Modelling of the arc reattachment process in plasma torches

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 40 (2007) 5635–5648 doi:10.1088/0022-3727/40/18/019

Modelling of the arc reattachment processin plasma torchesJ P Trelles, E Pfender and J V R Heberlein

Department of Mechanical Engineering, University of Minnesota, Minneapolis,MN 55455, USA

E-mail: jptrelles@me.umn.edu

Received 29 April 2007, in final form 27 July 2007Published 30 August 2007Online at stacks.iop.org/JPhysD/40/5635

AbstractThe need to improve plasma spraying processes has motivated thedevelopment of computational models capable of describing the arcdynamics inside plasma torches. Although progress has been made in thedevelopment of such models, the realistic simulation of the arc reattachmentprocess, a central part of the arc dynamics inside plasma torches, is still anunsolved problem. This study presents a reattachment model capable ofmimicking the physical reattachment process as part of a localthermodynamic equilibrium description of the plasma flow. The fluid andelectromagnetic equations describing the plasma flow are solved in afully-coupled approach by a variational multi-scale finite element method,which implicitly accounts for the multi-scale nature of the flow. Theeffectiveness of our modelling approach is demonstrated by simulations of acommercial plasma spraying torch operating with Ar–He under differentoperating conditions. The model is able to match the experimentallymeasured peak frequencies of the voltage signal, arc lengths and anode spotsizes, but produces voltage drops exceeding those measured. This finding,added to the apparent lack of a well-defined cold boundary layer all aroundthe arc, points towards the importance of non-equilibrium effects inside thetorch, especially in the anode attachment region.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Plasma spraying, one of the most versatile and widely usedthermal plasma applications, suffers from occasional limitedreproducibility. A major factor for this limited reproducibilityis the lack of understanding of the dynamics of the arc insidethe spraying torch, the main component of plasma sprayingprocesses. Figure 1 shows a schematic view of the flow insidea direct current (dc), non-transferred arc plasma torch. Afterthe working gas enters the torch, it is heated by an electric arcformed between a nozzle-shaped anode and a conical cathodeto form a plasma, which is ejected as a jet. This flow, despitethe axi-symmetry and steadiness of the geometry and boundaryconditions, is inherently three-dimensional and transient.

The dynamics of the arc inside a dc plasma torch are theresult of the imbalance between the drag force caused by theinteraction of the incoming gas flow over the arc attachment

Figure 1. Schematic representation of the flow inside a dc arcplasma torch.

and the electromagnetic (or Lorentz) force caused by the localcurvature of the arc [1, 2]. Because the length of the arc islinearly dependent on the voltage drop across the torch, thevariation of the voltage drop over time gives an indication ofthe arc dynamics inside the torch. This voltage drop signal has

0022-3727/07/185635+14$30.00 © 2007 IOP Publishing Ltd Printed in the UK 5635

J P Trelles et al

0 1 2 3 4

20

40

60

80

100

time [ms]

Vo

ltag

e D

rop

[V

]

steadytakeoverrestrike

Figure 2. Schematic voltage drop signals for the different modes ofoperation of an arc plasma torch.

allowed the identification of three distinct modes of operationof the torch [1–5], namely: steady, takeover and restrike. Thesemodes are schematically depicted in figure 2.

The need to improve plasma spraying processes hasmotivated the development of computational models capableof describing the arc dynamics inside dc plasma torches. Thesemodels need to describe basically two main features of thearc dynamics: the movement of the arc due to the imbalancebetween flow drag and electromagnetic forces and the processof formation of a new arc attachment, in what is called thereattachment process. Progress has been achieved in thedevelopment of such models [6–11], but no single modelcapable of describing the different modes of operation of thetorch has been developed yet.

There have been basically two approaches to model thedynamics of the arc: relying on the anode boundary model toaccount for the dynamics of the arc (i.e. [6–8]) and the useof an arc reattachment model (i.e. [9–11]). The first approachallows not only the movement of the anode attachment alongthe anode surface but also the formation of a new attachmentwhen the arc gets close to the anode surface. This approach,commonly used in steady-state simulations of thermal plasmas[12–14], is capable of describing the steady and takeovermodes of operation of the torch; but, as shown in section 6.3,it is not adequate for the modelling of the restrike mode.The second approach is capable of describing the restrikemode by mimicking the physical process by which a newarc attachment is formed in a region upstream of the originalattachment when the available voltage in the arc is largerthan the breakdown voltage [1]. The major challenge of thisapproach is that it needs to define where and how to apply thereattachment process in order to reproduce the effect of thephysical reattachment process on the flow field.

This paper describes the development and implementationof a modelling approach capable of describing the arc dynamicsin plasma torches operating in the restrike mode. Thisapproach combines the above techniques for describing thearc dynamics and has the potential to describe the steadyand takeover modes as limiting cases. Section 2 presentsthe mathematical model based on local thermodynamicequilibrium (LTE) approximation, section 3 its numericalimplementation by using a variational multi-scale (VMS) finiteelement method and section 4 describes the computationaldomain and boundary conditions. The developed reattachmentmodel is described in section 5, and numerical results ofits implementation applied to the realistic modelling of the

flow inside a commercial plasma torch, including comparisonswith experimental measurements, are shown in section 6.Conclusions are drawn in section 7.

2. Mathematical model

2.1. Model assumptions

Our model is based on the following assumptions: (1) theplasma is considered as a compressible fluid composed ofa mixture of perfect gases in LTE, hence characterized by asingle temperature for all its constituent species; (2) the quasi-neutrality condition holds; (3) the plasma is optically thin;(4) the chemical composition corresponds to the equilibriumcomposition of a perfectly-mixed Ar–He (vol 75–25%) plasma(i.e. demixing effects [15] are neglected); (5) Hall currents,gravitational effects and viscous dissipation are considerednegligible; and (6) the flow is considered laminar in the sensethat no turbulence model is included in the mathematicalformulation. Nevertheless, the numerical method is potentiallycapable of modelling turbulent characteristics in what is calleda ‘residual-based large eddy simulation’ if adequate spatialresolution is employed. This has not been pursued in thepresented work.

