The solidification process of metals, a rapid and opaque phase transformation, is nearly impossible to describe accurately through direct observation. In engineering practice, casting process parameters are often designed based on empirical knowledge, an approach that is not only imprecise but also consumes significant time and resources. The Cellular Automata-Finite Element (CAFE) method offers a powerful tool for visualizing and analyzing the metal solidification process, thereby providing a scientific reference for optimizing these parameters. While numerous studies worldwide have applied the CAFE method to simulate the solidification microstructures of various alloys with results closely matching experiments, research specifically focused on simulating the solidification organization of 16Cr20Ni14Si2 high-temperature resistant steel is scarce. This steel grade is crucial in aerospace and military applications, creating a pressing need for its investigation.
This study begins by constructing a CAFE mathematical model tailored for the 16Cr20Ni14Si2 alloy. Subsequently, the CAFE method is employed to simulate the microstructural evolution during solidification in a simple cylindrical specimen. The influence of key investment casting process parameters on the alloy’s final microstructure is systematically investigated. Building upon the optimal parameters identified from the specimen simulations, a gating system is designed for a complex thin-walled engine connection housing component. The applicability of the CAFE method to three-dimensional solid models is then verified through simulation and analysis of nucleation parameters. Furthermore, an appropriate heat treatment holding time is designed based on this methodology. Finally, pouring trials are conducted. The results show a remarkably low error of only 0.07% between the simulated and experimentally measured grain sizes, validating the correctness of the CAFE method for predicting the microscopic solidification structure of the 16Cr20Ni14Si2 alloy under the investment casting process from the perspective of grain size.
Summarizing the effects of various investment casting process parameters on the solidification microstructure of 16Cr20Ni14Si2 steel based on the CAFE method yields the following key insights:
- An increase in shell thickness from 7mm to 9mm leads to fewer grains, as thicker shells impede heat dissipation. Conversely, a higher shell preheat temperature (from 1143.15K to 1203.15K) and a lower pouring temperature (from 1933.15K to 1913.15K) promote a greater number of grains. Among different cooling methods (air, oil, water), water quenching, with its highest thermal conductivity, yields the finest grain structure. Increasing the pouring speed from 0.5 kg/s to 4.5 kg/s gradually increases the grain count and aligns the grain orientation closer to the axial heat flow direction, although the magnitude of this effect diminishes at higher speeds.
- Increasing the nucleation density results in a larger number of finer grains and a higher proportion of equiaxed grains. A larger undercooling facilitates nucleation, leading to a greater population of fine equiaxed grains.
- At a constant solution treatment temperature, longer holding times lead to significant grain coarsening. Notably, changes in holding time do not alter the fundamental grain morphology (e.g., triggering a clear columnar-to-equiaxed transition).
Mathematical Model Construction and Experimental Materials for CAFE Simulation
Temperature evolution is paramount in determining the final solidification microstructure. Therefore, a thermodynamic mathematical model is first established, encompassing the three fundamental modes of heat transfer: conduction, convection, and radiation.
Heat conduction, the transfer of energy through direct contact, is described by Fourier’s Law:
$$ \vec{q} = -\lambda \cdot \text{grad} T = -\lambda \frac{\partial T}{\partial n} \cdot \vec{n} = f(x, y, z, t) $$
where $\vec{q}$ is the heat flux density, $\lambda$ is the thermal conductivity (dependent on the material and temperature), $T$ is temperature, and $x, y, z, t$ are spatial coordinates and time.
Heat convection, involving energy transfer between a surface and a moving fluid, is governed by Newton’s Law of Cooling:
$$ q = h(T_w – T_f) $$
where $h$ is the convective heat transfer coefficient, $T_w$ is the wall temperature, and $T_f$ is the fluid temperature.
Heat radiation, the emission of electromagnetic waves, is modeled by the Stefan-Boltzmann Law. For high-temperature alloys like 16Cr20Ni14Si2, this mode cannot be neglected:
$$ q = \varepsilon \sigma_{\text{SB}} (T_w^4 – T_f^4) $$
where $\varepsilon$ is the emissivity (typically 0.8 for casting processes) and $\sigma_{\text{SB}}$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$).
