Thermodynamic Assessment of Liquid Mn–Fe–Si–C–Ca–P System by Unified Interaction Parameter Model

Abstract

The Unified Interaction Parameter Model was used to evaluate the thermodynamics of Mn–Fe–Si–C–Ca–P system. The calculated results of the activities and activity coefficients of different components as well as the solubility of C and Ca agree well with the experimental data in binary, ternary and quaternary system. These parameters could be applied to calculate the activities of Mn, Si, Fe, C, Ca and P and the equilibrium between Ca and P in carbon saturated Mn–Fe–Si–C–Ca–P melts during the dephosphorization process. They are also useful for understanding the process of ferromanganese and silicomanganese. Using the present UIPM parameters, the calculated results showed that temperature had a weak effect on the activity coefficient of manganese in Mn–C melt. The effect of temperature and Fe content on the carbon solubility in Mn–Fe melts could be expressed as: X_Csat=(−229.18/T+0.411)+(−186.54/T+0.014)X_Fe. Manganese activity monotonically decreased as the silicon content increased and slightly decreased as the temperature increased in the carbon saturated Mn–Si–C system.

1. Introduction

There is an increasing demand for low phosphorous manganese ferroalloys due to the development of high manganese steels (10–30 mass% Mn).¹⁾ The high phosphorus content of manganese ferroalloys is mainly originated from the manganese ores. It is not beneficial to remove phosphorus from manganese ores by the ore dressing techniques, although some works had been done.^2,3,4,5,6) Several investigations had been carried out about the oxidizing de-P of ferromanganese with slags containing BaO.^{7,8,9,10,11,12,13)} However, the conventional oxidizing dephosphorization (de-P) is not feasible for manganese ferroalloys, especially for silicomanganese ferroalloys, due to the inevitable prior oxidation of manganese and silicon. Thus, reducing de-P should be applied for manganese ferroalloys using alkaline earth metals or their compounds,^{14,15,16,17,18,19)} for example, Ca–Si alloy. Ca plays an important role during the reducing de-P. There were also some reports about the reducing de-P mechanism using CaO-based slags.^{20,21,22,23,24,25,26)}

In order to get a full understanding about the thermodynamics of the reducing de-P process, it is of great importance to know the activities of different components involved in the manganese ferroalloys, such as Ca, P, Si, Mn and C. Efforts have been devoted to describe the thermodynamic properties of Mn-based melts. A large amount of experimental works has been done for the systems Mn–Fe,^{27,28,29,30,31,32,33)} Mn–Si,^{34,35,36,37,38,39,40,41,42)} Mn–C,^{35,36,43,44,45,46,47,48,49,50,51)} Mn–Fe–Si,^52,53) Mn–Fe–C,^{45,46,47,49,54,55,56,57,58,59,60)} Mn–Si–C,^{34,35,45,46,47,56,61,62,63,64)} Mn–Fe–Si–C,^{35,56,57,63,64,65)} Mn–i–Ca (i=Fe, Si or C)^46,66,67) and Mn–P.^68,69,70) However, reliable thermodynamic data are still very short to calculate the activities of different components in manganese ferroalloys at present. Since experimental measurements are both time-consuming and difficult at high temperatures, it is not possible to provide thermodynamic data for slag compositions in the many metallurgical processes; hence the need for a reliable model to estimate is urgent. The Unified Interaction Parameter Model (UIPM) was proposed by Bale and Pelton,^71,72) which is thermodynamically exact in both dilute and nondilute composition regions. This model can also reduce to Wagner’s formalism at infinite dilution and to Darken’s quadratic formalism in dilute solutions. Some works has been done to optimize the UIPM parameters for Mn–Si,⁷³⁾ Mn–Fe–C,⁷⁴⁾ Mn–Fe–C–N⁷⁵⁾ and Mn–Fe–Si–C⁷⁶⁾ system. During the reducing de-P process, Mn–Fe–Si–C–Ca–P system is always involved, however, the activity calculation of this system cannot be completed by the current model.^73,74,75,76) In the present paper, UIPM was extended to the Mn–Fe–Si–C–Ca–P system by considering all relevant experimental data in the literatures.

2. Unified Interaction Parameter Model for Mn–Fe–Si–C–Ca–P System

The detail description about the UIPM was shown in the literatures.^71,72) When used to the Mn–Fe–Si–C–Ca–P manganese ferroalloy system, Mn is taken as solvent and Fe, Si, C, Ca and P are taken as solutes. Both the first-order and second-order interaction parameters are considered for Fe, Si and C, while only the first-order interaction parameters are considered for Ca and P due to their low contents. The activity coefficients are represented as follows:   

$$\begin{array}{l}ln{\gamma }_{\text{Mn}}=-\frac{1}{2}({\epsilon }_{\text{SiSi}}{X}_{\text{Si}}^2+{\epsilon }_{\text{FeFe}}{X}_{\text{Fe}}^2+{\epsilon }_{\text{CC}}{X}_{\text{C}}^2+{\epsilon }_{\text{CaCa}}{X}_{\text{Ca}}^2\\ +{\epsilon }_{\text{PP}}{X}_{\text{P}}^2+2{\epsilon }_{\text{SiFe}}{X}_{\text{Si}}{X}_{\text{Fe}}+2{\epsilon }_{\text{SiC}}{X}_{\text{Si}}{X}_{\text{C}}+2{\epsilon }_{\text{SiCa}}{X}_{\text{Si}}{X}_{\text{Ca}}\\ +2{\epsilon }_{\text{SiP}}{X}_{\text{Si}}{X}_{\text{P}}+2{\epsilon }_{\text{FeC}}{X}_{\text{Fe}}{X}_{\text{C}}+2{\epsilon }_{\text{FeCa}}{X}_{\text{Fe}}{X}_{\text{Ca}}+2{\epsilon }_{\text{FeP}}{X}_{\text{Fe}}{X}_{\text{P}}\\ +2{\epsilon }_{\text{CCa}}{X}_{\text{C}}{X}_{\text{Ca}}+2{\epsilon }_{\text{CP}}{X}_{\text{C}}{X}_{\text{P}}+2{\epsilon }_{\text{CaP}}{X}_{\text{Ca}}{X}_{\text{P}})-\frac{2}{3}({\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^3\\ +{\epsilon }_{\text{FeFeFe}}{X}_{\text{Fe}}^3+{\epsilon }_{\text{CCC}}{X}_{\text{C}}^3+3{\epsilon }_{\text{SiSiFe}}{X}_{\text{Si}}^2{X}_{\text{Fe}}+3{\epsilon }_{\text{SiSiC}}{X}_{\text{Si}}^2{X}_{\text{C}}\\ +3{\epsilon }_{\text{SiFeFe}}{X}_{\text{Si}}{X}_{\text{Fe}}^2+6{\epsilon }_{\text{SiFeC}}{X}_{\text{Si}}{X}_{\text{Fe}}{X}_{\text{C}}+3{\epsilon }_{\text{SiCC}}{X}_{\text{Si}}{X}_{\text{C}}^2\\ +3{\epsilon }_{\text{FeFeC}}{X}_{\text{Fe}}^2{X}_{\text{C}}+3{\epsilon }_{\text{FeCC}}{X}_{\text{Fe}}{X}_{\text{C}}^2)\end{array}$$(1)

