1.   Introduction

Mass flow, energy flow, and information flow are the three basic flows of the ironmaking and steelmaking process [1]. For the refining of the steel, complex mass flow occurs during the process as the system is a multiphase and a large number of chemical and physical reactions occur in the system. These mass transfer flow have a significant influence on the composition of inclusions, steel, and slag.

The control of nonmetallic inclusions is an important target during the steel refining process, as inclusions have detrimental effect on the property of steel products [2–3]. Not all inclusions are harmful to the performance of the steel. One of the pivotal factors deciding the influence of inclusions is their composition. Calcium aluminate with large size can cause the fatigue failure in the rolled product of the bearing steel [4]. The MgAl₂O₄ spinel inclusion and some other solid inclusions cause the nozzle clogging during the continue casting process [5]. Some inclusions containing Ti and Mg have positive effect on the toughness of steel [6]. Therefore, it is of a great importance for the accurate control of inclusions.

The slag has an important influence on the composition of the steel and inclusions [7–8]. Shin and Park [9] reported that the mass ratio of CaO/A₂O₃ in refining slag should be controlled between 1.5 and 2.5 to inhibit the formation of solid inclusions in Al deoxidized Mn–V alloyed steel which could ensure the absorption of inclusions by the slag. Ren et al. [10–11] proposed the optimized slag of CaO–Al₂O₃–SiO₂–MgO slag which was used to modify solid Al₂O₃ inclusions to liquid CaO–Al₂O₃–MgO inclusions. The refractory also has effect on the composition of inclusions in the steel [12], which is often neglected in previous studies [13–15]. Refractory could supply the Mg for the formation of spinel inclusions [16–17]. Reactions between inclusions and refractory have been reported by many researchers [18–21].

To sum up, during the refining process, the molten steel reacts with the slag and refractory continuously, resulting in the constant change of composition of the molten steel. The composition of inclusions also changes accordingly. Shin et al. [22] proposed a refractory–slag–metal–inclusions multiphase reaction model and evaluated the influence of slag and refractory on the composition of inclusions. In the model, the mass transfer coefficient of refractory and slag was regressed using the experiment data. Basing on the model proposed by Shin et al. [22], Wang et al. [23] also established a kinetic model and studied the influence of MgO/ZrO₂/Al₂O₃ refractories on the Ti-containing steel. Although other kineticmodels [24–27] have also been established, in these models reactions between the refractory and the molten steel have often been ignored.

In the current study, laboratory experiments were carried out to study the influence of refractory and slag on the mass transfer and composition evolution of the steel and inclusions. Thereafter, a mathematical kinetic model was established to predict the evolution transiently. The mass transfer phenomenon in the model was evaluated using a simplified natural convection heat transfer model.

2.   Laboratory experimental setup

Laboratory experiments were carried out using the Fe–1.0C–0.3Si–0.39Mn–1.563Cr–0.0003Mg–0.0015O (wt%) bearing steel and the 51.5CaO–24.3Al₂O₃–7.3SiO₂–6.1MgO–10.8CaF₂ (wt%) pre-melted slag. Two experiments were conducted, one with slag and another without slag. The steel, refractory, and slag reacted with each other for 180 min at 1600°C in a high purity (99.71%) MgO crucible (inner diameter: 42 mm; height: 100 mm) within a Si–Mo furnace underargon atmosphere. The porosity of crucibles was 1.03% according to the test report provided by the Beijing Dingsheng Brothers Technology Co., Ltd., China. After the temperature of the steel reached 1600°C, high purity aluminum (99.999%) was added into the molten steel to control the initial aluminum content to 0.03wt%. After 60 s for the homogenization of the steel, a water-cooled sample was taken out and 25 g pre-melted slag was added on the surface of the steel. Then, steel samples and slag samples were taken at specific intervals.

Steel samples were prepared for the composition and inclusions analysis. The content of total oxygen (T.O) was detected using a LECO ONH analyzer, and the dissolved aluminum ([Al]) and total magnesium (T.Mg) were determined using inductively coupled plasma atomic emission spectrometry (ICP-AES). The composition of slag was analyzed using X-ray fluorescence spectrometry (XRF). Inclusions analysis was using an automatic scanning electron microscope equipped with an energy dispersive spectrometer (SEM–EDS) with an acceleration voltage of 20 kV and a spot size of 35%. The detection limit of inclusion diameter was 1 μm.

