• Arc pressure model

(26)

(27)

<sup>9</sup>IjXj (28)

ij <sup>8</sup>VjXj þ β

ij

XL ¼ C<sup>1</sup>

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

XT ¼ C<sup>2</sup>

σij ¼ β ij <sup>0</sup> þ β ij <sup>1</sup>Ij þ β ij <sup>2</sup>Vj þ β ij <sup>3</sup>Xj þ β ij 4I 2 <sup>j</sup> þ β ij 5V<sup>2</sup> <sup>j</sup> þ β ij 6X<sup>2</sup> <sup>j</sup> þ β ij <sup>7</sup>IjVj þ β

2.2.1. Arc interaction model

2.2.2. Boundary conditions

• Arc heat source model

model.

IT IL � � l 2 L d � �

IL IT � � l 2 T d � �

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.

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

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

> 2πσ<sup>2</sup> AL

> > 2πσ<sup>2</sup> AL

2πσ<sup>2</sup> AT

> 2πσ<sup>2</sup> AT

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

2

2

2

2

!

!

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

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

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

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

! (29)

! (30)

the Gaussian asymmetric arc models with DC and AC welding signals.

if <sup>x</sup> <sup>≤</sup> x0 then, qaLð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>η</sup>aVLIL

if <sup>x</sup> <sup>&</sup>gt; x0 then, qaLð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>η</sup>aVLIL

if <sup>x</sup> <sup>≤</sup> x1 then, qaTð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>η</sup>aVTIT

if <sup>x</sup> <sup>&</sup>gt; x1 then, qaTð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>η</sup>aVTIT

The distribution of arc pressure model is the same with that of arc heat source model [10]; therefore, arc pressure model can be derived in equation (31) and (32).

$$\begin{aligned} \text{if } \mathbf{x} \le \mathbf{x}\_0 \text{ then, } P\_{aL}(\mathbf{x}, \mathbf{y}) &= \frac{\mu\_0 I\_L^2}{4\pi^2 \sigma\_{AL}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{RL}^2} - \frac{y^2}{2\sigma\_{AL}^2}\right) \\\text{if } \mathbf{x} > \mathbf{x}\_0 \text{ then, } P\_{aL}(\mathbf{x}, \mathbf{y}) &= \frac{\mu\_0 I\_L^2}{4\pi^2 \sigma\_{AL}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{FL}^2} - \frac{y^2}{2\sigma\_{AL}^2}\right) \end{aligned} \tag{31}$$

$$\begin{aligned} \text{if } \mathbf{x} \le \mathbf{x}\_1 \text{ then, } P\_{dT}(\mathbf{x}, y) &= \frac{\mu\_0 I\_T^2}{4\pi^2 \sigma\_{AT}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{AT}^2} - \frac{y^2}{2\sigma\_{AT}^2}\right) \\\text{if } \mathbf{x} > \mathbf{x}\_1 \text{ then, } P\_{dT}(\mathbf{x}, y) &= \frac{\mu\_0 I\_T^2}{4\pi^2 \sigma\_{AT}^2} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}\_0)^2}{2\sigma\_{ET}^2} - \frac{y^2}{2\sigma\_{AT}^2}\right) \end{aligned} \tag{32}$$

• Slag heat source model

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

• Droplet model

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 as shown in Figures 8 and 9.

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 signals [15, 16, 18, 21].
