st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
Multi-Fluid Modeling of an Electrothermal Radio-Frequency plasma Thruster
Minkwan Kim1, Christine Charles2, Rod Boswell2
1
University of Adelaide, Adelaide, SA, Australia
Australian National University, Canberra, Australia
2
Abstract: For micro-propulsion applications, a capacitively coupled RF (13.56 MHz) micro-discharge thruster is
proposed by Charles et al. which is known as Pocket-Rocket. To date, the performance of Pocket Rocket has been experimentally studied using standard optical and electrical methods and using a global plasma model based on power
balance. However, the detailed plasma motion and physics are not studied yet due to the lack of physical model to describe a RF micro-thruster. In this study, we develop a physical model of a RF micro-thruster to investigate neutral
heating and propulsive performances.
Keywords: Plasma thruster, Plasma modeling, Discharge.
1. Introduction
Plasma is an ionized gas that typically generated by applying energy to gas in order to reorganize the electronic
structure of the species and to produce the excited species
and ions [1]. Recently, plasma is being considered as an
alternative propulsion method for deep space mission and
small spacecraft because plasma propulsion can minimize
the required propellant mass. One of the proposed methods as plasma propulsion is a radio-frequency (RF) plasma thrusters. The concept of RF plasma thrusters is using
a radio-frequency capacitively coupled discharge
(RFCCD) to heat a propellant by creating glow discharge
plasma between coaxial electrodes.
For micro-propulsion applications, a capacitively coupled RF (13.56 MHz) micro-discharge thruster is proposed by Charles et al. which is known as Pocket-Rocket
[2]. Previously, the performance of Pocket Rocket is experimentally studied using standard optical and electrical
methods and demonstrated that the propulsive performance relies on RF power, propellant gas species, and
mass flow rate [2]. However, the detailed plasma motion
and physics are not studied yet due to the lack of physical
model to describe a radio-frequency micro-thruster. In this
study, we develop a physical model of a RF micro-thruster, Pocket Rocket, to investigate neutral heating
and propulsive capabilities for a comparison with mature
plasma propulsion systems such as resistojets, arcjet, and
magnetoplasmadynamic thrusters.
2. Numerical Modeling
In this study, the chemical nonequilibrium is considered
using a finite rate chemistry model with a multi-species
model [3][4]. Thermal nonequilibrium effects are accounted using multi-temperature model with Kim’s electron energy equation model [5]. Therefore, the governing
equations consist of the mass conservation equations of
each chemical species, the momentum conservation equations, and the energy conservation equations for each energy mode including translational, rotational, vibrational,
and electron energies. In this study, multi-fluid approach
is employed with drift-diffusion approximation in order to
consider the motion of ions, neutrals, and electrons, separately.
The governing equations of ions and neutrals can be
given by:
¶Q
+ Ñ × F - Ñ × FV = S f + S RF
¶t
(1)
where Q is the vector of conserved variables, F is the
inviscid flux vector, Fv is the viscous flux vector including diffusion, Sf it the source term due to the thermo-chemical non-equilibrium, and SRF is the source term
for the applied radio frequency voltage. The details of
each flux vectors can be found in [6]. Since ions are only
accelerated by the applied electric field, a radio-frequency
source term, SRF, can be given by:
S RF
ì
ï
ï
ï
ï
ï
ï
ï
=í
ï
ï
ï
ï
ï
ï
ï
î
ü
ï
ï
"ions
ï
å qsni,sE x ï
ï
s
ï
"ions
å qsni,sE y ïý
s
ï
"ions
q
n
E
å s i,s z ïï
s
ï
E × ji
ï
ï
0
ï
0
þ
0
0
(2)
where qs is a species charge, E is an electric field vector,
and ji is an ion electrical current density.
In RF discharge, the characteristic time between momentum transfer collisions is much smaller than the impulse frequency for pressures larger than 500 mTorr.