3.   Model mathematical formulation

3.1   Thermodynamic and kinetic models

The current model was based on the coupled-reaction model [25,27–28], and reactions between the steel and the refractory were taken into account. Reactions listed in Table 1 were considered at the steel/slag and steel/inclusion interfaces to predict the evolution of inclusions. At the steel/refractory interface, the equilibrium between the dissolved [Mg] and [O] in the steel and the solid magnesium oxide in the refractory ((MgO)_s) was also considered. It was assumed that the refractory only dissolved into the slag, and no chemical reaction occurred at the refractory/slag interface.

Table  1.  Reactions considered at the interfaces of steel/slag and steel/inclusion [29]

Reaction	$\Delta {G^{\ominus}}$ / (J·mol−1)
$\left[ {{\rm{Ca}}} \right]{\rm{ + }}\left[ {\rm{O}} \right]{\rm{ = }}\left( {{\rm{CaO}}} \right)$	$\Delta G_{{\rm{Ca}}}^{\ominus} = - 645200 + 148.7T$
${\rm{2}}\left[ {{\rm{Al}}} \right]{\rm{ + 3}}\left[ {\rm{O}} \right]{\rm{ = }}\left( {{\rm{A}}{{\rm{l}}_{\rm{2}}}{{\rm{O}}_{\rm{3}}}} \right)$	$\Delta G_{\rm{Al}}^{\ominus} = - 120{\rm{6220}} + 3{\rm{90}}{\rm{.39}}T$
$\left[ {{\rm{Si}}} \right]{\rm{ + 2}}\left[ {\rm{O}} \right]{\rm{ = }}\left( {{\rm{Si}}{{\rm{O}}_{\rm{2}}}} \right)$	$\Delta G_{{\rm{Si}}}^{\ominus} = - 581900 + 221.8T$
$\left[ {{\rm{Mn}}} \right]{\rm{ + }}\left[ {\rm{O}} \right]{\rm{ = }}\left( {{\rm{MnO}}} \right)$	$\Delta G_{{\rm{Mn}}}^{\ominus} = - 244300 + 107.6T$
$\left[ {{\rm{Mg}}} \right]{\rm{ + }}\left[ {\rm{O}} \right]{\rm{ = }}\left( {{\rm{MgO}}} \right)$	$\Delta G_{{\rm{Mg}}}^{\ominus} = - 89960 - 82.0T$
Note: $\Delta {G^{\ominus}}$—standard Gibbs free energy; T—temperature.

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The thermodynamic equilibrium was assumed at the interface, and the equilibrium was solved using the coupled-reactionmodel. A general expression for reactions at the steel/slag interface and steel/inclusion interface was as follows.

where M represents elements in the steel, Ca, Al, Si, Mn, Mg, etc.; n represents the stoichiometric number of O in MO_n.

When the reaction is at the equilibrium state, then:

where ΔG is the change of free Gibbs energy, J·mol⁻¹; R is the ideal gas constant, which is equal to 8.314 J·mol⁻¹·K⁻¹; T is the temperature, K; a is the activity of components.

The activity of components in the molten steel was calculated according to the Wagner Interaction Parameter Formalism (WIPF) [30–31], as shown in Eq. (3). A few second-order interaction parameters were employed in the calculation for more accurate calculated results, even though only a small number of parameters were available in the literature.

where f_M is the activity coefficient of M; e and r are the first-order and second-order interaction parameters, respectively, as listed in Table 2; j and l are components of the molten steel Fe–C–Si–Mn–Cr–O–Ca–Al–Mg, so n = 9; [%j] and [%l] are the content of j and l in the steel, wt%. The standard state was chosen as 1wt%.

Table  2.  Some activity interaction coefficients of elements in the molten steel [32–34]

