In modern foundry engineering, the continuous pursuit of higher quality and more reliable performance in sand casting parts has led to increasingly complex mold materials, alloy compositions, and auxiliary external fields such as forced air cooling, water cooling, centrifugal forces, negative pressure, and ultrasonic vibration. These developments make the solidification and cooling processes of castings significantly more difficult to predict. Numerical simulation has become an indispensable tool for optimizing casting processes, yet the accuracy of simulation results is strongly dependent on the availability of reliable physical parameters, especially the interfacial heat transfer coefficient (IHTC) between the casting and the mold. The IHTC directly governs the heat dissipation efficiency from the molten metal to the mold and therefore controls the cooling rate, temperature distribution, defect formation, residual stress, and final mechanical properties of sand casting parts. However, comprehensive studies on the IHTC under combined conditions such as forced air cooling remain limited. In this context, the present work systematically investigates the heat transfer laws of annular sand casting parts under both natural cooling and forced air cooling conditions, using an optimized Beck inverse algorithm, numerical simulation, and carefully measured temperature fields. The goal is to provide essential data and theoretical guidance for the application of forced cooling technologies in the production of high-quality sand casting parts.
The experiments were designed to cover two typical casting alloys, namely ZL101 aluminum alloy and 40 steel, with two kinds of sand mold materials: furan resin self-hardening sand and sodium silicate water glass self-hardening sand. The castings had annular geometry with different inner diameters, and the sand cores were either solid or internally equipped with steel pipes for forced air cooling. The measured temperature histories at several locations inside the casting and the sand mold were recorded using thermocouples. These data were then used as inputs for an inverse heat conduction problem solver based on the Beck nonlinear estimation method, which enables the time-dependent IHTC to be determined accurately. The following sections describe the experimental methodology, the mathematical formulation of the inverse algorithm, the validation of the calculated temperature fields, and the detailed results concerning the effects of casting alloy, casting size, mold material, and forced air cooling on the IHTC. This work highlights the critical role of the interfacial gap between sand casting parts and the mold, and explains the physical mechanisms behind the observed variation of the IHTC.
Experimental Methodology and Materials
The experimental program included eleven separate casting trials, each with a distinct combination of alloy, mold material, core geometry, and cooling condition. The detailed parameters are listed in Table 1. The annular casting geometry was characterized by its inner diameter \(r\), which was varied among 60 mm, 100 mm, and 140 mm for aluminum alloy castings. For steel castings, the inner diameter was fixed at 100 mm, while the diameter of the internal air-cooling pipe was varied as 40 mm, 60 mm, and 80 mm. The thermocouple layout inside the sand mold is shown in the original experimental setup. The casting temperature was measured by a thermocouple placed at 2 mm from the mold cavity surface. Three additional thermocouples were placed inside the sand mold at distances of 6 mm, 14 mm, and 22 mm from the mold surface, respectively, to record the transient temperature evolution in the mold.
| No. | Casting alloy | Sand mold material | Casting inner diameter r (mm) | Air-cooling pipe diameter (mm) |
|---|---|---|---|---|
| 1 | ZL101 | Furan resin self-hardening sand | 60 | None |
| 2 | ZL101 | Furan resin self-hardening sand | 100 | None |
| 3 | ZL101 | Furan resin self-hardening sand | 140 | None |
| 4 | ZL101 | Furan resin self-hardening sand | 100 | 60 |
| 5 | ZL101 | Sodium silicate self-hardening sand | 60 | None |
| 6 | ZL101 | Sodium silicate self-hardening sand | 100 | None |
| 7 | ZL101 | Sodium silicate self-hardening sand | 140 | None |
| 8 | 40 steel | Furan resin self-hardening sand | 100 | None |
| 9 | 40 steel | Furan resin self-hardening sand | 100 | 40 |
| 10 | 40 steel | Furan resin self-hardening sand | 100 | 60 |
| 11 | 40 steel | Furan resin self-hardening sand | 100 | 80 |
The mold materials were prepared from 50–100 mesh silica sand. For furan resin self-hardening sand, the binder was FFD-121 furan resin (1.2% of sand weight) and the hardener was p-toluenesulfonic acid solution (0.3% of sand weight). For sodium silicate self-hardening sand, the binder was sodium silicate (4% of sand weight) and the hardener was organic ester (0.4% of sand weight). The molds were cured at room temperature for about 24 hours before stripping. A zirconium-based alcohol coating was applied to the mold cavity surfaces to improve surface finish and prevent sand erosion.
