Effect of Al₂O₃ and TiO₂ Contents in the Refining Slag on Al and Ti Contents of Incoloy825 Alloy

Abstract

During the refining process of Incoloy825 nickel-based alloy, the reaction between the refining slag and Al and Ti in the alloy can result in the oxidation loss. Therefore, the effect of the TiO₂ and Al₂O₃ contents in the refining slag (CaO–SiO₂–Al₂O₃–MgO–CaF₂–TiO₂) on the final Al and Ti contents in Incoloy825 alloy was investigated by slag-metal reaction experiment in a 1-kg MoSi₂ resistance furnace. The thermodynamic model was established based on ion and molecular coexistence theory and the kinetic model was established based on two-film theory. The experimental results show that an increase in TiO₂ (Al₂O₃) content in the slag leads to an increase (decrease) in Ti content and a decrease (increase) in Al content in the final alloy. The thermodynamic model demonstrates that the activities of SiO₂, Al₂O₃, and TiO₂ in the slag are negatively correlated with the content of CaO and positively correlated with SiO₂, Al₂O₃, and TiO₂ in the slag. The kinetic model indicates that the slag-metal reaction reaches equilibrium within 10 min, and the model has good applicability for the reaction process of Incoloy825 alloy and slag refining.

1. Introduction

Incoloy825 alloy is a high-temperature corrosion-resistant nickel-based alloy that can resist alkali metal and acid corrosion and has strong resistance to stress-corrosion cracking. This alloy is widely used in aerospace, power generation, and marine engineering applications.^1,2,3) At present, nickel-based corrosion-resistant alloys are usually smelted by vacuum induction furnace, electroslag remelting. These processes are followed by casting, homogenization, forging, and rolling, and the final alloy is then formed into biscuits, rods, tubes, plates, and other profiles.⁴⁾ To reduce production costs and increase production capacity, the process flow of electric-arc furnace, argon oxygen decarburization, ladle furnace, vacuum oxygen decarburization, and continuous casting is used. However, due to the easy oxidation of molten steel, a large number of TiN, TiO₂, Al₂O₃, and other inclusions are generated, resulting in many process problems,^5,6,7) such as slag entrapment, nozzle blocking, and longitudinal surface cracks. Such challenges limit the successful continuous casting production of Incoloy825 alloy. Therefore, it is necessary to study and control the oxidation loss of Al and Ti elements in such alloys.

Some studies on the control of compositional uniformity of Al and Ti alloys in electroslag remelting have been published.^8,9) Shi et al.^10,11) found that the addition of a certain amount of TiO₂ to the slag increases the uniformity of the alloy composition, reduces the slag viscosity, and increases the fluidity during slag refining. Jiang et al.¹²⁾ found that when remelting high-titanium and low-silicon superalloys, slags with high CaO contents can better avoid the titanium loss caused by the reaction between Ti and SiO₂. Duan et al.¹³⁾ found that the presence of SiO₂ and FeO are the main factors influencing the oxidation loss of Al and Ti by changing the composition of the CaO–SiO₂–MgO–FeO–Al₂O₃–TiO₂–CaF₂ slag. Gao et al.¹⁴⁾ verified that the content of TiO₂ in ladle slag significantly affects the content of Ti in molten steel by laboratory and industrial experiments. Pateisky et al.¹⁵⁾ found that the addition of appropriate amounts of TiO₂ in electroslag reduces the oxidation loss of Ti and increases the content of Al in the alloy. For a fixed Al content, the Ti content in the alloy increased with increasing TiO₂ content in the slag. Hou et al.¹⁶⁾ obtained the change of Al and Ti contents in steel by experiment and thermodynamic model calculation, and found the suitable electroslag for smelting 1Cr21Ni5Ti stainless steel. Duan et al.¹⁷⁾ found that the kinetic model is very suitable for describing the relationship between the sulfur content in the alloy and time by comparing the results from experiments and kinetic calculation. Wei et al.¹⁸⁾ found that the activities of SiO₂, Al₂O₃, and TiO₂ play an important role in the content of Ti in electroslag.

Since there is little research on the reaction between the Incoloy825 alloy and that of the refining slag, thermal equilibrium experiments of the reaction between CaO–SiO₂–Al₂O₃–MgO–CaF₂–TiO₂ refining slag and Incoloy825 alloy were studied here. The loss laws of Al and Ti in the alloy were analyzed by thermodynamic and kinetic calculations.

2. Experiments

The metal raw material was commercial Incoloy825 corrosion-resistant alloy, which is produced by vacuum induction melting and electroslag remelting. The composition of the tested alloy is shown in Table 1. Refining slag was prepared with chemically pure reagents (CaO≥97.0%, SiO₂≥98.0%, Al₂O₃≥98.0%, MgO≥98.0%, CaF₂≥98.5%, and TiO₂≥98.0%).

Table 1. Composition of the Incoloy825 alloy (mass%).

C	Mn	Si	P	S	Al	Co	Cr	Cu	Mo	Ni	Ti	Fe
0.015	0.094	0.32	0.003	≤0.001	0.20	0.044	21.52	2.02	3.15	39.31	0.96	Bal.

The refining slag was heated to 873 K and held at this temperature for 4 h to remove moisture before the experiment, it was then configured proportionally. The prepared refining slag (65 g) was placed in a graphite crucible (ID: 40 mm; OD: 50 mm; H: 120 mm). The resistance furnace diagram for the experiment is shown in Fig. 1. A 1-kg MoSi₂ resistance furnace was heated under a protective argon atmosphere. After the temperature reached 1723 K, the graphite crucible was placed in the furnace to start premelting. To prevent the volatilization of the refining slag, the graphite crucible was air-cooled after 15 min of heat treatment to obtain the premelted refining slag. The cooled refining slag was crushed and divided into two portions of 50 and 15 g, where the former was used in experiments, and the latter was used for composition detection of the as-prepared slag.

Fig. 1.

Schematic diagram of experimental resistance furnace. (Online version in color.)

After the premelting experiment, the slag-metal equilibrium experiment was started, following the process shown in Fig. 2. The slag composition was analyzed by X-ray fluorescence spectroscopy (Rigaku ZSX Primus II). The composition of the refining slag before and after premelting is shown in Table 2. The content of Al and Ti in samples No. 0–4 was quantified by inductively coupled plasma atomic emission spectrometry (Plasma 2000).

Fig. 2.

Experimental process of slag-metal equilibrium.

Table 2. Slag composition before and after premelting (mass%).

