1. Introduction

Marine propellers are critical components that directly influence a vessel's propulsion efficiency and fuel consumption. Due to their significant impact on overall performance, a high degree of precision is required throughout the design and manufacturing stages. Dimensional deviations in propeller geometry can lead to reduced propulsion efficiency, increased vibration and noise, and, over time, can negatively affect the ship’s operational safety and maintenance costs. Accordingly, quality control across the entire production process — from design to casting and machining — is of paramount importance in propeller manufacturing.

Sand casting is widely employed to produce large metallic components with complex geometries, making it a suitable method for marine propeller manufacturing. Although sand casting offers cost advantages for producing large-scale components, it inherently involves dimensional changes due to thermal expansion and contraction as molten metal solidifies within the mold. If these thermal deformations are not properly accounted for, dimensional inaccuracies may occur in the final product, necessitating additional machining and rework, leading to material waste, increased production time, and higher manufacturing costs. Currently, most propeller production sites domestically and internationally rely on empirically determined casting allowances, typically applying a margin of 8–10%.

However, such empirical methods often result in excessive margins and inefficient material use, leading to increased machining and material waste (Park, 2012;Mun et al., 2008).

Recently, as the demand for manufacturing efficiency and cost reduction has increased, there has been growing interest in scientifically analyzing and optimizing casting allowances based on quantitative methods. CAD-based mold design and FEM-driven contraction simulation have been explored in recent research to enhance dimensional accuracy and reduce overdesign in large-scale castings (Park et al., 2016;Galles and Beckermann, 2016).

In response to these needs, this study aims to precisely predict the thermal expansion and contraction behavior during the sand casting process of marine propellers, thereby establishing optimal dimensional allowances for each section. The goal is to minimize additional machining and reduce production costs.

This research focuses on propellers made of aluminum bronze alloy (ALBC3), utilizing thermal expansion equations and finite element analysis (FEM) to quantitatively analyze deformations in key sections such as the blades, hub, and overall diameter.

Additionally, a cost analysis is conducted to demonstrate the material and labor savings achievable through the application of the optimal allowance design. Through this, the study aims to enhance the practical applicability of the proposed methodology to real-world manufacturing environments. According to Seo et al. (2016), small deviations in propeller geometry—whether from marine biofouling or manufacturing inaccuracies—can significantly degrade hydrodynamic efficiency by increasing drag and disturbing flow uniformity (Lee, 2018). emphasized that additional structures mounted on propellers, such as rope cutters, must be assessed for their influence on vibration and safety, reinforcing the importance of structural precision in rotating marine components.

Furthermore, Kim et al. (2021) applied FEM-based analysis to evaluate the stress distribution and effectiveness of an improved propeller-mounted device, supporting the role of simulation in optimizing propeller geometry and performance. The scientific approach proposed herein is expected to contribute to improving both the quality and cost-efficiency of marine propeller manufacturing processes.

2. Research Methodology

2.1 Research Target and Material Properties

In this study, a 1.1-meter diameter marine propeller made of aluminum bronze alloy (ALBC3) was selected as the research target to analyze the thermal expansion and contraction occurring during the sand casting process and to establish the optimal dimensional allowances. A photograph of the actual propeller installed on the vessel is shown in Fig. 1.

The main specifications and geometry of the propeller are as follows; key parameters including diameter, pitch, and projected area are provided in Table 1.

The material properties presented in Table 2 are based on values reported in the CDA (Copper Development Association UK). This material exhibits excellent resistance to cavitation and corrosion in seawater, making it particularly suitable for shipboard use (Hyun et al., 2013). According to the Copper Development Association’s engineering guide, casting simulation is strongly recommended for optimizing shrinkage allowances and improving yield in complex geometries like marine propellers (CDA Pub. No. 222).

2.2 Thermal Expansion and Contraction

In metal casting, dimensional deformation occurs as molten metal solidifies inside the mold. To estimate these changes quantitatively, the linear thermal expansion formula was applied

where ΔL is the length change, L₀ is the initial length, α is the coefficient of thermal expansion, and ΔT is the temperature difference. In this study, deformation during cooling from the pouring temperature of 1140°C to room temperature (25°C) was considered. As a representative example, the thermal expansion of the blade width (initial length = 0.3389 m) was calculated

This result corresponds to 6.4 mm of linear expansion. Similar calculations were applied to blade thickness, hub diameter, and overall propeller diameter, providing baseline values for casting allowance design.

However, this approach assumes uniform cooling, which rarely occurs in practice. Cooling rates vary by geometry and mold contact, often causing non-uniform contraction and internal stress.

