*4.2. Predicted Temperature-Dependent Parameters*

Both of the temperature gradient (G) and the growth rate (R) are of paramount importance for determining the microstructure of solidified welds. The analytical Rosenthal equation can be utilized to estimate these imperative parameters abruptly via derivation from Equation (5). The temperature gradients in the directions of the axis y- and the axis z- are computed using the following equations. Equations (9) and (10) demonstrate the temperature gradients in the direction of the axis y- and the

axis z-, respectively; both of these equations can be utilized through a section in the y-z plane where x is equal to zero. Moreover, for any point in the transverse section of the molten pool at *x* = 0, the cooling rate can be computed using Equation (11).

$$\frac{\partial T}{\partial y} = \left[1 + \frac{y}{\sqrt{y^2 + z^2}} + \frac{2\alpha y}{V(y^2 + z^2)}\right] \left[-\frac{\lambda P}{2\pi k} \frac{V}{2\alpha} \frac{1}{\sqrt{y^2 + z^2}}\right] \exp\left[-\frac{V}{2\alpha} \left(y + \sqrt{y^2 + z^2}\right)\right] \tag{9}$$

$$\frac{\partial T}{\partial z} = \left[1 + \frac{2\alpha}{V\sqrt{y^2 + z^2}}\right] \left(-\frac{\lambda P}{2\pi k} \frac{V}{2\alpha} \frac{z}{y^2 + z^2}\right) \exp\left[-\frac{V}{2\alpha} \left(y + \sqrt{y^2 + z^2}\right)\right] \tag{10}$$

$$\frac{\partial T}{\partial t} = \left[1 + \frac{y}{\sqrt{y^2 + z^2}} + \frac{2ay}{V(y^2 + z^2)}\right] \left(\frac{\lambda P}{2\pi k} \frac{V^2}{2a} \frac{1}{\sqrt{y^2 + z^2}}\right) \exp\left[-\frac{V}{2a} \left(y + \sqrt{y^2 + z^2}\right)\right] \tag{11}$$

From the work of Bontha et al. [31], temperature-dependent parameters, namely, temperature gradient, cooling rate, and growth rate vary considerably at different points inside the molten pool. It is of great importance that the fusion depth should be calculated by Equation (7) so that the temperature gradient and the cooling rate of solidified welds can be determined in this regard. In this case, the average fusion depth can be seen from Figure 5 as well. Using Equation (5), the points y and z for different heat inputs are computed from the bottom of the welds to the centerline. These values are utilized to investigate the heat transfer behavior within the molten pool using Equations (9)–(11). Then, the average amounts of the results obtained from the analytical method are compared for various locations. In order for the solidification rate within the melt pool to be determined, the values of temperature gradient and cooling rate achieved from the analytical Rosenthal equation are used in Equation (12) for each point in the melt pool.

$$R = \frac{1}{G} \frac{\partial T}{\partial t} = \frac{1}{\sqrt{\left(\frac{\partial T}{\partial \dot{y}}\right)^2 + \left(\frac{\partial T}{\partial z}\right)^2}} \frac{\partial T}{\partial t} \tag{12}$$

As to the numerical modeling, it is also of paramount importance that the fusion depth of solidified welds needs to be determined to evaluate the heat transfer behavior within the molten pool. Then, temperature-dependent parameters including temperature gradient and cooling rate are calculated at different points within the melt pool. In order for this case to be clearer, conceive an example of a laser beam with an average power of 80 W, welding velocity of 2 mm.s−1, and an absorption coefficient of 0.36. Performing these laser parameters, the obtained fusion depth using the numerical modeling is 576 μm. Temperature and temperature gradient, as thermal properties of the welded material, are computed in five different points from the bottom of the weld pools to the centerline: 76, 176, 276, 376, and 476 μm. By doing so, Figure 8a–c is attained in which the temperature and the temperature gradient in the direction of the axis z- are illustrated as functions of time and temperature, respectively. It is worthwhile to mention that the average amount of the temperature gradient is taken into account for the melting point and compared with the results obtained from the analytical method. For determining the solidification rate within the molten pool, the cooling rate is measured in the same way as the temperature gradient which was discussed before. Then, the temperature gradient and the cooling rate obtained from the numerical modeling are used in Equation (12), and ultimately, the values of the solidification rate are attained for different locations within the melt pool.

