Assessment of Inclusion Removal Ability in Refining Slags Containing Ce₂O₃

Abstract

The elimination of inclusions in steelmaking processes has been widely studied. The removal of inclusions by slags containing the rare earth oxide Ce₂O₃ are studied using an integrated numerical model. The integrated model involves the inclusion motion model, interfacial tension calculation model, surface tension calculation model of slag, and the mass action concentration model, based on ion and molecule coexistence theory. The motion behaviors of both solid Al₂O₃ inclusions and 50%wtAl₂O₃–50%wtCaO liquid inclusions of varied sizes at CaO-Ce₂O₃-SiO₂-Al₂O₃(-MgO) slag systems are evaluated. The results show that it is more difficult to remove the inclusions with smaller sizes and in slag with a higher viscosity. Liquid inclusions are more difficult to remove than solid inclusions. It is found that the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO refining slag shows a better ability to remove Al₂O₃ inclusions than that of the CaO-SiO₂-Al₂O₃-MgO slag. The reason for this is that the addition of the rare earth oxide Ce₂O₃ can decrease the viscosity of slags, as well as improving the wetting effects of slags on Al₂O₃ inclusions. For two slags systems, the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system shows a better ability to remove Al₂O₃ inclusions than the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system. The addition of 5% to 8% Ce₂O₃ in a CaO-SiO₂-Al₂O₃-MgO slag is an optimized case for industrial applications.

1. Introduction

Non-metallic inclusions are harmful to steel product quality [1,2,3], and they also influence the steelmaking process via clogging of the nozzles [4,5,6]. A great number of studies have focused on the analysis of inclusions during the whole steelmaking process [7,8,9,10,11,12,13,14,15]. Meanwhile, the inclusion formation and control in a steelmaking unit, for example, a ladle furnace [16,17,18], vacuum degassing [19,20,21,22], Ruhrstahl–Heraeus [23,24,25,26,27], and tundish [28,29,30,31] have also been widely studied.

Absorption by slag [32,33,34] and adhesion to refractories [35] are two common ways to remove inclusions. Adherence to rising bubbles is a good method for the floatation of inclusions. The process of inclusion absorption by the top slag can be divided into four steps [36]: (1) inclusions are transported to the turbulent boundary layer of the steel–slag interface; (2) inclusions transport through the boundary layer to the steel–slag interface; (3) inclusions separate to the slag; and (4) inclusions are dissolved in the slag phase.

For step (1), a great number of studies have focused on the nucleation, motion, collision, and transport of inclusions in secondary refining units by using fundamental water models [37,38,39,40,41,42], mathematical models [43,44,45,46,47], and Computational Fluid Dynamics (CFD) models [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66]. Both the Eulerian approach (the statistical evaluation of inclusion species) and the Lagrangian approach (the evaluation of particles) were used in the models. However, as reviewed by C. Chen et al. [66], steps (2)–(4) of the removal of inclusions were neglected in most CFD studies. This is challenging as it is a combination of all the steps. The set-up of inclusion removal in slag is according to the Stokes rising velocity in most Eulerian approaches, following the work of Tacke and Ludwig [67]. In Lagrangian approaches, the inclusion particles were commonly treated as being fully absorbed by the slag once the particle touched the interface. A novel approach was developed [68] by considering a critical velocity condition to evaluate the trap or to reflect the conditions of inclusions. A more complicated approach was developed recently [69].

As mentioned above, step (1) can be modeled elaborately in the CFD models. One attempt was made to combine steps (1)–(2) in a CFD model [66,70]. It was found that the interface slip velocity has a greater influence on the removal of small-sized inclusions. Step (2), i.e., the deposition of inclusions at the boundary layer of steel–slag interface, is dominant and cannot be neglected for small inclusions. In contrast, the influence of the boundary layer on large-sized inclusions is negligible since the inclusions are likely to penetrate the boundary layers. Thus, steps (3) and (4) are considered to be key steps for the removal of inclusions in general descriptions [32].

The studies on step (3) could be dated back to a pioneer work by Nakajima and Okamura [71] in 1992. They developed a model for single solid spherical inclusion motion at the steel–slag interface. This model could be categorized into a Lagrangian approach, which was based on Newton’s second law. The model has been continuously developed by introducing the turbulent burst theory to calculate the possibility of the re-entrainment of inclusions [72], by applying it to solid inclusions of other compositions [15,33,73] and liquid inclusions [74,75], by extending the model to different shapes of inclusions, such as octahedral and plate-like, etc. [76,77,78], by introducing the slag dissolution kinetics [79], by improving the theory of a slag film capture [80], and by applying it to a study on the water model of particles [81,82]. Recently, CFD models coupled flow–solid interaction [83,84] or the use of the dynamic overset grid technique [85] were developed to study the inclusion motion and separations. Furthermore, the motions of inclusions under a supergravity field were studied from a new perspective [86].

For step (4), inclusions dissolution in slags has been widely studied using experiments [34,87,88,89,90,91,92,93,94] and numerical models [79,95,96,97,98]. It worth mentioning that the oxide dissolution in slags have been studied using a novel in situ observation method [92,98]. In addition, phase field models have been developed to study the dissolution of inclusions [95,96,97,98]. Recently, a concept of the inclusion capacity of slag is put forward [94].

Typically, slag is formed in many processes [99,100,101,102,103,104,105,106,107]; for example, ironmaking, convertor steelmaking, refining, continuous casting, the electroslag remelting process, and ferroalloys production. The thermodynamic properties and physicochemical properties of many types of slags have been widely evaluated and modeled [108,109,110,111,112,113,114]. Typical refining slags are CaO-Al₂O₃-SiO₂ slag and CaO-SiO₂-Al₂O₃-MgO slag systems. The inclusions dissolution in two slag systems have been studied [90,91,92,115].

For rare earth (RE) oxides, it has been added in liquid steel, as mentioned, and reviewed by Hasegawa and Sakuma [116] in 1956. It is noted that the rare earth metal addition in liquid steel in ingot casting processes have been widely investigated in 1940–1970s [116,117,118]. The clogging of the submerged entry nozzles of RE oxide inclusions have hindered the application of RE element addition in steels in continuous casting routes [119]. Later, in the 1970s, the phase diagram of Al₂O₃-REO_X was systematically measured by Mizuno et al. [120,121]. An attempt on the activity measurement in steelmaking slags CaO-Al₂O₃-Ce₂O₃ was first performed by Ueda et al. [122]. The thermodynamic properties, including phase diagram [123,124,125], structure [126,127,128], the activity and mass action concentration [129,130,131]; and physicochemical properties, including the viscosity [127,128,131], melting point [132,133], and surface tension [134] of the slag were studied. According to the research results, the addition of rare earth oxides in refining slag can reduce the activity of Al₂O₃ in slag and increase the viscosity and the melting point of slag. Attempts were also made to evaluate the effects of rare earth oxides-containing slags on steel cleanliness [135], as well as the performance of the microalloying of rare earth elements in steels by reducing the rare earth oxides in slag [136,137]. The microalloying method may be an alternative way to add rare earth elements into steel. Specifically, the balance between the Ce₂O₃-containing refining slag and the molten steel-containing aluminum will lead to a small amount of cerium being dissolved into the molten steel. This rare earth element can perform a vital role in purifying, modifying, and micro-alloying steels [138], modifying the inclusions [139,140,141,142], refining the solidification structure [143], and improving the performance [144,145,146] of RE steels. The effects of rare earth elements on the modification of inclusions have been widely studied, both in lab experiments and within industrial scale production. The rare earth can react with O and S to form rare earth oxygen sulfide [139]. The sizes of inclusions are also decreased. However, the mechanism of inclusion absorbed by the rare earth oxide-containing slag is still unclear.

In this paper, the separation of inclusions to the rare earth oxide-containing slag are studied. The motion behaviors of both solid Al₂O₃ inclusions and the liquid inclusions of varied sizes at CaO-Ce₂O₃-SiO₂-Al₂O₃ and CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag systems are evaluated using an integrated numerical model. The integrated model involves the inclusion motion model, the interfacial tension calculation model, the surface tension calculation model of slag, and the mass action concentration model, based on the ion and molecule coexistence theory. This paper is organized as follows: all the model descriptions, validations, and calculation process parameters are presented in Section 2. The results and discussions are presented in Section 3. The results can be used to predict the movement of inclusions at the steel–slag interface, with different overall wettabilities of the steel–slag-inclusion system. It will provide hints for the design of the refining slag composition, and an evaluation method of inclusion removal ability in the slags.

