arc-driven rail gun research - nasa · 18 variation of electron and plasma density with the plasma...
TRANSCRIPT
ARC-DRIVEN RAIL GUN RESEARCH
PREPARED K)R LEWIS RESEARCH CENTER
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NAG 3-76
Pradosh K. Ray Department of ~echanical Engineering
Tuskegee Institute Tuskegee Institute, Alabama
.- . . . .
REPRODUCED BY U.S. DEPARTMENT OF COMMERCE
NATIONAL TECHNICAL INFORMATION SERVICE' SPRINGFIELD, VA 22161
. . - - . . .
https://ntrs.nasa.gov/search.jsp?R=19850013352 2020-02-20T08:29:57+00:00Z
I - - - - -- .- - . - 1 , lierro~t No 2. Goverrlnien~ Access~on No. 3. A c c ~ p ~ e n l ' s Ca~aloy No.
blAy\-CR 174816 4 11t11: l l l l c l S,IIIIIII~
L 5, I l ~ , ~ > ~ b r t L1,11,!
Uecciilber 1 Yd5 Arc 3 r i vcri Rai 1 Gun l !escarc t~ e G. t'crfor~n~ny - - - C r g a ~ ~ ~ z ~ ; ~ o n -- ~ o r l c
---- 7. A ~ r l l o r ~ s l
Pradosh li. I?dy
9. Purlorrnlng Oryan~~ilttori Nanie and Address
I . : c c l i ~ r ~ i c a l E l iq i l i e e r i rig Uepart i i .ent Tus ieyee I r ~ s t i tuLe Tuskcgee I n s t i t u l e , AL 36033
1 2 S~rorisorinq Ail~,firy Nzrilc arid , ~ ~ l t l t , ? s s
tla t i o r ~ a l i r e r o l i a u l i c s arld S;race hdiiii n i st1.s ticill
Lcw i s Kesedrch Cer l te r 2 1032 8r'oo~.1~al 'k Road rii.~;~;illauL~ 011 114135
15. S u ~ ~ l > l e n t u ~ i l a r y Pdolrs
0. Per(orn~.rir, O:gsn~zat,on Rer,orr P,o
- 10 %'ark VIIII No
YOS 2415 RTO? 506-55-22
1 1 . Con~ract or Grant No. f4AG 3 - 7 6
13. T y w of Repu-1 and Perlod Covered Ar~riua 1 R e p o r t t o Cec., 1954
1 4 . Sponsoring Agency Code
LeHC Code 5323
I
G r a r ~ t i h n i t o r : L h l i 11 i a l ; ~ I;cl-s 1aI:r. Iid2S/'~-Lch i s ?.ese;l~.cIl Cerltc!r C l e v e l a n d , 011 5 4 \ 3 5
IG Abrrracr
r l ic c r l ~ r d l i o ~ ~ s dt !sc l . ib i I I ~ l i ~ e l:cr.ror-ll,;l l~cc o r at1 i ~ ~ d u c t i v ~ l y - d r i v e n r a i i sun a r e alld l y r c d i i b ~ i l e r i c d l I;'. F t - i cL i o r ~ L ~ e t u e c n t i l e p r o j e c t i l e arid r s i 1s i s i n c l b d e d t n r o u y h a11 ei:ipi r i c a l f o r i i l ~ ~ l a t i o r i . The ec lua t ions a r e a p p l i e d t o t l i e e x p e r i ~ i e n t o f R a s h l e i q h a i ~ t l I . la1-sl1~11 t o o b t a i r i ai l e s t i~ l ia lc ! o f eriel.!l;y d i s t l - i b u t i o n i n r a i l qhl is as a f u n c t i o n o r t i111c. T t ~ e c!fFect O F f t - i c : t i o r i d l t ~ e i l t d i s s i l ~ a t i o ~ ~ 011 t h e b o r e o f t h e gbrl i s ca I c u l a t e d i n t h e e x l i e r i ~ ~ i e n l : o f Lau~!r- e t a1 .
Ttie ~ : c c l \ a n i s ~ n o f ; i ias~i .d and [ ! I - o j e c t i l e a c c e l e r s t i o n i n a dc r a i l gun i s dcsc r - ibed f1.01.i a ~ i : i c r ~ o s c o l ? i c ~ ! o i r ~ t o f v iew Ll~roucj i l t h e e s t a b l i s h ~ ~ i c n t o f t i l e Hal 1 f i e l c l . Ttip p1as111a cc> i~c luc t i v i t , ; i i s sliown t u he a t e r ~ s o r i n d i c a t i r l g t h a t t h e r e i s a sllla I I c o ~ ~ , [ l o l ~ e ~ i t o f c u r r e n L para1 l e l t o t r l c d i r e c t i o r ~ o f acceleration. Tllc p1asll;a c t ! d r d c t e r i s t i c s a r c eva lu ;~ tcc l i n t t ~ c c s p c r i ~ ~ ~ c n : o f 3auer e t a i . a s a f u n c t i o n o f l , I , ~ s ~ i a III,I~S t11t-uug11 a 5 i l , i l > l ~ r l u i ~ l I~IL-:~I,III i i i l l a r l a l y s i s o f t t > e 111as1~1a. Ey c q u s t i n g t l ~ c cllel-!ly d i s s i l ! a t e d I l l t l ~ e 111asl:ia w i L h t h e I - a d i d t i o n h e a t l o s s , t h e p r o p e r t i e s o f Llie p l as,i~a a r e de ten i i i r l ed . I
I
17. Kev Y~o~rlr iSil(l!~c,red t ~ i . Aurhor(s) l 18 D~str~but~on S~a~ernun: Rrl i 1 C,,III L Ii!c t ~ . o i l ~ d ! j r ~ e l i c ACCC I il1.d t u ~ ' U r ~ c l a s s i f i e d - - 9 1 1 1 i i ~ i i Led I l i g h - ' i c l o c i t y P r o j c c t i l c A c c c l e r a t i r l g P1as1,ia 1.
19 Sccurilr Class11 lo1 1h8s reporti 20. S e c u r ~ t r C I d s s ~ l . (of r t 1 1 5 s,lqci U r i i l i l s ~ i f i e d 1 JIIC I d s s i f i e d 5 3
L -
21. No c l Pagcs 2 2 . Pr,ce'
TABLE OF CONTENTS
Top ic
A b s t r a c t
Energy P a r t i t i o n i n g i n an I n d u c t i v e l y - D r i v e n R a i l Gun
I n t r o d u c t i o n R a i l Gun Systern Mathemat ica l F o r m u l a t i o n Energy P a r t i t i o n i n g R e s u l t s and D i s c u s s i o n Conc lus ions Nomenclature
F r i c t i o n i n R a i l Guns
I n t r o d u c t i o n O r i g i n o f t h e R e t a r d i n g Force R e s u l t s D i s s i p a t i o n o f Energy Concl u s i on Nomencl a t u r e
P l a s m a - P r o j e c t i l e I n t e r a c t i o n i n an A r c - D r i v e n R a i l Gun
I n t r o d u c t i o n Mechanism o f A c c e l e r a t i o n Plasma C o n d u c t i v i t y C h a r a c t e r i z a t i o n o f Plasma
Induced Magnet ic F i e l d P ressure Temperature and Degree o f I o n i z a t i o n D e n s i t y C o n d u c t i v i t y and Res is tance
R e s u l t s and D i s c u s s i o n Conc lus ion Nomenclature
Page
i
References
D i s t r i b u t i o n L i s t
LIST OF FIGURES
T i t l e Page
5
6
12
F igu re No.
1
2
3
4
Schematic diagram o f a r a i 1 gun system
E q u i v a l e n t 1 umped-parameter e l e c t r i c c i r c u i t
A c c e l e r a t i o n l e n g t h as a f u n c t i o n o f t ime
Percentage o f energy d i s t r i b u t i o n as a f u n c t i o n o f t ime
E f f e c t o f a c c e l e r a t o r l e n g t h on p r o j e c t i l e e x i t v e l o c i t y
E f f i c i e n c y and p r o j e c t i l e e x i t v e l o c i t y as a f u n c t i o n o f (L0 /L1X)
Plasma r e s i s t a n c e and c u r r e n t as a f u n c t i o n o f t i me
Percentage d i s t r i b u t i o n o f r e s i s t i v e losses versus t ime
A c c e l e r a t i o n l e n g t h as a f u n c t i o n o f t ime
Lo ren t z f o r c e and f r i c t i o n f o r c e as a f u n c t i o n o f a c c e l e r a t i o n l e n g t h i n t he exper iment o f Bauer e t a1 .
Loren tz f o r c e and f r i c t i o n f o r c e versus acce l - e r a t i o n l e n g t h i n t h e exper iment o f Rashle igh and Marsha l l
F r i c t i o n a l energy d i s s i p a t i o n as a percentage o f t o t a l energy d e l i v e r e d t o t h e gun versus a c c e l e r a t i o n l e n g t h i n t h e exper iment o f Bauer e t a l .
Charge sepa ra t i on a t t h e edges o f plasma due t o a magnet ic f i e l d
M i g r a t i o n o f e l e c t r o n s i n t h e e l e c t r i c and magnet ic f i e l d s
Geometry o f plasma f o r model ing
V a r i a t i o n o f p ressure a long t h e l e n g t h o f t h e plasma and average pressure
F i g u r e No. T i t l e Page
17 Average plasnia tempera tu re and average degree o f i o n i z a t i o n as a f u n c t i o n o f plasma mass 54
18 V a r i a t i o n o f e l e c t r o n and plasma d e n s i t y w i t h t h e plasma mass 55
19 E f f e c t o f v a r i a t i o n o f plasma mass on plasma l e n g t h and c o n d u c t i v i t y 5 6
Tab le No.
1 I n p u t d a t a
LIST OF TABLES
T i t l e Page
9
2 I n p u t d a t a 3 3
3 I n p u t d a t a f o r e s t i m a t i n g plasma c o n d i t i o n s 5 8
4 E s t i m a t e d plasma parameters i n t h e exper imen t o f Bauer e t a l . 59
ENERGY PARTITIONING I N AN
INDUCTIVELY-DRIVEN
RAIL GUN
I n t r o d u c t i o n
There a r e a number o f a p p l i c a t i o n s t h a t r e q u i r e m a c r o p a r t i -
c l e s moving a t v e r y h i g h speeds. E q u a t i o n o f s t a t e measurements
can be ex tended w i t h s u i t a b l e p r o j e c t i l e t r a v e l l i n g a t speeds i n
excess o f 10 km/s [I]. E f f e c t s o f m i c r o m e t e o r o i d impac t on space
v e h i c l e s can be s i m u l a t e d a t speeds exceed ing 20 km/s [2 ] . Payloads
can be d i r e c t l y l aunched f r o m t h e e a r t h ' s s u r f a c e t o e a r t h o r b i t o r
beyond a t speeds i n t h e range o f 15 t o 25 km/s [3 ] whereas impact
f u s i o n m i g h t be a c h i e v a b l e a t speeds g r e a t e r t h a n 150 km/s [4 ] .
A l s o t h e r e a r e space p r o p u l s i o n a p p l i c a t i o n s [5 ] and many m i l i t a r y
a p p l i c a t i o n s [6 ] o f p r o j e c t i l e s mov ing ' a t h i g h speeds.
