Quasi-Chemical Viscosity Model for Fully Liquid Slag in the Al₂O₃-CaO-MgO-SiO₂ System. Part II: Evaluation of Slag Viscosities

Abstract

A model is presented that enables viscosities to be predicted reliably over the whole range of compositions and temperatures in the Al₂O₃-CaO-MgO-SiO₂ slag system above liquidus in the temperature range from 1543 K to 2643 K (1270 °C to 2370 °C). Experimental procedures and data from the studies reported in the literature have been collected and critically reviewed with particular attention to the viscometry methods and possible contamination of slag samples to select reliable data points for further model development. Relevant revised formalism to describe the complex viscosity trends including charge-compensation effect of the Ca²⁺ and Mg²⁺ cations on the formation of tetrahedrally coordinated Al³⁺ was introduced. Parameters of the quasi-chemical viscosity model have been optimized to reproduce within experimental uncertainties most of the selected experimental data in the Al₂O₃-CaO-MgO-SiO₂ system and all subsystems. This study is part of the overall development of the self-consistent viscosity model of the Al₂O₃-CaO-FeO-Fe₂O₃- MgO-Na₂O-SiO₂ multicomponent slag system.

Introduction

A previous article[1] in this series of two presented the revised formalism of the quasi-chemical viscosity model.[2–7] This work is undertaken as part of the overall development of the self-consistent viscosity model of the Al₂O₃-CaO-FeO-Fe₂O₃-MgO-Na₂O-SiO₂ multicomponent slag system. The current article outlines development of the revised viscosity model[1] for the Al₂O₃-CaO-MgO-SiO₂ slag system, which is a key slag system for a number of metallurgical processes including ironmaking and steelmaking. A number of viscosity measurements has been carried out for this system[8–21] and its subsystems.[22–39] A critical review of experimental procedures and selection of reliable experimental data is an essential stage of the model development. The procedures involved in optimization of the model parameters are reported. The comparisons between the viscosities predicted with the current model and selected reliable experimental data demonstrate agreements within experimental uncertainties over a wide range of temperatures and compositions above liquidus.

The viscosities in this Al₂O₃-containing system have complex trends; in particular, they have the characteristic maximum at the (Meⁿ⁺)_2/n O/Al₂O₃ ratios around 1. Particular focus in the current article therefore was given to the description and analysis of this characteristic maximum, which is attributed to the so-called “charge-compensation effect” commonly explained by the ability of the Al³⁺ cation to take tetrahedral interstitial between the oxygen anions if the excess negative charge for Al³⁺ is compensated with alkali or alkaline earth cations such as Ca²⁺ and Mg²⁺, and therefore by the formation of the stronger (Al-O-Metal) covalent bonds. This characteristic maximum was described in the present formalism through the introduction of the probability function expressing proportion of the tetrahedrally coordinated Al³⁺-containing viscous flow structural units. The Eyring equation[40–42] was used in the current quasi-chemical viscosity model[1–7]; it proved to be adequate to develop a reliable viscosity model to reproduce complex viscous behavior of slag systems. The viscosity equation and parameters for the Al₂O₃-CaO-MgO-SiO₂ slag and its subsystems, and the detail optimization procedures of these parameters, are presented in our previous article[1] of this series of two articles. These model parameters have been optimized using the available experimental viscosity data and some correlations with other physical properties.

Review of Experimental Data

Viscosity measurements at high temperatures are difficult to undertake and are subject to a number of composition and temperature uncertainties related to the interaction of slag with crucible and sensor, and to the temperature measurements. A critical and systematic analysis of the experimental procedures therefore is essential to identify reliable experimental data for further model optimization. All existing experimental viscosity data for the quaternary Al₂O₃-CaO-MgO-SiO₂ system and its subsystems were carefully analyzed in this study with a focus on container and sensor materials, chemical analyses after viscosity measurement, temperature measurements, etc.; details of the critical review are given in previous publications.[1–3] Table I summarizes details of the experimental data selected in the current work.