Therefore, the drift-diffusion approximation can be employed to simplify electron equations. Under the
st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
drift-diffusion approximation, the continuity of electron
can be expressed as:
¶ne
+ Ñ × G e = we
¶t
(3)
where w e is electron production rate by chemical reactions and G e is the electron flux, deduced from a
drift-diffusion approximation of the momentum equation
as:
Ge = -neme E - De Ñne
(4)
where μe and De are electron mobility and diffusion coefficients, respectively. A standard finite-rate chemistry
model is employed to describe electron source term, w e ,
including recombination ionization and electron-impact
ionization. The chemical reactions can be represented
generically as:
åa S
åb S
s
(5)
s
where S represents chemical species including electron
and α and β are the stoichiometric coefficient. Therefore,
electron source term can be given by
a
b
é
ì
æ
æ
r ö
r ö
ï
w e = N A åêê( be,k - ae,k ) í103 k f ,k Õ çç10-3 j ÷÷ -103 kb,k Õ çç10-3 j ÷÷
Mjø
Mjø
ïî
k
j è
j è
êë
j ,k
j ,k
üù
ïú
ýú
ïþú
û
Ñ 2f = -
e
e0
(n - n )
i
(10)
e
A RF voltage is applied at an antenna. Therefore the
boundary condition at an antenna is given by
f (t ) = fdc + fRF sin ( 2p vRF t )
(11)
where φdc is an autobias voltage, φRF is the applied RF
voltage, and νRF is frequency applied at an antenna. The
dc voltage is obtained from the current balance to the
electrodes which are antenna and grounded electrodes.
3. Results
A capacitive RF (13.56 MHz) micro-discharge is simulated in the geometry of Pocket-Rocket. A schematic of
the Pocket-Rocket is shown in Fig. 1. As can be seen, it
consists of a 20 mm long, 4.2 mm inner diameter ceramic
tube which is surrounded by one 5 mm wide copper central RF ring and two 3 mm wide copper grounded electrodes. Each copper rings are separated by 3 mm and 4
mm, respectively. In this study, argon is employed as
working gas with the injection rate of 100 mg/sec. The
details of the flow conditions are listed at Table 1.
(6)
where NA is Avogadro’s number and CGS units are used
in reaction rates coefficients.
An electron energy equation also should be considered
because ionization reaction rate largely governed by electron temperature. Using an ohmic heating term, electron
energy equation can be written as:
æ
¶Ee
G ö
+ Ñ × çç Ee + pe e ÷÷ - Ñ × -q e = Se (7)
¶t
ne ø
è
(
( )
)
where Ee is the electron energy, pe is the electron pressure,
qe is the heat flux vector of electrons, and Se is the source
term. The source term of electron energy equation consist
of an ohmic heating term and a collisional loss term as:
Se = je × E -
"electron-impact ionization
å
we,k I k
(8)
k=1
where Ik is the energy to used to ionize neutral species,
w e,k is electron source term by electron-impact ionization, and je is an electron current density. The electron
current density can be obtained from Eq. (4) as:
(
je = e neme E + De Ñne
)
(9)
where e is an elementary charge, 1.602×10-19 C.
The electric field is deduced from Poisson’s equation
for the electric potential f :
Fig. 1. Schematic of the 'Pocket Rocket' RF cylindrical
device
Argon flow rate
100 mg/sec
Inflow temperature of argon, T∞Ar
293 K
Background pressure, pres
3 Torr
Frequency applied at an antenna, νRF
13.56 MHz
Applied peak RF voltage, φRF
1 kV
Table 1. Inflow and RF conditions of an electrothermal
Radio-Frequency plasma Thruster, Pocket-Rocket
Figure 2 shows the contour of the calculated electric
potential distribution near a RF thruster. As can be seen it
has a peak potential near an antenna. The potential distribution is obtained by solving Eq. (10) using a finite
volume method. At the end of antenna edge, it has discontinuity of potential distribution, which leads to an infinite
large electric field. Therefore, a shape function is em-
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21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
ployed at the edge of antenna in order to reduce a numerical instability issue due to the discontinuity of potential.
Fig. 2. Electric potential contours of the Pocket-rocket
Figures 3 and 4 show the distributions of electron temperatures and densities, respectively. Previously, we
measured electron temperatures for various operating gas
pressure over the range 0.3 ~ 3m Torr, and figured out that
electron temperature decreases as increase the reservoir
pressure of argon due to the changing ionization balance
[8]. In this study, the considered argon pressure is 3 Torr
in order to use a drift-diffusion approximation. Figure 5
shows calculated electron temperature contours without
RF source using Kim’s electron energy equation model
[5]. As can be seen, downstream electron temperature is
very close to ion temperature of argon due to high background pressure.
