Thermal Contact Conductance of Coated Multilayered Sheets J. W. Sheffield and T. N. Veziroglu, University of Miami, Coral Gables, Fla.; and A. Williams, Monash University, Clayton, Victoria, Australia AlAA 15th THERMOPHYSICS CONFERENCE July 14-16, 1980/Snowmass, Colorado For permission to copy or republish. contact the American Institute of Aeronautics and Astronautics.1290 Avenue 01 the Americas. New York. N.Y. 10019 I Downloaded by PURDUE UNIVERSITY on February 19, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.1980-1468
Transcript
Thermal Contact Conductance of Coated Multilayered Sheets J. W. Sheffield and T. N. Veziroglu, University of Miami, Coral Gables, Fla.; and A. Williams, Monash University, Clayton, Victoria, Australia
AlAA 15th THERMOPHYSICS CONFERENCE
July 14-16, 1980/Snowmass, Colorado
For permission to copy or republish. contact the American Institute of Aeronautics and Astronautics.1290 Avenue 01 the Americas. New York. N.Y. 10019 I
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THERMAI. CONTACT CONDUCTANCE OF COATED MUI.TlI.AYIIRED SIIEETS
.J.V!. S h e f f i e l d " , T .N. V e z i r o g l u + and A . V i l l i n m s # l l e r h a n i c a l h g i n c c r i n z Department
Schaol of E n g i n e e r i n g and A r r h i t e c t u r e University of Miami. C o r a l C a h l e s . F l o r i d a
,\hs t r n r t
S f e a d y - s t n t P tilermill C O " t i l C t c o n d r l r t n n c e s of m i i l t i l a y c r e d e l e c t r i c a l l y i n s u l a t e d s h e e t s are PX- m i n e d t h e o r e t i r a l l y and c x p c r i m e n t n l l y . Tests lzwe heen c o n d u c t e d f o r twelve s e l e c t e d combina- t i o n s of s h e r t m a t o r i a l and s u r f a c e coatings i n e n v i r o n m m t s of .vncuum a n t i low pressure h c l i u m . C o m p a r i s o n s of p r e v i o u s l y reported theories w i t h the new planar c o n t a r t thcory show good a e r e c m e n t f o r s n r l l l values of t h e r a t i o of t h e d i m e n s i o n l e s s c o n d u c t i v i t y number, !< = k f / k , , to t h e d i m m s i o n - less gap numbFr, R = & l a , c o r r e s p o n d i n g t o tlir
t h o s e e x i s t i n g under v n c ~ i i m c o n d i t i o n s . T h e DCW
circu!ar c o n t a c t t h e o r y i s used t o p r e d i c t t h e t h e r m a l c o n t a c t c o n d t i c t m r r and is cornporcd w i t h e n p c r i m m t n l r c s u l t s .
c a s e s of low f l u i d gap c n n d u c t j v i t i c s , such as
liornenc1 i t tu re -~___ constont c o n t a c t c l e m e n t r a d i u s o r h z l f wid t i ) gap number s o l i d c o n t a c t r a d i u s or h a l f w i d t h c o n s t r i c t i o n numhrr th ickness numhrr n d r n i s s i h l p f u n c t i o n f o r temperature a l o n g g a p e f f e c t i v e heat t r a n s f e r c o e f f i c i e n t f o r i n t c r s t i t i a l media z c r n t i i o rde r Ressel f u n c t i o n o f t l ic f i r s t k i n d f i r s t o r d e r Rrssel f r m c t i o n o f t h e f i r s t k i n d c o n d u c t i v i t y numher; t r a n s f o r m k e r n e l t h e r ~ i a l c o n d u c t i v i t y o f c o a t i n g t h e r m a l c o n d u c t i v i t y o f i n t e r s t i t i a l f l u i d thermal h a l f t l i j
c o n d u c t i v i t y of s o l i d ~ c k n e s s o f a s i n g l e s h e e t .
Eleyer hardness number of sheets i n a s t a c k a p p a r e n t c " n t a c t pressurc t c m p e r a t u r c t e m p e r a t u r e a t midplane c o a t i n g t h i c k n e s s t C
U=I&a/k, t h r r r n n l conductnncc number UC thermal con t r l c t ronduc t a n c e X,Y Cartesian c o o r d i n a t e s r,z cylindrical^ c o o r d i n a t e s 6 i n t e r s t i t i a l f l u i d t h i c k n c s s
- 1. I n t r o d u c t i o n
S t a c k s of f l a t , c o a t e d , t h i n s h e e t s o f m e t a l , clnrnped t o g e t h e r , a r e used i n most e l e c t r i c a l t r n n s f o r m e r s , g e n e r a t o r s an? motors . The s t e a d y s t a t e t e n p e r a t u r e s a t t a i n e d w i t h i n s u c h an elec-
__ "Research A s s i s t a n t P r o f e s s o r , Member A I A A . + P r o f e s s o r , Member A I A A . #Adjunct P r o f e s s o r , also Associate P r o f e s s o r , )!onash University, C l a y t o n . Victoria, A u s t r a l i a .
