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THERMAL STABILITY OF SUPERCONDUCTORS
C. Meuris
To cite this version:
C. Meuris. THERMAL STABILITY OF SUPERCONDUCTORS. Journal de Physique Colloques,
1984, 45 (C1), pp.C1-503-C1-510. �10.1051/jphyscol:19841103�. �jpa-00223571�
Colloque C I , supplement au no 1, Tome 45, janvier 1984 page C1-503
THERMAL S T A B I L I T Y OF SUPERCONDUCTORS
C. Meuris
C E A / S a c Z a y , L ) P ~ / P E - S T . ~ P E , 9 1 1 9 1 M f - s h r - Y v e t t e C e d e x , F r a n c e
Resume
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Nous analysons l e s p r i n c i p a u x r 6 s u l t a t s experimentaux obtenus s u r diff@rentes c o n f i g u r a t i o n s , l i e e sa
l a t e c h n o l o g i e a c t u e l l e des a i m a n t s supra-conducteurs, en c e q u i concerne l e u r s t a b i l i t e .
A b s t r a c t
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The l i m i t s o f a p p l i c a t i o n o f t h e commonly a p p l i e d s t a b i l i t y c r i t e r i a a r e discussed. Some p a r t i c u l a r c o n f i g u r a t i o n s connected t o t h e p r e s e n t t e c h n o l o g y o f s u p e r c o n d u c t i n g o i l s have been s t u d i e d e x p e r i m e n t a l l y . We a n a l y s e t h e p r i n c i p a l r e s u l t s o b t a i n e d .I
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INTRODUCTIONWhen a superconducting magnet i s quoted t o be " s t a b l e " , t h i s a l m o s t c e r t a i n l y means t h a t no-matter-what mechanical, thermal o r e l e c t r i c a l p e r t u r b a t i o n induces a t r a n s i t i o n f r o m t h e superconducting s t a t e t o t h e normal s t a t e , t h e s u r r o u n d i n g medium i s capable o f e x t r a c t i n g t h i s sudden q u a n t i t y o f energy t o g e t h e r w i t h t h e subsequent e n e r g y generated b y J o u l e e f f e c t , t h e magnet then b e i n g b r o u g h t back t o i t s o r i g i n a l s t a t e . However, t h e d i v e r s i t y o f cryomagnetic systems makes i t d i f f i c u l t t o develop a g e n e r a l model o f t h e s t a b i l i t y o f t h e o p e r a t i n g mode.
A f t e r a s h o r t o u t l i n e o f t h e c l a s s i c a l s t a b i l i t y c r i t e r i a , some s p e c i a l sys- tems f o r which these c r i t e r i a a r e n o t a p p l i c a b l e a r e presented. A few i m p o r t a n t p o i n t s a r e examined, some o f which a r e g e n e r a l , such as t h e MPZ concept and c r i t i c a l energy, and some o f which a r e more s p e c i f i c t o c e r t a i n c o i 1 and c o n d u c t o r c o n f i g u r a t i o n s c u r r e n t l y employed e.g. " t r a n s i e n t " s t a b i l i t y and s t a b l e normal zones.
I n t h e l i g h t o f a l l t h e s o l v e d problems, f u r t h e r work which can be madeissuggested.
I 1
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OPERATIONAL CONDITIONS SOUGHT. LIMITS OF APPLICATION OF GENERALLY ACCEPTED STABILITY CRITERIA1
.
S t a b i l i t y o f an e q u i l i b r i u m p o s i t i o n o f a system...
I t i s u s e f u l t o c o n s i d e r t h e problem o f t h e thermal s t a b i l i t y o f a super- c o n d u c t o r i n t h e general framework o f t h e s t a b i l i t y o f a system so as t o be a b l e t o d e f i n e an a p p r o p r i a t e language ( 1 , 2 ) . The s t a t e o f a superconductor i s c h a r a c t e - r i z e d b y i t s temperature d i s t r i b u t i o n T. A s t a t e o f e q u i l i b r i u m To i s a s t a t i o n a r y s o l u t i o n o f t h e h e a t e q u a t i o n w i t h boundary c o n d i t i o n s . A r e a l system i s exposed t o p e r t u r b a t i o n s , and t h e q u e s t i o n o f t h e s t a b i l i t y o f t h e e q u i l i b r i u m s t a t e t h u s a r i s e s : i f i t i s assumed t h a t t h e system i s i n an i n i t i a l s t a t e Ti
,
s u f f i c i e n t l y c l o s e t o To, o r t h a t i t i s s u b j e c t e d t o t h e a c t i o n o f a small p e r t u r b a t i o n , t h e problem i s t o know whether t h e system w i l l r e t u r n t oTo.
If t h i s i s t h e case, t h e To s t a t e i s ( a s y m p t o t i c a l l y ) s t a b l e o r a t t r a c t i v e . An e v i d e n t a t t r a c t o r i s t h e u n i f o r m temperature d i s t r i b u t i o n T=
T,
t h e b a t h temperature.2
. ~ttractlye_reglon-of-~-?ta!!1e-?tkte
I t i s p o s s i b l e t o have a s t a b l e system which does n o t behave c o r r e c t l y when p e r t u r b a t i o n s exceed a c e r t a i n magnitude. The system may t h e r e f o r e be u n s t a b l e i n p r a c t i c e . The q u e s t i o n o f t h e s i z e o f t h e r e g i o n o f a t t r a c t i o n o r t h e r e g i o n o f s t a b i l i t y o f t h e a t t r a c t o r t h u s a r i s e s . I f t h e superconducting s t a t e i s t h e o n l y a t t r a c t o r , g l o b a l s t a b i l i t y o c c u r s ; f o r a l l p e r t u r b a t i o n s , t h e c o n d u c t o r w i l l r e t u r n spontaneously t o t h e s u p e r c o n d u c t i n g s t a t e . I f t h e r e i s a n o t h e r a t t r a c t o r however, s t a b i l i t y i s l i m i t e d ; i t i s t h u s u s e f u l
to
know what p e r t u r b a t i o n s c o n s t i t u t e t h e boundary o f t h e a t t r a c t i v e r e g i o n o f t b e s u p e r c o n d u c t i n g s t a t e ;.such p e r t u r b a t i o n s a r e d e s i g n a t e d " c r i t i c a l p e r t u r b a t i o n s.
