Artificial Neural Network-Based Non-Linear Modeling and Simulation of CaO-SiO2-Al2O3-MgO Blast Furnace Slag Viscosity

Abstract

Blast furnace slags are formed by CaO-SiO₂-Al₂O₃-MgO systems and have several physical characteristics, one of which is viscosity. Viscosity is an important variable for the operation and blast furnace performance. This work aimed to model viscosity through linear and non-linear models in order to obtain a model with precision and accuracy. The best model constructed was a non-linear model by artificial neural networks that presented 23 nodes in the first hidden layer and 24 nodes in the second hidden layer with 6 input variables and 1 output variable named ANN 23-24. ANN 23-24 obtained better statistical evaluations in relation to 11 different literature equations for predicting viscosity in CaO-SiO₂-Al₂O₃-MgO systems. ANN 23-24 was also subjected to numerical simulations in order to demonstrate the validation of the non-linear model and presented applications such as viscosity prediction, calculation of the inflection point in the viscosity curve by temperature, the construction of ternary diagrams with viscosity data, and the construction of iso-viscosity curves.

1. Introduction

The viscosity of blast furnace slags has been investigated by several researchers [1,2,3,4,5], and these slags in conventional blast furnaces have CaO, SiO₂, Al₂O₃, and MgO mainly in their chemical composition. Viscosity is considered an important physical property for determining the stability and productivity in a blast furnace [1].

Zhou et al. [1] studied the viscosity of CaO-SiO₂-Al₂O₃-MgO slags with additions of TiO₂ and demonstrated that the action of Al₂O₃ generates the formation of complex silicates that help to increase viscosity and that the increase in basicity (CaO/SiO₂) also causes a decrease in viscosity.

Cheng et al. [2] demonstrated through the study of CaO-SiO₂-Al₂O₃-MgO slags with additions of TiO₂ and CaCl₂ that increasing the MgO/Al₂O₃ ratio helps improve the fluidity of these slags and improves thermal stability. The action of increasing the MgO/Al₂O₃ ratio acts by breaking complex silicate networks and consequently increasing the degree of depolymerization of these slags, which causes a decrease in viscosity.

Li et al. [3] investigated the viscosity of CaO-SiO₂-Al₂O₃-MgO-based slags through the influence of the addition of Al₂O₃ and TiO₂. The addition of Al₂O₃ between 10%–15% by mass increases the viscosity of this slag, and when there is the addition of Al₂O₃ above 18%, there is a decrease in viscosity due the formation of weak Si-O-Al bonds when an addition 18% occurs, which causes this decrease. The continuous addition of TiO₂ in these slags results in a decrease in viscosity because it denotes complex Si-O and Ti-O bonds, which corresponds to this variation in viscosity.

Tian et al. [4] analyzed the viscosity of CaO-SiO₂-Al₂O₃-MgO-based slags using the ratio (SiO₂ + Al₂O₃)/(CaO + MgO) and demonstrated that an increase in this ratio causes a decrease in the viscosity of these slags. In the same study, they demonstrated that there is a point where the viscosity of these slags presents an abrupt change and increases rapidly. This point is when the crystallization of the studied slags begins, which denotes an increase in the size of the crystals formed during cooling, which causes this loss of fluidity.

Zheng et al. [5] described that the increase in basicity (CaO/SiO₂) and the increase in the MgO/Al₂O₃ ratio in CaO-SiO₂-Al₂O₃-MgO slags with TiO₂ additions causes a decrease in Si-O bonds, which indicates an increase in the degree of depolymerization of these slags, resulting in a decrease in viscosity. They also built a mathematical model to predict the viscosity of these slags in a range of approximately 0.2 ≤ viscosity ≤ 0.4 Pa·s with an average percentage error of approximately 10%, considered a good, reliable, and highly accurate model.

The viscosity of CaO-SiO_2-Al₂O₃-MgO slags can be measured through linear modeling [2] and non-linear modeling using artificial neural networks [6,7].

