Modelling of Viscosity of Melts Containing Iron Oxide in Ternary Silicate Systems

Abstract

The motivation of this work is to show that the structural model, which was initially used to estimate the thermodynamic properties of binary silicate systems, can be also used to estimate the viscosity of binary and ternary silicate melts in terms of temperature and composition. The model links the viscosity to the internal structure of melts through the concentration of the oxygen bridges present in the slag. A previously proposed structural thermodynamic model was used to calculate the content of oxygen bridges. The viscosity model requires only three parameters to obtain a good agreement between experimental and calculated data for the SiO₂−FeO binary system and for the SiO₂−CaO−FeO, SiO₂−MgO−FeO and SiO₂−MnO−FeO ternary systems. The viscosity of ternary systems was calculated with the model while assuming a linear function of the parameters from binary systems; however, the content of the oxygen bridges was calculated using the thermodynamic model for ternary systems.

Keywords:
Viscosity; silicate structure; thermodynamic model

1. Introduction

The viscosity of molten slags is an important parameter for metal manufacturing since the loss of metal values may be in part attributed to mechanical entrainment in the slag phase¹. The viscosities of molten silicates are difficult to obtain experimentally due to the complexity of the procedures at a high temperature. Therefore, it is desirable to have reliable models for the prediction of viscosity in terms of the temperature and composition. The viscosity of liquid silicate systems has been studied and several models have been developed, among which some are empirical, such as the models by Riboud et al.² and Urbain³, and others are based on the structure of silicate melts.

Reddy and Hebbar⁴ developed a structure−based model and predicted the viscosities of SiO₂−MO (M=Ca, Mn, Mg, Fe) melts. Alternatively, Shu et al.⁵ combined the Temkin ionic theory with the Lumsden theory in order to consider the molten slag as a system with a matrix of oxygen ions with various cations (including Si⁴⁺). Furthermore, Le Losq and Neuville⁶ considered the viscous flow of silicate melts as governed by the cooperative re-arrangement of molecular sub-systems that involve the silicate Qⁿ units (n is the number of bridging oxygens). This model linked the Qⁿ unit fractions to the melt configurational entropy at the glass transition temperature and, finally, to its viscosity. Kondratiev and Jak⁷ used a model to link the slag viscosity to the internal structure of melts through the concentrations of various anion/cation structural units. The concentrations of structural units were equivalent to the second−nearest neighbor bond concentrations calculated by the quasi-chemical thermodynamic model.

The present model uses the structural model to estimate the concentration of the types of oxygen in binary and ternary silicate systems. This structural model has been used to calculate the thermodynamic properties and the phase diagrams for binary and ternary systems^8,9 as well as to estimate the sulphide capacity of binary silicate melts. The model has also been used to estimate the molar volume of binary and ternary silicates¹⁰ and the viscosity of the binary and ternary silicate of the system SiO₂−CaO−MgO−MnO−Na₂O¹¹. Recently, Wu et al.^12,13 developed a viscosity model where the oxygen partial pressure was taken into account, and the structural roles of FeO and Fe₂O₃ in determining the slag viscosity were assessed. In this model, the silicate structure was described by means of a non−ideal associate solution to describe the Gibbs energy of the liquid phase.

In the present work, we use the structural model to estimate the viscosity of the SiO₂−FeO binary system and the viscosity of the SiO₂−CaO−FeO, SiO₂−MgO−FeO and SiO₂−MnO−FeO ternary systems. It is worth mentioning that the SiO₂–FeO system is a base for many metallurgical slags, particularly steelmaking, copper smelting, converting and slag cleaning. The motivation of this work is to show that the structural model, which was initially used to estimate the thermodynamic properties and phase diagrams of binary SiO₂−MO systems (M=Ca, Mn, Fe, etc.), can be also used to estimate other physicochemical properties, such as molar volume and viscosity, which depend on the structure of the molten silicates.

2. The Model

2.1 Thermodynamic Model

The model, as it has been mentioned in a previous work¹¹, is based on the silicate structure where the basic building block is the Si–O tetrahedron, in which one Si⁴⁺ cation is surrounded by four O− ions. The silica network structure breaks down with the addition of basic oxides, which gives it a more depolymerized structure. These structural models consider three types of oxygen: (1) bridging oxygen bonded to two silicon atoms (O°), (2) non-bridging oxygen bonded only to one silicon atom (O⁻) and (3) free oxygen bonded to no silicon atom (O²⁻):

$\text{Si}-\text{O}-\text{Si}+\text{MO}=\text{Si}-{\text{O}}^{\text{M}}\text{O}-\text{Si}$ (1)

