ML20012D884
| 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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'U.iS~.i uclsdrlR;gu1Ictory Commiccieni N
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- March 120.J1990:-
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l-I 1xctJ_(w/ Attachments)c a=
p' Darl. S. Hood, Project Manager ~.
Office of Nuc1 car: Reactor-Regulation-7;,
U.E S.1 Nuclear Regulatory Commission:
Washington.1 D.C.-l20555 l
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- Dr. Kahtan
- Jabbour, Pt o' ject Manager '
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Office of Nuclear Reactor Regulation-
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I U.-S.: Nuclear = Regulatory Commiwston-
'One' White Flint' North.
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q.
' Mail'Stop'14H25-Washington. D.C. : 20555 h
'Mr; Robert C. Jones, Acting Branch Chief
~ Reactor Systems (Branch'...
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Office of Nuclear Reactor. Regulation U.;S.. Nuclear Regulatory. Commission.
4 Washington,fD.C. J20555 l
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(w/o Attachments) 14r. W. Ti Orders.
SeniorfResident' Inspector Catawba Nuclear. Station i
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
D
-E
- FE 4 (u) g g
g g
= (0)
( A 3-1) -
I 2
d E
D27~U 2 21 4 I"I 2
du l
where D a group g diffusion coefficient (cm) g E m gr up 2 macroscopic absorptier. cross section (cm"1) ~
2 e
vE a group g macroscopic fission cross section times nu-g I
the mean number of neutrons emitted per fission (cm"1)'
E m group'1 macroscopic removal (absorption plus outscatter) 3 cross aeetion minus 1 vE (cm-1) 7' f i E
a macr scopic transfer cross section from group i to 21 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
0-4 (u)
-B 0
4 (u) 2 g
g du (A3-2)
=
2 d
2 0
,y 6 (u) 0
-B d I")
2 2
du _
2 then B must satisfy the equation 9:
A3-2
y,. -,. _
L
-D 3
-E
- vE (g(u) g g
f 2
=[0].
' ( A 3-3)
E
-D 3 2_ _4 I"Ii
~
21-2 1
' For a nontrivial solution to exist to Eq. A3-3, the determinant of the 2
coefficient matrix must be identically zero. This implies that B must i
2 have very special values.
If the two values of B : which satisfy Eq. A3-2 2
2 are designated x and -u, their values are given by
/E E \\
fU U \\
I 21 1
1 2
2 1
2
" ~Y (q+ %j
+
(%" %j + 7D D 1 2 1
. ( A 3-4) l U
U U
~U 21 2 " Y(q+ gj +
(Yg:.rgj + yD D 1
I 2
2 1
2 8
12 2
2 has been chosen such that it will always be positive, and x can be 1
u i
, either positive or negative. With two '" slow-to-fast' flux ratios" defined j
y to be i
l E
ra 21' D"'*
2 2
( A 3-5) -
U 21 sa
-D " 2 + U 2
2 the general solution to Eq. A3-3 is then given by
~4 (u)~
1 l ~ "a sin x u + a ces x u "
~
~
1 g
2
=
( A 3-6) 4 I")
r s
a sh uu+ a cosh uu 2
3 4
O A3-3
t t
Likewise, the current vector is given by
~J (u)~
-D.
-D a x cos cu'- a x sin xu
~
~
~
~
3 3
1 g
2
. ( A 3-7)
J (u)
-rD
-sD u e sh uu + a u sinh uu 2
2 2_
.."3 4
0-With the definitions
~4 (u) ~
g 4 I")
2 (e(u))e J (u) g
_ 2 "I JI
~1 1
0 0
r s
0 0
(P] s 0
0
-D
-D 1
3 0
0
-rD.,
-sD2_
~
- ( 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 a4_
A 3-4 9
a
~i x-n' j';
X.,
f Y
<s Equations A3-6 and A3-7 can be expressed as (e(u)] = (P ][Q(u)][R ],
' ( A 3-9)
}
The inverses;of (P ] and (Q ] both exist and are given by 7
~
s
-1 0
O-
-r:
1 0
0 1
s f r,
( A 3-10)
(P)~1 =
0 0
i r
1 0
0 y
y 1
2_
and I
sin.zu 0
- ces xu 0
-jshzu 0
c s xu
'O
_g O
-sinh uu 0
1 cosh un 0
cosh uu' O
1 sinh uu M
Hence, the unknown coefficients of the general solution are 1
(R) = (Q(u)]~1[P)~1[e(u)].
( A 3-12)
L to u,1, (e(u )] can-l/
For a homogeneous region which extends from u j
j j
be expressed in terms of- (e(u,1)] by_ applying Eq. A3-9 at u =uj and f
Eq. A3-12 at u = u,1 and eliminating (R) to obtain i-f (e(u )] = (P][Q(u )][Q(u,1)]~1[P]-1(e(u,3)].
( A 3-13) j f
y j
A 3-5 l
l l
l
- ~
9:
3 y,
j-
-s n,,...
..c gm With the further definitions, h a u,g - u. and '(Q) a (Q(u )](Q(u,1))~1 j
j j
j (Q) can be expressed as
~
1 cos sh -
0-
- sin xh-0
-f sinh uh
'O cosh uh.
0 (Q) =
, '( A 3-14) -
a sin sh 0
cos sh 0
0
-u sinh uh 0
cosh uh _
s and Eq. A3-13 becomes
. t.
'[e(u )] = (P)(Q)(P]'1[e(u,3)).
( A 3-15) j j
In Appendix 2, an expression was derived which also related [e(u )) to.
