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UNCLASSIFIED
MIT-5000

REACTORS ~' RESEARCH AND POWER

 

NUCLEAR PROBLEMS
OF NON-AQUEOUS FLUID-FUEL REACTORS

 

 

October 1 1952

Authors

Clark Goodman
John L. Greenstadt
Robert M. Kiehn
Abraham Klein
Mark M. Mills
Nunzio Trallil

Consultants
-Harvey Brooks
Henry W. Newson

   

shington 25, D. C.

 

NUCLEAR ENGINEERING PROJECT
MASSACHUSETTS INSTITUTE OF TECHNOLOGY

 

 

" Manson Benedict, Director CLASSIFICATION CANCELLED

DATE  FEB 28 1957

U.S. ATOMIC ENERGY COMMISSION -Hr 2 )
NEW_YORK OPERATIONS OFFICE N h N\ QA lE

Chief, Declassification Branch

 

 

 

 

LEGAL NOTICE

This report was prepared as an account of Government sponsored work, Neither the
United States, nor the Commission, nor any person acting on behalf of the Commission:

 

A. Makes any warranty or representation, express or implied, with respect to the ac-
curacy, completeness, or usefulness of the information contained in this report, or that the
use of any information, apparatus, methed, or process disclosed in this report may not in-
fringe privately owned rights; or

B. Assumes any liabilities with respect to the use of, or for damages resulting from the
use of any information, apparatus, method, or process disclosed in this report.

As used in the above, ""person acting on behalf of the Commission” includes any em-
ployee or contractor of the Commission to the extent that such employee or contractor
prepares, handles or distributes, or provides access to, any information pursuant to his em-
ployment or contract with the Commission.

 

 

UNCLASSIFIED .

[ ]
e & e ' e
 

£y

TABLE OF CONTENTS

Chapter I General Considerations sseecceessccsnces
1.1 INtroduction eesecocecrocsassesce
1.2 Tast ReaCltors cevececcccvossacces
1.3 Thermal Converters cecesceesccceces

Glossary of Symbols Used in Chapter I ..

Chapter II Fast Reactors ® S0 5000 H P SO S 0SS OSSO B

l. Nuclear Constants O 0O O B0 O 0P OO R P OO SO EE NSRS E O PDE

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2. Calculation

2.1
2.2

2.3
2.4

Total Cross-SectionS.cccecccccses
Transport Cross=-SectionSeeecseees
Inelastic Cross~-Sectlons eceeseess
Fission Cross-Sections eseeesecoses
Captule Cross-Sections eeecececse
Scattering Cross-Sections eceeees
Neutron Yields .eeeccecscscccccnas
Neutron Spectra .ecseceesceescsscce
Recommendations eeeececccccecenens

MethOdS ® O & 0 0 &8 0 0 050 2P L BO e ST S SLSDS

Bare Reactor Multigroup Method ..

Multigroup Estimates for
Blanket * & 00 0" P 5" OO S Oe s 08N P

Two-Group Two-Region Method .....
One-Group C310Ulat10ns o¢ o8 e

3. Results of Fast Reactor Calculations seeesceee

3.1 Introduction .eeeceececcecccccccccsce

3.2 Bare Reacltor ececececccccsscccnces

3.3 Two-Group Two-Region Calculations

3.4 One-Group Three Region ceeecvvens

3.5 Special Multigroup Calculations .
.

N3

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66
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97
 
  

.‘,rvr Y
! -
n"
v

3. Results of Fast Reactor Calculations (contd.)
3.6 General TrendS cceceescscscscaves

3.7 Final Design of Fused Salt
ReaCtor o0 & 0 0 00 O 6P BSO OO0 s ODS

Y}, TFast Reactor PolsSoning sececescssccccssssocces
4.1 Introduction eeeececeeeescrcsccas
L.,2 Fission Products eceeccoeveccocses
4.3 Higher ISotODES ccereececcsenvcoas
4.4 Engineering Considerations ......

5. ContrOl MethOdS 5 O % B 0 080 %08 SO B S S e LN ON SN POSRDS

5.1 General Considerations .eceeeecees
5.2 Control Calculations sececcesccsce

5.3 Control of the Fused Salt
Reactor o & 5 5 & & ¢ 488 8 S5 80 S B2Ee T e S 0SS

Glossary of Symbols Used in Chapter II .
Acknowledgments cvesescecccncscsosnscncena

Chapter III Thermal Reaclors seceeececsscsocsoseccosse
1. Introduction ...cccevvvvvrenecccercnscescannns

2. Bare Homogeneous ReactOr ceeeccccsrecscccsccsss

2.1 Definitions and Basic Constants .
2.2 Analytical Expressions ..cecceceee
2.3 The Conversion Ratio, CiRe ceeose
2.4 Results of Calculations sceceecees

3. Poisoning Effects S 0 8 08 800 8800 00 5e BSOS e s seR

3.1 Uranium 236 S 9 9S00 H S PSSO N PR S SSDR

3.2 Tast Neutron Reactions of
Beryllium ..000‘00.0.0....0.....

3.3 PFission Products eceeeecevecceescne

 

 

104

111
127
127

127
128

130
131
131
132
140
145
150
151
151
153

153

159
164

165
166
166

173
174
 

4., Heterogeneous ReacCtorS eeeeccecccsscsssccsesss

4,1 The "Immoderate" Blanket ccececes

4.2 The Bffect of Be Lumping eecveeee

5. Final Design CalculationsS .eecececvecscccssssne
5.1 Reactor Constituents and

PrOPerties ® S 00 & 0P S P O OE PO eY S8 b

5.2 Calculation Procedure c.ecessseesee

5.3 Results .!.......‘...............
5.4 Critique of ReSultS cesececcccces

6. Recommendations ciceccescccscessccssscscancans
6.1 Design Studies cececeveccccsccces

6.2 Nuclear Data Studies cecececscces

Glossary of Symbols Used in Chapter III

Chapter IV  Comparison of Fast and Thermal Conver-
TerS ceeessctscccvcsscssonsscsnncsnnne
1. Core STruCture eceevvsvecsesesesvsesenssessscnce
2. Reflector Structure ..eeeceeevesvcocscsccsnscsns
3. Blanket Structure ceceecccecceccessecscsssossssscs
%, Critical Mass and INventory .eeeeeecceecessess
5. Conversion Ratio stesescececcsconssoscscnnncnscs
6. Parameters and Processing of POlSONS seeeecces
7¢ CONtrol ceeeeececccesvessacssccscascsosscncsse
Glossary of Symbols Used in Chapter IV
Appendix A Procedure for Homogeneous, Fast, Bare
Reacltorl teeesvncscccscccscosccccncsncncne

Appendix B Summary of Calculations on Bare Homo-
geneous Thermal Reacltors .sseececcssses

Appendix C Secular Equations eeeeesesceocssccescsss

Appendix D Effect of Fast Neutron Reactions of Be
on the Conversion Ratio ceeceececencens

Appendix E Two-Group, Two-Region Reactor Equations.

References ® 0 % O BB OO OO B USRS PRNETOSESNEPEES OO EER e
Acmow:—edgments ® 08 009 00 02T PO OE OO RO NSO S OO IYTEOIEOSORTEYOPRETS

r ¥

i

* s 9 0 e &
* - L J -
. .

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TABLE II-l.2-1

11-103-1
II-1.3-2

IT-1.4-1
II-lOS-l
II"lo 5-2

II"lo 6“"1
II"lo 9-1
II"3 02-1
II"3 . 2"'2
IT-3.2=3
II-3.2-4%
11-3 * 2-5
II"3 . 3"'1
II-3.5-1
II"B . 5"2

II-3.5-3
II-3.7-1

II"'307"‘2

 

   
      

;:.u : PTERT L L

LIST OF TABLES =

Transport Cross=Sections eeeecececes
Inelastic Cross=Sections .cceeceses

Assumed Spectral Distribution of
Inelastically Scattered Neutrons.

FiSSiOH CrOSS-SGCtionS S o0 P OO BSBLIEOGOEOETRTS
Capture CTOSS-SGCtiODS 2209 98 e0s 0000

Effect of Changing <(Bi) on C.R.
in U-Bi SYStems S e P e B IPIPOERTEOIRNSEY

Degradation: Product of Scattering
Cross=-Section and Mean Loga-
rithmic Energy Decrement (ficé)...

Resume of Nuclear Data Needed .ce..

System Constituents for Bare Re-
actor Multigroup Calculations ...

Breeders - Results of Multigroup
Bare Reactor Calculations ..cceee.

Converters - Results of Multigroup
Bare Reactor Calculations ,...e¢e.

Breeders - Neutron Balance ...cccee

Converters - Neutron Balance .eesee

Comparison of Special Multigroup
Calculations with Two-Group
Calculations * & 8 & 0 0 & & 09 PO Ve S e e 0o

System Constituents for Special
Multigroup Calculations ececeeeee

Results of Speclal Multigroup

Calculations cesccececcevcsccscnee

Special Systems - Neutron Balance .

System 24: System Constituents
and Results es e s e s e s s sOBREREEGEE

Systemp 24 - Neutron Balance escecese
e

2724 +05 : THNRE

 

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- e W x g | Page

No.

TABLE II-3.7-3 System Constituents of Fused Salt |
Re&CtOP ooooo & & ® ¢ & O " O S S8 B O B g ¢ e s 118

II-3.7-4 Results of Fused Salt Multi-
region Calculation .¢ccccseeccee 119

II1-3.7=-5 Neutron Balance - Fused Salt
Reactorj Multigroup Calculation . 120

II-5.4=1  Delayed Neutrons ......ecceeeeeees 137

I1I-2.1-1 Temperature Corrections for
Nuclear Datl ececeocosssessscessss 15U

ITI-2.1=2 Thermal Neutron Properties of
Thermal Reactor Constituents .... 155

111-2.1-3 Properties of Isotopic Uranium
Mixtures ® 8 5 % 0 80 S PO GBS HSE BB s 157

III-2.1-4 Resonance Escape Probability ...... 159
III“2.2—1 Age tO Therm&l ® 0 0 9 08 0606006 BB s e et oS 161
I1I-2.2=2 Age and Density of Bi-Be Mixtures . 163

III-3.1-1 Effect of U-236 on Conversion
Ratio in Thermal Reactors «...... 171

III-4,2-1A Results of Cell Calculations ...... 183
III-4.2-1B Required Data for Evaluation of p . 184
ITI-5.1-1A Reactor Constituents and Constants. 188
I1I-5.1-1B Reactor Constituents and Constants. 189
1II-5.2-1 Summary of Reactor Properties ..... 191
III-5.3-1 Results Qf Calculation ecscecesessces 192
III-5.3=2 Neutron BalanCe .....ceovesessescees 193
I11-5.3=3 Delayed Neutrons ...ccececessccsnecs 196
III-5.4-1 Homogeneous Reactor Results ...... 198

III-5.4=2 Gain in Production Ratio by Use of
Blanket S 2 5 8 65 86 00Ot O SO B OSSN SS 201

S | EY
;-;Q %x‘
TABLE B-l

B=-2
B-3
Bl
B-5
B-6
B-7
B-8
B-9
B-10
B-11
C-1

C=2

C-3
D-1

D=2

 

w Page

Summary of Calculations on Bare
Homogeneous Thermal Reactors ....

"

. & 88

"

® o e

® & & &

n

L 2 N

"

n

.‘.Q.

Numerical Values Used in Computa=-

tion *© 5 0 0 8 06 0SS S BRIV B0 SN OO G LS

U-236 Concentration and Bulldup

Time [ I N B BN B BN O BN NN BN R BN N NN NN CEE I S BN N B N BN B AR AN

Summary of Secular Results .......

Data for Fast Neutron Reactions of

Be.'.... ..... * o 0 0 5 0P OSSO B0 PSS e

L16 Concentration Factors .....c..

 

No.

232
232
233
233
23%
234
235
235
236
236
237

ol

245
246

252
253
 

FIGURE II-2.4-1

IT-3.2-1
IT-3.2=-2
IT-3.2=-3
II-3.2-k4
IT-3.2-5

IT-3.2-6
II-3.2=7

I1-3.2-8
II-302-9

IT-3.2=-10
II-3.2-11

IT-3.2-12
II-3.3-1
II-3.3=2
II-3.3=3

II-3.3-4

 

LIST OF FIGURES

Three Media-One Velocity Systems ..
Pu013-U01h Breeders S e s s 0ess s res e
PU**U-238——Bi Breeders "o 0000w
U-235'-U-238--Bi COnverterS ses s esee
UClh‘N&Cl Converters sss0sevesses s

System 1l4: Spectrum of Fissions
and Fraction of Fissions above u

System 14: Flux SpectITUlm seveeess.

System 15: Spectrum of Fissions
and Fraction of Fissions above u

System 15: Flux SpecCtrull ceeecseese

System 16: Spectrum of Fissions
and Fractlion of Fissions above u

System 16: Flux Spectrum ....ccc..

System 17: Spectrum of Flssions
and Fraction of Fissions above u

System 17: Flux SpecCctrul .eeseecees
Reflector Size vs. Core Size sccese

Reflector Size vs. Core Size eceecee

X.C.R.
X.C.R. (Bare Reactor) versus

__Core Size

Bare Core Size
X.C.R.

X.C.R.(Bare Reactor) versus

Core Size
Bare Core Size

=i
»

¢ & ¢ 0 &0 %" s OB P S E NS e

* 00 080000 P eSO PS

e

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92

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9l
FIGURE II-3.4=-1

II-305‘1

II-3.5-2

II"3 . 7-1

II-307'2
II"‘3 . 7-3
I1I-3.7-l

II-3.7-5
II-3.7-6
I11-2.1-1

II1-3.1-1
ITI-3.1=-2
III-‘-!-.2-1

III"L"Q 2"'2
IT1I-5.4=1

 

Reflector Thickness and X.Co
X.C.o

R BN N W NN N R I N A

versus Core Slize

System 52: Comparison of the
Spectrum of Fissions of a
Bare Reactor and a Blanketed
ReaCtor "TEEEEREEEEERE XN NN B I B B BN N I S B N N

Comparison of the Flux Spectrum
of a Bare Reactor and a
Blanketed Reactor

System 24: Spectrum of Fissions
and Fraction of Fissions above u ..

¢ 600 F O S0 s o080

System 24: TFlux Spectrum ..ceveceees
Fused Salt Reactor - Flux Spectrum ..

Fission Power Distribution in the
Fllsed Salt Reactor ® $ 6 0 6% &0 S OO OO e S

. Total Neutron Flux per Unlt Radial

Distance vs. Radial Distance ecc¢eeee

Normalized Flux Spectrum at Various
Radial Distances

® 5 5 €9 5O 0 e 8 s s 0 0

Experimental Resonance Absorption
Integral, A, versus "¢ /U" ........

Reaction Equations ...cceevveccscccosse
Higher Isotope Chains ....cecoeoeseos

f and p versus Radius of Be Lump
for Various N(Be)/N(BL) ....ccveeee

Product pf vs. Radius of Be Lump ,...

Critical Mass versus Production Ratio
for One-Zone and Two-Zone U-Bl-Be
Reactors

& ¢ & &6 & ¢ 8 058P0 0L eSS OSSN

 

_9-

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102

103

114
115
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199
-10-

 

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No.
FIGURE A-1 System 21: Spectrum of Fissions
| and Fraction of Fissions above u 224
A-2 System 21: Flux Spectrum ...ccceee 225
u
D-1 q)(u) =J:,J(f (u') du' as a

Function of ENergy .cccsccecoscns 254

D-2 3% = 7 {HEY . Number of Be
Atoms Undergoing (n,a) Reaction
per Unit Source Neutron vs. X. ... 255

D-3 g = Average Fraction of Final L16
Concentration as a Function of
M(U-235) ® & 5 & 90 @ s SO O PSSP LS 0O PSSP PSS 256
-11-

    

%

I. GENERAL CONSIDERATIONS
1.1 INTRODUCTION &l . .. .

The primary purpose of the Nuclear?Efigfifiéé}ffifii?roject
at M.I.T. during the summer of 1952 was to investigate the
problems of reactors using non-aqueous fluid fuels for the
production of plutonium and to recommend a program of research
and development to supply the information needed to provide a
sound basis for the engineering of this important type of
reactor. Aqueous fluld-fuels are receiving attention at 0Oak
Ridge and elsewhere.
The results of this Project are being described in
three companion reports.,
l. "Engineering Analysis of Non-Aqueous Fluid-Fuel
Reactors" - (MIT-5002),
2., Y"Chemical Problems of Non-Aqueous Fluid-Fuel
Reactors" - (MIT-5001), and
3. This report.

5

The first of these reports describes the objectives of
the Project, the lines of investigation pursued, and the main
conclusions drawn. It describes in detail the engineering
studies carried out by the Project and the bases for them. It
summarizes all recommendations for future research and de-
velopment.

The second of these réports describes the chemical
studies conducted by the Project and gives details of the
program of chemical and chemical engineering research re-
commended by it.

The present report treats the nuclear studies conducted
by the Project. The basic nuclear data and design methods
are described and the results of the nuclear studies are
given in detail. A research program on nuclear properties
of importance to non-aqueous fluid-fuel reactors is recommended.

==

¢
?,l
=12~

 

Chapter I of this report outlines the considerations
which led to the choice of two reactors for detailed study by
the Project:

(1) A fast converter using as fuel a solution of UCl,,

| in fused chlorides |

(2) A thermal converter using as fuel a liquid alloy

of U in Bi,
and lists the main characteristics of each reactor. Nuclear
studies on the fast and thermal reactors are described in
Chapters II and III, respectively. Chapter IV compares the
two reactors. The Appendices contain details of calculation
methods, and the results of nuclear studies not directly
related to the reactor processes given engineering study.

1.2 FAST REACTORS

The two main guestions regarding fast reactors asked
at the beginning of the Project were:
(1) Should the fast reactor to be investigated be a
converter or a breeder, and
(2) What fuel system should be chosen?

BREEDERS VS. CONVERTERS. = As shown in Chapter II of
thls report, fast, fluid-fuel breeders may yield a breeding
gain of the order of 0.6, whereas a fast converter using the
same type of fuel except for the interchange of U-235 for
Pu-239 will give a conversion ratio of around 1.15. Cost
analyses described in more detail in the engineering analysis
report show that plutonium can be produced more economically
in a fast converter than in the corresponding breeder, under
the cost bases adopted for this project.

The cost advantage of the fast fluid-fuel converter
compared with a feasible fast breeder arises from two main
causes:

1). The high unit cost of Pu compared with that of
U-2353 i.e., U=-235 is cheaper to burn or store than
Pu-239 on a gram for gram basis, because the

 
-13-

 

projected cost per gram of Pu-239 is still larger
than the present cost per gram of U-235,

2). The inventory charges on Pu-239 are based upon
the total critical mass of the breeder, but only
on the equilibrium concentration of Pu-239 in the
converter, which may be quite small in comparison.
Hence for the low specific powers considered for
the fused salt reactor (~ 300 watts/gm), an
equivalent breeder (also with a. specific power of
300 watts/gm) would have such a largé inventory
charge assessed against it, that it could not
compete with a converter in the economic production
of Pu-239. Higher specific powers, if possible
engineeringwise, would decrease the disparate
cost estimates for Pu-producing breeders and con-
verters.

Even if no charge was made for inventory, however, some cost
advantage would remain with the converter under the cost
bases adopted by this project.

SELECTION OF IFUEL. - Fast reactors must have high con-
centrations of uranium and low concentrations of absorbing
diluents which should not moderate excessively. An initial
survey of possible fluid core materials showed that few, if
any, fluid alloys with uranium have sufficiently satisfactory
nuclear and engineering properties to be practical for fast
reactors. Fused salts are an alternative to the fluid alloys.
The fluorides were eliminated on the basis of thelr excessive
moderation which would increase the overall a of the melt. The
bromides and iodides were eliminated on the basis of their un-
favorable capture cross-sections (associated with high atomic
weight). These decisions were based on the following useful
rule-of-thumb for estimating the upper limit for the micro-
scopic capture cross-section of diluents, <,(D), in fast con-
verters:

 
 

-14- BN
< (D) < N-%z-?- x 0.2% barns (1.2-1)
c N D . o

at 0.2 Mev mean neutron energy.

This relation assumes as a practical basis that the decrease
in the conversion ratio resulting from capture in diluents
should be less than one-tenth the conversion ratio for a non-
poisoned reactor. For example, in a hypothetical mixture:
U235Brh, which is certainly the lowest possible concentration
of Br in a bromide fused salt, S, (D) = 0.24/% = 0.06 barn

‘ max -
at 0.2 Mev., However, Br has an estimated ©_ = 0.1 barn at

this energy and<3;(iodine) is even greater.c Bromine and
iodine were ruled out for this reason, leaving chlorine as
the only possible halogen, T,(C1l) = 0.003% barn at 0.2 Mev.

The fused salt, uranium chloride, was chosen on the
basis of metallurgical, engineering and nuclear considerations
to be the subject of the fast reactor investigations by this
project. The corrosion problem remains as its most detri-
mental feature. To lower operating temperatures in the fused
salt, Pb Cl, and NaCl were added to the UCly. The optimum
ratio of U-238/U~-235 was then determined by minimizing the
estimated production cost of plutonium in this system, in-
cluding the inventory charges on external holdup.

Internal or external cooling is a major consideration in
the design of a fast reactor. Externally cooled systems have
the advantages of safety and replaceablility of heat exchangers,
and absence of parasitic loss of neutrons to cooling flulds
and internal structural materials. Internally cooled systems
have the important advantage of lower critical mass due to
absence of non-reacting inventory in external heat exchangers.
' For the fused salt system selected for detailed investigation,
an engineering study suggested that the externally cooled system
would be preferable. For a liquid-metal reactor, it is
probable that an internally cooled system would be preferred.

 
-15-

 

For fast reactors the highest possible Conversion
Ratios (and Breeding Ratios) occur for U-238/U-235 ratios
of ~%—§% equal to or less than 2.5. Essentially, the
internal conversion ratio depends only upon the ratio
”%_?% implying that the ratio‘ggég%% is a constant over
the reglon between 0.1 and 1 Mev., The upper limit of
g 38 is about 125 higher ratios will lead to either in-
finite critical mass or intermediate to thermal spectra,
and the systems are no longer considered to be fast re-
actors. For externally cooled reactors, it turns out that
slightly higher ratios of N 28 y 1loe. 3 to 5, will give the
minimum overall cost, even though some conversion ratio
has been sacrificed, because the higher dilution with U-238
decreases the holdup mass of U=235, |

Internally cooled systems will contain structural
material (Fe, say) and coolant (Na, say) to remove the
heat. Admittedly, external holdup is decreased, but with
a sacrifice in C.R. due to parasitic absorption in the
above materials.

The high densities of liquid metals imply small
critical masses of fissionable material, but most known
uranium alloys involve high operating temperatures when
used in the fluid form. |

Captures in fuel material and reactor poisons reduce .
the conversion ratio to about 1.15 in a practical fast con-
verter. The relatively low capture cross-sections of the
fission products and the feaslbility of removing them by
processing make the poisoning effects of fission products
negligible in fluid-fuel fast reactors. . In fact, large
fractional burnup - and hence large buildup of fission
products - may be tolerated in fast reactor designs. This
can be seen quite readily from Eq. (1.2-1). . When

 
 

-16~-

N(F.P.) = N(25), i.e. 33 per cent burnup, qE(F.P.) should

be less than 0.24% barn if the decrease in C.R. is to be

less than 10%, whereas, in fact, the estimated average value
UE(F'P') is 0.2 barn at 0.2 Mev.

U-236, formed by U-235 parasitic capture, is a trouble-
some poison in fast converters. Unfortunately, aé(U-236)
is not known in the fast region. It was assumed to be the
same as U-238 in all NEP calculations, i.e. g (U-236) = 0.22
barn at 0.2 Mev. On this basis, U-236 is about equivalent
to the average F.P., atom for atom. However, because it
1s isotopic with the primary fuel, U-235, the removal of
accumulated U-236 and its subsequent separation, presumably
by gaseous diffusion, would be quite expensive. Thus re-
processing costs dictate the upper limit, and the loss in
conversion ratio by parasitic capture in U-236 sets a lower
limit on the rate of reprocessing the fuel for U-236,

For Pu Eroducing reactors, one should note that
the ratio %%%52% 1s smaller in the fast region than in
the thermal. Therefore, Pu-240 contamination of the product
Pu-239 is of less importance in a fast converter than in a
thermal converter.

The fast converter structure studied consists of a semi-
spherical core, reflector, and blanket. The reflector is used
to reduce the inventory of critical mass. In general, the re=-
flector should be a dense material of high transport cross-
section and low capture cross-section, such as lead. Iron
could be used as a combination container-reflector provided
it were not too thick. Reflectors should not be designed
such that they more than replace fissionable material, for
large thickness of iron (or, to a lesser extent, lead)
seriously reduce the conversion ratio by decreasing the
leakage flux reaching the outer blanket,

 

<,
-17-

 

The blanket structure should be sufficiently thick
that it does not allow an appreciable number of neutrons
to leak out of it. Moreover, the blanket should completely
cover the core so that no leakage gaps are present. This
last requirement is difficult to achieve in practice,
since cooling pipes and control mechanisms must be in-
serted through the reactor. It is obvious that the blanket
should be processed often enough to keep the inventory of
Pu-239 to a minimum. It is well to keep in mind that the
fast fission effect in U-238 approximately balances the
parasitic captures in structural materials in a well de-
signed blanket.

Constructional problems may dictate that a suitable
nuclear reflector is impractical. It should then be
remembered that the blanket, the containing structural
materials (Fe), and the reflector control rods (Pb) will act
to some extent as an effective reflector if properly de-
signed.

As the generation time (<10~
reactor is much shorter than in a thermal reactor, fast
reactors are inherently harder to control. Moreocever, in
externally cooled fast reactors with their rapid fuel flow
rates there can be up to a 60% loss of delayed neutrons in
the heat exchanger, depending upon the relative fuel transit
times. Flinid fuels will however have large negative
temperature coefficients due to thermal expansion giving a
distinct safety advantage. The use of absorbers for
control mechanisms is not practical, as absorption cross-
sections are low in the fast region and because the addition
of absorbers will decrease the conversion ratio. Reflector
control is the most efficient method of controlling fast

6 seconds) in a fast

reactors, because its use will not significantly destroy

 
18-

 

the conversion ratio.

Fluld fuels have the additional feature that they
are not as sensitive as solid fuels to radiation damage.
Processing problems are also greatly reduced, enabling
the concentrations of fission products and Pu to be kept
small.

In general, as diluent materials are added to the
composition of the non-aqueous fluid-fuel reactor core, the
critical mass at first rises quite slowly since the diluent
merely replaces fissionable material (lower leakage loss and
number density of U-235 compensate for fuel dilution). The
neutron spectrum is rapidly degraded to a mean energy of
about 200 Kev. As more diluents are added, the neutron loss
by parasitic capture more than compensates for the improved
transport cross-section (less leakage). The neutron spectrum
falls into the 100 Kev region, or less, and the critical mass
Increases rapldly - in many cases it becomes infinite. As
further dilution of fissionable material 1s brought about,
especially with moderating material, the critical mass of the
reactor may again become finite as the neutron spectrum is
degraded to thermal energies, for in this region the fission
eross-section of U~-235 increases rapidly with decreasing
energy. Further dilution with good moderators causes the
critical mass to go through a minifium, until the uranium is
so diluted that the loss of neutrons to capture and leakage
is not balanced by the neutron production from U-235. The
critical mass again goes to infinity even though the
microscbpic fission cross-section 1s very largé in the
thermal region.

It is evident that only for particular combinations
-19-

 

of enriched fuel, moderator and other diluents may a reactor
become critical in the thermal region.

One notes that the critical size of a reactor depends
upon the buckling and atom density of the fuel material,
and that the critical mass is inversely proportional to the
square of the U=-235 atom density. As the conversion ratio
is decreased by parasitic capture, the best converters have
the following general design specifications.

1). small parasitic capture

2). large macroscopic transport cross-section
3). large U-235 atom density

4). high specific power

Items 1, 2 and 3 imply small critical mass (large buckling)
and item 1 implies large conversion ratio. To bring in the
last, important general design parameter for Pu-producing
reactors, we note that the inventory charges on the final
product are inversely proportional to specific power. To
reduce the effect of inventory on the cost of Pu-239, we
~add item 4 to our list,

Recommendations for future work involving nuclear
data which are important in fast reactors are to be found
in detail in Section II-1.9, The major deficlency in data
is to be found in the fast inelastic and capture cross-
sections. |

Paralleling this lack of information are a number of
engineering deficienciles which are to be found in detail in
the engineering analysis report. The major reason why
these deficiencies are important nuclearwise 1s that they
set an upper limit on the specific power of a fast reactor.

Until these deficlencles are removed by further basic
research, we can not be certain of the feasibility of fast,
non-aqueous, fluid-fuel reactors for the economical pro-

duction of Pu-239. | -
s TS
. e, "

-

-

4“("‘_'
=20~
M

 

l.3 TIHERMAL CONVERTERS

Preliminary analysis of the U-Bi system showed that
this was the best liquid alloy fuel for thermal reactors and
that a reactor using this fuel would have an acceptable con-
version ratio and critical mass. Since a fused salt system
had been chosen for the fast reactor, it was decided to
study the U-Bi system for a thermal reactor, in order that
l1iquid metals could also be investigated by the Project.

A limited amount of study was also given to fused salts
for thermal reactors, with the conclusion that solutions of
UF), in fused fluorides were the only ones competitive with
the U-Bi fuel chosen for detailed investigation. Fluoride
fuels merit further study. Of other fused salt possibilities,
ci, Se, Te, Bry, I, N and CN were eliminated because of un-
favorable cross-sectionss carbides, oxldes, sulfides,
silicides and arsenides because of high melting point;
phosphates, sulfates and nitrates because of poor thermal
and radiation stability. A more detailed report of the
search for fused-salt mixtures for thermal reactors 1s des-
cribed in the engineering analysls report.

The particular U-Bi thermal reactor studied consists
of a core in which U-Bi solution flows through holes in a Be
matrix. Some of the special problems which must be con-
sidered in choosing a particular design are conversion ratio,
critical mass and inventory, processing rates, internal vs.
external cooling, uniform or variable Bi/Be ratio, and
control.

The maximum possible C.R. in a thermal converter
employing U-235 is 1.10. This maximum is reduced by para-
sitic capture and leakage. A rough and ready criterion to
test whether parasitic capture by the basic constituents of
the reactor is excessive is

N(s)ag(s) N(s)gz(s)
N(25)0;(25) * N(DRoL(25) <.1, (1.3-1)

e
Z:: ————-_.. -

€
-21-

 

where N(s)<S(s) implies a summation over s, the elements
other than uranium. For the reactor design chosen, equation
(1.3-1) ylelds a parasitic loss of .08 for the Bi and Be
alone. To this value must be added the loss due to polsons.

The problem of poisons in the U-Bi-Be system studied
divides into three parts. Two of these, the formation of
fission products and of higher isotopes are quite general,
since they are directly tied to the presence of fissionable
material. The third, specific to the use of Be as moderator,
i1s the formation of Li~ as a consequence of an (n,a) reaction.
A fourth possible source of poisoning, absorption by the
products of the neutron reactions of Bl was not considered
because of the complete absence of information on the cross-
sections of its decay products.

The concentration of poisohs which are formed directly
in the liquid fuel can be kept to admissible values by
sufficiently rapid processing. In this reactor, the
Pu-239/U-238 ratio was determined by the requirement that
the Pu-240 content of the product Pu be kept below the
maximum allowable limit,

The fission product poisons where concentrations must
be kept low are Xe and Sm which have anomalously large
thermal absorption cross-sectionsj these are processed
separately. The loss in C.R. due to fission product capture
in the final reactor is .,017, and that due to higher
isotope capture .0Ol.

The loss of neutrons resulting from the high energy
neutron reactions of Be forms a particularly knotty problem.
Though the direct loss resulting from the (n,a) reaction is
compensated by an (n,2n) reaction, the 1i® which is a decay
product of the former reaction 1s lodged directly in the Be
matrix and is therefore unremovable by processing of the
1iquid fuel., The loss in C.R. resulting from the high
thermal capture of Li~ turns out to be .02, based on the

 

- ¥
&
22~

 

-

assumption of a six-year operating period for the Be.

The leakage of neutrons from the reactor constitutes as
serious a loss as parasitic capture. Of course these neutrons
could be all productively captured if one were willing to
consider the added complexity, processwise and engineering-
wise, of a substantially pure U-blanket. In the design of
the thermal converter this added complexity was ruled out a
priori by the requirement of structural and processing
simplicity. It might be pointed out here, however, that if
one attempted to design a two-region thermal converter, one
would undoubtedly move away from the type of design in which
one attempts to minimize the leakage, as required for a bare
reactor. External conversion would become more important.
Maximization of the leakage would effect a substantial saving
in eritical mass since the requirements of minimizing leakage
and critical mass are directly opposed.

An attempt was made, within the restrictions imposed by
the requirement of a single region reactor, to do something
about reclaiming the fast leakage from a bare reactor. The
basic strategem consists of having a specified zone on the
outside of the reactor with radically reduced spatial density
of Be, but with the same fuel mixture coursing through the Be
matrix. The basic idea 1s to provide a low multiplication
region of high resonance capture for the fast leakage and
thus to increase the C.R. There are two attendant facts
which nullify much of the gain thus achieved: first, that
the outer zone is one of higher spatial density of fuel and
one therefore pays in critical mass, and second, that the
outer zone has pérforce poor moderating power and that there-
fore a substantial fraction of the neutrons produced in this
region will leak out as fast neutrons. Calculations show
that there is still a net gain in C.R. compared to the homo-
geneous reactor with same critical mass, but that the saving
is not as substantial as initially conceived. Some further

 
’

-23=

 

benefit might be achleved by including a reflector of fast
neutrons on the outside of the reactor, but this possibility
was not investigated.

The question of internal vs. external cooling was
determined on the same basis as for the fast reactor; namely,
the criteria of reliability and replaceability dictated the
choice of external cooling. This decision had the effect of
essentially doubling the 1nventory of fissionable materisl,
since the external holdup is the same order of magnitude as
the critical mass. On the grounds of costs alone, engineer-
ing studies seem to show that there is little to choose
between the two methods, since the loss in C.R. due to the
inclusion of structural materials required for internal cool-
ing is balanced by the gain from decreased inventory.

Fianlly, the problem of control of the thermal con-
verter 1s not nearly as serious a one as for the fast reactor.
In particular, calculations of delayed neutron effectiveness
show that the value of the dollar is decreased by less than
10% of its no-flow value compared with the almost 50% loss
for the fast reactor.

 

 
-24-

B.R.

C.R.

F.P.
N(a)

 

as

GLOSSARY - CHAPTER I

Breeding Ratiog gross atoms of Pu-239 produced per
atom of Pu-239 destroyed (No U-235 fuel)

Conversion Ratioj gross atoms of Pu-239 produced
per atoms of U-235 and Pu-239 destroyed

Fission Products

Number of atoms of type a per cm3

- N(2
Atomic enrichment fraction = Fr5EyiRT5

Ratio of the number of parasitic neutron captures
to the number of neutron captures producing a
fission in fuel material, i.e.,

N@Mfé(a)
Nio s (Fuel)

for any element of type a, where the fuel is
U-235 or Pu-239

Macroscopic cross-section; the subscripts on
are defined as: tr = transporty a = absorption
(fission + capture)s s = elastic scattering;

i = inelastic scattering; f = fission

Microscopic cross-section (per atom)s; subscripts
same as for

 
-25-

 

IT. FAST REACTORS

1. NUCLEAR CONSTANTS

This chapter summarizes the calculations on fast re-
actors. Prior to presenting the results, an outline is given
of the calculation methods. Important details are included
in appendices. Since the reliability of the results depends
directly on the nuclear constants, we begin with a brief
resumé of the values used in the NEP calculations.

1.1 TOTAL CROSS-SECTIONS

Extensive data are available on total cross-sections,
o’y because of the relative simplicity of such measurements.
Total cross-sections do not enter reactor calculations per se
but are of value in establishing upper limits for a) transport
cross-sections o, for fast neutrons and, b) absorption cross-
sections Ty for slow neutrons. Total cross-sections also
provide some basis for the theoretical interpretation of
neutron interactions. In some cases theory has been used to
extend available experimental data and to estimate unmeasured
nuclear constants. ;

A very complete compilation of neutron cross-sections
(AECU-2040) has been published (August 1952) by the AEC
Neutron Cross-Section Advisory Group. A classified supple-
ment (BNL-170) summarizes the data available for heavy ele-
ments. We are indebted to the chairman, Dr. D. J. Hughes,
for prepublication copies of these reports.

1.2 TRANSPORT CROSS-SECTIONS

 

The transport cross-section, which determines the net
loss of forward momentum of a neutron, is defined as:

o =6 - o5 (0) oo dn (1.2-1)

 
tttttt

......

.......

.....

-----

......

......

......

e ;
(OB

 

PAS

TABLE II-1.2-1, Transport Cross—Sections

RN
MRS

3.75-k.5

 

These cross-sections (in barns) used in all NEP fast reactor calculations

 

)"‘o 5-50 5

!
N
o

t

 

 

 

 

" Fe

 

u e5=-1 1-2 2=3 3-3.75 5¢5=7 7-10
Group 1 2 3 L 5 6 7 8
Pu-239 5.0 5.0 6.0 7,0 8.5 10.0 12,5 18
U-238 %.0 4, 1% .86 6.27 8.0 10.6 13.8 14,6
U=-235 4,52 5.17 6.3 7.48 8.61 10. 24 12.6 17.8
Bi 3.7 4.0 4,7 6.0 8.7 11.5 12.5 13.0
Pb 3.7 4,0 4.7 6.0 8.7 11.5 12.5 13.0
F.P. 3.35 3.7 4.35 4,92 5.35 5.75 6.3 7.12

‘ 2.1 2.09 2,29 3.4 3.48 3.92 4.18 11.2
cl 1.8 2.0 2.8 3.5 3.7 3.9 4,2 8.2
Al 1.752 2.017  2.852 3.851 3.956 20911 1.118 1.496
Na 1.595  1.818  3.426  3.605 3.861 30922 4,134 5,222
Be 1,028 1.534% 3.045 3.755 4,272 4.936 5.181 5.42

These values obtained from KAPL except for the following: Pu-239, Bl and Cl extrapola-
tion and interpolation of KAPL values using Feshbach-Weisskopf theory (NY0-636) as
guide, F.P., average values obtained from theoretical curves (NY0-636) weighted by
fission yield (Steinberg and Freedman)
-27-

 

where a@(@)d!L is the scattering cross-section into the
solid angle dQ@ in direction 6. Clearly Tir < 9%°

Transport cross-sections enter all reactor calculations
significantly. Measurements of Tip have been made for only
a few of the elements of interest. Most of these data were
obtained some time ago when experimental techniques were pre-
sumably less reliable. Accordingly, a consistent set of
transport cross-sections can only be obtained by a Jjudicious
combination of theory and limited experimental data. The
cross-sections used in all NEP fast reactor calculations
were compiled by the group at KAPL under Dr. H. Hurwitz.
Based on additional information gathered by the NEP, it
suggests that some revisions in certain of the KAPL cross-
sections may be in order. However, the KAPL cross-sections
possess the virtue of having been consistent with a large
number of critical mass measurements. Hence, we concluded
that on the whole it would be prudent to use the complete
set of KAPL values rather than make indivlidual revisions.

