2021 Volume 62 Issue 3 Pages 461-467
Mg Cored wire method is a type of spheroidization treatment method. It has been introduced by many foundries from the viewpoints of improving the environment safety of factory workers, automation and productivity. However, the prediction of residual Mg content depends on experience because various factors complexly affect each other and the effects of each factor on Mg yield rate have not been clarified theoretically.
In this research, we regard the relationship between these factors of spheroidization and residual Mg content as a nonlinear optimization problem. Therefore, we attempted to predict residual Mg content in the ladle and the product after pouring, which was spheroidized by Mg Cored wire method using the layered neural network (ANN). In ANN learning, the method of using the actual measured data as it is, the method of utilizing engineering knowledge, and the method of correcting data noise were examined. In addition, we constructed ANN for the relationship between the operation conditions of the spheroidizing treatment and residual Mg content in the ladle, and the relationship between the residual Mg content of the ladle and pouring conditions and residual Mg content of the product.
As a result, the predicted residual Mg content in the ladle using ANN could be estimated with the correction coefficient R = 0.91. According to the constructed ANN, the higher the residual Mg content in the ladle, the lower is the yield rate of the Mg in the product. The yield rate of Mg could be improved to 54% by reducing the residual Mg content in the ladle to 0.042%. It was also clarified that the residual Mg content is the same in the ladle and the product. When this result was verified in an actual plant using 25% Mg wire, the yield rate of Mg improved by 53% at a monthly average and to about 65% at the highest value.
This Paper was Originally Published in Japanese in J. JFS 92 (2020) 408–415. The captions of Fig. 3–10, 12, 13 are slightly changed. Reference 11, 12) were removed due to no citation.
When industrially producing spheroidal graphite cast iron, approximately 0.03–0.10 mass% magnesium (hereinafter referred to as % Mg) is added to the base iron to exclude sulfur (S) and oxygen (O) that inhibit graphite spheroidization. By doing so, graphite becomes rounded.1) Mg shows a low boiling point of about 1100°C, does not form a solid solution with iron and its solubility in molten cast iron is low. For the reasons, special treatment is required to add Mg to the molten metal at around 1500°C and achieve an efficient yield rate, and various treatment methods such as the sandwich method, Tundish method and Plunger method have been implemented.2)
Mg Cored wire treatment,3,4) one of the graphite spheroidizing methods, has been introduced by many foundry factories, from viewpoints of worker safety, environmental improvement, automation and productivity improvement. The Mg Cored wire treatment shows the four main advantages as follows:
In order to determine the effects of such multidimensional factors, multiple regression analysis is used as often. However, the multiple regression analysis has a problem that the output cannot be expressed by a non-linear function, such as y = log(x) or sin(x). It also has a problem that the problem cannot be solved when the factors influence each other in a complex manner, such as an exclusive logical sum, where the result is true only when one of the two factors is true, and the result is irrelevant if both are true or false. For this reason, in Mg Cored wire treatment, a method to predict the yield rate of Mg that can answer the demand for low Mg in foundry field has not yet been found.
In this research, we consider a problem with many factors and complicatedly interacted between them as a non-linear optimization problem, and use a step type artificial neural network (hereinafter referred to as ANN). A neural network is a model that simplifies the way of the human brain processes. By using this ANN, nonlinear problems that cannot be solved with conventional multiple regression analysis can be solved. In this paper, as a learning method in ANN and as a data optimization preprocessing, the following methods are examined; a method of simply using existing raw data, a method of using theoretical data and a method of correcting data noise.
Finally, we will use ANN to calculate the influence of various factors on the ladle Mg yield rate and the product Mg yield rate. In addition, the aim is to improve the yield rate of ladle residual Mg and product residual Mg by inputting the ANN non-linear formula to the Mg Cored wire treatment equipment.
In this study, calculations were performed using the layered neural network shown in Fig. 1. This ANN had an input layer for inputting various factors, a medium layer (hidden layer) that adds weight to input data, and an outputting layer. This time, ANN was trained with teacher data of actual residual Mg value sampled at the foundry work field. With the non-linear model based on this training, the Mg residual content was predicted with this formula, after wire Mg treatment. As shown in Fig. 2, each layer consists of information processing elements called units (nodes), and the input/output relationship of the units is as follows; {xi} was a input to the unit, y is the output from a unit, and {wi} was the connection weight between the unit and the unit. Φ was an output function and θ was a threshold value for output judgement.5) Here, the output y was expressed as eq. (1). With substituting the value obtained by subtracting the threshold value of a certain unit from the total sum into the output function, the input/output relationship of a certain unit is expressed by eq. (1).
| \begin{equation} y = \varPhi\left(\sum w_{i}x_{i} - \theta\right) \end{equation} | (1) |
Schematic diagram of Layered Neural Network in this study.