${\text{O}}^{\text{o}}+{\text{O}}^{2-}=2{\text{O}}^{-}$ (2)

In a binary solution SiO₂−MO (M= Ca, Fe, Na₂, etc.), the parameters ${\text{N}}_{{\text{O}}^{2-}}$, ${\text{N}}_{{\text{O}}^{-}}$ and ${\text{N}}_{{\text{O}}^{\text{o}}}$ represent the number of moles of the various oxygen species per mole of solution (n_MO+n_SiO2). It is assumed that every silicon atom is bonded to four oxygen atoms. Thus, mass balance considerations require the following:

${\text{N}}_{{\text{O}}^{\text{o}}}=2{\text{X}}_{{\text{SiO}}_2}-\frac{{\text{N}}_{{\text{O}}^{-}}}{2}$ (3)

${\text{N}}_{{\text{O}}^{2-}}={\text{X}}_{\text{MO}}-\frac{{\text{N}}_{{\text{O}}^{-}}}{2}$ (4)

where ${\text{X}}_{{\text{SiO}}_2}$ and X_MO are the mole fractions of the oxides. The configurational entropy (S^c) is physically related to the number of ways in which particles themselves can be distributed in space. S^c is calculated through the multiplicity of the random distribution of the Si atoms and O²⁻ ions on the sites (Ω₁) and the multiplicity of the random distribution of the ${\text{N}}_{{\text{O}}^{\text{o}}}$ bridging oxygen atoms over the number of moles of neighboring Si-Si pairs positions (Ω₂).

$S^c=\frac{R}{N^o}\ln \left[{\Omega }_1{\Omega }_2\right]$ (5)

where R is the gas constant and N° is Avogadro’s number. The structural model assumes that the depolymerization reaction, Equation 2, is associated with the Gibbs energy change containing an enthalpic (ω) and entropic (ε) term:

$\Delta H-T{S}^{nc}=\left(\frac{N_{O^{-}}}{2}\right)\omega -T\left(\frac{N_{O^{-}}}{2}\right)\epsilon$ (6)

where ΔH is the change of enthalpy of the breaking bridge process, S^nc is the non-configurational entropy and T is the absolute temperature. Finally, ω and ε are expanded as polynomials:

$\omega ={\omega }_0+{\omega }_1{X}_{Si{O}_2}+{\omega }_2{X}_{Si{O}_2}^2+\dots$ (7)

$$\epsilon ={}_{{\epsilon }_0}+{\epsilon }_{{}_1}{X}_{Si{O}_2}{+}_{{\epsilon }_2}{X}_{Si{O}_2}^2+\dots$$ (8)

The coefficients ω_i and ε_i are the parameters of the thermodynamic model, which are obtained by the optimization of data. When given a composition ${\text{X}}_{{\text{SiO}}_2}$ and the values of the parameters ω_i and ε_i, the actual value of ${\text{N}}_{{\text{O}}^{-}}$ can be calculated by minimizing the Gibbs energy at constant ${\text{X}}_{{\text{SiO}}_2}$, ω and ε:

$\Delta G=\Delta H-T\left(S^c+S^{nc}\right)$ (9)

Table 1 shows the parameters ω_i and ε_i for SiO₂−FeO, SiO₂−CaO, SiO₂−MgO and SiO₂−MnO binary systems⁸, and Figure 1 shows the concentrations of oxygen bridges (O^O) calculated by the model at 1673 K for these systems. This Figure also shows that the efficiency in breaking oxygen bridges increases in the order FeO < MnO < MgO < CaO.

Table 1
Parameters for the thermodynamic model of binary silicate systems⁸.

System	Expression of ω and ε
SiO2−FeO	$\omega =6770-122724X_{SiO2}+183040{\text{X}}_{\text{SiO2}}^{\text{2}}+106539{\text{X}}_{\text{SiO2}}^{\text{4}}J/mol$
	$\epsilon =-34.59X_{SiO2}+87.366{\text{X}}_{\text{SiO2}}^{\text{3}}J/molK$
SiO2−CaO	$\omega =-57738-258739X_{SiO2}+262548{\text{X}}_{\text{SiO2}}^{\text{2}}+89325{\text{X}}_{\text{SiO2}}^{\text{6}}J/mol$
	$\epsilon =-35.5X_{SiO2}+70.71{\text{X}}_{\text{SiO2}}^{\text{4}}J/molK$
SiO2−MgO	$\omega =-51140–76333X_{SiO2}+245570{\text{X}}_{\text{SiO2}}^{\text{3}}J/mol$
	$\epsilon =-47.99X_{SiO2}+94.164{\text{X}}_{\text{SiO2}}^{\text{3}}J/molK$
SiO2−MnO	$\omega =-56143-1791X_{SiO2}+116820{\text{X}}_{\text{SiO2}}^{\text{3}}J/mol$
	$\epsilon =-15.11+18.638{\text{X}}_{\text{SiO2}}^{\text{3}}J/molK$

Figure 1
Concentration of oxygen bridges in the SiO₂−MO (M = Fe, Mn, Mg, Ca) system calculated at 1673 K by the structural model.

