The Suppression of the Natural Convection in the Directional Solidification Processing of Superalloy by the Introduction of the Traveling Magnetic Field: 2D and 3D Simulation

Electromagnetic field model

Sine alternating current was fed into the coil. The electromagnetic boundary conditions both on far field elements of the open boundary and at the axis of the model are assumed to be flux parallel. Maxwell equations to solve quasistatic electromagnetic field can be written as follows.

Electromagnetic constitutive equation in isotropic medium is

Lorentz Force can be expressed as

To simplify the solution, equations applied magnetic vector potential A→ and scalar potential φ is given by

where H→ is the magnetic intensity, B→ the magnetic flux density, E→ the electric field strength, J→ the current density, μ the magnetic permeability, σ the electric conductivity and F→ the Lorentz Force.

Solidification model

Continuity equation (mass conservation equation)

Navier–Stokes equation (conservation of momentum)

Energy equation (conservation of energy)

where u→ is the velocity of flow, P the pressure, g→ the gravity, τij the shear stress, T the temperature, ρ the density, F→ the Lorentz force and k the thermal conductivity.

Radiative transfer equation, the radiative transfer equation is solved for a set of n different directions, S⃗i, i=1, 2,…, n, and the integrals over these directions are replaced by numerical quadratures.

where I is the radiation intensity, r⃗ the position vector, s⃗ the direction vector, σ the Stefan–Boltzmann constant and a the wall absorptivity.

Boundary conditions and computational parameters

In this study, each TMF coil represents a 54×58 mm² bunch of 285 windings. The three co-coils, which are coupled in star connection to the variable frequency power source, are supplied by the standard three-phase alternating current. The distance between the coils is Δh=15 mm. The height of the TMF coil system and the inner diameter are h=204 mm and d=185 mm, respectively. The specific dimension of casting with variable section is shown in Figure 1. The Ni-based superalloy employed was CMSX-4(Cr 6.2, Co 9.4, Mo 0.7, W 5.7, Al 5.6, Ta 6.5, Ti 1.2, Re 3.0 and Hf 0.08 in wt pct). Tables 1 and 2 show physical parameters of the FEM model. In this process, density changes with temperature using Boussinesq assumption. As shown in Figure 1, the complete directional solidification system in traveling magnetic field includes the coil generating traveling magnetic field, graphite heating body and withdrawal unit. In a 2D model, the specific dimension of specimen with variable section is also shown in Figure 1. Schematic representation of the computational domains and the meshes for the furnace heat transfer is shown in Figure 2(a). In a 3D model, the position of the surface and central cross-sections in the casting is shown in Figure 2(b).

Figure 1:

Sketch of the setup (left half) and geometry of the specimen (right half).

Figure 2:

(a) Schematic representation of the computational domains and the meshes for the furnace heat transfer, (b) the position of the surface and central cross-sections in the 3D casting.

There are some assumptions in the DS process. (1) Displacement current is ignored and time-harmonic electromagnetic field is assumed as quasistatic electromagnetic field; (2) flow in melt is laminar flow; (3) electromagnetic force affects fluid flow, but fluid flow has no effect on electromagnetic; (4) alloy melt is incompressible Newton fluid.

Solution procedure

The distribution of magnetic field and Lorentz force field is calculated by ANSYS Emag. In the calculation of TMF, solid236 quadrilateral element is used for coil, graphite bush and casting. Solid236 triangular element is used for the near field. In addition, the far field dissipation was described by using INFIN111 far field element.

Then the date of force will be introduced into Fluent15.0 in the way of interpolation. During this process, some User-Defined Functions (UDFs) are developed to precisely describe the movement of casting and data transfer between FDM and FEM.

The calculations of temperature field and flow field are performed by applying the discrete ordinates (DO) model and laminar model in software FLUENT, respectively. DO model will be used to describe the radiative heat transfer between furnace wall and castings, and the flow in melt in the process will be calculated by laminar flow model. Solidification and melting model will be used to describe the DS process of alloy. In addition, the alloy will be treated as the first phase, while the air is the second phase in volume of fluid (VOF) model.

Magnetic fields

Figure 3(a) shows the 2D magnetic strength distribution in upward and downward traveling magnetic field, respectively. Figure 3(b) shows the 3D magnetic strength distribution in upward and downward traveling magnetic field, respectively. In 2D case, under the action of upward and downward traveling magnetic field, the magnetic strength at specimen top is 34.2 mT and 35.0 mT, respectively. With the reduction of specimen mid-height, the magnetic field strength also reduces correspondingly. In the same height, the magnetic field strength at both sides of specimen will be higher than that in middle. The comparisons of 2D and 3D models show that the intensity distributions of their magnetic fields are basically identical, but the intensity of magnetic field for 3D model tends to be smaller.

Figure 3:

Numerical results computed by Ansys Emag (I=50 A, f=50 Hz): (a) magnetic field intensity distribution: 2D upward TMF (up) and 2D downward TMF (down); (b) magnetic field intensity distribution: 3D upward TMF (up) and 3D downward TMF (down).

