THEORY OF ARC CLOGGING IN NOZZLES

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THEORY OF ARC CLOGGING IN NOZZLES

P. Kovitya, J. Lowke, A. Stokes

To cite this version:

P. Kovitya, J. Lowke, A. Stokes. THEORY OF ARC CLOGGING IN NOZZLES. Journal de Physique

Colloques, 1979, 40 (C7), pp.C7-299-C7-300. �10.1051/jphyscol:19797147�. �jpa-00219121�

JOURiQAL DE PHYSIQUE CoZZoque C 7 , suppZ4ment au no?, Tome 40, JuiZZet 1979, page C7- 299

THEORY OF ARC CLOGGING IN NOZZLES

P. Kovitya, J.J. Lowke and A.D. Stokes.

University of Sydney, Australia.

I n t r o d u c t i o n : P r e v i o u s l y a s i m p l e c h a n n e l model ,has been used t o s u c c e s s f u l l y p r e d i c t p r o p e r t i e s of low c u r r e n t a r c s i n f o r c e d flow i n a n o z z l e [ I ] . The p r e s e n t paper e x t e n d s t h i s t r e a t m e n t t o l a r g e c u r r - e n t s when mass flow t h r o u g h t h e a r c i s a n a p p r e c i - a b l e f r a c t i o n of t h e c o l d g a s flow s u r r o u n d i n g t h e a r c . It i s n e c e s s a r y t o s o l v e t h e a x i a l momentum and mass c o n t i n u i t y e q u a t i o n s t o o b t a i n t h e a x i a l p r e s s u r e and v e l o c i t y d i s t r i b u t i o n s . At s u f f i c i e n t - l y l a r g e c u r r e n t s t h e a r c d i a m e t e r e q u a l s t h e n o z z l e d i a m e t e r a t some a x i a l p o s i t i o n s . Then a b l a t i o n from t h e n o z z l e w a l l markedly i n c r e a s e s t h e p r e s s u r e w i t h i n t h e n o z z l e .

Theory: It i s assumed t h a t t h e a r c plasma c a n b e r e p r e s e n t e d a s a f u n c t i o n of a x i a l p o s i t i o n z by a r e a A(z) and t e m p e r a t u r e T ( z ) , t h e a r c b e i n g i s o - t h e r m a l w i t h r a d i u s . When t h e a r c a r e a a t t a i n s t h e n o z z l e a r e a i t i s assumed t h a t a l l of t h e i n p u t e l e c t r i c a l e n e r g y i s absorbed a t t h e n o z z l e w a l l e i t h e r by t h e r m a l c o n d u c t i o n o r by t h e a b s o r p t i o n of r a d i a t i o n . T h i s energy t h e n a b l a t e s w a l l m a t e r i a l which i s brought t o t h e a r c t e m p e r a t u r e and i n c r e a s - e s t h e plasma flow. The b a s i c c o n s e r v a t i o n equa- t i o n s and Ohm's law d e f i n e t h e q u a n t i t i e s E ( z ) , T(z), V(z), V ( z ) , P ( z ) , A(z) and rh(z) where E i s t h e e l e c t r i c f i e l d , V and V t h e v e l o c i t i e s of t h e plas- m a and c o l d g a s r e s p e c t i v e l y , P i s t h e p r e s s u r e and 6 i s t h e r a t e o f mass e n t r y i n t o t h e a r c i n gm s -1 cm -1 . ohm's Law d e f i n e s t h e e l e c t r i c f i e l d f o r any i n p u t c u r r e n t I where D(T) i s t h e e l e c t r i c a l con- d u c t i v i t y ,

I = oEA (1)

The energy b a l a n c e e q u a t i o n a t t h e a r c c e n t r e i s

p is t h e plasma d e n s i t y , C t h e s p e c i f i c h e a t , t P

t h e t i m e and U t h e r a d i a t i o n e m i s s i o n c o e f f i c i e n t g i v i n g n e t r a d i a t i o n e m i t t e d per u n i t volume a t t h e a r c c e n t r e .

d e f i n e d p r i m a r i l y by t h e coupled momentum and mass b a l a n c e e q u a t i o n , i.e.,

a ^{(pc (Q-A)} a (pcvc (Q-A)

= - - i

a t a z ⁽⁵⁾

E q u a t i o n s ( 4 ) and (5) e x p r e s s mass c o n t i n u i t y f o r t h e plasma and c o l d g a s r e g i o n s r e s p e c t i v e l y ; t h e s u b s c r i p t c r e f e r s t o t h e c o l d g a s s u r r o u n d i n g t h e a r c and Q is t h e n o z z l e a r e a . R a t h e r t h a n s o l v e a n a d d i t i o n a l e q u a t i o n f o r a x i a l momentum f o r t h e c o l d g a s we assume t h a t t h e Mach number of t h e plasma e q u a l s t h e Mach number of t h e c o l d gas.

