In my research, I focused on one of the most critical boundary parameters for the numerical simulation of sand casting parts: the interfacial heat transfer coefficient (IHTC) between the casting and the sand mold or sand core. The IHTC is not a simple material property; it evolves with time, local temperature, contact geometry, and the state of solidification. During the cooling of sand casting parts, the metal volume shrinks while the mold and core expand under heating. This creates a dynamic interface gap that controls the heat flow. Without accurate IHTC data, even the most advanced solidification software cannot reliably predict temperature fields, shrinkage porosity, hot tears, or microstructural evolution. Therefore, I designed an experimental and inverse-calculation framework to obtain the IHTC for typical plate-shaped and ring-shaped sand casting parts.
My main objective was to measure the transient temperature fields inside the casting, the sand mold, and the sand core, and then to use a Beck-type nonlinear inverse method to estimate the interfacial heat flux and the IHTC. The study includes a flat plate casting with a thickness of 50 mm and two groups of ring-shaped sand casting parts with different inner and outer radii. All castings were made of ZL101 aluminum alloy and poured at 705°C. The mold material was furan resin sand. I placed thermocouples at known distances from the interface and used the measured temperatures as input for inverse heat conduction calculations. The results show that the IHTC varies strongly with casting surface temperature, especially between the liquidus and solidus of the alloy.
1. Experimental Procedures
I carried out four pouring experiments. The first was a plate casting with dimensions 150 mm × 150 mm and a thickness of 50 mm. The other three were ring-shaped castings with a constant height of 150 mm and a constant wall thickness of 50 mm. Their inner and outer radii are listed in Table 1. This design allowed me to compare the effect of curvature and internal sand core constraint on the IHTC.
| Casting | Inner radius (mm) | Outer radius (mm) | Wall thickness (mm) | Height (mm) |
|---|---|---|---|---|
| Plate | — | — | 50 | 150 |
| Ring A | 30 | 80 | 50 | 150 |
| Ring B | 50 | 100 | 50 | 150 |
| Ring C | 70 | 120 | 50 | 150 |
For the plate casting, I installed eight thermocouples. The casting interface thermocouple was located 2 mm from the casting/mold interface. In the sand mold, thermocouples were placed 6 mm, 14 mm, and 22 mm from the interface on both sides. For the ring castings, I installed thermocouples on both the inner sand core side and the outer sand mold side. On the outer side, the casting-side thermocouple was 2 mm from the interface, and the mold-side thermocouples were at 6 mm, 14 mm, and 22 mm. On the core side, the same arrangement was used, as summarized in Table 2.
| Thermocouple | Position | Distance from interface (mm) |
|---|---|---|
| TC1 | Casting surface, outer sand mold side | 2 |
| TC2 | Sand mold, first internal point | 6 |
| TC3 | Sand mold, second internal point | 14 |
| TC4 | Sand mold, third internal point | 22 |
| TC5 | Casting surface, inner sand core side | 2 |
| TC6 | Sand core, first internal point | 6 |
| TC7 | Sand core, second internal point | 14 |
| TC8 | Sand core, third internal point | 22 |
The thermocouples were K-type with 100 mm long double-bore alumina protection tubes near the hot junction. I fixed the casting-side thermocouples with a mechanical locking device to prevent them from floating in the molten metal. A Toprie-700 data recorder collected all temperature data at a frequency of 2 Hz. The pouring temperature was 705°C and the initial mold temperature was about 25°C. After pouring, the top of the mold was covered with asbestos felt to reduce heat losses and to make the heat flow approximately one-dimensional through the side walls and the core. This is important because my inverse model assumes one-dimensional heat conduction.
2. Temperature Field Model
The heat transfer inside the sand mold and the sand core can be treated as one-dimensional transient conduction. For the plate casting, I used Cartesian coordinates along the thickness direction. For the ring-shaped sand casting parts, I used radial coordinates with sector-shaped control volumes to account for the change of heat-transfer area with radius. The general governing equation is:
$$ \rho(T)\,C_p(T)\,\frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k(T)\,\frac{\partial T}{\partial x}\right) + \dot{q}_v $$
where \(\rho\) is density, \(C_p\) is specific heat, \(k\) is thermal conductivity, and \(\dot{q}_v\) is the volumetric heat source. Inside the mold and core, there is no internal heat source, so \(\dot{q}_v = 0\). The initial condition is:
$$ T(x,0) = T_{ini}(x) $$
The boundary condition at the interface is the unknown heat flux. I treated the heat flux as positive when heat flowed from the casting into the mold or core:
$$ -k\left.\frac{\partial T}{\partial x}\right|_{x=0} = q(t) $$
The far boundary condition was taken from the measured thermocouple temperature at 22 mm from the interface. In the numerical calculation, the sand mold or core was divided into six equal control volumes, each 4 mm thick.
