Einstein-Roscoe Equation Research Articles

Influence of unburned pulverized coal on the viscosity of blast furnace slag

ABSTRACT In order to explore the influence of unburned pulverized coal in ironmaking on the viscosity of blast furnace slag, the impact of temperature, volume fraction of solid particles and the rotating speed on the viscosity of CaO–MgO–Al2O3–SiO2 heterogeneous slag were studied by experiment. The calculation model of relative viscosity of heterogeneous slag containing unburned pulverized coal is corrected. The results showed that the viscosity of heterogeneous slag increased with the increase of the volume fraction of the solid particles, and decreased with the increase of the rotating speed. For the heterogeneous slag with the same composition, the relative viscosity is not affected by the temperature and decreases with the increase in the rotating speed. The impact of unburned pulverized coal solid particles on the viscosity of the slag was much larger than the predicted value of the Einstein–Roscoe equation.

Viscosity of Slag Suspensions with a Polar Liquid Matrix

Under the operating temperatures employed in steelmaking, most slags and fluxes often contain solids, such as undissolved CaO and its reaction products; thus, they are more viscous than their fully liquid states. However, few studies have considered the dielectric interactions of solid particles with the liquid matrix in such systems. In the present study, the viscosity of suspensions of dispersed particles consisting of polyethylene beads in a matrix of silicone oil or aqueous glycerol at room temperature was measured. Then, empirical models for estimating the viscosity based on the Einstein–Roscoe equation were proposed. Furthermore, the viscosity of suspensions of CaO and MgO particles dispersed in a matrix of CaO–Al2O3–SiO2–MgO slag at 1773 K was measured, and the feasibility of the proposed viscosity equations was investigated. As expected, the viscosities of the suspensions of polyethylene beads dispersed in silicone oil and glycerol increased with an increasing bead volume fraction. Under comparable measurement conditions, the viscosities of the glycerol suspensions were higher than those of the silicone oil suspensions. The proposed viscosity models based on the Einstein–Roscoe equation and the capillary number reproduced the viscosity of the silicone oil suspensions but underestimated that of the glycerol suspensions. The trend of increasing viscosity of the molten slag suspensions with dispersed CaO and MgO particles was similar to that of the room-temperature suspensions, exhibiting Bingham non-Newtonian behavior. The viscosity model composed with the results from the glycerol aqueous suspensions underestimated the slag viscosity, which can be attributed to the repulsive forces in the high-polarity liquid matrix.

Effect of MnO on the fusing temperature and the viscosity of high titanium slag

The fusing temperature and the viscosity are important physical properties of high titanium slag. The Einstein-Roscoe equation was used to study the thermodynamic simulation of high titanium slag via FactSage software. The effects that the MnO had on the fusing temperature and the viscosity of high titanium slag were studied, and the reliability of the thermodynamic simulation method is verified by experiments. The results showed that the fusing temperature and the viscosity of high titanium slag decreased when the MnO content increased. The fusing temperature of high titanium slag was 1626.17∼1691.24 °C. After the high titanium slag was completely melted, the viscosity of high titanium slag was 86∼127 mPa · s. When the temperature was lower than the fusing temperature, the high titanium slag precipitated into solid-phase particles. The amount of precipitation increased when the temperature decreased, which in turn increased the viscosity of the high titanium slag.

Effect of slag composition on the cleanliness of drill rod steel

ABSTRACTIn order to obtain the 55SiMnMo drill rod steel with a high cleanliness, the slag refining has been simulated by laboratory experiments. More desired spherical-shaped complex inclusions with an average diameter of about 2.7 μm, total oxygen of 4 ppm and Mg of 10 ppm after refining were obtained with initial slag basicity of 2.1 and Al2O3 15 wt-%. The relationship between the slag composition and the melting temperature and viscosities of slag was achieved based on a calculation by Factsage Software and Einstein–Roscoe Equation. The refractory–slag–metal–inclusion multiphase reactions were investigated from the viewpoint of thermodynamics and kinetics by the estimation of viscosities, MgO solubility, Al2O3 activity in slag and sulphur capacity of slags. It is experimentally confirmed that the corrosion of MgO crucible by slag was affected by the MgO solubility and viscosity of slag. The factors facilitating to obtain low oxygen and control sulphur content were also analysed. Finally, the composition transformation of inclusions during slag refining and cooling process was discussed based on thermodynamic calculation.

