Prediction of anode arc root position in a DC arc plasma torch

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1 Prediction of anode arc root position in a DC arc plasma torch He-Ping Li 1, E. Pfender 1, Xi Chen 1 Department of Mechanical Engineering, Uniersity of Minnesota, Minneapolis, MN 55455, USA Department of Engineering Mechanics, Tsinghua Uniersity, Beijing 184, P. R. China Abstract Modeling results are presented concerning the 3-D flow and heat transfer inside a non-transferred DC arc plasma torch using two different electric potential boundary conditions, i.e. anode iso-potential condition and the Steenbeck minimum principle. The computed results show that although the predicted anode arc root positions by the two approaches are almost the same, using the Steenbeck minimum principle can predict more precisely the anode arc root shape and the local high temperature region near the anode arc spot. 1. Introduction Although DC arc plasma torches hae been widely used oer the past decades, understanding of the complex physical and chemical processes occurring inside plasma torches are still incomplete. The complex interactions between electromagnetic fields, gas flow and arc self-induced flow fields and temperature fields make modeling a formidable task. Transient behaior of the arc, non-local thermodynamic equilibrium (non- LTE) effects near electrodes, three-dimensional (3-D) effects inside the torch add further to this complexity [1]. In this paper, we will focus on the 3-D effects inside a non-transferred DC arc plasma torch. The complex physical phenomena inside a DC arc plasma torch can be described by a complete set of magnetohydrodynamic (MHD) equations. Preiously, many researchers used the two-dimensional (-D) MHD equations to simulate the temperature, flow and electromagnetic fields inside the torch [-7]. Although some useful information can be obtained by -D modeling, 3-D effects are entirely ignored. For example, the predicted arc oltage for the turbulent regime is much higher than measurement indicated, and the calculated anode arc root position at the anode surface is also much farther downstream than actually obsered [6, 7]. A 3-D MHD model was proposed in Refs. [7, 8] to simulate heat and flow patterns inside a DC arc plasma torch. And physically reasonable anode arc root positions at the anode surface were obtained for a few operating parameters [7, 8]. As indicated in Ref. [4], the Steenbeck minimum principle [9] is a ery useful tool to predict anode arc root positions at the anode inner surface. This principle is compatible with Maxwell s equations, and thus, it is not an independent law [1]. The goal of this paper is to compare the 3-D heat and flow patterns inside a non-transferred DC arc plasma torch by employing the Steenbeck minimum principle with those using the preious 3-D MHD model presented in Refs. [7, 8].. Mathematical model.1 Basic assumptions In this paper, the following assumptions are employed for the 3-D modeling: (1) The flow inside the plasma torch is quasi-steady, turbulent, haing temperature-dependent properties; () The plasma is in LTE state outside the electrode boundary layers and optically thin; (3) The iscous dissipation and the pressure work terms in the energy equation are neglected; (4) The working gas (argon) is injected into the plasma torch in axial direction, i. e., without any swirling elocity component of the gas at the torch inlet; (5) The induced electric field V B is negligible in comparison with the electric field intensity E r.. Goerning equations Based on the foregoing assumptions, the goerning equations for the 3D quasi-steady numerical simulations can be written in ( r,q, z) coordinates as follows: ( r V ) (1) t r VV - pi + t + j () ( ) ( ) B