2. Modeling Methodology and Model Details

The studied CaO-Ce₂O₃-SiO₂-Al₂O₃ and CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag systems are denoted as slag A and slag C, respectively. The scope and flow chart of the models are shown in Figure 1.

The inclusion motion model requires the input of the properties of slags and inclusions; for example, the density and viscosity of slags, and the overall wettability of the steel–slag-inclusion system. The density and viscosity data of the two slag systems are referred from [87,133]. The overall wettabilities of steel–slag inclusions require interfacial tension between each of the two components, i.e., the interfacial tension of steel–slag, steel–inclusion, and slag–inclusion. The estimation of the interfacial tension via the Girifalco-Good equation may further require the surface tension of steel, inclusion, and slag, respectively. Thereafter, the data of the inclusion and steel have been widely studied. The surface tension of rare earth oxide-containing slag is a challenging work. The estimation model using the Butler equation and the individual surface tension of the oxide component can be utilized. In the model, the molar fraction and the activity of the oxide component are required data. Alternatively, the mass action concentration (activity) of the oxide could be calculated using the well-known ion and molecule coexistence theory.

The previous analysis followed a sequence from the modeling strategy perspective. In this section, the inverse order of model descriptions will be presented. Specifically, the surface tension of slag will be described in Section 2.1. The ion and molecule coexistence theory, and the Butler equation-based model will be briefly introduced in Section 2.1.1 and Section 2.1.2, respectively. The specific surface tension of the CaO-Ce₂O₃-SiO₂-Al₂O₃ and CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag systems will be illustrated in Section 2.1.3 and Section 2.1.4, respectively. Secondly, the interfacial tension of steel–inclusion, slag–inclusion, and steel–slag will be calculated in Section 2.2.1, Section 2.2.2 and Section 2.2.3, under Section 2.2. The yellow dotted line box in Figure 1 shows the calculation flow in Section 2.2, and the arrow indicates that the parameter value obtained in the previous step is substituted into the formula for the next step. For example, the yellow arrow indicates substituting the value of the surface tension of solid Al₂O₃ or liquid 50%wtAl₂O₃–50%wtCaO inclusion and the surface tension of liquid steel into the Girifalco-Good equation. The interfacial tension between liquid steel and inclusion can be calculated by substituting the determined value of the interaction coefficient at the same time. After this, the overall wettability of the steel–slag-inclusions is obtained. The inclusion motion model will be described in Section 2.3. The results of the motion behavior of the solid and liquid inclusions will be presented in the Results and Discussion sections. It should be noted that the overall studied 11 slag compositions are presented in

Table 1. The viscosities of CaO-Ce₂O₃-SiO₂-Al₂O₃ of different compositions at 1773 K (mass fraction).

Serial	CaO %wt	Al2O3 %wt	SiO2 %wt	Ce2O3 %wt	Basicity C/A	Viscosity (Pa·s)
A1	54.64	30.36	10	5	1.8	0.416
A2	51.00	34.00	10	5	1.5	0.363
A3	46.36	38.64	10	5	1.2	0.398
A4	51.43	28.57	10	10	1.8	0.497
A5	48.21	26.79	10	15	1.8	0.289

,

Table 2. The mole fractions of studied CaO-Ce₂O₃-SiO₂-Al₂O₃ slag cases.

Serial	XCaO	XAl2O3	XSiO2	XCe2O3
A1	0.6705	0.2045	0.1145	0.0105
A2	0.6387	0.2338	0.1169	0.0107
A3	0.5962	0.2728	0.1200	0.0110
A4	0.6580	0.2007	0.1194	0.0218
A5	0.6444	0.1966	0.1248	0.0342

,

Table 3. Mass fraction composition and viscosity of CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system at 1823 K.

Serial	CaO %wt	Al2O3 %wt	SiO2 %wt	MgO %wt	Ce2O3 %wt	Basicity C/A	Viscosity (Pa·s)
C0	48	37.5	4.5	10	0	1.27	0.495
C1	45	37.5	4.5	10	3	1.2	0.307
C2	46.2	36.3	4.5	10	3	1.27	0.374
C3	45	35.5	4.5	10	5	1.27	0.225
C4	43.4	34.1	4.5	10	8	1.27	0.265
C5	45	32.5	4.5	10	8	1.385	0.264

and

Table 4. The mole fraction of studied CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag cases.

Serial	XCaO	XAl2O3	XSiO2	XMgO	XCe2O3
C0	0.5531	0.2371	0.0484	0.1613	0
C1	0.5338	0.2442	0.0498	0.1661	0.0061
C2	0.5445	0.2349	0.0495	0.165	0.006
C3	0.5386	0.2333	0.0503	0.1676	0.0102
C4	0.5313	0.2292	0.0514	0.1714	0.0167
C5	0.5461	0.2165	0.051	0.1699	0.0166

.

The studied composition and viscosity of the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system are listed in Table 1; the viscosity data are referred to from [133].

The composition of slag is converted to mole fraction and shown in Table 2.

The studied composition and viscosity of the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system are listed in Table 3; the viscosity data are referred from [87,88].

The composition of slag is converted to mole fraction and shown in Table 4.

2.1. Calculation Model of the Surface Tension of Slag

2.1.1. The Ion and Molecule Coexistence Theory

The ion and molecule coexistence theory of molten slag is based on the analysis of the structural units existing in molten slag. According to the equilibrium reaction equation and the equilibrium constant between the components, and from calculating the mass action concentration of each component, to determine the activity of each component in molten slag.

The ion and molecule coexistence theory describes the slag structure as [147]:

• Slag is composed of various ions, compounds (molecules), and electrons. For the slag system studied in this paper, slag contains Ca²⁺, O²⁻, Ce²⁺, Mg²⁺, Al₂O₃, SiO₂, and aluminate and silicate generated by their mutual reactions.
• The coexistence of ions and molecules is continuous in the slag system of any composition.
• Dynamic equilibrium reactions are achieved between ions and molecules.
• The chemical reaction in slag obeys the law of mass action.

It is considered that this theory can be extended from inside of the slag to the surface of slag, so that the surface tension of the slag can be calculated [148]. Recently, many scholars have used this theory to carry out theoretical calculation research, and the results are in good agreement with the experimental results [149,150,151,152].

2.1.2. The Butler Equation

The Butler equation proposes that the surface of the solution is regarded as the surface phase, and that the relationship between the surface tension and the thermodynamic parameters is deduced based on the assumption that the components in the surface phase and the components in the bulk phase are in thermodynamic equilibrium [153,154]. This model calculates the surface tension of slag based on the Butler equation, which can be expressed as:

$$\sigma ={\sigma }_i^0+\frac{RT}{A_i}ln\frac{N_i^{Surf}}{N_i^{Bulk}}$$

(1)

where σ_i⁰ is the surface tension of pure component i, R is a gas constant, T is the temperature, and N_i^Surf and N_i^Bulk are the mass action concentrations of component i in the surface phase and bulk phase, respectively. A_i is the surface area of the monolayer atoms of component i, which can be expressed as:

$$A_i=L\cdot N_A^{1/3}\cdot V_i^{2/3}$$

(2)

where L is the correction factor caused by the surface structure, which is related to the arrangement of monolayer atoms in the lattice. For a molten salt and ionic oxide mixture, it is usually set to 1.091. N_A is Avogadro’s constant, N_A = 6.02 × 10²³; V_i is the molar volume of component i.

Taking a multi-components slag system as an example, the calculation steps of the model are:

• The mass action concentration N_i^Bulk of each component in a multi-components slag system are calculated by using the ion and molecule coexistence theory;
• According to Equation (1), the expression of the mass action concentration N_i^Surf of each component on the surface of a multi-components slag system can be written;
• For a combination of the surface tension and the molar volume data of each component, the equation ∑N_i^Surf = 1 and the expansion of the Butler equation, the mass action concentrations N_i^Surf of each component on the surface of a multi-components slag system are obtained;
• The surface tension is acquired by solving the Butler equation.

2.1.3. Surface Tension Calculation of the CaO-Ce₂O₃-SiO₂-Al₂O₃ Slag System

(1)

Mass Action Concentration Calculation of Each Component

Based on the ion and molecule coexistence theory and phase diagram, the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system was analyzed. At a refining temperature of 1773~1873 K, the slag contains three kinds of ions Ca²⁺, O²⁻, Ce²⁺, two kinds of simple molecules, Al₂O₃ and SiO₂, and 13 kinds of complex molecules CaO·Al₂O₃, CaO·2Al₂O₃, CaO·6Al₂O₃, 3CaO·Al₂O₃, 12CaO·7Al₂O₃, CaSiO₃, Ca₂SiO₄, Ca₃SiO₅, Ce₂O₃·Al₂O₃, Ce₂O₃·11Al₂O₃, 2CaO·Al₂O₃·SiO₂, CaO·Al₂O₃·2SiO₂, and 3Al₂O₃·2SiO₂ [130].

