THEORETICAL MODEL OF CURRENT-ZERO BEHAVIOUR OF AXIALLY BLOWN ARC IN SF6

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THEORETICAL MODEL OF CURRENT-ZERO BEHAVIOUR OF AXIALLY BLOWN ARC IN SF6

M. Hrabovsky, M. Konràd, P. Skoda

To cite this version:

M. Hrabovsky, M. Konràd, P. Skoda. THEORETICAL MODEL OF CURRENT-ZERO BEHAVIOUR

OF AXIALLY BLOWN ARC IN SF6. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-293-C7-

294. �10.1051/jphyscol:19797144�. �jpa-00219118�

JOURNAL

PHYSIQUE CoZZoque C7, suppldment au n07, Tome

~ u i Z Z e t

page C7- 293

THEORETICAL MODEL OF CURRENT-ZERO BEHAVIOUR OF AXIALLY BLOWN ARC

IN

M.

M.

f.

Research I n s t i t u t e of EZectricaZ Engineering Computer Centre, +

I n t r o d u c t i o n

Much a t t e n t i o n has been r e c e n t l y devoted t o t h e s t u d y of c u r r e n t - z e r o behaviour of high p r e s s u r e , a-c a r c i n s u p e r s o n i c n o z z l e flow /1-5/. R a d i a l conduction enhanced by turbu- l e n c e i s t h e most e f f e c t i v e energy t r a n s - p o r t mechanism w i t h i n a r c column i n t h e c u r r e n t - z e r o region. I n s e v e r a l p a p e r s /3-

t r a n s i e n t behaviour of t u r b u l e n c e do- minated a r c w i t h f i x e d r a d i a l temperature p r o f i l e i s t h e o r e t i c a l l y s t u d i e d , i n f l u e n c e of e l e c t r i c c i r c u i t n o t being taken i n t o account. W e p r e s e n t t h e model of t r a n s i e n t a r c with time dependent r a d i u s 01 conducti- ve c o r e and v a r i a b l e temperature p r o f i l e . I n t e r a c t i o n of a r c w i t h connected e l e c t r i c c i r c u i t i s c o n s i d e r e d i n model c a l c u l a t i o n s Both t h e a r c - c i r c u i t i n t e r a c t i o n and t h e t e m p e r a t u r e - p r o f i l e changes a r e supposed t o have c o n s i d e r a b l e e f f e c t on r e s u l t i n g dyna- mic a r c behaviour.

Model Equations and R e s u l t s of C a l c u l a t i o n

where

i s t u r b u l e n t kinematic v i s c o s i t y and Prt i s t u r b u l e n t P r a n d t l number. To d e s c r i b e e f f e c t o f t u r b u l e n c e on t h e a r c , formulae f o r f r e e t u r b u l e n t s h e a r flow were adapted /3-6/. &

considered t o be pro- p o r t i o n a l t o t h e a x i a l v e l o c i t y c / 6 / , which i s e q u a l t o t h e v e l o c i t y o f sound a t given temperature. P r a n d t l number

i s c o n s t a n t and approximately e q u a l t o 0 . 5 f o r f r e e t u r b u l e n t flow /7/. Then from

1 we o b t a i n kt - ^A

^{c where h} i s c o n s t a n t w i t h t h e dimension of l e n g t h . I n F i g . P

t h e tem- p e r a t u r e dependence of kt/A i s given f o r SF6 a t

rlPa, c a l c u l a t e d from d a t a i n /a/.

For t h e s o l u t i o n of ( l ) , it i s convenient t o i n t r o d u c e t u r b u l e n t h e a t f l u x p o t e n t i a l

kt/A d ~ -

Equation ( 1 ) a c q u i r e s then form

Following assumptions a r e used t o f o r m u l a t e

current is given

law model e q u a t i o n s :

1.

