Non-transferred Arc Torch Simulation by a Non-equilibrium Plasma Laminar-to-Turbulent Flow Model
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Abstract
Non-transferred arc torches are at the core of diverse industrial applications, particularly plasma spray. The flow in these torches transitions from laminar inside the torch to turbulent in the emerging jet. The interaction of the plasma with the processing gas leads to significant deviations from local thermodynamic equilibrium (LTE) far from the arc core. The flow from a non-transferred arc plasma spray torch is simulated using a non-LTE (NLTE) plasma flow model solved by variational multiscale (VMS) and nonlinear VMS (VMS_{n}) methods, which are suitable for unified laminar and turbulent flow simulations. Non-plasma turbulent jet simulations indicate that the VMS_{n} method produces results comparable to those by the dynamic Smagorinsky method, often considered the workhorse for turbulent incompressible flow simulations. VMS and VMS_{n} approaches are applied to the simulation of incompressible, compressible, and NLTE plasma flows in non-transferred arc torch operating at representative conditions found in plasma spray processes. The NLTE plasma flow simulations reproduce the dynamics of the arc inside the torch together with the evolution of turbulence in the produced plasma jet in a cohesive manner. However, the similarity of results by both methods indicates the need for numerical resolution significantly higher than what is commonly afforded in arc torch simulations.
Keywords
incompressible–compressible flow non-transferred arc plasma torch two-temperature model turbulence turbulent free jet variational multiscaleIntroduction
Non-transferred Arc Plasma Torches
Direct-current (DC) non-transferred arc plasma torches are at the core of diverse technological applications such as metallurgy, spheroidization, chemical and particle synthesis, waste treatment, and particularly plasma spraying. Non-transferred arc torches have been the workhorses of plasma spraying processes for a wide range of applications, particularly for the deposition of thermal barrier (about 25% of the plasma spray market) and wear-resistant coatings. Even though a great amount is known about the operation of non-transferred arc plasma torches thanks to numerous computational and experimental means (e.g., Ref 1, 14), the need for improved spraying process performance, efficiency, as well as novel technological developments, such as liquid and suspension plasma spraying, prompts the need for greater understanding of these devices. Physics-based computational models can provide insight into process characteristics practically inaccessible through experimental means, as well as guide equipment design and process development.
Phenomena in Non-transferred Arc Torches
The plasma in non-transferred arc torches is typically considered as thermal plasma. Thermal plasmas are characterized by high collision frequencies among electrons and heavy species (molecules, atoms, ions), which cause them to be in a state of local thermodynamic equilibrium (LTE). Under LTE, all species share the same Maxwellian velocity distribution characterized by a single temperature (Ref 3). Nevertheless, the LTE state is generally found in the arc core only. The interaction of the plasma with its surrounding environment causes large variations in the degree of ionization, from non-ionized within the working gas to fully ionized in the plasma core, as well as large velocity, temperature, and density gradients. Such interactions cause significant deviations from LTE, leading parts of the flow to a state of thermodynamic non-equilibrium (non-LTE or NLTE). Under NLTE, electrons and heavy species have different Maxwellian velocity distributions (Ref 2-4) and therefore are characterized by different temperatures (Ref 5).
In addition to the occurrence of NLTE, the plasma flow in non-transferred arc torches experiences compound coupling among fluid dynamics, heat transfer, chemical kinetics, and electromagnetic phenomena. This multiphysics coupling leads to intricate flow dynamics and makes the flow inherently multiscale; that is, distinct phenomena are observed through different regions of the flow domain, e.g., from the thickness of boundary layers and plasma–gas interaction fronts, to the amplitude of instabilities along the jet. Moreover, the flow in arc torches is prone to the development of different types of instabilities, and it often transitions from laminar inside the torch to turbulent in the emerging jet.
Turbulent flow behavior enhances the exchange of species mass, momentum, and energy and therefore alters all intervening transport processes. The strength of turbulence in plasmas depends on their thermodynamic state (Ref 6); particularly, as indicated by Volkov (Ref 7), the amplitude of turbulence in plasmas increases as the plasma gets farther from thermodynamic equilibrium. From a practical perspective, turbulence in non-transferred arc torches can have a determining effect in the aimed process. Sometimes turbulence is desirable in processes such as surface treatments and nanoparticle synthesis (Ref 8) that benefit from the enhanced flow dissipation. For other processes, particularly for plasma spraying, the enhanced mixing from the surrounding environment caused by turbulence can lead to undesirable oxidation (Ref 9). Moreover, plasma–feedstock interactions are markedly important in emerging processes, particularly in liquid feedstock and suspension plasma spraying (Ref 10-13). The liquid-base droplets and resulting very fine particles are very sensitive to any perturbation in the jet, as they can drastically enhance mixing and heat transfer and therefore markedly affect the evolution of droplets and particles and their subsequent transfer by condensation into the coating. An overview of the challenges in understanding brought forward by novel and emerging plasma spray technologies, particularly of the interaction between the plasma flow and the feedstock, is discussed in the review article by Vardelle et al. (Ref 10).
