Desulfurization Behavior of FeNi-Based Expansion Alloy Melt Using CaO-SiO₂-MgO-Al₂O₃-CaF₂ Slag

Abstract

The effects of slag components on the sulfur distribution ratio {(wt.%S)/[wt.%S]} were analyzed using thermodynamic calculations. On this basis, the effects of the slag with different binary basicities (wt.% CaO/wt.% SiO₂ = 3.0, 5.0, 7.0, 9.0 and 11.0) and Al₂O₃ contents (wt.% Al₂O₃ = 12, 16, 20 and 24) on the desulfurization behavior were investigated. The results show that the binary basicity and Al₂O₃ content were the main factors that affected the desulfurization. And an increase in the binary basicity and a decrease in the content of Al₂O₃, increase the sulfide capacity of the slag. In the early stage of the smelting process, the desulfurization process was limited due to the high content of oxides, such as FeO and MnO in the slag, and the sulfur content in the alloy melt reached 35 ppm. After the final deoxidation of the Si-Ca alloy, the desulfurization rate was significantly increased, and the maximum desulfurization rate reached 44.12%. During the ladle standing, the sulfur content in the alloy melt changed little because of the limitation of kinetics. The rate-limiting step of the desulfurization process was the diffusion of sulfur in the alloy melt.

1. Introduction

Expansion alloys are widely used in fields such as the electronic industry and precision instruments due to their abnormal thermal expansion characteristics [1,2,3]. In general, expansion alloy ingots are obtained through vacuum or non-vacuum induction melting and then processed into the required product specifications via hot forging and cold rolling of the ingots. In the process of smelting and subsequent processing, some wastes will inevitably be produced, such as risers, sprues, chips, corner materials and unqualified castings. In addition, in order to reduce production costs, some expansion alloy manufacturers will purchase chip materials and scrapped parts from downstream enterprises. The sulfur content in these return scraps is high, which is easy to segregate at the grain boundary and generates low-melting-point sulfide in the solidification process of the expansion alloy, resulting in a decrease in the mechanical properties of the alloy [4]. Therefore, minimizing the sulfur content of the alloy melt in the smelting process is the key to improving the quality of alloy products.

Many metallurgical scholars have conducted extensive research on the desulfurization behavior of Ni-based alloys. Meyer et al. [5] determined the equilibrium partial pressure of sulfur vapor over Ni-S melts at temperatures from 1100 °C to 1600 °C. On this basis, Lin et al. [6] determined the thermodynamic activity of sulfur in the β₁-Ni₃S₂, β₂-Ni₄S₃, γ-Ni₆S₅ and δ-NiS phases. Degawa et al. [7] studied the desulfurization behavior of Inconel738 and Mar-M247 Ni-based superalloys smelted in crucibles of different materials and showed that a CaO crucible is most conducive to the desulfurization of an alloy melt. Kishimoto et al. [8] developed a desulfurization model by using solid CaO in molten Ni-based superalloys containing Al and found that S and Al in the alloy melt react with CaO to generate CaS and Al₂O₃ at any temperature, and Al₂O₃ reacts with CaO to generate calcium aluminate slag. In addition, they found that the rate-limiting step of the desulfurization reaction is S diffusion in the generated calcium aluminate slag at any temperature [9]. Horie et al. [10] showed that the liquidus temperature of a CaO-MgO mixture is lower than that of CaO, which is conducive to the diffusion of S. Therefore, the desulfurization effect of a CaO-MgO mixture is better than that of CaO.

Special metallurgical methods, such as vacuum induction melting and electroslag remelting, can effectively remove the sulfur content in the Ni-based alloy [11,12]. Zhang et al. [11] studied the desulfurization behavior of a Ni-based superalloy smelted in a vacuum induction furnace, in which Al was added as a desulfurizer during refining, and they found that Al can significantly decrease the content of [S] compared with a CaO-based crucible. Electroslag remelting has a strong desulfurization ability, which is affected by various factors, such as the oxygen content of the liquid metal [13], slag composition [14,15,16,17,18,19], remelting atmosphere [20,21] and sulfide inclusions [22,23]. In addition, the melting rate and electrical parameters of electroslag remelting have an influence on the desulfurization degree, and a low melting rate and reversed polarity are conducive to improving the desulfurization degree [14,24,25].

Most of the above studies used laboratory experiments to study the desulfurization behavior of alloy melts, and the final desulfurization rate could reach 90% or above. However, in actual industry, the desulfurization rate of alloy melts is much lower than this level. In addition, the research object in the above literature was mainly Ni-based alloys, with the nickel content exceeding 80 wt.%. There is little research on the desulfurization behavior of FeNi-based alloys with an iron content of about 57 wt.% and a nickel content of about 42 wt.%.