M	$e_{\rm{M}}^j$
	Al	C	Ca	Cr	Mg	Mn	O	Si
Al	63/T + 0.011	0.091	−0.047	0.012	−0.13	0	−34740/T + 11.95	0.056
C	0.043	158/T + 0.0581	−0.097	−0.023	−0.07	−0.012	−0.34	162/T − 0.008
Ca	−0.072	−0.34	−0.002	0.02	0	−0.0156	−2500	−0.097
Cr	0.023	−0.12	0.026	−0.0003	0.1	0.0039	−0.14	−0.0043
Mg	−0.12	−0.15	0	−0.0003	0	0	−289	−0.09
Mn	0	−0.07	−0.023	0.05	0	0	−0.083	−0.00026
O	−20600/T + 7.15	−0.45	−990	−0.04	−190	−0.021	−1750/T + 0.734	−0.131
Si	0.058	380/T − 0.023	−0.067	−0.0003	−0.105	0.002	−0.23	34.5/T + 0.089
M	$r_{\rm{M}}^{jj}$
	Al	C	Ca	Cr	Mg	Mn	O	Si
Al	0.17/T − 0.0011	−0.004	—	—	—	—	—	−0.0006
C	−0.0007	8.94/T + 0.0026	—	—	—	—	—	11.94/T − 0.0003
Ca	0.0007	0.012	—	—	—	—	2.6 × 105, 2.1 × 105*	0.0009
Cr	—	—	—	—	—	—	—	—
Mg	—	—	—	—	—	—	—	—
Mn	—	—	—	—	—	—	—	—
O	1.7	—	4.2 × 104, 2.1 × 105*	0.00037	—	—	—	—
Si	—	—			—	—	—	6.5/T − 0.0055
Note: the superscript * represents the value was $r_{\rm{M}}^{{\rm{M}}j}$.

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The activity of MO_n in the slag or inclusions was calculated according to the Ion and Molecular Coexistence Theory (IMCT) [35–37]. The activity of solid oxide at the steel/refractory interface was set to be 1.0. Although the slag used in the current study was CaO–Al₂O₃–SiO₂–MgO–CaF₂ system, which did not contain MnO, a slag consisted of CaO–Al₂O₃–SiO₂–MnO–MgO–CaF₂ was employed to calculated the activity. This is because the steel contained 0.39wt% Mn, and MnO might be formed during the reaction process. There are six simple structural units and 26 complex molecules in this slag system according to the IMCT, as listed in Table 3. The detailed calculation measure can be found in the authors’ previous work [34].

Table  3.  Structural units in the CaO–MgO–CaF₂–Al₂O₃–SiO₂–MnO slag according to the IMCT [34]

Simple compounds	Complex compounds
(Ca2+ + O2−); (Mg2+ + O2−); (Ca2+ + 2F−); Al2O3; SiO2; MnO.	3CaO·SiO2; 2CaO·SiO2; 3CaO·2SiO2; CaO·SiO2; 3CaO·Al2O3; 12CaO·7Al2O3; CaO·Al2O3; CaO·2Al2O3; CaO·6Al2O3; 2MgO·SiO2; MgO·SiO2; MgO·Al2O3; 2MnO·SiO2; MnO·SiO2; MnO·Al2O3; 3Al2O3·2SiO2; 3CaO·MgO·2SiO2; 2CaO·MgO·2SiO2; CaO·MgO·SiO2; CaO·MgO·2SiO2; 2CaO·Al2O3·SiO2; CaO·Al2O3·2SiO2; 3CaO·2Al2O3·CaF2; 11CaO·7Al2O3·CaF2; 3CaO·2SiO2·CaF2; 2MgO·2Al2O3·5SiO2.

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As reactions at the interface are at the equilibrium state, a concentration gradient occurs between the bulk and interface. This concentration gradient is the driving force of the components’ diffusion, which could be calculated according to Eqs. (4)–(5):

where J is the molar flux density, mol·m⁻²·s⁻²; k_M and ${k_{{{\rm{MO}}_n}}}$ are the mass transfer coefficients of the M and MO_n, respectively, m·s⁻¹; ρ_steel is the steel density, kg·m⁻³; ρ_slag is the slag density, kg·m⁻³; ρ_inclusion is the inclusion density, kg·m⁻³; M is the molecular weight, g·mol⁻¹; (%MO_n) is the content of MO_n in the slag or inclusion, wt%; the superscript b and * represent the bulk and interface, respectively.

Under the assumption that no material accumulates at the interface, the molar flux densities of the slag and steel are equal to each other, as shown in Eq. (6).

Moreover, Eq. (7) is the electric neutral equation, showing the flux density of cations is equal to that of anions.

Coupling Eqs. (2)–(7), the concentration of components at the interface could be solved, and then the mass transfer between the bulk and the steel/slag interface in time dt was calculated as Eqs. (8)–(9) [25,27,38].

where, t is the time, s; A_interface is the area of interface, m²; V is the volume, m³.

