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Results of heat flow calculated by Rosenthal's 3D equation are shown in Figures 2.20 and 2.21. They show the calculated thermal cycles and temperature distributions at the top surface (z = 0) of a thick plate of 1018 steel for two different sets of heat input and welding speed. The pool boundary is assumed to be at the liquidus temperature T _L = 1530 °C. The solid‐state transformation temperature responsible for the formation of the HAZ at T _H = 780 °C. The eutectoid temperature 727 °C of the binary Fe‐C phase diagram might have been a better choice for 1018 steel. However, since the isotherm at 780 °C has already been calculated, it will be taken as T _H as an approximation just for the purpose of illustration.

Figure 2.20 Calculated Rosenthal's three‐dimensional heat flow in 1018 steel: (a) thermal cycles; (b) isotherms. Welding speed: 2.4 mm/s; heat input: 3200 W; material: 1018 steel; liquidus temperature: 1530 °C; solid‐state transformation temperature: 780 °C.

Figure 2.21 Similar to Figure 2.20 but with faster welding speed of 6.2 mm/s and higher heat input of 5000 W, resulting in faster cooling rate and more elongated weld pool.

As can be seen in Figures 2.20a and 2.21a, the peak temperature of the thermal cycle at the centerline of the weld surface (i.e., at y = z = 0) is infinite. This error is caused by the singularity problem in Rosenthal's equations, that is, caused by the assumption of a point heat source at x = y = z = 0 (the origin of the coordinate system). Thus, near the center of the heat source (the origin), Rosenthal's equations are not reliable. Away from it, however, they can be more accurate. Because of their simplicity, Rosenthal's equations have been used despite the error.

In Figure 2.20 Q = 3200 W and V = 2.4 mm/s, and the Q/V ratio is 1333 J/mm. At the top surface of the workpiece, the weld pool is 9.5 mm wide and 12 mm long, and the length/width ratio is 1.26. In Figure 2.21 the power input is increased to Q = 5000 W and the travel speed increased to V = 6.2 mm/s. The Q/V ratio is 806 J/mm. At the top surface of the workpiece, the weld pool is 9.0 mm wide and 16 mm long, and the length/width ratio is 1.78.

Thus, as shown by Rosenthal's 3D equation, the weld pool becomes more elongated as Q and V are both increased. As shown in Figure 2.22, the pool shape in GTAW of a Ni‐base alloy IN718 does become more elongated when Q and V are both doubled [38]. In fact, the weld pool becomes teardrop‐shaped at higher Q and V. Figure 2.23 shows the end of a teardrop‐shaped weld pool in GTAW of 309 stainless steel, which was revealed by Kou and Le by quenching the weld pool during welding [39]. Although Rosenthal's equation can correctly predict that the weld pool becomes more elongated at higher Q and V, it cannot calculate a teardrop‐shaped weld pool because it neglects, among other things, heat of fusion and the presence of the liquid phase. A very sharp end of the weld pool like that in Figure 2.22 or 2.23 is likely caused by undercooling at high travel speed (i.e. high cooling rate).

Figure 2.22 Weld pool shapes in GTAW of IN718 sheets.

Source: Hunziker, Dye, and Reed [38]. © Elsevier.

Figure 2.23 Sharp pool end in GTAW of 309 stainless steel revealed by ice quenching during welding.

Source: Kou and Le [39]. © TMS.

The calculated results of Rosenthal's equation in Figures 2.20 and 2.21 show that the cooling rate is higher at a lower Q/V ratio. The thermal cycle at the centerline (y = 0 mm) of the fusion zone is steeper (i.e. higher cooling rate) in Figure 2.21, where Q/V = 806 J/mm, than in Figure 2.20, where Q/V = 1333 J/mm. Lee et al. [40] measured the cooling rate in GTAW of 2024 aluminum by sticking a thermocouple into the weld pool. They showed that decreasing Q/V increased the cooling rate (the slope). Kihara et al. [41] showed that the cooling rate increased with decreasing Q/V and preheating. Liu et al. [42] showed that the cooling rate in electroslag welding (ESW), which is known to have a very high Q/V, is much smaller than that in arc welding. Thus, the prediction of a higher cooling rate at a lower Q/V based on Rosenthal's equation is consistent with experimental results.

Kihara et al. [41] showed that the cooling rate increases with the thickness of the workpiece. This is because a thicker workpiece acts as a better heat sink to cool the weld down. Inagaki and Sekiguchi [43] showed that, under the same heat input and plate thickness, the cooling time is shorter for fillet welding (Figure 1.6d) than for bead‐on‐plate welding because of the greater heat sink effect in the former.