Rheological Evolution of Ti-Bearing Slag with Different Volume Fractions of TiN

The effect of the volume fraction of TiN (0 to 4.27 vol pct) on the rheological behaviors of Ti-bearing slag, including the viscosity and the constitutive equation, was investigated using a Brookfield rheometer at 1773 K. With the increasing TiN, the viscosity and the shear stress increased substantially, and the slag is converted from Newtonian to non-Newtonian. When the slag samples were dispersed with TiN (1.00 to 4.27 vol pct), the measured viscosities were greater than those estimated by the Einstein–Roscoe equation, and the complex internal structure was suggested to be responsible for the extra enhancement of the viscosity beyond that caused by TiN itself. The viscosity curves of the slag samples containing TiN (1.00 to 4.27 vol pct) were divided into three regions depending on the critical shear rate and the initial shear rate. In the Newtonian region, the apparent volume fraction of TiN was defined and used to develop a semiempirical model based on the Einstein–Roscoe equation to describe the viscosity. In the two non-Newtonian regions, the shear rate, the yield stress, and the viscosity differences between the non-Newtonians regions and the Newtonian regions were applied in developing the viscosity models.

Influence of TiC0.3N0.7 on viscosity of TiO2-containing slag

The influence of TiC0.3N0.7 on viscosities of CaO-SiO2-MgO-Al2O3-TiO2 slags was investigated by the rotating cylinder method in this work. From the viscosity experimental results, it was found that the viscosity of the two-phase suspension system increased with increasing the volume fraction of TiC0.3N0.7 particle and decreased with increasing the rotation speed. The viscosity increased with the temperature decreasing and the relationship between viscosity and temperature could be described by the Arrhenius equation. However, temperature has little influence on the relative viscosity which is the ratio of solid–liquid mixture viscosity to pure liquid viscosity. The modified Einstein–Roscoe equation based on the present experimental values could well estimate the viscosity of TiC0.3N0.7 containing melts. The apparent volume of TiC0.3N0.7 was calculated to be 5.0–6.0 times of its real volume.

Thermodynamic Simulation of Viscosity of TiO2-Ti2O3-CaO Ternary Slag

The viscosity of high titanium slag at high temperature is one of the key factors of slag-iron separation. Based on the Einstein-Roscoe equation, thermodynamic simulation of viscosity of TiO2-Ti2O3-CaO ternary slag is studied by using FactSage® software, and the effects of temperature, CaO content and solid-phase particles on the viscosity of slag were studied. The results show that the increase of CaO content has the effect of reducing melting temperature and viscosity of TiO2-Ti2O3-CaO ternary slag. After the TiO2-Ti2O3-CaO ternary slag is completely melted, the increase of temperature has little effect on viscosity of slag, and the viscosity is about 110~125mPa·s. When the temperature is lower than melting temperature, TiO2-Ti2O3-CaO ternary slag will precipitate solid-phase particles, and the precipitation process is carried out in stages, and with the decrease of temperature, the precipitation will increase and the viscosity will sharply increase. TiO2-Ti2O3-CaO ternary titanium slag has obvious characteristics of short slag.

