Two-Dimensional Cross-Section Sensitivity and Uncertainty ...

LA—9 23 2 - 7

DEC; Oi l DO J

LA-9232-T Thesis

UC-20d Issued: February 1982

Two-Dimensional Cross-Section Sensitivity and Uncertainty Analysis

For Fusion-Reactor Blankets

Mark Julien Embrechts

,1t W©

M Los Alamos National Laboratory Los Alamos.New Mexico 87545

CONTENTS

Nomenclature viii

List of Figures x

List of Tables xi

Abstract xiv

1. Introduction 1

1.1 Motivation 2

1.2 Literature Review A

2. Sensitivity Theory 7

2.1 Definitions 8

2.1.1 Cross-section sensitivity function, cross-section

sensitivity profile and integral cross-section sen

sitivity 8

2.1.2 Vector cross section 9

2.1.3 Geometry related terminology 10

2.2 Cross-Section Sensitivity Profiles 12

2.2.1 Introduction 12

2.2.2 Analytical expression for the cross-section sen

sitivity profile 14

2.2.3 Explicit expression for the cross-section sensi

tivity profile in discrete ordinates for a two-

dimensional geometry representation 20

2.3 Source and Detector Sensitivity Profiles 3A

iv

2.4 Sensitivity Profiles for the Secondary Energy Distribution

and the Secondary Angular Distribution 38


2.4.2 Further theoretical development 40

2.4.3 Secondary energy and secondary angular distribution

sensitivity profile 43

2.4.4 Integral sensitivities for SEDs and SADs 44

2.4.5 Explicit expressions for integral SED sensitivity

profiles in a two-dimensional geometry

representation 47

2.5 Design Sensitivity Analysis 48

3. Application of Sensitivity Theory to Uncertainty Analysis 52

3.1 Definitions , 52

3.2 Cross-Section Covariance Matrices 54

3.3 Application of Cross-Section Sensitivity Profiles and Cross-

Section Covariance Matrices to Predict Uncertainties 55

3.4 Secondary Energy Distribution Uncertainty Analysis 57

3.5 Overall Response Uncertainty 62

4. SENSIT-2D: A Two-Dimensional Cross-Section Sensitivity and

Uncertainty Analysis Code 64

4.1 Introduction 64

4.2 Computational Outline of a Sensitivity Study 66

4.2.1 Cross-section preparation module 70

4.2.2 The TRIDENT-CTR and TRDSEN block 71

4.2.3 The SENSIT-2D module 71

v

4.3 The SENSIT-2D Code 73

4.3.1 Flowcharts 73

4.3-2 Subroutines used in SENSIT-2D 78

5. Comparison of a Two-Dimensional Sensitivity Anlysis with a

One-Dimensional Sensitivity Analysis 83

5.1 Sample Problem #1 84

5.1.1 TRIDENT-CTR and ONEDANT results 92

5.1.2 SENSIT and SENSIT-2D results for a standard cross-

section sensitivity analysis 94

5.1.3 Comparison between a two-dimensional and a one-

dimensional cross-section sensitivity and

uncertainty analysis 99

5.2 Sample Problem i\l 104

5.2.1 Influence of the quadrature order on the sensi

tivity profile 104

5.2.2 Comparison between the two-dimensional and one-

dimensional analysis of sample problem #2 Ill

5.2.3 Comparison between the x's calculated from the

angular fluxes and the x's resulting from flux

moments 114

5.2.4 Evaluation of the loss term based on flux moments

in the case of low c 116

5.3 Conclusions 117

6. Sensitivity and Uncertainty Analysis of the Heating in the

TF-Coil for the FED 118

vi

6.1 Two-Dimensional Model for the FED 118

6.2 Two-Dimensional Sensitivity and Uncertainty Analysis for

the Heating in the TF-Coil due to SED and Cross-Section

Uncertainties 123

6.3 Comparison of the Two-Dimensional Model with a One-

Dimensional representation 129

7. Conclusions and Recommendations 133

8. References 138

Appendix A: SENSIT-2D Source and Code Listing 146

Appenlix B: TRIDSEN 190

vii

NOMENCLATURE

E reflects energy function

ft reflects angular distribution

0 microscopic cross section

1 macroscopic cross section, where x indicates the

material and/or the type of cross section

Z macroscopic scattering cross section for material x x,s

I _ total cross section for material x x, i

4> angular flus

• angular flux in discrete ordinates representation

4>0 spherical harmonics representation for the angular

flux

R» spherical harmonics function

P„ Legendre polynomials k

P. associated Legendre polynomials

L transport operator

L* adjoint transport operator

Q source

R response function

I integral response

V volume

F fractional uncertainty for the secondary angular

distribution

X part of the loss term in the sensitivity profile

¥ part of the gain term in the sensitivity profile

v m

F cross-section sensitivity function for cross section *• x

*x p | cross-section sensi t ivi ty prof i le for a cross-section

x lx for group g

Au8 lethargy width for group g

w quadrature veight

IX

LIST OF FIGURES

Figure 1 Illustration of the terminology: blank region,

source region and detector region

Figure 2a Coordinates in x-y geometry

Figure 2b Coordinates in r-z geometry

Figure 3 Definition of median energy and integral SED

sensitivity

Figure 4 Interpretation of the integral SED uncertainty

as spectral shape perturbations and definitions

of the spectral shape uncertainty parameter "f"

Figure 5 Computational outline for a two-dimensional sen

sitivity and uncertainty analysis with SENSIT-2D

Figure 6 Data flow for the SENSIT-2D module

Figure 7 Flow chart for SENSIT-2D

Figure 8 Cylindrical geometry representation for sample

problem #1

Figure 9 Two-dimensional (TRIDENT-CTR) representation

for sample problem #1

Figure 10 One-dimensional (ONEDANT) representation for

sample problem #1

Figure 11 Two-dimensional model for sample problem #2

Figure 12 Two-dimensional model for the FED

Figure 13 The TRIDENT-CTR band and triangle structure

for the FED

LIST OF TABLES

Table I Formulas for the sensitivity function

Table II Median Energies (E1, in MeV) and fractional

uncertainties (F) for secondary energy distributions

at incident neutron energies E n

Table III Summary of the features of SENSIT-2D

Table IV List of subroutines used in SENSIT-2D

Table V 30-group energy structure

Table VI Atom densities of materials

Table Vila Comparison of the heating in the copper region

calculated by ONEDANT and TRIDENT-CTR

Table Vllb Computing times on a CDC-7600 machine

Table Villa Comparison of the heating in the copper region


Table Vlllb Partial and net sensitivity profiles for the

two-dimensional analysis

Table IX Comparison between the \'s calculated from

angular fluxes and flux moments

Table Xa Predicted response uncertainties due to estimated

cross-section and SED uncertainties in a one-

dimensional analysis

Table XI) Predicted response uncertainites due to estimated

cross-section and SED uncertainties in a two-


Table XI

Table XII

Table XHIa

Table Illb

Table XlVa

Table XlVb

Table XIVc

Table XV

Table XVI

Cross section table used in sample problem #2

(the P-0, P-l, P-2, and P-3 tables are identical)

Integral response for sample problem #2

Standard cross-section sensitivity profiles calcu

lated by SENSIT-2D for the EQ-6 case (convergence

precision 0.0001, 4 triangles per zone) for sample

problem #2

Standard cross-section sensitivity profiles cal

culated by SENSIT-2D for the EQ-12 case (conver

gence precision 0.001, 4 triangles per zone) for

sample problem #2


culated by SENSIT for the S-12 case (convergence

precision 0.0C31, 4 intervals per zone) for sample

problem 112



precision 0.0001, 8 intervals per zone) for sample

problem #2


culated by SENSIT-2D for the EQ-12 case (con

vergence precision 0.0001) for sample problem 112

Comparison between the x's calculates from angular

fluxes and the x's calculated for flux moments

Atom densities for the isotopes used in the materials

x n

Table X I I Predicted Uncertainties (standard deviation) due

to estimated SED and cross-section uncertainties

for the heating of the TF-coil

Table XVIII Predicted SED and cross-section uncertainties

in the TF-coil due to uncertainties in the SS316

regions

Table XIX Comparison between the x's calculated from angular

fluxes and the x's resulting from flux moments for

region 11 (SS316)

Table XX Predicted uncertainties (standard deviation) due


in regions 1 and 3 for the heating in the TF-coil.

xiii

TWO-DIMENSIONAL CROSS-SECTION SENSITIVITY AND UNCERTAINTY ANALYSIS

FOR FUSION REACTOR BLANKETS

by


ABSTRACT

Sensitivity and uncertainty analysis implement the information ob

tained from a transport code by providing a reasonable estimate for the

uncertainty for a particular response (e.g., tritium breeding), and by

the ability to better understand the nucleonics involved. The doughnut

shape of many fusion devices makes a two-dimensional calculation capa

bility highly desirable. Based on first-order generalized perturbation

theory, expressions for a two-dimensional SED (secondary energy distri

bution) and cross-section sensitivity and uncertainty analysis were de

veloped for x-y and r-z geometry. This theory was implemented by deve]-

oping a two-dimensional sensitivity and uncertainty analysis code,

SENSIT-2D. SENSIT-2D has a design capability and has the option to cal

culate sensitivities and uncertainties with respect to the response

function itself. SENSIT-2D can only interact with the TRIDENT-CTR code.

A rigorous comparison between a one-dimensional and a two-dimen

sional analysis for a problem which is one-dimensional from the neu-

tronics point of view, indicates that SENS1T-2D performs as intended.

A two-dimensional sensitivity and uncertainty analysis for the heat

ing of the TF coil for the FED (fusion engineering device) blanket was

performed. The uncertainties calculated are of the same order of magni

tude as those resulting from a one-dimensional analysis. The largest un

certainties were caused by the cross section uncertainties for chromium.

xiv

1. INTRODUCTION TO SENSITIVITY THEORY AND UNCERTAINTY ANALYSIS

In a time characterized by a continuously growing demand for so

phisticated technology it should not be surprising that the production

of fusion energy might materialize more rapidly than commonly predicted.

With fusion devices going into «'• demonstration phase there is a need for

sophisticated nucleonics methods, tailored to the fusion community- In

a relatively short time frame fusion nucleonics has established itself

as a more or less mature subfield. In this context sensitivity theory

has beccne a widely applied concept which provides the reactor designer

with a deeper understanding of the information obtained from transport

calculations.

Under the term sensitivity theory usually algorithms based upon

classical perturbation and variational theory are understood. The scope

of this work will be limited to cross-section and design sensitivity

analysis with respect to fusion reactors. Since fusion nucleonics do

not involve eigenvalue calculations, the mathematical concepts utilized

will be simpler than those required by the fission community.

Sensitivity theory determii.es how a design quantity changes when

one or more of the design parameters are altered. Uncertainty analysis

1

http://determii.es

provides the error range on a design quantity due to errors on the de

sign parameters. Sensitivity information can easily be incorporated

into an uncertainty analysis by introducing covariance matrices.

Cross-section sensitivity and uncertainty analysis will give error

estimates of response functions (such as tritium breeding ratio, heating

and material damage) due to uncertainties in the cross-section data.

Such a study will reveal which partial cross sections and in what energy

range contribute most to the error and will recommend refinements on

cross-section evaluations in order to reduce that error. Although those

results will depend on the particular response and the particular de

sign, general conclusions can still be drawn for a class of similar

18 designs. Sensitivity theory is a powerful design tool and is commonly

1-3 applied to cross-section adjustment procedures. Design sensitivity

analysis is frequently used to reduce the many and expensive computer

runs required during the development of a new reactor concept.

1.1 Motivation

The purpose of this work is to assess the state of the art of sen

sitivity and uncertainty analysis with respect to fusion nucleonics,

fill existing gaps in that field and suggest areas which deserve further

attention.

At this moment the literature about sensitivity theory is scattered

between various journal articles and technical reports. Therefore, the

2

author considered it as one of his responsibilities to provide a con

sistent monograph which explains, starting from the transport equation,

how analytical and explicit expressions for various sensitivity profiles

can be obtained. Current limitations with respect to the applicability

of sensitivity theory are pointed out and the application of sensitivity

theory to uncertainty analysis is explained. At the same time the scope

has been kept limited to those algorithms which are presently used in

calculation schemes.

Due to the particular geometry of fusion devices (toroidal geom

etry, non-symmetric plasma shape, etc.), a one-dimensional transport

code (and therefore a one-dimensional sensitivity analysis) will gener

ally be inadequate. In order to mock-up a fusion reactor more closely,

a two-dimensional analysis is required. Although a two-dimensional

A 5 sensitivity code - VIP ' - already exists, VIP was developed with a

fission reactor in mind, and does not include an r-z geometry option,

nor a secondary energy distribution capability. To answer the needs of

the fusion community, a two-dimensional sensitivity and uncertainty

analysis code, SENSIT-2D, has been written.

A sensitivity code uses the regular and adjoint fluxes of a neutron

transport code in order to construct sensitivity profiles. SENSIT-2D

requires angular fluxes generated by TRIDENT-CTR.6,7 TRIDENT-CTR is a

two-dimensional discrete-ordinates neutron transport code specially

developed for the fusion community. Since SENSIT-2D incorporates the

essential features of TRIDENT-CTR, i.e., triangular meshes and r-z geom

etry option, toroidal devices can be modeled quite accurately. SENSIT-

3

2D has the capability of group-dependent quadrature sets and includes

the option of a secondary energy distribution (SED) sensitivity and un

certainty analysis. An option to calculate the loss term of the cross-

section sensitivity profile based on either flux moments or angular

fluxes is built into SENSIT-2D. The question whether a third-order

spherical harmonics expansion of the angular flux will be adequate for a

g 2-D sensitivity analysis has not yet been adequately answered. The

flux moment/angular flux option will help provide an answer to that

question.

As an application of the SENSIT-2D code, a two-dimensional sensi

tivity and uncertaintly analysis of the inboard shield for the FED

(fusion engineering device), currently in a preconceptual design stage

by the General Atomic Company, was performed.

1.2 Literature Review

The roots of cross-section sensitivitv theorv can be traced to the

9 10

work of Prezbindowski. ' The first widely used cross-section sensi

tivity code, SWANLAKE, * was developed at ORNL (Oak Ridge National Lab

oratory). In order to include the evaluation of the sensitivity of the

response to the response function, SWANLAKE was modified to SWANLAKE-UW

by Wu and Maynard. Already early in its history, sensitivity theory

1 ?-16 was applied to fusion reactor studies. It has now become a common

17-23 54 practice to include a sensitivity study in fusion neutronics. '

4

The mathematical concepts behind sensitivity theory are based on

24-29

variational and perturbation theory. The application of sensitiv

ity profiles to uncertaintly analysis was restricted not due to a lack

of adequate mathematical formulations, but due to the lack of cross-

section covariance data. An extensive effort to include standardized

covariance data into ENDF/B files has recently been made.

The theory of design sensitivity analysis can be traced to the work

of Conn, Stacey, and Gerstl. ' ' ' The current limitation of de

sign sensitivity analysis is related to the fact that the integral

response is exact up to the second order with respect to the fluxes, but

only exact to the first order with respect to design changes. There

fore, only relatively small design changes are allowed. The utilization

of Pade approximants might prove to be a valuable alternative to

higher-order perturbation theory, but has not yet been applied to design Co

sensitivity analysis.

A 5 The two-dimensional sensitivity code VIP ' was developed by

Childs. VIP is oriented towards fission reactors and does not include a

design sensitivity option, nor a secondary energy distribution capa

bility.

The theory of secondary energy distribution (SED) and secondary

angular distribution (SAD) sensitivity and uncertainty analysis was

originated by Gerstl and is incorporated into the SENSIT code.

The FORSS code package has been applied mainly to fast reactor stud

ies ' but can be applied to fuBion reactor designs as well, Higher-

order sensitivity theory * ' still seems to be too impractical to

5

be readily applied. Recently however, the French developed a code

52 system, SAMPO, which includes some higher-order sensitivity analysis

capability.

2. SENSITIVITY THEORY

In this chapter the theory behind source and detector sensitivity,

cross-section and secondary energy distribution (SED) sensitivity, and

design sensitivity analysis will be explained. Starting from the trans

port equation, expressions for the corresponding sensitivity profiles

will be derived. Those formulas will then be made more explicit and

applied to a two-dimensional geometry. The theory presented in this and

the following chapter is merely a consistent combination and reconstruc-

, , , 3,13,16,17,18,43-46,53

tion of several papers and reports. ' ' ' ' ' '

Since up to this time no single reference work about the various

concepts used in sensitivity and uncertainty analysis has been pub

lished, the author uses the most commonly referred to terminology. In

an attempt to present an overview with the emphasis on internal consist

ency, there might be some minor conflicts with the terminology used in

earlier published papers.

7

2.1 Definitions

2.1.1 Cross-section sensitivity function, cross-section sensitivity profile and integral cross-section sensitivity

Let I represent a design quantity (such as a reaction rate, e.g.,

the tritium breeding ratio), depending on a cross-section set and the

angular fluxes. The cross-section sensitivity function for a particular

cross section I at energy E, F 5 (E), is defined as the fractional

change of the design parameter of interest per unit fractional change of

cross section i , or

x x' x

In a multigroup formulation the usual preference is to work with a

sensitivity profile P| , which is defined by x

P| = MU- . J_ , (2) x 3I8/Z8 Au8

X X

where Au8 is the lethargy width of group g and I 8 is the multigroup

cross section for group g. The sum over all the groups of the sensi

tivity profiles for a particular group cross section I8, multiplied by

8

the corresponding lethargy widths, is called the integral cross-section

sensitivity for cross section I , or

S1 = I P| • Aug , x g x

= / dE IV (E) . (3) x

The integral cross-section sensitivity can be interpreted as the

percentage change of the design parameter of interest, I, resulting from

a simultaneous one percent increase of the group cross sections I* in

all energy groups g.

2.1.2 Vector cross section

The term "vector cross section" describes a multigroup partial

cross-section set with one group-averaged reaction cross section for

each group. Such a cross-section set can be described by a vector with

GMAX elements, where GMAX is the number of energy groups. The term

vector cross section was introduced by Gerstl to discriminate it from

the matrix representation of a multigroup cross-section set. Differ

ential scattering cross spctions can obviously not be described in the

form of a vector cross section.

9

2.1.3 Geometry related terminology

Under the term region we will understand a collection of one or

more zones. A zone will always describe a homogeneous part of the reac

tor. We will make a distinction between source regions, detector

regions and perturbed regions, and as a consequence between source,

detector and perturbed zones. We will introduce the term blank region

for a region that is neither a detector, source or perturbed region. A

zone will further be divided into intervals.

The source region will describe that part of the reactor which con

tains a volumetric source. The detector region indicates the part of

the reactor for which an integral response is desired. In the perturbed

region changes in one or more cross sections can be made.

A source or a detector regions can contain more than one zone, and

each zone can be made up of a different material. Due to the mathemat

ical formulations a perturbed region can still contain more than one

zone, but in this case all the zones have to contain identical materials.

If there is more than one perturbed region, all those regions should

contain the same materials.

The geometry-related terminology is illustrated in Fig. 1. In this

case, there are six regions; a source region, two perturbed regions, one

detector region and two blank regions. The source region contains three

zones (identified by a, b, and c). The first zone, a, is a vacuum,

while the other two zones are made up of iron. Note that both perturbed

regions satisfy the requirement that the zones in these regions contain

10

1

i -

A y.„.

1 £.

REGION I

Source

1

d j e

1 J SL

h | i_

1 1 Jt

— + .

1 J m

n

REGION I I REGION I I I REGION IV

B l a n k P e r t u r b e d B l a n k

°. J £

REGION V

3.

jr

——r— i i • l

s • t - 1 -

I f

REGION VI

P e r t u r b e d D e t e c t o r Region Region

MATERIALS ZONES

vacuum iron copper copper + iron beryllium

Figure 1. Illustration of the tenninology: blank region, source region, perturbed region and detector region

11

identical materials. This requirement does not have to be met for

source and detector regions.

2.2 Cross-Section Sensitivity Profiles

2.2.1 Introduction

Perturbation theory is most commonly applied in order to derive

analytical expressions for the cross-section sensitivity profile. We

11 25 therefore will follow in this work Oblow's approach. ' Based on the

analytical expression, an explicit formula for the cross-section sensi

tivity profile in discrete ordinates form for a two-dimensional geometry

will then be derived.

During the last few years there has been a trend towards using gen

eralized perturbation theory for sensitivity studies. ' ' General

ized perturbation theory has the advantage that it can readily be

applipd to derive expressions for the ratio of uilinear functionals and

that it can be used to study nonlinear systems. ' Also, higher-order

expressions, based on generalized perturbation theory, have been de-

. . 57,58,61 rived. ' '

Tha differential approach is closely related to generalized pertur

bation theory and has been applied to cross-section sensitivity analysis

28 by Oblow. A more rigorous formulation of the differential approach

was made b., Dubi and Dudziak. ' Although higher-order expressic Lons

12

for cross-section sensitivity profiles can be derived, ' the practi

cality of its application has not yet been proved. ' *

The evaluation of a sensitivity profile will generally require the

solution of a direct and an adjoint problem. Such a system carries more

information than the forward equation and it is therefore not surprising

that this extra amount of information can be made explicit (e.g.,

through sensitivity profiles).

The higher-order expressions for the cross-section sensitivity pro

files derived by Dubi and Dudziak involve the use of Green's func

tions. ' The Green's function - if properly integrated - allows one

to gain all possible information for a particular transport problem. It

therefore can be expected that higher-order sensitivity profiles can be

calculated up to an arbitrary high order by evaluating one Green's func

tion. For most cases, the derivation of the Green's function is ex

tremely complicated, if not impossible. It therefore can be argued that

the Green's function carries such a tremendous amount of information

that it is not surprising that higher-order expressions for the sensi

tivity profile can be obtained, and that while the use of Green's func

tions can prove to be very valuable for gaining analytical and physical

insight, they will not be practical as a basis for numerical evaluations.

78 From the study done by Wu and Maynard, it can be concluded that a

first-order expression allows for a 40% perturbation in the cross sec

tions (or rather the mean free path) and will still yield a reasonably

accurate integral response (less than 10% error). Larger perturbations

give rapiciy increasing errors (the error increases roughly by a powe~

13

of three). Expressions exact up to the second order allow a 65% per

turbation, and a sixth-order expression allows a 190% perturbation, both

for an error less than 10%. Also, for higher-order approximations, if

was found that the error on the integral response will increase drastic

ally once the error exceeds 10%. It can be concluded therefore that the

higher-order expressions do not bring a tremendous improvement over the

first-order approximation (unless very high orders are used), while the

computational effort increases drastically. Higher-order sensitivity

analysis can only become practical when extremely simple expressions for

the sensitivity profiles can be obtained, or when a suitable approxima-

79 tion for Green's functions can be found.

2.2.2 Analytical expression for the cross-section sensitivity profile

Consider the regular aud adjoint transport equations

!.• = Q , (4)

and

I".*"" = R , (5)

where 4> and 4> represent the forward and the adjoint angular fluxes, L

and L are the forward and adjoint transport operator, Q is the source,

14

and R is the detector response function. The integral response, I, can

then b^ written as

I = <R,4» (6)

or

I = <Q,* > , (7)

where the symbol < , > means the inner product, i.e., the integral over

the phase space. In a fully converged calculation I will be equal to

I. For the perturbed system, similar expressions can be obtained:

L * = Q , (8) P P

L ' V = R , (9 ) P P

I = <R,* > , (10) P P

and i" = <Q,<t>"> , O D p p

where

<t> = <J> + 6* , ( 1 2 ) P

15

•* = ** + 6** , (13) P

and I = 1 + 6 1 . (14) P

From Eqs. ( 9 ) , (13) , and (5) we have

L*.6** = (L* - L*).4>* - (15) P P

Further, we have from Eqs. (14), (11), (6), (12), and (9)

61 = Ip - I ,

= <R,4> - <t» , P

= <R,64» ,

or 61 = <L~* ,6*> (16) P P

Using the d e f i n i t i o n of the adjoint t r anspo r t operator and Eqs. (15) and

(16) transforms to

61 = < * , L 6* > ,

o r

61 = <• ,(L* - L*)**> . (17)

It is assumed that the perturbed differential scattering cross

section can be expressed as a function of the unperturbed differential

scattering cross section by

KJl*&Q.',W) = C.X (r.CMT.EÊ') , (18) °v p

and similarly for the total cross section

ITp(r,E) = C.I^r.E) , (19)

where C is a small quantity, which can be a function of E and Q. Defin

ing 6C = C - 1, we have

* Z„ (r,E) - 2T(r,E) I (r.sXT ,E-»E) - Z (r,fM)',EÊ')

fir - lP ~ l ~ = SP ~ 2 (2Q) °L ZjCr.E) Is(r,S>ST,E+E')

U U J

so that

61(E) = 6C J dr J dfi.*{-ZjCr.E).•"(£,«,£)

+ J dE' J dST I (r.fM}',EÊ').*"(r,n',E')} . (21) s

The cross-section sensitivity function Ey (E) is defined by x

Fi <E> = a r y r • (22)

X X X

I

17

X

and can be approximated by

Fz (E) = | J dr J dQ {-*(r,n,E).ZxJ(r,E).**(r,n,E)

+ / dfl' J dE•*(r>a,E).ZXJS(r,i>flÊÊ•).*"'(r,fiÊ,)} . (23)

The sensitivity function Fy (E) represents the dependence or sensi-x

tivity of a design parameter of interest to a particular cross section

Z at energy E. The first term is usually referred to as the loss term

27 and the second term is called the gain term.

The cross-section sensitivity profile P| is then defined as x

Pf = - ^ / dE F7 (E) . (24) x AuB E x

g

The scaling factor Au" is the lethargy width of group g and is intro

duced as a normalization factor in order to remove the influence of the

choice of the group structure.

Remarks

1. In the previous section Z represents a partial cross section for

a particular material. Z can be an absorption cross section, a

total cross section, a differential scattering cross section, a re

action cross section, etc. Therefore Z has a surpressed index

13

which indicates the specific partial cross section. When evaluat

ing the cross-section sensitivity profile for a partial cross sec

tion only the appropriate part, either the loss term or the gain

term, will have to be considered in Eq. (23). When the partial

cross section is not related to the production of secondary parti

cles (e.g., a differential scattering cross section) the sensitiv

ity profile in the multigroup form is referred to by Gerstl as a

vector cross-section sensitivity profile. Obviously such cross

sections contribute only to the loss term.

2. It is possible to define a net or a total sensitivity profile,

which can be obtained by summing the loss and the gain terms for

various partial reactions. The net sensitivity profile can be used

to determine how important a particular element is with respect to

a particular response.

3. Note that while deriving an expression for the cross-section sensi

tivity profile, we implicitly assumed that the response function

was independent from the partial cross section for which a sensi

tivity profile is desired. If this assumption does not hold, an

extra term has to be added to the previously obtained expressions.

When the response function is also the cross section for which a

sensitivity profile is sought, the sensitivity function will take

the form

19

<4>,L5 * > 3I/I = <M1 + * (25) 91 /Z I I • U J

x' x

where 1- represents that portion of the transport operator that x

contains the cross-section set {I }. In this expression the first

term is a direct effect and the second term is an indirect effect.

If the direct effect is present, the indirect effect will usually

be negligible. A summary of the various possibilities is given in

Table I.

4. The spatial integration in Eq. (23) has to be carried out over the

perturbed regions only.

2.2.3 Explicit expression for the cross-section sensitivity profile in discrete ordinates form for a two-dimensional geometry represen-tation

Coordinate system

The coordinate systems for x-y and r-z geometry are shown in Figs.

53 2a and 2b. In both geometries (J» was chosen to be the angle of rota-

2 2 2 tion about the (J-axis such that dQ = dp.d$, and since £ + (J + v = 1,

we have

20

TABLE I: FORMULAS FOR THE SENSITIVITY FUNCTION

Case

a. I = <R,*>, where 2. ^ R

b. I = <R,*>, where 2. = R

and Z i <f- L

c. I = <R,*>, where 2. = R

and I. c L l

Sensitivity Function

F z = <**,Lj*>/I i

F 2 = <R,4»/I i i i

1

F z = <R,*>/I + <*",LI *>/I : i i 1

direct indirect j effect effect <

The direct effect is usually • dominant J

< > indicates the inner product over the phase space £

L stands for the transport operator

I. represents that portion of the transport operator which contains cross-section {2.}

I

C means is included in

£ means is not included in

21

Figure 2. a. Coordinates in x-y geometry

22

n

- i

Figure 2. b. Coordinates in r-z geometry

23

I = (1 - |i2Asin* ,

n = (i - \i )^.cosi(i .

Therefore both the x-y and the r-z geometry representation will

lead to identical expressions for the sensitivity profile, with the

understanding that in x-y geometry the angular flux is represented by

<l>(x,y,|j,(j>), and by <l>(r,z,|J,<}») in the case of r-z geometry.

We now will derive an expression for the sensitivity profile in an

x-y or in an r-z geometry representation.

Method

Before deriving an expression in a discrete-ordinates formulation

and a two-dimensional geometry for Eq. (23), a brief overview of the

methods used is outlined.

Gain term:

I* order to represent the differential scattering cross section in

a multigroup format, the common approach to expand the differential

scattering cross section in Legendre polynomials is used. The num

ber of terms in the expansion is a function of the order of aniso

tropic scattering. The Legendre polynomials are a function of the

scattering angle u (Fig. 2). Introducing spherical harmonics

24

functions and applying the addition theorem for spherical har

monics, the dependence on |J can be replaced by y's and ty's. The

angular fluxes are expanded in flux moments. The integrals are

replaced by summations. Defining multigroup cross sections rn

expressions for the gain term can be obtained.

Loss term:

An explicit expression for the loss term can be derived based on

angular fluxes or based on flux moments. In order to check the

internal consistency in SENSIT-2D both methods will b- applied.

The derivation of an expression based on angular fluxes is

straightforward: the integrations are replaced by summations and

the appropriate multigroup cross sections are defined. An expres

sion as a function of flux moments can be obtained by expanding the

fluxes in flux moments, using spherical harmonics functions. The

orthogonality relation of spherical harmonics is applied, the

integrations are replaced by summations and appropriate multigroup

cross sections are defined. Finally an expression for the loss

term is the result.

Analytical derivations

Expand the differential scattering cross section in Legendre poly

nomials according to

25

LMAX

X x s«Wr,E^) = IX>S(M0,E^E') = I ?gl P£(po)ZS(fi(E.E-) , (26)

£=0

where the P « ( M )'S are the Legendre polynomials and LMAX the order of

anisotropic scattering. Here, the scattering angle (J can be written as

u = fi.Q' = fi Q' + fi fi' + fi Q' , ro - - xx yy z z

or

MO = w' + nn' + W ,

= MM' + ( l - M ^ U - H 1 2 ) ^ cosij cos(j>' + (1-M2)^(1-(J '2)^ sin<J) sin({j ,

or

Mo = W' + (l-^Al-M'2)'"5 cos(0-$') .

The spherical harmonics addition theorem states that (see e.g., Bell and

Glasstone )

?li[io) = P£(M)P£(M') + 2 Z H ^ j - PJCM)PJ(M')cos[k(«t-0') ] , (27)

where the P.(|j)'s are the associated Legendre polynomials. The above

expression can then be reformulated as

26

H (2-6. ) ( l - k ) ! . P£ ( Mo ) = I dSkj! Pj(M)P5(M')cos[k((l.-$')]

SL I

k=0

(2 -6 t e )U-k ) l1^ [(2-6 k o)(£-k)!]^ (I^kTl J [ oFkTI J PJJOOPJOJ' )

x ( c o s k<{> cos k $ ' + s i n k<j> s i n k<J>') (28)

We define

KJ(M,«) [(2-6ko)U-k)^ |_ (Wi J P£(fj)cos k(|) f29)

and

QJCM.O) = |-(2-6ko)(£-k)!^

[ U+kJ] J P£(p)sin k(ji (30)

so that

P£(po) = I {RJ(M,4.)RJ(M*,0') + QJCJJ.^QJCM',*')} k=0

(31)

The Q terms will generate odd moments which will vanish on integration,

thus the Q terms will be omitted in the following discussion The R.

terms are the spherical harmonics polynomials. Using the above expres

sion for P,(M ) in the expansion of the scattering cross section, we

have

27

LMAX . £ . R Zv (IWi'.E^E') = 2 ^|Iil (E*E') I R^(H,0)Rj(M',<f) , (32) x.s - - £=0 w s,x. k=0 x. x.

where LMAX is the order of anisotropic scattering.

The second term of the sensitivity profile, Eq. (24), becomes

V l » LMAX „ S. 1 7i - ± - J d E / d r / d E ' I ^ 1 .(E^E1) 1 2 J dp J d<»

IAuB E V 0 £=0 ' k=0 -1 o g

k l n k * Rj(|J,4>)»(r,n,E) . 2 / dp' J CKP'RJCM' , 4>' ) * (r.D'.E') . (33)

-1 o

Note that

^ dn J d« R*(M,0)R>,*) = 6 ^ , (34)

and therefore the angular flux can be expanded according to

OB £

4>(0,E) = I (2fi+l) I R W ( E ) , (35a) fi=0 k=0 * *

k 1 1 k

where 4^(E) = J d|J J d$ Rj*(fi,E)/2n , (35b) -1 p

28

and similarly for the adjoint angular flux

£ * (n,E) = I (2£-U) I Rp*!k<n , (36a)

£=0 k=0

.... i n where *^K(E) = J d J" d$ RJ *(n,E)/2n , (36b)

-1 o

Introducing these expansions in the sensitivity piofile, the gain term

becomes

, GttAX g'-l g-1 LMAX Pf = -25_ / dr 2 / dE' / dE I (2J?+1)2 Jl-Z') "x.gain lAu8 V g'=l E , E £=0 S'

6 g g

2 *h.(E)4- K(E-) , (37) k=0

where GMAX is the number of energy groups. Defining

E . E _j lTl *£8*£k8' " ^ dE' J" dE 1 s £(*>E')*£(E)*£

k(E') . (^ Eg' g

and discretizing over the spatial variable we have

29

. GMAX LMAX _ , £ IPERT . *.., Pf = -^- I I ^ l U f f I I V.*58(i)<l.:k8 (i) , (39)

x.gain IAu8 g'=l £=0 * k=0 i=l

where IPERT is the number of perturbed spatial intervals and i indicates

the spatial interval. If there is no upscattering, and introducing

9. IPERT j. M*88' = 4TI Z (2£+l) I V.*„8(i)**k8'(i) , (40) * k=0 i=l x * *

we have

. LMAX GMAX , , P| = -^- 2 2 i f | 4-f . (41) x.gain IAu8 £=0 g'=g s,Jt *

The loss term of the sensitivity profile is given by

E _ j

P | = —^r J dE J d r J .ffl{-*(r,n,E)I (E)**(r,fl,E)} , (42) x.loss IAus E V '

g

1 V 1 1 n * - 4 / dE / dr 2 J dM S W {-«KM,4»,E)ZV T ( E ) * (M.E)} , IAug E V -1 o X j l

g (43)

-Lr V l MM * - ^ J dE J dr Z (E) Z wm*(Mm,<fr_,E)* (p_ ,* ) , (44) IAu8 E V »' ra=lm m m

g

30

where $ = tan" ]( l - u2 - n2)^/P for M > 0 , (45) in "m' ' ""m ' rm ' v J'

• n = U n - ' d - n2 - *fch/vn + n f o r p m < 0 , (46)

and MM is the number of angular fluxes per quadrant.

Define

MM Eg-1 ^ MM 2 / dE 1 (E)-*(p 0 Ej-4» (p ,0 ,E) = if I * V 6 , (47) __ i r xi' m m m m x . i . m m m-1 E m=]

g

so that

. IPERT MM

V l o s s IAu8 X'T i=l 1 m=l m B "

Introducing

IFFRT MM f =4n I V . I w *8(i)*"8(i) , (49)

._ , l , m m m i= 1 m=l

we have

Pf = ~~ ll y\g • (50) x.loss IAuE *'x

Note that the gain term was expressed as a function of flux moments,

while the loss term was expressed in terms of angular fluxes. When the

gain term is expressed as a function of flux moments, a very useful

31

relationship between the Ws and the x's will be obtained. For this

case, substituting Eqs. (36) and (38) into Eq. (42), the loss term can

be expanded as

s| = - ~ J J dr I T J dji J difr I (2£+l) 1 R V x.loss IAug E V x,t -1 o £=0( k=0 )

" I £ k*kl I (2£+l) I Rp* * . (51)

£=0( k=0 * *)

Using the orthogonality relations Eq. (34) and defining the multigroup

total cross section for group g by

LMAX £ . ... LMAX £ Eg-1 , ... I l l * T*J

8(r)*:Kg(r) = 1 I J dE I T(E)*:(r,E)4.:k(r,E)

£=0 k=0 *' £=0 k=0 E ' g

(52)

we have after discretizing the spatial variable, r, and truncating the

summation over £,

2g LMAX IPERT

P| = ~4" *>T £ (2£+D 2 V.*{8(i)»{8(i) . (53)

Sc.loss IAu* £=Q .=1

32

Introducing

LMAX X8 = I Yf8 , (54)

£=0 *

the expression for the loss term reduces to Eq. (50) again.

Summary

. LMAX GMAX , Pf = - i — -X8 xE + 2 1 if f Vf8 , (55) x I.Au8 X>1 £=0 g'=g S'* *

where

o i T = total macroscopic cross section for reaction type x, x, l

1^ | = £'th Legendre coefficient of the scattering matrix element for ' energy transfer from group g to group g' , as derived from the

differential scattering cross section for reaction type x,

1PERT £ . .,, , H-88 = 4TI(2£+1) I I V.*„ 8(i)V 8 (i) (56) * i=l k=0 x "• *•

= spatial integral of the product of the spherical harmonics expansions for the regular and adjoint angular fluxes,

IPERT MM X8 = An I V. I *8(i)*"8(i)w (57) . , I , m m m i=l m=l

= numerical integral of the product of forward and adjoint angular fluxes over all angles and all spatial intervals described by i=l . . ., IPERT,

LMAX = 2 ¥§* . (58)

£=0 "•

33

Note that expression (55) is identical vith the expression for the

46 cross-section sensitivity profile in a one-dimensional formulation.

The flux moments can be expressed in terms of angular fluxes corre

sponding to

and

•J« = / dM J d<t.R>8(n)/2n = I ** R J ( M • )wB , (59)

-1 0 m=l

*^k8 = J dp J d<t>R£t>"8 ( 0 ) / 2 n = 2 ** Rj(Mm,Om)wm " ( 6 0 )

-1 o m=l

R„(Q) = spherical harmonics function

V. = volume of rotated triangles

Aug = lethargy width of energy group g

= In (E6/E6 ), where E 6 and E 6 are upper and lower energy group boundaries

= integral response as calculated from forward fluxes only

IDET IGM n

= I I V.Rf^8(i) i=l g=l

R. = spatially and group-dependent detector response function.

46 2.3 Source and Detector Sens .> vity Profiles

Source and detector sensitivity profiles indicate how sensitive the

34

integral response I or I is to the energy distribution of the source,

or to the detector response R. The integral response I can be calcu

lated from the forward flux, according to Eq. (63), or from the adjjint

flux, according to Eq. (64). When the integral response is calculated

from the adjoint flux it will be denoted as 1 . Ideally, I will be

equal to I .

The sensitivity of the integral response to the energy distribution

of the detector response function or the source can therefore be ex

pressed by the sensitivity profiles

E n

P| = J dr / dE J dQ R(r,E).*(r,fi,E) / I.Aug (61)

d g

and

g-1 a. .x. P* = J dr j dE J dQ Q(r,fi,E).4>"(r,fi,E) / l".Aug , (62) g V E

s 8

where R(r,E) is the detector response and Q(r,fi,E) is the angular source,

and V. and V are the volumes of the detector and the source region. I d s

was used in the denumerator of P2 and I was used in the denominator of K

P? for internal consistency. It is obvious that the integral source and

detector sensitivities, S» and SD, will be equal to one.

35

It is possible to derive an expression similar to Eq. (61) for the

sensitivity of the integral response to the angular distribution of the

source. The derivation of explicit expressions for P| and P° is

straightforward. The detector sensitivity profile as a function of the

scalar fluxes becomes

IDET n

P8 = I V..R8.*„8(i) / I.Au8 , (63) R . , 1 1 0

i=l

where the 4>f8(i) are the scalar fluxes for group g at interval i, IDET

is the number of detector intervals, and Rr is the detector response at l r

interval i for group g.

For the source sensitivity profile in case of an isotropic source

Eq. (62) transforms into

ISRS . P8 = I V..Q8.*|J8(i) / l".Au8 , (64) ^ i=l

where Q? is the voluminar source for group g at source interval i.

In the case of an anisotropic source we defined Q8(r,Q) by

Q8(r,fi).* 8(r,n) = J dE Q(r,Q,E).* (r,n,E) , (65) E 6

36

and expand the angular source according to

IQAN £ . . Q8(r,fi) = Q8(r,M,*) = Z (2£+l) I R£(M,<t>).Qp8(r)/27l , (66)

£=0 k=0 * *

where IQAN is the order of anisotropy of the source.

Substituting Eqs. (65) and (66) in Eq. (63), discretizing the

spatial variable and using Eq. (36), the expression for the source

sensitivity profile becomes

ISRS IQAN £ . ... . P8 = 2. I V. I (2£+l) I Q8K(i)*"8K(i)/l".Au8 W i=l x £=0 k=0

As in Eq. (61) we can also define an angular source sensitivity func

tion. The angular source sensitivity function indicates how sensitive JL

the integral response I is to the angular distribution of the source,

or

0D

FJ? = \ J dr J dE Q(r,n,E).*"(r,n,E)/I* . (68) w i V 0

s

37

2.4 Sensitivity Profiles for the Secondary Energy Distribution and the Secondary Angular Distribution

The theory of the secondary energy distribution (SED) and the sec

ondary angular distribution (SAD) sensitivity analysis was originated by

43-46 Gerstl. Physically the only difference between a secondary energy

distribution and a cross-section sensitivity profile is the way in which

the integration over the energy variable is carried out. The "hot-cold"

and the "forward-backward" concepts lead to a simple formulation of

secondary sensitivity theory and can easily be incorporated in an uncer

tainty analysis. Even when both those concepts are a rather coarse

approximation they have the advantage that they are simple and can be

physically understood.

A more rigorous formulation might be possible, but its simple

physical interpretation would be lost. The primary restriction on the

application of secondary energy distribution and secondary angular dis

tribution sensitivity profiles is the lack of cross-section uncertainty

information in the proper format.

2.4.1 Introduction

The expression for the sensitivity profile for the differential

scattering cross section is part of the gain term of the cross-section

sensitivity profile and takes the form

38

E °° Pz (CWT ,E^E') = — ~ / dr J" dfl J8"1 dE J dE' / dfi' x Au8.I E o

g

x Rz (r.CWV.E+E') , (69) x.gain

where K. (r ,Q->Q',E->E') is a shorthand notation for x.gain

R2 (r.fMi'.E^E1) = *(r,fi,E)Z (r.JW)',E-»E')* (r.fl',E') (70) x,gain '

and similarly,

R 2 (r,fi'-Q,E'-»E) = *(r,n,E)I (r ,Q'-»n,E->E)**(r ,Q,E) . (71) x,gain '

Equation (70) gives the contribution to the integral detector re

sponse, I, from the particles born at position r with energy E', travel

ing in direction Q', since

I = <*,L""V"> = <*"',L*> . (72)

Similarly, Ry (r,(M)',E'->E) gives the contribution to the integral x.gain

detector response from the particles born at position r_, with energy E,

traveling in direction Q.

As it turns out, up to this point there is no difference in the

physical interpretation of Eqs. (70) and (71). The way the integrations

39

are carried out will distinguish between the differential scattering

cross-section sensitivity profile and the secondary energy distribution

and secondary angular distribution sensitivity profile.

2.4.2 Further theoretical development

In this section we will elaborate ou the physics behind the deriva

tion of SEDs and SADs. Consider

F z (E,E') = j J dr J dn J dfl' 8 Z (r.fM^ ,E-»E') . (73) x,s x,gain

In this expression Fj represents the fractional change in the inte-x,s

gral response per unit fractional change in the differential scattering

cross section I (E-*E'); i.e., it is the fractional change in the in

tegral response when the number of particles that scatter from E into E'

is increased by one percent. Obviously this will always be a positive

effect and will therefore be included in the gain term.

Similar to Eq. (73),

~* 1 E 8 _ 1 °° Pf = j j dE J dE' J dfi J dfi'Rj. (r.fMT ,E-*E') (74) x,s E 0 x,gain

40

represents the fractional change in the integral response when the num

ber of particles that scatter from group g is increased by one percent.