W i t h c u r r e n t t e c h n o l o g y , p r o j e c t i l e speeds i n excess o f
10 km/s can be o b t a i n e d o n l y by t h e use o f e l e c t r o m a g n e t i c ene rgy
[7 ] . The s i m p l e s t o f t h e e l e c t r o m a g n e t i c l a u n c h i n g d e v i c e s i s t h e
dc r a i l gun where t h e a c c e l e r a t i n g f o r c e i s t h e L o r e n t z f o r c e
r e s u l t i n g f r o m a c u r r e n t f l o w i n g o r t h o g o n a l l y t o a s e l f - g e n e r a t e d
magne t i c f i e l d . There has been a resu rgence o f i n t e r e s t i n r a i l guns
s i n c e R a s h l e i g h and M a r s h a l l ( h e r e a f t e r r e f e r r e d t o as R & M) success-
f u l l y a c c e l e r a t e d a 3 gm p r o j e c t i l e t o a v e l o c i t y o f 5 .9 km/s i n a
5 m l o n g r a i l gun by u s i n g a plasma a rmatu re and an i n t e r m e d i a t e
s t o r a g e i n d u c t o r f o r p u l s e shap ing [8]. I n d u c t i v e l y - d r i v e n r a i l
guns were found t o have s u p e r i o r per formance c h a r a c t e r i s t i c s t o t h o s e
1
2
d r i v e n w i t h o t h e r t y p e s o f power sources. I n t h e i n t e r v e n i n g y e a r s ,
a c o n s i d e r a b l e amount o f work has been done t o e x p l o r e v a r i o u s aspec ts
o f r a i l gun systems. L i m i t s o f r a i l gun per formance have been e v a l u a t e d
by Hawke e t a l . and v e l o c i t i e s up t o 10 km/s have been o b t a i n e d f o r 3 gm
p r o j e c t i l e s i n a 1 .8 m l o n g r a i l gun 191. Even so, t h e exper imen ts b y
R & M remain one o f t h e most s u c c e s s f u l ones t o da te . I n t h i s
a r t i c l e , t h e energy p a r t i t i o n i n g i n an i n d u c t i v e l y - d r i v e n r a i l gun
i s ana lyzed as a f u n c t i o n o f t i m e a f t e r t h e gun i s e n e r g i z e d and t h e
parameters o f R & M exper imen ts a r e used i n t h i s numer i ca l s i m u l a t i o n .
From a fundamental p o i n t o f v iew, t h e n a t u r e o f f r i c t i o n between t h e
p r o j e c t i l e and r a i l s was found t o be r a t h e r complex [ l o ] . I n v iew
o f t h i s , i t was e x p l o r e d i n t h i s s t u d y i f t h e e f f e c t o f f r i c t i o n on
r a i l gun per formance can be a d e q u a t e l y expressed t h r o u g h an e m p i r i c a l
f a c t o r as a f u n c t i o n o f t h e v e l o c i t y o f t h e p r o j e c t i l e . T h i s can be
shown t o be p r o p o r t i o n a l t o t h e square o f t h e p r o j e c t i l e v e l o c i t y .
However, a r e c e n t s t u d y i n d i c a t e d t h a t r a i l gun per formances c o u l d be
e x p l a i n e d by t a k i n g i n t o accoun t an i n c r e a s e i n t h e mass o f t h e a r c
as i t a b l a t e s m a t e r i a l s f r o m t h e r a i l s [Ill.
Rai 1 Gun System
The r a i l gun c o n s i s t s o f a p a i r o f para1 l e l conduc to rs
separa ted b y a d i s t a n c e and connected by a movable c o n d u c t o r . A
l a r g e dc c u r r e n t ( k i lo -amperes) f l o w s i n a s h o r t b u r s t f r o m one r a i l
t o t h e o t h e r t h r o u g h t h e i n t e r c o n n e c t i n g conduc to r . The i n t e r c o n n e c t -
i n g c o n d u c t o r i s n o r m a l l y a t h i n m e t a l l i c f u s e wh ich becomes a plasma
when t h e l a r g e c u r r e n t i s d i s c h a r g e d t h r o u g h i t .
3
The c u r r e n t f l o w i n g i n t h e r a i l s genera tes a magne t i c f i e l d
B between t h e r a i l s and t h i s magne t i c f i e l d i n t e r a c t s w i t h t h e
c u r r e n t f l o w i n g i n t h e a rma tu re . The r e s u l t i n g L o r e n t z f o r c e J x - B
a c t i n g on t h e a rma tu re a c c e l e r a t e s t h e plasma a l o n g t h e r a i l s . I f
t h e plasma i s c o n f i n e d b e h i n d a p r o j e c t i l e made o f d i e l e c t r i c
m a t e r i a l s , t h e p r e s s u r e o f plasma w i l l a c c e l e r a t e t h e p r o j e c t i l e
a l o n g w i t h t h e plasma. The con f inemen t o f t h e plasma can be p r o v i d e d
by t h e c o n d u c t i n g r a i l s on two s i d e s and d i e l e c t r i c m a t e r i a l on t h e
o t h e r two s i d e s . D u r i n g l a u n c h o p e r a t i o n s t h e r e a r e h i g h peak ing
l o a d s wh ich can be met w i t h a s u i t a b l e energy s t o r a g e system. The
power sources t h a t a r e c u r r e n t l y b e i n g used t o s u p p l y t h e p r i m a r y
energy t o r a i 1 guns i n c l u d e homopolar g e n e r a t o r [8], c a p a c i t o r bank
[12] and magne t i c f l u x compress ion g e n e r a t o r [ l o ] .
The r a i l gun f u n c t i o n s e s s e n t i a l l y as a l i n e a r dc mo to r .
The plasma behaves as an a rma tu re w h i l e t h e p a r a l l e l r a i l s se rve as
a s i n g l e - t u r n f i e l d w i n d i n g i n s e r i e s w i t h t h e a rma tu re . Hence, t h e
r a i l gun r e q u i r e s a l o w v o l t a g e , h i g h c u r r e n t power sou rce . The
f o r c e F a c t i n g on t h e p r o j e c t i l e can be r e p r e s e n t e d by
I d e a l l y , f o r c o n s t a n t a c c e l e r a t i o n a c o n s t a n t c u r r e n t i s r e q u i r e d
and i f an i n d u c t o r i s used t o d e l i v e r t h e energy t o t h e gun, a
c l o s e approach t o t h i s i d e a l can be r e a l i z e d [ 5 ] . The i n d u c t o r i s
a l s o a b l e t o overcome t h e back emf w h i c h i n c r e a s e s w i t h t h e i n c r e a s e
i n v e l o c i t y o f t h e p r o j e c t i l e [ 6 ] .
4
A schemat ic d iagram o f a r a i l gun sys tem i s shown i n F i g . 1.
When s w i t c h e s S1 and S3 a r e c l o s e d and S2 i s open, a c u r r e n t i s
genera ted i n t h e s t o r a g e i n d u c t o r from an energy s t o r a g e d e v i c e .
A f t e r t h e d e s i r e d c u r r e n t i s e s t a b l i s h e d i n t h e i n d u c t o r , s w i t c h e s
S1 and S3 a r e opened and S2 i s c l o s e d . The p r o j e c t i l e i s l o c a t e d
a t t h e b reech o f t h e gun i n t h e b e g i n n i n g , i . e . , a t t = 0, x = 0 .
The f u s e a t t h e back o f t h e p r o j e c t i l e a l l o w s t h e c u r r e n t t o con-
t i n u e t o f l o w u n t i l i t v a p o r i z e s and e s t a b l i s h e s t h e a r c wh ich
a c c e l e r a t e s w i t h t h e p r o j e c t i l e a l o n g t h e r a i l s . J u s t p r i o r t o t h e
emergence o f t h e a r c f r o m t h e muzzle o f t h e gun, t h e s w i t c h S4 i s
c l o s e d t o l e t t h e c u r r e n t f l o w t h r o u g h a r e s i s t o r t o e x t i n g u i s h t h e
a r c .
Ma themat i ca l F o r m u l a t i o n
The e q u a t i o n s d e s c r i b i n g t h e per formance o f t h e r a i l gun a r e
d e r i v e d b y u s i n g t h e c o n v e n t i o n a l lumped-parameter c i r c u i t a n a l y s i s
t e c h n i q u e . The e q u i v a l e n t e l e c t r i c c i r c u i t o f a r a i l gun i s shown
i n F i g . 2. C i r c u i t r e s i s t i v e l o s s e s a r e assumed t o be n e g l i g i b l e
i n t h i s a n a l y s i s .
By a p p l y i n g K i r c h o f f ' s l aw , t h e d i f f e r e n t i a l e q u a t i o n govern-
i n g t h e c i r c u i t b e h a v i o r i s o b t a i n e d as
d - { ( L o + L 1 x ) 1 ) + (Ro + R ' x + R )I = 0 d t P
d I dx o r , ( Lo d t + L ' x ) - + ( R + R ' x + R + L ' -)I = 0
0 P d t ( 3
7
The mot ion o f t he p r o j e c t i l e i s desc r ibed by,
The mass o f t h e plasma i s neg lec ted as i t i s much sma l l e r than t h e
mass o f t he p r o j e c t i l e . I t i s assumed t h a t t he f r i c t i o n f o r c e can
be adequate ly represen ted by
The s k i n e f f e c t con f i nes t he c u r r e n t t o a t h i n sheet on t he r a i l
su r f ace d u r i n g t h e i n i t i a l p a r t o f t h e a c c e l e r a t i o n . By assuming
a s t ep -cu r ren t d i f f u s i n g i n t o a conductor , t he s k i n depth can be
represen ted by [13 ]
then,
where D i s equal t o t h e minimum o f d o r ~ ( t ) .
Equat ions 3 t o 7 r ep resen t a h i g h l y coupled n o n l i n e a r
system. The i n i t i a l c o n d i t i o n s a re g i ven by
and = 0
8
The e q u a t i o n s a r e s o l v e d n u m e r i c a l l y by u s i n g a f i n i t e d i f f e r e n c e
a p p r o x i m a t i o n w i t h a t i m e i n t e r v a l o f 20 bsec. The f i n i t e d i f f e r -
ence e q u a t i o n s a re ,
The v a l u e o f b and n a r e a d j u s t e d t o p r o v i d e a good f i t t o t h e
e x p e r i m e n t a l d a t a o f R & M. The muzzle v o l t a g e o f t h e gun was
found t o remain c o n s t a n t . Hence, t h e v a l u e o f plasma r e s i s t a n c e i s
c a l c u l a t e d f r o m
The va lues o f t h e parameters used i n s o l v i n g these e q u a t i o n s a r e
l i s t e d i n T a b l e 1 . I t was found t h a t va lues o f b = 2.26 x kg/m
and n = 2 p r o v i d e a good f i t t o t h e exper in ien ta l d a t a . The v a l u e s
o f b and n can t h e n be used i n e q u a t i o n 5 t o e s t i m a t e t h e f r i c t i o n
f o r c e .
Tab le 1
I n p u t Data
Parameter Value
22 pH
0.42 pH/m
3 !3
150 v o l t
160 pohm
1400 psec
-1 -1 0.58 x l o 8 ohm -rn
10
Energy P a r t i t i o n i n g
A t t he beg inn ing o f t h e launch c y c l e , t he s to rage i n d u c t o r
i s charged from the p r imary power supply . When t h e energy s t o r e d
i n t h e magnet ic f i e l d o f t he s to rage i n d u c t o r i s d e l i v e r e d t o t h e
gun, i t i s p a r t i t i o n e d i n f o u r ways;
a ) a p a r t o f i t i s d i s s i p a t e d r e s i s t i v e l y i n t he r a i l s ,
i n t he plasma and i n t he r e s i s t a n c e o f t h e s to rage i n d u c t o r ,
b ) a p a r t o f i t i s s t o r e d i n t h e magnet ic f i e l d o f t h e
r a i l s ,
c ) a p a r t o f i t i s s t o r e d i n t h e form o f t h e k i n e t i c
energy o f t he p r o j e c t i l e , and
d ) a p a r t o f i t i s l o s t i n f r i c t i o n between t he r a i l s
and t he p r o j e c t i l e .
T h e i r magnitudes a re g i ven by
and E f = E - (Er t Ei + Ek)
Lo 2 2 where E = - ( I - I f ) 2 0
11
Resu l t s and D iscuss ion
The p o s i t i o n o f t h e p r o j e c t i l e as a f u n c t i o n o f t ime i s
shown i n F i g . 3 i n s o l i d l i n e . For comparison, t h e exper imenta l
da ta p o i n t s o f R & M a re shown i n s o l i d c i r c l e s . The t h e o r e t i c a l
c a l c u l a t i o n by R & M w i t h o u t f r i c t i o n i s a l s o shown i n F i g . 3 i n
broken l i n e s . I t i s c l e a r t h a t t h e f r i c t i o n f o r c e can be adequate ly
represen ted by a s imple f u n c t i o n p r o p o r t i o n a l t o t he square o f t he
v e l o c i t y o f t h e p r o j e c t i l e .