To expand the model for ternary systems, we considered 5 kinds of oxygen⁹. Let us consider the general SiO₂−AO−BO ternary system where A and B are divalent cations, such as Ca²⁺ and Fe²⁺:

1. 1

   O^o: Oxygen bridge

2. 2

   ${\text{O}}_{\text{A}}^{2-}$: ‘Free oxygen ions’ introduced in the system by the AO species

3. 3

   ${\text{O}}_{\text{B}}^{2-}$: ‘Free oxygen ions’ introduced in the system by the BO species

4. 4

   ${\text{O}}_{\text{A}}^{-}$: Broken bridges for the AO species

5. 5

   ${\text{O}}_{\text{B}}^{-}$: Broken bridges for the BO species

There are two depolymerization reactions, which are given as follows:

$Si-O-Si+AO=Si–O^{A}O-Si{\left(\omega -\epsilon T\right)}_{AO}$ (10)

$Si-O-Si+BO=Si–O^{B}O-Si{\left(\omega -\epsilon T\right)}_{BO}$ (11)

The mass balance considerations now require the following:

${\text{N}}_{{\text{O}}^{\text{o}}}=2{\text{X}}_{{\text{SiO}}_2}-\frac{{\text{N}}_{{\text{O}}_{\text{A}}^{-}}+{\text{N}}_{{\text{O}}_{\text{B}}^{-}}}{2}$ (12)

${\text{N}}_{{\text{O}}_{\text{A}}^{2-}}={\text{X}}_{\text{AO}}-\frac{{\text{N}}_{{\text{O}}_{\text{A}}^{-}}}{2}$ (13)

${\text{N}}_{{\text{O}}_{\text{B}}^{2-}}={\text{X}}_{\text{BO}}-\frac{{\text{N}}_{{\text{O}}_{\text{B}}^{-}}}{2}$ (14)

The expression of the configurational entropy is obtained by making two statistical distributions:

$S^c={\text{S}}_{\text{I}}^{\text{c}}+{\text{S}}_{\text{II}}^{\text{c}}$ (15)

where ${\text{S}}_{\text{I}}^{\text{c}}$ is estimated through the distribution of ${\text{O}}_{\text{A}}^{2-}$, ${\text{O}}_{\text{B}}^{2-}$ and Si in a quasi-lattice and ${\text{S}}_{\text{II}}^{\text{c}}$ is estimated through the distribution of O° over the neighboring Si-Si pairs. The excess free energy expression for the ternary system is obtained with the addition of the interaction energy terms (ω-εT) for each bridge-breaking reaction, Equations 10 and 11, which are known in the two binary systems (SiO₂−AO and SiO₂−BO) from the binary optimizations. This expression must also include the contribution of the excess free energy for the AO−BO binary system, ${\text{G}}_{\text{AO}-\text{BO}}^{\text{E}}$, which is multiplied by the fraction of free oxygen ions in the quasi-lattice whose sites are occupied by O²⁻ ions and Si atoms.

2.2 Viscosity Model

Binary Systems

In the present study, the viscosity for binary SiO₂−MO systems (M = Ca, Mg, Fe, Mn, Na₂, etc.) is expressed as follows:

$\ln \eta =A+\frac{B}{T}+C{N}_{O°}$ (16)

The temperature dependence of viscosity is described by the Arrhenius equation, where η is viscosity in Pa⋅s, A is the natural logarithm of the pre-exponential term, B is the activation energy over the gas constant (E/R), and T is the absolute temperature. It has been shown in a previous work¹¹ that, at a given temperature, ln(η) is nearly a linear function of the concentration of oxygen bridges (N_O°). Parameter C in Equation 16 gives to the linear relationship between experimentally calculated ln(η) and the concentration of oxygen bridges (N_O°) calculated by the structural model.

It is noteworthy that even though there are three types of oxygens in the silicate structure (O°, O⁻ and O²⁻) in a binary system, the viscosity model in Equation 16 is expressed only in terms of the amount of the oxygen bridges (N_O°) since the other two types of oxygens are related directly to N_O° through the mass balance given in Equations 3 and 4.

