactor.

results


The

from


static


the


neutronic


general


analysis


purpose,


mainly


three-


dimensional


Monte


Carlo


transport


code,


MCNP


(3),


and


the


one-dimensional


discrete


ordinates


transport


code,


XSDRNPM


(4) .


The


static


neutronic


analysis


includes


the


effects


variation


inner


and


outer


moderator


thickness


and


the


PGCR


and


BPGCR


gas


loading


on the


system


neutron


multiplication


factor


(keff) ,


the


neutron


removal


lifetime


neutronic


().


Chapter


analysis


the


which


system


contains


includes


the


the


dynamic


effects


variation


inner


and


outer


moderator


thickness


and


gas


loading


on the


core


coupling


coefficients,


system











Chapter


6 draws


conclusions


from


the


results


obtained


this


study.


This


chapter


also


provides


some


recommendations


various


areas


future


research


to be


undertaken


towards


further


development


multiple


gaseous


core


nuclear


reactor


power


systems.


The


appendices


that


are


given


at the


end


this


dissertation


include


sections


on the


derivation


and


method


of solution


of coupled


core


point


reactor


kinetics


equations


as employed


research,


a discussion


the


basic


features


and


capabilities


the


MCNP


Monte


Carlo


code,


benchmark


neutronic


calculations


with


XSDRNPM


MCNP


codes


on reference


spherical


and


cylindrical


gaseous


core


reactor


systems,


and


comparative


evaluation


results


from


the


coupled


core


kinetics


program,


COUPKIN,


a single


core


the


point


reactor


kinetics


code,


ANCON


(5),


from


the


Alamos


National


Laboratory.















CHAPTER


PREVIOUS


STUDIES


ON COUPLED


CORE


NEUTRONICS


Introduction


The


spatial


and


spectral


coupling


effects


reactor


kinetics


were


treated


many


authors


(6-24)


during


the


1960s


and


1970s.


The


coupled


nuclear


reactor


systems


subjected


kinetics


studies


included


fast-thermal


reactors


, modular


cores


large


thermal


power


reactors,


clustered


rocket


reactors,


and


Argonaut


type


reactors.


From


the


point


view


of coupling,


that


the


amount


reactivity


supplied


the


mutual


neutronic


interactions


among


subsystems,


the


above


reactors


range


from


very


tightly


coupled


systems


instance,


the


fast-thermal


reactors)


very


loosely


coupled


systems


(like


the


coupled


rocket


reactors).


Except


the


coupled


core


nuclear


rocket


engines,


the


other


works


on coupled


core


kinetics


referenced


this


chapter


are


directed


towards


reactors


with


solid


fuel


cores


In view


the


existence


many


apparently


analysis,


different


seems


formulations


appropriate


coupled


compare


core


various


neutronic


models


br nriaoraBanni4 in ,


-tho


nouiitrnnni r


hph ~i nr


rnimli Ar


rr kalln~











Coupled


core


reactor


kinetics


was


initiated


1958


Robert


Avery


(6,7)


with


his


investigation


of dynamic


characteristics


the


coupled


fast-thermal


breeder


reactors.


The


investigated


fast-thermal


reactor


system


that


couples


a fast


and


thermal


assembly


was


designed


obtain


relatively


long


prompt


neutron


lifetime


that


characteristic


of a thermal


assembly


and


a high


breeding


gain


that


characteristic


fast


assembly


(Figure


The


coupled


system


considered


Avery


consisted


of a Pu-239


fueled


cylindrical


fast


core


surrounded


an annular


blanket


of natural


uranium,


coolant


and


structural


material


followed


an annular


beryllium


moderator


surrounded


outer


blanket


consisting


primarily


depleted


uranium.


The


inner


blanket


serves


as the


core


the


thermal


system,


a barrier


the


low


energy


neutrons


between


the


moderator


and


fast


core,


and


as a "reflector"


for


the


fast


core.


In Avery'


formulation,


each


core


or system


divided


into


nodes


the


point


model


kinetics


equations


are


constructed


coupling


effects


each node,

among the


taking

nodes.


into

The


account

integral


the

and


neutronic

coupling


parameters


the


coupled


core


point


kineti


formulation


are


obtained


from


the


steady


state


analysis


the


corresponding


system


cores


or subsystem.

















Shield


Outer Blanket


Inner Blanket


Inner Blanket

Moderator

Outer Blanket





Shield


Figure


2-1. Schemati


Representation of


Coupled


Fast


Thermal Power Breeder;











moderator/reflector


medium


Masaoki


Komato


(8) .


Komato


developed


two


models;


one


multi-node


kinetic


theory


model


with


time


dependent


coupling


parameters


and


a time


dependent


multi-core


parameters,


diffusion


such


theory


the


model


diffusion


which


nuclear


coefficient,


microscopic


cross


section


etc.,


are


invariable


with


respect


space


each


region


but


can


vary


with


time.


Modifications


and


alterations


the


methods


developed


Avery


and


Komato


have


been


achieved


many


authors


and


are


referenced


this


chapter


(6-24)


Except


Avery'


approach


modifications


others,


other


formulations


make


use


the


assumption


factorized


into


that


a time


neutron


dependent


angular


neutron


flux


can


angular


flux


an amplitude


(power


level)


function


*(r,E,n,t)


= N(t)


(r,E,n,t)


(2-1)


with


the


restriction


that


fd3rldEfd2f


w(r


n,E)


t(r,E,f,t)


(2-2)


where


w(r,n,E)


an arbitrary


weighting


function.


The


major


differences


the


various


coupled


core


neutronic


analyses


that


are


examined


this


work


can


classified


into


three


categories,


namely,


the


choice


a
a -


-
*1e n r~ 44--'. a -' r. a -


S*1 fl -'I


1.e.,


,,,: c, klh


.~CIL: ~Y


C-%A


nk a ~h,


AC











various


cores


that


include


the


coupling


coefficients


core-to-core


delay


times.


Choice


of Weiahtina


Functions


Three


procedures,


namely


, straight


averaging


the


neutron


flux


over


individual


cores


or reactor


regions,


importance


weighting,


and


weighting


an average


fission


density


can


be distinguished.


In the

formulation


straight


averaging


of space-time


reactor


technique,


kinetics


as

by


the

. Kaplan


based


on a one-group


diffusion


theory


approximation,


the


averaging


done


over


the


core


volume


fd3r


*(r,t)


(2-3)


where,


9(r,t)


the


space-time


dependent


flux,


and


the


core


volume


used


as a basis


the


averaging


process


leading


the


nodal


equations.


The


transport


theory


development


of Bellani-Morante


(10)


introduces


a space-angle


averaged


density,


as a basic


variable


Mj(t)


fd3r


fd2n


F(x,n,t)


(2-4)


where


F(x,n,t)


the


neutron


angular


flux.


4j(t)











choice


of importance


function


depends


on the


nature


coupled


core


array


i.e.,


whether


the


cores


are


loosely


tightly


coupled,


whether


or not


a buffer


zone


exists


between


the


regions,


and


whether


neutron


exchange


primarily


unidirectional


or multi-directional.


Another


criterion


the


selection


importance


function


whether


the


function


describing


the


importance


distribution


should


valid


over


entire


reactor


or the


importance


function


calculated


separately


each


core.


the


reactor


system


can


be analyzed


a single


importance


one-dimensional


function


geometry


entire


the


adoption


reactor


most


suitable


isolated


(as,


the


other


example


hand,


, the


the


KIWI


cores


cores)


are


nearly


weighting


the


neutron


distribution


with


an importance


function


associated


with


each


core


is much


more


reasonable


,17).


terms


the


degree


coupling


among


cores,


a single


importance


function


the


entire


reactor


suitable


a tightly


coupled


system,


whereas


a loosely


coupled


system


the


use


individual


core


weighting


functions


is required.


comparison


Cockrell


and


Perez


and


Hansen


(11)


approaches


illustrates


the


above


criteria


regarding


the


selection

treatment


the

the


importance

University


function.


Florida


Cockrell

Training


his


Reactor











the


entire


reactor.


On the


other


hand


, Hansen


s treatment


(11)


the


loosely


coupled,


clustered


KIWI


cores


employed


an importance


function


associated


with


each


the


individual


cores.


Although


each


case


must


be considered


separately,


these


examples


from


two


extreme


cases


bear


out


the


general


guidelines


suggested


above.


Avery


his


formulation


of coupled


core


kinetics


introduced


function

In the a


fission


the


application


source


determination


Avery


density


the


s method


as the

integr


Kom


weighting

al parameters.

lato, a time


dependent


ssion


source


density


was


used


as the


weighting


function.


Gyftopaulos


(18)


and


Gage


et al.


(13)


their


formulation


of coupled


core


kinetics


employed,


addition


to the


fundamental


mode


adjoint


function,


the


fission


source


density


fission


Sf(r,E,t)


density


as the


weighting


as a weighting


function.


function


means


The


that


use


the


dependent


variables


reflect


the


exact


growth


and


decay


tendencies


the


average


power


individual


cores.


