AECL-5180

ATOMIC ENERGY K&& L'ENERGIE ATOMIQUEOF CANADA LIMITED f £ 9 DU CANADA LIMITEE

A MODEL FOR ESTIMATING FLUX DISTRIBUTIONS IN

A (THORIUM FUELLED) THERMAL REACTOR

by

M. MILGRAM

Chalk River Nuclear Laboratories

Chalk River, Ontario

January 1976

ATOMIC ENERGY UF CANADA LIMITED

A MOOtL FOR ESTIMATING FLUX DISTRIBUTIONS IN A

(THORIUM FUELLED) THERMAL REACTOR

by

M. Mil gram

Chalk River Nuclear Laboratories

Chalk River, Ontario

January 1976

AECL-51SO

Mo del e_ pour e st_inier_ _l§s_rj£ a r t i t i ons de f lux dans un reacteur

j L j J_ combu_s_tj_b1_e__es_t__du..tho_ri_umi

par

M, Milgrarc

Resume

On montre que 1'equation bien connue de la d i f f u s i o n , lsquel leest non-1 inea i re , unidimensionnelle et ne comporte qu'un groupe, estapplicable a toute une gamme de problernes d'appauvrissement et def lux , part icul ierement dans le cas d'un r^acteur thermique employanldu thorium comme combustible et pouvant Stre recharge en cours demarch? dans un ensemble coniplet de canaux. Les solut icns axialessont comparees aux solutions obtenues de facon plus riroureuse et1'on constate un bon accord, ce qui permet d'epa/gner enormement detemps de ca lcu l . On presente quelques courbes conceptuelles et desdevis de s t a b i l i t e pour le cas axial general.

L' tnergie Atomique du Canada, LimiteeLaboratoires Nucleaires de Chalk River

Chalk River, Ontario

Janvier 1976

AECL-5180

ATOMIC ENERGY OF CANADA LIMITED

A MODEL FOR ESTIMATING FLUX DISTRIBUTIONS IN A

(THORIUM FUELLED) THERMAL REACTOR

by

M. Mil gram

ABSTRACT

A well-known one-group, one-dimensional non-l inear d i f fus ionequation is shown to be applicable to a range of deplet ion and f luxdependent problems, pa r t i cu la r l y to the case of a thorium fuel ledthermal reactor wi th f u l l channel on-power f u e l l i n g . The axial solut ionsare compared wi th solutions obtained in a more rigorous manner andare found to be i n adequate agreement, with considerable savings incomputation t ime. Some design curves, and s t a b i l i t y estimates forthe general axia l case are presented.

Chalk River Nuclear Laboratories

Chalk River, Ontario

January 1976

AECL-5180

TABLE OF CONTENTS

P_afj_e_ No.

Glossary v i

1. INTRODUCTION 1

2. MOTIVATION 4

3. EFFECT OF BURNUP 7

4. DIFFUSION EQUATION 7

5. PRACTICAL CONSIDERATIONS 9

6. AXIAL LEAKAGE 12

7. STABILITY 13

8. EXAMPLE 15

9. GENERAL APPLICATIONS 26

10. SUMMARY 30

ACKNOWLEDGEMENTS 32

FOOTNOTES AND REFERENCES 33

APPENDIX A 36

APPENDIX 6 40

GL0SSARY 0 £ RECURRENT SYMBOLS

Symbol First Defined MeaningAppearance

eq. 2.2 macroscopic absorption crosssection for material M, or forbundle if subscript is deleted

maximum flux in channel

coefficient of eigenvalue expansion

coefficient of eigenvalue expansion

slope of reactivity as a function offlux

slope of reactivity as a function ofirradiation

coefficient of >j>2 term in expansionof reactivity

ratio of diffusion coefficients at interface

constant

infinite cell multiplicationconstant

constant

reactivity per unit migration area

constant

derivative of reactivity with respectto irradiation at constant f lux

A

An

Bn

C l

C2

C 3 .

D

h

ko

ko

kM

k 'oo

k

4.

7.

7.

2.

3.

5.

4.

2.

2.

5.

5.

2

3

3

14

1

3

4

11

14

1

1

eq. 7.6

7.6

5.4

5.2

5.3

page 39

eq. A.2

2.11

5.1 derivative of reactivity with respectto flux at constant irradiation

k(JJW

5 - l second derivative of reactivity withrespect to irradiation at constantflux (f"" ' "" "

VI 1

Symbol First Defined MeaningAppearance

5.1 mixed derivative of reactivity withrespect to flux then irradiation{d2kj duty)

5.1 mixed derivative of reactivity withrespect to irradiation then flux

k2

L

£

M2

m

P

P

q

r

s

t

u(x)

Vn

X

X-,

A. 9

4.2

7.6

6.1

4.3

4.4

2.1

5.7

A.I

A.I

B.3

6.1

3.1

B.IO

7.1

4.1

4.4

A.11

6.1

A. 11

A.2

A.3

B.5

B.4

B.15

7.2

4.2

constant

total channel length

normalized length of fuelled regionof channel

fractional neutron leakage

migration area

constant

atomic number density of material M

total channel power

normalized eigenvalue

function of eigenvalue

function of eigenvalue

constant

irradiation time

function of position

n excited axial mode of thenormalized flux

position per unit channel length

position of near end of fuelledpregion of channel per unit channellength

VI 1 1

Symbol F i r s t Defined MeaningAppearance

x9 eq. 7.5 posit ion of fa r end of fuel led region

of channel per un i t channel length

y 4.1 eq. 4.2 normalized f lux

y 4.4 A.16 value of normalized f lux at near end0 of fue l led region of channel

YM 2 . 1 0 macroscopic y i e l d cross-section formaterial M, or fo r bundle i f subscriptis deleted

z 4.2

4.4

4.4

Y B. 6

B.9

7.4

B.13

•i B.13

^ 5.7 slope of flux/power as a function of

flux

A B.10 page 42 constant

^ 2.10 decay constant for material M

M 5.5 eq. 5.6 coefficient of cubic term in diffusionequation

v 4.1 4.3 coefficient of quadratic term in

diffusion equation

'•' B.I 5 B.14 constant

a B.I 5 B.14 constant

B.7

B.7

B.7

page

page

page

39

43

43

axial position along channel

root of algebraic

root of algebraic

root of algebraic

ratio of constant?

constant

constant

constant

equation

equation

equation

5 (V/S)

Symbol First Defined MeaningAppearance

2.1 absorption, fission or capture micro-scopic cross section depending onsuperscript T, for material M

average flux in the fuel

constant value of flux about whichexpansion coefficients are computed

n excited mode of the axial flux

total fuel irradiation

constant value of irradiation aboutwhich expansion coefficients arecomputed

eigenvalue corresponding to the nexcited mode of a linear diffusionequation

eigenvalue of non-1irear diffusionequation

eigenvalue corresponding to the nexcited mode of the axial flux

()o

vn

OJ

n

a

2.1

5.1

7 .3

3 .1

5.1

7.3

4.1

7.2

e q . 7 .3

3 . 2

7.5

4.4

7.7

A MODEL FOR ESTIMATING FLUX DISTRIBUTIONS IN A

fTHORIUM FUELLED) THERMAL REACTOR

1. INTRODUCTION

An in teres t ing problem in the examination of reactor designs

is to obtain estimates of the f l ux d i s t r i bu t i on and leakage in a

simple way. Usually these resu l ts are derived by combining ce l l

and core codes on a computer. The cel l code calculates (mult i-group)

physics parameters for a given fuel (bundle) design, assuming there

are no feedback effects caused by reactor condi t ions. These

parameters are then fed in to a core code which solves the (mu l t i -

group) d i f f us ion equations fo r the ent i re l a t t i c e .

