Numerical simulation of cellular automaton in vacuum arc remelting during the solidification process

In this paper, the finite element and cellular automaton coupling (CAFE) model is used to simulate the solidification process of a large ingot during vacuum consumable arc melting (VAR). The effects of melting temperature, melting rate, and mold cooling coefficient on temperature field distribution and solidification structure were studied by simulation. The results show that the microstructure predicted by the numerical method is consistent with the experimental results. As the melting temperature increases from 1500 °C to 1800 °C, the depth of the molten pool increases from 14 mm to 24 mm, the width of the mushy zone decreases from 10 mm to 9 mm, and the average radius of the grains increases from 584.3 μm to 679 μm. With the increase in melting rate from 6 kg min−1 to 12 kg min−1, the maximum depth of the molten pool increases from 4 mm to 32 mm, the width of the mushy zone increases from 8 mm to 13 mm, and the average grain radius decreases from 943 μm to 497 μm. As the cooling coefficient of the mold increases from 1000 W m−2·K−1 to 5000 W m−2·K−1, the depth of the molten pool decreases from 16.7 mm to 12 mm, the width of the mushy zone decreases from 7.3 mm to 5.9 mm, and the average radius of the grains increases from 630 μm to 1303.5 μm.


Introduction
As the last step of the three-step smelting process of vacuum induction melting (VIM), electroslag remelting (ESR), and vacuum arc remelting (VAR), the VAR process can effectively improve the purity of remelting metals, which has consequently been widely used to prepare special alloys such as superalloys [1]. Vacuum arc remelting (VAR) is a secondary smelting method used to create metal ingots [2]. The goal of the vacuum arc remelting method is to create ingots with high chemical homogeneity and lower impurity content by melting selfconsumptive electrodes under vacuum. This process is used for the industrial manufacturing of reactive metals (titanium and zirconium), nickel-base alloys, and special steels. Due to the high temperature in the smelting process, the experimental research was limited, and the best parameters to improve the quality of the ingot can be obtained through the VAR modeling research process. During use, when the tip of the self-consuming electrode comes in contact with the secondary ingot, a DC arc is generated, producing the temperature required to melt the electrode (figure 1). Droplets of liquid metal are created at the electrode's tip and then gravitationally moved to the liquid pool at the top of the ingot. In ingot solidifies upon coming into contact with the watercooled copper crucible wall. The furnace has an exterior horizontal coil surrounding the crucible that creates a vertical magnetic field and helps to homogenize the metal in the liquid pool by, among other things, creating electromagnetic stirring as a result of the interaction of the magnetic field with the melting current [3].
Vacuum arc remelting (VAR) process enables production of alloys based on reactive and refractory metals. The parameters which are crucial to maintain the melting process stable and to obtain good solidification structure are arc length, fill ratio and melt rate4. However, the ingot produced by VAR cannot remove high and low density inclusions. Therefore, it is necessary to improve the solidification structure of ingot by improving the VAR refining process [4]. Zhang et al [5] studied the formation of grain structure in Ti-6Al-4V ingots during vacuum arc remelting (VAR) was simulated using the finite element cellular automata (CAFE) method. By using numerical simulation, the effects of melting rate, heat transfer coefficient, and nucleation parameters on the distribution, ratio, and size of equiaxed and columnar crystals were investigated, and the transformation of columnar crystals to equiaxed crystals (CET) was predicted. Kou et al [6] simulated the top columnar grains of the ingots, which were simulated when the effect of thermal radiation was taken into account, were in good agreement with the experimental results when the temperature field, fluid flow, and solidification structure of Ti-6Al-4V alloy ingots during the VAR process were calculated. Fei et al [7] investigated the microstructural evolution and plastic flow characteristics of a Ni-based superalloy were investigated using a simulative model that couples the basic metallurgical principle of dynamic recrystallization (DRX) with the two-dimensional (2D) cellular automaton (CA). The grain topography, the grain size and the recrystallized fraction can be well predicted by using the developed CA model, which enables to the establishment of the relationship between the flow stress, dislocation density, recrystallized fraction volume, recrystallized grain size and the thermomechanical