JOURNAL OF APPLIED PHYSICS
VOLUME 84, NUMBER 7
1 OCTOBER 1998
Numerical model of a transferred plasma arc
S. M. Aithala) and V. V. Subramaniamb)
Center for Advanced Plasma Engineering, Department of Mechanical Engineering,
The Ohio State University, Columbus, Ohio 43210
J. Paganc) and R. W. Richardsond)
Department of Industrial, Systems, and Welding Engineering, The Ohio State University,
Columbus, Ohio 43210
~Received 23 March 1998; accepted for publication 9 July 1998!
This article presents results of two-dimensional simulations for a transferred plasma arc.
Calculations are performed for the internal plasma flow within the torch body, as well as the external
plasma jet impinging on a surface. The governing equations describing the plasma flow are
self-consistently coupled. These equations are the compressible Navier–Stokes equations for
conservation of mass and momentum ~in the radial and streamwise directions!, conservation of
energy, species continuity with finite-rate ionization and recombination kinetics, and the magnetic
diffusion equation for the electromagnetics. The unsteady forms of these equations are time marched
to steady state using the linearized block implicit method. This model does not employ any
adjustable parameters, and therefore enables direct comparison with experiments. Model predictions
are compared with experimental measurements of stagnation pressure distributions recorded on a
water-cooled copper plate ~workpiece!, and indicate good agreement, given the lack of adjustable
parameters. © 1998 American Institute of Physics. @S0021-8979~98!07419-2#
melt.1,2 Subsequent solidification of this molten region,
termed the weld pool, forms the weld or actual joint.
There are several electric arc-welding techniques. Of
these, plasma arc welding ~PAW! has several advantages
over others such as gas-tungsten arc welding ~GTAW! and
gas-metal arc welding ~GMAW!. First, PAW has a continuous forced flow of plasma gas, resulting in a high velocity
plasma jet which impinges on the workpiece ~transferred
arc!, thereby providing enough penetration to produce a deep
weld. In contrast, GTAW and GMAW are free-burning arcs,
where the flow ~if any! is induced by magnetohydrodynamic
~MHD! pumping from the arc. It is possible to operate the
plasma arc in the ‘‘key-holing’’ mode. In this mode, the
plasma jet penetrates the workpiece by melting through,
opening up a hole in front of the traversing jet that seals
upon cooling. The PAW process thus has the ability to form
a deep and relatively narrow weld bead in a single pass,
whereas GTAW requires multiple passes to achieve the
same. Secondly, the forced flow offers the ability to limit
entrainment of atmospheric oxygen from the ambient. This is
particularly important when a surface is prone to oxidation.
An important example of such an application is in the welding of large aluminum plates that comprise the external tank
of the space transport shuttle.3 Finally, PAW can be operated
in straight polarity, reverse polarity, or variable polarity
mode. The discharge is initiated by applying a high frequency voltage superimposed on a dc bias between the inner
electrode ~cathode! and the constricting nozzle ~anode! in
order to establish a pilot arc. After the pilot arc is established, the high frequency starter is shut off. The nozzle is
maintained positive ~by about 20 V! with respect to the electrode whenever the pilot arc is used. The main arc ~transferred arc! is subsequently struck between the grounded
I. INTRODUCTION
Electrical discharges with high currents on the order of
hundreds of amperes occur in many applications ranging
from industrial-scale arc furnaces and high power switches,
to materials synthesis and processing.1 Plasma torches are an
important class of electric arcs used in welding. Welding is
one of the basic industrial processes used widely in manufacturing, and in fabrication of components used in a myriad
of commercial applications. Despite their wide commercial
use, there are few detailed models describing high-current
electrical arcs from first principles. Modeling of these arcs
can enable in-depth understanding of the process itself, and
aid in the design of plasma torches and improvement of
plasma processes. This article describes a model of such a
welding arc, and compares predictions of the model with
experimental measurements.
Electric arc-welding processes, in general, consist of an
electrode and a workpiece with opposite polarities. An arc is
struck by applying an electric field between two electrodes
causing current flow through the ionized gas column ~established between the electrodes!. The heat generated within the
arc produces the high temperatures needed to sustain the gas
in its ionized state. Thermal energy is transferred to the
workpiece primarily due to particle fluxes causing it to
a!
Post-doctoral Research Associate, Robinson Laboratories, The Ohio State
University, 206 West 18th Avenue, Columbus, Ohio 43210.
b!
Associate Professor, Robinson Laboratories, The Ohio State University,
206 West 18th Avenue, Columbus, Ohio 43210; electronic mail:
subramaniam.1@osu.edu
c!
Graduate student, Baker Systems Laboratories, The Ohio State University,
Columbus, Ohio 43210.
d!
Associate Professor, Baker Systems Laboratories, The Ohio State University, Columbus, Ohio 43210.
0021-8979/98/84(7)/3506/12/$15.00
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© 1998 American Institute of Physics
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
FIG. 1. Schematic of plasma welding torch.
workpiece and the inner electrode by transferring the discharge. Such an arc is called a transferred arc, and a numerical model of the plasma flow in such an arc is the focus of
this article. The electrode is biased negative ~by about 30 V!
with respect to the workpiece in the transferred arc studied
here.
A typical PAW torch assembly consists of a central electrode concentric with two enclosing outer surfaces that confine the main plasma gas and shield gas flows, as shown
schematically in Fig. 1. These are called the constricting
nozzle and the shield gas nozzle, respectively. The constricting nozzle confines the plasma gas and acts as the anode for
supporting a pilot arc ~i.e., a nontransferred arc!. The shielding gas flows through the annulus between the constricting
nozzle and the shielding gas nozzle. Both the plasma and
shield gases are inert. Typically, argon is the plasma gas
while the shield gas is either argon itself or helium. In this
article, we model the transferred arc ~in what is referred to as
electrode negative operation! in the absence of the pilot arc
in order to gain a fundamental understanding of the relevant
physical processes. Predictions of the model are then compared with experimental measurements reported in Ref. 4.