The results show that only three parameters were needed to calculate the viscosity in terms of both composition and temperature in binary silicate systems. The values of these parameters for the SiO₂−MO (M = Fe, Mn, Mg and Ca) binary systems were obtained by the regression of viscosity data, and the results are shown in Table 2.

Table 2
Values of model parameters for different binary silicate systems.

System	A	B	C
SiO2−FeO	-8.33278	6 706.76	4.8261
SiO2−MnO	-12.4586	14 048.62	6.0765
SiO2−MgO	-14.5122	21 061.97	4.1265
SiO2−CaO	-13.2523	18 429.96	4.0069

Ternary Systems

The model is expanded for ternary silicate systems using a linear relationship of the model parameters of the binary silicate systems. That is, if Y represents any of the fitting binary parameters A, B, or C of Equation 16, Z in the SiO₂−AO−BO ternary system is obtained as follows:

$\text{Z}=\left[\frac{{\text{X}}_{\text{AO}}}{{\text{X}}_{\text{AO}}+{\text{X}}_{\text{BO}}}\right]{\text{Y}}_{{\text{SiO}}_2-\text{AO}}+\left[\frac{{\text{X}}_{\text{BO}}}{{\text{X}}_{\text{AO}}+{\text{X}}_{\text{BO}}}\right]{\text{Y}}_{{\text{SiO}}_2-\text{BO}}$ (17)

However, the concentration of oxygen bridges (N_O°) of Equation 16 is calculated with the structural model for ternary systems.

3. Modelling Results and Discussion

3.1 SiO₂–FeO Binary System

The viscosity model for the binary systems was used in a previous work for the SiO₂−MnO, SiO₂−MgO and SiO₂−CaO systems. In this work, we applied the model to the SiO₂−FeO system. Figures 2 and 3 show the experimental^14-20 and calculated values of viscosity of the SiO₂−FeO system at 1573 and 1673 K, respectively, where it can be observed that they are in good agreement.

Figure 2
Calculated (line) and experimental (points) viscosities at 1573 K of SiO₂−FeO melts.

Figure 3
Calculated (line) and experimental (points) viscosities at 1673 K of SiO₂−FeO melts.

The SiO₂−FeO is one of the systems with a considerable amount of viscosity experimental data. Myslevic et al.¹⁴ used the rotating-cylinder method to measure the viscosities of SiO₂–FeO slags using pure iron, crucibles and bobs, to minimize the chemical attack of slags with high FeO contents.

Zhang and Jahanshahi¹⁵ reported that unlike the viscosity of other binary silicate systems which increase monotonically with an increase in silica content, the viscosity of the FeO−SiO₂ system exhibits a maximum at about the fayalite (FeO⋅2SiO₂) composition, and the values of maximum decrease with increase in temperature. A huge peak was reported in the studies by Röntgen et al.¹⁶, and Shiraishi et al.¹⁷, which found small but sharp humps on viscosity near the fayalite composition.

Chen et al.¹⁸ measured the viscosity of the SiO₂–FeO system in equilibrium with iron using a rotational rheometer and Mo crucible and spindle under Ar gas. This work showed that there was no maximum viscosity in the fayalite composition in this system. Kucharski et al.¹⁹ measured the viscosity of SiO₂–FeO under a higher oxygen potential away from iron saturation and the peak at the fayalite composition was not observed.

Table 3 lists the calculated and experimental^14,17,19,20 results obtained at 1573 K between 0.201 and 0.423 mole fraction of SiO₂, whereas Table 4 shows the calculated and experimental results obtained at 1673 K by Shiraishi et al.¹⁷ and Urbain et al.²⁰ Even though Shiraishi et al.¹⁷ reported on an average of 3.5 wt.% Fe₂O₃ and 0.86 wt% elemental Fe, the values calculated by the present model were obtained by assuming that the melt was made up of only SiO₂ and FeO, i.e., FeO and small quantities of Fe₂O₃ were both considered as FeO. Tables 3 and 4 also show the concentration of oxygen bridges (N_O°) as calculated by the thermodynamic structural model, which was used in Equation 16 to estimate the viscosity of this binary system.

Table 3
Experimental and calculated viscosities of the SiO₂−FeO system at 1573 K, in Pa⋅s.