A medium-frequency induction furnace with a smelting capacity of 2 ton was used to smelt an FeNi-based expansion alloy in a special steel plant in China. During smelting and production, it is inevitable to produce some waste materials, such as risers, sprues, chips, corner materials and unqualified castings, and some scrapped parts will be returned by downstream enterprises. The content of impurity elements in these return scraps is high, such as a sulfur content of more than 60 ppm. Therefore, these return scraps need to be remelted in a medium-frequency induction furnace to make the masterbatch of FeNi-based expansion alloy, and then finally smelted together with new materials to make the finished alloy products.

In order to study the effect of slag components on the desulfurization of alloy melt, and obtain suitable slag components for the desulfurization of an FeNi-based expansion alloy. In the current study, the effect of CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag components on the sulfur distribution ratio was calculated using ion and molecule coexistence theory. On the basis of this, the masterbatch and new material were smelted together in a 2-ton medium-frequency induction furnace, and the effects of CaO-SiO₂-MgO-Al₂O₃-CaF₂ slags with different binary basicity (wt.% CaO/wt.% SiO₂) and Al₂O₃ content on the desulfurization behavior of the FeNi-based expansion alloy melt were analyzed by sampling throughout the whole smelting process.

2. Thermodynamic Calculation Theory and Material Preparation

2.1. Thermodynamic Calculation Theory

The ion and molecule coexistence theory assumes that slag is a high-temperature melt composed of ions and undissociated molecules. From this, the action concentration of slag components is derived such that the metallurgical reaction equilibrium follows the mass action law. The effect of each component in the CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag on the sulfur distribution ratio was calculated using ion and molecule coexistence theory, and the factors affecting desulfurization of FeNi-based expansion alloy melt were obtained. Based on this, the desulfurization thermodynamics and kinetics of the alloy melt were studied through industrial experiments.

2.2. Material Preparation

The FeNi-based expansion alloy was obtained via smelting in a 2-ton medium-frequency induction furnace (model GWT-2/1000, 1000 kW, Inductotherm Group China, Shanghai, China), and the alloy raw materials were composed of the masterbatch and new material. After the previous alloy melt tapping was completed, the alloy raw materials were charged into the medium-frequency induction furnace. At this time, the power supply was turned on and the smelting started. The whole smelting process lasted about 150 min, including 8 key steps, as shown in Figure 1. The smelting steps were described in detail in the previous study [26]. Here, the CaO-SiO₂-MgO-Al₂O₃-CaF₂ master slags were prepared by pre-melting the mixture of industrial-grade CaCO₃ (≥98 wt.%), SiO₂ (≥99 wt.%), MgO (≥95 wt.%), Al₂O₃ (≥99 wt.%) and CaF₂ (≥95 wt.%), and the composition of the slag is shown in

Table 1. Chemical composition of smelting slag (wt.%).

No.	Binary Basicity (R)	CaO	SiO2	MgO	Al2O3	CaF2
Exp. 1	3.0	37.5	12.5	10	20	20
Exp. 2	5.0	41.7	8.3	10	20	20
Exp. 3	7.0	43.7	6.3	10	20	20
Exp. 4	9.0	45.0	5.0	10	20	20
Exp. 5	11.0	45.8	4.2	10	20	20
Exp. 6	9.0	52.2	5.8	10	12	20
Exp. 7	9.0	48.6	5.4	10	16	20
Exp. 8	9.0	41.4	4.6	10	24	20

.

In order to observe the desulfurization behavior in the process of slagging and smelting the FeNi-based expansion alloy, slag samples and alloy samples were taken for composition analysis, in which sample #6 and sample #7 represented alloy samples of ladle standing for 0.5 min and 4.5 min, respectively. The slag composition was analyzed using an X-ray fluorescence spectrometer (XRF, model AXIOSmax, PANalytical B.V., Almelo, The Netherlands), and the sulfur content in the alloy sample was measured using a carbon sulfur analyzer (model CS600, LECO, Laboratory Equipment Corporation, San Jose, CA, USA).

3. Thermodynamic Calculation of the Effect of Slag Composition on Sulfur Distribution Ratio

The composition of slag has an important effect on desulfurization. Therefore, the effect of each component in the CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag on the sulfur distribution ratio was calculated using ion and molecule coexistence theory.

