Oxidizability characterization of slag system on the thermodynamic model of superalloy desulfurization

3.1 Model of mass action concentrations of structural units and ion couples

According to the ion and molecular coexistence theory, the phase composition of slag is very important for the establishment of the desulfurization thermodynamic model. According to the phase diagrams of CaO–SiO₂, CaO–Al₂O₃, TiO₂–Al₂O₃, TiO₂–CaO, CaO–Al₂O₃–SiO₂, CaO–SiO₂–CaF₂, CaO–Al₂O₃–CaF₂, CaO–SiO₂–TiO₂, CaO–FeO–SiO₂, CaO–MgO–SiO₂, and MgO–Al₂O₃–SiO₂ [22], the structural units from 1,473 to 1,973 K [23] are listed in Table 5.

The molten slag was regarded as an ideal solution in dynamic balance reactions between the simple ions and the simple molecules. The weight of the initial slag system was set to be 100 g. To represent the chemical composition of slags, the variables, b1=nCaO0, b2=nMgO0, b3=nCaF20, b4=nFeO0, b5=nSiO20, and b6=nTiO20, b7=nAl2O30, were assigned to represent the number of moles of CaO, MgO, CaF₂, FeO, SiO₂, TiO₂, and Al₂O₃ in the slag system CaF₂–CaO–Al₂O₃–SiO₂–TiO₂–MgO–FeO. The ion and molecule coexistence theory hold that each ion couple is electrically neutral and can be dissociated into separate cations and anions. Each ion pair’s balance number of moles is equal to the summation of the cations and the anions in the ion couple.

The total number of balance moles of all structural units in 100 g slag ∑ni can be expressed as follows:

According to the ion and molecule coexistence theory, the ratio of the balance number of moles of a structural unit i to the total number of moles of all the structural units in the balance system is the mass action concentration of the structural unit i in the molten slag, which is called N_i. It can be calculated as follows:

When the system reaches a dynamic balance, each simple molecule and complex molecule can only provide one structural unit. Therefore, the mass action concentrations of all structure units in the form of ions, simple molecules, and complex molecules in CaF₂–CaO–Al₂O₃–SiO₂–TiO₂–MgO–FeO slag can be calculated from equation (2). However, mass action concentrations of ion couples, such as (Ca²⁺ + O²⁻), should be calculated as follows:

Assuming the conservation of mass and dynamic balances for the reactions of the slag structure units, the activity model of slag can be established. The ions and structural units can form complex molecules through chemical reactions in the slag system CaF₂–CaO–Al₂O₃–SiO₂–TiO₂–MgO–FeO. The chemical reactions, the standard molar Gibbs free energies ΔrGm,ciθ, and the mass action concentrations of possibly formed complex molecules are shown in Table 6. The standard molar Gibbs free energies for the reactions were cited from the literatures [24,25] or calculated with the FactSage software.

Based on the balance number of moles and mass action concentration of each structural unit of slag, the mass conservation equation is established as follows:

Because the sum of the mole fractions of all structural units in slag is 1 under balance conditions, the following formula can be obtained:

Equations (4)–(11) govern the calculation of the mass action concentrations of structural units or ion couples in slag system CaF₂–CaO–Al₂O₃–SiO₂–TiO₂–MgO–FeO. There are eight unknowns as in N₁, N₂, N₃, N₄, N₅, N₆, and N₇ and ∑ni with eight independent equations. Using MATLAB 7.0 to solve these higher order nonlinear equations, the activity of each component in slag was obtained.

3.2 Verification of mass action concentration model

The mass action concentration model of slags is also called the activity model of slag components. Because the measured activity values of slag CaF₂–CaO–Al₂O₃–SiO₂–TiO₂–MgO–FeO have rarely been reported, another common slag system studied by researchers [26,27] was used to verify the new model. The components of the common slag system were brought into the model and then the calculated activity values were compared with previous experimental results, as seen in Figure 2.

Figure 2

Comparison between calculated values of the activity model and the tested values. (a) FeO, (b) CaO, and (c) Al₂O₃.

