Retraction: A Prediction Model of Phosphorus Distribution between CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags and Liquid Iron

Abstract

CaO−SiO₂−MgO−FeO−Fe₂O₃−P₂O₅ slags is typical in the basic oxygen steelmaking(BOS) process. Phosphate distribution between slags and liquid iron is an index of the phosphorus holding capacity of the slag, which determines the phosphorus content achievable in the finished steel. In this study, a thermodynamic model for calculating phosphorus distribution between CaO−SiO₂−MgO−FeO−Fe₂O₃−P₂O₅ slags and liquid iron, i.e., IMCT−L_p model, has been developed coupled with a developed thermodynamic model for calculating mass action concentrations of structural units, i.e., IMCT−N_i model, based on ion and molecule coexistence theory. Simple binary basicity R have a large effect on equilibrium mole number Σn_i of structural units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags, the formula of equilibrium mole number Σn_i against the simple binary basicity R of slags can be regressed as Σn_i = 2.604 – 3.029*exp(–R/2.339), and the fitting degree is 0.995. The comparison of the calculated phosphorous distribution by IMCT−L_p model with the measured phosphorous distribution reported from different literatures shows that the agreement between the calculated phosphorous distribution and measured phosphorous distribution is good. Meanwhile, some other L_p prediction models have also been taken into consideration for calculating phosphate distribution between CaO−SiO₂−MgO−FeO−Fe₂O₃−P₂O₅ slags and liquid iron, and the results shows that IMCT−L_p model have more accuracy compared with other L_p prediction models. The developed IMCT−L_p model can quantitatively calculate the respective contribution of Fe_tO, CaO+Fe_tO and MgO+Fe_tO in the slags. A significant difference of dephosphorziation abilities among Fe_tO, CaO+Fe_tO and MgO+Fe_tO can be found as approximately 0.00%, 99.98%, 0.01%. Meanwhile, the phosphorus in liquid iron can be effectively extracted by CaO+Fe_tO in slags to form complex molecules 3CaO·P₂O₅ which made the main contribution to dephosphorizaiton in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags.

1. Introduction

Phosphorus can make the steel prone to embrittlement during heat treatment and cause degradation of electrical properties. It’s therefore useful to study suitable metal-slag reactions with the objective to remove phosphorus from steel melts below up-to-date limits. Many studies were carried out to measure phosphorous distribution between various slags and liquid iron.^1,2,3,4,5,6) Colclough^1,2) have made experiments involving the dephosphorization from the liquid iron in a basic open-hearth furnace, and he proposed that the phosphorus in liquid iron was mainly extracted in the form of tetra-calcium phosphate (4CaO·P₂O₅). Herty³⁾ carried out a series of experiments using an experimental furnace as well as 200 T basic open-hearth furnace and came to the conclusion that P₂O₅ formed tri-calcium phosphate (3CaO·P₂O₅), but not tetra-calcium phosphate (4CaO·P₂O₅). Considering the fact that the measurement of phosphorous distribution between molten slag and liquid iron was costly and difficult in the operation, several attempts^4,5,6,7,8) were made to describe and predict phosphorus partition ratios as a function of slag composition. Healy⁴⁾ conducted a series of experiments involving the measurement of phosphorus distribution between slags and liquid iron, and published the famous correlation for equilibrium phosphorus partition in a wide range of slag compositions and temperatures as $lg\frac{(\%\text{P})}{[\%\text{P}]}=\frac{22\mathrm{\hspace{0.17em} }350}{T}+0.08(\%\text{CaO})+2.5lg(\%\text{T}\cdot \text{Fe})-16$ . Similarly, many phosphorous distribution prediction models have been developed based on some empirical regressions of the measured data, such as Suito’s models,^6,7) Sommerville’s model⁸⁾ and Balajiva’s model.⁹⁾ All the above-mentioned phosphorous distribution prediction models^4,6,7,8,9) have been evaluated in the following sections.

Many researchers^{5,6,12,35,36,37,38,39,40)} have worked towards estimation of γ_P2O5 in metallurgical slags with different chemical compositions. Turkdogan and Pearson⁵⁾ proposed an expression of activity coefficient of P₂O₅ as a function of slag compositions i.e., lgγ_P2O5 = –1.12(22X_CaO + 15X_MgO + 13X_MnO + 12X_FeO –2X_SiO2) – $\frac{42\mathrm{\hspace{0.17em} }000}{T}$ + 23.58. In 1981, on the basis of the measured phosphorus distribution between MgO–saturated CaO–MgO–FeO–SiO₂ slags, containing 30–40% CaO, in the temperature range from 1823 K to 1923 K, Suito et al.⁶⁾ modified Turkdogan’s correlation⁵⁾ as that lgγ_P2O5 = –1.01 (23X_CaO + 17X_MgO + 8X_FeO) – $\frac{26\mathrm{\hspace{0.17em} }000}{T}$ + 1.12. The correlation proposed by Suito et al.⁶⁾ indicates significantly lower temperature dependence of γ_P2O5, compared with Turkdogan’s expression.⁵⁾ Basu et al.¹⁰⁾ observed that the relationship between the measured γ_P2O5 and the calculated γ_P2O5 by Turkdogan’s γ_P2O5 prediction model⁵⁾ or Suito’s γ_P2O5 prediction model⁶⁾ was not good, and developed two γ_P2O5 prediction models as lgγ_P2O5 = –6.775N_CaO – 4.995N_FeO + 2.816N_MgO – 1.377N_SiO2 + $\frac{1\mathrm{\hspace{0.17em} }007}{T}-13.992\mathrm{\hspace{0.17em} }(r^2=0.77)$ and lgγ_P2O5 = – 8.172X_Ca²⁺ – 7.169X_Fe²⁺ + 1.323X_Mg²⁺ $-\text{1}\text{.377}{X}_{{\text{SiO}}_{\text{4}}^{\text{4}-}}+\frac{340}{T}$ –11.66 (r² =0.86). Both of Basu’s models¹²⁾ were satisfactorily applicable to the data of other previous workers.^5,6)

Suito et al.¹¹⁾ and Kobayashi et al.¹²⁾ made extensive work on the dephosphorizaion ability of steelmaking slags containing MnO, and they observed that the addition of MnO had a negative effect on dephosphorization. In 2007, Basu et al.¹⁰⁾ conducted experiments to establish for equilibrium phosphorus partition between CaO–SiO₂–MgO–FeO–P₂O₅ slags and molten Fe which is free from Al₂O₃ or MnO, and proposed the correlation for equilibrium phosphorus partition in terms of ionic.