The energy s u p p l i e d by t h e i n d u c t o r t o t he gun i s shown i n
F i g . 4 where t h e percentage o f o r i g i n a l s t o r e d energy remain ing i n
t he i n d u c t o r i s p l o t t e d a g a i n s t t ime . It i s found t h a t o n l y 37% o f
t he energy o f t h e i n d u c t o r i s supp l i ed t o t h e gun. A lso shown i n
F i g . 4 a re t h e percentage d i s t r i b u t i o n s o f t h e d e l i v e r e d energy as
a f u n c t i o n o f t ime as i t i s p a r t i t i o n e d i n f o u r ways. Dur ing t h e
i n i t i a l phase o f t h e a c c e l e r a t i o n , most o f t h e losses a re i n
r e s i s t i v e hea t i ng . The percentages o f energy t ha t . . go t o t h e p r o j e c -
t i l e and t o t h e magnet ic f i e l d o f t h e r a i l s inc rease a t t h e beginn ing,
then l e v e l o f f and f i n a l l y decrease. Th i s i n d i c a t e s an approach t o
v e l o c i t y s a t u r a t i o n . The energy l o s t i n f r i c t i o n inc reases mono-
t o m i c a l l y w i t h t ime . The maximum energy l o s s occurs i n t h e r e s i s -
tances, amounting t o 62% o f t h e t o t a l energy supp l i ed by t h e i n d u c t o r .
A t t h e end o f t h e a c c e l e r a t i o n , 16% o f t h e energy s u p p l i e d t o t h e gun
remains s t o r e d i n t h e magnet ic f i e l d o f t h e r a i l s , whereas o n l y 15%
o f t he energy i s conver ted i n t o t he k i n e t i c energy o f t h e p r o j e c t i l e .
The f r i c t i o n a l l o s s accounts f o r t h e remain ing 7% o f t he energy.
Fig. 3.
5- 'Experimental Data
I - 3 - I I- w Z W J
- I- 4 a W
T IME, MICRO-SECOND, x lo2
Acceleration length as a function o f time
0 b=
m W - arn <+ k?
5"
-* -A- INDUCTOR - 0 - RCCICT
-RAILS (MAGNETIC) --- PROJECTILE KE - -- - FRICTION
TIME, MICRO-SECOND ,XI 0*
Fig . 4. Percentage o f energy d i s t r i b u t i o n as a f u n c t i o n o f t ime
14
The v e l o c i t y s a t u r a t i o n process can be seen c l e a r l y i n F i g . 5,
where t h e e x i t v e l o c i t y o f t h e p r o j e c t i l e i s p l o t t e d a g a i n s t t h e
a c c e l e r a t o r l e n g t h . I t i s observed t h a t as t h e a c c e l e r a t o r l e n g t h
i s i n c r e a s e d , t h e f i n a l v e l o c i t y goes t h r o u g h a maximum, i n t h i s case
6.2 km/s.
The e f f i c i e n c y o f e l e c t r i c t o k i n e t i c energy c o n v e r s i o n based
on t h e energy s t o r e d i n t h e i n d u c t o r i s ' o n l y 5.32. T h i s i s due t o
t h e f a c t 63% o f t h e energy s t o r e d i n t h e i n d u c t o r remains u n d e l i v e r e d
t o t h e gun. Hence, i t i s d e s i r a b l e t o have l o w e r va lues o f Lo. I n
F i g . 6, t h e e f f i c i e n c y o f t h e gun i s p l o t t e d w i t h r e s p e c t t o
( L 0 / L 1 X ) . It i s observed t h a t t h e e f f i c i e n c y i s improved a t s m a l l e r
va lues o f Lo. However, t h e e x i t v e l o c i t y o f t h e p r o j e c t i l e a l s o
becomes somewhat s m a l l e r a t s m a l l e r v a l u e s o f Lo and i t s t a r t s t o
s a t u r a t e a t abou t ( L 0 / L 1 X ) = 10 ( F i g . 6 ) . Thus, a compromise has
t o be made between t h e e f f i c i e n c y o f t h e r a i l gun and t h e f i n a l
v e l o c i t y o f t h e p r o j e c t i l e .
The c u r r e n t i n t h e gun d rops f r o m an i n i t i a l v a l u e o f 300 KA
t o 239 KA ( F i g . 7 ) , a decrease o f 20X, wh ich has a1 so been observed
i n t h e exper imen ts . Cor respond ing ly , t h e plasma r e s i s t a n c e
i n c r e a s e s f r o m an i n i t i a l v a l u e o f 500 pohm t o 625 pohm ( F i g . 7 ) .
I t i s found t h a t t h e c u r r e n t d i f f u s e s c o m p l e t e l y i n t o t h e
r a i l s i n 260 gsec. I f s k i n d e p t h i s i g n o r e d , t h e t o t a l r e s i s t i v e
l o s s e s a t t h e end o f a c c e l e r a t i o n amount t o 55% o f t h e energy
s u p p l i e d b y t h e i n d u c t o r , as opposed t o 62% when s k i n d e p t h i s t a k e n
i n t o accoun t .
ACCELERATOR LENGTH, M
Fig. 5. Effect of accelerator length on projectile exit velocity
EFFICIENCY (010)
F i g . 7 . Plasma r e s i s t a n c e and c u r r e n t a s a f u n c t i o n o f t i m e
18
The r e s i s t i v e l o s s e s o c c u r i n t h e r a i l s , i n t h e plasma and .
i n t h e i n d u c t o r . The pe rcen tage d i s t r i b u t i o n s o f r e s i s t i v e l o s s e s
i n each o f t h e s e components a r e shown i n F i g . 8 . I t i s observed
t h a t i n i t i a l l y t h e r e s i s t i v e l o s s e s i n t h e plasma dominate ( -65% o f
t o t a l r e s i s t i v e l o s s e s a t 200 psec ) b u t w i t h t i m e t h e r e s i s t i v e l o s s
i n t h e r a i l s b e g i n s t o i n c r e a s e . A t t h e end o f a c c e l e r a t i o n , 85%
o f t o t a l r e s i s t i v e l o s s e s o c c u r i n t h e r a i l s .
Conc lus ions
I t i s observed t h a t t h e r a i l gun i s a l ow e f f i c i e n c y
a c c e l e r a t i o n d e v i c e . I n t h e exper imen ts o f R & M, o n l y 15% o f
t h e energy s u p p l i e d t o t h e gun i s t r a n s f o r m e d i n t o t h e k i n e t i c
energy o f t h e p r o j e c t i l e . Much o f t h e energy o r i g i n a l l y s t o r e d i n
t h e i n d u c t o r remains u n d e l i v e r e d t o t h e gun a l t h o u g h i n p r i n c i p l e
t h i s energy can be recove red . R e s i s t i v e l o s s e s accoun t f o r t h e
l a r g e s t amount o f energy d i s s i p a t i o n and t h i s a l o n e tends t o l i m i t
t h e l e n g t h of t h e r a i l guns and hence, t h e maximum v e l o c i t y o f t h e
p r o j e c t i l e .
100- - RAILS --- PLASMA --- INDUCTOR
80 - \ \ \
\ \ 1 - - -- ---
- - - - - - - - -
TIME, MICRO-SECOND, XI o2 Fig. 8. Percentage distribution o f resistive losses versus time
20
Nomenclature
b = P r o p o r t i o n a l i t y c o n s t a n t f o r t h e f r i c t i o n f o r c e
d = Th ickness o f t h e r a i l s
E = T o t a l energy s u p p l i e d by t h e i n d u c t o r
Ef = Energy l o s t i n f r i c t i o n between t h e r a i l s and t h e p r o j e c t i l e
Ei = Energy s t o r e d i n t h e magnet ic f i e l d o f t h e r a i l s
Ek = K i n e t i c energy o f t h e p r o j e c t i l e
Er = R e s i s t i v e energy l o s s i n t h e r a i l s
F = Force on t h e p r o j e c t i l e
F = F r i c t i o n f o r c e f
I = C u r r e n t i n t h e gun a t any t i m e
If = C u r r e n t i n t h e gun a t t h e end o f a c c e l e r a t i o n
I. = C u r r e n t i n t h e gun a t t h e b e g i n n i n g o f a c c e l e r a t i o n
Lo = Induc tance o f s t o r a g e i n d u c t o r
L ' = Induc tance g r a d i e n t o f t h e r a i l s
M = Mass o f t h e p r o j e c t i l e
MV = Muzzle v o l t age
Ro = Res is tance o f t h e s t o r a g e i n d u c t o r
R = Plasma r e s i s t a n c e P
R ' = Res is tance p e r u n i t l e n g t h o f t h e p a i r o f r a i l s
t = Time
te = Time o f e x i t o f t h e p r o j e c t i l e
v = V e l o c i t y o f t h e p r o j e c t i l e
w = Width o f t h e r a i l s
x = D i s t a n c e t r a v e l l e d by t h e p r o j e c t i l e
X = Leng th o f t h e r a i l gun
2 1
5 = Skin depth
p = Permeability of the rails
o = Conductivity of the rails
FRICTION I N RAIL GUNS
I n t r o d u c t i o n
The v e l o c i t y o f t h e p r o j e c t i l e i s t h e o r e t i c a l l y g i v e n b y
However, r a i l gun exper imen ts have y i e l d e d c o n s i s t e n t l y l o w e r p r o -
j e c t i l e v e l o c i t i e s t h a n a r e t o be expec ted f rom e q u a t i o n 19. T h i s
2 i n d i c a t e s t h a t t h e f o r c e F ( t ) = 4 L ' I ( t ) i s n o t f u l l y e f f e c t i v e i n
a c c e l e r a t i n g t h e p r o j e c t i l e . Plasma leakage around t h e p r o j e c t i l e
c o u l d be p a r t i a l l y r e s p o n s i b l e f o r t h i s . However, t h e m a j o r cause
o f reduced a c c e l e r a t i o n c o u l d be due t o f r i c t i o n between t h e
p r o j e c t i l e and b o r e s u r f a c e s .
A c c u r a t e d e s c r i p t i o n o f t hese r e l a t i v e l y l a r g e f r i c t i o n
f o r c e s i n t i m e p e r i o d s o f t h e o r d e r o f m i l l i seconds i s e x t r e m e l y
d i f f i c u l t . As shown e a r l i e r , t h e e f f e c t o f f r i c t i o n on r a i l gun
per formance can be a d e q u a t e l y exp ressed t h r o u g h an e m p i r i c a l formu-
l a t i o n as a f u n c t i o n p r o p o r t i o n a l t o t h e square o f t h e p r o j e c t i l e
v e l o c i t y . I t s h o u l d be n o t e d t h a t s e m i - e m p i r i c a l f o r m u l a t i o n s a r e
a l s o used i n t h e i n t e r i o r b a l l i s t i c a n a l y s i s o f c o n v e n t i o n a l chemica l
p r o p e l 1 a n t d r i v e n guns [I 41.
I n t h i s s e c t i o n , t h e exper imen t o f Bauer e t a l . i s s i m u l a t e d
u s i n g e q u a t i o n s 3 t h r o u g h 13. I n t h i s exper imen t , a 6 .3 mm x 3.8 mm
x 6.3 mm p a r a l l e l o p i p e d (0 .2 g mass), made o f po l yca rbona te , was
a c c e l e r a t e d t o a v e l o c i t y o f 2 km/s i n t h e f i r s t 0 . 4 m o f a 3 m l o n g
2 2
2 3
gun, when t h e a c c e l e r a t i o n o f t h e p r o j e c t i l e ceased a p p a r e n t l y due
t o e x c e s s i v e f r i c t i o n between t h e p r o j e c t i l e and t h e gun [12 ] .