Nyichomba and Campbell (1998) showed that aluminum alloy castings develop residual stress and nonlinear shrinkage due to temperature gradients and section-specific constraints.

Therefore, while the linear expansion formula offers a useful baseline, it must be refined by numerical simulation methods such as FEM, which capture temperature gradients, thermal inertia, and geometric constraints. These are discussed in Section 2.5. The computed values for key dimensions are presented in Table 3.

2.3 Heat Transfer Calculation

During the sand casting process, as molten metal is poured into the mold and cooled, the internal temperature distribution of the product changes non-uniformly over time. This temperature gradient induces differences in cooling rates among sections, which subsequently act as major factors causing uneven contraction and dimensional deviations. Therefore, to quantitatively evaluate the thermal behavior occurring inside the casting during the cooling process, the transient heat conduction equation was applied. The time- and position-dependent variation of temperature is expressed by the following heat conduction equation

where T is the temperature (K), t is the time (s), α is the thermal diffusivity (m²/s), and ∇² is the Laplacian operator. The thermal diffusivity α can be expressed as follows.

where k is the thermal conductivity, ρ is the density, and c is the specific heat. By applying the material properties of the aluminum bronze alloy (ALBC3), the thermal diffusivity was calculated.

This heat conduction equation and thermal diffusivity were then applied in the subsequent finite element method (FEM)-based numerical analysis to analyze the temperature distribution and cooling rate variations across different sections during the casting cooling process.

2.4 Stress-Strain Relationship

Thermal expansion and contraction occurring during the casting process induce stresses within the product. Particularly, if temperature variations and contraction magnitudes differ across sections during cooling, constraint forces between sections arise, leading to the formation of residual stresses. These residual stresses can negatively impact the structural stability of the product, thus making it important to quantitatively evaluate the stress and strain behavior induced by temperature changes.

The strain (ε) of the material is defined as follows.

The thermal expansion strain caused by temperature changes is expressed as

When temperature changes occur in a constrained state, thermal expansion is restricted, and internal stresses are generated in the material. The thermal stress generated under constrained conditions can be calculated using the following relation

The stress-strain relationships defined above are utilized later in the structural stability assessment and stress concentration analysis, considering the constraint conditions and thermal expansion of each section of the propeller.

2.5 Finite Element Method (FEM)

Finite element analysis (FEM) was conducted to evaluate the thermal expansion and contraction deformations, as well as the stress distribution induced by constraint forces during cooling in the casting process.

ANSYS software was used for the analysis, and a finite element model was constructed based on the CAD model of the target propeller. Thermo-mechanical coupled analysis was performed.

The analysis items were as follows.

The FEM results were cross-verified against the theoretical calculation results previously obtained, and were utilized to establish the optimal dimensional allowances.

3. Thermal Expansion Analysis

3.1 Comparison of Thermal Expansion and Contraction Deformations

To evaluate the deformations caused by thermal expansion and cooling contraction during the casting process, theoretical calculation results were compared with finite element method (FEM) analysis results. Both approaches exhibited similar deformation trends overall, although some discrepancies were observed in specific regions. For instance, FEM analysis revealed relatively greater contraction at the blade tip, which was interpreted as a result of enhanced heat dissipation due to close contact with the mold through radiation and conduction. In contrast, delayed cooling in the hub area, attributed to heat accumulation effects, was considered a major factor causing differences between theoretical calculations and FEM results. Such discrepancies suggest that FEM analysis, by reflecting localized thermal constraints and differences in cooling rates, can provide a more realistic basis for dimensional correction. Overall, theoretical calculations and FEM analysis demonstrated similar trends.

However, FEM results generally showed slightly smaller amounts of expansion, which is attributed to the influence of non-uniform thermal gradients and constraint-induced contraction during cooling. Specifically, variations in cooling rates due to mold contact at the blade tip and hub joint appeared to significantly affect the deformation magnitudes. Cooling rate differences between the blade and hub cause uneven shrinkage, leading to localized deformation and non-linear distortion patterns.

The stress distributions corresponding to these deformation patterns are shown in Fig. 2.

Based on this comparative analysis, theoretical values were adjusted using FEM results to finally establish optimal dimensional allowances for each section. To refine these theoretical values, the casting model was iteratively adjusted using FEM simulations— performed more than 30 times under varying cooling conditions and geometric configurations—to reflect more realistic boundary constraints and non-uniform temperature gradients during solidification. The resulting length changes and optimal allowances for each section were as follows a clearance rate of 1.9% for blade width and thickness, 1.5% for hub diameter, and 2.0% for overall diameter was derived, as summarized in Table 4.

