**1. Introduction**

Gas metal arc welding (GMAW) is the most widely used joining process due to its ability to provide high-quality welds for a wide range of ferrous and non-ferrous alloys at low cost and high speed. As shown in **Figure 1**, GMAW is an arc-welding process that uses arc plasma between a continuously fed filler metal electrode and the workpiece to melt the electrode and the workpiece. The melted filler metal forms droplets and deposits on the partially melted

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

workpiece to form a weld pool. The weld pool solidifies to bond the workpieces after the arc moves away. A shielding gas is fed through the gas nozzle to protect the molten metal from nitrogen and oxygen in the air. GMAW is also commonly known as metal inert gas (MIG) since inert gasses argon and helium are often used as a shielding gas. An active shielding gas containing oxygen and carbon dioxide is also used and thus the GMAW process is also called metal active gas (MAG). Direct current is usually used with the filler wire as the anode electrode to increase wire melting rate. GMAW can be easily adapted for high-speed robot‐ ic, hard automation, and semiautomatic welding applications.

In these simplified models, the droplet formation is considered as an isolated process in the electrode. The influence of the arc plasma is considered as boundary conditions with assumed distributions, such as linear current density distributions [1–3] or Gaussian distributions for

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The effects of droplet impingement on the weld pool have been significantly simplified as boundary conditions in the modeling of the weld-pool dynamics by many researchers [9–21]. The weld-pool surface was assumed to be flat [9–14] or modeled with boundary-fitted coordinates [13–15]. The dynamic impingement of a droplet onto the weld pool has been omitted [13], treated as a liquid column [14] or cylindrical volumetric heat source [15–19] acting on the weld pool in many weld-pool models. Only recent models [20–29] have simulated the dynamic interaction of droplets impinging onto the weld pool including both heat transfer and fluid flow effects and tracked the deformed weld pool free surface. However, they all applied assumed current, heat flux, and arc pressure boundary conditions at the weld-pool surface and also approximated the droplet impingement with the assumed droplet shape,

In almost all aforementioned studies, the interaction of arc plasma with electrode melting, droplet generation and transfer, and weld-pool dynamics was not considered. Linear or Gaussian current density and heat flux were assumed as boundary conditions at the electrode surface [1–6] and weld-pool surface [15–29]. However, the surface of the workpiece is highly deformable, and the profile of the electrode changes rapidly, which greatly influence the arc plasma flow and thus change the current, heat flux, and momentum distribution at the surfaces of electrode and workpiece. Furthermore, the arc plasma can be dramatically distorted when there are free droplets between the electrode tip and the surface of the weld pool as observed in experimental studies [30–32]. Several models [33–39] have been developed to study the dynamic interaction of the arc plasma with the droplet formation. However, the droplet was eliminated when it was detached from the electrode tip or when it reaches the workpiece. The weld-pool dynamics was also omitted and the workpiece was treated as a flat plate. Some recent models [40–43] included the arc plasma, the filler wire, and the workpiece to study the direct interactions of the three domains. However, they are not completely coupled models since the droplet transfer in the arc still relies on an empirical formulation to calculate the

plasma drag force in [43] or the droplet impingement is not simulated in [40–42].

detachment, transfer, and impingement, and the weld-pool dynamics.

The authors developed a fully coupled comprehensive GMAW model [44–52] to include the entire welding process—the arc plasma evolution, the electrode melting, the droplet formation and detachment, the droplet transfer in the arc, the droplet impingement onto the weld pool, and the weld-pool dynamics and solidification. The volume of fluid (VOF) technique was used to track the interface of the arc plasma and the metal. The temperature, pressure, velocity, electric, and magnetic fields are calculated in the entire computational domain, including the arc, filler wire, and the workpiece without using assumed heat, current, and pressure distri‐ butions at the interfaces. In the following sections, the comprehensive mathematical model is first presented to model the GMAW physics, and then the computational results are presented to show the evolution of the arc plasma and its dynamic interaction with the droplet formation,

volume and temperature, and impinging frequency and velocity.

the current density and heat flux [6–8].

**Figure 1.** Schematic representation of a GMAW system with the computational domain shown inside the frame.

GMAW is a complex process with three major coupling events: (1) the evolution of arc plasma, (2) the dynamic process of droplet formation, detachment, and impingement onto the weld pool, and (3) the dynamics of the welding pool under the influences of the arc plasma and the periodical impingement of droplets. The stability of the GMAW process and the weld quality depend on many process parameters, such as welding current, welding voltage, wire feed speed, wire material and wire size, arc length, contact tube to workpiece distance, workpiece material and thickness, shielding gas properties, shielding gas flow rate, welding speed, etc. Selection of these welding-processing parameters relies on extensive experimentation and is an expensive trial-and-error process. Therefore, tremendous research efforts have been devoted to developing mathematical models of the GMAW process in order to reveal the underlying welding physics and provide key insights of process parameters for process optimization and defect prevention. Due to the complexity of the welding process and the associated numerical difficulty, many numerical models in the literature have simplified the GMAW process and only focused on one or two events. Many works on droplet formation [1– 8] and weld-pool dynamics [9–29] have not included the arc plasma. More works now have been devoted to study the arc plasma and its influence on the metal transfer [30–41] and weldpool dynamics [41–49].

