*3.3. Process Definition and Boundary Conditions*

The main process parameters applied in the welding experiments are summarized in Table 2. Based on the mean welding voltage, *U* (V), the mean welding current, *I* (A), and the welding speed, *v* (m/min), the nominal energy input, *E* (J/mm), was calculated:

$$E = 0.06 \frac{III}{\upsilon} \tag{3}$$

The efficiency of the welding process, *η*, is in the range of about 0.8–0.9 for energy-reduced GMA welding processes with controlled dip transfer [65–67]. However, *η* tends to increase with decreasing arc power [67]. Since the modelled welding process was operated with comparatively low energy input or arc power, *η* was approximated as unity in the present model. The heat density distribution, *q* (W/mm3), describes the time-dependent movement of the heat source along the predefined welding trajectory. It was calculated according to the double-ellipsoidal heat source model of Goldak et al. [68,69]:

$$q(\mathbf{x}, y, z, t\_w) = \frac{6\sqrt{3}}{\pi\sqrt{\pi}} \frac{f\eta Ev}{abc} \exp\left(-\frac{3x^2}{a^2}\right) \exp\left(-\frac{3y^2}{b^2}\right) \exp\left(-\frac{3(z - vt\_w)^2}{c^2}\right) \tag{4}$$

In Equation (4), *x*, *y* and *z* (mm) are the coordinates of the fixed coordinate system of the model, and *tw* (s) is the welding time. The linear welding trajectory is oriented in *z*-direction. The heat source

moves with constant welding speed, *v* (mm/s), along the welding trajectory. The heat source center is located at the initial coordinates *x*<sup>0</sup> = *y*<sup>0</sup> = *z*<sup>0</sup> = 0 when welding starts at *tw*<sup>0</sup> = 0. In order to calculate *q <sup>f</sup>* and *qr*, the ellipsoidal heat density distributions at the front and rear sections of the weld pool, the factor *f* is replaced by *f <sup>f</sup>* or by *fr*, and the length *c* is replaced by *c <sup>f</sup>* or by *cr*, respectively. The heat fractions *f <sup>f</sup>* and *fr* are deposited at the front and rear sections of the weld pool, with *f <sup>f</sup>* + *fr* = 2. The lengths of the front and rear sections are *c <sup>f</sup>* and *cr*. Accordingly, the total length of the weld pool is *c <sup>f</sup>* + *cr*, with *c <sup>f</sup>* :*cr* = 1:2. The total width of the weld pool is 2*a* and the penetration depth is *b*.

The radiative heat transfer coefficient at the surfaces of the weld and of the sheets, *hr* (W/m2K), was calculated based on the Stefan-Boltzmann constant, *σ*, the thermal emission coefficient, *ε*, the predefined ambient temperature, *T*∞ (K), and the local surface temperature, *T* (K):

$$h\_r = \sigma \,\varepsilon \left( T + T\_{\infty} \right) \left( T^2 + T\_{\infty}^2 \right) \tag{5}$$

In order to quantify the total thermal losses the total heat transfer coefficient, *h* (W/m2K), was then calculated by adding both the radiative heat transfer coefficient, *hr*, and the convective or conductive heat transfer coefficients, *hc*:

$$h = h\_r + h\_c \tag{6}$$

The basic input parameters used for the simulation of the welding process are summarized in Table 4. They basically correspond to the conditions of the welding experiments conducted for validating the results of the simulations. However, in order to model the heat losses it was necessary to make some feasible assumptions. Since the temperature dependence of *ε* was actually unknown, *ε* = *ε*(*T*) was approximated as unity. This is particularly suitable for elevated temperatures, because *ε* of metals is known to increase markedly with rising temperature, and at elevated temperatures radiative thermal losses become also dominant. The convective and the conductive heat transfer coefficients, *hc* (W/m2K), were assumed to be constant over the entire temperature range. Since the metal sheets were clamped with massive metal bars on both sides of the weld butt, the conductive heat transfer between the sheets and the clamps was certainly higher than the convective heat transfer between the weld seam region and the ambient air.


**Table 4.** Input parameters used for the simulation of the welding process.

The dimensions of the weld pool, *a*, *b*, *c <sup>f</sup>* and *cr*, were calibrated based on micrographs of the joint, as exemplarily shown in Figure 10. Note that in the present work, only the narrow region with the locally molten aluminum sheet was considered for calibration, because the steel sheet remains solid in dissimilar CMT welding/brazing of aluminum alloys to steel. This reasonable simplification allowed the utilization of Goldak's symmetric heat source model according to Equation (4), even though the aluminum-steel joint had an asymmetric shape.