A number of groups have attempted to model plasma
jets and compare their numerical results with available experimental measurements. The experimental measurements
mostly comprise of temperature measurements in the plume
of a free jet. The earliest attempts to model plumes of plasma
torches5–7 required the specification of temperature and velocity profiles at the exit plane of the nozzle as boundary
conditions. Lee and Pfender7 studied the effects of prescribing two different temperature and velocity profiles at the
nozzle exit on the external flow of a free jet. The profiles
chosen were such that the overall mass and energy flow rates
in the two cases were identical. In spite of this, the largest
difference between calculated and measured temperatures in
the external plume domain was 3000 K and the largest velocity difference was 500 m/s. These are equivalent to about
a 30% discrepancy in temperature and about 100% in velocity. Since these models did not include the plasma flow
within the torch, the temperatures and velocities of the
plasma jet at the exit plane of the torch could be obtained
only by a gross specification of the mass and energy flow
rate. It is clear therefore that to obtain realistic information
about the flow characteristics in the plume or in an impinging external jet, it is essential to correctly model the plasma
flow in the interior of the torch.
Aithal et al.
3507
To overcome these deficiencies in the earlier models,
Scott et al.8 have modeled both the torch flow and external
free jet in a single calculation. The region behind the cathode, the arc region and the plasma plume region are all included in the model. Their model is a steady, 2D model
which includes the swirl component of the gas flow and a k-e
turbulence model. Westhoff et al.9 have attempted a similar
study, with no additional physics included in their model.
Murphy and Kovitya10 have extended the model of Scott
et al. to include the effects of mixing of the plasma gas with
a different ambient gas. However, these models do not include any shield gas flows. Furthermore, they also do not
include finite-rate ionization and recombination processes,
and ionization fractions are computed from equilibrium considerations, i.e., from the Saha equation. All the aforementioned models simulate free jets ~i.e., nontransferred arcs!,
and solve the governing equations at steady state.
In contrast to the literature on free plasma jets, very few
multidimensional numerical models exist for transferred
arcs.11 Coudert et al. conducted studies on a transferred
plasma arc with a flat anode at atmospheric pressure.12 While
the authors compare results of their experiments with model
predictions, very little information is presented about the
model itself. As with other investigations, comparison of
measured temperature versus predicted temperature was used
to validate their model. Further, since little detail is provided
about the model, no assessment can be made regarding its
predictive capabilities, especially when the number of adjustable parameters is not revealed.11 A review of existing studies on transferred arcs is given in Ref. 11.
In this article, we present a self-consistent model of
transferred plasma arcs from first principles. The governing
equations and problem formulation are described next in Sec.
II and are solved to obtain the flow characteristics and species ~electron, argon ion, and neutral argon atom! concentrations. The internal flow calculations within the torch are used
to obtain all the flow variables at the exit plane of the torch.
Characteristics of the external plasma jet impinging on a
planar surface are then calculated by using the exit plane
values obtained from the internal flow calculations. The numerical method used to solve the governing equations is
briefly described in Sec. III, followed by a discussion of the
results in Sec. IV. A brief description of experimental measurements of stagnation pressure profiles using a watercooled copper block, as well as the procedure used to compare model predictions with measurements are given in Sec.
IV. This work is summarized and concluded in Sec. V.
II. PROBLEM FORMULATION AND GOVERNING
EQUATIONS
We consider the plasma flow to be laminar and to consist
of neutral Ar atoms, singly ionized atoms, and electrons. The
presence of multiply ionized atoms (Ar11, Ar111, etc.) is
excluded since for the gas temperatures and densities of interest here, their number densities are expected to be much
smaller than the number density of singly ionized argon atoms. This assumption will be verified a posteriori. The assumption of a laminar flow is justified by the fact that the
maximum Reynold’s number for the transferred arc studied
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Aithal et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
in this article is less than 100. Such a low value of the Reynold’s number is due to the low flow rates of interest here
~6 l per minute versus the 60 l per minute used in the only
other work on a transferred arc, see Ref. 12!. The electronimpact ionization and three-body recombination processes
are described by the following overall reaction:
]
]
rv2
1 ]
~ ru !1
~ rru2!1
~ r uw ! 2
]t
r ]r
]x
r
52
1
Ar1e 2 Ar11e 2 1e 2 .
In the present work, emphasis is placed on the plasma jet
impinging on the workpiece. Phase change at the welding
surface is not taken into consideration. This work does not
couple modeling of the weld pool with the arc, but rather
focuses on the gas-phase processes in detail. The workpiece
is considered a boundary of the fluid domain. The plasma is
assumed to be quasineutral (i.e., n i 'n e ) since the representative Debye length is much smaller than the characteristic
dimensions that are involved. Sheath regions adjacent to surfaces or plasma-cold gas boundaries are not considered here,
and the standard MHD approximations are assumed to
hold.13 The geometry is two-dimensional and axisymmetric,
and it is assumed that only the azimuthal component of the
magnetic induction, B u , is significant. This is correct for a
transferred arc with no pilot arc present, where the current
mainly flows from the electrode ~cathode! to the workpiece
~anode!. Finally, it is assumed that a single temperature can
represent the plasma. This is justified by examining the
maximum deviation between electron and heavy particle
temperatures, which is given by ~see p. 166 of Ref. 1!
T e 2T p m H l 2e e 2 E 2
5
,
Te
24m e k 2B T 2e
]r 1 ]
]
1
~ rru !1
~ r w ! 50,
]t r ]r
]x
~1!
G
FS
]
]u ]w
1
h
]x
]x ]r
DG S
1
D
2h ]u u
2 ,
r ]r r
~2!
]
]
1 ]
~ rw !1
~ r r uw ! 1
~rw2!
]t
r ]r
]x
52
F
G
DG
]p
]
]w 2
1
2 h “.u
2h
]x ]x
]x 3
1 j rB u1
F S
]u ]w
1 ]
1
hr
r ]r
]x ]r
~3!
,
]ne 1 ]
]
1
~ un e r ! 1
~n w!