2.2. Thermal plasma equations

According to the above assumptions, the flow of a thermalplasma can be fully described by five independent variables.In our model, we chose as unknowns the primitive variables:p pressure, �u velocity, T temperature, φ electric potential and�A magnetic vector potential (a total of nine components in a

three-dimensional model). These variables form the vector Yof primitive variables:

Y = [p �u T φ �A]t, (1)

where the superscript t indicates transpose. For a given vectorof unknowns Y, the set of equations describing a thermalplasma flow can be expressed in a compact form as a system oftransient–advective–diffusive–reactive (TADR) equations as

R(Y) = A0∂Y/∂t︸ ︷︷ ︸transient

+ (A · ∇)Y︸ ︷︷ ︸advective

− ∇ · (K∇Y)︸ ︷︷ ︸diffusive

− (S1Y + S0)︸ ︷︷ ︸reactive

= 0,

(2)

where R represents the vector of residuals and A0, A, K, S1

and S0 are matrices of appropriate sizes (i.e for nv variablesand nd spatial dimensions, the matrix A is of size nv · nd

by nv). These matrices are fully defined by the specificationof the vector of unknowns Y and a set of five independentequations describing the plasma flow. Our model is based onthe equations of (1) conservation of total mass, (2) conservationof mass-averaged momentum, (3) conservation of thermalenergy, (4) conservation of electrical current and (5) themagnetic induction equation. These equations are shown intable 1 as a set of TADR equations.

In table 1, ρ represents mass density, t time, µ dynamicviscosity,

↔δ the identity tensor, �Jq current density, �B magnetic

field, Cp specific heat at constant pressure, κ thermalconductivity, he electron enthalpy, �Je mass diffusion flux ofelectrons, �E electric field, Qr volumetric radiation losses,

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Modelling of the arc reattachment process in plasma torches

Table 1. Thermal plasma equations: transient + advective − diffusive − reactive = 0.

i Yi Transient Advective Diffusive Reactive

1 p∂ρ

∂t∇ · ρ �u 0 0

2 �u ρ∂ �u∂t

ρ �u · ∇�u − ∇p ∇ · µ(∇�u + ∇�ut − 23 ∇ · �u↔

δ) �Jq × �B

3 T ρCp

∂T

∂tρCp �u · ∇T ∇ · (κ∇T − he �Je) �Jq · ( �E + �u × �B) − Qr + Wp

4 φ 0 0 ∇ · σ(∇φ − �u × ∇ × �A) 0

5 �A ∂ �A∂t

∇φ − �u × ∇ × �A (µ0σ)−1∇2 �A �0

Wp pressure work, σ electrical conductivity and µ0 thepermeability of free space.

The transport of the heavy species enthalpy by diffusionis accounted for in the definition of the thermal conductivityas a reactive component, which is added to the translationalconductivity (κ = κtranslational + κreactive). The transport ofelectron enthalpy by diffusion, which can dominate over theJoule heating term near the anode surface, is accounted forexplicitly in the energy equation (see table 1, diffusive columnfor i = 3) and approximated by

he �Je ≈ −5

2kBT

�Jq

e, (3)

where kB is Boltzmann’s constant and e the elementary charge.Volumetric radiation losses, approximated by the use of a netemission coefficient εr for pure argon [16], and the pressurework are given by

Qr = 4πεr, Wp = − ∂ ln ρ

∂ ln T

∣∣∣∣p

(∂p

∂t+ �u · ∇p

). (4)

The set of equations in table 1 is completed by thefollowing electromagnetic expressions relating the electric andmagnetic potentials to the electric and magnetic fields and thecurrent density:

�E = −∇φ − ∂ �A∂t

, �B = ∇ × �A, �Jq = σ( �E + �u × �B),

(5)

and by appropriate thermodynamic and transport properties.These properties are obtained from a computer code developedin our laboratory, which calculates them based on thekinetic equilibrium assumption (minimization of Gibbs freeenergy) [16].

As mentioned in section 2.1, the plasma is considered asa compressible fluid. Compressibility needs to be taken intoaccount because the Mach number inside the torch could varyfrom nearly zero at the torch inlet to a value as high as 0.7 atthe outlet. To account for compressibility, the plasma densityneeds to be modelled as a function of both temperature andpressure (not only function of temperature, as commonly usedin thermal plasma simulations). In our model, the plasma massdensity is calculated by

ρ = p

RgTwith Rg = kB

∑s ns∑

s nsms

, (6)

where Rg is the gas constant, which is proportional to thecompressibility factor and due to the reactive nature of theplasma is a strong function of temperature, ns and ms arethe number density and particle mass of species s, respectively,and the summations are for all the species in the plasma.Even though the number densities in (6) are functions of bothtemperature and pressure, they are calculated as a function oftemperature for a pressure of 1 atm. This is justified by thefact that the ratio of summations in (6) is a weak function ofpressure (i.e. varies by less than 30% for a pressure variationfrom 1 to 10 atm), and thus Rg can be well approximated as afunction of temperature only.

3. Numerical model

3.1. Preliminaries

Our numerical model is based on the solution of equation (2) forany set of independent TADR equations (i.e. the particular formof the matrices A0, A, K, S1, S0) and definition of the vector ofunknowns (i.e. a set of variables composing Y). Therefore, anyother set of independent equations (i.e. conservation of totalenergy instead of thermal energy) and/or independent variables(i.e. mass density instead of pressure) could be used in ourmodel. Our selection of variables has been strongly motivatedby the work of Hauke and Hughes [17] which indicates thesuperiority of the use of primitive variables over other sets ofvariables (i.e. conservative, entropy variables). The equationsused in our mathematical model are the ones that have beenpredominantly used in the thermal plasma literature [6–14].Furthermore, our numerical approach is strongly coupled, inthe sense that the solution of the fluid and electromagneticequations is sought simultaneously, which adds robustness andaccuracy to our model.

