Optimal Alkalinity Model of Ladle Furnace Slag for Bearing Steel Production Based on Ion–Molecule Coexistence Theory

Abstract

The fatigue life of bearing steel is closely related to the total oxygen content (T(O)) of the liquid steel. In order to stably and effectively control the T(O) during the ladle furnace (LF) refining process, we established a calculation model of optimal alkalinity for the refining slag CaO–SiO₂–Al₂O₃–MgO–FeO–CaF₂ at 1853 K based on ion–molecule coexistence theory (IMCT). Here, the influencing factors are discussed. The results show that the maximum value of $N_{\mathrm{FeO}}$ occurred when the optimal alkalinity was around five at varied FeO contents, and that the optimal alkalinity basically remained the same with changes in FeO content. With the increase of MgO content, the optimal alkalinity decreased. However, the change in the value of $N_{\mathrm{FeO}}$ against the higher alkalinity was not obvious at a given MgO content. The effect of Al₂O₃ content on the optimal alkalinity was opposite to that of MgO. With the increasing Al₂O₃ content, the optimal alkalinity obviously increased, while the maximum value of $N_{\mathrm{FeO}}$ occurred when the Al₂O₃ content varied from 35 wt% to 45 wt% at higher alkalinity. The higher w(CaO)/w(Al₂O₃) mass ratio had a distinct effect on the value of $N_{\mathrm{FeO}}$ against alkalinity, while the effect of alkalinity on the value of $N_{\mathrm{FeO}}$ was not obvious at a fixed CaF₂ level. This optimal alkalinity model based on IMCT can provide a certain guiding role in the process of refining slag composition optimization and is conducive to effectively controlling total oxygen content in the refining process.

1. Introduction

Bearing steel is regarded as “the heart of industry”, which is closely related to development of the national economy and widely used in all walks of life. Its quality directly affects the operational life, safety and reliability of equipment [1,2]. However, inclusion of brittle oxide in steel is extremely harmful to the fatigue life of bearing material. Studies have shown that reducing the total oxygen content (T(O)) in liquid steel can greatly improve these problems for bearing steel [3,4]. Controlling T(O) is a critical aspect that requires attention during the production process. In recent decades, the T(O) in high-chromium bearing steel has decreased from 10 ppm to 5 ppm [5]. However, in the actual production process, oxide inclusions still exist, such as Mg–Al spinel and calcium–aluminate inclusions, that affect the fatigue strength and impact toughness of bearing steel [6,7]. In 2016, China strengthened the inclusion control standard for high-carbon chromium bearing steel. A tracing experiment showed that large DS-size inclusions mainly come from Mg–Al spinel or calcium–aluminate inclusions, the formation of which are closely related to the T(O) during ladle furnace (LF) refining processes [8]. Therefore, determining how to stably and effectively control the T(O) in bearing steel to a very low level has become a focus of bearing steel enterprises. In the LF refining process, the properties of refining slag, such as alkalinity, oxidation and viscosity, play a key role in reducing the T(O). It has been proven that through optimizing the composition of refining slag, the T(O) in liquid steel can be effectively reduced, and the amount and size of nonmetallic inclusions in bearing steel can be controlled to an ideal level [9,10,11,12,13]. Therefore, it is very important to control the composition of refining slag in the refining process. At present, research on optimizing the composition of refining slag is mainly based on laboratory research, and the relevant parameters acquired at elevated temperatures are arduous and costly. Furthermore, due to the complexity of high-temperature slag in the actual production process, it is inevitable that there will be some discrepancies between the experimental and actual production. To date, there have been limited theoretical studies implemented on the alkalinity optimization of refining slag. It is essential to establish a thermodynamic model for calculating the optimum alkalinity of refining slag.

The concept of ions and molecules in metallurgical slag was originally proposed by the metallurgist N.M. Chuiko [14], and then developed to study the reaction ability of structure units in metallurgical slag system at elevated temperature [15]. Based on the coexistence of ions and molecules in slag, J. Zhang developed the theory model and proposed the ion and molecule coexistence theory (IMCT), which improved the understanding of slag structure [16]. In the IMCT, the defined mass action concentration is consistent with the classical concept of activity in the slag. Therefore, the reaction ability of structure units in the slag system has been expressed by the IMCT. There are many theoretical models for the calculation of activity in molten slag, such as ideal ionic solution model, regular ionic solution model, Masson model, quasichemical equilibrium solution model, Lin and Pelton model, etc. Meanwhile, due to the complexity of structural units in slag at high temperature, the relevant thermodynamic data and the theoretical basis of different models all have their own limitations [17]. However, the activity can be calculated directly from the Gibbs free energy of each equilibrium equation by the IMCT without using the activity coefficient. Until now, the IMCT model has been effectively applied to describe the manganese distribution, titanium distribution ratio, phosphate capacity, sulfide capacity, oxidation ability for metallurgical slags, and so on [18,19,20,21,22,23,24,25,26,27], as listed in

Table 1. The applications of IMCT model during ironmaking and steelmaking processes.

