Modeling of Viscosities of the Partly
Crystallized Slags
I.V. Nekrasov*, O.J. Sheshukov*, V.N. Nevidimov**, S.A. Istomin*
* Establishment of the Russian Academy of Sciences Institute of Metallurgy of the
Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
** Ural Federal University, Ekaterinburg, Russia
ABSTRACT
The opportunity of a prediction of physical and chemical properties of slags is an essentially important for development of new and perfection of existing industrial
schemes of metallurgy. One of the most significant properties of slags is viscosity as
the phenomena determined heat (mass) transport in slag lay in a basis of many metallurgical processes. As a rule, under production conditions slags contain solid particles;
therefore a problem of a prediction of viscosity of the heterogeneous slags gets a special urgency. In this article the technique of a prediction of amount of solid particles
and a chemical compound of a liquid phase of heterogeneous slags depending on temperature is presented. This technique in many respects is similar to a technique of calculations under phase diagrams. The known chemical composition of a liquid phase of
slag allows to calculate its viscosity on models for the homogeneous slags, therefore
the presented method allows to obtain data for calculation of effective viscosity of
heterogeneous slags with use of the dependences offered by A. Einstein and J. Frenkel
for viscosity of suspensions.
INTRODUCTION
The viscosity is one of the main physical properties of slags, largely determining the efficiency of the metallurgical technology (1). Practically, the slag required
viscosity is obtained by adjustment its heterogeneity level. In this case, empirical dependences, representing only quantitative factors of this process, and partial derivatives or laboratory conditions are used. The objective problems of the property description of heterogeneous slags make the development and the application of quantitative dependencies more difficult. Obviously, this decreases the possibility of control
actions and the technology efficiency. Therefore the development of the scientifically
grounded method of the heterogeneous slag viscosity prediction is still a topical and
essentially relevant problem.
THEORETICAL CONCEPT AND EXPERIMENTAL CONFIRMATION
The main difficult of the above problem solution is related to the heterogeneous
slag average chemical composition, non-defining its physical and chemical properties
completely. Therefore we must have the method of the quantitative estimation of the
phase distribution of components and of the phase amount, representing the physical
nature of heterogeneous slags, to simulate the appropriate model of their properties.
Particularly, such method will allow calculating the effective viscosity of heterogeneous slags based on their average chemical composition and temperature data, using
dependencies, submitted by A. Einstein and J. Frenkel for the suspension viscosity
(2).
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The possibility of the suspension viscosity formula application for the calculation of the same slag properties is observed in the number of articles (3-7). Particularly, in these articles the application of the Einstein equation (1) and its modification of
(2) and (3) for heterogeneous slags was verified:
(1)
η  η (1  2,5  ε )
о
η  ηо (1  5,5  ε)
(2)
η  ηо (1  а  ε)  n
(3)
where, η – suspension viscosity, ηо – viscosity of the suspension liquid phase, ε – volume ratio of solid particles, a and n – constants.
The authors (3) performed their studies as follows. The synthetic slag of the
CaO – MgO – Al2O3 – SiO2 system was prepared and heated up to its liquidus temperature, down that it became the saturated spinel MgAl2O4. Using a rotational viscosimeter the slag viscosity at this temperature was measured, allowing obtaining the
experimental value of the homogeneous slag viscosity ηо. Then, some amount of solid
MgAl2O4 was added to the slag and its viscosity was measured. As far as the solid
particles were added to slag at the temperature of its equilibrium with the solid
MgAl2O4, it was supposed that all the spinel added was in the undissolved form.
Therefore, in this case, the viscosity measured is an effective viscosity of the heterogeneous slag η, and the amount of MgAl2O4 added to the original homogeneous slag
is an amount of the solid phase ε. Comparing the viscosity values of the heterogeneous mixture η and of the original homogeneous slag ηо, taking the solid phase amount
ε into account, the feasibility of equations (1) - (3) was verified. The satisfying convergence of the experimental data and the calculation results from the equation (3) in
the following form:
(4)
η  η (1  1,35  ε )  2,5
о
was noted.
In the articles (4-6) the detailed study of the viscosity of slags of the CaO – SiO2
– FeOn system in the Ca2SiO4 and СаО saturation area was performed. In addition, the
authors of the articles (4, 6) checked the application possibility of CaO – SiO2 – FeOn
phase diagram for effective viscosity estimation by means of determination of the solid particle amount and the liquid phase composition of heterogeneous slags using geometric constructions. The known composition of the mixture allowed determination
of its viscosity ηо using homogeneous slag viscosity models, and, therefore, calculation the effective viscosity using equations (2) and (3). Comparing measured and calculated values of the heterogeneous slag viscosity, the feasibility of equations (2) and
(3) was verified. The satisfying convergence of the experimental data and the calculation results from the equation (2), and from the equation (3) in the following form:
(5)
η  η (1  ε )  2,5
о
was noted.
