Sand casting remains one of the most widely used manufacturing processes for producing complex metal components, particularly aluminum alloys. The quality of castings is strongly influenced by the solidification process, which in turn depends on the heat transfer at the casting–mold and casting–core interfaces. A precise knowledge of the interfacial heat transfer coefficient (IHTC) is essential for accurate numerical simulation of the casting process. However, the IHTC is not a constant; it varies with time, temperature, casting geometry, and the contacting materials. In sand casting, the situation is even more complicated because the mold and core are granular, permeable, and undergo thermal expansion and contraction during solidification. Moreover, the formation of air gaps at the interface due to casting shrinkage and mold expansion drastically alters the heat flow. This leads to common sand casting defects such as hot tears, shrinkage porosity, cold shuts, and misruns. Understanding the IHTC behavior helps predict and mitigate these sand casting defects. In this paper, I present a systematic study of the IHTC between ZL101 aluminum alloy castings and resin-bonded sand molds or cores, using both flat plate and annular geometries. The temperature fields were measured with embedded thermocouples, and a Beck nonlinear inverse estimation method combined with a finite volume heat conduction model was developed in MATLAB to extract the interfacial heat flux and IHTC. The results show that the IHTC varies significantly with the casting surface temperature, following an approximate “S” shaped curve during solidification. The effect of casting curvature and core constraint is quantified. Finally, the obtained IHTC curves are applied in ProCAST simulations to reproduce the cooling curves of a 60/110 mm ring casting, demonstrating that the inverse method offers reliable boundary conditions for casting simulation and helps reduce sand casting defects through better thermal design.
Introduction
Casting simulation has become an indispensable tool for optimizing casting processes and minimizing sand casting defects. The accuracy of such simulations depends on the correct description of boundary conditions, especially the heat transfer coefficient at the casting/mold interface. In sand casting, the mold is composed of sand grains bonded by resin or clay, and its thermal conductivity is relatively low compared with metal molds. The heat flow from the solidifying casting to the sand mold or core is controlled by the interfacial thermal resistance, which includes conduction through the gas gap, radiation, and possible points of contact between the casting surface and the mold surface. Because the casting contracts during solidification while the sand mold expands due to heating, a gap inevitably forms at the interface. This gap grows with time and strongly reduces the heat transfer. The gap size is influenced by the casting geometry, rigidity of the casting, and restraint from cores. For castings with internal sand cores, the core is surrounded by solidifying metal; the core expands while the casting shrinks, creating a complex interaction that may either open or close the gap depending on the relative displacements. Therefore, the IHTC is not only a function of material properties but also of the local geometry and boundary constraints.
Many researchers have studied the IHTC in metal mold casting using inverse heat conduction methods. However, for sand casting, especially with internal cores, fewer systematic data are available. The difficulty lies in the inability to measure the gap thickness directly and the low thermal diffusivity of sand, which makes the temperature response slow and requires careful treatment of the inverse problem. In this work, I focus on two fundamental geometries: a flat plate casting and annular castings with different inner and outer radii. The plate represents a simple, one-dimensional case with free contraction along the thickness direction. The annular castings introduce curvature and a central core that restricts contraction. By comparing the IHTC behavior among these geometries, I aim to quantify the influence of casting shape and core constraint on the interfacial heat transfer. Such data are crucial for the simulation of complex castings and the prevention of sand casting defects.
Experimental Setup
The casting material was ZL101 aluminum alloy with the composition given in Table 1. The mold and core material was furan resin-bonded silica sand. The sand mixture consisted of 50–100 mesh silica sand, 1.2% FFD-121 furan resin, and 0.3% p-toluenesulfonic acid solution as hardener. After mixing, the sand was immediately rammed around the patterns to form the mold and core. The cured molds were dried naturally and coated with alcohol-based zirconium powder paint.
| Si | Mg | Ti | Cu | Mn | Zn | Zr | Al |
|---|---|---|---|---|---|---|---|
| 6.5–7.5 | 0.25–0.45 | 0.08–0.20 | ≤0.10 | ≤0.10 | ≤0.10 | ≤0.20 | Balance |
Four casting configurations were designed:
- Plate casting: thickness 50 mm, width 150 mm, height 150 mm.
- Ring casting A: inner radius 30 mm, outer radius 80 mm, height 150 mm.
- Ring casting B: inner radius 50 mm, outer radius 100 mm, height 150 mm.
- Ring casting C: inner radius 70 mm, outer radius 120 mm, height 150 mm.