Slag Systems	Applications	Ref.
CaO–SiO2–FeO–MgO–MnO–Al2O3	A thermodynamic model for calculating and predicting manganese distribution ratio and manganese capacity in the slag system was built based on IMCT.	[18]
CaO–SiO2–MgO–FeO–MnO–Al2O3–TiO2– CaF2	A thermodynamic model for predicting manganese distribution ratio between the slag and carbon-saturated liquid iron was established based on IMCT.	[19]
CaO–SiO2–Al2O3–MgO–FeO–MnO–TiO2	A thermodynamic model for titanium distribution ratio calculation between the slag and liquid steel was developed based on IMCT combined with industrial measurements.	[20]
CaO–SiO2–MgO–FeO–Fe2O3–Al2O3–P2O5	A thermodynamic model for calculating the phosphorus distribution ratio between the slag and liquid steel was built based on IMCT.	[21]
CaO–based slags	A thermodynamic model for calculating the phosphorus partition between CaO-based slags and hot metal during hot metal dephosphorization pretreatment process was developed based on IMCT.	[22]
CaO–FeO–Fe2O3–Al2O3–P2O5	Thermodynamic models for predicting dephosphorization ability and potential of the slag during the secondary refining process was developed based on IMCT.	[23]
CaO–based slags	A further evaluation of the coupling relationship between dephosphorization and desulfurization abilities or potentials for CaO-based slags was built based on IMCT.	[24]
CaO–FeO–Fe2O3–Al2O3–P2O5	A thermodynamic model for calculating sulfide capacity of slag in various oxygen potential ranges was established based on IMCT.	[25]
CaO–FeO–Fe2O3–Al2O3–P2O5	Prediction model of sulfur distribution ratio between the slag and liquid iron over a large variation range of oxygen potential during secondary refining process was built based on IMCT.	[26]
CaO–SiO2–MgO–FeO–Fe2O3–MnO–Al2O3– P2O5	Representation of oxidation ability for metallurgical slags based on IMCT was verified by comparing with the reported activity in the slag systems.	[27]

.

In order to explore the internal relationship between the alkalinity of refining slag and the T(O) in liquid steel, a thermodynamic model for calculating the optimum alkalinity at 1853 K was established for the refining slag system CaO–SiO₂–MgO–Al₂O₃–FeO–CaF₂ according to the actual production process based on IMCT in this paper and the influencing factors on the optimum alkalinity are discussed, which can provide the necessary theoretical basis for effective control of T(O) in the refining process for bearing steel production.

2. Materials and Methods

2.1. Production Procedure and Materials

The following process was adopted for the production of high-chromium bearing steel in Daye Special Steel: basic oxygen furnace (150t BOF) → tapping → ladle furnace (120t LF) → Ruhstahl–Hausen vacuum degassing (RH) → soft blowing (SB) → continuous casting (CC 300 mm × 400 mm) → rolling. The schematic diagram is depicted in Figure 1.

During the tapping of BOF, FeSi and SiMn are added as deoxidizers and alloying elements. In the LF process, Al is employed to final deoxidation and about 1000 kg synthetic refining slags and 400 kg limes are used as refining slag composition adjustment. Deoxygenation, desulfurization, inclusion removal, composition and temperature homogenizing can be achieved through electric arc heating, slag refining, bottom argon blowing and alloy micro-adjustment operations. After LF refining for 70 min, degassing is carried out in RH for 30 min at a pressure of 67 Pa. Then, soft blowing is employed for about 30 min to remove inclusions. Finally, the molten steel is sent to continuous casting. At the end point of the LF refining process, 20 heats of the balanced slag are sampled at about 1853 K to ensure accuracy of the analysis results within an error range of ±2%. The average chemical components are as shown in

Table 2. The average orthonormal chemical components of slag (wt %).

CaO	SiO2	Al2O3	MgO	FeO	CaF2	Binary Alkalinity
44.50	6.25	37.13	3.29	0.74	8.09	7.12

.

