AIAA-98-3800

NUMERICAL RESEARCH OF DYNAMICS OF STREAM IN THE STATIONARYPLASMA THRUSTER CHANNEL OF A VARIABLE SECTION

Boris Arhipov,Design Bureau Fakel, Kaliningrad (obi.), Russia

Elena Goghaya,Kaliningrad state university, Kaliningrad (obi.), Russia

Nikolay Nikulin,Kaliningrad state university, Kaliningrad (obi.), Russia

ABSTRACT

The outcomes of numerical research of influenceof the change of geometry of SPT digit channel ondynamics of a plasma flow and on the main outputintegral parameters of a thruster are represented inthe paper. It's shown that the extension of a channelin the output part results in decreasing a flow of high-energy ions on walls in an output part of a channeland improving the integral characteristics of athruster: a force of a thruster and its life time. It'sshown the change of geometry of a digit channel byits extension in the output part is one from paths ofincreasing a thruster life time.

INTRODUCTION

The spent earlier numerical simulation of currentof a plasma flow in a SPT digit channel has shownthat in an output part of a channel the plasma flowdiverges and causes destruction of ceramic wallslimiting a channel and the speed of erosion of wallsdecreases with current of time at the expense ofdecreasing a flow of ions on walls near to an outputedge of the accelerator that is there where the ionshave energy E ~ lOOeV which has enough fordestruction of walls [1]. Life time of a thruster inmany respects also is determined by durability of adigit channel. Therefore one from paths of itsincrease is the change of geometry of a digit channelby its extension in the output part.

The purpose of this paper is the numerical researchof influence of change of geometry of a digit channelon dynamics of current of a plasma flow and onoutput parameters of a thruster - force of a thruster,life time, and also research of a capability ofmanagement of these parameters by profiling of theSPT channel walls.

l.THE DESCRIPTION OF THEMATHEMATICAL MODEL

considered. Density of plasma even in a zone ofionization is rather insignificant: characteristic densityof charged particles nj = 10 n -5- 1018nT3

10 •=• 10density of neutral atoms g= 10 •=• 10 m [2].In these conditions in a volume of a channel thecollisions an electron - atom, and electrons with wallsare dominating above all other kinds of collisions [2]:

0)

Probabilities of recombinations of ions involume of a channel, collisions of ions with eachother and with neutral atoms is neglectly small, asthe role of triple and quarter collisions is small also.The pair collisions - collisions of atoms of neutral gaswith electrons are taken into account only in thispaper.

At discharge voltage Ud = 220V a temperatureof electrons in volume of a channel is Te= 3 H- 20 eV[2], the Debye radius of electrons makes Dc =(0.4-r 1)-10 m,, electron plasma frequency - oc =.(2 4- 5) 1010c~', Larmor frequency of electrons COH

= 3-10 c"1 and accordingly a Larmor radius ofelectrons RLc = V& I at H = 1,5 • 10"3 m.For ions the Larmor radius RU will be in thousandstimes more Ru » L therefore influence of amagnetic field to motion of ions can be neglected.

In these conditions with a huge degree ofprobability the derivated ion escapes a channel of thethruster or perishes on a wall, and the motion ofseparate particles is described by an equation of theNewton - Lorenc [2]:

dt (2)

d t

At numerical simulation in view of azimuthaluniformity of plasma in a channel the quasistationarymotion of ionizing gas in two-dimensional area is

Copyright© 1998 by Design Bureau Fakel. Published by theAmerican Institute of Aeronautics and Astronautics, Inc. withpermission.

Taking into account that the characteristic valueof plasma is equal L ~10"2m it is possible to considerit quasineutral: De « L, HJ ~ n<..