W
t r i c a l d e v i c e w i l l u s u a l l y c o n f i n e t he c n e r g y con- version e f f i c i e n c y by l i m i t i n g t h e n l l o i r n b l e o u - p u t . Thcse e l e c t r i c a l d e v i c e s h a w i n t r i n s i c a l l y low, e f f e c t i v e thermal c o n d u c t i v i t i e s across t h e l a m i n a t i o n d u e t o t h e l n r p c numher o f c o n t a c t i n t e r f a c e s o c c u r r i n g i n s e r i e s , and a l s o d u e to t h e c l e c t r i c a l l y i n s u l a t i n g c o a t i n g . Such low t h e r m a l c o n d t i c t i v i t i r s cause i n c r e a s e i n t h e r m a l s l o p e s i n t r a n s f e r r i n g t h e h e a t produced by m a g n e t i c and e l e c t r i c a l l o s s e s w i t h i n the e l c r t r i c a l machinery .
l l i s t o r i c a l l y , t h e h e a t f l o w thronS!i s t a c k s of l a m i n a t i o n h a s been exp lo red 8s an a r l j u n c t t o stii-
d i c s of thermal c o n t a c t c o n d u c t a n c e . Tmpcrfec t c o n t a c t s bc twern a d j a c e n t f a c e s i n s t a c k s of Inm- i n a t i n n C I ~ U S ~ thermal c o n t a c t r c s i n t n n c c . T h c main parameters a f f e c t i n g the resistance arc t h e clamp- i n g prcssure, t h e h a r d n e s s of t h e mc tn l s , thr ge- om-try, t h e s u r f a c e r o u g h n e s n r s , t h c thermal p r o p - e r t i e s o f t h e m e t a l s i n c o n t a c t anti of tlir i n t c r - s t i t i n l f l u i d . I n t h c case of s t a c k s l a m i n a t e d w i t h t h i n s h e e t s of m e t a l having w r y t h i n n n d relatively s o f t c o a t i n g s , t h c true m e t a l - t o - m e t a l c o n t a c t s a r e d e s t r o y e d and lo r reducrd i n numhcr.
T h i s p a p c r p r e s e n t s a n e x p c r i n r n t n l a n d thCO- r e t i c a l e x a m i n a t i o n of t h e s tcndy s t a t r t k r m a l c o n t a c t c o n d u c t a n c e o f m u l t i l i i y e r r d e l e c t r i c a l l y i n s u l a t e d s h e e t s . The new t h e o r y d c v e l o p d l i r r f i n i s used t o p r e d i c t t h e thermal c o n d n c t n n c c s of s t a c k s o f lamination over ii w i d e r n n g r o f c o n t a c t pressures and w i t h v a r i o u s i n K ? r S t i t i n l f l u i d con- d i t i o n s . The v a l i d i t y of t h c new Circular t l i c o r y was conf i rmed w i t h r e c e n t l y o b t n i n c d r n p c r i m r n t n l r e s u l t s . Comparisons of e a r l i e r t h e o r i e s w i t h t h i s new t h e o r y show s a t i s f a c t o r y a g r e e m e n t .
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2 . L i t e r a t u r e S u r v e l
E a r l i e r s t u d i e s on c o n t a r t h e a t t r a n s f e r of m u l t i l a y e r r d s t a c k s of t h i n l a y e r s have a d d r e s s e d a p p l i c a t i o n s r a n g i n g from m c c h n n i c a l l y s t r o n g thermal i n s u l a t o r s f o r c r y o g e n i c t a n k s u p p o r t s ] to t h o s e i n e l e c t r i c a l machinery2-'. T h o m a s a n d Pro- b e r t l i n v e s t i g a t e d t h c thermal r e s i s t a n c e s of m11- t i i a y e r c d c o n t a c t s u n d e r s t a t i c l o a d s a s an appli- c a t i o n of c ryogen ic t h e r m a l i n s u l a t i o n . T h e i r con c l u s i o n s BuEgest t h a t f o r t h e d e s i g n o f i n s u l a t i o n s u p p o r t s o n e s h o u l d s e l e c t a m a t e r i a l p o s s r s s i n g low t h e r m a l c o n d u c t i v i t y . a low niimher of s u r f a c e a s p e r i t i e s p e r u n i t a r ea a n d n h i p h v a l u e of h a r d - n e s s . I n a d d i t i o n , a l a r g e number of l a y e r s was recommended f o r t h e i n s u l a t o r .