The Inverse Heat Conduction Problem and Solution Procedure
To determine the interfacial heat transfer coefficient from measured temperature histories, the present work adopts an optimized version of the Beck inverse algorithm. The heat conduction inside the sand mold is modeled as a one-dimensional transient problem in the radial direction of the annular casting. The governing equation is the classical Fourier heat conduction equation:
$$
\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) = \rho C_p \frac{\partial T}{\partial t}
$$
where \(\rho\) is density, \(C_p\) is the effective specific heat, \(k\) is thermal conductivity, \(T\) is temperature, \(t\) is time, and \(x\) is the spatial coordinate measured from the casting–mold interface. The initial condition is:
$$
T(x,0) = T_{in}(x)
$$
The boundary condition at the casting–mold interface is specified by a heat flux \(q(t)\):
$$
k(T)\left.\frac{\partial T}{\partial x}\right|_{x=0} = q(0,t)
$$
The temperature at the far boundary \(x=x_6\) is known from the measured thermocouple data:
$$
T(x_6,t) = T_{m3}(t)
$$
The sand mold region adjacent to the casting was discretized into six elementary volumes, as schematically described in the original model. The measured temperatures at the second, fourth, and sixth nodes were obtained from thermocouples \(T_{m1}\), \(T_{m2}\), and \(T_{m3}\), respectively. The front-face temperature of the mold and the heat flux were unknown and needed to be estimated. An implicit finite volume method was adopted for the discretization.
For each control volume \(j\), the energy balance can be written as:
$$
\Delta u = A_{in,j} q_{in,j} + A_{out,j} q_{out,j}
$$
where \(\Delta u\) is the change in internal energy during a time step \(\Delta t\), \(q_{in,j}\) and \(q_{out,j}\) are the incoming and outgoing heat fluxes, and \(A_{in,j}\) and \(A_{out,j}\) are the respective areas. The volume of element \(j\) is \(V_j\). For the first element adjacent to the interface, the incoming flux is the unknown interfacial heat flux \(q^{i+1}\), and the outgoing flux is due to conduction to the second element. The discrete equation becomes:
$$
(1+S_{out}(1))T_1^{i+1} – S_{out}(1)T_2^{i+1} = T_1^i + S_{in}(1)\Delta x q^{i+1}
$$
For internal elements (\(j=2\) to \(5\)), the energy balance leads to:
$$
-S_{in}(j)T_{j-1}^{i+1} + [1+S_{in}(j)+S_{out}(j)]T_j^{i+1} – S_{out}(j)T_{j+1}^{i+1} = T_j^i
$$
Here, the dimensionless coefficients \(S_{in}(j)\) and \(S_{out}(j)\) are given by:
$$
S_{in}(j) = \frac{A_{in,j}}{V_j} \cdot \frac{\Delta t k^i}{\rho C_p^i \Delta x}
$$
$$
S_{out}(j) = \frac{A_{out,j}}{V_j} \cdot \frac{\Delta t k^i}{\rho C_p^i \Delta x}
$$
For the annular geometry, the inflow and outflow areas must account for the curvature. The area-to-volume ratios were expressed as:
$$
\frac{A_{in,j}}{V_j} = \frac{r_{use} – j\Delta r + \Delta r}{\Delta r(r_{use} – j\Delta r + 0.5\Delta r)}
$$
$$
\frac{A_{out,j}}{V_j} = \frac{r_{use} – j\Delta r}{\Delta r(r_{use} – j\Delta r + 0.5\Delta r)}
$$
where \(r_{use}\) is the inner diameter of the annular casting and \(\Delta r\) is the thickness of the sand layer represented by each element. These geometric corrections are crucial for accurately predicting heat transfer in annular sand casting parts, since the heat flux area changes with radial position.
