THE UN IV ER SIT Y OF MI CHI GAN
COTLLEGE OF ENGINEERING
Department of Nuclear Engineering
Technical Report
THE EFFECT OF THE PORT VOID IN THE PREDICTION
OF THERMAL NEUTRON BEAM PORT CURRENT
Sanford C. Cohen
O R'A Proe'. t'.' ,:' 78:
under contract with:
NATIONAL SCIENCE FOUNDATION
RAT NO. GP1032
WASHINGTON. C.
administered throughOFFICE OF RESEARCH ADMINISTRATION ANN ARBOR
January 1964
This report was also a dissertation submitted in
partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of
Michigan, 19630
ACKNOWLEDGMENTS
I would like to express my appreciation for the numerous helpful
discussions with the members of the doctoral committee during the course
of the research. In particular, the encouragement and suggestions of
Professors John S. King and Paul F. Zweifel are gratefully acknowledged.
The complete cooperation of Mr. J. B. Bullock and the staff of the
Ford Nuclear Reactor is appreciated. I am grateful to Mro Hilding 01son and the staff of the Phoenix Memorial Laboratory for their cooperation. Utilization of The University of Michigan Computing Center and
the services of the staff are acknowledged.
A U. S. Atomic Energy Commission special fellowship in nuclear
science and engineering provided financial support during most of my
years of graduate work. A Phoenix Memorial Project fellowship made
possible by the Ford Motor Company helped me to continue my work. For
these financial grants I am most grateful.
Finally, the support provided under NSF Grant GP1032 for the
preparation of this manuscript is acknowledged.
ii
TABLE OF CONTENTS
Page
LIST OF TABLES v
LIST OF FIGURES vi
NOMENCLATURE x
ABSTRACT xiii
I. INTRODUCTION 1
A. Statement of the Problem and Motivation for the Study 1
B. Related Published Studies 2
C. Outline of This Study 3
II. ANALYSIS 6
A. Orientation 6
B. Infinite HalfSpace Limit 8
C. Diffusion Theory Boundary Condition 13
D. Angular Flux Distribution 16
E. Derivation of j+B 17
F. Derivation of j[ 19
G. Scalar Flux Distribution Along the Lateral Surfaces 22
IIIo EXPERIMENT 28
A. The Ford Nuclear Reactor 28
B. Insertion and Positioning of Ducts 28
C. Normalization 33
D. Detectors 34
E. Counting Equipment 37
F. Unperturbed Flux Distributions for the Water
Reflector 38
G. Preliminary Duct Experiments 46
H. Duct Experiments in Water as a Function of
Duct Diameter 50
I. Thermal Spectrum 55
Jo Measurements in Complete Graphite Reflector 59
iii
TABLE OF CONTENTS (Concluded)
Page
IV. DISCUSSION 68
Ao Unperturbed Thermal Flux Distributions 68
B. 11/2in. Duct Experiments 69
Co Thermal Flux Attenuation Within the Empty Ducts 70
D. Radial Flux Distributions at the Duct Mouth 71
E. Fast Flux Perturbation 72
Fo. Thermal Spectrum 75
G. Scalar Flux at the Duct Mouth 76
Ho Scalar Flux Distribution Along the Duct Walls 79
I. Reactivity Predictions 86
V.ow CONCLUSIONS 88
APPENDIX
A. A SURVEY FOR THE ENHANCEMENT OF THERMAL NEUTRON
LEAKAGE FLUX 90
1. Introduction 90
2, ThreeDimensional Simulation 91
3. Streaming Flux 94
4. Configurational Effects 95
5, The Effect of Reflector Materials lo09
6. Final Remarks 132
7. Subroutine STREAM 135
B. DESCRIPTION OF THE CODES 140
C. COMPOSITIONS AND GROUP CONSTANTS 144
D. COMPARISON OF CURRENT BOUNDARY CONDITION AND PSEUDO
BOUNDARY CONDITION AT VACUUM INTERFACE 150
E. DUCT MOUTH CONTRIBUTION TO THE RADIAL CURRENT AT THE WALL 154
F. NORMALIZATION FOR TIME VARYING FLUX 158
Go COMPUTER PROGRAM FOR EVALUATION OF ~/r  R 160
REFERENCES 163
iv
LIST OF TABLES
Table Page
III.1 Duct Specifications 32
III.2 Normalization Data 35
III.3 Foil Characteristics 36
IIIo.4 Prediction of Activation at Exit of
1l/2In. Duct in Water 49
III.5 Neutron Temperature Measurement 58
IVol First Estimate of the Radial Component of the
Gradients for Boundary Condition in TwoDimensional Calculation 85
IVo2 Duct ReactivityMeasured and Predicted 87
AoI Transverse Bucklings 93
AoII Critical Volume, Area, and Relative Leakage for
Bare Reactors of Three Geometries 97
A.III Relative Leakage from a Face for Bare Parallelepipeds of Various Shapes 100
A.IV ColdClean H20 Reflected Parallelepiped Cores
of Various Shapes 106
A.V Infinite ReflectorCylindrical Geometry; Various
Reflector Materials 116
AoVI TwoDimensional Calculations; Fixed Sized Core;
Inserts of Various Materials 120
A VII Three and Six Inch Reflectors of Various Materials 129
Co I Compositions 147
C oII Thermal Group Constants 148
C oIII Fast Group Constants 149
v
LIST OF FIGURES
Figure Page
2.1 Void configuration. 6
2.2 Infinite halfspace geometry. 9
2.3 Normalized angular flux for c = 1; exact and 12
P1 approximation.
2.4 Linear extrapolation at vacuum interface. 15
2.5 Vector diagram at vacuum interface. 16
2.6 Configuration for calculation of jA. 19
2.7 Current balance geometry. 23
2.8 Current balance geometry including front interface. 26
3.1 Ford Nuclear Reactor core configuration. 29
3.2 Streaming duct, foil holder, and duct support
with normalizer. 30
3.3 Photograph of 11/2in, duct encased in 4in,
graphite insert. 31
3.4 Decay of a gold foil counted in the well counter. 39
3.5 Thermal and epicadmium flux out to four feet
in water reflector and along axis of 11/2in. duct, 40
3.6 Thermal and epicadmium flux in the core and the
water reflector, 42
3.7 Eastwest traverse of thermal flux in the fuel element
bordering on the east of the control rod element, 43
3~8 Unperturbed thermal and epicadmium flux in water
reflector. 44
359 Reaction rate at exit of 1l/2.in. duct in water, 47
vi
LIST OF FIGURES (Continued)
Figure Page
3.10 Thermal reaction rate in 4in. graphite insert and at
the exit of 11/2in. duct encased in 4ino graphite
insert. 48
3.11 Perturbed radial flux distributions at duct mouths. 51
3.12 Perturbed axial flux distributions at duct mouths, 52
3.13 Collimator design. 53
3.14 at collimator exit.
OBth
3.15 Radial variation of flux at 5in, duct exit, 56
3.16 Thermal flux attenuation within 11/2 and 5in, ducts. 57
3.17 Ratio of reaction rate of Lu176 to reaction rate
of 1/v absorber. 60
3.18 Graphite insert. 61
3.19 Unperturbed thermal and epicadium flux in graphite
insert, 64
3.20 Radial flux distribution in graphite insert. 65
3.21 Perturbed thermal flux at center of duct mouths in
graphite. 66
3.22 Axial distribution of thermal flux at mouth of
5in, duct in graphite. 67
4.1 Angular distribution of fast neutrons. 73
4.2 Effect on predicted thermal flux of fast flux
depression. 74
4.3 Predicted Bth for graphite. 78
th
4,4 Radial thermal flux behavior at four positions
along 5in. duct wall. 80
4.5 Axial thermal flux distribution along walls of 5ino duct. 82
vii
LIST OF FIGURES (Continued)
Figure Page
4.6 Cylindrical geometry approximation to coreduct
configuration. 84
A.1 Core geometries for H20 reflected configuration
studies (coldclean fuel). 101
A.2 Computed flux distributions; H20 reflected core;
3 elements transverse. 102
A.3 Computed flux distributions; H20 reflected core;
4 elements transverse. 103
A.4 Computed flux distributions; H20 reflected core;
5 elements transverse. 104
A.5 Computed flux distributions; H20 reflected core;
7 elements transverse. 105
A.6 Computed flux distributions; comparison of depleted
fuel in center and on circumference of core. 108
A,7 Computed flux distributions; infinite H20 reflector;
cylindrical geometry. 110
Ao8 Computed flux distributions; infinite graphite
reflector; cylindrical geometry. 111
A,9 Computed flux distributions; infinite BeO reflector;
cylindrical geometry. 112
A.10 Computed flux distributions; infinite D20 reflector;
cylindrical geometry. 113
A.11 Computed flux distributions; 7.5 cm graphiteremainder
H20; cylindrical geometry. 114
A.12 Computed flux distributions; 2,5 cm H20remainder
graphite; cylindrical geometry'. 115
Ao13 Core geometries for material insert studies, 118
A.14 Core geometry for 6 in. BeO reflector. 119
viii
LIST OF FIGURES (Concluded)
Figure Page
Ao15 Twodimensional computed flux distributions; HaO
reflected. 121
A.16 Twodimensional computed flux distributions; graphite
insert. 122
Ao17 Twodimensional computed flux distributions; BeO
insert. 123
A,18 Twodimensional computed flux distributions; D20
insert. 124
A.19 Twodimensional computed flux distributions; 6in.
BeO reflector. 125
A.20 Core geometries for 3 and 6in, reflectors of
various materials. 127
A.21 Core geometries for H20 reflected configuration
studies (equilibrium Xe8,4 percent burnup fuel). 128
A.22 (S4S3)max for 3 and 6ino reflection. 130
A.23 (S4S1)max for 3 and 6ins reflection. 131
A.24 Measured and computed flux distributions along axis
of cylindrical DO0 insert. 133
C.o WignerWilkins and Maxwellian thermal spectra
for coldclean FNR core at 300~K. 146
E.1 Duct configuration for calculation of mouth
contribution to wall radial current. 155
ix
NOMENCLATURE
c Number of neutrons produced per collision
d Linear extrapolation distance
h Duct length
j Neutron current
k Multiplication constant
~ Neutron mean free path
r Radial position
r Position vector
s Neutron source
t Time
v Neutron velocity
z Axial position
Zo Extrapolated endpoint
B2 Buckling
D Neutron diffusion coefficient
E Neutron energy
L Neutron diffusion length
R Duct radius
S Surface area
T Neutron temperature
I Neutron streaming flux
a Largest eigenvalue of diffusion equation
NOMENCLATURE (Continued)
K Inverse of diffusion length (1/L)
A Radioactive decay constant
Cosine of angle between neutron direction and z axis
v Average number of neutrons born per fission
Average logarithmic energy decrement
a Microscopic neutron cross section
T Fermi age
Neutron flux
X Fraction of neutrons born in energy group i
Z Macroscopic neutron cross section
Unit vector denoting direction of travel
Subscripts
a Absorption
act Activation
as Asymptotic
eff Effective
epi Epithermal
f Fission
g Geometric
m Material
o Scalar
s Scattering
t Total
xi
NOMENCLATURE (Concluded)
th Thermal
tr Transport
R Removal
T Transverse
TC Thermal cutoff
Superscripts
f Fast
q Source
th Thermal
M Milne
N Normalized
xii
ABSTRACT
A method is presented for the prediction of the exit thermal neutron current from a reactor beam port. The results of duct experiments
at the Ford Nuclear Reactor afford confidence in the validity of the
analytical technique. The motivation for the study is provided by a
prevalent desire to enhance thermal neutron beam intensity for spectrometer investigations.
The analysis employs elementary P1 theory, which is inspired by
an examination of the infinite halfspace limit. A return current
boundary condition at the source plane of the port is developed by summing current contributions from the lateral surfaces. An idealization
of the scalar flux distribution along the lateral surfaces of the port
permits a diffusion theory solution for the source plane angular flux
as a function of the port radius and the unperturbed flux distribution
in the medium.
Activation measurements of the scalar flux distributions at the
source planes of 11/2, 3, and 5in. diameter aluminum ducts were
performed in light water and graphite. Measurements of the exit current from the ducts provided with an internal collimator to eliminate wall leakage contributions were carried out in light water. The
largest duct perturbation was observed for the 5ino duct in water.
In this case the source plane scalar flux was depressed by 50 percent
from the unperturbed flux, and the angular flux in the direction of
the duct exit was 61 percent of the unperturbed value. Comparison of
the data with analytical results indicates that the exit current is
predicted very well, The source plane scalar flux is predicted with
somewhat less accuracy.
A background survey for the enhancement of the thermal neutron
beam intensity at the Ford Nuclear Reactor is presented in an appendix.
Variations in reflector materials and corereflector configurations
were investigated. The results suggest the employment of a ID20reflected core in a slab configuration,
xiii
I. INTRODUCTION
A. Statement of the Problem and Motivation for the Study
This dissertation develops a method for predicting the exit thermal
neutron current from a reactor beam port. The necessary removal of a
portion of the diffusing medium to accommodate a beam port exerts a significant influence on the behavior of the thermal neutrons in the vicinity of the port. Inasmuch as these neutrons comprise a source for the
beam, the beam port perturbs the intensity and spectrum of the exit
beam. Experimental results are presented which test the analytical
method under realistic reactor conditions.
The motivation for this study arises from a demand for beam intensity enhancement. An estimate of the influence of the beam port on the
exit current intensity is necessary for any thorough enhancement study,
although the problem has a great deal of intrinsic interest as well.
The increasing importance of neutron spectrometer experiments has revealed the need for the examination of problems associated with neutron
beam extraction.l
Roughly, the neutron flux, ~(h), at the exit of a beam port of
length h and cross sectional area A is given by o
/(O) A >
where /(O) is the neutron flux at the source plane of the port. The
1
2
crosssectional area of the beam is genera.lly limilted by the resolu.tion
requirements of the experimento Thus the intense neutron densities
available in the reactor may be attenuated by orders of magnitude at the
beam port exit.
Bo Related Published Studies
Studies for the enhancement of thermal neutron beam intensity have
emphasized the reduction of fast neu.tron and gamma ray backgroun.d. The
split core concept,2 resulting in the design of the National Bureau of
Standards Reactor93 involves the removal of fuel from the midplane region
of a D20 moderated and' reflected core, The tangential beam port concept,
applied to the Brookhaven Hig:h Flu.x Beam Reactor4 JHFBR.) involves placement of the beam ports tangential to the core. Furthermore. the D20 moderated and reflected HFBR is undermoderated, so as to produace exceptionall.ly high tiherrr.aI flt.,x peaki:ng in the reflector.
The effects of d.uc.t intlArodiUctior.n. into a react.or system ha've been
examined primarily:fromT t.:e standpojrint of the inf l1uence on the multiplication constant of tPe reactor. Behrens7 revised the magnitLde of L
the diffusion area of a m.edium containing gas,ccoolant passages, to account for the additional streaming leakage. Reynolds et a.6 investigated the reactivity effects of large voids in tole nrefl.ector of the Pool
Critical Assembly at Oak Ridge National LaboratoryO In a theoretical examination of the effect on reactivity of a reflector d;cnt Baraff et a1O7
3
treated the problem by ordinary perturbation theory. These studies do
not yield information concerning the perturbed scalar flux in the vicin.
ity of the void or the current within the void.
Shielding considerations have prompted the examination of the transport of neutrons down long ducts.8 The influence of the duct walls on
the attenuation of neutrons by ducts was studied by Simon and Clifford9
and they concluded that the direct streaming contribution predominates
for h/R >> 1o PierceylO attempted to separate the contributions to the
streaming flux at various distances along the dusct axis in measurements
at the LIDO reactor. In none of these investigations, however, was consideration given to the duct influence on the streaming source.
C. Outline of This Study
In Chapter II a simple analytical method is developed to predict
the th.erma1, neutron current at the exit of an evacuated duct from a
knowledge of the unperturbed scalar flux distribution. in. the medium0
We consider a long:radial* duct introduced into the reactor reflector)
presenting a small reactivity perturbation to the mulwrt:iplying system.
The infinite halfspace problem is reviewed from the standpoint of the
anisotropic limit of the finite void. This permits comparison between
exact solutions and the P1 approximations and motivates a PIL approach
to the present problem. A pseudo boundary condition on the scalar flux
A railal duct refers to a duct whoseaxis lies along a radius of the reactor core0
4
is developed from the consideration of a finite return current to the duct
mouth. Utilization of this boundary condition. permits a diffusion theory
solution for the perturbed scalar flux distribution at the duct mouth.
The diffusion theory solution is used in conjunction with the P1 angular
distribution to predict the exit current. The return current to the
mouth is evaluated. by summing contributions from the lateral surfaces of
the duct. Finally, some consideration is given to the scalar flux distribution along the lateral surfaces of the duct; although the realistic
problem is not solved, a reasonable boundary condition is derived from
idealizations of the physical situation.
A number of activation experiments were conducted outside of the
core of the Ford Nuclear ReactorO These are described in Chapter III.
Initial comments concern the reactor, the positioning of ducts, the
normalization procedure, and the foil detection techniques. Measurements
of the unperturbed scalar flux distributions in the water reflector and
reactor core are presented. Preliminary l1/2ino duct experiments in
water and a small graphite insert are described0 The scalar flux data
at the mouth and the collimated exit current for 11/2, 3, and 5in0
ducts in water are displayed. Data arepresented for the attenuation
of the thermal flux within the i1/2 and 5inoducts. A measurement of
the neutron temperature perturbation at the mouth of the 5ino duct is
described, The final section describes the scalar flux measurements at
the mouth of the ducts in the graphite reflector~ All unperturbed scalar
flux measurements are compared with groupdiffusion calculations~ ScaLar
flux data at the duct mouths and exit currents are compared with predictions based upon the analysis presented in Chapter IIo
The discussion in Chapter IV considers the assumptions inherent in
the analysis. The limitations of diffusion theory for the prediction of
the unperturbed scalar flux in the reflector are pointed out, and the
seriousness of.these limitations from the point of view of the present
analysis is considered. An attempt to predict the realistic scalar flux
distributions along the duct walls is discussedo Duct reactivity evaluations obtained from twodimensional groupdiffusion calculations based
upon initial estimates of the wall gradients are compared with experiment. In addition9 items from previous chapters which require further
consideration or clarification are discussed in Chapter IV.
A survey directed toward enhancement of the thermal neutron beam
intensity at the Ford Nuclear Reactor is presented in Appendix A. The
investigation considers variations in reflector materials and corereflector configurations for radial beam portso The results are relegated
to an appendix because the work was in the nature of a background study
preliminary to the main body of this dissertation~ In this survey, consequently. the beam port influence on the exit current is neglected.
This can be considered a conservative omission from the standpoint of
the reflector materials study, since the longer diffusion lengths possessed by the most desirable reflectors give rise to lower beam port current depressionso The method described in Chapter II, however, can be
used as a calculational refinement for the conigurati.ons of particlar
interest and a specified beam port diameter,
II. ANALYSIS
A. Orientation
The configuration of interest is a long cylindrical void inserted
into a diffusing medium. This is sketched in Figure 2.1. Neutrons
Ri
A`~ B
Figure 2.1. Void configuration.
streaming from all points on the interface between the diffusing medium
and the void form a beam at B far from interface A.* Assume that the beam
is collimated, so that a detector placed on the axis at B optically views
only the front interface at A. We are interested in the number of neutrons
*Henceforth interface A will often be referred to as the mouth of the
duct and B as the duct exit.
6
per unit time passing through a unit area normal to the axis at B. This
quantity will be called JB(E), the partial current at B. JB(E) is
equivalent to the integral over the surface, SA, of the angular current with
energy, E, leaving z dSA directed into solid angle d2 about Q intercepting
dSB:
(2.1)
j+(E) = dSA. A(r,Q,E) dS (2.1)
B dSB
Then for any fragment of the energy spectrum:
~~+ B1 A — ~~d2
jB(EoHE1) dE dSA ZQ ~A(r,,E) , (2.2)
dSB
or the thermal current can be expressed as:
j =Bth = dE dSA z. A(rQ E) d = dSA Z )
Bth dSB dSB
The problem then reduces to that of solving for the angular current
at interface A. The formulation is initiated by writing the steady state
Boltzmann equation for the angular flux, ~(r,Q,E), in space, angle, and
energy:
Q.v ~(r,,E) + Z(r,E)X(r,s,E)  s(r,QE)
d2' dE' d(r,Q',E')Zs(r,Q'+Q,E'+E). (2.4)
8
The conventional symbols used in reactor physics are employed, and are
defined in the Nomenclature. Hypothetically a solution for the geometry
under consideration can be obtained after the source and scattering kernel are specified.
In light of the complexity of the geometry, the pursuit of an exact
solution for ~(r,,QE) is nearly futile. A numerical approach, utilizing
a high speed digital computer is feasible, but besides lacking in generality, would be prohibitively timeconsuming. Rather, we seek a technique which yields the angular flux to a sufficiently high degree of approximation to provide realistic predictions for jBth and which is simple enough to be useful for practical computationso To this end, P1
theory is examined.
B. Infinite HalfSpace Limit
In order to assess the validity of a Pl, or diffusion theory, approach consider the case which brings about the highest degree of anisotropy at interface A, that of a full. vacuum boundaryo Allowing R., the
void radius, to go to infinity and considering monoenergetic neutron
diffusion in an infinite halfspace of fixed composition, exact solutions
can be obtained for the steady state transport equation (assuming isotropic
scattering):.. + I s(z,l) + s) d+I (z,iL), (2.5)
9
and comparisons may be drawn with P1 theory results. Two cases are
particularly well known. These are the classical Milne problem, s = 0,
and the case of constant production, s = constant.
We shall examine the angular flux in the case of constant production
for a unit isotropic source per unit volume. The coordinate frame is
sketched in Figure 2.2. Davison expresses the angular flux at the
Figure 2.2. Infinite halfspace geometry.
boundary in the constant production problem in terms of the Milne solution:
9(oti)= tL([lz iM(o,1),for [ <. (2.6)
( ijM.l
10
For c = 1:*
IjM(0) = o(), (2.7)
and thus:
[ q((o,)=] = (OL,), for,u < 0. (2.8)
where [~(0,1)]N is the tabulatedl2 Milne angular distribution normalized
to unit scalar flux at the boundary.
The P1 equations for isotropic scattering and a constant isotropic
source are:
dz
(2.9)
3 d. o(Z) + Zt sd(z) = o
3 dz
The angular flux is given by:
1
(Z) = 2n+l in(z)Pn(G). (2.10)
The solution for the scalar flux is expressed as:
Sl (Z)= Ce KZ + s (2.11),,.l
*Our concern with reflector materials indeed directs attention to values
of c close to 1. For light water, c =.994.