Hargudon' measured t h e t h e r m a l i n t e r f a c e con- d u c t a n c e of m a t e r i a l s w e d i n t h e hardware of t h e r - m o e l e c t r i c g e n e r a t o r s . S p e c i a l tes ts were conduc- t e d t o e v a l u a t e t h e e f f e c t of i n t e r s t i t i a l m a t e r i a l on t h e i n t e r f a c e t e m p e r a t u r e g r a d i e n t .
W i l l i a m s 3 i n t r o d u c e d an a n a l v t i c a l model f o r v ~ ~~~~~ ,
t h e thermal behavior of s t a c k s of enameled s h e e t s . His model relies on t h e s e p a r a t i o n of t h e compon-
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e n t s of c o n t a c t conduc tance a t t r i b u t a b l e t o t h e s o l i d c o n t e c t s and t o t h e i n t e r f a c e f l u i d . Mea- s u r e m e n t s of t h e thermal. c o n d u c t i v i t i e s of s t a c k e d l a m i n a t i o n s were r e p o r t e d by Wi l l i ams4 i n s u p p o r t o f h i s model f o r a v a r i c t y of m a t e r i a l s i n e n v i r o n - ments of a i r , vacuum and he l ium. The clamping pressure was va r i ed over a r a n g e of 50-150 p s i ( 3 . 4 - 1 0 . 3 x IO5 ! i /m2) t o r e p r e s e n t c o n d i t i o n s t y p i c a l f o r t u r b o g e n e r a t o r s . Wi l l i ams concluded t h a t the smoothness of t h e l a m i n a t i o n p l ayed a prominen t p a r t i n t h e c o n t a c t conduc tance fo r nor- m a l a t m o s p h e r i c c o n d i t i o n s . Ifowever, 8s t h e con- d u c t i v i t y of t h e i n t e r s t i t i a l f l u i d i n c r e a s e d , t h e t h e r m a l l o a d was c a r r i e d ma in ly by t h e i n t e r s t i t i a l f l u i d r a t h e r t han by t h e s o l i d c o n t a c t s .
.i
Jhamnnni5 measurcd t h e t o t a l t h e r m a l r e s i s t - ance of s t a c k s of t h i n c l e c t r i c a l s t e e l l a m i n a t i o n s . lioth g r a i n o r i e n t e d and "on-o r i en ted e l e c t r i c a l s t e e l s wcre tes ted i n a i r env i ronmen t s and undc r h i g h VJCIIIIB. Although Jlramnani. examined s t a c k s c o n t a i n i n g onc , t h r e e , f i v e and s i x d i s c s i n lam- i n a t i o n , h e p r e s e n t e d h i s r e s u l . t s as t h e r m a l rc- s i s t a n c c pe r i n t e r f a c e . However, i n g e n e r a l , t h e r e s i s t a n c e per c o n t a c t depends upon t h e numher o f layers p r e s e n t . Hence, t h e r e s u l t s were n o t gen- e r a l i z e d . Jliamnani n o t e d a t endcncy f o r t h e t h c r m a l r e s i s t a n c e p e r c o n t a c t t o hecone l c s s de- penden t upon t h e number of l ayers as t h e number o f l a y e r s increased, e s p e c i a l l y for h i g h c o n t a c t pressure.
Al -As t rnbnd i6 ,* e t al. proposed n si.mple corrclatirn Tor r e s i s t a n c e s of s l a c k s of t h i n l ay - ers u n d e r compress ion . Thr l e a s t - s q u a r c s s t r a i g h t - l i n c power law f i t was made f o r a l l a v a i l a b l c p u b l i s h e d d a t a and t h e r e s u l t i n g r e l a t i o n s h i p had
\- A c o r r e l a t i o n c o e f f i c i e n t of 0 .95 .
