st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
Modeling study on the nonequilibrium expansion process of
plasma arc through a nozzle
Fu-Zhi Wei, Hai-Xing Wang, Wei-Ping Sun
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
(Corresponding author. E-mail: [email protected])
Abstract: A two-temperature thermal and chemical nonequilibrium model is applied to investigate the
expansion processes of an argon plasma arc through a Laval nozzle. It is found that the plasma is far
from thermodynamic equilibrium in the entire arc expansion process through a nozzle, especially in the
cooler outer region. Although the number density of excited argon atoms (Ar∗) is much lower than that
of other species in the argon plasma, Ar ∗ plays an important role in the arc attachment to the anode.
Keywords: nonequilibrium, expansion, argon plasma arc, modeling study.
1. Introduction
Expansion of a plasma arc through a nozzle is of particular interest due to its wide applications in plasma
chemistry, material processing and the aerospace industry,
such as low-pressure plasma spraying, nanostructured
film preparation using hypersonic plasma particle deposition, and arc-heated thrusters. In these applications, a DC
arc is struck between the cathode and nozzle (the anode),
passes through the constricted channel, and attaches in a
diffuse manner along the expanding portion of the nozzle.
When a plasma arc expands through a supersonic nozzle,
different modes of nonequilibrium can exist[1,2]. In
low-pressure regions, collisional coupling between electrons and heavy species may not establish thermal equilibrium between gas and electron temperatures. Since the
flow velocity within these kinds of devices is large, with
fluid residence times in the nozzle of the order of 1 μs,
ionization nonequilibrium occurs because the fluid-dynamic time scales are comparable to the relaxation
times for energy equilibrium and/or recombination of
electrons and ions.
Over the past several decades, although much effort has
been devoted to the development of comprehensive physical models to describe the arc expansion processes, questions remain concerning the thermodynamic and chemical
nonequilibrium features of the high velocity plasma flow
within plasma torches and arcjet thrusters operated under
low-pressure conditions.
The purpose of this study is to provide a better understanding of chemical non-equilibrium effects on arc expansion processes through a detailed numerical investigation. Departures from thermal and chemical equilibrium
in arc expansion are studied by direct two-dimensional
simulations using a two-temperature multicomponent
fluid dynamic model with chemical and excited-state kinetics.
trons and heavy species of argon, including ground atoms
(Ar), excited atoms (Ar*), and ions (Ar+). The main assumptions employed in the modelling study are as follows,
(i) the gas flow in the arcjet nozzle is steady, axisymmetric, laminar and compressible; (ii) each of the species has
a Maxwellian velocity distribution function; (iii) each of
the heavy species has the same temperature; however, the
heavy-species temperature Th, and the electron temperature Te may differ; (iv) the excited species have the same
cross sections as those of the ground-state atoms; (v) the
plasma is optically thin; (vi) the energy of electrons is
partially transferred to heavy species through elastic collisions with heavy species. Based on these assumptions, the
set of governing equations in the cylindrical coordinate
system can be written as follows:
U
t
E
x
F
y
Fv
y
Hv
u
u
u2
v
U
e
p
E
hu
ee
e
F
Ar*
Ar* u
u2
p
et
1
e
u
Ar
et
Ev
u
xy
he v
e
v
,
p,
h
,p
1
e
e
e e
pe
ph ,
pe
e e
xx
yx
yy
q ex he Da
e
yx
x
Fv
e
x
u
yy
v q ey q hy he D a
q ey he D a
Ar
x
Da
Ar
v
Ar* v
pe
e
e e
he v
Ar
xy
Ar*
hv
0
v q ex q hx he Da
Ar*
1
y
H
0
xx
v2
p
Ar* v
2
h
v
uv
Ar
v2
（1）
SC
v
hv
he u
Ar
S em
uv
v2
uv
et
Da
2. Modeling Approach
In this study, the plasma is considered to contain elec-
Ev
x
H
Ar
x
Ar*
Da
Ar
Ar*
Da
e
y
y
Ar
Ar
y
e
y
st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
0
0
jy B
yx
yy
yxu
jx B
qey
yy v
Hv
qey
qhy
y
S EM
jx Ex
y
jy Ey
0
Ar
0
y
Ar*
jy Ey
jx Ex
e
he Da
Da
Ar*
e
he Da
Ar
Da
y
Ar
0
0
SC
Ur
Ele
Ur
Ele Qech
0
T
5k
jx e
2e
x
Te
5k
jx
2e
x
m Ar n e
Elh
Te
y
Te
y
jy
jy
m Ar n Ar*
4 u
3 x
xx
4 v
3 y
yy
qex
e
2 v
,
3 y
xy
2
3
2 u
,
3 x
Te
, qey
x
e
u
y
yx
u
x
Te
, q hx
y
h
v
,
x
v
y
Th
, q hy
x
4 v
,
3 y
h
Th
y
Here x and y represent the axial direction and radial direction. u, v, p, pe, ph, ρ, ρs(s=e, Ar+, Ar*), Te, Th are the
axial velocity, radial velocity, pressure, electron partial
pressure, heavy species partial pressure, density, partial
density of species s, electron temperature and heavy species temperature, respectively. Bθ, Ex, Ey, jx, jy, Ele, Elh,
Qech and Ur are the azimuthal magnetic induction intensity,
axial electric field, radial electric field, axial current density, radial current density, chemical reaction energy
bound to electrons, chemical reaction energy bound to
heavy species, collisional energy transfer from electrons
to the heavy species[3] and radiation power, respectively.
