Modeling and Simulation of High-Current Constricted Vacuum Arcs
Driven by a Strong Magnetic Field in 3D
Kai Hencken1, Dmitry Shmelev 2, Oliver Fritz1
1
2
ABB Switzerland Ltd, Corporate Research, Segelhofstrasse 1K, 5405 Baden-Dättwil, Switzerland
Institute of Electrophysics, Russian Academy of Science, 106 Amundsena st., 620016 Ekaterinburg, Russia
Abstract: Constricted Vacuum arcs are metal vapor arcs occurring in vacuum
gaps at high currents (larger than 10kA). Such arcs are found typically in vacuum
interrupters making use of the TMF (transverse magnetic field) arc control, i.e., a
mechanism which moves the arc over the contact surfaces by the strong magnetic
field, which is generated from the current and the shape of the contacts. The
heating as well as the material erosion of the electrode surfaces are important
processes influencing the interruption capabilities of these vacuum interrupters.
We discuss the model underlying the simulation of these arcs. Especially the
microscopic processes occurring at the two arc-foots at the cathode and the
anode, that is, the attachment layer between plasma and metal surface, are
considered in detail. They are the basis to calculate the different fluxes at the
boundary of the gap for the simulation of the plasma. The model is able to
describe a self-sustained arc moving under the influence of the self-generated
magnetic field. We show results of the simulation in a 3D geometry. The motion
is dominated by the heating of the electrode surface in front of the arc, leading to
new arc-foots.
Keywords: 3D arc simulations, constricted vacuum arc, arc root modeling
1.Introduction
Vacuum interrupters are the most common
switching technology in the medium voltage area.
Their principle has been described in a number of
reviews, for example, see [1]. At high currents above
about 10kV the vacuum arc, that is, the metal vapor
arc formed from the material emitted by the two
electrodes, changes from a “diffuse mode” into a
“restricted mode”. In this mode the arc no longer
fills the full gap, but is restricted to a small column
with a diameter of about 1cm. The heating of the
contacts below the arc foots is rather strong, leading
to a melting of the contacts. Without an arc control
mechanism, this overheating of the contacts would
lead to an immediate reignition of the arc after
current zero.
One such mechanism is the “TMF arc control”.
The shape of the contacts are designed in such a
way, that a transverse magnetic field results from the
current flowing through them. This results in a
directed Lorentz force on the arc, which leads to a
movement of the arc over the outer periphery of the
contacts. In this way the heat to the contacts is
spread over a wider area. The surface temperature at
current zero is then reduced and the current can be
interrupted.
The modeling of this constricted arc and its
movement has been originally developed for the
case of 2D simulations. The simulation in 2D has
some limitation though. Especially the magnetic
field cannot be self consistently calculated. In
addition the stability of the arc in 3D is much more
demanding than in 2D. For details of the model, we
refer to [2-4]. In this paper we review the model
used and some modifications, which were done in
order to be able to simulate in 3D.
2. Details of the Model
A complete self-sustained simulation requires the
modeling of three different regions: The gap, that is,
the plasma region between the two contacts, the
contact itself, and the interface region between them.
The interface provides the boundary conditions for
the other two regions.
In order to calculate the magnetic field - essential
for the arc control - the electric current is calculated
throughout the whole domain (electrodes and
plasma). From this, the calculation of the magnetic
field is straightforward. As the vacuum interrupter
has a rather simple current path, the effects of the
sheath voltage drop as well as the time dependency
of the fields are neglected. This reduces our problem
to a standard DC conduction problem. This
calculation is done either using the built-in
electromagnetic solver of the corresponding CFD
program, or alternatively a separate FEM code was
used.
2.1. Electrode region
The description of the heating of the contacts is
straightforward. The heat equation is solved with
thermodynamic properties either as constants or as
functions of temperature. The heating due to the
electric current is taken into account as well, even
though it is in general smaller than the heat flux
coming from the plasma.
Due to the short time duration and the small
penetration depth of the heat into the contact surface
during the arcing time, one can model the heat
conduction as an one-dimensional problem. In this
way the strong heat flux at the surface is very well
resolved in the relevant direction while keeping a
reasonably low number of cells and a very fast
calculation time.
