A three-dimensional (3D) Cartesian model is presented to analyze the profile and convection in the weld pool following the TP-GMAW process. A laminar flow of incompressible and Newtonian fluid is thought in the computational domain to support non-isothermal phase change between solid and liquid phases. The effect of body forces, such as the electromagnetic and buoyancy forces, are considered in the molten pool, together with Marangoni convection. The effect of plasma heat on the droplets is simplified as a constant value of the droplet’s initial temperature. To visualize the characters of the fusion zone and track the flow, the Finite Difference Method (FDM) is employed besides three governing equations of mass continuity, energy conservation, and momentum (Navier–Stokes) [7]. In addition, the Lagrangian-based volume of fluid technique (VOF) is used for computing the free surface of the molten pool [39]. By neglecting the volume variation of the phase change, the porous media drag concept is applied for modelling the flow of the mushy zone. To handle the computational domain, the commercial code, Flow-3D (v.11.2u6 general package), is used while keeping high accuracy in tracking the moving free surface and the significant optimizations of the simulation cost. A simplified model of the electromagnetic force is used in double-ellipse mode by adding a correction factor [40]. The pressure boundary condition is expressed based on Newton’s viscosity law. Arc pressure in this study is assumed to have a double-ellipse distribution, where the magnitude and the effective radii are estimated using empirical measurement [14]. The plasma jet shear is defined by an analytical solution [41] where the velocity of the jet and the plasma Reynolds number are calculated by the maximum arc pressure employing Bernoulli’s equation.
The surface of the base material is exposed to electrode heat sources while losing heat in the form of convection and radiation. In the presence of trailing and leading heat sources, the mathematical expression of heat flux (\(q\)) at the surface is given as follows:
$$K\frac{\partial T}{\partial \overrightarrow{n}}={q}_{trail}+{q}_{lead}-{h}_{c}\left(T-{T}_{0}\right)-\epsilon {e}_{S-B}\left({T}^{4}-{T}_{0}^{4}\right)$$
1
where \(K,\overrightarrow{n}, {h}_{c}, {T}_{0}, \epsilon ,\)and \({e}_{S-B}\) represent the thermal conductivity, normal vector of the free surface, convection coefficient, ambient temperature, emissivity of the workpiece surface, and the Stefan-Boltzmann constant, respectively. The heat power of each source (\(Q\)) is assumed to be equal to the sum of the arc heat rate and the droplets’ heat content as follows:
$$\eta Q=\eta {Q}_{arc}+\eta {Q}_{drop}\to \eta \stackrel{-}{UI}=\eta \left(1-{\eta }_{d}\right)\stackrel{-}{UI}+\eta {\eta }_{d}\stackrel{-}{UI}$$
2
where the \(\eta\) is the welding efficiency, \(\stackrel{-}{UI}\) represents the instantaneous average power and the \({\eta }_{d}\) represents the ratio of the droplet heat rate and average power [42]. The heat flux on the free surface is modelled as a double-ellipse distribution of the modified fixed Gaussian relation with effective radii in the rear of the \(x\) direction (\({\sigma }_{qxr}\)), the front of the \(x\) direction (\({\sigma }_{qxf}\)), and the \(y\) direction (\({\sigma }_{qy}\)) [25]:
$${q}_{arc}=\frac{\eta \left(1-{\eta }_{d}\right)UI}{2\pi {\sigma }_{q\stackrel{-}{x}}{\sigma }_{qy}}exp\left\{-\frac{{\left(x-{x}_{c}-{V}_{w}t\right)}^{2}}{2{\sigma }_{qx}^{2}}-\frac{{\left(y-{y}_{c}\right)}^{2}}{2{\sigma }_{qy}^{2}}\right\}$$
3
$${\sigma }_{qx\left(trail \right)}=\left\{\begin{array}{c}{\sigma }_{qxf}, x\ge {x}_{c}-{V}_{w}t \\ {\sigma }_{qxr}, x<{x}_{c}-{V}_{w}t\end{array}\right., {\sigma }_{q\stackrel{-}{x}}=\left(\frac{{\sigma }_{qxf}+{\sigma }_{qxr}}{2}\right)$$
4
To estimate the heat energy transferred by the droplets to the workpiece (\({Q}_{drop}\)), the heat capacity equation is employed, assuming the spray transition mode as follows:
$${Q}_{drop}=\frac{4}{3}\pi {r}_{d}^{3}{f}_{d}\left\{{\rho }_{s}{C}_{s}\left({T}_{s}-{T}_{0}\right)+\rho {L}_{f}+{\rho }_{l}{C}_{l}\left({T}_{d}-{T}_{l}\right)\right\}$$
5
where \({r}_{d}\), \({f}_{d}\), \(\rho ,\) \(C,\) and \({L}_{f}\) are the initial radius of the droplet, frequency of droplet generation, density, specific heat capacity, and latent heat of fusion, respectively.
