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The influence of local absence of cooling on the cladding temperature of a fuel rod
NUCLEAR ENGINEER]:NG AND DESIGN 3 (1966) 1-10. NORTH-HOLLAND PUBLISHING COMPANY, AMSTERDAM
THE INFLUENCE OF LOCAL ABSENCE OF COOLING O N T H E C L A D D I N G T E M P E R A T U R E O F A F U E L R O D * ** Mrs. C. M. KALKER R e a c t o r C e n t r u m Nederland, The Hague, The N e t h e r l a n d s Received 5 N o v e m b e r 1965
The t e m p e r a t u r e r i s e at the position of support of the fuel r o d s which r e s u l t s f r o m the local a b s e n c e of cooling is calculated. The support of the fuel rod is r e p r e s e n t e d by a s m a l l c i r c l e , lying on the outer s u r f a c e of the cladding, which s i m u l a t e s the contact a r e a between fuel rod and supporting grid. The t e m p e r a t u r e r i s e is c o n s i d e r e d a s a p e r t u r b a t i o n t e r m s u p e r - i m p o s e d on the u n d i s t u r b e d t e m p e r a t u r e distribution. Two e x t r e m e c a s e s a r e c o n s i d e r e d , n a m e l y complete isolation of the p e r t u r b a t i o n t e r m and complete cooling of the p e r t u r b a t i o n t e r m outside the s m a l l c i r c l e . In o r d e r to c o m p a r e with e x p e r i m e n t a l r e s u l t s the s a m e c a l c u l a t i o n s a r e applied to a c a s e of heat production in a sl:eel tube, which is heated e l e c t r i c a l l y .
1. INTRODUCTION This p a p e r p r e s e n t s the r e s u l t s of c a l c u l a t i o n s p e r f o r m e d in connection with the NERO-project. T h i s p a p e r e n v i s a g e s the study by R.C.N. of a r e a c t o r for ship propulsion. In the p r e s e n t p a p e r we d e r i v e some e x p r e s s i o n s for the t e m p e r a t u r e r i s e o c c u r r i n g at the place of support of fuel rods which r e s u l t s f r o m the local a b s e n c e of cooling. As we have as yet no data a v a i l a b l e with r e g a r d to the local v a r i a t i o n in cooling, we c o n s i d e r only the l i m i t i n g case by a s s u m i n g that t h e r e is no cooling at all at the place of support of the fuel rods. The r e s u l t will t h e r e f o r e be m o r e p e s s i m i s t i c than the reality. E x p e r i m e n t s will have to b e done to get an idea how much t e m p e r a t u r e s have b e e n o v e r e s t i m a t e d . Such e x p e r i m e n t s g e n e r a l l y take place by the heating of t h i n - w a l l e d steel tubes. The place and the s p a c e - d e p e n d e n c y of the heat p r o d u c t i o n is in this way different. To t r a n s l a t e the e x p e r i m e n t to a r e s u l t for fuel r o d s , the t e m p e r a t u r e r i s e has also been c a l c u l a t e d for u n i f o r m heat p r o d u c t i o n in a t h i n - w a l l e d steel tube u n d e r fully identical conditions. C o m p a r i n g the e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s it is hoped that a good i m p r e s s i o n of the t e m p e r a t u r e r i s e of the fuel rod is obtained.
2. FORMULATION OF THE PROBLEM We c o n s i d e r a cyl:mdrical fuel rod p r o d u c i n g heat of q watt p e r unit volume. We neglect the gap between cladding and fuel. Let the r a d i u s of the fuel rod be a , that of both fuel and cladding b (see fig. 1). On the s u r f a c e of the c y l i n d e r t h e r e is a disk with r a d i u s c where t h e r e is no cooling at all. The t h e r m a l conductivity of the fuel is k i , that of the cladding kII. The t e m p e r a t u r e in I s a t i s f i e s P o i s s o n ' s d i f f e r e n t i a l equation: AT = -q/k I .
i.
I
b
(1) I
The t e m p e r a t u r e in B: s a t i s f i e s L a p l a c e ' s equation: AT = 0 .
I
(2)
Fig. 1.
* R e s e a r c h s p o n s o r e d by E u r a t o m , and originally i s s u e d a s r e p o r t R C N - I n t - 6 4 - 0 0 3 / 1 (EUR-2182.e). ** Accepted by M. Bogaardt.
I¢
2
C.M. KALKER
F o r r = a we have that the h e a t flux and t e m p e r a t u r e a r e c o n t i n u o u s , that i s : aT I ~T H k I - ~ - = klI ~ - -
T I = TII ,
for r = a .
(3)
F o r r = b we have that t h e h e a t c o n d u c t e d by the c l a d d i n g e q u a l s the h e a t t r a n s f e r r e d to the c o o l i n g l i q uid, that is:
a(T(b) -Tk) + kIi [OT/Or]r__b = 0 ,
(4)
w h e r e T k is the a v e r a g e t e m p e r a t u r e of the c o o l i n g liquid. S o l u t i o n of eqs. (1) a n d ( 2 ) w i t h b o u n d a r y c o n d i t i o n s (3) and (4) g i v e s a s o - c a l l e d u n d i s t u r b e d t e m p e r ature distribution. H o w e v e r , we a s k the t e m p e r a t u r e which r e s u l t s f r o m the l o c a l a b s e n c e of c o o l i n g , or: a =0
(5)
inside a small circle with radius c lying on the outer surface of the cladding, which is considered representative of the support of the fuel rod.
3. THE UNDISTURBED T E M P E R A T U R E D I S T R I B U T I O N We s h a l l now c a l c u l a t e the t e m p e r a t u r e in the s o - c a l l e d u n d i s t u r b e d s t a t e ; we n e g l e c t the " l o c a l c o n d i t i o n " (5) a n d s o l v e eqs. (1) to (4). B e c a u s e of the c y l i n d e r - s y m m e t r y the e q u a t i o n s a r e : in I:
i n II:
d2T 1 dT q dr2 + r-d-r = - k I '
(6)
d2T I dT dr 2 +rd-r =0"
(7)
The b o u n d a r y c o n d i t i o n s a r e : f o r ~ = a:
TI = TII '
dT I dTII kI - ~ - = kII d r
(8) dTII
for r = b:
a ( T - Tk) + kii - ~ - = 0 .
A p a r t i c u l a r s o l u t i o n of eq. (6) is: T = -qr2/4k I . A s o l u t i o n of the h o m o g e n e o u s e q u a t i o n (7) i s : T = A + B l o g r , w h e r e A a n d B a r e c o n s t a n t s . I n r e g i o n I we h a v e B = 0 , a s T s h o u l d be f i n i t e f o r r = 0 . We t h u s have the f o l l o w i n g s o l u t i o n s :
TI
=A -qr2/4ki,
TII =C + O l o g r .
S e t t i n g t h e t e m p e r a t u r e of~the c o o l i n g l i q u i d T k = O, we d e f i n e the c o n s t a n t s A , C and D b y m e a n s of the b o u n d a r y c o n d i t i o n s (8) a n d , f i n d f o r the u n d i s t u r b e d t e m p e r a t u r e :
-qa2
TOI - ~ I
(1
r2 2kI 2kI -a-2 + -~- +k-HHlog b )
qa2 lo
ToII = 2 k i i
biqa2
g r T-2ba
for ~" --< a ,
for a --< r -< b .
(9)
INFLUENCE OF LOCAL ABSENCE OF COOLING
3
4. EQUATIONS F OR THE P E R T U R B A T I O N T E R M Eqs. (9) g i v e the u n d i s t u r b e d t e m p e r a t u r e d is t r i b u t i o n . We c a l l it T o. On it we s u p e r i m p o s e a p e r t u r b a t i o n t e r m T, which r e s u l t s f r o m the disk on the s u r f a c e of the c y l i n d e r w h e r e t h e r e is no cooling at all. We a p p r o x i m a t e the neighbourhood of the disk, only f o r c a l c u l a t i o n of the p e r t u r b a t i o n t e r m by a cooling " " X h a l f - s p a c e z >I O. In this h a l f - s p a c e we work with a C a r t e s i a n c o o r d i n a t e s y s t e m x, y and z (see fig. 2). Let the total t e m p e r a t u r e be Tg, with Tg = = T+ T o. As the total t e m p e r a t u r e Tg in I and II s a t i s f i e s eqs. (1) r e s p e c t i v e l y (2) a n d T o is a s o lution, we have f o r T in both r e g i o n s : .~T = 0 . We s h al l now d e r i v e the boundary conditions f o r T. We a s s u m e that f a r f r o m the disk we have the u n d i s t u r b e d t e m p e r a t u r e d i s t r i b u t i o n , that is T ~ 0 f o r ( x , y , z ) - ~ ~o. Outside the disk, on the s u r f a c e z = 0, we have that: 0 :
Fig. 2. ~T
o r
a
TO
aT
+
f or z = 0, x 2 + y 2 > c 2, s i n c e we have f o r the u n d i s t u r b e d t e m p e r a t u r e To: aTo a T o - kii - ~ - = O . I n s i d e the disk we have that:
aTg az
O=
-
aT az
+
aT o az
We can w r i t e : aTo = _ FdToII~ az L dr J r=b' w h e r e ToII is the u n d i s t u r b e d t e m p e r a t u r e c a l c u l a t e d in c y l i n d r i c a l c o o r d i n a t e s in eq. (9). D i f f e r e n t i a tion y i e l d s :
aTo
qa 2
az
2bkii
and so:
T _ az
qa 2
f o r z = 0 , x 2 + y 2 < c 2.
