Dominant dimensionless parameters controlling solute transfer during electromagnetic cold crucible melting and directional solidifying TiAl alloys

International Communications in Heat and Mass Transfer 90 (2018) 56–66

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Dominant dimensionless parameters controlling solute transfer during electromagnetic cold crucible melting and directional solidifying TiAl alloys

T

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Yaohua Yang, Ruirun Chen , Qi Wang, Jingjie Guo, Yanqing Su, Hongsheng Ding, Hengzhi Fu School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Electromagnetic cold crucible Solute transfer Finite volume method Segregation Solid-liquid interface Uniformity

Electromagnetic cold crucible (EMCC) is widely applied to melt and solidify refractory and reactive materials, the electromagnetically driven flow in the melt leads to intensive transfer behaviors of solute and strongly affects crystal growth. In this paper, a 3-D numerical model for predicting the solute transfer behaviors in a square EMCC used for melting and directional solidifying was established and verified. A two-way coupling method was proposed to calculate the electromagnetically driven flow and its effects on solute transfer. Influences of the dominant dimensionless parameters on solute transfer behaviors in the EMCC were examined, those being the Hartman (Ha), magnetic Reynolds (Rω), coils-melt position (h) and the ratio of the melt height to length (H/L) numbers. Results demonstrate that the solute segregation tends to appear in the vicinity of solid/liquid (S/L) interface, the top of meniscus and the confluence of two eddies near the wall of meniscus. The solute segregation −C C degree (Se = max C min ) decreases with increasing Ha, which contributes to the augment of flow intensity in the 0

melt. Moreover, the Se decreases linearly with increasing Rω owing to the intensive oscillation of melt flow. The enhanced EM coupling with increasing H/L results in the decrease of Se in the melt. However, the solute segregation in the corner of the meniscus gradually aggravates with increasing h, which results from the enlarging of lower eddies in the bulk of the melt. The solute segregation in the vicinity of S/L interface is strongly influenced by the scale and the flow intensity of lower eddies, which could change the phase transition path and morphology of crystal during directional solidification process. Larger value of Ha, Rω and H/L, as well as smaller h are beneficial to alleviate the solute segregation in the vicinity of S/L interface.

1. Introduction γ-TiAl based alloys are considered for high-temperature applications in aerospace and automotive industries due to their outstanding properties, for instance high specific modules, low density, excellent corrosion and creep resistance, as well as good oxidation resistance at higher temperatures [1–4]. However, their poor room temperature (RT) ductility and fracture toughness limit their engineering applications. Directional solidification is a promising technique to process TiAl alloys, which could eliminate the transverse grain boundaries, improve their RT and high-temperature mechanical properties with controllable microstructures [5–7]. However, the contamination and size limitation of the directionally solidified TiAl alloys are main problems of the traditional directional solidification methods. Cold crucible directional solidification (CCDS) technique proposed by Fu et al. [8] is a novel method for growing industrial scale TiAl alloys with controllable microstructure, while has no contamination. The electromagnetic force could induce complicated melt flow during CCDS, which is strongly

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Corresponding author. E-mail address: [email protected] (R. Chen).

https://doi.org/10.1016/j.icheatmasstransfer.2017.10.013

0735-1933/ © 2017 Elsevier Ltd. All rights reserved.

influenced by the configuration parameters of the electromagnetic cold crucible (EMCC). Crystal growth is intensively affected by heat/mass transfer in the melt and especially in the vicinity of solid/liquid (S/L) interface [9–11]. During directional solidification process, the solute may be rejected from the solidified interface due to segregation effect, and piles up in the vicinity of S/L interface, subsequently increases the solute concentration in the melt and the constitutional supercooling. Lots of investigations have proved that the melt flow in the molten pool could affect the solute distribution that determine the solidification microstructure during directional solidification. Kartavykh et al. [12] found that the rejection of Al solute ahead of the concave growth interface, along with thermo-gravitational convective mass transfer result in the appearance of peritectic macro-segregation during unidirectional counter-gravity solidification of Ti-46Al-8Nb alloy. By in situ observation of the S/L interface evolution, Wang et al. [13] found the thermoelectric magnetic (TEM) flows at the front of S/L interface can transport the rejected solutes from the depressed region of interface to

