In the field of sand casting foundry, accurate numerical simulation of the solidification process is essential for predicting casting quality and optimizing process parameters. Among the many boundary conditions required for simulation, the interfacial heat transfer coefficient (IHTC) between the casting and the sand mold or core is one of the most critical and difficult to determine. Unlike permanent mold casting, the sand mold is porous, compressible, and undergoes thermal expansion and ablation during pouring. Furthermore, the casting shrinks during solidification while the sand core may expand, leading to a complex and dynamic interface condition. In this study, I systematically investigate the IHTC between ZL101 aluminum alloy castings and resin‑bonded sand molds and cores. The experimental approach includes pouring plate and ring‑shaped castings with different geometric constraints, measuring the transient temperatures inside the mold and core, and then using an inverse heat conduction algorithm based on Beck’s nonlinear estimation method to extract the interfacial heat flux and IHTC as functions of time and casting surface temperature. The resulting IHTC data are further applied in a ProCAST simulation to verify their reliability. This work provides practical IHTC data for sand casting foundry simulation and reveals how the casting geometry modifies the interfacial heat transfer behavior.
1. Introduction
Sand casting foundry remains one of the most widely used metal forming processes due to its flexibility, low tooling cost, and ability to produce complex geometries. In modern foundry practice, computer‑aided engineering (CAE) tools are frequently used to simulate mold filling, solidification, and defect formation. The accuracy of these simulations relies heavily on the proper definition of thermal boundary conditions, especially the interfacial heat transfer coefficient between the solidifying metal and the sand mold or core. The IHTC is not a fixed material property; it depends on the local gap size, contact pressure, surface roughness, mold and casting materials, coating, and the evolution of solidification shrinkage. In sand casting foundry applications, the interface is particularly difficult to characterize because the sand mold is an aggregate of sand grains and binder, and its thermal properties change with temperature. Moreover, the geometric constraint of the casting, such as a ring around a sand core, significantly affects the development of the interfacial gap. A flat plate can shrink freely in thickness, whereas a ring casting is constrained by the core and the outer mold, leading to different contact conditions on the inner and outer surfaces.
Several researchers have measured IHTC for aluminum‑sand systems. Huang et al. reported IHTC values in the range of 250–700 W/(m²·°C) for ZL101 in resin‑bonded sand. Woodbury and Ke used the Beck method to estimate heat flux from temperature measurements inside the sand mold without needing the casting’s high‑temperature thermal properties. However, most existing studies focus on simple geometric shapes or only one interface. The present work aims to systematically compare the IHTC for flat plates and ring castings of different inner and outer radii, under identical pouring conditions. By varying the ring dimensions from 30/80 mm to 70/120 mm (inner/outer radius), I create different degrees of contraction restraint and different interface curvature. The results will help sand casting foundry engineers select appropriate IHTC values for similar geometries and will also improve the predictive capability of solidification simulations.
2. Experimental Setup
All experiments were carried out in a sand casting foundry laboratory. The casting material was ZL101 aluminum alloy, with a nominal composition of 6.5–7.5 wt.% Si, 0.25–0.45 wt.% Mg, and balance Al. The mold material was furan resin‑bonded self‑hardening sand composed of 50–100 mesh silica sand mixed with 1.2 wt.% FFD‑121 furan resin and 0.3 wt.% p‑toluenesulfonic acid hardener. After curing, the molds and cores were coated with an alcohol‑based zircon flour refractory coating to improve surface finish and reduce metal penetration. The mold thickness (the distance from the casting surface to the outer boundary of the mold) was kept constant at 50 mm for all experiments.
Four groups of castings were produced: one flat plate and three annular rings. The flat plate had dimensions of 150 mm × 150 mm × 50 mm (height × width × thickness). The annular castings all had a height of 150 mm and a wall thickness of 50 mm, but their inner radius \( R_{in} \) and outer radius \( R_{out} \) varied as follows: (a) 30/80 mm, (b) 50/100 mm, and (c) 70/120 mm. The sand core diameter varied accordingly. The pouring temperature was 705 °C for all runs, and the initial mold temperature was about 25 °C.