The desulfurization reaction in the present study can be expressed by an ion exchange reaction, as follows:

$$\left[\mathrm{S}\right]+\left({\mathrm{O}}^{2-}\right)=\left({\mathrm{S}}^{2-}\right)+\left[\mathrm{O}\right]$$

(1)

In the CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag, only two basic oxides, namely, CaO and MgO, can react with sulfur in the melt. The desulfurization reactions of ion couples (Ca²⁺ + O²⁻) and (Mg²⁺ + O²⁻) in the molten slag can be expressed using Equations (2) and (3), respectively [27,28]:

$$\left({\mathrm{Ca}}^{2+}{+\mathrm{O}}^{2-}\right)+\left[\mathrm{S}\right]=\left({\mathrm{Ca}}^{2+}{+\mathrm{S}}^{2-}\right)+\left[\mathrm{O}\right]\Delta G_{\mathrm{CaS}}^{\mathsf{\theta }}=105784.6-28.723T (\mathrm{J}/\mathrm{mol})$$

(2)

$$\left({\mathrm{Mg}}^{2+}{+\mathrm{O}}^{2-}\right)+\left[\mathrm{S}\right]=\left({\mathrm{Mg}}^{2+}{+\mathrm{S}}^{2-}\right)+\left[\mathrm{O}\right]\Delta G_{\mathrm{MgS}}^{\mathsf{\theta }}=203604.6-35.023T (\mathrm{J}/\mathrm{mol})$$

(3)

According to the sulfur distribution ratio relationship shown in Equation (4), the sulfur distribution ratio of the ion couples (Ca²⁺ + O²⁻) and (Mg²⁺ + O²⁻) in the molten slag that equilibrated or reacted with alloy melt can be deduced from the equilibrium constant of Equations (2) and (3), as follows:

$$L_{\mathrm{S}}=\frac{\left(\mathrm{wt}.\%\mathrm{S}\right)}{\left[\mathrm{wt}.\%\mathrm{S}\right]}$$

(4)

$$L_{\mathrm{S},\mathrm{CaO}}=\frac{{\left(\mathrm{wt}.\%\mathrm{S}\right)}_{\mathrm{CaS}}}{\left[\mathrm{wt}.\%\mathrm{S}\right]}=\frac{16K_{\mathrm{CaS}}^{\mathsf{\theta }}{N}_{\mathrm{CaO}}{\displaystyle \sum n_i}}{\left[\mathrm{wt}.\%\mathrm{O}\right]}\times \frac{f_{\mathrm{S}}}{f_{\mathrm{O}}}$$

(5)

$$L_{\mathrm{S},\mathrm{MgO}}=\frac{{\left(\mathrm{wt}.\%\mathrm{S}\right)}_{\mathrm{MgS}}}{\left[\mathrm{wt}.\%\mathrm{S}\right]}=\frac{16K_{\mathrm{MgS}}^{\mathsf{\theta }}{N}_{\mathrm{MgO}}{\displaystyle \sum n_i}}{\left[\mathrm{wt}.\%\mathrm{O}\right]}\times \frac{f_{\mathrm{S}}}{f_{\mathrm{O}}}$$

(6)

Therefore, the total sulfur distribution ratio L_S between the CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag and the alloy melt can be expressed using Equation (7):

$$L_{\mathrm{S}}=L_{\mathrm{S},\mathrm{CaO}}+L_{\mathrm{S},\mathrm{MgO}}=\frac{16\left(K_{\mathrm{CaS}}^{\mathsf{\theta }}{N}_{\mathrm{CaO}}+K_{\mathrm{MgS}}^{\mathsf{\theta }}{N}_{\mathrm{MgO}}\right){\displaystyle \sum n_i}}{\left[\mathrm{wt}.\%\mathrm{O}\right]}\times \frac{f_{\mathrm{S}}}{f_{\mathrm{O}}}$$

(7)

where $N_i$ is the mass action concentration of compound i in the slag, which can be calculated using ion and molecule coexistence theory; ${\displaystyle \sum n_i}$ is the total mole number of structural units; and $f_i$ is the activity coefficient of element i, which can be calculated using Wagner’s equation [29].

Figure 2 shows the effect of slag components on the sulfur distribution ratio. It was found that the change in slag composition had no effect on L_S,MgO, and the total sulfur distribution ratio L_S was mainly controlled by L_S,CaO. In addition, as shown in Figure 2a, an increase in the content of CaO in the slag was beneficial to improve L_S. This was mainly because CaO is a basic substance, which is easy to combine with S in an alloy melt. While in this desulfurization system, SiO₂ and Al₂O₃ in the slag are acidic substances, which easily combine with CaO in the slag, leading to a decrease in the desulfurization ability of the slag. Therefore, an increase in the content of SiO₂ and Al₂O₃ reduces L_S, as shown in Figure 2b,d. The changes in the contents of MgO and CaF₂ in the slag had almost no effect on L_S, as shown in Figure 2c,e.

