22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Chemical and thermal non-equilibrium effects in free-burning arcs
M. Baeva
Leibniz Institute for Plasma Science and Technology, INP Greifswald e.V., DE-17489 Greifswald, Germany
Abstract: Effects caused by thermal and chemical non-equilibrium in free-burning arcs in
argon are presented and discussed. The results cover a range of arc currents between 100
and 200 A, interelectrode distance of 5 - 10 mm, and a variation of the electrode material
and the shape of the cathode tip. Non-equilibrium in the near-electrode regions and in the
arc fringes has a strong impact on the arc properties.
Keywords: arc plasma, thermal, ionization-recombination, non-equilibrium, modelling
1. Introduction
Arc discharges are being investigated over the years
experimentally and by means of modelling because of
their wide range of applications. As a part of complex
equipment, the arc discharge has to be well predictable in
order to save costs for consumables and repairing. For all
applications, the determination of plasma parameters such
as temperature and densities of the plasma components is
very important. Due to transport phenomena the arc
plasma is, in general, not in complete thermodynamic
equilibrium. The most methods developed for diagnostic
of thermal plasma are based on the assumption of local
thermodynamic equilibrium (LTE) implying small
changes in velocity distribution functions and population
densities over the distance of the mean free path for
elastic collisions between identical particles. Therefore,
the justification of the LTE model has been investigated
in tremendous amount of research work. Studies on freeburning arcs in argon [1] have declared LTE as
convincingly demonstrated. Criteria for validity of LTE
have been formulated from considerations accounting for
effects of diffusion, collisional excitation, ionization and
recombination [2, 3] – critical values of electron density
between 6.1022 m-3 and 1025 m-3 for argon arc at pressure
of 1 atm and electron temperature of 1 eV. Other works,
both experimental and theoretical, have reported
deviations from LTE [4-15]. Especially, the effect of
non-equilibrium near the arc electrodes is important to the
predictive description of the arc properties since current
density and temperature at the plasma edge have strong
influence on the arc properties [16].
It appears that the need for non-equilibrium description
of the free-burning is unavoidable. However, nonequilibrium models require great computational costs due
to the increasing complexity, the strong non-linearity and
coupling of the corresponding plasma equations,
especially in the case of realistic geometries and timedependent features. The present work presents results of
a non-equilibrium model underlying the effects of thermal
and chemical non-equilibrium observed in a free-burning
arc in argon. The non-equilibrium model has been
described in great detail in [17, 18]. The arc geometry is
characterized by axial symmetry so that a pie-slice of 10°
IN-21b
corresponds to a 2.5D model. The computational domain
includes the arc region and the electrodes. The cathode is
a cylindrical rod with a conical or semispherical tip made
of lanthanated or pure tungsten. The water cooled anode
is a plate made of copper. The arc is operated in argon at
atmospheric pressure with typically with a gas inflow of
12 slpm, an interelectrode distance of 5 - 10 mm, and arc
current in the range 100 - 200 A.
2. Effects of thermal non-equilibrium
For the sake of comparison, results of an equilibrium
model of the arc column will be presented along with
those of the non-equilibrium one. The equilibrium model
makes use of the LTE assumption for the bulk plasma and
involves a non-equilibrium description of the cathode
boundary layer similar to that of the nonlinear surface
heating [19] and the approach used in [20].
Fig. 1 presents the radial distribution of the electron
(T e ) and heavy particle (T) temperatures obtained in the
frame of the non-equilibrium model and the plasma
temperature from the LTE model in the midplane of an
free-burning arc of 10 mm length for two arc currents.
The cathode is a cylindrical rod made of pure tungsten
with a diameter of 2 mm and a hemispherical tip. The
non-equilibrium model applies a two-level representation
of the atomic argon energy structure. The results clearly
show that for radial positions up to approximately 5 mm,
T e and T are equal, i.e., the plasma is in thermal
equilibrium. Toward the arc fringes, a significant
departure from thermal equilibrium is observed.
Although in the arc core T e ≈ T, the common value
departs from the value obtained in the LTE model. The
latter predicts higher values in the arc core and narrower
radial profiles. Moreover, the LTE-profiles become
narrower with increasing arc current in contrast to the
prediction of the non-equilibrium model. As a result, the
profile of the non-equilibrium electric conductivity (Fig.