Because


this


approach


exact


treatment


of neutron


density


within


the


boundaries


of a particular


core,


the


only


assumption


required


are


therefore


the


source


term


that


provides


the


coupling


effects


from


the


other


cores


the


array.











based


on the


assumption


that


only


spatial


effects


were


significant


the


coupling


between


spatial


regions


or nodes


(cores)


the


reactor


system


considered.


the


case


KIWI


cores


with


a leakage


spectrum


characterized


intermediate


energy


range


or the


Argonaut


reactor


with


largely


kinetics

parameter


thermal


equations


are


neutron


are


properly


interaction,


considered

weighted.


energy


sufficient


cases


independent


provided


such


large


coupled


core


reactors


a better


accounting


the


spectral


effects


of leakage


neutrons


would


be desirable,


especially,


the


study


of short


time


transients


because


the


spectrum


any


the


cores


composed


of two


parts;


one,

which


a distribution


character


stic


fast,


the


intermediate,


material


and


thermal


composition


the


core,


and


two,


a distribution


characteristic


the


coupling


or leakage


neutrons


from


the


other


cores.


the


dynamics


a single


core,


a common


time


dependency


neutron


energy


groups


a good


approximation


since


the


neutron


spectrum


adjusts


any


nuclear,


mechanical,


thermal


changes


the


core


within


a few


neutron


generations


after


the


changes


are


introduced.


But


coupled


cores,


assumption


a common


time


dependency


neutron


groups


a bad


approach


because


of the


spectrum











contribution


the


spectrum


due


to leakage


neutrons


may


require


a longer


period


to adjust.


An example


coupled


thermal


cores


where


the


prompt


neutron


lifetime


may


be of


the


order


of 10


second.


For


a short


transient


which


appreciable


changes


occur


the


range


milliseconds,


assumption


a common


time


dependency


neutron


groups


would


ignore


the


important


spectral


coupling


effects.


Cockrell


Perez


(12),


their


formulation


which


they


applied


on the


Argonaut


type


UFTR,


accounted


spectral


effects


the


leakage


neutrons


explicitly.


In it


the


which


angular


flux,


originated


k^(r,4n, E)
in region \


refers


the


energy


neutron


flux


group


Multiplication


the


transport


equation


the


adjoint


angular


flux


I (r,n,E)


integration


over


j-th


spatial


region


and


over


the


a-th


energy


group


leads


an ordinary


differential


Njk
aB


equation


d2n


in terms


the


d3r


variable


k1 (r,f,IE, t).


2-5)


n E


The


quantity


Njk(t)


represents


the


neutron


number


density


the


j-th


spatial


region


and


a-th


energy


group


due


neutrons


born


k-th


spatial


region


and


fi-th


energy


-(E)


*(r,n,E)v











example


an m- group


n- region


problem,


this


formulation


requires


(mn) 2


basic


equations,


while


the


other


approaches


require


a much


smaller


number


equations.


This


formulation


may


be classified


as overly


descriptive


system;


i.e.


provides


more


equations


than


are


required


describe


the


neutronic


behavior


a reasonable


degree


accuracy.


Treatment


Couolinq


Effects


The


reactivity


contribution


due


to neutronic


coupling


effects


among


the


cores


introduced


a manner


consistent


with


the


treatment


of the


time


dependent


variables


selection


phase


space


regions,and


the


selection


weighting


functions.


A particular


technique


treating


the


coupling


effect


may


more


compatible


with


the


one


formulation


than


with


another.


Couolina


Coefficients


Methods


introducing


coupling


effects


can


classified


as falling


under


either


the


"reactivity"


approach


or the


"effective


source"


technique.


For


comparison


various


treatments


of coupling


effects


that


are


examined


this


chapter,


we can


classify


the


methods


of Cockrell


and


Perez


(12),


Hansen


(11),


Kaplan


1. (9),


Kohler


and


Plaza











Hansen


(21),


Gage


(13),


and


Bellani-Morante


(10)


the


"effective


source"


technique.


In the


generalized


reactivity


reactivity


approach,


expression


assumed


a given


that


core


the


contains


implicitly


the


contribution


due


coupling


from


the


other


cores.


In this


approach,


account


the


coupling


interaction


with


the


transport


theory


formalism


and


importance


weighting,


the


leakage


term


VI(r


fn,E,t)


transformed


surface


into


coupling


a surface


accounts


integral.


the


This


neutronic


concept


interaction


between


the


cores.


The


derivation


the


coupled


core


kinetics


equations


analogous


to that


of the


generalized


point


reactor


kinetics


equations


as developed


Henry


(22)


or Bell


and


Glasstone


(23);


the


adjoint


flux


an arbitrary


stationary


state


the


unperturbed


interacting


reactor


used


as a


weighting


function.


Cockrell


s coupled


core


kinetics


formulation


introduces


both


spatial


and


spectral


coupling


and


arrives


at generalized


coupled


point


reactor


kineti


equations


region


for


and


energy (t)


energy c


which


Iroup


the


tim


number


e t tha


of neutrons


t originated


region


and


energy


group


and


Cik(t)


the


precursor


a aI











both


the


spatial


and


spectral


effects.


Cockrell'


(12)


formulation


assumes


that


cores


have


common


interfaces


with


one


another


and


that


the


coupling


neutrons


travel


only


through


interfaces.


Cockrell


assumptions


are


too


restrictive,


and


exclude


many


realistic


physical


situations


which


coupling


between


isolated


regions


which


not


share


common


interfaces


important.


A second di

calculation


fficulty


the


with

source


Cockrell'

integral


s approach


the


that


the


formulation


but


the


simplest


geometrical


configurations


may


impractical.


dependent


Third,


coefficient


the


only


partial


under


lifetime


certain


O *


a time


circumstances.


Finally


from


a practical


standpoint,


the


large


number


equations


significant


introduced


Cockrell


disadvantage;


s formulation


a m-group,


n-region


presents


problem


this


formulation,


as already


indicated,


requires


(mn) 2


basic


equations,


while


other


approaches


require


a much


smaller


number


equations.


Plaza


and


Kohler


(19),


their


coupled


core


formulation


consider


a group


of reactors


or cores


interacting


with


each


other


through


a non-multiplying


coupling


medium.


The


point


reactor


kinetics


equations











contribution


using


the


surface


coupling


technique


(19).


Hansen

importance


(11)


weighting


ne-group

consists


transport


theory


of dividing


the


approach


total


using


flux


(Fs)


the


core


into


two


parts:


which


refers


neutrons


time


born


t and


the


which


core


refers


and


to all


remaining


other


there


neutrons


until


the


core


identified


with


neutron


leakage


into


core


Consequently


he developed


his


theory


starting


from


two


transport


equations


and


derived


two


sets


kinetics


equations


and


N' .


Coupled


core


formulations


Baldwin


(20),


Seale


and


Hansen


(21),


and


Gage


et al.


(12)


include


the


contribution


from


coupling


effects


as an effective


source;


the


fact


that


the


reactivity


coupling


requirement


recognized


but


not


each


core


explicitly


reduced


taken


the


into


account.


The


(tacit)


assumption


inherent


these


derivations


that


calculating


the


reactivity


associated


with


the


core,


the


contributions


due


"effective


sources

In the


on the


Baldwin


surface


(20)


that


formulation


core

for


must


the


be set


two


zero.


core


configuration


the


Argonaut


reactor,


which


two


slightly


subcritical


slabs


about


two


feet


apart


are


immersed


large


graphite


reflector,


the


coupling


effect


introduced











which


accounts


the


interaction


between


the


two


cores.


In this


expression


the


additional


source


component,


' is


the


delay


time


due


the


exchange


neutrons


between


two


cores


and


characteristic


the


graphite


reflector,


represents


the


neutron


flux


core


k and


the


interaction


probability


between


the


two


cores.


The


interaction


component


the


source


core


e12(t-r)


and


that


core


i.e.,


the


interaction


component


proportional


the


average


flux


the


other


core.


Seale


and


Hansen


(21)


uses


Baldwin


(20)


model,


generalizing


to M


cores


but


retaining


a source


component


the


above


form.


Gage


et al.


(13),


their


coupled


core


analyst


introduced


a source


stribution


function


Sj (r,E,t)


that


composed


a series


of terms


representing


the


power


coupling


with


the


other


cores


the


reactor.


Since


there


a finite


time


lapse


from


the


birth


of a neutron


one


core


capture


another


core


causing


fission,


coupling


power


growth


assigned


a retarded


argument


the


form


Sj (r,E,t)


= E ek(r,E,t)
k=lk


(2-6)


aI nr,1n 1 4 4. 1 Ja


*


6201(t-T),


*pj(r,E,t)Nk(t-Tjk)


r ir F +>


r.rk hra


C~~C


~L Al i C ZIIArr












energy


the


core.


