ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
Viscosity Measurement of Slags using Rotating Bob and Vibrating Finger Viscometer
Dmitry ChebykinHans-Peter HellerTobias DubbersteinIurii KorobeinikovOlena Volkova
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2017 Volume 57 Issue 8 Pages 1319-1326

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Abstract

Within the present investigation, results of the viscosity measurements using a rotating bob viscometer and a newly-developed vibrating finger viscometer have been presented. The knowledge of the viscosity property of metallurgical slags and fluxes is of major importance considering its tasks of fulfilment in the process of steel manufacturing and casting.

Due to the chemical stability of the slag and/or the measurement technology, the measurements were applied in the temperature range from 1300°C to 1600°C. Nowadays, the rotation bob method with a concentric cylinder and a wide gap is the state of the art of the viscosity measuring devices for slags, e.g. rheometers with air bearing. In the present investigation, the Anton Paar MCR 301 rheometer was used as recently reported in Ref. 1). Furthermore, a new vibrating finger viscometer was developed to provide a wider measurement range in high temperature regions from 1 mPa∙s up to 1 Pa∙s dynamic viscosity.2,3,4) In the current investigation, the viscosities of CaO–Al2O3, CaO–SiO2, CaO–SiO2–Al2O3–MgO slags and reference glass have been measured up to 1600°C using the two viscometer apparatuses. The results have been discussed in the context of the high temperature calibration errors, e.g. deviations in geometry of the crucible, the slag stability, the fluid flow, the atmospheric conditions, as well as for the difference in both measurement principles. Also, in the last section of this manuscript, the achieved results have been compared with the previous literature reports.

1. Introduction

Viscosity is one of the most important properties of the metallurgical melts (slag and steel) and an accurate information about this property is crucial for a better understanding, stabilization and modeling of the metallurgical processes.5) In addition, viscosity is also essential for the infiltration and gas atomization processes.6,7) Nowadays the most common method for the determination of viscosity at high temperature is the concentric cylinder method. Upon recent ten years, this method has been widely used for the determination of the slag viscosity.8,9,10,11,12,13,14,15,16) However, it is difficult to determine a precise low torque due to the mechanical friction inside the viscometer.17) Viscometers of this type consist of two concentric cylinders (a bob and a crucible) and the viscosity is determined from the calculations of the torque of bob at constant or different rotations. A few other studies were reported to use the different types of viscometers such as the rotation crucible viscometer and the oscillating plate method.17,18,19,20) The concentric cylinder method was primary used for the determination of slag viscosity. It is important to note that for the liquid metals another type of viscometer should be applied. In the following, several types of viscometer are summarized:21)

• Capillary method

• Falling body method

• Oscillating method (oscillating cylinder or plate, oscillating vessel method)

• Concentric cylinder (rotation) method

Each method is applicable for the different temperature and viscosity ranges. The capillary method has found an application in the medium temperature range. However, applying this method at higher temperatures brings about some complications related to the selection of a suitable crucible and a capillary material (i.e. dimensional stability and corrosion resistance).21) Another method is the falling body method which is used for the determination of the glass and slag melts viscosity. In addition, for the determination of viscosities in the centipoise (mPa·s) range e.g. CaF2 based slags, metals and etc. the oscillating methods have been used.21) In fact, during recent years the oscillating cylinder and plate methods were successfully applied for the determination of the steel and slag viscosity at high temperatures (≤ 1600°C).4,19,20,22,23)

The present study was performed to experimentally investigate an applicability of the vibrating finger viscometer for the measurements of slag viscosity up to 1600°C and to discuss the high temperature calibration errors such as deviations in geometry of the crucible, the slag stability, the fluid flow, the atmospheric conditions, as well as the difference in both measurement principles. For this purpose CaO–Al2O3, CaO–SiO2 binary systems and an CaO–SiO2–Al2O3–MgO industrial blast furnace slag and a glass as a reference system were chosen.