We include in our simulation the latent heat from
the melting of the material, but do not consider
effects coming from the movement within the
molten material. Also effects of the contact erosion
itself are not included, especially with respect to
changes in the geometry.
Even though the simulation of the temperature
inside the electrodes is not difficult, one should keep
in mind, that the surface temperature is the key
quantity to be used for assessing the interruption
capability of the breaker.
2.2 Plasma region
In the plasma region the MHD equations of the
metal vapor plasma needs to be solved. In our case
we simulate pure copper plasma. As the
temperatures inside the arc are rather high, ion states
up to Cu3+ have been taken into account.
In most of the calculations we have assumed local
thermodynamic equilibrium outside the interface
region. The particle densities of the different ion
species are calculated from the Saha equation
ni 1ne
ni
2 gi 1
exp
3
gi
Ei 1
.
kT
ni denotes the density of the ith ionization state and Ei
the ionization energy from the ith to the (i-1)th
ionization state and
is the thermal de Broglie
wavelength.
Alternatively we have used also a non-equilibrium
model using rate equations for the ionization and
recombination of the atoms, ions and electrons. In
most commercial CFD solver such equations for the
different species can be solved in parallel with the
fluid dynamics equation.
In addition we need the transport properties of the
Cu plasma. For the electrical and thermal
conductivity we use the Spitzer formula [6,7].
Modifications of the transport properties due to the
presence of strong magnetic fields have been
included as well.
The plasma dynamics is calculated by the MHD
equations. We use a single temperature model,
assuming the electron and the heavy temperatures
(atoms and ions) to be the same. We get
n
t
mn
(n u )
u
t
u
0
u
P
j
B
P
3
j2
Pu
P u
Qe
E ioniz E rad ,
2
t
where m is the mass of the atoms or ions and the
electrical conductivity of the plasma, P is the total
pressure, Qe is the energy flux of the electrons, Eioniz
the ionization energy density, Erad the radiation
losses. For Erad we have used a net emission
formulation.
Two different commercial solvers were used in
order to calculate these equations: CFD-Ace (ESI)
and Fluent (ANSYS) using their compressible-fluid
solvers. The dominant heating of the plasma is due
to the Joule heating. This and the Lorentz force
density are calculated using the result of the
electromagnetic calculation as discussed above.
The most detailed models are the one describing
the attachment areas between the contacts and the
plasma. From a CFD point of view mass,
momentum and energy fluxes are calculated for the
plasma and an energy flux into the electrodes. The
interface model itself is calculated taking into
account the microscopic processes in the sheath and
solving an algebraic set of equation. Due to the
different physical processes at work, cathode and
anode need to be treated differently.
The equations for the particle fluxes into and out
of the electrodes are of an algebraic nature. For the
cathode we calculate the electron current from the
McKeown model. The ion flux into the cathode is
given by the difference between the total current and
the cathode voltage drop from the energy
requirement to produce the ions in the sheath.
4 me e
(k bTs ) 2 exp
h3
3
e Ec
;
4 0
ji
Uc
j
Ec2
kbTs
4
0
meU c
m
ji
2e
zme
2 je k bTs
je
(1 z )k bT
m
4 me e
(kbTs ) 2 exp
h3
j
ji
j em ; U a
kbTs
; jth
kbT
j
ln e ,U a
e
jth
ne e
k bT
2 me
0.
Here ci denotes the fraction of the heavy particles,
which are ionized and
is calculated as for the
cathode.
In addition we have strong evaporation of copper
vapor from both arc-foots [5]. For this we use an
evaporation model given by
gv
2
Ps Ts
2 m k bTs
Ppl
mn u 2 ,
where PS is the vapor pressure. The same
evaporation model is used for cathode and anode.
From these particle fluxes between the electrodes
and the plasma mass, momentum and energy fluxes
are calculated by taking into account the energies of
each particle species. For example, the heating of the
electrodes is mostly given by the ions.
3. Simulation and results
je
je
ji Eioniz
0.61ci n e
jem
je
2.3 Interfaces
je
ji
U0 .