The entire set of equations can be assembled into a tridiagonal matrix system, which is efficiently solved by the Thomas algorithm. The unknown heat flux is estimated sequentially in time using the Beck nonlinear estimation method. For a given time step \(m\), the future heat flux is assumed to remain constant over a number of future time steps \(r\):
$$
q_m = q_{m+1} = \cdots = q_{m+r-1} = q_{set}
$$
The sand mold temperature field is then computed with this assumed heat flux, and the calculated temperatures at the sensor locations are compared with the measured values. The objective function is defined as:
$$
F(q_{set}) = \sum_{f=m+1}^{m+r}\sum_{j=1}^{J} \left[ T_j^f – T_{m,j}^f(q_{set}) \right]^2
$$
where \(J\) is the number of sensors used for comparison (here \(J=2\)). Minimizing \(F\) with respect to \(q_{set}\) yields the correction formula:
$$
q^{(k+1)} = q^{(k)} + \frac{\sum_{f=m+1}^{m+r}\sum_{j=1}^{J} \left[ T_j^f – T_{m,j}^f(q^{(k)}) \right] \phi_j^f}{\sum_{f=m+1}^{m+r}\sum_{j=1}^{J} \left( \phi_j^f \right)^2}
$$
where \(\phi_j^f\) is the sensitivity coefficient defined as:
$$
\phi_j^f = \frac{\partial T_j^f}{\partial q} \approx \frac{T_j^f(q_{set}+\varepsilon q_{set}) – T_j^f(q_{set})}{\varepsilon q_{set}}
$$
The iteration continues until the relative change of \(q\) satisfies:
$$
\left| \frac{q^{(k+1)} – q^{(k)}}{q^{(k+1)}} \right| < \varepsilon
$$
Once the converged heat flux \(q_m\) is obtained, the interfacial heat transfer coefficient \(h\) is calculated using the interface temperature difference between the casting outer surface and the mold surface:
$$
h_i = \frac{q_i}{T_{C,i} – T_{1,i}}
$$
where \(T_C\) is the casting temperature measured near the interface and \(T_1\) is the back-calculated mold surface temperature.
Validation of the Inverse Calculation
To verify the reliability of the inverse algorithm, the computed temperature histories at the sensor locations were compared with the experimentally measured values. For the ZL101 casting, the maximum deviation between the measured and calculated temperatures was about 5 °C, and the average deviation was 1.27 °C. The relative deviation, defined as:
$$
\delta = \frac{|T_{calc} – T_{meas}|}{T_{calc}} \times 100\%
$$
was found to be less than 5% for the entire cooling process, with an average value of approximately 1% and a variance of \(1.08 \times 10^{-4}\). For the 40 steel casting, the maximum deviation was 11 °C during the first 5 seconds, but it quickly reduced to about 0.1 °C afterwards, leading to an average error of 0.4 °C. The relative deviation for steel was below 3%, with an average of 0.13% and a variance of \(2.15 \times 10^{-4}\). These small deviations confirm that the one-dimensional inverse model with the geometric correction is adequate for the annular sand casting parts investigated here.
Interfacial Heat Transfer Coefficient without Forced Air Cooling
Behavior of ZL101 Aluminum Alloy
The measured temperature–time curves for the ZL101 casting under natural cooling are shown in Figure 1. The casting temperature \(T_C\) decreases rapidly at the beginning, then exhibits a plateau around 575 °C due to the release of latent heat during eutectic solidification, and finally decreases gradually. The sand mold temperatures first rise and then fall. The back-calculated mold surface temperature \(T_1\) lies between the casting temperature and the first measured mold temperature, showing the same trend as the measured values.