Table I Experimental Conditions of Viscosity Data for the Al₂O₃-CaO-MgO-SiO₂ and its Subsystems

Table II summarizes the temperature and composition ranges of the selected viscosity measurements for the Al₂O₃-CaO-MgO-SiO₂ quaternary system. Kim et al.,[8] Kita et al.,[9] Kim et al.,[10,11] and Liao et al.[12] focused on the blast-furnace slag compositions. Because Liao et al.[12] reported that some crystalline phases were detected in several slag samples after viscosity measurements and it may affect accuracies of measurements, their results were given low weights in the model optimization. Johannsen et al.[13] examined the effect of MgO on the slag viscosity at several compositions in the Al₂O₃-CaO-MgO-SiO₂ system by substituting CaO with MgO at fixed SiO₂ and Al₂O₃ concentrations. Forsbacka et al.[14] measured the viscosity of the synthetic slag with relatively low CaO concentrations of interest to the ferrochromium production. Song et al.[15] investigated viscosities for the highly basic slag used for secondary refining process in steelmaking. Machin et al.[16–19] investigated a wide range of compositions in the Al₂O₃-CaO-MgO-SiO₂ system focusing on the role of alumina and charge-compensation effect on viscosity. Scarfe et al.[20] and Taniguchi[21] performed measurements along the diopside (CaMgSi₂O₆)–anorthite (Al₂CaSi₂O₈) join investigating the relation of magmatic processes to the physical properties of the molten oxides.

Table II Summary of Composition and Temperature Ranges for the Viscosity Measurements in the Al₂O₃-CaO-MgO-SiO₂ System Selected in This Study

Comparisons of Predicted and Experimental Viscosities

The Al₂O₃-CaO and Al₂O₃-MgO Systems

Figure 1 shows the compositions and temperature ranges for the selected data points in the Al₂O₃-CaO system along with the phase diagram calculated using the FactSage computer package (CRCT/GTT Technologies, Montréal, Canada).[43,44] Most of the points were measured using the rotating bob or cup method with molybdenum or tungsten as a crucible and sensor materials, and an optical pyrometer for temperature control. As indicated in Table I, Urbain[35] and Kozakevitch[22] measured slag compositions after the viscosity experiments and confirmed that the slag contamination by the dissolution of crucible and sensor materials was negligible. Hofmaier’s[24] data were given lower weight because the chemical analysis after experiments was not reported. No experimental data for the Al₂O₃-MgO system were found in the literature.

Fig. 1

Compositions and temperature ranges for selected viscosity measurements in the Al₂O₃-CaO system. Liquidus lines are predicted by FactSage[43]

The predicted viscosities in the Al₂O₃-CaO and Al₂O₃-MgO systems are shown in Figure 2. Note that in Figure 2 and all further figures in this article, viscosities have also been predicted for the liquid compositions below the liquidus temperatures for hypothetical, metastable liquids only for theoretical analysis (this comment refers to all further figures and is not repeated). The viscosity at a fixed temperature changes nonlinearly in those aluminate systems as a function of the Al₂O₃ concentration with distinct maximum at intermediate compositions attributed to the charge-compensation effect and described by the special terms proportional to the proportions of the tetrahedrally coordinated Al³⁺ due to charge compensation by Ca²⁺ or Mg²⁺ cations, respectively. Good agreement between the predicted and experimental viscosities demonstrated in Figure 2 may be taken as an indication of the validity of the introduced formalism.

Fig. 2

Calculated viscosities of the Al₂O₃-CaO and Al₂O₃-MgO melt at 1973 K and 2073 K (1700 °C and 1800 °C); Liquidus compositions in the Al₂O₃-CaO system are calculated by FactSage[43]