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⇀ vs ⇀ n

Nomenclature B J ⇀ FEM t Re Rω Ha h hm ht hb H L p g ⇀ v

magnetic flux density vector (T) current density vector (A m− 2) Lorenz force vector (N m− 3) time (s) Reynolds number of fluid magnetic Reynolds number Hartmann number coils-melt position number height of the middle of coils (m) height of the meniscus peak (m) height of the meniscus bottom (m) meniscus height (m) characteristic length scale of fluid flow (m) pressure (N m− 2) gravitational acceleration (m s− 2) velocity vector of melt (m s− 1)

C D C0 S Se

velocity vector of interface (m s− 1) surface normal vector solute concentration of melt (at.%) solute diffusion coefficient (m2 s− 1) initial solute concentration of melt (at.%) segregation of solute in the melt uniformity of solute in the melt

Greek symbols σ υ ρ μ δ ω βT βS

electrical conductivity of melt (S m− 1) dynamical viscosity (kg m− 1 s− 1) density of melt (kg m− 3) permeability of fluid (H m− 1) skin layer thickness (m) angle frequency (rad s− 1) thermal expansion coefficient (K− 1) solute expansion coefficient (at.%− 1)

S/L interface to reduce or even avoid segregation, i.e., an axially and/or radially non-uniform distribution of solute in the crystals [25,26]. Growth of crystal from a well-mixed melt offers several options to increase the process yield, such as by reducing the time step necessary to reach a uniform solute distribution in the melt prior to solidification, or by an increase of the solidification rate with lower risk of constitutional supercooling or formation of precipitates. The main purpose of the present work is to examine in some detail the effect of electromagnetic stirring on the distribution of solute in the melt from an initially homogeneous melt with planar S/L interface in the EMCC. The dimensionless parameters were derived to analyze the process, and a 3-D numerical model was established and verified for solving the problem by using finite volume method (FVM). The numerical investigations were carried out for different governing parameters, such as the frequency and intensity of imposed current, the relative coils-melt position and the melt shape. Our investigations could

the protruding region, which could decrease the interface tilting degree. Bogno et al. [14] revealed that the lateral solute segregation induced by melt flow leads to a significant deformation of the S/L interface, moreover, the melt flow can also influence the growth velocity and the characteristic parameters of the solute boundary layer. In addition, the transverse solute gradient in the vicinity of S/L interface leads to a transverse microstructure gradient, as already reported in previous papers [15,16]. In order to obtain a good mixing of solute near the S/L interface and avoid the large chemical segregations, alternating magnetic field was used to generate convection in the melt during directional solidifying GaInSb alloys in the axisymmetric configuration with low current frequency [17]. The application of magnetic field is an effective method to control the fluid flow and further the heat and mass transfer [18–22]. The optimization of solute transport by the melt flow is of particular relevance for the growth of crystals [23,24]. Forced flow in this respect aims mainly at effective mixing of the melt in front of the

Fig. 1. (a) 3-D Geometry of EMCC; (b) mesh of the EMCC for electromagnetic field calculation; (c) mesh and boundary conditions of meniscus for flow field and solute transfer calculation.

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configuration parameters of the EMCC in order to identify the dimensionless parameters that characterize the process. The melt flow has the characteristic Reynolds number Re = ρvL/ υ~ 104, therefore, it is considered to be turbulent. The turbulent flow in the melt is described by the Navier-Stocks equations following the assumptions below: (a) melt flow is incompressible, turbulence in the bulk of the meniscus while laminar in the boundary layer; (b) melt is the Newtonian fluid and (c) the thermo-physical properties are assumed to be constant. Therefore, the conservation equations for convective transport process can be written as follows: Continuity equation:

provide prediction of influences of electromagnetic stirring on the crystal growth in the EMCC based on solute transfer behaviors. 2. Numerical model and experimental procedure 2.1. Geometry definition The geometry and mesh of the model were generated using ANSYS (distributed by ANSYS HIT) according to the experimental equipment, as shown in Fig. 1. Only a quarter of the square EMCC was established due to its symmetry. The electromagnetic pressure confines the melt and a meniscus is formed by the balance of EM pressure, gravity, surface tension and hydrodynamic pressure, meanwhile, a planar S/L interface at the bottom was established due to the axial temperature gradient, which compose the computational domain of melt flow in our case, as shown in Fig. 1(c). The computational domain was discretised using tetrahedron and hexahedron elements, and there are 326,508 elements in the whole domain for solving the electromagnetic field by ANSYS and 850,914 elements in the meniscus domain for solving flow field by Fluent. The material properties for magnetic field and flow field simulations are presented in Table 1.

