Thermodynamic models for predicting dephosphorisation ability and potential of CaO–FeO–Fe 2 O 3 –Al 2 O 3 –P 2 O 5 slags during secondary refining process of molten steel based on ion and molecule coexistence theory

Abstract

Thermodynamic models for predicting phosphorus distribution ratio L_P and phosphate capacity of CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags during secondary refining process of molten steel, according to the ion and molecule coexistence theory (IMCT), i.e. IMCT–L_P and IMCT– models, have been developed by coupling with the developed thermodynamic model for calculating the mass action concentrations N_i of structural units or ion couples in the slags, i.e. the IMCT–N_i model, based on the IMCT. The developed IMCT–L_P and IMCT– models have been verified with the experimental results from the literature and the predicted results by the summarised five L_P models and three models. Besides the total dephosphorisation ability and potential of the slags as L_P and , the respective phosphorus distribution ratio L_P,i and the respective phosphate capacity of six dephosphorisation products as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅ and 4CaO·P₂O₅ can also reliably be predicted by the developed IMCT–L_P and IMCT– models. 3CaO·P₂O₅ accounts for 95.00% of dephosphorisation products com- pared with 5% for 4CaO·P₂O₅. The comprehensive effect of CaO + Fe _t O, which can be described by the mass action concentration ratio , controls dephosphorisation ability and potential of the slags. An exponentially growing relationship of lg L_P against slag oxidisation ability and slag basicity as the simplified complex basicity (%CaO)/[(%P₂O₅) þ (%Al₂O₃)] or optical basicity can be established for the slags in a temperature range from 1811 to 1927 K. However, a linearly increasing relationship of lg against slag oxidisation ability and the aforementioned two kinds of slag basicity can be correlated for the slags in the lower temperature range from 1811 to 1828 K and the middle temperature range from 1870 to 1876 K; an inversely asymmetrical U type relationship of lg against the aforementioned parameters can be observed for the slags in the higher temperature range from 1918 to 1927 K. The influences of the aforementioned parameters on dephosphorisation ability are significantly different from those on dephosphorisation potential of the slags.

Introduction

Removal of phosphorus and sulphur from iron based melts is a long term challenge for metallurgical researchers and engineers to refine low or ultralow phosphorus and sulphur steel products with high performances. The technologies^1–23 on simultaneous dephosphorisation and desulphurisation of iron based melts have been developed using CaO based or Na₂CO₃ based or mixture of CaO and Na₂CO₃ fluxes or slags since the 1970s through hybridising advantages of single dephosphorisation and desulphurisation processes. Beside the widely applied hot metal pretreatment technologies as desiliconisation, dephosphorisation and desulphurisation, the simul- taneous dephosphorisation and desulphurisation process is also used in secondary refining process of molten steel. Compared with phosphorus distribution ratio L_P and sulphur distribution ratio L_S based on slag–metal equili- brium reaction for representing dephosphorisation and desulphurisation abilities of slags, the defined phosphate capacity by Wagner²⁴ and sulphide capacity by Richardson and Fincham^25,26 based on gas–slag equilibrium reaction, or the defined phosphate capacity index ; index and sulphide capacity index ; index by Yang et al.^27–29 based on slag–metal equilibrium reaction have been recommended to describe dephosphorisation and desulphurisation potentials of slags. The accumulated data from numerous researchers on dephosphorisation ability as L_P and desulphurisation ability as L_S between various metallurgical slags and iron based melts have chronologically been summarised and assessed by Young et al.²¹ before the 1990s based on experimental conditions and analytical errors.

In comparison with tremendous studies on kinetics,^{1–4,6–17,19,23} the thermodynamic study on simultaneous dephosphorisation and desulphurisation reactions of iron based melts has not widely been reported for slags in a large variation range of slag oxidisation ability.^5,18,20–22 The CaO–Fe _t O–Al₂O₃ slag system has been recommended by Ban-ya et al.²⁰ for simultaneous dephosphorisation and desulphurisation in secondary refining process of molten steel. The distribution equilibria of oxygen, phosphorus and sulphur between CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags and liquid iron were experimentally measured by Ban-ya et al.²⁰ The applied chemical composition of the slags by Ban-ya et al.²⁰ exhibits a large variation range of slag oxidisation ability corresponding to mass percentage of Fe _t O from 1.88 to 55.50%. However, the related information on dephosphorisation and desulphurisation reactions of the slags was only discussed by Ban-ya et al.²⁰ based on mole fraction of iron oxides and activity of iron oxides from the regular solution model.

To further elucidate dephosphorisation and desulphurisation mechanisms of CaO–FeO–Fe₂O₃– Al₂O₃–P₂O₅ slags in this study, it is important to determine the reaction abilities of all components and the formed complex molecules in the slags simultaneously. It should be pointed out that the results on desulphurisation ability and potential of the slags will not be involved in this paper. The related information on desulphurisation mechanisms of the slags will be reported in the next step. Although numerous activity data of components in various slags had been measured and compiled in some handbooks on chemical metallurgy, activities of all components in CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags cannot be found from the literature. With respect to experimental measurements of time consuming and requiring considerable expertise, calculating activities of all components in the slags by prediction models is a cost reduction alternative. Some activity prediction models have been proposed and summarised elsewhere,^30,31 in which the modified quasi-chemical model³² had been applied in commercial software of FactSage (jointly developed between Therm-fact/CRCT, Montreal, Canada, and GTT-Technologies, Aachen, Germany) and MTDATA (National Physical Laboratory, Teddington, Middlesex, UK) for equilibrium calculation³³ in multicomponent multiphase systems. The empirical coefficients should be more or less embodied in the summarised models^30,31 for predicting activities of components, rather than all structural units such as complex molecules in slags. Thus, none of the activity prediction models can be applied in all slags without any limitations.

The defined mass action concentrations N_i of structural units or ion couples in slags based on the ion and molecule coexistence theory (IMCT)^{30,31,34–41} for metallurgical slags can be applied to reliably represent reaction abilities of components, like the traditionally applied activities a_R,i of components relative to pure liquid or solid matters as standard state in the classical metallurgical physicochemistry. The feasibility and validity of replacing activities a_R,i of components by the mass action concentrations N_i of structural units or ion couples in slags based on the IMCT has been verified through predicting desulphurisation ability and potential of CaO–SiO₂–MgO–Al₂O₃ slags^34,35 in blast furnace ironmaking process as well as CaO–SiO₂–MgO–FeO–Al₂O₃–MnO slags^36,37 in ladle furnace refining process. Meanwhile, the validity of the mass action concentrations N_i has also been testified by predicting dephosphorisation ability and potential of CaO–SiO₂–MgO–FeO–Fe₂O₃–MnO–Al₂O₃–P₂O₅ slags^38,39 in top–bottom combined blown converter steelmaking process. Furthermore, it has also been verified by Zhang⁴¹ that the defined mass action concentrations N_i can be applied to successfully replace activities a_R,i of components in various slags.

To provide more detailed information on the dephosphorisation mechanisms of CaO–FeO–Fe₂O₃–Al₂O₃– P₂O₅ slags and to verify the accuracy of the mass action concentrations N_i of structural units or ion couples in the slags for representing reaction abilities, a thermodynamic model for predicting dephosphorisation ability of the slags, i.e. the IMCT–L_P model, has been developed by coupling with a thermodynamicmodel for calculating N_i of structural units or ion couples in the slags, i.e. the IMCT–N_i model, based on the IMCT.^{30,31,34–41} After that, a thermodynamic model for predicting dephosphorisation potential of the slags, i.e. the IMCT– model, has also been proposed through the correlated relationship between L_P and . The developed IMCT–L_P model has been verified by comparing the measured²⁰ values of L_P by Ban-ya et al. with the calculated ones by the developed IMCT–L_P model and the calculated ones by the summarised³⁸ five L_P models.^43–47 Meanwhile, the developed IMCT– model has also been validated by comparing with the determined values of after the original data from Ban-ya et al.²⁰ with the calculated ones by the developed IMCT– model and the calculated ones by the summarised³⁸ five L_P models^43–47 or the listed³⁹ three models.^21,48–50 The dephosphorisation mechanism of the slags has been elucidated. The influence of slag chemical composition including slag oxidisation ability, slag basicity and the comprehensive influence of iron oxides Fe _t O and basic oxide CaO on dephosphorisation ability and potential of the slags have also been discussed. Furthermore, the contribution of each dephosphorisation product containing P₂O₅ to dephosphorisation ability and potential of the slags is also predicted.