XSiO2	NO°	Shiraishi et al. 17	Myslevic et al. 14	Kucharski et al. 19	Urbain et al. 20	Calculated
0.2010	0.0662	−	−	−	0.022	0.0235
0.2520	0.1150	0.030	0.0296	−	−	0.0298
0.3000	0.1771	−	−	−	0.034	0.0402
0.3162	0.2017	−	−	0.0406	−	0.0452
0.3170	0.2029	0.043	0.0542	−	−	0.0455
0.3212	0.2096	−	−	0.0423	−	0.0470
0.3280	0.2207	0.052	0.0625	−	−	0.0496
0.3330	0.2290	0.064	0.0667	0.0476	0.068	0.0516
0.3440	0.2479	0.053	0.0675	−	−	0.0565
0.3490	0.2568	0.059	−	−	−	0.0590
0.3500	0.2586	−	−	−	0.047	0.0595
0.3558	0.2691		−	0.0576	−	0.0626
0.3600	0.2768	0.066	0.0563	−	−	0.0650
0.3676	0.2911	−	−	0.0640	−	0.0697
0.3690	0.2937	−	−	0.0683	−	0.0705
0.3710	0.2976	0.084	0.0813	−	−	0.0719
0.3789	0.3129	−	−	0.0714	−	0.0774
0.3892	0.3335	−	−	0.0822	−	0.0855
0.4068	0.3700	−	−	0.0996	−	0.1019
0.4230	0.4051	0.143	0.1750	−	−	0.1207

Table 4
Experimental and calculated viscosities of the SiO₂−FeO system at 1673 K, in Pa⋅s.

XSiO2	NO°	Shiraishi et al. 17	Urbain et al. 20	Calculated
0.201	0.0674	−	0.019	0.0183
0.252	0.1169	0.027	−	0.0233
0.300	0.1797	−	0.028	0.0315
0.317	0.2056	0.037	−	0.0357
0.328	0.2234	0.041	−	0.0389
0.333	0.2317	0.047	0.051	0.0405
0.344	0.2506	0.043	−	0.0444
0.349	0.2594	0.047	−	0.0463
0.350	0.2612	−	0.037	0.0467
0.371	0.3001	0.060	−	0.0564
0.400	0.378	−	0.06	0.0821
0.423	0.407	0.092	−	0.0944

Figure 4 shows a comparison between the estimated and measured values for system SiO₂−FeO at 1573 K and 1673 K. The mean deviation Δ, as calculated by Equation 18, is about 11.7%.

$\Delta =\frac{1}{N}{\displaystyle \sum }_{i=1}^N\frac{\left({\eta }_{cal}-{\eta }_{exp}\right)}{{\eta }_{exp}}100$ (18)

where η_cal and η_exp are the calculated and experimental viscosities, respectively, N is the total number of values.

Figure 4
Comparison of the calculated and experimental^(14,17-20) viscosities for the SiO₂−FeO system.

Dingwell²¹ has shown that an increase in the Fe³⁺/Fe²⁺ ratio led to an increased viscosity. Wu et al.^12,13 reported that the local viscosity maximum, around the fayalite composition in the SiO₂−FeO melts, is related to the charge compensation of FeO₂⁻ by Fe²⁺ and is dependent on temperature and oxygen partial pressure.

Like the other proposed models^4,15, the present model has a drawback as it does not show the peak in the viscosity composition curve for melts containing approximately 30 mol% of SiO₂. However, this phenomenon was not experimentally observed by all the researchers who studied this system and any other binary metal-oxide silicate system; thus, further experimental verification is needed in future works. There was a good agreement between the experimental and calculated data even though the model does not consider some intrinsic physicochemical properties of metal oxides explicitly, such as the electronegativity or ionic radii of metal ions.

The current model has been applied with reasonable success to describe the thermodynamic properties and phase diagrams of binary silicate systems SiO₂−MO (M= Ca, Mg, Fe, Na₂, etc.). However, one limitation of the model is that it cannot deal with systems with amphoteric oxides (Fe₂O₃ and Al₂O₃), which act as either network formers or network modifiers depending on the composition. Fe³⁺ and Al³⁺ require a charge compensation in the network to form a building block such as the Si–O tetrahedron, where one Si⁴⁺ cation is surrounded by four O− ions.

3.2 Ternary Systems

The viscosity of SiO₂−CaO−FeO melts was measured at 1573 K by Kucharski et al.¹⁹ and Shidar et al.²² and at 1673 K by Sridhar et al.²², Ji et al.²³ and Johannsen and Wiese²⁴. Most of these experiments were carried out at a relatively low oxygen partial pressure of 6x10^-11 atm (6.08x10^-6 Pa); thus, it was reasonable to treat the system as a ternary SiO₂−CaO−FeO. The viscosities predicted by the model were compared with these studies in Figures 5 and 6. These Figures show the limiting liquidus curve at the temperatures considered. The model reproduced the measured viscosities within the scatter of the experimental data from different authors. The mean deviation Δ that was calculated for the SiO₂−CaO−FeO was about 26% and 31% at 1573K and 1673 K, respectively.