11
where
K Zt a L (212)
Applying the Marshak boundary condition.
1
L 0,(O~ )dp 0 ~ (2.13)
0
and setting the source equal to unity., one obtains for the P1i angular
distribution at the boundary:
N
pl() 2 ( 2 1 4)
~' O 2
~p_1(0.,tl) is the familiar form of the normalized Pdl angular distribution,
and is given by:
rPl (~90) t Z 3 (2.15)`p1 4
The normalized P1 and Milne angular distributions are compared in
Figure 2035
A comparison of the scalar fluxes is obtained "by integrating Eqso
(208) and (2o14) over all angles. yielding:
(0)
9 1.a=. 155 (21 6)
L~
N
1.4 ANGULAR FLUX N (0,pL(NORMALIZED TO UNIT SCALAR FLUX) VS.
1.3
1.2 EXACT; MILNE PROBLEM AND CONSTANT PRODUCTION (C=I)
1~2~,,P1 THEORY
i. i7
7
1.0.97.8
=t.7~ ~ ~ ~ ~.7 701.6 7.5.4.3.2 7
1.0 9 .8 .7 .6 5 .4  3 2 .1 0.1.2.3.4.5.6.7.8.9 1.0
Figure 2.5. Normalized angular flux for c = 1; exact and Pi approximation.
13
angular flux close to A = 1o Setting 1t = 1 in Eqs. (2.8) and (2.14) we
obtain:
 ~' 1 =.993 (217)
The same result is obtained for the Milne problem (s = O) if both the P1
solution and the exact solution are normalized to the exact value of the
scalar flux at the boundary. Thus the P1 approach gives rise to an
angular flux for very small angles which differs by less than 1 percent
from the exact solution. It should be pointed out that the Pl1 solution
is free of transport corrections.
C. Diffusion Theory Boundary Condition
Since a Pl1 approach provides an. excellent approximation for the
angular flux in the limiting case of the infinite halfspace, we shall
examine the finite void using diffusion theory. The current returning to
the mouth of the void originates from neutrons leaving the lateral surfaces. This forms a basis for a boundary condition at A which is physically realistic and consistent with the general formulation of diffusion
theoryo
In addition to the assumption of low capture in the medium, c' 1,
and small angles to the detector, (R/h)2 << 1, the following two assumptions
are made"
lo The perturbed scalar flux is flat over the entire duct mouth.
This permits us to neglect the radial dependence at the mouth.
20 The fast flux is unperturbed by the. void. As a corollary to
this the assumption is made that the overall reactor fission source
perturbation is negligible.
Specification of the partial current at A, coupled with the continuity
requirements at the boundary between the multiplying medium and the reflector completely specifies the diffusion theory solution for the scalar
flux in the medi.um~ It is convenien.t however. to transform the current
condition into one on the scalar flux.9 o0 Proceeding analogously to the
familiar treatment of the limiting full vacuium interface1 and referring
to Figure 2o4o
jA 4 6 dz(28)
A.Assuming a linear decay of the scalar fl.ux past A.
OgZ) Zi d A 9 (2.19)
dzA
we solve for the distance, d, past the interface, at which o0(d) = 4 JA
4 jA =a d I + 1o A0 (2.20)
15
d4'0 (Z)
d+Oid
A
Figure 2.4. Linear extrapolation at vacuum interface.
Combining Eqs. (2.18) and (2.20) we obtain:
( ) = 4 A (2.21)
3 ~tr 4 jA
the wellknown diffusion theory extrapolation condition, applied here to
a vacuum boundary possessing a finite return curent.
It is shown in Appendix D that the application of this pseudo boundary condition on the scalar flux rather than the actual one involving the
partial current introduces negligible error for c Z 1. Furthermore, the
employment of a linear extrapolation distance, d, rather than the extra11
polated endpoint, zo, is valid for values of c close to one:
16
d _c 1/2 for c 1 (2.22)
zo
D. Angular Flux Distribution
The P1 approximation to the angular flux is needed to calculate
duct currents. It is instructive to derive it directly from the integral form of the transport equation. In the constant cross section
approximation the angular flux is given by (Eq.(416) of Reference 11):,(r, 0) = r 4+ s(r r) e dr). (2.23)
Figure 2.5 is a vector diagram pertinent to Eq. (2.23). The source term
in the equation is defined per unit solid angle.
Figure 2.5. Vector diagram at vacuum interface.
Expanding any function, f(rQ), about the point r in a Taylor series
gives~
f(r( a) = f(r) + ( k)f(r)+V ) f) + (2 24)
Accordingly expanding the scalar flux and source in the integral equation,
performing the integration over r, and retaining two terms yields:
C.
/(rQ) &[Oo(r) ~ 2\7,o(r)] + I[s(r) m>Vs(r)] (225)
4jc
Eo Derivation of j
Using the angular distribution presented in Eqo (2o25) and diffusion
theory solutions for the source and scalar flux distributions at the mouth,
we are in a position to derive an expression for jB, the partial current
at the exit of a long collimated duct. Restricting ourselves to the
geometry illustrated in Figure 201, and assuming polar symmetry:
V r + z (2026)
ar az
Q r + (227)
and:
dQ d SB Q (2 28)
The angular flux at A in the direction of a detector on the axis at B
is expressed as:
~A(rQ)' o (r)
+' { _ zI r ajs: (2.29)
Assume that,0o s, and their derivatives are constant over the emitting
surface, so that:
A(r L ) +  s _ n ds. (2.30)
Lrto p dz A dz
Substituting this expression for the angular flux into Eq. (23) and
integrating over the surface at A results in:
B 42+ 3 ( ) d0 A
R2 2 ds
+ g2 s h ds 231
For a long duct, R2/h2 ~ 1, SO that the expression reduces to:
jB @h dzj [s X A (2052)
19
The slowingdown source can be computed from the fast flux obtained in a
fewgroup analysis. Assuming fast flux isotropy:
f f
S ZR o
4ir
F. Derivation of JA
Equation (2.21) specifies the diffusion theory boundary condition to
be used in obtaining the distribution of the perturbed scalar flux in the
vicinity of the duct mouth. It is necessary to sum contributions from
all lateral surfaces of the duct in order to derive an expression for jA
in terms of the scalar flux distribution along the walls of the duct. The
geometry is illustrated in Figure 2.6.
Figre2..0 on f co
Figure 2.6. Configuration for calculation of jA.
20
Equation (2.25) is employed once again for the angular flux along
the lateral surfaces of the duct. Assuming that the slowingdown source
contribution is negligible, and that c' 1:
d(z,R,n)  4Lo(z,R)  ~ V BoI ~ (2.34)
Since the mouth is treated in one dimension, it will suffice to solve
for the return current at the center. Referring to Figure 2.6:
= r z (2355)
dQ dS, (2.36)
= R + z (2.37)
and:
dS = 2T Rdz. (2~38)
The return current at the mouth is expressed as:
jA = dS ( ))( R.,~ Q )(2239)
= dSA
Substituting the angular flux:
~(z,R,Q)  o(z,R) + + z, (2~40)
4~~~~~C _ ar'
21
into Eqg (239), and supposing that the void extends to infinity, we
obtain for the return current at the mouth:
r0 2C 00 z 4E
j =,, dz Z(Rz) +I dz A e R d dz 2
A 2 (R2z2)2o (+) (R2+z2) 92
(2. 41)
In order to assess the validity of this expression, let us examine
the limits as R+O and Roo. It is first convenient to transform variables
such that:
o00 Z 00 z nLR
_ _ of z~o(R, Rz) )r Rz.
=dz Z(o~+2)2)+ I dz A R Qt dz (12
A 2 (l0z)0 (i+z52 (1+z
(2042)
When R becomes small, the scalar flux and its derivatives are slowly varying with respect to the remainder of the integrand, and can be taken outside the integral sign. The integrations yield:
lim JA  o(,0) + IR i ~o A (2.43)
lmjA 4
which is the partial current at A in the absence of the void0
which is the partial current at A in the absence of the void.
In the limit of large R the rest of the integrands are slowly varying with respect to >o and its derivatives, and accordingly:*
*We assume here that the scalar flux is monotonically decreasing with z;
the unperturbed flux is of the form: S(Rz) = CieaiRz
ii
22
lim jA = 0, (2.44)
R>oo
the vacuum interface condition. Equation (2.41), then, has the desired
limits.
G. Scalar Flux Distribution Along
the Lateral Surfaces
The final task, and evidently the most formidable, is the evaluation
of the scalar flux distribution along the lateral surfaces of the void.
We expect the average perturbation of the angular flux along the duct walls
contributing most heavily to the return current to be less than that at
mouth. This is because of the considerably more favorable view factors
along the walls. For small void diameters it is reasonable to assume
that the angular flux along the walls is unperturbed.
A sound approach to the problem, however, consists of a current
balance over all surfaces. Such a procedure has been carried out for a
void of finite length along the axis of a cylindrical reactor.l4 Considering for the moment a long cylindrical void, the radial component of
the gradient at a point (R,zl) on the surface is obtained by equating the
outgoing (in the positive radial direction) partial current to the contributions from all surfaces of the void. The angular distribution is
given by Eq. (2.34). Figure 2.7 illustrates the geometry. Carrying out
23
Figure 2.7. Current balance geometry.
the angular integrations and solving for the radial component of the
gradient, one obtains:
~~~/~rl R
z _/r _ l + 3  dz o(R,z) fi(p) + dz o/Rz f2(p)
Xo(Rnzj) 2 2R (Rz) (Rz)
dz (Rz ) f ) 2.45)
where:
f(P) = {2(2+p )(lp2, (2.46)
f2(p) = k[2(lp4) + 3(2p2))E(P2)  (1_p2)(8+p2)K(p2fl
3p (2.47)
_(p) = 2(lp)l/ (2+p2)K(p2)  2(l+P )E(p), (2.48)
24
2 2l/2 zjz
pi(z,z)2 + ~ p;) = PW 1+/2 (2.49)
and K and E are the complete elliptic integrals of the first and second
kinds o15
The integral equation is not easily soluble~ Zimmerman assumed a
separable solution, in which case Eqo (2459) reduces to.
(25o0)
rigor, it possesses the correct limits (for the void along the axis of
a cylindrical reactor) o
The trivial case of the infinite cylinder with a constant flux
along the walls2 although. not directly applicable to our problem, is instructive and does permit a separable solution~ For this case ~o(Z)/~o(z
= 1,S and o/z = 0, and the i ntegrations are performed:from ca to c
The numerator in Eqo (200) vanishes, and we obtairn,e 8,/ z.4~..z = o 0, (2o51)
for all zthe axis of
ThIn the spirital case of the infinite cylinder with a constant flux of
25
the form ~o(z) = a+bz also leads to a zero gradient condition for the
infinitely long cylinder. This can be shown by examining the point
zl = 0. For this case ~o(z)/o(zl) = l+cz and o Z) = c, and the
numerator of the righthand side of Eq. (2.50) becomes:
1 dz(l+cz)fl(p) + dz cf3(p) (2.52)
00 00
The terms multiplied by the constant are odd functions, and their interals are zero. The remaining terms are identical to the constant flux
case, giving:
R
I 6/6/ra o 0 (2.53)
O(R,O)
This example, however, is somewhat of a mathematical artifice, in that
the result is not the same for zl f 0; and so the assumption of separability
is not strictly valid (moreover, the linear expression permits negative
fluxes).
The actual void geometry consists of a semiinfinite cylinder bounded
by an interface at z = 0 (See Figure 2.8). The current balance at any
point zi > 0 now contains additional terms from the contribution of the
mouth. The isotropic and z component of the gradient contributions are
derived in Appendix E and are of the form:
1 zl2
2 g o ) 2;0.' l+ R2 Z2 6, z2 /,Riz.
d,L~2 Z('O~~~'"'I~B; " " "r''"'u
26
R
Figure 2.8. Current balance geometry including front interface.
These additional terms must be added to the righthand side of Eq. (2.45).
Postulating as before a constant flux everywhere along the boundaries
such that O/kzo = o and ~o(O) = 1, and assuming separability, one
finds that the contribution from the mouth is equivalent to that from
the negative half of the infinite cylinder studied previously. This
leads once again to the condition:
R = O, for all zl >. (2.55)
~O ZZ
We have not solved the physically interesting problem, however. In
fact the argument for the R + 0 limit of Eq. (2.41) demonstrates that a
constant flux along the lateral surfaces does not give rise to a duct
perturbation at the mouth at all. The realistic unperturbed scalar flux
is likely to be a decaying exponential or sum of exponentials, in which
27
case a z dependent radial component of the gradient is expected along
the duct walls. Positive near the mouth, the radial component of the
gradient becomes negative at an axial distance from the mouth such that
the streaming contributions exceed the unperturbed flux in the medium.
In light of the complexity of the actual physical situation, the
pursuit of an accurate representation of the scalar flux distribution
at the duct walls will be avoided. Instead we assume that the zero
gradient boundary condition is a reasonable approximation. This is
equivalent to direct utilization of the unperturbed scalar flux if the
unperturbed flux distribution is radially flat in the vicinity of the
duct. The validity of this zero order perturbation treatment will be
assessed by its success in predicting the results of experiment.
III. EXPERIMENT
A. The Ford Nlfuclear Reactor
The experiments were conducted on the south face of the Ford Nuclear
Reactor~ The FNR is a swimming pool facility similar in design. to the
16
Bulk Shielding Reactor at Oak Ridge National Laboratory~ The fuel is
of the MTR type. Regular elements contain 140 grams of U235 in 18 plate
3 ino x 3 ino subassemblies~ Partial. elements_ which house the poison
235
rods9 contain 71 grams of U3 in 9 plate subassemblies. A more complete
description of the facility is contained in the literature.17
The core configuration adopted for the experiments is illustrated
in Figure 3lo. Occasional minor variations in the fuel loading were necessitated by criticality considerations, bult changes were confined to
the north extremities of the core. The specific elements on the south
face were left runchanged during the period of experimentso
B, Insertion anad Positioning of Ducts
A duct. the duct support, and a foil holder are sketched in Figure
3.2. Variable sized collars mounted on the aluminum plate of the support
accommodate the ducts for positioning~ The weighted plug bolted to the
bottom of the plate fits into a south matrix hole of the grid plate which
is furnished with an offset dowel pin to guarantee angular placement.
28
29
BEAM
TUBE 1 2 1R W
I (d
PNEUMATICA (
N /
I~,,,,,,, t,,,E /\ N I
)'
A,BC SHIM RODS
CR CONTROL ROD
G GRAPHITE
FC FISSION CHAMBER
Figure 3.1. Ford Nuclear Reactor core configuration.
30
it(Q,~)o I
L"W~~~ ~4'.1
Streaming Duct
Foil Holder
Duct Support with Normalizer
Figure 3.2. Streaming duct, foil holder,
and duct support with normalizer.
31
Figure 3.3 is a photograph of the 11/2in. duct encased within the 4in.
graphite insert positioned adjacent to the core.
4in. graphite insert.
32
Table III. 1 is a list of the dimensions of the three alumi:num ducts
employed in the water runso Also tabulated are the reactivity worths of
the ducts positioned 4 in. below the centerline of the core in the column pictured in Figure 3 l. The ducts were evacuated to eliminate attenuation of neutrons by air and water vapor.'TABLE III o 1
DUCT SPECIFTICATIO NS
Wall Front Reactivity bk/k
0OD.o Thick Window Length., Axial. Distance from Core
i.n. ness9 Thick ft 0 76 6 6 1 166
ino ness in. ____: crm cm. cm
11/2 00625 O020 4 0....7 007
5 o083,O.40 O.... o ooo
5 o187 0 0 6 2 4 1.% o o0%j < ol.
Runs were made at vJarious low power le'vels in the range of 500 watts
to 70 kilowatts Th.e duration of each run was ch.osen so that duct insertion and withdrawal, occupied a time interval, of less than. 11/2 percent of the total i rradiation. Spacing of runs was limited'by the induced
acti.vity of the m.etal wnich would h.ave decayed in. less th.an oneh.alf
h~our had t~h.e material been pure alumilnumo Th~e presence of ailloyi.ng con.
33
stituents in the aluminum extended the cooling time considerably for high
power activationso
C. Normalization
Because the experiments were performed at variable reactor power on
different days, it was necessary to employ a precise and reliable procedure to normalize the runs to one another. First attempts at normalization using the pneumatic tubes on the west face of the core proved unsatisfactory~ Although the activations in the pneumatic system agreed
within ~ 2 percent with a fixed monitor viewing the west face at the exit
of a beam port, the proportionality between the flux intensity on the
west face and at the duct position was destroyed by rod manipulations,
xenon buildup, and core movements,
Normalization was finally accomplished with the aid of the pinwheel
frame shown in Figure 3~2~ Foils were mounted at the four corners of
the diamond which were each 61/2 ino from. the duct axis and 1/2 ino from
the reactor core. That the duct itself exerted no influence on the normalizing foils was verified by comparing the normalization of consecutive
5 in. void and novoid runs with. the fission chambers positioned as in
Figure 3o1. The infl.u.ence of poison rods was minimized by holding B
shim. rod 100 percent withdrawn and the control rod greater than 40 percent
withdrawn du.ring an irradiationc Furtherm.ore, the duct axis was positioned 4 ino below the core midplaneo That the spectrum was not changing
54.
significantly for the pul.rpose of normali.zation. was confirm.ed by irradiating two cadmi.um co vered foi l.s si.mui.lta.rneou.sl.y wit:',t the bare normali zing
foils Typical. normalization data are displayed in Table T1::I:2 Com.parison. of'the roo'tmean..squ.are devi.ation between. the four positions set
a normalization error i.pper Limiit, of approximately: 13 percentL Repeated
ru.ns were reprodu1cible within. t.nere lm.i. tso
D, De't e tcto rs:Mevtallic foil acti vators were used for all of the measurements.
Goldd, copper9 man.ganese. and lutetirnf. were each u.tili zed to some extent.
Table'T1To3 iJS a list of the perti rnenrt foil characteristicso
Gold was used:for most of t"he mea....re.mients, f.t could be obtained
chemically' p;ure and possesses a hi.gh actiration cross section together
with. a long half1lifeo In. addition., te large 409 e' resonance cross
sectieon permi.ts epithe'rmal a... a r.c. a a welldefined en.ergyo bRare
and cadm. iu.'er.; d 1/ 6 i.n.x 1/1 6 in x fOC;1 i,,:foils were employed at
th:e duletl c olit:ho wit;h rno s.Tignifi carn.t pertim.bratio n.iduced in larger foi l
iLrradiated at the exi.t of2 th e di.nt, The small. foils at the mou.th. could
he coun.ted in. a well. co;aftter soo: 1fter i.r.radi.ati.on. at the flulx Level.s
necessary to acttivate thie foil s at the e:xito Weigh:i.ng was accomplished
on ar eiec tre osta tic a ba la.e.
Epi2thermial flu.xes were miea.ured by completely su:r:rosindirng the foi ls
with 20 mnils of cadmftnu Icomplete; cadmrm,.um enclosuare was shown. to lead
TABLE III.2
NORMALIZATION DATA
E2 4ww
Nominal
Duct Rms Deviation
Run Control Rod Reactor Mon. 2 Mon. 4
No Date S ize, Mon. 1..2 Mon. 3.....m..eviatio
DNo. te inze'vPosition, Power Bare Thermal Mon Bare Thermal }
32 9/4 11/2 40 5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 _
28 9/4 1120 42 1.200.204.201.210.207.211.205 2.5
31 9/4 H20 40 0.5.099.0995.097.099.099.102.99 1.5
39 9/12 11/2 58 70 10.75 10.9 11.05 11.2 10.6 10.9 1l1.0 1.5
40 9/12 11/2 40 70 8.35 8.25 8.13 8.32 8.40 8.15 8.12 2.0
44 9/14 5 80 35 4.15 3.96 3.81 3.90 4.o09 4.19 4.01 3.5
61 12/4 5 70 1.75 1.69 1.64 1.82 1. 71 1.73 3.5
*As recorded by the reactor instrumentation.
TABLE IIL5
FOIL CHARACTERISTICS
Ref. 18 Ref. 19
Princ,
Abundance, Thickness, A) f.act(E) EP
Detector i.Activity gact(.025 eN epi E Pur ity
4& ~~in. atepi B
Counted b. b.
Au 100.001.41 Mev 98 1558 Excellent
7
Mn 100.002.845 Mev 15.5 11.8 Poor
7 Cu and Ni
Impurities
CU63 69 Nat., 0oo5.~5 Mev 43 4L4 Excellent
Abund. Annih. y
from B
Lu176 2.6 Nat..o1o.208 Mev 360 b. Excellent
Abund. in 7 at.14
2.3 ev
LuAl Alloy Reson.
37
to large errors. Cadmium cutoff energies are derived by Dayton and
Pettuso20 A 20 mil covering of cadmium around any of the foils listed
in Table III3 affords a cutoff energy for an equivalent perfect filter
of ~448 ev in an isotropic flux and o342 ev in a monodirectional flux.
The region between these two energies is estimated to contribute less
than 1/2 percent to the total activation in an assumed Maxwellian
spectrum fitted with a 1/E tail at o18 evo
The foils were washed in acetone, weighed, and covered with a thin
layer of paper and tapeO it was determined that the covering material
introduced no perturbation in the foil activation by carrying out several
trial irradiations with various thicknesses of paper and tapeo Bare and
cadmiumcovered foils were placed at least one inch apart at the mouth
of the ducts. The perturbation induced by the cadmium on the bare foil
at the mouth and on the one at the exit of the duct was estimated to be
well within the normalization erroro
E. Counting Equipment
Except for occasional utilization of the multichannel analyzer,
counting was done in a well counter equipped with a 3 in. x 3 in. NaI (T1)
scintillation crystal backed with a Dumont No. 6292 photomultiplier tubeo
A Radiation Instrument Development Laboratory Model 4951 scaler provided
the high voltage to the photomultipliera The discriminator was set well
above the noise level and the high voltage was maintained in the center
of the counting plateau at 1275 volts. Reproducible positioning of foils
in the well was accomplished with the aid of a test tube device incorporating a centering rod~
Stability of the system over long intervals of time was important.