]t
r ]r
]x e
S
D
1
1 ]
rD e ]
2rn e m e E r8 2
~ n kT !
r ]r
kT ] r e
1
]
De ]
2n e m e E 8e 2
~ n kT ! 5ṅ e ,
]x
kT ] x e
S
D
~4!
]eT 1 ]
]
1
~@ e T 1 p # ur ! 1
~@ e T 1 p # w !
]t
r ]r
]x
5
S D S D
]T
]
]T
1 ]
kr
1
k
1 h F1 j r wB u 2 j x uB u
r ]r
]r
]x
]x
1
where T e is the electron temperature, T is the heavy particle
temperature, m H is the atomic mass of the heavy particles, l e
is the electron mean free path, e is the charge on the electron,
E is the magnitude of the electric field, m e is the mass of an
electron, and k B is Boltzmann’s constant. For an atmospheric
pressure, high-intensity argon arc with E51300 V/m, l e
5331026 m, m Ar /m e 573104 , and T e 530 000 K, all of
which are characteristics of the present problem, the deviation between T e and T computed from the above formula is
about 1%. This justifies the assumption of local thermodynamic equilibrium ~LTE! or thermal equilibrium. This last
assumption of a single-temperature plasma could be invalid
at the boundaries of the discharge, despite the fact that the
pressure is atmospheric. Although our formulation allows
thermal nonequilibrium effects to be considered, we nevertheless ascribe a single temperature to the heavy particles
~atoms and ions! and electrons in the interest of simplicity.
Given these assumptions, the governing equations for
conservation of mass, momentum ~neglecting the swirl component!, species continuity, and energy can be written as14
F
]p
]
]u 2
1
2 h ¹.u 2 j x B u
2h
]r ]r
]r 3
j 2r 1 j 2x
s
,
~5!
where r is the mass density, p is the pressure, u is the radial
component of velocity, w is the axial or streamwise component of velocity, t is time, r is the radial coordinate, x is the
axial or streamwise coordinate, B u is the azimuthal or tangential component of the magnetic induction vector, j r is the
radial component of the current density vector, j x is the axial
or streamwise component of the current density vector, F is
the viscous dissipation, E r8 5E r 2wB u , E 8x 5E x 1uB u , k is
the thermal conductivity, s is the electrical conductivity, and
h is the coefficient of viscosity. Note that the energy equation is written in terms of the total internal energy per unit
volume, defined as e T 5e1 r (u 2 1 v 2 1w 2 )/2, and the internal energy per unit volume, e, is given by
e53/2n T k B T1ṅ e e i ,
where n T is the total number density ~i.e., n T 5n A
12n e ), k B is Boltzmann’s constant, T is the temperature,
and e i is the ionization potential of the argon atom. The
detailed expression for F is given in Ref. 15. A twotemperature description of an atomic plasma can be easily
accomplished in the above formulation by defining an electron temperature T e that is different from the heavy particle
temperature T(ÞT e ). An additional conservation equation
describing the evolution of T e in space and time would then
have to be provided. The above equations are the compress-
Aithal et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
ible Navier–Stokes equations but including the Lorentz body
force terms ( j x B u andj r B u ) in the momentum equations and
additional terms in the energy equation representing ohmic
heating and work done by the Lorentz body forces. In the
above equations, the pressure is p5n T k B T, and ṅ e
5k f n e n A 2k r n 3e represents the net production of electrons
due to electron impact ionization ~with a rate constant k f )
and three-body recombination ~with a rate constant k r ) processes. The rate constants k f and k r are obtained from Ref.
16. The mass density r is related to the neutral atom number
density n A and electron ~or ion! number density n e through
the following algebraic relation:
r 5m A n A 1m i n i 1m e n e 'm A ~ n A 1n e ! 5m A n 0 .
~6!
In order to ensure that the above algebraic constraint is
satisfied at each point at all times, the species continuity
equation for the neutral atom number density is discarded.
n A is then evaluated using Eq. ~6!. The above Eqs. ~1!
through ~6! represent six equations for the seven unknown
quantities r , u, w, T, n e , n A , and B u . The seventh
equation is obtained from Maxwell’s equations, which can
be combined to yield a single wave equation for the magnetic induction. However, since only quasi-steady or steadystate solutions are of interest here, this wave equation can be
reduced to a parabolic or elliptic form.13 Inclusion of the
time derivative in the parabolic form via the m 0 s ( ] B/ ] t)
term would force us to resolve extremely small time scales
involving the inverse of the speed of light. This would be
very computationally intensive, and unnecessary when seeking a steady-state solution or when time scales of interest are
much longer than the time required for an electromagnetic
wave to traverse the domain. Using Ampere’s law @ “3B
5 m 0 s (E1v3B) # , expanding the resulting cross products,
and using the vector identity:
“3 ~ “3B! 52¹ 2 B1“ ~ “–B! 52¹ 2 B,
Maxwell’s equations can be written in strong conservation
form for zero space-charge density for the present problem,
as follows:
2¹ 2 B5 m 0 ~ “3j! .
By definition, plasmas exhibit the property of overall electrical neutrality on length scales much larger than the Debye
length. Therefore on these scales, the electron and positive
ion number densities can be taken to be approximately equal
or that the net space-charge density is zero. Using Ohm’s law
~a constitutive equation of the form j5 s (E1v3B), the
above equation can be written as follows for a single component of the magnetic induction, B u :
S D S D
SD
1 ] r ]Bu
Bu ] 1
] 1 ]Bu
1
1
r ]r s ]r
]x s ]x
r ]r s
2m0
]
]
~ wB u ! 2 m 0
~ uB u ! 50.
]x
]r
~7!
The current densities j r and j x appearing in Eqs. ~2!, ~3!, and
~5! are computed from Ampere’s law (“3B5 m 0 j):
j r 52
1 ]Bu
,
m0 ]x
j x 52
1 ]Bu
.
m0 ]r
3509
and
It is also possible to reformulate Maxwell’s equations in
terms of the scalar potential governed by a Poisson equation.
