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An electrical arc erosion model valid for high current:
Vaporization and Splash Erosion.
Frédéric Pons 1,2,3
and Mohammed Cherkaoui 1,2
(1) George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA (2) UMI 2958 Georgia Tech-CNRS, Georgia Tech Lorraine, Metz, France
(3) Laboratoire de Physique et Mécanique des Matériaux, Université Paul Verlaine, Metz, France
Abstract—Electrical arc erosion plays an important role in
limiting the life of power switching devices. Consequently, a
macroscopic arc erosion model, valid for high current, has been
developed. This is the first of its kind since it takes into account
two modes of contact material erosion: vaporization and splash
erosion. Specifically, this model is used to measure the amount of
contact material removed after one electrical discharge. It,
therefore, allows the comparison of electrical contact erosion
resistances between materials. Furthermore, this model is the
first step in determining the composition of a suitable AgCdO
substitute by identifying the most influential material properties
in the arc erosion process.
Keywords—Arc erosion, splash erosion, AgCdO
I. INTRODUCTION
AgCdO is one of the most widely used contact materials in
the world because of its outstanding performance.
Nevertheless, due to environmental considerations, it will soon
be forbidden by European environmental directives [1-3].
Therefore, finding a good substitute is of crucial importance.
Electrical arc erosion plays a crucial role in the reliability
and life of power switching devices. Depending on the contact
material behavior in response to an electrical arc, surface
damages can induce severe changes in contact material
properties that will impact the power switching device functioning.
Arc erosion process mainly results from two modes:
vaporization of contact material under heat energy coming
from the electrical arc and splash erosion induced by driving
forces such as electromagnetic force, and the Buoyancy and
Marangoni effects. Most of the models dealing with electrical
arc erosion only take into account the vaporization part [4-11].
But these can only be valid for low current since the splash
erosion contribution grows with current intensity [4-10].
Xixiu’s model [11], based on a kinetic energy approach, was
the first one to describe sputter erosion of electrical contact material.
The purpose of this paper is to present a complete
macroscopic arc erosion model that describes, for the first
time, the arc erosion process during a breaking operation
under DC conditions. This model, based on the coupling of
three different models, combines:
• an Arc energy transport model [4] to get the
energy brought by the electrical arc at time t
• a Thermal model to get the arc erosion due to
vaporization
• a Magneto-hydrodynamic model to get the arc
erosion due to splash erosion
Since this model we developed takes into account the two modes of arc erosion, it describes the erosion due to an
electrical arc for low and high currents.
II. ARC EROSION MODEL
The model that we propose in this study, is based on the
coupling (see fig.1) of an arc energy transport model, a
thermal model and a magneto-hydrodynamic model. It
calculates the amount of contact material removed after one
electrical arc corresponding to one breaking operation under
DC conditions assuming that all vaporized and ejected molten
materials have been removed. It does not take into account deposition mechanisms.
Figure 1. Diagram of the models outputs and their coupling
Fig. 1 describes the interactions between the three basic
models by showing outputs of one model used as inputs for
another one.
987-1-4244-1902-9/08/$25.00 IEEE. 9
A. Arc Energy Transport Model
The purpose of this model developed by Swingler and
McBride [4] is to give, function of input parameters related to
experimental conditions such as the current intensity, circuit
voltage and opening velocity, the amount of energy brought
by the electrical arc that will hit the electrode surface at time t
of the breaking process under DC conditions.
Figure 2. Power dissipation through plasma region
It considers that the arc is divided into three regions (fig.2):
the anode region, the plasma region and the cathode region.
The power dissipation within each of these three regions is
computed from the current through the region and the
potential drop across it. Then, power dissipation through the
arc, from region to region up to the contact surface, is
calculated considering two ways of energy transport processes that encompass all the mechanisms, at a microscopic scale,
involved in the energy transport process: radial and channel
transport processes. The power flux density out of the plasma
region transporting energy towards the cathode is given by:
( )1
22 2 22
p a
c
p
P P k rq k
r r rπ+
= ++
(1)
Where k1 is the proportion of energy radially dissipated
k2 = 50% related to the channel transport process
The power flux density out of the plasma region
transporting energy towards the anode is given by the same
equation as (eq.1) using k3 instead of k2.
