Numerical Prediction on Dependence of the Arc Current
and Swirl Gas Angle on Evaporation of Hafnium Cathode
in a Plasma Cutting Arc
Nguyen Phi Long1 , Y. Tanaka1 , Y. Uesugi1 , and Y. Yamaguchi2
1
Division of Electrical Engineering and Computer Science, Kanazawa University,
Kakuma, Kanazawa 920-1192, Japan
2
Komatsu Industries Corporation, Japan
Abstract: The effects of arc current and swirl gas angle on plasma cutting arc were investigated using a developed two-dimensional thermofluid model for arc plasma with
considering hafnium (Hf) cathode evaporation. Numerical modeling is very important
to predict the erosion amount of Hf cathode for different process parameters in the
plasma cutting arcs. Results showed that larger swirl gas angle caused a high temperature plasma area in front of the cathode surface shrunken in radial direction and
the flow patterns changed to opposite direction in front of the cathode at the swirl gas
angle from 15 degrees to 30 degrees. This change in gas flow pattern makes Hf vapor
transported along the center axis direction to the outlet of the nozzle. The total amount
of mass loss of Hf cathode evaporation was predicted to be enhanced with increasing
swirl gas angle and arc current.
Keywords: Plasma cutting arcs, Swirl gas, Hafnium Evaporation, Erosion of Cathode
1. Introduction
A plasma cutting arc is well known as a tool for cutting different metals with high-speed and high-accuracy in
the industrial fields. An arc plasma is established between
the electrode in the plasma torch and a work-piece which
is molten by heat flux from the arc plasma and heated by
oxidation reactions. Understanding details of interactions
between the electrode and the arc plasma under the different operating conditions is necessary to enhance the performance and the lifetime of plasma arc cutting devices.
Recently, some research works have investigated the
effect of process parameters inside plasma cutting arc torch.
V. Nemchinsky et al. [1] have shown that swirl flow makes
the evaporated atoms return to the cathode more probable.
In another paper [2], they confirmed that the plasma arc
is highly constricted by swirl gas. In addition, the experimental study of hafnium cathode for oxygen plasma arc
cutting have found that the temperature of the cathode surface reaches 3600 K during 200A operation [3] and the consumptions rate of hafnium cathode rapidly increases with an
increase in operating current [4].
In this paper, to understand further physical phenomena in plasma cutting arcs such as the interaction between
plasma and cathode evaporation, we have developed a numerical model. This developed model takes into account
the evaporation phenomena of hafnium cathode and the redeposition of hafnium vapor on the surface of the cathode.
It also accounts for the mass, momentum, and the energy
transfer from evaporated material to the arc plasma. Using
this model, the temperature field, gas flow field, and mass
flux of hafnium vapor inside of plasma cutting torch as well
as the total amount of mass loss of hafnium cathode due to
evaporation were obtained for different process parameters.
inlet
O2gas D
4.0
]3.5 E
m
m
[3.0
I
Current onti 2.5
source si2.0
op1.5 Cu
la1.0
id
Nozzle
aR0.5 Hf H
0.0 O
G 2
Work-piece
0
Electrode
Hf insert
Swirl gas
O2 gas
Arc
plate
C
Cu-nozzle
Gas/plasma
4
6
Axial position [mm]
outlet B
A
8
Fig. 1. Arc model and calculation space.
2. Mathematical model
2.1. Calculation space and assumptions
Fig. 1 illustrates the schematic diagram of the arc model
and the calculation space of the DC plasma cutting arc torch