Schematic diagram of calculation method in neural network.
(1) Expressing the input variables as several functional forms such as exponential function or hyperbolic function, and finding the one that shows with the highest correlation coefficient between the calculation result and the output variables. If found, it is normalized and expressed as input to the network. Here, normalization was a transformation that limits the range of data values to a certain range such as 0 to 1.
(2) The normalized input variables and the normalized hidden variables are represented by functions including thresholds and weights, and finally represented by sigmoid function or hyperbolic tangent functions (tanh). Here, Sigmoid function transformed the input data into a certain range as 0 to 1 and outputted them. It is calculated by eq. (2). On the other hand, tangent functions (tanh) transformed the input data into the range as −1 to 1 and output them. It was calculated by eq. (3). Expressing output variables in the same way as above.4)
| \begin{equation} f(x) = \frac{\text{e$^{x}$} - \text{e$^{-x}$}}{\text{e$^{x}$} + \text{e$^{-x}$}} \end{equation} | (2) |
| \begin{equation} f(x) = \frac{1}{1 + \text{e$^{-x}$}} \end{equation} | (3) |
(3) The normalized output value was converted to the actual scale and the final solution was displayed. In addition, sensitivity analysis8) was performed to quantitatively show the contribution of the input variables to the determination of the neural network output. Sensitivity analysis was to determine the degree of influence of various factors. In this case, it was to see which shows the greatest influence among the following factors; the added Mg amount, the Mg treated temperature, and the weight of the molten metal. The sensitivity of each input variable is calculated by the following formula.
The sensitivity of each input variable was first partially differentiated with respect to the input variable x = j of interest for the nonlinear model, and then the slope at that time was obtained in the entire region of the function, and the average value was calculated. The slope [a] was calculated by the following equation (4).
| \begin{equation} \text{a}_{j} = \frac{\partial f(x,y,z\ldots)}{\partial x}\bigg|_{x = j} \end{equation} | (4) |
| \begin{equation} \frac{1}{\text{n}}\sum\nolimits_{\text{k} = 1}^{\text{n}}\text{a}_{j}\bigg/\frac{1}{\text{n}}\sum\nolimits_{\text{k} = 1}^{\text{n}}(\text{a}_{j} - \overline{\text{a}_{j}})^{2} \end{equation} | (5) |
As the first step, we simply inputted raw data into ANN about the residual Mg in the ladle, and used it as a learning data. The input factors were (1) the specific surface area of the molten metal in the ladle (cm2), (2) the amount of Mg added (mass%), (3) the Mg treatment temperature (°C), (4) the feed speed of the Mg Cored wire (m/min), and (5) the composition of the base iron (mass%). The actual ladle Mg yield rate was used as teaching data for learning, and the number of data was 300. The standard range of spheroidization graphite cast iron was set as 6 types from JIS FCD400 to FCD800.
Theoretical data was inputted as a normalized method and the influence of various factors was re-examined. We assumed that the following reactions occurred during the Mg spheroidization treatment in the ladle.
| \begin{equation} \text{Mg} + \text{S}\to \text{MgS} \end{equation} | (6) |
| \begin{equation} \text{3Mg} + \text{N$_{2}$}\to \text{Mg$_{3}$N$_{2}$} \end{equation} | (7) |
| \begin{equation} \text{2Mg} + \text{O$_{2}$}\to \text{2MgO} \end{equation} | (8) |
Finally, the following factors were reconstructed as inputted factors of non-linear model. (1) specific surface of molten metal in ladle (cm−1), (2) added Mg amount in ladle (mass%), (3) Mg treated temperature (°C), (4) Wire sending speed (m/min), (5) carbon content in base iron (mass%), (6) silicon content in base iron (mass%), (7) Mg content reacting with sulfur (g), (8) Mg content reacting with nitrogen (g), (9) Mg content reacting with oxygen (g).
The output was set as Mg yield rate in ladle.