The ultimate aims of the seriate study can be summarised as to reveal and verify the accuracy of the mass action concentrations N_i for representing reaction abilities of structural units in slags, like the traditionally applied activities a_R,i of components relative to pure liquid or solid matters as standard state and to enrich the foundations of dephosphorisation mechanism by the slags in a large variation range of slag oxidisation ability.

Cited experimental data of oxygen, phosphorus and sulphur distribution equilibria between CaO–FeO–Fe₂O₃– Al₂O₃–P₂O₅ slags and liquid iron

The original chemical compositions of both CaO–FeO– Fe₂O₃–Al₂O₃–P₂O₅ slags and liquid iron for 31 test runs of equilibrium experiments of oxygen, phosphorus and sulphur between 10 g liquid iron without carbon and 15 g CaO–Fe _t O–Al₂O₃ slags from Ban-ya et al.²⁰ are summarised in Table 1 in a decreasing order or hierarchy of the calculated oxygen potential of the slags based on (Fe _t O)–{O₂} (J mol^–1) by^51–53

where denotes the calculated comprehensive mass action concentration of iron oxides by the developed IMCT–N_i model described in the section on ‘Validation of comprehensive mass action concentration of iron oxides in slags’, p^Θ depicts the standard pressure of gas at sea level and 273 K as 101 325 Pa. Obviously, the activity of [Fe] as solvent in liquid iron relative to pure liquid Fe(l) as standard state can be thought as 1.0. The original test run numbers of 31 experiments from Ban-ya et al.²⁰ are also listed in Table 1. It should be emphasised that the designed chemical composition of the slags by Ban-ya et al.²⁰ follows the CaO saturation line, but is not saturated with CaO. In addition, the oxygen content of liquid iron in test run nos. 30 and 31 was not analysed by Ban-ya et al.²⁰ To keep consistent, the applied CaO– Fe _t O–Al₂O₃ slag system by Ban-ya et al.²⁰ is represented as CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in this study.

Table 1

Original chemical compositions of slags and liquid iron after Ban-ya et al.,²⁰ calculatedmass action concentrations N_i of structural units as components, calculated total equilibrium mole numbers Σn_i of structural units in 100 g slags, calculated oxygen potentialpO2based on , measured²⁰ lg L_{P, measured} by Ban-ya et al., predicted lg from the developed IMCT–L_P model, determined lg after Ban-ya et al.²⁰ and calculated lg from the developed IMCT– model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1811 to 1927 K.

Thermodynamic model for calculating mass action concentrations N_i of structural units or ion couples in slags based on IMCT

Hypotheses

According to the classical hypotheses of the IMCT,^{30,31,34–41} the main assumptions for developing the thermodynamic model for calculating N_i of structural units or ion couples in CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags, i.e. the IMCT–N_i model, can briefly be summarised as follows. First, structural units in the slags are composed of Ca²⁺ , Fe²⁺ and O^2– as simple ions, Fe₂O₃, Al₂O₃ and P₂O₅ as simple molecules and calcium aluminates and so forth as complex molecules. Each structural unit occupies its independent position in the slags. Every cation and anion generated from the same component will take part in reactions for forming complex molecules in the form of ion couple (Me²⁺ + O^2–) with simple molecules by taking MeO type oxide as an example. Second, reactions of forming complex molecules ci areunder dynamic chemical equilibrium between the bonded ion couples (Me²⁺ + O^2–) and simple molecules. Third, structural units in the slags keep continuity in the range of the investigated concentration range. Fourth, chemical reactions of forming complex molecules ci conform to the mass action law^54–56 through by taking xMe1O·yMe2O as an example of complex molecule or associated molecule ci.

IMCT–N_i thermodynamic model for slags

Structural units in slags

The IMCT^{30,31,34–41} suggests that the extracted phosphorus from liquid iron into the slags can be bonded by ion couples (Fe²⁺ + O^2–) or (Ca²⁺ + O^2–) as well as iron oxides Fe _t O to form structural units as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅ and 4CaO·P₂O₅ with the proceeding of simultaneous dephosphorisation and desulphurisation reactions. Hence, the applied CaO–FeO–Fe₂O₃–Al₂O₃ slags by Ban-ya et al.²⁰ will change from an open slag system without phosphate at the initial stage to a close slag system containing phosphate at the final stage equilibrated with liquid iron because the IMCT can only be applied to a close slag system. Therefore, the applied CaO–FeO–Fe₂O₃–Al₂O₃ slags by Ban-ya et al.²⁰ as an open slag system will be transformed to CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags as a close slag system by omitting desulphurisation reactions.

It can reasonably be obtained from the aforementioned assumptions in the section on ‘Hypotheses’ that there are three simple ions as Ca²⁺ , Fe²⁺ and O^2– by ignoring S^2–, three simple molecules as Fe₂O₃,Al₂O₃ and P₂O₅ in the slags at metallurgical temperatures based on the IMCT. According to the reported^51,57 binary and ternary phase diagrams of CaO–FeO, CaO–Al₂O₃, CaO–Al₂O₃–FeO, CaO–Al₂O₃–Fe₂O₃ systems and so forth at the elevated temperatures, ∼ 13 kinds of complex molecules such as 3CaO·Al₂O₃, FeO·Al₂O₃, 2CaO·Fe₂O₃ and so forth can be formed in the slags as listed in Table 2. All simple ions, simple and complex molecules in the slags are also summarised on the left region of Table 2 with exclusive numbers. It should be stressed that ignoring sulphide S^2– in the slags cannot cause visible errors on the related results of the developed IMCT–N_i model due to very low content of S in the slags.

Table 2

Expression of structural units as ion couples, simple or complex molecules, their mole numbers n and mass action concentrations N_i, chemical reaction formulae of possibly formed 13 complex molecules ci, their standard molar Gibbs free energy changes Δ_r , standard equilibrium constants and mass action concentrations N_ci in CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron based on the IMCT

Development of IMCT–N_i thermodynamic model for slags

The mole number of the aforementioned five components as CaO, FeO, Fe₂O₃, Al₂O₃ and P₂O₅ in 100 g slags is assigned as respectively. The defined^{30,31,34–41} equilibrium mole number ni of the aforementioned 18 structural units in 100 g slags is also expressed on the left region of Table 2 respectively. The total equilibrium mole number Sni (mol) of all 18 structural units in 100 g slags can be expressed as^{30,31,34–41}

According to the defined^{30,31,34–41} N_i of structural unit i, which is a ratio of equilibrium mole number N_i of structural unit i to the total equilibrium mole number Σn_i of all structural units in a close slag system at a fixed amount, the mass action concentration N_i of structural unit i or ion couple(Me²⁺ + O^2–) in slags can be calculated by^{30,31,34–41}

All the defined^{30,31,34–41} N_i of the formed ion couples from simple ions, the simple and complex molecules in the slags are also listed on the left region of Table 2.

The chemical reaction formulae of the possibly formed 13 complex molecules, their standard molar Gibbs free energy changes of formation reactions as a function of absolute temperature T, standard equilibrium constants and representation of N_ci for all 13 complex molecules expressed by , N₁ (N_CaO), N₂ (N_FeO), N₃ ( ), N₄ ( ) and N5 ( ) based on the mass action law^54–56 are summarised on the right region of Table 2.

The mass conservation equations of five components in 100 g slags (mol) can be established from the definitions ^{30,31,34–41} of n_i and N_i in Table 2 and Σn_i in equation (2) as

According to the principle that the sum of equilibrium mole fractions for all structural units in a fixed amount of slags is equal to unity, the summation of N_i for all 18 structural units in the slags can be described by^{30,31,34–41}

The equation group of equations (4) and (5) is the governing equations of the developed IMCT–N_i model for calculating N_i of structural units or ion couples in CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags. However, no convergent solutions of the aforementioned six unknown variables can be obtained by solving the equation group because the solved values of are < 10^–20, in accordance with the reported^{2,4,16,64–67} values of in metallurgical slags < 10^–19. Thus, the P₂O₅ free slag system of CaO–FeO–Fe₂O₃–Al₂O₃ slags is applied to substitute CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags. The mass percentage of P₂O₅ in the slags is < 3.0; replacing CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags by CaO–FeO–Fe₂O₃–Al₂O₃ slags cannot generate inconceivable errors on the calculated values of N_i, Σn_i and N_i. Therefore, the equation group of equations (4a)– (4d) and (5) should be simplified by deleting N₅ ( ) and N_c9–N_c13 for CaO–FeO–Fe₂O₃–Al₂O₃ slags. The calculated values of N₁ (N_CaO), N₂ (N_FeO), N₃ ( ), N₄ ( ) and Σn_i in 100 g CaO–FeO–Fe₂O₃–Al₂O₃ slags after the original data from Ban-ya et al.²⁰ are summarised in Table 1 respectively. The values of the calculated Σn_i in 100 g slags do not show a large variation for the investigated 31 test runs as listed in Table 1 with 2.56 mol as an average datum.