The resulting equation for the electric potential f is analogous to Eq. ~7!.
The governing Eqs. ~1!–~7!, are nondimensionalized for
both internal flow and external flow using the same scales.
These are a length scale of 3.175 mm, a temperature scale of
10 000 K, a pressure scale of 1.08773105 Pa for the 100 A
case, a pressure scale of 1.16253105 Pa for the 150 A case,
and a velocity scale of 1861 m/s, corresponding to the isentropic speed of sound in argon at 10 000 K. The nondimensionalized equations are then solved numerically in conjunction with appropriate boundary and initial conditions to
obtain the distributions of velocity, temperature, magnetic
induction, and species concentrations both within the torch,
as well as in the external jet flow impinging on a workpiece.
The solution strategy adopted here is to split the domain into
two parts—one representing the internal flow up to the exit
plane of the torch, and the other representing the external
plasma jet issuing from the exit plane. Calculated values for
all the dependent variables from the internal flow are then
used as boundary conditions for the external flow. In the
interest of attaining high computational speeds, the magnetic
transport equation is not solved over the entire domain in
both internal and external flows. In the internal flow, Eq. ~7!
is solved only within the region denoted as BCDEFG in Fig.
2~a!. It has been observed experimentally that the spread of
the jet in the radial direction is negligible.4 Hence, the extent
of the computational domain for the electromagnetics calculation in the external flow is restricted to twice the extent of
the nozzle radius ~ACGH in Fig. 2~b! denotes the region over
which the magnetic transport equation is solved!. The boundary conditions necessary to complete the formulation of this
problem are given in Tables I and II, corresponding to the
internal flow and external flow domains shown in Figs. 2~a!
and 2~b!, respectively. The heat flux into the workpiece is
modeled as a convective boundary condition, and is given
explicitly in footnote 10 of Table II. The radial distribution
of gas temperature at the surface is then determined from this
boundary condition.
The unsteady form of Eqs. ~1! through ~5!, in conjunction with Eqs. ~6! and ~7! are solved using a time-marching
method which is described in Sec. III. Equation ~6! being
algebraic is imposed at each grid point at each time step.
Equation ~7! being elliptic is solved iteratively at each time
step. Consequently, initial conditions also need to be supplied in order to complete the formulation of the problem.
For the internal flow calculations, the radial velocity is
set to zero everywhere initially, while the axial velocity is
given a linear variation from a small value at the inlet ~10
m/s! to a higher subsonic value ~1675 m/s! at the exit. A
subsonic value at the exit is prescribed for the initial velocity
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Aithal et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
FIG. 2. ~a! Physical domain used for the internal flow simulations. The
plane DE denotes the exit plane of the torch. ~b! Physical domain used for
the external flow simulations. The torch exit plane is denoted as AB, the
shield gas exit plane is denoted as BD, DE is the torch body, EF is the
outflow boundary, FGH represents the surface of the water-cooled copper
plate, and AH denotes the centerline or axis of the torch.
regardless of whether the final solution is supersonic or subsonic. An initial linear temperature distribution is prescribed
over the whole domain ~ranging from 1000 K at the inlet to
14 500 K at the exit, along the centerline!. This temperature
distribution is then given a parabolic variation in the radial
direction. An initial pressure distribution is also prescribed,
with the pressure held constant (at 1.087310 5 Pa) until a
specified downstream distance (x52.7 mm), followed by a
linear decrease to a specified value of 1.0133105 Pa at the
exit. The total number density, n T at each axial section is
then calculated from these initial pressure and temperature
distributions. The initial electron number density is prescribed at a constant value of 7.831018 m23. The initial
number density profiles of the atoms and mass density are
then calculated from the algebraic relationship, Eq. ~6!. The
specific values used for these initial guesses do not impact
the final converged results, though the initial profiles and
magnitudes do need to be physically reasonable and selfconsistent in order for the time-marching procedure to be
numerically stable. No initial distributions are required for
the magnetic induction as it is governed by an elliptic equation @Eq. ~7!#.
For the external flow, values are initially prescribed for u
~20 m/s! and w ~20 m/s! throughout the domain. The static
pressure within the domain is given the ambient value of 1
atmosphere initially. The initial temperature is given the
value of 10 000 K throughout the domain. The electron number density is given a small constant value (7.831018 m23)
throughout the domain. The initial value of density is then
evaluated from the temperature and pressure using the equation of state ( P5n T k B T).
We evaluate the coefficient of viscosity, thermal conductivity, electrical conductivity, and the species diffusivities
using mean-free-path theory.17 The electrical resistivity is
calculated as the sum of an electron-neutral resistivity and a
Spitzer–Harm resistivity.17 In colder regions of a plasma
flow, the discretized form of Eq. ~7! can be confronted with
the loss of diagonal dominance. Hence, stable numerical solutions cannot be obtained below a minimum value of the
electrical conductivity. This minimum value is called the
TABLE I. Boundary conditions for the internal flow problem. Please refer to Fig. 2~a!.
ABCD
~Orifice or nozzle
wall!
r
u
w
T
ne
]r / ] r50
0
0
c
] 2 n e / ] r 2 50
DE
~Exit plane!
EF
~Centerline!
FGH
~Electrode or
cathode!
a
]r / ] r50
0
] w/ ] r50
] T/ ] r50
] n e / ] r50
] 2 r / ] r 2 50
0
0
] 2 T/ ] r 2 50
] 2 n e / ] r 2 50
] 2 u/ ] x 2 50
] 2 w/ ] x 2 50
] 2 T/ ] x 2 50
] 2 n e / ] x 2 50
HA
~Inlet!
b
0
] 2 w/ ] x 2 50
d
] 2 n e / ] x 2 50
Boundary conditions for B u , with electromagnetics calculation over GBCDEFG
Bu
GBCD
DE
~Exit plane!
EF
~Centerline!
FG
~along cathode surface!