The transport cross-sections used in all NEP fast re-
actor calculations are given in Table II-l.2-1l. The values for
Pu-239, Bi, F.P. (fission products) and Cl are NEP estimates.
The other values were obtained from KAPL.

1.3 INELASTIC CROSS-SECTIONS

In nearly all the NEP fast reactor caleculations, in-
elastic scattering is the predominant process in degrading
neutrons from above to below 0.5 Mev. Below this energy,
i.e. u>4, elastic scattering is the only process for de-
grading the neutron energy. The inelastic cross-sections
used are given in Table II-1l.3-1l. The yield of inelastically
scattered neutrons in other groups due to inelastic scatter-
ing in a given group is presented in the calculation sheet,
Appendix A, More generally the assumed energy distributions
of inelastically scattered neutrons as obtained from KAPL are
summarized in Table II~l.3-2.

&
*
.y -
an ee s i» . . s 4% § 448 4 sus e
- ¥ ’ & & o . * » * * o * » -

. es » " . . a aw .
. o . . . *ne . . % » s ¢
s & . | . e

-
«928-

 

TABLE ITI-1.3~1l. TInelastic Cross-Sections

These cross-sections (in barns) used in all NEP fast reactor
calculations

 

 

 

Group 1 2 3 L
Pu-239 7 o7 | o5

U-238 2.5 2.5 2.1 | .85
U=-235 1.2 1.2 9

Bi .8 .7 L

Pb .95 .88 oM .0
F.P. .4 78 31

Fe .8375 .7875 .3125 .0

* Negligible above 4th group

The values for U-238 are based on a re-analysis of the Snell
and similar experiments (see KAPL=-741).

.9
*
"RENS s
. .
-
- . -
a®ts
.
L] *
"
TN .
.
- -
- .
e
- » ’
. -
»
*ee ‘
- » ‘
L R ]
]
- ®
. .
-
*e e
»
» *
yee
\\\\\\

ooooo

.......

iiiii

 

N 7

——

TABLE II-1.3-2. Assumed Spectral Distribution of Inelastically Scattered Neutrons

The fractional yield(X
inelastic s

1

. =¥

é£in each of the energy groups (2-6) is shown due to

ttering in a given higher energy group.

 

 

 

 

 

 

 

 

Group| Range U-238 and U-236|Pu-239 and U-235|Fission Products and Fe|Pb and Bi|Cl
N ifl 123wl 2 | 3 1 2 3 1 12{3]1
2 1-2 A5 - - - }.10 - - L - - M| -] -
3 2-3 «351.35| =~ - 1.20 .20 - M .6 - 31.6] =
4 [3-3.75 [.2%].33].50 .30 | .33 .51 |. Rn 1.0 2.3 ]9
5 13.75-4.5].16|.22].3%|.80].20 22| W3 | - - - 1l.1.1
6 4,5-5.5 |.10{.10{.15|.20].20 25| .15 | - - - -] =] =

 

 

 

 

 

 

 

 

 

 

These data, except for Cl, obtained from KAPL. There is considerable uncertainty in
all values, since so few experimental measurements ofjxiflj have been made.

-68—
  

30 %.;\

1.4 TFISSION CROSS-SECTIONS

The fission cross-sections ar used are listed in

Table II-1l.4-1. These values are based on recent repeat -
measurements at Los Alamos which indicate a reduction of
about 10 per cent is necessary in the fast region. For ‘
example, @y (Pu-239) has been reduced from 1.9% to 1.75 .
barns, which is of considerable practical importance, since
it results in a significant reduction in estimated breeding
ratios. Similarly, the reduction in the accepted value of
0} (U-235) lowers practically attainable conversion ratios
in the production of plutonium.

This revision in g7 (Pu-239) has not been made in
BNL-170. Apparently the revision in g (U-235) has been
made in this recent publication. It is uncertain whether
the revision has been made in G} (U-238) and 0 (U-236).
The values for<jf(238) as used by NEP were obtained from
KAPL. The value, g7 (U-238) = 0.5805 for group 1 is about
10 per cent lower than the average taken from the curve in
BNL-170 for this group, indicating that the correction has ,
not been applied. KAPL had no values for gz (U-236), hence
the values used were taken from BNL-170 but without the
10 per cent revision. Because the concentration of U-236 -
1s less than that of other fissionable materials in all
of the reactors considered by NEP, this possible correction
in g7 (U=-236) would result in a negligible change in all
NEP calculations.

—

E

-
eeeee

. khsea

ooooo

ttttt

------

------

nnnnnn

111111

TABLE II""lo l‘""lo

3|

Fission Cross-Sections

These cross-sections (in barns) used in all NEP fast reactor calculations

 

 

 

 

 

 

 

 

 

 

 

u .5-1 1-2 2-3 3-3.75 | 3.75-4%.5 | &.5-5.5| 5.5-7 | 7-10
Group 1 2 3 b 5 6 7 8
Pu-239 | 1.75 1.75 1.75 1.75 1.75 1.79 1.95 6
U-238 . 5805 .3916 .0171 - - - - -
U-236 .86 - . 5% .1 - - - - -
U-235 1.2 1.2 1.22 1.43 1.7% 2.1 2.82 6.5%

»

Values for Pu-=239 from BNL—170 with 104 downward revision (see text). Values

for U-235 and U-238 from KAPL.

Values for U-236 from BNL-l?O.

 

1
o
it

‘
32~

 

1.5 CAPTURE CROSS-SECTIONS

It is evident from Table II-1.5-1 that the capture
cross-sections of the fissionable elements and of the fission
products predominate over those of the other elements present
except for unusually high concentrations of diluents. The
rapid increase in capture cross-section with decreasing -
energy for both U~235 and Pu-239 imposes serious limitations '
on the minimal energy allowable in the neutron spectrum in
fast breeders and converters. In general, fast reactors de-
signed primarily for the production of plutonium, have too
low breeding or conversion ratios to be considered practicable
if the neutron spectrum has an average value below ~ O.l1 Mev,

i.e. u > 5.

The capture cross-section of fissionable nueclei is
generally expressed as the ratio a = 6:/ G';.. Unfortunately,
the data on a in the fast region are so sparse and so un-
certain that only rough estimates can be made of the varia-
tion with energy. Unpublished values of a(Pu) obtained from
KAPL are:

En(kev): 0.15 l.2 3. 10. 122.

a (Pu) 0.7%0.1 0.6%0.15 0.52%0.17 .43%.09  o.1%0.1

9

For want of any better data, a smooth curve was drawn through
the mean values and extrapolated to a = .02 at 10 Mev. The
values of 0‘5(Pu—239) for the 8 energy groups listed in Table
I1-1.5-1 were obtained from this very approximate curve.
Similarly the values of O';(U-235) were obtained from
KAPL's unpublished curve of a(U-235) as a function of energy.

tered neutrons with a broad energy spread of mean value
15 kev.
 

-33-

 

Roughly the same uncertainty exists in «(U-235) as in
a (Pu-239).

Recent information from BNL and KAPL suggests that
the values of G‘é(Bi) given in Table II-1l.5-1 should be re-
vised. Measurements with the pile oscillator at ANL and by
activation using thermal neutrons have established that
Bizlo formed by B1209(n,Y) decays both by B and a emlssion
with a branching ratio of about 50 per cent. Hence the
single experimental value of 03 (Bi) = .003% barn at 1 Mev
(group 3) as measured by Hughes from induced § activity
should be doubled. Since the values for groups 1 and 2 are
based on Hughes value at 1 Mev, it might be concluded that
these should also be doubled. However, some unpublished ex-
periments at KAPL suggest that 07 (Bi), unlike its non-magic
neighboring nuclei, is essentially constant over the eight
energy groups. Transmission measurements at KAPL indicate
no detectable variation.of‘O;(Pb) with neutron energy.
Other experiments indicate no significant difference in this
respect between Pb and Bi. If these conclusions are correct
JZ(B1)2=i.OO7 barn for each of the 8 energy groups rather
than increasing with decreasing neutron energy as assumed in
Table II-1.5-1. Corrections of the fast reactor calculations
for Bi systems can be accomplished easily by reference to the
detailed balance sheets of Section II-3.2. However, an
estimate has been made based on the one-group method which
indicates that the correction in the conversion ratio is
generally less than 104. Table II-1.5-2 presents the effects
of changing the capture cross-section of Bi on the conversion
ratio for System 12.

 
¥

------

aaaaaa

oooooo

------

lllll

sssss

......

------

------

.

 

>4
TABLE II-1.5-1. Capture Cross=-Sections
These cross-sections (in barns) used in all NEP fast reactor calculations

 

u .5-1 1-2 2=3 3=3.75  3.75-k.5 %.5-5.5 5.5-7 7-10

 

 

Group 1 2 3 4 5 6 7 8
Pu-239 .05 .07 .088 .1k .23 M1 L7 3.3
U-238 015 .05 .13 .16 .22 32 .52 1.0
U-235 .065 .065 .075 .12 . 207 375 .726 2.k%

Bi .0025 .003 .003% .007 .009 015 .03 .07
Pb .0025 .0025 .0025 .0025 .0025 .0025 .0025 .0025
F.P. . 002 .006 .052 .125 .195 27 .38 . 576
Fe .001 .001 .0013 .0039 . 0062 . 0097 .01 .013
cl .0006 .0007 .001 .0018 .003k4 .0063 .0085 .0113
Al . 0004 . 000k . 000% . 000k% . 000k « 0004 LOoo0F  ,00092
Na .00027  .00027 .00027 .00027  .00027 .00028 00049  ..02211

For Pu~-239 and U-235 see text.
Values for U-238 from BNL-170, p. 35.

Valves for Cl and Bi based on single measurements by Hughes at 1 Mev extrapolated
to other energies with average slope obtained from measured values for nelghbor-
ing elements.

Estimates for F.P. based on yields and measured values when avallable. Considera-
tion given to low values for magic numbered isotopes; however, energy variation
assumed to be the same as for non-magic.

All other cross-sections from XAPL.

G (Pb) probably should rise with decreasing energy although magicity (2=82) for
all isotopes and (N=146) for Pb~208 may cause 0~ to remain essentially constant
as assumed by KAPL.

-vg-

 

 
 

TABLE II-1.5=-2

-35-

Effect of Changing 67 (Bi) on C.R. in U-Bi Systemss One-Group
Calculation, System 123 U-235:U-230sBiz1l:3s

 

 

 

 

Assumed value of
a_é(Bi) 1n fifth u group IQCOR. XOCOR. T.C.R.
.009 barns «339 .683 1,022
.00’+ barns 0339 0786 1.125

 

 

 
-36-

 

1.6 SCATTERING CROSS-SECTIONS

Since the scattering cross-sections 05 always enter
the calculations in the form &7, where § = the mean
logarithmic energy loss, this product, called the degradation,
is given for each of the elements in each energy group in
Table II-1.6-1. & is calculated from the atomic weight in
the usual manner. The values of Eflg are conslidered quite
reliable, except, possibly,., for that of Cl1 (which is the
principal source of elastic degradation unfortunately in the

fused salt reactor).
......

s e e

“eszew

.....

tttttt

------

TemeaN

......

TABLE II~1.6-1. Degradation: Product of Scattering Cross-Section and Mean
Logarithmic Energy Decrement EEO‘Sl: |

These values used in all NEP fast reactor calculations

 

 

sarrerat — g —
R—— M——

u .5-1 1-2 2-3 3-3.75 3.75-4.5 %.5-5.5 5e5=7 7-10

|

 

 

 

Group 1 2 3 L 5 6 7 8
Pu-239 .037 .037 . 046 .061 .071 .078 .08k .089
U-238 - .037 .037 .Ol6 .061 .071 .078 . 084 .089
U-235 . 037 .037 .O46 .061 .071 .078 .084 .089
Bi .038 .039 . 045 .063 .085 <111 .120 .125
Pb .038 .039 045 .063 .085 <111 .120 .125
F.P. .055 . 081 .106 «120 .122 122 .121 .119
Fe .103 .096 .09 122 122 0137 <147 o
cl »139 o 114 179 «197 .206 .217 234 L55
Al .176 .192 «256 .288 «303 .215 .082 .11
Na . 194 .209 347 .323 346 .351 .369 o5l
Be .213 <320 . 64 .787 2955 1.103 1.16 1.211

These values obtained from KAPL except for the following:
Pu-239 taken same as U-238 since 0t 's are in good agreement between 0.3 to 3.5 Mev
(BNL-170). The absence of data on Cl necessitated an interpolation of the KAPL

values for Na, Al and Fe. The values for F.P., obtained by plotting averaged £ o

for known F,P. versus energy (welghted for yield), averaging and interpolating. S

 

L
w
-

t
-38-

 

1.7 NEUTRON YIELDS

One of the most important parameters in the calcula-
tions of critical mass and conversion ratio 1s the number of
neutrons per fission v. The experimental values used in NEP
calculations were:

2.47
2.97 -

v(U=235)
v (Pu-239)

Assumed values were:

v(U=236) = v(U-238) = 2.50

The latter are more important in fast reactors than in
thermal reactors, since the fast effect (in U-236 and U-238)
contributes from about 8 to 15 per cent of the total number

of fissions in the fast reactors considered.

1.8 NEUTRON SPECTRA . -
, The fission spectrum for all but delayed neutrons
(which are substantially lower in energy(a)) is given in the

following table: R
Group u = 1n(E_/E) . :

1 0.5 = 1.0 .13k

2 1 -2 522

3 2 -3 .295Y%

L 3 - 3.75 .0807

5 3‘75 - L"‘S 00373

These are the KAPL ylelds of fission neutrons in each of the
indicated groups. They have been used for all fissionable

(a) No correction was made for the fact that the delayed .-
neutrons have a lower mean energy than the prompt
neutrons. |

 
   

-39-
v _

 

elements in all NEP calculations. While there 1is probably
some variation in‘J( between different fissionable nuclei,
this difference is probably not sufficient to affect seriously
any of the NEP calculations.

Since only limited attention was given to control
problems by the NEP, accurate values for the half-lives and
yield of delayed neutrons were not essential. The values
quoted by Glasstone and Edlund were used, namely:

Tfi in sec. Fraction By Energy in Mev
0.43 0.0008% 0.42
1.52 0.002% 0.62
4,51 0.0021 0.43
22,0 0.0017 0.56
55.6 0.00026 0.25

1.9 RECOMMENDATIONS

In the course of the NEP studies, the need arose for
new nuclear data and for improvements in existing measure-
ments., This section outlines these needs specifically, and
makes some suggestions regarding methods of satisfying them
in the future.

The fast reactors considered by NEP have neutron spectra
peaked in the range of 100~-300 kev., This energy region is
beyond the reach of velocity selectors and below the fission
spectrum. However, it is conveniently covered by electro-
static generators, for example, via the Li7(p,n) reaction.
By means of such sources the fast cross-sections needed can
probably be obtained most expeditiously. |

Fewer gaps appear in the thermal data. These can be
filled quite readily using pile sources (ANL, ORNL and BNL)
provided qualified personnel can be attracted to these prob-
lems.

 
 

CAPTURE CROSS-SECTIONS. - In the thermal region capture
cross-sections may be measured by transmission methods (large
G:) and by diffusion methods (small G;fi in addition to being
measurable in some cases (both large and small.G;) by activa-
tion methods. However, in the region of primary interest
(100 - 300 kev) for fast reactors, the capture cross~sections
of all elements are so much less than the scattering cross-
sections that transmission and diffusion methods cannot be
used. Hence, nearly all of the measurements in this region
have been made by bombardment of the elements with neutrons
of more or less well-defined energy followed by measurements
on the reaction productss either induced radiocactivity or
isotopic abundance.

These methods have a number of limitations. When
radiocactive isotopes are produced in even atomic-numbered
elements which often have several isotopes, there may be
some uncertainty as to which isotope is activated. 1If this
is not a serious rroblem, it may still happen that the
abundance of the isotope activated may be small and non-
representative of the element as a whole. The half-l1ife of
the radioisotope formed must be convenient for activation and
measurement. The radiations emitted must be sufficiently
energetic to be measured and must be readily distinguished
from radiations (if any) emitted by the target element. Many
elements, especially those of promise in reactor design, have
too low cross-sections to he measured except in large neutron
fluxes. The last limitation is especially critical when an
electrostatic generator is being used as the neutron source,

Despite these difficulties, enough data are available
to allow some fairly reliable general conclusions to be drawn.
As indicated in Section II-1l.5, experiments have established
that 0"; increases rapidly with decreasing energy below 1 Mev
for most elements. It is also well established that the
cross-section increases with increasing atomic weight and
that the magic nuclei (atomic number and/or neutron number

 
-41-~

 

2, 8, 20, 50, 82, 126) have unusually low capture cross-
sections. Some rough measurements at KAPL suggest for Pb
and possibly for Bi that O‘; is not nearly as dependent on
energy as for non-magic nuclei. This hypothesis is of
practical value and should be investigated experimentally.
If this proves to be true, certain of the fission products
will be less absorptive than has heretofore been assumed.
What is more important, the magic nuclei, Pb and Be, which
have good engineering properties, will also be shown to
have unusually good nuclear properties for fast reactors.
Measurements of capture cross-sections in U-235 and
Pu-239, obtained from the U-236 and Pu-240 formed, give only
rough indications of their energy dependence. Danger co-
efficient measurements are only of slightly better reliability.
Improvements in the knowledge of these capture cross-sections
are urgently needed.

CHLORINE. - If fused salt reactors continue to be of
practical interest, further information is needed on the
cross-sections of chlorrne.cr'(Cl) has been measured only up
to 280 ev. CT'(Cl37) = .74 mb has been obtained by Hughes et
al for flSSlon neutrons, (mean energy about 1 Mev). Assum-
ing this value applies to chlorine as a whole (25% C137),
0.74% mb was used as the single experimental point on a curve
whose slope was obtained from a combination of theory and
experiment (TMS-5). Average cross-sections for the other
seven energy groups were obtained from this curve (linear on
semi-log plot). The activation.cross-sectionuGE(Cl37) should
be extended below 1 Mev, and the capture cross-section of Cl
as a whole determined from 0.1 to 1 Mev if possible.

IRON GROUP (Mn, Fe, Co, Ni). - These elements are of
importance as structural materials. Only Fe has been con-
sidered in the NEP calculations. As indicated above, the
values of(Tg(Fe) in the fast region were obtained from KAPL.
These probably were derived by analogy with neighboring
42~

 

nuclel since there appear to be no measured values for Fe in
the literature.

Activation measurements on 0059 and Ni
made only at 1 Mev. Fairly complete data on Mn have been
obtained from .03 to 3 Mev. It would be desirable to have
similar data for Fe, Co and Ni.

64

have been

Pb AND Bi. -~ These two magic nuclei are so important
as possible constituents in fast reactors (coolants, re-
flectors, controls and blanket components) that a special
effort should be made to establish G;(Pb) and G;(Bi) over the
energy range 0.1 to 1 Mev.

U=236. - In order to keep down the reprocessing
costs, it is necessary to allow substantial buildup of U-236
(0.2 atoms U=-236 to 1 atom U-235) to occur before sending
the fluid fuel of the fast converter through the gaseous
diffusion plant for separation of U-235 and U-236. TFor this
reason accurate knowledge of(T;(U-236) is of importance.
In the NZP fast reactor calculations, it was assumed (purely
ad hoc) that U-236 has the same capture cross-section as U-238.
This was necessary since no data on,U;(U-236) in the fast
region appear to be available. Obviously this uncertainty
should be removed by experimental measurements of QZ(U-236).

U-235 AND Pu-239. - Better knowledge of the capture
cross-sections of U-235 and Pu-239 in both the slow and fast
regions is urgently needed. This lack results in serious un=-
certainties in the design of the most economical reactors for
the production of plutonium,

| Not only are these measurements difficult to make,
but the results are difficult to interpret unambiguously.
Weisskopf has reviewed the situation in a preliminary report
(NDA Memo-15B-1) received near the end of the summer (1952).
As he points out, the easiest region to measure is at low

‘. -:*:.. M

v
-43-

   

v
energles where scattering is negligible compared to fission

and capture. Under these conditions a is determined by
measuring the peak values of 7% and Opt

it = 0t
¢ = T )peak

He states, however: "Unfortunately the resolving power of
‘the relevant measurements is not adequate, and the observed
peak value corresponds to the actual one only for the first
few ev. At higher energles, the observed peak value is much
less than the actual one and in the case of the total cross-
section it contains a good deal of the potentlial scattering
taking place between rescnances". Reference 1s made to
KAPL-377 and KAPL-394 for an analysis of the U-235 levels
made at G.E. based upon very unrellable data. While theory
1s of value as a guide, what 1s really needed for the low
energy region is an improvement of resolving power in the
fission as well as the total cross-section measurements.

Two types of measurements have been made at higher
energles - both give only rough indications of the energy
dependence because of the broad spectral distributions of
the neutrons. In the first method (KAPL-183), samples of
U-235 and Pu-239 are enclosed in shields of different thick-
nesses and composition, and are irradiated at Hanford.

o (0-235) is obtained from mass spectrographic measurements

of the U-236 produced and ¢ (Pu-239) from measurement of

the spontaneous fission in the Pu-240 produced. The second
method consists in measurement of the danger coefficlent at
different locations in several reactors. The energy spectra

of the neutrons at these locations are known only approximately.
a can be calculated from the danger coefficients. This method
requires a correction for the thickness of sample irradiated -
a procedure which 1s not very reliable.
-44-

 

Thus it appears that improvement in the knowledge of
the a's in the fast region can be attained by these two
methods only with considerable further experimental effort.
This effort is amply Jjustified by the need for these data.

A possible technique which apparently has not been
tested is to determine a from the prompt gamma rays emitted
during fission and capture. Through the use of scintillation
counters (NaI-T1I) one might be able to distinguish between
the fission and capture gammas since the latter should have a
higher energy component. If this is possible, monoenergetic
neutrons from the Li7(p,n) reaction could be used to irradiate
a ccnical sample of U=-235 or Pu-239 surrounding the scintilla-
tion detector which is shielded from the direct beam by a cone
of boro-paraffin, |

TRANSPORT CROSS-SECTIONS. - Measurements of O, have
been made for only a few of the elements in the NEP calcula-
tions. Fortunately, except for Cl, values of U; are available
and these set upper limits for the values of OL,.. In addition
the theory is fairly reliable and can be used for interpola-
tion between measured elements. It would be of particular
value to have G¢.(Cl) or alternatively (T;(Cl) in the fast
region. This should not be difficult - C Clh might be used

as scatterer. At preseni;G;(Cl) has been measured only to

400 ev. G;r(Bi) has been assumed the same as G:;I.(Pb) (some
experimental data ) O"{r(F.P.) was obtained by interpolation
and (T;r(Pu-239) by extrapolation of the limited experimental
values for other elements. 'hile large differences are not
expected, reliable experimental measurements on the elements
of interest should be obtained as soon as possible.

SCATTERING CROSS-SECTIONS. - Values of Qg can be

- obtained with sufficient reliability from 0. Hence, the
only urgent data needed are for chlorine (see above).

 
-45-

 

FISSION CROSS-SECTIONS. - Except for the 10 per cent
downward revision, which appears to be in process of general
accertance, and the unknown value of S¢(240), the fission
cross-sections used by N.E.P., appear to be of adequate re-
1liability for such preliminary design calculations. For more

refined calculations, more accurate values of S in the region
of 100 to 300 kev wounld be regquired,

INELASTIC CROSS~SECTIONS. - The need for improvement
in inelastic crcss-sections is widely recognized. This has
been glaringly evident for some time in shielding studies
and is becoming of increasing urgency in reactor calculations.
These measurements are unusually difficult to make with
electronic detectors,and easy,but extremely, tedious with
photographic emulsions as detectors of the knock-on protons.

In the NEP calculations the KAPL values of g7 and
:[i*j‘were taken over without critical review even though
it was recognized from the outset that these data are
probably quite uncertain. These cross-sections and the
assumed energy distribution of inelastically scattered
neutrons enter the calculations very significantly in that
they determine to a great extent the average energy of the
neutron spectrum, and therefore the mean effective a and the
percentage of fast fissions.

In addition to adding to the theory of the nucleus,
the c¢ross-sections and energy distributions of inelastically
scattered neutrons by reactor fuels, diluentsyand structural
materials are urgently needed for accurate design calculations
of fast reactors.

NEUTRON YIELDS. - In all NEP calculations it was tacitly
assumed that the v of each fissionable element 1s independent
of energy. While this is very likely, experimental confirma-
tion is desirable. |

It was also assumed that v(U-238) = v(U-236) = 2.50

 
 

46~

and v(U=-235) = 2.47. This important parameter has only been
measured reliably for the last of these three isotopes. The
fact that in fast reactors, U-236 and U-238 can account for
15% of the total number of fissions and that the fast fission
effect in U-238 in the blanket helps to offset the neutron
loss due to parasitic capture in structural materials
emphasizes the importance of accurately determining v of
U-238.

RESUME_OF NUCLEAR DATA NEEDED:

J. Capture cross~sections.

a. Ascertain whether maglc nuclel are less energy
dependent than non-magic. |

b. Improved resclving power 1s required in the fission
and total cross-section measurements in order to
obtain reliable values of the capture cross-sections
of U-235 and Pu-239 up to several hundred electron

~ volts. |

c. Obtain.dz(01) and d;(U-236) as functions of energy
in the fast region.

d. Obtain U”C(Fe), O"C"(Co), O'E(Ni), O'é(Bi) and G’C(Pb)

as functions of energy in the fast region.

2. Transport cross-sections.

a. @3, (or CT;) for chlorine particularly needed in fast
region.

b. G, (B1) Q. (Pu-239) and U of intermediate
elements representative of the fission products
should be obtained.

3. Inelastic cross-sections.

a. Improvement should be made in the experimental
values of the inelastic cross-sections of U-238,
Pb, Bi and Fe.

b. Measurements ofG';_’ (Pu-239), OE’(U-235) and
O'i'_' (Cl) especially in the energy range 0.2 to 1

 
~47-

 

Meve.

c. Additional information is needed on the energy
distribution of inelastically scattered neutrons
even though it is recognized that these measure-
ments are difficult and tedious. |

4, Pission Cross-sections.
a. af(Pu-ZhO) as a function of energy.

The following table checks the most urgently needed
nuclear data in the fast region (100 to 3000 kev)

TABLE II-1.9-1. Resume of Nuclear Data Needed

 

 

Element | v dc Op o4 Gy Jg
U=-235 3* *

U-236 * 3# 3

U-238 #* #*

Bi * 3#* 3* 3#
Pb * * 3 *
Cl * # #* #*
Fe * 3*

Co 3# *

Ni # *

Na *

Pu-239 * * % *
Pu~-240 | & %* * * * 3#
Pu-241 | i * #* * * #*

 

 

 

 

 

 

anik A e SW AEL SIS SRS AU GE A GNS ORE VAN MW WEP SN OME EIG LS GSS AUl WEA WE TMD WN GID 4N GES WML GED I NS S SRR GUr  awm A

 
-48- m |

2. CALCULATION METHODS

2.1 BARE REACTOR MULTIGROUP METHOD

At steady state the rate of removal of neutrons from
each energy group a (leakage + absorption + inelastic scatter-
ing + degradation by elastic scattering) equals the rate of
addition (fission + elastic and inelastic scattering from
higher energy groups). The bare reactor multigroup approxima-
tion to the age equation for this steady state condition can
be written:

—(i—lfh)fl v‘@d + (z“)NQN + (z'i)u ix + (“f':'%)“ §°‘ =
p = oty (2.1=-1)

B=n
('flf)“ 1)‘.J:£F;.)fi @p +(_é_223_) §or -1 +£('x‘)p-£z5)p§9

These and all other symbols used are defined in the glossary
which is appended at the end of this report. In these equa-
tions it is implicitly understood that v, Z and the inelastic
spectra are summed over the appropriate nuclear species.

Since the flux distribution has the forms:

§°ffl= §—ill?-l-c—-1: <P°‘ | (2.1-2)
we can write eq. (2.1-1) as:
£2Z.
{p{“(--g'--z.-w)°t (2D, ¥ (Z). + ",?17’-),,,} P,
B= -1 (201"'3)

() % I (Zf)P‘PO e () et I (X)g e

For the solution of eq. (2.1-3) it is convenient to scale the
system to one source neutron, i.e. let:

i

) J; (,z"')‘* P = | (2.1-4)

 

____-_-.“.:-.:-'
- -49-

 

 

It is also convenient to take the macroscopic cross-sections
as per atom of fuel, i.,e., to divide the macroscopic cross-
sections by the atomic density of fuel. In this way only
atomic ratios enter the calculations. The conversion to

more usual units need then be considered only for the calcula-
tion of the critical mass.

The solution proper is substantially a matter of
detailed bookkeeping of the neutrons. A convenient form,
obtained at G.E., has been used. A typical example is given
in Appendix A together with a step-by-step procedure for
carrying through the calculation.

After a consistent calculation has been completed,
one can obtain the following information for the bare reactor:

a). Internal conversion ratio

b). External conversion ratio

¢). Mean effective ©

d). Critical radius and critical mass

e). Spectrum of fissions, i.e. percentage of
fissions due to neutrons in a given energy
range

f). Spectrum of neutron flux
g). Fraction of fast fissions
h). Detailed neutron balance

 
-50-

 

2.2 MULTIGROUP ESTIMATES FOR BLANKET

The method of calculation described in Sec. 2.1 re-
presents the basic method employed by this Project to obtain
reliable information about neutron spectra and conversion
ratios (See Section II-3) for a reasonable number of
different systems in a severely limited amount of time. The
actual system which consisted of, in addition to the core, a
blanket or a blanket and reflector will have properties which
may differ substantially from those predicted for a bare re-
actor; one can declde this question only on the basis of
multigroup-multiregidn calculations for which there was in-
sufficient time to complete more than two. The methods to
be described in this section may be viewed as complementary
procedures designed to investigate the possible range of
spectra in these outer regions, and the effect on the con-
version ratios, and to provide additional spectra for one-and
two-group calculations.

These methods have been called:
a). Equilibrium blanket (e.b.)

b). Driven blanket (d.b.)
c). Intermediate blanket (i.b.)
d). Strong coupling (s.c.)

We shall describe these methods in turn. In order to
compare results, the numerical calculations were made on the
same core-blanket system (Core: System ¥2 in Table II-3.2-1j
Blanket: UC].)_[_).

a). Equilibrium blanket. - The actual integration of
the core and blanket is such that "fast" leakage neutrons
from the core enter the blanket and are to some extent slowed
down by scattering before being absorbed., Some of these
slower blanket neutrons leak back and influence the core.
Thus, there will be a continuous degradation in the neutron
spectrum as the radial distance increases. However, one can
imagine a point in the blanket beyond which the neutrons
=51~

 

from the core source have practically disappeared and all
the neutrons are in spectral equilibrium; i.e. the spectrum
no longer changes appreciably with radial distance beyond
that point. It is this part of the neutron distribution
that one treats by the Mequilibrium blanket" method.

In a non-miltiplying medium, the spatial dependence
of the principal mode is _g;fr . This introduces into
the age-diffusion equation 2.1-1 the term -% ®atw
instead of +& F,¢) . The rest of the multigroup calcula-
tion is carried through in an exactly analogous fashion (of
course, the quantity }%“. above, being a "leakage source",
is subtracted when computing the quantity Aa)'

Unfortunately, this calculation, when applied to the
UCl, blanket, gave an unreasonably slow spectrum, one which
had its maximum in the seventh group with a very large de-
gradation out of the eighth and last group. The shape of
the spectrum indicated that the neutrons are degraded rapidly
by inelastic scattering through the first four groups into
the fifth and sixth, and that they then lose energy slowly
by elastic collisjons while waiting, as it were, to be
absorbed in groups where high absorption cross sections
prevail.

While the above situation may obtain, it must be true
only when there is speetral equilibriumj however, the actual
neutron density will then be very small. Therefore, the very
slow spectrum obtained by this method cannot be considered
to characterize the blanket realistically.

b). Driven blanket. - The e.b. neutron distribution
may also be conceived as determined by an additional source
term (to be added to Sa) which is proportional to the pre-
vailing neutron spectrum in the blanket. By contrast, we can
add to the source a term proportional to the bare core
(leakage) spectrum. (In analogy to oscillator problems, we
think of the blanket as being "driven" by the core) and drop
the diffusion term altogether. While a "“local" interpreta-
tion of this procedure must be somewhat strained (although
one might imagine a situation approximating this one to pre-
vail near the core-blanket interface), one can obtain exactly
the same equation by integrating equation 2.1-1 over the
entire blanket volume and assuming the net current at the
interface between core and blanket to have the core spectrum.
This in effect assumes a "one-way coupling" from the blanket
to the core, since the incoming neutrons from the core have
the same spectrum as the leakage from the unperturbed bare core.
The arbitrary parameter (equal to the number of externally
supplied neutrons per fission neutron required for steady
state) that multiplies the core source term is to be varied
until the normalization condition on the fission source is
satisfied. The blanket spectrum that results from this
calculation is slower than the core spectrum, but not nearly
as slow as the e.b. spectrum.

c). Intermediate blanket. - The e.b. and d.b. calcula-
tions may be considered extreme cases of a more general
approach. The e.b. method assumes no coupling at all from
core to blanket while the d.b. assumes no "feedback" at all
from the blanket spectrum to the blanket source. The more
general approach is to assume some arbitrary mixing of these
two cases with the proviso that neutron balance be maintained.
This has been done for one case by assuming a weaker core
source than in the 4.b. case and by varying %; until balance
was attained. The spectrum, as one might guess, was inter-
mediate between the e.b. and d.b. spectra. Since there has
been no way found of estimating the appropriate core source
strength, this type of calculation is too indefinite to yleld
useful results.

EEEEQEEEEEEEE§ o
4
el ;
5
e
-

 

-y
d). Strong coupling. - From the point of view of the
effect of the blanket on the core, all of the above approxima-
tions are extreme because they are based upon the assumption
of one-way coupling. At the other extreme is the "strong
coupling" approach, in which one assumes such complete inter-
action between core and blanket that they both have the same
neutron spectrum. Thus we sets

Py ) = Y ¢, (2.2-1)

and ¢, applies to both regions of the reactor. If we inte-
grate eq. (2.1-1) over the entire reactor volume, the
Laplacian term vanishes by Gauss' Theorem with the result:

C (), + B+ ()} 4. = g;.‘;.z;s)_‘%_,

- (2.2-2)

p=n p= ot ~
+ (X ? J: (Z)g o *+ L‘""’e-w‘z"’? Y,

where

2 = I‘Z(core) + Z(blanket)

L = I'Szg(core) + Sfls(blanket)

and — s *(") Avc’ore
- s '\’ (n dvblane‘f

To perform the computation, one chooses values for I,
and completes multigroup calculations, similar to those done
for the bare pile, until a consistent solution I is found
which satisfies the normalization condition on the total
source strength. With this method, the spectrum is between
the bare-pile and e.b. spectra, as one expects.

The breeding ratio is calculated by dividing the total
captures in 28 by the total absorptions in 49:
B.R., = J:‘ (Zen), 9a
P (Z, 40 9

Unfortunately, the result of this calculatlon in the
case considered was so much smaller(a) than the bare plle
result, that strong doubt is cast on the validity of the s.c.
method. The unfavorable result is due to the excessive de=-
gradation of the core spectrum which in turn increases E,
the mean capture-to-fission ratio of 49.

The disparate results obtained using the various
approximations-by-extreme indicate that the spectral varia-
tion with radial distance may be too complicated and too
strong to be computed by a smeared-out approximation.
Evidently one must do a multigroup-multiregion calculation
to obtain dependable results and to decide which one, if any,
of these pictures can be trusted.

(2.2-3)

¢). Other multigroup methods. - Reports available to
NEP outlining multigroup methods distinct from the KAPL
method used by us were: 1) LA-1391 (B. Carlson - Serber-
Wilson method)s and 2) R-233 and RM-852 (G. Safonov -~ his own
method) .
1). The Serber-Wilson method offers the possibility
of performing a multigroup-multiregion calculation
in a reasonable time by hand, but it is still
somewhat too complicated to be used on this project.
To achleve the same spectral accuracy as our bare-
pile calculations, it would be necessary to solve
two or three eighth order transcendental de-
terminantal equations. However, this is a
distinct advantage over having to solve a 16 or 32
order transcendental determinant in the "exact™®

 
2).

-55-

 

multigroup method. Moreover, the Serber-Wilson
method starts with the transport equation rather
than the diffusion approximation.

The Safonov method also starts from the transport
equation, but 1s far too complicated to be
feasible for this Project, since it involves step-
wise integration over a lattice in both configura-
tion and energy space and is in this respect a
more elaborate version of the KAPL multigroup-
multiregion method.

1
f.1
=
2.3 IWO-GROUP TWO-REGION METHOD

INTRODUCTION., -~ Some two-group two-region calcula=-

tions were performed for one or more of the following reasons:

(a) To determine reflector savings (for either
reflector or blanket).

(b) To determine loss in conversion ratio due to
presence of reflector.

(¢) To ascertain to what extent conversion ratios
were fixed by the cholce of average cross-
sections for core and blanket regions, 1i.e.
to determine what adjustment was provided by
the required satisfaction of the boundary
conditionsy to compare these results with those
obtained by the various types of blanket
calculations described in Section 2.2.

The dividing line in energy between the groups is most
conveniently chosen as the threshold of the fast fission
effect in U-238. This subsumes the first three groups of the
previous multigroup division under the present label of "“fast"
group. The remaining five groups constitute the "slow" group.
Cross-sections were chosen by averaging over the multlgroup
neutron spectrum when such was available for the system con-
sidered. The resulting eguations were solved exactly rather
than by the KAPL self-consistent method, since the former
turns out to be numerically less tedious in this special
case.