The Al₂O₃-CaO-MgO System

The model parameters determined for the Al₂O₃-CaO, Al₂O₃-MgO, and CaO-MgO binary subsystems were used to predict viscosities in the Al₂O₃-CaO-MgO ternary system; Figure 3(a) shows the calculated isoviscosity contours at 1973 K (1700 °C), with the corresponding liquidus line predicted by FactSage.[43] Although this ternary system is important for refining processes in iron productions, no reliable experimental viscosity data are available. Viscosities as a function of molar ratio MgO/(CaO + MgO) at a fixed Al₂O₃ concentration are shown in Figure 3(b), indicating that the viscosity of the melt is predicted to exhibit maximum at an intermediate compositions in this ternary system (note that predictions at Al₂O₃ = 0 mol pct and in other subliquidus conditions are performed for hypothetical, metastable liquid). This maximum of viscosities at Al₂O₃ mole fraction of ~0.3 and at intermediate MgO/CaO ratios is unexpected; it resulted from the parameters obtained through fitting experimental viscosities for the Al₂O₃-CaO-MgO-SiO₂ quaternary liquids at low SiO₂ concentrations.[14,15]

Fig. 3

Calculated viscosity in the CaO-MgO and Al₂O₃-CaO-MgO systems: (a) isoviscosity contours at 1973 K (1700 °C) (liquidus line was predicted by FactSage[43]), (b) viscosities at 0 or 40 mol pct Al₂O₃ as a function of molar CaO/MgO ratio at 1973 K, 2073 K, and 2173 K (1700 °C, 1800 °C, and 1900 °C)

The Al₂O₃-CaO-SiO₂ and Al₂O₃-MgO-SiO₂ Systems

The compositions of the selected experimental viscosity data points for the Al₂O₃-CaO-SiO₂ and Al₂O₃-MgO-SiO₂ systems are plotted in Figure 4 along with the liquidus isotherms predicted by FactSage.[43] Slag viscosities have been measured over wide composition and temperature ranges providing a firm basis for the model development. Various sections in ternary diagrams used for a comparison of predictions with experiments given below are illustrated with thick solid lines in Figure 4.

Fig. 4

Compositions of selected viscosity measurements in (a) Al₂O₃-CaO-SiO₂ and (b) Al₂O₃-MgO-SiO₂ systems along with the liquidus isotherms predicted using FactSage[43]

Figures 5 and 6 demonstrate agreement between the model predictions and experimental data in the Al₂O₃-CaO-SiO₂ and Al₂O₃-MgO-SiO₂ systems along different sections with fixed SiO₂ concentrations or fixed Al₂O₃/CaO (or MgO) molar ratios. The predicted viscosities along sections with fixed SiO₂ concentrations have characteristic maximum (upward curvatures) attributed to the “charge-compensation effect” and describe satisfactorily available experimental data.

Fig. 5

Comparison of predicted and experimental viscosities in the Al₂O₃-CaO-SiO₂ system: (a) 40 mol pct SiO₂, (b) 50 mol pct SiO₂, (c) 67 mol pct SiO₂, (d) 75 mol pct SiO₂, and (e) molar ratio Al₂O₃/CaO = 50/50 from 1673 K to 1973 K (1400 °C to 1700 °C). Liquidus is calculated by FactSage[43]

Fig. 6

Comparison of predicted and experimental viscosities in the Al₂O₃-MgO-SiO₂ system along different sections: (a) 50 mol pct SiO₂, (b) 67 mol pct SiO₂, (c) 75 mol pct SiO₂, and (d) molar ratio Al₂O₃/MgO = 50/50 from 1773 K to 1973 K (1500 °C to 1700 °C). Liquidus is calculated by FactSage[43]

Analysis of the “Charge-Compensation Effect”

The “hypothetical” properties $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{E}_{{a,\,{\text{no}}\,\,\,{\text{charge}}\,\,\,{\text{comp}}.}}$, $\varepsilon_{{V,\,{\text{no}}\,\,\,{\text{charge}}\,\,\,{\text{comp}}.}}$, $\eta_{{{\text{no}}\,\,\,{\text{charge}}\,\,\,{\text{comp}}.}}$, those without assumed contribution of the tetrahedrally coordinated Al³⁺ to the corresponding $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ energies and viscosity, were introduced to analyze the charge-compensation effects by making the probability to form the tetrahedrally coordinated Al³⁺,$p_{{{\text{AlO}}4}}^{{{\text{ch}},{\text{Me}}}}$, equal to zero in the corresponding expressions of the properties $F_{{{\text{Al}} - m}} = F_{{{\text{Al}} - m}}^{0} + \sum\limits_{{j\,\, = \,\,{\text{Ca,Mg}}, \ldots }} {\Updelta F_{{{\text{Al}} - m}}^{{{\text{ch}},j}} p_{{{\text{AlO}}_{4} }}^{{{\text{ch}},j}} }$ (where F corresponds to $\overline{{E_{a} }}$ or $\varepsilon_{V}$). The subsequent functions then are defined by the following equations expressing the contribution of the charge-compensation effect:

