*3.1. Temperature Profile, Maximum Temperature, and Surface Temperature*

In this section, the temperature profile, maximum temperature and surface temperature are predicted and compared to the experimental results. A moving heat source analysis is used in order to predict the temperature distribution associated with the dynamic heat deposition. The explicit and closed-form temperature solutions are calculated in Section 2.1. The general differential equation of heat conduction in the 2D plane is used. In order to consider the moving heat source, it is assumed that the coordinate system moves with the heat source by using a transformation as shown in Equation (3). Finally, using the separation of variables, the closed-form solution of temperature is obtained in Equation (5). The material properties are assumed to be temperature dependent. The melting/solidification phase change is also considered. The analytical and numerical analysis are conducted in this work.

In order to validate the proposed model, the experimental temperature data are used from the work of Pauzet [35]. The Ti-6Al-4V samples are manufactured using the DMD machine. The dimensions of the samples are 2 mm in width, 70 mm in depth and 80 mm in length. The temperature on the build part surface is measured using the thermocouple of type K. In order to control the experimental setup, the authors used a thermal-camera and a high-speed camera to provide comparison bases for the temperature and the melt-pool size. The DMD machine has used the laser with the wavelength of 1030 nm. The scanning speed of 0.2 m/min and 0.4 m/min and the laser power of 400 W and 600 W are studied. The initial temperature of each layer depends on the final temperature of the previous layer, as the process is multi-layered.

Figure 5 shows the temperature profile of the build part. The temperature is predicted using both the analytical model and the numerical model. The laser moves along the *x*-axis from left to right. The small red spot on top shows the laser location. The layer thickness is chosen to be 80 μm. The distance of the laser from the powder is 0.4 mm. For the same power, as the velocity is increased the maximum temperature is decreased since the powder has less time to absorb the energy. Different combinations of the process parameters are presented in Figure 5, specifically scanning speed and laser power.

**Figure 5.** Predicted temperature profile using (**a**–**d**) physics-based modeling and (**e**–**h**) numerical modeling.

The evolution of the surface temperature is plotted as a function of time for each case as shown in Figure 6. A study point will be chosen from the 2D geometry. When the laser is far away from the study point, the powder is at room temperature. As the laser approaches the study point, the temperature increases continuously. The maximum temperature on the curve corresponds to the moment that the laser is above the study point. After the laser passes the point, the temperature is decreased which shows that the material is cooling down. As shown in these plots, the cooling rate in the AM process is substantially high.

**Figure 6.** Evolution of surface temperature as a function of time for (**a**) P = 400 W, V = 0.4 m/min, (**b**) P = 600 W, V = 0.4 m/min, (**c**) P = 400 W, V = 0.2 m/min, (**d**) P = 600 W, V = 0.2 m/min.

In order to understand the influence of the process parameters on the maximum temperature, and surface temperature, a sensitivity study is designed to investigate both the scan speed and laser power. The short computational time associated with the analytical modeling approach allows for a better understanding of the influence of the process parameters as discussed previously. Figure 7 depicts the influence of the scan speed and laser power on temperature, as predicted by the analytical model and compared to the experimental results.

The results of the simulations from the analytical model illustrates that the maximum temperature decreases linearly as the scan speed increases since the material has less time to absorb the energy. On the other hand, for the fixed scanning speed, as the power increases the maximum temperature increases. The four experimental data are also pointed in Figure 7. The predicted temperature from the analytical model is slightly higher than the experimental values. This error is mainly because the temperature is measured using thermocouples which are a little below the surface.

**Figure 7.** Effect of scan speed and laser power on peak temperature.

Figure 8 represents the influence of the laser on the surface temperature. As the power increases from 200 W to 600 W, the surface temperature increases for a fix scanning speed. On the other hand, the surface temperature will decrease as the scanning velocity increases from 0.1 m/min to 0.6 m/min for a fix laser power as shown in Figure 9.

**Figure 8.** Comparison of evolution of surface temperature for (**a**) V = 0.3 m/min, and (**b**) V = 0.6 m/min.

**Figure 9.** Comparison of evolution of surface temperature for (**a**) P = 400 W, and (**b**) P = 600 W.

As explained, the proposed model considers the multi-layer aspects of metal additive manufacturing. The effect of considering the layer addition on peak temperature is compared to the obtained peak temperature without considering the layer addition, and also compared to the experimental results.

To further validate the proposed model, the peak temperature is plotted as a function of scanning speed for different laser powers. Two different values of laser power (400 W and 600 W) and scanning speed (0.2 m/min and 0.4 m/min) are chosen. The temperature considering the layer addition, the temperature not considering the layer addition, and also experimental values are compared. The values are listed in Table 3. The observations show that considering layer addition improves the prediction of temperature, as shown in Figure 10. For example, the predicted temperature for scanning velocity of 0.2 m/min and laser power of 400 W without considering the layer addition is 2042 ◦C, but when considering the layering aspect of AM, the predicted temperature reduces to 1802.8 ◦C which shows that it affects the heat transfer mechanisms.

**Table 3.** Comparison of temperature prediction among considering layer addition, not considering the layer addition, and experimental values.


**Figure 10.** Comparison of prediction of temperature with and without considering the layers with experimental values.

A comparison is also conducted among the analytical model considering the layer addition and dwell time, numerical model and experimental values as shown in Figure 11.

**Figure 11.** Comparison of predicted temperature among analytical model, experimental values, and FEA.

Overall, the temperature on the surface in terms of magnitude is well captured by both analytical and numerical approaches. The analytical model better approached the experimental measurements. This comparison shows the capability to accurately predict the temperature profile on the surface using the analytical modeling. The analytical approach also provides the power of a short computational time.

#### *J. Manuf. Mater. Process.* **2018**, *2*, 63

In order to illustrate the importance of considering the temperature dependent material properties, a sensitivity analysis is conducted to compare the predicted surface temperature with and without considering the property's temperature-sensitivity. The obtained results demonstrate a significant difference between them as shown in Figure 12. The thermal conductivity of the Ti-6Al-4V is 6.7 W/m· ◦C which results in a low rate of heat transfer in the build part. However, the thermal conductivity of Ti-6Al-4V varies from 6 to 35 W/m· ◦C with respect to temperature. The increase in heat transfer rate induced by the increase in thermal conductivity, causes the predicted surface temperature decrease. In the cases that the temperature sensitivity of the material properties is considered, as the velocity increases from 0.2 m/min to 0.4 m/min, the variation of predicted surface temperature is less than 100 ◦C. However, when the temperature sensitivity of material properties is not considered, the variation of temperature is more than 1000 ◦C. As it is shown in Figure 12 the predicted temperature can be quite unrealistic without considering the material properties sensitivity to temperature.

**Figure 12.** Comparison of predicted temperature considering the temperature sensitivity of material properties (WMTS), and without temperature sensitivity of material properties (WoMTS).