From the above discussion, it can be obviously observed that much of the previous work was carried out using artificial parameters, and most of the regressed parameters were valid only within certain limits of concentration. The present study developed a fresh model without artificial regressed parameters to establish phosphorus partition between CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags and liquid iron based on the ion and molecule coexistence theory. The measured data of phosphorus distribution of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1823 K to 1923 K reported by Nagabayashi et al.¹³⁾ and the measured data of phosphorus distribution of CaO–SiO₂–MgO–FeO–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1873 K to 1923 K by Basu et al.¹⁰⁾ were both selected to verify the accuracy of the developed IMCT–L_p prediction model. It should be pointed out that CaO–SiO₂–MgO–FeO–P₂O₅ slags was a special CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags with the assumption that the content of Fe₂O₃ equal to zero. However, the major aim of this study is to represent an investigation of true characteristics of dephosphorization in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags. Hence, only the reported data of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags by Nagabayashi et al.¹³⁾ was taken into consideration for representation of reaction abilities of structure units in the following sections.

2. Model for Calculating Mass Action Concentrations of Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

2.1. Hypotheses

In view of the fact that there are ions and molecules in molten metallurgical slags simultaneously,^{13,14,15,16,17,18,19,22)} the main viewpoints of the ion and molecule coexistence theory of slag structures in the developed thermodynamic model can be simply summarized as follows: 1) Molten slags of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ are composed of Ca²⁺, Mg²⁺, Fe²⁺, and O²⁻ as simple ions, SiO₂ and P₂O₅ as simple molecules, silicates, aluminates as complex molecules. Each structural unit has its independent position in the slags. Every cation and anion generated from the same component will take part in reactions of forming complex molecules in the form of ion couple as (Me²⁺+O²⁻). 2) Reactions of forming complex molecules are under chemically dynamic equilibrium between bonded ion couples from simple ions and simple molecules. 3) Structural units in the selected slags equilibrated with liquid iron bear continuity in the range of investigated concentration range. 4) Chemical reactions of forming complex molecules obey the mass action law.

2.2. Model for Calculating Mass Action Concentrations of Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

2.2.1. Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

According to traditional metallurgical physicochemistry, CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags is composed of CaO, SiO₂, MgO, FeO, Fe₂O₃ and P₂O₅, while the extracted phosphorus gradually enter into the slags with the proceeding of dephosphorization reactions until dephosphorization reactions reach equilibrium. However, the IMCT^{14,15,16,17,18,19,20,23)} suggests that the extracted phosphorus from liquid iron into the slags exists as P₂O₅ can be bonded with ion couples (Fe²⁺+O²⁻), (Ca²⁺+O²⁻) and (Mg²⁺+O²⁻) to form molecules as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅, 4CaO·P₂O₅, 2MgO·P₂O₅, and 3MgO·P₂O₅ in oxidizing slags containing iron oxides Fe_tO, respectively.

In view of the reported binary and ternary phase diagrams^20,21) of CaO–SiO₂ slags, CaO–P₂O₅ slags, FeO–P₂O₅ slags, MgO–P₂O₅ slags, CaO–MgO slags, CaO–MgO–SiO₂ slags, MgO–SiO₂ slags, and CaO–FeO–SiO₂ slags in a temperature range from 1673 K to 1986 K, about 20 kinds of complex molecules can be formed in the slags as listed in Table 1.

Table 1. Chemical reaction formulas of possibly formed complex molecules, their standard molar Gibbs free energy changes, mole numbers, mass action concentrations and equilibrium constants in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags.

2.2.2. Model for Calculating Mass Action Concentrations of Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

The mole number of above–mentioned six components, such as CaO, SiO₂, MgO, FeO, Fe₂O₃ and P₂O₅ in 100 g of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags is assigned as b₁ = $n_{\text{CaO}}^0$ , $b_2=n_{{\text{SiO}}_{\text{2}}}^{\text{0}}$ , $b_3=n_{\text{MgO}}^{\text{0}}$ , $b_4=n_{\text{FeO}}^{\text{0}}$ , $b_5=n_{{\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}}}^{\text{0}}$ and to b₆ = $n_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\text{0}}$ present chemical composition of the slags. The defined^{14,15,16,17,18,19,20,23)} equilibrium mole numbers n_i of all above-mentioned structural units in 100 g slags equilibrated with liquid iron at metallurgical temperatures have been also given out in Table 1. According to the ion and molecule coexistence theory,^{14,15,16,17,18,19,20,23)} each ion couple is electroneutral and can be electrolyzed into cation and anion based on electrovalence balance principle. Hence, ion couple (Me²⁺+O^2–) with a fixed amount under equilibrium condition can produce two times amount of structural units, i.e., $n_{{\text{Me}}^{\text{2+}}\text{,MeO}}+n_{{\text{O}}^{\text{2-}}\text{,MeO}}=2n_{\text{MeO}}$ . The total equilibrium mole number Σn_i of all structural units in 100 g slags equilibrated with liquid iron can be expressed as follows   

$$\begin{array}{c}\sum n_i=2n_{\text{CaO}}+n_{{\text{SiO}}_{\text{2}}}+2n_{\text{MgO}}+2n_{\text{FeO}}+n_{{\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}}}\\ \hspace{1em} +n_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}+n_{\text{c1}}+n_{\text{c2}}+\cdots +n_{\text{c20}}\hspace{1em}(\text{mol)}\end{array}$$(1)

With respect to the definition of mass action concentrations^{14,15,16,17,18,19,20,23)} N_i for structural units, which is a ratio of equilibrium mole number of structural units i to the total equilibrium mole numbers of all structural units in a close system with a fixed amount, N_i of structural units i and ion couples (Me²⁺+O²⁻) in molten slags can be calculated by   

$$N_i=\frac{n_i}{\sum n_i}\hspace{1em}\hspace{1em}\hspace{1em}(-)$$(2a)

  

$$\begin{array}{l}{N}_{\text{MeO}}=N_{{\text{Me}}^{\text{2+}}\text{,MeO}}+N_{{\text{O}}^{\text{2-}}\text{,MeO}}\\ \hspace{1em}\hspace{1em} =\frac{n_{{\text{Me}}^{\text{2+}}\text{,MeO}}+n_{{\text{O}}^{\text{2-}}\text{,MeO}}}{\sum n_i}=\frac{2n_{\text{MeO}}}{\sum n_i}\hspace{1em}\hspace{1em}(-)\end{array}$$(2b)

All definitions of N_i for formed ion couples from simple ions, simple and complex molecules in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags were listed in Table 1. Meanwhile, it should be pointed out that the physicochemistry meaning of N_i is almost consistent with the traditionally applied activity a_i of component i in slags, in which pure matter is chosen as standard state and mole fraction is selected as concentration unit.^{14,15,16,17,18,19,20,23)}

The chemical reaction formulas of 20 kinds of possibly formed complex molecules, their standard molar Gibbs free energy ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m,c}i}^{\mathrm{\Theta }}$ of formation reactions as a function of absolute temperature T, reaction equilibrium constant $K_{\text{c}i}^{\mathrm{\Theta }}$ and presentation of mass action concentration of all complex molecules N_ci expressed by using $K_{\text{c}i}^{\mathrm{\Theta }}$ , N₁ (N_CaO), N₂ ( $N_{{\text{SiO}}_{\text{2}}}$ ), N₃ (N_MgO), N₄ (N_FeO), N₅ ( $N_{{\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}}}$ ) and N₆ ( $N_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}$ ) based on the mass action law were summarized in Table 1.

