In the field of advanced manufacturing, lost wax investment casting plays a critical role in producing complex and high-integrity components, particularly for aerospace applications where titanium alloys are extensively used due to their superior strength-to-weight ratio and corrosion resistance. The quality of the final cast part heavily depends on the properties and behavior of the ceramic shell during the baking process. This study focuses on developing a comprehensive numerical model to simulate the temperature field during the shell baking stage of lost wax investment casting. We aim to address the challenges associated with thermal gradients, phase transformations, and structural integrity of the shell under high-temperature conditions.
The ceramic shell in lost wax investment casting is a multi-phase system composed of refractory materials such as Y₂O₃, ZrO₂, SiO₂, and alumina, bound together with adhesives. During baking, the shell undergoes significant changes in its thermophysical properties, which influence its performance during metal pouring. Accurate simulation of the baking process requires precise knowledge of these properties, both before and after baking. We conducted extensive measurements to determine the thermal conductivity, density, and specific heat capacity of the shell materials across a temperature range of 20°C to 1000°C. The results are summarized in the following tables, which provide essential input parameters for our numerical model.
| Temperature (°C) | Thermal Conductivity (W/m·K) | Density (kg/m³) | Specific Heat Capacity (J/kg·K) |
|---|---|---|---|
| 20 | 0.85 | 2100 | 850 |
| 200 | 0.92 | 2080 | 920 |
| 400 | 1.05 | 2050 | 980 |
| 600 | 1.18 | 2020 | 1050 |
| 800 | 1.30 | 1990 | 1120 |
| 1000 | 1.42 | 1960 | 1180 |
| Temperature (°C) | Thermal Conductivity (W/m·K) | Density (kg/m³) | Specific Heat Capacity (J/kg·K) |
|---|---|---|---|
| 20 | 1.10 | 2300 | 900 |
| 200 | 1.20 | 2280 | 950 |
| 400 | 1.35 | 2250 | 1000 |
| 600 | 1.50 | 2220 | 1080 |
| 800 | 1.65 | 2190 | 1150 |
| 1000 | 1.80 | 2160 | 1220 |
The heat transfer during the shell baking process involves conduction, convection, and radiation. The general three-dimensional unsteady heat conduction equation governs the temperature distribution within the shell:
$$\rho c_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( \lambda \frac{\partial T}{\partial z} \right) + Q$$
where $\rho$ is the density, $c_p$ is the specific heat capacity, $\lambda$ is the thermal conductivity, $T$ is the temperature, $t$ is time, and $Q$ represents any internal heat source. For convection at the shell surfaces, Newton’s law of cooling applies:
$$q = h (T_f – T_w)$$
where $h$ is the convective heat transfer coefficient, $T_f$ is the fluid temperature, and $T_w$ is the wall temperature. Radiation heat transfer is described by the Stefan-Boltzmann law:
$$q = \varepsilon \sigma_0 T_s^4$$
where $\varepsilon$ is the emissivity, $\sigma_0$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W/(m}^2 \cdot \text{K}^4)$), and $T_s$ is the absolute temperature.
To model the baking process, we developed a novel approach by reversing the traditional solidification cooling theory. In conventional lost wax investment casting, the focus is on cooling and solidification of the metal. Here, we treat the shell as the primary object and simulate its heating by the baking furnace, followed by cooling. The furnace temperature profile is assigned to the ambient air in the simulation, enabling heat exchange with the shell. The process includes heating to a target temperature, holding for a specified duration, and controlled cooling to room temperature. This逆向升温理论 (reverse heating theory) allows us to capture the transient thermal behavior of the shell during the entire baking cycle.
We implemented this model in a custom-developed solver for lost wax investment casting shell baking. The solver automates the generation of the shell geometry from the CAD model of the casting. For this study, we used a complex thin-walled titanium alloy skeleton component as a case study. The casting system was designed with a bottom gating approach, and the shell model was created by offsetting the casting surface to account for the ceramic thickness. The mesh generation process ensures that the irregular external features of the shell are accurately represented, which is crucial for realistic temperature field simulation.
The simulation results reveal significant spatial and temporal variations in temperature during the baking process. The total baking time was set to 120,000 seconds, including heating, holding, and cooling phases. We analyzed the temperature distribution at different time points and identified critical areas, such as sharp corners, where rapid temperature changes could lead to cracking or deformation. For instance, at the top sharp corner of the skeleton shell (labeled as Node A), the heating and cooling rates were substantially higher compared to other regions (Nodes B and C). The temperature difference between these nodes increased over time, highlighting the risk of thermal stress concentration.
The temperature evolution at Node A, Node B, and Node C was extracted and compared. Node A reached the target temperature faster and cooled more rapidly, with a maximum temperature gradient of approximately 50°C during the heating phase. The temperature rate of change, $dT/dt$, was calculated to quantify these variations:
$$\frac{dT}{dt} = \frac{T_{i+1} – T_i}{\Delta t}$$
where $T_i$ is the temperature at time step $i$, and $\Delta t$ is the time interval. The rate at Node A exceeded 0.5°C/s during initial heating, while Nodes B and C showed rates below 0.3°C/s. During cooling, Node A’s rate dropped to -0.4°C/s, compared to -0.2°C/s for the others. This disparity underscores the need for optimized baking protocols to minimize thermal shock in vulnerable areas.
Furthermore, we compared the shell temperature with the furnace set temperature. The shell exhibited a lag during heating due to the dominant radiation heat transfer mechanism. The furnace temperature, $T_{\text{furnace}}$, was always slightly higher than the shell temperature, $T_{\text{shell}}$, satisfying the heat flux condition:
$$q_{\text{rad}} = \varepsilon \sigma_0 (T_{\text{furnace}}^4 – T_{\text{shell}}^4)$$
During the holding phase, both temperatures converged, and during cooling, the shell temperature followed an exponential decay pattern influenced by convective and radiative losses to the environment.
In conclusion, our numerical model successfully simulates the temperature field during the baking process in lost wax investment casting. The integration of measured material properties, advanced heat transfer equations, and reverse heating theory provides a robust framework for predicting thermal behavior. The case study on the titanium alloy skeleton shell demonstrates the model’s capability to identify critical regions prone to defects. Future work will focus on coupling this temperature field with stress analysis to predict deformation and cracking, further enhancing the reliability of lost wax investment casting for high-performance applications.
The development of this solver and the insights gained contribute significantly to the optimization of shell baking parameters, ensuring higher quality and integrity in cast components. By continuously refining our approach, we aim to address the complex challenges in lost wax investment casting, ultimately advancing the manufacturing of titanium alloy parts for critical industries.