Estimation of mass action concentrations N_i of structural units or ion couples containing P₂O₅ in slags

Absenting values of cannot affect the development of the IMCT–L_P model as described in the section on ‘Thermodynamic models for calculating dephosphorisation ability and potential of slags based on IMCT’. However, getting the accurate values of is also an important task in this study compared with no values of for CaO–SiO₂–MgO–FeO–Fe₂O₃–MnO–Al₂O₃–P₂O₅ slags in a previous publication.38 According to the obtained values of N₁ (N_CaO), N₂ (N_FeO), N₃ ( ), N₄ ( ) and Σn_i in 100 g CaO–FeO–Fe₂O₃–Al₂O₃ slags, values of can be estimated by inserting the obtained data of N₁ (N_CaO), N₂ (N_FeO), N₃ ( ), N₄ ( ) and Σn_i into the mass conservation equation of P₂O₅ for the slags in equation (4e) as

According to the estimated values of NP₂O₅ in Table 1, values of Nci for structural units containing P₂O₅ such as 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅ and 4CaO·P₂O₅ can also be determined, but will not be reported in this study for the sake of concision.

Re-evaluation of IMCT–N_i model

Similarities and differences between applied IMCT and other structural theories for slags

Obviously, some assumptions of both molecular theory and ionic theory for metallurgical slags have been embodied in the IMCT. The similarities and differences among the IMCT, molecular theory and ionic theory can be summarised as three statements.⁴¹ First, molecular theory, which assumes all components as simple molecules and complex molecules in slags, can be applied to successfully elucidate some dephosphorisation and desulphurisation properties, although it cannot reasonably explain the electrical conductive characteristics as well as the electrolytic properties of slags. To obtain constant values of the standard equilibrium constants of dephosphorisation and desulphurisation reactions, some empirical coefficients should be applied in the molecular theory. Second, ionic theory, especially fully ionic theory, which assumes all structural units or components as cations or anions, cannot explain the experimental fact that some ions can be bonded as molecules due to the saturated covalent bonding. Third, the IMCT assumes that both ions and molecules can coexist in slags by hybridising the reasonable assumptions of both molecular theory and ionic theory for metallurgical slags. Additionally, a cation and an anion should be treated as an ion couple when participating in reactions for forming complex molecules.

Actually, the classical hypotheses of the IMCT^{30,31,34–41} are to some degree similar with those of the associated solutionmodel,70–75 especially the ideal associated solution model.^70,71

The defined Ni of the structural unit i or ion couple (Me²⁺ + O^2–) in slags by equation (3) based on the IMCT looks like the defined activity a_R,i by the molecular theory. As one of the most important structural theories for metallurgical slags, the molecular theory was first introduced to metallurgical slags by Schenck et al.⁷⁶ in the 1930s, and further applied by Chipman et al.,^77–80 Darken et al.^81,82 and so on since the 1940s. Zhang has verified in his book⁴¹ and one article⁸³ that the calculated Ni of the components in the five slag systems as MnO–SiO₂, FeO–Fe₂O₃–SiO₂, FeO–Fe₂O₃–TiO₂, FeO–Fe₂O₃–B₂O₃, CaO–FeO–Fe₂O₃–SiO₂ slags are more accurate than the calculated activities a_R,i of the components by the molecular theory, while the calculated N_i of the components in MnO–FeO–Al₂O₃ slags show the similar precision with the calculated activities a_R,i by molecular theory.

Validity of replacing activities a_R,i by mass action concentrations N_i of components in slags

It is well known that the sum of activities a_R,i for the components in slags is not absolutely equal to unity, i.e. , considering the fact that activity coefficients γ_i of components are usually not equal to 1.0 for most slags. Hence, the absolute values of the defined Ni of structural units are not absolutely equal to the defined activities a_R,i of components. Under this circumstance, the values of the defined Ni of structural units in slags can only be compared with activities a_R,i of components. However, it has been verified^30,40,41 that the aforementioned theoretical defect cannot largely affect the feasibility and validity of replacing activity a_R,i by N_ci and N_i considering the uncertainties of the reported standard molar Gibbs free energy changes of reactions for forming the complex molecule ci.

Results of IMCT–N_i model for slags

Relationship between mass percentage of five components and mass action concentration Ni of related structural units or ion couples in slags

The relationship between mass percentage and the calculated N_i of CaO, FeO, Fe₂O₃, Al₂O₃ and P₂O₅ for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in a temperature range from 1811 to 1927 K is shown in Fig. 1 respectively. The good linear relationship between mass percentage and the calculated N_i for CaO, FeO and P₂O₅ can be observed in Fig. 1a, b and e; while the non-linear corresponding relationship with higher fitting degree can be found for Fe₂O₃ and Al₂O₃ in Fig. 1c and d. The results in Fig. 1 indicate that the calculated Ni of all five components, like mass percentages of components, can be applied to represent chemical composition of the slags.

Relationship between mass percentage and calculated mass action concentration N_i of CaO, FeO, Fe₂O₃, Al₂O₃ and P₂O₅ by developed IMCT–N_i model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in a temperature range from 1811 to 1927 K respectively

Validation of comprehensive mass action concentration of iron oxides in slags

The IMCT suggests that all iron oxides in metallurgical slags are composed of ion couple (Fe²⁺ + O^2–), simple molecule Fe₂O₃ and complex molecule FeO·Fe₂O₃. Thus, the related structural units of iron oxides can dynamically equilibrate as^30,38–41

It can be derived from equation (7a) that the contribution of the simple molecule Fe₂O₃ to slag oxidisation ability is equivalent to three times that of the ion couple (Fe²⁺ + O^2–). Likewise, the contribution of the complex molecule FeO·Fe₂O₃ to slag oxidisation ability is four times that of the ion couple (Fe²⁺ + O^2–) as shown in equation (7b). Therefore, slag oxidisation ability expressed by the comprehensive mass action concentration of iron oxides can be defined as^30,38–41

The activity of Fe _t O in the slags can be calculated based on (Fe _t O)–[O] equilibrium (J mol^–1) as^51,52

The activity a_%,O of [O] in liquid iron mass percentage of element [O] as standard state can be determined through a_%,O = [%O] f_%,O according to the measured [%O] in liquid iron by Ban-ya et al.²⁰ as listed in Table 1. The involved activity coefficient f_%,O of [O] can be determined by Wagner's equation as^51,55,56

where A and B are two parameters related with the first order activity interaction coefficient . Because values of for element [j] to [O] have a little change with temperature at ∼ 1873 K, the value of for element [j] to [O] is chosen^51,84 as = 0.07, = –0.133 and = –0.2 respectively.

Comparison between the calculated activity of Fe _t O based on equation (9) and the calculated N_FeO of FeO as well as of iron oxides for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags is illustrated in Fig. 2. Obviously, magnitude of of iron oxides is predominated by N_FeO of FeO. The calculated values of of iron oxides give good correlation with the determined data of activity of Fe _t O in the slags. This means that activity of Fe_tO in the slags can accurately be substituted by the calculated of iron oxides for representing slag oxidisation ability of the slags.

Comparison between activity of and comprehensive mass action concentration of iron oxides or mass action concentration of FeO for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1811 to 1927 K respectively

Validation of mass action concentration NP₂O₅ of P₂O₅ in slags

It is indicated by numerous researchers^{2,4,16,64–69} that activity coefficient of P₂O₅ in various steelmaking slags is < 10^–17, as mentioned in the section on ‘Development of IMCT–N_i thermodynamic model for slags’. Three models for predicting in various steelmaking slags are chronologically summarised in Table 3. Comparison between the calculated activity of P₂O₅ by the summarised three prediction models and the calculated of P₂O₅ for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags is illustrated in Fig. 3.

Comparison between mass action concentration of and calculated activity of in logarithmic form for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1811 to 1927 K respectively

Table 3

Summary of prediction models of activity coefficient of P₂O₅ in various steelmaking slags

The calculated values of give good correlation with the calculated data of by Suito's model,⁴³ while the calculated values of by Turkdogan's no. 1⁶⁸ and no. 2⁶⁴ models are four orders of magnitude greater than the calculated data of . Compared with Turkdogan's no. 1⁶⁸ and no. 2⁶⁴ models, Suito's model⁴³ is suitable for predicting activity of P₂O₅ in the slags. Thus, the developed IMCT–N_i model and Suito's model⁴³ can be recommended to precisely predict the reaction ability of P₂O₅ in the slags.