In these simplified models, the droplet formation is considered as an isolated process in the electrode. The influence of the arc plasma is considered as boundary conditions with assumed distributions, such as linear current density distributions [1–3] or Gaussian distributions for the current density and heat flux [6–8].

workpiece to form a weld pool. The weld pool solidifies to bond the workpieces after the arc moves away. A shielding gas is fed through the gas nozzle to protect the molten metal from nitrogen and oxygen in the air. GMAW is also commonly known as metal inert gas (MIG) since inert gasses argon and helium are often used as a shielding gas. An active shielding gas containing oxygen and carbon dioxide is also used and thus the GMAW process is also called metal active gas (MAG). Direct current is usually used with the filler wire as the anode electrode to increase wire melting rate. GMAW can be easily adapted for high-speed robot‐

**Figure 1.** Schematic representation of a GMAW system with the computational domain shown inside the frame.

GMAW is a complex process with three major coupling events: (1) the evolution of arc plasma, (2) the dynamic process of droplet formation, detachment, and impingement onto the weld pool, and (3) the dynamics of the welding pool under the influences of the arc plasma and the periodical impingement of droplets. The stability of the GMAW process and the weld quality depend on many process parameters, such as welding current, welding voltage, wire feed speed, wire material and wire size, arc length, contact tube to workpiece distance, workpiece material and thickness, shielding gas properties, shielding gas flow rate, welding speed, etc. Selection of these welding-processing parameters relies on extensive experimentation and is an expensive trial-and-error process. Therefore, tremendous research efforts have been devoted to developing mathematical models of the GMAW process in order to reveal the underlying welding physics and provide key insights of process parameters for process optimization and defect prevention. Due to the complexity of the welding process and the associated numerical difficulty, many numerical models in the literature have simplified the GMAW process and only focused on one or two events. Many works on droplet formation [1– 8] and weld-pool dynamics [9–29] have not included the arc plasma. More works now have been devoted to study the arc plasma and its influence on the metal transfer [30–41] and weld-

ic, hard automation, and semiautomatic welding applications.

78 Joining Technologies

pool dynamics [41–49].

The effects of droplet impingement on the weld pool have been significantly simplified as boundary conditions in the modeling of the weld-pool dynamics by many researchers [9–21]. The weld-pool surface was assumed to be flat [9–14] or modeled with boundary-fitted coordinates [13–15]. The dynamic impingement of a droplet onto the weld pool has been omitted [13], treated as a liquid column [14] or cylindrical volumetric heat source [15–19] acting on the weld pool in many weld-pool models. Only recent models [20–29] have simulated the dynamic interaction of droplets impinging onto the weld pool including both heat transfer and fluid flow effects and tracked the deformed weld pool free surface. However, they all applied assumed current, heat flux, and arc pressure boundary conditions at the weld-pool surface and also approximated the droplet impingement with the assumed droplet shape, volume and temperature, and impinging frequency and velocity.

In almost all aforementioned studies, the interaction of arc plasma with electrode melting, droplet generation and transfer, and weld-pool dynamics was not considered. Linear or Gaussian current density and heat flux were assumed as boundary conditions at the electrode surface [1–6] and weld-pool surface [15–29]. However, the surface of the workpiece is highly deformable, and the profile of the electrode changes rapidly, which greatly influence the arc plasma flow and thus change the current, heat flux, and momentum distribution at the surfaces of electrode and workpiece. Furthermore, the arc plasma can be dramatically distorted when there are free droplets between the electrode tip and the surface of the weld pool as observed in experimental studies [30–32]. Several models [33–39] have been developed to study the dynamic interaction of the arc plasma with the droplet formation. However, the droplet was eliminated when it was detached from the electrode tip or when it reaches the workpiece. The weld-pool dynamics was also omitted and the workpiece was treated as a flat plate. Some recent models [40–43] included the arc plasma, the filler wire, and the workpiece to study the direct interactions of the three domains. However, they are not completely coupled models since the droplet transfer in the arc still relies on an empirical formulation to calculate the plasma drag force in [43] or the droplet impingement is not simulated in [40–42].

The authors developed a fully coupled comprehensive GMAW model [44–52] to include the entire welding process—the arc plasma evolution, the electrode melting, the droplet formation and detachment, the droplet transfer in the arc, the droplet impingement onto the weld pool, and the weld-pool dynamics and solidification. The volume of fluid (VOF) technique was used to track the interface of the arc plasma and the metal. The temperature, pressure, velocity, electric, and magnetic fields are calculated in the entire computational domain, including the arc, filler wire, and the workpiece without using assumed heat, current, and pressure distri‐ butions at the interfaces. In the following sections, the comprehensive mathematical model is first presented to model the GMAW physics, and then the computational results are presented to show the evolution of the arc plasma and its dynamic interaction with the droplet formation, detachment, transfer, and impingement, and the weld-pool dynamics.

