1.   Introduction

The quality of liquid steel during secondary refining plays an important role in the quality control of steel products. Deoxidation, as one of the refining reactions, plays a crucial role in determining the quality of liquid steel. This process affects important factors, such as cleanliness in relation to nonmetallic inclusion evolution [1] and blow hole formation [2]. Numerous approaches have been developed to predict deoxidation equilibria in liquid steel [3–11]. When coupled with computational fluid dynamics (CFD) calculations, deoxidation equilibrium calculations enable the simulation of composition evolution in the liquid steel within vessels (ladle, tundish, and mold). As a result, spatial and temporal distributions of the composition of the liquid steel as well as those of the non-metallic inclusions can be analyzed. This capability can assist steel plant operators in assessing the control of liquid-steel composition.

The reliability of a thermodynamic model determines the accuracy of deoxidation equilibrium calculations. Wagner’s interaction parameter formalism (WIPF) is popularly used in this context [12]. WIPF can be used to determine the solute activity coefficients of liquid steel as explicit functions of composition and temperature. According to this formalism, the activity coefficient of solute $i$ can be expressed as follows [12–14]:

on the mass percentage base or

on the mole fraction base, where ${e}_{i}^{j}$, ${r}_{i}^{j}$, ${\varepsilon}_{i}^{j}$, and ${\rho }_{i}^{j}$ are the first- and second-order interaction parameters on the mass percentage base between i and j, and those on the mole fraction base, resepctively [13–14]. These equations, which are collectively called formalism, provide activity coefficients as explicit functions of composition and temperature. However, this formalism does not originate from a comprehensive thermodynamic model and has certain limitations [7,15–18]. Notably, it lacks full thermodynamic consistency [15–17,19] and may be inapplicable to systems with highly remarkable interactions. Nonetheless, this type of formalism has gained popularity due to its simplicity. Additional calculations are unnecessary in determining the activity coefficient, easing the direct implementation of WIPF in computational codes such as CFD. Therefore, Eqs. (1) or (2) can be directly utilized in CFD or other computing codes to conduct thermodynamic calculations.

Alternatively, modern CALPHAD software tools, such as FactSage [20–21] and ThermoCalc [22], can be used to solve deoxidation equilibria using the available appropriate thermodynamic models and databases. These software packages also provide application programming interfaces (APIs), such as ChemApp [23] and TC-Python [22], which allow users to develop their own customized application programs. One possible application is the coupling of CALPHAD-API with CFD codes, in which iterative calculations between the two are performed. However, not only the accuracy and reliability of the selected thermodynamic model but also the computation cost associated with the required calculation steps must be considered when developing a user-specific application program. Successive iterations between CALPHAD-API and CFD code can considerably increase the computation time.

The present authors are currently developing a tundish process simulator that can accurately calculate the composition evolution of Al-killed liquid steel, with a unique focus on the concentration profiles of Al, O, and alumina inclusions in liquid steel, under time and space scales [24]. The extensive computational cost involved implies the need for the minimization of computation steps while ensuring reliability. One potential approach is deoxidation equilibrium calculations within the CFD code itself, which eliminates the need for external CALPHAD-API software and reduces the cross-calling time between two software components. However, as highlighted by Kang [18], sophisticated thermodynamic models are required in the accurate prediction of deoxidation equilibria. The modified quasichemical model (MQM) in the pair approximation, employed by FactSage software, has shown great promise in describing deoxidation equilibria in liquid steel; it has been successfully used in the treatment of deoxidation by Al and Mn [11,25]. These advanced thermodynamic models typically involve an additional step known as internal Gibbs energy minimization. Consequently, the Gibbs energy of the liquid phase cannot be expressed simply as a function of composition and temperature. An internal minimization routine involving Gibbs energy must be used to obtain the Gibbs energy for a given composition and temperature. As a result, the activity coefficients of solutes cannot be explicitly expressed as functions of composition and temperature but were rather obtained implicitly after the internal equilibrium is solved. This limitation restricts the direct incorporation of such thermodynamic models into the CFD code. Therefore, a thermodynamic model for deoxidation equilibria that does not require internal Gibbs energy minimization should be developed while maintaining a high level of accuracy.

The present study introduces a thermodynamic model that does not require the internal Gibbs energy minimization for deoxidation equilibria of liquid steel. The model demonstrates accuracy and reliability across a wide range of compositions and temperatures for deoxidation equilibria involving various elements in liquid steel. Furthermore, the model successfully handles complex deoxidation equilibria involving elements such as Al and Mn. Lastly, the present study presents an example showcasing the coupling of the proposed model with a CFD code to predict the composition profile of liquid steel in a tundish process.

