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

- 86 -

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.

- 87 -

•

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%.

- 91 -