When t h e p e r t u r b a t i o n 1 ~ e s i n s l d e t h e Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841103C1-504 JOURNAL DE PHYSIQUE
a t t r a c t i v e r e g i o n o f t h e s u p e r c o n d u c t i n g s t a t e , t h e c o n d u c t o r i s s t a b l e ; t h e normal zone i s absorbed. Otherwise, t h e temperature d i s t r i b u t i o n tends towards t h e o t h e r a t t r a c t o r ; t h e c o n d u c t o r i s u n s t a b l e ; t h e normal zone propagates u n t i l t h e second s t a b l e p r o f i l e i s e s t a b l i s h e d .
I n o r d e r t o e x a c t l y i d e n t i f y t h e s t a b i l i t y r e g i o n and t o p r e d i c t t h e t r a n s i e n t response o f t h e conductor, i t i s necessary t o use n u m e r i c a l models.However, appro- x i m a t e a n a l y t i c a l s o l u t i o n s a r e u s e f u l t o a c q u i r e an u n s d e r s t a n d i n g o f t h e
c o n d u c t o r ' b e h a v i o r , t o a p p r e c i a t e t h e e f f e c t o f v a r i o u s p h y s i c a l parameters and geometries on s t a b i l i t y and t o compare d i f f e r e n t conductors.
3
.
Staymt~-~rctfTI"
Several q u a n t i t a t i v e e x p r e s s i o n s f o r t h e s t a b i l i t y o f a superconductor c o o l e d b y a h e l i u m b a t h have been f o r m u l a t e d f r o m t h e b a s i c i d e a s developed b y Z.J.J. S t e k l y and J.L. Zar ( 3 ) and f r o m t h e equal areas theorem o f B.J. Maddock and a l . ( 4 ) . The l a t t e r a u t h o r s p r e s e n t a model e n a b l i n g a s t a b i l i t y c r i t e r i o n , which t a k e s c o n d u c t i o n a l o n g t h e c o n d u c t o r I n t o account, t o be d e f i n e d g r a p h i c a l l y . It i s a p p l i c a b l e when t h e power t r a n s f e r r e d t o t h e o u t s i d e and d i s s i p a t e d i n t h e c o n d u c t o r depends e x p l i c i t l y o n l y on t h e temperature o f t h e c o n d u c t o r . Furthermore, t h e c o n d u c t o r i s t r e a t e d as a one-dimensional i n f i n i t e l y l o n g w i r e .
The c r i t e r i o n d e f i n e s t h e c o l d end r e c o v e r y c u r r e n t Ir and a s s u r e s g l o b a l c o n d u c t o r s t a b i l i t y f o r a l l c u r r e n t s l e s s t h e n Ir. F o r c u r r e n t s i n excess o f Ir, t h e c o n d u c t o r can, i n f a c t , r e t u r n t o t h e s u p e r c o n d u c t i n g s t a t e p r o v i d e d t h a t t h e p e r t u r b a t i o n has a f i n i t e a m p l i t u d e . Conductor s t a b i l i t y i s l i m i t e d . M. N. Wilson and Y . Iwasa extended t h e equal areas theorem t o t h e normal f i n i t e zones case ( 5 ) . F o r a l l c u r r e n t s i n excess o f Ir, a s t a t i o n a r y temperature p r o f i l e e x i s t s f o r which t h e J o u l e power i s i n e q u i l i b r i u m w i t h t h e power e x t r a c t e d b y c o n d u c t i o n and b y t h e e x t e r n a l environment. T h i s s t a t i o n a r y s o l u t i o n i s d e s i g n a t e d "minimum p r o p o g a t i n g zone" (MPZ) ( 6 ) . As w i l l be seen l a t e r , t h e u n s t a b l e s t a t i o n a r y s t a t e MPZ g i v e s i n f o r m a t i o n about t h e a t t r a c t i v e r e g i o n of t h e s t a b l e s t a t i o n a r y s u p e r c o n d u c t i n g s t a t e .
The h y p o t h e s i s o f a c o n d u c t o r , which i s s e m i - i n f i n i t e i n t h e d i r e c t i p o f i t s a x i s , l e a d s t o two c o n d i t i o n s b e i n g imposed a t t h e i n f i n i t y l i m i t : T (x-,
-
) = T and aT/ a x = 0. These two c o n d i t i o n s e l i m i n a t e t h e s t a t i o n a r y s o l u t i o n f o r which a l m o s t t h e e n t i r e c o n d u c t o r 1 s i n t h e normal s t a t e . I t i s , i n f a c t , i n t u i t i v e l y c l e a r t h a t i f T 5 T i s s t a b l e and MPZ u n s t a b l e , t h e n a n o t h e r s t a b l e s o l u t i o n e x i s t s , o t h e r w i s e t k e c o n d u c t o r whould be g l o b a l l y s t a b l e i n a l l cases : a l l s o l u - t i o n s c o u l d o n l y converge t o t h e s t a b l e s u p e r c o n d u c t i n g s t a t e . T h i s o t h q r s t a b l e s o l u t i o n can be c a l c u l a t e d b y imposing a f i n i t e boundary and t h e T ( x =-
L ) = Tb c o n d i t i o n .I f t h e c o n d u c t o r e x h i b i t s t i m e dependent h e a t exchange c h a r a c t e r i s t i c s w i t h t h e o u t s i d e environment, t h e p r e v i o u s l y d e s c r i b e d c r i t e r i a a r e n o t a p p l i c a b l e . F o r example, f o r i n t e r n a l l y c o o l e d cables, t h e s t a b i l ~ t v concept i s d i f f e r e n t t o t h a t a p p l i c a b l e t o c o i l s c o o l e d b y h e l i u m channels a c t i n g as a c o l d source.