2. Experimental Procedures

2.1. Alloys and Conditions of Experiments

The determination of the viscosity of CaO–Al2O3, CaO–SiO2, CaO–SiO2–Al2O3–MgO slag systems and reference glass up to 1600°C was carried out in argon 5.0 (99.999 vol.% Ar, < 2 ppm O2) using the vibrating finger viscometer and the Anton Paar MCR 301 rheometer in molybdenum crucibles (Fig. 1).

Fig. 1.

Geometrical parameters of molybdenum crucibles in millimeters for: a) vibrating finger viscometer; b) rotating bob viscometer (Anton Paar MCR 3301 rheometer).

The slags and glass were pre-melted in molybdenum crucibles under argon atmosphere in an induction furnace and the chemical compositions were analyzed afterwards using a Bruker AXS S8 Tiger (XRF). The corresponding results presented in Table 1 are described as follows: CA (CaO–Al2O3), CS (CaO–SiO2), CASM (CaO–SiO2–Al2O3–MgO) and glass. The weight of the slags in the crucibles was ca. 30–35 g. For the main experiments a similar construction of induction furnaces for both viscometers were used (Fig. 2). The temperature control in the furnace and in the melt was accomplished with the help of the bottom thermocouples in both furnaces.

Table 1. The measured composition of slags by Bruker AXS S8 Tiger (XRF) after pre-melting.
NameChemical composition, mass.%
CaOSiO2Fe2O3MgOAl2O3MnOK2ONa2OTiO2
CA51.620.320.150.2447.880.000.000.000.00
CS40.7859.530.090.120.290.010.000.000.03
CASM38.8534.911.038.5011.840.170.380.180.56
Glass9.9473.040.182.01.000.000.2812.960.01
Fig. 2.

Scheme of the induction furnace for the determination of slag viscosity.

Viscosity of the slags was measured continuously using various cooling rates of 2, 5 and 10 K/min. After the complete melting of slags and obtaining the first measuring temperature, the holding time of ca. 30 minutes was used. A temperature field in molybdenum crucibles was controlled using two thermocouples (Type B). For the determination of viscosity in the vibrating finger viscometer and in the Anton Paar MCR 301 rheometer, two types of bobs were applied which dimensions are presented in Table 2 and Fig. 2.

Table 2. Dimensions of the bobs in millimeters for viscosity measurements.
MaterialBob (Anton Paar MCR 301), mmBob (Vibrating finger viscometer), mm
Molybdenum (99.9% Mo)diameter of bob12diameter of bob5
length of bob20length of bob30
diameter of shaft3diameter of shaft3

3. Methods of Viscosity Measurement

3.1. Rotating Bob Viscometer

The viscometer for the determination of slag viscosity at high temperature using the rotating bob method in the present work consists of the following parts: The Anton Paar MCR 301 rheometer, the cardan joint, the molybdenum rod, the melting crucible, the heating crucible (graphite), two thermocouples, the crucible positioning system and the measuring system (Fig. 2). The viscosity can be calculated using Eq. (1):   

η= M 8 π 2 n H b ( 1 R b 2 - 1 R mc 2 ) (1)
where η is the dynamic viscosity, M is the torque, n - number of rotation per minute, Hb - the height of the long side of the bob, Rb and Rmc - the radius of the bob and molybdenum crucible, respectively. Before measurements, only the rotation speed and torque of the Anton Paar MCR 301 rheometer have been calibrated. The immersion depth for measurements of the bobs in the slags system is calculated by Eq. (2):   
ΔH= H b ( 1- ( R b R mc ) 2 ) + H md ( 1- ( R rod R mc ) 2 ) (2)
where ΔH is the immersion depth, Hmd is the standard measuring depth (7 mm) and Rrod is the radius of rod.