All castings had a wall thickness of 50 mm. The mold thickness (sand thickness around the casting) was 50 mm for all configurations. Type K thermocouples were placed at carefully controlled positions. For the plate casting, thermocouple TC1 was embedded in the casting at a distance of 2 mm from the casting/mold interface. Thermocouples TC2, TC3, and TC4 were placed in the sand mold at distances of 6 mm, 14 mm, and 22 mm from the interface, respectively. The thermocouples on the opposite side were symmetric. For the ring castings, the outer interface had thermocouples TC1 (in casting) and TC2, TC3, TC4 in the sand mold at the same distances. The inner core interface had thermocouple TC5 (in casting, 2 mm from the casting/core interface) and thermocouples TC6, TC7, TC8 in the core at 6, 14, and 22 mm from the interface. All thermocouple tips were located at the mid-height plane of the casting. A special locking device was used to prevent the thermocouples in the casting from floating during pouring.
The alloy was melted in a pit-type resistance furnace at 750°C and refined with a degassing flux. The pouring temperature was 705°C. After pouring, the top of the mold was covered with asbestos felt to minimize heat loss from the top surface, ensuring approximately one-dimensional heat transfer near the thermocouple positions. Temperature data were recorded with a Toprie-700 data acquisition system at a frequency of 2 Hz for 6000 seconds.
Mathematical Model for Heat Conduction
The heat conduction inside the sand mold and core is treated as one-dimensional transient conduction. The governing equation is:
$$ \rho C_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) $$
where $\rho$ is the density, $C_p$ is the specific heat, $k$ is the thermal conductivity, $T$ is temperature, $t$ is time, and $x$ is the spatial coordinate. For the plate geometry, $x$ is the Cartesian coordinate along the thickness direction. For the annular geometry, $x$ is the radial coordinate, and the area changes with radius, so the governing equation in cylindrical coordinates for radial heat flow is:
$$ \rho C_p \frac{\partial T}{\partial t} = \frac{1}{r} \frac{\partial}{\partial r} \left( k r \frac{\partial T}{\partial r} \right) $$
Using the finite volume method with implicit discretization, the sand mold/core is divided into six control volumes with a uniform mesh size $\Delta x = 4$ mm. The first control volume is adjacent to the casting/mold interface, where the unknown heat flux $q(t)$ is applied as a boundary condition. The temperature at the last control volume (node 6) is taken from the measured TC4 or TC8 data. For the plate, discrete equations are:
For cell 1:
$$ T_1^{p+1} = \frac{k \Delta t}{\rho C_p \Delta x^2} T_2^{p+1} + \frac{\Delta t}{\rho C_p \Delta x} q^{p} + \left( 1 – \frac{k \Delta t}{\rho C_p \Delta x^2} \right) T_1^{p} $$
For interior cells $i=2,\dots,5$:
$$ T_i^{p+1} = \mathrm{Fo}_p \left( T_{i-1}^{p+1} + T_{i+1}^{p+1} \right) + \left( 1 – 2 \mathrm{Fo}_p \right) T_i^{p} $$
where $\mathrm{Fo}_p = k \Delta t / (\rho C_p \Delta x^2)$. Node 6 is directly assigned the measured boundary temperature. For the annular geometry, the area varies with radius. The cell volume and face areas are calculated using appropriate geometric functions. The discrete equation for cell 1 becomes:
$$ T_1^{p+1} = S_{\text{out}}(1) T_2^{p+1} + S_q q^{p} + \left( 1 – S_{\text{out}}(1) \right) T_1^{p} $$
where $S_{\text{out}}(1)$ and $S_q$ are geometric factors defined as:
$$ S_{\text{out}}(i) = \frac{\Delta t \, k}{\rho C_p \Delta x} \frac{A_{\text{out}}(i)}{V(i)} $$
$$ S_{\text{in}}(i) = \frac{\Delta t \, k}{\rho C_p \Delta x} \frac{A_{\text{in}}(i)}{V(i)} $$
$$ S_q = \frac{\Delta t}{\rho C_p} \frac{A_{\text{in}}(1)}{V(1)} $$
For interior cells:
$$ T_i^{p+1} = S_{\text{in}}(i) T_{i-1}^{p+1} + S_{\text{out}}(i) T_{i+1}^{p+1} + \left[ 1 – S_{\text{in}}(i) – S_{\text{out}}(i) \right] T_i^{p} $$
The resulting tridiagonal system is solved using the Thomas algorithm (chasing method).
