# An Overview of the Process Mechanisms in the Laser Powder Directed Energy Deposition
## 2 Laser Irradiation and Material Addition Mechanisms
### Powder Heating and Laser Attenuation
Figure 15 shows the comparison of numerical and experimental increases in powder temperature versus in-flight distance for different values of laser power. Figure 14: Illustration of the effects occurring during the laser-powder interaction. By comparing the curves, the authors showed that the model allows the temperature increase during the process to be analyzed from a qualitative point of view. Several authors have successfully applied the lumped capacitance method combined with a different solution approach. [PERSON] [59] solved Equation (1) analytically and reported that the powder was characterized by different temperatures within the flow due to different trajectories. In the focus plane, the maximum temperature was reached at the centre of the laser beam, whereas a uniform temperature distribution along the radial position was observed below the focus plane. In order to avoid the complex radiation problem involved in laser-powder interaction, several methods were used to study the power attenuation caused by the powder stream. The most used methods are (i) the Hagens-Rubens relationship, (ii) Bramson's equation, and (iii) the use of the Beer-Lambert attenuation model. Using the Hagens-Rubens relationship, defined by Equation (2), or the Bramson equation, described by Equation (3), a time-space-temperature averaged coefficient is obtained to estimate the laser attenuation. However, it should be noted that these relationships do not consider the powder flow rate. Typical values obtained for an Nd-YAG laser are 0.36 for Inconel 718 [80] and 0.28 for 316L stainless steel [81]. The laser-surface coupling coefficient, \(A(T)\), and the absorption coefficient, \(\eta\), are defined by the following equation:  
\[A(T)=\left[8\varepsilon_{0}\omega\rho_{e}(T)\right]^{1/2} \tag{2}\]  
\[\eta=0.365\Big{(}\frac{\rho_{e}}{\lambda}\Big{)}^{1/2}-0.067\Big{(}\frac{\rho _{e}}{\lambda}\Big{)}+0.006\Big{(}\frac{\rho_{e}}{\lambda}\Big{)}^{3/2} \tag{3}\]  
where \(\varepsilon_{0}\) is the permittivity of free space, \(\omega\) is the angular frequency of the laser irradiation, \(\rho_{e}(T)\) is the material temperature-variant electrical resistivity, and \(\lambda\) is the wavelength of the laser beam. Using the Beer-Lambert attenuation model, the radial distribution of the attenuated laser beam intensity at a distance z from the nozzle can be estimated as:  
\[I^{\prime}(r,z)=I_{0}(r)exp(-\sigma_{ext}Nz) \tag{4}\]  
where \(I^{\prime}(r,z)\) is the attenuated laser beam intensity, \(I_{0}(r)\) is the initial laser beam intensity, \(\sigma_{ext}\) is the powder extinction coefficient, and \(N\) is the number of powder particles in a unit volume. \(N\) is proportional to the powder flow rate [74]. [PERSON] and [PERSON] [82] observed that increasing the powder flow rate results in a higher value of laser beam absorption and, consequently, a lower value of useful laser power (Figure 16). Figure 15: (**a**) Numerical and (**b**) experimental temperature of the powder at different laser powers (reproduced with permission from [54]; Copyright © 2011 Taylor & Francis). The evaluation of laser absorption and, therefore, the absorption coefficient was also carried out by experimental tests. [PERSON] and [PERSON] [83] used a Thermoites Seebeck envelope calorimeter to estimate the process efficiency during the deposition of H13 tool steel powder on a substrate of the same material and copper powder deposited on an H13 tool steel substrate. Varying the laser power between 125 and 500 W and the powder flow rate between 0.08 and 0.33 g/s, they found that the efficiency varied between 0.3 and 0.5. Using the same experimental setup, [PERSON] [84] showed that process efficiency was about 0.42 during the deposition of Ti-6 Al-4V on a Ti-6 Al-4V substrate and about 0.34 depositing 316L stainless steel powder on a 304L stainless steel substrate. [PERSON], et al. [85] showed that the attenuation slightly increases by increasing the powder flow rate. Table 3 summarizes the main aspect involved in the powder flow process and the most influential factors.