In this work, for the determination of the slag viscosity, the bobs were immersed into the molybdenum crucibles by applying two restrictions. Firstly, the distance between the bottom of the molybdenum crucible and the bottom of the bob needs to be more than 10 mm. Secondly, the height of the slag layer above the bob should be 7 mm (Fig. 3). Previous investigations revealed that the mentioned height of the liquid above the bob (i.e. 7 mm) led to a negligible systematic error of 1.43% by changing this depth in the range of ± 3 mm.1) Before the start of each experiment, the inner diameter of molybdenum crucible (Rmc) and the diameter of the bob (Rb) were measured to determine the accurate height of the slag layer above the bob. The position of the molybdenum crucible relative to the axis of rotating the bob was centered by the crucible position system. Additionally, the bob at 6 rpm was recorded using special software NATIONAL INSTRUMENTS Vision Builder 2012 for Automated Inspection and the deviations from the central line of rotation were continuously measured and controlled. Consequently, the bobs with the deviations less than 1.5° and 1.5 mm from the axis of rotation were used. After the complete melting of slags and obtaining the first measuring temperature, the holding time of ca. 30 minutes was used. For the first 10 minutes, the bob was above the slags and then was gradually immersed in the slag within 5 minutes. The last 15 minutes were allocated to the holding time at a constant temperature and a constant rotation speed of the bob (30 rpm). Within the experiments, the temperature of the molybdenum crucible and the position of the molybdenum rod were continuously measured and recorded. All experiments were carried out under inert argon atmosphere (with flow rate 200 l/min (furnace) and 60 l/min (top cover) (Fig. 2)).

Fig. 3.

Schematic illustration of the bob immersion into the molybdenum crucible.

3.2. Vibrating Finger Viscometer

The viscosity measurement equipment was based on the recent invented patent of a new vibrating finger viscometer.2) The technical scheme of the viscometer is presented in Fig. 4 and the operation concept was recently described in reference.22) The molybdenum bob (indicated with No. 8 in Fig. 4) is set in an oscillatory motion with constant peak-to-peak amplitude of 0.625 mm. For the stable oscillatory motion an AC-powered field coil (No. 2) and neodymium magnet (No. 3) were used. The path of finger vibration is measured using a highly sensitive laser micrometer (No. 7) and a sharp-edged diaphragm (No. 5). Via a 12-bit ADC microcontroller, the field coil is powered in a rectangular mode using the bipolar transistors (No. 4). An absorber system (No. 1) is installed to avoid any external interferences of the viscometer oscillation. The vibration of the viscometer as a function of the oscillator weight and spring constant (No. 6) was set in its resonant frequency at approximately (26 ± 0.1) Hz. In contrast to a vibrating wire viscometer or an oscillating plate viscometer, the recorded amplitude is kept constant in the vibrating finger viscometer in all measurement states.

Fig. 4.

Schematic illustration of the vibrating finger viscometer which consists of 1: absorber, 2: field coil, 3: neodymium magnet, 4: microcontroller 12-bit ADC, 5: diaphragm, 6: spring and spring holder, 7: laser micrometer, 8: molybdenum finger.23)

In the case of the present viscometer, the following behavior of the measurement system was assumed:

➢ Newtonian behavior of investigated slags,

➢ Absence of induced turbulent flow within the melt container,

➢ Full stick condition of finger/melt interface,

➢ No slipping effects,

➢ Possible end effects of finger/melt interface are negligible small and are considered, within the pre-calibration using reference fluids,

➢ Absence of reflective wave interactions on the melt surface.

The viscometer was calibrated using the silicon reference oils of varying dynamic viscosity at 25°C. The product of viscosity and density ( ηρ ) is a function of the relative change of the field coil current (Irel) of the vibrating finger which is defined as follows:   

I rel = I fluid - I 2 - I 1 2 I 2 - I 1 2 (3)
where Irel is the relative change of the field coil current [dimensionless quantity], Ifluid is the average coil current at the fixed immersion depth (20 mm) of the finger in the liquid melt at a constant peak-to-peak amplitude of 0.625 mm, I1 and I2 are the average coil current after the achievement of a constant peak-to-peak amplitude of 0.625 mm in the atmosphere and after the immersion of the finger into the melt.

The relation between ηρ and Irel provided by using an exponential fit of calibration for the experimental data of silicon oils (Fig. 5) has been shown as Eq. (4):   

ηρ =-27.45exp( - I rel 1.77 ) -980   540.49exp( - I rel 216   055.06 ) +980   540.49 (4)
Fig. 5.

The exponential fit of calibration of ηρ as a function of Irel using different silicon oils at 25°C.