2.2. Establishment of the IMCT Model

According to the assumption that there are simultaneously ions and molecules in metallurgical slag at steelmaking temperature, the mass action concentrations of structural units or ion couples in CaO–SiO₂–MgO–Al₂O₃–FeO–CaF₂ slag based on IMCT can be summarized as follows:

• The structural units in CaO–SiO₂–MgO–Al₂O₃–FeO–CaF₂ at steelmaking temperature are composed of Ca²⁺, Mg²⁺, Fe²⁺ and F⁻ as simple ions, and SiO₂ and Al₂O₃ as simple molecules, silicates, aluminates, ferrites and so on as complex molecules.
• Complex molecules are formed by the reactions of simple ions and simple molecules under dynamic equilibrium.
• The chemical reactions obey the law of mass conservation.
• The activity of constituents in the slag equals the mass action concentration of the structural unit. The activity is relative to pure solid or liquid matter as the standard state according to the existing state at evaluated temperature.

The initial mole number for each composition in 100 g of CaO–SiO₂–Al₂O₃– MgO–FeO–CaF₂ slag was assigned as $a=n_{\mathrm{CaO}}^0$, $b=n_{{\mathrm{SiO}}_2}^0$, $c=n_{{\mathrm{Al}}_2{\mathrm{O}}_3}^0$, $d=n_{\mathrm{MgO}}^0$, $e=n_{\mathrm{FeO}}^0$, $f=n_{{\mathrm{CaF}}_2}^0$. The balanced mole number of each constituent unit in the slag was defined as $n_i$, and $N$_i denotes the mass action concentration of each constitutional unit, which can be expressed as follows:

$$N_i=\frac{n_i}{{{\displaystyle \sum }}^{ }{n}_i}$$

(1)

where ${{\displaystyle \sum }}^{ }{n}_i$ is the total balanced mole number of all structural units.

According to the IMCT model and the phase diagrams of CaO–SiO₂, CaO–Al₂O₃, Al₂O₃–SiO₂, MgO–SiO₂, MgO–Al₂O₃, FeO–SiO₂, FeO–Al₂O₃, CaO–Al₂O₃–SiO₂, CaO–MgO–SiO₂, MgO–Al₂O₃–SiO₂, CaO–Al₂O₃–CaF₂ and CaO–SiO₂–CaF₂ from 1573 K–1873 K, 31 types of complex molecules can be generated at steelmaking temperature [28,29], as listed in

Table 3. Parameters of structural units in the slag system.

