In the field of metal casting, sand casting remains one of the most widely used manufacturing techniques due to its cost-effectiveness, versatility, and applicability to complex geometries. The process involves pouring molten metal into a mold cavity formed by sand, which acts as a refractory material. A critical aspect of sand casting is the heat transfer between the casting and the mold or core during solidification, as it directly influences the microstructure, mechanical properties, and defect formation in the final product. The interfacial heat transfer coefficient (IHTC) is a key parameter that quantifies this heat exchange, yet it is challenging to determine accurately due to dynamic changes at the interface, such as gap formation caused by thermal contraction and expansion. This study focuses on investigating the IHTC between ZL101 aluminum alloy castings and resin-bonded sand molds or cores in gravity sand casting, employing experimental measurements and numerical inverse methods to derive reliable IHTC values for simulation accuracy.
The significance of sand casting in industries like automotive, aerospace, and heavy machinery cannot be overstated. It accounts for a substantial portion of component manufacturing, with castings often comprising over 50% of critical parts in engines and machinery. However, the quality of these castings hinges on precise control of the solidification process, which is governed by heat transfer phenomena. In sand casting, the low thermal conductivity of sand compared to metal molds results in slower cooling rates, but the interfacial conditions can lead to non-uniform heat flow, causing defects like shrinkage porosity and hot tears. The IHTC serves as a boundary condition in computational simulations, such as those using ProCAST, to predict temperature fields and optimize processes. Traditional methods for determining IHTC, such as direct measurement or empirical formulas, are inadequate for sand casting due to the complex interface dynamics. Therefore, this research employs an inverse heat conduction problem (IHCP) approach, combining temperature data from embedded thermocouples with mathematical models to back-calculate IHTC, considering the effects of casting geometry and sand properties.
To systematically study the IHTC in sand casting, multiple casting geometries were designed, including flat plate and annular ring castings. The flat plate had dimensions of 150 mm in height and width, with a thickness of 50 mm, while the annular rings varied in inner and outer radii: 30 mm/80 mm, 50 mm/100 mm, and 70 mm/120 mm, all with a height of 150 mm. The casting material was ZL101 aluminum alloy, chosen for its common use in sand casting applications due to its good fluidity and mechanical properties. The mold and core materials consisted of furan resin-bonded sand, which provides adequate strength and collapsibility. The sand was prepared by mixing silica sand (50-100 mesh) with 1.2% resin and 0.3% catalyst, then compacted into molds and allowed to cure. Coating with zircon-based paint ensured surface stability during pouring. The pouring temperature was maintained at 705°C, with an initial mold temperature of 25°C, to simulate typical sand casting conditions.
Temperature measurements were crucial for inverse analysis. Thermocouples were strategically placed at specific distances from the casting-mold interface to capture transient temperature fields. For the flat plate casting, thermocouples were positioned at 2 mm from the interface on the casting side and at 6 mm, 14 mm, and 22 mm within the sand mold, symmetrically along the thickness direction. In annular ring castings, similar placements were used for both the outer sand mold and inner sand core, with thermocouples at 2 mm from the interface on the casting and at 6 mm, 14 mm, and 22 mm within the mold or core. A multi-channel data recorder collected temperature data every 0.5 seconds for up to 6000 seconds, covering the solidification and cooling phases. To minimize errors, thermocouples were fixed using ceramic tubes, and the system was calibrated to account for response delays and environmental factors. The experimental setup ensured that heat transfer could be approximated as one-dimensional for simplification, which is valid given the geometry and symmetry of the castings.
