Composition Engineering on the Local Structure and Viscosity of the CaO-SiO₂-Al₂O₃-P₂O₅-FeO Slag by Machine Learning Methods

Abstract

Due to the high cost and low accuracy of high-temperature tests, the viscosity data for multicomponent slag systems is difficult to be obtained precisely. Therefore, it is important to fulfill the viscosity database of the multicomponent slag systems via reasonable methods with lower costs. In this study, a viscosity prediction method based on the machine learning method was proposed for the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ quinary slag system. To provide valid data for the machine learning model, the viscosity predicted by the molecular dynamic method and multiple semi-empirical models were compared to verify the applicability of these methods to the slag system. Different machine learning models were also developed. The results showed that the prediction results from the gradient boosting decision tree method were the most accurate for the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ quinary slag system. Based on this method, a color-map concerning the numerical effect of Al₂O₃ and P₂O₅ contents and slag viscosity is provided, which also provides assistance for the composition engineering to fulfill a certain demand on the viscosity design.

1. Introduction

Slag is widely applied in the ironmaking and steelmaking process for steel purification. Many scholars are studying the online modification of hot steel slag [1] to obtain slag with practical application effects by adjusting its physical phase and microstructure [2,3]. Among all the slag properties, viscosity is one of the most important properties concerning the smelting performances of the slag. The slag viscosity determines the fluidity of the slag, the speed of mass transfer in the slag, the effect of slag–steel separation, and the metal yield. Wang [4] investigated the changes in the CaO-SiO₂-Al₂O₃-MgO-Fe_tO-P₂O₅ slag system caused by the composition change of P₂O₅ and Fe_tO by Raman spectroscopy, Fourier transform infrared reflection, nuclear magnetic resonance, and viscosity measurements, reporting that P⁵⁺ copolymerizes with [SiO₄]. Therefore, the increase in P₂O₅ leads to an increase in polymerization degree and viscosity. Wang [5] also claimed that the slag polymerization and the viscosity of the CaO-SiO₂-Al₂O₃-MgO-FeO slag system increase with the increase in Al₂O₃ content based on Raman and magic angle spinning nuclear magnetic resonance.

However, the experiments still have limitations. Besides the difficulties to conduct experiments at such a high temperature, the instrumentation, fluid state, and homogeneity can all contribute to errors in the experimental results. Seshadri et al. [6] suggested that the most effective way to confirm viscosity is to repeat the experiment on two devices with overlapping shear rate ranges but with different modes of operation. As components in the slag system increase and the temperature rises, less experimental data are available. To numerically analyze the influencing factors on the slag viscosity, many researchers have developed reasonable models based on the limited amount of experimental data to predict viscosity over a wide range of components.

Bouhadja et al. [7] used the reverse non-equilibrium molecular dynamics (RNEMD) method to calculate the shear viscosity of (CaO–Al₂O₃)_1–x(SiO₂)_x liquid melts in calcium aluminosilicate glass system. Santhy et al. [8] revealed the relationship between the network character and viscosity of silicates that the viscosity decreases with increasing O/Si ratio. Tang et al. [9] established a model based on the effect of different oxides on the polymerization of slags, which was built according to the NBO/T model proposed by Mills [10] for estimating the viscosity of silica-aluminate melt and has a good prediction for CaO-SiO₂-MgO-Al₂O₃-R₂O system slag. However, the data coverage of the semi-empirical viscosity model is still limited.

With the development of computing power, machine learning (ML) provides a practical way to analyze complex data relationships. In contrast to physical models, ML models provide satisfying data extensibility. Jiang et al. [11] performed a PCA-KNN model for the prediction of blast furnace slag viscosity based on a large number of experimental data, which also showed accuracy in terms of data sets containing temperature and alkalinity information. Saigo et al. [12] proposed Einstein–Roscoe regression (ERR) regression, which employs a Gaussian process-based migration learning framework to improve the accuracy of high-temperature slag viscosity datasets using auxiliary measurement regression parameter estimates provided at a balanced reasonable cost. Cai et al. [13] proposed a Kriging interpolation geometric model with the introduction of an oxide property weight correction for multi-system slag viscosity prediction with less error than the empirical model and can be extended to the continuous physicochemical properties of multi-component slags. Huang et al. [14] automatically predicted the viscosity of the slag by an innovative two-stage predictive modeling approach and demonstrated its effectiveness on a collected unbalanced data set. A two-equation model with a six-order polynomial combined with the Arrhenius formula was developed accordingly.