Figure 5
Isoviscosity curves (Pa⋅s) of SiO₂−CaO−FeO melts at 1573 K.

Figure 6
Isoviscosity curves (Pa⋅s) of SiO₂−CaO−FeO melts at 1673 K.

Several models, such as Urbain’s model³, considered the viscosity of the ternary silicate system to be a linear function of the viscosity of the binary silicate systems, which may not be justified in all the systems and all the compositions. This is because, in these ternary silicate systems, there are two network modifying metallic oxides and the ideal mixing assumption is not enough to account for the property changes with composition.

The present viscosity model is quite simple with only three parameters for each binary system. The oxygen bridges parameter (N_O°) implicitly considered the effect of both temperature and composition. We did not include additional adjusted parameters for ternary systems, which is why the mean deviations for these systems were higher than those of the binary systems.

The present model for ternary systems dealt with the effect on viscosity of substituting one basic metal oxide with another. Figure 7 shows the estimated and experimental^19,22 viscosities in the SiO₂−CaO−FeO system at 1573 K and X_SiO2 = 0.327. Thus, it is clear that the viscosity was higher than that of the linear extrapolation.

Figure 7
Calculated and experimental^(19,22) viscosities in the SiO₂−CaO−FeO system at 1573 K and X_SiO2 = 0.327.

It has been reported¹¹ that this model predicts a maximum value in the viscosity of ternary systems. For the SiO₂−CaO−FeO system, this maximum can be explained because FeO is not as efficient in breaking the oxygen bridges as CaO; furthermore, when CaO is replaced by FeO, the amount of oxygen bridges (NO°) and viscosity increases as compared with the linear extrapolation results. Figure 8 shows the concentration of oxygen bridge (N_O°) in terms of composition for the SiO₂−CaO−FeO system at 1573 K and X_SiO2 = 0.327. Figures 7 and 8 show that the N_O° and viscosity of the ternary system cannot be linearly extrapolated from the results of the binary systems. However, additional experimental results are required to confirm the maximum viscosity values predicted by this model.

Figure 8
Concentration of oxygen bridges in the SiO₂−CaO−FeO system calculated by the structural model at 1573 K and X_SiO2 = 0.327.

Shidar et al.²² and Ji et al.²⁵ measured the viscosity values of the SiO₂−MnO−FeO system using the rotating cylinder method with an iron spindle and crucible to ensure all Fe was in valence of 2+. Comparisons between the estimated values and measured values for this system are shown in Figures 9 and 10 at 1673 K and 1723 K, respectively. The mean deviation Δ calculated for the SiO₂−MnO−FeO system was about 29% and 34% at 1673K and 1723 K, respectively.

Figure 9
Isoviscosity curves (Pa⋅s) of SiO₂−MnO−FeO melts at 1673 K.

Figure 10
Isoviscosity curves (Pa⋅s) of SiO₂−MnO−FeO melts at 1723 K.

Kucharski et al.¹⁹, studied the SiO₂−MgO−FeO in a rather limited composition range in the SiO₂–FeO rich side with X_SiO2 from 0.3 to 0.38, and a narrow temperature range of 1543 to 1623 K. In Figure 11, the experimental data at 1623 K and calculated results are shown; as can be seen, the liquid region is very small and measurements are very close to the liquidus. This Figure also presents some experimental values reported by Sridhar et al.²² and Ji et al.²⁵ The mean deviation Δ that was calculated for the SiO₂−MgO−FeO system was about 23% at 1623 K.

Figure 11
Isoviscosity curves (Pa⋅s) of SiO₂−MgO−FeO melts at 1623 K.

We tried to make the viscosity model as simple as possible and included only three parameters for each binary system, two parameters for the temperature function, and one for the composition function. This model estimates the viscosity of ternary systems by combining the results of the binary systems. The model considers that the viscosity of the ternary systems cannot be linearly extrapolated from the results of the binary system. However, a more complete model for ternary systems (SiO₂-AO-BO) would be obtained if the broken bridges for the AO and BO species and the free oxygen were taken into account besides the non-bridging oxygen.

The adjusted parameters used in this model were calculated using the temperatures (between 1573 K and 1723 K) and compositions (0.2 < X_SiO2 < 0.5) of the experimental results reported in the literature; following this, the slag viscosity can be properly described by the actual model in the range of temperatures and compositions above mentioned. More experimental data are needed for further assessment of the model parameters.