• To the integral activation energy:

$$ \Updelta \underline{E}_{{a,{\text{AlO}}_{4} }}^{{{\text{ch}},{\text{Me}}}} = \underline{E}_{{a,{\text{real}}}} - \underline{E}_{{a,{\text{ no charge comp}} .}} $$

(1)

• to the integral energy term $\Updelta \underline{E}_{V}$:

$$ \begin{aligned} \Updelta \varepsilon_{{V,{\text{AlO}}_{4} }}^{{{\text{ch}},{\text{Me}}}} = & \ln \left( {{\raise0.7ex\hbox{${\Updelta \underline{E}_{{V,{\text{real}}}} }$} \!\mathord{\left/ {\vphantom {{\Updelta \underline{E}_{{V,{\text{real}}}} } {\Updelta \underline{E}_{{V,{\text{ no charge comp}}.}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\Updelta \underline{E}_{{V,{\text{ no charge comp}}.}} }$}}} \right) \\ = \varepsilon_{{V,{\text{real}}}} - \varepsilon_{{V,{\text{ no charge comp}}.}} \\ \end{aligned} $$

(2)

• To the viscosity:

$$ \Updelta \ln \eta_{{{\text{AlO}}_{4} }}^{{{\text{ch}},{\text{Me}}}} = \ln \left( {{\raise0.7ex\hbox{${\eta_{\text{real}} }$} \!\mathord{\left/ {\vphantom {{\eta_{\text{real}} } {\eta_{{{\text{no charge comp}} .}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\eta_{{{\text{no charge comp}} .}} }$}}} \right) $$

(3)

The charge-compensation effect on the slag viscosity calculated with the Eyring equation[40–42] is the combination of the effects on the $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ energy terms.

Figure 7 shows the predicted charge-compensation effects on the integral activation energy, integral energy term $\Updelta \underline{E}_{v}$, and viscosity in the Al₂O₃-MeO-SiO₂ melts (Me = Ca, Mg) at 50 mol pct SiO₂. The composition dependence of the concentrations of the tetrahedrally coordinated Al³⁺ in the Al₂O₃-CaO-SiO₂ slag was derived from the magnetic-angle spinning nuclear magnetic resonance (MAS-NMR) experimental data[44] and was assumed to be similar in other Al₂O₃-MeO-SiO₂ systems (Me = Mg, Fe²⁺) due to the lack of experimental information.[1] The charge-compensation effects on the $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ energies and viscosity parameters in those systems were derived from available experimental viscosity data. Figures 7(a) through (c) show satisfactory agreement with experiments and indicate that all those functions are positive over the whole composition range with the maximum close to the Al₂O₃/MeO molar ratio of unity. The charge-compensation effect at a fixed SiO₂ concentration was found to be greater in the Al₂O₃-CaO-SiO₂ system compared to the Al₂O₃-MgO-SiO₂ system in agreement with previous experimental viscosity studies.[23,32]

Fig. 7

Predicted charge-compensation effects on (a) integral activation energy, (b) integral energy term $\Updelta \underline{E}_{V}$ and (c) viscosity in Al₂O₃-MeO-SiO₂ melts at 1773 K (1500 °C) where SiO₂ content is kept as 50 mol pct (Me = Ca, Mg)