A s t a t i o n a r y s t a t e does n o t , i n g e n e r a l , e x i s t f o r h e a t exchange. The k i n e t i c s o f t h e h e a t exchange between t h e c o n d u c t o r and c o o l a n t and t h e e n t h a l p y a v a i l a b l e i n t h e l i m i t e d volume o f h e l i u m a d j a c e n t t o t h e c o n d u c t o r a r e o f p r i m e importance f o r s t a b i 1 i t y ( s e e paragraph V ) .
Another c o n f i g u r a t i o n f o r which t h e h y p o t h e s i s o f t h e equal areas c r i t e r i o n a r e n o t v e r i f i e d i s t h e one i n which heat t r a n s f e r c h a r a c t e r i s t i c s a r e e x p l i c i t l y dependent on t h e x c o o r d i n a t e . T h i s case i s encountered, i n p a r t i c u l a r , w i t h magnets c o o l e d b y channels i n s i d e t h e w i n d i n g . The i n s u l a t i n g spacers d e f i n i n g t h e channel geometry r e s u l t i n a heterogeneous c o o l a n t d i s t r i b u t i o n a l o n g t h e c o n d u c t o r . Experimental and t h e o r e t i c a l s t u d i e s i n d i c a t e t h e e x i s t e n c e o f a r a t h e r unusual b e h a v i o r o f t h e superconductor (see paragraph I V ) .
F i n a l l y , a t t e n t i o n must be drawn t o t h e case o f compact magnets, c o o l e d by c o n d u c t i o n f r o m t u r n t o t u r n , which i s n o t s t u d i e d i n d e t a i l here. The minimum energy o f t h e l o c a l p e r t u r b a t i o n necessary t o quench an impregnated s u p e r c o n d u c t i n g c o i l i s t h e o r e t i c a l l y determined f r o m t h e t h r e e - d i m e n s i o n a l minimum p r o p a g a t i n g zone concept ( 7 ) . A h e a t c o n d u c t i o n model i s developed by c o n s i d e r i n g t h e c o i l t o be a c o n t i n u o u s a n i s o t r o p i c medium w i t h l o n g i t u d i n a l and t r a n s v e r s e thermal conduc-
s i z e i s g r e a t e r t h a n t h a t o f t h e c o n d u c t o r .
111
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ATTRACTIVE REGION OF THE SUPERCONDUCTING STATE. ENERGY OF CRITICAL PERTURBATIONSP e r t u r b a t i o n s can be c h a r a c t e r i z e d i n d i f f e r e n t ways. They may o r i g i n a t e e i t h e r o u t s i d e o r i n s i d e t h e superconductor. Furthermore, t h e y may o r may n o t be r e p e t i t i v e . T h e i r importance can, however, be deduced f r o m t h e h e a t which t h e y d e v e l o p i n a c o n d u c t o r . P e r t u r b a t i o n s can t h u s be c h a r a c t e r i z e d b y t h r e e parameters s p a t i a l d i s t r i b u t i o n , d u r a t i o n and energy o r power amp1 i t u d e s .
The g e n e r a l l y accepted h y p o t h e s i s i s t h a t an i n s t a n t a n e o u s p e r t u r b a t i o n , d e s c r i b e d b y a l o c a l i z e d e x t e r n a l i n p u t o f h e a t E,, l e a d s t o a p r o p a g a t i o n o f t h e normal zone i f E, exceeds t h e e n t h a l p y o f t h e MPZ d e f i n e d b y :
j x + - m / T b
where A i s t h e c r o s s s e c t i o n o f t h e c o n d u c t o r and C i t s volumic h e a t . I f t h e energy, E,, i s l e s s than EMpZ, c o n d u c t o r r e c o v e r y i s assured. EMpZ i s t h e r e f o r e t h e c r i t i c a l energy o f t h e c o n d u c t o r .