∇⋅ρ⇀ v =0

(4)

Momentum equation:

In skin layer:

v) ∂ (ρ⇀ v ⋅∇)⇀ v = υ∇2⇀ v − ∇p + ρ⇀ g βT ΔT + (ρ⇀ ∂t ⇀ + ρ⇀ g β ΔC + FEM S

In interior region:

2.2. Analysis of the dominant dimensionless parameters

2 μσω

∂C +⇀ v ⋅∇C = D∇2 C ∂t

(1)

2L2 δ2

(2)

The relative meniscus-coils position strongly influences the distribution and the intensity of electromagnetic forces, further, induces complex fluid flow in the melt. The melt flow in the meniscus is organized in cellular patterns, whose size mainly depends on the relative position of the meniscus and the coils, which can be expressed as:

h=

1 (ht − hm )2 + (hb − hm )2 hm

(7)

where υ is the viscosity, p is the pressure, g is the gravitational acceleration, βT is the thermal expansion coefficient, βS is the solute expansion coefficient, T is the temperature, C is the solute concentration, D is the solute diffusion coefficient and FEM is the Lorenz force. However, the importance of buoyancy forces depends strongly on the magnitude of the characteristic temperature and concentration differences in the melt. In our case, the turbulence flow induced by the high frequency electromagnetic field in the square EMCC could strongly homogenize the temperature and concentration field. As a consequence, the buoyancy is probably a secondary effect in the dynamics of the electromagnetic stirring [29]. In order to identify the dimensionless numbers that characterize the electromagnetic stirring process during directional solidification process by EMCC, the non-dimensional formulation of Eqs. (5) and (6) were derived. For this purpose, the following dimensionless variables are defined:

The magnetic Reynolds number, Rω, is a measure of the effective range of the magnetic field in the melt; it is the square of the radio of the characteristic length scale of melt to the skin layer thickness:

R ω = μσωL2 =

∂ (ρ⇀ v) + (ρ⇀ v ⋅∇)⇀ v = υ∇2⇀ v − ∇p + ρ⇀ g βT ΔT ∂t + ρ⇀ g βS ΔC (6)

Solute transport equation:

First of all, assuming that the melt is nonmagnetic with large electrical conductivity (displacement currents are negligible by the comparison with the current density J) and the accumulation of the electrical charge does not occur in the medium. In this case, the steady state time harmonic electromagnetic field is governed by Maxwell's equations. The action of the magnetic field is limited to the electromagnetic layer, its thickness, δ, depending on the current frequency ω,

δ=

(5)

p ⇀∗ y x z tU ⇀ ⇀ v , Y = , Z = , ∇∗ = L∇ , τ = ,V = ,P= ,B ρU 2 L L L L U ⇀ ⇀ B ⇀∗ J = ⇀, J = ⇀ B0 J0

X=

(3)

where hm is the height of the middle of coils (the maximum B appears), ht is the height of the meniscus peak, hb is the height of the bottom of meniscus. Therefore, h is a dimensionless number that reflects the deviation of the meniscus from the position of the maximum B, which can influence the flow pattern, its magnitude and the turbulence kinetic energy. Therefore, a smaller value of h means the meniscus is more close to the region where the maximum B locates. On the other hand, the increase of coils power leads to a more distinctive meniscus shape and reduced contact-zone between the melt and the water-cooled crucible. Therefore, the EM coupling between the inductor, the crucible and the melt changes, which finally determines the melt flow velocity distribution. Baake et al. [27,28] reported that higher H/D is more efficient for improving the superheat of the melt due to the enlargement of the lower flow eddy toroid. In our case, the ratio of meniscus height and length (H/L) is treated as a dimensionless configuration parameter of EMCC that affects the melt flow. Therefore, the dimensionless parameters of Rω, h and H/L were derived as the

where L, U and B0 are the length, velocity and magnetic field intensity reference scales respectively. Therefore, the dimensionless formulation are. Table 1 Material properties for magnetic field and flow field simulation.

58

Parts

Crucible and coils (Copper)

TiAl melt

Surroundings (air)

Relative permeability Resistivity (Ω·m) Permittivity Density (kg/m3) Dynamic viscosity (Pa s)

1

1

1

1.67 × 10− 8 – – –

1.79 × 10− 6 – 4000 0.00789

– 1.0059 – –

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In skin layer:

In interior region:

⇀ ∂V 1 ∗2 ⇀ Ha2 ⇀∗ ⇀∗ ⇀ ⇀ + ( V ⋅∇∗) V = −∇∗P + ∇ V + J ×B ∂τ Re Re (8) ⇀ ∂V 1 ∗2 ⇀∗ ⇀ ⇀ + ( V ⋅∇∗) V = −∇∗P + ∇ V ∂τ Re

Table 2 Input values of parametric simulations of solute transport.

