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Estimation of internal quantities of a BWR core using information on the exit void fraction
Progress in Nuclear Energy. 1985, Vo]. 15, pp. 233-240
(1079~553(1/85 $0.00 + .50 Copyright © 1985 Pergamon Press Ltd
Printed in Great Britain. All rights reserved.
ESTIMATION OF INTERNAL QUANTITIES OF A BWR CORE USING INFORMATION ON THE EXIT VOID FRACTION M . J.
DAMBORG*
AND
R. W . ALBRECHT**
*Department of Electrical Engineering, and **Departments of Electrical and Nuclear Engineering, University of Washington, Seattle, Washington, USA.
ABSTRACT A method i s d e v e l o p e d f o r u s i n g i n f o r m a t i o n on q u a n t i t i e s internal t o a EWE c o r e , s p e c i f i c a l l y the exit void fraction, t o i m p r o v e e s t i m a t e s of a d d i t i o n a l internal quantities using a thermal/ hydraulic model and e s t i m a t i o n theory. The method i s a p p l i e d t o an e x a m p l e u s i n g s i m p l e geomet r y but r e a l i s t i c data,
KCVgOROS BWR t h e r m a l / h y d r a u l i c s , estimating core estimation, minimum norm e s t i m a t i o n ,
internals,
estimating
power
distribution,
nonlinear
I NI'RODUCTI ON I n a n a l y z i n g BWR p e r f o r m a n c e , one i s i n t e r e s t e d in s detailed description of t h e c o r e power distribution and t h e c o o l a n t f l o w c o n d i t i o n s in individual rue1 b u n d l e s . Since limited in-core instrumentation is available, it is necessary to estimate these quantities f r o m models of t h e c o r e and f r o m m e a s u r e m e n t s or v a r i a b l e s external to the core. The c u s t o m a r y way t o f i n d such e s t i m a t e s i s t o s o l v e a s e t or n o n l i n e a r e q u a t i o n s w h i c h model t h e q u a n t i t i e s in the core subject to certain constraints. For e x a m p l e , t o t a l coolant flow is typically known and t h e p r e s s u r e d r o p s r r o m b o t t o m t o t o p o f a l l f u e l b u n d l e s must be e q u a l . A l l such e s t i m a t e s of BWR i n t e r n a l conditions a r e known t o be u n c e r t a i n t o at l e a s t some extent. Both the neutronic and t h e r m a l / h y d r a u l i c portions of t h e models l e a d t o i n a c c u r a c i e s associated with mathematical approximations and t h e use o f e m p i r i c a l data. As e r e s u l t , inacc u r a c i e s i n t h e computed power d i s t r i b u t i o n in local regions are often larger than desired. Hence, o p e r a t i n g margins are set v e r y c o n s e r v a t i v e l y . Additional m e a s u r e m e n t s can be used t o r e f i n e t h e s e e s t i m a t e s and r e d u c e t h e u n c e r t a n t l e s . One a p p r o a c h i s t o compute t h e s t e a d y s t a t e LPRM r e s p o n s e w h i c h a g r e e s w i t h t h e model and compare i t t o t h e a c t u a l r e s p o n s e ( E P E I , 19821. D i s c r e p a n c i e s can t h e n be r e d u c e d by a l t e r i n g model p a r a m e t e r s t o o b t a i n a computed r e s p o n s e w h i c h a g r e e s more c l o s e l y w i t h t h e m e a s u r e m e n t . The a p p r o a c h d i s c u s s e d i n t h i s paper uses t h e f l u c t u a t i n g c o m p o n e n t s of t h e LPRM s i g n a l s . These s i g n a l s have been shown t o be s e n s i t i v e to the thermal/ hydraulic conditions in the core. In particularj t h e t i m e d e l a y of t h e c r o s s c o r r e l a t i o n function between axially d i s p l a c e d LrHNs results i n a measured v e l o c i t y of t h e d e n s i t y wave of t h e two phase c o o l a n t ( g o s a l y , 1980) . From t h i s v e l o c i t y , one can d e t e r m i n e t h e c o o l a n t v o i d f r a c t i o n ( K o s a l y and c o l l e a g u e s , t982). Assuming t h a t two a x i a l l y s e p a r a t e d LPRM d e t e c t o r s a r e p l a c e d near t h e t o p of t h e c o r e i n each instrument string, one has a " m e a s u r e m e n t " of t h e e x i t v o i d f r a c t i o n of t h e c o o l a n t i n at l e a s t a few l o c a t i o n s in the core. I n t h i s p a p e r , we d e v e l o p a method f o r u s i n g i n - c o r e m e a s u r e m e n t s and a model o f t h e t h e r mal/hydraulic properties or t h e c o r e t o r e f i n e e s t i m a t e s of c o r e i n t e r n a l s . I t i s assumed t h a t t h e model i n v o l v e s t h e measured q u a n t i t y explicitely. I n our d i s c u s s i o n we w i l l a l s o assume that it is exit void fraction w h i c h i s measured but a s i m i l a r p r o c e d u r e would a p p l y t o u s i n g measurements of o t h e r q u a n t i t i e s , le first d e v e l o p t h e g e n e r a l a p p r o a c h d r a w i n g upon i d e a s
......
233
234
M.J.
DAMBORG and R. W. ALBRECHT
from linear and n o n l i n e a r estimation theory. The a p p r o a c h i s t h e n a p p l i e d t o a v e r y s i m p l e reactor c o r e model t o v e r i f y feasibility of t h e method. Ne f i n i s h with a discussion of t h e additional d e v e l o p m e n t w o r k t h a t w o u l d be r e q u i r e d for the operational use o f t h e s e i d e a s .
ANALYTICAL APPROACH Suppose
the
model
is
given
by t h e
expression
F(w,y,~
)
=
0
(~)
where F r e p r e s e n t s a v e c t o r of f u n c t i o n s , Fi, i=l,...,n; w i s s v e c t o r or measured q u a n t i t i e s external to the core, w., i:l,...,m; ~ i s the vector of exit void fractions at t h e l o c a t i o n s t h e LPRM i n s t r u m e n t strings, !~i' i = l , . . . , p ; and y i s t h e v e c t o r o f t h e r e m a i n i n g q u a n t i t i e s internal t o t h e c o r e w h i c h a r e t o be e s t i m a t e d , Yi' i=l,...,q. S i n c e t h e number o f such quantities w i l l e x c e e d t h e n u m b e r of m e a s u r e d e x i t v o i d f r a c t i o n s , we w i l l a s s u m e t h a t q) p,
of
I f no measurement o f ~ i s a v a i l a b l e , w h i c h i s t h e case i n p r e s e n t p r a c t i c e , then all core internals a r e e s t i m a t e d by s o l v i n g (I) f o r y and (~ w i t h w d e t e r m i n e d by measurement. In the e v e n t t h e model i s w e l l c o n d i t i o n e d and a s o l u t o i n exists f o r t h e g i v e n w, ( 1 ) can be s o l v e d . However, i t i s more l i k e l y t h a t an e x a c t s o l u t i o n does n o t e x i s t f o r t h e p a r t i c u l a r model parameters c h o s e n a n d f o r t h e v a l u e of .w. This expected l a c k o f an e x a c t s o l u t i o n i s due t o the approximate nature of t h e model a n d t h e u n c e r t a i n t y in the parameter values. In fact, the p r o b l e m may be o v e r d e t e r m i n e d w i t h more e q u a t i o n s t h a n unknowns. Thee one seeks t h e e s t i m a t e (ye,~e) which solves (I) as c l o s e l y as p o s s i b l e i n t h e sense t h a t IIF(w,y
)ll
,~ e
for
the
given
norm given
measurement
w,
=
min y,~
e
Here
the
liF(w,y,~)ll
norm o f
F,
iiPi!,
(2) is
defined
as
the
weighted
Euclidean
by n
I IFI
I
i IF i 12
=i~id
(3)
where t h e d i a r e p o s i t i v e real numbers. Hence an a c c e p t a b l e e s t i m a t e or t h e c o r e i n t e r n a l quantities ss a s s o c i a t e d w i t h an a c c e p t a b l y s m a l l v a l u e of I { P ! l . Note t h a t i t i s n e c e s s a r y t o use a w e i g h t e d norm so t h a t t h e d i f f e r e n t c o m p o n e n t s Fi c o n t r i b u t e equitably. For e x a m p l e , i f one c o m p o n e n t r e p r e s e n t s heat input with quantities on t h e o r d e r of 10 T BTU a n d a n o t h e r r e p r e sents void fraction, on t h e o r d e r of l , t h e t w o c o m p o n e n t s must be n o r m a l i z e d b e f o r e b e i n g summed i n t o a s i n g l e e x p r e s s i o n l l k e ( 3 ) i f a l l c o m p o n e n t s a r e t o have t h e same s i g n i f i c a n c e . Now s u p p o s e that is
a measurement
of
the
vector
*?h d e n o t e d
=
~
m Then i t
must
be
+
(~m' i s
I IF(w,y
e
,(~
,e( m
#1
) I I
f r o m ~e'
/1~
Z
(4)
I IF(w,y
)11
=
min y
m
le suggest t h a t such • y can be f o u n d u s i n g nonlinear i n y, we use an i t e r a t i v e process Given the original e s t i m a t e (ye,,J:e) l e t
e
,(~
y
expansion F(w,y
,e m
elements
of
or
+
A(~
=
y
+
Ay
the
minimizes
ILFli
)11
subject (6)
Since square
F is estimate.