The momentum of the workpiece vibration is another boundary condition applied in this investigation. The displacement, \({x}_{l}\), and instant velocity, \(v\), of the workpiece for a given frequency of sinusoidal vibration can be expressed as:
$${x}_{l}=\left(D/2\right)sin\left(2\pi ft\right)$$
6
$$v=D\pi fcos\left(2\pi ft\right)$$
7
where \(D\) is the peak-to-peak displacement and \(f\) is the frequency of workpiece vibration. The acceleration reaches a peak, \({G}_{p}\), when \(sin\left(2\pi ft\right)=1\), as shown in Eq. 8. The vibration is defined as the continuous change in velocity of the workpiece at fixed acceleration. The interaction of the workpiece vibration with the molten pool is investigated during the welding simulation.
$$\left|{G}_{p}\right|=2D{\left(\pi f\right)}^{2}$$
8
The 3D computational domain is divided into two regions, fluid and air-filled, where the enthalpy relation [14] is used to simulate the phase change of solid and liquid in the pseudo-fluid region. The cell’s temperature is calculated from its enthalpy, considering the convection and conduction. The total symmetric domain of 60x18x7.5 mm (parallel to X, Y, and Z directions, respectively) is assumed to be the solid phase at the start of the simulation by 5 mm thickness and 2.5 mm thickness of the void. The welding direction is parallel to the X direction. To reproduce a clear penetration zone, the domain is meshed by non-uniform rectangular cuboid cells of 0.15 mm in the central region and the utmost 0.2 mm in the surrounding. One of the most challenging parts of the tandem wire’s simulation is the impingement of the droplets from the two sources to the free surface, which occurs relatively closer than the initial distance between the two sources. The wire speed, torch angle, and arc interaction affect the droplet path. To improve the simulation efficiency, the arc interaction is simplified to only two conditions of the current mode: background-pulse and pulse-background, while the transitional effect of arc interaction is fairly ignored due to the 2.2 ms delay between pulse-charging. In addition, it is assumed that the spherical droplets generate at 1.5 mm above the top surface at a 6 mm distance versus an initial 8 mm, alongside the pseudo-parallel sources, as shown in Fig. 2(a). The TP-GMAW produces finger-type penetration, as illustrated in Fig. 2(b). The cross-sectional OM image shows that the penetration shape is not exactly symmetric, and a slight deviation can occur. Once the torch is rotated by 90 degrees around the Z axis, keeping other conditions the same (Fig. 2(c)), the deposited area is similar, while the penetration shape is completely different, and the effective distance between the leading and the trailing torches is estimated to be 6 mm. The mass balance of the feed rate (7.4 m/min) and the droplet generation frequency (150 Hz) is employed to estimate the initial radius of the droplet, 0.59 mm, while it keeps constant in all simulations. On the other hand, the current fluctuation is assumed to affect the superheated droplet’s temperature regardless of its size. The droplet generation frequency follows the current pulse frequency, which means one drop per pulse is produced in a semi-stable condition of the spray mode with respect to shielding gas [43].