The tilda in Eq. (74) is introduced to distinguish from a lethargy nor

malized sensitivity profile.

In the adjoint formulation the equivalent of Eq. (73) will be

F z (E' ,E) = FSED(E' ,E) = i J dr J dQ J dfi'Itj (r.Q'^.E'^E) , x,s x,gain

(75)

which represents the fractional change in the integral response per unit

fractional change in differential scattering cross section I (E'-»E),

i.e., it is the fractional change in the integral response when the num

ber of primary particles that scatter from E' to E is increased by one

percent, or for that matter that the number of secondary particles that

were scattered from E into E' were increased by one percent. Again,

this will always have a positive effect and will therefore constitute a

gain term in the sensitivity profile.

Define

Y i » y / dE J dE' J dn / dfi'Rj (r.fi'+Q,E'+E) . (76)

E 0 x.gain g

While there is no difference in the physical meaning of Eqs. (73)

and (75), the formulations (74) and (76) are different. Equation (74)

41

3 g =

SED


ber of secondary particles that were scattered into group g have been

increased by one percent.

It is clear from these examples that, depending on the way the

integrations are done, several different sensitivity profiles can be

constructed. In order to stuoy the secondary angular distribution, we

can introduce

F z (Q,E') = FSAD(fi,E') = i J dr / dE J dn'Rz (r,fi'^,E'-E) x.gain 0 x.gain

(77)

This expression gives the fractional change in the integral response

when the number of secondary particles scattered from initial energy E'

into final direction Q is increased by one percent. It will therefore

be clear that

PSAD(0)=J-dE'FSM)(n,E') (78)

is the fractional change in the response function when the number of

secondary particles which were scattered into direction Q was increased

by one percent.

42

2.4.3 Secondary energy and secondary angular distribution sensitivity profiles

A double secondary energy distribution (SED) sensitivity profile is

defined by

E . E , . e'e 1 8 _ 1 8 _ 1

?IA = — r - ^ J" dE J" dE- ; dr ; dn ; dn-R5 (r.Q'-c.E'-*) *LU IAu8Au8 E E , x.gain g g

(79;

The energy integrated SED sensitivity profile becomes

CD g-l PSED = ~^~ J" dE' J' dL J dr J dQ'J^ (r,fl'-»fl,E'-E) . (60)

IAug 0 E x.gain g

The differential sensitivity profile for the angular distribution

of secondary particles scattered from initial energy E' is

, Eg''l °° PLn(9) = — — J" dE' J dE / dr J dQ'R5 (r,Q'-»fl,E-E') (62) b A U IAu8 E 0 x.gain

g

An energy integrated SED sensitivity profile can be defined by

43

*WQ) = T J dE' / dE J" d£ / ^ 5 (r.fi'^E+E') . (82) b A U * 0 0 s.gain

2_.4.4 Integral sensitivities for SEDs and SADs

In order to make the sensitivity and uncertainty analysis for

secondary energy distributions and secondary angular distributions less

tedious, Gerstl introduced the concepts of the "hold-cold" SED and the

"forward-backward" SAD integral sensitivity:

5| = / dE P« (E) - J dE P* (E) , (83) btu H Q T btu C 0 L D bAU

and

SSAD = f I „ d° PSAD(^ " / « W ? > • ("> forward backward angles angles (M>0) ((J<0)

The forward-backward SAD integral sensitivity can be interpreted as

the fractional change in the integral response when the number of sec

ondaries which were scattered forward is increased by one percent, while

the number of secondaries that were scattered backwards (p<0) is de

creased by one percent. The integral SAD sensitivity is a positive

number which is labeled "forward" or "backward" depending whether the

first or the second term in Eq. (84) is the larger one. Physically,

44

that positive number indicates how much more sensitive the response

function is to forward scattered particles than to backward scattered

particles, or vice versa.

For the hot-cold integral SED sensitivity, the concept of the

median energy has to be introduced. In the multigroup formulation, the

median energy defines the energy boundary which roughly divides the

cross-section profile into two equal parts. The median energy and the

43 integral SED sensitivity are illustrated in Fig. 3. Note that the

median energy g' is a function of the primary energy group g'. For that

reason also the integral SED sensitivity will depend on g'.

The hot-cold integral SED sensitivity expresses the fractional

change in integral response when the number of secondaries which scatter

in the "hot" part of the secondary energy distribution is increased by

one percent while the number of secondaries scattered into the "cold"

part is decreased by one percent. The integral hot-cold SED sensitivity

is a positive number> labeled "hot" or "cold" depending on which term

dominates in Eq. (83). That number indicates how much more sensitive

the integral response is to particles scattered into the hot part of the

secondary energy distribution than to particles scattered into the cold

part, or vice versa.

45

„Fe(n,2n)_#

2 g E

15

14

13

'cold

I

12=g

"hot

11

10

9

^

' ho t - £ " g g=l

G

'cold = £

median

a s= a !=• 1/2 t hot co ld

out (MeV)

U"p

pb SED

15

14

12=gr

1 n 10

9

»

6, median E o u t ( N e V )

"HOT" or

"COLD"

. . 43 Figure 3 . D e f i n i t i o n of median energy and i n t e g r a l SED s e n s i t i v i t y

46

2.4.5 Explicit expressions for integral SED sensitivity profiles in a two-dimensional geometry representation

The expression for the double SED sensitivity profile, Eq. (79), is

similar to the gain term of the cross-section sensitivity profile, Eq.

(24). By comparing Eq. (79) with Eq. (24) and using Eq. (41), the ex

plicit expression for the double SED sensitivity profile becomes

LMAX p8 .8 = I 1 ?& "8 4.8 8 (85) SED TA gA g' /. S,£ £ * ( J

IAuBAu6 £=0 '

From Eqs. (85) and (80), it follows that the energy integrated SED sen

sitivity profile for the case of no upscattering can be represented by

g LMAX P8 = _ 2 _ i i Z8 ^g 4,8 8 ( 8 6) SFD P «; £ £ btU IAu8 g'=l £=0 S'* iL

Using the definition for the integral SED sensitivity (83), it becomes

clear that

8 m ( 8 ) GMAX „ o

8=8 8=8m(8 )+ 1

where g (g') is defined in Fig. 1.

47

2.6 Design Sensitivity Analysis

Design sensitivity analysis provides a method to estimate changes

in integral response for a slightly altered design. The results are

exact up to the second order with respect to the corresponding flux

changes, but only exact up to the first order with respect to design

changes. The theory presented in this section is applicable only when

the design changes can be expressed in terms of macroscopic cross-

section changes. Methods based on generalized perturbation theory have

been applied to design sensitivity analysis. '

The integral response for the perturbed system can be expressed by

35 Eq. (88) for the adjoint difference formulation,

*AD = <R'*> " ** ,AL<1,> = J ' 6IAD ' ( 8 8 )

and by Eq. (89) in the forward di f ference formulation

I r o = <Q,I > - <*,AL * > = I - 6 I F D . (89)

Proceeding in a manner similar to the derivation of the cross-

section sensitivity profile, the second-order term in the right hand

side of Eqs. (88) and (89) can be written as

48

and

6IAD = J" dE / dl S dn{*(r,n,E)6I T(r,E)*'"(r,fi,E) v vd

+ J dE' J dn'*(r,n\E')6I c(r,n'-»n,E'-»E)*"(r,n,E)} , (90)

Co

61 = J" dE J dr / dQ{*(rfQfE)6I (r,E)*"(r,n,E) 0 V x>1

s

+ J dE' J dn'*(r,n,E)6Z (r.JM)',E-E')*"~(r,fi',E') . (91) 0 x's

In the above expressions we used

6Ix,T = Zx,T " ^ T ' (92)

and

61 = Z - I , (93)

where Z refers to a perturbed cross section and Z to a reference cross

section.

A design sensitivity coefficient X can be defined as the ratio of

the integral response for the altered design over the integral response

49

for the original model. Depending whether the forward or the adjoint

difference method are used, the design sensitivity coefficient equals

or

XAD = W 1 = J * "A D / 1 • (94)

XFD = W1" = 1 " 61¥Dn" • (95)

Note that respectively, I and I were used in the denominator of Eqs.

(94) and (95) for internal consistency. Numerically 6l.n and 61™ are

* 35

identical; I and I , however, can be different. Gerstl and Stacey in

dicate that the adjoint formulation is more accurate for perturbations

closer to the detector, while the forward difference method gives better

results for perturbations closer to the source. If both reference

fluxes 4> and • are completely converged, Eqs. (94) and (95) will give

identical results.

Explicit expressions for Eqs. (94) and (95) can be formulated. The

procedure for the evaluations of 61.-, and 61™ is similar to the deriva-AD FD

tion of the cross-section sensitivity profile and leads to the equations

IGM ( LMAX g , , i

*" g=l ( x'1 £=0 g'=l s'* s»* J

50

and

6IFD =

IGM Z g=l

6Ix,T^

LMAX I

GMAX I

1=0 g'=g s,£

ugg (97)

51

3. APPLICATION OF SENSITIVITY THEORY TO UNCERTAINTY ANALYSIS

Sensitivity theory can be used to do an uncertainty analysis by

introducing the concepts of cross-section covariance matrices and frac

tional uncertainties for SEDs. In this chapter we will explain how sen

sitivity profiles can be used in order to calculate the uncertainty of a

reaction rate due to the uncertainties in the cross sections.

3.1 Definitions

Let I represent a design parameter depending on a multigroup cross-

section set {!}, so that

I = HI±) , (98)

where the index i can reflect a group, a partial cross section or a

material.

52

The variance of I is defined as the expected value of the square of

the difference between the actual value of I and the expected value of

I, or

Var(I) = E{(61)2} = E{(I - E{I})2] . (99)

The standard deviation of I is the square root of the variance,

AI E [Var(I)]^ . (100)

The covariance of a and b i s defined as

OS 00

Cov(a,b) = E{6a-6b) ^ J J da .db.(a - E{a}) . (b - E{b}) . f (a ,b) , _0O - 0 0

(101)

where f(a,b) is a joint probability density function. A nonzero covar

iance between the quantities a and b indicates a mutual dependence on

another quantity. Obviously we have

Cov(a.a) = Var(a) , (102)

since f(a,a) = 1.

A relative covariance element is defined by

53

R(a,b) z. Cov(a,b)/a.b (103)

3.2 Cross-Section Covariance Matrices

During the experimental evaluation of cross-section data, statis

tical errors arise from the fact that two similar experiments never

agree completely. Also a systematic error reflects the fact that no

equipment and no evaluation procedure is perfect, and that - among other

factors - reference standards are used.

Cross-section covariance data describe the uncertainties in the

multigroup cross sections and the correlation between those uncertain

ties. A nonzero nondiagonal covariance matrix element indicates that

there was a common reason why an uncertainty in two different (e.g.,

partial cross sections or energy range) cross section was introduced.

The evaluation procedure for cc'ariance data is tedious and requires a

2 30 31 sophisticated statistical analysis. ' '

Multigroup cross-section covariance data are ordered in covariance

matrices. Such a covariance matrix contains GMAX rows and GMAX columns,

where GMAX is the number of energy groups. A covariance matrix can

contain covariance data of a particular partial cross section with

itself over an energy range , with a different cross section for the

same element, or with a partial cross section of a different element.

It has become a common practice to include formatted uncertainty

data in the ENDF/B data files. Even though the uncertainty files are

54

still missing for many materials in ENDF/B-V, extensive work is under

way. Based on these uncertainty data, covariance libraries can be con-

32 33 structed. ' A 30-group covariance library based on ENHF/B-V which

contains most of the elements commonly used in reactor shielding has

33 been constructed by Muir and LaBauve. The covariance data in this

library were processed into a 30-group format by using the NJOY

code. ' In this particular library, called COVFILS, the multigroup

cross sections and the relative covariance matrices for H, B, C, 0,

Cr, Fe, Ni, Cu, and Pb are included. Another covariance library was set

32 up by Drischler and Weisbin.

3.3 Application of Cross-Section Sensitivity Profiles and Cross Section Covariance Matrices to Predict Uncertainties

Using first-order perturbation theory, the change in the integral

response I, 61, as a consequence of small changes in I. can be approxi

mated by

61 S I |J- 61. . (104) oi., l

l 1

We fur ther have

Var(I) = E{612} = E{ Z §§- § | - 61.61 } , (105) i>j i J

55

or

Var(I) = 2 ||- ||- Cov(Z ,1 ) . (106)

From Eqs. (100) and (106) it now becomes obvious that

Cov(Z.,Z.)

= i PZ PZ —ri~2~ • (107)

J x s i , j i j i j I II

where Pj and P, are sensitivity profiles, and the subscript xs refers

i j to reactor cross sections.

The concept of covariance data and sensitivity profiles leads to a

simple way to evaluate the error in I. The first part in the summation

requires sensitivity profiles and is highly problem dependent. The

second part requires cros^-section uncertainty information and is prob

lem independent.

When trying to apply the theory presented here, very often covari

ance data will be missing for certain materials. One way of going

around this problem would be to substitute the covariance file of the

missing material by a covariance file for another material for which the

45 cross sections are less well known. Other methods to eliminate this

problem would be to make very conservative estimates. '

The most conservative method would be to assume that the error in

the cross section is the same for all groups and equal to the largest

error for any one group. In that case it can be shown that '

56

AI.

max 1 1 I IPv I (108)

3.4 Secondary Energy Distribution Uncertainty Analysis

For evaluat: g uncertainties in the integral response due to un

certainties in the secondary energy distribution we will follow Gerstl's

44 46 approach ' and introduce the spectral shape uncertainty parameter for

the hot-cold concept.

When the total number of secondaries scattered from group g' are

held constant, then necessarily

51 HOT

"HOT

61 COLD

"COLD f , (109)

Therefore f , quantifies the uncertainty in the shape of the SEDs and is

g . -44

c a l l e d t h e s p e c t r a l shape u n c e r t a i n t y pa rame te r ( F i g . 4)

I t now becomes p o s s i b l e t o e x p r e s s t h e r e l a t i v e change in i n t e g r a l

r e sponse due t o t h e u n c e r t a i n t y in t h e secondary ene rgy d i s t r i b u t i o n in

a form s i m i l a r t o Eq. ( 1 0 7 ) :

^ = I P , 61 ,

8 8 6 ^E

ISED g .8 SED I

(110)

g "*g

57

median E ,. (MeV) out

S(Te +f if Eg

Figure 4. ITiterpretation of the integral SED uncertainty as spectrum shape perturbations Bnd definition of the spectral shape uncertainty paraneter "f" (ref. UU)

58

Substituting Eqs. (87) and (109) in Eq. (110), it follows that

= 2 SED g'

!«' f , JSED g (111)

Denote f , by f., where the index j refers to a particular nuclear o J V;

reaction, e .g . , (n,2n), at specific incident energy g ' , and^jlet f. rep

resent some different reaction/primary energy combination. Then the

uncertainty in integral response corresponding to correlated uncertain

t ies of a l l SEDs for a specific isotope is

J SED

Var(I)

I2 = E i (61)' = E< I S

SEDJSEDfi f j | (112)

n I S SED i.j

SED SED Cov(f.,f.)

i J (113)

If the spectral shape uncertainty parameters for a specific par

ticle interaction, identified by the subscript SL, are assumed to be

fully correlated, it can be shown that

Cov(f.,f .) . , x i' j cor(+l)

lCov(f.,f.)]' [Cov(fj,fj)]'» , (114)

so that

59

pL sh. ^ ' ^ " i g - v ^ l (115)

or,

n ll/J, ' ^ E D ' " ^ ^ ' ^ • (116)

If N independent measurements of the same SED are available, the

values for Var(f. ,) can easily be evaluated. For each cross-section

evaluation, weights, w , are assigned, then

n n ,n _ HOT " COLD , _ . „ ., , 1 1 7. f„. = FT7T* » f o r a ~ 1,2..-N (117) V EjcT

with

N E{f" } = I w f" = 0 . (118)

8 n=l g

The variance of f , will be g

2 2

Var(f ,) =E{f* } = I °HOT PCOLD ( n g ) 8 8 n=l n lE{a}]2

Var(f ,) is called the fractional uncertainty for the secondary energy

distribution and is identified by the symbol F. A short program which

evaluates the values of F has been written by Muir; the results for

the 30-group neutron structure is shown in Table II.

60

TABLE II

MEDIAN ENEUGIKS (E1, IN MEV) AND FRACTIONAL UNCERTAINTIES (F) FOR SECONDARY m

ENERGY DISTRIBUTIONS AT INCIDENT NEUTRON ENERGIES EQ

E 0

16.0

14.25

12.75

11 .00

8 .90

6 . 9 3

A. 88

3.27

2 .55

1.99

1.55

1.09

0 .66

0 .40

0 .24

0 .13

12C E'

m

14.71

13.00

11 .71

9 .77

7.35

5 .96

4 .46

2 .63

2 .16

1.73

1.34

0 .95

0 .57

0 .35

0 .21

0 .12

F _

0 .071

0 .059

0 .054

0 .060

0 .048

0 .035

0 .010

0 .010

0 .010

0 .005

0.005

0 .005

0 .005

0 .005

0 .005

0 .005

16

E* m

14.62

13 .33

11 .93

9 .82

7.90

6 .04

4 .57

3 .09

2 .31

1.79

1.35

0 .94

0 .60

0 .34

0 .22

0 .12

0

F

0 .088

0 .072

0 .062

0 .057

0 .050

0 .030

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 ,010

c E1

m

3.27

8 .65

11.42

10.48

8.79

6 . 8 3

4 .81

3.21

2 .48

1.93

1.50

1.05

0 .64

(r

r

F

0 .17

0 .15

0 .13

0 .11

0 .09

0 . 0 8

0.07

0 .06

0 .05

0 .04

0 .03

0 .02

0 .0?

:i'f. 4 5;

Fe F.'

IN

4 .49

5 .99

11.17

10.57

8.77

6 .86

4 .81

3.21

2 .49

1.93

1.51

1.06

n . 6 3

0 .39

)

F

0 .11

0 . 1 0

0 .10

0 .09

0 .08

0 .07

0 .07

0 .06

0 .06

0 .06

0 .05

0 . 0 3

0 .02

0 .02

Ni E'

m

14.95

13.97

12.67

10.91

8 .85

6 . 8 8

4 . 8 3

3.24

2 .49

1.94

1.50

1.06

* F

0 . 1 3

0 .11

0.11

1.10

0 .09

0 . 0 8

0 .07

0 .06

0 .05

0 .04

0 .03

0 .02

Ci E'

m

3 .42

3 .51

4 . 3 0

10.42

8.81

6 .86

4 . 8 2

3 .22

2 .50

1.94

1.51

1.06

0 .64

0 .39

0 . 2 4

0 .12

i

F

o.n 0 . 1 0

0 . 1 0

0 .09

0 .08

0 .07

0 .07

0 . 0 6

0 .06

0 .06

0 .05

0 . 0 3

0 .02

0 .02

0 . 0 2

0 .02

W E' m

1.86

2 .17

1.91

1.57

1.24

6 .66

4 . 8 3

3 .25

2 .51

1.96

1.51

1.06

0 .66

0 .38

0 . 2 2

0 . 1 2

F

0 . 1 2

0 .10

0 .10

0 .09

0 .08

0 .07

0 .07

0 . 0 6

0 .06

0.05

0 .05

0 .05

0 .04

0 . 0 3

0 . 0 2

0 .01

3.5 Overall Response Uncertainty

The overall response uncertainty will be of the form

= V B ] S E D + l ^ x s

(120)

where

'SED i L JSED,i (121)

and

[Ml2 = z [Ml2

L XS i , k L 1 J XS, i , k (122)

The index i reflects the uncertainties in the various materials.

It was assumed that the effects from SED uncertainties for all possible

reactions which generate secondaries are uncorrelated. It is also

assumed that the uncertainties due to the SEDs are uncorrelated with

other uncertainties due to reaction cross sections (XS), and that the

uncertainties between the reaction cross sections themselves are un

correlated.

62

Remarks

1. To be absolutely correct, a term reflecting the uncertainty in

the secondary angular distribution should be included. Due to

the difficulty in generating uncertainty data from ENDF/B-V in

the proper format, we do not include that term.

2. In order to evaluate the sensitivity profiles, we should keep

in mind that the form of the sensitivity profile will depend

on the particular reaction cross section for which a response

is desired (Table I).

53

4. SENSIT-2D: A TWO-DIMENSIONAL CROSS-SECTION AND DESIGN SENSITIVITY

AND UNCERTAINTY ANALYSIS CODE

4.1 Introduction

The theory explained in the previous chapters has been incorporated

in a two-dimensional cross-section and design sensitivity and uncertain

ty analysis code, SENSIT-2D. This code is written for a CDC-7600

machine and is accessible via the NMFECC-network (National Magnetic

Fusion Energy Computer Center) at Livermore. SENSIT-2D has the capa

bility to perform a standard cross-section and a vector cross-section

sensitivity and uncertainty analysis, a secondary energy distribution

sensitivity and uncertainty analysis, a design sensitivity analysis and

an integral response (e.g., dose rate) sensitivity and uncertainty

analysis. As a special feature in the SENSIT-2D code, the loss term of

the sensitivity profile can be evaluated based on angular fluxes and/or

flux moments.

64

SENSIT-2D is developed with the purpose of interacting with the

TRIDENT-CTR code, a two-dimensional discrete-ordinates code with tri

angular meshes and an r-z geometry capability, tailored to the needs of

the fusion community. Angular fluxes generated by other 2-D codes, such

a* DOT, TWODANT, TRIDENT, etc., cannot be accepted by SENSIT-2D due to

the different format. The unique features of TRIDENT-CTR (group de

pendent quadrature sets, r-z geometry description, triangular meshes)

are reflected in SENSIT-2D. Coupled neutron/gamma-ray studies can be

performed. In contrast with TRIDENT-CTR however, SENSIT-2D is re-

stricted to the use of equal weight (EQ ) quadrature sets, symmetrical

with respect to the four quadrants. Upscattering is not allowed.

Many subroutines used in SENSIT-2D are taken from SENSIT or

TRIDENT-CTR. SENSIT-2D is similar in its structure to SENSIT, but is an

entirely different code. Unlike SENSIT, SENSIT-2D does not use the

BPOINTR package for dynamical data storage allocation, but rather uses

a sophisticated pointer scheme in order to allow variably dimensioned

arrays. As soon as an array is not used any more, its memory space

becomes immediately available for other data. SENSIT-2D does not in

clude a source sensitivity analysis capability and cannot calculate

integral responses based on the adjoint formulation. This has the dis

advantage that no check for internal consistency can be made. There

fore, other ways have to be found in order to determine whether the

fluxes are fully converged. One way for doing so would be to calculate

the integral response based on the adjoint formulation while performing

65

the adjoint TRIDENT-CTR or the adjoint TRDSEN run, and compare with the

integral response based on the forward calculation.

SENSIT-2D requires input files which contain the angular fluxes at

the triangle midpoints multipled by the corresponding volumes, and tfc _•

adjoint angular fluxes at the triangle midpoints. A modified version of

TRIDENT-CTR, TRDSEN, was written by T. J. Seed to generate these flux

files. A summary of these modifications was provided by T. J. Seed and

is included as Appendix B. After a TRIDENT-CTR run, the TRDSEN code

will use the dump files generated by TRIDENT-CTR, go through an extra

iteration, and write out the angular fluxes in a form compatible with

SENSIT-2D. Both SENSIT-2D and TRDSEN use little computing time compared

with the time required by TRIDENT-CTR.

Tht Lc iLUT-e- of SENSIT-2D are summarized in Table III. The SENSIT-

2D source code is generously provided with comment cards and is included

as Appendix A.

4.2 Computational Outline of a Sensitivity Study

A flow chart (Fig. 5) illustrates the outline for a two-dimensional

sensitivity and uncertainty analysis. From this figure it becomes imme

diately apparent that a sensitivity analysis requires elaborate data

management. The data flow can be divided into three major parts: a

cross-section preparation module, in which the cross sections required

by TRIDENT-CTR and SENSIT-2D are prepared, a TRIDENT-CTR/TRDSEN block,

66

TABLE III: SUMMARY OF THE FEATURES OF SENSIT-2D (PART I)

SENSIT-2D: A Two-Dimensional Cross-Section and Design

Sensitivity and Uncertainty Analysis Code

Code Information:

* written for the CDC-7600

* typical storage, 20K (SCM), 80K (LCM)

* number of program lines, 3400

* used with the TRIDENT-CTR transport code

* typical ru/i times, 10-100 sec

Capabilities:

* computes sensitivity and uncertainty of a calculated

integral response (e.g., dose rate) due to input cross

sections and their uncertainties

* cross-S£Ction sensitivity

* vector cross-section sensitivity and uncertainty

analysis

* design sensitivity analysis

* secondary energy distribution (SED) sensitivity and

uncertainty analysis

67

TABLE III: SUMMARY OF THE FEATURES OF SENSIT-2D (PART 2)

SENSIT-2D

TRIDENT-CTR Features Carried Over into SENSIT-2D:

* x-yor r-z geometry

* group-dependent S order

* triangular spatial mesh

Unique Features:

* developed primarily for fusion problems

* group dependent quadrature order and triangular mesh

* can evaluate loss-term of sensitivity profile based

on angular fluxes and/or flux moments

Current Limitations:

* can only interact with TRIDENT-CTR transport code

* not yet implemented on other than CDC computers

* based on first-order perturbation theory

* upscattering not allowed

68

CROSS-SECTION

PREPARATION BLOCK

TRIDEHT-CTR ( f o r w a r d )

TRIDRN'T-rrR ( a d j o i n t )

- •» - [ MIMP2 J

L TRIRENT-CTR/TRDSEN BLOCK

TRIIS r.ti (oiljoinl )

SENSIT-2D MODULE

SENSIT-2D

Figure 5. Computational outline for a two-dimensional sensitivity analysis with SENSIT-2D

where the angula-. fluxes in a form compatible with SENSIT-2D are gener

ated, and a SENSIT-2D module, which performs the calculations and manip

ulations necessary for a sensitivity and uncertainty analysis.

4.2.1 Cross-section preparation module

There are many possible ways to generate the rnultigroup cr ss-

section tables required by SENSIT-2D and TRIDENT-CTR. The flow chart

of Fig. 5 illustrates just one of these possibilities. All the codes

mentioned here are accessible via the MFE machine. Basically, three

codes are required: NJOY, TRANSX, and MIXIT. Starting from the ENDF/B-V

64 data file, the NJOY code system generates a rnultigroup cross-section

library (MATXS5) and a vector cross-section and covariance library

(TAPE10). A covariance library can be constructed by using the ERROR

33 module in the NJOY code system.

From the multigroup cross-section library (MATXS5), the desired

isotopes can be extracts i by the TRANSX code' and will be written on

73

a file with the name XSLIBF5. The MIXIT code can make up new mate

rials by mixing isotopes from the XSLIBF5 library. The cross sections

used in SENSIT-2B have to be written on a file called TAPE4. The cross

sections used in TRIDENT-CTR and TRDSEN will be on file GEODXS. SENSIT-

2D and TRIDENT-CTR include the option to feed in cross sections directly

from cards.

70

4.2.? The TRIDENT-CTR and TRDSENS block

SENSIT-2D requires regular angular fluxes at the triangle center-

points, multipled by the corresponding volumes, and adjoint angular

fluxes at the triangle centerpoints. TRIDENT-CTR does not write out

angular fluxes. Therefore the TRDSEN version of TRIDENT-CTR was written

by SEFD. TRDSEN makes use of the flux moment dump files, generated by

TRIDENT-CTR. These dump files will be the starting flux guesses for

TRDSEN. TRDSEN will perform one more iteration and write out the

angular fluxes. In this discussion we will represent the dump file

families by DUMP! for the regular flux moments, and DOIP2 for the

angular flux moments. Except for a different starting guess option,

TRDSEN requires the same input as TRIDENT-CTR.

4.2.3 The SENSIT-2D module

The SENS1T-2D code performs a sensitivity and uncertainty analysis.

When vector cross sections and their covariances are required, they have

to be present on a file with the name TAPE10. If the cross section data

are read from tape, they have to be written on a file called TAPE4. The

regular angular fluxes at the triangle centerpoints multiplied by the

corresponding volumes (TAPE11, TAPE12,...) and the adjoint angular

fluxes at the triangle centerpoints (TAPE15, TAPE16,...) can be quite

voluminous. Writing out large files can be troublesome on the MFE

71

machine when there is a temporary lack of continuous disk space. There

fore TRIDENT-CTR and SENSIT-2D have the built-in option to specify the

maximum number of words to be written on one file. This limit has to

v « ~ ~ ^ L .* _ v. .. - L. •-- *.!._*. - I T * i n j _ * _ — _ i - *. „ j * _ . — ,*..,...-. kJC O t - C L l X g U C I J U U ^ U t-KJ C 1 I 3 U 1 C L U d L d l i L U C U U A UC3LCI 1 C X d L. C U «- \J uHl l - £ L l * U f >

can be written on one file. 1 000 000 words per file is usually a

practical size and is the default in TRIDENT-CTR.

SENSIT-2D can generate four more file families:

1. TAPE1, which contains the regular scalar fluxes at the triangle

centerpoints.

2. TAPE20, TAPE21,..., which are random access files and contain the

adjoint angular fluxes at the triangle centerpoints,

3. TAPE25, TAPE26,..., containing the regular flux moments at the

triangle centerpoints, multipled by the corresponding volumes,

4. TAPE30, TAPE31,..., which contain the adjoint angular fluxes at the

triangle midpoints.

SENSIT-2D has the option of not generating those file families, but

using those created by a former run. The flux momeats are constructed

from the angular fluxes according to the formula

2. r,z ., m 9. 'm'Mir m r.z ' m=l '

where the w 's are the quadrature weights, the R„"'s the spherical har

monics functions, and MN the total number of angular fluxes.

72

4.3 The SENSIT-2D Code

In this section the structure of the SENSIT-2D code, its options

and capabilities will be explained in more detail. SENSIT-2D is a

powerful sensitivity and uncertainty analysis code. The description of

this code from the user's point of view is given in the user's manual.

A.3.1 Flow charts

The overall data flow within the SENSIT-2D module is repeated in

Fig. 6. A simplified flow chart is illustrated in Fig. 7. The main

parts of the flow chart include these steps:

* The control parameters and the geometry related information are

read in.

* The quadrature sets and the spherical harmonics functions required

to generate the flux moments are constructed.

* The adjoint angular fluxes at the triangle centerpoints are written

on random access files, flux moments are generated and scalar

fluxes are extracted.

* A detector sensitivity analysis is performed; if desired an uncer

tainty analysis is done.

* The x's and (J>'s which form the essential parts of the cross-section

and secondary energy distribution sensitivity profiles are calcu

lated for each perturbed zone and for the sum over all perturbed

zones.

73

Vector cross-sections and crvarJane- data

(only required for vector cross-i.?" U n

•ensltlvlty and uncertainty analysis)

Cross sections in LASL format (only required If cross sections are intended to be read froir cross-«»ction file}

Angular fluxes at triangle midpoints multiplied by the corresponding voluoes

•Joint flux moments at triangle center-points

P 6* B adjoint angular regular flux ccrert5

*°d fluxes at triangle triangle centerrsir. t *

chi's aidpoints mltiplied by the

(random access file) corresponding volur.es

Figure 6. Data flow for the SENSIT-2D module

74

http://volur.es

*

*

*

*

*

Read

CALL

CALL

CALL

CALL

the input parameters .

EBK'D: Read in the neutron and garca-ray s t r u c t u r e and c a l c u l a t e the lethargy width/group.

CEOM: Read and e d i t the geometry.

S*:CON: Read quadrature information and c a l c u l a t e EO - s e t s .

TAPAS: Assign f i l e s t o the f l u x e s .

do for forward and adjoint fluxes

i >

* CALL PKGE!; : Calculate spheTical hamonics f u n c t i o n s .

* CALL FLl.'EKS'.: Calculate f lux eiDnents. Extract s c a l e r f l u x e s .

Continue

*

*

*

CALL

CALL

CALL

DETSEN'

CHIS

PSIS

—————^—_-——————- - ———^_ • Calculate d e t e c t o r response and d e t e c t o r

s e n s i t i v i t y p r o f i l e . If des i red a de tec tor uncerta inty a n a l y s i s i s pelformed.

: Calculate c h i ' s based on sngulaT f l u x e s i f d e s i r e d .

: Calculate the p s i ' s bassd or. f lux B D ^ K . : S and s t o r e in LOi. If des ired c h i ' s based on f lux moments v i l l be c a l c u l a t e d .

Figure 7. Flov? chart for SrTJSIT-2D (part 1)

75

0- Is a vector c r o s s - s e c t i o n s e n s i t i v i t y and uncerta inty a n a l y s i s desired?

• In the case that « Sm uncerta inty a n a l j s i s i s required. T#«H *o th* SEP ur.ccrtiir*!.} u » i a .

• CALL SUBS : Read In cross s e c t i o n s . Convert to nacroscoplc cross sec t ions v ia nunber d e n s i t i e s .

• CALL SUBS : Read In second c r o s s - s e c t i o n set In the case that a design s e n s i t i v i t y analys i s Is d e s i r e d ,

• CALL SUB6 ; Extract vector croas aect lons and s c a t t e r i n g matrix fror- the f u l l c ros s - sec t ion t a b l e . In case of a design s e n s i t i v i t y ana lys i s ca l cu la te d e l t a s l g a a . Calculate macroscopic scat ter ing cross s e c t i o n s .

K do for all perturbed zones and for the lun over all perturbed zones )

• CALL Tf.rr or ItX'A: Trint appropriate de f ln i t i j - i s wher. . h i s sect ion i s passed for tnt f i r s : tj.oe.

• CALL POINTS : Set pointers in order to c"hoos» proper c h i ' s and p s i ' s .

* CALL 5UB8 : Calculates and e d i t s f ina l Tesults of s e n s i t i v i t > analys i s If I t I s not a SEE a n a l y s i s .

* CALL SVE11 : Calculates and e d i t s f ina l r e s u l t s for a SHi s e n s i t i v i t > and uncerta inty analys i s (neutror. groups o n l y ) .

Read covarlance data and provide uncerta int icF in the integral response for the fu l l y correlaiec" and the non-corre lated case .

Stop

Figure 7. Flow chart for SENSIT-2D (part 2)

76

i This section performs a complete sen*1ri\Mry a«H nnrprta^ry analysis for vector cross sections. The code requires a covariance file to be given in LASL error-file format which contains pairs of vector cross sections with their corresponding covariance matrix.

< do for all successive cases

* CAlL SUB5V : Reads cross-section ID from input file. Reads number density from input file. Reads relative vovariance data (via COVAPJ)) Generates macroscopic cross sections. Reads cross sections (via COYARD).

* CALL POINTB: Set appropriate pointers for chi's and psi's. * CALL SUBBY : Cor.putes and edits sensitivity profiles and

folds ther; with the covariance matrix in order to obtain the relative integral response.

•C continue } * CALL SVB9V : Cor.putes p a r t i a l sues of individual response

variances. Reads SUMSTPT and SVJ—ND (variances to be su—ed) assuring no corre la t ion between individual vector cross-section e r r o r s , the to ta l variance and the re la t ive standard var ia t ion are cor.puted.

( stop )

Figure 7. Flow chart for SF.i;SIT-2I) (part 3)

77

Up to this point, all the subroutines used are different from those used

in the SENSIT code. The remaining calculations are done with SENSIT sub

routines.

* Cross sections are read in.

* Vector cross sections are extracted.

* Sensitivity profiles are calculated used in the appropriate t|t's and

X's.

* If desired to do so, an uncertainty analysis is performed.

* A vector cross-section sensitivity and uncertainty analysis can be

performed and partial sums of individual response variances can be

made.

4.3.2 Subroutines used in SENSIT-2D

Table IV summarizes the subroutines used in SENSIT-2D and indicates

their origin in case they were taken over or adapted from another code.

The essential difference between SENSIT and SENSIT-2D is the way that

the geometry is described and how the i|>'s and the x's are calculated.

Basically, all the subroutines are called from the main program with a

few exemptions when subroutines are called from other subroutines. The

subroutines for SENSIT-2D which were not taken over from other codes

will now be described. For the SENSIT subroutines we refer to the

, 46 user s manual.

78

TABLE IV: LIST OF SUBROUTINES USED IN SENSIT-2D

Name Subroutine Origin If Taken From Another Code, Were Changes Made?

EBND

GEOM

SNCON

TAPAS

PNGEN

FLUXMOM

DETSEN

CHIS

P0INT4B

PSIS

P0INT8

SUB5

SUB6

TEXT

TESTA

SUB8

SUB 11

SUBBY

SUB9

SUB9V

SUB5V

COVARD

SET ID

SENSIT-

SENSIT-

TRIDENT

SENSIT-

TRIDENT

SENSIT-

SENSIT-

SENSIT-

SENSIT-

SENSIT-

SENSIT-

SENSIT

SENS1T

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

2D

2D

-CTR

2D

-CTR

2D

2D

2D

2D

2D

•2D

-

-

yes

-

yes

-

-

-

-

-

-

yes

no

no

no

yes

yes

no

no

no

no

no

no

79

1. Subroutine EDNB. Neutron and gamma-ray energy group structures

are read in from cards and the lethargy widths for each group are

calculated.

2. Subroutine GEOM. Geometry related information is read in and

edited.

3. Subroutine SNCON. This routine was taken and adapted from the

TRIDENT-CTR code. The EQ cosines and weights are calcualted. The

quadrature information is edited whenever I0PT is 1 or 3.

4. Subroutine TAPAS. Files are assigned to the various flux data.

The filenames for the angular fluxes are read from the input file.

Those filenames will have to be of the form TAPEXY, where XY will

be the input information. Filenames in the same format will then

be assigned to the adjoint angular fluxes (on sequential files in

this case), and the flux moments. The maximum number of words to

be written on each file is controlled by the input parameter

MAXWRD. Groups will never be broken up between different files.

5. Subroutine PNGEN. This subroutine originates from the TRIDENT-

CTR code. Spherical harmonics functions, used for constructing

flux moments, are calculated. For the adjoint flux moment calcu

lation the arrays related to the spherical harmonics will be re

arranged to take into account the fact that the numbering of the

angular directions was not symmetric with respect to the four

quadrants in TRIDENT-CTR.

6. Subrout ine FLUXMOM. The adjoint angular fluxes will be re

written on a random access file. The direct and adjoint flux

80

moments are constructed and written on sequential files. In the

case that the input parameter IPREPl, it is assumed that those

manipulations are already performed in an earlier SENSTT-2D run.

In this casp one has tn make sure that the parameter MAXVRD was not

changed. While creating the regular flux moments, the scalar

fluxes will be extracted and written on a file named TAPE1.

7. Subroutine DETSEN. From the scalar fluxes, the integral re

sponse for each detector zone is read from input cards. The detec

tor sensitivity profile is calculated and edited. In the case that

the input parameter DETCOV equals one, a covariance matrix has to

be provided, subroutine SUB9 will be called and a detector response

uncertainty analysis is performed.

8. Subroutine CHIS. The x's are calculated for each perturbed

zone and for the sum over all perturbed zones based on angular

fluxes. In the case that the parameter ICHIMOM equals one, this

subroutine will be skipped and the x's will be calculated based

on flux moments via the iji's.

9. Subroutine P0INT4B. This subroutine sets LCM pointers for the

flux moments which will be used in SUB4B.

10. Subroutine PSIS. The (JJ'S are calculated for each of the per

turbed zones and for the sum over all perturbed zones based on

flux moments. In the case that ICHIMOM is not equal to zero also

the x's will °e calculated from flux moments. In the case that

parameter IPREP equals one, the iji's will be read in from file

TAPE3.

81

11. Subroutine P0INT8. This subroutine sets pointers for the

appropriate x's and i|)'s, used in subroutine SUB8.

5. COMPARISON OF A TWO-DIMENSIONAL SENSITIVITY ANALYSIS WITH A ONE

DIMENSIONAL SENSITIVITY ANALYSIS

Before applying SENSIT-2D to the FED (fusion engineering device)

inboard shield design, currently in development at the General Atomic

Company, it was necessary to make sure that SENSIT-2D will provide the

correct answers. One way for checking on the performance of SENSIT-2D

is to analyze a two-dimensional sample problem, which is one-dimensional

from the neutronics point of view, and then to compare the results with

a one-dimensional analysis. In this case ONEDANT and SENSIT are

used for the one-dimensional study, while TRIDENT-CTR, TRDSEN, and

SENSIT-2D are used for the two-dimensional analysis.

Two sample problems will be studied. The first sample problem uses

real cross-section data, while the second sample problem utilizes arti

ficial cross sections. Computing times, the influence of the quadrature

set order, and the performance of the angular fluxes versus the flux

moments option for the calculation of the chi's will be discussed.