S h e f f i c l d ' e t a1 . p r e s e n t e d t h e r c s r i l t s o f an c x p c r i m e n t n l i n v e s t i g a t i o n on t he rma l c o n t a c t con- d u c t a n c e of m u 1 t i l a y e r e d e l e c t r i c a l l y i n s u l a t e d s h e e t s . Tlic test n s s e m b l i e s were made from p i e c e s o f m a t e r i a l s u p p l i r d hy m a n u f a c t u r c r s of e l e c t r i c a l mnch in r ry so t h e test measurements are t r u l y r e p r e s e n t a t i v e o f c u r r e n t p r a c t i c e , w i t h r e s p e c t t o p l a t e t h i c k n e s s e s , s u r f a c e f i n i s h e s , r o u g h n e s s , c o a t i n g s , e t c . Thc measurements a t vacuum cond i - t i o n s conf i rmed flirlt t h c major p a r a m e t e r i n d e t e r m - i n i n g t h e c f f e c t i v f t h e r m a l c o n d u c t i v i t y across t h e l a m i n a t i o n s is t h e t h e r m a l c o n t a c t conduc tance .
3 . T h m
For t h e t h e o r e t i c a l s t u d y t h e c o n t a c t i n t e r - f n c r s i n n s t a c k of 1ami.nqtion are assumed t o be composed of a numher of s i m i l a r c o n t a c t e l e m e n t s . T ~ C S C c o n t a c t d e m e n t s have a c e n t e r e d , l oad h e a r - i n g c o n t a c t r e g i o n sur ror inded by a "on-con tac t re- g i o n . For s i m p l i c i t y , t h e c o n t a c t elements are nssumcd t o he mad? up of two s o l i d p l a t e s , of t h i c k n e s s 21. each, and h a v i n g a c i r c u l a r c o n t a c t i n t h e midd lc of t h e i r sur faces whtch f a c e each o t h e r , w i t h t h e gap between them f i l l e d w i t h a f l u i d of un i fo rm conductance (see P i g . I ) . T f t h e r - a n i s d i v i d e s t h e gap. 6 , i n t o two e q u a l gaps of t h i c k n e s s , 6 / 2 , t h e n the s y s t e m becomes symmetr i - c a l w i t h r e s p e c t t o t h e r - a x i s f rom t h e l ieat t r a n s - fe r p o i n t of view. S i rn i . l a r ly , t h e z - a x i s d i v i d e s t h ? c o n t a c t e l emen t w i d t h , 2a, i n t o two cqun l h a l v e s . Thus i t s u f f i c e s t o c o n s i d e r a one q u a r t e r r e g i o n o n l y , as i l l u s t r a t e d i n F ig . 2 .
v
Midplane I
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F i g . I C i r c u l a r c o n t n c l e l emen t
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I.
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Fig. 2 Q u a r t e r e l c m r n t used fo r e v a l u a t i o n of t e m p e r a t u r e d i s t r i b u t i o n
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The s t e a d y s t a t e t p m p e r a t u r e d i s t r i b u t i o n i n t h e p l n t c must s a t i s f y La l ' l nce ' s c q u n t i o n :
( 1 )
Thc boundary c o n d i t i o n s of t h c problem can he writ- t e n as fo l lo5rs :
ZIT ( 2 )
( 3 )
- a r ( 0 , ~ ) = D f o r n<z<i.
- :: (a,z) = o f o r wz<i.