The mass conservation equations of six components in 100 g of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron can be established according to the definition^{14,15,16,17,18,19,20,23)} of n_i and mass action concentrations N_i of all structural units listed in Table 1 as follows   

$$\begin{array}{l}{b}_1=(\frac{1}{2}{N}_1+3N_{\text{c1}}+2N_{\text{c2}}+N_{\text{c3}}+N_{\text{c7}}+N_{\text{c8}}+2N_{\text{c9}}+3N_{\text{c10}}\\ \hspace{1em}\hspace{1em}+2N_{\text{c14}}+3N_{\text{c15}}+4N_{\text{c16}})\sum n_i=(\frac{1}{2}{N}_1+3K_{\text{c1}}^{\mathrm{\Theta }}{N}_1^3{N}_2\\ \hspace{1em}\hspace{1em}+2K_{\text{c2}}^{\mathrm{\Theta }}{N}_1^2{N}_2+K_{\text{c3}}^{\mathrm{\Theta }}{N}_1^{}{N}_2+K_{\text{c7}}^{\mathrm{\Theta }}{N}_1{N}_{\text{3}}^{}{N}_{\text{2}}^{}+K_{\text{c8}}^{\mathrm{\Theta }}{N}_1{N}_{\text{3}}^{}{N}_{\text{2}}^{\text{2}}\\ \hspace{1em}\hspace{1em}+2K_{\text{c9}}^{\mathrm{\Theta }}{N}_{\text{1}}^{\text{2}}{N}_{\text{3}}^{}{N}_{\text{2}}^{}+3K_{\text{c10}}^{\mathrm{\Theta }}{N}_{\text{1}}^{\text{3}}{N}_{\text{3}}^{}{N}_{\text{2}}^{\text{2}}+K_{\text{c14}}^{\mathrm{\Theta }}{N}_1^{\text{2}}{N}_6\\ \hspace{1em}\hspace{1em}+3K_{\text{c15}}^{\mathrm{\Theta }}{N}_1^3{N}_6+4K_{\text{c16}}^{\mathrm{\Theta }}{N}_1^4{N}_6^{})\sum n_i=n_{\text{CaO}}^0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\text{mol})\end{array}$$(3a)

  

$$\begin{array}{l}{b}_2=(N_2+N_{\text{c1}}+N_{\text{c2}}+N_{\text{c3}}+N_{\text{c4}}+N_{\text{c5}}+N_{\text{c6}}+N_{\text{c7}}+2N_{\text{c8}}\\ \hspace{1em}\hspace{1em}+2N_{\text{c9}}+2N_{\text{c10}})\sum n_i=(N_2+K_{\text{c1}}^{\mathrm{\Theta }}{N}_1^3{N}_2+K_{\text{c2}}^{\mathrm{\Theta }}{N}_1^2{N}_2\\ \hspace{1em}\hspace{1em}+K_{\text{c3}}^{\mathrm{\Theta }}{N}_1{N}_2+K_{\text{c4}}^{\mathrm{\Theta }}{N}_2{N}_3^2+K_{\text{c5}}^{\mathrm{\Theta }}{N}_2{N}_3+K_{\text{c6}}^{\mathrm{\Theta }}{N}_2{N}_4^2\\ \hspace{1em}\hspace{1em}+K_{\text{c7}}^{\mathrm{\Theta }}{N}_1{N}_3{N}_{\text{2}}+2K_{\text{c8}}^{\mathrm{\Theta }}{N}_1{N}_3{N}_2^2+2K_{\text{c9}}^{\mathrm{\Theta }}{N}_1^2{N}_3{N}_2^2\\ \hspace{1em}\hspace{1em}+2K_{\text{c10}}^{\mathrm{\Theta }}{N}_1^3{N}_3{N}_2^2)\sum n_i=n_{{\text{SiO}}_{\text{2}}}^{\text{0}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\text{mol})\end{array}$$(3b)

  

$$\begin{array}{l}{b}_3=(\frac{1}{2}{N}_3+2N_{\text{c4}}+N_{\text{c5}}+N_{\text{c7}}+N_{\text{c8}}+N_{\text{c9}}+N_{\text{c10}})\sum n_i\\ \hspace{1em}=(\frac{1}{2}{N}_3+2K_{\text{c4}}^{\mathrm{\Theta }}{N}_2{N}_3^2+K_{\text{c5}}^{\mathrm{\Theta }}{N}_2{N}_3+K_{\text{c7}}^{\mathrm{\Theta }}{N}_1{N}_3{N}_2\\ \hspace{1em}\hspace{1em}+K_{\text{c8}}^{\mathrm{\Theta }}{N}_1{N}_3{N}_2^2+K_{\text{c9}}^{\mathrm{\Theta }}{N}_1^2{N}_3{N}_2^2\\ \hspace{1em}\hspace{1em}+K_{\text{c10}}^{\mathrm{\Theta }}{N}_1^3{N}_3^2{N}_2^2)\sum n_i=n_{\text{MgO}}^{\text{0}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\text{mol})\end{array}$$(3c)

  

$$\begin{array}{l}{b}_4=(\frac{1}{2}{N}_4+2N_{\text{c6}}+3N_{\text{c17}}+4N_{\text{c18}})\sum n_i\\ \hspace{1em}=(\frac{1}{2}{N}_4+2K_{\text{c6}}^{\mathrm{\Theta }}{N}_2{N}_4^2+3K_{\text{c17}}^{\mathrm{\Theta }}{N}_4^3{N}_6\\ \hspace{1em}\hspace{1em}+4K_{\text{c18}}^{\mathrm{\Theta }}{N}_4^4{N}_6)\sum n_i=n_{\text{FeO}}^{\text{0}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\text{mol})\end{array}$$(3d)

  

$$\begin{array}{l}{b}_5=(N_5+N_{\text{c11}}+N_{\text{c12}}+N_{\text{c13}})\sum n_i\\ \hspace{1em}=(N_5+K_{\text{c11}}^{\mathrm{\Theta }}{N}_1^2{N}_5^{}+K_{\text{c12}}^{\mathrm{\Theta }}{N}_4^{}{N}_5\\ \hspace{1em}\hspace{1em}+K_{\text{c13}}^{\mathrm{\Theta }}{N}_3^{}{N}_5)\sum n_i=n_{{\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}}}^{\text{0}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\text{mol})\end{array}$$(3e)

  

$$\begin{array}{l}{b}_6=(N_6+N_{\text{c14}}+N_{\text{c15}}+N_{\text{c16}}+N_{\text{c17}}+N_{\text{c18}}+N_{\text{c19}}+N_{\text{c20}})\sum n_i\\ \hspace{1em}=(N_6+K_{\text{c14}}^{\mathrm{\Theta }}{N}_1^2{N}_6+K_{\text{c15}}^{\mathrm{\Theta }}{N}_1^3{N}_6+K_{\text{c16}}^{\mathrm{\Theta }}{N}_1^4{N}_6\\ \hspace{1em}\hspace{1em}+K_{\text{c17}}^{\mathrm{\Theta }}{N}_4^3{N}_6+K_{\text{c18}}^{\mathrm{\Theta }}{N}_4^4{N}_6+K_{\text{c19}}^{\mathrm{\Theta }}{N}_3^2{N}_6\\ \hspace{1em}\hspace{1em}+K_{\text{c20}}^{\mathrm{\Theta }}{N}_3^3{N}_6)\sum n_i=n_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\text{0}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\text{mol})\end{array}$$(3f)