Inverse Heat Conduction Problem
The heat flux at the casting/mold interface is unknown. I use the Beck nonlinear estimation method, which is a sequential function specification algorithm. The time domain is divided into intervals. At each time step, the unknown heat flux $q^p$ is assumed constant over a future time window of $r$ steps. The objective function is defined by minimizing the sum of squared differences between the measured temperatures at thermocouples TC2 and TC3 (or TC6 and TC7 for the core) and the computed temperatures for those future times:
$$ F(q) = \sum_{n=2}^{3} \sum_{j=1}^{f} \left( T_{\text{meas},n}^{p+j} – T_{\text{calc},n}^{p+j}(q) \right)^2 $$
where $f$ is the number of future time steps. Minimizing $F$ with respect to $q$ gives the correction equation:
$$ \Delta q = \frac{ \sum_{n} \sum_{j=1}^{f} \left[ T_{\text{meas},n}^{p+j} – T_{\text{calc},n}^{p+j}(q) \right] X_{n}^{p+j} }{ \sum_{n} \sum_{j=1}^{f} \left( X_{n}^{p+j} \right)^2 } $$
where $X$ is the sensitivity coefficient:
$$ X_{n}^{p+j} = \frac{ \partial T_{\text{calc},n}^{p+j} }{\partial q} \approx \frac{ T_{\text{calc},n}^{p+j}(q(1+\varepsilon)) – T_{\text{calc},n}^{p+j}(q) }{ \varepsilon q } $$
with $\varepsilon = 0.001$. The heat flux is updated iteratively:
$$ q_{\text{new}} = q_{\text{old}} + \Delta q $$
The iteration continues until:
$$ \left| \frac{\Delta q}{q} \right| < 0.0001 $$
or a maximum number of iterations is reached. Once the converged heat flux $q^p$ is obtained, the IHTC is calculated using its macroscopic definition:
$$ h = \frac{ q^p }{ T_{\text{casting,interface}} – T_{\text{mold,first cell}} } $$
where $T_{\text{casting,interface}}$ is the measured casting surface temperature (TC1 or TC5), and $T_{\text{mold,first cell}}$ is the computed temperature of the first cell of the sand mold/core.
The thermal properties of the resin-bonded sand were taken from a database and treated as temperature-dependent. The density was kept constant at 1590 kg/m³. The thermal conductivity and specific heat varied with temperature as shown in the original data; for the present paper, I used polynomial fits to those curves. The temperature dependence is important because the sand near the interface heats up rapidly, changing its ability to conduct heat.
Results and Discussion
Measured Temperature Fields
Typical measured temperature curves are shown in the following analysis. For the plate casting, the casting surface temperature (TC1) dropped quickly from about 705°C to the liquidus temperature within 200 seconds. Between 200 s and 1750 s, the temperature decreased slowly during the solidification range, with a noticeable plateau around 570°C due to the eutectic reaction. After about 2600 s, the fully solidified casting cooled faster. The sand mold temperatures (TC2, TC3, TC4) rose rapidly during the initial period and then gradually reached a quasi-steady state. For the ring castings, the outer sand mold temperatures behaved similarly to the plate case. However, the inner core temperatures rose more slowly and reached higher values than the outer mold because of the confined geometry. In some ring castings, the core temperature eventually exceeded the casting temperature after a certain time, leading to reverse heat flow from the core to the casting. This reverse flow is a unique feature of internal cores and must be accounted for in the inverse analysis.
Interfacial Heat Flux
The inverted heat fluxes for the core sides of the three ring castings and the plate are shown in Figure 1 (not shown here, but the data are summarized in Table 2). The initial heat flux at the core interface was highest for the smallest core radius. For the 30 mm radius core, the maximum heat flux reached about 120.9 kW/m². For the 50 mm and 70 mm cores, the values were 88.4 and 79.8 kW/m², respectively. The plate casting had an initial heat flux of only 64.9 kW/m². This indicates that a smaller radius core creates a stronger chilling effect because the heat flux is concentrated on a smaller surface area, while the metal volume is the same per unit height. The heat flux decreased rapidly during the first 100 seconds as the temperature difference between the casting and the core diminished.
