Non-equilibrium effects in high pressure, high temperature plasmas
Ananth N. Bhoj and Mustafa Megahed
ESI US R&D Inc., 6767 Old Madison Pike, Huntsville, AL 35806 USA.
Abstract: High temperature plasmas generated at high pressures are generally modeled assuming local
thermal equilibrium. When non-equilibrium effects are important it is useful to consider fluid based two
temperature models, solving separately for gas and electron temperatures. In this paper, plasma kinetics
are addressed by calculating electron transport and rate coefficients and solving for the detailed
volumetric mechanism of electron and heavy particle reactions. The quasi-neutrality condition is assumed
for the discharge, while the electric conduction equation is solved to obtain electric fields. Results from
the 3-d simulations using this approach as applied to an example representative of technological
applications involving arc discharges are presented.
Keywords: arc discharge, non-equilibrium, plasma modeling, plasma kinetics
1. Introduction
Arc discharges are of central importance to
materials processing and high temperature chemistry
applications, including combustion. Advances in
modeling arc discharges in recent years have led to a
better understanding of the underlying physics [1].
The complex interaction of flow, electric, magnetic
and thermal effects along with gas chemistry and
electrode surface phenomena renders very difficult
generic, comprehensive analytical or computational
tools suitable for the industrial setting. Simplifying
assumptions are normally used to focus on particular
physics or engineering issues of interest [2, 3].
2. Model Description
In this paper, we discuss the computational
modeling of an arc discharge, typical of industrial
applications, using a generic fluid based approach to
allow for significant non-equilibrium conditions [4].
The model has been presented earlier [5] and is part
of the CFD-ACE+ modeling platform [6]. Electron
density is obtained from a quasi-neutrality condition.
Electron energy is computed by addressing the
solution of the electron energy balance equation.
The electric conduction equation is solved to
obtain the electric potential and electric field within
solids and the plasma. The conductivity of the
plasma can be modeled as either a function of gas
temperature and pressure or computed from the
electron density. The electron conduction current is
computed from the known quantities of conductivity
and electric fields.
The current boundary condition applied at
the electrode is known. The model is able to selfconsistently compute quantities such as the velocity
and pressure fields in the gas. The electron, gas and
solid temperatures are calculated. In addition, the
solution of reaction chemistry yields the number
density (or concentrations) of various species. This
includes the computation of electron transport and
rate coefficients and solving for the detailed
volumetric mechanism of electron and heavy particle
reactions. The output electric parameters include the
conductivity, electron density and electric potential.
In the next section, the application of this model to
an example is discussed.
3. Arc Discharge in Oxygen
In this example, a typical 3-d section is
considered as shown in Figure 1. The top powered
electrode tip is exposed to the gas, here Oxygen. The
sides are covered by dielectric. O2 enters through the
inlet at a slightly higher pressure and exits through
the outlet. The pressure is near atmospheric at 105
Pa. The lower solids, 14.5 mm from the powered
electrode, contain a metal and a dielectric and their
lower surfaces are grounded. The electrical
conductivity of the metal and dielectric are assumed
to be 106 and 10-6 1/ohm-m respectively. The mesh
with 13000 cells is locally refined near the electrode.
For the initial conditions, the initial gas
temperature field is about 1000 K. The top surface of
the electrode is assigned a fixed current boundary
condition of 2.5×106 A/m2. The initial electron
temperature is assumed to be 2 eV. The reaction
chemistry includes electron impact collisions
including
momentum
transfer,
dissociation,
excitation, ionization and recombination. Heavy
species reactions include neutral – neutral reactions,
electron - ion and ion – ion recombination, and
charge transfer. Surface chemistry at electrode and
grounded surfaces includes neutralization of charged
species and recombination of atoms.
lower at around 109 W/m3. Relatively more power
deposition occurs near the lower metal surface.
(a)
(b)
Figure 1. The 3-d sector geometry used in these simulations.
The distance from the electrode tip to the metal is 14.5 mm. In
the remaining figures results are shown on a section plane.
