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With the rapid development of computer technology, the calculation of heat transfer and fluid flow for a three-dimensional (3D) thermal plasma torch with axisymmetric geometries became feasible (Ref 2, 3, 15- 22). Models most frequently used for simulations of plasma spray torches rely on local thermal equilibrium (LTE) approximation, and regard plasma flow as a property-varying electromagnetic reactive fluid in the state of chemical equilibrium, in which the internal energy of the fluid can be characterized by a single gas temperature (Ref 2, 3, 15- 21). Selvan et al. developed a steady-state 3D LTE model to describe the temperature and velocity distributions inside a DC plasma torch. The length of the arc and the radius are also being discussed. However, the model over-estimated the local gas temperature near the anodic arc root with an assumption that all the electric current transferred into the anode can only go through a fixed arc root (Ref 3, 16). Klinger et al. also developed a steady-state 3D LTE model simulation of the plasma arc inside a DC plasma torch. The position of the arc root was determined arbitrarily (Ref 17) with the steady-state 3D LTE model. It is possible to predict the temperature and velocity distributions inside the plasma torch. The arc length and power can also be predicted. However, the fluctuation of the plasma arc cannot be determined. S. Paik, P.C. Huang, J. Heberlein, and E. Pfender, Determination of the Arc Root Position in a DC Plasma Torch, Plasma Chem. Plasma Process., 1993, 13(3), p 379-397 Zhu T, Wang HX, Sun SR et al (2019) Numerical simulation of constricted and diffusive arc–anode attachments in wall-stabilized transferred argon arcs. Plasma Sci Technol 21(12):125406 The variation of gas pressure inside the torch is so little that the effects of pressure on the thermodynamic and transport properties of plasma are negligible.

Wang HX, Sun SR, Sun WP (2015) Status and prospects on nonequilibrium modeling of high velocity plasma flow in an arcjet thruster. Plasma Chem Plasma Process 35(3):543–564Pascal Chambert and Nicholas Braithwaite (2011). Physics of Radio-Frequency Plasmas. Cambridge University Press, Cambridge. pp.219–259. ISBN 978-0521-76300-4. Geddes, C. G. et al. Plasma-density-gradient injection of low absolute-momentum-spread electron bunches. Phys. Rev. Lett. 100, 215004 (2008). Wang HX, He QS, Murphy AB et al (2017) Numerical simulation of nonequilibrium species diffusion in a low-power nitrogen–hydrogen arcjet thruster. Plasma Chem Plasma Process 37(3):877–895

Torch brand products are widely used in civil high-end fields such as communication equipment, industrial control equipment, precision instruments, medical equipmentBrijesh, P. et al. Tuning the electron energy by controlling the density perturbation position in laser plasma accelerators. Phys. Plasmas 19, 063104 (2012). Selvan B, Ramachandran K (2009) Comparisons between two different three-dimensional arc plasma torch simulations. J Therm Spray Technol 18(5–6):846–857 Electron density is one of the key parameters in the physics of a gas discharge. In this contribution the application of the Stark broadening method to determine the electron density in low temperature atmospheric pressure plasma jets is discussed. An overview of the available theoretical Stark broadening calculations of hydrogenated and non-hydrogenated atomic lines is presented. The difficulty in the evaluation of the fine structure splitting of lines, which is important at low electron density, is analysed and recommendations on the applicability of the method for low ionization degree plasmas are given. Different emission line broadening mechanisms under atmospheric pressure conditions are discussed and an experimental line profile fitting procedure for the determination of the Stark broadening contribution is suggested. Available experimental data is carefully analysed for the Stark broadening of lines in plasma jets excited over a wide range of frequencies from dc to MW and pulsed mode. Finally, recommendations are given concerning the application of the Stark broadening technique for the estimation of the electron density under typical conditions of plasma jets. E = U 2 π r = ω r H 0 2 sin ⁡ ω t {\displaystyle E={\frac {U}{2\pi r}}={\frac {\omega rH_{0}}{2}}\sin \omega t} , [6]

Numerical calculation provides a valid way to understand the arc behavior inside the plasma torch. The modeling of a DC arc plasma torch is an extremely challenging task because plasma flow is highly nonlinear and it presents strong property gradients. It is characterized by a wide range of time and length scales, and often includes chemical and thermodynamic non-equilibrium effects, especially near its boundaries (Ref 8). Despite the complexity of the subject, over the past few decades, many papers concerning numerical studies of the characteristics of DC arc plasma torches have been published (Ref 2, 3, 8- 22). At the initial stage, two-dimensional (2D) models were used in research to predict heat transfer and flow patterns inside the plasma torch (Ref 10- 14). The predicted arc voltage of the torch in the turbulent regime is much higher than the measured value. In addition, the predicted axial location of the arc attachment at the anode surface is also much farther downstream than in experimental observations (Ref 15). Faure, J. et al. Controlled injection and acceleration of electrons in plasma. Nature 444, 737–739 (2006).Ammosov, M. V., Delone, N. B. & Krainov, V. P. Tunnel ionization of complex atoms and atomic ions in a varying electromagnetic-field. Sov. Phys. JETP 64, 1191–1194 (1986).