In modern foundry practice, the continuous expansion of alloy compositions, mold materials, and auxiliary external fields such as air cooling, water cooling, centrifugal force, negative pressure, and ultrasonic vibration has made the solidification and cooling processes of castings increasingly complex and difficult to predict. Although simulation software provides convenient tools for predicting casting outcomes, the accuracy of numerical results is strongly limited by the lack of fundamental knowledge of solidification theory and thermophysical parameters under complex casting conditions. Among these parameters, the interfacial heat transfer coefficient between the casting and the mold cavity surface plays a decisive role in controlling the heat dissipation efficiency of the molten metal and the solidified high-temperature casting. This coefficient directly affects the cooling rate, and therefore determines the distribution of sand casting defects, residual stress, grain size, and mechanical properties. Consequently, studying the theoretical basis of the interfacial heat transfer coefficient and its variation under diversified casting methods is of great scientific and practical importance.
In this work, I systematically investigated the interfacial heat transfer coefficient of annular castings produced in sand molds under both natural cooling and forced air cooling conditions. The castings were made of ZL101 aluminum alloy and 40 steel, while the sand cores were prepared from furan resin no-bake sand and sodium silicate no-bake sand. Temperature fields at selected locations were measured experimentally during pouring and solidification. An optimized Beck inverse algorithm was used together with numerical simulation to estimate the interfacial heat transfer coefficient as a function of time and temperature. The study aims to provide reliable data and theoretical explanations for the application of forced air cooling technology in the casting industry, with a particular focus on how the interfacial heat transfer coefficient influences the formation of sand casting defects.
Experimental Procedures
Materials and Mold Preparation
The chemical compositions of ZL101 aluminum alloy and 40 steel used in this study are summarized in Table 1 and Table 2, respectively. ZL101 is an Al-Si-Mg alloy with approximately 6.5–7.5% Si and 0.25–0.45% Mg, while 40 steel is a medium-carbon steel containing 0.37–0.44% C, 0.17–0.37% Si, and 0.50–0.80% Mn.
| 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 | ≤0.10 | Balance |
| C | Si | Mn | Cr | Ni | Cu | Fe |
|---|---|---|---|---|---|---|
| 0.37–0.44 | 0.17–0.37 | 0.50–0.80 | ≤0.25 | ≤0.30 | ≤0.25 | Balance |
Two types of chemically bonded sand were prepared. The furan resin no-bake sand consisted of 50–100 mesh silica sand mixed with 1.2% FFD-121 furan resin and 0.3% p-toluenesulfonic acid solution as hardener. The sodium silicate no-bake sand was made by mixing silica sand with 4% sodium silicate and 0.4% organic ester hardener. The molds were hand-rammed and allowed to cure for about 24 hours before stripping. The thermophysical properties of the furan sand and sodium silicate sand are listed in Table 3 and Table 4, respectively.
| Temperature (°C) | Density (kg·m⁻³) | Heat capacity (kJ·kg⁻¹·K⁻¹) | Conductivity (W·m⁻¹·K⁻¹) |
|---|---|---|---|
| 20 | 1590 | – | 0.71 |
| 50 | 1590 | 0.73 | – |
| 100 | 1590 | 0.80 | – |
| 200 | 1590 | 0.92 | – |
| 232 | 1590 | – | 0.62 |
| 400 | 1590 | 1.00 | – |
| 500 | 1590 | 1.00 | – |
| 600 | 1590 | – | 0.50 |
| 708 | 1590 | – | 0.61 |
| 980 | 1590 | – | 0.78 |
| Temperature (°C) | Density (kg·m⁻³) | Heat capacity (kJ·kg⁻¹·K⁻¹) | Conductivity (W·m⁻¹·K⁻¹) |
|---|---|---|---|
| 50 | 1590 | 0.77 | 0.77 |
| 200 | 1590 | 0.72 | 0.84 |
| 500 | 1590 | 0.62 | 0.88 |
| 700 | 1590 | 0.58 | 0.92 |
| 900 | 1590 | 0.53 | 0.99 |
| 1100 | 1590 | 0.55 | 1.03 |
| 1350 | 1590 | 0.62 | 1.06 |
| 1450 | 1590 | 0.68 | 1.08 |
| 1500 | 1590 | 0.76 | 1.09 |
| 1550 | 1590 | 0.80 | 1.10 |
Experimental Setup
Figure 1 illustrates a typical experimental arrangement used in this investigation. A thermocouple designated as Tc was placed inside the casting cavity at 2 mm from the mold surface to measure the molten metal temperature. Three additional thermocouples, Tm1, Tm2, and Tm3, were embedded in the sand mold at distances of 6, 14, and 22 mm from the casting surface, respectively. This arrangement improved the measurement accuracy and avoided cracking of the sand mold during pouring. The thermocouple positions strongly affect the stability of the inverse calculation, as noted in previous studies.