83

LA-9232-T Thesis

UC-20d Issued: February 1982

Two-Dimensional Cross-Section Sensitivity and Uncertainty Analysis

For Fusion-Reactor Blankets

Mark Juhen Ernbrechts

." :t1-«n©

Los Alamos National Laboratory Los Alamos. New Mexico 87545

CONTENTS

Nomenclature viii

List of Figures x

List of Tables xi

Abstract xiv

1. Introduction 1

1.1 Motivation 2

1.2 Literature Review 4

2. Sensitivity Theory 7

2.1 Definitions 8

2.1.1 Cross-section sensitivity function, cross-section

sensitivity profile and integral cross-section sen

sitivity 8

2.1.2 Vector cross section 9

2.1.3 Geometry related terminology 30

2.2 Cross-Section Sensitivity Profiles 32


2.2.2 Analytical expression for the cross-section sen

sitivity profile 14

2.2.3 Explicit expression for the cross-section sensi

tivity profile in discrete ordinates for a two-

dimensional geometry representation 20

2.3 Source and Detector Sensitivity Profiles 34

iv

2.4 Sensitivity Profiles for the Secondary Energy Distribution

and the Secondary Angular Distribution 38


2.4.2 Further theoretical development W

2.4.3 Secondary energy and secondary angular distribution

sensitivity profile 43

2.4.4 Integral sensitivities for SEDs and SADs 44

2.4.5 Explicit expressions for integral SED sensitivity

profiles in a two-dimensional geometry

representation 47

2.5 Design Sensitivity Analysis 48

3. Application of Sensitivity Theory to Uncertainty Analysis 52

3.1 Definitions 52

3.2 Cross-Section Covariance Matrices 54

3.3 Application of Cross-Section Sensitivity Profiles and Cross-

Section Covariance Matrices to Predict Uncertainties 55

3.4 Secondary Energy Distribution Uncertainty Analysis 57

3.5 Overall Response Uncertainty 62

4. SENSIT-2D: A Two-Dimensional Cross-Section Sensitivity and

Uncertainty Analysis Code 64

4.1 Introduction 64

4.2 Computational Outline of a Sensitivity Study 66

4.2.1 Cross-section preparation module 70

4.2.2 The TRIDENT-CTR and TRDSEN block 71

4.2.3 The SENSIT-2D module 71

v

4.3 The SENSIT-2D Code 73

4.3.1 Flowcharts 73

4.3.2 Subroutines used in SENSIT-2D 78

5. Comparison of a Two-Dimensional Sensitivity Aniysis with a

One-Dimensional Sensitivity Analysis 83

5.1 Sample Problem ill 84

5.1.1 TR1DENT-CTR and OKEDANT results 92

5.1.2 SENSIT and SENSIT-2D results for a standard cross-

section sensitivity analysis 94

5.1.3 Comparison between a two-dimensional and a one-

dimensional cross-section sensitivity and

uncertainty analysis 99

5.2 Sample Problem #2 104

5.2.1 Influence of the quadrature order on the sensi

tivity profile 104

5.2.2 Comparison between the two-dimensional and one-

dimensional analysis of sample problem #2 Ill

5.2.3 Comparison between the x's calculated from the

angular fluxes and the x's resulting from flux

moments 114

5.2.4 Evaluation of the loss term based on flux moments

in the case of low c 116

5 .3 Conclusions 117

6. Sensitivity and Uncertainty Analysis of the Heating in the

TF-Coil for the FED 118

vi

6.1 Two-Dimensional Model for the FED 118

6.2 Two-Dimensional Sensitivity and Uncertainty Analysis for

the Heating in the TF-Coil due to SED and Cross-Sertion

Uncertainties 123

6.3 Comparison of the Two-Dimensional Model with a One-

Dimensional representation 129

7. Conclusions and Recommendations 133

8. References ! ih

Appendix A: SENSIT-2D Source and Code Listing ;^6

Appenflix B: TRIDSEN 190

vii

NOMENCLATURE

reflects energy function

reflects angular distribution

microscopic cross section

macroscopic cross section, where x indicates the

material and/or the type of cross section

macroscopic scattering cross section for material

total cross section for material x

angular flus

angular flux in discrete ordinates representation

spherical harmonics representation for the angular

flux

spherical harmonics function

Legendre polynomials

associated Legendre polynomials

transport operator

adjoint transport operator

source

response function

integral response

volume

fractional uncertainty for the secondary angular

distribution

part of the loss term in the sensitivity profile

part of the gain term in the sensitivity profile

vii i

F cross-section sensitivity function for cross section *• x

I x

p? cross-section sensitivity profile for a cross-section x

I x for group g

Au" lethargy width for group g

w quadrature weight

ix

LIST OF FIGURES

Figure 1 Illustration of the terminology: blank region,

source region and detector region

Figure 2a Coordinates in x-y geometry

Figure 2b Coordinates in r-z geometry

Figure 3 Definition of median energy and integral SED

sensitivity

Figure 4 Interpretation of the integral SED uncertainty

as spectral shape perturbations and definitions

of the spectral shape uncertainty parameter "f"

Figure 5 Computational outline for a two-dimensional sen

sitivity and uncertainty analysis with SENSIT-2D

Figure 6 Data flow for the SENSIT-2D module

Figure 7 Flow chart for SENSIT-2D

Figure 8 Cylindrical geometry representation for sample

problem ill

Figure 9 Two-dimensional (TRIDENT-CTR) representation

for sample problem #1

Figure 10 One-dimensional (ONEDANT) representation for

sample problem //l

Figure 11 Two-dimensional model for sample problem #2

Figure 12 Two-dimensional model for the FED

Figure 13 The TRIDENT-CTR band and triangle structure

for the FED

LIST OF TABLES

Table I Formulas for the sensitivity function

Table II Median Energies (E', in MeV) and fractional

uncertainties (F) for secondary energy distributions

at incident neutron energies E n

Table III Summary of the features of SENSIT-2D

Table IV List of subroutines used in SENSIT-2D

Table V 30-group energy structure

Table VI Atom densities of materials

Table Vila Comparison of the heating in the copper region

calculated by ONEDANT and TRIDE.VT-CTR

Table Vllb Computing times on a CDC-7600 machine

Table Villa Comparison of the heating in the copper region


Table Vlllb Partial and net sensitivity profiles for the

two-dimensional analysis

Table IX Comparison between the x's calculated from

angular fluxes and flux moments

Table Xa Predicted response uncertainties due to estimated

cross-section and SED uncertainties in a one-


Table Xb Predicted response uncertainites due to estimated

cross-section and SED uncertainties in a two-


Table XI Cross section table used in sample problem //2

(the P-0, P-l, P-2, and P-3 tables are identical)

Table XII Integral response for sample problem #2

Table XHIa Standard cross-section sensitivity profiles calcu

lated by SENSIT-2D for the EQ-6 case (convergence

precision 0.000], 4 triangles per zone) for sample

problem #2

Table Illb Standard cross-section sensitivity profiles cal

culated by SENSIT-2D for the EQ-12 case (conver

gence precision 0.00], 4 triangles per zone) for

sample problem #2

Table XlVa Standard cross-section sensitivity profiles cal


precision O.OCOl, 4 intervals per zone) for sample

problem #2

Table XlVb Standard cross-section sensitivity profiles cal


precision 0.0001, 8 intervals per zone) tor sample

problem i\2

Table XIVc Standard cross-section sensitivity profiles cal

culated by SENSIT-2D for the EQ-12 case (con

vergence precision 0.0001) for sample problem HI

Table XV Comparison between the x's calculates from angular

fluxes and the x's calculated for flux moments

Table XVI Atom densities for the isotopes used in the materials

xii

Table XVII

Table XVIII

Table XIX

Table XX

xiii

Predicted Uncertainties (standard deviation) due


for the heating of the TF-coil

Predicted SED and cross-section uncertainties

in the TF-coil due to uncertainties in the SS316

regions

Comparison between the x's calculated from angular

fluxes and the x's resulting from flux moments for

region 11 (SS316)

Predicted uncertainties (standard deviation) due


in regions 1 and 3 for the heating in the TF-coil.

TWO-DIMENSIONAL CROSS-SECTION SENSITIVITY AND UNCERTAINTY ANALYSIS

FOR FUSION REACTOR BLANKETS

by


ABSTRACT

Sensitivity and uncertainty analysis implement the information ob

tained "rom a transport code by providing a reasonable estimate for the

uncertainty for a particular response (e.g., tritium breeding), and by

the ability to better understand the nucleonics involved. The doughnut

shape of many fusion devices makes a two-dimensional calculation capa

bility highly desirable. Based on first-order generalized perturbation

theory, expressions for a two-dimensional SED (secondary energy distri

bution) and cross-section sensitivity and uncertainty analysis were de

veloped for x-y and r-z geometry. This theory was implemented by devel

oping a two-dimensional sensitivity and uncertainty analysis code,

SENSIT-2D. SENSIT-2D has a design capability and has the option to cal

culate sensitivities and uncertainties with respect to the response

function itself. SENSIT-2D can only interact with the TRIDENT-CTR code.

A rigorous comparison between a one-dimensional and a two-dimen

sional analysis for a problem which is one-dioiensionai from the neu-

tronics point of view, indicates that SENSIT-2D performs as intended.

A two-dimensional sensitivity and uncertainty analysis for the heat-

icg of the TF coil for the FED (fusion engineering device) blanket was

performed. The uncertainties calculated are of the same ort'rr of magni

tude as those resulting from a one-dimensional analysis. The largest un

certainties were caused by the cross section uncertainties for chromium.

xiv

1. INTRODUCTION TO SENSITIVITY THEORY AND UNCERTAINTY ANALYSIS

In a time characterized by a continuously growing demand for so

phisticated technology it should not be surprising that the production

of fusion energy might materialize more rapidly than commonly predicted.

With fusion devices going into a demonstration phase there is a need for

sophisticated nucleonics methods, tailored to the fusion community. In

a relatively short time frame fusion nucleonics has established itself

as a more or less mature subfield. In this context sensitivity theory

has become a widely applied concept which provides the reactor designer

with a deeper understanding of the information obtained from transport

ca]culations.

Under the term sensitivity theory usually algorithms based upon

classical perturbation and variational theory are understood. The scope

of this work will be limited to cross-section and design sensitivity

analysis with respect to fusion reactors. Since fusion nucleonics do

not involve eigenvalue calculations, the mathematical concepts utilized

will be simpler than those required by the fission community.

Sensitivity theory determines how a design quantity changes when

one or more of the design parameters are altered. Uncertainty analysis

1

provides the error range on a design quantity due to errors on the de

sign parameters. Sensitivity information can easily be incorporated

into an uncertainty analysis by introducing covariance matrices.

Cross-section sensitivity and uncertainty analysis will give error

estimates of response functions (such as tritium breeding ratio, heating

and material damage) due to uncertainties in the cross-section data.

Such a study will reveal which partial cross sections and in what energy

range contribute most to the error and will recommend refinements on

cross-section evaluations in order to reduce that error. Although those

results will depend on the particular response and the particular de

sign, general conclusions can still be drawn for a class of similar

18 designs. Sensitivity theory is a powerful design tool and is commonly

1-3 applied to cross-section adjustment procedures. Design sensitivity

analysis is frequently used to reduce the many and expensive computer

runs required during the development of a new reactor concept.

1.1 Motivation

The purpose of this work is to assess the state of the art of sen

sitivity and uncertainty analysis with respect to fusion nucleonics,

fill existing gaps in that field and suggest areas which deserve further

attention.

At this moment the literature about sensitivity theory is scattered

between various journal articles and technical reports. Therefore, the

2

author considered it as one of his responsibilities to provide a con

sistent monograph which explains, starting from the transport equation,

how analytical and explicit expressions for various sensitivity profiles

can be obtained. Current limitations with respect to the applicability

of sensitivity theory are pointed out and the application of sensitivity

theory to uncertainty analysis is explained. At the same time the scope

has been kept limited to those algorithms which are presently used in

calculation schemes.

Due to the particular geometry of fusion devices (toroidal geom

etry, non-symmetric plasma shape, etc.), a one-dimensional transport

code (and therefore a one-dimensional sensitivity analysis) will gener

ally be inadequate. In order to mock-up a fusion reactor more closely,

a two-dimensional analysis is required. Although a two-dimensional

4 5 sensitivity code - VIP ' - already exists, VIP was developed with a

fission reactor in mind, and does not include an r-z geometry option,

nor a secondary energy distribution capability. To answer the needs of

the fusion community, a two-dimensional sensitivity and uncertainty

analysis code, SENSIT-2D, has been written.

A sensitivity code uses the regular and adjoint fluxes of a neutron

transport code in order to construct sensitivity profiles. SENSIT-2D

requires angular fluxes generated by TRIDENT-CTR. ' TRIDENT-CTR is a

two-dimensional discrete-ordinates neutron transport code specially

developed for the fusion community. Since SENSIT-2D incorporates the

essential features of TRIDENT-CTR, i.e., triangular meshes and r-z geom

etry option, toroidal devices can be modeled quite accurately. SENSIT-

3

2D has the capability of group-dependent quadrature sets and includes

the option of a secondary energy distribution (SEO) sensitivity and un

certainty analysis. An option to calculate the loss term of the cross-

section sensitivity profile based on either flux moments or angular

fluxes is built into SENSIT-2D. The question whether a third-order

spherical harmonics expansion of the angular flux will be adequate for a

c

2-D sensitivity analysis has not yet been adequately answered. The

flux moment/angular flux option will help provide an answer to that

question.

As an application of the SENSIT-2D code, a two-dimensional sensi

tivity and uncertaintly analysis of the inboard shield for the FED

(fusion engineering device), currently in a preconceptual design stage

by the General Atomic Company, was performed.

1.2 Literature Review

The roots of cross-section sensitivitv theofv can be traced to the

9 10

work of Prezbindowski. ' The first widely used cross-section sensi

tivity code, SWANLAKE, was developed at ORNL (Oak Ridge Nat ional Lab

oratory). In order to include the evaluation of the sensitivity of the

response to the response function, SWANLAKE was modified to SWANLAKE-UW

by Wu and Maynard. Already early in its history, sensitivity theory

was applied to fusion reactor studies. It has now become a common

practice to include a sensitivity study in fusion neutronics. '

4

The mathematical concepts behind sensitivity theory are based on

2A-99

variational and perturbation theory. ~ The application of sensitiv

ity profiles to uncertaintly analysis was restricted not due to a lack

of adequate mathematical formulations, but due to the lack of cross-

section covariance data. An extensive effort to include standardized

30-34 covariance data into ENDF/B files has recently been made.

The theory of design sensitivity analysis can be traced to the work

of Conn, Stacey, and Gerstl. ' ' ' The current limitation of de

sign sensitivity analysis is related to the fact that the integral

response is exact up to the second order with respect to the fluxes, but

only exact to the first order with respect to design changes. There

fore, only relatively small design changes are allowed. The utilization

42 of Pade approximants might prove to be a valuable alternative to

higher-order perturbation theory, but has not yet been applied to design


4 5 The two-dimensional sensitivity code VIP ' was developed by

Childs. VIP is oriented towards fission reactors and does not include a

design sensitivity option, nor a secondary energy distribution capa

bility.

The theory of secondary energy distribution (SED) and secondary

angular distribution (SAD) sensitivity and uncertainty analysis was

43-45 46 originated by Gerstl and is incorporated into the SENSIT code.

47 The FORSS code package has been applied mainly to fast reactor stud-

48 49

ies ' but can be applied to fusion reactor designs as well. Higher-

order sensitivity theory ' ' still seems to be too impractical to

5

be readily applied. Recently however, the French developed a code

52 system, SAMPO, which includes some higher-order sensitivity analysis

capability.

6

2. SENSITIVITY THEORY

In this chapter the theory behind source and detector sensitivity,

cross-section and secondary energy distribution (SED) sensitivity, and

design sensitivity analysis will be explained. Starting from the trans

port equation, expressions for the corresponding sensitivity profiles

will be derived. Those formulas will then be made more explicit and

applied to a two-dimensional geometry. The theory presented in this and

the following chapter is merely a consistent combination and reconstruc-

, , , 3,13,16,17,18,43-46,53 tion of several papers and reports. ' ' ' ' ' '

Since up to this time no single reference work about the various

concepts used in sensitivity and uncertainty analysis has been pub

lished, the author uses the most commonly referred to terminology. In

an attempt to present an overview with the emphasis on internal consist

ency, there might be some minor conflicts with the terminology used in

earlier published papers.

7

2.1 Definitions

2.1.1 Cross-section sensitivity function, cross-section sensitivity profile and integral cross-section sensitivity

Let I represent a design quantity (such as a reaction rate, e.g.,

the tritium breeding ratio), depending on a cross-section set and the

angular fluxes. The cross-section sensitivity function for a particular

cross section I at energy E, Fj (E), is defined as the fractional x

change of the design parameter of interest per unit fractional change of

cross section 2 , or x'

X X X

In a multigroup formulation the usual preference is to work with a

sensitivity profile Pf , which is defined by

P| = JUL. . _JL , (2) x 3I8/I8 Aug

x x

where Au8 is the lethargy width of group g and I 8 is the multigroup

cross section for group g. The sura over all the groups of the sensi

tivity profiles for a particular group cross section I 8, multiplied by

8

the corresponding lethargy widths, is called the integral cross-section

sensitivity for cross section I , or

Sl = I P| • Au8 , x g x

= / dE F z (E) . (3) x

The integral cross-section sensitivity can be interpreted as the

percentage change of the design parameter of interest, I, resulting from

a simultaneous one percent increase of the group cross sections 1° in

all energy groups g.

2.1.2 Vector cross section

The term "vector cross section" describes a multigroup partial

cross-section set with one group-averaged reaction cross section for

each group. Such a cross-section set can be described by a vector with

GMAX elements, where GMAX is the number of energy groups. The term

vector cross section was introduced by Gerstl to discriminate it from

the matrix representation of a multigroup cross-section set. Differ

ential scattering cross sections can obviously not be described in the

form of a vector cross section.

9

2.1.3 Geometry related terminology

Under the term region we will understand a collection of one or

more zones. A zone will always describe a homogeneous part of the reac

tor. We will make a distinction between source regions, detector

regions and perturbed regions, and as a consequence between source,

detector and perturbed zones. We will introduce the term blank region

for a region that is neither a detector, source or perturbed region. A

zone will further be divided into intervals.

The source region will describe that part of the reactor which con

tains a volumetric source. The detector region indicates the part of

the reactor for which an integral response is desired. In the perturbed

region changes in one or more cross sections can be made.

A source or a detector regions can contain more than one zone, and

each zone can be made up of a different material. Due to the mathemat

ical formulations a perturbed region can still contain more than one

zone, but in this case all the zones have to contain identical materials.

If there is more than one perturbed region, all those regions should

contain the same materials.

The geometry-related terminology is illustrated in Fig. 1. In this

case, there are six regions; a source region, two perturbed regions, one

detector region and two blank regions. The source region contains three

zones (identified by a, b, and c). The first zone, a, is a vacuum,

while the other two zones are made up of iron. Note that both perturbed

regions satisfy the requirement that the zones in these regions contain

10

1

1 -

£ H-1 S.

1

d 1 e

- i *

_h | _i

i | Js

.—4

1 J m

ii 2. 1 £

.a

_r

— r — f i i i

» i

REGION I REGION II REGION III REGION IV REGION V REGION VI

Source Blank Perturbed Blank Region Region Region Region

Perturbed Detector Region Region

MATERIALS ZONES

vacuum iron copper copper + iron beryllium

Figure 1. Illustration of the terminology: blank region, source region, perturbed region and detector region

11

identical materials. This requirement does not have to be met for

source and detector regions.

2.2 Cross-Section Sensitivity Profiles

2.2.1 Introduction

Perturbation theory is most commonly applied in order to derive

analytical expressions for the cross-section sensitivity profile. We

11 25 therefore will follow in this work Oblow s approach. ' Based on the

analytical expression, an explicit formula for the cross-section sensi

tivity profile in discrete ordinates form for a two-dimensional geometry

will then be derived.

During the last few years there has been a trend towards using gen

eralized perturbation theory for senn'tivity studies. ' ' General

ized perturbation theory has the advantage that it can readily be

appljpd to derive expressions for the ratio of bilinear functionals and

that it can be used to study nonlinear systems. ' Also, higher-order

expressions, based on generalized perturbation theory, have been de-

, 57,58,61 rived.

The differential approach is closely related to generalized pertur

bation theory and has been applied to cross-section sensitivity analysis

28 by Oblow. A more rigorous formulation of the differential approach

ade by Dubi and Dudziak. ' Although higher-order expressions was mac

12

for cross-section sensitivity profiles can be derived, ' the practi

cality of its application has not yet been proved. ' '

The evaluation of a sensitivity profile will generally require the

solution of a direct and an adjoint problem. Such a system carries more

information than the forward equation and it is therefore not surprising

that this extra amount of information can be made explicit (e.g.,

through sensitivity profiles).

The higher-order expressions for the cross-section sensitivity pro

files derived by Dubi and Dudziak involve the use of Green's func

tions. ' The Green's function - if properly integrated - allows one

to gain all possible information for a particular transport problem. It

therefore can be expected that higher-order sensitivity profiles can be

calculated up to an arbitrary high order by evaluating one Green's func

tion. For most cases, the derivation of the Green's function is ex

tremely complicated, if not impossible. It therefore can be argued that

the Green's function carries such a tremendous amount of information

that it is tiot surprising that higher-order expressions for the sensi

tivity profile can be obtained, and that while the use of Green's func

tions can prove to be very valuable for gaining analytical and physical

insight, they will not be practical as a basis for numerical evaluations.

78 From the .study done by Wu and Maynard, it can be concluded that a

first-order expression allows for a A0% perturbation in the cross sec

tions (or rather the mean free path) and will still yield a reasonably

accurate integral response (less than 10% error). Larger perturbations

give rapidly increasing errors (the error increases roughly by a power

13

of three). Expressions exact up to the second order allow a 65% per

turbation, and a sixth-order expression allows a 190% perturbation, both

for an error less than 10%. Also, for higher-order approximations, if

was found that the error on the integral response will increase drastic

ally once the error exceeds 10%. It can be concluded therefore that the

higher-order expressions do not bring a tremendous improvement over the

first-order approximation (unless very high orders are used), while the

computational effort increases drastically. Higher-order sensitivity

analysis can only become practical when extremely simple expressions for

the sensitivity profiles can be obtained, or when a suitable approxima-

79 tion for Green's functions can be found.

2.2.2 Analytical expression for the cross-section sensitivity profile

Consider the regular aiid adjoint transport equations

L.* = Q , (4)

and

!*.** = R , (5)

where 4> and 4> represent the forward and the adjoint angular fluxes, L

and L are the forward and adjoint transport operator, Q is the source,

14

and R is the detector response function. The integral response, I, can

then be written as

I = <R,<t» (6)

or

I = <Q,4» > , (7)

where the symbol < , > means the inner product, i.e., the integral over

the phase space. In a fully converged calculation I will be equal to

I. For the perturbed system, similar expressions can be obtained:

LPV Q ' (8)

L * = R , (9) P P

I = <R,4> > , (10) P P

and I = <Q,<5>"> , 0 0 P P

where

4> = <{> + 6* , (12) P

15

<t>" - *" + 64>" (13) P

and I = 1 + 6 1 . (14) P

From Eqs. ( 9 ) , (13) , and (5) we have

-«. J . J. J. J.

L .6* = (L - L ).<*> . (15) P P

Fu r the r , we have from Eqs. (14) , (11) , ( 6 ) , (12) , and (9)

61 = I - I , P

= <R,<J> - *> , P

<R,64» ,

or 61 = <L <t> ,6<t» (16) P P

Using the definition of the adjoint transport operator and Eqs. (15) and

(16) transforms to

61 = <* ,L 6* > , P P

or

61 = «J> ,(L* - L*)**> . (17)

It is assumed that the perturbed differential scattering cross

section can be expressed as a function of the unperturbed differential

scattering cross section by

zs (r.fW}',E-»E') = C.I (r.tWl',E-»E') , (18)

and similarly for the total cross section

ITp(r,E) = C.LpCr.E) , (19)

where C is a small quantity, which can be a function of E and Q. Defin

ing 6C = C - 1, we have

lTJr,E) - ZT(r,E) Isp(r,ft*g' ,E->E) - Z^r.fMi' ,E-»E' )

X^r.E) = ^ ( r ^ ' ^ E 1 ) 6C = TP -' T -' = SP — ~ ' 1 s--*-- ( .

so that

61 (E) = 6C J dr / dO.* {-IT(r,E).4>''(r,fi,E)

+ J dV J dfl' 2 (r.JWT ,E->E').<J> (r,Q\E')} . (21)

The cross-section sensitivity function Fj (E) is defined by x

h (E) = aTTT • <22)

X X X

*

17

and can be approximated by

F2 (E) = | J dr / dfi {-*(r,n,E).I T(r,E).**(r,fi,E) x '

+ / dfl' / dE't(r,o,E).I (r.CWy.E-E'^'tr.n'.E')} . (23)

The sensitivity function F^ (E) represents the dependence or sensi-x

tivity of a design parameter of interest to a particular cross section

I at energy E. The first term is usually referred to as the loss term

27 and the second term is called the gain term.

The cross-section sensitivity profile P| is then defined as x

Pi = - ^ J dE F 7 (E) . (24) x Au8 E x

g

The scaling factor Au° is the lethargy width of group g and is intro

duced as a normalization factor in order to remove the influence of the

choice of the group structure.

Remarks

1. In the previous section 2 represents a partial cross section for

a particular material. X can be an absorption cross section, a

total cross section, a differential scattering cross section, a re

action cross section, etc. Therefore I has a surpressed index

18

which indicates the specific partial cross section. When evaluat

ing the cross-section sensitivity profile for a partial cross sec

tion only the appropriate part, either the loss term or the gain

term, will have to be considered in Eq. (23). When the partial

cross section is not related to the production of secondary parti

cles (e.g., a differential scattering cross section) the sensitiv

ity profile in the multigroup form is referred to by Gerstl as a

vector cross-section sensitivity profile. Obviously such cross

sections contribute only to the loss term.

2. It is possible to define a net or a total sensitivity profile,

which can be obtained by summing the loss and the gain terms for

various partial reactions. The net sensitivity profile can be used

to determine how important a particular element is with respect to

a particular response.

3. Note that while deriving an expression for the cross-section sensi

tivity profile, we implicitly assumed that the response function

was independent from the partial cross section for which a sensi

tivity profile is desired. If this assumption does not hold, an

extra term has to be added to the previously obtained expressions.

When the response function is also the cross section for which a

sensitivity profile is sought, the sensitivity function will take

the form

19

«J\Lj * > 9I/I _ <R,»> + » (25)

az /z " i i ' ( J x x

where L^ represents that portion of the transport operator that x

contains the cross-section set {Z }. In this expression the first

term is a direct effect and the second term is an indirect effect.

If the direct effect is present, the indirect effect will usually

be negligible. A summary of the various possibilities is given in

Table I.

4. The spatial integration in Eq. (23) has to be carried out over the

perturbed regions only.

2.2.3 Explicit expression for the cross-section sensitivity profile in discrete ordinates form for a two-dimensional geometry represen-tation

Coordinate system

The coordinate systems for x-y and r-z geometry are shown in Figs.

53 2a and 2b. In both geometries $ was chosen to be the angle of rota-

2 2 2 tion about the p-axis such that dfi = dp.d<J>, and since £ + (J + v = 1»

we have

20

TABLE I: FORMULAS FOR THE SENSITIVITY FUNCTION

a.

b.

c.

< >

L

h.

c

t

Case

I = <R,4», where I. f R

I = <R,4», where I. = R

and I. $ L 1

I = <R,*>, where I. = R ' ' 1

and I. C L

Sensitivity Function

F I = <**» L2* >/ 1

i

F z = <R,4»/I !

i i

F z = <R,*>/I + «t>*,Lz 4»/I : i i '

direct indirect ] effect effect i

The direct effect is usually \ dominant ]

1

indicates the inner product over the phase space £

stands for the transport operator

represents that portion of the transport operator which contains cross-section [I.]

means is included in

means is not included in

21

Figure 2. a. Coordinates in x-y geometry

22

/ Q

Figure 2. b. Coordinates in r-z geometry

23

i = (1 - Asin* ,

and

2 h r) = (1 - (J ) .cosij) .

Therefore both the x-y and the r-z geometry representation will

lead to identical expressions for the sensitivity profile, with the

understanding that in x-y geometry the angular flux is represented by

*(x,y,|J,4>), and by 4>(r,z,(J,<)>) in the case of r-z geometry.

We now will derive an expression for the sensitivity profile in an

x-y or in an r-z geometry representation.

Method

Before deriving an expression in a discrete-ordinates formulation

and a two-dimensional geometry for Eq. (23), a brief overview of the

methods used is outlined.

Gain term:

I» order to represent the differential scattering cross section in

a multigroup format, the common approach to expand the differential

scattering cross section in Legendre polynomials is used. The num

ber of terms in the expansion is a function of the order of aniso

tropic scattering. The Legendre polynomials are a function of the

scattering angle \i (Fig. 2). Introducing spherical ha ionics

24

functions and applying the addition theorem for spherical har

monics, the dependence on p can be replaced by (j's and <|)'s. The

angular fluxes are expanded in flux moments. The integrals are

replaced by summations. Defining multigroup cross sections an

expressions for the gain term can be obtained.

Loss term:

An explicit expression for the loss term can be derived based on

angular fluxes or based on flux moments. In order to check the

internal consistency in SENSIT-2D both methods will b_- applied.

The derivation of an expression based on angular fluxes is

straightforward: the integrations are replaced by summations and

the appropriate multigroup cross sections are defined. An expres

sion as a function of flux moments can be obtained by expanding the

fluxes in flux moments, using spherical harmonics functions. The

orthogonality relation of spherical harmonics is applied, the

integrations are replaced by summations and appropriate multigroup

cross sections are defined. Finally an expression for the loss

term is the result.

Analytical derivations

Expand the differential scattering cross section in Legendre poly

nomials according to

25

LMAX

IXiS«HJ',E*E') = 2XfS(M0.E-E') = 2 " ^ V o ^ s , * " ^ ' 5 , (26)

£=0

where the P_(p )'s are the Legendre polynomials and LMAX the order of

anisotropic scattering. Here, the scattering angle p can be written as

u = fi.fi' = fi fi' + fi Q' + fi fi* , o - - xx yy zz

M0 = W' + OH' + IV .

= pp' + (l-|J2)'s(l-p,2)'s cos$ cos(tl, + (l-(j2)'5(l-p'2)'s sin0 sinO ,

Mo = MM' + (l-H2)l!(l-H,2)'S cosf^-O') .

The spherical harmonics addition theorem states that (see e.g., Bel] and

Glasstone )

V Mo } = P£(M)P£(M') + 2 2 j g j j p5(^)P^M,)cos[k(0-(t.')] , 127)

k= 1

where the P.(p)'s are the associated Legendre polynomials. The above

expression can then be reformulated as

26

http://fi.fi'

£ (2-6 )(£-k)!

'W = ,in —(W! P,(M)P^(M')cos[k($-$')] K—U

£ I k=0 L

(2-6ko)(£~k)!

(£+k)1

(2-6ko)(£-k)!

(£+kTi PJ(|J)PJ(P')

x (cos k<() cos k<(i' + sin k<)> sin k<j)') (28)

We define

R£(H,<C) (2-6ko)(£-k)!

(£+k)! P£(M)cos k$ , (29)

and

Qj(M,$) = (2-6ko)(£-k)!

(£+kl! P^(M)sin k<}> (30)

so that

P£(^0) = 1 {Rj(H,4>)Rj(M,,«n + Qj(M,$)Q^(p',$')} k=0

(31)

The Q terms will generate odd moments which will vanish on integration,

thus the Q terms will be omitted in the following discussion. The R»

terms are the spherical harmonics polynomials. Using the above expres

sion for Pp(|J ) in the expansion of the scattering cross section, we

have

27

LMAX inn £ k k I (ft*Q\E->E') = 1 ^ Z . (E-E-) I R K ( v , m (»',*•) , (32) x,s £ = 0 ta s,z R_0 * x.

where LMAX is the order of anisotropic scattering.

The second term of the sensitivity profile, Eq. (24), becomes

E g - 1 » LMAX £ 1 n i - / d E / d r / d E ' I £ * - * ! ( £ - > £ ' ) 2 2 / dp J d«

IAu8 E V 0 £=0 UTL S , £ k=0 -1 o

k ! * k *

Rj(p,(|»)*(rtfi,E) . 2 J dp' J d^'RjCM' , • ' ) * (r.fi'.E') . (33) -1 o

Note that

\Y d" I d* *£<M,*)R>,*) = 2ITT V k n • <*>

and therefore the angular flux can be expanded according to

00 £ v • *(n,E) = 1 (2£+l) I R W ( E ) , (35a)

£=0 k=0 *

. 1 1 . where *J(E) = J dp J d<}> Rjt>(0,E)/2n , (35b)

-1 p

28

and similarly for the adjoint angular flux

* (0,E) = I (2£+l) I Rp*p K ( E ) » £=0 k=0

(36a)

.... 1 n where <^K(E) = J d^ / dO R^ *(Q,E)/2n ,

-1 o (36b)

Introducing these expansions in the sensitivity profile, th*3 gain term

becomes

.8 Un GMAX Eg'-1 Eg-1 LMAX

5 - - f dr I J dE" J dE 2 (2i+l) 2 ,(E-E') x,gain IAuB V g'=l E , E £-0

g g

i *J:(£K':k(E-) , k=0

(37)

where GMAX is the number of energy groups. Defining

s,£ *£ *£

E . . E .

= / dE* / dE Z .(£-£•)*;•(£)*/(£•) , E . E ' g g

(36)

and discretizing over the spatial variable we have

29

, GMAX LMAX , 2. IPERT . ,,. , Pf = - ~ I I (2£+l)lf8 I 2 V.4>Jg(i)*'p

kg (i) , (39) x,gain IAu8 g'=l £=0 s'* k=0 i=l

where IPERT is the number of perturbed spatial intervals and i indicates

the spatial interval. If there is no upscattering, and introducing

2. IPERT ., , t 8 8 = 4n I (2£+l) I V.4>„8(i)*'0

K8 (i) , (40) * k=0 i=l 1 * iL

we have

LMAX GMAX — I I I8 8 f88 (41)

Zx,gain IAu8 £=0 g'=g S'A Z

The loss term of the sensitivity profile is given by

E

Pf = - 1- J dE / dr J JQ{-<t>(r,fi,EU (E)<J>*(r ,fi,E)} , (42) x.loss IAu8 E V

8

i- J dE J dr 2 J dp J d<(> {-•(p,0,E)2. T ( E ) * (M,4>,E)} , IAu8 E V -1 o '

8 (43)

-4TT V l "M J dE / dr I _(E) 2 w * ( u , 6 ,E)4> (u ,A) , (44)

T. g i ,, - x.T , m VHm'vm' nn'Mn IAu° E V ' m=l

8

30

where 0 = tan_1(l - p2 - n2)'i/M for p > 0 , (AS) m m m m m

0 = tan'^l - p2 - H2)'i/P + n for p < 0 , (46) m m m m m

and MM is the number of angular fluxes per quadrant.

Define

MM Eg-1 ^ MM 2 / dE I T(E)-*(p ,0 ,E)-<Mp ,0 ,E) = 2

8 _ 2 * V 8 , (47) _, i x,T m' m' ' rm'vm x.T , m m ' m=l E ' m=l g

so that

,_ IPERT MM pf = I^ZL 2

8 _ I V. I v *8(i)«t>"E(i) . (46) 2 . .. g x.T . , I , m m m J

x,loss lAu6 ' i=l m=]

Introducing

IPERT MM X8 = 4n 2 V. 2 w <t>8(i)4>"8(i) , (45)

, l , m m m i= 1 m=l

we have

Pf = - ^ 2! TX8 • (50)

x.loss IAug X,J

Note that the gain term was expressed as a function of flux moments,

while the loss term was expressed in terms of angular fluxes. When the

gain term is expressed as a function of flux moments, a very useful

31

relationship between the f's and the x's will be obtained. For this

case, substituting Eqs. (36) and (38) into Eq. (42), the loss term can

be expanded as

g-1 1 n <* I £ ) pf = - ^ J ; dr i ; dM ; d$ 2 UZZ+D 2 R**k

x.loss IAu8 E V x,x -1 o £=0 k=0 1

00 ( £ ) 1 (2£+l) I RJ**J . (51) £=0( k=0 )

Using the orthogonality relations Eq. (34) and defining the multigroup

total cross section for group g by

LMAX £ , .c, LMAX £ Eg-1 , , I I 2 8

T<8(r)*7 8(r) = 1 I / dE I T(E)4»5(r,E)<J.'p

K(r,E) £=0 k=0 ' £=0 k=0 E

g (52)

we have after discretizing the spatial variable, r, and truncating the

summation over £,

zg LMAX IPERT

p| = -4JI X?T £ (2£+1) J^ V.^g(i)*^8(i) . (53) x.loss IAu8

£ = Q .=1

32

Introducing

LMAX X8 = 1 4-f8 , (54)

£=0 *

the expression for the loss term reduces to Eq. (50) again.

Summary

LMAX GMAX P8- = — - — -I 8 v8 + Z I Z 8^ 8 H*88 (55)

1 T . g x,TX / . f s,£ T£ ' l" J

x I.Au6 ' £=0 g =g '

where

Z T = total macroscopic cross section for reaction type x,

I 6 P = £'th Legendre coefficient of the scattering matrix element for ' energy transfer from group g to group g', as derived from the

differential scattering cross section for reaction type x,

1PERT £ . .... , Y 8 8 = 4TI(2£+1) Z I V.4>p8(i)4>"K8 (i) (56) * i=l k=0 * * *

= spatial integral of the product of the spherical harmonics expansions for the regular and adjoint angular fluxes,

IPERT MM X8

= 4 T I I V. Z *8(i)*m8(i)w (57)

. . I , m m m i=l m=l

= numerical integral of the product of forward and adjoint angular fluxes over all angles and all spatial intervals described by i=l . . ., XPERT,

LMAX = Z V 8 8 . (58)

£=0 *

i 3

Note that expression (55) is identical with the expression for the

46 cross-section sensitivity profile in a one-dimensional formulation.

The flux moments can be expressed in terms of angular fluxes corre

sponding to

and

, 1 n MM ,

- 1 0 m=l

..., , 1 71 , ... , MM , . •?* = / dp / d«Rj»-« mnn - I *J 4(,m,^m • (60)

-1 o m=l

R„(n) = spherical harmonics function

V. = volume of rotated triangles l °

Au° = lethargy width of energy group g

= In (E°/E8 ), where E g and E 8 are upper and lower energy group boundaries

= integral response as calculated from forward fluxes only

IDET IGM n

2 I ViR*(|)J8(i)

i=l g=l

R. = spatially and group-dependent detector response function.

46 2.3 Source and Detector Sensitivity Profiles

Source and detector sensitivity profiles indicate how sensitive the

34

integral response I or I is to the energy distribution of the source,

or to the detector response R. The integral response I can be calcu

lated from the forward flux, according to Eq. (63), or from the adjoint

flux, according to Eq. (64). When the integral response is calculated

from the adjoint flux it will be denoted as I . Ideally, I will be

equal to I .

The sensitivity of the integral response to the energy distribution

of the detector response function or the source can therefore be ex

pressed by the sensitivity profiles

Pf = J" dr J dE J dQ R(r,E) .4>(r,Q,E) / I.Au8 (61) R V, " E ~ "

<* g

and

0 g-1 J- JL

PS = J dr ; dE J dQ Q(r,Q,E).<f> (r,Q,E) / I .Aug , (62) ^ V E

s g

where R(r,E) is the detector response and Q(r,Q,E) is the angular source,

and V, and V are the volumes of the detector and the source region. I a s

E * was used in the denumerator of Pj? and I was used in the denominator of

Pi for internal consistency. It is obvious that the integral source and

detector sensitivities, Sn and SR, will be equal to one.

35

It is possible to derive an expression similar to Eq. (61) for the

sensitivity of the integral response to the angular distribution of the

source. The derivation of explicit expressions for P5 and P^ is

straightforward. The detector sensitivity profile as a function of the

scalar fluxes becomes

IDET n

P8 = I V..R8.*"8(i) / I.Aug , (63) K . , 1 1 U

1=1

where the 4> °(i) are the scalar fluxes for group g at interval i, IDET

is the number of detector intervals, and R? is the detector response at

interval i for group g.

For the source sensitivity profile in case of an isotropic source

Eq. (62) transforms into

ISRS P8 = I V..Qf.*^8(i) / T.Au 8 , (64)

i=l

where Q. is the voluminar source for group g at source interval i.

In the case of an anisotropic source we defined Q8(r,Q) by

Q8(r,n).**8(r,n) = / dE Q(r,fi,E).**(r,n,E) , (65) E g

36

and expand the angular source according to

IQAN I , , Q8(r,fi) = Q8(r,p,*) = I (2£+l) I R%,*).Q*8(r)/27i , (66)

£=0 k=0 * *

where IQAN is the order of anisotropy of the source.

Substituting Eqs. (65) and (66) in Eq. (63), discretizing the

spatial variable and using Eq. (36), the expression for the source

sensiti?ity profile becomes

ISRS IQAN S. , .,. . P 8= 2. I V. I (22+1) I QfK(i)*"gk(i)/l".Au8 g i--i x £=0 K=0 * ii

As in Eq. (61) we can also define an angular source sensitivity func

tion. The angular source sensitivity function indicates how sensitive

the integral response I is to the angular distribution of the source,

TO

F" = \ / dr J dE Q(r,Q,E).**(r,fl,E)/I* . (68) w 1 V 0

s

37

2.4 Sensitivity Profiles for the Secondary Energy Distribution and the Secondary Angular Distribution

The theory of the secondary energy distribution (SED) and the sec

ondary angular distribution (SAD) sensitivity analysis was originated by

43-46 Gerstl. Physically the only difference between a secondary energy

distribution and a cross-section sensitivity profile is the way in which

the integration over the energy variable is carried out. The "hot-cold"

and the "forward-backward" concepts lead to a simple formulation of

secondary sensitivity theory and can easily be incorporated in an uncer

tainty analysis. Even when both those concepts are a rather coarse

approximation they have the advantage that they are simple and can be

physically understood.

A more rigorous formulation might be possible, but its simple

physical interpretation would be lost. The primary restriction on the

application of secondary energy distribution and secondary angular dis

tribution sensitivity profiles is the lack of cross-section uncertainty

information in the proper format.

2.4.1 Introduction

The expression for the sensitivity profile for the differential

scattering cross section is part of the gain term of the cross-section

sensitivity profile and takes the form

38

P5 (fMT ,E-»E') = — — / dr / dfi J8"1 dE / dE' / dfi* x Au8.I E o

g

x R2 (r.fWJ'.E+E') , (69) x.gain

where R^ (r ,fW}',E-*E') is a shorthand notation for x.gain

\ (r,£MT ,E->E') = *(r,fi,E)I (r,fW)' ,E-E' )<t> (r,fi' ,E') (70) x.gain '

and similarly,

R2 (r.Q'-tf.E'+E) = 4>(r,fi,E)Ix s(r,n'^),E-E)* (r,.Q,E) . (7:j x.gain '

Equation (70) gives the contribution to the integral detector re

sponse, I, from the particles born at position r with energy E 1, travel

ing in direction Q' , since

I = < * , L ' V > = <4>",I/t» . (72)

Similarly, R^ (r,fWV ,E'->E) gives the contribution to the integral x.gain

detector response from the particles born at position r, with energy E,

traveling in direction Q.

As it turns out, up to this point there is no difference in the

physical interpretation of Eqs. (70) and (71). The way the integrations

39

are carried out will distinguish between the differential scattering

cross-section sensitivity profile and the secondary energy distribution

and secondary angular distribution sensitivity profile.

2.4.2 Further theoretical development

In this section we will elaborate oa the physics behind the deriva

tion of SEDs and SADs. Consider

Fz (E,E') = j J dr / dfi J dfi' R (r,Q*fl',E-»E') . (73) x,s x,gain

In this expression F^ represents the fractional change in the inte-x,s

gral response per unit fractional change in the differential scattering

cross section I (E-»E'); i.e., it is the fractional change in the in

tegral response when the number of particles that scatter from E into E'

is increased by one percent. Obviously this will always be a positive

effect and will therefore be included in the gain term.

Similar to Eq. (73),

1 E 8 _ 1 " P| =jj dE J dE1 J dfl J dfi'Rj (r,(M5',E->E') (74) x.s E 0 x.gain

g '6

40


ber of particles that scatter from group g is increased by one percent.

The tilda in Eq. (74) is introduced to distinguish from a lethargy nor

malized sensitivity profile.

In the adjoint formulation the equivalent of Eq. (73) will be

F1 (E' ,E) = FS£D(E' ,E) = i J dr J dfi J dfi'Rj (r,Q'-»fl,E'*E) , x,s x,gain

(75)

which represents the fractional change in the integral response per unit

fractional change in differential scattering cross section I (E'-*E),

i.e., it is the fractional change in the integral response when the num

ber of primary particles that scatter from E' to E is increased by one

percent, or for that matter that the number of secondary particles that

were scattered from E into E' were increased by one percent. Again,

this will always have a positive effect and will therefore constitute a

gain term in the sensitivity profile.

Define

Vl J dE J" dE' / dfl / dO'R^ (r.O'-^.E'-^E) . (76) E 0 x.gain g

While there is no difference in the physical meaning of Eqs. (73)

and (75), the formulations (74) and (76) are different. Equation (74)

41

•SED


ber of secondary particles that were scattered into group g have V .-en

increased by one percent.

It is clear from these examples that, depending on the way the

integrations are done, several different sensitivity profiles can be

constructed. In order to study the secondary angular distribution, we

can introduce

a, Fz (Q,E') = F (Q,E') = \ J dr J dE J dQ'R^ (r.Q'-Q.E'-EJ

x,gain 0 x,gain (77)

This expression gives the fractional change in the integral response

when the number of secondary particles scattered from initial energy E'

into final direction Q is increased by one percent. It will therefore

be clear that

W 2 ) = J^SADW.E') (78)

is the fractional change in the response function when the number of

secondary particles which were scattered into direction Q was increased

by one percent.

42

2. 4^ 3 Secondary energy and_secondary angular distribution sensitivity profiles

A double secondary energy distribution (SEDJ sensitivity profile is

defined by

E . E , . B'B 1 8-1 8 _1

Pern = r / dE / dE' J" dr / dft J dfi'R, (r,fi'-»fi,E'-»E) . IAu8Au8 E. E„, x,gain g g

(79)

The energy integrated SED sensitivity profile becomes

'g-1 — J" dE' J dE J dr J dfi'R, (r,fi'^,E'-E) .

S E D IAu8 0 E (80)

x,gain

The differential sensitivity profile for the angular distribution

of secondary particles scattered from initial energy E' is

Pg (Q)

Eg'-1 — J dE' / dE / dr J dfi'R

0 x.gain g

lAu8 E (r,0'-*Q,E->E') (82)

An energy integrated SED sensitivity profile can be defined by

43

PSAD(fi) = | / dE' J" dE J dr / dQ'I^ (r,fi'-^,E-E') . (82) 0 C s,gain

2 . 4 . 4 I_nt egra 1 sensi t_i_W_ti_es ior SEDs and SADs

In order to make the sensitivity and uncertainty analysis for

secondary energy distributions and secondary angular distributions less

tedious, Gerstl introduced the concepts of the "hold-cold" SED and the

"forward-backward" SAD integral sensitivity:

Sjj = / dE P|Fn(E) - J dE P|.n(E) , (83) L HOT b L U COLD b A U

and

SSAD = f J" „ *> P S A D ^ - J . d5 PSAD<0> " <84> forward backward angles angles (fJ>0) (M<0)

The forward-backward SAD integral sensitivity can be interpreted as

the fractional change in the integral response when the number of sec

ondaries which were scattered forward is increased by one percent, while

the number of secondaries that were scattered backwards (p<0) is de

creased by one percent. The integral SAD sensitivity is a positive

number which is labeled "forward" or "backward" depending whether the

first or the second term in Eq. (84) is the larger one. Physically,

44

that positive number indicates how much more sensitive the response

function is to forward scattered particles than to backward scattered

particles, or vice versa.

For the hot-cold integral SED sensitivity, the concept of the

median energy has to be introduced. In the multigroup formulation, the

median energy defines the energy boundary which roughly divides the

cross-section profile into two equal parts. The median energy and the

43 integral SED sensitivity are illustrated in Fig. 3. Note that the

median energy g' is a function of the primary energy group g1. For that

reason also the integral SED sensitivity will depend on g'.