parameters. Xu et al [8] established a mathematical model to describe the temperature distribution and the solidification structure of ESR hollow ingot of Mn18Cr18N steel has been established. The change of metal pool profile, pool cylinder part height, and grain growth angles during hollow ESR process are calculated based on the coupled technology of CAFE method and the moving boundary method, which are consistent with the experimental results. Liu et al [9] carried out the multi-scale modelling of macroscopic grain formation in large-scale titanium slab ingots during EBCHM was carried out, by combining the finite element (FE) method on the macroscale with a cellular automaton (CA) model on the microscale. The effects of bulk nucleation parameters on the formation of macroscopic grains were studied, and the predictions arising from simulations were compared with experimental results. Based on the nucleation parameters that were investigated, the process parameters could be optimised during EBCHM to achieve a solidification structure favourable for rolling. Md Perwej Iqbal et al [10] used a combined CA-LJ methodology is followed for simulating the microstructure of aluminium alloy via a 3-D thermo-mechanical model. The outcomes will be helpful to the researchers in academia and industry to predict the microstructure, and therefore mechanical and metallurgical properties of the FSW samples. Nucleation and grain growth have also been considered in the model. The results have been validated by comparing the experimentally obtained grain size results at the weld zones namely stir zone, thermo-mechanically affected zone and heat-affected zone; Mahmoud ABBASI et al [11] utilized a twodimensional cellular automaton (CA) model was utilized to analyze the effect of mechanical vibration on microstructure evolution of AZ91 alloy during friction stir welding (FSW). The simulated results, namely grain topology, grain size distribution, average grain size, and also the dynamic recrystallization (DRX) fraction were compared with measured data. It is concluded that the dislocation density during the friction stir vibration welding (FSVW) is higher than that in the FSW process and the process of nucleation and grain growth is faster for samples during FSVW compared to FSW. Peng et al [12] used the Cellular Automaton Finite Element (CAFE) method is used to numerically simulate the solidification structure of Ti-6Al-4V ingot. Firstly, the mathematical model is established with a numerical solution. Secondly, effects of process parameters including the pouring temperature and pulling speed on the solidification structure are revealed. The results show that the microstructures predicted by the numerical method match the experimental results. Wang et al [13] established a 3D vacuum arc remelting model coupling heat transfer, fluid flow, solidification, nucleation, and grain growth is established. The effect of melting rate and current intensity on the solidification structure, depth of the molten pool, and equiaxed crystal rate is investigated by the simulation. Nastac et al [14] proposed a stochastic model to describe some stochastic phenomena that should be considered during solidification, including the random distribution of nucleation positions, random orientation of grains, and the transition from columnar to equiaxed crystals.
The cellular automata (CA) method as a representative of stochastic methods, can be used to describe these stochastic phenomena in the solidification process, and the finite element (FE) method coupled with the cellular automata method has been widely used in the simulation of solidification organization. However, the nucleation and grain growth processes are not easily observed in the real solidification and it will lead to very much cost to obtain the effect of processing conditions on the microstructure of the large round ingot. In comparison with experimental method, the numerical simulation can make contribution to saving costs and reducing experiment cycles for investigating the microstructure evolution because it's beneficial to observe the formation of microstructure visually and give a better understanding of the coupled heat transfer involved in the crystallization [15].
In this paper, the process of vacuum consumable arc melting (VAR) (660 mm × 2000 mm) of a large steel ingot is studied. We know that the temperature field, the shape of the molten pool and the width of mushy zone are the key factors affecting the final quality of the steel ingot. In order to reveal the solidification characteristics and reduce the experimental period, the finite element and cellular automaton coupling (CAFE) model was used to study the effects of melting temperature, melting rate and crystallizer cooling coefficient on the shape of molten pool, the width of mushy zone, grain growth and grain distribution.