When the fur ther complication of d i f f e r e n t i a l burnup throughout

the l a t t i c e is considered, two approaches may be taken. In the f i r s t

of these, the physics parameters are calculated by the cel l code as

a function of to ta l bundle i r r a d i a t i o n at representative condit ions

of such things as neutron f l ux leve l . The core code then calculates

the f lux at mesh points in the reactor, in te rpo la t ing in tables of

physics parameters with respect to i r r a d i a t i o n , to determine the

physical propert ies of the reactor at each mesh po in t , at a given

time. This use of tables is j u s t i f i e d when the spectrum is not posi t ion

or time dependent so that l a t t i c e parameters can be expressed in terms

of i r r a d i a t i o n . An example^1' of th is approach is the LATREP-PERIGEE codes

used at Chalk River Nuclear Laboratories (CRNL). An a l ternat ive

approach is to calculate the f lux d i s t r i bu t i on at some time, and

then solve the burnup equations and calculate the physics parameters at

representative mesh points according to calculated conditions a t

each of those points. The f l ux d i s t r i bu t i on is then recalculated at(2

the next time step, as the process continues. The codev FEVER

is an example of th is approach.

Although both methods have advantages, they suf fer from the

fact mat they are unsuitable for suivey work because of the large

jr'ounts of computation involved. Add i t iona l l y , when feedback effects

are considered, the complexity increases markedly. For example, fuel

bundle properties can depend on the magnitude of the f l ux in which the

<• ..-'I is i r rad ia ted , due to the well-known xenon buildup in thermal

reactors, or tf.e presence of Pa"13 in a thorium fue l led bundle

pec t i on 2) , as opposed to being a function only of i r r a d i a t i o n .

In the case of the f i r s t approach, to calculate a f lux d i s t r i bu t ion

accurately, one would have to run the ce l l code at several typical

f lux levels and then interpolate in both f lux and i r r a d i a t i o n tables

to f ind properties at each mesh point . A variant involves in ter -

polat ion only with respect to i r r ad i a t i on at me<h po in ts , whose

properties are ind iv idua l l y calculated by the cel l code at a pre-

determined f lux l e ve l . There fol lows then an i t e ra t i ve scheme

whereby the cel l code is rerun several times at new f l ux levels

which resul t from the previous core code ca lcu la t ion. This method

enables one to handle problems in on-power fuel management, typical

of CANDU (Canada Deuterium Uranium) reactors, when feedback effects due

to f lux level and bundle history are important. An example of such a

code is FUMANCHU, under developmentw ' at Chalk River Nuclear Laboratories (CRNL).

The purpose of th is paper is to describe a simple and fast

method of estimating f lux d is t r ibu t ions and leakages f o r cases where

only the replacement of a f u l l channel is involved; the method is va l i d

when both depletion and f lux dependent effects occur. This diminishes

the number of cases that must be run as part of a survey study.

The method is pa r t i cu la r l y useful when a simple one-group one-

dimensional axial representation is required, although increasing

the number of dimensions or groups is theoret ica l ly s t ra ight forward.

- 3 -

1) 2 3 3u 2 3 4 2 3 5

THERMALLYFISSILE

27 DAYS

P a 2 3 3

THERMALLY* \ FISSILE

Pa 2 3 4

22 MINUTESKEY

DECAY

Th 2 3 2Th 2 3 3

(n ,y

Figure 1 - Transmutation chain leading to the formation of U233

from neutron captures in Th 2 3 2 .

- 4 -

Consider the case of fuel irradiated in some flux 4>. A well-

3he

g i ven by

Known phenomenon i s the b u i l d u p o f xenon to an e q u i l i b r i u m l e v e l

, - aN = <"Y. N . . f / ( l + \ / ( p o a ) ( 2 . 1 )

x x ,• . .x • • x ' * x ' '

where x subscripts refer to Xe1 3 5 , N. is the atomic number density

of element j , •J . ' refers to the f i ss ion (absorption) cross section

of element j , A . is the decay constant for element j , Y., is the•j * J

f ract ional y i e l d of element j from f iss ions in element i and thesum over . covers a l l f i s s i l e parent species. In simpler notat ion,

8 - P ' f i l l / ' - ^ [ n 91n - u / \ I +A / i{)U / \L. C)

where G is a constant and A =0 N is the macroscopic absorptionX X X X

cross section of xenon.

A similar form for a cross section comes about if the irradiated

fuel bundle conta', s Th232. In this case, the burnup equat-ons are

dNT

d NP c adt T T p p p

(2.5)

in the sama notation, where the subscript T refers to Th 2 3 2, p refers

to Pa 2 ? 3, u refers to U233and a c is a capture cross section. This trans-

mutation chain is illustrated in Figure 1. If equilibrium occurs, the

cross sections of Pa 2 3 3 and U233 are given byA a» a c., , , aAp = °pNP = V ° T N T / A p ( 1 + V A p ) (2-6)

Au = °uNu = aCTNT /(l+aVA ) (2.7)

- 5 -

Again, in simpler notation

A = G <J>/(l+n <p,'X ) (2.8)p p p p

Au = Gu ^ 1 + O p ' t / X ) ^2-9^

and

Yu = v u°uAu / ru = Hu / ( 1 + ° a *A ) <2-10)

fol lows as an expression for the t o ta l neutron y i e l d cross section from

f iss ions in U 2 J 3 , i f \> is the y i e l d of neutrons per f i ss ion of U ' 3 3 .

Now define an i n f i n i t e ce l l mu l t i p l i ca t i on constant

k = Y/A (2.11)

where Y is the t o t a l neutron y i e l d cross section and A is the to ta l

neutron absorption cross-section of the f ue l . Supposing that the

reaction is proceeding pr imar i ly as the resul t of U233 f i ss ions ,

Y=YU=HU/(1 + aJ*/A ), (2.12)

and

A=AT + Au + Ap + Ax + constant

= NT(af + o£) + GX4>/(1 + V * a x ) + constant (2.13)

I t follows from equation (2.11) tha t kro is of the form

- 6 -

IRRADIATION

Figure 2 - Idealized representation of kai as a function of irradiat ionfor two constant values of f lux.

