Simulation of arc root fluctuation in a DC non-transferred plasma torch with three
dimensional modeling
R. Huang, H. Fukanum, SaiWama/J, Y. Uesugi and Y. Tanaka, Kanazawa/Japan
It has been well known that the coating quality of plasma spraying is strongly influenced by instability of jets inplasma spray due to the arc root fluctuation. A three dimensional (3D) unsteady modeling was employed in the
research to analyze the arc root fluctuation in a DC non-transferred plasma torch. Numerical calculations on the
distribution of gas temperature and velocity in plasma torch were carried out using argon as plasma gas. The
electrical current density and potential were also discussed. The results indicate that the fluctuation of arc inside
the plasma torch is mainly induced by the movement of the arc root on the anode surface. The arc root moves
downstream with the flow of gas, and the arc will wraped caused by the electromagnetic force simultaneously.
While the arc wraped closed enough to anode boundary, a new arc root is formed somewhere upstream of theoriginal attachment. This article represents nature of fluctuation of arc root, also in this paper we will present thatthe voltage-drop calculated is larger than that measured experimentally based on the hypothesis of local
thermodynamic equilibrium.
1

Introduction

The plasma spraying is the injection of metal or
ceramic powder into hot gas plasma which melts andprojects the molten droplets at high velocity onto asubstrate to form coatings. Gases such as argon orhydrogen are passed through as electric arc inside a
torch [1]. Plasma spraying, one of the most widelyused in industrial fields based on thermal plasmas, iscommonly employed to provide coatings for protectionof materials against wear, erosion, corrosion, andthermal loads. Despite its versatility, the limited
reproducibility of the processes is a major limitation for
its wider application. A major factor for this limited
reproducibility is the lack of understanding and controlof the dynamic behaviours of the arc inside thespraying torch and, the effect of erosion of the anodeon the forcing of the plasma jet [2-6].
A conventional DC non-transferred plasma torch(more than 90% of industrial torches) with a stick type
cathode is shown schematically in Fig. 1 [7-8]. After
working gas enters into the torch, it is heated by anelectric arc formed between a nozzle-shaped anodeand a conical cathode, which is ejected as a jet. The
arc inside the torch has been characterized
experimentally [4, 6, 9] and numerically [1-3, 8].
Unfortunately, experiments have been limited byinvolvement of high cost equipments and lack ofunderstanding of the results obtained.

Fig. 1. Schematic of a conventional dc arc spraytorch.
Fortunately, the numerical calculation provides a valid
way to understand the arc behavior inside plasma
torch. The modelling of DC arc plasma torches is anextremely challenging task because the plasma flow ishighly nonlinear and presents strong property
gradients. It is characterized by a wide range of timeand length scales, and often includes chemical andthermodynamic non-equilibrium effects, especiallynear its boundaries [8]. Despite of the complexity ofthe subject, over the past few decades, many papers
concerning numerical studies of the characteristics ofDC arc plasma torches have been published [2-3, 8,
9-22]. At the initial stage, two-dimensional (2D)modeling method was employed in the research topredict the heat transfer and flow patterns inside theplasma torch [10-14]. The predicted arc voltage of thetorch in the turbulent regime is much higher than themeasured value; in addition the predicted axial
location of the arc attachment at the anode surface isalso much farther downstream than that observed inexperiments [15]. With the rapid development ofcomputer technology, the calculation of heat transfer
and fluid flow for a 3D thermal plasma torch withaxisymmetrical geometries become feasible [2-3, 1522]. The models most frequently used for simulation ofplasma spray torch rely on the LTE approximation,and regard the plasma flow as a property-varying
electromagnetic reactive fluid in chemical equilibrium
in which the internal energy of the fluid ischaracterized by a single parameter of gas
temperature [2-3, 15-21]. Selvan et al. developed asteady 3D LTE model to describe the temperature andvelocity distributions inside a DC plasma torch.
Moreover the arc length and radius were also
discussed. But the model overestimated the plasma
gas temperature near the arc-root due to theassumption that all the electric current transferred tothe anode only through a fixed arc-root [3, 16]. Klinger
L. et al. also developed a steady 3D LTE model
simulation of the plasma arc inside a DC plasma
torch. However, the position of arc-root wasdetermined arbitrarily [17]. A. Vardelle and J. P.
Trelles developed a time-dependent 3D LTE model
representing the fluctuant of plasma arc [2, 18-21].
The voltage drop for the LTE model was larger
compared with the experimental ones due to the