Mathematical simulation 2.1. Heat transfer equations
The simulation of a macroscopic temperature field is the premise of the microstructure simulation. In this paper, the finite element method (FE) is used to solve the heat conduction differential equation. The governing equation of the two-dimensional heat conduction macroscopic temperature field is [16]: are the heat flows in the x, y and z directions; r is the density, kg m −3 c is the specific heat capacity, J (kg·K) −1 ; l is the thermal conductivity W (m·K) −1 ; T is the temperature, K; t is the time, s; Q is the heat released per unit volume of a body per unit time, J. Then the above equation can be changed into: Where L is the latent heat, J kg −1 and f s is the solid fraction. The release of latent heat during solidification is a distinguishing characteristic of the metal solidification process that sets it apart from the general thermal conductivity process. It also serves as a link between macroscopic and microscopic phenomena and coupled macro and micro simulations. In the microscopic category, grain nucleation and development directly influence the release of latent heat, and dendritic growth feeds back the latent heat of crystallization released into the computation of the temperature field. The enthalpy method is used in this model to deal with the latent heat of crystallization [17].
Where H is the enthalpy, J mol −1 ; C p is the specific heat capacity, J (kg·K) −1 ; L is the latent heat, J kg −1 ; T is the temperature, K; and f s is the solid fraction.

CAFE model 2.2.1. Heterogeneous nucleation
Both homogeneous and non-homogeneous dendritic nucleation occur during solidification, and there are two ways to handle non-homogeneous nucleation in the numerical simulation of microstructure: transient nucleation and continuous nucleation. The continuous nucleation model suggested by Rappaz and Gandin [18,19] was utilized in this work to more properly depict the actual situation and to account for the impact of various solidification circumstances during the solidification process on the final grain size and grain shape distribution. Assuming that the nucleation process occurs at different locations, the distribution function D / ( ) dn d T , which reflects the change in grain density, is continuous rather than discrete, and increasing the subcooling degree DT leads to an increase in the grain density D ( ) n T .
The Gaussian distribution function determines the value of D / ( ) dn d T [20].
Where D s T is the deviation standard of the nucleation undercooling, K; DT max is the maximum nucleation undercooling, K; n max is the maximum nucleation density, surface nucleation density units for them , 2 volume nucleation density units for them ; 3 n is the grain density; DT is the undercooling.

Kinetics of dendritic tip growth
Crystal growth during actual metal solidification is affected by compositional subcooling as well as kinetic subcooling. The total undercooling of the dendrite tip DT consists of the following four energy [21] Where DT c is the compositional undercooling in degrees,°C; DT t is the thermodynamic undercooling in degrees,°C; DT r is the solid/liquid interface curvature undercooling in degrees,°C; DT k is the growth kinetic undercooling in degrees,°C. As the type of DT , t DT , r DT k item in most alloys as compared to the DT c is relatively small, so it can be neglected. Therefore, in the solidification structure simulation process, the model can be simplified. In the model, KGT [22] model is used as the dendrite tip growth kinetics model. First, the KGT model can describe the growth of the forced tip during the rapid solidification process and establish the relationship between the degree of undercooling and the dendrite growth rate. Second, the KGT model includes the relationship between solute collapse and temperature change of diffusivity at a strong solidification rate. Finally, dynamic effects and curvature effects on the magnitude of undercooling are taken into account.
The link between the nucleation undercooling degree DT and the dendritic tip growth rate D ( ) v T is derived by fitting the KGT model.
Where a 2 and a 3 represent growth kinetic coefficients.

Calculation procedure
The flow chart of the numerical simulation of the solidification structure in this work is shown in figure 2.
Firstly, the SolidWorks software should be used to establish the solid model, which should be imported into the Visual-Cast module for setting operation parameters, operation parameters and material parameters, and for data verification. Finally, the Mathematical model solver is used for calculation to obtain the temperature field results. Before configuring the CAFE module preprocessor, the nucleation and growth parameters must first be determined, and then the microstructure must be simulated by the CAFE module.

Geometric modelling
This study is based on the vacuum arc remelting (VAR) process of a domestic steel plant, using SolidWorks software to establish a three-dimensional model, and then using Mathematical model software to set parameters, and finally importing the CAFE module for solidification structure simulation. The model is a cylindrical geometry model, figure 3 shows the three-dimensional finite element geometric model of large ingot and the location of sample selection; it has a 660 mm diameter, a 2000 mm height, the established geometric model was import into the finite element software, and Mesh module was used for mesh division. The mesh was set as a hexahedron with a size of 10 mm. After mesh division, the total number of mesh was 100160, among which the number of surface mesh was 12160 and the number of body mesh was 88000. The microstructure calculation area (CAFE calculation area) is selected on the ingot, and the sample selection position is 330 mm away from the side of the ingot. The relevant parameters of the model are shown in table 1.
The following assumptions were made in order to simplify the model because the actual vacuum melting process is more complex. (1) the electromagnetic field generated by the arc between the electrode and the molten pool was ignored; (2) the molten droplet dropping process was reduced to a large number of molten droplets uniformly falling onto the top of the melt pool. (3) the molten drop's temperature was uniform and determined by the empirical formula [23]as depicted in equation (8).
Where T is the temperature of droplets,°C; T L is the liquidus temperature of the alloy,°C; J is the current intensity, KA; D i is the diameter of the ingot, m.

