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GAMMA,,KRAY STREAMING ALONG DUCTS
IN SHIELDS
By
Ian Roger Terry
ABSTRACT
- The problem of estimating gamma photon dose-rates along
the ducts of nuclear installations has been approached with
varying degrees of success, during the last decade, for a
limited range of photon energies and a limited range of duct
geometries. The following work contains a prediction method
that has been tested over a range of photon energies from
0.66 MeV to 6.13 MeV for mouth sources in ducts consisting
of two and three legs at right angles to each other. The
transport of photons along the duct is expressed in terms of
a multigroup energy reflection probability which is applied to
the centroids of each scattering area. The method is called
the kernel-albedo technique and permits the dose-rate from
multiply scattered radiation to be determined. Existing
experimental data has been supplemented where necessary
and two duct liner materials used, namely concrete and steel.
In all cases the agreements of theory wi th experiment is
excellent.
•
TABLE OF CONTENTS
Page ABSTRACT
1. INTRODUCTION 1
1.1 The Nature of the Problem
2. A COMPARISON OF THE AVAILABLE METHODS FOR CALCULATING GAMMA RADIA-TION TRANSMITTED' THROUGH DUCTS
2.1 Simple Ray Analysis 5 2.2 Single Scatter Methods 8 2.3 The Monte Carlo Method 1 n
2.4 Albedo Methods 12
3. AVAILABILITY OF ALBEDO DATA 15
3.1 Experimental Albedo Determinations 15 3.2 Theoretical Albedo Determinations 16 3.3 Semi-Empirical' Data Representation 20 3.4 Comparison of Experimental and Theoretical
Albedos 24 3.5 Spatial Dependence of the Generalised Albedo 25 3.6 Choice of Albedo Data 29
4. REVIEW OF EXISTING GAMMA-RAY STUDIES 30
4.1 The LeDoux-Chilton Method 32 4.2 Developments by Chapman 33
5. THE MULTISORD METHOD AND ITS SUCCESS IN APPLICATION TO GAMMA-RAY STREAMING PROBLEMS 37
5.1 Mathematical Model 37 5.2 Albedo Data 39 5.3 Overall Accuracy and Sources of Error 46 5.4 A Mouth Source in Two-Legged Ducts 49 5.5 A Mouth Source in Three-:Legged Ducts 54
6. CONCLUSIONS ARISING FROM THESE TESTS 70
7. THE LIDO EXPERIMENT 72
8. CONCLUSIONS AND RECOMMENDATIONS
REFERENCES 79
•
1. INTRODUCTION
1.1 .1.ne iNature or rile t-rooiern
The operational requirements of all nuclear
facilities normally demand their bulk shielding to be pene-
trated by gas;-filled slots and ducts. As radiations can
traverse such utility voids relatively unimpeded, these
irregularities present a problem to the shield designer•.
The radiation intensity at the exit of straight
ducts is usually prohibitively large owing to the high
flux environments contained by shielding. Consequently
attenuation is best affected by the introduction of one
or more right angled bends into the duct so preventing
a straight penetration through the shield. The radiation
collides with the surrounding shield walls so losing energy
until it is absorbed or escapes from the system. Escape
is achieved by multiple reflections along the duct which
forms a preferential transport path. The streaming
effect is an important consideration in any shielding design.
Much attention has been focused on predicting neutron
flux along the ducts of operating power stations. Acti-
vation gamma dose-rates are usually determined by the
neutron distribution in the region of interest. The
streaming of gamma rays in these circumstances has
usually been treated in a crude manner to give an upper
limit to the dose-rate estimates. This approach is valid
in typical mixed fields (for example, those found in gas-
cooled reactors) because neutrons have a higher
reflection probability (usually expressed in terms of
albedos) than photons. Gamma-ray streaming assumes
greater importance, however, in the design of shielding
for fuel handling and source caves, and during fuel transfer
°gyrations etc. It then becomes necessary to predict the
migration of photons which penetrate the void by virtue of
multiple scattering at the wall surfaces and give rise to
the predominant dose-rate contribution at the position of
interest.
For neutrons the reflection coefficients or albedos
are usually greatest at low energies and in consequence
the main neutron streaming problem is concerned with the
energy range below about 1 MeV. For these energies
isotropic scattering between the neutrons and wall materials
is usual with the result that the angular distribution of
neutron radiation reflected from a surface does not
depend markedly upon the incident direction. For the gamma-
ray problem, however, the important source energies are
above about 1 MeV and the scattering is predominantly into
the forward direction. As a result the emergent distri-
bution of the reflected gamma radiation becomes a function
of the incident direction and exhibits a marked azimuthal
dependence. The present study is concerned with an in-
vestigation of the importance of the anisotropy of the
•
•
albedo and the determination of a suitable calculation
model for the gamma-ray streaming problem.
Recently the neutron streaming problem has been
solved successfully and utilises the computer program
MULTISORD (5) which contains a kernel-albedo technique
for calculations.
The azimuthal angle dependence of the albedo is
small f or neutrons and is ignored for calculations based upon
kernel-albedo techniques. This approach results in a very
much faster method than the Monte Carlo Albedo technique,
as discussed in Section 2.4, which can make proper allowance
for the azimuthal variation. It would, therefore, be
desirable to implement the same method for gamma-ray
streaming calculations. If necessary a correction for
azimuth can be written into the program to account for
the azimuthal variation of the distributions.
Thus it was logical to investigate the possibilities
of using the MULTISORD Method for gamma-ray streaming
calculations, and the importance of the azimuthal dependence
of the albedo in practical situations encountered in power
reactor design.
To carry out this work it was necessary to:
(i) derive a tabulation of gamma-ray albedo data
suited for use in MULTISORD.
- 3 -
•
(ii) obtain experimental data for comparison purposes.
Published data, suitably manipulated, provides an
albedo tabulation and will be described in Section 3.
Experimental data is obtainable from the literature and
supplemented by independent measurements.
The following work consists of seven sections.
Section 2 examines the techniques and methods at present
used for dose-rate predictions through ducts. Section 3
appraises the availability of gamma-ray albedos. The next
section contains the tesults of previous prediction attempts
and Section 5 describes the application of MULTISORD to
the problem. The final sections present the conclusions
formed from the preceding work and supplement the
previous gamma-ray data associated with the streaming of
radiation along ducts.
•
•
2. A COMPARISON OF THE AVAILABLE METHODS FOR.
CALCULATING GAMMA RADIATION TRANSMITTED
THROUGH DUCTS.
Basic line-of-sight methods were developed originally for
both neutron and gamma streaming problems because:
(1)
for neutrons the complexity of material cross-
sections together with the complexity of the neutron
moderation process necessitated numerous semi-
analytical and empirical formulae.
(ii) for gammas although the fundamental processes
governing gamnia behaviour are fully understood (1)
the multiplicity of events when a gamma beam inter-
• acts with matter meant that full analytical predictions
were not possible.
Later methods, developed for more complex geometries and
to give better estimates, were facilitated by the advent of the
electronic computer. These methods outlined below are currently
used in shield design.
2.1. Simple Ray Analysis
The total dose at a point may be the sum of two
components, as is illustrated in Figure 1. Consideration of
the current distribution and duct geometry allows the
of-sight component to be determined quite easily. The
second component is radiation that has penetrated the duct walls
and has either travelled uncollided to the dose point or been
5
NE,
tine of sigh radiat ion
swag Plane h
•
F IG I SIMPLE RAY ANALYSIS SCHEME
•
scattered into the appropriate direction. For this corn-
ponent ray analysis makes the assumption that:
(1)
the straight line joining a source point to the
dose point is the path taken by the latter com-
ponent so representing it as a ray.
(ii) scattered radiation may be expressed in the form
of a build-up factor appropriate to each material
thickness traversed by the ray.
(iii) the attenuation is a property only of the material
encountered along the ray's path.
Such assumptiOns allow the flux at a point P due
to a point source of strength Np to be expressed by
Np C6r 4 —A-r2 K
K is an attenuation kernel due to the variation material
such that
( 1)
K = B (ti5 t2,---)exp. ) li
(2)
where r = magnitude of the ray path
= material build-up factor • 1i = mean free path in the ith material
ti = straight line path through the ith material
By summing the contributions from each source point an
estimate may be obtained for the total flux and dose. For
simple configurations as in Figure 1 the method can give
adequate pr edictions. Most practical ducts, as mentioned
in 1.1. are stepped. Complex configurations of this type
invalidate assumption (iii) above, for K is strongly
dependent on geometry in such cases.
2.2 Single Scatter Methods
These methods are commonly referred to in the
literature as the /1.-7 method and Stephenson's Method.
Each makes the assumption that photons undergo a single
scatter within the medium suffering an energy degradation
given by the well known Compton expression
Eo (1 - Cos er
-1
(3)
Eo m c2 0
• Where E, E0 are the Energies of deflection and incidence
respectively.
• Where the angle of scatter.
The duct is considered to be partitioned into scattering
areas each contributing to the dose point. The scatter
occurs at the centroid of each area.
The basis of the 247 method is that all photons are
• isotropically scattered at the surface of the material. It
makes no allowance for photon absorption. The contribution
A N to the flux point from a source strength S scattered
by an elementary areae. A is
4N = S Cos e'paA
(1+7 r l r2)2 (4)
- 8 -
d 0-- d dts.
r12 r22 (),( 1 +//r- 2 Cos61-2
+ Cos 3.015 x 10
23 S cos
A N.= (5)
•
Where r1, r2 are the respective distances of the source
and flux point from / A.
Where .6).1 is the angle of incidence of the radiation to A A.
Stephenson tried to allow for both absorption and
• anisotropic scattering by formulising an expression that
contained an absorption term and a scattering probability .
term. This expression .for a material of low atomic
number is given by
Where 1 and
2 A L are the macroscopic mass absorption
cross-sections in the Material for incident and emergent
photons respectively.
Where d 0— is the differential Klein-Nishina cross- d
section of the material.
Where is the angle between the emergent photons and 2
the normal to the surface.
Both these expressions may be integrated over the
total scattering area for the total flux spectrum.
The ifImethod will usually tend to overpredict as
no allowance is made for absorption, whilst the assumption
of single scatter in the medium will underpredict in the
latter method. An experiment by Clark et al (2), using a 6o
Co source and concrete medium, established these tendencies
•
9
and obtained dose-rates differing by a factor of approxi-
mately 20 about the experimental point.
Such uncertainties preclude these methods for
serious design studies but are capable of providing initial
estimates for simple geometries. More powerful techniques
arte needed for a full study.
Methods adopted in practice for gamma-ray streaming
consider the transmission of selected photons. These
are scattered to the dose point following a single reflection
and multiple scattering events are thus ignored.
2.3 The Monte Carlo Method
Monte Carlo techniques, using computers for speed,
produce full solutions to duct streaming problems by an
analogue of the physical processes which occur. Each
photon event is analysed until the initial photon is
absorbed or escapes. Such a single tracking scheme for
each photon allows the determination of both flux and
its associated variance. The accuracy of the solution is
limited by the number of photon histories traced, the
accuracy of the cross-section data and formulae, and
by the representation of physical processes considered.