used in this work. The electrode is made of copper with
hafnium tip insert of 1.27 mmϕ in diameter. Oxygen is used
for plasma gas and is supplied from inlet by swirling gas
flow. An arc plasma is constricted by the copper nozzle
with a nozzle outlet of 1.33 mmϕ in diameter. In this paper,
the developed two dimensional numerical model assumes
the followings: the gas flow is laminar, the plasma is in
steady state and axisymmetric, the plasma is in the local
thermodynamic equilibrium and optically thin.
2.2. Governing equations for gas and plasma
On the basis of the assumptions described above, the
gas and arc plasma are governed by the following equations:
- Mass conservation:
∇ · (ρu) = S pHf
(1)
- Momentum conservation:
(
)
∂u
∂p
Axial: ∇ · (ρuu) = −
+ ∇ · (η∇u) + ∇ · η
∂z
∂z
2 ∂
(η∇ · u)
−
3 ∂z
(2)
(
)
∂p
∂u
+ ∇ · (η∇υ) + ∇ · η
∂r
∂r
υ ρw2 2 ∂
−2η 2 +
−
(η∇ · u) − µ0 σEz Hθ
r
3 ∂r
r
Radial: ∇ · (ρuυ) = −
Swirl:
∇ · (ρuw) = ∇ · (η∇w) −
- Energy conservation:
∇ · (ρuh) = ∇ ·
(
ρυw ρw ∂
− 2
(rη)
r
r ∂r
(3)
(4)
)
λ
∇h + u · ∇p + σ|Ez |2
CP
−Prad − Lv S pHf
(5)
- Mass conservation for Hf vapor:
∇ · (ρuYHf ) = ∇ · (ρDHf−O2 ∇YHf ) + S pHf
- Ohm’s law:
Ez = ∫
I
2πrσdr
(6)
(7)
The thermionic emission current density was given by the
Richardson-Dushmann equation with Schottky effect:
√
(
) (
)
Wℓ
e
eE0
2
je = αℓ AT c exp −
exp −
(13)
kT c
2kT c πε0
jion = max(0, σEz − je )
(14)
where qK : heat flux between the surface cathode and plasma,
T p : the temperature of the plasma contacting with the cathode, T c : the temperature of the surface cathode contacting
with the plasma, e: elementary charge, k: Boltzmann constant, σsb : Stefan-Boltzmann constant, A: thermionic emission constant, ε0 : vacuum permittivity, Wℓ : the work function of Hf and Cu, ϵion : the ionization energy of oxygen ion,
δℓ : solid emissivity αℓ : material factor for thermionic emission, je : thermionic emission current density of electronic,
jion : ion current density, E0 : electric field at the cathode surface, λpc : thermal conductivity of cathode, ∆z: the distance
of surface cathode with the center control volume.
2.4. Governing equations for evaporation flux
The mass production rate due to evaporation S pHf was
1
Hθ =
σEz ξdξ
(8) approximately calculated as follows:
{
r 0
∆S
(neighbor to wall)
mHf (Γevp − Γdep ) ∆V
Hf
(15)
Sp =
where u: gas flow vector, r, z: radial and axial position, u,
0
(otherwise)
υ, w: axial, radial, and swirl velocity, ρ: mass density, h:
enthalpy, p: pressure, η: viscosity, σ: electrical conductiv- where mHf is the effective mass of Hf vapor, Γevp is the
ity, λ: thermal conductivity, CP : specific heat at constant mass flux of evaporated vapor, Γdep is the mass flux of repressure, Prad : radiation loss, Ez : electric field, Hθ : mag- deposition, ∆S is the surface of evaporated Hf, ∆V is the
netic field, YHf : mass fraction of Hf vapor, DHf−O2 : effective volume of the control volume.
diffusion coefficient of Hf vapor, Lv : Latent heat for evapThe mass flux of evaporated vapor Γevp was calculated
oration, S pHf : the mass production rate due to evaporation by the following Hertz-Knudsen relation:
√
√
1 Pv
8kT
mHf
2.3. Governing equations for electrode
mHf Γevp = mHf
= Pv
(16)
4 kT πmHf
2πkT
Inside the solid and liquid, the energy conservation equation is established as follows:
where Pv is the saturation vapor pressure of hafnium vapor.
(
)
(
)
The saturation vapor pressure Pv was evaluated using the
∂ λs ∂hs
1 ∂ λs ∂hs
0=
+
r
+ σs |Ez |2
(9) Clausius-Clapeyron’s relation:
∂z Cps ∂z
r ∂r Cps ∂z
[ (
)]