Significant understanding related to turbulence in arc plasma torches has been achieved to date. For instance, Pfender et al. (Ref 11) experimentally investigated the turbulent structures of DC thermal plasma jets. They observed that the plasma core is generally not turbulent (i.e., laminar), whereas shadowgraphs revealed significant turbulence in the jet enhanced by fluctuations of the jet core due to the arc dynamics. Hlína and Šonský (Ref 12-15) investigated the dynamic behavior of the core of a plasma jet using three-dimensional optical diagnostics and image-processing techniques. Complementing experimental observations, numerical findings have revealed that high-temperature plasma regions exhibited less turbulent structures with large eddies (i.e., coherent regions with high vorticity), whereas low-temperature regions tend to be more turbulent with numerous small eddies. Shigeta (Ref 8) showed that the turbulent nature of the emerging plasma jet depends on the dynamics of the arc inside the torch, which cause major fluctuations in the jet core, and that the high temperature of the plasma produces a re-laminarization effect. Therefore, comprehensive arc plasma torch simulations have to be able to reproduce the transition from laminar flow inside the torch to turbulent flow in the plasma jet naturally, that is, as a consequence of the combined effects of the arc dynamics and the inherent development flow instabilities, without artificial or ad hoc artifacts. An important review and perspective on the modeling of turbulence in thermal plasma flows, including pressing challenges, current efforts, and state-of-the-art simulations, is presented by Shigeta (Ref 8).
Turbulent Thermal Plasma Flow Simulation
The computational simulation of the operation of an arc plasma torch requires the coupled solution of the set of transport and electromagnetic equations that describe the plasma flow, together with adequate constitutive relations and material properties (Ref 16). The multiphysics nature of plasma flows, their inherent nonlinearity, and large variation of material properties, together with their propensity to unstable and turbulent behavior, make their computational simulation exceedingly challenging. The modeling and simulation of turbulent plasma flows have traditionally relied on the use of models developed for other types of flows, particularly incompressible flows. Turbulence modeling is one of the most important issues in a wide range of computational fluid dynamic (CFD) applications. The main methods for the modeling and simulation of turbulent flows are divided among direct numerical simulation (DNS), Reynolds-averaged Navier–Stokes (RANS), and large eddy simulation (LES) techniques.
DNS seeks to resolve all the characteristics of the flow throughout all intervening scales, up to the smallest turbulent eddies. DNS provides the greatest accuracy but also have the greatest computational cost. The exploration via DNS of turbulent thermal plasma flows is exceedingly expensive, or even unachievable, due to the large range of scales that need to be resolved, which has prevented its use for the analysis of thermal plasma applications, particularly of plasma spraying.
RANS approaches use model equations that describe the time-averaged distributions of turbulent flow fields. RANS have a drastically reduced computational cost with respect to DNS, but at the expense of the reliance of ad hoc models and parameters. The severe modeling approximations used by RANS models limit their applicability to very-well-understood and specific flow problems. Nevertheless, their lowest computational cost makes them the outmost preferred approach for industrial flow simulations. Despite their severe modeling limitations, RANS approaches have been extensively adopted for analyses of numerous thermal plasma flows, including arc plasma torches. By far the RANS model most favored by researches has been based on the k − ε (k-epsilon) turbulence model first developed by Launder and Spalding (Ref 17). Examples of the use of k − ε models for the analysis of arc plasma torches are the work of Bauchire et al. (Ref 18), McKelliget et al. (Ref 19), Li and Chen (Ref 20), Li and Pfender (Ref 21), Huang et al. (Ref 22, 23), Yuan et al. (Ref 24, 25), and Gu et al. (Ref 26). Other RANS models applied to thermal plasmas include the Spalart–Allmaras model, which was used by Sahai et al. (Ref 27) to simulate an arc heater.