Items	Constitutional Units	Balanced Mole Number	Mass Action Concentrations
Simple cations and anions	${\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}$	$n_1=n_{{\mathrm{Ca}}^{2+}}=n_{{\mathrm{O}}^{2-}}=n_{\mathrm{CaO}}$	$N_1=\frac{2n_1}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}}$
	${\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}$	$n_4=n_{{\mathrm{Mg}}^{2+}}=n_{{\mathrm{O}}^{2-}}=n_{\mathrm{MgO}}$	$N_4=\frac{2n_4}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{MgO}}$
	${\mathrm{Fe}}^{2+}+{\mathrm{O}}^{2-}$	$n_5=n_{{\mathrm{Fe}}^{2+}}=n_{{\mathrm{O}}^{2-}}=n_{\mathrm{FeO}}$	$N_5=\frac{2n_5}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{FeO}}$
	${\mathrm{Ca}}^{2+}+2{\mathrm{F}}^{-}$	$n_6=n_{{\mathrm{Ca}}^{2+}}=\frac{1}{2}{n}_{{\mathrm{F}}^{-}}=n_{{\mathrm{CaF}}_2}$	$N_6=\frac{3n_6}{{{\displaystyle \sum }}^{ }{n}_i}=N_{{\mathrm{CaF}}_2}$
Simple molecules	${\mathrm{SiO}}_2$	$n_2=n_{{\mathrm{SiO}}_2}$	$N_2=\frac{n_2}{{{\displaystyle \sum }}^{ }{n}_i}=N_{{\mathrm{SiO}}_2}$
	${\mathrm{Al}}_2{\mathrm{O}}_3$	$n_3=n_{{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_3=\frac{n_3}{{{\displaystyle \sum }}^{ }{n}_i}=N_{{\mathrm{Al}}_2{\mathrm{O}}_3}$
Complex molecules	$\mathrm{CaO}·{\mathrm{SiO}}_2$	$n_7=n_{\mathrm{CaO}·{\mathrm{SiO}}_2}$	$N_7=\frac{n_7}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·{\mathrm{SiO}}_2}$
	$3\mathrm{CaO}·2{\mathrm{SiO}}_2$	$n_8=n_{3\mathrm{CaO}·2{\mathrm{SiO}}_2}$	$N_8=\frac{n_8}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3\mathrm{CaO}·2{\mathrm{SiO}}_2}$
	$2\mathrm{CaO}·{\mathrm{SiO}}_2$	$n_9=n_{2\mathrm{CaO}·{\mathrm{SiO}}_2}$	$N_9=\frac{n_9}{{{\displaystyle \sum }}^{ }{n}_i}=N_{2\mathrm{CaO}·{\mathrm{SiO}}_2}$
	$3\mathrm{CaO}·{\mathrm{SiO}}_2$	$n_{10}=n_{3\mathrm{CaO}·{\mathrm{SiO}}_2}$	$N_{10}=\frac{n_{10}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3\mathrm{CaO}·{\mathrm{SiO}}_2}$
	$3\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{11}=n_{3\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{11}=\frac{n_{11}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$12\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{12}=n_{13\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{12}=\frac{n_{12}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{12\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{13}=n_{\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{13}=\frac{n_{13}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{14}=n_{\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{14}=\frac{n_{14}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$\mathrm{CaO}·6{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{15}=n_{\mathrm{CaO}·6{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{15}=\frac{n_{15}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·6{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$3{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2$	$n_{16}=n_{3{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2}$	$N_{16}=\frac{n_{16}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2}$
	$2\mathrm{MgO}·{\mathrm{SiO}}_2$	$n_{17}=n_{2\mathrm{MgO}·{\mathrm{SiO}}_2}$	$N_{17}=\frac{n_{17}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{2\mathrm{MgO}·{\mathrm{SiO}}_2}$
	$\mathrm{MgO}·{\mathrm{SiO}}_2$	$n_{18}=n_{\mathrm{MgO}·{\mathrm{SiO}}_2}$	$N_{18}=\frac{n_{18}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{MgO}·{\mathrm{SiO}}_2}$
	$\mathrm{MgO}·{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{19}=n_{\mathrm{MgO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{19}=\frac{n_{19}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{MgO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$2\mathrm{FeO}·{\mathrm{SiO}}_2$	$n_{20}=n_{2\mathrm{FeO}·{\mathrm{SiO}}_2}$	$N_{20}=\frac{n_{20}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{2\mathrm{FeO}·{\mathrm{SiO}}_2}$
	$\mathrm{FeO}·{\mathrm{Al}}_2{\mathrm{O}}_3$	$n_{21}=n_{\mathrm{FeO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$	$N_{21}=\frac{n_{21}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{FeO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$
	$2\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{SiO}}_2$	$n_{22}=n_{2\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{SiO}}_2}$	$N_{22}=\frac{n_{22}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{2\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{SiO}}_2}$
	$\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2$	$n_{23}=n_{\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2}$	$N_{23}=\frac{n_{23}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2}$
	$2\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2$	$n_{24}=n_{2\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$	$N_{24}=\frac{n_{24}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{2\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$
	$3\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2$	$n_{25}=n_{3\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$	$N_{25}=\frac{n_{25}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$
	$\mathrm{CaO}·\mathrm{MgO}·{\mathrm{SiO}}_2$	$n_{26}=n_{\mathrm{CaO}·\mathrm{MgO}·{\mathrm{SiO}}_2}$	$N_{26}=\frac{n_{26}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·\mathrm{MgO}·{\mathrm{SiO}}_2}$
	$\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2$	$n_{27}=n_{\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$	$N_{27}=\frac{n_{27}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$
	$2\mathrm{MgO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·5{\mathrm{SiO}}_2$	$n_{28}=n_{2\mathrm{MgO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·5{\mathrm{SiO}}_2}$	$N_{28}=\frac{n_{28}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{2\mathrm{MgO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·5{\mathrm{SiO}}_2}$
	$3\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2$	$n_{29}=n_{3\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2}$	$N_{29}=\frac{n_{29}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2}$
	$11\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2$	$n_{30}=n_{11\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2}$	$N_{30}=\frac{n_{30}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{11\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2}$
	$3\mathrm{CaO}·2{\mathrm{SiO}}_2·{\mathrm{CaF}}_2$	$n_{31}=n_{3\mathrm{CaO}·2{\mathrm{SiO}}_2·{\mathrm{CaF}}_2}$	$N_{31}=\frac{n_{31}}{{{\displaystyle \sum }}^{ }{n}_i}=N_{3\mathrm{CaO}·2{\mathrm{SiO}}_2·{\mathrm{CaF}}_2}$

.

The presentation of mass action concentration for 31 kinds of potentially formed complex molecules can be expressed by using the reaction equilibrium constant $K_i$, $N_1\left(N_{\mathrm{CaO}}\right)$, $N_2\left(N_{{\mathrm{SiO}}_2}\right)$, $N_3\left(N_{{\mathrm{Al}}_2{\mathrm{O}}_3}\right)$, $N_4\left(N_{\mathrm{MgO}}\right)$, $N_5\left(N_{\mathrm{FeO}}\right)$ and $N_6\left(N_{{\mathrm{CaF}}_2}\right)$ based on mass action laws, which are listed in

Table 4. Chemical reaction formulas of complex molecules, Gibbs free energy and mass action concentrations of the slag system [30,31,32,33].