Density temperature functions for the binary systems (CaO–Al2O3 and CaO–SiO2) and for the industrial blast furnace slag were taken from the maximum bubble pressure method measurements.24)

4. Results

The experiments were carried out on the four slags at different temperatures and were subsequently compared to the published literature data (Table 3). The information regarding to the blast furnace slag was taken (selected) from the researches of Han et al.,25) Kim et al.,26,27) Suzuki et al.28) and Tang et al.29) The measurements of blast furnace slags have covered the compositions of 40 to 43.3 mass. pct. CaO, 33 to 45 mass. pct. SiO2, 10 to 15 mass. pct. Al2O3, 5 to 10 mass. pct. MgO and the experiments were carried out in temperature range from 1300°C to 1575°C. Also, the binary slags systems (CaO–Al2O3 slag and CaO–SiO2 slag) were compared with the results obtained by Oliveira et al.,30) Urbain et al.31) and Wu et al.32,33)

Table 3. Comparison between the experimental data and the data published in literature.
Source of dataCompositions (mass%)Viscosity, Pa·s
CaOSiO2Al2O3MgOTemperature, K
157316731723174817731798182318481873
Temperature, °C
130014001450147515001525155015751600
Blast furnace slag
Rotating viscometer40.3036.2012.38.81.180.460.320.25
Oscillating viscometer0.360.240.18
Han et al.25)42.9033.001590.300.250.210.190.16
Kim et al.27)43.3036.701370.40.26
Suzuki et al.28)41.9038.1010100.27
Kim et al.26)40.0040.0010100.780.520.34
Tang et al.29)40.0045.001051.10.740.520.42
Chen et al.34)38.5035.0017.38.90.480.400.340.290.250.21
CaO–Al2O3 slag
Rotating viscometer51.510.3247.780.240.710.450.310.210.16
Oscillating viscometer0.740.530.350.250.18
Oliveira et al.30)52480.28
Wu et al.33)0.28
Wu et al.32)0.820.37
CaO–SiO2 slag
Rotating viscometer40.4459.030.290.121.400.940.660.480.39
Oscillating viscometer1.090.750.540.410.33
Urbain et al.31)37.662.41.701.19
Wu et al.33)40600.6
Wu et al.32)0.37
Wu et al.32)0.64

The results of present work for the industrial blast furnace slag are in a good agreement with the data reported by Han25) and Kim.27) In the work of Kim,27) a Brookfield digital viscometer (rotating cylindrical method) was employed for the viscosity measurements and holding time for reaching equilibrium state was more than 30 minutes at each temperature. The aforementioned condition can be the source of deviation from results of the vibrating finger viscometer and the Anton Paar MCR 301 rheometer. In present study for comparison of the results between the two types of viscometers, cooling rate of 10 K/min was chosen; therefore the equilibrium state could not be reached. As a result, the viscosity values for the two types of viscometers are lower than those in the work of Kim et al.27) The viscosity values reported by Han25) were collected from the other publications and contain no information about the procedure of viscosity measurement. Meanwhile, the significant deviation from the results of Tang et al.29) and Chen et al.34) can be associated with the considerable differences in the slag compositions as well as the measurement techniques. In these two mentioned researches, Brookfield digital viscometers were applied and the viscosity of slags was estimated with an approx. holding time of 30 minutes at each temperature to ensure the chemical equilibration. As previously noted, the equilibrium state of the slag viscosity during the continuous cooling may not be achieved which can cause the incompatibility of the different results. Besides, Chen et al.34) reported data for a slag that contains 17.3 mass% of Al2O3. This high concentration of aluminum oxide in slag can lead to the increase of viscosity as reported by Persson et al.8) and Yan et al.35) Additionally, the basicity (CaO/SiO2) of the slag in research done by Tang29) was less than 1 (this parameter is 1.12 in the present work). It is well known that the change of basicity of slag can cause essential changes in its viscosity.