2.   Thermodynamic approaches for deoxidation equilibria of liquid steel

2.1   Deoxidation equilibria

A well-known deoxidation reaction in liquid steel occurs due to the presence of Al:

where $\mathrm{\Delta }{G}_{\text{Al}_{2}\text{O}{}_{3}}^{\ominus ,'}$ is the standard Gibbs energy change of reaction (3). The equilibrium constant of this reaction is as follows:

When the reaction product is pure, e.g., ${a}_{\text{Al}{}_{2}\text{O}{}_{3}\text{(s)}}=1$, then the equilibrium constant $K'_{\text{Al}{}_{2}\text{O}{}_{3}\text{(s)}}$ is calculated using the following equation:

When the values of ${\gamma }_{\underline{\mathrm{A}\mathrm{l}}}$ and ${\gamma }_{\underline{\mathrm{O}}}$ (relative to the infinite dilution standard state or Henrian standard state) are known, they can be substituted into Eq. (5). The equation is then solved to establish the relationship between ${X}_{\text{Al}}$ and ${X}_{\text{O}}$ at equilibrium.

In general, a metal element ($\underline{\mathrm{M}}$) reacts with dissolved $\underline{\mathrm{O}}$ via the following:

where $\Delta {G}_{\text{M}{}_{x}\text{O}{}_{y}}^{\ominus ,'}$ is the standard Gibbs energy change of reaction (6), and

where

The Raoultian standard state can be used as an alternative to the infinite dilution standard state:

where

where ${G}_{i}^{\ominus}$ refers to the molar Gibbs energies of pure $i$.

The relation between ${G}_{i}^{\ominus}$ and ${G}_{\_{ i}}^{\ominus}$ is as follows:

where ${\gamma }_{i}^{\ominus}$ represents the activity coefficient at the infinite dilution of $i$.

2.2   Explicit expression of activity coefficient as a function of composition

${\gamma }_{\underline{\mathrm{M}}}$ and ${\gamma }_{\underline{\mathrm{O}}}$ are typically functions of composition (X_k, k = M or O) and temperature ($T$). According to WIPF [12,14], these functions can be expressed as follows:

where ${\varepsilon}_{i}^{j}$s are the first-order interaction parameters among solutes. ${K}_{\text{M}{}_{x}\text{O}{}_{y}}$ and ${\varepsilon}_{i}^{j}$s can be obtained from literature [14,26]. Substitution of these parameters into Eq. (5) along with Eqs. (13) and (14) can be used to determine the equilibrium composition of liquid steel. Examples of methods belonging to this category include the Unified Interaction Parameter Formalism and Darken’s quadratic formalism [15,17,27–28].

Activity coefficients can also be derived from the excess Gibbs energy of mixing of the liquid solution:

where total excess Gibbs energy (${G}^{\text{ex}}$ in J) can be expressed as an explicit function of $T$ and the number of moles of components (e.g., ${n}_{i}$). This relationship holds in cases involving a regular solution model or a random mixing model with polynomial expansions for excess Gibbs energy [8,17].

Notably, Eq. (5) does not provide the explicit form of composition (X_k), which indicates the need for a numerical approach to obtain a deoxidation equilibrium data point. The results can be readily expressed as mass percent ([%M]). Fig. 1 provides an overview of this approach.

Figure  1.  Overview of deoxidation equilibrium–calculation approaches.

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2.3   Implicit expression for the activity coefficient

When ${G}^{\text{ex}}$ is not an explicit function of composition, the activity coefficient obtained from Eq. (15) also lacks explicit dependence on composition. This situation often occurs during the deviation of a solution from random mixing behavior, such as in the case of an associate solution model featuring a strong chemical attraction (short-range ordering, SRO) [3,7,29]. Bouchard and Bale [29] reported the successful application of this model in the deoxidation equilibria of liquid steel; in this case, in addition to unassociated Al and O, Al*O and Al₂*O associates were assumed to be present in Al-deoxidized liquid steel. Jung et al. [7] further extended this approach to other systems and effectively described the deoxidation equilibria of M in liquid steel up to high M concentrations. However, these approaches require internal Gibbs energy minimization to determine the number of associates before the activity coefficient can be calculated, which adds an additional step to calculations. Other thermodynamic approaches requiring internal Gibbs energy minimization to address SRO, such as the MQM [30–31] and Generalized Central Atoms model [32], are applied in the same context.

In the present study, a novel model was presented to handle the deoxidation of liquid steel without internal Gibbs energy minimization. The model considers strong SRO under limiting conditions to eliminate the requirement for internal Gibbs energy minimization. In addition, the model is based on a simplified quasichemical model, which will be discussed in Section 4.

3.   MQM in the pair approximation for deoxidation equilibria

Pelton et al. proposed a modification of the classical quasichemical approach [33], known as the MQM in the pair [30] and the quadruplet approximations [31]. This model has been successfully applied in various solution systems, including liquid metals [34–36] and liquid oxides [37–40] using MQM in the pair approximation and liquid oxysulfides [41–42] and liquid oxyfluorides [43] using MQM in the quadruplet approximation. Specifically, MQM in the pair approximation has been successfully applied to model deoxidation equilibria in liquid steel involving elements such as Al [11], Mn [25], and Al–Mn [25].

