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Technical coacuttee Meeting on Methods ot neutron transport thexoy xn reactor cqlculations. Uologne, Italy, 3-i) November \97S

CLA-CONf— y J»/

NEW PEVEi.OPME.VTS IN THE ( MCTW0D

Paul GRANDJEAN and Alain KAVti\!0KY

Service d'Etudes ries Réacteurs et de Mathématiques Appliquées Centre d'Etudes Nucléaires de Soclay

O.P. n° 2 911D0 Gif sur Yvette France

/. ÎHTROVUCTJOM

The C method of solving the transport equation has been developed at Saclay during the last few years. The results in plane geometry with usual scattering laws had been extensively published [1], [2], [3]. This report is devoted to the most recent developments : treatment of the Rayleigh scattoring Kerne! in plane geometry [4] and of the cylindrical pro­blems with an isotropic scattering law [3], [5], [6].

Three problems are solved using the Rayleigh scattering Kernel in plane geometry : albedo and extrapolation length for the mine problem, al­bedo and transmission factors for slabs j comparison calculations performed using collision probabilities and Chandrasekhar's method are presented.

In cylindrical geometry 2nd with an isotropic scattering kernel some problems are solved : albedo of the inner and outer Hilns problem, de­termination of the extrapolation length, calculation of the critical radius.

Section 2. is devoted to the Raylcigh kernel and cylindrical cal­culations are presented in Section 3.

2. C w \MTW0V A W MŒ1GH SCATTERING KERNEL

Light in plaruitar atmosphères is scattered according to Rayleigh scattering kernel : the diffusion of light in thosfi atmospheres hd.3 to bs; determined by the resolution of the Boltzmann'o equation i the application of the C w mathod to this problem is straight-forward.

E 9S

Th8 Royleigh scattering Kernel is defined by

! f

• . : ' :

- Vf'

•4- , a. \ •*. * * . '

E (Ù.ÎI-) =• — 4*

[[1 -ij) • atn.i»') 2! (2.1)

2.1 Albedo e,4 -t/te Mi6tc. Picbtzm

2.1.?, The cciHptcflifaftt'u f/ equations The half space x>o is made of a r.sdiurr. absorbing and scattering

the particles acccrdinz to the Rayleigh kernel ; it is surrounded by a blacK body. An angular flux V Cu) is uniformely applied on ths surface x=o ; it is directed Lo x >o j the medium reflects the angular flux v~(u).

The application of the Placzek lerina to this problem had been often presentpd [1l, [2], [3] ; two equivalent integral equations are obtained :

y>o

y<o

p r o - I G(y,y') v (y') \i' d\i' * I G(y.y') v (y'

P • f° ) » I G(y,y') v Cy') y' dy' • I G(y.y') v fy*) y'

) y' dy' 12.2)

dy' (2.3)

In those equations G(y,y') is the infinite medium Green's function ; G(y,y') is the angular flux at the abscissa of a uniform source emitting nne neutron in the direction y'.

Each of these two equations is equivalent to the Boltzmann equa­tion of the problem » we apply now the C., method to these equations.

2.1.2. The C w titbolution

A variational derivation of the C^ approximation had yet been pu­blished j here we use the weighted remainder method.

Let us expand v (y) and v l\t) as :

v (y) - 2_JI v£ v ( 2 . 4 )

l*N v"(y) -2^v~ y*"1 (2 .5 )

£«1

And define the momenta of the deen ' s furction

ytf'l ' f ifdy /*G(y,y') y'Ady» (2 .6 )

Whare c arid e' may represent the segments [-• 1,0] or [0,1] .

-»<**.

3

The expansions (2.4) and (2.5, ara used in fq. (2.2) which is multiplied by u m and integrated on [0,1] ; a set of linear equation is obtained :

£»N 1 to M (2.7)

| X. +X X. -X. | £-1

The same treatment applied to t"q. (2.3) provides

0 1 - / 1 4-m \ 4- 4m I = 0 (m = 1 to N) (2.8Î £ <->' [4* • c) • *; c ]

Eqs (2.7) and (2.8) are the C^ equations for this problem, they are not equivalent for any fixed value of N but only in the limit U •*• c a .

2.2 Calculation o& the Gxe.en'4 function

ThB infinite medium Green's function is the solution of the Boltzmann equations :

o.gred f (r.in • zt f (~r,n) = f iJ&'Jn f (r.n*) • s(?,3) (2.9) •Ali

The Reyleigh scattering Kernel being defined by : E « M ) = — (1 - £) • atô.ft')2

411 L 3 J I s

We define c » r~ and ws use the mean free path as length unit. Zt

Assuming the plane symetry of the problem, the Green's function equation becomes : y fCx,y) • f(x,y) - § M - |i • |(1 - y 2) f f(x,y') dy'

• 3_£ ( 3 u2 . 1 } I f(x,y-) y» 2 dy' • S(x) 6i\i-Mo)

(2.10)

This equation in solved by use of a Fourier transformation :

f(K,y) (1 - i k y) - § (1 - §) + |(1 - y 2) | f(k,y«) oy« L 1 J -1 (2.11)

/ : • ~ p t3y2 - 1) I f(k,y') y'2 dp' • 6l\i-\iQ)

4

This in tngra l equation i s oF the Goursat type with a rtfi£f!n.-?rv->te'_ Kernel ; i ts; so lut ion i s :

G(K.- - - - ' ' " ^ ~ 1 c a f tp +PQJ (3c-3-V) - 3 i l - c ) - c k +

i .y.<j . c . a ) = . -—• — • DCK) 4 L ( 1 - i k u) ( 1 - i K p )

(2 .12)

• 3(1*ik u) [1*ik p ) f l - c ^ E Î 8 -H o V. k

D(k)

With the denominator D(k) defined by :

ArctE y.f,^,l 2 la c + 3(1-C) - ^ ( k 4 ( ^ * ~) • k2(2 - c) • 3(1 - c))

,e'm (2.13) The Fourier transforms of the momenta G^ £ cf th^ Grenn's function

may bo computed analytically » the inverse Fourier transformation is per­formed partly numerically partly analytically : the transient part of the momenta is obtained by a numerical integration in the complex plane and ths asymptotic part is obtained theoritically.

2.3 C», Hianwical KZAU&U .&oi pie, albedo o$ tht Mine. Vxobtm

The CNRAYL program had been written for the resolution at any order N of Eqs (2.7) and (2.6) ; the Ingoing flux may also be expanded ot the same order N. Results with various entering currents are to be pre­sented i the reference result Is obtained by a new development of the ChandraseKhar's method ["].

2.3.1. COL&Z oi an AAot/iopic entesUnç) &lux

Table 1 presents the results obtained for the approximations CL. to C.„ for the cases where c =• 0.8 a;id a = ± 0.5. 10

CN 0

U - - 0.5 a «= • 0.5

CN 0

Eq. (2.7) Eq. (2.8) Eq. (2.7) Eq. [2.8) CN 0 .348 616 495 4 .342 173 821 9 .349 247 238 4 .343 406 135 i> 1 .340 313 400 0 .341 231 252 5 .342 313 382 4 .342 550 993 6 2 .341 146 620 0 .341 266 223 8 .342 524 415 4 .342 526 971 9 3 .341 271 956 4 .341 264 782 5 .342 526 556 1 .342 525 117 7 4 .341 263 099 7 .341 264 627 0 .342 524 363 0 .342 524 915 1 5 .341 265 176 6 .341 264 599 3 .342 524 995 5 .342 524 878 5 6 .341 264 592 4 .342 524 869 3 7 .341 264 5S0 3 .342 524 866 5 6 .341 264 5«9 5 .342 524 6C5 7 g .341 264 589 1 .342 524 865 6 10 ,341 264 583 9 .342 524 865 3

Lhandranokhar 0.341 264 589 0 0.342 524 664 7

TABLE 1

•||-|iiliiiiiii mu I

K

The accuracy of the numerical resu l ts gi jrantees that the d iscre­pancies b'jtwnen tlu? resu l ts l.r, ttjtj to C. «DProximfihinns.

