ML20012D884

Forwards App 3 to Topical Rept DPC-NE-3001-P, Safety Analysis Physics Parameters & Multi-Dimensional Reactor Transients Methodology Consisting of EPRI Documents Describing Arrotta Computer Code

ML20012D884
Person / Time
Site:	Mcguire, Catawba, McGuire
Issue date:	03/20/1990
From:	Tucker H DUKE POWER CO.
To:	NRC OFFICE OF INFORMATION RESOURCES MANAGEMENT (IRM)
References
NUDOCS 9003290045
Download: ML20012D884 (14)

Text

-

$ ' ' . Duke'j%unt Cornpany : , fin B khn J *

'P O Bat 33198 ike hesident Charlotte, NC 2S242 :- Nuclear haduction (704)373 4531 :

' DUKE POWER

. March 20,-1990 4

U S. Nuclear Regulatory Commission ATTN: Document Control Desk Washington, D.C. .20555 -

Subjects- McGuire Nuclear Station Docket Numbers 50-369 and -370 Catawba Nuclear Station Docket Numbers 50-413 and -414 Topical Report'DPC-NE-3001-P;

" Safety Analysis Physics Parameters and Multidimensional Reactor (~

Transients Methodology" (

On' January 29,'1990, Duke submitted the subject Topical Report for NRC' review. Enclosed with the Topical Report were 3 EPRI documents'which described the ARROTTA computer code.

As1 discussed'in a telephone conversation between the NRC staff-(Darl llood.

-Stan Kirslis, and Dan Fieno) and Duke Power, (Scott Gewehr and Robert-Van Namen),:one'of the documents which described the ARROTTA Theory was missing an appendix. That appendix is attached.

If we can be of further assistance-in your review of this topical, please.

+: call = Scott Gewehr at (704) 373-7581.

Veryltruly yours, l .  % my llal B. Tucker SAG /215/lcs 9003290045 900320 m, h PDR- ADOCK 05000369 '

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. P:gY 2: O n,  : March 120.J1990:- .

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p' Darl . S. Hood, Project Manager ~ . .

7;, Office of Nuc1 car: Reactor-Regulation-U.E S.1 Nuclear Regulatory Commission: -;

. Washington.1 D.C.-l20555 l x ,

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Dr. Kahtan

Jabbour, Pt o' ject Manager '

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'One' White Flint' North. (

q. ' Mail'Stop'14H25-Washington. D.C. : 20555 h 'Mr; Robert C. Jones, Acting Branch Chief '

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c. Washington,fD.C. J20555 l (w/o Attachments) ,

14r. W. Ti Orders.

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Mr. P.: K'. VanDoorn -

Senior Resident Inspector ,

McGuire Nuclear Station- i

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i Appendix 3 j EVALUATION OF SPATIAL COUPLING MATRICES . -

Actual application of the Analytic Nodal M'ethod requires ' evaluation ,

of the matrices defined in Eq. A2-11 Each of these matrices is a

- 2G X 2G matrix whose elements depend only on the properties of a single node. The essential difficulty in' evaluating these matrices stems from the fact that the exponential of (N1,m,n), as' defined in Eq. A2-1, must be evaluated.. (N1,m.n] is block antidiagonal with its lower block

' being partially comprised of the GX G group-to group scattering matrix.

In the general multigroup case, it is not apparent how to obtain this ex-ponential. -If the matrix [N1,m,n) could be diagonalized in some fashion, the exponential of (Nf,m.n] w uld, of course, be readily obtainable.

-If the number of neutron energy groups is restricted.to a-small- [

number, direct evaluation of the matrices becomes feasible. Since this '

paper is primarily concerned with light water reactor analysis, in ,

which two group diffusion theory is commonly used, the matrices . will x be evaluated directly for the two group case.

The one-dimensional, source-free, two group, diffusion equation -;

for a nuclearly homogeneous region (uj<u<up t) can be expressed ns A 3-1

__- 2._ - . _ . . .

-1, ,

i d 1 Dg -E g - FE g 4g(u) ,

= (0) ( A 3-1) - I 2

d E

21 D27~U 2 42 I"I du l

where Dga group g diffusion coefficient (cm)

• E2m gr up 2 macroscopic absorptier. cross section (cm"1) ~

e -

vE g a group g macroscopic fission cross section times nu- ,

I the mean number of neutrons emitted per fission (cm"1)' -

E3m group'1 macroscopic removal (absorption plus outscatter) cross aeetion minus 1 vE f (cm-1) 7' i E

21 a macr scopic transfer cross section from group i to group 2 (em-1)

(g a grou'p g scalar neutron flux (cm-2 ,,c-1) y a critical eigenvalue of global static reactor problem, and it has been assumed that there exists'no upscatter and all fission l

neutrons are born in group 1 (i.e. , E 12 = 0, Xi " 1, X2 = 0). If a particular solution to Eq. A3-1 exists such that 2

2 0- 4g(u) -B 0 4g(u) du

=

(A3-2) 2 d 2 0 ,y 62(u) 0 -B d2I")

.. du _ _ _ _ _ _ _

2 then B must satisfy the equation

9

A3-2

y, . - , . _

L

-D g 3 -E g - vE f (g(u) ,

2

=[0]. ' ( A 3-3)

E -D 2 3 ~

21- 2_ _41 I"Ii

' For a nontrivial solution to exist to Eq. A3-3, the determinant of the coefficient matrix must be identically zero. This implies that B2 must i have very special values. 2 If the two values of B : which satisfy Eq. A3-2 2

are designated x and -u2, their values are given by 1 /E 1 E 2\ fU 2 U1 \ I 2

21

• +

" ~Y (q+ %j (%" %j + 7D 1D2 1

. ( A 3-4) l 1

U I

U 2

U 2

~U 1 2 21 8

2 " Y(q+ gj + (Yg:.rgj + yD 12 D '

2 u has been chosen such that it will always be positive, and x 2 can be 1

, either positive or negative. With two '" slow-to-fast' flux ratios" defined i

j

_ _ _ . , _ _ _ _ . . . . _ _ y to be .