O r i g i n o f t h e R e t a r d i n g Force I
As opposed t o t h e o r d i n a r y guns where t h e p r o j e c t i l e base
p r e s s u r e i s r e l a t i v e l y low, t h e p r e s s u r e i n t h e r a i l gun exceeds
t h e y i e l d s t r e n g t h o f t h e p r o j e c t i l e m a t e r i a l . Hence, t h e p r o j e c -
t i l e undergoes p l a s t i c d e f o r m a t i o n and expands a g a i n s t t h e b o r e o f
t h e gun. The r e s u l t a n t i n i t i a l c o n t a c t f r i c t i o n soon g i v e s way t o
t h e f o r m a t i o n o f a nu1 t i p h a s e ( s o l i d p a r t i c l e and gas) e r o s i v e
v i s c o u s boundary l a y e r where t h e f r i c t i o n a l ene rgy i s d i s s i p a t e d
110, 151.
Ano the r source o f t h e r e t a r d i n g f o r c e i s due t o t h e i n ipu l -
s i v e l o a d i n g o f t h e r a i l gun s t r u c t u r e by t h e magne t i c r e p u l s i v e
f o r c e s on t h e r a i l s f r o m t h e passage o f t h e c u r r e n t . The dynamic
s t r e s s genera ted i n t h e s t r u c t u r e causes d i sp lacemen ts o f t h e r a i l s
and t h e s i d e w a l l s ahead o f t h e p r o j e c t i l e [16 ] . The r e s u l t a n t
p i n c h i n g o f t h e p r o j e c t i l e c o u l d be severe enough t o s u f f i c i e n t l y
reduce t h e a c c e l e r a t i o n o f t h e p r o j e c t i l e .
R e s u l t s
The va lues o f t h e parameters used i n s o l v i n g t h e r a i l gun
c i r c u i t e q u a t i o n s a r e l i s t e d i n T a b l e 2 . I t was found t h a t a v a l u e
o f b = 8.65 x kg/m p r o v i d e s a good f i t t o t h e e x p e r i m e n t a l d a t a .
2 4
The p o s i t i o n o f t h e p r o j e c t i l e as a f u n c t i o n o f t i m e i s
shown i n F i g . 9 i n s o l i d 1 i n e . F o r compar ison, t h e e x p e r i m e n t a l
d a t a p o i n t s o f Bauer e t a l . a r e shown i n t h i s f i g u r e i n s o l i d
c i r c l e s . The t h e o r e t i c a l c a l c u l a t i o n w i t h o u t f r i c t i o n i s a l s o
shown i n t h i s , f i g u r e i n b roken l i n e s .
F i g u r e 10 shows t h e v a r i a t i o n o f t h e L o r e n t z f o r c e and t h e
f r i c t i o n f o r c e as a f u n c t i o n o f t h e d i s t a n c e t r a v e l l e d b y t h e p r o -
j e c t i l e . The two f o r c e s a r e dqua l a f t e r o n l y 0 .4 m o f p r o j e c t i l e
t r a v e l . Fo r compar ison, t h e f r i c t i o n f o r c e i s 66% o f t h e L o r e n t z
f o r c e a t t h e muzzle o f t h e gun i n R a s h l e i g h and M a r s h a l l ' s e x p e r i -
ment ( F i g . 11): I n t h e fo rmer case, t h e energy d i s s i p a t e d ' i n
f r i c t i o n i s 9% o f t h e energy s u p p l i e d t o t h e gun when t h e p r o j e c t i l e
has advanced by 0 . 4 m f r o m t h e b reech p o s i t i o n ( F i g . 1 2 ) . I n t h e
l a t t e r case, t h e f r i c t i o n l o s s amounts t o 7% o f t h e t o t a l energy . ,
d e l i v e r e d t o t h e gun.
D i s s i p a t i o n o f Energy
How i s t h e f r i c t i o n a l ene rgy d i s s i p a t e d ? By t a k i n g i n t o
accoun t t h e development o f an e r o s i o n p r o d u c t , C o u e t t e - l i k e v i scous
boundary l a y e r between t h e r a i l and t h e p r o j e c t i l e s u r f a c e , Buck ing -
ham has demonst ra ted t h a t t h e p r o j e c t i l e w i l l s u f f e r t h e l o s s o f
o n l y a sma l l f r a c t i o n o f i t s o r i g i n a l mass due t o d i s s i p a t i o n o f
d r a g energy [ l o , 151. Obv ious l y , t h e f r i c t i o n a l ene rgy has t o be
d i s s i p a t e d t h r o u g h t h e r a i l gun s t r u c t u r e .
Time, micro-second, x l o2
F i g . 9. A c c e l e r a t i o n l e n g t h as a f u n c t i o n o f t i m e . So l i d c i r c l e s r e p r e s e n t e x p e r i m e n t a l d a t a o f Bauer e t a1 . [12] . Broken 1 i n e s r e p r e s e n t c a l c u l a t i o n s w i t h o u t f r i c t i o n .
A c c e l e r a t i o n length, rn
5 -
F i g . 10, L o r e n t z f o r c e and f r i c t i o n f o r c e a s a f u n c t i o n o f a c c e l e r a t i o n l e n g t h i n t h e exper imen t o f Bauer e t a l .
4
3
2
I -
0,
- c
d @ # - 0
/ /
0
- /
/ I L o r e n t z f o r c e
/ I - - - - - F r i c t i o n f o r c e - f
I /
/ - I
I I
I I
I I
I 1 I 1
0 0. I 0.2 0.3 0.4
A c c e l e r a t i o n length, m
F i g . 11. L o r e n t z f o r c e and f r i c t i o n f o r c e versus a c c e l e r a t i o n l e n g t h i n t h e exper iment o f Rash le igh and Marsha l l
0 0.1 0.2 0.3 0 .4
A c c e l e r a t i o n length, m
F i g . 12. F r i c t i o n a l ene rgy d i s s i p a t i o n as a pe rcen tage o f t o t a l ene rgy d e l i v e r e d t o t h e gun ve rsus a c c e l e r a t i o n l e n g t h i n t h e exper imen t o f Bauer e t a l .
I n t h e e x p e r i m e n t o f Bauer e t a l . , t h e p r o j e c t i l e ceased t o
i n c r e a s e i n v e l o c i t y a f t e r a s h o r t p e r i o d o f a c c e l e r a t i o n . No s i g -
n i f i c a n t l eakage o f plasma t o t h e f r o n t o f t h e p r o j e c t i l e was
d e t e c t e d f r o m t h e magne t i c f l u x probe measurements. To e x p l a i n
t h e l o s s o f a c c e l e r a t i o n , one h y p o t h e s i s advanced was t h a t t h e
c u r r e n t must have commutated o u t o f t h e a rma tu re i n t o a secondary
c u r r e n t p a t h w i t h i n t h e l a u n c h e r . However, no c o n v i n c i n g p r o o f
was l a t e r f ound t o s u p p o r t t h i s h y p o t h e s i s . A l though , i t i s p o s s i -
b l e t h a t a s m a l l f r a c t i o n o f t h e c u r r e n t d e l i v e r e d t o t h e gun was
shunted f r o m t h e a rma tu re i n t o a s p u r i o u s pa th , i t i s d o u b t f u l
t h a t t h e e n t i r e c u r r e n t has done so.
The more l i k e l y e x p l a n a t i o n o f t h e l o s s o f a c c e l e r a t i o n i s
due t o t h e e x c e s s i v e d r a g between t h e p r o j e c t i l e and t h e l a u n c h e r
s u r f a c e s . A l t h o u g h t h e reasons a r e n o t e n t i r e l y c l e a r , s m a l l b o r e
r a i l guns a p p a r e n t l y o f f e r more r e s i s t a n c e s compared t o l a r g e b o r e
r a i l guns, as ev idenced by t h e l o s s o f a c c e l e r a t i o n i n o t h e r s m a l l
b o r e r a i l gun exper imen ts [I 71.
The e f f e c t s o f t h e f r i c t i o n a l ene rgy d i s s i p a t i o n on t h e b o r e
sur faces a r e ana lyzed i n t h i s paper f o r t h e exper imen t o f Bauer
e t a l . On ly an approx ima te p h y s i c a l d e s c r i p t i o n o f t h e p rob lem i s
a t t e m p t e d h e r e as t h e d e t a i l s a r e t o o c o m p l i c a t e d . F o r t h i s purpose,
i t i s assumed t h a t a q u a s i - s t e a d y s t a t e has been e s t a b l i s h e d i n t h e
p r o j e c t i l e r e s t f rame.
When t h e p r o j e c t i l e s topped a c c e l e r a t i n g , t h e d r a g f o r c e
3 was e s t i m a t e d t o be 4 x 10 N and t h e p r o j e c t i l e v e l o c i t y was
3 2 x 10 m/s. I f one i d e n t i f i e s a s p e c i f i c e lement o f t h e w a l l
30
.equal t o t h e l e n g t h o f t h e p r o j e c t i l e i n t h e d i r e c t i o n o f mo t ion ,
t h e energy d i s s i p a t e d i n t r a v e r s i n g t h i s l e n g t h i s 24 J . T h i s amount
o f ene rgy b y i t s e l f i s n o t s i g n i f i c a n t l y l a r g e ( t h e amount o f ene rgy
genera ted i n an e q u i v a l e n t l e n g t h o f r a i l by J o u l e h e a t i n g i s
8 x l o 4 J ) . B u t , because o f t h e e x t r e m e l y s h o r t p e r i o d o f t i m e
i n wh ich t h i s energy i s l i b e r a t e d , t h e h e a t f l u x on t h e b o r e s u r -
f aces becomes r a t h e r l a r g e . Assuming t h a t a l l t h e f r i c t i o n a l ene rgy
i s u n i f o r m l y d i s s i p a t e d t h r o u g h t h e s u r f a c e s , t h e h e a t f l u x i s found
2 t o be 6 .5 x 10'' W/m based on t h e i n s t a n t a n e o u s c o n t a c t a rea o f t h e
p r o j e c t i l e w i t h t h e b o r e s u r f a c e s .
The p e n e t r a t i o n o f h e a t i n t o a s u r f a c e can be r o u g h l y
e s t i m a t e d f r o m
The the rma l ene rgy d i s s i p a t e d i n t i m e t i s q " t . Thus, t h e s p e c i f i c
t he rma l energy o f t h e hea ted zone i s
C o n s e r v a t i v e l y assuming t h a t t h e h e a t f l u x i s a p p l i e d t o t h e s u r f a c e
e lement f o r a t i m e p e r i o d o f 3 .us t h a t i t t a k e s t h e p r o j e c t i l e t o
t r a v e r s e t h i s l e n g t h , t h e s p e c i f i c t he rma l energy o f t h e copper
r a i l s i s UCu = 1186 J / g and t h a t o f t h e f i b e r g l a s s s i d e w a l l s i s
4 = 6.1 x 10 J /g . Hence t h e su r face o f t h e r a i 1s w i 11 be
r a i s e d t o a tempera tu re wh ich i s beyond t h e m e l t i n g p o i n t o f copper .
The g l a s s s u r f a c e s w i l l v a p o r i z e .
3 1
A p p l y i n g t h e t h e o r y o f one d imens iona l m e l t i n g o f a h a l f -
space s u b j e c t e d t o a s t e p f u n c t i o n h e a t i n p u t a t t h e s u r f a c e [18]
The t h i c k n e s s o f t h e zone where t h e tempera tu re i s beyond t h e
m e l t i n g tempera tu re i s g i v e n b y
Fo r t h e copper r a i l s tm = 0.5 p s and z = 5 pm. However, m e l t i n g 1
i s a t ime- tempera tu re r e a c t i o n and w i l l o c c u r o n l y under e q u i l i b r i u m
c o n d i t i o n s . The s u r f a c e m a t e r i a l o f t h e r a i l s i s n o t n e c e s s a r i l y
m o l t e n . Under t h e s e c o n d i t i o n s , t h e r a i l s u r f a c e i s n o t expec ted
t o recede and i t s t e m p e r a t u r e w i l l be reduced once t h e h e a t source
moves beyond t h e s p e c i f i e d s u r f a c e e lemen t .