The resul ts obtained by resolut ion of the second Kind equation converge fas te r l i n n tJ'or-r? ohtMnsd t?y th_ ' i r s t Kind equation : th i s uîrjnomcnon i s the S.TII-J dj for '-he nth r;.- scu l l c r i ng Iz.-i:, tr-jot«d in plana geomfjtry. Nnvcrtnclpss an accuracy of 10~ J i s obtain wi th the L\. second Kind equation and C 4 f i r s t kind equation.

2.3.2. Ca&e. o& a tinzaxlij anl&o.txopic enfi&iing ilux

Table 2 presents the Sifnc resu l ts f o r V (u) = \i and the accuracy is the same.

C N

0

O = - 0.5 ct • • + 0.5

C N

0

Eu,. ( 2 .7 ) Eq. ( 2 . 6 ) Eq. (2 .7 ) Eq . (2 .8 ) C N

0 .330 130 604 9 .32" 603 800 7 .230 61D 286 3 .326 583 775 9 1 .323 427 614 9 .323 837 473 6 .325 636 071 2 .325 998 650 2 2 .323 83 / 825 1 .323 825 5fc8 h .325 9&? 341 3 .325 932 907 1 3 .323 821 571 8 .323 825 543 5 .325 952 084 2 .325 SS2 803 9 4 .323 820 793 0 .323 825 604 4 .325 963 263 8 .325 932 861 9 5 .323 825 112 6 .323 825 622 7 .325 232 786 9 .325 982 904 5 6 .323 825 623 1 .325 962 911 3 7 .323 825 629 9 .325 982 £13 6 0 .323 825 630 5 .325 382 914 4 S .323 825 630 0 .325 982 914 7

10 .323 825 630 8 .325 982 914 7

ChandreseKhar .323 8; 25 631 0 .325 982 915 1

TABLE 2

2.3.3. Tablz oi fiuvJUU

The accuracy of the comparison of the preceeding section batwean C N results and other methods allow us to present on TablB 3 rer.ults for various values of C and a.

2.4 Ittxapolatlpn length o& the. M<l*iz Viobtm

The extrapolation length characterizes the effect of a boundary on the flux i it can easily be computed U3ing the C^ method.

J.v»

&4~ ,

Y • *C**^U, «*£•»!**..-'*: ..--.

* » • * o * ' > 11"! * o

1-4 I I

* • * « k l - » * -«t k vn k o-* N

* * • * O * M * OC

• l-J • V I • M

4-1 M * * « « • C l

k l - 4 k < £ • H»

**•

« • • O

• M « t £ • tO • f ?

• • • • M I O • n • T i • o

n n • • • • ta • r « W • *• » I V » C

t - i M

• • • C3 • 4-• <A • I-» • U* • C>

I - I n • k

• - I M •

* • * Ci k - l # •"» • ' . i * r>

n i l * « • • * <•••

• N • • « >C k <-4 " ' .

• - « «-4

« • * T • Ou

k O • fj> f c " * I - *

l - l M

«

* * • • « f k H k < J * -• • o

l - l h *

» * • 4 i . l * r • • * » t» » v i * • • k H • o • * • » -«I 4 4-* r»

« • » • « 11 « O « " • "!» « r.-

t-t t-* k

* • « »-» « r • u « c • >c * • ;

* • • * M « 4» t A" * ro • M * Ce

l - i H M H « * « * k • « « « * ' » S # . - 4 >-J « ' - • • 4-1 k *-.i k re * «•• k «••«

l - i l - l 1-4 »- ) k k

k « • k N * vn k r k O

• • * t * t u k «o * <••)

* -v • cr

I -» l - l M #-4 k k k k k • k • •f • ' # f \ >

c io * .r • - J * v f ! » I v « V I

« k k • k t * k - ' k f < k / . » • -a

i - . 1 1 k k • » k M k * -• •-.-k *

« Mi k • - »

k k k • • .o k <_> • « * k ». • •:•

1-4 »-4 * • k • k * • k - o k > J

k ro 4 cr » v i

H H H H N k k k

• k • K l l i t .£> k 1 0

""» • «O k l V ^ • •> k i >

' H H H I - ' H

i • » i • k • k l i N k « 9 * • * ' t a k r o * • i f * %o * •"» i < • k "•»• k ' j > k o « ' « J l k a k

1 ^ H H H k * » k

• k k k k k • • • • k • k c i * * - t - o k «•- k «4 4 u j « C J k - J * f • -*i » r * » n « »-» k < k ' • » f t t- t o>

1-1 »-4 H * H l - l l - l k k k k k k

* k k • k <2 k rw » t& • •«» k : k tf

« « k

« I • • k o> « « k

H H H N k k

k <-k !/J k <* •» N

4 S.1 M M

k k « • » » J k m k * r v k u

•k . - • k > l > t v •• -v k * ' * r

K M k • r • » w k -N • M k v0 k VI k ro

k « i k o i k u « «•*.

» - i n k k k • k »-* k . > k « I • >0 k N k U

k • » k * • k x » N k C> « a\

t-4 M * k k • * • -k J-k VI » 0J k u k c

* )» » iv . o u n * e t k a . * k » k M * i * N * r » x

H H H M k k k k * • k • k k» « r o k «a k t a « u t u k i r k ~ l I U I N * t- * t*

k iM

» •--. k o k i * k f \ »

k k k »

4 l * k *" k k » * r t l > C -k vn

k -M k . 3 k ^ • «•" I U I VI k * • 4 t\> » ,(• t M

H H H M k k k k k • k • k -F « -M k t j « - J k "J k »-»

k - < k u.-k * -« >J1 k («I k vn

k • * + to k t * m * • « o k c> k w k • c » k <— *

M M M M k « k * k » « k t C k i k r u * • k t 0 « « • k > J k k l - * * k • - »

k * « • • c i

« M • O k «

• r l - l M

* • • k -m • f\J • t * • 0> k ta • 0=

k « k • k » * *• • V» k * -k u k w

k • * t . k * •

* 0 > k lv> • f ? • V »

« k k k * • • • k c- k »-» « ^ j • .•» k C J . k c * C * ç> * Co k * • k > i • ••>

n w » - i M

k k * • * . • k • k c - » »->

* N l ' . i « f * s k f k i*J * f i « U * »* • \n

« k k • k • • * « r-k v i k t i k f \ j k C9

M l - l * * * « •« t * « i-* C-k v ; • x-• -r

« « • k k * • t « k f k k » k a k o>

h t H

k k k • • H k t O k * -k - M '

« k k k • • k • k n j • o . k « n * -«-* C * t* « I » I U I k d k V I k •=•• • » c

r-« M M I H k k k k * • k • « I M « U k v i k * • k f j i » » -4 V I > » I O I f f • - » l k « 4

k k k k k • k -k * • k > l l t | l i O k « j k * -i l i n • v i • * • k tfl » »£> > I M H M k k k k k • k • • *-k > l k a » ~j k ro k * -

4 >J « O « -F k v i • e « • f-

* k k • k «O k ro k «u k *o * »-• k M

k « • • « ro k t a k X k k * k c g

k k * * I « • k ro k k k

• - • M

k * • k o k • k o k k k

k o • » _ * _ , • » - > k K - i l " k r o k i - k - * » ' . » r » ' £ k r o k - ^ k \ n 4 o » k - ^ i a k < r * i - » k r o • ' ! • * 1 4 * i \ i k l - « r o k r 4 v i k c r . k c > • o» • * - » fc * c c r ^ i * ^

k l \> k C M I f k ».n k * - k • > ! I - M 4 r o 4 o k u . * t* k H »

4 r i a k t f * r o k a > k o

I N l « I k t 0 • r o k • 4 *• k ,u k ro • m I N I ******* * vu k v i *

k k k • • Q • ro k u • v i « N k M

» - • M k « « • « o • ro • * • • i n • . T . k m

• • • o k ro • vn k c» • r-• K»

t - < 4 4 * k « • k u « r o • o> • 0 -• « k »'