In
p r a c t i c e , t h e temperature d i s t r i b u t i o n o f t h e c o n d u c t o r tends t o e v o l v e i n a p r e d i c t a b l e way o n l y when i t i s s u b j e c t e d t o an i n i t i a l temperature d i s t r i b u t i o n capable o f b e i n g compared t o t h e MPZ p r o f i l e ( 8 ) . A c o n d u c t o r exposed t o a p e r t u r b a t i o n r e s u l t i n g i n a temperature p r o f i l e l e s s t h a n t h e MPZ r e t u r n s t o t h e s u p e r c o n d u c t i n g s t a t e . I f t h e i n i t i a l t e m p e r a t u r e p r o f i l e i s g r e a t e r t h a n t h e MPZ, t h e normal zone converges t o a n o t h e r s t a b l e non supercon- d u c t i n g s t a t e .A n u m e r i c a l s i m u l a t i o n o f t h e temperature d s i t r i b u t i o n e v o l u t i o n a l o n g a composite superconductor, based on t h e h e a t e q u a t i o n , e n a b l e s t h e e f f e c t s of s p a t i a l and temporal d i s t r i b u t i o n s on c r i t i c a l energy t o be i n v e s t i g a t e d . F i g u r e 1 r e p r e - s e n t s t h e c r i t i c a l energy f o r a g i v e n c o n d u c t o r ( 8 ) as a f u n c t i o n o f t h e h a l f - l e n g t h upon which energy i s d e p o s i t e d i n s t a n t a n e o u s l y and homogeneously. F o r l a r g e l e n g t h s ( g r e a t e r t h a n 10 cm i n o u r case) t h e c h a r a c t e r i s t i c parameter i s no l o n g e r t o t a l energy, b u t energy p e r u n i t l e n g t h o f conductor. The s e m i - l e n g t h s c o n s i d e r e d s h o u l d be compared w i t h t h e c h a r a c t e r i s t i c thermal l e n g t h X = 2 cm and t h e MPZ s e m i - l e n g t h : xMpZ = 3.5 cm. F i g u r e 2 i l l u s t r a t e s t h e b e h a v i o r o f a c o n d u c t o r
exposed t o h e a t p u l s e s o f t h e same energy b u t o f d i f -
12 . T,=L 2K 1 = 5 0 0 ~ 0.51 f e r e n t d u r a t i o n s as w e l l as t h e v a r i a t i o n s i n t h e c r i t i c a l energy w i t h
11 - -
d u r a t i o n . For l o n g t i m e i n t e r v a l s ( i n o u r case
. t > 30 ms), t h e charac- t e r i s t i c parameter i s power. The e n e r g y o f t h e c r i t i c a l p e r t u r b a t i o n i s p r a c t i c a l l y c o n s t a n t f o r p u l s e d u r a t i o n t i m e s i n - t h e 0 t o 1 ms i n t e r v a l .
-
& -
#- U
7 .
6
5
j 1 k b
lo c a t e d b y a more generalHalf-lenght of conductor subjected t o the d~sturbance X, lcml a n a l y s i s o f t h e dependence o f c r i t i c a l energy on F i g . 1 - C r i t i c a l e n e r g y v e r s u s t h e l e n g t h c u r r e n t , c o o l i n g c o n d i t i o n s
o f t h e temperature d i s t u r b a n c e and t h e n a t u r e o f t h e p e r t u r b a t i o n s ( 9 ) . .
-
These d u r a t i o n s should be compared w i t h t h e thermal r e l a x a t i o n t i m e o f t h e c o n d u c t o r
2 / Dth = CA/hp= 2.5 ms- S i m i l a r t r e n d s a r e i n d i -
C1-506 JOURNAL DE PHYSIQUE
I V
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NORMAL STABLEEc PC ' STATIONARY ZONES
.
m
-
Tb:L 2K 1=500A B=ST STABILITY DEGRADATION3 E X,=O Scm
-,
30 . 15 Normal s t a b l e s t a t i o --
n a r y zones have been obser-a= ved b y v a r i o u s experimen-
L
0 .
3 t e r s . The reasons p u t
0 a
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f o r w a r d f o r t h i s b e h a v i o rrT a r e changes i n c o o l i n g
:
20 - '3L c o n d i t i o n s a l o n g t h e conduc-
t o r : non c o o l e d r e g i o n s underneath i n s u l a t i n g spacers d e f i n i n g t h e h e l i u m channels ( l o ) , l e n g h t s n o t immersed i n t h e l i q u i d ( l l ) , v o i d s
I between t h e m a t r i x and s o l d e r e d s t a b i 1 iz e r (12,13),
L v a r i a t i o n s i n t h e depths
" 0 - 1
.
o f t h e channels ( 1 4 ) .o 1 10 20 30 LO I t i s p r o b a b l e t h a t
Heat pulse durat~on Oirnsl parameters o t h e r t h a n
F i g . 2
-
C r i t i c a l energy versus t h e d u r a t i o n exchange can i n d u c e such o f t h e h e a t d i s t u r b a n c e phenomena, f o r example av a r i a t i o n i n maametic i n d u c t i o n a l o n g - t h e l e n g t h o f t h e c o n d u c t o r ( 1 5 ) o r a change i n r e s i s t i v i t y .
The s t a b i l i t y o f such normal s t a t i o n a r y zones has been shown f o r t h e case o f a c o o l e d r e g i o n ( 1 6 ) and f o r t h e case o f a h e t e r o g e n e i t y , which can be t r e a t e d as a s t e a d y p o i n t h e a t source (17,18, 19, 6 ) , s i m u l a t i n g , f o r example a b r e a k i n super- c o n d u c t i n g f i l a m e n t s o r t h e j u n c t i o n between two composites. The consequences o f t h e e x i s t e n c e o f such s t a b l e zones a r e , f i r s t l y , a r e d u c t i o n i n t h e c o l d end r e c o v e r y c u r r e n t and, secondly, a r e d u c t i o n i n t h e MPZ and t h e r e f o r e i n t h e energy o f t h e c r i t i c a l p e r t u r b a t i o n s .
An example o f such r e d u c t i o n s i s i l l u s t r a t e d i n F i g u r e 3 o f r e f e r e n c e ( 1 6 ) which shows how t h e c o l d end r e c o v e r y c u r r e n t v a r i e s as a f u n c t i o n o f t h e l e n g t h o f a non cooled zone. Another example o f s t a b i l i t y d e g r a d a t i o n i s i l l u s t r a t e d i n f i g u r e 3 o f r e f e r e n c e ( 1 8 ) which shows t h e v a r i a t i o n s o f t h e c r i t i c a l energy w i t h t h e power o f a s t e a d y p o i n t h e a t source.