2.1 Finite Volume Discretization for Plate Casting
For the plate casting, I used an implicit finite volume scheme. The Fourier number was defined as:
$$ Fo^p = \frac{k^p\, \Delta t}{\rho^p\, C_p^p\, \Delta x^2} $$
For the first control volume adjacent to the interface, the energy balance gives:
$$ T_1^{p+1} = \left(1 – Fo^p\right) T_1^p + Fo^p\, T_2^p + Fo^p\, \frac{\Delta x}{k^p}\, q^p $$
For internal control volumes \(i = 2\) to \(5\), the discretized equation is:
$$ -Fo^p\, T_{i-1}^{p+1} + \left(1 + 2 Fo^p\right) T_i^{p+1} – Fo^p\, T_{i+1}^{p+1} = T_i^p $$
The last control volume was set equal to the measured boundary temperature:
$$ T_6^{p+1} = T_{TC4}^{p+1} $$
I solved this tridiagonal system with the Thomas algorithm.
2.2 Finite Volume Model for Ring-Shaped Castings
For the ring-shaped sand casting parts, the cross-sectional area changes with radius. I therefore used sector-shaped control volumes. The ratio of the inner area to volume and the outer area to volume becomes a function of the element position. I introduced the following geometric coefficients:
$$ S_{in}(i) = \frac{\Delta t\, k}{\rho\, C_p\, \Delta x}\, \frac{A_{in}(i)}{V(i)} $$
$$ S_{out}(i) = \frac{\Delta t\, k}{\rho\, C_p\, \Delta x}\, \frac{A_{out}(i)}{V(i)} $$
where \(A_{in}(i)\) and \(A_{out}(i)\) are the inner and outer heat-transfer areas of element \(i\), and \(V(i)\) is its volume. The coefficient for the boundary heat flux at the first element is:
$$ S_q = \frac{\Delta t}{\rho\, C_p}\, \frac{A_{in}(1)}{V(1)}\, q $$
With these coefficients, the discretized temperature equations for the ring-shaped mold/core are:
$$ \left[1 + S_{out}(1)\right] T_1^{p+1} – S_{out}(1)\, T_2^{p+1} = T_1^p + S_q $$
for the first control volume,
$$ – S_{out}(i)\, T_{i-1}^{p+1} + \left[1 + S_{in}(i) + S_{out}(i)\right] T_i^{p+1} – S_{in}(i)\, T_{i+1}^{p+1} = T_i^p $$
for internal control volumes, and the boundary element was again set to the measured temperature. This model works for both the outer sand mold and the inner sand core, but the geometric factors are different because heat flows outward in the mold and inward in the core.
3. Inverse Heat Conduction Model
The interfacial heat flux was unknown and had to be estimated from measured temperatures at 6 mm and 14 mm from the interface. I used the Beck nonlinear estimation method with future time steps. The heat flux was treated as piecewise constant in each time interval. To reduce the ill-posed nature of the inverse problem, I assumed that the heat flux remained constant for several future time steps. In my calculations, I used six future time steps.
The objective function for the inverse problem was:
$$ F(q_p) = \sum_{n=2}^{3} \sum_{j=p}^{p+f-1} \left[T_{m,n}^j(q_p) – T_{c,n}^j\right]^2 $$
where \(T_{c,n}^j\) is the measured temperature at thermocouple position \(n\) and time \(j\), and \(T_{m,n}^j(q_p)\) is the calculated temperature using the assumed heat flux \(q_p\). The summation over \(n=2\) and \(3\) corresponds to the thermocouples at 6 mm and 14 mm from the interface.
The sensitivity coefficient was calculated by perturbing the heat flux by a small factor \(\epsilon\):
$$ X_{m,n}^j = \frac{T_{m,n}^j\left[(1+\epsilon)q_p\right] – T_{m,n}^j(q_p)}{\epsilon\, q_p} $$
I used \(\epsilon = 0.001\). The correction to the heat flux was then computed as:
$$ \Delta q_p = \frac{\sum_{n=2}^{3}\sum_{j=p}^{p+f-1} \left[T_{c,n}^j – T_{m,n}^j(q_p)\right] X_{m,n}^j}{\sum_{n=2}^{3}\sum_{j=p}^{p+f-1} \left(X_{m,n}^j\right)^2} $$
The updated heat flux became:
$$ q_p^{corr} = q_p + \Delta q_p $$
I iterated this procedure until the relative correction satisfied:
$$ \left|\frac{\Delta q_p}{q_p^{corr}}\right| < \varepsilon $$
After convergence, I calculated the IHTC from the macroscopic definition:
$$ h^p = \frac{q_p^{corr}}{T_{cast,1}^p – T_{mold,1}^p} $$
where \(T_{cast,1}^p\) is the measured casting surface temperature and \(T_{mold,1}^p\) is the calculated first-cell temperature of the sand mold or core. This calculation was repeated at each time step for the entire 6000 s of cooling.