Fo r t h e g l a s s s i d e w a l l s , t h e t i m e f o r t h e i n i t i a t i o n o f
v a p o r i z a t i on can be c a l c u l a t e d by assuming one d imens iona l h e a t
f l o w , a c o n t i n u a l l y v a p o r i z i n g s u r f a c e w i t h c o n s t a n t h e a t i n p u t
a t t h e s u r f a c e and c o n t i n u e d removal o f t h e v a p o r i z e d m a t e r i a l f r o m
t h e s u r f a c e [19]. Thus
The dep th o f m a t e r i a l removed by v a p o r i z a t i o n i s g i v e n by
For g l a s s , t a k i n g Tv = 3300°K and L = 20,000 J/g, one f i n d s
v M 4 ns and z M 3.4 urn. 2
Concl u s i on
The d r a g f o r c e i n a r a i l gun can be adequa te l y d e s c r i b e d
2 by an e m p i r i c a l r e l a t i o n Ff = bv . Obv ious l y , t h e va lues o f b w i l l
depend on t h e geometry and m a t e r i a l c o m p o s i t i o n as w e l l as t h e
s t r u c t u r a l i n t e g r i t y o f t h e r a i l gun. No genera l p h y s i c a l model
has y e t been deve loped f o r f r i c t i o n i n t h i s v e l o c i t y and p r e s s u r e
range t o i n t e r p r e t t h e observed da ta .
Table 2
I n p u t Data
Parameter Value
0 .2 g
200 v o l t
650 uohrn
250 psec
6 .35 mrn
4n x l o m 7 H/m
8 -1 -1 0 .58 x 10 ohm -m
34
Nomenclature
b = P r o p o r t i o n a l i t y c o n s t a n t f o r t h e f r i c t i o n f o r c e
c = S p e c i f i c h e a t
Ff = F r i c t i o n f o r c e
I = C u r r e n t i n t h e r a i l gun a t any t i m e
k = Thermal c o n d u c t i v i t y
L = L a t e n t h e a t o f v a p o r i z a t i o n
L ' = Induc tance g r a d i e n t o f t h e r a i l s
M = Mass o f t h e p r o j e c t i l e
q" = Heat f l u x
t = T i m e
T = Temperature
U = S p e c i f i c Thermal Energy
v = V e l o c i t y o f t h e p r o j e c t i l e
z1 = Th ickness o f me1 t
z2 = Th ickness o f m a t e r i a l v a p o r i z e d
a = Thermal d i f f u s i v i t y
p = D e n s i t y
Ax = Depth o f p e n e t r a t i o n o f h e a t
Subsc r i ~ t s :
o = Room c o n d i t i o n
m = M e l t i n g
v = V a p o r i z a t i o n
PLASMA-PROJECTILE INTERACTION I N AN
ARC-DRIVEN RAIL GUN
I n t r o d u c t i o n
I n a r a i l gun, t h e plasma i s formed b y p a s s i n g t h e c u r r e n t
t h r o u g h a t h i n me ta l f o i l a t t a c h e d t o t h e base o f t h e p r o j e c t i l e .
When t h e i n d u c t o r i s s w i t c h e d i n t o t h e r a i l gun c i r c u i t , t h e f o i l
b e g i n s t o h e a t up and e v e n t u a l l y m e l t s . S ince t h i s t a k e s p l a c e i n
a f r a c t i o n o f a microsecond, t h e p h y s i c a l shape o f t h e f o i l i s
m a i n t a i n e d by i n e r t i a and magne t i c p r e s s u r e . As h e a t i s f u r t h e r
added t o t h e s t i l l l i q u i d me ta l b y t h e c o n t i n u i n g f l o w o f t h e
c u r r e n t , i t s t empera tu re r i s e s t o t h e b o i l i n g p o i n t . However,
e q u i l i b r i u m b o i l i n g can n o t t a k e p l a c e , so s u p e r h e a t i n g occu rs
u n t i 1 t h e l i q u i d me ta l exp lodes and forms t h e plasma. Due t o t h e
i nduced magne t i c f i e l d and t h e p o s i t i o n i n g o f t h e p r o j e c t i l e ahead
o f t h e f o i l , t h e plasma i s c o n f i n e d t o a f i n i t e volume and a c c e l e r -
a t e s t o g e t h e r w i t h t h e p r o j e c t i l e . The mechanism o f t h e a c c e l e r a -
t i o n process and how t h e c u r r e n t i s p a r t i t i o n e d i n t h e plasma a r e
e x p l a i n e d i n t h i s s e c t i o n .
The c h a r a c t e r i s t i c s o f a r a i l gun plasma were f i r s t s t u d i e d
by McNab f o r t h e R a s h l e i g h and M a r s h a l l expe r imen t b y assuming a
s teady s t a t e and u n i form c o n d i t i o n s i n t h e plasma [20] . Subsequent ly ,
B a t t e h and Powel l c a l c u l a t e d plasma p r o p e r t i e s f o r a n o n - u n i f o r m
plasma i n t h e s t e a d y s t a t e b y u s i n g f i r s t , one d imens iona l and l a t e r ,
two d imens iona l magnetohydrodynamic e q u a t i o n s [21, 221. However,
t hese e q u a t i o n s t u r n o u t t o be v e r y complex and a p h y s i c a l u n d e r s t a n d i n g
3 5
3 6
o f t h e processes t h a t t a k e p l a c e i n t h e plasma i s n o t r e a d i l y ach ieved .
A computer s i m u l a t i o n code was deve loped by T h i o where t i m e v a r y i n g
p r o p e r t i e s o f t h e plasma as a component o f t h e r a i l gun c i r c u i t were
e v a l u a t e d [23]. I t was observed r e c e n t l y t h a t t h e a n a l y s i s can be
c o n s i d e r a b l y s i m p l i f i e d b y assuming u n i f o r m c o n d u c t i v i t y i n t h e plasma
[24] . To g a i n an i n s i g h t i n t o t h e p h y s i c s o f t h e plasma processes,
a s i m p l i f i e d a n a l y s i s i s p r e s e n t e d he re t o e s t a b l i s h t h e plasma c o n d i -
t i o n s i n t h e . e x p e r i m e n t o f Bauer e t a l . b y assuming u n i f o r m plasma
c o n d u c t i v i t y and by normal i z i ng t h e p r e s s u r e , magnet ic f i e 1 d and
v o l t age d r o p a c r o s s t h e p l asma t o t h e e x p e r i m e n t a l cond i t i ons [I 21.
Mechanism o f A c c e l e r a t i o n
. I t i s w e l l known f r o m Ampere's l aw t h a t a c o n d u c t o r c a r r y i n g
a c u r r e n t d e n s i t y - J i n a magne t i c f i e l d - B exper iences a body f o r c e
F g i v e n by
T h i s body f o r c e o r i g i n a t e s f r o m t h e sum o f t h e L o r e n t z f o r c e s on a l l
t h e mov ing charged p a r t i c l e s i n s i d e t h e c o n d u c t o r .
The plasma a c c e l e r a t i o n can be vi,ewed as a s i m p l e two s t e p
p rocess . The plasma i s assumed t o be f u l l y i o n i z e d . F i r s t , due t o
t h e passage o f t h e c u r r e n t i n t h e plasma an e l e c t r i c f i e l d E i s s e t Y
up under wh ich t h e e l e c t r o n s and t h e i o n s a c q u i r e d r i f t v e l o c i t i e s
v and vi r e s p e c t i v e l y ( v e >> v i ) Nex t , t h e f l o w o f c u r r e n t e
genera tes an i nduced magne t i c f i e l d BZ ( F i g . 1 3 ) . Due t o t h i s
CHARGE SEPARATION AT THE EDGES
OF PLASMA PROJECTIL NEUTRAL
Fig. 13. Charge separat ion a t the edges of plasma due t o a magnetic f i e l d
magnet ic f i e 1 d, t h e e l e c t r o n s and i o n s exper ience Loren tz f o r ces
eveBZ and evi Bz r e s p e c t i v e l y .
The a c c e l e r a t i o n o f t h e e l e c t r o n s due t o t he magnet ic f o r c e
i s much h i ghe r than those o f t h e i ons , as t h e mass o f t h e e l e c t r o n s
i s much sma l l e r than those o f t h e i o n s . Thus, an e l e c t r i c charge
sepa ra t i on occurs a t t he plasma boundar ies whereas t he main body o f
t h e plasma remains e l e c t r i c a l l y n e u t r a l . The excess e l e c t r o n s a t t h e
l e a d i n g edge o f t h e plasma adhere t o t h e base o f t he p r o j e c t i l e which
i s made o f a d i e l e c t r i c m a t e r i a l . Once t h i s i s achieved, t h e charge
d i s t r i b u t i o n a t t h e base o f t h e p r o j e c t i l e remains f i x e d and does n o t
move under t he i n f l u e n c e o f any e l e c t r i c a l f o r c e t h a t may be
exper ienced by i t .
Due t o t h e d isp lacement o f charges, an e l e c t r i c f i e l d Ex
( a l s o known as H a l l f i e l d ) i s e s t a b l i s h e d i n t h e plasma t o oppose
t he sepa ra t i on o f charges. Charge sepa ra t i on ceases when t h e e l e c t r i c
f o r c e on t h e e l e c t r o n s i s balanced by t h e magnet ic fo rce , i . e . ,
However, t h e i ons now exper ience a n e t force i n t h e x - d i r e c t i o n
The c u r r e n t d e n s i t y J i s d e f i n e d by, Y
E q u a t i o n 28 t h e n reduces t o ,
F = J B Y z
wh ich i s t h e same as e q u a t i o n 26. Bu t , as soon as t h e p o s i t i v e
i o n s b e g i n t o a c c e l e r a t e , t h e e l e c t r o n s a d h e r i n g t o t h e base o f t h e
p r o j e c t i l e t r a n s m i t t h e same f o r c e t o t h e p r o j e c t i l e t o m a i n t a i n t h e
p r e v i o u s l y a t t a i n e d cha rge s e p a r a t i o n . T h i s makes t h e plasma and
t h e p r o j e c t i l e a c c e l e r a t e t o g e t h e r i n t h e r a i l gun.
Plasma Conduct i v i t y
Fo r c a l c u l a t i n g t h e c o n d u c t i v i t y , each c o n s t i t u e n t spec ies
o f t h e plasma i s assumed t o be a c o n t i n u o u s f l u i d w i t h macroscop ic
p r o p e r t i e s d e r i v e d from a p p r o p r i a t e averages o v e r t h e p a r t i c l e s .
Thus t h e plasma i s c o n s i d e r e d t o c o n s i s t o f n f r e e e l e c t r o n s , n
s i n g l y charged p o s i t i v e i o n s and nn n e u t r a l atoms p e r u n i t volume.
The n e u t r a l atoms a r e assumed t o be c o u p l e d t o t h e main body o f t h e
plasma t h r o u g h c o l l i s i o n processes w i t h e l e c t r o n s and i o n s .
I n t h e r e s t f rame o f t h e f l u i d , t h e e l e c t r i c c u r r e n t i s
c a r r i e d p r e d o m i n a n t l y b y t h e e l e c t r o n s . I n an i d e a l , c o l l i s i o n l e s s
plasma t h e charged p a r t i c l e s g y r a t e around t h e l i n e s o f f o r c e as t h e y
deve lop d r i f t v e l o c i t i e s i n t h e - E x - B d i r e c t i o n . I n a r e a l plasma,
t h e r e a r e c o l l i s i o n s between e l e c t r o n s and i o n s and between e l e c t r o n s
and atoms. These c o l l i s i o n s produce a dampjng e f f e c t on t h e m o t i o n
o f t h e e l e c t r o n s . The damping e f f e c t can be expressed by a c o l l i s i o n
f requency veT i n te rms o f t h e r a t e a t wh ich an e l e c t r o n l o s e s i t s
4 0
momentum through a l l such c o l l i s i o n s . The response o f t he e l e c t r o n s
t o t h e e l e c t r i c and t h e magnet ic f i e l d i s then determined by t h e
r a t i o o f t h e i r gy ro f requency we t o t h e c o l l i s i o n f requency veT.
