Numerical Simulation of the Influence Mechanism of Melt Rate Variation on the Macrosegregation of 8Cr4Mo4V-Bearing Steel During Vacuum Arc Remelting

Abstract

In this study, 8Cr4Mo4V steel was selected as the research material to develop a numerical model of the macrosegregation phenomenon during vacuum arc remelting (VAR). The accuracy of the model was validated by comparing it with the literature and experimental results. According to the simulation results, molten steel flows down along the solidification front, resulting in positive segregation at the center and negative segregation close to the edge of the ingot. Solute enrichment reduces the undercooling of the alloy system, which in turn decreases the local solidification rate and causes a slight increase in steady-state molten pool depth. Notably, as the molten pool depth increases, the temperature gradient decreases, while the local cooling rate remains nearly constant, which leads to an increase in the local solidification rate again. Consequently, the positive segregation degree at the ingot’s center is gradually alleviated, and the depth of the molten pool gradually decreases. Furthermore, macrosegregation in VAR ingots becomes pronounced with an increase in melt rate. The main reason for this is due to the increased molten pool depth when the melt rate is increasing, which strengthens fluid flow and accelerates the migration of solute elements to the center. Additionally, due to the increase in the extent of solute enrichment when the melt rate is increasing, the degree of fluctuation in both the steady-state molten pool depth and positive segregation increases.

1. Introduction

Vacuum arc remelting (VAR) is a refining process in steelmaking that plays a decisive role in the performance of the final ingot. In the VAR process, the electrodes melt to form droplets under vacuum conditions, and these droplets then drop into the mold and solidify. This unique smelting method can effectively reduce gas contents (nitrogen, oxygen, etc.), improve homogeneity, and decrease defects, allowing us to obtain high-quality ingots [1,2]. Therefore, VAR is widely used in the production of high-quality special steel, such as high-temperature alloys, titanium alloys, etc. [3,4].

Macrosegregation is the phenomenon of chemical composition inhomogeneity in a macroscopic region caused by solute redistribution and solid–liquid convection during the solidification of alloys. This leads to great differences in the mechanical and physical properties of different parts of ingots and seriously affects the service life of materials. Until now, many studies have put forward thoughts of ways to mitigate macrosegregation in VAR ingots. Some studies have shown that adding magnetic field stirring during the VAR process can strengthen the mass transfer process in the molten pool, thereby effectively reducing the extent of macrosegregation in ingots [5,6,7]. Other studies have found that different arc distributions in the VAR process also affect the distribution of solute elements [8,9], and results show that a diffuse arc can mitigate macrosegregation in ingots [9]. Some studies have shown that an increase in melting current in the VAR process will lead to an increase in Lorentz forces, which will affect the flow of the molten pool and deepen the degree of macrosegregation in the ingot [5,10]. In addition, it has been found that enhanced cooling can mitigate positive segregation in the center area of VAR ingots and improve homogenization in the ingot [11]. Other scholars have examined other research ideas to resolve this issue, from the remelting period to the feeding period, and have found that extending the feeding time in the feeding period of the VAR process can decrease the positive segregation zone [12]. In addition to the above research, there is another important parameter in the VAR process: the melt rate, which is closely related to the solidification quality of the ingot. However, presently, few studies have focused on the influence of melt rate variations on the macrosegregation of VAR ingots, and the effect of melt rate variation on solute distribution is not clear. In addition, possible transverse cracks in the electrode can lead to uneven variations in material temperatures around the crack and then lead to a rapidly increasing melt rate at first, followed by a decrease, which is conducive to the formation of freckle, white spot, and segregation defects in VAR ingots [13,14,15]. This is also related to changes in solute distribution with melt rate. Therefore, it is necessary to reveal the effect mechanism of melt rate variations on solute distribution in VAR ingots.

The purpose of this paper was to investigate the influence of melt rate variations on ingot macrosegregation in the VAR process. Considering the prohibitively expensive cost of using a trial-and-error method for a VAR experiment, numerical simulations were applied to this work, and a two-phase model was developed to predict solute distribution in VAR ingots. The model was verified by comparing it with experimental results in the literature. Then, taking the carbon element distribution in 8Cr4Mo4V steel as an example, the distribution of macrosegregation in VAR ingots and the causes for its formation were analyzed. In addition, the variation in macrosegregation with melt rate was investigated, and the influence mechanism of melt rate on macrosegregation in the VAR process was revealed.

2. Numerical Model Description

A schematic sketch of the VAR process is shown in Figure 1. It can be seen that the smelting process mainly includes molten steel (liquid phase) and solidified ingots (solid phase), and the mushy zone is a mixture of liquid and solid phases. In order to describe the liquid phase flow, as well as the heat and mass transfer between the solid and liquid phases, an Eulerian two-phase model is established in this paper.