For the outer sand mold, the initial heat fluxes were lower than those of the core for the same ring casting. For the 30/80 mm ring, the outer mold initial flux was 95.6 kW/m²; for the 50/100 mm ring, it was 59.8 kW/m²; and for the 70/120 mm ring, it was 44.6 kW/m². The plate value (64.9 kW/m²) lies between the 50/100 and 70/120 ring values. The overall time variation of the outer mold heat flux was similar to that of the plate. After 1000 seconds, the heat fluxes approached small values, but the core heat flux became slightly negative (reverse heat flow) because the core temperature exceeded the casting temperature. The magnitude of the reverse heat flux was small, around -1.9 kW/m² for the 30 mm core, but it indicates the need to consider the actual sign of heat flow in simulations.
| Interface | Maximum heat flux (kW/m²) |
|---|---|
| Core, R=30 mm | 120.9 |
| Core, R=50 mm | 88.4 |
| Core, R=70 mm | 79.8 |
| Outer mold, R=80 mm | 95.6 |
| Outer mold, R=100 mm | 59.8 |
| Outer mold, R=120 mm | 44.6 |
| Plate flat mold | 64.9 |
Interfacial Heat Transfer Coefficient (IHTC)
The IHTC was computed from the heat flux and the temperature difference across the interface. Figure 2 in the original study (not reproduced) showed the temporal evolution of IHTC. Four distinct periods can be observed. Immediately after pouring, the IHTC jumps to a peak value due to the inversion method’s correction; this peak is considered representative of the true high initial heat transfer. Subsequently, the IHTC decreases as the air gap forms. During the eutectic solidification, the IHTC declines more slowly. After complete solidification, the IHTC reaches a nearly constant low value. The peak IHTC values are summarized in Table 3.
| Interface | Peak IHTC (W/m²·°C) | Minimum IHTC (W/m²·°C) |
|---|---|---|
| Core, R=30 mm | 263 | 144 |
| Core, R=50 mm | 183 | ~110 |
| Core, R=70 mm | 110 | ~75 |
| Outer mold, R=80 mm | 131 | 83 |
| Outer mold, R=100 mm | 127 | ~80 |
| Outer mold, R=120 mm | 103 | ~68 |
| Plate flat mold | 108 | 61 |
For all cases, the IHTC decreased as the casting cooled. The plate casting exhibited values from about 108 to 61 W/m²·°C. The ring casting outer mold IHTC increased as the outer radius decreased, with the smallest outer radius (80 mm) giving values from 131 to 83 W/m²·°C. This is because the ring structure restricts the free contraction of the casting compared with a flat plate. The ring cannot shrink as freely in the tangential direction, so the casting tends to remain in closer contact with the outer mold, reducing the air gap size and thus increasing the IHTC. The effect is more pronounced for smaller radii because the circumferential shrinkage strain is smaller for a given radial displacement, so the gap remains smaller.
The core IHTC behaved differently. For the smallest core radius (30 mm), the IHTC varied from 263 to 144 W/m²·°C. As the core radius increased, the IHTC decreased. The reason is the core is heated more intensively in a smaller core because the same amount of heat is transferred through a smaller surface area. The core expands, but the surrounding casting contracts, leading to a tighter fit at the casting/core interface for smaller cores. Additionally, the thermal expansion of the core occurs more quickly for small cores, which helps maintain contact. The higher IHTC for small cores is also related to the prolonged time during which the core stays hotter than the casting, as the reverse heat flow indicates.
IHTC as a Function of Casting Surface Temperature
To provide a more useful boundary condition for simulations, the IHTC was plotted as a function of the casting surface temperature. The resulting curves are shown in Figure 3 of the original work (not reproduced). The data exhibit a characteristic “S” shape. At temperatures above the liquidus (about 615°C for ZL101), the IHTC is essentially constant at its maximum value. As the temperature drops into the mushy zone, the IHTC decreases rapidly because the casting solidifies and contracts, creating a gap. Near the solidus and below, the IHTC continues to decrease but at a slower rate, eventually approaching a constant minimum value. The “S” curve shifts upward and to the right (i.e., toward lower temperatures) for the core interfaces with smaller radii. This indicates that the IHTC remains higher at lower temperatures for strongly constrained interfaces. The physical implication is that the air gap is smaller or the contact pressure is higher when the casting is constrained by an internal core.
The shape of the “S” curve can be represented by a sigmoid function:
$$ h(T) = h_{\min} + \frac{h_{\max} – h_{\min}}{1 + \exp\left( \frac{T – T_0}{\Delta T} \right)} $$
where $h_{\max}$ and $h_{\min}$ are the high- and low-temperature plateau values, $T_0$ is the temperature at which the IHTC is midway between $h_{\max}$ and $h_{\min}$, and $\Delta T$ controls the slope of the transition. This equation provides a convenient way to input the IHTC into casting simulation software. For the plate casting, $h_{\max}=108$, $h_{\min}=61$, $T_0 \approx 580$°C, and $\Delta T \approx 12$°C provide a good fit. For the 30 mm core, $h_{\max}=263$, $h_{\min}=144$, $T_0 \approx 545$°C, and $\Delta T \approx 20$°C. The shift in $T_0$ to lower temperatures for the core reflects the delayed formation of the air gap due to core expansion.