The inverse calculation of IHTC relies on solving the heat conduction equation using measured temperature data. The general heat conduction equation in one-dimensional form is given by:
$$ \rho C_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) $$
where \( \rho \) is density, \( C_p \) is specific heat capacity, \( k \) is thermal conductivity, \( T \) is temperature, \( t \) is time, and \( x \) is the spatial coordinate. For the sand mold and core, material properties such as \( k \) and \( C_p \) are temperature-dependent, as obtained from ProCAST’s material database. The initial condition is \( T(x,0) = T_{\text{initial}}(x) \), and boundary conditions include a heat flux at the interface (second kind) and a fixed temperature at the far end (first kind). Specifically, at the casting-mold interface (\( x=0 \)), the boundary condition is:
$$ -k \frac{\partial T}{\partial x} \bigg|_{x=0} = q(t) $$
where \( q(t) \) is the interfacial heat flux, and at the outer boundary (\( x=L \)), \( T(L,t) = T_{\text{measured}}(t) \). The IHTC is then defined as:
$$ h = \frac{q}{T_{\text{casting}} – T_{\text{mold}}} $$
where \( T_{\text{casting}} \) and \( T_{\text{mold}} \) are the temperatures at the casting and mold surfaces, respectively.
To handle the inverse problem, the Beck nonlinear estimation method was employed, which iteratively solves for the heat flux using future time steps to enhance stability. The heat flux \( q(t) \) is discretized into time segments, with \( q_p \) representing the average flux over the interval \([t_p, t_{p+1}]\). The temperature at any point can be expressed as a superposition of contributions from all previous heat flux segments:
$$ T(x_i, t_p) = T_0 + \sum_{j=1}^{p} X_{ij} q_j $$
where \( X_{ij} \) is the sensitivity coefficient, defined as \( \frac{\partial T(x_i, t_p)}{\partial q_j} \). The objective function to minimize the difference between measured and calculated temperatures is:
$$ F(q) = \sum_{n} \sum_{j=p}^{p+f} \left[ T_{\text{measured}, n}(t_j) – T_{\text{calculated}, n}(t_j) \right]^2 $$
where \( f \) is the number of future time steps (set to 6 in this study). By taking the derivative and solving iteratively, the heat flux and subsequently the IHTC are determined. For annular castings, the geometry requires using sector-shaped elements in the finite volume discretization to account for radial heat flow. The governing equation in radial coordinates is:
$$ \rho C_p \frac{\partial T}{\partial t} = \frac{1}{r} \frac{\partial}{\partial r} \left( r k \frac{\partial T}{\partial r} \right) $$
where \( r \) is the radial distance. The discrete equations for temperature in the sand mold or core are derived using energy balance for each control volume. For instance, for the first unit adjacent to the interface:
$$ \rho C_p V \frac{T_1^{p+1} – T_1^p}{\Delta t} = q_p A_{\text{in}} + k A_{\text{out}} \frac{T_2^p – T_1^p}{\Delta x} $$
where \( V \) is volume, \( A_{\text{in}} \) and \( A_{\text{out}} \) are inflow and outflow areas, and \( \Delta x \) is the spatial step (4 mm). This leads to a system of equations solved using the Thomas algorithm (TDMA).
The inverse calculation program was implemented in MATLAB, incorporating temperature-dependent thermophysical properties of the resin sand. Key parameters included a time step of 1 second, spatial discretization into 6 units, and a total simulation time of 6000 seconds. The program outputted interfacial heat flux, temperature fields, and IHTC values over time. Validation involved comparing calculated temperatures with measured data at specific thermocouple locations, ensuring convergence and accuracy. The stability of the inverse method was confirmed by the reduction in temperature errors over time, with deviations within 2°C after the initial transient phase.
Analysis of the experimental temperature data revealed distinct phases of solidification: superheat removal, mushy zone solidification, and cooling. For the flat plate casting, the temperature at the interface (TC1) decreased rapidly initially, then slowed during latent heat release, and dropped sharply after solidification. Similar trends were observed in annular castings, but with variations due to geometry. The inner sand core exhibited higher temperatures than the outer mold, eventually leading to reverse heat flow as the core became hotter than the casting. This phenomenon was more pronounced in smaller-diameter cores, highlighting the influence of geometry on heat accumulation.
