*4.1. Molten Pool Dimensions*

According to the Rosenthal equation, the fusion width maximizes as *dx*/*d*ξ = 0. To compute the fusion width in the current study, a simple equation was developed which is derived from the Rosenthal equation. This particular equation enabled the calculation of the path length between the laser beam and any location on the melt pool at a specific temperature. By doing so, computing of the fusion width is possible, and then, the fusion depth can be determined as well. Regarding the welding velocity of 2 mm/s, the laser beam will move 8 μm in the direction of *y*-axis for 4 ms (the pulse duration of laser beam). The abovementioned criteria are shown as Equation (7).

$$\frac{2\alpha}{V}\ln(r') + r' = -\frac{2\alpha}{V}\ln\left(\frac{2\pi k(T - T\_0)}{\lambda P}\right) - \xi \tag{7}$$

where *r* is the path length between any location on the weld pool with a specific temperature and the laser beam; *P* is the welding power; *T* is the temperature of any particular location on the melt pool; *T*<sup>0</sup> is the initial temperature of the material, which is about 293 K; α is the thermal diffusion coefficient at 293 *K*; *V* is the welding velocity; *k* is the thermal conductivity of the material at 293 K; and λ is the absorption coefficient of the substrate. In order for the weld pool width to be calculated, Equation (8) is used which is written as follows:

$$w = 2r = 2\sqrt{\left(r'\right)^2 - \left(V \times t\right)^2} \tag{8}$$

where *r* is the path length between any location on the weld pool with a particular temperature and the heat generator, *V* is the welding velocity, and *t* is the welding time. It should be taken into account that the analytical Rosenthal equation creates a semi-circular molten pool in which the fusion width (*w*) is two times bigger than the fusion depth (*d*).

Experimental results in terms of the fusion width were compared with fusion widths, fusion depths, and partially melted zones thickness achieved by numerical modeling and analytical method. To do so, Rosenthal equation and FE modeling were used in order to calculate these values with which the comparison of these results becomes available with regards to experimental results. It should be noted that the PMZ is the area in which solidification cracks can be produced during laser welding such that the study in terms of this particular area can be beneficial to predict the solidification microstructure. The parameters performed in the numerical modeling are similar to those utilized in experiments; these parameters are comprehensively mentioned in Section 2. In order to compare the validation of the numerical and analytical methods regarding experimental results, four distinct laser powers were selected in the present study.

Fusion widths obtained from experiments are compared with the estimated ones from the numerical modeling and the analytical method. This comparison is illustrated in Figure 5. The average fusion widths were measured from three different transverse sections perpendicular to the welding direction for each heat input. Based on these results, it is observed that fusion widths achieved from the numerical modeling are consistent with those from experiments up to an energy density of 30 <sup>J</sup>·mm<sup>−</sup>1. By approaching higher energy densities, the predicted fusion widths tend to overestimate the average fusion widths obtained from experiments. However, there is a gap between the results from the analytical method and ones from experiments; obviously, fusion widths achieved from the Rosenthal equation are bigger than those from experiments. It is apparent that the results from Rosenthal equation are not matched with the experimental results due to some of selected assumptions described in Section 3. According to the FE models, natural convection within the molten pool is neglected so that in smaller melt pools, the temperature is overestimated, compared with genuine melt pools. Therefore, radiation losses are increased in smaller fusion welds. In contrast to this idea, based on the assumptions for the analytical Rosenthal equation, radiation losses are not involved in this method; therefore, fusion widths attained from the analytical method are larger than those from the numerical modeling and experiments.

**Figure 5.** Comparison of the fusion width obtained from experiments, the numerical modeling, and the analytical Rosenthal equation.

Fusion depths obtained from experiments are compared with the estimated ones from the numerical modeling and the analytical method. This comparison is demonstrated in Figure 6. The average fusion depths were measured from three different transverse sections perpendicular to the welding direction for each heat input. According to these results, it is apparent that fusion depths achieved from the numerical modeling are consistent with those from experiments up to an energy density of 30 <sup>J</sup>·mm<sup>−</sup>1. By approaching higher energy densities, the predicted fusion depths incline to overestimate the average fusion depths obtained from experiments. On the other hand, the predicted results from the Rosenthal equation seem to overestimate the average fusion depths attained from the experiments. The main reason may lie in the assumptions of the Rosenthal equation. As discussed above, there is no convection during the simulation of the welding process such that the results can be overestimated. Furthermore, the radiative losses might be too high especially for smaller melt pools; therefore, the predicted results might be inaccurate. In contrast, in the analytical method, radiation losses and natural convection are not considered with which the predicted results could be overestimated in comparison to the predictions from FE modeling and the experimental results.

**Figure 6.** Comparison of the fusion depth obtained from experiments, the numerical modeling, and the analytical Rosenthal equation.

Figure 7 demonstrates a comparison made on the partially melted zones thickness obtained from the experiments, the FE modeling, and the Rosenthal equation as well as a cross-section of the molten pool, whose parameters were selected from A1 in Table 2, fitted by the FE model of the partially melted zone. As shown in Figure 6, the partially melted zone thickness obtained from the experiments increases when the heat input of the laser beam is augmented. Actually, the partially melted zone (PMZ) is an area formed between the eutectic (or solidus temperature for solutionized work pieces) and liquidus temperatures of materials [29]. It is reported that the energy density can only affect the PMZ length during welding [30]. According to the predicted results from the FE modeling, it is obvious that the partially melted zone thickness enhances with the heat input increment; however, these results overestimate the length of PMZ due to the simplifications described above, such as insulating of substrate walls and neglecting the existence of convection inside the melt pool. Regarding the results obtained from the analytical method, an overestimation of the experimental results can be seen from Figure 7. As described above, there were some assumptions performed to the Rosenthal equation with which the predicted results are not matched with the experimental ones in this regard.

**Figure 7.** (**a**) Comparison of the partially melted zone thickness obtained from experiments, the numerical modeling, and the analytical method; (**b**) fitted partially melted zone from the finite-element (FE) model on the cross-section of the melt pool using an average laser power of <sup>50</sup> *<sup>W</sup>*, welding velocity of 2 mm·s<sup>−</sup>1, and absorptivity of 0.36.