Figure 8 reports the predicted charge-compensation effects on the integral activation energy, integral energy term $\Updelta \underline{E}_{v}$ and viscosity in the Al₂O₃-MeO-SiO₂ melts (Me = Ca, Mg) as functions of SiO₂ concentration at fixed molar Al₂O₃/MeO ratios of 1.0. Figures 8(a) through (c) indicate that the resulting charge-compensation effect on the $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ energies and viscosity do not change significantly at a low SiO₂ concentrations but increase at high SiO₂ concentrations and exhibit maxima in the range of 60 to 70 mol pct SiO₂. These tendencies at high SiO₂ concentrations are related to the increasing concentrations of the (Al-O-Si) structural units. Note that the Al³⁺ cations taking the tetrahedral rather than octahedral coordination not only form the stronger covalent bonds but also do not weaken the silica tetrahedra covalent bonding, thus having an additional effect of increasing the viscosity.

Fig. 8

Predicted charge-compensation effects on (a) integral activation energy, (b) integral energy term $\Updelta \underline{E}_{V}$, and (c) viscosity in Al₂O₃-MeO-SiO₂ melts at 1773 K (1500 °C) where molar ratio Al₂O₃/MeO is kept as 1.0 (Me = Ca, Mg)

The Al₂O₃-CaO-MgO-SiO₂ System

Most of the selected experimental viscosity data[8–21] (see Tables I and II) in the Al₂O₃-CaO-MgO-SiO₂ system were grouped and compositions were projected into pseudoternary sections with approximately 50 mol pct SiO₂ (Figure 9(a)) or with fixed Al₂O₃ concentrations (Figures 9(b) and (c)). The liquidus isotherms calculated using FactSage[43] are also plotted in Figure 9. The viscosity model parameters were optimized to reproduce the experimental data in this quaternary system simultaneously with subsystems; the predicted viscosities in the Al₂O₃-CaO-MgO-SiO₂ system are compared with the corresponding experimental results along the following sections:

Fig. 9

Compositions of the selected viscosity measurements in the Al₂O₃-CaO-MgO-SiO₂ quaternary system with fixed (a) 50 mol pct SiO₂, (b) 15 mol pct Al₂O₃, and (c) 19 mol pct Al₂O₃ concentrations. Liquidus isotherms are predicted by FactSage[43]

• (a) 50 mol pct SiO₂, 25 mol pct CaO (diopside – anorthite join), Figure 10(a)

  Fig. 10

  

  Predicted slag viscosities of the Al₂O₃-CaO-MgO-SiO₂ system along different sections: (a) 50 mol pct SiO₂, 25 mol pct CaO, (b) 50 mol pct SiO₂, molar ratio Al₂O₃/MgO = 40/60, (c) 15 mol pct Al₂O₃, 15 mol pct MgO, and (d) 19 mol pct Al₂O₃, 8 mol pct MgO at 1673–1873 K (1400–1600 °C). Liquidus is calculated by FactSage.[43]

• (b) 50 mol pct SiO₂, molar ratio Al₂O₃/MgO = 40/60, Figure 10(b)

• (c) 15 mol pct Al₂O₃, 15 mol pct MgO, Figure 10(c)

• (d) 19 mol pct Al₂O₃, 8 mol pct MgO, Figure 10(d)

The predicted viscosities reproduce available experimental results as demonstrated in Figures 10(a) through (d). The predicted viscosities increase with increasing Al₂O₃/MgO molar ratio along the diopside–anorthite pseudobinary join (Figure 10(a)) and decrease with the addition of CaO when SiO₂ concentrations are fixed at 50 mol pct and at a constant Al₂O₃/MgO molar ratio of 40/60 (Figure 10 (b)). The latter trend may result from a decreasing charge-compensation effect due to the “excess” of basic oxides (most Al³⁺ took tetrahedral coordination). The predicted viscosities increase as the SiO₂ concentration increases at fixed Al₂O₃ and MgO concentrations (Figures 10(c) and (d)).