e
(ye,(~e) +
matrices
one w h i c h
(7)
) m
is,
(5)
least square estimation concepts. where each s t e p i s s l i n e a r least
?(w, ym, l~m) a b o u t =
) I I
m
(~ e
m
order
e
I IF(w,y,(~
= m
ij
differs
e
and we w o u l d l i k e t o f i n d a " b e t t e r " e s t i m a t e o f y, t h a t t o t h e known v a l u e s of w and t~. Hence, we seek Ym where
where t h e
which
that
I IF(w,y
Then a f i r s t
available
F(w,y
,e( e
) e
3y and 3~ a r e
JAy y
is +
J
.40( e~
(8)
I n f o r m a t i o n on the exit void fraction
235
~F. (J).. y ij
1
=
by. 8F.
(J)..
=
J (9)
1 J
Assuming (ym,,~m)
and ( y e , ~ e )
to " e s s e n t i a l l y "
JAy y
solve
+ J A~
F,
we have
= 0
(10)
w h e r e o n l y ~y i s unknown. I f 3y i s or maximum rank we can r i n d minimizes 1[3y~y, + 30: °"'~e~' by c o m p u t i n g ( L u e n b e r g e r , 196g)
Ay = - ( j T D T D j
)-IjTDToj
Y
Y
e
+Ayl,~
m
1
~
)11
= Y
or ( 1 0 )
which
(11)
A~
u s e d i n t h e norm. We w i l l d e n o t e t o t h e e s t i m a t e or y and i f t h e
ltF(u,y
,~
e
m
To f u r t h e r i m p r o v e t h e e s t i m a t e or y we can i t e r a t e on ~y, e s t i m a t e we can c o m p u t e a s e c o n d c o r r e c t i o n ~y2 Let
Y
solution
Y
where D:diag(d I,...,an), t h e m a t r i x or p o s i t i v e weights v e c t o r round i n ( 1 1 ) as ~ y l , t h e f i r s t order correction e s t i m a t i o n procedure i s well behaved we expect t h a t
IlF(w,y
the
e
+
)11
the
(12)
Indeed,
to rind
an i m p r o v e d
Ayl
(13)
and w r i t e
F(w,Yl+Ay2,e
m
) = F(w,yl,e
m
) + JAy y
2
w h e r e J i s e v a l u a t e d at ( y l , ~ i m ) . S i n c e we s e e k Ay 2 t o m i n i m i z e of ( 1 4 ) , y we u s e l i n e a r l e a s t square estimation a g a i n and c o m p u t e
AY 2 = - ( j T D T o j
)-IjTDTF(',J,y
y
If
y
1 ,e
(14) the
norm of
the
)
y
left
hand s i d e
(15)
m
we d e n o t e
y
2
= y
1
+
Ay2
(16)
we expect
IIF(w,y This
iteration
continues
according
Y k+l
Ay k + l where 3¥
k
indicates
2 ,~
that
=
Yk
+
m
at
least
a local
IIF( u,y 1 ,~
~
(17)
(18)
evaluated
minimum or
)It
~yk+l
= -(jkToTDjk)-ljkToTF(w,yk,~ y y y 3y i s
m
to
y gives
)11
m
at =
(yk,0: m)
lim k~
y
k
)
(i9)
m If
the i t e r a t i o n
converges,
we know t h a t
(20)
iiY(w, yk,~m) il.
APPLICATION TO SIMPLE GEOMETRY The a b o v e a p p r o a c h has b e e n a p p l i e d t o a BHR model u s i n g v e r y s i m p l e g e o m e t r y . The 1 e e l i s t o demonstrate the feasibility of c o m p u t i n g e s t i m a t e c o r r e c t i o n s ~y t h a t a r e c o n s i s t e n t with d e v i a t i o n s ~ between measured and e s t i m a t e d e x i t void f r a c t i o n s . We have a t t e m p t e d t o make this demonstration realistic by using model e q u a t i o n s i n t e n d e d f o r o n - l l n e m o n i t o r i n g and by using actual data f o r o p e r a t i n g BWR fuel bundles while r e d u c i n g the computational c o m p l e x i t y .
236
M . J . DAMBORG and R. W. ALBRECHT
Me assume t h e c o r e c o n s i s t s of two " M a c r o - B u n d l e s " , each r e p r e s e n t e d by t h e a v e r a g e q u a n t i t i e s of t h e t o u r f u e l b u n d l e s s u r r o u n d i n g an LPRM i n s t r u m e n t string, A model c o n s i s t i n g of two macro-bundles permits us t o r e p r e s e n t the constraints that nil fuel bundles experience an e q u a l pressure difference b e t w e e n t o p and b o t t o m and t h a t t h e c o o l a n t flow in individual b u n d l e s must sum t o t h e known t o t a l . The p h y s i c a l p r o p e r t i e s and o p e r a t i n g c o n d i t i o n s of t h e m a c r o - b u n d l e s are obtained u s i n g d a t a f r o m t h e M u h l e b e r g BilE. Macro-bundle number 1 (MB-I) has p r o p e r t i e s t h a t e r e an a v e r a g e of t h e f o u r f u e l b u n d l e s a r o u n d p o s i t i o n 1 2 - 1 3 a n d ME=2 i s d e t e r m i n e d by position 4=21. The power, q u a l i t y and c o o l a n t d e n s i t y d i s t r i b u t i o n s were o b t a i n e d f r o m t h e KKM process computer (personal communication, K. Behringer). The f o l l o w i n g assumptions were used in developing the ten macro-bundle model: 1) The a r e a of e a c h m a c r o - b u n d l e i s t h e sum of t h e a r e a s of t h e f o u r b u n d l e s i t r e p r e s e n t s . 2) The h y d r a u l i c d i a m e t e r and h e a t e d p e r i m e t e r a r e computed f o r t h e m a c r o - b u n d l e . 3) The a x i a l v o i d f r a c t i o n , steam q u a l i t y and r e l a t i v e heat flux distributions; the average heat flux; mass f l u x and n o n - b o i l i n g l e n g t h e r e o b t a i n e d by averaging over the four bundles. The model we used t o r e p r e s e n t t h i s two m a c r o - b u n d l e c o r e was d e r i v e d f r o m t h e model used i n t h e Power S h a p e M o n i t o r i n g S y s t e m (PSMS) ( E P R I , 1 9 8 1 ) . The PSMS was d e s i g n e d to estimate the l o c a l power d i s t r i b u t i o n of t h e c o r e . As an o n = l i n e e s t i m a t o r , i t was i n t e n d e d t o use r e a l time data to monitor core conditions. I t was a l s o t o be used o f f - l i n e to predict l o c a l power fluctuations due t o c o n t r o l action, The PSMS model c o n s i s t s o f two c o u p l e d modules: a neutronic module called Node-8 and a thermal/hydraulic module called Therm-g. The n e u t r o n i c module has a r e s o l u t i o n of 24 a x i a l nodes and models a s s e m b l y a r r a y s up t o 3 0 x 3 0 . Its outputs include three-dimensional arrays of power and m o d e r a t o r d e n s i t y . The t h e r m a l / h y d r a u l i c module computes the i n l e t flow distribution a c r o s s t h e a r r a y of a s s e m b l i e s . The i n p u t s t o t h e PSM5 consist of o p e r a t i n g d a t a f r o m t h e p l a n t c o m p u t e r s u c h as t o t a l core coolant flow in addition to descriptions of t h e p h y s i c a l , neutronic and t h e r m a l / h y d r a u l i c properties or t h e f u e l . The estimates of t h e c o r e i n t e r n a l quantities result f r o o an i t e r a t i v e procedure between the neutronic and t h e r m a l / h y d r a u l i c modules i n w h i c h t h e s e e s t i m a t e s a r e a d j u s t e d u n t i l they yield external conditions w h i c h match t h e m e e s u r e o e n t s . We u s e d o n l y t h e t h e r m a l / h