Reference case for 2D simulation

Under the condition of natural convection, uneven distribution of temperature and concentration easily lead to the morphology diversification of the solidification interface. Especially, the radial temperature gradient is closely related to the morphology of solidification interface. The vector diagram of velocity distribution, the distribution of solidification fraction and the temperature distribution in 820 s, 1,040 s and 1,200 s in the solidification process are shown in Figure 4(a)–4(c), respectively.

Figure 4:

Reference case with no TMF: (a1–a3) The corresponding velocity vectors; (b1–b3) the corresponding contours of solid volume fraction; (c1–c3) the corresponding contours of temperature; (a1–c1) t=820 s; (a2–c2) t=1,040 s; (a3–c3) t=1,200 s.

As shown in Figure 4(a1), when solidification proceeds to 820 s, the flow pattern in the large cross-section region is symmetrical. The left-side vortex at the bottom is clockwise, and the other vortex is nested in the right side of the mold and driven in a counter-clockwise motion. As the solidification proceeds, the flow pattern does not change, but the velocity decreases in the melt (Figure 4(a2)–(a3)). When the solidification interface stays in the small cross-section region, the interface maintains flat as shown in Figure 4(b1). When the solidification enters into large cross-section, the interface changes from flat to slightly concave at 1,040 s (Figure 4(b2)).

The concave degree of solicitation interface becomes intensified (Figure 4(b3)) at 1,200 s. In the later stage of solidification, the temperature distribution is non-uniform. The isothermals, which exhibit the similar trend with the change of solidification interface, become concave in the later stage of solidification (Figure 4(c3)). Both the concave solidification interface and non-straight isothermals cause large concentration gradient in the flat direction. According to literature [1–3], oblique solidification interface is more likely to generate macrosegregation. In the actual situation, freckle defects also appear in the position of the blade platform. Combing with the change rule of velocity in the front of solidification interface in Figure 5, it can be observed that when solidification enters into the large cross-section from the small cross-section, the velocity in the front of solidification interface increases significantly, and the flow pattern also changes greatly, which is one of main reasons of solidification interface changing to concave shape.

Figure 5:

Time history of V during solidification (V refers to the average velocity of the solidification front, no TMF).

Influence of magnetic field intensity for 2D simulation

When B=5 mT, to more systematically investigate the influence of downward TMF to different physical fields in different stages of solidification, the vector diagram of velocity distribution, the distribution of solidification fraction and the temperature distribution corresponding to 820 s, 1,080 s, 1,150 s and 1,320 s in the solidification process with downward TMF are shown in Figure 6(a)–6(d), respectively. From the evolution of flow pattern in the melt, it can be observed that the left-side vortex in the front of solidification interface expands to the right side gradually, causing the flow pattern changing from symmetry to non-symmetry. As a result, the solidification interface evolves to slightly concave interface. The corresponding isothermals also experience a flat-convex-concave transition process.

Figure 6:

The case with I=10 A, f=50 Hz, Δϕ=120°, TMF down: (a1–a4) The corresponding velocity vectors at different times; (b1–b4) the corresponding contours of solid volume fraction; (c1–c4) the corresponding contours of temperature; (a1–c1) t=820 s; (a2–c2) t=1,080 s; (a3–c3) t=1,150 s; (a4–c4) t=1,320 s.

The downward TMF with different intensities significantly affects the flow pattern in the melt and the solidification interface. The velocity profile in front of solidification interface, the vector diagram of velocity distribution and the distribution of solidification fraction in downward TMF with different intensities are shown in Figure 7(a)–7(d), respectively. Especially, the flow pattern and distributions of solidification fraction are shown in Figure 6(a) at 1,050 s. By comparing Figure 7(a1), 7(b1) and 7(c1), it can be observed that when maintaining frequency f to be constant and increasing the current at the same time, the velocity of front of solidification interface increases with the current significantly. The flow pattern at front of solidification changes from symmetry to non-symmetry by comparing Figure 7(a2), 7(b2) and 7(c2), but the direction does not change. The shape of the solidification interface also changes from the planar interface to concave interface. By comparing Figure 7 (b1) and 7(d1), it can be observed that when maintaining the current I to be constant, and increasing the frequency at the same time, the velocity of front edge of solidification interface also increases with the frequency. When the frequency changes from 50 Hz to 500 Hz, the maximum velocity in front of solidification interface increases from 14 mm/s to 65 mm/s. At the same time, the flow pattern in the melt also changes significantly, when the frequency increases, only one counter-clock wise vortex exists in the melt (Figure 7(d2)).

Figure 7:

Numerical results computed by Fluent (v=120 um/s, Δϕ=120°): (a1–d1) velocity profiles at the front of solidification interface; (a2–d2) the corresponding velocity vectors; (a3–d3) the contours of solid volume fraction; (a1–a3) I=10 A, f=50 Hz; (b1–b3) I=25 A, f=50 Hz; (c1–c3) I=50 A, f=50 Hz; (d1–d3) I=25 A, f=500 Hz.