According to the principle that the sum of mole fraction for all structural units in a fixed amount of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags under equilibrium condition is equal to 1.0, the following equation can be obtained   

$$\begin{array}{l}{N}_1+N_2+N_3+N_4+N_5+N_6+N_{\text{c1}}+N_{\text{c2}}+\cdots +N_{\text{c20}}\\ =N_1+N_2+\cdots +N_5+K_{\text{c1}}^{\mathrm{\Theta }}{N}_1^3{N}_2+K_{\text{c2}}^{\mathrm{\Theta }}{N}_1^2{N}_2^{}\\ +\cdots +K_{\text{c20}}^{\mathrm{\Theta }}{N}_3^3{N}_6=\sum N_i=1.0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(-)\end{array}$$(4)

The equation group of Eqs. (3) and (4) is the governing equations of the developed thermodynamic model for calculating mass action concentrations N_i of structural units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron. Obviously, there are seven unknown parameters as N₁, N₂, N₃, N₄, N₅, N₆ and Σn_i with 7 independent equations in the developed equation group of Eqs. (3) and (4). The unique solution of N_i, Σn_i and n_i can be calculated by solving these algebraic equation group of Eqs. (3) and (4) by combining with the definition of N_i in Eq. (2).

2.3. Definition of Mass Action Concentration for Iron Oxides Fe_tO in the CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

It’s well-known that high oxygen potential and basicity of the slags are needed to remove phosphorus from steel melts. In this study, the mass action concentration of iron oxides Fe_tO, i.e., $N_{{\text{Fe}}_t\text{O}}$ , is recommended to present slag oxidization ability. Meanwhile, it should be pointed out that the defined comprehensive mass action concentration $N_{{\text{Fe}}_t\text{O}}$ of iron oxides in slags can be applied to accurately replace activity of $a_{{\text{Fe}}_t\text{O}}$ relative pure liquid matter as standard state in previous publication.²³⁾

The IMCT^{14,15,16,17,18,19,20,23)} proposed that all iron oxides in metallurgical slags are composed of ion couple (Fe²⁺+O²⁻), simple molecule Fe₂O₃ and complex molecule FeO·Fe₂O₃, therefore, the related structural units of iron oxides can dynamically equilibrate among those structural units as follows   

$$({\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}})+[\text{Fe}]=3({\text{Fe}}^{2+}{\text{+O}}^{2-})$$(5a)

  

$$(\text{FeO}\cdot {\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}})+[\text{Fe}]=4({\text{Fe}}^{2+}{\text{+O}}^{2-})$$(5b)

Obviously, when the reactions of Eqs. (5a) and (5b) reach equilibrium, increase of single molcule of Fe₂O₃ will lead to the changing of balance with increase of one third ion couple since the equilibirum constant is kept constant. Therefore, the mass action concentration of free ion couple (Fe²⁺+O²⁻) can directly represent present slag oxidization ability. Meanwhile, it should be pointed out that the symbol of $N_{{\text{Fe}}_t\text{O}}$ is applied in this study to keep consistent with traditional symbol of iron oxides “Fe_tO” in the related reference. Therefore, the slags oxidization ability or $N_{{\text{Fe}}_t\text{O}}$ can be defined as   

$$N_{{\text{Fe}}_t\text{O}}^{}=N_{\text{FeO}}^{}\approx a_{{\text{Fe}}_t\text{O}}\hspace{1em}\hspace{1em}\hspace{1em}(-)$$(6)

According to IMCT, the defined $N_{{\text{Fe}}_t\text{O}}$ from IMCT²³⁾ has the similar meaning with $a_{{\text{Fe}}_t\text{O}}$ from viewpoint of traditionally metallurgical physicochemistry, which can be calculated according to (Fe_tO–O) equilibrium as   

$$\begin{array}{l}t[\text{Fe}]+[\text{O}]=\mathrm{F}{\mathrm{e}}_t\mathrm{O}\hspace{1em}\hspace{1em}\\ {\mathrm{\Delta }}_{\text{r}}{G}_{{\text{m, Fe}}_t\text{O}}^{\mathrm{\Theta }}=-116\mathrm{\hspace{0.17em} }100+48.79T{\mathrm{\hspace{0.17em} }}^{}\hspace{1em}\hspace{1em}\hspace{1em}(\text{J/mol)}\end{array}$$(7a) ²²⁾

  

$$K_{{\text{Fe}}_t\text{O}}^{\mathrm{\Theta }}=\frac{a_{{\text{Fe}}_t\text{O}}}{a_{\text{Fe}}^t{a}_{\text{O}}^{}}\hspace{1em}{a}_{{\text{Fe}}_t\text{O}}=K_{{\text{Fe}}_t\text{O}}^{\mathrm{\Theta }}{a}_{\text{O}}^{}=K_{{\text{Fe}}_t\text{O}}^{\mathrm{\Theta }}[\%\text{O}]f_{\text{O}}$$(7b)

where f_O is oxygen activity coefficient (–), and can be determined by   

$$lg{f}_{\text{O}}=e_{\text{O}}^{\text{O}}[\%\text{O}]+e_{\text{O}}^{\text{P}}[\%\text{P}]$$(8)

The related values of interaction coefficients are chosen as $e_{\text{O}}^{\text{O}}$ =–0.2, $e_{\text{O}}^{\text{P}}$ =–0.07.

3. Results of Mass Action Concentrations for Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

3.1. Relation between Mass Percent of Six Components and Mass Action Concentrations of Related Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

The relationship between mass percentage of CaO, SiO₂, MgO, FeO, Fe₂O₃ and P₂O₅ as components and the calculated mass action concentrations N_i of structural units, i.e., (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻), Fe₂O₃ and P₂O₅, in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags in a temperature range from 1823 K to 1923 K was shown in Fig. 1, respectively. The good scatter linear relationship can be observed for (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻) and Fe₂O₃ as components in Figs. 1(a), 1(b), 1(c), 1(d) and 1(e). The scatter corresponding relationship can be found for P₂O₅ in Fig. 1(f). The good scatter linear relationship for (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻) and Fe₂O₃ can be explained as follows 1) some basic oxides such as (Ca²⁺+O²⁻) and (Mg²⁺+O²⁻) can react with SiO₂ to form complex molecules 3CaO·SiO₂, 2CaO·SiO₂, CaO·SiO₂, 2MgO·SiO₂, MgO·SiO₂, CaO·MgO·2SiO₂, 2CaO·MgO·2SiO₂ and 3CaO·MgO·2SiO₂ as shown in Table 1; 2) (Fe²⁺+O²⁻) and Fe₂O₃ can also be consumed as iron oxides in the process of dephosphorization.