To guarantee the precise placement of the thermocouples, I prepared small pre-formed sand blocks with embedded thermocouples, following the method proposed by earlier researchers. The pre-formed blocks were then inserted into the mold at the desired positions during the ramming process. K-type thermocouples (NiCr-NiSi) were employed for aluminum castings, while B-type thermocouples (PtRh30-PtRh6) were used for steel castings. The temperature data were recorded with a TP700 multi-channel data logger at a frequency of 1 Hz, with an accuracy of ±0.5 °C.
Casting and Cooling Conditions
Eleven experiments were conducted. The variables included casting material (ZL101 or 40 steel), sand mold type (furan resin or sodium silicate), casting inner radius (60, 100, or 140 mm), and the presence or absence of forced air cooling. In the forced-air-cooling experiments, a steel pipe with an outer diameter of 40, 60, or 80 mm was embedded in the sand core. Compressed air was supplied through the pipe immediately after pouring was completed. The air velocity was adjusted to 6 m/s, 8 m/s, and 10 m/s for pipe diameters of 40, 60, and 80 mm, respectively. All experiments used the same pouring time of approximately 10 seconds. ZL101 was melted in a resistance furnace and poured at 730 °C, while 40 steel was melted in a medium-frequency induction furnace, deoxidized with 0.5% aluminum, and poured at 1560 °C.
| Experiment No. | Casting material | Sand mold material | Inner radius r (mm) | Air cooling pipe diameter (mm) |
|---|---|---|---|---|
| 1 | Al (ZL101) | Furan resin | 60 | None |
| 2 | Al (ZL101) | Furan resin | 100 | None |
| 3 | Al (ZL101) | Furan resin | 140 | None |
| 4 | Al (ZL101) | Furan resin | 100 | 60 |
| 5 | Al (ZL101) | Sodium silicate | 60 | None |
| 6 | Al (ZL101) | Sodium silicate | 100 | None |
| 7 | Al (ZL101) | Sodium silicate | 140 | None |
| 8 | Steel (40) | Furan resin | 100 | None |
| 9 | Steel (40) | Furan resin | 100 | 40 |
| 10 | Steel (40) | Furan resin | 100 | 60 |
| 11 | Steel (40) | Furan resin | 100 | 80 |
Inverse Heat Conduction Model
One-Dimensional Model Formulation
The heat transfer between the annular casting and the sand mold was simplified to a one-dimensional cylindrical heat conduction problem. The geometry was divided into six sectors in the sand mold starting from the casting-mold interface. The measured temperatures at the second, fourth, and sixth sector centers correspond to Tm1, Tm2, and Tm3. The temperature at the first sector is the unknown surface temperature of the mold, which is later used to compute the interfacial heat transfer coefficient.