Defining b₁ = ∑n_CaO, b₂ = ∑n_Ce2O3, a₁ = ∑n_SiO2 = 0.1, a₂ = ∑n_Al2O3, these are the mass fractions of CaO, Ce₂O₃, SiO₂, and Al₂O₃, respectively. The mass action concentrations of each component are defined as follows: N₁ = N_CaO, N₂ = N_Ce2O3, N₃ = N_SiO2, N₄ = N_Al2O3, N₅ = N_CaO·Al2O3, N₆ = N_CaO·2Al2O3, N₇ = N_CaO·6Al2O3, N₈ = N_3CaO·Al2O3, N₉ = N_{12CaO·7Al2O3}, N₁₀ = N_CaSiO3, N₁₁ = N_Ca2SiO4, N₁₂ = N_Ca3SiO5, N₁₃ = N_Ce2O3·Al2O3, N₁₄ = N_{Ce2O3·11Al2O3}, N₁₅ = N_{CaO·Al2O3·2SiO2}, N₁₆ = N_{2CaO·Al2O3·SiO2}, and N₁₇ = N_{3Al2O3·2SiO2}.

The chemical equilibriums in the temperature range of 1773~1873 K can be defined as follows [130]:

$$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+{\mathrm{Al}}_2{\mathrm{O}}_3={\mathrm{CaOAl}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-18120-19.62T,N_5=K_1\cdot N_1\cdot N_4$$

(3)

$$\left({\mathrm{Ca}}^{2+}+{\mathrm{O}}^{2-}\right)+2{\mathrm{Al}}_2{\mathrm{O}}_3=\mathrm{CaO}\cdot 2{\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-16400-26.8T,N_6=K_2\cdot N_1\cdot N_4{}^2$$

(4)

$$\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+6\mathrm{Al}}_2{\mathrm{O}}_3=\mathrm{CaO}\cdot {6\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-17430-37.2T,N_7=K_3\cdot N_1\cdot N_4{}^6$$

(5)

$$3\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{Al}}_2{\mathrm{O}}_3=3\mathrm{CaO}\cdot {\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-17000-32.0T,N_8=K_4\cdot N_1{}^3\cdot N_4$$

(6)

$$12\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+7\mathrm{Al}}_2{\mathrm{O}}_3=12\mathrm{CaO}\cdot {7\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-86100-205.1T,N_9=K_5\cdot N_1{}^{12}\cdot N_4{}^7$$

(7)

$$\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{SiO}}_2{=\mathrm{CaSiO}}_3$$

$$\Delta G^{\Theta }=-81416-10.498T,N_{10}=K_6\cdot N_1\cdot N_3$$

(8)

$$2\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{SiO}}_2{=\mathrm{Ca}}_2{\mathrm{SiO}}_4$$

$$\Delta G^{\Theta }=-160431+4.016T,N_{11}=K_7\cdot N_1{}^2\cdot N_3$$

(9)

$$3\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{SiO}}_2{=\mathrm{Ca}}_3{\mathrm{SiO}}_5$$

$$\Delta G^{\Theta }=93366-23.03T,N_{12}=K_8\cdot N_1{}^3\cdot N_3$$

(10)

$$\left({2\mathrm{Ce}}^{3+}{+3\mathrm{O}}^{2-}\right){+\mathrm{Al}}_2{\mathrm{O}}_3{=\mathrm{Ce}}_2{\mathrm{O}}_3\cdot {\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-58555.5-15.04T,N_{13}=K_9\cdot N_2\cdot N_4$$

(11)

$$\left({2\mathrm{Ce}}^{3+}{+3\mathrm{O}}^{2-}\right){+11\mathrm{Al}}_2{\mathrm{O}}_3{=\mathrm{Ce}}_2{\mathrm{O}}_3\cdot {11\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-49323-80.56T,N_{14}=K_{10}\cdot N_2\cdot N_4{}^{11}$$

(12)

$$\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{Al}}_2{\mathrm{O}}_3{+2\mathrm{SiO}}_2{=\mathrm{CaO}\text{×}\mathrm{Al}}_2{\mathrm{O}}_3\cdot {2\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-13816.44-55.266T,N_{15}=K_{11}\cdot N_1\cdot N_4\cdot N_3^2$$

(13)

$$2\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{Al}}_2{\mathrm{O}}_3{+\mathrm{SiO}}_2=2\mathrm{CaO}\cdot {\mathrm{Al}}_2{\mathrm{O}}_3\cdot {\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-61964.64-60.29T,N_{16}=K_{12}\cdot N_1{}^2\cdot N_4\cdot N_3$$

(14)

$${3\mathrm{Al}}_2{\mathrm{O}}_3{+2\mathrm{SiO}}_2{=3\mathrm{Al}}_2{\mathrm{O}}_3\cdot {2\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-4351-10.46T,N_{17}=K_{13}\cdot N_4{}^3\cdot N_3^2$$

(15)

The equilibrium constants K_i of all reactions satisfy the expression ΔG^Θ = −R·T·lnK_i; K_i can be calculated using the following formula:

$$K_i=\exp (\frac{\Delta G^{\Theta }}{RT})$$

(16)

where ΔG^Θ is the change of the reaction-free energy of each reaction.

The total equilibrium mole number of all structural units in the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system can be expressed as:

$$\begin{array}{l}{{\displaystyle \sum }}^{\text{}}{n}_i=2n_1+5n_2+n_3+n_4+n_5+n_6+n_7+n_8+n_9+n_{10}\\ +n_{11}+n_{12}+n_{13}+n_{14}+n_{15}+n_{16}+n_{17}\end{array}$$

(17)

where, n_i corresponds to the mole number of 17 substances existing in the slag, N_i = n_i/∑n_i, but for the ion-pairs (2Ce³⁺ + 3O²⁻) and (Ca²⁺ + O²⁻) existing in the slag, N_Ce2O3 = N_Ce³⁺ + N_O²⁻ = (2n_Ce³⁺ + 3n_O²⁻)/(∑n_i ) = 5n_Ce2O3/∑n_i, (Ca²⁺ + O²⁻) is calculated in the same way as (2Ce³⁺ + 3O²⁻).

According to mass equilibrium, the following formulas could be obtained:

$$b_1={{\displaystyle \sum }}^{\text{}}{n}_i\cdot \left(\begin{array}{c}0.5N_1+N_5+N_6+N_7+3N_9+N_{10}+2N_{11}\\ +3N_{12}+N_{15}+12N_{16}\end{array}\right)$$

(18)

$$b_2={{\displaystyle \sum }}^{\text{}}{n}_i\cdot \left(0.2N_2+N_{13}+N_{14}\right)$$

(19)

$$a_1={{\displaystyle \sum }}^{\text{}}{n}_i\cdot \left(N_3+N_{10}+N_{11}+N_{12}+2N_{15}+N_{16}+2N_{17}\right)$$

(20)

$$a_2={{\displaystyle \sum }}^{\text{}}{n}_i\cdot \left(\begin{array}{c}{N}_4+N_5+2N_6+6N_7+N_8+7N_9+N_{13}\\ +11N_{14}+N_{15}+N_{16}+3N_{17}\end{array}\right)$$

(21)

$$\begin{array}{l}{N}_1+N_2+N_3+N_4+N_5+N_6+N_7+N_8+N_9+N_{10}\\ +N_{11}+N_{12}+N_{13}+N_{14}+N_{15}+N_{16}+N_{17}=1\end{array}$$

(22)

Therefore, Equations (3)–(15) and (18)–(22) are the governing equations of the developed thermodynamic model for calculating the mass action concentrations N_i of the structural units or ion couples in the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag. Specifically, the values of N₁, N₂, N₃, and N₄ can be obtained by solving the Equation set (18)–(22), by taking Equations (3)–(15) and the specific values of T, b₁, b₂, and a₂ as the initial conditions. Then, N₅–N₁₇ can be solved by substituting N₁–N₄ into Equations (3)–(15). Newton’s iteration method is used in the process of solving the equation set.

The calculated results of the mass action concentration (activity) and the surface tension are identical to the calculation results of Wu et al. [130], which have been validated with experimental results. The total average deviation of the model was 4–9% [154,155,156].