Fig. 4. Electron number density contours of the Pocket-Rocket RF plasma thruster where the base pressure is 3
Torr and the argon injection rate is 100 mg/sec. The applied voltage at an antenna is 1kV.
Fig. 5. Electron temperature contours without RF
source where the base pressure is 3 Torr and the argon
injection rate is 100 mg/sec.
Fig. 3. Electron temperature contours of the Pocket-Rocket RF plasma thruster where the base pressure is 3
Torr and the argon injection rate is 100 mg/sec. The applied voltage at an antenna is 1kV.
Figure 6 shows the distributions of argon ion densities
around a RF thruster under a 1kV RF voltage. Inside of
the thruster it has a high plasma density at the center of
the thruster due to the viscous effect. Figure 7 shows argon ion density distributions along the radial direction at z
= 2.5 cm. As can be seen, argon ion has peak number
density at the edge of plume jet which is about r = 2mm.
It is because the electron temperature on the central axis is
very close to ion temperature at high background pressure
condition before a RF voltage is applied. When a RF
source is applied, the downstream has low electron temperature compared to low background pressure operating
condition.
4. Conclusion
The developed RF thruster model is employed to simu-
st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
late the Pocket Rocket. In order to consider the detailed
motion of plasma, a multi-fluid approach is employed
which is based on a drift-diffusion approximation. In this
study, electrons are considered separately because an
electron has a mass that is approximately 1/1836 that of
the ion. Ions and neutrals are considered to have a same
velocity because of high ion-neutral collisional frequency
and ion-neutral diffusion rate. The thermo-chemical
non-equilibrium effect is considered using a fluid approach that is based on Navier-Stokes equations. The employed fluid model is extended to describe a RF plasma
thruster using drift-diffusion approximation. Therefore,
the developed RF model can used to analyze the performance of a RF thruster in terms of different operating
conditions.
5. References
[1] Lieberman, M. A. and Lichtenberg, A. J., Principles
of Plasma Discharges and Materials Processing,
Wiley-interscience, 2nd ed., 2005.
[2] Charles, C and Boswell, R. W., “Measurement and
Modelling of a Radiofrequency Micro-thruster,”
Plasma Sources Sci. Technol., Vol. 21, 2012
[3] Park,
C.,
Nonequilibrium
Hypersonic
Aerothermodynamics, Wiley-Interscience, 1990.
[4] Candler, G. V. and MacCormack, R. W.,
“Computation of Weakly Ionized Hypersonic Flows
in Thermochemical Nonequilibrium,” Journal of
Thermophysics and Heat Transfer, Vol. 5, No. 3,
1991, pp. 266-273.
[5] Kim, M., Boyd, I. D., and Gulhan, A., “Modeling of
Electron Energy Phenomena in Hypersonic Flows,”
Journal of thermophysics and heat transfer, Vol. 26,
No. 2, 2012, pp. 244-257.
[6] Kim, M. and Boyd, I. D., “Effectiveness of a
Magnetohydrodynamics System for Mars Entry”,
Journal of Spacecraft and Rockets, Vol. 49, No. 6,
2012, pp. 1141-1149
[7] Passchier, J. D. P. and Goedheer, W. J., “A
two-dimentional fluid model for an argon rf
discharge”, Journal of Applied Physics, Vol. 74, No.
6, 1993, pp. 3744-3751
[8] Takahashi, K., Charles, C., Boswell, R., Lieberman,
M. A., and Hatakeyama, R., “ Characterization of the
temperature of free electrons diffusing from a
magnetically expanding current-free double layer
plasma”, Journal of Applied Physics, Vol. 43, 162001,
2010, pp. 3744-3751
Fig. 6. Argon ion density contours of the Pocket-Rocket
RF plasma thruster where the base pressure is 3 Torr and
the argon injection rate is 100 mg/sec. The applied voltage at an antenna is 1kV.
Fig. 7. Argon ion density distributions along the radial
direction at z = 2.5 cm where the base pressure is 3 Torr
and the argon injection rate is 100 mg/sec. The applied
voltage at an antenna is 1kV.