EvaluatIon of carBIdE capacIty In cao-BaSEd tErnary SyStEMS at 1773K for rEfInIng procESS

Refining slags is widely used in the production of high-value-added alloys and special steels. The removal of impurities depends on the mass transfer between the slag-metal interface, and the carbide capacity of the refining slags is crucial to control the carbon content in the final products. A phenomenological model is introduced in this article for the calculation of carbide capacity of different CaO-based ternary refining slags. The contour lines of carbide capacity in CaO-Al2O3CaF2, CaO-SiO2-CaF2 CaO-SiO2-MnO systems are calculated based on limited experimental data by the present model. The experimental data within the calculation boundary are compared with the predicted values and satisfactory agreements are observed with the mean deviation being 1.4%, 2.3%, and 1.6%, for these three systems respectively. The present model is powerful and flexible in the calculation of carbide capacity of CaO-based ternary refining slags and can be applied to other systems.


Introduction
The key to the production of special steels or superalloys lies in the removal of impurities in the molten metal [1,2]. The existence of carbon in steels causes brittle fractures of steel in extreme conditions, which need to be removed through the refining process [3]. Carbon removal by slag refining isn't the priority in steelmaking, but can be more important in other alloys, such as ferrochrome [4,5] and ferromanganese [6] alloys. The removal of carbon depends on the mass transfer between the slag-metal interface when the hot metal drops cross the slag layer and fall into the molten pool [7,8]. As a result, the carbide capacity of the refining slags is crucial to the control of the carbon content in the final products. However, most of the refining slag systems are with high melting points and can corrode the crucible at high temperatures, so it is impossible to obtain all these carbide capacity data for practical use just through experimental methods.
On the contrary, the model prediction is an alternative which may address this issue in a more effective way. The geometrical model [9][10][11] is a powerful tool to predict the properties of slags and has been widely used in the calculation of sulfide capacities [12][13][14], phosphate capacities [15], and other physicochemical properties of slags [15][16][17][18][19][20][21][22][23][24][25][26][27]. Traditional geometrical models [9,11,[28][29][30][31][32] can predict the properties of ternary systems only when the binary data are abundant. However, most of the slag systems are with miscibility gaps at their application temperature, where the limited solubility area doesn't even intersect with the binary boundaries. So, on this condition, these systems are not suitable to be calculated by traditional geometrical models for the lack of corresponding binary data.
Recently, Chou [33][34][35] has extended the geometrical model to the systems with limited solubility area by proposing a new method. The new method can predict the physicochemical properties of ternary systems just based on limited experimental boundary data, which will enlarge the application range of geometrical models in practical systems. In this article, our new model will be first applied into the calculation of carbide capacity of ternary slags for refining process. The predicted results show good agreements with the experimental data in CaO-Al 2 O 3 -CaF 2 , CaO-SiO 2 -CaF 2 CaO-SiO 2 -MnO slag systems. The contour lines of carbide capacity for different systems are also calculated by the present model.

Model description
According to the new model [33][34][35], there are three conditions: limited binary boundary data, limited ternary boundary data, and even only three experimental data, which can be used for calculation of the systems with limited solubility. A short description of the second condition that calculation based on limited ternary boundary data will be given here.
As illustrated in Fig. 1, the gray part is the limited solubility area, which has no interaction with the binary boundary. The dash line along the limited solubility area is the calculation boundary, which is not a real boundary and can be arbitrarily selected according to the distribution of the experimental points. The physicochemical properties can be calculated within this boundary when choosing three points ("A" "B" "C") on this calculation boundary for the calculation, and the selection rule of the three calculation points (A" "B" "C") can just follow Chou's method [28]. Then, the properties of point "O" are calculated by a combination of the three calculation points ("A" "B" "C") according to the equations below. (1) where is the physicochemical property of point "i", and is the corresponding weight probability of different calculation points (A" "B" "C"). In the present model, the weight probability of each calculation point can be expressed as the following equations.
where is the composition of "j, at point "i", and the weight probabilities of the calculation points are expressed as the component coordinates of each point. In Fig.1, the distance between point "O" and points "A" "B" "C" are OA= , OB= , and OC= . It is easy to find that the weight probabilities of the calculation points just depend on their distances from point "O". The nearer the calculation point ("A" or "B" or "C") is to the calculated point "O", the larger the weight probability is. Combined with the equation (1), as a result, the property between the calculation point ("A" or "B" or "C") and point "O" will be more similar, which is reasonable and consistent with the reality. The detailed theoretical derivation process can be referred to our previous article [35].