- 7 -

where C may be found from the constants in equations (2.12) and

(^.13). This form is particularly useful if Xe13' effects are minor

compared to those of Pa'33.

If an accurate value of C, can be determined, effectively

equation (2.14) is an analytic continuation of k from one repre-

sentative value of flux at which a cell code may be run, to otner

values of flux. This eliminates the necessity of computations of

fuel (bundle) parameters at many levels of flux.

3. EFFECT OF BURNUP

In an enriched fuel bundle in many cases, kOT shows the ideal

behaviour depicted in Figure 2 as a function of irradiation. This

suggests that k^ be parametrized in the form

where

u)(x,t) = /^(x,T)dT=<Kx)t (3-2)

at every time t for constant flux. This approximation that OJ = 4>t

will be justified in Sectiot; 8.

4. DIFFUSION EQUATION

With approximations (2.14) and (3.1) as background, consider

for simplicity' ' a one-group, one-dimensional diffusion equation

on the axial (z) direction

- 8 -

where the s u b s t i t u t i o n s

y = :/A and x - z/L (4 .2 )

normalize the flux $ to the maximum flux A in a channel of length ...

^ere we hive written

v = %- (C, + C9t) (4.3)

wi.ere the migration area M*1, C, and Co are dll (known) "constants'

(at any time t ) . Setting 3y/nt = 0 is in accord with the assumption

_j=ct and physically means that ; ( x , t ) describes the evolution of a

sequence of steady states. The one-group eigenvalue

k -1D. - -S— L is found by imposing boundary conditions, and physically

corresponds to a homogeneous poison (boron for example) distributed

throughout the reactor.

Equation (4.1) is a standard non-linear di f ferent ial

equation^ 'whose solution in terms of the Jacobi e l l i p t i c function^ '

sd(ujm) is discussed in Appendix A, for the case of both a bare

and a reflected reactor. The result is the eigenvalue equation

(4.4)((1 - B)(a - 1) 2

for Q with the f lux y(x) being given by

|m) ( 4 . 5 )

- 9 -

in terms of the constants •<•, tf, m, h, x1 and y f j defined in Appendix

A. These equations are quickly solved numerically by zero-

searching techniques u t i l i z i ng step-sizes discussed in the

appendices.

5. PRACTICAL CONSIDERATIONS

The analysis up to this point has hinged on the assumption

that the coef f ic ient v may be readi ly computed based on the

considerations outl ined in Section 2. In pract ice, this is not

so, and several reasons are offered here.

(1) In a fuel bundle which u t i l i zes cluster geometry, the f lux

depression guardfiLees that in each ring of f u e l , xenon and U233

w i l l come to an equilibrium concentration dependent on the f lux

in that r ing . This means that i t is d i f f i c u l t to f ind a

representative value of the macroscopic cross sections that

relate to the fuel average f lux <f> in the simple way suggested by

equation (2.10).

(2) In addit ion to Xe135, other f iss ion products ex ist (e.g.

Sin1"9, Cd113, Sm151, Eu155, Gd157 and Rh105) that have individual ly

small, but co l lec t ive ly measureable ef fects.

(3) Because of depletion of the f i s s i l e parent species in the

case of f iss ion products, and the Th232 precursor in the case of

U 2 3 3 , the assumption that the atom number densities are in true

equilibrium is not s t r i c t l y j us t i f i ed .

(4) In the case of a thorium fuelled bundle, other nuclides such

as U23" and U235 w i l l be present; depending on relative concentrations,

these may or may not contribute to f lux dependent effects.

- 10 -

(5) As a funct ion of i r rad ia t i on ••, k,, is rare ly s t r i c t l y l i nea r ,

which makes i t d i f f i c u l t to estimate the coe f f i c ien t C^.

(6) For f lux values in the range of in terest , rewr i t ing equation

(2.2) in the form of equation (2.14) is not v a l i d , since Xe135 is

usually saturated.

(/', The cross sections vary to second order because of the d i f f e ren t

sp?ctrum at d i f f e ren t f lux leve ls , so that estimates of nuclide

concentrations are not quite exact.

The resu l t of these considerations is to make the simple

estimate of v in Section 2 inexact. This is i l l u s t r a t e d in

-some detai l in Section 8. Notwithstanding t h i s , the analysis

of Sections 3 and 4 may s t i l l be appl ied, by reca l l i ng that the

expansion motivated by these sections is formally equivalent

to a Taylor's series expansion about some point U'> , OJ ).

That is

9k

+•X,

where miscellaneous constants are collected into k' (* ,w ).°° XTo o

At every time t then, using n = <H> the parameter C2 may be

obtained from

C 2 = -

- 11 -

by a simple running f i t to k(;- ,.,.) near some value of u = -., . By

choosing (J, = frt, the greatest value i n the channel, errors in the

f i t are shi f ted to the region near the ends (UJ o) where the i r e f fec t

is smallest. A t h i r d parameter C, ma.v also be introduced at this stage,

via the expression

C , = ( t ' k + t ( k t + k ) ) / 2 ( 5 . 3 )

to cope wi th n o n - l i n e a r i t i e s i n k{$ , LJ ) . And i f an important f l u x

dependent term e x i s t s , i t is also necessary to ob ta in k j ^ , ^ ) at two

values o f f l u x to determine k . The c o e f f i c i e n t C, is then simply

I (j) 0 OJ<J) ()XJJ

by taking differences at u : u . I have found that using $ %A, the

maximum channel flux, and a second flux $ = $12 works well; if flux

dependent effects are severe, a third evaluation at a level of flux

still lower than $12 and a second order expansion in $ may be called

for.

In analogy with Section 4, the diffusion equation is now

y " + fiy-vy2 + yy3 = 0 (5.5)

where

y - C 3 A V / t T (5.6)

together with boundary conditions. The solution to this equation is

discussed in Appendix B, and methods for the evaluation of u and v are

described in Section 8.

A second effect must also be considered. In most cases, the

absolute magnitude of the maximum flux A is not known, since reactor

calculations are generally normalized to total (channel) power. Taking

equation (2.12) into account, for a U 2 3 3 fuelled bundle, we might

expect that the power distribution along a channol will be related tc

m e flux via an expression uf the form

?! . - h/( i t • ;) (5.7)

where • - 0 if no LT '' is present. By compariny power/flux ratios at

equal tines between cases run et two flux levels, values of - and H

can be found. Although no simple expression exists to relate total

channel power

P = 2H / ** r/(l + -;,)dx

to the expression (B.16) for y ( x ) , simple numerical integrat ion

produces a value for P and hence a new value for A. This necessitates

an i te ra t i ve loop to set t le on a correct value fo r A because of the

non- l inear i ty , but the convergence is swi f t , and can be accelerated' V ,

by using the results from the previous step - to f ind A correct to

within .]% requires about three iterations. An unnormalized calcula-

tion of y(x) for given values of y and v averages .11 seconds of

central processor time on a CDC 6600 computer.