Reaction Formulas	$\Delta {\mathit{G}}^{\mathit{\theta }}/\left(\mathbf{J}\mathit{·}{\mathbf{mol}}^{-1}\right)$	${\mathit{N}}_{\mathit{i}}$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(\mathrm{CaO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-21757-36.819T$	$N_7=K_1{N}_1{N}_2$
$3\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{SiO}}_2\right)=\left(3\mathrm{CaO}·2{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-236972.9+9.6296T$	$N_8=K_2{N}_1^3{N}_2^2$
$2\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(2\mathrm{CaO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-102090-24.267T$	$N_9=K_3{N}_1^2{N}_2$
$3\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(3\mathrm{CaO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-118826-6.694T$	$N_{10}=K_4{N}_1^3{N}_2$
$3\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)=\left(3\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=-21757-29.288T$	$N_{11}=K_5{N}_1^3{N}_3$
$12\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+7\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)=\left(12\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=617977-612.119T$	$N_{12}=K_6{N}_1^{12}{N}_3^7$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+{\mathrm{Al}}_2{\mathrm{O}}_3=\left(\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=59413-59.413T$	$N_{13}=K_7{N}_1{N}_3$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)=\left(\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=-16736-25.522T$	$N_{14}=K_8{N}_1{N}_3^2$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+6\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)=\left(\mathrm{CaO}·6{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=-22594-31.798T$	$N_{15}=K_9{N}_1{N}_3^6$
$3\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)+2\left({\mathrm{SiO}}_2\right)=\left(3{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-4351-10.46T$	$N_{16}=K_{10}{N}_2^2{N}_3^3$
$2\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(2\mathrm{MgO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-56902-3.347T$	$N_{17}=K_{11}{N}_2{N}_4^2$
$\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(\mathrm{MgO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=23849-29.706T$	$N_{18}=K_{12}{N}_2{N}_4$
$\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)=\left(\mathrm{MgO}·{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=-18828-6.276T$	$N_{19}=K_{13}{N}_3{N}_4$
$2\left({\mathrm{Fe}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(2\mathrm{FeO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-9395-0.227T$	$N_{20}=K_{14}{N}_2{N}_5^2$
$\left({\mathrm{Fe}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)=\left(\mathrm{FeO}·{\mathrm{Al}}_2{\mathrm{O}}_3\right)$	$\Delta G^{\theta }=-59204+22.343T$	$N_{21}=K_{15}{N}_3{N}_5$
$2\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)+\left({\mathrm{SiO}}_2\right)=\left(2\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-116315-38.911T$	$N_{22}=K_{16}{N}_1^2{N}_2{N}_3$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)+2\left({\mathrm{SiO}}_2\right)=\left(\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-4148-73.638T$	$N_{23}=K_{17}{N}_1{N}_2^2{N}_3$
$2\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{SiO}}_2\right)=\left(2\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-73638-63.597T$	$N_{24}=K_{18}{N}_1^2{N}_2^2{N}_4$
$3\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{SiO}}_2\right)=\left(3\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-205016-31.798T$	$N_{25}=K_{19}{N}_1^3{N}_2^2{N}_4$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{SiO}}_2\right)=\left(\mathrm{CaO}·\mathrm{MgO}·{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-124683+3.766T$	$N_{26}=K_{20}{N}_1{N}_2{N}_4$
$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{SiO}}_2\right)=\left(\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-80333-51.882T$	$N_{27}=K_{21}{N}_1{N}_2^2{N}_4$
$2\left({\mathrm{Mg}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)+5\left({\mathrm{SiO}}_2\right)=\left(2\mathrm{MgO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·5{\mathrm{SiO}}_2\right)$	$\Delta G^{\theta }=-14422-14.808T$	$N_{28}=K_{22}{N}_2^5{N}_3^2{N}_4^2$
$3\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)+\left({\mathrm{Ca}}^{2+}+2{\mathrm{F}}^{-}\right)=\left(3\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2\right)$	$\Delta G^{\theta }=-44492-73.15T$	$N_{29}=K_{23}{N}_1^3{N}_3^2{N}_6$
$11\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+7\left({\mathrm{Al}}_2{\mathrm{O}}_3\right)+\left({\mathrm{Ca}}^{2+}+2{\mathrm{F}}^{-}\right)=\left(11\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{CaF}}_2\right)$	$\Delta G^{\theta }=-228760-155.8T$	$N_{30}=K_{24}{N}_1^{11}{N}_3^7{N}_6$
$3\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+2\left({\mathrm{SiO}}_2\right)+\left({\mathrm{Ca}}^{2+}+2{\mathrm{F}}^{-}\right)=\left(3\mathrm{CaO}·2{\mathrm{SiO}}_2·{\mathrm{CaF}}_2\right)$	$\Delta G^{\theta }=-255180-8.2T$	$N_{31}=K_{25}{N}_1^3{N}_2^2{N}_6$

.