**Figure 8.** *Cont*.

**Figure 8.** (**a**) Variation of temperature versus welding time for different points within the molten pool; (**b**) variation of temperature gradient versus welding time for different points within the molten pool; (**c**) variation of temperature gradient versus temperature for different points within the molten pool obtained from the numerical modeling using a laser beam with an average power of 80 W, welding velocity of 2 mm·s<sup>−</sup>1, and absorption coefficient of 0.36.

Temperature gradients are determined for different heat inputs using the numerical modeling and the analytical method, and the results obtained from the numerical modeling are compared with those attained from the analytical Rosenthal equation as shown in Figure 9, where the welding velocity is constant and its value is about 2 mm·s<sup>−</sup>1. As shown in Figure 9a, the temperature gradient is decreased by increasing heat input of the laser beam; this trend is the same for both approaches including the numerical modeling and the analytical method. The reason may lie in that by increasing the heat input of the laser beam, the molten pool becomes larger, which deteriorates the temperature gradient within the weld pool. According to the parameters used in this study, the results for temperature-dependent parameters such as temperature gradient, cooling rate, and growth rate obtained from the numerical modeling are slightly bigger than those from the analytical Rosenthal equation, as shown in Figure 9a–c.

**Figure 9.** *Cont*.

**Figure 9.** (**a**) Temperature gradient, (**b**) cooling rate, and (**c**) solidification rate achieved from the numerical modeling and the analytical method versus heat input of the laser beam and their comparison.

Additionally, as can be seen from Figure 9b, the difference between the values of cooling rates obtained from the numerical modeling and the analytical method is increased by enhancing the energy density of the laser beam. The major reason may lie in that by increasing the heat input, radiation loss is increased as well so that this plays an integral role in increasing the difference between the results achieved from the numerical modeling and the analytical method at higher energy densities. Furthermore, as to the analytical method, heat losses are not considered through the substrate such that the temperature is raised considerably within the molten pool which makes it bigger in comparison to the FE model. Therefore, the cooling rate attained from the analytical method is lower than that achieved from the numerical modeling. Moreover, the more heat input is increased, the more the cooling rate from the analytical method is decreased due to an increase in weld pool dimensions. In addition, as to the FE model, the thermal conductivity of the material is a function of temperature. Therefore, by approaching the temperature near to the melting point of the material, the thermal conductivity is increased so that this can help the cooling process in the simulation. However, this cannot be a positive point for the cooling rate achieved from the analytical method because materials properties are not dependent on the temperature of the material using in the analytical Rosenthal equation, as discussed before. Goldak et al. [32] conducted a research in terms of the cooling rate and a comparison of its value obtained from the numerical modeling, the analytical Rosenthal equation, and experiments during welding. Based on the results from the numerical modeling, the analytical method, and experiments, they reported that the cooling rate attained from the FE model was underestimated by 5%, while the cooling rate computed from the Rosenthal equation was overestimated by 41%. Quite contrary, the results obtained from this study differ from those achieved by Goldak et al. because the cooling rate attained from the numerical modeling is higher than that achieved from the analytical method in the present study due to the reasons explained above. It should be considered that Goldak et al. [32] used low-carbon steel as the base metal that has lower thermal conductivity at higher temperatures so that the cooling rate calculated by the numerical modeling became lower than that computed by the analytical Rosenthal equation.

While investigating the heat transfer behavior within the molten pool, it is noteworthy to mention that the initial temperature of the substrate is considered to be constant, which is equal to room temperature. However, in genuine environments, the accumulation of heating in the substrate may affect the initial temperature so that it cannot be considered to become constant. There are some factors that can influence on the initial temperature of the substrate, including a minute time length between pulses with which the time needed for the cooling process is decreased; a small welding path with which the temperature of the molten pool is raised considerably after each pulse; and small features in the geometry with which the heat conduction is become weak due to the reason that oxide films on the substrate have low thermal conductivity. Without considering the local temperature of the substrate, the analytical Rosenthal equation is not applicable to evaluate the heat transfer behavior of the welded material; nevertheless, particular geometries can be performed in the numerical modeling for investigation of the heat transfer during welding.