### **2. Mathematical model**

#### **2.1. Governing equations**

The computational domain is shown in **Figure 1**, which has an anode region, an arc region, and a cathode region. The governing equations for the arc, the electrode, and the workpiece can be written in a single set based on the continuum formulation given by Diao and Tsai [53]:

Mass continuity

$$\frac{\partial}{\partial t}(\rho) + \nabla \cdot (\rho V) = 0 \tag{1}$$

where *h* is the enthalpy, *k* is the thermal conductivity, *c* is the specific heat, *SR* is the radiation heat loss, *kb* is the Stefan-Boltzmann constant, σ*e* is the electrical conductivity, and *e* is the

> <sup>1</sup> () 0 *<sup>r</sup> rr r z* f

¶¶ ¶ Ñ= + =

, *re ze J J r z* f

> 0 0 *r B J rdr <sup>z</sup> <sup>r</sup>*

The continuum model [53] included the first- and second-order drag forces and the interaction between the solid and liquid phases due to the relative velocity in the mushy zone (0< *f <sup>l</sup>* <1 and 0< *f <sup>s</sup>* <1), which are represented by the corresponding third to fifth terms in the right-hand side of Eqs. (2) and (3). The energy flux due to the relative phase motion in the mushy zone is represented as the second term in the right-hand side of Eq. (4). The enthalpy method is used for phase change during the fusion and solidification processes. The enthalpy for the solid and

Continuum density (*ρ*), specific heat (*c*), thermal conductivity (*k*), velocity (*V*), and enthalpy

, , *s s l l ss ll s s l l*

 r

> r

= + =+ = + *g g c fc fc k k g kg*

, *s s l l*

*l s g g f f* r

r

m

s

q

where *ϕ* is the electrical potential and *μ0* is the magnetic permeability.

2

f

2

 f

2

 f

 s

¶¶ ¶ (5)

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¶ ¶ =- =- ¶ ¶ (6)

<sup>=</sup> ò (7)

, () *s s l l s ls h cT h cT c c T H* = = +- + (8)

= = (9)

electronic charge.

Current continuity

Maxwell's equation

liquid phases can be expressed as

where *H* is the latent heat of fusion.

rr

 r

(*h*) are defined as follows:

Ohm's law

Momentum

$$\begin{split} \frac{\partial}{\partial t}(\rho u) + \nabla \cdot (\rho V u) &= \nabla \cdot (\mu\_l \frac{\rho}{\rho\_l} \nabla u) - \frac{\partial p}{\partial r} - \frac{\mu\_l}{K} \frac{\rho}{\rho\_l} (u - u\_s) - \frac{C \rho^2}{K^{12} \rho\_l} |u - u\_s| \left( \mu - u\_s \right) \\ - \nabla \cdot (\rho f\_s f\_s' V \mu\_r) - J\_z \times B\_\theta \end{split} \tag{2}$$

$$\begin{split} \frac{\partial}{\partial t} (\rho \mathbf{v}) + \nabla \cdot (\rho \mathbf{V} \mathbf{v}) &= \nabla \cdot (\mu\_l \frac{\rho}{\rho\_l} \nabla \mathbf{v}) - \frac{\partial p}{\partial z} - \frac{\mu\_l}{K} \frac{\rho}{\rho\_l} (\mathbf{v} - \mathbf{v}\_s) - \frac{C \rho^2}{K^{12} \rho\_l} |\mathbf{v} - \mathbf{v}\_s| \left(\mathbf{v} - \mathbf{v}\_s\right) \\ - \nabla \cdot (\rho f\_s f\_l' V\_r \mathbf{v}\_r) + \rho \mathbf{g} \, \beta\_l (T - T\_0) + J\_r \times B\_\theta \end{split} \tag{3}$$

where *V* is the velocity vector, and *u* and *v* are the velocities in the *r* and *z* directions, respec‐ tively; *Vr* is the relative velocity vector between the liquid phase and the solid phase. The subscripts *s* and *l* refer to the solid and liquid phases, respectively, and the subscript *0* represents the initial condition. *g* is the gravitational acceleration, *p* is the pressure, *ρ* is the density, *μ* is the viscosity, *β<sup>T</sup>* is the thermal expansion coefficient, *T* is the temperature, *Jr* and *Jz* are current densities in the respective r and z directions and *Bθ* is the self-induced electro‐ magnetic field. *K* is the permeability function, *C* is the inertial coefficient, and *f* is the mass fraction.