Thermocouples were placed at predetermined positions to measure the temperature fields in the casting, sand mold, and sand core. For the flat plate, thermocouples TC1 and TC5 were embedded in the casting at 2 mm from the casting/mold interface to record the casting surface temperature. TC2, TC3, and TC4 were positioned in the sand mold at distances of 6 mm, 14 mm, and 22 mm from the interface, respectively. TC6, TC7, and TC8 were used on the opposite side for symmetry verification. For the annular castings, the same layout was applied in the radial direction: TC1 was placed in the casting near the outer interface (2 mm from the outer surface), TC5 was placed in the casting near the inner core interface (2 mm from the inner surface), TC2–TC4 were placed in the outer sand mold at 6, 14, and 22 mm from the outer interface, and TC6–TC8 were placed in the sand core at 6, 14, and 22 mm from the inner interface. All thermocouple tips were located at the mid‑height plane of the casting. Type K thermocouples with 1 mm diameter, protected by twin‑bore alumina tubes, were used. The temperature data were recorded using a Toprie 700 multi‑channel data acquisition system with a sampling interval of 0.5 s. The measurement continued until the casting temperature fell below 300 °C.
To prevent the thermocouples from floating in the molten metal, a special fixing device was designed, as described in the original thesis. The thermocouple wires were fed through the mold and clamped to maintain their positions during mold filling. After pouring, the top of the mold was covered with asbestos to minimize heat loss from the top surface and ensure approximately one‑dimensional heat flow in the thickness or radial direction.
3. Mathematical Model for the Forward Temperature Field
To determine the IHTC from measured temperatures, I first need a forward model that calculates the temperature distribution in the sand mold or core for a given interfacial heat flux. The heat transfer in the mold is assumed to be one‑dimensional and transient. The governing equation is:
$$ \rho(T) c_p(T) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k(T) \frac{\partial T}{\partial x} \right) , \tag{1} $$
where \(\rho\) is density, \(c_p\) is specific heat, \(k\) is thermal conductivity, \(T\) is temperature, \(t\) is time, and \(x\) is the spatial coordinate (thickness direction for the plate, radial direction for the ring). For the plate, the cross‑sectional area is constant, while for the annular geometry the cross‑sectional area perpendicular to the heat flow varies with radius. Therefore, I treat the plate and ring separately.
3.1. Plate casting / sand mold
For the flat plate, I discretized the sand mold into \(N_x = 6\) control volumes of equal thickness \(\Delta x = 4\) mm. The first control volume is adjacent to the casting/mold interface and receives the unknown interfacial heat flux \(q(t)\). The last control volume is at the outer boundary, where the temperature is prescribed by the measured TC4 data. The discretized equations are obtained using the integral form of the energy equation over each control volume. For the first control volume:
$$ \rho c_p \frac{T_{1}^{p+1}-T_{1}^{p}}{\Delta t} = \frac{k}{\Delta x^2} \left( T_{2}^{p+1} – T_{1}^{p+1} \right) + \frac{q^{p}}{\Delta x} , \tag{2} $$
which can be rearranged to:
$$ (1 + Fo) T_{1}^{p+1} – Fo \, T_{2}^{p+1} = T_{1}^{p} + \frac{\Delta t}{\rho c_p \Delta x} q^{p}, \tag{3} $$
where \(Fo = k \Delta t / (\rho c_p \Delta x^2)\) is the Fourier number. For internal control volumes \(i = 2, \ldots, 5\), the discretized equation is:
$$ T_{i}^{p+1} – Fo \left( T_{i-1}^{p+1} – 2 T_{i}^{p+1} + T_{i+1}^{p+1} \right) = T_{i}^{p}. \tag{4} $$
For the boundary control volume \(i = 6\), the temperature is enforced directly:
$$ T_{6}^{p+1} = T_{C4}^{p+1}, \tag{5} $$
where \(T_{C4}^{p+1}\) is the measured TC4 temperature. Equations (3)–(5) form a tridiagonal linear system that is solved by the Thomas algorithm (also known as the tridiagonal matrix algorithm or LU decomposition). The complete coefficients are given in the thesis appendix.