The total power dissipated in the arc is used as an input for
the thermal model. Model results about the total power
dissipated in the arc for I=9A, shown on fig.3, show good
agreement between experimental data [4] and the model
results.
Figure 3. Arc power output for I=9A
B. Thermal Model
This model, using the same approach as Swingler and
McBride in [4], consists on a transient axi-symmetric thermal
conduction model (see eq.2) within the electrode based on
Fourier’s law whose heat energy input is the power flux
density output from the arc energy transport model acting at the center of the electrode, as it can be seen on fig.4.
Figure 4. Meshing scheme
Meshing scheme and the boundary conditions can be
seen on fig.4. Eq.2 has been solved using finite differences.
10
1
'''r z
T T Tk r k g c
r r r z z tρ∂ ∂ ∂ ∂ ∂+ + =
∂ ∂ ∂ ∂ ∂ (2)
Where rk and zk are the directional conductivities
'''g is the generation rate per unit volume and is only
nonzero for the element hit by the electrical arc
The thermal model gives the temperature distribution (see
fig.5) at each time t, and so, it allows us to deduce the molten
pool size at each time step and the total amount of material
vaporized after one breaking operation. We will consider that all vaporized material is part of the arc erosion.
Figure 5. Temperature distribution at the arc extinction (I = 9 A, V = 64
VDC, S = 1 m/s)
Temperatures of all the elements which have been vaporized
during the breaking process are set to 0 K (see fig.5).
C. Magneto-hydrodynamic Model
This model considers the formation of a molten metal jet, at
the contact surface center of the molten pool and along the z-
axis, under the influence of driving forces induced by the
electrical arc during the breaking operation. Whenever this
molten metal jet reaches critical conditions of stability, a
droplet comes off the molten metal jet tip and is ejected. All
the ejected droplets after one breaking operation are assumed
to constitute the splash erosion part of the arc erosion.
To develop our model, we use the same approach as J. N. Anno used in the case of a free, viscous inertial jet [12].
• Molten pool driving forces Molten metal is driven by gravity, buoyancy, surface
tension and electromagnetic forces [13-26]. We consider to be
in the “worst erosion case” where only surface tension
prevents the molten metal jet from growing and the three other
driving forces are working together to make it grow.
The buoyancy force, induced by the density change with
the spatial temperature distribution, is given by:
( )Buoy mF g T Tρ β= − (3)
Where g is the acceleration of gravity
is the density
is the thermal expansion coefficient
mT is the melting temperature
The buoyancy force at time t depends on the value of the jet
tip temperature at time t, output of the thermal model.
The electromagnetic force, resulting from the interaction
between the current flow and its self-induced magnetic field,
is given by:
ElecF J B= × (4)
Based on the assumptions of Kumar and DebRoy [27], the electromagnetic force at the center of the contact material
surface is:
2
2 20 2
m
Elecr
arc
I dF
r c
µπ→
= (5)
Where mµ is the magnetic permeability of the material
arcr is the arc radius
c is the thickness
d is the current density distribution factor
The electromagnetic force at time t depends on the value of
the intensity at time t, output of the energy transport model.
• Axial velocity distribution The momentum equation for the molten metal cylindrical jet
can be developed by evaluating the forces acting on a differential element of the jet (see fig.6)[12]. Considering this
element as a control volume and the case of steady flow,
Newton’s second law projected on the z-axis gives:
.z z
controlvolume
F v v dAρ= ∆ (6)
This gives the following differential equation:
( )
3 / 2
0 0
2
2 2
3 1 1
2
11 ( )
2
z z z
z z
m
m
zarc
u u u
z z u z zut w
I dg T T
ur c
µ γρ ρ
µβρπ
∂ ∂ ∂∂= +∂ ∂ ∂ ∂
+ + − +
(7)
11
Where zu is the axial velocity
µ is the molten metal viscosity (=0.01kg.m-1.s-1)
γ is the surface tension (=1N.m-1)
0t is the molten pool radius
0w is the velocity at the contact surface
On fig.6, where we can see the molten liquid jet at time t,
we assume that the molten pool surface (at z = 0) corresponds
to the contact surface of the electrode under study.