Figure 3~4 illustrates the measured decay of a gold foil over a 7day
periodo Subsequent count rates were kept below the maximum in the figure
to eliminate dead time corrections and to avoid jamming of the mechanical
register. An absolute activity determination on the multichannel analyzer
set an efficiency of 52 percent for the.41, Mev gold gamma rays~
Fo Unperturbed Flux Distributions for the Water Reflector
The first experiment was done to verify that the current at the
exit of the evacuated 11/2 in. duct was far in excess of the unperturbed
flux in the vicinity of the duct exit. If this were the case, activations
measured at the duct exit could be attributed solely to streaming contributions from'the region close to the mouth of the duct. Thermal* and
epicadmium gold reaction rates were measured in the water bordering the
south face of the reactor core out to four feet. The measurements were
performed at the midplane of the core one row east of the position displayed in Figure 3l.o The relative reaction, rates are plotted in Figure
*The thermal reaction rate is taken to mean the difference between bare
and cadmi mcovered measurements.
59
106
Activity versus Time For a Gold Foil in the Well Counter
105
2 3 4 5 6
Time In Days
Figure 3.4. Decay of a gold foil counted in the well counter.
40
7 Relative Gold Activation versus Distance from Core
10 \ 0 Thermal Reaction Rate in Water
\ x A Epithermal Reaction Rate in Water
XThermal Reaction Rate Along Axis of 1 1/2" Duct
6
I0 \
IQ5
I0 \
0
U
0 3
>I 0
L \
r \
0 10 20 30 40 50
Distance From Core (in.)
10
Thermal reaction rate data for foils placed within the 11/2 in.
duct positioned 5/16 in. from the reactor core are shown in the same
figure. The bare and cadmiumcovered measurements were made separately
to prevent severe mutual shielding. The results demonstrate that the
streaming current in the duct exceeds the unperturbed flux in the medium
by greater than two orders of magnitude 34 in. from the duct mouth.
Of peripheral interest is the flux distribution in the core. Foil
insertions were made between the plates of the fuel using 10 mil aluminum
venetian blind slats for positioning. The disadvantage factor in the
fuel plates of the FNR is nearly unity,22 and for this reason measurements in the channels between plates are representative of the homogenized
medium. Since the gold cadmium ratio is low within the core, copper
foils were used for this measurement. The data are shown in Figure 3.6.
The flags indicate the uncertainty in position and normalization. The
results for groups three and four of a fourgroup, onedimensional machine calculation are normalized to the data.
The flux peak in the two offcenter fuel elements can be explained
by the close proximity to the water channels serving the poison rods,
which were nearly 100 percent withdrawn during the course of the experiment. Since access to the partial element was limited, the peaking
effect in the vicinity of a water channel is demonstrated in Figure 3.7
by an eastwest flux distribution in the element bordering the east side
of the control rod element (see Figure 3.1).
Flux Distribution in Reactor Core
OMeas. Thermal Rcn. Rate
I Meas. Epicadmium Rcn. Rate (X5)
3.0
3.0 N Predicted by 2Gp.
I IN 1D. Group Diffusion Calculation
o! I
2.O
I.0
H20 Reflector F.E.#34 F.E. 31 F.E.#29 F.E.433 F.E.# 413 Graphite Reflector
Core
Figure 3.6. Thermal and epicadmium flux in the core and
the water reflector.
43
1.50
Cu Reaction Rate versus Distance
From West Boundary of Fuel Element
1.40 
@ 1.30
1.20' 1.10
1.00
9Q' i I
0 I 2 3
Distance From West Boundary of Fuel Element (in.)
Figure 3.7. Eastwest traverse of thermal flux in
the fuel element bordering on the east of the control rod element.
Most of the streaming measurements in water were performed with the
duct axis positioned 4 in. below the core midplane in the column illustrated in Figure 3.1. This position was chosen to minimize the perturbing influence of the control rod. The unperturbed duct axis flux distribution at this position is plotted in Figure 3.8 out to 22 cm from the
core. The data are normalized at 2.16 cm to the results of a twogroup,
twodimensional groupdiffusion code.
The data in Figure 35.8, as well as most of the subsequent results,
are plotted on the basis of detector reaction rates. This presents the
UNPERTURBED FLUX VERSUS DISTANCE FROM CORE IN WATER
70
0 MEASURED THERMAL REACTION RATE
o 0 A MEASURED EPICADMIUM REACTION RATE
0 60 0 CALCULATED BY 2GROUP DIFFUSION
F  0 0 CODE (NORM. AT 2.16 CM.)
03 50 ESTIMATED POSITIONING ERROR ~ 3mm
ESTIMATED REACTION RATE ERROR+2%' 40
z30
0
< 20
b.I
o 10
2 4 6 8 10 12 14 16 18 20 22
DISTANCE FROM CORE (CM.)
Figure 3.8. Unperturbed thermal and epicadmium flux in water reflector.
45
data in its raw form, yet it is a convenient form for interpretation in
the present study. The normalization technique discussed earlier permits
direct comparison of the results of measurements employing identical detectorso
Corrections for flux perturbations induced by the foils must be made
before the ratios of epithermal to thermal fluxes can be evaluated.
23
From the results of Dalton and Osborn we estimate a low correction to
the thermal flux measured'by small l mil gold foils in water:
~'th
th = lo04 o (3.1)
meas.
The selfshielding induced by the 4~9 ev gold resonance brings about a
24
considerably larger correction to the epicadmium reaction rate:
Lepi= 22 (3A2),/ oa(E) dE
epi measo
for 1 mil gold foils.
Applying these corrections, a comparison can be drawn with the resulting ratios of epithermal to thermal flux predicted by the fourgroup codeo
Assuming a 1/E epicadmium spectrum and adopting the epithermal absorption
integral in Table IIIo3 we get:
46
5.53 Kev.625 iev (E)dE
o.625 ev
/(E)dE
at 2.16 cm in the water reflector (Figure 3.8). The fourgroup code predicts that ~3/'4 =.195, a 5 percent deviation from the measured ratio.
G. Preliminary Duct Experiments
With the evacuated 11/2in. duct positioned at various distances
from the core, bare and epicadmium reaction rates were measured at the
duct exit. The data are presented in Figure 3.9. The thermal data are
compared to predictions for the angular flux, obtained by using the unperturbed flux distribution in Eq. 2.41 to compute the boundary condition,
a groupdiffusion calculation to obtain the perturbed flux distribution
at the duct mouth, and Eq. 2.30 for the angular flux. It is noteworthy
that the peak occurs at a duct position closer to the core than the position of the unperturbed flux peak in the medium.
The same measurement was performed in a 4 in. x 4 in. x 28 in. graphite insert bored with a 11/2in. axial hole to accommodate the duct.
The insert was positioned flush against the reactor core as shown in
Figure 3.3, and the duct was moved in steps of 1 in. by introducing graphite plugs into the hole in front of the duct. The thermal flux data in
the solid insert as well as the streaming data at the duct exit are plotted
in Figure 3.10. The dotted lines represent the previous water data for comparison.
47
MEASURED REACTION RATE AT 1 1/2" DUCT
EXIT VS. DISTANCE OF DUCT MOUTH FROM CORE
o MEASURED THERMAL REACTION
RATE (CADMIUM BACKED)
a MEASURED EPICADMIUM.008 REACTION RATE
0 _PREDICTED THERMAL
ANGULAR FLUX
..007 (NORMALIZED AT 1.16 CM.)
E.006.005.0 0
H.004.0035
J
0.002 ~..001
I i I I I I
1.0 2.0 3.0 4.0 5.0 6.0 7.0
DISTANCE OF DUCT MOUTH FROM CORE (CM.)
Figure 3.9. Reaction rate at exit of 11/2in. duct in water.
48
60
6 Unperturbed Thermal Flux versus Distance From Core
//'O In Graphite Insert
5 0 / X ___.Previous Water Data of Figure 3.8
50 / \
O! \\ I
x 40 
E
0
0 20
oI I
U
0
I0
10
I I I I I I
Thermal Reaction Rate at 1 1/2"Duct Exit versus.007 O Distance of Duct IMouth From Core
O In Graphite Insert
  Previous Water Data of Fi gure 3.9.00 6 t j Ax (Shifted for no cadmium backing)
E
E.:.005
E\'.004
0
o.003
_., I I I I,
2 3 4 5 6 7
Distance From Core (in.)
Figure 5.10. Thermal reaction rate in 4in. graphite insert and at the
exit of ll/2in. duct encased in 4in. graphite insert.
To compare the data with predictions for the actual magnitude of
j+Bt at the duct exit it was necessary to measure the leakage contriBth
butions from the duct walls and also the effect of activation from backscattering at the exit. Measurements with the duct positioned at 2.16 cm
in water resulted in the compilation of Table III4. A 23 percent backTABLE IIIo4
PREDICTION OF ACTIVATION AT EXIT OF 11/2 INCH DUCT IN WATER
Exit Measurement
Bare reaction rate 0O0691 x 1010 gm min10 It
Cdobacked reaction rate o00531 x 10
Cd.covered reaction rate o00142 x 1010'I.0I x101 0
Thermal monodirectional reaction rate.00389 x 10
Reaction rate from walls of duct.00082 x 101
Net thermal monodirectional reaction rate
from mouth of duct.00307 x 1010
Net thermal monodirectional reaction rate
from mouth of duct'B (corro for
cntgo 1/4 in. foil) o00322 x 10
Mouth Prediction
Direct streaming from ~6875 in.of radius
of duct oO0286 x 10 1
Transmission through wall, from last
10.0625 inoof radius o00035 x 10
Net direct streaming prediction. from
mouth Bth2.00321 x 1010
scattering contribution was revealed by backing the bare foils with
cadmium. A wall contribution of 21 percent was measured by covering the
mouth of the duct with cadmium A 5 percent correction originated from
53
47.61
24. 61
4',, z1/26
1 1/2.252",SPACER 426 S; CADMIUM
DUCT MOUTH CENTERER POSITION
Figure 3.13. Collimator design.
neutron detector at the exit to view.905 sq. in. of the mouth regardless of the duct diameter.
The high epithermal content of the spectrum at the collimator exit
necessitated employment of a detector possessing a relatively low resonance
integral. Manganese combined this attribute with a high activation rate.*
The bare and cadmiumcovered foils were irradiated simultaneously by offsetting the covered foils by 3/8 in. from the centerline. A copper impurity in the manganese required that the foils be counted in the single
channel well counter within one manganese halflife.
The results of the collimator measurements are plotted in Figure 3.14.
*The reactivity perturbations induced by the introduction of the larger
ducts caused power fluctuations in the reactor system. Since the measuring foils and normalizing foils possessed different decay constants,
the transient was not automatically corrected for by the normalizing
foils. It was necessary to evaluate the normalization by the analysis
presented in Appendix F.
51
PERTURBED FLUX AT DUCT MOUTH VS DISTANCE
FROM DUCT AXIS
IN WATER AT 2.16 CM.
x NO DUCT
680Co 80 1/2" DUCT
o + 3" DUCT
x A 5" DUCT
E 70 PREDICTED THERMAL
ot) FLUX AT MOUTH
60 x IX Xxxl xe5 X X xxx oxx I
~II I I
<. +1 o o
50 al i 0
50A 1 0/2" PRED.
0 + I THERMALI
I +I
a 1 1+ I 5" PRED.
C) 3 I' Ir I A A
3 2 I 0 2 3
EA ST WEST
DISTANCE FROM DUCT AXIS (IN.)
Figure 3.11. Perturbed radial flux distributions at duct mouths.
52
DUCT INTERFACE
THERMAL /
50 ' I5~01 A~d,/dz =.08 CM,
PREDICTED =.I I CMZ>9 + =.24 CM
T70~~~~~~~~ ~PREDICTED =.26 CM
z
k d /dz 60M
Q 30.60 CM
PREDICTED = 42 CMI
0
ct3, /X/1PERTURBED FLUX VS
DISTANCE FROM CORE
9 IN WATER
20
EPICADMIUM'z _,9 _+~~~ O 11/2" DUCT
10 — z + 3'DUCT
~z~*A 5'DUCT
/ /UNPERTURBED
FLUX.2 4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
DISTANCE FROM CORE (CM.)
Figure 3512. Perturbed axial flux distributions at duct mouths.
53
24. 61 "
XI I 1/2 1/252
SPACER 46" CADMIUM
FOIL
DUCT MOUTH CENTERER POSITION
Figure 3.13. Collimator design.
neutron detector at the exit to view.905 sq. in. of the mouth regardless of the duct diameter.
The high epithermal content of the spectrum at the collimator exit
necessitated employment of a detector possessing a relatively low resonance
integral. Manganese combined this attribute with a high activation rate.*
The bare and cadmiumcovered foils were irradiated simultaneously by offsetting the covered foils by 3/8 in. from the centerline. A copper impurity in the manganese required that the foils be counted in the single
channel well counter within one manganese halflife.
The results of the collimator measurements are plotted in Figure 3.14.
*The reactivity perturbations induced by the introduction of the larger
ducts caused power fluctuations in the reactor system. Since the measuring foils and normalizing foils possessed different decay constants,
the transient was not automatically corrected for by the normalizing
foils. It was necessary to evaluate the normalization by the analysis
presented in Appendix F.
1.00
Bth AT COLLIMATOR EXIT (FIXED SOURCE PLANE AREA)
VS. VOID DIAMETER. NORMALIZED TO I FOR NO.90 [ \ VOID. IN WATER AT 2.16 CM.80.70 
+m
w.60
I I
n.50
0 RELATIVE REACTION RATE AT.40 COLLIMATOR EXIT
PREDICTED RELATIVE jBth.30
I I I I I I I
1.0 2.0 3.0 4.0 5.0 6.0 70 8.0
VOID DIAMETER (IN.)
Figure 3.14. JBth at collimator exit.
55
The solid line presents the prediction for jBth The data and the predictions for j are normalized to a value of unity for no void.
Bth
3. Thermal Flux Attenuation Within the Empty Ducts
The behavior of the flux within the empty duct is related to the
main topic of this study, and a series of experiments was performed to
examine the attenuation of the forward component of the thermal flux.
Bare and cadmiumcovered measurements were performed separatelyo The
detectors consisted of 1/16 ino square 1 mil gold foils. The bare foils
were backed with cadmium, which posed the formidable problem of mutual
shieldingo In order to avoid the laboriousness of separate runs for each
foil position, the five foils were positioned to avoid shadowing by spacing them 600 apart. Investigations within the 5 ino duct demonstrated
that the radial flux was reasonably flat near the exit of the 5 in. duct
(Figure 3.15)o
The thermal reaction rates within the 11/2 in. and 5 in. ducts as
a function of axial distance from the mouth are plotted in Figure 3.16.
The measured slopes at 44 in, are recorded in the figure.
I The rmal Spectrum
A question of some concern was the influence of the void on the
thermal spectrum, and consequently the spectral effect on the apparent
perturbations measured. Results published by Walton et alo25 indicate no
56.07.06
Flux versus Distance From Axis at Exit of 5" Duct
0 Measured Thermal Reaction Rate.05 A Measured Epicadmium Reaction Rate
o.
E
*.04
E
c
CLI_
o.03
I A.02.0 I I
1.5 1.0 .5 0.5 1.0 1.5
Radial Distance (in.)
Figure 3.15. Radial variation of flux at 5in. duct exit.
57
1.0
Thermal Flux versus Distance From Mouth of Duct
O 5 "Duct
A 11/2" Duct.1 
Slope 2.48
i,
ro
o
a
U
0 I
o.01
Slope= 2.70.001 I
20 30 40 50
Axial Distance From Mouth of Duct (in.)
Figure 3.16. Thermal flux attenuation within 11/2 and 5in. ducts.
58
hole effect on the spectrum in polyethylene with voids up to 2 in. in
diameter. The effect of the 5 in. duct on the thermal neutron temperature
at the mouth was measured using the.14 ev resonance in the Lul76 absorption cross section as a spectral index. Bare and cadmiumcovered 10 mil,
21/2 percent LuAl foils were irradiated simultaneously with manganese*
at the mouth of the voided 5 in. duct and at the same positions with the
duct filled with water. The activities were measured on a multichannel
analyzer corrected for drift with a Co59 standard. The resultant activities are listed in Table III.5.
TABLE III.5
NEUTRON TEMPERATURE MEASUREMENT
H20 Medium
Bare Lul76 21.82 x 105 gmlmin1
Cd. covered Lu176...
Bare Mn 11250 x 105 "
Cd.covered Mn 356 x 105
Ath (Lul76 )/Ath(Mn).00200
5 in. Void
Bare Lul76 10.99 x 105 gmlmin1
Cd. covered Lu76...
Bare Mn 5561 x 105
Cd.covered Mn 276 x 105
Ath (Lu76 )/Ath(Mn).00208
*Manganese was used as the l/v detector because Lu175 possesses a large
resonance integral, giving a cadmium ratio of only 1.59 for the voided duct.
59
The ratio of the reaction rate of Lu176 to that of l/v detector is
calculated using the form of the resonance given by Schmid and Stinson:26
a(E\FE =Const. (3.4)
1 + 1108(E.142)'
The calculated ratio, normalized to unity at 0~C, is plotted up to 100~C
in Figure 3.17. In this range of temperatures, the observed ratios suggest a temperature rise at the mouth of the voided duct of 10~ + 5~C. A
10OC temperature rise at 3000K would account for a 1.7 percent depression
in the average activation cross section of a 1/v detector.
J. Measurements in Complete Graphite Reflector
Graphite was chosen as a second medium in which to examine duct
effects. The selection of graphite was based upon diffusing properties
widely different* from those of light water, as well as accessibility at
the FNR. The south face of the core illustrated in Figure 3.1 was covered
with three rows of 3 in. x 3 in. graphite reflector elements, except for
two columns at the center, which were left open to accommodate the solid
graphite (reactor grade) insert sketched in Figure 3.18. The axis of the
5 in. hole bored through the graphite was along the centerline of the
reactor core. Graphite plugs were designed to accommodate 11/2 and 3 in.
aluminum ducts.
*The thermal diffusion coefficient for light water is.155 cm and for pure
graphite.915 cm.
6o
1.50
Ratio of Reaction Rate of Lu176 to Reaction Rate of a (I/V)
Absorber versus Neutron Temp
0~ 100~ C
(Norm. to 1 at T= 0~ C)
1.40 
1.30
I
_ /._.2
I. IO
00 i I I I I I
I0 20 30 40 50 60 70 80 90.100
Temperature ~ C
Figure 3.17. Ratio of reaction rate of Lu176 to reaction
rate of l/v absorber.
61
6"  NORMALIZERS
HANDLE' 9"
CORE
INTERFACE
MATRIX
PLUGS
Figure 3.18. Graphite insert.
62
Measurements were made with the insert positioned against the south
face of the reactor core. The side clearance was such as to allow no
more than a 0050 ino water gap between the insert and the columns of reflector elements. Time limitations allowed only scalar fluxes at the
duct mouth to be measured~ For this reason, the duct lengths were limited
to 11/2 ft, which is effectively a void of infinite length from the
standpoint of the perturbation at the mouth of the largest duct. The
ducts were evacuated and positioned 3 in, from the coregraphite interfaceo
It was anticipated that water absorption might be a serious problem.
To avoid the consequences of water penetration the graphite insert was
coated with a thin layer of an Epoxy resin. It was verified that thlis
was an effective sealer by soaking coated and uncoated samples; of graphite in water and weighing shortly after removal.o
Since the insert represented a large amount of reactivity, *the reactor was brought critical with the insert in place on the south face of
the core. The normalization was accomplished with four foils extending
into the water from lucite brackets. Comparing the normalization results
with counts from the fission chamibers revealed that normalization was
not influenced by the presence of the ductso
Inasmuch as the thermal neutron mean free path is long in graphite,
simultaneous bare and cadmiumcovered measurements were avoided. A cadmium ratio of 25 in t:he graphite permitted the use of bare foil measure
ments exclusively. If the epicadmium flux had been perturbed by the
5 ino duct to the same extent as the thermal, flux, such a procedure could
lead to an error of no more than 11/2 percent.
Figure 3o19 presents the results of unperturbed flux measurements
in the solid graphite insert. The results are normalized at 7~62 cm to
the distributions obtained by a twogroup, onedimensional diffusion
calculation. A radial traverse is shown in Figure 3020~
The flux measurements at the duct mouth. were made at two positions
3/8 in. from the centerline, and agreed within 1 percent. The data are
plotted in Figure 3o210 The solid line is the predicted scalar flux at
the mouth obtained byusing the measured unperturbed flux distribution
to compute the'boundary condition~ Figure 3~22 displays the axial distribution at the mouth of the 5 ino duct and compares the data with. the
diffusion theory resultO
64
24
GRAPHITE H2 0
22 UNPERTURBED FLUX
VERSUS DISTANCE
FROM CORE IN
GRAPHITE
o MEASURED THERMAL
REACTION RATE
1 8_ \n z MEASURED
EPICADMIUM
REACTION RATE (XIO)
CALCULATED BY
r16\\ t 2GP DIFFUSION
o \ CODE(NORM. AT
7.62 CM.)
_ 14t\ X ESTIMATED POUJ \ \ SITIONING ERROR
1o I —' 2mm
12
ESTIMATED REAC_ 0 \o \TION RATE ERROR
< 10 \
0
8
w
cr
bJ
w 0
a.
u 4 o
2 0Io
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
DISTANCE FROM CORE (CM.)
Figure 3. 19. Unperturbed thermal and epicadmium flux in graphite insert.
65
Unperturbed Flux versus Distance from Duct Axis In Graphite
2 2 0 Measured Thermal Reaction Rate
A Measured Epicadmium Reaction Rate
20
18,o 16
C)
* 14
E
12
o
U 10
0
U 8
6
4
2
Ah AA AA
3I0 2 3
Distance From Axis (in.)
Figure 3.20. Radial flux distribution in graphite insert.
66
24
23
22
21 MEASURED THERMAL
REACTION RATE
0.... PREDICTED THERMAL FLUX
X 19
(3 18
17
wI 16cr 15
zo 14
13Cr 12
0L
o PERTURBED THERMAL FLUX VS. VOID DIAMETER
10
9
8
6
1.0 2.0 3.0 4.0 5.0 6.0 70
VOID DIAMETER (IN.)
Figure 3.21. Perturbed thermal flux at center of duct mouths in graphite.
Perturbed Thermal Flux versus Distance From Core
5" Duct in Graphite
0 Measured
22 ___Predicted.
20
18
c 16
. 14
0 12
o
u
u s F
1 0
U
8
6
I2 3 4 5 6 7 8
Distance From Core (Cm)
Figure 3.22. Axial distribution of thermal flux at mouth
of 5in. duct in graphite.
IV. DISCUSSION
A. Unperturbed Thermal Flux Distributions
The measured unperturbed thermal flux distributions, rather than
the computed ones, were used to obtain the predictions presented in
Chapter III. Utilization of the computed unperturbed thermal flux distributions, however, leads to no more than 3 percent disparity in the
prediction of the duct mouth thermal flux at the positions studied.