(2)

Surface tension calculation

Extending the ion and molecule coexistence theory from inside of the slag to the surface, the surface mass action concentrations are defined as follows: N₁^S = N^S_CaO, N₂^S = N^S_Ce2O3, N₃^S = N^S_SiO2, N₄^S = N^S_Al2O3, N₅^S = N^S_CaO·Al2O3, N₆^S = N^S_CaO·2Al2O3, N₇^S = N^S_CaO·6Al2O3, N₈^S = N^S_3CaO·Al2O3, N₉^S = N^S_{12CaO·7Al2O3}, N₁₀^S = N^S_CaSiO3, N₁₁^S = N^S_Ca2SiO4, N₁₂^S = N^S_Ca3SiO5, N₁₃^S = N^S_Ce2O3·Al2O3, N₁₄^S = N^S_{Ce2O3·11Al2O3}, N₁₅^S = N^S_{CaO·Al2O3·2SiO2}, N₁₆^S = N^S_{2CaO·Al2O3·SiO2}, and N₁₇^S = N^S_{3Al2O3·2SiO2}.

According to the mass conservation:

$$\begin{array}{l}{N}_1^S+N_2^S+N_3^S+N_4^S+N_5^S+N_6^S+N_7^S+N_8^S+N_9^S+N_{10}^S\\ +N_{11}^S+N_{12}^S+N_{13}^S+N_{14}^S+N_{15}^S+N_{16}^S+N_{17}^S=1\end{array}$$

(23)

A new equation containing only N₁^S, N₂^S, N₃^S, and N₄^S is obtained by substituting the surface mass action concentration into the equilibrium constant expression in Equations (3)–(15), and then substituting the results into Equation (23).

According to the Equation (1), the Butler equation corresponding to the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system can be written in the following form:

$${\sigma }_{CaO-C{e}_2{O}_3-Si{O}_2-A{l}_2{O}_3}={\sigma }_{CaO}^0+\frac{RT}{A_{CaO}}ln\frac{N_1^s}{N_1}$$

(24)

$$={\sigma }_{C{e}_2{O}_3}^0+\frac{RT}{A_{C{e}_2{O}_3}}ln\frac{N_2^s}{N_2}$$

(25)

$$={\sigma }_{Si{O}_2}^0+\frac{RT}{A_{Si{O}_2}}ln\frac{N_3^s}{N_3}$$

(26)

$$={\sigma }_{A{l}_2{O}_3}^0+\frac{RT}{A_{A{l}_2{O}_3}}ln\frac{N_4^s}{N_4}$$

(27)

Three new equations are obtained by subtracting (25)–(27) from (24), respectively, and the equation simplified by (23) is formed into a set of equations. The values of N₁^S, N₂^S, N₃^S, and N₄^S can be gained by solving the equations. The surface tension of the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system can be acquired by substituting the values of N₁^S, N₂^S, N₃^S, and N₄^S into one of the Equations of (24)–(27).

To calculate the surface tension of this slag system, the surface tension and molar volume data of each component are also required, which can be found in references [109,148], and are listed in

Table 5. The relationships between surface tension and molar volume of pure components and temperature.

Component	Relationship between Surface Tension (×10−3 N/m) and Temperature (K)	Relationship between Molar Volume (m3/mol) and Temperature (K)
CaO	791 − 0.0935 T	20.7 [1 + 1·10−4·(T-1773)]·10−6
Al2O3	1024 − 0.177 T	28.3 [1 + 1·10−4·(T-1773)]·10−6
SiO2	243.2 + 0.031 T	27.516 [1 + 1·10−4·(T-1773)]·10−6
MgO	1770 − 0.636 T	16.1 [1 + 1·10−4·(T-1773)]·10−6

.

The surface tension and molar volume of Ce₂O₃ cannot be found in the available literatures. However, the data of rare earth lanthanide elements can be found in references [134,147], and they are listed in

Table 6. Surface tension and molar volume data for several lanthanide oxides.

Element	Oxide	Surface Tension (×10−3 N/m, 1873 K)	Surface Tension (×10−3 N/m, 2573 K)	Relationship between Molar Volume (m3/mol) and Temperature (K)
La	La2O3	686 (Solid)	560 (Liquid)	50.1 [1 + 1·10−4·(T-1773)]·10−6
Ce	Ce2O3	To be estimated	Unknown	To be estimated
Nd	Nd2O3	677 (Solid)	Unknown	46.62 [1 + 1·10−4·(T-1773)]·10−6
Sm	Sm2O3	Unknown	815 (Liquid)	47.93 [1 + 1·10−4·(T-1773)]·10−6
Gd	Gd2O3	664 (solid)	Unknown	49.09 [1 + 1·10−4·(T-1773)]·10−6

.

According to the data in Table 6, the relationship between the molar volume (m³/mol) of Ce₂O₃ and temperature (K) is reasonably estimated to be 49.0 [1 + 1·10⁻⁴·(T-1773)]·10⁻⁶, and the surface tension of Ce₂O₃ at 1873 K is estimated to be 680 × 10⁻³ N/m. The surface tension at 1773 K is 690 × 10⁻³ N/m.

After calculating via MATLAB software, the surface tension data of the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system are obtained and listed in

Table 7. Surface tension of CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system at 1773 K.

Serial	CaO %wt	Al2O3 %wt	SiO2 %wt	Ce2O3 %wt	Basicity C/A	Surface Tension (N/m)
A1	54.64	30.36	10	5	1.8	0.5693
A2	51.00	34.00	10	5	1.5	0.5690
A3	46.36	38.64	10	5	1.2	0.5693
A4	51.43	28.57	10	10	1.8	0.5682
A5	48.21	26.79	10	15	1.8	0.5670

. It should be noted that the surface tension varies within a limited range.

2.1.4. Surface Tension Calculation of the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO Slag System

Compared with the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system, the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system increases by one more ion Mg²⁺, one more simple molecule MgO, and five more complex molecules MgO·Al₂O₃, MgO·SiO₂, Mg₂SiO₄, CaO·MgO·2SiO₂, and 2CaO·MgO·2SiO₂. The mass action concentration of each component was defined as follows: N₁₈ = N_MgO, N₁₉ = N_MgO·Al2O3, N₂₀ = N_MgO·SiO2, N₂₁ = N_Mg2SiO4, N₂₂ = N_{CaO·MgO·2SiO2}, and N₂₃ = N_{2CaO·MgO·2SiO2}. The following five equilibrium reactions are required:

$$\left({\mathrm{Mg}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{Al}}_2{\mathrm{O}}_3=\mathrm{MgO}\cdot {\mathrm{Al}}_2{\mathrm{O}}_3$$

$$\Delta G^{\Theta }=-35530-17.39T,N_{19}=K_{14}\cdot N_{18}\cdot N_4$$

(28)

$$\left({\mathrm{Mg}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{SiO}}_2=\mathrm{MgO}\cdot {\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-41089.4-6.10T,N_{20}=K_{15}\cdot N_{18}\cdot N_3$$

(29)

$$2\left({\mathrm{Mg}}^{2+}{+\mathrm{O}}^{2-}\right){+\mathrm{SiO}}_2=2\mathrm{MgO}\cdot {\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-67130.8-2.508T,N_{21}=K_{16}\cdot N_{18}^2\cdot N_3$$

(30)

$$\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right)+\left({\mathrm{Mg}}^{2+}{+\mathrm{O}}^{2-}\right){+2\mathrm{SiO}}_2=\mathrm{CaO}\cdot \mathrm{MgO}\cdot {2\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-162602+18.81T,N_{22}=K_{17}\cdot N_1\cdot N_{18}\cdot N_3^2$$

(31)

$$2\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right)+\left({\mathrm{Mg}}^{2+}{+\mathrm{O}}^{2-}\right){+2\mathrm{SiO}}_2=2\mathrm{CaO}\cdot \mathrm{MgO}\cdot {2\mathrm{SiO}}_2$$

$$\Delta G^{\Theta }=-73688-63.639T,N_{23}=N_1^2\cdot N_{18}\cdot N_3^2$$

(32)

The surface tension of the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system is calculated by using the existing data and calculation methods described in Section 2.1.3. The results are shown in

Table 8. Surface tension of CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system at 1823 K.

Serial	CaO %wt	Al2O3 %wt	SiO2 %wt	MgO %wt	Ce2O3 %wt	Surface Tension (N/m)
C0	48	37.5	4.5	10	0	0.5992
C1	45	37.5	4.5	10	3	0.6725
C2	46.2	36.3	4.5	10	3	0.6735
C3	45	35.5	4.5	10	5	0.6736
C4	43.4	34.1	4.5	10	8	0.6740
C5	45	32.5	4.5	10	8	0.6754

.

2.2. Calculation Model of Interfacial Tension

According to Girifalco-Good equation, the interfacial tension between inclusions and molten steel, between inclusions and slag, and between slag and steel are calculated.