To conclude, the model proposed in this study links melt composition, structure and thermodynamic properties. The structural model used in this work has been extended to predict other properties, such as molar volume, phase diagrams, sulphide capacity and all the thermodynamic properties of binary and ternary silicate systems.

4. Conclusions

A structure−related model for the viscosity of silicate melts has been extended to FeO containing systems by considering a directly dependent viscosity with the oxygen bridges (O°), which was calculated by a structural thermodynamic model. The viscosity of the system SiO₂−FeO was estimated in the present work. A good agreement with mean deviation less than 12% was achieved for the comparison of the estimated and available experimental values.

The model is capable of predicting the viscosity of the ternary systems SiO₂−CaO−FeO, SiO₂−MnO−FeO and SiO₂−MgO−FeO by using binary parameters. The model also considers the effect of the content of the different metal oxides in the silicate structure through the value of the oxygen bridges calculated with the thermodynamic model for ternary systems. The present model provides a good representation for most of the experimental data in these systems.

5. Acknowledgements

The authors wish to thank the Institutions CONACyT, SNI, COFAA and IPN for the support of this research.

6. References

• ¹
  Keskinkilic E, Pournaderi S, Geveci A, Topkaya YA. Use of colemanite in ferroniquel smelting. J. Min. Metall. Sect. B-Metall. 2019;55(1):1-8. http://dx.doi.org/10.2298/JMMB181009007K
  » http://dx.doi.org/10.2298/JMMB181009007K
• ²
  Riboud PV, Roux Y, Lucas D, Gaye H. Improvement of continuous casting powders. Fachber. Huttenprax. Metallweiterverarb. 1981;19:859-67.
• ³
  Urbain G. Viscosity estimation of slags. Steel Res. 1987;58(3):111-6. http://dx.doi.org/10.1002/srin.198701513
  » http://dx.doi.org/10.1002/srin.198701513
• ⁴
  Reddy RG, Hebbar K. Viscosity of FeO-SiO₂ slags. Miner Metall Process. 2001;18:195-9.
• ⁵
  Shu Q, Wang L, Chou KC. Estimation of viscosity for some silicate ternary slags. J. Min. Metall. Sect. B-Metall. 2014;50:139-44.
• ⁶
  Le Losq C, Neuville DR. Molecular structure, configurational entropy and viscosity of silicate melts: link through the Adam and Gibbs theory of viscous flow. J Non-Cryst Solids. 2017;463:175-88. http://dx.doi.org/10.1016/j.jnoncrysol.2017.02.010
  » http://dx.doi.org/10.1016/j.jnoncrysol.2017.02.010
• ⁷
  Kondratiev A, Jak E. A quasi-chemical viscosity model for fully liquid slags in the Al₂O₃−CaO−‘FeO’-SiO₂ system. Metall Mater Trans, B, Process Metall Mater Proc Sci. 2005;36(5):623-38. http://dx.doi.org/10.1007/s11663-005-0053-9
  » http://dx.doi.org/10.1007/s11663-005-0053-9
• ⁸
  Romero-Serrano A, Pelton AD. Extensions of a structural model for binary silicate systems. Metall Mater Trans, B, Process Metall Mater Proc Sci. 1995;26(2):305-15. http://dx.doi.org/10.1007/BF02660973
  » http://dx.doi.org/10.1007/BF02660973
• ⁹
  Gutiérrez J, Romero-Serrano A, Plascencia G, Chávez F, Vargas R. Thermodynamic model for ternary silicate systems. ISIJ Int. 2000;40(7):664-9. http://dx.doi.org/10.2355/isijinternational.40.664
  » http://dx.doi.org/10.2355/isijinternational.40.664
• ¹⁰
  Romero-Serrano A, López-Rodríguez J, Hernández A, Pérez-Labra M, Cruz-Ramírez A. Evaluation of molar volume of silicate systems using a structural model. Glass Phys Chem. 2019;45(3):195-201. http://dx.doi.org/10.1134/S1087659619030088
  » http://dx.doi.org/10.1134/S1087659619030088
• ¹¹
