Numerical analysis of the arc in pulsed current gas tungsten arc welding using a boundary-fitted coordinate

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1 Journal of Materials Processing Technology 72 (1997) Numerical analysis of the arc in pulsed current gas tungsten arc welding using a boundary-fitted coordinate H.G. Fan a, *, Y.W. Shi b, S.J. Na a a Department of Mechanical Engineering, Korea Adanced Institute of Science and Technology, Kusong-dong 373-1, Yusong-gu, Taejon , South Korea b Welding Research Institute, Beijing Polytechnic Uniersity, Beijing, People s Republic of China Received 8 September 1996 Abstract A two-dimensional transient model is introduced to describe the heat transfer and fluid flow in pulsed current gas tungsten welding arc (GTA). In the model, a boundary fitted coordinate system is adopted to describe precisely the cathode shape. The current-continuity equation has been solved with the combined arc plasma-cathode system, independent of the assumption of the current density distribution on the cathode surface, which was essential in previous studies of the arc plasma. The temperature distribution in pulsed GTA welding has been described and the transition processes of temperature contours and arc pressures at the anode center have been studied. Moreover, the effects of pulsed welding parameters on the dynamic processes of the arc pressure at the anode center have been studied using the developed model, the results being compared with experimental data measured by the micro-pressure sensor Elsevier Science S.A. Keywords: Numerical model; Pulsed gas tungsten welding arc; Arc plasma; Boundary fitted coordinate 1. Introduction * Corresponding author. Present address. 805 State Street, Rolla, MO 65401, USA. hgfan@umr.edu Since a multi-variable control system has as many input variables as the functions to be controlled, the pulsed-current gas tungsten arc welding (PC-GTAW) process has been found to have a number of advantages over the conventional direct current gas tungsten arc (DC-GTA) welding process, these advantages include: Arc stability, lower heat input requirement, better penetration control and grain refinement of the weld pool [1 3]. To improve the quality and productivity of the PC-GTA welding process, an improved physical understanding of the welding process is needed. The literature describing DC-GTAW systems is quite extensive and comprehensive mathematical models have been available for representing this welding process [4 8]. In most previous studies of the heat transfer and fluid flow in arc plasma, a current density profile has been assumed over the surface plane of the cathode, although it should be emphasized that the theoretical predictions are sensitive to the current density at the cathode [6]. Also, in pulsed current arc welding, there seems to be no mathematical model of the heat transfer and fluid flow reported in the open literature. In the present paper the model addresses a pointedtip cathode that is precisely described by a boundaryfitted coordinate. The distribution of current density, which is determined primarily by the welding current and cathode shape, is calculated with the combined arc plasma-cathode system, independent of the assumption of the current density distribution on the cathode surface. On the basis of this, the temperature distribution of the arc and the transition process between pulsing currents are analyzed using the finite difference method. Finally, the effects of pulsing current, electrode vertex angle, pulse duty ratio, frequency and arc length on the distributions of the arc pressure at the anode center are determined and compared with the experimental data /97/$ Elsevier Science S.A. All rights reserved. PII S (97)

2 438 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Modelling of arc 2.1. Conseration equations In GTA welding, the process usually uses a directcurrent of straight polarity. In this model, a tungsten electrode with a diameter of 3.2 mm is used. The assumptions adopted for modelling are summarized as follows: 1. The arc is assumed to be pure argon in local thermodynamic equilibrium (LTE) and is assumed to be optically thin to radiation. 2. The arc is radially symmetrical and its flow is laminar. 3. The heating effect of viscous dissipation is negligible. 4. The welding current wave form is assumed to be an ideal DC square-wave pulse. A free burning arc should be analyzed with a combined model of fluid mechanics and electromagnetics and satisfy a group of magneto-hydro-dynamics (MHD) equations. With the above-mentioned assumptions, the conservation equations are expressed in terms of cylindrical coordinates as follows: Equation of mass continuity: t +1 r () + 1 t r (r) r + (u) =0 (1) z Conservation of radial momentum: r r + r r z 2 = P r +1 r (u) + 1 t r r r r r 2+ z Conservation of axial momentum: u ru r + r r z + u = P z +1 r r r z z u z u r jz B (2) u uu z z +jr B +g (3) For low currents of less than 30 A, the buoyancy force rather than the magnetic force dominates the convective flow and arc properties [6]. Since the background current selected to maintain the arc is usually less than 30 A, the buoyancy force has been considered in the momentum equation. Conversation of energy: k rh r (h) + 1 t r r = j 2 r +j 2 z S R + 5 K b 2 e + k uh z h h C p r C p z j z h C P z + j r h (4) C P r Enthalpy h is defined as: T h= C p dt (5) T ref Further, there are Lorentz force terms in the momentum equations and the energy equation contains the Joule-heating term and an additional term which represents the transport of electron enthalpy due to the drift of electrons. Therefore, it is necessary to solve Maxwell s equations for the electromagnetic field. According to Ohm s law: j r = r j z = z In the axisymmetric cylindrical coordinates, the current continuity can be written in terms of the potential as follows: 1 r r r r + z (6) z =0 (7) Finally, the self-induced magnetic field B is calculated using Ampere s law: r B = 0 j r z r dr (8) 0 The above partial differential equations (Eqs. (1) (8)) are solved together to obtain the distributions of arc plasma temperature, velocity, current density, etc. The plasma properties for this analysis are treated as temperature dependent and taken from published data [5] Calculation domains and boundary conditions The calculation domains and boundary conditions used are shown in Fig. 1 and Table 1. A large calculation domain of AGFEA including the cathode is used for solving the electrical potential equation. At the surface of the anode (GF), the electrical potential is assumed to be constant. /r=0 has been set on EF to represent the condition that no current flow crosses this boundary. Over the cathode cross AD a uniform Fig. 1. Computational domain of the pulsed GTA welding arc.