It is known that the most general equationcircumscribing a kinetics of environment in

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rarefied plasma is the classical Bolcman equationdescribing evolution of a cumulative distributionfunction owing to free motion of particles which byconsideration of pair elastic collisions has a kind [3]:

d t d r m 3V

r.,f ,t)f ,(?,¥',, t ) - f ( f ,

(3)

^V, t,)\dD.

where g = Vi - V - relative speed, %- scattering

angle, cr - effective section of dispersion, Q - body

angles, cr>/; = The difficulties of the

solution of a Bolcman equation in main are stipulatedby complexity of an integral of collisions. Neglectinglatching of radiation, multistage processes andrecharge processes, change of speed of heavyparticles at their collisions with electrons (Vz &VZ)we obtain an integral of collisions as follows [2]:

dV (4)

where <j \m\V j - full section of ionization.The system of kinetic equations for neutral atoms

and ions then has a kind [2]:

intowhere (ov}imne = \feaim\V\dV . Takingaccount a quasineutral condition n; = r^ in (5) insteadof iritis necessary to substitute:

(6)

Thus function /^enters in a system only through(aV} which depends on some mean energy ofelectrons. Using quasistationary approximation -approximation of a constant drain [2] we supposethat terms d-f ld-t=Q.As in the thruster spread on speeds is small incomparison with a flow velocity it is possible to

consider a cumulative distribution function fproportional to 8- function of Dirac [2]:

(7)

It is considered that neutral atoms move alonq achannel with a constant speed VQa .and their densityon a radius not vary in the whole channe and at theextension of a channel they continue to move along anaxis of a channel and their density variesproportionally to the profiling factor kp

Dynamics of electrons in a channel of the thrusteris rather difficult. In activity the approximation of anelectron background [2] is used. In thisapproximation for obtaining distribution of anelectrical field in a volume of a channel theequation of a termolized potential obtained in thesupposition of infinite conductivity of plasma andconstancy of a temperature of electrons T,. onmagnetic lines is used:

n(r)(8)

where O (7) - a termalized potential, valueconstant along a magnetic line, y - "number" of amagnetic line. For it it is necessary to set adistribution of electron temperature T"e{j} andelectrical potential <&*( y ) along some referenceline intersecting all magnetic lines. The ratio (8)means presence of Bolcman distribution of electronsalong them.

As in SPT the static electrical and magneticfields are used the ratio is executed:

(9)

where E - electric field strength, O.- potential of anelectrical field.

Thuse for simulation of current of a plasma flowin ExH fields with the registration of processes ofionization, acceleration and change of geometry of adigit channel is used the following set of equations:

dr £, 3V $H &

Qa^- = -^]ftndr;n= \fdV;J= \VfdVdt (, I J,

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«,. = ne = n; E = -V®

= n

S z=o = £ o z = o = no'*

(10)

= [arccos(l-<?)]/(l-z0)

= r2(L)-r2(0) = rl(0)-rl(_r2(0)-r.(0)-

_where £ =

e<E>dimensionless

parameters of a similarity, analogs of Frud andKnudsen numbers accordingly; L - length of achannel; n1=n0+go;

.(ID

_/J,_/^ - cumulative distribution function of ions andneutral atoms; e- charge of an electron; g- density ofneutral atoms; j3 = /3(Te)- factor of ionization(known function of electron temperature T,,;¥(/•,£) - function of a magnetic flow; / -index of amagnetic line; G - two-dimensional area, in which isconsidered of problems; Zy section, since which isset change of profile of a channel; 5- value ofassigned change of width of a channel in an outputsection; kp- a profiling factor (fig.l).

n (p

———t

On lateral walls of a channel r,(z) and r2(z) thecondition of an absorption of ions is put:

(12)

For closing of a set of equations (10) the modelrelations of a potential O(z) and a temperature ofelectrons" T=(z) on an axis of a channel (fig.2) areused. As it is considered T5 as a constant on amagnetic line, on this line a factor of ionization /?will be constant too. The diagram of relation J3 fromtemperature of electrons Tc in case of Maxwelldistribution of electrons on speeds used in the paper isshown in a fig.3.

0,4 0,6z/L

Fig.2: Distributions of a potential O anda temperature of electrons

i//sec

8 12 16 T.eV

Fig.3 :Relation of a factor of ionizationfrom a temperature of electrons

The external magnetic field in a digit channel isapproximated by a following ratio:

= Z+ A

Fig.l

(13)

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where A, r0 - any parameters, y -is proportional to amagnetic flow: y = ^f0- *?(/%::) and is connected toa strength of a magnetic field by the followingratioes:

\_dy_r & (14)

Parameter A characterizes a curvature of magneticlines and parameter r0 - determines an axis wherelongitudinal making of magnetic field Hz is equal tozero. At concurrence of this axis with an axis of asymmetry of a digit channel the picture of magneticlines is symmetric relatively this axis. By determiningexperimentally Hr and Hz it is possible to selectparameters A, TO for the model description of amagnetic field.