The modeling of the viscosity of CaO-SiO₂-Al₂O₃-MgO slags at different temperatures through linear and non-linear methods is proposed in this work, with subsequent statistical evaluations and numerical simulations.

2. Materials and Methods

2.1. Data

2.1.1. Database

The database was taken from Duchesne et al. [6], where there are more than 4000 viscosity datapoints and datapoints on chemical composition, temperature, and other variables. In total, more than 50 different references were used in which more than 750 compositions were represented.

With the chemical composition and temperature data, the FactSage^® software (version 6.2) [8] was used to calculate the volume fraction of solids and liquid slag. The acquisition of the volume of solids and liquids in a given system for the calculation of viscosity is necessary to perform the viscosity correction, as described in the work of Roscoe [9].

2.1.2. Decisions

From the database, some filters were performed in order to obtain data on slags with the CaO-SiO₂-Al₂O₃-MgO system (mass fraction) with the related temperature (K), the volume of solids in the system at the given temperature (volume fraction), and the measured viscosity (log η = log₁₀ η, η in Pa·s).

The filters were as follows:

1. The values of CaO, SiO₂, Al₂O₃, and MgO should be greater than zero.

2. The sum of the values of CaO, SiO₂, Al₂O₃, and MgO should be between 0.9999 and 1.0001.

After the filters, the database presented a total of 197 datapoints. The data were subsequently submitted to the feature engineering process.

2.2. Preprocessing

2.2.1. Feature Engineering

Feature engineering can be described as the process that transforms raw data into features that improve the prediction efficiency of an algorithm. This technique can help in the interrelation between variables and a better representation of the phenomena involved, increasing the algorithm’s ability to identify patterns and dependencies between the data [10]. Examples of techniques used in feature engineering are feature extraction [11] and feature selection [12].

As the rheology and consequently the viscosity of the slag behave according to its molecular structure and the slag has SiO₂ as one of its main components, the CaO-SiO₂-Al₂O₃-MgO slag can be classified as a silicate melt where the random network model can be used to describe its viscosity [7,13].

Thus, the CaO, SiO₂, Al₂O₃, and MgO data were converted to Ca, Si, Al, Mg, and O data. According to the random network model, the components of silicate melts can be described as network formers (e.g., Si⁴⁺) that act by stabilizing the network and increasing viscosity, and network modifiers (e.g., Ca²⁺, Mg²⁺) that act by breaking the bonds between the components and decreasing viscosity. Another description is network amphoterics, such as Al³⁺, which can act as network formers or network modifiers depending on the system [7]. Oxygen, depending on its valence, can act as a network modifier with another network modifier or act directly on a SiO bond [14] (Equation (1)).

≡Si—O—Si≡ + O²⁻ → 2 ≡Si—O⁻(1)

Then, the data considered outliers were removed through the normal distribution of each variable. If a datapoint from a variable had an absolute value above 3 standard deviations in relation to the mean, it would be removed. With this, each variable presented approximately 99.7% [15] of its data, considering that the remaining 0.3% are outliers.

Data that presented a solids fraction value equal to 1 were removed because, according to Roscoe [9], liquid physical systems with spheres of various sizes with 100% solids would tend to have an infinite viscosity. The fact that the solids may not be strictly spherical does not significantly affect the formula developed by Roscoe (Equation (2)).

η = η_L(1 − c)^−2.5(2)

η is the viscosity of the liquid-solids system, η_L is the viscosity of the liquid, and c is the volume fraction of solids.

After the feature engineering processes, the database presented a total of 190 datapoints.

The feature engineering process resulted in two types of data: non-linear data and linear data. The non-linear data consist of data for Ca, Si, Al, Mg, and O, and the linear data consist of non-linear data for Ca, Si, Al, and Mg converted to their respective oxides, CaO, SiO₂, Al₂O₃, and MgO. Both contained the same amount of data with the respective temperature and viscosity.

