In the production of high-performance steel casting parts, such as those made from GCr15SiMn bearing steel, achieving uniform solute distribution is critical to ensure mechanical properties, wear resistance, and fatigue resistance. However, macrosegregation—the non-uniform distribution of solutes across a casting part—remains a prevalent defect, particularly in high-carbon steels where elements like carbon tend to segregate during solidification. This phenomenon can lead to detrimental issues like carbide networks, reduced impact toughness, and increased hot cracking susceptibility. Understanding and predicting macrosegregation is essential for optimizing casting processes and improving the quality of casting parts. In this study, I employ a continuum macroscopic transport model to investigate the effects of multi-component solute convection and mold structure on solute distribution in GCr15SiMn steel casting parts. The model accounts for thermal-solutal buoyancy forces, which are primary drivers of fluid flow and solute migration in casting parts, and I explore how factors like mold taper and three-dimensional (3D) effects influence the final segregation patterns. By coupling multiple solute elements, I aim to provide a more accurate prediction of macrosegregation, which is vital for designing casting parts with enhanced homogeneity.
The continuum model, originally developed by researchers in the 1980s, treats the solid, mushy, and liquid zones as a homogeneous medium, allowing for the simulation of macrosegregation induced by thermal-solutal buoyancy. This approach is computationally efficient and has been validated against experimental data for various casting parts. For multi-component alloys like GCr15SiMn steel, which contains carbon (C), chromium (Cr), silicon (Si), and manganese (Mn), the model must incorporate the contributions of each solute to the buoyancy forces and solidification path. The governing equations include continuity, momentum, energy, and solute transport equations, solved under assumptions such as steady-state laminar flow, constant densities, and local thermodynamic equilibrium at the solid-liquid interface. The momentum equation features a thermal-solutal buoyancy term that drives convection in the casting part, expressed as:
$$F_i = -\rho g_i \left[ \beta_T (T – T_{ref}) + \sum_{m}^{N} \beta_{c_m} (c^l_m – c_{m,ref}) \right]$$
where \( F_i \) is the buoyancy force in the \( i \)-direction, \( \rho \) is density, \( g_i \) is gravitational acceleration, \( \beta_T \) is the thermal expansion coefficient, \( T \) is temperature, \( T_{ref} \) is reference temperature, \( \beta_{c_m} \) is the solutal expansion coefficient for solute \( m \), \( c^l_m \) is the liquid-phase mass fraction of solute \( m \), \( c_{m,ref} \) is the reference mass fraction, and \( N \) is the number of solute elements. This equation highlights how multiple solutes collectively influence fluid flow in the casting part, with each element contributing based on its expansion coefficient and concentration gradient. The permeability in the mushy zone is described using the Kozeny-Carman model:
$$K = \frac{\lambda_2^2 f_l^3}{180(1 – f_l)^2}$$
where \( K \) is permeability, \( \lambda_2 \) is the secondary dendrite arm spacing, and \( f_l \) is the liquid fraction. The energy equation incorporates an enthalpy-porosity approach to handle phase change, while solute transport accounts for diffusion and convection effects. The liquid fraction \( f_l \) and interface temperature \( T^* \) are derived iteratively using lever rule-based relationships, ensuring accurate tracking of solidification progression in the casting part.
To simulate the solidification of GCr15SiMn steel casting parts, I set up two-dimensional (2D) and three-dimensional (3D) computational domains based on experimental casting conditions. The casting part dimensions correspond to a top diameter of 54 mm and height of 70 mm, with varying mold tapers (0°, 3.68°, and 5°) to assess structural effects. The mesh size is set to 1 mm after a grid independence study, as finer meshes showed minimal variation in solute distribution predictions. The initial pouring temperature is 1784 K with a superheat of 70 K, and boundary conditions include heat transfer coefficients between the casting part and mold walls. Thermal properties and computational parameters are summarized in the table below, which includes data for binary, ternary, and quaternary alloy systems to compare the impact of solute complexity.