  

$$\begin{array}{l}ln{\gamma }_{\text{Si}}\text{=}ln{\gamma }_{\text{Si}}^0+ln{\gamma }_{\text{Mn}}+({\epsilon }_{\text{SiSi}}{X}_{\text{Si}}+{\epsilon }_{\text{SiFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{SiC}}{X}_{\text{C}}\\ +{\epsilon }_{\text{SiCa}}{X}_{\text{Ca}}+{\epsilon }_{\text{SiP}}{X}_{\text{P}})+({\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^2+2{\epsilon }_{\text{SiSiFe}}{X}_{\text{Si}}{X}_{\text{Fe}}\\ +2{\epsilon }_{\text{SiSiC}}{X}_{\text{Si}}{X}_{\text{C}}+{\epsilon }_{\text{SiFeFe}}{X}_{\text{Fe}}^2+2{\epsilon }_{\text{SiFeC}}{X}_{\text{Fe}}{X}_{\text{C}}+{\epsilon }_{\text{SiCC}}{X}_{\text{C}}^2)\end{array}$$(2)

  

$$\begin{array}{l}ln{\gamma }_{\text{Fe}}\text{=}ln{\gamma }_{\text{Fe}}^0+ln{\gamma }_{\text{Mn}}+({\epsilon }_{\text{FeSi}}{X}_{\text{Si}}+{\epsilon }_{\text{FeFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{FeC}}{X}_{\text{C}}\\ +{\epsilon }_{\text{FeCa}}{X}_{\text{Ca}}+{\epsilon }_{\text{FeP}}{X}_{\text{P}})+({\epsilon }_{\text{FeFeFe}}{X}_{\text{Fe}}^2+2{\epsilon }_{\text{FeFeSi}}{X}_{\text{Fe}}{X}_{\text{Si}}\\ +2{\epsilon }_{\text{FeFeC}}{X}_{\text{Fe}}{X}_{\text{C}}+{\epsilon }_{\text{FeSiSi}}{X}_{\text{Si}}^2+2{\epsilon }_{\text{FeSiC}}{X}_{\text{Si}}{X}_{\text{C}}+{\epsilon }_{\text{FeCC}}{X}_{\text{C}}^2)\end{array}$$(3)

  

$$\begin{array}{l}ln{\gamma }_{\text{C}}\text{=}ln{\gamma }_{\text{C}}^0+ln{\gamma }_{\text{Mn}}+({\epsilon }_{\text{CSi}}{X}_{\text{Si}}+{\epsilon }_{\text{CFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{CC}}{X}_{\text{C}}\\ +{\epsilon }_{\text{CCa}}{X}_{\text{Ca}}+{\epsilon }_{\text{CP}}{X}_{\text{P}})+({\epsilon }_{\text{CCC}}{X}_{\text{C}}^2+2{\epsilon }_{\text{CCSi}}{X}_{\text{C}}{X}_{\text{Si}}\\ +2{\epsilon }_{\text{CCFe}}{X}_{\text{C}}{X}_{\text{Fe}}+{\epsilon }_{\text{CSiSi}}{X}_{\text{Si}}^2+2{\epsilon }_{\text{CSiFe}}{X}_{\text{Si}}{X}_{\text{Fe}}+{\epsilon }_{\text{CFeFe}}{X}_{\text{Fe}}^2)\end{array}$$(4)

  

$$\begin{array}{l}ln{\gamma }_{\text{Ca}}\text{=}ln{\gamma }_{\text{Ca}}^0+ln{\gamma }_{\text{Mn}}\\ +\hspace{0.17em}\hspace{0.17em}({\epsilon }_{\text{CaSi}}{X}_{\text{Si}}+{\epsilon }_{\text{CaFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{CaC}}{X}_{\text{C}}+{\epsilon }_{\text{CaCa}}{X}_{\text{Ca}}+{\epsilon }_{\text{CaP}}{X}_{\text{P}})\end{array}$$(5)

  

$$\begin{array}{l}ln{\gamma }_{\text{P}}\text{=}ln{\gamma }_{\text{P}}^0+ln{\gamma }_{\text{Mn}}\\ +\hspace{0.17em}\hspace{0.17em}({\epsilon }_{\text{PSi}}{X}_{\text{Si}}+{\epsilon }_{\text{PFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{PC}}{X}_{\text{C}}+{\epsilon }_{\text{PCa}}{X}_{\text{Ca}}+{\epsilon }_{\text{PP}}{X}_{\text{P}})\end{array}$$(6)

where γ_i is the activity coefficients; ${\gamma }_i^0$ is the activity coefficient of solute i at infinite dilution; ε_ij and ε_ijk are the first-order and second-order interaction parameters respectively. It should be noted that ε_ij=ε_ji and ε_ijk=ε_ikj=ε_jik=ε_jki=ε_kij=ε_kji. The values of $ln\mathrm{\hspace{0.17em} }{\gamma }_i^0$ , ε_ij and ε_ijk obtained in the present paper were shown in Tables 1, 2, 3 respectively. The standard states for Fe, Si, C, Ca and P were pure iron, pure silicon, graphite, liquid pure Ca and liquid pure P, respectively. The evaluation process was given in the next section.