1 The accuracy varies as approximately N2 , where N
is the number of histories traced, which is directly
proportional to machine time. The design constraints
-10 -
•
•
•
select the appropriate optimum between accuracy and
• . . III 111110
Generally such a technique considers the photoelectric
absorption, pair production, and Compton scattering pro-
cesses. It is usual not to consider Rayleigh scattering
as it may be assumed negligible compared to Compton
scattering. This assumption is valid from consideration
of the respective total cross-sections. Rayleigh
scattering is from all the electrons of the atom,
coherent in the forward direction implying a small total
cross-section. The total Compton cross-section is
very much larger than the total ,Rayleigh cross-section,
even in the middle Z elements for low e nergy photons
where the latter cross-section is maximal.
Secondary gamma radiation sources, Bremsstrahlung
and fluorescence, are not considered in the Monte Carlo
technique and may be assumed to be of negligible effect.
The Monte Carlo Method is the most sophisticated
tool available to the shield designer. Exorbitant comp-
utation is reduced by using acceleration techniques which
place greater importance upon photons moving towards the
regions of interest. This may take the form of a
splitting routine, for example, which ceases to track
photons entering unimportant regions which allows more
time to be alloted to the tracking of photons contributing
to the dose-rates at positions in question.
The use of Monte Carlo methods for radiation
design calculation is generally impracticabletof the geometric
complexity of typical duct layouts or the difficulty in
estimating the importance functions. For example, a
typical attenuation encountered is of degree 105. To obtain
an overall statis\tical accuracy of less than 30% at least
106 histories must be traced, which uses several hours
machine time on an IBM 360/75 computer.
2.'4. Albedo Methods
A simplified albedo technique is usually employed for
streaming calculations and the more rigorous methods are
reduced to the secondary role of providing albedo or
reflection probability data.
The mathematical basis for the method may be
expressed in the following manner.
The transmission of radiation in a void using the
albedo approximation can quite generally be described by a
transport equation of form:
(E,r,) =
S i3(1,E0->E, st_es2),3-o (Ecor,-0-)dEod -Col-
E 2?
+31 ( E,r (6)
Where J ( E ,r 2.11)dEda is the differential current entering the
- 12 -
void in the energy range E to E+dE and direction (1.0__
about-n- at r; 30( Eo ,r ,.115)dEd_Dois the differential
current entering the wall in the energy range E0 to
Eo dEo and direction cul about Si_ at r;
t_71 (E,r )dEd.11. is the initial leakage current of
• quanta which are difussing through the surrounding wall
material and enter the void for the first time at r
with energy in the range E to E+dE and direction d_0._
about For mouth sources this term is zero
except at the mouth.
(r,E0 E In- 0 > -n-)dEdiL is the differenti.al
current albedo which describes the probability of
reflection into energy range E to E + dE and direction
•
d-C1- about SZ for radiation incident with energy E0 and
direction o I at r .
In practice the gamma albedo will also contain
a spatial dependent quality since the reflected photon does
not emerge from the surface at the point of incidence.
This effect, which is discussed in Section 3.5. is
small, however, and is usually ignored. A second pseudo-
spatial effect may also occur when photons are transmitted
between the legs of a duct following penetration through
the wall material. These transmissions are most
important for penetration between adjacent legs when
they are referred to as "corner-lip" effects and are
discussed below in Section 4.1.
- 13 -
Two distinct methods of solving the albedo-transport
equation are in general use. In the first of these, the
Monte. Carlo Albedo method (3) (4) the calculation is
performed in an analogous manner to conventional Monte
Carlo with the differential cross-section data for the
wall material replaced by differential albedo data thus
confining the tracking procedure to the void and wall
surfaces. Detailed albedo data may be used but although
. the method is more rapid than conventional Monte Ca...1o,
it is still too slow at present for practical use. The
second method, the so-called kernel-albedo method, as used
in MULTISORD (5) describes the penetration in terms of
the transmission probabilities between wall areas. MULTI-.
SORD will be described more fully later, but at this
stage it may be useful to illustrate briefly the method to
gain an overall understanding of available methods. As
with the methods of 2.2. the duct is partitioned into
scattering areas but here each area is linked by a
scattering probability and an area to area transmission
expressed as a kernel. The number albedo is written as
the numerical fraction emerging from the incident point on a
semi-infinite_ wall. (Appendices A and 13 contain a full
description of the albedo concept.)
-14-
3. AVAILABILITY OF ALBEDO DATA
The accuracy of an albedo streaming calculation depends
ultimately upon the precision of the differential albedo data
available. The determination of such data which depends upon
five energy and angular variables for each shield material is,
however, a formidable task which can only be performed for a
very restricted range of the variables. In order to reduce
the amount of computation involved it has become common
practice to determine integral quantities such as the emergent
angular distribution of the dose or the angular distribution
averaged over all emergent energies.
A survey of available data was therefore carried out to
obtain albedo values for the present work.
3.1. Experimental Albedo Determinations
Reliable experimental data are difficult to obtain for
two reasons:
(i) High energy monoenergetic sources are required to
give values over a wide energy range and it is diffi-
cult to obtain suitable sources above 1.5 MeV;
-A large amount of shielding is necessary to reduce
the background and collimate the incident beam and
the accuracy of the published experiments has often
been impaired by so-called "bad geometry".
- 15 -
The most comprehensive sets of experimental data
avanaote are those or teyn and Anctrews (o), 1:3aarli (7)
Clifford (8) and Haggmark et al (9). These are briefly
summarised in Table 1 of which Haggmark's results are
the most comprehensive.
Difficulties in assessing the experimental corrections
are likely to be the main reason for the discrepancies
between the published experiments which are generallj.
greater than those between different calculations.
Tables 2 and 3 illustrate the disagreement found between
different experimental results.
3.2. Theoretical Albedo Determinations
During the last decade the most comprehensive
theoretical albedo data sets available are those calculated
using Monte Carlo methods. Differential tabulations have
been compiled by Raso (10) (11) (12) and by Davisson and
Beach (13).
Raso calculated the number, energy and dose albedos
for concrete. Incident energies were 0.1, 0.5, 1.0,
2.0, 4.0, 6.0, and 10.0 MeV and the incident angles were
defined by cosine values of 1.0, 0.75, 0.50, 0.15 and 0.10.
In each case 5000 photon histories for each incident energy
and direction were traced and the azimuthal dependence of
the albedo determined.
- 16 -
•
TABLE 1
SUMMARY OF AVAILABLE AL,BEDO EXPERIMENTS
Author Reference Materials Incident Energy ( MeV )
Albedo differential in Albedo Type Quantity No. of
Divisions
Steyn. 6 Graphite 1.25 Normal Incidence Number Aluminium 0.662 Scatter Angle 6 Current Concrete 0.1{.10 Iron Exposur e Nickel - Current Tin Lead Uranium
Baarli 7 Iron 1.25 Emergent Polar Angle Number 0.662 Emergent Energy Current 0 . 410
Expo sur e Current
Clifford 8 Iron 0.662 Incident Angle 3 Dose Concrete Emergent polar angle 6 Flux Lead Emergent azimuthal angle 6
Haggmark 9 Steel 1.25 Incident angle 3 Dose Concrete 0.662 Emergent polar angle 8 Current Aluminium Emergent azimuthal angle 13
TABLE 2
I✓X1r J J 11V11J1V'1'1-!L '1 (.J _city r•Jr L I L U FLiN T ALBEDOS FOR IRON
REFERENCES INCIDENT ENERGY (MeV)
Perpendicular Incidences 0.41 0.66 1.25
32 ' 0.086 0.025
33 0.087 0.026
7 0.062 0.021
Isotropic • Incidence .
17 0.15 0.12
18 0.10 0.077
34- 0.105
- 18 -
•
TABLE 3
EXPERIMENTAL TOTAL EXPOSURE CURRENT ALBEDOS FOR IRON (PERPENDICULAR INCIDENCE)
REFERENCES INCIDENT ENERGY (MeV)
0.41 0.66 1.25
0.045 0.038 0.028 6 0.042 0.055 0.033
TABLE I+ CALCULATED TOTAL CURRENT ALBEDOS FOR
IRON (ISOTROPIC INCIDENCE)
AUTHOR Energy (MeV)
0.2 1.0
Davisson and Beach (13) 0.147 0.105
Raso (10) 0.150 0.105
•
- 19 -
Davisson and Beach have published similar compu-
tations for water, concrete, iron and lead. Their
values contain an azimuthal dependence and are differential
albedos for the current weighted with the dose response
function. The photons were of energies 0.2 , 0.662 ,
• 1.0, 2.5 and 6.13 MeV, incident at polar angles 0°,
220 !iJ 66°, and 88°. The case of a point source
located on the surface was also scanned for these
energies. In each case 20,000 photon histories wer_
traced giving a greater accuracy than the Raso compilation.
The measure of agreement between the two sets is
illustrated in Table 14.1 which compares the albedos for the
a total photon current at two representative energies for
iron.
3.3. Semi-empirical Data Representation
It is apparent that any useful set of differential
albedo data in five dimensional phase space amasses a
considerable amount of information for each material.
Monte Carlo techniques are subject to statistical
•
fluctuations which require smoothing for interpolation
purposes. Chilton, and Huddleston (1L.) derived a semi-
empirical formula that reduces a data set to a number of
simple parameters. The differential dose albedo is then
given by:
- 20 -
QC ( E 0
• e, = c(EcoK(es )1026 + ct(E )
(7) 3
•
•
Where K(es ) = the Klein-Nishina differential energy
scattering cross-section per electron, for a scattering
angle 0 s given by
Cos Bs = Sin Go sin 0 cos /- cos a cos 0o
eo and u are the polar angles of incidence and reflection
respectively. /is the azimuthal angle of reflection.
C and C' are simple adjustable parameters which depend
only upon the energy of the incident radiation (E0 ).
See Figure 2.
This expression is a sum of two components; a
single scatter component and a component for multiple
scattered radiation.
Chilton and Huddleston (1L.) presented some rep-
resentative comparisons between their formula and the
Monte Carlo results of Raso. For the case of normally
incident radiation emerging at glancing angles the fit is
uncertain. The values should be independent of the azimuthal
emergent angle but due to statistical fluctuations this is
not apparent. Raso's values fluctuated from the fitted
value by a factor as great as two. This poor result is
to be expected as the probability of reflection in this
direction is extremely small producing large statistical
uncertainties. At high incident energies the fit is not
- 21 -
0
FIG 2 CHILTON FIT TO THE DAVISSON AND BEACH ALBEDO DATA FOR CONCRETE REF(14)
- 22 -
exact. Low azimuthal angles -produce a deviation from
the formula as great as a factor of 1.5. This again
limits its range of application because at higher energies
the scattering is predominantly into low azimuthal
angles. For ducts containing right angled bends,
important areas are the planar surfaces at the inter-
section which are common to both legs because these areas
directly scatter incident photons from one leg to the
next. As a consequence the albedos appropriate for
inter-leg transmission by these areas assume greater
importance and any general albedo formula must yield
values that are accurate. A representative duct made
of steel with a 6oCo(1.25 MeV effective) point source
at the mouth allowed a comparison of the Chilton and
Huddleston fitted albedos for these important areas with
values interpolated from the data of Davisson and Beach.