This is significant in that it suggests that duct perturbations can be
predicted from diffusion theory flux distributions without the need for
experiment. It should be pointed out, however, that the unperturbed
flux shape in the immediate vicinity of the duct mouth must be accurately described by theory~
It is generally recognized that fewgroup diffusion theory has
limitations when used to predict accurate flux distributions in reflectorso The cross section averaging procedure27 and the high degree
28
of fast flux anisotropy give rise to deviations'between theory and
experiment. Theory and experiment are compared in Figures 3~8 and 319.
The percentage deviation increases with distance from the core, although
for the graphite reflector the situation is complicated by the existence
of.interfaces between highly dissimilar media. Utilization of more than
two groups in the fewgroup scheme does not improve the agreement No
attempt has been made to include spectral. effects close to the core68
69
reflector interface, nor has the curvature of the fuel element, which
introduces up to 1/4 in. of water in front of the graphite insert, been
considered. Infinite medium, Maxwellianaveraged cross sections were
used in the reflector region. Presumably accurate predictions for the
flux distribution far into the reflector region can be attained provided
that appropriate transport and multigroup techniques are utilized. A
thorough examination of this problem is beyond the scope of this study.
B. 11/2In. Duct Experiments
The agreement between the experimental data shown in Figure 3.9 and
the prediction based upon the perturbed angular flux at the duct mouth
is good. It should be recalled, however, that unlike the calculations
the measurements include the leakage contribution from the walls of the
duct. This implies that the wall leakage contribution in the region examined is either a small or a constant percentage of the mouth contribution. Measurements with the duct at 2016 cm indicate a 21 percent contribution from the walls. This number compares well with the results of
10
similar measurements reported by Pierceyo Summing contributions from
the lateral surfaces based upon the unperturbed thermal flux in the
medium, a wall leakage contribution of 15 percent is predicted. The
disparity of 6 percent could be accounted for by an albedo* contribution
*The albedo contribution is taken to mean neutrons of all energies which,
having entered the duct through the mouth, make at least one scattering
collision in the wall before they diffuse back into the duct and are detected as thermal neutrons.
70
of the fast neutrons passed by the cadmium.
Subtraction of the wall leakage contribution permits the prediction
f Bth The excellent agreement presented in Table III.4 suggests that
any thermal neutron albedo contribution from the mouth at the exit of
the 11/2 in. duct lies within the experimental error.
The 11/2 in. duct was provided with the internal collimator sketched
in Figure 3..i3 to completely eliminate duct wall and albedo considerations.
A measurement was performed with identical copper foils at the duct mouth
and the collimator exit. This permitted a convenient comparison between
the experimental value and the prediction for the mouth contribution to
the exit thermal activation. The difference between the two was within
the uncertainty in the collimator dimensions. Unfortunately no data:comparable to the.uncollimated data of Figure 3.9 were obtained.
C. Thermal Flux Attenuation.
Within the Empty Ducts
The "duct streaming" measurements within the empty 11/2 in. and
5 in. ducts were motivated more out of academic interest than from any
direct bearing on the main theme of this study. It was of interest to
determine whether the thermal flux far from the mouth of the duct obeyed
the l/r2 geometrical attenuation predicted by Simon and Clifford.9 The
cadmiumbacked foils were allowed to view all surfaces of the duct directly in front of them in order to include thet albedo contribution. For
the 11/2 in. duct h/R = 60 at the farthest measured distance from the
71
mouth., and for the 5 in. duct h/R = 18o
The experiment did not correspond to the configuration studied by
Simon and Clifford in that they treated a point isotropic source at the
duct mouth. The distributed source available experimentally at the duct
mouth does not alter the results of their analysiso The presence of
direct streaming sources in the duct walls, however, is estimated to gire
rise to a slightly more rapid attenuation than l/r2o
If we assume an. attenuation of the form 1/rn, the measurements
(Figure 3516) yield exponents of n 2 2.7 for the 11/2 in. duct and n ~ 2.5
for the 5 ino duct. The large value of the exponent for the 11/2 in.
duct is surprising, since the thermal albedo contribution from the mouth
was estimated from other measurements (see Section IV. B) to be insignificant. In spite of the precautions taken, however, mutual shadowing between foils might have taken place.
D. Radial Flux Distributions at the Duct Mouth
We have assumed in the analysis that the flux is flat over the entire duct mouth. Examination of the distributions in Figure 31.11 reveals that this is not entirely correct, but that most of the recovery
in the flux occurs close to tlhe walls of the ducto The skewing of the
perturbed thermal flux distributions is caused by the asymmetry of the
unperturbed radial distribution. In any case, the analytical technique
treats only the center of the duct moutho The degree of accuracy achieved
72
in applying the analysis to the entire source plane, then, depends upon
the extent of collimation utilized in a practical situation~ The average
perturbed scalar flux over the entire duct mouth. is approximately 10 percent higher than the flux minimum at the center.
Eo Fast Flux Perturbation
In assuming a negligible fast flux perturbation at the duct mouth
it was anticipated that even significant changes would affect the thermal
flux i.n only a minor way. It is conceivable that the epithermal region
of the spectrum (for example, ~625 ev to 5 Kev) could be treated by the
analytical method applied to the thermal flux, but no attempt was made
to predict duct perturbations in. the epithermal regionO Epicadmium gold
activations, which primarily measure the 4.9 ev flux, indicate a 26 percent scalar flux depression at the mouth of the 5 in. duct, Epicadmium
manganese activations, constitutinhg absorption in a 340 ev resonance
plus a significant portion of the 1/v portion of the spectrum, indicate
a 22 percent scalar flux depression. It is reasonable to assume that the
duct perturbation tothe higher energy, more nearly monodirectional* components of the spectrum is smallero The integrated effect from the point
of view of slowing down sources into the thermal region is difficult to
*Figure 4o1 displays angular distributions for the very fast flux (~82 to
10 Mev) derived from an S8 calculation.,
73
2100 2000 190~ 1700 1600 1 50~
150~ 1600 170~ 1800 190~ 200~ 2100
2200 Angular Distribution of Fast Neutrons 1400
1400 — (.82 to 10 Mev) Calculated by S8 Approx. 2200
(a) 1.5 Cm. from Core i
(b) Core  H2 0 Interface
(c) 5 Cm. into H2 0 Reflector
2300 130~
1300 2300
2400 1200
1200 2400
~~~~~~~~~~2500 1~~~~~~~~~ I l1100
1100 250
2 6 O' 1000
I00~ 2600
2700 g
900 2700
2800 80~
800 280~
2900, 1 700
700 1 290~
3000!) ~. 60~
60~0  300~
3100o ((b) 50
50' I 310~
3200  400
400 32 00
3300 3400 3500 O 10~ 20' 30 o
300 200 10~ 3500 3400 330~
Figure 4.1. Angular distribution of fast neutrons.
74
evaluate. Figure 4.2 presents the calculated effect of assuming various
group 1 (fast) flux depressions at the mouth on the group 2 (thermal)
flux for the 5 in. duct. This is computed on the basis of a constant
thermal return current to the duct mouth.The results suggest that a 25
percent depression in the integrated fast flux at the duct mouth only
leads to a 3 percent depression in the thermal flux.
Computed Thermal Flux and Fast Flux.34  versus Distance From Core
For 5" Duct at 2.16 cm in Water
O5U Thermal FluxforjAth constant.32  Fast Flux.3 Unperturbed Fast Flux.\ 4.81 O/ \.30  \\ \.28.66 \ \.2  6a
\ \ \
\ \
\.24
E. 22
\%.8\\.16
db/dZ
2 b =.515.14 ( 5  d/dZ 445
02(R) =.153
d C/dZ.12 42(R)=.149.388
I 0 I 2 3
Z Distance From Core (Cm.)
Figure 4.2. Effect on predicted thermal flux of fast flux depression.
75
The analysis is inapplicable to ducts which appreciably perturb the
overall fission source in the multiplying mediumo The assumption of
small fission source perturbations should. of course, be realized for
small ratios of the void area to the core face area, or for voids inserted
sufficiently far from the reactor core (i.eo. for small reactivity effects).
The crosssectional area of the 5 in. duct in the experimental configuration is of the order of 5 percent of the total area of the reactor face
normal to the duct axis~ Epicadmium flux measurements along the walls
of the 5 ino duct indicate no significant depression in the fast flux.
Ducts which appreciably perturb the reactor fission source may be amenable
to an iteration analysis involving one and twodimensional groupdiffusion codest
Fo Thermal Spectrum
The 100~ 5~C hardening of the thermal spectrum measured at the
mouth of the 5 in. duct is insignificant in the present study (the effect
was estimated to lower the average activation cross section of a l1/v detector by less than 2 percent). The fact that a measurable effect was
observed, however, suggests that spectral hardening induced by the void
may be important in experimental determinations of the neutron spectrum
at the exit of a beam port.
Measurements at the Ford Nuclear Reactor using a crystal spectrom~29
eter at the exit of the 6 ino beam port sketched in Figure 301 determined
76
a neutron temperature of 40~ + 5~C. The temperature of the pool water is
28~C at a reactor power level of one megawatt. It is reasonable to assume that the 3 in. of graphite and 11/2 in. of water between the mouth
of the port and the core are sufficient to bring the core neutrons into
equilibrium with the diffusing medium. Using the Brown prescription30
for the hardening effect in water:
Teff = To +.876 (T ~)(41)
we expect a neutron temperature in the unperturbed medium of less than
320C. The difference of 8~ + 50C between the estimated unperturbed
neutron temperature and the measured value is presumably due to spectral
hardening induced by the port void.
G. Scalar Flux at the Duct Mouth
The thermal flux measurements at the center of the 5 in. duct mouth
lie approximately 7 percent below the predicted values in both media, as
seen in Figures 3.11 and 3.21. Assuming for the moment that the unperturbed
flux is a correct representation for the distribution at the duct walls,
attenuation by the aluminum walls of the duct, which is difficult to
evaluate, in addition to the fast flux depression and hardening of the
thermal spectrum, which have been discussed earlier, all act to depress
slightlythe measured thermal flux at the mouth. The current measurement
77
at the exit of the collimated duct inwater, however, shown in Figure 3.14,
agrees well with the predicted valueo The arguments presented in Section
II.B indicate that in the limit of the infinite halfspace (jAth = 0)
P1 theory affords an adequate prediction for the angular current at the
boundary (for A = 1), but the P1 scalar flux prediction is 15 percent
higher at the boundary than the exact quantity. Since the experimental
data conform to this trend (in water jA is less than 50 percent of the
th
unperturbed return current to the 5 in. duct mouth), it is reasonable to
ascribe the discrepancy in the scalar flux to measurable deviations from
diffusion theory. Further evidence for deviations from diffusion theory
is afforded by the comparison of the measured and predicted gradients
at the mouth. In water the measured gradient is 40 percent higher than
the predicted value from diffusion theory, and in graphite the measurement is 70 percent higher. Examination of the Milne problem reveals that
the exact solution for the gradient possesses a logarithmic infinity at
the moutho
A measurement of jBth was not performed in the complete graphite reflectoro However, we infer from the results of the perturbed scalar flux
measurements at the duct mouths that the analysis applied to the water
medium should be equally valid in graphite. Figure 4.3 is a plot of the
prediction for j. in graphite, It should be pointed out that since the
Bth
constant cross section approximation used to obtain Eq. (2.25) is not
strictly valid here, this equation is not rigorously applicable. The results of a numerical integration of Eqo (2.23), however, agree to within
1.0
JBtk
versus Void Diameter.9
Prediction in Graphite.8.7
+d.6.5.4.3
1.0 2.0 3.0 4.0 5.0 6.0
Void Diameter (in.)
Figure 4.I5. Predicted JBth for graphite.
79
one percent with predictions based upon the constant cross section approximation.
H. Scalar Flux Distribution Along the Duct Walls
An idealization of the scalar flux distribution along the duct walls
has been incorporated into the framework of the analysis. Predictions
within this framework have compared favorably with the experimental results. Nevertheless, it would have been more satisfying to use the realistic flux distribution in Eq. (2.41) to compute the return current. Up
to this point, however, an examination of the behavior of the realistic
scalar flux distribution along the duct walls has been avoided.
Information concerning the realistic duct wall flux distribution
was accessible experimentally. The configuration which gave rise to the
largest flux perturbation at the mouth, the 5 in. duct in water, was examined. Foil strips fastened to thin lucite brackets were mounted perpendicular to the duct wall at 1, 3, 5, and 8 cm from the mouth. The
relative thermal reaction rates for strips mounted on the east and west
sides of the duct are plotted versus radial distance from the duct wall
in Figure 4.4. The ratio of the radial component of the gradient at the
wall to the scalar flux,, is recorded by each curve.*
Figure 4.5 is a plot of the perturbed scalar flux distribution along
*The fact that the gradients are higher on the east side of the duct originates from the asymmetry of the unperturbed radial distribution.
1 cm. from Mouth 3 cm. from Mouth 5 cm. from Mouth 8 cm. from Mouth
East East East East
/2.0 Thermal Reaction Rate versus Radial
2.0 // Distance From Duct Wall
1.8 / At 1, 3, 5, 8 cm. from
Mouth of 5"Duct in Water
1.6 
1.4  d#/dr
1.2  d dr
@1 ~~~~~~~~.24 d/r_ _ _ _ _
do/dr
1.0 4,
Eg d~~~~~~~~~~~~~~~~~d/dr.8 d / .04
We st West West West.02.0
1.8 /
1.6 
1.4 d4/dr.3 2 do =.3
1.2 46 d//dr
= .606
1.0 4,dd4d/dr
i~~~o C I I ~~~~d4,/d r Q, .0
10.84,o. _
0 1.0 2.0 0 1.0 2.0 0 1.0 2.0 0 1.0 2.0
Radial Distance From Wall of 5"Duct (Cm.)
Figure 4)4. Radial thermal flux behavior at four positions along 5in, duct wall.
the 5 in.o duct wall compared with the unperturbed scalar flux distribution
in the medium out to 8 cm from the mouth. As anticipated, the scalar
flux is initially depressed close to the mouth, but recovers rapidly and
eventually exceeds the unperturbed flux where the radial component of the
gradient has become negative. Unforititnately, no data were obtained beyond
8 cmo However, assuming for the moment that the perturbed distribution
is identical to the unperturbed distribution beyond 7 cm, utilization of
the perturbed distribution in the calculation of JAth predicts a value for
the duct mouth scalar flux which is 10 percent lower than that obtained
from the unperturbed distribution4 Since we have concluded earlier that
the unperturbed distribution predicts the correct Pl1 value of the duct
mouth scalar flux, the enhancement of the perturbed distribution beyond
7 cm must compensate for the depression close to the mouth,
An attempt to predict the realistic behavior of the thermal flux
distribution along the duct walls was made by the following method. An
initial estimate ofrthe quantity ~/ R was obtained for several disz
crete axial positions along the duct wall using the unperturbed axial
flux distribution in Eq4 (2050) (revised to include the two terms given in
expression (2454) to account for the mouth contribution), the separated
form of the current balance relationship4 A computer program written to
evaluate the initial estimate of / R is presented in Appendix Go
The initial estimate of the relative radial component of the gradient at
the mouth, I R was obtained in the usual wayo The relative gradients
t~~~~~~~~~
Thermal Flux versus Distance From Duct Mouth
O Unperturbed Reaction Rate
XMeasured Reaction Rate Along Duct Walls
70 ~For 5" Duct 2.16 cm From Core in Water
~ 60
50
40 _
30
20
I 2 3 4 5 6 7 8 9 10
Distance From Duct Mouth (Cm.)
Figure 4.5. Axial thermal flux distribution along walls of 5in. duct.
were used for a z varying boundary condition along the waterduct interfaces in a twodimensional groupdiffusion code.* The cylindrical
geometry approximration to the actual confi.guration is sketched in Figure
4~6. The calculations were performed with two energy groups, and the
fast flux was assumed unperturbed.
It was intended that the resulting perturbed flux distribution be
used in the unseparated current balance relationship, Eqo (2o45) (revised to include the terms in expression (2o54))X to obtain improved estimates for g rRo The procedure forms a basis for an iteration
i lz
technique which was expected to converge rapidlyo The initial estimates
for R however, did not permit convergence of the groupdiffusion
Z
code for the 5 ino or 5 in. ducts. Even when the computation was forced
into convergence by diminishing the absolute magnitudes of / R far
Z
from the mouth, the computed flux distributions were unrealistic, tending
to increase with distance from the coreo The initial estimates and altered values (to achieve code convergence) of R for the 11/2, 3 and
5in. ducts i.n water are recorded in Table IVoilo Comparisons with the
measured gradients at the walls of the 5 in. duct recorded in Figure 404
*The TWENTY GRAM code, described in Appendix B, provides for a logarithmicderivative or rodregiond' boundary condition, specified by
for energy group io Ci is supplied as input data for any region specified as a "rodregion r O
6".H20
I, \
I I~~~~~~d
Ii'
II:'i I
Fuel Q, Graphite
Core Plane of Symmetry
2.16 cm.
Figure 4.6. Cylindrical geometry approximation to coreduct configuration.
85
TABLE IV.1
FIRST ESTIMATE OF THE RADIAL COMPONENT OF THE GRADIENTS FOR BOUNDARY
CONDITION IN TWODIMENSIONAL CALCULATION
z 11/2 In. Duct 3 Ino Duct 5 In'. DuctDistance Alrt ReAltered Altered
from  IR 8Z/6r R 6:  //r
Duct Mouth c cm l Z
(cm) 1 cm 1 cm 1
(cm) cm cm cm
0.120 Unaltered.240.240.400.400
1.150 3 40 3 40 547.547
2.104.265.265.450.450
3.051.16 164.310.310
4.000.057.057.149.o 149
5 .047 .050 o050 .020 .020
6 .090 .154 .154 .193 .193
7 .130 .254 .254 .365 .365
8 .165 .350 .350 .536 .450
9 .198 .442 .396 .704 ~450
10 .531 o396 .870 o450
11 1..033 ~450
12 lo 195 .450
13 1o355 .450
14 1.516 .450
15 1.678 .450
16 lo 841 .450
17 2.006 .450
18 2175 o 450
86
indicate that the estimated gradients close to the duct mouth are not
unreasonable. The apparent deadlock is attributed to the large negative
estimates of the gradients far from the duct mouth. Either these values
or the treatment of the negative gradients by the code (as a negative
absorption in the duct region) is unrealistic.
I. Reactivity Predictions
A far more encouraging prediction emerges from an examination of
the eigenvalues computed by the twodimensional code. The duct reactivity
calculated by using the altered estimates of the duct wall gradients listed
in Table IV.1 compare favorably with experiment. This is not surprising,
since the fundamental eigenvalue of the diffusion equation is relatively
insensitive to the flux distribution. However, inasmuch as the altered
gradient estimates for the 5 in. duct provided a highly unrealistic flux
distribution, an additional calculation was performed that utilized only
the positive gradients (FiR was set equal to zero past 4 cm). Considering the degree of approximation to the realistic geometry, the result of this calculation was also in reasonable agreement with experiment.
The reactivity results are presented in Table IV.2o
87
TABLE IV.2
DUCT REACTIVITYMEASURED AND PREDICTED
DUCTS AT 2.16 CM FROM CORE IN WATER
bk/k bk/k
Predicted Predicted
Predicted
using
Duct bk/k using usint
Diameter (in.) Measured, altered positive
z, I.....(Zero past 4 cm),
11/2 . 007 .007
53 .03. .04
~~~~~5 .10 . 11 . 16
Vo CONCLUSIONS
The main results of the analysis and measurements discussed in Chapters II, III, and IV may be summarized as followso
l P1 theory can be used to predict adequately the thermal. neutron
current at the exit of a collimated beam port. The source plane scalar
flux can be predicted using Phi theory with somewhat less accuracy. The
method has been applied by computing the return current at the duct source
plane in terms of the measured di.stribution of the unperturbed scalar flux
in the reflector. The return current is used in. a fewgroup diffusion
calculation in the form of a pseudo boundary condition on the scalar flux.
Predicted exit thermal neutron. currents for ducts up to 5 in.. in diameter
in the water reflector at the Ford Nuclear Reactor agree within + 3 percent with activation measurements. Predicted source plane scalar fluxes
in water and graphite reflectors agree within 7 percent with the measurements
2. The method can be applied satisfactorily if the calculated distributiorl of the unperturbed scalar f7.L:Ux in the reflector as obtained
from fewgroup diffusion theory is used to compute the return current.
The unperturbed scalar flux distributions in the reflector deviate significantly from the fewgroup diffusion theory results close to boundaries and
far from the reactor core. Predictions for the beam port exit currents
obtained from the calcul.ated distributions, however, deviate by only 3
percent from those obtained from the measured distributions.
88
89
3. An iteration technique was proposed in an attempt to predict the
realistic distributions of the scalar flux along the lateral surfaces of
the ducts. Unfortunately, the initial estimates of the radial component
of the gradients (obtained by separation of variables) failed to produce
convergence of the twodimensional groupdiffusion code. Further work is
needed to explore this difficulty~
4. The results of a survey to optimize thermal neutron leakage flux
from the Ford Nuclear Reactor clearly indicate thato
a, The optimum geometry is a slab reactor with the beam port
external to the core face normal to the smallest core dimension.
b. An increase in power density by improved reflection is
more desirable than an increase in integrated power
because of a reduction in the ratio of fast to thermal
flux.
co D2O is the most effective reflector material for this
purpose.
APPENDIX A
A SURVEY FOR THE ENHANCEMENT OF THERMAL
NEUTRON ILEAKAGE FLUX
lo Introduction
This survey is directed toward an optimization of the thermal
neutron flux intensity and the neutron spectrum in a reflector region
of the Ford Nuclear Reactor (FNR) for beam port applications. The
FNR was briefly described in Chapter III. Although the results of the
investigation pertain to a swimming pool reactor facility of the FNR
type, they should apply at least qualitatively to a more general type
of thermal reactor~
For purposes of efficiency, as well as compliance with economic
and physical limitations at the existent reactor facility, the following
constraints were established:
(a) The core composition was held constant, For this the FNR
regular fuel element was adopted (partial elements and poison
rods were disregarded).
(b) The integrated reactor power was held constant.
(c) The vertical core dimension was fixed at 24 in., the height
of the "meat" portion of the FNR fuel element~
(d) The beam port was aligned with a radius of the core outside
of the fuel, region~
90
91
(e) The beam port influence on the exit current was neglected in.
this phase of the studyo
In keeping with these constraints, the survey considers variations in
reflector materials and corereflector configurations.