Slag	Before premelting						After premelting
	CaO	SiO2	Al2O3	MgO	CaF2	TiO2	CaO	SiO2	Al2O3	MgO	CaF2	TiO2
S1	45.00	2.00	30.00	8.00	10.00	5.00	43.63	2.54	31.16	7.92	9.14	4.35
S2	45.00	2.00	25.00	8.00	10.00	10.00	44.35	2.45	24.73	7.94	9.37	9.26
S3	45.00	2.00	20.00	8.00	10.00	15.00	44.89	2.43	19.32	7.87	9.23	15.06

3. Results and Discussion

3.1. Evolution of the Al and Ti Contents Over Time

The variation in the Al, Ti and Si contents in the alloy with increasing TiO₂ content in the slag is shown in Fig. 3. Composition of refined slag after slag-metal reaction as shown in Table 3. During the slag-metal reaction process, with the addition of refining slag, Al, Ti and Si elements began to undergo redox reactions, with the change of time, the Ti content decreased, while the Si content increased. After the reaction reached equilibrium, the content of SiO₂ in the slag decreased, while the TiO₂ increased. Figure 3(d) shows that with the increase of TiO₂ content and the decrease of Al₂O₃ in the refining slag, the content of Ti in the alloy increased when the slag-metal balance reached equilibrium, which significantly inhibited the oxidation loss of Ti. In the S1 slag, the slag-metal reaction process oxidized Ti and increased the Al content. In the case of S2 and S3, with increasing TiO₂ content, the oxidation loss of Al increased, indicating that TiO₂ can inhibit the oxidation loss of Ti and increase the oxidation loss of Al.

Fig. 3.

Effect of refining slag with different compositions on the oxidation loss of Al, Ti and Si. (a) TiO₂: 4.35 mass%; (b) TiO₂: 9.26 mass%; (c) TiO₂: 15.06 mass%; (d) The contents of Al, Ti and Si in the alloy at 40 min. (Online version in color.)

Table 3. Composition of refined slag after slag-metal reaction (mass%).

Slag sample	CaO	SiO2	Al2O3	MgO	CaF2	TiO2
S1	42.04	1.19	27.84	10.14	8.40	7.80
S2	42.34	1.20	25.13	9.64	8.66	12.55
S3	40.15	1.21	21.74	9.45	8.57	18.22

3.2. Thermodynamic Analysis of Slag-metal Reaction

During the slag-metal reaction, the changes in Al and Ti contents are mainly determined by the following six equations.¹²⁾ It can be seen from the reaction equation that after reaching equilibrium, the main factors influencing the Al and Ti contents in the alloy are the contents of Al, Ti, and Si and the activities of SiO₂, Al₂O₃, and TiO₂ in the slag.   

$$4\left[\mathrm{A}\mathrm{l}\right]+3\left(\mathrm{T}\mathrm{i}{\mathrm{O}}_2\right)=3\left[\mathrm{T}\mathrm{i}\right]+2\left(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\right)$$(1)

  

$$\begin{array}{l}lg{K}_1=lg{\displaystyle \frac{a_{\text{Ti}}^3\cdot a_{{\text{Al}}_2{\text{O}}_3}^{\text{2}}}{a_{\text{Al}}^{\text{4}}\cdot a_{{\text{TiO}}_2}^3}}\\ \hspace{1em}\hspace{1em}=lg{\displaystyle \frac{f_{\text{Ti}}^{\text{3}}\cdot {[\%\text{Ti}]}^3}{f_{\text{Al}}^{\text{4}}\cdot {[\%\text{Al}]}^4}}+lg{\displaystyle \frac{a_{{\text{Al}}_2{\text{O}}_3}^{\text{2}}}{a_{{\text{TiO}}_2}^3}}={\displaystyle \frac{35\mathrm{\hspace{0.17em} }300}{T}}-\mathrm{9.94.}\end{array}$$(2)

  

$$4\left[\mathrm{A}\mathrm{l}\right]+3\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_2\right)=3\left[\mathrm{S}\mathrm{i}\right]+2\left(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\right)$$(3)

  

$$\begin{array}{l}lg{K}_2=lg{\displaystyle \frac{a_{\text{Si}}^3\cdot a_{{\text{Al}}_2{\text{O}}_3}^{\text{2}}}{a_{\text{Al}}^{\text{4}}\cdot a_{{\text{SiO}}_2}^3}}\\ \hspace{1em}\hspace{1em}=lg{\displaystyle \frac{f_{\text{Si}}^{\text{3}}\cdot {[\%\text{Si}]}^3}{f_{\text{Al}}^{\text{4}}\cdot {[\%\text{Al}]}^4}}+lg{\displaystyle \frac{a_{{\text{Al}}_2{\text{O}}_3}^{\text{2}}}{a_{{\text{SiO}}_2}^3}}={\displaystyle \frac{35\mathrm{\hspace{0.17em} }840}{T}}-5.86\end{array}$$(4)

  

$$\left[\mathrm{T}\mathrm{i}\right]+\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_2\right)=\left[\mathrm{S}\mathrm{i}\right]+\left(\mathrm{T}\mathrm{i}{\mathrm{O}}_2\right)$$(5)

  

$$\begin{array}{l}lg{K}_3=lg{\displaystyle \frac{a_{\text{Si}}\cdot a_{{\text{TiO}}_2}}{a_{\text{Ti}}\cdot a_{{\text{SiO}}_2}}}\\ \hspace{1em}\hspace{1em}=lg{\displaystyle \frac{f_{\text{Si}}\cdot [\%\text{Si}]}{f_{\text{Ti}}\cdot [\%\text{Ti}]}}+lg{\displaystyle \frac{a_{{\text{TiO}}_2}}{a_{{\text{SiO}}_2}}}={\displaystyle \frac{180}{T}}+1.36\end{array}$$(6)

Here, K_i, a_i, a_ixOy, f_i, T, and [%i] are the equilibrium constant of alloy component i, activity of i, activity of component i_xO_y in the slag, activity coefficient of i, temperature (K), and mass fraction of i, respectively. The f_i of the alloying elements was obtained by the Wagner Eq. (7). The interaction coefficients used here are shown in Table 4.^{19,20,21,22,23)}   

$$lg{f}_i=\sum \left(e_i^j[\%j]+{\gamma }_i^j{[\%j]}^2\right)$$(7)

Here, $e_i^j$ is the first-order activity interaction coefficient, ${\gamma }_i^j$ is the second-order activity interaction coefficient.

Table 4. Interaction coefficients used in this study at 1873 K.

$e_i^j$	Ni	Cr	Mn	Al	Ti	Cu	Mo	Si
Al	−0.037620)	0.04519)	0.03419)	0.0419)	–	–	–	–
Ti	−0.016621)	0.02522)	−0.1222)	–	0.04821)	0.01422)	0.01622)	–
Si	−0.00923)	−0.02121)	–	0.05820)	–	–	–	0.13219)
$r_{\text{Al}}^{\text{Ni}}$ = 0.000164,20) $r_{\text{Ti}}^{\text{Ni}}$ = 0.0005,20) $r_{\text{Si}}^{\text{Cr}}$ = 0.0004320)