Fig. 1.

Relationship between mass percent of CaO, SiO₂, MgO, FeO, Fe₂O₃ and P₂O₅ as components and calculated mass action concentrations of (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻), Fe₂O₃ and P₂O₅ as structural units in 100 g CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron at elevated temperatures, respectively.

3.2. Relation between Mass Percent of Six Components and Equilibrium Mole Numbers of Related Structural Units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

The relationship between mass percentage of CaO, SiO₂, MgO, FeO, Fe₂O₃ and P₂O₅ as components and the calculated equilibrium mole numbers n_i of structural units, i.e., (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻), Fe₂O₃ and P₂O₅, in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags in a temperature range from 1823 K to 1923 K was illustrated in Fig. 2, respectively. Obviously, the calculated equilibrium mole numbers n_i of all 6 structural units have some relation with mass percentage of the corresponding components. The good linear relationship can be found for SiO₂ and FeO in Figs. 2(b) and 2(d), this phenomenon can be explained as that not so many molecules of SiO₂ or FeO can be bounded to form complex molecules. The scatter relationship can be observed for (Ca²⁺+O²⁻), (Mg²⁺+O²⁻), Fe₂O₃ and P₂O₅ as components in Figs. 2(a), 2(c), 2(e) and 2(f). However, a linear relationship can be correlated for (Ca²⁺+O²⁻), (Mg²⁺+O²⁻), Fe₂O₃ and P₂O₅ in Figs. 2(a), 2(c), 2(e) and 2(f) although some mass of above mentioned components can be found as various complex molecules. Therefore, the calculated equilibrium mole number n_i of structural units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags can be applied to represent chemical composition of the slags.^{14,15,16,17,18,19,20,23)}

Fig. 2.

Relationship between mass percent of CaO, SiO₂, MgO, FeO, Fe₂O₃ and P₂O₅ as components and calculated equilibrium mole numbers of (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻), Fe₂O₃ and P₂O₅ as structural units in 100 g CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron at elevated temperatures, respectively.

3.3. Relation between Equilibrium Mole Numbers and Basicity in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

The relationship between the calculated equilibrium mole numbers n_i and simple binary basicity R of slags, i.e., ((%CaO)/(%SiO₂)) was illustrated in Fig. 3. Total equilibrium mole number Σn_i has a very good linear relationship with simple binary basicity R of slags, i.e., ((%CaO)/(%SiO₂)). The formula of equilibrium mole number Σn_i against the simple binary basicity R of slags can be regressed as Σn_i = 2.604–3.029*exp(–R/2.339), and the fitting degree is 0.995. This reliable fitting degree results suggest that changing the simple binary basicity R of slags have a large effect on Σn_i.

Fig. 3.

Relationship between binary basicity R and calculated total equilibrium mole number of structural units Σn_i in 100 g CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron at elevated temperatures.

4. Model for Calculating Phosphorus Distribution between CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags and Liquid Iron

According to the ion and molecule coexistence theory that only free ion couples (Ca²⁺+O²⁻), and (Mg²⁺+O²⁻), which can be combined with iron oxides Fe_tO in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags, can take roles in dephosphorization reactions in terms of forming 8 dephosphorization molecules as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅, 4CaO·P₂O₅, 2MgO·P₂O₅ and 3MgO·P₂O₅ according to IMCT^{14,15,16,17,18,19,20,23)} as follows   

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)=\left({\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{{\text{m, P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-122\mathrm{\hspace{0.17em} }412+312.522T\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9a)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+3({\text{Fe}}^{2+}+{\text{O}}^{2^{-}})=\left(3\text{FeO}·{\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-552\mathrm{\hspace{0.17em} }816+405.23T\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9b)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+4({\text{Fe}}^{2+}+{\text{O}}^{2-})=\left(4\text{FeO}·{\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-504\mathrm{\hspace{0.17em} }243+359.889T\hspace{1em}\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9c)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+2({\text{Ca}}^{2+}+{\text{O}}^{2-})=\left(2\text{CaO}·{\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-484\mathrm{\hspace{0.17em} }372-26.569T\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9d)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+3({\text{Ca}}^{2+}+{\text{O}}^{2-})=\left(3\text{CaO}·{\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-832\mathrm{\hspace{0.17em} }302+318.672T\hspace{1em}\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9e)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+4({\text{Ca}}^{2+}+{\text{O}}^{2-})=\left(4\text{CaO}·{\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-783\mathrm{\hspace{0.17em} }768+309.049T\hspace{1em}\hspace{1em}\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9f)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+2({\text{Mg}}^{2+}+{\text{O}}^{2^{-}})=\left(2\text{MgO}·{\text{P}}_2{\text{O}}_5\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=45\mathrm{\hspace{0.17em} }957-26.835T\hspace{1em}\hspace{1em}\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9g)

  

$$\begin{array}{l}2\left[\text{P}\right]+5\left({\text{Fe}}_t\text{O}\right)+3({\text{Mg}}^{2+}+{\text{O}}^{2-})=\left(3\text{MgO}·{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}\right)+5t\left[\text{Fe}\right]\\ {\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, 3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=-390\mathrm{\hspace{0.17em} }053+197.318T\hspace{1em}\hspace{1em}\hspace{1em}\left(\text{J/mol}\right)\end{array}$$(9h)

The corresponding equilibrium constants of Eq. (9) can be expressed according to IMCT^{14,15,16,17,18,19,20,23)} as   

$$\begin{array}{c}{K}_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{P}}^2}=\frac{N_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}\hspace{1em}(-)\end{array}$$(10a)

  

$$\begin{array}{c}{K}_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{FeO}}^3{a}_{\text{P}}^2}=\frac{N_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{FeO}}^3{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{FeO}}^3{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10b)

  

$$\begin{array}{c}{K}_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{FeO}}^4{a}_{\text{P}}^2}=\frac{N_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{FeO}}^4{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{FeO}}^4{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10c)

  

$$\begin{array}{c}{K}_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{CaO}}^2{a}_{\text{P}}^2}=\frac{N_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^2{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^2{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10d)

  

$$\begin{array}{c}{K}_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{CaO}}^3{a}_{\text{P}}^2}=\frac{N_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^3{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^3{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10e)

  

$$\begin{array}{c}{K}_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{CaO}}^4{a}_{\text{P}}^2}=\frac{N_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^4{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^4{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10f)

  

$$\begin{array}{c}{K}_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{MgO}}^2{a}_{\text{P}}^2}=\frac{N_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{MgO}}^2{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{MgO}}^2{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10g)