Girifalco-Good equation [157]:

$${\sigma }_{ab}={\sigma }_a+{\sigma }_b-2{\phi }_{ab}\sqrt{{\sigma }_a\cdot {\sigma }_b}$$

(33)

where the subscripts a and b represent the two phases, σ_ab is the interfacial tension between phases a and b, σ_a and σ_b are the surface tensions of each phase, and φ_ab represents the interaction coefficient between a and b.

2.2.1. Interfacial Tension between Inclusion and Molten Steel

This section mainly refers to the research results of Nakajima [158]. According to the Girifalco-Good equation, the interfacial tension between inclusions and molten steel can be expressed as:

$${\sigma }_{IM}={\sigma }_I+{\sigma }_M-2\cdot {\phi }_{IM}\sqrt{{\sigma }_I\cdot {\sigma }_M}$$

(34)

The surface tension σ_I of Al₂O₃ inclusions is 0.75 N/m. To calculate the surface tension σ_M of molten steel, as a simplification, only a certain content of O and S are considered in steel. Then, the surface tension of liquid steel is expressed as follows:

$${\sigma }_M=\left[\begin{array}{c}1910-825\cdot {log}_{10}\left(1+210\cdot \left[O\right]\right)\\ -540\cdot {log}_{10}\left(1+185\cdot \left[S\right]\right)\end{array}\right]\cdot {10}^{-3}\left(\mathrm{N}/\mathrm{m}\right)$$

$$\left(\left[S\right]\le 0.18wt\%,\left[O\right]\le 0.015wt\%\right)$$

(35)

In this study, [S] = 0.005%, [O] = 0.0025%, and σ_M = 1.6052 N/m.

The interaction coefficient between solid inclusions and molten steel:

$${\phi }_{IM}=0.351-0.507\cdot \left(R-1\right)$$

(36)

where R is the surface roughness of the inclusion, and its value ranges over 1.0~1.36.

Ideally, the surface roughness of the inclusion is 1, so that φ_IM = 0.351. Substituting all the data into Equation (35), we can obtain σ_IM = 1.585 N/m.

2.2.2. Interfacial Tension between Inclusion and Slag

The interfacial tension between inclusion and slag can be expressed as follows:

$${\sigma }_{IS}={\sigma }_I+{\sigma }_S-2\cdot {\phi }_{IS}\sqrt{{\sigma }_I\cdot {\sigma }_S}$$

(37)

The surface tension between the inclusion and slag has been obtained. The interaction coefficient φ_IS between the inclusion and slag is required.

The interaction coefficient between the Al₂O₃ inclusion and the CaO-SiO₂-Al₂O₃-MgO-CaF₂-Na₂O slag system was studied in reference [158]; φ_IS is expressed as:

$$\begin{array}{l}{\phi }_{IS}=0.947\times X_{A{l}_2{O}_3}+0\times X_{Si{O}_2}+0.602\times X_{Ca{F}_2}+0.601\times X_{MgO}\\ +0.545\times X_{CaO}+0.475\times X_{N{a}_{2}O}+0.705\times X_{A{l}_2{O}_3}\times X_{CaO}\\ -0.609\times X_{A{l}_2{O}_3}\times X_{Si{O}_2}+1.564\times X_{Si{O}_2}\times X_{CaO}\\ +0\times X_{Ca{F}_2}\times X_{CaO}+0.755\times X_{Ca{F}_2}\times X_{Si{O}_2}\\ +1.292\times X_{Si{O}_2}\times X_{MgO}+1.156\times X_{Si{O}_2}\times X_{N{a}_{2}O}\end{array}$$

(38)

where X_CaO, X_SiO2, X_Al2O3, X_MgO, X_CaF2, and X_Na2O are the molar fractions of each component.

The interaction coefficient between the slag system and the Al₂O₃ inclusion was studied in the research results of Nakajima [158], as shown in

Table 9. Measured data of interaction coefficient between slag system and Al₂O₃ inclusion.

XCaO	XAl2O3	XSiO2	XMgO	φIS
0.453	0.125	0.423	0	0.630
0.486	0.059	0.454	0	0.701
0.342	0.125	0.532	0	0.564
0.673	0.287	0.035	0.005	0.820
0.673	0.287	0.035	0.005	0.821

.

According to the data in the literature and theoretical analysis, φ_IS = 0.8 is estimated for the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system.

For the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system, the interaction coefficient is calculated by Equation (38), ignoring the effect of Ce₂O₃. The calculation results are shown in

Table 10. Interaction coefficient between inclusion and slag (ignoring the effect of X_Ce2O3).

Serial	XCaO	XAl2O3	XSiO2	XMgO	XCe2O3	φIS
C0	0.5531	0.2371	0.0484	0.1613	0	0.7603
C1	0.5338	0.2442	0.0498	0.1661	0.0061	0.7588
C2	0.5445	0.2349	0.0495	0.165	0.006	0.7542
C3	0.5386	0.2333	0.0503	0.1676	0.0102	0.7499
C4	0.5313	0.2292	0.0514	0.1714	0.0167	0.7424
C5	0.5461	0.2165	0.051	0.1699	0.0166	0.7361

.

Based on the calculated data, considering the stronger interaction between slag containing Ce₂O₃ and Al₂O₃, φ_IS = 0.75 is used for the inclusion–slag interfacial tension of slag C0, and φ_IS = 0.78 is used for the inclusion–slag interfacial tension of slag C1–C5.

By substituting σ_I, σ_S, and φ_IS of slags with different compositions into Equation (37), σ_IS is calculated.

2.2.3. Interfacial Tension between Slag and Steel

The interfacial tension between slag and steel can be expressed as:

$${\sigma }_{MS}={\sigma }_M+{\sigma }_S-2\cdot {\phi }_{MS}\sqrt{{\sigma }_M\cdot {\sigma }_S}$$

(39)

For the CaO-SiO₂-Al₂O₃-MgO-CaF₂-Na₂O slag system studied in reference [158], the interaction coefficient between slag and steel is expressed as:

$$\begin{array}{l}{\phi }_{MS}=0.576\times X_{Si{O}_2}+0.399\times X_{A{l}_2{O}_3}+0.326\times X_{Ca{F}_2}+0.687\times X_{MgO}\\ +0.607\times X_{CaO}-0.675\times X_{Si{O}_2}\times X_{CaO}+0.338\times X_{Si{O}_2}\times X_{A{l}_2{O}_3}\\ -0.391\times X_{A{l}_2{O}_3}\times X_{CaO}-0.892\times X_{A{l}_2{O}_3}\times X_{Ca{F}_2}\\ -1.701\times X_{MgO}\times X_{Si{O}_2}-0.27\times X_{MgO}\times X_{A{l}_2{O}_3}\end{array}$$

(40)

The relevant data of Ce₂O₃ cannot be found in the literature, so that the interaction coefficient between the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system and molten steel needs to be reasonably estimated. The CaO-SiO₂-Al₂O₃-MgO slag system is referred, and the interaction coefficient between slag and steel in reference [158] is listed in

Table 11. Measured data of interaction coefficient between CaO-SiO₂-Al₂O₃-MgO slag system and steel.

XCaO	XAl2O3	XSiO2	XMgO	φMS
0.536	0.246	0.168	0.050	0.439
0.604	0.277	0.068	0.051	0.433
0.481	0.287	0.165	0.067	0.432
0.514	0.306	0.109	0.072	0.453
0.544	0.324	0.056	0.076	0.454
0.505	0.233	0.164	0.098	0.438
0.572	0.262	0.067	0.099	0.428
0.466	0.215	0.160	0.159	0.435
0.531	0.243	0.065	0.161	0.429

.

It is observed that the interaction coefficient between the CaO-Al₂O₃-SiO₂-MgO slag system, which is similar to the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system studied in this paper, and steel, ranges from 0.428 to 0.454. The average value of the interaction coefficient between CaO-Ce₂O₃-SiO₂-Al₂O₃ slag and liquid steel is about 0.43. In order to improve the accuracy of the interaction coefficient, it is modified according to the following analysis.

In Section 2.3, the motion model of inclusion at the slag–steel interface, the overall wettability cosθ_IMS of the steel–slag inclusion system is required:

$$cos{\theta }_{IMS}=\frac{{\sigma }_{IM}-{\sigma }_{IS}}{{\sigma }_{MS}}$$

(41)

The relationship between σ_IM, σ_IS, σ_MS, and θ_IMS is shown in Figure 2. There are two wetting states between slag and inclusion, as shown in Figure 2a; cosθ_IMS > 0, which means the inclusion shows a good wettability for slag. If cosθ_IMS <0, as shown in Figure 2b, this means that the inclusion shows a poor wettability for slag.