3.2.1. Mass Action Concentration (Activity) of Each Component in the Slag

A series of models has been developed to simulate and elucidate the complex characteristics of slag. The most widely used theoretical model in the iron and steel and metallurgical industries is ion and molecular coexistence theory (IMCT). According to IMCT, the structural units in the CaO–SiO₂–Al₂O₃–MgO–CaF₂–TiO₂ slag system can be divided into simple ions, simple molecules, and complex molecules. A mathematical model was established according to the dynamic equilibrium between each structural unit (see Tables 5 and 6^24,25,26,27)). In order to represent the mole number of components in 100 g slag, the mole number of CaO, SiO₂, Al₂O₃, MgO, CaF₂ and TiO₂ in slag system is expressed as b₁ = n′_CaO, b₂ = n′_SiO2, b₃ = n′_Al2O3, b₄ = n′_MgO, b₅ = n′_CaF2, b₆ = n′_TiO2, which represents the chemical composition of slag. Since MgO ions can be divided into Mg²⁺ and O²⁻ structures, and the equilibrium molar number can be expressed as n₄ = n′_Mg²⁺(MgO) = n′_O^2-(MgO), the total equilibrium molar number of all structural units is defined as: $\sum n_i$ = 2n₁ + n₂ + n₃ + 2n₄ + 3n₅ + n₆ + n_c1 + n_c2 + … + n_c31. The mass action concentration is defined^28,29) as the ratio of the equilibrium mole number of structural units to the total equilibrium mole number of all units, so the action concentration can be expressed as $N_i=\frac{n_i}{\sum n_i}$, where i values are defined in Table 5. The N_i of complex molecules can be expressed by the standard molar Gibbs free energy ($\mathrm{\Delta }{G}_i^{\theta }$), reaction equilibrium constant (K_ci), and the corresponding N_i of simple ions. For example, CaO·Al₂O₃ can be expressed by N₁, N₃, and K_c5. Because (Ca² + O²⁻) + (Al₂O₃) = (CaO·Al₂O₃), $\mathrm{\Delta }{G}_5^{\theta }$ = −RTlnK_c5 = $RTln\frac{N_{c5}}{N_1\cdot N_3}$ = 59413–59.413T, so the action concentration is N_c5 = K_c5N₁N₃, K_c5 = $exp\left(-\frac{\mathrm{\Delta }{G}_5^{\theta }}{RT}\right)$.

Table 5. Structural units in CaO–SiO₂–Al₂O₃–MgO–CaF₂–TiO₂ slag system.