Pahoehoe to `a`a transition of Hawaiian lavas: an experimental study

Basaltic lavas collected at the Muliwai a Pele lava channel, built during 1974 as part of Mauna Ulu’s eruption on Kilauea’s east rift zone, have been studied to investigate the effect of cooling and crystallization on the rheological properties of the lava. We have quantified the viscosity-strain-rate dependence of lava during cooling and crystallization, using concentric cylinder viscometry. We measured the viscosity of the crystal-free liquid between 1600 and 1230 °C, where we observed a deviation from the expected viscosity trend, marking the liquidus. We then made rheology measurements at subliquidus temperatures of 1207, 1203, 1183, 1176, and 1169 °C, varying the applied strain rates at each temperature. While the crystal-free liquid behaved as a Newtonian fluid, crystallization changed the rheological response to pseudo-plastic behavior, even at the lowest crystal volume fraction of 0.025. Pseudo-plastic behavior was observed down to a temperature of 1183 °C, with a crystal fraction of 0.15. Between 1183 and 1176 °C, the two-phase suspension transitioned from a power-law fluid to a Herschel-Bulkley fluid. At temperatures of 1176 and 1169 °C, with crystal fractions of 0.33 and 0.42, respectively, we observed lobate surface textures on the experimental samples, which remained preserved until the end of the experiments. Measurements at these temperatures indicated yield strengths of 82 ± 16 and 238 ± 18 Pa, respectively. The yield strength resulted from the development of an interconnected crystal network of diopside and enstatite by 1176 °C. By 1169 °C, diopside and plagioclase microcrystals had also appeared, and the effective viscosity was between 2000 and 5000 Pa s, depending on the strain rate. Further cooling to 1164 °C resulted in a rapid viscosity increase, to an effective viscosity in excess of 105 Pa s that exceeded the measurement range of our apparatus. The yield strength varies with crystallinity in an exponential fashion, with yield strength in Pa given by σ y = 1.25e12.93Φ c, where Φ c is the crystal volume fraction. The physical effect of crystals on the relative viscosity of magma was assessed by removing the effects of changing residual liquid viscosity, due to changing composition and temperature. To do this, we analyzed, synthesized, and measured the most evolved residual liquid from the subliquidus experiments. The effect of crystals was best captured by the Einstein-Roscoe equation for polydisperse spherical inclusions. We also measured the viscosity of the same crystal-liquid mixtures at low temperatures and strain rates using parallel-plate viscometry. The effect of crystals on magma viscosity was slightly greater at low strain rates, in agreement with theoretical models, although no single model reproduced these results well. In our experiments, the transition from pahoehoe to `a`a occurred between 1200 and 1170 °C, at viscosities between 100 and 1000 Pa s, depending on strain rate.

Influence of alumina on physical properties of an industrial zinc-copper smelting slagPart 1 – viscosity

The rotating cylinder method was applied to measure the viscosities of an industrial iron silicate slag and mixtures of this slag with 5, 10 and 15 wt-% alumina addition, in temperature range 1100–1300°C. The measured viscosities were compared with the predicted values using two of the commercially available software products for viscosity calculations, namely Thermoslag®1·5 and FactSageTM6·2. As the models can only predict viscosities for a solid free melt, obtained values by FactSageTM6·2 were modified using the Einstein–Roscoe equation. Results show that aluminium behaves as a network former cation in this type of slag, and by increasing the alumina concentration, the melt becomes progressively polymerised. Consequently, the viscosity of the slag increases at a given temperature, which is supported by thermodynamic predictions. According to the modified FactSageTM6·2 calculations, the viscosity of the solid containing slag increases from 2·1 to 5·5 poise at the industrial operating temperature (∼1250°C).