In this study, the ML viscosity prediction model was developed based on CaO-SiO₂-FeO-(P₂O₅)-(Al₂O₃) slag. Molecular dynamics (MD) simulations of the slag system were firstly carried out in order to investigate the influence of different compositions of P₂O₅ and Al₂O₃ in the structure of the slag system. The viscosity of the slag was predicted by the RNEMD method to investigate the correlation between the slag structure properties and the physical properties. Several semi-empirical models were also adopted for viscosity prediction. The results were compared based on the Pearson correlation coefficient with the contents of P₂O₅ and Al₂O₃. ML model was further developed for viscosity prediction of the CaO-SiO₂-FeO-(P₂O₅)-(Al₂O₃) slag system. The results of the study can provide a theoretical basis for subsequent studies of CaO-SiO₂-FeO-(Al₂O₃)-(P₂O₅) slag systems with specific compositions.

2. Calculation Methods

2.1. Model Construction and Parameter Selection

The simulation system was established based on the CaO-SiO₂-FeO-(P₂O₅)-(Al₂O₃) slag system. The basicity of CaO/SiO₂ was fixed at 1.5. The FeO content was fixed at 30%. The absolute temperature was kept at 1873 K. The mass fraction of Al₂O₃ and P₂O₅ in the slag system varied in the range of Al₂O₃: 0–30%; P₂O₅: 0–30%. A total of 64 sets of simulations were conducted. The Al₂O₃ and P₂O₅ contents in each set of simulations and the corresponding case names are shown in Figure 1. Detailed compositions and density of different slag systems can be referred to in Table S1.

In this study, a cubic box of approximately 5000 atoms was built. The size of the box was determined by Equations (1) and (2).

$$\frac{1}{{\rho }_{\mathrm{slag}}^0}=0.45\omega {\left({\mathrm{SiO}}_2\right)}_{\%}+0.285\omega {\left(\mathrm{CaO}\right)}_{\%}+0.204\omega {\left(\mathrm{FeO}\right)}_{\%}+0.48\omega {\left({\mathrm{P}}_2{\mathrm{O}}_5\right)}_{\%}\left(1400°C\right)$$

(1)

$${\rho }_{\mathrm{slag}}={\rho }_{\mathrm{slag}}^0+0.07\left(\frac{1400-T}{100}\right)$$

(2)

where ω(MO)% is the mass fraction of the oxide composition in the slag, T is the actual temperature of the system, and ρ_slag is the density of slag melt, g/cm³.

The Born–Mayer–Huggins interatomic potential function and Lennard–Jones two-body potential function were adopted in the present simulations. These two potential functions have been widely used in silicate systems and showed good results [15,16,17]. The functions are shown in Equations (3) and (4), and the parameters are shown in

Table 1. The potential function parameters used in this calculation [15,16,17].

Ion i	Ion j	Aij (gÅ2/fs2)	βij (1/Å)	Cij (gÅ8/fs2)
P	P	4.56 × 10−22	7.0600	0
P	Ca	2.64 × 10−21	12.5000	0
P	Si	1.73 × 10−23	12.5000	4.49 × 10−25
P	Fe	2.05 × 10−22	6.2500	0
P	O	3.04 × 10−23	3.4500	0
Ca	Ca	5.27 × 10−21	6.2500	6.9501 × 10−26
Ca	Si	4.28 × 10−22	6.2500	0
Ca	Fe	3.53 × 10−23	6.2500	0
Ca	O	1.15 × 10−20	6.0600	1.39 × 10−25
Si	Si	3.47 × 10−23	6.2500	0
Si	Fe	9.22 × 10−23	12.9001	0
Si	O	1.01 × 10−21	6.0600	0
Fe	Fe	4.7 × 10−23	3.4500	0
Fe	O	6.41 × 10−22	5.1600	0
O	O	2.4 × 10−20	5.8800	2.78 × 10−25
Al	Al	6.6302 × 10−23	6.2500	0
Ca	Al	5.9095 × 10−22	6.2500	0
Si	Al	4.7906 × 10−23	6.2500	0
O	Al	1.3775 × 10−21	6.0606	0
Ion i	Ion j	Dij	xij
Fe	Al	0.0036	3.71
P	Al	0.017018	4.323