The range of input data were as follows: (1) 6.0–12.0 × 10−3 (cm−1), (2) 0.085–0.14 (mass%), (3) 1420–1470 (°C), (4) 32–40 (m/min), (5) 3.0–3.9 (mass%), (6) 1.1–2.4 (mass%), (7) 75–488 (g), (8) 77–535 (g), (9) 12–96 (g). The output factors are set as 40.3–53.4 (%). Data number was 300. With randomly dividing at a ratio 7:3 (learning data:test data = 7:3), a nonlinear model was constructed for each of the three factories with different product sizes by the Cascade correlation method. We improved the Mg yield rate by inputting the non-linear model for the correlation between the ladle residual Mg by ANN and various factors to the Mg Cored wire treatment device. As a result, the ladle Mg yield rate with 25% wire Mg is improved by 53% at a monthly average and is improved to around 65% at the highest value.
In this way, a non-linear model for investigating the relationship between the residual Mg content of the ladle and various factors was obtained by ANN. With inputting the non-linear model to the wire Mg treatment device, we tried to improve the Mg yield rate in the ladle. With investigating same ladles of three factories, and from the tendency of the three factories, we could determine the added Mg amount which provide the highest Mg yield rate in Mg Cored wire treatment.
Since the prediction of the ANN for the residual Mg amount in the ladle finished, we next constructed the ANN model for the residual Mg amount in the product for every three factories. With regard to the Mg yield rate of the product, various possible factors were inputted to the ANN, and the factors judged to have a relatively high degree of influence (sensitivity) by the ANN were selected. Among these factors, the factor that had a good correlation between the residual Mg amount of the actual product and the residual Mg amount of the product predicted by ANN was set as the final input factor.
In other words, we used (1) the amount of residual Mg in the ladle, (2) the product weight, (3) the casting temperature, and (4) the time from the end of the spheroidizing process to the pouring, as input factors, and learned the actual Mg yield rate value of the product as teacher data. The residual Mg content of the product was obtained by chemical analysis from the attached TP to the product. The number of data was 100. The effects of various elements on the molten metal had already been taken into consideration during the spheroidization process and were included in the data of the amount of residual Mg in the ladle. Similar ANN predictions were performed for the residual Mg content of the products of the three factories, and the residual Mg content of the ladle, where the residual Mg content of the product and the residual Mg content of the ladle were the same, was calculated. This was set as the target residual Mg content of the ladle.
In this study, 70% of all data was used as a learning data for ANN to derive nonlinear model. The remaining 30% of the data was used to test how correct the model derived by ANN were. Regarding the evaluation of the ANN model, the higher the correlation between the actual value and the predicted value is (correlation coefficient R), the better the ANN is. Finally, we investigated the changes in the added Mg amount to the ladle and the yield rate, before and after inputting ANN into the wire Mg device at the three factories.
As the first step, Fig. 3 shows the relationship between the actual ladle Mg yield rate (teacher data) and the predicted ladle Mg yield rate by ANN. As for the ANN network, the number of inputting layer unit is 7, that of hidden layer unit is 11. As for outputting layer, Sigmoid function is selected. The conditions of PC are as follows: 2.5 GHz, 2 core, memory 8 GB and learning time takes about 10 seconds. When the correlation coefficient between learning data and test data is best, various coefficient and function are determined and the learning finishes. The correlation coefficient is as low as R = 0.55, in the case of inputting raw data. Generally, it is judged that there is a strong correlation when the correlation coefficient R is 0.7 or more, and that there is a very strong correlation when the correlation coefficient R is 0.9 or more.10)
Relationship between actual Mg yield rate in ladle and predicted Mg yield rate with input raw data.
As shown in Fig. 4, from the sensitivity of Mg yield rate in ladle, the influence of various factors was classified on the following order by ANN: Mg addition > Ladle specific surface area > Mg treatment temperature > Base iron C amount > Base iron N amount > Base iron S amount.
Sensitivity of several factors on Mg yield rate in ladle with input raw data.
The field workers comment that the larger the product size is, the longer it takes to solidify and the fading proceeds, so the residual Mg content must be increased naturally. However, the ANN concludes contrary to the field workers’ feeling that regardless of the product size, the higher the amount of Mg added is (the higher the target residual Mg value is), the lower the Mg yield rate is, which is different from the field workers’ comment.
ANN also shows that the higher the C value is, the higher the Mg yield rate is. This is consistent with the foundry workers’ feeling that at the same carbon equivalent (CEL) value, the pearlite system with a low Si value and high C value has a better yield rate than the ferrite system with a high Si value. It is convinced under experience and theoretically that the shallower the depth in the ladle is and the higher the spheroidization temperature is, the lower the yield rate of residual Mg. In the raw data range, the yield rate of Mg shows a strongest correlation with sulfur or oxygen among various element in base iron, but the effect of nitrogen (N) is greater in this study. If the base iron amount before spheroidizing treatment is 0.01%S (100 ppm), 0.0020%O (20 ppm) and 0.0052%N (52 ppm), then the amount of Mg consumed by these elements is 134 ppm by N, 75 ppm by S and 30 ppm by O, and N shows the strongest influence on predicted Mg yield rate in ladle. In addition, the total amount of Mg consumed by these elements is as high as 0.024% (240 ppm).