The comparison, expressed as a number albedo, is made
in Table 5. For cases B and C the differences between
the fitted and true albedos are not of significance, the
variances of the fitted values (about 60%) enfold the true
values. This is not so for Case A. These angular
parameters are those of normal incidence and glancing
emergence mentioned in the previous paragraph. It is
fortunate that the smallness of this albedo, which gives
rise to the statistical uncertainty, implies a smaller con-
tribution from this area contained by the configuration.
- 23 -
It may be concluded that the Chilton-Huddleston
equation is valid for most practical situations but each
problem should be examined for the extreme cases noted
above and appropriate corrections made if the relevant
contribution is considered to be of importance.
TABLE 5
COMPARISON OF FITTED NUMBER ALBEDO TO. THE TRUE VALUES FOR A RIGHT-ANGLED DUCT
Area Location
Planar Area parallel to the second leg
Planar Area parallel to the first leg
Planar Area for-ming the ceiling and floor
Designation A B C
Fitted Albedo
0.0028 0.018 0.013
True Albedo
0.0011 0.025 0.008
3.4. Comparison of Experimental and Theoretical Albedos
The Chilton and Huddleston formula provides a con-
venient basis for comparing the published values of theo-
retical and experimental gamma-ray albedos. Such a study
has been carried out by Huddleston (15) who compared
Raso's calculations with the experimental results of
— 24- -
Clifford ( 8 ) and a preliminary set from LT .S .N .R .D .L
given by Hurley (lb). The results examined were all
quoted for radiation incident normally upon concrete.
Huddleston obtained a reasonable agreement between
theory and experiment as illustrated in Table 6. It should be noted that the energy range is limited
for this comparison by the experiments. Also the
experimental results themselves are not in such good
agreement as might be expected.
Comparison of the theoretical albedos as cal-
ciliated by Raso and by Davisson and Beach (13) are
presented in Table 7. The Raso results have been
interpolated and the variances estimated by the author.
The agreement between these two sets is superior to
any comparison between experiments.
3.5. Spatial Dependence of the Generalised Albedo
A basic assumption of the albedo concept is that
the photons emerge from a point on the scattering
surface close to the point of entry. Hyodo (17) and
+ Bulatov (18) have performed experiments to investigate
this assumption for gamma rays. Davisson and Beach (13)
and Clifford (8) have also examined it using a Monte Carlo
technique. The simplest experimental case of an isotropic
emitter in contact with a plane scatter was chosen for
this purpose.
- 25 -
• • • • •
TABLE 6
COMPARISON BETWEEN THEORETICAL AND EXPERIMENTAL PARAMETERS IN THE CHILTON AND HUDDLESTON EQUATION OF THE AL,BEDO FOR NORMAL INCIDENCE
PHOTONS ON CONCRETE
rn
AUTHOR
ENERGY (MEV)
0.662 .1■■••••••■•■
1.250
C CI CI
RASO (10)
HURLEY (16)
CLIFFORD (8)
0.0404
0.0669
0.0545
+ 0.0020
+ 0.0023
+ 0.0018
0.0172
0.0091
0.0083
+ 00012
+ 0.0010
+ 0.0012
0.0645
0.0706
+ 0.0022
+ 0.0045
0.0090
0.0123
± 0.00(.7
+ 0.00D9
• •
TABLE 7
PARAMETERS IN THE CHILTON-HUDDLESTON REPRESENTATION OF THE ALBEDO FOR CONCRETE AS CALCULATED BY RASO AND BY DAVISSON AND BEACH
ENERGY (MeV)
RASO (10) DAVISSON AND BEACH (13)
C C' C C,
0.2. 0:0221 + 0.0018 0.0356 + 0.0033 0.0023 + 0.0033 0.0737 + 0.0065
0.662 0.0404 + 0.0019 0.0172 + 0.0030 0.0347 + 0.0050 0.0197 + 0.0035
1.00 0.0547 + 0.0020 0.0111 + 0.0025 - 0.0503 + 0.0056 0.0118 + 0.0025
2.50 0.0980 + 0.0030 0.0077 + 0.0003 0.0999 + 0.0078 0.0051 + 0.0011
6.13 0.154 + 0.006 0.0075 + 0.0003 0.1717 + 0.0103 0.0048 + 0.0005
All workers are agreed that emergence points can be
represented by a distribution of the form: 2 e-ar a
where 'r' = distance between entry point and exit point;
and 'a' is a parameter given in the report of Davisson and
Beach (13) such that Nr = - e-ar Ntotal
In this expression Nr is the number of photons emerging
within a circle of radius 'r' from the entry point. Davisson
and Beach have shown by using this formula that the dis-
tance in which half the back-scattered photons have
emerged is always less than one mean free path at the
source energy in the common shielding materials. Clifford's
results for 0.662 MeV gamma-rays incident upon concrete,
reproduced in Table 8 confirm these findings. For ducts with
dimensions which are large in comparison with the mean free
path of the radiation in the wall material it is therefore valid
to ignore the spatial dependence term.
TABLE 8
THE SPATIAL DEPENDENCE OF 0.662 MeV PHOTONS _INCIDENT:UPON CONCRETE AS A_FUNCTION OF THE
INCIDENT ANGLE (1 m.f.p. = 4.75 ems.)
Fraction back-scattered with surface range>R (8) R cm
= 0° 00 = 30° go = 60° ua = 750 o I
2 4 6 8
0.677 0.452 0.289 0.183
0.737 • 0.481
0.320 0.214
0.822 0.592 0.407 0.283
0.841 0.641 0.469 0.341
- 28-
3.6. Choice of Albedo Data
The only comprehensive albedo data available are the
Carlo compilations obtained by Davisson and Beach (13) and
by RaS0 (10). The two compilations are substantially' in
agreement and the Davisson and Beach values were chosen
for the present work because of their greater statistical
accuracy and the availability of data for materials other than
concrete. In order to reduce the discontinuities and to
assist interpolation between energies and directions the
results were used in the form of Chilton and Huddleston
expression given by Chilton et al (19). The parameters
used in this equation are given in Table 9.
TABLE 9
VALUES OF C-H PARAMETERS FITTED FROM DAVISSON AND BEACH'S RESULTS
E(MeV) C
CONCRETE
0.2 0.0023 + 0.0033 0.0737 + 0.0065 0.662 0.0347 + 0.0050 0.0197 + 0.0035 1.00 0.0503 + 0.0056 0.0118 + 0.0025 2.50 0.0999 + 0.0078 0.0051 + 0.0011 6.13 0.1717 + 0.0103 0.0048 + 0.0005
IRON 0.2 0.0272 + 0.0033 -0.0100 + 0.0062 0.662 0.0430 + 0.0045 0.0063 + 0.0030 1.00 0.0555 + 0.0049 0.0045 + 0.0021 2.50 0.1009 0.0073 0.0044 + 0.0010 6.13 0.1447 + 0.0101 0.0077 + 0.0006
- 29 -
4. REVIEW OF EXISTING GAMMA-RAY STREAMING
STUDIES
The emphasis in most of the existing work on gamma-ray
streaming, which has been carried out in the U.S.A. under
Civil Defence Contracts, has been concerned with the design of
shelter entrance-ways. It has therefore concentrated upon
low energy sources below about 1.3 MeV situated at the mouth
of the ducts ; and integral quantities such as dose have been
calculated and measured. For convenience a point source at
the centre of the duct mouth was usually chosen rather than a
distributed source. The available experimental results are sum-
marised in Table 10.
In . only one instance was a source energy used greater than
that of 60Co. This was in the work of Terrell (20) who used
a 24Na source which produces two photons per disintegration
having energies of 1.368 MeV and 2.75 MeV. Terrell (21) has
also carried out the only published experiment into three-legged
ducts which included both a U-shaped and a Z-shaped configuration.
Similar work in a three-legged system has been performed by
Chapman and Grant (22) but the report describing this work is
so far unobtainable in this country. The theoretical methods of
calculation which have been developed reflect this emphasis and
often amount to little more than fitting procedures. It is,
however, useful to survey briefly the more important of those
methods, especially that of LeDoux and Chilton (23) which
- 30 -
•
•
TABLE 10
SUMMARY OF EXPERIMENTAL DUCT MEASUREMENTS
1 CONCRETE DUCT
Author Ref. Source W H Li, L2 Duct Energy W 1 W Shape (MeV).
Terrell 24 0.662 6 ft. 6 ft. 2.0 .17 r - 1.25 6 ft. 6 ft. 1.33 .17 t L
1.66 .17 L
2.0 .17 L
Terrell 20 1.368 & 2.75 6 ft. 6 ft. 1.66 .17 L
2.17 3.17 L
2.84 .17 L
Terrell 21 0.662 6 ft. 6 ft. 2.17 2.33 1.67 2 '
6 ft. 6 ft. 2.17 2.33 1.67 U 1 ft. 1 ft. 3.5 4.0 3.5 Z
1 ft. 1 ft. 3.5 4.0 3.5 U
1.25 6 ft. 6 ft. 2.17 2.33 1.67 2
6 ft. 6 ft. 2.17 2.33 1.67 U 1 ft. 1 ft. 3.5 4.0 3 .5 Z
1 ft. 1 ft. 3.5 4.0 3.5 U
Eisenhauer 27 1.25 11.1 in. 7.56 in. 3.45 3.45 L
Chapman 36 0.410 3 ft. 3 ft. 2.0 2.67 L, 11 in. 11 in. 4.1253.27 L
Chapman 32 0.662 3 ft. 3 ft. 2.0 2.67 L 2.5 2.5 L
1.25 3 ft. 3 ft. 2.0 2.33 L 2.5 2.5 L
Green 37 1 .25 , 11 in. 11 in. 1.9 3.68 L 3.58 3.68 L
STEEL DUCT
Chapman 38 0.41 11 in. 11 in. 4.13 .68 L 0.662 11 in. 11 in. 4.13 .27 L 1.25 11 in. 11 in. 4.13 .68 L
31 - •
introduces the concepts of the corner-lip effects. Early theo-
retical albedo methods were attempted by Terrell (24) and
Ingold (25) for straight ducts but their accuracy was poor.
More recently. Chapman (26) has extended the I..eDoux-Chilton
method and removed many of its irregularities and his work
• represents the limit of a simple ray analysis model.
4.1. The LeDoux-Chilton Method
LeDoux and Chilton constructed an analytical
technique to compare their calculated values with the
measured values of dose-rate. The method consisted of
summing the dose contributions from a number of scattering
areas assuming single scattering; They compared their
theoretical results with the measurements of Terrell (24)
and Eisenhauer (27). The comparison was made for a
cobalt source (1.25 MeV effective energy) located at the
mouth of a two-legged duct with a square cross-section.
They obtained good agreement using isotropic differential
albedos. The more exact cosine distribution for differential
albedo gave a less favourable agreement with experiment (23).
These results were fortuitous, however, and it was shown
• in later work that the apparent agreement was due to a
cancellation of errors (28) (25). The use of isotropic (416(1.,
differentialn this situation is incorrect, since they were
obtained from the Berger and Raso (11) data which is
based upon reflected photon current and not flux.