The numerical solution of general systems of TADRequations represents an enormous challenge when theequations are strongly coupled, highly non-linear and presentmultiple scales, as it is the case for the set of equationsdescribing thermal plasma flows. The strongly coupled andnon-linear nature of equation (2) is evident from the inspectionof the terms in table 1. The multi-scale nature of thisequation arises when a given term dominates over the othersin different spatial and/or temporal regions. This imbalanceamong terms produces strong spatial and/or temporal gradientsin the interfaces between regions and the predominance ofdistinctive flow features in different regions of the domain.

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J P Trelles et al

A clear example of the multi-scale nature of (2) is flowturbulence, which occurs when the advective part dominatesover the diffusive and reactive parts in the bulk flow, whereasthe diffusive part dominates in the regions near the boundaries,where vorticity is produced. Other manifestations of the multi-scale nature of equation (2) are shock waves, reaction layers,boundary layers and plasma sheaths.

3.2. Sub-grid scale finite element method

Variational multi-scale (VMS) methods have been developedby Hughes and collaborators [18, 19] to model multi-scalephenomena in realistic technological applications. The VMSmethod uses a variational decomposition of a given fieldinto a large scale component (which is described by thecomputational mesh) plus a small or sub-grid scale component,whose effect into the large scales is modelled. VMS methodsare promising for the modelling of complex multi-scaleflows in complex domains, as typically found in industrialapplications, offering an improved alternative to traditionallarge eddy simulation techniques [20]. The interested readeris referred to [19] for a recent review of VMS methods.

The algebraic sub-grid scale finite element method (SGS-FEM) is one of the simplest formulations of the VMS approach,in which large and small scales are approximated with the samecomputational grid [21]. The SGS-FEM model of the TADRsystem given by equation (2) is expressed by

∫

W · R(Y)︸ ︷︷ ︸Galerkin

+∫

′P(W) · τR(Y)︸ ︷︷ ︸

sub-grid scale

= 0, (7)

where W = W(X) is the weight function typical of every finiteelement method and X the vector of spatial coordinates;

represents the spatial domain; the Galerkin term represents thepart of the problem solved by the mesh (large scale); the sub-grid scale term represents the modelling of the small scales; ′

represents the spatial domain minus the skeleton of the meshformed by the elements boundaries; P is a differential operatorapplied to W and defined as the negative of the adjoint of thedifferential operator applied to Y in equation (2) (P = −L ∗

and L is defined from R(Y) = L Y − S0) and τ is the matrixof intrinsic time scales, the model parameter.

The formulation of the problem given by (7) is consistentwith equation (2) as the sub-grid scale term becomes negligibleas the discretization mesh becomes finer, and hence more‘small scales’ are resolved by the computational mesh. To addrobustness to the formulation in regions with large gradients, adiscontinuity-capturing-operator is added to (7) [22]. Moredetails of our implementation of the SGS-FEM applied tothermal plasmas are found in [7].

3.3. Solution approach

Spatial discretization of (7) leads to the formation of a verylarge system of time-dependent non-linear algebraic equationsexpressed as

R(th, Xh, Yh, Yh) = 0, (8)

where R represents the global vector of residuals, the subscripth indicates the discrete counterpart of a given continuous

variable, Xh the computational domain, Yh the global vectorof unknowns and Yh its time derivative.

The time discretization of system (8) allows us to expressthe vector R as a function of Yh only. The multi-scalenature of the plasma equations makes equation (8) very stiff,which makes the use of time-implicit solution algorithmsmandatory. We discretize (8) using a second-order, fully-implicit, predictor multi-corrector α-method [23], whose highfrequency amplification factor is the sole parameter, togetherwith an automatic time stepping strategy. The generalizedα-method requires the solution of a non-linear system ofequations at each time step. The solution of this non-linearsystem is accomplished by an inexact-Newton method togetherwith a line-search globalization procedure [24, 25]. Thesetechniques form an iterative procedure in which the valuesof an approximate solution and its residual vector at a giveniteration k are updated according to

‖Rk + Jk�Yk+1h ‖ � γ k‖Rk‖ (9)

and

Yk+1h = Yk

h + λk�Yk+1h , (10)

where the superindex k indicates the iteration counter, J isan approximation of the Jacobian matrix of the global system(8) (J ≈ ∂R/∂Yh), �Yh represents a correction to the actualsolution, γ is a parameter controlling the accuracy requiredfor the linear solve in the current iteration and λ is the line-search parameter. Equation (9) expresses the inexact-Newtoncondition whereas equation (10) expresses the updating of thesolution according to the line-search procedure.

The approximate solution of a linear system requiredby the inexact Newton condition is accomplished by thegeneralized minimal residual (GMRES) method with a block-diagonal preconditioner. The use of an inexact-Newtonmethod together with a GMRES solver greatly improves thecomputational efficiency (in terms of memory requirementsand computational time) of the solution of the non-linearsystem of equations. The globalization strategy in (9) togetherwith adaptive time stepping has proven mandatory for attainingconvergence when the reattachment model to be described insection 5 is applied.

4. Computational domain and boundary conditions

The geometry studied corresponds to the commercial plasmatorch SG-100 from Praxair Surface Technology, Concord, NH.The computational domain is formed by the region inside thetorch limited by the cathode, the inflow region, the anodenozzle and the torch exit (see figure 1) and is shown in figure 3together with its finite element discretization using hexahedralelements. The mesh is structured in the z-axis and unstructuredin the x–y plane and refined around the solid boundaries;its total number of nodes and elements is 68482 and 65044,respectively. We have run preliminary simulations of theresults shown in section 6 with a coarser grid (similar to theone used in [7], with 41800 nodes) and, although the resultswith the finer grid are better resolved, the dynamics of the arcare essentially the same.