.

$$U_{ij}\left(r\right)=\frac{q_i{q}_j}{r_{ij}}+A_{ij}{e}^{\left(-B_{ij}\left(r\right)\right)}-\frac{C_{ij}}{r_{ij}^6}$$

(3)

$$E_{vdw}=D_{ij}\left\{\left.-2{\left[\frac{x_{ij}}{x}\right]}^6+{\left[\frac{x_{ij}}{x}\right]}^{12}\right\}\right.$$

(4)

where U_ij(r) is the interatomic pair potential, q_i,q_j is the ionic charge, r_ij is the inter-ion distance, E_vdw describes the van der Waals force interactions, D_ij is the well depth, and x_ij is van der Waals bond length.

The simulation process was carried out with the melt quenching method. In the beginning, each sample was relaxed for 150 ps at 5000 K in a regular system synthesis (NVT), followed by a cooling rate of 10¹³ K/s to 1873 K in 312.7 ps. Data were collected in a holding time of 1873 K for more than 0.3 ns. All the MD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package [18]. The integration time step was set to 1 fs with a cut-off value of 10 Å for short-range interactions and a cut-off value of 10 Å for Coulomb interactions. Finally, the radial distribution functions, coordination numbers, atomic structure, and viscosity were analyzed with the software VMD (1.9.3, University of Illinois at Urbana-Champaign, USA, open access), Ovito (basic, 3.5.4, Germany, open access), and ISAACS (V2.1, Central Michigan University, USA, open access) [19,20,21].

2.2. RDF and CN Calculations

The radial distribution function (RDF) is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized to the ideal gas. The expression is shown in Equation (5). In the RDF value curve, the first extreme value is the distance between the atom and its neighbors. The area under the extreme value curve is calculated as the coordination number (CN), with the expression in Equation (6).

$$g\left(r\right)=\frac{V}{N}\frac{n\left(r,\Delta r\right)}{4\pi r^2}$$

(5)

$$N=4\pi {\displaystyle {\int }_{r0}^{r1}{r}^{2}g\left(r\right)\rho dr}$$

(6)

where N/V is the number density of particles within dr, n(r,Δr) is the average number of other atoms around the central atom, r0 is the rightmost position starting at r = 0 where g(r) is approximately zero, and r₁ is the first minimal value.

2.3. Viscosity Calculation Method

The viscosity of a fluid is commonly known as a measure of the internal friction that impedes the flow of the fluid. In MD studies, the calculation of viscosity is essential for the analysis of fluid transport properties and thermal properties.

Muller-Plathe [22] proposed a method to calculate shear viscosity in terms of RNEMD, which was achieved by exchanging momentum and artificially constructing the shear field; the formulas are shown in Equations (7) and (8). The velocity gradient and shear rate can be obtained from the simulation accordingly. This method has been applied to many fields, such as nanofluids, for the calculation of thermally conductive viscosity [23,24]. In this simulation, the RNEMD method was also adopted to calculate the viscosity of different slag systems. During the calculation, the two ends were fixed, and the system was sliced along the Z-axis. 50 slices were cut. The momentum components at the center and both sides were exchanged to construct the velocity gradient and form the shear field to calculate the viscosity. The schematic diagram is shown in Figure 2. During the calculation, the slag was considered as a liquid state. The solid fraction was not introduced in the model, which may cause some inaccuracies.

$$j_z\left(p_x\right)=-\eta \frac{\partial v}{\partial x}$$

(7)

$$\eta =-\frac{\underset{transfer}{\Sigma }\left(p_{x,1}-p_{x,11}\right)}{2\Delta t{L}_x{L}_y\langle \frac{\partial v_x}{\partial Z}\rangle }$$

(8)

Moreover, the viscosities of different slag systems in a full liquid state were also predicted by several semi-empirical models, including the NPL model [25], Urbain model [26,27], and Kondratiev and Jak model [28]. The predicted results were compared with those from RNEMD methods. The relation between the structure of different slag systems and the predicted viscosities was further analyzed for the applicability assessment of these perdition methods in terms of the slag systems in this study.

