THE MASS FLOW FIELD OF THE FULL CIRCLE ARC

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Submitted on 1 Jan 1979

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THE MASS FLOW FIELD OF THE FULL CIRCLE ARC

W. Tiller

To cite this version:

W. Tiller. THE MASS FLOW FIELD OF THE FULL CIRCLE ARC. Journal de Physique Colloques,

1979, 40 (C7), pp.C7-313-C7-314. �10.1051/jphyscol:19797154�. �jpa-00219129�

JOURNAL

PHYSIQUE CoZZoque C7, suppZ6ment

n07, Tome

JuiZZet

page C7- 313

THE

THE

W.

Tiller.

HochschuZe der Bundemehr, Miinchen,

Introduction

As reported i n former papers /1,2/ a radial f r e e f u l l c i r c l e a r c was operated in an argon atmosphere.

Due to the experimental conditions and with respect t o cylindrical coordinates r, 9 , z the statements

a / 3 t

0 and 3 /3p = 0 hold f o r t h i s arc. The mass flow in the a r c cross section as a r e s u l t should prove Waecker's theory /3/ of a r c motion anc displacement. Therefore one must know the tempera- t u r e d i s t r i b u t i o n . I t can be measured spectroscopi- c a l l y as described i n /2/. One has t o consider t h a t those measurements a r e correct only i n the case of local thermal equilibrium. For a r c currents g r e a t e r than 50 A t h i s condition i s f u l f i l l e d .

Results of Temperature Measurements

In continuation t o the measurements reported in /2/

the a r c was investigated a t ' c u r r e n t s between 50 W and 100 A and a r c r a d i i from 35 mm t o 45 mm. As an example f i g u r e 1 shows the temperature f i e l d f o r an a r c current I

75 A and a radius of 4G mm.

r l rn rn

f i g . 1 temperature d i s t r i b u t i o n

111 a d d i t i o n t o former r e s u l t s f i g u r e 2 g i v e s t h e c o r r e l a t i o n of t h e maximum temperature i n t h e a r c core with t h e a r c c u r r e n t and t h e c u r v a t u r e o f t h e a r c .

f i g . 2 temperature maxima vs. c u r r e n t and c u r v a t u r e of t h e a r c

The Mass Flow Field

By use of the known temperature d i s t r i b u t i o n the velocity of the mass flow can be evaluated from the convective term of the energy equation:

% 3.0s = G E ' * v ~ $

and from the continuity equation: v . ~ ? = 0 The expressions

and v 2 S (S = heat f l u x poten- t i a l ) a r e derived from the measured temperature f i e l d T ( r , z ) . The c o e f f i c i e n t s A(T)

thermal con- d u c t i v i t y , 6 (T)

e l e c t r i c a l conductivity, u(T)

s p e c i f i c radiation and y (T)

mass density a r e t a - ken from the l i t e r a t u r e . The e l e c t r i c a l f i e l d strength E i s measured by means of probeetech- niques. One has t o s t a r t the evaluation in the cen- t e r plane (z=O) which i s a l s o the plane of symmetry of the a r c . There, the velocity has only a radial component vr, t h a t means v,(,,~) = 0. Using t h e ve- l o c i t y d i s t r i b u t i o n v r ( r ) a t z

0 a s an i n i t i a l value one can compute the QV, and pvr components i n small steps

solving a l t e r n a t e l y the two equations ventioned above. Introducing the vector potential 3 f o r t h e mass flow by

, the evaluation of stream 1 ines including a constant mass flow becomes possible. Because of 3 /3p

0 , the vector? has only an azimuthal component, and so one has only t o solve a plane problem. The d i - stance (z2-zl) between two stream l i n e s a t a spe- c i a l radius ro i s given by the integral

IL

LYi,zL-Yr,,&,

have t o be chosen i n a way t h a t t h e difference ~ 5 6 , ~ ~ - y6,% i s constant.This evaluation was done f o r a constant mass flow of 0.05 mg/s per cm a r c column. As a r e s u l t one gets a symmetric quadrupol whirl in the a r c cross sec- tion as shown i n f i g u r e 3 .