Figure 5 shows the result, where raw data is converted into theoretical data and non-linear model was reconstructed. In other word, the relationship between the actual and the predicted ladle Mg yield rate is shown when the Mg amount of MgS, that of Mg3N2, and that of MgO were used as new factors. As for the ANN network, the number of inputting layer unit is 7, that of hidden layer unit is 13. As for outputting layer, Sigmoid function is selected. The conditions of PC are as follows: 2.5 GHz, 2 core, memory 8 GB and learning time takes about 10 seconds. When the correlation coefficient between learning data and test data is best, various coefficient and function was to be determined and learning finishes.
Relationship between actual Mg yield rate in ladle and predicted Mg yield rate with input logical data.
By adding the theoretical data, the correlation coefficient R increased from 0.55 to 0.76. In this way, the accuracy of ANN analysis is improved by replacing the raw data with theoretical data, using a relational expression known from engineering. This theoretical method is especially effective for analyzing small data.
As there is a difference between the actual Mg yield rate and the predicted Mg yield rate, it is thought that there is tacit knowledge that cannot be understood yet in the wire Mg treatment, such as the Mg evaporation and the reaction with air after spheroidization treatment.
Therefore, we tried to improve the accuracy by correcting the data for factors that may not be taken into consideration. Figure 6 shows the relationship between the actual ladle residual Mg and the predicted residual Mg obtained from the raw data and theoretical data.
Prediction of Mg yield rate with ANN and further analysis with human.
As shown in Fig. 6, it is thought that the prediction accuracy is improved while the actual Mg rate of the raw data and the predicted Mg rate result of theoretical data rotate. From this, it can be said that by correcting the unknown factors with θ2 in Fig. 6, the actual ladle residual Mg and the predicted ladle residual Mg by ANN can be matched. In other words, by rotating the individual values of the theoretical data in Fig. 5 by θ2 and correcting the prediction value, the actual ladle residual Mg by ANN and the predicted ladle residual Mg by ANN can be matched. By using this technique, even if unknown factors are unknown, it is possible to obtain the predicted ladle residual Mg from various factors. In this ANN analysis, correction was performed by rotation, but there may be cases of parallel movement that changes intercept.
As shown in Fig. 7, the difference between the actual ladle Mg yield rate and the predicted ladle Mg yield rate become even smaller and high accuracy is obtained (high correlation coefficient R = 0.91), due to the correction of unknown factors and the input of theoretical data.
Relationship between actual Mg yield rate in ladle and predicted Mg yield rate in ladle.
In this study, the effect of various factors on the output can be relatively quantified with the sensitivity analysis. This function is especially effective when the numbers of data and input factors are large. Figure 8 shows the relationship between the amount of Mg added to the ladle of three factories with different product sizes and the ladle Mg yield rate, obtained by applying this function.
Relationship between added Mg amount and Mg yield rate in ladle.
As ANN concluded that the effect of the amount of Mg added is the highest in Fig. 4, we investigated the relationship between the amount of Mg added to the ladle and the Mg yield rate. From Fig. 8, assuming that it is possible to apply outside the learning range as long as the relationship between the trained data is small, if the amount of Mg added to the ladle is reduced to about 0.078%, the ladle Mg yield rate in wire Mg treatment becomes the highest as 54%. This indicates that the yield rate becomes the best when the residual Mg content of the ladle is 0.042% (0.078 × 0.54).
3.3 Consideration of product residual Mg in three factoriesSince the prediction of the ladle residual Mg and the appropriate amount of Mg addition amount can be determined, we conducted ANN analysis for the product residual Mg, with using the ladle residual Mg as a factor. Figure 9 shows the relationship between the actual product Mg yield rate and the predicted product Mg yield rate with using ANN. As for construction of ANN, the number of the inputting unit is 4, medium layer is single one and its unit number is 6.
Relationship between actual Mg yield rate and predicted Mg yield rate in product.
As for outputting layer, Sigmoid function is selected. The conditions of PC are as follows: 2.5 GHz, 2 core, memory 8 GB and learning time takes several seconds. When the correlation coefficient between learning data and test data becomes best, various coefficients or functions are determined and the learning finishes.