The application possibility of equations (1) – (5) to calculating the heterogeneous slag viscosity can be approved by comparing the data of CaO – SiO2 – FeO viscosity diagram and calculation results of CaO – SiO2 – FeOn phase diagram for slags
with the same composition (8). The viscosity values on CaO – SiO2 – FeOx diagram
are at 1400°С and are mainly related to the Ca2SiO4 saturation area (8). By selecting
in the phase diagram a slag composition having the measured viscosity value at
1400°С, we can determine its liquid phase composition and the amount of Ca2SiO4
undissolved at 1400°С using geometric constructions. If we calculate the viscosity of
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this liquid phase ηо using available models (9), taking Ca2SiO4 undissolved into account in the form of ε, then effective viscosity values, obtained by equations (2) and
(5), will be satisfying converged with measured values of the viscosity diagram. Generally, the references describe the key possibility of the application of equations (1) –
(5) to calculating the slag effective viscosity (3-7).
The phase diagram being the solubility diagram of some compound into a melt
specifies the possibility of the phase diagram application to calculation the effective
viscosity. Liquidus surfaces are the surfaces of the limit solubility. The phase diagram
shows solid phases and their amounts being in the equilibrium with the melt of fixed
composition at the fixed temperature. Therefore, the phase diagram can be considered
as a “saturation graphical model” for two-, three- or, maximum, four-component system. Assume that, there is a possibility to change the “saturation graphical model” to a
purely analytic multidimensional model. Then, we will have the possibility to determine the amount and the composition of phases at the equilibrium with a multicomponent melt at various temperatures. In this case, the calculation method of the relative amount of phases and their compositions is the same as the available calculation
method for phase diagram, and is its generalization for a multicomponent system (1).
In this article, we use the updated version of the slag structure polymer model to
estimating the equilibrium composition of solid and liquid phases in slags (9). The
polymer theory of the slag structure is extension of the ion theory. Its feature is a consideration the presence into the melt not only simple ions, but some large complex anions, as well, that can joint together, i.e. to polymerize. On this basis, the polymer
model calculates a number of slag properties and its component activities. The cause
for complex anions formation is an essential difference between ion potentials of slag
cations, measured by the ratio between the cation charge and the cation size. This difference results in the most “strong” cations, primarily, of silicium, form around themselves a sufficiently stable enclosure from oxygen anions. The binding force inside
these complexes exceeds the same force between these anions and catinos surrounded,
essentially allowing extracting complexes as some separate structure units of the melt.
Furthermore, the simple oxygen anion has a charge “-2”, therefore, it can form two
bonds with the nearest cations. For example, one bond is with the silicium cation, and
another – with the calcium cation; or both bonds with the silicium cations. As a result,
the various developments of the polimerization process of silicium anions are possible, depending on the slag composition, for example, depending on calcium cation
concentration.
Let us consider our algorithm developed to predicting the composition and the
phase amount using the example showing the similarity of this algorithm with the calculation method by phase diagrams (see Fig.1). In the point 1, set above the liquidus
line L, the original melt has the corresponding temperature and concentration of MeO
oxide. The melt proximity to the saturation of МеО in the polymer model is estimated
by the ratio between the following two parameters: the thermodynamic activity of
ТD
МеО in the melt, а МеО
, mainly being the function of the melt composition, and the
SAT
saturation activity of МеО, а МеО
, being the function of temperature Т (9):
ТD
а МеО
 f (melt composition)
(6)
а  f (Т)
(7)
In the point 1, at the corresponding temperature Т1, the inequality, indicating the
nonsaturation the melt of MeO, is true (see Fig. 1b):
SAT (Т1 )
ТD
(8)
аМеО
 аМеО
SAT
МеО
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With the temperature decreasing from the point 1 to the point 2 (excepting the
SAT
SAT
point 2 itself, lying on the liquidus line L) а МеО
decreases, however а МеО
still exceeds
ТD
а МеО
. In the point 2, the saturation activity of MeO, specified by the temperature Т2,
reaches the thermodinamical activity of МеО in the melt, indicating the saturation of
the melt of МеО (9):
SAT (Т 2 )
ТD
(9)
аМеО
 аМеО
Fig. 1. The basic diagram of calculation of solid particle amount in the heterogeneous
slag and in its liquid phase, showing the similarity of this method with the phase diagram calculation method: а – the area of the hypothetical phase diagram of the binary
ТD
SAT
melt in the “composition” – “temperature” coordinates; b – the ration а МеО
and а МеО
in the same melt during changing the temperature and the melt composition
If during the further temperature decreasing from the point 2 to the point 3, we
suppose that the original melt composition does not change (the МеО concentration
SAT (Т 3 )
ТD
does not decrease), then а МеО
becomes more than, for example, а МеО
. It is indicated, that the homogeneous melt having the original МеО concentration cannot exist at
the temperature below the liquidus. Therefore, while the temperature decreases below
the liquidus, МеО begins to extract from the homogeneous melt in such amount that is
ТD
SAT
necessary to maintain the equality а МеО
and а МеО
true at every temperature, i.e. during
decreasing the temperature from the point 2 to the point 3, the homogeneous melt
composition changes along the line 2-4 (Fig. 1а).