2 5 k B ( r c pvt) ( GT T) + j s + j T - S (3) R e s f (4) ( ) In the preceding equations, V, p, T and f are the elocity, pressure, temperature of the gas and the electric potential, respectiely. I t and t are unit tensor and shear stress tensor, respectiely. The current density ector j relates to the electric potential f as j -s f (5) B is the magnetic induction, which is obtained by B A (6) and by using this equation in the actual modeling work speeds up the iteration process [7]. In Equation (6), A is the magnetic ector potential, which satisfies the following equation A -m j (7) with m j( x, y, z ) A( x, y, z) Ú dv (8) 4p r V as the boundary conditions, where m is the permeability in acuum. The boundary conditions for the magnetic ector potential employed in this paper are ery different from those employed in Ref. [11]. The standard K - e two-equation turbulence model is employed in this study. The turbulent kinetic energy and its dissipation rate equations are ( rvk) ( GK K) + G - re (9) e ( rve) ( Ge e) + ( c1g - c re) (1) K The turbulent generation term G appearing in Equations (9) and (1) is expressed as ÔÏ ÈÊ ˆ Ê ˆ Ê ˆ Ê ˆ Ê ˆ Ê ˆ Ô G m z r q r z r q z r q q ÌÍÁ + Á + Á + + Á + + Á + + Á + - (11) t Ë Ë ÔÓ ÍÎ Ë z Ë r Ë r q r Ë r z z r q r q r r Ô where r, q and z are the components of the elocity ector V in r-, q- and z-directions, respectiely. All the physical quantities in the foregoing equations are their time-aeraged alues. r, c p, s and S are the R temperature-dependent mass density, specific heat at constant pressure, electric conductiity and the radiation power per unit olume of an argon plasma, respectiely. k B is the Boltzmann constant, e is the elementary charge. The effectie diffusion coefficients are the combination of laminar and turbulent alues in the transport equations..3 Calculation domain and boundary conditions Figure 1. Calculation domain. The calculation domain, as shown in Figure 1, is formed by the rotation of the region ABCDEFGHA about the torch axis with the indicated geometric dimensions. In this paper, the copper anode, the thickness of

3 which is in order of 1 mm, is included in the computational domain for getting a reasonable prediction of the temperature distribution on the anode. The boundary conditions are listed in Table 1, where Q is the olumetric flow rate of the working gas, A is the inlet cross-section area of the torch, while F i are the alues of the ariable F at the grid points on the small circle nearest to the centerline ( F r, z, T, K, e, f ). At the cathode surface, the temperature (T c) and current density ( j ) distributions are specified as shown in Figure according to the experimental data presented in Ref. [1]. Then, the potential gradient is calculated as f (1) j (r ) -s n where j (r ) is the current density ector at the location of the cathode arc root. This ector is perpendicular to the cathode surface and a function of the radial distance (r) and the axial distance from the cathode tip ( z c ), while n indicates the normal direction of the cathode surface from the inner side to the outer side of the cathode. On the symmetry plane [ q (p )], a periodic boundary condition is employed as follows [7]: 1 (13) F b1 F b ª F a1 + F a where F r, q, z, T, K, e, f. The subscripts b1 and b indicate the grid points located on the symmetry ( ) plane, while a1 and a indicate the nearest inner grid points to the symmetry plane located on both sides of this plane in the calculation domain. Because the outer side of the anode is immersed in the cooling water, the third kind of boundary condition is employed in the present paper, i. e., T (14) -k h(tw - T ) r w where the subscript w indicates the interface between the anode and the cooling water, h is the heat transfer coefficient, which is in order of 1 5 W m K, T is the temperature of the cooling water. ( ) Table 1. Boundary conditions for 3D modeling of non-transferred DC arc plasma torch. q (p ) ABC CD DE EFGH HA r r q q r z T Assumed Fig. (b) T K e f e f f j (r ) -s n Â r,i r z r r q z q q z z z z Q A T z Eq. (14) 3 K K z /.5 z e z /.1K f z See Section.3 f z i 1 N z Wall Function Method N rq z K 1 N 1 Â z,i N i 1 Â Ti N i 1 Â Ki N i 1 Âei N i 1 Â fi N i 1 On the symmetry plane, periodic boundary conditions, [Eq. (13)], are employed. In this paper, two approaches are employed to deal with the electric potential boundary condition on the anode surface, which are described as follows: CASE I: Specified arc-root position boundary condition. With this approach, the anode arc-root position is specified in adance with f as boundary condition, and no current passes through the remaining parts of the anode. Thus, the total arc oltage can be obtained through soling the goerning equations mentioned aboe. In our modeling work, we can obtain different arc oltages