The change in sulfur distribution ratio with slag composition in Figure 2 is beneficial to optimize the composition of smelting slag, which has important theoretical value. Therefore, based on the optimization of the smelting slag, the effects of CaO-SiO₂-MgO-Al₂O₃-CaF₂ slags with different binary basicity and Al₂O₃ content on the desulfurization behavior of the FeNi-based expansion alloy melt were studied, and the chemical composition of the slag is shown in Table 1. Among them, the binary basicities of the slag were 3.0, 5.0, 7.0, 9.0 and 11.0, and the Al₂O₃ contents were 12 wt.%, 16 wt.%, 20 wt.% and 24 wt.%.

4. Results and Discussion

4.1. Desulfurization Thermodynamics of FeNi-Based Expansion Alloy

In the smelting process, the desulfurization reaction between a slag and alloy melt can be expressed using Equation (1). Therefore, the sulfide capacity of the slag can be expressed using Equation (8):

$$\lg C_{{\mathrm{S}}^{2-}}=\lg \left(\frac{\left(\mathrm{wt}.\%{\mathrm{S}}^{2-}\right)a_{\mathrm{O}}}{a_{\mathrm{S}}}\right)=\lg \left(\frac{\left(\mathrm{wt}.\%{\mathrm{S}}^{2-}\right)a_{\mathrm{O}}}{\left[\mathrm{wt}.\%\mathrm{S}\right]f_{\mathrm{S}}}\right)=\lg L_{\mathrm{S}}+\lg a_{\mathrm{O}}-\lg f_{\mathrm{S}}$$

(8)

where $a_{\mathrm{O}}$ is the activity of O in the alloy melt and $f_{\mathrm{S}}$ is the activity coefficient of S in the alloy melt.

Due to the lack of thermodynamic data, e.g., interaction parameters for all solute elements in FeNi-based alloy melts, the values of $a_{\mathrm{O}}$ and $f_{\mathrm{S}}$ cannot be calculated using the Wagner model. Yang et al. [30] proposed a sulfide capacity prediction model, which combines the slag–metal equilibrium desulfurization reaction shown in Equation (1) and the slag–gas equilibrium desulfurization reaction shown in Equation (9) to obtain the gas–metal equilibrium desulfurization reaction, as shown in Equation (10):

$$\frac{1}{2}{\mathrm{S}}_2(\mathrm{g})+\left({\mathrm{O}}^{2-}\right)=\frac{1}{2}{\mathrm{O}}_2(\mathrm{g})+\left({\mathrm{S}}^{2-}\right)$$

(9)

$$\left[\mathrm{S}\right]+\frac{1}{2}{\mathrm{O}}_2(\mathrm{g})=\frac{1}{2}{\mathrm{S}}_2(\mathrm{g})+\left[\mathrm{O}\right]\hspace{1em}\Delta \mathrm{r}{G}_{\mathrm{m}}^{\mathsf{\theta }}=17907.96-26.34T$$

(10)

The relationship between the $C_{{\mathrm{S}}^{2-}}$ and L_S of the slag can be obtained, as shown in Equation (11) [30]:

$$\lg C_{{\mathrm{S}}^{2-}}=\lg L_{\mathrm{S}}+\lg a_{\mathrm{O}}-\lg f_{\mathrm{S}}+\frac{935.45}{T}-1.3757$$

(11)

Therefore, for the CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag, the $C_{{\mathrm{S}}^{2-}}$ can be obtained by inserting Equations (6)–(11), as shown in Equation (12):

$$\lg C_{{\mathrm{S}}^{2-}}=\lg C_{{\mathrm{S}}^{2-},\mathrm{CaO}}+\lg C_{{\mathrm{S}}^{2-},\mathrm{MgO}}=\lg \left[16\left(K_{\mathrm{CaS}}^{\mathsf{\theta }}{N}_{\mathrm{CaO}}+K_{\mathrm{MgS}}^{\mathsf{\theta }}{N}_{\mathrm{MgO}}\right){\displaystyle \sum n_i}\right]+\frac{935.45}{T}-1.3757$$

(12)

However, in the smelting process of the FeNi-based expansion alloy in this study, Mn and Fe in the alloy oxidize to form MnO and FeO, as shown in

Table 2. Chemical composition of slag sample (wt.%).