« •

• - * k o * k * * • c= • t i : k • 1 - k • t O k

H U M k k • •

* r i *• r * k 0 3 « * t O * k O 4 k l - «

h - ' H n « k * k « • • • o • •k - t - • k c : 4 • i C k • f\> 4 k ro

k k • «

• k • k o « • * « N I C < 0< k a k C I U « - 0 k 4 * • f * t^ * M h l H 4 H

k k « k

i 4 « « C » 4 ( J t • - J k c * ff « i t < vO I V I • U « i f i >J « - 4 « l - l * - t 1-4 4-1

k k k k

• k • k o k I-» k - M k M k N l o l C K I C I 4 M k N k N

k k k • k »-• 4 vu 4 v : « «* • X-• ro

k • « «Ni • V I • - 0 k f k «T

k i • k O . k . * -k rv 4 U t « ro k vn

k k k 4 « -NI k vD

<7< 4 I H K 4 «

k k

4» * r v t û k v » C k - M r- k » o x- 4 v n r o * 4T *-* *-* 4-t

k k • k • • k » • r j

• t f k v . 4 - 4 4 CT • «" 4 4 "

k k • « *-f N

f C : t J • u i n » - J 4 & * u3 4 VI

1-4 | - « | -4 4-4 k k k •

k k k k k • k k t f • * I M • « I C « f * -4 * f ! - • • * c k

• u i r I N • f I V « 1 Û I I M | o . k i ­l o » 4 > n 4 c k - 0 4 N I • » I N f t - I l \>

U» 4 M • J ~ 4-4 1 4 »-4 • - ' l « H H H M H l - l

* * 4 I » 4 * k k • k k

k k k • k 4 « o j • * • k « • I N » o» * c ; I W I N .. t » N 4 N f O

• N • VC 4 4 -4 a-4 C l 4 4 -

I • -» »-4 1-4 1-4 1-4 4 4 1 4 4 I I I

k k k k k • * l i T I I r o k • i t o i c r k - 4 k < f « - N «

1 4 W H M k k k « k i k i «p • • rc « • • t û * c o « > 4 k 4 1 - 4 k CC k

» 4 4-1 4-4 l - l k • • •

1 1 - I C 4 • » 4. • - 4 4 -• V I 4 N 4 4 * 4 t : . 4 >.:\ k « • c i f k» l e . i -si k t C i -N I I'M I < - k -M • U 4 N 4 -M 4 VJI I IM • (!• 4 » l • VI k C 7 «3

H H M H I - I M I 4 H M M 1-4 l - l 1-4 4-4

4 I M I 0 . 4 4 * k - s i • • V 4 X - 4 -~J 4 %T> > O I ' 4 < CE i 1 i a i l u * * su • o* * »C I M 4 O." I 4T-

• V*S I •• I -C H I I I 4 M H H

4, 4 • <L i ro • «o i N I I M 4 t '

4-4 4-4 4 4 M M

3

7

2.4.1. C w equation* tfo* tkt extn.apcn..ition length

The gsomst-y is the same as in $ 2.1. but the entering current ir> canceled ; a source S is place 1. insido t!ie medium, at n large Jistîir.ne L from the boundary ; S(y) is the infinité medium angular flux dus to this source

Using Placzek's lemma two integral equations can ba obtained far the transmitted angular flux v~(u) t

u>o 0 «= S(u) • / G(y.w') v'tp'J y' du' (2.14)

p<o v'lp) - S(u) * / Gtji.p') p' dji* (2.15)

Solving one of these two equivalent equations, v (u) can be deter­mined ; the asymptutic flux inside the medium is a sum of two terms one due to the source S tha other to the current v"(y). The extrapolation length is the limit :

•o L as J X = lim

*+o *

A variational argument can be developed to obtain an approximate expression of X.

r Let us define : s •

m / S(u) y dp

The two C pquation: for the extrapolation length are :

0 = s* • 7 ^ G+™ v" (m - 1 to n) (2.16) AM

1-N

s~ - V* C-)1"1 v j ~ + G + T (m » 1 to n) (2.17) m L-i % [_A+m +JLJ

&=1 £.4.2. Numc/Ucal xeMiJUi

The CNRAYL program computes the extrapolation length as well as the albedo. Table 4 présents the convergence of C N extrapolation length obtained by resolution of Eq» (2.16) and (2.17). The case is the 6wne as in S 2.3.1. (c = 0.8, a " ± 0.E).

8

a = - C5 a ' = + 0.5 C N 0

Eq. ( >./) Eq. (2.6) Eq. ( 2.7) Eq. (2.8) C N 0 .784 342 370 7 .763 429 196 7 .796 345 213 3 .77 5 480 527 1 1 .783 239 577 7 .763 790 682 2 .791 250 005 4 .791 859 985 6 2 .763 2K9 142 1 .763 251 737 7 .791 204 580 •j .791 1f>3 924 1 3 .783 268 263 9 .783 268 170 4 .791 205 010 7 .791 203 867 5 4 .793 265 261 5 .783 268 399 0 .791 204 007 i .731 204 161; 1 5 .783 2B8 261 4 .783 265 242 0 .791 204 007 0 .791 203 984 1 6 .783 268 261 4 .783 268 263 6 .791 204 007 0 .7'J1 204 013 4 7 .763 268 261 4 .783 268 261 1 .791 204 006 S .791 204 006 7 6 .783 7UU 261 4 .791 2J4 COB 9 .791 204 007 e. 9 .783 2G8 261 4 .791 204 005 9 .791 204 007 2 10 .783 2SS 261 4 .791 204 007 0

TABLE 4

As it had been shown in [ 3] the first Kind C^ solution is a clas sical variational approximation : its accuracy is much better than that of the second Kink which is only a Kahan and Rideau approximation.

Table 5 presents the Z-u results with 6 exact figures for various values of c and a.

2.5 Albedo end tiamnu^siion jaxJjonJ^OK a 6tab

4 A slat) of thickness T is considered : an ingoing entering flux V (y) 1B placed on the left side ; the slab reflects v~(y) and transmits V^(y). These two functions are to be determined by the C N method : the Placzek's lemna is used for the two faces of the slab and two equations are obtained in each side :

2.5.1. Equatcoiu

Left side :

y>o 0 » / v*(y') G(o,y,y») y' du' • / v'fu'î

f Jo

G(o,y,y') y' du'

(2.18) v (y') G(-T.y.y') y du'

y<o v (y) J v+(y') G(o,y,y') y' dy' • / v"(y') 'G(o,y,y') y

f (2.19)

T v (y') G(-T»y,y') y' dy'

t t-J&ËÊ

:4

•r

,y;^'

M M » 4 M l- l M 1 » 1-4 n n K4 !-• rt H p-4 l-« 1-4 M I r t 1-4 t-« H 4-4 M 1-4 1 CM » N. i » e\j » f^ » r j » CM 4 r-. 4 CM » IA « CM * <0 4 • J - » a> i » <•> » SL> * W 4 r«- 4 U> 4 r- 4 -» • •ii 4 r- 4

» «-< » Ps. < • is> » « 4 .* • vO 4 fM 4 - t 4 IN. 4 <o 4 r » 4 * O i » CM » m i t- "> » ^ * «M 4 (O 4 «O 4 . * 4 N • .» 4 CM 4 * • 1 • •-4 4 » i » n » O» » .X • t> 4 rt 4 J - 4 • 0 4 fN 4 «O 4 * «H > » r - 4 P. » •< t K. » cr> » "O 4 V 4 a> 4 CP 4 0» » 0> 4

» • * t » • •

• i r • » • 4 t * » •

• 4 • 4 4 4

• 4 • 4

• • » 4

» 4 4 4

• 4 » 4

• *

1 CM • S 1 rt » M » a» 4 f * 4 . * 4 l i t 4 «3 m •o 4 >0 4 • i n • e> • J1 • O 0 :ft « J> 4 r-- 4 ri 4 •O 4 -r 4 rt 4 » o » ro t i\» t ^> • a » X> * r o 4 •O * •o * <j» 4 . - I •

* CO « 1 ri » i n • M 1> lA » a» » rt 4 3 4 O 4 •SI 4 û> 4 CO 4 4 • 1 » ri 4 rt » m » o> « K> 4 «O » •ri 4 J » . £ 4 r-. 4 co 4

• N. » P*. 4 IV K. » N . 4 *» » (O 4 0> 4 0» » o» 4 3» 4 J> 4 • • 4 • 4 • »

• I * • • » 4 t 4

• 1 • 4 4 »

• 4 4 4

4

• • 4

4 4

• 4 4 4

• 4

r o » W < • ^t » rS- * U3 4 . * 4 <y> 4 c: * CM 4 PO 4 •t 4 I- \D » (A) t I U » o » > oo 4 r o 4 O» 4 C » <o 4 o> 4 o » • m » O I » o> » r o » M • ^< 4 ir> 4 J 4 J - 4 PO 4 PO 4