In the deoxidation of liquid steel by M (Fe–M–O liquid solution), the following pair-exchange reactions are considered:

where $\left(i-j\right)$ and Δg_ij (which is usually a function of composition and temperature) refer to the first-nearest-neighbor (FNN) pair $i-j$ and pair formation energy, respectively.

The number of moles of each pair (${n}_{ij}$) is related to the number of moles of each component (${n}_{i}$) through the FNN coordination numbers (${Z}_{i}$) :

The Gibbs energy of the liquid solution is then expressed as follows [30]:

where ${g}_{i}^{\ominus}$ is the molar Gibbs energy of pure liquid $i$, and

where ${X}_{ij}$ and ${Y}_{i}$ indicate the pair fraction of $i-j$ pair and the coordinate equivalent fraction of $i$, respectively:

Further details regarding the development of the model can be found elsewhere [30].

Given that Eq. (23) is not an explicit function of composition (e.g., ${X}_{\text{M}}$, ${X}_{\text{O}}$), differentiation through internal Gibbs energy minimization is required to obtain internal equilibrium:

This process yields three equilibrium constants for quasichemical reactions (16)–(18):

For a given set of ${n}_{\text{Fe}}$, ${n}_{\text{M}}$, and ${n}_{\text{O}}$, six unknowns (${n}_{\text{FeFe}}$, ${n}_{\text{MM}}$, ${n}_{\text{OO}}$, ${n}_{\text{FeO}}$, ${n}_{\text{MO}}$, and ${n}_{\text{FeM}}$) and six equations (Eqs. (26)–(29)) are involved. These equations are solved using numerical methods to obtain the values of ${n}_{ij}$ at internal equilibrium. The obtained values can be back-substituted into Eq. (23) using Eqs. (19)–(21) to calculate G^L. Substitution of Eq. (23) into Eq. (15) yields the activity coefficients of M (${\gamma }_{\text{M}}$) and O (${\gamma }_{\text{O}}$).

4.   Model development in the present study

This section proposes a new model for the deoxidation of liquid steel based on the MQM in pair approximation.

4.1   Assumptions

Among the six FNN pairs in the Fe–M–O liquid solution, (Fe–Fe), (M–M), and (Fe–M) are pairs between metallic elements, (Fe–O) and (M–O) are pairs between a metal and O, and (O–O) is a pair between two O atoms. The following assumptions were considered:

(1) To simplify the model, it was assumed the coordination numbers of all components in the Fe–M–O liquid solution are equal:

Eq. (25) is reduced to the following:

From Eqs. (19) to (21), the total number of moles of all FNN pairs is computed as follows:

where $n={n}_{\text{Fe}}+{n}_{\text{M}}+{n}_{\text{O}}$.

The excess Gibbs energy is then expressed as follows:

(2) If Fe, M, and O mix randomly, then the probabilities of finding an $i-i$ pair and an $i-j$ pair ($i\ne j$) are ${{X}_{i}}^{2}$ and $2{X}_{i}{X}_{j}$, respectively. However, given the presence of strong SRO between metals (Fe and M) and O [18], ${X}_{\text{FeO}} > 2{X}_{\text{Fe}}{X}_{\text{O}}$ and ${X}_{\text{MO}} > 2{X}_{\text{M}}{X}_{\text{O}}$. In the presence of a highly pronounced SRO between metals and O, almost all O in the liquid steel forms Fe–O and M–O pairs, and the O–O pair becomes scarce. Fig. 2 illustrates a limiting case of the M–O binary solution, where all O atoms in the solution form M–O pairs. Whenever an O atom enters the liquid, a new M–O pair forms, which leads to the consumption of M–M pairs. Therefore, in the M–O binary solution, the following can be obtained:

Figure  2.  Pair fraction X_ij in M–O binary liquid solution. Dashed lines represent the (X_ij)s in the random mixing solution: (X_MM) = ${\boldsymbol X}_{\bf M}^{2}$ = (1 − X_O)², (X_OO) = ${\boldsymbol X}_{\bf O}^{2}$, and (X_MO) = 2X_MX_O = 2(1 − X_O)X_O. Solid lines indicate those in the strongly ordered solution. The present model can be defined up to ${X}_{\text{O}}$ = 0.5 but is meaningful in the dilute region of O.

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Notably, this assumption is meaningful only when ${X}_{\text{O}}$ is low.

Similar to the SRO observed between M and O, a strong SRO was assumed between Fe and O in the Fe–M–O ternary solution. However, random mixing of Fe and M was also assumed. As a result, the following approximations can be made in the dilute region of O:

Eqs. (35)–(37) are an extension of Eq. (34) to the ternary solution.