B u 5 m 0 I/2p r
] B u / ] x50
0
2
B u 5r m 0 I/2p r cathode
n T is obtained from P exit51 atmosphere, and extrapolated value of T exit . n e is obtained by implicit extrapolation, and r is calculated from r 5m A (n e 1n A ).
b
n T is obtained from the equation of state, P5n T k B T. P is obtained from the prescribed total pressure P 0 and
total temperature T 0 , assuming isentropic flow conditions exist between the stagnation state and the inlet, and
T is obtained from the prescribed total temperature T 0 and the extrapolated value of w at the inlet. n 0 is
obtained by implicit extrapolation, and r is calculated from r 5m A (n e 1n A ).
c
T is computed from the boundary condition ] T/ ] r u (r5r 0 )52(h/k) @ T(r5r 0 )2T coolant# , where h/k53.47
31022 m, and T coolant5300 K.
d
Obtained from prescribed total temperature T 0 at the inlet, and the extrapolated value of w.
a
Aithal et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
3511
TABLE II. Boundary conditions for the external flow problem. Please refer to Fig. 2~b!.
AB
~Exit
plane of
torch!
BD
~Exit plane
of shield
gas flow!
a
b
b
b
a
c
0
0
500 K
] 2 n e / ] r 2 50
] 2 u/ ] r 2 50
] 2 w/ ] r 2 50
] T/ ] r50
] 2 n e / ] r 2 50
r
u
w
T
ne
Bu
a
c
a
500 K
a
a
AB
BC
~Torch body!
a
B u 5 m 0 I/2p r
DE
~Torch
outerbody!
EF
~outflow!
CG
B u 5 m 0 I/4p r cathode
FH
~Plane of
workpiece!
]r / ] x50
0
0
d
] n e / ] x50
HA
~Axis or
centerline of
torch!
]r / ] r50
0
] w/ ] r50
] T/ ] r50
] n e / ] r50
GH
~along workpiece!
AH
~Centerline!
] B/ ] x50
0
a
Obtained from the solution of the internal flow problem.
n T is obtained from the equation of state ( P5n T k B T), after setting the p51 atmosphere and T is obtained
from its respective boundary condition. n e is obtained by implicit extrapolation, and r is calculated from r
5m A (n e 1n A ).
c
Quadratic distribution is prescribed for w at the exit of the shield gas channel ~surrounding the outer surface of
r
orifice! such that * r 2 2w p rdr yields a volume flow rate of 25 ft3/h ~CFH!. The radial velocity component is
1
then prescribed as u52w tan@a(r)#, where a (r) changes depending on the angles that the respective nozzle
walls make with the streamwise ~x! direction.
d
Boundary condition for temperature is ] T/ ] x u (x5L)52(h/k) @ T(x5L)2T plate# , where h/k
575e 25r/r 0 m21, r 0 512.7 mm, L is the standoff distance, and T plate5500 K.
b
conductivity ‘‘floor.’’ The value of this ‘‘floor’’ is not an
adjustable parameter for a given set of operating conditions
~i.e., flow rate, total current, and electrical power input!. A
specific value can be assigned to the conductivity floor based
on the input electrical power, consistent with Poynting’s
theorem. This will be addressed in more detail in Sec. IV.
III. NUMERICAL METHOD
The linearized block implicit ~LBI! method developed
by Briley & McDonald is used to solve Eqs. ~1!–~6!
numerically.18 This method was originally developed for the
solution of the compressible Navier–Stokes equations for
nonreacting flows, with some limited extension to reacting
flows.19 We have previously extended the LBI method to
include state-specific kinetics20,21 and plasma flows.22,23 The
governing equations are written in strong conservation form
and nondimensionalized. They are then discretized in time
using the Crank–Nicolson scheme, and then linearized about
the previous time step using a Taylor series first-order accurate in time. These linearized equations ~and boundary conditions! are then discretized in space to second-order accuracy by central differencing, after transforming from a
uniform grid in the physical domain to a uniform grid in the
computational domain.14 The two-dimensional operator is
then split consistently into two one-dimensional operators
using the Douglas–Gunn alternating direction implicit ~ADI!
scheme.24 This procedure results in a series of block tridiagonal matrices ~with block sizes equal to the number of
unknowns at each point!, which can be solved efficiently
using standard LU decomposition methods.25 The magnetic
transport equation is solved separately at each time step of
the time-marching process, using the odd-even Jacobi iterative scheme.
The LBI method is an implicit method that correctly
preserves the coupling between the different governing equations, but suffers from the need to add artificial or numerical
dissipation.18 Such addition of artificial dissipation is necessary for overall stability of the numerical scheme, and when
inaccuracies arise from the way boundary conditions are
prescribed.18 This difficulty is due to the central differencing
used by the method to discretize the convective terms in the
governing equations. It has been observed that numerical solutions exhibit nonphysical behavior if the cell Reynold’s
number is greater than 2.18 This numerical instability can be
avoided by using upwind differencing for the convective
terms, or by adding a certain amount of second-order or
fourth-order dissipation explicitly.26 We have found that, in
practice, the amount of artificial dissipation that is added
based on the cell Reynold’s number criterion, is overly restrictive. Consequently, we employ the second-order dissipation term used by Briley and McDonald18 but multiplied by a
constant less than 1, and a second-order dissipation term resembling that of Rusanov ~see p. 235 of Ref. 27!. Several
methods do exist ~e.g., flux-corrected transport or FCT, total
variational diminishing or TVD schemes! that add and subsequently remove numerical dissipation at various stages of
the calculation.28 However, the LBI method allows the governing equations to be efficiently solved on vector and parallel computers and for the dissipation level to be controlled
during the time-marching process.