LES is a coarse-grained turbulent modeling and simulation approach that seeks to resolve the large-scale features of the flow and model the small-scale ones, which are expected to depict somewhat universal characteristics. LES provides a level of accuracy and cost somewhere in between DNS and RANS approaches, and is nowadays considered de facto standard for the exploration of turbulent flows. The vast majority of LES techniques rely on two basic assumptions, namely the suitability of spatial filters to separate scales into large and small, and the adequacy of the use of an eddy viscosity to model the redistribution of momentum from the small scales. The effects of anisotropy and nonlinearity, as well as the interaction of different physical phenomena in plasmas generally, invalidate those assumptions. Therefore, traditional LES approaches cannot, in general, be expected to provide comprehensive descriptions of turbulent plasma flows. Nevertheless, due to the absence of comprehensive LES approaches for thermal plasmas, traditional LES strategies have been adopted in thermal plasma flow simulations. For example, Colombo et al. used LES to investigate the complex unsteady 3-D turbulent flow in an inductively coupled thermal plasma torch together with its attached reaction chamber (Ref 27). Colombo et al. (Ref 26) and Ghedini and Colombo (Ref 28) employed LES to study the flow in a DC non-transferred arc plasma torch including the effect of particle injection. Furthermore, Caruyer et al. (Ref 29) used LES for the modeling of the unsteadiness and turbulence in the plasma jet from plasma torches.
A comprehensive coarse-grained plasma flow modeling and simulation method should be complete and consistent; that is, it should be free of ad hoc parameters or procedures and should be able to reproduce laminar-to-turbulent transitions naturally, and its solution has to approach that by DNS as the numerical accuracy is increased. Recently, Modirkhazeni and Trelles (Ref 30, 31) proposed the variational multiscale-n (VMS_{n}) approach for the coarse-grained simulation of general transport problems, which is particularly suited for highly nonlinear problems such as turbulent plasma flows. VMS_{n} is based on the VMS framework (Ref 30, 32, 33), which has proven to be very versatile and robust for the solution of diverse multiphysics problems. The method uses variational scale decomposition together with a residual-based approximation of the small scales, circumventing the need for empirical parameters. The VMS_{n} method addresses up-front the nonlinear inter-dependence between large and small scales due to the high nonlinearity in plasma flow models, distinctly exemplified in NLTE models. In this paper, the VMS and VMS_{n} methods are evaluated for the comprehensive simulation of the flow from a DC non-transferred arc plasma spray torch.
Non-equilibrium Plasma Flow Model
Balance Equations
Fluid models, which are based on the continuum assumption and describe the evolution of the moments of the Boltzmann transport equation (Ref 34), provide appropriate descriptions of plasmas when the constituent particles experience relatively high collision frequencies, as typically found under atmospheric pressure conditions. In this paper, the plasma is considered as a nonrelativistic, non-magnetized, quasi-neutral fluid in chemical equilibrium and thermodynamic non-equilibrium (two-temperature NLTE).
Set of fluid—electromagnetic equations comprising the non-equilibrium plasma flow model; for each equation: Transient + Advective − Diffusive − Reactive = 0
Equation | Transient | Advective | Diffusive | Reactive |
---|---|---|---|---|
Conservation total mass | \(\partial_{t} \rho\) | \({\mathbf{u}} \cdot \nabla \rho + \rho \nabla \cdot {\mathbf{u}}\) | 0 | 0 |
Conservation momentum | \(\rho \partial_{t} {\mathbf{u}}\) | \(\rho {\mathbf{u}} \cdot \nabla {\mathbf{u}} + \nabla p\) | \(\begin{aligned} \nabla \cdot \mu (\nabla {\mathbf{u}} + \nabla {\mathbf{u}}^{\text{T}} ) - \hfill \\ \nabla \cdot (\tfrac{2}{3}\mu (\nabla \cdot {\mathbf{u}}){\mathbf{\delta )}} \hfill \\ \end{aligned}\) | \({\mathbf{J}}_{q} \times {\mathbf{B}}\) |
Thermal energy heavy species | \(\rho \partial_{t} h_{h}\) | \(\rho {\mathbf{u}} \cdot \nabla h_{h}\) | \(\nabla \cdot (\kappa_{hr} \nabla T_{h} )\) | \(\begin{aligned} \partial_{t} p_{h} + {\mathbf{u}} \cdot \nabla p_{h} + K_{eh} (T_{e} - T_{h} ) \hfill \\ - {\mathbf{\uptau}}:\nabla {\mathbf{u}} \hfill \\ \end{aligned}\) |