y d r a u l l c portion o f t h e PSMS model f o r o u r two m a c r o - b u n d l e core. Hence, i n our e x p r e s s i o n f o r t h e model, F ( w , y , ~ ) , w r e p e r s e n t s a l l g i v e n d a t a on t h e p r o c e s s which affects the thermal/hydraulic model s u c h a s t o t a l coolant flow and the heat flux distribu tion. The y v e c t o r r e p r e s e n t s t h e 22 t h e r m a l / h y d r a u l i c quantities estimated by t h e model e x c e p t f o r t h e two e x i t v o i d f r a c t i o n s w h i c h a r e r e p r e s e n t e d by 5. Table I lists the elements of y and ~ and g i v e s t h e v a l u e s f o r ( y e , ! ~ e ) , t h e s o l u t i o n which minimizes iiF]i. Some of t h e s e variables such as i n l e t velocity are self explanatory w h i l e o t h e r s such t h e a c c e l e r a t i o n loss coefficient may be s p e c i f i c to this model. The t o t a l pressure drop is the pressure loss minus the buoyant pressure. The 24 PSMS e q u a t i o n s which are necessary to represent our core ere given in the appendix. Briefly, t h e y c o n s i s t of I1 e q u a t i o n s f o r each m a c r o - b u n d l e and two c o n s t r a i n t equations, one representing t h e f a c t t h a t a l l r u s t b u n d l e s e x p e r i e n c e t h e same p r e s s u r e d r o p and t h e o t h e r t h a t t h e two b u n d l e c o o l a n t f l o w s must sum t o a known q u a n t i t y . Table 2 lists the results of t h e e s t i o e t i o n approach developed in the previous section. This t a b l e shows t h e d e v i a t i o n s ~y i n t h e e s t i m a t e Ye w h i c h c o r r e s p o n d t o each of s i x d e v i a t i o n s ~ i n t h e e s t i m a t e Re. I t can r e a d i l y be seen t h a t t h e r e s u l t s ere reasonable. For e x a o p l e , t h e inlet velocity i n c r e a s e s when ~¢~ d e c r e a s e s and c o n v e r s e l y . Comparing cases 2 and 3 shows t h a t t h e p r e s s u r e l o s s e s have t h e p r o p e r s i g n r e l a t i o n s h i p with respect to ~. The p r e s s u r e l o s s f o r case 6 i s a p p r o x i m a t e l y t h e sum o f t h e l o s s e s f o r c a s e s I and 2. And c a s e s 5 and 6 e r e approximately equal i n m a g n i t u d e and o p p o s i t e i n s i g n f o r n i l v a r i a b l e s . Me a l s o see t h a t k ! P t l b e h a v e s as e x p e c t e d . The v a l u e f o r ( y e , ~ e ) i s t h e minimum o f e l l c a s e s and t h e v a l u e s o f (ye,~o) are significantly greater than (ya,~m) for nil cases.
CONCL USI ONS Our b a s i c r e s u l t i s t h a t one can e x p e c t t o use m e a s u r e m e n t s of e x i t v o i d f r a c t i o n to improve e s t i m a t e s of o t h e r i n t e r n a l core quantities u s i n g a model and e s t i o n t i o n concepts. Plausible end c o n s i s t e n t quantitative results can be c a l c u l a t e d . U s i n g t h e PSMS model and o u r s i m p l e g e o m e t r y , t h e c o m p u t a t i o n s were w e l l b e h a v e d and c o n v e r g e d r a p i d l y , fie s u g g e s t t h a t i t i s feasible t o d e v e l o p t h i s a p p r o a c h i n t o an o n - l l n e operations aid. That i s , i f m e a s u r e m e n t s of ere developed from fluctuating signals of t h e LPRMs, t h e y c a n be u s e d t o i o p r o v e t h e knowl e d g e of q u a n t i t i e s internal to the core. To e x t e n d t h i s approach from our simple geometry to the full core requires n significantly l a r g e r c o m p u t a t i o n t h a n was p e r f o r o e d h e r e . A c o r e any have a p p r o x i m a t e l y 40 LPRM i n s t r u m e n t strings so t h e nodal w o u l d c o n s i s t o f 40 s a c r a - b u n d l e s . W l t h Jl e l e m e n t s of t h e y v e c t o r per
Information on the exit void fraction
237
macro-bundle, the problen is large. Re can s u g g e s t two a p p r o a c h e s t o t h e c o m p u t a t i o n but have developed neither. One c o u l d r e t a i n t h e two m a c r o - b u n d l e a p p r o a c h but l e t one m a c r o - b u n d l e represent t h e f o u r b u n d l e s a r o u n d an i n s t r u m e n t string and t h e t h e o t h e r r e p r e s e n t the balance of t h e c o r e , Then t h e d e t a i l e d macro-bundle c o u l d be i t e r a t e d around the core until consistent quantities ere obtained throughout. The o t h e r a p p r o a c h i s t o e x t e n d our p r e s e n t a p p r o a c h and represent t h e 40 m a c r o - b u n d l e s , A potential computational difficulty results f r o m t h e need to invert a 4 4 0 x 4 4 0 a r r a y which i s u s u a l l y dirficult and t i m e consuming. However, t h i s a r r a y i s v e r y s p a r s e and t h e i n v e r s i o n of l a r g e s p a r s e m a t r i c e s is often very reasonable ( T i n n o y and lalker, 1977). A further necessary extension f o r HWR a p p l i c a t i o n s is to include the neutronic model i n t h e estimation process. The power d i s t r i b u t i o n of t h e n e u t r o n i c model can be compared t o t h e DC f l u x measured by t h e LPRMs. To get a b e s t e s t i m a t e of c o r e p e r f o r m a n c e , it is necessary to iterate b e t w e e n t h e two models.
APPENDI X Our PSMS based model c o n s i s t s of 24 e q u a t i o n s . Each e q u a t i o n i s g i v e n b e l o w i n t h e f o r m Fk=O and i s w r i t t e n using the symbols in Table l for the quantities i n t h e y v e c t o r and t h e p h y s i c a l constants i n T a b l e 3. The f i r s t It expressions represent two e q u a t i o n s each, one r o t MB-I (i:l) and one f o r MB-2 ( i = 2 ) . The l a s t two e q u a t i o n s represent t h e equal p r e s s u r e and t o t a l coolant flow constraints. The e q u a t i o n s for k:ll-14 were o b t a i n e d f r o m r e g r e s s i o n s while the o t h e r s a r e as r e p r e s e n t e d i n t h e PSM8 model.
Pressure
loss i n
channel:
in~i
Fk = P l i Buoyant
oi
~g¢ -
+
oi
Oe
+ 2ri
I
L L + i IO[e,(Z)pg+(l-e (z))Pl]dZ PI i i
,:~:i(z) = a x i a l Non-bioling
length
vs,
inlet
void
= 0
k=lo2 i=1.2
qi(z) vs.
heat
= channel
fraction
in
heat
)V
in,i
-
~0 Lp ~ l ( z ) d z 1
flux
input:
L
H. -H in sat hfg
= 0
k=7.8 i=1.2
relationship:
F
= cx k i
Q. 1
-
pgC°i Co vs. v o i d
k=5,6 i=1,2
= 0
Fk = Q. 1 Io p q i ( z ) d z i 3600API hi=gVin, i Void-quality
k=3,4 i=1,2
= 0
velocity:
Fk = 3600Ap l(Hsat-H.