The data obtained for the CaO–Al2O3 system was compared with the both calculated and experimental data. The experimental data presented in the research performed by Oliveira et al.30) were estimated using Brookfield LVTD viscometer at 1550°C. Results of the present work for the CaO–Al2O3 system are in a good agreement with the data reported by Oliveira et al.30) Comparison of the experimental data of the present work using two viscosity measurement techniques with the data of Oliveira et al.30) reveals that the obtained results for the vibrating (oscillating) finger viscometer is more compatible with the data reported by Oliveira et al.30) However, there are considerable differences between the experimental results of the present work and the calculated data presented by Wu et al.32,33)

Moreover, the obtained viscosity values for CaO–SiO2 system have been compared with both the rotation cup viscometer results reported by Urbain et al.31) in the temperature range of 1552 to 1903°C and with the viscosity data obtained by simulations reported by Wu et al.32,33) Surprisingly, computed results of Wu et al.32,33) depict a better agreement with the present research as compared to the experimentally obtained values of Urbain et al.31)

5. Discussion

It is generally accepted that the rotational viscometer is the most validated and used viscosity measurement technique comparing to the other techniques.21) At the same time, the oscillating methods (oscillating cylinder or plate and vessel method) are more suitable for the materials with low viscosities within the viscosity range of 10−5 to 10−2 Pa·s, such as liquid steel and metals.2,23,36,37)

In this research, the vibrating finger (cylinder) viscometer for measurement of the slag viscosity up to 5 Pa·s was applied at high temperatures. In order to study the applicability of this method for the wider range of viscosities, the vibrating finger viscometer was calibrated with the silicon oils up to viscosity of 10 Pa·s. The measuring range of the vibrating viscometer is limited to the capability of a coil field to maintain the stable oscillation motion. At the viscosity of 10 Pa·s a problem with the stable oscillatory motion with the constant peak-to-peak amplitude of 0.625 mm arises. During experiments, voltage of the field coil is always 5 V and the current can be changed by a 12-bit ADC microcontroller to up to value of 10 A. Thus, the measuring range of the oscillating viscometer is essentially limited by the current in the field coil. Therefore, before the start of experiments three types of cylinders: 1. cylinder, 2. cylinder with the spherical nose and 3. cylinder with the conical nose (Fig. 6) were tested in silicon calibration oil (WACKER SILICONEOEL AK 200) to study the influence on accuracy and measuring range of the vibrating finger viscometer. The corresponding results are illustrated in Fig. 7.

Fig. 6.

Geometrical parameters of the investigated bobs in millimeters.

Fig. 7.

Influence of the bobs on the current of the field coil.

The test program was carried out within approx. 2 min and was divided into three operations. Firstly, the vibrating finger was immersed into the calibration silicon oil to the depth of 20 mm. At this depth a relative current change within approx. 35 seconds for each type of cylinder was measured. After determination of the relative current change, the vibrating cylinders were lifted.

It has been concluded that cylinders with the spherical or conical nose and low weight should be applied for the experiments. In the first instance, the spherical or conical nose provides a relative minor change of the current in the field coil (in comparison with the molybdenum or aluminum cylinder). At the conical and spherical nose, at the aluminum cylinder and at the molybdenum cylinder the relative current changes reach the approximate values of 0.62, 0.77 and 0.84, respectively (Fig. 7). This trend can be explained by the lower turbulence and lower weight of the fingers. In addition, the mentioned types of noses decrease the probability of so-called “end effect” which has been discussed in literature.38,39) Furthermore, the oscillating motions of cylinder can cause the formation of secondary flow and the production of flow instability.

Moreover, all types of applied cylinders (bodies) were calibrated in silicon oils using varying viscosities from 200 to 12500 mPa·s at room temperature (25°C). Estimated viscosity values for cylinder with the spherical or conical nose can be better described than for the cylinder by mathematical functions. In fact, the mathematical function for estimation of viscosity is more accurate in the case of spherical and conical nose than for cylinder. The determined correlation coefficients are: 1. conical nose - 0.99989, 2. spherical nose - 0.99985, 3. aluminum cylinder - 0.99983, 4. molybdenum cylinder - 0.99947.

Another critical aspect for the selection of cylinders is the weight of the cylinders as the light ones are necessary for the broadening of measuring range. Heavier cylinders, molybdenum cylinder with the approx. weight of 5.6 g, require a higher level of relative current change of ca 0.84 in the field coil for the stable motion than the aluminum cylinder with the approx. weight of 5.0 g (~ 0.77) (Fig. 7).