Lv

1
1

P
exp
−

1atm

RHf T boil
Tc

where hs : enthalpy of solid/liquid, Cps : specific heat of



(T c ≥ T melt )
Pv = 
(17)
solid, λs : thermal conductivity of solid, σs : electrical con

T v −T melt +∆T


ductivity of solid. The enthalpy of solid is related to specific


∆T
 Pmelt

heat as:
(T c < T melt )
∫ Ts
eff
hs (T s ) =
Cps dT s
(10) where RHf is the gas constant of hafnium vapor, P1atm is
T0
the standard pressure, T melt is the melting temperature of Hf
 Lm
with ∆T = 0.1 K. The mass flux of re-deposition vapor Γdep


 ∆T (T melt − ∆T ≤ T s ≤ T melt )
eff
(11) was calculated as follows:√
Cps = 

√
 Cps (otherwise)
8kT
kT
1 ρYHf
= ρYHf
(18)
mHf Γdep = mHf
4 mHf
πmHf
2πmHf
where T s : temperature of solid/liquid, T 0 : reference temeff
perature (T 0 =300 K), T melt : melting temperature, Cps
: ef- 2.5. Thermodynamic and transport properties of solid
fective specific heat, Lm : latent heat for melting of solid,
material, transport properties of oxygen plasma
∆T : step temperature for melting (∆T =5 K).
and Hf vapor
In this model, the following additional heat flux is added
Tab.
1 shows the thermodynamic properties of solid mato the energy conservation equation for the surface of the
terial
for
Hf and Cu. Thermodynamic and transport propcathode:
(
)
erties
of
oxygen
thermal plasma with Hf vapor were caljion 3
qK =
kT p + ϵion − Wℓ
culated using the equilibrium composition and the collision
e 2
integrals between species. These transport properties of Hf
λpc
je
4
4
(T c − T p ) − δℓ σsb (T c − T p )
(12) vapor were obtained by the first order approximation of the
− Wℓ −
e
∆z
Chapman-Enskog method [5].
- Ampere’s law:
∫
r
Tab. 1. Thermodynamic properties of solid material
Solid
Cu
Hf
ρs
[kg/m3 ]
8930
13310
Cps
[J/kg/K]
385.62
140
λs
[W/m·K]
381
23
T melt
[K]
1356
2506
T boil
[K]
2855
4876
Lm
[MJ/kg]
0.206
0.1347
5
[K]
4
35000
3
10 deg
]2
m
[m
no 1
iti
so 0
pl
iad 1
aR
33000
19306
14339
10650
10000
7909
5874
4363
θ=15 deg
θ=10 deg
0
0.0
3240
3000
0.2 0.4 0.6 0.8 1.0
Radial position [mm]
Fig. 3. Radial distribution of surface temperature of Hf cathode.
2407
1787
1328
2
Wℓ
[eV]
4.65
3.53
θ=30 deg
θ=20 deg
]4
K
[k
er 3
ut
ar
ep 2
m
e1
T
25995
Lv
[MJ/kg]
4.815
3.211
986
1000
3
30 deg
732
18
16
544
404
4
300
0
2
4
6
Axial position [mm]
θ=30 deg
θ=20 deg
] 14
kK
[ 12
er 10
tua
re 8
p 6
m
e 4
T
300
8
Fig. 2. Temperature distribution of arc plasma with swirl angle of
10 deg and 30 deg.
θ=15 deg
θ=10 deg
2
0
2.6. Boundary conditions
0.0 0.2 0.4 0.6 0.8 1.0
Radial position [mm]
Fig. 1 also shows the computational domain used in this
work. On the axis OA, axial symmetry condition is applied. Fig. 4. Radial distribution of arc plasma temperature just near the
surface of cathode.
Non-slip condition was considered on all the boundary wall
between solid and gas GH, HI, EI and BD. The velocity
]4
m
inside the solid was fixed at 0 m/s. At copper wall around
[m3
no
inlet OD and CD, temperature was fixed at 300 K. At the
itis
op 2
outlet AB, the axial gradients of physical parameters such
la
id 1
as enthalpy and velocity were set to zero. The oxygen gas
aR
is injected from the inlet ED with a swirl component which
0
0
2
4
6
8
can be expressed by the swirl gas angle (θ) as follow:
Axial position [mm]
win = uin (r/R)tanθ
(19)
where: uin , win : the axial velocity and swirl velocity at the
inlet, R: the radius of nozzle OD, r: the radial position between E-D. The boundary shape is assumed not to change
by melting and evaporation. The SIMPLE method after
Patankar [6] was used for the calculation scheme to solve
the governing equations described in the previous section.
3. Results
3.1. Effect of the swirl gas angle on the arc plasma
characteristics
For purpose of studying the effect of swirl gas flow on
arc plasma characteristics, the swirl angles are chosen with
values of 10, 15, 20, and 30 degrees. The arc current of 100
A and gas flow rate of 20 slm are fixed for different swirl
angles. Pressure is also fixed at 0.9 MPa.
Fig. 2 shows the temperature distribution of arc plasma
with swirl angle of 10 deg and 30 deg. This figure indicates
that the temperature along the center axis reaches a maximum temperature above 33000 K at nozzle throat by the
constriction of nozzle. Moreover, the arc root near the cathode surface is shrunken in radial direction in case of large
swirl gas angle.
Fig. 5. Gas flow field with swirl angle of 10 deg.
]4
m
[m3
no
iti
so 2
pl
iad 1
aR
0
0
2
4
6
8
Axial position [mm]
Fig. 6. Gas flow field with swirl angle of 30 deg.
The radial distribution of the surface temperature of Hf
cathode is shown in Fig. 3. It is clear that the temperature
of the surface cathode is decreased at the center of cathode
tip with a swirl angle increase from 10 deg to 30 deg. This
drop in the temperature is mainly caused by high energy
consumption of hafnium evaporation from the molten cathode surface. In addition, Fig. 4 represents that large swirl