T ( ~ , L ) = T~ f o r n<rca ( 4 )
T ( r , O ) = 0 f o r O<rcb ( 5 )
T ( r . 0 ) = f ( r ) f o r bcr'a ( 6 )
1111crc To i s t h e un i fo rm temperature a t t h e midp lane o f t h e s o l i d p l a t e and f ( r ) is t h e unknown tempera- ture p r o f i l e a lone t h e gap . Equa t ion ( 6 ) i s n new rnotl~l r e p l a c i n g t h e c o n v e n t i o n a l c o n v e c t i v e hound- a r y c o n d i t i o n . T h u s , t b e new f o r m u l a t i o n e l i m i n - a t e s t h r mined boundary c o n d i t i o n s which are more d i f f i c u l t t o h a n d l e . To so lve e q u a t i o n (1) s u b j e c t t o houndary c o n d i t i o n s g iven by e q u a t i o n s (2 -61 , t h e t c m p e r a t u r e is f i r s t s c a l e d by To s u c h t h a t
0 ( r , z ) = T(r.7.) - 'L'" ( 7 )
An i n t e g r a l t r a n s f o r m is used t o remove t h e p a r t i a l d e r i v a t i v e w i t h respec t t o "r". Thc i n t e g r a l t r a n s f o r m a t i o n and i n v e r s e t r a n s f o r m a t i o n are Riven by :
O(B,,z) = r ' K ( G m , r ' ) O ( r ' , z ) d r ' ( 8 ) 1 .-
n
where K(fl,",r) is t h e t r a n s f o r m k e r n e l and B,,'s are t h e c o r r e s p o n d i n g e i g e n v a l u e s . The k e r n e l i s g i v e n by:9
f o r e = n (10)
where Bm's are t h e p o s i t i v e r o o t s of
8, J1(Gma) = 0 for = n. i .2, . . . (11)
The g e n e r a l s o l u t i o n f o r t h i s f o r m u l a t i o n i s g i v e n bv:
(12) b
V a r i o u s a d m i s s i b l e f u n c t i o n s f o r t h e unknown temp- e r a t u r e , f ( r ) , were assumed. A f t e r comparison of t h e s o l u t i o n s w i t h p r e v i o u s l y r e p o r t e d t h e o r i e s , a Bessel t e m p e r a t u r e p r o f i l e was s e l e c t e d f o r
f u r t h e r a n a l y s i s . L ikewise t h e a n a l o g o u s p l a n a r c o n t a c t problem was f o r m u l a t e d . s o l v e d and cvn lua - t e d . The unknown c o n s t a n t of t he Scssel tempern- t u r e p r o f i l e f o r t h e c i r c u l a r c o n t a c t problem was s e l c c t c d a s ' W
f ( r ) = A.(J"(c) - .J,,(:)) ( 11 )
where A is a c o n s t a n t . It s h o u l d he n o t e 6 t h a t t h i s t c m p e r a t u m distribution s n t j s f i e s t h r end c o n d i t i o n o f f ( r ) = 0 when r = b . The c o n s t a n t A is de tp rmined by n h r a t f l ow b a l a n c e acrcss t h c f l u i d gao as f o l l o w s :
wliere hf i s t h e gap conduc tance and k, i s t h e s o l i d the rma l c o n d u c t i v i t y . The c o n t a c t c o n d u c t a n c e , U,, was t hcn found t o be g i v e n by:
(15 )
where om =
& ( C ) J 1 (3d>) -ab f l .I h h . ) J 1 ( C ) + ~ 2 B , J o ( 8 m ~ ~ ) . J ~ (1) nQm?flR,2 n i7 1 )
(16)
and C , the c o n s t r i c t i o n numhr.r, i s g i v e n b y :
C = h l a (17) L
To o b t a i n R c l o s e d form s o l u t i o n , t h e unknorm c o n s t a n t "A" was d e t e r m i n e 6 by u t i l i z i n g e q u a t i o n ( 1 4 ) f o r t h e l i m i t i n g c a s e as t h e c o n s t r i c t i o n number, C , app roached zero. The r e s u l t i n g c o n t a c t conduc tance was o b t a i n e d a s :
I n p r a c t i c a l a p p l i c a t i o n s t h e c o n s t r i c t i o n number t e n d s t o h a v e a s m a l l v a l u e j u s t i f y i n g t h i s l i m i t - i n g case.
For comparison t h e s o l u t i o n f o r t h e planar c o n t a c t conduc tance f o r m u l a t i o n is:
f o r t h e s p e c i a l rase of s m a l l c o n s t r i c t i o n number, C , and a l i n e a r t e m p e r a t u r e p ro f i1 . e f o r f ( x ) .
b
__ 4 . Exper imen t s
A d e t a i l e d d e s c r i p t i o n of t h e t es t a p p a r a t u s and t h e e x p e r i m e n t a l p r o c e d u r e used t o measure t h e t h e r m a l c o n t a c t conduc tance is g i v e n i n Ref . 7 . For each t e s t m a t e r i a l , measurements of s u r f a c e roughness and waviness u s i n p a p r o f i l a m e t e r were made. T h i c k n e s s e s of t h e i n s u l a t i n g s u r f a c e c o a t - i n g s were checked u s i n g a t h i c k n e s s gauge. In an
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attempt to find an effective material hardness of the coated electrical steel sheets, a micro-hard- ness tester was used. Numerous loads were applied so that the effective hardness as a function of penetration depth could he estimated. Reduced loads were used in order to obtain penetration depths less than the surface coating thickness.
U
Circular discs were punched from the samples. The diameters of the discs were measured using a micrometer after filing the burrs produced during the punching process. Fifty-one discs formed a typical stack of lamination. Three thermocouples were inserted at Equal axial spacings. The stack was then wrapped with several layers of thin asbes- tos cloth making a smallsub-assembly for testing.