This agreement indicates that FEM-based thermal deformation modeling can serve as a sufficiently reliable basis for dimensional prediction and compensation in real-world sand casting operations, even in the presence of minor process variations. These optimal allowances can serve as practical criteria for minimizing post-casting grinding operations and reducing rework and material waste.

3.2 Analysis of Stress Distribution and Concentration Regions

During the cooling process, thermal stresses were induced within the casting due to differences in shrinkage and constraint forces among sections. In particular, stress concentration was prominent in regions with significant thermal gradients and geometric variations. Finite element analysis (FEM) results confirmed stress concentration phenomena at the blade root and hub joint regions, as shown in Fig. 3.

This observation aligns with the findings of Motoyama et al. (2012), who developed an in-situ measurement system to track contraction forces and internal stresses forming during sand casting, confirming the critical influence of non-uniform cooling on stress development.

These stress concentrations were attributed to increased constraint forces in areas with relatively large thickness variations and geometric discontinuities during thermal contraction.

Specifically, thickness variations across the propeller structure lead to differential cooling rates during solidification. Thicker regions retain heat longer and contract more slowly compared to thinner adjacent sections, resulting in strain mismatch. This mismatch generates tensile stresses in the rapidly contracting thin sections, while compressive stresses accumulate in the thicker zones, forming internal stress gradients across the interface.

Additionally, geometric discontinuities such as sharp junctions between the blade root and hub create localized constraint conditions. These discontinuities impede uniform thermal contraction, acting as stress intensification zones where deformation is geometrically restricted. The FEM analysis confirmed these effects, showing Von-Mises stress peaks at the blade-hub transition area, which corresponds to regions with pronounced curvature and abrupt changes in cross-sectional geometry (Fig. 3).

The maximum stress was observed at the hub joint, and it was evaluated to be within a safe range compared to the material’s yield strength.

Regions with confirmed stress concentration are considered zones where quality issues are more likely to arise during production, with potential risks of micro-cracks or defects.

Therefore, at the design stage, geometric optimization such as curvature smoothing and thickness distribution should be considered, and after casting, quality verification through nondestructive testing (NDT) or high-precision dimensional inspection should be prioritized. Such stress distribution analysis results can be utilized to preemptively identify risks in process design and to reflect stress sensitivity due to geometric changes.

However, because stress concentration zones have a relatively higher possibility of micro-crack occurrence during casting and post-processing, additional quality management and strengthened inspection measures are deemed necessary for these regions.

These thermal stress distribution results can serve as a basis for applying stress mitigation designs, such as easing thickness variations and incorporating smooth curvature designs, at future casting process design stages. Given that typical casting tolerance standards allow deviations up to ±3%, the observed deviation within ±1.5% validates the practical applicability of the proposed shape design method without requiring further post-processing.

3.3 Stability Assessment through Vibration Analysis

Shape deformations occurring during the casting process can affect not only the dimensional accuracy of the product, but also its dynamic characteristics (such as natural frequencies and resonance behavior) under operational environments.

Accordingly, in this study, the natural frequencies and the potential for resonance occurrence of the propeller were evaluated through modal analysis and the use of a Campbell Diagram. The natural frequencies from the first to the sixth modes were derived through finite element analysis (FEM).

Additionally, as illustrated in Fig. 4, Campbell Diagram analysis showed that across all analyzed modes, the natural frequencies and rotational speeds did not intersect, indicating that resonance risks under actual operating conditions are essentially negligible. The specific values of the natural frequencies obtained through modal analysis are summarized in Table 5, clearly showing their separation from the operational speed range and reinforcing the absence of resonance. This suggests that shape errors and dimensional deformations were well controlled within acceptable levels after sand casting, and that the current casting process maintains sufficient reliability in ensuring rotational stability.

Nonetheless, since material fatigue accumulation and micro-wear over long-term operation could cause changes in dynamic characteristics, it is necessary to not only secure dynamic stability at the initial design stage, but also to establish regular vibration monitoring and shape inspection systems.

3.4 Comparison and Optimization Review of Theoretical Calculations and Analysis Results

In this study, theoretical calculations and finite element analysis (FEM) results were compared and analyzed to predict the thermal expansion and contraction deformations occurring during the sand casting process. Both approaches exhibited similar overall deformation trends, although FEM analysis showed slightly smaller deformation magnitudes compared to the theoretical calculations.

This discrepancy is attributed to the effects of localized thermal gradients and constraint-induced contraction, which were not considered in the theoretical calculations but were reflected in the FEM analysis. The summarized comparison shows that for blade width, the theoretical calculation yielded 6.4 mm while the FEM result was 6.2 mm; for thickness, the values were 0.87 mm and 0.85 mm respectively; and for overall diameter, 20.8 mm and 20.5 mm were observed.