4.2   Model formulae

With the assumption of random mixing of metallic elements, the following equations can be derived:

At equilibrium, rearranging Eqs. (28) and (29) yields the following:

Substituting Eqs. (38) and (40) into Eq. (41) gives

The relationship described above bears similarities to that proposed by Alcock and Richardson in their quasichemical approach [44]. However, the detailed development of their model differs from that used in the present study.

Along with Eq. (35), the above equation gives the following:

where $Q\equiv {X}_{\text{Fe}}+{K}_{\text{MO}}{X}_{\text{M}}$. Therefore, with Eq. (31), the following equations are obtained:

Substitution of the above relationships, along with all ${X}_{ij}$s (Eqs. (38)–(40), (43), and (44)) yields the following:

Through the substitution of Eq. (48) into Eq. (22), the complete expression of the Gibbs energy of the Fe–M–O ternary liquid solution can be obtained in the limiting case, which features an extreme SRO between the metal and oxygen (O), and random mixing exists between metallic elements.

4.3   Deoxidation equilibrium calculation

The activity coefficients of M and O (${\gamma }_{\text{M}}$ and ${\gamma }_{\text{O}}$) are obtained using Eq. (15):

The derived expressions for ${\gamma }_{\text{M}}$ and ${\gamma }_{\text{O}}$ represent the explicit forms that show dependence on the composition and temperature of the solution. Thus, ${\gamma }_{\text{M}}$ and ${\gamma }_{\text{O}}$ can be calculated directly as functions of composition and temperature, which eliminates the need for additional steps such as internal Gibbs energy minimization.

Substitution of these calculated values of ${\gamma }_{\text{M}}$ and ${\gamma }_{\text{O}}$ into the equilibrium-constant equation (Eq. (10)) determines the deoxidation equilibrium conditions (${X}_{\text{M}}$ and ${X}_{\text{O}}$). The conversion of these equilibrium compositions into mass percentages ([%M] and [%O]) is a straightforward process.

Determining the deoxidation equilibria for element M requires the parameters Δg_FeM and Δg_MO, which represent the Gibbs energy differences of the reactions involving Fe–M and M–O species, respectively. Δg_FeO is common in all systems containing Fe and O. In addition, the coordination number $Z$ can be set as a constant value for all elements.

The incorporation of these parameters and values into the model equations enables the accurate prediction of the deoxidation equilibria for element M, providing insights into the deoxidation process of liquid steel.

5.   Deoxidation equilibrium calculations using the present model

The present model was applied to describe the deoxidation equilibria of liquid steel. Here, all parameters (Δg_FeM, Δg_FeO, and Δg_MO) were optimized through the study, and their values are listed in Table 1. Table 2 lists the equilibrium constants (${K}_{\text{M}{}_{x}\text{O}{}_{y}}$) for the deoxidation reactions. These equilibrium constants were determined by considering the Gibbs energy of oxidation involving pure elements from databases, such as FactSage FTOxid database [20–21] for oxides based on Barin’s compilation [45] and SGTE database [46] for metal elements.

Table  1.  Model parameters optimized in the present study

ΔgFeO = −130000 J·mol−1
M	ΔgFeM / (J·mol−1)	ΔgMO / (J·mol−1)	$\mathrm{\mathrm{\mathit{\mathrm{l}}}g}\; K_{\rm{M}O}\left(=-\dfrac{\Delta g_{\text{MO}}-\Delta g_{\text{FeO}}}{2RT}\right)$
Al	−50000	−260000	7818/T
B	−43000	−225000	5713/T
C	−77000	−170000	2406/T
Ca	10000	−450000	19245/T
Ce	−40000	−350000	13231/T
Cr	8000	−200000	4210/T
La	−30000	−450000	19245/T
Mg	10000	−400000	16238/T
Mn	−1000	−180000	3007/T
Nb	7000	−240000	6615/T
Si	−95000	−190000	3608/T
Ti	−40000	−270000	8420/T
V	−15000	−240000	6615/T
Zr	−60000	−360000	13832/T

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Table  2.  Equilibrium constant of deoxidation reactions used in the present study [20–21,45–46]. ${\bf {lg}}\;{\boldsymbol K}_{{\bf M}_{\boldsymbol x}{\bf O}_{\boldsymbol y}}=-{\boldsymbol \Delta} {\boldsymbol G}_{{\bf M}_{\boldsymbol x}{\bf O}_{\boldsymbol y}}^{\ominus}$/(2.303RT)