2bkii
S u m m a r i z i n g we find f o r the p e r t u r b a t i o n t e m p e r a t u r e T the d i f f e r e n t i a l equation: ~T=O with b o u n d a r y conditions: for ( x , y , z ) ~ o o ,
T~O
aT a T - kii ~ = O
forz=0,
aa -TY
f o r z = 0 , x 2 + y 2 < c 2.
x2 + y2 > c 2,
(lO) = -
2 -qa2 6-~,
=
-A
In the following s e c t i o n s we will s o l v e t h e s e e q u at i o n s f o r s o m e s i m p l i f i e d c a s e s :
4
C.M. KALKER
1) We a s s u m e t h a t the p e r t u r b a t i o n t e m p e r a t u r e i s not n o t i c e a b l e on the b o u n d a r y b e t w e e n u r a n i u m and c l a d d i n g , i . e . , ( b - a ) is p r a c t i c a l l y i n f i n i t e w h e r e T is c o n c e r n e d . T h u s the h a l f - s p a c e z >/ 0 has a t h e r m a l c o n d u c t i v i t y kii. With t h i s a s s u m p t i o n we c a l c u l a t e two l i m i t i n g c a s e s l a and lb: l a ) T h e p e r t u r b a t i o n t e m p e r a t u r e i s not c o o l e d at a l l , i . e . , a = 0 and s o a T / a z = 0 o u t s i d e the disk. lb) C o m p l e t e c o o l i n g of the p e r t u r b a t i o n t e m p e r a t u r e o u t s i d e the d i s k , i . e . , a = oo o r T = 0 o u t s i d e the disk. 2) T h e p e r t u r b a t i o n t e m p e r a t u r e h a s a c e r t a i n v a l u e on the b o u n d a r y of the c l a d d i n g and u r a n i u m , i . e . , the h a l f - s p a c e z >/ 0 c o n s i s t s of a s l a b w i t h t h i c k n e s s d = b - a and t h e r m a l c o n d u c t i v i t y kii and a r e g i o n with t h e r m a l c o n d u c t i v i t y k I. We c a n h e r e a l s o c o n s i d e r c o m p l e t e i s o l a t i o n and c o m p l e t e c o o l i n g of the p e r t u r b a t i o n t e r m . A s the l a t t e r c a s e p r e s e n t s u n s u r m o u n t a b l e d i f f i c u l t i e s f o r the p r e s e n t , we c a l c u l a t e only the c a s e a = 0 and a T / a z = 0 o u t s i d e the disk. In any c a s e that w i l l b e m o r e p e s s i m i s t i c t h a n the t r u t h .
5. S O L U T I O N IN A H O M O G E N E O U S H A L F - S P A C E O F T H E R M A L C O N D U C T I V I T Y kI I WITH COMPLETE ISOLATION We w i l l now s e e k a s o l u t i o n of e q s . (10) f o r c a s e l a ) , a = 0 and a T / a z = o o u t s i d e the disk. thus become: AT = 0 ,
H e r e a T / a z is p r e s c r i b e d
T ~ 0
for (x,y,z)
(10)
~ oo ,
a__TT= 0 az
on z = 0, x 2 + y 2 > c 2,
aT --A ~z
onz =0, x2+y2
on the b o u n d a r y of the h a l f - s p a c e .
Eqs.
(11)
Now it is known t h a t R -1, with
R 2 = (x-~)2 + (y_~)2 + (z-~)2 , satisfies
Laplace's
equation everywhere
e x c e p t in x, y , z . We c a n show t h a t the c o n n e c t i o n b e t w e e n T
and ~ T / a z i s g i v e n by: T(x,y,z)
1
= --~
f-~aT < ~ , y ) lpd v
dy
,
p2 = (x _~)2 + (y _~)2 + z 2
In o u r e a s e t h i s b e c o m e s : 1 f Ad~d~ T ( x , y , z) = - ~ d i s k p
"
W e s e e t h a t T ( x , y , z ) ---' 0 f o r ( x , y , z ) ---. 00. F r o m g e o m e t r i c a l c o n s i d e r a t i o n s it i s e a s y to s e e t h a t the t e m p e r a t u r e h a s its m a x i m u m in the c e n t e r of the d i s k , n a m e l y :
f
A 1 dx dy , T(0, 0, 0) =~-~ d i s k r Integration yields:
We c a n c a l c u l a t e the t e m p e r a t u r e
r 2 = x2 + y2 .
qa2c T(O, O, O) = A c = 2b---6-~ "
(12)
on t h e b o u n d a r y of t h e disk. T h i s g i v e s : T(O, O, c) = 2Ac/Tr
(13)
T(0, 0, 0) = ½7r . T(0, 0, c)
(14)
and so:
INFLUENCE OF LOCAL ABSENCE OF COOLING
6. SOLUTION IN A HOMOGENEOUS H A L F - S P A C E OF T H E R M A L CONDUCTIVITY kli WITH C O M P L E T E COOLING We now c o n s i d e r the e x t r e m e c a s e of c o m p l e t e cooling of the p e r t u r b a t i o n t e r m outside the disk, i . e . , c a s e lb) m e n t i o n e d in s e c t i o n 4, w h e r e we have T = 0 on z = 0 outside the disk. Eqs. (10) a r e in this c a s e :
AT=O,
f o r ( x , y , z ) - * ~o ,
T -~ 0
onz =0, x2+y 2 >c2,
T=O aT Oz
(15)
o n z = 0 , x 2 + y 2 < c 2.
--=-A
We now i n t ro d u ce c o o r d i n a t e s in which the L a p l a c e equation s e p a r a t e s . T h e s e a r e the oblate s p h e r o i d a l coordinates: x = c ~ ~ c o s g ~ ,
y = c ~ / 1 + 7
2 sin~,
z =ewe.
~? = 0 is the s m a l l c i r c l e , ~0 = 0 is z = 0 outside the c i r c l e , w >/ 0 is the h a l f - s p a c e z >/ 0. Rotation s y m m e t r i c a l solutions of the potential equation a r e given by the functions:
T = B n Pn(w) Vn(i~) , w h e r e B n is a constant, n an i n t e g e r , Pn(¢O) is the L e g e n d r e function of the f i r s t kind, Qn(b?) is the L e g e n d r e function of the second kind with p u r e l y i m a g i n a r y a r g u m e n t , defined in r e f . [1], vol. 1, s e c t i o n 3.16. The condition outside the disk is:
B n Pn(O) Vn(i~) = 0 . In view of the p r o p e r t i e s of the L e g e n d r e functions n has to be an odd n u m b e r to s a t i s f y this condition. On the disk we have: aT Bn r d V ( i ~ a-z = cw L ~ ? = 0
Pn(w) = -A .
We w r i t e qn for [dQn(iy)/dy]y= O. As Pl(W) = w, we have B l q l / c = -A, so that S 1 =-~
cA
and
T =-
cA~o
q--~-Ql(iT) •
On the disk we have :7 = 0 and with p 2 = x 2 + y 2 we have in C a r t e s i a n c o o r d i n a t e s :
T(p,O,O) =
-cAQl(iO) ~1 p2 ql ---~"
The m a x i m u m v al u e o c c u r s when p = 0: T(0,0,0) = -
cAQ 1 (i0) -
ql
A f t e r c a l c u l a t i o n of Ql(i0) and q l we finally find: T(p, 0,0) =
2cA ~11 ~
p2
c2 ,
p2 = x 2 + y 2 ,
2
Tma x = T(0,0,0) =~ A c .