- 32 -
The method included an allowance for photon trans-
mission across the inside corner--lips formed by bends in
the ducts. Two effects were considered:
Direct penetration across the lip by source
radiation which contributes to the incident current
along wall areas in the second leg (LeDoux and
Chilton proposed that a cut-off path length through
the corner of 1 m.f.p. should be taken);
(ii) In-scatter of radiation by the lip which effectively
gives rise to an isotropic line source of radiation
located at the lip.
The importance of these effects to dose-rate measure-
ments in the second leg of a duct is typically less than 20%
and varies approximately in the ratio of the duct height to
cross-sectional area.
L..2. Developments by Chapman
Chapman (26) programmed the LeDoux-Chilton method
and included allowance for:
O _multiple scattering in the corner-lip (LeDoux and
Chilton assumed single scattering at this position);
(ii) double scattering along a two-legged duct.
Chapman found that the former effect could contribute
up to 12% of the total dose-rate and the latter as much
- 33 -
•
as 30%. His technique improved the accuracy of the
results which are tabulated in Table 11, and it appears
that his values compare well with the experiments performed.
by Terrell, Eisenhauer, Green and also with his own set of
measurements. In most cases his agreement is excellent
and within + 40% over a wide range of geometries for two-
legged ducts of square cross-section. Chapman also
applied this technique to his measurements in steel and
obtained agreement to within 20%. These findings fk r
steel were later confirmed by Monte Carlo techniques (29).
(Table 12).
The disadvantage of Chapman's method is that it
a employed several different sub-programmes and was
restricted to two-legged ducts. Although the albedo
representation was improved, and some allowance made
for azimuthal variation, the method did not take proper
account of multiple scattering effects which become more
important in ducts with three or more legs.
-34+-
• TABLE 11
SUMMARY OF THE CHAPMAN CODE PREDICTIONS
Source W LW1
L2 Rate (rnr/hr) To Difference
Ref Calculated Measured
60Co 11 in. 1.90 1.65 2.06
87.3 44.5
125 61
-30 -27
7
2.46 27.1 30.5 -11 3.68 8.46 7.31 +16
3.58 2.06 6.17 7.3 -15 2.86 2.61 2.7 -3 3.68 1.30 1.3 0
60 11.111.1 in, 3.54 1.72 17.4 15.6 +11 27 2.79 4.94 3.7 +33 3,51 2.65 2.02 +31
6oCo 12 in. 3.50 2.0 916 852 +8 3.0 317 243 +3o 4.0 140 110 +28
6o 3 ft. 2.0 1.67 20.6 17.5 +18 39 Co 2.0 12.6 12.1 +4 2.34 8.35' 7.1 +18
2.5 1.50 14.5 13.5 +7 1.83 8.42 9.1 -8 2.0 6.70 6.4 +5 2.5 3.79 3.7 +2
6oc 6 ft. 1.33 1.83 15.4 11.8 +31 24 o 2.50 6.56 4.75 +38 3.17 3.47 2.42 +43
1.66 1.83 7.85 7.30 +8 2.50 3.46 2.73 +27 3.17 1.85 1.39 +33
2.0 1.83 4.71 4.56 +3 2.50 2.12 1.79 +18 3.17 1.14 o.935 +21
137 12 in. 3.5 2.0 606 430 +41 21 Cs 3.0 208 132 +58
4.0 90 90 +41 6 ft. 2.17 1.83 36,5 35.5 +3
2.33 19.7 19.6 0
198 3 ft. 2.0 1.83 0.858 0.714 +20 36 u 3.17 0.207 0.186 +11 2.0 1.67 55.8 37.6 +48
2.0 3.4 23.3 +46 2.67 15.7 11.1 +41
11 in. 4.125 2.04 85 46.7 +83
I 2.86 3.27
35.2 24
19.0 13.5
+85 +78
•
•
•
- 35 -
• • • •
TABLE 12
A COMPARISON OF TERRELL'S EXPERIMENTAL RESULTS WITH CHAPMAN'S CALCULATIONS FOR A 24Na SOURCE
W L, 1 W
- L2 W
; Dose Rate (mr/hr)
Difference —
-
Reference Calculated Measured
6 ft. , 1.66 1.83 8.77 6.78 +29 20
2.50 3.84 2.80 +37
3.17 2.05 1.50 +37
2.17 1.83 4.17 3.64 +15
2.50 1.88 1.67 +13
3.17 1.02 0.912 +11
2.84 1.83 2.02 1.94 +9
2.50 0.931 0.828 +12
3.17 0.475 0.462 +3
5. THE MULTISORD METHOD AND ITS SUCCESS IN
APPLICATION TO GAMMA-RAY STREAMING PROBLEMS
MULTISORD is the program currently used in the U.K. (30)
for predicting neutron streaming in multiple-legged slots and is
based on the kernel-albedo method using a simplified represent-
* ation of the albedo. The most recent version of this program )
MULTISORD II (31) extends the capabilities to include rec-.
tangular-section ducts and it is the prototype of this program
which has been used for the present work.
5.1 Mathematical Model
The one group photon current emerging into a void
from any position along the wall'can be represented by a
reduced form of equation 6 such that:
J(r) K(r,rl )3(r9ds o
(8)
•
where describes the reflection probability;
K(r,r') is a kernel describing the current that enters
the wall at r per unit current emitted by an elemental
area ds at r', and J(r/) is the emergent current at r' .
The kernel term may be simplified by the assumption
that the duct can be divided into areas over which the
spatial distribution of current is assumed to be constant.
Thus:
Cyr) K•J - j 1 (9)
where ,3 K1 • • is a kernel describing phe current incident
on area i for uniform current emitted by area j.
j =
- 37 - •
Ja = g1
gt (10)
L' =
where g' g) is the albedo describing the probability
of current incident in energy group gT being reflected
within energy group g.
In practice the .albedo is a function also of the
incident angle. If the energy and angular dependence
functions of the albedo are assumed separable and the
azimuthal dependence negligible, the albedo may be written;
A(go -› g ) =A1( g0 f ([b) f (i)
where f o(110 ) is a function describing the dependence of
of the albedo upon the incident polar angle, and 4
f( )3.) is the emergent angular distribution and
(go g) the energy group dependence.
For neutron calculations fo (lao ) and f(i) are
assumed to obey relationships of the form
f0(/tio) 2(2-n)(3-n) 12 - 5n r-0
-n (12)
f( r ) 2-m 2K 1
1-m
(13) •
where m and n are parameters derived from the differential
albedo data.
The incident angular dependence terms are incorporated
into the kernels of equation 10 to produce a hybrid kernel
• - 38 -
•
used in the MULTISORD calculation. Whilst similar
expressions for fo( )_10 ) and f p) are not essential for
gamma-ray streaming calculation it is convenient to retain
the albedo in this form so that the single program could
perform both the neutron and gamma calculations.
5.2. Albedo Data
For the MULTISORD calculation it is necessary to
process the Davisson and Beach data as expressed by
the Chilton-Huddleston equation into the multigroup form
of equation 11; and an equi-spaced 15 group scheme was
chosen with an upper energy of- 7.5 MeV. The Chilton-
Huddleston representation evaluates the albedo for given
incident energies and 'directions and for a specified
emergent direction. For the present determination it is
therefore necessary to make assumptions concerning the
incident distribution. In order to reduce the computational
requirements it was decided to discretise both the angular
variable and the energy variable, and to assume that the
value at the mean of the interval was representative
of the average over that interval. A fine subdivision
of variables is thus required and the incident and emergent
polar angles were divided into 10 equi-spaced intervals over
the cosine of the angle whilst the azimuthal angle was
divided into 18 equi-spaced intervals.
Since the Chilton-Huddleston equation gives an estimate
of the emergent dose, the number albedo and the emergent
• - 39 -
angular distribution were obtained by making the assumption
that the emergent energy correspond. to photons undergoing
a single Compton scatter event.
The incident angle dependence of the albedo for each
energy group was determined using the derived values of the -n
total number albedo. A fit of the form Au.0 as used r -
for neutron calculations, was initially attempted and found
to be acceptable. This is shown in Fig. 3., where the
logarithm of the albedo for iron is plotted as a function
of 73 o for an energy of 2.75 MeV. The fit is moderately
good and the mean of the indicated n values was chosen.
MULTISORD calculations performed using the upper and
lower limits of n were not found to vary significantly from
calculations using the mean value. The values of n were
also found to be sensibly constant with incident energy.
The emergent energy and angular (azimuthally averaged)
distributions were calculated for a cosine incident distribution.
The emergent angular fit was again expressed in the form
Brja fn and the fit is shown in Fig. 4. The emergent
- energy distribution (g' g) was determined by summing
• the reflected energy (the energy albedo) within energy
group g and dividing by the mean energy of the group.
The group energy albedo was calculated by taking the product
of the energy of the photons that emerge into the group
with the associated number albedo, both characterised by
the angle of scatter, and then summing the products.
- 40
i N
blINN,
7
9 7
imetica5
FIG 3 DETERMINATION OF THE ANGULAR PARAMETER FOR IRON .
• ' I •
. ! ,
. : •
1. ... . ..
...
• • 4 • •
J I 2
CE- ilkticu5 9
FIG J. DETERMINATION OF THE ANGULAR PARAMETER t M f FOR IRON
•
The angular parameters used in the albedo expressions
(12) and (13) are:
Concrete n = 0.66 m = 0.66
n = 0.67 m =0.67
and the energy group variations of the albedos are given
in Tables 13 and 14. With these assumptions the gamma-ray
differential albedo can be described in an identical form
to that used in the neutron streaming problem. A
modification was, however, necessary to the prototype
MULTISORD II program since all existing experimental
data referred to a point source at the mouth of the duct
whilst MULTISORD assumes a distributed mouth source.
The KMOU 1, KMOU 2, FMOU 1 and FMOU 2 routines
of MULTISORD were therefore modified to accept this
source condition.
MULTISORD calculations were performed for all
the published experiments available to the author. Corner
penetrations of the radiation were allowed for track lengths
of less than one mean free path through the corner lip.
Corner "in-scatter" effects cannot be included using
MULTISORD but were - evaluated using the LeDoux-Chilton
technique as programmed by Chapman (26). The effect
of this additional source was negligible. For one case an
investigation was made of the effects of:
(i) varying the albedo;
(ii) omitting the corner penetrations.
- 43 -
• • • •
TABLE 13
-g-
INTERGROUP ALBEDO SCHEME FOR CONCRETE USED IN MULTISORD
INCIDENT ENERGY GROUP
EMERGENT ENERGY GROUP .