The


term


,E,t)


represents


the


new


shape


function


that


accounts


the


power


distribution


arising


from


the


effective


surface


source


of coupling


neutrons.


In introducing


such


effective


source


term,


the


contributions


the


reactivity


expression


each


core


due


to the


coupling


neutrons


from


other


cores


must


be set


zero.


Avery


formulation


of coupled


cores


treats


the


system


terms


integral


parameters


which


explicitly


characterize


the


individual


reactors


and


the


coupling


between


them.


The


total


fission


source


in the


core


set


equal


to the


fission


source


contribution


arising


that


core


and


the


other


(M-l)


cores


(2-7)


The


coupling


coefficient


defined


Sjk


(2-8)


kjk


where


kjk


the


expectation


value


or probability


that


fission


neutron


core


(reactor)


k gives


rise


to a next


generation


of fi


ssion


neutrons


The


equations


describing


the


kinetic


behavior


the


n Al' Wi A mw a M 7W


= 2


nrr


Ejk(r,E,t)pj(r


^ i ^rAn


~rrrrCArn











dCki


and


(2-10


AiCki.


In Equations


(2-9)


and


(2-10)


and


are


the


effective


delayed


neutron


fraction


and


decay


constant,


respectively,


the


group


delayed


neutron


precursors,


the


total


effective


delayed


neutron


fraction,


Cki


is a properly


weighted


measure


delayed


neutron


precursors


the


group


core


ND is


number


of delayed


neutron


groups,


M is


the


number


cores


the


reactor,


kjk


the


expectation


value


or probability


that


a fi


ssion


neutron


core


k gives


rise


to a next


generation


fission


neutrons


core


and


the


prompt


neutron


lifetime


the


process.


The


term


a measure


kjk


the


cross


coupling


from


core


k to


core


The


Equations


(2-9)


and


(2-10)


can


be rewritten


terms


the


time


rate


change


of Njk(=ljkSjk)


neutrons


equal


the


difference


between


the


production


(including


delayed


as well


as prompt


neutrons)


and


the


loss


these


neutrons


(1- )


M Nkm

m=l 1km


(2-11)


Nj
1jk

ljk


+ E
i=1


dCki


and


=-i


m=l


Ckm.


(2-12)


ikm


dN
jk
dt


= i


j+k


= kjk


Nkm











produce


the


next


generation


of neutrons


The


term


dNjk/dt


represents


the


difference


in production


and


loss


rate


Njk


type


neutrons


The


first


term


on the


right


hand


side


Equation


(2-11) ,


the


kjk(l-3) Sk,


production


rate


the


Njk


type


of neutrons


prompt


emission.


The


term


DiSk


represents


the


precursor


type,


so that


this


term


represents


Cki.


The


the


term


production


XiCki


rate


represents


term


the


Equation


loss


(2-12)


rate


The


total


number


delayed


source


neutrons


in core


k is


Cki
the


so that


the


production


neutron


last


rate


emission.


term


of Njk
Finally


equation


type


Njk/ljk


(2-11)


of neutrons


represents


kjk


SXiCki


delayed


the


loss


rate


type


neutrons


equation


(2-11).


The


power


level


each


core


the


reactor


system


proportional


to the


sum


the


partial


fission


neutron


sources:


a S
k=l


Sjk'


(2-13)


a S Nijk
k=l


In applying


this


theory


the


coupled


fast-thermal


breeder


rector


experiments,


a spectral


definition


the


interacting


thermal


and


fast


cores


terms


thermal


and


fast


sources


was


chosen.


The


definitions


these


sources


are


given


Nj k











fma 9()
system (rv)
system vc


d3r


(2-15)


where


the


thermal


cutoff


speed


and


Vmax


the


speed


at the


maximum


neutron


energy


considered.


Since


the


dependent


variables


Avery's


kinetics


equations


are


partial


neutron


densities


(Njk) ,


this


formulation


requires


equations


where


M i


the


number


cores

Delay


the


Time


system.


Effects


In Avery's


formulation,


the


delay


times


associated


with


the


transfer


neutrons


between


cores


are


not


introduced


explicitly.


When


considering


pulsed


experiments


with


coupled


cores


or other


aspects


the


behavior


of loosely


coupled


systems,


desirable,


sometimes


necessary


introduce


the


delay


times


explicitly


into


the


coupled


core


kinetics


formalism.


One


possible


representation


the


delay


times


can


the


probability


jk(t-t


that


neutron


born


the


a th


energy


group


in the


k th


core


time


' enters


the


core


within


dt at a time


This


probability


should


be chosen


so as to represent


the


physical


and


nuclear


characteristics


the


system


being


considered.


In Baldwin


s formulation


(20)


two


loosely


coupled


core


loadings


the


Argonaut


type,


the


interaction


term











the


interaction


term


the


delay


time


the


neutron


transfer


process


between


the


cores.


Chezem


model


and


to analyze


Helmick


the


(24)


developed


Alamos


Coupled


a general


core


theoretical


experiment


which


two


cores


were


separated


beryllium


reflector.


pulse


of neutrons


was


introduced


into


the


core


time


and


the


neutron


response


each


the


core


was


obtained


from neutron

The general


detectors


model


located


developed


the


Chezem


center


of each


Helmick


core.


consists


a single


neutron


balance


equation


terms


the


neutron


population


Ni(t)


the


i th


core


of a loosely


coupled


array


cores


given


dt
dt


P(T)dT


+ A6il(t)


ai4JN0(t-T)
0


(2-16)


where

when


isolated


the excess

from the


prompt


reactivity


environment


and


the

the


core


neutron


lifetime.


The


source


term


A6il(t)


represents


a delta


pulse


of neutrons


core


1 and


time


t=0.


The


term


aijNj (t-T)P(T)dT


source


contribution


core


from


fissions

coupling


core


coefficient


times


t-T


core


from


Quantity


fission


neutrons


the

the


Ni(t)+











where


an "effective


interaction


time"


neutrons


entering


core


from


core


This


expression


P(t)


will


lead


to a simple


form


transfer


function


and


consequently


a simple


analytical


form


the


time


dependent


flux.


The


interaction


time


which


may


also


called


delay


time


does


really


provide


the


function


P(T)


represent


the


minimum


time


delay


corresponding


the


minimum


neutron


transit


time.


For


example


, in


the


case


an internally


pulsed


"driver"


core


coupled


to a "passive"


core,

time


the


response


the


external


the

pulse


passive


core


no matter


would


how


close


start


or far


the

apart


the


two


cores


are.


This


simple


model


would


lead


a simple


exponential


behavior


the


driver


core


while


there


are


indications


that


a quasi-oscillatory


response


possible


and


may


have


occurred


the


KIWI-Transient


Nuclear


Test


(KIWI-TNT)


experiments


(25,26).


For


these


reasons


desirable


introduce


slightly


more


complicated


representation


P(T)


that


will


include


a finite


delay


time.


Hansen


(11)


his


neutronics


studies


a cluster


of reactors,


gives


the


source


term


the


i th


core


due


interaction


the


other


cores


= ij


N.
(- -


(t-T)


(2-18)


Pij (r)


C,










effectiveness


fission


neutron


that


moves


from


core


core


i in


inducing


fission


and


eventual


neutron


population


and


Pij(r)


the


time


distribution


function


this


transfer.


the


cluster


array


of coupled


cores,


Hansen


(11)


assumed


that


= 6 (t-Tij)


(2-19)


where


the


effective


mean


drift


time


of neutrons


between


that


the


the


sum


boundaries


first


three


moment


mean


cores


the


times;


and


distribution


the


time


to be noted


function


fission


Pij () i

neutron


to leak


out


core


from


birth,


the


drift


time


from


the


boundary


of j


the


boundary


, and


the


time


from


entrance


absorption


In Avery


and


Komato


formulations


of coupled


core


through


neutronics


a partial


core-to-core


lifetime,


ljk(t)


delay

, the


time

mean


introduced


lifetime


neutrons


the


process


(j,k) ,


.e.,


of neutrons


born


core


k which


diffuse


the


system


ultimately


cause


fission


core


the


formulation


Gage


et al.


(13)


nonlinear


stability


of a coupled


core


reactor


system,


the


coupling


term


that


accounts


the


contribution


from


remaining


cores


to a specific


core


introduced


with


retarded


time


Pij (v)











diffuse


from


core


k to


core


through


any


intervening


medium


which


separates


the


cores.


Comments


Salient


features


the


major


formulations


coupled


reactor


kinetics


are


presented


this


chapter.