No.	Sampling No.	CaO	SiO2	MgO	Al2O3	CaF2	MnO	TiO2	FeO
Exp. 1	#1	35.18	13.61	8.77	18.85	8.65	3.94	2.34	8.66
	#2	35.46	11.59	9.5	20.66	10.99	3.01	1.77	7.02
	#3	42.52	7.74	7.78	22.87	11.46	1.89	1.05	4.69
	#4	45.49	5.52	6.64	22.93	15.67	0.85	0.66	2.24
	#5	51.07	6.87	1.65	20.35	17.74	0.42	0.24	1.66
Exp. 2	#1	38.71	10.7	8.51	18.96	9.15	3.55	2.39	8.03
	#2	36.54	9.66	9.77	20.97	11.41	3.12	1.91	6.62
	#3	42.47	5.68	8.05	23	13.69	1.63	0.96	4.52
	#4	46.83	6	5.9	21.28	16.76	0.74	0.61	1.88
	#5	52.04	7.25	2.38	19.01	17.38	0.28	0.27	1.39
Exp. 3	#1	41.43	8.12	8.95	19.32	10.22	3.22	2.06	6.68
	#2	42.07	7.79	9.72	21.17	9.41	2.86	1.77	5.21
	#3	43.59	4.6	8.64	23.22	14.16	1.29	0.81	3.69
	#4	47.47	3.45	6.67	21	19.01	0.42	0.34	1.64
	#5	53.96	4.87	2.84	18.86	17.95	0.2	0.21	1.11
Exp. 4	#1	44.31	7.19	8.8	19.55	10.61	2.59	1.54	5.41
	#2	45.28	6.9	10.93	21.54	9.77	1.49	0.91	3.18
	#3	49.61	4.26	6.7	24.78	11.91	0.91	0.57	1.26
	#4	50.13	2.36	7.67	21.02	18.22	0.14	0.11	0.35
	#5	56.57	4.39	2.6	18	17.22	0.21	0.14	0.87
Exp. 5	#1	44.72	6.66	9.65	19.56	11	2.31	1.23	4.87
	#2	46.32	6.08	10.36	22	10.23	1.24	0.77	3
	#3	50.92	3.77	7.74	25	10.25	0.79	0.49	1.06
	#4	51.99	2.16	6.1	22.05	16.98	0.15	0.12	0.45
	#5	55.39	4.21	3.05	18.52	17.88	0.17	0.12	0.66
Exp. 6	#1	54.58	6.75	9.24	11.66	9.54	2.12	1.25	4.86
	#2	55.25	6	10.32	13.05	10.54	1.19	0.77	2.88
	#3	57.84	3.74	7.87	14.17	14.32	0.63	0.52	0.91
	#4	59.15	3.32	5.99	12.99	17.83	0.26	0.21	0.25
	#5	63.74	3.96	3.12	10.93	17.65	0.16	0.13	0.31
Exp. 7	#1	48.82	6.3	9.02	15.6	11.4	2.33	1.41	5.12
	#2	50.46	5.78	10.65	17.23	10.68	1.32	0.84	3.04
	#3	56.02	3.46	7.14	18.36	12.65	0.78	0.6	0.99
	#4	56.11	2.12	6.52	16.98	17.69	0.2	0.12	0.26
	#5	59.16	3.88	3.46	14.44	18.21	0.19	0.12	0.54
Exp. 8	#1	41.47	6.32	8.95	23.12	10.12	2.69	1.67	5.66
	#2	41.45	5.46	9.63	26.67	10.45	1.58	1.05	3.71
	#3	44.19	3.72	7.62	27.98	13.21	1.12	0.71	1.45
	#4	46.75	2.11	6.89	25.34	17.76	0.32	0.21	0.62
	#5	51.07	3.62	3.45	22.92	18.19	0.21	0.13	0.41

. The generated MnO and FeO combine with the S in the alloy melt, as shown in Equations (13) and (14), respectively. Therefore, in the actual smelting process, the sulfide capacity of the slag should be changed as shown in Equation (15).

$$\left({\mathrm{Mn}}^{2+}{+\mathrm{O}}^{2-}\right)+\left[\mathrm{S}\right]=\left({\mathrm{Mn}}^{2+}{+\mathrm{S}}^{2-}\right)+\left[\mathrm{O}\right]\hspace{1em}\Delta G_{\mathrm{MnS}}^{\mathsf{\theta }}=83956-30.78T (\mathrm{J}/\mathrm{mol})$$

(13)

$$\left({\mathrm{Fe}}^{2+}{+\mathrm{O}}^{2-}\right)+\left[\mathrm{S}\right]=\left({\mathrm{Fe}}^{2+}{+\mathrm{S}}^{2-}\right)+\left[\mathrm{O}\right]\hspace{1em}\Delta G_{\mathrm{FeS}}^{\mathsf{\theta }}=115526-33.352T (\mathrm{J}/\mathrm{mol})$$