7. 2 3 4 5 6 7 8 9 10 11 12 13 :4 15
1. .0004 .0000 .0016 .0000 .0006 .0018 .0006 .0030 .0042 .0080 .000 .2082 2. .0003 .0007 .0008 .0006 .0012 .0011 .0029 .0037 .0082 .0291 .2061 3. .0003 .0000 .0015 .0005 .0012 .0011 .0018 .0049 .0076 .0'g'93 .2040 4. .0003 .0007 .0008 .0017 .0010 .0017 .0044 .0071 .0';:92 .1988 5. .0003 .0014 .0005 .0022 .0014 .0034 .0077 .0;:93 .1897 6. .0003 .0007 .0013 .0016 .0017 .0036 .0072 .0;:80 .1807 7. .0003 .0014 .0016 .0011 .0048 .0069 .0'i:70 .1717 8. .0003 .0019 .0022 .0038 .0071 .061 .1593 9. .0015 .0019 .0032 .0071 .051 .1502
10. .0019 .003). .0068 .046 .1441 11. .0002 .0035 .0066 .033 .1392 12. .0010 .0060 .018 .1417 13. .0019 .0:.81 .1442 14. .0(76 .1765 15. .1741
-__
• •
TABLE 14
Ui
INTERGROUP ALBEDO SCHEME FOR IRON USED IN ..MULTISORD
NCIDENT ENERGY GROUP
EMERGENT ENERGY GROUP
1 2 3 4 5 6 7 8 9 10 11 12 13 :4 15
1. .0002 .0000 .0012 .0000 .0004 .0014 .0006 .0026 .0040 .0088 .01.32 .4006 2. ' .0002 .0006 .0006 .0004 .0010 .0010 .0026 .0034 .0084 .0:74 .3404 3. .0002 .0000 .0012 .0004 .0010 .0010 .0016 .0044 .0074 .0=26 .2730 4. - .0002 .0006 .0006 .0014 .0010 .0016 .0040 .0066 .0:06 .2428 5. .0002 .0012 .0004 .0020 .0012 .0030 .0070 .0'4'88 .2070 6. .0002 .0006 .0012 .0014 .0016 .0032 .0066 .Ti.66 .1830 7. .0002 .0012 .0014 .0010 .0044 .0062 .052 .1638 8. .0002 .0018 .0020 .0036 .0066 .0;42 .1478 9. .0014 .0018 .0030 .0068 .036 .1396
10. .0020 .0034 .0066 .036 .1324 11. .0002 .0036 .0068 .0;:34 .1314 12, .0012 .0066 .0;:30 .1342 13. .0024 .0;:16 .1368 14. .oc86 .1404 15.
i .1006
For each case the effect of assuming pure cosine-emitted
radiation and an albedo which was independent of the incident
angle was calculated by putting m=n=0. The theoretical
predictions so obtained were normalised to the same experi-
mental source strength by using the experimental points
closest to the source.
The results are presented in Figures 5 to 13 and a
compendium of these checks is given in Figure 1L.. Tables
16 to 24 contain a comparison of the experimental points
with MULTISORD. Sections 5.4 and 5.5 discuss these
results.
5.3. Overall Accurac and Sources of Error
The accuracy of any calculational work is normally a
straightforward academic exercise using well defined formulae.
Usually it is an overall error of which the derivation assumes
common-sense margins of error on each physical parameter
contained in a function which describes a physical event.
The same cannot be stated in shielding work. Indeed the
difficulty is in deriving the descriptive functions for the
provision of theoretical predictions that may be checked
against bench-mark data experiments. Shielding methods
describe multiple events, occurring in the realm of atomic
dimensions, by macroscopic formulae which are not always
concise for every situation encountered. It has been
recognised that an accuracy within a factor of two can
be considered good for a shielding prediction which deals
with attenuations of several orders of magnitude.
- 46-
It is improper then to attempt an evaluation of an
overall error in this work. The goodness of the results
can only be tested by comparison with experiment. If the
accuracy achieved is similar to the required accuracy then
it may be stated that the method is good within the
problem bounds.
However, inherent errors may be examined to a
certain degree. These are to be found here in the albedo
fitting expressions, namely the m and n parameters. These
paraineters were stated above as being sensibly constant
with incident energy and the 2.75 MeV values were used in
calculations. Albedo data has been provided for energies
upto 6.25 MeV and a smiliar procedure was employed as that
for fitting 2.75 MeV differential albedo data. The angular
parameters then derived for iron are
= 0.47 m = 0.41.
MULTISORD calculations performed using both sets
of m and n were not significantly different in result. The
high energy results along the first leg were 15% higher and
15% lower along subsequent legs than for the low energy
results. (See Table 15).
In conclusion an overall error may not be determined
easily, but should the achieved accuracy not be sufficient then
some indication has been made as to likely areas for investi-
gation.
- 47 -
• • •
TABLE 15
DOSE-RATES OBTAINED FROM MULTISORD FOR A REPRESENTATIVE STEEL DUCT TO COMPARE THE EFFECT OF DIFFERING ANGULAR PARAMETERS M & N
L,1 = 31.5" L = 33" 2 L3 = 27.511 • w = 101 H =
co
Position point along the duct
MULTISORD
Dose-rate using 2.75 MeV m and n
parameters mr/hr
Dose-rate using 6.-25 MeV m and n
parameters mr/hr
Difference from the result using lower energy parameters
• 1 1.37 1.45 6 2 0.73 0.80 9 3 0.22 0.25 14 4 0.198 0.225 13 5 5.63(-3) .5.19(-3) -3 6 1.82(-3) , 1.64(-3) -10 7 1.12(-3) 9.77(-4) -13 8 - 9.12(-4) 7.90(-4) -13 9 5.17(-4) 4.81(-4) -7
10 5.80(-5) 5.36(-5) -7 11 2.94(-5) 2.61(-5) -11 12 9.03(-6) 7.31(-6) -19 13 6.74(-6) 5.38(-6) -20
N.E. All dose-rates are normalised to a unit source.
5.4. A Mouth Source in Two-Legged Ducts
Reference to Figures 5 and 6 demonstrates that
agreement obtained between experiment and theory is
within + 20% except in the regin,-, of intersections where
the deviation is larger. This is due to assumptions made
• concerning multiple scattering necessary for the appli-
cation of MULTISORD and to the breakdown of the wall
areas chosen for these regions. The agreement can be
considered reasonable throughout the whole duct.
The predictions of Figure 5 are made for the only
case of a duct with a rectangular cross-section which has
been traced in. the literature. Those of Figure 6 for
the highest source energy used.
Figure 7 represents the only comparison in a duct
through a material other than concrete. As may be seen
the agreement is excellent for such a large attenuation
down a steel duct.
-1+9-
•
•
• • 4 • •
TABLE 16
0
COMPARISON OF 'EXPERIMENTAL AND CALCULATED POINTS ALONG THE SECOND LEG OF A TWO-LEGGED CONCRETE DTJCT_
6oCo point source - L =3.28' L2 = 3.281 W = 0.9521 H 0.631
Distance from source (cms)
Experimental Dose Rate
mr/hr
Multisord Difference from Experiment
% *
Multisord m = n = o
Difference from Experiment
e /0
99.90 1155 1155 1155 114.60 142 1096 +700 1111 . 119.80 75 72 -4 42.7 -43 124.70 49 45 -8 28 -43 129.50 33 29 -12 15 -55 139.60 15 13.3 -11 7.1 -53 148.70 8.4 8.1 . -4 3 -64 149.60 7.80 7.78 0 3.52 -55 158.80 4.80 5.2 ' +8 2.2 -54 159.70 4.4 5.0 +14 2.0 -55 168.50 3.4 3.29 +8 1.20 -61 178.60 2.07 2.28 +10 0.77 -63 188.70 1.46 1.70 +16 0.55 -62
* Percent difference = (calculated - measured) 100 measured
Ui
TABLE 17
COMPARISON OF EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS OF A TWO-LEGGED CONCRETE DUCT
24Na point source L,1 = 171 L2 = 191 H = 61 W = 6?
Distance from source (ems)
Experimental Dose Rate
mr/hr
Multisord Difference from Experiment
%
Multisord m = n = o
Difference from Experiment
60.9 18,000 18,000 18,000 91.4 8,760 8,016 • -8 8,130 -7
152.4 3,306 2,904 -12 3.054 -8 243.8 1,332 1,224 -8 1,230 -8 335.3 744 655 -12 648 -13 1+26.7 475 406 -15 396 -17 472.4 399 330 -17 324 -19 518.1 331 274 . -17 267 -19 563.8 328 271 -17 263 -20 609i 5 300 259 •.14 253 .--15 640.0 44 12.9 -71 11.4. -74 670.5 16.6 8.5 -49 7.2 -57 701.0 9.8 5.2 -47 4.2 -57 731.4 6.3 4.2 -33 3.2 -49 853.4 1.94 1.58 -19 1.02 -47 975.3 0.83 0.77 -7 0.40 -52
I0
0.630x 0.95 CONCRETE DUCT 1-25(EFFECTIVE)MeV POINT SOURCE
L2
-
—MULTISORD - 0.66 0.3/ p/Sg1.1)0•72 'Jo 0-67 ./..1
--MULTISORrift611-.01/N • EISENHAUER (REF27)
- \ - \
\
-
_ \- -
\
- \ \
-
\\.
N.. e
120 130 140 ISO 160 170 DISTANCE ALONG CENTRE LINE (cms)
'FROM DUCT MOUTH FIG 5 COMPARISON OF MULTISORD PREDICTIONS WITH
EXPERIMENT - 52 -
3 . 10
100 HO 180
—I-1 LI L2 1
. — MULT1SORD _ 0.66 0.
01-4 0.72 1Ji 0 0•67p 1
-- MULTISORD P@Lb-e) IA
® TERREL (REF 2o)
e
\
4 I0
I0
I0
GA
MM
A D
OS
E R
ATE
2 I0
61; 6 CONCRETE DUCT 24Na 1.368&2.75 MeV POINT SOURCE I I
400 600 12 00 DISTANCE ALONG CENTRE LINE ms)
FROM DUCT MOUTH FIG 6 COMPARISON OF MULTISORD PREDICTIONS
WITH EXPERIMENT - 53 -
5.5. A Mouth Source in Three-Legged Ducts
General agreement within + 40% is to be observed
in Figures 8 and 9. It is noticeable that the effect of
diff erent incident energies annears to be negligible.
The shape along the duct is almost identical for both
energies with the angular dependence parameters set
to zero.
The accuracy of the dose measurements has been
assessed as -I- 10% error (32). The apparent failure
to predict the shape at the second intersection of iew
Figure 9 is therefore • significant since all these points
lie within the 10% range. Additionally, a slightly different
choice of output points to give a more detailed dose-rate
profile could well have yielded the observed shape.
The effect of geometry and energy becomes more
noticeable for ducts of more than 1 ft. square-section
as in Figures 10 and 11. Agreement is generally within
± 40%. Figure 11 illustrates the effect of a 10% reduction
in albedo and no corner penetrations.
Figures 12 and 13 furnish the strongest support for
the method. All previous three-legged results are for
Z-shaped ducts. These figures show attenuations down
U-shaped ducts of 1, square cross-section in which the
agreement is generally within 50%.
-; -
to
J1
♦ • •
TABLE 18
COMPARISON OF EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS OF A TWO-LEGGED DUCT
60Co point source L1 = 45.4" • L2 = 40.5" = 11" TId = 11"
Distance from source (cms)
Experimental Dose Rate
mr/hr .