Thermodynamic models for calculating dephosphorisation ability and potential of slags based on IMCT

Thermodynamic model for calculating dephosphorisation ability of slags based on IMCT

According to the IMCT,^{30,31,34–41} the dephosphorisation reactions between CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags and liquid iron can be represented by the ion couple (Fe²⁺ + O^2–) or (Ca²⁺ + O^2–) with iron oxides Fe _t O to form six dephosphorisation products or molecules as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅ and 4CaO·P₂O₅ (J mol^–1) as³⁸

The corresponding of dephosphorisation reactions for forming six dephosphorisation products in equation (11) can be determined from the reported data as summarised by Yang et al. elsewhere.³⁸ The corresponding of dephosphorisation reactions in equation (11) can be expressed through replacing activities a_R,i of components by the calculated N_ci, , N_FeO and N_CaO for the slags as

where is the relative molecular mass of P₂O₅ as 141.94. It should be stressed that the mass action concentrations of dephosphorisation products containing P₂O₅ by taking 3CaO·P₂O₅ as an example can be defined as based on the IMCT.^{30,31,34–41} According to the expression of the related in equation (12), the respective phosphorus distribution ratio L_P,i of the generated dephosphorisation product i containing P₂O₅ in the slags can individually be described by

The total phosphorus distribution ratio L_P between the slags and liquid iron can be derived from equation (13) as

The equation (14) is the developed IMCT–L_P model based on the defined^30,38–41 of iron oxides in equation (8) for representing slag oxidisation ability. It can be obtained from Table 1 that values of the involved of iron oxides in equation (14) are predominated by those of N_FeO of FeO, while values of both and N_FeO·Fe₂O₃ are very small compared with those of N_FeO. Meanwhile, the calculated values of of iron oxides give good correlation with the determined data of activity of Fe _t O in the slags as shown in Fig. 2. Therefore, the determined values of oxygen potential of the slags by equation (1) based on (Fe _t O)–{O₂} equilibrium reaction through replacing activity by of iron oxides are also listed in Table 1.

The involved activity coefficient f_{%, P} of [P] in liquid iron in equation (14) can be calculated by Wagner's equation in equation (10). The value of for element [j] to [P] in liquid iron at a temperature of 1873 K is summarised^51,84 as = 0.062, = –0.028 and = 0.13 respectively. Besides values of the measured²⁰ L_P, measured by Ban-ya et al., data of the calculated by the developed IMCT2–L_P model are also listed in Table 1 for further evaluation.

Thermodynamic model for calculating dephosphorisation potential of slags based on IMCT

To establish the IMCT– model, the relationship between L_P and should first be applied. Three kinds of phosphorous distribution ratio as and have widely been applied to depict dephosphorisation ability of slags. To distinguish L_P, and , the applied L_P in this study is defined to be equivalent to . The relationship between or or and has been derived by Yang et al. elsewhere.³⁹ The previously proposed relationship between or or and by Yang et al.³⁹ has been made a minor modification for some coefficients in this study. It should be emphasised that the proposed relationship between or or and in this study can cause omitted difference for the predicted values of compared with the predicted ones of by the previously proposed39 relationship between or or and .

Relationship between phosphorus distribution ratio and phosphate capacity for slags

Relationship between phosphate capacity and phosphate capacity index for slags

It has been verified by Sano et al.^85,86 that oxygen potential p_O2 of slags >10⁻¹³ Pa results in the formation of phosphate in slags, whereas under more reducing conditions, phosphide P³⁻ will account for most of the phosphorus in slags. The dephosphorisation reaction through slag–metal reactions is usually expressed with P₂O₅ or as dephosphorisation products. The definition of by Wagner²⁴ based on gas–slag dephosphorisation equilibrium reaction can be expressed as^85,86

The definition of phosphate capacity index by Yang et al.^27,29,39 based on slag–metal dephosphorisation equilibrium reaction can be described as^85,86

Phosphorus in slags is usually analysed as P₂O₅ by routine chemical analysis methods. In the view of the main dephosphorisation products as 3CaO·P₂O₅,³⁹ the mole number of 3CaO·P₂O₅ can be considered to equal that of the analysed total P₂O₅. According to the P and O equilibrium in phosphate , the mass percentage of can reliably be estimated from the mass percentage of P₂O₅ by

Comparing between the gas–slag dephosphorisation reaction in equation (15a) and the slag–metal dephosphorisation reaction in equation (15b), the gas–metal reaction of P₂ dissolving into Fe based melts (J mol⁻¹) can be expressed as⁵³

Likewise, the gas–metal reaction of O² dissolving into Fe based melts (J mol⁻¹) can be described by⁵³

The defined phosphate capacity by Wagner²⁴ in equation (15a) can be rewritten by combining the expressions of both phosphorus potential in equation (17b) and oxygen potential in equation (18b) as³⁹

The relationship between and for slags can be derived from equation (19a) in logarithmic form as³⁹

It can be obtained from equation (19b) that of slags is much greater than . It should be emphasised that the measured of various slags by some researchers^21,48–50 was nominated as . Certainly, not distinguishing between and can lead to misunderstanding the magnitude of both and for a fixed slag. The correlated³⁹ relationship between and for slags in equation (19b) can be used to link and for slags at a fixed temperature.

To describe precisely, phosphate capacity index of the studied slags will not be paid more attention because , rather than , for various slags can easily be found in the related literature.

Relationship between phosphorus distribution ratio and phosphate capacity for slags

The dephosphorisation reaction through slag–metal reaction (J mol⁻¹) is usually described as^53,87

The expression of standard equilibrium constant of dephosphorisation reaction in equation (20b) can further be rewritten by combining the relationship between mass percentage of P₂O₅ and in equation (16), the definition of in equation (15b), and the relationship between and for slags in equation (19b) as

Consequently, the relationship between L_P and for slags can be established by inserting the definition of L_P=(%P₂O₅)/[%P]² into equation (20c) as

The relationship between L_P and for slags can be expressed by inserting in equation (16b) and in equation (18b) into equation (21) as

Likewise, the relationship between and for slags can be described fromequation (22a) as

Additionally, the relationship between and for slags can be derived from equation (22a) as

The equation group of equations (22a)–(22c) depicts the relationship between or or and for slags. Consequently, the phosphate capacity of slags can be determined by phosphorous distribution ratio or or through equation (22) after knowing the chemical composition of iron basedmelts, especially [%O], [%P] and temperature T under slag–metal equilibrium. Fortunately, the required parameters were measured by Ban-ya et al.²⁰ except for the oxygen content of liquid iron in the test run nos. 30 and 31 as mentioned in the section on ‘Cited experimental data of oxygen, phosphorus and sulphur distribution equilibria between CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags and liquidiron’.Toobtain the oxygen contentof liquid iron in the test run nos. 30 and 31, activity of oxygen at the slag/metal interface can be calculated based on the [O]–(Fe _t O) equilibrium in equation (9) through replacing activity as

Establishment of thermodynamic model for calculating phosphate capacity of slags based on IMCT

The IMCT– model for the slags can be derived by inserting the developed IMCT–L_P model in equation (14) and the relationship of activity of oxygen at the slag/metal interface against in equation (23) into the relationship between or or and for slags in equation (22) as

It can be obtained from the developed IMCT– model in equation (24) that the dephosphorisation potential in logarithmic form lg can correlate with 5/2 lg . However, it can also be found from the developed IMCT–L_P model in equation (14) that dephosphorisation ability in logarithmic form lg L_P can establish with 5 lg . Thus, the effect of slag oxidisation ability expressed by of iron oxides on is smaller than that on L_P because and L_P are both distinct thermodynamic quantities and related to each other provided that the entire slag system is correctly defined.

It should be emphasised that the claimed determined after the original data from Ban-ya et al.²⁰ can be obtained from the relationship between and for slags in equation (22a) based on the determined activity a_{%, O} of oxygen in bulk liquid iron through . However, the calculated by the developed IMCT– model in equation (24) is based on the calculated , and by the developed IMCT–N_i model for the slags. Thus, the determined lg after original data from Ban-ya et al.²⁰ as well as the predicted by the developed IMCT– model for the slags are also listed in Table 1 for further evaluation.