2.4. Machine Learning (ML) Methods

In this paper, four ML methods, namely artificial neural network (ANN) [29], random forest (RF) [30], Support Vector Machine (SVM) [31], and gradient boosting decision tree (GBDT) [32] were used based on the MATLAB toolbox to calculate the corresponding properties of the models for prediction. ANN is an information processing system that mimics the structure and function of neural networks in the human brain by simulating neuronal activity with a mathematical model. ANN usually shows better performance for fuzzy systems [33]. RF is an integrated algorithm consisting of decision trees. RF achieves the prediction of properties by building a large number of decision trees to form a forest. SVM performs non-linear classification by a kernel method and uses a hinge loss function to calculate empirical risk. A regular term to optimize structure risk in the solution system is usually added in this method. GBDT consists of multiple decision trees and accumulates the conclusions of all trees as the final result. GBDT performs well on low-dimensional and non-linear data. In this study, the P₂O₅ and Al₂O₃ contents were used as independent variables. The viscosity calculated by Urbain’s method was used as the target variable. The optimal ML model is evaluated by Root Mean Squared Error (RMSE).

3. Results and Discussion

3.1. Slag Structure Analysis

In the basic CaO-SiO₂-FeO converter slag system, the Si⁴⁺ in the system acts as a network-former and has a strong adsorption capacity for surrounding oxygen atoms due to its high charge and small radius. Si atoms can combine with O and form σ-bonds. When the p-orbitals of oxygen atoms are filled, the d_x-p_π bonds can be formed with the d-orbitals of Si, thus enhancing the Si-O bonds. These characteristics lead that silicate structures are easy to form large irregular short-range ordered aggregates. A certain proportion of P₂O₅ and Al₂O₃ exhibit similar properties when brought into the system. The increase in these compositions helps increase the degree of the system polymerization. Figure 3 shows the structures of CaO-SiO₂-FeO-Al₂O₃ (A model), CaO-SiO₂-FeO-P₂O₅ (P model), and CaO-SiO₂-FeO-Al₂O₃-P₂O₅ (AP model). The [SiO₄], [AlO₄], and [PO₄] tetrahedral structures of O combined with Si⁴⁺, Al^3+, and P⁵⁺ are present in Figure 3a–c, respectively, as the main body of the grid structure. While Ca²⁺ and Fe²⁺ in the slag exist as free states in the interstices of the structure.

Figures S1–S3 show the RDFs and CNs of the main atomic pairs in the P model, A model, and AP model, respectively. Based on the RDF curves and characteristic positions of the different models, the detailed bond lengths are shown in Table S2. The bond lengths barely change with the slag compositions. There are no obvious coordination plateaus in the CN curves of Ca-O and Fe-O. While the CNs of Si-O all remain around 4, indicating that the tetrahedral structure of Si-O is quite stable, as shown in Table S3. The coordination plateau of P-O is also stable and constant at 4. The CNs of Al-O have a tendency to increase, indicating that with the increase in Al content, Al³⁺ and O^2- have a tendency to form an octahedral structure with a coordination of 6.

3.2. Oxygen Network Structure Analysis

The variation of different oxygen types for the A model and the P model is shown in Figure 4. In the A model, with the increase in Al₂O₃ content, the content of bridging oxygen increases, and the content of non-bridging oxygen and free oxygen decreases. When Al₂O₃ content is less than 10%, OA-type non-bridging oxygen increases rapidly, and AOA-type bridging oxygen grows slowly. When Al₂O₃ content exceeds 10%, the growth rate of SOA-type bridging oxygen slows down, and AOA-type bridging oxygen increases rapidly. The OA-type non-bridging oxygen increases more slowly when the Al₂O₃ content is above 20%. In the P model, the increase in P₂O₅ content led to an increase in PO-type non-bridging oxygen and SOP-type bridging oxygen content in the slag. The FO in the slag system decreased under the aggregation caused by P₂O₅. Based on the approximate relationship between the content of bridging oxygen proposed by Diao [15] and the local structure changes in this simulation, which are shown in Equations (9)–(11), the Si-O-P connection is more stable than SOS and POP at higher P₂O₅ content. The growth rate of AOA is greater than the growth rate of SOA with the increase in Al₂O₃ content when the Al₂O₃ content is greater than 10% due to the influence of Si in the slag.

