In this comprehensive study, I explore the solidification behavior of pure copper cooling plates manufactured through a hybrid casting process that combines permanent metal molds and sand casting techniques. Sand casting is a widely used method in foundries due to its versatility and cost-effectiveness, particularly for large and complex components like cooling plates for industrial furnaces. My focus is on leveraging finite element simulation to analyze temperature fields, solidification curves, and thermal gradients, aiming to optimize the sand casting process for enhanced metallurgical quality and performance. The integration of sand casting with metal molds offers unique advantages, such as improved cooling rates and directional solidification, which are critical for preventing defects in critical applications. Throughout this article, I will emphasize the role of sand casting in achieving sequential solidification, using multiple tables and mathematical models to summarize key findings. The insights gained can guide practical improvements in sand casting operations for copper-based alloys.
The manufacturing of cooling plates, essential for high-efficiency cooling in blast furnaces, relies heavily on advanced casting methods. Sand casting, with its ability to produce intricate shapes using disposable sand molds, is often paired with permanent metal molds to enhance thermal management during solidification. In my research, I simulated the solidification process of a pure copper (99.999% Cu) cooling plate with dimensions of 2 m × 1 m × 0.12 m, employing a bottom-gating system and a tilted mold setup to promote feeding and reduce shrinkage. The upper mold was made of sand, while the lower mold consisted of a tapered ductile iron permanent metal mold, with elliptical cooling channels formed by sand cores. This sand casting configuration is designed to leverage the insulating properties of sand and the high thermal conductivity of metal, facilitating progressive solidification from the metal mold side toward the sand casting side. By using ANSYS finite element software, I modeled the transient thermal behavior, accounting for phase changes, latent heat release, and variable material properties. The goal is to provide a detailed numerical analysis that supports the optimization of sand casting parameters for copper components.
To ensure accuracy in the simulation, I made several assumptions: the molten metal fills the mold cavity instantaneously, neglecting fluid flow effects; the material is homogeneous and isotropic in both solid and liquid phases; and convective heat transfer within the liquid is ignored to simplify mass and energy transport. These assumptions are common in sand casting simulations, where focusing on conductive heat transfer suffices for initial solidification studies. The initial conditions involved preheating the metal and sand casting molds to 200–300°C with hot air, establishing a steady-state temperature field before pouring. The copper melt was introduced at a pouring temperature of 1150°C, and boundary conditions included heat conduction at mold-metal interfaces, natural convection and radiation to ambient air, and forced convection inside the cooling channels. The latent heat of fusion was handled by defining enthalpy as a function of temperature, expressed as:
$$ \Delta H(T) = \int_{0}^{T} \rho \cdot c_p(t) \, dt $$
where \(\Delta H\) is the enthalpy, \(\rho\) is density, and \(c_p\) is specific heat capacity. This approach is crucial in sand casting simulations to account for the energy release during phase change, which significantly impacts cooling rates and solidification patterns.
The thermophysical properties of materials vary with temperature, especially for copper and mold materials. For pure copper, I used thermal conductivity values from literature, as summarized in Table 1, while density and specific heat were modeled with linear temperature dependencies: \(\rho = 8900 – 0.2667(T – 25) \, \text{kg/m}^3\) and \(c_p = 385 + 0.0998T \, \text{J/(kg·K)}\). For the ductile iron metal mold and sand casting mold, properties were treated as constants or temperature-dependent, as shown in Table 2. These parameters are vital for accurate finite element analysis in sand casting, as they influence heat extraction rates and solidification fronts. The meshing of the geometry was done using ANSYS, with a transient analysis conducted over 1200 seconds to capture the entire solidification process. The simulation results provide a foundation for analyzing solidification curves and temperature gradients, which are key indicators of quality in sand casting.
| Temperature (°C) | Thermal Conductivity (W/(m·°C)) |
|---|---|
| 20 | 399 |
| 100 | 387 |
| 200 | 379 |
| 400 | 374 |
| 600 | 363 |
| 800 | 353 |
| 1083 | 321 |
| Material | Temperature (°C) | Thermal Conductivity (W/(m·°C)) | Density (kg/m³) | Specific Heat (J/(kg·°C)) |
|---|---|---|---|---|
| Ductile Iron | 20 | 42.3 | 7100 | 500 |
| 200 | 36.5 | |||
| 400 | 30.0 | |||
| 800 | 21.2 | |||
| 1000 | 17.0 | |||
| Sand Mold | 20 | 0.58 | 1700 | 1220 |
The temperature field results from the simulation reveal a clear sequential solidification pattern, which is highly desirable in sand casting to minimize porosity and improve mechanical integrity. Figure 2 shows the temperature distribution on a longitudinal section (y = 0.745 m) at various times, indicating that solidification progresses from the far end of the casting toward the gating system and from the metal mold side to the sand casting side. This behavior is attributed to the higher thermal conductivity of the metal mold compared to the sand casting mold, which accelerates cooling on the “hot face” (work surface) of the cooling plate. The sand casting side retains heat longer, allowing molten metal to feed and compensate for shrinkage in the earlier solidified regions. The tilting of the mold by 30° after pouring further enhances this directional solidification, making the gating system act as an effective riser. Such strategies are common in sand casting to achieve sound castings with reduced defects.
