Three-dimensional FEM simulation of the solidification process was conducted using the finite element code, THERCAST
®. Thermo-mechanical phenomena associated with the various phases of mold filling and the process of cooling in and out of the mold were described using an arbitrary Lagrangian–Eulerian (ALE) formulation based on the volume-average two-phase model [
23,
24]. The fluid flow, temperature, and distribution of solute in the solidifying material were analyzed using the coupled solutions of equations denoting the conservation of mass, momentum, energy, and solute, as indicated in
Table 2. Heat transfer equations were solved for each subdomain (mold, hot-top, ingot, sideboard, etc.) using appropriate boundary conditions, including the average convection, radiation, external imposed temperature, and external imposed heat flux. A Fourier-type equation was used for the evaluation of contact resistance at the interfaces of metal and mold components. The convection and radiation equations are also reported in
Table 2. At first, thermal computation was conducted using a specified time step, which was based on the rate of cooling during solidification, followed by mechanical computation. The thermal problem was treated by the resolution of the heat transfer equation (energy conservation), the solute conservation equation governed the redistribution of each solute element, and the equation for momentum conservation governed the mechanical equilibrium. The determination of macrosegregations involved, in part, the resolution of the energy conservation and solute conservation equations for each alloying element. Additionally, a microsegregation model was incorporated into these equations, allowing for the characterization of the non-uniform distribution of chemical elements between the solid and liquid phases. The microsegregation model employed in this study was the Brody–Flemings model [
23,
25]. The model was based on a one-dimensional solute redistribution model. This followed a decreasing parabolic pattern based on a one-dimensional solute redistribution. The transient solute transfer in the solid, which occurred due to diffusion, was contingent upon the dimensionless back-diffusion Fourier number α (
Table 2). In this study, the nominal liquidus and solidus temperatures were determined utilizing JmatPro software version 11.0 [
4,
23,
25], taking into account the nominal chemical composition of the alloy. These calculated values served as input parameters for simulations, defining the alloy’s properties. Nevertheless, it was crucial to recognize that the temperature, liquid fraction, and solute composition of alloy elements were not static; they varied at each time step during the solidification process. This model iteratively computed the local chemical composition, local liquid fraction, and local temperature at each node throughout the progression of solidification time. The local temperature was expressed as a function of the liquid composition and the liquidus slope (refer to
Table 2). The determination of the local liquidus temperature for each node was achieved by analyzing the liquid fraction and temperature at the conclusion of the solidification process. In the liquid phase, gravity-driven natural convection loops were formed due to local density variations. Just as reported by Ludwig et al. [
5], Lesoult et al. [
6], and Beckermann et al. [
7], such local density variations induced local natural convection loops within the liquid phase, driven by gravity. These convective flows primarily consist of two types: (i) thermal convection flows, triggered by thermal expansion and temperature gradients, and (ii) solutal convection flows, induced by solutal expansion and concentration gradients [
5,
6,
7,
23]. In fact, the metal goes through three separate phases during solidification: the liquid state, the mushy state, and the solid state. A hybrid constitutive model is then used to simulate the transition from the liquid state, to the mushy state, and then to the solid state [
4,
23,
25,
26,
27,
28]. The behavior of the liquid metal (above the liquidus temperature) requires thermo-Newtonian treatment with the Navier–Stokes equation using an arbitrary Lagrangian–Eulerien approach. Below the solidus temperature, a thermo-elasto-viscoplastic Lagrangian formulation is used to treat the behavior of the solid metal. The mushy state, a transition between solid and liquid, is assumed to be a single-continuum non-Newtonian fluid and is characterized by a so-called ‘coherency temperature’, which corresponds to when a solid skeleton is formed and supports solidification stresses [
23,
25,
29]. Above the coherency temperature, a Norton–Hoff type law is used with a Newtonian prolongation used when the scenario is above the liquidus temperature [
23,
25,
30]. Below the coherency temperature, the material behavior is modeled based on the approach proposed by Perzyna, who assumed that the semi-solid metal had thermo-elasto-viscoplastic (TEVP) constitutive behavior. Finally, at the solid–liquid interface, thermodynamic equilibrium is assumed. This makes it necessary to assume the continuity of the flow stress at the coherency temperature (i.e., consistency between material behavior for both viscoplastic and elasto-viscoplastic conditions). In the present work, the mushy zone is conceptualized as an isotropic, porous solid medium saturated with liquid and characterized by the condition
, where
and
represent the volumetric fractions of solid and liquid, respectively. The permeability components are contingent on the model that establishes the connection between macroscopic equations and microscopic effects. The isotropic permeability (
) is defined using Carman–Kozeny analysis [
23,
25] (
Table 2), which depends on the secondary arm spacing (SDAS) value and local liquid fraction. It must also be noted that, in the Carman–Kozeny equation, the assumption is made that a liquid fraction is always present, corresponding to a
value that is a real number. When the liquid fraction is zero and the material is completely solidified, its permeability is negligible or zero, indicating that there is no flow of fluid through the solid material. As the solidification progresses, the mushy zone’s thickness evolves as a function of the interaction between various factors such as thermal conditions and solidification kinetics. The thickness of the mushy zone is calculated in each time step based on the calculation of the volume fraction of liquids and solids, temperature, solute composition, and local solidification time. Following the simulation of the initial design, the width of the mushy region was calculated in the radial direction at the bottom, middle, and top positions of the ingot in the 90° model.