The inverse calculation yields the interfacial heat flux, the temperature difference, and the IHTC as functions of time. For ZL101, the heat flux is initially very high and then drops quickly, followed by a slow decline. The temperature difference follows a similar pattern. The IHTC initially increases with time, exhibits a small drop around 200 s, then rises to a maximum, after which it decreases sharply and finally stabilizes. When plotted against the casting surface temperature, the IHTC follows an “S”-shaped curve. This is shown in Figure 2 for the casting with \(r = 100\) mm. The IHTC increases steadily as the temperature decreases from about 650 °C to 614 °C, then experiences a slight reduction, increases again up to 575 °C, and then drops abruptly. Below 575 °C, the IHTC remains nearly constant or decreases very slowly.
The observed behavior can be explained by the formation and growth of an interfacial air gap between the solidifying alloy and the mold. Above 575 °C, the metal is mostly liquid and remains in good contact with the mold surface. The latent heat released during solidification maintains the interface temperature, while the mold temperature increases, leading to a decreasing temperature difference and an increasing IHTC. The small drop at 614 °C corresponds to the first formation of a solid shell, which creates micro-gaps. However, the thin shell cannot sustain the metallostatic pressure of the remaining liquid, so the gaps do not grow significantly. As the solid fraction increases and approaches the critical solid fraction (CSF) around 575 °C, the liquid feeding between dendrites becomes difficult, and the solid shell becomes thick enough to support the casting weight. As a result, the air gap expands rapidly and the IHTC drops steeply.
Behavior of 40 Steel
For the 40 steel casting, the casting temperature remains above 1450 °C initially and then decreases quickly. The inverse calculation reveals that the IHTC varies with time in a more complex manner than for aluminum. As shown in Figure 3, the IHTC-time curve exhibits a distinct “double-peak” behavior. The initial IHTC is about 449 W/(m²·°C). In the first stage (0–200 s), it decreases slightly due to the rapid formation of a solid shell and the appearance of small air gaps. In the second stage (200–1425 s), the IHTC rises to a first maximum of 493 W/(m²·°C). This increase is attributed to the mold expansion caused by the temperature rise. During this period, the mold sand undergoes phase transformation from β-quartz to α-quartz, leading to a volume increase. Since the external steel sleeve constrains the mold from expanding outward, the mold expands inward, reducing the air gap thickness and thereby increasing the IHTC. In the third stage (1425–2710 s), the IHTC decreases sharply to a minimum of 186 W/(m²·°C). Both the casting and the mold are now cooling, causing both to shrink, which widens the air gap. In the fourth stage (2710–4100 s), the IHTC rises again to a second maximum of 372 W/(m²·°C). This occurs because the casting temperature approaches the eutectoid transformation temperature (A₁), and the pearlite transformation causes the steel to increase in volume. This expansion reduces the air gap thickness and enhances the heat transfer. In the final stage (4100 s onwards), the IHTC decreases, initially rapidly and then slowly. The rapid decrease is associated with the α-to-β quartz transformation in the sand at about 573 °C, which causes a sudden shrinkage of the mold and an increase in the gap width.
The double-peak behavior is a unique feature of steel castings in sand molds, and it highlights the importance of thermal expansion and solid-state phase transformations of both the casting and the mold in controlling the interfacial heat transfer in sand casting parts.
Effect of Casting Alloy
The direct comparison between ZL101 and 40 steel castings under identical mold and geometry conditions demonstrates that the alloy type has a decisive influence on the IHTC. Although the heat flux and temperature difference curves are qualitatively similar, the IHTC values and their variation patterns are remarkably different. The S-curve for aluminum is governed by the gradual solidification over a wide temperature range and the final abrupt air gap formation at the critical solid fraction. The double-peak curve for steel is governed by the solid-state phase transformations and the thermal contraction/expansion behavior. This comparison emphasizes that a single constant IHTC cannot be used to accurately model the solidification of sand casting parts when different alloys are involved.
Effects of Casting Size and Mold Material on the IHTC
The effect of casting size was investigated by varying the inner diameter of the annular ZL101 castings from 60 mm to 100 mm and 140 mm. The overall S-shaped variation with temperature remained the same, but the quantitative values changed significantly. The maximum, average, and stabilized values of the IHTC first increased and then decreased with increasing radius, resulting in a non-monotonic relationship. This is different from the behavior reported for plate or cylindrical castings, where the IHTC generally increases with casting thickness or diameter. The annular geometry introduces a curvature effect, which modifies the air gap evolution. The slope of the IHTC drop at the critical solid fraction increased with radius, indicating that larger annular castings experience faster gap growth after the solid fraction exceeds 0.5.