6. AXIAL LEAKAGE

In) -v

In standard fashion^ 'we identify the normalized current J bythe relationship

dj _ /[Tdx " ~\L) dx2

Then the relative neutron leakage from the channel is given by

normalizing to the total number of absorptions and assuming M is

constant. So the fractional leakage (excess react iv i ty) represented

by equations (5.5) and (B.3) is

X .p^.n&h, fa

- 13 -

which I'-eeps the channel c r i t i c a l . This c o n d i t i o n i s cons i s t en t

w i t h the assumption n f the s e p a r a b i l i t y of equat ion (4 .1 ) - there

is no rad ia l leakage.

7. STABILITY

A useful question to consider when evaluating any proposed

channel design is the stability of the flux distribution against

the excitation of higher harmonics. In the cases being evaluated,

we have a differential equation

where v = <f> /A is the normalized f l ux in the channel, and we have

a model for the pos i t ion dependence of kOT(y,t), since y is known as

a funct ion of z. The equation becomes

v^ + U n - vy + uy 2 )v n = 0 (7.2)

which is a standard l inear Sturm-Liouvi i le system when boundary

conditions are imposed. In the case n = 0, (7.2) reduces to

equation (5.5) and v = y corresponds to the fundamental mode.

A simple means of f ind ing the eigenvalues n is to use

var iat ional techniques, employing the orthonormal functions

$ = A sino) x + B cosoi x (7.3)

to approximate the solution, because as v,y -»• 0, v n •+ $n and the

method becomes exact. Furthermore, this choice of functions has

simple symmetry properties for odd and even values of n.

As in equation (A.IS) we have the boundary conditi on

f ( x ) - •-: ' ( x ) = 0 ( 7 . 4 )H i n i

and a s i m i l a r e q u a t i o n h o l d s a t t h e o t h e r end x. = l - x , . Sub-

s t i t u t i n g e q u a t i o n ( 7 . 3 ) i n t o boundary c o n d i t i o n ( 7 . 4 ) r e s u l t s i n

t h e t r a n s c e n d e n t a l e q u a t i o n

(1 + • • ,) )2tamo x - (1 - •^••n)2tanuinx ( ( 7 . 5 )

f o r , . n , which is e a s i l y solved by zero-searching techniques.

Demanding that * be normalized over the i n t e r v a l 1 = x2 - x , ,

i d e n t i f i e s

where

A 2 = 2/{ .e( l + f 2 ) + — ^ (2f sinui I - (1 - f , _.

fn " _JL_" t a n %x, (7.6)

and

B = f A .n n n

Ga le rk in ' s (Ray le igh-R i tz ) method( ' ° ) then i d e n t i f i e s the eigenvalues

^ n from the s o l u t i o n of

det | a , , + (n - ai.ffl , . | = 0

i , j , k = 0, . . .N (7.7)

- 15 -

wherey „

« . . = I (v>v - [jy )< t . t .dx (7 8)i.J j[ " i J

The integrals (7.8) may be evaluated by Filon quadrature^ '

for i , j < 10, and because of the symmetry, a. . = 0 i f i and j are

separated by an odd integer. This means that the solutions to equation

(7.7) can be found by evaluating two N/2 x N/2 matrices instead of one

NxN matrix. Using N=2n gives good convergence to n from above.

The reactivity separation between the fundamental and nth excited

axial mode is then given by

kn ( t ) - k°(t) =

This expression is an approximation to the case in which the

position dependence of k^ doesn't change, and the effects of delayed

neutrons are unimportant. That i s , the channel remains c r i t i c a l ,

the xenon and U233 concentrations do not change and the w(x,t)

distr ibution remains constant. Since changes in the xenon concentration

are characterized by half-l ives of the order of hours, I)233 by ^27

days, and over a short period the irradiat ion distr ibution w should not

be perturbed by $ , the model is expected to be valid as an approx-

imation to the medium-term s tab i l i ty of axial flux distr ibutions.

This effect is evaluated in Section 9. The s tab i l i ty over longer

periods where the atomic number densities can change with perturbations

in flux shape is a more complex problem.

8. EXAMPLE

To gain some insight into the accuracy of the model, a samplecase was run on FUMANCHIT 'to obtain detailed results for comparison with

K (*,«) VERSUS u, AT TWO VALUESM OF 4

0 0.4 0.8 1.2

IRRADIATION (n/Kb)

x 1013

. . j

Figure 3 - Calculated shape of reactivity per unit migration areaas a function of i rradiat ion u and flux c .

- 17 -

answers from the code FLOTSAM which is the embodiment of this model.

The case consisted of a "standard" CANDU fuel bundle irradiated in

a thermal spectrum typical of a D?0 moderated Bruce-type reactor.

The fuel channel was taken to be 600 cm in length with no reflector,

operating at a constant 7 megawatts per channel and containing 12

fuel bundles each of 50 cm length. The fuel i t se l f consisted of

1.6 atom percent U^31 in TnO,, with appropriate amounts of the other

uranium isotopes that characterize this self-sustaining lT3i-Th

fuel cycle. Burnup continues for 310 days between channel loadings.

In the FUMANCHU run, a separate LATREP case was done for each

of six fuel bundles (symmetry about the channel centre) at represent-

ative flux levels <|>(t), and the two-group f in i te difference code

PERIGEE was run in i ts "one-dimensional" mode, consisting of a 2x2

channel superceii, with 24 mesh points along each channel. Since

there are two mesh points per bundle in the core code at each time

step, a simple arithmetic average of two flux values was passed back

to LATREP as representative of the flux in that particular bundle..

The convergence cri terion imposed was that <j)(x,t) not vary by more

than 2% for a l l x and t between two successive i terat ions. Because

of the way the iterations in FUMANCHU are set up, the irradiation

w(x,t) = .f$(x,x)dT provides a means of testing the approximationp

that (u = <j)t.

Consider Figure 3, where kM(c}>o,w) = km(<t>o,w)/M ($Q,w) is shown

as a function of w for different values of A . I t can be seen that

Figure 1 is indeed idealized. (Because the results are to be compared

with FUMANCHU which uses the four factor k^ in i t s formulation, this

reactivity scale was chosen instead of the yield/absorption scale.2

Although M ($,w) is nearly a constant, kM instead of kro was alsoused in order not to introduce extraneous effects.) The flux A = <$>o

- 6.6 x 1013 n.cm"2- Sec"1 was estimated and held constant for the

duration of the i rradiat ion.