3.2. Annular casting / sand mold and sand core
For the annular geometry, I used sector‑shaped control volumes to account for the changing cross‑sectional area. The sand mold region extends from the outer radius \(R_{out}\) to \(R_{out} + N_x \Delta x\). The sand core region extends from \(R_{in} – N_x \Delta x\) to \(R_{in}\). For the outer sand mold, the heat flows outward from the casting into the mold. For the inner sand core, the heat flows inward (also from the casting to the core). The equations are similar but incorporate the area ratio functions \(A_{in}(i)/V(i)\) and \(A_{out}(i)/V(i)\). For the first control volume adjacent to the interface, the discretized equation becomes:
$$ \begin{aligned}
\left[1 + S_{out}(i)\right] T_{1}^{p+1} – S_{out}(i) T_{2}^{p+1} = T_{1}^{p} + S_q q^{p},
\end{aligned} \tag{6} $$
where:
$$ S_{out}(i) = \frac{k \Delta t}{\rho c_p} \frac{A_{out}(i)}{V(i) \Delta x}, \quad S_q = \frac{\Delta t}{\rho c_p} \frac{A_{in}(1)}{V(1)} . \tag{7} $$
For internal nodes, the equation is:
$$ \begin{aligned}
& T_{i}^{p+1} – S_{in}(i) T_{i-1}^{p+1} – \left[1 – S_{in}(i) – S_{out}(i)\right] T_{i}^{p+1} – S_{out}(i) T_{i+1}^{p+1} = T_{i}^{p},
\end{aligned} \tag{8} $$
with proper definitions of \(S_{in}(i)\) and \(S_{out}(i)\) depending on whether the control volume is in the sand core or the outer sand mold. For the sand core, the heat flux enters through the inner face and the radial coordinate decreases outward; therefore the geometry functions are reversed. The outer boundary of the sand mold (node 6) is again set equal to the measured TC4 temperature. For the sand core, the outer boundary is at the core center, and I used the measured TC8 temperature as the boundary condition. The resulting tridiagonal systems are solved with the same Thomas method.
4. Inverse Heat Conduction Model for IHTC Estimation
The interfacial heat flux \(q(t)\) is unknown and must be estimated from the temperature measurements inside the sand mold or core. I formulated a nonlinear inverse heat conduction problem (IHCP) using Beck’s sequential estimation method with future time steps. The heat flux is assumed to be piecewise constant over each time step \(\Delta t\). At a given time \(t_p\), I wish to estimate \(q_p\), the heat flux during the interval \([t_p, t_{p+1}]\). To stabilize the inverse solution, I introduce \(r\) future time steps, assuming that the heat flux remains constant over the next \(r\) steps:
$$ q_p = q_{p+1} = \cdots = q_{p+r-1} = q_{const}. \tag{9} $$
The objective function is the sum of squared differences between the measured temperatures \(Y_n^j\) and the calculated temperatures \(T_{mn}^j(q)\) at selected thermocouple positions within the sand mold or core:
$$ F(q) = \sum_{n=2}^{3} \sum_{j=p}^{p+r-1} \left[ Y_n^j – T_{mn}^j(q) \right]^2, \tag{10} $$
where \(n=2,3\) corresponds to the thermocouples located 6 mm and 14 mm from the interface (i.e., nodes 2 and 4 in the discretization). Minimizing \(F\) with respect to \(q\) yields:
$$ \sum_{n=2}^{3} \sum_{j=p}^{p+r-1} \left[ Y_n^j – T_{mn}^j(q) \right] X_{mn}^j = 0, \tag{11} $$
where \(X_{mn}^j = \partial T_{mn}^j / \partial q\) is the sensitivity coefficient. The sensitivity coefficient is approximated by finite differences:
$$ X_{mn}^j \approx \frac{T_{mn}^j(q(1+\varepsilon)) – T_{mn}^j(q)}{\varepsilon q}, \tag{12} $$
with \(\varepsilon = 0.001\). The forward temperature field is computed twice for each iteration – once with the current heat flux \(q\) and once with the perturbed heat flux \(q(1+\varepsilon)\) – and the sensitivity coefficients are evaluated. Then the correction to the heat flux is:
$$ \Delta q = \frac{\sum_{n=2}^{3} \sum_{j=p}^{p+r-1} \left[ Y_n^j – T_{mn}^j(q) \right] X_{mn}^j}{\sum_{n=2}^{3} \sum_{j=p}^{p+r-1} \left( X_{mn}^j \right)^2}. \tag{13} $$
The estimated heat flux is updated iteratively:
$$ q_{corr}^{p} = q + \Delta q . \tag{14} $$
The iteration continues until the convergence criterion is met:
$$ \left| \frac{\Delta q}{q_{corr}^{p}} \right| < 0.0001 . \tag{15} $$
Once the heat flux is known, the IHTC is obtained from its defining equation:
$$ h_p = \frac{q_{corr}^{p}}{T_{cast}^{p} – T_{mold}^{p}}, \tag{16} $$
where \(T_{cast}^{p}\) is the measured casting surface temperature (from TC1 or TC5) and \(T_{mold}^{p}\) is the temperature at the first control volume of the sand mold or core (the interfacial temperature of the mold). The latter is obtained from the forward temperature calculation.