After performing the calibration and preliminary tests with the cylinders, the main experiments using four different slags were carried out. The results are presented in Fig. 8.

Fig. 8.

Comparison of the measured viscosities of the slags using the vibrating finger viscometer and the rotating bob viscometer (Anton Paar MCR 301 rheometer).

Comparing the experimental data of the vibrating (oscillating) finger (cylinder) viscometer with the results obtained for the rheometer, some differences in the value of viscosity are revealed. These differences can be attributed to the:

i) differences in the geometry of the molybdenum crucibles,

ii) different chemical composition of the slags,

iii) different temperature fields occurring in two molybdenum crucibles

iv) Taylor vortex in the rotation bob viscometer.

i) During experiments, two types of molybdenum crucibles with different external geometrical parameters but similar inner volume geometries (Fig. 1) were used. Obviously, the hardness of the metal decreases as the applying temperature approaches the melting point. Temperature of experiments (max. 1600°C) was substantially below the melting point of molybdenum – 2617°C. Therefore, upon the experiments the hardness limit of molybdenum should not be surpassed.

Another point to be noted is that the geometrical parameters of the molybdenum crucibles and the cylinders can be physically changed at high temperatures due to the thermal expansion. However, this statement is not valid for molybdenum as this metal has a small thermal expansion like tungsten, chromium and zirconium.40) The values of constants A1 and A2 for the pure molybdenum in the temperature range from 25 to 2341°C amount to 3.91·10−6 and 1.81·10−9, respectively.41)

As the chemical reactions can change the geometrical parameters of crucibles, the inner diameters of them were controlled before and after each experiment. Since this effect was negligible or was not observed in our experiments, it is possible to conclude that the deviations in geometry didn’t have an influence on the accuracy of the measurements.

The main experiments were carried out in an inert argon gas with the flow rate of 200 l/min (bottom) and 60 l/min (top). Such a high level of flow rate is essential to protect the graphite heater and molybdenum crucibles from high temperature oxidation. Using an insufficient flow rate of argon , the graphite heaters and the molybdenum crucibles may react with oxygen at high temperatures which can lead to both the changes of slags chemical composition and to a temperature field change in molybdenum crucibles. Nevertheless, the molybdenum oxide has not been found in slag and the geometry of heaters faced only minor changes. Thus, the chemical composition of slags and the geometrical parameters of heaters after experiments demonstrate the sufficiency of flow rate of argon in our experiments.

ii) Chemical stability of slags may affect the accuracy of measurements. To investigate that, the chemical composition of slags after the experiment was analyzed by (XRF) and the corresponding results are presented in Table 4.

Table 4. Chemical composition of slags before and after experiments.
Compositions (mass%)
CaOSiO2Al2O3MgO
Blast furnace slag
Original (before experiment)40.2936.2112.288.82
Rheometer (after experiment)40.3336.2912.318.78
Oscillating viscometer (after experiment)40.1335.9512.178.66
CaO–Al2O3 slag
Original (before experiment)51.510.3247.780.24
Rheometer (after experiment)51.800.1147.680.23
Oscillating viscometer (after experiment)51.740.1547.690.21
CaO–SiO2 slag
Original (before experiment)40.4459.030.290.12
Rheometer (after experiment)39.6959.240.640.18
Oscillating viscometer (after experiment)39.8558.740.780.17

The chemical composition of the CaO–Al2O3 and CaO–SiO2 slag systems are stable, while for the industrial blast furnace slag an insignificant change of the chemical composition is observed. It can be concluded that the small differences of the chemical composition within experiments should not considerably affect the estimated viscosity.

iii) The difference between the experimental data or high temperature errors can mainly be explained by the different temperature fields in molybdenum crucibles and an uncertainty in estimation of the density functions for the vibrating finger viscometer. It has been turned out that the other factors have a minor influence on the estimated values. The applied thermocouples were calibrated using high purity copper and nickel references and the deviations from the real temperature were taken into account. In addition, the density functions were determined using the maximal bubble pressure method which is a fundamental method for the precise determination of thermophysical properties.

iv) Starting of Taylor vortex can influence the accuracy of results of the rotating bob viscometer. For the given geometry of the system crucible-melt-rotating bob the the conditions Taylor vortex initiations were calculated. It was estimated that with the typical slag density of 2.70 g/cm3 and the viscosity of 100 mPa·s Taylor vortex can arise only at 745 rpm of the bob. The experiments were accomplished under 30 rpm, what is very far from necessary preconditions of vortex initiation.