The non-linear and linear data were partitioned into training and testing data with 70% and 30%, respectively. The testing data were used after training the models and were not used in training. The use of training and testing data partition helps against unexpected issues and overfitting that may arise when applying a given model [16].

2.2.2. Normalization

The normalization process was performed using the rescaling method [17] (Equation (3)).

(3)$x_{new}={\displaystyle \frac{x_i-X_{min}}{X_{max}-X_{min}}}$

X = [x₁, x₂, …, x_i, …] x_i is the i-th value of X, X_min is the minimum value of X, and X_max is the maximum value of X.

The normalization process results in more accurate results compared to non-normalized data, playing a fundamental role in helping to obtain data with greater accuracy [18]. Normalization also helps to stabilize the modeling process and is essential for efficient modeling [19].

2.3. Modeling

For modeling, the training database was used to adjust each model and estimate its hyperparameters. The training database is part of the total database because if the entire database were used during training, it would overfit the data, which could lead to low-quality predictions in different scenarios [16].

2.3.1. Linear Modeling

The modeling of the viscosity η of the CaO-SiO₂-Al₂O₃-MgO slag at a temperature T was performed by approximating the Laurent series expansion [20] of the linearized form of the VFT equation [21] (Equation (4)).

(4) $\log \eta =A+{\displaystyle \frac{B}{T-C}\approx {Dx}_1+{Ex}_2}$

η is the viscosity, A, B, C, D, and E vary depending on the chemical composition of the system, T the temperature, x₁ = 1/T, and x₂ = 1/T².

For this modeling, the least squares technique was used [22] considering b as a function of chemical composition and temperature and d as a function of the chemical composition using linear data. Thus, the viscosity is a function of five variables, considering η = f(CaO, SiO₂, Al₂O₃, MgO, T).

2.3.2. Artificial Neural Networks

The construction of artificial neural networks for viscosity prediction was performed by varying the number of nodes per hidden layer and by varying the number of hidden layers depending on the number of predictor variables. From a value k ≤ n + 4 with n equal to the number of predictor variables, the number of nodes per hidden layer (width) is w ≤ k1.5 and the number of hidden layers (depth) is d ≤ k + 2 when the artificial neural networks use the ReLU activation function [23].

Using a minimum k and n equal to 6 (Ca, Si, Al, Mg, O, and T), the value of w ≤ 31 and the values of d = 1 and d = 2 were used, considering that artificial neural networks with one hidden layer can approximate continuous functions [24] and that the use of two hidden layers also approximates continuous functions [25], usually have a better generalization in the modeling in relation to one hidden layer [26], and that the increase in depth implies a greater oscillation in the modeling, and this denotes a better approximation in continuous functions [27].

The optimization algorithm used in the construction of each artificial neural network was Adam [28]. The EarlyStopping [29] and the adaptive learning rate [30] were used.

Considering the variation in width and depth (w, d ∈ N), 992 different artificial neural networks were developed.

2.4. Predictions

In the predictions, the test database was applied to calculate the accuracy of each developed model. The test database helps to protect against unexpected problems and to estimate the model’s performance before its implementation [16].

2.4.1. Model Selection

The selection of the best model among the developed artificial neural networks was carried out through the variance (Equation (5)) of the deviations.

(5)$Var={\displaystyle \frac{\sum {\left(D_i-D_{mean}\right)}^2}{N}}$

Xt = [xt₁, xt₂, …, xt_i, …] as the test database, P = [p₁, p₂, …, p_i, …] as the predicted data and p_i = f(xt_i), d₁ = xt₁ − p₁, d₂ = xt₂ − p₂, …, d_i = xt_i − p_i, D = [d₁, d₂, …, d_i, …], D_i is the i-th value of D, D_mean is the mean of D and N the number of data in the test database.

The artificial neural networks with the lowest variance were then ordered up to the artificial neural networks with the highest variance.

