Arc Plasma Torch Modeling
Abstract
Arc plasma torches are the primary components of various industrial thermal plasma processes involving plasma spraying, metal cutting and welding, thermal plasma CVD, metal melting and remelting, waste treatment, and gas production. They are relatively simple devices whose operation implies intricate thermal, chemical, electrical, and fluid dynamics phenomena. Modeling may be used as a means to better understand the physical processes involved in their operation. This article presents an overview of the main aspects involved in the modeling of DC arc plasma torches: the mathematical models including thermodynamic and chemical nonequilibrium models, turbulent and radiative transport, thermodynamic and transport property calculation, boundary conditions, and arc reattachment models. It focuses on the conventional plasma torches used for plasma spraying that include a hot cathode and a nozzle anode.
Keywords
arc reattachment chemical equilibrium electrode local thermodynamic equilibrium nonequilibrium plasma jet plasma spraying plasma torch thermal plasmaNomenclature
Variables
- A
magnetic vector potential
- a_{s,r}
stoichiometric coefficient in the forward direction for the reaction r
- B
magnetic field
- b_{s,r}
stoichiometric coefficients in the backward direction for the reaction r
- C_{μ}
constant of k-ε model
- C_{ε1}
constant of k-ε model
- C_{ε2}
constant of k-ε model
- D_{s}
effective diffusivity of species s
- D_{sj}
binary diffusion coefficient species s and j
- E
real electric field
- E_{b}
critical electric field
- E_{p}
effective electric field
- e
elementary electric charge
- G_{k}
generation of turbulent kinetic energy
- h
plasma-specific enthalpy
- h_{e}
electron-specific enthalpy
- h_{h}
heavy-particle-specific enthalpy
- h_{w}
heat transfer coefficient
- I
arc current intensity
- I_{λ}
spectral intensity
- I_{bλ}
spectral black body intensity
- J_{cath}
current density over the cathode surface
- J_{q}
arc current density
- J_{s}
mass diffusion flux of species s
- k
turbulent kinetic energy
- k
total thermal conductivity
- k_{B}
Boltzmann constant
- k_{e}
electron translational thermal conductivity
- k_{h}
heavy-particle thermal conductivity
- k_{r}
reactive thermal conductivity
- k_{λ}
spectral absorption coefficient
- k_{f,r}
forward reaction rate for reaction r
- k_{b,r}
backward reaction rate for reaction r
- M_{s}
molecular weight of species s
- m_{e}
electron mass
- n_{e}
electron number density
- ns
number of species
- nr
number of reactions
- p
pressure
- \( \dot{Q}_{\text{eh}} \)
electron-heavy-particle energy exchange
- \( \dot{Q}_{\text{r}} \)
volumetric net radiation losses
- q_{a}
heat transferred to the anode surface
- q_{r}
radiative heat flux
- q_{wall}
heat transferred to the wall
- q′
total heat flux
- r
radius
- R_{s}
gas constant of species s
- T
temperature
- T_{w}
wall temperature
- T_{c}
critical temperature
- T_{e}
electron temperature
- T_{h}
heavy-particle temperature
- t
time
- u
mass average velocity
- u_{s}
velocity of species s
- W_{a}
work function of the anode material
- x
spatial coordinate
Greeks
- δ
Kronecker delta
- δ_{es}
inelastic collision factor
- ε
turbulent dissipation rate
- ε_{r}
effective net emission coefficient
- θ
departure from thermal equilibrium
- \( \upphi_{p} \)
effective electric potential
- λ
wavelength
- μ
molecular dynamic viscosity
- μ_{0}
permeability of free space
- μ_{t}
turbulent dynamic viscosity
- ρ
mass density
- ρ_{s}
mass density of species s
- \( \dot{\uprho }_{s}^{c} \)
volumetric production rate of species s
- σ
electrical conductivity
- σ_{k}
constant of k-ε model
- τ
stress tensor
- ν_{es}
collision frequency between electrons and species s
- ω
reaction molar rate
- ψ
conserved property
- \( \dot{\upvarpi }_{r} \)
progress rate of reaction r
- Δ
finite change in quantity
Subscripts
- a
anode
- e
electron
- h
heavy particle
- j
specie
- k
turbulent kinetic energy
- r
reactants or reactions
- s
species
- t
turbulent
- w
wall
- ε
turbulent dissipation rate
Abbreviations
- CFD
computational fluid dynamics
- DC
direct current
- DES
detached eddy simulation
- SCEBD
self-consistent effective binary diffusion
- DNS
direct numerical simulation
- DOM
discrete ordinates methods
- RANS
Reynolds-averaged Navier-Stokes
- EEDF
electron energy distribution function
- LES
large eddy simulations
- LTE
local thermal equilibrium
- NEC
net emission coefficient
- NLTE
nonlocal thermal equilibrium
- RTE
radiative transfer equation
- HVOF
high-velocity oxygen fuel
Introduction
Thermal plasma processes have proven their technological advantage in a wide variety of fields for over 40 years. The features that make thermal plasmas attractive are a high energy density (~10^{6}-10^{7} J/m^{3}) that comes with high heat flux density (~10^{7}-10^{9} W/m^{2}), high quenching rate (~10^{6}-10^{8} K/s), and high processing rates.
Direct current (DC) arc plasma torches are, generally, the primary component of these processes that include plasma spraying, ultra fine particle synthesis, metal welding and cutting but also, extractive metallurgy, waste treatment, and biogas production. These torches operate as thermal, chemical, and electrical devices in processes that achieve material modifications which often cannot be achieved, or are not economically feasible, with other devices. A distinctive example of an application that relies on DC arc plasma torches is plasma spraying that has become a well-established and widely used technology to manufacture coatings resistant to wear, corrosion, and temperature and generate near-net shapes of metallic and ceramic parts. For instance, plasma-sprayed coatings make possible turbine blades to withstand temperatures up to 1200 °C and provide unparalleled wear resistance to prosthetic implants. The continuous development of thermal plasma-based technologies stresses the need for a better understanding of the operation of arc plasma torches.