Comparison of the experimental data of two types of viscometer reveals the following phenomenon: for three slags with SiO2-content the experimental viscosity values obtained by the vibrating finger viscometer are lower than the data obtained by Anton Paar MCR 301 rheometer, whereas for the CaO–Al2O3 system (without SiO2) the experimental viscosity values achieved by the vibrating finger viscometer are higher than for Anton Paar MCR 301 rheometer (Fig. 8). This phenomenon can be explained by the different measuring principle. Using the oscillating finger viscometer, the viscosity has been estimated using an oscillatory motion with constant peak-to-peak amplitude of 0.625 mm. Existence of SiO2 in the CaO–SiO2 system, in the blast furnace slag and in the glass leads to the formation of polymeric structure. In polymeric systems, the real viscosity of the complex dynamic viscosity is inversely proportional to the frequency of oscillations.

An alternative frequency-dependent viscosity, so-called dynamic viscosity, has been introduced by Ferry42) and can be calculated by Eq. (65):   

| η * |= ( η 1 2 + η 2 2 ) 1/2 = ( G 1 2 + G 2 2 ) 1/2 /ω. (5)

Where η*, η1 and η2 are the dynamic viscosity, real part of real viscosity and imaginary part of complex viscosity, respectively. G1 is the shear storage modulus, G2 is the shear loss modulus and ω is frequency.

As the proof of this theory can be mentioned the researches of Shin et al.43,44) which were mainly focused on the viscoelastic properties of calcium silicate-based mold fluxes at 1623 K and presents the required values for the storage modulus (elastic property) and the loss modulus (viscosity property) of the SiO2-contained slags.

Thus, it is important to emphasis that another source of discrepancy between the data obtained by two viscometers can be the elastic properties of slags that reduce the measured viscosity. That is, the three cases of CaO–SiO2 system, the blast furnace slag and the glass at high temperature may turn into viscoelastic liquids whose viscosities depend on the frequency.

6. Conclusion

In the current investigation, the viscosities of CaO–Al2O3, CaO–SiO2 and CaO–SiO2–Al2O3–MgO slags and reference glass are measured up to 1600°C using the two viscometers. The results are discussed in the context of high temperature calibration errors, e.g. deviations in geometry of the crucible, the slag stability, the fluid flow, the atmospheric conditions, as well as the differences in both measurement principles. The main results can be summarized as follows:

• The measured data of viscosity for the binary slag systems (CaO–Al2O3 and CaO–SiO2) and for the blast furnace slag were in good agreement with the literature data.

• Vibrating finger viscometer was successfully applied for the viscosities of up to 5 Pa·s.

• Measuring range of the vibrating finger viscometer essentially depends on the current in the field coil.

• Increasing the mass of bob or finger decreases the measuring range of the vibrating finger viscometer.

• Type of the bob or finger nose influences the accuracy of the viscosity data.

• Spherical or conical nose can be preferably applied for the measurements of slag viscosity using the vibrating finger viscometer.

• Changes of the geometrical parameters of the molybdenum crucibles at high temperature were negligible.

• Chemical composition of slags was stable and thus had a minor influence on the viscosity data.

• The relative measurement difference between the two types of viscometers at up to 1 Pa·s and up to 1400°C was less than 30%.

• Viscoelastic properties of the CaO–SiO2 system, the blast furnace slag and the glass at high temperature can be the main reason for the dissimilarities between the two groups of experimental data obtained by two different viscometers.

Acknowledgment

The authors acknowledge the funding of the German Research Foundation (DFG) within the Collaborative Research Centre CRC 799 (TRIP-Matrix-Composite) part project A2 steel design liquid phase.

References
 
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