(9)

Note that, for the derivation of Eqs. (8) and (9), all the physical properties were considered constant. There are three dimensionless numbers are identified, which represent the relative magnitudes of the different force terms in Eqs. (5) and (6). The Reynolds number (the ratio of inertia, (v·∇)v, to viscous forces, υ∇2v) dominates in the whole meniscus is defined as:

Re = U Lρ / υ

(11)

(12)

Evidently Ha represents the effects of the electromagnetic force on melt flow. 2.3. Boundary conditions For solving the electromagnetic field, the eddy current problem consists of a vortex region (cold crucible, coils and TiAl charge) and a non-eddy region (air and cooling water). A boundary condition gives the normal flux density ΓB:

⇀⇀ B ⋅n = 0

A boundary condition gives the tangential magnetic field strength (14)

(15)

There are two types of fluid boundaries used to simplify the mathematical models: a smooth solid wall of crucible and a free surface of the fluid. No fluid may pass through the wall and therefore the velocity component normal to the boundary must be equal to the velocity of wall:

⇀ v ⋅⇀ n =⇀ vs ⋅⇀ n

400 20 50 17/14 1.8 6.13 0.35 1.21

600 50 30 17/12 2.2 15.48 0.42 1.42

1000 160 60 20/14 2.5 49.64 0.56 1.43

1200 240 10 24/14 2.7 74.10 0.91 1.71

(16) 3. Results and discussion

where ⇀ v is the velocity of fluid, ⇀ n is the vs is the velocity of wall and ⇀ surface normal vector. The tangential velocity component on the wall must be zero (no-slip boundary condition). For the fluid next to the boundary, which allows existence of the tangential velocity component, can be described as:

⇀ v ×⇀ n =⇀ vs × ⇀ n

I (A) f (kHz) ht (mm) H/L (mm) Ha Rω h H/L

The as-cast ingots with nominal composition of Ti-48Al alloy were prepared by induction skull melting technology and remelted for three times for homogenization. Then the ingot was cut into many smaller bars (Φ20 mm) as the primer and feeding rod. The alloying of Ti-48Al alloy with addition of tungsten has been carried out in the EMCC, and the experimental procedure was presented in our previous investigation [32]. Directional solidification experiments were carried out under a certain pulling velocity and coil current, the specific details of CCDS process have been described by Nie et al. [33]. The solidified TiAl ingots were cut into two halves longitudinally for microstructure observation. The morphology of crystal in the vicinity of S/L interface was observed by scanning electron microscopy (SEM) in back-scattered electron (BSE) mode, while the concentration distribution was detected by energy-dispersive X-ray spectroscopy (EDS).

The boundary between vortex and non-vortex regions Γ12:

⇀ ⇀ n1 + B2⋅⇀ n2 = 0 ⎧ B1⋅⇀ ⎯ ⎯⇀ ⎯ ⇀ ⇀ ⎨ ⎯⇀ H × n + H 1 1 2 × n2 = 0 ⎩

S1 S2 S3 S4 S1 S2 S3 S4

2.5. Experimental procedure

(13)

ΓH:

⎯⇀ ⎯ H ×⇀ n =0

Input values of parametric simulations

In order to gain knowledge with respect to the relative influence of each of the important dimensionless numbers on the solute transfer behaviors in the EMCC, a parametric study for simulating the solute transfer was carried out by considering the key aspects of the solute segregation in the melt. The parametric analysis was performed by taking into account the processing parameters, such as the intensity and frequency of the current, the relative coils-melt position and the melt shape (height and length of melt). The following cases were considered and the calculation for various input values of the dimensionless parameters is shown in Table 2. The upper part of Table 2 reports the dimensional input data used in the study, while the bottom part shows the relevant dimensionless quantities to each simulation. A two-way coupling method was used to calculate the flow field in the melt, the electromagnetic field problem was firstly solved by using finite element method, then the Lorenz force was transferred into fluid dynamics differential equations via the source term and solved by using finite volume method [30,31]. The RNG modification of the k-ε twoequation model was used to solve the turbulent flow, for each simulation, convergence to steady state was usually accomplished in 3000 iterations and 8 h was required on a single EV5 processor of a Dell T7810 workstation.