Based on the law of mass conservation in the slag system, the balanced relationships can be built as follows:

$$\begin{gathered} a={{\displaystyle \sum }}^{ }{n}_i\left(0.5N_1+N_7+3N_8+2N_9+3N_{10}+3N_{11}+12N_{12}+N_{13}+N_{14}+N_{15}+2N_{22}+N_{23}+2N_{24}+3N_{25} \\ +N_{26}+N_{27}+3N_{29}+11N_{30}+3N_{31}\right) \end{gathered}$$

(2)

$$\begin{gathered} b={{\displaystyle \sum }}^{ }{n}_i\left(N_2+N_7+2N_8+N_9+N_{10}+2N_{16}+N_{17}+N_{18}+N_{20}+N_{22}+2N_{23}+2N_{24}+2N_{25}+N_{26}+2N_{27} \\ +5N_{28}+2N_{31}\right) \end{gathered}$$

(3)

$$c={{\displaystyle \sum }}^{ }{n}_i\left(N_3+N_{11}+7N_{12}+N_{13}+2N_{14}+6N_{15}+3N_{16}+N_{19}+N_{21}+N_{22}+N_{23}+2N_{28}+2N_{29}+7N_{30}\right)$$

(4)

$$d={{\displaystyle \sum }}^{ }{n}_i\left(0.5N_4+2N_{17}+N_{18}+N_{19}+N_{24}+N_{25}+N_{26}+N_{27}+2N_{28}\right)$$

(5)

$$e={{\displaystyle \sum }}^{ }{n}_i\left(0.5N_5+2N_{20}+N_{21}\right)$$

(6)

and

$$f={{\displaystyle \sum }}^{ }{n}_i\left(\frac{1}{3}{N}_6+N_{29}+N_{30}+N_{31}\right)$$

(7)

According to the principle that the total mole fraction for each constitutional unit in the CaO–SiO₂–Al₂O₃–MgO–FeO–CaF₂ slag system with a fixed amount is equal to unity, then the following equation can be derived:

$$\sum _{i=1}^{31}{N}_{\mathrm{i}}=1$$

(8)

Equations (2)–(8) represent the governing equations of the IMCT thermodynamic model for calculating the mass action concentration in the CaO–SiO₂–Al₂O₃– MgO–FeO–CaF₂ slag system. The activity of each constituent in the slag at the refining temperature can then be obtained by solving the algebraic equation groups by using MATLAB software (Matlab 9.0, MathWorks, Natick City, MA, USA, 2016).

2.3. Characterization of the Optimal Alkalinity

The activity of Fe_tO a(Fe_tO) is an important index to characterize the oxidation ability of molten slag or steel liquid [34,35,36]. Metallurgists have studied the a(Fe_tO) of different slag systems and established a corresponding prediction model [37,38,39]. However, due to the limited scope of the application, it cannot be extended to all metallurgical slag systems. Li et al. [40] showed that the mass action concentration of iron oxides $N_{\mathrm{FeO}}$ based on IMCT could more effectively characterize the oxidation capacity of slag than a(Fe_tO). According to the IMCT model, the oxidation capacity of the refining slag system can be expressed as follows [16]:

$$N_{\mathrm{FeO}}=\frac{2n_{\mathrm{FeO}}}{{{\displaystyle \sum }}^{ }{n}_i}$$

(9)

where $n_{\mathrm{FeO}}$ is the mole number of FeO in the free state when slag is in equilibrium and ${{\displaystyle \sum }}^{ }{n}_i$ is the total number of structural units. The larger the value of $N_{\mathrm{FeO}}$, the stronger the oxidation capacity of the slag. When the slag has the maximum oxidation capacity, the oxygen content in the molten steel that is balanced with the refining slag is the highest. In this case, the oxygen in the molten steel can be effectively removed as soon as possible by using aluminum precipitation deoxidation or vacuum carbon deoxidation. Studies have shown that when w(FeO) in slag is lower than 1 wt%, the FeO has little effect on the T(O) in liquid steel [16,41,42], which provides a reliable guarantee for the production of low-oxygen, clean bearing steel, and the secondary oxidation of molten steel caused by high FeO content in the refining slag can be ignored. Therefore, the corresponding alkalinity when the refining slag has the maximum oxidation capacity at the condition of FeO content less than 1 wt%, is defined as the optimal alkalinity.