Since two-group fast reactor equations differ in
content from the better-known two-group formulation of the
thermal problem, it may be worthwhile to consider their
derivation briefly. This derivation could be carried out
directly from the age-diffusion equation. We prefer, how-
ever, to proceed from the multigroup equations, (2.1-1), as
a more realistic basis, since the accuracy of the two group
treatment 1s predicated on the availability of multigroup
spectra which are used in the derivation of average cross-

- *,
— &
=]

- 4
g
i
!
M -«
T A .
* ass & sa s .
* . . » .« 8 .
+ a e ¢8s = - - * ®
* 3 . o - tew
. . 0 .« o 9 . s @
.

P
-57-

 

sections.

DERIVATION OF EQUATIONS. - Let the subscript 1 refer
to the "fast" group and the subscript 2 to the "slow" group.
If we now sum eqs. (2.1-1) from a = 1 to a = n = 3 and from
cz=n+ltoa=n-= 8, we obtain the two equations

~(35) Ve + (203« (2)E + )&
- (2.3-1)

260, {E)E + EVE] + §§ ), (5

and

(35). V& + (2.8 +(zi),§,. = (£&) 8,
- n (2.3-2)
+ VEA{ENEEEf ] S () (L) d

where, for example,

h
_(Z), 2

(Zfl)| = JQ&—&‘—L_&__ (203-3)
Ja:l ip

is a number derived from a bare pile multigroup calculation,

or guessed. The other average cross-sections are similarly
defined, except that

£ _ *
(Es) - Aguk s - 3fi“" (2.3-1)
v £=t ifi §='§@

In the summation over the finer group divisions, it 1s
clear that only the degradation appropriate to the gross
division into two groups survives. A similar expectation
leads one to rewrite the inelastic terms. Consider first
the summation

S::l S,Ps GH(ZL) (Z )9 i@ - f(::l Lzfil(ri)fi"“ (Z;)P%

¥
-58-

 

{ B=1 I p+\— f§=| S’: %Uzap*fl' (Zi)a

C(Z)E - § 0, (2), 8, =GB, - @,

In deriving eq. (2.3-5), we have in turn interchanged the

é
& (2.3-5)

order of summation, added and subtracted the f - .and
=4

recognized the meaning of the resulting summations. Thus

(X ;)g,, 1s the fraction of those neutrons which are in-

elastically scattered from the fine energy group g and

into the gross group 2. 2; is an average inelastic cross-

section defined by the last two sides of eq. (2.3-95).
Similarly we find

jwjp (1o (20T = J" {j“* I“;H}(xt)p (23,

A=A+ =t

" —
CnE e [ [, (28, =T, +E)F @30

@hfl «

By using the results of egs. (2.3-5) and (2.3-6), we
can now write eqs. (2.3-1) and (2.3-2) in the forms

VEM - &+ fIEM=0 (559

vfl §;(‘") - q: ig(fl + e:- ilc"):"o (2.3-8)

where

12 (33),{(Z), + (f-oz-:"). +2; ‘“*)n”(z“g(z.a-%

gi= G2y, { ()2 (B, @3

 
 
    

: "‘_\ o =590~

o = (33,), i(xa\,_ — (%), ? (Z;\z} (2.3-11)

and
6 = 62, {(3B), + L+ 2B e

are numbers obtainable from the known cross-sections and
composition, and from the assumed energy spectrum.

Several additional remarks are in order. We have a
pair of equations such as (2.3-7,8) for each homogeneous
region., A basic assumption In their derivation is that it /)
i1s a good approximation to assume a single energy spectrum o
for each homogeneous region, an assumption that is certainly
not true as soon as we couple two regions together. Finally
the particular dividing line between fast and slow regions
proves particularly convenient in the case of a blanket con-
taining only U-238 as fissile material. In this case
5312 = 0, since (ZL;), = O, and the solution of the problem
is in every way analogous to that of the thermal two-group
problem. For a non-fissioning reflector this 1is true no
matter where we make the group division.

SOLUTION OF EQUATIONS. = It suffices merely to cilte
a standard reference, say CF-51-9-127, for the solution procedure
for the two-region problem each of which is governed by a set
of equations such as (2.3-7,8).

2.4 ONE-GROUP CALCULATIONS

ONE VELOCITY - SINGLE MEDIUM. - One-velocity calcula-
tions give approximate results for critical size and conversion
ratios for homogeneous fast reactors, if appropriate cross-
sections are chosen. The cholce of average cross-sections
implies a previous knowledge of the neutron spectrum, which
1s obtained from experience or by judicious guessing.

Neglecting elastic scattering and inelastic scattering,
-60-

gty

 

the neutron diffusion equation becomes

2
-5 vzie =z (2.4-1)

wheré the neutron production is assumed to be proportional
to the fisslon cross-section.
For a multiplying region, eq. (2.4%=1) becomes

V¢ + &14’:0 (2.4=2)
where the buckling k is defined as
g =32, (92, -Z.) (2.14-3)

For a non-multiplying medium, eq. (2.4-1) becomes

V’"P - Ksz = O (2.4=-k)

where Klis defined as

K* = 3 2y, (B -2 2) (2.%-5)
The solution of eg. (2.4-2) in spherical coordinates is

(P . A sig kr (2.4-6)

subject to the boundary condition that the flux vanish at the
extrapolated bare pile radius. Hence, the critical radius is

T
b = -k- (20"""’7)

and the critical mass of extrapolated reactor is:
CoMc - L'-/3 L b3 pf (2."""8)

where P = density of fissionable material in the reactor
core. '
If one assumes that all the leakage neutrons can be

==

YA w
-61-

 

used to produce Pu, the external conversion ratio is the
ratio of the number of leakage neutrons to the number of
neutrons absorbed by U=235 in the core:

-F2 Jv
X,C.R = J- I
f { Z'+(zs)+z;:(zfl7] £\ (2.4-9)

_ D Zs = Za
Z. () +Z )

The internal conversion ratio i1s the ratio of the number of
neutrons captured in U-238 to the number absorbed in U-235
in the core:

JZ.eanpdv &L
f {Z.69) ¢ 2, Cw); ¢dV 2y (5) +2.09)

These equations are in agreement with a detailed
neutron balance. The added information given by eq. (2.4%-2)
is the buckling, kfi and hence the critical size.

Experience dictates the choice of the average neutron
energy at which the neutron cross-sections are chosen. For

I.GR, = (2.4-10)

 

the systems considered, the third and fourth energy groups
were appropriate for fast systems: 1 - 2 atoms U-238/U_235;
£ 8 diluent atoms/U-235. The fifth and sixth energy groups
were appropriate for more dilute systems: 3 - 6 atoms
U"238/‘U-2355 8 - 35 atoms diluent/U-235.

For most cases the one-velocity techniques were used to
determine the initial value of k°/3 to be used in the multi-
group calculations., Table 2.4-1 summarizes the comparison
between the one-velocity approximation and the multigroup
treatment for system# 21, in which the fifth energy group
was chosen.

 

 
==
~-62- Pf

TABLE II=2,4-1

 

 

b(em) C.M. X.C.R. I.C.R. T.C.R.
(metric tons)
One-velocity 125.1 2.693 .813 2339 1.152
estimate
8 group 125.1 2.693 .771 .35%  1.125
system 21 |

 

For rapid estimates, one-group calculations can easily
give 10% accuracy.

ONE VELOCITY - MULTIREGION. - For multiregion systems,
the one velocity egs. (2.4-2) and (2.4-4) are solved subject
to the boundary conditions of continuity of neutron flux and
neutron current. For the three-reglon system consisting of a

core, reflector, and blanket, shown in Fig. II-2.4-1, the
criticality equation is:

. K, (a- b\ L 1-Ka — (K cothks(x-a)+1§ Z L \]

[ | +K,a — {K;a cotly Ky (%) 'H "3 ] (2.4-11)

Gfl
(fl
L (kb c.dflz\o"n%% + | kb
tr
From eq. (2.4-11), reflector savings and pipe holdups may be
estimated, using the bare pile calculations to determine k.
The equation is strictly true if ratios of cross-sections in

the various regions are independent of neutron energy; other-
wise, the flux consists of additional spatial harmonics.

 

e

o O s
-63-

 

Region I Region IT Region IIT
?‘ = A_S_AM__’?_!._P ‘fz= B'e-x': Bz.efl(“ QB =C .'MM.L Kx(x-v)
v — e

 

Core Reflector Blanket

FIGURE II-2.4-1

THREE MEDIA - ONE VELOCITY SYSTEMS
Y

64 =

-64- - E :

The X.C.R. will decrease somewhat as the reflector is
made thicker due to its small, but not negligible, absorption.
For those systems satisfying eq. (2.4-11), X.C.R. may be
calculated froms:

 

 

X.C. R. - gbt«nkt‘l‘z‘fig\ ‘P dv
Lo"_ {Z.; (N + Z‘(w)ztpcw'
: KaX
= £ a8 ( & \“‘- {HK;«Q&M*‘G\" swhks(x-a) { (2.4-12)
thzm-zcm Ky | = kbt beb

 

. (2)
. g Mkz(fi-b\ + w‘f’-‘fa- b’[;—;—;(&bd@b")"'}
2 t

The first term in brackets takes into account the finite
thickness of the blanket, and the second term in brackets
represents the loss due to absorptions in the reflector.

On a one~velocity basis, the I.C.R. remains the same
as for the bare pile.

Eq. (2.4-12) may be used to compare the relative
merits of reduced inventory and reduced X.C.R. for varilous
thicknesses of reflector. Such a process was used to
optimize the nuclear constituents in the fast fused-salt
reactors considered by this Project.

DETAILED NEUTRON BALANCE

Neutrons can be lost by parasitic capture, fission
or leakage from a chain reacting system. Assume that the
probability per unit event for each of the processes 1is
represented by a macroscopic cross-section sultably averaged
over the spectrum of the reactor.

 

-
»

00000
.
%, - ..

Zflc = probabllity of a neutron being captured
lflf = probabllity of a neutron producing a fission
L, = probability of a neutron leaking out of the

system

The production of neutrons is:

For a critical assembly, the number of neutrons produced per
unit event must be equal to the number of neutrons lost per
unit event. Hence, the leakage 1is:

L ::‘i’zgf.“'zgc'_ 2:;

Assuming that all the leakage neutrons can be used to produce
Pu=239:

 

X.C.R. = D3¢ -2 - %
Z’{.(zs) 1"2((?-5')
I.C.R. = Z¢c (28)

 

L) + Z 5)

These equations, coupled with the oné-velocity results,
may be used to predict the effect of small additional
quantities of different materials added to an original re-
actor system.
"66- .,-‘N

3. RESULTS OF FAST REACTOR CALCULATIONS

3.1 INTRODUCTION

Eight-group calculations based upon the age-diffusion
theory were carried out by the methods outlined above to
determine the following quantities for the mixtures of
materials considered by NEP to be practical:

a). Internal conversion ratio

b). External conversion ratio

c). Mean effective @

d). Critical radius and critical mass

~e). Spectrum of fissions, i.e. percentage of

fissions due to neutrons in a given energy
interval

f). Spectrum of the neutron flux

g). Fraction of fast fissions

h). Detalled neutron balance

Terms and symbols are defined in the glossary.

3.2 BARE REACTOR

The followihg methods were used for the bare reactor
calculations.
Critlical radius: The extrapolated critical radlus was
calculated by the methods outlined in 2.1

Critical mass: The critical masses were computed using
the extrapolated radius as the dimension of the
volume to facilitate comparisons between different
systems.

External conversion ratio: The external conversion
ratio was computed on the basis of one neutron
absorbed in U-235 in the core, and assuming that
all leakage neutrons would produce Pu-239 in the

%
A

€724 66 = =
§ | -67-

blanket. This assumption was based upon the ex-
perience of the KAPL physics group, and the test
methods of section 2.2. This assumption that the
fast fission effect in U-238 effectively counter-
balances the non-productive captures by other
blanket materials is slightly optimistic for the
slower spectra, possibly introducing a five to ten
per cent correction in the external conversion
rdatio.

Internal conversion ratio: The internal conversion
ratio was computed on the basis of one neutron
absorbed in U-235.

Fraction of fast fissions: The fraction of fast
fissionsindicates the contribution made by fast
fissions in U-238 and U-236.

Neutron spectrum: The bare pile multigroup calculation
gives an indication of the neutron spectrum in the
actual reactor. This spectrum served as the basis
for making suitable cross-section averages which
were later used in the less precise two-group and
one-group calculations of multi-region problems.

Cross-sections: The cross-sections for the atomic
constituents have been discussed in Section II-l1.

Densities: Although the calculations for the conversion
ratios are independent of the density of the mixture,
the density will determine the critical mass of the
assembly. For the fused salts, the macroscopic
densities used were the best estimates for the
average operating temperatures given by the
engineering group. The number density of fission-

able material was determined by dividing the
macroscopic density by the relative atomic volumes
of the various constituents:

. o "'f {; e fi.-
M&m -5 67

Mm
-68-

 

6.03 x 1023
N(25) = L=
235 + IE22 (238) + ML (35.5) +...

The densities of the liquid metals were assumed to
be those of a volumetric mixture.

As more accurate density determinations are
made, corrections to the critical mass may be made
quite readily, as the density is a common factor
in the bare pile calculation.

 

Interpolations: For chemically similar systems, inter-
polation methods were utilized to optimize various
core mixtures with respect to theilr atomic ratios.
Generally, chemical and metallurgical considera-
tions eliminate the problem of many independent
atomic ratios for a complete survey.

For the fused salts, the main variable of
interest was the ratio of U-238 atoms to U-235
atoms. Eutectic compositions which gave tlie lowest
melting points fixed the other atom ratios.

TABLES AND GRAPHS

Table 3.2-1: System Constituents for Bare Reactor
Multigroup Calculations
This table describes the atomic constituents and
their relative atomic ratios, and the macroscopic
density used in the calculations of the bare reactor
assemblies considered by NEP. Table 3.2-1 also
gives a code number to each reactor system which is
used as a means of reference throughout this report.
Table 3.2-2: Breeders - Results of Multigroup Bare
Reactor Calculatlions
The results to be found in Table 3.2-2'include:

:
P 68 SSETEESR"

i

lllllll
-
L e
. -69-
.%»

a). Number density of Pu
b). Critical Mass (in metric tons)
c). Mean effective a
d). Total Breeding Ratio
e). External Breeding Ratio
f). 1Internal Breeding Ratio
£). Buckling (k°/3 with number density factored out)
h). Fraction of fast fissions in U-238
i). 50% fission energy, i.e., 50% of fissions occur
below this energy. -
Table 3.2-3: Converters - Results of Multigroup Bare
Reactor Calculations
Table 3.2-3 is identical to Table 3.2-2, except
for the fact that U-235 is the primary fuel,
instead of Pu-239.

Table 3.2-4: Breeders - Neutron Balance
Table 3.2« gives a detailed neutron balance for
the breeder reactors lnvestigated by this project.
The baslis of the balance is one neutron absorbed
by fission and capture in Pu-239,
The degree of balance is an indication of the
calculation accuracy, not of the accuracy of the
results.

Table 3.2-5: Converters - Neutron Balance
Table 3.2=5 is similar to Table 3.2-% except that
the balance basis is one neutron absorbed in U-235.

Figures 3.1-1,2,3 and 4: These figures present the re-
sults of the multigroup calculations for chemically
similar systems graphically. Trends of the varlous
parameters as functions of different atomic ratios
are obvious from this presentation. When external
holdup 1s added to each of these plots, a definite
inventory minimum in the vieinity of N(28)/y(25)= 3
.
“TLA

S

is evident. TFor optimalizations, the interpola-
tions indicated by these graphs may be used to save
many manhours of calculation.

Figures 3.2-5 to 12: Spectra
These figures are presented as belng representa-
tions of the flux and fission spectrums in the
fast reactors considered by this project. The
graphs are chosen for the chemically similar
UCL) -NaCl fused salt systems, and show how the
neutron spectrum changes with varying atomic
ratios.
TABLE II-3.2-1. System Constituents for Bare Reactor Multigroup Calculations

(atom basis)

System P
Number | U-235 | U-236| U=238 | Pu-239| Bi Cl |[Na Pb |Fe | F.P. gm/cc

4,578

------
3 -
caws ®
aaaaaa
' *

......

ooooo

......

  

'
-l
f—

i
~, _ 72

vy
-ZL-

TABLE I1I1-3.2-2. Breeders

 

 

 

 

 

 

 

 

 

 

 

 

 

™ Results of Multigroup Bare Reactor Calculations
Ly Lem | N (ig)2tons C-M-Emi§£§°§ 3 | T.B.R. X.B.R. I.B.B,(Efézgzizgggizz 50§nzizsion
"""" 1. |7.982x10"%Y|  0.516 | 0.065/1.786 | 1.786 [0.0 | u5. | - 910 Kev
2. |1.687 1.113 0.130{1.768 | 1.461 | 0.307 | 214. | .11 333
3. |0.992 1.725 0.176|1.678 | 1.015 { 0.663 | 326. | .13 140
. [0.595 14,36 0.232|1.570 | 0.250 | 1.320 | 157, .155 130
5. |0.626 3.43 0.254| 1.%1k% | 0.711 {0.703 | 380. | .095 111
....... 6. |1.083 0.75Y% 0.101|1.593 {1.593 |{0.0 | 503. - 287
7. 10.992 0.890 0.131|1.622 | 1.191 {0.431 | 505.8| .085 224
8. |0.621 2.33 0.161| 1.493 | 0.677 {0.816 |499.4| .10 174
23, |1.316 1.139 0.143] 1.57% | 1.57% | 0.0 296. - 368

 

 

 

 

 

 

 

 

 

 
93

TABLE II-3.2-3. Converters.
Results of Multigroup Bare Reactor Calculations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fraction
pystem y(agyacoms c.M.gmfgfiigf- 2 |T.C.R.| X.C.R. 1.c.3.(§z£35)2 | Fast 50%Eg駧;on

0.996x107%t 1,882 0.13% 1.156 | 0.697 | 0.459 | 303.% | .075 150 Kev
1.863 1.090 0.121 {1.301 | 0.892 | 0.409 | 189. .10 202
2.537 1.095 0.09351.217 | 1.217 | 0.0 125.3 - 318
0.63% 2.69 0.14%4 {1,007 | 0.655 | 0.352 | 435.0 .OU5 143
0.362 206. 0.169 [0.776 | 0.047 | 0.729 | 50.7 .055 100
4,943 1.355 0.08641.267 | 1.267 | 0.0 4y, 5 - 550
1.229 2.36 0.161 1.211 | 0.853 | 0.358 [196.9 .085 143
0.702 6.48 0.187 1.161 | 0.450 | 0.711 |211.% .11 87
0.491 1790. 0.213 1.107 | 0.017 | 1.089 8. .12 61
3.302 1.596 0.092 1.190 |1.190 | 0.0 68.1 - 420
1.026 2.45 0.155 [1.200 | 0.853 | 0.347 | 243.2 .075 134
3.391 1.502 0.095 1.244 | 1.24% | 0.0 68.6 - 401
10.845 2. 64 0.176 1.137 | 0.781 | 0.356 |300. .07 103
1.687 2.13 0.138 1.290 | 0.945 | 0.345 | 139. .10 180
.825 2.918 0.173 1.126 | 0.744+ | 0,382 | 288.7 .08 111

 

1
-3
w

1
)4

¥ e a0

 

¥
5 '
=3
4 TABLE 1I-3.2-4, Breeders - Neutron Balance T
-+ ‘ Basis: 1 neutron absorbed in Pu-239

System Number 1, 2, 3. v, | s, 6. 7. 8. 23,

 

 

Neutron Production

v(49) «Fissions of Pu-239| 2.7885 [2.6293 | 2.5248 | 2.4113 | 2.3695 | 2.6941 | 2.6258 | 2.5631 | 2.5987
v(28)¢Fissions of U-238 0.2713 | 0.334%5 | 0.3733| 0.2147 0.2070 | 0.2445

iiiiii

 

R q’i Total Source 2.7885 [2.9006 | 2.8593 | 2.7846 | 2.5842 | 2,6941 | 2.8328 | 2.8076 | 2.5987

------

 

 

  
   
  

 

 

: o ’ “ Neutron Consumption
P R - Figsions of Pu-239 0.9389 (0.8853 | 0.8501 | 0.8119 | 0.7978 | 0.9071 | 0.8841 | 0.8630 | 0.8750
,,,,, wiil|Fissions of U-238 0.1085 | 0.1338 | 0.1493 | 0.08587 0.08279{ 0.09781

.....

 

 

..... - Captures by Pu-239 0.06112 [0.11%7 | 0.1499 | 0.1881 | 0.2022 | 0.0929%| 0.1158 | 0.1370 | 0.1250 ,
LL Captures by U-238 | 0.3074% | 0.6625 | 1.3196 | 0.7034 0.4312 | 0.8161
X e Captures by Bi 0.1007 | 0.120% | 0.2170
""" Captures by Cl 0.002289 0.02376| 0.04755 | 0.06538/ 0.04826 0.02029
""" o Captures by Na 0.01016 0.004075
Captures by Fe 0.02668
Leakage 1.7862 |1.4606 | 1.0154 | 0.2503 | 0.7107 | 1.5932 | 1.1916 | 0.6766 | 1.5743
Total Consumption 2.7885 [2,9003 | 2.8593 | 2.7846 | 2.5850 | 2.6939 | 2.8259 | 2.8075 | 2.5987

 

 

 

 

 

 

 

 

 

J
 

 

LE II" .2“

75

Converters — Heulron Balance

 

1 Beutron absorbed 1n 0-235

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Systes Number 9 10 11 12 1 15 16 17 18 19 20 2 22
XHeoutron Production
Y (25). Fisgions of U-235 | 2.1771 | 2.2040 | 2.2581 | 2.1600 2.2736 2,1272 | 2.0805 |2,0365 | 2.2628 | 2,1400 | 2.2549 | 2.0999 | 2.1709
V(28)e Fissions of U-238 | 0.2803 | 0.2552 0.1071 0.2024 | 0.2597 |0,2827 0.1789 0.,1560 | 0.2429
.rotal Source 23574 | 2.4592 | 2.258) | 2.2671 2.2736 2,329 2.3402 2,3192 2.2628 2,3199 2.2549 R.2559 2.4137
Neutron Consmption
Fissions of U-235 0.8814 | 0.8923 | 0.9242 | 0.8745 0.9205 0.8612 10.8423 [0.8245 [0.9261 | 0,868 | 0.9129 | 0.8502 | 0.8789
Fisgions of U-238 0.07212! 0,1021 0.04283 0.08097 |0,1039 |0.1131 0.07156 0.06239 | 0.09715
Captures by U~235 0,1186 | 0.1077 | 0.08580 | 0.1255 0.07954, 0.1388 |0.1577 |0,1755 |[0.08387 i 0,1331 | 0,08709 | 0.1498 | o0.1211
_Captures by U-238 0.4597 | 0.4085 0.3515 0.3582 |0,7107 |1.0899 0.3473 0.3557 | 0.3447
_Captures by Bi 0.1289 | 0.05705| 0.04087 | 0.2169
_Captures by C1 0.006365 | 0.03732 | 0,07156 |0,09230 | 0.009889] 0,04353 | 0,008970! 0.05109 | 0.02708
Captures by Na 0.000177 | 0.,001067 | 0.003491 | 0,007018 | 0.000167 0,000174| 0,001701
Leptures by Fe |
Captures by Fb 0.004733| 0,001567| 0,004397
Capturea by F.P, 0.06209
Leskage 0.6966 | 0.8916 | 1.2170 | 0.6547 1,2667 0.8531 [ 0,4504 [0.01760 |1.1903 | 0.8529 | 1,2442 | 0,7807 | 0.9448
Total Captures 2,3573 | 2.4592 | 2.2580 | 2.2659 2.2733 2.3306 [2.,3400 [2,3199 [2.2624 | 2.3199 | 2.2549 | 2.2559 | 2.4137

 

 

 

 

 

 

 

 

 

 

-gL-
-76- |
FIGURE II-3.2-1. PuCl3-fiblu BREEDERS
SYSTEMS 1 TO Y4

Total Breedi

Breeding Ratio

28154

22
20

I8

Breeding Ratio——

Critical Mass

         

 

Il 2 3 4 5 6 7 8 9 10 N 12
N@8)
N(49) ¢

Bare Reactor Critical Pu Mass, metric tons
Breeding Ratio ——

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.8 — e
G -3,2-2. Pu--U-238--Bi BREEDERS
- SYSTEMS 6 TO 8 i
L6 — \ { — 32
Total Breeding \
Ratio
\ ™~ 30
14 \>\ 28
I—N 26
Exfernal Breeding
Ratio
1.2 29
\ 22
1.O \\ 20
/ 18
18
\
8 16 f16
, t-"‘ 14
/ \ ,fi; 12
6 /“ l —12
.// "/’5 / /\ \ 10
\ —=110
Internal Breeding \
Ratio
4 Xle
06
6
2 ' - - -04 4
/ Critical Mois
| 02
A ) 2
- 1 0
o I 2 3 4 5 6 7 8 9
Ned) ot ——
N(49) ¥ ~ "o
- - - T |
 ENEORFE—= i 7y

 

Bare Reactor Critical Pu Mass, metric tons ————>

 

-7 =
-78-

Conversion Ratio ——»

 

73

.5/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T T T T 1 1 T 30
FIGURE II-3.2=3. U-235--U-238--Bi
B CONVERTERS |, ¢
SYSTEMS 11 TO 13
26
'Totol Conversion .24 24
-\\\\ Ratio
\\\/ .22 5
\ 20 20
\\ \\ 18|18
CExieran \ »T’ 6| 6
 Conversion :
Ratio /
/ 1 14! 14
(/\ T 12|,
/ = \ /
| 10
4 / 10
Internal \ / 0
"Clgn\’r'ersion 8
atio
\/ X 08 .
Y
/ A N a
| / \ fi 2
' [ Critical \
Mass
] 0
l 2 3 4 5 6 7 8
N(28)_~A
N(ZS)-F

Bare Reactor—Critical U-235 Mass ; metric tons

 
 

_—= -79-

FIGURE II-3.2-4, UCl),-NaCl CONVERTERS
SYSTEMS 1% TO 17

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Conversion Ratio —=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30
28
Total Conversion
Ratio 26 26
\
s | 24154
N 22 |,,
w
\ External Conversion 5
Rotio -
| ] .20 o
l / I8 E
/ —18  _
— &
ol 1 116 g
| 16
|, 8
/ 14 a4 o
/ N\ / 3
12 x
£ 2 5
|
Internal /\ / 10 0 §
Conversion / \ , 5
Ratio / / S
N oglsg «
o
@
/ / Nk
/ / \Critical N\
y / Mass. \ 02l 5
v N

 

 

 

 

 

 

 

 

N
o
- p
W
o))

7 8 9 10

 
-80-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e,
FIGURE II-3,2-5
SYSTEM 14: SPECTRUM OF FISSIONS AND
FRACTION OF FISSIONS ABOVE u
30
128
26
/Spectrum of Fissions
24
| *
20 20
© ——\
>
o
0
o 9 18
» \
et
o 8 16
m \
»
w 7 14
=z \ 550 Kev.
o /
- .6 1 112
o
<
c 5 10
4 08
3 06
2 04
N D2
% 2 8 9 10

FISSIONS/ UNIT u —
e T E A T il i T

 

-81-~

 

 

 

 

 

 

 

 

 

G =3.2-6
SYSTEM 1lk4: FLUX SPECTRUM

 

 

30

 

 

.28

 

 

26

 

 

24

 

22

 

 

 

 

—— Flux Spectrum

 

 

@

 

 

 

UNIT y————
o

 

X

FLU

 

—
N

 

 

 

 

 

 

06

 

o4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
-82-

FRACTION FISSIONS above u——

 

 

 

 

 

 

 

 

 

 

 

FIGURE II=3.2=7

SYSTEM 15: SPECTRUM OF FISSIONS AND
FRACTION OF FISSIONS ABOVE u

 

 

 

 

 

 

 

10

 

 

 

 

 

 

 

¥ S;;ectrum of Fissions

 

 

N L7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

S0

26

24

.22

20

18

12

28

— FISSIONS AJNIT y —————
26

24}

22

 

 

-83-

 

 

 

 

 

 

 

FIGURE II-3.2-8

 

 

 

SYSTEM 15: FLUX SPECTRUM

 

 

 

 

 

 

 

 

 

 

 

®

FLUX/UNIT U
= .

 

 

 

 

 

 

 

 

Flux Spectrum

 

 

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ty Rl g 3
- 1

10

23
-84-

FRACTION FISSIONS obove u——=

S

@

é )

 

FIGURE II-3.2-9

SYSTEM 16s: SPECTRUM OF FISSIONS AND
FRACTION OF FISSIONS ABOVE u

.30

 

 

 

 

 

 

 

 

 

 

 

@

 

0

 

 

/

 

\ Spectrum of Fissions

o

=

 

~

 

ro

 

 

87 Kev

 

o

 

 

 

 

106

 

 

02

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—_ FlSSIONS/UNIT u
FLU&/’UNH'U

 

-85~

 

 

 

 

 

 

 

 

 

FIGURE IT-3,2-10

SYSTEM 16:

 

FLUX SPECTRUM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flux Spectrum

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 4 S

 

 

 
-86-

FRACTION FISSIONS above u——

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE II-3,2-11
SYSTEM 17: SPECTRUM OF FISSIONS AND
FRACTION OF FISSIONS ABOVE u
~ 30
28
26
24
22
1.0 20
18
Spectrum of Fissions
8 4 16
7 14
12
6 61Kev
i / o
4 ‘ \\ 08
X \ 06
2 04
N 02
A \
o | 2 3 4 5 6 7 8 9 10
~ — U —a%
C6 . ’1
SE T A
RSRSEEE RN

FISSIONS/ UNIT u

¥
FLUX/ UNIT u

ity
s

-87~-

 

 

 

 

SYSTEM 17:

 

 

 

 

FIGURE II-3,2-12

 

FLUX SPECTRUM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.30
28

26

24
22

20

A

Flux Spectrum,

e

14

12

10

.08

06

.04

.02

0 3 4 5 6 7 8 9 o)
8"{ = y———--=
.
SSECRET
‘.,,,q
%

+
(B mrerariony

'l
|

-88-

3.3 IWO-GROUP TWO-REGION CALCULATIONS

The several two-group calculations that were carried
out can be divided into two distinct sets:

(a) Computations were performed for a given core and
infinite blanket to determine reflector savings and to com-
pare breeding ratios with those obtained by means of special
multigroup methods for the blanket. The results are given in
Table II-3.3-1, which compares the result of a given two-
group calculation with the multigroup calculation which was
taken as the basis of cross-sectlion averaging in the given
case, It is seen that the requirement of satisfying boundary
conditions in the two=-group case leads to a small adjustment
of breeding ratios away from the extreme values given by the
special multigroup methods. However, this adjustment 1s not
sufficient to warrant an extensive number of such calcula-
tions for present purposes. It is further seen that the re-
flector savings are not great compared to those obtainable
with a good reflector because of the low density of the
blanket.

(b) Core size as a function of reflector size was in-
vestigated for system No. 1l coupled to a Pb reflector. We
have also obtained the X.C.R. as a function of the same para-
meters. This was calculated as

X.C.R. = Leakage from reflector
*vett T Pission + Capture of U-235 in core ’?
consistent with the assumption made in the bare pile multi-
group calculations that all neutrons which leak out of the
core are avallable for conversion.

Three distinct calculations were performed. Two of
these were two-group calculations, which differed in the
choice of reflector cross-sections. The results for the
critical size are compared in Figure 3.3-1, where we have
plotted reflector size as a function of core size, both
given in units of the bare core size, a = 56 cm. The overall

 
TABLE I1-3.3-1, Comparison of Special Mult]
tions with Two-Group Calcul

 

 

-89-

Lgroup Calcula-
lations

 

 

System Number

Multigroup Two-Group

52: Strong Coupling

53¢

Driven Blanket

Multigroup Two-Group

 

 

 

 

 

T.B.R. 1.556 1.58% 1,728 1.717
X.B.R. 1.170 1.233 1.421 1.403
I.B.R. 0.386 0.351 0.307 0.314%
Core Radius

- 1.000 0.681 1.000 0.688
Bare Core Radius

 

 

e
rmnr oL

ae*ee . ,‘1
ase ;
- <
ews
.
@
-*
-
*
- *
adee
- 9 O - o~ — o M

size of the reflected reactor relative to the bare reactor is
then obtained by adding abscissa and ordinates. C.M. can
then be directly obtained from the value given in the previous
section for the system in question (see Table 3.3-1). The
curve labeled "Fast Reflector Spectrum" was obtained on the
extreme assumption that the reflector has the same spectrum
as the bare core, whereas the one marked "Slow Reflector
Spectrun" resulted from the assumption of a substantially
slower energy distribution. In Figure 3.3-3 the same compar-
ison is made for the X.C.R.'s., In all cases, the points
actually computed are indicated. |

In addition to the results found above, we have also
performed a one-group calculation for the same system, using
the fast spectrum to determine average cross-sections. The
results of this computation are compared with the two-group
fast spectrum result in Figures 3.3-2,k.

As was to be expected, the slower reflector spectrum
predicts smaller critical masses because it results in the
assumption of larger transport cross-sections for the re-
flector than in the fast spectrum case. The results for the
conversion ratio quite likely underestimate the conversion
loss, since there is no information contained in the calcula-
tions made about the degradation of the leakage spectrum
relative to bare core leakage. The actual rise of the X.C.R.
as seen in Figure 3,3-3 may be understood on the basis of 'the
remark that for small reflector sizes, the overall size of
the reflected reactor is actually smaller than that of the
bare pile. The most important result to be gleaned from the
graphs, however, is that with little loss in conversion ratio,
one can approach optimum reflector savings.

 
,"

 

 

 

 

 

 

 

 

 

 

 

 

 

L] _91_
1.8 l ‘
Legend
16 A Fast Reflector Spectrum __
' ® Slow Refiector Spectrum
Bare Core Size=56cm.

14

L2
Q®
~N
B
£ FIGURE II-3,3-1
o0 —
;c:': Reflector Size vs. Core Size

 

$—a)=
!1?#_

 

 

 

 

 

 

 

 

 

 

 

 

7
kd _ .
- Core Size

1.0
-92-

L6

e TR

e CRET

2 Group Asymptotes % @e=——r_
/? Group

5

bl

|

 

 

Legend

 

@ | Group

 

 

@ 2 Group

Bare Core Size= 56 cm.

 

 

 

 

 

 

 

 

 

FIGURE II-3,3-2

 

Reflector Size vs. Core Size

 

 

i\
X

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

FIGURE II-3.3-

 

 

 

 

m ©w O =

X.CAXC) bare

[

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-93-

X.C.R. vs Core Size
X.C.R. (Bare Reactor) * Bare Core Size
Legend
A Fast Reflector Spectrum
@ Slow Reflector Spectrum.
o ) s H
5 .6 7 8 9 1.0
kd
n.
83
v—-:______,.,_—-—-
e R
-94- s N
2 Gro |
| G:ogg} Asymptotes

 

FIGURE II-3.3-4

 

X.C.R. vs Core_Slze
X.C.R. lfiEre Reactor) * Bare Core Slze

 

]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Legend
® | Group
A 2 Group
" T ‘TL
.9
2
2 8
O
25' 7
~
O §
>
5
4
-
2
Jd
)
4 5 6 7 8 9
K
=
-95-

 

3.4 ONE GROUP - THREE REGION

The methods developed in Section II-2.4 were used to
estimate the loss of conversion ratio and the reduced critical
core mass which would be obtained for various thicknesses of
a Pb or Fe reflector. Equation (II-2.4-3) was applied to the
systems of interest.

As an example, the results of applylng this method to
a system # 15 core, Pb reflector and an infinite UCl, blanket
are represented graphically in Figure 3.4l. The fifth
energy group was used to determine cross-sections for all
three regions.

If the core buckling is low, loss of external con-
version ratio does not permlt the utilization of the maximum
possible reflector savings. Economic considerations will de-
termine how much loss of X.C.R. can be tolerated, or traded
for decreased inventory.

The one velocity -~ three region techniques are
applicable for comparing and optimizing similar reactor
systems. For an actual reactor, however, these results
indicate trends, and are not expected to give precise values
as a detalled account of the neutron distribution as a func-
tion of space and materlals was neglected.
~-96-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

| I
Reflector Thickness and X.C. vs. Core Size __|
8 X.CCO
7
1.3
] 6 \ 1.2
@ Il
Q
: \
Ig S e/l.0
S 9
§ / \xc
g 4_ \ XCO .8
’_.f I XC.
B \ X.Co
x| 3 6 |
\ _ 5
\ Reflector
2 \<£ Thickness 4
‘\\\\\\‘ J
| \ i
\\.I

D 6

kb .
— Core Thickness —

 
97~

 

3.5 SPECIAL MULTIGROUP CALCULATIONS

Additional multigroup calculations were carried out to
investigate specific problems encountered. For the bare re-
actor in most cases, interpolations proved satisfgctory for
intermediate values of parameters.

System 18 examines the effects of Fission Product
poisoning on conversion ratio and critical mass.

System 51 examines the effects of mixtures of Pu-239
and U-235.

Systems 52 and 53 are self-consistent calculations which
were wndertaken to evaluate the assumptions made concerning
the effect of a blanket on the spectrum and conversion ratios
of the bare reactor, see Section II-2.2.

System 54 examines the effect of cross-section varia-
tions on critical mass and breeding ratio.

TABLES AND GRAPHS
Table 3.5-1: System Constituents for Special Multi-
group Calculations
Table 3.5-1 lists the atomic constituents and gives
a code number for each of the special multigroup
calculations.
Table 3.5-2: Results of Special Multigroup Calcula-
tions
Table 3.5=2 summarizes the results of the multi-
group calculations for each special reactor system
considered.

Table 3.5-3: Speclal Systems, Neutron Balance
Table 3.5-3 gives a detailed neutron balance for
each special system considered. The basis for
these balances was 1 neutron absorbed in U-235
and/or Pu=-239 in the core.
[ T

 

Figures 3.5-1,2: Comparison of neutron spectrums by
strong-coupling theory.

Figures 3.5-1,2 compare the neutron spectrum of
System 2 with the spectrum given by the strong-
coupling theory. One notes that the mean fission
energy is lower and a larger in the strong-coupling
case.

o
o
......

lllllll

ssssss

ooooo

-----

......

''''''

......

tttttt

nAada®E

TABLE IT-3.5-1.

