2.1. CFD modeling for single electrode

Wen et al. [1] modeled multi-wire SAW of thick-wall line pipe and calculated the thermal distributions under various welding conditions. Sharma et al. [2] predicted the temperature distributions and angular distortions in single-pass butt joints using three-dimensional simulations. Mahapatra et al. [3] suggested and validated a volumetric heat source model of twinwire SAW by using different electrode diameters and polarities. Kiran et al. [4] simulated a three-dimensional heat transfer of a V-groove tandem SAW process for various welding conditions using FEM. However, these studies with FEM only considered the heat conduction transfer in the welding process, which is insufficient to explain the curve weld bead such as

4 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

To overcome these disadvantages, computational fluid dynamics (CFD) is widely used to investigate molten pool flows and final weld beads because it makes it possible to approach the welding process more realistically [5]. Considering the importance of weld pool convection in the welding process, numerous researchers have attempted to analyze the heat transfer and fluid flow. Kim et al. [6] calculated the convective heat transfer and resultant temperature distributions for a filet gas metal arc welding (GMAW) process. Kim et al. [7] obtained the thermal data and analyzed the molten pool flows for various driving forces in stationary gas tungsten arc welding (GTAW). However, these studies assumed that the welding process was in a quasi-steady-state. Thus it was very difficult to approximate the droplet impingent and arc variation with alternating current (AC). Therefore, it is necessary to apply a transient analysis to the welding simulation because it can detect the free surface variation during the simulation time. One transient analysis method is the volume of fluid (VOF) method, which can track the molten pool surface; therefore, the variable models from arc plasma could be implemented in the simulations. Cho et al. [8] calculated the electromagnetic force (EMF) with mapping coordinates in V-groove GTAW and GMAW, and then applied it to the numerical simulation to obtain the dynamic molten pool behavior and final weld bead using the commercial software, Flow-3D. With the advantage of VOF transient simulation, Cho et al. [9] could calculate unstable molten pool flow patterns such as humping and overflow in V-groove positional GMAW. Cho et al. [10] obtained the heat flux distribution of the arc plasma in gas hollow tungsten arc welding (GHTAW) using the Abel inversion method and applied it to the VOF model to predict the molten zone area. Additionally, a more complex welding process can also be calculated by VOF. Cho and Na [11] conducted a laser welding simulation that included the multiple reflection and keyhole formation. Moreover, Cho and Na [12] conducted the threedimensional laser-GMA hybrid welding, which adopted the laser welding and GMAW. Han et al. [13] compared the driving forces for the weld pool dynamics in GTAW and laser welding. The VOF method could also be applied to describe the alloying element distributions and pore

The modeling and the molten pool flow analysis of SAW process are mostly conducted by Cho et al. [15–21]. Cho et al. [15] conducted molten pool analysis of SAW for single electrode for high-current (I > 500 A) condition with spray metal transfer droplet impingement. They considered electrode angle and wave form and modeled to analyze the molten pool behavior for single electrode direct current (DC) and alternative current (AC) welding signals. It was found that the penetration of weld bead is closely related with electrode angle and waveform of welding signal. Cho et al. [16] also found that droplet impingement of low-current (I < 500)

generation in the laser-GMA hybrid welding process [14].

fingertip penetration.

Figure 1 shows a schematic diagram of SAW to allow the following characteristics to be understood [15]: (a) the flux and molten slag cover the overall weld bead and, (b) the fabricated flux wall protects the flux cavity.

Although it is very difficult to observe the metal transfer of SAW, some previous studies succeeded in capturing the motion of a droplet in SAW. Franz [22] and Van Adrichem [23] observed the metal transfer through a ceramic tube using a X-ray cinematography and found that drops travel in free flight to the weld pool, or they may project sideways to collide with the molten flux wall. This metal transfer in SAW is the so-called flux-wall guided (FWG) transfer, as shown in Figure 2.

During the SAW process, a small portion of the flux is melted and consumed. Chandel [24] found that the flux consumption relies upon three sources: (a) conduction from the molten metal, (b) radiation from arc and (c) resistance heating of the slag. However, their individual 6 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

contributions to flux consumption are still unclear. In any case, the total flux consumption can be calculated by measuring the mass of the flux used. Renwick and Patchett [25] analyzed the relations between welding parameters and the flux consumption and found that flux consumption initially increased with increasing current, reached a maximum, and then decreased. Chandel [24] also measured the flux consumption of SAW with various welding parameters and showed that the flux consumption reached a peak value at 500 A and decreased at higher currents, as shown in Figure 3 [15]. This decrease at a high current is a result of the increasing current causing the droplet size to decrease. Therefore, the contact area between the droplet and the flux-wall could be decreased, as shown in Figure 4. In short, FWG transfer is difficult to observe at high current and the spray mode of transfer can be expected to be considered in

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high current SAW [15].