2Al(l) + 3O(l) = Al2O3(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{Al}{}_{2}\text{O}{}_{3}}=86839.9/T-11.7$
2B(l) + 3O(l) = B2O3(l)	$\mathrm{l}\mathrm{g}\;{K}_{\text{B}{}_{2}\text{O}{}_{3}}=67721.0/T-7.6$
C(l) + O(l) = CO(g)	$\mathrm{l}\mathrm{g}\;{K}_{\text{CO}}=12007.2/T+4.8$
Ca(l) + O(l) = CaO(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{CaO}}=33025.9/T-4.0$
2Ce(l) + 3O(l) = Ce2O3(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{Ce}{}_{2}\text{O}{}_{3}}=100194.1/T-14.5$
Fe(l) + 2Cr(l) + 4O(l) = FeCr2O4(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{FeCr}{}_{2}\text{O}{}_{4}}=79025.9/T-12.4$
2Cr(l) + 3O(l) = Cr2O3(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{Cr}{}_{2}\text{O}{}_{3}}=60410.0/T-9.0$
2La(l) + 3O(l) = La2O3(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{La}{}_{2}\text{O}{}_{3}}=93488.5/T-10.0$
Mg(l) + O(l) = MgO(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{MgO}}=31306.7/T-4.2$
Mn(l) + 2Al(l) + 4O(l) = MnAl2O4(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{MnAl}{}_{2}\text{O}{}_{4}}=109703/T-14.8$
Mn(l) + O(l) = MnO(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{MnO}}=21560.9/T-3.2$
2Nb(l) + 5O(l) = Nb2O5(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{Nb}{}_{2}\text{O}{}_{5}}=102568.3/T-15.6$
Nb(l) + 2O(l) = NbO2(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{NbO}{}_{2}}=41217.3/T-5.4$
Si(l) + 2O(l) = SiO2(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{SiO}{}_{2}}=49445.9/T-7.2$
2Ti(l) + 3O(l) = Ti2O3(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{Ti}{}_{2}\text{O}{}_{3}}=78680.3/T-8.9$
3Ti(l) + 5O(l) = Ti3O5(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{Ti}{}_{3}\text{O}{}_{5}}=127859.1/T-14.4$
Ti(l) + O(l) = TiO(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{TiO}}=28206.2/T-3.2$
Fe(l) + 2V(l) + 4O(l) = FeV2O4(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{FeV}{}_{2}\text{O}{}_{4}}=82231.3/T-12.4$
2V(l) + 3O(l) = V2O3(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{V}{}_{2}\text{O}{}_{3}}=62455.8/T-7.5$
V(l) + O(l) = VO(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{VO}}=22621.1/T-2.8$
Zr(l) + 2O(l) = ZrO2(s)	$\mathrm{l}\mathrm{g}\;{K}_{\text{ZrO}{}_{2}}=56890.1/T-6.3$

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O activity (${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$) is often reported to be relative to the 1wt% standard state (${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$). This activity is related to the activity relative to the Raoultian standard state (${a}_{\text{O}}$) through the following relationship:

5.1   Al deoxidation in liquid Fe: Fe–Al–O system

The deoxidation equilibria of Al in liquid Fe–Al–O alloys have been extensively investigated through experimental studies [4,11,47–54] and thermodynamic analyses [3,7,11,55]. The data obtained from these studies are typically reported in two ways: O content ([%O]) and ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$ in the 1wt% standard state. The [%O] is often determined using methods such as the inert-gas fusion infrared absorption technique [56–57], and ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$ is measured using solid-oxide electrolyte methods [47–48,58]. In the case of [%O], the total [%O] (soluble and insoluble) in the quenched specimen is usually measured and assumed to represent the “soluble” [%O]. Figs. 3 and 4 present the data measured at 1600°C.

Figure  3.  Deoxidation equilibria of Al in liquid steel at 1600°C: [%O] as a function of [%Al]. Experimental data were obtained from Refs. [4,11,47–54]. Thermodynamic calculations reported in the literature are also shown [7,11,55].

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Figure  4.  Deoxidation equilibria of Al in liquid steel at 1600°C: ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%})}$ (with respect to 1wt% standard state) as a function of [%Al]. Experimental data were taken from Refs. [47–49]. Thermodynamic calculation reported in the literature is also shown [55].

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Various thermodynamic analyses were conducted on Al deoxidation equilibria, and one of the present authors applied the MQM in pair approximation [11] (Section 3). The figures display the [%O] calculated using the MQM as a thin blue dotted line. Thus far, this approach is considered the most accurate for the estimation of Al deoxidation equilibria among different thermodynamic analyses owing to its accuracy over a wide composition range. Further details on model calculations can be found in the work of Paek et al. [11], whereas the related working principles have been discussed elsewhere [18]. For comparison, the results calculated using other approaches, such as the WIPF, along with parameters suggested by JSPS [55] and the associate solution model [3,7,29], were also included.