V
-
TRANSIENT STABILITYDesign t r e n d s o f l a r g e h i g h c u r r e n t d e n s i t y magnets towards c o n f i n e d geometries u s i n g f o r c e d cooled c a b l e s , c o n d u c t o r s i n t e r n a l l y c o o l e d by s a t u r a t e d h e l i u m a t 4. 2 K o r c o n d u c t o r s c o o l e d b y a l i m i t e d amount o f s u p e r f l u i d helium, lead, as a l r e a d y mentioned t o a new concept o f s t a b i l i t y . I n these c o n f i g u r a t i o n s , t h e h e l i u m f l o w s a l o n g t h e c o n d u c t o r and does n o t e a s i l y communicate w i t h a l a r g e
r e s e r v o i r . Only p a r t o f t h e h e l i u m a d j a c e n t t o t h e c o n d u c t o r p a r t i c i p a t e s i n c o o l i n g . I t i s t h e r e f o r e necessary t o f i r s t analyze t h e c o o l i n g medium i n o r d e r t o e s t a b l i s h a r e l a t i o n between t r a n s i e n t t r a n s f e r , u s a b l e e n t h a l p y and t h e temporal d i s t r i b u t i o n o f t h e thermal p e r t u r b a t i o n . The dynamics o f t h e s u p e r c o n d u c t i n g system can t h e n be s t u d i e d , account b e i n g t a k e n o f t h e e n t h a l p y r e s e r v o i r l i m i t e d i n such a way as t o c h a r a c t e r i z e t h e s t a b i l i t y o f t h e c o n d u c t o r by i t s c r i t i c a l p e r t u r b a t i o n s .
1
.
Forced-cooled-cable?The s t a b i l i t y q f c a b l e - i n - c o n d u i t t y p e c o n d u c t o r s c o o l e d by s u p e r c r i t i c a l h e l i u m i s s t u d i e d from b o t h e x p e r i m e n t a l and t h e o r e t i c a l p o i n t s o f v i e w i n r e f e r e n c e ( 2 0 ) . The s t a b i l i t y margin, .i .e. t h e h i g h e s t energy p e r u n i t volume o f m e t a l f r o m which t h e c o n d u c t o r can r e c o v e r , has, under c e r t a i n c o n d i t i o n s , s e v e r a l values.
F i g u r e 3 i s a . t y p i c a l r e p r e s e n t a t i o n o f t h e s t a b i l i t y m a r g i n ,as a f u n c t i o n o f t r a n s p o r t c u r r e n t . I t i s suggested t h a t r e c o v e r y i n a l o n g t u b e m i g h t be enhanced by t r a n s i e n t h e l i u m f l o w due t o thermal expansion induced b y t h e i n i t i a l h e a t i n g of t h e h e l i u m . T r a n s i e n t f l o w c o n s i d e r a b l y i n c r e a s e s h e a t t r a n s f e r i n t h e subsequent
A 5 0 0 power induces a h i g h e r h e l i u m f l o w and i n c r e a s e s h e a t t r a n s f e r . The c o n d u c t o r can t h u s more r a p i d l y r e c o v e r even i f t h e i n i t i a l h e a t p u l s e i s l a r g e r . A q u a l i t a t i v e model i s developed f r o m these c o n s i d e r a t i o n s ; t h i s model c o r r e c - t l y e x p l a i n s t h e m u l t i v a l u e d beha- 200 v i o r o f t h e s t a b i l i t y m a r g i n .
I n p r a c t i c e , t h e u s e f u l opera-
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t i n g c u r r e n t s a r e l e s s t h a n t h e
l i m i t i n g c u r r e n t , 1 lim, below OE
.
uwhich o n l y t h e upper boundary
-,
between r e c o v e r y and non r e c o v e r y
-
E (00 e x i s t s . T h i s upper s t a b i l i t y Ia
m a r g i n i s 1 im i t e d because t h e quan- t i t y o f h e l i u m a v a i l a b l e i n a .- C g i v e n c r o s s s e c t i o n o f c o n d u c t o r i s l i m i t e d . The he1 ium and t h e conduc- E t o r must t e n d a t r e c o v e r y towards 50 a temperature l o w e r t h a n t h e .
- -
c u r r e n t s h a r i n g t h r e s h o l d tempera- 5
t u r e T c S The h e a t a b s o r p t i o n capa- N b T i SINGLE TRIPLEX c i t y o f he1 i um between t h e b a t h t e n
+,,,
= 4.0rnm. pob, = 5.0orrnp e r a t u r e and Tcsdetermines t h e t o t a l
T , , = 46.7 i n s , v,, = 0 q u a n t i t y o f h e a t (sum o f t h e i n i t i a l
h e a t p u l s e ~ H and o f J o u l e h e a t A H j = 3.8 rn 8 = 6.OT qenerated d u r i n q r e c o v e r y ) t h a t 7
-
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can be absorbed-during r e c o v e r y . 340 3 6 0 3 8 0 4 0 0 4 2 0 4 4 0
It should be n o t e d , t h a t i n o r d e r
t o b e n e f i t f r o m t h i s upper l i m i t , C u r r e n t IS ( A i t i s necessary t o ensure a s u f f i -
ciently high heat transfer rate so F i g . 3
-
S t a b i l i t y b e h a v i o u r o f a c a b l e i n t h a t a l l t h e h e l i u m i n t h e v i c i n i - a c o n d u i t , a f t e r ( 2 1 )-~ ~
t y o f t h e heated s e c t i o n p a r t i c i p a t e s
i n t h e r e c o v e r y o f t h e superconducting s t a t e .