Thermal energy electrons | \(\rho \partial_{t} h_{e}\) | \(\rho {\mathbf{u}} \cdot \nabla h_{e}\) | \(\nabla \cdot (\kappa_{e} \nabla T_{e} )\) | \(\begin{aligned} \partial_{t} p_{e} + {\mathbf{u}} \cdot \nabla p_{e} - K_{eh} (T_{e} - T_{h} ) - 4\pi \varepsilon_{r} \hfill \\ + {\mathbf{J}}_{q} \cdot ({\mathbf{E}} + {\mathbf{u}} \times {\mathbf{B}}) + \tfrac{{5k_{B} }}{2e}{\mathbf{J}}_{q} \cdot \nabla T_{e} \hfill \\ \end{aligned}\) |
Conservation charge | 0 | 0 | \(\begin{aligned} \nabla \cdot (\sigma \nabla \phi_{p} ) - \hfill \\ \nabla \cdot (\sigma {\mathbf{u}} \times (\nabla \times {\mathbf{A}})) \hfill \\ \end{aligned}\) | 0 |
Magnetic induction | \(\mu_{0} \sigma \partial_{t} {\mathbf{A}}\) | \(\begin{aligned} \mu_{0} \sigma \nabla \phi_{p} - \hfill \\ \mu_{0} \sigma {\mathbf{u}} \times (\nabla \times {\mathbf{A}}) \hfill \\ \end{aligned}\) | \(\nabla^{2} {\mathbf{A}}\) | 0 |
In the energy conservation equations, κ_{h} and κ_{e} denote the heavy species and the electron translational thermal conductivities, respectively, and \(\kappa_{hr}\) is the translational-reactive thermal conductivity; K_{eh} is the electron–heavy-species kinetic energy exchange coefficient; and ε_{r} is the effective net emission coefficient. The NLTE model is described in greater detail in Ref 1, 3, 30, 35-37.
The description of the plasma flow model as a generic TADR transport system is particularly appealing due its simplicity and the uniform handling of different physical phenomena (Ref 39). The TADR system describing the NLTE plasma flow model is expressed in so-called residual form by:
Other sets of variables could have been chosen, such as the set of conservation fluid variables \([\begin{array}{*{20}c} \rho & {\rho {\mathbf{u}}} & {\rho h_{h} } & {\rho h_{e} } & {\phi_{p} } & {\mathbf{A}} \\ \end{array} ].\) The use of primitive variables has been motivated by prior work (Ref 38) that demonstrated their advantages, especially for the unified handling of incompressible and compressible flows by a single formulation. Particularly, the fully coupled approach used here does not require the modification of the mass conservation equation as an equation for pressure p (e.g., by its modification as a Poisson equation, or by the addition of penalty terms, or artificial sound speeds).
Material Properties and Constitutive Relations
The equations in Table 2 are complemented with the definition of thermodynamic (e.g., ρ, p_{h}, p_{e}, h_{h}, h_{e}) and transport (e.g., κ_{h}, κ_{e}) properties, which add further coupling among model variables and are highly nonlinear and computationally expensive to compute, especially for NLTE models (Ref 39, 40). Thermodynamic properties link thermodynamic variables to the set of primitive variables, whereas transport properties are required for the modeling of diffusive fluxes. In typical arc plasma torches, temperatures vary from ≈ 300 K within the stream of working gas to ≈ 25 kK or higher within the arc core. Such large range implies that material properties typically vary by several orders of magnitude within a given thermal plasma flow (Ref 1, 41). In this work, argon is used as the working gas and gas environment.
The calculation of the thermodynamic properties requires knowledge of the thermochemical state of the plasma. Given the chemical equilibrium and thermodynamic non-equilibrium assumed here, the plasma state is defined in terms of the temperatures T_{h} and T_{e}, and the total pressure p. The composition is determined from them using the mass action laws, Dalton’s law of partial pressures, and the quasi-neutrality condition. Knowledge of the chemical state allows the evaluation of the thermodynamic properties ρ, h_{h} and h_{e} following standard procedures from thermodynamics.
Transport properties viscosity μ, thermal conductivities κ_{hr}, κ_{e}, and electrical conductivity σ for argon are computed by a table lookup procedure as function of T_{e} and T_{h} assuming p = 1 atm using data from Ref 42. Radiative energy transfer has a primary role in plasma flows, e.g., it is the dominant energy transport mechanism for temperatures above 30 kK (Ref 3). The direct solution of the radiative transport equation (RTE) is exceeding expensive, 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 photons with the intervening media (inside and outside the torch). This last characteristic jeopardizes the use of models that rely on strong coupling like diffusion-like models 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).