77

M

System Constituents for Special Multigroup Calculations

 

System

 

 

 

 

 

il U-235|U-238|Pu-239| Cl |Na Remarks

51. 1 4,5 .25 |28.75| 6 | Bare Reactor calculation (Compare with 15.)

- Mixture of Pu=-239 and U=-235

52. core 3 1 15 Strong Coupling Blanket Calculation
Blanket 1 L (Compare with #2.)
53. core 3 1 15 Driven Blanket Calculation
Blanket 1 L (Compare with #2.)

5, 3 1 15 10% Revision of ¢, and ¢, for Pu, S R)=19%b.

 

 

 

 

 

 

 

 

-66-
e
e
: .
; R
e
.

-001-

 

 

 

 

 

 

]7. ¢
;gz TABLE II-3,5-2. Results of Special Multigroup Calculations
System| Atom Density (metric) - ]
----- : 51. |N(25)=.852x10°% | .M. (25)=1.847 4,520.161/1.302 | 0.888 | 0.414 ¥°/3=375.4| Bare Reactor Calculation
Ll - C.M. (49)=0.470 E§9=0.176
3 52, @ = .235 [1.556 |1.170 {0.386 | I = .11 |Strong Coupling Blanket
: Calculationy Compare
with No. 2.
53. 1.728 |1.%21 | 0.307 | A = 12.% | Driven Blanket Calcula-
...... tionj; Compare with No. 2.
T.B.R. X.B.R. I.B'R.
5, N(u9)=1.687x1021 CeM. (49)=20.926|3=0.125 [1.778 |1.512 | 0.266 k2/3=2h2 <, and o, of Pu-239

 

 

 

 

 

 

 

 

increased by 10%.
......

......

lllll

flflflflfl

......

------

oooooo

rassaw

sresan

 

 

TABLE I1I-3,5-3,

by

Special Systems - Neutron Balance

 

 

 

 

 

 

 

 

 

B

 

 

 

 

 

 

 

 

 

 

 

 

 

System Number 51, 524 53 54,
Core Blanket
Neutron Production
v(25)-Fissions U-235 1.6974
v(28)+Fissions U=-238 0.2276 0.1045 0.3167 0.1174 .272
v(49) Fissions Pu=239 0.5105 2.4892 2.639
‘Leakage into blanket 1.4606
Total source 2.4355 2.9104 1.5780 2.911
Neutron Consumption
Fissions in U-235 0.6872
Fissions in U=-238 0.09103 0.04180 00,1267 0.04697 .1088
Fissions in Pu-239 0.1719 0.8381 . 8886
Captures by U-235 0.1106
Captures by U-238 0.4141 0.3999 1.,2115 1.434] _. 2664
Captures by Pu-239 0.03032 0.1974 <1114
Captures by Cl 0,04121 0.05724%  0.07689 0.08806 . 0202
Captures by Na 0.001490
Leakage 0.8876 1.5116
Total 2.4355 2.9495 1.5691 2907

 

 

 

 

!
ok
o
ek

1
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

~-102-~ :_:W
FIGURE II-3.5-1
SYSTEM 52: COMPARISON OF THE SPECTRUM OF
FISSIONS OF A BARE REACTOR AND
A BLANKETED REACTOR
Legend
30 — Bare Reactor
~ ——— Strong Coupling
2
26 1
.24
22
20 J 1
S e I"“'u_.fi
- .8
: :
E 16 ] ;
5 | |
w
o 14 1 !
L | |
2} | |
al ] J j
| I~ f|
10 1 |
] |
og) —- '
! |
0 — |
O4
1 1T 1
|
0 |
|
o | 2 3 4 5 6 7 8 9 0
102 e

 
NEUTRON FLUX/UNIT y—»

=2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-103-
vv—T
Legend
Bare Pile
30 ~——Strong Coupling
28 FIGURE II-3.5-2
*! SYSTEM 52: COMPARISON
26 ! OF THE FLUX ]
I SPECTRUM OF
24 | A BARE RE-
‘ I ACTOR AND A
| BLANKETED
22 | ~ REACTOR
I
I
20 |
I8 I
I
16 ;
I
| L.
.41 ——
12 !
10 4
I
.08 %
0 ;
|
|
04 —
I
.02
T
l I
% 7 8 9 10

 
 

-104-

 

 

3.6 GENERAL TRENDS

TRENDS OF BREEDING OR CONVERSION RATIO. - The breed-
ing(a) characteristics of a reactor are determined by the
average final fate of its neutrons. If we assume the blanket
large enough to capture practically all the neutrons that
leak into it from the core, then the breeding ratio will
depend on the outcome of the competition between radiative
capture by 28 on the one hand, and parasitic capture by 49
or by reactor contaminants and diluents on the other. While
degradation by collision and leakage cannot be considered
the "final fate" of a neutron, the extent to which they in-
fluence the neutron spectrum will have an important effect
on the absorption competition referred to, principally
because of the tendency with slower spectra for capture
cross-sections to increase relative to the fission cross-
section of 49.

The following discussion attempts to explain the
general trends of breeding ratio and critical mass for
chemically similar systems as variations are made in the
relative atomic ratios of the reactor constituents. The
results of such an investigation will appear as Figures 1 to
4 of Section II-3.2. An understanding of these results en-
ables one to generalize fast reactor characteristics to such
an extent that optimalizations may be made without lengthy
calculations.

a). Internal Breeding Ratio (I.B.R.)

The internal breeding ratio is given by:

IT.B.R = S Zs‘““?d“dv / .L,‘z“-.‘w?‘“w (3.6-1)

Corg
It happens that the ratio z““-”/z*qis not strongly

energy dependent in the range of energy where all the in-

"Conversion" and "25" may replace "Breeding" and "49%
in the subsequent discussion.

__04 A gt
~105~

 

 

vestigated neutron spectra are concentrated. Therefore, if
an average value for é%%gg 1s taken outside the integrals,

the remaining integrals cancel, and we have:

B.R. = Z G = EEESLEEEED .6~
T Zi.. (4) N (Q4) 0z 49) (3.6-2)

Thus we see that the I.B.R. is proportional to the atomic
ratio of 28 to 49, and is roughly independent of the neutron
spectrum.

b). External Breeding Ratio (X.B.R.)

In the case of internal breeding, we noted that
the competition for neutrons between 28 and 49 takes place
"locally"; i.e., in each small volume element of the core,
the number of neutrons available to 28 and 49 is the same
(for all energies), since the flux is the same for both.
Hence, the I.B.R. does not depend on geometry, other con-
stituents, parasitic absorption, etec. The situation in the
case of the X.B.R. 1is entirely different. Here the competi-
tion is between 49 in the core and 28 in the blanket.

Between the time that a neutron is emitted in the core and

leaks into the blanket to be captured by 28, it has to run a

gamut of parasitic absorption by 49, diluents and contaminants

and fast fission absorption by 28. The effect of this gamut

on decimating the blanket-bound neutron population varies with

core and blanket density, spectrum composition, and geometry.
One can always wrlte for the core

 

Leakage + Absorption = Production (3.6-3)

The assumption underlying our blanket breedling estimates from
bare reactor calculations is that the fast fission effect in
28 compensates for non-productive absorption in and possible
leakage from the blanket. On this basis, all the neutrons
that leak into the blanket are captured by 28 to form 49
| - S,

"106" M—.—-.

finally. Therefore, the rate of external breeding is Just
equal to the leakage rate divided by the rate of absorption
in (49).

The production is given by

P = 9(49)[2*(4«\¢dml\" + Dlzs)yz*wypduow (3.6-1)

and the absorption by:

A= ffl.;(«)cpcluow' + S\Z“(z?)tfdualv-{- gflc(p\cpduc“-

where Zi (f) expresses the parasitic capture by diluents and
contaminants. We rewrite the balance equation:

L = N (44) { 7)(44)!@'(«) cpduow' + %'r_f(_?_:.z\z)(z'i) jq;(z&) P cl::j]g )

_ f oz @ ¢dudY - %%a% Yo; 29) @dudV - flN;((% S'GE ) ‘fdudv:}

For the fused salt systems, the parameter of interest
1s the ratio N(28)/N(49); therefore, denoting § 23 by ¥
~ and Sd'?mby ¢V , we have: |

L = KOO FY | D) )~ 69 N .m0

_ M Tan —26)T e §

In the range of energy where most of our spectra fell,
the various cross-section averages were roughly constant,
though not exactly so., Therefore, the changes in the neutron
spectrum due to changing vy were reflected in relatively small
changes in the mean cross-sections. Moreover, one notes that
if %%E%% is a linear function of y, where N(Cl) is the number
density of chlorine, L is a linear function of vy, with
negative slope.

We may write

L = N(49)-(G - Hy) (3.6-7)

- T

. .06
———-_...‘__ E -107-

M___.___—

and |
X.B.R. I I/ZM9) V = Silos— = Aly,-y)  (3.6-8)

and we remember that A is a slowly varylng function of ¥y 1in
virtue of the small changes in G caused by spectral shifts
with increasing dilution. These small variations cause, in
fact, a slight downward curvature in the plot of XBR vs. vy.

¢). Total breeding ratio (T.B.R.)

The T.B.R., being the sum of the I.B.R. and
X.B.R., will also be an approximately linear function of y.
Moreover, the leakage does not play a direct part in the
overall neutron economy, since we have assumed that whatever
neutrons leak out of the core are captured by 28. Therefore,
any changes in T.B.R. will be due to differential parasitic
capture with changing y. This can occur because of the shift
in spectrum that takes place with dilution. a increases, as
well as the relative parasitic capture by diluents and
contaminants. Therefore, the T.B.,R. falls off somewhat with
greater dilution although not as strongly as the X.B.R.
Moreover, for the dilutions at which the T.B.R. does begin to
drop more sharply, the critical mass becomes infinitej for
feasible masses, the whole variation of T.B.R. is about 15%,

TRENDS OF SIZE AND CRITICAL MASS. -

a). Trends of size
Formula (3.6-7) gives the leakage L as an
approximately linear function of the 28 to W49 ratio y. For
a bare pile, the following formulae are true:

 

La K
= 3 Ztr (3’6"9)

kb - T (306"10)

 

07 F=ETEEr
-108~ e emea!

T T ———

where k2 i1s the buckling and b is the radius of the bare

plle. For chloride systems (PuCl3-UClh), the transport cross-
section,‘Z:tr, is approximately independent of y since the
Tip of 28 and 49 are approximately equal(a) and the total
number density of heavy atoms(b) is practically independent

of vy, being 20% less for y =z 10 than for vy = O, In fact, we
can write: N(49) + N(28) = N = const.j therefore:

No
N49) §1+v§ = N5 N(49) = 5o (3.6-11)
Combining formulas, we obtain for b2: '
2
2 = 1ty 1 1ty
bT = e——— . . = C (3.6=12)
304p N, Ay,-v) YooY

Strictly speaking, the constants in this formula vary weakly
with y. However, (3.6-12) still displays the main feature
of the critical size: slow variation for small y, rising
rapidly to infinity for y -+ Yoo One notes that the X.B.R.
goes to zero as the critical mass goes to infinity.

b). Trends of mass.
The mass of 49 in the bare pile is given by:

M = % 3« N(49) » m(49) (3.6-13)

where m(49) 1s the mass of the 49 atom. Substituting for b
gives:

\ 3/2 N
M = -31& mm(49) c3/2(%3_fi\,) . T (3.6-1%)
T (y,-m)3/2

According to this formula, the critical mass varies
relatively slowly for small y, then increases to infinity for
Y Yoo (Yo 1s of the order of 10). Therefore, there is an
extensive range of y wherein inventory 1s roughly the same

(8) nis is even truer for 25 mixtures.
(b) As well as that of Cl atoms

 

<08 | o —
e
sl

LTI L - Y a——— - 1 0 9 -

.

 

and therefore can be a secondary consideration in choosing ry.

c). Temperature effects on density.
The bare-reactor multigroup method of calc¢ula-

tion shows that the spectrum is independent of density; there-
fore, one need only consider density changes in analyzing
critical size and mass changes with temperature. Since the
proportions of a chloride mixture are independent of
temperature, the temperature variation of N (number density) -
for each component - will be the same. If we assume:

p=p, £ (t) (t = centigrade (3.6-15)
temperature)

then we can write:

N(49) = N (49) £ (t)
(3.6-16)

N(28) = N_(28) £ (t), ete.

where £(0) = 1.
Now we rewrite the formulas for b° and M, keeping
track this time of all number densities. We have:

 

 

L = N(#9) }G - H v} (3.6-17)
2
e L1 (3.6-18)
5552m'N
Therefore,
2
p? g ——T (3.6-19)
3Ty, N-N(9) | G-Hy}
2
b2 - —EE—_————— . “ufl_;—_—_—_ . 1 :El_gg__fi
3T 36-H% Ny * Ny (%9) £(8)%  £(t)

®ocsoece s (306‘20)
10 ——=

7 bd e N(49)

b3 (3.6-21)
w2« N (49)£(E)
£(t)

WIE  WIF

Mo
Mz—7 | (3.6-22)
£(t)
Equation (3.6-22) shows that M is more strongly dependent on
temperature than the critical radius. Estimates of f(%t),

made by chemical engineers for the fused salt systems, gave:

F(E) ™1 - .25 x 1073 « & (3.6-23)

These formulas are of use in calculating the temperature co-
efficient of reactivity due to density changes.,

TRENDS OF &. - The only effect of y on a is the change
it makes on the neutron spectrum, viz., the spectrum will
become slower as y increases. However, this effect will be
less marked for larger vy it is found that the spectral
shift becomes very slow for y's within several units of the
point of infinite critical mass. Therefore, although a(%9)
changes rapidly with energy, the median spectral energy of
the reactor changes slowly for this range of y so that a(49),
the average over the spectrum also changes slowly.

For small values of y (0O to 3 or 4), a follows the
relatively rapid shift in neutron spectrum. For interesting
values of y (those which lead to reasonable critical mass
and inventory} a is less than 0.2.
N Lo -, .-
“m" b 150 . .
ol

. -111-

3,7 FINAL DESIGN OF FUSED SALT REACTOR

BARE REACTOR CALCULATION. - A bare reactor calcula-
tion was made for the system described in Table II-3.7-1.
The atomic constituents were chosen by optimalization of pro-
cessing and production costs. One notes that the fission
product concentration is quite low, and the U-236 content
relatively high (see Section II-4).

Table II-3.7=2 is a detalled neutron balance of system

24 based on 1 neutron absorbed in Pu-239 and U-235. Excess
degradation is that small amount of neutrons which are de-
graded below the 8th y group, and which are assumed to be
lost by paraslitic capture.

Figures II-3.7-1,2 are the fission and flux spectrums
for the prescribed core.

This calculation was used as a preliminary basis for the desigr
and evaluation of the fused salt reactor discussed in the
engineering analysis report. The one-velocity methods
of Section II-2.4 were used to compute the reflected critical
mass and the Ak assoclated with the molten Pb control rods.
-112-

 

System 2k

System Constituents and Results

 

 

System No. U25 U26 28 Pu239 Na

Cl

Pb

 

2L 1 | 0.2 |3.25 | 0.0048 |1.81

 

 

 

 

 

0.173
1.126

0. 744
0.382

(125' -
T.C.R.

N

XoCoRo -
I.C.R.

Fraction
Median Fission Energy = 1ll1 Kev
Critical Radius = 129.39 cm
Critical Mass (Bare)=z 2.918
N(25) = .825 x 10°! atoms U-235
p = 4.2 gm/ce

28.75

of Fast Fissions = 8.2%

/ce

 

4. 57

 

0.017

metric tons of U=235
 

- IABLE IT-3.7-2, System 24 - Neutron Balance

Basis:

1 neutron

Neutron Production

absorbed in U-235 and Pu-239

v(25)+*Fissions of U-23%5
v(#9)-Fissions of Pu-239
v(28)+Fissions of U-238
v(26)+Fissions of U-236

Fissions

of

Total Neutron Production

Neutron Consumption

U-235

Fissions of Pu-239

Fissions
Fissions

Captures

Captures

Captures
Captures
Captures
Captures
Captures
Captures

of
of

by
by
by
by
by
by
by
by

U-238
U-236

U-235
Pu-239
U-238
U=236
Pb

Na

Cl
r.P.

Excess Degradation

Leakage

=13

- Total Consumption

 

lllll

2.094728
0.011942

0.173925 -
0.017257

2.297852°

0.848068
0.004021
0.069570
0.006903

0.147120

0.000790

0.382281
0.023525
0.005060
0.000735
0.052288
0.001480
0.011659

- 0.74%356

2.297856

-~113-
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-114- e ————
FIGURE II-3.7-1 .
[ SYSTEM 24: SPECTRUM OF FISSIONS AND — 130
FRACTION OF FISSIONS ABOVE u
28
.26
124
| ’ 22
=
® 10 20
o
£
z \ |
D Total Fissions
w
“ .8 1 6
u
> \
e .7 14
- \
<
x 6 12
L.
1 Kev.
.5 10
4 N 08
3 \ 06
/Fost Fissions \
2 04
A 02
O —~1_ o
0 2 3 4 5 6 7 8 9 IO

 

FISSIONS/ UNIT u —= .
-115-

 

 

 

 

 

 

 

 

 

FIGURE II-3.7-2
SYSTEM 24:; FLUX SPECTRUM

 

 

30

 

.28

 

 

26

 

.24

r
N

 

 

 

 

 

 

N
O

/Flu

x Spectrum

 

®

 

o

 

B

 

FLUX /UNIT u—e

o

 

o

 

 

 

Q
@

 

06

04}

 

 

 

02

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
-116-

 

 

MULTIREGION CALCULATION. - Through the cooperation of
Dr. Ehrlich at KAPL, NEP obtained two machine calculations
for the three region multigroup problem of the fused salt re-
actor. The calculation was made to determine the correct re-
flected critical mass, and to determine the decrease in con-
version ratio brought about by absorptions in the reflector.
Such a calculation by hand was not considered practical.

The reactor was approximated by a spherical core en-
closed by a reflector and a blanket. To make the calculation
for the blanket and reflector easier, these regions were
homogenized with their containing structural materials, i.e.,
the core container, the inside edge of the blanket contalner,
the two-inch void, and the Pb control rods were homogenized
into one region called the reflector; similarly, the rest of
the blanket container and the blanket material (UCL,,) were
homogenized into one region called the blanket. In the machine
calculation, the reflector uranium was assumed to be pure
U-238, and the blanket uranium was assumed to be depleted
uranium containing 0.3% U=235. The neutron balance achieved
by the machine computation was .984 produced neutrons for
1.000 source neutron, using v = 2.50 for U-235.

Table II-3.7-3 summarizes the results of the homogeniza-
tion and gives other facts concerning this reactor. The re-
actor constituents were changed slightly from the bare reactor
calculation above. |

~ Table II-3.7-5 gives the neutron balance based on the
machine calculations, adjusted to v = 2.47, to be consistent
with the rest of this report.

Results of perturbation calculations to force the neutron
balance, and to include the effects of small changes in Pu-239
and U-235 concentrations in the reflector and blanket are to
be found in the engineering analysis réport.

SR

b
 

 

-117-

Table II-3.7-4 presents the results of the multigroup
multiregion calculation, and by comparison with Table II-3.7-1
indicates that the assumptions made using the bare reactor
calculations and one-group theory were satisfactory. The
only significant change brought about by the more elaborate
calculation was a 5 per cent reduction in reflected core
radius. The blanket was designed to be four diffusion
lengths thick, on the basis of one velocity theory, to reduce
the leakage loss out of the blanket to a minimum. The multi-
region calculation substantiates this decision.

Figure II-3.7-3 compares the averaged flux spectrum in
the three regiohs. By comparison with Figure II-3.7-2 one
sees that the core spectrum is only weakly coupled to the
‘blanket. |

Figure II-3.7-4 represents  the powér distribution within
the reactor as a function of the radius.

Figures II-3.7-5,6 portray the integrated flux vs.
radius, and the shift in spectral distribution of the neutron
flux with radius for the fused salt reactor. The areas under
each curve in Figure II-3.7-6 have been normalized to unity.

Only one spectral curve has been drawn for the interior
of the core, Figure 3.7-6, because the shape is practically
invariable almost all the way out to the core-reflector
interface. This effect, the truth of which was assumed in
the bare-reactor multigroup calculation, demonstrates why the
bare reactor calculations give good results. |

The rather small shift in spectrum at the outer surface
of the core shows that the’back-couplipg from the reflector
to the core is weak, and hence that the "strong-coupling"
method outlined in Section 2.2 would not be expected to
give reliable results. On the contrary, the "driven-blanket"
method, which assumes the blanket to be coupled to the leakage
spectrum of the core and to exert no reciprocal influence on

-

the core, rests on reliable assumptions and is the calculation

 
oot

f—

)

------

......

oooooo

------

[1&

TABLE II-3.7-3. System Constituents of Fused Salt Reactor

 

 

 

 

 

Region | U-235 | U-236|0-238 | pu-239 | Na | C1 Pp | Fe | F.E. N o
atom/ce gm/ce
Core 1.000 0.20 | 3.2% |.oou8 1.89| 29.20 | 4.77 0.016 |N(25)z.810x10°T 4.2
Reflector .035 0.141 | 0.271]1.000 N(Fe)=23.798x10°T -
Blanket .003012 .996988 4,000 0.736 N(28) = 6.043x10°Y 4.05

 

 

 

 

 

 

 

 

 

 

 

-81i1-

 

 
117

w

3¢

Results of Fused Salt Multiregion Calculation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-

e

m TABLE II-3 07-’+o
...... _ C.M. (25) Reflector | Blanket | Fission (Including U-235)
?:;: . T.C.R.{ X.C.R. | I.C.R. a25 netric bons Core Radius Thickness | Thickness | Power (1n blanket)

g Core Blanket and
o ' Reflector

;.:.:' (given) (given)

ot 5 1.126 0.7%5 | 0.381 | 0.1381 1.168 95.9 cm 15.25 cm 150 cm 97.13% 2.874

* From Table II-3.7-5

i

—
—
v
o

027

s ¥

......

......

------

aaaaaa

nnnnn

nnnnn

000000

000000

¥

 

TABLE II-3.7-5.

 

/120

w

Neutron Balance - Fused Salt Reactor:

Multiregion Calculation

 

 

 

Basis: 1 neutron absorbed in U-235 and Pu-239 in the core
Core Reflector . Blanket All Regions =
Neutron Production ®
*v(25)+Fissions of U-235 2.080600 0.033276
v(26)+Fissions of U-236 0.015642
v(28) «Fissions of U-238 0.157183 | 0.005895 0.028103
v(49)-Fissions of Pu-239 | 0.011740
Total Production 2.265165 0.005895 0.061379 2.332439
Neutron Consumption ,
Fissions: U-235 0.842348 0.013472
U-236 0.006257 ’
U-238 0.062873 0.002358 0.011241
Pu-239 0.003953
Captures: U-235 0.152868 0.003437
U-236 10.023531
U-238 0.381222 '0.040077 0.704953
Pu-239 0.000830
Na 0.000905
c1l 0.052761 0.002442 0.042803
Pb 0.005033 0.002192
Fe - 0.024205 0.010443
F.P. 0.001383
Net Leakage 0.795872 | =-0,071486 -0.71901k
Total Consumption 2.329836 -0.000212 0.067335 2.396959

 

 

 

 

- - - ”
- - - - - - - -l - - - - - - - - - - - - - - - - - L] — - - - - - - - - - - - -

* v(25) = 2,47 was used in this table to be consistent with rest of report. v=2.50
was used in original machine calculation.

1 1
*

1

L
Flux /Unit u

-121-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.30 Fused Salt Reactor - Flux Spectrum
28
1
26
. Reflector
Spectrum.
24
P
22 =T ="
2 e==i— '
i J Blanket
| .. ' Spectrum
18
16 —core
Spectrum y
I
14 i
12 ! —
| I
E Lo —
10 - — ]
|
I
08 1
06
04 i____
02— _____l L.................. |
L]
| 2 3 4 5 6 7 8 9 10 I
U —
~21 PTTI T
J 22~

 

 

¢z
}6
-221-

 

 

......

oooooo

cccccc

 

......

6 -
\ —— ~— Reflector

 

......

Relative Fission Power

 

oooooo

 

 

 

 

 

......

 

 

 

 

+————— Core - — Blanket

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30 60 20 120 150 180 2l0 240

r (cm)—

 

 

 

FIGURE II-3.7-4., Fission Power Distribution in the Fused Salt Reactor
/23

FIGURE II-3.7-5. Total Neutron Flux per Unit Radial Distance vs. Radial Distance
(r2(P, arbitrary units)

 

0 AN

 

 

 

-------

 

 

 

- \ Reflect
3 | / \ eflector

lllll

 

 

 

 

 

 

 

nnnnnn

......

......

 

DDDDDD

 

 

 

 

 

,,"__
P
—_— "'“P , Arbitrary units ——=
p
[ —
/

Core Blanket
. // \\
L/ . \\
\
20 40 60 80 00 120 140 160 180 200 220 240 260

r i
(cm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

€21~

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
ek
/24 0
; S
i
}-2
o
>
40
T D
...... h V
o
| -
...... o
S
------ L.
@
...... ‘J
...... = o
e e C .
+ = By
i de — - °
e b
IIE. Q :
i3 5 2 ’
: L
o
S
5 15
b
2
...... o |
o~
I .05
0 9

 

c
Y

 

FIGURE II-3.7=6. Normalized Flux Spectrum at Various Radial Distances
*»

method of choice for computing the blanket spectrum.

The spectrum in the blanket degrades slowly with in-
creasing radius as the flux gradually shifts to the
fundamental mode (the "equilibrium blanket"). However, it
is evident that the blanket ends long before that stage.

To evaluate the core radius for an exact neutron
balance, an extrapolation procedure gives 95.9 cm. One will
note that the neutron balance is slightly inconsistent due
to round-off error and the slightly non-critical multiplica-
tion constant assumed. The figures in Table II-3.7-5 are
given to 6 figures only to allow intercomparisons to be made
by the reader, and not as an indication of the accuracy of
the calculation.

Based on the preceding results, the following calcula-
tion procedure appears appropriate in future work:

a)., Surveys: Use the one-velocity techniques
described in section II-2.4 to compute con-
version ratios and critical size.

b). Design parameters for particular systems:

Utilize the multigroup technique for the
bare reactor to compute conversion ratios and
core bucklings next, use the core spectrum to
choose appropriate average cross-sections to
be used in a one-velocity calculation for the
three-region problem giving reflected core
radiussy then use the driven blanket technique
to determine the characteristics of the blanket.

¢). Final design: Use multi-region, multigroup
calculation methods.

A second calculation for the fused salt reactor was
undertaken to determine the effectiveness of the Pb reflector
control rods. The second calculation substantiated (within
15%) the one-velocity estimates (Section II-5.2) for the
effectiveness of the Pb control rods. The change of

 
-126- : """‘"—-—-h.._

reactivity according to the multigroup multi-region calcula-
tion is

Ak
=5 = 0.027

if all of the Pb is removed from the reflector.

126 =
4, FAST REACTOR POISONING

4.1 INTRODUCTION

Fast reactor poisons are detrimental because they
decrease the conversion ratio and increase the critical mass
of fuel. The two principal types of fuel poisons are the
fission products and the higher isotopes. The features of
these two types are different in important respects.

The concentrations of both types of polisons are, how=
ever, governed by secular equations of standard form: The
rate of change of element "a" (= —2) is equal to its pro-
duction by fission (= Z'f?fiY(a)), by neutron absorption by
element "3" (= N(3) c%(fi)i?) and by radiocactive decay of a
predecessor a' (= A(a') N(a')), minus its loss by neutron
absorption (= g% (a)$N(a)), by radicactive decay
(= A(a) N(a)) and by processing (=« R(a) N(a)). (Here Zs is
the macroscopic fission cross-section, @ is the neutron flux
integrated over the core, Y(a) is the yield function for
fission products, N(x) is the number of atoms of element "xV
in the core, A is the reciprocal mean 1life and R(a) is the
processing rate.)

The balance equation (for each "“a") is then:

igfltal = 2, P¥(a) +Po(8) N(&) + n(a') N(a') -§a%(a) N(a)

- A(a) N(a) - R(a) N(a). (4,1-1)

4,2 FISSION PRODUCTS

As the fission product capture cross-sectlions are
small in the fast region, loss of fission product atoms by
neutron capture 1s a relatively small item in comparison with
the high chemical extraction rates possible in fluid fuel
systems. In equation (h.l-l), processing is assumed to be
done at a constant rate R.

Moreover, the small capture cross-sections (relative

* R

RS RET—

i e c vt

27
-128-

 

to the fission cross-section of U-235) make it possible to
allow large concentrations of fission products to build up
without too adverse an effect on the conversion ratio. In
fact,

_ (PP .
a (F.P.) = E%TE?TH- < .12 4,2=1)

Accordingly, if a 10% loss of neutrons to fission products
1s tolerable, the fractional burnup allowed is 53%, for:

fract. burnup v-1-0 (tolerable neutron loss)

ceees  (W2-2)

o
g
-

1.27
5%. 15 X +1 = 53

The direct effect of fission product capture on conversion
ratio is actually as follows:

2 (F.P.)
Loss iI'l XOC.R. = Z__(-gr- (L"02-3)
a

If the radiocactive decay half-life is long, the ex-
traction rate is much larger than the decay rate; on the
other hand, if the decay half-life is short, one can consider
the atom to decay immediately into its daughter isotope, (in
effect, not to exist at all). Then, competing events (viz.,
burnup, processing) are negligibly probable by comparison
with decay. Fortunately, the decay half-lives of the fission
products are either much longer than or much shorter than the
processing half-life, which leaves the processing rate the
dominant quantity in determining fission product concentra-
tions.

4.3 HIGHER ISOTOPES
In contrast to fission products, higher isotopes
have larger capture cross-sections and fission cross-sections

W
T
AR gyt

R

A
0,\ L
»
— -129-

as well. Moreover, the role of capture-decay chains is
much more important in the formation of higher isotope
poisons. (These chains are described in detail in Section
III-3.1.) Therefore, the neutron capture and radioactive
decay half-lives are of the same order of magnitude as
economically feasible processing half-lives. This means
that higher-isotope equilibrium concentrations depend more
strongly on nuclear quantities than do fission product con-
centrations.

For converters, in particular, the formation of U-236
is of prime importance, because chemical processing techniques
cannot be used to remove it from the reactor. Hence, its
concentration will build up to a much greater degree than
those of fission products, thus rendering its effect on the
conversion ratio much more deleterious than fission products,
in view also of its higher capture cross-section (on an
atom-for-atom basis).

A mitigating feature of higher isotopes is their non-
vanishing fission cross-sections. However, this effect does
not nullify the loss due to capture.

If one denotes the fission cross-section of these
elements averaged over the neutron spectrum as Zf(H.I.) and
assumes v neutrons per fission, the loss in X.C.R. is:

(H.I.) = (H.I.)
Loss in X.C.R. = za Z‘ a(2‘5";2f (11'03"'1)

 

Except for meager data in BNL-170, little is known
about the higher uranium isotopes. U-236 was assumed to be
similar to U-238 for all neutron processes except fission.,

In breeders, the principal higher isotope poison 1s
Pu~-240, Experiments at L.A. indicate that

gp(2k0)
< . L"o -
and one may claim on the basis of general nuclear considerations
| L TR
_ —

 

 

P TR
- N
that
(24+0)
%—2—3-87 ~ 1 (%.3-3)
a

4.4 ENGINEERING CONSIDERATIONS

A more detailed discussion of the secular equations
and their solution is contained in the engineering analysis
report, It is evident from Eq. (4.1-1l), however, that
the relations between concentrations under steady-state
conditions will depend only on the ratios of cross-sections.
In general, the ratios of cross-sections vary much less with
energy than the cross-sections themselves. Therefore, it 1s
possible to make estimates of steady-state concentrations
using values of cross-sections taken at the mean neutron
energy of the core. It remains only to choose processing
rates.

Because the processing cost is strongly dependent on
the processing rate, economic factors, as well as design
feasibility factors,come into play in the choice of pro-
cessing rates. These must be balanced against the loss in
conversion ratio and the increased inventory that attend
high poison concentrations. The arguments above show that
this balance will be well on the side of low processing cost
for fission products, because of the ease of processing as
well as the high allowable concentrations. ¥For higher
isotopes, this balance is not so readily struck, because it
is expensive to extract them and expensive to leave them in.
Thus, the higher isotopes are much the more serious poisons
for fast liquid-fuel reactors.

217¢ 130 S

sSsw B oo .e * . . 23cC e
* . e - . e =
 

 

~131-

5. CONTROL METHODS

5.1 GENERAL CONSIDERATIONS

When considering the kinetics of fast reactors, one
is always struck by the disastrously short periods which
would result by even small Instantaneous increases in k.
However, in nature nothing happens instantaneously, and, 1in
fact, in a reactor the speed at which the multiplication
constant of the reactor may change is to a considerable
extent under the control of the designer.

The possible methods of control consist in adding
absorbers, removing fissionable material and changing leakage
by moving parts of the reflector.

As the absorption cross-section of most elements is
less than 1 barn in the fast region, absorption control "
implies moving large amounts of material in short periods of
time. One must also consider that additions of absorbing
material upset the neutron balance and lower the conversion
ratio.

Control by removing amounts of fuel is more effective
in fast reactors, but one is then faced with the problems
of removing large amounts of heat generated within the
control element. U-238 control rods would not destroy the
conversion ratio, but then the rods must be processed from
time to time to recover the valuable product produced.

For large fast reactors, reflector control implies
moving a large mass of reflector in short periods of time, a
difficult mechanical accomplishment. However, reflector
control has the advantage that it does not destroy conversion
ratio, as those neutrons which leak out are captured by the
blanket to produce useful Pu.

One type of fast reactor control which has not been
examined thoroughly but has some merit, is moderation
control. Some moderating material is added to the reactor,
=

-132" M

slowing down the reactor spectrum, and decreasing the reactor
reactivity by essentially raising the overall a. This method
of control would also destroy conversion ratio, and in this
light is not as advantageous as reflector control.

5.2 CONTROL CALCULATIONS

REACTOR KINETICS. - We consider now the time behavior
of a reactor., If n 1s the number of fissions per unit time
occurring in a reactor, then the equations describing the

secular behavior of n are(a)z

éi - (5.2-1b)

§

i
>J

|ul.
Q

e
+

=
=

where B; is the number of delayed neutrons of 1 th type arising
from each fission divided by the number of prompt neutrons so
arising, €{ is the "effectiveness" of the ith type neutron,

S is a neutron source term, L 1s the prompt generation time,

and Cy is related to number of delayed neutron emitters of type 1.

If one calculates the complementary function of (5.2-1)
ot wt

by assuming n = ne" ", ¢y = Cy, € 9 One obtains for w:
hfiili
L -
w=A+F TR (5.2-2a)
or, substituting for A:
B1€12y
Lo = k_ =1+ 24 ———— (5.2-2b)
k@ 1 w + ki

In general, if there are N delayed groups, there wlll
be N+1 solutions of (5.2-2), of which only one can be positive.
The reactor reproduction constant k, is obtained by setting
w = 0 and requiring that k = 1 for this case (steady state).

(a) Equations (5.2-1) and, (5.2-2) are based on: H. Hurwitz,
Nucleonics, %, p. 61 (July 1949).

AR

EC R Pi—

. i

w o~ 29 see v see W . (X}
j . s .. . e v s v @ o ® .
"y ) e @ ae o . * . g - . e
- e 2 s » . o @ » ; - . » -
. . + . s e & * . e .
e an3 s . . ¢ sea . e

 

 

ates

s g
% -133-

Then we have:

0 = Xk, "1‘*23-3%61 (5.2-3)
and thus
ky + 2By € =k, Kzl (5.2-4)

Eliminating kp between (5.2-2b) and (5.2-4) yields:

A
w = k -1+ Zsiei[—-——w,rii - 1% (5.2-5)
L

Now, let us consider a reactor with a fixed k, and
change the effectiveness €1 of the 1% delayed neutron
group. (That this can happen will be shown in the next
section). We wish to find the induced change in o resulting
from changing €; . Therefore, we differentiate equation
(5.2-5) with respect to €; to obtain:

w A\; Bi€: \ )
4 ——J—- — -# _&)_ ° -
L ééej = fi"iwfl\j '} (;Z(wwli)z) IE; (5.2-6a)

or, rearranging:

{L + 7 P& f W = - Bl g,
( (w+A)* 963 a.J+AJ
Now consider a rising period (k» 1l). The brace in (5.2-6b)
is always positive as are all the factors on the right-hand
side. Therefore, the derivative )w/aij 1s negative, so that
if €) decreases, w will increase. A similar argument for
the case when w< 0 (k<1) shows that g-lé";—'<0; thus, the
absolute magnitude of w always increases when ej decreases,
and vice versa.,
Now consider the steady state of the reactor for k<« 1.
(For k » 1, there is no steady state). We set all time-

TR

S Ee—

—
——
——

. . - i 4 e . dee 4 -« -
L & 9 Y @ s 3 s 8 [
(1] . e o » . . . e . . . @
. -y . s ..
« s » @ . s 0
€ . are & 2 re e e
-134- i EEEE

derivatives equal to zero in equation (5.2-1) and eliminate
ci. We obtain:

oo (7

or, after substitution:

 

Pt'E,; S .
™ )h + T =0 (5.2-7a)

n = g (5.2-7Db)
This relation will be useful in calculating heat generation
under subcritical conditions.

We shall now derive some approximate consequences of
(5.2-1) by dropping delayed neutron and spontaneous fission
terms. This procedure is to the point when the reactor 1is
prompt supercritical (an "accident"). Since Ajcy 1s positive,

we have:
n=An:t Z)‘i éi + %) An (5.2-8a)
¢
or
* -1
Bsa- EP_T_ (5.2-8b)

We integrate this equation with respect to time from © to T
(maintaining the inequality), and obtain:

T
In n(T) - In n_ > %j‘(l&)-l) dt (5.2-9)
©
We assume now that:

k,=1+Rt (5.2-10)

i.e., kp is increasing linearly with time and at t - O, the
reactor has just become prompt critical. Equation (5.2=-9)
becomes, for this case:

 

134 | ST

 

 
 

-135-
1n{nnT >-23—I: 2 (5.2-11)

Now, when t = T, we assume that the reactor is well
into the super-prompt critical condition and that delayed
neutrons can be neglected. Then, from (5.2-8a),

a | k -1 RT
Lo (= _E_L - —L') (5.2=12)
By definition, %E% where © 1s the reactor period. There-
fores
fclt'T‘S' = %2 (5.2-13)

Combining this with (5.2-11) (by eliminating T), we obtain:

mifi%fl 7%5 ’ (%{}2 = %"fi : 1—2' (5.2-14%a)

T

or

y
q:>[ L ] = (5.2-14b)
°R In { i

n\e

Finally, assume that a control element which moves
along a coordinate x has a linear coefficlent of reactivity
%}1—: equal to s Assume also that the distance x covered in
time t (from a standstill) under the influence of gravity
will be + g t2. The reactor period will then vary (from

(5.2=12)) as follows:

k--l-Sm-w,i;gt2

1_ & D
- = ENA(t) = T, (5.2-15a)

 

Using (5.2-10), we have, finally,

 

 

 

0
1, k-1 me-3% 2 1, Rt-3Ppt?
T L L = To L
s 000 s (5'2-15b)
~-136-

 

These formulae will be used in Section 5.3.