+
= 0
k=9,10
i=1.2
g g],i 3600PIVin, i
PI
fraction:
g k = Col Drift
+ Ki
pressure:
Fk = P b i
Quality
ioi
e
velocity
va. v o i d
1.006
- 0.4090~.1
+ 0"39Z18~21 = 0
k=ll,12 i=1,2
fraction:
Fk = Vgj, i -
0.~262
+ 0.3728~z
= 0
k=13,1~ i=1,2
238
M . J . DAMBORG and R. W. ALBRECHT,
Friction
loss coefficient:
gk = foi Two-phase friction
- 0.19 loss
Fk = R i Acceleration
e P]
in, i
k=15,16 i=1,2
= 0
multiplier:
1 +
k=17,18 i=1,2
= O
loss coefficient:
( l_Qi )2 Fk = r :
Q2pl
cx p
i-~. 1
k=19,20 i=1,2
+ i =0
1 g
Ford l o s s c o e f f i c i e n t :
F k = K.i - kl - k2 - k3 LI_% jqa,,:~a : averages Equal
pressure
o
k=21,22 i=1,2
the g r i d spacers
constraint:
F23 = P l l Constant
at
- k 4 Ll_e ]
- Pbl
- P12 +Pb2 = 0
total
flow constraint:
F2~
= 3600APl(Vin,l
+ Vin,2)
- GT = 0
REFERENCES EPRi ( 1 9 8 1 ) . Power Shape M o n i t o r i n g System, Vol. I: Overview and Performunce, Vol. 2: Technical Description, Report NP-1660, E l e c t r i c Power R e s e a r c h I n s t i t u t e , Paid A l t o . EPRI ( 1 9 8 2 ) . Technical D e s c r i p t i o n and E v a l u a t i o n of BWR Hybrid Power Shape M o n i t o r i n g System, Report NP-2234, E l e c t r i c Power Research I n s t i t u t e , PaiD A l t o . Kosuly, G. ( 1 9 8 0 ) . Noise i n v e s t i g a t i o n s in b o i l i n g - w a t e r and p r e s s u r i z e d - w a t e r r e a c t o r s . r r o s r e s s i n N u c l e a r Energy, ~ pp. 1 4 5 - 1 9 9 . Kosaly, G., J. N. Fuhley, K. Behringer and R. D. Crowe ( 1 9 8 2 ) . I n v e s t i g a t i o n or the a x i a l void p r o p a g a t i o n and v e l o c i t y p r o f i l e in EWR fuel bundles. NUREG/CP-O034. Tiuney, W. F. and 3. N. l u l k e r (°1977). D i r e c t s o l u t i o n s of sparse network e q u a t i o n s by o p t i m a l l y ordered t r i a n g u l a r factorizatlon, Proceedings IEEE, ~ pp. 500-535. Luenberger, D, G. ( 1 9 6 9 ) . O p t i m i z a t i o n b~ Vector Space Methods, l i l e y .
I n f o r m a t i o n on the exit void fraction
TABLE 1
SYMBOL P1
Lo CO V j Kg f R° r
TABLE 2
VARIABLE
MB
The E l e m e n t s o{ the ¥ and e Vectors and their Nominal or E s t i m a t e d Values DESCRIPTION P r e s s u r e loss (ft) Buoyant pressure (~t) Inlet velocity (ft/s) Exit quality N o n - b o i l i n g length (~t) C-zero Drift velocity (ft/5) Form loss c o e f f i c i e n t Friction loss coefficient 2-ph f r i c t i o n l o s s m u l t i p l i e r A c c e l e r a t i o n loss c o e f £ i c i e n t Exit void fraction
Kg j
1 2 1 2 1 2 1 2 1 2 1 2 1
K
2 1 2
R r r
2 1 2 1 2
Case 1 ~el= .00
.01
0.600 0.578 0.0002 -0.0002 0.0339 -0.0413 -0.00087 0.00151 0.0148 -0.0151 -O.OOlO
-0.0067 0.0 -0.00372 -0.002 0.511 -2.0E-5 2.5E-5 -0.0005 0.3903 -0.0020 0.0863
NORM I IF(w.ye,~e)ll I IF(w.ye,~m)ll IIF(w,ym~em)l I
VALUE MB-2 MB-I 65.454 65.059 4.5667 4.9533 6.6695 6.4351 0.13333 0.15265 3.0537 2.9490 1.0993 1.0971 0.15957 0.16908 71.234 67.818 0.018797 0.018917 5.9439 6.7835 1.9443 2.1830 0.6897 0.7152
O e v i a t i o n s o f y from the Nominal Values in Table 1 and Values o f I I F I I Correspondin@ t o D e v i a t i o n s o f e
Ae2= P1 Pl Pb Pb Vin Vin O Q Lo ko Co Co V . Vg~
239
VALUES OF Ay FOR GIVEN Ae Case 2 Case 3 I Case 4 ~ e l = .01 Ael=-.O1 I ~el=-.O1 Ao2= .00 ~ 2 = .00 I ~ = .01 0.496 0.522 -0.0001 0.0001 -0.0368 0.0287 0.00128 -0.00087 -0.0160 0.0106 -0.0066 -0.0007 -0,00373 0.0 0.421 -0.003 2.2E-5 -1.9E-5 0. 3066 -0.0006 0.0757 -0.0024
-0.452 -0.480 0.0002 -0.0002 O. 0350 -0.0268 -0.00123 0.00081 0.0153 -0.0098 0.0065 0.0007 0.00373 0.0 -0.386 0.003 - 2 1E-5 1 7E-5 - 0 2805 0 0007 - 0 0710 0 0022
Case 5 A~1=-.01
Case 6 , l e l = .01 ~oc2= .01
0.140 0.092 0.0004 -0.0004 0.0689 -0.0681 -0.00209 0.00233 0.0301 -0.0249 0.0056 -0.0060 0.00373 -0,00372 -0.389 0.514 -4.1E-5 4.2E-5 -0.2811 0.3909 -0.0732 0.0884
-0.994 -0.999 0.0001 -0.0001 0.0036 0.0121 -0.00041 -0.00063
1. 103 1.108 0.0 0.0 -0.0029 -0.0126 0.00040 0.00063 -0.0012 -0.0045 -0.0076 -0.0075 -0.00373 -0.00372 0.420 0.511 O. 2 E - 5 0.6E-5 0.3061 0.3897 0,0739 0.0841
0.00171 0.10482 0.00785
0.00171 0.10004 0.01345
~{=-.ol
0.0016
0.0045 0.0073 0.0074 0.00373 0.00373 -0.381 -0.461
-0.3E-5 -0.7E-5 -0.2800 -0.3533 -0,0690 -0.0780
VALUE OF I I F I I 0.00171 0.08252 0.00675
0.00171 0.06947 0.00905
I I I
0.00171 0.06466 0.00765
0.00171 0.10786 0.01284
240
M . J . DAMBORG and R. W. ALBRECHT
TABLE 3 SYMBOL
~
e
A p gc Pl
Pg
~
sat in
k1 k2 k3 4 GT
Physical
Constants
and t h e i r V a l u e s
DESCRIPTION Equivalent
Channel
diameter
length
VALUE
(ft)
(~t)
C h a n n e l a r e a (ft ~) Channel p e r i m e t e r (ft) Gravitation constant L i q u i d d e n s i t y ( l b / f ~ 3) V a p o r d e n s i t y ( ] b / f t ~) Liquid viscosity (lb/s-ft) Saturated entha]py Input e n t h a l p y H e a t of v a p o r i z a t i o n F o r m loss - lower tie F o r m loss - o r i f i c e F o r m loss - s p a c e r s F o r m loss - u p p e r tie Total c o o l a n t 9 l o w
0.04645 12.0 0.4431 32.37 32.2 47.6 2.343 6.667E-5 548.6
516.7 642.8 7.56 29.58
8.88 0.801 9.943E+5
(1079~553(1/85 $0.00 + .50 Copyright © 1985 Pergamon Press Ltd
Printed in Great Britain. All rights reserved.