To simulate the macroscopic flow and solidification, a Finite Element (FE) model is built upon the conservation laws of mass, momentum, and energy. The governing equations, incorporating gravity for the investment casting process, are:
Mass Conservation (Continuity Equation):
$$ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0 $$
Momentum Conservation (Navier-Stokes Equation, x-direction shown):
$$ \rho \left[ \frac{1}{f_l} \frac{\partial u}{\partial t} + \frac{1}{f_l^2} \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) \right] = -\frac{\partial p}{\partial x} + \rho g_x + \frac{\partial}{\partial x} \left( \frac{\mu}{f_l} \frac{\partial u}{\partial x} \right) + \frac{\partial}{\partial y} \left( \frac{\mu}{f_l} \frac{\partial u}{\partial y} \right) + \frac{\partial}{\partial z} \left( \frac{\mu}{f_l} \frac{\partial u}{\partial z} \right) – \left( \frac{\mu}{K} \right) u $$
Energy Conservation:
$$ \rho \frac{\partial H}{\partial t} + \rho \left( u \frac{\partial H}{\partial x} + v \frac{\partial H}{\partial y} + w \frac{\partial H}{\partial z} \right) = \frac{\partial}{\partial x} \left( k_T \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( k_T \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( k_T \frac{\partial T}{\partial z} \right) $$
where $H(T)$ is the enthalpy, defined as:
$$ H(T) = \int_0^T C_p(T) dT + L (1 – f_s) $$
In these equations, $u, v, w$ are velocity components; $f_l, f_s$ are liquid and solid fractions; $p$ is pressure; $g_x$ is the gravitational component; $\rho$ is density; $\mu$ is dynamic viscosity; $k_T$ is thermal conductivity; $K$ is permeability; $C_p$ is specific heat; $L$ is latent heat of fusion; and $H$ is enthalpy.
While the FE model captures macroscopic phenomena, it cannot simulate the mesoscale mechanisms of grain nucleation and growth. This requires coupling with a Cellular Automaton (CA) model.
The CA model simulates nucleation and growth. Heterogeneous nucleation, dominant in practical casting, is modeled using a continuous nucleation model based on a Gaussian distribution. The density of nuclei activated at a given undercooling $\Delta T$ is:
$$ n(\Delta T) = \int_{0}^{\Delta T} \frac{dn}{d(\Delta T’)} d(\Delta T’) $$
with the nucleation distribution function:
$$ \frac{dn}{d(\Delta T)} = \frac{n_{max}}{\sqrt{2\pi} \Delta T_{\sigma}} \exp \left( -\frac{(\Delta T – \Delta T_{max})^2}{2 \Delta T_{\sigma}^2} \right) $$
where $n_{max}$ is the maximum nucleation density, $\Delta T_{max}$ is the mean nucleation undercooling, and $\Delta T_{\sigma}$ is the standard deviation of the undercooling. Separate parameters are defined for surface nucleation (at mold walls) and volume nucleation (within the melt).
Following nucleation, grain growth is simulated. The growth kinetics of dendritic tips, crucial for microstructure prediction, is described for multi-component alloys like 16Cr20Ni14Si2. The relationship between the dendrite tip growth velocity $v$ and the total undercooling $\Delta T$ (dominated by constitutional undercooling $\Delta T_c$ for this alloy) can be simplified to a polynomial:
$$ v = a_2 \Delta T^2 + a_3 \Delta T^3 $$
where $a_2$ and $a_3$ are growth kinetics coefficients specific to the alloy.
The CAFE method couples the macroscopic FE temperature field with the microscopic CA rules. Interpolation functions transfer temperature data from FE nodes to CA cells, and the latent heat released during solidification is fed back to the FE calculation, ensuring a fully coupled thermal-microstructural simulation.
For the experimental validation, materials suitable for the investment casting process were selected. A green, recyclable pattern wax was chosen. Silica sol was used as the binder. For the ceramic shell, a zircon flour/sand face coat was applied, followed by backup layers of mullite. This combination ensures good surface finish, dimensional accuracy, and sufficient high-temperature strength for the 16Cr20Ni14Si2 alloy.