The


various


major


formulations


core-to-core


differ


coupling


mainly


effects


that


their

include


treatment


the


the


coupling


coefficients

approaches h


and


delay


handling


the


times.

couplin


Although

ca effects


the

can


various


broadly


classified


into


"effective


source"


and


"reactivity"


techniques,


the


various


approaches


differ


their


actual


implementation


during


the


derivation


the


coupled


core


kinetics

these ap


equations.


preaches,


Based


three


on the


distinct


actual im

categories


plementation


can


identified


the


introduction


the


coupling


effects


the


coupled


core


kinetics


equations:


introduction


these


additional


sources


into


the


basic


neutron


balance


equation


(14,15,20);


(ii)


introduction


coupling


sources


during


the


derivation


the


coupled


point


reactor


kinetics


equations


from


neutron


balance


equations


replace


the


contribution


from


the


surface


integral


(11,19,24);


and


(iii)


the


partitioning


the


neutron


flux


and


hence


the


neutron


source


into


partial


sources


that


contain


the


contribution











The


method


introduction


time


delay


also


differs


among


the


various


formulations.


Delay


time


can


introduced


as a probability


distribution


function


time


finite


time


a retarded


time


argument


the


power


levels


the


remaining


cores.


The


delay


times


can


neglected


a situation


where


the


delay


times


are


very


short


compared


to the


prompt


neutron


generation


time.


The


expected


influence


delay


times


coupled


core


kinetics


a decrease


the


systems


stability


(13).


However,


the


maximum


delay


time,


'max


so short


that


N(t)-N(t-rmax)


<< N(t)


or N(t)N (t- rmax)


times


of practical


interest,


then


the


delay


times


can


be expected


to have


very


little


or no practical


effect


on the


time


behavior


the


system.


Most


the


coupled


core


reactor


kinetics


formulations


that


have


been


previewed


this


chapter


are


applied


solid


systems


one-dimensional


plane


geometry


with


very


restrictive


conditions.


the


system


has


a complex


geometry,


the


case


the


bimodal


gaseous


core


reactor


system,


the


calculation


of precise


coupling


coefficients


delay


times


becomes


complicated


and


hence


should


be obtained


from


Monte


Carlo


methods.















CHAPTER


PREVIOUS


REACTOR


STUDIES


AT THE


ON PULSED
UNIVERSITY


GASEOUS


CORE


OF FLORIDA


Introduction


Pulsed


cyclic


gaseous


core


reactors


have


undergone


extensive


theoretical


and


experimental


investigation


the


University


Florida


One


the


earliest


conc


epts


that


was


proposed


Kylstra


et al


consists


a pulsed


plasma


core


reactor


enclosed


a cylinder


and


a piston


and


analogous


to a gasoline


internal


combustion


engine


(Figure


3-1)


The


fuel-working


fluid


a UF6


mixture.


The


reactor


made


critical


during


the


compare


ssion


stroke


and


the


neutron


flux


allowed


to build


to a significant


power


level


The


neutronic


and


energy


calculations


have


shown


that


significant


power


(MWs)


per


cylinder


with


high


efficiencies


(40%


-50%)


can


be achieved


Kylstra


feasibility


studi


on pulsed


gas


core


reactors


have


indicated


such


good


performance


potential


that


Dugan


. (28)


were


encouraged


to undertake


extens


theoretical


and


experimental


neutronics


and


energetic


studies


on a variety


gas


core


reactor


concepts.


These


a~I~I a mi l.. a 4t h1&n Itt, ~e'e


'DUnun\


T.d-li t


nl I*C


4- A


Thtln A A.^


It^^I A 1 k


nrrcrCI hnr


























































Figure


3-1.


Schematic


Representation


Piston


Engine.











piston


assembly


which


operates


on a thermodynamic


cycle


similar


the


internal


combustion


engine


and


the


Pulsed


Gas


Generator


(PGG)


that


employs


a core


of fixed


dimensions.


These


compact


, cyclic,


fission


driven


systems


have


been


found


to have


unique


power


producing


characteristics


are


capable


of generating


Piston


10 to 100


Enaine


MW(e)


Kvlstra


power.


et al.


The


piston


engine


developed


Kylstra


et al.


(27)


consists


of a cylinder


a pi


ston


surrounded


graphite


moderator


with


a nickel


lining


protection


from


the


fuel-


working


fluid,


UF6


During


the


intake


stroke,


the


UF6


fuel-working


fluid


drawn


into


the


cylinder


and


during


exhaust


stroke


the


gas


and


the


fission


products


are


ejected


from


the


cylinder.


The


reactor


made


critical


during


compares


sion


stroke


before


the


piston


reaches


the


top


dead


center


(TDC).


To minimize


the


fission


heat


release


after


piston


has


already


passed


the


power


stroke,


the


reactor


required


to shut


down


rapidly.


The


reflector


surrounding


the


cylinder


piston


graphite.


The


graphite


has


an inner


nickel


lining


because


corrosion


resistant


properties


a UF6


atmosphere.


The


graphite


chosen


as the


reflector


low


neutron


UF6










thermodynamic


properties


the


working


fluid.


External


equipment


to be


used


remove


the


fission


products,


cool


gas,


and


recycle


back


to the


cylinder.


The


reflector


thickness


varied


during


the


operation


of the


engine


to simulate


the


desired


time


sequence


the


subcritical-supercritical-subcritical


state


the


reactor.


the


compression


stroke


starts,


the


reflector


starts


as a thin


reflector,


then


increases


thickness


slowly


until


a step


increase


reflector


removed


going


back


a thin


reflector;


reflector


continues


to decrease


thickness


until


the


piston


bottoms.


The


engine


designed


to operate


at a high


graphite


temperature


approximately


1000


-1200"


so as to minimize


heat


loss


from


core.


The


compression


ratio


the


engine


with


a clearance


volume


of 0.24 m3


The


engine


shaft


speed


rpm


and


the


UF6


100%


enriched.


Neutronic


Analysis


For


steady


state,


two-group,


two


region


diffusion


theory


analysis,


the


following


assumptions


were


made.


No fast


neutron


interaction


the


core.


The


fast


neutron


core


equation


replaced


a boundary


condition


the


fast


neutron


current


into


the


moderator.











iii)


No delayed


neutrons.


assumed


that


the


delayed


neutron


precursors


are


swept


out


the


cylinder


before


they


exert


any


influence.


No time


No angular


dependence.


dependence.


simplicity,


the


cylindrical


geometry


was


replaced


the


analysis


a spherical


geometry


with


the


core


volume


and


reflector


thickness


conserved


times;


but


they


change


time


simulating


the


motion


the


piston


within


the


cylinder.


The


set


of equations


based


on the


above


assumptions


are


solved


average


thermal 1


core


and


moderator


flux


and


the


neutron


multiplication


factor


(keff)


as a function


the


position


the


piston


the


cylinder.


the


multiplication


factor


approaches


and


exceeds


unity,


a one


group,


point


reactor


kinetics


equation


used


to solve


the


time


dependent


thermal


flux


the


core.


This


time


dependent


solution


method


adopted


because


the


slow


variation


system


configuration


(due


to piston


motion)


compared


to the


diffusion


speed


or neutron


cycle


time.


Enerav


Model


The


conservation


energy


equation


a non-flow


system


is used


the


energy


model


the


system.











fission


fragments


negligible


and


that


the


heat


transferred


to the


walls


also


negligible.


The


energy


equation


balances


the


rate


of variation


the


internal


energy


the


working


fluid


(UF6+He)


against


the


rate


fission


heat


release


net


work


done


the


gas


the


piston.


This


equation


solved


the


bulk


temperature


the


core.


Results


Figure


(3-2)


shows


the


multiplication


factor,


keff


a system


with


an infinite


graphite


reflector


as a function


UF6


pressure


partial


pressure.


of greater


than


seen


atmosphere,


that


the


a UF6


core


partial


becomes


black


the


neutrons


that


additional


uranium


ineffective.


Figure


(3-3)


shows


the


relative


thermal


neutron


flux


distribution


a UF6


partial


pressure


0.67


atmosphere


and


a core


volume


of 0.44


These


two


figures


imply


that


low


UF6


loading


required


minimize


the


nonuniform


flux


core


so that


the


fission


heat


deposited


a thin


shell


next


to the


reflector.


The


average


core


thermal


flux


and


neutron


multiplication


through


the


factor


cycle


(keff)
Figure


as a


function


shows


that


of percent


the


time


keff


increases


to a value


greater


than


when


the


thick


reflector
































1.25






1.0






0.75






0.5


0.3 1.0 3.0 10


Pressure


(Gas


Temperature


(atm)

= 400K)


Figure


3-2.


Neutron Multiplication Factor


Vers


Pn Tnf nH-io lrrnh nb i'


us UF6 Par
of I or. rr


'tial


DyoC cnrro




























1.5
u
0
N
E
U




rl.0

tol

E
0)
C


20 40 60 80


Radius,


Figure


Relative
Reflector


Thermal


Flux


the UF6


in Core
Piston


and Graphite
Engine






























T 1
J' 1015
Cu




N-
I
rz 110

0
X:

4C
'a


4I
00

C


0.9 5
0
*)
-4

r'-
a
HI
0.8 S



0.7 c



0.6 Z



0.5



0.4


< 5
0


25 50 75


Percent

Initial
Initial


U-235


Travel


Through


Pressure


Mass


Temperature
= 2.15 ka


Cycle


= 400K


Figure


3-4.