- 18 -

From the two curves at ; and 4 /2, it is simple to evaluate

k by taking differences at any value of *>. Because of the large

curvature at small values of 0, the evaluation of k , and k is

difficult though. Considerable experimentation has shown that

setting k .=kk is the only method that will produce reliable results,

although this relationship is only true in the limit of very small

irradiation and flux intervals. Accurate results thus depend on

the accurate estimate of k .

Three methods of evaluating this derivative were investigated.

The first two methods involve the use of n ' order difference

formulae1* "' and the third method utilizes a quasi-hermite interpolating

spline . The latter method was found to lack the accuracy of the

difference methods and is not recommended. Because the best expansion

of kM(4),:o) is the double Taylor's series (5.1) about the largestn —

values of w and $ at a given time (which tends to shift errors towards the

cnannel ends -smaller . and ;• - where thev are less important), only

relatively low order difference formula are recommended. This report

considers a first order formula which is equivalent to the "eyeball"

evaluation of a derivative by straight-line differencing, and a

third order, backward difference formula which does not place undue

weight on points far from the point of expansion.

In addition to the choice of order, a second problem can occur

if running estimates of u and u are desired at each irradiation step

in order to feed such things as instantaneous axial leakage back to

a burnup calculation. For the first few steps,where the curvature

is greatest, only a few points are available so that the order of

the derivative evaluation, and hence the accuracy, is limited. It

has been found that when only three points are available some

spurious derivatives may be generated by using a second order formula—

the highest available. Once eight or nine points define the curve

.80-

AXIAL POWER FORM FACTOR (FUMANCHU)

AND PERCENT ERROR (FLOTSAM)

— 3rd ORDER

- - 1st ORDER (EYEBALL)

o POINTS AT WHICH DERIVATIVESARE EVALUATED

3Dm

3

TO3D

o3D

-1.0

.62550 75

PERCENT OF DURATION OF

100

IRRADIATION

Figure 4 - Average/maximum power rat io as a function of time ascalculated by FUMANCHL) and relative error in this parameteras calculated by FLOTSAM using different techniques toevaluate the derivative k .

- 20 -

though, the order of the formula makes l i t t l e difference to the

accuracy of the f inal answer, irrespective of whether the evaluation

is a running one, or al l points on the curve are used. I f the

former option is employed, only the previous four points are

required; i t should also be noted that the greatest errors associated

with difference formulae are distributed at the endpoints of the

intervar ' so that a higher order running f i t when only a few points

are available should be regarded with some apprehension.

Figure 4 i l lustrates the variation in axial power form factor

with i r radiat ion, as predicted by FUMANCHU. The lower curves on

this figure show the relative error in this parameter when calculated

by FLOTSAM i f the derivative k is based on f i r s t or third order

formulae. Calculating k using a l l points on the k(u>,<j> ) curve or

just the last four points (running f i t ) leads to the errors shown

also.

In summary, when the curvature of kU,ib ) has decreased to a

small value, results are best, although they are acceptable over

the range, and the error in the predicted axial form factor is

about 1-1/2%. Most of this error is attributed to a combination

of several effects:

1) The expression (5.5) is a one-group model, but FUMANCHU

is a two-group code.

2) For small values of w, the expansion (5.1) is more than

simply linear in <(>, and quadratic in u>.

3) The assumption u)(x,t) = <|>(x)t expressed in equation (3.2)

is not exact.

4) The relationship between P(x,t) and ^(Xjt) given by

equation (5.7) is also approximate.

- 21 -

. . ' • / . •

! . • • • • , ' , . ' •

Figure 5a) - Axial variation of w(x,t)/<p(x,t) at three times.

Figure 5b) - Flux to power ratio as a function of absolute fluxlevel at three times, showing slight non-linearity.

1.040-

1.030-

1.020-

1.010

1.000

.990

55n/M>

I.l6n/kb

«.|.77 ti/Kb

20

•18

FLUX

Figure 5c) - Reactivity as a function of flux at three irradiations,and the Xe135 component of reactivity.

- 22 -

eu

FUMANCHU

FLOTSAMc o s i n e

100 IbO 200 250

DISTA N C E ALONG CHANNEL (cm

300

Figure 6 - Axial flux distribution as calculated by FUMANCHU andFLOTSAM and a normalized cosine curve for comparison.

- 23 -

Although the f i r s t effect cannot be investigated without a considerable

loss of simplicity, the accuracy of the approximations 2) , 3) and 4)

may be tested using the FUMANCHU output. Figure 5a shows the

axial variation of w(x,t)/<i>(x,t) at three times using w(x,t) =

/<j)(x,T)dT. This shows that near the channel ends, where the fluxQ

is changing most, the approximation (D=((>t is least accurate. Similarly,

Figure 5b shows that <j>/P is not s t r i c t l y a linear function of <j>

with slope K/H as suggested by equation (5.7), and Figure 5c shows

that the same is true of both k(<j>,u> ) and i ts contributing Xe135

part. Figure 6 i l lustrates that the difference between <j>(x)

calculated by FUMANCHU and FLOTSAM for the worst case is very small

on an absolute scale. A normalized cosine (using the same P/<|> ratio)

is also included for the sake of comparison.

A further effect discussed in Section 5 is i l lust rated by

Table 1. Listed there are the contributions to yield Y and absorption

A from al l the nuclides, at two levels of f lux. Although the Pa233

and U233 contributions to A are to f i r s t order in agreement with the

estimate (2.13), when differences are taken to establish a guess

for C,, the agreement is poor. From the simple considerations of

Section 2, one might guess C, ^ aj?/A = 4.2 mk/<t>, whereas the trueI a

value established from Table I is 6.8 mk/$, an error of ^ 60%.

(The curves in Figure 5c that have a slope of about 8.9 mk/<|>

relate to the react iv i ty k^epryfj not k^Y/A.and so are expected to

be somewhat di f ferent, but here, too, a naive estimate of k, is seen

to be incorrect.)

For the purpose of survey work, the model seems adequate

however. This is exemplified by Figure 7 where the axial form

factor as a function of time has been computed by both FLOTSAM and

FUMANCHU, but for a channel power of 5 instead of 7 Mw. Using the

previous results for the 7 Mw case as an i n i t i a l guess, FUMANCHU

was able to converge to a <i>(x,t) surface in 3 iterations using

- 24 -

Th'"3R

U233

U235

D20

Total

7.81

0.10

7.64

0.771

1 .866

1 .0972

7

7

.76

.20

.33

.777

.866

.0948

A ~[ <A/A=-A/(A.\!t

- . 0 5

.1 i

- . 3 1

.006

-

- . 0 0 2

TABLE ' Lomponent i o n t r i b u t i o n s t o k a t ..> = 1 . 7 7 n / k b

: ' 3 . 1 ! 6 . 6

Th- '•

P a - ' '

U- ' '

u- ••

U "

u - ' •' •

PFP-1

PFP-2

PFP-3

S m 1 ^

Xe 1 3 5

Rh 1 0 5

Qlt,

H:0

D20

Zr91

Total

.357

.487

.0917

.412

.00447

.031

.099

.176

.454

.341

.475

-.016

-.012

.0884. .003i

.431 .019

.0078 .003

.020

.099

.172

.449

.011

-.004

-.005

19.4032 19.1286 -.274

.228

17.517

1.771

.122

19.6388

.220

16.818

1.770

.117

18.9269

AY

-.008

-.699

-.001

-.004

- .7119

- 4 . 3

5Y/Y=AY/(YA«iO

5Y _ 6A\= . 6

Y A /

- 1 1 . 0

NOTE: * i n units of 1013 n.cm'2 -Se c - i j s the constant f l ux during i r r a d i a t i o n .