The one-dimensional heat conduction equation in cylindrical coordinates can be written as:
$$
\frac{1}{r}\frac{\partial}{\partial r}\left(k r \frac{\partial T}{\partial r}\right) = \rho C_p \frac{\partial T}{\partial t}
\tag{1}
$$
where ρ, Cp, and k are the density, effective specific heat, and thermal conductivity of the sand, respectively, T is temperature, r is the radial coordinate, and t is time. The initial condition was the experimentally measured temperature field at time zero:
$$
T(r,0) = T_{\text{in}}(r)
\tag{2}
$$
The boundary condition at the casting-mold interface (r = 0) was the unknown heat flux q(0,t):
$$
k \left. \frac{\partial T}{\partial r} \right|_{r=0} = q(0,t)
\tag{3}
$$
The boundary condition at the location of Tm3 (r = x6) was the measured temperature:
$$
T(x_6,t) = T_{m3}(t)
\tag{4}
$$
Discretization with the Finite Volume Method
The implicit finite volume method was used to discretize the governing equation. The spatial step was Δx. For each control volume j, the energy balance can be expressed as:
$$
\Delta u = A_{\text{in},j} q_{\text{in},j} + A_{\text{out},j} q_{\text{out},j}
\tag{5}
$$
where Δu is the change of internal energy of volume j from time i to i+1, q_in is the heat flux entering the volume, and q_out is the heat flux leaving the volume. For the first volume (j=1), which is at the interface, the energy balance leads to:
$$
(1 + S_{\text{out}}(1)) T_1^{i+1} – S_{\text{out}}(1) T_2^{i+1} = T_1^i + S_{\text{in}}(1) \Delta x q^{i+1}
\tag{6}
$$
For the internal volumes (j = 2 to 5):
$$
– S_{\text{in}}(j) T_{j-1}^{i+1} + [1 + S_{\text{in}}(j) + S_{\text{out}}(j)] T_j^{i+1} – S_{\text{out}}(j) T_{j+1}^{i+1} = T_j^i
\tag{7}
$$
where the dimensionless coefficients S_in and S_out account for the cylindrical geometry:
$$
S_{\text{in}}(j) = \frac{A_{\text{in},j}}{V_j} \frac{\Delta t k^i}{\rho C_p^i \Delta x}
\tag{8}
$$
$$
S_{\text{out}}(j) = \frac{A_{\text{out},j}}{V_j} \frac{\Delta t k^i}{\rho C_p^i \Delta x}
\tag{9}
$$
For an annular geometry with inner radius r_use and thickness δ, the area-to-volume ratios are:
$$
\frac{A_{\text{in},j}}{V_j} = \frac{r_{\text{use}} – j\Delta r + \Delta r}{\Delta r (r_{\text{use}} – j\Delta r + 0.5\Delta r)}
\tag{10}
$$
$$
\frac{A_{\text{out},j}}{V_j} = \frac{r_{\text{use}} – j\Delta r}{\Delta r (r_{\text{use}} – j\Delta r + 0.5\Delta r)}
\tag{11}
$$
The sixth volume is a temperature boundary condition:
$$
T_6^i = T_{m3}^i
\tag{12}
$$
Equations (6), (7), and (12) form a tridiagonal matrix system that can be solved efficiently with the Thomas algorithm. This formulation explicitly accounts for the curvature of the annular casting, which is a key improvement over the classical Beck method that assumed a flat geometry.
Beck Inverse Algorithm
To determine the unknown interfacial heat flux q(t), the Beck nonlinear estimation method was employed. Future time steps were introduced to improve the stability of the solution. During the calculation of the flux at time m, the heat flux was assumed to remain constant for r future time steps:
$$
q_m = q_{m+1} = \cdots = q_{m+r-1} = q_{\text{set}}
\tag{13}
$$
The objective function was defined as the sum of squared differences between the calculated and measured temperatures at the thermocouple locations:
$$
F(q_{\text{set}}) = \sum_{f=m+1}^{m+r} \sum_{j=1}^{J} \left[ T_j^f – T_{mj}^f(q_{\text{set}}) \right]^2
\tag{14}
$$
Minimizing F with respect to q_set yields:
$$
\frac{\partial F(q_{\text{set}})}{\partial q_{\text{set}}} = \sum_{f=m+1}^{m+r} \sum_{j=1}^{J} \left[ T_j^f – T_{mj}^f(q_{\text{set}}) \right] \frac{\partial T_j^f}{\partial q_{\text{set}}} = 0
\tag{15}
$$
Using a Taylor expansion and the sensitivity coefficient φ, defined as:
$$
\phi_j^f = \frac{\partial T_j^f}{\partial q} \approx \frac{T_j^f(q_{\text{set}} + \varepsilon q_{\text{set}}) – T_j^f(q_{\text{set}})}{\varepsilon q_{\text{set}}}
\tag{16}
$$
The iterative formula for the heat flux becomes:
$$
q^{(k+1)} = q^{(k)} + \frac{\sum_{f=m+1}^{m+r} \sum_{j=1}^{J} \left[ T_j^f – T_{mj}^f(q_{\text{set}}) \right] \phi_j^f}{\sum_{f=m+1}^{m+r} \sum_{j=1}^{J} (\phi_j^f)^2}
\tag{17}