Table 1. The values of $ln\mathrm{\hspace{0.17em} }{\gamma }_i^0$ in liquid Mn.

i	$ln\mathrm{\hspace{0.17em} }{\gamma }_i^0$
Fe	679.33/T
Si	−13431.02/T
C	3582.26/T−3.12
Ca	10159.98/T
P	−24491.39/T+4.93

Table 2. The values of first-order interaction parameters for liquid Mn–Fe–Si–C–Ca–P system.

ij	Fe	Si	C	Ca	P
Fe	−1358.62/T	8931.29/T	10606.95/T	6183.63/T	188.76/T
Si		29582.57/T	20134.76/T	−7573.57/T	28223.97/T
C			−43864.22/T+25.31	17055.78/T	23160.21/T
Ca				−57342.38/T	44437.74/T
P					9791.13/T+10.43

Table 3. The values of second-order interaction parameters for liquid Mn–Fe–Si–C–Ca–P system.

ijk	εijk
FeFeFe	0
SiSiSi	5752.92/T
CCC	182756.04/T−67.47
FeFeSi	8909.77/T
FeSiSi	−19300.95/T
FeFeC	−3842.88/T
FeCC	−11299.88/T
SiSiC	22845.70/T
SiCC	32789.92/T
SiFeC	62008.42/T−48.70

3. Results

3.1. Evaluation of $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{F}\mathrm{e}}^0$ , ε_FeFe and ε_FeFeFe for Mn–Fe Binary System

Generally, Fe–Mn melts were treated as an approximately ideal solution all over the composition range.^{35,69,74,75,76,77)} However, it seemed likely that Fe–Mn solutions might show slight non-ideality from the viewpoint of the release of 3d electrons for valence bonding.⁷⁸⁾ The activities of Mn in Fe–Mn melts were measured by the EMF method (electromagnetic force method) using liquid electrolyte galvanic cells²⁷⁾ or solid electrolyte galvanic cells,^28,29) the vapor transport method^30,31,32) and the isobaric method.³³⁾ The results of different works were shown in Fig. 1.

Fig. 1.

Activity coefficient of manganese in Fe–Mn melts.

The results of EMF might be affected by the concentration polarization at the alloy electrode, penetration of Mn through the wall of thoria electrolyte tube and the reaction between Mn and the molten oxide in the electrolyte. The results obtained by the vapor transport method might be influenced by the decrease of surface concentration of Mn in Mn–Fe melts and the condensation of Mn on the cooler surface of the ceramic enclosure. Therefore, in the present paper, the work of Mukai et al.³³⁾ measured by isobaric method was used to evaluate the UIPM parameters for Mn–Fe system. The work of Mukai et al.³³⁾ was also recommended by the Japan Society for the Promotion of Science.⁷⁹⁾ The parameters at 1843 K were derived:   

$$ln{\gamma }_{\text{Fe}}^0=0.37;\hspace{1em}\hspace{1em}{\epsilon }_{\text{FeFe}}=-0.74;\hspace{1em}\hspace{1em}{\epsilon }_{\text{FeFeFe}}=0$$(7)

Figure 2 shows the comparison of the calculated ln γ_Mn with the experimental data. The triangle points denote the experimental data of ln γ_Mn by Mukai et al.³³⁾ respectively, and the dash line denote the calculated ln γ_Mn by the present parameters. The calculated results agree well with the experimental data.

Fig. 2.

Comparison of model calculation with experiments of ln γ_Mn in Mn–Fe melt.

Since the temperature under concern is not only 1843 K, it is desirable to assume a temperature-dependent interaction coefficient based on the regular solution.⁸⁰⁾ The relationship can be written as:   

$$T_{1}ln{\gamma }_i^0{\hspace{0.17em}}_{({\mathrm{T}}_1)}=T_{2}ln{\gamma }_i^0{\hspace{0.17em}}_{({\mathrm{T}}_2)};\hspace{1em}{T}_1{\epsilon }_{ij({\mathrm{T}}_1)}=T_2{\epsilon }_{ij({\mathrm{T}}_2)};\hspace{1em}{T}_1{\epsilon }_{ijk({\mathrm{T}}_1)}=T_2{\epsilon }_{ijk({\mathrm{T}}_2)}$$(8)

Thus, the following parameters were gotten:   

$$ln{\gamma }_{\text{Fe}}^0=678.9612/T;\hspace{1em}{\epsilon }_{\text{FeFe}}=-1\hspace{0.17em}\hspace{0.17em}357.92/T;\hspace{1em}{\epsilon }_{\text{FeFeFe}}=0$$(9)

The assumption of regular solution was also adopted by Li and Morris,⁷⁶⁾ in whose work UIPM was applied to evaluated the Mn–Fe–C–Si system. This approximate treatment was widely used to evaluate the effect of temperature when there are only experimental data at a specific temperature. This assumption was also applied to other binary system in the following sections.

3.2. Evaluation of $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{S}\mathrm{i}}^0$ , ε_SiSi and ε_SiSiSi for Mn–Si Binary System

Activities of manganese in Mn–Si melts could be measured by different authors employing the vapor transport method,^34,35,36) torsion-effusion method^37,38) and EMF,³⁹⁾ and the activities of silicon were obtained from the equilibrium between MnO bearing slags and Mn–Si melts,^40,41,73) or derived from the measured manganese activity and the Gibbs-Duhem relationship. The results obtained by different authors are in good agreement with each other.

Firstly, interaction parameters ε_SiSi and ε_SiSiSi were evaluated from the activity data of Mn for compositions with the silicon content less than 0.5. According to Eq. (10), the values of ε_SiSi and ε_SiSiSi around 1700 K were optimized to be 17.40 and 3.38 respectively. With these two parameters, the calculated activity coefficients of Mn at 1700 K are shown in Fig. 3, which agree well with the experimental data.   

$$ln{\gamma }_{\text{Mn}}=-\frac{1}{2}{\epsilon }_{\text{SiSi}}{X}_{\text{Si}}^2-\frac{2}{3}{\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^3$$(10)

Fig. 3.

Comparison of model calculation with experiments of ln γ_Mn in Mn–Si melt.