2) is wider. The arc fringes possess electric conductivity
which is higher than the equilibrium one in several orders
of magnitude. Therefore, the electric current density is
also higher, which in turn leads to higher heating of the
electrons in the outer arc region. Since the total arc
current must be the same, the corresponding values of the
1
electron and heavy particle temperature, and the current
density are lower in the arc core in comparison to the
equilibrium plasma temperature and current density.
Similar behaviour has been observed in other nonequilibrium models of free-burning arcs; see e.g., [21-24].
Fig. 1. Temperatures of electrons T e and heavy particles
T from the non-equilibrium model (neq) and the
equilibrium temperature T in the midplane of the arc.
Fig. 2. Electric conductivity in the midplane of the
free-burning arc for arc current of 200 A.
Fig. 3 shows the calculated arc voltage in comparison
with experimental findings presented in [25].
For a
better reading, the following notations are introduced.
U arc (LTE) and U arc (neq) denote the total voltage over the
arc axis obtained in the framework of the LTE- and the
non-equilibrium description of the arc column,
respectively. These values of the arc voltage consist of
the voltage drop in the bulk plasma U pc (LTE) and
U pc (neq) , and the voltage drop in the space-charged
sheath U sd (LTE) and U sd (neq). Notice that the nonequilibrium arc model includes the ionization layer in the
bulk plasma, so that the value U pc (LTE) is composed by
the contributions of the non-equilibrium ionization layer
and the equilibrium bulk plasma. The experimental
values from [25] are given as solid symbols and are
2
denoted by U arc (exp). Since both the anode voltage drop
and the voltage drop over the cathode body were found to
be about 0.4 - 0.6 V but with opposite signs, the total of
the voltage drop over the arc column and the cathode
boundary layer was interpreted as the arc voltage and
compared with the experimental results.
Fig. 3. Calculated sheath voltage drop, arc column
voltage and the total of both in comparison with
experimental data.
The results obtained show that U arc (neq) agrees very
well with the experimental data U arc (exp) whereas the
values of U arc (LTE) even following the experimental
observed course are in about 2 V higher. The deviation of
maximum 14% can be seen as satisfactory in the
framework of the LTE description. The contribution of
the arc column U pc (LTE) (already containing the voltage
drop over the ionization layer) is at least about 25% lower
than the experimental values. The discrepancy would rise
to at least 35% if only the net contribution of the arc
column is considered. Furthermore, the discrepancy
increases for lower arc current values. The arc column
voltage alone cannot fit the experimental data even for
higher arc currents, the contribution of the cathode
boundary layer is significant. Therefore, a no-sheath
approach must be considered as improper. The higher
values of U arc (LTE) in comparison with U arc (neq) need
some more attention despite the relatively good
agreement between U arc (LTE) and U arc (exp). The voltage
drop over the arc column U pc (LTE) obtained from the
equilibrium model is significantly higher than the value
U pc (neq) obtained with the nonequilibrium model as a
result of the effect of thermal non-equilibrium discussed
above. The discrepancy is nearly by a factor of two.
Apparently, the enhanced electric conductivity predicted
by the LTE-model compensates in some way the
contribution of the cathode boundary layer for the region
of arc currents under consideration. Nevertheless, the
results clearly show that the physically justified treatment
necessarily includes a non-equilibrium description of the
cathode boundary layer.
A good agreement with experimental data is observed
IN-21b
in a free-burning arc configuration with a tungsten
cathode with a conical tip, anode made of steel, and
various interelectrode distance over the range of arc
currents 50 - 250 A (Fig. 4).
Fig. 4. Arc voltage and total of both in comparison with
experimental data.
3. Effects of chemical non-equilibrium
The populations of the excited states are essential for
the evaluation of emission spectra and the spectroscopic
validation of arc plasma models. Excited atoms play an
important role in the production and loss of charged
particles. Under a wide range of conditions, the total
particle density of excited states is much less compared
with the ground state and the electron number density.