The sub-assemhly was then placed in the therm- a l contact conductance testing apparatus and the expcriments were conducted in a vacuum and low pressure helium using different loads.
5. Comparison of Present Solutions with - Existing ~~ Theories
For comparison of the solutions for the therm- al conductances of contacts having interstitial fluids, the earlier theories given in Ref. 10, 11, 12 and 13 Ihave been rewritten in terms of the di- mensionless numbers. The percentage of deviation of earlier theories ( U ) from the present theories ( U o ) , defined as 100.(U - U,)/U,, has been calcn- lnted for several combinations of constriction numbcr, C, and the ratio of the conductivity num- ber, K = kf/k,, to the dimensionless gap number, H = a l a . The results a r e presented in Tables 1 and
i/ 2. A study of Table 1. shows that the earlier the"-- ries l ime small deviations for small constriction numbers. An important note to be made here is that the results from Ref. 11, 12 and 13 are solutions f o r semi-infinite circular contact elements, i.e., D = - and those from Ref. 10 are for finite cir- c u l a r contact elemcnts applicable for laminated stacks. For comparison the current theories were evaluated for D = 2 showing satisfactory agreement of the planar contact formulation. The circular contact solutions, however, have significant devi- ations from the results of Ref. 12 and 13.
Thc utility of modelling the mixed boundary conditions of the contact plane by an admissible temperature distribution at the fluid gap surface is confirmed by thc satisfactory agreement shown in Tab1.e 1. The circular contact element formu- lation will be shown in a later section to have good predictive capabilities for the problem posed in this investigation. Thus, this circular theory was selected as the most appropriate formulation.
Table I Comparison of Planar Contact Solutions ____ with Ex~>~i~.~h!leories --
% deviation from present solution = 100(U - U,)/U,
Figures 3-8 show the measured and predicted valoes using the circular contact formulation of the thermal contact conductance for stacks of lam- inations of electrically insulated sheets of elec- trical steels and indicate good agreement. The experimental value of contact conductance was de- rived from the temperaturc drop directly attribut- able to contacts by subtracting from the overall temperature d r o p across the test specimen that por- tion due to the heat flow conducted through the solid components. Each set of experimental values was measured at steady state thermal conditions for each loading increment.
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0 I 2 I 4 5
CONllCl P R I S S U R I l I 0 1 X / M 2 1
~ i g . 3 Con tac t conduc tance v s p r e s s u r e (Idest inghouse E l e c t r i c Co., o r i e n t e d c o r e / 0 . 0 1 4 " , onidelf l .0002")
10
0 1 1 3 4 I
CONllCl V R t S S U R l I 1 0 1 H / M 2 1
~ i g . 4 C o n t a c t conduc tance vs pressure (TJest inghouse E l e c t r i c Co., 11-2210.0175", ox ide /< f l . 0001" )
C O X M I P R l S S U R t 1101H/M21
F i g . 5 C o n t a c t conduc tance vs pressure (West inghouse E l e c t r i c C o . , M-2210,0175". c a r l i t e I 0 . O O f l 4 " )
0 I 2 3 4 5
COnllCl PRlSSURt lI01N/M21
~ i g , 6 C o n t a c t conduc tance vs pressure ( B e l o i t Power Systcrns, M-45 /0 .024" , C-310. f lO04" )
2 0
I 0 1 1 3 4 5 I-,
COXllCI PRIISURt II01N/M21
Fig . 7 C o n t a c t conduc tance vs p r e s s u r c (~irna E l e c t r i c C o . , P!-4510.023", c-510.0005")
O I 2 I 4 I
COnllCl P R l S S U R l IIO'H/M21
F i g . 8 ContacC conduc tance vs pressure (West inghouse E l e c t r i c Co., o r i e n t e d c o r e l o . 0 1 4 " . c a r l i t e i f l . 0004")
