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BY 4.0 license Open Access Published by De Gruyter Open Access November 16, 2022

Numerical simulation and mathematical modeling of the casting process for pearlitic spheroidal graphite cast iron

  • Abdullah Tahir Şensoy EMAIL logo
From the journal Open Chemistry


Fluidity, an essential parameter in casting operations, is influenced by the thermal properties of the alloy and the mold, pouring temperature, modification, and inoculation of the alloy. In this work, pearlitic spheroidal graphite cast iron (PSGI) was studied in terms of fluidity characteristics. The sand mold used had different section thicknesses. The alloy was cast at pouring temperatures changing between 1,360–1,480°C. Liquid metal quality ranged between 10 and 90% where the section thickness was changed from 1 to 5 mm. FlowCast simulation tool was used for the modeling of the tests. The analysis of variance results of the response surface model constructed between certain casting parameters and fluidity length have shown that the reduced regression equation is very accurate in terms of statistical indicators (predicted R 2: 0.99). The sensitivity analysis has shown that the section thickness is the most dominant parameter on the fluidity, while metallurgical quality is the least. The proposed model was also compared with the studies in the literature and it was seen that the results are well-matched. Therefore, it was suggested to use the proposed equation in order to estimate the fluidity results of PSGI without the need for real casting operations.

1 Introduction

Pearlitic spheroidal graphite cast iron (PSGI) consists of carbons in the form of spheroidal shape due to the addition of Mg or Ce to the alloy during the casting process [1]. This alloy has been used as an indication of the growth rate in developing countries. Amongst all the castable alloys, spheroidal cast iron has a large portion where it mainly finds its application area in machine components in the industry [2]. This alloy has a low melting point, good fluidity, and machinability characteristics. It also reveals similar tensile strength, toughness, and ductility as steel and thereby it is a preferred choice of material which is also known as ductile iron [3,4,5]. The matrix of spheroidal cast iron can be subjected to heat treatment to obtain bainitic or martensitic structure to have high tensile strength and behave as a brittle material. On the other hand, pearlitic, ferritic and/or pearlitic–ferritic structure can also be formed to exhibit lower strengths with high ductility. Thus, a wide range of properties can be found in this alloy. The common one is the pearlitic ductile iron that finds a commercially wide range of application areas [2,6].

PSGI, also known as ductile iron, is a structural material which can be used to cast thin wall parts. Its enhanced properties, particularly the high ratio of strength over density, help this alloy to reveal a higher advantage over aluminum alloys making it the economical alloy of choice [7,8,9,10,11]. The superior combination of such features of ductile iron makes it an alternative alloy to be selected and used in different applications [10,12,13,14]. Górny and Tyrała [15] remark that the numerical modeling methods are of great importance in terms of the solidification of thin-wall ductile iron castings as well as forecasting the microstructure of the manufactured part.

The structure of ductile iron is known to be affected by cooling rate, chemical composition, heat treatment, and cutting speed. Cooling rate, section thickness, pouring temperature, and mold material are the functions of microstructure. Therefore, it is important to note that the cooling rate is in direct relationship with the solidification in thin sections [15,16].

Fluidity, in the case of foundry terminology, is the length of the flow of liquid metal until it solidifies. It determines the filling ability of the liquid alloy that flows through the runners, mold cavity, and feeders, and thereby when no incomplete filling occurs, the cast part can be solidified without any defect formation. Since cast parts are in many different geometries, the fluidity of the alloy plays an important role in terms of the determination of its ability to fill the mold cavity. The alloys with low fluidity, therefore, have the potential of not filling the cavity which may result in misruns [17]. The fluidity characteristic of liquid alloys is an essential property of cast alloys [18]. Particularly for ductile iron, it depends on casting conditions and microstructure [19,20]. For example, metallurgical factors that affect fluidity can be listed as follows: viscosity, composition, superheating, latent heat, melting point, oxide content, density, surface tension, and solidification conditions [21,22]. Mold/metal reactions are the other factors that need to be considered such as conductivity, mold temperature, and heat transfer coefficient [20,23].