2.1. General Assumptions

• This research focuses on the ingot, without considering the electrode, vacuum arc and mold [16].
• The ingot is assumed to be a completely symmetrical cylinder, and the numerical model is simplified to a 2D axisymmetric model.
• The molten steel is considered as an incompressible fluid, and the effects of thermal and solutal buoyancy are calculated using the Boussinesq approximation [17,18].
• The solid–liquid interface is assumed to be in a thermodynamic equilibrium condition during solidification [19,20].
• The gap between the ingot and mold induced by solidification shrinkage is ignored in the electromagnetic field boundary [6,21].

2.2. Electromagnetic Field

The governing equation of the electric potential (φ) is written as:

$${\nabla }^2\phi =0$$

(1)

The relationship between the current density ($\overrightarrow{j}$) and the electric potential is described as:

$$\overrightarrow{j}=-\sigma \nabla \phi$$

(2)

where $\sigma$ denotes the electrical conductivity of the ingot, S/m.

In order to calculate the magnetic flux density ($\overrightarrow{B}$), an intermediate variable, magnetic vector potential ($\overrightarrow{A}$), is introduced, and its governing equation is as follows:

$${\nabla }^2\overrightarrow{A}=-{\mu }_0\overrightarrow{j}$$

(3)

where ${\mu }_0$ is the magnetic permeability, and its value is set to be constant (1.26 × 10⁻⁶ H/m).

Finally, the magnetic flux density is obtained by the magnetic vector potential:

$$\overrightarrow{B}=\nabla \times \overrightarrow{A}$$

(4)

2.3. Mass Transfer

The mass conservation equations of the two solid–liquid phases during the solidification process are as follows [22]:

$${\displaystyle \frac{\partial \left(f_l{\rho }_l\right)}{\partial t}}+\nabla \cdot \left(f_l{\rho }_l\overrightarrow{u_l}\right)=-M_{\mathrm{ls}}$$

(5)

$${\displaystyle \frac{\partial \left(f_s{\rho }_s\right)}{\partial t}}=M_{\mathrm{ls}}$$

(6)

where $f_{\mathrm{l}}$ and $f_{\mathrm{s}}$ are the liquid fraction and solid fraction, respectively; ${\rho }_{\mathrm{l}}$ and ${\rho }_{\mathrm{s}}$ are the density of the liquid phase and solid phase, respectively, kg/m³; t is the time, s; $\overrightarrow{u_{\mathrm{l}}}$ is the velocity of the liquid phase, m/s; and $M_{\mathrm{ls}}$ is the mass transfer rate from the liquid to solid phase, which is obtained by the following equation [21,23]:

$$M_{\mathrm{ls}}=v_{\mathrm{s}}\cdot S_{\mathrm{A}}\cdot {\rho }_{\mathrm{s}}\cdot {\Phi }_{\mathrm{imp}}$$

(7)

where $v_{\mathrm{s}}$ is the dendrite growth speed in the radial direction, m/s; $S_{\mathrm{A}}$ is the surface area concentration, m⁻¹; and ${\Phi }_{\mathrm{imp}}$ is the dendrite growing surface impingement. These parameters are calculated as follows:

$$v_{\mathrm{s}}={\displaystyle \frac{D_{\mathrm{l}}}{R_{\mathrm{s}}}}\cdot {\displaystyle \frac{c_{\mathrm{l}}^{*}-c_{\mathrm{l}}}{c_{\mathrm{l}}^{*}-k_0{c}_{\mathrm{l}}^{*}}}\cdot {\ln }^{-1}\left({\displaystyle \frac{R_{\mathrm{f}}}{R_{\mathrm{s}}}}\right)$$

(8)

$$S_{\mathrm{A}}={\displaystyle \frac{4\pi R_{\mathrm{s}}}{\sqrt{3}{\lambda }_1^2}}$$

(9)

$${\Phi }_{\mathrm{imp}}=\left\{\begin{array}{cc}1& R_{\mathrm{s}}\le 0.5{\lambda }_1\\ {\displaystyle \frac{2\sqrt{3}{f}_{\mathrm{l}}}{2\sqrt{3}-\pi }}& R_{\mathrm{s}}>0.5{\lambda }_1\end{array}\right.$$