1,0SS OF DELAYED NEUTRONS. ~ In externally cooled
fluild fuel reactors, the fuel circulates through the core
and the heat exchanger. Accordingly, some of the delayed
neutrons are produced in the core and some are produced in
the heat exchanger. To estimate the control dollaf*hevalua-
tion due to this loss of delayed neutrons, one may assume
uniform progressive flow through the core and heat exchanger.
The fuel spends a time T1 in the core and a time T2 in the
_heat exchanger.

If one assumes that the delayed neutron emitters are
produced uniformly throughout the core, one may write the
fractional number of delayed neutrons produced within the
core during the continuous cycling process at constant power

as:s

(- Yi-&™)

 

6,‘_ = | 3,
Z, |- e"(fo“'zz) (5.2-16)
where Z, = -'.1_-5. and Z, = I.‘%—
L L

T, = mean life of delayed neutron emitter i.

Table I1-5.4-1 summarizes the result of applying
equation (5.2-16) to each delayed neutron emitter, assuming
representative values of Ty = 3.36 seconds and T, = 5.96
seconds.

One dollar =%;eifli’ For all delayed neutrons emitted in
the core@eifii = 0.0073 in Table II-5.4-1 (data from
ORNL-1099).

=36
A .‘ ‘
e == -137-

TABLE II-5.4-1, Delayed Neutrons

 

Half Life (sec) Bi(no flow) €;B4 (flow) €;
0.43 0.0008% . 000687 .818
1,52 | 0.0024 .001236 515
4,51 0.0021 . 000808 .385

22,0 0.0017 . 000614 .361
55.6 0.00026 . 000094 .360
0.0073 .003439

One sees from the table that the delayed neutron dollar
has been deflated to 47 per cent of its original (no flow)
value. In view of the discussion in the section on kinetics,
one notes that for a given k, the reactor period has become
smaller,

One may also compute the generation time from equation
(5.2-5) when w&«X;, as

E ’-‘-‘Z%é‘-:- [, = .035% secs. (5.2-17)

for the fast reactor for small Ak changes. If no delayed
neutrons were lost, the generation time would be

L. = Z_ IG‘;+ |_, = 0942 secs. (5.2-18)

One notes that for reactivity changes greater than one
delayed neutron dollar, the pile period becomes the order of
the mean lifetime of a fast neutron in the reactor. For a
fast reactor

| ‘
L.""viiz‘ 2 10'"6 secs.

It is obvious that such reactivity changes could not be
tolerated in a fast reactor. Furthermore, one notes that the

 

e . s |

e pp——pr—

-
!
- #4 ee 4 ess & ww .
v« @ s P . . .
. " » . . ow . -
nee - s = @ .
* n * -

 
loss of delayed neutrons lowers the upper limit of controllable
reactivity changes by about a factor of 2.

~-138-

DENSITY CHANGES. = It is of some interest to estimate
the amount of reactivity change due to thermal expansion
which will be found in the fused salt reactor. It is ex-
pected that such a reactor will have a large negative
temperature coefficient.

From the one-velocity relation (II-2.4=3)

Oa
) o 0 o — N? ¥ 'b% ’ (5.2-19)

If one assumes that the reactivity, é% is approximately<%3 )

then, approximately

 

8k ody o M2 % ng_@.Ii (5.2-20)
kK~ v b N ~=T
zlt-erf

For the fused salt reactor
NN (1~ .25x 1073T) (5.2-21)

where T 1s in degrees centigrade.

"Iz

fi,dv

Hence ~ 2 = 0,25 x 103 4T (5.2-22)

and the negative temperature coefficient of the fused salt
reactor 1is approximately

2.5 x 1o‘“/°c (5.2-23)

For metallic systems the negative temperature coefficient
due to thermal expansion is of the order of 10'5/°C; while
for an aqueous reactor, the coefficient 1s of the order of
1073 /°c.

It 1s evident that the fused salt reactor has a large
inherent stablilizing feature due to its large and negative

 

L —

Z 3R e

O A TV AT

 
“_:—--_._;___. E | ~-139~

————

density temperature coefficient. It is expected that the
nuclear temperature coefficient will be much smaller.

REFLECTOR CONTROL. -~ To estimate the effectiveness of
reflector control, it is assumed that the removal of a slug
of reflector may be represented by a homogeneous macroscopilc
density change over the entire reflector.

From the results of section II-3.4, it is obvious that
the core radius depends upon the thickness of the reflector.
Accordingly, the effectiveness of the reflector control also
depends upon the thickness of the reflector.

The reflected critical radius is first computed for a
given core buckling (kg fixes the dimensions of the reactor).
Using these same dimensions a new fictitious core buckling,
k2, is computed assuming the reflector density has decreased
by an amount equivalent to the removal of a control rod of
reflector. The change in core buckling can then be con-
veniently expressed as an equivalent change in v.

The change of v so computed will give an approximate
result for the effectiveness of the reflector control, as

Ak ~2 Ay
k v

As an examplé, a lead reflector 2.6 cm thick on core system
No. 15 will give a reactivity change of

Ok o, Ov
kfi-’- vN0.025

if it is completely removed from the reactor. (See Figure
II-3.4-1). |

 

 

 
e Ml ket

 

-140-
M

5.3 CONTROL OF THE FUSED-SALT REACTOR

The fast fused-salt reactor, being composed of a
homogeneous melt, can stand rather considerable overloads in
temperature for short times. This is a distinet advantage
inasmuch as it renders control more feasible and permits of
the use of the quenching action of a temperature rise (via
the negative temperature coefficlent of the reaction
demonstrated in Section 5.2) as a safety mechanism.

CONTROL DURING CHARGING. - Let us consider what sorts
of accldents may occur in the operation of reactor and what
thelr consequences will be. At some interval not known, it
will be necessary to f11ll the core with the uranium chlorilde
melt, and it will also be necessary to start up the reactor
by inserting the liquld lead reflector which is to be installed
to act as a control rod. The lead reflector occuples a space
about 2 feet high and about 2 inches thick over the surface
of the core. The total effect on k of this amount of lead
will be about 3%(8). If we are therefore able to insert
this lead in about 5 minutes, the time rate of change of k
will be .01% sec'lg or R will be 10'” sec™T. (See eq.
(5.2=10)). The rate of filling the core may be calculated
as follows: approximately 1 liter of melt will be equivalent
to the addition of a Sk of .0001. If, therefore, 1 liter of
melt is added per second, the rate of change of k will be
.01% sec™t and R will be 10" sec™! which is the same as
when the lead reflector is moved. The total time for filling
the core will then be 10,000 seconds, or roughly 3 hours.

Using equation (5.2-14b) with the above value of R, and
assuming L to be 10 -~ seconds and g g to be e3° (a rather
pessimistic estimate), we obtain a runaway period of %5 SeC.,
l.e.y, 11 milliseconds.

(a) See the analysis in Section 5.2. The effect derived
there (2.5%)1s for a reflector thickness of 1 inch. A
solid-angle factor has also been included here to take
account of Incomplete covering of the core by the reflector.

 

- A D

 

s 3

 
 

-141-

Under normal circumstances, the intensity of the reactor
and the rate of change of intensity will be observed on
suitable instruments, and the insertion of melt or lead will
be stopped when the reactor is on a stable period of about 10
seconds. However, if by some maloperation the reactor should
run away, we have the problem of stopping its rise before
damage to 1ts structure can occur, when the period is 11
milliseconds. For this purpose valves will be opened which
allow the lead to flow out of the reflector at the same rate
as if it were falling free under the influence of gravity.

To this situation we apply formula (5.2-15b). If the total
movement of the lead (2 feet) i1s equivalent to 3% in k, then
the value of S is 5 x 10~ cm'l. Equation (5.2-15b) shows
that the power will rise to a maximum and then fall rapidly.
The time tm at which it will reach the maximum may be obtained
by setting the left side of equation (5.2-15b) equal to zero
and solving for t.. This gives a time of < .0l seconds
between the instant that the lead starts to fall and the
instant that the reactor power starts to decrease, The
fractional rise during this time may be calculated by
integrating equation (5.2-15b) from t = 0 to t = tm (computed
as above). This gives a factor of rise of about 2. However,
one must consider the fact that after the instruments have
detected a runaway period, there will still be some time

delay while a solenoid valve is operated to drop the lead.
This delay may be estimated as about 30 milliseconds, and
during this time the reactor will rise by a factor of e |(~ 20).
However, instrumentation is now well developed which will
detect a runaway period at a power level which is a thousandth
of full power. A short-period runaway (as calculated above)
may be checked and the reac¢tor shut off by means of the moving
‘lead reflector in a much shorter time than that in which
appreciable heating of the melt would occur. Consequently,
reasonable times for inserting the lead or filling the core seem
to be consistent with safety. '

 

S

|9

*e s e - . * . . . LN ] - * -
L I ) ;e . - w9 & > * & . .
e 3 anm t“i e 2 a * . ¢ o . . « @
e s 9 a *‘,}a XX . « o @
- - - g - - . * . . .
. - » qes - » * & ad . &
———

- It may at some time be necessary to fill the blanket
while the core is already full, and since the blanket has a
powerful reflector action, the possibility of overshooting
criticality during the filling will exist. In order to fill
the blanket safely under these conditions, it should be
arranged that the blanket and the core cannot be filled
simultaneously and that the times to fill the blanket be
about the same. We have shown above that convenient operat-
ing times are quite safe, and in fact, if it were desirable
to be more conservative, considerably longer times for filling
core, blanket, and reflector would probably be operatlionally
tolerable.

CONTROL DURING OPERATION. - We should now consider
accidents which are not connected with the start-up or the
filling of the reactor. A serious possibility of accident
might be anticipated if the reactor were run with a partially
filled core. The rapid rate of circulation of the core
material and the large pump connected therewith might be ex-
pected to shift this level appreciably and thereby cause a
runaway accident at the operating power level. To avert this
possibility, careful precautions should be taken to bleed the
top point of the core so that no pockets of gas can exist
within the core volume itself.

The only other accident which seems at all likely
would be a precipitation of 25 in the heat exchanger as, say,
a scale, and then a rather sudden break-off of this scale,
which, when swept into the core,would cause an increase of k.
In order to change the quantity of U-235 within the core by an
amount sufficient to exceed the delayed neutron fraction,
about 5 liters of break-off scale would have to be trans-
ferred from the heat exchanger to the core. If a sufficlent
amount of this scale were to be swept into the core to drive
the reactor over prompt critical, the time to do so would be
roughly the circulation time of the core, or about one second.
This would give a rate of change of k (i.e., R) of
-143-

 

5 x 1073 sec™l. 1If we now substitute this into equation
(5.2-14b) and take 1n(2$4) as only about 1 (since this
accident would be most dangerous if it occurred at full
power), we agaln get a runaway period of about .0l sec.

This would cause a power surge to about 50 times full(power;
but since its duration would only be .01 sec., the total
surge would be equivalent to an operation at twice power for
one second. This would raise the temperature of the melt by
about 200°C. Such an increase of temperature of the melt
would cause boiling. However, an increase of only a few
degrees in the témperature of the melt would be sufficlent

to pull the k of the reactor below prompt critical,and there-~
by to decrease the intensity and duration of this surge by a
large factor. Under these assumed conditions, the pile would
probably survive this accident without damage. This type of
accident, however, is very difficult to characterize quanti-
tatively, and an analysis in terms of equation (5.2-14b) must
be regarded as very rough indeed. Nevertheless, it indicates
that even a rather large amount of scale formation might be
tolerable as far as this accident is concerned. Moreocever,
even such an approximate analysis as this points clearly to
the desirablility of éontaining materials which will not form
corrosion products which precipitates clearly also, precau-
tions must be taken to remove solid materials as soon as
possible after they are formed,

Naturally, before one attempted a more specific design
for such a reactor, the accident possibilities and con-
sequences would be analyzed in much greater detail than has
been done above. However, a preliminary consideration such
as this indicates that, with reasonable design points and
operating conditions, the Operafion of this reactor would be
quite comparable in safety to that of such thermal reactors
as the Naval reactor or the MIR.

In case of signs of maloperation, it is proposed that
the melt from the core would be drained into a sump. Since

———
—

O s
S .
SU it LT S ————_y
by

€3 et s oFs & tea s
. M I ..

¢ s .

e ® . @

.. . .

» . * . . « o
» * s ] « & B
| . .
-144-

 

there is a considerable cooling problem in the sump, and
since the reason for draining the core might well be mal-
operation of the heat exchangers, it 1s important to estimate
the times that it would take to stop the chain reaction. Any
maloperation in pumping the core would of course scram the
reactor and drop the lead reflector out in the manner des-
cribed before. This would reduce k by 3% during the total
time of fall of the lead, which would be about 1/3 second.
The power due to all sources would decrease by about a factor
of 10, since the power due to fission can drop to much less
than that due to the decay of fission products, at least

over short intervals of time. This can be shown as followss
Equation (5.2-7b) gives the total rate of fissions in a sub-
eritical reactor induced by a source of neutrons. For the
purpose of this calculation, we can consider the delayed
neutrons as originating from a constant source (even though
this source really decays with the mean 1life of delayed
neutrons). The reactor under this assumed circumstance was
suberitical by about .3% when it was running steadily. When
the lead is dropped out, the reactor is then subcritical by
about 3%. The ratio of the fission rates in the two condi-
tions (remembering that there is the same source for both) is,
according to (5.2-7b), just equal to the reciprocal of the
ratio of their "criticality defects", i.e.:

 

N. (after shutdown) 1 -k
= —-————-9-%-1 (5.3-1)
VNb‘Tbefore shutdown) = 1 -k, = 3 = *

This shows that the power will drop to about 1/10th its value
(and subsequently decay exponentially).
B.R.

C.M.
av
' F.P.

~145-

 

GLOSSARY - CHAPTER II

Neutron absorptions in the neutron balance
equations

A(t) 1is related to the reactivity of the reactor in
the reactor kinetlec equations:

1
A(t) = ETE;:TT

One=-velocity neutron flux coefficients as defined
in Figure II-2.4-1

Sum of all neutron losses, except the degradation,
in u group a

Critical radlus of bare critical pile or reactor.
Actually this is the extrapolated critical radius,
no account being made for difference between
extrapolated radius and radius of actual material
to achieve bare critical

Breeding Ratio (see T.B.R.)

Variable related to the number of delayed neutron
emitters, my3 i.e.,

 

€114
C; =
i L vp
where v_ is the number of prompt neutrons per
fissionP

Critical mass of fissionable material
Differential volume element

Fission Products

Ratio of average slowing down density g in energy

group a to slowing down density at bottom of
energy interval qy: -

P
= Fy q'; '(—_FL)a§a

Acceleration of gravity

 

ot Lo

P
dwanee
dhvae
> .
ot
»
-
-
-146-

H.I.

I.B.R.,

I.C.R.

W

&

l‘(a)

K%a)

t

R e e

m(a)

N(s)

R(a)

  

'!’;

*é;-fihuun;-..'

Higher l1lsotopes

:£!K£l.§!;£2£§__- the eigenvalue in the "strong

{ ¢tr) av blanket ’

coupling" blanket calculatlons

Internal Breeding Ratio =
Gross atoms of Pu-239 produced in the core per
neutron absorbed in Pu-239 by fission and
capture in the core

Internal Conversion Ratio =
Gross atoms of Pu=239 produced in the core per
neutron absorbed in U-235 by fission and capture
in the core

Effective reproduction constant

Prompt reproduction constant

Buckling, with atom number density factored out,
for multiplying media in multigroup calculations

One-velocity buckling for multiplying region a

One-velocity buckling for non-multiplying reglon a

Fictitious cross~section representing neutron
leakage

Neutron leakage in the neutron balance equations
Prompt geheration time

Generation time including delayed neutrons

Mass of atom a

Number of fissions per unit time in reactor
Atom number density of material s, atoms per cm3
Neutron production in the neutron balance equations

Time coefficient of prompt reproduction constant

Processing rate, fraction of (a) inventory pro-
cessed per unit time

Source in reactor, i.e., spontaneous fission source

.

- .
sy 38 ¢ & s .e . » * are *e
. . . * F @ » * ® - L 2

 

a0

*soRPS
o
I

X.B.R,

X.C.

X.C.R.

X.C.o
Y(a)

Z

a

o s fi;E“ " -147-

———

A neutron source term in u group a
Centigrade temperature

Time (seconds)

A definite value of time, t

Time spent by circulating fuel in core

Time spent by circulating fuel in pipes and heat
exchangers

Total Breeding Ratio = I.B.R. + X.B.R.

Total Conversion Ratio = X.C.R. + I.C.R.

In (E/E) where B, = 107 e.v.

width of energy interval a 1n lethargy units
Neutron velocity

Coordinate x

External Breeding Ratio =
Gross atoms of Pu~239 produced in the blanket
per neutron absorbed in Pu-239 by fission and
capture in the core

External Conversion Ratio with reflector in two
and three region calculatlons

External Conversion Ratio =
Gross atoms of Pu-239 produced in the blanket
per neutron absorbed in U-~235 by fission and
capture in the core

External Conversion Ratio without reflector in two
and three region calculations

Yield function: atoms of (a) formed per atom
fissioned

T/t5, a dimensionless parameter

Ratio of capture to fission in fissionable nuclei,
for example a(Pu) = o-. (Pu)/g%(Pu). From the
context this is easil? distinguishable from
the following second definition of «a.

 
-148- | @ -

a Index labeling the energy group. In the present
calculations a runs from 1 to 8 (n = 8), the
groups being selected on a logarithmic energy
scale as follows:

as 1 2 3 L 5 6 7 8
us 05"1 1"2g 2"333"3-75 3-75"‘"05,?"'- 5"50 535- 5'7,7‘10
: 4.87,2. 51,.92#,.367, 173, .076, .025,.0048

- _ _fi (),
£ B (5 as)

in maltigroup calculations

, capture to fission ratio for U-235

=2
N
W

!

By Fraction of delayed neutrons of type 1
Y = N@8)/N(M9)
€4 "effectiveness" of ith delayed neutron emitter on
reactivity
3(2 Buckling, with atom number density factored out
for non-multiplying media in multigroup calc
tions

Radiocactive decay constant (sec'l)
Ratio of externally supglied neutrons to blanket

(by leakage from core) per fission source neutron
in the blanket for the driven blanket calculatlion

v(a) Number of neutrons per fission of element a

v Number of neutrons emitted per fission (must be
summed over all fissionable materials).
vo = calculated value of v.

g - Mean logarithmic energy decrement per elastic
collision = average change in lethargy

q

Microscopic cross-section (per atom)s subscripts
same as for

Z Macroscopic cross-section; the subscripts on X
are defined as: tr = transporty a = absorption
(fission + capture); s = elastic scatterings
is= inelastic scatterings f =« fission

Jr Summation sign (to avoid confusion withZ))

 
\“ o
T E > -149-
M—

Reactor Period (sec); defined by Equation II-5.2-13
Mean life of neutron emitter 1 (sec)

One-velocity neutron flux, if subscript a is
missing

Neutron flux integrated over the reactor;-—uSCPdV

Neutron flux at point r integrated over the energy
interval denoted by a

Spatial part of neutron flux, §
Fission yleld function
Fractlion of fission neutrons emitted in energy

range a, crall aXq =1

Probability that a neutron inelastlcally scattered
in group g will be scattered into group a,

ofa()ti)fi—*a =1

A root of the inhour equation
-—_rt -
TETe—

-150- Pt s

ACKNOWLEDGMENTS

 

The physics group of the Nuclear Engineering Project
is greatly indebted to Dr. H. Hurwitz, Dr. R. Ehrlich and
the staff of the Knolls Atomic Power Laboratory for the
assistance, advice, and data they gave for use in the fast
reactor calculations. Through their generosity, the N.E.P.
received invaluable aid in the form of

1). Nuclear Data,
2). Fast Reactor Calculation Methods, and
3). Machine Calculation Assistance

g

15p e
 

-151~
ITI. THERMAL REACTORS
1. INTRODUCTION

In the earlier stages of the project, calculations
were made for fast reactors using Uranium-Bismuth solutions.
It soon became clear, however, that the concentrations of
Uranium in Bismuth available at reasonable temperatures would
not provide a sufficiently fast neutron spectrum. Only by
going to slurries could such a fast spectrum be obtained.
Because of the large number of uncertainties in the behavior
of slurries under reactor operating conditions, in particular,
in regard to their stability, it was concluded that further
investigation of the nuclear properties of such systems would
be of academic interest only.

The available concentrations, as well as the low thermal
capture cross-section of Bi made a U-Bl fuel mixture a strong
contender when the decision to study a non-aqueous thermal
converter was made. At the time no fused salt thermal system
seemed more promising. It was therefore decided to in-
vestigate the nuclear properties of a Uranium-Bismuth-Beryllium
reactor.

A large number of bare homogeneous thermal converters
were studied. The variable parameters were the U to Bi and
Be to Bi ratios as well as various U-235 atomlic enrichment
fractions. The calculations are described in Section 2.
While the results of these early calculations (See Appendix B)
were crude and approximate, they described trends correctly
so that a choice of the optimum reactor could be made on both
physical and economic bases. '

The modification of the results of Section 2 due to the
effects of polsons 1s investigated in Section 3. There we
have studied higher isotope buildup, in particular that of
U-236, the fast neutron reactions of Beryllium and the
poisoning due to fission products.
 

~152- \ |

Actually the conversion ratios of the bare homogeneous
reactors with sufficiently low critical mass turn out too
small for economically feasible operation. This is due to
the loss of neutrons which leak from the reactor. Both the
fast and slow leakage could be reclaimed by the use of a sub-
stantially pure U blanket. It is generally conceded, how-
ever, that a truly multiple region thermal reactor is out of
the question because the increase in conversion ratio does
not compensate for the increased inventory, additional
chemical processing, and the added engineering problems. The
fast leakage, however, can be reclaimed by the simple ex-
pedient of reducing the Be to Bi ratio in a peripheral zone
of specified thickness. The manner by which this is
accomplished is described in Section 4%.1. In Section 4.2,
the effect of Be lumping is briefly considered.

Finally, in Section 5, we report in detall the results
of calculations on the final reactor design.

In connection with the latter, there are some problems,
both nuclear and enginéering, left unfinished, which might
alter the significance of our specific study. There are also
some uncertainties in the nuclear data which might modify
both our particular results as well as the general status of
the type of reactor considered in this chapter. These are
discussed and specific recommendations for further study
made in Section 6.

 

!
*y

-153-

 

2. BARE HOMOGENEOUS REACTOR

The computations for unreflected thermal reactors
utilized simplified approximate expressions. The reactors
were assumed to be spherical and free of poisons. The effect
of poisons 1s considered in Section 3. The results of the
calculations reliably show trends but absolute values (e.g.
critical mass) are uncertain. The calculated values of con=-
version ratios and thermal utilizations are considered re-
liable. The computations are made for neutrons of 760°C.
thermal energy (see Table III-2,1-1).

2.1 DEFINITIONS AND BASIC CONSTANTS
The important nuclear data used in the calculations

of thermal reactors are summarized in Table III-2.1-2.
Thermal neutron properties at room temperature (2.2 km/s.
neutron velocity) have been converted to the assumed operating
temperature of 760°C by means of the conversion factors given
in Table III-2.1-1.

The factors which determine the value of the multi-
plication constant, kbo’ are

a). The number of neutrons produced per neutron
absorbed in uranium, "M s defined by:

— (25)
e fl%‘ gfifi (2.1-1)

where

o= 2e@) L X es) =68+ 26s) ;. Zy(U)=265)+DuGs)
Z 4 (5)
By usling the values of the basic constants given in Table
ITI-2.1-1 and the definition of the atomic enrichment fraction,

namely,
N§2§%

the expression for V] at 760°C may be reduced to:

63R
N = 2.10 3337 + ':1i.1?911-37 (2.1-3)

*n se & F43 & o4& &
. .8 » s s & T
- . . . *e « -
. 408 .0 ¢ e v 33
. v 0 L L ’ [
- * shn gh
-154-

 

TABLE I11-2.1-~1 Temperature Corrections for Nugléar;pata

The computations were made for neutrons of 76000.
thermal energy since a % per cent U in Bi solution
was initially contemplated. In BNL-170 cross-
sections are given for monoenergetic (not Maxwell)
neutrons at 2.2 km./sec. These are thermal
(Maxwell) for 20°C.,, if the cross-section is 1/v.
BNL-170 gives factors for cross-sections not 1/v.

Energy Comparison

Ordinary Thermal in
Thermal Bl Reactor
Temperatures 20°c¢. 760°C.
293°K 1033°K
68°F. 1400°F
527°R 1859°R
Neutron velocity, km./sec. 2,20 .13
Neutron energy, €.V. 0.0250 0.0881
1/v factor 0.5326
(1.04)(3) x 1/v factor 0.5539

-----‘--------_fl----‘—fi---—--fl-

(a) Extr%polated from BNL-170 factor-temperature curve, for
U“"23 -

.54

-

 
hhhhh

""""

......

lllll

lllll

......

oooooo

......

aaaaaa

AN

AR
TABLE 111-2.1-2, Thermal Neutron Properties of Thermal Reactor Constituents,

The cross-sections o are in barns,N in
Values at 760°C calculated from thermal data by means of

factors given in Table III-2.1-1.

(atoms/cm3) x 10~2,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nuclear U-235 | U-238 Bi Be
Parameters), 2 um/s. 760°C || 2.2 km/s. 760°C| 2.2 km/s.| 760°C | 2.2 km/s.| 760°C
Op 549 303
o, 101 60 2.8 1.%91 || 0.016 0.00852 [ 0.009 0.00%473
op + 0, 650 363 2.8 1.%91 [ 0.016 0.00852 | 0.009 0. 00473
o 9.3 9.3 || 9.2 9.2 6.1 6.1
Oy 9.3 9.3 || 9.2 9.2 5.67 5.67
(Za,) 0.088 0.088 [1.23 1.23
a 0.183 0.197
v 12.51 2.51
n 2,12 2.10
p gm/ce 9.#6(a) 1.85(b) 1.798(°)
N 0.0272 0.1199
App(cm) 3.997 [|1.3'P) | 1um
xg (cm) 3.997 1.368
2 (em™ | 0,250 0.731
(a)

(b)  prom BNL-170
(c) Corrected density assuming 3.66 x 10'5/°C. volumetric expansion, Esbach.

Interpolated, Table I, p. 31, "Liquid Metals Handbook."

-GG1-

. iy
;“‘ . ) . .
~156~ *\ *

The average absorption cross-section per uranium atom
is

dz(U) = 363 R + 1.49 (1-R) (2.1=4)

In Table III-2.1-3 1 and o are tabulated for various
values of R.

b). The thermal utilization, f, is defined ass

. Z.W) (2.1-5)
f - 2o o (4at)

where: Zzgtot.) = ZJfU) + Z(structure, coolant, moder-
ator).

The cross-sections used ln the calculations ofvland f
are quite sensitive to the thermal neutron temperature. In
general the data for Bi are less reliable than those for Be
and U. In particular the value of o, (Bi) = 0.016 barn,
accepted during the early stages of the Project and used in
most of the thermal calculations, is probably low by a factor
of two. The higher and more reliable value Gb(Bi) = 0,032 |
barn was obtalned later and was used only in the final design
calculations of the thermal reactor.

¢). The resonance escape probability, p, was determined
from the expression

In (1/p) « BP0 (2.1-6)
where A is the experimental resonance absorption integral:

A =(fSa (238) 4)
eff

and EE?s represents the quantity &; N, (G‘s)1 summed over all
the 1 atoms. The scattering cross-sections were evaluated
near 7 ev, the largest, lowest lying U-238 resonance.

Two curves of A vs. "o_/U", the total scattering cross-
section per atom of U-238, are plotted in Figure III-2.1-1.
The optimistic values of p thus derived and used in the
calculations slightly underestimate the conversion ratlos.
However, they more stroggly affe¢t the multiplication constant
and the critical sizes &nd masses.

e asm o
6 .. < . o‘ cow :
. ® L I . s & @
.. . .W . . s 2
PA 8 i Nty "-W"Mn o ..

'
il

»t!
TABLE IIT-2.1-3.

R
atom fract.

0.01
0.02
0.03
0.0k
0.05
0.06
0.07
0.08
0.09
0.10

1.00

 

(For Maxwell neutrons, 760°)

Ta(U)

barns

5.106

8.721
12.336
15.951
19.566
23.182
27.797
30.412
34,027
36.642

363

L I

Ny

1.4929
1.7482
1.8538

1.9116

1.94%80
1.9730
1.9913
2.0053
2.,0163
2.0251

 

. .
iiiiiii
-

Zio(U) /Z3(235)

-157-

Properties of Isotopic Uranium Mixtures

(for conversion ratio)

1.4+066
1.2013
1.1328
1.0986
1.0780
1,064k
1.0546
1.0472
1.0415
1.0370

1.00
1
o

vy

¥ *
Teeaey

/S¥

 

 

~100 -

| This optimistic curve was

i used in the calculations. ’
£

b 3 -
s

L
3,
I<|

=10

 

T T 1T T 1T

FIGURE III-2.1-1l. Experimental Resonance Absorption Integral, A, versus "a—s/U".
The value A = 9.25 barns for U-238 metal is from CL-697.

and the asymptotic value for infinite dilution, 240
For "rst" > 2000 barns, the curve was faired into the asymptote.

The lower curve, based on"the more recent asymptotic value given in BNL-170, was used
The straight line portion of this curve is from ci'-51-5-98.
For "r’/U"> 2000 barns, this curve was also faired into the asymptote.

also from CL-697.
in the NEP calculations.

The straight line portion

barns, of the upper curve are

-8g1-

 

 

 

 

 

 

1
iu- METAL, 9.25b

 

 

1 1.1

 

1 y o3 1111

 

1 L1 & 111

 

L1l

 

100

"0 /U" barns

10000

00000

e A
4y

 

i,
S T ——

 

 

 

 

 

-159-
N ——"
TABLE I11-2,1-k4
Resonance Escape Probabillgz
No/ig, = 0.01 0.02 0.03 0.0k
NBe/NBi | - Resonance Escape Probability, p
1 0.541 0.410 0.329 0.273
2 .69 . 583 . 510 455
L .80 .722 .656 .612
6 . 849 . 776 . 721 677
8 .876 .802 . 764 «733
10 .893 .836 791 752
12 « 907 .852 .81k .780
14 .916 .867 .831 . 784
20 « 936 . 894 . 864 .838

In Table III-2.1-k are given the values of p correspond-
ing to different values of the ratio NBe/NBi for NU/NB1 =
= 0.01, 0.02, 0.03, 0.0%. These p values correspond to an
enrichment fraction of 0.05. Since the enrichment has little
effect on the values of p, the same values were used for
R = 0.03, 0.05, 0.07.

d). The fast fission factor,€ , was assumed to be
equal to unity in the calculations.

2.2 ANALYTICAL EXPRESSTONS

a). The thermal diffusion area, 12 1s given by the

expression

L = |
'E.' 3 Z:tr Z“’ - (2.2=1)
in which Z,. and Za represent, respectively, the quantities
Ni(d'{;r)i and Ni(aa)12summed over all the i atoms.
The value of L= 1s dependent on the energy of the
thermal neutrons. In the calculations thils energy corresponded
o e e

» *en * » * .s s & T a2 »
[ L +« ¢ @ *
. [
R

-160~ --"-'--—--

g

to 760°C. (T0 = 1033°K) or 0.0883 ev. Through L the
critical radil R, and masses Mc will vary with the temperature
T according to the relations

R(D)~ R, (7)) {1 - + 4 123 (2.2-2)

M (D) a2 M (1) {1 - 32 18P} (2.2-3)

in which A L2 = L2(T0) - 1%(T). If it is assumed that the

absorption cross-sections vary as 1/v, L(T) = L(To) {T/Tag%.
Hence

A L2 - L2(TO) {1- T/1.} (2.2-}4)

b). The age to thermal, 1.=- In Table III-2.2-1 are
given the values of « which were used in the calculations
together with additional values, denoted by Tqs obtained by
a detailed calculation.

The values used in the calculations were obtained from
the crude expression

(Ezgtrzzs)Bi
(EztrzsjBe

in which t(Be), the age to thermal (760°C) in Be, was taken
to be 86.5 cm2.

In the detailed calculations Tq wWas determined by
numerical integration of

E
S m d(1nE)
3L T,

(2.2=5)

T = T(Be)

Ta
Etn
in which

=
"
N

MeV, Eth = 000883 eV,

(NEOo)g, + (N E G )py
= (NG, + (N

?Er)Bi’
-}

r)Be

.

LIRS

!
£y

   

e -161-

TABLE IIT-2.2=-1

Age to Thermal

Be/NBi T T4
1 895cm2 779cm2
2 472 431
Y4 266 256
6 202
8 171
10 133
12 141
14 13
20 11

Check calculation: +
td(Be) to indium resonance = 78.1 (= 80-2, BNL-170)

In these calculations g3 for both Be and Bi was assumed
equal to @ and the curves of o vs. E in SENP I, (33), pp.#02-403,
h98-h99, were used. As a check on the calculations the value
of ©t(Be) to the indium resonance was calculated. The value
obtained agreed with the value 802 cm® given in BNL-170.
Although the values of t used in the calculations are
somewhat higher than those obtalned by the detailed calcula-
tions, the discrepancy is appreciable only for low values of
the ratio NBe/NBi’ Furthermore, any error in v will affect
the critical radii and masses but will alter the conversion
ratio only slightly. Better values of the critical radii and
masses than those used in the calculations may be obtained
from the relations

Ryg=R, (1 - % At/M) (2.2-7)
M, (1 - 3 Oc/M2) (2.2-8)

where Rc and Mc are given in the tables included as Appendix B,
and

Mg =

M2 = 12 + T 3 b=z -oT4
‘162“ ARy - «

¢). Density.- Since in the reactor there is a mass of
Be in Bi rather than a solution, the densities were calculated
on the assumption that the volumes of Be and Bl were additive.
It was further assumed that the uranium (1 - 4%4) dissolved in
Bi without increasing the volume.

The densities of Be and Bl given in Table III-2.2-2 are
for a neutron temperature of 760°C. which corresponds to a
solution of 4 atomic per cent U in Bi. For lower U to Bi
ratios the densitlies will be somewhat highéf than those
tabulated because of the lower eutectic temperature.

d). Critical radii and masses. - The critical radiil
R, given in Appendix B, Tables B-1 to B-ll were calculated
from the relation

R, =7 (22 + ¥opnh (2.2-9)

The critical masses Mc were calculated using (2.2-9) and the
appropriate density and composition.

The expression for the c¢ritical radii is crude and
optimistic. Better values of R, and M, than those used in
the calculations may be obtained by using the results of
two-group calculations. Denoting by Bc(a) and MC(Z),
respectively, the critical radii and masses obtained from

the two-group calculations, 1t is found that

Rc(a) = B, §2 u™t ({170 - 1)} o (2.2-10)
and ,
M ?) o n {2 v (VL - 1)} -3/2  (2.2-11)
where
u o= % (k1) B2 (1-gD)

.52-_.1._1'_
= 2

L+t - M2 ’

 

o 3 — —
—r—— .‘WM‘

,,,,,,
 

 

 

 

4

¥ mgtoms" = No. of atoms x 10~

2k

153

————— -163=-
TABLE III-2.2-2. Age and Density of Bi-Be Mixtures
(assume additive volume, 760°C.)
Atom ratio Age Be Bi
N(Be)/N(Bi) <t cm? _ gm/cc Matoms"/oc¥ gm/cc  "atoms'/cc
1l 895 0.333 0.0222 7.71 0.0222
2 Y72 0.561 0.0374 6.51 0.0187
4 266 0.856 0.0571 4.96 0.0142
6 202 1.036 0.0691 4.00 0.0115
8 171 1.160 0.0773 3.36 0.00966
10 153 1.248 0.0832 2.90 0.00832
12 141 1.316 0.0878 2.54 0.00731
14 133 1.366 0.0911 2.26 0.00650
20 118 1.474% 0.0983 1.71 0.00492
0 - - - 9.46 0.0272
Qo 86.5 1.798 0.1199 - -
Ry
-164- '

2.3 THE CONVERSION RATIO, C.R.

The conversion ratio consists of two parts, the
contribution due to resonance capture and that due to thermal
capture. The former is merely

Neutrons produced
Neutrons absorbed in U-235 * fast leakage escape probability

X resonance capture probability

 

2
v 1 v v
= T3g 1:;;(2'( (;--pl)c = Tro (1-p) ; Tra i:% (1-p)
U
v La 0 v_ =X (1o -
= Ta "5 (35 T " Tra @ P (2.3-1)
in which

X = b(fizu“l (1 - 1)} bfz S

the latter 1s
Neutrons absorbed in U-238 _ 22(238) 3, (0
Neutrons absorbed in U-23 jf‘Cng) w‘E§;T§§§T
"0 e (2.3-2)

The conversion ratio can then be written in two useful forms:
First by adding the initial expression in (2.3-1) to (2.3-2),

v 1 Za (U)
C.R. = i;;';:;gg— (1-p) + 3521533) -1, (2.3=3)

we have a direct expression in terms of U-238 captures second,
by employing the last form of (2.3-1) to rewrite (2.3-3), we
easily find

C.R. = =0— = 1 = Za(U) (..].-._1) - Za(U) M
*t = Jta Za<23§5 f Zaz2335 £
2
v_z¥® 4 5. 3.7

The difference of the terms

v
Co:m-l:l.lo
~ -165~

and

>, (0 1
P - 35215537 ( - 1)

represents the conversion ratio for an infinite, clean
thermal reactor at steady-state critical. The last two
terms of (2.3-3) represent, respectively, the slow leakage
loss and the fast leakage loss.

2.4% RESULTS OF CALCULATIONS

The results of the calculations on the poison-free
reactors are summarized in Appendix B, Tables B-1 to B-1ll.
The cases treated are itemized below.