(16)
We s ee that indeed T(c, 0, 0) = 0, which fact was i m p l i c a t e d in the boundary condition.
7. SOLUTION IN A H A L F - S P A C E CONSISTING OF URANIUM AND CLADDING WITH C O M P L E T E ISOLATION
7.1. Derivation o f an integral r e p r e s e n t a t i o n We now t r y to find a solution of the p e r t u r b a t i o n t e r m by s e p a r a t i n g the v a r i a b l e s . We c o n s i d e r c a s e 2) of s e c t i o n 4, w h e r e the w a l l - t h i c k n e s s is d = b - a . As we have c y l i n d e r - s y m m e t r y b e c a u s e of the disk we i n t ro d u ce c y l i n d e r c o o r d i n a t e s in the h a l f - s p a c e (see fig. 3).
6
C.M. KALKER z=-d
We s e e k a s o l u t i o n of the f o r m :
T ( r , z ) =f(r) g(z) . z=O
Laplace's equation AT = 0 gives: _l d_ 2_ f 4. 1 . d f. r dr 2 rfdr
. 1 .d2g
.
_A2 ,
g dz2
w h e r e A is an a r b i t r a r y c o n s t a n t . T h i s g i v e s a s s o l u t i o n s : f(r) = Jo(Ar), w h e r e J o is the B e s s e l f u n c t i o n of the f i r s t kind of o r d e r z e r o d e f i n e d in r e f . [1], vol. 2.7.2.1.(4) and
Fig. 3.
g(z) =A e -Az + B e Az . The b o u n d a r y c o n d i t i o n s g i v e that in I:
g(z) = C e -~z ,
a s T - ~ 0, f o r z ~
¢o.
On the b o u n d a r y - p l a n e z = 0 we h a v e :
dgI gi = g I I ,
dgII
~-a-¢ =ku d~ '
so that:
A+B=C,
-)~CkI = k I I ( - k A + A B ) .
T h i s g i v e s f o r the c o n s t a n t s A , B and C: B = (kn - k i ) B ,
A = (bi + k I I ) ~ ,
C = 2kli~,
w h e r e /~ i s an a r b i t r a r y c o n s t a n t . T h u s a s o l u t i o n in II is:
[3go(Xr){(kI+ kii) e -Xz +
(kii -ki) eXZ}.
H e r e the c o n s t a n t ~ m a y a s s u m e a r b i t r a r y v a l u e s and f~ m a y d e p e n d on ~. A s u p e r p o s i t i o n g i v e s a l s o a p e r m i s s i b l e t e m p e r a t u r e , so that in II we h a v e : oO
T =f
Jo(kr) /~(A){(kii+ ki) e -~z + ( k i i - kI) e }'z} dA .
(17)
0 We now s e e k a /~(X) w h i c h m a k e s T s a t i s f y the b o u n d a r y c o n d i t i o n s in e q s . (10). e q u a t i o n s is the p l a n e z = - d in the c o o r d i n a t e s u s e d in t h i s s e c t i o n . T h e s o l u t i o n (17) now g i v e s f o r z = -d:
T h e p l a n e z = 0 in t h o s e
cO
T(-d) = f
Jo(~r) /3(X) {(k H + ki) e ~d + (kii - ki) e -Ad} dA ,
(18)
0 oo
r...O~_[j = f Jo(Ar) ~(X) {z=-d 0
(kli + ki)
e xd +
(kli - k i ) e-Xd} d~ .
We ~ve a b l e to find a BOO w h i c h s a t i s f i e s the b o u n d a r y c o n d i t i o n s f o r d T / d z f o r c o m p l e t e i s o l a t i o n of the p e r t u r b a t i o n t e r m , v i z . : aT az = 0
onz =-d,
r>c,
OT az - - ' 4
onz =-d,
r
(19)
With the a i d of the w e l l - k n o w n d i s c o n t i n u o u s i n t e g r a l ( s e e r e f . [1], v o l . 2, 7.7.4.(29)):
f
0
oo
t c-1
Jo(Xr) J l ( C k ) dh : ~
0
for r c,
w h e r e J1 i s the B e s s e l f u n c t i o n of t h e f i r s t o r d e r , we find t h a t eq. (18) s a t i s f i e s the b o u n d a r y c o n d i t i o n s of c o m p l e t e i s o l a t i o n (19), if:
INFLUENCE OF LOCAL ABSENCE OF COOLING
7
~(k) {-k(kii +ki) eXd+ ( k l i - k i ) e -kd} = -Jl(Ck)Ac, so that the solution (17) b e c o m e s : oO
T(z,r) = f Jo(kr) cnJl(CX) {(kII+kI) e-kZ + (kII -kI) eXz}dk . 0
X{(kII + kI) ekd - (kII - ki) e-;td}
7.2. Analytical evaluation of the integral We a r e e s p e c i a l l y i n t e r e s t e d in the value of the above integral on the boundary z = -d, where we have T( - d , 7)
cA
Jo(Xr) gl (cA) {(kii + ki) e ;td + (kii - ki) e-;td} 30
X{(kll + ki) e )td - (kli - kI) e-~td}
d~t
D
(20)
We evaluate this integral by splitting off a s o - c a l l e d "known part" T h f r o m the right-hand side, namely the integral a r i s i n g when kI = kii. This integral T r e p r e s e n t s the t e m p e r a t u r e of a homogeneous h a l f - s p a c e with complete isolation of the p e r t u r b a t i o n t e r m , which we a l r e a d y found in section 5, eq. (12). We have thus for eq. (20):
T(-d,r) = T s+ T h , with
oO
Ts = cA :
T h = cA f J°(kr) gl(c~t) ~t dk 0 Jo(~r)_~l (C~t) l (kI +kII) e~td + (kII -kI) e-~d
0
I
(kI + kII) e;td - (kII - kI) e-~td " 1 d~t.
(21)
which b e c o m e s after s o m e calculations:
T s = 2cAp f ~ Jo(~tp) Jl(;t) 0
e-6X 1 - /~ e -6~
d~t
with r
p =-~ ,
kii - kI
2d
5 =-c- '
P - kii + k I"
We can evaluate this integral f o r p = 0, i.e., for the m a x i m u m t e m p e r a t u r e in the center of the disk. S e r i e s expansion in the integrand gives: ao
rs(O,O,O)--2cA n--o: 0 f Jl(X) T h e s e integrals can be e x p r e s s e d in t e r m s of the h y p e r g e o m e t r i c function. We thus get with ref. [1], vol. 2, 7.7.3. (16): 1 r s ( 0 , 0 , 0 ) -- 2cAp ~ : 2F1(~; 1; 2; ) n---0 2(n+ 1)5 (n+ 1)252 " With the aid of a ~Lown integral r e p r e s e n t a t i o n f r o m ref. [1], 2.12.(1) for the h y p e r g e o m e t r i c function this b e c o m e s :
cA, ~ ,n : Ts(O,O,O)=~n=O-n--~-~O
(1+
t -½ (n + 1)252 ~dt.