The physical properties μ, κe, κh, γ, γe and Da, obtained
using the Chapman–Enskog theory [3], are the viscosity,
electron thermal conductivity, heavy species thermal
conductivity, specific heat ratio, electron specific heat
ratio and ambipolar diffusion coefficient, respectively.
Rewriting Ohm’s law by making use of Maxwell’s
equations for steady conditions, one obtains an equation
for the magnetic induction intensity in the following form
(vB )
(uB )
1 ( yB )
1 ( yB )
(2)
y
y
y
x
y
x
0
y
ter-cooled arcjet thruster designed by NASA Lewis Research Center[4]. Due to the axisymmetry of the thruster
nozzle, only the upper half is taken into account in the
computation. The computational domain used in the model is denoted as B-C-D-E-F-G-H-B in Fig. 1, in which
C-D, D-E and E-F are respectively the inner surfaces of
the convergent segment, constrictor and divergent segment of the anode/nozzle.
The conditions at the inlet of the computational domain
are taken to be those of a subsonic uniform flow. The
mass flow rate is specified based on the parameters used
for a particular case. The boundary conditions along the
centerline are set to ensure axisymmetry. At the inner wall
of the arcjet, no slip conditions are maintained, and the
species concentrations have zero gradient. Along the inner
surface of the anode, a zero gradient is imposed on the
electron temperature, while the heavy-species temperature
is held constant at 600 K. The outflow boundary conditions at the exit are assumed that gradients of the variables
are zero. The electromagnetic fields boundary conditions
are identical to those in Ref. [5].
x
where σ and μ0 are the electric conductivity and vacuum
permeability, respectively. After Bθ has been obtained, the
current density and electric field can be determined by
Ampere's law and Ohm’s law.
A schematic diagram of the low-power arcjet device
under study is presented in Fig. 1, showing the main dimensions and the domain adopted in calculation. The design and dimensions are almost the same as the wa-
Fig. 1 Schematic diagram of the arcjet nozzle under study.
Table 1 summarizes the chemical reactions in the plasmas, which account for electron-impact excitation from
the ground state (Process 1), electron-impact de-excitation
(Process 2); electron-impact ionization (Process 3), excited-species collisional ionization (Process 4), electron–electron–ion collisional recombination (Process 5),
electron–atom–ion collisional recombination (Process 6),
electron–ion radiation recombination (Process 7), atom–atom collisional excitation (Process 8) and atom–atom collisional de-excitation (Process 9).
3. Results and Discussion
Typical modelling results are presented for the argon
plasma arc expansion through the arcjet nozzle for a fixed
inlet mass flow rate of 115 mg/s and arc current of 8 A.
Figures 2 and 3 show the calculated Te and Th fields and
the thermal nonequilibrium parameter Te/Th in an argon
plasma. It is seen that the electron temperature is higher
than the heavy-species temperature throughout the whole
flow field. A high degree of thermal nonequilibrium, Te/Th
≈36, is noted at the current attachment location, where the
heating creates a ‘hot spot’ on the anode surface. The
heavy-species temperature in this region remains close to
the anode wall temperature since the ionization fraction is
too low to provide any significant thermal coupling to
electrons. This thermal nonequilibrium largely controls
the electron densities near the electrode, which in turn
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21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
Table 1. Chemical reactions considered in the model
Reaction
1
e
2
Ar
e
e
Ar
e
e
Ar
e
4
Ar
Ar
Ar
5
e
6
e
7
Ar
e Ar
8
Ar
9
Ar
Ar
Ar
Ar
Ar
Ar
Ar
5 10
e
Ar *
8.75 10
Ar *
hv
1.68 10
Ar
26
r, m
0
-0.001
Th
-0.003
0.004
0.006
0.008
0.01
39
21
[6]
-11.65
[6]
4.11
[7]
-7.54
[7]
(m s )
-4.11
[6]
(m 6 s -1 )
-4.11
[7]
-15.76
[8]
11.65
[9]
-11.65
[6]
(m 3s -1 )
(m 3s -1 )
4.5
6 -1
3 -1
(m s )
3 -1
(m s )
fluid residence time in the nozzle of order 0.2 μs, which is
comparable with the ionization relaxation time of argon.