Molecular Ion Terms	Structural Units	Number	Mole Number of Structural Unit ni (mol)	Action Concentration of Structural Unit Ni
simple ion	Ca2++O2−	1	$n_1=n_{{\text{Ca}}^{2+}(\text{CaO)}}=n_{{\text{O}}^{2-}(\text{CaO)}}$	$N_1=\frac{2n_1}{\sum n_i}=N_{\text{CaO}}$
	Mg2++O2−	4	$n_4=n_{{\text{Mg}}^{2+}(\text{MgO)}}=n_{{\text{O}}^{2-}(\text{MgO)}}$	$N_4=\frac{2n_4}{\sum n_i}=N_{\text{MgO}}$
	Ca2++2F−	5	$n_5=n_{{\text{Ca}}^{\text{2}+}{\text{(CaF}}_{\text{2}}\text{)}}=2n_{{\text{F}}^{-}{\text{(CaF}}_{\text{2}}\text{)}}$	$N_5=\frac{2n_5}{\sum n_i}=N_{{\text{CaF}}_{\text{2}}}$
simple molecules	SiO2	2	$n_2=n_{{\text{SiO}}_2}$	$N_2=\frac{n_2}{\sum n_i}=N_{{\text{SiO}}_{\text{2}}}$
	Al2O3	3	$n_3=n_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$	$N_3=\frac{n_3}{\sum n_i}=N_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$
	TiO2	6	$n_6=n_{{\text{TiO}}_{\text{2}}}$	$N_6=\frac{n_6}{\sum n_i}=N_{{\text{TiO}}_{\text{2}}}$
Complex molecule	CaO·SiO2	c1	$n_{c1}=n_{\text{CaO}\cdot {\text{SiO}}_2}$	$N_{c1}=\frac{n_{c1}}{\sum n_i}=N_{\text{CaO}\cdot {\text{SiO}}_{\text{2}}}$
	2CaO·SiO2	c2	$n_{c2}=n_{\text{2CaO}\cdot {\text{SiO}}_{\text{2}}}$	$N_{c2}=\frac{n_{c2}}{\sum n_i}=N_{\text{2CaO}\cdot {\text{SiO}}_{\text{2}}}$
	3CaO·SiO2	c3	$n_{c3}=n_{3\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2}$	$N_{c3}=\frac{n_{c3}}{\sum n_i}=N_{\text{3CaO}\cdot {\text{SiO}}_{\text{2}}}$
	3CaO·2SiO2	c4	$n_{c4}=n_{\text{3CaO}\cdot 2{\text{SiO}}_{\text{2}}}$	$N_{c4}=\frac{n_{c4}}{\sum n_i}=N_{\text{3CaO}\cdot 2{\text{SiO}}_{\text{2}}}$
	CaO·Al2O3	c5	$n_{c5}=n_{\text{CaO}\cdot {\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$	$N_{c5}=\frac{n_{c5}}{\sum n_i}=N_{\text{CaO}\cdot {\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$
	3CaO·Al2O3	c6	$n_{c6}=n_{\text{3CaO}\cdot {\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$	$N_{c6}=\frac{n_{c6}}{\sum n_i}=N_{\text{3CaO}\cdot {\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$
	12CaO·7Al2O3	c7	$n_{c7}=n_{\text{12CaO}\cdot {\text{7Al}}_{\text{2}}{\text{O}}_{\text{3}}}$	$N_{c7}=\frac{n_{c7}}{\sum n_i}=N_{\text{12CaO}\cdot 7{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$
	CaO·2Al2O3	c8	$n_{c8}=n_{\text{CaO}\cdot {\text{2Al}}_{\text{2}}{\text{O}}_{\text{3}}}$	$N_{c8}=\frac{n_{c8}}{\sum n_i}=N_{\text{CaO}\cdot 2{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$
	CaO·6Al2O3	c9	$n_{c9}=n_{\text{CaO}\cdot {\text{6Al}}_{\text{2}}{\text{O}}_{\text{3}}}$	$N_{c9}=\frac{n_{c9}}{\sum n_i}=N_{\text{CaO}\cdot 6{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}$
	CaO·TiO2	c10	$n_{c10}=n_{\text{CaO}\cdot {\text{TiO}}_{\text{2}}}$	$N_{c10}=\frac{n_{c10}}{\sum n_i}=N_{\text{CaO}\cdot {\text{TiO}}_{\text{2}}}$
	3CaO·2TiO2	c11	$n_{c11}=n_{\text{3CaO}\cdot {\text{2TiO}}_{\text{2}}}$	$N_{c11}=\frac{n_{c11}}{\sum n_i}=N_{\text{3CaO}\cdot 2{\text{TiO}}_{\text{2}}}$
	4CaO·3TiO2	c12	$n_{c12}=n_{\text{4CaO}\cdot {\text{3TiO}}_{\text{2}}}$	$N_{c12}=\frac{n_{c12}}{\sum n_i}=N_{\text{4CaO}\cdot 3{\text{TiO}}_{\text{2}}}$
	MgO·SiO2	c13	$n_{c13}=n_{\text{MgO}\cdot {\text{SiO}}_{\text{2}}}$	$N_{c13}=\frac{n_{c13}}{\sum n_i}=N_{\text{MgO}\cdot {\text{SiO}}_{\text{2}}}$
	2MgO·SiO2	c14	$n_{c14}=n_{\text{2MgO}\cdot {\text{SiO}}_{\text{2}}}$	$N_{c14}=\frac{n_{c14}}{\sum n_i}=N_{\text{2MgO}\cdot {\text{SiO}}_{\text{2}}}$
	MgO·Al2O3	c15	$n_{c15}=n_{\text{MgO}\cdot {\text{Al}}_2{\text{O}}_{\text{3}}}$	$N_{c15}=\frac{n_{c15}}{\sum n_i}=N_{\text{MgO}\cdot {\text{Al}}_2{\text{O}}_{\text{3}}}$
	MgO·TiO2	c16	$n_{c16}=n_{\text{MgO}\cdot {\text{TiO}}_{\text{2}}}$	$N_{c16}=\frac{n_{c16}}{\sum n_i}=N_{\text{MgO}\cdot {\text{TiO}}_{\text{2}}}$
	MgO·2TiO2	c17	$n_{c17}=n_{\text{MgO}\cdot 2{\text{TiO}}_{\text{2}}}$	$N_{c17}=\frac{n_{c17}}{\sum n_i}=N_{\text{MgO}\cdot 2{\text{TiO}}_{\text{2}}}$
	2MgO·TiO2	c18	$n_{c18}=n_{\text{2MgO}\cdot {\text{TiO}}_{\text{2}}}$	$N_{c18}=\frac{n_{c18}}{\sum n_i}=N_{\text{2MgO}\cdot {\text{TiO}}_{\text{2}}}$
	3Al2O3·2SiO2	c19	$n_{c19}=n_{{\text{3Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot 2{\text{SiO}}_{\text{2}}}$	$N_{c19}=\frac{n_{c19}}{\sum n_i}=N_{{\text{3Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot 2{\text{SiO}}_{\text{2}}}$
	Al2O3·TiO2	c20	$n_{c20}=n_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot {\text{TiO}}_{\text{2}}}$	$N_{c20}=\frac{n_{c20}}{\sum n_i}=N_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot {\text{TiO}}_{\text{2}}}$
	CaO·MgO·2SiO2	c21	$n_{c21}=n_{\text{CaO}\cdot \text{MgO}\cdot {\text{2SiO}}_{\text{2}}}$	$N_{c21}=\frac{n_{c21}}{\sum n_i}=N_{\text{CaO}\cdot \text{MgO}\cdot {\text{2SiO}}_{\text{2}}}$
	2CaO·MgO·2SiO2	c22	$n_{c22}=n_{\text{2CaO}\cdot \text{MgO}\cdot {\text{2SiO}}_{\text{2}}}$	$N_{c22}=\frac{n_{c22}}{\sum n_i}=N_{\text{2CaO}\cdot \text{MgO}\cdot {\text{2SiO}}_{\text{2}}}$
	3CaO·MgO·2SiO2	c23	$n_{c23}=n_{\text{3CaO}\cdot \text{MgO}\cdot {\text{2SiO}}_{\text{2}}}$	$N_{c23}=\frac{n_{c23}}{\sum n_i}=N_{\text{3CaO}\cdot \text{MgO}\cdot {\text{2SiO}}_{\text{2}}}$
	CaO·Al2O3·2SiO2	c24	$n_{c24}=n_{\text{CaO}\cdot {\text{Al}}_2{\text{O}}_{\text{3}}\cdot {\text{2SiO}}_{\text{2}}}$	$N_{c24}=\frac{n_{c24}}{\sum n_i}=N_{\text{CaO}\cdot {\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot {\text{2SiO}}_{\text{2}}}$
	2CaO·Al2O3·SiO2	c25	$n_{c25}=n_{\text{2CaO}\cdot {\text{Al}}_2{\text{O}}_{\text{3}}\cdot {\text{SiO}}_{\text{2}}}$	$N_{c25}=\frac{n_{c25}}{\sum n_i}=N_{\text{2CaO}\cdot {\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot {\text{SiO}}_{\text{2}}}$
	3CaO·3Al2O3·CaF2	c26	$n_{c26}=n_{\text{3CaO}\cdot 3{\text{Al}}_2{\text{O}}_{\text{3}}\cdot {\text{CaF}}_{\text{2}}}$	$N_{c26}=\frac{n_{c26}}{\sum n_i}=N_{\text{3CaO}\cdot 3{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot {\text{CaF}}_{\text{2}}}$
	CaO·MgO·SiO2	c27	$n_{c27}=n_{\text{CaO}\cdot \text{MgO}\cdot {\text{SiO}}_{\text{2}}}$	$N_{c27}=\frac{n_{c27}}{\sum n_i}=N_{\text{CaO}\cdot \text{MgO}\cdot {\text{SiO}}_{\text{2}}}$
	11CaO·7Al2O3·CaF2	c28	$n_{c28}=n_{\text{11CaO}\cdot 7{\text{Al}}_2{\text{O}}_{\text{3}}\cdot {\text{CaF}}_{\text{2}}}$	$N_{c28}=\frac{n_{c28}}{\sum n_i}=N_{\text{11CaO}\cdot 7{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot {\text{CaF}}_{\text{2}}}$
	CaO·SiO2·TiO2	c29	$n_{c29}=n_{\text{CaO}\cdot {\text{SiO}}_{\text{2}}\cdot {\text{TiO}}_{\text{2}}}$	$N_{c29}=\frac{n_{c29}}{\sum n_i}=N_{\text{CaO}\cdot {\text{SiO}}_{\text{2}}\cdot {\text{TiO}}_2}$
	3CaO·2SiO2·CaF2	c30	$n_{c30}=n_{\text{3CaO}\cdot 2{\text{SiO}}_{\text{2}}\cdot {\text{CaF}}_{\text{2}}}$	$N_{c30}=\frac{n_{c30}}{\sum n_i}=N_{\text{3CaO}\cdot 2{\text{SiO}}_{\text{2}}\cdot {\text{CaF}}_2}$
	2MgO·2Al2O3·5SiO2	c31	$n_{c31}=n_{\text{2MgO}\cdot 2{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}\cdot 5{\text{SiO}}_{\text{2}}}$	$N_{c31}=\frac{n_{c31}}{\sum n_i}=N_{\text{2MgO}\cdot 2{\text{Al}}_{\text{2}}{\text{O}}_3\cdot 5{\text{SiO}}_2}$

Table 6. Chemical reactions forming complex molecules.