3 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Table 1 Boundary conditions for welding arc AG or BG GF FE ED DA or BCD u u/r=0 u=0 u/r=0 u=u given u=0 =0 =0 (r)/r=0 =0 =0 h h/r=0 h=h const h=h const h=h const h=h const T=1000 K T=1000 K T=1000 K T BH =20000, T HCD =3000 K /r=0 =0 /z=0 /z=j 0 /z=j 0 current density of j 0 = /z is assumed, defined as the input current divided by the area of cathode cross section ( j 0 =I/(R 2 1)). Zero gradients are assumed for the electric potential along the boundary of DE and the center-line of AG. A mesh generation for a 3 mm long arc with a 60 cathode vertex angle is shown is Fig. 2, in which 50 nodes in the r-direction and 45 nodes in the z-direction are located in the entire domain. Boundary-fitted nodes are generated for describing the cathode shape. A finer mesh is used near to the cathode and the anode, because a steeper gradient of dependent variables is expected along these surfaces. For solving the momentum and energy equation, a smaller domain of BCDEFGB excluding the cathode is used. Basically, a no-slip condition at the solid surface and a symmetric condition along the center line is adopted. A temperature of 1000 K is assumed on the anode surface of GF. Previous works indicated that the temperature close to the cathode in the cathode spot mode is about K and does not appear to vary greatly with arc current up to 400 A [9], therefore, K is adopted as the temperature of the cathode spot. In the process of welding, the apex of a sharpened electrode usually tends to experience some melting and rounding up. The value of the cathode spot radius BH is set to be 0.9 mm for an arc current of 100 A according to experiment and is assumed to be proportional to I 1/2 [8], whilst the temperature at the cathode cone surface HCD is assumed to be 3000 K. Across the boundary EF, a constant mass flow rate and 1000 K temperature condition is used. Along the boundary DE, the radial velocity component is neglected and the axial velocity component is determined from the equation of pipe flow as follows [10]. R 2 u=2 Q 2 r 2 +(R 2 2 R 2 1) ln (r/r 2) n ln (R 2 /R 1 ) R 4 2 R 4 1+ (R (9) 2 2 R 2 1) n 2 ln (R 2 /R 1 ) where Q is the inflow rate of shielding gas, R 1 is the radius of the electrode and R 2 is the internal radius of the shielding nozzle. The temperatures of the inflow shielding gas with a flow-rate of 10 l min 1 was assumed to be 1000 K. 3. Numerical procedure 3.1. Transformation Fig. 2. Mesh generation with boundary-fitted coordinates. Fig. 3. Geometric transformation and grid structure with notation. For solving the governing equations with the finitedifference method, the physical domain represented by a boundary-fitted coordinate system must be transformed into a rectangular domain. Before applying the transformation, the governing equations can be expressed in a generalized equation using a general dependent variable as follows: () + 1 t r r r r r + z u z =S (10) where is the general diffusion coefficient and S is the corresponding source term. Then Eq. (10) is trans-