Figs. 3 and 4 display the results obtained using the present model (blue lines). The present model demonstrated good agreement with the measured [%O] up to a certain aluminum content ([%Al] = $\sim$10 or $\mathrm{l}\mathrm{g}$ [%Al] = $\sim$1). It also reasonably described the measured ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$ within the range of experimental measurements (Fig. 4). The present model performed as well as the MQM in describing deoxidation equilibria up to [%Al] = $\sim$10, and this range is sufficient to cover the [%Al] of most practical steel products.

To obtain these results at 1600°C, the following model parameters were deduced (kJ⋅mol⁻¹):

along with $Z=2$. The deoxidation equilibria for a given set of model parameters were calculated and compared with experimental data. The procedure was repeated until a satisfactory result was obtained. Then, the optimized model parameters were finally obtained. Furthermore, ${K}_{\text{Al}{}_{2}\text{O}{}_{3}}$ in Eq. (10) was provided as a function of temperature (Table 2). Notably, Δg_FeO was optimized to describe not only Al deoxidation but also the deoxidation equilibria of other elements M. This will be shown in subsequent sections.

Although the parameters Δg_FeO, Δg_FeAl, and Δg_AlO were newly optimized in the present study, they showed very similar sizes of excess Gibbs energy for each binary system, which is proportional to the product between the coordination number $Z$ and Δg_ij (Eq. (33)).

Fig. 5 displays the calculated Al deoxidation equilibria ([%O] and ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$ over the temperature range of 1550–1850°C, along with available experimental data [11,59–60]. Within the composition range of Al ([%Al] $< \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\sim$10), the model demonstrated good agreement with experimental data across the entire temperature range.

Figure  5.  Deoxidation equilibria of Al in liquid steel at various temperatures: [%O] and ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%}) }$ (with respect to 1wt% standard state) as a function of [%Al]. Experimental data were taken from Refs. [11,59–60]. Lines represent the results calculated in the present study.

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5.2   Deoxidation of liquid Fe by other elements relevant to steel production: Fe–M–O system (M = Si, Ti, Mn, Cr, and C)

The deoxidation equilibria of other deoxidizing elements (M) in Fe–M–O alloys was described using the present model. Figs. 6–9 illustrate the deoxidation equilibria ([%O] and ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$) of Si, Ti, Mn, and Cr, respectively. Comparison was conducted between experimental data reported in the literature [9,47,49,61–80] and model calculations. Similar to the approach presented in Section 5.1, a coordination number ($Z$) of 2 was assumed for all elements, and the Δg_FeO was maintained at −130 kJ⋅mol⁻¹. The calculation for each Fe–M–O solution required two additional parameters, namely, Δg_FeM and Δg_MO, which were set as constants, respectively. The equilibrium constants (${K}_{\text{M}{}_{x}\text{O}{}_{y}}$) in Eq. (10) were obtained from reliable sources [20–21,46], so these are not the model parameters. As a result, the advantage of the present model in requiring a minimal number of model parameters in calculations has been demonstrated.

Figure  6.  Deoxidation equilibria of Si in liquid steel at 1600°C: [%O] and ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%}) }$ (with respect to 1wt% standard state) as a function of [%Si]. Experimental data were taken from Refs. [61–65]. Thermodynamic calculation reported in the literature is also shown [55].

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Figure  7.  Deoxidation equilibria of Ti in liquid steel at 1600°C: [%O] and ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%}) }$ (with respect to 1wt% standard state) as a function of [%Ti]. Experimental data were obtained from Refs. [9,47,49,66–69]. Thermodynamic calculation reported in the literature is also shown [55].

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Figure  8.  Deoxidation equilibria of Mn in liquid steel at 1600°C: [%O] and ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%}) }$ (with respect to 1wt% standard state) as a function of [%Mn]. Experimental data were taken from Refs. [63,69–73]. Thermodynamic calculation reported in the literature is also shown [55].

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Figure  9.  Deoxidation equilibria of Cr in liquid steel at 1600°C: [%O] and ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%}) }$ (with respect to 1wt% standard state) as a function of [%Cr]. Experimental data were taken from Refs. [49,74–80]. Thermodynamic calculation reported in the literature is also shown [55].

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In addition to the deoxidation equilibria of deoxidizing elements, the investigation focused on the solubility of O in liquid Fe–C–O alloys. As shown in Fig. 10, the [%O] in liquid Fe–C–O alloys was calculated at various temperatures and ${P}_{\text{CO}}=1$ atm and presented along with the reported experimental data [81–87].

Figure  10.  Deoxidation equilibria of C in liquid steel at 1600°C: [%O] as a function of [%C]. Experimental data were taken from Refs. [81–87].

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5.3   Deoxidation of liquid Fe by strong deoxidizing elements

In the thermodynamic modeling of deoxidation equilibria, the deoxidation of liquid steel by strong deoxidizing elements, such as Ca, presents a challenge. The deoxidation curve of Ca deoxidation exhibited an unusual shape when the random mixing of atoms was assumed [88]. However, the accuracy of the reported experimental data is also a concern due to limitations in chemical analysis techniques [5,88–90]. The overestimation of [%O] in the experimental data is possible due to analysis limitations [91–93]. Therefore, caution must be exercised in the performance assessment of any model in the absence of reliable experimental data on deoxidation equilibria.