CAFE Simulation of 16Cr20Ni14Si2 Solidification Microstructure in a Specimen
The chemical composition of the 16Cr20Ni14Si2 alloy used is shown in Table 1.
| Element | C | Si | Mn | P | S | Cr | Ni | Fe |
|---|---|---|---|---|---|---|---|---|
| Content (wt.%) | 0.100 | 1.612 | 0.55 | 0.005 | 0.009 | 20.235 | 14.028 | Bal. |
Based on the alloy’s thermophysical properties and phase diagram data from the ProCAST database, the key parameters for the CAFE model were determined. The dendrite tip growth kinetics coefficients were calculated as $a_2 = 6.631854 \times 10^{-8}$ and $a_3 = 1.181568 \times 10^{-6}$. Nucleation parameters were set with a maximum surface nucleation density $n_{s,max} = 8.55 \times 10^5 \, \text{m}^{-2}$ and a corresponding volume nucleation density $n_{v,max} = 6.3247 \times 10^8 \, \text{m}^{-2}$ (following the ASTM relationship $n_v \propto n_s^{3/2}$).
A simple cylindrical specimen (30mm diameter x 80mm height) was modeled to investigate the influence of key investment casting process parameters. An orthogonal experimental design (L9 array) was employed to efficiently study four factors at three levels: Shell Thickness (7, 8, 9 mm), Shell Preheat Temperature (1143.15, 1173.15, 1203.15 K), Pouring Temperature (1913.15, 1923.15, 1933.15 K), and Cooling Method (Air, Oil, Water). The number of grains in a cross-section was used as the primary metric to assess microstructural refinement, as a higher count correlates with a larger proportion of fine equiaxed grains.
The simulation results, analyzed via range analysis, revealed the degree of influence and optimal combination. The cooling method had the most significant effect (largest range), followed by shell temperature, shell thickness, and pouring temperature. The optimal combination for maximizing grain count was found to be: 7 mm shell thickness, 1173.15 K shell preheat, 1923.15 K pouring temperature, and water cooling. The trends were clear: thinner shells, lower preheat temperatures, lower pouring temperatures, and faster cooling rates all promoted grain refinement by increasing cooling rates, undercooling, and nucleation rates while limiting grain growth time.
Furthermore, the influence of pouring speed on microstructure and grain orientation was studied. Simulations were run at pouring speeds of 0.5, 1.5, 2.5, 3.5, and 4.5 kg/s with other parameters at their optimal values. The results are summarized in Table 2.
| Pouring Speed (kg/s) | Number of Grains | Average Orientation Deviation from <001> (°) |
|---|---|---|
| 0.5 | 1125 | 32.740 |
| 1.5 | 1177 | 32.400 |
| 2.5 | 1203 | 32.171 |
| 3.5 | 1221 | 31.999 |
| 4.5 | 1225 | 31.994 |
The data shows that increasing pouring speed leads to a greater number of grains and a gradual reduction in the average deviation of grain growth directions from the preferred <001> thermal axis, promoting more aligned growth. This is attributed to increased fluid turbulence causing dendrite arm fragmentation, which acts as new nucleation sites, refining the structure. However, the effect diminishes at higher speeds. A pouring speed of 3.5 kg/s was selected as a balance between microstructural benefits and the risk of turbulence-related defects in complex thin-walled castings.
CAFE Simulation and Process Design for a Complex Thin-Walled Casting
The target component was a complex thin-walled engine connection housing with dimensions 216mm x 180mm x 240mm, featuring a minimum wall thickness of 3.5mm over 70% of its surface. Based on the specimen study results, the gating system was designed for top-pouring to promote directional solidification. Key parameters for the investment casting process were set as shown in Table 3.
| Parameter | Value |
|---|---|
| Shell Thickness | 7 mm |
| Shell Preheat Temperature | 1173.15 K (900 °C) |
| Pouring Temperature | 1923.15 K (1650 °C) |
| Pouring Speed | 3.5 kg/s |
| Cooling Method | Water Quench |
Macroscopic FE simulations of mold filling, solidification sequence, and defect formation (shrinkage porosity, cold shuts) were conducted first. The results confirmed smooth filling, a desirable sequential solidification pattern (bottom-up, with the riser solidifying last), and predicted shrinkage porosity to be confined to the gating system riser, validating the gating design.
