The evolution of the foundry industry towards specialization, intelligence, and green manufacturing has positioned lost foam casting as a transformative technology. Compared to traditional sand casting, lost foam casting offers significant advantages in achieving high-quality, high-precision, and high-performance castings. It is often hailed as a “green revolution in the foundry industry” and a “21st-century casting technology.” As components produced via lost foam casting find increasing applications in critical sectors such as thermal power equipment, automotive, aerospace, and defense, the demands for development cost, casting quality, and economic efficiency have intensified. Numerous defects in the casting process are intrinsically linked to the design and optimization of the process parameters. With the advancement of simulation technologies, computer-aided numerical simulation has become an indispensable tool in modern foundry practice for predicting and eliminating defects. This article employs a comprehensive approach, integrating numerical simulation with statistical experimental design, to systematically optimize the lost foam casting process for a carbon steel valve body, aiming to eliminate shrinkage porosity and minimize hot tearing tendency.
The core of lost foam casting simulation lies in accurately modeling the complex interfacial phenomena between the molten metal and the vaporizing foam pattern. The flow must satisfy the fundamental laws of fluid dynamics. Assuming the liquid metal is an incompressible Newtonian fluid, the governing equations include the continuity equation for mass conservation, the Navier-Stokes equations for momentum conservation, and the energy equation.
The continuity equation is given by:
$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$
where \( u, v, w \) are the velocity components in the \( x, y, z \) directions, respectively.
The momentum equations (Navier-Stokes) are expressed as:
$$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} = -\frac{1}{\rho}\frac{\partial P}{\partial x} + g_x + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) $$
$$ \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} = -\frac{1}{\rho}\frac{\partial P}{\partial y} + g_y + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) $$
$$ \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} = -\frac{1}{\rho}\frac{\partial P}{\partial z} + g_z + \nu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) $$
where \( t \) is time, \( P \) is pressure, \( g_x, g_y, g_z \) are gravity components, \( \nu \) is kinematic viscosity, and \( \rho \) is fluid density.
The energy equation describing temperature change during filling and solidification is:
$$ \rho C_p \frac{\partial T}{\partial t} + \rho C_p \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} \right) = \lambda \nabla^2 T + \rho L \frac{\partial f_s}{\partial t} $$
where \( C_p \) is specific heat, \( T \) is temperature, \( \lambda \) is thermal conductivity, \( L \) is latent heat of fusion, \( f_s \) is solid fraction, and \( \nabla^2 \) is the Laplace operator.
A unique aspect of modeling lost foam casting is the gas gap that forms between the advancing metal front and the decomposing foam. A pressure model within this gap is crucial for accurate simulation. One proposed model describes the pressure \( P_{i+1} \) at time step \( i+1 \) as:
$$ P_{i+1} = \frac{\alpha_p \Delta t V_p T_{i+1}(T_i – T_m) P_0}{L_P T_m \delta_{i+1}} – \frac{\delta_i F \Delta t K T_{i+1} P_i (P_i + P_0)}{x_c S T_i \delta_{i+1}} + \frac{\delta_i T_{i+1}(P_i + P_0)}{T_i \delta_{i+1}} – P_0 $$
The change in gap size \( \Delta \delta \) is given by:
$$ \Delta \delta = u \Delta t – \frac{\alpha_p \Delta t (T_i – T_m)}{L_p \rho_p} $$
where \( \alpha_p \) is the effective heat transfer coefficient, \( V_p \) is gas generation rate, \( L_P \) is latent heat of pyrolysis, \( T_m \) is foam temperature, \( P_0 \) is atmospheric pressure, \( K \) is sand permeability, \( x_c \) is coating layer thickness, \( S \) and \( F \) are cross-sectional area and perimeter of the casting, \( \rho_p \) is foam density, and \( \delta_i, \delta_{i+1} \) are gap thicknesses at successive time steps.
The boundary condition at the metal-foam interface, neglecting internal foam conduction, can be expressed as:
$$ k \frac{\partial T}{\partial n} ds \Delta t = \rho_p L_s u_n \Delta t ds $$
where \( \partial T / \partial n \) is the temperature gradient normal to the interface, \( L_s \) is the latent heat of the foam, \( u_n \) is the metal velocity normal to the interface, and \( ds \) is an elemental area.
The valve body selected for this study is made of carbon steel ZG230-450, known for its good toughness, plasticity, and castability. The 3D model has an envelope dimension of 300 mm × 170 mm × 82 mm, a volume of approximately 2.91e6 mm³, and a maximum local wall thickness of 51 mm. Its structure is complex and asymmetrical. The model was meshed using tetrahedral elements, resulting in 41,558 surface elements and 529,446 volume elements for the simulation domain.