The exact form of artificial dissipation is often undisclosed in the literature, and therefore we describe it in detail
here. It must be made clear at the outset that the addition of
such a term to the discretized form of the governing equations is important for stability of the particular timemarching scheme used. The fully converged solution is independent of the magnitude of artificial dissipation beyond a
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J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
certain minimum amount. In this work, terms of the form
m r ] 2 f / ] r 2 1 m x ] 2 f / ] x 2 are added to the discretized and
nondimensionalized forms of the governing Eqs. ~1!–~5!, resulting in additional implicit and explicit terms in the final
linearized difference equations. The specific expressions for
the non-dimensional coefficients, m r and m x for the internal
flow within the torch are
H
0.00025r ~ Dr ! for T<2000 K
m r u explicit5 0.005 r ~ Dr ! for 2000 K<T<10000 K
0.025 r ~ Dr ! for T>10000 K,
m x u explicit5
H
m r u implicit5
531026
,
Dr
m x u implicit5
H
0.05AT ~ Dx ! for T<10000 K
0.025 AT ~ Dx ! for T>10000 K,
0.0025AT ~ Dx ! for T<2000 K
0.00025 AT ~ Dx ! for T>2000 K.
Similarly, the coefficients of the artificial dissipation terms
for the external flow are
H
0.0005r ~ Dr !
m r u explicit5 0.005r ~ Dr !
0.025r ~ Dr !
m x u explicit5
m r u implicit5
H
AT ~ Dx ! for
0.05 AT ~ Dx !
0.05 AT ~ Dx !
for T<2000 K
for 2000 K<T<10000 K
for T>10000 K,
T<2000 K
for 2000 K<T<10000 K
for T>10000 K,
2.531023
Dr
H
0.1AT ~ Dx ! for T <2000 K
m x u implicit5 0.0005AT ~ Dx ! for 2000 K<T<10000 K
0.00025 AT ~ Dx ! for T>10000 K.
The model for m x resembles that of Rusanov’s to some
extent.27 This model, in which the flow regimes are categorized into three regions, is found to be very stable numerically. The cold regions of the flow, i.e., regions close to the
inlet plane in the internal flow and regions close to the torch
body in the external flow, are well represented by T
<2000 K. Regions close to the core of the arc are characterized by temperatures between 2000 and 10 000 K, both in
the internal and external flows. Temperatures above 10 000
K are typical of the arc core. In the colder regions, where the
mass density is high, m r has a smaller constant multiplicative
value as compared with the hotter regions where the mass
density is lower. Constant multiplicative factors for the implicit and explicit terms in the axial directions also scale with
temperature. They are obtained by trial and error by monitoring the location of the maximum and minimum values of
temperature, pressure, and velocity throughout the domain
during the time-marching process. The values of the coefficients are chosen based upon the idea that insufficient dissipation leads to nonphysical solutions. The above dissipation
model is used to simulate the case of a transferred arc for a
total current of 100 A. The coefficients in the implicit and
Aithal et al.
explicit terms have to be increased by a factor of 2 or 3 for
the higher current case of 150 A. This shows that the above
artificial dissipation model is general in form, requiring only
minor changes in the coefficients to yield stable solutions for
different operating currents.
IV. RESULTS, COMPARISON WITH MEASUREMENTS,
AND DISCUSSION
The governing equations in Sec. II have been solved
numerically according to the procedures outlined in Sec. III.
The physical domain for the internal flow calculations consists of a portion extending about 8 mm inwards from the
exit plane of the torch. The cathode tip is approximately 3
mm from the exit plane. This portion of the torch was included since it is important to model the region before the
plasma gas encounters the arc. This internal flow calculation
does not include the outer concentric channel, carrying the
shield gas. The radial extent of the domain for the external
flow is taken to be eight times the radius of the torch nozzle
~1.5875 mm!, and the shield gas velocity and temperature
distributions exiting the outer channel are prescribed as inflow boundary conditions for the external flow calculation.
The domain for the external flow is thus a rectangle of dimensions 9.5 by 12.7 mm, where 9.5 mm is the stand-off
distance. A grid consisting of 50332 points was used for the
internal flow calculations, and a grid of 30380 grid points
was used for calculation of the external plasma jet. These
grids were found to adequately resolve the gradients in the
problem. These calculations were carried out on a silicon
graphics challenger L 8-processor work station. The challenger comprises 8 IP19 200 MHz processors. Using a nondimensional time step of 8.031024 ~corresponding to a dimensional time step of 1.36 ns!, a solution converged to four
decimal places was obtained for the internal flow calculation,
in about 12 h of CPU time. The external flow calculation
consumed a similar amount of computing resources. The results of these calculations are discussed here, and are compared with available experimental measurements for this specific torch geometry.4
We refer again to Figs. 2~a! and 2~b! for details of the
geometry of the torch and its location relative to the workpiece. Although the dimensions are indicated in the figures,
we provide the critical dimensions here for convenience. The
electrode diameter is 3.175 mm, and has a conical shape at
its tip that extends 1.42875 mm beyond the straight portion
of the cathode. The orifice or constricting nozzle is 6.35 mm
in diameter, and converges down to 3.175 mm in diameter at
the orifice. This orifice @surface CD in Fig. 2~a!# extends
between 0.9525 and 3.7825 mm from the cathode or electrode tip, the latter length defining the location of the exit
plane relative to the tip of the electrode. The standoff distance ~i.e., the distance from the torch exit to the workpiece!
is 9.525 mm.
The hostile nature of the hot arc environment limits the
number of experimental measurements that can be made. No
diagnostics could be performed on the internal flow. The
apparatus consisted of a B&B Precision Machine HEC-300A
plasma torch ~jointly developed in collaboration with the
Materials and Processing Laboratory of NASA Marshall
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
Space Flight Center!, a Hobart model VP-300-S variable polarity power supply, gas manifold, and associated electrical
and cooling connections.4 Some of the key features of the
B&B torch are a self-centering electrode design, repeatable
electrode setback adjustment, hermetic sealing, and capacity
to carry 300 A while operating in the variable polarity mode,
and 360 A while operating in the electrode-negative ~i.e.,
transferred arc! mode. A cutaway of this torch is shown in
Fig. 1. The Hobart power supply was capable of delivering a
constant current output of 3–299 A at a voltage of up to 50
V for each polarity. The welding torch was mounted on a
motion stage driven by a superior slo-syn stepper motor,
which was computer controlled in the three coordinate directions. The transferred arc current was monitored using two
nonobtrusive Hall effect PI-600 current/W sensors ~from F.
