PROPAGATION OF THE LEADER OF A LONG SPARK IN AIR WITHOUT PARTICIPATION OF THERMAL IONIZATION PROCESSES

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PROPAGATION OF THE LEADER OF A LONG SPARK IN AIR WITHOUT PARTICIPATION OF

THERMAL IONIZATION PROCESSES

Henryk Ryżko

To cite this version:

Henryk Ryżko. PROPAGATION OF THE LEADER OF A LONG SPARK IN AIR WITHOUT PARTICIPATION OF THERMAL IONIZATION PROCESSES. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-427-C7-428. �10.1051/jphyscol:19797208�. �jpa-00219188�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, page C7- 427

PROPAGATION OF THE LEADER OF A LONG SPARK I N AIR WITHOUT PARTICIPATION OF THERMAL IONIZATION PROCESSES

Henryk Ryiko.

RoyaZ I n s t i t u t e o f TeehnoZoau. S-100 44 StoekhoZm, a e d e n .

We consider the i n i t i t a l stage o f t h e l e a d e r propa- g a t i o n i n a p o i n t - t o - p l a n e a i r gap a t s w i t c h i n g surge voltage (impulse v o l t a g e w i t h f r o n t d u r a t i o n o f some hundreds o f micro-seconds). I n t h i s stage corona ahead o f t h e l e a d e r has n o t reached the plane e l e c t r o d e so t h a t t h e c u r r e n t o f t h e gap i s s t i l l small.

The e l e c t r o n i c c u r r e n t o f t h e leader channel i s

where r - r a d i u s o f t h e channel crosssection, i n cm

q - elementary charge i n As

v - 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 n cm/s Ne - average e l e c t r o n d e n s i t y i n cm -3, On t h e o t h e r hand i = Q v where Q - e l e c t r o n i c charge o f t h e channel i n As/cm.

Hence

The value o f $ depends mainly on the d i f f u s i o n o f t h e e l e c t r o n s , on the i o n i z a t i o n due t o t h e r a d i a l f i e l d o f t h e channel and on gas expansion i t h e channel. As we a r e n o t a b l e t o take i n t o account a l l thes processes, we . c o n s i d e r o n l y t h e r a d i a l d i f f u s i o n o f e l e c t r o n s which d r i f t along t h e chan- n e l a x i s . I n t h i s way t h e lower l i m i t o f F w i l l be c a l c u l a t e d .

We assume as a f i r s t approximation t h a t F a t t h e cathode end o f t h e channel equals zero, f u r t h e r t h a t e l e c t r o n s are moving i n a u n i f o r m f i e l d and t h a t a Maxwellian d i s t r i b u t i o n o f e l e c t r o n v e l o c i - t i e s i s t h e case. Then t h e r a d i u s o f t h e channel d e f i n e d as t h e average r a d i a l displacement o f e l e c - t r o n s from t h e channel a x i s i n time t i s

where D-electron d i f f u s i o n c o e f f i c i e n t .

v e

We s u b s t i t u t e D =

$

where 7 - mean random energy o f e l e c t r o n s i n V ,

u - m o b i l i t y o f e l e c t r o n s i n cm2/vs

k - d i s t a n c e o f t h e considered c r o s s - s e c t i o n o f the channel from i t s cathode end, i n cm

E

-

Hence f i n a l l y

We assume t h a t t h e gas pressure i n t h e channel i s equal t o atmospheric pressure ( j u ~ t i f i c a t i o n -

see l a t e r ) . Using datal r e l a t i n g E and E values we o b t a i n nummerical r values presented i n Table 1 Tab& 1 . R a b od t h e channnel ctionn -n e d o n

(&owen. LiXX), i n cm.

I n o r d e r t o determine t h e range o f Ne values we assume t h a t Q i n equation (1) i s equal t o the p o s i - t i v e charqes per u n i t l e n q t h o f t h e channel, measu- r e d i n 2. The average value o f t h i s charge i s 13-7 As/cm. I n s e r t i n g t h i s value o f Q and F values from Table 1 i n equation $1) one o b t a i n s t h a t Ne i s ir

t h e range 101 1 - 101 electrons/cm3. The a c t u a l r values a r e l a r g e r than those o f Table 1, because we have taken i n t o account o n l y t h e d i f f u s i o n o f e l e c t r o n s . I f f o r i n s t a n c e t h e a c t u a l r values are t h r e e times l a r g e r than those i n Table 1, t h e ac- t u a l d e n s i t y of e l e c t r o n s i s 1010

-

L e t us now e s t i m a t e t h e temperature o f the channel under assumption t h a t t h i s d e n s i t y o f e l e c t r o n s i s produced by t h e thermal i o n i z a t i o n . I f t h e r a t i o o f t h e e l e c t r o n d e n s i t y t o t h e p a r t i c l e d e n s i t y i s low and i f t h e e l e c t r o n s are i n thermal e q u i l i b r i u m w i t h gas p a r t i c l e s , Saha's equation can be w r i t t e n i n the form

where N - d e n s i t y o f e l e c t r o n s i n cm::, N~ - d e n s i t y o f molecules i n cm ,

P - gas pressure i n t o r r , T - gas temperature i n OK,

Vi - i o n i z a t i o n p o t e n t i a l o f t h e gas i n V, q - elementary charge i n As,

k - Boltzmann's constant i n WS/'K.