--..- rl mm

f i g . 3 mass flow f i e l d i n t h e a r c c r o s s s e c t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797154

In addition t o the stream l i n e s several isothermes and a l s o the curve y$.vS

0 a r e shown i n t h i s figure. This curve i s the locus of a l l pointswith- out convective energy transport, i . e . there occurs only thermal conduction. By forming the 1 imi t i n g value, i t can be shown t h a t the temperature maximum as well as the whirl centers and the stagnation point a r e extraordinary points of t h i s curve.

Interpretation of the Mass Flow Field

Discussing the conservation equation of momentum, one has a p o s s i b i l i t y t o i n t e r p r e t the mass flow f i e l d and t o v e r i f y the r e s u l t s . Neglecting the small temperature dependence of viscosity 7 i n the a r c , the equation of conservation of momentumreads:

pd.od = j'* d ^-vp+ ,Z ( $ & - v x v ~ $ )

j =

e l e c t r i c a l current density, B

magnetic f i e l d , p

pressure. Taking the curl of the l e f t and r i g h t hand term and neglecting the curl of the i n e r t i a l force (low Reynolds number), one gets:

o = v x ( ] x P ) -q(v%v'v'f 1 .

The introduction of the v o r t i c i t y by:

= v x 3

leads t o the Poisson equ.

v2&=-$

_4i 5 ) .

The curl of the Lorentz force

_{( i x} 8 ) which has azimuthal direction can be evaluated as f o l l ows: From the temperature d i s t r i b u t i o n and the measured e l e c t r i c a l f i e 1 d strength one knows the e l e c t r i c a l current density. Soluiion O F the Gio t Savart equation y i e l d s the magnetic f i e l d of the a r c . The external magnetic f i e l d i s known from the experiment. Therefore a1 1 terms- of the curl

v%@.(;.)

= ; f ( ~ r ( a ~ - k)+ ~~k ^J

can he coiputed. Because 7 i s constant, one can t a k e - as an analogon t o the d i f f e r e n t i a l equation of the e l a s t i c membrane -, the curl of the Lorentz force as an area force on such a membrane. The de- f l e c t i o n of the membrane i s then proportional t o the value of the v o r t i c i t y & . The curl of t h e Lo- rentz force as well a s i t s two terms as functions of the radius f o r a plane z

0.86 mm, which con- t a i n s approximately the whirl centers and the s t a - gnation point, i s shown in f i g u r e 4.

For the same plane, figure 4 shows a l s o the d i s t r i - bution of the v o r t i c i t y , calculated from t h e mass flow f i e l d ( f i g . 3 ) . A comparison of the two curves by use of the membrane model mentioned above

A 4

( (

)

area force, ' e p w

^de-

f l e c t i o n ) shows qua1 i t a t i v e l y ( i .e. without respect t o the boundary values of ) the r o l e of t h e curl

11

rlmm

, . 1 . 1 . 8 , 1 , 1 . I . I . 1 . , . ,

35 '0 65

f i g . 4 curl of the Lorentz force and v o r t i c i t y of the Lorentz force in generating the quadrupol whirl. Horeover one r e a l i z e s , evidently shown by the unsymmetry of the term Br ( a j / a r - j / r ) which i s caused by t h e curvature of the a r c , t h a t j u s t t h i s curvature i s the reason f o r a quadrupol whirl contrary t o a double whirl as formerly expected.

Conclusion

The experiment shows, t h a t a magnetically deflected curved a r c can be fixed i n a equilibrium position.

This equilibrium i s determined by the f a c t t h a t there i s a magnetohydrodynamic mass flow through the a r c core which compensates the a r c motion to- wards the center of curvature due t o thermodynami- cal e f f e c t s .

/1/

T i l l e r , Proc, of the Xlrh Int. Conf. on Phen. in Ionized Gases (1973) /2/ W . T i l l e r , Proc. of the XIIth I n t . Conf. on

Phen. in Ionized Gases (1975) /3/ H. Maecker, Proc. of t h e IEEE 59, 439 - 449

(1971)