In sand castings, not just the chemical composition, but pouring temperature and flow rate of the molten metal in the mold cavity is also important [24]. There are several works in the literature on the fluidity of ductile iron in sand castings [25,26,27,28]. Górny [29] studied the fluidity of ductile iron in 1, 2, and 3 mm thickness channels in different mold materials. In a similar work [30], it was reported that the heat loss in thin sections was significant and thus high heat transfer of iron had led to faster cooling and mold filling was impeded due to a drop in temperature below eutectic; and the solidification and microstructure were affected. Han and Xu [31], showed that the fluidity was inversely proportional to the increased solidification range of an alloy. As the difference between solidus and liquid temperatures is increased, fluidity is decreased. However, under high-pressure castings, this feature becomes irrelevant. Haque [32] added Si and Al to ductile iron. 1 m length of the mold with different section thicknesses was used to characterize the fluidity. Experimental work had shown that Fe–C–2Si ductile iron revealed higher fluidity than Fe–C–2Al ductile iron regardless of the channel thickness. Jafari et al. [33] measured the fluidity lengths changing from 1 to 8 mm thickness with and without coating. It was found that when the coating was applied to the mold, the fluidity lengths were decreased.

Aslandoğan [34] used a spiral mold to evaluate the fluidity of AISI 1040 steel. The change in fluidity at different ratios of Cr and Ni addition was studied. It was reported that the most significant factor that affected the fluidity was the pouring temperature, and the addition of Cr and Ni increased the fluidity of steel. Fraś et al. [35] showed that the fluidity of cast iron was increased with pouring temperature more pronouncedly than the carbon equivalence. Yang et al. [36] added 3 wt% Si and 2–6 wt% Ni to aluminum alloys and found that the fluidity length in the spiral mold increased with the increase in the Si and Ni content.

Although there are various studies in the literature about the effect of different gating systems, grit sizes, mold materials, and other variables on flowability, to the best of the authors’ knowledge, there is no response surface study available that can provide numerical data for casting the part with varying casting temperatures and metallurgical qualities in different section thicknesses. On the other hand, as an important industrial material, PSGI has been far less investigated in terms of fluidity in the foundry applications. Therefore, this study focused on developing a mathematical model for the PSGI casting process by constructing a relationship between certain casting parameters and the corresponding response as the fluidity length.

2 Materials and methods

2.1 Simulation

In the scope of the study, the sand-casting process of the PSGI was simulated under varying casting conditions. Thus, the feed distance of the liquid metal has been determined depending on the selected process parameters which are section thickness, casting temperature, and metallurgical quality. Their ranges were set as 1–5 mm, 1,360–1,480°C, and 10–90%, respectively. Metallurgical quality was based on the type, quality, quantity, and effectiveness of the inoculant within the experience of the foundry. These parameters were selected considering the manual intervention capability in the foundry during the manufacturing and their impact rate on the selected response as the fluidity of the alloy. Another selection criterion was the impact rate of the parameters on the fluidity properties of the alloy in the sand mold casting method. Model geometry was obtained by implementing the fluidity test model on a rectangular-shaped prism (20 mm wide, 500 mm long, and 1 to 5 mm height) based on a similar study in the literature [26]. The liquid metal canal length was deliberately kept long in order to prevent the liquid metal from full feeding under varying conditions, and thus, it was aimed to measure the distance of liquid flow. In addition, using the model developed within the scope of this study, it was aimed to determine the thickness of the parts that can be successfully manufactured for selected process parameters.

The numerical casting simulation process is a valuable computational tool to reliably forecast the parameters of the actual pouring operation [37]. The FlowCast® program uses calculations of the finite difference method for the numerical casting simulations. Using the above program, under user-defined boundary conditions, casting process can be simulated for any casting geometry, for various casting and mold materials. The software database will describe the material properties for the simulation process. In addition, any substance can also be specified using its thermo-physical parameters manually.

For the simulation process, the fluidity test model geometry constructed in a 3D modeling software was saved as a *.STL format. Thereafter, the model was transferred to a commercial software, SolidCast® as preferred in many casting simulation studies [38,39,40], in order to perform the casting simulations. The thermo-physical properties of the mold and the casting material are described as shown in Table 1, where the data were obtained from the commercial simulation software database. The simulation analysis was conducted using the FlowCast® which is a subroutine of this software. As a matter of theory, this software simulates liquid metal flow in the mold cavity as follows from the principles of fluid mechanics founded on Darcy’s law:

(1) Q = K A μ   ( Pb Pa ) L ,

where K is the permeability, Q is the flow rate, A is the overall cross-sectional thickness, μ is the viscosity, L is the flow rate, and Pa and Pb are inlet and outlet pressures, respectively. In addition, in order to eliminate undesirable situations, many restrictions are considered for simulations such as turbulence, underfill, cold shut, etc.