If we consider this example as a part of the phase diagram (see Fig. 1а), then the
mass fraction of МеО crystals and the slag liquid phase composition can be calculated
using graphical constructions (line 5-3-4) – this possibility is specified by the known
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location of the liquidus line in the diagram. However, actually the presence of the
graphical liquidus line in the “composition” – “temperature” coordinates is not essenТD
SAT
tial, since this liquidus line is described by the equality of а МеО
and а МеО
, fairly defining these coordinates. Therefore, the liquidus line location in the “composition” –
“temperature” coordinates can be defined by computational method (see Fig.1b). This
results to possibility of application the above algorithm to multicomponent slags
without existing graphical phase diagrams – the polymer model equations equally
change the graphically represented liquidus surfaces.
Therefore, this method of determination of the undissolved crystal amount and
the liquid phase composition of the heterogeneous slag includes the following:
- determination of the minimum temperature at which thermodinamical activities for all slag components are no more than activities of saturation (the liquidus
temperature is defined);
- decreasing of concentration of such components in the melt for which while
SAT
ТD
the temperature decreases below liquidus, а МеО
becomes equal а МеО
; decreasing of
concentrations of these components is preformed in such manner that at every temТD
SAT
perature а МеО
is equal а МеО
(the slag liquid phase composition is calculated);
- determination of the mass fraction of undissolved crystals using the balance of
the original melt components at the liquidus temperature and of the liquid phase composition during its heterogenization.
The known liquid phase composition at the temperature T specifies the possibility of determination its viscosity η0 using available models for homogeneous systems
(9). For conversion the mass fraction of the solid phase to its volume fraction, we
shall have the following data available: the density of solid particles and the density of
the slag liquid phase (10). The liquid phase density can be defined using the following
formulas, submitted by V.I. Yavoisky for basic and semiacid slags (11):
1
V1400 
1400
 0,45  SiO 2  0,286  CaO  0,286  FeO  0,35  Fe 2 O 3  0,237  MnO 
 0,367  MgO  0,48  P2 O5  0,402  Al 2 O3
 
T
1400
 0,07 
1400  T
100
,
( 10 )
( 11 )
V1400 – slag specific volume 1400 °С (sm3/100 g);
ρ1400 and ρТ – slag density at 1400 °С and at temperature Т, (100 g/sm3);
SiO2…A2O3 – concentration of corresponding component (mass %).
Using formulas of (2) – (5)-type at the available viscosity values of the liquid
phase η0 and at the volume fraction of undissolved crystals ε, we can exactly determine the effective viscosity of the slag η at the values of ε up to 0,2 – 0,3.
The correlation of the experimental results with computational data obtained by
the above method is shown in the Figure 2 for the typical secondary steelmaking slag
(СаО = 55,8 wt. %; SiO2 = 23,6 wt. %; MgO = 7,7 wt. %; FeO = 1,5 wt. %; A2O3 =
11,4 wt. %).
where,
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Fig. 2. Temperature dependencies of the viscosity η experimental results on its precomputational data: а – СаО concentration in the slag liquid phase (1); b – MgО concentration in the slag liquid phase (2); c – volume fraction ε of solid particles of CaO
(3) and of MgО (4) in the slag; d – viscosity values, calculated using the polymer
model, taking the heterogenization into account (5); viscosity values, calculated using
the polymer model without heterogenization (6); experimental viscosity values (7)
According to polymer model calculations, while the temperature decreases below the liquidus (1565°С) the CaO concentration in the melt becomes to decrease (see
Fig. 2а), resulting in the occurrence the corresponding quality CaO crystals in the slag
volume (see Fig. 2с). During decreasing of CaO concentration in the slag liquid
phase, the concentration of the all other slag components, including MgO, in the liquid phase is accordingly increased (see Fig. 2b). At the temperature of 1440°С the
liquid phase saturates with MgO, the further temperature decreasing results in decreasing of MgO concentration in the slag liquid phase (see Fig. 2b) and in the occurrence of the corresponding amount of MgO crystals in the slag volume (see Fig. 2с).
Therefore, while the temperature decreases below liquidus, the slag viscosity increases faster, than we can suppose from the calculations without taking the heterogenezation into account (see Fig. 2d).
REMARK
We shall note, that precipitated crystals of CaO and MgO can capture iron oxides from the slag liquid phase. These processes shall affect the experimental results
for heterogeneous slag viscosity measuring. For example, the more particles of CaO
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and MgO in the slag and the more duration of their existence, the more expected iron
oxides captured by them from the slag liquid phase, and, thus, the more viscosity of
the liquid phase and the slag itself is affected by this process. The development of
such processes in the above method is not yet considered.
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