As a consequence, the ratio of the Lorentz force to viscous forces in skin layer can be expressed as a hybrid of Re and N. It is

Ha2 = N Re = B2L/ μUυ

Parameters

2.4. Numerical procedure

(10)

The radius of the crucible can be selected as the characteristic length scale L, the characteristic velocity U can be estimated as the average velocity of the flow for induction crucible furnaces. In skin B layer, J is primarily driven by B in Ohm's law, and so |J |~ μL . In such a case, N represents the ratio of the Lorenz force, J × B/ρ, to inertia, (v·∇) v.

N = B2/ μρU 2

Simulation

3.1. Electromagnetically driven flow and mass transfer behaviors in the meniscus From a metallurgical point of view, the electromagnetic stirring is an efficient method to achieve a homogeneous melt. In our case, the calculated time-averaged 3-D flow pattern induced by the high frequency electromagnetic field in a square EMCC was shown in Fig. 2(a) using velocity vectors. It can be seen that two eddies with different flow directions appear at the half meridian plane of the meniscus, the core of eddies locate close to the free surface within the skin layer due to skin

(17)

For the simulation of solute transport in the bulk of meniscus, the melt was assumed from an initially homogeneous melt with 48 at. % Al solute, and non-flux boundary condition for solute transport equations are applied for all walls. 59

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Fig. 2. Calculated time-averaged velocity vectors in the meniscus during directional solidification by square EMCC (a) 3D flow field in meniscus, (b) flow field in a vertical cut through the melt center perpendicular to the side wall.

effect of high frequency magnetic field, as shown in Fig. 2(b). The mass transfer behaviors by the meridional flow in the EMCC have been investigated by tracer method and the results are shown in Fig. 3 [32]. Because of the tangential velocity near the surface of the melt, tungsten particles would flow downwards near the surface of meniscus after they entered into the molten pool, as shown in Fig. 3(a). In the bulk of melt, tungsten particles move following the mean eddies and emerges preferentially in the skin layer; when the tungsten is transported to the region between the upper and the lower eddies, which is homogenized in axial by the strong oscillations of convection. In the vicinity of S/L interface, the downwards flow in the center of meniscus takes the tungsten from the bulk of melt to the S/L interface, then the left to right flow transport them to the corner of meniscus in the skin layer, where the stronger upwards flow brings the tungsten to the confluence of two eddies near the wall of meniscus, as shown in Fig. 3(b). The recirculation of tungsten in the melt following the meridional flow efficiently promote the uniformity of melt. Due to the similar flow pattern except the flow velocity in the longitudinal section of the meniscus, the following cases were just considered to investigate the melt flow and solute transfer behaviors in the meridian plane perpendicular to the side wall in the meniscus. Two dimensionless numbers were derived to evaluate the influence of electromagnetically driven flow on solute distribution in melt. The first one represents the segregation of solute in the vicinity of S/ L interface with an initial solute concentration of C0 at steady state, which can be defined as: S = | C − C0 |/C0. The solute segregation degree in the bulk of the meniscus can be expressed as:

Se =

Cmax − Cmin C0

(18) that Se is a dimensionless number that reflects the uniformity of solute in the bulk of melt. A smaller value of Se yields a more uniform solute distribution in the melt. 3.2. Influence of Ha on solute distribution Al solute distribution contours and velocity vectors of the flow field in the meridian plane of meniscus for different values of Ha are shown in Fig. 4. Mean flows observed consist normally of two eddies in the half-meridian plane of meniscus, and the dimensions of two eddies are very sensitive to the parameter of Ha. The lower mean eddies gradually decrease and confine to the corner of meniscus with increasing Ha, meanwhile, the upper eddies extend from the wall of meniscus to the symmetry axis. The maximum velocity (vmax) in melt and the velocity in the vicinity of S/L interface increases with increasing Ha. For the binary Ti-48Al melt in our case, the distribution of solute Al at steady state is almost homogeneous in the half-meridian plane of meniscus, except slight segregation at the top, the corner and the vicinity of S/L interface, as shown in the right part of Fig. 4. The smaller velocity and turbulent kinetic energy at the top and the confluence of two eddies near the wall of meniscus leads to the concentration of solute Al when Ha < 2.2. Further, the segregation of solute Al disappears with increasing Ha owing to the increased intensity of melt flow there. The oscillating velocity between the upper and lower eddies plays a main role in convective solute transfer between the upper and lower part of meniscus, which results in the homogeneous distribution of solute in the bulk of meniscus [34,35]. As shown in Fig. 5(a), Se decreases with increasing Ha, which contributes to the augment of melt flow intensity and turbulent kinetic energy. It indicates that the high frequency electromagnetic stirring in the melt promotes the homogenization of melt with increasing Ha. However, the nonuniform melt flow in the vicinity of S/L interface and

(18)

where Cmax is the maximal solute concentration in the melt, Cmin is the minimal solute concentration in the melt. It can be observed from Eq.