Average


Core


Thermal


Neutron


Flux











before


the


thick


reflector


removed


slightly


subcritical,


causes


majority


of heat


to be added


during


the


last


i.e.,


the


compression


stroke.


While


this


long


cycle


time


permits


easier


control


the


reactor,


a little

at this


larger

point;


uranium


this


loading


results


yields


the


a keff


majority


greater


the


than


heat


being


added


a 15-25


ms period


as the


piston


passing


top


dead


center


(TDC).


Figure


(3-5)


shows


efficiency,


power,


and


pressure


versus


U235


loading


an engine


with


the


thick


reflector


imposed


at the


cycle


position


and


removed


the


position


(40%


cycle


available


buildup


neutron


flux)


A maximum


keff


of 1


to 1


reached


these


as the


TDC


systems


with


the


approached.


keff
This


dropping


behavior


0.99


keff


from


1.01


greatly


increases


the


control


safety


since


the


time


cons


tant


larger


at higher


pressure


Increasing


U235


loading


leads


to larger


keff


but


also


reduces


the


helium


the


mixture,


the


same


initial


pressure.


Thus,


efficiency


and


the


power


curves


are


concave


downward


reflect


the


lower


average


specific


heat


and


hence


the


poorer


thermodynamic


properties


the


gas


mixture


as more


UF6


added


the


expense


of helium.



















2.10


2.20


2.25


2.30


2.35


-r4
e-

>4
l0
a)

0
00

&4
"o
n


2.15


2.30


1
101
2.35


U-235 Mass, KG











researchers


at the


University


of Florida


to embark


on more


refined


and


comprehensive


studies


on gaseous


core


reactor


systems


with


and


without


movable


pistons.


Pulsed


Nuclear


Piston


and


Pulsed


Gas


Generator


Systems


Two


basic


alternative


pulsed


gaseous


core


reactor


(PGCR)


systems


that


have


undergone


extensive


theoretical


and


experimental


investigations


since


Kylstra


s work


are


the


Pulsed


Nuclear


Piston


(PNP)


and


Pulsed


Gas


Generator


(PGG)


systems

concept


(29).

except


These

that


two

the


reactors


PNP


are


consists


based


on a similar


of a pulsed


gaseous


core


reactor


enclosed


a moderating


reflector


and


a piston


assembly


while


the


employs


a core


fixed


dimensions.


These


studies


have


establi


shed


a basic


scientific


understanding of the

conceptual operation


physical

of pulsed


phenomena

gaseous


assoc

core s


iated with

systems and


the

have


resulted


favorable


response


from


the


scientific


and


engineering


Results


community


detailed


on their


theoretical


use


and


space


experimental


applications.


neutronic


analyses


descriptions


processes


used


the


selection


of optimum


configuration


the


PNP


and


can


be found


28-31).


This


section


presents


a brief


description


the


4-ileonic a


ranri-rtT"


nnncnrm't -


PNP


and


Pa.a


and


ynr 1 c~rl


-t.th


IrnrLS











Pulsed


Nuclear


Piston


System


The


pulsed


Nuclear


Piston


(PNP)


system


consists


pulsed


gaseous


core


reactor


enclosed


a moderating-


reflector


cylinder


a piston


assembly


(Figure


3-6)


which


operates


on a thermodynamic


cycle


similar


to the


internal


combustion


engine.


The


primary


working


fluid


a mixture


uranium


hexafluoride


(UF6)


and


helium;


highly


enriched


UF6


the


reactor


fuel


and


helium


gas


enhances


the


thermodynamic,


transport,


and


heat


transfer


characteristics


of the


working


fluid


mixture.


The


energy


released


the


fissioning


gas


can


extracted


both


as mechanical


power


and


as heat


from


the


circulating


gas


which


recycles


back


the


core


the


pressure


differential


established


the


power


chamber.


Mechanical


power


can


be derived


from


the


engine


means


a conventional


crankshaft


operating


at low


speeds.


addition


the


mechanical


power,


a significant


amount


energy


from


the


hot


gas


can


removed


an external


heat


removal

exhaust


loop.


gas


can


The


high


be cooled


temperature


(1000


an UF6/He


o-1300


-to-He


heat


UF6/He


exchanger


and


the


heated


helium


gas


can


be used


a suitable


power


conversion


cycle


drive


a turbine.


Summary


PNP


Results









































BeO
Moderating-
Reflector
Region

DO
20"
Moderating-
Reflector
Region


He/UF6























BeO Piston











research


at the


University


Florida.


Two-stroke


engine


analysis


considered


only


the


compression


and


power


strokes


while


the


four-stroke


engine


analysis


explicitly


examined


the


intake


and


exhaust


phases


operation


addition


the


compression


and


power


strokes.


While


the


earlier


studies neglected

subsequent studies


delayed

include


neutron

d first,


and photoneutro

the influence


effects,


of delayed


neutrons


and


then,


the


influence


both


the


delayed


and


photoneutrons


on the


PGCR


s performance.


Graphite


was


found


to be unacceptable


as the


moderating


reflector


since


the


dimen


sions


the


system


with


graphite


as a moderator


were


found


to be


too


large


to be practical.


was


determined


that


BeO)


and


D20


were


the


most


desirable


moderating-reflector


materials


from


the


standpoint


of neutron

neutron li


economy


fetimes


and

for


size

the b


the


eryllium


system.


The


moderated


prompt


system


were


found


be half


as great


as for


the


D20


reflected


system.


Although


D20


has


a significantly


lower


thermal


absorption


cross


section


than


beryllium


, the


Be reflected


system


has


smaller


critical


mass.


This


due


to the


large


(n,2n)


production


that


occurs


in beryllium.


It has


been


estimated


that


the


(n,2n)


production


leads


an effective


increase


the


average


number


neutrons


per


fission


(30).











found


to be


large


negative


.5x10


Ak/k


per


whereas


the


moderator


temperature


coefficient


of reactivity


Be reflected


systems


was


found


to be small


and


positive


1x10


Ak/k


per


Although


this


made


or BeO


undesirable


as a moderator


material


was


suggested


that


composite,


with


inner


or BeO


region


surrounded


would


ideal


moderator


arrangement


with


an overall


moderator


temperature


coefficient


of reactivity


large


negative


-2x10


Ak/k


per


Moreover


with


the


(n,2n)


production


beryllium,


this


configuration


will


lead


reduced


critical


mass.


Fuel


temperature


coefficients


of reactivity


100%


enriched


UF6


were


found


to be small


and


positive


5x10


Ak/k


per


the


fuel


enrichment


was


reduced


the


fuel


temperature


coefficient


reactivity


was


found


to decrease


and


at 80%


enrichment


the


coefficient


was


found


be small


and


negative


-1.2x10


Ak/k


per


The


effects


delayed


neutrons


photoneutrons


the


system


behavior


were


examined


different


combinations


of procedures.


One


procedure


involved


increasing


the


fuel


loading


to compensate


the


absence


of delayed


neutrons


and/or p

neutrons


hotoneutrons.


n.. *U


nhotoneutrons


example,


were


when


nealecte


both

. the


delayed

maximum


U










were


present)


to 1.091


order


to maintain


the


engine


performance


level.


When


photoneutrons


were


neglected,


order


to maintain


the


system


s performance


level,


the


maximum


neutron


multiplication


factor


had


to be


increased


1.061.


When


the


delayed


neutrons


and


photoneutrons


were


neglected


the


analysis,


the


power


output


dropped


relative


to the


case


where


only


photoneutrons


were


ignored.


Thus,


the


delayed


neutrons


and


photoneutrons


both


exert


noticeable


influence


on system


behavior


with


the


delayed


neutron

The


influence

pulsed


being


gaseous


the

core


stronger

system


of the


two.


capable


of rapid


startups


the


presence


delayed


neutrons


The


startup


procedures


used


the


referenced


work


(30)


include


variation


circulation


inlet


time


gas


Changes


pressure


the


and


loop


the


loop


circulation


time


change


the


fraction


of delayed


neutrons


which


undergo


decay


while


the


external


circulation


loop.


Those


precursors


which


do not


decay


the


loop


, but


decay


upon


reentering


the


core


after


pas


sing


through


the


loop,


provide


an extra


neutron


source


and


hence


can


be used


to adjust


the


power


level.


Figure


gives


the


delayed


neutron


and


photoneutron


precursor


densities


as a function


time


during


the


PNP






























1011




21010


0 100 200 300 400 500 600 700 800


Time


(sec)


Figure


3-7.