$$
The iteration continued until the convergence criterion was satisfied:
$$
\frac{|q^{(k+1)} – q^{(k)}|}{|q^{(k+1)}|} < \varepsilon
\tag{18}
$$
The interfacial heat transfer coefficient h was then calculated from the heat flux and the temperature difference between the casting surface and the sand mold surface:
$$
h_i = \frac{q_i}{T_{c}^i – T_1^i}
\tag{19}
$$
Results and Discussion
Verification of the Inverse Model
To validate the reliability of the inverse calculation, I compared the computed temperature at the second sensor location with the measured Tm1. Figure 2 shows the comparison for a typical ZL101 casting. The calculated and measured temperatures agree very well, with a maximum error of 5 °C in the initial period and an average deviation of 1.27 °C. The relative deviation, computed as:
$$
\delta = \frac{|T_{2} – T_{m1}|}{T_{2}} \times 100\%
\tag{20}
$$
was less than 5% throughout the process, with an average of 1.017% and a variance of 0.000108. For steel castings, the maximum error was 11 °C during the first 5 seconds, decreasing rapidly to about 0.1 °C, with an average error of 0.4 °C. The relative deviation for steel castings was below 3%, which confirms the accuracy of the inverse method. The larger early errors are attributed to the finite response time of the thermocouples and the time lag caused by the protective sleeves.
Interfacial Heat Transfer Coefficient for ZL101 Castings
Figure 3 shows the heat flux, temperature difference, and heat transfer coefficient for a ZL101 casting without air cooling (Experiment No. 2). The heat flux and temperature difference show similar trends: they reach very high initial values and then drop rapidly, followed by a slow decline. The initial high values are caused by the large temperature difference between the molten metal (~700 °C) and the sand mold (near room temperature), which produces a strong driving force for heat conduction.
The interfacial heat transfer coefficient for ZL101 follows an “S-shaped” curve as a function of casting temperature. Initially, h increases as the metal cools, until the temperature drops to about 614 °C, where a slight drop occurs. This drop is caused by the initial formation of a solid shell, which creates a small air gap at the interface. Between 614 °C and 575 °C, h increases again because the latent heat released during solidification keeps the interface temperature relatively high. At approximately 575 °C, which corresponds to a solid fraction near 0.5, h drops sharply. This is the critical solid fraction (CSF) at which liquid metal can no longer feed the shrinkage between dendrites, leading to a sudden enlargement of the air gap. Below 575 °C, h tends to remain stable or decrease slowly.
Forces air cooling significantly changes the magnitude and evolution of h for ZL101 castings. Figure 4 compares the heat transfer coefficient for experiments with and without air cooling. The air-cooled casting shows higher heat transfer coefficients throughout the solidification process, with an average value increasing from 94 W/(m²·°C) to 143 W/(m²·°C), a 52% improvement. The “S-shaped” trend is preserved, but the initial increase is shorter and the decrease begins earlier. This behavior is related to the higher cooling rate caused by forced air, which accelerates the formation of a solid shell and promotes more rapid air gap growth.
The effect of casting geometry on h for ZL101 was also investigated for annular castings with inner radii of 60, 100, and 140 mm. As shown in Figure 5, the general shape of the h versus temperature curve remains unchanged, but the maximum, average, and final steady-state values of h first increase and then decrease as the radius increases. This non-monotonic behavior differs from flat or cylindrical castings, where h generally increases with casting size. For annular castings, the radius-to-thickness ratio determines the curvature and the amount of contraction. The slope of h at the CSF point increases with the inner radius, indicating that larger annular castings develop air gaps more rapidly due to greater linear contraction.