And then, $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{S}\mathrm{i}}^0$ could be evaluated to be −7.90 from the activity data of Si by Eq. (11).   

$$ln{\gamma }_{\text{Si}}\text{=}ln{\gamma }_{\text{Si}}^0+ln{\gamma }_{\text{Mn}}+{\epsilon }_{\text{SiSi}}{X}_{\text{Si}}+{\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^2$$(11)

Figure 4 shows the comparison of calculated ln γ_Si with the literature data. It should be noted that the evaluated parameters was valid only when the silicon content is less than 0.5. The activity coefficient of Si approached unity as the silicon content is greater than 0.8.

Fig. 4.

Comparison of model calculation with experiments of ln γ_Si in Mn–Si melt.

The values of $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{S}\mathrm{i}}^0$ , ε_SiSi and ε_SiSiSi mentioned above were valid around about 1700 K. The effects of temperature on the parameters were assumed as following according to Eq. (8):   

$$\begin{array}{l}ln{\gamma }_{\text{Si}}^0\text{=}-13\hspace{0.17em}\hspace{0.17em}431.02/\text{T};\hspace{1em}\hspace{1em}{\epsilon }_{\text{SiSi}}=29\hspace{0.17em}\hspace{0.17em}582.57/T;\\ {\epsilon }_{\text{SiSiSi}}=5\hspace{0.17em}\hspace{0.17em}752.92/T\end{array}$$(12)

3.3. Evaluation of $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}}^0$ , ε_CC and ε_CCC for Mn–C Binary System

The activity of Mn in Mn–C melts was measured by the vapor transport method,^34,35,36) or calculated according to the equilibrium established between Mn–C melts and Ar–CO atmosphere.^48,49) In the latter case, the activity of C was also obtained. The solubility of C in Mn melt was measured in a graphite at different temperatures in an inert atmosphere. The inert atmosphere should not be nitrogen due to the high solubility of N in Mn-based melt.⁸¹⁾ Consequently, the work of Turkdogan et al.⁴³⁾ was not considered in the present paper. The data of carbon solubility^{35,36,44,45,46,47,48,49,50,51,62)} were considered as the saturated carbon content in manganese melt, and the relationship between carbon solubility and temperature was given as   

$$ln{X}_{\text{C}}sat=-867.64/T-0.78$$(13)

The values of ln γ_C from the work of Kim et al.⁴⁹⁾ were used to evaluate $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}}^0$ , ε_CC and ε_CCC. And then, the value of $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}}^0$ was revised by the values of ε_CC and ε_CCC using the carbon content at saturation. When carbon is saturated in Mn–C melts, the activity of carbon is unity. That means   

$$X_{\text{C}}sat=\frac{1}{{\gamma }_{\text{C}}sat}$$(14)

  

$$\mathrm{l}\mathrm{n}{X}_{\text{C}}sat=-ln{\gamma }_{\text{C}}sat$$(15)

  

$$\begin{array}{l}ln{\gamma }_{\text{C}}^0\text{=}-\mathrm{l}\mathrm{n}{X}_{\text{C}}sat-{\epsilon }_{\text{CC}}{X}_{\text{C}}sat-{\epsilon }_{\text{CCC}}{X}_{\text{C}}^{2}sat\\ +\frac{1}{2}{\epsilon }_{\text{CC}}{X}_{\text{C}}^{2}sat+\frac{2}{3}{\epsilon }_{\text{CCC}}{X}_{\text{C}}^{3}sat\end{array}$$(16)

Therefore, the parameters of Mn–C binary system were optimized to be:   

$$ln{\gamma }_{\text{C}}^0=3\hspace{0.17em}\hspace{0.17em}582.26/T-3.12$$(17)

  

$${\epsilon }_{\text{CC}}=-43\hspace{0.17em}\hspace{0.17em}864.22/T+25.31$$(18)

  

$${\epsilon }_{\text{CCC}}=182\hspace{0.17em}\hspace{0.17em}756.04/T-67.47$$(19)

Figure 5 shows the comparisons of calculated ln γ_C with the literature data. The calculated values agree well with the work of Kim et al.⁴⁹⁾ It should be noted that the values by Katsnelson et al.⁴⁸⁾ at carbon content less than 0.114 were extrapolated from their model whose parameters were obtained in the carbon content range of 0.114 to 0.269. It means that their data in the low carbon range might not be accurate. Figure 6 shows the comparison of calculated ln γ_Mn at 1673 K and 1773 K with the literature data in Mn–C melt. The calculated results showed that temperature had a weak effect on the activity coefficient of manganese in Mn–C melt.

Fig. 5.

Comparison of model calculation with experiments of ln γ_C in Mn–C melt.

Fig. 6.

Comparison of model calculation with experiments of ln γ_Mn in Mn–C melt.

3.4. Evaluation of ε_FeSi, ε_FeFeSi and ε_FeSiSi for Mn–Fe–Si Ternary System

The activity of Mn increases as the ratio of X_Fe/X_Si increases when X_Mn keeps constant.^52,53) Because the manganese content is less than 0.3 in the work of Zaitsev et al.,⁵³⁾ only the work of Gee and Rosenqvist⁵²⁾ was taken into consideration to evaluate ε_FeSi, ε_FeFeSi and ε_FeSiSi. In Mn–Fe–Si system, the activity coefficient of Mn could be represented as:   

$$\begin{array}{l}ln{\gamma }_{\text{Mn}}=-\frac{1}{2}({\epsilon }_{\text{SiSi}}{X}_{\text{Si}}^2+{\epsilon }_{\text{FeFe}}{X}_{\text{Fe}}^2+2{\epsilon }_{\text{SiFe}}{X}_{\text{Si}}{X}_{\text{Fe}})\\ -\frac{2}{3}({\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^3+{\epsilon }_{\text{FeFeFe}}{X}_{\text{Fe}}^3+3{\epsilon }_{\text{SiSiFe}}{X}_{\text{Si}}^2{X}_{\text{Fe}}+3{\epsilon }_{\text{SiFeFe}}{X}_{\text{Si}}{X}_{\text{Fe}}^2)\end{array}$$(20)

Substituting the experimental data at 1700 K from Ref. 52) and the values of ε_FeFe, ε_FeFeFe, ε_SiSi and ε_SiSiSi into Eq. (20), the values of ε_FeSi, ε_FeFeSi and ε_FeSiSi are obtained. Considering the effect of temperature, the parameters were assumed as following according to Eq. (8):   

$$\begin{array}{l}{\epsilon }_{\text{FeSi}}=8\hspace{0.17em}\hspace{0.17em}931.29/T;\hspace{1em}\hspace{1em}{\epsilon }_{\text{FeFeSi}}=8\hspace{0.17em}\hspace{0.17em}909.77/T;\\ {\epsilon }_{\text{FeSiSi}}=-19\hspace{0.17em}\hspace{0.17em}300.95/T\end{array}$$(21)

The comparison of calculated ln γ_Mn with the literature data is shown in Fig. 7. The present calculations agree well the experimental data.