Probably one of the best radiation transfer simulations applied to a thermal plasma flow is the work of Menart et al. (Ref 43) who used a DOM for a large set of wavelengths. Because their work was focused on analyzing radiative energy transfer, Menart et al. did not solve the radiative transport coupled to the plasma flow model. Instead, they used a pre-calculated temperature field to solve the RTE. Their approach is justified by the enormous computational cost required to solve the plasma flow together with radiative transport. An alternative approach, valid mainly when the plasma can be assumed optically thin, 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 44) for the simulation of a free-burning arc.
A detailed description of radiative transport requires solving the RTE together with proper consideration of emission and absorption processes, which make radiation modeling exceedingly involved and computationally expensive (Ref 45). Nevertheless, the form in which radiation directly interacts with the plasma flow (i.e., as a reactive term) suggests that detailed description of radiation transfer is not essential and that direct approximations of the \(\nabla \cdot {\mathbf{q}}_{r}\) term could be used, where q_{r} is the radiative heat flux. In this regard, the most common approximation in thermal plasma modeling is the use of the effective net emission approximation \(\nabla \cdot {\mathbf{q}}_{r} \ne 4\pi \varepsilon_{r}\) (Ref 46-48), where the coefficient ε_{r} is a function of the state of thermodynamic state of the plasma and a given effective absorption radius. The latter represents the radius of a sphere in which the emitted radiation can be re-absorbed; outside of this sphere, the emitted radiation leaves without further interaction with the plasma. (Hence, the optically thin approximation implies that this radius is zero). The net emission approach is particularly appealing for plasma flow simulation because ε_{r} can be treated as any other material property. For the argon plasma studied in this paper, the net emission coefficient for argon was computed as ε_{r} = ε_{r}(T_{e}) using a lookup table procedure based on the data reported in Ref 46 assuming optically thin plasma (i.e., absorption radius equal to zero), which can overestimate radiative energy losses and hence significantly reduce the temperature within the core of the plasma, but should have a minor effect within the jet due to its significantly lower temperature.
Numerical Model
Variational Multiscale-n Formulation
The complex coupling among model variables, together with the high nonlinearity and large variation of material properties, makes numerical counterparts of the NLTE plasma flow model very stiff. Numerical stiffness is a somewhat broad concept related to largely disparate temporal and/or spatial characteristics of the solution and is more directly related to the multiscale nature of the problem. The numerical solution of the NLTE plasma flow model given by Eq 1 is based on variational multiscale (VMS) framework (Ref 33, 34). VMS methods effectively address the challenges associated with multiscale problems, which makes them ideal for the solution of transport problems. Particularly, the VMS_{n} method constitutes an extension of the classical VMS approach to directly account for the intricate nonlinear coupling among large and small scales, as particularly found in turbulent flows.
Using the scale decomposition, the weak form of the transport problem leads to two separate problems: one for the large scales and another one for the small scales, i.e.,
Solution of the Small-Scale Problem
The VMS_{n} formulation has been implemented in the TPORT (TransPORT) solver (Ref 56). TPORT is designed for solving general systems of TADR equations of the form of Eq 1. TPORT is written in C++ and performs parallel processing in shared-memory architectures using Open Multi-Processing (OpenMP) and in distributed-memory systems using PETSc. In TPORT, the solution to Eq 8 and 9 is perused by using a version of a second-order generalized-alpha predictor multicorrector time-stepper method developed by Jansen and collaborators (Ref 57) together with the globalized inexact Newton–Krylov nonlinear solver approach of Eisenstat and Walker (Ref 58). The TPORT code has been extensively used for the simulation of scalar transport, incompressible and compressible flow problems, radiative transport, and equilibrium (LTE) and non-equilibrium (NLTE) plasma flows (e.g., see Ref 30-33, 41 and references therein).
Validation of VMS_{n} Method with Turbulent Incompressible Free Jet
Turbulent Free Jet
The boundary conditions are defined as follows: for the wall boundary: ∂_{n}p = 0 and u = 0; for inlet: ∂_{n}p = 0 and u = [0 0 U_{in}]; and for outlet: p = p_{atm} and ∂_{n}u = 0. U_{in} is the average velocity at the inlet and p_{atm} = 1.01325 [10^{5} Pa] represents atmospheric pressure. The working fluid is argon. In order to investigate the capabilities of the VMS_{n} method under different levels of turbulence, two values of Reynolds number \(Re = \rho U_{in} D/\mu\) are considered, i.e., Re = 5142 and Re = 20,000. Both values of Re have been extensively studied in the literature, and according to Ref 59, 60, both Re lead to turbulent jets. For each Re, simulations are performed using VMS_{n}, VMS, and the dynamic Smagorinsky [LES, presented by Germano et al. (Ref 61) and Lilly (Ref 62)] methods. The dynamic Smagorinsky method is often considered the workhorse for turbulent incompressible flow simulations; its use here is intended to provide a benchmark against the VMS and VMS_{n} results. It is to be noted that the dynamic Smagorinsky simulations here were run using the commercial software ANSYS Fluent (Ref 63) (in contrast to TPORT for the VMS and VMS_{n} results).
