## 4 Results Discussion
### The Dimension of Melt Pool from Experiment, FE and Analytical Solution  
In order to validate the analytical result, the width and depth of the melt pool is predicted by FE, analytical solution proposed in this paper, and analytical solution by Rosenthal equation, which are compared with the experimental data published from [PERSON] [12]. Usually, heat input or energy density is used to analyze the influence of process parameters on the size of the melt pool, but it can be found that the same heat input or energy density can be obtained from different process parameter combinations. Therefore, in order to provide process designers with the optimal process parameters required for printing, we separately discuss the influence of laser power and scanning speed on the size of the molten pool. The size of the melt pool is affected by laser power and scanning speed. The effect of laser power on the size of the melt pool is analyzed. In an FE simulation, process parameters are set (the laser scanning speed is set to 1000 mm/s, the substrate preheating temperature is set to 80 degrees Celsius, the layer thickness is set to 40 \(\upmu\)m and the hatch spacing is 0.1 mm), the laser power is 50 W, 100 W,150 W, 200 W, 300 W, respectively. The effect of laser power on the dimension of the melt pool is shown in the Figure 4.  
Figure 3: Cont. Figure 3: Melt pool temperature contour at top view and longitudinal view for In718 with a laser power of 100 w, scanning speed of 1 mm/s, laser spot radius of 50 \(\upmu\)m and different absorptivity ((**a**,**b**) A = 0.3, (**c**,**d**) A = 0.87). It can be found from Figure 4 that the analytical method proposed in this paper is more consistent with the experimental results and FE simulation results. At the same time, it can be found that with the increase of laser power, the width of the melt pool obtained through Rosenthal equation and FE simulation is smaller than the analytical method proposed in this paper. Because the point source is used as a heat source and the temperature dependence of the material, surface convection and radiation are ignored in the Rosenthal equation, so radiation loss is completely ignored, the width of the melt pool is smaller than the analytical method proposed in this paper. Figure 4: The dimension of melt pool of In718 by selective laser melting (SLM) processing at different laser power from the experiment by [PERSON] [12], the finite element (FE) modeling and the analytical solution (equation from this paper and the Rosenthal equation). (**a**) Depth of melt pool, (**b**) width of melt pool. of the melt pool predicted by the Rosenthal equation is overestimated. However, for the FE model established in this paper, the influence of the top surface radiation is considered. \[h_{rad}=\varepsilon\alpha\Big{(}T^{2}+T_{\infty}^{2}\Big{)}(T+T_{\infty}) \tag{13}\]  
where \(h_{rad}\) is the heart transfer coefficient for radiation, \(\sigma\) is the Stefan-Boltzmann constant, \(\varepsilon\) is the emissivity. In the FE modeling, the radiative losses are overestimated because the convection of the melt pool is neglected, so the melt pool predicted by FE modeling is narrower. In addition, it can be found from the figure that under the condition of a certain scanning speed, as the laser power increases, the deviation between the Rosenthal equation simulation results, the numerical simulation results and the experimental values gradually increases. However, the error between the analytical method proposed in this paper and the experimental results is significantly smaller than other methods, so it can be used to predict the size of the molten pool in the keyholing area. When the energy deposited in the powder layer is high enough, the evaporation of the liquid metal creates cavity in the molten area, resulting in a narrow and elongated molten pool. We know that input energy can be defined as the ratio of input power to scanning speed.