"Multisord Difference from Experiment
/
Multisord m = n = o
Difference froir. Experiment
G'
58.2 9990 9990 9990 69.6 6800 6942 +2 6822 0
103.9 3230 3053 -5 2949 -9 115.3 2770 2472 -11 2401 -13 161.0 30.5 18.1 -41 9.6 -69 172.5 17.3 11.3 -35 5.2 -70 183.9 10.2 6.7 •-34 2.9 -72 195.3 6.33 4.67 . -26 1.86 -71 206.8 4.25 3.44 -19 1.30 -70 218.2 2.86 2.58 -10 0.94 -67
• 4 •
TABLE 19 COMPARISON OF EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS
OF A THREE-LEGGED DUCT 13705 point source = 131 L2 = 141 L3 = 10' = 6 H = 61 Distance from source (cms)
Experimental Dose Rate
mr/hr
Multisord Difference from Experiment
%
Multisord m = n = o
'Difference from Experiment:
% 121.9 24492 24492
1 1
1 1
1 1
1 1
1 1
I 1
I *1
1 1
I ÷
W iv 1
v 1v
W
kJ.)
kJ) 00 Z1-)
-I-
U-t U
l O
• -F
"" C7‘
U-t
tN) \
D
I-1
CO
ON
0 k_r
c 0 24492
182.9 10356 11399 11404 +10 243.8 6300 6607 6527 +4 304.8 4284 4288 4161 -3 350.5 3061 3254 3128 +2 396.2 2376 2562 2452 . +3 442.0 2067 2504 2399 +16 518.2 1978 221 ' 215 -89 548.7 267 162 156 -42 609.6 115 67 60 -48 670.6 60.8 44.3 34.2 -44 731.5 33.1 24.8 19.0 -43 792.5 24.1 17.2 12.3 -49 823.0 20.3 14.8 10.3 -49 853.0 20.6 14.3 9.8 -52 884.0 18.2 13.6 9.1 -50 914.5 17.3 14.5 10.3 -40 945.0 14.1 7.9 6.7 -52 975.4 7.56 . 4.81 3.79 -50
1006.0 1.99 1.69 1.53 -23 1036.0 1.44 0.94 0.89 -38 1067.0 1.02 0.75 0.67 -34 1097.0 0.76 0.60 0.50 -34 1128.0 0.61 0.47 0.37 -39 1250.0 0.26 0.17 .0.10 -62 --...%
IC
10
4 10
1 1 1r STEEL DUCT; 60 Co 1.25 (EFFECTIVE) MeV POINT SOURCE
e CHAPMAN \-1 (REF 38 p I)
—0.67 0.33 JJo 0.67,U
Phr
— MULT1SOR 1%.01.-9) 072 --MULTISORD
P (91-.1)
•
.. . ...
I
\
\
\ \
• \
\
\ \
100 200 300 DISTANCE ALONG CENTRE LINE Cc ms)
F ROM DUCT MOUTH FIG 7 COMPARISON OF MULTISORD PREDICTIONS WITH
EXPERIMENT _ 57 _
I
.......– 2 ...
— MULTISORDr 10
134.01-0. 0.72)J 0 0.6 p
-- MULT1SbRD
P(51-9 Wr
• TERREL (REF 21 ))
•
.
\®
o 0
e
\ e
IC
10
• 10
2 IC (.9
10
10
w • cc 2 10 0
ixd CONCRETE DUCT i37Cs 0.662 MeV POINT SOURCE
500 1000 1500 DISTANCE ALONG THE CENTRE LINE
FROM DUCT MOUTH (CMS) FIG 8 COMPARISON OF MULTISORD PREDICTIONS
WITH EXPERIMENT - 58 -
4 •
TABLE 20 COMPARISON OF i EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS
OF A THREE-LEGGED DUCT 60Co point source Li 1 = 13' L2 = 14' L3 = 10' H = 6' w =
,Distance from source (ems)
Experimental Dose Rate
mr/hr
Multisord Difference from Experiment
%
Multisord m = n = o
Difference from Experimen-:
ol p 121.9 54000 54000 54000 182.9 24630 24791 0 24769 0 243.8 13572 14234 +5 14099 +4 304.8 9090 9196 +1 9000 ...1 350.5 7524 6971 - -7 6777 • -10 396.2 5839 5481 -6 5312 -9 442.0 5952 5368 -10 '5207 -13 518.2 4687 332 -93 325 -93 548.7 474 242 -49 236 -50 609.6 190 98 -48 88 -54 670.6 105 65 ?-38 50 -52 731.5 61.0 36.0 ' -41 27.5 -55 792.5 37.2 24.9 -33 17.9 -52 823.0 34.9 21.4 -39 15.0 -57 853.0 36.0 . 20.8 -42 14.3 -44 884.0 36.6 19.6 -46 13.2 -64 914.5 31.4 21.3 -32 14.8 -53 945.6 25.1 . 12.2 -51 9.5 -62 975.4 12.6 7.3 -42 5.3 -58
1006.0 3.31 2.26 -31 2.04 -38 1036.0 2.37 1.24 -48 1.20 -49 1067.0 1.68 1.00 -40 0.90 -46 1097.0 1.24 0.79 -36 0.67 -46 1128.0 1.00 0.63 -37 , 0.50 -50 1250.0 0.37 0.23 -38 -
.0.13
0 -- -65
TABLE 21
COMPARISON OF ,EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS OF A THREE-LEGGED DUCT
60 Co point source L1 = 3.8' L2 =
L3 = 3.5'
W = H. = 1'
Distance, from source (cms)
Experimental Dose Rate
mr/hr
Multisord Difference from Experiment
%
Multisord m = n = o
Difference f?om Experiment
07 /0
76.2 177915 179642 106.7 89820 89820- 89820 137.2 6996 2776 —60 - 1979 -72 167.6 828 555 —33 319 —61 198.1 239 210 -12 92 —49 228.6 109 107 -2 42 —61 259.1 15.9 8.6 --46 3.o -81 289.6 2.11 2.05 - —3 0.58 —73 335.3 0.57 0.46 —19 0.09 —84
4
1-1 rn
4 • s
TABLE 22
COMPARISON OF EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS OF A THREE-LEGGED DUCT
137Cs point source = 3.5'
L,2 = = 3'5? W = 1' H = 1'
Distance from source (cms)
Experimental Dose Rate
Multisord
mr/hr
Difference from Experiment
Multisord m = n = o
Difference from Experiment
76.2 72780 72780 •72780 106.7 37572 36601 -3 35939 -4 137.2 2918 1630 -42 1162 -59 167.6 434 338 -22 . 283 -35 198.1 134 127 -5 56 -58 228.6 65.3 64.3 - ' -3 48 -26 259.6 6.82 5.58 -18 1.98 -71 289.6 1.18 1.34 +14 0.38 -68 335.3 0.41 0.31 -24 0.06 -85
r-- L I
'
— MULTISORD, 0.72 jk
_ 0.66 024 01-0.6) 410 0.o7p
-- MULT1SORD P Cg -'9i) p/,
0 TE(RREL (REF 21
9
0000 0 0
N. -,
0
‘
\ \
I0
- I0
5 1
4 10
2
GA
MMA
DO
SE
RAT
E
10
6x6 CONCRETE DUCT 60Co 1.25 (EFFECTIVE) MeV POINT SOURCE
500 1000 1500 DISTANCE ALONG THE CENTRE LINE
FROM DUCT MOUTH (cMS)
FIG 9' COMPAPISON OF MULTISORD PREDICTIONS WITH EXPERIMENT - 62 -
300 FROM DUC T
100 200 DISTANCE. ALONG CENTRE LINE
MOUTH (cms) FIG 10 COMPARISON OF MULTISORD PREDICTIONS WITH
EXPERIMENT -.63
Ix 1 CONCRETE DUCT 137CS 0.66 I I
5 I0
4 10
3 10
.c
w
2 10
0
2 2
10
10
10
® TERREL (IR- EF 2 I — MULTISORD_ 0°66 0 .34
1-4. g) 0.72 po 06 ).1 7c -- – MULTISORD
p (gl 4-0 44c 1
- \ \
I k \
\ 1
■
360.
• a •
TABLE 23
COMPARISON OF EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS OF A THREE-LEGGED U-SHAPED . DUCT •
60 o point source L1 ' = 3 5' = 4.0' =5.5' W = 1' H = 1'
Distance from source (cms)
Experimental Dose Rate
Multisord
mr/hr
Difference from Experiment
Multisord m = n = o
Difference f:-o Experimer.t
. . 76.2 179182 180989 106.7 90480 90480 . 90480 137.2 6960 2796 -60 1993 -71 167.6 874 559 -36 321 -63 198.1 247 212 -14 93 -62 228.6 111 107 -4 42.3 -62 259.1 11.5 9.3 . ...19 3.3 -71 289.6 1.60 2.73 +71 0.86 -46 320.0 0.56 0.79 +41 0.16 -71 350.5 0.19 0.40 +110 0.066 -63
TABLE 24
COMPARISON OF EXPERIMENTAL AND CALCULATED POINTS ALONG THE AXIS OF A THREE-LEGGED U-SHAPED DUCT
137 Cs point source Li = 3.5' L2 = 4.0' L3 = 5.5' W = 1' H=
rn Ui
Distance from source (cms)
Experimental Dose Rate
mr/hr
Multisord Difference from Experiment
e /0
Multisord m = n = o
Difference from Experiment
°70
76.2 73932 75679 106.7 37170 37170 37170 137.2 2870 1656 -42 1199 -58 167.6 424 343 -19 201 -53 198.1 129 129 0 57 -56 228.6 62.7 65 - +4 26 -59 259.1 5.99 6.10 • +18 2.2 -63 289.6 1.63 1.79 +74 0.57 -45 320.0 0.34 0.52 +53 0.11 -68 350.5 0.15 0.26 +73 0.04 -73
1 _ _
10
LLD
2 2 < IC
10
10
_____,_t_
— MULTISORD -0-66 34 al--... 0.72)J 0 0-orp
— — MULTISORD P C91°- i'Y'R
--MULTISORD -0.66 0.34 0.9401-.-g)0.72)J0 0.67)J
-MULTISORD _066 r4 ps..g) 0 .7 2g, 0- 67
--g e TERREL(REF 21 )
‘
kk
V 1 1 .
V _ \\ 0 V\
-I-
`RNs o\\ \ \
\ `.,,
i- \
\
\
\ \
\ % \ \\\ ‘
\ ‘ \
\
It
lid CONCRETE DUCT 60Co 1.25 (EFFECTIVE) MeV POINT SOURCE
100 200 DISTANCE ALONG CENTRE LINE (cms)
FROM DUCT MOUTH FIG II COMPARISON OF MULTISORD PREDICTIONS
WITH EXPERIMENT - 66 -
300 360
3 I0
cc 2 10
if) 0
10
10
10
lid U-SHAPED CONCRETE DUCT 60Co1.2S(EFFECTIVE MeV POINT SOURCE L1--I-1--i- I_2-- F----{
--
— MULTISORD – j3/1\01-0.9) 0.721 0.66.0• 0.66.
MULT!SORD
PEgis-- PA
TERREL(REF 21 )
34 7p
•
\ 0
\
\ \ . \\
\
\
\ 0
100 200 300 DISTANCE ALONG THE CENTRE LINE (cms)
FROM DUCT MOUTH FIGI2 COMPARISON OF MULTISORD PREDICTIONS
'WITH EXPERIMENT - 67 -
4 10
2 10
V)°
2 2
IC
L2 L ---1
— MULTISORD 06a ,Li p34 oi,g) o.72,u 0
- - m I; MDR C .13 (91-61) iy7c 1
® TERREL REF 21 ) I . . . .