Reference case for 3D simulation

For comparing with the 2D model mentioned above, the 3D model is constructed under the same boundary conditions. The number of grids in 3D model is much greater than that of 2D model. It can be seen from Figure 7(a)–7(b) that the evolution of solidification interface morphology in 3D model is similar to that in 2D model. Yet, their flow forms are different from each other (Figure 7(c). As can be found in Figure 7(a), in the early stage of solidification, large flow rate can be found in the places where geometry changes, which is the same with 2D model in terms of flow form but has a greater flow rate. The comparison results show that, the flow rates on the casting surface and in the middle section also differ a lot. As the solidification progresses (Figure 7(b)), compared to the flow rate in the middle section, the flow rate on the casting surface gradually increases. Thus, segregation defects are more likely to appear on the surface of the casting samples, which is consistent with the actual experimental findings. When it comes to the end of the solidification (Figure 7(c)), the difference between flow rates on the casting surface and in the middle section gradually increases. These evolutions of flow form cannot be reflected in the simulation of 2D model.

For describing the complicated flow status in the liquid, two longitudinal sections chosen from different places are used: the middle longitudinal section and the surface longitudinal section [10] (as shown in Figure 2(b)). It is found that the both of the flow intensity and the pattern abruptly change when the solid grows into the large section (Figure 8). (1) When the solid/liquid interface is in the small section (Figure 9(a1)), the flow intensity at the interface front is relatively weak at 824 s. The average values on the middle longitudinal section and the surface longitudinal section are about 0.95 mm/s and 0.82 mm/s (Figure 9(a2)–(a3)), respectively. The flow vortex is driven in a counter-clockwise motion. (2) As the solid grows into the large section (Figure 9(b1)), the solid/liquid interface was macroscopically curved and convex toward the melt at 900 s. The corresponding flow intensities (Figure 9(b2)–(b3)) were enhanced to 3.21 mm/s (middle longitudinal section) and 2.02 mm/s (surface longitudinal section). Furthermore, the direction of the flow vortex was found to be reversed to anti-clockwise on both the surface and middle longitudinal sections. Additionally, the flow pattern also undergoes significant changes.

Figure 8:

The case with no TMF: (a1–a3) The contours of temperature; (b1–b3) the corresponding streamlines; (a1–b1) t=805 s; (a2–b2) t=1,010 s; (a3–b3) t=1,320 s.

Figure 9:

The case with no TMF at different times: (a1–c1) 3D velocity vectors; (a2–c2) 2D velocity vectors in the middle plane; (a3–c3) 2D velocity vectors in the surface plane; (a1–a3) t=805 s; (b1–b3) t=1,010 s; (c1–c3) t=1,320 s.

Influence of magnetic field intensity for 3D simulation

The above computations show that, under the action of downward TMF with appropriate intensity, the natural convection within the melt can be suppressed and a flat solidification interface can also be obtained in regions with cross-section variation. Based on 3D model, the influences of downward TMFs with different intensities on melt flow form and solidification interface morphology are studied. As B changes from 0, 6.3 mT to 32.4 mT, the morphologies of the solidification interfaces are shown in Figure 10(a), respectively. When B=6.3 mT, a flat solidification interface can be obtained in regions with geometric changes (Figure 10(a2)); the natural convection within the melt will also be suppressed to a certain extent. However, with the increase of intensity of magnetic field, the melt flow rate rises rapidly (Figure 10(b3)) and the solidification interface becomes out of flatness again (Figure 10(a3)).

Figure 10:

(a1–a3) The solidification interface at 1,013 s; (b1–b3) the corresponding streamlines at 1,013 s; (a1–b1) no TMF; (a2–b2) downward TMF (I=10 A, f=25 Hz, Δϕ=120°); (a3–b3) downward TMF (I=50 A, f=50 Hz, Δϕ=120°).

The comparison between 2D and 3D simulation

The simulation results obtained from 2D and 3D models are similar with each other in characterization of the temperature field and solid–liquid interface morphology. Additionally, when the solid grows into the large section, the abruptly changes of the flow status are found in both of the above two models. However, the flow intensity and pattern conducted from the two are significantly different. The flow intensity in 2D model is relatively weaker than that in 3D model. Furthermore, during the solidification at the large section, there are two symmetric vortexes at the front of solidification interface in 2D model (pointed by the arrows in Figure 4(a1)) but only one in 3D model (pointed by the arrows in Figure 9(b2)). At the final stage of the solidification, the flow pattern remains unchanged in 2D model but undergoes marked changes in 3D model. In this case, a vortex parallel to the solid/liquid interface is found in 3D model (Figure 8(b3)), substituting the former vortex perpendicular to the solid/liquid interface. In summary, 2D model may not work well to illustrate the flow status in the final stage of the solidification.