Measurement and simulation of flow properties of coal ash slag in coal gasification

AbstractThe viscosities of 45 coal ash slag samples at high temperature have been measured under different temperatures and shear rates. The computer thermodynamic software package FactSage has been used to predict liquidus temperatures, volume fractions of crystallized solid particles (ϕ), and the compositions of remaining liquid phase for 45 coal ash slag samples. The flow properties of completely liquid and partly crystallized coal ash slag samples have been predicted by three viscosity models. The Urbain formalism has been modified to describe the viscosities of fully liquid slag and homogeneous remaining liquid phase in coal ash slag samples. The modified Einstein equation and Einstein–Roscoe equation have been used to describe the viscosities of heterogeneous coal ash slag samples of ϕ < 10.00 vol % and ϕ ≥ 10.00 vol %, respectively. These three models provided a good description of the experimental data of fully liquid and heterogeneous coal ash slag samples. The new models also predicted flow properties of mixtures of coal ash slags with CaO, Fe2O3, MgO, SiO2, and Al2O3. © 2010 American Institute of Chemical Engineers AIChE J, 2011

Experimental investigation of magma rheology at 300 MPa: From pure hydrous melt to 76 vol.% of crystals

The rheological behaviour of synthetic crystal-bearing magmas containing up to 76 vol.% of crystals (0 ≤ ϕ S ≤ 0.76) has been investigated experimentally at a confining pressure of 300 MPa and temperatures between 475 and 1000 °C at shear rates between 10 − 4 and 2 × 10 − 3 s − 1 . Starting hydrated crystal-bearing glasses were synthesized from a dry haplogranitic glass (Qz 36Ab 39Or 29) and 2.5 wt.% water mixed with 0 (pure hydrous melt), 16, 34, 54, 65 or 76 vol.% of Al 2O 3 sieved (45 < Ø < 90 µm) crystals. Shear viscosity measurements were performed in torsion (simple shear) in a Paterson gas-medium apparatus. For pure hydrated melt and for 16 vol.% of crystals, the rheology is found to be Newtonian. At higher crystal contents, the magmas exhibit shear thinning behaviour (pseudoplastic). The Einstein–Roscoe equation adequately estimates viscosities of the crystal-bearing magmas at low crystal contents ( ϕ S ≤ ~ 0.25), but progressively deviates from the measured viscosities with increasing crystal content as the rheological behaviour becomes non-Newtonian. On the basis of a power–law formulation, we propose the following expression to calculate the viscosity as a function of temperature, crystal content and applied stress (or shear rate): γ . = A 0 ( 1 − Φ / Φ m ) K τ ( 1 + K 1 Φ K 2 ) exp ( − Q R T ) , where γ. is shear rate (s − 1 ), τ is shear stress (MPa), Φ is the crystal volume fraction, T is temperature (K), Φ m is the relative maximum packing density, R is the gas constant, Q = 231 kJ mol − 1 is the activation energy of the viscous flow and A 0, K, K 1 and K 2 are empirical parameters.

Non-Newtonian rheological law for highly crystalline dome lavas

Volcanic eruption models are hampered by the lack of multiphase magmatic flow laws. Most rheological models estimate the viscosity of multiphase lavas via the Einstein-Roscoe equation, but this simplification cannot be used for high crystallinity and it does not consider the non-Newtonian strain-rate dependence of viscosity. We carried out parallel plate experiments on natural samples to simulate multiphase lava deformation under various stresses and strain rates. Multiphase lavas exhibit an important component of shear thinning, and appear to invalidate the adequacy of Einstein-Roscoe–based formulations for highly crystalline lava rheology. The remarkable singular dependence of viscosity (η) on strain rate (γ) yields a novel universal rheology law at eruptive temperatures ( T ), i.e., log η = −0.993 + 8974/ T −0.543·log γ Our work reveals the importance of considering microcracking and viscous dissipation at very high strain rate (>10 −3 s −1 ), explaining the occurrence of seismic swarms along the conduit margins, and consequently supporting plug-like magma ascent models.