Energy

$$\begin{aligned} \frac{\partial}{\partial t}(\rho h) + \nabla \cdot (\rho V h) &= \nabla \cdot \left(\frac{k}{c\_s} \nabla h\right) + \nabla \cdot \left(\frac{k}{c\_s} \nabla (h\_s - h)\right) - \nabla \cdot (\rho (V - V\_s)(h\_l - h)) \\ - \Delta H \frac{\partial f\_l}{\partial t} + \frac{J\_r^2 + J\_z^2}{\sigma\_s} - S\_\kappa + \frac{5k\_b}{e} \frac{J\_r}{c\_s} \frac{\partial h}{\partial r} + \frac{J\_z}{c\_s} \frac{\partial h}{\partial \overline{z}} \end{aligned} \tag{4}$$

where *h* is the enthalpy, *k* is the thermal conductivity, *c* is the specific heat, *SR* is the radiation heat loss, *kb* is the Stefan-Boltzmann constant, σ*e* is the electrical conductivity, and *e* is the electronic charge.

Current continuity

$$\nabla^2 \phi = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial \phi}{\partial r}) + \frac{\partial^2 \phi}{\partial z^2} = 0 \tag{5}$$

Ohm's law

**2. Mathematical model**

The computational domain is shown in **Figure 1**, which has an anode region, an arc region, and a cathode region. The governing equations for the arc, the electrode, and the workpiece can be written in a single set based on the continuum formulation given by Diao and Tsai [53]:

() ( ) 0

1/2 ( ) ( ) ( ) ( ) | |( )

() ( ) ( ) ( ) | |( )

 mr

q

( ) ( ) ( ) ( ( )) ( ( )( ))

*k k h h h hh h h*

*V V V*

<sup>5</sup> ( )

¶ +Ñ× =Ñ× Ñ +Ñ× Ñ - -Ñ× - -

*s s*

*p C vv v vv vv vv <sup>t</sup> zK K*

¶ ¶ +Ñ× =Ñ× Ñ - - - - - -

*l*

where *V* is the velocity vector, and *u* and *v* are the velocities in the *r* and *z* directions, respec‐ tively; *Vr* is the relative velocity vector between the liquid phase and the solid phase. The subscripts *s* and *l* refer to the solid and liquid phases, respectively, and the subscript *0* represents the initial condition. *g* is the gravitational acceleration, *p* is the pressure, *ρ* is the density, *μ* is the viscosity, *β<sup>T</sup>* is the thermal expansion coefficient, *T* is the temperature, *Jr* and *Jz* are current densities in the respective r and z directions and *Bθ* is the self-induced electro‐ magnetic field. *K* is the permeability function, *C* is the inertial coefficient, and *f* is the mass

*l ll*

 mr

*p C uu u uu uu uu <sup>t</sup> rK K*

¶ ¶ +Ñ× =Ñ× Ñ - - - - - -

*l*

*l ll*

*l s s s*

*l s s s*

rr

rr

¶ +Ñ× =

 r

*V* (1)

(2)

(3)

(4)

2

2 1/2

 r

*s s l*

r

 r

*t* r

¶

r

r

0

r

r

**2.1. Governing equations**

Mass continuity

80 Joining Technologies

Momentum

fraction.

Energy

( )

rr

*V*

r

r

¶


r

rr

*sl rr z*

*ff u J B*

*V*

*V*

¶ ¶

( ) ()

2 2

s

 r

*lrz br z R*

*f J J kJ h J h H S t ecr cz*

¶ + ¶¶ -D + - + + ¶ ¶ ¶

*t cc*

*e s s*

 rb

¶ ¶ -Ñ × + - + ´

*sl rr T r*

*ff v g T T J B*

 m

q

 m

$$J\_r = -\sigma\_\circ \frac{\partial \phi}{\partial r}, J\_z = -\sigma\_\circ \frac{\partial \phi}{\partial z} \tag{6}$$

Maxwell's equation

$$B\_{\theta} = \frac{\mu\_0}{r} \int\_0^r J\_z r dr \tag{7}$$

where *ϕ* is the electrical potential and *μ0* is the magnetic permeability.

The continuum model [53] included the first- and second-order drag forces and the interaction between the solid and liquid phases due to the relative velocity in the mushy zone (0< *f <sup>l</sup>* <1 and 0< *f <sup>s</sup>* <1), which are represented by the corresponding third to fifth terms in the right-hand side of Eqs. (2) and (3). The energy flux due to the relative phase motion in the mushy zone is represented as the second term in the right-hand side of Eq. (4). The enthalpy method is used for phase change during the fusion and solidification processes. The enthalpy for the solid and liquid phases can be expressed as

$$h\_s = \mathbf{c}\_s T,\\ h\_l = \mathbf{c}\_l T + (\mathbf{c}\_s - \mathbf{c}\_l) T\_s + H \tag{8}$$

where *H* is the latent heat of fusion.