The hot-cold integral SED sensitivity expresses the fractional

change in integral response when the number of secondaries which scatter

in the "hot" part of the secondary energy distribution is increased by

one percent while the number of secondaries scattered into the "cold"

part is decreased by one percent. The integral hot-cold SED sensitivity

is a positive number^ labeled "hot" or "cold" depending on which term

dominates in Eq. (83). That number indicates how much more sensitive

the integral response is to particles scattered into the hot part of the

secondary energy distribution than to particles sea ;tered into the cold

part, or vice versa.

45

„ F e ( n , 2 n ) = r

2 g E

15

14

13

' ' c o l e

12=g r

"hot

11

10

Q

i » -

"8. 'median out

'hot •= £ *g

G

cold

SED

Z- ug ! -c +1 ' g"£

14

15

a = <r *=• 1 / 2 ff hot cold

12-£„

13

"Ui. 10

JL<6'> in

s6 =5"

e 'median

P6

SED *- SED

G

, ,,+1 SED 8m(E >

Eout ( M e V )

"HOT" or

"COLD"

43 Figure 3. Definition of median energy and integral SED eensitivity

46

2.4.5 Explicit expressions for integral SED sensitivity profiles in a two-dimensional geometry representation

The expression for the double SED sensitivity profile, Eq. (79), is

similar to the gain term of the cross-section sensitivity profile, Eq.

(24). By comparing Eq. (79) with Eq. (24) and using Eq. (41), the ex

plicit expression for the double SED sensitivity profile becomes

LMAX D8 >8 - l 7 58 8 1118 8 fco 'SED " t t1 s 2 { ' l " J

From Eqs. (85) and (80), it follows that the energy integrated SED sen

sitivity profile for the case of no upscattering can be represented by

g LMAX

IAu6 g'=l £=0 '

Using the definition for the integral SED sensitivity (83), it becomes

clear that

8m ( 8' ) GMAX SSED= * , Au8 -PIED- \ M+1

A U 8- P1ED • («) 8=8 g=8m(8 )+l

where g (g') is defined in Fig. 1.

47

2.6 Design Sensitivity Analysis

Design sensitivity analysis provides a method to estimate changes

in integral response for a slightly altered design. The results are

exact up to the second order with respect to the corresponding flux

changes, but only exact up to the first order with respect to design

changes. The theory presented in this section is applicable only when

the design changes can be expressed in terms of macroscopic cross-

section changes. Methods based on generalized perturbation theory have

been applied to design sensitivity analysis. '

The integral response for the perturbed system can be expressed by

35 Eq. (88) for the adjoint difference formulation,

1 ^ = <R,4» - «t> ,AL*> = 1 - 6 1 ^ , (88)

and by Eq. (89) in the forward difference formulation

I F D = <Q,I > - <*,AL * > = I - 6I F D . (89)

Proceeding in a manner similar to the derivation of the cross-

section sensitivity profile, the second-order term in the right hand

side of Eqs. (88) and (89) can be written as

48

and

6I,n = / dE / dr J dn{*(r,fi,E)6I T(r,E)* (r.fi.E) AU Q v x,i a

+ J" dE" / dn'*(r,0',E')6Z (r,fl'-»fl,E'-*E)* (r.Q.E)} , (90) 0 x's

61 = J dE J dr J dQ{*(r,n,E)6I T(r,E)4> (r,Q,E) 0 V '

+ / dE' J dQ;*(r,fi,E)6I (r,fMT ,E-E' )<J> (r,fi' ,E') . (91) 0 x's

In the above expressions we used

5Zx,T = Zx,T " Zx,T > (92>

and

62 = 1 - I , (93)

where I refers to a perturbed cross section and I to a reference cross

section.

A design sensitivity coefficient X can be defined as the ratio of

the integral response for the altered design over the integral response

49

for the original model. Depending whether the forward or the adjoint

difference method are used, the design sensitivity coefficient equals

or

XAD = W1 = ! " 6 IAD / : > (9A)

XFD = W1 = l ' 6 IFD 7 1 • (95)

Note that respectively, I and I were used in the denominator of Eqs.

(94) and (95) for internal consistency. Numerically 61,- and 61 — are

* 35

identical; I and I , however, can be different. Gerstl and Stacey in

dicate that the adjoint formulation is more accurate for perturbations

closer to the detector, while the forward difference method gives better

results for perturbations closer to the source. If both reference

fluxes <t> and 4> are completely converged, Eqs. (94) and (95) will give

identical results.

Explicit expressions for Eqs. (94) and (95) can be formulated. The procedure for the evaluations of 61._ and 6IT,_ is similar to the deriva-r AD FD

tion of the cross-section sensitivity profile and leads to the equations

IGM ( LMAX g . ] 6l A n = I [6ItTxB- * I 61* C

B f* 0 ^ > ( 9 6 ) *AD" g = 1| x,T* £ = 0 gt=1 s,£ s,£

50

aad

IGH f LMAX GMAX 6IFD= \ 6Ix,T^- , \ ,* «!,! i g • (97)

g=l ' £=0 g'=g J

51

3. APPLICATION OF SENSITIVITY THEORY TO UNCERTAINTY ANALYSIS

Sensitivity theory can be used to do an uncertainty analysis by

introducing the concepts of cross-section covariance matrices and frac

tional uncertainties for SEDs. In this chapter we will explain how sen

sitivity profiles can be used in order to calculate the uncertainty of a

reaction rate due to the uncertainties in the cross sections.

3.1 Definitions

Let I represent a design parameter depending on a multigroup cross-

section set {I.}, so that

I = HI±) , (98)

where the index i can reflect a group, a partial cross section or a

materia]..

52

The variance of I is defined as the expected value of the square of

the difference between the actual value of I and the expected value of

I, or

Var(I) = E{(61)2} = E{(I - E{I})2} . (99)

The standard deviation of I is the square root of the variance,

AI = [Vard)]3* . (100)

The covariance of a and b is defined as

00 00

Cov(a,b) E E{6a-6b] = J J da.db.(a - E{a]).(b - E{b}).f(a,b) , -00 -05

(101)

where f(a,b) is a joint probability density function. A nonzero covar

iance between the quantities a and b indicates a mutual dependence on

another quantity. Obviously we have

Cov(a.a) = Var(a) , (102)

since f(a,a) = 1.

A relative covariance element is defined by

53

R(a,b) :. Cov(a,b)/a.b (103)

3. 2 Cross-Sectioo Covariance Matrices

During the experimental evaluation of cross-section data, statis

tical errors arise from the fact that two similar experiments never

agree completely. Also a systematic error reflects the fact that no

equipment and no evaluation procedure is perfect, and that - among other

factors - reference standards are used.

Cross-section covariance data describe the uncertainties in the

multigroup cross sections and the correlation between those uncertain

ties. A nonzero nondiagonal covariance matrix element indicates that

there was a common reason why an uncertainty in two different (e.g.,

partial cross sections or energy range) cross section was introduced.

The evaluation procedure for covariance data is tedious and requires a

sophisticated statistical analysis. ' '

Multigroup cross-section covariance Hat.a are ordered in covariance

matrices. Such a covariance matrix contains GMAX rows and GMAX columns,

where GMAX is the number of energy groups. A covariance matrix can

contain covariance data of a particular partial cross section with

itself over an energy range , with a different cross section for the

same element, or with a partial cross section of a different element.

It has become a common practice to include formatted uncertainty

data in the ENDF/B data files. Even though the uncertainty files are

54

still missing for many materials in ENDF/B-V, extensive work is under

way. Based on these uncertainty data, covariance libraries can be con-

32 33 structed. ' A 30-group covariance library based on ENDF/B-V which

contains most of the elements commonly used in reactor shielding has

33 been constructed by Muir and LaBauve. The covariance data in this

library were processed into a 30-group format by using the NJOY

code. ' In this particular library, called COVFILS, the multigroup

cross sections and the relative covariance matrices for H, B, C, 0,

Cr, Fe, Ni, Cu, and Pb are included. Another covariance library was set

32 up by Drischler and Weisbin.

3.3 Application of Cross-Section Sensitivity Profiles and Cross Section Covariance Matrices to Predict Uncertainties

Using first-order perturbation theory, the change in the integral

response I, 61, as a consequence of t>niall changes in I. can be approxi

mated by

61 S Z U- 6Z. . (104) az.. l l l

We further have

Vai(I) = E{612} = E{ I §|- §|- 61.61 } , (105)

55

or

Var(I) = I ||- ||- Cov(I.,Z.) . (106)

From Eqs. (100) and O06) it now becomes obvious that

rM-|2 Cov(I I )

L J x s i,j 1 j x j

I II

where P^ and P^ are sensitivity profiles, and the subscript xs refers

i j to reactor cross sections.

The concept of covariance data and sensitivity profiles leads to a

simple way to evaluate the error in I. The first part in the summation

requires sensitivity profiles and is highly problem dependent. The

second part requires cross-section uncertainty information and is prob

lem independent.

When trying to apply the theory presented here, very often covari

ance data will be missing for certain materials. One way of going

around this problem would be to substitute the covariance file of the

missing material by a covariance file for another material for which the

45 cross sections are less well known. Other methods to eliminate this

problem would be to make very conservative estimates. '

The most conservative method would be to assume that the error in

the cross section is the same for all groups and equal to the largest

error for any one group. In that case it can be shown that '

56

AI.

max 1 1 1

(108)

3.4 Secondary Energy Distribution Uncertainty Analysis

For evaluating uncertainties in the integral response due to un

certainties in the secondary energy distribution we will follow Gerstl's

44 46 approach ' and introduce the spectral shape uncertainty parameter for

the hot-cold concept.

When the total number of secondaries scattered from group g' are

held constant, then necessarily

51 HOT

61 COLD

"HOT "COLD = f - (109)

Therefore f , quantifies the uncertainty in the shape of the SEDs and is

44 called the spectral shape uncertainty parameter (Fig. 4)

It now becomes possible to express the relative change in integral

response due to the uncertainty in the secondary energy distribution in

a form similar to Eq. (107):

I"—1 = 1 P8'8 e'"e I L r n T SED 1 .

I JSED g' ,g g'-*g (110)

57

'median

Scr. +f if g<gr

-f if g>g

Figure U. Interpretation of the integral SED uncertainty as spectrum shape perturbations and definition of the spectral shape uncertainty paraneter "f" (ref. 64)

58

Substituting Eqs. (87) and (109) in Eq. (110), it follows that

:8

SED g' JSED xg f , (111)

Denote f , by f., where the index j refers to a particular nuclear

reaction, e.g., (n,2n), at specific incident energy g', and\£et f. rep

resent some different reaction/primary energy combination. Then the

uncertainty in integral response corresponding to correlated uncertain

ties of all SEDs for a specific isotope is

M2 Var(I) (61)'

SED .Z. SSEDSSEDfi fj[

(112)

or

M ~ = i SED i,J

SSEDSSED Cov(f.,f.) (113)

If the spectral shape uncertainty parameters for a specific par

ticle interaction, identified by the subscript £, are assumed to be

fully correlated, it can be shown that

Cov(f . ,f .) ,,,.. i' jycor(+l)

|Cov(fi,f.)] [Cov(f..,f..)]* (114)

so that

59

AI \, S S F J ' [ C O V ( V ' V ) ] ! S (115)

or,

h] *• g* 'SffiD'HV-^g.J^

(116)

If N independent measurements of the same SED are available, the

values for Var(f„ ,) can easily be evaluated. For each cross-section

evaluation, weights, w , are assigned, then

^

n n HOT " °COLD "ElcT

for n = 1,2.. .N (117)

with

E{fn,} = I w fn, = 0 . 8 n=l n 8

(118)

The variance of f , will be g

2 2 v fe w r f f 2 , _ ? (°H0T " CTCOLD) Var(f , ) = E{f ,} - I w 5

8 8 n=l ° mo})2 (119)

Var(f ,) is called the fractional uncertainty for the secondary energy

distribution and is identified by the symbol F. A short program which

evaluates the values of F has been written by Huir; the results for

45 the 30-group neutron structure is shown in Table II.

60

TABLE II

MEDIAN ENERGIES (E', IN MF.V) AND FRACTIONAL UNCERTAINTIES (F) FOR SECONDARY m

ENERGY DISTRIBUTIONS AT INCIDENT NEUTRON ENERGIES E

E 0

16.0

14.25

12.75

11.00

8.90

6 .93

4 .88

3.27

2 .55

1.99

1.55

1.09

0 .66

0 .40

0 .24

0 .13

12C E'

m

14.71

13.00

11.71

9 .77

7.35

5.96

4 .46

2 .63

2 .16

1.73

1.34

0 .95

0 .57

0 .35

0.21

0 .12

F

0 .071

0 .059

0 .054

0 .060

0 .048

0 .035

0 .010

0 .010

0 .010

0 .005

0 .005

0 .005

0 .005

0 .005

0 .005

0 .005

16

E' m

14.62

13 .33

11 .93

9 . 82

7 .90

6 .04

4 .57

3 .09

2 .31

1.79

1.35

0 .94

0 .60

0 .34

0 .22

0 .12

0

F

0 .088

0 .072

0 .062

0 .057

0 .050

0.030

0 . 0 ) 0

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

0 .010

C E'

m

3 . 2 7

8.65

11.42

10.48

8.79

6 . 8 3

4 .81

3.21

2 .48

1.93

1.50

1.05

0 .64

(1

r

F

0 .17

0 .15

0 . 1 3

0 . 1 1

0 .09

0 .08

0 .07

0 .06

0 .05

0 .04

0 . 0 3

0 .02

0 . 0 ?

U-f. 4 5)

Fe E' m

4 .49

5 .99

11.17

10.57

8.77

6 .86

4 .81

3 .21

2 .49

1.93

1.51

1.06

0 . 6 3

0 .39

0 .11

0 .10

0 .10

0 .09

0 .08

0 .07

0 .07

0 .06

0 .06

0 .06

0 .05

0 .03

0 .02

0 .02

Ni E'

m

14.95

13.97

12.67

10 .91

8 .85

6 . 8 8

4 . 8 3

3 .24

2 .49

1.94

1.50

1.06

F

0 . 1 3

0 . 1 1

0 . 1 1

1.10

0 .09

0 . 0 8

0 . 0 7

0 .06

0 .05

0 .04

0 . 0 3

0 . 0 2

Cu E'

m

3 .42

3 .51

4 . 3 0

10.42

8.81

6 .86

4 . 8 2

3 .22

2 .50

1.94

1.51

1.06

0 .64

0 .39

0 .24

0 .12

F

0 .11

0 .10

0 . 1 0

0 .09

0 .08

0 .07

0 .07

0 .06

0 .06

0 .06

0 .05

0 .03

0 .02

0 .02

0 .02

0 .02

W E' m

1.86

2 .17

1.91

1.57

1.24

6 .66

4 . 8 3

3 .25

2.51

1.96

1.51

1.06

0 .6 6

0 . 3 8

0 . 2 2

0 . 1 2

F

0 .12

0 .10

0 .10

0 .09

0 .08

0 .07

0 .07

0 .06

0 .06

0 .05

0 .05

0 .05

0 .04

0 . 0 3

0 . 0 2

0 .01

3.5 Overall Response Uncertainty

The overall response uncertainty will be of the form

#C7M (120)

where

L JSED i :i21)

SED,i

and

-All2

1 (122)

L J XS i , k L J XS,i,k

The index i reflects the uncertainties in the various materials.

It was assumed that the effects from SED uncertainties for all possible

reactions which generate secondaries are uncorrelated. It is also

assumed that the uncertainties due to the SEDs are uncorrelated with

other uncertainties due to reaction cross sections (XS), and that the

uncertainties between the reaction cross sections themselves are un

correlated .

62

Remarks

1. To be absolutely correct, a term reflecting the uncertainty in

the secondary angular distribution should be included. Due to

the difficulty in generating uncertainty data from ENDF/B-V in

the proper format, we do not include that term.

2. In order to evaluate the sensitivity profiles, we should keep

in mind that the form of the sensitivity profile will depend

on the particular reaction cross section for which a response

is desired (Table I).

53

4. SENSIT-2D: A TWO-DIMENSIONAL CROSS-SECTION AND DESIGN SENSITIVITY

AND UNCERTAINTY ANALYSIS CODE

4.1 Introduction

The theory explained in the previous chapters has been incorporated

in a two-dimensional cross-section and design sensitivity and uncertain

ty analysis code, SENSIT-2D. This code is written for a CDC-7600

machine and is accessible via the NMFECC-network (National Magnetic

Fusion Energy Computer Center) at Livermore. SENSIT-2D has the capa

bility to perform a standard cross-section and a vector cross-section

sensitivity and uncertainty analysis, a secondary energy distribution

sensitivity and uncertainty analysis, a design sensitivity analysis and

an integral response (e.g., dose rate) sensitivity and uncertainty

analysis. As a special feature in the SENSIT-2D code, the loss term of

the sensitivity profile can be evaluated based on angular fluxes and/or

flux moments.

64

SENSIT-2D is developed with the purpose of interacting with the

TRIDENT-CTR code, a two-dimensional discrete-ordinates code with tri

angular meshes and an r-z geometry capability, tailored to the needs of

the fusion community. Angular fluxes generated by other 2-D codes, such

a<= DOT, TWODANT, TRIDENT, etc., cannot be accepted by SENSIT-2D due to

the different format. The unique features of TRIDENT-CTR (group de

pendent quadrature sets, r-z geometry description, triangular meshes)

are reflected in SENSIT-2D. Coupled neutrun/gamma-ray studies can be

performed. In contrast with TRIDENT-CTR however, SENSIT-2D is re-

stricted to the use of equal weight (EQ ) quadrature sets, symmetrical

with respect to the four quadrants. Upscattering is not allowed.

46 Many subroutines used in SENSIT-2D are taken from SENSIT or

TRIDENT-CTR. SENSIT-2D is similar in its structure to SENSIT, but is an

entirely different code. Unlike SENSIT, SENSIT-2D does not use the

69 BPOINTR package for dynamical data storage allocation, but rather uses

a sophisticated pointer scheme in order to allow variably dimensioned

arrays. As soon as an array is not used any more, its memory space

becomes immediately available for other data. SENSIT-2D does not in

clude a source sensitivity analysis capability and cannot calculate

integral responses based on the adjoint formulation. This has the dis

advantage that no check for internal consistency can be made. There

fore, other ways have to be found in order to determine whether the

fluxes are fully converged. One way for doing so would be to calculate

the integral response based on the adjoint formulation while performing

65

the adjoint TRIDENT-CTR or the adjoint TRDSEN run, and compare with the

integral response based on the forward calculation.

SENSIT-2D requires input files which contain the angular fluxes at

the triangle midpoints multipled by the corresponding volumes, and the

adjoint angular fluxes at the triangle midpoints. A modified version of

TRIDENT-CTR, TRDSEN, was written by T. J. Seed to generate these flux

files. A summary of these modifications was provided by T. J. Seed and

is included as Appendix B. After a TRIDENT-CTR run, the TRDSEN code

will use the dump files generated by TRIDENT-CTR, go through an extra

iteration, and write out the angular fluxes in a form compatible with

SENSIT-2D. Both SENSIT-2D and TRDSEN use little computing time compared

with the time required by TRIDENT-CTR.

The features of SENSIT-2D are summarized in Table III. The SENSIT-

2D source code is generously provided with comment cards and is included

as Appendix A.

4.2 Computational Outline of a Sensitivity Study

A flow chart (Fig. 5) illustrates the outline for a two-dimensional

sensitivity and uncertainty analysis. From this figure it becomes imme

diately apparent that a sensitivity analysis requires elaborate data

management. The data flow can be divided into three major parts: a

cross-section preparation module, in which the cross sections required

by TRIDENT-CTR and SENSIT-2D are prepared, a TRIDENT-CTR/TRDSEN block,

66

TABLE III: SUMMARY OF THE FEATURES OF SENSIT-2D (PART I)

SENSIT-2D: A Two-Dimensional Cross-Section and Design

Sensitivity and Uncertainty Analysis Code

Code Information:

* written for the CDC-7600

* typical storage, 20K (SCM), 80K (LCM)

* number of program lines, 3400

* used with the TRIDENT-CTP. transport code

* typical run times, 10-100 sec

Capabilities:

* computes sensitivity and uncertainty of a calculated

integral response (e.g., dose rate) due to input cross

sections and their uncertainties

* cross-S£Ction sensitivity

* vector cross-section sensitivity and uncertainty

analysis

* design sensitivity analysis

* secondary energy distribution (SED) sensitivity and

uncertainty analysis

67

TABLE III: SUMMARY OF THE FEATURES OF SENSIT-2D (PART 2)

SENSIT-2D

TRIDENT-CTR Features Carried Over into SENSIT-2D:

* x-yor r-z geometry

* group-dependent S order

* triangular spatial mesh

Unique Features:

* developed primarily for fusion problems

* group dependent quadrature order and triangular mesh

* can evaluate loss-term of sensitivity profile based

on angular fluxes and/or flux moments

Current Limitations:

* can only interact with TRIDENT-CTR transport code

* not yet implemented on other than CDC computers

* based on first-order perturbation theory

* upscattering not allowed

68

Figure 5. Computational ou t l i ne for a two-dimensional. s e n s i t i v i t y ana lys i s with SENSIT-2D

where the angular fluxes in a form compatible with SENSIT-2D are gener

ated, and a SENSIT-2D module, which performs the calculations and manip

ulations necessary for a sensitivity and uncertainty analysis.

4.2.1 Cross-section preparation module

There are many possible ways to generate the multigroup cross-

section tables required by SENSIT-2D and TRIDENT-CTR. The flow chart

of Fig. 5 illustrates just one of these possibilities. All the codes

mentioned here are accessible via the MFE machine. Basically, three

codes are required: NJOY, TRANSX, and MIXIT. Starting from the ENDF/B-V

64 data file, the NJOY code system generates a multigroup cross-section

library (MATXS5) and a vector cross-section and covariance library

(TAPE10). A covariance library can be constructed by using the ERROR

33 module in the NJOY code system.

From the multigroup cross-section library (MATXS5), the desired

72 isotopes can be extracted by the TRANSX code and will be written on

73

a file with the name XSLIBF5. The MIXIT code can make up new mate

rials by mixing isotopes from the XSLIBF5 library. The cross sections

used in SENSIT-2D have to be written on a file called TAPE4. The cross

sections used in TRIDENT-CTR and TRDSEN will be on file GEODXS. SENSIT-

2D and TRIDENT-CTR include the option to feed in cross sections directly

from cards.

70

4.2.2 The TRIDENT-CTR and TRDSENS block

SENSJT-2D requires regular angular fluxes at the triangle center-

points, multipled by the corresponding volumes, and adjoint angular

fluxes at the triangle centerpoints. TRIDENT-CTR does not write out

angular fluxes. Therefore the TRDSEN version of TRIDENT-CTR was written

by SEFD. TRPSEN makes use of the flux moment dump files, generated by

TRIDENT-CTR. These dump files will be the starting flux guesses for

TRDSEN. TRDSEN will perform one more iteration anu write out the

angular fluxes. In this discussion we will represent the dump file

families by DUMP! for the regular flux moments, and DUMP2 for the

angular flux moments. Except for a different starting guess option,

TRDSEN requires the same input as TRIDENT-CTR.

4.2.3 The SENSIT-2D module

The SENS1T-2D code performs a sensitivity and uncertainty analysis.

When vector cross sections and their covariances are required, they have

to be present on a file with the name TAPE10. If the cross section data

are read from tape, they have to be written on a file called TAPE4. The

regular angular fluxes at the triangle centerpoints multiplied by the

corresponding volumes (TAPE11, TAPE12,...) and the adjoint angular

fluxes at the triangle centerpoints (TAPE15, TAPE16,...) can be quite

voluminous. Writing out large files can be troublesome on the MFE

71

machine when there is a temporary lack of continuous disk space. There

fore TRIDENT-CTR and SENSIT-2D have the built-in option to specify the

maximum number of words to be written on one file. This limit has to

U - . *- U . _ L . . . U * _ . . . . . 4 . L . * . - I T » I r i J - * - 1 . » _ J t- ~ , L/e OCL. U l g l l CTMWUgl l L O C N S U 1 C L j l d l O X X L U C 1 1 U A U d L d I C l f l t L U *. v> ^ » i « . ^ l ^ u j - '

can be written on one file. 1 000 000 words per file is usually a

practical size and is the default in TRIDENT-CTR.

SENSIT-2D can generate four more file families:

1. TAPE1, which contains the regular scalar fluxes at the triangle

centerpoints.

2. TAPE20, TAPE21,..., which are random access files and contain the

adjoint angular fluxes at the triangle centerpoints,

3. TAPE25, TAPE26,..., containing the regular flux moments at the

triangle centerpoints, multipled by the corresponding volumes,

4. TAPE30, TAPE31,..., which contain the adjoint angular fluxes at the

triangle midpoints.

SENSIT-2D has the option of not generating those file families, but

using those created by a former run. The flux moments are constructed

from the angular fluxes according to the formula

, MN 4>*(X,Y) = I w R*(JJ ,«|i ) * (X'y) , & r,z n m £v,m'Mir m r,z '

' m=l '

where the w 's are the quadrature weights, the R^'s the spherical har

monics functions, and MN the total number of angular fluxes.

72

4.3 The SENSIT-2D Code

In this section the structure of the SENSIT-2D code, its options

and capabilities will be explained in more detail. SENSIT-2D is a

powerful sensitivity and uncertainty analysis code. The description of

this code from the user's point of view is given in the user's manual.

A.3.1 Flow charts

The overall data flow within the SENSIT-2D module is repeated in

Fig. 6. A simplified flow chart is illustrated in Fig. 7. The main

parts of the flow chart include these steps:

* The control parameters and the geometry related information are

read in.

* The quadrature sets and the spherical harmonics functions required

to generate the flux moments are constructed.

* The adjoint angular fluxes at the triangle centerpoints are written

on random access files, flux moments are generated and scalar

fluxes are extracted.

* A detector sensitivity analysis is performed; if desired an uncer

tainty analysis is done.

* The x's and ip* s which form the essential parts of the cross-section

and secondary energy distribution sensitivity profiles are calcu

lated for each perturbed zone and for the sum over all perturbed

zones.

73

Vector cross-sect ions »nd covariance data

(only required for vector cross-sect ion

•ensitivlty and uncertainty analysis)

Cross sections in LASL format (only required If cross sections are int ended to be read from cross-a-ction file)

Angular fluxes at triangle midpoints

multiplied by the corresponding volumes

Adioir: ar.pi-lar i luxes at triangle midpoints

/ I

ajoint flux moments pel's

at triangle center- and

points chi's

adjoint angular regular flux merer.;?

fluxes at triangle triangle centerpcir.t

•idpoints multiplied by the

(random access file) corresponding volur.e

Figure 6. Data flow for the SENSIT-2D module

74

Start

* Read the input parameters.

* CALL EEND: P.ead in the neutron and gar_-.a-ray structure

and calculate the lethargy width/group.

* CALL CEOM: Read and edit the geometry.

* CALL S1X0X: P.ead quadrature information and calculate EQ -sets, n

* CALL TAPAS: Assign files to the fluxes.

do for forward and adjoint fluxes

1 >

* CALL Pt.'CES : Calculate spherical hamonics functions.

* CALL FLUXMOM: Calculate flux moments. Extract scalar fluxes.

•^ Continue \

* CALL DETEEN: Calculate detector response and detector sensitivity profile. If desired a detector uncertainty analysis is performed.

* CALL DUE : Calculate chi's based on angular fluxes if desired.

* CALL PS1S : Calculate the psi's based on flux moments and store in LCM. If desired c h i ' s based on flux moments v i l l be calculated.

Figure 7. Flow char t for SENSIT-2D (par t 1)

75

o- Is a vector cross-section sensitivity and uncertainty analysis desired"1

* In the case that a SIT) uncertainty anal\sis is required, T*mA ir< r*» SEP -"CTtiir.lj u*i«.

* CALL SUB5 ; Read In cross sections.

Convert to nacroscopic cross sections via

nucber densities,

* CALL SUi5 : Peid in second cross-section set in the case that a design sensitivity analysis is desired.

* CALL SUB6 : Extract vector cross sections and scattering

matrix fror the full cross-sect lor. table.

In case of a design sensitivity analysis

calculate delta sigsa.

Calculate macroscopic scattering cross sections.

dc for all perturbed zones and

over all perturbed 2Dnes

for the sun A

*

*

*

*

CALL

CALL

CALL

CALL

irn or

POINTS

sine

SIT 11

TEXT A: Print appropriate definitions vner. this

section is passed for the firs: tine.

Set pointers in order to choose proper chi's

and psi 'B.

Calculates and edits final results of sensitivity

analysis if it is net a ST.D analysis.

Calculates and edits final results for a SZt sensitivity and uncertainty analysis (neutron

groups only).

* CALL 5^E9 : Read cDvariance data and provide uncertain! ic< in the integral response for the fully crrrtlated and the non-correlated case.

Figure 7. Flow chart for SENSIT-2D (part 2)

76

This section performs a complete sensUwity ?nH imrprfainry analysis for vector cross sections. The code requires a covariance file to be given in LASL error-file format which contains pairs of vector cross sections with their corresponding covariance matrix.

< do for all successive cases >

* CALL SUB5V

CALL POINTS: CALL SUB8V :

Reads cross-section ID from input file. Reads number density from input file. Reads relative vovarlance data (via COVARD) Generates macroscopic cross sections. Reads cross sections (via COYARD). Set appropriate pointers for chi's and psl's. Computes and edits sensitivity profiles and folds them vith the covariance matrix in order to obtain the relative integral response.

< cont mue

* CALL SUE9V : Cor.putes partial sums of individual respor.se variances. Reads SUMSTP.T and SUMT>:D (variances to be surged) assur.ing no correlation between individual vector cross-section errors, the total variance and the relative standard variation are computed.

C Stop

Figure 7. Flow chart for SF.KSIT-2D (part 3)

77

http://respor.se

Up to this point, all the subroutines used are different from those used

in the SENSIT code. The remaining calculations are done with SENSIT sub

routines .

* Cross sections are read in.

* Vector cross sections are extracted.

* Sensitivity profiles are calculated used in the appropriate tjj's and

X's.

* If desired to do so, an uncertainty analysis is performed.

* A vector cross-section sensitivity and uncertainty analysis can be

performed and partial sums of individual response variances can be

made.

4.3.2 Subroutines used in SENSIT-2D

Table IV summarizes the subroutines used in SENSIT-2D and indicates

their origin in case they were taken over or adapted from another code.

The essential difference between SENSIT and SENSIT-2D is the way that

the geometry is described and how the I|J'S and the x's are calculated.

Basically, all the subroutines are called from the main program with a

few exemptions when subroutines are called from other subroutines. The

subroutines for SENSIT-2D which were not taken over from other codes

will now be described. For the SENSIT subroutines we refer to the

user s manual.

78

TABLE IV: LIST OF SUBROUTINES USED IN SENSIT-2D

Name Subroutine Origin If Taken From Another Code, Were Changes Made?

EBND

GEOM

SNCON

TAPAS

PNGEN

FLUXMOM

DETSEN

CHIS

P0INT4B

PSIS

P0INT8

SUB5

SUB6

TEXT

TESTA

SUB8

SUB11

SUB8V

SUB9

SUB9V

SUB5V

COVARD

SETID

SENSIT-

SENSIT-

•2D

•2D

TRIDENT-CTR

SENSIT-•2D

TRIDENT-CTR

SENSIT-

SENSIT-

SENSIT-

SENSIT-

SENSIT-

SENSIT-

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

SENSIT

•2D

•2D

•2D

•2D

•2D

•2D

-

-

yes

-

yes

-

-

-

-

-

-

yes

no

no

no

yes

yes

no

no

no

no

no

no

79

1. Subroutine EDNB. Neutron and gamraa-ray energy group structures

are read in from cards and the lethargy widths for each group are

calculated.

2. Subroutine GEOH. Geometry related information is read in and

edited.

3. Subroutine SNCQN. This routine was taken and adapted from the

TRIDENT-CTR code. The EQ cosines and weights are calcualted. The

quadrature information is edited whenever I0PT is 1 or 3.

4. Subroutine TAPAS. Files are assigned to the various flux data.

The filenames for the angular fluxes are read from the input file.

Those filenames will have to be of the form TAPEXY, where XY will

be the input information. Filenames in the same format will then

be assigned to the adjoint angular fluxes (on sequential files in

this case), and the flux moments. The maximum number of words to

be written on each file is controlled by the input parameter

MAXWRD. Groups will never be broken up between different files.

5. Subroutine PNGEN. This subroutine originates from the TRIDENT-

CTR code. Spherical harmonics functions, used for constructing

flux moments, are calculated. For the adjoint flux moment calcu

lation the arrays related to the spherical harmonics will be re

arranged to take into account the fact that the numbering of the

angular directions was not symmetric with respect to the four

quadrants in TRIDENT-CTR.

6. Subroutine FLUXMOM. The adjoint angular fluxes will be re

written on a random access file. The direct and adjoint flux

80

moments are constructed and written on sequential files. In the

case that the input parameter IPREP1, it is assumed that those

manipulations are already performed in an earlier SENSTT-2D run.

In this case onp has to make sure that the parameter MAXVRD was not

changed. While creating the regular flux moments, the scalar

fluxes will be extracted and written on a file named TAPE1.

7. Subroutine DETSEN. From the scalar fluxes, the integral re

sponse for each detector zone is read from input cards. The detec

tor sensitivity profile is calculated and edited. In the case that

the input param.-ter DETCOV equals one, a covariance matrix has to

be provided, subroutine SUB9 will be called and a detector response

uncertainty analysis is performed.

8. Subroutine CHIS. The x's are calculated for each perturbed

zone and for the sum over all perturbed zones based on angular

fluxes. In the case that the parameter ICHIMOM equals one, this

subroutine will be skipped and the x's will be calculated based

on flux moments via the t|>'s.

9. Subroutine POINTAB. This subroutine sets LCM pointers for the

flux moments which will be used in SUB4B.

10. Subroutine PSIS. The tj/'s are calculated for each of the per

turbed zones and for the sum over all perturbed zones based on

flux moments. In the case that ICHIMOM is not equal to zero also

the x's will be calculated from flux moments. In the case that

parameter IPREP equals one, the tji's will be read in from file

TAPE3.

81

11. Subroutine P0INT8. This subroutine sets pointers for the

appropriate x's and t|>'s, used in subroutine SUB8.

5. COMPARISON OF A TWO-DIMENSIONAL SENSITIVITY ANALYSIS WITH A ONE

DIMENSIONAL SENSITIVITY ANALYSIS

Before applying SENSIT-2D to the FED (fusion engineering device)

inboard shield design, currently in development at the General Atomic

Company, it was necessary to make sure that SENSIT-2D will provide the

correct answers. One way for checking on the performance of SENSIT-2D

is to analyze a two-dimensional sample problem, which is one-dimensional

from the neutronics point of view, and then to compare the results with

74 46

a one-dimensional analysis. In this case ONEDANT and SE"SIT are

used for the one-dimensional study, while TRIDENT-CTR, TRDSEN, and

SENSIT-2D are used for the two-dimensional analysis.

Two sample problems will be studied. The first sample problem uses

real cross-section data, while the second sample problem utilizes arti

ficial cross sections. Computing times, the influence of the quadrature

set order, and the performance of the angular fluxes versus the flux

moments option for the calculation of the chi's will be discussed.

83

5.1 Sample Problem #1

The first sample problem is a mock-up of a cylindrical geometry

(Fig. 8). There are four zones present: a source zone (vacuum), a per

turbed zone (iron), a zone made up of 40% iron and 40% water, and a

detector zone (copper). The reaction rate of interest is the heat gen

erated in the copper region. The source was assumed isotropic and had a

neutron density of one neutron per cubic centimeter (1 neutron/cm3).

The source neutrons are emitted at 14.1 MeV (group 2). The left bound

ary is reflecting, and on the right there is a vacuum boundary condition.

Thirty neutron groups were used with a third order of anisotropic scat-

72 tering. The cross sections were generated using the TRANSX code. The

energy group boundaries are reproduced in Table V.

In the two-dimensional model (TRIDENT-CTR) two bands--each 0.5-cm

wide--are present. In order to be consistent with the one-dimensional

analysis the upper and the lower boundaries were made reflective (Fig.

9). Each band is divided into 35 triangles (5 triangles for the source

zone, 10 triangles for each of the other three zones). The automatic

mesh generator in TRIDENT-CTR was used. The convergence precision was

-3 -3

set to 10 . A convergence precision of 10 means here that the aver

age scalar flux for any triangle changes by less than 0.1% between two

consecutive iterations. A similar criterion is used in 0NEDANT. The

calculation is performed with the built-in EQ -8 (equal weight) quad

rature set. The mixture densities are given in Table VI. For the ad

joint calculation the source is in zone IV and consists of the copper

84

10cm

Figure 8. Cylindrical geometry representation for sanple problem i'l

85

TABLE V: 30-GROUP ENERGY STRUCTURE

N e u t r o n * E-Upper Croup ( H r V j

1.700*01

1.500*01

1.350*01

1.200*01

1.000*01

7.790*00

6.070*00

3.680*00

2.865*00

2.232*00

1.738*00

1.353*00

8.230-01

i.000-01

3.030-01

1.840-01

6.760-02

2.480-02

9.120-03

3.350-03

1.235-03

4.540-04

1.670-04

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

E-Lower E -Uppr r ( r t fV ) (MrV I

1.500*01

1.350*01

1.200*01

1.000*01

7.790*00

6.070*00

3.680*00

2.865*00

2.232*00

1.738*00

1.353*00

8.230-0!

5.000-01

3.030-01

1.840-01

6.760-02

2.4S0-02

9.120-03

3.350-03

1.235-03

4.540-04

1.670-04

6.140-05

« •

2

8

3

1

4

1

.140-05

.260-05

.320-06

.060-06

.130-06

. 140-07

.520-07

Group E-Lower

24 2 . 2 6 0 - 0 5

25 8 . 3 2 0 - 0 6

26 3 . 0 6 0 - 0 0

27 1.130-06

28 4 .140-07

29 1 . 5 2 0 - 0 7

30 I . 3 9 0 - 1 0

86

c

E-U w •J l-t: Li

10 cm

\ZO/E i\ I WURCE W

/VAVLUI/ \

' 5 t n f a n g l e g

REFLECTING 10 cm

ZONE //II

PERTURBED

Fe

10 t r i a n g l e s

BOUNDARY 10 cm

ZONE 0111

U0% Fe


10 cm

ZONE #IV

DETECTOR

Cu


u

E CJ 5 -

m < • C

° £

t:

REFLECTING BOUNDARY

30 neutron groups

neutron source: 1 neutron / cm in group2 (14.1 Me")

P-3 EQ -8 : third-order of anisotropic scattering ' n

8th-order equal weight quadrature set

response function:copper kerma factor in zone /!IV

-3 convergence precision : 10

Figure 9. Two-dimensional (TRIDENT-CTR) representation for sample problem i'l

87

TABLE VI. ATOM DENSITIES OF MATERIALS

ZONE //l

ZONE #11

ZONE #IIIb

ZONE #IV

Vacuum

Fe

Fe

H

0

Cu

Atoms/m

—

8.490 + 28a

3.396 + 28

4.020 + 28

1.900 + 28

8.490 + 28

a 8.490 + 28 = 8.49 * 1028

b 40 vol % Fe and 40 vol % water.

88

kerma factors. The response is calculated in that case in zone I. It

was found that the adjoint calculation required more iterations and time

in order to reach convergence. Originally the forward calculation was

done using 20 triangles per band. The adjoint problem, however, did not

converge. In the evaluation process of the kerma factors, the kermas

for some groups are made negative in order to satisfy energy balance.

Making those negative sources zero in the TRIDENT-CTR run did not lead

to any improvement. Subsequently, 35 triangles per band were used.

When the negative sources were set to zero convergence was reached.

Ignoring the negative kerma factors leads to a 20% increase in the total

heating. The forward calculation required about 11 minutes cpu time

(central processor unit time on a CDO7600), while the adjoint cal

culation required about 13.5 minutes. Generating the angular fluxes

using the TRDSEN code required about 20 seconds of cpu time for each

case.

TRDSEN does on extra iteration in order to generate the angular

fluxes. The convergence criterion in TRIDENT-CTR is based on the scalar

fluxes, and therefore one extra iteration in TRDSEN should be adequate.

However, restarting TRIDENT-CTR with the flux moments s„ starting

guesses, revealed that for some groups two extra iterations were neces-

_3 sary to reach a convergence precision of 10 . No explanation for this

could be found.

The one-dimensional model (ONEDANT) contains 35 intervals (5 for

the source zone, 10 intervals for each of the remaining zones). The

one-dimensional description for the forward problem is summarized in

89

<

o u z

e 3

10 cm

ZONE #1 SOURCE vacuum

5 intervals

10 cm

ZONE /'II PERTURBED

zFe 10 intervals

10 cm 10 cm

ZONE /'III

40%Fe+60%H,

ZONE i'lV DETECTOR

Cu 10 intervals 110 intervals

o co

< >

3 .

30 neutron groups

neutron source: 1 neutron / cm"' in group 2 (14.1MeV0

P-3, S-8 : third-order of anisotropic scattering

8th-order Gaussian quadrature set

detector response: copper kerma factor in zone /'1V

convergence precision: 10

Figure 10. One-dimensional (ONEDANT) -^presentation for sample problem //l

90

Fig. 10. Again it was found that the use of negative sources in the

adjoint calculation caused difficulties with respect to the convergence.

In that case, groups IB and 19 triggered the message "TRANSPORT FLUXES

BAD"; groups 4, 5, 6, 7, and 19 did not converge (max. number of inner

iterations 300/group). However, the overall heating in the copper

region was within 0.1% of the heating calculated by the forward run. A

coupled netnron/gamma-ray calculation (30 neutrons groups and 12 gamma-

ray groups) in the adjoint mode led to some improvement. In that case,

only group 2 did not converge. The required convergence precision in

-4 the ONEDANT runs was set to 10 . The built-in S-8 Gaussian quadrature

sets were used. In order to be consistent with the TRIDENT-CTR calcula

tions, the negative sources in the adjoint case were set to zero, even

though this did not seem to be necessary. Each run required about six

seconds of cpu time.

A standard cross-section sensitivity analysis (the cross sections

in zone II are perturbed) was performed using the SENSIT code and the

SENSIT-2D code. A comparison between the SENSIT and the SENSIT-2D

results revealed that SENSIT does not rearrange the angular fluxes

correctly (in cylindrical geometry). To correct this error, a shuffling

routine which takes case of this deficiency was then built into SENSIT.

The SENSIT results are in good agreement with those obtained from

SENSIT-2D. The flux moments versus the angular flux option was tested

out for the calculation of the loss term. Again there is good agreement

Finally, an uncertainty analysis was performed for the heating in the

copper zone. The SENSIT-2D analysis matches the SENSIT analysis.

91

5.1.1 TRIDENT-CTR and ONEDANT results

A comparison of the heating in the copper region (zone IV) between

TRIDENT-CTR (and SENSIT-2D) and ONEDANT (and SENSIT) is summarized in

Table VII a. The adjoint calculations yield a 20% higher heating rate

due to the fact that the negative kerma factors were set equal to zero.

The one-dimensional and the two-dimensional analysis are in agreement.

The computing times for those various runs are given in Table VII b.

Each ONEDANT run requires about 8 seconds of total computing time

(LTSS time), whereas it takes about 12 minutes to do the TRIDENT-CTR

runs. The TRIDENT-CTR runs were done with a convergence precision of

-3 -4

10 , whereas for the ONEDANT runs a convergence precision of 10 was

specified. In order to obtain the same convergence precision in

TRIDENT-CTR about eight additional minutes of cpu time are required. It

was found that a forward coupled neutron/gamma-ray calculation (30 neu

tron groups and 12 gamma-ray groups) required only 8 minutes of comput-

ing time with TRIDENT-CTR (convergence precision 10 ). An explanation

for this paradoxial behavior is related to the fact that O /o_ has a s T

different (smaller) value in a coupled neutron/gamma-ray calculation.

The flux moments generated by TRIDENT-CTR and ONEbANi were com

pared. In the ONEDANT geometry the angular fluxes are assumed to be

symmetrical with respect to the z-axis, so that the odd flux moments

(0j, 02, <J>3, and $ ) vanish. Since TRIDENT-CTR performs a real two-

dimensional calculation the odd moments will not be zero in that case.