Reactions occurred at the steel/inclusions interface were similar to those at the steel/slag interface. The difference lied in the characteristics of the interface. The mass transfer between the bulk and the steel/inclusions interface was calculated as Eqs. (10)–(11).

where, P_3D is number density of inclusions in three dimension, m⁻³; d_inclusion is the diameter of inclusion, m.

As for reactions at the steel/refractory interface, only the generation and decomposition of MgO and Al₂O₃ were considered. The MgO crucible used in the current study was quitedense, so that the absorption of inclusions by the crucible was ignored. What’s more, the spalling of the crucible maybe occurred at the steel/refractory interface [39–40], which was not considered due to the high purity of the crucible. Only the spalling or dissolution of refractory at the slag/refractory interface was taken into account, and an overall variation of slag composition caused by the slag/refractory reaction was estimated, no matter with the dissolution or the spalling.

3.2   Determination of mass transfer coefficients

Owing to the shape difference among the steel/inclusion interface, the steel/slag interface, the steel/refractory interface,and the slag/refractory interface, the mass transfer at these phases are different, implying different models to calculate the mass transfer coefficient (MTC) within these phases.

The surface renewal theory [41] was employed to calculate the MTC at steel/inclusion interface. According to the surface renewal theory, there are numerous fluid eddies with different residence time at the interface. The substance at the interface was constantly replaced and updated in the process of fluid flow. The MTC at steel/inclusion interface was calculated from Eq. (12), suggested by Harada et al. [27].

where, k_{M_inc} is the mass transfer coefficient of M at the steel/inclusion interface, m·s⁻¹; D_M is the diffusivity coefficient, m²·s⁻¹, as listed in Table 4; u_slip is the slip velocity between the inclusion and the molten steel, which was calculated using Eq. (13) [27], m·s⁻¹.

Table  4.  Diffusivity coefficients of components in the molten steel and slag

Component	D / (10−9 m2·s−1)
Mg, Al, Si	22.36
Mn, Ca	18.83
MgO, CaO, MnO	2.03
SiO2, Al2O3	2.08
Note: values were from published references [42–44]

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where, g is acceleration of gravity, g = 9.8 m·s⁻²; μ_steel is the viscosity of steel, kg·m⁻¹·s⁻¹.

For the other three types interfaces, the MTC was calculated as Eq. (14) [45–46] shows.

where, c is a constant, c = 0.4 at the refractory/slag and refractory/steel interface, and c = 2.0 at the steel/slag interface; ε is the turbulent dissipation rate, m²·s⁻³; ν is the kinematic viscosity, m²·s⁻¹.

To obtain the value of ε, a simplified natural heat transfer model was established. The geometry of the computational domain is shown in Fig. 1. In the model, the effect of temperature on the density of molten steel was described using the Boussinesq model, as shown in Eq. (15). The heat transfer rate at the top surface was −15 kW·m⁻², and that at the bottomsurface was −1.4 kW·m⁻². The temperature at the side wall was set to be a constant, which was equal to 1873 K. Other parameters used in the simulation are listed in Table 5. The commercial CFD software, Fluent 17.0, was used to simulate the distribution of temperature and turbulent properties under steady state. Calculated results are shown in Fig. 2. The temperature of the steel was range from 1860 to 1873 K. There was a coolest region at the center of top surface, which was below 1870 K. The temperature increased gradually with this coolest as the center. A similar condition was at the bottom surface. Fig. 2(b) shows the contour of turbulent dissipation rate. The average stirring energy of the steel at the top surfacewas 8.28 × 10⁻⁸ m²·s⁻³, and that at the bottom and the side were 1.06 × 10⁻⁸ m²·s⁻³ and 3.35 × 10⁻⁸ m²·s⁻³, respectively.

Figure  1.  Geometry and mesh of the computational domain: (a) geometry; (b) mesh.

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Table  5.  Parameters employed in the simulation

Property	Value
Density of the molten steel, ρsteel / (kg·m−3)	7000
Viscosity of the molten steel, μsteel / (Pa·s)	0.0067
Heat capacity of the molten steel, Cp,steel / (J·kg−1·K−1)	823.6
Thermal conductivity of the molten steel, λsteel / (J·m−1·s−1·K−1)	40.3
Thermal expansion coefficient of the molten steel, βsteel / K−1	4.4 × 10−6

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Figure  2.  Calculated temperature and kinetic turbulent energy of the molten steel in the heated crucible: (a) temperature contour; (b) turbulent kinetic energy dissipation rate contour.