Continuum density (*ρ*), specific heat (*c*), thermal conductivity (*k*), velocity (*V*), and enthalpy (*h*) are defined as follows:

> r

r

$$
\rho = \rho\_s \mathbf{g}\_s + \rho\_l \mathbf{g}\_l,\\
\mathbf{c} = f\_s \mathbf{c}\_s + f\_l \mathbf{c}\_l,\\
k = k\_s \mathbf{g}\_s + k\_l \mathbf{g}\_l
$$

$$
f\_l = \frac{\rho\_s \mathbf{g}\_s}{\sigma}, \quad f\_s = \frac{\rho\_l \mathbf{g}\_l}{\sigma} \tag{9}
$$

$$\mathcal{V} = f\_s \mathcal{V}\_s + f\_l \mathcal{V}\_{l'} \quad h = f\_s h\_s + f\_l h\_{l'}$$

where *g* is the volume fraction of the solid or liquid phase.

The permeability function is assumed to be analogous to fluid flow in porous media employing the Carman-Kozeny equation [54, 55]

$$K = \frac{\text{g}\_{l}^{\text{3}}}{\text{c}\_{1}\text{(l}-\text{g}\_{l})^{2}}, \quad \text{c}\_{1} = \frac{180}{d^{2}}\tag{10}$$

**2.4. Forces at the interface of the arc plasma and metal regions**

surface

where *n*

where *s*

surface and is given by [7]

from the results in the arc region.

k

The molten metal is subject to body forces and surfaces forces at the interface of the arc plasma and metal regions. The body forces include gravity, buoyancy force, and electromagnetic force. The surface forces include arc plasma shear stress, arc pressure, surface tension due to surface curvature, and Marangoni shear stress due to temperature difference. The surface forces are included as source terms to the momentum equations according to the CSF (continuum surface force) model [59–61]. Using *F* of the VOF function as the characteristic function, the surface

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forces are transformed to the localized body forces and added in the free surface cells.

*ps*

Surface tension pressure is normal to the free surface and can be expressed as [60]

*<sup>s</sup> p* = gk

where *γ* is the surface tension coefficient. The free surface curvature *κ* is given by

<sup>→</sup> is the surface normal, calculated as the gradient of the VOF function

*n F* = Ñ

*Ms*

t

<sup>→</sup> is a tangent vector of the local free surface.

The temperature-dependent Marangoni shear stress is in a direction tangential to the local free

*T T s* g

¶ ¶ <sup>=</sup> ¶ ¶

 m

¶ = ¶ r **V**

t

The arc plasma shear stress is calculated from the velocities of the arc plasma cells at the free

*s*

where *μ* is the viscosity of the arc plasma. The arc pressure at the metal surface is also obtained

<sup>1</sup> ( ) )| ( ) || || || *n n n n n nn*

é ùæ ö é ù = - Ñ× = ×Ñ - Ñ× ê ú ê ú ç ÷ ë ûè ø ë û

r (13)

(14)

r r r r r rr (15)

<sup>r</sup> (16)

<sup>r</sup> <sup>r</sup> (17)

where *d* is proportional to the dendrite dimension. In this study, it is assumed to be a constant and is on the order of 10−2 cm.

The inertial coefficient, *C*, is calculated from [56]

$$C = 0.1 \,\text{kg}^{-3/2} \tag{11}$$

#### **2.2. Arc region**

The arc region includes the arc plasma column and the surrounding shielding gas. The arc plasma is assumed to be in local thermodynamic equilibrium (LTE) [57]. The plasma proper‐ ties, including enthalpy, density, viscosity, specific heat, thermal conductivity, and electrical conductivity, are calculated from an equilibrium composition [57, 58]. The influence of metal vapor on plasma material properties [37–42] is not considered in the present study. The plasma is also assumed to be optically thin, thus the radiation may be modeled as a radiation heat loss per unit volume represented by *SR* in Eq. (4) [57, 58].

#### **2.3. Metal region and tracking of free surfaces**

The metal region includes the electrode, droplet in the arc, and the workpiece. The dynamic evolution of the droplet formation of the electrode tip, the droplet transfer in the arc, and the weld-pool dynamics require precise tracking of the free surface of the metal region. The volume of fluid method is used to track the moving free surface [59]. A volume of fluid function, *F*(*r,z,t*), is used to track the location of the free surface. This function represents the volume of fluid per unit volume and satisfies the following equation:

$$\frac{dF}{dt} = \frac{\partial F}{\partial t} + (V \cdot \nabla)F = 0\tag{12}$$

The average value of *F* in a cell is equal to the volume fraction of the cell occupied by the metal. A zero value of *F* indicates that a cell contains no metal, whereas a unit value indicates that the cell is full of metal. Cells with *F* values between zero and one are partially filled with metal.