(14)

$$\begin{array}{l}\lg C_{{\mathrm{S}}^{2-}}=\lg C_{{\mathrm{S}}^{2-},\mathrm{CaO}}+\lg C_{{\mathrm{S}}^{2-},\mathrm{MgO}}+\lg C_{{\mathrm{S}}^{2-},\mathrm{MnO}}+\lg C_{{\mathrm{S}}^{2-},\mathrm{FeO}}\\ =\lg \left[16\left(K_{\mathrm{CaS}}^{\mathsf{\theta }}{N}_{\mathrm{CaO}}+K_{\mathrm{MgS}}^{\mathsf{\theta }}{N}_{\mathrm{MgO}}+K_{\mathrm{MnS}}^{\mathsf{\theta }}{N}_{\mathrm{MnO}}+K_{\mathrm{FeS}}^{\mathsf{\theta }}{N}_{\mathrm{FeO}}\right){\displaystyle \sum n_i}\right]+\frac{935.45}{T}-1.3757\end{array}$$

(15)

where the values of $K_{\mathrm{CaS}}^{\mathsf{\theta }}$, $K_{\mathrm{MgS}}^{\mathsf{\theta }}$, $K_{\mathrm{MnS}}^{\mathsf{\theta }}$ and $K_{\mathrm{FeS}}^{\mathsf{\theta }}$ can be calculated using the standard molar Gibbs free energy of the desulfurization reactions, as shown in Equation (16); the mass action concentrations $N_{\mathrm{CaO}}$, $N_{\mathrm{MgO}}$, $N_{\mathrm{MnO}}$ and $N_{\mathrm{FeO}}$ can be calculated using the thermodynamic model based on the ion and molecule coexistence theory [30].

$$K_i^{\mathsf{\theta }}=\exp \left(-\Delta G_i^{\theta }/RT\right)$$

(16)

Figure 3 shows the effect of the binary basicity and Al₂O₃ content on the sulfide capacity of the slag, in which the comparison values were calculated using the relationship between the sulfide capacity and optical basicity, as shown in Equation (17) [31]:

$$\lg C_{{\mathrm{S}}^{2-}}=\frac{22690-54650{\Lambda }_{\mathrm{melt}}}{T}+43.6{\Lambda }_{\mathrm{melt}}-25.2$$

(17)

where ${\Lambda }_{\mathrm{melt}}$ is the optical basicity of the slag, which can be calculated using Equation (18):

$${\Lambda }_{\mathrm{melt}}=\frac{{\displaystyle \sum x_i{n}_i{\Lambda }_i}}{{\displaystyle \sum x_i{n}_i}}$$

(18)

where x_i is the mole fraction of oxide component i; n_i is the number of oxygen atoms in each oxide i; ${\Lambda }_i$ is the optical basicity of pure oxide component i; and the optical basicity values of CaO, SiO₂, MgO, Al₂O₃, MnO, TiO₂, FeO and CaF₂ are 1.0, 0.48, 0.78, 0.60, 0.59, 0.61, 0.51 and 0.20, respectively.

It can be seen from Figure 3 that the sulfide capacity of the slag calculated using the two formulas was different, but the change trend was consistent. An increase in the binary basicity and a decrease in the content of Al₂O₃ in the slag are beneficial for increasing the sulfide capacity. The results are consistent with the thermodynamic calculation results shown in Section 2. In addition, with the progress of the smelting process, the sulfide capacity of the slag increased. Although the Al₂O₃ generated by the Al deoxidation process in the smelting process was not conducive to the desulfurization process, Al reduced FeO, MnO and SiO₂, which reduced the oxidizability and improved the basicity of the slag, thus increasing the sulfide capacity of the slag.

4.2. Desulfurization Kinetics of FeNi-Based Expansion Alloy

Table 3. Sulfur content in alloy melt (ppm).

No.	Sam. #1	Sam. #2	Sam. #3	Sam. #4	Sam. #5	Sam. #6	Sam. #7	Sam. #8
Exp. 1	35	35	35	33	29	25	23	22
Exp. 2	34	34	35	34	27	22	19	19
Exp. 3	35	35	35	33	26	19	17	16
Exp. 4	35	35	34	31	24	16	14	14
Exp. 5	34	34	33	31	22	15	14	13
Exp. 6	34	34	33	29	19	14	10	9
Exp. 7	34	35	35	31	22	16	13	12
Exp. 8	35	36	35	33	25	18	17	15

shows the change in sulfur content in the alloy melt under different binary basicities and Al₂O₃ contents of the slag. It was found that an increase in the binary basicity and a decrease in the content of Al₂O₃ in the slag reduced the sulfur content of the alloy ingot (i.e., sample #8). And exp. 6 had the highest desulfurization rate, reaching 73.53%, which met the needs of the alloy finished product. The results are consistent with the variation trend of sulfide capacity with binary basicity and Al₂O₃ content calculated using thermodynamics. In addition, before taking the intermediate sample (i.e., sample #4), the sulfur content in the alloy melt had almost no change. This was because the content of oxides, such as FeO and MnO, in the slag was high, which is not conducive to the desulfurization process. After the final deoxidation of the Si-Ca alloy, the desulfurization rate increased significantly, and the maximum desulfurization rate of exp. 6 reached 44.12%. In the process of ladle standing, due to the limitation of desulfurization kinetics, the sulfur content in the alloy melt changed little.