Fig. 3. Mass transfer behaviors in the half median plane of meniscus in the EMCC with tungsten tracer in the melt for different times (a) △t = 2 s, (a) △t = 5 s [32].

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Fig. 4. Al solute distribution contours and velocity vectors of the flow field in the meridian plane of meniscus for different values of Ha.

solidification segregation of TiAl alloys [36]. It can be seen that the segregation of Al and W solute alleviates in the melt with increasing operating current, which agrees well with the calculated results above.

the lighter Al solute lead to the segregation in front of the interface, where the concentration of Al solute is lower than C0. In the vicinity of S/L interface, the left to right melt flow brings the Al to the corner of meniscus, where the intensive upward flow in the skin layer transport the lighter Al to the confluence of two eddies near the wall of meniscus. Al segregation in the vicinity of S/L interface for different values of Ha are shown in Fig. 5(b). It can be found that relatively serious Al segregation appears at the center and periphery of meniscus when Ha = 1.8. Then, the segregation at the center of meniscus decreases and tends to zero with increasing Ha, and it is observably weakened near the periphery due to the augment of melt flow intensity there. The typical quasi-stationary distribution of W in the TiAl melt under different operating powers are displayed in Fig. 6. The W has been dissolved for long time interaction with the TiAl melt, and formed B2 phase (bright phase in BSE mode) after solidification. EDS in line presented in Fig. 6(a) indicates that the dendrite/equiaxed core is rich in W while Al is rich in the interdendritic liquid, which is known as the

3.3. Influence of Rω on solute distribution With increasing Rω, the electromagnetic forces are confined to the wall and the corner of the meniscus due to skin effect. As shown in Fig. 7, there are two mean eddies exist in the half meridian plane of the meniscus, the typical variation scale of the mean flow is of order L for low Rω and decreases close to the wall for larger Rω. As a consequence, the core of the eddies moves to the wall and downwards to the corner of the meniscus with increasing Rω, meanwhile, the lower eddies gradually decrease and even disappear when Rω is > 50. With augment of Rω, vmax increases first and then reaches the maximum when the skin layer thickness is approximately equal to 0.2 L (Rω = 50), after that it decreases. It reveals that the segregation at the top, confluence of two eddies

0.06

0.04

(a)

(b)

Ha=1.8 Ha=2.2 Ha=2.5 Ha=2.7

0.05 0.03

0.04

S

Se

0.03

0.02

0.02 0.01

0.01

0.00 1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.00 0.0

0.2

0.4

0.6

0.8

1.0

x/L

Ha

Fig. 5. Al solute distribution in the meridian plane of meniscus for different values of Ha (a) variation of Se in the melt, (b) Al solute segregation in the vicinity of S/L interface.

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Fig. 6. Quasi-stationary distribution of W in TiAl melt (a) 30 kW, (b) 40 kW and (c) 45 kW [32].

solute tends to be uniform in the vicinity of S/L interface with increasing Rω, which results from the gradually disappeared lower eddies and increased flow velocity there. It can be concluded that the melt is more homogeneous with increasing Rω, which results from the increased flow intensity near the wall of meniscus and S/L interface along with disappearing of lower eddies.

near the wall of meniscus and S/L interface front extensively exist for lower value of Rω = 6. With increasing Rω, the segregation at the confluence of two eddies decreases and finally disappears when Rω ≥ 50. However, the lower flow velocity in the core of upper eddies results in slight segregation near the wall of meniscus when Rω = 50. Meanwhile, the upper eddies gradually move downwards to the corner of meniscus with increasing Rω, which is beneficial to eliminate the segregation there, while the segregation at the top of meniscus still exists due to the lighter Al solute and smaller flow velocity there. When Rω = 74, the upper eddies are more confined close to the meniscus wall due to the smaller skin layer thickness, therefore, the segregation in the core of eddies disappears. However, under the limitation of small skin depth, the domain may be split into two regions: the skin layer and an interior region where the effective Reynolds number remains large. Therefore, there still exists slight segregation at the symmetry axis of S/ L interface (x/L = 0) due to the insufficient electromagnetic stirring there in our case. As discussed above, It can be seen that the Se decreases with increasing Rω linearly, which means that the melt is more homogeneous for larger Rω, as shown in Fig. 8(a). Fig. 8(b) reveals that relatively serious Al solute segregation emerges at the center and periphery of S/L interface when Rω ≤ 16, then it decreases and the distribution of Al