Delayed Neutron and


Photoneutron


Precursor


Concentration


Buildup


During


Startup from


- r I


-- I I -


--


I


































1200




900




600


0 100 200


Time


(sec)


Figure


3-8.


Peak Gas


Temperature


and Mechanical


Power


Versus


Time


During


Startup


from Shutdown


Pulsed Neutron


Piston


Engine.































2000


1750




1500


W1250
(0
(U
'CU
0


1.8 'O


1.6 u


1.4


300 400 500 600 700 800


Time


sec


Figure


. Peak


Temperature


and Mechanical


Power


(continued)


Versus


Time


During
Engine.


Startup


from Shutdown











with


a loop


circulation


time


11.3


seconds


was


found


to be


S13 minutes.


Dugan


included


(30)


both


the


a simple


PNP


point


PGG


reactor


neutron


kinetics


kinetics


model


analysis


as well


as a more


sophisticated


adiabatic


kinetics


model.


Changes


during


the


engine


cycle,


the


flux


shape,


and


hence


the


neutron


lifetime


and


source


weighting


functions


were


thoroughly


examined


and


found


to be significant.


The


point


reactor


kinetics


solution


procedure


was


found


inadequate


handle


the


significant


variations


these


parameters.


correct


the


inadequacy


a simple


point


reactor


kinetics


model,


a more


complex


adiabatic


model


was


implemented.


This


method


involves


an iterative


procedure


between


the


point


reactor


kinetics


program


and


other


independent


static


neutronic


analysis


codes


to obtain


the


integral


parameters


various


selected


points


throughout


the


cycle


of operation


the


system.


Figures


3-10,


which


give


the


average


core


thermal


neutron


flux


as a


function


of cycle


time


from


the


adiabatic


model


and


the


point


reactor


kinetics


model,


respectively,


illustrate


the


inadequacy


the


point


reactor


kinetics


model


the


neutronic


analysis.


Results


from


the


point


reactor


kinetics


model


were


























1015



1014


1013


1012
10



1011


1010


0.00 0


.15 0.45 0.60 0.75 0.90 1.05


1.20


Cycle Time


(sec)


Figure


3-9.


Average Thermal
Cycle Time for
Kinetics Model


Neutron Flux
the PNP Engin
Ti Used.


in the Core


e When


Versus


the Adiabatic



























1015


2 o14
be
Cu
C:
0
O
4 13
110
0)
z




0

La 1
010
U
0f

4

adO10
O
u 1


0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05


1.20


Cycle


Time


(Sec)


Figure


3-10.


Average Thermal


Neutron


Flux


the Core


Versus


cycle Time


Point


Kinetics Model


P Engine
used with


when


.557


msec.











adiabatic


analysis


were


used.


However


was


found


to be


possible


to obtain


an adjusted


"equivalent"


prompt


neutron


lifetime


to be used


the


point


reactor


kinetics


calculations.


The


results


from


such


a calculation


were


found


to be


very


close


to those


obtained


from


the


more


complex


adiabatic


calculations.


The


variation


the


average


core


thermal


neutron


flux


with


cycle


time


obtained


from


a point


reactor


kinetics


calculation


with


"equivalent"


prompt


neutron


lifetime


of 1.


milliseconds


(ms)


shown


Figure


3-11.


These


results


are


much


closer


the


results


from


the


adiabatic


kinetics


model


(Figure


3-9)


than


those


obtained


from


the


point


reactor


kinetics


model


with


a prompt


neutron


lifetime


2.557


msec


(Figure


3-10)


which


the


cycle-averaged


prompt


neutron


lifetime


value


obtained


from


the


adiabatic


analy


sis.


Results


from


these


three


different


models


a specific


configuration


are


given


Table


3-1.


Generator


Pulsed


Gas


Core


Nuclear


System


The


pulsed


gas


generator


(PGG)


gas


core


nuclear


concept


similar


to the


pulsed


nuclear


piston


concept


except


that


the


gas


generator


employs


a core


of fixed


dimensions


(Figure


3-12).


The


gas


generator


system


simpler


operation


design


than


the


nuclear


piston


concept


and


offers


a more
































x



O
sr 11


~10D
10
(U
Z
rid



10

o 1$0
1
Eio12
0)
XC
ES-
o(


C-)
0lO'


to0
0


0.00


0.15


0.30


0.45 0.60 0.75 0.90


1.05


1.20


Cycle


Time


(Sec)


Figure


3-11.


Average


Cycle


Thermal


Neutron


Time


Kinetics


Model


Is Used


Flux
Engine
with


Core


When the


=1.718


Versus


Point


msec.



















rfl
CM w


-r r-
*r 0
C o
C
k


S00

r0
Z a

s E

0
'C
*c
90


()
0
H
rl
r6 k
O
,C
0
U
*rlW0
c e
. %


*


U
O
M C

co 0-
0
ri
u-QI


X
0)
#I
a
e


w














































Inner
Moderating-
Reflector
Region


Outer


Moderating-


Reflector
Region


Rotating
Absorber


He/UF
-~6
Fuel
Gas


























Core


Region

Rotating
Absorber


I











pulsing


to supercriticality


achieved


means


of a pile


oscillator


with


a rotating


absorber


the


reflector


surrounding


the


core.


The


system


has


the


advantage


mechanical


simplicity


over


the


PNP


concept


while


maintaining


the


advantages


of a pulsed


system


mentioned


elsewhere


this


dissertation.


These


include


the


capacity


attaining


high


peak


gas


temperature


2000


and


high


neutron


fluxes


neutrons/cm2


sec


while


maintaining


relatively


average


gas


temperatures


(800-900


degree


Tables


and


compare


typical


operating


conditions


the


PGG


and


PNP


systems,


respectively,


and


reveal


that


gas


generator


capable


operating


at lower


enrichments


than


the


nuclear


piston


system.


Figures


3-13


3-14


which


illustrate


the


variation


core


average


thermal


neutron


flux


with


cycle


time


the


PGG


and


PNP


systems,


respectively,


reveal


that


the


neutron


flux


gas


generator


peaks


late


cycle


relative


the


piston


driven


system.


The


system


can


provide


average


exhaust


gas


temperatures


that


are


closer


the


peak


gas


temperature


than


can


the


PNP


system.


UF6


dissociation


considerations


limit


the


peak


gas


temperature


to around


2000


then


the


piston


driven


system


where


gas


temperature


peaks


earlier


cycle


can


supply


helium


gas


turbines


temperatures


of 1015











Table


3-2.


Typical Equilibrium Cycle
Generator System Operating


Pulsed Gas
Conditions.


Core
Intak
Intak
Heate
Syste


Vol
e L
:e
d G
Tm C


Loop Cir
Average
Core Hei
Core Rad
Uranium
Helium M
Helium M
Uranium
Maximum
Maximum
Maximum
Flux (n/
Average
Average
Cycl
Cycle Av
Cycle Av
Flux (
Fission


ume (
mine G
Line
as Av
ycle
culat
Promp
ght (
ius (
Mass


as
;ol


Enr
Gas
Gas
Cor
cm
Exh


Exhau
e Ave
erage
erage
n/cm


Energy


m3)
as Pressu
Gas Tempe
erage Mas
Time (mse
ion Time
t Neutron
cm)
cm)
in Core (
n Core (k
reaction i
hment (wt
emperatur
pressure (
Thermal N
c)
st Gas Te
st Gas Pr
rage Gas
Gas Pres
Core The
sec)


Release


Chamber (Mw)
Net Turbine Power
Chamber (Mw)
Efficiency (%)


Output


(atm)
ture (K)
Flow Rate


)
sec)
Lifetime


kg)
g)
n Gas
%)
e (K)
atm)
eutron


(kg/sec)


(msec)


Mixture


mperature (
essure (atm)
Temperature
sure (atm)
rmal Neutron


Rate


K)

(K)


Per


Per


X 1
14
31
74
20


X 10

16.4


The
is

The
is


inner
30 cm

outer
70 cm


BeO
thick

D20
thick


region
and at

region
and at


of the moderating-reflector
an average temperature of

of the moderating-reflector
an average temperature of


regions
620K.

region
490K.











Table


3-3.


Typical Equilibrium Cycle
Operating Conditions.