A validation was applied to evaluate the order relationship from the artificial neural network with the lowest variance. Olden and Jackson [31] established the response curve, which indicates the contribution of an input variable by its variation in the output variable in different types of relationships. In practice, one input variable is varied, and all other variables are kept fixed at previously established percentiles in order to analyze the contribution of this variable that is varying in the output variable.

For the relationship between viscosity and temperature, a model must demonstrate an exponential response curve with a monotonically decreasing behavior according to the VFT equation (Equation (4)), considering a constant chemical composition.

Applying simulations, the relationships between viscosity and temperature were analyzed, and the artificial neural network with minimum variance and monotonically decreasing behavior of viscosity in relation to temperature was considered the best artificial neural network.

2.4.2. Statistical Evaluation

Each model, the best artificial neural network and the linear model, was evaluated using the mean absolute error (MAE) [32] (Equation (6)) and correlation coefficient (R) [33] (Equation (7)) metrics.

(6)$MAE={\displaystyle \frac{\sum \left|{Xt}_i-P_i\right|}{N}}$

(7)$R={\displaystyle \frac{Cov\left(Xt,P\right)}{S_{Xt}{S}_P}}$

Cov( ) is the covariance, S_Xt is the standard deviation of Xt, and S_P is the standard deviation of P.

The MAPE metric [34] was not used because some compositions, at a given temperature, present the logarithm of their viscosity close to or equal to 0.

Subsequently, the same test data were applied to the ANNLiq, BBHLW, Bomkamp, Duchesne, Kalmanovitch–Frank, Riboud, Shaw, Streeter, S2, Urbain, and Watt–Fereday models, described in the work of Vargas et al. [35] and Duchesne et al. [6]. The viscosity results of these 11 models were also subjected to MAE metrics.

2.4.3. Best Model

The best model, considering the best artificial neural network, the linear model, the ANNLiq, BBHLW, Bomkamp, Duchesne, Kalmanovitch–Frank, Riboud, Shaw, Streeter, S2, Urbain, and WattFereday models, was chosen as the one that presented the lowest MAE. The MAE is a metric capable of presenting the average error of a model.

The correlation coefficient presents the degree of linear relationship between two variables and varies between −1 and 1, with three types of classifications [36]:

• −0.3 ≤ R ≤ 0 or 0 ≤ R ≤ 0.3: There is a weak or low linear correlation;

• −0.7 ≤ R < −0.3 or 0.3 < R ≤ 0.7: There is a moderate linear correlation;

• −1 ≤ R < 0.7 or 0.7 < R ≤ 1: There is a strong or high linear correlation.

There is also a very high classification when the correlation coefficient has a value between 0.9 and 1 [37]. A negative correlation coefficient value indicates a monotonically decreasing correlation, and a positive correlation coefficient indicates a monotonically increasing correlation.

The evaluation through the correlation coefficient is based on the fact that the coefficient R(X, X) = 1 by definition. If two vectors, X and Y, have a high correlation coefficient (−1 ≤ R(X, Y) < 0.7 or 0.7 < R(X, Y) ≤ 1) and one vector is a function of the other vector, Y = f(X), then the vector Y can predict the vector X.

2.4.4. Numerical Simulation

Numerical simulations were performed to demonstrate the validation and application of the best model. In addition to the temperature variation, with fixed values of the chemical composition, in relation to viscosity, the variation in different chemical compositions, and therefore, different NBO/T [38], at different temperatures was also performed.

In order to demonstrate the applications of the developed model, simulations were performed to demonstrate the prediction of viscosity in chemical compounds at a given temperature, the viscosity inflection point in relation to temperature that can indicate different temperatures [39], and the construction of iso-viscosity curves [40] at different viscosity values and in different MgO compositions. There is also a description of other variables that have a correlation with viscosity, such as liquidus temperature [41], glass transition temperature [42], and electrical conductivity [43].

The flowchart of materials and methods can be seen in Figure 1.