T h e s e integrals can easily be integrated and we finally get the convergent s e r i e s :
Ts(0,0,0)=4~ ~ ~ (n+1): (Vi+ n=O
I
(n + 1)262
1)
"
8
C.M. KALKER
We thus have for the i n t e g r a l (20) for r = 0:
T(O,O,O)=Ts(O,O,O)+Th(O,O,O)=cA+4dAp with
2a c ,
~ (n+l)pn n=O
+
1 (n+ 1 )252
(22)
kii -ki ~ -~i+Ktcji/,¢.I
7.3. Calculation of some limiting cases We will now i n v e s t i g a t e the b e h a v i o u r of T(0, 0, 0) for asymptotic values of the t h i c k n e s s d of the cladding. F o r d ~ 0 we have: V
1 1 1 + (n+ 1)262 ~ (n+ 1)6 '
so that: l i m T(O,O,O) = cA + 2cA#
d~O
~ Un = cA + 2cAp = cA--kII . n=O ~ hi
As A = qa2/2bkii, this is the r e s u l t of s e c t i o n 5 for a t h e r m a l conductivity kI. Thus for d ~ 0 we find indeed the t e m p e r a t u r e r i s e in a homogeneous h a l f - s p a c e with t h e r m a l conductivity kI. F o r d --* oo we have: VI+
1 -1~ (n+ 1)26 2
1 2(n+ 1)262 '
so that:
c2A ~ /~n+l c2A l i m T(0,0,0) = cA + log (1 _ p ) - I = cA(1 + 2---dlog c = cA + (1 - / ~ ) - 1 ) d--.oo ~-d- n= o ~ + 1 2--d-
(23)
F o r d ~ oo we find l i m T(0, O, O) = cA, which is the t e m p e r a t u r e r i s e of a homogeneous h a l f - s p a c e with d--. oo t h e r m a l conductivity kIi. When we calculate the t e m p e r a t u r e r i s e for some p r a c t i c a l c a s e s a c c o r d i n g to eq. (22), and for the same v a l u e s of the p a r a m e t e r s a c c o r d i n g to eq. (23), these f o r m u l a e give the s a m e r e s u l t with a d i s c r e p a n c y of about 10%. So we can use f o r m u l a (23) which is e a s i e r to handle as a rough a p p r o x i m a t i o n of eq. (22) for the v a l u e s of the c o n s t a n t s which we have used here. M o r e o v e r it a p p e a r s f r o m c a l c u l a tion of eq. (22): l i m T(0, 0, 0) < [T(0, 0, 0)] c e r t a i n d < l i m T(0, 0, 0) , d-oO d~oo and m o r e p r e c i s e l y : lira T(0, 0, 0) ~ ½T(0, 0, 0) , d~0
lira T(0, 0, 0) ~ 3T(0, 0, 0) . d--~oo
8. HEAT-PRODUCTION IN STEEL TUBE SIMULATING THE CLADDING In o r d e r to c o m p a r e with e x p e r i m e n t a l r e s u l t s we c o n s i d e r the case of a c o n s t a n t heat p r o d u c t i o n of q watt per unity of volume in the cladding (in fact the s t e e l tube). We thus have the following d i f f e r e n t i a l equations for the t e m p e r a t u r e (see fig. 1): in I:
AT = 0 ,
in II:
AT = -q/kii .
The fact that d i f f e r e n t i a l equations in I and II have b e e n i n t e r c h a n g e d , r e s u l t s in a different t e m p e r a t u r e d i s t r i b u t i o n in the u n d i s t u r b e d state. With the s a m e b o u n d a r y conditions as in s e c t i o n 2, we find a c o n stant t e m p e r a t u r e in I, in the cladding 11 we have for the u n d i s t u r b e d t e m p e r a t u r e d i s t r i b u t i o n :
INFLUENCE OF LOCAL ABSENCE OF COOLING
Ton
a2q
r
= 2-~II l n ~ +
~
9
q(b 2 - a 2)
-IA (b2-r2) +
~
fora
F o r t h e p e r t u r b a t i o n t e r m t h e s a m e c o n s i d e r a t i o n s hold, o n l y the b o u n d a r y c o n d i t i o n f o r 3T/Sz, d e r i v e d in s e c t i o n 4, b e c o m e s :
]
3__T = ~ T I I ! old s y s t e m : -q(b2 -a2) L d r [r=bJ 2kII
Oz
The r e s u l t i s t h a t in the c a s e of h e a t - p r o d u c t i o n in the c l a d d i n g we h a v e f o r the c o n s t a n t A in the p r e ceding sections:
A - q(b2 - a2) 2kiib If we c a l l t h e t e m p e r a t u r e r i s e with h e a t p r o d u c t i o n in the c l a d d i n g Tq c l a d d i n g and the t e m p e r a t u r e with h e a t p r o d u c t i o n in the f u e l Tqfuel, t h e n we h a v e :
rise
T q c l a d d i n g = b2 _ a 2
Tq f u e l
a2
9. S U M M A R Y AND C O N C L U S I O N We s u m m a r i z e t h e r e s u l t s of the p r e c e d i n g s e c t i o n s in t a b l e 1. We m e n t i o n f i r s t t h e p a r a m e t e r s o c c u r r i n g in t h e f o r m u l a e (see fig. 4):
ljd I
c q a b d M kli
= = = = = = =
t h e r a d i u s of t h e d i s k w h e r e no c o o l i n g t a k e s p l a c e , the h e a t p r o d u c t i o n p e r u n i t v o l u m e , the r a d i u s of the u r a n i u m p a l l e t s , t o t a l r a d i u s of f u e l and c l a d d i n g , b - a = the t h i c k n e s s of the c l a d d i n g , t h e r m a l c o n d u c t i v i t y of the f u e l , t h e r m a l c o n d u c t i v i t y of the c l a d d i n g . kII-kI P
-
kiI+k
5-2d I '
IT
:_I¢ I-
I,<
(1
b
2(b-a)
c
-
e
It s e e m s r e a s o n a b l e to s u p p o s e t h a t the r e a l c a s e w h e r e a c e r t a i n a m o u n t of c o o l i n g of the p e r t u r b a t i o n t e r m t a k e s p l a c e , l i e s b e t w e e n both e x t r e m e s . In c a s e 1) we h a v e that:
Fig. 4.
Ta-¢o(O, O, O) =2 Ac = 0.636Ac = 0.636Ta_0(0 , 0, 0) --
7T
--
'
s o that:
0.636Ta=0(0 , 0, 0) --< T r e a l ( 0 , 0, 0) ~< Ta=o(O, O, O) . We c a n s a y t h a t T r e a l ( 0 , 0, 0) h a s b e e n c a l c u l a t e d with an a c c u r a c y of 20%. N o t e t h a t t h e real. t e m p e r a t u r e in t h e c e n t e r of the d i s k e q u a l s T O + T r e a l + T c o o l i n g l i q u i d , t h a t is in c a s e 1)
qa 2
qa2c
T t o t a l = T c o o l i n g l i q u i d + - 2 ~ + 0 2kii b w i t h 0.636 < O < 1. In c a s e 2) we c a l c u l a t e d only the u p p e r l i m i t f o r T(0, 0, 0), n a m e l y the c a s e a = 0. A s s u m i n g that the l o w e r l i m i t , the c a s e a = co, i s a f a c t o r 2 / ~ s m a l l e r , l i k e in the s i m p l i f i e d c a s e , we g e t
Treal(0, 0, 0) = 0 with 0.636 < 0 < 1.
+ 4d.4~
~
n=O
(n+l)p n \ I+
(n + 1)252
- I
c. M. KALKER
10
Table 1 Heat p r o d u c t i o n of q watt p e r unity of v o l u m e in fuel Temperature rise
A = qa2/2kiib No cooling of p e r t u r b a t i o n t e r m , a = 0 outside the i s o l a t e d disk
C o m p l e t e co o l i n g of p e r t u r b a tion t e r m , a =¢o outside the i s o l a t e d disk.
T(O, O, O) = A c = qa2c/2bkii
T(0, 0, 0) = (2/y)Ac
T(O, O, c) = (2/~)Ac
T(O,O,c) = 0
1) H o m o g e n e o u s h a l f - s p a c e c o n s i s t i n g of cladding (kii) p e r t u r b a t i o n t e r m = 0 on bound a r y of cladding and fuel. 2) H a l f - s p a c e , c o n s i s t i n g of fuel and cladding with t h i c k n e s s d
T(O, O, O) = cA + +4dAp
~
n=O
Not known (n+l)/~ n ×
1 - 1) × (Vl + (n+ l)262 Same as 2), ro u g h a p p r o x i m a t i o n
Idem,
T(O,O,O) = A c = qa2c/2bkIi
l i m i.e. c a s e 1) d-.~o
I d e m , lira i.e. c a s e 1) f o r fuel (ki) d-~0
I0. NUMERICAL
cA(1 + ~ d log (1-~t) - 1 )
T(O,O,O) = qa2c/2bk I
RESULTS
When the fuel r o d has a r a d i u s of 0.5 c m and the t h i c k n e s s of the cladding is 0.085 c m , the r a d i u s of the disk 0.I cm, we find with: kUO 2 = 2.9 W / m ° C ,
k z i r c a l o y = 12.8 W / m ° C ,
q = 600 W / c m 3 ,
in c a s e 1) a t e m p e r a t u r e r i s e of 100oc in the c e n t e r of the disk in the c a s e a = 0, and a t e m p e r a t u r e r i s e of 63°C in the c a s e a = ¢¢. C a l c u l a t i o n of c a s e 2) g i v e s with the s a m e c o n s t a n t s a t e m p e r a t u r e r i s e of 150 ° f o r a = 0; f o r a = we can a s s u m e , as s t a t e d in the p r e c e d i n g s e c t i o n ,
Ta=~o(O, O, O) = 0.636 × 150oc ~ 95°C. In the c a s e of heat p r o d u c t i o n in the cladding t h e s e r e s u l t s should be m u l t i p l i e d by: (b 2 ~a2)/a 2 ~ 0 . 3 , usi ng the s a m e q and kip
REFERENCE [1] Erdelyi, Bateman Manuscript Project, Higher Transcendental Functions, Vols. 1 and 2.