4. Measured Temperature Fields
The measured temperature curves showed three distinct regions for all sand casting parts in my experiments. In the first region, the liquid metal was above the liquidus and cooled rapidly because of the large temperature difference between the melt and the mold. In the second region, the metal released latent heat and the cooling curve flattened. In the third region, below the solidus, the solid casting cooled more quickly again. For the ring-shaped sand casting parts, the inner sand core heated up more slowly than the outer sand mold, and in some cases the core temperature reached a plateau close to the casting temperature. This is important for understanding the reverse heat flow observed later.
For the internal sand core, especially in the ring with a 30 mm inner radius, the core temperature eventually became higher than the casting surface temperature. This caused a negative interfacial heat flux. The larger ring castings delayed this temperature crossing because the larger core volume had a higher thermal capacity. In contrast, the outer sand mold never became hotter than the casting in any of my experiments, so the interfacial heat flux at the outer mold remained positive for the entire cooling process.
5. Estimated Interfacial Heat Flux
Table 3 summarizes the maximum initial heat flux values for each interface. I removed the very early values caused by filling turbulence and thermocouple lag, and used the first converged peak as the initial heat flux.
| Geometry | Interface | Initial heat flux (kW/m²) |
|---|---|---|
| Plate | Sand mold | 64.9 |
| Ring 30/80 | Sand core | 120.9 |
| Ring 30/80 | Sand mold | 95.6 |
| Ring 50/100 | Sand core | 88.4 |
| Ring 50/100 | Sand mold | 59.8 |
| Ring 70/120 | Sand core | 79.8 |
| Ring 70/120 | Sand mold | 44.6 |
I found that for the ring-shaped sand casting parts, the initial heat flux decreased as the inner core radius increased. The strongest heat extraction occurred at the 30 mm inner sand core, where the curvature was greatest and the casting contraction was strongly constrained by the core. The outer sand mold showed a similar trend with the outer radius: the smaller the outer radius, the larger the initial heat flux. The plate casting had an initial heat flux between those of the 100 mm and 120 mm outer molds, which shows that the curvature and geometric constraint are both significant factors.
During the later stages, the interfacial heat flux at the sand cores became negative when the core temperature exceeded the casting temperature. The negative heat flux was small compared with the positive peak, but it indicates a reversal of heat flow. For the 30 mm inner core, the reverse heat flux appeared at about 2100 s and reached about \(-1.9\) kW/m². For larger cores, this reversal was delayed and weaker. The outer mold interfaces never showed negative heat flux because their temperature remained lower than the casting temperature at all times.
6. Interfacial Heat Transfer Coefficient Results
The estimated IHTC values followed the expected physical behavior. At the beginning of solidification, the metallic liquid was in good contact with the sand mold and core, so the IHTC was high. As solidification progressed, the casting shrank away from the mold and the core expanded, forming an air gap that reduced the heat transfer. The IHTC therefore dropped from a maximum at high temperature to a minimum at low temperature.
Table 4 lists the maximum and minimum IHTC values that I extracted from the stable high-temperature and low-temperature regions of the curves. The minima for the largest rings and the outermost molds were not explicitly reported in my analysis, but their trends were consistent with the values shown.
| Geometry | Interface | Maximum IHTC (W/m²°C) | Minimum IHTC (W/m²°C) |
|---|---|---|---|
| Plate | Sand mold | 108 | 61 |
| Ring 30/80 | Sand core | 263 | 144 |
| Ring 30/80 | Sand mold | 131 | 83 |
| Ring 50/100 | Sand core | 183 | — |
| Ring 50/100 | Sand mold | 127 | — |
| Ring 70/120 | Sand core | 110 | — |
| Ring 70/120 | Sand mold | 103 | — |
The effect of geometry was very clear. For the internal sand core, the IHTC increased significantly as the inner radius decreased. The smallest core, with an inner radius of 30 mm, produced a maximum IHTC of 263 W/m²°C. The reason is that a smaller core has a more convex surface and creates a stronger geometrical constraint on the solidifying casting. The casting cannot shrink away from the core as freely as it can from a flat plate, so the interfacial contact remains better and the heat transfer is more intense. The same trend was observed on the outer mold side, but the effect was weaker. The outer mold interface behaved more like a flat interface because the heat flow is outward and the shrinkage acts to separate the casting from the mold.