(10)

where $D_{\mathrm{l}}$ is the solute diffusion coefficient in the liquid phase, m²/s; $c_{\mathrm{l}}^{*}$ is the equilibrium concentration of the liquid phase at the liquid/solid interface, wt.%; $k_0$ is the solute equilibrium partitioning coefficient; $c_{\mathrm{l}}$ is the species concentration, wt.%; ${\lambda }_1$ is the primary dendrite arm spacing, m; and R_s and R_f are the radius and far field radius of the primary dendrite arm trunk, respectively, m, which are described as [24]:

$$R_{\mathrm{s}}=0.5{\lambda }_1\cdot {\left({\displaystyle \frac{2\sqrt{3}{f}_{\mathrm{s}}}{\pi }}\right)}^{1/2}$$

(11)

$$R_{\mathrm{f}}={\displaystyle \frac{{\lambda }_1}{\sqrt{3}}}$$

(12)

2.4. Fluid Flow

The flow field of the liquid phase is determined by the momentum equation:

$${\displaystyle \frac{\partial (f_{\mathrm{l}}{\rho }_{\mathrm{l}}\overrightarrow{u_{\mathrm{l}}})}{\partial t}}+\nabla \cdot (f_{\mathrm{l}}{\rho }_{\mathrm{l}}\overrightarrow{u_{\mathrm{l}}}\otimes \overrightarrow{u_{\mathrm{l}}})=-f_{\mathrm{l}}\nabla p+\nabla \cdot \left({\mu }_{\mathrm{l}}\left(\nabla \cdot \left(f_{\mathrm{l}}\overrightarrow{u_{\mathrm{l}}}\right)+\nabla \cdot {\left(f_{\mathrm{l}}\overrightarrow{u_{\mathrm{l}}}\right)}^T\right)\right)+\overrightarrow{F_{\mathrm{B}}}+\overrightarrow{F_{\mathrm{L}}}-\overrightarrow{U_{\mathrm{ls}}^{\mathrm{M}}}-\overrightarrow{U_{\mathrm{ls}}^{\mathrm{D}}}$$

(13)

where p is the pressure, Pa; ${\mu }_{\mathrm{l}}$ is the viscosity of the liquid phase, Pa s; $\overrightarrow{F_{\mathrm{B}}}$ and $\overrightarrow{F_{\mathrm{L}}}$ are the buoyancy force and Lorentz force of the liquid phase, respectively, kg/(m²·s²); and $\overrightarrow{U_{\mathrm{ls}}^{\mathrm{M}}}$ and $\overrightarrow{U_{\mathrm{ls}}^{\mathrm{D}}}$ are the momentum exchange due to the phase change and drag force, respectively, kg/(m²·s²). These momentum source terms are obtained by the following formulas [21,25,26]:

$$\overrightarrow{F_{\mathrm{B}}}=f_{\mathrm{l}}{\rho }_{\mathrm{l}}\left({\beta }_{\mathrm{T}}\left(T_{\mathrm{ref}}-T_{\mathrm{l}}\right)+{\beta }_{\mathrm{C}}\left(c_0-c_{\mathrm{l}}\right)\right)\overrightarrow{g}$$

(14)

$$\overrightarrow{F_{\mathrm{L}}}=\overrightarrow{B}\times \overrightarrow{j}$$

(15)

$$\overrightarrow{U_{\mathrm{ls}}^{\mathrm{M}}}=\overrightarrow{u_{\mathrm{l}}}\cdot M_{\mathrm{ls}}$$

(16)

$$\overrightarrow{U_{\mathrm{ls}}^{\mathrm{D}}}={\displaystyle \frac{f_{\mathrm{l}}^2{\mu }_{\mathrm{l}}}{K}}\overrightarrow{u_{\mathrm{l}}}$$

(17)

where ${\beta }_{\mathrm{T}}$ is the thermal expansion coefficient, K⁻¹; ${\beta }_{\mathrm{C}}$ is the solutal expansion coefficient; $T_{\mathrm{ref}}$ is the reference temperature, K; $T_{\mathrm{l}}$ is the temperature of the liquid phase, K; $c_0$ is the initial concentration, wt.%; $\overrightarrow{g}$ is the gravity acceleration, m/s²; and $K_{\mathrm{m}}$ is the permeability in the mushy zone, m², which is defined as [27,28]:

$$K_{\mathrm{m}}={\displaystyle \frac{{\lambda }_2^2{f}_{\mathrm{l}}^3}{180{\left(1-f_{\mathrm{l}}\right)}^2}}$$

(18)

where ${\lambda }_2$ is the secondary dendrite arm spacing, m.