Boundary conditions
As for the boundary conditions, the contact surface between the ingot and the mold, the contact surface between the ingot and the bottom cooling water, and the melt pool interface at the top of the ingot should be considered.  The thermal boundary conditions involved in the VAR process are relatively complex, including heat conduction, heat convection, and heat radiation.
(1) Convection heat between the ingot and the bottom cooling water, Q c Where h c is the heat transfer coefficient between the ingot and the bottom cooling water W (m 2 ·K) −1 ; T and T w are the ingot's surface temperature and the temperature of the cooling water, K, respectively.
Where h t is the heat transfer between the ingot and the crystallizer, W (m 2 ·K) −1 T and T m are the surface temperature of the ingot and the temperature of cooling water, K, respectively.
(3) Radiant heat transfer on the surface of a molten pool, Q r [25].
Where T is the surface temperature of the ingot, K; T a is the ambient temperature, K; ε is the emissivity of the material; σ is the Stefan-Boltzman coefficient.
The heat transfer coefficient of the section between the ingot and the cooling water at the bottom was set at 3000 w/m 2 ·K, and the temperature at the top of the molten pool was set at 1500°C. The heat transfer coefficient between the ingot and the mold was controlled by C language which was developed by the finite element software. In this paper, the influence law of main process parameters (melting temperature, melting speed, mold cooling coefficient) on the temperature field and solidification structure of large ingot was studied. The following three working conditions were set respectively: Under the first condition, the melting temperature is controlled at 1500°C and the cooling coefficient of the mold is 3000 w/m 2 ·K, and the melting speed is set at 6 kg min −1 , 8 kg min −1 , 10 kg min −1 and 12 kg min −1 respectively. Under the second condition, the cooling coefficient of the mold is 3000 w/m 2 ·K, and the melting temperature is set to 1500°C, 1600°C, 1700°C and 1800°C respectively. Under the third working condition, the melting speed is 6 kg min −1 and the melting temperature is 1500°C, and the mold cooling coefficient is 1000 w/m 2 ·K, 2000 w/m 2 ·K, 3000 w/m 2 ·K, 4000 w/m 2 ·K, and 5000 w/m 2 ·K. As shown in table 3.

Calculation of the nucleation parameters
The CAFE model in the mathematical model program was used to simulate the solidification structure based on the computed solidification temperature field. According to the KGT growth dynamics model, in order to simulate the solidification structure of the ingot, it is necessary to calculate the growth coefficient of the large steel ingot, as follows: In the equation, α and β are the growth coefficient, m is the liquidus slope, C 0 is the initial concentration of the alloy, Γ is the Gibbs-Thompson coefficient, D is the solute diffusion coefficient in the liquid phase, K is the solute  The size of the simulation's crystallizer matches that of the actual production crystallizer, and table 3 lists the process parameters utilized for the structure simulation.

Comparison of solidification structure
To verify the feasibility and reliability of the model in simulating the microstructure of ingots, the same boundary conditions as in [27] were set and the solidification structure of the alloy round ingots was simulated. Figure 4 shows the simulation and experimental results of the longitudinal section of the ingot. Figure 4(a) is the experimental diagram, figure 4(b) is the simulation diagram, and h/R represents the ratio of the depth and radius of the molten pool, the h/R of the simulation results is 1.25, the h/R of the experimental results is 1.31, and it is found that the h/R of the experimental results and the simulation results are basically the same.   In addition, there are shrinkage defects at the center of the top of the ingot in both the simulation results and the experimental results. This is mainly due to the change in the molten pool structure in the final melting and casting stage. Some of the molten steel in the molten pool gradually solidified into a solid ingot shell, and the center part formed a hole, so before rolling, the top position should be removed.