Inside the torch, the working gas flows around the cathode and through a constricting tube or nozzle. The plasma is usually initiated by a high-voltage pulse which creates a conductive path for an electric arc to form between the cathode and anode (the torch nozzle in nontransferred arc torches or the working piece in transferred ones). The electric heating produced by the arc causes the gas to reach very high temperatures (e.g., >10,000 K), thus to dissociate and ionize. The cold gas around the surface of the water-cooled nozzle or constrictor tube, being electrically nonconductive, constricts the plasma, raising its temperature and velocity. Most of the commercial plasma spray torches operate at atmospheric pressure with electric power levels ranging between 10 and 100 kW, arc currents between 250 and 1000 A, arc voltages between 30 and 100 V, and flow rates between 20 and 150 slpm (standard liters per minute). Common gases used in thermal plasma processing are Ar, He, H_{2}, N_{2}, O_{2}, and mixtures of these.
The modeling of DC arc plasma torches is extremely challenging because the plasma flow is highly nonlinear, presents strong property gradients, is characterized by a wide range of time and length scales, and often includes chemical and thermodynamic nonequilibrium effects, especially near its boundaries. Moreover, the modeling of the torch is frequently part of the modeling of a given plasma process or application (e.g., plasma spraying, plasma cutting). In that case, the description of the plasma flow needs to be coupled with suitable models of the processing material (e.g., sprayed powder, work piece) and its interaction with the plasma flow. Hence, the domain of analysis is typically extended beyond the plasma and across the torch components (e.g., to model the electrical characteristics across the electrodes, or the dissipation of heat to the cooling water) and/or the working material (e.g., to describe heat transferred and phase changes in the processing powder or work piece).
This article presents an overview of the main aspects involved in DC arc plasma torch modeling as well as some examples that typify the current state of the art. Particular emphasis is given to conventional nontransferred arc plasma torches with thermionic (hot) cathodes as those employed in plasma spraying. Section 2 describes the physical/mathematical models used to describe the plasma flow, including turbulent and radiative transport models. Section 3 presents calculation procedures for the gas thermodynamic and transport properties that are fundamental for a realistic and accurate description of thermal plasmas. Boundary conditions, which seek to represent the physical phenomena dominating the interactions between the computational domain of the plasma flow and its surroundings, are described in Section 4. Section 5 describes models of the arc reattachment process, a complex phenomenon which severely alters the arc dynamics and is inherently present in the flow inside nontransferred arc torches. Finally, in Section 6, conclusions are drawn and some of the developments required for furthering the current understanding of plasma torch operation and achieving truly predictive DC arc plasma torch models are presented.
Plasma Flow Models
Fluid Models
The plasma formed in DC arc torches is an example of a thermal plasma. Among other things, thermal plasmas are characterized by high electron density (ranging between 10^{21} and 10^{24} m^{−3}) and high collision frequencies among its constituents (i.e., molecules, atoms, ions, and electrons) (Ref 1). High collision frequencies lead to a state close to local thermodynamic equilibrium (LTE) in which the kinetic energy of the constitutive species can be characterized by a single temperature. The LTE approximation is often violated near the plasma boundaries, either as the plasma interacts with solid walls, the working material, or with the cold working gas or atmosphere.
Thermal plasmas, due to their relatively high densities and pressures, and hence small mean free paths of the constitutive species, are most appropriately described by fluid models. Fluid models describe the evolution of the moments of the Boltzmann equation for each species in the plasma, and provide direct measures of macroscopic flow properties, such as temperature and pressure.
Conservation equations of the thermodynamic and chemical equilibrium thermal plasma model
Conservation | Accumulation | Net flux | Net production |
---|---|---|---|
Total mass | \( {\frac{\partial \uprho }{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}}\uprho ) \) | 0 |
Mass-averaged momentum | \( {\frac{{\partial \uprho {\mathbf{u}}}}{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}} \otimes \uprho {\mathbf{u}} + p{\varvec{\updelta}} + {\varvec{\uptau}}) \) | \( {\mathbf{J}}_{q} \times {\mathbf{B}} \) |
Internal energy | \( {\frac{\partial \uprho h}{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}}\uprho h + {\mathbf{q}}^{\prime } ) \) | \( {\frac{Dp}{Dt}} - {\varvec{\uptau}}:\nabla {\mathbf{u}} + {\mathbf{J}}_{q} \cdot ({\mathbf{E}} + {\mathbf{u}} \times {\mathbf{B}}) - \dot{Q}_{\rm r} \) |
Conservation equations of the thermodynamic and chemical nonequilibrium thermal plasma model
Conservation | Accumulation | Net flux | Net production |
---|---|---|---|
Total mass | \( {\frac{\partial \uprho }{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}}\uprho ) \) | 0 |
Species mass | \( {\frac{{\partial \uprho_{s} }}{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}}\uprho_{s} + {\mathbf{J}}_{s} ) \) | \( \dot{\uprho }_{s}^{c} \) |
Mass-averaged momentum | \( {\frac{{\partial \uprho {\mathbf{u}}}}{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}} \otimes \uprho {\mathbf{u}} + p{\varvec{\updelta}} + {\varvec{\uptau}}) \) | \( {\mathbf{J}}_{q} \times {\mathbf{B}} \) |
Int. energy heavy species | \( {\frac{{\partial \uprho h_{\text{h}} }}{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}}\uprho h_{\text{h}} + {\mathbf{q}}_{\text{h}}^{\prime } ) \) | \( {\frac{{Dp_{\text{h}} }}{Dt}} - {\varvec{\uptau}}:\nabla {\mathbf{u}} + \dot{Q}_{\text{eh}} \) |
Int. energy electrons | \( {\frac{{\partial \uprho h_{\text{e}} }}{\partial t}} \) | \( \nabla \cdot ({\mathbf{u}}\uprho h_{\text{e}} + {\mathbf{q}}_{\text{e}}^{\prime } ) \) | \( {\frac{{Dp_{\text{e}} }}{Dt}} + {\mathbf{J}}_{q} \cdot ({\mathbf{E}} + {\mathbf{u}} \times {\mathbf{B}}) - \dot{Q}_{\rm r} - \dot{Q}_{\text{eh}} \) |
In Tables 1 and 2, ρ represents the total mass density, ρ_{s} the mass density of species s, u the mass-averaged velocity, J_{s} the mass diffusion flux, and \( \dot{\uprho }_{s}^{c} \) the volumetric production rate of species s; p represents the pressure, δ the Kronecker delta, τ the stress tensor, J_{q} the current density, B the magnetic field, J_{q }× B the Lorentz force; h, h_{h}, and h_{e} the equilibrium, heavy-species, and electron enthalpy, respectively (no subscript indicates an equilibrium or total property, while the subscripts “h” and “e” stand for heavy particle and electron properties, respectively); q′ the total heat flux; Dp/Dt is the pressure work with D/Dt as the substantive derivative; the term J_{q} · (E + u × B) represents the Joule heating, \( \dot{Q}_{\rm r} \) the volumetric net radiation losses, and \( \dot{Q}_{\text{eh}} \) the electron-heavy-particle energy exchange term, which couples the two energy equations in the NLTE model. In Table 2, only ns − 1, where ns is the number of species, species mass conservation equations are required because the total mass conservation equation is included in the system.
