A comparison i s presented of transient heat flow i n. He I 1 as measured experimentally and as predicted by analysis based on the Gorter-Mellink equation.
I
1508
IEEE TRANSACTIONS ON MAGNETICS, VOL. 25, NO. 2, MARCH 1989 TWO DIMENSIONAL TRANSIENT HEAT TRANSFER I N HE I 1 J. M. Pfotenhauer and X. Huang A p p l i e d S u p e r c o n d u c t i v i t y Center U n i v e r s i t y o f Wisconsin 1500 Johnson D r i v e Madison, W I 53706 Abstract
A comparison i s presented o f t r a n s i e n t heat f l o w i n He I 1 as measured e x p e r i m e n t a l l y and as p r e d i c t e d by a n a l y s i s based on t h e G o r t e r - M e l l i n k equation. The geometry i s t h a t envisioned f o r t h e He I 1 c o o l i n g o f a SMES system, v i z , an annular l a y e r o f He I 1 i n d i r e c t c o n t a c t w i t h one l a y e r o f a s o l e n o i d and extending t h e f u l l h e i g h t o f t h e c o i l . A normal zone o v e r a f r a c t i o n o f a t u r n provides f o r two dimensional heat f l o w i n t h e annular l a y e r o f He 11. The comparison i s g i v e n f o r b o t h a d i a b a t i c and isothermal boundary c o n d i t i o n s a t t h e ends o f t h e channel. We f i n d good agreement between t h e a n a l y s i s and t h e experimental data, and thereby v e r i f y t h e usefulness o f t h e a n a l y s i s f o r l a r g e s c a l e systems. I n a d d i t i o n , d i s c r e p a n c i e s between t h e a n a l y s i s and d a t a p r o v i d e i n s i g h t i n t o t h e s t a b i l i t y process o f t h e He I 1 cooled superconductor.
I nt roduc t ion
computations, i n o r d e r t h a t t h e same computational method may be c o n f i d e n t l y used f o r l a r g e s c a l e magnets. I n a d d i t i o n , i t i s hoped t h a t parametric i n v e s t i g a t i o n s u s i n g t h e computations can p r o v i d e general i n f o r m a t i o n about geometry dependent heat f l o w i n He I 1 which can be communicated w i t h o u t t h e use o f computer codes. Features o f Model A d e t a i l e d d e s c r i p t i o n o f t h e computer model has been given elsewherel; however, a few o f t h e i m p o r t a n t f e a t u r e s a r e worth r e p e a t i n g here. F i r s t , heat t r a n s f e r i n He I 1 i s governed by t h e Gorter M e l l i n k r e 1a t i o n
where
Heat t r a n s f e r i n He I 1 has been analyzed and researched e x t e n s i v e l y s i n c e t h e d i s c o v e r y o f s u p e r f l u i d h e l i u m i n 1940. However, whereas t h e aspects o f heat t r a n s f e r i n one dimension have r e c e i v e d abundant a t t e n t i o n from b o t h t h e p h y s i c s and e n g i n e e r i n g communities, aspects o f heat t r a n s f e r i n two o r t h r e e dimensional geometries have been r e l a t i v e l y untouched. The l a r g e e f f e c t i v e thermal c o n d u c t i v i t y and r a p i d thermal d i f f u s i o n o f He I 1 r e s u l t i n t r a n s i e n t heat f l o w which i s q u i t e s e n s i t i v e t o t h e geometrical boundary c o n d i t i o n s , y e t t h e nonl i n e a r n a t u r e of t h e Gorter M e l l i n k heat conduction equation p r o h i b i t s a n a l y t i c a l descriptions o f t h a t heat f l o w f o r a l l b u t t h e one dimensional l i n e a r geometry. On t h e o t h e r hand, t h e use o f He I 1 i n engineering applications - notably, cooling o f superconducting magnets - commonly i n v o l v e s geometries which e a s i l y d i v e r g e from simple one dimensional channels. One s o l u t i o n t o t h i s problem i s t o develop f i n i t e d i f f e r e n c e computational techniques t o address t h e problem. I n t h i s r e p o r t we address t h e problem o f t r a n s i e n t heat f l o w i n an annular l a y e r o f He I 1 which i s used t o cool a superconducting solenoid. The geometry i s t h e same as t h a t envisioned f o r a Superconducting b u t on a s m a l l e r Magnetic Energy Storage (SMES) c o i l In s c a l e - t h a t o f a l a b o r a t o r y s i z e magnet. p a r t i c u l a r , t h e annulus i s t h i n when compared w i t h i t s r a d i u s and extends v e r t i c a l l y over t h e f u l l h e i g h t o f t h e magnet. I n t h i s i n v e s t i g a t i o n heat i s generated i n t h e magnet i n a h o r i z o n t a l s t r i p a t t h e mid h e i g h t o f t h e annulus and along a l e n g t h which i s some f r a c t i o n o f t h e circumference. Thus t h e heat f l o w i s two dimensional, f l o w i n g from t h e h e a t e r towards t h e v e r t i c a l ends o f t h e annulus, and i s symmetric v e r t i c a l l y about t h e inid h e i g h t and h o r i z o n t a l l y about t h e mid p o i n t o f t h e heater.