ESTIMATION OF INTERNAL QUANTITIES OF A BWR CORE USING INFORMATION ON THE EXIT VOID FRACTION M . J.
DAMBORG*
AND
R. W . ALBRECHT**
*Department of Electrical Engineering, and **Departments of Electrical and Nuclear Engineering, University of Washington, Seattle, Washington, USA.
ABSTRACT A method i s d e v e l o p e d f o r u s i n g i n f o r m a t i o n on q u a n t i t i e s internal t o a EWE c o r e , s p e c i f i c a l l y the exit void fraction, t o i m p r o v e e s t i m a t e s of a d d i t i o n a l internal quantities using a thermal/ hydraulic model and e s t i m a t i o n theory. The method i s a p p l i e d t o an e x a m p l e u s i n g s i m p l e geomet r y but r e a l i s t i c data,
KCVgOROS BWR t h e r m a l / h y d r a u l i c s , estimating core estimation, minimum norm e s t i m a t i o n ,
internals,
estimating
power
distribution,
nonlinear
I NI'RODUCTI ON I n a n a l y z i n g BWR p e r f o r m a n c e , one i s i n t e r e s t e d in s detailed description of t h e c o r e power distribution and t h e c o o l a n t f l o w c o n d i t i o n s in individual rue1 b u n d l e s . Since limited in-core instrumentation is available, it is necessary to estimate these quantities f r o m models of t h e c o r e and f r o m m e a s u r e m e n t s or v a r i a b l e s external to the core. The c u s t o m a r y way t o f i n d such e s t i m a t e s i s t o s o l v e a s e t or n o n l i n e a r e q u a t i o n s w h i c h model t h e q u a n t i t i e s in the core subject to certain constraints. For e x a m p l e , t o t a l coolant flow is typically known and t h e p r e s s u r e d r o p s r r o m b o t t o m t o t o p o f a l l f u e l b u n d l e s must be e q u a l . A l l such e s t i m a t e s of BWR i n t e r n a l conditions a r e known t o be u n c e r t a i n t o at l e a s t some extent. Both the neutronic and t h e r m a l / h y d r a u l i c portions of t h e models l e a d t o i n a c c u r a c i e s associated with mathematical approximations and t h e use o f e m p i r i c a l data. As e r e s u l t , inacc u r a c i e s i n t h e computed power d i s t r i b u t i o n in local regions are often larger than desired. Hence, o p e r a t i n g margins are set v e r y c o n s e r v a t i v e l y . Additional m e a s u r e m e n t s can be used t o r e f i n e t h e s e e s t i m a t e s and r e d u c e t h e u n c e r t a n t l e s . One a p p r o a c h i s t o compute t h e s t e a d y s t a t e LPRM r e s p o n s e w h i c h a g r e e s w i t h t h e model and compare i t t o t h e a c t u a l r e s p o n s e ( E P E I , 19821. D i s c r e p a n c i e s can t h e n be r e d u c e d by a l t e r i n g model p a r a m e t e r s t o o b t a i n a computed r e s p o n s e w h i c h a g r e e s more c l o s e l y w i t h t h e m e a s u r e m e n t . The a p p r o a c h d i s c u s s e d i n t h i s paper uses t h e f l u c t u a t i n g c o m p o n e n t s of t h e LPRM s i g n a l s . These s i g n a l s have been shown t o be s e n s i t i v e to the thermal/ hydraulic conditions in the core. In particularj t h e t i m e d e l a y of t h e c r o s s c o r r e l a t i o n function between axially d i s p l a c e d LrHNs results i n a measured v e l o c i t y of t h e d e n s i t y wave of t h e two phase c o o l a n t ( g o s a l y , 1980) . From t h i s v e l o c i t y , one can d e t e r m i n e t h e c o o l a n t v o i d f r a c t i o n ( K o s a l y and c o l l e a g u e s , t982). Assuming t h a t two a x i a l l y s e p a r a t e d LPRM d e t e c t o r s a r e p l a c e d near t h e t o p of t h e c o r e i n each instrument string, one has a " m e a s u r e m e n t " of t h e e x i t v o i d f r a c t i o n of t h e c o o l a n t i n at l e a s t a few l o c a t i o n s in the core. I n t h i s p a p e r , we d e v e l o p a method f o r u s i n g i n - c o r e m e a s u r e m e n t s and a model o f t h e t h e r mal/hydraulic properties or t h e c o r e t o r e f i n e e s t i m a t e s of c o r e i n t e r n a l s . I t i s assumed t h a t t h e model i n v o l v e s t h e measured q u a n t i t y explicitely. I n our d i s c u s s i o n we w i l l a l s o assume that it is exit void fraction w h i c h i s measured but a s i m i l a r p r o c e d u r e would a p p l y t o u s i n g measurements of o t h e r q u a n t i t i e s , le first d e v e l o p t h e g e n e r a l a p p r o a c h d r a w i n g upon i d e a s
......
233
234
M.J.
DAMBORG and R. W. ALBRECHT
from linear and n o n l i n e a r estimation theory. The a p p r o a c h i s t h e n a p p l i e d t o a v e r y s i m p l e reactor c o r e model t o v e r i f y feasibility of t h e method. Ne f i n i s h with a discussion of t h e additional d e v e l o p m e n t w o r k t h a t w o u l d be r e q u i r e d for the operational use o f t h e s e i d e a s .
ANALYTICAL APPROACH Suppose
the
model
is
given
by t h e
expression
F(w,y,~
)
=
0
(~)
where F r e p r e s e n t s a v e c t o r of f u n c t i o n s , Fi, i=l,...,n; w i s s v e c t o r or measured q u a n t i t i e s external to the core, w., i:l,...,m; ~ i s the vector of exit void fractions at t h e l o c a t i o n s t h e LPRM i n s t r u m e n t strings, !~i' i = l , . . . , p ; and y i s t h e v e c t o r o f t h e r e m a i n i n g q u a n t i t i e s internal t o t h e c o r e w h i c h a r e t o be e s t i m a t e d , Yi' i=l,...,q. S i n c e t h e number o f such quantities w i l l e x c e e d t h e n u m b e r of m e a s u r e d e x i t v o i d f r a c t i o n s , we w i l l a s s u m e t h a t q) p,
of
I f no measurement o f ~ i s a v a i l a b l e , w h i c h i s t h e case i n p r e s e n t p r a c t i c e , then all core internals a r e e s t i m a t e d by s o l v i n g (I) f o r y and (~ w i t h w d e t e r m i n e d by measurement. In the e v e n t t h e model i s w e l l c o n d i t i o n e d and a s o l u t o i n exists f o r t h e g i v e n w, ( 1 ) can be s o l v e d . However, i t i s more l i k e l y t h a t an e x a c t s o l u t i o n does n o t e x i s t f o r t h e p a r t i c u l a r model parameters c h o s e n a n d f o r t h e v a l u e of .w. This expected l a c k o f an e x a c t s o l u t i o n i s due t o the approximate nature of t h e model a n d t h e u n c e r t a i n t y in the parameter values. In fact, the p r o b l e m may be o v e r d e t e r m i n e d w i t h more e q u a t i o n s t h a n unknowns. Thee one seeks t h e e s t i m a t e (ye,~e) which solves (I) as c l o s e l y as p o s s i b l e i n t h e sense t h a t IIF(w,y
)ll
,~ e
for
the
given
norm given
measurement
w,
=
min y,~
e
Here
the
liF(w,y,~)ll
norm o f
F,
iiPi!,
(2) is
defined
as
the
weighted
Euclidean
by n
I IFI
I
i IF i 12
=i~id
(3)
where t h e d i a r e p o s i t i v e real numbers. Hence an a c c e p t a b l e e s t i m a t e or t h e c o r e i n t e r n a l quantities ss a s s o c i a t e d w i t h an a c c e p t a b l y s m a l l v a l u e of I { P ! l . Note t h a t i t i s n e c e s s a r y t o use a w e i g h t e d norm so t h a t t h e d i f f e r e n t c o m p o n e n t s Fi c o n t r i b u t e equitably. For e x a m p l e , i f one c o m p o n e n t r e p r e s e n t s heat input with quantities on t h e o r d e r of 10 T BTU a n d a n o t h e r r e p r e sents void fraction, on t h e o r d e r of l , t h e t w o c o m p o n e n t s must be n o r m a l i z e d b e f o r e b e i n g summed i n t o a s i n g l e e x p r e s s i o n l l k e ( 3 ) i f a l l c o m p o n e n t s a r e t o have t h e same s i g n i f i c a n c e . Now s u p p o s e that is
a measurement
of
the
vector
*?h d e n o t e d
=
~
m Then i t
must
be
+
(~m' i s
I IF(w,y
e
,(~
,e( m
#1
) I I
f r o m ~e'
/1~
Z
(4)
I IF(w,y
)11
=
min y
m
le suggest t h a t such • y can be f o u n d u s i n g nonlinear i n y, we use an i t e r a t i v e process Given the original e s t i m a t e (ye,,J:e) l e t
e
,(~
y
expansion F(w,y
,e m
elements
of
or
+
A(~
=
y
+
Ay
the
minimizes
ILFli
)11
subject (6)
Since square
F is estimate.