3. Results and Discussion

The distribution of viscosity data can be seen in Figure 2. Figure 2 shows the difference between training data and test data, since the total data is the sum of the training and test data.

The minimum and maximum values of the linear data and non-linear data are shown in Table 1.

Applying the least squares technique to the linearized VFT equation demonstrated the following parameters:

• D = −50.38T + 10⁶(15.08%CaO + 20.61%SiO₂ + 17.62%Al₂O₃ − 8.44%MgO);

• E = 10⁹(−8.99%CaO − 23.41%SiO₂ − 13.83%Al₂O₃ + 33.80%MgO);

Regarding the test data, an MAE = 0.7581 log η and an R = 0.5078 were calculated when applied to the linearized VFT equation. The relationship between the test data and the data predicted by the linear model can be seen in the inset of Figure 5.

The construction of the artificial neural networks by varying the width and depth resulted in Figure 3, computed by the variance of the deviations.

The best artificial neural network with minimum variance of deviations and with monotonically decreasing relationship of viscosity in relation to temperature was the artificial neural network with 23 nodes in the first hidden layer and 24 nodes in the second hidden layer (ANN 23-24), considering the six input variables (Ca, Si, Al, Mg, O, and T) and one output variable (log η). It can be noted that some points in Figure 3, considering the values close to the maximum variance of deviations, presented a local minimum [44] and therefore did not obtain a good approximation in the modeling.

The training curve of ANN 23-24 can be seen in Figure 4.

As shown in Figure 4, the loss calculated from the mean squared error metric [32] has a value close to 0 when the number of epochs in the training of ANN 23-24 is increased and the maximum value of epochs is less than 10⁴, which demonstrates the action of the EarlyStopping technique.

The calculated MAE = 0.1879 log η and R = 0.9284. The MAE results of the linear model, ANN 23-24, ANNLiq, BBHLW, Bomkamp, Duchesne, Kalmanovitch–Frank, Riboud, Shaw, Streeter, S2, Urbain, and Watt-Fereday models can be seen in Table 2.

Table 2 indicates that ANN 23-24 has the lowest MAE in relation to the 11 literature models and the linear model developed. Therefore, ANN 23-24 was considered the best model to predict the viscosity of Cao-SiO₂-Al₂O₃-MgO slags.

The other models mentioned in Table 2, in addition to the Linear and ANN23-24 models, probably demonstrated higher MAE because they were modeled in chemical compositions with a different range than the Linear and ANN 23-24 models.

The MAPE was not computed in ANN 23-24 because the training data had values equal to or close to zero, but it can be estimated considering the MAE metric. The absolute average difference between the real viscosity (η_R) and the predicted viscosity (η_P) has the value of 0.1879 (|log η_R − log η_P| = 0.1879). It can be considered by the property of the logarithm that |n_R/η_P| = 1.5414, which indicates an approximate error of 54.14%. Mills et al. [45] indicate that models with an error of 25–35% are reasonable, but in the work, the viscosity data vary approximately between −1 ≤ log η ≤ 1, which is a much smaller range when compared with the range in the present work (−1 ≤ log η ≤ 6).

The result of the correlation coefficient between the test data and the data predicted by ANN 23-24 indicates that there is a high correlation between the data, considering its collinearity and therefore the precision and accuracy of ANN 23-24.

The graph that relates the test data and the data predicted by ANN 23-24 can be seen in Figure 5.

From Figure 5, it is possible to see that there are points far from the predicted data = test data line. Among these points, two of them have a standard deviation greater than three in relation to the deviations, and if they were removed from the test data, ANN 23-24 would result in an MAE equal to 0.1136 log η and an R equal to 0.9864.

The deviations of ANN 23-24 can be seen in Figure 6.

The MAE line is drawn in Figure 6 to demonstrate the accuracy of ANN 23-24. A total of 77.19% of the data predicted by ANN 23-24 demonstrated a deviation smaller than the MAE (0.1879 log η) from the test data. The inset of Figure 6 indicates that the deviations of the data predicted by ANN 23-24 from the test data are most frequent near the deviation equal to 0, with a few points above the deviation equal to 1.