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Modelling of the arc reattachment process in plasma torches

Figure 3 shows that the boundary of the computationaldomain is divided into five different sides to allow thespecification of boundary conditions. Table 2 shows theboundary conditions used in the simulations, where pin

represents the inlet pressure equal to 111.325 kPa (10 kPaoverpressure), �uin the imposed velocity profile equal to theprofile of a fully developed laminar flow through an annulus,Tin the inlet temperature equal to 500 K, Tcath the cathodetemperature which is approximated by a Gaussian profile alongthe z-axis varying from 500 K at the inlet to 3000 K at thecathode tip, hw the convective heat transfer coefficient at theanode wall equal to 105 W m−2 K−1 [9], Tw a reference coolingwater temperature of 500 K and Jcath the imposed currentdensity over the cathode defined by

Jcath = Jcath0 exp(−(r/Rc)nc), (11)

where r is the radial distance from the torch axis (r2 = x2 +y2)

and Jcath0, Rc and nc are parameters that specify the shapeof the current density profile. Typically, Jcath0 and nc arechosen to mimic experimental measurements [26], whereasRc is calculated to ensure that integration of Jcath over thecathode tip equals the total applied current. As an example,for a specified current of 800 A, Jcath0 = 3.0 × 108 A m−2,nc = 4 and Rc = 0.913 mm.

Due to the thermal equilibrium assumption, the valueof the electron temperature is equal to the heavy particletemperature, which is low near the electrodes, especiallynear the anode surface (i.e. less than 3000 K). Hence theequilibrium electrical conductivity, being mostly a functionof the electron temperature, is extremely low (less than0.01 S m−1), which limits the flow of electrical current through

Figure 3. Computational domain, finite elements mesh andboundary sides.

Table 2. Boundary conditionsa,b.

Variable p �u T φ �ASide 1: inlet p = pin �u = �uin T = Tin ∂nφ = 0 Ai = 0Side 2: cathode ∂np = 0 ui = 0 T = Tcath ∂nφ = 0 ∂nAi = 0Side 3: cathode tip ∂np = 0 ui = 0 T = Tcath −σ∂nφ = Jcath ∂nAi = 0Side 4: outlet ∂np = 0 ∂nui = 0 ∂nT = 0 ∂nφ = 0 ∂nAi = 0Side 5: anode ∂np = 0 ui = 0 −κ∂nT = hw(T − Tw) φ = 0 ∂nAi = 0

a ∂n = ∂/∂n differentiation in the direction of the (outer-) normal to the boundary.b i = x, y, z Cartesian coordinates defined in figure 3.

the electrodes. To alleviate this, an alternative used in[9–11] is to let the temperature remain high enough (i.e.above 7000 K) at the electrodes. The clear advantage of thisapproach is that it is consistent with the LTE assumptionin the sense that the heavy particle temperature remainsequal to the electron temperature, which remains high all theway up to the electrodes. Unfortunately, this approach notonly requires unrealistically high heavy particle temperaturesat the anode but also produces anode attachment spots anorder of magnitude larger than what has been observedexperimentally [9].

A common technique to avoid the above problems consistsof specifying an artificially high electrical conductivity in theregion immediately adjacent to the electrodes [6–8, 12–14].The value of this ‘artificial’ electrical conductivity used inthe literature is somewhat arbitrary. The only requirementfor its value is that it needs to be high enough in order toensure the flow of electrical current from the plasma to theanode. Typically it has been defined as the harmonic meanbetween the plasma electrical conductivity and that of theanode material [13] or as the electrical conductivity of theplasma at a given electron temperature (i.e. 15 000 K) [7]. Thislatter approach has been used in the results shown in section 6.Our experience indicates that a value greater than or equal to thevalue of the plasma electrical conductivity for a temperature of∼12 000 K allows smooth current continuity. Higher values,even values of the order of the conductivity of copper, do notperceptively affect the results, except for a slight decrease inthe size of the anode spot. Lower values do not allow enoughcurrent continuity and make the numerical solution diverge.

5. Modelling of arc reattachment processes

5.1. Arc reattachment in steady and takeover modes

The high electrical conductivity technique described aboveallows not only the continuity of the electrical flow betweenplasma and electrodes but also the free movement of the arcattachment along the anode surface and the formation of a newarc attachment if some part of the arc gets ‘close enough’ to theanode surface. In this type of reattachment process the requiredproximity of the arc is given by the thickness of the region infront of the anode specified with high electrical conductivity.As typically this thickness is very small (i.e. less than 0.1 mm[7, 13]), the dynamics of the arc are not greatly affected byits specific value, the difference between the locations of theattachment for two different values of this thickness beingin the order of the difference between the thicknesses used(and therefore typically less than 0.1 mm). This type of arc

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A B

C D

Figure 4. Arc reattachment model: (A) arc deflected by incoming flow and location where the breakdown condition is satisfied; (B) insertionof an electrically conductive channel; (C) formation of a new attachment and (D) predominance of the new attachment over the old one.

reattachment does not depend perceptively on the specificvalue of σ in front of the electrodes, as long as this valueallows smooth current continuity.

From the numerical results to be shown in section 6.3,this approach works well for modelling the dynamics of thearc in a plasma torch operating in steady or takeover mode(as shown in [7, 8]), but it is not adequate for simulating therestrike mode of operation, as it leads to solutions displayingexceedingly long arcs and high voltage drops.