• 10 • » W 4 Ift > • o » M » t». 4 e - 4 » 4 0» 4 rt » i n 4 PO 4 4 • • r ri 4 rt > r tft • c e t <n 4 T> » rt 4 . f 4 P» « •o 4 O» 4

• IN . 4 Is. i » rv » r - * CO 4 50 4 a» 4 5» * C 4 co 4 <r :; h • 4 • 4 • *

• 1 k • » » 4 » 4 » 4

• h • 4 4 4

• 4 4 4

• 4 • 4

• 4 4 4

• 4 • 4

• 4

» J - » ro i » fs- * -f » m 4 M 4 m 4 ri » O 4 .> 4 O 4 • 3> • •J) t rt » •-> * I - - 4 < • 4 O 4 =0 * 0" 4 «0 4 r~ 4 r CM 4 » -O » <SJ » r - 4 (O 4 «> 4 IA 4 •o » ï - 4 U l 4

» J - 1 • rt 4 •7 • co • rt • J - 4 CO 4 » 4 CM 4 >o 4 o 4 P>- 4 4 * ' • >H • •w* » - * • » <J> 4 I O 4 r- 4 rt 4 i n 4 r- 4 o> 4 0» 4

r P-. * p». 1 • r * » r^ » <o 4 «> 4 CP * CP » 0» * £ \ 4 0» 4

• . 4 • ' 4

• ; • ^ • 4 » 4 t 4

4 » 4

• 4 * 4

• 4 4 4

• 4 » • 4

r ^ * I t • - f » 3» 4 j - 4 0» * C I 4 M3 4 * * J 4 GO * > "0 4 *4 » tv> » a» » (M 4 J - 4 » 4 J» 4 » 4 • 0 4 * 4 r <i> 4 PO < t f > » a 4 fO 4 <t> 4 PO 4 rt> t » 4 •T» 4 IN 4

« CM ! • W 4 J - 1 » « » a» * CM 4 O 4 C» 4 O 4 M 4 . 0 4 O 4 * • I » rt » •ri » -» ( « t I O • I s - 4 •ri 4 tr* 4 CO 4 a» 4 0-» 4

r P- * »N 1 r t^- » |S- 4 « • T> 4 <r» 4 0 ' 4 fP* 4 7* 4 C» 4 r • * ^ * • 4

• 1 ^ • » - 4 4 * » 4

• I • 4 4 4

• 4 4 4

• 4 * 4

• 4 4 4

• 4 4 4

• 4

• ST* » O M » N . » a 4 r o 4 l » 4 rt 4 <0 4 • 0 4 0> 4 o 4 • K » •O i » 90 • r - * »0 » m 4 »0 4 r- 4 t n 4 • > 4 O 4 r J» 4 •J» » ^ » r.« 4 IT> » 30 4 T> * •* 4 *jr> 4 • •») 4 r » 4

• O i » C) 4 •o i r t^- » t». 4 C» 4 -» * •3» 4 ••• 4- « 4 T 4 O 4 4 • : • •rt » •ri 1 » J - » « 4 CM 4 r- 4 y-i 4 «0 » co 4 tP 4 O 4 4 f^- i C- 4 Ps. 1 t ^ t S < "O 4 co 0 0"> 4 3> 4 C» 4 7 4 r 4

• • * * 4 • • • « • | • » • 4 » 4 • 4 • 4 • jt 1 • 1 4

» 4 » 4 4

4 4

4 4

4 4 •

rt 4

> iH * tO 1 » «O > O * r o » - f 4 t - 4 CM » CM 4 IN. 4 > U \ 4 U> 1 i m » t») » •n » .* 4 r » 4 Cr 4 cO 4 0» 4 1 If, * \C l • o » rt 4 •O 4 CM 4 rv 4 >0 » . C> 4 3» 4

* CM : : o 4 « > i o i • i r i » K » ro 4 •-J 4 m 4 ^f 4 C» 4 4 * ' rt » rt • » -r t «o * CM 4 O- 4 CM 4 v0 4 CT> 4 0> 4 » t i » r - * r » J r »w » !>>. 4 » 4 CO 4 a> 4 0> 4 O» 4 9» 4

• • 4 » 4

4 ^ • » * 4

» 4 1 4

• ^ • 4 « » • 4

4 • • »

4 * • 4

• 4

• 4 « 4

• I N . » M < » <o f cr? » rt » »~> 4 C 4 ( 0 4 CM4 3 4 • C i 4 J 1 } -O 1 «O 4 I»> » -o • PO 4 •5» 4 CM 4 C 4

*l * ft 1 & t M • PO « IX< 4 «O 4 r» 4 rt 4 • ^ t 4 » - t i e» 4 o . t J » r > 4 ••r» * rt 4 •ri 4 rt 4 C 4 O 4 4 * 1 • rt 4 «H 1 • T » <r> » CM 4 r~ 4 «AI 4 f » * en 4 o 4 1 1 ! » N . 4 K l > K » N . » «f> « cr 4 ?» 4 f » 4 ç\ * "" 4

4 4 IV 4 4 » » 4 ri »

» 4 * 4 4 4 * 4 4

> rt 4 ^t i r «J» 1 N * I O I P0 4 r o 4 •O * r- 4 4 » <D 4 •T < 1 (M » IV 4 rt 4 * 4 '-•» 4 Ut 4 C» 4 4

o 4 * ^ 1 > C> • -f * rt t O 4 M » M * c 4 4 * kP 1 rj •» M t l*> • rt » PO } G* 4 r o 4 O 4 o> f 4 4 • ' r Ti » rt < » -» » «r * CM » tO 4 CM 4 r*. 4 0» 4 4 4 1 * h o- 4 K 1 r f » » N . 4 eo • " 4 fj> « 9> » «o 4 4

* 4

» « 1 4

4 4

4 4

4 4

4-4

« 4 f

» J 4 N • 1 •*> » W 4 rt 4 0 - 4 s 0 4 C> 4 Cs 4 4 > tft 4 ^ J ~> • - t • •a 4 »'l 4 •* * J » r> * 4

<"» 4 K 1 l C » f~ * 9> • W • tf»4 CCI » « -4 4 * « i • <J» 4 (M i » <v> t O» 4 *:» » TV • » •r 4 K 4 W* 4 » 4 • * > i f 4 rt ) ( -f 1 S * M 4 vd 4 M • •O » O 4 4 4 1 4 h f - 4 *. i »- t r*. 4 * i m 4 a» 4 T 4 c-. 4

4 4 4

> 4 1 «

» 4 I 4

4 4 4

» 4

rt 4 » 4 4

3» 4 W i » N • K • -t » -T 4 r» 4 l!» 4 » 4 > .T 4 «0 ^ r» » N . 4 i p f J3 » * £ > • <M 4 4 4 » -O * •f l 1 3> » -• * <o 4 K 4 rt 4 c » 4 4 * 4 O < c » I V I 1 •-< • f » 4 <o » >0 4 P* 4 O 4 4 t

4 • * I •:• 4 rt i - t t K * rt • v£> 4 CVJ 4 C 4 4 4 4 rt 1 r P» 4 r» • * • • t*. 4 •* » » 4 C P 4 0> » « 4 * • « • • 4 >

• 4 • 1 • » • 4

1 4 t 4

» I • • 4 4

• • 4 4

• 4 4 4

4 4

4 4 t

M >-4 1 1 M »-l »-i » ( M H I - ' M M «c n l-f f - l »•* 1-4 M M 1-4 1-4 r-4 4-1 1-4

* / K» 4 O 1 r < ^ » O 4 U 4 O » r < 4 O » CV 4 O 4 r . »

:s/ r t . 4 u» 1 > « ' t O 4 O » ( J 4 •-t 4 V» 4 S3 4 C» 4 - V 4 :s/ r S T » O 1 » './ 1 O 4 Vê 4 t^r 4 0 4 <-> • W 4 •V 4 u 4 4 / < • C/> 4 0» 1 ' T l . ; 4 rs f '•£ » T 4 O 4 r i 4 r« 4 r^ jf

* / ' a» 4 en i • et » « 4 N 4 >0 4 m 4 . » 4 'O 4 CM 4 rt »

1/1 • • » • 1 • 1 • • » 4 4 •

» 4 • » 4 4

• 4 4 *

• 4 » 4

• » 4 4

• 4 4 *

v 4

H h H W M M M M M M M H W U M M I - I H M M W I - I M M M M

03

<

ci ri n O ri L O a c t. a a C J£

ri •ri « t C

rt O (4

JC e 4J * >

+» <>- a o o

M

n ta o C rt

o a «H rt c a o rr o r. n *• o a r» a o 14 «»-• » x

UJ

> y y « * # •

. ^ M M

10

riight sidn :

y<o 0 = I v iu') G(T»y»y') P' du' • I v (y'j Gd.y.y'j y* dp'

1 1 (2.20) v T(u') G{o,y,y') y* dy'

y>o v (u) = I v*(y') GCt,y.y') y' dy» • / v~(y') G(T,y.y'î y* dp'

I (2.21) V (y') GCo,y,y») y' dy1

For the C^ resolution, we have to choose one equation on the left side and ons on the right one ; the expansions are performed as in the other cases.