In the present study, the developed model was applied to calculate the deoxidation equilibria of liquid steel by Ca, and the results are shown in Fig. 11, including those at conventional logarithmic scale (upper panel) and normal scale (lower panel) [88,94–98]. The calculated [%O] in the logarithmic scale differs from that in previous figures (Figs. 3 and 5–10). The lower panel in Fig. 11 presents the same results but in a more dilute concentration region in the normal scale. A minimum [%O] was observed, as discussed by one of the present authors [99].

Figure  11.  Deoxidation equilibria of Ca in liquid steel at 1600°C: [%O] as a function of [%Ca]. Experimental data were taken from [88,94–98].

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Fig. 12 displays the deoxidation equilibria of liquid steel by cerium (Ce). A lower [%O] was calculated compared with the reported experimental data [100–101]. However, the calculated ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$ shows very good agreement with the data from the work of Janke and Fischer [102].

Figure  12.  Deoxidation equilibria of Ce in liquid steel at 1600°C: [%O] and ${\boldsymbol a}_{\underline{\bf{O}}({\bf {wt}} \text{%}) }$ (with respect to 1wt% standard state) as a function of [%Ce]. Experimental data were collected from [100–101]. Thermodynamic calculation reported in the literature is also shown [55].

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Appendix A in Supplementary Information provides the model calculations for deoxidation equilibria by other strong deoxidizers, such as Mg, La, and Zr.

5.4   Deoxidation of liquid Fe by other elements relevant to steel production: Fe–M–O system (M = Nb, V, and B)

Nb, V, and B are also commonly added to liquid steel as alloying elements. Appendix B in Supplementary Information presents the model calculations for the deoxidation equilibria involving these elements.

6.   Discussion

6.1   Application of the present model in complex deoxidation

The present model can be readily extended to multicomponent liquid-steel systems, where multiple deoxidizing metallic elements coexist with Fe and O. Appendix C in Supplementary Information provides the expressions for the activity coefficients of any metallic element (M) and that of O (Eqs. (C.3) and (C.4)). To demonstrate the applicability of the model to complex deoxidation equilibria, a case study involving high-Mn and high-Al steel was conducted.

Fig. 13 displays the calculated oxide stability diagram of the Fe–Al–Mn–O system; this system represents the equilibrium conditions for oxide formation, along with the reported experimental data [25,70,103–106]. The construction principles observed in the oxide stability diagrams have been discussed by Kang and Jung [107]. The calculated results obtained from the present model agree well with experimental data, highlighting the effectiveness of the model in describing complex deoxidation equilibria in high-Mn–high-Al steel systems. However, contrary to the oxide stability diagram proposed by one of the present authors’ previous studies [25], the present calculations disregarded the stable region of liquid oxide. This omission was due to the characteristic calculation logic used in the present study, as pointed out by the present author’s other article [107]. Therefore, the present calculation approach can only be used in solid equilibrium-oxide phases with a low nonstoichiometry, such as Al${}_{2}$O${}_{3}$, MnO, or MnAl${}_{2}$O${}_{4}$.

Figure  13.  Oxide stability diagram of Fe–Al–Mn–O system calculated using the present thermodynamic model. Experimental data were obtained from [25,70,103–106].

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6.2   Coupling to CFD simulation

The present thermodynamic model was utilized to develop a reoxidation simulator for tundish liquid steel during the casting process. The designed simulator calculates the extent of reoxidation, referring to the consumption of the deoxidizing element caused by the presence of a tundish open eye (TOE). The thermodynamic model was incorporated into a user-defined function in the CFD software Ansys Fluent (version 18.2) to determine the deoxidation equilibria of Al-killed ultra-low carbon steel.

Thermodynamics was coupled to fluid dynamics to simulate reoxidation conditions, including the concentrations of soluble Al, soluble O, and suspended alumina inclusions in liquid steel within the tundish at each time and location. A reoxidation simulator of liquid steel in a tundish was developed using the present thermodynamic model to calculate the extent of reoxidation (consumption of deoxidizing element) due to a TOE during the casting process. The thermodynamic model was programmed into a CFD user-defined function of Ansys Fluent (version 18.2) [108] to determine the deoxidation equilibria of Al-killed ultra-low C steel. Nonsteady–state fluid dynamic calculations were performed using the CFD code. The direct coupling of thermodynamics to fluid dynamics enabled the simulation of the extent of reoxidation (concentrations of soluble Al, soluble O, and suspended alumina inclusion in liquid steel in the tundish) at each time and location. The reoxidation conditions of the liquid steel are as follows: (1) The liquid steel consisted of Fe–Al–O and represented ultra-low carbon steel with initial compositions of [%Al] = 0.052 and [%O] = 0.0003. The steel initially filled the tundish and was covered by a tundish flux, which was assumed to be a nonreactive liquid phase with a thickness of 20 mm. (2) Liquid steel of the same initial composition flowed into the tundish through the shroud nozzle, creating a TOE. The air entered through the TOE and oxidized the liquid steel via reaction (3). (3) The flow rates of the incoming steel through the shroud nozzle and the outgoing steel through the submerged-entry nozzle were balanced, resulting in a constant liquid-steel depth in the tundish.