#### **2.4. Forces at the interface of the arc plasma and metal regions**

, *ss ll ss ll VVV* =+ =+ *f f h fh fh*

The permeability function is assumed to be analogous to fluid flow in porous media employing

2 2 1

*cg d* = = - (10)


*V* (12)

<sup>180</sup> , (1 )

where *d* is proportional to the dendrite dimension. In this study, it is assumed to be a constant

The arc region includes the arc plasma column and the surrounding shielding gas. The arc plasma is assumed to be in local thermodynamic equilibrium (LTE) [57]. The plasma proper‐ ties, including enthalpy, density, viscosity, specific heat, thermal conductivity, and electrical conductivity, are calculated from an equilibrium composition [57, 58]. The influence of metal vapor on plasma material properties [37–42] is not considered in the present study. The plasma is also assumed to be optically thin, thus the radiation may be modeled as a radiation heat loss

The metal region includes the electrode, droplet in the arc, and the workpiece. The dynamic evolution of the droplet formation of the electrode tip, the droplet transfer in the arc, and the weld-pool dynamics require precise tracking of the free surface of the metal region. The volume of fluid method is used to track the moving free surface [59]. A volume of fluid function, *F*(*r,z,t*), is used to track the location of the free surface. This function represents the volume of

( )0 *dF F <sup>F</sup>*

¶ = + ×Ñ =

The average value of *F* in a cell is equal to the volume fraction of the cell occupied by the metal. A zero value of *F* indicates that a cell contains no metal, whereas a unit value indicates that the cell is full of metal. Cells with *F* values between zero and one are partially filled with metal.

*dt t*

¶

3/2 0.13 *C gl*

3

*l l <sup>g</sup> K c*

1

where *g* is the volume fraction of the solid or liquid phase.

the Carman-Kozeny equation [54, 55]

and is on the order of 10−2 cm.

**2.2. Arc region**

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The inertial coefficient, *C*, is calculated from [56]

per unit volume represented by *SR* in Eq. (4) [57, 58].

fluid per unit volume and satisfies the following equation:

**2.3. Metal region and tracking of free surfaces**

The molten metal is subject to body forces and surfaces forces at the interface of the arc plasma and metal regions. The body forces include gravity, buoyancy force, and electromagnetic force. The surface forces include arc plasma shear stress, arc pressure, surface tension due to surface curvature, and Marangoni shear stress due to temperature difference. The surface forces are included as source terms to the momentum equations according to the CSF (continuum surface force) model [59–61]. Using *F* of the VOF function as the characteristic function, the surface forces are transformed to the localized body forces and added in the free surface cells.

The arc plasma shear stress is calculated from the velocities of the arc plasma cells at the free surface

$$
\stackrel{\cdot}{\sigma}\_{\mu\circ} = \mu \frac{\partial \mathbf{V}}{\partial \mathbf{s}} \tag{13}
$$

where *μ* is the viscosity of the arc plasma. The arc pressure at the metal surface is also obtained from the results in the arc region.

Surface tension pressure is normal to the free surface and can be expressed as [60]

$$
\mathbf{p}\_s = \mathbf{\hat{x}}\mathbf{\hat{x}}\tag{14}
$$

where *γ* is the surface tension coefficient. The free surface curvature *κ* is given by

$$\boldsymbol{\kappa} = -\left[\boldsymbol{\nabla} \cdot (\boldsymbol{\frac{\vec{n}}{|\boldsymbol{\vec{n}}|})} \right] = \frac{1}{|\boldsymbol{\vec{n}}|} \left[ \left( \frac{\vec{n}}{|\boldsymbol{\vec{n}}|} \cdot \boldsymbol{\nabla} \right) |\boldsymbol{\vec{n}}| - (\boldsymbol{\nabla} \cdot \boldsymbol{\vec{n}}) \right] \tag{15}$$

where *n* <sup>→</sup> is the surface normal, calculated as the gradient of the VOF function

$$
\vec{n} = \nabla F \tag{16}
$$

The temperature-dependent Marangoni shear stress is in a direction tangential to the local free surface and is given by [7]

$$
\stackrel{\rightharpoonup}{\tau}\_{Ms} = \frac{\stackrel{\rightharpoonup}{\mathcal{O}}\mathcal{Y}}{\stackrel{\rightharpoonup}{\mathcal{O}}\stackrel{\rightharpoonup}{\mathcal{O}}\mathcal{S}}{\stackrel{\rightharpoonup}{\mathcal{O}}\mathcal{S}}\tag{17}
$$

where *s* <sup>→</sup> is a tangent vector of the local free surface.