In our sample problems there is still symmetry with respect to the

92

TABLE. Vila. COMPARISON OF THE HEATING IN THE COPPER REGION CALCULATED BY ONEDANT AND TRIDENT-CTR

FORWARD ADJOINT

ONEDANT

ONEDANT

TRIEENT-CTR

SENSIT

SENSIT-2D

2.37382 + 7

2.01189 + 7

2.01175 + 7

2.01011 + 7

2.01098 + 7

2.40541 + 7

2.01882 + 7

2.39263 + 7

2.4054* + 7

negative KERJ1A factors set to zero

TABLE Vllb. COMPUTING TIMES ON A CDC-7600 MACHINE

CPU-TIME' I/O TIME

ONEDANT FORWARD

ONEDANT ADJOINT

TRIDENT-CTR FORWARD

TRIDENT-CTR ADJOINT

SENSIT

SENSIT-2D

5.80 s e c .

6.09 s e c .

4.92 s e c .

8.50 s e c .

1.87 sec .

1.82 sec .

0.55 sec .

9.02 sec .

LTSS TIME

7.65 sec.

7.97 sec.

1 ^ ^ »r>T ^ > i f at?

X~/>~J I I U l l U L C b

11.1 minutes

6.08 sec.

17.84 sec.

central processor unit time

input/output time

Livermore time sharing system t 'me (total computing time)

93

z-axis. For that reason, the odd moments in TRIDENT-CTR will have oppo

site signs in band one and band two. For some zones and some groups

this was not completely the case. There was about 30% difference in the

absolute values of some flux moments in band one and band two, which

indicates that the problem was not ir J sense truly converged. The con

vergence criteria in ONEDANT and TRIDENT-CTR test only for the scalar

fluxes between two consecutive iterations. Even when the convergence

criteria are satisfied in both codes, a true convergence of the angular

flux is not guaranteed. The even moments in band one are exactly the

same as those for band two. Because the contribution of the odd moments

is small compared to the contribution of the even moments (about one

thousandth), the problem can be considered fully converged.

The scalar flux moments calculated by TRIDENT-CTR and ONEDANT are

in very good agreement. The higher-order moments are different. Since

TRIDENT-CTR and ONEDANT do not use the same coordinate system, they do

not calculate the same physical quantity for the higher-order flux

moments. As long as TRIDENT-CTR is consistent with SENSIT-2D, and

ONEDANT consistent with SENSIT, the results from the one-dimensional

sensitivity analysis should match those obtained from a two-dimensional


5.1.2 SENSIT and SENSIT-2D results for a standard cross-section sensitivity analysis

A standard cross-section sensitivity analysis was performed using

94

SENSIT and SENSIT-2D. The sensitivity of the heating in zone IV to the

cross sections in zone II was studied. SENSIT-2D requires about three

times more computing time than SENSIT in this case (Table VII b). The

main part of the calculation involves the evaluation of the ijj's (gain

term). A complete sensitivity and uncertainty analysis may involve

several SENSIT (or SENSIT-2D) runs. Thus an option which allows one to

save the i|f's has been built into SENSIT-2D. It is obvious that the com

puting time required in SENSIT-2D is negligible compared to the comput

ing time required for the forward and adjoint TRIDENT-CTR calculations.

The partial and the net sensitivity profiles calculated by SENSIT

and SENSIT-2D are reproduced in TABLES VIII a and VIII b. It can be

concluded that the SENSIT-2D results are in good agreement with those

obtained by SENSIT. Note that the absorption cross section is negative

for groups 2 and 3. A negative absorption cross section does not nec

essarily indicate that errors were made during the cross section pro

cessing. There are various ways to define an absorption cross section,

and a controversy about a commonly agreed on definition is currently in

progress. What is called an absorption cross section in a transport

code is not truly an absorption cross section but the difference between

the transport cross section and the outscattering (o8 = cr - I a ^ 8 ). 3 tr g'

Note that groups 2 and 3 are the main contributors to the integral

sensitivity.

It was mentioned earlier that the x's can be calculated based on

flux moments or based on angular fluxes according to

95

TABLE V i l l a : PARTIAL AND NET SENSITIVITY PROFILES FOR THE ONE-DIMENSIONAL ANALYSIS

( P a r t 1)

DCFIHITIOHS OF S E H S I T SENSITIVITY PROFILE NOrtNCLRTURE

OfUS - SENSITIVITY PKOFILE HCR I>ELTn-U FOR TIE RUSDRP1IUII CKOSS-SEET10H ITfiHiM FROJI PO5ITI0N 1IIR IN INPUT CRUSS-SECTIUN ThULES). PUKE LUSS TERTI

MU-FISS - SENSITIVITY PkOFILE PER DELTn-U FUR THE CRDSS 5CCTI0II IN POSITION IHiVrt IN INPUT XS-TRBLES. UI1CH 15 USURLL, HU-TlrCS THE FISSION CRUSS SCCT1UH. PURt LUSb TERI1

S)S • PRRTIRL SENSITIVITY PROFILE PtR ICLTn-U FUR THE SCHTTERHC. ERli:^-3ECTI0N (CQIPUTED FOR EACH ENERGY GROUP RS A OlfiuUHRL bun FUUrl INI'UC XS-1RULCS>. LUSS lEkl l DULY

TXS • SENSITIVITY PROFILE PEP DELTH-U FOS THE TOTKL CROSS SEETIOH ((lb GIVEN IN POSITION IHT IK INPUT CRDSS-'-XCTlOH l f l L L t b ) . HURL LOLS lERrl

M-GfllN • mPTlRL SENSITIVITY PROrlLI Pr:.' I L L T . I - U FOR llir NLU1IUU bEfiTTERItlG C n j S j - S r c T l f l H . COIN TECH FOR SCH51TIVI7Y Lftlh= DUE 1U U.HIN.R1RG UUI UF EtCI.LY ul.UUR G IHIU (ILL L u X H HLUIitJil

ENERGY GROUPS. CDrFUTED FROH FORU1RB DIFFERENCE FORrtJLRriON.

6-GaiN - PPPTIflL SENSITIVITY Pl'OFILE PER DTLTfl-U FUR TIIL GC l « SCnTTERItT, r.CQiS-^CC MOI I . G'lIH U P H FUR SENSITIVITY GRINS DUL 1U ..TnTILT'lllG UIIT UF L i t , IB EI ILRGY LKUUP G IHID RLL LDULR GArlt) ENERGY COWilTED FROM FOI.URRD DIFFERENCE FUKIULfiriOH.

N-GfllHlSED) - rr-arOCRED PnttTML SEHSlT IV i r * W UTILE PLn ICLTn-U FOR SCRTTi;:llIG E n ; . ' . -SLCTIDil. Cnlll TERM FOR SENSITIVITY GnlllS HUE 1U bLnilEKlllT. IHTU U.'UUP G TROI1 IU.L1IIUU.K NLU N.UN thi..:GC GROUPS. CfiivuTEIl H.'UII HDJUINr Dllt-LRIIU. FURI IJI Nl IUM. COrrESPOHDS TU SIHGLE-DIITLRLIIf l l lL bLU m i b l T I V I T Y PROFILE. PSLDlG-QUTl PER I tLU-OUT. INTEGRATED OVER RLL IHCIDLHT ENERGY GRUUHb.

HC-CfilN - PARTIAL SENSITIVITY PROFILE PCI! IiLLlfi-U IOR MIL. t r . l t l i PI OUUCTIOH r r U i ' . - S T T l U H flT t'CUTDOH EIILl.'GY GRIIUP G. I'UI.L U1IM KRr l FOrt 5 C I L , I T l V n Y GMNS DUE TO TRniWLK FROM NEUTRON GROUP G INTU ALL G H I t * GKUUPj.

SEH • NET SENSITIVITY PROFILE HER DELTR-U FUR THE SOirTLKIHG CRUiS-bCCTIOH lSEN-S>£»NCillN)

SENT • NET SEMSITIVITY PROFILE PER DCLTR-U FOR THE TUTRL CROSS-SECT IUN CSLHT-TSCiHUfllH)

SEW - SENSITIVITY PRUF1LE HER DELTR-U FUR TIC DETCC1UR REGHUItE FUNCTION RIG)

SENS • SENSITIVITY PROFILE PER DELTR-U FUR THL SOURCE DlblRIUUTION FUNCTION 0<G)

y6

(Part 2)

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• . - - . - . . , • • • • - • • • • • • * • - • • 2.10C92E»07

» J . J - J PURE CS1N TFRrE H-CB1N

B. Q.432C<0B 7 .523E-U1 1.3GSE-01 I .UU2E-BI 1 .GIBE-BI I . 7 1 3 E - B 1 2 . 3 2 1 L - B 1 3 . I 1 1 E - C 1 3 . 1 ' j J l > B l 3 . b l l C - 8 l 4 . 3 ' J1L -01 6 . 3 3 J L - 0 I 7 . 5 2 ^ E - 0 1 I . 9 U ; E - U 1 2 . C J . E - B 1 3 .191E-B1 1.9- IJE-L2 4.G3i ; - -Ci2 2 . ;u ' JE-G2 i . ^ j r s - 0 2 1.G37C-02 l . i 3 1 L - 0 2 I . U O ; E - D 3 6 . 2 3 3 L - L 3 4.3-11E-E3 2 .509C-03 I .3J3E-G3 6 . 3 6 ; r . - 0 - 1 1 . IG3E-03

3.275E-K.0

N-GniN(5ES) 0 . 4.033E-KW l.b'JJE»G9 2.01DE-OI 3 . l 2 9 r Ol 2.4( '0C-BI 2 .303L-B1 2.C33E-B1 3 .6P iE -O l 3 .GG:E-B I 3.SGGE-01 4 .37VL-01 7.C63E-01 0.0<!'. 'L-l, l 2 .01LC-B1 3 .0^3E-U l 3 . 2 I ! E - B l 2 .C l i :E -02 4 . & J E - G 2 2.13. iE-B2 l . u 2 ; C - 0 2 l . 6 - r jE -02 I .3G1E-02 l . t ) l l ;E - t !2 B . 9 3 r t - 0 3 4 . 4 U E - D 3 2.S3CE-B3 l .3 ' j 'JE-B3 6 .5G-E-04 1. IB1E-03

3.275E«0B

w t i r r a NG-G(llN

a. e. 0 . 0 . a. B. 0 . 0 . a. B. u. 0 . B. 0 . u. 0 . B. u. 0 . 8 . a. 0 . 0 . 0 . 0 . B. 0 . B. 0 . 0 .

8 .

98

http://iicuTr.au

http://-3.rA.E-US

http://-3.2iC.E-H2

http://-4.94.C-n3

LMAX £ . LMAX X 8 = 1 I ¥$ g g = Z T « .

£=0 k=0 £=0

Table IX provides a comparison between the x's calculated from angular

fluxes and flux moments. There is a very good agreement. It was found

that this relationship is also true in the one-dimensional analysis.

For £ = 0 and £ = 1, the 4^ s calculated in SENSIT and SENSIT-2L are

different. However, the f.'s defined by

£ 4,88 = z yKgg ( ] 2 3

are in agreement.

5-1.3 Comparison between a two-dimensional and a one-dimensional cross-section sensitivity and uncertainty analysis

A cross-section sensitivity and uncertainty analysis was done lor

the heating in the copper region, using SENSIT and SENSIT-2!). In this

analysis the effects of the uncertainties in the secondary energy dis

tribution were included. Six separate SENSIT (or SENSIT-2D) runs were

required:

99

TABLE IX: CO-MPAPaSON BETWEEN THE CHI'S CALCULATED FROM ANGULAR FLUXES AND FROM FLUX MO'ENTS

J T T _ ^

1

?

3

h

r

6

7

9

9

1C

11

1?

13

l i

13

16

17

18

If

20

21

2?

?3

2u

2?

26

27

?6

29

30

sT\sn-

ehi (anr . f luxes 1

0 . 0

2 .5059-9

2.1,390*7

6.1301,*6

8.3365*6

5 .6969-6

9 .3953-5

5.6893*6

7.151i3*6

7.331,9*6

7 .9619 -6

J.1571*7

2.9571*7

2.51i'9*7

6 .3191-6

1.57 9 j -7

6 .5769-6

1 .7636-6

1.21i63*6

7.6B60-S

3.7761.-5

3 . 7 U S - 5

2.9673*5

2.20L1-5

1 .5067-5

9 . l63 l i - l ,

5.U090-1,

?.3Ji76»li

1.3J21i»ii

?.3951*li

n

chi (flux me-ie-.ts)

0 .0

2.5053*9

2.li36?*7

6.1?6li-6

8.3329*6

5.69514-6

9.39i,l,*6

5.6991-6

7.151,3*6

7.33L9-6

7.9619-6

2.1571*7

2.9571-7

2.51,29-7

6.3L91-6

1.57?3-7

6.5269*6

1.7536-6

1.21i63-6

7 .6860-5

3 .776t-5

3.7U15-5

2.9673-5

2.?01il-5

1.5067-5

9.1i631i-li

5.1,091-ii

J.81i76-li

1.33?5-li

?.391i9-li

c I («ng. r iures )

o.o

J.li933-8

2.1ili93-7

6.1829-6

8.U135-6

$,7L2li*6

9.h"*75**

5.7312-6

7,2079-6

7.3991i-6

7 .9321* '

7.1793-6

2.9322-6

2.6076-6

6.5029-6

l .«3«5*7

7. l39l i -6

1.9060-6

1,2862*6

7.7152-5

3.91mO*5

3.9705*5

3.0?u5-5

j.71,06-5

l.SfP'S

9.5li05*li

S.lijoo-li

2.81|99-li

1.3318-1,

2.1,910*1,

c*.. ( f l u * w m - n t s )

2.50O3-9

100

three runs for the vector cross-section sensitivity and uncertainty

analysis (one for the cross sections of zone II, one for the cross

sections of zone III, and one for the cross sections of zone IV),

three runs for the SED sensitivity and uncertainty analysis.

Oxygen was not included in the vector cross-section sensitivity and un

certainty analysis, and hydrogen was ignored in the SED sensitivity and

uncertainty analysis.

The procedure for an uncertainty analysis has been discussed by

45 Gerstl. The results from the one-dimensional analysis are reproduced

in Table Xa, while those from the two-dimensional study are given in

Table Xb. The studies are in good agreement. Sensit required a total

of 89 seconds of computing time, while SENSIT-2D required 90 seconds on

a CDC-7600 machine. The uncertainty of the heating rate due to all

cross-section uncertainties is 30%. The iron in zone II is the largest

contributor to that uncertainty. The contribution of the SED uncer

tainty is smaller than that from the vector cross sections. Gerstl

points out that the results obtained from the SED analysis might have

been underestimated due t<"> the simplicity of the "hot-cold" concept and

due to the fact that the partial cross sections which contribute to the

secondary energy distribution were not separated into individual partial

45 cross sections.

101

TABLE Xa. PREDICTED RESPONSE UNCERTAINTIES DUE TO ESTIMATED CROSS SECTION AND SED UNCERTAINTIES IN A ONE-DIMENSIONAL ANALYSIS

CROSS SECTION

Fe

Fe

0

H

Cu

All*

ZONE

II

III

III

III

IV

RESPONSE UNCERTAINTIES DUE TO SED UNCERTAINTIES, IN %

AR" R x-sect

zone

8.18

2.50

0.78

-

4.02

AR R

zone

L

1 i

8.18

2.61

4.02

9.48

RESPONSE UNCERTAINTIES DUE TO CROSS-SECTION UNCERTAINTIES, IN %

AR R

x-sect "zone

23.80

10.33

-

AR" R

*

zone

23.80

10.52

1.96 !

11.72 11.72

28.54

Overall uncer ta in ty = (9 .48 2 + 28 .54 2 ) ' s = 30.0%

* quadrat ic sums

102

TABLE Xb. PREDICTED RESPONSE UNCERTAINTIES DUE TO ESTIMATED CROSS SECTION AND SED UNCERTAINTIES IN A TWO-DIMENSIONAL ANALYSIS

CROSS SECTION

Fe

Fe

0

H

Cu

All*

ZONE

II

III

III

IV


AR R element

zone

8.17

2.50

0.79

r —

A.02

AR R

zone

8.17

2.62

a.02

9.47

RESPONSE UNCERTAINTIES DUE TO CROSS-SECTION UNCERTAINTIES, IN %

AR R

element zone

23.88

10.27

-

1.96

ARI*

zone

23.88

10.46

11.68 i 11.68 i

28.57

Overall uncertainty = (9.472 + 28.b l 2 ) ^ = 30.1%

* quadratic sums

103

5.2 Sample Problem #2

A simple one-band problem will be analyzed to study the influence

of the mesh spacing, quadrature order, convergence precision, and the

c-factor (mean number of secondaries per collision) on the sensitivity

profile. The band is l-cm high and 20-cm wide. There are ten distinct

zones, each l-cm wide (Fig. 11), and all zones are made of the same

material. A three-group artificial cross-section set with a third-order

anisotropic scattering is used (Table XI). The P., P2, and P. compo

nents of the scattering cross-section tables were chosen to be identical

with the P. component. A volumetric source with a source density of

1 neutron/cm3 in group 1 is present in the first zone. A standard

cross-section sensitivity analysis will be performed, in which the cross

sections in zone IV are perturbed, and the detector response is calcu

lated in zones IX and X for a response function of 100 cm in each

group.

5.2.1 Influence of the quadrature order on the sensitivity profile

The detector response calculated by TRIDENT-CTR using EQ,, EQ _,

and EQ.., quadrature sets are compared in Table XII. For the first three

cases, the pointwise convergence precision was set to 10 and each zone

contained four triangles (using automatic meshes). Five additional

cases are included in Table XII:

104

RSFLFCTIW B O " X W Y

o fit s

Sk

sm-BCE ZONE

tone 0 I

—

tone

# n ton* ' III

PKHT. ZONE

tone • IV

tone » V

tone * VI

tone

# v:: tone * VIII

DrT. 7O:;E

* I

tone • I I

D T , Z0KE * II

tone * I

— i

I

^

f 'i

— 1 0 .0 1 . 0 2 . 0 3 .0 li.O 5 . 0 6 .0

REFl.r?Tn.-3 BtWNDART

e.c S.C 10 .0 c*.

r - z geometry

All zones contain identical materials

3 neutron groups

Neutron source: 1 neutron / cm in zone I and group 1

Response function: 100 cm (all groups)

Figure 11. Two-dimensional model for sample problem "2

105

TABLE XI: CROSS SECTION TABLE USED IN SAMPLE PROBLEM 112 (THE P0, P , P2, AND P TABLES ARE IDENTICAL)

Cross Section

Croup g = l g = 2 g = 3

I 8

e d i t

* !

* !

* !

-

0.02

0.0

0.1

-

0.05

0.0

0.2

-

0.1

0.0

0.3

yg->g

5g-l-g

5g-2^g

0.05 0.1 0.2

0.0 0.02 0.05

0.0 0.0 0.01

106

TABLE XII: INTEGRAL RESPONSE FOR SAMPLE PROBLEM #2

Transport Quadrature Converge Code Set Precisi

TRIDENT-CTR

TRIDENT-CTR

TRIDENT-CTR

TRIDENT-CTR

TRIDENT-CTR

ONEDANT

ONEDANT

ONEDANT

EQ-62

EQ-12

EQ-16

EQ-12

Eq-12

C-123

S-32

S-32

10"3

10"3

10"3

10"4

10"4

10'4

10"4

ID'4

1 if spatial intervals for ONEDANT.

2 equal-weight quadrature sets.

3 Gaussian quadrature sets.

Forward Adjoint // Triangles1 Response Response

40

40

40

40

80

40

40

80

593.968

592.326

553.659

593.659

593.688

593.855

591.814

592.055

592.256

591.659

592.476

593.148

593.208

590.370

590.883

590.900

1. integral response using an EQ 1 2 quadrature set with conver-

gence precision 10 ;

2. integral response using an EQ1? quadrature set with conver-

-4 gence precision 10 and eight triangles per zone;

3. integral response calculated by ONEDANT using an S 1 ? quadra-

-4 ture set, four intervals per zone and a 10 convergence pre-

4. integral response calculated by ONEDANT, using an S~„ quadra-

-4

ture set, four intervals per zone and a 10 convergence pre

cision;

5. integral response calculated by ONEDANT, using an S„_ quadra-

-4

ture set,^eight intervals per zone and a 10 convergence pre

cision.

The response functions in Table XII are in good agreement (maximum dif

ference 0.6%). The standard cross-section sensitivity profiles for the

EQ,, EQ.-, and EQ... calculations are reproduced in Tables Xllla, XII lb,

and XlIIc. The integral sensitivity for the EQ, case is 5% different

from the E0.„ case for AXS (absorption cross-section sensitivitv nro-

file) and 5% different for N-GAIN (outscattering cross-section sensi

tivity profile). The results obtained from the EQ,? calculation are in

good agreement with those obtained from the E Q ^ calculation. The sen-

-4 sitivity profiles for the EQ._ case (10 convergence precision) and the

-4 EQ1? case (10 convergence precision, eight triangles per zone) are not

shown. They are nearly identical with Table XHIb.

108

TABLE I l i a : STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT-2D FOR THE EQ-6 CASE (CONVERGENCE PRECISION 0 . 0 0 1 , U TRIANGLES PER ZONE) FOR SAMPLE PROBLEM i'2

GROUP UPPl'o-EIEV)

2 s.oBaE-ee 3 I . B B B E - H - B

JNTEGRBL

DELTfl-U 6 . 9 2 E - 8 1 1.61E^9C

6 .33E-B1

m«™«» F l l t E L 0 5 BXS NU-FISS

-4 .E32E-B2 B. - 4 . 5 i 2 E - B 3 B. - 1 . U 2 E - B 2 6 .

-4 .732E-B2 B.

S T E l n S « » « » . » • * » SXb TXS

-1.B53E-B1 -2.31EE-B1 -J.3t9E-B2 -I.S17.E-B? -2 .26JC-B2 -3.<S2oE-B2

•« PURE Gain TEP-E h-GolN h-GMMSEIi] NG-r,c

1.549E-B1 I.I5M-BI 6. 1.JB6E-B2 2.1SLE-3:- B. 2.2eJE-B2 3.677E-B2 8.

-I.E6IE-B1 -2.I35E-BI 1.423E-B1 l . « 3 L - e i

GROUP UPPEP-E'EV) D E L T B - U 1 I.6BBE*Bi 6.93E-BI 2 S.BBBE-HJB i .6 I E - 6 0 3 l.BBBE- BB 6.93E-B1

INTEGRAL

»*»» NET PROFILES « « « • SEN SENT

-3.B39F-B2 - 7 . b ? ! E - e 2 -1 .731C-B3 -6 .2E3E-B3

3.252E-B6 - ] . i a ? E - B 2

-2.3B5E-B2 - 7 . I 1 C E - 8 2

TABLE Xlllb: STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT-2D FOR THE EQ-12 CASE (co:rvERCFNCE P R E C I S I O N o . o o i , u TRIANGLES

G^nUP UPPFP-E(EV1 i i.eoiv-ei 2 5.BB":~-H!B 3 i.eoec-ee

INTEGRA^

GROUP UPPt' -EfEV) 1 l.BB"E->«l 2 5 .BfV : -BB 3 l.BBBE-BB

INTEGRAL

BELTft-U 6 . 9 3 E - 6 1 i KiE+ee 6 .93E-E1

BCTB-U fc.93t-El 1 -6 l t * i3B 6 .93E-B1

n o * > p u BXE

- a . j : - 9 t - B 2 -a .36oE-B3 - I . 1B3E-B2

-4.£E,BE-B2

R E L O S S NU-FISS B. - : B. - 1 6 . - 2

B. - 1

w * » NET PROFILES « » SEN

-2.731C-B2 - l . 6 b 8 E - 8 3

B.B481-B7

-2 .161E-B2

SENT -7.BSBl.-B2 -6 .834E-B3 - I . I B 3 E - B 2

-6.62-SE-B2

T E U S • i i w i i i SXE TXS

. r j I E - B l - 2 . l t . 4 E - B l

.3 IBE-02 - I . 7 4 E E - B 2

.2B6E-B^ -3 .318E-B2

.564E-6 ; -2.B11E-B1

* W M * M PUKE GP1N TER^, » ] « • « » . • N - G C >

1 .4ME-B1 1 . 1 4 i E - e ~ 2 .2B-E-B?

I .34BE-B1

N-GP!* fSEl> H G - G P I N I.B7SE-Ci B. 2 . i r s E - P 2 e . 3 . 7 E I E - ; : e .

l . i 46E -B l B.

109

http://-7.BSBl.-B2

TABLE X I I I c : STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT-2D FOR THE EQ-16 CASE CCONGRUENCE PRECISION 0 . 0 0 1 , A TRIANGLES PER ZONE) FOR SAMPLE PROBLEM #2

GROUP UPPEP-E(EV) 1 l . B B B ' « e i 2 5.8aaE*ue 3 i.eoaE^ea

1HTCGRM.

CROUP UPPEB-EtE." i i.BaaE-oi 2 S.BBBE-BB

DEI.TB-U 6.93E-B1 l.6IE->«B 6.93E-B1

DELT L" 6.93E-B1 1.61E*flB 6.93E-BI

U S E RXS NU'

- 4 . 2 9 3 E - B 2 6. -U.33BE-B3 B. - 1 . B 9 4 E - B 2 B

L D 5 E T E » B S ISS SXS TXS

-1 .71EE-B1 -2 .145E-B1 -1 .295E-B2 - I . 7 3 J E - 0 2 -2 .18BE-B2 -3 .2E1E-B2

PURE COIN TERP6 « * * » — » . . h-GBIN N-GBINtSEDJ NG-&AIN

1.4J3E-BI 1.BG2E-B1 B. 1.133E-B2 2.1B9E-B2 6 . 2.1BPE-B2 3.?32E-B2 B.

INTEGRBL

- 4 . 4 2 9 C - B 2 B.

» * • » HET PROFILES «*»» S.EH 5ENT

-2.734E-B2 -7.824E-B2 - l . t L S E - a i -5.9B3E-B3 -1.4B4E-B6 -1.B94E-B?

-2.1E.2E-B2 -6.5°!E-B2

-1.55BE-BI -I.993E-B1 1.334E-B1 1.334E-B1 B.

110

Note that the net sensitivity profiles SEN (= SXS + N-GAIN) for

group 3 are respectively 3.252 x 10 , 8.040 x \i)~7, and 1.484 x io"6

for the EQ,, the EQ , and the EQ.., case. The large discrepancies here

can be attributed to the fact that those quantities result from sub

tracting two numbers that are nearly equal in magnitude.

It can be concluded from Tables XII and XIII that even when the

integral responses differ by less than 0.4%, the sensitivity profiles

can differ by as much as 5% between an EQ, and an EQ „ calculation. The

close agreement between the results from the EQ ? and the EQ., calcula

tion suggest that this difference is probably due to the fact that the

SI angular fluxes in the EQ, calculation are not yet fully converged.

Indeed, choosing the higher-order anisotropic scattering cross sections

equal to the isotropic components is unphysical. The convergence cri

teria used in ONEDANT and TRIDENT-CTR do guarantee > -jnvergence for the

scalar fluxes, but not for the higher-order flux moments.

5.2.2 Comparison between the two-dimensional and one-dj.m^nsional ar.aVy-sys of csmp 1 e problem j/_2

The cross-section sensitivity profiles resulting from a one-dimen-

-4 sional analysis (S.„ quadrature set, 10 convergence precision and four

-4 intervals per zone; S .„ quadrature set, 10 convergence precision and

eight intervals ptr zone) are compared with those obtained from a two-

-•4 dimensional analysis (EQin quadrature set, 10 convergence precision

1 i-

and eight trianglts per zone in Tables XlVa, .IVb, and XIVc.

Ill

TABLE XlVa: STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT FOR THE S-12 CASE (CONVERGENCE PRECTSION 0 . 0 0 0 1 , U INTERVALS PER ZONE) FOR SAMPLE PROBLEM «2

• • • • - • • m u * • » - • • • « < • * * * • " • » i ^ C § * ) » • TBMws » • • « • • > •

« - * * ! • • h - 6 » I « < . i l t ' • • « - » » ! • • 1 i . bUU("0l fc.9j«-01 -« .CB*€-&C

-01 l . i ' l i - O i ^ . 9 r D * - t c 0. C 5 . Ubbi»£iL I . e i l ' t C - 4 . j ^ , « - L 3

- b e I . 1 4 L - I - U a .0J=pt -Cc D. 3 1. GbC«*bO 6 . V j t - O l - 1 . J t l « - u c

-Oc c . r . j * « - 0 t 3 . ? 3 J B - 0 c C.

•

b.

b.

C.

L D S 1 t t • «

- 1 . 71 j f - u l

- 1. 3L r t - u;

- c . t . c l - w t

I

•* -S

- t . 1 4 £ |

- 1 . ?4JV

- J . Ji-JC

I - T a l l i n . - « . 4 j j i - u e C. - 0 1 l . J 7 « « - 0 l l . E ? « « - C . l 0 .

M 3 U * v » ^ « » ~ B - « * - ' » « c » « - 0 • • • • K B - ' 1 l . O l l ' C * ! . ] fe.Sji-Cl - 3 . b C B l - L c - ? . ^ : c t - l > C £ i .Ol*<«*LC 1 . £ : • - - £ - l . b - l « - 0 j - t . O c f c l - C j 3 >.Glibc- l .C t . V j l - t ; j . s . J £ « - U « - ! . Irt *.! - Lc

-c.?*CW-\tc - ~ . l V l i - 0 c

STANDARD CRCSH-SF.CTION SENS 17IVTTV PP.'FILEF CALC7LA7EIJ EV SENSIT FOP THE S-32 CAST (CONVERGENCE PRT.riSTCN 0 . 0 0 3 1 , 8 INTERVALS PEP. ZONE) FOP SAMPLE PROBLEM i'2

T E B f 5 « • » - » * * » wwwww*. PDBC GCJN TrO-^ n o

- 1 . A 6 L . E - ? : - 2 . I B ^ E - e i 1 .2r'ai . -Bl 9 . S 1 7 E - B : B . - 1 . 2 6 3 1 - i W - I . 7 S I E - B 2 l . l . b L - B : 1 . 9 9 I E - 6 2 B. - 2 . 1 6 1 E - 0 2 - 3 . 2 J 2 E - 0 2 ?. . . ' . 'IE-82 3.EBSE-U2 B.

- I . 5 2 5 E - B I - l . H l E - B I 1 .276F-B1 1 .236E-B1 B.

U E O J ^ L « W E = - E ' E V I

I i be?- . - ; ; 2 5 Be.-: -SB 3 i eeeE-ce

ixn;=a.

0 ( 0 ^ l*>PC5-tiEvi i i.eeet-ei 2 5 BS'- -ee 3 i.Baof«e

luTtseaL

i 93E-B! : fciE-ee f 93i - 6 :

T'ElTn-U t . 9 3 c - B l ! .6!E»flB S.S3E-B]

- 4 - 2 : 5 F - B ; - a . 2 ' * E - e ; - 1 UBIE -U '

-<: 359E-B:

f i L C 5 S NJ-C i = E 0 . e.

B.

« » • MET P P D ^ I L E S « • • S.S" 5 f - "

- 3 - 6 - I E - B ? - § . B ^ I - S 7 - > . 5 i i £ - B 3 •'. o . ' i t - e ;

5 -SB9 i -BJ - I . B 2 I E - B 2

- 2 B91E-B2 -?.258E-B2

112

http://-1.A6l.E-

TABLE XIVc: STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCU CALCITLATED BY SENSIT-2D FOR THE EQ-12 CASE (CONVERGENCE PRECISION 0 . 0 0 0 1 , 8 TRIANGLES PER ZONE) FOR SAMPLE PROBLEM 1)2

GPDjP uPPEP-6'EV)

1 l.BOu-->ei

2 I.BBUE-»BB

1NTEGRBL

UPPEP-E(EV) l.BuSE'Ol

I.BBBE*BB

BELT9-U 6.93E-B1 l .eiE*60 6.S3E-B1

BELTO-U 6.93E-B1 l.61E««B 6.93E-B1

• * * > - » P U R E L O S S T E R n S fiXS NJ-FISS SXS TX5

-4.32dE-B2 U. -I.73BE-B1 -Z.ltJE-Bl -4.36JE-B3 B. -1.3B9E-D2 -l.TzU-e: -I .I62E-B2 t. -2.283t.-B2 -3.3ei.E-B2

PUKE GO IN TEPrt «*«»*»». h-GMN h-GMN(SED) NG-ni"

I .IS'E-BI i.e7<s-ei e. l.I«2E-B2 2.123E-C2 B. 2.2HXE-B2 3.?S5E-B2 B.

-4.4c3E-B2 B.

« « « . NET PROFILES «-»» 5EN SENT

-2.72?E-B2 -7.B51E-B2 - I .b6 iE-63 -C.B3.JE-B3

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113

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Note that the N-GAIN integral sensitivity differs by about 6% be

tween Table XIV b and XIV c. The integral net sensitivity shows a 35%

difference for SEN (= SXS + N-GAIN) and a 10% difference for SENT

(= TXS + N-GAIN) between the one-dimensional and the two-dimensional

analysis. The bulk part of this large difference for the integral net

sensitivity results from the subtraction of two numbers that are nearly

equal in magnitude. A comparison of N-GAIN (integral) in Tables XIV a,

XIV b, and XIV c suggests that - even with an S.„ quadrature set - the

one-dimensional calculation is not yet fully converged.

5.3.2 Comparison between the x's calculated from angular fluxes and the X's resulting from flux moments

The x's (or the loss term of the cross-section sensitivity profile)

can be evaluated based on flux moments (Eq. 58) or based on angular

fluxes (Eq. 57). A calculation based on flux moments requires less com

puting time, less computer memory, and less data transfer. To have an

idea of the order of expansion of the angular fluxes in flux moments

necessary to reach a reasonable accuracy, the x's resulting from angular

fluxes are compared with those obtained from a P-0, P-l, P-2, . . . ,P-17

spherical harmonics expansion of the angular fluxes (Table XV). It is

found that for any expansion of order greater than P-0, there is good

agreement (less than 1% difference for 2 xg) • For very high spherical

g

harmonics expansions (P-15 and higher) there is divergence. This diver

gence can be avoided by doing the computations in quadruple precision.

114

TABLE XV: COMPARISON BETWEEN THE CHI'S CALCULATED FROM ANGULAR FLUXES AND THE CHI'S CALCULATED FROM FLUX MOMENTS

Order of Expans ion of x t h e Angular F lux

r fluxes

0

1

2

3

4

5

6

7

8

9

10

11

12

14

16

17

889.88

796.63

880.43

868.78

884.11

884.58

886.82

888.82

889.16

889.78

889.74

889.91

890.01

889.89

898.27

938.59

951.85

^ o m " Xang^

83.382

76.811

83.755

S3.054

83.374

83.388

83.385

83.389

83.381

83.381

83.382

83.383

83.385

83.382

83.573

83.100

85.617

45.352

41.898

45.524

45.326

45.353

45.364

45.357

45.354

45.353

45.352

45.352

45.352

45.351

45.351

45 431

46.203

46.416

-

103.275

8.905

21.454

5-777

5.282

2.052

1.051

0.720

0.101

0.140

0.031

0.133

0.009

8.610

51.279

65.269

X mean x f° r grouP

115

The small differences in Table XV indicate that the loss term of

the sensitivity profile can indeed be calculated based on a low-order

spherical harmonics expansion of the angular fluxes.

5.2.4 Evaluation of the loss term based on flux moments in the case of low c

The question whether the x's can be computed with adequate accuracy

from Eq. (58) in the case of low c (mean number of secondaries per col-

g

lision) was raised. Based on an analytical one-dimensional analysis of

the half-space problem (one group) with a mono-directional boundary

source, it was found that for c less than 0.8, a low-order spherical

harmonics expansion of the angular flux would lead to erroneous results

in the x's.

In order to confirm the analytical study, sample pmhlem #2 was re

examined with a different cross-section table. The corresponding c's

were 0.5 for the high-energy group, 0.4 for the second group, and 0.33

for the low-energy group. The x's calculated based on flux moments were

still in agreement with those obtained from the angular fluxes (even for

a P-l expansion). An explanation for this paradoxical behavior is prob

ably related to the use of a distributed volumetric source in sample

problem #2, whereas the conclusions drawn in the analytical evaluation

were based on the presence of a mono-directional boundary source.

116

5.3 Conclusions

*

The rigorous study of the two sample problems indicates that there

is good agreement between the one- and two-dimensional analysis. Wher

ever discrepancies appear, a plausible explanation can be provided. Ul

timately, the comparison between a one- and two-dimensional study proves

to be a sound debugging procedure for SENSIT-2D as well as for the

SENSIT code.

For the flux moments versus the angular fluxes comparison for the

evaluation of the x'st there is a strong indication that the loss term

can be calculated from lower-order flux moments (P-l) as well as from

angular fluxes. By the same token, a P-l sensitivity and uncertainty

analysis seems to provide sufficient accuracy.

The study of the influence of the quadrature sets on the sensi

tivity profiles reveals the importance of the angular-flux convergence

in ONEDANT and TRIDENT-CTR. Furthermore, some doubts about the meaning-

fulness and practicality of the net sensitivity profile (SEN) can be

raised.

117

6. SENSITIVITY AND UNCERTAINTY ANALYSIS OF THE HEATING IN THE TF COIL

FOR THE FED

In this part a secondary energy distribution and a vector cross-

section sensitivity and uncertainty analysis will be performed for the

heating of the TF coil in the inner shield of the FED. The results ob

tained from the two-dimensional analysis will be compared with selected

results from a one-dimensional model. The blanket design for the FED is

82 8*} currently in development at the General Atomic Company. '

6.1 Two-Dimensional Model for the FED

The two-dimensional model for the FED in r-z geometry is illus

trated in Fig. 12, and is documented in more detail in reference 84.

The material composition is shown in Table XVI. In the forward TRIDENT-

84 CTR model, which was set up by W. T. Urban, the standard Los Alamos

85 42 coupled neutron/gamma-ray group structure was used. There are 30

84 neutron groups and 12 gamma-ray groups. The TRIDENT-CTR model (Fig.

118

TABLE XVI. ATOM DENSITIES FOR THE ISOTOPES USED IN THE MATERIALS (atom/b cm)

MATERIAL

Isotope

H-1 He-4 B-10 B-11 C 0-16 Al-27 Si Ca Cr Mn-55 Fe Ni Cu Nb-93

I

1 1 5 1

SS316

.67E-2

.75E-3

.44E-2

.15E-2

TFCOIL

3.79E-3 6.67E-3 2.98E-5 1.20E-4 1.90E-3 2.17E-3 1.81E-4 5.59E-4 2.42E-4 5.97E-3 6.27E-4 1.95E-2 4.12E-3 2.11E-2 2.44E-4

SS304

1.77E-2 1.67E-3 6.06E-2 7.40E-3

CNAT

8.03E-2

IHDLC

5.03E-2

2.51E-2

4.18E-3 4.38E-4 1.36E-2 2.88E-3

IHDLB

1.34E-2

6.70E-3

1.34E-2 1.40E-3 4.35E-2 9.20E-3

IHDLA

1.68E-3

8.38E-4

1.63E-2 1.71E-3 5.30E-2 1.12E-2

SS312

3.34E-3 3.50E-4 1.09E-2 2.30E-3

Mo 1.51E-3 5.41E-4 3.78E-4 1.21E-3 1.47E-3 3.02E-4

V ! Void SS : S t a i n l e s s Steel

C : Carbon IDSL : S t a in l e s s S tee l + HO

NJ O

178 666 cm

84 Figure 12. Two-dimensional model for the FED

The numbers within each zone indicate the zone number.

3 0 0 . *

2IK

100.

0.

+

wm^'m

1\

3$3iBJj j::Mllir- "SF^i:^^7^^ ^••M-Vv

?uu.

J V ' J T "

r -.i f M * t

Figure 13. The TRinFNT-CTR band and trianfilr structure for the FED

13) utilizes 2062 triangles, divided over 27 banas. The response func

tions for calculating the heating in the TF coil, were prepared by the

72 TRANSXX code. Those response functions will be the sources for the

adjoint calculation. It was noted earlier that negative sources can

introduce instabilities in the sweeping algorithm for the adjoint

TRIDENT-CTR calculation. The negative kerma factors are therefore set

to zero. This will have a minor effect on the total heating calculated

in the TF-coil (less than 1%).

EQ-2 and EQ-B quadrature sets are used for groups 1 and 2 respec

tively, EQ-3 is used for groups 3, 4, and 5, while an EQ-4 quadrature

set is utilized for the remaining groups. The convergence precision is

-3

specified to be 10 . The gamma-ray groups contribute most to the heat

ing in the TF coil (93%). The total heating in the TF coil is 823 *

10"6 MW.

The heating calculated by the adjoint TRIDENT-CrR calculation is

found to be 3% smaller than the heating resulting from the forward run.

The forward calculation required about one hour of c.p.u. time on a

CDC-7600 computer, while the adjoint run took about four hours. Groups

11 to 23 required significantly more inner iterations in the adjoint

mode than the other groups. No explanation of this behavior could be

found. Experience with other neutronics codes indicates that the ad

joint mode for this type of calculation requires usually no more than

30% extra calculation time.

122

6.2 Two-Dimensional Sensitivity and Uncertainty Analysis for the Heating in the TF Coil due to SEP and Cross-Section Uncertainties

A secondary energy distribution and vector cross-secticn uncertain

ty analysis was performed with SENSIT-2D using the forward and adjoint

angular flux files created by TRDSEN. A separate SENSIT-2D run is re

quired for each zone. Because separate runs are necessary for a cross-

section and a SED analysis, a total of 22 SENSIT-2D cases were analyzed.

A total of 15 minutes c.p.u. time was used by SENSIT-2D. The bulk of

this time is consumed during input/output manipulations.

The median energies and fractional uncertainties for the SED unter-

45 tainty calculations were taken from Table II. A special cross-section

33 table was created - using TRANSX - for the SED analysis. COVFILS data

were used for generating the covariance matrices utilized in the cross-

section uncertainty evaluation. Only 0-16, C, Fe, Ni, Cr, and Cu were

considered for the SED uncertainties, while H, Fe, Cr, Ni, B-10, C, and

Cu were included for the cross-section uncertainties. With the excep

tion of oxygen, no important materials were left out. It was found in

an earlier study that the cross-section uncertainties for oxygen caused

an 8% uncertainty in the heating. The current version of SENSIT-2D

does not include the option to extract the covariance data for oxygen

from COVFILS.

The gamma-ray cross sections are generally better known than the

neutron cross sections. Therefore, only th- uncertainties resulting

123

from uncertainties in neutron cross sections are calculated. Throughout

this analysis a third order of anisotropic scattering is used.

The predicted uncertainties in the heating of the TF-coil are sum

marized in Table XVII. It was assumed that the uncertainties for a par

ticular element in the various SS316 zones (1, 3, 7, 11, and 12 in Fig.

12) are fully correlated, while all other uncertainties wt ssumed to

be noncorrelated. This implies that the uncertainties for a particular

element can be added over all SS313 zones, while all the other uncer

tainties are added quadratically. The approach of either assuming full

correlation or assuming noncorrelation is rather simplistic. Trans

lating the physics of this particular problem into a more sophisticated

correlation scheme would be a major study by itself. The uncertainties

resulting from the uncertainties in the cross sections for Cr, Fe, and

Ni in the SS316 zones are reproduced in Table XVIII.