THE INFLUENCE OF LOCAL ABSENCE OF COOLING O N T H E C L A D D I N G T E M P E R A T U R E O F A F U E L R O D * ** Mrs. C. M. KALKER R e a c t o r C e n t r u m Nederland, The Hague, The N e t h e r l a n d s Received 5 N o v e m b e r 1965
The t e m p e r a t u r e r i s e at the position of support of the fuel r o d s which r e s u l t s f r o m the local a b s e n c e of cooling is calculated. The support of the fuel rod is r e p r e s e n t e d by a s m a l l c i r c l e , lying on the outer s u r f a c e of the cladding, which s i m u l a t e s the contact a r e a between fuel rod and supporting grid. The t e m p e r a t u r e r i s e is c o n s i d e r e d a s a p e r t u r b a t i o n t e r m s u p e r - i m p o s e d on the u n d i s t u r b e d t e m p e r a t u r e distribution. Two e x t r e m e c a s e s a r e c o n s i d e r e d , n a m e l y complete isolation of the p e r t u r b a t i o n t e r m and complete cooling of the p e r t u r b a t i o n t e r m outside the s m a l l c i r c l e . In o r d e r to c o m p a r e with e x p e r i m e n t a l r e s u l t s the s a m e c a l c u l a t i o n s a r e applied to a c a s e of heat production in a sl:eel tube, which is heated e l e c t r i c a l l y .
1. INTRODUCTION This p a p e r p r e s e n t s the r e s u l t s of c a l c u l a t i o n s p e r f o r m e d in connection with the NERO-project. T h i s p a p e r e n v i s a g e s the study by R.C.N. of a r e a c t o r for ship propulsion. In the p r e s e n t p a p e r we d e r i v e some e x p r e s s i o n s for the t e m p e r a t u r e r i s e o c c u r r i n g at the place of support of fuel rods which r e s u l t s f r o m the local a b s e n c e of cooling. As we have as yet no data a v a i l a b l e with r e g a r d to the local v a r i a t i o n in cooling, we c o n s i d e r only the l i m i t i n g case by a s s u m i n g that t h e r e is no cooling at all at the place of support of the fuel rods. The r e s u l t will t h e r e f o r e be m o r e p e s s i m i s t i c than the reality. E x p e r i m e n t s will have to b e done to get an idea how much t e m p e r a t u r e s have b e e n o v e r e s t i m a t e d . Such e x p e r i m e n t s g e n e r a l l y take place by the heating of t h i n - w a l l e d steel tubes. The place and the s p a c e - d e p e n d e n c y of the heat p r o d u c t i o n is in this way different. To t r a n s l a t e the e x p e r i m e n t to a r e s u l t for fuel r o d s , the t e m p e r a t u r e r i s e has also been c a l c u l a t e d for u n i f o r m heat p r o d u c t i o n in a t h i n - w a l l e d steel tube u n d e r fully identical conditions. C o m p a r i n g the e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s it is hoped that a good i m p r e s s i o n of the t e m p e r a t u r e r i s e of the fuel rod is obtained.
2. FORMULATION OF THE PROBLEM We c o n s i d e r a cyl:mdrical fuel rod p r o d u c i n g heat of q watt p e r unit volume. We neglect the gap between cladding and fuel. Let the r a d i u s of the fuel rod be a , that of both fuel and cladding b (see fig. 1). On the s u r f a c e of the c y l i n d e r t h e r e is a disk with r a d i u s c where t h e r e is no cooling at all. The t h e r m a l conductivity of the fuel is k i , that of the cladding kII. The t e m p e r a t u r e in I s a t i s f i e s P o i s s o n ' s d i f f e r e n t i a l equation: AT = -q/k I .
i.
I
b
(1) I
The t e m p e r a t u r e in B: s a t i s f i e s L a p l a c e ' s equation: AT = 0 .
I
(2)
Fig. 1.
* R e s e a r c h s p o n s o r e d by E u r a t o m , and originally i s s u e d a s r e p o r t R C N - I n t - 6 4 - 0 0 3 / 1 (EUR-2182.e). ** Accepted by M. Bogaardt.
I¢
2
C.M. KALKER
F o r r = a we have that the h e a t flux and t e m p e r a t u r e a r e c o n t i n u o u s , that i s : aT I ~T H k I - ~ - = klI ~ - -
T I = TII ,
for r = a .
(3)
F o r r = b we have that t h e h e a t c o n d u c t e d by the c l a d d i n g e q u a l s the h e a t t r a n s f e r r e d to the c o o l i n g l i q uid, that is:
a(T(b) -Tk) + kIi [OT/Or]r__b = 0 ,
(4)
w h e r e T k is the a v e r a g e t e m p e r a t u r e of the c o o l i n g liquid. S o l u t i o n of eqs. (1) a n d ( 2 ) w i t h b o u n d a r y c o n d i t i o n s (3) and (4) g i v e s a s o - c a l l e d u n d i s t u r b e d t e m p e r ature distribution. H o w e v e r , we a s k the t e m p e r a t u r e which r e s u l t s f r o m the l o c a l a b s e n c e of c o o l i n g , or: a =0
(5)
inside a small circle with radius c lying on the outer surface of the cladding, which is considered representative of the support of the fuel rod.
3. THE UNDISTURBED T E M P E R A T U R E D I S T R I B U T I O N We s h a l l now c a l c u l a t e the t e m p e r a t u r e in the s o - c a l l e d u n d i s t u r b e d s t a t e ; we n e g l e c t the " l o c a l c o n d i t i o n " (5) a n d s o l v e eqs. (1) to (4). B e c a u s e of the c y l i n d e r - s y m m e t r y the e q u a t i o n s a r e : in I:
i n II:
d2T 1 dT q dr2 + r-d-r = - k I '
(6)
d2T I dT dr 2 +rd-r =0"
(7)
The b o u n d a r y c o n d i t i o n s a r e : f o r ~ = a:
TI = TII '
dT I dTII kI - ~ - = kII d r
(8) dTII
for r = b:
a ( T - Tk) + kii - ~ - = 0 .
A p a r t i c u l a r s o l u t i o n of eq. (6) is: T = -qr2/4k I . A s o l u t i o n of the h o m o g e n e o u s e q u a t i o n (7) i s : T = A + B l o g r , w h e r e A a n d B a r e c o n s t a n t s . I n r e g i o n I we h a v e B = 0 , a s T s h o u l d be f i n i t e f o r r = 0 . We t h u s have the f o l l o w i n g s o l u t i o n s :
TI
=A -qr2/4ki,
TII =C + O l o g r .
S e t t i n g t h e t e m p e r a t u r e of~the c o o l i n g l i q u i d T k = O, we d e f i n e the c o n s t a n t s A , C and D b y m e a n s of the b o u n d a r y c o n d i t i o n s (8) a n d , f i n d f o r the u n d i s t u r b e d t e m p e r a t u r e :
-qa2
TOI - ~ I
(1
r2 2kI 2kI -a-2 + -~- +k-HHlog b )
qa2 lo
ToII = 2 k i i
biqa2
g r T-2ba
for ~" --< a ,
for a --< r -< b .
(9)
INFLUENCE OF LOCAL ABSENCE OF COOLING
3
4. EQUATIONS F OR THE P E R T U R B A T I O N T E R M Eqs. (9) g i v e the u n d i s t u r b e d t e m p e r a t u r e d is t r i b u t i o n . We c a l l it T o. On it we s u p e r i m p o s e a p e r t u r b a t i o n t e r m T, which r e s u l t s f r o m the disk on the s u r f a c e of the c y l i n d e r w h e r e t h e r e is no cooling at all. We a p p r o x i m a t e the neighbourhood of the disk, only f o r c a l c u l a t i o n of the p e r t u r b a t i o n t e r m by a cooling " " X h a l f - s p a c e z >I O. In this h a l f - s p a c e we work with a C a r t e s i a n c o o r d i n a t e s y s t e m x, y and z (see fig. 2). Let the total t e m p e r a t u r e be Tg, with Tg = = T+ T o. As the total t e m p e r a t u r e Tg in I and II s a t i s f i e s eqs. (1) r e s p e c t i v e l y (2) a n d T o is a s o lution, we have f o r T in both r e g i o n s : .~T = 0 . We s h al l now d e r i v e the boundary conditions f o r T. We a s s u m e that f a r f r o m the disk we have the u n d i s t u r b e d t e m p e r a t u r e d i s t r i b u t i o n , that is T ~ 0 f o r ( x , y , z ) - ~ ~o. Outside the disk, on the s u r f a c e z = 0, we have that: 0 :
Fig. 2. ~T
o r
a
TO
aT
+
f or z = 0, x 2 + y 2 > c 2, s i n c e we have f o r the u n d i s t u r b e d t e m p e r a t u r e To: aTo a T o - kii - ~ - = O . I n s i d e the disk we have that:
aTg az
O=
-
aT az
+
aT o az
We can w r i t e : aTo = _ FdToII~ az L dr J r=b' w h e r e ToII is the u n d i s t u r b e d t e m p e r a t u r e c a l c u l a t e d in c y l i n d r i c a l c o o r d i n a t e s in eq. (9). D i f f e r e n t i a tion y i e l d s :
aTo
qa 2
az
2bkii
and so:
T _ az
qa 2
f o r z = 0 , x 2 + y 2 < c 2.