The computer codes utilized for the numerical computations are
described in Appendix B. Compositions and group constants are tabulated in Appendix C. Fourgroup diffusion theory was adopted. The
fourgroup scheme, with breakpoints listed in Appendix C, has'been
highly successful in treating small, enriched, lightwater reactors.
Moreover, a fourgroup study affords some useful information concerning
the neutron spectrum.
2o ThreeDimensional Simulation
Threedimensional groupdiffusion calculations would be prohibitively timeconsuming for an inquiry of this nature. Twodimensional
calculations were performed whenever possible, but they were limited in
usefulness, since criticality search routines were not included~ Therefore most of the computations were performed using onedimensional
simulation of the threedimensional geometry. This was accomplished by
holding two of the dimensions, y and z, constant and searching for the
critical size and flux distribution in a variable third dimension, x.
Leakage in the y and z directions was incorporated as a transverse buckli ng
92
The transverse buckling was obtained in the following manner. Onedimensional flux distributions inthe y and z directions were acquired
for each of the four groups for constant y and z dimensions (leakage in
in the x direction was accounted for by allowing the code to search for
the critical transverse buckling). Then for each group the bucklings
transverse to the x direction were derived from the following relationships:
1i(y)d,
<(B2)>i core (A.1)
/ i (y)dy
core
and:
* v2i(z)dz
<(B )>i core (A 2)
0' "(z)dz
core
Transverse buckling components which were used in the acquisition of
the subsequent results are listed in Table A.Io* The total buckling
transverse to the xdirection for the ith group is given by:
<(B <(Bt)>i + <(B>i )>i (A.3)
*Transverse bucklings obtained for the core were used in all regions
Refinements to some of the subsequent results could be achieved by using
regiondependent as well as groupdependent bucklingsO
TABLE A. I
TRANSVERSE BUCKLINGS
(cm2)
ColdClean Core. Equil. Xe, b.4% BU Core
(B) T2 (B2)T2 (B3)2 (B4)T2 (BT2 B2)T (B. T (B4) T
Vertical.00296.00253.00153 .00472.00298.00253.oo56 . oo448
3 Rows
Horiz..00933. 00oo493.00o499 .00026  
Total.01229.00746.00652 . 00oo498
4 Rows
Horiz. oo635337.00341 .000202.00651.00346.o00348 .000057
Total.00931.oo00590..00oo494 .00492.00949.00599.00504 .00oo454
5 Rows
Horiz..00oo464.00248.00252 . 00016..00254.00255 .000042
Total.00760.00501.0oo405 . oo00488 00773.0007.00411 .00452
7 Rows
Horiz..00283.00152.00155 .000117.00289.00155.00156 .000035
Total.00579.00oo405.oo308. 00oo484.00587.00oo408.00312 .00451
8 Rows
Horiz.    .00235.00126.00127 .000032
Total    .00533.00379.00283 .o00o451
94
3. Streaming Flux
The scalar flux distributions, which together with the fundamental
eigenvalue constitute the principal output of the groupdiffusion codes,
are not the quantities of foremost interest from the point of view of
the present study. Rather, we are interested in the angular flux at
the source plane in the direction of the beam port exit. The angular
flux is given by Eq. (2.23) in the constant cross section approximationo
If the constant cross section restriction is removed:
oo.s rl
(r_ 1 / df[Zs(rr)Wo)(rQ9) + s(r —RQ)]e
(A. 4)
s(rRQ) is defined here as the source integrated over all angles.
For long ports, (h/R)> 1, it is the forward flux, close to i = 1,
which is detected at the exito In the limit of a collimated port of
infinite length, Eq. (A.4) can be approximated by the onedimensional
integration:
XI
X d Z (xx" )dx"
(X=1) _ 4; d/x [s (xx')Oo(xx,)+ s(xx')]e
(A.5)
Then for any position x in the reflector and energy group i, we define
the streaming flux:
95
J'(x)o~~ =~ ~ CZ (x xlT)dx"
(dx'[Z5(xx'V)(xxT) + si(xx?)]e
0
(Ao6)
The streaming flux is emphasized in the subsequent presentation
of results. Since beam port perturbations were neglected in this survey, the unperturbed scalar flux and source distributions were used to
compute this quantity. Slowing down sources were neglected. Subroutine
STREAM, written to compute Ji(x) and the partial current, ji(x), in
conjunction with the FOG code, is listed together with the revised main
program of FOG in Section A.7.
4. Configurational Effects
BARE REACTOR EXAMINATION
In addition to allowing a simple analysis, the bare reactor exhibits fast neutron leakage behavior which is similar to that of the
reflected core. Examination of the behavior of fast neutron leakage
should provide considerable insight into the influence of the reactor
configuration on the thermal neutron. flux intensity in a reflected
systemn inasmuch as the removal of fast neutrons constitutes a thermal
neutron source in the reflector.
The flux in a bare homogeneous reactor satisfies the Helmholtz
equation:
96
72 + Bg = oo (A.7)
The total leakage is given by:
D(r)VY2(r)d3r Bg2 D(r)(.r)d3r, (A.8)
and the integrated power may be expressed as:
Power = consto tZf (r)(r)d3r (A.9)
For a fixed integrated power and spatially independent properties:
Total Leakage B2 (A lO)
A critical bare reactor, however, must satisfy the condition:
2
k~ e
k i(A. 1)
keff 1 (A+ L2B)
2
The material buckling, 9 is a function only of the reactor material
composition, and for a, critical. system it must be equal to the smallest
eigenvalue, the geometrical buckli.ng, of Eqo (Ai?7). Thus:
2 2
Total Leakage Bg Bm B (A l2)
which is a constant for a critical bare reactor of homogeneous composition operating at a fixed power level, regardless of the core geometry~
The average leakage per unit area, however, is a function of the
97
total surface area of the core. The minimum critical volume and surface area for bare reactors of three geometries are listed in Table
AoIIo
TABLE A.II
CRITICAL VOLUME, AREA9 AND RELATIVE LEAKAGE
FOR BARE REACTORS OF THREE GEOMETRIES
Minimum Critical Relative Average
Geometry
Critical Volume Surface Area Leakage/Area
Sphere 130/B3 125/B2 1.0
Finite Cylinder 148/B3 155/B2.806
Rectangular
Parallelepiped 161/B3 179/B2.698
The fourth column of the table does not express the complete story,
however. The leakage is strongly peaked at the center of a face of the
parallelepiped, whereas it is uniform over the entire surface of the
sphere. Using Gauss's theorem, the total leakage from the reactor can
be expressed as:
 D d3r V2 (r)  D dS nv _ _ (A.1l).vo s urf r R
The flux distribution in a bare cube of linear dimension c is given by
(assume the reactor is large enough so that c cd )
98
((x,yz) =A cos cos Y cos T (A.14)
c c c
The average leakage per unit area is given byo
D c/2 c/2 [D
D 44DA
dy dz (x y) Z (A.l1)
c/2 Cc/2 c
x = c/2 (CC
whereas the peak leakage per unit area occurs at the center of the face,
and is equal to:
 D n V (r  R  D a (x,y,z DAK (A.16)
z o
= c/2
Thus for a cube, Peak Leakage/area 2~ 46, and
Average Leakage/area
Peak Leakage/area for a cube This result indicates that
Average leakage/area for a sphere
the bare parallelepiped possesses an advantage over the more efficient
reactor geometry from the point of view of maximum leakage.
Now let us estimate the effect of varying two of the dimensions of
a rectangular parallelepiped, holding the third dimension constant. We
designate the fixed dimension by c, and the variable dimensions by a
and b. For a bare critical parallelepiped operating at constant power,
the geometrical buckling is given by:
Bg = +a () +() = constant = 3 A.7)
99
which is the buckling for a cube. The total leakage from one face of
the reactor is expressed by:
4DA bc
 c 2 b2. (A. 18)
cdz dy b/ 6 (xyz) I~ a
c/2 b/2 x a/2
Using Eq. (A.17), we obtain for the leakage out of a face normal to
dimension a:
4~DAc 1 a >.707
Leakage = 1 > 707 ~ (A.l9)
fCa2 1/2 c
Dividing by the area of the face, the average leakage/area is given by:
Average Leakage 4DA 1 (A.20)
Area xIc (a/c)
Table A.III is a list of the total leakage and average leakage per unit
area(normalized to unity for a cube) from a face of the parallelepiped
normal to dimension a.
100
TABLE A.III
RELATIVE LEAKAGE FROM A FACE FOR BARE
PARALLELEPIPEDS OF VARIOUS SHAPES
a/c Total Leakage Leakage/Area.8 1.89 1.25
~9 1.27 1.11
1.0 1.00 1.00
1.1.84.909
1.2 73.83 4
Beam extraction from a face which is normal to the slender core dimension
is suggested.
THE WATER REFLECTED PARALLELEPIPED
One group of neutrons is insufficient to study the reflected parallelepiped, thus it is convenient to solve the fourgroup equations on
the computer. The configurations examined are shown in Figure A.1. Reactor cores with 3, 4, 5, and 7 transverse elements were considered.
The critical x dimensions recorded in Figure A.1 were obtained from
dimension searchs using the threedimensional simulation technique discussed earlier. The fuel (coldclean composition) was surrounded by
light water except for symmetric rows of graphite normal to the yaxis.
Figures A.2A.5display the xaxis scalar flux and streaming flux distributions. Three of the four computed groups are presented in these and
subsequent plots.
Table A.IV is a list of pertinent information from Figures A.2A.5.
101
z
3 Elements 9/ 1 4 Elements 12'
24"
/ 19.3 "./ 13"
(a) (b)
H2 0 Surrounding All Cores
5 Elements 15" 7 Elements 21"
/ 1~ e/945
(c) (d)
Figure A.1. Core geometries for O20 reflected configuration
studies (coldclean fuel).
102
1.2
1.0  
Core H20
0.9  
Computed Flux versus Distance from Core Midpoint
_ Slab Geometry
~~~~0.8   3 Elements Transverse
Scalar Flux
0 .625 ev
0.7.625 ev  5.53 Key v.821 Mev  10 Mev
Streaming Flux
06 0  1.625 ev
0.6~mmm. —. 625 ev  5.53 Kev
 .821 Mev  10 Mev
0.5
0.4
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Distance from Core Midpoint (Cm)
Figure A.2. Computed flux distributions; H20 reflected core;
3 elements transverse.
103
Core H O
1.0
0.9b Geometry
0.8
0.1.625
F ACo mputed l Fluxdsr uin H relc t or
4eElements transverse.
I N O .625 ey \ A _
0.5
BW I.625 ev  5.53 Kev
"821 Mev 10 Me \ 
0. I'.625 ev.821 Mev 1 1 1,,, I Mev.
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Distance from Core Midpoint (Cm)
4 elements transverse.
104
1.2
Core H20
0.8
0.7
0.C
0.5
0.4
Computed Flux versus
0.3 Distance from Core Midpoint \
Slab Geometry I
5 Elements Transverse
Scalar Flux \
0.2 0 .625 ev 
 ~ ~.625 ev  5.53 Kev.821 Mev  10 Me
O. I Streaming Flux
inin O0 .625 ev
I_ i.,..625 e v  5.53 Kev
—.821 Mev 10 Mev
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Distance from Core Midpoint (Cm)
Figure A.4. Computed flux distributions; H20 reflected core;
5 elements transverse.
105
1.2
Core H20
1.0 
0.9
0.8
\ 06 I I V; 17 Elements Transverse
it~~~~L \: O .625 eV.625 ev; 5;53 Ke
Computed Flux versus
l I\  \.625 ev  5.53 Key
4  <    _  .821 Mev  10 Me
0.5 \ Streaming Flux 
\. __[.1 lMev1 OMev.
0.4
0.3
0.2
0.1
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Distance from Core Midpoint (Cm)
Figure A.5. Computed flux distributions; H20 reflected core;
7 elements transverse.
1o06
TABLE A. IV
COLDCLEAf BH20 REFLECTED PARALLELEPIPED CORES OF
VARIOUS SHAPES
1/v
ISm 1 (s4_s3) (S4max S )m/ (S3 x) (SI/Simax) in 3
ax max Slmx.. in. 
3 Elemo
Trans~
oldClean 448 38 256 202 7 00024
Core
4 Elemo
Trans",.811 ~647 0478,2`0 434.00027
ColdClean4 27
Core
5 Elem
Trans.
Cora1s l.o6 0794 o r99,o 180 o 3 & 00025
ColdClean 3
Core
7 Elem.
Trans~
1.21:: 208.432.00021
ColdClean995 2
Core
7 Elem.
Trans~
Eq. Xe.965 770 96.215 9422.ooo16
Core
The quantities chosen.here and in t:hle u:bsequent studies to be of major
interest are~ (i) the maximum. value of the thermal streaming flux.j;
max
(ii) tthe maximum. difference between the thermal. and epithermal and the
thermal and fast streaming fl'uxes, (W )max and (i  )max;
(iii) the ratio of epithermal. to thermal and fast to thermal. streaming
fluxes at the location of t:he maximum thermal streaming flux (1 3 4ma
and (/1 /Imax); (iv) the inrerse volume of the core, 1/V, a measure of
the average power density of the reactor~ Arn additional listing in, Table AoIV
107
compares the coldclean, sevenelement transverse core with a core containing equilibrium Xe and 844 percent fuel burnupo
Table AoIV strongly suggests beam extraction adjacent to a face
which is normal to a slender dimension of the core~ Quantitative justification is presented for deforming the core to the optimum configuration (maximum power density), and going past this point if possible.
LOCALIZED CORE ALTERATIONS
It was of interest to examine the effects of an uneven fuel distributiono Figure Ao6 presents the results of a calculation comparing
the leakage flux of a core with clean fuel in the center and 8.4 percent
fuel burnup on the outside with that of a core endowed with the converse
loading. In the latter core the clean and burnedup fuels were loaded
in approximately even proportions, while in the former core the clean
fuel was moved to the central zone while enough burnedup fuel was loaded
on the circumference to maintain criticality4 The calculation was accomplished in cylindrical geometry, and the burnedup composition is that
listed in Table CoI (minus the xenon). The slightly reduced size of the
configuration with the clean fuel in the center gives rise to higher
thermal streaming flux in the reflector; however, the increase is only
5 percent.
Localized fuel additions in the vicinity of a beam port would merely
act to depress the thermal flux while increasing the fast fluxo The
effect of small water gaps at the center of the core was explored, but
o08' ~Clean Fuel 8.4 % B. U. Fuel H20
*O 10
Computed Flux versus
Distance from Core Center
Cylindrical Geometry
u....005'Clean Fuel in Center
Scalor Flux
2 416220.625e 1 1 1 1.625 ev  5.53 Ke C' l \\
.821 Mev  10 Mev
Streaming Flux v
Cle 0 .625 e v
_ . _.62 5 ev  5.53 Kev
.821 Mev  10 Mev
2 4 6 8 10 1 2 14 16 18 20 22 24 26 28 30 32 34
i...8.4 % B.. U, Fuel Clean Fuel (Cm
i \.0100
Computed Flux versus
Distance from Core Center
x 005L I Cylindrical Geometry \
005 Clean Fuel on Circumference
Scalar Flux
0..625 ev.625 ev  5.53 Ke.821 Mey  10 Mey
Streaming Flux..625 ey  5.53 Kev. —...821 Mey  10 Me'y
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Radial Distance (Cm)
Figure A.6. Computed flux distributions; comparison of
depleted fuel in center and on circumference of core.
109
yielded no net improvement. (A 3 in. gap, while giving rise to a 250
percent local flux peak, would increase the fuel loading by 12 percent,
and decrease the reflector thermal flux by 14 percent.)
5. The Effect of Reflector Materials
NEARLY INFINITE REFLECTORS IN CYLI[TDRICAL GEOMETRY
The initial examination of reflector materials was carried out in
cylindrical geometry for a coldclean core composition. The core was
surrounded by a thickness of at least one migration length of each material to achieve the effect of a nearly infinite reflector. Group independent transverse Bucklings were used for simplicity3* The results
of the computations are plotted in Figures Ao7A.12. Results for H20,
graphite, BeO, D20, and combinations of graphite and H20 are presented.
BeO results are presented because the calculations indicated that this
material is at least as effective as pure Be.
Table A.V, composed of significant information from Figures Ao7Ao.l2
points to the near linear dependence of the thermal streaming flux on
the average reactor power density (tabulated as the inverse core volume).
BeO, clearly the best reflectors increases the thermal streaming flux by
a factor of 3,2 while raising the average power density by a factor of
B =.00165 cm'2 was adopted here for the core and BS =.00125 cmr2 for
the reflectoro These numbers were obtained from epicadmium flux traverses in the FNR17:
110.020
Computed Flux versus Distance from Core Center
Cylindrical Geometry I
Infinite H20 Reflector
Scalar Flux
0 .625 ev.625 ev  5.53 Kev.821 Mev  10 Mev.015 Streaming Flux
iR0  s.625e v
 ' —.625 ev  5.53 Kev
  — _.821 Mev 10 Mev H20
Core
2 4 6 8 I 0 12 14 16 18 20 22 24 26 28 30 32 34
Radial Distance (Cm)
Figure A.7. Computed flux distributions; infinite H20
reflector; cylindrical geometry.
111
Core Graphite.020
Computed Flux
versus Distance from Core Center
Cylindrical Geometry.005 60 cm Graphite Reflector
Scalar Flux
0 .625 ey v.625 e v  5.53 Key.. — .821 Mev  10 Mev "
Streaming Flux._ __...625ev  5.53 Key
' —.821 Mev  10 Mev
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32'34
Radial Distance (Cm)
Figure A.8. Computed flux distributions; infinite graphite
reflector; cylindrical geometry.
112
\
Core BeO.020 \\\.010  
Cylindrical Geometry\.005 40 cm BeO Reflector _
Scalar Flux
0 .625\ e
4 .625 ev  5.53 Ke \.821 Mev  10 Me\
Streaming Flux.625 ev  5.53 Key.0 0 40.821 Mev  10 Mev
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Radial Distance (Cm)
Figure A.9. Computed flux distributions; infinite BeO
reflector; cylindrical geometry.
113
Core D20.020.015
/1 X I,2 waft.010o! I
Computed Flux versus
Distance from Core Center \
I Cylindrical Geometry I.005 190 cm D20 Reflector
Scalar Flux
0 .625 ev.625 ev  5.53 Key.821 Mev  10 Mev
Streaming Flux
ininm 0 .625 ey._ m _.625 ev  5.53 Kev
— ~ —.821 Mev  10 Mev
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Radial Distance (Cm)
Figure A.O10. Computed flux distributions; infinite D20
reflector; cylindrical geometry.
114
Core Graphite H20.020.0 15.010 \ \
Computed Flux versus
O .625 e\.625 ev  5.53 Ke Y
inderi.821 Meyv 10 Mev o
Streaming Flux.625 ev  5.53 Key
  ..821 Mev  10 Mev
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Figure A Computed flux distribtions; 7. cm graphite
"5.020
Core H20 Graphite1.0 1 5 I
I I I I I 1 11/,,O.1.1
I'
Computed Flux versus
Dista ce from Core Center:
2.5 cm H20 Remainder Graphite \.005 I Cylindrical Geometry
Scalar Flux
0 .625 ev.625 eRv  5.53 Kev (
.821 Mey  10 Me \
Streaming Flux
m ~~ 0 .625 ev \
. ,..625 ev  5.53 Kev
 .821 Mey  10 Me v
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Radial Distance (Cm)
Figure A.12. Computed flux distributions; 2.5 cm H20remainder graphite; cylindrical geometry.
116
TABLE A. V
INFINITE REFLECTOR —CYLINDRICAL GEOMETRY
VARIOUS REFLECTOR MATERIALS
SM (S4S3) (S4_S1) S I3 q(S l/S41ax) I
(si/SmYax) ino.s
Ho20
Rfo.0119.0095 o006i. 218. 546.00025
Refl.
Graph.
Reflo o 0151 0071 o og8 o 642 6o 69 0 o 00oo46
BeO
R 0379 0258.0309. 368. 258. ooo83
Reflo _
D20
Refio o 0255.0191 o 0224 o279 o 183.00050
7~5 cm
Graphder0118.0088.0070 305.4 32 o00034
Remainder]
H20
2.5 cm
H20
Remainde o 0161 o0108 oo0068.329.609.00033
D20 in extending the thermal streaming flux peak farther from the core
than in H20 is a reason for the improved spectral. ratifoK~/4 This
fact presents a strong motivation for increasing the average reactor
power density by improved reflection rather than integrated power boost.
Infinite reflection by D20 increases the average power density by a factor
of 2 over the H20 reflected core, whereas (J ) goes up by a
factor of 35.7
Infinite graphite reflection shows up poorly in this set of calculationso Graphite used in conjunction with water demonstrates some im
117
provement. The effect of 1 in. of water located between the core and
the graphite is conspicuous. This configuration gives rise to significant
gainsins in 4 and (f  )max
max max
REFLECTOR INSERTS
The effect of reflector inserts of graphite, BeO, and D20 iwas examined for the equilibrium.,Xei. 84 percent, fuel burnup composition. The
geometry is sketched in Figure A.13. The fundamental core dimensions
shown in Figure A.13a were established from a onedimensional simulation study. Calculations were performed in XY geometry, and the vertical
transverse bucklings recorded in Table A.I were used to account for leakage in the third dimension. The insert widths were chosen to enable the
inserts to accommodate a six inch port. The reactivity contributions,*
6k/k, are recorded in Table A.VI along with the other significant quantities. Figures Aol5A.18 present the scalar flux and streaming flux
distributions along the xaxes of the inserts. Figure A.o19 and the
bottom row of Table A.VI present the results of a calculation of the
core displayed in Figure A.14, possessing full 6 in. layers of BeO,
The insert study permits a comparison of the effects of H20, graphite,
D20, and BeO independent of power density influences (a somewhat artificial situation). Light water excels from the standpoint of suppression
*The core dimensions were held constant, therefore all of the inserts,
possessing better reflector properties than H20, constituted positive
reactivity perturbations. Enhancement of the thermal flux arising from
an increase in power density, then, does not appear here.
118
I —' 19.2" 
12E g ox
3 —
rI
(a)
H20 Surrounding all Cores
73.94 X
BeO ~r~t~xruSO L~n"N ~ Be.~  7.08" o x
(b)
5.50"
D20 i' ~a~F~&S~jt~S~f~S~f~ ~D 20 7.08" — x
Graphite Graphite
(c)
Figure A.13. Core geometries for material insert studies.
119
3/,
x CORE: BeOH2
12" BeO CORE BeO
GRAPHITE
Figure A.14. Core geometry for 6 in. BeO reflector.