~
y (e(u,g)). This expression, from Eq. A2-6, is j
+ (N Jh
-(et, m n "l)) *
- k 'O m,n(*1+1)) ' -
I which (with the node subscripts dropped since each node is' treated -
separately) becomes
[e(u))=e
)h (@(u,1)).
j j
(A3-16) l Comparison of Eqs. A3-15 and A3-16 indicates that lh = [P) (Q) (P)~1 e
~ ( A3-17)
The matrices (P) and (Q) depend only on the nuclear properties and mesh spacing of each node; hence, the exponential of (N)h is com-i pletely specified by Eq. A3-17 With this expression for e[N]h,
l A 3-6 o
U
n w:
1
- y..
m I
a identities for each of the spatial coupling matrices of Eq. A2-11 can be--
1 derived. Making use of the _ definitions of the hyperbolic functions, one ~
'i
~
L finds that s
)
l 0
0 1 csc xh 0
0 0
0 1 esch uh' (sinh (N]h) 1 = (P)
(P)~1
~
-n esc xh' O
0-0 0-
-p each uh
'O O
( A3-18a)
~ 1 - coa xh 0
0 0
0 1 - cosh uh 0
0 1
~(P]7 '.
(I) - co'sh (N]h = (P) 0 1 - coa xh
.0 1
0
[
0 0
.0 1 - cosh uh _
F
( A 3-18b)
The matrices defined in Eq. A2-11 involve only certain blocks of the full-a (4 X 4) matrices; hence, only certain blocks need to be evaluated.
Several identities which prove very useful in simplifying the matrices are 1
i i
2 (N]h = h
[p)2 (A 3-19a)
+
r s
A3-7
fi r.; _,
q
(' sinh (N)h)h (N) (sinh (N)h)12 = (N)N (A3-19b)
(sihh (N)h)
[N)j ([I)- cosh (N)h)22 = -(tanh (N) h)
(N'I)12 *
( A3-19 c)
The latter two identities are easily proven from the fact that-f((N})g([N)) = g([N))f((N])
for any functions f and g which can be expressed as series involving q
powers of (N).
~ Evaluation of the matrices in Eq. A2-11 is by no means a trivial exercise, but once the algebra has been performed, the following simple j!
expressions are obtained:
l
.a
.g -
( A)12 (P) h
=
_-ra
-sS-4 l
y 6- ~s
-1
~
(B]11 1
=
s-r
_ ry s 6 _ _-r 1_
e g
2
[C*]12 =
(P]
h
_-rc st _
g o
(D+)12 2
(P)~2 h
=
,_-rf so.
-n e~
i 2
[D"]12
{ P)~2h
=
_ rn se_
A3-8
_ _ = _ --_-_.
p 4
? ; ;'. g.c u
y.
[E+]i2 =
(P)~f h 2
_ -re.
sr_
2
[z 112 =
(P)~2 h (A3-20)
,_-rr sp _
a where tanh(
h/2) ; x2<0 I
a a h tah(xh/2);
2>0 1
x m hh S ad tanh(uh/2) 1m~ ch esc (ch)";
2>0 m h h esch(
h) ;
= < 0 x
6 mluh esch (ph) en (7-1)
(*
(6-1)~
e = p({-s)
({+a-1)
L (m
c.g.(4.s-1)
" " h($- 2c) l A 3-9 I
g; 7
- p j
- (,:
o i
' ~
i
.-l (h+ 2tf.
9*
( 2S - 1)
-( A 3-21) -
When vE - is identically zero,- a is idinite, and l'Hospitol's rule g
2 must be used to obtain,
.J
]
a 0
B'1
[ A]12 =
h B
i
_ - (a-s) f y
1 2_
a b
. [B)13 l
=
.. r( 7-6 ) -
6 U
F 1
9
[ C* }12 =
h~
(c+()
-h
[1 2 _,
.0
[o+112 "
h' f (e + o)
- g-
_1 2_
s-A 3-10
J
- . :..f
.')
4 64 i
s
\\
Y I
2
( D )12 '-
h r-9-
y (71+6) y.
1 2
i 0
l y
h' (E+112 "
r 7
y (c + t)
-[y 1
2 r
T
~
(E )12 h
(A3-22a) :
=
v v
p y(r+ 9) y 1
2_
2 9
and. x,-. u" are given by the simple expressions -
U U
2 I
2 x = max y,y 1
2.
(A3-22b)
'U E
2= max t
2 u
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 U
'. -. L'.
+
4 i
J a = q w (ch)2 + m(ch)4
/1-
\\
24 fg- +... )
9 S, q/ 1=(uh)2 + m(uh)4
\\
24 yg- +... )
Y = (1 + "
- )
+
360 8 = [\\ 1 - (uh)2,7(uh)4,***)/
6 360 g.,=[I,7(xh), 31 ( ch)4 T
1
\\
360 15120,***/
1 7(uh)2, 31(uh)4 p,
,, I 360 15120,***
4 0 = (g + 4(xh) + 73 (ch)4 1
720 1209 6 0 + * *
- R + 4(uh) _ 73 (ub)4 I
6=
720-120060-2 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
- " [\\~ I6 + 36(uh)15120 '1814400 + ***)/
- * (\\T6 + 162 (xh)2 + 2008(xh)4 +
- -)/
- 1 15120 1814400
, (.L,162 (uh)2,,,200 8 (uh)4
+ * * *
(A3-23) 7 10 15120 1814400
(
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
I
- that all of the matrices depend on the eigenvalue'of the global static 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
=D.
i 3
1 G
l k -
' 41 y
4 0
9 4
5 l
l A3-13
- m *
,.,