2
. Irenstent-stab~llt~-0f~ca!1e1-~001eIr~!~~~~~!~aL~Ir-~sl~!~-~!-~~-~-5
The t r a n s i e n t s t a b i l i t y o f superconducting c a b l e s c o o l e d by s t a t i c b o i l i n g h e l i u m i s d e s c r i b e d i n r e f e r e n c e ( 2 2 ) . The aim o f t h e work i s t o improve h e a t t r a n s - f e r between t h e c o n d u c t o r and t h e l i q u i d , i n p a r t i c u l a r , by i n c r e a s i n g t h e s u r f a c e c o o l e d p e r u n i t volume o f m e t a l , so as t o t a k e advantage of t h e h i g h energy absorp- t i o n c a p a c i t y o f l i q u i d h e l i u m a t 4. 2 K. S t a b i l i t y w i t h r e s p e c t t o p e r t u r b a t i o n s o f s h o r t d u r a t i o n a f f e c t i n g a l a r g e volume o f c o n d u c t o r i s s t u d i e d .
Conductor r e c o v e r y must t a k e p l a c e b e f o r e t h e t r a n s i t i o n o f h e l i u m i n t o f i l m b o i l i n g . T h i s t r a n s i t i o n o c c u r s when s u f f i c i e n t energy Ef i s absorbed t o v a p o r i z e t h e h e l i u m i n t h e l i q u i d d i f f u s i o n l a y e r a d j a c e n t t o t h e s u r f a c e o f t h e c o n d u c t o r . I t i s shown t h a t t h e i n s t a n t when t h e t r a n s i t i o n o c c u r s i s i n v e r s e l y p r o p o r t i o n a l t o t h e square o f t h e h e a t f l u x t r a v e r s i n g t h e i n t e r f a c e ( 2 3 ) . Conductor r e c o v e r y t h u s depends on two parameters : t h e t o t a l q u a n t i t y o f h e a t E t h a t must be t r a n s - f e r r e d t o t h e l i q u i d ( e n e r g y o f t h e i n i t i a l p e r t u r b a t i o n and J o u l e energy) and t h e t o t a l t r a n s f e r t i m e ( d u r a t i o n o f p e r t u r b a t i o n + r e c o v e r y t i m e ) . The t r a n s i e n t h e a t e q u a t i o n i s s o l v e d by a numerical method. I f t h e E ( t ) > E ( t ) c o n d i t i o n o c c u r s p r i o r t o r e c o v e r y o f t h e superconducting s t a t e , i t i s assuXed t h a t t h e c o n d u c t o r cannot r e c o v e r . I t should be n o t e d t h a t t h e Ef ( t ) f u n c t i o n i s o n l y known f o r t h e s t e p power changes, whereas i t i s used, here, f o r h e a t t r a n s f e r w i t h two power l e v e l s which a r e , a p r i o r i , d i f f e r e n t : t h e p e r t u r b a t i o n and t h e J o u l e e f f e c t .
S t a b i l i t y measurements were c a r r i e d o u t ; t h e r e s u l t s a r e i n good agreement w i t h t h e t h e o r e t i c a l model ( f i g u r e 4 ) .
JOURNAL DE PHYSIQUE
3
. ---
S t a b i l i t y of superconductors cooled by a limited g u a n t i t y of--- ... --- --- ---
s u p e r f l u i d helium
-- ---
The s t a b i l i t y of a superconductor cooled by a l i m i t e d volume of s u p e r f l u i d helium a t a pressure of 1 atmosphere i s studied in reference ( 2 4 ) . The p e r t u r b a t i o n s considered a f f e c t a l a r g e length of conductor.
As long a s t h e temperature of the f l u i d a t t h e conductor i n t e r f a c e i s l e s s than T A , heat t r a n s f e r i s governed by t h e conductance of t h e interface.The heat f l u x d i f f u s e s i n t h e f l u i d , t h e helium temperature increasing u n t i l Th i s reached near t h e heated s u r f a c e . The q u a n t i t y of energy absorbed by t h e s u p e r f l u i d helium up t o Th was determined both experimentally and t h e o r e t i c a l l y a s a function of the heat f l u x a p p l i e d . A f t e r the occurence Of
TA,
and a s a r e s u l t of the formation of a normal helium l a y e r heat t r a n s f e r diminishes considerably. In g e n e r a l , a thermal runaway occurs f o r t h e heated s u r f a c e ( i f the f l u i d remains a t atmospheric pressure).The f l u x e x t r a c t e d by the helium I1 i s c a l c u l a t e d using a mathematical model t o solve t h e t r a n s i e n t heat t r a n s f e r equations f o r the s u p e r f l u i d h e l i u
.
In the t r e a t e d c a s e , the p e r t u r b a t i o n i s assumed t o be
07
l a r g e amplitude and s h o r t d u r a t i o n , i . e . t o d e p o s i t a l a r g e q u a n t i t y of energy i n a time which i s s h o r t i n comparison with the c h a r a c t e r i s t i c heat t r a n s f e r process time. A mathema- t i c a l r e p r e s e n t a t i o n of t h i s hypothesis i s given in ( 2 4 ) . The temperature of the helium a t t h e i n t e r f a c e s e p a r a t i n g the l i q u i d from the conductor exposed t o such a p e r t u r b a t i o n reaches T A almost instantaneously. Recovery occurs when the time i n t e g r a t e d cooling f l u x becomes equal t o t h e i n t e g r a t e d heat f l u x generated by t h e Joule e f f e c t . The l i m i t i n g c a s e occurs when t h e cooling f l u x equals t h e Joule f l u x a t t h e time of recovery. The f r a c t i o n of t h e t o t a l energy a v a i l a b l e in t h e helium 11, which i s a c t u a l l y used by t h e Joule e f f e c t and the p e r t u r b a t i o n diminishes with i n c r e a s i n g t r a n s p o r t c u r r e n t , a s a r e s u l t of a reduction in t h e volume of helium p a r t i c i p a t i n g in the exchange process.The agreement obtained between theory and experiment i n r e f e r e n c e (25) i s shown i n f i g u r e 5
.