2bkii
S u m m a r i z i n g we find f o r the p e r t u r b a t i o n t e m p e r a t u r e T the d i f f e r e n t i a l equation: ~T=O with b o u n d a r y conditions: for ( x , y , z ) ~ o o ,
T~O
aT a T - kii ~ = O
forz=0,
aa -TY
f o r z = 0 , x 2 + y 2 < c 2.
x2 + y2 > c 2,
(lO) = -
2 -qa2 6-~,
=
-A
In the following s e c t i o n s we will s o l v e t h e s e e q u at i o n s f o r s o m e s i m p l i f i e d c a s e s :
4
C.M. KALKER
1) We a s s u m e t h a t the p e r t u r b a t i o n t e m p e r a t u r e i s not n o t i c e a b l e on the b o u n d a r y b e t w e e n u r a n i u m and c l a d d i n g , i . e . , ( b - a ) is p r a c t i c a l l y i n f i n i t e w h e r e T is c o n c e r n e d . T h u s the h a l f - s p a c e z >/ 0 has a t h e r m a l c o n d u c t i v i t y kii. With t h i s a s s u m p t i o n we c a l c u l a t e two l i m i t i n g c a s e s l a and lb: l a ) T h e p e r t u r b a t i o n t e m p e r a t u r e i s not c o o l e d at a l l , i . e . , a = 0 and s o a T / a z = 0 o u t s i d e the disk. lb) C o m p l e t e c o o l i n g of the p e r t u r b a t i o n t e m p e r a t u r e o u t s i d e the d i s k , i . e . , a = oo o r T = 0 o u t s i d e the disk. 2) T h e p e r t u r b a t i o n t e m p e r a t u r e h a s a c e r t a i n v a l u e on the b o u n d a r y of the c l a d d i n g and u r a n i u m , i . e . , the h a l f - s p a c e z >/ 0 c o n s i s t s of a s l a b w i t h t h i c k n e s s d = b - a and t h e r m a l c o n d u c t i v i t y kii and a r e g i o n with t h e r m a l c o n d u c t i v i t y k I. We c a n h e r e a l s o c o n s i d e r c o m p l e t e i s o l a t i o n and c o m p l e t e c o o l i n g of the p e r t u r b a t i o n t e r m . A s the l a t t e r c a s e p r e s e n t s u n s u r m o u n t a b l e d i f f i c u l t i e s f o r the p r e s e n t , we c a l c u l a t e only the c a s e a = 0 and a T / a z = 0 o u t s i d e the disk. In any c a s e that w i l l b e m o r e p e s s i m i s t i c t h a n the t r u t h .
5. S O L U T I O N IN A H O M O G E N E O U S H A L F - S P A C E O F T H E R M A L C O N D U C T I V I T Y kI I WITH COMPLETE ISOLATION We w i l l now s e e k a s o l u t i o n of e q s . (10) f o r c a s e l a ) , a = 0 and a T / a z = o o u t s i d e the disk. thus become: AT = 0 ,
H e r e a T / a z is p r e s c r i b e d
T ~ 0
for (x,y,z)
(10)
~ oo ,
a__TT= 0 az
on z = 0, x 2 + y 2 > c 2,
aT --A ~z
onz =0, x2+y2
on the b o u n d a r y of the h a l f - s p a c e .
Eqs.
(11)
Now it is known t h a t R -1, with
R 2 = (x-~)2 + (y_~)2 + (z-~)2 , satisfies
Laplace's
equation everywhere
e x c e p t in x, y , z . We c a n show t h a t the c o n n e c t i o n b e t w e e n T
and ~ T / a z i s g i v e n by: T(x,y,z)
1
= --~
f-~aT < ~ , y ) lpd v
dy
,
p2 = (x _~)2 + (y _~)2 + z 2
In o u r e a s e t h i s b e c o m e s : 1 f Ad~d~ T ( x , y , z) = - ~ d i s k p
"
W e s e e t h a t T ( x , y , z ) ---' 0 f o r ( x , y , z ) ---. 00. F r o m g e o m e t r i c a l c o n s i d e r a t i o n s it i s e a s y to s e e t h a t the t e m p e r a t u r e h a s its m a x i m u m in the c e n t e r of the d i s k , n a m e l y :
f
A 1 dx dy , T(0, 0, 0) =~-~ d i s k r Integration yields:
We c a n c a l c u l a t e the t e m p e r a t u r e
r 2 = x2 + y2 .
qa2c T(O, O, O) = A c = 2b---6-~ "
(12)
on t h e b o u n d a r y of t h e disk. T h i s g i v e s : T(O, O, c) = 2Ac/Tr
(13)
T(0, 0, 0) = ½7r . T(0, 0, c)
(14)
and so:
INFLUENCE OF LOCAL ABSENCE OF COOLING
6. SOLUTION IN A HOMOGENEOUS H A L F - S P A C E OF T H E R M A L CONDUCTIVITY kli WITH C O M P L E T E COOLING We now c o n s i d e r the e x t r e m e c a s e of c o m p l e t e cooling of the p e r t u r b a t i o n t e r m outside the disk, i . e . , c a s e lb) m e n t i o n e d in s e c t i o n 4, w h e r e we have T = 0 on z = 0 outside the disk. Eqs. (10) a r e in this c a s e :
AT=O,
f o r ( x , y , z ) - * ~o ,
T -~ 0
onz =0, x2+y 2 >c2,
T=O aT Oz
(15)
o n z = 0 , x 2 + y 2 < c 2.
--=-A
We now i n t ro d u ce c o o r d i n a t e s in which the L a p l a c e equation s e p a r a t e s . T h e s e a r e the oblate s p h e r o i d a l coordinates: x = c ~ ~ c o s g ~ ,
y = c ~ / 1 + 7
2 sin~,
z =ewe.
~? = 0 is the s m a l l c i r c l e , ~0 = 0 is z = 0 outside the c i r c l e , w >/ 0 is the h a l f - s p a c e z >/ 0. Rotation s y m m e t r i c a l solutions of the potential equation a r e given by the functions:
T = B n Pn(w) Vn(i~) , w h e r e B n is a constant, n an i n t e g e r , Pn(¢O) is the L e g e n d r e function of the f i r s t kind, Qn(b?) is the L e g e n d r e function of the second kind with p u r e l y i m a g i n a r y a r g u m e n t , defined in r e f . [1], vol. 1, s e c t i o n 3.16. The condition outside the disk is:
B n Pn(O) Vn(i~) = 0 . In view of the p r o p e r t i e s of the L e g e n d r e functions n has to be an odd n u m b e r to s a t i s f y this condition. On the disk we have: aT Bn r d V ( i ~ a-z = cw L ~ ? = 0
Pn(w) = -A .
We w r i t e qn for [dQn(iy)/dy]y= O. As Pl(W) = w, we have B l q l / c = -A, so that S 1 =-~
cA
and
T =-
cA~o
q--~-Ql(iT) •
On the disk we have :7 = 0 and with p 2 = x 2 + y 2 we have in C a r t e s i a n c o o r d i n a t e s :
T(p,O,O) =
-cAQl(iO) ~1 p2 ql ---~"
The m a x i m u m v al u e o c c u r s when p = 0: T(0,0,0) = -
cAQ 1 (i0) -
ql
A f t e r c a l c u l a t i o n of Ql(i0) and q l we finally find: T(p, 0,0) =
2cA ~11 ~
p2
c2 ,
p2 = x 2 + y 2 ,
2
Tma x = T(0,0,0) =~ A c .
(16)
We s ee that indeed T(c, 0, 0) = 0, which fact was i m p l i c a t e d in the boundary condition.