4 440 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Technique of solution The calculation domain for u, and h excludes the solid cathode. This is carried out by rendering the control volume of the solid cathode inactive, so that the remaining active control volume forms the desired irregular domain (BCDEFGB). For solving the potential equation, the field is specified by employing the electrical conductivity of the solid cathode and the plasma fluid in their respective regions. As a result, the problem is solved as a conduction problem throughout the entire calculation domain. The transformed Eq. (11) is discretized using the control volume approach [11] in the transformed domain, as shown in Fig. 3, which results in an equation of the following form: a P P =a E E +a W W +a T T +a B B +S (13) where the general dependent variable can be any of u,, h, P or. The values of a p, etc. are the coefficients which result from Eqs. (1) (7) and S is the source term from Eqs. (1) (7). The variables of the equations are solved iteratively. The overall solution procedure is shown in Fig. 4. The arc is easier to be ignited during the peak pulse Fig. 4. Flow chart of calculations. formed into a generalized curvilinear coordinate system as shown in Fig. 3, resulting in the following equations: () + 1 t r r r r r r + r n r + 1 r r + n r r + u z + n z z + u z + n z z =S (11) where: r =1 z J, r = 1 z j, z = 1 r J z =1 r J, r z J= r z (12) The transformation coefficients were calculated numerically using the second-order central difference method. In the transformed domain the grid size was set to be unity for simplifying the calculation. Fig. 5. Schematic diagram of the experimental arrangement: 1, water cooled copper anode; 2, gating orifice; 3, communication channel; 4, micro-pressure sensor; 5, power source; 6, signal amplifier; 7, function recorder. Fig. 6. Temperature contours of a low frequency pulsed arc: I p =100 A; I b =25 A; l=2 mm; f=1 Hz, =60 ; =0.5.

5 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Fig. 7. Effect of current on dynamic process of the arc pressure at the anode center: (a) Calculation; and (b) experiment ( f=1 Hz; =60 ; =0.5; l=3 mm). duration, so that the input current is given as a peak current at the beginning of the calculations. Since the maximum calculation time t max is set to 2(t p +t b ), there is no fundamental difference between setting the input current as I p or I b at the beginning. Convergence is declared when the following condition is satisfied: L i=1 N k=1 m i,k m 1 i,k max, =0.01 (14) where L and N are the total number of nodes in the r- and z- directions, respectively and m is the number of iterations. 4. Experimental Fig. 5 illustrates the arrangement used to measure the arc pressure during this investigation. The basic units of this device are a receptor, a communication channel and a micro pressure sensor. The receptor is represented by a gating orifice made in a water-cooled copper plate that is the anode of the arc. The communication channel connects the output of the gating orifice with the input of the micro-pressure sensor, the output signal of the micro-pressure sensor being recorded by a function recorder. 5. Results and discussion 5.1. Low frequency pulsed current GTAW Temperature distribution The temperature distribution in the arc, during the peak and background phase, is influenced by the instantaneous power which varies with time. According to the calculations, the temperature contours are similar between successive time steps during the peak pulse duration t p and background duration t b in low-frequency pulsed GTA welding. The results imply that the arc can become steady immediately after the welding current changes to the peak current from the background phase and vice versa. Fig. 6 shows the temperature contours for the pulsed arc ( f=1 Hz, =0.5,

6 442 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Fig. 8. Effect of the arc length on the dynamic process of the arc pressure at the anode center: (a) Calculation; and (b) experiment (I p =100 A; I b =25 A; f=1 Hz; =60 ; =0.5). Fig. 9. Effect of the cathode vertex angle on the dynamic process of the arc pressure at the anode center: (a) Calculation; and (b) experiment (I p =100 A; I b =25 A; l=2 mm; f=1 Hz, =0.5). l=2 mm, =60, I p =100 A and I b =25 A). Compared with the arc in the background duration, the arc temperature in the peak pulse duration is higher and the arc column is wider, which is in fair agreement with the observation that the pulsed arc is bright and dark, relaxed and contracted by turns. As shown, different arc shapes have been observed with the pulsing current: The arc is bell-shaped during the peak pulse duration whilst not having the characteristics of a bell-shaped arc during the background duration. The reason is that the velocity of the plasma jet is not high enough owing to a smaller electromagnetic force during the background duration, since it is the strong plasma jet impinging on the anode surface that forms the typical bell shape of the arc due to the flow stagnation Influencing factors on the distribution of the current density at the anode center There are many factors affecting the distribution of arc pressure. In this study, the effects of pulsing current, arc length, electrode vertex angle and pulse duty ratio on the dynamic process of the current density at the anode center are discussed as follows. The basic welding conditions used in calculations are: f=1 Hz, =0.5, l=2 mm, =60, I p =100 A and I b =25 A. The effect of different pulsing currents on the dynamic process of the arc pressure at the anode center for a 3 mm long, 60 cathode vertex angle arc is shown in Fig. 7, where (a) and (b) express the calculated and experimental results for the current values considered, respectively. As shown, the arc pressure at the anode