_ _ --- - -
\ 9
i 1 ••■ N I 1
\
A \ A
\e
\ 0
\ e
I0
10
'XI' U-SHAPED CONCRETE DUCT 13.7Cs 0.662 MeV POINT SOURCE .......
100 200 300 DISTANCE ALONG THE CENTRE LINE
FROM DUCT MOUTH (CMS)
FIG 13 COMPARISON OF MULTISORD PREDICTIONS WITH EXPERIMENT w 68 -
los 104 100
10 1 102 103
EXPERIMENTAL DOSE RATE mr/hr
FIG 14 COMPENDIUM OF THEORETICAL RESULTS USING . MULTISORD VERSUS EXPERIMENTAL RESULTS FOR ALL ENERGIES AND DUCT GEOMETRIES
Y,
•
0
•
a
et.'"
•
•
•
- 0 0
0
•
0
i. ,1. a 0
G
i
I
-
11 111
a 0 a 0
I I
a o,
a
111111 1 I
.
ilit it' I Ira' tit I um t I hutt 1 I I
- 69 -
0 10
--1 10
' 4 10
3 I0
TH
EOR
ET
ICA
L
10
2 10
6. CONCLUSIONS ARISING FROM TE-ESE TESTS
There is excellent agreement in all the test comparisons
between MULTISORD predictions and experiment which must be
regarded as fortuitous to a certain degree because:
(a) low energies only have been considered where forward
scattering is not pronounced;
(b) the configuration of the experimental ducts minimise the
effects dependences: (nnR of the
most important scattering areas at the first intersection
receives radiation normally incident and so contains no
azimuthal dependence. The second area reflects with
normal emergence which utilises 'an albedo near the mean
for these conditions.) (See Appendix E).
The experimental data do not, however, provide an adequate
test of the method under conditions appropriate to power reactor
design, and the objectives of the experimental program at LIDO
were therefore to supplement the existing data in order to
provide a complete range of measurements for checking the 6 ,W
MULTISORD method. The following features were/included:
(1) monoenergetic source energies extending up to 6 MeV;
(ii) three—legged slots and rectangular section ducts;
(iii) wall materials composed of concrete, steel and lead;
(iv) configurations with distributed wall sources as opposed to
the mouth sources which have been used exclusively in the
published work;
—
(v) measurements of the angular distribution and spectrum
of the flux in addition to dose-rate distributions.
If the accuracy achieved in these limited preliminary cases
. can be maintained over a more representative selection of
source energies, materials and duct geometries then it is anti-
cipated that MULTISORD will become the standard tool for
gamma-ray streaming calculation.
— 71 —
7. THE LIDO EXPERIMENT
The test comparisons above utilised only several elements
of the albedo array provided and it is necessary to use higher
energy sources to check the higher elements in the arrays.
The sources themselves are in point geometry which does not
typicalise situations encountered in power reactor design.
Similarly absent from published experiments are data on three-
legged steel ducts.
The aim of the LIDO experiment was to obtain good
data in known geometries from measurements along a three-
legged duct rectangular section using non-point geometry
sources. Such measurements were then compared with
MULTISORD predictions for varied source energies. The use
of high energy sources checked additionally whether this
azimuthally independent code suited a strongly azimuthal situation,
whilst lower energy sources supplied a set of results that were
absent from the published literature.
Such an experiment required:
(i) monoenergetic sources giving a choice of energy;
(ii) a suitable detector;
(iii) a three-legged steel duct.
The experimental results obtained are to be read in
Table 25 and displayed in Figure 15; they are for a constant
10 inch width along the three legs using the 6.13 MeV pipe
source. This result with its extended source is more
- 72 -
representative of a situation encountered in power reactor
design. As is observed there is excellent agreement between
the MUL,TISORD predictions and this independent experimental
data. Agreement is generally within 50%. The MTJL,TISORD
source strength Was normalised to that of the experiment
using the dose-rates measured 304.8 mm along the centre
line .
- 73 -
• • •
TABLE 25 COMPARISON OF SOME EXPERIMENTAL DOSE-RATES WITH THOSE PREDICTED BY MTJLTISORD FOR A RECTANGULAR SECTION STEEL DUCT WITH TWO RIGHT
ANGLE BENDS, USING A 6.13 MeV SOURCE
N16 Pipe Source L,1 = 31.5" = 35" L3 = 2.7.5" W = 10” H =
Position relative to the source (cms)
Experimental Dose- Rate (mrihr/kw)
Multisord Dose--Rate Difference from Exaeriment (%)
27.31 6.95, -1 6.7 , -1 --3 32.3 5.7 , -1 5.6 , -1 --2 47.3 3.39, -1 3.47, -1 +2 57.3 2.57, -1 . 2.68, -1 +4 67.3 1.98, -1 2.13, -1 +8 92.3 1.04, -2 6.21, -3 -40 97.3 5.36, -3 5.23, -3 +2.5 102.3 3.68, -3 3.85, -3 +5 107.3 2.72, -3 3.28, -3 +20 117.3 1.79, -3 2.09, -3 , +17 127.3 1.24, -3 • 1.60, -3 +29 152.3 1.03 , -3 1.42, -3 +98 149.3 6.24, -4 9.84, -4 +58 154.3 6.16, -4 9.16, -4 +49 159.3 6.01, -4 8.31, -4 +38 179.3 7.98, -5 3.77, -5 -53 199.3 1.43,-5 1.38 , -5 -.3 209.3 7.84, -6 1.05, -5 +34 219.3 5.79, -6 ' 8.26, -6 +243
-------...:!-... -,--- --------- ---'._. __ -....>_----_._.-.
! I
I I
() • '. ~ .' 0' J!;1 0 r b' 0 I • './ 'b.. • """ to ."'t>
.-.-.r-r-7'~-r-7'-1' ~ NI6pIPE, ,.: I • I-.,!.;=~==o=:::t..- E ~ C
t> : 0 \ SOURC ,. \ • 0' "" I
,;, '~',!.I,,'\O: " \' _ ,,11 I '. ,
IS. 'Q. 6' ~ _ ••
IO-II----------I---~· ~ - .0 'b
~ . " o ~ '\ '\ #~, -lo... ............. ,--..,.
• • V 0 .t; 0 '0, t) , ,
• • A • • I) 0 I 0 ~ ~ () '.
1621--__ ~ ________ ~~~,-.CONCRETE • tJ, ... '\0
o (1" • ~ ~ " LAYOUT OF DUCT
, " ~.
IOem STEEL WALL
L
~ IO·3~-----------------------~~------~-------~ c::-e UJ !;( a: I
UJ (/)
MULTISORD PREDICTION • o. MEASUREMENTS
O· o 164~-----------------------------_+------------~
165~----------------------------------~~----~
166~~=-~~~~~~--~~~~~~1~~~~~~~~~ 60 80 100 120 140 160 180 200 220 240 CENTRE-LINE DISTANCE FROM MOUTH
FIG.IS COMPARISON OF MULTISORD PREDICTIONS WITH MEASUREMENTS OF GAMMA-RAY DOSE-RATE DUE TO A 6 MeV SOURCE IN A RETANGULAR SECTION DUCT WITH TWO RIGHT ANGLE BENDS.
- 75 -... - .... -._._-_ .. __ .... _----
8. CONCLUSIONS AND RECOMMENDATIONS
nas been snown tnat -cne lvlU L., yr
predictions, compares well with all the experimental data
available, including that supplied by the author. It may be
concluded on this evidence that
(i) the albedo data of Davisson and Beach are an adequate
basis for kernel albedo predictions.
(ii) the Chilton-Huddleston' formulisation of the albedo data
presents such albedos in a form amenable to manipulation.
(iii) the albedo array in Table 14 for steel is correct and it
is reasonable to accept its companion concrete albedo
array also (Table 13).
(iv) MULTISORD gives excellent results for all tests
attempted in this work. Neglect of an azimuth correction
to the albedo form has not resulted in prediction failure
even in the 6.13 MeV case.
(v) Experimental data has been supplied for 6.13 MeV photons
streaming along a three-legged steel duct having rectangular
cross-section for comparison with MULTISORD. In
addition, this data fills a vacancy found in the published
literature.
The MULTISORD method does not, however, give good
predictions in the vicinity of an intersection. This is partly
..re . due to the finite peitIon of the wall areas into sub-areas.
The dose points may view a considerable amount of a sub-area
at one extremity of the total visible area but not see its
- '76 -
centroid, so causing an underestimation. Secondly, the
difficulty in specifying the corner lip component indicates a
likely reason for disagreement, although this contribution was
discounted earlier as being small it may be that the method of
estimation was not sufficiently rigorous. This can be tested
by putting distributed wall sources at the intersection (in both
MULTISORD and experiment) so mitigating the importance of
the corner lip component. Such a comparison would furnish
a different type of test for the general method, as so far
only mouth sources have been used.
The overall accuracy of MULTISORD is maintained for
source configuration types to be found in general reactor
design work and furnishes a valuable tool for design calculations
on gamma-ray streaming.
- 77 -
ACKNOWLEDGEMENTS
The author is indebted to Dr. U 1:3utler or the .t.adiation
Physics and Shielding Group , AERE , Har well for suggesting
this work and extending the facilities to make it possible.
The generous and helpful cooperation of his group and others
too numerous to name is gratefully acknowledged. In particular
the valuable assistance provided by P.C. Miller, writer of
MULTISORD, and his ceaseless encouragement are gratefully
appreciated.
- 78 -
REFERENCES
1. B .T. Price, C .0 . Horton and K.T. Spinney,
"Radiation Shielding", 1957.
2. E.T. Clark and J.F. Batter,
"Gamma-ray Scattering by Concrete Surfaces".
Nuclear Science and Engineering 17, p.125 - 130, 1963.
P.L.3. Maerker and V.P.
AMC. "A Monte Carlo Code Utilising the Albedo Approach".
ORNI, - 3964.
P.C. Miller
RANSORD - Unpublished.
5. P .0 . Miller
"MULTISORD - A Fortran Code for Calculating the
Streaming of Neutrons in Slots using the Interative-
Albedo Method".
JNPC/SWP/N88 1966.
6. J.J. Steyn and D.G. Andrews.
"Experimental Differential Number, Energy and Exposure
Albedos for Semi-Infinite Media for Normally Incident
Gamma Photons".
Nuclear Science and Engineering: 27, p.318 - 327, 1967.
O. Baarli
"Archly for Mathematik og Naturvidenskap (Oslo)".
B.LV.Nr.8. , 1961.
- 79 -
8. C.E. Clifford
"Differential Dose Albedo Measurements for 0.bb MeV
Gammas Incident on Concrete, Iron and Lead".
D.R.C.L. R-412, 1963.
9. L.G. Haggmark, T.H. Jones, N.E. Scofield, and
W.J. Gurney.
"Differential Dose-Rate Measurements of Backscattered
Gamma-Rays from Concrete ; Aluminium and Steel".
Nuclear Science and Engineering: 23, p.138-149, 1965.