2.5. Heat Transfer

The energy equations for the liquid and solid phases are as follows:

$${\displaystyle \frac{\partial (f_{\mathrm{l}}{\rho }_{\mathrm{l}}{h}_{\mathrm{l}})}{\partial t}}+\nabla \cdot (f_{\mathrm{l}}{\rho }_{\mathrm{l}}\overrightarrow{u_{\mathrm{l}}}{h}_{\mathrm{l}})=\nabla \cdot \left(f_{\mathrm{l}}{k}_{\mathrm{l}}\nabla T_{\mathrm{l}}\right)+Q_{\mathrm{J}}+Q_{\mathrm{l}}^{\mathrm{M}}-Q_{\mathrm{ls}}^{\mathrm{D}}$$

(19)

$${\displaystyle \frac{\partial (f_{\mathrm{s}}{\rho }_{\mathrm{s}}{h}_{\mathrm{s}})}{\partial t}}=\nabla \cdot \left(f_{\mathrm{s}}{k}_{\mathrm{s}}\nabla T_{\mathrm{s}}\right)+Q_{\mathrm{J}}+Q_{\mathrm{s}}^{\mathrm{M}}+Q_{\mathrm{ls}}^{\mathrm{D}}$$

(20)

where h_l and h_s are the enthalpy of the liquid and solid phases, respectively, J/kg; $k_{\mathrm{l}}$ and $k_{\mathrm{s}}$ are the thermal conductivity of the liquid and solid phases, respectively, W/(m K); $Q_{\mathrm{J}}$ is the Joule heating generated by the current, W/m³; $Q_{\mathrm{l}}^{\mathrm{M}}$ and $Q_{\mathrm{s}}^{\mathrm{M}}$ are the energy exchange due to the phase change, W/m³; and $Q_{\mathrm{ls}}^{\mathrm{D}}$ is the heat transfer between the two phases, W/m³. These energy source terms are obtained by the following formulas [23]:

$$Q_{\mathrm{J}}={\displaystyle \frac{\overrightarrow{j}\cdot \overrightarrow{j}}{\sigma }}$$

(21)

$$Q_{\mathrm{l}}^{\mathrm{M}}=\left(L_{\mathrm{m}}\cdot f_{\mathrm{l}}-h_{\mathrm{l}}\right)\cdot M_{\mathrm{ls}}$$

(22)

$$Q_{\mathrm{s}}^{\mathrm{M}}=\left(L_{\mathrm{m}}\cdot f_{\mathrm{s}}+h_{\mathrm{s}}\right)\cdot M_{\mathrm{ls}}$$

(23)

$$Q_{\mathrm{ls}}^{\mathrm{D}}=H^{*}\cdot \left(T_{\mathrm{l}}-T_{\mathrm{s}}\right)$$

(24)

where L_m is the latent heat of solidification, J/kg, and H^* is the volume heat transfer coefficient between the two phases, which is set as 10⁹ W/(m³·K) to balance the temperature between the two phases [24].

2.6. Solute Transport

The solute conservation equations for the liquid and solid phases are written as [25,29,30]:

$${\displaystyle \frac{\partial (f_{\mathrm{l}}{\rho }_{\mathrm{l}}{c}_{\mathrm{l}})}{\partial t}}+\nabla \cdot (f_{\mathrm{l}}{\rho }_{\mathrm{l}}\overrightarrow{u_{\mathrm{l}}}{c}_{\mathrm{l}})=\nabla \cdot \left(f_{\mathrm{l}}{\rho }_{\mathrm{l}}{D}_{\mathrm{l}}\nabla c_{\mathrm{l}}\right)-k_0\cdot c_{\mathrm{l}}^{*}\cdot M_{\mathrm{ls}}$$

(25)

$${\displaystyle \frac{\partial (f_{\mathrm{s}}{\rho }_{\mathrm{s}}{c}_{\mathrm{s}})}{\partial t}}=\nabla \cdot \left(f_{\mathrm{s}}{\rho }_{\mathrm{s}}{D}_{\mathrm{s}}\nabla c_{\mathrm{s}}\right)+k_0\cdot c_{\mathrm{l}}^{*}\cdot M_{\mathrm{ls}}$$

(26)

where $D_{\mathrm{s}}$ is the solute diffusion coefficient in the solid phase, m²/s. In addition, the mixed concentration ($c_{\mathrm{mix}}$) is defined as [31,32]:

$$c_{\mathrm{mix}}={\displaystyle \frac{f_{\mathrm{l}}{\rho }_{\mathrm{l}}{c}_{\mathrm{l}}+f_{\mathrm{s}}{\rho }_{\mathrm{s}}{c}_{\mathrm{s}}}{f_{\mathrm{l}}{\rho }_{\mathrm{l}}+f_{\mathrm{s}}{\rho }_{\mathrm{s}}}}$$

(27)