5.2. Arc reattachment in restrike mode

In the restrike mode, flow drag forces dominate overelectromagnetic forces, causing the arc to be draggeddownstream; the arc is very constricted and a thick coldboundary layer can be observed around the arc. At the sametime as the arc moves downstream, its length increases and sodoes the total voltage drop and the magnitude of the electricalfield around the arc. If the available voltage across the arcexceeds the local breakdown voltage, a new arc attachmentforms somewhere upstream of the original attachment [1, 2].Even though the exact mechanisms driving the reattachmentprocess are not known, the relatively high electric fields, addedto the abundance of excited species around the arc (i.e. dueto UV excitation), and the short time scale of the processsuggest that the arc reattachment is initiated by a streamer-likebreakdown. This streamer connects the arc with the anode ina region somewhere upstream of the existing arc attachment,creating a conducting channel that allows the establishment of anew arc attachment. The detailed modelling of this process isunfeasible with actual computational methods and computerpower, especially when the reattachment process is part ofthe modelling of a realistic plasma application. Therefore,approximate models are needed which imitate the effects of thereattachment process within the flow field of a given plasmaapplication.

To mimic the physical reattachment process, a modelmainly faces the questions of where and how to introduce thenew attachment. Where to locate the new attachment translatesinto the definition of an adequate breakdown condition,whereas how to introduce the reattachment translates into the

definition of adequate modifications of the flow field in orderto mimic the formation of an attachment.

Probably the most advanced reattachment model to dateis the one developed by Moreau et al [11]. Their model solvesthe above questions by (1) using as breakdown criterion thecomparison between the voltage drop between the isosurface of7000 K around the arc and the anode and a pre-specified criticalelectric field above which a breakdown occurs and (2) imposinga 8000 K column that connects the arc and the anode at thelocation where the maximum in (1) is located. A limitationof this model is that the imposition of a high temperaturechannel connecting the arc and the anode violates global energyconservation, because the amount of energy contained in thechannel is added instantaneously to the flow. In this paperwe develop a reattachment model that tries to circumvent thislimitation. This model is illustrated in figure 4.

Depicted in figure 4(A) is the dragging of the arc by theflow field, which causes the local electric field around the arcto increase. The breakdown condition is satisfied when, at alocation �xb, the maximum of the magnitude of the local electricfield in the direction normal to the anode En,max exceeds thepre-specified breakdown electric field Eb(En,max > Eb), with

En,max = max(| �En|), �En = �E · �na, (12)

and �na as the normal to the anode. Centred on �xb, a cylindricalchannel with radius Rb and connecting the anode surface andthe arc in the direction parallel to �na is introduced (figure 4(B)).Within this breakdown channel, the value of the electricalconductivity is modified according to

σ ← max(σ, σb), (13)

where the σ inside the parentheses is the equilibrium electricalconductivity of the plasma and σb is an electrical conductivitythat characterizes the breakdown process, which is assumed tobe of the form

σb = σb0 exp(−βb(rb/Rb)nb), (14)

where rb is the radial coordinate having as origin the axisof the cylindrical conducting channel (see figure 5). The

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Modelling of the arc reattachment process in plasma torches

Figure 5. Distribution of the electrical conductivity σb inside thecylindrical channel connecting the arc and the anode.

conducting channel stimulates the growth of a plasma column(figure 4(C)), which by reaching the anode surface forms anew attachment that remains over the old attachment, whichdies out (figure 4(D)). The shape of the σb(rb) profile in (14)is defined by the parameters Rb, βb and nb. Values usedin our simulations are Rb = 1.25 mm, βb = 6, nb = 4and σb0 = 104 S m−1. The cylindrical channel with σb ismaintained until the maximum electric field within the domaindecreases to a value below 0.8Eb. The reattachment modelis mainly driven by the specification of Eb; the parametersdefining σb only control the form of the reattachment process(i.e.σb0 controls the speed of the reattachment process, whereasRb the width of the reattachment column). Moreover, if thephysical reattachment process is indeed triggered by the valueof the local electric field exceeding a certain ‘breakdown’voltage (the Eb used in our model), it is reasonable to expectthat the value of this breakdown voltage will be a functionof the gas composition and probably only a weak function ofthe torch characteristics and operating conditions (i.e. nozzlediameter, mass flow rate, total current).

The use of σb to drive the formation of a new attachmentmodifies the energy equation only weakly, because the Jouleheating term will be artificially high only in the regionwhere E is large (i.e. around �xb). Application of thereattachment model simultaneously with the satisfaction ofenergy conservation requires a rapid accommodation of energythroughout the domain. This causes severe difficulties forattaining convergence when the reattachment model is justapplied. In this regard, utilization of the globalizationprocedure within the inexact-Newton method (equation (9))and an adaptive time stepping procedure have been mandatoryfor obtaining the results shown in section 6.

The reattachment described above is complemented withthe use of a high electrical conductivity in the region ofthickness ∼0.08 mm in front of the anode surface, as describedin section 5.1. This reattachment model can be applied togeneral geometries and flow configurations since its onlyrequirement is that the normal to the anode surface needs tobe well defined (i.e. it is not suitable for anode surfaces withsharp corners or discontinuities).

Because of the free parameter Eb, the describedreattachment model cannot predict the operating mode of thetorch. The model can only predict the arc dynamics inside atorch operating under given operating conditions and a givenvalue of Eb. If the model is used to simulate a torch operatingunder conditions leading to a takeover mode, a high enoughvalue of Eb should be used in order to ensure that a restrike-like reattachment does not occur. Otherwise, the voltage signalobtained could resemble that of the restrike mode.

Table 3. Cases simulated.

FlowCurrent rate Eb

Case Gas (A) (slpm) Injection (V m−1)

A Ar–He (75–25%) 800 60 Straight ∞a

B Ar–He (75–25%) 800 60 Straight 5 × 104

C Ar–He (75–25%) 800 60 45◦ swirl 5 × 104

D Ar–He (75–25%) 800 60 45◦ swirl 2 × 104

E Ar–He (75–25%) 800 30 45◦ swirl 5 × 104

a No reattachment model.