 

 

 

Table Nyy/Npy _JE_
B-1 0.0k 0.03
B-2 . O4 .05
B-3 Ol .07
B-4 0.03 0.03
B-5 .03 .05
B=6 .03 | .07
B-7 0.02 0.03
B-8 .02 .05
B-9 .02 .07
B-~10 0.01 0.03
B-11 .01 .07

 
~-166- o o

3. POISONING EFFECTS

3.1 URANIUM-236

U~-236 1s generated in thermal reactors by non-fission
-~ neutron capture in U-235. It has a half-life of 107 years.,
‘In converters operating at 10 to 105 watts/gm. it reaches
equilibrium in approximately 10 to 1 years, respectively
(see Aprendix C). At equilibrium there are 10 to 15 atoms of
U~-236 to 1 atom of U-235. This concentration of U-236, if it
does not shut down the reactor, will reduce the conversion
ratio substantially. .

U-236 generates a chain leading to Pu(238) which has a
spontaneous fission rate 6.8 times that of Pu(240). The re-
action equations for this chain, together with those of the
U-238 chain leading to Pu(240) are given in Figure III-3.1-1.
The general chain scheme is outlined in Figure III-3.1-2.

The secular equations for both chains are considered in
Appendix C.

The effect of the U-236 on the conversion ratio is two=-
fold: It introduces a factor 1 - N(2§é?3E)N(238) in the
contribution due to the resonance capture (Section 2.3), if
we assume the same resonance structure for U-236 as for
U-238, and adds the term.-%%éégggé to the contribution from
thermal capture. The resu %ng expression for the conversion

ratio is then

z, (0 Kol v (1-—921:)&2
C.R. = CO - P -W —F = T+a l+'l:3&2 + S(C.R.)

s o000 e (301-1)
in which |

Zg(230) 3. N(236
$er) = - Py - 125 wme) SNGTE (3012)
TS

R . e
‘ Y v
. ¥ P .

FIGURE IIT1-3,1-1

Reactlon Equations

Lower chain Upper chain
/U(236)
U(235)-+11\‘ U0(238) + n—1U(239)

fission fragments

U(239) 23220, Ny (239)

U(236) + n —~U(237) U(239) + n —U(240)

u(237) 284, Np(237) B(2koy ¥ 1 oy (B)
U(237) + n —U(238) (&)
Np(237) + n —s Np(238) Np(239) 239, pu(239)
Np(239) + n-Np(240) (¢)
Np(238) —2:12,_ py(238)
Np(238) + n—fission fragments Pu(239) + n<Pu(2’+O)
fission
/ Pu(239) fragments
Pu(238) + n
\fission fragments Pu(240) + n — Pu(241)
(a) Chain is stopped here since U(238) 1is loaded into the re-

actor for conversion

Chain 1s stopped here since no data are available for
Np (240)

(c) Chain i1s stopped here since no data are available for
Np(239) 3
-168-

 

 

 

 

 

 

239
23 m

 

 

 

 

 

238
109 y

 

 

 

start oven

 

)

 

 

237
6.8 d

 

 

 

 

 

 

 

 

 

 

 

 

. e Lt e
" PR .t s -
— - v

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:
M
ome——— .  FIGURE III-3.1-=2
Higher Isotope Chains
1+ h ?
240 c 241
X ' /1l 12y
F
23.5 m * | \\ ? l
} ~
C 239 c 240
F
F
* )
| ?
i
c | 238 o 239
; 2.1 d ; 10)+ y
F F
6.8 4 T 2.1 d 1
C 237 C 238
'e 106 /7 90
s y s~ y
F F
N,Y
Cc
B
C = Chemical Removal
F = Fission Removal
X, = = - Small or Uncertain
93 ol
N Pu

p
e e e o e
M : .

 

 
- — -169-
- A"
It should be noted, however, that the effect of the U-236 is
not merely to add the correction term (3.1-2) to the formula
(2.3-3) but also to change the values of p and f in the entire
expression.
When the effect of the U-236 is included, the defini-
tion of the thermal utilization becomes

— _ Z, (U)
= Za(0) + I, (236) + Z, (structure,coolant, moderator)

f
in whichZ'a(U) :Z‘a(235) +Z‘a(238) only (i.e. U-236 is con-
sidered as a parasite). The change in the thermal utilization
due to the presence of U-236 is approximately

 

(oL 2Ta®0) 7,39 Z,(36)
=Tt @ T n @ x,@3%

EEEEEREN (301-3)

where f is given in the tables of Appendix B and

2,(235)/ 72, (U) in Table III-2.1-3.
The expression for the resonance sscape probability

In (vp) - MEGHE . A (3.1-4)

where A, the experimental resonance absorption integral, 1is

now computed by summing the number of U-23§ and U-238 atoms
s s

(i.e. A is considered as a function of N(238) + N(236) ).

An approximate expression for the change in p due to U-236 1is

Jp = —H 536 P lnp (3.1-5)

where p is given in the tables of Appendix B.

For a first approximation to the reduction of the con-
version ratio due to the presence of U-236 consider (3.1-2).
A pessimistic approximation is obtained by taking 1 + 1:3[2

becomes

 

. . . . os o® 5 s4® & ang o t
. . @ . L - . * * . & [ .‘i.‘
» - . s .
. (X1}

¢ @ o « se 4§
¢« v "'..‘..:b
o ===

as equal to unity. Then

o, (236) N(236
§(C.R.) = -Zfi—@gy - 2.1 (1-p) §{(336) + W(23D)

ceees (3.1-6)

where v/l1+a = 2.1. Expanding 1ln p in (3.1-5) by'means of
the expression

-1)° -1)3
Inps(p-1) - (921) + (p31)

and combining the resulting expression with (3.1-6) yields

Z, (236) N(236 N(236
S(C.R.) = -W - 2.1 m (1l-p)(1+p N(23 ))

s o000 e (301-7)

which is somewhat more pessimistic than (3.1-6). In Table
III-3.1-1, - §(C.R.) is given as a function of p and the
ratio N(236)/N(235) for various atomic enrichment fractions.
The assumed low values of the ratio are less than saturation
(see Appendix C). They must be maintained by diluting the
reactor effluent with diffusion plant stream. 1In a few cases
the loss exceeds the upper limit, 0.155% (see equation
(3.1-11), below)., This is because (3.1-7) is rather crude
and pessimistic. The table shows trends satisfactorily.

The upper limit to the change in the C.R. due to U-236
may be found as follows: Write

b’ N(236) a-'eff(236)
‘ (C.RJ) = - N(235) a-, (235) (3.1=-8)

 

where <réff(236) is the effective absorption cross-section.
It depends on the resonance structure of U-236 which, un-
fortunately, is not known. This upper limit pertains to a
reactor from which no uranium is being purged, and is un-
realistic since any practical reactor will have such a purge
stream,

The secular equation for U-236 buildup in the absence
of a purge stream is:

 

Wt
ffeet of U-236 on Conversion Ratio in Thermal Reactors

- W e % S = ap o= B S TR S o T =W o s N A M O S - W = @ =-.

.05

« 07

 

m mmg

TABLE ITIT-3.1-1

0.7 0.8 0.9
- § (C.R.)
.0287 .0225 .0159
.0570 o Ollely .0317
.1130 .0884 . 0634
L0419 .0312 . 0204
.0835 .0623 . 0408
. 1640 .1220 .0811
.0558 . 0405 .0251
.1100 .0803 .0501
2140 .1580 .0992
W

-171-
 

-172- —_—

QIQ*

= N(236) = N(235) 05 (235) @ - N(236) gop,(236) ¢

s s e os 00 (301“9)

Hence, at equilibrium,

N(236) _ €c(235)
N(235) = g2, (238) (3.1-10)

Substitution of (3.1-10) into (3.1-8) then gives

-6 (235)
YGRS "'a:im = - 7%= = - 0.155% at equilibrium

cssassee (3-1“"11)

where ¢ = ¢ (235)/ g% (235).

 
et Y " L LT

i

'

-173-

3.2 FAST NEUTRON REACTIONS OF BERYLLIUM

Beryllium has two fast neutron reactions which affect
the conversion ratio of the U-Be-Bi thermal reactors, namely,
the (n,a) and (n,2n) reactions.

The (n,a) reaction has a threshold at 0.8 Mev. The
cross-section rises to a maximum at 2.6 Mev and then remains
roughly constant with a value of approximately 0.045 barns
up to the largest energy measured (about 4 Mev). For a plot
of 0(n,a) vs. energy, see R. K. Adair, Rev. Mod. Phys. 22,
257 (1950), or AECU-2040,

The threshold of the endothermic (n,2n) reaction is
1.63 Mev., The cross-section data are unfortunately both
sparse and contradictory. Measurements using Ra-Be and Po-Be
sources yield values ranging from 0.3 to 3.6 barns. In order
to obtain a rough estimate of the effect of thls reaction,
the cross-section was assumed to rise linearly with energy
from the threshold to a value of 0.3 barns at 4 Mev. This
assumption undoubtedly underestimates the rapidity with
which the cross-section rises with energy.

The (n,a) reaction constitutes a negative fast effect
while the (n,2n) reaction is a positive fast effect. It
turns out that the two effects Just about compensate one
another. In fact, there is a likelihood of a small net
positive effect, although accurate predictions are impossible
with the available cross-section data. Hence, at the worst,
there is no loss in the conversion ratio which 1is directly
attributable to the fast neutron reactions. The pertinent
calculations are given in Appendix D.

The (n,a) reaction, n + Be9 -+ a + He6, indirectly pro-
duces a reduction in the conversion ratio because of the de-
cay of He® to 118, He® B1(0:85 sec.) 136, 13% nas a thermal
(0.025 ev) absorption cross-section of 910 barns and is
potentially a serious poison. The average loss in C.R. due to
this source is

 

 

 

 
-174-

 

S(CoRo) - - Z 23 | (302-1)

where the bar indicates a time average over the operating
period for the Be in the reactor. The considerations of
Appendix D show that (3.2-1) may be written

S (C.R) = - g 7%k %"U (3.2-2)

where g is less than unity and J < 0.027.

3.3 FISSION PRODUCTS

It was shown in Section II-L4.1 that the net rate of
growth of concentration of a poison "a" is given by

N(a) = Z, Y(2)@ + o5(8) NBYP + n(a') N(a')

Q-:IQ‘-
c+

- 0(a) N(a)P - A(a) N(a) - R(a) N(a).  (II-%,1-1)

The poisons consist of direct fission products, nuclei
resulting from their decay and higher isotopes. The worst
of the higher isotopes, namely, U-236, has been discussed in
Section 3.1. Most of the direct fission products, e.g.,
Zirconium and Molybdenum, have capture cross-sections which
are small compared to the Uranium-235 fission cross-section
at thermal energies, and it is not difficult to maintain the
steady-state concentration of these fission products low
enough by continuous processing so that their poisoning effect
is negligible. In many cases the time required for these
fission products to bulld up to steady-state concentration is
s0 long that their poisoning effect is negligitle during the
operating period of the reactor.

Certain of the nuclei resulting from the decay of
fission products, notably Xenon-135 and Samarium-149, have
relatively high thermal capture cross-sections. These decay
products with high cross-sections usually build up relatively
soon to steady-state concentration, and this concentration
must be maintained low enough to reduce the effect on
criticality and conversion ratio by asproperrchoice of the

n74

 
   
 
Wy e -

e -175-
processing rate (R in Equation II-4.1-1).
As an example, consider Xenon-l35. This is found in a
direct yield of 0.002 atoms per atom fissioned, but a much

larger amount results from the two-stage decay of the direct
product Tellurium-135, whose yield is 0.065. Since the half-

1ife of Tellurium-135 is only 2 min. and that of Iodine-135,
the first decay product, is only 6.7 hr., it is assumed for
simplicity that Xenon-135, the second decay product, is a
direct fission product with a yleld of 0.067.

The net rate of formation of iodine-135 is given by

%;I_ = Z,Y, 9 - PT-ATX (3.3-1)

Setting B = A,+6; 9, the solution of (3.3-1) is
Tw= NZe¥ (™) (3.3-2)

where it has been assumed that I(o) = O.
The net rate of formation of xenon-135 is given by

dX = AT +ZY,? - A X - X -RX

It

~+

or

Q_

__x_ +AX = A|I- +Z§YZ¢
dt | (3.3-3)

where A= R + }‘2 + 0-22? » The solution of (3.3-3) is

 

X(t) = 2-%?%2 * il‘;l - (- 'i:'lg'l'fi) -At
o a(Mh e-Bt}
B\L - B
O
L -
e —

- * -
* 4k.e -« a s
s e
i
-4

-

; n'-’(‘nffi--m :
~-176~- M‘é
Z. P A Y -(R+A+0L,P) T
X() = Forespte - Yz'iaT-l-xla-—qSe e
2109 2=t oy
(R+J\2+o'2 ¢ ) -2t }

- W (3.3=k4)

where 0;25 has been neglected in comparison to A; in

B=X; +67@ . It is clear from expression (3.3-4) that by
making R large enough the concentration of Xe-~1l35 may be
kept as low as desired. The poison calculations are carried
out in the engineering analysls report.

 

},;Srh’v‘“\"'!f
o
' i

L8
 

— -177~

. HETEROGENEOUS REACTORS

4.1 THE "IMMODERATE" BLANKET

The loss in conversion ratio due to both fast and
slow leakage constitutes a serious problem for the one-region
thermal reactor. Yet it is generally conceded that a
multiple region thermal converter cannot compete economically
with a fast converter because of the intrinsically smaller
conversion ratio of the former. The difficulty is that the
leakage loss increases rapidly with increasing multiplication
constant, i.e., with decreasing reactor size.

In order to achieve a deeper understanding of the
problem of obtalining a nice economic balance between in-
ventory and conversion ratio, consider the expression for
the conversion ratio (2.3-20). From Table I1I-2.1-3 it is
clear th%E ?%? contribution to the C.R. from the thermal U-238

a
capture,zi-z§§§7 - 1, is less than O.% in general, and less

than 0.2 f%r the enrichment ratios of interest., Clearly,
most of the C.R. comes from resonance capture. The problem,
therefore, is to maximize the resonance capture in a manner
consistent with the other requirements of reactor operation,

The only way to reclaim both thermal and fast leakage
is to use a pure U blanket. Since thls has been assumed
economically unfeasible, the following practical expedient
was suggested: Since the resonance capture increases as the
moderating material is removed, it 1s possible to establish
a high resonance capture, low multiplication zone on the out-
side of the reactor, using the same fuel mixture, merely by
radically decreasing the Be/Bi ratio in this region. ©Such a
region has been termed the "immoderate" blanket.
-178- L —

The requirements for the "immoderate" blanket are as
follows:

(a) It must be such as to maximize the resonance
capture, i.e. p<<1l. This can be accomplished by decreasing
the Be/Bi ratio. It follows, then that k _<<1 so that the
blanket is a non-multiplying region.

(b) If the thickness of the blanket is denoted by v,
then y22F in order to minimize the fast leakage from the
reactor. As may be seen from Figure 4.1-1, the age of a
Be-Bi mixture (and, hence, the thickness of the blanket) in-
creases rapidly as the Be/Bi ratio decreases below unity.
However, since the immoderate blanket does not provide much
in the way of reflector savings, it cannot be made too thick
because of the excessive inventory. There is therefore a
practical lower limit on the Be/Bi ratio.

Since the immoderate blanket contains fissionable
material, it was necessary to generalize the usual two-group,
two-region treatment. There are two equations of the form

TV, - ¢ o+ va ‘P;"O
2ee - 49+ P‘z';‘:"Pz =0

(+.1-1)

for each region. 1In these equations the fast fission effect,
&, has been neglected, CPLis the flux in the 1D energy
group (1 = 1 represents the thermal group and i = 2 the epi-
thermal group), :Z; 1s the total thermal absorptiofi cross-
section and Z,_"‘- 'fl%(éjrfifi s with Ej = 2 Mev, is the total
"fast absorption" cross-section. The solution of the equa-
tions is carried out in Appendix E,

The expression for the conversion ratio in the two-

AL

|

e T aileele
- -
"o - i
- - » c
. » i ¥
seemn
- . 5
aasave ¥
- -
sendas -k
. . 3,
- -
cane
. )
- L I
AeP B ge
thae
. -
sedaes i
e deE

 

|
 

-179-
f_"&-.\.

region reactor 1s also derived in Appendix E. It may be
written

CR = CG-P—-§ +§ -5, +§;

 

2« (235) oc 3 S"‘
(o) ) o @

S = ZG(U) koo - S? = e S P -
° Zia (235) § (o) S(¢,\+ 'S‘m

The superscripts 0, 1 refer to the core and blanket,
respectively. S represents the (fast) neutron source and
Lf the fast neutron leakage. The other symbols have their
usual meaning.

Expression (2.1-20) for the conversion ratio of a bare
homogeneous reactor may be written

v T’b‘z

C.R.(bare reactor) = C, - P - So - Tva W(l-p).

ceceeess (WH.1-3)

Note that the first three terms of (%.1-2) and (%.1-3)
are identical. It turns out that for all cases considered
the last three terms of (4.1-2) contribute less than 1% to.

the conversion ratio. This result depends on the cholce of
blanket size which was taken for engineering reasons as two
g;{
I:‘ [ e e .
T p——n e —

-180~
ra———

feet, somewhat less than 2{t. For such a blanket then, an
excellent approximation and a lower limit to the conversion
ratio can be obtained from the formula

C.R. & C - P =~ § (. 1-4)

Comparing (%.1-4) with (4.1-3), we see that the net
- effect of the blanket is to reclaim the fast leakage loss
from the bare reactor. It must be recalled, however, that
this gain is achieved at the expense of inventory and that
the blanket has some fast leakage of its own. Whether there
is a net gain in inventory over the larger homogeneous
reactor with the same conversion ratio is a priori a moot
point. In Section 5, we demonstrate that the blanket

achieves an actual saving.

W F %
§ ———— - AR 0—-——"-5

g. f'—':"“:’*"" s e v G i
% s | o 3

L0
 

-181-

 

4,2 THE EFFECT OF BE LUMPING

In the discussion thus far, the reactor has been con-
sidered as a homogeneous mixture of U, Bi and Be. Lumping
the Be into rod form affects the multiplication constant
Ko zneps-

The effect of Be lumping on the thermal utilization f
has been investigated by means of the standard cellular
approximation and the application of diffusion theory as
described by Weinberg in "The Science and Engineering of
Nuclear Power", Vol. II, Chap. 6 and CF-51-5-98, Vol. I,
Chap. IV. The results of calculations for several reactors
in Table B-7 are summarized in Table III-4.2-1. In Figure
ITI-4.2-1,f has been plotted as a function of r_, the radius
of the Be rod, for different ratios x = Ny /Ng,.

As ry O, the values of f should approach those for
the homogeneous distribution. This transition takes place as
soon as the dimensions of the lumped regions become of the
order of magnitude of the transport mean free path. How-
ever, the calculated values of f formed smooth curves which
gave for r, - O values greater than those for the homogeneous
distributions. This behavior is characteristic of the diffu-
sion theory approximation which does not sufficiently accentuate
the depression of the flux in the fuel mixture relative to
that in the Be and which, therefore, gives values of f
greater than the true values. In order to obtain more re-
liable results, it was assumed that the shape of the curves
of f vs. r, was correct so that the correct curves could be
obtained by renormalizing the curves for the correct value
of £ at r, = 0. It is these renormalized values of f which
are given in Table III-4.2-1 and plotted in Figure III-L4,2-1,

The resonance escape probability, p, was calculated
from the expreSsion

N(U)
b eXp{- 8% efr AS

 
in which A is the effective resonance absorption integral
and

(EZ,) pp = E(Be)Z ((Be) —ig,— + £(B1) Z(B1)

|._.I

where

Vo  N(Be Bi
7, N%Big p%Be%

average flux in Be rod

o

|
"

average flux in Bi-U solution

Figure III-4.2-2, summarizing the results of the calculations,
gives curves of the product pf as a function of the radius of

the Be rod.

 

 
- TABLE TIT

 

I-4.2-1A. Results of Cell Calculations

-183-

 

 

 

 

 

 

 

 

 

- X = H%%%% 1 2 3 IR
- f: ry =0 cm - .932 .899 . 869
2 .961 .949 .905 842

I . 957 945 .901 .836

6 .950 .938 . 895 .830

f(renormalized) - .932 .899 .869

- .915 . 880 . 842

- .911 0871 0836

- - 0905 .868 0830
) p: Tz O - .583 722 .776
2 L66 .626 o« 743 «799
b . 502 o 642 751 « S04

6 . 552 665 . 762 .810
pf: vy = O - . 543 .649 674
2 48 «573 «654 673
L 480 . 585 . 654 672
6 . 524 . 602 .661 .672
N(U

1.8533
iiiiii

* L
OOOOOO

......

lllll

-
llll

.....

 

 

TABLE III-4.2-1B, Required Data for Evaluation of p
. . . 4| ==
N(Be)/N(B1)| Z, (Be) Xo  |op(BD) | EOL(BY) | (M) G (BD)| 47| T (V) X,
1 0.0276 om~t|.237 em™t|.00852 b]0.088 b[9.3 b|9.2 b |62 bl 9.69 b| .0671 em
2 0.0276 om™1|.237 em™1|.00852 b/0.088 b[9.3 b|9.2 b |72.4{11.31 | .0719
1t 0.0276 em~t|.237 em™*|.00852 b|0.088 b|9.3 b|9.2 b |89.0{13.91 | .07%0
6 0.0276 cm~t|.237 em™t|.00852 b|0.088 b[9.3 b|9.2 b |100 [15.63 | .083%

 

 

 

 

 

 

 

 

 

!
P—l
oo
g

'

 
@
2 -t F

——

e

-185-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1‘[1

sSssee
o o MRt
» . .
-
z=e I .
-
. . ) '
4 .
rvoene
.
.
*eePed
- -
esaPe
- -
. = &
.e . 1%
. v 9
a .
- . 8
. s
eneaen 1
Ko es p
. s
*

 

¢

l L.egend
f= Thermal utilization
p = Resonance escape probability
o= Radius of Be tump
X = N(Be)/ N(Bi)
—————
\ '
X=6
X=6 — —— 1
e
X:=4 ?
P p
__/}‘
7&-—"’
f and p versus Radius of Be Lump
for Various N(Be)/N(Bi).
U/Bi - 0002
R = 0.03
i 1
2 4 6
o —Cm
"o we H‘M
-' fim

 

 

 

 

 

 

 

 

 

 

 

 

 

-186-
FIGURE III-4,2-2
Product pf vs. Radius of Be Lump.
U/Bi = 0.02
R - 0003
8
X=6
7 .
X=4
.6 X=2 —
/-/—
w .5
Legend
f = Thermal Utilization
4 p= Resonance Escape Probability
= Radius of Be lump-
x= N{(Be)/N(B1)
.2

 

 

 

 

fo-—cm
— -187-

5. FINAL DESIGN CALCULATIONS

5.1 REACTOR CONSTITUENTS AND PROPERTIES

In this section we shall describe the calculations
made in connection with the final reactor design, chosen as
the result of an approximate cost optimalization in which all
poisoning effects were included in the nuclear calculations.
The concentrations and assumed thermal absorption cross-
sections of the constituents are given in Table III-5.1-1.
The Be/Bi, U/Bi, and enrichment ratios were determined by
the optimalization procedure. The fission product and
higher isotope concentrations are fixed by the processing
rates. The L16 concentrations in core and blanket were de-
termined by the application of the considerations of Section
3.2, assuming a 1000 Mw reactggg a six-year operating life
for Be in the reactor, and approximate values for fuel inven-
tory in core and blanket. The most reliable current value
for G;(Bi) of 0.032 at 0.025 ev. was used in these calculations.

 

 

 

5.2 CALCULATION PROCEDURE

The reactor consists of two zones differing only in
Be concentration. The calculation method employed has there-
fore been that described in Section %.1 in the discussion of
the "immoderate" blanket scheme. It is thus essentially a
two-region two-group calculation, though one cannot emphasize
too strongly that there is no container wall between the two
regions.

The relevant thermal and epithermal properties that
enter this calculation are listed in Table III-5.2-1. 1In
the following discussion we shall consider only those data
and formulas which either supplement or replace the previous
considerations of this chapter. For example, the multiplica-
tion constant, |

kCD = ‘lf P (502“1)
-188-~

TABLE I1I-5.1-1

 

Table III-5.1-1A

Reactor Constituents and Constants

 

 

 

 

Isotope Concentration Relative Assumed Og
to Total U Content at .0881 ev-barns(a)

B1-209 50 .0170%(P)
U-235 0.04726 363

Pu-239 0.00365% 595.45
Np-239 0.000232 79.89
Pu-240 0.0001615 600 (c)
U-236 0.00319% 341956
U-237 2.39x107° 31.956 (@
Cd-112 0.0950x10‘5 0.1598x10“
Cd-113 0.155::10‘6 1.0386x10“
Ca-115 0.0865%10™? 0.1598x10
Ag-109 0.175x10™ 55.92
Rh-105 5.605x10™ 79.87
1-129 6l Bux10™" 7.989
Sm-149 0.260x10™7 2.503x10 "
Xe-135 0.0595x10~6 1.86k4x10°
Xe-133 0.071x10~° 0.5326x10" ()
Other F.P. 0.01063 0.994%

 

(a) Corrected for 1/v unless otherwise specified.

(b) New value which is twice that previously used. Private
commanication from I. Kaplan, Brookhaven,

(¢)  Agsumed same as at .025 ev because near resonance region
Of Pu-21+0 *®

(d) Arbitrarily assumed ten times value for U=-236.
(e) Interpolated between Xe-131 and Xe-135.

BT sty 7., A~ -"‘“q

|38 'g ~
:-E ..23 -“0‘:7"'""-":1 .. ses .
e » e .' :.: : - an 8
 

TABLE III-5.1-1B

 

 

 

Quantity Value
N(Be)/N(Bi), core 2.3
N(Be)/N(Bi), blanket 0.5
N(Lié)/N(U), core 4,037 x 107
N(L1%)/N(U), blanket 9.52 x 10~

o(U) 18.5758 b

2 (0)/x, (235) 1.0828
v (Pu-239) 2.97
a (Pu-239) 0.47

 

 
 

~190-
————"

has been defined in terms of absorption in both U-235 and
Pu-239, Thus we define n as

Jia (25) T (119)
\: TR, RO T Ty, 7,0
sevscoss (502‘2)

and for f we write

Za(U) + 2, (%9)
f= Z, (0) + 7, ThI) + ;} {everything else). (5.2-3)

 

It should also be noted that resonance absorption by U=-236
has been included in the evaluation of p under the assumption
that it has the same resonance structure as U-238.

Perhaps the only other point that has not been pre-
viously mentioned is the cholice of fast scattering cross-
sections for Be and Bi, which enter the calculation of the
quantities 2 . and 2Z,, the latter representing the fast
"absorption" cross-section, in reality the degradation. It
is incorrect to use the resonance values which help determine
p. Rather we have chosen values, a’(Be) 5 b, and
cré(Bi) = 8 b, determined by a rough average with respeect to
logarithmic energy over the energy interval from 5 Mev to

10 ev. These yield va%u?s of the age, t, here defined by
i
D

1) =f'§'(‘r)' , (5.2-14)
in close accord with values obtained previously from more
detailed calculations.

Finally, we should note that the value of N(Be)/N(Bi)
chosen for the blanket zone yields results which satisfy
the requirements of Section 4%.1. There was not time enough,
however, to investigate how the results would be affected by
a different choice of the Be concentration. It is expected
that our cholce is close to optimum.

==

pmw o7 ————l

 

 

940
vt

W

T IJE - 02-1

N(U)/N(Bi) = .02 j

R - 00""726
Dimensions: D in cm, £in em~i, L

5 in ;Lfi

-191-

ummary of Reactor Properties

1.946965

and ©t in cm.

 

 

(a)

 

 

 

 

 

 

 

Property Core Blanket
N(Be) /N (B1) 2.30 0.5
p(Bi)-atoms/cc 0.017873 0.0214429

£ 0.904983 0.937018

p 0.61730 0.221773

K 1.08766k 0.404590

Thermal Group
Z{1) _ absorption 0.00819667 0.010820
Zi(i) - transpcrt 0.400838 0.229983
D%fi) - diffusion constant 0.831591 1.449382
Lc - diffusion area 101.455 133.954

Epithermal Group
'Zégiz 0.0026172 0.000864
1
Z, 0.333293 0.251979
p§1¥ 1.000121 1.322861
(1) _ ape 382,134 1531.09
5.3 RESULTS

The basic results of the calculation are summarized
in Table I1II-5.3-1l. One 1s immediately struck by the large
inventory in the blanket. This is a consequence both of the
large volume assoclated with the blanket and the higher
spatial density of fuel than in the core zone. The question
arises whether the homogeneous reactor with the same pro=-
duction ratio has as high an inventory as the reactor
contemplated here. The answer 1s that it has a larger in-
ventory as will be demonstrated in the next section.

The P.R. (production ratio) is defined as

P.R, - 122 8 consumed - Pu-239 consumed
e o= U-235 consumed
Net Pu produced
= 0-239% consumed (5.3-1)

TABLE I1I-5.3-1. Results of Calculation*

 

 

 

 

 

Quantity Core Blanket Total
Radius of Region-cm 205.76 60.96 266.72
C.M. (U-235) - kg 240, 64 387.49 628,13

P.R. 0.8601%

 

¥ These caleculations are for a spherical reactor and will
not agree exactly with the actual cylindrical reactor.

 
 

 

 

 

 

 

 

 

 

 

T
~193-
TABLE IIT-5.3-2. Neutron Balance
Basis: 1 neutron absorbed in U-235 and Pu=-239
Neutron Production Core Blanket Total
U-235 fission - P(235) 1.756050 | 0.107601 | 1.863651
Pu-239 fission- 7/ (239) 0.213683 0.013093 0.226776
Total fission 1.969733 | 0.12069% 2.090427
Neutron Consumption
U~-235 fission 0.699621 | 0.042869 0.742490
Pu=-239 fission 0.071947 0.004408 0.076355
U~-235 capture 0.136593 | 0.008370 0.144963
Pu-239 capture 0.034103 | 0.002090 0.036193
U=238 capture (thermal) 0.069243 O. 00'-!-211-3 0.073486
U=-238 capture (res.) 0.718250 | 0.084150 0.802400
U-236 capture (res.) 0.00241% | 0.000282 0.002697
Be capture 0.026 5'11|- 0.000353 0.026867
Bi capture 0.041530 | 0.002545 0.044075%
Xe capture 0.905460 | 0.000335% 0.005795
Sm capture 0.003170 | 0.000194 0.003364
Other F.P. capture 0.005490 | 0.000336 0.005826
H.1I. capture 0.006129 | 0.00037% 0.00650%
14% capture 0.017907 | 0.000026 | 0.017933
Fast leakage 0.086625 | 0.012564% 0.099189
Thermal leakage 0.044656 |-0,042137 0.002519
Total Neutron Consumption | -1.969653 | 0.121003 2.090656

gim
ST
. b

 

 

o
tw W
AT A
_194_ TSR

In the present case the value given was computed
directly from Table III-5.3-2, which contains a detailed
breakdown of the neutron economy. The same result can be
found by means of the formula (cf. equation (4.1-2))

C.R. = C,' - P' - 56' + glt - 52' + 53;

sosesanee (5-3"2)

in which
7, (49)
Co' = (Trg = Vo5 * (155 - Dug §_25)
o F O+ (W9) 4

 

Ta.mn - Gy -

7o (W1 (49)  xOL 1

o' = T3 RONE

/ .
and one obtains S,', J; ‘53 from S':S 5 )53 respectively by
making the replacement

7. (49)
"1:‘57;)25* {“1‘1‘&‘ 25 + (13 49 z'_("'s’ia }

To this formula one must still apply small corrections for:
the U-236 resonance capture, as described in Section 3.1.
The sum of the first three terms of equation (5.3-2) is an
excellent approximation to the P.R. as one easily verifies
for this particular case.

An examination of the neutron economy reveals several
salient features:

a). The fast leakage from the blanket 1s even larger
than that from the core into the blanket. This i1s accounted
for by the poor moderating power of the blanket and is more
than compensated on a Pu production basis by b).

- —
'\;\Xfl
be s e *e * - » 'S .e
T & . » & @ L I * & »
- - * . 8 - - as » 0
* v 0 - "e s . .
 

-195-

 

 

b). The relatively high resonance capture in the
blanket.

¢). It is seen that of the poisoning effects, the L16
absorption is the most deleterious, followed by the F.P. and
the H.I. (higher isotopes) capture respectively.

d). The current higher and presumably correct thermal
cross-section of Bi results in a seriocus drain on the neutron
econony.

We shall complete this section by reporting the results
of some delayed neutron calculations. It was demonstrated in
Section II-5.2 that, assuming the delayed neutron emitters
are produced uniformly throughout the reactor, the fractiocnal
number of delayed neutrons, €45 produced within the reactor
by the ith delayed neutron emitter during the continuous
eycling process at constant power is given by

-Z -Z
c1-1 QL= NG ) -5.2-
€ =1 7 o I T (II-5.2-16)

in which 2, =t£l/1i and Z, = Ta/'c1 where t; 1s the mean life-
time of the 1™ emltter (v, = 7,(3)/0.693), T; 1s the time
spent by the fuel mixture inside the reactor and T2 the time
it spends outside.

Table I11-5,3-3 summarizes the results of applying
(IX-5.2-16) to each of the delayed neutron emitters, using
representative values of Ty = 175.8 seconds and T, = 86.5
seconds. These results show that the delayed neutron dollar
has been reduced only 6.18% of its original (no flow) value.
Since so small a fractlion of the delayed neutrons is emitted
outside of the reactor, they have no significant effect on
the neutron balance or conversion ratio.

 

 

il
0
 

 

 

 

-196-
TABLE IITI-5.3-3. Delayed Neutrons
Half-1ife %4 of fission c % of fission
Ty (%) neutrons (no flow) i neutrons (flow)
€, By

0.05 sec, 0.029 0.9996 0.0290
0.43 0.084% 0.9965 0.0837
1.52 0.2% 0.9875 0.2370
k.51 0.21 0.9630 0.2022
22,0 0.17 0.8315 0.141h
55.6 0.026 0.7220 0.0188
Total = 0.759 Total = 0.7121

 

 

{
d’\
g l
5.4 CRITIQUE OF RESULTS

The calculations suffer first of all from a lack of
accurate nuclear data in certain sensitive areas. We may
especially note the following dubious assumptions or omissionss

a). The resonance structure of U-236 was assumed to be
the same as that of U-238 in the absence of any more rellable
information. Thils principally affects the conversion ratio.

b)., Inelastic scattering by Bismuth was lgnored. 1In
particular, this will modify the age calculations and there-
fore the calculated sizes and critical masses.

¢). Values of the effective resonance integral given
in the literature are not all consistent. The values assumed
may underestimate resonance capture.

d). No account was taken of the decay products of

B1210. According to the reaction scheme

-197-

6

- 210 a 20
qu,BHPo T40 days goFP

qx 81T1206

~ 50% branching ratio

209 21
83Bi + n -+ 83B1

there exists the possibility of further serious poisoning of
the reactor 1f the absorption cross-sections of Po-210 and
T1-206 are appreclable. The existence of the a-decay of
Bi-210 has only recently been substantiated.

e). No information is available on the fission cross-
section of U=237.

In addition to the uncertainties arising from the
nuclear data, there are further questions associated with the
incompleteness of the nuclear calculations. The most important
of these is the problem of lumping the Be in the reactor.

In the time available no real headway could be made in de-
termining its effect on reactor properties. Assoclated

 

Y
u..__‘j
zl

~-198-

with this problem is that of the fast fission effect which
was completely ignored and which might yield a significant
neutron premium.

One source of doubt which can be removed is that
connected with the choice of the "immoderate" blanket design.
The specific question which we have set out to answer is the
following: Given N(U)/N(Bi), enrichment, and poison con-
centrations, does the homogeneous reactor with a ratio
N(Be)/N(B1) adjusted in such a way as to yield the same pro-
duction ratio as in the two-region design have a smaller or
a larger critical mass than the latter? The results justify
the choice of design. In Table 11I-5.4-1 we have given the
results of several bare-homogeneous reactor calculations with
N(Be)/N(Bi) varying from 2.0 to 2.5. The essential results
are also plotted in Figure III-5.4-1. According to these
results, the homogeneous reactor with the same P.R. as our
design has a critical mass which 1s over 100 kg larger than
the latter. If one includes external holdup, the difference
is even greater. Similarly the homogeneous reactor with
the same critical mass has a P.R. which is ,015 smaller.

The differences, though real, are not as substantial as
initially contemplated. One might hope to achieve a further
gain in conversion by using a reflector of fast neutrons on
the outside of the reactor, but this has not been investigated.

TABLE ITI-5.4-1, Homogeneous Reactor Results

 

N(Be)/N(Bi) Core Radius C.M.(U=-235) P.R.
' cm Kge

2.5 190 195 o734

2.3 | 233 349 0797

2.1 306 815 .865

2.0 364 1475 ¢895

 

 

 

Reactor constituents are listed in Table III-5.1-1A.

%

oo TRy
. 'J?u' m-nmmw -
. MW .

 
Kgm. U-235

Mass,

Critical

 

 

 

 

 

 

 

 

 

“M -199-’
—
150C
EI_G‘URE III"'S.""‘:!._
Critical Mass vs. Production Ratio
for One-Zone and Two-Zone U-Bi-Be
Reactors
100(
2—-Z0ONE
50(
C
070 080 090

Production Ratio

 

 

& 0 . &
* & =

SERE— | K724-799
~-200-

 

 

The reactor designed consists of one region from the
cooling and processing point of view. The design of a
multiregion thermal converter was ruled out a priori by the
requirement of simplicity. The question of what further in-
crease in P.R. is thus obtainable 1s nevertheless interesting
from the nuclear point of view. For the purposes of this
discussion we define the leakage losses as follows:

Fast leakage production loss = fi%%%%“%%%%%fi%&

2
Tfi e ———
[(1"'“ 25 * () 49 23(2%%} 1+7 ¥ 2

ecseps e BB (5.""-1)

 

Thermal leakage production loss = §h§§mai°;§§§2§e

Y Y Z_%_%}“ } o, 12%2
{(1"“)25+ (1*'“)492'2 14X % 141202

H(:i“"-_-&- s (35) v Z’{_%l] g. 12 X2 .

coreeees  (5.4-2)

 

The sum of (5.4-1) and (5.4-2) would be the increase in
production ratio obtainable by the use of an infinite U-238
blanket. The results of calculations performed for reactors
previously considered are given in Table II1I-5.4-2. These
demonstrate the tendency of the blanket to equalize production
ratios and further shows that if at all feasible economically,
the use of a U-238 blanket is most gainful for the smaller
reactors.
TABLE TTI-5.4-2,

20 I

Gain in Production Ratio by Use of Blanket

 

 

Loss in P.R. due to Leakage

 

P.R.