carbide capacity prediction of various systems
The measurement of carbide capacity is difficult due to the high melting points of the refining slags and the experimental data varies when it is measured by different researchers, which brings difficulties for model prediction. Park et al. [36][37][38] measured the carbide capacity of different systems and determined the content of carbon in the slags by a LECO (CS-300) analyzer. All the carbide capacity data measured by Park et al. were chosen for the calculation in the following examples. First, the outmost layer experimental data were selected as the calculation boundary (Bi, i≥3) and the inner data as the calculated points (Ci) for the comparison. All the calculation data of the present model are along the virtual ternary boundary, and all the calculated contour lines of carbide capacity are with the same range of notation to have an intuitive comparison of different systems.
The relative deviation (R.D.) and mean deviation (Δ) were used to compare the inner experimental data (Ci) with the calculated values, as expressed in the equation below: . . % and n arbide arbide

CaO-Al 2 O 3 -CaF 2 ternary system
CaO-Al 2 O 3 -CaF 2 slag is the most basic ternary system for ESR process, and has been widely used in the production of special steels. Park et al. [36] measured the carbide capacity of this system in a wide composition range within the limited solubility area at 1773K. Seventeen experimental data were chosen as the calculation boundary points, as set out in Table 1. All these calculation points were located at the outmost layer of the composition triangle among all the experimental data and constituted a virtual calculation boundary (Fig. 2). Combined with the equations (1-4), the carbide capacity could be predicted within this calculation boundary. As shown in Fig. 2, the contour lines of carbide capacity in CaO-Al 2 O 3 -CaF 2 system at 1773K were calculated by the present model.
The experimental data [36] within the calculation boundary (C1-C7) were selected to compare with the calculated values. As illustrated in Fig. 2, the comparison points (C1-C7) are marked in red and uniformly distributed inside the calculation boundary. The detailed comparison results between the experimental carbide capacity and the calculated values of each point are listed in Table 2. The relative deviation (R.D.) ranges from -0.4% to -2.5%, with the mean deviation (Δ) being 1.4% for seven groups of data.
Besides, the predicted results along three component lines of CaO-Al 2 O 3 -CaF 2 ternary system were chosen to show the influence of different additions on the carbide capacity of this system. As depicted in Fig. 3, when the content of CaF 2 or CaO is 0.3 (30 mol.%), the addition of Al 2 O 3 decreases the carbide capacity of CaO-Al 2 O 3 -CaF 2 ternary system, especially when the content of CaF 2 is constant (red points in Fig. 3). However, the addition of CaF 2 shows a slight effect on the carbide capacity of CaO-Al 2 O 3 - Yu Fig. 3). It can be concluded that, if one wants to increase the carbide capacity of CaO-Al 2 O 3 -CaF 2 slag system for ESR, the content of Al 2 O 3 needs to be reduced.

CaO-SiO 2 -CaF 2 ternary system
Silicate slags are extensively used to control the removal process of boron from molten silicon, and CaO-SiO 2 -CaF 2 is a basic silicate slag system in the refining process of metallurgical-grade silicon [39][40][41]. Park et al. [37] also measured the carbide capacity of this system within the limited solubility area at 1773K. Because the 20 experimental points were not evenly distributed, just 6 of them located at the outmost layer of the composition triangle were chosen as the calculation boundary points (B1-B6). The calculation boundary points are listed in Table 1 and constitute the calculation boundary in Fig. 2. Combined with the above equations (1-4), the contour lines of carbide capacity in CaO-SiO 2 -CaF 2 system at 1773K were predicted, as shown in Fig. 2. The contour lines are much denser than that of CaO-Al 2 O 3 -CaF 2 system in Fig. 2, which means the carbide capacity changes more dramatically in CaO-SiO 2 - The comparison results between the experimental carbide capacity [37] and the calculated values are shown in Table 3. As illustrated in Fig.4, all the comparison points (C1-C14) are within the calculation boundary, marked in red. Although the calculation boundary points are very limited, the predicted results still agree well with the experimental ones, with the mean deviation (Δ) being 2.3% for 14 groups of data. This result proves the reliability of the present model, even under the conditions with very limited experimental data. It reveals the potential application value of the present model because the experimental data of most slag systems are scarce and scattered.
In addition, the predicted results along three component lines of CaO-Al 2 O 3 -CaF 2 ternary system are also illustrated in Fig.5 to show the influence of different additions on the carbide capacity of this system. It can be seen from Fig. 5 that the addition of SiO 2 decreases the carbide capacity dramatically when the content of CaF 2 or CaO is constant (0.3 CaF 2 or 0.5CaO), while the addition of CaF 2 increases the carbide capacity of CaO-SiO 2 -CaF 2 ternary system slightly, when the content of SiO 2 is constant (0.3SiO 2 , black points in Fig. 5). Combined with the results of CaO-Al 2 O 3 -CaF 2 system in Fig.3, it is easy to find that SiO 2 and Al 2 O 3 show the similar influence on the carbide capacity of CaO-CaF 2 -based ternary system, which can be explained that SiO 2 is a kind of acid oxides and Al 2 O 3 , which is a kind of amphoteric oxides, exhibits the characteristics of acidic oxides with high content CaO. As illustrated in Fig. 5, the reduction of SiO 2 results in the increase of carbide capacity of 0.3CaF 2 -x(SiO 2 ) and 0.5CaO-x(SiO 2 ) systems.