It can be seen that cosθ_IMS increases with the decrease in σ_IS or σ_MS, so that the slag–steel interaction coefficient should be reduced to a smaller value to compensate for the deficiency where σ_IS may be too large, which is beneficial for improving the accuracy of the calculation model. Therefore, for CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system, φ_MS = 0.4 is used in the calculation.

By substituting σ_M, σ_S, and φ_MS into Equation (39), the interfacial tension σ_MS between the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system and molten steel can be obtained.

For the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system, the slag composition is close to the slags in Table 11. Thus, the interaction coefficient between slag and steel is referred to in Table 11. Therefore, φ_MS = 0.43 is used in the calculation of the steel–slag interfacial tension of slag C0 without Ce₂O₃. Considering that the interaction between slag containing Ce₂O₃ and molten steel will be strengthened, φ_MS = 0.45 is estimated in the calculation of slag C1–C5. In the same way, the interfacial tension σ_MS between the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system and molten steel can be obtained.

2.2.4. Interfacial Tension Summary

All of the calculation results of the two slag systems are summarized in

Table 12. Interfacial properties of the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system at 1773 K.

Serial	σIM (N/m)	σIS (N/m)	σMS (N/m)	Overall Wettability cosθIMS
A1	1.585	0.2738	1.4097	0.9301
A2		0.2738	1.4096	0.9302
A3		0.2738	1.4097	0.9301
A4		0.2737	1.4094	0.9304
A5		0.2736	1.4090	0.9307

and

Table 13. Interfacial properties of the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system at 1823 K.

Serial	σIM (N/m)	σIS (N/m)	σMS (N/m)	Overall Wettability cosθIMS
C0	1.585	0.3436	1.3610	0.9121
C1		0.3146	1.3426	0.9462
C2		0.3148	1.3429	0.9459
C3		0.3148	1.3430	0.9458
C4		0.3149	1.3431	0.9457
C5		0.3151	1.3435	0.9452

.

2.3. Motion Model of Inclusion at the Steel–Slag Interface

The interfacial tension data calculated in the above sections will be used in the motion model of inclusions that will be described in this section.

2.3.1. Assumptions of This Model

The basic assumptions of this mathematical motion model of inclusions at the steel–slag interface are listed as follows:

(1) The inclusions are spherical in constant volume;

(2) No chemical reaction occurs between the phase interfaces;

(3) All fluids are incompressible and isothermal;

(4) The slag phase exists in liquid form;

(5) The steel–slag interface is smooth;

(6) The movement process of inclusions depends on the force balance between the buoyancy force, rebound force, drag force, and the fluid added mass force;

(7) The surface tension is uniform at the interface.

As proposed by Nakajima and Okamura [71], when the inclusion approaches the slag–steel interface, a steel film forms when Re ≥ 1, otherwise, the inclusion contacts with the slag directly without steel film formation. The Reynolds number of inclusions is defined as:

$$Re=\frac{2{\rho }_M{u}_{\infty }{R}_I}{{\mu }_M}$$

(42)

where R_I is the radius of inclusion, ρ_M is the density of molten steel, μ_M is the viscosity of the molten steel, and u_∞ is the velocity of inclusion.

In the studied small size of the inclusion cases, for example, if the diameter is less than 150 μm, the Reynolds number is less than 1. Thus, the model of non-steel film could be used. In this paper, the motion behavior of the inclusion is studied at the steel–slag interface. Specifically, the inclusion is at an initial position below the slag; that is, when t = 0, the displacement from the center of inclusion to the interface Z = 0, as shown in Figure 3. Then, there are three kinds of interfacial motion behaviors of the inclusions: Pass (P), Oscillate (O), and Remain (R). “Pass” means that the displacement of the inclusion is higher than two times the radius, the motion path of inclusions is 1→2→3, as shown in Figure 3a. “Oscillate” means that the inclusions oscillate but do not penetrate the slag–steel interface, and the motion path of the inclusions is 1→2→3→4, as shown in Figure 3c. “Remain” means that the displacement of inclusion is less than two times the radius, and the motion path of the inclusions is 1→2→3, as shown in Figure 3b. When it moves up by two times their radius 2R_I, as shown in Figure 3a, it is considered to be completely separated from the molten steel into the slag.

The initial floating velocity of inclusions is expressed by the Stokes velocity, and the Stokes velocity of liquid inclusions (such as 50%wtAl₂O₃–50%wtCaO) can be expressed as:

$$u_{\infty }=\frac{2}{3}{R}_I^2\left({\rho }_M-{\rho }_I\right)\frac{g}{{\mu }_M}\frac{\left(1+\kappa \right)}{\left(2+3\kappa \right)}$$

(43)

where ρ_I is the density of inclusions; g is the acceleration of gravity; κ = μ_I/μ_M, μ_I is the viscosity of inclusion, and μ_M is the viscosity of molten steel.

In this study, ρ_M = 7000 kg/m³, μ_M = 0.006 Pa·s. ρ_I = 2814 kg/m³, μ_I = 0.118 Pa·s, and κ = μ_I/μ_M =19.67 for liquid inclusion 50%wtAl₂O₃–50%wtCaO.

For solid inclusions (such as Al₂O₃), κ→∞, ρ_I = 3990 kg/m³, the Stokes velocity of this inclusion can be expressed as:

$$u_{\infty }=\frac{2}{9}{R}_I^2\left({\rho }_M-{\rho }_I\right)\frac{g}{{\mu }_M}$$

(44)

2.3.2. Motion Equation of Inclusion

When the inclusions move across the slag–steel interface, four forces act on the inclusions, i.e., the buoyancy force F_b, rebound force F_r, drag force F_d, and fluid added mass force F_f.

The buoyancy force is in an upward direction, while the rebound force, fluid added mass force, and drag force are either upward or downward, depending on the behavior of the inclusions at the interface.

According to Newton’s second law:

$$\frac{4}{3}\pi R_I^3{\rho }_I\frac{d^{2}Z}{d{t}^2}=F_b-F_d-F_r-F_f$$

(45)

where Z is the displacement of the inclusion and t is the movement time of the inclusion.

The rebound force is defined as:

$$F_r=2\pi R_I{\sigma }_{MS}H\left(Z^{*}\right)$$

(46)

where Z* is the dimensionless displacement of the inclusion and H is a function of Z*.

$$H\left(Z^{\ast }\right)=Z^{\ast }-1-\cos {\theta }_{IMS}$$

(47)

The buoyancy force is defined as:

$$F_b=\frac{4}{3}\pi R_I^{3}g\left({\rho }_S\cdot J\left(Z^{*}\right)-{\rho }_I\right)$$

(48)

where J(Z*) is the density-varying term, which describes the degree of the inclusion’s entering into the slag.

$$J\left(Z^{*}\right)=\frac{1}{4}\left(\frac{{\rho }_M}{{\rho }_S}-1\right)Z^{*}{}^3-\frac{3}{4}\left(\frac{{\rho }_M}{{\rho }_S}-1\right)Z^{*}{}^2+\frac{{\rho }_M}{{\rho }_S}$$

(49)

The drag force is defined as:

$$F_d=4\pi \sqrt{g{R}_I^3}{\mu }_{S}A\cdot I\left(Z^{*}\right)\frac{d{Z}^{*}}{d{t}^{*}}$$

(50)

where t* is the dimensionless movement time of the inclusion, A is a function of κ, which is related to the viscosity of inclusions and molten steel; I(Z*) is the viscosity-varying term, which is related to the position of the inclusion relative to the interface.

$$A=\frac{2+3\kappa }{2\left(1+\kappa \right)}$$

(51)

$$I\left(Z^{*}\right)=\left(\frac{{\mu }_M}{{\mu }_S}-1\right)Z^{*}{}^2-2\left(\frac{{\mu }_M}{{\mu }_S}-1\right)Z^{*}+\frac{{\mu }_M}{{\mu }_S}$$

(52)

The fluid added mass force is defined as:

$$F_f=\frac{2}{3}\pi R_I^{3}g{\rho }_{S}J\left(Z^{*}\right)\frac{d^2{Z}^{*}}{d{t}^{*}{}^2}$$

(53)

Substituting Equations (46), (48), (50) and (53) into (45), the dimensionless motion equation of inclusion can be written as follows:

$$\frac{d^2{Z}^{*}}{d{t}^{*}{}^2}=\frac{2\left({\rho }_{S}J\left(Z^{*}\right)-{\rho }_I\right)}{\left({\rho }_{S}J\left(Z^{*}\right)+2{\rho }_I\right)}-3*D\left(Z^{*}\right)*H\left(Z^{*}\right)-\frac{6A}{E\left(Z^{*}\right)}I\left(Z^{*}\right)*\frac{d{Z}^{*}}{d{t}^{*}}$$