3.4. Influence of h on solute distribution The relative coil-melt position determines the distribution of electromagnetic force and its magnitude, further, it affects the flow pattern and solute transfer behaviors. The dimension of mean flows in the half meridian plane are very sensitive to the parameter of h, as shown in the left of Fig. 9. Two equivalent vortices appears in the half meridian plane with small value of h (h = 0.35 and h = 0.42), which results in more homogeneous solute distribution in the bulk of meniscus, except slight segregation at the corner of meniscus. One of the vortices will dwindle with increasing h as the change of electromagnetic force distribution, the lower vortices lessen when hb > hm (h = 0.56) while the upper vortices lessen when ht < hm (h = 0.91). It indicates that the enlarged upper eddies (h = 0.56) promotes the homogenization of the upper

Fig. 7. Al solute distribution contours and velocity vectors of the flow field in the meridian plane of meniscus for different values of Rω.

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Y. Yang et al. 0.06

0.08

(a)

Calculation Curve fitting

0.05

0.07

(b)

R ω=6 R ω=16 R ω=50 R ω=74

0.06 0.05 0.04

0.03

S

Se

0.04

0.03 0.02

0.02 0.01

0.01 0.00

0.00 0

10

20

30

40

50

60

70

80

Rω

0.0

0.2

0.4

x/L

0.6

0.8

1.0

Fig. 8. Al solute distribution in the meridian plane of meniscus for different values of Rω (a) variation of Se in the melt, (b) Al solute segregation in the vicinity of S/L interface.

3.5. Influence of H/L on solute distribution

part of meniscus, however, it aggravates the segregation at the corner and vicinity of S/L interface due to the lower flow velocity there. However, with enlarging of lower eddies (h = 0.91), the flow velocity in the skin layer increases at the upper part of meniscus while decreases at the corner, which aggravates the segregation near the wall of meniscus especially at the corner. It can be seen from Fig. 10(a) that Se increases with an increase of h. The nonuniform distribution of Al solute results from the unbalanced distribution of two eddies in the half median plan of the meniscus, which results in heterogeneous flow velocity and turbulence kinetic energy distribution. With increasing h, the Al solute segregation in the vicinity of S/L interface aggravates gradually especially at the periphery as presented in Fig. 10(b), which results from the enlarged lower eddies and decreased fluid velocity there.

Al solute distribution contours and velocity vectors of the flow field in the meridian plane of meniscus for different values of H/L are presented in Fig. 11. With increasing H/L, the maximum velocity of melt flow increases gradually. It can be explained as the increased H/L leads to stronger EM coupling of the melt in the EMCC. However, smaller value of H/L = 1.21 limits the lower eddies and aggravates the segregation at the corner of meniscus, while the enlarged upper eddies could eliminate the segregation at the top and the confluence of two eddies near the wall of meniscus as discussed above. The segregation at the top and the confluence of two eddies near the wall of meniscus appears when H/L = 1.42, which results from the expansion of lower eddies and the smaller velocity there. After that, the increased EM coupling with augment of H/L decreases the segregation at the top, the confluence of two eddies near the wall of meniscus and the S/L interface gradually, which leads to the decrease of Se, as shown

Fig. 9. Al solute distribution contours and velocity vectors of the flow field in the meridian plane of meniscus for different values of h in EMCC.

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0.16

(a)

Calculation Curve fitting

0.20

0.14

(b)

h=0.35 h=0.42 h=0.56 h=0.91

0.12 0.10 0.08

0.12

S

Se

0.16

0.06

0.08 0.04

0.04

0.02

0.00 0.2

0.00

0.4

0.6

0.8

1.0

h

0.0

0.2

0.4

x/L

0.6

0.8

1.0

Fig. 10. Al solute distribution in the meridian plane of meniscus for different values of h (a) variation of Se in the melt, (b) Al solute segregation in the vicinity of S/L interface.