All the simulations in this work used spatial discretizations based on trilinear basis functions (i.e., 6-node hexahedra). For simulations with Re = 5142, the computational grid consists of ~ 230 k nodes and ~ 235 k elements with D = 0.0088 m, L = 0.3 m, and W = 0.088 m, whereas for Re = 20,000, the grid is made of ~ 245 k nodes and ~ 251 k elements, D = 0.0088 m, L = 0.35 m, and W = 0.08 m. Both grids are structured with a non-uniform distribution of elements. In order to reduce the computational time to observe the evolution of the flow into a turbulent state, the inflow velocity profile is modified with random spatial perturbations with a standard deviation of 5% of the magnitude of the average inlet velocity (5%U_{in}), which was constant in time for the VMS and VMS_{n} simulations, and time dependent for the dynamic Smagorinsky simulations (default setting for LES approaches in Fluent).
The instantaneous results in Fig. 3 show that for both Re numbers, the jet appears to be significantly more laminar for the VMS method compared to those for the dynamic Smagorinsky and VMS_{n} methods. The amount of turbulent dissipation produced by each method is better contrasted by the time-averaged results of Fig. 3, which indicates that the locations where the axial velocity decays to 80% of U_{in} are significantly lower for the VMS_{n} method than for VMS, for both Re = 5142 and Re = 20,000; the results for VMS_{n} are very similar to those for the dynamic Smagorinsky. Therefore, for the same computational domain and Re, VMS_{n} is more capable of capturing the turbulent characteristics of the flow than VMS, and its accuracy is comparable to that by the dynamic Smagorinsky method. As expected, more turbulent behavior in the flow is observed for the larger values of Re. Figure 3 also shows that the difference between the locations in which the axial velocity decays to 80%\(U_{\text{in}}\) for the VMS and VMS_{n} methods is approximately conserved for different values of Re. It has to be mentioned that the results are expected to be dependent on the resolution of the mesh, e.g., the results using VMS should approach those by VMS_{n} as the computational grid is refined.
Incompressible and Compressible Gas Flow in a Non-transferred Torch
Incompressible Gas Flow in a Non-transferred Torch
As a preliminary step to the VMS_{n} simulation of the NLTE plasma flow in an arc plasma torch, the incompressible flow through a non-transferred arc plasma torch is investigated using both VMS and VMS_{n} approaches. The domain geometry and boundaries that are shown in Fig. 2(b) have D = 0.0088 m (diameter of the torch outlet), L = 0.3 m (length of the outflow domain), and W = 0.08 m (width of the outflow domain), and have been discretized using ~ 386 k nodes and ~ 375 k trilinear hexahedral elements. Simulations used argon as the working fluid with boundary conditions defined by: wall, cathode, and anode boundaries: ∂_{n}p = 0 and u = 0; inlet boundary: ∂_{n}p = 0 and u = [0 0 U_{in}]; and outlet boundary: p = p_{atm} and ∂_{n}u = 0. The average velocity, U_{in}, is defined such that that if the Re number at the exit of the torch was computed based on the average velocity of the jet, the value of Re equals 5142.
Compressible Gas Flow in a Non-transferred Torch
The compressible flow simulation of the flow from a non-transferred torch using a computational domain discretized by ~ 351 k trilinear hexahedral elements and ~ 362 k nodes with D = 0.0088 m, L = 0.25 m, and W = 0.08 m using argon as the working gas is considered next. The mesh is finer near the cathode tip and stretched toward solid boundaries to better resolve boundary layers. Based on the obtained locations for the start of turbulence in the incompressible flow cases, for the compressible and plasma flow simulations, the length of the domain (L) was shortened. Nevertheless, the number of elements is approximately conserved; that is, the mesh is finer than those used in the simulations in Fig. 5.