TABLE A. VI
TWODIMENSIONAL CALCULATIONS;FIXED SIZED COREINSERTS OF VARIOUS MATERIALS
1/V
Smaxa ( (4_S3) max (S4S) maX (S3/Sjax) (S /Slax) 6k/k in 3
Full
0.0316.0234.0173.266.506 .00018
H20
Graph..0288.0225.0211.271.347 +1.92%.00018
Insert
BeO l
Insert.0288.0205.0212.389.375 +2.3 %.00018
Insert
D420
Insert.0361.0279.0289. 266.266 +1.86%.00018
6 in..o980.o649 50759 I.330.212 .00041
BeO J J
.05
Core H2 0.04.03
x~~~~~~~~~~~~~~~~~~~~
Computed Flux versus k.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~02.02 Distance from Core Midpoint
Two  Dimensional Calculation
Full H20 Reflector
Scalar Flux
0.625 ev
.625 ev  5.53 Key _ _.O I.821 Mev  10 Mev
Streaming Flux
~ .625 e6 v
 " —.625 ev  5.53 Key
 —.821 Mev  10 Me,
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Distance from Core Midpoint (Cm)
Figure A.15, Twodimensional computed flux distributions; H20 reflected.
Core Graphite H 20.04.030.02 Computed Flux versus __
Distance from Core Midpoint
Two Dimensional Calculation
14 cm x 18 cm Graphite Insert
Scalar Flux
0 .625 eyv.O I L.625 ev  5.53 Kev.821 Mev Y10 Mev
Streaming Flux I ""
0  0. 625 e v._  ~1,.625 ~v  5.53 Kev
—' —.821 Mev  10 Mev
2 4 6 8 I 0 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Distance from Core Midpoint (Cm)
Figure A.16. Twodimensional computed flux distributions; graphite insert.
.05.......Core BeO H20.04.03.02 Computed Flux versus
TwoDimensional Calculation I t L \
IO cm x 18 cm BeO Insert
Scalar Flux.01 1 *.625 eY  5.53 Kev
Streaming Flux
0 .625 ev.625 ev  5.53 Key
~821 Mevy 10 Mev
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Distance from Core Midpoint ( Cm )
Figure A.17. Twodimensional computed flux distributions; BeO insert.
.05
Core D20 H20.04.01
Computed Flux versus
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Distance from Core Midpoint (Cm)
Figure A.18. Twodimensional computed flux distributions; D20 insert.
Figure A.18. Twodimensional computed flux distributions; D20 insert.
.12
Core BeO H20. 1 I _ _ _ _ _
Computed Flux versus
J1O Distance from Core Midpoint
TwoDimensional Calculation.09 _ ___ 6" BeO Reflection
Scalar Flux.08 0 .625ev
 II 1.625 ev  5.53 Key.821 Mev  10 Mev.077
Streaming Flux
UI i I I I I I ~'  i.5.625 e3.06
Y  —.~~~~~~~~~~~~~~~~~~~~~~~~~~~625 ev  5.53 Key
, —.821 Mev  10 Mev.05.04.03.02.0 I
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Distance from Core Midpoint (Cm)
Figure A.19. Twodimensional computed flux distributions; 6 in. BeG reflector.
126
of the epithermal content of the spectrum, exemplified by comparison
of the quantities (2>/1 ) and _J ) for all four materials.
max max
Every reflector insert diminishes the ratio, (S1/S 4ax)O As before, we
attribute this effect to the longer diffusion lengths possessed by the
insert materials, extending the high thermal flux region farther into
the reflector. The D20 insert results are particularly conspicuous. In
addition to a 65 percent increase in the quantity (  4 )max, the
D20 insert brings about a 14 percent enhancement in the thermal flux.
REFLECTOR LAYERS IN PARALLELEPIPED GEOMETRY
The final study constitutes a comparison of the streaming flux for
3 in. and 6 in. layers of reflector materials completely covering two
faces of a rectangular parallelepiped reactor core. Figure A.20 is an
illustration of the geometry. The fundamental water reflected core is
sketched in Figure A.20ao Equilibrium Xe, 8.4 percent fuel burnup was
chosen for the core composition. Once again, H20, graphite, BeO, and
D20 were examined. The configurations were simulated with onedimensional computations. The results are tabulated in Table A.VII. Together with the quantities previously tabulated, an additional dimension,
d, is listed, which is the distance from the core to the reflector location appropriate to the quantity alongside it o Figures A.22 and Ao23
present the quantities (J4  )max and ( 4 1)max plotted versus
127
the dimension a.* Dimension a is an inverse measure of the relative
average power density.
19.2
I x,/ X
(a) 12/
C C0RE
3,
3"
(b)
a A 6", H20 Surrounds all cores.
(c)
Figure A.20. Core geometries for 3 in. and 6 in. reflectors
of various materials.
*Additional points for 41/2 in. of the reflector materials were computed to obtain the shapes of the curves.
128
The consequence of shrinking dimension a by extension of the transverse dimension, illustrated in Figure A.21, is compared in Table A.VII
and Figures A.22 and A.23 with the results of reflection by 3 in. and
6 in. layers of reflector materials. Although this calculation was presented in Section A.4 for the coldclean core composition, it is of interest to compare this configurational effect with the effect of reflection for the fuel composition presently under consideration.
4N' I
Cu4,{a)(c)
(d)
Figure A.21. Core geometries for H20 reflected configuration
studies (equilibrium Xe8.4 percent burnup fuel).
The results appearing in Table A. VII are too numerous to discuss
in detail. The geometry considered in this portion of the study has
practical application for ports located 180~ apart. Graphite makes a
far better showing here than in the cylindrical geometry study. The
maximum thermal streaming flux, (S ), in this configuration increases
more rapidly than the average power density with improved reflection.
The results for reflection by 6 in. of D20 are striking.* Figures A.22
*Due to the calculational simplification of regionindependent bucklings
(thermal, in particular), the results for the 6 in. reflectors may be
highly unrealistic.
129
TABLE A.VII
THREE AND SIX INCH REFLECTORS OF VARIOUS MATERIALS
S4 d s3/S4 (S4S3) d (4 1) d a/V
max S max i.max max ( m m 3
in. in. in. in. in.
3 in.
Graph.
Refl..614 3.254.324.44.4o8 7 13.5.000223
4 Elem.
Trans.
n in.
Graph.
Refl. 1.28 4.3.186.174 1.08 5.6 1.09.6 lo.9.000317
4 Elem.
Trans.
3 in.
BeO
Refl..773 2.1.333.307.582.0.6oo0 3.0 12.9.000268
4 Elem.
Trans.
6 in.
BeO
Refl. 1.65 3.1.250.140 1.29 3.9 1.6.9 8..ooo4o6
4 Elem.
Trans.
5 in.
Do20
Refl..853 3.254.232.636 3.66 14.4.00024
4 Elem.
Trans.
6 in.
D20
Refl. 3.4 4.3.144.074 2.99 4.9 3.22 4.9 5.45 ooo635
4 Elem.
Trans.
H20
Refi..485.9.216.464.385 1.1.282 1.3 19.2.00018
4 Elem.
Trans.
H20
Refl.
5 Ele.714.9.216.444.569 1.1.428 1.3 15.0.00018
5 Elem.
Trans.
H20
Refl.
Ele..965.9.216.422.770 1.1.596 1.3 12.2.0001ooo63
7 Elem.
Trans.
H20
Refl.
8 Ele. 1.039.9.216.416.830 1.1.648 1.3 11.6.00015
ElemTrans.
Trans.
(54_53 Max versus Core Width
3.0
For Graphite, BeO, D2 0, H2 0
EquiI.Xe8.4 % B. U. Core
Slab Geometry
4 Elements Transverse
GRAPHITE
2.0O
20 ~~~~~~~~~~~~~~~~~~~~~~~~~BeO
0 H2 RefI.
@ 3" Refl.
* 6" Refl.
0 4 1/2"RefI. 0
1.0
ZI~~~~hI
10 20 30 40 50
WIDTH OF CORE (Cm.)
S4 3
Figure A.22. (SS )max for 5 and 6in. reflection.
(S4 S )Max versus Core Width
3.0 For Graphite, BeO, D2 0, H 2 0
Equil. Xe  8.4 % B. U. Core
Slab Geometry
4 Elements Transverse
2.H
2.0   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Graphite
BeO
0 H20 Refl.
M 3" RefI. H
* 6" Refl.. 4 1/2"Refl.
1.0
10 20 30 40 50
WIDTH OF CORE (Cm.)
Figure A.23. (S'KS1)max for 3 and 6in, reflection.
132
and A.23 compare the reflectors on the basis of constant average power
density (excluding the curves for H20 reflection obtained by altering
the ratio of the core dimensions). On this basis, D20 excels, followed
in order by graphite and BeO.
6. Final Remarks
Although the group constants used in the calculations were obtained
in a consistent manner, a question may be raised regarding the validity
of the cross sectionso The source of the cross section library is cited
in Appendix C. Much of the analysis reported in the literature3 is
based upon this set of cross sections or slight variations of ito The
reasonable agreement between the predicted and measured distributions
of the thermal and epicadmium fluxes presented in Figures 3.6, 3o8, and
3519 tends to corroborate the validity of the group constants for H20,
graphite, and the coreo
In order to examine the validity of the D20 group constants, a
measurement was made with a limited quantity of D20, and the experimental
results were compared with a groupdiffusion calculation. A flux traverse was performed in a cylindrical D20 insert 5 in. long and 51/2 in.
in diameter placed against the center of the south face of the FNR.32
The twodimensional analysis utilized 4 groups in cylindrical geometry.
A comparison of the measured and calculated thermal and epit~hermal f lux
distributions along the axis of the insert is presented in Figure A2 24)
8.0
Flux in D2 0 Insert versus Axial Distance From Core
X Measured Relative Thermal Activation in D2 0
7.0
OMeasured Relative Epicadmium Activation in D2 0
i \  Computed Flux Distributions in D2 0 Insert
/~~~ ~ ~~~~~~~~~ \  ~~~Thermal
6.0 x X \Epithermal
\r0  X a Computed Flux Distributions in H2
2.  _ _ _Thermal
5.0
2'\ D20 I H20
4.0
3.0
2.0
0
1.00
0
2 4 6 8 10 12 14 16 18 20 22
Distance From Core'"m.)
Figure A.24. Measured and computed flux distributions along axis of cylindrical D20 insert.
134
Both the measured and calculated distributions were normalized to distributions obtained along the same axis in the complete light water
medium (also plotted in Figure A.24)o
Many of the results reported here were based upon the technique
described in Section Ao2, simulating threedimensional geometry using
onedimensional computations. An examination of the validity of this
method is afforded by comparing the results of actual twodimensional
calculations with the results of onedimensional simulations. Since
fourgroup, twodimensional computer studies are extremely timeconsuming, few comparisons were made. The first comparison was carried
out for the core illustrated in Figure A.14, possessing symmetric 6 in.
layers of BeOo The twodimensional calculation resulted in an eigenvalue (See Eqo B.1).6 percent lower than that obtained in the onedimensional simulation. The water reflected core of Figure Ao13a was
examined next, the twodimensional calculation yielding an eigenvalue
3 percent lower than the onedimensional simulationo Finally, the deviation for the coldclean, water reflected core shown in Figure Aolb
was 2 percent. None of the discrepancies are serious, considering the
disagreement between the results of groupdiffusion calculations and
precise criticality determinations for clean, homogeneous assemblies
reported by Feinero33 It is curious, however, that the smaller cores
yield better agreemento It was anticipated that the deviation between
the one dimensional simulation results and the twodimensional results
would be inversely proportional to the core size, owing to corner effectso
135
Apparently a reflector composition influence exists, as wello
A question may be raised concerning the accuracy of the calculated
fast flux magnitudes in the reflector, i1 and _2o P1 theory tends to
overestimate the attenuation of the forwardpeaked, high energy components of the spectrum. Experimental fast flux determinations are
scarceo Since the reflector regions of greatest interest lie within 1
to 3 fast neutron mean free paths from the core, P1 errors in the fast
flux are presumed to be small.
Most of the core alterations examined in this appendix involved
increases in the reactor power density~ Practical considerations call
for a remark concerning core heat removal capabilitieso A cursory examination of this problem was carried out with an existent computer
34
program, originally written to evaluate the FNR two megawatt performanceo The results of the calculation, assuming a constant coolant mass
flow rate through the core and an integrated reactor power of one megawatt, indicate that in going from 24 to 4 regular fuel elements a
maximum fuel cladding temperature rise of only 7~F is anticipated~
7o Subroutine STREAM
Subroutine STREAM, used in conjunction with the FOG groupdiffusion
code (control program 3), computes the partial current, defined by.
J (x) 4 a, A2dx1)
136
and the streaming flux, defined by Eq. (Ao6), at each mesh point in a
designated region. The streaming flux integral is performed over the
designated region plus the region before it. Thus for reasonable
accuracy the width, w, of the region immediately before the designated
region must be defined sufficiently large so that w >> 1 for each
energy group. The Fortran program listing of the revised main program
of FOG and subroutine STREAM are recorded hereo
In order to call the subroutine L63 of the input data must be
set different from zero. Then a master card and one card for each
energy group is expected at the end of the FOG input deck. The card
arrangement is as follows:
Master Card (Format 4I12)
N(1) = 0 Bypass Subroutine
= 1 Utilize Subroutine
N(2)  0 Do not compute j
1 Compute jo
N(3) = 0 Do not compute M
= 1 Compute /
N(4) = I I is the FOG region designated
for the computation.
Cross Section Card* (Format 2F12.0)
F(l) = ES Scattering cross section in region I
F(2) = 7,s Scattering cross section in region I1
*One card is required for each energy group.
C FOG CODE MAIN PROGRAM  WRITTEN AT NORTH AMERICAN AVIATION
C REVISED AT UNIVERSITY OF MICHIGAN, 1962
$COMPILE FORTRANPRINT OBJECTgPUNCH OBJECT.DUMP*EXECUTE MAIN3000
C CONTROL PROGRAM NO.3. THIS PROGRAM CONTAINS THE CRITICALITY SEARCHFOG23080
C ES AND ADJOINT FLUX CALCULATIONS. FOG23090
C FOG2 3100
DIMENSION LM(40) LB(40) NPO(3) LBS(40) MR(40) L(250),A(5000),PHI (FOG23110
1239,4),APHI(239,4),SOU(239)SOP(40),SOUP(239)gBSA(40,4),SOPP(40) FOG23120
2,R(239),RI(239),NN(5),C(5),D(239),DP(40)tG(239),GP(40)G1l(239),G1PFOG23130
3(40),BETA(239),DELTA(239),DELR(40),T(4),DIF(40,4),TRANS(40, FOG23140
44), SIGT(40,4),CIH(4),OMEGA(4),OMEG1(4),FLP(239),BUCK(40),BUCK2(4FOG23150
50,4),VUSIG(4094),SIGPT(40)tFL(239),SIGA(40,4),ACl(40),AC2(40),AC3(FOG23160
640,4),AC4(40,4),AC5(4094),BUCK3(4),AC6(40,4),AC7(40,4), AC8(40O4),FOG23170
7AC9(40),AC10(40),CHI(40,4),BL4B2( 4),83(4).B4(4),B5(4),RW(40) FOG23180
DIMENSION NFU(40),SOUS(239),SOPS(40),SIP(4) FOG23190
COMMON L,A,APHI,D,G,G1,AC3,AC4,AC5,AC6,AC7,AC8,DPNFU,C,NN FOG23200
EQUIVALENCE (L(249),N)t(L(248),LOB) FOG23210
EQUIVALENCE (L(1),NOG) (L(2),N1) (L(3),N2),(L(4),N3) (L(5) N4) (L(FOG23220
16),N5),(L(7),M),(L(8),LM)(L(48),N6),(L(49),N7),(L(50),N8),(L(51),FOG23230
2N9),(L(52),NlO),(L(53),Nll),(L(54),Nl2)9(L(55),N13),(L(56)N14)(LFOG23240
3(57),N15),(L(58),N16),(L(59),N17),(L(60),NPO),(L(63)tN18).(L(64),NFOG23250
419),(L(65),N20),(L(66)9N21),(L(67),LBS),(L(107),N22)*(L(108),N23),FOG23260
5(L(i09),MA),(L(110).,MR),(L(150)tMICT),(L(151),MICT2),(L(152).LB) FOG23270
EQUIVALENCE (A(4956),LB1),(A(4957),LB2),(A(4958),LBT),(A(4959),LB3FOG23280
1),(A(4960),SOPS),(A(3340),SIP),(A(3344),LB5),(A(3345),GAM1). FOG23290
EQUIVALENCE (A(1),ESP1),(A(2),ESP2),(A(3.),E. IGEN2),(A(4),ESP3),(A(5FOG23300
1),THET),(A(6),FMUL),(A(8),RW)t(A(48),C1),(A(49),C2),(A(50),BUCK,BUFOG23310
2CK1,'BUCK3),(A(368)9C3),(A(369),C4),(A(395),CIH),(A(400),BUCK2), FOG23320
3(A(560),TRANS,SOP),(A(720),AC1),(A(760),AC2),(A(800).VUSIG),(A(960FOG23330
4),DELR),(A(1000),SIGA),(A(1160),AC9),(A(1200),DIF),(A(1360),GlP),(FOG23340
5A(1560),AC1O),(A(1400),CHI),(A(1600),OMEGA).(A(1604).OMEG1),(A(160FOG23350
68), T),(A(16'12),SIGPT),(A(1652),PG), 9(A(1653),PG2),(A(1654),PG3),(AFOG23360
7(1655),PG4),(A(165&),N100),(A(1657),EIGEN),(A(1658),EIGEN1) FOG23370
EQUIVALENCE (A(1659),P),(A(1660),GP),(A(1700)SOU),(A(1939),A1),( FOG23380
1A(1940),SOPP),(A(198Q),LT1),(A(1981),LT2),(A(1982),B1),(A(1983),BLFOG23390
2),(A(1987) B2),(A(1991) B3),(A(1995) B4) (A(1999),W)(A(2085),B5)FOG23400
3(A(2009),W1),(A(2004),W2),(A(2014),PSI1),(A(2019),PSI2),(A(2024),DFOG23410
4ET),(A(2025),PIM),(A(2026),ERR),(A(2027).SUM),(A(2028),GAM),(A(202FOG23420
59),NM1)(A(2034),NM2),(A(2038),MM1),(A(2039),NAA),(A(2044),NSKP), FOG23430
6(A(2049),ICT),(A(2054),ICT1),(A( 2055),ICT2),(A(2056.),ICT3),(A(205FOG23440
77),DELRT),('A(2058),BUCKT),(A(2059),C10),(A(2064),SGPT) FOG23450
EQUIVALENCE (A(2600),SOUP),(A(2850),R),(A(3100),RI),(A(3350)tFL),(FOG23460
1A(3600),FLP),(A(3840),SIGT),(A(4000),PHI),(APHISOUS.BSA),(GBETA)FOG23470
2,(G1,DELTA),(A(2068),CON),(A(2069),CON1),(A(2074),FAC) FOG23480
CALL ZERO
MICT=35 FOG23490
MICT2=10 FOG23500
5 READ INPUT TAPE 7,6 FOG23510
6 FORMAT (72H FOG23520
1 FOG23530
WRITE OUTPUT TAPE 6,6 FOG23540
10 CALL DATIN FOG23550
20 CALL OMCAL FOG23560
IF (N1001) 30,595 FOG23570
30 NSKP=O FOG23580
40 CALL DAPR FOG23590
IF (N221) 50,50,110 FOG23600
50 CALL CONS FOG23610
60 CALL FLUX FOG23620
IF (N20) 70,90,65 FOG23630
65 IF (N203) 70,70,80 FOG23640
70 CALL CRIT1 FOG23650
GO TO 90 FOG23660
80 CALL CRIT2 FOG23670
90 CALL ANPR FOG23680
100 IF (N22) 120,120,110 FOG23690
110 CALL AFLUX FOG23700
120 IF (N18) 140,1409130 FOG23710
130 CALL CHECK FOG23720
CALL REPR FOG23730
CALL STREAM FOG23685
140 IF (L(250)) 5,5,150 FOG23740
150 CONTINUE FOG23750
END FOG23760
138
C SUBROUTINE STREAM WRITTEN AT U OF M TO BE USED IN CONJUNCTION WITH FOG
S COMPILE FORTRAN9PRINT OBJECT.PUNCH OBJECT SUBSTREAM
SUBROUTINE STREAM 1
DIMENSION LM(4Q),LB(40)tNPO(3),LBS(40).MR(40),L(250),A(5000).PHI( 1
1239,4),APHI(239.4)tSOU2 39) P(40)0UP(239)*BSA(404)SOPP(40) 2
2.R(239),RI(239).NN(5)C(5).D(239).DP(40)tG(239)tGP(40).Gl(239)GIP 3
3(40).BETA(239),DELTA(239)DELR(40),T(4)DIF(40,4)tTRANS(40. 4
44), SIGT(404).CIH(4).OMEGA(4).OMEG1(4).FLP(239).BUCK(40).BUCK2(4 5
50.4).VUSIG(40,4),SIGPT(40).FL(239).SIGA(40.4)AC1(40).AC2(40).AC3( 6
640,4).AC4(40,4).AC5( 40,4),BUCK3(4).AC6(40.4),AC7(40.4). AC8(40.4). 7
7AC9(40).AC10(40),CHI(40,4),BL(4).82(4) B3(4).84(4)B85(4).RW(40) 8