When plotted against casting surface temperature, the IHTC showed an approximate “S”-shaped curve. The IHTC was almost constant above the liquidus and below the solidus. In the mushy zone between the liquidus and solidus, the IHTC decreased rapidly with temperature. For the smaller sand cores, the “S” curve shifted upward and extended toward the solidus. This happened because the casting contraction was blocked by the core and the core remained hot for a longer time, preserving better contact.
7. Inverse Calculation Stability and Accuracy
To verify my inverse model, I compared the measured temperatures at the thermocouple positions with the calculated temperatures obtained from the inverse heat flux. The differences were largest during the first 100 seconds because of the initial filling disturbance, thermocouple thermal lag, and the low thermal diffusivity of the resin sand. After about 200 seconds, the inverse calculation became stable and the temperature differences were consistently below 2°C.
| Casting | Thermocouple position | Maximum absolute difference before 100 s(°C) | Maximum absolute difference after 200 s(°C) |
|---|---|---|---|
| Plate | TC2 in sand mold | 8.6 | < 2 |
| Ring 50/100 | TC2 in sand mold | 17 | < 2 |
| Ring 50/100 | TC6 in sand core | 12 | < 2 |
These results show that the inverse program converged quickly and that the calculated temperature field matched the measured field closely in the long-time cooling stage. This indirectly confirms that the estimated interfacial heat flux and IHTC are reliable for the sand casting parts examined in this study.
8. ProCAST Simulation Validation
I further validated the inverse IHTC data by simulating the solidification of a ring-shaped casting with an inner radius of 60 mm and an outer radius of 110 mm. This geometry was not used in the inverse calculation, so it provides an independent check. I created the three-dimensional model in UG and imported it into ProCAST. The finite-element mesh contained 980,680 elements. The casting material was ZL101 aluminum alloy and the mold/core material was furan resin sand. The pouring temperature was 705°C and the initial mold temperature was 25°C. The time step was 0.5 s. The boundary conditions used the inverse-calculated IHTC as a function of casting surface temperature.
| Parameter | Value |
|---|---|
| Geometry | Ring, inner radius 60 mm, outer radius 110 mm |
| Mesh elements | 980 680 |
| Casting material | ZL101 |
| Mold/core material | Furan resin sand |
| Pouring temperature | 705°C |
| Initial mold temperature | 25°C |
| Interface condition | Inverse-calculated IHTC vs temperature |
| Time step | 0.5 s |
| Termination temperature | 300°C |
I selected a node in the middle of the ring wall corresponding to the casting-side thermocouple TC1. The simulated temperature was compared with the measured temperature from the 60/110 mm casting. The results showed excellent agreement during the solidification stage. After complete solidification, the simulated temperature was slightly lower than the measured temperature, but the maximum difference was only about 17°C. This small deviation is likely caused by the assumption that the interface gap remains constant below the solidus, whereas in reality the gap continues to change slightly during cooling. For the purpose of predicting solidification defects, this error is acceptable.
The ProCAST simulation therefore confirms the reliability and accuracy of my inverse methodology. It also demonstrates that using the inverse IHTC data can significantly improve the simulation accuracy for sand casting parts.
9. Conclusions
In this research, I determined the interfacial heat transfer coefficients for sand casting parts with different geometries by combining experimental temperature measurements, finite volume heat conduction modeling, and the Beck nonlinear inverse algorithm. The main conclusions are:
First, the IHTC of the investigated ZL101 alloy sand casting parts is not constant. It reaches a maximum shortly after pouring, remains nearly constant at temperatures above the liquidus, decreases rapidly in the mushy zone, and becomes nearly constant again below the solidus. The overall shape of the IHTC versus temperature curve can be described as an “S” curve.
Second, the IHTC on the internal sand core side is much higher than that on the external sand mold side for the same ring-shaped casting. This is caused by the different geometric constraint and by the opposite directions of casting shrinkage and core expansion. The smallest core with a 30 mm inner radius produced the highest IHTC, with a maximum of 263 W/m²°C and a minimum of 144 W/m²°C. For the plate casting, the maximum and minimum IHTC values were 108 W/m²°C and 61 W/m²°C.
Third, the geometry of sand casting parts strongly influences the interfacial heat transfer. As the inner radius of the core decreases, the core constrains the casting contraction more strongly, leading to better interfacial contact and higher heat transfer. As the outer radius increases, the mold side behavior approaches that of a flat plate.
Finally, the inverse procedure was verified by comparing measured and calculated temperatures, and by simulating the solidification of a ring casting in ProCAST. The maximum temperature difference between simulation and measurement was 17°C, which proves that the inverse-calculated IHTC data can be used as a reliable boundary condition for the numerical simulation of sand casting parts.
Overall, this work provides a practical methodology for obtaining geometry-dependent IHTC data for sand casting parts, and it highlights the importance of using accurate boundary conditions in casting simulation.