| Parameter | Fe-1.05% C | Fe-1.05% C-1.65% Cr | Fe-1.05% C-1.65% Cr-0.65% Si-1.2% Mn |
|---|---|---|---|
| Initial C mass fraction \( c_{C0} \) /% | 1.05 | 1.05 | 1.05 |
| Initial Cr mass fraction \( c_{Cr0} \) /% | — | 1.65 | 1.65 |
| Initial Si mass fraction \( c_{Si0} \) /% | — | — | 0.65 |
| Initial Mn mass fraction \( c_{Mn0} \) /% | — | — | 1.2 |
| Liquidus slope for C \( m_C \) /(K·%-1) | -72.46 | -70.13 | -73.54 |
| Liquidus slope for Cr \( m_{Cr} \) /(K·%-1) | — | -1.31 | -1.12 |
| Liquidus slope for Si \( m_{Si} \) /(K·%-1) | — | — | -17.31 |
| Liquidus slope for Mn \( m_{Mn} \) /(K·%-1) | — | — | -3.8 |
| Partition coefficient for C \( k_C \) | 0.35 | 0.35 | 0.35 |
| Partition coefficient for Cr \( k_{Cr} \) | — | 0.86 | 0.86 |
| Partition coefficient for Si \( k_{Si} \) | — | — | 0.5 |
| Partition coefficient for Mn \( k_{Mn} \) | — | — | 0.75 |
| Solutal expansion coefficient for C \( \beta_C \) /%-1 | 1.1×10-2 | 1.1×10-2 | 1.1×10-2 |
| Solutal expansion coefficient for Cr \( \beta_{Cr} \) /%-1 | — | 3.97×10-3 | 3.97×10-3 |
| Solutal expansion coefficient for Si \( \beta_{Si} \) /%-1 | — | — | 1.19×10-2 |
| Solutal expansion coefficient for Mn \( \beta_{Mn} \) /%-1 | — | — | 0.2×10-2 |
| Density \( \rho \) /(kg·m-3) | 7071 | 7067 | 7020 |
| Dynamic viscosity \( \mu_l \) /(Pa·s) | 5.68×10-3 | 5.68×10-3 | 5.68×10-3 |
The simulation solves the governing equations using the SIMPLE algorithm, with convergence criteria set to residuals below 10-4 for momentum and solute equations and 10-5 for energy. Time steps are adjusted based on numerical stability, and the solidification process is tracked until the liquid fraction reaches zero. This approach allows me to predict macrosegregation patterns in the casting part, focusing on the distribution of key solutes like carbon, which is critical for the performance of GCr15SiMn steel casting parts.
The influence of multi-component solutes on macrosegregation in casting parts is profound, as each element contributes differently to thermal-solutal buoyancy and solidification kinetics. For the GCr15SiMn steel casting part, I compare binary (Fe-1.05% C), ternary (Fe-1.05% C-1.65% Cr), and quaternary (Fe-1.05% C-1.65% Cr-0.65% Si-1.2% Mn) alloy systems to assess how solute interactions affect segregation. The total buoyancy force per unit liquid concentration change for each solute is given by:
$$\sum_{m} \left( \beta_T \frac{\partial T}{\partial c^l_m} + \beta_{c_m} \right) \Delta c^l_m$$
where \( \frac{\partial T}{\partial c^l_m} \) is the liquidus slope for solute \( m \). The contributions of C, Cr, Si, and Mn are summarized in the table below, indicating the direction and magnitude of buoyancy in the casting part during solidification.
| Element | Fe-1.05% C | Fe-1.05% C-1.65% Cr | Fe-1.05% C-1.65% Cr-0.65% Si-1.2% Mn |
|---|---|---|---|
| C | -0.003492 ↓ | -0.003026 ↓ | -0.003708 ↓ |
| Cr | — | 0.003708 ↑ | 0.003728 ↑ |
| Si | — | — | 0.008438 ↑ |
| Mn | — | — | 0.00124 ↑ |
In the binary alloy casting part, carbon dominates the buoyancy, creating a downward force along the solidification front that leads to negative segregation near the bottom walls and positive segregation at the top center. For the ternary alloy casting part, chromium adds an upward buoyancy contribution, but carbon’s larger concentration gradient results in a net downward force, reducing the severity of negative segregation. In the quaternary alloy casting part, silicon provides a strong upward buoyancy, reversing the net force direction upward, which transports enriched solute to the top of the casting part, forming a thin enriched layer. This shift in flow patterns directly impacts the macrosegregation in the casting part, as shown by the carbon segregation ratio \( c/c_0 \) distributions. The maximum negative segregation ratio improves from 0.96 in binary to 0.965 in ternary, while positive segregation increases to 1.07 in quaternary systems, highlighting the need to include multiple solutes for accurate predictions in casting parts.
The flow velocity and liquid fraction fields further illustrate these dynamics. In the binary alloy casting part, at 250 s, hot melt flows downward along the walls, forming two symmetric vortices with maximum velocities around 0.005 m/s. By 500 s, the flow direction persists downward, carrying solute-depleted melt to the bottom. In contrast, the quaternary alloy casting part exhibits upward flow along the solidification front after 500 s, with velocities up to 0.004 m/s, promoting solute accumulation at the top. These variations underscore how multi-component buoyancy alters convection patterns, which in turn govern solute redistribution in the casting part. For other solutes like chromium, silicon, and manganese, segregation trends mirror carbon but vary in intensity due to differing partition coefficients. For instance, in the quaternary casting part, manganese shows a segregation ratio range of [0.99, 1.026], while chromium ranges [0.995, 1.015], with lower partition coefficients like silicon leading to more severe segregation. This comprehensive analysis emphasizes that for casting parts made from complex alloys, modeling must account for all major solute elements to capture the true segregation behavior.
