## 3.1.1 Governing equation

Laser cladding process is a typical transient heat transfer process. The transient heat source control equation satisfies the first law of thermodynamics and the Fourier heat equation[37]:

$$\rho c\frac{\partial T}{\partial t}=k\left(\frac{{\partial }^{2}T}{\partial {x}^{2}}+\frac{{\partial }^{2}T}{\partial {y}^{2}}+\frac{{\partial }^{2}T}{\partial {z}^{2}}\right)+{Q}_{laser}$$

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Where *ρ* (kg\(\cdot\)m−3) is the material density, *k* (W\(\cdot\)m−1\(\cdot K\)−1) and c (J\(\cdot\)kg−1\(\cdot\)*K*−1) respectively represent the thermal conductivity and specific heat capacity of the material, \({Q}_{laser}\)(W\(\cdot\)m2) represents the input laser energy.

## 3.1.2 Initial and boundary conditions

When the cladding process is not performed, the substrate has a uniform room temperature, which is the initial temperature.

$$T\left(x,y,z,t=0\right)={T}_{0}$$

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\({T}_{0}\) stands for room temperature and the default value is 25°C.

During laser cladding process, the heat conversion mainly includes: the heat absorbed by the metal powder, the heat lost and radiated by the convective heat exchange between the workpiece and the surrounding environment. According to the law of conservation of energy, the boundary conditions are:

-\(k\frac{\partial T}{\partial n}\)=\(h(T-{T}_{0})\) (6)

In the formula, \(T\) \(\text{a}\text{n}\text{d} {T}_{0}\) represent the boundary temperature and the room temperature, respectively, and h represents the comprehensive coefficient considering the effects of convection and radiation. The formula for calculating the comprehensive coefficient is as follows[38]:

$$h=24.1\times {10}^{-4}\epsilon {T}^{1.61 }$$

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where ε is the surface emissivity.

The top surface is set to:

-\(k\frac{\partial T}{\partial n}\)=\({Q}_{laser}-{h}_{1}\left(T-{T}_{0}\right)-\sigma \epsilon \left({T}^{4}-{T}_{0}^{4}\right)\)(8)

Where \({h}_{1}\)=100 W\(\cdot\)(m2\(\cdot \text{K}\))−1represents the convection coefficient between the molten pool and the surrounding environment, \(\sigma\)= 5.67×10− 8W\(\cdot\)(m2\(\cdot\)K4)−1represents the Stefan Boltzmann constant.

The boundary conditions at the bottom of the substrate are:

-\(k\frac{\partial T}{\partial n}\)=\({h}_{2}\left(T-{T}_{0}\right)\)(9)

Where\({ h}_{2}\) represents the convection coefficient between the substrate bottom and the worktable, \({h}_{2}\)=30 W\(\cdot\)(m2\(\cdot\)K)−1[39].

The rest of surfaces are set to the following boundary conditions:

-\(k\frac{\partial T}{\partial n}\)=\({h}_{3}\left(T-{T}_{0}\right)\)(10)

where \({h}_{2}\) is the natural convection coefficient, \({h}_{3}\)=15 W\(\cdot\)(m2\(\cdot\)K)−1[37]