The variation in viscosity as a function of temperature considering the data of Ca, Si, Al, Mg, and O constants can be seen in Figure 7.

Figure 7 shows the constant data for Ca, Si, Al, Mg, and O in relation to the 25th, 50th, and 75th percentiles, in mass, of each component in relation to the training data. Figure 7 demonstrates the validation of ANN 23-24 in the training interval between 1498 and 1995K.

As described by the VFT equation (Equation (4)), there is a monotonically decreasing relationship between viscosity and temperature. In other words, there is an indication that viscosity decreases as temperature increases throughout the training data domain of ANN 23-24 and the derivative ∂(log η)/∂(T⁻¹) remains with negative values.

A simulation was also performed to obtain the inflection point between viscosity and temperature (Figure 8). In the work of Min and Tsukihashi [39], different inflection points are referenced depending on the cooling of the slag. When there is cooling at equilibrium, the inflection point is identified as the critical temperature, which indicates the division in which the viscosity is affected or not by the formation of crystals [46], and when there is rapid cooling of the slag, the inflection point is the break temperature, which can indicate when the slag behaves like a non-Newtonian fluid [47].

A slow cooling of the slag indicates the fictive temperature [39].

It is also indicated in the inset of Figure 8 that the derivative ∂(log η)/∂(T⁻¹) remains throughout the training data domain of ANN 23-24 with values below 0.

Figure 9 demonstrates different NBO/T by varying the viscosity with respect to temperature. The NBO/T was calculated by Equation (8).

(8)$NBO/T={\displaystyle \frac{\sum 2\left(X_{MO}+X_{M_{2}O}-X_{M_2{O}_3}\right)}{X_{{MO}_2}+2X_{M_2{O}_3}}}$

X_y is the molar fraction of y, MO, M₂O, M₂O₃, and MO₂ are the chemical species in which M has valence +2, +1, +3, and +4, respectively.

The NBO/T indicates the degree of depolymerization of a chemical composition. The higher the NBO/T, the greater the depolymerization and consequently the lower the viscosity [2,3].

The application of ANN 23-24 on chemical composition data in the work of Shankar et al. [48] was also performed to demonstrate the accuracy of the model in relation to literature data (Figure 10). The chemical composition data are CaO = 39.39%, SiO₂ = 34.27%, Al₂O₃ = 21.31%, and MgO = 5.03%.

The exponential relationship and the linear relationship between viscosity and temperature can be seen in Figure 10. The ANN 23-24 metrics in relation to the data of Shankar et al. [48] resulted in an MAE of 0.3001 Pa·s and a correlation coefficient of 0.9921.

The variation in viscosity through a CaO-SiO₂-Al₂O₃ ternary diagram with the construction of different iso-viscosity curves considering different iso-viscosity curves with MgO = 5% and a temperature of 1500 °C (Figure 11) and varying the MgO (10% and 20%) at a temperature of 1450 °C and viscosity equal to 10^0.5 Pa·s (Figure 12) were explained.

Figure 11 and Figure 12 describe the behavior through numerical simulations using the Monte Carlo method. Monte Carlo simulation is a numerical method for solving mathematical problems by simulating random variables [49] developed by Metropolis and Ulam [50].

The variation in viscosity equal to 1 Pa·s and 10³ Pa·s by an iso-viscosity curve in Figure 11 and the influence of MgO on a chosen viscosity in Figure 12 are examples of applications of ANN 23-24.

In addition to the applications previously demonstrated, the predictions of ANN 23-24 can denote physical properties such as the liquidus temperature and the glass transition temperature, in addition to the electrical conductivity of CaO-SiO₂-Al₂O₃-MgO slags.

The liquidus temperature can be defined as the temperature at which a slag becomes completely liquid [38] and is directly related to viscosity through Equation (9).