Accurate thermophysical property data is essential for reliable simulation. The properties for the foam pattern and the permeable sand mold used in this lost foam casting analysis are summarized below.
| Property | Value | Unit |
|---|---|---|
| Density | 25 | kg/m³ |
| Specific Heat | 3.7 | kJ/(kg·K) |
| Latent Heat | 100 | kJ/kg |
| Thermal Conductivity | 0.15 | W/(m·K) |
| Solidus Temperature | 330 | °C |
| Liquidus Temperature | 350 | °C |
| Property | Value | Unit |
|---|---|---|
| Density | 1520 | kg/m³ |
| Thermal Conductivity | 0.53 | W/(m·K) |
| Specific Heat | 1.22 | kJ/(kg·K) |
| Permeability | 1×10⁻⁷ | cm² |
The relationship between viscosity and temperature for ZG230-450 steel was defined as a critical input for the fluid flow calculations. Initial simulation parameters were set as follows: pouring temperature of 1580°C, mold and foam initial temperature of 25°C, pouring speed of 89 mm/s, and a vacuum level of 0.05 MPa. The simulation type was specifically set to the lost foam mode.
The design of the gating system in lost foam casting is paramount. For the valve body, two primary schemes were investigated: a top-gating system and a side-gating system. The initial top-gating design, while simple, led to a non-uniform temperature distribution upon filling. The region near the upper positioning hole remained at a significantly higher temperature, disrupting directional solidification. This non-uniform cooling inevitably resulted in scattered shrinkage porosity and cavities within the thicker sections of the casting, as these areas, cooling last, lacked sufficient feeding.
The side-gating system was designed with three horizontal runners feeding into the valve body, complemented by three risers placed at potential hot spots identified from the geometry. Simulation of this system showed a marked improvement. The temperature field was more uniform overall, with the risers and their adjacent areas being the last to solidify, effectively establishing a directional solidification pattern from the casting towards the risers. Consequently, shrinkage defects were successfully isolated within the riser heads, yielding a sound casting free from internal porosity. This comparative analysis underscores the critical importance of gating design in lost foam casting for achieving soundness.
With a sound gating system established, the focus shifted to optimizing process parameters to mitigate another major defect: hot tearing. Hot tears occur when the thermal stresses developed during solidification exceed the strength of the partially solidified material. The ProCAST software employs a hot tearing indicator (HTI) based on a stress-strain criterion to qualitatively assess the propensity for this defect. A higher HTI value signifies a greater risk. Three key process parameters for lost foam casting were selected for optimization: Pouring Temperature (T), Vacuum Level (M), and Pouring Speed (V). Their influence was first studied individually.
• Pouring Temperature: Simulations at 1560°C and 1600°C showed that higher pouring temperatures increased the maximum effective stress in the casting (from 230 MPa to 232 MPa). Stress concentrated near the riser connections. Higher temperatures lead to greater thermal gradients and longer solidification times, exacerbating stress development.
• Vacuum Level: Vacuum level significantly affects the removal of pyrolysis gases. Simulations at 0.04 MPa and 0.06 MPa showed a substantial difference in stress, with the lower vacuum (0.04 MPa) resulting in lower maximum stress (210 MPa vs. 232 MPa). Lower vacuum facilitates easier gas evacuation, potentially reducing back-pressure and allowing for more stable filling and less disturbance during solidification.
• Pouring Speed: Varying the speed between 74 mm/s and 104 mm/s revealed that a higher pouring speed could reduce the stress amplitude (202 MPa vs. 221 MPa at 104 mm/s and 74 mm/s, respectively). Faster filling might reduce temperature losses and thermal gradients during the initial stage.
To systematically explore the interactions between these parameters and find the global optimum, a Box-Behnken Response Surface Methodology (RSM) design was employed. This design is efficient for fitting a second-order polynomial model. The factors and their levels are defined below.
| Factor | Symbol | Level (-1) | Level (0) | Level (1) |
|---|---|---|---|---|
| Pouring Temperature | T | 1560 °C | 1580 °C | 1600 °C |
| Vacuum Level | M | 0.04 MPa | 0.05 MPa | 0.06 MPa |
| Pouring Speed | V | 74 mm/s | 89 mm/s | 104 mm/s |
The Box-Behnken design generated 17 simulation runs. The response variable (Y) was the maximum Hot Tearing Indicator (HTI) value predicted in the casting. The simulation results for each run are presented in the following table.