CaO-SiO 2 -MnO ternary system
The manganese-rich slag, generated from the high carbon ferromanganese production process, has the advantage of low phosphorus content, and CaO-SiO 2 -MnO slag is a good alternative to producing silicomanganese alloys. The carbide capacity of this system, which was measured by Park et al. [38] at 1773K, is critical to the carbide content of the final products. We selected 11 of the experimental points as the calculation boundary (B1-B11 in Table 5), and 8 of them as the comparison points (C1-C8 in Table 6). All the experimental points are shown in Fig. 6,   We predicted the contour lines of carbide capacity in CaO-SiO 2 -MnO system at 1773K, according to equations (1)(2)(3)(4). As shown in Fig. 6, the contour lines are not as regular as those of CaO-Al 2 O 3 -CaF 2 or CaO-SiO 2 -CaF 2 system above, which means that the carbide capacity changes may not be linear with the addition of the oxide. Also, the predicted results along three component lines of CaO-SiO 2 -MnO ternary system just prove this inference. The influences of different additions on the carbide capacity of this system are depicted in Fig.7, where the existence of MnO results in irregular changes of the carbide capacity (black and blue points in Fig.7). When the content of MnO is restricted to 0.1(10mol.%), the addition of SiO 2 decreases the carbide capacity regularly (red points in Fig. 7). The irregular effect of MnO on the carbide capacity of CaO-SiO 2 -MnO can be attributed to the abnormal-valence of Mn at high temperature. The reduction of the content of SiO 2 is easier to control if increasing the carbide capacity of CaO-SiO 2 -MnO slag system is wanted.
The detailed comparison of the predicted results with the experimental values of the inner comparison points [38] are listed in Table 6. The relative deviation (R.D.) varies from -4.8% to 0.5%, with the mean deviation (Δ) being 1.6% for 8 groups of data, where the calculated results agree well with the experimental carbide capacity.

discussion
In the three examples above, the distribution of the experimental points varies, but all the predicted results by the present model agree well with the Z.-G. Yu   experimental values, which means the present model is powerful in the prediction of the carbide capacity of slags and can be applied to more systems. Theoretically, the denser the calculation boundary points are, the higher the prediction accuracy is, which was proved by the results in the examples above. Besides, the selection of the calculation boundary is flexible and has no need to be restricted to the binary boundaries, which enlarge the application range of geometrical models extensively. We calculated the contour lines of carbide capacity within the calculation boundary and the carbide capacity along different component lines to show the influences of diverse components. It can be concluded that the reduction of SiO 2 or Al 2 O 3 will be of help for the increase of carbide capacity of CaObased systems, while the CaF 2 shows slight influence. Besides, the addition of MnO shows irregular influence due to the abnormal-valence of Mn at high temperature.

conclusion
In this article, an extended geometrical model is applied to the calculation of the carbide capacity, of CaO-Al 2 O 3 -CaF 2 , CaO-SiO 2 -CaF 2, and CaO-SiO 2 -MnO systems for the refining process. Satisfactory agreements between the experimental data and the calculated values are observed. In addition, the contour lines of carbide capacity in all three systems are predicted based on limited experimental data by the present model. The selection of the calculation boundary is flexible and has no need to be restricted to the binary boundaries, which enlarges the application range of geometrical models extensively.