 

 

 

 

 

 

N(Be)/N(Bi)| Character :
------ Fast Leakage|Thermal Leakage|Total|¥1'h U Blanket
2.5 homogeneous . 204 .035 .239 «973

,,,,, 2.3 homogeneous .151 .026 177 <974

;’ 2.3 "immoderate” blanket .112 .003 115 975
...... 2.1 homogeneous .098 .01k .112 «977
el
..... Izg;; ' 2.0 homogeneous .075 « 009 .08% . 979
R
u...:‘k

 

Reéctor constituents are listed in Table III-5.1-1A.

 

!
o
o
—

1
_f,u;;.unf
-202- ST

W""’

6. RECOMMENDATIONS

The material to be compiled in this section 1s already
largely contained in the foregoing sections, in more diffuse
form. This is especially true of the previous Section 5.4
which should be studied in connection with many of the
allegations to be set down below.

6.1 DESIGN STUDIES

 

The following questions, which have been considered
either partially or not at all in the course of our design
studies are deemed worthy of further investigation:

a). The effect of lumping of the Be on critical size,
inventory, and conversion ratioj the optimalization of cell
dimensionsy the fast fission effect. Because of lack of time,
no real solution of these problems was attempted.

b). Engineering analysis of a two-region reactor
consisting of core and separated U-238 blanket. It has been
demonstrated that such a system can provide significantly
higher production ratios than the one-region reactor (compare
Tables III-5.3-1, III-5.k-1, and III-5.4-2), that the overall
production ratio becomes a much more slowly varying function
of N(Be)/N(Bi), and therefore that if a separate blanket is
at all feasible, it is so for reactors which are smaller than
the one chosen for the final design in this report.

¢c). The addition of a reflector of fast neutrons on
the outside of the reactor designed with "immoderate" blanket
should be investigated. What 1s desired.is a mechanism for
degrading the fast leakage into the resonance region and re-
turning some of it to the blanket.

6.2 NUCLEAR DATA STUDIES

Some of the nuclear data found wanting in the design
of the present thermal converter parallel those mentioned in
Section II-1.9, which contains the recommendations for fast

-
- -203-

cross-section studies. In that section will also be found
suggested approaches to some of the problems. We shall
therefore content ourselves with a resume of the required
nuclear information. Many of these have already been dis-
cussed in Section 5.k.
a). The need for more accurate information on the
ratio
o—é(fuel)
a = 5§T?§3T7

as a function of energy for the available nuclear fuels
cannot be too strongly stressed. In particular, the complete
absence of any information on a (U=-233) in the intermediate
energy region (below 100 kev) 1s to be deplored. U-233 re-
presents the remaining possibility for breeding or converting
in the intermediate region.

b). Resolved measurements of the U-238 resonance
structure would go a long way toward refining the predictions
of neutron economy for a thermal converter.,

¢)., Information on the resonance structure of U-236
is completely lacking. The need for such information will
increase in the future as one designs reactors of higher and
higher power levels.

d). Because of the increasing concern with Bi as a
diluent in both fast and thermal reactors, it is essential
that the absorption cross-sections of decay products of the
bismuth neutron reactions, namely Po-210 and T1-206 be
measured, especially in the thermal region (see Section 5.4%),

e). The age of Bi should be measured. By comparison
with the value computed on the basis of elastic scattering
alone, one could obtain the integral effect of inelastic
scattering.

f). Information on the fission cross-section of U-237
would be desirable, since it occurs in the higher isotope
chain emanating from U-236.

 

g

%
W
) o ;

B e O
.

e R ]

saafdisa
“ay e -
DTS
» .

. -
e e gRT
Lok
- -y
-204- _._————-—""""'-!

g). More accurate measurements of the (n,2n) reaction
of Be would yield information needed for the evaluation of
the fast effect premium in Be moderated reactors.

va
o
) i
 

-205-

GLOSSARY - CHAPTER III

Experimental resonance absorption integralj also
used as atomic weight - distinction clear from
context

Temperature, degrees Centigrade

Critlical Mass of fissionable material (U-235
and/or Pu-239) of a bare reactor (extrapolated) in
metric tons

Vv

Virgin Conversion Ratio = Tta - 1l

Conversion Ratio

Neutron Diffusion Constant

Neutron Energy

Thermal utilization = 2, (U)/ 2o, (total)

Temperature, degrees Fahrenheit

Fission Products

Higher Isotopes

Number of Iodine atoms

Reproduction constant for infinite medium

Temperature, degrees Kelvin

Therma% diffusion length (L2 = thermal diffusion
area

Migration length (M2 = migration area = L2+T)
Critical mass of bare reactor at temperature T.
Avogadro's number (atoms/mole) = 6.023 x 10°3
Atom number density of material s, atoms per em3
Resonance escape probability
oM
Za (235 13-4

[ o—————

= s v s e

———
-206- e
P.R. Production ratio defined as net Pu-239 production

per atom of U-235 consumed. Net Pu-239 pro-
duction = U-238 absorption less Pu-239 burnup.

r Be rod radius (cm)
N(2
R Atomic enrichment fraction = N5 +N (2
°R Temperature, degrees Rankine
R(a) Processing rate (sec'l), unit volume basis, of atoms a
R, (T) Critical radius of bare reactor, temperature T,
T Absolute temperature (°Kelvin); reference
temperature used was T, = 10339K,
T1 Time spent by circulating fuel in core
T2 Time spent by circulating fuel in pipes and heat
exchangers
u A function defined in Equation III-2.2-10.
v Neutron velocity
X Number of Xenon atoms (unit volume basis)
N(Be
X N(B1
Y(a) Yield function for fission products of type a, per
fission
Z %— s & dimensionless parameter
i
. (25)
a a =
| Z',f(255
Bi Fraction of delayed neutrons of type i.
B A function defined in Equation ITI-2.2-11
d Increment operatory with subsceripts, implies
conversion ratio increments
€ ‘Fast fission factor
Gi Neffectiveness" of ith delayed neutron emitter on
emitter on reactivity
1 Number of neutrons produced per neutron absorbed

in uranium
Jeff

n o-s‘/U!l

Z

Ty

-

— -207-

A parameter defined in Equation III-2.3-1

Mean free path (A., for transport m.f.p.j; etec.)

Radiocactive decay constant (sec'l)

Number of neutrons per fission

Mean logarithmic energy decrement per elastic
collision = average change in lethargy

Density of material. g = density of fissionable
material (grams/cm 5

Microscopic cross-section (per atom)s subseripts
same as for Z, (barns)

Effective thermal absorption cross-section which
usually includes the contribution of fast or
resonance capture, according to the definition

N(s)Oerr(8)Pinerma1 = 7

where J is the number of neutrons absorbed by
the fast or resonance process

Notation commonly used to designate Zg/N(238) in
which contribution of 238 in Z, is usually
neglected

Macroscopic cross-sectiong the subscripts on 22
are defined as: tr = transportsy a = absorption;
(fission + capture); s « elastic scattering;

i = inelastic scatteringy f = fission

Neutron age, i.e. slowing down distance squared(cmz)

< values calculated in detail (cm?)

Mean life of delayed neutron emitters (sec)

Neutron flux integrated over the reactor;~J(P§V
_208- s

IV. COMPARISON OF FAST AND THERMAL CONVERTERS

In this chapter a detalled comparison in tabular
form is made of the nuclear properties of the two-reactor
systems which were chosen for a detailed analysis by NEP.
The systems ares a) A fast fused salt converter, and,
b) A thermal U-Bi liquid metal converter. Engineering
and cost comparisons are to be found in the engineering
analysis report.

Fast Converter Thermal Converter

l. Core Structure

Dimensions:
Core radius = 95.9 cm. 205.76 cm.
Fuel:
Homogeneous fused salt of Solution of U in molten
proportionss Bi flowing in solid Be
1 UC1y-1.125 Pb Clo-.Wh6 matrix

NaCl, (The chlorides of

lead and sodium are in-

cluded to depress the NCO) _ _ o2
melting point) N(B1) = °

%{%%} - 3.2k %{%%% = 20.28

Mean Operating Temperature:

 

4180°¢ 260°C
Container:
Iron shell No container shell
 

-209-

 

Fast Converter Thermal Converter
2. Reflector Structure

Dimensions and Materials:

L. The reflector is effectively: The thermal reactor has
. - the core container wall (Fe no reflector.

- of thickness #"); void (of

thickness 2")s the blanket

container wali (Fe of thick-

ness 1")s liquid Pb in Fe

pipes immersed in UCl), of

bl%nket for thickness of

2. tl.

Operations:

The ligquld lead is pumped in
and out of the pipes to act
as control mechanism.

3. Blanket Structure
Dimensicns:

- 150 em. thick 61 cm. thick
Composition:

" UCly, (uranium is depleted U-Bi solution (same as core)
diffusion plant tallings)j in Be matrix, but:
Fe ribbing and container
shell. N{Be) _ 5
N(Bi) = °

4, Critical Mass and Inventor

Critical Mass: (25)

 

1290 kg.(in core); tailings 273 kg.in Core
only, in blanket, 473 kg.in Blanket
’ Total Inventory:
. ~ 3,550 kg. ~1148 kg.
2 nieW

-210- ———
PR

Fast Converter

A fast reactor inventory

need not be go large. The

low density of the fused
salt makes the critical
mass large, and the ex-
ternal cooiing leads to a

large holdup. The external
holdup is about 63.6% of the

inventory.

Pu Content:

N (&
fii%}?-oool"a
Relative Cost of U=235:

The fast reactor uranium

(in core) is highly enriched

Multiplication Effect of Blanket:

 

Thermal Converter

The external cooling in-
creases the inventory
considerably. The external
holdup is about 50% of the
total inventory.

The fuel 1s slightly en-
riched natural uranium and
is not as expensive as in
the fast reactor.

The blankets contribute to the multiplication to the extent

of:

3% of power generated in

blanket. The blanket serves
as a partial reflector, but
full reflector savings are

not obtalned due to the low

density of UClu.

10% of power generated in
blanket. The enriched
blanket serves as an -
excellent, though expensive,
"reflector®,

5. Conversion Ratio

Conversion Ratio:

I.c.RC - 0382
T.C.R. = 1.126

Production Ratio:
PoRo - 1.126

C-Ro - 0876

P.R. - 0860
Fast Converter

- -211-

Thermal Converter

Effect of lLosses on C.R.: (All losses expressed in units of
neutrons lost per neutron absorbed in U-235 and Pu-239).

Parasitic Capture loss:

N(26) . .2, loss = .023

 

Ni262 - .068, loss = .003

(U-236 resonance capture removes neutrons).

No Be

Fission product loss

.P. - 0016

2=
N

Pu Captures:

N4 -
Tagt = 0048, 2.z +2L

6
%%%%Tl - .0085, loss = .018

The L:l6 comes from neutron

absorption Ry and subsequent
decay of Be/. There 1is

also an (n,2n) reaction of
Be”7 which cancels the imme-
diate loss to absorpt%on by
Be?. However, the Li6 then
absorbs further.

Fission product loss = .015

'§{§§§=l'= .3745 (pessimistic)

NCIL - 077, Q= b7k

While the presence of Pu(240) resulting from capture by
Pu(239) causes extra capture losses, it has the more serious
effect of degrading the quality of the Pu produced.

Leakage Loss:
The blanket is designed to

contribute about 2/3 of the

conversiony the intrinsic
small core size and con-
sequent high leakage

demands the use of a blanket
for a large percentage of the

conversion.

The blanket 1s designed for
high resonance capture of
fast leakage from core. At
the same time there is high
leakage of 1ts own source
neutrons because of poor
moderating power. Slow
leakage losses are sub-
stantially reduced.

 

—
 

 

e et
-212- v —
Fast Converter Thermal Converter
Total leakage from Total leakage from
reactor - 0 reactor = .102

(predominantly fast)

6. Parameters and Processing of Poisons

The nuclear properties of the polsons have a direct
bearing on the extent to which buildup can be permitted, and
hence, on processing rates.

Pu-240:

The longer Pu is left in the reactor, the higher the
Pu-240 contamination will be. The‘gp-aho content in the
product is determined by the ratiocfiségg%. One notes that

o, (#9) <, (49)
‘ <, (28) fast = 1.47 Cr‘cZ285therm = 11.3

showing that the fast reactor has a declided advantage on this
point. Moreover, the processing system (distillation) used in
the fast reactor is much more selective than the thermal pro-
cessing scheme, enabling the fast reactor to operate econom-
ically at a much lower Pu/U-238 ratio.

U-236: :
The rate of formatlion of U-236 is determined by a(25)

“fast(zs) = .18 atherm.(ZS) - ';95

Therefore, the buildup of U-236 per 25 atom is approximately
the same for both reactors.

Fission Products:
There are no known anomalously Xe and Sm have anomalously

large capture cross-sections high capture cross-sections
in the fast region. The in the thermal region.
equivalent a of the average The relative captures due to
fission product relative to these F.P. alone are:
fission of 25 is:

ci’c (Xe) o

a (Xe) =15§(§gy = 6,153,

ogSm
a.(Sm) =¢f(25) - 82.""

 

cré(F.P.) -
a(F.P.) =EET§?)— - .12
Fast Converters Thermal Converters

These elements must be
extracted by processing
and their concentrations
kept at a low level.

It is of interest to compare the macroscopic alpha of the
fission products in these two reactors including the effects
of processing.

7. (F.P.) of
A(F.P.) =7 Ty = .0016 d (Xe) Eneilk .0078
d.(Sm) = 0045

d(other F.P.) = .0078

Cooling Times:
The main cooling problem is to keep the U-237 activity
below 1 uc/gm of U before the core material can be processed

for U=236. (This number is the limit set by K-25). Long

cooling times (~ 125 days) impose excess inventory charges

on the product. '
Additional cooling is desired to allow the fission product

activity to dle out enough so that radiation damage to the

Redox process is negligible.

2.3 4

 

Np-239 __7;_——b Pu-~239
h‘;"m__,-_
e by

 
-214-

Fast Converters

Natural Stability:

Negative temperature co-
efficient of k due to
thermal expansion of the
melt makes this reactor
stable against not-quite
sudden changes in re-
activity. However, the
delayed neutron fraction
= .0034, which makes for
less stability.

Power Level Control:

Pb reflector rods (pipes
filled with molten Pbg
will be used for control.
This method does not
result in a loss in con-
version., Shim control
may be handled as in the
thermal reactor.

==

7. Control

Thermal Converters

Doppler broadening of
fast capture resonance in
28, together with longer
prompt generation time
makes this reactor in-
herently more stable than
the fast reactor. The
delayed neutron fraction
is .007.

.Control will be accome

plished by variation in
the 28 concentration by
regulating processing
rates.

i&i
C.R.

FOP.
I.C.R.

P.R.

T.C.R.
. X.C.R.

X

tx

o8 (o x)
‘o) 3,80,

TSRS

-215~
GLOSSARY - CHAPTER IV

Ratio of Z, of fission product to Zf of U-23%

Conversion Ratio, gross Pu atoms produced per Pu
and U-235 atoms destroyed

Fission Products

Internal Conversion Ratio =
Gross atoms of Pu-239 produced in the core per
neutron absorbed in U-235 and Pu-239 by fission
and capture in the core

Production Ratio defined as net Pu-239 production
per atom of U-235 consumed. Net Pu-239 pro-
duction =z U-238 absorption less Pu-239 burnup

Total Conversion Ratio = X.C.R. + I.C.R.
External Conversion Ratio =
Gross atoms of Pu~239 produced in the blanket
per neutron absorbed in U-235 and Pu-~239

Element of type x

(x)
g;TET y ratio of capture to fission cross-sections
f

y ratio of number of captures to

number of fisslons in multigroup calculations

Beta decay process

Macroscopic cross-section; the subscripts on 2.
are defined as: tr = transport; a = absorption
(fission + capture); s = elastic scattering;

1 = inelastic scattering; f = fission

Microscopic cross-section (per atom); subscripts
same as for

>
’216' PR

APPENDIX A
Procedure for Homogeneous, Fast, Bare Reactor

The accompanying calculation sheet gives the computed
values of the indicated quantities for each of the eight
energy groups. In order to assist those unfamiliar with
multigroup calculations of this type, a step by step
description of the procedure is presented below.

1. The relative proportions of the reactor core materials
are chosen primarily from engineering considerations.
The example to be considered was system =I""21, one of
the fused salt converters. It was assumed that the core
material consists of a homogeneous mixture of
102351, + 3u238cy, + ‘+PbCl§ + 4NaCl which is equivalent

to atomic proportions: 10235, 3U230, 4Pb, 4Na and 28Cl.

2. From experience with these calculation methods a
judicious guess of k2/3 = 295 is made. This guess in-
volves choosing the correct energy group in which the
average of the neutron spectrum lies. For dense systems
(1ittle 238 and/or other materials) the 3rd group 1s
likely; for more dilute systems (3-6 atoms 238 to 15-30
diluent atoms) the 4th or 5th group is appropriate; for
very dilute systems the 6th or 7th group should be
selected.

In the present example since there are 3U
36 diluent atoms per U235, the 5th energy group was
chosen. TFrom accompanying data on cross-sections and

238

and

other nuclear parameters we obtain:
2.47 N(25) = .8hh579x10-3x102u atoms/cm3

Z,. = 186.454 N(25)  Zp(25) = 1.74N(25)
L, = 2.71328 N(25)  Z,(25) = .207N(25)

<
"
-217-

 

- One=-velocity diffusion theory yields the relation:

K2/3 = Ty, (vE~Z)N25) = 295.6N(25)

Since we normalize all cross-sections to N(25) = 1 for
convenience, the value k2/3 = 295 was selected for use
in the trial calculation. As shown in this example, a
judicious choice of the energy group enables a balance
between sources versus absorption and leakage to be
attained in only one or two sets of calculations. Since
each set represents several man-hours of work, there 1s
a distinct advantage in reducing the number to a minimum.
In the present case the balance achieved on the first
calculation, after making the minor corrections dis-
cussed below, was sufficliently accurate.

The foregoing method of estimating k2/3 usually
yields results which check the assumed source strength
within 10%. The second value of k2/3 is then computed
by means of the formula:

 

2 2
3 3 1l - F, - Cl
where, 5, =z computed source strength for k2 = kl2
Fl = number of fissions using k2 = k12
Cl = number of captures using k2 = kl2

- This formula has been found to give 1 per cent accuracy

or better in all cases.

In thé above list of cross-sections many figures are
included beyond the significant place. The same is true
in the calculation sheet. It was found convenient in
using hand calculating machines to standardize on six
figures and not attempt to round off the last signifi-
cant figure until the final step in the calculation.

The fraction of fission neutrons emitted in each of the
first five energy groups,Xf, was obtained from KAPL.

A

3 w

T

SRy - T ki,
_218_ M;M

Above the 5th group, 1l.e. up>k4.5, and below the 1lst group,
i.e.u 0.5,the contribution from the fission spectrum
1s considered negligible.

The tabulated values of (xi)aéa given for groups 2-6
inclusive under the heading "fraction" were likewlse
obtained from KAPL, except for chlorine (Cl) in which
case our own estimates were made. All of these values
are only approximate since reliable information on the
energy distribution of inelastically scattered neutrons
is not yet available. No appreciable inelastic scatter-
ing is expected above the 6th group, since the first
excited states of even the heavy nuclei appear to be
greater than this energy. The U23 data are based on a
re-analysis of the Snell and similar experiments by KAPL,
reported in KAPL-741.

The figures entered under “sources" opposite Group I
are the contributions to each of the lower energy groups
from energy group 1l resulting from inelastic scattering
by each of the reactor constituents which contribute.
Similarly those entries opposite Group II are contribu-
tions from inelastic scattering in energy group 2, etc.
Sodium does not contribute appreciably, chlorine
contributes only from Group I and 238 is the only impor-
tant inelastic scatterer from Group Iv.

Another contribution to the total source of neutrons 1is
from elastic scattering from group (a~l) to group «.
These sources are entered opposite "previous gq+". The
total source Xy in each energy group is obtained by
simple addition.

The first entries in the lower half of the calculation
sheet, where the losses are summarized, are in the left-
hand column of each energy group. These figures are the
average macroscopic cross-sections normalized to one

T3
O

L
atom of 235 for each energy group. The leakage L is re-
presented by a fictitious cross-section k2/3 2 tp entered
opposite L in the left hand column for each energy group.

The degradation resulting from elastic scattering is

~ also represented by a fictitious cross-section EZS/FU.

The product EZS for each energy group 1ls determined by

standard methods.

The width of each energy group, U,

is determined by the arblitrary selection of the energy

intervals.
A modification, used by us, in the manner of
ing F  in the degradation term {Z,/FU saves time

 

‘choos~

as

compared with the graphical method used by KAPL. By
assuming that (linear approximation)s:
+ +
q + q
- a-1 a
qa - 2 (A"'2)
and using the definition of Fa (see Glossary) we can
eliminate q)a from eq. 3 and obtain
+
(Lt 951
F, = T (A-3)
2 (%) - (Ay/By) a3
where (x1\¢ = right hand side of eq. 1I-2.1-3, sec. 1.1
2
AU. - (k /3z|tr a (Z ) (Zi)a
B = 5 s
a ( U )q
This is an adeguate expression for Fa in all but the
last group or possibly the last few groups where the
choice of F is of least relative significance. Some=
what better approximations, based on an 1mproved
approximation for qa, have been tested. The results

show a negligible correction for a representative

system,

The next step is the stepwise computation by groups

-2,
-

kA
&

[E
Loy )
¥
-220- i ———
b ——

10.

11.

12,

13.

R Ir s-ve - SRR

beginning with group 1. Since there 1s no contribution
to this group from higher energies, q:_l for the first
group is zero. Hence from Egq. (4-3) above,Fa = 0.5 as
entered at the bottom of the group 1 column.
From this value of F the degradation (& ZS/FU) = 19.872
which is entered together with the other cross sections.
The total cross section 1s obtained by addition of all
cross sections, real and fictitious.
The total source Yy is obtained by addition of the
entries in the source column., In group 1 this is simply
the fission source Xf. The relative flux in group 1,
P = (XT/a-'T), is then obtained by division.

Each of the cross-sections is multiplied by ¢ and these
products, which are the fractional leakage and losses,
are entered in the right hand column. The sum of these
should equal X, e.g. 1.34340 = .13%4, the difference
simply resulting from the rounding off of the figures.
This completes the entries under group 1.

The source terms in group 2 consist of x?, the group I
inelastic sources,obtained from the product of the
fraction for each component times the fractional in-
elastic loss from group l)plus the loss by degradation
from group 1l.

The calculation of F for group 2 and subsequent groups
is obtained through Eq. (A-3). (Aa/Ba) is computed
separately. In some calculations F may be negative in
groups 7 and 8. In which case it is assumed that the
degradation out of the group 1s zero.

The procedure for groups 2 - 8 follows the same steps as
for group 1. The only source terms for groups 7 and 8
are by degradation from higher energy groups. The
calculation must be internally consistent on two counts,
The fraétional leakage and absorption terms summed over
can

 

-221-

 

all groups should = 1 (see last column)., If this is not
the case, within the rounding off error, there must be a
numerical mistake in the calculation . Iikewise the
total fractional fission term for each fissionable
component is multiplied by its v, and the sum of these
products, the fission source strength, should also = 1.
If this is not the case the calculations do not represent
the critical condition and must be repeated using a new
k2/3 estimated from Eq. (A-l) employing the above fission
source strength. ‘

After a consistent calculation has been completed, one
obtain the following information for the bare reactor
a). Internal conversion ratio
b). External conversion ratio
¢). Mean effective @
d). Critical radius and critical mass

e). Spectrum of fissions, i.e. percentage of
fissions due to neutrons in a given energy
range

f). Spectrum of neutron flux
g). Fraction of fast fissions
h). Detailed neutron balance

 

) Sy

- -
LI a ¢
.

-
EE XK ]
~222-

Typical Tabulation. - The following tables and graphs
exemplify the procedure for system 21.
1s corrected by changing the fission cross-sections slightly.
The resulting calculation error in conversion ratios is then
less than the original error in the neutron balances in this

case 0.6%.
a).

b).
c)e.

d).

e).

£).

g)e

¥
§ -~

 

I.C.R. = rfi%@%%%am = 0.355681

X.C.R. = 5o3oed 15062 = 0.780702

- 0. 06640

ayg - 5733333% = 0.176203

b = EEETE?T = 125,1 em

C.M. « 4/3 wb3 = 2.693 tons U-235
assuming N(25) = .8ul46 x 10+2l atoms/cc.

See Figure A-1l

See Flgure A-2
Fraction of fast fissions by U-238

= .0276 - -
0-396865:0.007887 068369 = 6.8%

ke 29w 5 088 9§ e L] . - 2 e 40
. *

The neutron balance
-—-_“—* _223_

Detalled Neutron Balance: basis - 1 source neutron

 

 

 

 

 

 

 

 

 

 

Neutron Production Uncorrected Corrected
v(25) ¢+ Fissions in U-235 0.936471 0.930857
v(28) + Fissions in U-238 0.069560 0.069143

Total Production 1.006031 1.000000

Neutron Consumption
Fissions in U-235 0.379138 0.376865
Fissions in U-238 0.027824% 0.027657
Captures by U-235 0.066405 0.066405
Captures by U=-238 0.157663 0.157663

~ Captures by Pb 0.001949 0.001949
Captures by Cl 0.0226%45 0.022645
‘Captures by Na 0.00075% 0.00075%
Leakage 0. 343494 0. 346062

Total Consumption 0.999872 1.000000

LT TR T
L 02k
-224-

FRACTION FISSIONS above u ~——»—

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE A-1l

SYSTEM 21: SPECTRUM OF FISSIONS AND
FRACTION OF FISSIONS ABOVE u

 

 

 

 

 

 

 

 

Spectrum of Fissions

 

 

e

 

 

 

14

 

 

 

 

 

 

Fraction Fissions
\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12

.10

08

04

102

——FISSIONS/ UNIT u—>
 

 

-225-

 

 

 

 

 

 

 

 

 

FIGURE A-2

SYSTEM 21:

 

FLUX SPECTRUM

 

 

 

 

 

Flux Spectrum

 

 

 

 

 

 

 

 

 

 

N

o

 

 

 

 

FLUX/UNIT u—
O ‘ :
Q0

 

o
&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
-2926- M

DERIVATION OF "EXPONENTIAL"™ Fa' - A somewhat improved
approximation for_F;, which is expected to be more accurate
than A-3 when ¢ varies strongly within a group, may be derived

as follows: We start with the age diffusion equation:

 

& “ '
(-5‘;‘1‘& + 2w L) s 34 =7.=’+ LK(u,u')z,.pru (A-1)

We shall denote the parenthesis on the left by Z,.and the
right hand side br S (the total source). We replace (f by
A CB and denote Zr by h. We now have

325 25

hg +%& = S (A-5)

uWe can integrate (A-95) by multiplying by
expg Su_ h(u')du'} and integrating over u. The final result
is:

q{u) = e 4 q(u-) + €

U(, “:,u u,uuu
_s_fldu - u_elduj-cs“'fia S‘Ju'

@

ceeees  (A=6)

Now let u~ be the lower limit of a lethargy group and u be
the upper limit. If we make the assumption that both h(u)
and S(u) are constant throughout the lethargy interval
[u', u+_] , then we have:

~hU
at = e g7+ g — s (A=7)
where U = u’ - u~. Now we define T =US and hU = X, and we
have
- .
gt ceFqT e - ¥ (A-8)

Returning now to (A-5), we integrate it another way by
assuming h and S to be constant and we obtain the multigroup

equation:
-227-

 

hgU+q -q =8 (A-9a)
or |

Ix +q -a" = 8 (A-9b)
We now assume that:

q = aq” + bq+ | (A-10a)

q = Fq© | (A-10Db)

Equation (A-10b) is, in a sense, superfluous, since it only
fulfills the function of defining ¥. However, in the actual
multigroup calculation procedure, F is the useful quantity.

From (A-9b) and (A-10a), we eliminate g and solve for
q+, with the result:

+ l=ax - 1
T = T5px T T Tox © B | (A-11)

In the multigroup calculation procedure, q and S are known,
as 1s x. If we wish formula (A-1l1l) to agree with formula
(A-8), we must have:

 

%i%% - e X (A-122a)
-X
I%Ek = 1'2 (R-12b)

These equations can be solved for a and bs

 

 

1 e~*
a= = - - (A-13a)
X 1-e-%
1 1
b=-=4% - (A-13Db)
X q_gX

3,
.
o
| -
-228- e

Note that a + b = 1, rigorously. Now, we can eliminate q+
and ¢~ from (A-9b), (A-10a) and (A-10b) to obtain for F:

 

b Xy + & q
F - m (A-1k4)
where J(Ta. 2+ q . We can write this formula for the o.th
group as follows:
b, (Xp), + 2, q,
F_o- S a1 (A-15)

a - (XT) - X, a, q -1

where we have replaced q_ by qZ_l.
Going back now to the definition of x (z hU), and re-
placing h by 2 /EE we haves

£ s
U

 

X, = (Hg), < . (A-162a)
We can now recognize x, :
XC& = ACL / Ba (A-16Db)

In an actual calculation, a(x) or b(x), given by (A-13),
can be tabulated or plotted accurately and then read off for
various X, If a is obtained, b is just l-a, and vice versa,
This operation, together with formula (A-15), replaces step 6
in the procedural outline above.

Examination of particular cases shows that this method
of estimating ¥, is much better than the one which assumes
a, = b, = %, for groups wherein q and q are very different.
Moreover, Fa can never become negatlve wzth this procedure,
which is a more realistic behavior than if it can be negative.
On the other hand, calculations on a representative case gave
the result that, while the spectrum of the bare pile was
changed somewhat, the differences between the calculated

rexr aw
L e Y

R —229'

critical sizes, conversion ratios and other neutron balances
were negligibvle. However, for a spectrum which varies much
more sharply than the one tested, there might be a non-
negligible difference. 1In any case, there is very little
more work involved in using the method with variable a and b.

 
2 3¢
NUCLEAR ENGINEERING PROJECT

Calculation Sheet

 

System No.2I
Atomic Proporfions
u"“ U®*Pb Na CI

-0€¢~-

K/3 =295 (estimated from one velocity approximation)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 4 4 28
u interval 5= | -2 2-3 3-375 |375-4.5 45-55 5.5 -7 7-10 Summa-
Group | 2 3 4 5 6 7 8 tion
Fraction|Sources |Fraction Sources| Fraction| Sowces |Fraction| Sources | Fraction|Sources |FractioSources | Fraction Sources|FractionSources
Fission Source Xe | 1344 :-'tfaa .39 5% Logo? loa73 | ! 1
| | |
l Gl’OUp I 235 l .l }-9001“‘ .2 |.0“483 3 }.00072" A :,009483 ‘6 }.“04’3 I !
NS 238 I s r o016 .35  |.o0suma] a4 l.oo361.o de ecadid | .to l.oo,s-as ; 1
EO Pb l -4 |r 00305y - 3 [+002293( .2 |.oo|fas .l [-eeoted ! — I [ |
LU Cl | 2 |sonaez|-e  lozanis| .2 renaenl — | — | I
AR Group II 235 | l 2 |ooues1| .33 | o0T700| 122 |-e05133 | .28 | 005234 | [
SC 238 | |' a5 liosios] .33 lodwiaci.ze  e3a034| .10 |.o14s3e | ]
TE Pb | | o otoez| .3 :.ozoss'i I |.oobyds| — ] |
I'S  Group III 235 | f | 51 013179 34 |.oos136 | AE 100370 | I
C 238 | | | 51 otaasy 34 |.obisos | oIS 023t | |
Pb | * ' (4 LT[l Lests|—— | |
Group IV 238 1 ! ! | 8 loviesz|.n loisam ! |
Previous g+ =(¥/FU), 4., : | o3v9¢3 et 174 i-mm' : 365 303 I.‘il‘o!'.-. l.aflm o
o | = .
Total Source Xr i:sw | so¥qgs 1595 197 {.5‘[1230 l.s9¢ o |.494¢q2 - 3T4éN |0 TRAY |
(Xr/e3) = N 1. poaol | 019945 loag 713 .039903 |.oa7193 .o'llffla 029779 00 5397 s
LEAKAGE : L=k¥/3e, [5.39¥%tl. cosr3s potssag | osvaus 2,209 |, oudaré [Lir304s |.os4ars r.:qm.qo|.osqmz Layswet | o8 091237787 | 037014 |, ti6ads) 004778, ad 344y o
. . Fissions 23512 l.eeatis [z loasasy |f,21 j.o35030| 143 }.oiriu 4 oHATIe | 2 .o88054 |33 l.owm L5y lowse].579)87
0 3x 238,42 l.oozsos|hits  |-earyy| o513 |.ootqn3| — —_ 027324
crecen S Captures 235 ].065 lioooisi .05 Lootaey).oms |.ooaiss) iz :-»35'87 07 007649 375 IolS‘?.'ts' Py TR i.oub‘ia a4 lLeiss |. 0be dos
S 3x238 |- ot5 |ovooso|.i5 ooy |30 lowif| 4E  Lewszi.be  |oatsN7| 'Iaqaasa wse  hotéen | 3.0 lemen|isyiey
g 4x Pb|.el | oec0ael. ol |.oaoHl| 0 |.oooal’? ol .000aq9| .0l |:000372| 0| |.oao'm 0l ‘I.Oooaqq 1ol i.uoo.fl WYILLL]
28 x Cl |.ot68 |oocosr|.0i% joeo0dPt| c02? .ooo¥e¥| .oJ0Y I.ools‘o‘? (0452 |.0036Y] | L1764 [,007347 | 38O 00 TIII| 316Y |i00186L], 02RENT
4 x Na |.o0i08 lLooceoz|: ool0¥ |.oe00l | .0010¥ |1000031 | . 00108 |.000032 00108 | :000041| .pon2 I.oaoc'#'] 00I% l.oooosq 103%%4 | 000521 |, 000T3Y
Inelastics 235 1z ooy 3| ha  J.o®33t|. 9  |.ea5¥4] — l l i
3x238(7ns  Loisew3|hs  |Msyag[é3 Lisosaz| A.ss |0%af3| T | | | |
4x Pb |3z |.oo7b4z 352 |.o0éryd| he l.oqm; — — 1 | | |
28x Cl [240  Lo56308|— | ' | - l | | |
Pegradation$%/FU |i11.9% .o 39963(v. 341953 : et v.w'iml.nv:sr 1.R78507]. 365203 |K. 195757 416 032 |4.56706| 2746 1. 5762810 TTCHY —— |
. 1 i 1 | ] 1 | —
Totai Cross Section o3 6. Boe3 7, 1343 46, 175702 |.5°08 917 e tagéial. SIs180 e 3ss0at].suv 2 ISMI7a7] .S109 bE5¥600 1934690 HT8375 127689 ;a..ums}.ms;; .999%712
F .
os %a3763 T5velo L1985 426533 7 269 k% ¥39 ©0
Mi | -231-

M

APPENDIX B
SUMMARY OF CALCULATIONS ON BARE HOMOGENEQUS THERMAL REACTORS

Tables B-1 to B-ll contain the results of calculations
of a number of bare, spherical, critical, thermal reactors
composed of mixtures of enriched uranium, beryllium and
bismuth. Each of the eleven tables gives the results for a
specified U/Bi ratio and an assumed enrichment for a range
of Be/Bi ratios.

These tables have been computed for a Bi capture
cross~-section of 0.016 barns at 0.025 ev., the value
accepted at the beginning of the Project. The final calcula-'
tions, reported in Chapter III use a cross-section of 0.032
barns, the value now considered more reliable. Results of
this Appendix may be expected to show trends correctly, but
are inconsistent numerically with the final calculations,
......

......

......

.....

ooooo

......

......

QQQQQQ

seasey

 

T

&

132

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE Bl
Ny/Ng; = O Ok R = 0.03 M = 1.853 0
0
;B.Q , . N -~ L2 T M2 Rc Vc MBi MU M25 MBe |
Bi (en?) [(ea®)| (en®) | (em) |(10%em| (108m) | (1050m) 110°cm) (mégm)
1 - - - - - - - - - - - - -
2 | 0.455 | 0.965|0.813| - - |- - - - - - - -
4 612 | .947|1.074| 0.903| 98.8| 266 | 365 | 221 | 44.95 |223. |10.2 | 306. 38.5
6 617 .930 1| 1.166 718 109.7| 202 | 312 136 | 10.6 42.2 1.933 | 58,0 10.9
8 733 | W91311.241| .605|121.2f 171 | 292 | 109 | 5.48 | 18.4 | 0.842| 25.3 | 6.36
TABLE B-2
Ny/Ngy = 0.04 R = 0.0 "= 1.94798
Ny L T M R, Ve Mps My Mos Mpe
B | P LT T P (ad)| (@B ead) | (e |08y (10%m) | (108em) | (10%m)| (108em)
1 |0.273 |0.9833|0.5229| - - - | - - - ~ - - -
R | W455 | J9774| .8663] - - - | - - - - - - -
4 | 612 | .9658|1.151 |0.801 63.54| 266 |329.5| 146.7| 13.22 | 65.57 {3.000 |150., |11.32
6 | 677 | .9546[1.259 | .632| 71.04| 202 |273. | 102. | 4.444 | 17.78 |0.8136 | 40.68 | 4.604
8 | J733 | J9435|1.347 | .500{ 78.97| 171 |250. | 84.32| 2.511 | 8.43 | .3861 | 19.31 | 2.913

 

 

 

 

 

 

 

 

 

 

 

 

 
.....