Set of boundary conditions for the non-transferred arc plasma torch simulation
Boundary | Variable | |||||
---|---|---|---|---|---|---|
p | u | T _{ h} | T _{ e} | ϕ _{ p} | A | |
Cathode | ∂_{n}p = 0 | u = 0 | T_{h} = T_{cath}(z) | ∂_{n}T_{e} = 0 | –σ∂_{n} \(\phi_{p}\) = J_{qcath}(z) | ∂_{n}A = 0 |
Anode | ∂_{n}p = 0 | u = 0 | –κ_{h}∂_{n}T_{h} = h_{w}(T_{h} − T_{w}) | ∂_{n}T_{e} = 0 | \(\phi_{p}\) = 0 | ∂_{n}A = 0 |
Inlet | ∂_{n}p = 0 | u = u_{in} | T_{h} = T_{in} | T_{e} = T_{in} | ∂_{n} \(\phi_{p}\) = 0 | A = 0 |
Outlet | p = p_{out} | ∂_{n}u = 0 | ∂_{n}T_{h} = 0 | ∂_{n}T_{e} = 0 | \(\phi_{p}\) = 0 | ∂_{n}A = 0 |
Wall | ∂_{n}p = 0 | u = 0 | T_{h} = T_{w} | ∂_{n}T_{e} = 0 | ∂_{n} \(\phi_{p}\) = 0 | ∂_{n}A = 0 |
The similarity in the location of the onset of turbulent behavior in the compressible flow results is largely due to the increased temperature of the flow. The higher-temperature distributions are due to the temperature boundary condition over the cathode (Table 2). Higher temperature affects the value of density and viscosity of the fluid, which can affect the value of Re of the exiting jet from the torch. Although the results in Fig. 5 and 6 correspond to simulations for different Re, the representation of non-dimensional velocity distributions makes it possible to compare them. Contrasting the location of the arrows using the VMS_{n} method in Fig. 5 and 6, it is concluded that the onset of turbulence for incompressible and compressible simulation is approximately the same (i.e., L/D ~ 13). This observation has important implications in the interpretation of the NLTE plasma flow results, as presented next.
Turbulent Non-equilibrium Flow from a Non-transferred Arc Plasma Torch
Problem Setup
Simulation of the NLTE plasma flow in a non-transferred arc torch is carried out using the geometry and domain shown in Fig. 2(b) and the set of boundary conditions in Table 2. The computational domain is the same as the one used for the compressible flow simulations.
In Table 2, ∂_{n} ≡ n·∇, where n is the outer normal to the surface, denotes the derivative in the direction of the outer normal to the surface, p_{out} is set equal to the atmospheric pressure (1.01325 × 10^{5} Pa). The volumetric flow rate Q_{in}, similarly as for the compressible flow simulations, is set to 40 slpm using straight (no swirl) gas injection. Hence, the inflow velocity profile, assumed as uniform, is such that \(Q_{\text{in}} = \int_{{S_{\text{inlet}} }} {{\mathbf{u}}_{\text{in}} {\text{d}}S} = U_{\text{in}} S_{\text{inlet}} \hat{z}\), where S_{inlet} is the inlet surface and \(\hat{z}\) the unit vector along the torch axis.
Simulation Results
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 of order 10^{4} K mm^{−1}. 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 over 2 orders of magnitude (e.g., from O(10) to O(1000) ms^{−1}). Such trend can be observed in the simulation results in Fig. 8. The large gas acceleration and shear velocity and temperature gradients inside the torch, together with the electromagnetic forcing, cause the flow to become unstable and turbulent. According to Fig. 8, turbulence is further enhanced when the plasma flow leaves the torch and interacts with the cold environment, which is consistent with experimental observations. Furthermore, the distribution of \(\phi_{p}\) shows a maximum voltage drop of 35 V. The distribution of magnetic vector potential shows that the magnetic field is self-induced, and the approximately linear gradient of ||A|| along the z-axis indicates that the magnetic field should act as constricting the arc radially. The distributions of \(\phi_{p}\) and ||A|| in Fig. 8 show marked resemblance with computational observations of the dynamics of plasma jets such as those reported in Ref 33.
The balance between the Lorentz force and the drag caused by the cold working gas is the main factor determining the dynamics of the arc inside DC non-transferred torches. The ratio between the total flow rate Q_{in} and the total current I_{tot} provides insight into the stability of the flow. If the ratio Q_{in}/I_{tot} is small, then the arc develops a steady, uniform, and axisymmetric attachment along the anode surface. For larger ratios, despite the axisymmetry and constancy of boundary conditions, the arc develops a constricted anode attachment and a quasi-periodic movement of the arc is established. For even larger ratios, the arc displays chaotic dynamics (Ref 66-68). These characteristics are often referred as the steady, takeover, and restrike modes of operation of the torch (Ref 69, 70).
