A steady-state fluid model of the coaxial plasma gun

Volume hA, number 1

PHYSICS LETTERS

16 April 1979

A STEADY-STATE FLUID MODEL OF THE COAXIAL PLASMA GUN G. HERZIGER, H. KROMPHOLZ, W. SCHNEIDER and K. SCHONBACH Institut fOrAngewandte Physik, Technische Hochschule Darmstadt, Germany

Received 7 August 1978 Revised manuscript received January 1979

The plasma layer in a coaxial plasma gun is considered as a shock front driven by expanding magnetic fields. Analytical steady-state solutions of the fluid equations yield the plasma properties, allowing the scaling of plasma focus devices.

A two-dimensional fluid model is considered, describing a current carrying plasma layer travelling along a tube between two coaxial electrodes [1]. The plasma conductivity is assumed to be infinite. Hence the current is flowing in a thin sheet separating plasma and magnetic field. Driven by the/ plasma layer propagates axially withXaB-force speed uthe far exceeding the acoustic speed in the undisturbed gas. Hence a thin shock layer is bounding the plasma layer

at its front. The gas enters the layer across the shock interface and its motion becomes parallel to the plasma layer.

(3) the equation of energy conservation: r~’(d/dr) [p i.~ru

5(h+

~

v~)1= p0u (h0 + u~) ~

,

(3)

(4) the pressure balance equation at the current sheet: 2 cos2O ; (4) Pm = p0 + p0U r is the distance from the axis, i~is the angle of the normal to the shock interface with the axis. The plasma layer has a mass density p(r), a pressure p(r) and an enthalpy h(r). The undisturbed ambient gas is characterized by p 0,p0, h0. The thickness of the layer is i~(r),v5(r) is the velocity of the gas flow along the

Experimental observations indicate that all relevant properties of the plasma layer are approaching stationary state in short time [2,31. Hence the running down phase of a coaxial gun can be described by stationary

2 2 ‘ Pm =I~LOI /8ir r

equations. Analytical solutions of stationary fluid equations have been reported by several authors [1,4,5].

Ispace. being the current and b

In this paper more general analytic solutions are pre-

In most cases of practical interest p 0 ~ Pm and 2. Hence both terms can be neglected. In addih0 ~ U tion, the second term in eq. (2a) can be neglected if r> 1.2 r 0, r0 being the radius of the center electrode.

sented which plasma parameters during the runningfrom down phaseallcan be derived. The stationary equations of a plasma layer in a reference frame moving axially with the speed u are (1) the equation of mass conservation: 1(d/dr) (pL~rv

r

5)= p0u

(1)

,

(2) the equation of momentum conservation, transverse and normal to the layer: 2 sin (2a) 1 2 ~,

r p

(d/dr) (pz~rv5+ par) = p0u 2cos2t~—p~cosiI(v~+p/p)di~/dr—pm— p 0u

54

current sheet. Pm is the magnetic pressure:

1o

the permeability of free

‘~

Assuming ideal gas conditions in the layer the enthalpy

is given by h

=

[71(7 1)1 (p/p) —

(6)

,

‘y being the ratio of the specific heats. With the additional assumption z~(r0)= 0 analytical solutions for the particle density n and the temperature T can be obtained [5]: (x2 1)2 2(x2lnx2—x2i-1) = = (7) fl0 00 71 x —~

0,(2b)

(5)

-~

-~-

—~--—

————-

,

Volume 71A, number 1

PHYSICS LET1~ERS

5,___,~~

I

I

:

16 April 1979

2~

,—~-o 5

N

‘,a’rO.5

1

—

-J

~0 0~

~

a~10 ____________________

I

3

T I

—

I

02 I

—

~0.1

‘.~

•—

~ uJ

Cr

w I

w ~

C

2

3

t

Fig. 2. Image converter photograph showing contour of the plasma layer in a coaxial gun. The profilethe (dashed lines) is calculated using eq. (1 1) witha = 0.95.

1 w

z (_)

a

_____________________

1

2

53

T I—

of the layer u exceeds a minimum speed umin:

RADIUS r/r0

112 Fig. the and 1. (a) mass Theper normalized ualt area ~p/r profile of the plasma layer z(r)/ro

versus normalized radius r/ro for different values of a; (b) normalized temperature 0p0

T/T1, density n/n0 and thickness of the plasma layer ~/ro versus radius r/ro.