To describe the temperature-dependent heat transfer coefficient, I fitted the inverse results to a logistic function:
$$
y = h_{\min} + \frac{h_{\max} – h_{\min}}{1 + \exp\left( \frac{x – a}{b} \right)}
\tag{21}
$$
where x is the casting surface temperature, y is the interfacial heat transfer coefficient, h_min and h_max are the minimum and maximum values, a is the temperature corresponding to the critical solid fraction (575 °C), and b is a size-dependent coefficient. The fitted parameters for the three annular castings are listed in Table 6. The fitted curves show excellent agreement with the estimated data.
| Inner radius (mm) | h_min (W/(m²·°C)) | h_max (W/(m²·°C)) | a (°C) | b |
|---|---|---|---|---|
| 60 | 60 | 109 | 575 | 1.5 |
| 100 | 83 | 127 | 575 | 0.36 |
| 140 | 45 | 91 | 575 | 0.44 |
Interfacial Heat Transfer Coefficient for 40 Steel Castings
Without air cooling, the heat transfer coefficient for 40 steel castings exhibits a distinctive “double-peak” behavior, as shown in Figure 6. The evolution of h with time can be divided into five stages:
Stage I (0–200 s): h decreases slightly from an initial value of 449 W/(m²·°C) due to the rapid solidification of a thin metal shell and the formation of small air gaps.
Stage II (200–1425 s): h increases sharply to a maximum of 493 W/(m²·°C). This is mainly caused by the thermal expansion of the sand mold, which is heated by the hot casting. The sand undergoes a phase transformation from β-quartz to α-quartz, leading to an increase in volume. The mold expansion reduces the air gap thickness, thereby increasing h.
Stage III (1425–2710 s): h drops rapidly to 186 W/(m²·°C). In this stage, both the casting and the sand mold cool down and contract. The shrinkage of the casting creates a larger air gap, increasing the interfacial thermal resistance.
Stage IV (2710–4100 s): h rises again to 372 W/(m²·°C). This second peak is attributed to the pearlite transformation in the steel at temperatures around the A1 point. The transformation causes the casting to expand, reducing the air gap thickness and improving the interfacial contact.
Stage V (4100–10000 s): h decreases gradually. The sand mold undergoes a phase transformation from α-quartz to β-quartz at about 573 °C, causing a significant volume reduction and an increase in air gap thickness. Later, as the temperature approaches room temperature, the heat transfer driving force diminishes and h declines slowly.
Forced air cooling changes the magnitude and timing of these peaks. With air cooling, the interfacial heat transfer coefficient remains of the “double-peak” form, but the average value can be more than twice that of the natural cooling case. As the air cooling pipe diameter increases from 40 mm to 80 mm, the first peak appears earlier and becomes higher. The second peak also appears earlier, but its height decreases with the pipe diameter due to the more rapid cooling of the casting, which modifies the pearlite transformation kinetics.
Effect of Sand Mold Material
Experiments with ZL101 castings using sodium silicate no-bake sand showed the same “S-shaped” h curve as with furan resin sand. However, the magnitude of h was consistently higher for the furan resin sand. This is because the furan resin sand has a smoother surface and lower gas evolution, resulting in a smaller initial air gap at the casting-mold interface. In contrast, the sodium silicate sand has a rougher surface and higher moisture content, promoting the formation of larger air gaps. The difference was most evident at the 614 °C transition point: for sodium silicate sand, the h drop was clearly visible, while for furan resin sand, only a slight perturbation was observed. In the later stages of solidification, the h for furan resin sand remained almost constant, whereas for sodium silicate sand it decreased gradually, probably because of the different temperature dependence of the specific heat capacity of the two sands.
Effect of Forced Air Cooling on Temperature Field and Sand Casting Defects
Forced air cooling is an effective method to increase the cooling rate and reduce the formation of sand casting defects such as shrinkage porosity, hot tears, and coarse grain structures. In my experiments, the forced air cooling shortened the solidification time of ZL101 castings by 25% (from 1655 s to 1224 s). For steel castings, the time to reach 400 °C decreased from 1119 s without air cooling to 864 s, 744 s, and 493 s for pipe diameters of 40, 60, and 80 mm, respectively. These reductions correspond to increases in cooling efficiency of approximately 50%, 75%, and 111%. A faster cooling rate is generally beneficial for refining the microstructure and reducing the severity of sand casting defects, provided that the thermal stresses do not lead to cracking.