To facilitate practical use in numerical simulations, a sigmoid correlation was developed to describe the IHTC of ZL101 castings as a function of casting surface temperature:
$$
y = h_{min} + \frac{h_{max} – h_{min}}{1 + \exp\left( \frac{x – a}{b} \right)}
$$
where \(x\) is the casting surface temperature, \(y\) is the IHTC, \(h_{min}\) and \(h_{max}\) are the minimum and maximum IHTC values, \(a\) corresponds to the critical solid fraction temperature (575 °C), and \(b\) is a geometry-dependent coefficient. The fitted parameters for the three casting sizes are listed in Table 2.
| Casting size | \(h_{min}\) (W/m²·°C) | \(h_{max}\) (W/m²·°C) | \(a\) (°C) | \(b\) |
|---|---|---|---|---|
| r = 60 mm | 60 | 109 | 575 | 1.5 |
| r = 100 mm | 83 | 127 | 575 | 0.36 |
| r = 140 mm | 45 | 91 | 575 | 0.44 |
The mold material also plays an important role. In the present study, furan resin self-hardening sand and sodium silicate self-hardening sand were compared. The IHTC curves for both sands followed the same S-shaped trend, but the absolute values were higher for the furan resin sand. The main difference occurred near 614 °C, where the IHTC for the sodium silicate sand showed a pronounced drop, while the furan resin sand only exhibited a slight fluctuation. This is related to the surface roughness and gas evolution of the molds. The sodium silicate sand has a rougher surface and higher moisture content, which promotes the formation of larger initial air gaps. Below 575 °C, the IHTC for the furan resin sand remained stable, whereas that for the sodium silicate sand gradually decreased, likely because the specific heat of sodium silicate sand decreases more significantly at low temperatures.
Effect of Forced Air Cooling on the Interfacial Heat Transfer Coefficient
For ZL101 Aluminum Alloy Castings
Forced air cooling was applied by passing compressed air through a steel pipe embedded in the sand core. The measured temperature fields for the ZL101 casting with and without air cooling are compared in Figure 4. The air cooling accelerates the solidification process. The total solidification time defined as the time for the casting surface temperature to reach the solidus temperature (556 °C) was 1224 s with air cooling versus 1655 s without air cooling, representing a 25% reduction. The cooling rate after solidification increased from 0.05 °C/s to 0.2 °C/s due to air cooling. Interestingly, once the casting temperature dropped below about 550 °C, the cooling rates became similar, both around 0.1 °C/s. The forced air cooling also modified the temperature distribution in the sand mold. The mold surface temperature rose faster because the increased heat flux from the casting exceeded the heat removed by the air flow through the core. However, the deeper mold temperatures increased more slowly because a larger portion of the heat was extracted through the core.
The inverse calculation results are presented in Figure 5. The IHTC for the air-cooled ZL101 casting still follows an S-curve with temperature, but the values are substantially higher. The average IHTC increased from 94 W/(m²·°C) for natural cooling to 143 W/(m²·°C) for forced air cooling, an increase of about 52%. The time at which the IHTC reached its maximum was shortened, and the decreasing phase after the peak was extended. The air cooling promotes earlier and larger air gap formation because the faster solidification rate produces a thicker solid shell at an earlier stage. This is reflected in the IHTC curve, where a slow decline appears before the casting temperature reaches 575 °C, which is not observed under natural cooling.
For 40 Steel Castings
For the steel castings, four different cooling conditions were tested: no air cooling, and air cooling through pipes with diameters of 40 mm, 60 mm, and 80 mm. The IHTC remained in the double-peak form for all conditions. The average IHTC values increased significantly with air cooling. As the pipe diameter increased from 0 to 80 mm, the maximum IHTC increased by approximately 50%, 75%, and 120%, respectively. The first peak appeared earlier for larger pipe diameters, while the second peak time shifted only slightly. The minimum IHTC between the two peaks also increased with air cooling intensity. The representative IHTC curves are plotted in Figure 6.