Figure 11 shows the predicted viscosities of the Al₂O₃-CaO-MgO-SiO₂ slag as functions of CaO/SiO₂ or Al₂O₃/SiO₂ mass ratios under fixed contents of MgO, compared with some experimental results in the composition ranges used in ironmaking and steelmaking.[10–12,16–19] In Figure 11(a), it was demonstrated that slag viscosities decrease with increasing CaO/SiO₂ mass ratio. The predicted viscosities reproduce well the measured values by Machin et al.[16–19] but are lower than values reported by Kim et al.[10,11]; the viscosities values reported by Kim et al. are higher than those by Machin et al. at identical slag compositions. Note that (Si-O-Ca) is the major structural unit in this composition range of the Al₂O₃-CaO-MgO-SiO₂ slag system; the predicted slag viscosities mainly depend on the partial activation energy of the (Si-O-Ca) structural unit determined using the data from the CaO-SiO₂ binary system.

Fig. 11

Predicted viscosities of the Al₂O₃-CaO-MgO-SiO₂ slag at 1773 K (1500 °C); (a) as a function of CaO/SiO₂ mass ratio under 10 mass pct MgO and (b) as a function of Al₂O₃/SiO₂ mass ratio under 15 mass pct MgO. Liquidus is calculated by FactSage[43]

Figure 11(b) demonstrates that the predicted viscosities decrease as the CaO content increases, decrease with the increase of the Al₂O₃/SiO₂ ratio at low CaO concentrations, and remain almost constant or even slightly increase with the increase of the Al₂O₃/SiO₂ ratio at high CaO concentrations; these differences in trends may be attributed to the charge-compensation effect. The predicted viscosities reproduce well the experimental data by Machin et al.[16–19]; however, those by Liao et al.[12] are high and do not agree with the results by Machin et al.[16–19] Liao et al. reported that the precipitation of crystalline phase was observed in some slag samples after measurements; therefore, such discrepancies could be caused by the presence of crystalline phases during viscosity measurements.

Figures 12 and 13 present the $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ energies and viscosities, including the charge-compensation effects on the corresponding functions, in the sections of 50 mol pct SiO₂, 15 mol pct Al₂O₃ (Figure 12), 50 mol pct SiO₂, mole CaO: MgO = 1:1 (Figure 13), respectively.

Fig. 12

Calculated results of (a) integral activation energy, (b) integral energy term $\Updelta \underline{E}_{V}$ and (c) viscosity in Al₂O₃-CaO-MgO-SiO₂ melt at 1773 K (1500 °C) (50 mol pct SiO₂, 15 mol pct Al₂O₃)

Fig. 13

Calculated results of (a) integral activation energy, (b) integral energy term $\Updelta \underline{E}_{V}$, and (c) viscosity in Al₂O₃-CaO-MgO-SiO₂ melt at 1773 K (1500 °C) (50 mol pct SiO₂, mole CaO:MgO = 1:1)

In the constant alumina and silica section (50 mol pct SiO₂, 15 mol pct Al₂O₃, see Figure 12), the charge-compensation effects by Ca²⁺ and Mg²⁺ cations on the $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ energies and viscosities are compared: The effect by Ca²⁺ is stronger than by Mg²⁺ cation for the same concentration of the tetrahedrally coordinated Al³⁺. The $\underline{E}_{a}$ and $\Updelta \underline{E}_{v}$ values therefore gradually decrease as the MgO/CaO ratio increases with downward curvatures following the corresponding charge-compensation trends for Ca²⁺ or Mg²⁺.

In the section with constant 50 mol pct SiO₂ and constant mole CaO:MgO ratio of 1:1 (Figure 13), the$\underline{E}_{a}$,$\Updelta \underline{E}_{v}$and viscosities have reached maxima at intermediate Al₂O₃ concentrations because little or no Al³⁺ cations are available to be compensated at low Al₂O₃/(Al₂O₃ + CaO + MgO) ratios, and little or no Ca²⁺ and Mg²⁺ cations are available to compensate at high Al₂O₃/(Al₂O₃ + CaO + MgO) ratios.