P-O-P+ Si-O-Si→2(Si-O-P)

(9)

Al-O-Al+ Si-O-Si→2(Al-O-Si)

(10)

P-O-P+ Al-O-Al→2(P-O-Al)

(11)

Figure 5 shows the changing trends of BO, FO, and NBO in the AP model. For the BO content, when the Al₂O₃ content is fixed, the BO content barely increases with the increase in P₂O₅ content. When the P₂O₅ content is fixed, the BO content increases significantly with the increase in Al₂O₃ content. For the NBO content, when the Al₂O₃ content, the increase in P₂O₅ content impels the NBO content to increase slightly. When the Al₂O₃ content is fixed, the content of NBO decreases as the content of Al₂O₃ increases. Some of the NBOs transfer to BO with the assistance of Al³⁺. For the FO content, the increase in both Al₂O₃ and P₂O₅ content decreases the FO content in the slag. The covariances between the different oxygen types and the content of Al₂O₃ and P₂O₅ were analyzed. The results are shown in

Table 2. Analysis of covariances for different types and components of oxygen.

Covariance	Al2O3	P2O5
BO	0.181464	0.026783
NBO	−0.13103	0.026341
FO	−0.05388	−0.05367

. There is a strong positive correlation of Al₂O₃ with BO but a strong negative correlation with NBO. The Al₂O₃ and P₂O₅ contents also showed the same negative correlation for FO.

Figure 6 shows the effects of Al₂O₃ and P₂O₅ on different types of NBO and BO in the slag. For the NBO, the content of OP increases with the increase in P₂O₅ content. The increase in Al₂O₃ content hardly affects the content of OP. The increase in Al₂O₃ content leads to the increase in OA content, but the increase in P₂O₅ content slightly decreases the content of OA. Combined with the changing trend of BO, it can be obtained that when the P₂O₅ content increases, the P atom combines with OA to form POA-type bridging oxygen. The depolymerization effect of Ca²⁺ and Fe²⁺ on the structure of AOA [34] also results in a higher OA content at low P content and high Al content in the slag. The increases in Al and P content also have the same reducing effect on the OS content. These OS structures will polymerize into structures such as SOA and SOP accordingly. As for BO, POP hardly appears. The AOA content increases with the increase in Al content. The AOA structure can be influenced by the depolymerization effect of Ca²⁺ and Fe²⁺, while the P₂O₅ content barely affects the AOA structure. In the slag system with low Al and high P contents, the SOS-type BO is easier to be formed. While high Al and low P content reduce the SOS content on the other hand. The POS-type bridging oxygen is only observed in the slag system with higher P₂O₅ content. When the Al content increases, Al content promotes the production of POS-type BO. The POA-type BO exists in the slag system with high Al and P contents. The SOA-type BO increases with the increase in Al content. Figure 7 shows the reaction mechanism of the oxygen network structures under the effects of P and Al atoms.

3.3. Distribution of Bond Angles

Figure 8 and Figure 9 show the bond angle distributions in the [SiO₄], [PO₄], and [AlO₄] structures at 1873 K for the A model, P model, and AP model. In A model and P model, it can be seen that the bond angles of O-Si-O are around 109.0–109.1°, which is close to the standard tetrahedral structure of 109.5°. The bond angles of O-P-O fluctuate in the range of 109–116°. The bond angle of O-Al-O stabilizes after the Al content becomes larger and forms a stable structure. In the AP models, both O-P-O and O-Si-O bond angles are more stable and close to the ideal tetrahedral structure bond angles. The O-Al-O bond angles fluctuate in the range of 102°–113°. Combined with the analysis of oxygen types in the previous section, it can be seen that the presence of both P and Al will affect the degree of polymerization to some extent. However, these two atoms have little effect on the bond angles of O-P-O, O-Al-O, and O-Si-O.

3.4. Viscosity Prediction with MD Models and Semi-Empirical Models

The Viscosity prediction results with the NPL model [25], Urbain model [26,27], Kondratiev, and Jak model [28] were used and analyzed along with RNEMD simulation results, which are shown in Figure 10.

It can be found that the viscosity of the slag increases with the increase in P content and Al content. Based on the NBO/T theory proposed by Mills [10], the amount of non-bridging oxygen possessed by a single polymer particle contributes significantly to the variation of viscosity. The increase in the Al and P content leads to a decrease in NBO/T, which also increases the slag polymerization and the viscosity.

