Thermodynamic properties

The equilibrium constants of the studied reactions were calculated at the temperature of the reaction interface:

K=exp

−∆G^◦ RT^∗

where ∆G^◦=∆H^◦−T^∗∆S, (287) where∆G^◦,∆H^◦, and∆S^◦are the changes in standard Gibbs free energy, enthalpy, and entropy of reaction, respectively. The Henrian standard state was employed for the species in the liquid metal phase (point B in Fig. 27), while the Raoultian standard state was employed for species in the slag phase (point A in Fig. 27). The non-ideality of the slag and gas species were taken into account by making use of available activity models;

a more detailed description of these models is provided in the following sections. The detailed treatment of the thermochemical properties is described in Appendix 1.

0 1

01 ↓

↓

↓

F

D L

E K

C BR a o u l t ' s l a w

Activity

M o l a r f r a c t i o n N o t t o s c a l e

H e n r y ' s l a w

A

A c t u a l a c t i v i t y A R a o u l t i a n s t a n d a r d s t a t e

B H e n r i a n s t a n d a r d s t a t e C 1 w t - % s t a n d a r d s t a t e K 1 a t o m - % s t a n d a r d s t a t e

Fig 27. Thermodynamic standard states. Adapted from [404].

Gas partial pressures

The gas phase was assumed to obey the ideal gas law. Therefore, the partial pressures of species at the reaction interfaces can be obtained using Dalton’s law [555]:

pi=x^∗_ipG, (288)

wherepGis the total gas pressure (in atm). The pressure changes in a free gas jet are sufficiently small to be ignored [492] and therefore the total pressure at the cavity and metal droplet interfaces was defined as equal to the pressure of the surrounding atmosphere,pG,cav=pG,md=pamb. In the case of the reduction model, it was assumed that the ferrostatic pressure near the metal–slag interface is negligible and that the total pressure at the slag droplet interface is equal to the ambient pressure,pG,sd=pamb. Activities of dissolved species

The Henrian activities of species dissolved in liquid iron can be expressed in terms of the mole fraction multiplied by the Henrian activity coefficient:

a^H_i =γ_i^Hxi. (289)

The Wagner formalism represents the MacLaurin series expansion of the activity coefficient in terms ofεinteraction parameters [556]. Consequently, the Henrian activity coefficient of speciesican be expressed according to [556]:

lnγ_i^H=ln

whereγ_i^Ris the Raoultian activity coefficient,γ_i^◦is the activity coefficient at infinite dilution,εis the molar first-order interaction parameter,ρis the molar second-order interaction parameter, andR(x³)represents the sum of the terms higher than the third order. Because suitable higher-order parameters are rarely available, it is more common to truncate the MacLaurin series after the first-order terms [556]. As a result, the following well-known expression is obtained [556]:

lnγ_i^H=lnγ_i^R

Although Eq. 291 is applicable for activity coefficients in the dilute region, the thermodynamic inconsistency of the truncated MacLaurin series can induce inaccuracy at high solute concentrations [254, 556, 557]. For this reason, the Wagner formalism is not recommendable for stainless steel melts [556]. Attempts have been made to extend theεapproach for concentrated solutions. Pelton and Bale [254] proposed the unified interaction parameter(UIP) formalism, which combines the Wagner-Lupis-Elliott(WLE) formalism [558], Darken’s quadratic formalism [559, 560], and Margules formalism [561]; it reduces to WLE formalism at infinite dilution and to Darken’s quadratic formalism in solutions. The Henrian activity coefficient of speciesi is expressed according to [254]:

Maet al.[557] proposed anεformalism, which is thermodynamically consistent with the Gibbs-Duhem relationship. The Maet al.[557] formalism has been employed particularly for stainless steel melts [556, 562, 563]. In comparison to the UIP formalism, it provides a more rigorous description for the effect of solvent on the activity of solutes [557, 564]. The Raoultian activity coefficient of the solvent and the Henrian activity coefficient of the solutes are expressed according to Eqs. 293 and 294.

lnγ_solvent^R =

lnγ_i^H=ln In this work, the activities of the dissolved species were calculated with the UIP formalism [254] based on the interfacial composition and hencexwas replaced byx^∗in Eq. 292. The employed molar first-order interaction parameters were obtained from the literature [548, 565–568] and are tabulated in Appendix 1. In order to find possible differences, the Wagner, UIP, and Maet al.formalisms were compared for stainless steel melts.Ceteris paribus, the deviations in the predicted activities of the main species were found to be small even in the case of concentrated solutions such as liquid stainless steels.