Alloy melt desulfurization is a liquid phase reaction that is controlled by the diffusion of reactants and products and an interfacial chemical reaction. Since the reaction rate is very fast at high temperatures and the sulfide capacity is high due to the relatively high slag basicity, the interface chemical reaction and the transfer of products in the slag phase are not rate-limiting, and the desulfurization process is mainly controlled by mass transfer of sulfur in the alloy melt [17,32].

The mass transfer of sulfur in the alloy melt and slag phase can be expressed using Equations (19) and (20):

$$J_{[\mathrm{S}]}=-\frac{dC}{dt}\frac{V_{\mathrm{m}}}{A}=k_{\mathrm{m}}\left(C_{[\mathrm{S}]}^{*}-C_{[\mathrm{S}]}\right)$$

(19)

$$J_{(\mathrm{S})}=-\frac{dC}{dt}\frac{V_{\mathrm{s}}}{A}=k_{\mathrm{s}}\left(C_{\left(\mathrm{S}\right)}-C_{\left(\mathrm{S}\right)}^{*}\right)$$

(20)

where $J_{\mathrm{i}}$ is the molar flux of sulfur, $V_{\mathrm{i}}$ is the volume, $k_{\mathrm{i}}$ is the mass transfer coefficient, $C_{\mathrm{i}}^{*}$ is the molar concentration of sulfur in the alloy melt and slag phase, and $C_{\mathrm{i}}$ is the molar concentration of sulfur at the reaction interface.

When the interfacial chemical reaction reaches equilibrium, all the sulfur consumption by the desulfurization reaction in the alloy melt diffuses into the slag, as described using Equation (21):

$$\frac{d[\mathrm{wt}.\%\mathrm{S}]}{dt}{W}_{\mathrm{m}}=-\frac{d(\mathrm{wt}.\%\mathrm{S})}{dt}{W}_{\mathrm{s}}$$

(21)

where $W_{\mathrm{m}}$ and $W_{\mathrm{s}}$ are the weights of the alloy melt and slag phase, respectively.

Expressing Equations (19) and (20) in terms of a mass fraction results in Equations (22) and (23):

$$\frac{\mathrm{d}[\mathrm{wt}.\%\mathrm{S}]}{\mathrm{dt}}=-\frac{A}{W_{\mathrm{m}}}{\rho }_{\mathrm{m}}{k}_{\mathrm{m}}\left\{{[\mathrm{wt}.\%\mathrm{S}]}^{*}-[\mathrm{wt}.\%\mathrm{S}]\right\}$$

(22)

$$\frac{\mathrm{d}(\mathrm{wt}.\%\mathrm{S})}{\mathrm{dt}}=-\frac{A}{W_{\mathrm{s}}}{\rho }_{\mathrm{s}}{k}_{\mathrm{s}}\left\{(\mathrm{wt}.\%\mathrm{S})-{(\mathrm{wt}.\%\mathrm{S})}^{*}\right\}$$

(23)

where ${\rho }_{\mathrm{m}}$ and ${\rho }_{\mathrm{s}}$ are the densities of the alloy melt and slag phase, respectively.

Equation (24) can be derived by substituting Equations (22) and (23) into Equation (21), as follows:

$${\rho }_{\mathrm{m}}{k}_{\mathrm{m}}\left\{[\mathrm{wt}.\%\mathrm{S}]-{[\mathrm{wt}.\%\mathrm{S}]}^{*}\right\}={\rho }_{\mathrm{s}}{k}_{\mathrm{s}}\left\{(\mathrm{wt}.\%\mathrm{S})-{(\mathrm{wt}.\%\mathrm{S})}^{*}\right\}$$

(24)

Substituting the sulfur distribution ratio formula into Equation (24) results in Equation (25):

$${[\mathrm{wt}.\%\mathrm{S}]}^{*}=\frac{{\rho }_{\mathrm{m}}{k}_{\mathrm{m}}[\mathrm{wt}.\%\mathrm{S}]-{\rho }_{\mathrm{s}}{k}_{\mathrm{s}}(\mathrm{wt}.\%\mathrm{S})}{{\rho }_{\mathrm{m}}{k}_{\mathrm{m}}-{\rho }_{\mathrm{s}}{k}_{\mathrm{s}}{L}_{\mathrm{S}}}$$

(25)