6. Modelling results

6.1. Simulation conditions

In this section we present results of our model applied to thesimulation of a typical dc arc plasma torch (figure 3) for thecases shown in table 3. Cases A and B allow the comparisonof the arc dynamics with and without the reattachment model.The operating conditions of cases C and D are typical ofplasma spraying processes and the different values of Eb

permit observing the dominant effect of Eb on controlling thearc dynamics. Case E allows seeing the change in the arcdynamics as the torch operates closer to the takeover mode(see section 1 and figure 2). In all cases, a value of electricalconductivity equal to 104 S m−1 has been specified in the regionof thickness ∼0.08 mm immediately in front of the electrodes(according to section 5.1). In all the simulated cases the timestep varied adaptively between 0.1 and 1.2 µs. Comparisonsof our modelling results with experimental measurements arepresented in section 6.4.

6.2. Flow fields inside an arc plasma torch

The instantaneous temperature distribution inside the torch forcase B is presented in figure 6. In the figure the following canbe observed: the marked three-dimensionality of the flow, theformation of a cold flow boundary layer around the plasma(figures 6(2) and 6(3)) and the extent of the anode attachment,whose maximum temperature is around 1400 K (figure 6(4)).The location and size of the anode spot shown in figure 6(4) isconsistent with experimental observations [27] and numericalsimulations [13, 27]. As explained in section 5, the lack of areattachment model produces excessively long arcs, and hencesimulations without a reattachment model produce anodespots further downstream than the one shown in figure 6(4).The excessive lengthening of the arc in the absence of areattachment model will be shown in the following section.

Figure 7 shows the distributions of electric potential andelectric field normal to the anode surface at the same timeinstant of the results in figure 6. Due to the cylindrical shapeof the anode, the normal to the anode surface is approximated inthe whole domain by the radial coordinate, therefore Er ∼ En.This approximation reduces the computational time of thesimulations considerably. It can be observed that the electricfield is negligible within the arc and it is largest in the arcfringes. The arrow indicates the location of the maximumelectric field En,max. If this location satisfies the conditionEn,max > Eb, then it will correspond to the value of �xb

described in section 5.2, and subsequently the reattachment

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1

3 4

2

Figure 6. Instantaneous temperature distribution inside a dc arc plasma torch: (1) through a vertical plane, (2) horizontal plane, (3) atdifferent axial cross sections and (4) over the anode surface, where the location of the anode spot is circled. (Colour online.)

Figure 7. Distribution of electric potential and radial electric field (which approximates the electric field in the direction normal to the anodesurface) inside the torch at the same time instant of figure 6. The arrow indicates the location of the maximum electric field (En,max).(Colour online.)

model would be applied. It can also be noticed that the locationof the anode attachment (at z ∼ 12 mm in the top part of theanode surface, see figure 6(4)) can hardly be deduced fromthe distribution of electric potential, whereas it can be clearlyobserved in the distribution of the radial electric field.

6.3. Arc reattachment processes

The time evolution of the temperature distribution and the totalvoltage drop during the formation of a new arc attachmentwhen straight gas injection is used are shown in figure 8 forEb = ∞ and in figure 9 for Eb = 5 × 104 V m−1 (cases Aand B). Figure 8 shows that, at the same time as the arcis dragged downstream, the voltage drop increases (from 1

to 2). The expansion of the plasma downstream added to theeffect of the anode jet causes the plasma to get in contactwith the anode surface at a location opposite to the originalattachment (frame 2). This new connection between theplasma and the anode surface causes a sudden increase in thelocal temperature, creating a new arc attachment. Frames 2–4 in figure 8 show that two arc attachments coexist, butbecause the new one is closer to the cathode (and hence has alower voltage drop, which makes it more thermodynamicallyfavourable) will remain while the old attachment is draggedaway by the flow. Furthermore, frame 3 shows that thenewly formed attachment, having an increased curvatureand correspondingly a larger electromagnetic force, strongly

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Figure 8. Arc reattachment process inside plasma torch for case A (no reattachment model): time sequence of temperature distributioninside the torch and evolution of the voltage drop across the torch. (Colour online.)

Figure 9. Arc reattachment process inside plasma torch for case B (with reattachment model): time sequence of temperature distributioninside the torch and evolution of the voltage drop across the torch. (Colour online.)

opposes the flow drag and causes a slight movement of theanode attachment upstream. Frames 5–7 show the draggingof the new attachment accompanied by an almost proportionalincrease in the total voltage drop. The formation of a newattachment at the opposite side of the original one has beenobserved experimentally in [3] and numerically in simulationsof the takeover mode in [7].

The form of the voltage drop signal in figure 8 resemblesthe voltage signal of the takeover mode of operation of the torch(see figure 2). Unfortunately, the arc length and voltage dropare larger than what is measured experimentally. It seems clearthat the described arc reattachment process can be used only tosimulate the steady and takeover modes (as done in [7,8]), butnot the restrike mode. For the restrike mode, a mechanism that

limits the length of the arc is needed. This is accomplished bythe use of a reattachment model.

Figure 9 shows the effect of applying the reattachmentmodel in the simulation of the flow inside a plasma torch.When the local electric field is below the pre-specified valueof Eb, the arc is dragged by the flow, and its length andthe total voltage drop increase almost linearly (from 1 to 2).As soon as the breakdown condition is met (En,max >

Eb), the reattachment model described in section 5.2 isapplied automatically. Frame 2 shows the growing of ahigh temperature appendage in the direction towards theanode, mimicking the formation of a new attachment. Thisarc reattachment process is in qualitative agreement withexperimental observations [1, 2]. In frame 3, the new

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attachment forms, and as seen by the voltage drop between 3and 4, it rapidly overcomes the old attachment, which isdragged away by the flow. Frames 5–7 show the draggingof the new attachment accompanied by a proportional increasein the total voltage drop.

The rapid decrease of voltage drop from 3 to 4 resemblesthe restrike mode of operation (figure 2). Furthermore, it canbe observed that the total voltage drop in figure 9 is lower thanin figure 8 and that the voltage signal is sharper in figure 9 thanin figure 8. This suggests that, by tuning the value of Eb, theprincipal parameter in the reattachment model, one could beable to match the voltage signal obtained experimentally. Thispoint is discussed in the following section.