e
(ye,(~e) +
matrices
one w h i c h
(7)
) m
is,
(5)
least square estimation concepts. where each s t e p i s s l i n e a r least
?(w, ym, l~m) a b o u t =
) I I
m
(~ e
m
order
e
I IF(w,y,(~
= m
ij
differs
e
and we w o u l d l i k e t o f i n d a " b e t t e r " e s t i m a t e o f y, t h a t t o t h e known v a l u e s of w and t~. Hence, we seek Ym where
where t h e
which
that
I IF(w,y
Then a f i r s t
available
F(w,y
,e( e
) e
3y and 3~ a r e
JAy y
is +
J
.40( e~
(8)
I n f o r m a t i o n on the exit void fraction
235
~F. (J).. y ij
1
=
by. 8F.
(J)..
=
J (9)
1 J
Assuming (ym,,~m)
and ( y e , ~ e )
to " e s s e n t i a l l y "
JAy y
solve
+ J A~
F,
we have
= 0
(10)
w h e r e o n l y ~y i s unknown. I f 3y i s or maximum rank we can r i n d minimizes 1[3y~y, + 30: °"'~e~' by c o m p u t i n g ( L u e n b e r g e r , 196g)
Ay = - ( j T D T D j
)-IjTDToj
Y
Y
e
+Ayl,~
m
1
~
)11
= Y
or ( 1 0 )
which
(11)
A~
u s e d i n t h e norm. We w i l l d e n o t e t o t h e e s t i m a t e or y and i f t h e
ltF(u,y
,~
e
m
To f u r t h e r i m p r o v e t h e e s t i m a t e or y we can i t e r a t e on ~y, e s t i m a t e we can c o m p u t e a s e c o n d c o r r e c t i o n ~y2 Let
Y
solution
Y
where D:diag(d I,...,an), t h e m a t r i x or p o s i t i v e weights v e c t o r round i n ( 1 1 ) as ~ y l , t h e f i r s t order correction e s t i m a t i o n procedure i s well behaved we expect t h a t
IlF(w,y
the
e
+
)11
the
(12)
Indeed,
to rind
an i m p r o v e d
Ayl
(13)
and w r i t e
F(w,Yl+Ay2,e
m
) = F(w,yl,e
m
) + JAy y
2
w h e r e J i s e v a l u a t e d at ( y l , ~ i m ) . S i n c e we s e e k Ay 2 t o m i n i m i z e of ( 1 4 ) , y we u s e l i n e a r l e a s t square estimation a g a i n and c o m p u t e
AY 2 = - ( j T D T o j
)-IjTDTF(',J,y
y
If
y
1 ,e
(14) the
norm of
the
)
y
left
hand s i d e
(15)
m
we d e n o t e
y
2
= y
1
+
Ay2
(16)
we expect
IIF(w,y This
iteration
continues
according
Y k+l
Ay k + l where 3¥
k
indicates
2 ,~
that
=
Yk
+
m
at
least
a local
IIF( u,y 1 ,~
~
(17)
(18)
evaluated
minimum or
)It
~yk+l
= -(jkToTDjk)-ljkToTF(w,yk,~ y y y 3y i s
m
to
y gives
)11
m
at =
(yk,0: m)
lim k~
y
k
)
(i9)
m If
the i t e r a t i o n
converges,
we know t h a t
(20)
iiY(w, yk,~m) il.
APPLICATION TO SIMPLE GEOMETRY The a b o v e a p p r o a c h has b e e n a p p l i e d t o a BHR model u s i n g v e r y s i m p l e g e o m e t r y . The 1 e e l i s t o demonstrate the feasibility of c o m p u t i n g e s t i m a t e c o r r e c t i o n s ~y t h a t a r e c o n s i s t e n t with d e v i a t i o n s ~ between measured and e s t i m a t e d e x i t void f r a c t i o n s . We have a t t e m p t e d t o make this demonstration realistic by using model e q u a t i o n s i n t e n d e d f o r o n - l l n e m o n i t o r i n g and by using actual data f o r o p e r a t i n g BWR fuel bundles while r e d u c i n g the computational c o m p l e x i t y .
236
M . J . DAMBORG and R. W. ALBRECHT
Me assume t h e c o r e c o n s i s t s of two " M a c r o - B u n d l e s " , each r e p r e s e n t e d by t h e a v e r a g e q u a n t i t i e s of t h e t o u r f u e l b u n d l e s s u r r o u n d i n g an LPRM i n s t r u m e n t string, A model c o n s i s t i n g of two macro-bundles permits us t o r e p r e s e n t the constraints that nil fuel bundles experience an e q u a l pressure difference b e t w e e n t o p and b o t t o m and t h a t t h e c o o l a n t flow in individual b u n d l e s must sum t o t h e known t o t a l . The p h y s i c a l p r o p e r t i e s and o p e r a t i n g c o n d i t i o n s of t h e m a c r o - b u n d l e s are obtained u s i n g d a t a f r o m t h e M u h l e b e r g BilE. Macro-bundle number 1 (MB-I) has p r o p e r t i e s t h a t e r e an a v e r a g e of t h e f o u r f u e l b u n d l e s a r o u n d p o s i t i o n 1 2 - 1 3 a n d ME=2 i s d e t e r m i n e d by position 4=21. The power, q u a l i t y and c o o l a n t d e n s i t y d i s t r i b u t i o n s were o b t a i n e d f r o m t h e KKM process computer (personal communication, K. Behringer). The f o l l o w i n g assumptions were used in developing the ten macro-bundle model: 1) The a r e a of e a c h m a c r o - b u n d l e i s t h e sum of t h e a r e a s of t h e f o u r b u n d l e s i t r e p r e s e n t s . 2) The h y d r a u l i c d i a m e t e r and h e a t e d p e r i m e t e r a r e computed f o r t h e m a c r o - b u n d l e . 3) The a x i a l v o i d f r a c t i o n , steam q u a l i t y and r e l a t i v e heat flux distributions; the average heat flux; mass f l u x and n o n - b o i l i n g l e n g t h e r e o b t a i n e d by averaging over the four bundles. The model we used t o r e p r e s e n t t h i s two m a c r o - b u n d l e c o r e was d e r i v e d f r o m t h e model used i n t h e Power S h a p e M o n i t o r i n g S y s t e m (PSMS) ( E P R I , 1 9 8 1 ) . The PSMS was d e s i g n e d to estimate the l o c a l power d i s t r i b u t i o n of t h e c o r e . As an o n = l i n e e s t i m a t o r , i t was i n t e n d e d t o use r e a l time data to monitor core conditions. I t was a l s o t o be used o f f - l i n e to predict l o c a l power fluctuations due t o c o n t r o l action, The PSMS model c o n s i s t s o f two c o u p l e d modules: a neutronic module called Node-8 and a thermal/hydraulic module called Therm-g. The n e u t r o n i c module has a r e s o l u t i o n of 24 a x i a l nodes and models a s s e m b l y a r r a y s up t o 3 0 x 3 0 . Its outputs include three-dimensional arrays of power and m o d e r a t o r d e n s i t y . The t h e r m a l / h y d r a u l i c module computes the i n l e t flow distribution a c r o s s t h e a r r a y of a s s e m b l i e s . The i n p u t s t o t h e PSM5 consist of o p e r a t i n g d a t a f r o m t h e p l a n t c o m p u t e r s u c h as t o t a l core coolant flow in addition to descriptions of t h e p h y s i c a l , neutronic and t h e r m a l / h y d r a u l i c properties or t h e f u e l . The estimates of t h e c o r e i n t e r n a l quantities result f r o o an i t e r a t i v e procedure between the neutronic and t h e r m a l / h y d r a u l i c modules i n w h i c h t h e s e e s t i m a t e s a r e a d j u s t e d u n t i l they yield external conditions w h i c h match t h e m e e s u r e o e n t s . We u s e d o n l y t h e t h e r m a l / h