------

-------

Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

253  TABLE B-3

Nyy/Ngg = 0.0k R = 0,07 N = 1.991
N, ¥y | YR Ve M3 Yo | M | Mg
— P £ k C.R.
"1 . (cw®)| (eu®) | (ca®) | (cm) | (20%en’)| (10°em)| (10%m) | (10%gm) | (1084m)
1 | 0.273 |0.988] 0.537 | - - - - - - - - _ -
2 455 | 984 .891 | - - N - - _ _ - _
4 612 | J976|1.189 |0.751 |46.8 | 266 | 313 |128. | 8.75 | 43.4 |1.986 | 139. | 7.49
6 677 | .968|1.304 | .583 |52.6 | 202 | 255 | 90.9]| 3.15 | 12.6 |0.577 40.4 | 3.26
8 2733 | J960 | 1.400 | .452 |58.6 | 171 | 230 | 75.3| 1.786 | 6.00 | .275 19.2 | 2.07

TABLE B-l

NU/NBi - 0003 R - 0.03 Q - 1085380
NB I..2 T M2 Rc vc MBi MU M25 MEe
e f 1 k | CR, 2.0, 2 2 6 3.0, 6 6 3 6
Bi (cw®)| (em®) | (en®)| (em) |(10°cw’)|(10%gm)| (10°%m) (10°gm) (10%am)

1 }0.329/0.9653]0.5887| - - - - - - - - - -

2 0510 09534 09014 - - - - - - - - - -

4| 656 .9304{1.131 |0.792 | 129.4| 266 | 395.4| 172.6| 21.54 |106.8 | 3.649 |109.5 |18.44

6 | 721 .9085[1.214 | ".639 | 143.0| 202 | 345. | 126.1| 8.399| 33.60 | 1.148 | 34.44| 8.700

8| .764| .8877|1.257 | .548| 157.1| 171 | 328.1| 112.2] 5.915| 19.87 | 0.6790 | 20.37| 6.861

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
N
o
W

!
.......

tttttt

------

------

00000

......

......

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE B-
NU/NB:I. - 0.03 R = 0.05 "N = 1,94798
e 2| | R | Ve | M| M| s |
W | 2] L5 T | (a@d | () ()| (em) |(108en®) (10%m) (20%m)| (10%gm) (20%m)
1 10.32910.9778|0.6266| - - - - - - - - - -
2| .510| .9701| .9637| - - - - - - - - - -
4 | 656 .9550|1.2204| 0.686| 83.75 | 266 | 349.8(125.1 | 8.202 | 40.68 [1.390 |69.50 | 7.024
6 | 721 .9403|1.3207| .531] 93.28 | 202 | 295.3| 95.30 3.625 | 14.50 [0.4955 | 24.78 | 3.755
8 | 764 .9261|1.3783| .438|103.34 | 171 | 274.3| 84.57) 2.534 8.514 | 2909 | 14.55 | 2,940
TABIE B-6
NU/NB:I. = 0.03 R = 0.07 Vt - 1.99133
12 T | R Ve Mpy My Mos Mo
':_:f P () () (@) (o) |0Pa?) (108 (10%m) | (10%am) | (20°em)
1 1 lo.329)0.983700.6445 - | - | - | - | - - - - - -
| 2 | 510| .9780| 9932 - - - - - - - - - -
L | 656 .9667|1.263 |0.637 | 61.91 266 |327.9(110.9 | 5.714 | 28.34 |0.9684 | 67.79 | 4.892
6 | J721| 9557 (1.372 | 482 | 69.24|202 |271.2 84.83| 2.557 | 10.23 3496 | 24.47 | 2.649
8 | 764 | 9449 |1.438 2386 | 77.00|171 | 248.0| 74.77| 1.751 5.883 | .2010 | 14.07 |2.023

 

 

 

 

 

 

 

 

 

 

 

 

 

-$EC-

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE B-
Ny/Ngg = 0402 R = 0.03 v = 1.8533
e p | £ | x | c.n e el Yo 1 | % M5 | MBe
%1 (en®) | (ea®)| (en®)| (cm)|(20%’) (10%m) |(10%gm) | (10°em)| (10%gm)
...... 1l =~ - - = - - - - - .- | - ~ -
------ 2| 0.5630.932|1.007 } 1.003 | 175 | 472 | 647 | 983 |3980. |2590. |so2. | 17800. |2234.
,,,,,, 41 J722| .899{1.203 | 0,641 | 187 | 266 | 453 | 156 | 15.9 78.9| 1.81 54.3| 13.64
e 6 76| .869|1.249 | .531| 205 | 202 | 407 | 127 | 8.59| 34.4| 0.789| 23.7| 10.88
8| 802 8411250 | .486 | 223 | 1TL| 394 | 124 | 8.04| 27.0| .618| 185| 9.3%
B TABLE B-8
s Ny /Mgy R = 0.05 M = 1.94798
“Be P £ | x |c.R L: T2 Mz R_c Z" 3 :Bi 35 H2°
N1 (en®) [(em®)| (em®) | (em) [(10°en”)| (207gm) | ( (10°gm) | (10°gm)
ool - |- | - - | -] - |- - - - - -
2| .583|0.9558)1.085(0.897| 113.38| 472 | 585.4 | 260.8 74,31 |483.8 | 11.07 |553.5 |41.76
4| o722| .9339|1.313| .533| 122.91| 266 | 388.9 | 110.7 5.684 | 28.19 | 0.6450| 32.25 | 4.868
::l 6| 76| 9130 1.380| .418| 135.94| 202 | 337.9| 93.67| 3.443 | 13.77 |  .3151) 15.76 | 3.566 'g
8| .202| .8932/1.395| .375| L49.54| 171 | 320.5 | 89.47| 3.000 | 10.08 | .2306| 11.53 | 3481

 

 

 

 

 

 

 

 

 

 

 

 
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23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE B-9
NU/NBi - 0002 R = 0.07 fl - 1.9907
2 .
lde L T Mz Rc ¢ . MBi Ml] M25 MBe
Nps (em™) | (cm™)| (cm™)| (em) |(107cm”)| (107 gm)| (10 gm) | (107gm) |(10 gm)
‘\A,}
-

<r; 1 {0.410|0.9766|0.841( - - - - - - - - - -

“ 2 | .583| .96851.124(0.754 | 80.8 | 472 553 | 209.8 | 38.7 252. 5.76 403.3 21.7

. L | J722| .9525|1.369| .481 | 88.2 | 266 354 | 97.27] 3.86 19.2 0.438 30.6 3.30

6 | JTT6| 9371]1.448] 362 | 94.1 | 202 296 | 80,74 2.20 8.82 o212 14.8 Re29

TABELE B~1l0
- - 1. 8 - 80
NU/NB1 = 0,01 R = 0.03 'Q = 53
2

NBe L T Rc vc: MBi Ml:l M25 MBe
N1 (em) | (em™) | ( (em) | (107cm’) (10°gm)| (10°gm)| (10”gm) (10°gm)

1 0-5406 009026 009045 - - - - - - - - - -

2 6937 | .8721|1.1215(0.728 [327.9 | 472 | 799.9|254.8 | 69.28 | 451.0 |5.137 |154.1 | 38.94

4 LB040| .8168|1.2174| .492 [340.9 | 266 | 606,9|165.9 | 19.13 94.88 | 1.081 32.43} 16.38

6 LB487] J7680|1.2083| 414 |362.5 | 202 | 564.5(163.5| 18.31 73.24 | 0.8342 | 25.03| 18,97

3 L7763 J7248|1.1774| 368 |384.9 | 171 | 555.9(175.9 | 22.80 76.61 | 8726 | 26.18| 26.45

 

 

 

 

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--------

PR eS

AR ee

=h & Tt

AABN e

......

llllll

!!!!!!

......

 

 

g

237

 

 

 

TABLE B-1l
Ny/Ng, = 0.01 R = 0.07 w = 1.99133
_1?]_3_9_ £ X C.R L2 T MZ Rc vc MB:l. MlJ M25 MBe
My | (e |(?)| (@d)| () | o) oam)| 10%em) (10%em)] (10%m)
1 | 0.5406(0,9527 1.0256{ 0,999 | 162.0 8951057, [638.1 {1088, 8388, 95.54 | 6688, 362.
2 | 6937| .9367|1.2940| .562 [162.1 | 472| 634.1|145.9 | 13.01L| 84.70|0.9647 | 67.53 | 7.312
4| .8040| .9064|1.4511| .319 |174.1 | 266| 440.1| 98.14| 3.959 | 19.64| .2237 | 15.66 |3.391
6 | .8487| .2779|1.4837| .236 |190.8 | 202 392.8| 89.53| 3.006 | 12.02| .1369|9.583 |3.113
8 | .8763| .8512|1.4853| .194 |208.1 | 17| 379.1| @7.81| 2.836 | 9.529| .1085|7.595 |3.290

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
o
o
-3

i
kb b &

-23‘8- M

APPENDIX C

SECULAR EQUATIONS

In writing the secular equations, the problem of branch-
ing fraction is encountered. This is best handled by a nota=-
tion. In general, a given element is removed by: chemical
processing, natural decay, the neutron flux. The next step
jn the chain is made by the flux or natural decay alone. It
is best to isolate the removed rate as a single time factor,
T, since the sum of the times from the start of the chain
gives an estimate of the time required to reach equilibrium
for a given element. The branching fractions, §, then
propagate through all succeeding members of the chain affect-
jng final equilibrium level but not time required to build up.

Since a given element can continue the chain only by
flux or natural decay, at most two 8's are needed for a given
step. This makes it convenient to define the p in terms of
the element starting the step, rather than the logical 8,
showing initial and final elements in the step. A bar over
one of the B8's is used when branching occurs.

The notation is:

C = conversion ratio
(p= flux
0= cross-section (= total absorption, except

0'25= G"F(25))

t = natural decay time

T

net chemical processing cycle time, and 1is
meant to include any efficlency effects.
(i.e., if half core processed in 10 days, and
concentration reduced by half in one pass,

T = 40 days).

It is assumed that uranium is not removed but all Np and
Pu is affected by chemical processing.

 

)
A
————tem——— -239-
n— e,

a =a'(n,Y)/0"F

:U'(n,Y)/(O'F + O'(n,Y))

= (92)2L+O = (20) for isotope notation,
see Table I, third line.

el

(27) = number U237 atoms/unit volume of
reactor, etc.

A, Lower chaln

d_d_(:?l = o, (25) oo 4 ~ (3¢) 03; ?:
¢
or d‘(ac) = %, (25)%s ) — %—%

d;, = "effective" cross section;j (26)
presumably has a resonance
absorption.

4

@Y _ @

 

dt Tae Taz
L = L + 03
'T57 Tfin J7¢P
d(z7) - (27) _ (37
= i
L
Py = T
L
fifi? + Q;qu

L 4

The formation of (28) by (27) + n - (28) is neglected
because the reactor normally will contain (28) for conversion.

- e S8 EB e e mp S5 W W S 4 W S - S A S T W W @ ap - -
 

~240-~

B3y =

|._I
1
Haf
...l-
|._.:
+
q'

T38

note: 038 assumed all fission

 

 

d(48) (38) M8 ¢
a® TTg T T (48 Tyg @
a8) o 38 (4+8)
dt = 38 T38 T48
_1
Fig = T:3L8
36 = 1 . 1L
T ™ T38 * 0-38 "P

B. Upper Chain

a(2 2
429) | c(atayg) (20,5 - é,gg—l - (2903 ¢
d(29)

C(1+a,5) (25) 0,5 @ - %;-’—l

¢ = conversion ratio

1 . l.g
Tog ~ 029¢

 

fT& i~
CHah en L
-241-

 

The reaction (29) + n - (20) is neglected since the
half life of (29) is only 23 minutes.

M_@l_%&l_%_(_gg)g}gcp

 

a(39) _ 29) _ (39)
at = Pag Tog  T3g
1
T
Bog = —=2— (= 1, all fluxes
%-5-9- + 0-2'9? here considered)
L - 1 + L + g,
T39 T T3g 39"P
ax9)  (39) —~ 4
& T (48)ay, g8 ¢ - ek (9 Tig P
dat = F39 T 6 Tyg  Tyg
1
B39 = T 139
+ == + O
T* T, 399
48 Iug8 @
Pug = T
48 ¢

Neglect effects (39) + n -+ (30), no cross-sections known.
a(4o - 40
1) | 50 - GO - 0o g

amo) | o (49) _ (40)
'+9TL|_9 %

Sememnenecyusneiiily
sy

 
 

"242"‘ m

B = %9 Tyo P
9- 1
T+ Tuofp
1 1
To =T+ %ol
3#9 - (rad. capt./total abs.)

C. Egquilibrium Values, Time Scale

. By setting %E = O, the equilibrium values are:

7_ Lower chain

(27) = Tpn(25)ay5035 ¢

(37) = By T37 (23)e5035¢

(38) = By B3y T3g(25)apsT35¢

48) = 527 537 538 TL|-8(25)G25°§5(P
Upper chain

(39) = Bpg T3g (25) C (L+azs) ga5¢p

(49) =

§329 Byg + Boy B3y F3g Pug Gliw §+a)§'

g (25) ¢ (+ayg) a5

i‘p——"'—'—'—
SRR ETE
: o

-
Py
 

-243-
The second term is negligible relative to first, hence:

(49) = Bog B3g Tyg(25)C(1+e,5)004 ¢
(#0) = Bpg B3g Bug T%(25)C(1+a25)q;_,5¢
Through a chain, the e~time required to reach a certain
level is approximately the sum of the times for each step

(including the time for the final step). If this total time
is indicated by a bar, this gives:

Lower chain

Tog

Tog + Top

T26 + T27 + T37

Tog * Top * T37 + T3g

Tog + T27 + T37 + T38 + T).g

3 =Bl i
Fww
O O 3 NN O
[} i 1l i 1

Upper chain

3]
[ §!

* T3

* 139 * Tyg
T39 4 Th9 + Tho

i 3
£
O O
f i i1
3
2 m'-fl = W3
O O
+

The values of the nuclear constants appearing in the
above relations are given in Table C-1l. The secular results
are summarized in Table C-2. The U-236 concentration and
build-up time is given in Table C-3.

These results are predicated on there being no purging
of U-236 by isotopic separation. The results for both U-236
buildup and Pu-238 contamination are therefore more pessimis-
tic than is the case for the reactors given engineering study
by this project.

 
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TABLE C-1, Numerical Values Used in Computation
- Uranium— Neptunium Plutonium
Mass munber 235 236 237 238 239 240 237 238 | 239 | 240 238 239 240 241
‘Symbol (25) (26) (27) | (=8) (29) (20) (37) (38) | (39) |(30) (48) | (49) (40) (41)
aF, b 538 0 0 0 0 0(19mb)| 1600 18 757 0 980
aln, ¥), b 99 6 (60) 2.8 22 150 (0) ? 425 361 600 350
__g(sbs), b 637 6 (60) 2.8 22 150 1600 443 111318 600 1430
2 2.51 2.5 2.5 2.96 (3.3)
n 2,12 0(10‘"4) (2.5) 0,115 | 2,01
o 0.183 10* | (0) 23.6 | 0.474 0,357
T (1/2, =) 10% | 107y | - 10%y - - laby | - - |- |9oy2 [10% | 10%y | 12y
T (1/2, B) - - 6.8d - 23.5m | 14h 2 - 2.4 |2.,3¢a | - -2 - - -
T (esf) 9.8d 33.8m | 20.7h - |3.02d |3.31d| - - - - -
0,0234d| 0.8634 |
s.f. no./gm.hr. |1.3% 0.6 | 6001000 24.8%0.9 20 1.1x07 | 35 | 1.6200°

 

 

() Indicate values used of considerable uncertainty.

#  Values generally from BNL-170,

0.025 e.v, Maxwell neutrons.

? for Pu-241 from Chalk River, s.f. rates from LA-1484, all data are for

-¥¥e-
 

-245-

IABLE C-2. U-236 Concentration and Buildup Time

 

 

£= 0 0.5 1.0 1.5

N(26)/N(25) 16.5 11.0 8.25 6.60

T,gs vears, (in body of table below)

for w = 10 233 155 116 93.2

-"%% 10* 23 15.5 11.6 9.3
107 2.3 1.6 1.2 0.93

€ = ratio: resonance capture in (26), to
thermal capture in (26),.

N(26)/N(25) = equilibrium atom ratio.

e-l1life for buildup to equilibrium

T
26 value of (26).

Loss in conversion ratio at eguilibrium is:

a
§C = - —h—ig; = -0.1545,

independent of assumption about & .
cccccc

00000

OOOOOO

vvvvvv

\\\\\\

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Sy Nn"

“i.f’{
& L

_omantlh )

246

 

 

 

 

 

 

i
TABLE C-3. Summary of Secular Results E
—_ 1
Process | Spec, (49) | T (38) {48) 8)+ T, T T 8)+(48)+
tyole | power | (3 | G| X\ THHWIY| T9)+(9Y| Go+a9)| 2B | 4B | (50)e(4s) fr | P
Days | Units(40) | Units(40) | Units(40) Ty | -Tpg| ¥res. Units{40) Fract.| Fract,
(days) | w/em Atomic | Atonmic Atomic Atomic Atoumic Atomic
Ratio Ratio Ratio Ratio Power | Days| Days Ratio
1. | 20° | 0003 | .00016 |2.76 | .000074 | .000074 | .000301 |11.5 |12.5 | 116 | .00046 00029 | 247
10% | .00292 | .o0158 |2.64 | .00304 | .000747 | .00379 [11.4 |12.4 | 11.6| .00537 L0029 | .243
10° | .0246 | .0125 |2.50 | .o188 00562 | .0244 - 110.4 | 11.5 | 1.2| .0369 028 | .201
10 | 10° | .00955 | .00507 |22.2 | .000715 | .00233 | .00305 . |22.1 |32.0 | 116 | .00812  |.0029 | .763
1% | o798 | .0450 |20.8 | .00742 | .0226 | .0300 |21.6 |31.8 | 11.6| .0750 028 | 720
| 10° | 313 | .156  |10.3 | L0645 | .14 | 78 (17.9 {23.3 | L.2| .33 26 | 44T
100 10° | .103 0608 | 173 | 00006 | .0315 | .0325 |10 | 202 | 116 | .0933 0282 | .962
10% | o7 291 [78.2 | .0183 .32 342 89.6 | 143 | 11.6| .633 225 | .893
100 | 57 304 |12.6 | .154 .517 671 35.9 | 46| 1.2| .975 Jh2 | 507
Footnote
1 2 3 4 5 6 7 8 9 10 11 12 13 14

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1, Net chemical process cycle time,
2. Based on U-235 alone, Pu fission adds to healt removal problem, not to conversion ratio,
3. From secular equation assuming conversion ratio of unity (entire table).
L. Usual Pu~2/0 buildup,
5. Time (e-time) to reach equilibrium up to and including Pu~240 in upper chain.
6., This is atomic ratio in column heading multiplied by 6.8, the ratio of spontaneous fission rates of Pu~-238 to Pu-240.

It represents contamination of product.

7. See footnote 6,
8. See footnote 6.

9. Time after (26) equilibrium to reach (38) equilibrium, both e-timea.

10.
1.
12,
13.
4.

4

Time after (26) equilibrium to reach (48) equilibrium, both e-times,
Time for (26) to reach equilibrium, e-time,
Total contamination of product by s.f. isotopes in Pu-240 equivalent units.

First reduction in propagation of lower chain by chemical removal of Np-237.
Reduction in propagation of lower chain beyond Np-238 by large fission cross section of Np-238.
 

——— _247-

APPENDIX D

EFFECT OF FAST NEUTRON REACTIONS OF RE ON
THE_CONVERSION RATIO

The‘average loss in C.R. due to L16 absorption is

I‘\f Li6 Oy L16
(C.R.) = - N(LL) Ol ) (D-1)
S 2,.(235).

where the bar indicates a time average over the operating
period for the Be in the reactor. The formulas required for
the evaluation of (D-1) are derived as follows: If the
thermal flux ?%h is normalized to one fission neutron, an
effective thermal cross-section, 0 ,py for the (nya) reaction

may be defined by the relation
N(Be) Ourr BAn= I - (D-2)

where J is the numbér of neutrons absorbed by the (n,a) re-
action. If ‘Z& denotes the total thermal absorption cross-
\ - section and p the resonance escape“probability, then

so that (D-2) becomes

J &t »
Gerr = “N(Be)p | (D-L)

From the secular equation

lQ-n

e N (118 = N(Be) o0 - N(1®) QiSYP  (D-5)

2.

it follows that at equilibrium

 
: - -
B L .

-248- | ——

6 6
N(Ii™) G(Ii") = N(Be) O e (D-6)
On the average, however,

fi(Li6) Q.'(L16) = g N(Be) o ¢p (D=7)

where g is the time average fraction of the final L16 con=-

centration., g depends both on the flux and on the operating
period.
Combination of (D-4%) and (D-7) yields

g T I,
p

T(1i®) a;(L16) = (D-8)

so that (D-1l) becomes

1 At Q)
J(C.R.):-gJE 23)=~gJ'}%_RTz‘2$§—5'%.

sesee e (D“g)

The evaluation of g will be considered in connection with the
secular equations for the Li6 buildup. The quantity J is
defined by the integral

J = N(Be)j G(n,a) ‘Pf . du , (D-10)
as

where the integration is taken over the region of non-
negligible flux and cross-section. This region was taken as
extending from E_ = Lt Mev to E = 0.892 Mev corresponding to
UzO0touwl.5 (EwEj e™™. G(n,a) 1s given as a function
of 1 1in Table D-1. In the absence of detalled information
concerning f?ast(u)’ 1t was assumed that

(w) ¢ —2W____ u (D-11)
‘Pfast r,[zs(u)Ja (u‘)] £ NG |5,

?1'

s
- -
el
-

=

 

o= " -249-
S Piaiing,
where O-E.Be) =Log +o;]B y and
N g
q(u) = fx (ut) exp J N%_Bi " du! (D=-12)
Be
pu,x) P, x = N /N, (D-13)
Here U
Y = S Kt aur
-0
and the fission spectrum g is normalized so that
+ oo
X(w) du =1, (D-1%)

and B(u) is an attenuation factor obtained by an examination
of the approximate behavior of the exponential in (D-12),
g(u) =+ 1 as x + o,

Substitution of (D=-11) and (D=-13) into (D-10) then gives

J(x) = Sdu Tr@ls_fi_l B (u,x) Y () (D-15)

Be

0f course J(0) = 0. Figure D-1 gives the variation of J with
X. Figure D-2 shows ‘Y(u) as a function of energy. This curve
was obtained by integrating the fission spectrum given in
LA-140-A., TFinally, Table D-1 lists the cross-section values
and other data required for the evaluation of the integrals
(D=15) as well as the analogous integrals (D-16) considered
below,

In analogy with (D-15) the excess neutrons provided by
the (n,2n) reaction per unit source neutron is approximately

gilven by |
0.9

K "-igdu -(érl'%—l)- Y (w) = 0.028 (D-16)

@

?::;=;=;======

e
. : L W A
:mh“g

-250- AT

Comparison of this number with J(m) &£ 0.027 shows that
there is no net fast effect when account is taken of both the
(n,a) and (n,2n) reactions.

To evaluate the guantity g in (D-9) consider first the
secular equations for Li~ buildup,

g—g N(Be) = - N(Be) [a;ff + aq:h]cp , (D-17)

4. N(11%) = N(Be) o

e - N(11® 011y (0-18)

With the definition
T(Be) = Cpp + Oy, (p-19)
the solution of (D-17) is
N(Be) = N°(Be) e"a-tBe)‘Pt (D-20)

Substituting (D-20) into (D-18) and solving the resulting
equation ylelds

+
N(Lié) - ¥°(Be) e~ G-(Lis)‘PtS a.f;f)era—-(Lié)‘OTBefl‘Pt'dv
e

o
eGSO EO (D"‘gl)

For further computations ¢ (Be) was ignored compared to
d‘(Li6) = 910 barns, since Gopr 18 Of the same order of
magnitude as 03y = 0.00473 barns. Then, to a good approxima-
tion,

6
N(Li6) = N°(Be) _I?_flé {1 - e-fl'(Li )‘Pti (D-22)
| ai®) ‘ -

1% @ - e~¥T)

Nequi. ) (D-23)
 

-251-

 

where

11®) = ¥O(Be) ToLL

N T118
L
1 3.48 x 10t
T = = years
1118y ¢

when (P is in neutrons/cmz/sec.
Evaluation of fi(LiG) from (D-23) and substitution into

(D-7) gives
To
€= 7 S (1 - &7 at
0 o

-T
=1-4 @-e T | (D-24)
o
where T  1is the length of time that the Be is left in the reactor.
Assuming T° equals six years, the cost write-off period,
and calculating the average flux from the relation

1,221 x 10”+1 5

¢ - ag¢(23

5
where G}(235) = 538 barns and S = Magslgf %Yé3 in Xa for

a 500 Mw reactor, the values of ¢, T and g corresponding 1o
various assumed masses of U-235 have been tabulated in Table
D-2. In Figure D-3 g is plotted as a function of the mass of
U=235 in the reactor.

 

.
\\\\\

lllll

-
Thede

28 2

TABLE D-1, Data for Fast Neutron Reactibns of Be.

~¢S¢C-

 

 

 

 

 

 

 

 

 

 

 

v B O n, a(a) tBe Be @) c"_n,zn(b) T (28) () X () () B (u,x),x=2
0 4% Mev | .O45 barns 416 barns | .30 barns .60 barns | .055 1
15 | 3.h4 . 043 37 .22 «55 .090 .902
.30 | 2.96 .02 .62k .16 .51 .150 .888
L5 2.55 .O5 . 520 .10 L6 .210 .811
.60 2.20 .033 <37k .06 3 .275 +699
.75 | 1.89 .030 . 354 .025 .39 .365 .638
.90 | 1.63 .026 .395 0 .31 475 657
1.05 | 1.%0 . 020 . 520 - .11 550 .701
1.20 | 1.20 .015 . 624 - .033 .625 o« 79k
1.35 | 1.04% .012 . 686 - .012 «705 . 886
1.50 .892 01 .728 - . 0066 755 . 900
(2) éECU—%O#O Neutron Cross-Sections (Compilation of Neutron Cross-Section Advisory
roup
(b) Guess based on meager data
()

Fission spectrum from LA-140 4.
 

Li6 Concentration Factors

 

 

TABLE D=2,
Reactor Power = 500 MW
M(25) ‘-P T g
neuts/cm2—sec
50 ke 2.27 x 10" 1.53 yrs. .75
100 1.1k " 3.01 .52
200 . 568 n 6.13 .36
400 284 12.25 .21

 

 

 

-253-
-254-

 

|

FIGURE D=1

U
VY = f X, ()

- 00
as a Function of Energy

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y

 

 

 

 

7

6

5 4

E(MEV)

e

  

3

 

 

 

 

 

 

 

 
J(X) x 102

 

 

-255-

FIGURE D-2

J(X) -’J{g gi }: Number of Be

Atoms Undergoing (n,a) Reaction
per Unit Source Neutron vs. X.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.0
b
"
2.0
10
|
0" > 7 — 5 10 12 14

X = N(Be) /N(Bi)

 
-256-

L.OK I I 1 i
FIGURE D-3

g = Average Fraction of Final

9 - 11® Concentration as a Function |
of M(U-235)
Reactor power = 500 MW

8

7

6

s\

o

4 \

3 \\

2 \‘\,

R

O 50 400

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

-257-

APPENDIX E

TWO-GROUP, TWO-REGION REACTOR EQUATIONS

E.1 EQUATIONS FOR THE FLUXES

Let

‘?1 = flux in the ith energy group (i1 = 1 repre-

sents the thermal group and i = 2 the epithermal)

A= 1 = transport mean free paths
17 (2,4

{2,

— - " '
22 = m = total "fast absorption'
(with Ej = 2 Mev),

 

Z& = total thermal (true) absorption
Ay
T=3x, =88°
2 M
L™= a2 thermal diffusion area
I35,

Superscripts (0 for the core and 1 for the blankét) will be
used to designate the region of the reactor to which any of
the above quantities belongs.

There are two equations of the form

,T V2¢2 - ¢2 +nt % ¢l =0 (E.1-1)
12 v2 ¢, - ¢+ p % $p =0 (E.1-2)

for each region. The fast fission effect, € , has been
ignored. '
On assuming a solution of the form

SEEn , -
Fal w oo
-258-

(E.1-3)

 

and substituting in (E.l-1) and (E.1-2) there results the
well-known secular equation

1+ 1°%%) @+ <R?) = pip = Ky (B.1-4)

Two cases may be distinguished:
a). Core. k> 1. The two roots of (E.1-4) are

)(2 = }(02 2u 1§ \}1+u - 1} (E.1-5)

where

k -1 k -1
Ko —I?':: = :I% y U = (koo-l) L|-fi‘2(1-[32), 52 = ;2-'!’- .

 

2
Here, one root x‘ = /u"" is positive while the other z(:_' = -v2

is negative.

b). Blanket. koo< l. Here

€ = Wov ‘1{ i-v - 1} (E.1-6)

where

xoz = %Q y V = (1'kco_) ""fia (1-32), 82 = ? .

2
In thls case both roots )(3 ,+ a"?\l 2 9 respectively, are

negative.
The equations for the fluxes are

(o)
¢ = A Snsr, g ..S.i__nh.;.;?_.l: (E.1~7)
" 4 s sinpr,cg slabvr (E.1-8)
 

 

 

 

e et— T -259-
N‘_—H
o sinh 24 (r-b) sinh A, (r-b)
¢=F r + G T (E.1-9)
al sinh )\, (r-b) sinh A, (r-b)
¢"°=F 5y = + G 8, —

>0 0088 (Eol“lO)

where b is the outer radius of the spherical reactor and the
coupling coefficients Si'are given by

2
S - = Eol"ll)
1= W2 /4, P 1+I.-2-z.(12 (

The constants A, C, F and G are determined from the boundary
conditions which state the continuity of flux and current at
the boundary, r = a, between the core and blanket. Deriva-
tion of convenient expressions for the critical determinant
and for these constants 1s straight forward, though tedious.

E.2 THE CONVERSION RATIO

The quantities required for the derivation of the ex-
pression for the C.R. are

Total source(fast) =ylf(°)2{°)5¢l(°) d(core) +

s Z{l)jfil) d(blanket)

 

. s(@) 4+ (D | (E.2-1)
~ U-235 consumed = 28 (5(®) + (1)) | (E.2-2)
: (o) d (o)
. Fast leakage from core = I.f(o) = -1l-1ra2 A2 ¢2 l

3 _ ar lraa

) eosces e (E02-3)
-260- M —
———————

A1) g4iL)
(1) _ 222 9%
Fast leakage from reactor = Ls s-Hrb 3 ar

 

r=b
ee s e e (E02""+)

The method of derivation of the expression for the
C.R. of the two-region reactor is the same as that employed
for the bare reactor except that the contributions from both
the core and blanket must now be considered. The contribu-
tion from the resonance capture in the core is

(S(o) - Lf(O)) (1 - p(o))

 

— (E.2-5)
Lo (glo) 4 (D),
The corresponding contribution from the blanket is
D 41,0 _ 1 Dy - 1)
t e i 2 (E.2-6)

 

Te (5@, gD,

The contribution due to U-238 thermal capture is

250 €0 2 o0 acore + 200 ¢ FHRfDnzman
_1-_‘;';}, (S(O) + ST]')) —

v 2a238) 4 Z (0
2T @ § 25 - L (E.2-7)

This simple result is a consequence of the fact that
L, (238)/7, (U) 1is independent of the region of the reactor.

The total C.R. is the sum of (E.2-5), (E.2-6) and
(E.2-7). After a simple rearrangement of terms, there results

the expression
c.R. =C, =P~ § + 51 -4, + 55 (4.1=2)

of the text.
1.

9.

10.

11.

12.

130

| tememeetet—— -261-

REFERENCES

A-4315, R. Ehrlich, H. Hurwitz, and J. R. Stehn,
"A Malti-Group Method for Computing Critical Masses
of Intermediate Piles", May 9, 1947.

AECU-2040, D. J. Hughes, et al, "Neutron Cross-Sections:
A Compilation of the AEC Neutron Cross-Section Group",
August 1952.

AERE- T/R/108, H. Melvin-Melvin, "A Method of Cal=-
culating the Critical Dimensions of Spherically
Symmetrical Reactors-I", November 30, 1950.

BNL-170, D. J. Hughes, "Neutron Cross-Sections - A
Compilation of the A.E.C. Neutron Cross-Section
Advisory Group", May 15, 1952.

BNL~-Log No. C-5754 and BNL-170, D. J. Hughes, et al,
"Classified Neutron Cross-Sections Compila%ion",
January 25, 1952. (To be reissued as part of another
report).

Cr-2881, F. L. Friedman and A. T. Monk, "Macroscopic
Theory of Breeders and Converters", March 26, 1945,

CF-51-5-98, A. M. Weinberg, L. C. Noderer, "Theory of
Neutron Chain Reactions - Volume I -~ Diffusion and
Slowing Down of Neutrons", May 15, 1951.

CL-697, A. Turkevich, H. H. Goldsmith, P. Morrison,
Volume II, Chapter 4, "Neutron and Fission Physics",
Figure 7.

CpP-1121, G. Young, "Displacement of Delayed Neutrons
in Homogeneous Piles", December 7, 1943,

Cp-1456, G. Young, Murray and Castle, "Calculations for
Some Pile Shapes of which the Boundaries are Partly
Spherical®, February 25, 194k, |

KAPLQI#, H. Hurwitz, "Pile Neutron Physics I", December L4,
1947, |

KAPL-33%, H. Brooks, “Secular Equations for Breeding",
April 28, 1950. | |

KAPL-346, R. Ehrlich, "Multi~Group and Adjoint Calcula-
tions for SAPL-5", May 10, 1950.
 

 

-262-

14+. KAPL-611, "The Plutonium -Power. Breeder Reactor",

15. KAPL-§3%, "A Discussion of Possible Homogeneous Reactor
Fuels". |

16. KAPL-741, R. Ehrlich and S. E. Russell, "Fast Neutron
Cross-Sections of U-235 and U-238", May 19, 1952.

17. LA-140 and LA-140A, "Los Alamos Handbook of Nuclear
Physics", September 30, 1944,

18. LA-994, W. Nyer, "Summary of Fast Fission Cross-Sections",
December 2, 1549,

19. LA-1391, B. Carlson, "Multi-Velocity Serber-Wilson
Neutron Diffusion Calculations", March 24, 1952.

20, LEXP-1, "“Nuclear Powered Flight", September 30, 1948.

2l. Memo PLH-3, P. L. Hofmann, "Multi-Group Procedures as
used at KAPL", March 27, 1952.

22, MonP-4%28, G. Young and L. Noderer, "Distribution
Fufictions and Fission Product Poisoning', November 6,
1947,

23. NAVEXOS P-733, R. N. Lyon, "Ligquid Metals Handbook",
June 1, 1950.

24, NDA Memo-15B-1, V. F., Weisskopf, "A Preliminary Dis-
cugsion of the Ratio of Capture to Fission", July 23,
1952. .

25. NYO0-636, B. T. Feld, et al, "Final Report of the Fast
Neutron Data Projeet®, January 31, 1951.

26. ORNL-1099, S. Glasstone and M. C. Edlund, "The Elements
of Nuclear Reactor Theory - Part I", October 26, 1951.

27. ORNL-51-9-126, S. Glasstone and M. C. Edlund, ¥The
Elements of Nuclear Reactor Theory - Part iI“, October

19, 1951.

28. ORNL-51-9-127, S. Glasstone and M. C., Edlund, "The .
Elements of Nuclear Reactor Theory - Part III", October
26, " 19510 -

29. ORNL-51-9-128, S. Glasstone and M. C. Edlund, "The
Elements of Nuclear Reactor Theory - Part IV,

Elements of Nuclear Reactor Theory - Part V", October

30, 1951.
—-—/

AV
S
31.
32.

33.

3k,
35-
36.
37.

38.
- 39.

SEo e

*——-—-—“ _263_

R-233, G. Safonov, "Notes on Multi-Group Techniques for
the Investigation of Neutron Diffusion'.

RM-852

. Safonov, "A Note of Fast Breeding in Mixtures
of U238 and Th232,

February 19, 1952.

SENP-I, "Science and Engineering of Nuclear Power",
Ygéfig§ I - edited by C. Goodman, Addlison-Wesley Press

TID-70(Rev), "Journal of Metallurgy and Ceramics",
January, 1951.

TID-389, R. L. Shannon, "Nuclear Breeding, A Literature
Search", November, 1§50.

TMS-5, T. M. Snyder, "Fast Neutron Radiative Capture
Cross-Sections of Some Reactor Materials',

Y-F10-98, W. K. Ergen, "Physics Considerations of
Circulating Fuel Reactors", April 16, 1952.

H. Hurwitz, Nucleonics, 5, 61 (July 1949).

E. P. Steinberg and M. S. Freedman, "Summary of Fission-
Yield Experiments”, pp. 1378-1389. Book 3 - "Radio-
chemical Studies™ by C. D. Coryell and N. Sugarman.
MeGraw-Hill Book Co. (1951). o

 

s
-264- S —————————
ACKNOWLEDGMENTS

The short-term nature of this project made i1t imperative
that we draw on exlsting information at other installations
wherever possible. We are very grateful for the active co-
operation in this which we encountered in every case.

The assistance of the New York Operations Office of the
AEC in many phases of our work proved invaluable. The Divlision
of Reactor Development and the Operations Analysis Staff of the
AEC in Washington contributed much-needed advice and data. We
also wish to express our appreclation of the generous amounts
of time and information extended to members of our staff at
the following AEC installations and Contractors! offices:

Argonne National Laboratory Dow Chemical Company
Battelle Memorial Institute Iowa State College
Brookhaven National Laboratory Knolls Atomic Power Lab.
Detroit Edison Company North American Aviation Co.

University of California

Advance reactor information obtained from other groups
saved a great deal of time here and served to expedite
comparison with project reactors. We are indebted to the
Washington Office of the AEC for data on the Jumbo and Aqueous
Homogeneous Reactors and to the KAPL staff for information
about their Pin-Type Fast Reactor.

The members of the Project's Advisory Committee were
Harvey Brooks, John Chipman, Charles D, Coryell, Edwin R.
Gilliland, and Harold S. Mickley. This committee, in addition
to its assistance 1n shaping the general course of the Project,
made many valuable, specific contributions to the work.

Among the Project's consultants, Henry W. Newson
prcvided many useful suggestions, particularly in the field
of reactor controls. Warren M. Rohsenow spearheaded the
mechanical design activities. David D. Jacobus helped
ecrystallize structural designs of the reactors. George

2724 -265 %
i

-

S —————
— E ~265

T T ——

Scatchard and Walter Schumb contributed to and coordinated
the work in chemistry. Herbert H. Uhlig provided valuable
interpretation of corrosion problems. _

We are grateful to Karl Cohen of the Walter Kidde
Nuclear Laboratories for allowing two physicists from his
staff to join the Project for the summer. Lee Haworth of
the Brookhaven National Laboratory also helped at a critical
time when he made it possible for Dr. Jacobus to spend a week
with the Project.

As Executive Officer, William E. Ritchie facilitated
the work of the Project in many ways and helped it to work
effectively.