(54)

where both D and E are functions of Z*.

$$D\left(Z^{*}\right)=\frac{{\sigma }_{MS}}{g{R}_I^2\left({\rho }_{S}J\left(Z^{*}\right)+2{\rho }_I\right)}$$

(55)

$$E\left(Z^{*}\right)=\frac{\sqrt{g{R}_I^3}\left({\rho }_{S}J\left(Z^{*}\right)+2{\rho }_I\right)}{{\mu }_S}$$

(56)

The dimensionless form of displacement, time, velocity, and acceleration are defined as:

$$Z^{\ast }=\frac{Z}{R_I}$$

(57)

$$t^{*}=t\sqrt{\frac{g}{R_I}}$$

(58)

$$\frac{d{Z}^{*}}{d{t}^{*}}=\frac{1}{\sqrt{g{R}_I}}\frac{dZ}{dt}$$

(59)

$$\frac{d^2{Z}^{*}}{d{t}^{*}{}^2}=\frac{1}{g}\frac{d^{2}Z}{d{t}^2}$$

(60)

The densities of slag ρ_s are 2700 kg/m³ and 2750 kg/m³ for the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system and the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system, respectively.

By solving Equation (54), the displacement, velocity, and acceleration changes of the inclusions with time can be solved, and the force changes of the inclusions can be calculated by using the values of these parameters.

The methods for solving the force balance and the motion equation of inclusions are as follows: the Equations (54) and (46), (48), (50) and (53) are the governing equations of the developed mathematical model for calculating the dimensionless displacement Z*(t) and the force on inclusions. Specifically, by initializing the inclusion size, the governing Equation (54) of Z*(t) is solved using the ode45 function (Runge kutta algorithm) in MATLAB software with input of the calculated physical properties, as mentioned in previous sections. After the dimensionless displacement is obtained, the force can be calculated using Equations (46), (48), (50) and (53) with specific sizes of inclusions. This model is originally validated using experimental data by Nakajima and Okamura [71]. Their modeling work on inclusions movement in slags were reproduced, and the results are in good agreement with the original paper [71]. This ensures the reliability of the present simulation.

2.3.3. Model Validation

Based on the similarity theory, a water model experiment was established to verify the mathematical model. A schematic diagram of the experimental equipment is shown in Figure 4. In the experiment, silicon oil was used to simulate slag, water was used to simulate molten steel, and hollow alumina balls with 2.1 mm diameter were used to simulate inclusions. A cylindrical container with a particle release port at the bottom was filled with water, and the height of the water was 40 cm. In order to eliminate the refraction generated by the cylindrical camber, a square container was added around the cylindrical container, and the water level was higher than 40 cm. The water was covered by silicon oil with a thickness of more than 5 mm. Hollow alumina balls were released from the bottom release port stationary. A high-speed camera (FuHuang AgileDevice Revealer 5KF20S) was used to record the frame of the alumina balls touching the water–oil interface until the motion stopped.

The dimensionless displacement curve of the alumina ball moving at the water–oil interface could be obtained by processing the video shot using the high-speed camera. Meanwhile, the initial conditions of the water model in

Table 14. Water model parameters required in calculation.

Substance	Density (kg·m−3)	Viscosity (Pa·s)	Interfacial Tension (N·m)			cosθIMS
			Water-Oil σMS	Al2O3-Water σIM	Al2O3-Oil σIS
Water	997	0.001	0.055	0.614	0.635	−0.382
Silicon oil	963	0.096
Al2O3	710	-

were substituted into the mathematical model for calculation [159]. The comparison between the calculated results and the experimental results is shown in Figure 5. It can be seen from the figure that the overall trend of the calculated value is consistent with the experimental value. After the particle reaches the water–oil interface, it floats up for a certain distance and is bounced back. The final dimensionless displacements for the experiment and model are 0.594 and 0.655, respectively. It is in good agreement, which verifies the feasibility and accuracy of the model.

3. Results and Discussion

In this section, the motion behavior of solid Al₂O₃ inclusions and liquid inclusions in two slag systems are presented. The evolution of force on inclusions, and the displacement of inclusions as a function of movement time are studied. By identifying the displacement of inclusion, the motion behavior of the inclusions can be categorized as Remain, Oscillate, and Pass. Using a large number of studied cases, the force and displacement results of two typical slag systems, A5 and C3, with respect to different sizes of inclusions, are given as examples. Finally, the inclusion removal abilities of the two slag systems are compared.

3.1. Motion Behavior of Solid Al₂O₃ Inclusions in the CaO-Ce₂O₃-SiO₂-Al₂O₃ Slag System

The force of the solid Al₂O₃ inclusions with different sizes in molten steel are shown in Figure 6. The maximum value of the t axis is the time taken for the inclusions to reach force equilibrium. The force equilibrium for the inclusions is a function of moving time with different sizes. As can be seen from the figure, with the increase in the inclusion size, the time for inclusions to reach the force equilibrium in the process of motion is increasing. For all of the inclusions, the rebound force and drag force play a major role during the movement of the inclusion. With the increase in the size of inclusion, the rebound force, drag force, and fluid added mass force are increased.

The displacements of solid Al₂O₃ inclusions with different sizes at the steel–slag interface in slag A5 are shown in Figure 7.

According to the dimensionless displacement shown in Figure 7 and the classifications of Pass, Oscillate, and Remain, the inclusions with radiuses of 50 μm, 80 μm, and 140 μm can be categorized as the “Pass” group, and the time of inclusions passing through the slag–steel interface is prolonged with the increase in inclusion size. The inclusions with a radius of 10 μm and 20 μm can be categorized as the “Oscillate” group, and the inclusion with a radius of 5 μm can be categorized as the “Remain” group.

The motion behaviors of inclusions with different sizes in slags A1–A5 were calculated, and the results are shown in

Table 15. Motion behaviors of solid Al₂O₃ inclusions with different radiuses in slags A1–A5.

Radius	Slags with Different Compositions
	A4	A1	A3	A2	A5
	51%CaO 29%Al2O3 10%SiO2 10%Ce2O3	55%CaO 30%Al2O3 10%SiO2 5%Ce2O3	46%CaO 39%Al2O3 10%SiO2 5%Ce2O3	51%CaO 34%Al2O3 10%SiO2 5%Ce2O3	48%CaO 27%Al2O3 10%SiO2 15%Ce2O3
5 μm	R	R	R	R	R
10 μm	R	R	R	R	O
20 μm	R	O	O	O	O
50 μm	O	O	O	P	P
80 μm	P	P	P	P	P
140 μm	P	P	P	P	P

Note: R = Remain, O = Oscillate, and P = Pass.

.

From Table 15, the smaller the size of the inclusion is, the more difficult it is to remove. The inclusions with a radius that is smaller than 20 μm cannot be removed in all the A series slag systems, and the inclusions with a radius of larger than 50 μm can be removed perfectly. The inclusion removal abilities of the five slags are ranked as A5 > A2 > A3 > A1 > A4. It should be noted that the viscosity of A5 is the smallest among the slags A1 to A5, indicating that the reduction in viscosity is more conducive to improving the inclusion removal abilities of the slags.

3.2. Motion Behaviors of Liquid 50%wtAl₂O₃–50%wtCaO Inclusions in the CaO-Ce₂O₃-SiO₂-Al₂O₃ Slag System

The force of liquid 50%wtAl₂O₃-50%wtCaO inclusion in molten steel is basically similar to that of solid Al₂O₃, and it is not shown here. The displacements of the liquid 50%wtAl₂O₃–50%wtCaO inclusion at the steel–slag interface with different sizes in slag A5 are shown in Figure 8. The motion behaviors of liquid 50%wtAl₂O₃–50%wtCaO inclusions are similar to those of solid Al₂O₃ inclusions. Specifically, the inclusions with radiuses of 5 μm and 10 μm show Remain behaviors at the steel–slag interface. The 20 μm-radius inclusion exhibits oscillatory behavior. The inclusions with radiuses of 50 μm, 80 μm, and 130 μm can pass the interface and can be removed, but the time for liquid inclusions passing through the slag–steel interface is longer than for solid inclusions under the same slag condition.

The motion behaviors of liquid 50%wtAl₂O₃–50%wtCaO inclusions with different sizes in slags A1–A5 were calculated, and the results are shown in

Table 16. Motion behaviors of liquid 50%wtAl₂O₃–50%wtCaO inclusions with different radiuses in slags A1–A5.