Summary of the Comparison of Predictions with the Experimental Data

The calculated viscosities were compared with all selected experimental data for each temperature and composition in the Al₂O₃-CaO-MgO-SiO₂ system and its subsystems, as summarized in Figure 14. The Round Robin project[45] previously evaluated the recommended slag viscosity values and the accuracies of viscosity predictions by representative models. In this project, experimental uncertainties of the recommended viscosity measurements were determined as approximately 30 pct. As shown in Figure 14, the current model reproduces almost all available experimental data within experimental uncertainties.

Fig. 14

Comparison between the calculated slag viscosities and all available experimental data in the system Al₂O₃-CaO-MgO-SiO₂ and its subsystems

The agreement of the calculated viscosities with experimental values was evaluated by taking the average of relative errors between the calculated and experimental results, described by the following equation:

$$ {\text{Average}}\;{\text{of}}\;{\text{relative}}\;{\text{errors}}\;({\text{pct}}) \, = \frac{100}{N}\sum\nolimits_{i = 1}^{N} {\left| {\frac{{\eta_{{{\text{Calc}},i}} - \eta_{{{\text{Expe}},i}} }}{{\eta_{{{\text{Expe}},i}} }}} \right|} $$

(4)

where N denotes total number of acceptable experimental data points. Figure 15 summarizes the results of analyzing the average of relative errors for the quaternary system Al₂O₃-CaO-MgO-SiO₂ and its subsystems.

Fig. 15

Average of relative errors between the calculated viscosities and experimental data for the quaternary system Al₂O₃-CaO-MgO-SiO₂ and its subsystems

The experimental viscosities in the Al₂O₃-CaO-MgO-SiO₂ quaternary system and its subsystems have been accurately reproduced by the revised quasi-chemical viscosity model within experimental uncertainties.

Examples of Model Application to Analysis of Viscosities of the Industrial Slags

Viscosity is one of the key properties important for a number of processes in industrial operations. A careful analysis of the compositional and temperature trends of viscosities is essential for improvements and optimization of the industrial processes. The following example demonstrates how the current viscosity model can be used for the analysis of viscosities in the iron blast furnace slags.

Figure 16(a) shows the predicted isoviscosity contours in the Al₂O₃-CaO-MgO-SiO₂ system with the constant Al₂O₃ concentration of 5 mass pct. This figure demonstrates good agreement of predictions with experimental results reported by Machin et al.[16–19] The isoviscosity contours in Figure 16(a) indicate that substitution of MgO with CaO at the constant SiO₂ concentration in slag increases the slag viscosity.

Fig. 16

Predicted (a) isoviscosity contours of Al₂O₃-CaO-MgO-SiO₂ slag at 1773 K (1500 °C) where Al₂O₃ mass fraction is kept as 5 mass pct. The liquidus line is calculated by FactSage.[43] (b) The viscosity of Al₂O₃-CaO-MgO-SiO₂ slag as a function of Al₂O₃ concentration at 1723 K (1450 °C) under different mass ratios of CaO/SiO₂, where all compositions are included in corresponding liquidus lines obtained by FactSage

Figure 16(b) shows the viscosity of Al₂O₃-CaO-MgO-SiO₂ iron blast-furnace slag compositions with fixed mass ratios of CaO/SiO₂ = 0.8, 1.0, or 1.2 and constant mass fraction of MgO = 10 mass pct. This figure demonstrates that viscosity increases with the increase of the Al₂O₃ concentration in the range of iron blast-furnace slags. The increase of the mass ratio of CaO/SiO₂ significantly decreases the slag viscosity; this decrease at lower CaO/SiO₂ ratios between 0.8 and 1.0 is stronger compared to the higher CaO/SiO₂ ratios between 1.0 and 1.2.

Conclusions

The revised quasi-chemical viscosity model that was developed is capable of predicting the viscosities in the Al₂O₃-CaO-MgO-SiO₂ quaternary system and its subsystems over the wide composition and temperature ranges above liquidus within experimental uncertainties. The model describes complex viscosity trends including the so-called charge-compensation effect in this Al₂O₃-containing system. The model can now be used to analyze viscosity trends in the complex industrial slags and thus assist in improvements and optimizations of the current industrial operations and development of new metallurgical processes.