4.2.
Change regulation of molten pool shape at different times 4.2.1. Change regulation of temperature field in smelting The metallurgical quality of the vacuum consumable ingot was closely related to the shape of the molten pool and the size of the mushy zone. When the molten metal pool is flat, the dendrites grow along the axial direction of the ingot, which facilitates the exclusion of gas and non-metallic inclusions from the metal and the replenishment of liquid metal to the area of volume shrinkage during solidification. When the metal melt pool is  deep, the ingot crystallization is in the horizontal direction, making it easy to produce a variety of defects [28]. The size of the melt mushy zone during solidification has a large influence on the degree of segregation of the ingot. Generally speaking, when the mushy zone during solidification was narrower, the degree of segregation of the ingot was smaller [29]. Numerical simulations of the stabilization phase (excluding the arc initiation phase and the head replenishment phase) during vacuum self-consumption melting with different process parameters were carried out in combination with the thermal physical parameters of the alloy. Figure 6 shows the characteristics of the temperature field distribution at different moments of the VAR process. It is clear that at the beginning of the melting phase, the bottom of the ingot relies mainly on cooling water since there is not much molten metal and the melt pool has not yet formed a recognizable shape (see figure 6(a). Since the columnar crystals develop along the direction of significant temperature gradients, it was assumed that the bottom columnar crystals first grow vertically upward. As the electrode melts to form a molten drop, the metal melt pool moves upward, and the melt near the bottom solidifies first. The cooling effect of the cooling water on the melt pool gradually decreases, and a flatter, shallower melt pool is gradually formed (see figure 6(b)). The temperature at the ingot's bottom progressively decreases as it increasing in height, away from the cooling influence of the water. The cooling effect of the crystallizer was strengthened, and the depth of the metal melt pool gradually deeper. The heat accumulation was most noticeable and the depth change was quickest when the ingot height reached 776 mm, causing the melt pool shape to progressively shift from flat to funnel-shaped (see figure 6(c)). There was a large temperature gradient in the radial direction, and the columnar crystal was inferred to grow at an oblique angle. The effective heat dissipation area and the contact area between the ingot and its crystallizer side wall both increase when the ingot height exceeds 1164 mm, causing more heat to be removed by cooling water. When the melting process reaches the 19404s, the ingot's internal heat absorption and dissipation progressively attain equilibrium, and a stable solution pool is created (see figure 6(d)), that is, reaching the steady-state melting stage. With the growth of columnar crystals, heat was gradually dissipated, the melt temperature in the melt pool of the ingot was decreasing, and heat dissipation had lost its direction, so the growth of grains in the melt finally formed equiaxed crystals. After that, the melt pool depth and shape do not continue to change (see figure 6(e)) until the end (see figure 6(f)). Due to the heat preservation effect near the end of the molten pool, the ingot's top slowly solidified. Since the top's heat dissipation was mostly dependent on thermal radiation, the ingot's ultimate solidification produced columnar crystals that grew downward. Figure 7 shows the results of the distribution of the fraction solid rate in the melting process obtained from the simulation. As can be observed, the metal molten pool was located between the liquid phase rate 1 and the top, and the cooled and solidified portion was located below the metal molten pool. As shown in figure 7, the ingot height increases from 2500s to 12936s, increasing the depth of the molten metal pool. However, after the 19404s, the shape of the metal melt pool does not change sharply with the increase in ingot height, indicating that the ingot has reached a steady state.

4.3.
Influence of different process parameters on the shape and solidification structure of molten pool Figure 8 shows the temperature field distribution diagram of large ingot. Under the working conditions of the melting temperature of 1500°C, the temperature field at steady state is reached, and the lines between different color regions are isotherms. As shown in the enlarged diagram on the right, the red area at the top is the liquid phase area, the orange area below is the solid-liquid two-phase coexistence area (the mushy region), and the other area is the solid phase area. The two lines marked in the diagram are the liquid phase line and the solid   phase line. Therefore, we measured the liquidus depth h 1 , solidus depth h 2 and mushy zone width h 2 -h 1 to observe the change of molten pool morphology of large ingot.