3. Results and Discussion

According to the actual composition of the refining slag for the production of high-chromium bearing steel in Daye Special Steel, as listed in Table 2, the influence of w(FeO), w(MgO), w(Al₂O₃), w(CaO)/w(Al₂O₃) and w(CaF₂) on the optimum alkalinity of the refining slag system is discussed based on IMCT.

3.1. Impact of FeO

When the components of MgO, Al₂O₃ and CaF₂ in the refining slag remain unchanged, the influence of FeO content on the optimum alkalinity is as shown in Figure 2. As can be seen, the value of $N_{\mathrm{FeO}}$ is significantly enlarged with the increase of FeO content. When the FeO content ranges from 0.4 wt% to 1.2 wt%, the refining slag shows the strongest oxidation capacity at an alkalinity of about five. The optimum alkalinity remains unchanged with increasing FeO content. Thus, if the composition of MgO, Al₂O₃ and CaF₂ in the refining slag is fixed, the optimum alkalinity of the slag can be given a specific reference value. However, the increasing FeO content will inevitably lead to the enhancement of oxidation capacity for the refining slag. Therefore, in order to avoid the secondary oxidation of liquid steel caused by excessive oxidation, the FeO content should be controlled at a low level. In general, the content of FeO in the refining slag is less than 1 wt%.

3.2. Impact of MgO

The impact of MgO content on the optimum alkalinity is shown in Figure 3 when the components of FeO, Al₂O₃ and CaF₂ in the refining slag are fixed. This indicates that the value of $N_{\mathrm{FeO}}$ is significantly decreased with the increase of MgO content. When the content of MgO ranges from 1 wt% to 13 wt%, the optimum alkalinity corresponding to the highest oxidation capacity of the refining slag system is not the same. The optimum alkalinity decreases slightly with the increase of MgO content. However, the change in value of $N_{\mathrm{FeO}}$ against the higher alkalinity at a given MgO content is not obvious. Therefore, reasonable regulation of alkalinity is particularly important at different levels of MgO in the slag. At present, slag with high alkalinity is mainly used in the LF refining process of bearing steel [7,43,44]. In order to ensure that the refining slag has a stable higher oxidation capacity to achieve a better refining effect under the condition of high alkalinity, the content of MgO should be controlled at an appropriate low level, which can also effectively reduce the Mg supply from refining slag to liquid steel [45]. Meanwhile, the effect of MgO content on the fluidity of refining slag also should be considered.

3.3. Impact of Al₂O₃

When the contents of MgO, FeO and CaF₂ are fixed, the effect of Al₂O₃ content on the optimum alkalinity is as depicted in Figure 4. As shown in Figure 4, when the Al₂O₃ content varies from 15 wt% to 25 wt%, the value of $N_{\mathrm{FeO}}$ against alkalinity decreases directly, the optimum alkalinity corresponding to the highest oxidation capacity of the refining slag system tends to occur at a lower alkalinity, and the value of $N_{\mathrm{FeO}}$ with 25 wt% Al₂O₃ content is higher than that of 15 wt% Al₂O₃ content at varied alkalinities. Meanwhile, the value of $N_{\mathrm{FeO}}$ first increases and then decreases at the Al₂O₃ content of 35 wt%, the optimum alkalinity occurs when the alkalinity is between 4 and 5, and the fluctuation value of $N_{\mathrm{FeO}}$ against alkalinity is not significant. When the Al₂O₃ content varies from 45 wt% to 55 wt%, the value of $N_{\mathrm{FeO}}$ against alkalinity shows an upward trend. The value of $N_{\mathrm{FeO}}$ with 45 wt% Al₂O₃ content is higher than that of 55 wt% Al₂O₃ content at varied alkalinities, indicating that with further increase of Al₂O₃ content, the value of $N_{\mathrm{FeO}}$ against alkalinity decreases obviously. According to Figure 4, when alkalinity is about 7.2, the appropriate range of Al₂O₃ content is 35–45 wt% within the scope discussed, which is consistent with the actual composition of refining slag. Therefore, to achieve a better refining effect, the Al₂O₃ content should be controlled at an appropriate high level at higher alkalinity. Meanwhile, the high Al₂O₃ content on the desulfurization ability of refining slag should also be considered [46].