-
i s t h e heat f l u x ,
i s t h e temperature
g r a d i e n t i n t h e d i r e c t i o n o f a streamline, and f - l ( T ) i s an e f f e c t i v e thermal c o n d u c t i v i t y f u n c t i o n as g i v e n b y Van Sciver.2
F u r t h e r i t i s assumed t h a t
;I i s
.
always p a r a l l e l t o t h e temperature g r a d i e n t $T Beyond t h i s t h e model assumes t h a t t h e temperature a t t h e heated s u r f a c e i s clamped a t T A ( = 2.163 K), t h e r e a r e no temperature g r a d i e n t s across t h e w i d t h o f t h e annulus (i.e. t h i s i s a 2-D heat f l o w model r a t h e r t h a n 3-D), and t h a t heat f l o w i n t h e h o r i z o n t a l d i r e c t i o n a t e x a c t l y one h a l f t u r n away from t h e h e a t e r ’ s c e n t e r i s zero. Boundary c o n d i t i o n s a t t h e v e r t i c a l ends o f t h e annulus can be s e t f o r a d i a b a t i c o r isothermal c o n d i t i o n s . F i n a l l y i t should be mentioned t h a t t h e numerical i n s t a b i l i t y i n h e r e n t t o an i n t e g r a t i o n o f t h e Gorter i d e l l i n k e q u a t i o n (see Dresne?) was avoided by keeping t h e incremental t i m e s t e p small enough t o a v o i d o s c i l l a t i o n s i n t h e temperature c a l c u l a t i o n s g r e a t e r than 5 x
K.
Experimental Parameters Temperature measurements have been gathered d u r i n g
P r e d i c t i o n s o f f i n i t e d i f f e r e n c e computations a r e compared w i t h temperature measurements i n t h e l a o s c a l e experiment. The obvious purpose o f t h i s comparison i s t o v e r i f y t h e accuracy o f t h e
a superconductor s t a b i l i t y e ~ p e r i m e n it n~ which ~ ~ ~ a~ s i n g l e l a y e r s o l e n o i d i s cooled on i t s i n s i d e diameter by an annulus o f subcooled He I 1 a t 1.8 K and 0.1 MPa. The h e l i u m annulus has dimensions o f 31.1 mm mean r a d i u s , 0.7 mm t h i c k n e s s and a h e i g h t o f 100 mm ( a d i a b a t i c c o n d i t i o n ) o r 114 mm ( i s o t h e r m a l c o n d i t i o n ) . C a l i b r a t e d A l l e n Bradley carbon r e s i s t o r s , i n c o n t a c t w i t h t h e i n n e r diameter surface o f t h e annulus, g a t h e r t h e temperature d a t a w i t h a response t i m e o f 1-3 msec and a f f o r d a temperature I n t h e experiment, a p r e c i s i o n o f a few m i l l i k e l v i n . heater located a t the v e r t i c a l midpoint o f t h e s o l e n o i d ( a l s o t h e v e r t i c a l m i d p o i n t o f t h e annulus) d i s s i p a t e s a 10 nsec p u l s e o f energy i n t o t h e superconductor which subsequently generates I L ,irn ~ heat o v e r t h e l e n g t h o f conductor which i s d r i v e n normal.