Fig. 7.

Comparison of model calculation with experiments of ln γ_Mn in Mn–Fe–Si melt.

3.5. Evaluation of ε_FeC, ε_FeFeC and ε_FeCC for Mn–Fe–C Ternary System

The solubility of C in Mn–Fe melts was used to evaluate ε_FeC, ε_FeFeC and ε_FeCC for Mn–Fe–C system. Due to the high solubility of N in Mn-based melt,⁸¹⁾ the works of Turkdogan et al.⁵⁵⁾ and Chen et al.⁶⁰⁾ were not taken into consideration because of the use of nitrogen as the protecting gas. According to Eqs. (15) and (22), the values of ε_FeC, ε_FeFeC and ε_FeCC were obtained with the known parameters $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}}^0$ , ε_CC, ε_CCC, ε_FeFe and ε_FeFeFe and the solubility data from Ni et al.,⁴⁵⁾ Ma et al.,⁴⁶⁾ Kim et al.,⁴⁹⁾ Chipman et al.,⁵⁴⁾ Paek et al.⁵⁶⁾ and Fenstad.⁵⁹⁾   

$$\begin{array}{l}ln{\gamma }_{\text{C}}\text{=}ln{\gamma }_{\text{C}}^0+({\epsilon }_{\text{CFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{CC}}{X}_{\text{C}})+({\epsilon }_{\text{CCC}}{X}_{\text{C}}^2+2{\epsilon }_{\text{CCFe}}{X}_{\text{C}}{X}_{\text{Fe}}\\ +{\epsilon }_{\text{CFeFe}}{X}_{\text{Fe}}^2)-\frac{1}{2}({\epsilon }_{\text{FeFe}}{X}_{\text{Fe}}^2+{\epsilon }_{\text{CC}}{X}_{\text{C}}^2+2{\epsilon }_{\text{FeC}}{X}_{\text{Fe}}{X}_{\text{C}})\\ -\frac{2}{3}({\epsilon }_{\text{FeFeFe}}{X}_{\text{Fe}}^3+{\epsilon }_{\text{CCC}}{X}_{\text{C}}^3+3{\epsilon }_{\text{FeFeC}}{X}_{\text{Fe}}^2{X}_{\text{C}}+3{\epsilon }_{\text{FeCC}}{X}_{\text{Fe}}{X}_{\text{C}}^2)\end{array}$$(22)

The obtained parameters were given as follows:   

$$\begin{array}{l}{\epsilon }_{\text{FeC}}\text{=}10\hspace{0.17em}\hspace{0.17em}606.95/\text{T};\hspace{1em}\hspace{1em}{\epsilon }_{\text{FeFeC}}\text{=}-3\hspace{0.17em}\hspace{0.17em}842.88/T;\\ {\epsilon }_{\text{FeCC}}\text{=}-11\hspace{0.17em}\hspace{0.17em}299.88/T\end{array}$$(23)

The validity of parameters was tested by calculating the carbon solubility. The comparisons of calculated values with the measured ones are shown in Fig. 8. It could be seen that present calculations reproduced the experimental data reasonably well. The carbon solubility had a linearly decreasing relationship with Fe content at different temperature. The effect of temperature and Fe content on the carbon solubility in Mn–Fe melts could be expressed as following:   

$$X_{\text{C}}sat=(-229.18/T+0.411)+(-186.54/T+0.014)X_{\text{Fe}}$$(24)

Fig. 8.

Effect of Fe content on solubility of C in Mn–Fe–C melt.

3.6. Evaluation of ε_SiC, ε_SiSiC and ε_SiCC for Mn–Si–C Ternary System

The evaluation of ε_SiC, ε_SiSiC and ε_SiCC in Mn–Si–C system was similar to the evaluation of ε_FeC, ε_FeFeC and ε_FeCC in Mn–Fe–C ternary system. The values of ε_SiC, ε_SiSiC and ε_SiCC were calculated according to Eqs. (15) and (25).   

$$\begin{array}{l}ln{\gamma }_{\text{C}}\text{=}ln{\gamma }_{\text{C}}^0+({\epsilon }_{\text{CSi}}{X}_{\text{Si}}+{\epsilon }_{\text{CC}}{X}_{\text{C}})+({\epsilon }_{\text{CCC}}{X}_{\text{C}}^2+2{\epsilon }_{\text{CCSi}}{X}_{\text{C}}{X}_{\text{Si}}\\ +{\epsilon }_{\text{CSiSi}}{X}_{\text{Si}}^2)-\frac{1}{2}({\epsilon }_{\text{SiSi}}{X}_{\text{Si}}^2+{\epsilon }_{\text{CC}}{X}_{\text{C}}^2+2{\epsilon }_{\text{SiC}}{X}_{\text{Si}}{X}_{\text{C}})\\ -\frac{2}{3}({\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^3+{\epsilon }_{\text{CCC}}{X}_{\text{C}}^3+3{\epsilon }_{\text{SiSiC}}{X}_{\text{Si}}^2{X}_{\text{C}}+3{\epsilon }_{\text{SiCC}}{X}_{\text{Si}}{X}_{\text{C}}^2)\end{array}$$(25)

With the known parameters $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}}^0$ , ε_CC, ε_CCC, ε_SiSi and ε_SiSiSi and the solubility data from Ni et al.,⁴⁵⁾ Ma et al.,⁴⁶⁾ Paek et al.,⁵⁶⁾ Abraham et al.,⁶¹⁾ Tang et al.,⁶²⁾ Turkdogan et al.⁶³⁾ and Ding et al.,⁶⁴⁾ ε_SiC, ε_SiSiC and ε_SiCC were obtained.   

$$\begin{array}{l}{\epsilon }_{\text{SiC}}=20\hspace{0.17em}\hspace{0.17em}134.76/T;\hspace{1em}\hspace{1em}{\epsilon }_{\text{SiSiC}}=22\hspace{0.17em}\hspace{0.17em}845.70/T;\\ {\epsilon }_{\text{SiCC}}=32\hspace{0.17em}\hspace{0.17em}789.92/T\end{array}$$(26)

The validity of these parameters was checked by calculating the carbon solubility in Mn–Si–C melts at different temperatures and silicon contents. As shown in Fig. 9, the present parameters reproduced the experimental data well.