Results from the simulation of this base case
are shown starting in Figure 2. Contours of electrical
characteristics of the discharge are plotted with
minimum and maximum limits on the respective
scales. The power deposition in the arc is high, on
the order of a few 1010 W/m3. Further, the region of
peak power deposition of 1012 W/m3 is localized in a
thin region near the electrode and the maximum
limit is deliberately lower so as to resolve the power
deposition in the bulk of the discharge, which is
(c)
Figure 2. Contours of electrical characteristics for the arc
discharge in O2. From top to bottom are (a) Power Deposition
(b) Electric Conductivity and (c) Electric Potential. Contour
levels below the minimum limit may have been removed.
The spatial variation of the electrical
conductivity is a direct consequence of the shape of
the discharge. Here it is computed using the spatial
distribution of electron density and mobility. One of
the difficulties involved in such models stems from
the large variation in electrical conductivity at
interfaces between material domains, which can be
several orders of magnitude apart. The greater the
width of this range, the harder it is to obtain
convergence in numerical solutions.
The discharge is sustained at low voltages
typical of high current arc discharges as in Figure
2(c). The discontinuities in the lower solids arise due
to their differing conductivities. The voltage at the
negatively powered electrode is about -11.4 V.
(a)
(b)
(a)
(c)
(b)
Figure 3. Contours of temperatures for the arc discharge in O2.
Shown from top to bottom are (a) temperature of the gas and in
solid materials in Kelvin and (b) electron temperature in eV.
Contour levels below the minimum limit may have been
removed.
Figure 4. Contours of charged species for the arc discharge in
O2. From top to bottom are number densities of (a) O2+ (b)
electrons and (c) O+ in m-3. Contour levels below the minimum
limit may have been removed.
The powered electrode heats up to 3500 K
(Figure 3) due to its high thermal conductivity. The
peak gas temperature is around 11000 K near the
region of peak power deposition. There is a sharp
gradient as gas temperature falls off several
thousand Kelvin at short radii from the central axis
of the arc. While peak electron temperature is about
1.1 eV near the region of peak power deposition, it
decreases more gradually throughout the plasma
domain. The conditions in the plasma region near
the central axis are close to local thermal equilibrium
(LTE) with reduced difference between electron and
gas temperatures. Departures from LTE are noted
farther out where the gas is cooler than electrons.
The dominant ion in the discharge is O+ for
these conditions (Figure 4) and as expected the peak
ion and electron densities occur close to the region
of peak power deposition. Sharp gradients exist in
the contours of charged species density. The space
charge region is very thin for these conditions and
the arc is largely quasi-neutral.
The spatial distribution of neutral species
densities are shown in Figure 5. The contours of the
background gas, O2, depict the degree of
dissociation resulting from the arc, with virtually no
dissociation away from the arc. The region of the arc
discharge nearer the central axis is abundant in
reactive O radicals resulting from the decomposition
of O2 and the high gas temperature. Given the role of
O atoms in the chemical reaction mechanism, it is
expected that this distribution critically determines
the shape and strength of the arc discharge.
Figure 5. Contours of number densities of (above) background
O2 and (below) O radicals in m-3. Contour levels below the
minimum limit may have been removed.
[2] J. J. Beulens, D. Milojevic, D. C. Schram and P.
M. Vallinga, Phys. Fluids B, 3, 2548 (1991).
4. Summary
In this work, the example of a typical arc
discharge configuration using O2 at near atmospheric
pressure was used to illustrate the discharge
characteristics that can be computed using a two
temperature fluid based modeling approach with
CFD-ACE+. The limits of application for this model
are being extended to address more complicated
chemistries over a wider range of pressures.
[3] C. Delalondre, A. Bouvier, A. Caruso, N.
Mechitoua, O. Simonin and J.-C. Verite, Pure &
Appl. Chem., 70, 1163 (1998).
[4] S. C. Synder, A. B. Murphy, D. L. Hofeldt and
L. D. Reynolds, Phys. Rev. E, 52, 2999 (1995).
[5] N. Zhou, “A Two Temperature Model for High
Pressure High Temperature Plasmas”, Presented at
the 58th GEC, San Jose, CA, October 2005.
http://meetings.aps.org/link/BAPS.2005.GEC.LT2.5
References
[1] M. S. Benilov, J. Phys D, 41, 133001 (2008).
[6] CFD-ACE+ User Manual (2010). CFD-ACE+
can be licensed from the ESI Group.
http://www.esi-cfd.com