Here Ts is the surface temperature, Eioniz the
ionization energy of the copper plasma and U0 is a
free tuning parameter, chosen to be 5V in the present
simulation.
At the anode we have an influx of both electrons
and ions. The Bohm criterion provides us with the
ion current. The thermo-field emission of electrons
from the electrode is calculated in the same way as
for the cathode. The electron current into the plasma
is then calculated from the total current and the
anode voltage drop follows from it.
As mentioned in the introduction, a full 3D
geometry is required in order to simulate the
dynamics of the arc. The contact system has no
intrinsic symmetry that would allow treating it in
two dimensions. In addition the movement of the arc
requires a simulation space large enough to cover the
whole area traveled by the arc. A rather finely
resolved mesh (in the sub-millimeter range) is
required close to the electrodes in order to cover the
heating up of the plasma close to the interface. We
have made use of two commercially available
software packages: CFD-Ace (ESI) and Fluent
(ANSYS). Whereas CFD-Ace already integrates an
electromagnetic solver, we have used Fluent only for
the hydrodynamic calculations and used a
proprietary code for solving the electromagnetic
equations and also the heat equation in the
electrodes.
Due to the large flow velocities, which can be up
to several thousands of meters per second, the time
step of the simulation was in the range of tens of ns.
Therefore, simulation time requirements were rather
demanding and only a few microseconds could be
calculated per day.
Acknowledgment
Fig. 1 and 2 show two typical examples of the arc.
Parameters like arc diameter and pressure are
comparable to the results of earlier simulation [2-4].
In addition we show the magnetic field as generated
by the current itself. A movement of the arc over
almost one full circle was simulated. The speed
found in the simulation is comparable to the one
found in experiments.
References
4. Conclusions
We have presented a model to simulate the
moving constricted arc in 3D in a vacuum
interrupter. The coupled system of electrode heating,
plasma dynamics and the attachment between them
has been discussed. The computation effort
necessary to simulate a significant length of time (a
few ms) was found to be rather high, and therefore
systematic studies cannot be done at the moment.
We expect that with additional improvements in the
understanding of the physics of the TMF controlled
constricted arc, its modeling and the increase of the
available computational resources simulations will
in the future complement the experimental and
theoretical investigations.
Figure 1: The results of the simulation for one time instant are
shown. The arc temperature is shown for an iso-pressure
surface of 10bar for a constricted arc with 40kA.
Various discussions with colleagues
physical and simulation aspects are
acknowledged. We would like to thank
Thomas Christen, Michael Schwinne,
Ostrowski.
on both
gratefully
especially
and Jörg
[1] P. G. Slade, “The Vacuum Interrupter: Theory,
Design, and Application”, CRC Pr Inc, 2007.
[2] T. Delachaux, O. Fritz, D. Gentsch, E. Schade,
D.L. Shmelev, “Numerical simulation of a moving
high-current vacuum arc driven by a transverse
magnetic field”, IEEE Transactions on Plasma
Science 35, p. 905, 2007.
[3] T. Delachaux, O. Fritz, D. Gentsch, E. Schade,
D.L. Shmelev, “Simulation of a High Current
Vacuum Arc in a Transverse Magnetic Field”,
IEEE Transactions on Plasma Science 37, p. 1386,
2009.
[4] D.L. Shmelev and T. Delachaux, “Physical
Modeling and Numerical Simulation of
Constricted High-Current Vacuum Arcs Under the
Influence of a Transverse Magnetic Field”, IEEE
Transactions on Plasma Science, vol. 37, 2009, pp.
1379-1385.
[5] S.I. Anisimov, “Vaporization of Metal Absorbing
Laser Radiation”, Soviet Physics JETP, vol. 27,
1968.
[6] A. Anders, A Formulary for Plasma Physics,
Wiley VCH, 1990.
[7] L. Spitzer, “The physics of fully ionized gases”,
Interscience, 1965
Figure 2: The magnetic field generated by the current
path is shown together with the pressure inside the arc
in the center of the gap. The asymmetry of the pressure
inside the arc provides the driving force of the arc.