The o n l y e f f e c t i v e energy l o s s mechanism

- 2mE J r r d r

0

i s r a d i a l h e a t conduction due t o t h e turbu- Equations ( 3 ) and ( 4 ) w i t h unknowns

( r , t ) , l e n c e . The t u r b u l e n c e o r i g i n a t e s i n t h e

t

( t ) can be solved u s i n g d a t a of s h e a r flow a s s o c i a t e d with t h e e x i s t e n c e of t r a n s p o r t p r o p e r t i e s o f t h e g a s

c(G) s t r o n g r a d i a l temperature g r a d i e n t w i t h i n t o g e t h e r w i t h c i r c u i t e q u a t i o n s . The mate-

t h e a r c column. r i a l f u n c t i o n s r(S)and c ( G ) , which determi-

2. Arc column h a s c y l i n d r i c a l symmetry. ne t h e e f f e c t of gas p r o p e r t i e s on t h e a r c

Conductive a r c c o r e of r a d i u s r c ( t ) i s behaviour, a r e g i v e n i n Fig. 2 f o r SF^ a t surrounded by t h e zone of i n t e r m e d i a t e tem- p -

^!4Pa.

p e r a t u r e w i t h f i x e d r a d i u s

f o r

t h e C a l c u l a t i o n s were made f o r t h e

w i < h temperature r e a c h e s t h e temperature o f c o l d s e r i a l inductance L and p a r a l l e l c a p a c i t y g a s . The same energy t r a n s p o r t mechanism,

and r e s i s t i v e - c a p a c i t y

branches.

a s w i t h i n t h e a r c c o r e , i s e f f e c t i v e i n t h e Equations(3),

together with cifcuit

i n t e r m e d i a t e zone. e q u a t i o n s were normalized t o reduce number

The energy balance e q u a t i o n h a s t h e n a form of independent c o n s t a n t s . Nondimensional

1

a (1) energy e q u a t i o n and

law a r e g i v e n by

where e f f e c t of t u r b u l e n c e i s d e s c r i b e d by t u r b u l e n t thermal. c o n d u c t i v i t y kt

Conductivity kt i s given by /7/

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797144

A A

where G,

$ are functions normalized to their axial value at t-0, i-i/i

, ^&v/v

-

=

E/Eo, x-r/R and 2 =t/a0, where Zo is characteristic time constant. The behaviour of arc in a given circuit is determined by two nondimensional parameters

further by the initial profile S(x,t=O)and initial axial value Gaol whfch define also the value of parameter x = ( f ^x P(x, t-0) d x ) ; ' The Bessel profile of

within conductive core and logarithmic profile in interme- diate region are supposed to occur at time t-0

6 ( x ,

=

ch +(l-Ch) J ~ ( ~ ~ x ) for =xo (91

A

G(x,'Z-SO) = % ^{In x} for xCO< x

(10) where xco is a normalized radius of con- ductive core at t-0, Gh=Gh/Gao, where Gh

is minimum value of G for which r ) O . For given Gao the values of eh, x ~ ~ , C ( ~ and0(2 can be evaluated from the condition of con- tinuity of heat flux ( i. e. a2/ a x) at x=xco.

Calculated waveforms of current, voltage, axial value Za and radius of conductive co- re xc are shown in Fig. 3 for the case near the boundary for thermal reignition of arc.

Fig. 4 presents the results in the ~ayr's plot giving information about character of dynamic behaviour of the arc. The calcula- ted curve is compared with the results of measurements on SF6 arc in the same elec- tric circuit. It can be seen that observed typical rapid changes of relative derivati- ve of conductivity in the vicinity of cur- rent zero can be properly described by the model.

References

/3/

Swanson

.W. , IEEE Trans. PAS-96 (1977) ,

1697

/4/ Hermann

Ragaller

IEEE Trans.

PAS-96 (1977), 1546

/5/ El-Akkari P.R., Tuma D.T., IEEE Trans.

PAS-96 (1977), 1784

/6/ Thiel H.G. , Proc. IEEE 59 (1971), 508 /7/ Schlichting H., Boundary Layer Theory,

New York, McGraw-Hill, 1960

/8/ Bart1 J. et al., Report O S P E - ~ ~ , 1977, Technical University Brno

Rq.

fiq. 2

/1/ Hermann W. et al., IEEE Trans. PAS-95 (19761, 1165

/2/ Topham D.R., Proc. IEE 120 (1973), 1568