.- i l

E ra 21' '  !

D"'*

2 2 '

( A 3- 5) -

U 21 sa *

-D 2" 2 + U 2 the general solution to Eq. A3-3 is then given by

~

~41(u)~ 1 l~ "a g sin x u + a2 ces x u " ~

=

, ( A 3-6) 42I") r s a3 sh uu+ a cosh 4 uu O

A3-3

t t

Likewise, the current vector is given by *

- ~ ~ ~

~J3(u)~ -D.3 -D 1 agx cos cu'- a 2x sin xu ~'

. . ( A 3-7)

J 2(u) -rD 2 -sD u e sh uu + a4u sinh uu

- - _ 2_ .

."3 ,

0- With the definitions >

~4g(u) ~

42I") *

(e(u))e J g(u)

JI

_ 2 "I _

~1 1 0 0 r s 0 0 (P] s 0 0 -D 1

-D 3 0 0 -rD., -sD

- ~

2_

~

- ( A3-8)

~

sin zu ces xu O O 0 0 sinh un . cosh uu x ces xu -x sin su O O

• 0 0 u cosh uu u sinh uu _

,1 2

[R ) a- ,

"3 a

- 4_

A 3-4 9

a

~i ... , .

x- .:.

n' j';

X. , f Y

0 m tanh( h/2) ; x2<0 I hh .

S ad tanh(uh/2) 1m~ ch esc (ch)"; x 2>0 m h h esch( h) ; = < 0 6 mluh esch (ph) en (7-1)

(* (6-1)~ .

1 e = p({-s)

L _ (m ({+a-1) c.g.(4.s-1)

" " h($- 2c) l l

l A 3-9 I

g; ,

7 .., :p j

- (,:

o i '~

1 i

.-l 9* (h+ 2tf.

1 1

e a e + m(2a- 1) x7h- i 1'

7mp+ ( 2S - 1) . -( A 3-21) -

When vEg - is identically zero,- a is idinite, and l'Hospitol's rule 2

must be used to obtain, .

.J ]

a 0

B'1

[ A]12 = -

h  ;

B i

_f - (a-s)y2_

1 a b

= l

. [B)13 6

. . r( 7- 6 ) -

U F1 9 h~

[ C* }12 =

[1 (c+() -h 2 _,

.0

[o+112 " - h' f (e + o) - g-

_1 2_

s-A 3-10

> 4 -

J

. :. .f .')

64 s i

\

YI *

" 2

( D )12 '- h r- 9-y (71+6) y.

1 2

. i y 0 l (E+112 " h' r 7 y (c + t) -[y -

1 2 r

T *

~

= h (A3-22a) : .;

v (E )12 v p y(r+ 9) y 1 2_

9 and. x2 ,-. u" are given by the simple expressions -

2 U I U2 x = max y,y

. 1 2.

(A3-22b)

'U t E 2

u 2= max y,y .

1 2.

2 When x h or u h approach zero, many of the leading terms in the Taylor's series expansions of a, J . . . t cancel, and it becomes-important to use the expansions rather than Eq. A2-21. The expansions for small ch and uh are A3-11 l

1 l

, _ _ _ _ . . _ . _ _ _ _ _ _ _ . . . _ _ _ _ _ _ _ _ _ . _ _ _ _ _ U

L'.

+

4 i

J a = /1-q w (ch)2 24

+fg-m(ch)4

+...)

\

9

\ ' '

S , q/ 1=(uh)2 24 +yg-m(uh)4

+...)

+

, Y = (1 + " 360 ****)

8 = [\ 1 - 6(uh)2 360 ,7(uh)4 ,***)/

1 g.,=[I

\

,7(xh) 360 , 31 (15120ch)4 ,***/T ,

p, 1 7(uh)2 , 31(uh)4

- ,, I 4

360 15120 ,*** ,

1 0 = (g + 4(xh) 720 + 73 (ch)4 1209 6 0 + * *

• I 6= R + 4(uh) 720- _120060-73 (ub)4 ***

2 4 -

f = (g + +

0****)-

175(uh)4 .

"I~I14(uh)T 120060 ' -).

1 . 478 (xh)4

• (66 + 36(ch)15120 ' 1814400.* ***)/

1 478(uh)4 I

1. " [\~ I6 + 36(uh)15120 '1814400 + ***)/

- 1

1. * (\T6 15120 + 162 (xh)2 1814400+ 2008(xh)4 *-)/ +

7 , ( .L 10,162 15120 (uh)2 ,, ,200 8 (uh)4 1814400 + * * * * (A3-23) 1

(

A3-12

p (

~

l J* . - ,

c; ..

~

6

'-- Equations A3 A3-23 completely specify the spatial coupling 2

matrices. From the definitions of x and u in Eq. A3-4, it is apparent -1 1

l

that all of the matrices depend on the eigenvalue'of the global static I '

reactor problem. ,

l The matrices required for a one group model are equal to the (1,1)

elements of the matrices in Eq. A3-22a, with'D 3 =D. i 1

G l

k -

' 41

. y 4

0 9

4 I

l l

5 l

l A3-13

-,-