AXIAL POWER FORM FACTOR

function

o-htime

gure

i

Axialfacto

O> =1—I 01

n —'c : - " •—> Noi n>

m Ln. o

_^*-*

§§9-S3

3

en

- 26 -

470 seconds of computer time, requiring 594 evaluations of la t t ice

parameters, and the solution of 99 one-dimensional f in i te difference

equations. FLOTSAM produced the lower curve in 4 seconds, using

the data for ku(<j>,w) which had been previously generated for the

curves of Figure 3 - about 90 seconds/curveu ' . The agreement is

adequate, and raises the possibi l i ty that once the coefficients of

the expansion (5.1) have been found for a given fuel design, properties

of that fuel irradiated in dif ferent channels at varying power levels

may be found without rerunning the cell code. This implication is

discussed in the next section.

9. GENERAL APPLICATIONS

According to equations (4.3) and (5.6) the coefficients v and

v may be estimated for a given fuel and channel design. I t is then

sometimes of interest to vary some of the parameters such as channel

length or power while maintaining a constant fuel design, as

i l lustrated in Section 8. By evaluating v and p as functions of

these parameters, one can solve equation (5.5) without the necessity

of rerunning the cell code, even when f lux dependent factors are

involved. In the case of an unreflected channel, unnormalized to

the total channel power, simple design curves can be generated.

Figures 8 and 9 respectively present the variation in axial form

factor and leakage^ as a function of y and v over the widest

possible range of interest.

Usually \x -y 0 as the i rradiat ion progresses and k(u>,<}> )

becomes more l inear, so i t is also interesting to plot this case for

independent v. The two curves in Figure 10 a are cross-sectional

views of Figures 8 and 9 for the case u = 0, and Figure 10 b is an

i l lust rat ive crossplot of leakage as a function of axial form

factor for this same case. Note that although both leakage and

0

Figure 8 - Lines of constant axial form factor of an unreflectedchannel as a function of the independent variabless and v in the unnormalized case K = 0.

1000

ro03

1000

Figure 9 - Lines of constant unweighted leakage^ in mk.cm2/unitmigration area of an unreflected channel as a functionof the independent variables s and v in the unnormalizedcase K = 0. Dividing by 10" gives approximately the rat ioof actual leakage to that in the same channel with a cosinef lux . To convert to leakage in mk, mul t ip ly j^ by M2/L2

for a given design.

- 29 -

95

,91

25

x

.74.

.67

I "' ' I

FORM FACTOR AND LEAKAGEVERSUS V

I 1

30 50 100 300 500 1000

55000

45000 o

iI

35000 %

25000

15000 2UJ

5000

Figure 10a) - A) Axial form factor of an unreflected channel as afunction of v in the case u = 0.

B) Leakage^" in mk.cm2/unit migration area of anunreflected channel as a function of v in thecase \i - 0.

30000

1

ti

LEAKAGE VERSUS AXIAL FORM FACTOR

LINE OF SLOPE EQUAL TO ONE

.63 .67 .71 .75 79

AXIAL FORM FACTOR

Figure 10b) - Leakage^ as a function of axial form factor for thecase u = 0.

- 30 -

form factor increase with increasing v, the leakage increases

more quickly. Figure 11 illustrates how the reactivity separation

between axial modes as given in equation (7.9) decreases with

increasing axial form factor, again for the case p = 0.

10. SUMMARY

A model has been presented which permits the swift evaluation

of one-group axial flux distributions in reactors when flux dependent

and/or depletion effects are important. By carefully tai loring a

series expansion to the reactivi ty of a bundle as a function of both

irradiation and flux level, the evolution of the flux distr ibution

in a channel may be simply estimated when only f u l l channel fuel l ing

is contemplated. These estimates w i l l prove part icularly useful for

survey studies when flux dependent effects are important, such as in

a thorium fuelled thermal reactor, where design curves may be cal-

culated for some simple cases. In addition, the s tab i l i ty of the

flux distributions in the medium term can be estimated.

Although the general model is multi-dimensional, and a mult i -

group formulation can be devised, in this paper only the one-group,

one-dimensional version has been solved. (This is equivalent to

the central flattened region of a thermal reactor.) The solution

in this case may be found analyt ical ly, and leads naturally to a

set of functions - the Jacobi e l l i p t i c functions - which appear to

play an important r o l e ^ ' i n reactor physics. I f the multi-dimensional

form of the non-linear eigenvalue equation ( in several groups) were

to be solved numerically, these functions, and their associated

eigenvalues would be natural choices as i n i t i a l guesses in the

axial direction. I t seems reasonable to assume also that they may

prove useful as t r i a l functions in 3-dimensional diffusion calcula-

tions when di f ferent ial burnup is an important factor.

- 31 -

80

2QZ3

6 0 .o

MODAL SEPARATION OF AXIALHARMONICS FROM FUNDAMENTALVS AXIAL FORM FACTOR

SECOND AXIAL MODE

240-

20

FIRST AXIAL MODE

64 68 72

AXIAL POWER FORM FACTOR

76

Figure 11 - Reactivity separation of fundamental from first andsecond axial harmonics using migration area M2 = 400for the unreflected, unnormalized case with y = 0, K = 0.

- 32 -

ACKNOWLEDGEMENTS

The first insight and the subsequent derivation of the non-

linear equation is due to M. Duret. Further discussions with

W. Selander, F. McDonnell, F. Barclay and J. Blair were invaluable

in helping to define and solve the problem. Of course, I must also

thank the many members of the CRNL computation centre who provided

most of the mathematical library routines, and P. Kumli who did much

of the computer programming, and extend my gratitude to J. Veeder,

and P. Garvey who commented on the original manuscript.

- 33 -

FOOTNOTES AND REFERENCES

1) "LATREP Users Manual",, G.J. Ph i l l i ps and J . G r i f f i t h s , Atomic Energy

of Canada Limited repor t , AECL-3857;

"A Guide to LATREP (1975)", M.S. Milgram, Atomic Energy of Canada Limited

Report, AECL-5O36;

"PERIGEE Computer Codes fo r Reactor Simulation in 3 Dimensions Using

1 or 2 Neutron Veloci ty Groups", A.P. Olson, Atomic Energy of Canada

Limited Report, AECL-1901.