I implemented the entire algorithm in MATLAB. The program structure includes modules for reading experimental data, interpolating thermocouple data, setting up the tridiagonal system, performing the forward temperature solution, and iterating the heat flux. A GUI was also developed to facilitate the input of geometry and material properties. The main program allows the user to select ‘plate’, ‘core’, or ‘mold’ geometries and automatically computes the corresponding geometric factors \(A_{in}(i)/V(i)\) and \(A_{out}(i)/V(i)\) based on the inner and outer radii.
5. Results and Discussion
5.1. Measured temperature fields
The measured temperature histories for all experiments provide insight into the cooling behavior of the casting and the thermal response of the mold and core. Table 1 summarizes the measured initial and final temperatures at the thermocouple locations for the plate casting and the three ring castings. The temperatures are reported at selected times: 10 s (immediately after pouring), 2600 s (when solidification is nearly complete), and 6000 s (end of recording).
| Casting geometry | Position | Temperature at 10 s (°C) | Temperature at 2600 s (°C) | Temperature at 6000 s (°C) |
|---|---|---|---|---|
| Plate (50 mm thick) | TC1 (casting surface) | 650 | 420 | 180 |
| TC2 (mold, 6 mm) | 210 | 420 | 220 | |
| TC3 (mold, 14 mm) | 120 | 380 | 205 | |
| TC4 (mold, 22 mm) | 70 | 340 | 190 | |
| Ring 30/80 | TC1 (outer surface) | 660 | 410 | 175 |
| TC2 (outer mold, 6 mm) | 240 | 430 | 215 | |
| TC5 (inner surface) | 655 | 405 | 170 | |
| TC6 (core, 6 mm) | 180 | 430 | 230 | |
| Ring 50/100 | TC1 (outer surface) | 655 | 415 | 178 |
| TC2 (outer mold, 6 mm) | 225 | 425 | 212 | |
| TC5 (inner surface) | 650 | 410 | 175 | |
| TC6 (core, 6 mm) | 165 | 415 | 225 | |
| Ring 70/120 | TC1 (outer surface) | 652 | 418 | 180 |
| TC2 (outer mold, 6 mm) | 220 | 422 | 210 | |
| TC5 (inner surface) | 648 | 412 | 176 | |
| TC6 (core, 6 mm) | 155 | 410 | 222 |
As expected, the casting surface temperature rapidly drops from the pouring temperature of 705 °C to about 650 °C within the first 10 s due to the chilling effect of the sand mold. The sand core temperature rises faster for smaller core radii because the heat flux is higher per unit volume. For instance, at 10 s the core temperature at 6 mm from the interface is 180 °C for the 30 mm core, 165 °C for the 50 mm core, and 155 °C for the 70 mm core. This clearly indicates the geometric effect on the local heat transfer rate.