2.7. Influence of Metal Droplets

The effect of metal droplets is added to equations in the form of a mass source term (S_M), energy source term (S_Q) and solute source term (S_C), and these source terms only act on the first layer of grid cells [33,34]:

$$S_{\mathrm{M}}={\displaystyle \frac{m}{V_{\mathrm{m}}}}$$

(28)

$$S_{\mathrm{Q}}={\displaystyle \frac{m}{V_{\mathrm{m}}}}\cdot \left(C_{\mathrm{p}}\cdot \left(T_{\mathrm{d}}-T_{\mathrm{i}}\right)+L_{\mathrm{m}}\right)$$

(29)

$$S_{\mathrm{C}}={\displaystyle \frac{m}{V_{\mathrm{m}}}}\cdot c_0$$

(30)

where m is the melt rate, kg/s; $V_{\mathrm{m}}$ is the volume of the first layer of grid cells at the top of the ingot, m³; $C_{\mathrm{p}}$ is the specific heat, J·kg⁻¹ K⁻¹; $T_{\mathrm{d}}$ is the temperature of droplets entering the molten pool, K; and $T_{\mathrm{i}}$ is the initial temperature of the electrode, K.

3. Benchmark Configuration and Numerical Procedures

3.1. Numerical Implementation

The model was solved numerically using FLUENT software (ANSYS, Inc., Canonsburg, PA, USA), with user-defined scalars and user-defined functions. The SIMPLE algorithm was utilized for the pressure–velocity coupling. The second order upwind scheme was applied to discretize the momentum and energy equations, while the quadratic upwind interpolation scheme was applied to discretize the volume fraction and solute transport equations. For each time step, the normalized residuals of velocity, solute and phase fraction must be below 10⁻⁴, and that of energy must be below 10⁻⁶. The time step was 10⁻³ during the simulation. During the numerical simulation process, the dynamic mesh technique was used to describe the ingot growth in a 2D axis symmetrical model, as shown in Figure 2, and the 1/2 computational domain was beneficial to increase the calculation efficiency. The initial height of the computational domain was 40 mm, which was divided into 3600 quadrilateral cells with a size of 1 mm. Finally, the ingot grew to 500 mm, and the number of meshes reached 45,000.

3.2. Boundary Conditions

The top of ingot represents a free-slip boundary condition, and the electric current density ($j_{\mathrm{t}}$) changes with the radial position (r) [35]:

$$j_{\mathrm{t}}={\displaystyle \frac{I\left(1-f_{\mathrm{side}}\right)\exp \left(-\frac{r^2}{R_{\mathrm{a}}^2}\right)}{{\displaystyle {\int }_0^{R_{\mathrm{i}}}2\pi r\exp \left(-\frac{r^2}{R_{\mathrm{a}}^2}\right)dr}}}$$

(31)

where I is the total current, A; $f_{\mathrm{side}}$ is the fraction of current passing through the crown, which has a value of 0.5 in this paper [35]; $R_{\mathrm{a}}$ is the arc radius, m; and $R_{\mathrm{i}}$ is the ingot radius, m. The temperature at the top of the ingot (T_t) also changes with the radial position [36,37,38]:

$$\begin{array}{l}{T}_{\mathrm{t}}=T_{\mathrm{liq}}+\Delta T\left(J,D_{\mathrm{i}}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}r\le {\displaystyle \frac{D_{\mathrm{e}}}{2}}\\ T_{\mathrm{t}}=T_{\mathrm{liq}}+\Delta T\left(J,D_{\mathrm{i}}\right){\displaystyle \frac{D_{\mathrm{m}}-2r}{D_{\mathrm{m}}-D_{\mathrm{e}}}}\hspace{1em}\hspace{1em}{\displaystyle \frac{D_{\mathrm{e}}}{2}}<r\le {\displaystyle \frac{D_{\mathrm{m}}}{2}}\end{array}$$

(32)

where $T_{\mathrm{liq}}$ is the liquidus temperature, K; $D_e$ is the electrode diameter, m; and $D_{\mathrm{m}}$ is the mold diameter, m. $\Delta T\left(J,D_{\mathrm{i}}\right)$ is the melt overheat which can be described as:

$$\Delta T\left(J,D_{\mathrm{i}}\right)=400e^{-12{\displaystyle \frac{D_{\mathrm{i}}}{J}}}$$

(33)

where $D_{\mathrm{i}}$ is the ingot diameter, m, and $J$ is the intensity of the current, kA.