(9)${\displaystyle \frac{1}{R}}{\displaystyle \frac{{\partial }^{2}Q}{\partial T^2}}={\displaystyle \frac{{\partial }^2\left(\frac{\partial log\eta }{\partial T^{-1}}\right)}{\partial T^2}}$

R is the universal gas constant, Q is the activation energy, T is the temperature, and η is the viscosity.

The application of Equation (9) is consistent with experimental values in CaO-SiO₂ slags [41].

The glass transition temperature can be defined as the temperature at which a slag has a viscosity of 10¹² Pa·s [42] and is related to viscosity through Equation (10).

(10)$m=\underset{T\to T_g}{\mathit{lim}}{\displaystyle \frac{\partial \left(log\eta \right)}{\partial \left(T_g/T\right)}}$

m is the fragility index, T is the temperature, T_g is the glass transition temperature, and η is the viscosity.

The fragility index was specified through the paper of Angell [51], in which he describes “strong” and “weak” liquids through the fragility index m and the capacity of the liquids to form crystalline and/or non-crystalline solids.

The liquidus temperature and the glass transition temperature can be considered the most important physical parameters in relation to the crystalline behavior of the slags [52].

In addition to the temperatures previously described, viscosity has a direct relationship with the electrical conductivity of slags [43] (Equation (11)).

log K = a log η + b(11)

K is the electrical conductivity a and b constants and η is the viscosity.

The electrical conductivity of slags as well as their viscosity are influenced by their structure [53].

In addition to the properties previously demonstrated, viscosity is directly related to the diffusivity of slags and also to the slag foaming index [39].

Viscosity is an important variable in the operation of blast furnaces [54] it also affects the smelting process, influences the quality of pig iron in blast furnaces [55], has a great influence on slag granulation [56], and can optimize the stability and fluidity properties of blast furnace slags [57].

Technical Use of the Viscosity Prediction Model

The computational efficiency of artificial neural networks can be understood as the processing of input data to determine output data through a process that can be called recall through a computer cycle. Through the training of the artificial neural network, recall is performed when the input data are submitted to the artificial neural network, and the artificial neural network determines the output data through internal processing [58]. Considering any computational process, artificial neural networks are dependent on the processing, where they will be installed and can establish a real-time prediction [59,60,61,62,63,64].

4. Conclusions

Viscosity data of CaO-SiO₂-Al₂O₃-MgO slags at different temperatures were pre-processed and subjected to linear modeling processes by the linearized VFT equation and non-linear modeling by artificial neural networks.

The modeling of the viscosity of CaO-SiO₂-Al₂O₃-MgO slags at different temperatures by artificial neural networks demonstrated better statistical evaluation of mean absolute error (MAE), and the best artificial neural network with minimum MAE and monotonically decreasing relationship of viscosity in relation to temperature was the artificial neural network, with 23 nodes in the first hidden layer and 24 nodes in the second hidden layer, considering six input variables (Ca, Si, Al, Mg, O, and T) and one output variable (viscosity) named ANN 23-24.

ANN 23-24 demonstrated better MAE in relation to 11 different literature models for viscosity calculation. ANN 23-24 was used to demonstrate the variation in viscosity in relation to temperature in different chemical compositions and to demonstrate the inflection point between viscosity and temperature that can indicate different physical properties. ANN 23-24 was also submitted to demonstrate the relationship between viscosity and temperature at different NBO/T and to predict viscosity in a defined chemical composition at different temperatures.

In addition, the ANN 23-24 predictions were submitted to Monte Carlo simulation to demonstrate different iso-viscosity curves considering CaO-SiO₂-Al₂O₃ ternary diagrams at different points of chemical composition and temperature and at different viscosity values.

Finally, different physical properties that have a direct relationship with viscosity were demonstrated, such as liquidus temperature, glass transition temperature, electrical conductivity, and other properties, and the importance of viscosity in blast furnace slag in relation to blast furnace operation and its impact on different processes was presented.