The correlation coefficient is 0.74. Figure 10 shows the results of an investigation into the effect of various factors on the Mg yield rate in products.
Sensitivity of several factors on Mg yield rate in product.
ANN also calculates that the remaining amount of Mg in the ladle has the greatest effect on the Mg yield rate in products. In addition, within the factor range of the product weight (200 kg to 23000 kg), higher the product weight, the lower the residual Mg content, but the time from spheroidization to pouring has a greater effect on the residual Mg. It is thought that because the Mg in the molten metal decreases due to evaporation, reaction with oxygen and nitrogen in the air, or reaction with refractories. Therefore, it is considered that the time from spheroidization treatment to pouring becomes important.
ANN analyzes that the pouring temperature has a small effect, but the higher the pouring temperature, the higher the product residual Mg. As well as ANN, machine learning does not think the meaning of given data. As the most optimized answer is obtained only based on the input data and its range, we need to think deeply about the data numbers, the variation and the range of the answer.
Figure 11 shows the relationship between ladle residual Mg and product Mg yield rate at three factories. We focused only on the effect of the ladle Mg content that shows the greatest effect on the Mg yield rate in products. From this, assuming that it can be applied outside the training range if the relationship between the trained data is small, it can be said that when the ladle residual Mg value becomes about 0.042%, the Mg yield rate in the product becomes almost 100%.
Relationship between added Mg amount into ladle and Mg amount in product.
In other words, Fig. 11 shows that if the ladle residual Mg is reduced to about 0.042%, the ladle residual Mg and the product residual Mg become equal. At present, the results shown by ANN cannot be explained theoretically. However, it is one example of using ANN to find out what was not known before.
From a theoretical point of view, on the condition of within the factor’s training range and foundry working state, the amount of Mg that can be stably contained in the molten spheroidal graphite cast iron is less than about 0.042%, as a result of ANN analysis of the residual Mg in the ladle and in the product. The residual Mg exceeding 0.042% is considered to decrease even after casting, because it is easy to evaporate. In addition, even for large products, if the molten metal near the black skin is solidified, the evaporation of Mg and the reaction with the mold are hindered, so the effect of the product size is not as large as expected.
3.4 Reduction of ladle residual Mg with using ANN analysisWe found that it is desirable to reduce the ladle residual Mg to about 0.042% on December 2017. After that, the product residual Mg was investigated, and the ANN calculation formula was implemented to the Mg Cored wire treatment device around the end of February 2019. With determining the input factors except Mg addition amount and target residual Mg in the non-linear model, the optimum Mg addition amount is calculated and input length of Mg Cored wire is gained. Figure 12 shows the residual ladle Mg from March 2018 to October 2019.
Change of added Mg in ladle from March 2018 to October 2019.
The Mg Cored wire at that time was Fe–25%Mg alloy. In the past, we had tried many times to reduce the residual Mg value to prevent dross and shrinkage cavity. However, there was no concept that the higher the residual Mg content is, the lower the yield rate of Mg is. Therefore, once poor nodularity occurred, the lower ladle residual Mg content was often returned to the original high ladle residual Mg content. From Fig. 12, it can be shown that at the S factory which produces 0.2 to 8 ton small items, and at the O factory which produces 5 to 30 ton large items, the Mg addition amount decreased slightly. With adopting nonlinear model, the added Mg amount decreases remarkably from the end of February 2019. The medium G factory is carefully reducing the residual Mg.
Figure 13 shows the change in Mg yield rate in a wire Mg device. In the beginning of 2018, the Mg yield rate in the ladle in the three factories was about 44.0%. However, after adopting nonlinear model, in the latter half of 2019, the Mg yield rate in the ladle was improved to 47.0–54.0%. As the prediction is enough, the residual Mg yield rate in the ladle has become stable from July 2019. At the time of October 2019, the residual Mg in the ladle of S plant is 0.040% (0.079 × 0.51), that in the ladle of G plant is 0.041% (0.087 × 0.47), that of O plant is 0.043% (0.082 × 0.53), which achieves target value 0.042%. With inputting the nonlinear model, it is possible to control the residual Mg content and to add the necessary minimum Mg amount and achieve the highest Mg yield rate in the ladle.
Change of Mg yield rate from March 2018 to October 2019.
After Mg Cored wire treatment, the ladle residual Mg and the product residual Mg were predicted by ANN. As a result, the following results are obtained.