Thus the above mentioned set of equations (10) ata specific magnetic field (13) and boundaryconditions on a channel input allows to describeprocesses of ionization and acceleration of an ionicflow.

For the solution of a set of equations (10) theeffective numerical algorithm combining finitedifference methods with a method of large particles isused. Coordinates and speed of large particles aredetermined by an integration of equations of motion(2). Concentrations and ion flow densities aredetermined under the formulas:

4?)=(15)

where H-nucleus of an integral transformation, Q-weights of particles.

At simulation of a process of interaction of a flowof accelerated ions with walls of a digit channel thespeed of motion of a sputtered surface is describedby an equation [2]:

— =dt " V

where v =

(16)

-;J0 —ion flow density atP t

input, hw- width of channel walls, Tmax- life time,Ma, p - mass of atoms and density of a material of

walls, Kv((p,E^ - volumetric factor of sputtering of

the given material, j], • n - projection of density of adropping flow on a normal to a surface (fig.l).

Thus for calculation of a process of degradationit is necessary to know in each instant distribution jjand Kv((p,E-) . In paper the distribution of the ion •flow density j, (z),obtained hi an outcome ofnumerical simulation of current of plasma in achannel is used. At numerical simulation of processof walls degradation it is necessary that a structureand the parameters of an ionic flow which are fallingout on initial boundary of a channel do not vary withcurrent of time:

j\z,(t) = const; Et\z,(/) = const (17)

Volumetric sputtering factor Kv(ip,E,f can besubmitted as product [4]:

where K1(.E'/j - factor of sputtering at a normal

falling of an ion on a sputtered surface; K (cp)-dimensionless function which is taking into accountinfluence of deviation of an angle of falling of an ionfrom normal on a value of a sputtering factor. Therelations of factors K . ( E . ) and K ( ( O ) , used incalculations,

K,at/ion0,25

0,20,15

0,10,05

04

are taken from [5](fig.4,a,b).

^

^^

^•^

"^^

^

0 80 120 160 200 240 280E,eVa

1.2

0,8

0,4

18 36 54b

72 90

Fig.4: Relation of a factor of sputtering a)-

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from energy at <p = 0; b) from an angleOn basis of datas about a flow of plasma on a

shear of a channel the integral characteristics arecalculated on which the activity of the thruster isestimated: a thruster force, efficiency of a thrusterforce TIF = Fp / 2fhND , a thruster life time Tdefined on erosion of walls of a channel. The thrusterforce is calculated according to [2]:

FF = (17)

where s - is an output section of a channel, V - meanspeed of a going out flow, Vn - projection of a speedon a normal to an output section.

2.OUTCOMES OF NUMERICALCALCULATION AND THEIR ANALYSIS

The calculations were conducted for SPT modelwith the following parameters: length of a channelL= 4 • 10~~ m, external radius of a channelr2=6- \Q~2m, an internal radius ^=3,6- 10~2/«,gasflow density on input q0 = 3,46-lQ2lm~2 • sec"',a degree of ionization of gas on an input a - 0,01,parameter of a curvature of a magnetic lines A=10,r0=(ri+r2)/2.Geometry of digit channel walls was set in agree with(10). Parameters Z0 and 8 (fig.l) were varied. Asworking substance Xe was used.