Fig. 9.

Effect of Si content on solubility of C in Mn–Si–C melt.

3.7. Evaluation of $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}\mathrm{a}}^0$ , ε_CaCa, ε_CaFe, ε_CaSi and ε_CaC for Mn–i–Ca Ternary System

When Ca is saturated in Mn–i–Ca (i=Fe, Si or C) melts, the activity of carbon is unity. That means   

$$X_{\text{Ca}}sat=\frac{1}{{\gamma }_{\text{Ca}}sat}$$(27)

  

$$\mathrm{l}\mathrm{n}{X}_{\text{Ca}}sat=-ln{\gamma }_{\text{Ca}}sat$$(28)

  

$$\begin{array}{l}\mathrm{l}\mathrm{n}{X}_{\text{Ca}}sat\text{=}-ln{\gamma }_{\text{Ca}}^0-{\epsilon }_{\text{CaCa}}{X}_{\text{Ca}}sat-{\epsilon }_{\text{Ca}i}{X}_i\\ +\frac{1}{2}({\epsilon }_{ii}{X}_i^2+{\epsilon }_{\text{CaCa}}{X}_{\text{Ca}}^{2}sat+2{\epsilon }_{\text{Ca}i}{X}_{\text{Ca}}{X}_i)+\frac{2}{3}{\epsilon }_{iii}{X}_i^3\end{array}$$(29)

The Mn–i–Ca system was treated in the following way. 6.26⁶⁶⁾ was taken as the value of $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}\mathrm{a}}^0$ at 1623 K. And the saturated solubility of Ca in Mn melt X_Casat is 2.055×10⁻³ at 1623 K.^46,66,67) Thus, the value −35.33 of ε_CaCa was obtained according to Eqs. (28) and (29) with the known $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}\mathrm{a}}^0$ and X_Casat. Treated as regular solution, the effects of temperature on the parameters were assumed as following according to Eq. (8):   

$$ln{\gamma }_{\text{Ca}}^0=10\hspace{0.17em}\hspace{0.17em}159.98/T;\hspace{1em}\hspace{1em}{\epsilon }_{\text{CaCa}}=-57\hspace{0.17em}\hspace{0.17em}342.38/T$$(30)

And then, parameters of ε_CaFe, ε_CaSi and ε_CaC are obtained according to Eqs. (28) and (29) with the known $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}\mathrm{a}}^0$ and ε_CaCa, and the experimental data from Ma et al.,⁴⁶⁾ Cheng⁶⁶⁾ and Wei et al.⁶⁷⁾ Treated like ε_CaCa, the effects of temperature on these parameters were   

$$\begin{array}{l}{\epsilon }_{\text{CaFe}}=6\hspace{0.17em}\hspace{0.17em}183.63/T;\hspace{1em}\hspace{1em}{\epsilon }_{\text{CaSi}}=-7\hspace{0.17em}\hspace{0.17em}573.57/T\\ {\epsilon }_{\text{CaC}}=17\hspace{0.17em}\hspace{0.17em}055.78/T\end{array}$$(31)

The validity of these parameters was checked by calculating the Ca solubility in Mn–i–Ca melts. As shown in Figs. 10 and 11, parameters in the present paper could reproduce well the experimental data.

Fig. 10.

Effect of Fe content on solubility of Ca in Mn–Fe–Ca melt.

Fig. 11.

Effect of Si content on solubility of Ca in Mn–Si–Ca melt.

3.8. Evaluation of $\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{P}}^0$ , ε_PP, ε_PFe, ε_PSi, ε_PC and ε_PCa for Mn–i–P Ternary System

There are only few experimental investigations about the thermodynamic properties of Mn–P system.⁶⁹⁾ However, thermodynamic parameters were not given in these reports. Lee⁸²⁾ derived the solution properties of Mn–P system based on the available thermodynamic information and phase diagram. Therefore, the results of Lee⁸²⁾ were used to evaluate $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{P}}^0$ and ε_PP for Mn–P system at the phosphorus content less than 0.005. Generally, X_P is less than 0.005 in manganese ferroalloys. The parameters of $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{P}}^0$ and ε_PP are   

$$ln{\gamma }_{\text{P}}^0=-24\hspace{0.17em}\hspace{0.17em}491.39/T+4.93;\hspace{1em}{\epsilon }_{\text{PP}}=9\hspace{0.17em}\hspace{0.17em}791.13/T+10.43$$(32)

The other P related parameters ε_PSi and ε_PCa were taken from Ref. 68), while the parameters ε_PFe and ε_PC were taken from Ref. 83). Using the regular solution concept, the effects of temperature on the parameters were assumed as following according to Eq. (8):   

$${\epsilon }_{\text{PSi}}=28\hspace{0.17em}\hspace{0.17em}223.97/T;\hspace{1em}\hspace{1em}{\epsilon }_{\text{PCa}}=44\hspace{0.17em}\hspace{0.17em}437.74/T$$(33)

  

$${\epsilon }_{\text{PC}}=23\hspace{0.17em}\hspace{0.17em}160.21/T;\hspace{1em}\hspace{1em}{\epsilon }_{\text{PFe}}=188.76/T$$(34)

3.9. Evaluation of ε_FeSiC for Mn–Fe–Si–C Quaternary System

For Mn–Fe–Si–C system, it needs 17 parameters to calculate the activity coefficient of carbon. From the evaluation mentioned above, parameters of $ln\mathrm{\hspace{0.17em} }{\gamma }_{\mathrm{C}}^0$ , ε_CC, ε_CCC, ε_FeFe, ε_FeFeFe, ε_SiSi, ε_SiSiSi, ε_SiC, ε_SiSiC, ε_SiCC, ε_FeC, ε_FeFeC, ε_FeCC, ε_FeSi, ε_FeFeSi and ε_FeSiSi has been evaluated, and only ε_SiFeC is unknown. The value of ε_SiFeC was calculated according to Eqs. (15) and (35), with the known parameters and experimental data from Paek et al.,⁵⁶⁾ Turkdogan et al.,⁶³⁾ Ding et al.⁶⁴⁾ and Swinbourne et al.⁶⁵⁾ The expression of ε_SiFeC was shown in Eq. (36).   