  López-Rodríguez J, Romero-Serrano A, Hernández-Ramírez A, Pérez-Labra M, Cruz-Ramírez A, Rivera-Salinas E. Use of a structural model to calculate the viscosity of liquid silicate systems. ISIJ Int. 2018;58(2):220-6. http://dx.doi.org/10.2355/isijinternational.ISIJINT-2017-483
  » http://dx.doi.org/10.2355/isijinternational.ISIJINT-2017-483
• ¹²
  Wu G, Seebold S, Yazhenskikh E, Hack K, Müller M. Viscosity model for oxide melts relevant to fuel slags. Part 3: The iron oxide containing low order systems in the system SiO₂−Al₂O₃−CaO−MgO−N₂O−K₂O-FeO−Fe₂O₃ Fuel Process Technol. 2018;171:339-49. http://dx.doi.org/10.1016/j.fuproc.2017.09.002
  » http://dx.doi.org/10.1016/j.fuproc.2017.09.002
• ¹³
  Wu G, Seebold S, Yazhenskikh E, Tanner J, Hack K, Müller M. Slag mobility in entrained flow gasifieres optimized using a new reliable viscosity model of iron oxide-containing multicomponent melts. Appl Energy. 2019;236:837-49. http://dx.doi.org/10.1016/j.apenergy.2018.11.100
  » http://dx.doi.org/10.1016/j.apenergy.2018.11.100
• ¹⁴
  Myslevic T, Wozniak J, Verny V. Sbornik Vedeckych Praci Vysoke Skoly Banske v Ostrave. Rada Hutnicka. 1974; 20(2): 57-67.
• ¹⁵
  Zhang L, Jahanshahi S. Review and modeling of viscosity of silicate melts: Part II. Viscosity of melts containing iron oxide in the CaO−MgO−MnO−FeO−Fe₂O₃ system. Metall Mater Trans, B, Process Metall Mater Proc Sci. 1998;29(1):187-95. http://dx.doi.org/10.1007/s11663-998-0021-2
  » http://dx.doi.org/10.1007/s11663-998-0021-2
• ¹⁶
  Röntgen P. Struktur und Eigenschaften von Schlacken der Metallhütten-prozesse, Winterhager H, Kammel R. Erzmetallurgy. 1956;9:207-14.
• ¹⁷
  Shiraishi Y, Ikeda K, Tamura A, Saitô T. On the viscosity and density of the molten FeO−SiO₂ system. Trans. JIM. 1978;19(5):264-74. http://dx.doi.org/10.2320/matertrans1960.19.264
  » http://dx.doi.org/10.2320/matertrans1960.19.264
• ¹⁸
  Chen M, Raghunath S, Zhao B. Viscosity measurements of “FeO”−SiO₂ slags in equilibrium with metallic Fe. Metall Mater Trans, B, Process Metall Mater Proc Sci. 2013;44(3):506-15. http://dx.doi.org/10.1007/s11663-013-9810-3
  » http://dx.doi.org/10.1007/s11663-013-9810-3
• ¹⁹
  Kucharski M, Stubina NM, Toguri JM. Viscosity measurements of molten Fe−O−SiO₂, Fe-O-CaO-SiO₂, and Fe-O-MgO-SiO₂ slags. Can Metall Q. 1989;28(1):7-11. http://dx.doi.org/10.1179/cmq.1989.28.1.7
  » http://dx.doi.org/10.1179/cmq.1989.28.1.7
• ²⁰
  Urbain G, Bottinga Y, Richet P. Viscosity of liquid silica, silicates and alumino-silicates. Geochim Cosmochim Acta. 1982;46(6):1061-72. http://dx.doi.org/10.1016/0016-7037(82)90059-X
  » http://dx.doi.org/10.1016/0016-7037(82)90059-X
• ²¹
  Dingwell DB. Redox viscosimetry of some Fe-bearing silicate melts. Am Mineral. 1991;76:1560-2.
• ²²
  Sridhar S, Sichen D, Seetharaman S, Mills KC. Viscosity estimation models for ternary slags. Steel Res. 2001;72(1):3-10. http://dx.doi.org/10.1002/srin.200100073
  » http://dx.doi.org/10.1002/srin.200100073
• ²³
  Ji F-Z, Sichen D, Seetharaman S. Experimental studies of the viscosities in the CaO−Fe_nO−SiO₂ slags. Metall Mater Trans, B, Process Metall Mater Proc Sci. 1997;28(5):827-34. http://dx.doi.org/10.1007/s11663-997-0010-x
  » http://dx.doi.org/10.1007/s11663-997-0010-x
• ²⁴
  Johannsen F, Wiese W. Erzmetallurgy. 1958;11:1-15.
• ²⁵
  Ji F-Z, Sichen D, Seetharaman S. Experimental studies of viscosities in Fe_nO−MgO−SiO₂ and Fe_nO−MnO−SiO₂ slags. Ironmak Steelmak. 1998;25:309-16.

Publication Dates

• Publication in this collection
  03 Apr 2020
• Date of issue
  2020

History

• Received
  02 Sept 2019
• Reviewed
  31 Jan 2020
• Accepted
  04 Feb 2020