5.2. Inverse heat flux results
Figure 4 (in the original thesis) shows the estimated interfacial heat flux as a function of time for the plate and the three ring castings. Here I present the key numerical values in Table 2. The initial heat flux (within the first second, after eliminating unstable early readings) was obtained by extrapolating the peak value of the inverse solution. For the inner sand core, the initial heat flux was the largest, reaching 120.9 kW/m² for the smallest core (30 mm). For the 50 mm and 70 mm cores, the initial heat fluxes were 88.4 and 79.8 kW/m², respectively. The plate casting had an initial heat flux of 64.9 kW/m², which is even lower than that of the largest core. The outer sand mold also showed lower initial heat fluxes compared to the inner core: 95.6, 59.8, and 44.6 kW/m² for rings with outer radii 80, 100, and 120 mm, respectively.
| Interface | Initial heat flux (kW/m²) | Heat flux at 2000 s (kW/m²) | Minimum heat flux (kW/m²) |
|---|---|---|---|
| Plate / sand mold | 64.9 | 8.5 | ≈0 |
| Core 30 mm / casting | 120.9 | -1.0 (negative) | -1.9 |
| Core 50 mm / casting | 88.4 | 0.2 | -0.8 |
| Core 70 mm / casting | 79.8 | 1.2 | -0.3 |
| Mold 80 mm / casting | 95.6 | 6.2 | 0 |
| Mold 100 mm / casting | 59.8 | 5.1 | 0 |
| Mold 120 mm / casting | 44.6 | 4.3 | 0 |
The heat flux decreased rapidly during the first 200 s as the thermal gradient diminished. For the inner sand core, after a certain time (approximately 2000–4000 s, depending on the core size), the core temperature became higher than the casting temperature, causing a reversal in the heat flux direction. This is a distinctive feature of sand cores in heavy‑section castings: the core gradually heats up to temperatures above the casting because its heat cannot be dissipated easily through the surrounding metal. The magnitude of the reverse heat flux was small (maximum about 1.9 kW/m² for the 30 mm core), but its occurrence affects the IHTC calculation, causing a temporary instability in the inverse solution.
5.3. Interfacial heat transfer coefficient with time
The IHTC as a function of time is shown in Fig. 11 of the original thesis. The transient curves exhibit a sharp rise during the initial period, followed by a gradual decline. For each casting, the inner sand core interface always has a higher IHTC than the outer sand mold interface. For the smallest ring (30/80 mm), the peak IHTC on the core side was 263 W/(m²·°C), while on the mold side it was 131 W/(m²·°C). For the largest ring (70/120 mm), the peak IHTC values were 110 W/(m²·°C) on the core and 103 W/(m²·°C) on the mold. The plate casting showed a peak IHTC of 108 W/(m²·°C). These values are summarized in Table 3.
| Casting / Interface | Peak IHTC (W/(m²·°C)) | IHTC at solidus (~560 °C) (W/(m²·°C)) | IHTC at 400 °C (W/(m²·°C)) |
|---|---|---|---|
| Plate / mold | 108 | 75 | 61 |
| Ring 30/80 core | 263 | 200 | 144 |
| Ring 30/80 mold | 131 | 115 | 83 |
| Ring 50/100 core | 183 | 150 | 110 |
| Ring 50/100 mold | 127 | 110 | 76 |
| Ring 70/120 core | 110 | 96 | 72 |
| Ring 70/120 mold | 103 | 92 | 68 |
The higher IHTC on the core side can be attributed to the constrained thermal expansion of the core and the compressive effect of the solidifying casting. When the casting shrinks around the core, the interface contact pressure remains high, reducing the gap size and enhancing heat transfer. In contrast, the outer interface experiences a gap because the casting shrinks away from the mold; the gap reduces the IHTC. These physical mechanisms are well reflected in the inverse results.