When u ~ / ' I ~ ~ << 1, as i s t h e case i n t h e r a i l gun plasmas, t he e l e c t r o n s
seldom complete one c y c l e o f t h e i r d r i f t mot ion be fo re c o l l i s i o n and
hence develop l i t t l e c ross f i e l d mot ion.
The average v e l o c i t y of t h e e l e c t r o n s - v i n t h i s plasma can
be g i ven by t h e Langevin equa t i on [25, 261
The f l o w o f e l e c t r o n s corresponds t o a c u r r e n t d e n s i t y - J g i ven by
S u b s t i t u t i n g t h e va lue o f - v f rom equa t i on 32 i n t o equa t ion 31 and
assuming steady s t a t e c o n d i t i o n one ob ta i ns ,
n e 2 - - where oo - .G
e eT
a can be i n t e r p r e t e d as t he dc c o n d u c t i v i t y o f t h e plasma. I f t h e 0
d i r e c t i o n o f t he induced magnet ic f i e l d i s taken as t he z - d i r e c t i o n ,
i .e., - B = (O,O,B,z), equa t ion 33 can be w r i t t e n as
-eB z where o = - e m e
i s t h e e l e c t r o n gyromagnet ic f requency and aZ i s t h e u n i t v e c t o r - i n t h e z - d i r e c t i o n . I f t h e e l e c t r i c f i e l d i n t h e plasma i s g i v e n b y
E = (E , E , O ) , t h e n e q u a t i o n 35 can be broken down i n t o i t s compo- - x Y nen ts as
W
and . . \ ,
e J x + J = o ~ "eT Y O Y
Equa t ions 37 and 38 can be reduced t o
and J = ulEy - 7, EX Y
3 V L
where o = o eT
U 2 + V 2 e eT
0 id '\J
and - - o e eT l + .\J
2 e eT
I n t he r a i 1 guns, veT >> ue, hence o,, << a Thus t he p r imary compo- I '
nent o f t h e c u r r e n t i s p a r a l l e l t o E which e s s e n t i a l l y rep resen ts Y
a s c a l a r conduct ion i n t h e plasma. A smal l c u r r e n t element i s a'dded
t o i t i n t h e pe rpend i cu l a r d i r e c t i o n ( F i g . 14 ) .
The f l o w o f c u r r e n t i n t h e r a i l gun can then be represen ted
where - a i s t h e c o n d u c t i v i t y t enso r and i s g iven by
C h a r a c t e r i z a t i o n o f Plasma
I n t h e r a i l gun plasmas veT >> , so t he c u r r e n t can be e
cons idered t o f l o w i n t he y - d i r e c t i o n o n l y . The plasma c o n d u c t i v i t y
i s assumed t o be un i fo rm. The assumption o f un i f o rm c o n d u c t i v i t y
does n o t i n t r o d u c e any s i g n i f i c a n t e r r o r , b u t cons ide rab l y s i m p l i -
f i e s t h e c a l c u l a t i o n s [24]. The plasma i s cons idered t o be a
r e c t a n g u l a r p a r a l l e l o p i p e d o f d imension 2 x h x w ( F i g . 15 ) . I t
i s f u r t h e r assumed t h a t t he plasma i s s i n g l y i on i zed .
The c u r r e n t d e n s i t y and t h e c u r r e n t pe r u n i t h e i g h t o f t he
r a i l a re g i ven by
Fig. 14. Migration o f electrons in the electric and magnetic fields
L m u L s PLASMA
F i g . 15. Geometry o f plasma f o r mode l ing
Equa t ions 45 and 46 a l s o i n d i c a t e t h a t
Induced magnet ic f i e l d
The induced magnet ic f i e l d - B can be e v a l u a t e d f rom Max-
w e l l ' s e q u a t i o n
Assuming t h a t t h e magnet ic f i e l d v a r i e s o n l y i n t h e x - d i r e c t i o n ,
e q u a t i o n 48 reduces t o
A t t h e l e a d i n g edge o f t h e plasma, t h e induced magnet ic f i e l d i s
ze ro . Wi th t h i s c o n d i t i o n , e q u a t i o n 49 can be i n t e g r a t e d t o y i e l d
The magnet ic f i e l d a t t h e t r a i l i n g edge o f t h e plasma i s then
g i v e n by
4 6
However, i t was found t h a t t h i s express ion overes t imates t he mag-
n e t i c f i e l d and hence, t he p r o p u l s i v e f o r c e by a f a c t o r o f 2 o r
more, as equa t ion 51 p e r t a i n s t o r a i l s o f i n f i n i t e h e i g h t [24] .
To es t ima te t h e va lue o f t h e magnet ic f i e l d more accu ra te l y , a
c o r r e c t i o n f a c t o r fl i s i n t r oduced here, such t h a t
The va lue o f fl can be found by n o r m a l i z i n g t he - J x - B f o r c e w i t h
t h a t c a l c u l a t e d f rom the exper imenta l da ta . The average magnet ic
f i e l d i s then
Pressure
The p ressure i n t h e plasma i s assumed t o va ry i n t h e
x - d i r e c t i o n . his i s a reasonable assumption because p ressure w i 11
va ry s i g n i f i c a n t l y i n t h e x - d i r e c t i o n only,due t o t he mechanical
e f f e c t o f t h e a c c e l e r a t i o n .
The equa t ion r e p r e s e n t i n g t h e conserva t ion o f momentum i n
t he plasma can be w r i t t e n as [27 ]
Ana l ys i s o f equa t i on 54 can be cons ide rab l y s i m p l i f i e d by d ropp ing
t h e f i r s t term f rom the equa t ion as i t i s an o rde r o f magnitude
47
s m a l l e r t h a n t h e e l e c t r o m a g n e t i c body f o r c e . E q u a t i o n 54 t h e n
reduces t o
S u b s t i t u t i n g t h e v a l u e s o f J and B f r o m e q u a t i o n s 47 and 52 i n t o
e q u a t i o n 55, one o b t a i n s
Fo r a p u r e e l e c t r o m a g n e t i c a c c e l e r a t i o n o f t h e plasma, t h e p r e s s u r e
a t t h e t r a i l i n g edge i s z e r o . W i t h t h i s c o n d i t i o n , e q u a t i o n 56 can
be i n t e g r a t e d t o y i e l d
The p r e s s u r e e x e r t e d b y t h e plasma a t t h e base o f t h e p r o j e c t i l e
i s t h e n
,,
I t has been observed t h a t t h e f o r c e on t h e p r o j e c t i l e a t t h e
b e g i n n i n g o f a c c e l e r a t i o n can a l s o be r e p r e s e n t e d by [8]
48
Equat ing t h e two f o r ces , one o b t a i n s
fld 2 L ' 12 (hw) = -
2h2 2
L ' h g i v i n g f = - 1 r-lw
I n t h e exper iment o f Bauer e t a l . , h = 3.85 mm, w = 6.35 mm and
L ' was measured t o have a va lue of 0.5 pH/m. S u b s t i t u t i n g these
va lues i n t o equa t i on 61, t h e va lue o f fl i s found t o be equal t o
0.24.
The average pressure i s g iven by
2 - p .'j
2 P ( x ) dx
S u b s t i t u t i n g t h e va lue o f P (x ) f rom equa t ion 57 i n t o equa t ion 62
and i n t e g r a t i n g , one ob ta i ns
The v a r i a t i o n o f p ressure a long t h e l e n g t h o f t h e plasma and t h e
average p ressure i s shown i n F i g . 16.
Temperature and degree o f i o n i z a t i o n
The average temperature o f t h e plasma can be ob ta i ned f rom
the average p ressure by assuming t h a t t h e e l e c t r o n s , t he i o n s and
t he n e u t r a l atoms a re a t t h e same temperature. Th i s i s a reasonable
0 . 5 ~ - PRESSURE --- AVERAGE PRESSURE
0.4 -
0.3 -
0.00 0.25 0.5 0.7 5 1 .OO
FRACTION OF PLASMA LENGTH
F i g . 16. V a r i a t i o n o f p r e s s u r e a l o n g t h e l e n g t h o f t h e plasma and average p r e s s u r e
50
assumption because due t o t h e h i g h c o l l i s i o n f requenc ies i n r a i l
gun plasmas, t h e t ime sca le i n which t h e p a r t i c l e s a t t a i n an
e q u i l i b r i u m v e l o c i t y d i s t r i b u t i o n i s niuch sma l l e r than t h a t o f t h e
a c c e l e r a t i o n i n t h e gun. Hence, P and T a re r e l a t e d by
where a i s t h e average degree o f i o n i z a t i o n and i s d e f i n e d by
The degree o f i o n i z a t i o n can be ob ta ined f rom the Saha equa t ion
[ 201
where K(T) i s g i ven by
Thus,
D e n s i t y
N e g l e c t i n g t h e mass o f t h e e l e c t r o n s , t h e average d e n s i t y
o f t h e plasma i s g i v e n by
S u b s t i t u t i n g t h e v a l u e o f a f rom e q u a t i o n 65 i n t o e q u a t i o n 69,
one o b t a i n s
Conduct i v i t y and r e s i s tance
Due t o t h e h i g h c o l l i s i o n f requency, t h e plasma can be
assumed t o have a s c a l a r c o n d u c t i v i t y . The plasma c o n d u c t i v i t y
can be c a l c u l a t e d f r o m e q u a t i o n 34, where
.> = " e ~ 'e i + "en (71
v and a r e g i v e n by [25] e i
and 4 2 1 /2 v = 2.60 x 10 6 nnT en
The Coulomb c u t o f f parameter A r e p r e s e n t s t h e e x t e n t t o wh ich
c o l l e c t i v e plasma e f f e c t s dominate o v e r i n d i v i d u a l p a r t i c l e
phenomena and i s g i v e n b y [ 25 ]
7 3/2n-1/2 A = 1.23 x 10 T
The r e s i s t a n c e o f t h e plasma i s g i v e n by
The l e n g t h o f t h e plasma i s c a l c u l a t e d f r o m
The v o l t a g e d r o p a c r o s s t h e plasma i s
R e s u l t s and D i s c u s s i o n
The amount o f plasma t h a t i s genera ted f rom t h e e x p l o d i n g
f o i l i s n o t de te rm ined f r o m t h e exper imen ts . I n v iew o f t h i s , t h e
plasma mass i s v a r i e d i n t h i s s t u d y and t h e parameters t h a t
c h a r a c t e r i z e t h e plasma a r e e v a l u a t e d as a f u n c t i o n o f t h e plasma
mass. The magne t i c f i e l d and hence, t h e p r e s s u r e a r e o f cou rse
de te rm ined p r i m a r i l y by t h e c u r r e n t i n t h e r a i l gun.
The c a l c u l a t i o n s a r e pe r fo rmed as f o l l o w s . F i r s t , a plasma
mass i s assumed. Next , t h e plasma tempera tu re i s assumed and t h e
degree o f i o n i z a t i o n , t h e plasma d e n s i t y , t h e e l e c t r o n d e n s i t y ,
t h e c o l l i s i o n f requency, t h e plasma c o n d u c t i v i t y , t h e plasma l e n g t h ,
t h e plasma r e s i s t a n c e and t h e v o l t a g e d r o p ac ross t h e plasma a r e
c a l c u l a t e d . The c a l c u l a t e d v o l t a g e d r o p ac ross t h e plasma i s t h e n
checked a g a i n s t t h e measured muzz le v o l t a g e . I f t h e two va lues a r e
d i f f e r e n t , a new tempera tu re i s s e l e c t e d and t h e i t e r a t i o n p rocess
5 3
i s c o n t i n u e d u n t i l t h e two v a l u e s do n o t d i f f e r by more t h a n 1 v o l t .