Through the conducting channel, temperature andpressure increase very rapidly, causing a radial expansion ofthe neutral gas, resembling the streamer breakdown processesstudied by Marode and collaborators [28]. This sudden gasexpansion is observed in figure 10, which shows the velocitydistribution in the region where the reattachment model isapplied. It can be seen that there is a net flow from the inside of

Figure 10. Velocity distribution in the region near the reattachmentcolumn at the time instant shown in figure 9(2). The new attachmentforms at z ∼ 2 mm, whereas the old attachment is located atz ∼ 12 mm. (Colour online.)

1 2

3 4

Figure 11. Arc reattachment process inside a dc plasma torch for case C: electric potential distribution over the 14 kK isosurface at t =(1) 360, (2) 362, (3) 364 and (4) 368 µs. The arrows indicate the locations of the arc attachments; O refers to the old attachment and N to thenew one. (Colour online.)

the reattachment column outwards. This is a consequence ofat least two effects: the gas expansion due to the rapid heatingof the reattachment column and the formation of an anode jetdue to the local constriction of the current path near the newattachment spot. Furthermore, it can be observed that, dueto the cathode jet, the flow from the reattachment column isdeflected as it approaches the cathode tip.

The problems attaining convergence when the reattach-ment model is applied (due to the sudden gas expansion andrapid energy accommodation throughout the domain) are tobe expected if conservation (of mass/momentum/energy) isenforced within consecutive time steps. Our code, being fullyimplicit and fully coupled, enforces global conservation in eachtime step as well as within steps, which made mandatory theuse of the globalization technique and adaptive time steppingdescribed in section 3.3 in order to attain convergence. Thesetechniques may not be needed if a semi-implicit code is used,because global conservation within consecutive time steps isusually not enforced in such a code.

The time evolution of the temperature isosurface of14 000 K (which resembles the shape of the arc) and the voltagedistribution over it during the formation of a new arc attachmentfor case C are shown in figure 11. The formation of the newattachment occurs in a similar manner as in figure 9, the cleardifference being that due to the swirl, the reattachment processis no longer confined to a single plane (y–z plane in figures 8and 9). The swirl causes the arc to remain more centred on thetorch axis than when straight injection is used. Furthermore,the stabilizing effect of the swirl causes the arc to be longerthan when straight injection is used. The growing of the hightemperature appendage from the column towards the anoderesembles the high speed camera pictures reported in [1, 2].

6.4. Comparison with experimental measurements

Figure 12 shows the voltage drop signals and their powerspectra for the torch operating with swirl injection obtainedexperimentally and numerically for cases C and D. (The

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Figure 12. Voltage drop signals and their power spectra: experimental (Exp.) and numerical (case C) reattachment model withEb = 5 × 104 V m−1 and (case D) with Eb = 2 × 104 V m−1.

Figure 13. Voltage drop signals and their power spectra for cases C and E.

operating conditions of the experiments are the same as forcases C and D.) The peak frequency fp in the spectra ofthe results from the simulation using the reattachment modelwith Eb = 5 × 104 V m−1 is in good agreement with theexperimentally measured (fp ∼ 3.2 kHz numerically and∼ 3.3 kHz experimentally). But the magnitude of the voltagedrop clearly exceeds the experimental results. Furthermore,the lack of adequate modelling of the plasma sheaths needs tobe considered, which makes the obtained voltage drops differeven more from those measured. Reducing the breakdownvoltage to 2 × 104 V m−1 causes the length of the arc andconsequently, the voltage drop, to decrease significantly, butat the same time the peak frequency increases considerably(fp ∼ 9 kHz). From this result it seems clear that thetuning of the parameter Eb allows the matching of either thepeak frequency or the magnitude of the voltage drop, butnot both simultaneously. By tuning the critical field (themodel parameter), Moreau’s et al model [11] is able to matchapproximately the voltage drop and peak frequency, but at theexpense of producing an anode spot an order of magnitudelarger than what is observed experimentally.

The effect of changing the flow rate is appreciated infigure 13, where it can be seen that decreasing the flow ratecauses the voltage drop signal to resemble the takeover modeof operation (a smoother and less chaotic signal). As the flowrate decreases from 60 to 30 slpm, the peak frequency increasesfrom 3.23 to 4.29 kHz. Case E shows lower minimum voltagesdue to a shorter arc. There is no appreciable difference betweenthe maximum lengths of the arcs for cases C and E.

The fact that our reattachment model with Eb = 5 ×104 V m−1 is able to reproduce the experimentally observedpeak frequency, arc length and size of the anode spot, butnot the total voltage drop, indicates that the total resistanceof the arc to the current flow is larger in our model than whatthe experiments suggest. This fact is a clear indication of thelimitation of using a LTE model to describe the dynamics ofthe plasma flow inside dc plasma torches.

The greater electrical resistance of a LTE model, giventhat the length and width of the arc agree with experimentalobservations (approximated by the location and size of theanode spot), can be explained by a low electrical conductivityin the arc fringes, especially through the anode attachment.Because σ is mostly a function of the electron temperature andgenerally increases with the degree of thermal non-equilibriumθ (θ = Te/Th, with Te as the electron temperature and Th asthe heavy particle temperature), it can be deduced that theanode attachment may be characterized by a wider region withhigh Te (and hence high θ ) than the region currently describedby the LTE model (T = Te = Th). The above argumentis also suggested by the comparison of end-on images of thetorch with the temperature distribution through different crosssections from our simulation results, as presented in figure 14.The experimental end-on image can be considered typical ofsimilar studies, i.e. [29].

It is evident in figure 14 that the numerical results donot display a well-defined cold boundary layer all around thearc, as observed in the end-on image. The high temperatureof the anode attachment in a LTE model is needed to allow

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Figure 14. End-on image of the arc inside a dc torch [3] (left) and numerical temperature distribution at different axial locations for case C(right). The arrow indicates the location of the anode attachment. In the numerical results the anode attachment is located at z ∼ 13 mm(∼13 mm from the cathode tip). (Colour online.)