DVS 276hypothesis of LTE, resulted in less estimation of theelectrical conductivity especially near the electrodeboundary. A non-equilibrium (NLTE) model wasdeveloped for the non-transferred arc plasma torch,which showed better agreement with the experimental
results [22]. However, to solve the NLTE model isextremely difficult due to the fact that the twotemperature chemical equilibrium need to beconsidered compared with the LTE mode.
In this research, an 3D LTE model was developed to
mimic the non-transferred DC plasma torch. The
plasma gas temperature and velocity distributions
were obtained with the LTE model. The fluctuation ofarc inside the torch was also presented.
2

Description of the Mathematical Model

2.1

Model Assumptions

The model developed in this study is base on thefollowing main assumptions for simulating the heattransfer and flow patterns inside a plasma torch.
(1) The continuum assumption is valid and theplasma can be considered as a compressible,
perfect gas in Local Thermodynamic Equilibrium(LTE).
(2) The plasma is optically thin.
(3) Gravitational effect and viscous dissipation are
considered negligible.
(4) The induced electric field is negligible incomparison with the applied electric field intensity
in the plasma arc region.
(5) The transport properties of plasma gas were only
determined by the plasma gas temperature.
(6) Because of the lower electric conducitivity near
the cold boundary of electrode, the vicinity ofanode, distance of 0.1 mm, is artificially4considered as a high electrical conductivity of 10
S/m, so that a new arc-root can be formed if thearc is closed enough to the inside surface ofanode.
2.2

Maxwell electromagnetism equations:

Ohm law:

Where
is gas mass density, time,
velocity,electric current density,
magnetic induction vector,
gas pressure,dynamic viscosity,
strain rate
tensor,specific heat at constant pressure,electric field,
volumetric net radiation losses,
electric conductivity,
gas thermal conductivity,
electric potential,
magnetic vector potential andpermeability of free space.
For the gas flow calculation, the K- ˢ model is
employed in this study. The thermodynamic andtransport properties of the plasma gas are taken from
Ref. [23,24].
2.2
Computational Domain and Boundary
Conditions
The geometry used in the current study corresponds
to the SG-100 plasma torch from Praxair. The
computational domain formed by the region inside thetorch limited by the cathode, the gas flow inlet, theanode and the outlet as shown in Fig. 2. The
computational domain is meshed using 217600
hexahedral cells with 224567 nodes. The governing
equations are solved by the SIMPLE algorithm using
the commercial CFD software of FLUENT 6.3.

Governing Equations

Based on the forgoing assumptions, the governing
equations for the 3D time-dependent for the arc
plasma can be written as follows:
Conservation of mass:

Conservation of momentum:

Fig. 2. Geometry of the computational domain
As seen in Fig. 2, the boundary of the computational
domain is divided into 4 different faces to allow thespecification of boundary conditions. Table 1 shows
the boundary conditions used in the simulation, where
Pin represents the inlet pressure equal to 111325 Pa(10 kPa overpressure), hw the convective heat transfer
coefficient at the anode wall equal to 1105 W.m-2.K-1[19-22], Tw a reference cooling water temperature of
500 K. The current density of cathode was defined by:Conservation of energy:
Where r is radial distance from the torch axis
, and Jcath0 and nc are parameters thatspecify the shape of current density profile. The Rc is

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calculated to ensure that integration of j(r) over thecathode equals the total applied current. According to
the reference 20 and 23, Jcath0 of 2.08108 A/m2, nc of
4 and Rc of 0.913 mm were used in this study for theapplied electric current of 500 A.
Table 1. Boundary conditions

gas velocity inside plasma torch has also a significant
fluctuation with the arc-root movement and transition
as shown in the Fig. 5 (b). The fluctuations of plasma
gas temperature and velocity are the main reason tocaused the plasma jet fluctuated, consequently
influencing on the reproductivity of coatings. The
maximum gas temperature more than 30000K andvelocity more than 1000 m/s are obtained under thecurrent spray conditions of argon gas, 500 A electric
current and 50 SLM gas flow rate.

Argon gas was employed as the plasma gas in thisstudy. As the table 1 showed, the spray conditions is
500 A of current and 50 SLM of gas flow rate.
3

Results and Discussions

3.1

Flow fields inside the torch

The distribution of electric field strength inside plasma
torch calculated by the LTE model is shown in the Fig.
3. The maximum electric field strength is about 0.78
510 V/m near the anode boundary. Figure 4 shows
the time-evolution of the electric current distributions.
It reveals that the arc-root moved downstream for thetime of 858 µs to 868 µs. At the same time, a newelectric current “path” will be formed if the electric fieldstrength is strong enough to break down between thearc and anode boundary. Consequently, electric
current will go through the old arc-root and the new
one simultaneously. With the time elapsed, the old
arc-root will be disappeared and only the new one
maintained as shown in the Fig. 4.
Fig. 4. Electric current distribution inside plasma torch
at different time.