From Table XVII it can be concluded that the cross-section uncer

tainties (predicted to be 113%) tend to be more important than the SED

uncertainties (20%). Even when the overal uncertainty seems to be rela

tively large (115%), the blanket designer is able to set an upper bound

for the heating in the TF coil. The largest uncertainties are due to

uncertainties in the Cr cross sections. A more detailed look at the

computer listings generated by this analysis reveals that the largest

uncertainties are produced by uncertaintier in the total Cr and the

elastic Cr scattering cross sections. The heating is less sensitive to

Cr than to Fe. This indicates that the calculated uncertainty is large

ly due to the fact that Cr has very large covariances. A re-evaluation

124

TABLE XVII: PREDICTED UNCERTAINTIES (STANDARD DEVIATION) DUE TO ESTIMATED SED AND CROSS-SECTION UNCERTAINTIES FOR THE HEATING IN THE TF COIL (part 1)

Cross Section

Material Zone

Cr SS316 TFCOIL SS304 SS312 ISDLC ISDLB ISDLA

Fe SS316 TFCOIL SS304 SS312 ISDLC ISDLB ISDLA

Ni SS316 TFCOIL SS304 SS312 ISDLC ISDLB ISDLA

SED Uncertainties in %

[1] L AJMat

3.8 0.2 0.1 0.0 0.2 0.8 3.0

14.8 0.1 0.0 0.2 0.5

i 2- 7

I 10.8

i ]- 5

: 0.7 0.0

, 0.0 I 0.0 ! 0.4

i 1-2

region ™M.t 1

4.9

18.4

i

j \ 1 4.3 | ! i i

XS Uncertaint

[1] 1 'Mat l

60.0 34.5 4.5 1.1 2.2 33.3 58.5

18.9 10.4 2.2 0.7 4.4 23.6 34.5

18.6 11.8 0.9 0.4 1.3 13.4

. 18.0

region

ies in %

AR" R

L "-Mat

:

'l 96.7

|

1

[

47.3 •

31 .4

, :

Quadratic Sums

1J5

TABLE XVII: PREDICTED UNCERTAINTIES (STANDARD DEVIATION) DUE TO ESTIMATED SED AND CROSS-SECTION UNCERTAINTIES FOR THE HEATING IN THE TF-COIL (part 2)

Cross Section

Material Zone

B

Cu

TFCOIL ISDLC ISDLB ISDLA

TFCOIL ISDLC ISDLB ISDLA

SED U n c e r t a i n t i e s i n % *

ARI

XS U n c e r t a i n t i e s i n % *

Total

TFCOIL C-region

rotal uncertainty due to cross-section uncertainties and SEDs = 114.6%

Quadratic Sums

126

TABLE XVIII: PREDICTED SED AND CFOSS-SECTION UNCERTAINTIES IN THE TF COIL DUE TO UNCERTAINTIES IN THE SS3iS ZONES

Cross Section

Material Zone

Cr

Fe

1 3 7 11 12

1 3

11 12


Ml •I Mat,region

0.1 0.7 0.0 3.0 0.0

0.5 3.1 0.1 11.0 0.1

(AR R Mat

3.8

XS Unceitainties in %

H] I N O * *• 'Mat , r e g i o n 1 J M a t

14.8

12 .0 4 5 . 5

0 . 8 4 . : 0.4

2 . 3 11 .3 0 . 7 4 . 3 0 . 3

60 .0

16.9

Ni 1 3 7

11 12

0.0 0.3 0.0 1.2 0.0

1.5 i :

3.6 12.8 0.3 1.8 1.1

18.6

127

of the covariance data for Cr is highly recommended. If new covariance

data would not reduce the predicted uncertainty, new experiments for

measuring the Cr cross sections are suggested. The conclusions drawn

45 here are consistent with an earlier study of a similar design.

The SED uncertainties, although less relevant to overall predicted

uncertainty, tend to become more important in the outboard shield

(region 11 in Table XVIII). An explanation for this behavior is related

with the fact that the heating in the TF coil will be very sensitive to

backscattering in this region. An SAD (secondary angular distribution)

sensitivity and uncertainty analysis might lead to very interesting

results.

The x's for the region near to the plasma in the outboard shield

are calculated for each group based on angular fluxes and based on flux

moments (Table XIX). Both methods lead generally to the same x's. The

difference for the upper neutron groups might indicate that a third-

order spherical harmonics expansion of the angular flux tends to become

inadequate, due to the peaked shape of the angular flux close to the

source region. In this particular study no serious error in the calcu

lation of the uncertainties would have been introduced if the loss term

of the sensitivity profile would have been calculated from flux moments.

For a situation where the angular flux would have a pronounced peaked

behavior, it would be highly desirable to evaluate the x's based on

angular fluxes.

128

It is obvious from Table XIX that some fluxes in the lower gamma-

ray groups (groups 41 and 42) are negative. Since only neutron sensi

tivity profiles are utilized to calculate uncertainties, this will not

affect the results.

6.3 Comparison of the Two-Dimensional Model with a One-Dimensional Representation

The results obtained from the two-dimensional sensitivity and un

certainty analysis will be compared with those of a one-dimensional

analysis in selected regions (Table XX). The uncertainties in the heat

ing in the TF coil due to the uncertainties in the Cr, Ni, and Fe cross-

sections and secondary energy distributions will be calculated with

ONEDANT and SENSIT in zone 1 and zone 3 (Fig. 12). The one-dimensional

model for ONEDANT is straightforward. The total heating calculated in

the TF-coil is 1043 x 10 MW (compared to 823 * 10~ MW for the two-

dimensional model). In this comparison the uncertainties calculated by

SENSIT will be normalized to the response calculated in the two-dimen

sional model.

It can be concluded from Table XII that the calculated uncertain

ties agree reasonably well for zone 3. There are substantial differ

ences for the results in zone 1. The reason for those differences is

probably related with the fact that the one-dimensional model is not

adequate for calculating the overall heating in the TF coil (especially

129

TABLE XIX: COMPARISON BETWEEN THE X 's CALCULATED FROM ANGULAR FLUXES CUPPER PART) AND THE X ' s RESULTING FROM FLUX MOMENTS (LOWER PART) FOR REGION 11 (SS316)a

* • • TEST l?»lHTOUT FDD Tut CHI'S • • * • • • K « !•••

0. .21326e-06 .22361C-05 .&6886c-oe .494336-09 .24385c-10 .130961-10

.13195c-05

.191666-06

.14093c-05

.68169E-08

.26407c-09

.71906c-!0

.22933c-11

.17957c-06

.32I75C-06

.18633c-U6

.28276c-Ufa

. 12122C-0*

.12091E-U9

.14850E-12

.67647c-07

.40693e-06

.37795E-06

.17686c-08

.48579C-10

.t>6533c-10 •£2953e-16

.86776E-07

.48797c-06

. 10788c-06

.12639C-08

.16319C-10

.49747C-10 ' .5 l897e-23

.79271C-07

.15589c-05 • 176C'?e-07 .54i43c-09 .47290E-U .26&69E-1 0

-. 28794c-48

• • • • • • C H I ' S OEHE»*»TCIl HW1M FLU> M D H C M T S *****

• • • TCST rciNTouT roii **** B I***

0. .21316C-06 •22366c-05 .868856-08 .494246-09 .244386-10 .13135F-10

.10555C-05

.191646-06

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.681676-06 •26401C-09 .71918C-10 .29 048c-11

THE C H I ' S •

.17003C-U6

.321fe7c-06

. 18630E-U6

.28273E-08

.12118E-09

.120686-09

.186316-12

• • • 59837c-

H >

•07 .40669c-06 • 37797c-•17886c-

-06

-oe -48553C-10 .66705c--10 .£624 06-16

.61854c-

.487986-

.10788E-

.12638c-

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-.45116c-

-07 -05 -07 -09 11

•10 -46

The X ' s are ordered by group (h igh neutron energy to low neutron energy; high gamma-ray energy to low gamma-ray energy)

130

TABLE XX: PREDICTED UNCERTAINTIES (STANDARD DEVIATION) DUE TO ESTIMATED SED AND CROSS-SECTION UNCERTAINTIES IN ZONES 1 AND 3 FOR THE HEATING IN THE TF-COIL

Cross Section

Material Zone

Cr 1

2

Fe 1

3

Ni 1

3


f-1 f-1* L RJMat,zone L RJnat

1-D 2-D

0.1 0.1

0.6 0.7

0.8 0.5

2.6 3.1

0.0 0.0

0.2 0.3

XS Uncertainties in %

H] M * L JMat,zone L KJMat

1-D 2-D

29.3 12.0

44.8 42.5

4.5 2.3

9.6 11.3

8.3 3.6

13.1 12.8

131

the source region is poorly simulated in the one-dimensional representa

tion). A more relevant sensitivity analysis would be to consider the

heating calculated at the hottest spot in the TF coil. The hottest spot

is in the center plane of the toroid. We would expect that the one-

dimensional model would be an adequate representation in this case.

132

7. CONCLUSIONS AND RECOMMENDATIONS

Expressions for a two-dimensional SED (secondary energy distribu

tion) and cross-section sensitivity and uncertainty analysis were de

veloped. This Llicory was implemented by developing a two-dimensional

sensitivity and uncertainty analysis code SENSIT-2D. SENSIT-2D has a

design capability and has the option to calculate sensitivities and

uncertainties with respect to the response function itself. A rigorous

comparison between a one-dimensional and a two-dimensional analysis for

a problem which is one-dimensional from the neutronics point of view,

indicates that SENSIT-2D performs as intended. Algorithms for calculat

ing the angular source distribution sensitivity and secondary angular

distribution sensitivity and uncertainty are explained.

The analysis of the FED (fusion engineering device) inboard shield

indicates that, although the calculated uncertainties in the 2-D model

are of the same order of magnitude as those resulting from the 1-D model,

there might be severe differences. This does not necessarily imply that

the overall conclusions from a 1-D study would not be valuable. The

more complex the geometry, the more compulsory a 2-D analysis becomes.

133

The most serious source of discrepancies between a 1-D and a 2-D

study are related to the difficulty of describing a complex geometry

adequately in a one-dimensional model. However, several neutronics

related aspects might introduce differences. The use of different quad

rature sets - especially when streaming might be involved - could lead

to different results. When the angular fluxes have a pronounced peaked

behavior, the angular flux option for calculating the loss term of the

sensitivity profile will provide a better answer than the flux moment

option. The different sweeping algorithms and code characteristics used

by the 1-D and 2-D transport codes might be another cause of discrep

ancies in the results. Needless to say, a meaningful transport calcu

lation is compulsory in order to obtain reliable results from a sensi

tivity and uncertainty analysis.

The results from the FED study suggest that the SED uncertainties

tend to be smaller than those generated by cross-section uncertainties.

45

It has been pointed out that, because all secondary particle produc

tion processes for a particular element are presently treated as one

single process, the simplicity of the hot-cold concept for SED sensi

tivity might mask several causes of a larger uncertainty than calculated

by SENSIT or SENSIT-2D. A more elaborate algorithm for a SED analysis,

as an alternative to the hot-cold concept, a separate treatment for the

various particle production processes involved, or a combination of

both, would eliminate this deficiency. Even with the hot-cold model,

which might underestimate SED uncertainties, the SEDs might become the

dominant cause of the calculated uncertainty in the case that the

134

response function is a threshold reaction or in the case that backscat-

teriug becomes important. In this latter situation, an SAD (secondary

angular distribution) analysis might also contribute significantly to

the overall uncertainty estimate. At present, the required cross-

section data are not arranged in the proper format to do this type of

study.

Sensitivity and uncertainty analysis estimates the uncertainty to a

calculated resfonse. It would be more meaningful to be able to imple

ment those uncertainties with a confidence level. In order to do this,

we have to know how reliable the covariance data are, what the effects

of errors resulting from the transport calculations will be, and what

the limits of first-order perturbation theory are. It was assumed in

this study that the uncertainties, resulting from uncertainties in

different regions, were either fully correlated or not correlated at

all, depending on whether these regions have the same or a different

material constituency. The evaluation of reliable correlation coeffi

cients would be a major effort by itself.

The validity of an uncertainty analysis is often limited more due

to the lack of the proper cross-section covariance data, than due to the

lack of representative mathematical formalisms. Covariance data for

33 several materials are still missing, or just guesstimates (e.g., Cu)

The fractional uncertainties required for an SED analysis are evaluated

for just a few materials and are not available for the various indi

vidual particle production processes.

135

The current version of SENSIT-2D cannot yet access all the covari-

33

ance data available in COVFILS, but will be able to do so in the

future. Even when SENSIT-2D does i;ot require a lot of computing time,

the extra amount of c.p.u. time required by the adjoint TRIDENT-CTR run

makes a two-dimensional sensitivity and uncertainty analysis demanding

when it comes to computer resources. The development and implementation

of acceleration methods for TRIDENT-CTR are therefore desirable. A

sensitivity analysis involves a tremendous amount of data management. A

mechanization of the various steps required, by the development of an

interactive systems code, would provide a more elegant procedure for

sensitivity and uncertainty analysis.

The algorithms to perform a higher order sensitivity analysis have

been developed, but are still too complicated to be built into a com

puter program for general applicability. The increasing number of

transport equations to be solved prohibits the incorporation of present

higher order sensitivity schemes in a two-dimensional code. An effort

to develop simple algorithms for higher order sensitivity can certainly

be justified, however.

It becomes obvious that several flaws can be found in the state of

the art of sensitivity and uncertainty analysis. Removing any one of

them would require a major commitment.

136

ACKNOWLEDGEMENTS

The author would like to thank the chairman of his committee, Dr.

J. R. Thomas, for ''he guidance received during this pursuit for a higher

education, the other members of his committee for their interest in this

work and their contribution to its completion, and Dr. D. J. Dudziak

from Los Alamos National Laboratory for his daily supervision and invalu

able advice.

The author is obliged to Dr. T. J. Seed, Dr. S. A. W. Gerstl, Dr.

W. T. Urban, Dr. E. W. Larson, Mr. D. R. Marr, Mr. F. W. Brinkley, Jr.,

and Dr. D. R. McCoy for their assistance with Los Alamos codes, and the

various discussions that were essential for the completion of this work.

ThanJ's go to Dr. C. S. Caldwell and Dr. W. G. Pettus from Babcock and

Wilcox for pointing out the need for sensitivity analysis and their con

tinued interest.

The encouragement received from his parents, Professor W. Geysen,

Professor and Mrs. W. Glasser, Mr. and Mrs. C. Kukielka, Mr. R. Florian,

Mr. and Mrs. M. Goetschalckx, and Mr. and Mrs. M. Denton, is highly

appreciated.

The author acknowledges Professor D. P. Roselle, the Dean of the

graduate School at Virginia Polytechnic Institute, for the Pratt Presi

dential fellowship, Dr. D. I. Walker from Associated Western Universi

ties, for the A. W. U. fellowship, which made his stay at the Los Alamos

National Laboratory possible, and the Los Alamos National Laboratory for

the use of their facilities. Special funding of the AWU fellowship by

the DOE Office of Fusion Energy is gratefully acknowledged. Thanks go

to Mrs. Mary Plehn, who prepared the manuscript.

137

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144

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145

APPENDIX A

SENSIT-2D SOURCE CODE LISTING

In this appendix a source listing of the SENSIT-2D code is repro

duced. The source listing is documented by many comments.

A source listing of the SENSIT-2D code can also be obtained from

the NMFECC by typing the command

FILEM$READ 50^3 .SENS2D SSSSSEND

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179

5.1 Sample Problem ffl

The first sample problem is a mock-up of a cylindrical geometry

(Fig. 8). There are four zones present: a source zone (vacuum), a per

turbed zone (iron), a zone made up of 40% iron and 40% water, and a

detector zone (copper). The reaction rate of interest is the heat gen

erated in the copper region. The source was assumed isotropic and had a

neutron density of one neutron per cubic centimeter (1 neutron/cm3).

The source neutrons are emitted at 14.1 MeV (group 2). The left bound

ary is reflecting, and on the right there is a vacuum boundary condition

Thirty neutron groups were used with a third order of anisotropic scat-

72 tering. The cross sections were generated using the TRANSX code. The

energy group boundaries are reproduced in Table V.

In the two-dimensional model (TRIDENT-CTR) two bands--each 0.5-cm

wide--are present. In order to be consistent with the one-dimensional

analysis the upper and the lower boundaries were made reflective (Fig.

9). Each band is divided into 35 triangles (5 triangles for the source

zone, 10 triangles for each of the other three zones). The automatic

mesh generator in TRIDENT-CTR was used. The convergence precision was

-3 -3

set to 10 . A convergence precision of 10 means here that the aver

age scalar flux fu* any triangle changes by less than 0.1% between two

consecutive iterations. A similar criterion is used in ONEDANT. The

calculation is performed with the built-in EQ -8 (equal weight) quad

rature set. The mixture densities are given in Table VI. For the ad

joint calculation the source is in zone IV and consists of the copper

84

^ ^

10cm 10cm 10cm

Figure 8. Cylindrical geometry representation for sample problem i>l

85

TABLE V: 30-GROUP ENERGY STRUCTURE

N e u t r o n s E - U p p e r C r o u p E - L o w e r (Ht-VJ (*1eV)

1.700*01

1.500*01

1.350*01

1.200*01

1.000*01

7.790*00

6.070*00

3.660*00

2.865*00

2.232*00

1.738*00

1.353*00

S.230-01

5.000-01

3.030-01

1.840-01

6.760-02

2.480-02

9.120-03

3.350-03

1.235-03

4.540-04

1.670-04

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

1.500*01

1.350*01

1.200*01

1.000*01

7.790*00

6.070*00

3.680*00

2.865*00

2.232*00

1.738*00

1.353*00

8.230-01

5.000-01

3.030-01

1.840-01

6.760-02

2.430-02

9.120-03

3.350-03

1.235-03

4.540-04

1.670-04

6.140-05

E - U p p p r G r o u p t - L o w e r (W.-V J ( t i l l I

6 . 1 4 0 - 0 5 24 2 26J -05

2 . 2 6 0 - 0 5 25 & .320 -06

8 . 3 2 0 - 0 6 26 3 . 0 6 0 - 0 0

3 0 6 0 - 0 6 27 1 . 1 3 0 - 0 6

1 . 1 3 0 - 0 6 28 4 . 1 4 0 - 0 7

4 . 1 4 0 - 0 7 29 1 .520 -07

1 .520 -07 30 1 . 3 9 0 - 1 0

86

10 cm

\zOjfE il\ 1

Wl'RCE \l

/vABVLiy \

' 5 t ryang lesS

REFLECTINC BOUNDARY

10 cm 10 cm

ZONE #11

PERTURBED

Fe


ZONE #111

40% Fe


10 cm

ZONE 1>I\'

DETECTOR

Cu


T

E u

m >-< c O g

E 5

0.5

<

1

REFLECTING BOUNDARY

30 neutron groups

neutron source: 1 neutron / cm in group2 (14.1 MeV)

P-3. EQ -8 : third-order of anisotropic scattering n

8th-order equal weight quadrature set

response function: copper kerma factor in zone /-'IV

_3 convergence precision : 10

Figure 9. Two-dimensional (TRIDENT-CTR) representation for sample problem #1

87

file:///zOjfE

TABLE VI. ATOM DENSITIES OF MATERIALS

ZONE in

ZONE 1)11

ZONE / / I I I b

ZONE 111V

Vacuum

Fe

Fe

H

0

Cu

Atoms/m

- -

8.490 + 28a

3.396 + 28

4.020 + 28

1.900 + 28

8.490 + 28

a 8.490 + 28 = 8.49 x 1C28

b 40 vol % Fe and 40 vol % water.

88

kerma factors. The response is calculated in that case in zone I. It

was found that the adjoint calculation required more iterations and time

in order to reach convergence. Originally the forward calculation was

done using 20 triangles per band. The adjoint problem, however, did not

converge. In the evaluation process of the kerma factors, the kermas

for some groups are made negative in order to satisfy energy balance.

Making those negative sources zero in the TRIDENT-CTR run did not lead

to any improvement. Subsequently, 35 triangles per band were used.

When the negative sources were set to zero convergence was reached.

Ignoring the negative kerma factors leads to a 20% increase in the total

heating. The forward calculation required about 11 minutes cpu time

(central processor unit time on a CDC-7600), while the adjoint cal

culation required about 13.5 minutes. Generating the angular fluxes

using the TRDSEN code required about 20 seconds of cpu time for each

case.

TRDSEN does on extra iteration in order to generate the angular

fluxes. The convergence criterion in TRIDENT-CTR is based on the scalar

fluxes, and therefore one extra iteration in TRDSEN should be adequate.

However, restarting TRIDENT-CTR with the flux moments as starting

guesses, revealed that for some groups two extra iterations were neces-

-3 sary to reach a convergence precision of 10 . No explanation for this

could be found.

The one-dimensional model (ONEDANT) contains 35 intervals (5 for

the source zone, 10 intervals for each of the remaining zones). The

one-dimensional description for the forward problem is summarized in

89

10 cm

ZONE ill SOURCE vacuum

5 intervals

10 cm

ZONE m PERTl'REED

zFe 10 intervals

10 cm

ZONE /'III

A0?:Fe+60::H 0 10 intervals

10 cm

ZONE /'IV DETECTOR

Cu 10 intervals

30 neutron groups

3

neutron source: 1 neutron / cm in group 2 (14.1MeV0

P-3, S-8 : third-order of anisotropic scattering

8th-order Gaussian quadrature set detector response: copper kerma factor in zone /'IV

- U convergence precision: 10

Figure 10. One-dimensional (ONEDAKT) .^presentation for sample problem ill

90

Fig. 10. Again it was found that the use of negative sources in the

adjoint calculation caused difficulties with respect to the convergence.

In that case, groups 18 and 19 triggered the message "TRANSPORT FLUXES

BAD"; groups 4, 5, 6, 7, and 19 did not converge (max. number of inner

iterations 300/group). However, the overall heating in the copper

region was within 0.1% of the heating calculated by the forward run. A

coupled neturon/gamma-ray calculation (30 neutrons groups and 12 gamma-

ray groups) in the adjoint mode led to some improvement. In that case,

only group 2 did not converge. The required convergence precision in

-A the OKEDANT runs was set to 10 . The built-in S-8 Gaussian quadrature

sets were used. In order to be consistent with the TRIDENT-CTR calcula

tions, the negative sources in the adjoint case were set to zero, even

though this did not seem to be necessary. Each run required about six

seconds of cpu time.

A standard cross-section sensitivity analysis (the cross sections

in zone II are perturbed) was performed using the SENSIT code and the

SEKSIT-2D code. A comparison between the SENSIT and the SENSIT-2D

results revealed that SENSIT does not rearrange the angular fluxes

correctly (in cylindrical geometry). To correct this error, a shuffling

routine which takes case of this deficiency was then built into SENSIT.

The SENSIT results are in good agreement with those obtained from

SENSIT-2D. The flux moments versus the angular flux option was tested

out for Lhe calculation of the loss term. Again there is good agreement.

Finally, an uncertainty analysis was performed for the heating in the

copper zone. The SENSIT-2D analysis matches the SENSIT analysis.

91

5.1.1 TRIDENT-CTR and ONEDANT results

A comparison of the heating in the copper region (zone IV) between

TRIDENT-CTR (and SENSIT-2D) and ONEDANT (and SENSIT) is summarized in

Table VII a. The adjoint calculations yield a 20% higher heating rate

due to the fact that the negative kerma factors were set equal to zero.

The one-dimensional and the two-dimensional analysis are in agreement.

The computing times for those various runs are given in Table VII b.

Each ONEDANT run requires about 8 seconds of total computing time

(LTSS time), whereas it takes about 12 minutes to do the TRIDENT-CTR

runs. The TRIDENT-CTR runs were done with a convergence precision of

-3 -4

10 , whereas for the ONEDANT runs a convergence precision of 10 was

specified. In order to obtain the same convergence precision in

TRIDENT-CTR about eight additional minutes of cpu time are required. It

was found that a forward coupled neutrou/gamma-ray calculation (30 neu

tron groups and 12 gamma-ray groups) required only 8 minutes of comput--3

ing time with TRIDENT-CTR (convergence precision 10 ). An explanation

for this paradoxial behavior is related to the fact that a /o_ has a

different (smaller) value in a coupled neutron/gamma-ray calculation.

The flux moments generated by TRIDENT-CTR and ONEDANT were com

pared. In the ONEDANT geometry the angular fluxes are assumed to be

symmetrical with respect to the z-axis, so that the odd flux moments

(<t> , (J)., 4>o> and <J) ) vanish. Since TRIDENT-CTR performs a real two-

dimensional calculation the odd moments will not be zero in that case.

In our sample problems there is still symmetry with respect to the

92

TABLE Vila. COMPARISON OF THE HEATING IN THE COPPER REGION CALCULATED BY ONEDANT AND TRIDENT-CTR

FORWARD ADJOINT

ONEDANT3 2.37382 + 7 2.40541 + 7

ONEDANT 2.01189 + 7 2.01882 + 7

TRICENT-CTR 2.01175 + 7 2.39263 + 7

SENSIT 2.01011 + 7 2.40541 + 7

SENSIT-2D 2.01098 + 7

negative KERMA factors set to zero

TABLE VIlb. COMPUTING TIMES ON A CDC-7600 MACHINE

CPU-TIME3 I/O TIMEb LTSS TIMEC

ONEDANT FORWARD 5.80 sec. 1.87 sec. 7.65 sec.

ONEDANT ADJOINT 6.09 sec. 1.82 sec. 7.97 sec.

TRIDENT-CTR FORWARD 13.5 minutes

TRIDENT-CTR ADJOINT 11.1 minutes

SENSIT 4.92 sec. 0.55 sec. 6.08 sec.

SENSIT-2D 8.50 sec. 9.02 sec. 17.84 sec.

a . . . central processor unit time

b . input/output time

Livermore time sharing system t 'me (total computing time)

93

z-axis. For that reason, the odd moments in TRIDENT-CTR will have oppo

site signs in band one and band two. For some zones and some groups

this was not completely the case. There was about 30% difference in the

absolute values of some flux moments in band one and bai,d two, which

indicates that the problem was not in a sense truly converged. The con

vergence criteria in ONEDANT and TRIDENT-CTR test only for the scalar

fluxes between two consecutive iterations. Even when the convergence

criteria are satisfied in both codes, a true convergence of the angular

flux is not guaranteed. The even moments in band one are exactly the

same as those for band two. Because the contribution of the odd moments

is small compared to the contribution of the even moments (about one

thousandth), the problem can be considered fully converged.

The scalar flux moments calculated by TRIDENT-CTR and ONEDANT are

in very good agreement. The higher-order moments are different. Since

TRIDENT-CTR and ONEDANT do not use the same coordinate system, they do

not calculate the same physical quantity for the higher-order flux

moments. As long as TRIDENT-CTR is consistent with SENSIT-2D, and

ONEDANT consistent with SENSIT, the results from the one-dimensional

sensitivity analysis should match those obtained from a two-dimensional


5.1.2 SENSIT and SENSIT-2D results for a standard cross-section sensitivity analysis

A standard cross-section sensitivity analysis was performed using

94

SENSIT and SENSIT-2D. The sensitivity of the heating in zone IV to the

cross sections in zone II was studied. SENSIT-2D requires about three

times more computing time than SENSIT in this case (Table VII b). The

main part of the calculation involves the evaluation of the ty's (gain

term). A complete sensitivity and uncertainty analysis may involve

several SENSIT (or SENSIT-2D) runs. Thus an option which allows one to

save the IJJ'S has been built into SENSIT-2D. It is obvious that the com

puting time required in SENSIT-2D is negligible compared to the comput

ing time required for the forward and adjoint TRIDENT-CTR calculations.

The partial and the net sensitivity profiles calculated by SENSIT

and SENSIT-2D are reproduced in TABLES VIII a and VIII b. It can be

concluded that the SENSIT-2D results are in good agreement with those

obtained by SENSIT. Note that the absorption cross section is negative

for groups 2 and 3. A negative absorption cross section does not nec

essarily indicate that errors were made during the cross section pro

cessing. There are various ways to define an absorption cross section,

and a controversy about a commonly agreed en definition is currently in

progress. What is called an absorption cross section in a transport

code is not truly an absorption cross section but the difference between

the transport cross section and the outscattering (a° = a - 1 a ° ° ). 3 tr g'

Note that groups 2 and 3 are the main contributors to the integral

sensitivity.

It was mentioned earlier that the x's c a n be calculated based on

flux moments or based on angular fluxes according to

95

( P a r t 1)

DEFINITIONS OF SEf:J IT SENSITIVITY PROFILE HOrE NCLATURE

BflJS - SE!;GITIVITY PROFILE fil' DELTO-U FOR Tlin AKO.TPl lu;< CKOSS-SECTItjH (TIliLN FRjl 1 POSITION MP. IN INPUT CRUSS-SLCTIUN ThULESl. PUKE LUSS TEHtl

HJ-F1SS - SENSITIVITY PkOTILE PER EELTn-U PUD THE EHUSS SECTIOH IN POSITION !H,i»l IN INPJT XS-TABLES. Ul lCH IS USUHLL. HU-TlrE5 THE FISulDH CIJUSS 5CCTIUH. Pul:t LUSS TERM

SX5 • PARTIAL SCNSITIVITY PROFILE PtR MXTO-U FOR TilE SCMTTEI.'IHH CRiOS-GEETIGN (CO:PUTEH FOR EACH ENERGY GROUP AS A Di.'uUHAU =UH Fk'Uri iNI'Uf XS-1AULCS). LOSS 1EKI1 ONLY

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S-GflIN - PPPTIflL SENSITIVITY PrOFILE PER DELTn-U FCI' THE Gi'.. > Ft SCATTERlt'r, ERO.S-SCCTIOH. G i lN TERM Ful' SENSITIVITY GHIMS MJL 1U ^ m r i E r i l l G OUT UP Ui i l l f i ENLRGY GROUP G IHIU ALL LOU.R GATIfl ENERGY COrPUTEB FROrl FOkuWill JUFFLI.CNL'E FUKrU.fi HOM.

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NG-GAIN - PARTIAL SENSITIVITY PI.'OFILE PEI! I iLLlA-U 101' MIL Gi : l t» PIODUETION CrUS'VS'CTIUM FIT CCuTROH EllLI.'GY CRllUP C. l"UI.'t GftlH 1LI.71 ^0K S E M - I T I V P Y GT.MS DUE Tl) ll.'/lJIJI L"R FROM NEUTTOM GHOUP G INTU ALL GHITW GKUUPi.

6EH • NET SENSITIVITY PROFILE PER DELTA-U FUR TIIC SCnnLRIHG tRUSS-SCCTIOH ISEN-SAS-1+ .GAJN)

SCUT • NET SENSITIVITY PROFILE PER MX7A-U FIW TIIC TOTAL CIHKS-SECTJUH (SLHT-F^,;HGAJJ(J

SEW - SENSITIVITY PROFILE PER D C L T H - U FOR TIC DETCC1UI: RESPONSE FUIICTIUH l.'(G)

SENO • SENSITIVITY PROFILE PER D E L T H - U FOR TKL SOURCE DIS1R1UUTI0H FUIICTIOII QCG)

96

http://FUKrU.fi

TABLE Villa: PARTIAL AND NET SENSITIVITY PROFILES FOR THE ONE-DIMENSIONAL ANALYSIS

(Part 2)

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TElTO- ' j j . 2 i r - y i i . E " . : - a i 1.1CE-01 i .e?r-ui 2 . 3 3 E - 9 I 2 4 ^ F - r n 5 . 0 i . - 0 l 2 . : - : : - o i 2 . o c : - i u 2 . E . - - C 1 2 . 5 ^ - 0 1 4 . 9 7 E - 0 1 4 .92E-0J 5 . 0 : 5 - 0 1 4 . 9 : E - t l l . t ; : f j J . P." L -tjD l .OD.-OO l . i L E H I O I . 5 : E - 3 1 1 .EL ;»30 I .C ; :»3J 1 L1L-B0 ?. : : -E-BI 9 .9?E-01 1.33E-03 9.!>6E-Q1 1.C2E«L.B i.e-.E-ca 1 . 11E-K1B

- * . - ^ » - r » ;

j - I L L S P . i . : SECTlUt l i . <H

K M 3 X * P U ft^S

c. 2 . 4 4 - C - 0 0 I.B3CE-U2

- i . 9 : ' : E - I : - I . 3 : ; L - ' . V - 5 . 7 > 3 E - O :

- i . r E - . t - o . . -9 .G23E-04 - b . y 3 7 E - B 4 -3 .C97E-84 -2.C4QL-L14 -S.9LCE-B4 - I . 53 -1C-23 - I . B S 7 E - B 3 - 3 . 8 . J E - U 4 -7 .023E-O4 - 4 . b S 3 E - « 4

-S .S32F-03 - 7 . : J ? L - L " J

-1 .C3--E-05 - - . \ G ~ . I L - D 4

- 3 . 6 L 7 E - L 3 - 4 . 7 I 2 E - 0 3 -5 .913E-U3 - £ . [ IGE-LL. - 7 . r C r . L - L S -G.673E-U3 -S .720E-C3 -•l.tZ-l.-K - ^ . B > u E - u 4

2.4SSE-B1

' I - .1 O^-1 ' L T.,- ,1. - n ; fiio

R E L MU-FI3 e. 0 . B . 0 . 0 .

G .

0 . 0 . 0 . 0 . 0 .

0 . 0 .

B .

e. 8 .

e.

e. 0 . 0 .

L .

0 . 0 .

c. 0 . V.

0 . 0 .

0 .

e.

0 .

• '_' n i - t .

0 S 5

•xa-J- MIT PRC-l^CS «=<*>• i LN

B. - 1 .3S f . r » f l l - I .OJ--C<D'J - 1 . 5 9 - U - 0 1 - l . i ' J . L - 0 1 _ ! F11V-OI - 7 . 0 - j . L - 0 2 - ^ . 2 C : . i - C 2 - 4 . r y . : - t . 2 - 3 . 1E.-C-02 - i . L G - : : - t 2 - 2 . 7 s e t - 0 2 - 1 . 2 S 3 E - 0 2 - C . 9 : 0 L - 0 3 -4 . ; . - r .E-G3 - 2 . 3 . J E - C 3 -S.2L'CL-U4 - 5 . 0 i r E - D 4 -2 .C9DC-01 - 3 . B . ' ^ E - ' J J

5 . 3 0 ' J E - 0 3 - 6 . 4 ^ 0 1 - 0 3 - 7 . E 3 f i . - 0 3 -S .9 -1 .L -B3 - 2 . - C 7 C - 0 J -O.eOL'L-OG

1 .30U - 0 3 2 . 5 I 3 L - U 3 2 . i 9 b E - D 5 I . 0 3 3 L - 3 4

-1 .743L» i ; e

SEMT 0 .

- : . : : 2f. - I -EJ~!L -1 .797E - I . 7 3 C . : - ! . l ; i?E -7 .2 .ME -5.-176E -4.r?jL - ^ . l 3 ' C - l . L . IL -2 . . -70L - ! . 0 7 t -7 .3U. -_ —1.;'27L -2 .0G3r - a . 3 7 - T -0 .370C -3 .G321 - 3 . . U 7 L -2.UP3L - 1 . 2 L LIE - i .3 -s : - 1 . ILL.!. - I .U - - IE - 7 . 0 3 0 : - 3 . 3 . - - 1 : - 3 .21 ' iC - I . B 3 9 ' . -2 .D32L

- 0 1

* L u - 0 1 - 0 1 - f t - c . -0"" -0 . " - 0 2 - D : -OL' - 0 2 - 0 3 - M 3 - O J - 0 4

-6/1 - 8 4

- b -- 0 4 - 8 4

-0-1 - I . . ' - 0 4

- C 5

- 0 3 -CS - 0 3 0 3

- ) . ^ U 4 £ « b 0

i L PE I TL „ : : . . I2EI ' TJ 11 . . ' I f .

5 f F. p n ^ L.

- 2 . : ' " - P ' - i . : i : t ' . . u - i . ' . i - . L -31 - " . • " ' - ' . < .

- * . L . . - L I -L.-"-' r-:: - L . 3 3 - L 1 - i .e.": ' -ui - i . L L { - 0 1 -l.?f C-01 - 4 . G I : : - L I -r.c.'.-E-ci - 7 . - I ; - [ . ) - I . S : 7 E - L I -3.t;rE-ci - 3 . 4 S C t - « H

- 2 . 0 3 2 f - 0 ? - • l . L . . -- '-.2 -2 . - : : L : - W 2 - 1 . ' «" . -L2 - l . C L l E - i 2 - 1 . • .. - L"? - 1 . ; - t : -

- r .c : . . -f.3 — 1 . ^ - • • -L.3 - 1 .'. . . - * 3 - i . - i . L i - ' ; - G . 1 ILL- . .4 - i . L 3 i . E - C :

-S.OL-^L-uO

roMit Pi l l • IK ,PHI

S «a .-.-T»JJ«

T > £ D .

- i . ? - ~ : - r . i

-;. -": - 3 . 13 ! L - ' . -i b . : - L I - 2 . . -H.L-U1 - 2 . ^ ' , " L - C 1

- r . C 3 - " - ' . i -3 .C; . 'L -OI - 3 . i i : t - u i - 3 . 7 4 s r - o . - 4 , G 4 j ; ; - j i - 7 . 1 1 0 . - C 1 - 7 . 7 b . E - 0 1 -2 .B1 K - 0 1 -3.e:of-ci - 3 . i L 2 E - B l

-2.B:c"-e2 -4 .04 . - . : - r .2 - 2 . 4 : 2 - -Z. - 1 . 3 : : : - . 2 - i . 6 : - - : - : 2 - 1 . 2 . : . - ' . - i . C : L : - L .

- 7 . C . . : - L : -£.c; • -LZ

-1 . : " . i - c : - I . : C . L - O . - G . S i . : c - o . i - 1 . 2 - i r L - L :

- 4 . 0 4 - E ^ ; 2

^ - « < -. _• -• 2 . 1 ! . ^ .

„. ,. , H - L - . : H

0 . c : 'Z •>'.': t..' '. - : 1 . r . v - i i i . r : -• : 1.L.--J! - . 1 1 . 7 : : - _ ! 2 . 3 - . . - 31

: . i " L i - r.; z.z Lf-:: 4 . : • • : - . ! L . : . ' : - ' j i 7 . 7 : ? : - L l I .?• . - • : - c i 2 . 9 ' > - D l 3 . 4 3 2 L - 0 :

1 .9-).-" -02 4 . _ 2. - r i . ' .. - r . i... r-L.' i . : • i - . . : i . . • • .

c. • • i - . . 4 . . . - :

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^:L:;:j:' 3 . 3 - ; L * i . 2

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~ p

; z A

J

r r 2 . 3 .

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-. ; i ; -Ll<

~" t * f ^ t -K-t

e i - C i iw. - c i • : . : - L J

J - - ' " - C :

" : : L - D I ~L, r*:-u] c:::-oi r * ' i - C ! ^ " . ^ -C ' i : - '_- t ) j

13JL-01 S ^ l t - t l

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"-',.' . ' -i' l - L .

. > ' 1 -*. . r - „ j

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TABLE VIIlb: PARTIAL AND NET SENSITIVITY PROFILES FOR THE TWO-DIMENSIONAL ANALYSIS

.-•I r„ r. I T T ! , ' ! • - . i r i w r ^ p r - o f i t c . i-cn M L I A - U . HO ri,- t I Z E C TO P.R - ( R . P H D M I U T . ' J H I N I E P I I C T I O N CPOSS SECTIUH5: (H-H) Willi i n -GAi lX )

2.10592E-O7

5 S 7 e 9

le II 12 13 I I 15 16 1? 1G 19 2J 21 2 ' ?3 24 25 25 2? 28 29 30

JPPEf-FrEV) I . . 1 I T &l 1. V . ' . t >07 l.3r>',E»i37 l . 2 : i " E « 7 i . e : 3 r * e 7 7.T9 JL-06 6.07b. r»6G 3 . 6 ; ' - r * 0 6 2 . c o t •or, 2.23.'E*€f> I . 7 ; C E + O G I . J5^C-03 e . ?.,'.c tas

"if -05 3 . L : . - : » € 5 I . c o n - a s 6.7t&:->3d 2 . T ' : 3 1 9.1206 -03 3.0o rEV)3 1.233C03 4.5"-E-^0? i . f - o r >:<2 6 . i<-o::>0i 2 . ? c r r , 0 ! 3 . ; : o : o a 3.C0-..E-H.3 I. ]30E<00 1. I-JOE-BI 1.5Z0E-0J

I .2 . .E-BI I . 0 0 E - 0 I

ICE-BI •C-BI

50E-8I 40E-BI er iE-ei 5 t r > e i 5 E - 8 I .GC-OI I / : - O I

9-E-B1 4 .9FC-GI 5 . 8 1 C - 0 I 4 . 0 - rc -a i i.rL r-><j3 i .n-.:»t.3 i .oor . io3

OOE-KIB OOC-BI OOE-OS

*03 GO'-iOO D-C-01 0?E-01

9.5CE-61 l . o o r i o a 1.C0E-OJ l . H E f O B

fIXS 0 . 2.4G4E I .797E

- I . 9 7 7 C - 1 . 3 0 0 f -5.G07E - I . 7 7 U C - 5 . VILC - 5 . eui-c - 3 .073E -2 .624E -5.92.1E -1 .509E -1 .032E -3.72'JE - 6 . 0 0 I E -4.20UE -5.3' jOL -7 .D07E - 1 . 0 2 2 E - 2 . 5 0 > ; E - 3 . r . ' L E -4.G2G0 - 5 . 0 I B E -6 .71-7t -7.GAZE' S . CAGE' S . 720C - 4 . J ~ I E - 2 . 0 2 ; E

P U R E L 0 NU-FISO 8 .

•OB 0 . - 0 2 B. -B2 0 . 02 B. -03 B. -03 0. 04 0.

-04 6. -04 U. U4 U. 04 0. -03 0. -03 0. 04 0. •04 B. •U-1 B . GO 0. 00 0. 00 0. 04 0. 00 B. 05 0. 05 U. II5 0. 05 0. 05 U. 05 0. 05 0. 04 0.

i T E R M SXS

e . - 2 . 2 I 5 E » B I - l . 7 9 I E " d O - 2 . 9 5 5 E - U I -3 .35 'JE-BI -2.GG2E-BI - 2 . 4 2 7 E - 0 I -2.BG<-E-B1 - 3 . 5 9 4 E - B I - 3 . 4 U I E - 0 1 - 3 . 7 1 I E - 0 1 -1 .50 'JE-BI - 6 . i . 2 0 [ - 0 1 -7 .6D1E-U1 - I . ' M J L - U I - 2 . 9 I 4 E - 0 I -3 .19GE-OI - 2 . L 0 o L - 0 2 -4 .u : : - * : -02 -2 .33GE-b2 - 1 . 4 . T C - G 2 - 1 . 6 4 5 L - 0 2 - 1 . 3 5 0 E - 0 2 -J .812C- I , 2 -G.927F-G3 -4 .3 ' 'U( - 0 3 -2 .40UE-03 - l . 3 0 - £ - B 3 - 5 . I IL ' i>04 -9 .990E-D4

u •»-•»-.. r PUCE CPIM TTRrE T>1

B. -1.9C.LC+0I - l . 7 7JC-»M> - 3 . I 5 2 E - B 1 - 3 . 5 3 / E - B I -2 .71CC-BI - 2 . 4 4 5 E - B I - 2 . 0 7 4 E - B I - 3 . G 0 u E - 0 l -3.40.1C-B1 - 3 . 7 K E - O I -4 .50 - lL -O I -6.9'J'--E-Bl -7 .G1- -E-0 I - I .OO- 'E-Ul -2.92L.C-01 - 3 . 2 0 C F - 0 I - 2 . LOOT-02 -4 .CJ1E-02 -2.M2C-U2 - I . 5G2E-U2 - I .G/- . ;C-u2 - I . 3 C 2 E - 0 2 - I . B I C E - 0 2 -6 .934E-G3 - -4 .4 ICL-03 -2 .5G2E-03 - I . 3 3 I E - 0 3 -G .5C11-04 - I . 2 B J E - B 3

N-CPIN B. 0.452E10B 7 .52JE-UI 1.3G0E-0I 1.B02E -BI 1 .C I6E-0 I I . 7 I 9 E - 0 1 2 . 3 2 1 L - 6 1 3 . I I I E - B I 3 . 1 0 J I > B I 3 . 5 I 1 C - B I 4 .3 'J IL -OI 6 . 0 3 J L - O I 7 .52L 'E - i l 1.9U0E-OI 2 . C J i E - B I 3 . I31E-C1 I .S -^C-02 4 . G 0 0 - - L 2 2 .3o5E-L2 I . ' : J : E - 0 2 I .G3^E-02 1.551L-02 1.007E-02 6.CD5L-03 4.341E-E.3 2 .509E-03 1.3-3E-C3 6.3o3C-C-1 l.!G5'E-B3

N-GnlN(SED) 0 . 4 .033E*0B 1.5'JJE-KiB

2 . 0 I O E - O I 3 . I 2 9 E - 0 1 2 . 4 / 0 ^ - B I 2 . 3 0 3 L - B I 2 .C55E-BI 3 . 6 P . E - B I 3.GG5E-0I 3.SGCE-0I " 5 r ' , L - 0 l 7 .563C-01 O.OCL'L-bl 2 . 0 1 ' . L - 0 1 3.0: . '52-0l 3 . 2 i ; t - 0 l 2 . C 0 - C - 0 2 4 . 6 5 J C - 0 2 2 . i i . . E - 0 2 l . u ; : E - D 2 1.615E-02 I .3G1E-02 1.010E-02

4 . 4 I I E - B 3 2 . t 5 3 E - B 3 1.35yE-03 e .SC-E-B4 1 . I 81E -03

HG-Qfl IN 0 . B. B. 0 . B. B. 0 . B. B. a . B. B. B. 8 . B. B. B. U. 0 . B. B. B. 0 . 0 . a . a . • . B. 0 . B.