  

$$\begin{array}{c}{K}_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}=\frac{a_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{a}_{\text{Fe}}^{5t}}{a_{{\text{Fe}}_t\text{O}}^5{a}_{\text{MgO}}^3{a}_{\text{P}}^2}=\frac{N_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\times 1}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{MgO}}^3{[\%\text{P}]}^2{f}_{\text{P}}^2}\\ =\frac{\left(\stackrel{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}/M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{}/\underset{\sum n_i}{}\right)}{N_{{\text{Fe}}_t\text{O}}^5{N}_{\text{MgO}}^3{[\%\text{P}]}^2{f}_{\text{P}}^2}\hspace{1em}(-)\end{array}$$(10h)

where $M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}$ is molecular mass of P₂O₅ as 141.94 (–). According to Eq. (10), the respective phosphorus distribution of structural units as basic components in the slags L_P,i can be expressed by   

$$L_{{\text{P, P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}=M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)$$(11a)

  

$$\begin{array}{l}{L}_{\text{P, 3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{FeO}}^3{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11b)

  

$$\begin{array}{l}{L}_{\text{P, 4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{FeO}}^4{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11c)

  

$$\begin{array}{l}{L}_{\text{P, 2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^2{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11d)

  

$$\begin{array}{l}{L}_{\text{P, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^3{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11e)

  

$$\begin{array}{l}{L}_{\text{P, 4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{CaO}}^4{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11f)

  

$$\begin{array}{l}{L}_{\text{P, 2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{MgO}}^2{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11g)

  

$$\begin{array}{l}{L}_{\text{P, 3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{K}_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{{\text{Fe}}_t\text{O}}^5{N}_{\text{MgO}}^3{f}_{\text{P}}^2\sum n_i\hspace{1em}(-)\end{array}$$(11h)

where f_P is activity coefficient of the dissolved phosphorus in liquid iron (–), and can be calculated as follows   

$$lg{f}_P=\sum e_{\text{P}}^j[\%j]\hspace{1em}\mathrm{\hspace{0.17em} }\hspace{1em}(-)$$(12a)

  

$$e_{\text{P}}^j\mathrm{\hspace{0.17em} }T=\text{const}\hspace{1em}\hspace{1em}\hspace{1em}(-)$$(12b)

The related values of interaction coefficients of [P]^[22] are chosen as $e_{\text{P}}^{\text{P}}$ =0.062, $e_{\text{P}}^{\text{O}}$ =0.13 at 1873 K. Therefore, the total phosphorus distribution between CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags and liquid iron can be obtained from Eq. (11) as follows   

$$\begin{array}{l}{L}_{\text{P}}=L_{{\text{P, P}}_{\text{2}}{\text{O}}_{\text{5}}}+L_{\text{P, 3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}+L_{\text{P, 4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}+L_{\text{P, 2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\\ \hspace{1em}\hspace{1em}+L_{\text{P, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}+L_{\text{P, 4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}+L_{\text{P, 2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}+L_{\text{P, 3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}=\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\\ \hspace{1em}\hspace{1em}\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\\ \hspace{1em}\hspace{1em}\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}+\frac{{(\%{\text{P}}_{\text{2}}{\text{O}}_{\text{5}})}_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}}{{[\%\text{P}]}^2}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}=M_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}{N}_{{\text{Fe}}_t\text{O}}^5{f}_{\text{P}}^2(K_{{\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}+K_{\text{3FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{FeO}}^3+K_{\text{4FeO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{FeO}}^4+\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}{K}_{\text{2CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{CaO}}^2+K_{\text{3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{CaO}}^3+K_{\text{4CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{CaO}}^4+\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}{K}_{\text{2MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{MgO}}^2+K_{\text{3MgO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\mathrm{\Theta }}{N}_{\text{MgO}}^3)\sum n_i\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(-)\end{array}$$(13)

Therefore, the developed L_P prediction model by $N_{{\text{Fe}}_t\text{O}}$ to present slag oxidizing ability is composed of Eqs. (11) and (13). According to the calculated N_i and Σn_i in Section 3, $K_i^{\mathrm{\Theta }}$ by Eq. (10) and f_P by Eq. (12), the total phosphorus distribution L_P of the slags which equals to the sum of respective phosphorus distribution L_{P, i} of basic ion couples in the slags can be calculated. The standard molar Gibbs free energy ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ of dephosphorization reactions in Eq. (9) for forming dephosphorization products i is determined from the reported data and summarized in Table 2.

Table 2. Calculation of standard molar Gibbs free energy for 8 dephosphorization reactions from the reported data of standard molar Gibbs free energy.

Dephosphorization reactions	Resource reactions	${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ (J/mol)
2[P]+5(FetO)=(P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–122412+312.522T	$\stackrel{1}{}/\underset{2}{}$ P2=[P]	–157700+5.4T30)
	$\stackrel{1}{}/\underset{2}{}$ O2=[O]	–117110–3.39T30)
	2[P]+5[O]=(P2O5)(l)	–702912+556.472T31)
	P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(P2O5)(l)	–1603862+550.322T30,31)
	t[Fe]+[O]=(FetO)	–116100+48.79T32)
2[P]+5(FetO)+3(Fe2++O2−)=(3FeO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–552816+405.23T	3(FeO)+(P2O5)=(3FeO·P2O5)	–430404+92.708T9)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T
2[P]+5(FetO)+4(Fe2++O2−)=(4FeO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–504243+359.889T	4(Fe2++O2−)+(P2O5)=(4FeO·P2O5)	–381831+47.367T9)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T
2[P]+5(FetO)+2(Ca2++O2−)=(2CaO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–606784+285.953T	2(CaO)+P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(2CaO·P2O5)(s)	–2189069+585.76T33)
	(2CaO·P2O5)(s)= (2CaO·P2O5)(l)	100834.4–62.0069T16)
	P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(P2O5)(l)	–1603862+550.322T30,31)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T
2[P]+5(FetO)+3(Ca2++O2−)=(3CaO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–832302+318.672T	3(CaO)+P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(3CaO·P2O5)(s)	–2313752+556.472T33)
	P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(P2O5)(l)	–1603862+550.322T30,31)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T
2[P]+5(FetO)+4(Ca2++O2−)=(4CaO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–783768+309.049T	4(CaO)+(P2O5)(l)=(4CaO·P2O5)(l)	–661356–3.473T34)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T
2[P]+5(FetO)+2(Mg2++O2−)=(2MgO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =45957–26.835T	2(Mg2++O2-)+(P2O5)=(2MgO·P2O5)	168369–339.357T9)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T
2[P]+5(FetO)+3(Mg2++O2−)=(3MgO·P2O5)+5t[Fe] ${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ =–390053+197.336T	3(MgO)+P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(3MgO·P2O5)(s)	–1992839+510.0296T33)
	(3MgO·P2O5)(s)= (3MgO·P2O5)(l)	121336–74.8936T32)
	P2+ $\stackrel{5}{}/\underset{2}{}$ O2=(P2O5)(l)	–1603862+550.322T30,31)
	2[P]+5(FetO)=(P2O5)+5t[Fe]	–122412+312.522T

5. Results and Discussion

5.1. Comparison of Measured Phosphorous Distribution L_{P, measured} and Calculated Phosphorous Distribution L_{P, calculated}

In order to examine the applicability of the IMCT−L_P model developed in the present study based on the ion and molecule coexistence theory, both methods have been adopted to evaluate the accuracy of the calculated $L_{\text{P, calculated}}^{\text{IMCT}}$ . One is directly comparing the measured L_{P, measured} from previous references^10,13) and the calculated $L_{\text{P, calculated}}^{\text{IMCT}}$ based on the ion and molecule coexistence theory; the other is comparing the IMCT−L_P model with some other reported phosphorous distribution prediction models,^4,6,7,8,9) such as Healy’s model,⁴⁾ Suito’s models,^6,7) Sommerville’s model,⁸⁾ and Balajiva’s model.⁹⁾

The comparisons between the measured phosphorous distribution L_{P, measured} for CaO–SiO₂–MgO–FeO–P₂O₅ slags equilibrated with liquid iron at 1873 K and 1923 K reported by Basu et al.¹⁰⁾ or the measured phosphorous distribution L_{P, measured} for the CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags at 1823 K, 1873 K and 1923 K reported by Nagabayashi et al.¹³⁾ and the calculated phosphorous distribution $L_{\text{P, calculated}}^{\text{IMCT}}$ based on ion and molecule coexistence theory was illustrated in Fig. 4, respectively.