10. D .3 . Raso
"Monte Carlo Calculations on the Reflection and Trans-
mission of Scattered Gamma-Rays".
Nuclear Science and Engineering: 17, p.411-418, 1963.
11. M.J. Berger and D.J. Raso
"Monte Carlo Calculations of Gamma-Ray Backscattering".
Radiation Research: 12, p.20-37, 1960.
12 . D .J . Raso
"Monte Carlo Calculations on the Reflection and Trans-
mission - of - Scattered Gamma Radiation".
Technical Operations Inc. Report TO-B 61739(Rev.), 1962.
13. C.M. Davisson and L.A. Beach
"Gamma-Ray Albedos of Iron"
NRL Quarterly on Nuc. Sci. & Tech. for the period
October - December 1959 (Jan. 1966).
- 80 -
14. A.B. Chilton and C.M. Huddleston.
"A Semi-Empirical Formula for Differential Dose
Albedo for Gamma-Rays on Concrete"
Nuclear Science and Engineering: 17, p.419-2424, 1963.
15. C.M. Huddleston
"Comparison of Experimental and Theoretical Gamma-
Ray Albedo"
NCEL TN-567, 1964.
16. J.P. Hurley
Private Communication to C.M. Huddleston
17. T. Hyodo
"Backscattering of Gamma-Rays"
Nuclear Science and Engineering: 12, p.178-184, 1962.
18. B.P. Bulatov
"The Albedos for Various Substances for Gamma-Rays
from Isotropic 60Co 137Cs, 51Cr"
Reactor Science 13, p.82-84, 1960.
19. A.B. Chilton, C.M. Davisson, and L.A. Beach
"Parameters for C-H Albedo Formula for Gamma-Rays
Reflected from Water, Concrete, Iron and Lead"
Trans. American Nuclear Society: 8(2) , p.656, 1965.
20. C.W. Terrell and A.3 . Jerri
"Radiation Streaming in Shelter Entranceways".
Armour Research Foundation Report ARF.1158-AO1-5, 1961.
- 81 -
21. C.W. Terrell, A.J. Jerri, and R.O. Lyday
"Radiation Streaming in Ducts and Shelter Entranceways"
Armour Research Foundation Report ARF.1158-A02-7, 1962.
22. J.M. Chapman and T.S. Grant
"Gamma-Ray Attenuation in Coplanar .and Non Coplanar
Three-legged Ducts"
NCEL-Tech. Note 658, 1964.
23. J .0 LeDoux and A.13 . Chilton
"Gamma-Ray Streaming through Two-Legged Rectangular
Ducts".
Nuclear Science and Engineering: 11, p.362-368, 1961.
24. C.W. Terrell A .J . Jerri R.O. Lyday and D. Sperber
"Radiation Streaming in Shelter Entranceways"
Armour Research Foundation Report ARF.1158-12, 1960.
25. W.C. Ingold
"Some Applications of a Semi-Empirical Formula for
Differential Dose Albedo for Gamma-Rays on Concrete".
NCEL-TN-469, 1962.
26. J.M. Chapman and C .M. Huddleston
"Dose Attenuation in Two-Legged Concrete Ducts for
Various Gamma-Ray Energies"
Nuclear Science and Engineering: 25, p.66-74, 1966.
27. C. Eisenhaur
"Scattering of 6oCo Gamma Radiation in Air Ducts"
NRS-TN-74, 1960.
- 82 -
28. C.E. Clifford
"Gamma Shielding provided by Ducts"
DRCT_, 370, 1962.
29. L.B. Gardner
"Neutron and Gamma-Ray Streaming through a Two-
Legged, Thick Wall Steel Duct"
NC FL R-558, 1968.
30. Proceedings of the Conference on the Physics Problems
of Reactor Shielding.
AERE R 5773, Vol. 3., September 1967.
31. P.C. Miller
"MULTISORD II - a programme specification"
JNPC/SWP/N.155
32. O.M. Chapman and T.R. Tree
"Dose Measurement of Gamma Radiation Streaming
through Concrete Ducts with and without Lead Liners
and through Corrugated Steel Ducts"
NCEL,-TR-590, 1968.
33. B.P. Bulatov and O.I. Leipunskii
Soviet Journal of Atomic Energy: 7, 1015, 1961.
34 . B .P . Bulatov and E .A. Garusov
Soviet Journal of Atomic Energy: 5, 1563, 1958.
35. H. Fajita, K. Kobayashi and T. Hyodo
"Backscattering of Gamma-Rays from Iron Slabs"
Nuclear Science and Engineering: 19, p.437-440, 1964.
- 83 -
36. J.M. Chapman
"The Variation of Dose Attenuation of Two-Legged
Concrete Ducts with Incident Gamma-Ray Energy"
NCEL-TN-N707, 1965.
37. D .W . Green
"Attenuation of Gamma Radiation in Two-Legged
11" Rectangular Ducts"
NCF,T,-R-7195 ;, 1962.
38. J.M. Chapman
"Attenuation of Au-198 Gamma-Rays in an 11" Steel
Diact"
NCEL-TN-N864, 1 966.
39. J.M. Chapman
"Gamma Dose-Rates and Energy Spectra in a 3 foot
square Duct"
NCEL-N-LJ 1962.
APPENDIX A
The Albedo Concept
Consider a photon of energy E incident upon a medium
with polar angle 0o , azimuthal angle V' at position (x Y ) o o" o' (E3-ee—Figurre--9-2--e-) After multiple scattering with decrease
in energy and change of direction the photon will emerge
with polar and azimuthal angles of Q and / respectively and
possess an energy E.
If the points (x0 , yo , o) and (x, y, o) are not too
far apart then the process can be reviewed as one reflection.
This reflecting power of the medium can be described as
the probability per unit area, energy ,` and solid angle that the
photon emerges with the above parameters: •
dP dE dx dy Sin ed ed( = A E,e E0 ,eo ,,,„...,yp.
for a reflective process 2S.x = x-x0 = y-y0
The differential albedo becomes, writing 8/ = 0
A e A/4 x Eo , e This depends on two parameters, E0 and Bo and on
•
five distributed variables E, 0 ,Lx,y,a),/. For practical
situations the dependence of the albedo on 46, x and y is
of minor importance (25) . Integrating out these two variables
and writing „cre" for p)a :
- 8 5 -
27C
NE
cy A(E, evi E0 00) c1.4)..x dAyA(E,0 10,L x,A3r I Eoe 0 )
-L/C. then dropping, for brevity E0 and Q0 , the albedo is written
simply as
A E , , )
This modified form of the generalised albedo can be used
to define three types of the albedo.
Numb-r albe,lo (ratio of the rriml-,e," o-P tn thP
number of incident photons).
N sin edo AcE,e
Energy albedo (ratio of the reflected to the incident energy)
sin ed e A(.,0,1) E dE
0
Dose Current albedo 7S.
27c 2 Eo
sin Eh-3( E ) A E , e , d E
Eofid E0 )
This latter definition, though useful is not of physical
significance. The true dose is proportional to flux, not current.
However, the dose passing through a surface can be considered
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as that energy absorbed per gram per second measured by an
infinitesimal detector, embedded in the surface, responding to
radiation flowing in the direction 9. That is, the source plane
is projected as in Appendix C. By reference to the Appendix
it is apparent that the true dose albedo (ratio of reflected dose
to the incident dose) is given by
rx 2 r sine d9 Eiad(E) 0 cos
j 0 'old( Eo) cos e
Ndt A(E, 0,51) dE
The 'use of the dose current albedo may be illustrated
by consideration of a scattering area dA in a surface located
r i from a source. The dose rate at a point located r2 from
dA may then be expressed by
dD = Di Nd (E0 ,(90 , el sin cos eo dA
2 2 r i r2
where D1 = dose in air at unit distance from the source
, 90 = polar angle of incidence of radiation.
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•
APPENDIX B
Full Monte Carlo and Albedo Concept Comparison
In the Monte Carlo process a photon may be incident
. upon a surface and enters at point 'A', scatters and finally
leaves at point 'B'. At each collision point the immediate
future history of the photon is determined using the micro-
scopic differential properties at the medium.
In contrast the Albedo process permits the incident' photon
to be reflected from the same point. All flux estimates at
each reflection point are made by basing them on a priori
macroscopic reflective properties of the medium.
Clearly in the former case, if the points at incidence
and emergence have small relative displacement then the albedo
concept should apply well. For then 'A' and 1131 become
coincident points and the latter case applies.
- 88 -
APPENDIX C
Clarification of Current and Flux Quantities
It is apparent from Appendix A that the differential
albedo is defined per unit surface area. Subsequently the
albedo so defined is a current quantity.
A particle emission rate is a current whereas the flux ,
of particles is always referred to a unit area normal to the
particle direction. To correct for the different orientation
of the flux passing through that area, at angle e to the
normal, by cos e in order to project the reference area back
to the source plane.
i.e. CURRENT = Cos B x flux
In earlier works the albedo is sometimes not defined and
the values presented are flux values, where
Cos e AFlux (7) x ACurrent Cos e
gives the required transformation.
It is a general guide to use only reports and tabulations
of albedo that define the quantities used.
- 89 -
+12 Cos G o Sec 9 )e " ao = 133 K ( es) 0< d ( o B2 ÷ 1.112 Cos
where P = E E
1 Eo
rnoc (1 - Cos es )
1
APPENDIX D
Chilton and Huddleston
The preliminary expression that Chilton and Huddleston
obtained for differential dose albedo is given by
where Ili = Effective attenuation coefficient at the incident energy
= Effective attenuation coefficient after a single scatter
it = Effective attenuation coefficient of the multiple
scattered radiation assumed to be emitted iso-
tropically.
On the assumption that the effective attenuation coefficient
is not greatly energy dependent for light materials in the photon
energy range of interest the above formula reduces to that
of section 1+.3 by cancelling out the rs) .
Consequently, whenever this equation is used this
assumption must be recalled.
Useful formulae are 2
r K( es) = p (1 - P Sin 2 es + P2 )
2
r2
= 7.941 x 10-26 cms; moc2 = 0.511 NWJ.
- 90 -
0
APPENDIX E
Mean Azimuth
The intersection wall area in the first leg, that is directly
viewed by the second leg is a major scattering contribution.
Unlike the case of perpendicular incidence there is azimuthal
dependence upon the emergent angle. It was assumed (in
Section 2) that the apparent unimportance of this dependence
is because the albedo utilised is very close to the azimuthal
angle mean. A simple check from the Chilton and Huddleston
tabulation of the Davisson and Beach data is obtainable to verify
this because the albedo values suitable for Multisord have been
obtained by modifying their data (Section 5.2.)
To check the assumption, the mean albedo from the
tabulation mentioned in Section 5.2 can be compared with the
albedo as by Chilton and Huddleston for the appropriate wall
area in a particular case.
For the steel duct employed in Chapman's measurements (38)
the angular parameters for the area 0 = 0° '0 = 83° relative to
the normal. By averaging over the nine azimuthal intervals of
the albedo data for these parameters one obtains the mean
= 0.0440
This can be compared with the albedo value for this area,
thus,
/3= 0.0465.
It is shown that the mean albedo is indeed near the albedo
appropriate to this area to within 5%.
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