Fig. 14 presents an example of the simulator calculation, specifically 200 s after the casting process was initiated. The depicted tundish represents a quarter section of an actual operational tundish in a casting shop at a 1:1 scale. The simulation yielded spatial distributions of the mass fractions of soluble aluminum, soluble oxygen content, and alumina content in the liquid steel. The TOE was observed near the liquid-steel surface between the shroud nozzle and the weir. The mass fraction of soluble Al noticeably decreased, and the [%Al] increased near the TOE.

Figure  14.  Snapshots of liquid steel in the tundish (a quarter of 1:1 scale tundish) simulated by the CFD coupled with the present deoxidation model: (a) Al (soluble) content, (b) O (soluble) content, and (c) Al₂O₃ (inclusion) content.

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The simulation was performed through the direct coupling of the CFD code to the thermodynamic model for deoxidation within the CFD framework. Further details on simulator development will be published separately.

6.3   Applicable range of the present model and its limit

The MQM in the pair approximation has been successfully applied in the accurate description of deoxidation equilibria across a wide range of compositions and temperatures [11,25].

In the present study, a new model approach based on the MQM was developed to provide a simpler but still accurate method of calculating deoxidation equilibria. The present model was successfully applied overall, and unsatisfactory results were obtained in a few cases, particularly at high concentrations of deoxidizing elements (Figs. 3, 5, and 7). The assumption on a constant pair formation energy (Δg_FeM in Eq. (16)) may be responsible for these discrepancies. Previous thermodynamic modeling research on Fe–M binary systems using MQM often considered Δg_FeM to be composition dependent, either through composition variables or pair fraction variables. In the present study, the complexity associated with composition-dependent Δg_FeM was intentionally avoided for simplicity. However, expressing Δg_FeM as a function of composition will lead to intricate expressions for the activity coefficients of M and O (Eqs. (49), (50), (C.3), and (C.4)), which were not pursued in the present study.

6.4   Accuracy of the measured [%O]

Challenges have been observed in the thermodynamic modeling of deoxidation equilibria of liquid steel by strong deoxidizing elements, such as Ca, Mg, Zr, La, and Ce, not only in the present study but also in previous literature [88,90]. Figs. 12 (Ce), A.2 (La), and A.3 (Zr) exhibit good agreement between the model calculations and experimental data for ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$), indicating a successful prediction of ${a}_{\underline{\mathrm{O}}\left(\mathrm{w}\mathrm{t}\%\right)}$. However, a discrepancy was observed in the [%O] values between the model calculations and experimental data. This discrepancy has been discussed by several researchers [88–90]. Hillert and Selleby [89] pointed out potential systematic errors in the measured solubilities of Ca-deoxidized liquid steel. Huang [90] also expressed skepticism regarding the accuracy of the measured [%O] in Zr-deoxidized steel. Gustafsson and Malmberg [88] cautioned about the interpretation of the currently used experimental techniques.

The measurement of [%O] in steel specimens through the gas-fusion infrared absorption method [56–57,91] is typically intended to determine the total [%O], including soluble and insoluble O. The overestimation of the soluble [%O] is possible due to the presence of oxide inclusions (insoluble O). Recent advancements in analysis methods allow for the simultaneous measurement of soluble and insoluble [%O], which can help in the refined determination of [%O] in steel specimens [91–93].

7.   Conclusions

In the present study, thermodynamic modeling of deoxidation equilibria of liquid steel considered the strong interaction between metal and O, avoiding the assumption of simple random mixing. The model is a limiting case of the quasichemical model, with the assumption of constant coordination numbers, constant pair-exchange energies, complete SRO between metal and O, and random mixing of metallic components. The equilibrium constant equation was solved to determine the deoxidation equilibria. The activity coefficients of M and O were expressed as explicit functions of composition and temperature, accounting for nonrandom mixing between metal and O. Importantly, the present model skips the minimization of complex internal Gibbs energy, resulting in a computationally efficient approach for the high-accuracy calculation of deoxidation equilibria of liquid steel.

The present model can be easily coupled with other simulation codes, such as CFD, for fluid flow simulations that involve deoxidation phenomena. This coupling considerably reduces the computation time by directly incorporating the deoxidation equilibria calculation into the simulation code.