Several assumptions and approximations are implied in the equations in Tables 1 and 2. In particular, closure of the moments of Boltzmann equation is taken into account in the specification of diffusive fluxes and/or transport coefficients. Furthermore, there are different forms to express the conservation equations in Tables 1 and 2, e.g., one could use conservation of total energy instead of internal energy. The most important requirement when formulating equilibrium or nonequilibrium plasma fluid models is self-consistency, which implies consistency with the moments of Boltzmann equation. In this regard, up to the specification of diffusive fluxes and source terms \( \dot{\uprho }_{s}^{c} , \), \( \dot{Q}_{\text{eh}} , \) and \( \dot{Q}_{\rm r} , \) the above models are self-consistent. Moreover, the LTE and NLTE models above are consistent with each other in the sense that the NLTE model gets reduced to the LTE model if thermal and chemical equilibrium are assumed (i.e., if one enforces T_{h} = T_{e} in the equations in Table 2 and if the plasma composition is determined only as function of the thermodynamic state of the fluid). Furthermore, the addition of the electron and heavy-species energy equations in Table 2 produces the total internal energy conservation equation in Table 1.
Diffusion Fluxes and Source Terms
The systems of equations in Tables 1 and 2 are closed with the specification of diffusive fluxes J_{s}, τ, and q′, and the source terms \( \dot{\uprho }_{s}^{c} , \)\( \dot{Q}_{\text{eh}} , \) and \( \dot{Q}_{\rm r} . \)
Electromagnetic Equations
Maxwell’s equations for thermal plasmas
Name | Equation |
---|---|
Ampere’s law | \( \nabla \times {\mathbf{B}} = \upmu_{0} {\mathbf{J}}_{q} \) |
Faraday’s law | \( \nabla \times {\mathbf{E}}_{p} = - {\frac{{\partial {\mathbf{B}}}}{\partial t}} \) |
(Generalized) Ohm’s law | \( {\mathbf{J}}_{q} = \upsigma ({\mathbf{E}}_{p} + {\mathbf{u}} \times {\mathbf{B}}) \) |
Gauss’ law (charge conservation) | \( \nabla \cdot {\mathbf{J}}_{q} = 0 \) |
Solenoidal constraint | \( \nabla \cdot {\mathbf{B}} = 0 \) |
Turbulence Models
In DC arc plasma torches, the working gas is typically at ambient temperature when it enters the torch. The temperature of the gas, as it interacts with the arc, increases by a rate in the order of 10^{4} K/mm. This rapid heating causes the sudden expansion of the gas and consequently its rapid acceleration. The velocity of the gas across the torch often varies by 2 orders of magnitude (e.g., from O(10) to O(1000) m/s). The large gas acceleration and shear velocity and temperature gradients inside the torch, together with the electromagnetic forcing (Ref 11, 12), cause the flow to become unstable and turbulent. Turbulence is further enhanced when the plasma flow leaves the torch and interacts with the cold and, thus, denser environment.
The accurate modeling of turbulent flows, due to their large range of length and time scales, represents a great challenge. The most faithful numerical description of turbulent flows is given by the approach known as direct numerical simulation (DNS), which seeks to resolve all the scales of the flow without any approximation (i.e., by definition, no physical, e.g., eddy viscosity, see below, or numerical, e.g., upwinding, dissipation mechanisms are employed). DNS of large Reynolds number (Re) flows and for industrial geometries are almost unfeasible with current computing power, as the range of length scales to be solved (i.e., the size of the grid needed) scales as Re^{3}. DNS of nonequilibrium plasmas can be found in the literature (e.g., Ref 13) but, to the best knowledge of the authors, no DNS of a thermal plasma has been performed to date.
The large cost of DNS has motivated a variety of alternative approaches to simulate turbulent flows. The main approaches are grouped in what are known as large eddy simulations (LES), which seek to model only the small scales of the flow, and Reynolds-averaged Navier-Stokes (RANS) models, which seek the solution of approximations of the time-averaged Navier-Stokes equations. LES is typically far more accurate than RANS, but often an order of magnitude or more expensive. Recently, the technique known as detached eddy simulation (DES) has been gaining more acceptance, especially in the modeling of external flows, as it mixes the LES and RANS approaches: LES is used where it is most needed (e.g., in highly unsteady zones) and RANS in the rest of the domain or where the use of LES would be prohibitive (e.g., near walls, where vorticity is mostly created).