Table 1

Thermo-physical properties of the casting material and the mold [26]

Material Thermal conductivity (W/m.K) Specific heat (J/kg K) Freezing range (°C) Density (kg/m3) Latent heat of fusion (J/kg)
Casting material PSGI 25.944 460.24 44.63 7176.06 230115.6
Mold Silica sand 0.59 1075.288 1521.71

After defining the material properties, the solid model transferred to the casting simulation software was meshed and the boundary conditions were defined. For the meshing process, 12 million mesh was created considering that the thinnest section includes at least three cubic elements. The aforementioned casting simulation program simulates the pouring of liquid metal into the mold cavity according to fluid dynamics criteria.

The filling time was taken as 5 s for each event. A sample simulation (cross-sectional thickness: 5 mm, pouring temp: 1,480°C, metallurgical quality: 50%; Table 3) was represented in Figure 1. For all measurements, an image analysis program was used which has been introduced by Clemex Vision Lite (Meyer Instruments Inc., Houston, US). The canal length was described as 500 mm; this reference was followed up by all measurements (Figure 1).

Figure 1 
                  Simulation and measurement of the 1st trial.
Figure 1

Simulation and measurement of the 1st trial.

2.2 Mathematical modeling

This study aimed to get a regression equation between the response parameter (fluidity distance) and the selected casting parameters. For this purpose, a commonly used experimental design method [41], Box–Behnken design (BBD), combined with the response surface methodology was used to define the input parameter values for the experiments. Output parameter of the regression equations was chosen as the fluidity distance values obtained from simulations. Linear response surface model (LRSM), equation (2), and a full-quadratic response surface model (FQRSM), equation (3), were considered as follows:

(2) y = β 0 + i = 1 k β i X i + ε ,

(3) y = β 0 + i = 1 k β i X i + i = 1 k β i i X i 2 + i = 1 k 1 j = i + 1 k β i j X i X j + ε ,

where y is the corresponding response, which is the fluidity length in this study, β 0 denotes a constant term, ε is the error term, β i , β i i , and β i j are the coefficients for linear, quadratic, and interaction effects, respectively.

3 Results and discussion

Analysis of variance (ANOVA) results have shown that LRSM has relatively low coefficient values (R 2: 0.92, adjusted R 2: 0.90, and predicted R 2: 0.85) compared to FQRSM. Even though the full-quadratic model presents satisfactory results in terms of statistical indicators of R 2 (0.999), adjusted R 2 (0.999), and predicted R 2 (0.997), model reduction may be useful in terms of constructing a simpler equation for the computational efficiency. Therefore, a stepwise reduction of the model was performed by removing non-significant terms whose determined p-values were greater than 0.05 in each step. After 3 steps, all terms are found to be statistically significant and the reduced-quadratic response surface model (RQRSM) has been obtained. Statistical indicators of RQRSM were determined to be 0.999, 0.998 and 0.996 for R 2, adjusted R 2, and predicted R 2, respectively. Table 2 presents the ANOVA results for LRSM and RQRSM for the fluidity length.

Table 2

ANOVA results for LRSM and RQRSM and the standardized effects of the factors

Source Degrees of Freedom Adj Sum of Squares Adj Mean Squares F-value P-value Remarks
LRSM 3 30853.9 10284.6 43.95 0.000 Significant
x(1) 1 25185.5 25185.5 107.62 0.000 Significant
x(2) 1 5376.8 5376.8 22.98 0.001 Significant
x(3) 1 291.5 291.5 1.25 0.288 Non- significant
Error 11 2574.3 234.0
Lack-of-fit 9 2574.3 286.0 * *
Pure error 2 0.0 0.0
Total 14 33428.2
RQRSM 5 33395.0 6679.0 1814.19 0.000 Significant
Linear 3 30853.9 10284.6 2793.58 0.000 Significant
x(1) 1 25185.5 25185.5 6841.06 0.000 Significant
x(2) 1 5376.8 5376.8 1460.49 0.000 Significant
x(3) 1 291.5 291.5 79.18 0.000 Significant
Square 1 2250.4 2250.4 611.28 0.000 Significant
[x(1)]2 1 2250.4 2250.4 611.28 0.000 Significant
2-Way interaction 1 290.7 290.7 78.96 0.000 Significant
x(1) * x(2) 1 290.7 290.7 78.96 0.000 Significant
Error 9 33.1 3.7
Lack-of-fit 7 33.1 4.7 * *
Pure error 2 0.0 0.0
Total 14 33428.2

As can be seen from the pareto chart of the standardized effects of the factors on fluidity, cross-sectional thickness is the most dominant factor, while metallurgical quality is the least.