2) D. Hamel, K. Dormuth (pr ivate communication) - also published reportsGA-6612, GA-8286.

3) M.S. Milgram, unpublished.

4) "The Elements of Nuclear Reactor Theory", S. Glasstone and M. Edlund,

D. Van Nostrand Co. Inc. (1952), Chapters 5 and 11.

5) S t r i c t l y speaking, equation (4.1) could be wr i t ten as a m u l t i -

group, multi-dimensional d i f fus ion equation and solved by the usual

numerical techniques. However, i n t h i s paper, we take the simpler

tack of studying a one-dimensional form to gain ins igh t in to the

physical nature of the solut ions. Although the three-dimensional

form is obviously non-separable, assuming the separab i l i ty gives a

s ta r t i ng approximation to a mult i-dimensional so lu t ion . The physical

s i t ua t i on best described by equation (4.1) is the centre channel(s)

of a thermal reactor operating on an enriched and/or thorium fuel

cyc le , with f u l l channel r e f u e l l i n g .

6) This equation has been invest igated by various authors ( c f .

W. Kastenberg in "Advances in Nuclear Science and Technology", 5_,

51 , Academic Press, New York, (1969), and references c i t ed there in) ,

i n par t i cu la r J . Chernick (BNL-126), who in 1951 solved th i s equation

in a c losely pa ra l l e l manner, but w i th a d i f fe ren t viewpoint and

application.

- 34 -

7) Properly, equations (3.1) and (2.14) should be expansions in

the variable kMU,io) = k U , wJ/M2^^) but as M has such a

weak dependence on 6 and to, I have neglected this complication

in the formalism.

8) "Introduction to Nonlinear Differential and Integral Equations",

H.T. D<u'is, Dover Publications, New York (1962), Chapter 7,

Section 11.

9) "Handbook of Mathematical Functions", M. Abramovitz and

I. Stegun, Dover Publications, New York (1965), Chapters 16 and

17.

10) "Variational Methods for Eigenvalue Problems", S.H. Gould,

Mathematical Expositions No. 10, University of Toronto Press,

Toronto (1957), p. 75.

11) Reference 9), Chapter 25.

12) M. Carver, "Subroutine DUDX and DUDXX to calculate the derivatives

of an Array", Atomic Energy of Canada Limited, unpublished internalreport CRNL-486.

13) H. Akima, "A New Method of Interpolation and Smooth Curve

Fi t t ing Based on Local Procedures", JACM, 17, 4, 589

(Oct 1970); Subroutine IQHERU, 1MSL l ib rary , 1MSL L1B3-0004.

14) C Lanczos, "Applied Analysis," Prentice Hal l , Inc. N.J.,

(1956), Chapter 5.

15) A special faster version of LATREP which runs about seven times

more swift ly than the stand-alone version has been incorporated

into FUMANCHU. Thus the computer times quoted are not as

disparate as they might be.

- 35 -

16) "Handbook of E l l i p t i c In teg ra l s " , P. Byrd and M. Friedman,

Springer-Verlag, Berl in (1970), Equation (259.000).

- 36 -

APPENDIX A

Equation (4 1) is a standard non-linear d i f fe ren t ia l equation(8)

which may be solved by rewrit ing i t in the fomr '

y l 2 = h2(y3-py2 + q)

where

h2 = 2v/3

and

p = ft/h2 (A.2)

We seek solutions which are synmetric about the point x=l/2;

this identifies

q = P-1 (A.3)

because

y ' ( l / 2 ) = 0 (A.4)

Integrate equation (A.I) between tha l imits xx and x, where

xx defines a reflector-fuel interface at an extreme end of the

channel. This gives

±h(x-Xl) = / ( x ) (A.5)

y 0 <fr _(y-B) (o-y)

- 37 -

where we have written y = y (x i ) , and a and Bare roots of the

equation

y -qy-q = 0 (A.6)

The factor (1-y) comes about as a result of equation (A.4} and the

boundary conditions

=y 0

=1

(A.7a)

(A.7b)

These imply that a and 3 cannot l i e between y and 1 because that

would be inconsistent with y' ^ 0 anywhere in the range y iy< l .

Suppose

V >P =Jo1 <a =

The substitution

,2 _ a-1Z = 1 y

(A.8)

soon yields the result

(A.9)

where

and

(A.10)

- 38 -

(9)nc(ulm) being the Jacobi e l l i p t i c function1 ' of argument u and

modulus

m = k2 / ( l+k2)

w i t h (A.11)

k2 = ( l -e ) / (a - l )

Solving equations (A.9) and (A.10) for y(x) results in

y(x) = l-(l-8)cn2(nc"1(V|f|- N * J ? htx-xjlm) (A.12)

which satisfies boundary condition (A.7a). Impose the second boundarycondition (A.7b) in (A.12) and the result is a new eigenvalue equation

i_i _ v /ct-BMil ~ l\ "" n

which may be simplified (K is the period of the function nc) to read

from which p and eventually fi may be found. The flux y(x) is thengiven by

y(x) = 1- < c t " | ^ 4 ) ' 3 ) scj2 < " ? h(x-]/2) I1") ('

In the limit m->o, one recovers the usual eigenvalue equations forcosine solutions from equations (A.13) and (A.14); for casesconsidered here m is usually too large to allow simple perturbativeapproaches^ ' to the solution of these equations.

- 39 -

To find y i f there exists an interface at the position x=x,,

define

<5 = ? - tanh K R X IK

where D=D /DD is the ratio of diffusion coefficients at the core-t K 1

reflector interface and KD = =— is the inverse migration length inRthe reflector. The boundary condition (A.7a) is satisfied but

must be supplemented by the (usual) condition (continuity of current)

y ( x j - fiy'fx,) = 0 (A.15)

Substituting equation (A.15) into equation (A.I) gives the require-

ment that y satisfy

yo3-(p + l/h262)yo

2 + q = 0 (A.16)

In practice, equations (A.13), (A.14) and (A.16) are quitesimple to solve. The elliptic functions are easily computed (one

(g\

method is to use the alternating-geometric-mean algorithnr ' which

converges very swi f t ly) . Equation (A.13) can be solved by standard

zero searching techniques, particularly i f one uses the constraint

a > 1 to derive a lower bound

p > 3/2 (A.17)

as an initial guess. Employing step sizes as discussed in Appendix Busually results in convergence to machine precision in about 8-10evaluations of the eigenvalue equation. (Rarely, the routines mayconverge to a higher eigenvalue, so that as a precaution, equation(A.14) must be checked for negative values of y(x).) A new valueof y is determined from equation (A.16), and the process is repeated.Convergence to a true value of y (error of 1.01%) takes about threeiterations.