7 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Fig. 10. Effect of the duty ratio on the dynamic process of the arc pressure at the anode center: (a) Calculation; and (b) experiment (I p =100 A; I b =25 A; l=2 mm; f=1 Hz; =60 ). center is pulsed with pulsing current between the peak and background arc pressure. The peak and background arc pressure increase with the increasing peak and background current respectively, whilst for the same peak current a higher background current results in a higher peak arc pressure according to the results of the experiment. The main reason for the above result is that with increasing current the arc temperature, and consequently the current density of the arc plasma, increases. The arc pressure increases due to the increasing current density, which causes the increase of the electromagnetic force, since the electromagnetic force is the driving force of the fluid flow of the arc plasma. Further, the result indicates that the arc pressure is somewhat lower at the first time step, which implies that there exists a transition process for the arc to become steady after ignition. According to calculation, the arc becomes steady rapidly for the 150 A peak current after ignition. The dynamic process of the arc pressure at the anode center for an arc with a 60 cathode vertex angle and different arc lengths (2, 3 mm) are shown in Fig. 8, from which it is indicated that the peak and background arc pressure increase with decreasing arc length. In GTA welding, the cathode is a rod and the anode is a plate, so that the arc spread widely with the increase of arc length. The decrease of current density, which is primarily from the expansion of the current-carrying zone with the increasing arc length, will result in the decrease of the electromagnetic force and consequently of the arc pressure. Fig. 9 shows (a) the predicted and (b) the experimental dynamic process of the arc pressure at the anode center with 30 and 90 tip angles for a2mmlong arc. It can be seen that the smaller the cathode vertex angle, the bigger the peak and background arc pressure. The reason for this is that the smaller cathode tip angle results in a reduced radius of the cathode tip conductive section, which causes increase of the electromagnetic force with increasing cathode current density. Furthermore, the sharper cathode vertex angle can decrease the resistance of additional gas flow, so that the velocity of the arc plasma flow is increased: The increasing velocity should bring about an increase of the arc pressure. The effect of the pulse duty ratio on the dynamic process of the arc pressure at the anode center is shown Fig. 11. Transition process of the temperature contour for a pulsed arc: (a) I p to I b ; and (b) I b to I p (I p =100 A; I b =25 A; l=2 mm; f=100 Hz; =60 ; =0.5). Fig. 12. Transition process of the arc pressure at the anode center: I p =100 A; I b =25 A; l=2 mm; f=100 Hz; =60 ; =0.5.

8 444 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) Fig. 13. Dynamic process of arc pressure at the anode center for a high frequency arc: I p =100 A; I b =25 A; l=2 mm; f=10 KHz; =60 ; d=0.5. in Fig. 10, from which it is seen that the peak arc pressure duration increases with the increasing pulse duty ratio. The result further indicates that the duration time is sufficient for the low frequency pulsed arc to become steady during the peak pulse and background phase. As shown in Fig. 10(b), spiking is observed only for the smaller duty ratio arc, where the mechanical inertia of the function recorder has to be considered Transition process of the pulsed arc Arc plasma is a gas which has been heated to a condition where it is ionized and capable of conducting electric current. A pulsed arc is a phenomenon of gas discharge under changing current, so that the thermal behavior of the pulsed arc will also change with the pulsing current. The change of the thermal field of the arc, which depends on Joule heating and heat transfer between the charged particles, should be slower than the change of the welding current. In other words, the arc has the characteristics of thermal inertia. It was found that a transition process exists at the period of the current level change. The transition process of a pulsed arc (I p =100 A, I b =25 A, l=2 mm, f=100 Hz and =0.5) has been studied as follows. To describe the thermal inertia at the period of current level change, a smaller time step (dt=t p(b) /100) is set in the transition time of t p(b) /10 corresponding to the upslope and downslope edge of pulses, whilst the other time steps are set to 9t p(b) /40. The arc shape can be described by the use of temperature contours. The transition process of K isotherms for the pulsed arc at the period of t p(b) /10 is shown in Fig. 11. When the welding current is changing from the peak to the background level, the arc column width decreases with a bigger interval between the temperature contours at first, but the interval becomes progressively smaller. Finally, the arc column width achieves a stable value corresponding to I b. Identically, when the welding current moves from I b to I p, the arc column width increases rapidly at first, but becomes steady gradually. The anode arc behavior is closely linked with the arc column. Fig. 12 shows the transition process of the arc pressure at the anode center. When the welding current jumps, there exists a sluggishness in the change of the arc pressure. At the start of transition, the arc pressure at the anode center changes greatly, but arrives at a stable state very quickly, which is similar to the abovementioned transition process of the temperature contours. In low-frequency pulsed welding, the arc column can change periodically with the pulsing current, but as the frequency increases, it becomes more and more difficult for the arc column to catch up with the change of the welding current due to the thermal inertia of the arc, i.e. with the increase of frequency the duration of the pulsing current is not long enough for the pulsed arc to become stable. For high frequencies, therefore, the arc is gradually transiting from the transient unsteady state to the steady state, which is expressed by the arc column having been frozen [3]. The dynamic process of the arc pressure at the anode center for an arc with a frequency of 10 khz is shown in Fig. 13, in which the current density of the DC arc with a current of the average value of the high frequency pulse is expressed by the dotted line, for comparison. It is shown that the dynamic process of the anode arc pressure of a high frequency arc is as stable as the DC arc, whilst the arc pressure value of a high frequency arc is larger than that of the DC arc having the same current value as average current of the high frequency arc. 6. Conclusions A transient model has been presented to describe the heat transfer and fluid flow in a pulsed GTA welding arc. In the model, a boundary fitted coordinate system is adopted to describe precisely the cathode shape. The distribution of current density, which is determined primarily by the welding current and the cathode shape, is calculated with a combined arc plasma-cathode system. The numerical simulation of low frequency pulsed GTA welding has indicated that the arc can become steady during the peak pulse and background duration. In the peak pulse duration, the arc is typically bellshaped, whilst the arc is not bell-shaped in the background duration. The peak arc pressure at the anode center increases with increasing peak current and the background arc pressure increases with increasing background current. The peak and background arc pressure at the anode center increase with decreasing arc length and also with decreasing cathode vertex angle. With the increase of the pulse duty ratio, the period of the peak arc pressure is prolonged.