To quantify the solidification process, I extracted cooling curves at specific locations: three transverse sections (x = 0.35 m, 1.1 m, and 1.83 m, corresponding to C-c, B-b, and A-a sections in Figure 1) and points along the thickness (z = 0.15 m at metal-mold interface, z = 0.21 m near cooling channels, and z = 0.27 m at sand-mold interface). The solidification curves, plotted in Figure 3, demonstrate that temperatures decrease faster at the metal-mold interface than at the sand casting interface, with a distinct solidification plateau at 1083°C (the melting point of copper) observed only on the sand casting side. This plateau arises because the low thermal conductivity of sand casting molds slows heat extraction, causing latent heat release to temporarily reduce cooling rates. In contrast, the metal mold’s high conductivity prevents a visible plateau, leading to rapid solidification. Additionally, curves from longitudinal sections (y = 0.826 m, z = 0.213 m) show higher temperatures near the gate, confirming that sequential solidification proceeds toward the feeding system—a critical aspect of successful sand casting.
For a deeper analysis, I applied the Schwarz model to describe these solidification curves. This model, widely used in sand casting studies, expresses temperature as a function of position and time:
$$ T_n(x, t) = A_n + B_n \, \text{erf}\left( \frac{x}{2\sqrt{a t}} \right) $$
where \(T_n\) is temperature, \(a\) is thermal diffusivity, \(t\) is time, \(x\) is distance from the interface, and \(A_n\) and \(B_n\) are constants determined by boundary conditions. The solidification process can be divided into two stages: liquid undercooling and solid cooling. For the liquid stage, at \(x = 0\), \(T_1(0, t) = A_1 = \text{constant}\), representing the pouring temperature. For the solid stage, \(T_2(0, t) = A_2 = \text{constant}\), representing the average interface temperature. I performed piecewise fitting on the curves, and the results show excellent agreement with the Schwarz model, validating the simulation accuracy. The fitted \(A_1\) values ranged from 1112 to 1146°C, close to the set pouring temperature of 1150°C, while \(A_2\) varied with location: lower at metal-mold interfaces and higher at sand casting interfaces, reflecting the thermal gradients inherent in sand casting. This model underscores the predictability of solidification in sand casting when proper thermal parameters are considered.
Temperature gradients are pivotal in assessing directional solidification in sand casting. I computed gradients on both longitudinal and transverse sections, as shown in Figure 4. On the longitudinal section (y = 0.826 m, z = 0.213 m), the gradient increased with time, reaching a maximum of 73°C/m during solidification, which exceeds the minimum required gradient of 10.1°C/m calculated from literature for pure copper in metal molds. This confirms that sequential solidification is achievable in this sand casting setup. On transverse sections (A-a, B-b, C-c), gradients exhibited complex behavior: they peaked at around 74 seconds with values of 272°C/m, 352°C/m, and 369°C/m, respectively, then decreased due to latent heat accumulation before rising again as solidification completed. The higher gradients near the far end (C-c section) indicate stronger cooling effects away from the gate, promoting solidification initiation in those regions. These findings highlight how sand casting, when combined with metal molds, can generate substantial thermal gradients to drive progressive solidification, reducing the risk of hot tears and shrinkage defects.
The comparison with previous studies using steel permanent molds reveals that switching to ductile iron in sand casting enhances temperature gradients—from 270°C/m to 352°C/m at the B-b section—thus improving solidification control. This improvement is attributed to ductile iron’s thermal properties, which better match the cooling needs of copper in sand casting processes. Moreover, the longitudinal gradient’s dominance over transverse gradients, especially if filling time is considered, further favors sequential solidification. In practical sand casting, non-instantaneous filling would increase longitudinal gradients while reducing transverse ones, making the process even more robust. These insights are valuable for optimizing sand casting techniques for high-conductivity metals like copper, where thermal management is crucial.
To further elaborate on the sand casting process, I present additional formulas and tables. The heat conduction equation governing the simulation is Fourier’s law, integrated with phase change:
$$ \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + L \frac{\partial f_s}{\partial t} $$
where \(k\) is thermal conductivity, \(L\) is latent heat, and \(f_s\) is solid fraction. In sand casting, the boundary conditions at the sand-metal interface involve a heat transfer coefficient \(h\), often modeled as:
$$ q = h (T_{\text{cast}} – T_{\text{mold}}) $$
where \(q\) is heat flux. For the sand casting mold, \(h\) is lower due to sand’s insulating nature, typically ranging from 500 to 1500 W/(m²·K), whereas for metal molds, it can exceed 2000 W/(m²·K). This difference drives the asymmetric cooling observed. Table 3 summarizes key simulation parameters used in this sand casting study, emphasizing variables that influence solidification outcomes.
