# Numerical and Experimental Investigations of Laser Metal Deposition (LMD) Using STS 316L
## 3 Numerical Model  
In this study, a finite element (FE) model was developed to calculate the solidification parameters. The laser heat source has a Gaussian beam and is mathematically expressed as follows [16; 17; 18]:  
\[I=\frac{P}{\pi r^{2}}exp\left[-\frac{\left(x-vt\right)^{2}+y^{2}}{r^{2}}\right], \tag{1}\]  
where \(P\) is the power of the laser; \(r\), the radius of the laser beam; \(v\), the scanning velocity of the laser beam; and \(t\), the scanning time.  
The deposition process is modeled as transient heat transfer with a moving heat source; the three-dimensional heat transfer equation is expressed as follows:  
\[\frac{\partial\left(\rho C_{p}T\right)}{\partial t}=\frac{\partial}{\partial x }\left(k\frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial y} \left(k\frac{\partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(k \frac{\partial T}{\partial z}\right)+a\times I, \tag{2}\]  
where \(\rho\) is the mass density; \(C_{p}\), the heat capacity; \(T\), the unknown temperature; \(t\), the time; \(k\), the conductivity; and \(a\), the laser absorptivity. The initial temperature, \(T_{0}\), at time \(t=0\) is a room temperature of 27 \({}^{\circ}\)C.  
The boundary between the deposit and the substrate surface was considered to be conductive, whereas the rest of the surface boundary was considered to be radiative and convective [6]. Boundary conditions such as heat conduction, radiation, and convection are applied as follows:  
\[k_{s}\frac{\partial T}{\partial S}=q-h(T-T_{0})-\sigma\epsilon\left(T^{4}-T_{ 0}^{4}\right), \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**STS 316L**} & \multicolumn{8}{c}{**Element (wt\%)**} \\ \cline{2-10}  & **C** & **Si** & **Mn** & **P** & **S** & **Cu** & **Ni** & **Cr** & **Mo** \\ \hline Substrate & 0.016 & 0.50 & 1.25 & 0.030 & 0.001 & 0.26 & 10.09 & 16.63 & 2.05 \\ Powder & 0.01 & 0.94 & 0.05 & 0.028 & 0.001 & 0.21 & 12.56 & 16.66 & 2.38 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Chemical composition of STS 316L substrate and powder.  
Figure 1: Schematic diagram of experimental setup for the laser metal deposition (LMD) process.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & **Value** & **Parameters** & **Value** \\ \hline Laser power [W] & 630 & Spot diameter [mm] & 1 \\ Scan speed [mm/s] & 6 & Shielding gas (Ar) flow rate [L/min] & 10 \\ Powder feed rate [g/min] & 16 & Delivery gas (Ar) flow rate [L/min] & 13 \\ Interlayer time [s] & 0 & Pyrometer emissivity & 0.7 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Process parameters for LMD.  
where \(k_{s}\) is the thermal conductivity at the surface of the base metal, \(q\) is the input heat flux, \(h\) is the convection heat transfer coefficient, \(\sigma\) is the Stefan-Boltzmann constant, and \(\varepsilon\) is the surface emissivity. The thermophysical properties of STS 316L, including the density, specific heat, and thermal conductivity, can be found elsewhere [14; 19]. To minimize the computational load for designing the process parameters, such as the scanning path, the flow of the molten pool and metal transfer are not considered.  
Figure 2 shows the mesh of the full-scale numerical model used in this study. To reduce the simulation time, a symmetric half model of the deposit to the X-Z plane was applied. Further, a non-uniform mesh was applied to increase the simulation accuracy and reduce the simulation time [13; 20]. The deposit part was composed of a fine hexahedral mesh with a size of 0.0375 mm. The mesh in the base metal zone gradually became coarser as it moved away from the deposit part. The total number of elements in the model was 800,000.