W. Bell Corp.!. Plasma and shielding gas flows were monitored using a Omega FMA-5609 mass flow meter ~range of
0–5 LPM!, and a Cole–Palmer GMF-1710 mass flow meter
~range of 0–15 LPM!, respectively. Argon was used as the
plasma gas as well as the shield gas. Measurements of stagnation pressure were performed on a water-cooled copper
block equipped with a pressure tap. The pressure was recorded using a Omega PX150 series pressure transducer. By
scanning the torch past the stationary pressure tap, a radial
profile of the stagnation pressure across the plasma arc could
be measured for a given standoff distance. It is important to
note that this is considerably different from the typical situation encountered during welding, wherein a keyhole is
formed. Nevertheless, these pressure measurements serve to
provide data against which the model can be compared.
Earlier works have validated calculations using limited
experimental measurements. Scott et al.8 and Murphy et al.10
have compared the results of their calculations to isotherms
obtained experimentally from free jets. Using measured values of temperature as an indicator of the reliability of the
model is questionable since the measurement techniques
make inherent assumptions about the radiative state of the
plasma @i.e., that the plasma is in local thermodynamic equilibrium ~LTE! as far as radiation is concerned#. Fincke
et al.29 discuss the results obtained by two methods—
enthalpy probe and laser light scattering experiments. Murphy et al.10 point out that comparisons reported by Scott
et al.8 are suspect since they used emission spectroscopy to
obtain experimental temperature profiles, a technique which
relies on the gas radiative properties being in LTE. On the
other hand, the laser light scattering technique for temperature measurement used by Murphy et al.10 requires a precise
knowledge of the local gas composition.
The present model has been validated in prior work, using detailed measurements of velocity, temperature, and
atomic hydrogen concentration profiles in a low-pressure hydrogen arcjet flow.23 Unfortunately, detailed experimental
measurement of temperature and ionization fraction profiles
for transferred arcs are unavailable. Therefore, in the present
work, predictions of this validated model are compared
against experimental measurements of radial profiles of stagnation pressure. These measurements were performed on a
water-cooled copper plate as the plasma jet impinges on it,
by scanning the torch.4 Model predictions are compared with
Aithal et al.
3513
FIG. 3. Calculated velocity vectors depicting flow within the plasma torch.
Long arrows indicate high speeds.
experimental measurements at two different current levels,
100 and 150 A.
It is important to compare the experimental conditions
against the quantities prescribed in the model and numerical
calculations. In the experiment, the total current is prescribed
~100 or 150 A!, the total volume flow into the torch is known
~;13 ft3/h or CFH!, and the total power ~product of terminal
voltage V and total current I! into the torch is known. The
pressure is measured well upstream of the region taken as the
inlet in the model, and the gases enter the torch at room
temperature. In order to enable comparisons between our
model and experiment, we ensure that comparable conditions
are imposed on the model. In the model formulation and
subsequent numerical calculation, the total current is also
prescribed. This defines the magnitude of the magnetic induction outside the current-carrying arc. Also prescribed in
the calculation are the stagnation pressure and the stagnation
temperature at the inlet to the torch. This is equivalent to
setting a flow rate experimentally, although the mass flow
rate is a predicted quantity in our calculation. We prescribe a
value of 30 in. of water ~gage pressure! for 100 A, and 60 in.
of water ~gage pressure! for 150 A for the total pressure. The
mass flow rate is then calculated at steady state. If the calculated value is below ~or above! that in the experiment, the
total pressure is raised ~or lowered!, and the entire calculations are repeated until the mass flow rate matches that of the
experiment. Similarly, the value of the conductivity floor is
set such that * ; (E–j)d;5VI, where ; represents the computational volume, V is the experimentally measured terminal voltage difference between electrode and workpiece, and
I is the measured total current. To recapitulate, in the experiment, I ~total current!, Q̇ in ~volume flow rate of entering gas!,
and power5VI ~electrical power input! are fixed and known.
In our model, B u on the outer boundary, P 0 and T 0 at the
inlet, and the conductivity floor are specified. Therefore, it
can be seen that there are no adjustable parameters in our
model, and this enables a direct comparison to be made between calculations and experiment.
Figure 3 shows a plot of calculated velocity vectors for a
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J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
Aithal et al.
FIG. 4. Radial variation of the streamwise component of velocity at the exit
plane of the torch.
FIG. 6. Radial profile of ionization fraction at the exit plane of the plasma
torch.
total current of 100 A and serves to visualize the flow inside
the torch body. Figures 4–6 show the radial variations of
streamwise component of velocity, temperature, and ionization fraction at the exit of the torch, for total currents of 100
and 150 A at a mass flow rate of 0.2 g/s. As can be seen the
plasma is fully ionized in the region near the centerline, and
is partially ionized at the fringes, as expected. This is consistent with the radial variation of temperature shown in Fig.
5. Note that maximum temperatures around 32 000 K are
observed near the centerline for the higher current case of
150 A. At these temperatures, equilibrium calculations indicate that the presence of doubly ionized atoms may not be
negligible as assumed in the present calculation.30 However,
these maximum temperatures are also observed at a location
where the flow velocity is greatest ~almost 1600 m/s, see Fig.
4!. At these velocities, the convective time scale is less than
a microsecond and comparable with the characteristic time
for second ionization ~i.e., ionization of singly ionized atoms
to form doubly ionized atoms!. Hence, the resulting compo-
sition of doubly ionized atoms will be determined by finiterate kinetics and be lower than that predicted by an equilibrium calculation.30 Moreover, this region of maximum
temperature encompasses only a small region near the centerline. The neglect of presence of doubly ionized atoms in
our calculation is therefore not likely to adversely impact the
accuracy of our solution. Figures 7 and 8 display axial profiles of temperature and streamwise velocity along the centerline, and show how the relatively cold, slowly moving gas
is accelerated and heated as it encounters the arc region.