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E v a l u a t i o n of t h e c o n t a c t c o n d u c t a n c e from e q u a t i o n (18) r e q u i r e s f o u r p a r a m e t e r s , C , h f . k,, and I,. The a v e r a g c h a l f t h i c k n e s s of a s i n g l c s h e e t was d e t e r m i n e d by h a l v i n g t h e measured t h i c k - ness of a s h e e t . The t h e r m a l c o n d u c t i v i t i e s of t h e s o l i d s were o b t a i n e d d i r e c t l y from t h e manufac tur - er's s p e c i f i c a t i o n of t h e e l e c t r i c a l s t ee l cores. The e f f e c t i v e c o e f f i c i e n t of h e a t t r a n s f e r f o r t h c i n t e r s t i t i a l media was d e f i n e d as:
L/
where kf i s t h e i n t e r s t i t i a l f l u i d c o n d u c t i v i t y , 6 i s t h e i n t e r s t i t i a l f l u i d t h i c k n e s s , k, i s t h e t h e r m a l c o n d u c t i v i t y o f t h e c o a t i n g and t C is t h e c o a t i n g t h i c k n e s s . The t h e r m a l c o n d u c t i v i t y of t h e i n t e r s t i t i a l a i r was c a l c u l a t e d from t h e f o l l o w i n g e x p r e s s i o n g i v e n i n Ref. 14 :
kf = 2 2 8 . 4 PVac <!I#'? (21)
whcrc P,,, i s t h e vnciium pressure i n l b f / f t 2 , T is t h e f l u i d t e m p e r a t u r e i n O R and 6 i s t h e gap t h i c k - ness. Reference 1 5 g i v e s a s i m p l e r e l a t i o n s h i p f o r "6" a s rl f u n c t i o n o f s u r f a c e r o u g h n e s s e s . The con- s t r i c t i o n number i s e q u a t e d t o t h e s q u a r e r o o t o f t h e r a t i o of t h e a c t u a l c o n t a c t area t o t h e a p p a r - e n t c o n t a c t area, and f o r i n c r e a s i n g pressure is g i v e n by t h e f o l l o w i n g e x p r e s s i o n l 5 :
By m e a s u r i n g t h e h a r d n e s s a t v a r i o u s p e n e t r a t i o n d e p t h s , an, e f f e c t i v c Meycr h a r d n e s s , M , was d e f i n e d a s t h e a p p r o x i m a t e value o b t a i n e d a t a p e n e t r a t i o n d e p t h e q u a l t o t h e c o a t i n g t h i c k n e s s .
\.-
7 . C o n c h L i z
A t h e o r c t i c a l s o l u t i o n h a s b e e n o b t a i n e d f o r t h c problem of t h e r m a l c o n t a c t c o n d u c t a n c e o f c o a t e d m u l t i l a y e r e d s h e e t s i n l a m i n a t i o n by model- l i n g t h e mixed boundary c o n d i t i o n a t t h e c o n t a c t p l a n e as a p r e s c r i 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 . By a j u d i c i o u s se1ec t i .o" o f t h e c o n t a c t t e m p e r a t u r e d i s t r i b u t i o n , a c l o s e d form s o l u t i o n f o r c i r c u l a r c o n t a c t s i s s u g g e s t e d f o r p r a c t i c a l a p p l i c a t i o n s . Comparisons of ea r l ie r t h e o r i e s w i t h t h i s new con- t a c t t h e o r y show s a t i s f a c t o r y agreement . Compari- sons w i t h e x p e r i m e n t a l r e s u l t s i n d i c a t e good pre- d i c t i v e c a p a b i l i t i e s . For improving t h e p r e d i c - t i o n , an e f f e c t i v e m i c r o h a r d n e s s i s recommended f o r t h e c o a t c d s h e e t s .
8. A c k n o w l e d E m n
The a u t h o r s g r a t e f u l l y acknowledge t h e s u p p o r t of t h e N a t i o n a l S c i e n c e F o u n d a t i o n , D i v i s i o n of E n g i n e e r i n g . We are e s p e c i a l l y g r a t e f u l t o Dr. Win Aung f o r h i s c l o s e i n t e r e s t i n t h e p r o j e c t and f o r h i s v a l u a b l e s u g g e s t i o n s . We acknowledge t h e c o o p e r a t i o n of t h o s e companies s u p p l y i n g s a m p l e s of t e s t m a t e r i a l . S p e c i a l t h a n k s are g i v e n t o A. Mentes , R. Samudra la , D. Cunninghan, F. D'Aquino and L. Engel f o r t h e i r i n t e r e s t and a s s i s t a n c e .
9. R e f e r e n c e s
1. Thomas, T.R. and P r o b e r t , S . D . , "Thermal Resis- t a n c e s of Some M u l t i l a y e r C o n t a c t s u n d e r S t a t i c Loads ," I n t e r n a t i o n a l J o u r n a l o f Heat and Mass T r a n s f e r , 9, No. 7 , 1 9 6 6 , pp . 739-754.
2 . Hareadon. J . M . . Jr . . "Thermal I n t e r f a c e Conduc- " tance of T h e r m o e l e c t r i c G e n e r a t o r P a r d w a r e , " ASME P a p e r No. h6-WAINE-2, 1966.