Thus Vc/ac = V/a

where "a" i s t h e s o n i c v e l o c i t y .

The energy b a l a n c e e q u a t i o n i n t e g r a t e d o v e r a c r o s s s e c t i o n of t h e a r c column i s

where h i s t h e e n t h a l p y of t h e plasma. Because h

<< h t h e term i h c a n b e o m i t t e d . Equation (7) p r i n c i p a l l y d e f i n e s a r c a r e a when A < Q. However when t h e n o z z l e i s c l o g g e d , i . e . , when A

Q, t h i s e q u a t i o n p r i m a r i l y d e t e r m i n e s t h e p r e s s u r e i n t h a t f o r t h e s t e a d y s t a t e , P i n c r e a s e s u n t i l phVQ a t t h e e x i t e q u a l s t h e i n p u t e l e c t r i c a l power. It i s assumed no a b l a t i o n o c c u r s b e f o r e t h e a r c c o r e d i - ameter a t t a i n s t h e n o z z l e d i a m e t e r .

M a n i p u l a t i n g e q u a t i o n s (21, ( 4 ) and (7) t o g e t h e r w i t h t h e i d e n t i t y a h l a t = C a ~ / a t i t i s p o s s i b l e t o d e r i v e P

i = UA/ (h-h ) - ^{U A / ~} (8)

Thus we c a n e l i m i n a t e rh from t h e e q u a t i o n s and a v o i d u s i n g e q u a t i o n (7).

The @a1 v e l o c i t y and p r e s s u r e d i s t r i b u t i o n s a r e (3-5) and ( 7 ) are expressed in the time

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797147

dependent form b o t h t o e n a b l e t i m e dependent calcu- l a t i o n s t o be made and a l s o t o p r o v i d e a means of i t e r a t i o n on a r b i t r a r y i n i t i a l v a l u e s of T ( z ) , A(z) and V(z) t o o b t a i n s t e a d y s t a t e s o l u t i o n s .

I n t h e numerical r e s u l t s t h a t f o l l o w , we have a s s w e d t h a t 0 , C h, a a r e independent of pres-

p 7

s u r e and taken v a l u e s f o r a i r a t 1 atmosphere. The v a l u e s of U and p a r e assumed t o b e p r o p o r t i o n a l t o p r e s s u r e . For c a s e s where a b l a t i o n i s s i g n i f i c a n t t h e m a t e r i a l f u n c t i o n s should b e t h o s e of t h e a b l a t e d n o z z l e m a t e r i a l b u t we have found t h a t t h e y d i f f e r only s l i g h t l y a t high t e m p e r a t u r e s from our c a l c u l a t i o n s of m a t e r i a l f u n c t i o n s of t e f l o n , PVC and perspex.

Axial Position

R e s u l t s : I n F i g . 1 a r e shown c a l c u l a t e d a r c r a d i i Fig. 2 Axial p r e s s u r e a s a f u n c t i o n of c u r r e n t a s a f u n c t i o n of a x i a l p o s i t i o n f o r v a r i o u s d . c .

3-

c u r r e n t s , I n Fig. 2 t h e c a l c u l a t e d p r e s s u r e d i s - t r i b u t i o n s a r e shown a s a f u n c t i o n of a x i a l p o s i -

t i o n . Above 2 kA, t h e a r c r e s t r i c t s flow i n t h e _{2 -} n o z z l e and t h e l o c a l p r e s s u r e i n c r e a s e s . I n F i g . 3

t h e c a l c u l a t e d a x i a l Mach number d i s t r i b u t i o n s a r e shown f o r v a r i o u s c u r r e n t s . A t 3 0 kA t h e r e i s a

s t a g n a t i o n p o i n t w i t h i n t h e n o z z l e and a flow of

n

plasma back i n t o t h e high p r e s s u r e tank.

z

5

The c a l c u l a t e d volt-ampere c h a r a c t e r i s t i c s a r e

_u shown i n Fig. 4 , t o g e t h e r w i t h t h e temperature a t

t h e n o z z l e t h r o a t . Also shown i s a curve c a l c u -

l a t e d w i t h t h e n o z z l e r e v e r s e d , t h e t h r o a t t h e n being n e a r t h e e x i t .

Reference:

[l] Lowke, J.J. and Ludwig, H.C. J.Appl.Phys.,

4 6 , 3352 ( 1 9 7 5 ) .

Fig. 3 Axial Mach number d i s t r i b u t i o n

-

0 2 4 6 8 10

Axial Position : cm

Fig. 1 C a l c u l a t e d a r c r a d i i a s a f u n c t i o n of c u r r e n t F i g , 4 C a l c u l a t e d V - I c h a r a c t e r i s t i c s

.I

^{' 1}

1 10 100

ARC CURRENT , K A