Due to the absence of experimental data in the slag system in this study, it is difficult to measure the applicability of these semi-empirical models and the RNEMD method on the CaO-SiO₂-FeO-(P₂O₅)-(Al₂O₃) slag systems. Considering that the positive relationship between the BO number and the contents of Al₂O₃ and P₂O₅ has been confirmed by existing studies [4,5], the Pearson correlation coefficient between BO number and the contents of Al₂O₃ and P₂O₅ was introduced for the applicability assessment of these models of the viscosity prediction, which is shown in Figure 11. Based on Table 2, the covariances between BO and the contents of Al₂O₃ and P₂O₅ have provided proof that the positive relationship between BO and Al₂O₃ is stronger than that between BO and P₂O₅. Hence, the positive value of Pearson correlation coefficients of Al₂O₃ should be larger than that of P₂O₅. Only the viscosities predicted by Urbain’s model are in accord with these inferences. Other viscosity prediction models and RNEMD methods are not suitable for the viscosity prediction of the slag system in this study. Therefore, Urbain’s model is determined to be the most applicable for the viscosity prediction of the slag system in this study. The predicted viscosity data of Urbain’s model was also adopted in the ML models developed in the following section.

3.5. ML Model Development for Viscosity Prediction

To improve the accuracy of viscosity prediction by the ML models, different ML models were developed in this section, including ANN, RF, SVM, and GBDT. Before performing the prediction, the different model parameters were optimized for implementation. ANN used two hidden layers with a maximum of 100 iterations and a tolerance of 0.01. RF used 20 decision trees with several bins of 32. GBDT used a loss function with squared error and 20 decision trees with a convergence tolerance of 0.1. SVM normalized the data with a regularization parameter is 0.01, and a polynomial kernel function is used with a kernel parameter p of 1. The prediction results of different ML models are shown in Figure 12. The values of RMSE between Urbain model data and the prediction results for each ML model were calculated accordingly, which is shown in

Table 3. RMSE of different ML models.

ML Methods	GBDT	ANN	RF	SVM
RMSE	0.000424539	0.013676	0.003173	0.011409

.

According to the values of RMSE in Table 3, GBDT and ANN have the best and worst predictive power for the viscosity prediction of the slag system in this study, respectively. To further prove this conclusion, the Pearson correlation coefficients of viscosity perdition results from the four ML models were also calculated, which is shown in Figure 12b. Both Al₂O₃ and P₂O₅ contents appear positively correlated with viscosity when we adopted the GBDT predictor. While in the prediction results of the ANN model, the Al₂O₃ content appears to have a negative correlation with viscosity, which is contrary to the rules provided by the semi-empirical models.

Based on the ML model with GBDT methods, a color-map concerning the numerical effect of Al₂O₃ and P₂O₅ contents and slag viscosity is provided in Figure 13a. The blue color represents the smaller value of viscosity. The red color represents the larger value of viscosity. The iso-viscosity lines with a viscosity of 0.15Pa·s, 0.175Pa·s, 0.2Pa·s, 0.225Pa·s, and 0.25 Pa·s were also extracted in Figure 13b. According to this figure, the slag viscosity of the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ slag system in terms of different Al₂O₃ and P₂O₅ contents can be predicted and it could also be used for composition engineering to fulfill a certain demand on the viscosity design.

4. Conclusions

(1)

With the increase in the mass fraction of Al₂O₃ and P₂O₅, the concentration of BO increases, and the concentration of FO decreases, which complicates the melt structure of the slag system and increases the polymerization degree.

(2)

In the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ slag system, the content of BO was positively correlated with the content of Al₂O₃ and P₂O₅. The positive correlation was stronger for the content of Al₂O₃.

(3)

Different semi-empirical models and RNEMD methods were used to predict the viscosity of the three slag systems. The reliability of the Urbain model in predicting the viscosity of the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ slag system has been proved according to the Pearson correlation coefficient analysis.

(4)

Among all the ML methods in this study, GBDT has the best predictive power for the viscosity prediction of the slag system in this study, building a credible correlation between the structure of the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ slag system and viscosity prediction. Iso-viscosity lines of the CaO-SiO₂-FeO-Al₂O₃-P₂O₅ slag system were provided accordingly.