As t h e i o n i z a t i o n p o t e n t i a l s o f N2 and 02 a r e 15,5 V and 12,2 V r e s p e c t i v e l y , we assume i n t h i s approxi- mate c a l c u l a t i o n V i = 14,5 V. Because t h e i n p u t c u r r e n t o f t h e l e a d e r i s low, t h e r a t e o f r i s e o f t h e channel temperature should be a l s o low. As t h e channel w i l l expand, i t s gas pressure cannot d i f f e r appreciably from t h e atmospheric one. Therefore we s u b s t i t u t e i n (3) p = 760 t o r r . With these nummeri- c a l value one o b t a i n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797208

Ne = 2.7 x 10'' w i t h u n i t s cmm3 and OK.

As i t i s shown i n Fig. 1 t h e e l e c t r o n d e n s i t i e s 1010

-

F i g . 1

.

We question now i f the considered channel can reache these temperatures. L e t us t h e r e f o r e compare t h e i n p u t energy t o t h i s channel w i t h t h e l o s s o f energy by r a d i a t i o n . As a degree o f i o n i z a t i o n i n t h e channel i s very low we consider these channel as unionized and consequently r a d i a t i n g as a b l a c k body. The l o s s by r a d i a t i o n p e r u n i t l e n g t h from t h e channel i s

where o-Stefan-Bol zmann ' s constant.

A t t h e temperature 4 9 0 0 ~ ~ ( r = 0.36 cm) t h e power l o s s i s about 7 kW/cm. The corresponding power l o s s a t a temperature o f 4 4 0 0 ~ ~ ( r = 1 cm) i s about 13 kW/cm.

The i n p u t power p e r u n i t l e n g t h i s

With nummerical values: Q = As/cm, E = 3 kV/cm and corresponding v = 25 x 105 cm/s3 t h e i n p u t power p e r u n i t l e n g t h i s 750 W/cn.

This rough c a l c u l a t i o n i n d i c a t e s t h a t i n t h e consi- dered c o n d i t i o n s t h e thermal i o n i z a t i o n as a source o f e l e c t r o n s i s r u l e d out.

dO E Q V

m = nr2 c p ~

where - d6 dt - r a t e o f temperature r i s e o f t h e chan- n e l i n OC/s,

c - s p e c i f i c h e a t o f a i r i n Ws/g OC,

G~ - s p e c i f i c weight o f a i r i n g/cm3.

With nummerical values: Q = As/cm, E = 3 kV/cm, cp = 1 Ws/g OC, G = 1.3 x 10-3 g/cm3, one o b t a i n s

* =

This temperature r i s e i s s u f f i c i e n t t o lower gas d e n s i t y i n t h e channel s i n c e each new s t e p d i s - charge r u n i n i t i a l l y along the same path.

We b e l i e f t h a t i n t h e considered case t h e l e a d e r c o n d u c t i v i t y undergoes l a r g e f l u c t u a t i o n s . The e l o n g a t i o n o f t h e l e a d e r occurs when t h i s anduc- t i v i t y i s reduced d u r i n g the s t e p discharge mani- f e s t e d by a r e i ll u m i n a t i o n o f t h e channel.

Conclusion: Mechanism o f t h e l o n g spark i n a i r a t p o s i t i v e s w i t c h i n g surge v o l t a g e d i f f e r s i n a i n i 4 t i a l stage from t h a t a t p o s i t i v e impulse v o l t a g e by d e f i c i e n c y o f t h e thermal i o n i z a t i o n .

References :

( 1 ) L.G.H. Huxley, A.A. Zaazou, "Experimental and t h e o r e t i c a l s t u d i e s o f the behaviour o f slow e l e c t r o n s i n a i r . " Phoc. Roy. Soe. A 196, pp. 402-426, 1949.

( 2 ) B. Ganger, " E l e k t r i s c h e F e s t i g k e i t von L u f t - i s o l i e r s t r e c k e n b e i hohem Schaltspannungen."

B&. SEV., pp. 227-236, 1971.

( 3 ) H. Rytko, " D r i f t v e l o c i t y o f e l e c t r o n s and i o n s i n d r y and humid a i r and i n water vapour."

Phoc. P h y ~ . S o c . , v o l . 85, pp. 1253-1295, 1965.

( 4 ) R.T. !.laters, "Streak photography and o t h e r s t u d i e s o f t h e l o n g spark i n a i r . " Pmc. I&.

Cond. Gas Dhch. and E l k & . Supp. lad., Leatherhead, London: Butterworth, 1962, pp. 38,53.

L e t us estimate t h e r a t e o f r i s e o f channel tempera- t u r e when t h i s temperature does n o t d i f f e r conside- r a b l y from t h e temperature o f t h e surrounding a i r . Because under these c o n d i t i o n s the l o s s by r a d i a - t i o n and by heat conduction a r e r e l a t i v e l y low, we assume t h a t t h e whole i n p u t energy i s converted i n - t o h e a t energy o f t h e channel. I f t h e qas pressure i n t h e channel i s unchanged, we w r i t e