The ingot sidewall has a non-slip boundary condition, and the electric potential is zero. The air gap formed by the solidification shrinkage between the ingot and the mold will lead to variations in the thermal boundary condition. When the ingot is in good contact with the mold, the side of the ingot experiences convective heat transfer, represented by the heat transfer coefficient. However, the formation of an air gap will cause radiative heat transfer at the thermal boundary on the side of the ingot [39,40], and this heat transfer coefficient is described as [41,42]:

$$h_{\mathrm{s}}={\displaystyle \frac{{\sigma }_{\mathrm{b}}\left(T_{\mathrm{is}}^4-T_{\mathrm{mi}}^4\right)}{\left(\frac{1}{{\epsilon }_{\mathrm{i}}}+\frac{1}{{\epsilon }_{\mathrm{m}}}-1\right)\left(T_{\mathrm{is}}-T_{\mathrm{mi}}\right)}}$$

(34)

where ${\sigma }_{\mathrm{b}}$ is the Stefan–Boltzmann constant, set to 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴; $T_{\mathrm{is}}$ and $T_{\mathrm{mi}}$ are temperatures at the ingot’s lateral surface and the mold’s inner surface, K; ${\epsilon }_{\mathrm{i}}$ is the emissivity of the ingot’s lateral surface; and ${\epsilon }_{\mathrm{m}}$ is the emissivity of the mold’s inner surface.

The bottom of ingot is considered to have a non-slip boundary condition, and the electric potential here is zero due to the contact between the ingot bottom and the mold. The thermal boundary condition in this area is convective heat transfer.

3.3. Properties and Parameter Settings

The process parameters of the VAR furnace used for smelting 8Cr4Mo4V-bearing steel are shown in

Table 1. Operating conditions.

Process Parameters
Height of final ingot	500 mm
Diameter of ingot	180 mm
Diameter of electrode	130 mm
Current	2.4 kA
Melt rate	1.2 kg·min−1

, and the physical properties of the ingot are listed in

Table 2. Physical properties.

8Cr4Mo4V
Density	7419 kg·m−3
Viscosity	0.006 Pa·s
Specific heat	663 J·kg−1·K−1
Thermal conductivity	32 W·m−1·K−1
Slope of the liquidus	−9211 K
Partitioning coefficient of the C element	0.31
Solute diffusion coefficient in the liquid phase	2 × 10−8 m2·s−1
Solute diffusion coefficient in the solid phase	1 × 10−9 m2·s−1
Latent heat	213,739 J·kg−1
Thermal expansion coefficient	0.000273 K−1
Solutal expansion coefficient of the C element	1.1
Primary dendrite arm spacing	350 μm
Secondary dendrite arm spacing	100 μm

.

4. Results and Discussion

4.1. Model Validation

D. J. Hebditch and J. D. Hunt have studied the macrosegregation of Pb–Sn alloys during solidification in a square mold using an experimental method [43], which has been used in many studies to verify the accuracy of the macrosegregation model [27,44,45,46], including studies of the vacuum arc remelting process [7,9]. In this paper, the solidification process of a Pb-48 wt pct Sn alloy was simulated by the two-phase model established in our study. Figure 3a shows the schematic diagram of the Pb–Sn alloy solidification experiment. The square mold was filled with the Pb-48 wt pct Sn alloy, and the molten metal solidified from left to right due to thermal insulation on all the surfaces except for the left surface. Figure 3b is the segregation index ($\mathrm{SI}=\left(c_{\mathrm{mix}}-c_0\right)/c_0$) of the Sn element predicted by our model. It can be seen that negative segregation appears in the lower left part of the square cavity, while positive segregation appears in the upper right. A comparison of the simulation results with the experimental results found in the literature [43] at different heights (H = 5, 25, 35 mm) is shown in Figure 3c. A relatively good agreement is observed, which proves that our model is reliable at predicting macrosegregation in the solidification process.

The non-uniform distribution of elements in the ingot will affect the formation of the precipitated phase. Typically, regions with a higher element content are more prone to form a precipitated phase. Therefore, the area of the precipitated phase can reflect the extent of macrosegregation [47]. Due to the problem of coarse carbides in 8Cr4Mo4V steel, this paper mainly studies the distribution of carbon elements in the ingot. In order to further verify the accuracy of the macrosegregation model, the carbides of an 8Cr4Mo4V VAR ingot were analyzed. The process parameters of the VAR furnace used in the numerical simulation in this paper are all from the production process of this ingot. Figure 4 illustrates the sampling position of the VAR ingot. Several 10 mm × 10 mm × 10 mm specimens were cut from a height of 280 mm above the bottom of the ingot. Four specimens, positioned at distances of 5, 25, 45 and 65 mm from the ingot’s edge, were selected for the experiment. The morphology of carbides was observed using an Apreo 2C field emission scanning electron microscope (Thermo Fisher Scientific Inc., Waltham, MA, USA), and the chemical compositions of carbides were analyzed with an Ultim Max 65 energy dispersive spectrometer (Oxford Instruments, Oxford, UK). Figure 5 presents the surface scanning results of the carbides, which indicates that the carbides in the ingot are network carbides enriched in the Mo and V elements.