#### **2.5. Energy terms at the interface of the arc plasma and metal regions**

#### *2.5.1. Plasma-anode interface*

The anode sheath region at the plasma-electrode interface is a very thin region, about 0.02-mm thick [57], and is at nonlocal thermal equilibrium. The very thin region is treated as a special interface by adding energy source terms, *S*a in the metal region and *Sap* in the arc region:

$$\delta S\_a = \frac{k\_{\text{eff}}(T\_{\text{ave}} - T\_a)}{\delta} + J\_a \phi\_w - \varepsilon k\_b T\_a^4 - q\_{\text{av}} H\_{\text{av}} \tag{18}$$

$$S\_{ap} = -\frac{k\_{eg}(T\_{av} - T\_a)}{\delta} \tag{19}$$

*2.5.2. Plasma-cathode interface*

of the cathode and is taken as 0.1 mm.

**2.6. External boundary conditions**

terms:

Similarly, energy source terms *S*c and *S*cp are added to the corresponding metal and arc regions at the plasma-cathode interface, taken into account the conduction, radiation, and evaporation

> <sup>4</sup> ( ) *eff arc c c ev ev b c*

> > ( ) *eff arc c*

d

where *Tc* is the metal surface temperature at the cathode surface, *k*eff is the harmonic mean of the thermal conductivities of the arc plasma and the cathode materials, and *δ* is the thickness

The computational domain for a two-dimensional (2D) axisymmetric GMAW system is shown as ABCDEFGA in **Figure 1**. The external boundary conditions are listed in **Table 1**. Symmet‐

**AB BC CD DE EF FG GA**

*h T* = 300 K *T* = 300 K *T* = 300 K *T* = 300 K *T* = 300 K *T* = 300 K ∂ *T*

<sup>∂</sup> *<sup>z</sup>* =0 <sup>∂</sup>*<sup>ϕ</sup>*

The velocity boundary takes into account the wire feed rate at AB, shielding gas inlet at BC, open boundaries at CD and DE, and non-slip wall condition at EF. The inflow of shielding gas from the nozzle at BC is represented by a fully developed axial velocity profile for lami‐

*kT T*

e


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<sup>∂</sup> *<sup>r</sup>* =0 <sup>0</sup> <sup>0</sup> <sup>0</sup>

<sup>∂</sup> *<sup>r</sup>* =0 *<sup>ϕ</sup>* =0 *<sup>ϕ</sup>* =0 <sup>∂</sup>*<sup>ϕ</sup>*

<sup>∂</sup> *<sup>r</sup>* =0

<sup>∂</sup> *<sup>r</sup>* =0

<sup>∂</sup> *<sup>r</sup>* =0

<sup>∂</sup> *<sup>z</sup>* =0 <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>∂</sup> *<sup>v</sup>*

*S q H kT*

*kT T*

*cp*

*S*

rical boundary condition is assigned along the centerline AG.

*u* 0 0 0 ∂ (*ρu*)

∂*ϕ*

**Table 1.** Boundary conditions on the outer boundaries.

nar flow in a concentric annulus [64]:

<sup>∂</sup> *<sup>z</sup>* =0 <sup>∂</sup>*<sup>ϕ</sup>*

*v vw* Eq. (24) ∂ (*ρv*)

*<sup>ϕ</sup>* <sup>−</sup>*<sup>σ</sup>*

∂*ϕ* <sup>∂</sup> *<sup>z</sup>* <sup>=</sup> *<sup>I</sup> πRc* 2 d

where *T*arc and *Ta* are the respective arc plasma and metal temperature at the plasma-anode interface, *k*eff is the harmonic mean of the thermal conductivities of the arc plasma and the anode materials, *δ* is the thickness of the anode sheath region and is taken as 0.1 mm according to the maximum thickness observed by experiments [62], *ϕw* is the work function of the anode material, *Ja* is the anode current calculated as the square root of *Jr* <sup>2</sup> and *<sup>J</sup> <sup>z</sup>* 2 , *ε* is the metal surface emissivity and *kb* is the Stefan-Boltzmann constant, *H*ev is the latent heat of vaporization of metal vapor, and *q*ev is the mass rate of evaporation at the metal surface. The mass rate of evaporation of metal, *q*ev for steel can be expressed as [63]

$$
\log(q\_{\rm ev}) = A\_{\rm v} + \log P\_{\rm am} - 0.5 \log T \tag{20}
$$

$$
\log P\_{atm} = 6.121 - \frac{18836}{T} \tag{21}
$$

The four terms in Eq. (18) take into account thermal conduction, electron heating associated with the work function of the anode material, black-body radiation heat loss, and evapora‐ tion heat loss, respectively, at the metal surface. The energy equation for the plasma region only considers the cooling effects through conduction.

#### *2.5.2. Plasma-cathode interface*

**2.5. Energy terms at the interface of the arc plasma and metal regions**

*kT T*

d

*ap*

*S*

material, *Ja* is the anode current calculated as the square root of *Jr*

evaporation of metal, *q*ev for steel can be expressed as [63]

only considers the cooling effects through conduction.

