2.4.1. Thermal Boundary Condition

The initial temperature of the workpiece, the tool, and the environment is defined as 25 ◦C. During the welding process, the heat transfer between the workpiece, tool, and the environment includes convection and radiation heat transfer, as shown in Equation (15). This heat transfer mode is defined on all surfaces except the root and butt surfaces of the workpiece. The heat transfer between the workpiece and the tool is defined as convective heat transfer, as shown in Equation (16). The heat transfer between the workpiece and the bottom plate is also defined as convection, as shown in Equation (17).

$$Q\_{\rm uva} = \sigma\_b \varepsilon\_b \left( T\_w^4 - T\_a^4 \right) + h\_a \left( T\_w - T\_a \right) \tag{15}$$

$$Q\_{\rm wt} = K\_{\rm wt} \frac{\partial T}{\partial z} = h\_{\rm wt} (T\_{\rm w} - T\_t) \tag{16}$$

$$Q\_{wb} = K\_{wb} \frac{\partial T}{\partial z} = h\_{wb} (T\_w - T\_b) \tag{17}$$