Figure 3. Current vs. flux consumption rate in single DC [15].

Figure 4. Expected metal transfer in single SAW process [15].

Figure 1. Schematic of SAW [15].

Figure 2. FWG transfer in SAW [15].

contributions to flux consumption are still unclear. In any case, the total flux consumption can be calculated by measuring the mass of the flux used. Renwick and Patchett [25] analyzed the relations between welding parameters and the flux consumption and found that flux consumption initially increased with increasing current, reached a maximum, and then decreased. Chandel [24] also measured the flux consumption of SAW with various welding parameters and showed that the flux consumption reached a peak value at 500 A and decreased at higher currents, as shown in Figure 3 [15]. This decrease at a high current is a result of the increasing current causing the droplet size to decrease. Therefore, the contact area between the droplet and the flux-wall could be decreased, as shown in Figure 4. In short, FWG transfer is difficult to observe at high current and the spray mode of transfer can be expected to be considered in high current SAW [15].

Figure 3. Current vs. flux consumption rate in single DC [15].

Figure 1. Schematic of SAW [15].

6 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

Figure 2. FWG transfer in SAW [15].

Figure 4. Expected metal transfer in single SAW process [15].

#### 2.1.1. Governing equations

The governing equations in the CFD simulation of a weld pool involve the mass conservation equation, momentum conservation equation (Navier-Stokes equations), and energy conservation equation. The time step used in the numerical simulation is 0.00001 s. To describe the molten pool behavior, the commercial package, Flow-3D was widely used [15, 16, 18, 21] (Table 1).

• Momentum equation

$$\frac{\partial \vec{V}}{\partial t} + \vec{V} \cdot \nabla \cdot \vec{V} = -\frac{\nabla p}{\rho} + \nu \nabla^2 \vec{V} + \frac{\dot{m}\_s}{\rho} \left( \vec{V}\_s - \vec{V} \right) + f\_b \tag{1}$$

• Mass conservation equation

$$
\nabla \cdot \overrightarrow{V} = \frac{\dot{m}\_s}{\rho} \tag{2}
$$

Symbol Nomenclature Symbol Nomenclature r Density, (solid:7.8, liquid:6.9, g/cm3) pA Arc pressure

ν Kinematic viscosity γ Surface tension

hs Enthalpy source of droplet WFR Wire feed rate

f <sup>b</sup> Body force Cp Specific heat

m\_ <sup>s</sup> Mass source of droplet rw Radius of wire, 2.0 mm h Enthalpy rd Radius of droplet, 2.1 mm

hs Enthalpy of solid To Room temperature, 298 K hsl Enthalpy between solid and solid η<sup>s</sup> Slag efficiency of SAW Ts Solidus temperature, 1768 K η<sup>d</sup> Droplet efficiency of SAW Tl Liquidus temperature, 1798 K m\_ <sup>f</sup> Flux consumption (g/s)

! Velocity vector Rc Radius of the surface curvature

! Velocity vector for mass source Cs Specific heat of liquid, 7.32x 106 erg/g s K

Modeling and Analysis of Molten Pool Behavior for Submerged Arc Welding Process with Single and Multi-Wire…

F Fraction of fluid x0, y<sup>0</sup> Location of the electrode center in x and y

! Normal vector to free surface <sup>μ</sup><sup>0</sup> Permeability of vacuum, 1.26 � <sup>10</sup><sup>6</sup> H/m qa Heat input from arc plasma <sup>μ</sup><sup>m</sup> Material permeability, 1.26 � 106 H/m qd Heat input from droplet Jz Vertical component of the current density qslag\_input Heat transfer from slag to molten pool Jr Radial component of the current density qslag\_loss Heat transfer from slag to molten pool B<sup>θ</sup> Angular component of the magnetic field η<sup>a</sup> Arc efficiency of SAW σx, σ<sup>y</sup> Effective radius of the arc in x-direction and

I Current Γ<sup>s</sup> Surface excess at saturation V Voltage R Universal gas constant

<sup>m</sup> Surface tension of pure metal at melting point a<sup>1</sup> Weight percent of sulfur

XL, XT Arc center position of x-direction for leading and

BθL, Bθ<sup>T</sup> Angular component of the magnetic field for leading and trailing electrodes