The total efficiency of 68% is adopted for the GMAW process, while the ratio of the droplet heat rate per average power is estimated at 36% and consequently, the droplet efficiency, \(\eta {\eta }_{d}\), and the arc heat efficiency, \(\eta \left(1-{\eta }_{d}\right)\), are set to 24.5% and 43.5%, respectively [44]. The metal evaporation is ignored in this study due to the maximum temperature remains below the boiling temperature (2800 K) during the welding procedure. As one of the most important effects of workpiece vibration, decreasing current values with decreasing the vibration frequency was detected in both background and the pulse mode (Fig. 1(b)). To consider this fluctuation, the average background and pulse current are measured using a histogram of recorded current values during the welding. The initial temperature of the droplets is estimated at around 2500 K for W/O vibration cases based on previous studies [44]. Adopting the current change in Eqs. 2 and 5, the initial temperature of the droplets is set to 2370, 2400, and 2440 K for 50, 250, and 450 Hz of vibration frequency, respectively. The arc efficiency is fixed for both W/ and W/O vibration cases as the initial velocity of the droplets is set to 0.7 m/s equally [45]. The voltage varies from 25 to 39 V, for which the arc length of each electrode is considered to be 4.8 mm in the stable welding mode [21]. In order to estimate the double-ellipse effective radii, the arc interaction phenomenon is employed in this study introduced by Ueyama et al. [46]. Using the current values of the leading electrode at the background (96 A) and the trailing at pulsed mode (330 A), the maximum deflection of the pulsed arc is calculated at 0.56 mm. The thermophysical properties of ASTM A1011 steel applied in this simulation are listed in Table 2.
Table 2
Thermophysical properties of A1011 and the welding environment. |
Property | Value |
Ambient and the initial material temperature | 300 (K) |
Convection coefficient | 100 (W/m2.K) |
Density* | 6.4–7.8 (g/cm3) |
Density of plasma argon (10,000 K, 1 atm) | 4.6\(\times\)10−5 (g/cm3) |
Drag coefficient constant | 1 (1/s) |
Emissivity of weld material | 0.5 |
Latent heat of phase change (fusion) | 2.75×105 (J/kg) |
Magnetic permeability of material | 1.26\(\times\)10−6 (H/m) |
Maximum plasma arc pressure (330 A) | 908 (Pa) |
Solidus and Liquidus temperatures | 1698, 1778 K |
Specific Heat* | 0.50–0.95 J/g.K |
Thermal Conduction* | 26–50 (W/m.K) |
Viscosity* | 2.0-5.5 (g/m.s) |
Viscosity of plasma argon (10,000 k) | 0.05 (g/m.s) |
* Temperature-dependent | |
To improve the simulation accuracy, the physical properties are considered temperature-dependent [14]. The surface tension of the free surface is defined in a pseudo-binary Fe-S system [47] as follows:
$$\gamma =1.94-4.30\times {10}^{-4}\left(T-{T}_{m}\right)-RT\times 1.30\times {10}^{-5}\times ln\left[1+3.18\times {10}^{-3}\times {a}_{s}{e}^{\left(1.66\times {10}^{5}/RT\right)}\right]$$
9
The average sulfur of 0.028 wt% is approximated in terms of the dilution corresponding to the base metal and the filler material content, 0.018 and 0.035 wt%, respectively. Note that the gradient of surface tension \(\left(\partial \gamma ∕\partial T\right)\) changes its sign from positive at lower temperatures to negative at elevated temperatures, making a clear change in result compared to the model with a constant value of the negative gradient. The oxygen effect on the surface tension gradient is ignored due to the high amount of manganese and silicon in the filler composition [48]. A range of vibration frequencies of 50 to 450 Hz at a fixed acceleration of 1.2 m/s2 was applied to the workpiece. For generating the pulsed waveform in compliance with recorded results, a trigonometric relationship is suggested as follows:
F \(\left(I,t\right)={I}_{1}+\left(\frac{{I}_{2}-{I}_{1}}{2}\right)(1+sin(\frac{2\pi }{{P}_{d}}t-\frac{\pi }{2}\left)\right)\) (10)
where the \({I}_{1}\), \({I}_{2}\), and \({P}_{d}\) are the background current, the pulsed current, and the pulse duty of the total waveform, respectively. A similar relation is applied to generate the voltage waveform in the simulation. The recorded average power per electrode, 4.96 kJ/s, compared with the simulated average power of 4.74 kJ/s, admits the model’s accuracy within 5% error. The generation time of the droplets is synchronized with the waveform once the current pulse ends up at each cycle during the total time of the simulation. The generalized minimum residual (GMRES) method is employed as the pressure-velocity solver (with the subspace of 30 and the split Lagrangian VOF technique) is used with the explicit solver for the viscous stress. The implicit solver was considered for the heat transfer with a 1.5 factor of over-relaxation. The maximum time step is set to \({10}^{-5}\) s is limited by the convergence criteria of both surface tension and advection. Fluid fraction clean-up is the other numerical option used to decrease the possible divergence created by tiny droplets with a volume less than 5% of cell volume in the air-filled region.