The ground and the continuum states can be considered as
particle pools and the particle flow between them goes
through the excited states. The distribution of atoms over
their excited states in the argon arc plasma has been
obtained in the frame of the non-equilibrium model by
means of incorporated collisional and radiative processes
between states included to represent the discrete atomic
structure. For the present analysis, the extended level
scheme applied in [18] is considered. In these scheme,
the four lowest excited levels 4s (1s 5 , 1s 4 , 1s 3 , 1s 2 in
Paschen notation) are treated as individual species, the
levels 2p 10 –2p 5 are grouped in an effective state, as well
as the levels 2p 4 –2p 3 , but 2p 2 and 2p 1 are taken as
individual, and an effective state (hl) includes further
higher excited levels. Then, a transport equation for the
ions and the excited states is solved in the arc model
(
)


∇ ⋅ ρVYi = ∇ ⋅ J i + S i ,
(1)
accounting for convection, diffusion, and production/loss
in collisional and radiation processes. The following

V is the

mass-averaged velocity, Y i is the mass fraction, J i - the
notations are used: ρ is the total mass density,
diffusive flux, and S i - the productive term of species of
kind „i“.
IN-21b
Plots of the reaction rates in a free-burning arc of 8 mm
length as a function of the radial and axial position are
shown in Figs. 5 and 6, respectively. Additionally, the
distribution of the electron and heavy particle
temperature, and the electron density obtained from the
non-equilibrium model as well as the corresponding
equilibrium values are shown. The arc is burning
between a tungsten rod with a radius of 2 mm (the
cathode) with a conical tip and a water cooled flat copper
anode at an arc current of 200 A and a gas flow rate of
12 slpm. The axial position of the cathode tip is
0.005 mm. The anode plate crosses the axis at 0.013 mm.
The results show that the ionization from the excited
states is the dominant process for ion production. The
dominating mechanism for loss of electrons in the arc
plasma is the three-body recombination. The reaction rate
for radiative recombination in which the resulting atom is
in the ground state is significantly lower. The total
reaction rate for radiative recombination of the ions to the
excited states is about half of that to the ground state. It
should be noted that in the radiative recombination to the
ground state a radiation trapping has been accounted for,
whereas the continuum radiation resulting from radiative
recombination to the excited states has been considered as
optically thin. In radial direction, the total rate being in
the order of 1 - 2 kmolm−3 s−1 is low compared to the
reaction rate for ionization itself but interesting is that the
total reaction rate is changing its sign. Close to the axis
(radial position zero), the arc plasma has a net
recombination, and then it shows a net ionization and
turns again to net recombination before it comes to
ionization–recombination equilibrium in the outer region.
Moreover, the departure from ionization–recombination
equilibrium appears even in the region of thermal
equilibrium, i.e., equal electron and heavy particle
temperatures. However, in the outer arc region where the
plasma exhibits ionization–recombination equilibrium, a
departure from thermal equilibrium is observed. Over the
arc axis, a departure from ionization–recombination
equilibrium appears in the near-electrode regions. In the
near-cathode region (the axial position of the cathode tip
corresponds to 0.005 m), the plasma is strongly ionizing
whereas near the anode (the axial position of the anode is
at 0.013 m) the plasma is close to
ionization–
recombination equilibrium but even recombining. In the
near electrode regions, a strong deviation from thermal
equilibrium is observed as clearly shown in the zoomed
plots. Along the arc column, the plasma state can be
characterized as that of LTE. The balance in the general
species equation (1) is ensured by the transport terms
describing the processes of convection and diffusion. The
effects of thermal and chemical non-equilibrium cause the
difference in the electron number density observed by
means of the non-equilibrium model in comparison with
the equilibrium values obtained from the Saha formula
with the temperature of the heavy particles T, and the
values obtained from the two-temperature Saha formula.
3
regions and the arc fringes. Deviation from thermal
equilibrium in the arc fringes influences the electric
conductivity of the plasma and therefore the arc current
density. Chemical and thermal non-equilibrium in the
near-electrode regions have a strong impact on the arc
properties. Especially, the contribution of the nearcathode layer to the arc voltage is significant.
Fig. 5. Reaction rates, temperature and electron density in
the midplane of the arc.
Fig. 6. Reaction rates, temperature and electron density
along the arc axis.
4. Conclusions
The present study gives an analysis of effects of thermal
and chemical non-equilibrium in free-burning arcs in
argon. The effects appear mainly in the near-electrode
4
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