σRL, σFL Rear and front effective radii of leading arc in x-

trailing electrodes

direction

Fb Buoyancy force k<sup>1</sup> Constant related to the entropy of segregation

βij

A Negative surface tension gradient for pure metal IL, IT Current of leading and trailing electrodes ΔH<sup>0</sup> Standard heat of adsorption lL, lT Arc length of leading and trailing leading

<sup>s</sup> Volume source of droplet x1, y<sup>1</sup> Location of the arc center in x and y directions k Thermal conductivity J0 First kind of Bessel function of zero order

directions

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y-direction

electrodes

<sup>n</sup> Coefficients of effective radius model

d Distance between leading and trailing electrode

JzL, JzT Vertical component of the current density for leading and trailing arcs

V

\_

Vs

F\_

n

γ0

• Energy equation

$$\frac{\partial h}{\partial t} + \vec{V} \cdot \nabla h = \frac{1}{\rho} \nabla \cdot (k \nabla T) + \dot{h}\_{\text{s}} \tag{3}$$

where

$$\begin{aligned} hh &= \rho\_s \mathbf{C}\_s T & (T \le T\_s) \\ hh &= h(T\_s) + h\_{sl} \frac{T - T\_s}{T\_l - T\_s} & (T\_s < T \le T\_l) \\ h &= h(T\_l) + \rho\_l \mathbf{C}\_l (T - T\_l) & (T\_s < T \le T\_l) \end{aligned} \tag{4}$$

• VOF equation

$$\frac{\partial F}{\partial t} + \nabla \cdot \left(\overrightarrow{V} \, F\right) = \dot{F}\_s \tag{5}$$

#### 2.1.2. Boundary conditions

There is no heat loss from the radiation, convection and evaporation on the molten pool surface because slag and flux cover the overall weld bead as shown in Figure 1. In SAW, the heat is input from the slag to the molten pool and lost from the molten pool to the slag. However, the summation of the heat input and heat loss can be regarded as the slag heat transfer (qs), and the energy boundary condition in equation (6) is used [15, 16, 18, 21].

$$k\frac{\partial T}{\partial \cdot \vec{n}} = q\_a + q\_{slag\\_input} - q\_{slag\\_loss} = q\_a + q\_s.\tag{6}$$

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2.1.1. Governing equations

• Momentum equation

• Energy equation

• VOF equation

[15, 16, 18, 21].

2.1.2. Boundary conditions

where

• Mass conservation equation

∂V ! <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>V</sup> ! �∇ V ! ¼ � <sup>∇</sup><sup>p</sup> r

(Table 1).

The governing equations in the CFD simulation of a weld pool involve the mass conservation equation, momentum conservation equation (Navier-Stokes equations), and energy conservation equation. The time step used in the numerical simulation is 0.00001 s. To describe the molten pool behavior, the commercial package, Flow-3D was widely used [15, 16, 18, 21]

8 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

<sup>þ</sup> <sup>ν</sup>∇<sup>2</sup> <sup>V</sup> ! þ m\_ s r V ! <sup>s</sup>� V !

∇� V ! <sup>¼</sup> <sup>m</sup>\_ <sup>s</sup> r

�∇<sup>h</sup> <sup>¼</sup> <sup>1</sup> r

h ¼ rsCsT Tð Þ ≤ Ts

T � Ts Tl � Ts

h ¼ h Tð Þþ<sup>l</sup> rlClð Þ T � Tl ð Þ Ts < T ≤ Tl

There is no heat loss from the radiation, convection and evaporation on the molten pool surface because slag and flux cover the overall weld bead as shown in Figure 1. In SAW, the heat is input from the slag to the molten pool and lost from the molten pool to the slag. However, the summation of the heat input and heat loss can be regarded as the slag heat transfer (qs), and the energy boundary condition in equation (6) is used

þ ∇ � V ! F 

<sup>∇</sup> � ð Þþ <sup>k</sup>∇<sup>T</sup> \_

<sup>¼</sup> <sup>F</sup>\_

ð Þ Ts < T ≤ Tl

! <sup>¼</sup> qa <sup>þ</sup> qslag\_input � qslag\_loss <sup>¼</sup> qa <sup>þ</sup> qs: (6)

∂h ∂t þ V !

h ¼ h Tð Þþ <sup>s</sup> hsl

<sup>k</sup> <sup>∂</sup><sup>T</sup> ∂ n ∂F ∂t

þ f <sup>b</sup> (1)

hs, (3)

<sup>s</sup> (5)

(2)

(4)



Table 1. Properties and constants used in simulations.