Figure 6. Differential element of the jet
By putting (eq. 7) into dimensionless form, it gives:
'
'' ' 1W
W W WW W
λ= + − − (8)
with ( )1/ 3
2
2 2
31 ( )
2
mz m
arc
I du g T T W
r c
µµ βρ ρπ
= + − +
( )1/ 32 / 3 2
2 2
31 ( )
2
mm
arc
I dz g T T
r c
µµ β ξρ ρπ
−
= + − +
( )2 / 32 / 3 2
2 2
0 0
1 11 ( )
2 3 2
mm
arc
I dg T T
t r c W
µγ ρλ βρ µ ρπ
−
= + − +
In the dimensionless non linear second order differential
equation (Eq.8), parameter depends on the current intensity I
at time t, output of the arc energy transport model and on the
jet tip temperature and the molten pool radius at time t,
outputs of the thermal model. Eq.8 has been plotted on fig.7
for two significant values of lambda.
Figure 7. Plot of the dimensionless form of the velocity differential equation
As it can be seen in Eq.8, W is related to the axial
velocity of the molten metal jet and ksi is related to the axial
coordinate z.
• Stability conditions During the arcing process, the molten liquid jet keeps
growing up to reach a critical length after which a droplet will
form. These stability conditions are derived from a mechanical
energy balance for a closed, stationary control volume of
volume V and surface area S [12]:
2
2
1
2
10
2
i i ij j i j j
V S S
V V
X u dV n u dS u n q dS
q dV RdVt
ρ τ ρ
ρ
+ −
∂− − =∂
(9)
Where iX is the generalized body force
ijτ is the stress tensor
iu is the velocity component in the ith direction
jn is the projection of the outward unit normal to the
surface in the jth direction
i iq u u= R is the viscous energy dissipation, given by
( ) ( ), , , ,2
i j j i i j j iR u u u uµ= + + (10)
12
This leads to the following stability condition,
corresponding to the jet breakup time:
072
crit
tT
µγ
= (11)
Where 0
t is the molten pool radius
µ is the molten metal viscosity
γ is the surface tension
• Droplet size From [12], we assume that the droplets are spherical and
their radii R0 are:
0
2 cR t= (12)
Where ct is the jet radius at breakup
Consequently, the amount of material removed by splash
erosion is given by:
3
0
4
3spl
Droplets
m Rπρ= (13)
D. Model results
Total masses of material removed by arc erosion during the
breaking process for different intensities from the model are
plotted on fig.8 and compared with experimental data [4].
Figure 8. Experimental data [4] and model results of AgCdO cathode arc
erosion
Fig.8 shows good agreement between model results and
experimental data (Polynomial (Model results) corresponding
to the curve of best fit polynomial of the model results).
III. CONCLUSION
A complete macroscopic arc erosion model which takes into
account the two arc erosion modes has been developed. This
model will be used to compare the arc erosion ability of
different contact material in order to find the best substitute to AgCdO. It will also be used to evaluate the most influential
contact material properties on the arc erosion process.
Other experiments will be conducted to finish validating our
model. White light interferometry will be used to get the total
amount of material removed due to the electrical arc from
each sample (see fig.9 & 10).
In our analysis of composite contact material such as
AgCdO, we assumed homogeneous material properties
(Surface tension, viscosity, etc..) reflecting the global
behaviour of the contact material under study.
Effects related to particle size, their distribution and percentage of CdO on the material properties will be the
object of future studies, where Molecular Dynamics will be
used to develop a model.
Figure 9. White light interferometry picture of a crater after one breaking
operation (400 A, 28 VDC)
Figure 10. White light interferometry picture of a crater after one breaking
operation (400 A, 28 VDC)
13
ACKNOWLEDGMENT
The authors would like to acknowledge Idriss Ilali (Project Manager at Leach Int.), Adeline Aubouin and Stephane
L’Attention (Research Engineers at Leach Int.), Alain Boyer
(Leach Int. Research Manager), Jean Michel Stocklouser
(Leach Int. R&D Manager), Dominique Bauthian (Leach Int.
General Manager), Jean-Michel Sigaud (Leach Int. President)
for their support.
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