y d r a u l l c portion o f t h e PSMS model f o r o u r two m a c r o - b u n d l e core. Hence, i n our e x p r e s s i o n f o r t h e model, F ( w , y , ~ ) , w r e p e r s e n t s a l l g i v e n d a t a on t h e p r o c e s s which affects the thermal/hydraulic model s u c h a s t o t a l coolant flow and the heat flux distribu tion. The y v e c t o r r e p r e s e n t s t h e 22 t h e r m a l / h y d r a u l i c quantities estimated by t h e model e x c e p t f o r t h e two e x i t v o i d f r a c t i o n s w h i c h a r e r e p r e s e n t e d by 5. Table I lists the elements of y and ~ and g i v e s t h e v a l u e s f o r ( y e , ! ~ e ) , t h e s o l u t i o n which minimizes iiF]i. Some of t h e s e variables such as i n l e t velocity are self explanatory w h i l e o t h e r s such t h e a c c e l e r a t i o n loss coefficient may be s p e c i f i c to this model. The t o t a l pressure drop is the pressure loss minus the buoyant pressure. The 24 PSMS e q u a t i o n s which are necessary to represent our core ere given in the appendix. Briefly, t h e y c o n s i s t of I1 e q u a t i o n s f o r each m a c r o - b u n d l e and two c o n s t r a i n t equations, one representing t h e f a c t t h a t a l l r u s t b u n d l e s e x p e r i e n c e t h e same p r e s s u r e d r o p and t h e o t h e r t h a t t h e two b u n d l e c o o l a n t f l o w s must sum t o a known q u a n t i t y . Table 2 lists the results of t h e e s t i o e t i o n approach developed in the previous section. This t a b l e shows t h e d e v i a t i o n s ~y i n t h e e s t i m a t e Ye w h i c h c o r r e s p o n d t o each of s i x d e v i a t i o n s ~ i n t h e e s t i m a t e Re. I t can r e a d i l y be seen t h a t t h e r e s u l t s ere reasonable. For e x a o p l e , t h e inlet velocity i n c r e a s e s when ~¢~ d e c r e a s e s and c o n v e r s e l y . Comparing cases 2 and 3 shows t h a t t h e p r e s s u r e l o s s e s have t h e p r o p e r s i g n r e l a t i o n s h i p with respect to ~. The p r e s s u r e l o s s f o r case 6 i s a p p r o x i m a t e l y t h e sum o f t h e l o s s e s f o r c a s e s I and 2. And c a s e s 5 and 6 e r e approximately equal i n m a g n i t u d e and o p p o s i t e i n s i g n f o r n i l v a r i a b l e s . Me a l s o see t h a t k ! P t l b e h a v e s as e x p e c t e d . The v a l u e f o r ( y e , ~ e ) i s t h e minimum o f e l l c a s e s and t h e v a l u e s o f (ye,~o) are significantly greater than (ya,~m) for nil cases.
CONCL USI ONS Our b a s i c r e s u l t i s t h a t one can e x p e c t t o use m e a s u r e m e n t s of e x i t v o i d f r a c t i o n to improve e s t i m a t e s of o t h e r i n t e r n a l core quantities u s i n g a model and e s t i o n t i o n concepts. Plausible end c o n s i s t e n t quantitative results can be c a l c u l a t e d . U s i n g t h e PSMS model and o u r s i m p l e g e o m e t r y , t h e c o m p u t a t i o n s were w e l l b e h a v e d and c o n v e r g e d r a p i d l y , fie s u g g e s t t h a t i t i s feasible t o d e v e l o p t h i s a p p r o a c h i n t o an o n - l l n e operations aid. That i s , i f m e a s u r e m e n t s of ere developed from fluctuating signals of t h e LPRMs, t h e y c a n be u s e d t o i o p r o v e t h e knowl e d g e of q u a n t i t i e s internal to the core. To e x t e n d t h i s approach from our simple geometry to the full core requires n significantly l a r g e r c o m p u t a t i o n t h a n was p e r f o r o e d h e r e . A c o r e any have a p p r o x i m a t e l y 40 LPRM i n s t r u m e n t strings so t h e nodal w o u l d c o n s i s t o f 40 s a c r a - b u n d l e s . W l t h Jl e l e m e n t s of t h e y v e c t o r per
Information on the exit void fraction
237
macro-bundle, the problen is large. Re can s u g g e s t two a p p r o a c h e s t o t h e c o m p u t a t i o n but have developed neither. One c o u l d r e t a i n t h e two m a c r o - b u n d l e a p p r o a c h but l e t one m a c r o - b u n d l e represent t h e f o u r b u n d l e s a r o u n d an i n s t r u m e n t string and t h e t h e o t h e r r e p r e s e n t the balance of t h e c o r e , Then t h e d e t a i l e d macro-bundle c o u l d be i t e r a t e d around the core until consistent quantities ere obtained throughout. The o t h e r a p p r o a c h i s t o e x t e n d our p r e s e n t a p p r o a c h and represent t h e 40 m a c r o - b u n d l e s , A potential computational difficulty results f r o m t h e need to invert a 4 4 0 x 4 4 0 a r r a y which i s u s u a l l y dirficult and t i m e consuming. However, t h i s a r r a y i s v e r y s p a r s e and t h e i n v e r s i o n of l a r g e s p a r s e m a t r i c e s is often very reasonable ( T i n n o y and lalker, 1977). A further necessary extension f o r HWR a p p l i c a t i o n s is to include the neutronic model i n t h e estimation process. The power d i s t r i b u t i o n of t h e n e u t r o n i c model can be compared t o t h e DC f l u x measured by t h e LPRMs. To get a b e s t e s t i m a t e of c o r e p e r f o r m a n c e , it is necessary to iterate b e t w e e n t h e two models.
APPENDI X Our PSMS based model c o n s i s t s of 24 e q u a t i o n s . Each e q u a t i o n i s g i v e n b e l o w i n t h e f o r m Fk=O and i s w r i t t e n using the symbols in Table l for the quantities i n t h e y v e c t o r and t h e p h y s i c a l constants i n T a b l e 3. The f i r s t It expressions represent two e q u a t i o n s each, one r o t MB-I (i:l) and one f o r MB-2 ( i = 2 ) . The l a s t two e q u a t i o n s represent t h e equal p r e s s u r e and t o t a l coolant flow constraints. The e q u a t i o n s for k:ll-14 were o b t a i n e d f r o m r e g r e s s i o n s while the o t h e r s a r e as r e p r e s e n t e d i n t h e PSM8 model.
Pressure
loss i n
channel:
in~i
Fk = P l i Buoyant
oi
~g¢ -
+
oi
Oe
+ 2ri
I
L L + i IO[e,(Z)pg+(l-e (z))Pl]dZ PI i i
,:~:i(z) = a x i a l Non-bioling
length
vs,
inlet
void
= 0
k=lo2 i=1.2
qi(z) vs.
heat
= channel
fraction
in
heat
)V
in,i
-
~0 Lp ~ l ( z ) d z 1
flux
input:
L
H. -H in sat hfg
= 0
k=7.8 i=1.2
relationship:
F
= cx k i
Q. 1
-
pgC°i Co vs. v o i d
k=5,6 i=1,2
= 0
Fk = Q. 1 Io p q i ( z ) d z i 3600API hi=gVin, i Void-quality
k=3,4 i=1,2
= 0
velocity:
Fk = 3600Ap l(Hsat-H.