The outcomes of numerical simulation have shownthat the extension of a digit channel of a thruster up tovalues 8 = 0,18 that corresponds to an angle oferosion a = 25° does not result in essential changesof a structures of an ionic flow (fig.5a,5b) that is inagree with the experimental data [5], but is observedin accordance with the extension of a channelsignificant decreasing of ions flows on walls in outputparts (fig.6) connected that at the extension of achannel flows of ions which were dropping out on theinitial boundary leave a channel. In accordance withincreasing of throwing out mass from a thruster aionic current I|, a thruster force FF and efficiency of athruster force t]F increases (fig.7,8,9) at all values ofdischarge voltage Ud (fig. 10). The thruster life timecalculated on erosion of channel walls (16) dependsof a density of dropping ion flow on channel wallsj • n , an energy of ions E and a sputtering factor. Asthe energy of a flow of ions in the area near the wallsvaries poorly (fig. 11), a life time increase at theextension of a channel (fig. 12) is stipulated first of allby sharp decreasing of ionsflows on walls connectedto increase of falling angles (p in accordance withthe extension of a channel. From fig. 13 it is clear that

the best value of parameter Zo corresponds to amiddle of a channel Zo=O,5 and is stipulated by aminimum flow of ions on walls of a channel in thiscase (fig. 14).

CONCLUSIONS

The spent numerical simulation of current of aplasma flow in a digit channel of SPT has shown thatthe extension of a channel results in decreasing a flowof high-energy ions on walls in an output part of achannel and improving the integral characteristics ofa thruster: a thruster force and its life time. The bestvalue of parameter ZQ is obtained.

REFERENCES

1 E.Foacaa, H.HHKyjmH. "HncjieHHoe Moaejmpo-BaHHe flHHaMHKH HOTOKa BKanajiecrauHOHapnoro miasMeHHoro flBHraxejia",ABHanjiOHHaa TexHHKa,,1992.-.N24.-

2 A.Mop03OB."<l>H3HqecKHe OCHOBBI KOCMKraec-KHX sjieKjpopeaKTHBHBix flBHraiejieH",M. ATOMH3«ax, 1978,T. 1

3 .Pe3n6yaM, fle.JIeHep, "KnaccmecicaaKHHexHHecKaa Teopaa Hcmncocreii H rasos",M.MHp,1980.

4 B.KnM,"AHajna saKOHOMepHocreH H3HocaH3OJWTOpa,orpaHOTHBaiomero cKopirreji&HHH

YSflll", HCTOHHHKH HycKopHiejm2,Btm.6,C.7-17.

A.EHmaeB,B.KHM, " HccjieflOBainre BJIHHHHHKOH<j)HrypaiiKK creHOK ycKopinejitHoroKanana na pacnpeflaneHHe jiOKantHtixnapaMerpoB B YSflll." vHcroHHHKH HycKopurejin roia3MHrXAH,1981,Bi>in.5,C.3-8

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Jz/Jo

. 1.5

0,5

p»1.2

r=1,08

Zo=0. i

\C ,96

2

1,6

o1,2

"'O.S

/

0.15 0,3 0,45 0,6 0,75 0,9

2/La

0,72 a84 0,96 1,03 1,2 1,32 1,44 1,55 1,68irt.

Fig. 5: Distributions of an ion flow density in a digit chant...

0,?

0,1/

/

,

/-"*

•*

' i

£^'

"\'•>'•S' ' * ,*

zo=

^^

''••4

•\

-^

• • -

1,3

o1'2

1,1

4

Zo=0,3;0,5

rr^T.--•̂̂ Zo=0,05

jS^'1

/0 ————————————————————————— i ——————————————————————

0,45 0.55 0,65 0,75 0,85 0,95 0 0,06 0,12 0,18

- Z/L exters'cnvalus

Fig.6: Distributions of an ion flow density on walls: Fig.7:Relations of an ionic current from 81-<5=0,06;2-

10,80,60,4

0,2

Zo=0, D5

0 0,06 0,12 0,18

extension value

Fig.8:Relations of a thruster force from 8

2

0,06 0,12 0,18extension value

Fig.9:Relatios of TJF from 8

LL 1,5

1120 140 160 180 200 220

ud)vFig. 10:

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140

0,15 0,3 0,45 0,6 0,75 0,9z/L

Fig. 11: Distributions of an energy in a channel

0)

To=1020k

0 0,15 0,3 0,45 0,6 0,75 0,9Zo/L

Fig.13:

0,06 0,12extension value

0,18

Fig. 12: Relations of life time from 5

0,45 0,6 0,75 0,9

Z/L

Fig.l4:Distributions of an ion flow densityat £=0,12

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