Manuscript r e c e i v e d August 22, 1988.
The l o c a t i o n o f t h e therinometers and t h e h e a t e r a r e shown i n F i g u r e 1. Here t h e annulus o f He 11 i s
0018-9464/89/0300-1508$01 .WO 1989 IEEE
-
.
I
--
--
1509
5 cm
9
7
5
1
t h e annulus i n c l u d e s l i p s a t t h e v e r t i c a l ends w h i c h The a r e 7.0 mm t a l l b y a p p r o x i m a t e l y 0.7 mm t h i c k . r e m a i n i n g c r a c k s between t h e s e l i p s and t h e s o l e n o i d It i s i m p o r t a n t a r e f i l l e d w i t h RTV s i l i c o n e s e a l a n t . t o n o t e here t h a t t h i s seal i s n o t a p e r f e c t thermal b a r r i e r as t h e RTV c r a c k s when c o o l e d t o h e l i u m temperatures. I n a d d i t i o n t h r e e small holes ( t o t a l
,.
0.06 cm2) p e n e t r a t e t o t h e t o p cross sectional area o f t h e annulus i n o r d e r t o a l l o w h e l i u m t o f i l l t h e annulus. Thus t h e a d i a b a t i c c o n d i t i o n s a r e n o t i d e a l . I n t h e i s o t h e r m a l case t h e l i p s a r e removed l e a v i n g t h e 0.7 mm annulus open a t t h e v e r t i c a l ends t o a s u r r o u n d i n g b a t h o f He 11. Resul t s and D i s c u s s i on
Thermometer Grid F i g u r e 1: Thermometer and h e a t e r l o c a t i o n s . A l l thermometers a r e mounted on t h e i n n e r - r a d i u s s i d e o f t h e h e l i u m annulus and a r e i n d i r e c t c o n t a c t w i t h t h e he1 ium. "unwrapped" and p r e s e n t e d as a two d i m e n s i o n a l s u r f a c e . The h o r i z o n t a l dimension i s e q u i v a l e n t t o one f u l l t u r n and as can be seen, t h e h e a t e r used i n t h i s i n v e s t i g a t i o n i s approximately a q u a r t e r t u r n i n l e n g t h . The e x a c t l e n g t h i s 5.2 cm. Temperatures a r e measured i n o n l y one quadrant o f t h e channel s i n c e we e x p e c t t h e t e m p e r a t u r e p r o f i l e s i n t h e channel t o be symmetric v e r t i c a l l y about t h e m i d h e i g h t o f t h e channel and h o r i z o n t a l l y about t h e m i d p o i n t o f t h e heater. The p h y s i c a l parameters o f t h e e x p e r i m e n t a r e used as i n p u t t o t h e computer program. One dimension w h i c h i s n o t p r e c i s e l y known f r o m t h e e x p e r i m e n t a l d a t a and w h i c h i s i n c l u d e d i n t h e i n p u t t o t h e computer program i s t h e l e n g t h o f t h e normal zone subsequent t o t h e t h e r m a l d i s t u r b a n c e . V o l t a g e t a p measurements i n d i c a t e t h a t i t e x t e n d s beyond t h e l e n g t h of t h e h e a t e r , b u t t h e e x a c t l e n g t h i s b o t h d i f f i c u l t t o d e t e r m i n e and may i n f a c t be c h a n g i n g d u r i n g t h e s t a b i l i t y process.