$$\begin{array}{l}ln{\gamma }_{\text{C}}\text{=}ln{\gamma }_{\text{C}}^0+({\epsilon }_{\text{CSi}}{X}_{\text{Si}}+{\epsilon }_{\text{CFe}}{X}_{\text{Fe}}+{\epsilon }_{\text{CC}}{X}_{\text{C}})+({\epsilon }_{\text{CCC}}{X}_{\text{C}}^2\\ +2{\epsilon }_{\text{CCSi}}{X}_{\text{C}}{X}_{\text{Si}}+2{\epsilon }_{\text{CCFe}}{X}_{\text{C}}{X}_{\text{Fe}}+{\epsilon }_{\text{CSiSi}}{X}_{\text{Si}}^2+2{\epsilon }_{\text{CSiFe}}{X}_{\text{Si}}{X}_{\text{Fe}}\\ +{\epsilon }_{\text{CFeFe}}{X}_{\text{Fe}}^2)-\frac{1}{2}({\epsilon }_{\text{SiSi}}{X}_{\text{Si}}^2+{\epsilon }_{\text{FeFe}}{X}_{\text{Fe}}^2+{\epsilon }_{\text{CC}}{X}_{\text{C}}^2\\ +2{\epsilon }_{\text{SiFe}}{X}_{\text{Si}}{X}_{\text{Fe}}+2{\epsilon }_{\text{SiC}}{X}_{\text{Si}}{X}_{\text{C}}+2{\epsilon }_{\text{FeC}}{X}_{\text{Fe}}{X}_{\text{C}})-\frac{2}{3}({\epsilon }_{\text{SiSiSi}}{X}_{\text{Si}}^3\\ +{\epsilon }_{\text{FeFeFe}}{X}_{\text{Fe}}^3+{\epsilon }_{\text{CCC}}{X}_{\text{C}}^3+3{\epsilon }_{\text{SiSiFe}}{X}_{\text{Si}}^2{X}_{\text{Fe}}+3{\epsilon }_{\text{SiSiC}}{X}_{\text{Si}}^2{X}_{\text{C}}\\ +3{\epsilon }_{\text{SiFeFe}}{X}_{\text{Si}}{X}_{\text{Fe}}^2+6{\epsilon }_{\text{SiFeC}}{X}_{\text{Si}}{X}_{\text{Fe}}{X}_{\text{C}}+3{\epsilon }_{\text{SiCC}}{X}_{\text{Si}}{X}_{\text{C}}^2\\ +3{\epsilon }_{\text{FeFeC}}{X}_{\text{Fe}}^2{X}_{\text{C}}+3{\epsilon }_{\text{FeCC}}{X}_{\text{Fe}}{X}_{\text{C}}^2)\end{array}$$(35)

  

$${\epsilon }_{\text{SiFeC}}=62\hspace{0.17em}\hspace{0.17em}008.42/T-48.70$$(36)

4. Discussions

4.1. Comparisons with Other Modelling Works

There were some studies about the optimizations of UIPM parameters in Mn–Si,⁷³⁾ Mn–Fe–C,⁷⁴⁾ Mn–Fe–C–N⁷⁵⁾ and Mn–Fe–Si–C⁷⁶⁾ systems. In the works of Lee⁷⁴⁾ and Li et al.,⁷⁶⁾ Fe and Mn were assumed to form ideal solutions. Following this assumption, both the activity coefficients of Mn and Fe will be unit for all the compositions in Mn–Fe system. But, Fe–Mn solutions showed slight positive deviation from deal solutions, which were confirmed by the works of Sanbongi et al.,²⁷⁾ Schwerdtfeger,²⁸⁾ Steiler et al.,³¹⁾ Arita et al.³²⁾ and Mukai et al.,³³⁾ as shown in Fig. 1. After the modelling work of Li et al.,⁷⁶⁾ many experimental works were published, such as the works of Kim et al.,⁴⁹⁾ Dashevskii et al.,⁵⁰⁾ Fenstad et al.,⁵¹⁾ Paek et al.,⁵⁶⁾ Fenstad⁶¹⁾ and Tang et al.⁶²⁾ In the present paper, UIPM was extended to the Mn–Fe–Si–C–Ca–P system by considering much more available experimental data to get a better estimation effect. Taking the activity coefficient calculation of C in Mn–C melts for example, the present calculation results agreed well with the measured values by Kim et al.⁴⁹⁾ However, the calculation results by the parameters of Li et al.⁷⁶⁾ were was not so good, as shown in Fig. 12.

Fig. 12.

Comparison of model calculations of ln γ_C in Mn–C melt.

4.2. Test of Evaluated Parameters

4.2.1. Activity of Manganese in Carbon Saturated Mn–Si–C and Mn–Fe–Si–C System

The data of manganese activity in carbon saturated Mn–Si–C and Mn–Fe–Si–C system were not adopted to optimize the UIPM parameters in the present paper. Therefore, those data were used to check the validity of the model parameters.

Figure 13 shows the effect of Si content on activity of Mn in carbon saturated Mn–Si–C melt. In the work of Tanaka,³⁵⁾ activity of manganese rapidly increased initially as the silicon content increased, and decreased after reaching a maximum value. However, activity of manganese monotonically decreased as the silicon content increased in the present calculation. The trend is similar to the works of Aida et al.,²³⁾ Gee and Rosenqvist,³⁴⁾ Abraham et al.⁶¹⁾ and Tang et al.⁶²⁾ Activity of manganese slightly decreased as the temperature increased, as shown in Fig. 13.

Fig. 13.

Effect of Si content on activity of Mn in carbon saturated Mn–Si–C melt.