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where, ρ and ρ₀ are the density of the molten steel at temperature of T and T₀, kg·m⁻³; β is the thermal expansion coefficient of the molten steel, K⁻¹.

3.3   Other parameters

Using the models established above, the mass transfer of elements during the refining process was analyzed. Other parameters used in the model are summarized in Table 6, where the saturated solubility of MgO in the slag was assumed to be 17wt%. Two laboratory situations were calculated to analyze the mass transfer flow. One was the refining system without slag, and the other was the system with slag.

Table  6.  Additional parameters used in the model

Property	Value
Density of the slag, ρslag / (kg·m−3)	3000
Density of the inclusion, ρinclusion / (kg·m−3)	3500
Density of the MgO refractory, ρref / (kg·m−3)	3400 [23]
Saturated solubility of MgO in the slag, (MgO)sat / wt%	17
Kinematic viscosity of the slag, νslag / (m2·s−1)	3.5 × 10−5
Diameter of the inclusion, dinclusion / m	2.0 × 10−6
Two-dimensional number density of inclusions, P2D / m−2	1.09 × 107

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4.   Dynamic mass transfer in the system of steel–lining refractory–nonmetallic inclusions

Fig. 3 shows the evaluation of inclusions composition over time in the system without slag. Primary inclusions in the steel had a high content of Al₂O₃, and there was also a small amount of CaO–Al₂O₃–MgO inclusions with melting point lower than 1873 K. When reacting for 60 min, pure Al₂O₃ inclusions disappeared in the steel, and some MgO·Al₂O₃ inclusions with small diameter generated. Increasing the reacting time to 120 min, the content of Al₂O₃ in inclusions was further reduced, and at the same time, a more amount of spinel was formed in the steel. The size of inclusions in the sample of “180 min” had a sharp decrease. The maximum diameter of inclusions in the steel after holding for 180 min was as small as 3.2 μm, implying an efficient removal of large inclusions within 180 min.

Figure  3.  Evolution of inclusions composition in the system of steel–inclusion–refractory after different reaction times: (a) 0 min; (b) 60 min; (c) 120 min; (d) 180 min.

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As the raw material used in the study was taken from the bloom, where inclusions could be assumed to be in thermodynamic equilibrium with the steel, the increase of content of MgO in inclusions was because of the increase of [Mg] in the steel. And there was no slag in the system, so reactions between the molten steel and the refractory supplied the [Mg], which was often neglected in the literature [13–14]. In order to study quantitatively the transfer process of [Mg] from the refractory to the molten steel, the calculated kinetic results are shown in Fig. 4. Fig. 4(a) is the evolution of steel composition, and Fig. 4(b) is that of inclusions composition. With the increase of holding time, the content of Al₂O₃ in inclusions decreased gradually, while the content of MgO had an opposite rise trend. The experimental results were accurately reproduced by the current kinetic model. As to the composition of the molten steel, the content of total Al (T.Al) had a decrease, which was thought to be replaced by the MgO in the refractory, as Eq. (16) shows. Fig. 5 shows the element distribution at the interface between the molten steel and the MgO refractory when holding for 180 min. MgO·Al₂O₃ inclusions and inclusions with high content of Al₂O₃ were observed at the interface, which also verified the occurrence of Eq. (16) at the interface. The absorption of inclusions by the refractory at the interface was not considered as the content of Mg in inclusions was quite higher than that at the interface where the content of Al was high.

Figure  4.  Evolution of composition of (a) steel and (b) inclusions.

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Figure  5.  Elements distribution at the steel/refractory interface when holding for 180 min.

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5.   Dynamic mass transfer in the system of steel–slag–lining refractory–nonmetallic inclusions

To study the influence of the slag on the mass flow during the refining process, a laboratory experiment and numerical simulation on the system with the slag was also conducted. The experimental evolution of inclusions composition over time is shown in Fig. 6. With the increase of holding time, inclusions in the molten steel transformed from Al₂O₃ into MgO·Al₂O₃ and then pure MgO. Most of inclusions changed into pure MgO after holding for 60 min.

Figure  6.  Evolution of inclusions composition in the system of steel–slag–inclusion–refractory after different reaction times: (a) 0 min; (b) 15 min; (c) 30 min; (d) 60 min; (e) 120 min; (f) 180 min.