Evaluation of developed IMCT–L_P and IMCT – models for slags based on IMCT

Two steps have been applied to evaluate and verify the accuracy of the developed IMCT–L_P model in equation (14) as well as the developed IMCT–

model in equation (24). The first step is to compare the measured²⁰ L_{P, measured} by Ban-ya et al. or determined

after Ban-ya et al.²⁰ with the calculated

by the developed IMCT–L_P or IMCT–

models. The second step is to compare the measured²⁰ L_{P, measured} by Ban-ya et al. or determined

after Ban-ya et al.²⁰ with the calculated

by the listed five L_P models^43–47 or the predicted

by aforementioned five L_P models^43–47 and the listed three

models^21,48–50 in Table 4.

Table 4

Summary of five phosphorus distribution ratio L_P models and three phosphate capacity models for various slags from related literature

Evaluation of developed IMCT– L_P model for slags

Comparison between measured L_{P, measured} and calculated by developed IMCT–L_P model for slags

Comparison between the measured²⁰ lg L_{P, measured} by Ban-ya et al. and the calculated lg for the slags is illustrated in Fig. 4a. Obviously, the developed IMCT–L_P model can give good correlation with the measured²⁰ values of L_P in the middle temperature range from 1870 to 1876 K as well as in the higher temperature range from 1918 to 1927 K, while the developed IMCT–L_P model gives slightly higher values than the measured²⁰ ones of L_P in the lower temperature range from 1811 to 1828 K. Thus, the developed IMCT–L_P model in equation (14) can be applied to predict dephosphorisation ability of the slags at temperatures over 1870 K.

Comparison between measured²⁰ lg L_{P, measured} by Ban-ya et al. and lg by developed IMCT–L_P model a or calculated lg by other five L_P prediction models b–f for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in temperature range from 1811 to 1927K respectively

Comparison between measured L_{P, measured} and calculated by other five L_P models for slags

Evaluation of other five L_P models

Comparing between the measured²⁰ values of L_P by Ban-ya et al. and the predicted ones by the reported L_P models is also of importance to verify the feasibility of the developed IMCT–L_P model. Obviously, the comprehensive effects of temperature T and slag composition such as CaO, MgO, MnO, SiO₂ and Fe _t O or T·Fe or FeO on L_P have been considered in the listed five L_P models in Table 4. All the listed five L_P models in Table 4 are empirical thermodynamic models from mathematical regression of experimental data. The main conclusions of the listed five L_P models in Table 4 can be summarised as follows:

increasing contents of basic components such as CaO, MgO and MnO can improve dephosphorisation ability; most L_P models in Table 4 except for Healy's model⁴⁴ indicate that contribution of CaO on dephosphorisation ability is greater than that of MgO and MnO respectively

the positive effect of iron oxides expressed as Fe _t O or T·Fe or FeO on dephosphorisation ability is smaller than that of each basic component as CaO, MgO and MnO with the same content

increasing temperature directly results in a decreasing tendency of dephosphorisation ability of slags due to the temperature dependence of the dephosphorisation reaction; a further decreasing trend of dephosphorisation ability of basic components such as CaO, MgO and MnO is seen by Sommerville's model^46,47

SiO₂ displays a very small contribution to dephosphorisation ability of slags

most models indicate that P₂O₅ can reduce dephosphorisation ability of slags

the individual effect of iron oxides Fe _t O and basic oxide CaO has been embodied in the listed five P models, while the comprehensive of iron oxides Fe _t O and basic oxide CaO on dephosphorisation ability is not explicitly expressed as one independent term in the listed five L_P models.

Comparison between measured L_{P, measured} and calculated by other five L_P models for slags

Comparison between the measured²⁰ lg L_{P, measured} by Ban-ya et al. and the calculated by the listed five L_P models in Table 4 for the slags is shown in Fig. 4b–f respectively. It should specially be emphasised that the calculated by the listed five L_P models in Table 4 has been transferred into (%P₂O₅)/[%P]², instead of other expressions like (%P)/[%P] in Healy's model⁴⁴ or (%P₂O₅)/[[%P]²(%Fe _t O)⁵] in Suito's no. 2 models^43,45 or (%P₂O₅)/[[%P]²(%T·Fe)⁵] in Suito's no. 3 model^43,45 or (%P₂O₅)/[%P] in Sommerville's model^46,47 as listed in Table 4.

Generally, all the models in Fig. 4b–f can be applied to reliably predict dephosphorisation ability of the slags, although none of the listed five L_P models in Table 4 was developed for the studied slags without SiO₂. The calculated values of L_P by Healy's model⁴⁴ in Fig. 4b or Suito's no. 2model^43,45 in Fig. 4c or Suito's no. 1model^43,45 in Fig. 4f are more or less greater than the measured²⁰ ones by Ban-ya et al. The calculated values of L_P by Suito's no. 3 model^43,45 in Fig. 4d and Sommerville's model^46,47 in Fig. 4e are more accurate than the calculated ones by other three L_P models based on the measured²⁰ ones by Ban-ya et al. as criteria. Thus, the developed IMCT–L_P model in Fig. 4a, Suito's no. 3 model^43,45 in Fig. 4d and Sommerville's model^46,47 in Fig. 4e can be used to reliably predict dephosphorisation ability of the slags.

The listed five L_P models in Table 4, especially Suito's no. 3 model^43,45 and Sommerville's model^46,47 can only correlate with contents of components and temperature T coupled with the regressed coefficients from experimental data.However, thedevelopedIMCT–L_P model in equation (14) can correlate with the determined N_i and Σn_i by the developed IMCT–N_i model for the slags, the standard equilibrium constant and activity coefficient f_{%, P} of [P] in liquid iron. This means that none of the empirical coefficients from the experimental data is required in the developed IMCT–L_P model in equation (14). Hence, the developed IMCT–L_P model has clearly physicochemical meanings comparedwith the listed five Pmodels, especially Suito's no. 3 model^43,45 and Sommerville's model^46,47 in Table 4. The measured²⁰ values of L_P by Ban-ya et al. and the calculated ones by the developed IMCT–L_P model in equation (14) will be applied as the basis for further evaluation in the section on ‘Influence of slag chemical composition on dephosphorisation ability and potential of slags’.

Evaluation of developed IMCT– model for slags

Comparison between determined and calculated by developed IMCT– model for slags

Comparison between the determined lg after the original data from Ban-ya et al.²⁰ and the calculated lg for the slags is shown in Fig. 5a. Obviously, values of the calculated lg are in good agreement with the determined ones after Ban-ya et al.²⁰ for the slags in the middle and higher temperature ranges. However, the developed IMCT– model can give significantly higher values than the determined ones after Ban-ya et al.²⁰ for the slags in the lower temperature range from 1811 to 1828 K. It can be deduced that the aforementioned conclusions on L_P in the section on ‘Comparison between measured L_{P, measured} and calculated by developed IMCT–L_P model for slags’ are also valid for . The higher values of the calculated dephosphorisation ability are the reason for the greater ones of dephosphorisation potential by the developed IMCT– model in the lower temperature range. This means that the accuracy of L_P is very important to obtain precise of slags through the relationship between L_P and for slags in equation (22a).

Comparison between determined lg after Ban-ya et al.²⁰ and calculated lg by developed IMCT– model a or calculated lg by five L_P prediction models b–f as well as three models g–i for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in temperature range from 1811 to 1927K respectively

Comparison between determined and calculated by some L_P or models for slags

Evaluation of other three phosphate capacity models

According to the relationship between L_P and in equation (22a), the listed five L_P models for various slags in Table 4 can also be transferred to be prediction models.Meanwhile, the summarised three models for various slags in Table 4 are also selected to verify the feasibility of the developed IMCT– model. Obviously, only optical basicity and temperatureTare included in the listed two models as Mori's model⁴⁹ and Suito's model,^48,50 while optical basicity andmass percentages of Fe _t O, MnO and P₂O₅ are contained in Young's model.²¹ Certainly, most L_P models^43–47 and models^21,48–50 in Table 4 can be sorted as empirical models from mathematical regression of experimental data.