Manganese activity in carbon saturated Mn–Fe–Si–C system was determined by Tanaka³⁵⁾ at 1673 K using the vapor pressure measurement method. The saturated carbon contents were calculated firstly according to the iron and silicon contents taken from the work of Tanaka³⁵⁾ and the present UIPM parameters because they were not given in Ref. 35). And then, manganese activities were calculated with the parameters and iron, silicon and carbon contents. In this work, the relative deviation between the calculated and measured activity, $\left(a_{\mathrm{M}\mathrm{n}}^{\mathrm{C}\mathrm{a}\mathrm{l}}-a_{\mathrm{M}\mathrm{n}}^{\mathrm{E}\mathrm{x}\mathrm{p}}\right)/a_{\mathrm{M}\mathrm{n}}^{\mathrm{E}\mathrm{x}\mathrm{p}}$ , was used to test the validity of the UIPM parameters. As shown in Fig. 14, although the points were rather scattered, most of the points (52/63) fell in the range of ± 40% and more than half the points (39/63) fell in the range of ± 20% in the following composition ranges: X_Si = 0.003 to 0.217, X_Fe = 0 to 0.232, carbon saturated, X_Mn = 0.475 to 0.725. The errors lager than ± 40% were mainly fell in the composition ranges: X_Si = 0.003 to 0.05, carbon saturated and X_Fe close to zero. The discrepancies in this composition ranges were similar to that in the carbon saturated Mn–Si–C system, as shown in Fig. 13.

Fig. 14.

Comparison of calculated a_Mn in carbon saturated Mn–Fe–Si–C melts with the literature data.

4.2.2. Equilibrium between Ca and P in Mn–Si–Ca–P Quaternary System

Effects of Si and Ca on P in Mn–Si–Ca–P melts equilibrated with CaO–CaF₂–Ca₃P₂–Ca slags were investigated by Ma et al.⁶⁸⁾ at 1623 K. When Ca and P were in equilibrium, there was:   

$${\text{(Ca}}_3{\text{P}}_2\text{)=3[Ca]}+2[\text{P]}\hspace{1em}\hspace{1em}{K}_{37}\times a_{{\text{Ca}}_3{\text{P}}_2}=a_{\text{Ca}}^3\times a_{\text{P}}^2$$(37)

Because Mn–Si–Ca–P melts containing different Si content shared the same slag, the product of $a_{\mathrm{C}\mathrm{a}}^3$ and $a_{\mathrm{P}}^2$ was constant. Figure 15 shows the Ca and P activity in Mn–Si–Ca–P melts using the experimental data and the present UIPM parameters. It could be seen that loga_P had a linearly decreasing relationship with 1.5loga_Ca. It meant that the product of $a_{\mathrm{C}\mathrm{a}}^3$ and $a_{\mathrm{P}}^2$ were constant. Points in Fig. 15 were somewhat scatter. It might be caused by the experimental data or the present UIPM parameters. The UIPM parameters involving Ca and P might not be optimized, because works were not enough about the thermodynamic of Ca and P in manganese based melts. It needed further optimization when there were more experimental data.

Fig. 15.

Relationship between Ca and P activities in Mn–Si–Ca–P melts.

4.3. Application of Evaluated Parameters

When Ca and its compounds were used for reducing de-P, the most important reaction was that:   

$$\begin{array}{l}\text{3[Ca]}+2[{\text{P]=(Ca}}_3{\text{P}}_2)\\ \hspace{1em}\hspace{1em}\hspace{0.5em}\mathrm{\Delta }{G}_{37}^0=-5\hspace{0.17em}\hspace{0.17em}124\hspace{0.17em}\hspace{0.17em}987.22+99.28T\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}\end{array}$$(38)

where the standard Gibbs energy was the extrapolation of the data from Ref. 84). When the reaction was in equilibrium, the relationship between Ca and P was that   

$$3ln{\gamma }_{\text{Ca}}+3ln{X}_{\text{Ca}}+2ln{\gamma }_{\text{P}}+2ln{X}_{\text{P}}-\mathrm{\Delta }{G}_{37}^0/RT=0$$(39)

where the activity of Ca₃P₂ was considered as unity. The activity coefficients of Ca and P were calculated from Eqs. (5) and (6). For carbon saturated Mn–Fe–Si–C system containing about 68 mass% Mn and 18 mass% Si, X_Si, X_Fe and X_C were about 0.2894, 0.1024 and 0.0494 respectively at 1673 K. For a given Ca content, P content was calculated with Si, Fe, C and Ca contents mentioned above using the present UIPM parameters. And then, Si, Fe, C and P contents were adjusted appropriately to meet the constraints that Mn and Si contents were 68 mass% and 18 mass% respectively. Figure 16 shows the equilibrium between Ca and P in carbon saturated Mn–Fe–Si–Csat–Ca–P melts containing about 68 mass% Mn and 18 mass% Si at 1673 K and 1773 K. P content decreased as the increase of Ca content. To keep the P content below 0.05 mass%, the Ca content was at least about 0.26 mass% at 1673 K, and correspondingly, it was 0.48 mass% at 1773 K. Then, the decrease of P content became slow as the Ca content further increased. Thus, a further increase of Ca content was not very economical for reducing de-P. Compared with 1773 K, 1673 K was more suitable for reducing de-P.

Fig. 16.

Equilibrium between Ca and P in carbon saturated Mn–Fe–Si–Csat–Ca–P melts (Mn: 68 mass%, Si: 18 mass%).

5. Conclusions

The UIPM was used to evaluate the Mn–Fe–Si–C–Ca–P system. Totally 30 parameters were obtained by reasonable simplification. These parameters include the activity coefficient of solute i at infinite dilution, first-order and second-order interaction parameters. The regular solution model was also used to modify the relationships between parts of the parameters and temperature. These parameters could reproduce the carbon solubility in Mn, Mn–Fe and Mn–Si melts and Ca solubility in Mn–Fe, Mn–Si and Mn–C melts. The calculated results of the activity coefficient of Mn, Si and C agree well with the experimental data respectively. The validity of these parameters was also tested by the activity of manganese in carbon saturated Mn–Si–C and Mn–Fe–Si–C system and equilibrium between Ca and P in Mn–Si–Ca–P quaternary system.

These parameters could be applied to calculate the activities of Mn, Si, Fe, C, Ca and P and the equilibrium between Ca and P in carbon saturated Mn–Fe–Si–C–Ca–P melts during the de-P process. They are also useful for understanding the process of ferromanganese and silicomanganese.

Acknowledgement

The authors gratefully acknowledge the research funding from the National Natural Science Foundation of China (No.51274030), and the Fundamental Research Funds for the Central Universities of China (FRF-TP-15-009A3).

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