Figure 15. Instantaneous (Inst.) and time-averaged over four reattachment periods (Ave.) velocity, temperature, and mass flow rate per unitarea profiles at the torch outlet. The instantaneous profiles correspond to the same instant of the numerical results shown in figure 14.(Colour online.)

the electrical current flow from the anode to the plasma.A thermal non-equilibrium model (a model that considersTe �= Th) should be able to match experimental observationsmore accurately, by displaying a Th distribution resembling theend-on image and a Te distribution resembling the numericaltemperature profiles in figure 14. To the best knowledge of theauthors, a non-equilibrium model adequate for the simulationof the arc dynamics inside dc arc plasma torches has not yetbeen reported in the literature.

Figure 14 also shows the elliptical shape of the crosssection of the arc column, with the main axis of the ellipse alongthe direction of the anode attachment (dashed line). This shapeof the arc cross section is caused in part by the effect of theanode jet and by the balance of forces over the arc [1,2] and ismore pronounced when straight injection is used, as observedin [7]. The elongated shape of the arc shown in the numericalresults of figure 14 is not observed in the outlet temperatureprofile. The continuous expansion of the arc along the torchaxis causes the cross section of the arc to change from beingelongated to being more axi-symmetrical, as can be observed

in figure 15. Furthermore, figure 15 shows a clear correlationbetween the temperature and the velocity distribution at theoutlet. This correlation has been observed experimentally[3–5]. The distribution of the mass flow per unit area indicatesthat most of the flow exits the torch through the ‘cold’ boundarylayer. The complexity of the instantaneous outlet profiles isevident, indicating that the observed complex nature of plasmajet is in a significant part due to the forcing caused by the arcdynamics; this complexity is not observed in the time-averagedprofiles (figure 15(Ave)). The time-averaged profiles are notcompletely axi-symmetric because of the limited amount oftime used in the averaging process (∼4 reattachment periods)and indicate that the reattachment process has taken placemostly in the direction of the dashed line in figure 14. Thelarge computational cost of the simulations presented makesour ability to study the long-term behaviour of the torch verylimited. The time-averaged profiles along the x and y axesare shown in figure 16. These results are consistent withthe measurements reported in [30] for the same operatingconditions used here, but using a nozzle of 5.5 mm diameter,

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Figure 16. Time-averaged velocity, temperature and mass flow per unit area at the torch outlet.

where temperatures of ∼15 000 K are reported, and with resultsfrom [31] for the same torch used here operating with Ar, 450 Aand 23.6 slpm, that show temperatures of ∼12 000 K, and withthe numerical results in [10] for a different torch of 7 mmdiameter operating with Ar–H2, 600 A and 60 slpm, whichshow temperatures of ∼13 000 K and maximum velocities of∼2200 m s−1.

7. Conclusions

The reattachment process is a central part of the arc dynamicsin plasma torches. The detailed modelling of this process isunfeasible with actual computational methods and computerpower, especially when it is required as a part of the realisticmodelling of the plasma flow in an industrial application.Therefore, approximate models that mimic the effect of thereattachment process in the arc dynamics are needed. Tomimic the physical reattachment process, a model facesthe questions of (1) where and (2) how to introduce thenew attachment. In this paper we presented a reattachmentmodel that answers these questions by (1) forming a newattachment at the location where the local electric field exceedsa certain pre-specified value and (2) introducing there a highelectrical conductivity channel between the arc and the anode.This model has been incorporated into a three-dimensionaland transient description of the plasma flow based on theLTE assumption and implemented numerically in a fully-coupled approach using a variational multi-scale finite elementformulation, which implicitly accounts for the multi-scalenature of the flow.

Simulations of the flow inside a commercial plasmaspraying torch show the effectiveness of our modellingapproach. By tuning the breakdown electric field Eb, themain parameter of our model, the calculated peak frequenciesof the voltage signal, arc lengths and anode spot sizes agreereasonably well with experimental observations, while thetotal voltage drops exceed those measured. It is observedthat the reattachment model can be tuned to match either theexperimentally measured peak frequency or the total voltagedrop, but not both simultaneously. This finding, added tothe apparent lack of a well-defined cold boundary layer allaround the arc, points towards the importance of thermal non-equilibrium effects inside the torch, particularly in the anodeattachment region and therefore shows clear evidence of thelimitations of the LTE assumption to describe the plasma flowinside plasma torches.

It is foreseen that a non-equilibrium model will providebetter agreement with experimental measurements over a LTEmodel. But, because a thermal non-equilibrium model willstill not be able to describe the restrike mode of operationof the torch (unless detailed excitation mechanisms, chargeaccumulation, etc are taken into account), a reattachmentmodel will still be needed. A non-equilibrium modelcomplemented with a reattachment model should be able tosimulate the different modes of operation of the torch andapproximately match the measured peak frequency, voltagedrop, arc length and anode spot size, while displaying acold flow boundary layer all around the arc. A promisingalternative to realistically model the reattachment process aspart of the simulation of the arc dynamics inside the torch isthe combination of a thermal and non-thermal plasma modelsimilar to the one developed by Papadakis et al [32], butsuitable for the modelling of atmospheric pressure discharges.But still the limited understanding of the physics driving thereattachment process will limit the development of such amodel.

Our future work includes the development of a non-equilibrium model adequate for the simulation of the arcdynamics in plasma torches and for devising a better criterionfor identifying the breakdown (i.e. a criterion based on the localelectric field and the local electron temperature).

Acknowledgments

This research has been partially supported by the NSFGrant CTS-0225962. Computing time from a grant fromthe University of Minnesota Supercomputing Institute andthe support for the first author by a Doctoral DissertationFellowship from the University of Minnesota are gratefullyacknowledged.

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