angle allows a local maximal presence in radial distribution
of arc plasma temperature near the surface cathode. This
highest temperature occurs at radial position off axis and
increases with an increase in swirl angle.
The gas flow field in arc plasma for swirl gas angle of
10 deg was presented in Fig. 5. It seems to exist a small
2
)]s 0.9
0.8
m
/(g
[k 0.7
xu 0.6
lf
ssa 0.5
m 0.4
no
it 0.3
ar
op 0.2
vae 0.1
te
N 0.0
0.9
2
θ=30 deg
θ=20 deg
θ=15 deg
θ=10 deg
-0.1
0.0 0.1 0.2 0.3 0.4 0.5
Radial position [mm]
Fig. 7. Radial distribution of net evaporated mass flux from Hf
cathode for different swirl angles.
]/s 0.20
g
m
[s
so 0.15
ls
sa 0.10
m
fo
tn 0.05
uo
m
A0.00
without re-deposition vapor
with re-deposition vapor
10
15 20 25
Angle swirl θ [deg]
30
)]s 0.8
(m
g/k 0.7
[ 0.6
xu
lf 0.5
ssa 0.4
m
no 0.3
tia 0.2
ro
apv 0.1
et 0.0
eN
-0.1
I=200 A
I=150 A
I=100 A
I=50 A
-0.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Radial position [mm]
Fig. 9. Radial distribution of net evaporated mass flux from Hf
cathode for different arc currents with swirl angle of 10 deg.
s]/ 0.35
g 0.30
m
[s
so 0.25
ls 0.20
as 0.15
m
fo
tn 0.10
uo 0.05
m0.00
A
50
100
150
Arc current [A]
200
Fig. 8. Amount of mass loss of hafnium cathode evaporation for
different swirl angles.
Fig. 10. Amount of mass loss of hafnium cathode evaporation for
different arc currents with swirl angle of 10 deg.
counterflow towards the cathode tip. This counterflow may
then transport the Hf vapor in radial direction. In case of
30 deg as shown in Fig. 6, the flow pattern in front of the
cathode turns to opposite direction near the cathode, and
made Hf vapor transported along the center axis direction.
The reason of this change in gas flow direction is that the
hafnium vapor near the axis is ejected by evaporation at
large swirl angle.
along the cathode tip from 0.12 mm to 0.46 mm in radius as
arc current increases from 50 A to 200 A. The expansion of
the positive net evaporation flux area is due to the expanding
of the arc plasma in radial direction at high arc current. In
addition, the negative net evaporation flux around the fringe
of the arc has a significant rise with increasing arc current.
The result in Fig. 10 predicted that total amount of mass loss
of hafnium cathode is enhanced by increasing arc current.
3.2.
4. Conclusions
We have developed a two dimensional thermofluid model
for plasma cutting arcs including the effect of swirl gas flow
angle and arc current on evaporation of hafnium cathode.
As the results, we found that the high amount of Hf
vapor decreases the temperature of the surface cathode at
center of cathode tip. In the vicinity of cathode surface,
the large swirl gas angle causes maximum temperature arc
plasma at the radial position off axis and the arc root is
shrunken in radial direction. The flow patterns changed to
opposite direction in front of the cathode and made hafnium
vapor transported along the center axis direction at large
swirl angle. In addition, the amount of mass loss and mass
flux of Hf cathode evaporation was enhanced with increasing swirl gas angle and arc current.
Effect of the swirl gas angle on evaporation amount
of Hf cathode
Fig. 7 represents the mass flux of Hf vapor ejected from
the cathode. The net evaporation means mHf (Γevp − Γdep ).
The mass flux is nearly uniform in the range of 0.3 mm in
radius, where net evaporated mass flux markedly increases
with increasing swirl angle. Also, the negative net evaporation flux appears in radial range of 0.3-0.4 mm where the
evaporated atoms return to the cathode surface.
Fig. 8 shows total amount of mass loss of the cathode
with consideration of re-deposition and without that of redeposition for different swirl angles. It is found that the
re-deposition flux decreases the mass loss of Hf cathode.
On the other hand, total amount of mass loss of the cathode
evaporation linearly increases when increasing swirl angle.
3.3.
Effect of the arc current on evaporation amount of References
[1] V. A. Nemchinsky et al., J. Phys. D: Appl. Phys., 39, R423-R438
Hf cathode
(2006)
The similar calculations were made to investigate the [2] Q. Zhou et al., J. Appl. Phys., 42, 095208 (2009)
dependence of the arc current on evaporation of hafnium
[3] F. Yin, J. Schein, C. Hackett and J. Heberlein, Proc., ISPC, vol 14, p
cathode. The swirl angle of 10 deg is fixed for different
49 (1999)
arc currents. Fig. 9 shows the net evaporation mass flux for [4] Y. Yamaguchi et al., Quarterly Journal of JWS, 28, 311-318 (2010)
different arc currents. The net evaporation mass flux from [5] Y. Tanaka et al., Electr. Eng. Japan, 163, 18-29 (2008)
Hf cathode markedly increases with increasing arc current. [6] S. V. Patankar, Numerical Heat Transfer and fluid flow, Hemisphere
The area with the positive net evaporation flux is expanded
Publishing (1980)