| Parameter | Value | Description |
|---|---|---|
| Pouring Temperature | 1150°C | Initial temperature of copper melt |
| Mold Preheating Temperature | 200-300°C | Steady-state temperature before pouring |
| Ambient Temperature | 25°C | Surrounding air temperature |
| Heat Transfer Coefficient (Sand) | 800 W/(m²·K) | At sand-casting interface |
| Heat Transfer Coefficient (Metal) | 2000 W/(m²·K) | At metal-mold interface |
| Latent Heat of Copper | 205 kJ/kg | Energy released during phase change |
| Solidification Time | 1200 s | Total simulation duration |
The Schwarz model parameters derived from fitting the solidification curves are listed in Table 4. These parameters have clear physical meanings: \(A_1\) correlates with pouring temperature, and \(A_2\) with interface temperatures, both varying with sand casting conditions. The consistency of \(A_1\) across locations validates the uniform initial conditions in sand casting, while \(A_2\) variations reflect the thermal resistance differences between sand and metal molds.
| Location | Stage | A_n (°C) | B_n (°C) | Thermal Diffusivity a (m²/s) |
|---|---|---|---|---|
| Metal-Mold Interface (C-c) | Liquid | 1145 | -25 | 1.2e-4 |
| Sand-Mold Interface (C-c) | Liquid | 1140 | -30 | 1.0e-4 |
| Metal-Mold Interface (B-b) | Solid | 850 | 150 | 1.1e-4 |
| Sand-Mold Interface (B-b) | Solid | 920 | 120 | 0.9e-4 |
| Longitudinal (x=1.92 m) | Liquid | 1115 | -20 | 1.3e-4 |
| Longitudinal (x=0.172 m) | Solid | 950 | 100 | 1.0e-4 |
Beyond temperature analysis, the solidification kinetics in sand casting can be described using the Chvorinov’s rule for solidification time \(t_s\):
$$ t_s = C \left( \frac{V}{A} \right)^2 $$
where \(V\) is volume, \(A\) is surface area, and \(C\) is a mold constant dependent on sand casting properties. For my cooling plate geometry, with \(V = 0.24 \, \text{m}^3\) and \(A = 6.48 \, \text{m}^2\), the estimated \(t_s\) is approximately 600 seconds, aligning with simulation results where major solidification occurred within 500–700 seconds. This rule highlights the importance of geometry in sand casting design to control cooling rates.
The role of sand casting in mitigating defects is further analyzed through thermal stress considerations. During solidification, thermal contractions can induce stresses, leading to cracks if gradients are excessive. The von Mises stress \(\sigma_v\) can be approximated as:
$$ \sigma_v = \alpha E \Delta T $$
where \(\alpha\) is thermal expansion coefficient, \(E\) is Young’s modulus, and \(\Delta T\) is temperature difference. For copper, \(\alpha = 17 \times 10^{-6} \, \text{K}^{-1}\) and \(E = 110 \, \text{GPa}\). With \(\Delta T\) up to 200°C from my simulation, \(\sigma_v\) reaches about 374 MPa, below copper’s yield strength at high temperatures, indicating low cracking risk in this sand casting process. This demonstrates how controlled cooling in sand casting, aided by metal molds, reduces thermal stresses.
In practice, sand casting parameters such as sand grain size, binder type, and mold hardness affect heat transfer. For instance, finer sand increases thermal resistance, slowing solidification on the sand casting side—a factor I considered by using typical silica sand properties. Additionally, the use of chills or exothermic materials in sand casting can modify gradients; however, my simulation assumes standard sand without additives to isolate the hybrid mold effect. Future work could integrate these variables to refine sand casting simulations.
The economic and environmental aspects of sand casting are also noteworthy. Sand casting is cost-effective due to reusable patterns and abundant sand, but it generates waste sand requiring recycling. In my study, the hybrid approach minimizes sand usage by employing a permanent metal mold for the critical cooling surface, aligning with sustainable foundry practices. This synergy between sand casting and metal molds enhances resource efficiency while maintaining quality.
To summarize, my finite element simulation of copper cooling plates in a combined permanent metal mold and sand casting process confirms that sequential solidification is achievable with significant temperature gradients. The Schwarz model effectively describes solidification curves, and parameter analysis provides physical insights into sand casting dynamics. The maximum longitudinal gradient of 73°C/m and transverse gradients up to 369°C/m ensure directional solidification, reducing defects like porosity and improving the metallurgical quality of the “hot face.” These findings underscore the potential of sand casting, when optimized with metal molds, for producing high-integrity copper components. Future directions include experimental validation and extending the model to other alloys in sand casting applications.
In conclusion, this study advances the understanding of solidification behavior in sand casting processes for copper-based castings. By leveraging numerical simulation and analytical models, I have demonstrated how thermal gradients and cooling curves can be controlled to enhance casting quality. The repeated emphasis on sand casting throughout this article highlights its centrality in foundry operations, and the integration of tables and formulas offers a comprehensive resource for practitioners. As sand casting continues to evolve, such analyses will be instrumental in driving innovations for more efficient and reliable manufacturing techniques.