Figure 2 illustrates the average thickness of the mushy zone at various solidification times. The average widths of the mushy zone were determined to be 48 mm, 83 mm, 112 mm, 117 mm, and 45 mm at the end of filling, 1 h post-pouring, 2 h post-pouring, 3 h post-pouring, and 4 h post-pouring, respectively (
Figure 2). As depicted in
Figure 2, the mushy zone’s thickness varies dynamically during solidification. Initially, at the end of filling, the mushy zone is thinner due to the higher cooling rate and faster solidification. As the solidification process progresses from 1 h to 3 h post-pouring, the rate of solidification reduces, and the mushy zone becomes thicker. After 3 h, the thickness of the mushy zone decreases as heat is continuously extracted from the molten metal, causing the mushy zone to shrink as more material solidifies. The model has the capability to predict the formation of shrinkage cavities. This prediction is grounded in the analysis of the change in the liquid fraction within the liquid and mushy zones. The underlying principle of this estimation involves calculating the loss of metal volume incurred during each simulation increment. This loss in volume is attributed to both thermal contraction and the transition from liquid to solid states. The distribution of this loss in volume is determined by the progression of the solidification front [
23,
25,
31].
To substantiate the accuracy of numerical computations, an assessment of the elemental concentration balance was conducted at various time steps following the solidification. The presentation of the mass balance during the simulation involved calculating the cumulative sum of both local and nominal concentrations for all elements at the total node after the simulation period. The outcomes, specifically those relating to the concentration balance for chromium (Cr), manganese (Mn), and molybdenum (Mo), are systematically provided in
Table 3. Additionally,
Figure 3 graphically depicts the temporal evolution of carbon (C) concentration during the solidification process at the location indicated by a red circle. As depicted in
Figure 3, the hot liquid reaches the indicated location in the ingot at around 1000 s after pouring. Following this point, fluctuations in the concentration of carbon atoms are observed until approximately 14,000 s after pouring. At this juncture, the curve depicting the concentration of carbon stabilizes, which is indicative of the complete solidification of the liquid. The solid front reaches a specific point (as indicated in
Figure 3) at 5800 s after pouring, coinciding with the local temperature reaching the local liquidus temperature. After this point, solidification initiates. The concentration of carbon at the indicated point (
Figure 3) is influenced by the complex interplay between solidification kinetics, segregation phenomena, and fluid flow dynamics within the molten metal. Fluid flow within molten metal during solidification, such as natural convection or turbulent mixing, can influence the distribution of solute elements. Variations in fluid flow patterns can result in non-uniform mixing of carbon throughout the ingot, contributing to differences in carbon concentration at the indicated location. Additionally, in the mushy state, thickness increases as the cooling rate slows down. According to
Figure 2, it then starts to decrease. In the mushy state, due to interdendritic liquid movement in the mushy zone, solute transfer, and the diffusion of solute elements in the solid state at each time step, there is a non-uniform distribution of carbon atoms between the solid and liquid phases, resulting in fluctuations in carbon concentration.