Using MINITAB software, LRSM and RQRSM equations were obtained as follows, respectively:

(4) y LRSM = 539 + 28.05 x 1 + 0.4321 x 2 + 0.151 x 3 ,

(5) y RQRSM = 194.4 109.7 x 1 + 0.2190 x 2 + 0.1509 x 3 + 6.138 [ x 1 ] 2 + 0.07104 x 1 x 2 .

The results of the test parameters and the resulting liquid metal flow distance are given in Table 3 below. According to the results, RQRSM can better estimate the simulation results.

Table 3

Experimental design based on BBD and the response values

Experiment no. Cross-sectional thickness (mm) (x 1) Casting temperature (°C) (x 2) Metallurgical quality (x 3) Fluidity distance (mm) (y)
Simulation Estimated %Error
1 5 1,480 50 269.89 248.31 267.91 8.00 0.73
2 5 1,420 10 224.43 216.34 227.42 3.60 1.33
3 5 1,420 90 238.64 228.42 239.50 4.28 0.36
4 3 1,420 50 153.41 166.28 152.90 8.39 0.33
5 3 1,360 90 133.52 146.40 133.01 9.64 0.38
6 3 1,420 50 153.41 166.28 152.90 8.39 0.33
7 5 1,360 50 201.70 196.46 199.01 2.60 1.34
8 1 1,420 10 116.48 104.14 115.40 10.59 0.92
9 3 1,360 10 119.32 134.32 120.93 12.57 1.35
10 3 1,420 50 153.41 166.28 152.90 8.39 0.33
11 3 1,480 90 184.66 198.25 184.86 7.36 0.11
12 1 1,480 50 139.20 136.11 138.84 2.22 0.26
13 1 1,420 90 125.00 116.22 127.48 7.02 1.98
14 1 1,360 50 105.11 84.26 104.04 19.84 1.02
15 3 1,480 10 173.30 186.17 172.79 7.43 0.30

As can be seen from the response surface plots, as the cross-sectional thickness and casting temperature increase, the fluidity increases (Figure 2a). For a fixed flow distance and metallurgical quality, there is a negative correlation between casting temperature and cross-sectional thickness. In other words, the combination of the selected two parameters can be adjusted in terms of operational flexibility. When all graphs are carefully examined together, it can be easily deduced that the less significant factor affecting the fluidity is the metallurgical quality which is a previous finding of the study from Table 1. Another interesting finding is the difference between the significance levels of the terms between linear and nonlinear models. ANOVA results given in Table 2 show that metallurgical quality is found to be non-significant for LRSM. However, variance analysis of RQRSM indicates that all main terms are significant. Even though metallurgical quality has the least effect on fluidity, its effect should not be neglected. Examination of Figure 2b reveals that for a constant casting temperature, there is a slight increase in the flow distance for each level of cross-sectional thickness.

Figure 2 
               Response surface plots for the interaction effect of (a) cross-sectional thickness and casting temperature, (b) cross-sectional thickness and metallurgical quality, and (c) casting temperature and metallurgical quality.
Figure 2

Response surface plots for the interaction effect of (a) cross-sectional thickness and casting temperature, (b) cross-sectional thickness and metallurgical quality, and (c) casting temperature and metallurgical quality.