9 H.G. Fan et al. / Journal of Materials Processing Technology 72 (1997) At the start of the transition from the peak to the background current or vice versa there exists a sluggishness in the change of the arc temperature contours and arc pressure. The dynamic process of the anode arc pressure of a high frequency arc (10 khz in this study) has been found to be as stable as that of a DC arc. 7. Nomenclature B azimuthal magnetic field (Wb m 3 ) C heat capacity (J kg 1 K 1 p ) e electronic charge (C) f current frequency h plasma enthalpy (J kg 1 ) I arc current (A) I av average current I av =*I p +(1 )*I b (A) I p peak current (A) I b background current (A) j axial, radial current density (A m 2 z, j r ) k thermal conductivity (W m 1 K 1 ) K Boltzman s constant (J K 1 b ) l arc length (m) L, N total number of nodes in the - and - directions of grid pressure (Pa) Q inflow rate of shielding gas (m 3 s 1 ) R 1 cathode radius (m) R 2 internal radius of shielding nozzle (m) S c constant part of source term S p linear part of source term radiative heat loss term (W m 3 ) S R T temperature (K) t b background duration (s) t p peak pulse duration (s) T ref reference temperature (K) u, axial, radial velocity (m s 1 ) z, r axial, radial coordinates Greek letters general diffusion coefficient error viscosity (kg m 1 s 1 ) 0 permitivity of free space (H m 1 ), transformed coordinate system density (kg m 3 ) electrical conductivity (1 ohm 1 m 1 ) potential (V) general dependent variable pulse duty ratio =t p /(t b +t p ) cathode vertex angle ( ) Subscripts exp experimental cal calculated References [1] G.R. Stoeckinger, Res. Suppl. Weld J. 52 (1973) 558s 567s. [2] E.P. Vilkas, Res. Suppl. Weld. J. 49 (1970) [3] H.R. Saedi, W. Unkel, Res. Suppl. Weld J. 64 (1988) 247s 255s. [4] K.C. Hsu, K. Etemadi, E. Pfender, J. Appl. Phys. 54 (1983) [5] R.T.C. Choo, J. Szekely, R.C. Westhoff, Met. Trans. 23B (1992) [6] J.J. Lowke, P. Kovitya, H.P. Schmidt, J. Phys. D: Appl. Phys. 25 (1992) [7] J. Mckelliget, J. Szekely, Met. Trans. 17A (1986) [8] P. Kovitya, L.E. Cram, Res. Suppl. Weld J. 64 (1986) 34s 39s. [9] J.F. Lancaster, The Physics of Welding, Pergamon, Oxford, [10] S.Y. Lee, S.J. Na, Proc. Inst. Mech. Eng. Part B: J. Eng. Manu. vol. 209, 1995, pp [11] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington DC,