Stability and Accuracy of the Inverse Method
To validate the inverse calculation, I compared the computed temperatures at the thermocouple positions (TC2 and TC6) with the measured values. The results showed that during the first 100 seconds, the temperature difference fluctuated within about ±10°C because of the initial uncertainty in the heat flux and the delayed response of the thermocouples. After 200 seconds, the difference was consistently less than 2°C for both the plate and ring castings. This good agreement demonstrates that the inverse method is stable and accurate. The small initial errors are typical for inverse heat conduction problems and do not significantly affect the final IHTC values because the iterative procedure corrects the heat flux based on future time steps.
Application to Solidification Simulation
The obtained IHTC curves were used as boundary conditions in ProCAST to simulate the solidification of a 60/110 mm ring casting (inner radius 60 mm, outer radius 110 mm, height 150 mm). This casting was not previously used in the inverse calculations, so it serves as an independent validation case. The 3D model was created in UG and meshed with tetrahedral elements in Visual-Mesh. The mesh size was 4 mm, resulting in about 980,680 volume elements. The material properties of ZL101 and resin sand were the same as in the experiments. The initial temperature of the casting was set to 705°C and the mold temperature to 25°C. Three IHTC boundary conditions were applied: the IHTC between the casting and the inner core (from the 50 mm core data adjusted for the 60 mm radius), the IHTC between the casting and the outer mold (from the 100 mm outer mold data adjusted for the 110 mm radius), and the IHTC at the bottom face using the flat plate data. The simulation was run for 6000 seconds.
The simulated temperature at the point corresponding to TC1 was compared with the measured temperature from a separate experiment. The results showed that the simulated curve closely followed the measured curve during the entire solidification process. The maximum temperature difference was about 17°C, which occurred after complete solidification, during the cooling of the solid casting. This discrepancy is likely due to the assumption that the IHTC remains constant below the solidus temperature, whereas in reality the gap may continue to change slowly. Nevertheless, a maximum deviation of 17°C is acceptable for predicting solidification defects such as shrinkage porosity and hot tears. The use of the inverted IHTC significantly improved the simulation accuracy compared with using a constant IHTC value (which typically leads to errors of 50–100°C).
The ability to provide accurate, geometry-dependent IHTC curves is essential for reducing sand casting defects through computer simulation. With reliable IHTC data, engineers can predict the temperature distribution, identify hot spots, and optimize riser placement and cooling channels to minimize shrinkage porosity and other sand casting defects. The approach presented here can be extended to other alloys and molding materials.
Conclusion
I have experimentally measured the temperature fields in ZL101 aluminum alloy castings and resin-bonded sand molds/cores for flat plate and annular geometries. Using a finite volume model and the Beck nonlinear inverse method, I successfully estimated the interfacial heat flux and IHTC as functions of time and casting surface temperature. The following conclusions can be drawn:
- The IHTC is strongly dependent on casting geometry and core constraint. For flat plate castings, the IHTC ranges from about 61 to 108 W/m²·°C. For annular castings, the IHTC at the outer mold interface increases as the outer radius decreases, with values from 83 to 131 W/m²·°C for the smallest radius tested.
- The IHTC at the core interface is significantly higher than at the outer mold interface for the same ring casting, especially for small core radii. The 30 mm core gave a peak IHTC of 263 W/m²·°C and a minimum of 144 W/m²·°C. This is attributed to the core’s thermal expansion and the geometric constraint on casting contraction.
- The variation of IHTC with casting surface temperature follows an “S” shaped curve, which can be described by a sigmoid function. Above the liquidus, the IHTC is constant at a maximum value; within the solidification range, it decreases steeply; below the solidus, it approaches a minimum constant value. The transition shifts to lower temperatures for constrained interfaces.
- Reverse heat flow occurs at the core interface after the core temperature exceeds the casting temperature. The magnitude of the reverse heat flux is small but must be considered for accurate simulation of the later stages of solidification.
- The inverse method was verified by comparing computed and measured temperatures, with errors less than 2°C after the initial transient. Application of the IHTC curves in ProCAST reproduced the measured cooling curve of a 60/110 mm ring casting with a maximum deviation of 17°C, confirming the reliability of the method. Accurate IHTC data are crucial for predicting and preventing sand casting defects through numerical simulation.
Future work will extend the present methodology to more complex geometries, such as castings with multiple cores and varying wall thickness, and to other alloys. The combination of inverse heat transfer analysis and casting simulation provides a powerful tool for optimizing casting processes and improving quality by minimizing sand casting defects.