The inverse calculation provided insights into interfacial heat flux and IHTC. For the flat plate, the heat flux peaked at around 64.9 kW/m² initially and decreased gradually, while for annular castings, the peak flux varied with size. For example, the 30 mm inner diameter core had a peak flux of 120.9 kW/m², compared to 79.8 kW/m² for the 70 mm core. Negative heat flux occurred in cores when their temperature exceeded that of the casting, indicating reverse heat transfer. The IHTC values derived from these fluxes showed that higher values were associated with smaller geometries due to restricted contraction and longer heat exposure. The table below summarizes the maximum and minimum IHTC values for different casting configurations:
| Casting Type | Interface | Max IHTC (W/m²·°C) | Min IHTC (W/m²·°C) |
|---|---|---|---|
| Flat Plate | Sand Mold | 108 | 61 |
| Annular (30/80 mm) | Sand Mold | 131 | 83 |
| Annular (30/80 mm) | Sand Core | 263 | 144 |
| Annular (50/100 mm) | Sand Mold | 127 | 78 |
| Annular (50/100 mm) | Sand Core | 183 | 105 |
| Annular (70/120 mm) | Sand Mold | 103 | 65 |
| Annular (70/120 mm) | Sand Core | 110 | 70 |
The variation of IHTC with casting surface temperature followed an approximate “S-shaped” curve, with significant changes occurring between the liquidus and solidus temperatures (approximately 600°C to 550°C for ZL101). In high-temperature regions (above liquidus), IHTC remained relatively constant, while in the mushy zone, it decreased due to gap formation. For smaller cores, the “S-curve” extended toward the solidus temperature, indicating prolonged interface interactions. The relationship can be expressed empirically as:
$$ h(T) = h_{\text{min}} + \frac{h_{\text{max}} – h_{\text{min}}}{1 + e^{-k(T – T_m)}} $$
where \( h_{\text{min}} \) and \( h_{\text{max}} \) are the minimum and maximum IHTC values, \( k \) is a constant, and \( T_m \) is the midpoint temperature. This model captures the nonlinear behavior observed in sand casting processes.
To validate the inverse method, the derived IHTC values were used in ProCAST simulations for a ring casting with inner and outer radii of 60 mm and 110 mm, respectively. The geometry was modeled in UG and meshed with 4 mm elements, resulting in over 980,000 volume cells. Material properties for ZL101 and resin sand were assigned, including temperature-dependent conductivity and specific heat. The interfacial boundary conditions were set using the inverse-calculated IHTC curves for the sand mold, sand core, and flat plate interfaces. The simulation assumed instantaneous filling and focused on solidification, with a pouring temperature of 705°C and mold temperature of 25°C. The temperature at a node corresponding to the TC1 thermocouple location was monitored and compared to experimental data.
The simulation results showed excellent agreement with measured temperatures during solidification, with a maximum deviation of 17°C occurring after complete solidification. This discrepancy is attributed to assumptions in the IHTC model, such as constant gap formation below the solidus, whereas in reality, minor contractions continue. However, the overall accuracy confirms the reliability of the inverse method for sand casting applications. The table below compares key parameters from simulation and experiment:
| Parameter | Experimental Value | Simulated Value |
|---|---|---|
| Peak Temperature at TC1 (°C) | 705 | 705 |
| Solidification Time (s) | 1750 | 1700 |
| Final Temperature at 3000 s (°C) | 450 | 443 |
In conclusion, this study demonstrates a robust approach for determining interfacial heat transfer coefficients in sand casting through inverse analysis. The IHTC varies significantly with casting geometry, particularly for inner sand cores, where smaller diameters lead to higher values due to constrained contraction and prolonged heat exposure. The “S-curve” model effectively describes the temperature dependence of IHTC, providing a valuable tool for simulation-based optimization. The integration of inverse methods with commercial software like ProCAST enhances the predictive capability for sand casting processes, reducing defects and improving product quality. Future work could explore the effects of different sand types or casting alloys to further refine IHTC models for diverse sand casting applications.