Radius	Slags of Different Compositions
	A4	A1	A3	A2	A5
	51%CaO 29%Al2O3 10%SiO2 10%Ce2O3	55%CaO 30%Al2O3 10%SiO2 5%Ce2O3	46%CaO 39%Al2O3 10%SiO2 5%Ce2O3	51%CaO 34%Al2O3 10%SiO2 5%Ce2O3	48%CaO 27%Al2O3 10%SiO2 15%Ce2O3
5 μm	R	R	R	R	R
10 μm	R	R	R	R	R
20 μm	R	R	R	R	O
40 μm	R	O	O	O	O
50 μm	O	O	O	O	O
80 μm	O	O	O	P	P
130 μm	O	P	P	P	P

Note: R = Remain, O = Oscillate, and P = Pass.

.

For liquid inclusions, the removal abilities in slags A1 to A5 are not as good as those of solid inclusions. The inclusions with a radius smaller than 50 μm are difficult to remove, and the inclusions with a radius larger than 80 μm can be removed in some of the slags. Similar to the results of solid inclusions, the inclusion removal abilities of the five slags are ranked as A5 > A2 > A3 > A1 > A4; the lower the viscosity of the refining slag is, the better the removal of inclusions.

Compared with Table 15 and Table 16, from the perspective of the separation behavior of inclusions at the steel–slag interface, the solid Al₂O₃ inclusion with a radius of 20 μm showed Remain behavior in slag A4, whose viscosity is the highest of the five slags, and it showed Oscillate behavior in other slags. For the liquid inclusion with the same radius of 20 μm, it showed Oscillate behavior merely in slag A5, which posed the lowest viscosity among the five slags. In addition, the solid Al₂O₃ inclusion with a radius of 80 μm can pass the interface of all five slags, while the liquid inclusion with the same radius can only pass the interfaces of slags A2 and A5. Obviously, the size range in which the solid inclusions can pass the interface is larger than that of liquid inclusions. Compared with Figure 7 and Figure 8, it is easy to find that the removal time of liquid inclusions is longer than that of solid inclusions. All of these are indicating that liquid 50%wtAl₂O₃–50%wtCaO inclusions are more difficult to remove than solid Al₂O₃ inclusions.

3.3. Motion Behaviors of Solid Al₂O₃ Inclusions in the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO Slag System

The displacement of solid Al₂O₃ inclusions with different sizes at the steel–slag interface in slag C3 is shown in Figure 9. As can be seen from Figure 9, the inclusions with a radius of 20 μm, 50 μm, 80 μm, and 140 μm can be removed from the slag. The inclusions with a radius of 5 μm and 10 μm showed oscillatory behavior. Compared with Figure 7 and Figure 9, the solid inclusions removal ability of slag C3 is improved compared to that of slag A5. Furthermore, the removal time is shorter.

Similarly, the motion behaviors of inclusions in slag C0–C5 were calculated, and the results are shown in

Table 17. Motion behaviors of Al₂O₃ inclusions with different radiuses in slags C0–C5.

Radius	Without Ce2O3	Slags containing Ce2O3 in Different Compositions
	C0	C2	C1	C4	C5	C3
	48%CaO 37.5%Al2O3 4.5%SiO2 10%MgO	46.2%CaO 36.3%Al2O3 4.5%SiO2 10%MgO 3%Ce2O3	45%CaO 37.5%Al2O3 4.5%SiO2 10%MgO 3%Ce2O3	43.4%CaO 34.1%Al2O3 4.5%SiO2 10%MgO 8%Ce2O3	45%CaO 32.5%Al2O3 4.5%SiO2 10%MgO 8%Ce2O3	45%CaO 35.5%Al2O3 4.5%SiO2 10%MgO 5%Ce2O3
5 μm	R	R	R	O	O	O
10 μm	R	O	O	O	O	O
20 μm	R	O	O	P	P	P
30 μm	R	O	P	P	P	P
50 μm	O	P	P	P	P	P
80 μm	O	P	P	P	P	P
140 μm	P	P	P	P	P	P

Note: R = Remain, O = Oscillate, and P = Pass.

.

The inclusions with a radius of smaller than 10 μm cannot be removed in all of the C series slag systems, and the inclusions with a radius of larger than 20 μm can be removed perfectly in slags C3 to C5. This C series slags showed a better performance than the A series slags for solid inclusion removal.

By comparing the removal of the Al₂O₃ inclusions between slag C0 and slags C1–C5, it can be seen that the refining slag containing rare earth oxide Ce₂O₃ shows a better ability to remove inclusions than ordinary refining slag. The inclusion removal ability of the six slags is ranked as C3 > C5 > C4 > C1 > C2 > C0. By comparing the physical properties of slag C0 and slags C1–C5, it can be found that slags C1–C5 have a higher wettability, as well as a better performance of the wetting effect on the Al₂O₃ inclusion than slag C0. However, the overall wettability varies within a limited range. As shown in Table 3, the viscosity of the six slags is ranked as C3 < C5 < C4 < C1 < C2 < C0. This is the same order as the removal ability of the slags. Thus, the viscosity may be a crucial factor. This is the reason for why the slag containing rare earth oxide Ce₂O₃ shows a better ability to remove Al₂O₃ inclusions.

Compared with Table 15 and Table 17, the removal ability of solid Al₂O₃ inclusions in the C series slag system is significantly higher than that in the A series slag system. The limiting size of the inclusions that can be removed is reduced in the C series slag system.

3.4. Discussion

According to the analysis above, the ability of slags containing Ce₂O₃ to remove inclusions is better than the slag without Ce₂O₃, and the viscosity plays a vital role in the ability of the two slag systems to remove inclusions. In the C series slags, the addition of MgO can effectively reduce the viscosity. The sizes of the inclusions that can effectually be removed is related to the statement of the inclusion. In this paper, for solid inclusions, it can be removed with a size of more than 20 μm in a special refining slag with a specific composition. For liquid inclusions, the size of the inclusion needs to be larger than 80 μm. That is to say, the liquid inclusions more easily remain in the molten steel. After the analysis, the ability of the refining slag to remove inclusions of specific compositions can be evaluated.

Nevertheless, the model is used to predict the motion of single particle inclusions at the slag–steel interface in a relatively ideal state, but there are some shortcomings. First, the liquid steel and slag are assumed to be static, without considering the real flow state. Second, the inclusion is a single particle floating at the interface, without considering the collision aggregation between the inclusions. Third, the inclusion may be accelerated and the rising velocity may be higher than the Stokes velocity. Although the results cannot be compared with industrial operations, the integrated model can predict a tendency for the different compositions of the slags. For a specific slag, the composition of the slag could influence the viscosity and surface tension, and further, the overall wettability, and many other issues. In this study, the C series slag with a lower viscosity shows the best performance of inclusion removal. The 45%wt CaO–35.5%wt Al₂O₃–4.5%wt SiO₂–10%wt MgO–5%wt Ce₂O₃ refining slag is optimized in this study. It is noted that a systematic evaluation of the inclusion removal ability of selected slag systems, as well as the effect of the tailored slag on slag eye in ladles [160], are ongoing work.

4. Conclusions

After summarizing and analyzing the motion behaviors of solid and liquid inclusions with different sizes at the steel–slag interface of the two slag systems, the following conclusions can be obtained:

(1) In the refining process, whether solid or liquid inclusions, large inclusions are easier to remove from the steel–slag interface than small inclusions. Reducing the viscosity of the slag is more conducive to the removal of the inclusions. For the same CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system, liquid 50%wtAl₂O_3-50%wtCaO inclusions are more difficult to remove than solid Al₂O₃ inclusions.

(2) The CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO refining slag shows a better ability to remove Al₂O₃ inclusions than that of CaO-SiO₂-Al₂O₃-MgO slag. The reason for this is that the addition of rare earth oxide Ce₂O₃ can decrease the viscosity of slags, as well as improving the wetting effect of slags on Al₂O₃ inclusions.

(3) For two slags systems, the CaO-Ce₂O₃-SiO₂-Al₂O₃-MgO slag system shows a better ability to remove Al₂O₃ inclusions than that of the CaO-Ce₂O₃-SiO₂-Al₂O₃ slag system. The addition of 5% to 8% Ce₂O₃ in the CaO-SiO₂-Al₂O₃-MgO slag is an optimized case for industrial applications.

(4) The integrated model mainly involves an inclusion motion model and the slag properties models, including the interfacial tension, surface tension, and mass action concentration model, based on ion and molecule coexistence theory. However, the predictions are limited to the force balance of inclusions. A combined model considering the transport of inclusions by fluid mechanics, the interaction of inclusions with a turbulent boundary layer, and the movement of inclusions at the steel–slag interface will be a future work.