Such numerical comparisons suggest that FEM analysis can present more realistic results by accurately reflecting actual physical conditions during cooling (such as heat dissipation conditions, thermal constraints, and position-dependent cooling rates).

Based on these results, optimal dimensional allowances of 1.9% for blade width and thickness, 1.5% for hub diameter, and 2.0% for overall diameter were established, providing more quantitative and reliable standards compared to conventional experience-based allowances. By cross-verifying FEM-based analysis results with theoretical calculations, the necessity and validity of analysis-based optimization were demonstrated.

Additionally, stress distribution analysis revealed stress concentration at the hub joint and blade root regions, which was interpreted as resulting from constraint stresses arising from significant thickness changes and geometric discontinuities during the casting process. Although the maximum stress was evaluated to remain below the material’s yield strength, the possibility of micro-crack formation at these critical areas suggests the need for additional quality management and inspection efforts.

Moreover, modal analysis and Campbell Diagram evaluation showed that the likelihood of resonance within the actual operating rotational speed range was low, although minor shape deformations during casting could still have potential impacts on dynamic stability. Therefore, continuous management to secure shape and dimensional accuracy was identified as necessary. By comprehensively reflecting the comparative analysis of theoretical calculations and FEM results, the final optimal dimensional allowances of 1.9% for blade width and thickness, 1.5% for hub diameter, and 2.0% for overall diameter were established.

These allowances are expected to minimize post-casting grinding work, reduce rework and material waste due to dimensional errors, and ultimately enhance the quality stability of the cast products.

4. Conclusion

This study proposed a methodology for accurately predicting thermal expansion and cooling contraction deformations occurring during the sand casting process and for establishing optimal dimensional allowances to compensate for these deformations.

In marine propeller manufacturing, deformations arising during the casting process are directly linked to dimensional precision and quality assurance, and excessive finishing work and rework due to dimensional errors have been major causes of productivity degradation and cost increases. Thus, precise deformation prediction and optimal allowance setting are essential technical factors for producing high-quality propellers in the shipbuilding and marine industries.

In this study, a 1.1-meter class marine propeller made of aluminum bronze alloy (ALBC3) was analyzed using both thermal expansion formulas and finite element analysis (FEM) to quantitatively evaluate thermal expansion and contraction deformations during the casting process.

Theoretical calculation results indicated expansions of 6.4 mm in blade width, 0.87 mm in thickness, 3.36 mm in hub diameter, and 20.8 mm in overall diameter. FEM analysis results showed slightly smaller deformations: 6.2 mm, 0.85 mm, 3.32 mm, and 20.5 mm, respectively. Both approaches exhibited similar trends, but FEM results reflected the effects of thermal gradients and constraint-induced contraction during cooling, which were not considered in the theoretical approach. Based on comprehensive comparison and analysis, optimal dimensional allowances of 1.9% for blade width and thickness, 1.5% for hub diameter, and 2.0% for overall diameter were established. These allowances can serve as practical standards to minimize post-casting grinding, reduce rework and material waste, and enhance the quality stability of the final propeller products. Stress distribution analysis confirmed stress concentration at the hub joint and blade root regions.

Although the maximum stress was found to be within the material’s yield strength, additional quality control through nondestructive testing (NDT) is necessary due to the risk of micro-crack formation in these concentrated areas. Modal analysis and Campbell Diagram evaluation revealed that the risk of resonance within the actual operational speed range was low.

However, as minor deformations during the casting process could still impact dynamic characteristics, continuous management of dimensional accuracy during design and production stages was identified as essential. By comparing and verifying theoretical and FEM-based approaches, this study presented a reliable method for setting optimal dimensional allowances applicable to marine propeller casting processes. The findings are expected to serve as a fundamental reference for controlling dimensional deformations and ensuring quality stability in various casting products beyond propellers. The application of optimized dimensional allowances contributed not only to improved dimensional precision but also to overall manufacturing process efficiency. Material input was reduced, and post-processing operations were simplified, resulting in significant savings in both material costs and working time. Moreover, the transition from a conventional 2–3-person manual casting process to a 1-person-centered operation was realized.

As shown in Table 6, material consumption was reduced by up to 23 kg, leading to cost savings of over 300,000 KRW. Post-processing time was reduced by more than half, and labor costs were cut by up to 60%. These quantitative results demonstrate that the dimensional allowance design method proposed in this study is not only technically feasible but also highly economical.

Future work will involve conducting actual casting experiments to verify the theoretical and FEM results obtained in this study. Furthermore, the effects of casting process variables such as mold material, cooling rate, and external constraints will be additionally analyzed, leading to the establishment of more precise design standards and process management strategies.