The effect of air cooling on the steel casting temperature is shown in Figure 7. The time required for the casting to cool to 400 °C was 1119 s without air cooling, and it was reduced to 864 s, 744 s, and 493 s for pipe diameters of 40 mm, 60 mm, and 80 mm. This corresponds to cooling efficiency improvements of 50%, 75%, and 111%, respectively. The forced air cooling also raised the maximum sand mold temperature, except for the smallest pipe diameter, where the heat removal through the core was insufficient to compensate for the increased heat flow from the casting.
Mechanism of Forced Air Cooling on the Interface Heat Transfer
The main physical mechanism linking forced air cooling to the IHTC is the modification of the air gap between the casting and the mold. Air cooling increases the cooling rate of the casting, which in turn accelerates the temperature changes of both the casting and the sand mold. These rapid temperature changes cause faster thermal contraction of the casting and faster expansion or contraction of the mold, thereby altering the air gap thickness more quickly and more dramatically. For aluminum castings, the faster cooling promotes the formation of a thicker solid shell at an early stage, which enables the air gap to survive against metallostatic pressure. As a result, the IHTC begins to decrease earlier than in natural cooling. For steel castings, the air cooling intensifies the volumetric changes associated with phase transformations. The faster cooling accelerates the pearlite transformation and the quartz phase changes in the sand, leading to sharper peaks and valleys in the double-peak curve. Nevertheless, the overall effect of forced air cooling is to enhance the average IHTC, which improves the heat exchange between the casting and the core and shortens the cooling time.
The relationship between the IHTC and the solid fraction for ZL101 is illustrated in Figure 8. Under natural cooling, the IHTC continues to rise until the solid fraction reaches about 50%, then drops abruptly. Under forced air cooling, a decline begins already at a solid fraction of about 10%, because the earlier formation of a robust solid shell creates larger micro-gaps. At the critical solid fraction, both curves show a steep drop, but the magnitude of the drop is smaller for the air-cooled case since many gaps already exist.
Conclusion
This work systematically investigated the interfacial heat transfer coefficient for annular sand casting parts under natural and forced air cooling conditions. The major conclusions can be summarized as follows:
(1) For ZL101 aluminum alloy castings, the IHTC as a function of casting temperature follows an S-shaped curve. It increases with decreasing temperature above 575 °C, then drops abruptly at 575 °C, and finally remains stable or slowly decreases. The abrupt drop corresponds to the critical solid fraction where the air gap expands rapidly.
(2) For 40 steel castings, the IHTC evolves with time as a double-peak curve. The first peak arises from the mold expansion due to the β-to-α quartz transformation, while the second peak is caused by the pearlite transformation of the steel. The peaks and valleys are strongly affected by the thermal contraction and phase transformations of both the casting and the sand mold.
(3) The casting size and mold material significantly influence the IHTC. The annular geometry leads to a non-monotonic dependence on the inner radius, with the maximum IHTC occurring for the intermediate radius. Furan resin self-hardening sand provides a higher IHTC than sodium silicate self-hardening sand, mainly because of differences in surface roughness and gas evolution.
(4) Forced air cooling enhances the average IHTC for both aluminum and steel castings. The IHTC of aluminum castings remains S-shaped but with higher values and an earlier decreasing phase. The IHTC of steel castings remains double-peaked, but the peaks become sharper and appear earlier. The average IHTC under forced air cooling can be more than twice that of natural cooling.
(5) The enhancement effect of forced air cooling is attributed to the accelerated growth and more rapid variation of the interfacial air gap. Air cooling increases the cooling rate, promotes earlier solid shell formation, and intensifies the volumetric changes of the casting and the mold, thereby increasing the heat transfer capability across the interface.
These findings provide valuable data for numerical simulations of sand casting parts produced with forced cooling technologies. The proposed inverse algorithm and the obtained IHTC correlations can be used to improve the prediction accuracy of casting solidification, defect formation, and final mechanical properties in industrial applications.