The electric field rapidly heats the electrons, while the
degree of ionization usually lags behind the values that
would correspond to the rising electron temperature. The
computed mass fraction contours of excited argon atoms
within the arcjet nozzle are shown in Fig. 5. It is seen that
the concentration of excited argon atoms is lower than
0.01 in most regions of the arcjet nozzle; which is significantly lower than the concentrations of electrons and
ions.
0.003
0.001
0
-0.001
-0.002
-0.003
0
0.002
Te / Th
r, m
0
-0.001
-0.002
-0.003
0.002
0.004
0.006
0.008
0.01
0.004
0.006
0.008
0.01
0.012
Fig. 4 Computed ionization fraction contours within the arcjet.
0.003
0.012
z, m
Fig. 3 Computed thermal nonequilibrium parameter (Te/Th) distribution
in argon arcjet.
The ionization fraction contours, i.e., those of the ratio
of the electron number density to the heavy-species number density, are shown in Fig. 4. It is interesting to find
that the ionization degree in arc core presented here is
quite low, about 0.24 in the constrictor of arcjet nozzle. It
is generally understood that working gases are be highly
ionized at the arc core with temperatures in excess of
about 20000 K, but it is important to note that these numbers are based on equilibrium considerations. In the case
of our study, partially-ionized gases are not in a state of
complete equilibrium. It is found that the computed axial
velocity at the constrictor exit center is 2245 m/s, while
the length of the constrictor is 0.5 mm. Thus one obtains a
0.01200
0.01084
0.00968
0.00852
0.00736
0.00620
0.00504
0.00388
0.00272
0.00156
0.00040
nAr*/(nAr +nAr*+nAr)
+
0.001
r, m
36
32.7
29.4
26.1
22.8
19.5
16.2
12.9
9.6
6.3
3
0.001
0.24
0.217
0.194
0.171
0.148
0.125
0.102
0.079
0.056
0.033
0.01
ne/(nAr ++nAr*+nAr)
0.002
0.002
0
(m s )
z, m
Fig. 2 Comparison of computed electron and heavy species temperature
distributions in argon arcjet.
0.002
3 -1
0.012
z, m
0.003
Refs.
11.65
(m s )
Th [K] 0.5
)
300
Te [eV]
(m s )
3 -1
1.5
(K)
20967
18977
16987
14997
13007
11017
9027
7037
5047
3057
1067
Te
0.002
(
3.0 10
0.001
0
16
3 -1
135300
135300
Th [K] (2
) exp(
)
Th [K]
Th [K]
Ar
0.003
-0.002
Te [eV]
0.5
T [K ] 2.5
1.5 10 40 ( h
)
300
17
2.174 10 /(Te [K]) 0.5
determine the electrical conductivity of the plasma. These
processes give a non-zero electrical conductivity near the
electrodes, even though heavy-species temperatures are
low. The arc region within the constrictor is near thermal
equilibrium, due to relatively high ionization fraction ensuring efficient coupling of the electron and
heavy-species temperatures through Coulomb collisions.
Since most of the heating occurs inside the constrictor,
after which the flow expands and cools in the arcjet nozzle, both the electron and heavy-species temperatures
axially decrease from the constrictor to the nozzle exit.
0.002
16
6.2
2 10 13 exp(
)
Te [eV]
Ar
e
Ar
4.8 10
Ar
e
Ar
e Ar
11.65
4.9 10 15 Te [eV]0.5 exp(
)
Te [eV]
Ar
3
Ei (eV)
Rate coefficient Ki
r, m
No.
0
-0.001
-0.002
-0.003
0
0.002
0.004
0.006
0.008
0.01
0.012
z, m
Fig. 5 Computed excited atom mass fraction contours within the arcjet.
The role of the excited species in determining the arc
attachment to the anode is often overlooked by the scientific community. This is largely because of the low concentration of excited species and their negligible effect on
macroscopic flow quantities. It is instructive to compare
simulations performed using the same geometry size and
run parameters including and neglecting the influence of
excited species, to identify the effect of excited processes
and species on arc attachment to the anode. Therefore, an
additional numerical test neglecting the excited-state kinetics in the model was performed. In this test, the following chemical reactions[9], instead of reactions listed in
Table 1, are included in the model.
e Ar
e e Ar
Ar Ar
e Ar Ar .
st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
Figure 6 compares the computed enclosed current contours in the arcjet thruster for the case without excited
species and excitation processes to those for the full model. Corresponding comparisons of net ionization rate contours are shown in Fig. 7. As seen in upper semi-plane of
Fig. 6, without the presence of the excited species, the
current is concentrated within a very short distance downstream of the constrictor exit. For the case with excited
species, the current attachment region extends much further, i.e., a diffuse arc-root attachment appears at the anode surface when excited species are included in the simulation. The comparison of the net ionization rates shown
in Fig. 7 further demonstrates the effects of the excited
species on the extent of the ionization region. Similarly to
the current distribution on the anode, the presence of the
excited species greatly increases the area of the ionization
region near the anode. This is mainly because the excited
atoms represent the primary source of electrons over a
broad range of conditions; on the other hand, it should be
noted that recombination occurs primarily through excited
states.