Chemical reaction	$\mathrm{\Delta }{G}_i^{\theta }$ (J·mol−1)	References	Action concentration of structural unit Ni
(Ca2+ + O2−) + (SiO2) = (CaO·SiO2)	−21757 − 36.819T	24	Nc1 = Kc1N1N2
2(Ca2+ + O2−) + (SiO2) = (2CaO·SiO2)	−102090 − 24.267T	24	Nc2 = Kc2N12N2
3(Ca2+ + O2−) + (SiO2) = (3CaO·SiO2)	−118826 − 6.694T	25	Nc3 = Kc3N13N2
3(Ca2+ + O2−) + 2(SiO2) = (3CaO·2SiO2)	−236814 + 9.623T	25	Nc4 = Kc4N13N22
(Ca2+ + O2−) + (Al2O3) = (CaO·Al2O3)	59413 − 59.413T	24	Nc5 = Kc5N1N3
3(Ca2+ + O2−) + (Al2O3) = (3CaO·Al2O3)	−21757 − 29.288T	24	Nc6 = Kc6N13N3
12(Ca2+ + O2−) + 7(Al2O3) = (12CaO·7Al2O3)	617977 − 612.119T	24	Nc7 = Kc7N112N37
(Ca2+ + O2−) + 2(Al2O3) = (CaO·2Al2O3)	−16736 − 25.522T	24	Nc8 = Kc8N1N32
(Ca2+ + O2−) + 6(Al2O3) = (CaO·6Al2O3)	−22594 − 31.798T	24	Nc9 = Kc9N1N36
(Ca2+ + O2−) + (TiO2) = (CaO·TiO2)	−79900 − 3.35T	26	Nc10 = Kc10N1N6
3(Ca2+ + O2−) + 2(TiO2) = (3CaO·2TiO2)	−207100 − 11.35T	26	Nc11 = Kc11N13N62
4(Ca2+ + O2−) + 3(TiO2) = (4CaO·3TiO2)	−292880 − 17.573T	26	Nc12 = Kc12N14N63
(Mg2+ + O2−) + (SiO2) = (MgO·SiO2)	23849 − 29.706T	24	Nc13 = Kc13N2N4
2(Mg2+ + O2−) + (SiO2) = (2MgO·SiO2)	−56902 − 3.347T	24	Nc14 = Kc14N2N42
(Mg2+ + O2−) + (Al2O3) = (MgO·Al2O3)	−18828 − 6.276T	24	Nc15 = Kc15N3N4
(Mg2+ + O2−) + (TiO2) = (MgO·TiO2)	−26400 + 3.14T	26	Nc16 = Kc16N4N6
(Mg2+ + O2−) + 2(TiO2) = (MgO·2TiO2)	−27600 + 0.63T	26	Nc17 = Kc17N4N62
2(Mg2+ + O2−) + (TiO2) = (2MgO·TiO2)	−25500 + 1.26T	26	Nc18 = Kc18N42N6
3(Al2O3) + 2(SiO2) = (3Al2O3·2SiO2)	−4354.27 − 10.467T	24	Nc19 = Kc19N22N33
(Al2O3) + (TiO2) = (Al2O3·TiO2)	−25270 + 3.924T	26	Nc20 = Kc20N3N6
(Ca2+ + O2−) + (Mg2+ + O2−) + 2(SiO2) = (CaO·MgO·2SiO2)	−80333 − 51.882T	24	Nc21 = Kc21N1N22N4
2(Ca2+ + O2−) + (Mg2+ + O2−) + 2(SiO2) = 2(CaO·MgO·2SiO2)	−73638 − 63.597T	24	Nc22 = Kc22N12N22N4
3(Ca2+ + O2−) + (Mg2+ + O2−) + 2(SiO2) = 3(CaO·MgO·2SiO2)	−205016 − 31.798T	24	Nc23 = Kc23N13N22N4
(Ca2+ + O2−) + (Al2O3) + 2(SiO2) = (CaO·Al2O3·2SiO2)	−4184 − 73.638T	24	Nc24 = Kc24N1N22N3
2(Ca2+ + O2−) + (Al2O3) + (SiO2) = (2CaO·Al2O3·SiO2)	−116315 − 38.911T	24	Nc25 = Kc25N13N2N3
3(Ca2+ + O2−) + 3(Al2O3) + (Ca2+ + 2F−) = (3CaO·3Al2O3·CaF2)	−44492 − 73.15T	27	Nc26 = Kc26N13N33N5
(Ca2+ + O2−) + (Mg2+ + O2−) + (SiO2) = (CaO·MgO·SiO2)	−124683 + 3.766T	25	Nc27 = Kc27N1N2N4
11(Ca2+ + O2−) + 7(Al2O3) + (Ca2+ + 2F−) = (11CaO·7Al2O3·CaF2)	−228760 − 155.8T	27	Nc28 = Kc28N111N37N5
(Ca2+ + O2−) + (SiO2) + (TiO2) = (CaO·SiO2·TiO2)	−114683 + 7.32T	26	Nc29 = Kc29N1N2N6
3(Ca2+ + O2−) + 2(SiO2) + (Ca2+ + 2F−) = (3CaO·2SiO2·CaF2)	−255180 − 8.2T	24	Nc30 = Kc30N13N22N5
2(Mg2+ + O2−) + 2(Al2O3) + 5(SiO2) = (2MgO·2Al2O3·5SiO2)	−14422 − 14.808T	24	Nc31 = Kc31N25N32N42

According to the mass balance and a molar fraction of all structural units of 1, seven equations were established, as follows:   

$$N_1+N_2+\dots +N_6+N_{c1}+N_{c2}+\dots +N_{c31}=\sum N_i=1$$(8)

  

$$\begin{array}{l}{b}_1 =(0.5N_1+N_{c1}+2N_{c2}+3N_{c3}+3N_{c4}+N_{c5}+3N_{c6}\\ +12N_{c7}+N_{c8}+N_{c9}+N_{c10}+3N_{c11}+4N_{c12}+N_{c21}\\ +2N_{c22}+3N_{c23}+N_{c24}+2N_{c25}+3N_{c26}+N_{c27}+11N_{c28}\\ +N_{c29}+3N_{c30})\sum n=n{\prime }_{\text{CaO}}\end{array}$$(9)

  

$$\begin{array}{l}{b}_2 =(N_2+N_{c1}+N_{c2}+N_{c3}+2N_{c4}+N_{c13}+N_{c14}\\ +2N_{c19}+2N_{c21}+2N_{c22}+2N_{c23}+2N_{c24}+N_{c25}\\ +N_{c27}+N_{c29}+2N_{c30}+5N_{c31})\sum n_i=n{\prime }_{{\text{SiO}}_{\text{2}}}\end{array}$$(10)

  

$$\begin{array}{l}{b}_3 =(N_3+N_{c5}+N_{c6}+7N_{c7}+2N_{c8}+6N_{c9}+N_{c15}+3N_{c19}\\ +N_{c20}+N_{c24}+N_{c25}+3N_{c26}+7N_{c28}+2N_{c31})\sum n_i=n{\prime }_{{\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}}\end{array}$$(11)

  

$$\begin{array}{l}{b}_4 =(0.5N_4+N_{c13}+2N_{c14}+N_{c15}+N_{c16}+N_{c17}+2N_{c18}\\ +N_{c21}+N_{c22}+N_{c23}+N_{c27}+2N_{c31})\sum n_i=n{\prime }_{\text{MgO}}\end{array}$$(12)

  

$$b_5 =(1/3N_5+N_{c26}+N_{c28}+2N_{c30})\sum n_i=n{\prime }_{{\text{CaF}}_{\text{2}}}$$(13)

  

$$\begin{array}{l}{b}_6 =(N_6+N_{c10}+2N_{c11}+3N_{c12}+N_{c16}+2N_{c17}+N_{c18}\\ +N_{c20}+N_{c29})\sum n_i=n{\prime }_{{\text{TiO}}_{\text{2}}}\end{array}$$(14)

Equations (8), (9), (10), (11), (12), (13), (14) were solved by the Matlab Newton iterative method, and the concentration of each element was obtained.