The desulfurization rate is described by Equation (26), which is obtained by substituting Equation (25) into Equation (22):

$$\frac{\mathrm{d}[\mathrm{wt}.\%\mathrm{S}]}{\mathrm{dt}}=-\frac{A}{W_{\mathrm{m}}}\frac{1}{\frac{1}{{\rho }_{\mathrm{s}}{k}_{\mathrm{s}}}-\frac{L_{\mathrm{S}}}{{\rho }_{\mathrm{m}}{k}_{\mathrm{m}}}}\left\{L_{\mathrm{S}}[\mathrm{wt}.\%\mathrm{S}]-(\mathrm{wt}.\%\mathrm{S})\right\}$$

(26)

The apparent desulfurization rate constant ($k_0$) can be expressed as Equation (27):

$$k_0=\frac{1}{\frac{1}{{\rho }_{\mathrm{s}}{k}_{\mathrm{s}}}-\frac{L_{\mathrm{S}}}{{\rho }_{\mathrm{m}}{k}_{\mathrm{m}}}}$$

(27)

By substituting Equation (27) into Equation (26), Equation (26) can be simplified to Equation (28):

$$\frac{\mathrm{d}[\mathrm{wt}.\%\mathrm{S}]}{\mathrm{dt}}=-\frac{A}{W_{\mathrm{m}}}{k}_0\left\{L_{\mathrm{S}}[\mathrm{wt}.\%\mathrm{S}]-(\mathrm{wt}.\%\mathrm{S})\right\}$$

(28)

According to the law of conservation of mass, the sulfur content in the alloy melt at some point can be expressed using Equation (29):

$$W_{\mathrm{m}}[\mathrm{wt}.\%\mathrm{S}]=W_{\mathrm{m}}{[\mathrm{wt}.\%\mathrm{S}]}^{\mathrm{int}.}-W_{\mathrm{s}}(\mathrm{wt}.\%\mathrm{S})$$

(29)

where $W_{\mathrm{m}}{[\mathrm{wt}.\%\mathrm{S}]}^{\mathrm{int}.}$ represents the initial sulfur content in the alloy melt and $W_{\mathrm{s}}(\mathrm{wt}.\%\mathrm{S})$ is the sulfur content in the slag phase at some point.

The evolution in time of the sulfur content in the alloy melt can be expressed by substituting Equation (29) into Equation (28) and integrating:

$$[\mathrm{wt}.\%\mathrm{S}]=\frac{{[\mathrm{wt}.\%\mathrm{S}]}^{\mathrm{ini}.}·L_{\mathrm{S}}}{L_{\mathrm{S}}+\frac{W_{\mathrm{m}}}{W_{\mathrm{s}}}}{e}^{-\frac{A}{W_{\mathrm{m}}}{k}_0(L_{\mathrm{S}}+\frac{W_{\mathrm{m}}}{W_{\mathrm{s}}})t}+\frac{{[\mathrm{wt}.\%\mathrm{S}]}^{\mathrm{ini}.}·\frac{W_{\mathrm{m}}}{W_{\mathrm{s}}}}{L_{\mathrm{S}}+\frac{W_{\mathrm{m}}}{W_{\mathrm{s}}}}$$

(30)

Figure 4 shows the experimental and fitting results of the sulfur content in alloy melt at different smelting stages, in which the solid lines were obtained by fitting Equation (30) with Matlab software (R2019b Update9), and the points are the experiment results. From Figure 4a, it can be seen that both the fitting results and experimental results confirm that increasing the binary basicity of the slag was beneficial to desulfurization, and under the conditions of different binary basicities of the slag, the fitting results are consistent with the experimental results. In addition, as shown in Figure 4b, both the fitting results and experimental results indicate that reducing the Al₂O₃ content in the slag was beneficial to desulfurization, and the fitting results are in good agreement with the experimental results under different Al₂O₃ content conditions. The results in Figure 4 show that the fitting function describes the experimental results well, indicating that the rate-limiting step of the desulfurization process was indeed the diffusion of sulfur.

5. Conclusions

• The sulfur distribution ratio was mainly controlled by L_S,CaO, and an increase in the binary basicity and a decrease in the content of Al₂O₃ of the slag improved the sulfur distribution ratio. Meanwhile, the content of MgO and CaF₂ in the slag had no effect on the sulfur distribution ratio.
• The variation trend of sulfide capacity with binary basicity and Al₂O₃ content of CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag obtained using the experimental results was consistent with that of sulfur distribution ratio with binary basicity and Al₂O₃ content calculated using thermodynamics.
• With the progress of the smelting process, the sulfide capacity of CaO-SiO₂-MgO-Al₂O₃-CaF₂ slag increased and the sulfur content in the alloy melt decreased. After the final deoxidation, the maximum desulfurization rate reached 44.12%, and the final desulfurization rate reached 73.53%. And the rate-limiting step of the desulfurization process was the diffusion of sulfur in the alloy melt.