The plasma mass i s t h e n changed t o a new v a l u e and t h e who le p rocess
i s repea ted .
I n t h e exper imen t o f Bauer e t a l . , t h e average mass o f t h e
copper f o i l s was a p p r o x i m a t e l y 30 mg [28]. I n t h i s s tudy , t h e plasma
mass was v a r i e d f r o m 10 mg t o 50 mg. F i g u r e 17 shows t h e plasma tem-
p e r a t u r e and degree o f i o n i z a t i o n as a f u n c t i o n o f plasma mass. I t i s
observed t h a t t h e tempera tu re decreases w i t h t h e i n c r e a s e o f plasma
mass. S ince t h e v o l t a g e d r o p a c r o s s t h e plasma i s n o r m a l i z e d t o t h e
measured muzz le v o l t a g e , t h e plasma r e s i s t a n c e remains c o n s t a n t and
thus , t h e ohmic h e a t i n g i n t h e plasma i s c o n s t a n t f o r a g i v e n c u r r e n t .
More plasma r e p r e s e n t s more p a r t i c l e s t o abso rb t h e energy d i s s i p a t e d
i n t h e plasma and hence, t h e plasma tempera tu re decreases. The degree
o f i o n i z a t i o n o f cou rse decreases w i t h an i n c r e a s e i n plasma mass as
i t s tempera tu re i s l owered .
The v a r i a t i o n o f e l e c t r o n and plasma d e n s i t y w i t h t h e plasma
mass a r e shown i n F i g . 18. The plasma d e n s i t y i n c r e a s e s w i t h plasma
mass as t h e l e n g t h o f t h e plasma and hence, i t s volume i n c r e a s e s a t a
much s l o w e r r a t e w i t h t h e i n c r e a s e o f plasma mass ( ~ i g . 1 9 ) . However,
t h e e l e c t r o n d e n s i t y i s i n i t i a l l y f ound t o i n c r e a s e and t h e n decrease
w i t h an i n c r e a s e i n plasma mass. T h i s b e h a v i o r i s due t o t h e f a c t t h a t
t h e degree o f i o n i z a t i o n a l s o decreases w i t h t h e i n c r e a s e o f plasma mass.
The e f f e c t o f v a r i a t i o n o f plasma mass on i t s c o n d u c t i v i t y i s a l s o shown
i n F i g . 19. As t h e plasma mass i s i n c r e a s e d i t s t empera tu re decreases
wh ich i n t u r n i n c r e a s e s t h e c o l l i s i o n f requency . S ince c o l l i s i o n s
r e p r e s e n t r e s i s t a n c e t o t h e m o t i o n o f t h e e l e c t r o n s , an i n c r e a s e
10 2 0 3 0 4 0 5 0
PLASMA MASS, MG
F i g . 17 . Average plasma tempera tu re and average degree o f i o n i z a t i o n as a f u n c t i o n o f plasma mass
PLASMA MASS, MG
F i g . 18. V a r i a t i o n o f e l e c t r o n and plasma d e n s i t y w i t h t h e plasma mass
PLASMA MASS, MG
F i g . 19. E f f e c t o f v a r i a t i o n o f plasma mass on plasma l e n g t h and c o n d u c t i v i t y
i n c o l l i s i o n f requency i s accompanied w i t h a decrease i n conduc-
t i v i t y .
S ince t h e a c t u a l mass o f t h e plasma c o u l d n o t be de te rm ined
f rom t h e exper imen t , t h e plasma c o n d i t i o n s can o n l y be e s t i m a t e d .
For t h e plasma t o a t t a i n an e q u i l i b r i u m c o n d i t i o n , t h e energy d i s s i -
p a t e d i n i t b y ohmic h e a t i n g must be ba lanced by t h e h e a t l o s s f r o m
t h e plasma. Because o f t h e h i g h plasma tempera tu re , t h e h e a t l o s s
i s p r i n i a r i l y by r a d i a t i o n . Thus,
The tempera tu re o b t a i n e d from t h i s e q u a t i o n i s t h e n matched w i t h
t h e plasma t e m p e r a t u r e c a l c u l a t e d t h r o u g h t h e i t e r a t i o n p rocess .
I n t h i s way, t h e plasma mass and hence, t h e o t h e r plasma parameters
can be e s t i m a t e d .
The two tempera tu res a r e matched when t h e plasma mass i s
28 mg. T h i s means t h a t o v e r 90 p e r c e n t o f t h e f o i l was c o n v e r t e d i n t o
p lasma. More r e s e a r c h i s needed i n t h e a rea o f e x p l o d i n g f o i l s t o -
de te rm ine i f such a l a r g e p o r t i o n o f t h e f o i l can be c o n v e r t e d i n t o
plasma. T a b l e 3 p r o v i d e s t h e i n p u t d a t a f o r t h e exper imen t o f Bauer
e t a l . and i n T a b l e 4 t h e e s t i m a t e d plasma c o n d i t i o n s a r e l i s t e d .
The plasma t e m p e r a t u r e i s found t o be 25,600°K. T h i s i s
somewhat l o w e r t h a n t h e e s t i m a t e o f 44,000 5 13,000°K by McNab f o r
t h e plasnia i n R a s h l e i g h and M a r s h a l l ' s exper imen t . However, t h i s
i s expec ted as e q u a t i o n 78 tends t o u n d e r e s t i m a t e t h e plasma tempera-
t u r e . The plasma i s found t o be 68% i o n i z e d , hence t h e assumpt ion
o f s i n g l y i o n i z e d plasma i s j u s t i f i e d . The r e s i s t a n c e o f t h e plasma
5 8
T a b l e 3
I n p u t Data f o r E s t i m a t i n g Plasma C o n d i t i o n s
Parameter Val ue
3.85 mm
135 KA
0.50 pH/m
200 : v o l t
6.35 mm
3 i s 1.48 mR and i t s d e n s i t y i s 22.1 kg/m . The l e n g t h o f t h e plasma
i s e s t i m a t e d t o be 52 mm wh ich i s abou t h a l f t h e l e n g t h assumed i n
2 6 3 McNab's c a l c u l a t i o n s . The e l e c t r o n d e n s i t y i s 1 .4 x 10 p e r m .
The r a t i o o f c o l l i s i o n f requency t o gyromagnet ic f requency i s 1246,
wh ich j u s t i f i e s t h e assumpt ion o f s c a l a r c o n d u c t i o n i n t h e plasma.
C o n c l u s i o n
I n a r a i l gun, charge s e p a r a t i o n occu rs a t t h e edges o f
t h e plasma t o produce a H a l l f i e l d . T h i s e l e c t r i c f i e l d f o r c e s t h e
p o s i t i v e i o n s and hence, t h e plasma and t h e p r o j e c t i l e , t o a c c e l e r -
a t e . The plasma c o n d u c t i v i t y i s shown t o be a t e n s o r , b u t because
o f t h e h i g h r a t i o o f veT t o we, a s c a l a r c o n d u c t i o n i n t h e plasma
can be assumed w i t h o u t much e r r o r . Plasma p r o p e r t i e s a r e e s t i m a t e d
as a f u n c t i o n o f plasma mass t h r o u g h a s i m p l e model. I t i s obse rved
t h a t t h e magne t i c f i e l d and hence, t h e p r e s s u r e depends on t h e
Tab le 4
E s t i m a t e d Plasma Parameters i n t h e Exper iment o f Bauer e t a l .
Parameter Va 1 ue
28 rng
26 3 1 .4 x 10 /m
1 . 2 x l o 8 ~ / m '
29.7 rn/s
0 . 6 8
22.1 kg/m3
-1 -1 21,510 ohm m
1 . 8 x 1014/sec
12 4.6 x 10 /sec
60
c u r r e n t i n t h e r a i l gun. The temperature, t h e degree o f i o n i z a t i o n
and t h e plasma c o n d u c t i v i t y decreases w i t h an inc rease i n plasma
mass whereas t h e plasma d e n s i t y , t h e plasma l e n g t h and t he e l e c t r o n
d e n s i t y increases as t h e plasma mass i s increased. F u r t h e r s tudy
i s needed t o understand t h e phys ics o f t he processes t h a t take
p lace i n an exp lod ing f o i l so t h a t t h e plasma mass can be c a l c u l a t e d
w i t h a h i g h degree o f p r e c i s i o n .