Most LES and RANS models rely on the Boussinesq hypothesis, which consists of modeling the turbulent stresses in a similar manner as the viscous stress and, hence, reduces the formulation of the turbulence model to the specification of an appropriate turbulent viscosity μ_{t} (the total stress τ is still given by Eq 4 but μ is replaced with μ + μ_{t}). More sophisticated models exist (Ref 14), which seek to model the whole turbulent stress with very few or no empirical approximations, such as the residual-driven model of Bazilevs et al. (Ref 15). But these models, although they are potentially the best approaches for the modeling of complex turbulent flows, are not widely used yet.
Diverse RANS and few LES models are often offered in commercial computational fluid dynamics (CFD) software, which has driven the increasing use of these models. Turbulent viscosity models for LES are usually far simpler than models for RANS. But, sound LES simulations require highly accurate spatial and temporal discretizations. These requirements are usually hard to satisfy using commercial software because robustness, probably the most important feature in commercial software, is very often in opposition to accuracy (e.g., it is harder to obtain convergence using more accurate, high order, discretizations).
The use of turbulence models in thermal plasma flows is significantly more involved than for most other industrial applications due to their inherent characteristics (i.e., reactivity, large property variations, and electromagnetic effects). The use of standard turbulent models for thermal plasma simulation often implicitly neglects several of these characteristics.
Equations of the standard k-ε turbulent model
Cons. | Accumulation | Net flux | Net production |
---|---|---|---|
Turbulent kinetic energy | \( {\frac{\partial \uprho k}{\partial t}} \) | \( \nabla \cdot \left( {{\mathbf{u}}\uprho k - \left( {\upmu + {\frac{{\upmu_{t} }}{{\upsigma_{k} }}}} \right)\nabla k} \right) \) | G_{k} − ρɛ |
Rate of dissipation | \( {\frac{\partial \uprho \upvarepsilon }{\partial t}} \) | \( \nabla \cdot \left( {{\mathbf{u}}\uprho \upvarepsilon - \left( {\upmu + {\frac{{\upmu_{\text{t}} }}{{\upsigma_{\upvarepsilon } }}}} \right)\nabla \upvarepsilon } \right) \) | \( c_{\upvarepsilon 1} G_{k} {\frac{\upvarepsilon }{k}} - c_{\upvarepsilon 2} \uprho {\frac{{\upvarepsilon^{2} }}{k}} \) |
To summarize, the use of turbulence models for the modeling of the flow in DC arc torches has to be approached with care and weighting the assumptions and approximations involved. Particularly, for the flow in nontransferred torches, which is highly unsteady, a LES approach is more appropriate, whereas for the modeling of transferred torches, especially when the flow is steady, the use of RNG k-ε or similar models may provide an adequate description of the flow. Nevertheless, validation with experimental measurements is required, and when possible should be pursued.
Radiative Transport
Radiative transfer in thermal plasmas involves line and continuum radiation, including bremsstrahlung and recombination radiation (Ref 1).
The direct solution of the RTE is exceedingly expensive due to the dual s to x dependence, and consequently diverse types of approximations are often employed. The detailed description of the radiative transport in thermal plasmas represents an enormous challenge not only because of the complex absorption spectra of the species present but also due to the weak interaction of the photons with the surrounding media. This last characteristic jeopardizes the use of models that rely on strong coupling (optically thick media), like diffusion-like models such as the P1 approximation, and makes mandatory the use of more computationally expensive techniques like Direct simulation Monte Carlo or directional transport methods, like ray-tracing techniques and discrete ordinates methods (DOMs). The DOM consists of solving the RTE along few ordinate directions transforming the RTE in a (typically large) set of transport equations. The P1 approximation consists of the formulation of a transport equation (of reaction-diffusion form) for the first-order term of the expansion of the radiative intensity in spherical harmonics. The P1 method is a good approximation of the radiation transport in optically thick media, and hence is not suitable for most DC arc plasma torch modeling.
Probably one of the best radiation transfer simulations applied to a thermal plasma flow is the work of Menart et al. (Ref 20) who used a DOM for a large set of wavelengths. Because their work was focused on analyzing the radiative transfer, Menart et al. did not solve the radiative transport coupled to a plasma flow model. Instead, they used a precalculated temperature field to determine κ_{λ}(T, λ) to solve the RTE. Their approach is justified by the enormous computational cost required to solve the plasma flow together with radiative transport. More recently, Iordanidis and co-workers (Ref 21, 22) compared the DOM and P1 methods and performed simulations of the plasma flow in circuit breakers using the DOM due to its greater accuracy. An alternative approach is the use of view factors to determine the exchange of radiative energy among the domain boundaries. Such approach has been successfully used by Lago et al. (Ref 23) for the simulation of a free-burning arc.
Currently, net emission coefficient (NEC) are available for the following pure gases (Ar, O_{2}, H_{2}O, Air) and mixtures (Ar-H_{2}, Ar-Fe, Ar-Fe-H_{2}, air-metallic vapors) (e.g., see Ref 28-32).
Gas Properties
In plasma simulations, mass, momentum, and energy equations, together with electromagnetic field equations (see Section 2.1), have to be solved with a coupled approach and the accuracy of results depends strongly on the use of suitable thermodynamic and transport properties (see Section 2.2).
LTE Thermodynamic and Transport Properties
The determination of thermodynamic and transport properties requires first the calculation of plasma composition that can be obtained either from a chemical nonequilibrium model (e.g., see Table 2) or chemical equilibrium models based on mass action laws or minimization of Gibbs free energies. Thermodynamic properties are directly calculated from the particle number densities of the various species forming the plasma and previous knowledge of the internal partition functions.