Comparing with the main conclusions from the listed five L_P models in the section on ‘Evaluation of other five L_P models’, the main statements from the listed three models in Table 4 can be summarised as follows:

increasing contents of basic components such as CaO and decreasing contents of acidic components such as SiO₂ and Al₂O₃ increase dephosphorisation potential and result in greater values of optical basicity of slags

greater optical basicity leads to larger values of dephosphorisation potential of slags in the summarised three models

Fe _t O and MnO show small negative effects on phosphate capacity of slags in Young's model²¹

P₂O₅ has a small promotive effect on phosphate capacity of slags as showninYoung's model²¹

increasing temperature results in a decrease in phosphate capacity of slags, which is included in Suito's model^48,50 and Young's model.²¹

Comparison between determined and calculated by different L_P or or models for slags

Comparison between the determined lg after the original data from Ban-ya et al.¹² and the calculated lg by the listed five L_P models as well as the summarised three models in Table 4 for the slags are shown in Fig. 5b–i respectively. It should specially be emphasised that the calculated values of dephosphorisation potential have been transferred to phosphate capacity , rather than the defined phosphate capacity index in Suito's model^48,50 and Young's model²¹ as listed in Table 4. The calculated values of dephosphorisation potential by the listed five L_P modelswere obtained fromthe relationship between L_P and for slags in equation (22a) based on the calculated activity of oxygen at the slag/metal interface through equation (23). In addition, the applied optical basicity of the slags for the summarised three models was determined by taking = 1.0 and = 0.75,⁸⁸ similar to the recommended = 1.0 and = 0.77 by Young et al.²¹ It can be observed in Fig. 5b–i that Healy's model⁴⁴ in Fig. 5b, Suito's no. 2 model^43,45 in Fig. 5c, Sommerville's model^46,47 in Fig. 5e and Young's model²¹ in Fig. 5i can be applied to reliably predict of the slags. Other models such as Suito's no. 3 model^43,45 in Fig. 5d, Suito's no. 1 model^43,45 in Fig. 5f, Mori's model⁴⁹ in Fig. 5g and Suito's model^48,50 in Fig. 5h show larger scatter and deviation from the perfect fit. It can be concluded from Fig. 5 that the developed IMCT– model, Healy's model⁴⁴, Suito's no. 2 model^43,45, Sommerville's model^46,47 and Young's model²¹ can be applied to reliably predict dephosphorisation potential of the slags. Therefore, the determined values of phosphate capacity after the original data from Ban-ya et al.²⁰ and the calculated ones by the developed IMCT– model will be applied as basis for further discussion in the section on ‘Influence of slag chemical composition on dephosphorisation ability and potential of slags’.

Influence of slag chemical composition on dephosphorisation ability and potential of slags

Influence of slag oxidation ability on dephosphorisation ability and potential of slags

The common consensuses^{21,42,44–48,50} on dephosphorisation reactions from the accumulated industrial experiences and experimental studies indicate that greater slag oxidation ability and lower temperature are two key factors to promote dephosphorisation from liquid iron or molten steel to slags. Three parameters as mass percentage of Fe _t O through (%Fe _t O) = (%FeO) + 0:9(%Fe₂O₃)^54–56 based on mass conservation of iron, the calculated of iron oxides, and the determined oxygen potential of slags can be applied to represent slag oxidation ability. The relationship of the calculated of iron oxides against mass percentage of FeO or Fe₂O₃ or Fe _t O or the determined of the slags is shown in Fig. 6 respectively. A good corresponding relationship of the calculated of iron oxides against mass percentage of FeO or Fe _t O can be observed in Fig. 6a, while a scattered corresponding relationship of the calculated of iron oxides against mass percentage of Fe₂O₃ can also be correlated. In addition, the determined of the slags can establish a good corresponding relationship with the calculated of iron oxides in the second power function as shown in Fig. 6b. Thus, good corresponding relationships among the aforementioned three parameters can be correlated for representing slag oxidation ability. To describe concisely, mass percentage of Fe _t O as the easy to measure parameter is applied to represent slag oxidation ability in this study. The influence of mass percentage of Fe _t O on the measured20 lg by Ban-ya et al. or the calculated lg for the slags in a temperature range from 1811 to 1927 K is illustrated in Fig. 7 respectively. Obviously, an exponentially growing tendency of dephosphorisation ability of the slags in the aforementioned temperature range can be observed with increasing mass percentage of Fe _t O from 1.88% to 55.50%, which corresponds to the calculated of iron oxides from 0.019 to 0.558 or the determined pO2 of the slags from 8.18 × 10^–8 to 3.23 × 10^–4 Pa. In addition, lower temperature is beneficial to dephosphorisation ability of the slags, and vice versa.

Relationship between calculated comprehensive a mass action concentration of iron oxides and mass percentage of FeO, Fe₂O₃ or Fe _t O or b determined oxygen potential of slags for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in temperature range from 1811 to 1927K respectively

Relationship between mass percentage of Fe _t O and measured20 ratio lg by Ban-ya et al. or calculated lg by developed IMCT–L_P model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in temperature range from 1811 to 1927K respectively

The influence of mass percentage of Fe _t O on the determined lg after the original data from Ban-ya et al.²⁰ or the calculated for the slags is shown in Fig. 8 respectively. Increasing slag oxidation ability cannot result in a visible effect on the determined values of dephosphorisation potential after Ban-ya et al.,²⁰ but can lead to a slightly increasing tendency of the calculated ones by the developed IMCT– model for the slags in the lower and middle temperature ranges as shown on the first and second layers of the subfigures in Fig. 8. However, increasing slag oxidation ability can cause an inversely asymmetrical U type trend of both the determined values after Ban-ya et al.20 and the calculated ones by the developed IMCT– model for the slags in the higher temperature range from 1918 to 1927 K as shown on the third layer of Fig. 8 respectively. The finding in Fig. 8c is similar as the obtained result of against mass percentage of Fe _t O for CaO–FeO– SiO₂–P₂O₅ slags at 1873 K by Turkdogan.⁸⁹ Therefore, a temperature >1918–1927 K can bring about an obvious change in influence pattern of slag oxidation ability on of the slags. The competitive effect between temperature and slag oxidation ability on of the slags can be observed in the higher temperature range from 1918 to 1927 K. The negative influence of higher temperature on of the slags will counteract the positive effect of slag oxidation ability on of the slags in the higher temperature range.

Relationship between mass percentage of Fe _t O and determined lg after Ban-ya et al.²⁰ or calculated lg by developed IMCT– model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in temperature range from 1811 to 1927K respectively

Certainly, the results in Fig. 8 by the developed IMCT– model more conform to the aforementioned common consensuses^{21,42,44–48,50} of the comprehensive effect of both slag oxidisation ability and temperature T on the dephosphorisation reaction based on the determined values of the dephosphorisation potential after Ban-ya et al.²⁰ as criteria. It is well known that and L_P are both distinct thermodynamic quantities. Thus, it is easily understood that the influence of slag oxidisation ability on L_P is different from that on for the slags.

Influence of slag basicity on dephosphorisation ability and potential of slags

The widely applied binary basicity (%CaO)/(%SiO₂) cannot be used for the slags without SiO₂. The more complex basicity [(%CaO) + 1:4(%MgO)]/[(%SiO₂) + (%P₂O₅) + (%Al₂O₃)]^38,51,54,55 can be applied. In the investigated slags, it was simplified to form (%CaO)/[(%P₂O₅) + (%Al₂O₃)]. The influence of the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] on the measured²⁰ lg P, measured by Ban-ya et al. and the calculated lg for the slags is illustrated in Fig. 9a.

Relationship of simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] a or optical basicity taking _FeO = 1.0 and = 0.75 b against measured²⁰ lg L_{P, measured} by Ban-ya et al. or calculated lg by developed IMCT – L_P model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in temperature range from 1811 to 1927K respectively

Three groups of values of optical basicity for FeO and Fe₂O₃ have been recommended as FeO = 0.51 and Fe₂O₃ = 0.48, which are measured from Pauling electronegativity;⁹⁰ FeO = 0.93 and Fe₂O₃ = 0.69, which are derived from average electron density;91 and FeO = 1.0 and Fe₂O₃ = 0.75, which are mathematically regressed88 from numerous experimental data. A higher content of Fe _t O in the slags can make a different magnitude of optical basicity for the slags with various values of optical basicity for FeO and Fe₂O₃. According to the evaluation results of the aforementioned three groups of values of optical basicity for FeO and Fe₂O₃, the obtained FeO = 1.0 and Fe₂O₃ = 0.75 from mathematical regression88 are applied to represent optical basicity for FeO and Fe₂O₃ in the slags, similar with the recommended FeO = 1.0 and Fe₂O₃ = 0.77 by Young et al.²¹ The effect of optical basicity on the measured²⁰ lg by Ban-ya et al. and the calculated lg for the slags is illustrated in Fig. 9b. Increasing the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] from 1.31 to 28.26 or optical basicity from 0.78 to 0.95 can result in an exponentially growing tendency of lgL_P. This is a simple and precise conclusion.

