2.3. Welding modeling
To describe the HAZ thermal cycles, the physical welding thermal cycle simulators used simplified analytical models such as Rosenthal [21] and Rykalin 2D [22]. These models simplified some factors; for example, the physical properties are temperature-independent, the metals are solid during welding, no solid phase transformation occurs, there is no heat loss by radiation and convection, and the heat source has zero volume (point or line) [23]. However, these simplifications lead to errors in the HAZ thermal cycle simulation. Numerical simulation methods (e.g., FEM) have been used to overcome these limitations and make the physical simulation more realistic [24–25]. The commercial FEM software Sysweld® was used to accurately describe the CGHAZ and ICCGHAZ thermal cycles. The weld metal elements were simulated using the birth and death technique. The procedures for executing, documenting, and validating the welding simulations followed the ISO 18166 standard.
The welding was modeled using the principle of energy conservation [26]. The fundamental equation of heat conduction in multiphase solids is expressed as Eq. 2, where the temperature distribution (T) is a function of the time (t) and position (x, y, z). k is the thermal conductivity, cp is the specific heat, ρ is the density, T is the temperature distribution, P is the phase proportion, and\(\dot{Q}\) is the heat generated per volume. i and j are phases indexes, Aij is the proportion of phase transformation per unit time, and Lij(T) is the latent heat of the phase transformation. cp, k, and ρ (Table 4) are obtained from the chemical composition-based thermodynamic simulation of the pipe using the JMatPro® software. The same properties were used for the base and weld metals during the simulation because their chemical compositions were similar.
$$\frac{\partial }{\partial x}\left(\left(\sum _{i}{P\left(T\right)}_{i}{k\left(T\right)}_{i}\right)\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(\left(\sum _{i}{P\left(T\right)}_{i}{k\left(T\right)}_{i}\right)\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left(\left(\sum _{i}{P\left(T\right)}_{i}{k\left(T\right)}_{i}\right)\frac{\partial T}{\partial z}\right)+\dot{Q}-\sum _{icript>$$
2
,
Table 4
Density (ρ), thermal conductivity (k), and specific heat (cp) of the HSLA API 5L X70 pipe at a constant pressure and various temperatures (T).
Temperature, T
[°C]
|
Density, ρ
[g∙. cm− 3]
|
Thermal conductivity, k
[W. (m∙. k)−1]
|
Specific heat, cp
[J. (g∙. K)−1]
|
2000
|
6.58
|
42.92
|
0.82
|
1900
|
6.67
|
41.07
|
0.82
|
1800
|
6.75
|
39.23
|
0.82
|
1700
|
6.84
|
37.38
|
0.82
|
1600
|
6.92
|
35.54
|
0.82
|
1500
|
7.25
|
35.45
|
0.73
|
1400
|
7.33
|
34.33
|
0.69
|
1300
|
7.38
|
33.11
|
0.67
|
1200
|
7.44
|
31.89
|
0.66
|
1100
|
7.49
|
30.66
|
0.64
|
1000
|
7.55
|
29.45
|
0.63
|
900
|
7.60
|
28.23
|
0.61
|
800
|
7.59
|
30.38
|
0.96
|
700
|
7.61
|
32.81
|
0.98
|
600
|
7.64
|
35.01
|
0.82
|
500
|
7.68
|
37.54
|
0.69
|
400
|
7.72
|
40.03
|
0.62
|
300
|
7.75
|
42.03
|
0.56
|
200
|
7.78
|
43.15
|
0.52
|
100
|
7.82
|
43.10
|
0.48
|
25
|
7.84
|
42.21
|
0.45
|
The initial condition of the model assumed an IT of 245°C at 25 mm away from the groove (a common measurement location in welding). The IT is heterogeneous in the joint owing to the localized heating induced by the electric arc [27]; hence, the model considers the temperature distribution as the initial condition. The boundary condition considers the heat flow due to convection and radiation in the welded joint according to Eq. 3, where q is the energy supplied by the heat source, ε is the emissivity coefficient (0.8), σ is the Stefan–Boltzmann constant (5.6704∙10− 8 Wm− 2K− 4), T0 is the room temperature (20°C), and h is the convective heat transfer coefficient (25 Wm− 2K− 4). The latent heat of the phase transformation is also considered.
$$-k\left(\frac{\partial T}{\partial x}+\frac{\partial T}{\partial y}+\frac{\partial T}{\partial z}\right)=q-\epsilon \sigma \left({T}^{4}-{T}_{0}^{4}\right)-h\left(T-{T}_{0}\right)$$
3
,
The moving Goldak double-ellipsoid model [28] was adopted to represent the welding heat source. This model is a combination of two ellipsoids, one in front and the other at the rear, as described by Eq. 4. a, b, and ci are the semi-axes of the double ellipsoid (Table 5); ν, qi, and t are the welding speed (Table 3), power density distribution in the double ellipsoid, and time, respectively. U is the electric arc voltage, η is the thermal efficiency of welding (0.86), and I is the welding current (Table 3). r and f are the rear and front quadrants of the Goldak double ellipsoid; ff and fr are the fractions of the heat at the rear (fr, Table 5) and front (ff = 2 − fr), respectively. Table 5 presents the dimensions of Goldak's heat source (I and II) used to simulate the weld passes 7–8 (I) and 5–6 (II). Figure 2 shows the weld pass numbers 5–8.
$${q}_{i}\left(x,y,z,t\right)= \frac{6\sqrt{3}{f}_{i}\eta UI}{ab{c}_{i}\pi \sqrt{\pi }}exp\left(-3\frac{{x}^{2}}{{a}^{2}}-3\frac{{ y}^{2}}{{b}^{2}}-3\frac{{\left(z- \nu t\right)}^{2}}{{c}_{i}^{2}}\right), i=f and r$$
4
Table 5
Goldak's heat source parameters.
Heat source
|
a
|
b
|
cf
|
cr
|
ff
|
I
|
3.50
|
2.50
|
6.00
|
6.00
|
1.09
|
II
|
3.00
|
1.00
|
1.15
|
3.85
|
1.20
|
The accuracy of an FEM model is influenced by the mesh density used in the physical analysis. Thus, it is necessary to perform mesh refinement in the study regions (CGHAZ and ICCGHAZ) because they experienced severe temperature gradients during welding. The adequate meshing for the problem was based on the literature [29] and the authors’ simulation experience. Figure. 3 presents the FEM model mesh and a macrograph of the welded joint. Figure. 4 shows the thermal profile during the welding simulation of the sixth pass (forming a CGHAZ – left) and the eighth pass (forming a CGHAZ and ICCGHAZ – right). The ICCGHAZ was formed because of the intercritical reheating (900–773°C) of the previous CGHAZ.
2.5. Thermodynamic simulation
Thermodynamic simulation was performed using the JMatPro® software to obtain the physical properties of the API 5L X70 pipe (Table 4) and the continuous cooling transformation (CCT) diagram (Fig. 5). To predict the microstructures, simulations of the phase contents during cooling for all ITs (CGHAZ and ICCGHAZ) using the average cooling rate as the input data were performed. The average cooling rate was calculated as 1350–300°C (CGHAZ) for IT = 300, 350, 400, and 425°C (9, 4.6, 2.9, and 2.6°C/s, respectively) from the simulated thermal welding cycles (Sect. 3.1). For the ICCGHAZ, the average cooling rate was similar for all conditions (1°C/s, owing to the lower peak temperature, 850°C); hence, only a single simulation was performed for all ITs. All the previously mentioned simulations were based on the chemical composition of the base metal (Table 1).