Previous studies found that the thermal efficiency of SAW is between 0.90 and 0.99, and many studies have used a thermal efficiency of 0.95 in numerical simulations [15, 16, 18, 21]. It is reasonable to use total thermal efficiency of 0.95 with the heat transfer from the arc plasma, droplets, and molten slag to the weld pool [15, 16, 18, 21].

$$
\eta\_a = \frac{\eta\_a + \eta\_d + \eta\_s}{VI} \approx 0.95\tag{7}
$$

<sup>η</sup><sup>s</sup> <sup>¼</sup> <sup>m</sup>\_ fCpf Tm,flux � <sup>T</sup><sup>0</sup>

Modeling and Analysis of Molten Pool Behavior for Submerged Arc Welding Process with Single and Multi-Wire…

<sup>2</sup> σ<sup>y</sup> σx � �<sup>2</sup>

ð Þ x � x<sup>0</sup>

re ¼

if Ra < re ≤Rb then,

Figure 5. Schematic of droplet impingement on the wall to describe FWG transfer [16].

qs <sup>¼</sup> <sup>η</sup>sVI π R<sup>2</sup> <sup>b</sup> � <sup>R</sup><sup>2</sup> a

Figure 6. Process acquiring the arc plasma image [15].

s

� �

þ y � y<sup>0</sup> � �<sup>2</sup>

� � , where Ra <sup>¼</sup> <sup>3</sup>σq, Rb <sup>¼</sup> Ra <sup>þ</sup> <sup>3</sup>:0mm (14)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

VI (12)

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(13)

11

The pressure boundary on the free surface is applied as follows:

$$p = p\_A + \frac{\mathcal{V}}{R\_c}.\tag{8}$$

#### • Droplet model

The droplet efficiency relies on the wire feed rate and it is possible to be calculated using Eqs. (9) to (11). The droplet efficiency can be varied from the wire feed rate and welding signals [15, 16, 18, 21].

$$f\_d = \frac{3r\_w^2 WFR}{4r\_d^3} \tag{9}$$

$$q\_{d} = \frac{4}{3}\pi r\_{d}^{3}\rho[\mathbb{C}\_{s}(T\_{s} - T\_{o}) + \mathbb{C}\_{l}(T\_{d} - T\_{s}) + h\_{sl}]f\_{d'} \tag{10}$$

$$
\eta\_d = \frac{q\_d}{VI}.\tag{11}
$$

For the high current (I > 500 A), some studies proved that spray mode of metal transfer, which is very similar to droplet impingement of GMAW, can be considered as a droplet impingement model [15, 18]. However, the metal transfer of the low current (I < 500 A) can be assumed as FWG metal transfer as shown in Figure 5 [16].

• Slag heat source model

Cho et al. [15, 16, 18]. used a slag heat source model that considers the flux consumption rate, and they assumed that the distribution of the slag heat input to the material surface would be an elliptical ring as shown in Figure 6. The slag heat source model can be calculated from equation (12) to (14).

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$$\eta\_s = \frac{\dot{m}\_f \mathbb{C}\_{pf} \left( T\_{m,flux} - T\_0 \right)}{VI} \tag{12}$$

$$r\_e = \sqrt{(\mathbf{x} - \mathbf{x}\_0)^2 \left(\frac{\sigma\_y}{\sigma\_x}\right)^2 + \left(y - y\_0\right)^2} \tag{13}$$

$$\begin{aligned} \text{if } R\_a &< r\_c \le R\_b \text{ then,} \\ \eta\_s &= \frac{\eta\_s V I}{\pi \left(R\_b^2 - R\_a^2\right)}, \text{where } R\_a = 3\sigma\_{q\prime} R\_b = R\_a + 3.0mm \end{aligned} \tag{14}$$

Figure 5. Schematic of droplet impingement on the wall to describe FWG transfer [16].

Figure 6. Process acquiring the arc plasma image [15].

Previous studies found that the thermal efficiency of SAW is between 0.90 and 0.99, and many studies have used a thermal efficiency of 0.95 in numerical simulations [15, 16, 18, 21]. It is reasonable to use total thermal efficiency of 0.95 with the heat transfer from the arc plasma,

qaL, qaT Arc heat flux for leading and trailing electrodes PaL, PaT Arc heat flux for leading and trailing electrodes

VI <sup>≈</sup> <sup>0</sup>:<sup>95</sup> (7)

JrL, JrT Radial component of the current density for leading and trailing arcs

: (8)

(9)

<sup>η</sup><sup>a</sup> <sup>¼</sup> qa <sup>þ</sup> qd <sup>þ</sup> qs

<sup>p</sup> <sup>¼</sup> pA <sup>þ</sup> <sup>γ</sup>

The droplet efficiency relies on the wire feed rate and it is possible to be calculated using Eqs. (9) to (11). The droplet efficiency can be varied from the wire feed rate and welding signals