0.002
0.5
r, m
1.5
2.5
3.5
4.5
5.5
6.5
7.5
without Ar*
0.001
0
with Ar*
-0.001
4. Conclusions
A two-dimensional nonequilibrium plasma model has
been employed to investigate the thermodynamic and
chemical nonequilibrium characteristics of the argon
plasma arc expansion through an arcjet nozzle. Numerical
results show that considerable thermal nonequilibrium
exists in the entire arc expansion process through a nozzle,
especially in the near anode/nozzle wall region. The
thermal nonequilibrium increases as the radial distance
toward to the anode wall increases. Strong departures
from excitation and ionization equilibrium are found in
high-velocity plasma arc expansion processes. The ionization degree in arc core is about 0.24 in the constrictor of
arcjet nozzle, much lower than that in chemical equilibrium. The role of excited species in the arc attachment on
anode has been studied. It is found that the presence of the
excited species promotes a diffuse type of attachment,
extending the arc root further downstream, while the absence of excited species would lead to the formation of a
constricted arc root on the upstream end of the nozzle.
Although the number density of excited species of argon
(Ar∗) is much lower than those of other species of the
argon plasma, it plays an important role in determining
the arc attachment mode to the anode.
Enclosed Current (A)
-0.002
0.002
0.004
0.006
0.008
z, m
Fig. 6 Comparison of predicted enclosed current contours without and
with the excited species within the arcjet.
0.002
r, m
0.001
Ionization Zone
without Ar*
0
-0.001
-0.002
0.002
with Ar*
Ionization Zone
0.004
z, m
0.006
0.008
Fig. 7 Comparison of net ionization rate contours without and with the
excited species within the arcjet.
The predicted diffuse arc root attachment on anode in
the presence of excited species agrees with the experimental observation of Curran and Manzella[10]. In their
experiment, the anode of a 1 kW radiation-cooled arcjet
was separated by thin boron nitride insulating spacers into
five electrical segments, without changing the nozzle
geometry. This modification permitted the investigation of
the current density distribution and plasma at the anode
surface. Interestingly, they discovered that the first segment (consisting of the converging section up to the end
of the throat) collected little or no current while the remaining four segments drew nearly equal currents. Further, Tiliakos et al. studied energy transfer to the anode of
a 1 kW arcjet using electrostatic micro-probes in the anode boundary. In their experiment, the current was also
found to attach diffusely to the anode/nozzle wall[11].
This finding indicates that the arc attached to the anode in
a diffuse mode and a significant fraction of the current
extended into the low density region of the flow.
5. References
[1] C. Park, J. Plasma Phys. 9, 187 (1973).
[2] V. Rat, A. B. Murphy, J. Aubreton, et al., J. Phys. D:
Appl. Phys. 41, 183001 (2008).
[3] K. C. Hsu, A self-consistent model for the high intensity free-burning argon arc. PhD thesis (University of
Minnesota Press, Minnesota, 1982).
[4] M. Moeller, D. Keefer, and R. P. Rhodes, in AIAA/
SAE/ASME/ASEE 28th Joint Propulsion Conference
and Exhibit, Nashville, US, 6-8 July 1992.
[5] H. X. Wang, F. Z. Wei, and A. B. Murphy, et al., J.
Phys. D: Appl. Phys., 45, 235202 (2012).
[6] M. Baeva, A. Bosel, and J. Ehlbeck, et al., Phys. Rev.
E, 85, 056404 (2012).
[7] X. M. Zhu, and Y. K. Pu, J. Phys. D: Appl. Phys., 43,
015204 (2010).
[8] J. W. Bond, Physical Review, 105, 1683 (1957).
[9] M. I. Hoffert, and H. Lien, Phys. Fluids, 10, 1769
(1967).
[10] F. M. Curran, D. H. Manzella, in AIAA/SAE/ASME/
ASEE 25th Joint Propulsion Conference and Exhibit,
Monterey, CA, 10-13 July 1989.
[11] N. T. Tiliakos, R. L. Burton, and H. Krier, in
AIAA/SAE/ASME/ASEE 32th Joint Propulsion Conference and Exhibit, Lake Buena Vista, FL, 1-3 July,
1996.
6. Acknowledgment
This study was supported by the National Natural Science
Foundation of China (No. 11275021, 11072020,
50836007)