Figure 7 illustrates the influence of forced air cooling on the temperature field of the sand core. In the absence of air cooling, the core that is directly surrounded by molten metal becomes thermally saturated: its temperature rises above the casting temperature, thereby slowing down the casting cooling. Forced air cooling removes heat from the core through the embedded pipe, preventing thermal saturation and maintaining the core at a lower temperature than the casting. This sustains a higher temperature gradient, which promotes faster heat extraction and more homogeneous cooling, thus reducing the risk of residual stress and distortion, which are common sand casting defects.
Mechanism of Enhanced Heat Transfer by Forced Air Cooling
The interfacial heat transfer coefficient is controlled primarily by the evolution of the air gap between the casting and the mold. The air gap thickness is determined by the relative volume changes of the casting and the mold, which in turn depend on their temperature histories. Forced air cooling modifies both temperature fields.
For aluminum castings, forced air cooling increases the cooling rate and accelerates the formation of the initial solid shell. The shell becomes thick enough to support the liquid metal pressure earlier than in natural cooling. Consequently, the air gap forms sooner and grows faster. The h curve still follows the “S-shaped” pattern, but with a notable slow decrease before the casting temperature reaches 575 °C. This is a direct consequence of the earlier air gap growth.
For steel castings, forced air cooling intensifies the volumetric changes of both the casting and the sand mold. The cooling rate is enhanced, so the casting contracts more rapidly, while the sand mold expands and contracts more sharply. The combined effect leads to faster and more pronounced changes in the air gap thickness, explaining the observed increase in the intensity of the double-peak h curve. The average h value under forced air cooling reaches more than twice the natural cooling value, indicating a significantly improved interfacial thermal contact during most of the solidification and subsequent cooling process.
The comparison between ZL101 and 40 steel under forced air cooling highlights the importance of the solidification path. ZL101 exhibits a long freezing range and releases a large amount of latent heat near the eutectic temperature, making the h evolution more sensitive to the solid fraction. Steel, on the other hand, solidifies over a smaller temperature range but then undergoes a long solid-state cooling path. The phase transformations in both the casting and the sand mold play a decisive role in shaping the h curve. These differences must be accounted for in numerical simulations to accurately predict cooling curves, residual stress fields, and the formation of sand casting defects.
Conclusions
- The interfacial heat transfer coefficient of ZL101 castings follows an “S-shaped” curve as a function of temperature. It increases with decreasing temperature until the casting reaches approximately 575 °C, where a sharp drop occurs due to the formation of an air gap at the critical solid fraction. Below 575 °C, the heat transfer coefficient remains relatively stable.
- For 40 steel castings, the interfacial heat transfer coefficient exhibits a “double-peak” behavior. The five stages correspond to the initial air gap formation, sand mold expansion, simultaneous contraction, pearlite transformation, and quartz transformation in the sand. These stages are governed by the competing volume changes of the casting and the sand mold.
- Forced air cooling enhances the interfacial heat transfer coefficient for both aluminum and steel annular castings. For ZL101, the average heat transfer coefficient increased by 52%. For 40 steel, the average value more than doubled. The general shape of the h curves was preserved, but the air cooling accelerated the transitions and increased the peak intensities.
- The casting geometry affects the interfacial heat transfer coefficient. For annular castings with inner radii of 60, 100, and 140 mm, the maximum, average, and final steady-state h values first increase and then decrease as the radius increases. A logistic function was proposed to describe h as a function of temperature, with size-dependent coefficients.
- The sand mold material influences h through the initial surface condition and the gas evolution. Furan resin no-bake sand provides a smoother interface and yields higher h values than sodium silicate sand. This finding suggests that the choice of mold material can be used to control the cooling rate and mitigate sand casting defects.
In summary, this study provides quantitative data and mechanistic insights into the interfacial heat transfer of annular castings under forced air cooling. The results are directly useful for optimizing cooling strategies in sand casting production, reducing the occurrence of sand casting defects, and improving the reliability of process simulations.