- 40 -

APPENDIX B

As described in Section 5, we seek solutions to the equation^ '

y " + Qy - vy •' + t ! y ' = 0 ( B . I )

together with the boundary conditions

= 1 (B.2a)

, = y0 (B.2b)

and

y ( x , ) - 6 y ' ( x , ) = 0

i n a n a l o g y t o e q u a t i o n s ( 4 . 1 ) , ( A . 7 a ) , ( A . 7 b ) and ( A . 15 ) . R e w r i t i n g

e q u a t i o n ( b . l ) i n t h e f o r m o f e q u a t i o n ( A . I ) g i v e s

y ' 2 = h 2 ( y 3 - p y 2 + r - s y 4 ) ( B . 3 )

2 ?where h = 2v/3 and p = fi/h as in equation {^2 )•We also identify

s = W h 2 = 3y/4v ( B.4)

as w e l l as

r = q + s = p - 1 + s ( B.5)

by imposing boundary c o n d i t i o n (E.2a) t o g e t h e r w i t h the

symmetry requ i remen t y ' ( 1 / 2 ) = 0 .

- 41 -

In e x a c t a n a l o g y to e q u a t i o n ( A . & ) , i n t e g r a t e e q u a t i o n( B.3) b e t w e e n x : and x to o b t a i n

y ( x )

± h ( x - x . ) = / dy= /{ V(l "y)(Y - sy)(a - y)(y - B)

w h ( ; r e a , S , and Y / S a r e r o o t s o f t h e c u b i c e q u a t i o n

y 3 + ( 1 - l / s ) y 2 + r / s ( y + 1) = 0 ( B . 7 )

To s o l v e e q u a t i o n ( B . 7 ) , i f ( i ) s £ . 0 1 and p < 5 . ,

an e x c e l l e n t a p p r o x i m a t i o n i s t o l e t

a = p - 1 + s p 2 + s 2 ( 2 p 3 - p + 1 ) + s 3 ( 5 p 4 - 4 p ( p - 1 ) - 1)

+ s 4 p ( 1 4 p 4 - 1 5 p ( p - 1 ) - 4 )

b = a ( ( a + l ) s - 1 ) / ( 1 - a s )

T h e n

Y = 1 - ( a + l ) s

a = J-2(a + y]a2 - 4b ) (B.8)

3 = >g(a - yja2 - 4b )

(ii) otherwise use one of the many polynomial zero-findingrouti nes.

F o r s m a l l v a l u e s o f s , Y S = Y / s >> a , so we c o n s i d e r

t h e s u b s t i t u t i o n

- 42 -

- a) (y - Y,

valid for

Y s >> a > 1

(B.9)

6 < y 0

which follow from the boundary conditions that imply y'(x) f 0

anywhere in the open interval (x,, 1/2).

Proceeding as in Appendix A soon yields the result that

y(x) = (Ytnu2 - As)/(smu 2 - As) (B.1O)where

u = sd(V(ci'- B ) " ( Y ~ ) \ (x - 1/2) |m)

= (a - Y S ) / ( 1 " «

and

m - 0 - B)(Y - as)m ~ (a - S)(Y - sT

together with the eigenvalue equation

, o (a - 3)(y - s)(yr

- 43 -

In the limit as 5 •» 0, Y -* 1 and equations ( B. 10) and (B.ll)

reduce to equations ( A.13) and ( A . 1 4 ) . And in exact analogy

to equation ( A.16 ) > we find that y Q must satisfy

y 03 - (P + l / h 2 6 2 ) y 0

2 + r - s y Q4 = 0 (B.12)

For larger values of s and p, equation (B.9) no

longer holds, as y/s and a both approach 1 and then become

complex conjugates. In this case equation (B.6) becomes

y(x)

+ \JTh (x - x, ) = f ,y)(y - e)(U - y ) 2 + n z)

using

a = c + in

Y / S = Y S = x, - in

Equation (B.13) is well known in terms of elliptic functions,

and evaluation of the integral eventually yields the result

+ hVspicf(x - X-.) = en" (u(x)|m) - en" (u(x-,)|m)

where

P 2 = (1 -r, )2 + n2

a 2 = (p -5 ) 2 +

(B.14)

U ( X ) = (a + P)y(x) - (o + pg) (6.15)

(a - ) ( ) )

- 44 -

andm = ({1 - i3)2 - (o - c)2)/4po

In this case, equation (B.10) is replaced by

using

u(x) = cn(h/sp"0(^ - x)|m)

The eigenvalue equation (B. 11) becomes

(o + p)yQ - (a + &>)U(X

1) = (p - a)y0 + (o - 3P) (B.17)

As in Appendix A, either of the eigenvalue equations (B.ll)

or (B.17) is easy tc solve by zero searching techniques if one

distinguishes the cases:

a) 0 < s < 3/8 means that if the roots a and ys are real, then

condition (B.9) is true provided that

p > f - 2s (B.18a)

although a and y complex is not precluded for large enough p.

However, this provides a lower bound (L ) for an initial

guess at p.

b) s >. 3/8 means that two roots of equation (B.7) are complex.

This provides the following curve (L-)

- 45 -

P 1 gs I1 " 12s " 16s2 + < 8 s + D 3 / 2 l > 0 (B.18b)

as a lower limit for initial guesses at p.

c) for s < 0, the roots of equat i o n (B.7) satisfy

Y / S < e < y Q < i < a

and all the above results are valid for this case bythe principles of analytic c o n t i n u a t i o n .

The lower limits indicated by equations (B.l8a) and(13.18b) are also useful in providing estimates

2for the eigenvalue Q = -* pv which approaches these bounds forlarge enough values of v. Solving the eige n v a l u e equations( B . 1 1 ) o r ( b. 1 7 ) n e a r s ~ 0 , s . J-2 a n d s ~ 1 f o r v

i n d i c a t e s t h a t t h e v a l u e o f E = ft - L , p ( t h e d e v i

ft f r o m i t s l o w e r l i m i t 2 / 3 v ( L ] o r L 2 ) ) i s :

a ) s ~ 0 , v •+ 0°

[- 1 + 3/ys( l - ^1 " 2ys/3 ) ]e =

w h e r e

y s = ( 1 - y o ) ( l - s ( l - y Q ) )

b ) s ~ h> v

c) s ~ 1 , v

e =

e ~ 4 . 8 x 1 0 4 ( s v ) " 3

T h e s e e s t i m a t e s o f ; p r o v i d e e x c e l l e n t g u e s s e s f o r

s t e p s i z e s t o b e u s e d f o r t h e z e r o - f i n d i n g r o u t i n e s w h i c h

s o l v e t h e e i g e n v a l u e e q u a t i o n s f o r m o s t v a l u e s o f s , v o f

i n t e r e s t , p a r t i c u l a r l y f o r t h e c a s e y • • w h e r e a s e c o n d

e i g e n v a l u e a p p r o a c h e s t h e d e s i r e d s o l u t i o n v e r y c l o s e l y .

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