Once the composition is known, the computation of heat, mass, and momentum fluxes (see Section 2.2) requires the knowledge of transport properties (see Tables 1 and 2). The calculation is based on solving the Boltzmann integrodifferential equation describing the evolution of the electron energy distribution function (EEDF) by the Chapman-Enskog (CE) method (Ref 35) applied to complex mixtures. This methodology has been analyzed exhaustively by Hirschfelder et al. (Ref 36). Although established for weakly ionized gases, this method has been demonstrated to be valid for thermal plasmas (Ref 37). The distribution function for different species is assumed to be Maxwellian with a first-order perturbation function which is developed in the form of a series of Sonine polynomials: This reduces the Boltzmann equation to a set of linear equations whose solution makes it possible to obtain the gas transport properties. The coefficients of the set of linear equations depend on collision integrals which take into account the binary interaction between colliding species. The computation of these data requires the knowledge of either the interaction potential, which describes the collision dynamics, or the transport cross-sections, which can be derived from differential cross-sections, quantum phase shifts, or experimental data.
Thermal conductivity (see Fig. 4b) is written as the sum of four components (see Eq. 5 in Section 2.2): one term due to the translation of heavy particles, a second due to the translation of the electrons, a third representing the internal thermal conductivity, and the last term corresponding to the reaction thermal conductivity (Ref 38). When H_{2} is added to Ar, the thermal conductivity of the mixture increases with the percentage of H_{2}, especially near the dissociation and ionization temperatures due to the reactive contribution as illustrated in Fig. 4(b).
The numerical treatment of mass diffusion is particularly complicated because a large number of diffusion coefficients [ns(ns − 1)/2 ordinary diffusion coefficients and ns − 1 thermal diffusion coefficients for ns species] has to be considered. To simplify this calculation, Murphy (Ref 6) introduced the treatment of diffusion in terms of gases instead of species (for example, Ar and H_{2} gases, instead of considering Ar, Ar^{+}, Ar^{2+}, H_{2}, H, H^{+}, and e^{−}). Total diffusion coefficients taking into account ambipolar diffusion and temperature or pressure gradients have been proposed by Devoto (Ref 39). The combined diffusion coefficients, very useful for gas mixture modeling, have been computed by Murphy (Ref 40).
NLTE Thermodynamic and Transport Properties
Despite the usefulness of the LTE assumption, one must realize that deviation from LTE is much more the rule than the exception in plasma-based processes. For example, deviations from LTE occur close to the electrodes of the electric arc or in the boundary layer insulating electrically the arc column from the anode wall of a plasma spraying torch. Deviation from LTE also occurs when liquid or solid precursors are injected inside the plasma jet to treat powders or coatings. In that case, the energy distribution function (EEDF) of each kind of particles remains Maxwellian, but the mean kinetic energy may be different for the electrons and heavy particles. The electron temperature T_{e} is, then, higher than the heavy-particle temperature T_{h}, and departure from thermal equilibrium may be characterized by the parameter θ defined as θ = T_{e}/T_{h}.
Similar to LTE assumptions, calculation of the thermodynamic and transport properties requires prior calculation of the two-temperature plasma composition. Nevertheless, generalization of the mass action law and/or Gibbs free energy minimization to NLTE plasma has a long and tumultuous history as evidenced by the different approaches found in the literature (Ref 50).
Figure 8 shows the dependence of viscosity on the electron temperature of an Ar-H_{2} (50 mol%) mixture at atmospheric pressure using compositions calculated by the steady-state kinetic calculation and van de Sanden’s method for θ = 1.6 and θ = 2. Due to the delay in ionization introduced by the kinetic method, large discrepancies can be observed between 8000 and 14,000 K; viscosity continues to increase until the ionization regime is reached. The maximum is therefore shifted to higher temperature with respect to van de Sanden et al.’s method. As in the case of plasma composition, calculations of transport coefficients would also require experimental validation, which is not available yet.
Currently, transport properties in NLTE are available for some pure gases (Ar, O_{2}, N_{2}, H_{2}) and their mixtures (Ar-H_{2}, Ar-O_{2}, Ar-N_{2}) with a simplified theory (Ref 41, 56, 60, 61); the application of the theory proposed by Rat et al. has been already presented for Ar, Ar-He, Ar-Cu, and Ar-H_{2}-He plasmas (Ref 38, 62-64). Air, oxygen, and oxygen-nitrogen transport properties in NLTE have been reported by Gupta et al. (Ref 65) and Ghorui et al. (Ref 60, 66), whereas Colombo (Ref 67) performed calculations for O_{2}, N_{2}, and Ar for electron temperature up to 45,000 K.
Boundary Conditions
Inflow
Inflow boundary conditions are probably the simplest to implement in a DC arc plasma torch simulation. Inflow conditions are typically specified by imposing values of known properties, typically velocity and temperature, over the region where the gas enters the computational domain (e.g., the left-hand side region in Fig. 1). Nevertheless, care must be taken when imposing inflow conditions as the type and number of these has to be consistent with the type and number of outflow conditions, as required for the well-posed formulation of compressible flow problems. This implies that the modeler needs to know/assume beforehand the state (subsonic or supersonic) of the inflow(s) and outflow(s). In some cases, the gas near the inflow and outflow regions can be considered incompressible, which simplifies significantly the imposition of boundary conditions in those regions.
The specification of pressure as inflow or outflow condition in arc plasma torch simulations is particularly cumbersome, especially if the simulation domain only covers the interior of the torch. For incompressible internal flows, pressure is often imposed as an outflow condition, whereas specified velocity is imposed as inflow condition. For compressible flows, pressure can be used as an inflow or outflow condition depending if the flow is subsonic or supersonic. The inflow in a DC arc torch is often incompressible (very-low Mach number), whereas the flow that leaves the torch is certainly compressible, either subsonic or supersonic. The simulation of DC arc plasma torch flows frequently requires to experiment with different sets of inflow/outflow conditions to find the most appropriate and physically sound conditions.