=(

~oi~

= (PmfrO))

U >~

2p 8~

0~ )l/2 (9) u is a free parameter, varying between umin and infinite. Instead of u in the following we shall use a generalizing parameter a, which varies between one und zero:

withxr/r

T

—

0,2lnx2 — x2 + 1 y—1 7 x (x2— 1)2

(8)

1Pm(’o) 1/2 11012 1/2 2 ) = (8~2~POU2) a=(~p0u

(10)

By use of eqs. (2b), (10) and the relation az/ar = —tg ~9 we get an expression for the geometry of the current

with T 1

Pm(ro)Iflok, k being the Boltzmann constant.

Density and temperature are drawn as a function of

the radius in fIg. lb assuming 7 = 5/3. Independent from current and gun geometry the value of the density at the center electrode is 2/(7 1) times the density of the ambient gas n energy density on the other hand, which0.atThe r = thermal r 0 is equivalent to the

sheet z(x): ~ — ~—

current I and radius r0. By neglecting the centrifugal term in eq. (2b) anaiy-

tical solutions for the profile z(r), the velocity v(r) of particles inside the plasma layer, the mass per area m(r) L~(r),and the layer thickness ~(r) can be derived. This neglection admissible in a rangethe 1
[x~x2

~+

—

magnetic energy density, depends quadratically on

2)1!2 —(1— a2)1/2

—

—_

—

—

a a2)1!2

2 in —

a

1 + (1 — a2)1!2] (11) Profiles at different values a are shown in fig. 1 a. All

possible steady-state profiles are located between a = and a = 0, the latter curve is coinciding with the ordinate Another quantity following from eq. (1) is the velocity of the gas flow along the current sheet: ~s 2a z(x) 2—i’o~ (12) II x The mass per unit area is derived from eqs. (1), (12): — —

= — (x2

—

1)2 r 0

r0~0

4ax

~(~)

(13)

55

Volume 71A, number 1

PHYSICS LETTERS

As shown in fig. 1 a the mass is slightly depending on a. To get an analytical expression for the thickness L~(x)we have to make use of eq. (7) where ideal gas conditions have been supposed: ~

=

—

2lnx2 y—1 —~---—x(x

—

X +

~ 2az(x) To

(14)

Graphs of eq. (14) with a = 0 and a = 1 are shown in fig. lb. From eqs. (7)—(14) all plasma parameters describing the running down phase at plasma focus devices can be calculated. A selection of frequently used parameters is plotted in fig. 1. With the exception of the profile of the plasma layer the individual curves do not depend strongly on the parameter a. Comparing literature data it appears that all focus devices are operated in a range 0.5
56

16 April

1979

files to the contour of the plasma layer the parameter a was determined to be 0.95 (fig. 2). On the other hand, a is defined by current I, axial speed u, the radius of the center electrode r 0 and the speed U = 9 X 106 cm/s has been measured by means density (eq. (10)). With A current I = 8 X l0~A and a of probep0techniques. r 0 = 0.8 cm and P~= 2.5 Torr we get a 0.9 from eq. (10). The agreement with results obtained from the image converter measurements demonstrates the applicability of the model. The shock wave model applied for obtaining an analytical solution for the equations treating the running down phase is not necessarily a complete description in the sense that it fails to include diffusion effects. However, it proved to be a satisfactory approximation for obtaining the gross dynamical variables describing the essential features of the running down phase.

References [11 F.J. Fishman and H. Petschek, Phys. Fluids 5 (1962) 632; F. Gratton and M. plasma Vargas,physics Proc. 7th Europ. Conf. Contr. fusion and (Lausanne, 1975)onp. 64. 121 J.C. Keck, Phys. Fluids 5 (1962) 630. [31 J.W. Mather, Methods of experimental physics, Vol. 9B (Academic Press, New York, 1971) p. 187. [41 T.H. Butler, J. Henins, F.C. Jahoda, J. Marshall and R.L. Morse, Phys. Fluids 12 (1969) 1904. [5} G.J. Pert, Brit. J. Appl. Phys. 1(1968)1487; 2 (1969) 429.