The r e s u l t s o f t h i s comparison f o r a sample o f t h e In t h e r m o n e t e r s a r e p r e s e n t e d i n F i g u r e s 2 t h r o u g h 6. g e n e r a l t h e agreement i s good, and disagreements become obvious where t h e computer code i s u n a b l e t o i n c l u d e t h e c o m p l e x i t i e s o f t h e p h y s i c a l processes I n t h i s r e s p e c t a number of'comments a r e occurring. worthwhile.
2. 8
5
t i m e ( A t * ) a s s o c i a t e d w i t h t h e f o r m a t i o n o f t h e vapor l a y e r i s given by
P
2.6 2.4
-I
-0
W
c 2.2 aJ a.
5
AJ
2.0
1.8
0
0. 04 0. 08 0. 12 0. 16
0.2
time (SI
The magnitude o f t h e power d i s s i p a t e d i n t h e h e a t e r T h i s same power, i s between 70 W/cm2 and 170 W/cm2. communicated t o t h e c o n d u c t o r - h e l i u m i n t e r f a c e r e s u l t s i n t h e r a p i d f o r m a t i o n o f a h e l i u m vapor l a y e r . The
I
F i g u r e 2 : Temperature response o f thermometer 1. D i s t a n c e f r o m c e n t e r o f h e a t e r : h o r i z o n t a l , 0 cm; v e r t i c a l , 0 cm. F i g u r e 2 demonstrates t h a t f o r t h e a d i a b a t i c ( o r c l o s e d c h a n n e l ) case t h e t e m p e r a t u r e d i r e c t l y a c r o s s t h e annulus f r o m t h e h e a t e r ' s c e n t e r r i s e s w e l l above Ta, and s u b s e q u e n t l y f a l l s t o a s t e a d y v a l u e somewhat b e l o w Ta.
T h i s b e h a v i o r r e f l e c t s t h a t t h e phase
e q u i v a l e n t amount of time. I n a l l o f t h e d a t a shown below, t h e comparison i s l i m i t e d i n t i m e t o a v o i d temperature p r o f i l e s r e a l i z e d a u r i n g t n e c o l l a p s e o i t h e normal zone.
boundary s e p a r a t i n g t h e He I 1 f r o m t h e He I i n i t i a l l y moves a c r o s s t h e annulus and beyond t h i s thermorneter, b u t r e t u r n s p a s t i t t o some s t e a d y p o s i t i o n between t h e h e a t e r and t h i s thermometer. I n t h e isothermal ( o r open c h a n n e l ) d a t a , t h e i n t i a l t e m p e r a t u r e r i s e r e c o r d e d b y TH1 i s n o t as g r e a t as i n t h e a d i a b a t i c case. It i s b e l i e v e d t h a t t h i s i s due t o t h e g r e a t e r freedom w i t h w h i c h t h e h e a t can move i n t h e v e r t i c a l d i r e c t i o n w i t h t h e i s o t h e r m a l boundary c o n d i t i o n s . I t i s a l s o assumed t h e r e f o r e t h a t t h e warm He I r e g i o n sppeads f u r t h e r i n t h e t w o d i m e n s i o n a l p l a n e o f t h e annulus w i t h t h e i s o t h e r m a l c o n d i t i o n t h a n w i t h t h e Indeed, a s i g n i f i c a n t i m p r o v e w n t i n a d i a b a t i c one. t h e i s o t h e r m a l c o m p u t a t i o n s t o r TH2, TH4, THb, THv, THlO and T H l l i s o b t a i n e d b y assuming t h a t t h e T A
The a d i a b a t i c and i s o t h e r m a l c o n d i t i o n s a r e r e a l i z e d e x p e r i m e n t a l l y as f o l l o w s : I n the adiabatic case t h e m i c a r t a c y l i n d e r d e f i n i n g t h e i n n e r r a d i u s o f
boundary i s v e r t i c a l l y p o s i t i o n e d 2 mm f u r t h e r f r o m t h e m i d h e i g h t o f t h e channel t h a n i n t h e a d i a b a t i c case.