Inflow boundary conditions often involve the description of the gas injection process. Gas can be injected straight (in the direction parallel to the torch axis), radially, tangentially (i.e., with swirl), and often using a combination of the above. Different forms of gas injection seek to impose different characteristics on the plasma flow, e.g., enhance arc constriction or increase gas mixing. The detailed modeling of the gas injection process is highly desirable, but it is often avoided to reduce the computational cost of the simulation. Gonzalez and co-workers presented in Ref 68 an important analysis of the effects of the accurate versus approximated simulation of the gas injection process in the context of a transferred arc torch simulation.
Outflow and Open Boundaries
The outflow boundary in arc torch simulations is typically the torch exit or some other region downstream the arc, e.g., a region within the extent of the plasma jet. Simulations of the jet produced by nontransferred arc torches, which is characterized by complex dynamics due to the arc movement and large velocity and temperature gradients, require special care in the imposition of outflow conditions.
By far, the most frequently used outflow condition in DC arc plasma torch modeling is the imposition of zero gradient of the transported variable. This condition is probably the easiest to implement, but unfortunately it is often too reflective, especially when the flow approaching the boundary varies significantly in time and/or space. Other typically used conditions are the imposition of a constant velocity gradient or zero second-order derivative. Outflow conditions need to ensure the uninterrupted transit of the flow characteristics out of the domain. Typical effects of the use of inadequate outflow conditions are unphysical heating, pressure build-up, and wave reflection. The use of physically sound boundary conditions often prevents the first two effects. But, to prevent wave reflection, typically more sophisticated numerical techniques need to be employed.
Figure 10 shows a time sequence of the dynamics of the arc inside the torch and the plasma jet obtained numerically with a NLTE model, represented by iso-contours of heavy-species temperature, as well as high-speed images of the plasma jet for the same torch and similar operating conditions. The plasma jet presents large-scale structures due to the dynamics of the arc inside the torch (to be explained in Section 5.1), whereas the fine-scale structures are a consequence of the interaction of the jet with the cold surrounding gas. No turbulence model has been employed in those simulation results. The imposition of a sponge zone (i.e., Eq 26) allows the uninterrupted transit of the large and small structures formed by the jet through the boundary.
Walls
For two-temperature models, it is not evident how to define the boundary condition for nonconducting walls for the electron energy conservation equation. Probably the simplest, and the most often used, approach is to define a zero electron temperature gradient condition (e.g., Ref 75).
The specification of electromagnetic boundary conditions is relatively straightforward as it basically consists on specifying the wall as a nonconducting surface, i.e., zero current density.
Anode
The specification of boundary conditions for the anode surface follows the descriptions of the previous section (i.e., no-slip condition), except for the treatment of the energy and electromagnetic boundary conditions.
Plasma flows typically develop what are called plasma sheaths near the electrodes. There are large property variations within these regions that often are negligible within the bulk plasma, like charge accumulation and thermodynamic nonequilibrium. The anode sheath thickness is on the order of a few Debye lengths, where the Debye length is a measure of the charge screening felt by a charged particle due to the other particles (Ref 1, 76). For thermal plasmas, the Debye length is often very small compared to the characteristic length of the flow (e.g., the torch diameter). This causes that the anode influence on the flow is localized very close to the anode surface.
The boundary conditions at the anode surface for the electromagnetic fields often consist of imposing a reference value of electric potential (e.g., \( \upphi_{\text{p}}=0 \) along the anode surface), whereas the total amount of current transferred is determined by the cathode boundary condition, as explained in the next section. An improved approach consists of including part of the electrodes in the computational domain, and hence solving the energy conservation and electromagnetic equations through the domain conforming the electrodes (e.g., see Ref 77).
When no sheath model is used in a LTE model, due to the thermodynamic equilibrium assumption, the electron temperature is equal to the heavy-particle temperature, which is low (i.e., less than 1000 K) near the electrodes due to the intense cooling they experience, especially near the anode surface. Hence, the equilibrium electrical conductivity of the plasma, being mostly a function of the electron temperature, is extremely low (i.e., less than 0.01 S/m for most gases), which limits the flow of electrical current through the plasma-electrode interface. To allow current continuity through the plasma-anode interface without the use of a sheath model or a NLTE model, an alternative used in Ref 74, 80, 81 is to let the temperature remain high enough (i.e., above 7000 K) all the way up to the anode surface. The clear advantage of this approach is that it is consistent with the LTE assumption in the sense that the heavy-particle temperature remains equal to the electron temperature (both assumed equal to the equilibrium temperature), which remains high all the way up to the anode surface. Another approach consists of specifying an artificially high electrical conductivity in the region immediately adjacent to the electrodes (Ref 72, 82-84). The value of this “artificial” electrical conductivity used in the literature is somewhat arbitrary. The only requirement for its value is that it needs to be high enough to ensure the flow of electrical current from the plasma to the electrodes. This latter model lets the arc reattach whenever it gets in contact, or “close enough”, to the anode surface at an axial location which is more thermodynamically favorable, i.e., a location that produces a configuration of the arc with a lower total voltage drop. However, this type of formation of a new attachment is different from the reattachment process described in Section 5. Important studies of the effect of the anode modeling in an arc plasma flow were performed by Lago et al. (Ref 23) and later expanded to three-dimensional modeling in Gonzalez et al. (Ref 85). Their models included the effect of metal vapor on the arc and melting of the anode, detailed heat transfer between plasma and anode (similar to Eq 28), and radiative transfer using view factors.