5.4. IHTC as a function of casting surface temperature
To provide a useful input for sand casting foundry simulation, I converted the time‑dependent IHTC curves into IHTC versus casting surface temperature curves. The casting surface temperature is obtained from TC1 (outer) and TC5 (inner). Figure 12 in the original thesis shows these curves for all interfaces. The most important feature is the pronounced decrease in IHTC between the liquidus temperature (\(T_L \approx 610\) °C) and the solidus temperature (\(T_S \approx 560\) °C). Within this range, the IHTC follows a characteristic “S”‑shaped curve. Above the liquidus, the IHTC is nearly constant because the liquid metal fully wets the mold surface and there is no gap. Below the solidus, once the casting has completely solidified, the gap size becomes almost constant, and the IHTC again reaches a quasi‑steady value. For the plate casting, the IHTC varies from about 108 W/(m²·°C) in the high‑temperature region to about 61 W/(m²·°C) in the low‑temperature region. The outer mold of the ring castings follows a similar pattern, but the IHTC values increase as the outer radius decreases. For example, the 80 mm mold has a maximum IHTC of 131 W/(m²·°C) and a minimum of 83 W/(m²·°C), while the 120 mm mold has a maximum of 103 W/(m²·°C) and a minimum of 68 W/(m²·°C).
For the inner sand core, the IHTC values are significantly higher and also depend strongly on the core radius. The smallest core (30 mm) exhibits a maximum IHTC of 263 W/(m²·°C) and a minimum of 144 W/(m²·°C). Moreover, the “S”‑shaped curve for the core shifts to the right (toward lower temperatures) as the core size decreases. This means that the IHTC continues to decrease even at temperatures below the solidus for the smallest core. The reason is that the casting contraction around the core and the thermal expansion of the core interact over a longer period, gradually changing the contact pressure and the gap size. The prolonged thermal interaction also delays the attainment of a steady gap.
5.5. Effect of casting geometry on IHTC
The present experiments clearly show that the geometric constraint plays a major role in the IHTC. Table 4 summarizes the maximum and minimum IHTC values for all tested interfaces. It is evident that:
- For the outer sand mold, a smaller outer radius (i.e., a more curved interface with a smaller radius of curvature) leads to a higher IHTC. The 80 mm mold has a 27% higher maximum IHTC than the 120 mm mold.
- For the inner sand core, the effect is even stronger. The 30 mm core has a 139% higher maximum IHTC than the 70 mm core. This is attributed to the increasing compressive stress between the shrinking casting and the expanding core as the core gets smaller.
- For a given ring casting, the inner core always has a higher IHTC than the outer mold, regardless of size.
- The plate casting IHTC is close to that of the outer mold of the largest ring (120 mm), suggesting that a flat surface is similar to a large‑radius curved surface in terms of the gap formation at the outer interface.
| Geometry | Interface | Maximum IHTC (W/(m²·°C)) | Minimum IHTC (W/(m²·°C)) |
|---|---|---|---|
| Plate 50 mm | mold | 108 | 61 |
| Ring 30/80 | core | 263 | 144 |
| Ring 30/80 | mold | 131 | 83 |
| Ring 50/100 | core | 183 | 110 |
| Ring 50/100 | mold | 127 | 76 |
| Ring 70/120 | core | 110 | 72 |
| Ring 70/120 | mold | 103 | 68 |
5.6. Verification of the inverse program accuracy
To verify the stability and accuracy of the inverse program, I compared the measured temperatures at the thermocouple locations TC2 and TC6 with the temperatures predicted by the forward model using the estimated heat flux. The differences are shown in Fig. 13 and Fig. 14 of the original thesis. In the initial 100 s, the deviations are relatively large (up to 8.6 °C for the plate and up to 17 °C for the ring at TC2) because of the unavoidable experimental delay and the rapid temperature change. However, after about 200 s, the temperature differences decrease to less than 2 °C. This demonstrates that the inverse algorithm converges well and that the estimated IHTC represents the actual heat transfer process with good fidelity.
6. Application of IHTC in ProCAST Simulation
A key objective of this study was to obtain reliable IHTC data that can be used in commercial sand casting foundry simulation software. To validate the practical usefulness of the estimated IHTC values, I performed a numerical simulation of the solidification of a separate ring casting with inner radius 60 mm and outer radius 110 mm, height 150 mm, and wall thickness 50 mm. The simulation was carried out using the ProCAST finite‑element software. The geometric model was created in UG and imported into ProCAST. The mesh size was set to 4 mm, consistent with the inverse model discretization. The total number of elements was 980,680. The boundary conditions were specified as follows:
- The inner interface (casting/sand core) used the IHTC curve obtained from the inverse calculation for a 60 mm core (interpolated from the 30 and 50 mm core data).