<sup>f</sup> <sup>d</sup> <sup>¼</sup> <sup>3</sup>r<sup>2</sup>

<sup>η</sup><sup>d</sup> <sup>¼</sup> qd

For the high current (I > 500 A), some studies proved that spray mode of metal transfer, which is very similar to droplet impingement of GMAW, can be considered as a droplet impingement model [15, 18]. However, the metal transfer of the low current (I < 500 A) can be assumed as

Cho et al. [15, 16, 18]. used a slag heat source model that considers the flux consumption rate, and they assumed that the distribution of the slag heat input to the material surface would be an elliptical ring as shown in Figure 6. The slag heat source model can be calculated from

Rc

<sup>w</sup>WFR 4r<sup>3</sup> d

<sup>d</sup>r Csð Þþ Ts � To Clð Þþ Td � Ts hsl ½ � f <sup>d</sup>, (10)

VI : (11)

droplets, and molten slag to the weld pool [15, 16, 18, 21].

σRT , σFT Rear and front effective radii of trailing arc in x-

Table 1. Properties and constants used in simulations.

direction

qd <sup>¼</sup> <sup>4</sup> 3 πr 3

FWG metal transfer as shown in Figure 5 [16].

• Slag heat source model

equation (12) to (14).

• Droplet model

[15, 16, 18, 21].

The pressure boundary on the free surface is applied as follows:

Symbol Nomenclature Symbol Nomenclature

10 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

• Arc heat source model

The actual arc plasma shape of SAW is very difficult to determine. Therefore, several studies assume some conditions to obtain the arc heat flux distributions [15].


Therefore, it is reasonable to apply the elliptically symmetric Gaussian arc heat flux model in Eq. (15).

$$q\_A(\mathbf{x}, y) = \eta\_A \frac{V l}{2\pi \sigma\_x \sigma\_y} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_1)^2}{2\sigma\_x^2} - \frac{(y - y\_1)^2}{2\sigma\_y^2}\right) \tag{15}$$

Fx ¼ �JzB<sup>θ</sup>

Modeling and Analysis of Molten Pool Behavior for Submerged Arc Welding Process with Single and Multi-Wire…

Fy ¼ �JzB<sup>θ</sup>

The surface tension and buoyance force models are not affected by the arc plasma distribution [14]. The buoyancy force can be modeled by the Boussinesq approximation and then expressed

A surface tension model that Sahoo et al. [27] developed for a binary Fe–S system is used to

<sup>m</sup> � A Tð Þ� � Tm RTΓsln 1 þ k1aie

For the better productivity, many industries applied two-wire or multi-electrode tandem SAW process. When the multi-wire electrodes are used, the arc shapes and the arc center positions

model the Marangoni flow. Thus, the surface tension can be expressed in Eq. (25)

are changed due to electromagnetic forces of arc plasma as shown in Figure 7.

<sup>γ</sup>ð Þ¼ <sup>T</sup> <sup>γ</sup><sup>0</sup>

Figure 7. Schematic representation of the arc images acquisition method [17].

2.2. CFD modeling for multi-wire electrodes

2.1.4. Other models

in Eq. (24).

x ra

y ra

Fz ¼ JzBθ: (23)

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Fb ¼ rgβð Þ T � T<sup>0</sup> (24)

�ΔH<sup>o</sup>

<sup>=</sup>RT (25)

(21)

13

(22)

• Arc pressure model

Due to the physical relationship, the effective radii of the arc heat flux and arc pressure are the same each other [10]. The resultant arc pressure model can be described in Eq. (16).

$$P\_A(\mathbf{x}, \mathbf{y}) = \frac{\mu I^2}{4\pi^2 \sigma\_x \sigma\_y} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_1)^2}{2\sigma\_x^2} - \frac{(\mathbf{y} - \mathbf{y}\_1)^2}{2\sigma\_y^2}\right) \tag{16}$$

#### 2.1.3. Electromagnetic force

In the arc welding process, Kou and Sun [26] found that the current density and self-induced magnetic field should be used to calculate the EMF in the molten pool. In the molten slag of a high current SAW process, however, Cho et al. [15, 16, 18] ignored the current flow effect in the molten slag because the magnitude of the current in the molten slag is tiny compared to the total current. Therefore, due to the physical relationship, the effective radius of electromagnetic force (EMF) model could be the same with that of the arc pressure and arc heat flux models [10]. For the elliptically symmetric distribution, EMF model can be calculated as follows:

$$k\_1 = \frac{\sigma\_y}{\sigma\_{\infty}}, \ (\mathbf{x} - \mathbf{x}\_1)^2 + \frac{\left(y - y\_1\right)^2}{k\_1^2} = r\_a^2 \tag{17}$$