( s e e Van S c i v e r 7 )
, from
which we f i n d t h a t A t * i s l e s s
than 5 x sec. Between t h e vapor l a y e r and t h e He I 1 t h e r e e x i s t s a t h i n l a y e r o f He I and t h e r e f o r e a l s o a phase boundary w i t h i t s t e m p e r a t u r e l o c k e d a t TA. F o r t h e d a t a i n c l u d e d i n t h i s r e p o r t , t h e d u r a t i o n o f t h e s t a b i l i t y p r o c e s s and t h e t i m e d u r i n g w h i c h t h e r e e x i s t s a normal zone ( a s measured w i t h v o l t a g e t a p s ) i s g r e a t e r t h a n 200 msec. As i s shown below, t h e e x i s t e n c e o f t h e T a phase boundary l a s t s an
1510
observed i n our s t a b i l i t y data6, which i s t h a t t h e normal zone appears t o be growing i n l e n g t h w h i l e i t i s cooling. The t i m e dependence o f t h i s growth i s d i f f i c u l t t o model. The i n c r e a s e i n heat f l u x r e s u l t a n t from t h i s growing normal zone o r more s i g n i f i c a n t l y , t h e e x t e n s i o n o f t h e TA boundary i n t h e
2. 2 n
s
2. 1
a
L
3 3 W
h o r i z o n t a l d i r e c t i o n , a1 so produces d i s c r e p a n c i e s between t h e model and d a t a f o r TH8, THlO and TH11. T h i s i s e s p e c i a l l y v i s i b l e i n t h e isothermal comparison o f T H l l i n F i g u r e 5. Here t h e computational r e s u l t s denoted b y l i n e i a r e d e r i v e d w i t h a T A boundary h a l f l e n g t h f i x e d a t 6 cm (as a r e
2.0
L
a 1.9 n E
a
c,
1. 8
I
t h e isothermal and a d i a b a t i c r e s u l t s shown f o r a l l o t h e r thermometers) w h i l e t h e l i n e i' represents t h e computations w i t h a T A boundary h a l f l e n g t h f i x e d a t 9
TH7 I
I
I
I
I
I
I
I
I
cm.
time (SI F i g u r e 3: Temperature response o f thermometer 7. Distance from c e n t e r o f heater: h o r i z o n t a l , 3 cm; v e r t i c a l , 0 cm. I n b o t h d a t a t r a c e s o f TH1, t h e i n i t i a l t r a n s i e n t g i v e s way t o a steady temperature as assumed i n t h e computational model. The f a c t t h a t a steady temperature approximately equal t o T A i s recorded b y
-
Y
TH1 (and TH5 and TH7) i n d i c a t e s t h e e x i s t e n c e o f t h e f i x e d temperature phase boundary i n t h e nearby v i c i n i t y . F u r t h e r , t h e f a c t t h a t t h e steady temperature as recorded b y TH1 (and TH5 and TH7) i s below T A suggests t h e e x i s t e n c e o f a temperature g r a d i e n t across t h e 0.7 mm annulus. T h i s suggestion i s supported b y observations o f s m a l l e r t!ermal disturbances r e s u l t i n g i n s l i g h t l y l o w e r steady" temperatures as recorded by TH1 (and TH5 and TH7). Thus, a 3-D d e t a i l o f t h e heat f l o w , n o t modelled b y t h e computer, appears i n t h e experimental data.
Y
The a c t u a l h a l f l e n g t h o f t h e TA boundary grows
w i t h t i m e between these two values. The a d i a b a t i c comparison shown i n F i g u r e 5 a l s o r e v e a l s a discrepancy, b u t i n t h i s case t h e d a t a c o n s i s t e n t l y f a l l s below t h e computations. T h i s i s due t o t h e non i d e a l a d i a b a t i c boundary c o n d i t i o n s mentioned above.
2. 00
a 1. 95 L
3 3 W
1. 90
L
E 1. 85 E a c, 1. 80
1.751 0
l
2.21
a 2.1 L
3 3 W
L 2.0 a 1.9 n E
c
tc-
I
I
a 1
TH9 0
'
' ' ' ' I ' ' ' 0. 04 0. 08 0. 12 0. 16 0. 2
time ( s ) F i g u r e 5: Temperature response o f thermometer 11. L i n e i represents isothermal c a l c u l a t i o n w i t h f i x e d TA h a l f l e n g t h of 6 cm; l i n e i' represents i s o t h e r m a l c a l c u l a t i o n w i t h f i x e d T A h a l f l e n g t h o f 9 cm.