+
= 0
k=9,10
i=1.2
g g],i 3600PIVin, i
PI
fraction:
g k = Col Drift
+ Ki
pressure:
Fk = P b i
Quality
ioi
e
velocity
va. v o i d
1.006
- 0.4090~.1
+ 0"39Z18~21 = 0
k=ll,12 i=1,2
fraction:
Fk = Vgj, i -
0.~262
+ 0.3728~z
= 0
k=13,1~ i=1,2
238
M . J . DAMBORG and R. W. ALBRECHT,
Friction
loss coefficient:
gk = foi Two-phase friction
- 0.19 loss
Fk = R i Acceleration
e P]
in, i
k=15,16 i=1,2
= 0
multiplier:
1 +
k=17,18 i=1,2
= O
loss coefficient:
( l_Qi )2 Fk = r :
Q2pl
cx p
i-~. 1
k=19,20 i=1,2
+ i =0
1 g
Ford l o s s c o e f f i c i e n t :
F k = K.i - kl - k2 - k3 LI_% jqa,,:~a : averages Equal
pressure
o
k=21,22 i=1,2
the g r i d spacers
constraint:
F23 = P l l Constant
at
- k 4 Ll_e ]
- Pbl
- P12 +Pb2 = 0
total
flow constraint:
F2~
= 3600APl(Vin,l
+ Vin,2)
- GT = 0
REFERENCES EPRi ( 1 9 8 1 ) . Power Shape M o n i t o r i n g System, Vol. I: Overview and Performunce, Vol. 2: Technical Description, Report NP-1660, E l e c t r i c Power R e s e a r c h I n s t i t u t e , Paid A l t o . EPRI ( 1 9 8 2 ) . Technical D e s c r i p t i o n and E v a l u a t i o n of BWR Hybrid Power Shape M o n i t o r i n g System, Report NP-2234, E l e c t r i c Power Research I n s t i t u t e , PaiD A l t o . Kosuly, G. ( 1 9 8 0 ) . Noise i n v e s t i g a t i o n s in b o i l i n g - w a t e r and p r e s s u r i z e d - w a t e r r e a c t o r s . r r o s r e s s i n N u c l e a r Energy, ~ pp. 1 4 5 - 1 9 9 . Kosaly, G., J. N. Fuhley, K. Behringer and R. D. Crowe ( 1 9 8 2 ) . I n v e s t i g a t i o n or the a x i a l void p r o p a g a t i o n and v e l o c i t y p r o f i l e in EWR fuel bundles. NUREG/CP-O034. Tiuney, W. F. and 3. N. l u l k e r (°1977). D i r e c t s o l u t i o n s of sparse network e q u a t i o n s by o p t i m a l l y ordered t r i a n g u l a r factorizatlon, Proceedings IEEE, ~ pp. 500-535. Luenberger, D, G. ( 1 9 6 9 ) . O p t i m i z a t i o n b~ Vector Space Methods, l i l e y .
I n f o r m a t i o n on the exit void fraction
TABLE 1
SYMBOL P1
Lo CO V j Kg f R° r
TABLE 2
VARIABLE
MB
The E l e m e n t s o{ the ¥ and e Vectors and their Nominal or E s t i m a t e d Values DESCRIPTION P r e s s u r e loss (ft) Buoyant pressure (~t) Inlet velocity (ft/s) Exit quality N o n - b o i l i n g length (~t) C-zero Drift velocity (ft/5) Form loss c o e f f i c i e n t Friction loss coefficient 2-ph f r i c t i o n l o s s m u l t i p l i e r A c c e l e r a t i o n loss c o e f £ i c i e n t Exit void fraction
Kg j
1 2 1 2 1 2 1 2 1 2 1 2 1
K
2 1 2
R r r
2 1 2 1 2
Case 1 ~el= .00
.01
0.600 0.578 0.0002 -0.0002 0.0339 -0.0413 -0.00087 0.00151 0.0148 -0.0151 -O.OOlO
-0.0067 0.0 -0.00372 -0.002 0.511 -2.0E-5 2.5E-5 -0.0005 0.3903 -0.0020 0.0863
NORM I IF(w.ye,~e)ll I IF(w.ye,~m)ll IIF(w,ym~em)l I
VALUE MB-2 MB-I 65.454 65.059 4.5667 4.9533 6.6695 6.4351 0.13333 0.15265 3.0537 2.9490 1.0993 1.0971 0.15957 0.16908 71.234 67.818 0.018797 0.018917 5.9439 6.7835 1.9443 2.1830 0.6897 0.7152
O e v i a t i o n s o f y from the Nominal Values in Table 1 and Values o f I I F I I Correspondin@ t o D e v i a t i o n s o f e
Ae2= P1 Pl Pb Pb Vin Vin O Q Lo ko Co Co V . Vg~
239
VALUES OF Ay FOR GIVEN Ae Case 2 Case 3 I Case 4 ~ e l = .01 Ael=-.O1 I ~el=-.O1 Ao2= .00 ~ 2 = .00 I ~ = .01 0.496 0.522 -0.0001 0.0001 -0.0368 0.0287 0.00128 -0.00087 -0.0160 0.0106 -0.0066 -0.0007 -0,00373 0.0 0.421 -0.003 2.2E-5 -1.9E-5 0. 3066 -0.0006 0.0757 -0.0024
-0.452 -0.480 0.0002 -0.0002 O. 0350 -0.0268 -0.00123 0.00081 0.0153 -0.0098 0.0065 0.0007 0.00373 0.0 -0.386 0.003 - 2 1E-5 1 7E-5 - 0 2805 0 0007 - 0 0710 0 0022
Case 5 A~1=-.01
Case 6 , l e l = .01 ~oc2= .01
0.140 0.092 0.0004 -0.0004 0.0689 -0.0681 -0.00209 0.00233 0.0301 -0.0249 0.0056 -0.0060 0.00373 -0,00372 -0.389 0.514 -4.1E-5 4.2E-5 -0.2811 0.3909 -0.0732 0.0884
-0.994 -0.999 0.0001 -0.0001 0.0036 0.0121 -0.00041 -0.00063
1. 103 1.108 0.0 0.0 -0.0029 -0.0126 0.00040 0.00063 -0.0012 -0.0045 -0.0076 -0.0075 -0.00373 -0.00372 0.420 0.511 O. 2 E - 5 0.6E-5 0.3061 0.3897 0,0739 0.0841
0.00171 0.10482 0.00785
0.00171 0.10004 0.01345
~{=-.ol
0.0016
0.0045 0.0073 0.0074 0.00373 0.00373 -0.381 -0.461
-0.3E-5 -0.7E-5 -0.2800 -0.3533 -0,0690 -0.0780
VALUE OF I I F I I 0.00171 0.08252 0.00675
0.00171 0.06947 0.00905
I I I
0.00171 0.06466 0.00765
0.00171 0.10786 0.01284
240
M . J . DAMBORG and R. W. ALBRECHT
TABLE 3 SYMBOL
~
e
A p gc Pl
Pg
~
sat in
k1 k2 k3 4 GT
Physical
Constants
and t h e i r V a l u e s
DESCRIPTION Equivalent
Channel
diameter
length
VALUE
(ft)
(~t)
C h a n n e l a r e a (ft ~) Channel p e r i m e t e r (ft) Gravitation constant L i q u i d d e n s i t y ( l b / f ~ 3) V a p o r d e n s i t y ( ] b / f t ~) Liquid viscosity (lb/s-ft) Saturated entha]py Input e n t h a l p y H e a t of v a p o r i z a t i o n F o r m loss - lower tie F o r m loss - o r i f i c e F o r m loss - s p a c e r s F o r m loss - u p p e r tie Total c o o l a n t 9 l o w
0.04645 12.0 0.4431 32.37 32.2 47.6 2.343 6.667E-5 548.6
516.7 642.8 7.56 29.58
8.88 0.801 9.943E+5