3.4. Impact of w(CaO)/w(Al₂O₃) Mass Ratio

The mass ratio of w(CaO)/w(Al₂O₃) is also intrinsically related to the mass fraction of oxygen in the molten steel [47,48]. When the contents of MgO, FeO and CaF₂ are fixed, the effect of w(CaO)/w(Al₂O₃) mass ratio on the optimum alkalinity is as shown in Figure 5, indicating that the value of $N_{\mathrm{FeO}}$ decreases with the increase of alkalinity when the mass ratio of w(CaO)/w(Al₂O₃) is 1.5 or 2.0, whereas, the effect of alkalinity on the value of $N_{\mathrm{FeO}}$ is not obvious when the w(CaO)/w(Al₂O₃) mass ratio is 1.0. In addition, with the increase of w(CaO)/w(Al₂O₃) mass ratio, the value of $N_{\mathrm{FeO}}$ decreases at a fixed alkalinity with a value greater than 4.5. Thus, in order to achieve a better refining effect at the condition of high alkalinity, a lower w(CaO)/w(Al₂O₃) mass ratio is preferred. According to the field production practice, when the mass ratio of w(CaO)/w(Al₂O₃) is controlled at about 1.2–1.4 [49,50], the total oxygen content in bearing steel can be controlled at a lower level. This is another reason for the high Al₂O₃ content in high-alkalinity refining slag at present, which also further explains that this model has a certain guiding role in optimization of refining slag composition. Meanwhile, the formation behavior of non-metallic inclusions in steel and the inclusion absorption ability of the refining slag affected by the w(CaO)/w(Al₂O₃) mass ratio should also be considered [51,52].

3.5. Impact of CaF₂

CaF₂ is added as an auxiliary material in slag, which can decrease the apparent viscosity and increase the flow ability of slag [53,54]. The effect of CaF₂ content on the optimal alkalinity when the contents of MgO, FeO and Al₂O₃ are fixed is depicted in Figure 6. This shows that the value of $N_{\mathrm{FeO}}$ first increases and then decreases at a given CaF₂ content with a value greater than 8 wt%, while the value of $N_{\mathrm{FeO}}$ against alkalinity decreases directly with a lower CaF₂ content. However, when the CaF₂ content is fixed, the relative percentage of variation value for $N_{\mathrm{FeO}}$ is not more than 5% at varied alkalinities. Furthermore, the value of $N_{\mathrm{FeO}}$ increases slightly with the increasing CaF₂ content at a fixed alkalinity. Comparatively speaking, a higher CaF₂ content is preferred in high-alkalinity refining slag, but the degree of influence is not significant. In addition, the content of CaF₂ must be strictly controlled due to the fact that CaF₂ has different degrees of evaporation at an elevated temperature, which is extremely harmful to the environment and health.

The above analysis shows that the change trend based on the IMCT model is consistent with actual high-alkalinity refining slag, indicating that this model can be used as a guide in the process of refining slag composition optimization and is conducive to realizing the effective control of total oxygen content in the refining process.

4. Conclusions

In this article, a calculation model of optimal alkalinity of the refining slag CaO–SiO₂–Al₂O₃–MgO–FeO–CaF₂ at 1853 K for bearing steel production was established based on IMCT, and the influencing factors were discussed. The change trend based on the IMCT model is consistent with actual high-alkalinity refining slag. Specific conclusions are drawn as follows:

• The maximum oxidation capacity value of the refining slag occurs when the alkalinity is about five at varied FeO contents. The optimum alkalinity remains the same with changes in FeO content.
• The content of MgO has an obvious effect on the optimum alkalinity. The results show that the optimum alkalinity decreases with increasing MgO content. However, the value of $N_{\mathrm{FeO}}$ against the higher alkalinity changes subtly at a given MgO content.
• Al₂O₃ content has a significant effect on the oxidation capacity of slag. Its effect on the optimum alkalinity is opposite to that of MgO. With increasing Al₂O₃ content, the optimal alkalinity obviously increased, while the maximum value of $N_{\mathrm{FeO}}$ occurs when the Al₂O₃ content varied from 35 wt% to 45 wt% at higher alkalinity.
• The higher w(CaO)/w(Al₂O₃) mass ratio has an obvious effect on the value of $N_{\mathrm{FeO}}$ against alkalinity. With the increase of w(CaO)/w(Al₂O₃) mass ratio, the value of $N_{\mathrm{FeO}}$ decreases at a fixed alkalinity with a value greater than 4.5. In order to achieve a better refining effect, a lower w(CaO)/w(Al₂O₃) mass ratio is preferred.
• The effect of alkalinity on the value of $N_{\mathrm{FeO}}$ is not obvious at a fixed CaF₂ content; however, when the alkalinity remains the same, the value of $N_{\mathrm{FeO}}$ increases slightly with increasing CaF₂ content.