Similarly, the relationship between the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] or optical basicity and the determined lg after Ban-ya et al.²⁰ or the calculated lg for the slags in a temperature range from 1811 to 1927 K is shown in Fig. 10 respectively. Compared with the determined values of dephosphorisation potential after Ban-ya et al.,²⁰ increasing the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] from 1.31 to 28.26 or optical basicity from 0.78 to 0.95 results in a slightly increasing tendency of the calculated ones by the developed IMCT– model for the slags in the lower temperature range from 1811 to 1828 K and the middle temperature range from 1870 to 1876 K. An inversely asymmetrical U type relationship of the applied two kinds of slag basicity against the determined values of dephosphorisation potential after Ban-ya et al.²⁰ and the calculated ones by the developed IMCT– model can be observed in the higher temperature range from 1918 to 1927 K.

Relationship between a simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] or b optical basicity by taking _FeO = 1.0, = 0.75 and determined lg after Ban-ya et al.²⁰ or calculated lg by developed IMCT – model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in temperature range from 1811 to 1927K respectively

Comprehensive effect of Fe _t O and CaO on dephosphorisation ability and potential of slags

Besides the individual effect of iron oxides Fe _t O or basic oxide CaO, the comprehensive effect of iron oxides Fe _t O and basic oxide CaO plays a pivotal role on dephosphorisation ability and potential of the slags. It has been verified byYang et al.^38,39 that themass percentage ratios or the mass action concentration ratios of various iron oxides to basic oxide CaO can be applied to elucidate the comprehensive effect of iron oxides Fe _t O and basic oxide CaO on dephosphorisation ability and potential of slags. Themass percentage ratio (%Fe _t O)/(%CaO) of iron oxides Fe _t O to basic oxide CaO can correlate good linear relationship with the corresponding mass action concentration ratio /N_CaO for the slags as shown in Fig. 11. Thus, the mass action concentration ratio /N_CaO can replace the corresponding mass percentage ratio (%Fe _t O)/(%CaO).

Relationship between mass percentage ratio (%Fe _t O)/(%CaO) and corresponding mass action concentration ratio =N_CaO for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in temperature range from 1811 to 1927K respectively

The relationship between the mass action concentration ratio /N_CaO and the measured20 lg by Ban-ya et al. or the calculated lg for the slags is shown in Fig. 12a. Similarly, the relationship between the mass action concentration ratio /N_CaO and the determined lg after Ban–ya et al.²⁰ or the calculated lg for the slags is also displayed in Fig. 12b. It can be observed in Fig. 12a that increasing /N_CaO can cause an exponentially growing tendency of L_P. However, it can be observed in Fig. 12b1 and b2 that increasing /N_CaO can result in a slightly promotive effect on of the slags in the lower and middle temperature ranges respectively. The inversely asymmetrical U type relationship between /N_CaO and can be observed for the slags in the higher temperature range from 1918 to 1927 K as shown in Fig. 12b3. Obviously, the influence of /N_CaO on in Fig. 12b is largely different with that on L_P in Fig. 12a. Temperatures > 1918–1927 K can only cause a large change of the influence pattern of /N_CaO on , rather than on L_P, for the slags.

Relationship between a mass action concentration ratio =N_CaO and measured20 lg L_{P, measured} by Ban-ya et al. or calculated lg by developed IMCT – L_P model or b determined lg after Ban-ya et al.²⁰ or calculated lg by developed IMCT – model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags in a temperature range from 1811 to 1927K respectively

Thus, attaining reasonable conditions such as slag oxidisation ability, slag basicity, temperature and so forth is very important to strengthen dephosphorisation ability and potential of the slags. The optimal range of chemical composition for the slags with good dephosphorisation ability and potential can be summarised as follows:

mass percentage of Fe _t Oas 35.0%corresponding to of iron oxides as 0.30 or oxygen potential of the slags as 1.14×10^–4 Pa is recommended for providing necessary slag oxidisation ability

the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] should be controlled at about 3.0, which corresponds to optical basicity in a range of 0.90–0.95

the mass action concentration ratio NFetO=NCaO should be maintained at 0.45 corresponding to the mass percentage ratio (% FetO)/(% CaO) also as 0.45.

In addition, the lower temperature range from 1811 to 1828 K is also recommended for enhancing dephosphorisation ability and potential of the slags under the condition of maintaining required fluidity of the slags.

Quantitative contributions of dephosphorisation products to dephosphorisation ability and potential of slags based on IMCT

Contribution ratios of various dephosphorisation products to dephosphorisation ability of slags

The respective phosphorus distribution ratios P,i of six dephosphorisation products as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅ and 4CaO·P₂O₅ in the slags equilibrated with liquid iron can be determined by the developed IMCT–L_P mode in equation (14). The relationship between the calculated respective phosphorus distribution ratios of the aforementioned six dephosphorisation products and the calculated total phosphorus distribution ratio for the slags can be correlated by linear functions as shown in Fig. 13a and summarised in Table 5. The slope of the regressed linear function in Table 5 can be defined as the contribution ratio of each dephosphorisation product in the slags to the calculated for the slags. The average contribution ratio of each dephosphorisation product to the calculated total phosphorus distribution ratio, i.e. / , is also summarised in Table 5. Obviously, the average contribution ratios of P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅ and 2CaO·P₂O₅ to L_P of the slags can be ignored compared with the average contribution ratio of 3CaO·P₂O₅ as 95.00% and 4CaO·P₂O₅ as 5.00%. Thus, the developed IMCT–L_P model in equation (14) can also be simplified as

The product term in equation (25) can be applied to represent the comprehensive effect of iron oxides Fe _t O and basic component CaO on the respective phosphorus distribution ratio L_P, 3CaO·P₂O₅ of 3CaO·P₂O₅. Similarly, the product term in equation (25) can be used to describe the comprehensive effect of iron oxides Fe _t O and the basic component CaO on the respective phosphorus distribution ratio of 4CaO·P₂O₅. Thus, the simplified expression of the developed IMCT–L_P model in equation (25) can be applied to theoretically describe the comprehensive effect of iron oxides Fe _t O and basic component CaO on L_P of the slags.

Contribution of a calculated lg of six structural units containing P₂O₅ to calculated lg by developed IMCT – L_P model or b to calculated lg by developed IMCT – model for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in temperature range from 1811 to 1927K respectively

Table 5

Regressed formulae of calculated respective phosphorus distribution ratio of six dephosphorisation products against calculated total phosphorus distribution ratio by the developed IMCT–L_P model and average contribution ratio of six dephosphorisation products to for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1811 to 1927K based on the IMCT

It can be obtained from Table 5 that the contribution ratio of iron oxides Fe _t O and CaO + Fe _t O to L_P for the slags is ∼0.0 and 100% respectively. The formed 3CaO·P₂O₅ accounts for 95.00%, and 4CaO·P₂O₅ contributes to 5.00% of the dephosphorisation products in the slags respectively. This result is similar with the obtained^38,39 conclusion as that 3CaO·P₂O₅ accounts for 96.01%, and 4CaO·P₂O₅ contributes to 3.97% of the dephosphorisation products in the CaO–SiO₂–MgO–FeO–Fe₂O₃–MnO–Al₂O₃–P₂O₅ slags during the top– bottom combined blown converter steelmaking process.

Contribution ratios of various dephosphorisation products to dephosphorisation potential of slags

Using the similar method described in the section on ‘Contribution ratios of various dephosphorisation products to dephosphorisation ability of slags’, the relationships between the calculated respective phosphate capacity

of the aforementioned six dephosphorisation products and the calculated total phosphate capacity

for the slags canbe correlated by the linear function as shown in Fig. 13b and summarised in Table 6. The average contribution ratio of each dephosphorisation product to the calculated total phosphate capacity, i.e.

, is also summarised in Table 6.

Table 6

Regressed expressions of calculated respective phosphate capacity of six dephosphorisation products against calculated total phosphate capacity by developed IMCT– model and average contribution ratio of six dephosphorisation products to for CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags equilibrated with liquid iron in a temperature range from 1811 to 1927K based on IMCT

It can be obtained from Table 6 that the contribution ratio of iron oxides Fe _t O to of the slags is ∼0.0%, while the comprehensive contribution ratio of CaO+Fe _t O to of the slags is ∼100.0%. The formed 3CaO·P₂O₅ accounts for 94.60% of of the slags, while 4CaO·P₂O₅ contributes to 5.35% of of the slags. Therefore, the developed IMCT– model in equation (24) can also be simplified as

The product term in equation (26) can be applied to represent the comprehensive effect of iron oxides Fe _t O and the basic component CaO on the respective phosphate capacity of 3CaO·P₂O₅. Similarly, the product term in equation (26) can be used to describe the comprehensive effect of iron oxides Fe _t O and the basic component CaO on the respective phosphate capacity of 4CaO·P₂O₅. Thus, the simplified expression of the developed IMCT– model in equation (26) can be applied to theoretically describe the comprehensive effect of iron oxides Fe _t O and the basic component CaO on of the slags.