The carbide area in each sample was measured using Image Pro Plus 6.0 image analysis software. The segregation ratio (SR) of the carbon element is defined as follows:

$$\mathrm{SR}={\displaystyle \frac{S_{\mathrm{n}}}{\overline{S}}}$$

(35)

where $S_{\mathrm{n}}$ represents the ratio of carbide area to the total area in specimen n, and $\overline{S}$ represents the ratio of total carbide area to the total area of all four specimens. If the SR value exceeds 1, positive segregation occurs at that position. Otherwise, it is negative segregation. Figure 6 presents a comparison between the SR values obtained experimentally and the simulation results of the segregation index (SI). It is evident that the distribution in the macrosegregation simulation results is in good agreement with the experimental results.

4.2. Distribution of Physical Fields in Ingot

VAR is a multi-physics coupling process, so an analysis of each physical field’s distribution is helpful to clarify the causes of the macrosegregation defects in ingots. Figure 7a illustrates the distribution of magnetic flux density (B) when the ingot grows to 500 mm. It is evident that the magnetic flux density near the top edge of the ingot is very large, and the magnetic field weakens with a decrease in radial distance. Figure 7b is the temperature field (T) of the ingot. During the VAR smelting process, heat is introduced from the top of the ingot, resulting in a high-temperature region located above the ingot. In this situation, a V-shaped molten pool is formed on the top of the ingot, as shown in the liquid fraction (LF) in Figure 7c. In addition, the flow streamlines in the molten pool show that the molten steel moves from the center to the edge of the ingot, indicating that thermal buoyancy dominates the flow in the molten pool. This flow behavior in the molten pool has been analyzed in detail in our previous research [48], which is in agreement with the results reported in the literature [49,50].

4.3. Macrosegregation of the Carbon Element in a 8Cr4Mo4V Ingot

Figure 8a is the simulation result of the carbon segregation index (SI) distribution when the ingot grows to 500 mm and shows a negative segregation close to the edge of the ingot, while a positive segregation is observed in the center of the ingot. The SI values at different positions in the solidified ingot were extracted for analysis. Figure 8b is the value of the SI in the center of the ingot, which shows a slight negative segregation near the bottom of the ingot. Moreover, the SI value at a height of 300 mm in the ingot is illustrated in Figure 8c. It can be seen that a serious positive segregation occurs in the center of the ingot, and the degree of positive segregation decreases with the increase in radial distance.

The distribution of segregation is closely related to the heat transfer and molten pool flow during VAR process. The molten steel flows down along the solidification front and transfers to the center of the molten pool, resulting in the aggregation of carbon. Consequently, positive segregation occurs in the center of the ingot. On the other hand, during the initial smelting stage, the low height of the ingot causes rapid solidification of the molten steel due to cooling from the bottom mold, which hinders the transfer of solute. As a result, the solute distribution near the bottom is relatively uniform and the degree of segregation is less pronounced. Additionally, the distribution coefficient of carbon is less than one, which leads the equilibrium concentration of the liquid phase at the solid–liquid interface to be greater than that of the solid phase [11]. As the ingot solidifies from the edge to the center, the solute element in the liquid phase is continuously enriched, and the solute content in the ingot center, as the final solidification region, is the highest.

It is worth noting that in Figure 8b, the degree of positive segregation at different heights in the center of the ingot fluctuates. In order to explain this phenomenon, the solidification process parameters were analyzed and discussed. Undercooling, as the driving force of the metal solidification process, is related to the solute concentration in the liquid phase, as shown in Equation (36):

$$\Delta T=T_{\mathrm{m}}+m_{\mathrm{l}}{c}_{\mathrm{l}}-T_{\mathrm{l}}$$

(36)

where $\Delta T$ is the undercooling, K; $T_{\mathrm{m}}$ is the melting point of pure iron, K; and $m_{\mathrm{l}}$ is the slope of the liquidus line (less than 0), K. The solute enrichment in the central region reduces the undercooling of the alloy system, thereby slowing the solidification rate. As a result, the molten pool depth in the steady-state stage increases slightly due to the solidification slowing down, as shown in Figure 9. It is generally considered that the molten pool almost does not change in the steady-state stage. However, the uneven distribution of solute disrupts this balance, which is less of a concern in previous publications. Furthermore, variations in the molten pool depth will, in turn, affect the solidification rate. The temperature gradient and local cooling rate at the solidification front in the center of the molten pool were analyzed as an example, as shown in Figure 10. During the growth of the ingot from approximately 180 mm to about 250 mm, the temperature gradient decreases gradually with the increase in molten pool depth, while the local cooling rate remains relatively constant. The relationship between the solidification rate $v_{\mathrm{s}}$, temperature gradient $G$ and cooling rate $v_{\mathrm{c}}$ is as follows:

$$v_{\mathrm{s}}={\displaystyle \frac{v_{\mathrm{c}}}{G}}$$

(37)

According to Equation (37), when the local cooling rate remains nearly constant, a decrease in the temperature gradient leads to a continuous increase in the solidification rate. In this situation, as the ingot continues to grow, the degree of positive segregation in the center is gradually alleviated, and the molten pool depth also decreases progressively.