4.3.1.
Influence of melting temperature on molten pool shape and solidification structure When the influence of melting temperature on solidification structure is studied, the influence of melting temperature on solidification structure can be attributed to the temperature of the droplet according to equation (8) when the melting rate and cooling intensity of the crystallizer are constant. When the current intensity is 7.8 KA, the droplet temperature is 1804.5°C, which means that the droplet temperature for simulation is 1500°C ,1600°C ,1700°C and 1800°C. When the melting temperature increases from 1500°C to 1800°C, the depth of the molten pool increases from 14 mm to 24 mm. The paste zone width was reduced from 10 mm to 9 mm. The width of the mushy zone was related to the temperature gradient. When the temperature gradient was small, the width of the mushy zone was large. However, when the melting temperature increases from 1500°C to 1800°C, the temperature gradient gradually becomes larger (see the slope of the curve in figure 9), resulting in a lower width of the mushy zone at 1800°C than at 1500°C (see figure 10). The shape of the melt pool changes as the temperature increasing. When the temperature is low, the energy input is small and the  melt pool is flat, and the higher the temperature, the more the melt pool gradually forms a V-shape. A flat melt pool was beneficial to the discharge of gas and the complementary shrinkage of the metal liquid, which can reduce the tendency of porosity and shrinkage, and was generally not recommended at too high a temperature. When the structure was coarser, the equiaxed crystals area was smaller and the columnar crystal area was substantially greater because the melting temperature was higher, which caused more small grains to be remelted (see figure 11). The equiaxed crystal area shrinks and the solidification structure gets coarser as the temperature increasing, with an average grain radius of around 584.3 μm at melting temperatures of 1800°C and 679 μm at melting temperatures of 1500°C, respectively (see figure 12). The number of columnar crystals and grain size may both be successfully reduced by appropriately adjusting the melting temperature.

4.3.2.
Influence of melting rate on molten pool shape and solidification structure When the melting temperature and crystallizer cooling intensity are known, increasing the melting rate from 6 kg min −1 to 12 kg min −1 causes a considerable shift in the temperature gradient in the direction of the melt pool depth (see the slope of the curve in figure 13. The melt pool depth increases from 4 mm to 32 mm and the mushy zone width increases from 8 mm to 13 mm when the melting rate is increased from 6 kg min −1 to 12 kg min −1 (see figure 14). The temperature gradient affects the width of the mushy zone, and when the temperature  gradient varies more, the width of the mushy zone decreases. The melt pool progressively transforms from a flat bowl shape to a V shape as the melting rate increases. The key factor influencing the metal pool's depth and form is the heat input into the molten metal. Reducing the melting rate has a similar result as lowering the melting temperature. A flat molten pool can be achieved by reducing the heat input to the metal molten pool per unit of time. Unlike the effect of temperature, a decrease in melting rate raises the temperature gradient and thus reduces the width of the mushy zone. Due to the narrow width of the mushy zone, the corresponding complementary shrinkage of the metal solution becomes easier, which also helps to reduce the segregation of the alloy. Melting rate is an important process parameter in the VAR process. The change in melting rate will affect the ingot temperature field and molten pool morphology, and finally the solidification structure of the ingot. It can be seen from figure 15 that with the increase of melting rate, the equiaxed crystal area becomes more and more obvious and the area becomes larger. When the melting speed was 6 kg min −1 , the average radius of grains was 943 μm. When the melting speed was 12 kg min −1 the average radius of grains was 497 μm. As shown in figure 16, the average grain radius was decreasing and the number of grains was increasing. With the increase in melting rate, the depth of the molten pool and the width of the paste zone increase, the temperature gradient in the molten pool decreases, and the grains tend to be refined. In addition, the increased of melting rate expands  the equiaxed crystal zone in the ingot core, which promotes the transformation of CET. Therefore, a high melting rate can improve production efficiency and the axial crystal rate.