3 . W i l l i a m s , A , . "Heat Flow Across S t a c k s of S t e e l . . L a m i n a t i o n s , " J o u r n a l o f ~&checal E n g i n e e r i n g S c i e n c e , 1 3 , No. 3 , 1971, pp. 217-233.
4. W i l l i a m s , A . , "Exper iments on t h e Flow o f Heat Across S t a c k s of S t e e l I . amina t ions ," of M e c h a n i c a l E n g i n e e r i n g S c i e n c e , 1 4 , No. 2 , 1 9 7 2 , pp. 151-154.
5. Jhamnani , J . , "Thermal and E l e c t r i c a l P rope r - t i e s of T r a n s f o r m e r I .amina t ions ," M.Sc. T h e s i s , C r a n f i e l d I n s t i t u t e of TechnoloRy, S c h o o l of M e c h a n i c a l E n g i n e e r i n g , 1974.
6 . A l - A s t r a b a d i , F.R., P r o h e r t , S . D . , O ' C a l l a g h a n , P.W. and Jones . A . M . . "Thermal R e s i s t a n c e s on S t a c k s of Thin Layers u n d e r Compress ion ," J o u r n a l o f M e c h a n i c a l E n g i n e e r i n g S c i e n c e , 1 9 , No. 4 , 1977, pp. 167-174.
7 . S h e f f i e l d , J . W . , V e e i r o g l u , T.N. and W i l l i a m s , A , , "An E x p e r i m e n t a l I n v e s t i g a t i o n of Thermal C o n t a c t Conductance of M u l t i l a y e r e d E l e c t r i c - a l l y I n s u l a t e d S h e e t s , " A I A A P a p e r N O . 79- 1067, 1979. ( t o appear i n &-ogress i n Astro- n a u t i c s and A e r o n a u t i c s S e r i e s ) .
8 . A I - A s t r a b a d i , F.R., O ' C a l l a g h a n , P.W., P r o b e r t , S.D. and Jones, A . X . , "Thermal C o n t a c t Conduct- ance C o r r e l a t i o n f o r S t a c k s of Thin L a y e r s i n High Vacuums," . Journa l o f B e a t T r a n s f e r , 99, F e b r u a r y 1977, pp. 1.39-142.
9 . O z i s i k , M . N . , Heat C o n d u c t i o n , John Ig i ley and Sons, I n c . , New York, 19x0.
IO. V e e i r o g l u , T.N., Williams, A . , Kakac, S. and Nayak, P., " P r e d i c t i o n and Measurement o f t h e Thermal Conductance of Laminated S t a c k s , " a- t e r n a t i o n a l Journal of Heat and Mass T r a n s f e r , - 2 2 , No. 3 , 1 9 7 9 , pp . 447-459.
11. V e z i r o g l u , T.N., H u e r t n , )!.A. and Kakac, S . , "Exac t S o l u t i o n s f o r Thermal Conductances of P l a n a r and C i r c u l a r C o n t a c t s , " i n Thermal Con- d u c t i v i t y , =, e d i t e d by Klemens, P.G. and Chu, T.K., pp. 435-448, Plenum P u b l i s h i n g Cor- p o r a t i o n , New York, 1976.
12. Fenech , H . and Rohsenow, l d . E I . , " P r e d i c t i o n of Thermal Conductance of E l e t a l l i c Surfaces i n C o n t a c t , " J o u r n a l o f Heat T r a n s f e r , 8 5 , Febru- ary 1963, pp. 15-24.
13. Laming, L .C. , "Thermal Conductance of Nachined M e t a l ~ Contac ts ," i n I n t e r n a t i o n a l Developments i n Heat T r a n s f e r , American S o c i e t y of Mechani- c a l Engineers, New York, 1963.
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1 4 . G.E.Hfat Transfer and F l u i d Flow Daeta&&, edited by Roecker, R.H., G e n e h l Electric Company, Schenectady, New York, 1976.
15. Veziroglu, T.N., "Correlation of Thermal Con- tact Conductance Experimental Results," in - P r s = in Astronautics and Aeronauticj, Eo& - 20, pp. 879-907, Academic Press Inc., Yew York, 1967.
15. Veziroglu, T.N., "Correlation of Thermal Con- tact Conductance Experimental Results," in - P r s = in Astronautics and Aeronauticj, Eo& - 20, pp. 879-907, Academic Press Inc., Yew York, 1967.