-4+ H-cl- 1 - - --
-0
1.7
'
0.04
0.08
0. 12 0. 16
0.2
time (SI F i g u r e 4: Temperature response o f thermometer 9. D i s t a n c e from c e n t e r o f heater: h o r i z o n t a l , 6.51 cm; v e r t i c a l , 0 cm. Demonstrates growth o f normal zone p a s t t h i s thermometer. The second obvious discrepancy i n t h e comparison between t h e d a t a and t h e computational model appears i n F i g u r e 4. I n t h i s case our computer i n p u t s e t t h e h a l f - l e n g t h o f t h e normal zone s h o r t e r t h a n t h e d i s t a n c e between TH1 and TH9; however, t h e experimental d a t a p o r t r a y a normal zone rrhich .ro.vs beyond t h a t length. T h i s scenario i s s i m i l a r t o t h a t
One f u r t h e r p h y s i c a l c o m p l e x i t y observed i n t h e d a t a i s e v i d e n t i n t h e f i r s t 20 m i l l i s e c o n d s o f F i g u r e s 4, 5, and 6. Here t h e channel temperatures appear t o decrease b e f o r e growing. We b e l i e v e t h i s behavior i n d i c a t e s a l o c a l i z e d pressure i n c r e a s e due t o a s h o r t l i v e d h e l i u m vapor bubble a t t h e heated surface. E l e c t r i c a l l y o r m a g n e t i c a l l y induced t r a n s i e n t s a r e n o t l a r g e enough t o account f o r t h e magnitude o f t h i s dip. However, l i q u i d h e l i u m does have n e g a t i v e values o f dT/dP i n t h i s temperature range. Indeed, t h e J o u l e Thompson c o e f f i c i e n t pj =
($)
a t s a t u r a t e d vapor pressure a r e reported8
t o be negative. I n a d d i t i o n , c a l c u l a t i n g p j f o r t h e subcooled He I 1 from
(3) Niert:
Cp and V are t h e s p e c i f i c heat and s p e c i f i c
volume r e s p e c t i v e l y a t 1 atm, r e v e a l s t h a t ~j =
x
KPa-l.
-
2.27
The teniperature changes observed a r e
1511 References
[I] Y. M.
Eyssa, X, Huang and J. Waynert, "Heat T r a n s f e r i n Helium I1 f o r Two-Layer Energy Storage Magnets," IEEE Trans. Mag., v o l . 23, p. 561, March
1987. New York:
[2] S . W. Van S c i v e r , Helium Cryogenics. Plenum Press, 1986, ch. 5, p. 144.
[3] L. Dresner, N o n l i n e a r D i f f e r e n t i a l Equations.
Oak
Ridge N a t i o n a l L a b o r a t o r y r e p o r t DRNL/TM-10655, 1988, ch. 7, pp. 115-117.