The exit plane profiles of T, w, u, r , and n e , obtained from the internal flow solution are used as boundary
conditions for modeling the external plasma jet impinging on
a water-cooled copper workpiece. Experimental measurements indicate stagnation pressures between 15 and 40 in. of
water ~gage!. Hence, it is necessary to ensure convergence of
the nondimensional quantities to at least four decimal places
FIG. 5. Radial variation of temperature at the exit plane of the torch.
FIG. 7. Axial profile of temperature along the centerline within the plasma
torch. Note the dramatic increase in temperature as the relatively cold,
slowly moving gas encounters the arc near the electrode tip at about 3 mm
downstream from the inlet.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
FIG. 8. Axial profile of streamwise component of velocity along the centerline within the plasma torch.
FIG. 9. Velocity vectors depicting flow in the external plasma jet issuing
from the plasma torch, and impinging on a water-cooled copper plate.
FIG. 10. Axial profile of streamwise component of velocity along the centerline in the plasma jet.
Aithal et al.
3515
FIG. 11. Axial profile of temperature along the centerline in the plasma jet.
FIG. 12. Temperature contours in the plasma jet. The innermost contour
represents 8000 K and the outermost, 2000 K.
FIG. 13. Profiles of radial component of velocity vs axial distance is shown
here at various radial locations. Note the presence of a wall jet on the far
right, and its growth as the flow turns from the axial or streamwise direction
to the radial direction.
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Aithal et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
FIG. 14. Comparison of predicted and experimentally measured stagnation
pressure profile along the workpiece, for a total current of 100 A. Experimental measurements are from Ref. 4.
FIG. 15. Comparison of predicted and experimentally measured stagnation
pressure profile along the workpiece, for a total current of 150 A. Experimental measurements are from Ref. 4.
since the differences between the stagnation and static pressures ~1 atmosphere! are so small. Such a requirement usually places a tremendous burden on numerical calculations,
and is a testament to the extreme accuracy of our computational model.
Figure 9 shows calculated velocity vectors, enabling the
external plasma jet to be visualized. Figures 10 and 11 show
the velocity and temperature profiles along the centerline,
respectively. Temperature contours are shown in Fig. 12.
The plasma jet is seen to spread only slightly due to the
relatively short standoff distance and high jet velocity. The
innermost contour shows the high temperature regions ~temperatures above 8000! in the arc. Figure 13 displays the
growth of the wall jet. It can be seen that with increasing
distance radially from the centerline, the maximum radial
velocity increases up to a point and then decreases thereafter.
The location of the maximum velocity in each of the radial
velocity profiles also occurs further away from the workpiece
~stagnation plane! with increasing radial distance from the
centerline. These are typical characteristics of wall jets.31
Figures 14 and 15 show comparisons between the stagnation pressure profiles obtained experimentally ~the only
quantity measured in the experiments! and those obtained
from the present simulations. The pressure profiles obtained
experimentally show that the plasma jet tends to remain collimated. It is seen that the pressure ~gage! drops sharply to
near-zero values within a distance of about three nozzle radii
from the centerline. It is seen that our model predicts a maximum stagnation pressure ~gage! of 15.0 in. of water for a
total current of 100 A. The experimental measurements indicate a peak value of 13 in. of water. As shown in Fig. 15, our
calculations for a total current of 150 A predict a maximum
stagnation pressure ~gage! of 37.7 in. of water, as against the
experimental value of 36 in. of water. The simulations predict a slightly broader pressure profile as compared to the
experiment. Overall, the agreement with experiment is very
good. As a measure of sensitivity, the upstream stagnation
pressure ~the experimental measurement of which has the
most uncertainty! was varied, in order to find its effect on the
maximum stagnation pressure at the workpiece. It was found
that a difference of 10 in. of water ~gage! in the upstream
stagnation pressure gave about a 10% difference in the peak
values of the predicted stagnation pressure at the
workpiece.14 Given the uncertainty in the upstream stagnation pressure in the experiment, this may explain the differences between model predictions and experimental measurements.
The slight overprediction of the width of radial variation
of arc stagnation pressure in our calculation could have its
origins in the model itself ~such as heat transfer model employed at the plate surface or inclusion of radiative heat loss
from the arc!. However, these have been systematically ruled
out. In order to assess the impact of neglecting radiation
from the plasma on its temperature, we included radiative
energy transfer under the optically thin approximation in the
energy equation @Eq. ~5!#. No significant difference was
found in the calculated temperatures or in any of the other
quantities. The only remaining issue is that of the influence
of artificial dissipation, which has already been minimized to
levels that will still yield stable solutions. Hence, there are
no quantities that can be adjusted to improve agreement between model and experiment. This is satisfying, as it reflects
the correctness of our formulation of the problem and that
the correct physics has been modeled.
V. SUMMARY AND CONCLUSIONS
A self-consistent, one-temperature model of a transferred plasma arc has been developed, and compared with
limited experimental measurements of stagnation pressure on
a water-cooled work-piece. Such arcs are used in welding
applications, but little information regarding detailed modeling of transferred arcs exists in the literature. In the numerical calculations presented in this article, the computational
domain has been decomposed so that the internal flow within
the plasma torch and the external plasma jet flow impinging
Aithal et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
on the water-cooled workpiece are solved separately. Comparisons between predictions of numerical calculations of
stagnation pressure profiles and experimental measurements
indicate good agreement for two different current levels,
with no adjustable parameters. This has been achieved without the need to invoke extraneous plasma radiation and
plasma turbulence effects, which are often wrongly blamed
for poor agreement between theory and experiment in the
literature. If there are circumstances where these effects may
be important, these can be added with relative ease in our
model. Our formulation is thus general enough to serve as a
predictive tool.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Contract No.
NAS8-39930 from NASA Marshall Space Flight Center, and
Grant No. PAS745-6 from the Ohio Supercomputer Center.
This work was also partially supported by Grant Nos. F49620-94-1-0015 and F-49620-96-1-0184 from AFOSR, and
benefited from equipment provided by a Grant from the Ohio
Board of Regents Investment Fund Program.
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