Cathode
Cathodes are the source of electrons in thermal plasma torches. The cathode in DC arc plasma torches used for plasma spraying is thermionic; that is, the electrons are emitted as a consequence of the high temperature of the cathode. The region in front of the cathode can be divided into two distinctive parts: the ionization region and the space charge sheath. Similar to the anode region, these regions are very small compared to the characteristic length of the flow. Typically, there is a considerable voltage drop in this thin layer and considerable power is deposited in it. This power is a consequence of the balance between the energy flux of ions and electrons from the plasma to the cathode surface and the heat removed by the electrons leaving the cathode (Ref 88).
The accurate modeling of the cathode region is quite involved due to the variety of chemical and electrical phenomena taking place. Furthermore, it has been shown that evaporation of the cathode material can have a significant effect on the plasma flow dynamics (Ref 89). Indeed, the metal vapor increases significantly the electrical conductivity of the plasma in front of the cathode, which causes constriction of the arc. These effects have to be added to the stability of the cathode spot (the region with highest current density), which is often of primary importance in cathodes whose geometry does not favor a preferred spot (see Ref 90 for detailed modeling of cathode spot stability).
The large computational cost associated to the self-consistent modeling of the electrode regions and the plasma flow has prevented them to be widely used in arc plasma torch simulation. A distinctive example of the coupled modeling of electrodes and thermal plasma flow in an industrial application is the work of Paul et al. (Ref 91) of the simulation of the arc discharge in a HID lamp. In their model, the cathode region is modeled using the nonlinear surface heating model of Benilov and Marotta (Ref 88). The work by Li and Benilov (Ref 92) of the coupled simulation of the arc and cathode region revealed that the electric power deposited into the cathode region is transported not only to the cathode but also to the arc column.
The need to reduce the computational cost of the modeling of the electrode regions in industrial thermal plasma flows has motivated the development of different sheath models. These models try to describe in a simplified manner the main physical effects that dominate the electrode-plasma interface. Particularly significant is the unified approach developed by Lowke et al. (Ref 93) which, when applied to the modeling of the cathode, does not require the specification of a current density profile.
As boundary conditions for the energy conservation, often a specified equilibrium or heavy-species temperature distribution is imposed over the cathode surface (where the highest temperature is usually assumed close to the melting point of the cathode material, e.g., ~3600 °C for tungsten), whereas a zero-gradient condition is imposed for the electron temperature. These conditions are rough approximations, and when possible, an adequate cathode region model should be employed.
Arc Reattachment Models
Operating Modes in Nontransferred Arc Torches
Steady: Characterized by negligible voltage fluctuations and, correspondingly, an almost fixed position of the arc attachment. This mode is not desirable due to the rapid erosion of the anode.
Takeover: Characterized by (quasi-) periodic fluctuations of voltage drop and a corresponding (quasi-) periodic movement of the arc. The spectrum of the voltage signal presents several dominant frequencies. This operating mode is the most desirable because it allows an adequate distribution of the heat load to the anode, and produces well-defined arc fluctuations.
Restrike mode: Characterized by a highly unstable, relatively unpredictable movement of the arc and quasi-chaotic, large amplitude, voltage fluctuations. An arc operating in this mode is very unstable and relatively unpredictable; the arc reattachment phenomenon plays a dominant role in the arc dynamics.
For a given torch, the flow can change from the steady mode, to the takeover, and then to the restrike mode as the mass flow rate is increased or as the total current is decreased; or more precisely as the enthalpy number N_{h} increases (i.e., N_{h} is proportional to the mass flow rate and inversely proportional to the total current squared). Therefore, the operating conditions determine the dynamics of the arc inside the torch.
Arc Reattachment Process
To mimic the physical reattachment process, a model mainly faces the questions of where and how to introduce the new attachment. Where to locate the new attachment translates into the definition of an adequate breakdown condition, whereas how to introduce the reattachment translates into the definition of adequate modifications of the flow field to mimic the formation of an attachment.
The detailed modeling of this process is unfeasible with actual computational methods and computer power, especially when the reattachment process is part of the modeling of a realistic plasma application. The work by Montijn et al. (Ref 99) is a notable example of the simulation of streamer propagation. A similar model would have to be integrated to an arc flow simulation to realistically simulate the arc reattachment phenomenon. Therefore, for the simulation of industrial thermal plasma applications, approximate models are needed which imitate the effects of the reattachment process within the flow field. Sections 5.3 and 5.4 describe two approaches that have been successfully applied to the simulation of commercial plasma spray torches.
Conducting Channel Reattachment Model
Because of the free parameter E_{b}, this reattachment model cannot predict the operating mode of the torch. The model can only predict the arc dynamics inside a torch operating under given operating conditions and a given value of E_{b}. If the model is used to simulate a torch operating under conditions leading to a takeover (or steady) mode, a high enough value of E_{b} should be used to ensure that a restrike-like reattachment does not occur. Otherwise, the voltage signal obtained could resemble that of the restrike mode.
Hot Gas Column Reattachment Model
- (i)
The thickness δ of the boundary layer that covers the anode surface is defined by the thickness between the region of the flow with an electrical conductivity lower than 150 S/m and the anode surface. For Ar-H_{2} plasma-forming gas, the value of 150 S/m corresponds to a critical temperature T_{c} (see Section 3.1, Fig. 4) under which the plasma gas acts as an insulating layer.
- (ii)
The electric field between the edge of the arc column and the anode wall \( E_{\ell } = (\upphi_{0} - V_{\text{a}} )/\updelta \) (where \( \upphi_{0} \) is the potential in the plasma and V_{a} is the potential at the anode surface) is calculated in the whole boundary layer and compared with a critical field E_{b}, under which no breakdown process can occur. A value of E_{b} of around few 10^{4} V/m had been chosen because it is now well established that the critical electric field is decreased by one order of magnitude when temperatures higher than 3000 K are encountered.
- (iii)
When the value E_{b} is reached at a particular location M, a short circuit occurs and a new arc attachment at the nozzle wall appears.
