2.2.2. Melt Pool Dimensions-Density Relationship

To establish the relationship between the previously defined dimensionless melt pool metrics and the density of manufactured parts, 10 mm-diameter 15 mm-height cylindrical coupons of IN625 alloy were printed to cover a *D/t* ratio ranging from 1 to 3.5, *W/h*, from 0.5 to 3 and *L/W*, from 3 to 6. To find the LPBF parameters resulting in these melt pool dimensions, the following ranges of printing parameters were reversely computed using the melt pool model presented in Section 2.1: the laser power varying from 160 to 350 W, the scanning speed, from 560 to 2800 mm/s, and the hatching space, from 30 to 550 μm; the layer thickness t was kept constant at 40 μm (see Table 2 for the selected values of the LPBF processing parameters). Two specimens were printed for each set of printing parameters.


**Table 2.** Imposed melt pool metrics and calculated processing parameters (plan of experiments).

After processing, the printed coupons were cut off the build plate and their densities measured using the Archimedes' technique (ASTM B962-15). Each density measurement using a SARTORIUS Secura 324-1s scale (Sartorius, Goettingen, Germany), having a precision of ~0.001 g, was repeated at least 3 times.

The results of this experiment are collected in Table 3 (Appendix A) and plotted in Figure 3 in the *D/t*-*W/h*-density coordinates. It can be seen from this figure that the density of IN625 coupons exceeding 99.5% (this value was selected arbitrarily to limit the amount of experimental data, while leaving enough space for optimization) was obtained for a *D/t* ratio ranging from 1.5 to 2.75 and a *W/h* ratio ranging from 1.8 to 2.8. The corresponding *L/W* ratio ranged from 3.8 to 4.6 (not shown on this diagram). Note that the calculated values fall close to the ranges recommended in the literature, which are 1.5 < *D/t* < 2, 1.5 < *W/h* < 2.5 and *L/W* < 2Pi [28].

From Figure 3, assuming that the calculated *D/t*, *W/h* ratios correspond to the effectively obtained melt pool dimensions, the materials density can be expressed as their function as follows:

$$\rho = a0 + a1 \cdot \left(\frac{D}{t}\right) + a2 \cdot \left(\frac{W}{h}\right) + a3 \cdot \left(\frac{D}{t}\right)^2 + a4 \cdot \left(\frac{D}{t}\right) \cdot \left(\frac{W}{h}\right) + a5 \cdot \left(\frac{W}{h}\right)^2\tag{6}$$

where *a*0 = 0.512, *a*1 = 0.212, *a*2 = 0.225, *a*3 = −0.027, *a*4 = −0.384, and *a*5 = −0.031.

**Figure 3.** Density of the printed coupons as a function of the *D/t* and *W/h* ratios; the calculated *D/t*-*W/h* area corresponds to the measured density of the printed material exceeding 99.5 ± 0.1%.

## *2.3. Energy Density-Build Rate Processing Map*

In this work, the LPBF processing conditions were expressed by a combination of two metrics: the volumetric laser energy density *E* (J/mm3) (7) and the material build rate *BR* (cm3/h) (8); the product of both corresponds to the laser power *P* (Watts).

$$E\left(I/\text{mm}^3\right) = \frac{P}{\upsilon \cdot \hbar \cdot t} \tag{7}$$

$$BR\left(cm^3/h\right) = \upsilon \cdot h \cdot t\tag{8}$$

Next, the analytical model (1–5) and Table 1 were used to map three *E–BR* areas corresponding to the experimentally obtained optimal ranges of the melt pool metrics (Figure 4a): *D/t* = 1.5–2.75, *W/h* = 1.8–2.8, and *L/W* = 3.8–4.6. These maps are calculated by varying the laser power from 20 to 380 W; the scanning speed, from 100 to 4000 mm/s; the hatching space, from 30 to 200 μm, and the layer thickness, from 20 to 80 μm.

Three *E–BR* areas of Figure 4a were then superposed in Figure 4b to schematically delimit a common processing window, which must guarantee the maximum density of printed IN625 parts. Next, the densities of the printed coupons (Figure 3 and Table 1 in the Annex) were superposed on this

processing window, and it can be seen that the coupons with a density ≥99.5% are indeed located, within a certain margin of error, in the numerically predicted optimal processing window. By refining the scanning steps and using calibration Equation (6), the described approach can then be used to build a more detailed processing map for IN625 powder (Figure 4c).

**Figure 4.** (**a**) Optimal areas for the *D/t*, *W/h*, and *L/W* ratios; (**b**) superposition of the numerically optimized processing window; (**c**) experimentally calibrated processing map.