I t appears t o confirm t h e v a l i d i t y of t h e exchange model used.I t should be noted t h a t t h i s s t a b i l i t y c r i t e r i o n a p p l i e s to, very severe p e r t u r b a t i o n s . I f the conductor i s subjected t o p e r t u r b a t i o n s of t h e same energy, but of longer d u r a t i o n , heat exchange may remain in t h e Kapitza regime and be very e f f i c i e n t . A higher s t a t i l i t y can thus be obtained, e s p e c i a l l y with low c u r r e n t s f o r which t h e t o t a l energy a v a i l a b l e i n t h e helium I1 can be completely used.
I I I I I 8 I 4 I I I I I I I
5.87 -
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x MeaSw33 T*=120psec -Corrprted LC * 1450A1 0 ~ ~ ~ ~ ~ ~ ~ ~ ' ~ " ~ ' ~ ~ '
500 1000 I500 0 500 1000 1500
CONOUCTCR CURRENT AMPS Transport current I, (A1
Fig. 4
-
Transient s t a b i l i t y of NbTi Fig. 5-
S t a b i l i t y of a superconductor conductor cooled by helium a t cooled by a l i m i t e d volume4. 2 K, a f t e r ( 2 2 ) of s u p e r f l u j d h e l i u m , a f t e r ( 2 5 )
The t r a n s i e n t s t a b i l i t y o f t h e t h r e e c o n f i g u r a t i o n s p r e v i o u s l y s t u d i e d depends on two parameters : h e a t t r a n s f e r a t t h e c o n d u c t o r
-
l i q u i d i n t e r f a c e and t h e h e a t absorpt,on c a p a c i t y o f f l u i d a d j a c e n t t o t h e c o n d u c t o r .When h e a t i n g begins, h e a t t r a n s f e r i s o n l y l i m i t e d by t h e r e s i s t a n c e o f t h e i n t e r f a c e . The temperature o f t h e s u r f a c e remains l o w as t h e h e a t t r a n s f e r c o e f f i - c i e n t i s l a r g e . A f t e r a s h o r t t i m e ( t a k e o f f t i m e ) , i f t h e f l u i d i s b o i l i n g t h e s u r f a c e goes o v e r t o f i l m b o i l i n g and t h e s u r f a c e temperature i n c r e a s e s . When t h e h e l i u m i s s u p e r c r i t i c a l , a l o w d e n s i t y h e l i u m f i l m o c c u r s i n s t e a d o f t h i s vapor f i l m and when t h e h e l i u m i s i n t h e s u p e r f l u i d s t a t e , a l a y e r o f normal h e l i u m i s formed when t h e f l u i d temperature reaches TA
.
The t i m e f o r t h e t e m p e r a t u r e t o t a k e o f f depends on t h e h e a t f l u x a t t h e i n t e r f a c e . . T h e t o t a l energy t h a t must beabsorbed b y t h e l i q u i d i s t h e sum o f t h e p e r t u r b a t i o n energy and t h e J o u l e energy generated d u r i n g r e c o v e r y . I f a l l t h i s energy can be t r a n s f e r r e d b e f o r e t h e t a k e o f f t i m e , t h e s t a b i l i t y w i l l depend on t h e t o t a l h e a t a b s o r p t i o n c a p a c i t y o f t h e f l u i d volume a d j a c e n t t o t h e heated conductor, as t h e h e l i u m cannot be r e p l a c e d w i t h i n t h e t i m e necessary f o r r e c o v e r y . T h i s s i t u a t i o n i s r e a l i z e d i n p r a c t i c e f o r l o w c u r r e n t s and energy p e r t u r b a t i o n s , which may be l a r g e , b u t o f l o n g d u r a t i o n . I n o r d e r t o i n c r e a s e t h e c r i t i c a l energy, i t i s n e c e s s a r y t o improve t h e h e a t t r a n s f e r a t t h e i n t e r f a c e ( b y i n c r e a s i n g , f o r example, t h e p/A r a t i o ) so as t o d i m i n i s h r e c o v e r y t i m e . F o r h i g h c u r r e n t s o r e n e r g i e s o f h i g h power, t a k e o f f t i m e g e n e r a l l y o c c u r s b e f o r e r e c o v e r y t i m e . S t a b i l i t y i s t h u s reduced because o n l y a s m a l l percen- tage o f t h e l i q u i d i s i n v o l v e d i n s t a b i l i z a t i o n . The f r a c t i o n o f t h e l i q u i d p a r t i - c i p a t i n g i n t h e process can be determined f r o m h e a t t r a n s p o r t i n t h e l i q u i d . I n o r d e r t o i n c r e a s e t h e c r i t i c a l energy, i t i s t h u s necessary t o improve t h e h e a t t r a n s p o r t i n t h e f l u i d ( u s i n g f o r example s u p e r f l u i d helium].
V I
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STUDIES REQUIRING FURTHER WORKA l t h o u a h t h e work c a r r i e d o u t UD t o t o d a y c o v e r s a v e r y l a r a e f i e l d o f
appl i c a t i o n s - a c t u a l 1 y encountered i n supercond;ctor technology, ii i s n o t e x h a u s t i v e . I t i s w o r t h w h i l e s u g g e s t i n g f u t u r e areas i n which work i s necessary, w h i l e a t t h e same t i m e emphasizing t h e d i f f i c u l t i e s i n v o l v e d .