Nomenclature
a z = U n i t v e c t o r i n t h e z - d i r e c t i o n
B = Induced magne t i c f i e l d -
BZ = Magnet ic f i e l d i n t h e z - d i r e c t i o n
e = Charge o f an e l e c t r o n
E = E l e c t r i c f i e l d -
Ex = E l e c t r i c f i e l d i n t h e x - d i r e c t i o n
E~ = E l e c t r i c f i e l d i n t h e y - d i r e c t i o n
fl = C o r r e c t i o n f a c t o r
F = Force
h = H e i g h t o f t h e r a i l s
I = C u r r e n t
j = C u r r e n t p e r u n i t h e i g h t o f t h e r a i l
J = C u r r e n t d e n s i t y -
J x = C u r r e n t d e n s i t y i n t h e x - d i r e c t i o n
Y = C u r r e n t d e n s i t y i n t h e y - d i r e c t i o n
k = Bo l t zmann 's c o n s t a n t
k = Length o f t h e plasma
L ' = Induc tance g r a d i e n t o f t h e r a i l s
me = Mass o f e l e c t r o n
"'n = Mass o f n e u t r a l atom
m P
= Mass o f plasma
P = P ressure - P = Average p r e s s u r e
R = Plasma r e s i s t a n c e P
t = Time
T = Average temperature
u = Plasma v e l o c i t y
v,ve = D r i f t v e l o c i t y o f e l e c t r o n s
i = D r i f t v e l o c i t y o f i o n s
Vi = I o n i z a t i o n p o t e n t i a l
P = Vol tage drop across plasma
w = R a i l sepa ra t i on
a = Average degree o f i o n i z a t i o n
& = Radius o f n e u t r a l atom
0 0
= DC conduc t i v i ty
0 - = C o n d u c t i v i t y t enso r
P = Average d e n s i t y
i~ = Magnet ic p e r m e a b i l i t y
LIJ e = E l e c t r o n gyromagnet ic f requency
v e i = E l e c t r o n - i o n c o l l i s i o n f requency
V en = E l e c t r o n atom c o l l i s i o n f requency
v e T = E l e c t r o n c o l l i s i o n f reuqency, t o t a l
A = Coulomb c u t o f f parameter
REFERENCES
1 KREISLER, M.N.: 'How t o make t h i n g s move v e r y f a s t , ' American S c i e n t i s t , 1982, - 70, pp. 70-75
2 IGENBERGS, E.B., JEX, D.W., and SHRIVER, E.L . : 'New two-s tage a c c e l e r a t o r f o r h y p e r v e l o c i t y impac t s i m u l a t i o n , ' AIAA J . , 1975, - 13, pp. 1024-1030
3 KERSLAKE, W.R., and CYBYK, B.Z.: ' R a i l a c c e l e r a t o r r e s e a r c h a t Lewis Research Cen te r , ' NASA T e c h n i c a l Memorandum 83015, 1982
4 RIBE, F.L., and VLASES, G . C . : 'Scope o f impac t f u s i o n and r e v i e w o f m a c r o p a r t i c l e a c c e l e r a t o r s , ' Repor t LA-8000-C, 1979, pp. 1-19
5 BARBER, J .P. : ' E l e c t r i c r a i l gun a p p l i c a t i o n t o space p r o p u l s i o n , ' AIAA paper no. 79-2091, 1979
6 KOLM, H., MONGEAU, P., and WILLIAMS, F.: ' E l e c t r o m a g n e t i c l a u n c h e r s , ' IEEE Trans. Magns., 1980, MAG-16, pp. 719-721
7 MARSHALL, R.A.: 'The A u s t r a l i a n N a t i o n a l U n i v e r s i t y r a i l gun p r o j e c t , ' A tomic Energy, 1975, - 1, pp. 16-19
8 RASHLEIGH, S.C., and MARSHALL, R.A.: ' E l e c t r o m a g n e t i c a c c e l e r a t i o n o f m a c r o p a r t i c l e s t o h i g h v e l o c i t i e s , ' J . A p p l . Phys., 1978, 49, pp. 2540- 2542
9 HAWKE, R.S., BROOKS, A.L., DEADRICK, F.J., SCUDDER, J .K. , FOWLER, C.M., CAIRD, R.S., and PETERSON, D.R.: ' R e s u l t s o f r a i l g u n exper imen ts powered by magne t i c f l u x compress ion g e n e r a t o r s , ' IEEE Trans. Magns., 1982, MAG-18, pp. 82-93
10 BUCKINGHAM, A.C.: ' E l e c t r o m a g n e t i c p r o p u l s i o n : d r a g and e r o s i o n mode l i ng , ' AIAA J . , 1981, - 19, pp. 1422-1428
11 PARKER, J . V . : Communicated by W . R . Ke rs lake , November, 1984
12 BAUER, D.P., MCCORMICK, T.J., and BARBER, J .P. : ' E l e c t r i c r a i l gun p r o j e c t i l e a c c e l e r a t i o n t o h i g h v e l o c i t y , ' AIAA paper no. 82-1939, 1982
13 YOUNG, F.J.. HOWLAND. H.R., HUGHES, W.F., and FIKSE, D.A.: ' I n t e r a c t i v e e l e c t r o m a g n e t i c l a u n c h e r s i m u l a t i o n , ' IEEE Trans. ~ a ~ n s . , 1982, MAG-18, pp. 29-32
14 BAER, P.G.: ' P r a c t i c a l i n t e r i o r b a l l i s t i c a n a l y s i s o f guns, ' Prog. A s t r o . Aero . , 1979, - 66, pp. 37
63
15 BUCKINGHAM, A.C.: ' P r o j e c t i l e and r a i l l a u n c h e r d e s i g n a n a l y s i s f o r e l e c t r o m a g n e t i c p r o p u l s i o n t o v e l o c i t i e s exceed ing 10 km/s,' AIAA paper no. 81-0750, 1981
16 WANG, S.Y.: 'The s t r u c t u r a l response o f a r a i l a c c e l e r a t o r , ' IEEE Trans. Magns., 1984, MAG-20, pp. 356-359
17 CLARK, G.A., and BEDFORD, A.J . : 'Per formance r e s u l t s o f a s m a l l - c a l i b r e e l e c t r o m a g n e t i c l auncher , ' IEEE T rans . Magns., 1984, MAG-20, pp. 276-279
18 COHEN, M . I . : ' M e l t i n g o f a h a l f - s p a c e s u b j e c t e d t o a c o n s t a n t h e a t i n p u t , ' J. F r a n k l i n I n s t i t u t e , 1967, - 283, pp. 271
19 LANDAU, H.G.: 'Hea t c o n d u c t i o n i n a m e l t i n g s o l i d , ' Q u a r t . A p p l . Math., 1950, - 8, pp. 81
20 MCNAB, I . R . : ' E l e c t r o m a g n e t i c m a c r o p a r t i c l e a c c e l e r a t i o n by a h i g h p r e s - su re plasma,' J. App l . Phys., 1980, 51, pp. 2549
21 POWELL, J .D., and BATTEH, J .H. : 'Plasma dynamics o f an a r c - d r i v e n e l e c t r o - magne t i c p r o j e c t i l e a c c e l e r a t o r , ' J . A p p l . Phys., 1981, - 52, pp. 2717
22 POWELL, J.D., and BATTEH, J.H.: 'Two-d imens iona l plasma model f o r t h e a r c - d r i v e n r a i l gun,' J . A p p l . Phys., 1983, - 54, pp. 2242
23 THIO, Y . C . : 'PARA: A computer s i m u l a t i o n code f o r plasma d r i v e n e l e c t r o - magnet ic l aunchers , ' Repor t no. MRL-R-873, Department o f Defense, A u s t r a l i a , 1983
24 BATTEH, J.H.: ' A n a l y s i s o f a r a i l gun plasma a c c e l e r a t o r , ' Repor t no . DAAKll-80-C-0102, B a l l i s t i c Research L a b o r a t o r y , December, 1981
25 TANENBAUM, B.S.: 'Plasma Phys ics , ' McGraw- t i i l l , New York, 1967
26 JAHN, R . G . : ' P h y s i c s o f E l e c t r i c P r o p u l s i o n , ' McGraw-Hi l l , New York, 1968
27 CHEN, F.F.: ' I n t r o d u c t i o n t o Plasma Phys ics , ' Plenum Press, New York, 1974
28 MCCORMICK, T .J . : P r i v a t e Communication, November, 1984
6 5
DISTRIBUTION LIST
Copies
N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Washington, DC 20546 A t t n : R S / M r . D e l l W i l l i a m s , I 1 1
RSE/Mr. Van Landingham MT/Mr. I v a n Bekey
N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Lewis Research Cen te r 21000 Brookpark Road Cleve land, OH 44135 A t t n : Research Suppor t Procurenlent S e c t i o n
M r . S teve Szabo, MS 501-11 Technology U t i l i z a t i o n O f f i c e , MS 3-19 Repor t C o n t r o l O f f i c e , MS 5-5 L i b r a r y , MS 60-3 D r . M. G o l d s t e i n , C h i e f S c i e n t i s t , MS 5-3 M r . J. Stone, MS 501-7 M r . D. Byers , MS 501-7 M r . W. Ke rs lake , MS 501-7 M r . H. A l l e n , MS 7-3
N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Lyndon B. Johnson Space Cen te r Houston, TX 77058 A t t n : M r . Hu Dav is
N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n M a r s h a l l Space F l i g h t Cen te r H u n t s v i l l e , AL 35812 A t t n : M r . George Von Tiesenhausen, P501
M r . Rober t B e c h t e l
Research and Technology D i v i s i o n W r i g h t - P a t t e r s o n AFB, OH 45433 A t t n : (AFWAL/NASA-PO) M r . E v e r e t t B a i l e y
(FTD/TQTD) Capt. Dav id G. Hal 1
NASA S c i e n t i f i c and Techn ica l I n f o r m a t i o n F a c i l i t y P.O. Box 8757 B a l t i m o r e , MD 21240 A t t n : Access ion ing Dept.
Copies
DARPA/TTO 1400 Wilson Boulevard A r l i n g t o n , VA 22209 A t t n : D r . Har ry D. F a i r , J r .
System P lann ing Co rpo ra t i on ATM Department 1500 Wi lson Boulevard A r l i n g t o n , VA 22209 A t t n : M r . Donald E. Shaw
ARDC, AMCCOM B u i l d i n g 382 Dover, NJ 07801 A t t n : Dr. Thaddeus Gora
D r . P. J. Kenlrney
U.S. Army ARRADCOM B a l l i s t i c Research Lab., DRDAR-BLB Aberdeen Prov ing Ground, MD 21005 A t t n : D r . Don Ecceshal l
D r . J. D. Powel l M r . K. Jamison
USAF Rocket P ropu l s i on Labora to ry Edwards AFB, CA 93523 A t t n : LKC/Maj. Nord ley
LKCJ/Dr. Robert Vondra LKCS/Mr. Frank Mead
USAF Armament Lab (AFATL/DLDB) E g l i n A i r Force Base, FL 32542 A t t n : L t . R ichard Wal ley
Boeing Aerospace Co. P.O. Box 3999 S e a t t l e , WA 98124 A t t n : M r . James F. Kenney, MS 8C-23
E lec t romagnet i c Launch Research, I nc . 625 Putnam Avenue Cambridge, MA 021 39 A t t n : D r . Henry H. Kolm
D r . Pe te r Mongeau D r . W i l l i a m Snow
GA Technologies P.O. Box 85608 San Diego, CA 92138 A t t n : Dr. S i b l e y C . B u r n e t t
D r . Michael M. Ho l l and
Copies
General Dynamics P .O . Box 2507 Pamona, CA 91769 A t t n : Mr. J . H . Cuadros
General Research C o r p o r a t i o n 5383 H o l l i s t e r Avenue Santa Barbara , CA 93111 A t t n : W i l l i a m M. I s b e l l
GT Dev ices , I n c . 5705 General Washington D r i v e A l e x a n d r i a , VA 22312 A t t n : Dr . Derek Tidman
IAP Research, I n c . 2763 C u l v e r Avenue Dayton, OH 45429-3273 A t t n : Dr . John B a r b e r
Phys i cs I n t e r n a t i o n a l Company 2700 Merced S t r e e t San Leandro, CA 94577 A t t n : Dr . Ed Goldman
LT Aerospace and Defense Company P.O. Box 225907 D a l l a s , TX 75265 A t t n : Mr. C h a r l e s H. H a i g h t
Sandi a L a b o r a t o r i e s P.O. Box 5800 A1 buquerque, NM 871 15 A t t n : Dr. M. Cowan
Dr. T. J . Burgess
Sc ience A p p l i c a t i o n s , I n c . 1503 Johnson F e r r y Road S u i t e 100 M a r i e t t a , GA 30067 A t t n : Dr . Jad B a t t e h
U n i v e r s i t y o f Cal i f o r n i a Lawrence L i v e r m o r e Lab P.O. Box 805 L ivern iore , CA 94550 A t t n : Dr . Rona ld Hawke, L-355
Copies
Los Alamos N a t i o n a l S c i e n t i f i c L a b o r a t o r y P.O. Box 1663 Los Alamos, NM 87544 A t t n : D r . J . V . Pa rke r
D r . Max Fowler D r . W i 11 Fox, MS 6787
Colorado S t a t e U n i v e r s i t y F o r t C o l l i n s , CO 80521 A t t n : P r o f . P. J . W i l b u r
P r o f . H. R . Kaufman
Westinghouse E l e c t r i c C o r p o r a t i o n 1310 Beulah Road P i t t s b u r g h , PA 15235 A t t n : M r . C. J . Mole
M r . Dan ie l W . De is D r . Y . C . T h i o
Westinghouse E l e c t r i c C o r p o r a t i o n 401 E. Hendy Avenue Sunnyvale, CA 94088 A t t n : Dr. I a n McNab, MS ED-5
M r . Ronald A. R i n d a l , MS ED-2
U n i v e r s i t y o f Texas a t A u s t i n Center f o r E lec t romagne t i cs T a y l o r Hal 1 227 A u s t i n , TX 78712. A t t n : D r . John H. G u l l y
U n i v e r s i t y o f Texas a t A u s t i n T a y l o r Hal 1 167 A u s t i n , TX 78712 A t t n : Dr. W i l l i a m E. Weldon
P r i n c e t o n U n i v e r s i t y P r ince ton , NJ 08540 A t t n : Dean R. G. Jahn
D r . R i c h a r d A. Marsha! 1 P.O. Box 50 Ascot Vale, V i c t o r i a 3032 A u s t r a l i a
Brookhaven N a t i o n a l Lab. Upton, NY 11973 A t t n : Dr. James Powel l , 802M
Copies
BDM Corp. 2227 Drake Ave. H u n t s v i l l e , AL 35805 A t t n : M r . S teve Mann
Mr. J i m Schaaf
BDM Corp. One 1 s t N a t i o n a l P laza , S u i t e 1010 Dayton, OH 45402 A t t n : M r . K e i t h J . Maxwel l
A u s t i n Research Assoc. 1901 R u t l a n d D r i v e A u s t i n , TX 87858 A t t n : D r . W i l l i a m E. Drummond
ASEA AB, Dept. KYEB S-721 83, Vas te ras Sweden A t t n : Dr . S t e f a n Toader
General E l e c t r i c Co. ARD No. 3, P l a i n s Road Ba l l s t o n Spa, NY 12020 A t t n : M r . Wendell Neugebaur