Activities of slag species

Various models have been proposed for calculating activities in molten slags; these include the regular solution model [569], modified quasi-chemical model [570–574], reciprocal ionic liquid model [575], cell model [576], polymerisation model [577–580], the molecular interaction volume model [581, 582], and various empirical models [49, 583–585].³⁹Regular solution models constitute one of the most simplistic types of slag models. The main assumptions of regular solution models are that the atoms are distributed randomly on the sites of a three-dimensional lattice (which has no vacancies) and that the energy of the system is defined as the sum of pairwise interactions [588]. In the regular solution model proposed by Ban-Ya [569], the Raoultian activity coefficients of the slag species are calculated as follows [569]:

RTlnγ_i^R=

39More detailed reviews of the activity models are available in the literature [586, 587].

whereα is the interaction energy between cations,Xis the cation fraction, and

∆Gconvis the conversion energy of the activity coefficient between a hypothetical regular solution and a real solution. Although the parameters for chromium-containing slags are available in the literature [178], they are not applicable to the high basicity ratios typical for AOD processing.

In this work, the activity coefficients of slag species were calculated according to the model employed by Wei and Zhu [49]. The Raoultian activity coefficients of FeO, Cr2O3, MnO, and SiO2are given by Eqs. 296, 297, 298, and 299, respectively [49].

log₁₀γ_FeO^R = ε1

T^∗(x^∗_CaO+x^∗_MgO)(x^∗_SiO2+0.25x^∗_AlO1,5) +ε2

T^∗x_MnO^∗ (x^∗_SiO2+0.45x^∗_CrO1.5) + ε3

T^∗x^∗_AlO1.5x^∗_SiO2 +ε4

T^∗x_MnO^∗ x^∗_AlO1.5+ ε5

T^∗x^∗_CrO1.5x^∗_SiO2, (296) log₁₀γ_Cr^R_2O3=log₁₀γ_FeO^R −ε6

T^∗(x^∗_CaO+x^∗_MgO)− ε7

T^∗x^∗_MnO−ε5

T^∗x^∗_SiO2, (297) log₁₀γ_MnO^R =log₁₀γ_FeO^R −ε2

T^∗(x^∗_SiO2+0.45x^∗_CrO1.5)−ε4

T^∗x_AlO^∗ _1.5, (298) log₁₀γ_SiO^R ₂=log₁₀γ_FeO^R −ε1

T^∗(x^∗_CaO+x^∗_MgO)

−ε2

T^∗x_MnO^∗ −ε3

T^∗x^∗_AlO1.5−ε5

T^∗x_CrO^∗ _1.5, (299)

whereε1. . .ε7are the interaction coefficients of the model. Table 21 shows the interaction coefficients reported by Wei and Zhu [49] for early and later periods of refining. The parameters for early refining are employed in the stages involving top-blowing, while the parameters for the later period of refining are employed in other stages,e.g.during the reduction stage.

Table 21. Interaction coefficients of the slag model. Adapted from [49].

Interaction coefficients

Stage ε1 ε2 ε3 ε4 ε5 ε6 ε7

Early period of refining 3540 1475 1068 36 593 1594 664

Later period of refining 4130 1720 1246 42 692 1859 774

In the top-blowing model it was assumed similarly to Wei and Zhu [49] thata^R_Cr2O3= 1 if the interfacial Cr2O3content is greater than the maximum solubility of Cr2O3in the slag. For this purpose, a simple regression equation for the solid fraction of the top slag

was derived with the help of the FactSage 7.0 computational thermodynamics software [589] (see Appendix 1).