$$J\_z = \frac{I}{2\pi} \int\_0^\infty \lambda I\_0(\lambda r\_d) \exp\left(-\lambda^2 \sigma\_r^2 / 4d\_d\right) \frac{\sinh\left[\lambda(c-z)\right]}{\sinh\left(\lambda c\right)} d\lambda \tag{18}$$

$$J\_r = \frac{I}{2\pi} \int\_0^\infty \lambda I\_1(\lambda r\_a) \exp\left(-\lambda^2 \sigma\_r^2 / 4d\right) \frac{\cosh\left[\lambda(c-z)\right]}{\sinh\left(\lambda c\right)} d\lambda \tag{19}$$

$$B\_{\theta} = \frac{\mu\_m I}{2\pi} \left[ I\_1(\lambda r\_a) \exp\left(-\lambda^2 \sigma\_a^2 / 4d\right) \frac{\sinh\left[\lambda(c-z)\right]}{\sinh\left(\lambda c\right)} d\lambda \right] \tag{20}$$

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$$F\_{\mathbf{x}} = -J\_z B\_\theta \frac{\mathbf{x}}{r\_a} \tag{21}$$

$$F\_y = -J\_z B\_\theta \frac{y}{r\_a} \tag{22}$$

$$F\_z = f\_z B\_\theta. \tag{23}$$

#### 2.1.4. Other models

• Arc heat source model

• Arc pressure model

2.1.3. Electromagnetic force

Eq. (15).

The actual arc plasma shape of SAW is very difficult to determine. Therefore, several studies

a. The shape of the arc plasma inside the flux (A) is very similar to the arc plasma outside the

Therefore, it is reasonable to apply the elliptically symmetric Gaussian arc heat flux model in

Due to the physical relationship, the effective radii of the arc heat flux and arc pressure are the

In the arc welding process, Kou and Sun [26] found that the current density and self-induced magnetic field should be used to calculate the EMF in the molten pool. In the molten slag of a high current SAW process, however, Cho et al. [15, 16, 18] ignored the current flow effect in the molten slag because the magnitude of the current in the molten slag is tiny compared to the total current. Therefore, due to the physical relationship, the effective radius of electromagnetic force (EMF) model could be the same with that of the arc pressure and arc heat flux models [10]. For the elliptically symmetric distribution, EMF model can be calculated as follows:

, xð Þ � x<sup>1</sup>

<sup>λ</sup>J0ð Þ <sup>λ</sup>ra exp �λ<sup>2</sup>

<sup>λ</sup>J1ð Þ <sup>λ</sup>ra exp �λ<sup>2</sup>

<sup>J</sup>1ð Þ <sup>λ</sup>ra exp �λ<sup>2</sup>

exp � ð Þ <sup>x</sup> � <sup>x</sup><sup>1</sup>

same each other [10]. The resultant arc pressure model can be described in Eq. (16).

2 4π<sup>2</sup>σxσ<sup>y</sup>

<sup>k</sup><sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>y</sup> σx

Jz <sup>¼</sup> <sup>I</sup> 2π ð ∞

Jr <sup>¼</sup> <sup>I</sup> 2π ð ∞

<sup>B</sup><sup>θ</sup> <sup>¼</sup> <sup>μ</sup>m<sup>I</sup> 2π ð ∞

0

0

0

exp � ð Þ <sup>x</sup> � <sup>x</sup><sup>1</sup>

2

2

!

2σ<sup>2</sup> x

<sup>2</sup> <sup>þ</sup> <sup>y</sup> � <sup>y</sup><sup>1</sup> � �<sup>2</sup> k 2 1

<sup>r</sup> <sup>=</sup>4<sup>d</sup> � � cosh ½ � <sup>λ</sup>ð Þ <sup>c</sup> � <sup>z</sup>

<sup>a</sup>=4<sup>d</sup> � � sinh ½ � <sup>λ</sup>ð Þ <sup>c</sup> � <sup>z</sup>

σ2 <sup>r</sup> =4da � � sinh ½ � λð Þ c � z

σ2

σ2

!

� <sup>y</sup> � <sup>y</sup><sup>1</sup> � �<sup>2</sup> 2σ<sup>2</sup> y

� <sup>y</sup> � <sup>y</sup><sup>1</sup> � �<sup>2</sup> 2σ<sup>2</sup> y

> ¼ r 2

sinh ð Þ λc

sinh ð Þ λc

sinh ð Þ λc

<sup>a</sup> (17)

dλ (18)

dλ (19)

dλ (20)

(15)

(16)

2σ<sup>2</sup> x

assume some conditions to obtain the arc heat flux distributions [15].