7. SOLUTION IN A H A L F - S P A C E CONSISTING OF URANIUM AND CLADDING WITH C O M P L E T E ISOLATION
7.1. Derivation o f an integral r e p r e s e n t a t i o n We now t r y to find a solution of the p e r t u r b a t i o n t e r m by s e p a r a t i n g the v a r i a b l e s . We c o n s i d e r c a s e 2) of s e c t i o n 4, w h e r e the w a l l - t h i c k n e s s is d = b - a . As we have c y l i n d e r - s y m m e t r y b e c a u s e of the disk we i n t ro d u ce c y l i n d e r c o o r d i n a t e s in the h a l f - s p a c e (see fig. 3).
6
C.M. KALKER z=-d
We s e e k a s o l u t i o n of the f o r m :
T ( r , z ) =f(r) g(z) . z=O
Laplace's equation AT = 0 gives: _l d_ 2_ f 4. 1 . d f. r dr 2 rfdr
. 1 .d2g
.
_A2 ,
g dz2
w h e r e A is an a r b i t r a r y c o n s t a n t . T h i s g i v e s a s s o l u t i o n s : f(r) = Jo(Ar), w h e r e J o is the B e s s e l f u n c t i o n of the f i r s t kind of o r d e r z e r o d e f i n e d in r e f . [1], vol. 2.7.2.1.(4) and
Fig. 3.
g(z) =A e -Az + B e Az . The b o u n d a r y c o n d i t i o n s g i v e that in I:
g(z) = C e -~z ,
a s T - ~ 0, f o r z ~
¢o.
On the b o u n d a r y - p l a n e z = 0 we h a v e :
dgI gi = g I I ,
dgII
~-a-¢ =ku d~ '
so that:
A+B=C,
-)~CkI = k I I ( - k A + A B ) .
T h i s g i v e s f o r the c o n s t a n t s A , B and C: B = (kn - k i ) B ,
A = (bi + k I I ) ~ ,
C = 2kli~,
w h e r e /~ i s an a r b i t r a r y c o n s t a n t . T h u s a s o l u t i o n in II is:
[3go(Xr){(kI+ kii) e -Xz +
(kii -ki) eXZ}.
H e r e the c o n s t a n t ~ m a y a s s u m e a r b i t r a r y v a l u e s and f~ m a y d e p e n d on ~. A s u p e r p o s i t i o n g i v e s a l s o a p e r m i s s i b l e t e m p e r a t u r e , so that in II we h a v e : oO
T =f
Jo(kr) /~(A){(kii+ ki) e -~z + ( k i i - kI) e }'z} dA .
(17)
0 We now s e e k a /~(X) w h i c h m a k e s T s a t i s f y the b o u n d a r y c o n d i t i o n s in e q s . (10). e q u a t i o n s is the p l a n e z = - d in the c o o r d i n a t e s u s e d in t h i s s e c t i o n . T h e s o l u t i o n (17) now g i v e s f o r z = -d:
T h e p l a n e z = 0 in t h o s e
cO
T(-d) = f
Jo(~r) /3(X) {(k H + ki) e ~d + (kii - ki) e -Ad} dA ,
(18)
0 oo
r...O~_[j = f Jo(Ar) ~(X) {z=-d 0
(kli + ki)
e xd +
(kli - k i ) e-Xd} d~ .
We ~ve a b l e to find a BOO w h i c h s a t i s f i e s the b o u n d a r y c o n d i t i o n s f o r d T / d z f o r c o m p l e t e i s o l a t i o n of the p e r t u r b a t i o n t e r m , v i z . : aT az = 0
onz =-d,
r>c,
OT az - - ' 4
onz =-d,
r
(19)
With the a i d of the w e l l - k n o w n d i s c o n t i n u o u s i n t e g r a l ( s e e r e f . [1], v o l . 2, 7.7.4.(29)):
f
0
oo
t c-1
Jo(Xr) J l ( C k ) dh : ~
0
for r
w h e r e J1 i s the B e s s e l f u n c t i o n of t h e f i r s t o r d e r , we find t h a t eq. (18) s a t i s f i e s the b o u n d a r y c o n d i t i o n s of c o m p l e t e i s o l a t i o n (19), if:
INFLUENCE OF LOCAL ABSENCE OF COOLING
7
~(k) {-k(kii +ki) eXd+ ( k l i - k i ) e -kd} = -Jl(Ck)Ac, so that the solution (17) b e c o m e s : oO
T(z,r) = f Jo(kr) cnJl(CX) {(kII+kI) e-kZ + (kII -kI) eXz}dk . 0
X{(kII + kI) ekd - (kII - ki) e-;td}
7.2. Analytical evaluation of the integral We a r e e s p e c i a l l y i n t e r e s t e d in the value of the above integral on the boundary z = -d, where we have T( - d , 7)
cA
Jo(Xr) gl (cA) {(kii + ki) e ;td + (kii - ki) e-;td} 30
X{(kll + ki) e )td - (kli - kI) e-~td}
d~t
D
(20)
We evaluate this integral by splitting off a s o - c a l l e d "known part" T h f r o m the right-hand side, namely the integral a r i s i n g when kI = kii. This integral T r e p r e s e n t s the t e m p e r a t u r e of a homogeneous h a l f - s p a c e with complete isolation of the p e r t u r b a t i o n t e r m , which we a l r e a d y found in section 5, eq. (12). We have thus for eq. (20):
T(-d,r) = T s+ T h , with
oO
Ts = cA :
T h = cA f J°(kr) gl(c~t) ~t dk 0 Jo(~r)_~l (C~t) l (kI +kII) e~td + (kII -kI) e-~d
0
I
(kI + kII) e;td - (kII - kI) e-~td " 1 d~t.
(21)
which b e c o m e s after s o m e calculations:
T s = 2cAp f ~ Jo(~tp) Jl(;t) 0
e-6X 1 - /~ e -6~
d~t
with r
p =-~ ,
kii - kI
2d
5 =-c- '
P - kii + k I"
We can evaluate this integral f o r p = 0, i.e., for the m a x i m u m t e m p e r a t u r e in the center of the disk. S e r i e s expansion in the integrand gives: ao
rs(O,O,O)--2cA n--o: 0 f Jl(X) T h e s e integrals can be e x p r e s s e d in t e r m s of the h y p e r g e o m e t r i c function. We thus get with ref. [1], vol. 2, 7.7.3. (16): 1 r s ( 0 , 0 , 0 ) -- 2cAp ~ : 2F1(~; 1; 2; ) n---0 2(n+ 1)5 (n+ 1)252 " With the aid of a ~Lown integral r e p r e s e n t a t i o n f r o m ref. [1], 2.12.(1) for the h y p e r g e o m e t r i c function this b e c o m e s :
cA, ~ ,n : Ts(O,O,O)=~n=O-n--~-~O
(1+
t -½ (n + 1)252 ~dt.
T h e s e integrals can easily be integrated and we finally get the convergent s e r i e s :
Ts(0,0,0)=4~ ~ ~ (n+1): (Vi+ n=O
I
(n + 1)262
1)
"
8
C.M. KALKER
We thus have for the i n t e g r a l (20) for r = 0:
T(O,O,O)=Ts(O,O,O)+Th(O,O,O)=cA+4dAp with
2a c ,
~ (n+l)pn n=O
+
1 (n+ 1 )252
(22)
kii -ki ~ -~i+Ktcji/,¢.I
7.3. Calculation of some limiting cases We will now i n v e s t i g a t e the b e h a v i o u r of T(0, 0, 0) for asymptotic values of the t h i c k n e s s d of the cladding. F o r d ~ 0 we have: V
1 1 1 + (n+ 1)262 ~ (n+ 1)6 '
so that: l i m T(O,O,O) = cA + 2cA#
d~O
~ Un = cA + 2cAp = cA--kII . n=O ~ hi
As A = qa2/2bkii, this is the r e s u l t of s e c t i o n 5 for a t h e r m a l conductivity kI. Thus for d ~ 0 we find indeed the t e m p e r a t u r e r i s e in a homogeneous h a l f - s p a c e with t h e r m a l conductivity kI. F o r d --* oo we have: VI+
1 -1~ (n+ 1)26 2
1 2(n+ 1)262 '
so that:
c2A ~ /~n+l c2A l i m T(0,0,0) = cA + log (1 _ p ) - I = cA(1 + 2---dlog c = cA + (1 - / ~ ) - 1 ) d--.oo ~-d- n= o ~ + 1 2--d-
(23)
F o r d ~ oo we find l i m T(0, O, O) = cA, which is the t e m p e r a t u r e r i s e of a homogeneous h a l f - s p a c e with d--. oo t h e r m a l conductivity kIi. When we calculate the t e m p e r a t u r e r i s e for some p r a c t i c a l c a s e s a c c o r d i n g to eq. (22), and for the same v a l u e s of the p a r a m e t e r s a c c o r d i n g to eq. (23), these f o r m u l a e give the s a m e r e s u l t with a d i s c r e p a n c y of about 10%. So we can use f o r m u l a (23) which is e a s i e r to handle as a rough a p p r o x i m a t i o n of eq. (22) for the v a l u e s of the c o n s t a n t s which we have used here. M o r e o v e r it a p p e a r s f r o m c a l c u l a tion of eq. (22): l i m T(0, 0, 0) < [T(0, 0, 0)] c e r t a i n d < l i m T(0, 0, 0) , d-oO d~oo and m o r e p r e c i s e l y : lira T(0, 0, 0) ~ ½T(0, 0, 0) , d~0
lira T(0, 0, 0) ~ 3T(0, 0, 0) . d--~oo
8. HEAT-PRODUCTION IN STEEL TUBE SIMULATING THE CLADDING In o r d e r to c o m p a r e with e x p e r i m e n t a l r e s u l t s we c o n s i d e r the case of a c o n s t a n t heat p r o d u c t i o n of q watt per unity of volume in the cladding (in fact the s t e e l tube). We thus have the following d i f f e r e n t i a l equations for the t e m p e r a t u r e (see fig. 1): in I:
AT = 0 ,
in II:
AT = -q/kii .
