*2.2. Heat Transfer Equation*

The welded sample is viewed as a solid subjected to conduction heat transfer, with boundary conditions to model heat transfer between the sample and the surrounding environment. Then, the three-dimensional transient thermal equation is given as follows [20]:

$$
\rho C\_{\text{P}} \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (k \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (k \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (k \frac{\partial T}{\partial z}) + Q\_{\text{v}} \tag{1}
$$

where *T* is temperature, *ρ* is material density, *C*<sup>P</sup> is specific heat, *k* is thermal conductivity, and *Q*<sup>v</sup> represents volumetric heat flux in W/m3. To solve the heat equation, the thermal conductivity, density, and specific heat are required.