Piston-Driven


System


Compression Ratio
Clearance Volume (
Intake Line Gas Pr
Intake Line Gas Te
Engine Speed (rpm)
Piston Cycle Time
Loop circulation T
Average Prompt Neu
Core Height at TDC
Core Radius at TDC
Uranium Mass in Co
Helium Mass in Cor
Uranium Enrichment
Maximum Gas Temper
Maximum Gas Pressu
Maximum Core Therm
Flux (n/cm sec)
Average Exhaust Ga
Average Exhause Ga
Cycle Average Gas
Cycle Average Gas
Cycle Average Core
Flux (n/cm sec)
Fission Energy Rel
Chamber (Mw)
Mechanical (Shaft)
Per Chamber (Mw)
Net Turbine Power
Chamber (Mw)
Total Power Output
Overall Efficiency


m3)
essure (atm)
mperature (' K

(sec)
ime (sec)
tron Lifetime
(cm)
(cm)
re (kg)
e (kg)
(wt%)
ature (K)
re (atm)
[al Neutron


(msec)


s Temperature (K)
s Pressure (atm)
Temperature (K)
Pressure (atm)
Thermal Neutron


ease


Power


Output


Per
(%)


Rate


Per


Output

Per


Chamber


(Mw)


-to-i
.180
9.5
50
00
.2
3.3
.90
8.1
1.4
.503
.876
3
011
29.9


1.065


15
X 10





X 1014


5.222

1.670

1.446
3.116
59.7


Adiabatic Neutron Kientics
Simple Harmonic Motion for
D20 Moderator-Reflector at


Model Employed
Connecting Rod
490K and 100 cm thick



























1015


114


C
10
1-
r*l

o

a)
z
" 13
S10

0
0)
z


SC


.l011
C)

4
(U
0.11
810'

0



I-3
Ool
o]


0.00


0.09 0.18 0.27 0.36 0.45 0.54 0.63


Cycle


Time


0.72


(Sec)


Figure


3-13.


Average


Cycle


Core


Time


Pulsed Gas


Thermal


for
Core


Neutron


a Typical
Nuclear


Flux


Versus


Gas Generator
System.
































x 14
S101


0 1
S013
10

210

l 11

o
0)
C

< 10

0


0.15


0.30


0.45 0.60 0.75


0.90


1.05


1.20


Cycle


Time


(seconds)


Figure


3-14.


Average


Core


Cycle Time
P sied Gas


Thermal


for
Core


Neutron


a Typical
Nuclear


Flux


Versus


ston-Driven


System.










helium,


would


require


to exceed


temperatures


where


significant


degree


of UF6


dissociation


occurs.


Comparison


between


the


PGG


and


PNP


systems


from


Table


and


indicates


that


the


PGG


concept


has


the


disadvantage


of lower


overall


efficiency


relative


the


PNP


concept


due


to the


fact


that


only


turbine


power


and


mechanical


or shaft


power


extracted


from


the


system.


Comments


The


promising


results


from


the


gaseous


core


reactor


research


efforts


undertaken


at the


University


of Florida


Dugan


and


others


(28)


have


encouraged


the


engineering


research


community


the


University


Florida


to continue


extensive


research


on various


advanced


gas


core


reactor


concepts


and


different


power


conversion


systems


especially


space


applications.


The


gas


core


reactor


neutronics


energetic


research


have


indicated


that


cyclic


PGCR'


are


capable


attaining


high


peak


gas


temperatures


2000-2500


and


high


thermal


neutron


fluxes


neutrons/cm


-sec)


while


maintaining


relatively


low


cycle


averaged


gas


temperatures


1000


K to


1200


The


pulsed


gaseous


core


systems


are,


thus,


able


to utilize


very


high


temperature


and


pressure


working


fluid


while


the


structural


stress


and


thermal


shock


are


kept


to a minimum.


1015











mode


of operation.


The


detailed


description


the


system


and


static


and


dynamic


neutronic


characteristics


are


given


Chapters


4 and

















STATIC


NEUTRONIC


CAVITY


CHAPTER
STUDIES


GASEOUS


CORE


ON COUPLED
REACTOR SY


MULTIPLE


STEM


Introduction


This


chapter


examines


the


bimodal


gaseous


core


reactor


system


detail


predicts


the


results


from


the


steady


state


reactor


neutronic


system.


analysis


Steady


state


the


bimodal


neutronic


multiple


analysis


cavity


the


system


includes


examining


the


effects


variations


chamber


dimen


sions,


inner


and


outer


beryllium


moderator/reflector


thicknesses,


the


BPGCR


gas


loading


and


of equal


and


unequal


gas


loading


among


the


PGCR


chambers


on the


neutron


system


removal


neutron


multiplication


lifetime


system


factor


neutron


(keff)


system


generation


time


(A),


and


core-to


-core


neutronic


coupling


coefficients


(ajk)


The


power


requirements


space


based


systems


under


consideration


range


from


s of KW(e)


to about


10's of


MW(e)


the


station


keeping


or surveillance


mode


operation


and


100'


of MW(e)


few


GW(e)


the


burst


( ),


I











low


power


pulsed


gas


core


reactor


system


were


employ


only


one


or two


small


chambers,


compressor (


would


needed


to provide


the


necessary


driving


pressure


not


only


during


the


system


startup,


but


also


during


the


steady


state


operation.


With


the


multiple


chamber


design


poss


ible


arrange


the


chambers


the


system


so as to "optimize


neutronic


coupling


and


reduce


the


system


critical


mass


requirement.


Therefore,


proposed


employ


a PGCR


system


high


with


power


multiple


chamber;


chambers


multiple


surrounding


chamber


the


large


design


central


the


low


power


PGCR


system


would


eliminate


the


need


compressor(


, except


during


system


startup,


and


would


reduce


the


fluid


pulsation


effects


or flow


discontinuiti


the


external


loop


compared


to single


chamber


design)


With


the


multiple


cavities


, the


system


can


be designed


assure


that


any


given


time,


one


or more


chambers


are


the


exhaust


phase


provide


a relatively


continuous


source


of pressurized


gas


for


power


generation.


The


multiple


cavity


design


should


have


a favorable


impact


on system


reliability


since


power


can


be generated


even


a few


reactor


chambers


fail.


The


Bimodal


Gas


Core


Reactor


System


Description










annular


ring


power


pulsed


gaseous


core


reactor


(PGCR)


chambers


(Figures


and


1-2) .


The


central


cylindrical


chamber


and


the


annular


ring


of PGCR


s are


separated


inner


beryllium


moderating/reflecting


material.


The


ring


PGCR


surrounded


an outer


beryllium


moderator/


reflector


region.


The


identical


upper


and


lower


halves


the


BPGCR


are


separated


a common


moderator


slab


the


mid-plane


and


the


MHD


disk


generator


regions


the


top


bottom


central,


high


power


chambers.


The


bimodal


reactor


system


also


provided


with


top


and


bottom


moderator/


reflectors


of appropriate


thickness.


The


PGCR


chambers


are


intended


to provide


low


power


station


keeping/surveillance


purposes.


The


primary


working


fluid


a mixture


of highly


enriched


uranium


hexafluoride


(UF6)

added


which


the


to enhance


reactor


the


fuel


and


thermodynamic,


helium


gas


transport,


which


and


heat


transfer


characteristics


the


primary


working


fluid.


The


fissile


gaseous


fuel


(UF6


-He)


mixture


cyclically


injected


into


PGCR


chambers,


which


are


individually


pulsed


near-critical,


critical,


or supercritical


state


and


the


heated


gas


then


discharged


energy


conversion;


each


chamber


capable


of providing


a few


MW(e)


power.


Neutronically


the


PGCR


cores


from


far-subcritical


state











core


will


have


achieved


an adequate


neutron


flux


level


sufficient


duration


to generate


a hot


gas


the


energy


conversion


system.


The


fissionable


gaseous


fuel


whose


nuclei


undergo


fission


primarily


thermal


neutrons


neither


moderates


nor


absorbs


fast


epithermal


neutrons


any


significant


degree. Fast

chamber, slow

moderator, and


neutrons


down


released


to thermal


diffuse


fission


energies


as thermal


neutrons


leave


the


the


beryllium


thereafter


until


they


leak


or are


absorbed


either


the


chambers


or in


the


reflector.


With


multiple


PGCR


chambers


and


proper


timing


operation


the


individual


chambers


among


their


intake,


power,


and


exhaust


(discharge)


phases


the


system


expected


to provide


a relatively


continuous


source


of pressurized


gas


any


suitable


power


conversion


system.


During


power


PGCR


operation


voided,


have


system,


a constant


the


non-fuel


central


gas


cavity


flow,


will


or have


either


a low


density


fuel


gas


mixture


flow.


The


transition


from


station


keeping


burst


power


operation


can


be achieved


circulating


a uranium


bearing


fuel


gas


through


the


central


BPGCR


cavity.


The


neutronic


coupling


effects


of the


PGCR


chambers


on the


gas


fuel


bearing


central


cavity


provide


the


necessary


initial











power


provided


the


central


BPGCR


chamber;


the


surrounding


PGCR's


continue


to provide


low


or station


keeping


power.


The


energy


conversion


the


BPGCR


chamber


occurs


the


disk


MHD


generator.


The


fuel


gas


then


passes


through


a radiator


heat


exchanger


before


being


circulated


back


into


the


central


chamber


a compressor.