Fig. 4.

Comparison between calculated and measured phosphorus distribution of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1823 K to 1923 K.^10,13)

As shown in Fig. 4, an excellent 1:1 agreement between phosphorous distribution lg L_{P, measured} reported by Basu et al.¹⁰⁾ and $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ based on the ion and molecule coexistence theory can be obtained, meanwhile, a good 1:1 linear relationship between phosphorous distribution lg L_{P, measured} reported by Nagabayashi et al.¹³⁾ and $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ based on the ion and molecule coexistence theory can also be found. It should be pointed out CaO–SiO₂–MgO–FeO–P₂O₅ slags was treated as a special CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags in this paper with the assumption that the content of Fe₂O₃ equal to zero. This result indicates that the developed IMCT–L_P prediction model can be applied to reliably predict the phosphorous distribution L_P of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron.

5.2. Comparison of Calculated Phosphorous Distribution $L_{\text{P, calculated}}^i$ by Different Models

The comparisons between the measured L_{P, measured} and calculated $L_{\text{P, calculated}}^i$ in logarithmic form by various models listed in Table 3 were shown in Fig. 5. Obviously, all prediction models have good agreement with the measured L_{P, measured} for CaO–SiO₂–MgO–FeO–P₂O₅ slags at 1873 K and 1923 K reported by Ban–ya et al.¹⁰⁾ except Suito’s No. 2 L_P model.^5,6) However, only $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ by IMCT–L_P model have 1:1 reliable linear relationship with the measured L_{P, measured} for CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags at 1823 K, 1873 K and 1923 K reported by Nagabayashi et al.,¹³⁾ compared with other prediction models such as $lg{L}_{\text{P, calculated}}^{\text{Healy}}$ by Healy’s L_P model,⁴⁾ $lg{L}_{\text{P, calculated}}^{\text{Suito's No}\text{.1}}$ by Suito’s No. 1 L_P model,^6,7) $lg{L}_{\text{P, calculated}}^{\text{Suito's No}\text{.2}}$ by Suito’s No. 2 L_P model,^6,7) $lg{L}_{\text{P, calculated}}^{\text{Sommerville}}$ by Sommerville’s L_P model⁸⁾ and $lg{L}_{\text{P, calculated}}^{\text{Balajiva}}$ by Balajiva’s L_P model.⁹⁾

Table 3. Formulas of phosphorus distribution prediction models reported in related literatures.

Fig. 5.

Comparison between the measured phosphorus distribution lg L_{P, measured} by previous literatures^10,13) and the calculated phosphorus distribution $L_{\text{P, calculated}}^i$ by various phosphorus distribution prediction models^4,6,7,8,9) of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron at elevated temperatures.

It should be specially emphasized that the calculated $L_{\text{P, calculated}}^i$ by various models listed in Table 3 has been transferred into (%P₂O₅)/[%P]², rather than the defined phosphorus distribution as (%P)/[%P] in Healy’s model,⁴⁾ or (%P₂O₅)/[%P]²(%Fe_tO)⁵ in Suito’s No. 1 and No. 2 models,^6,7) or (%P₂O₅)/[%P] in Sommerville’s model⁸⁾ as listed in Table 3.

5.3. Relation between Mass Action Concentrations of Components and lg L_{P, measured} or $\mathrm{l}\mathrm{g}\mathrm{\hspace{0.17em} }{L}_{\text{P, calculated}}^{\text{IMCT}}$

The influence of the calculated mass action concentrations of components upon lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ for the CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron at 1823 K, 1873 K and 1923 K was shown in Fig. 6, respectively. It was seen in Figs. 6(a) and 6(c) that increasing N_CaO and N_MgO can lead to an increase of phosphorus distribution lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ ; improving N_SiO2 will result in decreasing of phosphorus distribution lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ from Fig. 6(b). Meanwhile, clearly corresponding relationship between lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ and N_FeO or $N_{{\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}}}$ can be observed from Figs. 6(d) and 6(e). This phenomenon can be explained as that dephosphorization process is controlled by the comprehensive effect of basic oxides and iron oxides. Therefore, under circumstance of adequate basicity oxides, increasing iron oxides can effectively increase dephosphorization ability of slags. However, increasing iron oxides without enough basicity oxides can only lead to a slightly increasing of phosphorus distribution.

Fig. 6.

Effect of calculated mass action concentrations of ion couples or simple molecules of (Ca²⁺+O²⁻), SiO₂, (Mg²⁺+O²⁻), (Fe²⁺+O²⁻), and Fe₂O₃ on lg L_{P, measured} and $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ at elevated temperatures.

There are some extreme proofs to support this results as follows: 1) A slags with high iron oxides but very low CaO, which is applied in desiliconization pretreatment of liquid iron, can only extract silicon but not phosphorus; 2) A slags with high CaO but very low Fe_tO, which is applied at reduction period during electric arc furnace steelmaking process, can only extract sulfur but not phosphorus. Therefore, the comprehensive effect of basic components, especially CaO and Fe_tO, can make the main controlling contribution to dephosphorization in the slags.

5.4. Relationship between Slag Basicity and lg L_{P, measured} or $\mathrm{l}\mathrm{g}\mathrm{\hspace{0.17em} }{L}_{\text{P, calculated}}^{\text{IMCT}}$

No Al₂O₃ exists in the selected slags, the commonly applied complex basicity ((%CaO)+1.4(%MgO)/(%SiO₂)+ (%P₂O₅)+/(%Al₂O₃))²²⁾ can be simplified as ((%CaO)+ 1.4(%MgO)/(%SiO₂)+(%P₂O₅)). The relationship between lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ and binary basicity ((%CaO)/(%SiO₂)) or complex basicity ((%CaO)+1.4(%MgO)/(%SiO₂)+(%P₂O₅)) were illustrated in Figs. 7(a) and 7(b). It can be observed from Fig. 7(a) that increasing binary basicity from 1.25 to 2.7 can effectively improve lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ , further increasing binary basicity from 2.7 to 9 cannot bring an obvious increasing tendency of lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ due to saturation with solid phase; as shown in Fig. 7(b), increasing complex basicity from 1.5 to 2.5 can lead to an obvious increasing tendency of L_P, further increasing complex basicity from 2.5 to 3.5 have little influence on lg L_{P, measured} or $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ owing to saturation with solid phase;.