Fig. 11. Al solute distribution contours and velocity vectors of the flow field in the meridian plane of meniscus for different values of H/L.

boundary layer with Al solute higher than 48 at.% at steady state. Further, the heterogeneous distribution of Al solute in front of the S/L interface will change the phase transition path during crystal growth. As shown in Fig. 13(b), the primary β (bcc) phase with average Al solute of 48.12 at. % can be observed at the center of S/L interface (x/L = 0), meanwhile, the primary α (hcp) phase with average Al solute of 50.89 at. % can be observed at the midpoint of S/L interface (x/ L = 0.5), as shown in Fig. 13(c). The accumulated Al in the vicinity of S/L interface leads to the change of phase transition path during solidification process. The phase transition can be explained by the binary phase diagram of TiAl alloys, as shown in Fig. 13(a). For Ti-48Al alloy,

in Fig. 12(a). However, the Se reaches steady state when H/L > 1.43 due to the existence of lower eddies and their smaller flow velocity still results in Al solute segregation in the vicinity of S/L interface especially at the periphery. Fig. 12(b) indicates that the segregation in the vicinity of S/L interface decreases with increasing value of H/L, which results from the augment of EM coupling. The Ti-48Al alloy was directionally solidified by the EMCC under different operating powers (30 kHz) and pulling velocity of 1 mm/min, the binary phase diagram of TiAl alloys is shown in Fig. 13(a). During directional solidification process, the rejected Al solute from the S/L interface will be transported by the melt flow and formed solute 64

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0.30

(a)

0.30

H/L=1.21 H/L=1.42 H/L=1.43 H/L=1.71

0.25

0.25

0.20

S

0.20

Se

(b)

0.15

0.15

0.10

0.10 0.05

0.05 0.00

0.00 1.2

1.3

1.4

H/L

1.5

1.6

0.0

1.7

0.2

0.4

x/L

0.6

0.8

1.0

Fig. 12. Al solute distribution in the meridian plane of meniscus for different values of H/L (a) variation of Se in the melt, (b) Al solute segregation in the vicinity of S/L interface.

concentration at the downstream side of the dendrites can inhibit the formation of secondary branches completely. The calculated results indicate that the melt flows from the center to the periphery at the front of S/L interface (the red arrows displayed in Fig. 13(c)), which results in the preferential growth of the secondary arms of α primary phase in the vicinity of S/L interface, as presented in Fig. 13(c). Under the operating power of 45 kW, there are completely β primary phase in the vicinity of S/L interface shown in Fig. 13(d) and (e), which results from the increased flow intensity and decreased lower eddies with augment of Ha.

the solidification and phase transformation sequence is [37]:

L → L + β → L + β + α p → L r + α p → α p + αL → lamellar (α2 + γ) + γs where Lr is the residual melt after peritectic reaction, αp comes from the peritectic reaction, while αL is directly precipitated from the residual melt and γs is transformed by the solid α phase. When the concentration of Al is higher than 49.4 at.% at the solidification front, the primary phase changes from β phase to α phase, and the solidification path can be expressed as:

L → L + α → α + γ → lamellar (α2 + γ) + γs

4. Conclusions

During directional solidification process of Ga-In alloys, Shevchenko et al. [38] have found that the forced flow in radial direction could provoke a preferential growth of the secondary arms at the upstream side of the primary dendrite arms, whereas the high solute

1. The dimensionless form of electromagnetically driven flow governing equation and the configuration parameters of EMCC were derived to identify the dimensionless parameters that characterize the solute transfer behaviors in the EMCC, those being the Hartman,

Fig. 13. (a) binary phase diagram of TiAl alloys and the dendritic morphology in the vicinity of S/L interface (b) x/L = 0 and (c) x/L = 0.5 for current of 30 kW; (d) x/L = 0 and (e) x/ L = 0.5 for current of 45 kW.

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magnetic Reynolds, coils-melt position and H/L numbers. 2. A 3-D model was established for calculating the electromagnetically driven flow and the solute transfer behaviors in the EMCC, the solute distribution in the melt was well verified by the experiment, which indicates that the 3-D model can be used to predict the flow field and solute distribution in the EMCC. 3. The solute segregation degree (Se) decreases with increasing Ha and H/L, which contributes to the augment of flow intensity in the melt. Moreover, the Se decreases with increasing Rω linearly owing to the intensive oscillation of melt flow. However, the Se aggravates with increasing h as the enlarged lower eddies in the bulk of meniscus. 4. The solute segregation in the vicinity of S/L interface is strongly influenced by the scale and flow intensity of the lower eddies, and further could change the phase transition path and morphology of crystal during directional solidification process. Larger value of Ha and Rω decrease the scale of lower eddies, while larger H/L and smaller h increase the flow intensity, which are beneficial for alleviating the solute segregation at the front of S/L interface.

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