In addition


to providing


station


keeping


power


, the


PGCR's


support


burst


power


operation


the


central


chamber


the


following


reasons:


The


neutronically


coupled


PGCR


chambers


surrounding


central


chamber


are


near-critical,


critical,


supercritical


during


a significant


portion


their


power


for


cycle.


the


They


central


relax


BPGCR.


the

This


reactivity

provides


requirements

greater


flexibility


selecting


the


BPGCR/MHD


system


working


fluid


compo


sition.


The


PGCR


system


s provide


and


maintain


a base


the


level


system


neutron


"hot"


flux


so that


the


a more


rapid


transition


to burst


power


mode


possible


than


the


PGCR


s were


absent.


The


pulsed


nature


of the


system


greatly


minimizes


the


thermal


stresses


as compared


to those


typically


experienced


comparable


performance


steady


state


systems.


Also


the











Steady


State


Neutronic


Analysis


Steady


state


neutronic


analysis


the


multiple


cavity


bimodal


gaseous


core


reactor


system


has


been


performed


primarily


using


the


Monte


Carlo


Neutron


Photon


(MCNP)


transport


code


and


discrete


ordinates


(SN)


transport


code,


XSDRNPM


The


complex


geometry


of the


bimodal


system


and


the


tenuous


nature


of the


fuel-gas


mixture


core


require


multidimensional


or Monte


Carlo


transport


type


calculations


the


static


neutronic


analysis.


One-


dimensional


static


neutronic


analysis


of the


system


performed


using


an "equivalent"


sphere


which


the


volume


of the


core


and


thickness


the


moderator


are


preserved.


the


later


results


show,


this


can


lead


considerable


errors


in the


values


parameters


such


as the


system


neutron


multiplication


factor,


neutron


removal


lifetime,


etc.


Even


though


a two-dimensional


transport


calculation


can


be expected


to yield


good


results


system


neutron


multiplication


factor


and


neutron


removal


lifetime,


cannot


be expected


to provide


an accurate


estimate


themselves


the


and


coupling


between


coefficients


central


among


chamber


the


and


PGCR


the


cores


PGCR


cores.


The


complex


geometry


the


bimodal


system


requires


- 51 - -


-lur a a-


a a 1 at'I 4.-


LA a a 4 .


Ir 1


1


C1











probabilities


interaction


can


be simulated


the


MCNP


code


provides


the


capability


estimate


the


integral


kinetics


parameters,


especially


the


coupling


coefficients,


with


a high


degree


of confidence.


MCNP


a general


purpose,


continuous


energy,


generalized


three-dimensional


geometry,


time-dependent,


coupled


neutron-photon


Monte


Carlo


transport


code


which


has


the


capability


calculating


the


keff


eigenvalues


fissile


systems.


has


an elaborate


tally


structure


and


user


interface


that


allow


a user


to calculate


almost


anything


conceivable


and


has


elaborate


and


complete


cross


section


data.


has


a rich


collection


variance


reduction


techniques.


The


tally


structure


MCNP


capable


of producing


cell-averaged


particle


flux,


surface


neutron


current


and


flux,


and


neutron,


photon


and


fission


heating


a cell.


The


cell


flagging


and


surface


flagging


capabilities


of MCNP


allow


one


to determine


the


tally


contribution


of particle


tracks


from


flagged


cell


surface


system.


any


This


other


specified


capability


cell


of MCNP


or surface


used


the


to determine


the


neutronic


coupling


coefficients


which


are


a measure


the


degree


influence


the


k-th


core


neutron


flux


on the


j-th


core


neutron


flux.


Both


free


gas


thermal


and


S(a,$)


(ajk)











implicitly


included


the


MCNP


treatment


considering


total


the


average


number


neutrons


generated


per


fission.


more


detailed


description


the


MCNP


code,


associated


nuclear


cross


section


data


library,


capabilities


and


features,


and


sample


inputs


a single


chamber


PGCR


and


multiple


chamber


bimodal


system


are


given


Appendix


Neutronic


studies


the


bimodal


reactor


system


have


utilized


the


generalized


three


dimensional


geometry


capability


the


MCNP


code.


However,


because


the


gaseous


nature


the


cores,


the


convergence


the


source


from


initial


guess


to a distribution


fluctuating


around


the


eigenmode


solution


has


been


found


to be


too


slow


Therefore,


order


to get


a reasonably


converged


value


the


eigenvalue


(keff)


and


the


neutron


removal


lifetime


has


been


necessary


run


MCNP


many


more


cycles


generations


with


a larger


source


distribution


per


cycle


than


would


have


been


required


with


a denser,


solid


fueled


core.


XSDRNPM


a one-dimensional,


time


independent,


screte


ordinates


transport


code


that


capable


calculating


neutron


multiplication


factor


and


angular


and


spatial


neutron


flux


(4).


XS DRNPM


used


the


"equivalent"


one


dimen


sional


spherical


mock-ups


the


three











four


region


(Figure


4-1)


spherical


configuration


with


equivalent


The


volume


spherical


core


mock-up


and

has


equivalent

a central


thickness


spherical


moderator.


chamber


(BPGCR)


surrounded


an annular


ring


beryllium,


followed


an annular


finally


ring


an outer


gaseous


moderator


fuel

ring.


(PGCR

The


and


's) ,

keff


then


values


from


XSDRNPM

guesses


have

for


mainly


the


been


respect iv


used

e MCN


as the

P runs


required


order


initial

to have


keff
the


MCNP


eigenvalue


and


neutron


flux


converge


faster.


XSDRNPM


keff


values


have


also


been


used


to explain


trends


the


neutronic


behavior


different


configurations


the


BGCR


system.


Steady


State


Calculations


Procedure


A seri


preliminary


calculations


were


made


with


MCNP


code


various


configurations


the


bimodal


gas


core


reactor


system


(Figure


1-2) .


These


initial


steady


state


calculations


were


done


system


configurations


with


inner


beryllium


thicknesses


(TIBE)


of 10


and


cm during


which


the


outer


beryllium


thickness


(TOBE)


was


maintained


at 50


cm.


For


each


value


the


inner


beryllium


thickness


the


PGCR


gas


fuel


(UF6


-He)


mixture


pressure


was


varied


from


20 atmospheres


to 60 atmospheres.


























PGCR


Cylindrical


Geometry


MCNP


Calculation


PGCR


I











mixture


and


the


central


BPGCR


cavity


was


maintained


pressure


of 20 atmospheres


helium


gas.


For


each


fuel


gas


loading


and


system


geometry


configuration


the


MCNP


code


provides


the


the


system


system


neutron


neutron


removal


multiplication


lifetime


factor


system


(keff),


neutron


generation


time


(A),


and


the


track


length


averaged


neutron


flux


each


the


PGCR


chambers


and


cells


the


system.


The


cell


flagging


capability


the


MCNP


code


which


the


particle c

individual

to compute


contribution


cell

the


from


or a group

neutronic c


a flagged

of cells


oupling


cell


can


any


tallied


coefficients


other


used

the


among


PGCR


cells.


To study


the


variation


the


system


neutron


multiplication


factor


with


the


outer


beryllium


moderator


thickness,


calculations


were


done


with


the


MCNP


code


which


the


outer


beryllium


thickness


was


changed


from


cm for


two


cases


inner


beryllium


thicknesses


of 20


cm and


cm.


The


gaseous


fuel


mixture


was


maintained


constant


pressure


50 atmospheres


each


the


PGCR


chambers.


MCNP


calculations


were


then


repeated


for


two


configurations


the


system


with


inner


beryllium


thicknesses


cm and 30


Th a-


cm but


with


unequal


fuel-eras


( ),











situation


which


a fraction


the


total


number


PGCR


chambers


will


always


intake,


power,


or exhaust


(discharge)


phases


operation


so that


the


reactor


will


capable


delivering


a continuous


supply


pressurized


to the


PGCR


power


conversion


system.


Steady


state


calculations


were


also


done


with


central


central


burst


chamber


power


chamber


fuel-gas


fueled.


mixture


this


pressure


was


case


the


varied


from


atmospheres


atmospheres.


This


series


of steady


state


calculations


examined


system


configurations


with


inner


beryllium


thicknesses


of 10


and


cm.


The


outer


beryllium


thickness


was


maintained


at 50


cm.


The


fuel-gas


of the


pressure


individual


was


PGCR


maintained


chambers,


10 atmospheres


this


each


series


calculations.


For


these


calculations,


addition


the


system


neutron


multiplication


factors,


neutron


removal


lifetimes,


and


the


fluxes


various


BPGCR


fuel-gas


mixture


pressures,


the


neutronic


coupling


effect


the


central


chamber


on the


PGCR


chambers,


the


neutronic


coupling


effects


the


individual


PGCR


chambers


on the


central


BPGCR


chamber,


and


the


coupling


effects


among


the


PGCR


chambers


themselves


were


calculated


separately.


Static


neutronic


calculations


were


also


performed