4.4. Variation in Carbon Macrosegregation with Melt Rate

Figure 11 illustrates the molten pool’s shape when the height of the ingots reaches 500 mm at different melt rates. As the melt rate increases, the molten pool depth also increases. This is due to an increase in heat input with the increase in melt rate, which results in an increase in sensible heat stored in the ingot [48]. Figure 12 shows the segregation index distribution of ingots with a height of 500 mm at different melt rates, showing that the positive segregation in the center of the ingot becomes more serious with the increase in melt rate. Furthermore, the SI values at the same ingot position under different melt rates were extracted for comparison. Figure 13a presents the SI values along the central axis of ingots under different melt rates, showing that the maximum positive segregation indexes are 0.07, 0.13 and 0.21 for melt rates of 1.2, 1.3 and 1.4 kg/min, respectively. Figure 13b shows the SI values at a height of 300 mm in ingots under different melt rates. It can be observed that positive segregation in ingots under different melt rates decreases with the increase in radial distance. Additionally, Figure 13b shows that the degree of negative segregation near the edge of the ingot increases with the increase in melt rate. This occurs because the lower melt rate results in reduced heat input, and the solidification rate near the ingot’s edge becomes faster. As a result, the solute distribution is relatively more homogeneous.

In order to reveal the mechanism influencing melt rate variations in macrosegregation in the VAR process, firstly, the variation in molten pool flow was analyzed and discussed. Figure 14 presents the numerical simulation results of the velocity vector distribution in a molten pool under different melt rates. As the melt rate increases, the molten pool depth increases, and the velocity of the molten steel flowing down along the solidification front also increases. In this situation, the transfer rate of the solute from the edge to the center becomes faster, leading to an increase in the degree of solute enrichment in the center. Secondly, according to Equation (36) in Section 4.3, the more serious the solute enrichment, the smaller the undercooling of the alloy system, resulting in a slower solidification rate. Therefore, with an increase in the melt rate, the fluctuation range of the molten pool depth during the steady-state stage becomes larger, as shown in Figure 15. The temperature gradient significantly decreased with the substantial increase in molten pool depth, and according to Equation (37) in Section 4.3, the solidification rate $v_{\mathrm{s}}$ should also significantly increase. Figure 16 illustrates variations in the local solidification rate $v_{\mathrm{s}}$ at the solidification front in the center of the molten pool under different melt rates, showing that our findings areconsistent with the previous analysis. These findings confirm that variations in the melt rate of the VAR process lead to variations in the solidification process parameters (undercooling and temperature gradient), thus affecting segregation distribution in the ingot.

In summary, a lower melt rate in the VAR process is preferable from the perspective of macrosegregation. An increase in the melt rate not only deteriorates the homogeneity of the ingot but also causes fluctuations in the molten pool depth, affecting the stability of the remelting process.

5. Conclusions

In this article, a 2D axisymmetric Eulerian two-phase model was established to investigate the solute distribution in VAR ingots, and this model was validated using experimental results found in the literature. The distribution of macrosegregation in VAR ingots under different melt rates was predicted. The main conclusions are as follows:

• Under the operating conditions of the VAR process described in this paper, thermal buoyancy dominates the flow in the molten pool, causing the molten steel to flow down along the solidification front. This leads to positive segregation in the center and negative segregation close to the ingot’s edge.
• Solute enrichment reduces the undercooling of the alloy system, thereby decreasing the local solidification rate, resulting in a slight increase in the depth of the steady-state molten pool. This increase in molten pool depth reduces the temperature gradient, which in turn accelerates the local solidification rate. Consequently, the positive segregation degree in the ingot center is gradually alleviated, and the molten pool depth gradually decreases.
• The molten pool depth increases with the increase in melt rate, resulting in a higher velocity of molten steel flow. The accelerated migration of solute elements to the center exacerbates positive segregation at the ingot’s center. The maximum positive segregation index at melt rates of 1.2, 1.3 and 1.4 kg/min are 0.07, 0.13 and 0.21, respectively.
• The more serious the solute enrichment, the greater the decrease in undercooling, resulting in a greater increase in the molten pool depth. With an increase in melt rate, both the fluctuation range of the steady-state molten pool depth and the positive segregation at the ingot’s center increase.