As can be deduced from Figure 2b that for each level of casting temperature, metallurgy quality has a positive contribution to the fluidity. This finding is well-matched with the literature [42]. However, since the other parameters dominate the regression equation, metallurgical quality gives a slight change for response surface plots. Similar findings were also reported in refs [5,33]. As can be seen from Figure 2c, casting temperature is inversely proportional to metallurgical quality for a constant value of flow distance. In our study, as the cross-section thickness increased, castability increased, as expected [24,29]. When all response plots are carefully examined, it can be seen that the cross-sectional thickness is the most dominant factor between the selected input parameters. This finding also supports the previous studies conducted on the fluidity of casting alloys [43]. Jafari et al. [33] reported that the minimum section thickness that could fill 500 mm length was 5 mm. And below 3 mm, no filling was observed in ductile iron. When their values are substituted into equation (5), it can be seen that the results coincide. While their fluidity result is nearly 500 mm for a thickness of 8 mm under the pouring temperature of 1,450°C, for the same input values the response value determined by equation (5) is 476.02 mm with a difference of 4.7%. The flow distance-thickness trend is also consistent with the findings of Borowiecki [24]. However, as shown in Figure 3, the gap between the current study is increased when compared to ref. [33]. The possible reasons can be specified as the cross-section geometry of the canal and the difference in the test procedure followed.

Figure 3 
               Comparison of the flow distance results of the current study with the literature [24].
Figure 3

Comparison of the flow distance results of the current study with the literature [24].

Casting temperature which is another important parameter of the casting process gives a positive contribution to the fluidity of PSGI. For a constant value of cross-sectional thickness, pouring temperature is almost linearly proportional to the flow distance. This result is also well-matched with the existing studies [30,44].

In another similar study conducted on determining the flow distance of ductile iron by using artificial neural networks (ANN), 108 experiments were done for training and testing the model. Since the material properties of the casting metal and the three selected input parameters except filling time are the same with this study, the proposed regression equation is used for determining the sample design points given in Table 4. As the filling time of each simulation is presumed as 5 s, the closest design points (6 s) in the ANN model considered are used as input parameters in the model conducted in this study (RQRSM) and the results are well-matched as shown in Table 4. However, if filling time dramatically decreases, in other words pouring speed remarkably increases, the flow distance of the molten metal notably increases. When other parameters are fixed, reducing the filling time by 50% results in an increase in flow distance by 68% even metallurgical quality reduces by one-fifth [26]. Therefore, the difference percentage increases with the deviation of pouring speed. For instance, for 2nd experiment, the difference percentage is about 57% since the filling time decreases to 3 s.

Table 4

Comparison of the results of the present study with the ANN model [26]

Experiment no. Cross-sectional thickness (mm)(x 1) Casting temperature (°C)(x 2) Metallurgical quality (x 3) Flow distance (mm)(y)
RQRSM ( y 1 ) (this study) ANN model ( y 2 ) [26] % Difference | y 1 y 2 | y 1 × 100
1 5 1,400 10 215.93 200.54 7.13
2* 5 1,500 90 285.43 448.10 56.99
25 5 1,400 50 221.97 224.05 0.93
50 3 1,450 90 171.89 161.66 5.95
107 5 1,450 10 244.64 235.39 3.78
108 1 1,500 10 138.60 129.34 6.68

*Gap is particularly attributed to pouring speed.

By using only nearly one-eighth of the experiments required for ANN, a stronger predictive model is achieved by RQRSM. The difference between the results can be attributed to the filling time which is not taken into account as a parameter for the current study. On the other hand, the difference between the prediction accuracy of the compared models may be a sub-factor having little effect on the gap marked in Table 4.

4 Conclusion

This study offers a mathematical model for an accurate prediction of the fluidity of PSGI using only a few experiment points, unlike ANN. The fluidity length (y) of the PSGI has been found to be in relation with section thickness ( x 1 ), casting temperature ( x 2 ), and metallurgical quality ( x 3 ) in accordance with the equation below:

y RQRSM = 194.4 109.7 x 1 + 0.2190 x 2 + 0.1509 x 3 + 6.138 [ x 1 ] 2 + 0.07104 x 1 x 2 .

Even though LRSM provides an acceptable mathematical model with less computational expense, RQRSM gives more accurate results in terms of R 2 values as well as the errors determined between simulation results and the estimated results. Therefore, the reduced nonlinear model proposed in this study is recommended for real foundry applications without the need for any simulation or experiment. For future studies, the effects of different section types and other factors affecting the fluidity may be investigated to determine the best topology of the runners as well as to offer a more generalized mathematical model for casting applications.

  1. Funding information: This research received no external funding.

  2. Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.

  3. Ethical approval: The conducted research is not related to either human or animal use.

  4. Data availability statement: The authors confirm that the data supporting the key findings can be made available upon request to the corresponding author.


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Received: 2022-10-11
Revised: 2022-10-22
Accepted: 2022-10-26
Published Online: 2022-11-16

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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