The anode sheath region at the plasma-electrode interface is a very thin region, about 0.02-mm thick [57], and is at nonlocal thermal equilibrium. The very thin region is treated as a special interface by adding energy source terms, *S*a in the metal region and *Sap* in the arc region:

( ) *eff arc a*

d

where *T*arc and *Ta* are the respective arc plasma and metal temperature at the plasma-anode interface, *k*eff is the harmonic mean of the thermal conductivities of the arc plasma and the anode materials, *δ* is the thickness of the anode sheath region and is taken as 0.1 mm according to the maximum thickness observed by experiments [62], *ϕw* is the work function of the anode

emissivity and *kb* is the Stefan-Boltzmann constant, *H*ev is the latent heat of vaporization of metal vapor, and *q*ev is the mass rate of evaporation at the metal surface. The mass rate of

The four terms in Eq. (18) take into account thermal conduction, electron heating associated with the work function of the anode material, black-body radiation heat loss, and evapora‐ tion heat loss, respectively, at the metal surface. The energy equation for the plasma region

*kT T*



<sup>2</sup> and *<sup>J</sup> <sup>z</sup>* 2

log( ) log 0.5log *ev v atm qA P T* =+ - (20)

<sup>18836</sup> log 6.121 *Patm <sup>T</sup>* = - (21)

, *ε* is the metal surface

<sup>4</sup> ( ) *eff arc a a a w b a ev ev*

*S J kT q H* f e

*2.5.1. Plasma-anode interface*

84 Joining Technologies

Similarly, energy source terms *S*c and *S*cp are added to the corresponding metal and arc regions at the plasma-cathode interface, taken into account the conduction, radiation, and evaporation terms:

$$S\_c = \frac{k\_{\rm eff}(T\_{\rm arc} - T\_c)}{\delta} - q\_w H\_{\rm av} - \varepsilon k\_b T\_c^4 \tag{22}$$

$$S\_{cp} = -\frac{k\_{eg}(T\_{ac} - T\_c)}{\delta} \tag{23}$$

where *Tc* is the metal surface temperature at the cathode surface, *k*eff is the harmonic mean of the thermal conductivities of the arc plasma and the cathode materials, and *δ* is the thickness of the cathode and is taken as 0.1 mm.

#### **2.6. External boundary conditions**

The computational domain for a two-dimensional (2D) axisymmetric GMAW system is shown as ABCDEFGA in **Figure 1**. The external boundary conditions are listed in **Table 1**. Symmet‐ rical boundary condition is assigned along the centerline AG.


**Table 1.** Boundary conditions on the outer boundaries.

The velocity boundary takes into account the wire feed rate at AB, shielding gas inlet at BC, open boundaries at CD and DE, and non-slip wall condition at EF. The inflow of shielding gas from the nozzle at BC is represented by a fully developed axial velocity profile for lami‐ nar flow in a concentric annulus [64]:

$$\text{cov}(r) = -\frac{2Q}{\pi} \frac{R\_u^2 - r^2 + (R\_u^2 - R\_u^2) \frac{\ln(r \ / R\_u)}{\ln(R\_u \ / R\_u)}}{R\_u^4 - R\_u^4 + \frac{\left(R\_u^2 - R\_u^2\right)^2}{\ln(R\_u \ / R\_u)}} + V\_w \frac{\ln(R\_u \ / r)}{\ln(R\_u \ / R\_u)}\tag{24}$$

the solid and liquid mild steel taken from [7] and other parameters used in the computa‐

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**Figure 2.** Temperature-dependent material properties of argon and the volume radiation heat loss taken from [58].

**Nomenclature Symbol Value (unit)** Constant in Eq. (20) *Av* 2.52

Specific heat of solid phase *cs* 700 (J kg‒1 K‒1) Specific heat of liquid phase *cl* 780 (J kg‒1 K‒1) Thermal conductivity of solid phase *ks* 22 (W m‒1 K‒1) Thermal conductivity of liquid phase *kl* 22 (W m‒1 K‒1) Density of solid phase *ρ<sup>s</sup>* 7200 (kg m‒3) Density of liquid phase *ρ<sup>l</sup>* 7200 (kg m‒3) Thermal expansion coefficient *β<sup>T</sup>* 4.95×10‒5 (K‒1)

Radiation emissivity *ε* 0.4

Dynamic viscosity *μl* 0.006 (kg m‒1 s‒1) Latent heat of fusion *H* 2.47×10<sup>5</sup>

(J kg‒1)

tion.

where *Q* is the shielding gas flow rate, *Vw* is the wire feed rate, *Rw* and *Rn* are the radius of the electrode and the internal radius of the shielding gas nozzle, respectively.

The temperature boundaries along AD, DE, and EG are set as the room temperature. The boundary conditions for current flow include a zero voltage at the bottom of the workpiece FG, uniform current density along AB specified as *J <sup>z</sup>* = −*σ<sup>e</sup>* ∂*ϕ* <sup>∂</sup> *<sup>z</sup>* <sup>=</sup> *<sup>I</sup> πRw* <sup>2</sup> , and zero current flow along the other surfaces.