To verify the CAFE method’s applicability to the 3D component, simulations focusing on nucleation parameters were performed. The volume nucleation density ($n_v$) and mean undercooling ($\Delta T_{v,n}$) were varied. The results, analyzed on a thin-wall section, are summarized in Table 4.
| Model Set | $n_v$ (m⁻³) | $\Delta T_{v,n}$ (K) | Grain Count | Equiaxed Grain Ratio (%) |
|---|---|---|---|---|
| High $n_v$ | 1.5438×10⁹ | 10 | 3040 | 94.72 |
| 1.5438×10⁹ | 8 | 2843 | 92.58 | |
| 1.5438×10⁹ | 6 | 2649 | 90.97 | |
| Mid $n_v$ | 6.3247×10⁸ | 10 | 1383 | 84.53 |
| 6.3247×10⁸ | 8 | 1259 | 83.42 | |
| 6.3247×10⁸ | 6 | 1106 | 82.37 | |
| Low $n_v$ | 3.3077×10⁸ | 10 | 377 | 69.87 |
| 3.3077×10⁸ | 8 | 346 | 69.33 | |
| 3.3077×10⁸ | 6 | 303 | 68.81 |
The simulations clearly demonstrate that increasing either nucleation density or undercooling promotes a finer, more equiaxed microstructure. This physically aligns with the effects of the optimized investment casting process parameters (e.g., faster cooling increases effective undercooling and nucleation rate), confirming the CAFE model’s consistency in 3D.
Finally, CAFE simulation was used to determine the solution heat treatment holding time. The component was simulated after a solution treatment at 1100°C for 20, 30, and 40 minutes. The results showed significant grain coarsening with increased time, with the average grain radius increasing from approximately 11.91 µm (20 min) to larger sizes. No change in fundamental grain morphology was observed. Based on this, a 20-minute holding time was selected to achieve the desired carbide dissolution while minimizing excessive grain growth.
The casting was manufactured using the full optimized investment casting process chain: pattern production with the green wax, shell building with silica sol/zircon/mullite, dewaxing, mold firing, pouring with the specified parameters, shakeout, and final solution heat treatment at 1100°C for 20 minutes. The produced castings were sound, free from visible macro-defects like cold shuts or major shrinkage, and met dimensional requirements.
To quantitatively validate the CAFE model, a section from the actual casting was compared with the corresponding simulated microstructure. Metallographic analysis at 200x and 500x magnification revealed an austenitic matrix with some high-temperature ferrite and carbides. The average grain size was measured to be approximately 12.8 µm. The CAFE simulation for the same location predicted an average grain size of 11.91 µm. The error between simulation and experiment was a mere 0.89 µm, representing a negligible 0.07% deviation. This excellent agreement conclusively validates the accuracy of the CAFE method in predicting the solidification grain size of 16Cr20Ni14Si2 alloy under the designed investment casting process.
Conclusion
This research successfully applied the CAFE method to simulate and analyze the solidification microstructure of 16Cr20Ni14Si2 high-temperature steel in the investment casting process. A robust mathematical framework integrating macroscopic heat transfer with mesoscopic nucleation and growth kinetics was established. Through simulation of a simple specimen, the effects of key process parameters were quantified, leading to an optimized parameter set: 7 mm shell, 1173.15 K preheat, 1923.15 K pouring, water cooling, and 3.5 kg/s pouring speed. These parameters were effectively used to design the gating system and process for a complex thin-walled engine housing.
Further 3D CAFE simulations on the actual component provided critical insights: 1) Increasing nucleation density and undercooling parameters directly leads to a finer, more equiaxed grain structure, fundamentally explaining the effects of the optimized process parameters. 2) Solution treatment holding time primarily controls grain coarsening without altering the basic solidification morphology. The final casting trials produced high-quality components, and the remarkable agreement between simulated and experimental grain sizes (0.07% error) provides strong validation for the CAFE method’s predictive capability for this alloy and process. This study demonstrates that CAFE simulation is a powerful tool for guiding the optimization of the investment casting process to achieve desired microstructures and properties in critical high-temperature components.