12 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

flux (B) that just escaped (within 50 ms) as shown in Figure 6.

VI 2πσxσ<sup>y</sup>

b. The metal vapor in the arc plasma root is neglected.

qAð Þ¼ x; y η<sup>A</sup>

PAð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>μ</sup><sup>I</sup>

The surface tension and buoyance force models are not affected by the arc plasma distribution [14]. The buoyancy force can be modeled by the Boussinesq approximation and then expressed in Eq. (24).

$$F\_b = \rho g \beta (T - T\_0) \tag{24}$$

A surface tension model that Sahoo et al. [27] developed for a binary Fe–S system is used to model the Marangoni flow. Thus, the surface tension can be expressed in Eq. (25)

$$\gamma(T) = \gamma\_m^0 - A(T - T\_m) - \overline{R}T\Gamma\_s \ln\left(1 + k\_1 a\_l e^{-\Lambda H''/\overline{R}T}\right) \tag{25}$$

#### 2.2. CFD modeling for multi-wire electrodes

For the better productivity, many industries applied two-wire or multi-electrode tandem SAW process. When the multi-wire electrodes are used, the arc shapes and the arc center positions are changed due to electromagnetic forces of arc plasma as shown in Figure 7.

Figure 7. Schematic representation of the arc images acquisition method [17].

$$X\_L = \mathbb{C}\_1 \left(\frac{I\_T}{I\_L}\right) \left(\frac{l\_L^2}{d}\right) \tag{26}$$

• Arc pressure model

• Slag heat source model

as shown in Figures 8 and 9.

wire SAW process can be followed [18]:

JzL <sup>¼</sup> <sup>I</sup> 2π ð ∞

JrL <sup>¼</sup> <sup>I</sup> 2π ð ∞

<sup>B</sup>θ<sup>L</sup> <sup>¼</sup> <sup>μ</sup>mIL 2π ð ∞

0

0

0

signals [15, 16, 18, 21].

2.2.3. EMF model

process.

• Droplet model

The distribution of arc pressure model is the same with that of arc heat source model [10];

Modeling and Analysis of Molten Pool Behavior for Submerged Arc Welding Process with Single and Multi-Wire…

2 L 4π<sup>2</sup>σ<sup>2</sup> AL

> 2 L 4π<sup>2</sup>σ<sup>2</sup> AL

2 T 4π<sup>2</sup>σ<sup>2</sup> AT

> 2 T 4π<sup>2</sup>σ<sup>2</sup> AT

Slag heat source model in the two-wire SAW process are the same as that of single wire SAW

Kiran et al. [17] found that the molten droplet is directed to the arc center when it is just detached. Moreover, the direction of the droplet could not be changed during the free flight. Cho et al. [18] consider that physical phenomena and then applied to the numerical simulation

The droplet efficiency relies on the wire feed rate and it is possible to be calculated using equation (9) to (11). The droplet efficiency can be varied from the wire feed rate and welding

EMF can be induced from the two different arc plasmas; therefore EMF model used in the two-

σ2 AL=4da � � sinh ½ � λð Þ c � z

σ2 AL=4da � � cosh ½ � λð Þ c � z

σ2 AL=4da � � sinh ½ � λð Þ c � z

<sup>λ</sup>J0ð Þ <sup>λ</sup>ra exp �λ<sup>2</sup>

<sup>λ</sup>J1ð Þ <sup>λ</sup>ra exp �λ<sup>2</sup>

<sup>J</sup>1ð Þ <sup>λ</sup>ra exp �λ<sup>2</sup>

exp � ð Þ <sup>x</sup> � <sup>x</sup><sup>0</sup>

exp � ð Þ <sup>x</sup> � <sup>x</sup><sup>0</sup>

exp � ð Þ <sup>x</sup> � <sup>x</sup><sup>0</sup>

exp � ð Þ <sup>x</sup> � <sup>x</sup><sup>0</sup>

2σ<sup>2</sup> RL

> 2σ<sup>2</sup> FL

2σ<sup>2</sup> RT

> 2σ<sup>2</sup> FT

sinh ð Þ λc

sinh ð Þ λc

sinh ð Þ λc

dλ (33)

dλ (34)

dλ (35)

2

2

2

2

AT !

AL !