3.2.2. Relationship between Composition and Activity of SiO₂, Al₂O₃, and TiO₂ in Slag

By IMCT, the curves of SiO₂, Al₂O₃, and TiO₂ activities as a function of slag composition are shown in Fig. 4, and in order to facilitate the comparison, we made some vertical lines in this figure. The composition of refining slag used for calculation is shown in Table 7. Figure 4(a) shows that with increasing CaO content, $a_{{\text{Al}}_2{\text{O}}_3}$, $a_{{\text{SiO}}_2}$, and $a_{{\text{TiO}}_2}$ decreased. This was due to the formation of (x)CaO·(y)Al₂O₃, (x)CaO·(y)SiO₂, (x)CaO·(y)TiO₂, and other complex compounds by the combination of CaO with Al₂O₃, SiO₂, and TiO₂. With the increase in CaO content, the relative change rate of activity was $\mathrm{\Delta }{a}_{{\text{SiO}}_2}$ > $\mathrm{\Delta }{a}_{{\text{TiO}}_2}$ > $\mathrm{\Delta }{a}_{{\text{Al}}_{\text{2}}{\text{O}}_3}$ due to the optical alkalinity ${\mathrm{\Lambda }}_{{\text{SiO}}_2}$ > ${\mathrm{\Lambda }}_{{\text{TiO}}_2}$ > ${\mathrm{\Lambda }}_{{\text{Al}}_{\text{2}}{\text{O}}_3}$. As shown in Fig. 4(b), with increasing SiO₂ content, $a_{{\text{SiO}}_2}$ significantly increased. In addition, SiO₂ and CaO generate complex compounds, resulting in a decrease in CaO content, so that $a_{{\text{Al}}_2{\text{O}}_3}$ and $a_{{\text{TiO}}_2}$ increase. As shown in Fig. 4(c), with increasing Al₂O₃ content, $a_{{\text{Al}}_2{\text{O}}_3}$ increased significantly. Thereby, more Al₂O₃ was able to combine with CaO to form complex compounds, reducing the CaO content and increasing $a_{{\text{SiO}}_2}$ and $a_{{\text{TiO}}_2}$. Figure 4(d) shows that with increasing MgO content, $a_{{\text{SiO}}_2}$, $a_{{\text{Al}}_2{\text{O}}_3}$, and $a_{{\text{TiO}}_2}$ slightly decrease because MgO and CaO are alkaline oxides, which can combine with the three oxides to form complex compounds. However, MgO is less capable of forming compounds than CaO, resulting in a slight decrease in the activity of the three studied oxides. As shown in Fig. 4(e), with increasing CaF₂ content in the slag, CaF₂ reacts with trace Al₂O₃ and CaO to form complex compounds, resulting in the decrease of $a_{{\text{Al}}_2{\text{O}}_3}$, while the changes of $a_{{\text{SiO}}_2}$ and $a_{{\text{TiO}}_2}$ are not obvious. Figure 4(f) shows that with increasing TiO₂ content in the slag, $a_{{\text{TiO}}_2}$ significantly increased. Therefore, more TiO₂ was available to combine with CaO, resulting in a decrease in the CaO content and increase in $a_{{\text{SiO}}_2}$ and $a_{{\text{Al}}_2{\text{O}}_3}$.

Fig. 4.

Relationship between composition and activity of SiO₂, Al₂O₃, TiO₂ in slag. (Online version in color.)

Table 7. Composition of refining slag used in thermodynamic calculation (Ratio).

Number	CaO:SiO2:Al2O3:MgO:CaF2:TiO2
a	X:2:30:8:10:5
b	45:X:30:8:10:5
c	45:2:X:8:10:5
d	45:2:30:X:10:5
e	45:2:30:8:X:5
f	45:2:30:8:10:X

3.2.3. Relationship between Al and Ti Contents and Slag Composition at Equilibrium

By rearranging Eqs. (2), (4), and (6), expressions for the Al and Ti contents can be expressed as shown in Eqs. (15) and (16), respectively.   

$$\begin{array}{l}lg[\%\text{Al}]={\displaystyle \frac{1}{\text{8}}}\left(lg{\displaystyle \frac{a_{{\text{Al}}_2{\text{O}}_3}^4}{a_{{\text{SiO}}_{\text{2}}}^3\cdot a_{{\text{TiO}}_2}^3}}+lg{\displaystyle \frac{f_{\text{Si}}^3\cdot {[\%\text{Si}]}^3\cdot f_{\text{Ti}}^3\cdot {[\%\text{Ti}]}^3}{f_{\text{Al}}^8}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \frac{8\mathrm{\hspace{0.17em} }892.5}{T}}+1.975\end{array}$$(15)

  

$$\begin{array}{l}lg[\%\text{Ti}]={\displaystyle \frac{1}{6}}\left(-lg{\displaystyle \frac{a_{{\text{Al}}_2{\text{O}}_3}^2\cdot a_{{\text{SiO}}_{\text{2}}}^3}{a_{{\text{TiO}}_2}^6}}+lg{\displaystyle \frac{f_{\text{Al}}^{\text{4}}\cdot {[\%\text{Al}]}^4\cdot f_{\text{Si}}^3\cdot {[\%\text{Si}]}^3}{f_{\text{Ti}}^6}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{5\mathrm{\hspace{0.17em} }793.5}{T}}-2.335\end{array}$$(16)

The relationship between Al and Ti content and composition in slag was also calculated by using the composition in Table 7, the results were shown in Fig. 5. In this calculation, the mass balance of slag-metal reaction was ignored, and the activity coefficients were treated as constant values at the initial composition. Figure 5(a′) shows that with increasing CaO content, due to its optical alkalinity mentioned in Section 3.2.2, CaO combines more easily with SiO₂ than with Al₂O₃ and TiO₂, $a_{{\text{SiO}}_2}$ is significantly reduced, and the degree of reaction between Al/Ti and SiO₂ decreases. As shown in Fig. 5(a′′), the ΔG change with variations in CaO content in the reaction equilibrium of Ti with SiO₂ or Al₂O₃was calculated using Eqs. (1) and (5), respectively. As indicated by ΔG, the reaction between Ti and SiO₂ becomes weaker, so the Ti content increases, and the reaction between Ti and Al₂O₃ becomes stronger, resulting in the decrease of Ti content.

Fig. 5.

Relationship between composition and Al, Ti content in alloy at equilibrium. (Online version in color.)