A simple model is used for the ignition of a new arc root attachment by re-arcing. It consists of imposing a high-enough gas column temperature that connects the arc column fringe to the anode wall at the location where the electric field was found to be greater than E_{b}.
Each breakdown is associated with a negative jump of the voltage. In the case shown in Fig. 16, the latter was found to be 20 ± 5 V. The peak occurs at intervals of about 80 μs, i.e., at an average frequency of about 13 kHz. The predicted time-average torch voltage is about 65 V, close to the actual one of 60 V experimentally measured. The results reveal that decreasing the arc current intensity or increasing the plasma gas flow rate results in an increase of the average boundary layer thickness δ, favoring higher voltage jump amplitude \( \Updelta\upphi\) (Ref 81, 100). As mentioned by Trelles et al. (Ref 84), tuning the value of E_{b}, the principal parameter in the reattachment model, permits to either match the frequency or the voltage jump amplitude, the other quantity moving in the opposite manner.
Reattachment in Nonequilibrium Models
Interestingly, the NLTE simulations (which use no reattachment model) display the growth of a high-temperature appendage (see arrows in the T_{h} and T_{e} plots in Fig. 17) in a region upstream after the point of maximum total voltage drop (which could be correlated with the maximum value of electric field). The formation of the high-temperature appendage seems to be driven by high values of the local effective electric field and high values of electron temperature. Even though swirl injection is used, due to the relatively short time scale of the reattachment process, the arc reattaches at almost the opposite side of the original attachment (Ref 101). The reattachment process occurs in a natural manner mimicking the steady and/or takeover modes of operation of the torch. It must be noted that arcs in pure argon as simulated here rarely display a restrike behavior because the boundary layer is usually rather thin. It is expected that nonequilibrium simulations of the restrike mode will require the use of a reattachment model to produce more accurate results.
As explained previously (Section 4.4), in a LTE model, a reattachment can occur either due to the application of the reattachment model (i.e., when the breakdown condition is satisfied) or due to the arc dynamics causing the arc to get “close enough” to the anode. The growth of a high-temperature region from the arc column toward the anode can be observed in the LTE results in Fig. 17, which eventually initiates the formation of a new attachment. Moreover, the application of the reattachment model clearly disrupts the flow significantly.
Conclusions and What’s Next
Great progress has been achieved in the simulation of DC arc torches. We have reached a state in which three-dimensional and time-dependent simulations with detailed geometry description of industrial torches are reaching widespread use. These simulations have helped to achieve a better understanding of the operation of DC arc torches and have sometimes led to improved torch designs and plasma processes. However, these simulations still lack of complete predictive power, especially for the simulation of nontransferred arc torches. Several improvements could be achieved with higher computing power and massive use of parallel computing. But most importantly, the complexity of the models needed to describe the different processes that take place has been the limiting factor for more detailed and accurate plasma torch simulations. The ready availability of different “physics modules” in commercial CFD software has eased the incorporation of several physical processes and a wider range of models into plasma torch modeling. Unfortunately, the models in commercial CFD software are most often not implemented with thermal plasma flow applications in mind, and therefore, often rely on unrealistic assumptions (e.g., turbulence models that assume constant thermodynamic and transport properties, neglect of electromagnetic forces, etc.). Nevertheless, users of commercial CFD software frequently need to develop user-defined routines to be integrated into their simulations to account for the missing physical models. In this regard, the use of research, in-house, developed software generally provide the most faithful models for plasma torch simulation (e.g., electrode boundary models, electromagnetic equations). To the previous exposition, we have to add that several processes involved in the description of the plasma flow are not yet understood at the level that accurate models are available. A paramount example of this is the initiation of arc reattachment phenomena.
Widespread use of thermodynamic and chemical nonequilibrium models. These nonequilibrium models necessarily need to use sound values of nonequilibrium thermodynamic and transport properties. The accurate modeling of these properties still represents a big challenge, both in terms of implementation and on computational cost.
Incorporation of the electrodes into the computational domain. This would have a significant effect on the boundary conditions for electromagnetic equations.
Detailed modeling of electrodes and electrode processes, particularly heat transfer mechanisms and electric current flows. Furthermore, surface chemistry and phase change phenomena (e.g., electrode material evaporation, anode erosion) should be incorporated into these models.
Use of more faithful geometry representations. This is particularly important for the analysis of commercial plasma torches and for their design optimization.
In the case of nontransferred arc torches, more physical and mathematically sound models of the arc reattachment process are needed. It is reasonable to expect that the incorporation of such models into a thermodynamic nonequilibrium plasma torch simulation will be able to reproduce the steady, takeover, and restrike modes of operation without the need for tuning parameters.
Regarding the modeling of turbulence, DNSs would be highly desirable, especially to understand the mechanisms for turbulence formation inside the torch, particularly the role of fluid-dynamic, thermal, and electromagnetic instabilities and the arc reattachment process. DNS data would also guide the development of turbulence models (LES, RANS, and DES) suitable for thermal plasma flow simulations. Furthermore, detailed turbulence modeling would be of great benefit for the understanding of plasma-powder interaction, especially for ultrafine and nano-scale powders, as these processes are influenced by the fine-scale structures of the flow. Additionally, detailed comparison of simulation results against experimental measurements of turbulent flow characteristics (e.g., correlations, mean quantities, dissipation rates) is required to validate any turbulent thermal plasma flow model.
Rigorous validation with experimental data. The recent availability of high-fidelity three-dimensional and often time-dependent experimental data, such as the analysis of the anode attachment region in Ref 102, or of the plasma jet in Ref 103, raises the quality, validity, and resolution expected from numerical simulations.
We expect that direct current arc plasma torch modeling will be playing an increasingly important role in the design of thermal plasma processes. Several industrial applications will obtain better yields, higher efficiencies, and improved economical advantage thanks to the systematic use of numerical simulations to guide and/or aid the design and optimization of their processes.
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