4 by specifying different anode arc-root locations. And then, according to Steenbeck s minimum principle, the physically reasonable position of the anode arc-root should correspond to the position with minimum alue of the arc oltage for a certain set of operation parameters. CASE II: Iso-potential boundary condition. It means that the alues of the electric potential are equal on the whole outer surface of the anode. This boundary condition is physically reasonable because of the ery large electric conductiity of the anode material (copper), and was employed in Refs. [7, 8]. 3.x1 +8 I4 A 35.5x1 +8.x1 +8 j (A/m ) 1.5x x x1 +7 I6 A J (r)-j,max Exp(-b r) I (A) J,max b x z c (mm) Tc (K) I4 A I6 A I (A) T max (K) z c (mm) (a) (b) Figure. The assumed current density (a) and temperature (b) distributions on the cathode surface. 3. Modeling results and discussions In this paper, the SIMPLE algorithm [13] is employed to sole the non-linear equations (1)~(4), (7), (9) and (1) simultaneously with the new ersion of the non-commercial software FAST-3D (Fluid Analytical Simulation Tools Three Dimensional) [14]. A 3-D, body-fitted, non-uniform mesh 3 (z) ( r) 11( q ) is adopted for the 3-D modeling of a plasma torch in the present paper [7]. 3.1 Modeling results and discussions In CASE II, we assume that the anode arc-root positions are as L Arc.3, 3.9, 4.47, 5.86, 7.53 and 9.54 mm. The calculated arc oltages as a function of the arc length L Arc are plotted in Figure 3, which shows that for a gien set of operation parameters (e. g., I4 A, Q. STP m 3 /hr), the arc oltage V Arc aries with the increase of the arc length L Arc as a V-shaped cure, and there exists a position corresponding to the minimum alue of the arc oltage among the different assumed anode arc-root positions along the anode surface. Thus, according to Steenbeck s minimum principle, this position should be the real position of the anode arc-root attachment during torch operation. In our calculation, the arc length determined by Steenbeck s minimum principle is L Arc 5.86 mm. VArc (V) I4 A, Q. STP m 3 /hr L Arc (mm) Figure 3. Arc oltage ariations with different assumed anode arc-root positions. The corresponding temperature distributions inside the plasma torch at - p plane and the iso-contours of the radial current density component at the inner surface of the anode are plotted in Figures 4(a) and 5(a), and compared with their counterparts for CASE I as shown in Figures 4(b) and 5(b), respectiely. From Figures 4 and 5, we can find that although the predicted anode arc-root attachments in CASE I and CASE II are almost the same ( L Arc ª 5.9 mm ), the temperature distributions near the anode arc-root attachment in Case I are ery different from that in Case II. In CASE I, the anode arc root is confined to a certain spot, and

5 as a consequence, there is a higher temperature region near this spot, related to the anode jet formation as described in Ref. [15]. While in CASE II, such local characteristics of the arc cannot be predicted, and the maximum local current density is also much smaller than that in CASE I because the anode arc root distributes oer a larger area in CASE II. The calculated aerage maximum alues of the temperature and axial elocities at the torch exit are 1,65 K, 46 m/s and 1,738 K, 461 m/s, for CASE I and CASE II, respectiely. The corresponding relatie discrepancies for the aeraged maximum temperature and axial elocity component between these two cases are.7% and 7.6%, respectiely (both smaller than 1%). Similar results are also obtained for the case I6 A, Q. STP m 3 /hr. T: r (mm) 8 4 CATHODE CASE I -4-8 I4 A, Q. STP m 3 /hr (a) T: r (mm) 8 4 CATHODE CASE II -4-8 I4 A, Q. STP m 3 /hr (b) Figure 4. The temperature distributions inside the plasma torch for CASE I (a) and CASE II (b). q q - - CUR: CASE I I4 A, Q. STP m 3 /hr (a) CUR: CASE II I4 A, Q. STP m 3 /hr (b) Figure 5. the iso-contours of the radial current density component at the inner surface of the anode for CASE I (a) and CASE II (b). 3. Comparison with experimental obserations As discussed in Section 3.1, Steenbeck s minimum principle used in this study can predict the anode arc-root position and the arc shape more precisely than the method with an iso-potential boundary condition. The typically calculated arc contours for CASE I and II (I4 A, Q. STP m 3 /hr) and a photograph of the arc in steady mode of operation [15] are shown in Figure 6. The calculated arc shape (CASE I in Figure 6) is similar to the experimentally obsered arc shape (lower part in Figure 6), while by using a uniform potential

boundary condition at the outer surface of the anode [7, 8], instead of using Steenbeck s minimum principle, the local characteristics of the arc near the anode arc-root cannot be well predicted. 4.