Fig. 3. Electric field strength distribution inside plasma
torch at the time of 858 µs.
With the arc-root movement and transition, the arc
should be fluctuated. The time-evolution of gas
temperature and velocity distributions inside plasma
torch are showed in the Fig. 5. While the attachmentof arc to the anode at the underside surface at thetime of 858 µs as shown in the Fig. 4, the arc waswarped and deviated to the contrary side. The
deviation of arc resulted to rise of the electric fieldstrength at the fringe of the arc so that a new arc-root
formed. As the old arc-root disappeared, the arc will
deviate to the other side too in order to generate thenext attachment as shown in the Fig. 5 (a). The gas
velocity distributions inside plasma torch show that the

DVS 276(a)(b)

Fig. 5. Plasma gas temperature (a) and velocity (b)distributions inside plasma torch at different timeFigure 6 shows the gas temperature and velocity at
different cross sections inside plasma torch. It can beseen that the gas temperature and velocity
distributions are asymmetric even though thegeometry of plasma torch is axisymmetric due to thefluctuations of arc.

Fig. 6. Plasma gas temperature and velocity
distribution at different cross sections inside plasma
torch.

3.2

Gas flow at the torch exit

The plasma jet are mainly determined by the gas flowat the torch exit. The parameters of plasma jet can bepredicted by the distributions of gas temperature andvelocity of torch outlet. Therefore, it is extremely
important to mimic the outlet temperature and velocity
of gas in order to understand the fluctuations of jet ofplasma spray. The calculated distributions of gas
temperature and velocity at the nozzle exit are shownin Fig. 8. The maximum temperature of about 14000
K and velocity of about 1000 m/s are obtained at thenozzle exit. It can be founded that the value calculated
is well agree with the results measured experimentallyaccording to the references [25-27].

The electric potential distribution of plasma arc isshowed in Fig. 7 (a). The voltage drop of the arc
column should be lower than the voltage of power
supply theoretically because the voltage drop of
sheath exists. However, the arbitrary higher electric
conductivity nearby electrode and LET assumptions
result to a lower accuracy for the electric potential
distribution especially the vicinity of cathode.
Therefore,the sheath voltage drop cannot be
observed and the voltage drop of the arc column
calculated is much higher than the voltage of power
supply measured experimentally. Figure 7 (b) shows
the time-evolution of average voltage on the cathode.
It reveals the frequency of plasma arc fluctuation isabout 11 kHz.

(a)

(b)
(a)

Fig. 8. Plasma gas temperature (a) and velocity (b)distributions at the torch exit

50
Average Voltage of Cathode Calculated
Average Voltage of Power Supply Measured

45

The gas temperature and velocity at the nozzle exit
fluctuate with the time elapsed too as shown in the Fig.
8. The fluctuations result to the vibration of the gas
flow rate of torch outlet around 50 SLM, the inlet gas
flow rate, as shown in the Fig. 9.

40
35
30

4

Conclusions

25
20

600

800

1000

1200

1400

Time (Ps)(b)
Fig. 7. The electric potential distribution inside plasma
torch (a) and the time-evolution of cathodevoltage.

A LTE model has been developed and applied to thethree-dimensional and time-dependent simulation of
the flow inside a DC arc plasma torch. This mode well
mimiced the arc fluctuation inside a plasma torch. The
temperature and velocity distribution of arc gas inside
the torch were calculated. The electric current mainly
conducts through the arc-root to the anode. The
concentration of electric current causes the warp ofarc resulting to the rise of electric field strength at

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the contrary side of arc-root. When the arc is closed
enough to the anode boundary and the elecrricfield strength is strong enough, the old acr-root willtransfer to a new one. The movement andtransiation of arc-root result to the fluctuation ofplasma arc inside plasma torch with the frequency of
11 kHz. A gas temperature of about 14000K andvelocity of about 1000 m/s were obtained at the torch
exit. A higher voltage drop of arc column was obtained
compared to the one measured experimentally due tothe LTE assumption underestimated the electric
conductivity of plasma gas inside the torch, especiallythe regions around the arc-root.
65
60
55
50
45
40
35

600

800

1000

1200

1400

Time (Ps)

Fig. 9. The time-evolution of gas flow rate to the torch
exit.
5

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