The fact that d i f f e r e n t i a l equations in I and II have b e e n i n t e r c h a n g e d , r e s u l t s in a different t e m p e r a t u r e d i s t r i b u t i o n in the u n d i s t u r b e d state. With the s a m e b o u n d a r y conditions as in s e c t i o n 2, we find a c o n stant t e m p e r a t u r e in I, in the cladding 11 we have for the u n d i s t u r b e d t e m p e r a t u r e d i s t r i b u t i o n :
INFLUENCE OF LOCAL ABSENCE OF COOLING
Ton
a2q
r
= 2-~II l n ~ +
~
9
q(b 2 - a 2)
-IA (b2-r2) +
~
fora
F o r t h e p e r t u r b a t i o n t e r m t h e s a m e c o n s i d e r a t i o n s hold, o n l y the b o u n d a r y c o n d i t i o n f o r 3T/Sz, d e r i v e d in s e c t i o n 4, b e c o m e s :
]
3__T = ~ T I I ! old s y s t e m : -q(b2 -a2) L d r [r=bJ 2kII
Oz
The r e s u l t i s t h a t in the c a s e of h e a t - p r o d u c t i o n in the c l a d d i n g we h a v e f o r the c o n s t a n t A in the p r e ceding sections:
A - q(b2 - a2) 2kiib If we c a l l t h e t e m p e r a t u r e r i s e with h e a t p r o d u c t i o n in the c l a d d i n g Tq c l a d d i n g and the t e m p e r a t u r e with h e a t p r o d u c t i o n in the f u e l Tqfuel, t h e n we h a v e :
rise
T q c l a d d i n g = b2 _ a 2
Tq f u e l
a2
9. S U M M A R Y AND C O N C L U S I O N We s u m m a r i z e t h e r e s u l t s of the p r e c e d i n g s e c t i o n s in t a b l e 1. We m e n t i o n f i r s t t h e p a r a m e t e r s o c c u r r i n g in t h e f o r m u l a e (see fig. 4):
ljd I
c q a b d M kli
= = = = = = =
t h e r a d i u s of t h e d i s k w h e r e no c o o l i n g t a k e s p l a c e , the h e a t p r o d u c t i o n p e r u n i t v o l u m e , the r a d i u s of the u r a n i u m p a l l e t s , t o t a l r a d i u s of f u e l and c l a d d i n g , b - a = the t h i c k n e s s of the c l a d d i n g , t h e r m a l c o n d u c t i v i t y of the f u e l , t h e r m a l c o n d u c t i v i t y of the c l a d d i n g . kII-kI P
-
kiI+k
5-2d I '
IT
:_I¢ I-
I,<
(1
b
2(b-a)
c
-
e
It s e e m s r e a s o n a b l e to s u p p o s e t h a t the r e a l c a s e w h e r e a c e r t a i n a m o u n t of c o o l i n g of the p e r t u r b a t i o n t e r m t a k e s p l a c e , l i e s b e t w e e n both e x t r e m e s . In c a s e 1) we h a v e that:
Fig. 4.
Ta-¢o(O, O, O) =2 Ac = 0.636Ac = 0.636Ta_0(0 , 0, 0) --
7T
--
'
s o that:
0.636Ta=0(0 , 0, 0) --< T r e a l ( 0 , 0, 0) ~< Ta=o(O, O, O) . We c a n s a y t h a t T r e a l ( 0 , 0, 0) h a s b e e n c a l c u l a t e d with an a c c u r a c y of 20%. N o t e t h a t t h e real. t e m p e r a t u r e in t h e c e n t e r of the d i s k e q u a l s T O + T r e a l + T c o o l i n g l i q u i d , t h a t is in c a s e 1)
qa 2
qa2c
T t o t a l = T c o o l i n g l i q u i d + - 2 ~ + 0 2kii b w i t h 0.636 < O < 1. In c a s e 2) we c a l c u l a t e d only the u p p e r l i m i t f o r T(0, 0, 0), n a m e l y the c a s e a = 0. A s s u m i n g that the l o w e r l i m i t , the c a s e a = co, i s a f a c t o r 2 / ~ s m a l l e r , l i k e in the s i m p l i f i e d c a s e , we g e t
Treal(0, 0, 0) = 0 with 0.636 < 0 < 1.
+ 4d.4~
~
n=O
(n+l)p n \ I+
(n + 1)252
- I
c. M. KALKER
10
Table 1 Heat p r o d u c t i o n of q watt p e r unity of v o l u m e in fuel Temperature rise
A = qa2/2kiib No cooling of p e r t u r b a t i o n t e r m , a = 0 outside the i s o l a t e d disk
C o m p l e t e co o l i n g of p e r t u r b a tion t e r m , a =¢o outside the i s o l a t e d disk.
T(O, O, O) = A c = qa2c/2bkii
T(0, 0, 0) = (2/y)Ac
T(O, O, c) = (2/~)Ac
T(O,O,c) = 0
1) H o m o g e n e o u s h a l f - s p a c e c o n s i s t i n g of cladding (kii) p e r t u r b a t i o n t e r m = 0 on bound a r y of cladding and fuel. 2) H a l f - s p a c e , c o n s i s t i n g of fuel and cladding with t h i c k n e s s d
T(O, O, O) = cA + +4dAp
~
n=O
Not known (n+l)/~ n ×
1 - 1) × (Vl + (n+ l)262 Same as 2), ro u g h a p p r o x i m a t i o n
Idem,
T(O,O,O) = A c = qa2c/2bkIi
l i m i.e. c a s e 1) d-.~o
I d e m , lira i.e. c a s e 1) f o r fuel (ki) d-~0
I0. NUMERICAL
cA(1 + ~ d log (1-~t) - 1 )
T(O,O,O) = qa2c/2bk I
RESULTS
When the fuel r o d has a r a d i u s of 0.5 c m and the t h i c k n e s s of the cladding is 0.085 c m , the r a d i u s of the disk 0.I cm, we find with: kUO 2 = 2.9 W / m ° C ,
k z i r c a l o y = 12.8 W / m ° C ,
q = 600 W / c m 3 ,
in c a s e 1) a t e m p e r a t u r e r i s e of 100oc in the c e n t e r of the disk in the c a s e a = 0, and a t e m p e r a t u r e r i s e of 63°C in the c a s e a = ¢¢. C a l c u l a t i o n of c a s e 2) g i v e s with the s a m e c o n s t a n t s a t e m p e r a t u r e r i s e of 150 ° f o r a = 0; f o r a = we can a s s u m e , as s t a t e d in the p r e c e d i n g s e c t i o n ,
Ta=~o(O, O, O) = 0.636 × 150oc ~ 95°C. In the c a s e of heat p r o d u c t i o n in the cladding t h e s e r e s u l t s should be m u l t i p l i e d by: (b 2 ~a2)/a 2 ~ 0 . 3 , usi ng the s a m e q and kip
REFERENCE [1] Erdelyi, Bateman Manuscript Project, Higher Transcendental Functions, Vols. 1 and 2.