DIMENSION CURR(239).NNN(4) AA(2).SUMM(239.4)
DIMENSION NFU(40),SOUS(239).SOPS(40),SIP(4) 10
COMMON LA.APHID,G,GG1AC3,AC4,A C5.AC6,AC79AC8tDPNFU.CNN 11
EQUIVALENCE (L(249),N)*(L(248),LOB) 12
EQUIVALENCE (L(1),NOG),(L(2),N1),(L(3),N2),{L(4),N3) (L(5),N4)*(L( 13
16) NS5).(L(7).M),(L(8),LM) (L(48)9N6).(L(49).N7).(L(50).N8)t(L(51)# 14
2N9),(L(52),NlO), (L(53), Nll),(L(54),N12),(L(55),N13) (L(56),N14),(L 15
3(57) N15),(L(58),N16).(L(59).N17),(L(60) NPO).(L(63) N18),(L(64),N 16
419)*(L(65).N20).(L(66).N21)t(L(67?).LBS).tL(107) N22).(L(108)N23). 17
5(L(109),MA),(L(11O).MR). (L(150).MICT),(L(151),MICT2).(L(152).LB) 18
EQUIVALENCE (A(4956),LB1),(A(4957),LB2),(A(4958),LBT),(A(4959),LB3 19
1).(A(4960),SOPS).(A(3340).SIP).(A(3344),LB5).(A(3345),GAMl) 20
EQUIVALENCE (A(1),ESP1),(A(2).ESP2).(A(3),EIGEN2).(A(4),ESP3) (A(5 21
1).THET).(A(6).FMUL).(A(8).RW).(A(48). Cl1)(A(49).C2).(A(50),BUCKBU 22
2CK1.BUCK3). (A(368)tC3).(A(369).C4) (A(395)*CIH)*(A(400)BUCK2), 23
3(A(560),TRANSSOP)9(A(720).AC1)*(A(760).AC2)*(A(800)*VUSIG).(A(960 24.4),DELR) (A(1000) SIGA) (A(1160) AC9)t(A(1200)tDIF)*(A(1360),G1P).( 25
5A(1560),AC10),(A(1400),CHI),(A(1600),OMEGA),(A(1604)*OMEG1),(A(160 26
68).T).(A(1612),SIGPT),(A(1652).PG1).(A(1653))PG2),(A(1654),PG3),(A 27
7(1655) PG4).(A(1656).N100),(A(1656)A1657)EIGEN)*(A(1658),EIGEN1) 28
EQUIVALENCE (A(1659),P),(A(166Q),GP),(A(1700)*SOU).(A(1939),Al).( 29
1A(1940)SOPP A198981)LT2)(A(1982)B)(A(1983)BL 30
2),(A(1987).B2),(A(1 91),B3),(A(1995),B4),(A(1999).W),(A(2085),B5)t 31
3(A(2QO9).W1).(A(2004).W2).(A(2014).PSI1)*(A(2019),PSI2),(A(2024).D 32
4ET),(A(2025).PIM),(A(2026),ERR),(A(2027).5UM).(A(2028).GAM),(A(202 33
59).NMlI,(A(2034).NM2).(A(2038).MM1),(A(2039).NAA),(A(2044).NSKP), 34
6(A(2049),ICT)9(A(2054),ICT1),(A( 2055),ICT2),(A(2056),ICT3)9(A(205 35
77),DELRT),(A(2058)sBUCKT),(A(2059),C10O)(A(2064),SGPT) 36
EQUIVALENCE (A(2600),SOUP),(A(2850),R),(A(3100),RI),(A(3350),FL),( 37
1A(3600),FLP),(A(3840),SIGT),(A(4000),PHI)9,APHISOUSBSA),(GBETA) 38
2,(G1,DELTA)9(A(2068),CON)9(A(2069),CON1)*(A(2074).FAC) 39
READ INPUT TAPE 7,59,(NNN(I),I=14) 101
59 FORMAT (4112) 201
N24=NNN(1) 301
N25=NNN(2) 401
N26=NNN(3) 501
N27=NNN(4) 601
IF (N24) 19,19.1 2
1 IF(N25) 9,9,30
30 DO 31 J=1,NOG
NF=1 4
LOL=1 5
DO 5 I1,M 6
NF=NF+LM(I) 7
DO 2 K=LOLNF 8
IF (IM) 6,7.7 9
6 CURR(K) = (PHI(K.J)/4.0)((DIF(I,J)/2.O)*(PHI(K+1,J)PHI(KJ
1))/DELR(I)) 11
GO TO 2 12
7 CURR(K) = (PHI(KJ)/4.O)  ((DIF(I*J)/201) * (PHI(K.J)PHI(K1.J
1))/DELR(I)) 14
2 CONTINUE 15
IF (I1) 3,3,4 16
3 LOL=LOL + LM(I) + 1 17
GO TO 5 18
4 LOL z LOL+LM(I) 19
5 CONTINUE. 20
WRITE OUTPUT TAPE 6,60,J
60 FORMAT(35H RADIUS PLUS CURRENT GROUPI3) 23
WRITE OUTPUT TAPE 6,61,(R(L),CURR(L),L=1,NF) 24
61 FORMAT (2E16.8) 25
31 CONTINUE
139
9 IF (N26) 19,19,10 26
10 I=N27
WRITE OUTPUT TAPE 6,629I
62 FORMAT (55H1 STREAMING FLUX IN REGION
113)
DO 18 J=1,NOG
READ'INPUT TAPE 7964,(AA(K),K=1,2)
64 FORMAT(2F12*0) 226
SIGMS1=AA(1) 326
SIGMS2=AA(2) 426
WRITE OUTPUT TAPE 6,659SIGMS1.SIGMS2,J
65 FORMAT(23H SIGMS IN REGION I1 ISE16,8, 626
121H SIGMS IN REGION I ISE16.8.8H GROUPI3)
LOTT = O 0 28
I2=I2 128
IF(I2)28.28,27
27 DO 20 N=11I2 29
LOTTT = LM(N) 30
LOTT = LOTTT + LOTT 31
20 CONTINUE 32
28 LOT = LOTT + 1 3 33
I3=LM(f) 133
DO 17 K=1,I3 34
SUMM(K#J)= 0.0
14=LOT+LM(I1) 135
DO 13 L=LOT,I4 36
FL2=K 236
FL3=LM(I1)+LOTL 336
IF(((SIGA(Ii 1J)+SIGMS1)*DELR(I1)*FL3+(SIGA(I,J)+SIGMS2)* 436
1DELR(I).*FL2)500)23,13,13 536
23 IF(L14)25,11911 63625 IF (LLOT) 11,11,12 37
11 STR=(((SIGMS1*PHI(LJ)+SOUP(L)*CHI(I1,J))/2.O)*DELR(I1))*EXP(
1((SIGA(I1,J) + SIGMS1) * DELR(I1) * FL3 39
2)  ((SIGA(IJ)+SlGMS2 ) * DELR(I) * FL2)) 40
GO TO 21 41
12 STR = ((SIGMS1*PHI(L,J)+SOUP(L)*CHI(I1,J))*DELR(I1))*EXP(((
ISIGA(I1,J) +'IGMS1 )*DELR(I1)*FL3)((SIGA(19J)+SIGMS2
2 )*DELR(I)*FL2)) 44
21 SUMM(K.9J) = SUMM(KJ) + STR
13 CONTINUE 46
N=LOT+LM(I1) 47
N1=N+K 147
DO 16 L=NN1 48
FL4=N+KL 148
IF(((SIGA(IJ)+SIGMS2)*DELR(I)*FL4)50.O)24916916 248
24 IF(LN)15,15,26 348
26 IF (LNK) 14915,15 49
14 STR=((SIGMS2*PHI(LJ)+SOUP(L)*CHI( IJ))*DELR(I))*EXP((SIGA(IJ)+S
lIGMS2 )*DELR(I)*FL4) 51
GO TO 22 52
15 STR = ((SIGMS2*PHI(LJ)+SOUP(L)*CHI(IJ))/2.0) * DELR(I)
1*EXP((SIGA(IJ)+SIGMS2)*DELR(I)*FL4) 153
22 SUMM(KJ) = SUMM(KJ) + STR
16 CONTINUE 55
17 CONTINUE 56
18 CONTINUE 61
IF (NOG3) 35,35,32
32 WRITE OUTPUT TAPE 6,33
33 FORMAT(69HO PT. IN REGION GROUP 1 GROUP 2 GROUP 3
1 GROUP 4)
WRITE OUTPUT TAPE 6,34,(K,SUMM(K,1),SUMM(Kt2),SUMM(K,3),SUMM(K,4),
1K=1,13)
34 FORMAT(I7,E22.8,3E14.8)
GO TO 19
35 IF (NOG2) 39,39,36
36 WRITE OUTPUT TAPE 6,37
37 FORMAT(55HO PT. IN REGION GROUP 1 GROUP 2 GROUP 3)
WRITE OUTPUT TAPE 6,38,(KSUMM(K1),SUMM(K,2),SUMM(K,3),K=1,13)
38 FORMAT (I7,E22.8,2E14*8)
GO TO 19
39 WRITE OUTPUT TAPE 6.40
40 FORMAT(41HO PT. IN REGION GROUP 1 GROUP 2)
WRITE OUTPUT TAPE 6,41,(K,SUMM(K,1),SUMM(K,2),K:II3)
41 FORMAT(I7tE22.8,EE14.8)
19 RETURN 62
END 63
APPENDIX B
DESCRIPTION OF THE CODES
Computer calculations were performed on the IBM709, and later on
the IBM7090. Some revision of the Fortran language codes were necessary for compatibility with The University of Michigan executive system. Program revisions involved tape number changes, function name
revisions, memory zeroing, and occasional function deletions. In addition, since the executive system monitor occupies several thousand locations of the 32 K memory, it was often essential to shorten or break
down programs written for the scope of the entire memory.
Groupdiffusion codes solve the following set of coupled equations:
DiV2/i + [Di B i i i+ Z.Ei
~= xi (V4) + E (B.1)
Finitedifference methods along with a variety of convergence accelerators are employed in obtaining solutions. Flexibility is generally built
into a code in the form of an assortment of boundary conditions and specific geometrical configurations~
FOG is a onedimensional fewgroup diffusion code providing for
as many as four energy groups and 239 space points. The code permits
a choice of three geometries, nine sets of boundary conditions, and five
T40
141
types of criticality searches. Downscattering between adjacent groups
only is allowed.
TWENTY GRAND37 is a twodimensional groupdiffusion code based upon
the Equipoise convergence technique. Three thousand mesh points are accommodated, and the code was revised to handle as many as four energy
groups (the original form accommodated six energy groups). Both XY and
RZ geometries are available and neutron transfer between any of the
groups is permitted. Zero flux, zero derivative, and logarithmicderivative boundary conditions are utilizable.
38
Thermal neutron group constants were computed with TEMPEST, the
Fortran version of the SOFOCATE39 code. Cross sections, supplied in the
form of a library deck, can be averaged over a thermal spectrum determined from (i) the WignerWilkins equation for light moderators, (ii)
the Wilkins equation for heavy moderators, or (iii) a Maxwellian distribution.
FORM,40 the Fortran version of MUFTIV,41 a Fourier transform slowingdown code, was used to obtain fast group constants. Fast spectra
are generated from the spatially Fouriertransformed sl owingdown distribution by the P1 or B1 approximation, with or without the SelengutGoertzel approximation. Cross section data are supplied on tape in the
form of a 54group cross section library from o625 ev to 10 Meva The
memory size limitations of The University of Michigan system required alteration of the main program, a listing of which appears at the end of
this appendix~
142
C FORM (FORTRAN MUFT) PROGRAM  WRITTEN AT NORTH AMERICAN AVIATION
C REVISED UNIVERSITY OF MICHIGAN, 1962'
$COMPILE FORTRAN, PRINT OBJECT, PUNCH OBJECT MAIN1000
DIMENSION FILE3.(7776) PROB(325,18) D3(644)g D1(599) D2(1641), 0
1DUMMY(5215)
COMMON FILE 3,D 1,PROB,D2 NSD, IP, ANED.D3,NADD,DUMMY NFIRST
CALL ZERO
REWIND 3 80
NFIRST=1 90
CALL DELSET 100
CALL SEQPGM
END
C SUBROUTINE DELSET
$ASSEMBL E PUNCHOBJECT ZERO 0000
ENTRY ZERO
ZERO LXA *+4,1
8STZ 777'77 1
TIX * s 1 + 1
TRA 1,4
OCT 52661
END
$COMPILE FORTRAN, PRINT OBJECT, PUNCH OBJECT MAIN2000
DIMENSION FILE3(7.776),PROB(325.,18),D3(644), D1(599),D2(1641), 0
1DUMMY(5215)
C QMON F I.LE3D1 PROB,D2 NSDI P A,NED,D3,NADD,DUMMYNFIRS T
1NTIMENTESTgIPtNSD.NDNSNINULNALPHANINNRA
1 CALL INPUT 110
CALL INPi 120
IF (NFIRST1 )22,4 130
2 NTEST=O 140
NTIME 1 150
CALL READLB(IPNSDNTIMENTEST) 160
IF(NTEST)12,3,12 170
12 CALL SELPGM(l)
3 iF(NADD)13913,11 180
11 CALL ADD 190
13 WRITE TAPE 3,LFILE3(I),I1=,7776). 200
CALL SELPGM(4)
4 READ TAPE 3tFILE3(I),I=1,7776[ — 220
CALL SEQPGM
END
C SUBROUT I NE INPUT
C SUBROUTINE INP1
C SUBROUTINE READLB
C SUBROUTINE ADD
$COMPILE FORTRAN, PRINT OBJECT. PUNCH OBJECT MAIN3000
DIMENSION F ILE3(7776 ) PROB(325,18),D3 (644), D( 599 ) D2(1641 ) 0
143
1DUMMY( 5215)
COMMON FILE3,D1,PROBD2,NSD, IP,ANEDD3,NADDDUMMY.NFIRS.T,
1NTIMENTEST
CALL GRUCON (NF IRST)
CALL SEQPGM
END
C SUBROUTINE GRUCON
$COMPILE FORTRAN, PRINT OBJECT, PUNCH OBJECT MAIN4000
DIMENSION FILE3(7776),PROB(325,18),D3(644), Dl(599),D2(1641), O
1DUMMY( 5215 )
COMMON FILE3,DI,PROB8D29NSD, IP,ANED,D3,NADDDUMMYtNFIRST,
1NTIMENTEST, IPNSD*NSNI NUL.NALPHANINtNRA
IF(NFIRST1 )6,6,7 240
6 NTIME=2 250
CALL READLB lPNSD,N TIME NTEST)' 260
IF(NADD/10) 15915,14 270
14 CALL ADD 280
15 WRITE TAPE 3,PROB 290
7 CALL SEQPGM
END
C SUBROUTINE READLB
C SUBROUTINE ADD
$COMPILE FORTRAN9 PRINT OBJECT, PUNCH OBJECT MAIN5000
DIMENSION FILE3(7776),PROB(325,18), D3(644)t Dl(599),D2(1641), 0
1DUMMY(5215)
COMMON FILE3,D1 PROB D2*NSD, IPA,NEDD3,NADDDUMMYg NFIRST
CALL SLODON(NFIRST) 300
REWIND 3 310
CALL EDIT12 320
IF(NED2) 10,108 330
8 CALL EDIT3 340
IF(NED3) 1091099 350
9 CALL EDIT4 360
10 NFIRST=NFIRST+I 370
CALL SELPGM(3)
12 CALL SELPGM(1) 06000390
END 400
C SUBROUTINE SLODON
C SUBROUTINE EDIT12
C SUBROUTINE EDIT3
C SUBROUTINE EDIT4
APPENDIX C
COMPOSITIONS AND GROUP CONSTANTS
Table CoI is a listing of all compositions considered for this
study. Two classes of core compositions were examined~ The first is
the coldclean initial fuel loading of 140 gm/element. The second is
appropriate to a core with equilibrium xenon and 8f4% fuel burnup, representative of the FER operating at one megawatt during the fall of 1962
Core compositions are presented for a homogenized regular fuel element
with the water between subassemblies averaged into the composition.
The homogenization of the subassembly is valid since the disadvantage
factor of the fuel is o98822  The existence of partial elements and
poison rods was neglectedo
Thermal group constants were evaluated by averaging cross section
data over spectra computed by the TEMPEST code from 0~625 evo The
TEMPEST library of cross sections, most recently revised in February,
1961, was used throughout for consistency~ A temperature of 3000K and
a material buckling of o0130 cm2 were used as input data. Core constants were evaluated using the WignerWilkins light moderator approximationo Figure Col shows the WignerWilkins spectrum computed for the
coldclean core composition compared to a Maxwellian spectrum at 3000K.
Both spectra were normalized in the following manner:
~625 ev
S /(E)dE = o. (Cdl)
144
145
Reflector constants were averaged over a Maxwellian spectrum at 300OK
(the buckling was taken to be zero). Table C.II lists the thermal group
constants for the compositions in Table C.I.
Fast group constants, evaluated by the FORM code, are entered in
Table C.III. The FORM library tape, updated to May, 1962, was obtained
directly from the North American Aviation Corporation, and was used as
the source of all cross section data. The fourgroup scheme had the
following breakpoints: o625 ev5.53 Kev; 5o53 Kev.821 Mev; o821 Mev10 Mev. Constants were evaluated for a full U235 fission source spectrum using the Bl approximation and the selfconsistent age approximation. A buckling of.0130 cm2 was used to obtain core constants and.00001 cm2 for the reflector. Resonance selfshielding factors (Lfactors) of unity were used for both U235 and U238o
146
100
Thermal Spectrum O(E) versus E
As Calculated by Tempest  Cold Clean FNR Core
WignerWilkins 300~K
Maxwellian 3000 K
10
0.1
10 3 102 101
Energy (ev)
Figure C.1. WignerWilkins and Maxwellian thermal spectra
for coldclean FNR core at 300~K.
TABLE C. I
COMPOSITIONS
Reflector
ColdClean Equil. Xe Refector
Core 8.4% BU Core Homogenized Graph. H20 D20 BeO Be
Graph.
Volume Nuc. 1024 Volume Nuc. x 24 Volume Nuc. 1024 Nuclei x 1024
Fract ion cm3 Fract ion cm3 Fract ion cm3 cm3
H20.5845 . 5845 .0694
H .0392 .0392 .00464 .067 
0 .0196 .0196 .00232 .0335.0331.0728
A1.4133.0249.4133.0249.0774.00oo466   
U235.00197 9.46 x 105.00181 8.67 x 105      
u238 2.26 x 104 1.08 x 105 2.26 x 104 1.08 x 105 
B10'* _ .20 x.27x106  
Xel35   __.85 x 109..
C   .84.68.80   
D20      _ _  _.0331 
EBe to ac n — c for n X' fission.0725.1236
*Equivalents to account for nonXel35 fission product poisoning.
TABLE C. II
THERMAL GROUP CONSTANTS
Reflector
ColdClean Equil. Xe Homognize
Homogenized
Core 8.4% BU Core Graph. 1H20 D20* Be BeO
< Za > cm1.0629.0622.00256.000263. 0195.000029.00108.000651
< D > cm.2993.2989.693.915.1546.831.4155.4354
< VZf > cm1.0996.0915 0.0 0.0 0.0 0.0 0.0 0.0
< Ztr > cm 1.377 1.378.490.364 2.485 .802.7655
< Zs > cm 1.313 1.313 .384 3o6.375.741.720
*Not available in TEMPEST library; obtained by averaging BNL325 cross section data over a
Maxwellian spectrum.
149
TABLE C. III
FAST GROUP CONSTANTS
Reflector
Cold.Clean Equil. Xe*
Homogenized
Core BU Core Graph. H20 D20 Be BeO
Graph..... e
< Za >1 cm 1.o000898.000885.000056.oooo000003.00138.oo00144.00453.00449
< Za >2.000285.000267.000018 0.0.000013 0.0.000002.000001
< Ea >3.00oo485.oo454.000124 0.0.ooo00954 o0.0o. 000034. 1ooool
< a >.0019.00173.000073 0.0.000787.000148.ooo000729.00ooo549
< ZR >1 cm1.07429.07445.0321.0275.1087.0835.0419.0462
< ZR >2.08509.o86.0199.0113.1494.0372.0275.0230
< ZR >3.08252.0804.0161.00657.1520.0196.0169.0134
< ZR >.0254.0236.00734.00372.0o495.0115.00885.00746
< VZf >1 cm'.000306.000281 0.0 0.0 0.0 0.0 0.0 0.0
< Vif >2.000380.000346 0.0 0.0 0.0 0.0 0.0 0.0
< vwf >3.00593.00539 0.0 0.0 0.0 0.0 0.0 0.0
< vkf >.00207.00180 0.0 0.0 0.0 0.0 0.0 0.0
< D >1 cm 2.017 1.920 2.369. 2.410 2.048 2.175 1.908 1.442
< D >2 1.156 1.144 1.085 1.049 1.057 1.230.581.549
< D >3.905.907.950.934.595 1.242.486.502
< D > 1.365 1.341 1.245 1.122 1.245 1.334.7364.6287
< Zs >1 cm1.201.201 .163.261.178.209.285
< ZS >2.597.597 .342.761.248.636.641
< Zs >3.910.910 .377 1.385.240.736.695
Subscripts: 1.821 Mev to 10 Mev
25.53 Kev to.821 Mev
3.621 ev to 5.53 Kev
Nonsubscripted constant denotes.621 ev to 10 Mev.
*Xel35 cross sections were not available on the library tape, thus it was neglected here.
APPENDIX D
COMPARISON OF CURRENT BOUNDARY CONDITION AND PSEUDO
BOUNDARY CONDITION AT VACUUM INTERFACE
We desire to compare solutions to the
twogroup diffusion equations utilizing (i)
the realistic boundary condition on the par/o H tial current and (ii) the pseudo boundary condition on the scalar flux. We examine a
system which is similar to the one studied
in the text but deviates sufficiently to
O x
allow an analytic solution. Consider a slab of width a, with fast neutrons entering the medium at x = 0. We write the twogroup diffusion
equations:
2,
di 2  1/ 0 (D.1)
dx
d2 2
D2  ZaO2 + ZRi 0 (D.2)
The solution for the flux in group 1 is:
Ae + Be
~l(x) = Aelx + Be' (D.3)
and for group 2:
2(x) = Ce2x + De 2x + Sll(x); (D.4)
150
151
where:
2 ZRD
1 = (D.5)
D1
K 2 =__(D.6)
D2
and:
S Z = R 22 (D.7)
D2 ~C2 21
Adopting the approach of Chapter II, we assume that the fast flux
decays as it would in the infinite medium unperturbed by the void at a,
allowing us to set B = O. Then the thermal flux is given by:
A2(X) = Ce 2X + DeK2X + SiAeKlx, (DO8)
where the constant A is determined by the source strength, qo. Another
constant is eliminated by a boundary condition on the thermal flux at
x = O. For the sake of argument assume that:
() = 2(0) tr2 d/2 =0 (Do9)
j2(o) = 6 ax
which specifies the thermal flux within one arbitrary constant, C;giving:
1 + 2tr2K2 Kz2x
2(X) = C eK2X  e
(l  3 trz2) f
+ S1A  i1 K2 o (D.10)
152
Applying the realistic boundary condition at x = a:
jz(a)(a) (a) dx za jA (D.ll)
the constant becomes:
2  a + 32 ~tr2K ( 2 KZa
4jASA  3 tr2 e a  (12/3 tr2) + ~tr2 2 e
Ci 3=/3 Itr2 3
2
(  tr23 ee K2a 2 (+ trK2 2 e(2a
(l3 2tr22) (D.12)
Applying the pseudo boundary condition:
B(a+d) = 4jA' (D.13)
the constant becomes:
42rl(a+d
4jASA] e la d 3tr2rL eK2(a+d)
iiAe (~(1 2/3 ~tr2K2)
eK2(a+d) _ (1 + tr2K2 eK2(a+d) (D.14)
dt nc e i n tr2K2 he
Inasmuch as we have used the uncorrected diffusion theory extrapolation
distance in the analysis:
~ (a+d) + K+ 2 a +
e<~ = e 37 e e tr3 tr (D.15)
Then for ~tra << 1:
e+~(a+d) _ ea e~ 5a~ trK) (D.16)
1553
It follows that:*
Ci  Cii (Do17)
*For light watei 2/3 ~tr2z2 = 0095 and 2/3 ~tr2a1 = ~O52~
APPENDIX E
DUCT MOUTH CONTRIBUTION TO THE RADIAL CURRENT AT THE WALL
Consider the contribution to the partial current in the radial direction at (R,zl) from the mouth of a cylindrical void. The number of
neutrons per second passing through dS1 emanating from the mouth is
given by:
j (Rz1)dS1 = SdSz(~(r~0n)dO o (Eol)
SA
The angular flux at the mouth, neglecting source contributions, is expressed by:
p(r,O, Q) 1 "O7(rn)12 Vo r (Eo2)
Assuming polar symmetry:
Vo lr = l (Es3)
Referring to Figure Eol:
dSA = rdrdp, (Eo4)
dS1 = 2Rdgdz, (Eo5)
and:
Sh2p 1a. (E.6)
154
i~! Z155
Figure E.1. Duct configuration for calculation
of mouth.contribution to wall radial current.