2.5.2. Nume-'Ucal KO>ulti>

We present only the results for a simple case a slab of one mean frr:e path of thickness and c = 0.8 and a = ± 0.5.

Table 6 presents the results.

For this difficult case, the CJJ method provides a good accuracy : the best results are obtained when t!ia second kind equation is used on both sides of the slab.

2.6 ConcZuiion

The Cjv, method is a powerful tool for the calculation of light dif-fusior in planetar atmospheres : it may oe easily applied to Rayleigh scat­tering in spherical or cylindrical geometries.

In the same way the highly isotropic Demi-scattering law very also be treated.

3. THt CN HETHOV IN C/LÏMPRICAL GEOMETRY 3.1 albedo o& the. o\xWi problem

Let us consider an infinite medium absorbing and scattering neu­trons (in one-velocity theory) end a cylindrical rod made of a black body of radius r (a medium that absorbs any neutron entering it) placed inside it

An angular /lux v (r,R) directed to the outside medium is applied on the blacK body surface j the outnide medium reflects, at the radius r, an angular flux v"(r,ft) » the determination of v"(r,îî) will be named the albedo of ths ou tar problem.

s i v - • • ; . . , *• - _

»_ -:

11

en o v r vr ro a «r en o cn t o « f CM vr T - CM x- vT r s ro i n v - CM O CO U ) CO v—

• • i n v - r> t u r~ • • m CM CM CO CO CM CM rs t n tt> o i i CM CM r > LO «j- T "Ï -

CI •»-» CM t o m t o t o m +> CO t o tr> CO t o u o» o T - \- v~ *- cr CO o V x-- «— *—

an

UJ sr vr " v r •r l U vr *r ' j - vr vr

an o n o cn vr O T - n P I IS .

o t o * - CO T -o f - CM CO rs . r s CO to v - CM o CM i n x— O

• • CO i n r s rs rs • . •y* x— CO CO CO CM CM vr o O a o CM CM t n «r *r vr vr

to 91 4-» vr t o t o t o CO a 4 ' xt CO •si CO CO

eo t r o) v- T— T - v- v 1 I T 01 x- T - x- x- r - CO eo U i vr vr vr vr v r I U xt « • xr vr vr M

s:

CO o M

s: r s a

CO vr

M

s: vr rs CM cn v r

r s a n CO CO CO rs

CO vr

CO en %~ t o O ) T - CO CO x~ CM vr t o r s to t o r - v - CM n CJ i n x~ O x—

z • • •ti­ r s rs rs r s v f • • to P ) CO t o CO vr z CM CM rs O O O O • CM CM rs 1 - v sT vr • < «1 - M i n CO en CO t o CO *» m CO CO CO i n

or rr o> x~ * r— x- v~ cr e x~ T - T - x~ ~ or UJ «r vr VT vr vr LU XT V V xt vr i -i -

CM m n n vr CM CO x~ CM vr CO O «O C r - CM vr CM rs CO i n v - CM n r - en CO T—

• * «*• cn o CO IS. • . m n CM CO CO CM CM i n t n CO o o CM CM 0 ) i n xt vr vr to * J «- CO in CO CO n +> *— t o CO cn to a to o x— v- T - *— cr eg o T * x~ v - v -

UJ <• vr vr VT vr LU vr sr XT vr vr

en o o «J- vr eo. CO en o co CM CO CO m

v CM «r . 3 i n T vr v - CM to i n rs . CO CO

• • co CO i n m i n • » n n t o CO CO CM CM en CO o CO <o CM CM CO m •* vr vr CD 4-> n en cn cn en in +> T - r> n O o CT 0 ) oo rs rs rs rs ex a CO CO t o CO CO

UJ CM CM CM CM CM tu CM CM CM CM CM

CO v - rs CO rs CO CJ) x~ vr CO CO « - to r-T - CM CO rs. T - CO CM x- CM T tn rs r-» i n

• » CO T * CD i n m • . r~ cn to CO CO CM CM CN CO rs CO CD CM CM i n v- vr vr vr

o a) + J r s . CO CO cn a a -p r s o o o o CT CO CO rs. rs rs rs t n or a> CO t o CO CD 0 3 vr

o I U CM CM CM CM CM vr

LU CM CM CM CM CM CD

tu t n CO

CO vr

tu t n CO

CO vr CO cn r- o CO CO CO CO

o> O l r -r. CO en CO i n

o _ J « - CM o CO CO vr vr rs v - CM i n n CO CO CO CO

CO rs i n i n t n CM • • CO cn CO CO CO CM

< CM t N cn CO CO CO co 9 CM CM t o *r xt vr vr CO + » t o cn cn en cn CO -M O ) n o o o cr. co r s rs f s rs rs a cu rs CO CO cn t o LU CM CM CM CM CM LU CM IM CM CM CM

K~ '> rs en rs «*• en T - vr CO to o CO O T - CM co O o CO CM r - CM CO * CO rs i n

vi­ CO 3) i n i n • • cn T - co CO CO CM, CM r­ CO rs CO CO CM CM CO CM vr vr vr cn v er) to cn cn o> 0 ) +•> en O o CT o cr cu GO rs f s rs rs. cr a CO CO 0 3 CO CO LU CM CM CM CM CM I U CM CM CM CM CM

0.5

sion

ility

0.5 >»

c •:• O -rl

•ri r-t HI -ri

i O « - CM CO «T •H a • O T - CM CO V •ri Xi

e

O u u u O

Coll

rcba

* 0

u U U C J c_> rH ta r - l JO

o o a. a.

<

'•n5j3BFi^''***''(rt->f

^^m^ #i * I -. t.v». ".A-;«,-'.:,

«M» L-lTi--Tn 11Ï-' -^I'tHiiiiniiii r t^amtUi '•-iiiiiEi'i

MMMVMM fMpMH •* - •> ' -^ j " f —H • i miiiiiiiiiriM r - T H irnv m r in tin r

12

3./.?. Equation*

Definitions : - For point datemination, semi polar coordinates are used : radius r and azlrnut . - Around a point r. 'Ù is defined by the cdatitude 0 and tha polar angle $ with rer.pect to the radius r.

Angular Green's function : Wu consider a cylindrical shell source, located on the cylinder of radius r', emitting one neutron in the direction ft', defined by G' and <J>'.

The total angular flux G(r,5,r*,$') due to this source is the Green's function ; not taking into account the first flight neutrons it is G(r,$5,r\ft').

First Kind equation : Let us apply the Placzek lemma [2], [3], [7] to this problem • the black body gives place to the scattering medium cmi an additional source $?.n v~(r,ïî) is placed on the cylinder r.