CROUP UPPER-E(EV) BCLTB-U I l .?88E*e7 1.25E-B1

«»»» HET PROFILES « « SEN SENT

e. e.

-5.B41EK1B -4 .733E»C0 3.275E-Ki0 3.275C+0B B.

2 3 4 5

e 7 B 9 IB i i 12 13 14 15 IE 17 16 19 28 21 22 23 2-1 25 2G 27 29 29 33

i . s m O 0 7 i . V i ' , c --Br 1 . 1 5 / . < 0 7 l.G0'J5»07 7.7?'JE»BS 6.Br!iE-KiG 3.GC'.E-K,G 2.CC5EAG 2.232E-.35 i . 7 3 " ; L + B G 1.35 iE i9G 8.2;-,C-i05 5.E.'.i',E<35 3.C3'iC+03 l .n^'F. IFJJ 6 . 7G0HH4 2.0"riE»9-4 9 . I 20E -O3 3 . 3 ' C / > J 3 1 .2-5r»U3 ' ! . 5"5n*32 l.C.-0rHU2 G . I . I : * O I 2.2f~.EiDI e.3?l,E-.3'3 3.8i5E-»0H I . 12'.C->JB 4 . 1 'OE-BI 1 .520E-0 I

i . e s r - e i I . IL~.-!)1 I .B2. : .-BI 2 . 5 ' , C - 0 I 2 . 4 : r . - a i 5.0GC-01 2 . 5 f C - B I 2 .50C-B1 2 . 5 3 C - 8 I Z .bOE-0] 4 .9?C-B1 4.SCE-B1 5 .U1E-01 4 . D : E - B I I.OnEiOO i .ooEtca 1.05,- ioa i.coE-.ag 9.9IX-01 i . e i - c i r j i l . 8 ' X < 0 3 I.O'yiMBO 9.9'JE-OI 9.91'C-Bl

9.9GC-01 l.O'.C-iao 1.B5E+06 I . I ; E - * 3

-1.3GK-»0I - I . 0 L 3 r < 0 0 -1.5u'JC-Ul - l . b U C E - B l - 1 . B 4 . IE-01 - 7 . 0 0 I L - U 2 - 5 . 4 ^ 5 E - 0 2 - 4 . 0 3 2 E - 0 2 -3 .2 i 'GE-B2 -2 .U07E-02 -2.C0CE-D2 - 1 . 4 I 7 E - 0 2 - B . 1 9 - E - 0 3 - 4 . 5 7 2 C - 0 3 -2 .G25C-03 -5 . 'W5F.-04 -S -CVSt -BJ - 2 . G I 5 E - U 4 - 3 . U 10E-04

5.50'.'C-IJ5 -7 .725C-B5 -7.00CJE-05 -5 .45JC-U5 -3 . l i ; .L" -05 -S .522E-0G

1.32-IE-U5 2 . 4 V C - 0 5 2.45HE-U5 I .GOut-04

- 1 . 1 2 3 E f 6 1 -1.B2IE-HK1 - 1 . 7 0 , C - U I - l . 7 i - 1 E - 0 l - 1 . 1IJ1E-UI - 7 .253E-H2 - 5 . 5 3 1 E - B 2 - 4 . 0 9 H - B 2 - 3 . 2 5 J E - 0 2 - 2 . 0 3 ^ E - 0 2 -Z.'J2"iE-02 - l . 5 G l . E - 0 2 -9.22<>E-03 -4 .<J4.£-n3 - 3 .3 I . 1E -03 - 9 . 7 1 K - B 4 -G.21. IE-U4 - 3 . 3 2 5 E - 0 4 - 3 . 2 2 2 E - 0 4 -2.B25L"-04 - 1 . M 5 C - U 4 - l . i r . l . E - M -1 .12GE-B1

- 9 . u j y i . - 0 j -7 .G75E-f .S - 5 . 3 2 2 C - 0 5 - 3 . ; v t L - n 5 - l . yO- iE -05 -3.2-lUE-OJ

-1.7GCE-KJB - l . 5 I B E < « B

98

http://-l.5Gl.E-02

http://-9.ujyi.-0j

LMAX £ , LMAX X8 = 1 I % 8 8 = I Y 8 8

£=0 k=0 £=0

Table IX provides a comparison between the x's calculated from angular

fluxes and flux moments. There is a very good agreement. It was found

that this relationship is also true in the one-dimensional analysis.

For £ = 0 and £ = 1, the W^ s calculated in SENSIT and SENSIT-2D are

different. However, the Yp's defined by

Y 8 8 = I 4<J8g 023) k=0

are in agreement.

5.1.3 Comparison between a two-dimensional and a one-dimensional cross-section sensitivity and uncertainty analysis

A cross-section sensitivity and uncertainty analysis was done for

the heating in the copper region, using SENSIT and SENSIT-2.D. In this

analysis the effects of the uncertainties in the secondary energy dis

tribution were included. Six separate SENSIT (or SENSIT-2D) runs were

required:

99

TABLE I X : CO>JPARISON BETWEEN THE C H I ' S CALCULATED FROM ANGULAR FLUKES AND FROM FLUX MOMENTS

> 3 - ;

1

?

3

h

c

6

1

8

9

i :

l i

_i?

13

l a

15

16

V

IB

1?

?c

21

22

23

?u

25

26

27

28

29

30

S r T>I7-

C-.l f a n r . flu****

0 . 0

?.505«i«9

2.1,350*7

6.130u*6

8.3365»6

5.6869*6

9.3953-6

5.6853*6

7.151,3*6

7.331,9*6

7.6619*6

2.1571*7

2.9571*7

?.51)'9«7

6.31,51*6

1.571>3*7

6.5?69*6

l.''636*6

1.2i,63*6

7.6B60*5

3.7761i*5

3.7U5*5

2.9673*5

2.201,1*5

1.5067*5

9.l631,*li

5.1,090*1,

2.51,76*1,

1.33?li»li

? . 3951*1,

n

chi (Tjur fr:-p-ts )

0 . 0

?.5053*3

2.L362-7

6.1261,*6

8.3329*6

5.6S51i*6

9.39ua*6

5.6331*6

7.151,3*6

7.331,9-6

7.3619*6

?.1571*7

2.5571*7

?.5L?9«7

6.3a°l*6

1.5763-7

6.5269*6

1.7636*6

1.21,63*6

7.6660*5

3. ,76u*5

3.71,15*5

?.9673*5

2.'0L1*S

1.5067*5

9.ii631i*U

5.1,091*1,

2.81,76*1,

1.3325*1,

2.391,9*1,

C : I ; - T T

ch i (mn;. ' lv-c«s)

0 . 0

2.1,533*8

2.ld,93*7

6.1B29*6

8.U35*6

5.7i,?li*6

9.1/'75*6

5.7312*6

7.2073*6

7.399u*6

7.9321*6

2.1793*6

2.9022*6

2.6026*6

6.5023*6

l.*3u5*7

7.135k*6

1.9060-6

1.2562*6

7.71,52*5

3.3kuO*5

3.5705*5

3.021,5*5

2.'W6*S

\.itl-"S

9.5US*!.

5.1.300*1,

2.81,99*1,

1.331B*!,

2.1j?10*l,

c*. (flu* P9i«nts)

2.5003*3

100

three runs for the vector cross-section sensitivity and uncertainty

analysis (one for the cross sections of zone II, one for the cross

sections of zone III, and one for the cross sections of zone IV),

three runs for the SED sensitivity and uncertainty analysis.

Oxygen was not included in the vector cross-section sensitivity and un

certainty analysis, and hydrogen was ignored in the SED sensitivity and

uncertainty analysis.

The procedure for an uncertainty analysis has been discussed by

Gerstl. The results from the one-dimensional analysis are reproduced

in Table Xa, while those from the two-dimensional study are given in

Table Xb. The studies are in good agreement. Sensit required a total

of 89 seconds of computing time, while SENSIT-2D required 90 seconds on

a CDC-7600 machine. The uncertainty of the heating rate due to all

cross-section uncertainties is 30%. The iron in zone II is the largest

contributor to that uncertainty. The contribution of the SED uncer

tainty is smaller than that from the vector cross sections. Gerstl

points out that the results obtained from the SED analysis might have

been underestimatpd HUP to the simplicity of the "hot-ccld" concept and

due to the fact that the partial cross sections which contribute to the

secondary energy distribution were not separated into individual partial

cross sections.

101

TABLE Xa. PREDICTED RESPONSE UNCERTAINTIES DUE TO ESTIMATED CROSS SECTION AND SED UNCERTAINTIES IN A ONE-DIMENSIONAL ANALYSIS

CROSS SECTION

Fe

Fe

0

H

Cu

All*

ZONE

II

III

III

III

IV


TAR" R x-sect

zone

8.18

AR" R

*

zone

8.18

2.50 !

0.78 2.61

4.02 1 4.02

c .48

RESPONSE UNCERTAINTIES DUE TO CROSS-SECTION UNCERTAIN TIES, IN %

AR" R x-sect

zone

23.80 ;

10.33 i

!

1.96 i

11.72

AR" R

-,v

zone

23.80

10.52

11.72

2? .54

Overall uncertainty = (9.482 + 28.542)'s = 30.0%

* quadratic sums

102

TABLE Xb. PREDICTED RESPONSE UNCERTAINTIES DUE TO ESTIMATED CROSS SECTION AND SED UNCERTAINTIES IN A TWO-DIMENSIONAL ANALYSIS

CROSS SECTION

Fe

Fe

0

H

Cu

All*

ZONE

II

III

III

iv !

1


[AR R element

zone

8.17

2.50

0.79

-

4.02

AR R

_»-

zone

8.17

L

c

!.62

».02

.47

RESPONSE UNCERTAINTIES DUE TO CROSS-SECTION UNCERTAIN TIES, IN %

AR R

element zone

23.88

10.27

-

1.96

11.68

AR R

,v

zone

23.88

10.46

11.68

21 1.57

Overall uncertainty = (9.472 + 28.S72)"5 = 30.1%

quadratic sums

103

5.2 Sample Problem ='/2

A simple one-band problem will be analyzed to study the influence

of the mesh spacing, quadrature order, convergence precision, and the

c-factor (mean number of secondaries per collision) on the sensitivity

profile. The band is 1-cm high and 20-cm wide. There are ten distinct

zones, each 1-cm wide (Fig. 11), and all zones are made of the same

material. A three-group artificial cross-section set with a third-order

anisotropic scattering is used (Table XI). The P., P„, and P_ compo

nents of the scattering cross-section tables were chosen to be identical

with the Pn component. A volumetric source with a source density of

1 neutron/cm3 in group 1 is present in the first zone. A standard

cross-section sensitivity analysis will be performed, in which the cross

sections in zone IV are perturbed, and the detector response is calcu

lated in zones IX and X for a response function of 100 cm in each

group.

5.2.1 Influence of the quadrature order on the sensitivity profile

The detector response calculated by TRIDENT-CTR using EQ,, EQ17»

and EQ , quadrature sets are compared in Table XII. For the first three

-3 cases, the pointwise convergence precision was set to 10 and each zone

contained four triangles (using automatic meshes). Five additional

cases are included in Table XII:

104

RIFLFCTI'R BO"VHDT

II sm-RCE ZONE

tone » I

zone

» n tone * I I I

PKRT. ZONE

tone * IV

zone f V

ton* » VI

tone

» v:i tone * VIII

D-T. 7 0:;-

» I

tone • I I

DVT. ZOf.'E * II

tone * I

• "

C

' 0.0 1.0 2.0 3.0 li.O 5.0 6.0

REFI^CTIN:; BOUNDARY

?.o e.c 9.C 10.C or

r -z geometry

All zones contain identical materials

3 neutron groups

3 Keutron source: 1 neutron / cm in zone I and group 1

Response function: 100 cm (all groups)

Figure 11. Two-dimensional model for sample problem "2

105

TABLE XI: CROSS SECTION TABLE USED IN SAMPLE PROBLEM i\2 (THE P P P2, A.ND P TABLES ARE IDENTICAL)

Cross Section

Group g = 1 g = 2 g = 3

I 8

edit

0.02 0.05 0.1

0.0 0.0 0.0

*i 0.1 0.2 0.3

zs

I8"

2g" S

>g

1-*

0.05 0.1 0.2

0.0 0.02 0.05

0.0 0.0 0.01

106

TABLE XII: INTEGRAL RESPONSE FOR SAMPLE PROBLEM y/2

Transport Quadrature Convergence Forvard Adjoint Code Set Precision // Triangles1 Response Response

593

592

593

593

593

593.

591.

592.

.968

.826

.659

.659

.688

.855

.814

055

592

591

592

593

593.

590.

590.

590.

.256

.659

.476

.148

.208

.370

883

900

1 it spatial intervals for ONEDANT.

2 equal-weight quadrature sets.

3 Gaussian quadrature sets.

TRIDENT-CTR EQ-62 10_3 40



TRIDENT-CTR EQ-12 10~A 40

TRIDENT-CTR Eq-12 10"4 80

ONEDANT S-123 10~~* 40

ONEDANT S-32 10_A 40

ONEDANT S-32 10'4 80

107

1. integral response using an EQ15 quadrature set with conver-

gence precision 10 ;

2. integral response using an EQ.„ quadrature set with conver-

-A gence precision 10 and eight triangles per zone;

3. integral response calculated by ONEDANT using an S 1 7 quadra-

-4

ture set, four intervals per zone and a 10 convergence pre

cision ; ,

A. integral response calculated by ONEDANT, using an S„9 quadra-

-A ture set, four intervals per zone and a 10 convergence pre-

5. integral response calculated by ONEDANT, using an S__ quadra-

-A

ture set;..eight intervals per zone and a 10 convergence pre

cise on.

The response functions in Table XII are in good agreement (maximum dif

ference 0.6%). The standard cross-section sensitivity profiles for the

EQ, , EQ _, and EQ , calculations are reproduced in Tables Xllla, XHIb,

and Xlllc. The integral sensitivity for the EQ. case is 5% different

from the E0._ case for AXS (absorption cross-section sensitivity rro-• i z • - '

file) and 5% different for N-GAIN (outscattering cross-section sensi

tivity profile). The results obtained from the EQ.„ calculation are in

good agreement with those obtained from the EQ., calculation. The sen--A

sitivity profiles for the EQ..~ case (10 convergence precision) and the

-A EQ „ case (10 convergence precision, eight triangles per zone) are not

shown. They are nearly identical with Table Xlllb.

108

TABLE Ilia: STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT-2D FOR THE EQ-6 CASE (CONVERGENCE PRECISION 0.001, A TRIANGLES PER ZONE) FOR SAMPLE PROBLEM i>2

CR3JP UPPEP-E'EV) 1 1.0uOt*Ol 2 5.aen- »ea 3 l .&bi)L*Ce

INTEGRAL

DELTB-U 6 . 9 3 E - B 1 I.6IE-H3C 6 . 5 3 E - B I

G^F UPFEF-E'EV) DELTfi-U 1 I . E i - t E - e ; 6 . 9 3 E - B I 2 5.«:eE-fie I.6IE-H30 3 i .eeoE-ee 6.93E-ei

P U R E P.XS NU-F1S

- 4 . E 3 2 E - 8 2 B. - 4 . S i ? E - e 3 8 . - I . I 4 2 E - 8 2 B.

O S S T E R M S « * « . r » . * » SXS TVS

- I . 8 S 3 E - B 1 -2 .31CE-B I - l . 3 i 9 E - 8 ? - I . b ) ? E - 8 2 - 2 . 2 6 4 C - 8 2 - 3 . 4 i . i E - B ?

« PURE G«!N TER-E N-CP1N h-G^!MSED) N 0 - ' . ^ >

1.5-J5E-81 l . l S i E - B I e . I .I&fcE-e2 2 . J S t E - 3 2 I. 2 .2bAE-B2 3 . b ? ? E - B 2 8 .

-4 .732E-B2 B.

• * » » NET PROFILES * » « • SEN SENT

- 3 . e 3 9 F - 8 2 - 7 . b ? ] E - B 2 - I . 7 3 I E - 8 3 - 6 . 2 t J £ - 8 3

3.252E-Bb - I . I4:?E-82

-2 .3B5E-B2 - 7 . I 1 C E - 0 2

- i . E e i E - B l - 2 . 1 3 5 E - B I

TABLE v u i b : STA-NDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT-2D FOR THE EQ-12 CASE (CONVERGENCE PRECISION 0 . 0 0 1 , 4 TRIANGLES PER-ZONE) FOR SAMPLE PROBLEM r'2

2 5 . B e l . ; * e 0 3 l . B O i v - e e

INTEGRA,.

GROUP uP"l'-['£Ji i i . eaa>. >e i 2 5 . 8 * r = - f l B 3 1.BBbE - 8 8

INTEGRAL

TFL TS-U 6 . S 3 J - B l i . 6 i E * e a 6 . 9 3 E - e i

»E^TB-U 6 . 9 3 - - e I l.fc,E*<JB 6 .93E-B1

»M*M*.* P U

-4 .329E-B2 -4 .3S6E-B3 -1 .1B3E-B2

-4 .4 fc8E-82

R E L O S ND-FJSS e. 8 . B.

e.

«»*« NET PROFUEC «»*» SEN SE'<T

- 2 . 7 3 1 I - B 2 - 7 . 8 6 0 E - 8 2 -1 .6C9E-B3 - £ . 8 3 4 t - 8 3

B.84BE-B? -1 .1B3E-B2

-2 .161E-B2 - 6 . 6 2 9 E - 8 2

S T E « n EXE

- : . 7 J ! t - e i - I . 3 1 B E - 0 2 - 2 . 2 0 6 E - 8 2

- : . 5 6 4 E - e :

s •»

- 2 - 1 - 3

-2

TXE . ! b 4 t - 6 1 .?4££-82 .3 :0E-B2

,811E-B1

MM****** PJR£ GO IN YER*E N-GftjN N - G C ! N ( S E & )

1.4 ' jaE-BI ! . e?5E- t - i I .143E-B2 2 . i r b E - 0 2 2 . 2 B - E - 6 2 3 . 7 6 I E - 3 2

I .348E-B I l . 3 j e £ - e i

N G - G M N e. e. e.

8 .

109

http://-3.4i.iE-B

TABLE XI H e : STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCULATED BY SENSIT-2D FOR THE EQ-16 CASE (CONVERGENCE PRECISION* 0 . 0 0 1 , U TRIANGLES PER ZONE) FOR SAMPLE PROBLEM ill

GP3JP UPPEP-ECEV) DEI. T«-U i i .eee_*ei 2 5.eee£»ue 3 KBOBE^BB

6 . 9 3 E - B I 1 .6IE»BB 6 . 9 3 C - 8 I

T E R PI 5 « * * « * £ * * * x»r***«r

- a . i r 9 ; - B 2

- I .71E.E-B1 - 2 . U 5 E - B 1 -1 .29SE-B2 - l . 7 3 . - t -02 -2 .16EE-B2 - 3 . 2 6 1 E - B 2

PURE GO IN TERrE N-GMNISEB) HG-Gfl lh

1.J43C-B1 1.BG2E-8! B. 1.133E-82 2 . i e s E - B 2 B. 2 . I B 7 E - 8 2 3 .722E-82 B.

-1.9S3E-B1 1.334E-B1

GRDJP UPPEP-EfEVl 1 1.89BE-01 2 S.BBBE-ee 3 1.8BBE««e

INTEGR«L

DELTO-u 6.93E-B1 1.6!E*flB 6.93E-BI

NET PROFILES «« Eli SENT

- 2 . 7 ; 1 t - B 2 - l .C^ .aE-33 - 1 . 4 8 i £ - 8 6

- 7 . 8 2 J E - B 2 -5 .SSSE-83 - 1 . B 9 4 E - 8 2

- 2 . I 6 2 E - B 2 -6 .591E-B2

110

http://-l.73.-t

Note that the net sensitivity profiles SEN (= SXS + N-GAIN) for

-6 -•' -6 group 3 are respectively 3.252 x 10 , 8.040 x 10 ' , and 1.484 x 10

for the EQ,, the EQ.,„, and the EQ_, case. The large discrepancies here o 12 lb

can be attributed to the fact that those quantities result from sub

tracting two numbers that are nearly equal in magnitude.

It can be concluded from Tables XII a.nd XIII that even when the

integral responses differ by less than 0.4%, the sensitivity profiles

can differ by as much as 5% between an EQ, and an EQ „ calculation. The

close agreement between the results from the EQ.? and the EQ., calcula

tion suggest that this difference is probably due to the fact that the

S ] angular fluxes in the EQ, calculation are not yet fully converged.

Indeed, choosing the higher-order anisotropic scattering cross sections

equal to the isotropic components is unphysical. The convergence cri

teria used in 0NEDANT and TRIDENT-CTR do guarantee .onvergence for the

scalar fluxes, but not for the higher-order flux moments.

5.2.2 Comparison between the two-dimensional an<i one-dimensiona1 analv-sys of sample jirjDblem H2

The cross-section sensitivity profiles resulting from a one-dimen-

-4 sional analysis (S „ quadrature set, 10 convergent precision and four

-4 intervals per zone; S_„ quadrature set, 10 convergence precision and

eight intervals pt_r zone) are compared with those obtained from a two-

-4 dimensional analysis (EQ 1 9 quadrature set, 10 convergence precision

and eight trianglts per zone in Tables XlVa, ,1X\>, and XIVc.

Ill

TABLE X l V a : STANDARD CROSS-SECTION £F.':S I I I VI TV PROFILES CALCULATED EY SENSIT POP TV.?. S-12 CATE (CONVERGENCE P PJL C T F I nN O.OCj] i INTERVALS PEP. Z'JNE) FOP FA-' TI.E PP.l'LLF..^ i;2

*-•* « • »

-D ;

- Oc

- U

* - • • • * • • " • * ^ < * « * J » 1 * . ' I » - l

• . ' . • I N • > - » • > . { ! (

I J . L . - f • C J e . ^ ^ e - C j

l . J ! . ; I - « : V . * c C »»w

t .

c.

i.

*. C

- ' 3 * *

S *

-;

t I • -

S ' X

. " * ; „ • * - . ;

J . K " - -

i - i I ~ vt.

* • * - • * • -

- • &

- < . : * £ t

-1. ~* J *

- J . J - J I

1 . £ ? 4 f - C i I. i'**-!.:

£•• t * * - ^ fc.s-ji-c;

j . t : « - - -t . • T J I - - :

• ^ • « • • « • • # •

i i -- J . t C 5 l " t £

- 1 . t. - u - i : j . e. j £ l - U *

• O * I L I I *"* S I - *

- * . V . I - t - £,£ C I - I . i/t ^1

-. £

L C

- c . ?&<r* - Lc

. - \ . . _ > . - L J - . .

:>:: C . O O M , £ :NTF.:-.Y.

. : jOPES-E F^: i-.-- -? t 53- - e : s - • -ee . t . • -?? B*reE-ce f 5;-: e

C » 0 . - J»-< = t S3- - e i

s e-.'.-- - P ? ; . 6 : L * e e i . eu~:. -ee 6 . 5 K - e i

»•» ' . = F " 5 t - r * - - c :.: : . ' , r s\ e. . " i -^ e. LB £-L. e

B'-S'i-C- e .

• N?~ PPOf : . f c „ . .

- 6 . e « - : -?? - ' v. •-. e.< - . e. :[ -e-'

-7 e 9 : E - e ? -7.2S9E-e:?

-3 fe~:f-e;

5 9 - ! ^ - e ^

T E e r«

I t f • ? . : 6 '.i - e . '.fc:£-o:

L?^ i - e :

Til -: I C E

- ; <*t . s

- e :

-er

e :

i . p - - : e ; i . i . = : - e ; v „•. ,i - B :

i . : ' f c f - e i

*-:.-:>.,bf:. n 5 . ' : ? E - e ; e . i . s ? i E - t : e . 3.tB5E-U? e .

i . ?3cs -e i e .

112

TABLE XIVc: STANDARD CROSS-SECTION SENSITIVITY PROFILES CALCU CALCULATED BY SENSIT-2D FOR THE EQ-12 CASE (CONVERGENCE PRECISION 0 . 0 0 0 1 , 8 TRIANGLES PER ZONE) FOP. SAMPLE PROBLEM ff2

P U R E L O S S T E R M S mnam*** ****..*** PuPE G°!N TEPr£ CPOJP UPPEP-E'EV)

2 b. 6CSE -K3B 2 i .eeuE-Kie

INTEGRAL

CP3JP UPPEP-E(EV) ! l . e ^ E - o i 2 5 .eaoC 'Be 3 I . B B B E - B B

IN'ESRAL |

»ELTa-U 6 .93E-81 1.61E-MJB 6.S3E-B1

DELTO-U 6 . 9 3 E - 8 1 i . f c i f » e e 6 . 9 3 E - 0 I

fiXS - 4 . 3 2 4 E - B 2 -<S.3b?E-B3 - 1 . 1B2E-B2

- d . 4 t 3 E - 8 2

NU-fTSS u . B. e.

e.

* » * » NET PROFILES »*»» SEN SENT

- 2 . ? 2 r f - 8 2 - 7 . 8 5 I E - B ? -1.bfcLE-B3 - £ . . 8 3 ^ - 8 3

7.916E-B7 - 1 . 1 8 J E - 6 2

- 2 . I 5 8 E - B 2 - 6 . 6 2 1 E - 8 2

SXS - 1 . 7 3 8 E - B I - 1 . 3e9E-B2 - 2 . ? e 3 t - 6 2

- I . 5 6 2 E - B I

TXS -2 .1C2E-B1 -i.ri'ji-e: -3.SVz.l-B2

- 2 . B 8 9 E - 8 ]

N-GAJN l . ^ ^ E - B l I . u . ' t - e 2 2 . 2 a : t - C 2

1.346E-BI

N - G M N ( S E P ) i . e 7 4 E - p i 2 . I 2 3 E - C 2 3.75SE-D2

1 . 3 f c E - e i

N G - r i ! " e . e . e .

e .

113

http://-3.SVz.l-B2

Note that the N-GAIN integral sensitivity differs by about 6% be

tween Table XIV b and XIV c. The integral net sensitivity shows a 35%

difference for SEN (= SXS + N-GAIN) and a 10% difference for SENT

(= TXS + N-GAIN) between the one-dimensional and the two-dimensional

analysis. The bulk part of this large dilference for the integral net

sensitivity results from the subtraction of two numbers that are nearly

equal in magnitude. A comparison of N-GAIN (integral) in Tables XIV a,

XIV b, and XIV c suggests that - even with an S^? quadrature set - the

one-dimensional calculation is not yet fully converged.

5.3.2 Comparison between the x' s calculated from angula r fluxes and the X's resulting from flux moments;

The x's (or the loss term of the cross-section sensitivity profile)

can be evaluated based on flux moments (Eq. 58) or based on angular

fluxes (Eq. 57). A calculation based on flux moments requires less com

puting time, less computer memory, arid less data transfer. To have an

idea of the order of expansion of the angular fluxes in flux moments

necessary to reach a reasonable accuracy, the x's resulting from angular

fluxes are compared with those obtained from a P-0, P-l, P-2, . . .,P-17

spherical harmonics expansion of the angular fluxes (Table XV). It is

found that for any expansion of order greater than P-0, there is good

agreement (less than 1% difference for 2 x )• For very high spherical

g

harmonics expansions (P-15 and higher) there is divergence. This diver

gence can be avoided by doing the computations in quadruple precision.

114

TABLE XV: COMPARISON BETWEEN THE CHI'S CALCULATED FROM ANGULAR FLUXES AND THE CHI'S CALCULATED FROM FLUX MOMENTS

Order of Expansion of the Angular Flux g

• x g )

om ang

angular fluxes

0

1

2

3

4

5

6

7

8

9

10

11

12

14

16

17

889.88

796.63

880.43

868.78

884.11

884.58

886.82

888.82

889.16

889.78

889.74

889.91

890.01

889.89

898.27

938.59

951.85

83.382

76.811

83.755

S3.054

83.374

83.388

83.385

83.389

83.381

83.381

83.382

83.383

83.385

83.382

83.573

83.100

85.617

45.352

41.898

45.524

45.326

45.353

45.364

45.357

45.354

45.353

45.352

45.352

45.352

45.351

45.351

45.431

46.203

46.416

103.275

8.905

21.454

5.777

5.282

2.052

1.051

0.720

0.101

0.140

0.031

0.133

0.009

8.610

51.279

65.269

a 1 X mean x for group 1

115

The small differences in Table XV indicate that the loss term of

the sensitivity profile can indeed be calculated based on a low-order

spherical harmonics expansion of the angular fluxes.

5.2.4 Evaluation of J.he loss term based on flux moments in the case of low _c

The question whether the x's Cdr' be computed villi adequate accuracy

from Eq. (58) in the case of low c (mean number of secondaries per col-

p lision) was raised. Based on an analytical one-dimensional analysis of

the half-space problem (one group) with a mono-directional boundary

source, it was found that for c less than 0.8, a low-order spherical

harmonics expansion o± the angular flux would lead to erroneous results

in the x' s •

In order to confirm the analytical study, sample problem #2 was re

examined with a different cross-section table. The corresponding c's

were 0.5 for the high-energy group, 0.4 for the second group, and 0.33

for the low-energy group. The x's calculated based on flux moments were

still in agreement with those obtained from the angular fluxes (even for

a P-l expansion). An explanation for this paradoxical behavior is prob

ably related to the use of a distributed volumetric source in sample

problem #2, whereas the conclusions drawn in the analytical evaluation

were based on the presence of a mono-directional boundary source.

116

5.3 Conclusions

•

The rigorous study of the two sample problems indicates that there

is good agreement between the one- and two-dimensional analysis. Wher

ever discrepancies appear, a plausible explanation can be provided. Ul

timately, the comparison between a one- and two-dimensional study proves

to be a sound debugging procedure for SENSIT-2D as well as for the

SENSIT code.

For the flux moments versus the angular fluxes comparison for the

evaluation of the x's, there is a strong indication that the loss term

can be calculated from lower-order flux moments (P-1) as well as from

angular fluxes. By the same token, a P-1 sensitivity and uncertainty

analysis seems to provide sufficient accuracy.

The study of the influence of the quadrature sets on the sensi

tivity profiles reveals the importance of the angular-flux convergence

in ONEDANT and TRIDENT-CTR. Furthermore, some doubts about the meaning-

fulness and practicality of the net sensitivity profile (SEN) can be

raised.

117

6. SENSITIVITY AND UNCERTAINTY ANALYSIS OF THE HEATING IN THE TF COIL

FOR THE FED

In this part a secondary energy distribution and a vector cross-

section sensitivity and uncertainty analysis will be performed for the

heating of the TF coil in the inner shield of the FED. The results ob

tained from the two-dimensional analysis will be compared with selected

results from a one-dimensional model. The blanket design for the FED is

Do DO

currently in development at the General Atomic Company. '

6.1 Two-Dimensional Model for the FED

The two-dimensional model for the FED in r-z geometry is illus

trated in Fig. 12, and is documented in more detail in reference 84.

The material composition is shown in Table XVI. In the forward TRIDENT-

84 CTR model, which was set up by W. T. Urban, the standard Los Alamos

85 42 coupled neutron/gamma-ray group structure was used. There are 30

84 neutron groups and 12 gamma-ray groups. The TRIDENT-CTR model (Fig.

118

TABLE XVI. ATOM DENSITIES FOR THE ISOTOPES USED IN THE MATERIALS (atom/b cm)

Isotope

H-1 He-4 B-10 B-11 C 0-16 Al-27 Si Ca Cr Mn-55 Fe Ni Cu

c

1 1 5 1

5S316

.67E-

. 75E-

.44E--15E-

i

•2 •3

•2

•2

TFCOIL

3.79E-3 6.67E-3 2.98E-5 1.20E-4 1.90E-3 2.17E-3 1.81E-4 5.59E-4 2.42E-4 5.97F.-3 6.27F.-4 1.95E-2 4.12E-3 2.11E-2

SS304

1.77E-2 1.67E-3 6.06E-2 7.40E-3

MATERIAL

CNAT

8.03E-2

IHDLC

5.03E-2

2.51E-2

4.18E-3 4.38E-4 1.36E-2 2.88E-3

IHDLB

1.34E-2

6.70E-3

1.34E-2 1.40E-3 4.35E-2 9-20E-3

IHDLA

1.68E-3

8.38E-4

1.63E-2 1.71E-3 5.30E-2 1.12E-2

SS312

3.34E-3 3.50E-4 1.09E-2 2.30E-3

Nb-93 2.44E-4 Mo 1.51E-3 5.41E-4 3.78E-4 1.21E-3 1.47E-3 3.02E-4

v : Void SS : Stainless Steel C : Carbon

IDSL : Stainless Steel + H20

O

178 666 cm

84 Figure 12. Two-dimensional model for the FED

The numbers within each zone indicate the zone number.

t-

M M •'

V\l A . V ; H I \ ' / ! \ / | / \ l /

JUJ

£X

f-

J L

DC

121

13) utilizes 2062 triangles, divided over 21 bands. The response func

tions for calculating the heating in the TF coil, were prepared by the

72 TRANSXX code. Those response functions will be the sources for the

adjoint calculation. It was noted earlier that negative sources can

introduce instabilities in the sweeping algorithm for the adjoint

TRIDENT-CTR calculation. The negative kerma factors are therefore set

to zero. This will have a minor effect on the total heating calculated

in the TF-coil (less than 1%).

EQ-2 and EQ-8 quadrature sets are used for groups 1 and 2 respec

tively, EQ-3 is used for groups 3, 4, and 5, while an EQ-4 quadrature

set is utilized for the remaining groups. The convergence precision is

-3

specified to be 10 . The gamma-ray groups contribute most to the heat

ing in the TF coil (93%). The total heating in the TF coil is 823 *

10"6 MV.

The heating calculated by the adjoint TRIDENT-CfR calculation is

found to be 3% smaller than the heating resulting from the forward run.

The forward calculation required about one hour of c.p.u. time on a

CDC-7600 computer, while the adjoint run took about four hours. Groups

11 to 23 required significantly more inner iterations in the adjoint

mode than the other groups. No explanation of this behavior could be

found. Experience with other neutronics codes indicates that the ad

joint mode for this type of calculation requires usually no more than

30% extra calculation time.

122

6.2 Jwo-\Dimensional Sensitivity and Uncertainty Analysis for the Heating in the TF Coil due to SED and Cross-Section Uncertainties

A secondary energy distribution and vector cross-section uncertain

ty analysis was performed with SENSIT-2D using the forward and adjoint

angular flux files created by TRDSEN. A separate SENSIT-2D run is re

quired for each zone. Because separate runs are necessary for a cross-

section and a SED analysis, a total of 22 SENSIT-2D cases were analyzed.

A total of 15 minutes c.p.u. time was used by SENSIT-2D. The bulk of

this time is consumed during input/output manipulations.

The median energies and fractional uncertainties for the SED uncer-

tainty calculations were taken from Table II. A special cross-section

33 table was created - using TRANSX - for the SED analysis. COVFILS data

were used for generating the covariance matrices utilized in the cross-

section uncertainty evaluation. Only 0-16, C, Fe, Ni, Cr, and Cu were

considered for the SED uncertainties, while H, Fe, Cr, Ni, B-10, C, and

Cu were included for the cross-section uncertainties. With the excep

tion of oxygen, no important materials were left out. It was found in

an earlier study that the cross-section uncertainties for oxygen caused

45 an 8% uncertainty in the heating. The current version of SENSIT-2D

does not include the option to extract the covariance data for oxygen

from COVFILS.

The gamma-ray cross sections are generally better known than the

neutron cross sections. Therefore, only the uncertainties resulting

123

from uncertainties in neutron cross sections are calculated. Throughout

this analysis a third order of anisotropic scattering is used.

The predicted uncertainties in the heating of the TF-coil are sum

marized in Table XVII. It was assumed that the uncertainties for a par

ticular element in the various SS316 zones (1, 3, 7, 11, and 12 in Fig.

12) are fully correlated, while all other uncertainties wt ssumed to

be noncorrelated. This implies that the uncertainties for a particular

element can be added over all SS313 zones, while all the other uncer

tainties are added quadratically. The approach of either assuming full

correlation or assuming noncorrelation is rather simplistic. Trans

lating the physics of this particular problem into a more sophisticated

correlation scheme would be a major study by itself. The uncertainties

resulting from the uncertainties in the cross sections for Cr, Fe, and

Ni in the SS316 zones are reproduced in Table XVIII.

From Table XVII it can be concluded that the cross-section uncer

tainties (predicted to be 113%) tend to be more important than the SED

uncertainties (20%). Even when the overal uncertainty seems to be rela

tively large (115%), the blanket designer is able to set an upper bound

for the heating in the TF coil. The largest uncertainties are due to

uncertainties in the Cr cross sections. A more detailed look at the

computer listings generated by this analysis reveals that the largest

uncertainties are produced by uncertainties: in the total Cr and the

elastic Cr scattering cross sections. The heating is less sensitive to

Cr than to Fe. This indicates that the calculated uncertainty is large

ly due to the fact that Cr has very large covariances. A re-evaluation

124

TABLE XVII: PREDICTED UNCERTAINTIES (STANDARD DEVIATION) DUE TO ESTIMATED SED AND CROSS-SECTION UNCERTAINTIES FOR THE HEATING IN THE TF COIL (part 1)

Cross Section

Material Zone

Cr SS316 TFCOIL SS30A SS312 ISDLC ISDLB ISDLA


JAR]" R

Mat, region Mat

3.8 0.2 0.1 0.0 0.2 0.8 3.0

Fe

Ni

SS316 TFCOIL SS30A SS312 ISDLC ISDLB ISDLA

SS316 TFCOIL SS30A SS312 ISDLC ISDLB ISDLA

1A.8 0.1 0.0 0.2 0.5 2.7 10.8

1.5 0.7 0.0 0.0 0.0 O.A

: 1.2

A.9

18.A

A.3

XS Uncertainties in %

AR R Mat,region

AR R Mat

60.0 3H.5

A.5 1.1 2.2 33.3 58.5

18 10 2 0 A 23 3A

18 11 0 0 1

9 A 0

7 A 6 5

6 8 9 H

3

96.

AT.3

18.0

Quadratic Sums

123

TABLE XVII: PREDICTED UNCERTAINTIES (STANDARD DEVIATION) DUE TO ESTIMATED SED AND CROSS-SECTION UNCERTAINTIES FOR THE HEATING IN THE TF-COIL (part 2)

Cross Section

Material Zone

H TFCOIL ISDLC ISDLB ISDLA

0 TFCOIL ISDLC ISDLB

SED

AR R

Uncertainties in %

Mat

---

0.1 0.2 0. 1

ISDLA , 0.1

C TFCOIL '' 0.0 C-region 0.3

B TFCOIL

Cu TFCOIL 2 9

region

r 1 *

ARl

RL L ' Mat

i 0.3

0.3

-

2.9

XS Uncertainties

AR'

Mat,region

1-7 !

AR" R

in % .A.

Mat

6.0 ' 7.2 3.7 i 0.5 i

_

---

0. 1 3.2 j 3.2

i

0.0 ' 0.0

10. 1 10 1

Total 19.7 112.9

rotal uncertainty due to cross-section uncertainties and SEDs = IIA.6%

Quadratic Sums

126

TABLE XVIII: PREDICTED SED AND CROSS-SECTION UNCERTAINTIES IN THE TF COIL DUE TO UNCERTAINTIES IN THE SS3i5- ZONES

Cross Section

Material Zone

Cr

Fe

Ni

1 3 7 11 12

1! 12

3 7 11 12


JAR' " Mat, region >• -JMat

XS L'ncei tainties in %

0.1 0.7 0.0 3.0 0.0

0.5 3. 1 0.1 11.0 0.1

0.0 0.3 0.0 1.2 0.0

[AR

R Mat, region

FAR'

' RjMat

3.8

12.0 45.5 0.8 4.3 0.4

14.8

1.5

2 11 0 4 0

3 12 0 1 1

3 3 7 3 3

6 8 3 8 1

i 60.0

IS.9

18.6

127

of the covariance data for Cr is highly recommended. If new covariance

data would not reduce the predicted uncertainty, new experiments for

measuring the Cr cross sections are suggested. The conclusions drawn

45 here are consistent with an earlier study of a similar design.

The SED uncertainties, although less relevant to overall predicted

uncertainty, tend to become more important in the outboard shield

(region 11 in Table XVIII). An explanation for this behavior is related

with the fact that the heating in the TF coil will be very sensitive to

backscattering in this region. An SAD (secondary angular distribution)

sensitivity and uncertainty analysis might lead to very interesting

results.

The x's f°r the region near to the plasma in the outboard shield

are calculated for each group based on angular fluxes and based on flux

moments (Table XIX). Both methods lead generally to the same x's- The

difference for the upper neutron groups might indicate that a third-

order spherical harmonics expansion of the angular flux tends to become

inadequate, due to the peaked shape of the angular flux close to the

source region. In this particular study no serious error in the calcu

lation of the uncertainties would have been introduced if the loss term

of the sensitivity profile would have been calculated from flux moments.

For a situation where the angular flux would have a pronounced peaked

behavior, it would be highly desirable to evaluate the x's based on

angular fluxes.

128

It is obvious from Table XIX that some fluxes in the lower gamma-

ray groups (groups 41 and 42) are negative. Since only neutron sensi

tivity profiles are utilized to calculate uncertainties, this will not

affect the results.

6.3 Comparison of the Two-Dimensional Model with a One-Dimensional Representation

The results obtained from the two-dimensional sensitivity and un

certainty analysis will be compared with those of a one-dimensional

analysis in selected regions (Table XX). The uncertainties in the heat

ing in the TF coil due to the uncertainties in the Cr, Ni, and Fe cross-

sections and secondary energy distributions will be calculated with

ONEDANT and SENSIT in zone 1 and zone 3 (Fig. 12). The one-dimensional

model for ONEDANT is straightforward. The total heating calculated in

the TF-coil is 1043 * 10 MW (compared to 823 * 10 MW for the two-

dimensional model). In this comparison the uncertainties calculated by

SENSIT will be normalized to the response calculated in the two-dimen

sional model.

It can be concluded from Table XII that the calculated uncertain

ties agree reasonably well for zone 3. There are substantial differ

ences for the results in zone 1. The reason for those differences is

probably related with the fact that the one-dimensional model is not

adequate for calculating the overall heating in the TF coil (especially

129

TABLE XIX: COMPARISON BETWEEN THE X 's CALCULATED FROM ANGULAR FLUXES (UPPER PART) AND THE X ' s RESULTING FROM FLUX MOMENTS CLOV.'ER PART) FOR REGION 11 (SS316)a

• • • TEST • ' • • INTDUT FOX T H t C H I ' S • • • • • • K « 1 * * *

0. .13195E-05 . 1 7 9 ^ 7 E - 0 6 .67647E-07 .86778E-07 , 7 9 C 7 1 E - 0 7 .21326E-06 . I9166E-06 .3217SE-06 .40693E-06 . 4 8 7 9 7 E - 0 6 . l ' ? t 9 E - 0 * .2236IE-05 .14093E-05 .16633E-U6 . 3 7 7 9 5 E - 0 6 .10786E-06 . 1 7 6 0 7 E - 0 7 .e«86eE-08 .68189E-08 .c,ee76E-U8 .17886E-08 .12639E-08 .54i45E-09 .49433E-09 .264076-09 .12 I22E-U* .4&579E-10 .16319E-10 . 4 ? 2 9 0 E - 1 1 .2438..E-10 .71906E-10 .12091E-U9 .b6533E-10 . 4 9 7 4 ? E - 1 0 .2te69E-10 .13096E-10 .22933E-11 .14850E-12 . C £ 9 5 3 E - 1 6 - . 5 ie97E-23 - .28794E-46

• • • * • • C H I ' S G E N E M T E C rmOM F L U * M D H E H T S * * * » * »

• • • TEST PP1MTOUT F D " THE C H I ' S • • •

0. . ] 0 5 T ; 5 E - 0 5 .1700JE-Ub .59837E-07 .61854E-07 .78798E-07 .21318E-06 . I91b4E-06 .32167E-Ub .40b69E-06 .46798E-06 .1S592E-05 .22366E-05 . MO^E-OS .16630E-Ub .37797E-06 .10766E-06 . 17605E-G7 .86885E-08 .68187E-08 .28273E-08 .17886E-08 .12636E-08 .54232E-09 -49424e-09 .26401E-09 . I c l l B i - W .48553E-10 .16304E-10 .4?171e - l l -24438E-10 .71918E-10 . 1 2 0 6 6 E - 0 9 .b670"E-10 .49849E-10 .26296E-10 .13135E-J0 .2904&E-11 .18631E-12 .£6240E-lfc - .52915E-23 - . 4 5 U 6 E - 4 6

a The % 's are ordered by group (high neutron energy to low

neutron energy; high ganma-ray energy to low gamir.a-ray energy)

130

TABLE XX: PREDICTED UNCERTAINTIES (STANDARD DEVIATION) DUE TO ESTIMATED SED AND CROSS-SECTION UNCERTAINTIES IN ZONES 1 AND 3 FOR THE HEATING IN THE TF-COIL

Cross Section

Material Zone

Cr

Fe

Ni


L JMat,zone L JMat 1-D

0 .1

0 .6

0.8

2.6

0.0

0.2

2-D

0.1

0 .7

0 .5

3 .1

0.0

0.3

XS U n c e r t a i n t i e s i n %

AR~] R i

Mat ,zone "- JMat 1-D

2 9 . 3

4 4 . 8

4 . 5

9 .6

8 .3

13.1

2-D

12.0

4 2 . 5

2 . 3

11 .3

3 .6

12 .8

131

the source region is poorly simulated in the one-dimensional representa

tion) . A more relevant sensitivity analysis would be to consider the

heating calculated at the hottest spot in the TF coil. The hottest spot

is in the center plane of the toroid. We would expect that the one-

dimensional model would be an adequate representation in this case.

