*2.4. Validation Strategy for the Proposed Processing Optimization Approach*

We hypothesize that such a combined processing optimization approach is valid for any material processed by a given LPBF system. In our case it is the EOS M280 LPBF system. To verify this hypothesis, the results of such an optimization were compared with the numerical and experimental data found in the literature. This comparison was carried out in two phases: first, the melt pool dimensions were calculated for AlSi10Mg and 316L powders using a simplified analytical model of this work (Equations (1)–(5)), and then compared against those obtained for the same feedstock material, but using more powerful finite element models (FEM). These FEM models take into account the optical penetration depth, the mass transfer-related phenomena, such as the Marangoni convection and the Rayleigh capillary flow, and the heat losses to the environment [11,29]. Secondly, the numerically predicted densities for pure iron and Ti-6Al-4V alloy powders are compared with their experimentally measured equivalents from [6] and [18]. The optimal processing windows for all these validation studies were obtained using the algorithm previously presented. The data used for these calculations were taken from the corresponding literature sources.

#### **3. Validation**

#### *3.1. Melt Pool Dimensions (Single Track)*

As experimental validation of the melt pool model used in this study (Equations (1)–(5)) has already been carried out in our previous work [20], a decision was made to extend the validation experiments to literature data. With this objective in mind, the single track melt pool dimensions calculated by the said analytical model were compared with those calculated by two different finite element models. Note that each of these models was experimentally validated by their authors for two different alloys: 316L [11] and AlSi10Mg [14]. For ease of understanding and because the width and the depth of the melt pool are the most important characteristics impacting the density of the printed material, the geometric validation was limited to these two characteristics.

Regarding 316L, the simulations were carried out with an initial powder bed temperature of 296 K (first track in [11]), a fixed laser power of 110 W and a scanning speed ranging from 80 to 150 mm/s. For AlSi10Mg, the simulations were realized for a laser power ranging from 150 to 300 W and a fixed scanning speed of 200 mm/s [14]. In both cases, computations using (Equations (1)–(5)) were carried out using the physical properties taken from the corresponding literature sources (Table 3). In other words, the electrical resistivity, the thermal conductivity and the specific heat capacity values used in our calculations were set identical to those used in [11] and [14] and recalculated for a 60% powder bed density. This last value was selected on the basis of our previous results because no such information was provided in the literature sources.



The melt pool profiles calculated by the model of this study and those found in the literature are plotted in Figure 5 for different sets of printing parameters. The mean deviations between the results of the analytical and the finite element models are 4.3 ± 1.2% for 316L and 10.5 ± 5.8% for AlSi10Mg. (Note that the numerically estimated impact of a 5.8% deviation in the AlSi10Mg melt pool dimensions would have introduced only ~0.2% variation in the predicted density values; the last number being calculated by introducing a value of 5.8% in the density Equation (6)).

**Figure 5.** Comparison of the melt pools profiles for (**a**) 316L [11] and (**b**) AlSi10Mg [14] alloys.

#### *3.2. Density of the Same Alloy Printed with Two Different Layer Thicknesses*

Using a combination of the analytical modeling of melt pool dimensions (Equations (1)–(5)) and Equation (6), representing the correspondence between the melt pool dimensions and the density of the printed parts, processing windows can be calculated for multi-track LPBF. To validate this approach, processing maps for Ti64 alloy printed with two layer thicknesses (30 and 60 um) are plotted in Figure 6. The physical properties used for these calculations are collected in Table 4.

Note that the powder bed densities (evaluated using the method of encapsulated samples [25]) in these two cases are not identical: it is higher in the former than in the latter case, with ϕ = 70% for *t* = 30 μm and ϕ = 60% for *t* = 60 μm. These changes in the powder bed density are due to the

differences in the powder spreading conditions for different layer thicknesses as demonstrated in [31]. If a layer thickness is smaller than the D90 value of the powder particle distribution [32], the density increases because the biggest particles are kept at the top of the layer and finally swept out by the recoater [33], which increases the powder bed density. It can be seen from Figure 6 that the higher the powder bed density (70%, Figure 6b, instead of 60%, Figure 6a), the higher the optimal laser power density, while the lower the build rate of the process. Similar results were reported in [31].

**Figure 6.** Processing maps for Ti64 alloy for layer thicknesses of (**a**) 60μm and (**b**) 30μm (EOS M 280).


Specific heat capacity, J/(kg·K) 570 342 405 Electrical resistivity, 10−<sup>8</sup> <sup>Ω</sup>·<sup>m</sup> 170 283 239

**Table 4.** Physical properties of Ti64 powders used for melt pool modeling [26,34].

The experimentally measured densities of Ti64 coupons printed with the layer thicknesses of *t* = 60 μm (Figure 6a) and 30 μm (Figure 6b) were then superposed on the calculated processing maps (these coupons were printed using EOS Ti64 powder and the EOS M280 system of this study). The mean porosity deviations for *t* = 60 μm corresponded to 0.8%, while for *t* = 30 μm, it was 0.4%.