When the CaO content is low, the increase in the Ti content is less than that in the Ti loss by oxidation. However, when the CaO content increases, the increase in the Ti content is greater than that in the Ti loss by oxidation. Therefore, the Al content increased continuously with increasing CaO content, whereas the Ti content first decreased and then increased. Figure 5(b) shows that with increasing SiO₂ mass fraction, the reaction degree of Ti/Al and SiO₂ increased due to the rapid increase in $a_{{\text{SiO}}_2}$, so the contents of Al and Ti in the alloy decreased significantly. It can be seen from Fig. 5(c) that with increasing Al₂O₃ content, the content of Al in the alloy increased significantly, whereas the content of Ti decreased. The reason for this is that during the slag-metal process, $a_{{\text{Al}}_2{\text{O}}_3}$ increases and the reaction degree between Ti and Al₂O₃ increases, whereas the reaction degree between Al and SiO₂ decreases. Therefore, Al is generated, whereas Ti is consumed. Figures 5(d), 5(e) shows that with increasing MgO and CaF₂ contents, the Al and Ti contents did not change significantly; therefore, the physical and chemical properties of the refining slag can be changed by adjusting the contents of MgO and CaF₂. It can be seen from Fig. 5(f) that with increasing TiO₂ content, $a_{\mathrm{T}\mathrm{i}{\mathrm{O}}_2}$ increases, and the reaction degree of Ti with Al₂O₃ and SiO₂ decreases due to the reverse reaction direction of chemical equilibrium. Therefore, the Ti content in the alloy increased, and the Al content decreased.

3.3. Kinetic Analysis of Slag-metal Reaction

3.3.1. Establishment of a Kinetic Model

To further analyze the change in Al and Ti contents over time during the reaction process, the kinetic model was established based on two-film theory. As shown in Fig. 6, in the slag-metal reaction process, there are mainly Al and Al₂O₃, Ti and TiO₂, and Si and SiO₂ interactive reaction systems; the chemical reactions are shown in Eqs. (1), (3), and (5). Since the equilibrium of thermodynamic reactions is instantaneously achieved at the slag-metal interface, and to facilitate the calculations, the above reactions are simplified into three linear independent reactions and their equilibrium constant expressions shown in Eqs. (17), (18), (19), (20), (21), (22).   

$$\left[\mathrm{A}\mathrm{l}\right]+1.5\left[\mathrm{O}\right]=\left(\mathrm{A}\mathrm{l}{\mathrm{O}}_{1.5}\right)$$(17)

  

$$\begin{array}{l}lg{K}_{\text{Al}}=lg{\displaystyle \frac{a_{{\text{AlO}}_{1.5}}^{*}}{a_{\text{Al}}^{*}\cdot {(a_{\text{O}}^{*})}^{1.5}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}=lg{\displaystyle \frac{{\gamma }_{{\text{AlO}}_{1.5}}\cdot X_{{\text{AlO}}_{1.5}}^{*}}{f_{\text{Al}}\cdot {[\%\text{Al}]}^{*}\cdot {(a_{\text{O}}^{*})}^{1.5}}}={\displaystyle \frac{32\mathrm{\hspace{0.17em} }000}{T}}-10.29\end{array}$$(18) ³⁰⁾

  

$$\left[\mathrm{T}\mathrm{i}\right]+2\left[\mathrm{O}\right]=(\mathrm{T}\mathrm{i}{\mathrm{O}}_2)$$(19)

  

$$\begin{array}{l}lg{K}_{\text{Ti}}=lg{\displaystyle \frac{a_{{\text{TiO}}_{\text{2}}}^{*}}{a_{\text{Ti}}^{*}\cdot {(a_{\text{O}}^{*})}^2}}\\ \hspace{1em}\hspace{1em}\hspace{1em}=lg{\displaystyle \frac{{\gamma }_{{\text{TiO}}_2}\cdot X_{{\text{TiO}}_2}^{*}}{f_{\text{Ti}}\cdot {[\%\text{Ti}]}^{*}\cdot {(a_{\text{O}}^{*})}^2}}={\displaystyle \frac{34\mathrm{\hspace{0.17em} }458}{T}}-11.96\end{array}$$(20) ¹⁷⁾

  

$$\left[\mathrm{S}\mathrm{i}\right]+2\left[\mathrm{O}\right]=\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_2\right)$$(21)

  

$$\begin{array}{l}lg{K}_{\text{Si}}=lg{\displaystyle \frac{a_{{\text{SiO}}_{\text{2}}}^{*}}{a_{\text{Si}}^{*}\cdot {(a_{\text{O}}^{*})}^2}}\\ \hspace{1em}\hspace{1em}\hspace{1em}=lg{\displaystyle \frac{{\gamma }_{{\text{SiO}}_2}\cdot X_{{\text{SiO}}_2}^{*}}{f_{\text{Si}}\cdot {[\%\text{Si}]}^{*}\cdot {(a_{\text{O}}^{*})}^2}}={\displaystyle \frac{30\mathrm{\hspace{0.17em} }410}{T}}-11.59\end{array}$$(22) ³¹⁾

Fig. 6.

Mass transfer model of slag-metal interface. (a) Mass transfer process. (b) Component mass transfer diagram. (Online version in color.)

The parameters with the superscript “*” in the equations are the values at the slag-metal interface, $X_{i{\text{O}}_x}$ is the molar fraction of the constituent elements in the slag, and ${\gamma }_{i{\text{O}}_x}$ is the activity coefficient of the constituent elements in the slag phase with the pure matter as the standard state. The component $X_{i{\text{O}}_x}$ in the slag can be converted into a mass fraction (%iO_x) by Eq. (23), which further simplifies the calculation of the equilibrium constant in Eq. (24).   

$$X_{i{\text{O}}_x}={\displaystyle \frac{{\displaystyle \frac{(\%i{\text{O}}_x)}{M_{i{\text{O}}_x}}}}{{\displaystyle \frac{(\%{\text{CaF}}_{\text{2}})}{M_{{\text{CaF}}_{\text{2}}}}}+\sum {\displaystyle \frac{(\%i{\text{O}}_x)}{M_{i{\text{O}}_x}}}}}$$(23)

  

$$\begin{array}{l}{K}_i={\displaystyle \frac{{\gamma }_{i{\text{O}}_x}}{f_i\cdot \left\{{\displaystyle \frac{(\%{\text{CaF}}_{\text{2}})}{M_{{\text{CaF}}_{\text{2}}}}}+\sum {\displaystyle \frac{(\%i{\text{O}}_x)}{M_{i{\text{O}}_x}}}\right\}\cdot M_{i{\text{O}}_x}}}\cdot {\displaystyle \frac{{(\%i{\text{O}}_x)}^{*}}{{[\%i]}^{*}\cdot {(a_{\text{O}}^{*})}^x}}\\ \hspace{1em}\hspace{0.5em}=E_i\cdot {\displaystyle \frac{{(\%i{\text{O}}_x)}^{*}}{{[\%i]}^{*}\cdot {(a_{\text{O}}^{*})}^x}}\end{array}$$(24)

According to two-film theory, the mass-transfer flux of elements on the metal side is equal to that on the slag side, so the mass-transfer flux of each substance can be obtained.   

$$J_i=F_i\cdot \left\{{[\%i]}^b-{[\%i]}^{*}\right\}=F_{i{\text{O}}_x}\cdot \left\{{(\%i{\text{O}}_x)}^{*}-{(\%i{\text{O}}_x)}^b\right\}$$(25)

  

$$J_{\text{O}}=F_{\text{O}}\cdot \left\{{[\%\text{O}]}^b-{[\%\text{O}]}^{*}\right\}$$(26)