^ At the inside surface of this cylinder, the angular flux for H.n>0 is equal to zero :

ïî.n>0 0 = / G ( r , 6 , r . ï r ) v * ( r , f t ' ) $ ' . n dft'

• * I . G ( r , ï î , r , â ' ) v~(r , f t 'J j î ' . n d$' (3.1)

v~(r,5) Second kind equation : Ths angular f l ux f o r fl.*n<0 i s equal to

2irr and we obtain :

ft.fco £ & & . f G ( r . f i . r . f r ) v + ( r . 3 ' ) fr.S df r '2V

f J2n

(3.2) Glr.îl.r.fr) v~(r,0') ft- .n dSS"

2TT

3.1.2. Calculation o& tht albedo

Knowing v ( r ,Q ) , ws define the albedo 0 by

/ _ v~(r,$î) iî.n d$

f + v +(r,$) i$.n dfi

For convenience, WB normalize v*(r,ft) by : / v*(r,ft) jî.n d& - 1. 2ir

A variationnal schema [3], [5] had been developped for the albedo calculation : the stationaritj condition of a Roussopoulos's type functional is equivalent to the basis Eqs (3.1) or (3.2). If the unknown angular flux is expanded over a basis and if the expansion is truncated at the order N, a bilinear form dopending on the expansion coefficients is obtained. The statlonarity condition of this form gives o system of linear equations for the expansion » thin system of equation can be obtained in a simple way.

M n* ——-—. «,*&«<*%.,..-**.>*>*.*, .*m~ *

•srs"f- U Â A J nfcrifc i m —•'•• -, "^»»

y '•"*

\S

% *»# ?»

P m W:

S

m À

^•w^^w^*-"*^»»^ '^^ ' fr*#»t^j*'<*y'ft*'iK "•< Ml riMirtiO

S S K j j a j i g

14

G 6 ' ^ = / - f tS) 3 . n dS / f „ m t ô ) G ( r , f t , r , f t ' ) S ' . n d s i ' (3.G)

Whore c and E ' can have the va lues • t>r - .

For Eq. (3 .1 ) , named f i r s t kind equation, we obtain the l inear system of equations :

N Jk """\ / + + n r i — - A n n \

(3 .7) y jun **m jun -xm/ m=0 1=0

N U

0 . y y / ; G * p q • v" i*pq\ La JLJ \ itm +Am Am -£my

And f o r Eq. ( 3 . 2 ) , named second kind equation, we obtain

- f Z - f v i m *im VAm I -Am I o t-o L 2 i r r / J

(3 .8)

3.2 Mbtdo of, tiie. -inneA problem Lot us consider a cyli'.drical rod of radius r, surrounded by a

black body t the cylindrical rod can scatter and absorb one-velocity neu­trons. An angular flux v~(r," , for w 2ÏÏ", i3 applied on the surface of the rod which reflects and transmits v*(r,fi).

3.2.7. equation* oi thu pxoblm

As in i 3.1. we use the Placzek lemma and obtain two integral equations.

Fi r s t kincf equation :

S t 2ir" 0 « / G(r,ft,r,ft ') v~(r.ft') ( î ' .n dft' J21C

* f + G(r,fi,r,ft») v*(r»$S') fi'.n &'

(3 .9 )

Second kind equation :

ft t 2TT+ - - ^ - f G ( r , â . r , 8 ' ) v~(r,(5») îî'.n &'

• / G(r,{S,r,Ô') v + ( r , 6 ' ) jî'.n oft»

(3.10)

The treatment, previously presented for the outer problem is applied to Eqs (3.9) and (3.10) and two linear systems of equations are obtained :

t r

* « • • •

7$ f-'

WJ." " jF** "m*" '• - . , •

15

first kind equation :

N £>

m=0 l^C

Second Kind equation : N Jl

» - E È k *z • % f :s • °A] m=0 £,=0 L * 2n r/J

C3.12)

The flux Green's function will b 3 determined first : this is the lit* * U L t f \ a ? i 5 0 ^ r ° P l c s h e 1 1 s o , j r c e . The angular flux is calculated by use of the integral transport equation, and the angular Green's function is obtained by use of the reciprocity theorem. 3.3.1. TTie &lux Giceji'i function

An isotropic source, located on ths cylinder of radius r„. emits one neutron per unit height ; its analytical representation is : °

SCr.fi) - —i_fi( r - r ) 6-n2 r °

+ . ™ n The wen-Known Fourier's transform of the three-dimensional Green'. function [7] is used and we obtain :

, i /* Arctg K * C r ' r J ' ^T I J J k r ) J (Kr) dK (3 13]

ïï Jo ° ° 1-c Arctg k u.idJ k

3.3.2. Calculation o<J the. angular ilux port equations " S S ° f i 8 0 t r o p l c s c o t t e r * n g the integral form of the trans-

l o a o t n n T h ^ 1 ! 1 r S t S ™ ° f t h e r~ h~ s l 3 d u e t 0 neutrons havinS undergona at least one collision (f), the second to first flight neutrons (f ).

1G

Calculation of f :

0 end $ determine, at the point r, the direction ft. At "?' ths flux depends only on the distance to the axis to, by use of a slight change .-.a write :

f (r,e,<|>) 4*Jo . S i n % ( U ) d X

sin 8

The expression of $(r) given by Eq. (3.13) and the Dessel functions addition formula are used, we obtain :

7cr.e*i - -^ Y) "** r 8* ± i J°

Arctg k J {kr ) J (kr) - — T — ~ o o m 1-c Arctg k

k x

dk

f m sin 6

For m > o, the use of the Lipschitz integral [9] of the Bessel functions gives :

lb (k ' 9 ) " l • 8 i n ° J (kx) J X 1

m sin 6 / • k 2 sin2 6

f/+k 2 sin 8 - 1 k sin 6

Lm

For m < o

And finally

,m *(k»6) * (-) ib „ (k,0)

f(r,e,$) 81T E"° in,* r" Arctg k

e i m v I j (Kr ) J (kr) é (k,6) < . . . dk l o o m vm 1-c Arctg k

HI"-» (3.14)

Calculation of f(r,Q,<ft) : For the calculation of f_, we uso the representation of S(r,$) ;

S(r,fo - —j / Sir ./o

J (kr ) J (kr) dk o o o

i i

1?

The sarncj treatment as above i s used and we f inal ly obtain tha angu­lar flu;' dun to an i so t ropic source :

1 » - . /•«> kdk f ( r . 0 . * . r o J - - 7 22 B * / W V K r l * n ( K ' G ) ^ T T - E M T

8Ti ~~ JO .- f J

(3.15) 3.3.3. CaXculatÂ.on ojj -t/ie GVieen'a function Gl*,S,*' ,f t ' )

The angular flux due to an isotropic source (G(r',£$,r') ) is known from Eq. (3.15) and by use of the general reciproci ty theorem we Know $ t r ' # r , -^ ) which Is the t o t a l flux due to a source emitting in the direction

$ ( r ' , r ,"fi) « 4ir G(r , - f t . r ' ) o o

• ( r ' . r ,ST) ~ V * C-)m e 1 " ^ ' f" J ( K r ' ) J m ( k r ) * (k .0 ' ) , o 2TI - ^ I o m o m 1-c Arctg k

m=-" k We usa the sams treatment as previous?/ and obtain the Green's

function G(r,$,r* ,$?' ) which does not take into account the first flight neutrons :

B l r . 3 . r . . 3 . ) - ~ V V ( - ) m ***' e i n * BIT

m " " e 8 n 7 S " " (3.16) /•"» kdk

x I J (kr 'J J (kr) f (k ,9 f ) \\> (k,6) m n rrn "n 1-c Arctg k K

An other derivat ion of th i s Green's function i s given in [3 ] .

3.3.4. UomntA oi thz Gteen'a function

The moments of the Green's function are defined in Eq. (3.G) j using the expression (3.16) of the Green's function, these moments can be represented as a double sum of in tegrals j the t ransient part of the i n t e ­grals i s comput&d by numerical contour integrat ion and the asymptotic part is obtained ana ly t ica l ly .

3,4 C.. mmeMcal KtevJUtA JOK the. aJLbtdoi o& iht inneA and ovuteA ptobtemi

The CNCYL program was written for the CN solution of one-radius cylindrical problème and primarily, for the solution of the inner and outer problems. The order N of the expansion i s not limited and a l l the integrals are simultaneously computed to take advantage of the recurrence r e l a t ions .

18

3.4.?. Atbzdo o$ tilt oate-t pKobiom

We have carried out the numerical comporion for the case c = 0.0 r = 1 ; the C, results depend on the ordfir of truncation far the expsniiion. Tabler. 7 and 8 give respectively the recuits obtained for the approximation C n to L A , with a truncation order between 10 and 40 for the first and seconr kind equations.