In short, ANN 23-24 is a model that performs precision and accuracy in modeling the viscosity of CaO-SiO₂-Al₂O₃-MgO slags at different temperatures and has a range of applications to aid in decision making, data acquisition for process control, and the optimization and improvement of processes and products.

Author Contributions

Conceptualization, P.d.A.; methodology, P.d.A.; validation, P.d.A. and M.L.P.M.; formal analysis, P.d.A.; writing—original draft preparation, P.d.A.; writing—review and editing, J.L.C., E.J., F.F.G. and M.L.P.M.; visualization, P.d.A.; supervision, J.L.C., E.J., F.F.G. and M.L.P.M. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Restrictions apply to the availability of these data. Data were obtained from the Slag viscosity modeling toolbox paper and are available at https://doi.org/10.1016/j.fuel.2012.03.010 (accessed on 1 August 2024) with the permission of Ing. Arne Bronsch. The Python programming language was used to perform the analyses in this article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Flowchart of Materials and Methods used in this work.

Figure 2. Histogram of viscosity in the training database and the test database.

Figure 3. Variance of deviations of artificial neural networks.

Figure 4. ANN 23-24 training curve.

Figure 5. Plot between test data and data predicted by ANN 23-24 (white dots). The inset is the relationship between the test data and the data predicted by the linear model (black dots).

Figure 6. Deviations of ANN 23-24 with respect to the test data. The inset is the distribution of deviations.

Figure 7. Variation in predicted viscosity data by ANN 23-24 in relation to temperature with constant chemical composition at the 25th, 50th, and 75th percentiles.

Figure 8. Inflection point of viscosity data predicted by ANN 23-24 with respect to temperature at constant composition at the 75th percentile. The inset is the ratio of the derivative of log η by 1/T (∂(log η)/∂(1/T)) to the inverse of temperature.

Figure 9. Variation in viscosity data predicted by ANN 23-24 with respect to temperature at different NBO/T values.

Figure 10. Application of ANN 23-24 for predicting viscosity at different temperatures with a given chemical composition. The inset demonstrates the linear relationship between log η and T−1. (Shankar et al. [48]).

Figure 11. Iso-viscosity curves of 1 Pa·s (solid line) and 103 Pa·s (dashed line) considering the CaO-SiO2-Al2O3 ternary diagram (* Temperature (T) = 1500 °C and MgO = 5%).

Figure 12. Iso-viscosity curves of 100.5 Pa·s considering the ternary diagram CaO-SiO2-Al2O3 with MgO equal to 10% (solid line) and 20% (dashed line) and (* Temperature (T) = 1450 °C).

Table 1

Minimum and maximum data for each variable used in the linear model and non-linear models.

log η	T (K)	MgO	Al2O3	SiO2	CaO *	Variables	Linear Data
−0.9914	1498	1.99	3.75	15	25.75	Min.
5.349	1995	20	40	53.25	58	Max.
	O	Mg	Al	Si	Ca **	Variables	Non-linear Data ***
	0.3688	0.012	0.0198	0.0701	0.184	Min.
	0.443	0.1206	0.2117	0.2489	0.4146	Max.

* CaO, SiO₂, Al₂O₃, and MgO in %weight. ** Ca, Si, Al, Mg, and O in molar fraction. *** The temperature and viscosity data of the non-linear data are the same as those of the linear data.

Table 2

MAE calculated for the linear model, for ANN 23-24, and for the literature equations.

MAE (log η)	Model
0.7581	Linear
0.1879	ANN 23-24
5.7198	ANNLiq
3.2502	BBHLW
1.6275	Bomkamp
0.9434	Duchesne
2.1425	Kalmanovitch-Frank
2.7944	Riboud
2.0048	Shaw
2.1288	Streeter
1.462	S2
1.4938	Urbain
1.8177	Watt–Fereday

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