Figure 2(a) shows that the FeO activities calculated by the new model are in good agreement with the measured values from Zhou et al. [27]. Meanwhile, our calculation results of the FeO activities are slightly different from Chen’s measurement results [26]. The slight difference might be generated by the differences in thermodynamic values used in the calculation process. The results described in Figure 2(b) and (c) show that the CaO and Al₂O₃ activities calculated by the model are in good agreement with the results by Chen, which further proved that the model can be used to calculate the activities of slag components accurately.

3.3 Thermodynamic model of desulfurization between slag and metal interface

In the process of establishing the governing equation for calculating the mass action concentration of slag components in the CaF₂–CaO–Al₂O₃–SiO₂–TiO₂–MgO–FeO system, the structure unit S²⁻ was ignored because the balance number of moles of S²⁻ as a structure unit was very small. Therefore, the calculation of mass action concentration of other structural units was not affected by the number of moles of structured unit S²⁻.

According to the ion and molecule coexistence theory, only free (Ca²⁺ + O²⁻) and (Mg²⁺ + O²⁻) ion couples have the ability to desulfurize and participate in the desulfurization reaction. The reaction is expressed as follows:

The equations’ balance constants are expressed as follows:

The sulfur distribution ratio of (Ca²⁺ + O²⁻) and (Mg²⁺ + O²⁻) ion pairs can be calculated from equations (14) and (15) as follows:

In the electroslag remelting (ESR) process, the oxygen content in the molten metal pool is related to the content of FeO or Al₂O₃ in the slag [13]. According to the following reactions:

The deformations are expressed as follows:

The activity coefficient of the elements in superalloy can be calculated from equation (25). Because Inconel 718 alloy possesses high alloying element contents, the calculated results of the element activity coefficient could be much more accurate if the secondary interaction coefficient between elements was considered. However, the secondary interactions between elements in nickel-base superalloy activity coefficient have been rarely reported, thus this study only considers the primary activity interaction coefficients [28,29], as shown in Table 7.

Based on the [Fe]–[O] balance, the oxygen content in alloys is described as:

Based on the [Al]–[O] balance, the oxygen content in alloys is described as:

Based on [Fe]–[Al]–[O] balance, the thermodynamic model of oxygen content in alloys is described as:

The total sulfur distribution ratio in the slag system can be calculated using equations (26)–(28). The variable f_S represents the activity coefficient of dissolved sulfur in the nickel-based alloy. A 1% mass fraction was selected as the standard state, and the sulfur content of our experiment was less than 10 ppm. According to the previous studies of nickel-based alloy solutions, the activity coefficient f_S of dissolved sulfur is assumed to be 1. Since the entire experiment process was under the protection of argon atmosphere, and the added superalloy had no oxide layer, the oxygen in the molten metal can only be supplied from the slag. The components in slag that can affect the oxygen content in the molten metal are mainly FeO and Al₂O₃, and the effects of these two components are independent of each other. So the equilibrium oxygen content based on [Fe]–[Al]–[O] can be calculated as the sum of that of [Fe]–[O] and [Al]–[O] balance. In addition, KCaSθ and KMgSθ can be calculated using equation (29).

where R is the gas constant (8.314 J mol⁻¹ K⁻¹) and T is the absolute temperature (K).

4.1 Experimental results

The sulfur distribution ratio of experimental slags is shown in Table 8. It has been experimentally determined that as the content of FeO and Al₂O₃ increases, the desulfurization distribution ratio gradually decreases. By changing the content of FeO and Al₂O₃ in the experiment, slag desulfurization is affected. The effect of FeO will be more significant than that of Al₂O₃, which mainly affects the desulfurization effect by affecting the slag oxygen content.

4.2 Verification of the desulfurization model

Figure 3 shows the effect of FeO and Al₂O₃ on the oxygen content in the alloy through theoretical calculation based on ion and molecule coexistence theory and experimental data. It is obvious that the increase in FeO and Al₂O₃ will increase the oxygen content in the alloy, and the influence of FeO is obviously much greater than that of Al₂O₃.