Influence of crystallizer cooling strength on molten pool shape and solidification structure
The crystallizer cooling coefficient improves from 1000 W (m 2 · K) −1 to 5000 W (m 2 · K) −1 , the melt pool depth falls from 16.7 mm to 12 mm, and the mushy zone width decreases from 7.3 mm to 5.9 mm while the melting rate and melting temperature are constant (as shown in figures 17 and 18). he width of the mushy zone may be greatly reduced by increasing the crystallizer's cooling coefficient, and the shape and depth of the melt pool were also altered. Increasing the cooling coefficient increases the heat removal rate during the melting process and shifts the bottom of the melt pool upwards. As shown in figure 19, the proportion of the columnar crystal zone increases with the cooling coefficient, while the proportion of the equiaxed crystal zone in the core decreases. The average grain radius was about 630 μm when the cooling coefficient was 1000 W (m 2 ·K) −1 , and it was around 1303.5 μm when the cooling coefficient was 5000 W (m 2 ·K) −1 (as shown in figure 20). In general, the heat transfer coefficient increases with the increase in cooling water flow rate. For discussing the effect of the cooling intensity of the crystallizer on the shape and solidification of the melt pool, it was equivalent to studying  the effect of the heat transfer coefficient on the shape and solidification of the melt pool in the numerical simulation of the ingot flanks. When the heat transfer coefficient is lower, it takes longer for the solution to solidify and form crystals, and the unbound grains prevent further columnar crystal formation. The nucleation period is considerably decreased for higher heat transfer coefficients, and development into columnar crystals toward the core then occurs. When the interface heat transfer coefficient is increased, the heat dissipation rate increases and the temperature gradient in the molten pool increases, which on the one hand reduces the subcooled area in the molten pool and leads to a decrease in the equiaxed nucleation rate and, on the other hand, promotes the growth of columnar crystals. The increase of columnar crystal regions inhibits CET transformation during solidification of the ingot. Therefore, proper adjustment of the cooling intensity can effectively control the grain size, number, and molten pool of the ingot.
The flat form of the melt pool in the vacuum arc remelting process facilitates the release of gas from the melt pool and the corresponding shrinkage of liquid metal, whilst the smaller mushy zone facilitates the suppression of alloy segregation. The higher the cooling intensity of the crystallizer, the lower the temperature and the shallower the depth of the molten pool, and the average grain size will increase with the increase of the cooling intensity. The larger the melting temperature is, the deeper the molten pool is, the more the columnar crystals are in quantity and size. The higher the melting rate, the depth of molten pool becomes significantly deeper and the equiaxed crystal rate increases.

Conclusion
In the current study, a three-dimensional VAR model was established to study the solidification structure, the depth of the molten pool, and the width of the mushy zone. By simulating the effects of melting temperature, melting speed, and mold cooling coefficient on the solidification structure, the depth of the molten pool, and the width of the mushy zone, the following conclusions can be obtained: (1) As the melting temperature increases from 1500°C to 1800°C, the depth of the molten pool increases from 14 mm to 24 mm, the width of the mushy zone decreases from 10 mm to 9 mm, and the average radius of the grains increases from 584.3 μm to 679 μm. With the increase in melting rate from 6 kg min −1 to 12 kg min −1 , the maximum depth of the molten pool increases from 4 mm to 32 mm, the width of the mushy zone increases from 8 mm to 13 mm, and the average grain radius decreases from 943 μm to 497 μm. As the cooling coefficient of the mold increases from 1000 W m −2 ·K −1 to 5000 W m −2 ·K −1 , the depth of the molten pool decreases from 16.7 mm to 12 mm, the width of the mushy zone decreases from 7.3 mm to 5.9 mm, and the average radius of the grains increases from 630 μm to 1303.5 μm.
(2) In this paper, the software is used to simulate the temperature field and solidification structure process in a vacuum consumable arc melting process based on the CAFE method. By comparing the simulation results with the experimental results, the accuracy of the simulation is verified, and the shape of the molten pool and the grain growth under different process parameters are simulated.
(3) In order to obtain high-quality ingots, the process parameters should be reasonably adjusted. It is recommended to adopt an appropriate cooling intensity, and the heat transfer coefficient should not be greater than 3000 W/m 2 ·K. In addition, the melting temperature should not be too high, higher than 80°C-100°C of the liquidus of the alloy ingot. If the melting speed is particularly high, it will have a bad impact on the ingot. High melting rates can improve production efficiency and increase the equiaxed crystal rate, so low melting rate casting should be used as much as possible.