Here, J_i is the mass-transfer flux of Al, Ti, and Si, J_O is the mass transfer flux of oxygen, F_i and $F_{i{\text{O}}_x}$ are the modified mass-transfer coefficients, F_i = $\frac{k_i\cdot {\rho }_i}{100M_i}$, $F_{i{\text{O}}_x}$ = $\frac{k_s\cdot {\rho }_s}{100M_{i{\text{O}}_x}}$, ρ_i is the density of metal, ρ_s is the density of slag, k_i and k_s are the mass-transfer coefficients of the component on the metal-side and slag-side boundary layers, respectively. The parameters with the superscript “b” are the values of the component in the bulk of the metal or slag side. Combining the mass-transfer flux and equilibrium constant equations, the component content at the slag-metal interface is expressed as follows:   

$${(\%i{\text{O}}_x)}^{*}=\frac{K_i}{E_i}\cdot {[\%i]}^{*}\cdot {(a_{\text{O}}^{*})}^x$$(27)

  

$${[\%i]}^{*}={\displaystyle \frac{{\displaystyle \frac{F_i}{F_{i{\text{O}}_x}}}\cdot {[\%i]}^b+{(\%i{\text{O}}_x)}^b}{{\displaystyle \frac{F_i}{F_{i{\text{O}}_x}}}+{\displaystyle \frac{K_i}{E_i}}\cdot {(a_{\text{O}}^{*})}^x}}$$(28)

  

$$J_i=F_i\cdot \left\{{[\%i]}^b-{\displaystyle \frac{{\displaystyle \frac{F_i}{F_{i{\text{O}}_x}}}\cdot {[\%i]}^b+{(\%i{\text{O}}_x)}^b}{{\displaystyle \frac{F_i}{F_{i{\text{O}}_x}}}+{\displaystyle \frac{K_i}{E_i}}\cdot {(a_{\text{O}}^{*})}^x}}\right\}$$(29)

  

$$J_{\text{O}}=F_{\text{O}}\cdot \left\{{[\%\text{O}]}^b-\frac{a_{\text{O}}^{*}}{f_{\text{O}}}\right\}$$(30)

From the oxygen balance equation, we obtain:   

$$J_{\text{O}}=1.5J_{\text{Al}}+2J_{\text{Ti}}+J_{\text{Si}}$$(31)

By substituting Eqs. (27), (28), (29), (30) into Eq. (31), where there is only one unknown quantity, $a_{\text{O}}^{*}$, we used a Matlab program to calculate this parameter. Then, we calculated the percentage of each component at the interface so that the concentration of the metal component i varies with time:   

$$-V_M\cdot \frac{d[\%i]}{dt}=A\cdot k_i({[\%i]}^b-{[\%i]}^{*})$$(32)

Where V_M is the volume of the alloy, A is the interface area of slag-metal reaction.

3.3.2. Determination of Model Parameters

The activity coefficient γ in the slag can be solved according to the ratio of the concentration and molar fraction of each component in the slag (Eq. (33)). The solution of the activity coefficient f of each element on the liquid-steel side was calculated by the Wagner formula described in Section 3.2. The mass action concentration of each component in the slag was solved by the IMCT model described in Section 3.2.1.   

$${\gamma }_i=\frac{N_i}{X_i}$$(33)

The mass-transfer coefficient k_i and k_s in the slag-metal reaction process has been studied extensively. It is generally considered that the order of magnitude of the mass-transfer coefficient of elements in steel is 10⁻⁴ m/s, and that of components in the slag is 10⁻⁵ m/s.^32,33) The values assumed here are shown in Table 8.

Table 8. Mass transfer coefficient (m/s) of components.

component	Al	Ti	Si	O	Al2O3	TiO2	SiO2
mass transfer coefficient	3×10−4 32)	3×10−4 32)	3×10−4 32)	5×10−4 32)	2×10−5 32)	2×10−5 32)	2×10−6 33)

3.3.3. Calculation Results of the Model

Figure 7 is the kinetic model and experimental results about the change trend of Al, Ti and Si contents in alloy with time. The model calculation results showed that the slag-metal reaction reached equilibrium within 10 min. For a TiO₂ content in the slag of 4.35 mass% (Fig. 7(a)), the content of Ti in the alloy decreased over time, and the content of Al increased. This indicates that Ti reacts with Al₂O₃ by redox reactions, and 4[Al] + 3(TiO₂) = 3[Ti] + 2(Al₂O₃) shows a leftward trend, resulting in oxidation of Ti and an increase in the fraction of Al. It can be seen from Figs. 7(b), 7(c) that when the TiO₂ content in the slag was 9.26 mass% or 15.06 mass%, respectively, both Ti and Al were oxidized, indicating that with the increase of TiO₂ content, the trend of rightward reaction of the above equation increases. With the increase in TiO₂ content from 4.35 mass%, to 9.26 mass%, to 15.06 mass%, the oxidation loss rate of Ti decreased from 46.8 mass% to 35.3 mass%, and finally to 28.8 mass%, respectively, whereas Al content changed from an initial increase to an oxidation loss. In general, the calculated dynamic values are in good agreement with the experimental results. The dynamic model can be used to predict the trend of Al and Ti content in the alloy with time.

Fig. 7.

Experimental and calculated results of Al, Ti and Si contents in the alloy. (a) TiO₂: 4.35 mass%. (b) TiO₂: 9.26 mass%. (c) TiO₂: 15.06 mass%. (Online version in color.)

4. Conclusions

(1) Under equilibrium in the slag-metal reaction process, with increasing TiO₂ content and decreasing Al₂O₃ content, the content of Ti increases, whereas that of Al gradually decreases in the final alloy.

(2) With increasing content of CaO, the activities of the Al₂O₃, SiO₂, and TiO₂ in the slag decreased, whereas with increasing contents of these oxides, the corresponding activities increased. However, changing the content of MgO and CaF₂ in the slag resulted in relatively small changes in the activities of Al₂O₃, SiO₂, and TiO₂.

(3) The results of the thermodynamic model showed that with increasing CaO content in the slag, the content of Al in the alloy increased continuously when the reaction reaches equilibrium, whereas the content of Ti first decreased and then increased. With increasing SiO₂ content in the slag, the contents of Al and Ti in the alloy decreased significantly. With increasing Al₂O₃ content in the slag, the Al content in the alloy increased, whereas that of Ti decreased. With increasing MgO and CaF₂ contents in the slag, the contents of Al and Ti in the alloy did not change significantly. With increasing TiO₂ content in the slag, the content of Ti in the alloy increased, whereas the Al content decreased.

(4) The results of the kinetic model showed that the slag-metal reaction reaches equilibrium within 10 min, and the calculation results were in good agreement with the experimental ones. The mass-transfer coefficient and other parameters used in the model are considered reasonable. Therefore, the model has good applicability for evaluating the relationship between the slag refining reaction and the final Incoloy825 alloy.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. U1908223), the Fundamental Research Project of the Ministry of Education of China (Grant No. N2125038), and Program of Introducing Talents of Discipline to Universities (No. B21001).

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