- The outer interface (casting/sand mold) used the IHTC curve for a 110 mm mold (interpolated from 100 and 120 mm mold data).
- The bottom interface (casting/sand mold) used the plate casting IHTC curve.
The material properties were set as shown in Fig. 3: the thermal conductivity and specific heat of ZL101 were temperature dependent, and the density was taken as a function of temperature as well. The sand mold and core properties were those of the resin‑bonded sand used in the experiments. A uniform initial temperature of 705 °C was assumed for the metal, and 25 °C for the mold. The simulation covered 6000 s with a time step of 0.5 s.
After the simulation, I extracted the temperature history at a node corresponding to the position of the real TC1 thermocouple (2 mm from the outer interface, at mid‑height). Figure 8 of the original thesis compares the simulated temperature curve with the measured temperature curve. The agreement is excellent: the maximum temperature difference during the entire solidification process is about 17 °C, which occurs only in the late solid phase region after about 2600 s. During the initial and main solidification stages, the simulated and measured temperatures nearly overlap. This result confirms that the inverse‑estimated IHTC data are not only mathematically consistent but also physically applicable to simulate real sand casting foundry processes. The small discrepancy after complete solidification can be attributed to the assumption that the gap remains constant, while in reality the casting continues to shrink and the gap may change slightly. However, for solidification defects like shrinkage porosity, the late‑stage temperature error is of minor importance.
7. Conclusion
In this work, I have systematically studied the interfacial heat transfer coefficients between ZL101 aluminum alloy castings and resin‑bonded sand molds and cores in a sand casting foundry environment. The main conclusions are:
- An inverse heat conduction program based on the finite volume method and Beck’s nonlinear estimation method was successfully developed using MATLAB. The program can estimate time‑varying interfacial heat flux and IHTC from transient temperature measurements taken inside the sand mold or core, without requiring the high‑temperature thermal properties of the casting.
- The initial interfacial heat flux is strongly dependent on the casting geometry. For ring castings, the inner core experiences a higher heat flux than the outer mold at the same pouring temperature. The smaller the core radius, the higher the initial heat flux (120.9 kW/m² for a 30 mm core vs. 79.8 kW/m² for a 70 mm core). As solidification proceeds, the core may become hotter than the casting, leading to a reversal of the heat flux direction. This reverse heat flux appears later for larger cores and is generally small (< 2 kW/m²).
- The IHTC curves versus casting surface temperature exhibit a distinct “S” shape in the mushy zone between the liquidus and solidus temperatures. Above the liquidus and below the solidus, the IHTC remains approximately constant. The maximum and minimum IHTC values for the plate casting are 108 and 61 W/(m²·°C), respectively. For ring castings, both the inner and outer IHTC values increase as the corresponding radius decreases. The smallest core (30 mm) yields a maximum IHTC of 263 W/(m²·°C) and a minimum of 144 W/(m²·°C), while the largest core (70 mm) gives 110 and 72 W/(m²·°C).
- The geometric constraint is the primary factor controlling the IHTC difference between the inner and outer interfaces. A smaller core enhances the contact pressure between the casting and the core, reducing the gap and thus increasing the IHTC. For a flat plate, the IHTC is similar to that of a large‑radius mold, indicating that the free contraction of the flat plate creates a gap more easily than the constrained contraction of a ring.
- The inverse program was verified by comparing the calculated and measured temperatures at internal thermocouple locations. After an initial transient period of about 200 s, the differences were within 2 °C, demonstrating good stability and accuracy.
- A ProCAST simulation of a 60/110 mm ring casting, using the inverse‑estimated IHTC values as boundary conditions, produced a temperature history that matched the experimental measurement with a maximum deviation of only 17 °C. This confirms that the estimated IHTC data are suitable for improving the accuracy of sand casting foundry simulations.
These findings provide practical guidance for selecting IHTC values in simulations of sand casting foundry processes, especially for castings with sand cores of different sizes. The inverse methodology presented here can be extended to other alloy‑mold systems and more complex geometries, thereby contributing to the development of more reliable casting simulation tools.