| Run | T (°C) | M (MPa) | V (mm/s) | HTI (Y) |
|---|---|---|---|---|
| 1 | 1580 | 0.06 | 104 | 0.01263 |
| 2 | 1580 | 0.05 | 89 | 0.01303 |
| 3 | 1560 | 0.06 | 89 | 0.01598 |
| 4 | 1560 | 0.05 | 74 | 0.01576 |
| 5 | 1580 | 0.05 | 89 | 0.01310 |
| 6 | 1580 | 0.04 | 74 | 0.01167 |
| 7 | 1580 | 0.05 | 89 | 0.01310 |
| 8 | 1580 | 0.04 | 104 | 0.00984 |
| 9 | 1600 | 0.05 | 74 | 0.01339 |
| 10 | 1600 | 0.04 | 89 | 0.01336 |
| 11 | 1600 | 0.05 | 104 | 0.01321 |
| 12 | 1580 | 0.05 | 89 | 0.01383 |
| 13 | 1560 | 0.05 | 104 | 0.01410 |
| 14 | 1560 | 0.04 | 89 | 0.01244 |
| 15 | 1580 | 0.05 | 89 | 0.01320 |
| 16 | 1600 | 0.06 | 89 | 0.01408 |
| 17 | 1580 | 0.06 | 74 | 0.01420 |
The data was fitted to a second-order polynomial model using Design-Expert software. The resulting regression equation in terms of coded factors is:
$$ Y = 8.47507 – 0.010795T + 6.30918M – 0.00161172V – 0.003525TM + 1.2333\times10^{-6}TV + 4.3333\times10^{-4}MV + 3.42875\times10^{-6}T^2 – 6.585M^2 – 2.26\times10^{-6}V^2 $$
Analysis of Variance (ANOVA) was performed on the model. The model F-value was significant with a very low p-value (0.0008), and the “Lack of Fit” test was not significant (p-value = 0.1353 > 0.05), indicating the model adequately fits the data. The coefficient of determination R² was 0.9516, demonstrating that the model explains 95.16% of the variability in the hot tearing response. This confirms the high reliability of the regression model for prediction and analysis within the studied factor space.
The response surface plots generated from the model provide a visual understanding of parameter interactions. The most pronounced interaction was observed between Vacuum Level (M) and Pouring Speed (V). The corresponding surface plot was steeper, with denser contour lines, indicating that their combined effect significantly influences the hot tearing tendency. The hot tearing index decreased with both lower vacuum and higher pouring speed within the studied range. In contrast, the interaction between Pouring Temperature (T) and Vacuum Level (M) was relatively mild, suggesting a lesser combined impact on hot tearing for this specific lost foam casting case.
The optimization function within the RSM software was used to minimize the hot tearing indicator (Y). The numerical optimization predicted the following optimal parameter set: Pouring Temperature = 1576.01°C, Vacuum Level = 0.04 MPa, Pouring Speed = 104 mm/s. The predicted minimum hot tearing index at this point was 0.01011.
A final simulation was conducted using the rounded optimal parameters: T = 1576°C, M = 0.04 MPa, V = 104 mm/s. The simulated hot tearing index was 0.01074, which aligns closely with the model prediction, validating the regression model’s accuracy. The filling process under these conditions was smooth and stable, without signs of mist runs or cold shuts, thanks to the appropriate vacuum setting facilitating gas removal.
The solidification sequence was analyzed. At 22.3% solid fraction, the thin runners began to solidify while the main casting and risers were still largely liquid. At 50.1% solidification, the risers remained above the liquidus temperature, effectively feeding the casting, with solidification progressing from the casting body towards the risers. By 80.7% solid fraction, the casting surface was mostly solid, with no isolated liquid pools, and the risers continued their feeding role. This pattern confirms effective directional solidification. As expected, shrinkage porosity was completely relegated to the risers, confirming the soundness of the casting itself.
Stress analysis at the optimized parameters showed that the region of highest stress remained at the junction between the valve body and the riser, a typical hot spot. However, the magnitude of the stress was effectively reduced compared to non-optimized conditions. The evolution of stress at specific nodes showed an initial rise, a slight drop during phase transformation (due to volumetric expansion), followed by a final increase as austenite transformation completed. Crucially, the maximum stress values were well below the high-temperature strength limit of the steel, indicating a low probability of hot tear initiation.
The final hot tearing prediction map for the optimized process showed the indicator primarily in cool colors (blue/purple), representing a low propensity for hot cracking. The areas with warmer colors were minimal and confined to non-critical regions. This is a significant improvement over initial process conditions.
In conclusion, this integrated approach combining advanced numerical simulation for lost foam casting with statistical Response Surface Methodology provides a powerful and systematic framework for process optimization. The study successfully transitioned from a defect-prone top-gating system to a sound side-gating system with risers, eliminating internal shrinkage. Furthermore, through Box-Behnken design and analysis, the key process parameters—Pouring Temperature, Vacuum Level, and Pouring Speed—were optimized to minimize the hot tearing tendency. The analysis revealed that Vacuum Level had the most significant influence, followed by Pouring Speed and Pouring Temperature. The optimal parameters identified (1576°C, 0.04 MPa, 104 mm/s) resulted in a cast valve body predicted to be free from shrinkage porosity and with a minimized risk of hot tearing. This methodology demonstrates significant potential for improving quality, reducing trial-and-error costs, and enhancing the reliability of complex components produced via lost foam casting.