From geometrical considerations:
n =r cos 29 + y sin 29, (E7)
p = R sin 2, (E.8)
and'
h = r + R cos 2. (E.9)
Accordingly:
dS1 ~ dS1
nd = Sn) S (R+r cos 2, (E.7)
4r2 ~
156
and:
2 2 2 h2 2 R2 2
= zl2+p + = z + 2rR cos 2 + r (Ell)
Using Eqs. (E.3) and (E.6), the angular flux takes on the form:
0(r,0,Q) & o(r, ) —  z (E12)
and the current is obtained by substituting the above expressions into
Eq. (E.1):
j (Rzl) = 2dRdzdr d d rdr 0
1j!(E13)
_ X 0L1 O h aL1]zi 2Rdz (R+r cos 2O)
6 zr 6az rr tr
0 0
Assume that 'i is negligible in comparison with 0o(r,O) and
ar r
_ , which are taken to be constant over the entire mouth. Then Eq.
az r
(E.o3) reduces to:
R T/2
+ (R,zl) = (O) rdr dO (Rtr Qos 2)
j (Rz): o(r 0 [zl+R2+r2+2rR cos 20]2
0 0
(Eo14)
 r rr d0 [z1+R2+r2+2rR cos 205 o2

Or equivalently:
R R
+ (R,zl) = Q(0) rd rdri1(r,R,zl)zl rdrI2(rRz )
(Eo 5)
157
The integration to obtain I1 is easily performed, giving:
2 2 r2
Il(r,R,zl) = R (zl+R r) ) (E.16)
2 a( )/+R +r2 4r
The second integration is considerably more difficult, but finally reduces to:
I2 4 (R+rR2zl) 2 K( 2)
I2(Tr,R,zj) =96(rR)5/2(l )2 (_q2 )R+r) cI (R+3r) i +2
 2(R+r)q42(2R+r)q+2r] E(2)} (E.17)
where:
q~ 4rR
2 2 )2, (E.18)
and K and E are the complete elliptic integrals of the first and second
kinds.15
The first integral with respect to r, giving the isotropic contribution to the current at (R,zl), can be done analytically, yielding:
R i z1
4R2
r rdrIl(rRnzl)  4zR 2 + *.
The second integral, representing the contribution of the z component of
the gradient to the current, must be done numerically. Combining terms,
we write the final form for the contribution to the partial current at
(R,z1) from the mouth:
2
1+ f
4R2 2' Z1
+ Rz1 1~R Z ___ ~ o rdrI2(rR,zl)
APPENDIX F
NORMALIZATION FOR TIME VARYING FLUX
Consider the normalization of an experiment conducted in a time
varying flux, /(t). For purposes of discussion assume that the appropriate integrations over energy have been performed.
The time rate of change of the concentration of a species i is expressed as:
dCi i
dC1 Ri(t)  Xci (F.1)
dt
where R(t) is the time dependent reaction rate:
Ri(t) = Zi~(t). (F.2)
The activity at time t, the termination of an irradiation, is obtained
by solving Eq. (F.1), and is given by:
t
xi(t t)
Ai(t) = Ai dt'Ri(t' )ek (t't) = ii dt')(tt)e
~~.0 0(F.3)
We define the time averaged value of the flux:
t
<tf >(t')dt' (F.4)
Setting the time dependent flux equal to the product of a steady state
flux magnitude and a function of time:
/(t) = Of(t), (F.5)
158
159
Eq. (F.3) becomes:
A (t) = A fdt'f(t')e (tt) (F.6)
We desire to obtain the average value of the flux measured by a detector of species a normalized to the average flux measured at another
position and at the same time by a detector of species b. Then:
1 ~a ~ f(t')dt'
< / >a <$a=l (F.7)
a 1 $b f(t')dt'
t
and substituting from Eq. (F.6):
A (t)\a~a dtlf(t')e
< ________pa_______. (~F.8)
N tb
Ab (t ) Xb> dtlf~t~(t,_t)
0o
Since we are interested only in relative normalized quantities:
t~?t'
< N Ab(t) be ( a b)t (F.9)
< a A a (t) e~ Xb) dt'f(t' )e (F9
For the trivial case of a and b identical Eq. (F.9) reduces to:
a Aa(t)
<'N Ab(t) (F.)
APPENDIX G
COMPUTER PROGRAM FOR EVALUATION OF R
a / 1r
The Fortran program for the computation of R along the duct
wall is listed here. The calculation assumes separation of variables
in the current balance relationship and evaluates Eq. (2.50) (revised to
include the terms in Expression (2X54) to account for the duct mouth
contribution) at discrete points along the duct wall for an unperturbed
axial flux distribution of a designated shape. Exponential, cosine,
linear, and flat flux shapes are permissible. The input data are of the
following form (Format I2, 8F8.4):
N(1) = 1 Linear flux shape
= 2 Flat flux shape
= 3 Cosine flux shape
= 4 Exponential flux shape
F()
F(2)
F(53)F() 01 Parameters for flux shape
F(4)
F(5) = zo Position of duct mouth (cm)
F(6) = R Duct Radius
F(7) = ~ Neutron mean free path in medium
F(8) = 5 Interval of computation
160
161
C WALL INTERFACE SEPARABLE CALCULATION  WRITTEN U OF M t 1962
$COMPILE FORTRAN,PRINT OBJECT.PUNCH OBJECTEXECUTEDUMP BOUND
C COMPUTATION OF SEPARABLE BOUNDARY CONDITION FOR TUBE IN REFLECTOR 1
DIMENSION TEG(3)9 SUM(3)9 GET(3),TEG1(3) 2
C DEFINE FUNCTIONS USED IN PROGRAM 3
PHI1F(ZZ) = AA+A*ZZ 4
PHI3F(ZZ) = AA*COS(A*ZZ+B) 5
PHI4F(ZZ) = AA*FXP(A*ZZ) + BB*EXP(B*ZZ) 6
DPHI3F(ZZ) = AA*A*SIN(A*ZZ+B) 7
DPHI4F(ZZ) = AA*A*EXP(A*ZZ) + BB*B*EXP(B*ZZ) 8
PAINF(ZZ) = 1*O/SQRT(1O+(Z2ZZ)*(Z2ZZ)/(4O0*R*R)) 9
QUEERF(Y) = SQRT(4.0*R*Y/((Z2ZO)*(Z2ZO)+(R+Y)*(R+Y))) 10
C READ AND PRINT INPUT DATA 11
C NOTE. INDIC FOR FLUX. 1lLINEAR. 2'FLAT, 3'COS, 4'EXP 12
1 READ INPUT TAPE 7.2tINDICAAABBBtZOR.AMBDAtDECRE 13
2 FORMAT(I2,8F8.4) 14
WRITE OUTPUT TAPE 6,3 15
3 FORMAT(66H1COMPUTATION OF SEPARABLE BOUNDARY CONDITION FOR TUBE IN 114
1 REFLECTOR) 115
WRITE OUTPUT TAPE 694eAAgABBPB 16
4 FORMAT(26HOPARAMETERS FOR FLUX SHAPE/4HOAA=F8B.44H A=F8B4,4H BB=F 17
18*494H B=F8*4) 18
WRITE OUTPUT TAPE 695*.Z 19
5 FORMAT(23HOPOSITION OF ZO IN CM,=F8,4) 20
WRITE OUTPUT TAPE 6,6,R 21
6 FORMAT(23HORADIUS OF TUBE IN CM*=F8~4) 22
WRITE OUTPUT TA.PE 6.79 AMBDA, DECRE 23
7 FORMAT(32HOTRANSPORT M.F6Po LAMBDA IN CM.=F8*4/ 24
127HOINCREMENT FOR INTEGRATION=F8.4) 25
C SET Z2,COMPUTE PHIZ2, PHIZO, DPHIZOt'PRINT FLUX SHAPE 26
Z2=ZO+DECRE 27
8 IF(INDIC3) 9,1011O 28
9 IF(INDIC1) 12,12t13 29
10 PHIZ2=PHI3F(Z2) 30
PHIZO=PHI3F(Z0) 31
DPHIZO=DPHI3F(ZO) 32
IF(Z2(ZO+DECRE)) 14,14,24 33
11 PHIZ2=PHI4F(Z2) 34
PHIZO=PHI4F(ZO) 35
DPHIZO=DPHI4F(ZO) 36
IF(Z2(ZO+DECRE)) 15,15,24 37
12 PHIZ2=PHI1F(Z2) 38
PHIZO=PHILF(ZO) 39
DPHIZO=A 40
IF (Z2(ZO+DECRE)) 16,16924 41
13 PHIZ2=1.0 42
PHIZO=1.O 43
DPHIZO=OO 44
IF(Z2(ZO+DECRE)) 17,17.24 45
14 WRITE OUTPUT TAPE 6918 46
GO TO 22 47
15 WRITE OUTPUT TAPE 6,19 48
GO TO 22 49
16 WRITE OUTPUT TAPE 6.20 50
GO TO 22 51
17 WRITE OUTPUT TAPE 6921 52
18 FORMAT(24HOFLUX IS COSINE IN SHAPE) 53
19 FORMAT(29HOFLUX IS EXPONENTIAL IN SHAPE) 54
20 FORMAT(24HOFLUX IS LINEAR IN SHAPE) 55
21 FORMAT(22HOFLUX IS FLAT IN SHAPE) 56
22 WRITE OUTPUT TAPE 6923 57
23 FORMAT(72HO Z2 DENOM Fl F2 F3 F4 L.D 58
IPHI/PHI DPHI/PHI) 59
C BEGIN COMPUTATION OF DENOMFlF2 60
24 Zl=ZO 61
J=3 62
DO 25 I 1,3 63
TE'G( I) =OO 64
25 CONTINUE 65
GO TO 27 66
26 Z1=Z1+DECRE 67
27 DO 28 I=1.3 68
GET( I)=TEG( I) 69
28 CONTINUE 70
162
29 IF(INDIC3) 30,31,32 71
30 IF(INDIC1) 33,33,34 72
31 PHIZ1=PHI3F(Z1) 73
DPHIZ1=DPHI3F(Z1) 74
GO TO 35 75
32 PHIZ1=PHI4F(Z1) 76
DPHIZ1=DPHI4F(Z1) 77
GO TO 35 78
33 PHIZI=PHI1F(Z1) 79
DPHIZ1=A 80
GO TO 35 81
34 PHIZ1=1iO 82
J=2 83
35 X=PAINF(Z1) 84
CALL IEFl(15707XEFFG) 85
IF(G1.0) 102,102a100 86
100 WRITE OUTPUT TAPE 6,101,Z1 87
101 FORMAT(47H ELLIPTIC INT. OUTSIDE OF RANGE *Zl=F8*4) 88
E=l1O 188
F=10.O 288
102 TEG(1) = PHIZ1*((2O0*(1OX*X*X*X)+3.0*(2O0X*X))*E(l.oX*X) 89
1*(8.0+X*X)*F)/(PHIZ2*X) 90
TEG(2) = PHIZ1*(2.0(2.O+X*X)*SQRT(1*0X*X))/PHIZ2 91
IF(INDIC2) 36,37,36 92
36 TEG(3) = (DPHIZ1*(Z2Z1)*((2e0+X*X)*F2.0*(11*+X*X)*E))/ 93
1(PHIZ2*SQRT((Z2Z1)*(Z2Z1)+4*O*R*R)) 94
37 IF(Z1ZO) 38938,40 95
38 DO 39 1=1,3 96
TEG1(I) = TEG(I) 97
SUM(I) = 0*50*TEG(I) 98
39 CONTINUE 100
GO TO 42 99
40 DO 41 I=1,3 101
SUM(I) = SUM(I) + TEG(I) 102
41 CONTINUE 103
42 DO 44 I=1.J 104
IF(ABSF(TEG(I)IA8SF(GET(I))f 43,26,26 105
43 I.F (A&ISF(TEG(I))O0005*ABSF( tEG( f )) 44,26,26 106
44 CONTINUE 107
DEINOM.= 0:666.*.(l1.O+DECRE*SUM(1)/(6.2832*R)) 108
F1 = (10DECRE*SUM(2)/(4.0*R))/DENOM 109
 2 =.DECRE*SUM03)/(9*4248*R*DENOM) 110
C COMPUTE F3 AND F4 111
F3 = (PHIZO/(2.0*R*PHIZ2*DENOM)* (( IZ2ZO)*(Z2ZO)+2.0*R*R)/ 112
1SQRTL4._Q*tRL*R+IZ2ZO)*(Z2ZO) )(Z2ZO)) 113
DEL = R/20.0 114
SUM = 0.0 115
Y=O.O 116
TEG=O.O 117
IF (INDIC2) 47,45,47 118
47 Y=Y+DEL 119
Q=QUEERF(Y) 120
CALL IEF1(1.5707, Q.EFtG) 121
IF (G1.O) 105.105,103 122
103 WRITE OUTPUT TAPE 6,104.Y 123
104 FORMAT(48H ELLIPTIG INT. OUTSIDE OF RANGE,RAD=F8.4) 124
105 TEG = (Q*Q*Q/(96O0*SQRT(Y*Y*Y*Y*Y*R*R*R*R*R)*(1lOQ*Q)*(l.OQ*Q))) 125
1*Y*(((R+Y)*Q*Q*Q*Q(R+3.0*Y)*Q*Q+2.0*Y)*F(2.O*(R+Y)*Q*Q*Q*Q2.0*
2(2*0*+Y)*Q*Q+2.O*Y)*E) 127
IF ((RY)0.O0001) 45,45,46 128
46 SUM = SUM + TEG 129
GO TO 47 130'45 SUM = SUM+O*50*TEG 131
F4 = ( 4O*(Z2ZO)*(Z2ZO)*DPHIZO*DEL*SUM)/(3e1416*PHIZ2*DENOM) 132
C COMPUTE FINAL VALUE AND PRINT 133
BOUND = F1 + AMBDA*F2  F3 + AMBDA*F4 134
GRAD = BOUND/AMBDA 135
WRITE OUTPUT TAPE 6,48,Z2,DENOM,F1,F2,F3,F4,BOUND GRAD 136
48 FORMAT (8F9.4) 137
IF (INDIC2) 50,49,50 138
49 IF ((Z2ZO)12.0) 51.52.52 139
50 IF (PHIZ2O0.25*PHIZO) 52,51,51 140
51 Z2 = Z2 + DECRE 141
GO TO 8 142
52 GO TO 1 143
END 144
$DATA
REFERENCES
1. Egelstaff, P. A. (editor), "Tailored Neutron Beams," Special Issue, J. Nucl. Energy, Parts A & B, 17, Nos. 4/5 (1963).
2. Carter, R. S and Jablonski, F E., "A Study of Split Cores for
Research Reactors," Nucl, Sci. Eng., 5, 257 (1959).
3. Carter, R. S., Landon, H. H., and Muehlhause, C. O., "The National Bureau of Standards Reactor Facility," Trans. Amo Nucl.
Soc., 5, No. 2, 423 (Nov. 1962).
4. Kouts, H.,'"BeamTube Design for the HighFlux Beam Reactor,"
see Ref. 1.
5. Behrens, D. J., "The Effect of Holes in a Reacting Material
on the Range of Neutrons," Proc. Phys. Soc. (London), 62, Part
10, No. 358A, 607 (1949).
6. Reynolds, A, B.,et al., "Reactivity Effects of Large Voids in
the Reflector of a LightWaterModerated andReflected Reactor,
Nucl. Sci. Eng., 7, 1 (1960).
7. Baraff, G. A., Murray, R o L., and Menius, A. C., "Perturbation
Integrals for TwoGroup Calculations and Application to Reflector Ducts," Nucl. Sci. Eng., 4, 623 (1958).
8, Spinney, K. T., "Radiation Streaming Through DuctsA Survey of
the Present Situation," Trans. Am, Nuclo Soc,, 5, No. 2, 390
(Nov. 1962).
9. Simon, A., and Clifford, C. E,, "The Attenuation of Neutrons
by Air Ducts in Shields," Nucl, Sci. Eng., 1, 156 (1956).
10. Piercey, D. C., "The Transmission of Thermal Neutrons Along Air
Filled Ducts in Water," AEEWR70 (1962).
11. Davison, B., "Neutron Transport Theory," Oxford at the Clarendon Press (1957).
12, Case, K. M., DeHoffman, F., and Placzek, G., "Introduction to
the Theory of Neutron Diffusion," Vol. I, Los Alamos Scientific
Laboratory (1953),
164
REFERENCES (Continued)
13. Glasstone, S., and Edlund, M. C., "The Elements of Nuclear Reactor
Theory," D. Van Nostrand Company, Inc. (1952).
14. Zimmerman, E. L., "Boundary Values for the Inner Radius of a Cylindrical Annular Reactor," ORNL2484 (1958).
15. Jahnke, E. and Emde, F., "Tables of Functions," Dover Publications (1945).
16. "Research Reactors," Chapter 2, U. S. Atomic Energy Commission,
McGrawHill Book Company, Inc. (1955).
17. Shapiro, J. L., et al., "Initial Calibration of the Ford Nuclear
Reactor," MMPP110l (1958 ).
18. Hughes, D. J., and Schwartz, R. B., "Neutron Cross Sections,"
BNL325, Second Edition (1958),
19. Macklin, R. L, and Pomerance, H. S., "Resonance Capture Integrals," First Intern. Conf. Peaceful Uses of Atomic Energy,
Geneva, 1955P/833.
20. Dayton, I. E., and Pettus, W. G., "Effective Cadmium Cutoff Energy," Nucleonics, 15, No. 12, 86 (1957).
21. Dalton, G. R., private communication (1963).
22. Reynolds, A. B., "Reactivity Effects of Large Voids in the Reflector of a LightWaterModerated and Reflected Reactor," thesis,
AECU4391 (1959)5
235 Dalton, G. R., and Osborn, R. K., "Flux Perturbations by Thermal
Neutron Detectors," Nucl. Sci. Eng., 9, 198 (1961).
24~ Fastrup, B., "On Cadmium Ratio Measurements and Their Interpretation in Relation to Reactor Spectra," Riso' Report No, 11 (1959).
25. Walton, R. B., et al.,, "Measurements of Neutron Spectra in Water,
Polyethylene, and Zirconium Hydride," Proc. Symp. on Inelastic
Scattering of Neutrons, IAEA (1960).
26. Schmid, L. Co, and Stinson, W. Po, "Calibration of Lutetium for
Measurements of Effective Neutron Temperatures," Nucl. Sci, Enge
Letters, 7, 477 (1960).
165
REFERENCES (Continued)
27. Klahr, C,, "Limitations of Multigroup Calculations," Nucl, Scio
Eng., 1, 253 (1956).
28. Cantwell, M., and Goldsmith, M,'"The Effect on Calculated Clean
Critical Activation Shapes of the Use of Transport Approximations
in the Fast Groups," Nucl. Sci, Eng., 12, 490 (1962).
29. Donovan, J. L., King, Jo S., and Zweifel, P. F., "A Thermal Neutron Spectrum Measured by a Crystal Spectrometer," Univ. of Mich.,
ORA Report 036713T, Ann Arbor, January, 1963.
30. Brown, H. D., "Neutron Energy Spectra in Water," DP64 (1956),
31. Goldsmith, M,, et al., "Theoretical Analysis of Highly Enriched
Light Water Moderated Critical Assemblies," Second Intern. Conf.
Peaceful Uses of Atomic Energy, Geneva, 1958 P/2376.
32. Daniels, Eo, forthcoming master's thesis, The University of Michigan (1963).
33. Feiner, Fo., et al., "Precise Criticality Determinations in the
Solid Homogeneous Assembly," Trans. Am. Nucl, Soc., 5 No. 2,
344 (Nov. 1962).
34. Bullock, J. B., "Calculation of the Maximum Fuel Cladding Temperatures for Two Megawatt Operation of the Ford Nuclear Reactor,"
unpublished, Phoenix Memorial Laboratory Memo Report No. 1, The
University of Michigan (1962).
35. "University of Michigan Executive System for the IBM7090 Computer," unpublished, The University of Michigan (1963).
36. Flatt, H. P., "The Fog OneDimensional Neutron Diffusion Equation Codes," unpublished, North American Aviation Corp. (1961).
37. Tobias, M. L., and Fowler, T. B., "The Twenty Grand Program for
the Numerical Solution of FewGroup Neutron Diffusion Equations
in Two Dimensions," ORNL3200 (1962).
38. Shudde, R. H,, "Tempest II," unpublished, North American Aviation
Corp. (1960),
39. Amster, Ho, and Suarez, R., "The Calculation of Thermal Constants
Averaged Over a WignerWilkins Flux Spectrum; Description of the
Sofocate Code," WAPDTM39 (1957).
166
REFERENCES (Concluded)
40. McGoff, D, J., "Form, A Fourier Transform Fast Spectrum Code for
the IBM709," NAASRMemo5766 (1960).
41, Bohl, H., Gelbard, E. M., and Ryan, G. H., "Muft4; Fast Neutron
Spectrum Code for the IBM704," WAPDTM72 (1957).
UNIVERSIT OF MICHIGAN
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