6 boundary condition at the outer surface of the anode [7, 8], instead of using Steenbeck s minimum principle, the local characteristics of the arc near the anode arc-root cannot be well predicted. 4. Conclusions In this paper, the 3-D heat transfer and flow patterns inside a non-transferred DC arc plasma torch are simulated. Two different electric potential boundary conditions are employed to determine the position of the anode arc root. The main conclusions are as follows: (1) The 3-D effects inside the non-transferred DC arc plasma torch are significant. () The predicted anode arc root positions by either the iso-potential boundary condition or the application of Steenbeck minimum principle are almost the same. (3) The predicted anode arc root shape by using Steenbeck s minimum principle is more precise than that predicted by using an iso-potential boundary condition. Figure 6. Comparison of the typically calculated arc temperature distribution inside the DC arc plasma torch (upper part) with the photograph of argon arc in steady mode of operation (lower part, [15]). Acknowledgement This work was partly supported by the National Natural Science Foundation of China (grant No ). The first author has been supported through a postdoctoral fellowship by the Department of Mechanical Engineering, Uniersity of Minnesota. The support through a supercomputer grant by the Uniersity of Minnesota Supercomputing Institute is gratefully acknowledged. References [1] P. Fauchais, A. Vardelle - Plasma Physics and Controlled Fusion 4, B365 (). [] D. A. Scott, P. Koitya, G. N. Haddad - Journal of Applied Physics 66, 53 (1989). [3] R. Westhoff, J Szekely - Journal of Applied Physics 7, 3455 (1991). [4] S. Paik, P. C. Huang, J. Heberlein, E. Pfender - Plasma Chemistry and Plasma Processing 13, 379 (1993). [5] J. M. Bauchire, J. J. Gonzalez, A. Gleizes - Plasma Chemistry and Plasma Processing 17, 49 (1997). [6] Peng Han - Ph. D. Thesis, Department of Engineering Mechanics, Tsinghua Uniersity (1999, In Chinese). [7] He-Ping Li - Ph. D. Thesis, Department of Engineering Mechanics, Tsinghua Uniersity (1, In Chinese). [8] He-Ping Li, Xi Chen - Journal of Physics D: Applied Physics 34, L99 (1). [9] W. Finkelnburg, H. Maecker - Electric Arcs and Thermal Plasmas in Encyclopedia of Physics, Vol. XXII, Springer-Verlag, Berlin (1956). [1] Ming-Lun Xue, Yun-Ming Chen - Acta Mechanica Sinaca 11, 96 (1979, In Chinese). [11] P. Freton, J. J. Gonzalez, A. Gleizes, F. C. Peyret, G. Caillibotte, and M. Delzenne - J. Phys. D: Appl. Phys. 35, 115 (). [1] Xin Zhou - Ph. D. Thesis, Department of Mechanical Engineering, Uniersity of Minnesota (1995). [13] S. V. Patankar - Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, Taylor and Francis, (198). [14] J. Zhu - An Introduction and Guide to the Computer Program FAST-3D, Institute for Hydromechanics, Uniersity of Karlsruhe, Report No. 691 (199). [15] S. A. Wutzke - Ph. D. Thesis, Department of Mechanical Engineering, Uniersity of Minnesota (1967).

Effect of an axial magnetic field on a DC argon arc

Vol 17 No 2, February 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(02)/0649-06 Chinese Physics B and IOP Publishing Ltd Effect of an axial magnetic field on a DC argon arc Li Lin-Cun( ) and Xia Wei-Dong