TABLE 7 Albedo of the outer problem (c = 0.8 r = 1.)

10 20 30 40 CO .2293724 .2293217 .2293164 .2293151 C1 .2275367 .2275339 .2275385 .2275384 C2 .2279722 .2279774 .2279772 .2279771 C3 .2270998 .2279042 . .279045 .2279046 C4 .2279314 .2279332 .2279385 .2279335

TABLE 8 Albedo of the outer problem (c » 0.8 r » 1.)

10 20 30 40

CO .2285958 .2284043 .2203863 .2233820 C1 .2280145 .2278369 .2278203 .2278163 C2 .2281071 .2279376 .2279217 .2279180 C3 .2261143 .2279472 .2279317 .2279280 C4 .2281212 .2279526 .2279373 .2279337

For this problem it is rather difficult to obtain a result for comparison with an other method because we have to treat an infinite medium. We used the COLINE [10]program to calculate the albedo of a cylindrical an-nulus limited by a radius (r^ « 10.) with a boundary condition of reflexion (the albedo being calculated by the C method).

We consider the 40-term results as converged and we compare on Table 9 the results of the two equations » the C. of the second kine equa­tion is taken as reference.

19

I i

-Vsty*,',;

TABLE 9 Convergence and relative error for the

different C approximations (c - 0.8 r - 1.]

Approximation First Kind equation

Relative error

Second Kind equation

Relative error

CO C1 C2 C3 C4

0.2293151 0.2275384 0.2279771 0.2279045 0.2279385

6. .10"i?

2. 10 1. io : 2. 10 b

0.2283820 0.2278163 0.2279100 0.2279280 0.2279336

2. ia"j? -4 5. 10 6. 10~j* 2. 10

COLINE 0.22791 0.22721

The convergence of the second kind equation is faster than that of the first kind. For th« second Kind equation en accuracy of the order of 10~ 5 is obtained for the C3 approximation.

In this difficult case, the medium being strongly absorbing, the convergence of th C N method is satisfactory.

3.4.2. Albzdo oi the. innoJi pnoblw

The comparison is done for the case c » 0.8, with a radius equal to 1. and for the second Kind equation ; Table 10 presents the results obtained for the approximations C Q to C with a truncation order betwesn 10 and 40. 4

TABLE 10 Albedo of the inner problem (c » 0.8 r - 1.)

10 20 30 40 CO .6932245 .6934319 .6934507 .6934550 C1 .6923638 .6925482 .6925649 .6925688 C2 .692G906 .6S28E25 .6928779 .6928815 C3 .6927060 .6928768 .6920919 .6928954 C4 .6927112 .6926840 .6928990 .6929025

* *

l — IMKiri 1 i i TOlrfM) i^ n • • « E U * — ni i i . É l ' l à m O m «ml il KIII1> >li I I I ' i l ' W — l l l ilîlll I [-- —

- ' V . •

20

TABLE 11 Convergence and relative error for the different C approximations

Approximation Second kind equation

Relative error

CO c:i C2 C3 C4

0.6934550 0.6925507 0.6926815 0.6920954 0.6923025

8. 10"i? 5. 10 I 3. 10~^ 1. 10

COLINE 0.6929031

On Table 11, we compare ths converged C N results with that of the collision probabilities : the C is the reference.

We set that the convergence is good in this difficult case : with a highly anisotropic angular flux, we have obtained very gooci results.

3.5 fxtAapoiation lenijtii &oi the. cqtindAlcal Mint Vtioblm

The extrapolation length characterizes the effect of a boundary on the flux : in cylindrical geometry it depends of the radiu3 of the boundary.

3.5.1. The Itilne. Vioblw

W e consider the same geometry as for the albedo of the outer pro­blem, but a shell source is placed on the cylinder of radius r 0 j r D is much larger than r and than the mean free path of the neutrons.

S(r,$?) is the angular flux due to the shell source. By use of the Placzek lemma, w e obtain two equations :

ft.n > 0 0 - SCr

v"(r,3) *•"<» Ttff- "s(r

,fo • f v~(r.$5») G(r,$$,r,j$'î fi'.ndft» J2i~ (3.17)

.& • / v"(r.$') GCr,$,r,ft') 3'.n d{T ^2TI C3.18)

Calculation of the extrapolation length : v Cr,ft) can be detsr-mined by use of one of Eqs (3.17; and (3.1B) s the asymptotic flux inside the medium is a sum of two terms, one for the S(r,H) source, one due to v"(r,G) :

V r > ) "*es ( r , ) + C ( r , )

» a s( r ' >

The extrapolation length is tho limit s A « lim & , , ,\ r'-»r a s 1 '

•few** '

21

The problem is the same as f o r the albedo : a var ia t iona l argument has been developed ; i t shows that the* C^ solut ion of the f i r s t kind equation. i s var ia t io ' ina l i n the Kouigoncff *s c lasainol sunie and t l u t of the second Kind i s var ia t iona l i n the usual Rouasopouloa sense. The equation can also be obtained i n the sir ipln way p>'eviout-.ly presented.

3.5 .2 . Approximation o£ -t/ie extrapolation Zejigtk

We define S . and obtain :

elm

Firs t kind equation :

• / SCr.nl f , Û) ft.n dti Le £m

n p5

0 = S * y ^ y ^ G + r S

v" tr) •rs /LJL~J -PP pq

(3.19)

». i

r

q=o p=o

Second Kind equation

0 = S • -rs " ^ r r s a . r s l

q=o p=o 13.20)

3.5.3. Nme/û.ca.1 KWJJUU

_ By the resolution of Eq.(3.19) or Eq.(3.20) the outgoing angular flux V [rM) is calculated, then the asymptotic flux $ v and the extrapolation length ore determined.

The CNCYL code can calculate the extrapolation length in any C approximation. N

We shall begin by a presentation of the convergence of the C N re­sults for the cases r • 1., c » 0.9999 ; ws then give a table of extrapola­tion lengths for c « 1.

Convergence of the extrapolation length for the case c » 0.999 r » 1. : Table 12 presents the results obtained for the approximation CQ to C/j, with a truncation order between 5 and 40 for the first kind equation extrapolation length and Table 13 the same results for the second kind equation : Table 14 compares the first and second kind results : the rela­tive error is determined by assuming that the C4 result of the first kind equation is the most accurate.

« I i

•i

il l»1 . i r Il WlWtHii I.III.I.IIIII, .-»-lA-:,^t>i^^.,.-. ^J. i faiUiLa.1, . , a | : Y^f!).»,,! . - -^iff-riiif jflï r*r ' • * " ' ' " " * * !!•!•> I» 1

22

TABLL 12

Extrapolat ion le.igth of the Milne problem (c = 0.9999 r r 1.)

3*

10 20 30 40

CO .9175325 .9180251 .9180632 .9180719 C1 .9157917 • 91613UÎJ .9151607 .9161676 C2 .915â46B .9158747 .9159032 .9159098 C3 .9155327 .9158603 .9156890 .9158955 C4 .9155282 .9158576 .9153853 .9158924

TABLE 13

Extrapolat ion length of the Milne problem (c = 0.9999 r = 1.)

'if*

'pi

10 20 30 40

CO .9160451 .9180747 .9130761 .9180763 C1 .9156747 .9157217 .9157246 .9157255 C2 .9157874 .9158266 .9158290 .9158295 C3 .9156275 .9158598 .9158615 .9158618 C4 .9155511 .9158844 .9158857 .9158859

TABLE 14 Convergoncs and relative error for the different C^ approximations

of the extrapolation length (c = 0.9999 r » 1.)

%'

Approximation First Kind equation

Relative error

Second Kind equation

Relative error

CO C1 C2 C3 C4

0.9180719 0.9161676 0.9159098 0.9158056 0.915B924

2. W~3

2. 10 I 2. 10~* 3. 10" 6

0.9180763 0.9157255 0.9158295 0.9158618 0.915Û059

2. Uf? 2. 10 Z 7. 10"? 3. 10"p 7. 10 _ B

By inspection of Tables 12 and 13, the C^ results with 40 terms in the expansion seem to be converged with five figures % the results ob­tained by the firBt Kind equation are the best. This interesting feature is due to the classical variational sense of this equation.