132

7. CONCLUSIONS AND RECOMMENDATIONS

Expressions for a two-dimensional SED (secondary energy distribu

tion) ?:id cross-section sensitivity and uncertainty analysis were de-

v?ir<red. This Lncury was implemented by developing a two-dimensional

sensitivity and uncertainty analysis code SENSIT-2D. SENSIT-2D has a

design capability.' and has the option to calculate sensitivities and

uncertainties with respect to the response function itself. A rigorous

comparison between a one-dimensional and a two-dimensional analysis for

a problem which is one-dimensional from the neutronics point of view,

indicates that SENSIT-2D performs as intended. Algorithms for calculat

ing the angular source distribution sensitivity and secondary angular

distribution sensitivity and uncertainty are explained.

The analysis of the FED (fusion engineering device) inboard shield

indicates that, although the calculated uncertainties in the 2-D model

are of the same order of magnitude as those resulting from the 1-D model,

there might be severe differences. This does not necessarily imply that

the overall conclusions from a 1-D study would not be valuable. The

more complex the geometry, the more compulsory a 2-D analysis becomes.

133

The most serious source of discrepancies between a 1-D and a 2-D

study are related to the difficulty of describing a complex geometry

adequately in a one-dimensional model. However, several neutronics

related aspects might introduce differences. The use of different quad

rature sets - especially when streaming might be involved - could lead

to different results. When the angular fluxes have a pronounced peaked

behavior, the angular flux option for calculating the loss term of the

sensitivity profile will provide a better answer than the flux moment

option. The different sweeping algorithms and code characteristics used

by the 1-D and 2-D transport codes might be another cause of discrep

ancies in the results. Needless to say, a meaningful transport calcu

lation is compulsory in order to obtain reliable results from a sensi

tivity and uncertainty analysis.

The results from the FED study suggest th3t the SED uncertainties

tend to be smaller than those generated by cross-section uncertainties.

45

It has been pointed out that, because all secondary particle produc

tion processes for a particular element are presently treated as one

single process, the simplicity of the hot-cold concept for SED sensi

tivity might mask several causes of a larger uncertainty than calculated

by SENSIT or SENSIT-2D. A more elaborate algorithm for a SED analysis,

as an alternative to the hot-cold concept, a separate treatment for the

various particle production processes involved, or a combination of

both, would eliminate this deficiency. Even with the hot-cold model,

which might underestimate SED uncertainties, the SEDs might become the

dominant cause of the calculated uncertainty in the case that the

134

response function is a threshold reaction or in the case that backscat-

tering becomes important. In this latter situation, an SAD (secondary

angular distribution) analysis might also contribute significantly to

the overall uncertainty estimate. At present, the required cross-

section data are not arranged in the proper format to do this type of

study.

Sensitivity and uncertainty analysis estimates the uncertainty to a

calculated response. It would be more meaningful to be able to imple

ment those uncertainties with a confidence level. In order to do this,

we have to know how reliable the covariance data are, what the effects

of errors resulting from the transport calculations will be, and what

the limits of first-order perturbation theory are. It was assumed in

this study that the uncertainties, resulting from uncertainties in

different regions, were either fully correlated or not correlated at

all, depending on whether these regions have the same or a different

material constituency. The evaluation of reliable correlation coeffi

cients would be a major effort by itself.

The validity of an uncertainty analysis is often limited more due

to the lack of the proper cross-section covariance data, than due to the

lack of representative mathematical formalisms. Covariance data for

33 several materials are still missing, or just guesstimates (e.g., Cu)

The fractional uncertainties required for an SED analysis are evaluated

for just a few materials and are not available for the various indi

vidual particle production processes.

135

The current version of SENSIT-2D cannot yet access all the covari-

33

ance data available in COVFILS, but will be able to do so in the

future. Even when SENSIT-2D does not require a lot of computing time,

the extra amount of c.p.u. time required by the adjoint TRIDENT-CTR run

makes a two-dimensional sensitivity and uncertainty analysis demanding

when it comes to computer resources. The development and implementation

of acceleration methods for TRIDENT-CTR are therefore desirable. A

sensitivity analysis involves a tremendous amount of data management. A

mechanization of the various steps required, by the development of an

interactive systems code, would provide a more elegant procedure for

sensitivity and uncertainty analysis.

The algorithms to perform a higher order sensitivity analysis have

been developed, but are still too complicated to be built into a com

puter program for general applicability. The increasing number of

transport equations to be solved prohibits the incorporation of present

higher order sensitivity schemes in a two-dimensional code. An effort

to develop simple algorithms for higher order sensitivity can certainly

be justified, however.

It becomes obvious that several flaws can be found in the state of

the art of sensitivity and uncertainty analysis. Removing any one of

them would require a major commitment.

136

ACKNOWLEDGEMENTS

The author would like to thank the chairman of his committee, Dr.

J. R. Thomas, ior the guidance received during this pursuit for a higher

education, the other members of his committee for their interest in this

work and their contribution to its completion, and Dr. D. J. Dudziak

from Los Alamos National Laboratory for his daily supervision and invalu

able advice.

The author is obliged to Dr. T. J. Seed, Dr. S. A. W. Gerstl, Dr.

V. T. Urban, Dr. E. W. Larson, Mr. D. R. Marr, Mr. F. W. Brinkley, Jr.,

and Dr. D. R. McCoy for their assistance with Los Alamos codes, and the

various discussions that were essential for the completion of this work.

Thanks go to Dr. C. S. Caldwell and Dr. W. G. Pettus from Babcock and

Wilcox for pointing out the need for sensitivity analysis and their con

tinued interest.

The encouragement received from his parents, Professor V. Geysen,

Professor and Mrs. W. Glasser, Mr. and Mrs. C. Kukielka, Mr. R. Florian,

Mr. and Mrs. M. Goetschalckx, and Mr. and Mrs. M. Denton, is highly

appreciated.

The author acknowledges Professor D. P. Roselle, the Dean of the

graduate School at Virginia Polytechnic Institute, for the Pratt Presi

dential fellowship, Dr. D. I. Walker from Associated Western Universi

ties, for the A. W. U. fellowship, which made his stay at the Los Alamos

National Laboratory possible, and the Los Alamos National Laboratory for

the use of their facilities. Special funding of the AWU fellowship by

the DOE Office of Fusion Energy is gratefully acknowledged. Thanks go

to Mrs. Mary Plehn, who prepared the manuscript.

137

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50. A. DUBI and D. J. DUDZ1AK, "High-Order Flux Perturbations," Nuclear Science and Engineering, 7_7, 153 (1981).

51. A. DUBI and D. J. DUDZIAK, "High-Order Terms in Sensitivity Analysis via a Differential Approach," Los Alamos Scientific Laboratory report LA-UR-79-1903 (1979).

52. J. C. EST10T, G. PALMIOTTI, and M. SALVATORES, "SAMPO: Un Systeme de Code pour les Analyses de Sensibilite et de Perturbation a Differents Ordres d'Approximation," to be published. Private communication with R. ROUSSIN.

53. T. J. SEED, W. F. MILLER, Jr., and F. W. BRINKLEY, Jr., "TRIDENT: A Two-Dimensional Triangular Mesh Discrete Ordinates Explicit Neutron Transport Code," Los Alamos Scientific Laboratory report LA-6735-M (1977).

142

54. Y. SEKI, R. T. SANTORO, E. M. OBLOW, and J. L. LUCIUS, "Comparison of One- and Two-Dimensional Cross-Section Sensitivity Calculations for a Fusion Reactor Shielding Experiment," Nuclear Science and Engineering, 73, 87 (1980).

55. A. GANDINI, " Method of Correlation of Burnup Measurements for Physics Prediction of Fast Power-Reactor Life," Nuclear Science and Engineering, 38, 1 (1969).

56. A. GANDINI, "Implicit and Explicit Higher Order Perturbation Methods for Nuclear Reactor Analysis, Nuclear Science and Engineering, 67, 347 (1978).

57. A. GANDINI, "High-Order Time-Dependent Generalized Perturbation Theory," Nuclear Science and Engineering, 67, 347 (1978).

58. A. GANDINI, "Comments on Higher Order Perturbation Theory," Nuclear Science and Engineering, 7_3, 293 (1980).

59. D. G. CACUCI, C. F. WEBER, E. M. OBLOW, and H. MARABLE, "Sensitivity Theory for General Systems of Nonlinear Equations," Nuclear Science and Engineering, IJ^, 88 (1980).

60. A. GANDINI, "Generalized Perturbation Theory for Nonlinear Systems from the Importance Conservation Principle," Nuclear Science and Engineering, 77, 316 (1981).

61. E. GREENSPAN, D. GILAI, and E. M. OBLOW, "Generalized Perturbation Theory for Source Driven Systems," Nuclear Science and Engineering, 68, 1 (1978).

62. G. I. BELL and S. GLASSTONE, Nuclear Reactor Theory, p. 608, Van Nostrand Reinhold Company (1970).

63. S. GERSTL, private communication.

64. R. E. MacFARLANE, R. J. BARRET, D. W. MUIR, and R. M. BOICOURT, "The NJOY Nuclear Data Processing System: User's Manual," Los Alamos Scientific Laboratory report LA-7584-M (1978).

65. R. E. MacFARLANE and R. M. BOICOURT, "NJOY: A Neutron and Photon Cross-Section Processing System," Transactions American Nuclear Society, 221, 720 (1975).

66. C. I. BAXMAN and P. G. YOUNG (Eds.), "Applied Nuclear Date "Research and Development, April 1-June 30, 1977," Los Alamos Scientific Laboratory report LA-6971-PR (1977).

143

67. W. C. HAMILTON, Statistics in Physical Science, p. 31, Ronald Press, New York (1964)."

68. B. G. CARLSON, "Tables of Symmetric Equal Weight Quadrature EQ Over the Unit Sphere," Los Alamos Scientific Laboratory report LA-4734 (1971).

69. L. C. JUST, H. HENRYSON II, A. S. KENNEDY, S. D. SPARCK, B. J. TOPPEL, and P. M. WALKER, "The System Aspects and Interface Data Sets of the Argonne Reactor Computation (ARC) System," Argonne National Laboratory report ANL-7711 (1971).

70. T. J. SEED, private communication (June 1981).

71. M. J. EMBRECHTS, "SENSIT-2D: A Two-Dimensional Cross-Section and Design Sensitivity and Uncertainty Analysis Code," Los Alamos National Laboratory report to be published.

72. R. E. MacFARL, -E and R. J. BARRETT, "TRANSX," Los Alamos Scientific Laboratory memorandum T-2-L-2923 to Distribution, August 24, 1978.

73. F. W. BRINTCLEY, "M1XIT," Los Alamos Scientific Laboratory internal memo (1980).

74. R. D. O'DELL, F. W. BRINKELY, Jr., and D. R. MARR, "User's Manual for ONEDANT: A Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport," Los Alamos Scientific Laboratory internal report (October 1980).

75. T. R. HILL, "ONETRAN: A Discrete Ordinates Finite Element Code for the Solution of the One-Dimensional Multigroup Transport Equation," Los Alamos Scientific Laboratory report LA-5990-MS (1975).

76. D. J. DUDZIAK, "Transport and Reactor Theory, July 1-September 30, 1981, Progress Report," Los Alamos National Laboratory report, to be published. S. A. W. GERSTL, private communication.

77. T. WU and C. W. MAYNARD, "Sensitivity and Uncertainty Analysis for a Fusion-Fission Hybrid Systems: SOLASE-H," University of Wisconsin report UWFDM-401 (1981).

78. T. WU and C. W. MAYNARD, "Higher-Order Sensitivity Theory and Nonlinear Optimization in Fusion Neutronics Studies," University of Wisconsin report UWFDM-402 (1981).

144

79. T. E. ALBERT, "Introduction of the Green's Function in Cross-Section Sensitivity Analysis, Trans. Am. Nucl. Soc. J_7, 547 (3973.).

80. H. GOLDSTEIN, "A Survey of Cross-Section Sensitivity Analysis as Applied to Radiation Shielding," 5th Int. Conf. on Reactor Shielding, Science Press (1977).

81. D. W. McCOY, Private communication. See also "Transport and Reactor Theory, July ]-September 30, 1981," Los Alamos National Laboratory report, to be published.

82. B. A. ENGHOLM, The General Atomic Co., Letter to W. T. I';ban, 12 January 1981.

83. W. T. URBAN, Los Alamos National Laboratory, Letter to B. A. Engholm, 17 February 1981.

84. W. T. URBAN, Los Alamos National Laboratory report to be published, 1981.

85. W. T. URBAN, T. J. SEED, D. J. DUDZIAK, "Nucleonics .Analysis of the ETF Neutral-Beam-Injector Duct and Vacuum-Pumping-Duct Shields," Los Alamos National Laboratory report LA-8830-MS, 1981.

145

APPENDIX A

SENSIT-2D SOURCE CODE LISTING

In this appendix a source listing of the SENSIT-2D code is repro

duced. The source listing is documented by many comments.

A source listing of the SENSIT-2D code can also be obtained from

the tWTECC by typing the command

FILEM$READ 5043 .SENS2D SSSS$END

146

Los Alamos I d e n t i f i c a t i o n No. LP-1390.

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3041 f»Enp ( N i m ] 0> !&• BCNlf BCM£ 304£ I 0 F O - N P T ( i f c , 6 « . £ c l £ . S > 3043 CALL c a v w i ( t r . 0 . f » « l > i ' » * c . c i K - t ICP>1> 3044 I « < I T C S T . M C . 3 J SO TO 4b 3045 N P I T E ( N D U T I E O ) MCPVMAT! 3 0 4 6 £ 0 r D W t » T U H » • NULT ISPCUP C O L ' A A I A M C C giATA I N * | 3 i 3047 1 • » rau»s POP. MATS « * i 4 J 3 0 4 6 N P I T E ( M D U T i S O ) N T | 3 0 4 9 5 0 F D I M T l l " • • N I C A O S C D P I C C P O t l S C C T I O N S PDA A T I « »t3t 3 0 5 0 M P I T C ( N O L i T » 3 0 ) l l / » S l tN> I M C l l M i * ) 3 0 5 1 N P I T E I M O U T . 6 0 > W T £ 3 0 5 £ 6 0 r D M A T i I H • • M I C R O S C O P I C C A O S S H C T I O N S P O A N T £ P A I 3 J 3 0 5 3 HPJTC ( M D U T f 3 0 J ( i / | r S £ l * ) ) l N ' l f H t » > 3 0 5 4 N P I T E l N D U T l ? 0 > N A T l i H l l v H T c f 3 0 5 5 7 0 F O N M A T l l M ? • A C L A T | I / C COfARIMNCC A M T A I V - NAT 1 • * l 4 t 3056 1 • H T 1 « » i 3 . » « T £ « *l~J> 3057 N P I T C ( ~ D u ' . 3 C x ( c o i ' i n j ) i j " l i > * » ) i i » l i » « » ) 3 0 5 6 N P I T C I N D U T I B O ) N P T l l N T l f N T £ 3 0 5 9 6 0 F D A N A T O N t * ABSDLUTC COfAAIAMCC N A T A I k r i w n | C a O I C D * I C * t 3 0 6 0 1 • CROSS SCCTIONS OP N A T l " » l 4 i 3061 1 • m l " * I 3 I * N T £ • »|3> 306£ N P I T C ( M D u T i J D X C c t e d i j l i i ' l i H t r J i i ' l i m M 3063 3 ( f O « « M 3064 40 CO»TI«uf 3065 C • • • 1»*KS«P« HICAO MS INTO NACAOSCOPIC Kl

3 0 6 6 s o 9 0 N ' l i m l 3 0 6 7 f x s l ( N J P DEMI • M i l W 3 0 6 6 f » s £ ( N j • H N C • f * s £ ( N J 3069 9t c o x i l N u i 3070 AETUAN 3071 CMS 307£ c 3073 c 3074 c 3075 c 3076 c 3 0 7 7 SUBROUTINE CDv-APB i ^ F i i * t D M i » I A i l f | i n I i l l « l ) 3 0 7 6 C PQUTINE A f i A I CDfc'APlANCC l A T A I N i M t f - ^ I f C r p « l l « T DU?»UT * T 3 0 7 9 C NJOr MMt T A A N S P D A N S I T TO T ~ l r D f t M T . 3 0 6 0 e N P . * T - l i H » t i r i t > 3 0 6 1 C NDN - M I . C O t . P L A C . « 0 I T C S • J . N O . 306c c 30E3 LlflL £• ccl

3064 BINENEION KS« ' 1> iKSB «U iCCl llCnl • 1> 3 0 8 5 C D - N D * E M L * ' M * T l M r i N T f N & T l J A a . n P T ] . M t ] m P £ • NT 1 • NTC t H * T c • MOUT t N 3066 co>«»D«.'«TN> .-co~<50>50>iccci5li-MiJ • > s l (£00>> 3067 1 xsc I £ 0 0 J •mmviiOCi I X I O J I M J > 3066 c 3069 N P . P N . P 3090 j » » » 3 0 9 1 CALL S E T I B 3092 c 3093 C SETIB SETS LP* INDCvCE TO SET BESIACB NA» SCT.

3094 c

3 0 9 5 C T P » L E T N X i C ' I N I T I O N DT 1S-NOS I N TEAMS OP I M C I " ICAT I ON C*P 3 0 9 6 C CNDSS S E C T I O N C O L A A I A P * C C S . H D T | | N T N I S f l « l lONt t » T 1 P N A T £ 3097 c 3 0 9 6 C I D - N O PMtTl NATE N T ] MTC CADSS S C C T I D N C D v N U V M C C

3100 c 1 305 305 1 1 I 1 U T D T H L H I T N B 1 0 T O T A L 3101 c E 305 305 1 i * I 0 T O ' N . N I T H »10 I L O I T I C 310c c 3 305 305 1 107 a l b T O T A L N I T N »10 I N . A L A M ) 3103 c 4 305 305 £ £ BlU C L A S T I C N I T N B I C I L M T I C 3104 c 5 305 305 £ 107 B i b C L A S T I C N I T N B 1 0 I > , N . » — ' 3105 c 6 305 305 107 107 B i b > • . .« .> •« ) N I T N B ] 0 I N > A L P _ P 3106 c 7 306 306 I 1 c T D I » L N I T N c TOTAL 3107 c 6 306 306 1 £ c T O T A L M I T N C C L A S T I C 3106 c 9 306 306 E £ c C L A S T I C N I T N C E 4 . N E T I C 3109 c 10 306 306 4 4 c I N E L A S T I C N I T N C I N E L A S T I C 3110 c I I 306 306 107 107 c i m n i H i x N I T N C « . . « L N A ) 3111 C 1£ 3£4 3£4 1 1 CA TOTAL NITN CA TOTAL

311c c 13 3£4 3£4 1 £ C A T O T A L N I T N C A C L A S T I C

3113 c 14 3£4 3£4 £ c C A C L A S T I C N I T N C A C L A S T I C

3114 c 15 3£4 3E4 E 4 C A C L A S T I C N I T N C A I N E L A S T I C

3115 c 16 3£4 3£4 4 4 C A I N E L A S T I C N I T N C A I N E L A S T I C

3116 c 17 3£4 3£4 4 1 be C A I N E L A S T I C N I T N C A CAPTLN»C

3117 c 16 3£4 3£4 10£ 1 ue CA C A P T U A C N I T N CA C A P T U M

3116 c 19 3£6 3£6 1 1 n T O T A L N I T N * C TOTAL

3119 c £0 3£6 3£6 I £ PC T O T A L N I T N PC C L A S T I C

31c0 c £1 3£6 3£6 1 1 be PC T O T A L N I T N P E C N P T U A C

185

http://E4.net

3121 c ss 326 3£6 S £ re I L M I I C HITH re CLASTIC 3122 c 23 3£6 326 £ 4 re CLASTIC KITH re IN I L M I I C 3123 c 24 326 326 8 102 re I L K T I C «ITN re C » ' U H 312* c S3 3£6 3£6 4 4 re lanafiic W T H re IHBLASTIC 3123 c £6 326 326 4 102 re i«it«mc un« re CA 3126 c ~ " " " 3127 c 3126

re iNikRiTic HITH re re IMCLASTIC KITH re INI*L»« re CATTU»C HITH re C A T T U M

3125, c 30 3£6 326 103 103 re (H.T) HITH re inir) 3130 c 31 3?6 326 107 107 re I«II»I.»M«J MIT» re I m M f n w 3131 e 32 326 326 1 1 HI I D W L «IT« HI TOTAL 3132 C 33 326 326 2 2 HI CLASTIC KITH HI CLASTIC 3133 c 34 326 326 4 4 HI I M L W T I C HIT- HI IHSLASTIC 3134 c 35 326 326 102 102 HI C » " I « HITH HI CArTur* 3135 c 36 326 326 103 103 HI (H.A) HITH HI (n.r) 3136 c 37 329 329 1 1 cu TOTAL HITH CU TOTAL 3137 c 36 329 329 1 2 cu TOTAL HITH CU CLASTIC 3136 c 39 329 329 £ e cu CLASTIC HITH CU CLASTIC 3139 c 40 329 329 £ 4 Cu ILOITIC HITH CU IMCLASTIC 3140 c 41 329 329 4 4 cu l«i.»nic HITH CU IHCLASTIC 3141 c 42 329 329 4 102 Cu IMCLASTIC HITH CU CAATUAA. 3142 c 43 329 329 4 1U3 cu I N I L M T I C HITM CU CH.T) 3143 c 44 329 329 4 107 cu IHBLASTIC HITH CU iH.MLrHA) 3144 c 45 329 329 102 102 cu CAATUAB HITH CU CAATUAB 3145 c 46 329 329 103 103 Cu (H.A) HITH CU IH.TJ 3146 c 47 329 329 107 107 Cu lNi«ir.«u HITH eu 'H.ALAHA) 3147 c 46 362 362 1 1 ra TO»»L HITH AS TOTAL 3146 c 49 362 382 1 2 »» TOTAL HITM AB CLASTIC 3149 c 50 362 362 1 102 ri TOTAL HITH AB CAATIAHJ 3150 c 51 362 362 £ 2 »» I L M I I C HITH ra IL»IIIC 3151 c 32 362 362 £ 4 ra I L M T I C HITH ra INILOIIIC 3152 c 53 362 362 4 4 ra IMCLASTIC HITH ra IHCLASTIC 3153 c 54 362 362 4 102 ra I M I L U T I C HITH ra CAATUAC 3154 c 55 362 362 102 10c ra c w i u « HITH ra CATTUHC 3155 c 56 1301 1301 1 I n TOTHL HITH H TOTH. 3156 c 57 1301 1301 1 £ H TO'M. HITH H ILHITIC 3157 c 56 1301 1301 £ e n I L M T I C HITM H CLASTIC 3156 C 3139 C HFl-3iHr£»33 3160 c H T 1 " M T - H O ram n i H i - l * H T f « N T H D ADA * ! • • * » - £ . 3161 c 3162 n r o r l 3163 H T A » 4 5 I 3164 ••Mint HB£ 3165 c • ! » ! . SHOUT i T a u C T u w 3166 10 ABAB t«Bc*£0) i A i i > i i « l < 7 j t M T t M r i i i i i H l l l 3167 20 roHMHT ( 6 o l 0 . H 6 . l 4 . l 2 . l 3 . i l > 3166 ir ( m t . t T . 1301) MAT»MAT-1000 3169 i r I H A T . C B . H A T 1 ) mo T D 30 3170 i r ( H A T . L T . H M T 1 ) ao TO 10 3171 HHITB ( H O U T » 4 0 > MAT 1 f NBTltMT 317£ STOA 3173 30 COHTiHU* 3174 40 AOAMAT ( I M O I * I D H l l H I r i i T I B HAT • * l 4 t * HOT DM T H * | * l 3 . 3175 1 • LAST HAT d l n l HAS A|4 ) 3176 A I A B ( M B 2 « 3 0 ) C 1 I C C I L I > L £ I L ^ . L 4 I H H T . H * I H T . H | » 3177 50 roAHAT (eel1.4.4ill.l4.ic.13.IS) 3176 ir iM.ii.»a> ao TD 70 3179 HA1TK (HDUT»60) HBTtHTlHT 3180 60 rOAMATdH » • SOAATtTAAC * I 3 . « SCABu«a UA HT««|3 l • MT»*I4J 3181 iTor 3162 70 ir (HT.ca.riTH> ae TO 60 3 1 8 3 H A I T B ( H O U T F 6 0 > N B T H A I K T 3184 STDT 3185 80 coHTiMua 3166 H»»-L1 3187 HBB'4.3 3166 ABAB (HB£I90) («aa(i> iirlfHaa) 3189 90 roAHHT (6B11.4) 3190 c ABAS HIBC ror HTI M B HT£ 3191 100 >CAB IMB£«£0) < A ( D . 1-1.7) .MAT.H».HT.H»B» 3192 ir (nT.LT.Ml) ao TO 100 3193 ir (Ht.n.-rl) «o TO 110 3194 HAITC (HOUT»60) HBT»HT»HT 3195 BTOA 3196 110 coHTiMua 3197 ir (MT.LT.HT1) SO TO 100 3196 ir (HT.aa.HTl) ao TO 120 3199 HAITI (HOUT«60) HBT.MA.MT 3200 STDA

186

http://6ol0.H6.l4.l2.l3.il

http://ihat.cb.hat1

http://hat.lt.hmt1

http://HT.ca.riTH

http://mT.LT.HT1

3201 120 C O H T I H U C 3202 « • » > H B £ > 9 0 ) u s l I I > U < 1 I B V ) 3203 i» ( H T 1 . N C . H T 2 ) CO TO 130 3204 BO 125 1*1.««• 3205 * « 2 ' i > « » » l n > 3206 125 C I M T I X K 3207 60 TO ISO 3208 130 PCH» (MB?f£0> < A ( l } i | * l i 7 J i M A T i M t » M T i H f f l 3209 I* MT.I.Y.MT2) so TO 130 3210 IF (HT.CV.HT2J «o TO 140 3211 M»ITC IHOUT«60? NPTlMTfHT 3212 140 C P H T I N U C 3213 >(•>» i « c c . 9 0 ) 1»«£<|> r i « I •«• •»; 3214 ISO C O H T I H U C 3215 so l » K ' I O W 3216 no 155 H'ltNCr 3217 COH(OIM>'0. 3216 C«1<.*.H»-0. 3219 C«2C«IH>"0. 3220 155 cpNTiNut 3221 c •«»!. COK. P»T». 3222 U t H m I N I > T I 2 0 > t « U ) > i « l •?) I H H T I H V . H T I H C C C 3223 i r w r . L T . w J ) so TO 160 3224 ir W T . a l . m i l CO TO 160 3225 I ' I H ' . C C . H » 2 > CO TO 170 3226 HHITK < H D U T I 6 G J NSTlMTfUT 3227 CTOP 322B 170 C O H T I H U C 3229 » • * » C H B T I S O ) c l I C 2 I L 1 . L C . L J . L < . « « T — . m m i i i 3230 I» I « T . L T . « T | ) ce TO 160 3231 I * i m . w . p T ] ) cc TO 160 3 2 3 2 WM1TC < H O U T L 6 0 > M T I M T I H T 3233 » T D » 3234 160 C O N T I H U C 3235 H T » « I . 2 3236 •o»»i.« 3237 C lJNO.'e6-'"PQL4.QKlN« Tw»H L l > 4 1 1 * > & I » T I E Af » ( • K*Hpai FOUMl »* 323B c sec L I T T M t * T i i 13t-Dv80 « n t M C ' I B C N C C T - £ - i . - 3 6 4 5 . 323% BO 250 ~ - 1 . H * » 32*0 BD £50 t -al iMP 3241 250 C D * » 3£43 HKA£(HCTl50' CllCSitllLClLJoL* 3£4« u6»l"i-2 3245 i.c»2«i-£-»i.3-l 3246 H O M D » L 4 3247 «L«I-4 3246 NCMZ. lHBT»90> (COHlKLlL)lL>4.4*l»t.M>c; 3249 I* iNCHD.ii.iit*) co TO 200 3250 190 C O N T I N U E 3251 £00 ir I B T » . L T . H T £ J cp T O 170 3252 l» l«..i«.nT£j so TO £10 3253 MMITC <HOuTt60J HPTfHTlHT 3254 | T O > 3255 210 C O H T I H U C 3256 CO £3t K*lt*HV-3257 » • . » • » - * > • 1 3256 K f k t M ^ j « i K ] ( K J 3259 » l | [ « » l ' i » c ^ ) 3260 BO ££0 H - 1 I M » 3261 H H * H S * - H * 1 3262 CEl ( K P I M N ' « C » I > I H ) 3263 £2 0 C O H T I H U C 2 f64 £30 C D H T I H U C 3265 i r ( H O H . S T . C J W T U H . 3266 BO £4 0 | " 1 > M > 3267 BO 24 0 H - 1 . H C « 3266 etc <•• » H J " C C 1 (^iN)*Fi»m)»xll IN> 3269 24 0 CONTIHUC 3270 HCTUMW 3271 CHB 3272 c 3273 c 3274 e 3275 c 3£76 c 3277 B U C H O U T I H C C C T X B

3276 e 3279 C IUMOUTINC S t T I COWCT HHT»HFpHT C l fCH MMa 3260 c

187

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3281 coNnoH'mHO*'"*" ttw »HT •«»*>J"«i"«Tlt"»l ••»£••"!••«£ •»«TC:I ««uT!».»e

3282 c 3£83 «•••.!•» 3284 nrl-=3 • HF2«33 S M » M > « 6 3285 1* vnni.sT.6) «e TO £0 3266 »»Tl«305 328? ir <«»..ttt.3) «0 TD 10 3288 MT]«1 • *T2"1 3289 IF v»»«.«».e> »IT2«2 3290 I' im-.M. 31 HTS'107 3291 M T U W . 3292 10 CflNTiNut 3293 H T 1 * 2 S mtS'S 3294 I F ( n * > . M . S > W T 2 " 1 0 7 3295 IF ( " » » . « • . 6 > H T 1 » 1 0 7 3296 « T U » » « 3297 20 c o « T | n m 3298 1 * < n » . . « T . l l > «o TO 40 3299 «•»•» I "306 3300 IF I - » . . « T . 0 8 ) «o TO 30 3301 n T l » l » « T £ " 1 330S • • < » • • . » • . 6> F«T2«2 3303 wcrumM 3304 30 cDNTiuuC 3305 H T 1 « £ t « « T 5 « 2 3306 I F < « » . . « » . 10J » T l « « 3307 I F W ' . l l , 10-* t»T8«4 3306 I F <•>»•.IB. 1 U «T l«107 3309 IF < * » . . • • . 11> * T C » I 0 7 3310 • I T I ^ X 3311 4b E O M T I V U C 3315 I F I P W - . S T . 1 8 J co TO 70 3313 M».T1«324 3314 I F > « - . . • T . 1 3 ) so TO SO 3215 » T l « l t » T 2 - 1 3316 I F ! • * » . . « • . 13> X T £ « S 3317 W T u f c 3318 50 cO~Tl«ua 3319 I F I M I P . S T . 1 S J «o To 60 3320 P>T1»2 S »rr2»2 3321 1« ip«»i.Ba. 15> F * T 2 * 4 3322 MTiBm 3323 60 C O N T I N U I 2324 «T I»4 t H T 2 « 4 3325 I F < • * » « . « . 17> - T 2 « 1 0 2 3326 IF <- *»« . • • .18J P*T1»1C2 3327 H T u a N 3328 70 COMTIMUC 3329 ir <•••..ST.31> so To 110 3330 »«»Tl«326 3331 IF • •••».ST.£IJ CD To 80 333c «Tl«l t WT2«1 3333 IF (MB».».20) WT?«£ 3334 I F I « » . . H . J I J F I T 2 > 1 0 2 3335 I I T U N M 3336 80 C D X T I N U I 3337 I F I P . » . . « T . 2 4 > so TO 90 3338 F IT1«C t « T 2 « 2 3339 IF < » • . . « • . 2 3 . 1 F I T 2 > 4 334 0 I F (Mii i i .B*.£4> H T 2 B 1 0 £ 3341 K T I M N 3342 90 COuTtuuc 3343 IF (nSa.ST.28> SO TO 100 3344 « ? l - 4 f - T C . 4 3345 I F ( « • • « . K » . £ 6 J F * T 2 « 1 0 2 3346 IF ( > • > • • • . 2 7 > "T2 -103 3347 IF < « . . « « . 2 8 > XT2.1C7 3348 P f T u w 3349 100 ciMTimx 3350 M T 1 « 1 0 2 f M T 2 - 1 0 S 3351 I F <•.»».«». 30^ H T | « I 0 3 3352 I F < •< • . .«» .30 ) « T 2 - 1 0 3 3253 I F ( m i . i t . 3 D «T2«107 3354 IF ( M i i . | | . 3 | ) HT1.J07 3355 « T U M 3356 110 CDMTIMUC 3357 ir <•.».. ST. 36) so TO 120 3358 M«T]«326 3359 HTI-1 ( nTc.l 3360 IF <•••«.••. 33> HT1>2

188

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IF < M P N . C V . 3 3 > H T £ B £ 336E ir <nm..»c.34> P T C - 4 3363 I F M * P - . C C . 3 4 > « T l » 4 3364 I F ( M P » . C C . 3 5 > K T I « I O £ 3365 I F I K . . . m . 3 D M T £ « I O £ 3366 IF i n n • . c p . 3 6 J H T £ K 1 0 3 3367 I F tn»M.ca.36> »-r l"103 3366 K T U M 3369 1£0 cONTlMjf 3370 I F « . *<«• .CT.47J CD TO 16b 3371 H»T |«329 3372 I F < » » . . C T . 3 6 > co TO 130 3373 MT1«1 t MT£«1 3374 I F l n p i . ( c . 3 6 > M T £ P £ 3375 a i T u m 3376 130 cONTimjt 3377 IF C«a„.CT.40> CO TO 140 3376 f . T l - 2 I P T £ « £ 3379 I F l«»« .»B.40> « T £ » 4 3360 H T U M 33S1 140 C O N T I N U C 336E IF M P ' . C T . 4 4 > CD TO ISO 3383 X T 1 ' 4 t H T i M 3364 I F ( - » . . ! • . 4 c ) W T £ " 1 0 £ 3385 I F I H B . . « C . 4 3 > « T £ - 1 0 3 3366 i r < - » . . • » . 4 4 ; P < T £ « 1 0 7 3367 N T U M 3366 ISO C D M T I M U C 3389 nT l«10c t X T C - 1 0 2 3390 I F C . . . ( • . « « ) x T l - 1 0 3 3391 I F c . . . i e . 4 * j H T £ ' 1 0 3 339£ I F i « p . . « c . 4 7 ) K T 1 > 1 0 7 3393 I F M P » . C C . 4 7 J » > T £ « 1 0 7 3394 F I T U M 3395 160 CIMTINUI 3396 I F i n v i . c T . S S ) CD TO 190 3397 p>»-tl.36£ 3396 I F I W . X _ C T . 5 0 > CO TO 170 3399 «Tl«l I » T £ » 1 3400 IF (MP. .CB.49) M T £ F £ 3401 I F < M P « . C C . 5 0 > p i£«10c 340c CETUP** 3403 170 CDNTIMUI 3404 IF VM*».CT.5£> CO TO 160 34 05 « T I » C • « T J - C 3406 IF (MBB.CC.5C') HT£"4 3407 txTucm 3406 i e 0 c t w i i N u I 34 09 H t l M f X T £ > 4 34!0 IF -»P. .E(.54> « . T £ » 1 0 £ 3411 : F ( H « « . C B . 5 5 ; KTlFlOc 3 4 1 2 CKTUMMM 3413 190 C D M T I M U C 3414 IF 1PI«..CT.56.> CD T O £00 3415 -»Tl«1301 3416 PJTI«1 t «T£-1 3417 IF r»»..ct.57) mi~2 3416 IF <-•..«•.56) «.tl»2 34 19 w u * 34c0 £00 C D N T I H U I 34£1 IF ( H l v . | T . t « r M l ) M* 1 T" <*aDUT*SlO> 34c£ 510 " D " ' * T C|x t * **»»-»*l5f« t a i A ^ f * T M « 3423 I T D * 34£4 C M

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APPENDIX B

TRDSEN

This appendix was provided by T. J. Seed and is a summary of the

changes made in TRIDENT-CTR in order to obtain angular fluxes compatible

with SENS1T-2D. In order to make a distinction between this version of

TRIDENT-CTR and the normal version, it was renamed TRDSEN.

190

First UPDATE 1 »IB SENSIT 2 • ! SEEKTUI.2 3 C SENSIT 4 COTON / S E H S V FNSEN(ZB).IHDLTM«SJ 5 C SENSIT £ »p CD2.4 7 C SEMSJT B 2I.T0H. IPXS.LTC. IPCT.LTCT.LTJS. IPFSM. IPFSrB.LTFS. IPSEN.LT5EN 9 C SENSIT

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APPENDIX B

TRDSEN

This appendix was provided by T. J. Seed and is a summary of the

changes made in TRIDENT-CTR in order to obtain angular fluxes compatible

with SENSIT-2D. In order to make a distinction between this version of

T'<]DENT-CTR and the normal version, it was renamed TRDSEN.

190

F i r s t UPDATE I 2 3 4 5 e 7

s s

le I I 12 13

u IS IE 17 ie 19 2e 21 22 23 24 25 26 27 26 29 30 31 32 33 34 35 36 37 3B 39 46 41 42 43 44 45 46 47

49 56 51 52 53 54 55 56 57 SB 59 66 61 62 63 64 65 66 67 66 69 7B 71 72 73 74 75 76 77 76 79 M

• I B SENSIT • 1 SEEKTLB.2 C SENSIT

COtrON /SENSTV FNSEN!28).IH0LTHC23) C SENS IT »D CD2.4 C SENS IT

2LT0H.IPXS.LTC.IPCT.LTtT.LTJS. IPFSH.IPFSrB.LTFS.IPSEN.LTSEN C SENS IT • I TKID6D.26 C SENS1T

DATA FNSENtHSNSTB1.6HSNSTB2.6HSNST83.6x5hSTB4.6HSNSTBS.6HSNSTB6. 1 6HSN2TB-.eHS'ISTBB.6HSNSTD9.6MSNSTlB.6HSNSTll.6HSNSTir.l.HSNSTI3. 2 6MSNSTM.6MSNSTI5.6HSNST16.6HSNSTI7.6HSNSTie.6HSN' I l;\6HSNST28 /

C SENS IT • 1 INPUT!1.B4 C SENS IT

EOUIVRLENCE ( IO(16J) .LSEH) C SENSIT »!• INPUTI1.23B C SENSIT

1H0LTNU) • 4HTSID IH3i_THC2) - 4N-SEN IHDLTH(3) • 4HSIT IH0|_TH(4) . 4HLINK 1 H 0 L T H ( 5 ) - 4H

C SENSIT *1 IHPUT11.242 C SENSIT

IFCK.NE. I ) CO TO 158 DO 155 1 • 1 . IB IH0LTH[|«5l • IDUSE(I)

155 CONTINUE 159 CONTINUE C SENSIT •D INPUT 11.682 C SENSIT

LSEN . LFL • 3 » KM • ITMBX LTLM - LSEN < 3 • ITnBX

C SENSIT •C INPJT1 I .B I7 C SENSIT

LTSEN - 3 • HTC » ITN IPSEN . IPFSnB • NCF5B » LTFS LOSTEC • IPSEN •• LTSEN « 512 I F C I T M . E O . B ) IPSEH • IPPI

C SENSIT • S 1 N P U T I 1 . S I 2 . I H P U T I I . S I 3 C SENSIT 528 F0Pr*,Tf7BH THIS COSE l * S PROCESSES BY THE TRIDENT-CTR SENSIT P

1POCESSOS OH .2X.R18) C SEHSIT • 0 INPUT]1.1124 C SENSIT 756 F0RmT("!X.37HTRIBEHT- nTR SENSIT PROCESSOR. B«TE - . «1BX> C SENSIT • I CEOCDN. 14 C SENSIT

EOUIVRLENCE ( i n c i ) . I T H ) C SENSIT • 1 CE0C0N.S9 C SENSIT

IFMTH.EO.B) P.ETURN DO 126 J • 1 . JT CBLL L R E E D l f i I L I P ) . » ( L I P C ) . P I . J . 1 . 3 . I P P 1 . J T ) infix • ] T ( J )

»o lie i • 1. imx VI • PI C I . 1) • P H 2 . i l • P I C 3 . I ) DO I I B K • 1 . 3 PICK, I ) • P M K . l ) / V I

I I B CONTINUE CfiLL L R i T E ( « I L ) P ) . f i t L I P C ) , P l . J . I . 3 . I P S E N . J T )

I2B C0NT1HUE C SENSIT J l ml GPINC2B.S«SRll<D2B.f3 C SENSIT • 1 OUTER.1» C SENSIT •CALL INSTAL C

191

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• HtSD • rues

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113 114 115 C SENSIT l i b • ! OUTER.181 117 f SENSIT 1 IB IHSD" - rc - 1) ' NQSD • 1 119 I ' - i I I 'Stx-.EC.ICDtIS) CO TO I3B 128 I F d t S W . E O . I J CO TO 137 121 C«.L FI^^UI l.F»SEN( I D S D ' - n . F N S E N t l D S I W - D . M l X 122 C C L - S P I T E I N S N S * . i T E T P . e . e . e . « . J S E N ) 123 CO-L SEEKtFNSF.N<IIISD«-]>. 1MSS.NSNST.4) 124 137 C D N T I N J E 1'5 NJ&SI - NJDS !26 IF UDSOr.EO HSDKI NLDS1 • HJBLt 127 1VECS - 1DSW 126 j p a e m 9 i . NUSSI 129 JP-PTr lB) - H S l r 138 C-''.- F lLLUCl .FNi fNf lES&ri .FHSENtlSSBtl .WUDSK 131 CC.L FILLU[2.FNSENUSSi> ) . F N S E N : I D 5 » I 0 . 8 ) 132 Cfc.i SIE> < F N 5 £ N ( I I I S D « J . I V E R S . N S N S T . I ) 133 JSEN . 8 134 Ca-L Sc:TFrNSNST . IHO. T H . B . 8 . 2 3 . l . J S E N l 135 CR^L s 6 ; n i N 5 N S T , j p « > n . e . i e . e . i . j s £ N i 136 130 CONTINUE 137 C SENSIT 13B ml OUTER.3B3.OUTER.321 139 C SENSIT 148 ml OUTER.324.OUTER.334 141 C SENSIT 142 ml OUTER.337.OUTER.379 143 C SENSIT 144 m& INNER.78 145 C SENilT 146 ml INNER.81 147 C SENSIT 148 sC IK:iEJJ.3-4 149 C SEN51T ISB »P INNER.»7.1N»C«.nS 151 C SENSIT 152 JFS • 1 153 C SENSIT 154 »D I N N E R . 2 8 1 . 1 ( * C > . £ X 7 155 C SENS IT 156 r m w HE 1 *6 . ABSORB 157 » I SLEEP. 32 158 C SENSIT 159 E0U1WUMCE [1A( I«4) .LSEN> 168 C SENSIT 1EI «t> S L E E P . M 162 C SENSIT 163 CftU. L*SNST<«F<1.2>.SiS.*<LSEM>.IT» 164 C SENSIT 165 •» SLEEP.148 166 C SENSIT !E7 CBLl t » S N S T [ 8 F ( | . 2 ) . « S . » ( L S E H ) . I T ) 166 C SENSIT

192

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177 I7B I7S IBB i e : I B ? 163 I B * IBS 1B£ IB7 IBB 169 196 191 192 153 154 ISS

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