Mold structure, particularly taper, plays a significant role in controlling macrosegregation in casting parts. In practice, molds are designed with a taper to mitigate air gap formation and enhance heat transfer. I investigate the effect of mold taper (0°, 3.68°, and 5°) on solute distribution in 3D round casting parts of the quaternary alloy. The results show that increasing taper from 0° to 5° reduces the maximum carbon segregation ratio along the centerline from 1.041 to 1.035, a decrease of 0.576%. This improvement is attributed to more uniform cooling and modified fluid flow patterns in the casting part. The carbon distribution on cross-sections at heights of 0.015 m, 0.035 m, and 0.060 m reveals that higher tapers promote homogeneity, shrinking the positive segregation ring near the top. The table below summarizes the impact of taper on key segregation parameters for the casting part.
| Mold Taper θ | Max C Segregation Ratio on Centerline | Min C Segregation Ratio on Centerline | Uniformity Index* |
|---|---|---|---|
| 0° | 1.041 | 0.975 | 0.85 |
| 3.68° | 1.038 | 0.978 | 0.88 |
| 5° | 1.035 | 0.980 | 0.91 |
*Uniformity Index is defined as the ratio of area with segregation ratio between 0.99 and 1.01 to total area, higher values indicate better homogeneity.
For other solutes in the casting part, such as chromium, silicon, and manganese, similar trends are observed. With a 5° taper, chromium segregation ratios range from 0.995 to 1.018, silicon from 0.985 to 1.065, and manganese from 0.988 to 1.030. The reduced segregation at higher tapers is due to enhanced thermal contact between the casting part and mold, which minimizes localized hot spots and stabilizes fluid flow. This finding suggests that optimizing mold taper is a practical strategy for improving the quality of casting parts, especially for high-carbon steels prone to segregation. In industrial applications, a taper of 5° is recommended for GCr15SiMn steel casting parts to balance ease of casting with segregation control.
Comparing 2D and 3D simulations reveals significant dimensional effects on macrosegregation predictions for casting parts. While 2D simulations are computationally efficient, they may not accurately represent the 3D reality of round casting parts due to asymmetric heat and mass transfer. In my study, 2D simulations of the quaternary alloy casting part show a maximum carbon segregation ratio of 1.04 at the top and a minimum of 0.99 at the bottom. In contrast, 3D simulations for a 0° taper mold predict a wider range from 0.975 to 1.041. This discrepancy arises because the 3D casting part has a circular cross-section where the distance from the center to the mold wall varies radially, unlike the constant distance in 2D rectangular approximations. The flow patterns in 3D casting parts involve complex vortices that enhance solute transport, leading to more pronounced segregation. The solidification rate is also faster in 3D due to additional cooling surfaces, as evidenced by liquid fraction fields at 250 s and 500 s. In 3D, the maximum flow velocity decreases to 0.0027 m/s by 500 s, compared to 0.004 m/s in 2D, indicating dampened convection but more efficient heat dissipation. These differences highlight that for round casting parts, 3D simulations are essential to capture the true macrosegregation behavior, and 2D simplifications should be used with caution, considering the 3D effects on thermal-solutal convection.
The energy equation in the continuum model for casting parts is expressed as:
$$\frac{\partial (\rho H)}{\partial t} + \frac{\partial (\rho H u_j)}{\partial x_j} = \frac{\partial}{\partial x_j} \left( k_s \frac{\partial T}{\partial x_j} \right)$$
where \( H \) is the mixture enthalpy, given by \( H = h + h_f \), with \( h \) as sensible enthalpy and \( h_f \) as latent heat. The liquid fraction \( f_l \) is updated iteratively using:
$$f_l = f_l^{n-1} – \lambda \frac{a_p (T – T^*) \Delta t}{\rho V L – a_p \Delta t \frac{\partial T^*}{\partial f_l}}$$
and the interface temperature \( T^* \) is calculated as:
$$T^* = T_f + \sum_{m}^{N} m^l_m \frac{c_m}{k_m + f_l (1 – k_m)}$$
These equations ensure accurate tracking of phase change and temperature distribution in the casting part. For solute transport, the equation accounts for diffusion and convection:
$$\frac{\partial (\rho c_m)}{\partial t} + \frac{\partial (\rho u_j c_m)}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \rho D^l_m \frac{\partial c_m}{\partial x_j} \right) + \frac{\partial}{\partial x_j} \left[ \rho D^l_m \frac{\partial}{\partial x_j} (c^l_m – c_m) \right] – \frac{\partial}{\partial x_j} \left[ \rho (u_j – u^s_j)(c^l_m – c_m) \right]$$
where \( D^l_m \) is the liquid diffusivity of solute \( m \), and the last term represents interphase interactions. By solving these equations, I can predict the evolution of solute fields in the casting part, providing insights into segregation mechanisms.
In summary, the continuum model effectively predicts macrosegregation in GCr15SiMn steel casting parts by coupling multi-component solute effects, mold geometry, and 3D transport phenomena. Key findings include: (1) Incorporating multiple solutes, especially carbon and silicon, is crucial for accurate segregation predictions, as they dominate thermal-solutal buoyancy forces. (2) Increasing mold taper from 0° to 5° reduces maximum segregation by 0.576%, recommending a 5° taper for improved homogeneity in casting parts. (3) 3D simulations are necessary for round casting parts due to asymmetric flow and heat transfer, as 2D simplifications may underestimate segregation severity. These insights contribute to better design and optimization of casting processes, ensuring higher quality casting parts with minimized defects. Future work could expand the model to include additional factors like grain sedimentation and solidification shrinkage, further enhancing its predictive capability for industrial casting parts.