The dynamics of the arc inside the torch play a primary role in the flow of the plasma jet. The arc dynamics are a consequence of the unstable imbalance between the Lorentz force exerted over the arc due to the distribution of current density connecting the anode and cathode attachments, and the drag caused by the relatively cold and dense stream of inflow working gas over the hot and low-density arc plasma. The force imbalance causes the dragging of the anode attachment, the elongation of the arc, and the increase in arc curvature until the arc gets in close proximity to another location over the anode. If part of the arc reaches a location near the anode that is closer to the cathode than the current anode attachment, the arc will tend to re-attach, hence forming a new attachment. This phenomenon is known as the arc re-attachment process (Ref 37, 69, 71, 72). This re-attachment is most likely characteristic of the takeover mode of operation of the torch; arc re-attachment in the restrike mode of operation may be driven by different mechanisms, such as breakdown of the cold gas boundary layer or electron avalanches (Ref 69, 70).
Kolmogorov’s theory indicates the existence of a so-called inertial sub-range in fully developed turbulent incompressible flows. The scales of the flow within this sub-range are smaller than the ones directly affected by macroscopic parameters (i.e., domain geometry, boundary conditions), but larger than the scales dominated by molecular dissipation (i.e., viscosity). If the grid is selected such that it is able to resolve the scales within the inertial sub-range, then the results are primarily dependent on the accuracy of the small-scale model. The fact that VMS and VMS_{n} produced qualitatively the same results for the simulation of the flow from an arc plasma torch may be due to either the computational grid used was too coarse to resolve the inertial sub-range, or the grid was fine enough such that the small scales modeled were well at the end of the inertial sub-range such that the specific model of the small scales is of small relevance. The analysis by Shigeta (Ref 14) indicates that the core of the plasma jet is likely laminar while its surroundings can likely develop instabilities and turbulence. The results in Fig. 10 show a laminar jet core and the development of shear instabilities in the jet periphery, but not the level shown in Fig. 5. Therefore, the accurate description of the flow surrounding the jet requires a significantly finer mesh suitable to resolve the inertial sub-range. Further investigation of the effect of varying spatial and temporal discretization in the results by both methods is needed to verify this result and to further establish the degree of resolution required by comprehensive laminar-to-turbulent flow torch simulations.
Summary and Conclusions
Non-transferred arc torches are at the core of diverse applications, particularly plasma spray. The flow in these torches, particularly at industrially relevant conditions, often transitions from laminar within the torch to turbulent in the emerging jet, and presents significant deviations from local thermodynamic equilibrium (i.e., non-LTE or NLTE). The turbulent nature of the flow, which augments cold-flow entrainment and gas mixing, can drastically affect the degree of non-equilibrium. This fact prompts desirable the use of NLTE models as comprehensive modeling approaches according to their capability to capture both, laminar and turbulent flow regimes. The multiphysics nature of non-equilibrium plasma flows, their inherent high nonlinearity, and the large range of intervening scales make their direct computational simulation, following what is known as direct numerical simulation, practically unfeasible. This fact prompts the need for comprehensive coarse-grained modeling techniques that seek to resolve the large scales of the flow only, while modeling the small scales. Large eddy simulation (LES) techniques are considered the established standard for the exploration of turbulent incompressible flows. Nevertheless, the characteristics of non-equilibrium plasma flow models invalidate the major assumptions in traditional LES models. The flow from a non-transferred torch is simulated using a NLTE plasma flow model solved by a classical variational multiscale (VMS) method and a nonlinear VMS (VMS_{n}) method. Whereas both methods are suitable for the description of multiscale phenomena, the up-front handling of inter-scale coupling by the VMS_{n} makes it potentially more suitable for the description of turbulent flows without the need for empirical model parameters. Incompressible (non-plasma) turbulent jet simulations indicate that the VMS_{n} method produces results comparable to those obtained with the dynamic Smagorinsky method, typically considered the workhorse for turbulent incompressible flow simulations, and significantly more accurate than those by the VMS method. The VMS and VMS_{n} approaches are applied to the simulation of incompressible, compressible, and NLTE plasma flows in a non-transferred arc torch. The NLTE plasma flow simulations using VMS and VMS_{n} methods are able to reproduce the dynamics of the arc inside the torch together with the evolution of turbulence in the produced plasma jet in a cohesive manner. However, the similarity of results by both methods indicates the need for numerical resolution significantly higher than what is commonly afforded in arc torch simulations.
Notes
Acknowledgment
The authors gratefully acknowledge the support from the US National Science Foundation, Division of Physics, through award number PHY-1301935.
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