Figure 3

Influence of FeO and Al₂O₃ on the oxygen content in the alloy.

Taking the S4 slag system as an example, the calculated oxygen contents according to [Fe]–[Al]–[O] balance, [Fe]–[O] balance, and [Al]–[O] balance are shown in Figure 4. It can be seen from the figure that the influence of [Fe]–[O] balance on oxygen content is much greater than that of [Al]–[O] balance, but the influence of [Al]–[O] balance cannot be ignored. The [Fe]–[Al]–[O] balance is more comprehensive in characterizing the oxygen content. Figure 5 shows the comparison of sulfur distribution ratios at different FeO contents obtained through three characterization methods and experiments. In Figure 5, L_{s,[Fe]–[O],calculated} line is the sulfur distribution ratio calculated based on the desulfurization thermodynamic model established by [Fe]–[O] balance; L_{s,[Al]–[O],calculated} line is the calculated sulfur distribution ratio based on [Al]–[O] balance; L_{s,[Fe]–[Al]–[O],calculated} line is the calculated sulfur distribution ratio based on [Fe]–[Al]–[O] balance; L_s,tested line is the sulfur distribution ratio of slag system S0, S1, S2, and S3 measured by experiment. The sulfur distribution ratio determined by any characterization method or experiment decreases with the increase in FeO content at the basicity of 6.67. When the content of FeO in the slag is less than 0.4%, the variation ratio of sulfur distribution is larger. When the FeO content is between 0.4% and 0.7%, the variation range of sulfur distribution ratio is significantly reduced. With the increase in FeO content, the effect of FeO content on the sulfur distribution ratio becomes smaller, which is mainly because some FeO in the slag combines with Al₂O₃, SiO₂, and CaO in the slag to form FeO·Al₂O₃, 2FeO·SiO₂, CaO·FeO·SiO₂, and so on. The molecule [17] reduces the activity of FeO, so that the sulfur distribution ratio tends to be stable with the change in FeO. In Figure 5, the fluctuation range of Al₂O₃ content is in the range of 19.96–20.04%, so the change trend of sulfur distribution ratio calculated according to [Al]–[O] balance is not obvious. Line L_{s,[Fe]–[O],calculated} and Line L_{s,[Al]–[O],calculated} have an intersection point when the FeO content is 0.014%. On the left side of the intersection point, the sulfur distribution rate calculated based on the [Al]–[O] balance is closer to the experimental value, while the sulfur distribution rate calculated based on [Fe]–[O] balance is closer to the experimental value on the right side of the intersection point. Therefore, we can conclude that the free oxygen released by Al₂O₃ is higher than the free oxygen released by FeO when the FeO content is less than 0.014%. At this time, the [Al]–[O] balance is better than the [Fe]–[O] balance, but [Fe]–[Al]–[O] balance calculation result is closest to the measured value.

Figure 4

Calculated oxygen contents based on three oxidizability characterization methods.

Figure 5

Comparison of sulfur distribution ratios at different FeO contents obtained through three characterization methods and experiments.

Figure 6 shows comparison of the sulfur distribution ratios at different Al₂O₃ contents obtained through three characterization methods and experiments. In Figure 6, the L_{s,[Al]–[O],calculated} line is the sulfur distribution ratio calculated based on the desulfurization thermodynamic model established by [Al]–[O] balance; the L_{s,[Fe]–[O],calculated} line is the sulfur distribution ratio based on [Fe]–[O] balance; the L_{s,[Fe]–[Al]–[O],calculated} line is the sulfur distribution ratio based on [Fe]–[Al]–[O] balance; and the L_s,tested line is the sulfur distribution ratio in slag system S1, S4, S5, S6, S7, and S8 measured by the experiment. The slope of line L_{s,[Al]–[O],calculated} is obviously greater than the line L_s,[Fe]–[O] and line L_{s,[Fe]–[Al]–[O]}, which indicates that the effect of [Al]–[O] balance increases gradually with the increase in Al₂O₃. This further illustrates that when the Al₂O₃ content is higher than 20%, the decomposition of Al₂O₃ has a greater impact on desulfurization, which cannot be ignored. It can still be seen from the figure that the sulfur distribution ratio obtained by [Fe]–[Al]–[O] balance is closest to the experimental value.