1
. Work-townr~?-e-generalltheorr-of- ?:?if 1 I t ~
The thermal s t a b i l i t y o f a superconductor i n a g i v e n c r y o g e n i c system must a l m o s t always be t r e a t e d as a s p e c i a l case. I t i s n o t t h e r e f o r e p o s s i b l e , today, t o e n v i s a g e a g e n e r a l t h e o r y f o r t h e s t a t e o f a s u p e r c o n d u c t i n g system s u b j e c t e d t o an u n d e f i n e d p e r t u r b a t i o n . However, i t i s perhaps u s e f u l t o c o n s i d e r g e n e r a l methods f o r t r e a t i n g s t a b i l i t y problems and t o s t u d y concepts employed f o r o t h e r systems, b u t as y e t , n o t employed f o r superconductors. I n t h i s way, t h e o r i e n t a t i o n o f f u t u r e work can be e s t a b l i s h e d t o g e t h e r w i t h t h e t y p e o f s o l u t i o n most appro- p r i a t e t o t h e problems s t u d i e d .
An example, i s r e s e a r c h i n t o upper and l o w e r s o l u t i o n s e n a b l i n g s t a b l e s o l u t i o n s t o be found f o r t h e e q u a t i o n g o v e r n i n g t h e e v o l u t i o n o f systems ; a p p l i c a t i o n s t o e s t i m a t i n g t h e s i z e o f t h e a t t r a c t i v e r e g i o n f o r s t a b l e s o l u t i o n s a r e f o r e s e e n ( 2 6 ) .
The second method o f Liapunov enables t h e s t a b i l i t y o f a system t o be d e t e c t e d i n d i r e c t l y u s i n g a Liapunov f u n c t i o n V ( T ) b e g i n n i n g w i t h t h e d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g t h e e v o l u t i o n o f t h e system, and t h u s non d i r e c t l y f r o m a knowledge o f t h e s o l u t i o n s (1, 2). Such a f u n c t i o n a l s o enables p a r t o f t h e a t t r a c t i v e r e g i o n o f t h e s t a b l e s t a t e c o n s i d e r e d t o be e s t a b l i s h e d , s i n c e t h e r e g i o n bounded by t h e l a r g e s t s u r f a c e V ( T ) = c o n s t a n t l i e s i n s i d e t h e a t t r a c t i v e r e g i o n . The d i f f i c u l t y i s t o f i n d t h e Liapunov f u n c t i o n s . However, a much more complete i d e n t i f i c a t i o n o f t h e a t t r a c t i v e r e g i o n o f t h e superconductor can be e s t a b l i s h e d once t h e y a r e found.
2
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The c o n d u c t o r ( o r c o i 1 ) i s , i n g e n e r a l , c o n s i d e r e d t o be a homogeneous medi um. Homogenization o f t h e thermal c o n d u c t i v i t y , s p e c i f i c h e a t and e l e c t r i c a l r e s i s t i v i t y i s achieved u s i n g mean v a l u e s t a k i n g t h e d e s i g n l a y o u t n f t h e d i f f e r e n t components of t h e c o n d u c t o r i n t o c o n s i d e r a t i o n . Mathematical homogenization t e c h -
C1-510 JOURNAL DE PHYSIQUE
n i q u e s e x i s t s which would p r o b a b l y enable t h e v a l u e s o f t h e p t y s i c a l parameters o f a homogenized c o n d u c t o r t o be r e f i n e d ( 2 7 ) . Furthermore, such t e c h n i q u e s would p e r m i t more general a p p l i c a t i o n s o f c e r t a i n r e s u l t s ( f o r example, t h e r e d u c t i o n i n s t a b i 1 i t y due t o t r a n s v e r s e e l e c t r i c a l and thermal r e s i s t a n c e ) t o c o n d u c t o r s f o r which i t i s d i f f i c u l t t o determine e q u i v a l e n t t r a n s v e r s e thermal c o n d u c t i v i t i e s and e l e c t r i c a l r e s i s t i v i t i e s ( f o r example, c o n d u c t o r s e x h i b i t i n g r e s i s t i v e b a r r i e r s ) .
A t t e n t i o n i s a l s o drawn t o t h e f a c t t h a t c e r t a i n t y p e s o f c o n d u c t o r cannot be homogenized : c o n d u c t o r s welded t o l a r g e s t a b i l i z e r s . I n such cases, i t i s necessary t o t a k e t h e t r a n s v e r s e d i f f u s i o n o f c u r r e n t d u r i n g t h e t r a n s i t i o n i n t o account. A f i r s t approach t o such s t u d i e s has been undertaken b y d e c o u p l i n g t h e thermal and e l e c t r o m a g n e t i c e q u a t i o n s ( 2 8 ) .
As d e s c r i b e d i n p a r t V, t h e a b s o r p t i o n o f a p e r t u r b a t i o n b y h e l i u m s i t u a t e d i n t h e immediate v i c i n i t y o f a heated c o n d u c t o r has been e x t e n s i v e l y s t u d i e d . When t h e s u p e r c o n d u c t i n g r e c o v e r y s t a t e t e r m i n a t e s , t h e temperature o f t h e conduc- t o r and t h e h e l i u m d i f f e r s f r o m t h e i n i t i a l o p e r a t i n g temperature. The a b s o r p t i o n o f a second p e r t u r b a t i o n o f t h e same a m p l i t u d e i s o n l y p o s s i b l e a f t e r t h e e n e r g y d e p o s i t e d by t h e f i r s t p u l s e has been removed. Very l i t t l e work has been c a r r i e d o u t on t h i s second phase, which c o n s i s t s o f t r a n s f e r r i n g energy t o t h e c o l d
sourc% and which as a r e s u l t o f t h e c o i l geometry can l a s t s e v e r a l seconds o r even a few minutes, whereas t h e r e c o v e r y t i m e i s cf t h e o r d e r o f a few m i l l i s e c o n d s .
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