Obviously, the obtained contribution ratio of 3CaO·P₂O₅ or 4CaO·P₂O₅ to as 94.60 or 5.35% by the developed IMCT– model is similar as that of L_P of the slags as 95.00 or 5.00% by the developed IMCT–L_P model.

Conclusions

The thermodynamic models for predicting phosphorus distribution ratio L_P and phosphate capacity of CaO–FeO–Fe₂O₃–Al₂O₃–P₂O₅ slags, i.e. the IMCT–L_P and IMCT– models, have been developed by coupling with the developed thermodynamic model for calculating the mass action concentrations Ni of structural units or ion couples in the slags, i.e. the IMCT–N_i model, based on the IMCT. The developed IMCT–L_P and IMCT– models have been verified with the experimental results from Ban-ya et al.²⁰ and the predicted results by the summarised five P models and three models. The main concluding remarks can be summarised as follows.

The calculated mass action concentrations N_i of structural units or ion couples based on the IMCT can be applied to accurately represent reaction abilities of components in the slags, like activities a_{R, i} of components relative to pure liquid or solid components as standard state. Furthermore, the calculated comprehensive mass action concentration of iron oxides Fe _t O as can be applied to precisely describe the oxidation ability of the slags.

The measured phosphorus distribution ratio L_{P, measured} of the slags can reliably be predicted by the developed IMCT–L_P model and other two reported models as Suito's no. 3 model and Sommerville's model. The determined phosphate capacity of the slags after the original data from Ban-ya can also reliably be predicted by the developed IMCT– model and other four models as Healy's model, Suito's no. 2 model, Sommerville's model and Young's model. The developed IMCT–L_P and IMCT– models have a close relationship with the dephosphorisation mechanism. Greater values of the predicted dephosphorisation ability by the developed IMCT–L_P are the reason for the larger ones of dephosphorisation potential by the developed IMCT– model in the lower temperature range from 1811 to 1828 K.

An exponentially growing relationship of phosphorus distribution ratio in logarithmic form lg L_P against slag oxidisation ability and the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] or optical basicity can be established for the slags, respectively.

The linearly increasing relationship of phosphate capacity in logarithmic form lg against slag oxidisation ability and the simplified complex basicity (%CaO)/[(%P₂O₅) + (%Al₂O₃)] or optical basicity can be correlated for the slags in the lower temperature range from 1811 to 1828 K and the middle temperature range from 1870 to 1876 K respectively. However, the inversely asymmetrical U type relationship can be observed for the slags in the higher temperature range from 1918 to 1927 K.

The respective phosphorus distribution ratio L_{P, i} and the respective phosphate capacity values of six dephosphorisation products as P₂O₅, 3FeO·P₂O₅, 4FeO·P₂O₅, 2CaO·P₂O₅, 3CaO·P₂O₅ and 4CaO·P₂O₅ can be predicted by the developed IMCT–L_P and IMCT– models. Concerning the structural units or ion couples of the IMCT–L_P and IMCT– models, 3CaO·P₂O₅ accounts for 95.00% of all the dephosphorisation products and 4CaO·P₂O₅ for 5.00% respectively, whereas the other structural units have much meaningless contribution.

Not only the individual effect of various iron oxides and basic oxide CaO but also the comprehensive effect of CaO + Fe _t O, which can be described by the mass action concentration ratio /N_CaO, controls dephosphorisation ability and potential of the slags.

Footnotes

Acknowledgement

This work is supported by the National Natural Science Foundation of China (NSFC) by grant no. 51174186.

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Thermodynamic models for predicting dephosphorisation ability and potential of CaO–FeO–Fe 2 O 3 –Al 2 O 3 –P 2 O 5 slags during secondary refining process of molten steel based on ion and molecule coexistence theory

Abstract

Introduction

Cited experimental data of oxygen, phosphorus and sulphur distribution equilibria between CaO–FeO–Fe2O3– Al2O3–P2O5 slags and liquid iron

Thermodynamic model for calculating mass action concentrations Ni of structural units or ion couples in slags based on IMCT

Hypotheses

IMCT–Ni thermodynamic model for slags

Structural units in slags

Development of IMCT–Ni thermodynamic model for slags

Estimation of mass action concentrations Ni of structural units or ion couples containing P2O5 in slags

Re-evaluation of IMCT–Ni model

Similarities and differences between applied IMCT and other structural theories for slags

Validity of replacing activities aR,i by mass action concentrations Ni of components in slags

Results of IMCT–Ni model for slags

Relationship between mass percentage of five components and mass action concentration Ni of related structural units or ion couples in slags

Validation of comprehensive mass action concentration of iron oxides in slags

Validation of mass action concentration NP2O5 of P2O5 in slags

Summary of prediction models of activity coefficient of P2O5 in various steelmaking slags

Thermodynamic models for calculating dephosphorisation ability and potential of slags based on IMCT

Thermodynamic model for calculating dephosphorisation ability of slags based on IMCT

Thermodynamic model for calculating dephosphorisation potential of slags based on IMCT

Relationship between phosphorus distribution ratio and phosphate capacity for slags

Relationship between phosphate capacity and phosphate capacity index for slags

Relationship between phosphorus distribution ratio and phosphate capacity for slags

Establishment of thermodynamic model for calculating phosphate capacity of slags based on IMCT

Evaluation of developed IMCT–LP and IMCT – models for slags based on IMCT

Summary of five phosphorus distribution ratio LP models and three phosphate capacity models for various slags from related literature

Evaluation of developed IMCT– LP model for slags

Comparison between measured LP, measured and calculated by developed IMCT–LP model for slags

Comparison between measured LP, measured and calculated by other five LP models for slags

Evaluation of other five LP models

Comparison between measured LP, measured and calculated by other five LP models for slags

Evaluation of developed IMCT– model for slags

Comparison between determined and calculated by developed IMCT– model for slags

Comparison between determined and calculated by some LP or models for slags

Evaluation of other three phosphate capacity models

Comparison between determined and calculated by different LP or or models for slags

Influence of slag chemical composition on dephosphorisation ability and potential of slags

Influence of slag oxidation ability on dephosphorisation ability and potential of slags

Influence of slag basicity on dephosphorisation ability and potential of slags

Comprehensive effect of Fe t O and CaO on dephosphorisation ability and potential of slags

Quantitative contributions of dephosphorisation products to dephosphorisation ability and potential of slags based on IMCT

Contribution ratios of various dephosphorisation products to dephosphorisation ability of slags

Contribution ratios of various dephosphorisation products to dephosphorisation potential of slags

Conclusions

Footnotes

Acknowledgement

References

Cited experimental data of oxygen, phosphorus and sulphur distribution equilibria between CaO–FeO–Fe₂O₃– Al₂O₃–P₂O₅ slags and liquid iron

Thermodynamic model for calculating mass action concentrations N_i of structural units or ion couples in slags based on IMCT

IMCT–N_i thermodynamic model for slags

Development of IMCT–N_i thermodynamic model for slags

Estimation of mass action concentrations N_i of structural units or ion couples containing P₂O₅ in slags

Re-evaluation of IMCT–N_i model

Validity of replacing activities a_R,i by mass action concentrations N_i of components in slags

Results of IMCT–N_i model for slags

Validation of mass action concentration NP₂O₅ of P₂O₅ in slags

Summary of prediction models of activity coefficient of P₂O₅ in various steelmaking slags

Evaluation of developed IMCT–L_P and IMCT – models for slags based on IMCT

Summary of five phosphorus distribution ratio L_P models and three phosphate capacity models for various slags from related literature

Evaluation of developed IMCT– L_P model for slags

Comparison between measured L_{P, measured} and calculated by developed IMCT–L_P model for slags

Comparison between measured L_{P, measured} and calculated by other five L_P models for slags

Evaluation of other five L_P models

Comparison between measured L_{P, measured} and calculated by other five L_P models for slags

Comparison between determined and calculated by some L_P or models for slags

Comparison between determined and calculated by different L_P or or models for slags

Comprehensive effect of Fe _t O and CaO on dephosphorisation ability and potential of slags