Figure 6

Comparison of sulfur distribution ratios at different Al₂O₃ contents obtained through three characterization methods and experiments.

From the analysis of the reaction equation of desulfurization, it can be known that when the slag shows low oxidizability, it will promote the desulfurization reaction. The FeO content in the slag will directly affect the dissolved oxygen content, which in turn affects desulfurization. The experimental and theoretical calculation values of the sulfur distribution ratio in the slag decrease with the increase in the FeO content. When the FeO content is higher than 0.014%, Al₂O₃ will play a weak role. When the FeO content is lower than 0.014%, Al₂O₃ will play a stronger role than FeO. As the content of Al₂O₃ increases gradually, the sulfur distribution ratio changes greatly. However, the sulfur distribution ratio calculated by the desulfurization thermodynamic models has a large deviation from the experimental value, and the theoretical calculation values are higher than the experimental measurement value. In addition, during the experiment, the S0 slag which does not contain FeO still has desulfurization ability and a large sulfur distribution ratio. However, the desulfurization thermodynamic model based on the [Fe]–[O] balance cannot explain this phenomenon. This indicates that the dissolved oxygen content in the sample is affected not only by the FeO content but also by other factors such as Al₂O₃. Taniguchi et al. [13] has used the activity of FeO to characterize the dissolved oxygen content in the steelmaking process, but the calculated values of the model and the experimentally determined values are quite different, which further confirms this view. Based on the desulfurization thermodynamic model established by [Fe]–[Al]–[O] balance, L_{s,[Fe]–[Al]–[O],calculated} is lower than L_{s,[Fe]–[O],calculated} and closer to the experimental measurements. In addition, when the FeO content is 0, L_{s,[Fe]–[Al]–[O],calculated} is 746.03, and this model can explain the phenomenon that the S0 slag system has a large sulfur distribution ratio. And as shown in Figure 6, this also verifies that L_{s,[Fe]–[Al]–[O],calculated} is closer to the measured value. Therefore, using the desulfurization thermodynamic model established by [Fe]–[Al]–[O] balance is more reasonable and feasible compared to other models. However, the sulfur distribution ratio calculated by [Fe]–[Al]–[O] balance is still higher than that of the experimental value. This may be due to the insufficient protection of the atmosphere during the experiment. The element Fe in the alloy is oxidized, making the actual FeO content higher than the FeO content used in the model calculation, resulting in a large deviation. It can be seen from Figure 6 that when the FeO or Al₂O₃ content is high, the gap between the model calculated value and the experimental value becomes smaller.

When smelting ultra-pure alloys during electroslag remelting, the sulfur content of the alloy must be strictly controlled. Therefore, the content of FeO and Al₂O₃ must be strictly controlled to be low. In addition, the slag temperature in the electroslag remelting process is usually higher than 1,923 K, and that Al₂O₃ content in the electroslag remelted slag system usually exceeds 25%, so it is necessary to consider the effect of the oxygen released by the decomposition reaction of Al₂O₃ on desulfurization. The desulfurization thermodynamic model established by [Fe]–[Al]–[O] balance is more accurate than the desulfurization thermodynamic model constructed by [Fe]–[O] and [Al]–[O] balance; therefore, this method is more suitable for oxygen content characterization in the process of electroslag remelting. Superalloys have very strict requirements for oxygen, sulfur, and other components. Generally, it is required to control the oxygen content to less than 0.0012% and the sulfur content to less than 0.0020%. Therefore, FeO must be strictly controlled during the electroslag remelting process. According to the calculation results in Figure 5, it is recommended to control the FeO content to less than 0.014%.