Fig. 7.

Effects of binary basicity (%CaO)/(%SiO₂) (a) and complex basicity ((%CaO)+1.4(%MgO)/(%SiO₂)+(%P₂O₅)) (b) on lg L_{P, measured} and $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ at elevated temperatures.

5.5. Contribution of Basic Oxides to Dephosphorization Ability of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ Slags

According to IMCT,^{14,15,16,17,18,19,20,23)} the structural unit P₂O₅ is generated from Fe_tO in the slags or [O] in liquid iron, meanwhile, three components as CaO, MgO, and FeO in the slags can react with P₂O₅ to generate 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅, 4CaO·P₂O₅, 2MgO·P₂O₅, and 3MgO·P₂O₅ as complex molecules. As shown in Eq. (11), the respective phosphorus distribution L_{P, i} of above-mentioned eight molecules can be calculated from the developed IMCT−L_P model.

The relationship between the calculated respective phosphorus distribution $lg{L}_{\text{P, }i\text{, calculated}}^{\text{IMCT}}$ of 8 structural units containing phosphorus and the measured $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ of the slags was illustrated in Fig. 8(a), respectively. Obviously, the contribution of P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2MgO·P₂O₅, and 3MgO·P₂O₅ to the total dephosphorization ability is very small, therefore, the contribution of above mentioned complex molecules can be ignored compared with the contribution of 3CaO·P₂O₅ as 97.88%, 4CaO·P₂O₅ as 2.10% and 2CaO·P₂O₅ as 0.0013%.

Fig. 8.

Contribution of 8 structural units containing P₂O₅ in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags to the total dephosphorization ability (a), and the relationship between $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ and $lg{L}_{\text{P, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\text{IMCT}}$ at elevated temperatures (b).

Since 3CaO·P₂O₅ made the main contribution to dephosphorizaiton in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags, and the 1:1 corresponding relationship between $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ and $lg{L}_{\text{P, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\text{IMCT}}$ can be found in Fig. 8(b), meanwhile, the formula of $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ against $lg{L}_{\text{P, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\text{IMCT}}$ was regressed as $lg{L}_{\text{P, calculated}}^{\text{IMCT}}$ = 0.00697 + 0.99577 $lg{L}_{\text{P, 3CaO}\cdot {\text{P}}_{\text{2}}{\text{O}}_{\text{5}}}^{\text{IMCT}}$ . Hence, it can be concluded that P₂O₅ formed tri-calcium phosphate (3CaO·P₂O₅), but not tetra-calcium phosphate, (4CaO·P₂O₅) in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags.

6. Conclusions

A thermodynamic model for calculating phosphate distribution between CaO−SiO₂−MgO−FeO−Fe₂O₃−P₂O₅ slags and liquid iron, i.e., IMCT−L_P model, has been developed coupled with a developed thermodynamic model for calculating mass action concentrations of structural units, i.e., IMCT−N_i model, based on ion and molecule coexistence theory. The main information can be summarized as follows:

(1) The calculated results from IMCT−N_i model shows that the calculated mass action concentrations of structural units, rather than mass percentage of components, are recommended to represent reaction ability of components in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags. Simple binary basicity R of slags have a large effect on total equilibrium mole number Σn_i of structural units in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags, the formula of total equilibrium mole number Σn_i against the simple binary basicity R of slags can be regressed as Σn_i = 2.604–3.029*exp(–R/2.339), and the fitting degree is 0.995.

(2) The developed IMCT–L_P prediction model can be applied to reliably predict the phosphorous distribution L_P of CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags equilibrated with liquid iron. Meanwhile, some other phosphorous distribution prediction models have also been taken into consideration for calculating phosphate distribution between CaO–SiO₂−MgO−FeO−Fe₂O₃−P₂O₅ slags and liquid iron, and the results shows that IMCT−L_P model have more accuracy compared with other phosphorous distribution prediction models.

(3) The developed IMCT−L_P model can quantitatively calculate the respective contribution of Fe_tO, CaO+Fe_tO and MgO+Fe_tO in the slags. A significant difference of dephosphorziation abilities among Fe_tO, CaO+Fe_tO and MgO+Fe_tO can be found as approximately 0.00%, 99.98%, 0.01%. Meanwhile, the phosphorus in liquid iron can be effectively extracted by CaO+Fe_tO in slags to form complex molecules 3CaO·P₂O₅ which made the main contribution to dephosphorizaiton in CaO–SiO₂–MgO–FeO–Fe₂O₃–P₂O₅ slags.

Nomenclature

A: Constant, (–);

a_i: Activity of components i in liquid iron or in slags, (–);

B: Constant, (–);

b_i: Mole number of component i in 100 g slags, (mol);

$e_i^j$ : Interaction coefficient of component j on component i in liquid iron, (–);

f_i: Activity coefficient of component i in liquid iron, (–);

${\mathrm{\Delta }}_{\text{r}}{G}_{\text{m, }i}^{\mathrm{\Theta }}$ : Standard molar Gibbs free energy of forming complex molecule i in slags, (J/mol);

(%i): Mass percentage of component i in the slags, (mass%);

[%i]: Mass percentage of component i in liquid iron, (mass%);

$K_i^{\mathrm{\Theta }}$ : Equilibrium constant of chemical reaction for forming component i or structural unit i, (–);

L_P: Phosphorus distribution between slags and liquid iron, (–);

L_{P, i}: Calculated respective phosphorus distribution of generated structural unit i containing P₂O₅ in slags based on slag oxidization ability by IMCT model, (–);

$L_{\text{P, calculated}}^{\text{IMCT}}$ : Calculated total phosphorus distribution between slags and liquid iron based on slag oxidization ability by IMCT model, (–);

L_{P, measured}: Measured phosphorus distribution, (–);

$L_{\text{P, }i\text{, calculated}}^{\text{IMCT}}$ : Calculated respective phosphorus distribution between generated structural unit i containing P₂O₅ in slags and liquid iron based on slag oxidization ability by IMCT model from calculated data, (–);

$L_{\text{P, calculated}}^i$ : Calculated phosphorus distribution between slags and liquid iron by model i, (–);

Me: Metal, (–);

MeO: Oxide component in slags, (–);

M_i: Molecular mass of element i or component i, (g/mol);

$n_i^0$ : Mole number of component i in 100 g slags, (mol);

n_i: Equilibrium mole number of structural unit i in 100 g slags, (mol);

N_i: Mass action concentrations of structural unit i in the slags, (–);

Σn_i: Total equilibrium mole number of all structural units in 100 g slags, (mol);

R: simple binary basicity, (–);

R: Gas constant, (8.314 J/(mol⋅K));

T:  Absolute temperature, (K);

Subscripts

ci: Complex molecule i, (–);

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