[4] J. M. Pfotenhauer and S. W. Van S c i v e r , " S t a b i l i t y I
0
l
l
l
1
'
l
'
'
'
j
0.04 0.08 0. 12 0. 16 0.2
C5l J. M. Pfotenhauer, "Experimental I n v e s t i g a t i o n s on
time (SI F i g u r e 6. : Temperature response o f thermometer 6. D i s t a n c e f r o m c e n t e r o f h e a t e r : h o r i z o n t a l , 1 cm; v e r t i c a l , 1 cm. a p p r o x i m a t e l y 12.5 mk i m p l y i n g a p r e s s u r e r i s e of 5506 Pa, o r a 5.5% i n c r e a s e . Note t h a t t h e speed o f f i r s t sound i n He I1 i s 200 m/sec, t h u s t h e p r e s s u r e r i s e appears a t a l l t h e thermometers a t e s s e n t i a l l y t h e same time. The t e m p e r a t u r e r e c o r d e d t h e n by each thermometer i s t h e sum o f t h e t h e r m a l l y i n d u c e d t e m p e r a t u r e r i s e and t h e p r e s s u r e induced t e m p e r a t u r e decrease. Those thermometers c l o s e t o t h e h e a t e r show no n e t t e m p e r a t u r e decrease because o f t h e l a r g e and immediate thermal s i g n a l , w h i l e t h o s e thermometers f a r from t h e h e a t e r d i s p l a y t h e immediate p r e s s u r e induced t e m p e r a t u r e decrease f o l l o w e d by t h e s l o w e r t h e r m a l l y induced t e m p e r a t u r e i n c r e a s e . A n a l y s i s o f 1-0 h e a t f l o w i n He I1 p r e d i c t s a "thermal d i f f u s i o n " t i m e TD which i s r o u g h l y g i v e n by
- DT L4l3
AT''^
where L i s a
l e n g t h and DT i s an e f f e c t i v e d i f f u s i v i t y . C a l c u l a t i o n s based on t h e e x a c t form o f t h e 1-D a n a l y s i s r e v e a l t h a t f o r L = 5 cm, T D = 10 msec f o r a
1% o f TA - Tb and TD = 54 msec f o r a t e m p e r a t u r e r i s e e q u i v a l e n t t o 10% of
temperature r i s e equivalent t o TA
-
Tb. Conclusion
The c o m p u t a t i o n a l model o f Eyssa e t a1 p r o v i d e s an accurate t o o l f o r p r e d i c t i n g temperature p r o f i l e s r e a l i z e d i n two dimensional h e a t f l o w i n He 11. The temperature data gathered d u r i n g a s t a b i l i t y experiment i n a small l a b o r a t o r y s c a l e magnet agrees w e l l w i t h t h e p r e d i c t i o n s o f t h e model i n most D i s c r e p a n c i e s become obvious when t h e respects. computer model i s unable t o i n c l u d e t h e c o m p l e x i t i e s o f t h e s t a b i l i t y process. Two n o t a b l e c o m p l e x i t i e s a r e t h e growth o f a normal zone, and t h e p r e s s u r e i n c r e a s e r e s u l t a n t f r o m a t r a n s i e n t bubble o f h e l i u m vapor. With t h e s e c o n s i d e r a t i o n s t a k e n i n t o account, we a r e q u i t e pleased w i t h t h e accuracy o f t h e computer model and conclude t h a t i t i s a w o r t h w h i l e t o o l f o r p r e d i c t i n g t e m p e r a t u r e p r o f i l e s and h e a t f l o w i n t h e two dimensional geometry o f l a r g e s c a l e SMES systems. F u r t h e r r e f i n e m e n t s o f t h e model t o i n c l u d e aspects o f superconductor s t a b i l i t y a r e i n progress. Acknowledgement T h i s dark rlas c o i i l p l e t d under E P R I C o n t r a c t No.
RP22572-2.
Measurements o f a Superconductor Cooled by a TwoDimensional Channel o f He 11," Adv. Cryo. Eng., v o l . 31, pp. 391-398, 1986. t h e S t a b i l i t y o f a He I1 Cooled Superconducting S o l e n o i d , IEEE Trans. Mag., v o l . 23, pp. 926-929, March 1987.
[6] R. W. Boom, p r i n c . i n v e s t i g a t o r , "Superconducting
-
Final Magnetic Energy Storage: Basic R&D" Report t o E P R I c o v e r i n g t h e p e r i o d Feb. 1, 1987 t o Jan. 31, 1988, ch. 3, pp. 3.10-3.13.
[7] S . W. !an S c i v e r , " T r a n s i e n t Heat T r a n s p o r t i n He 11, Cryogenics, v o l . 19, pp. 385-392, J u l y 1979. [8] B. J. vuang, "Joule-Thomson E f f e c t i n L i q u i d He 11, Cryogenics, v o l . 26, pp. 475-477, August 1986.