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Comparisons of the composition evolution of the slag, inclusions and the steel between experimental results and predicted ones are shown in Fig. 7. The change of the slag composition was mainly owing to the dissolution of MgO from the refractory. The composition of inclusions changed quite faster in the system with slag than that without slag. It took about 30 min for reactions in the system with slag to reach the equilibrium, while reactions in the slag free system hardly reached the equilibrium even after 180 min. The slag had an apparent acceleration effect on the mass transfer during the refining, as shown in Fig. 8. The transformation rate of inclusions in the system with slag was about 5 times faster than that without slag. The slag absorbed MgO from the refractory, and then transferred Mg to the molten steel at the steel/slag interface. Besides, the slag could also affect the final equilibrium state of reactions in the system. In the current refining system, the aluminum kept transferring from the steel to the refractory and slag, while the magnesium kept transferring from the refractory to the steel with the form of [Mg] and to the slag with the form of (MgO). The MgO in the slag then reacted with the steel, transferring Mg to the moltensteel. The content of T.Mg in the steel had a slow decrease after holding for 60 min, as shown in Fig. 7(c), which was not predicted by the kinetic model. This is because the floatation removal of inclusions was not considered in the current model.With floatation removal of inclusions, it is apparent that the T.Mg and T.Al in the steel declined with the holding time.

Figure  7.  Comparisons between experimental results and the predicted ones: (a) slag composition; (b) inclusions composition; (c) steel composition.

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Figure  8.  Acceleration of the slag on the mass transfer: (a) composition of inclusions; (b) composition of the steel.

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It was concluded that there were two ways for the entrance of Mg into the molten steel. The mass transfer process of Mg ([Mg] or MgO) is shown in Fig. 9. The dissolution rate of MgO from the refractory to the slag was the fastest, which was larger than 3 × 10⁻³wt%/s within the first 20 min. The mass transfer rate of [Mg] from the slag to the steel in the first 20 min was approximately 7.04 × 10⁻⁷wt%/s in average. And that from the refractory to the steel and from inclusions to the steel were about 3.54 × 10⁻⁷wt%/s and −3.16 × 10⁻⁷wt%/s, respectively.

Figure  9.  Mass transfer process of Mg ([Mg] or MgO).

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The overall mass transfer of Mg and Al are shown in Fig. 10. The mass transfers of Mg and Al through direct steel–refractory reactions were a little slow. The slag phase in the system played a role like a catalyzer, accelerating both the dissolution of MgO at the slag/refractory interface and steel–slag reactions, resulting the faster mass transfer rate of Mg from the refractory. In the first 20 min of the refining, if the mass transfer rate of magnesium from the steel to inclusions was taken as x, then the rates of magnesium mass transfer at the steel/refractory interface and at the steel/slag interface were 1.1x, and 2.2x, respectively.

Figure  10.  Overall mass transfer of Mg and Al during the refining.

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6.   Conclusions

In the current study, the mass transfer flow in the refining process of a bearing steel was analyzed using laboratory experiments and numerical kinetic prediction. The following conclusions were derived.

(1) During the refining process, magnesium transferred from the refractory into the molten steel through two ways. One was reactions between the refractory and the steel, and another was steel–slag reactions, where the MgO was mainly originated the dissolution of refractory. Most of the aluminum involved in the transport entered the slag and a small part of the aluminum transferred to refractory, forming the Al₂O₃ or MgO·Al₂O₃.

(2) The slag had an apparent acceleration on the mass transfer. The mass transfer rate in the system with the current slag was about 5 times faster than that without slag. It took about 30 min for reactions in the system with slag to reach the equilibrium, while reactions in the slag free system hardly reached the equilibrium even after 180 min.

(3) A kinetic model for predicting the mass transfer and composition transformation of the refractory–steel–slag–inclusion multiphase was established based on the coupled reaction model. The experimental results were well reproduced using the model.

(4) The mass transfer rate of magnesium varied with the interface. In the first 20 min of the refining, the rates of magnesium mass transfer at the steel/inclusion interface, steel/refractory interface, and the steel/slag interface were x, 1.1x and 2.2x, respectively.

Acknowledgements

This work was financially supported by the National Natural Science Foundation China (Nos. U1860206, 51725402, and 51874032), the Fundamental Research Funds for the Central Universities (Nos. FRF-TP-19-037A2Z and FRF-BD-20-04A), the S&T Program of Hebei, China (No. 20311006D), the High Steel Center (HSC) at Yanshan University, China, and the High Quality Steel Consortium (HQSC) at University of Science and Technology Beijing, China.