� <sup>y</sup><sup>2</sup> 2σ<sup>2</sup>

http://dx.doi.org/10.5772/intechopen.76725

15

� <sup>y</sup><sup>2</sup> 2σ<sup>2</sup> AL ! (31)

> � <sup>y</sup><sup>2</sup> 2σ<sup>2</sup>

� <sup>y</sup><sup>2</sup> 2σ<sup>2</sup> AT ! (32)

therefore, arc pressure model can be derived in equation (31) and (32).

if <sup>x</sup> <sup>≤</sup> x0 then, PaLð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>μ</sup>0<sup>I</sup>

if <sup>x</sup> <sup>&</sup>gt; x0 then, PaLð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>μ</sup>0<sup>I</sup>

if <sup>x</sup> <sup>≤</sup> x1 then, PaTð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>μ</sup>0<sup>I</sup>

if <sup>x</sup> <sup>&</sup>gt; x1 then, PaTð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>μ</sup>0<sup>I</sup>

$$X\_T = \mathbb{C}\_2 \left(\frac{I\_L}{I\_T}\right) \left(\frac{l\_T^2}{d}\right) \tag{27}$$

$$\sigma\_{\vec{\eta}} = \beta\_0^{\vec{\eta}} + \beta\_1^{\vec{\eta}} I\_{\vec{\}} + \beta\_2^{\vec{\eta}} V\_{\vec{\}} + \beta\_3^{\vec{\eta}} X\_{\vec{\}} + \beta\_4^{\vec{\eta}} I\_{\vec{\}}^2 + \beta\_5^{\vec{\eta}} V\_{\vec{\}}^2 + \beta\_6^{\vec{\eta}} X\_{\vec{\}}^2 + \beta\_7^{\vec{\eta}} I\_{\vec{\}} V\_{\vec{\}} + \beta\_8^{\vec{\eta}} V\_{\vec{\}} X\_{\vec{\}} \tag{28}$$

#### 2.2.1. Arc interaction model

Figure 7 illustrates the procedure followed to capture the arc images. The procedure to capture the arc images of multi-wire SAW process are very similar to single wire SAW process. Initially, the leading and trailing arcs are completely submerged under the granular flux, and a CCD camera starts to record side images of the arcs at a sampling rate of 1 kHz from the instant both the arcs come out of the flux. Kiran et al. [17, 21] considered the physical models for the arc center displacement and arc shape factors in two-wire and three wire tandem SAW process. From Eqs. (26) and (27), it is possible to expect that the higher current arc plasma is more stable than the lower current arc plasma. Additionally, the lower current arc plasma can be shifted more due to the arc interaction effect with AC welding signal. Kiran et al. [17] also proposed the effective radius of arc plasma model (27), where the welding current and voltage values are used.

#### 2.2.2. Boundary conditions

The boundary conditions of single wire SAW from equation (6) to (8) are the same with that of multi-wire SAW process. However, two arc plasmas were used in the simulation so the arc heat source models, arc pressure, EMF models are different from those of the single wire model.

• Arc heat source model

As two arc plasmas were used in the simulations, two Gaussian asymmetric arc heat sources model, which contains different temperature distributions for the front and rear part in Eqs. (29) and (30). Therefore, Cho et al. [18] adopted the resultant effective radius to describe the Gaussian asymmetric arc models with DC and AC welding signals.

$$\begin{aligned} \text{if } \mathbf{x} \le \mathbf{x}\_0 \text{ then, } q\_{a\mathcal{L}}(\mathbf{x}, \mathbf{y}) &= \frac{\eta\_a V\_L I\_L}{2\pi \sigma\_{AL}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{RL}^2} - \frac{\mathbf{y}^2}{2\sigma\_{AL}^2}\right) \\\ \text{if } \mathbf{x} > \mathbf{x}\_0 \text{ then, } q\_{a\mathcal{L}}(\mathbf{x}, \mathbf{y}) &= \frac{\eta\_a V\_L I\_L}{2\pi \sigma\_{AL}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{\rm FL}^2} - \frac{\mathbf{y}^2}{2\sigma\_{AL}^2}\right) \end{aligned} \tag{29}$$

$$\begin{aligned} \text{if } \mathbf{x} \le \mathbf{x}\_1 \text{ then, } q\_{aT}(\mathbf{x}, \mathbf{y}) &= \frac{\eta\_a V\_T I\_T}{2\pi \sigma\_{AT}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{RT}^2} - \frac{y^2}{2\sigma\_{AT}^2}\right) \\\ \text{if } \mathbf{x} > \mathbf{x}\_1 \text{ then, } q\_{aT}(\mathbf{x}, \mathbf{y}) &= \frac{\eta\_a V\_T I\_T}{2\pi \sigma\_{AT}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{FT}^2} - \frac{y^2}{2\sigma\_{AT}^2}\right) \end{aligned} \tag{30}$$