\

23

Extrapolation lonrrth for the caae c » 1. : The extrapolation length in cylindrical gtMrrntry has long been irvjsrfectlv known. The first results were published by Davison -and KushnariuK [11], than a variation! calculation was carried out by ^aretsky [12] ; a new approximation was présente^ in las'* by KushnariuK ?"•! "r.x Kay [13j and variational resu.'ts by flex Kay [14] in 13 GO.

B. Pellôud [15] published in 1960 a spherical harmonics calcula­tion of the Extrapolation length and a table of mort probable values. As W3 hav« snsn previously thj accuracy of the C^ results allows values extra­polated for c • 1. to be obtained. Table 15 is reprinted frcia Pellaud and we have addad our results.

TABLE 15 Comparison of C w and Pellaud results for c • 1.

Approxi­ Approxi­ "Pellaud" r mation mation Mac Kay Davison Most probable CN

P11 P13 value CN

0.1 1.201 1.206 1.205 1.205SS D.2 1.132 1.142 1.136 1.13503 0.3 1.041 «.082 1.102 1.084 1.08571 0.5 0.990 0.993' 1.011 1.066 1.011 1.01474 0.7 0.950 0.952 0.963 - 0.933 0.96610 1. 0.906 0.907 0.912 - 0.912 0.91581 2.5 0.805 0.608 0.808 0.821 0.805 0.81D31 5. 0.761 0.761 0.761 0.764 0.761 0.76225 6. 0.743 0.743 0.741 0.743 0.743 0.74284

The spherical harmonics method gives very poor results for the small radii » Davison's expansions are accurate for small and large values of the radius Î the C N method seems to give at least four exact figures in each case. This extrapolation length can be represented with good accuracy for standard calculations :

w , 0.7104r2 • 0.6939r • 0.01147 X(r) = 5 r • 0.5416r • O.ODOGO

Î.6 CnJAlaxl naduui joh. multiplyinci cytindeM

3.6.1. Equation*

We consider the Bame geometry as for the Inner albedo problem, but the cylindrical rod multiplies tho neutrons (K» > 1). This cylinder will be critical if a solution exists for tho transmitted angular flux v lr,U) with on incoming angular flux v"(r,ft) equal to zero. Ths first kind equatior. is thus obtained from tq. (3,9) :

6 « 2ir" 0 « / G(r,fi,r,8') v+(r,JH Ô'.n d3' (3.21)

24

And the second k ind equat ion is ob ta ined front L'q. (3 .10) :

ft « 2ir* - v ( r ' " } = A G(r,£.r.3') vV.3' ) ft'.n dft' (3.22) ïï r J2n,

These equations are homogeneous integral equations, solutions exis' only for discrete values of r.

An eigenvalue problem is associated with this problem ; the same expansion as previously is used and we obtain linear systems of equations :

First Kind equation m=o Z=a

JL l> Second kind equation : V ^ V ^ G +? q + a*?q (-J A) v

JLJJLJ *«m *im 2ir r v

m=o £=o L J

Sm

(3.23)

= 0 (3.24)

3.6.2. NumesUcaZ KzioluXJjon

The matrix elements used in these equations are the same as those used in the previous sections with the difference that the medium multiplies the neutrons ; the Fourier integrals are taken in the Cauchy sense.

The CNCYCRI ^oda was designed for the calculation cf the critical radius of a multiplying cylinder : it solves at any order the first and second kind Cjy equations. We present results in two cases c « 1.1 and c " 2.

Convergence of the critical radius for the case c » 1.1 : Table 18 presents the results obtained by the resolution of the first and second kind equations, with 40 terms in the series. The reference result is obtained by the collision probability code COLINE fio] and a result recently published obtained by the Case method [16]. For the relative error, the reference result Is the second kind C*.

TABLE 16 Convergence and relative error for the different C N

approximations of the critical radius (c a 1.1)

Approximation First kind equation

Relative error

Second kind equation

Relative error

CO C1 C2 C3 C4

3.57526 3.577383 3.577476

B' 1 0-R 2. 10 I 2. 10 D

3.58923 3.576286 3.577380 3.577391 3.577384

3. 10~3 3. 10 g 7. 10 I 2. 10" 6

' COLINE 3.577394 CASE 3.577391

25

The CQ and Cj approximations of the first Kind equation are more accurate than those of the second Kind ; thu first Kind C-, is ICGÔ accurate dua to nurr.iHrJcal difficulties. The convergence of the second Kind equation is sood, and the accuracy of the result is fair compared to the reference results.

Convergence of the critical radius for the ca >e c = 2. : Table 17 presents the results obtained by the solution of the first Kind equation, because, as we have seen in plane geometry the convergence of the secjnd Kind oquation is rather slow. The results ere obtained with series limited to 40 terms ; for the relative error the reference result is that cbtainejd by COLINE. In this difficult case the convergence of the C N result is goad.

TABLE 17 Convergence of the first kind equation C.f results

for the critical radius (c = 2.)

Approximation First kind equation

Relative error

CO C1 C2

0.668130 0.6S8656 0.668614

-4 8. 10 6. 10 'I 1. 10"6

COLINE 0.6SB6130

CASE 0.668613

3.7 ConcJLaUon

Comparison of C N results in cylindrical geometry to reference results shows the accuracy of this method for the calculation of albedos and critical radii. For the extrapolation length the results seem to be more accurate than the previous ones.

The next step could be the treatment of an heterogeneous cell taking into account the spectrum of the neutrons.

^•ra

MC1«V".»4MB1M

26

[1] Bcnoist P., Kavenoky A. - The C^ method of approximating the Boltzmann equation in cylindrical gnometry. Nucl. Sci. Eng. 32» 27C1 (1938).

[2] Benoist P., Kavenoky A. - La méthode C.. do résolution du l'équation de Boltzmann. Journées d'études sur les calculs numériques de réacteurs. IAEA/SM 154/30 (17-21 Janvier 1972).

[3] KavenoKy A. - La méthode C». de résolution de l'équation du transport. Thèse de Doctorat - ORSAY (1973).

[4] Grandjean P. - Application de la méthode C N aux problèmes à loi de dif­fusion anisotrope. Thèse de 3ème Cycle (à paraître).

[5] KavenoKy A. - The C N method in cylindrical geometry and one-velocity theory. Submitted for publication in Nucl. Sci. Eng.

[6] KavenoKy A. - The C^ method in cylindrical geometry and one-velocity theory. Meeting ANS - April 15-17, 1974. Charleston (USA) - CONF 750413 Vol. II , III.67 à 63.

[7] Case K.M., de Hoffmann F., PlaczeK G. - Introduction to the theory of neutron diffusion. Vol. 1 - Los Alamos (1953).

[8] Magnus W., Oberhettinger F., Soni R.P. - Formulas and theorems for the special functions of mathematical physics. Springer Verlag (19S6).

[9] Petiau G. - La théorie des fonctions de Beasel. Centre National de la Recherche Scientifique (1953).

[10] Brun A.M., KavenoKy A. - An acceleration technique for the solution of the Boltzmann integral equation. ANS Idaho Falls Meeting (March 1971), CONF 710302-2 Vol. 2 p. 739.

[11] Davison B., Kushnoriuk S.A. - Linear extrapolation length for a blacK sphere and a blacK cylinder. National Research Council of Canada -Report MT 214 (1946).

[12] ZaretsKy D.F. - Proc. First Intern. Conf. Peaceful Uses Atom. Energy 5_, 525 (1955).

[13] KushneriuK S.A., Mac Kay C O . - Neutron density in an infinite non-cap­turing medium sun funding a long cylindrical body which scatters and captures neutrons, Atomic Energy of Canada Limited - Reaport AECL 1o7 (1954).

[14] Mac Kay C D . - The extrapolation length of blacK and grey cylinders in a purely scattering medium. Atomic Energy of Canada Limited - Report AECL 1250 (1960).

[15] PBllaud B. - The extrapolation distance for a blacK or grey cylindrical neutron absorber by the spherical harmonics method. Nucl. Sci. Eng. 33, 169 (1968).

[16] Wastfall R.W., Metcalf D.R. - Singular eigenfunction solution of the monoenergetic transport equation for finite rodiolly reflected critical cylinders. Nucl, Sci, Er.j. 52, 1, (1973).