Viscosity of Highly Basic Slags

The viscous behavior of CaO–SiO2–FeO–MgO slags saturated with dicalcium silicate (2CaO·SiO2) with 8 mass% MgO at 1873 K, and CaO–SiO2–FeO–Al2O3–MgO slags are measured under the conditions of highly basic slag compositions with regard to the fluidity of BOF slag and slag coating technology. The experiment shows that slag viscosities increase with decreasing temperature for almost all the slags that have been investigated over a range of temperature range, and the slag viscosity has exhibited different behavior in the temperature range lower and higher than 1773 K. The viscosity of the present slag systems has decreased with increasing FeO content. The viscosities depended on Al2O3 and FeO contents more strongly than on MgO content. The slag basicity has a more significant effect on increasing slag viscosity. A thermodynamic approach for the constants of the Einstein–Roscoe equation has resulted in a reasonable relationship between the relative viscosity and the volume fractions of solid phases estimated from the slag compositions. Thus, the present study confirms that the viscosities of highly basic slags can be estimated in terms of the slag compositions by predicting the fraction of solid phases in the slags for the stable BOF operation and the reliable coating of vessel lining.

Viscosity measurements of subliquidus magmas: Alkali olivine basalt from the Higashi-Matsuura district, Southwest Japan

We carried out viscosity measurements and sampling of a crystal suspension derived from alkali olivine basalt from the Matsuura district, SW Japan, at subliquidus temperatures from 1230 °C to 1140 °C under 1 atm with NNO oxygen buffered conditions. Viscosity increased from 31 to 1235 Pa s with a decrease in temperature from 1230 to 1140 °C. On cooling, olivine first appeared at 1210 °C, followed by plagioclase at 1170 °C. The crystal content of the sample attained 31 vol.% at 1140 °C (plagioclase 22%, olivine 9%). Non-Newtonian behaviors, including thixotropy and shear thinning, were pronounced in the presence of tabular plagioclase crystals. The cause of such behavior is discussed in relation to shear-induced changes in melt–crystal textures. Relative viscosities, η r (= η s / η m, where η s and η m are the viscosities of the suspension and the melt, respectively), were obtained by calculating melt viscosities from the melt composition and temperature at 1 atm using the equation proposed by Giordano and Dingwell [Giordano, D., Dingwell, D.B., 2003. Non-Arrhenian multicomponent melt viscosity: a model. Earth and Planetary Science Letters, 208, 337–349.]. The obtained relative viscosities are generally consistent with the Einstein–Roscoe relation, which represents η r for suspensions that contain equant and equigranular crystals, even though the crystal suspension analyzed in the present experiments contained tabular plagioclase and granular olivine of various grain sizes. This consistency is attributed to the fact that the effect of crystal shape was counterbalanced by the effect of the dispersion of crystal size. The applicability of the Einstein–Roscoe equation with respect to crystal shape is discussed on the basis of the present experimental results. Our experiments and those of Sato [Sato, H., 2005. Viscosity measurement of subliquidus magmas: 1707 basalt of Fuji volcano. Journal of Mineralogical and Petrological Sciences, 100, 133–142.] show that the relationship between relative viscosity and crystal fraction is consistent with the Einstein–Roscoe relationship for axial ratios that are smaller than the critical value of 4–6.5, but discrepancies occur for higher ratios.

Viscosity measurement of subliquidus magmas: 1707 basalt of Fuji volcano

We have installed a new furnace equipped with a rotational viscometer for use in viscometry at high tempera- ture (one atmosphere) under controlled oxygen fugacities. This furnace allows sampling at the time of the vis- cosity measurement. Utilizing this furnace, we examined the relationship between textural parameters (crystal- linity and the apparent aspect ratio of crystals) and viscosity during cooling of a basaltic magma. The 1707 basalt from Fuji volcano was used for the experiment and the result shows a drastic increase in viscosity from the liquidus temperature (1230°C; η = 52 Pa s), through 1170°C (crystallinity = 11%; η = 350 Pa s), to 1130°C (crystallinity = 23%; η = 1950 Pa s). The increase in viscosity during cooling is affected by both the increase of the crystallinity and the change of melt composition as a result of crystallization. For the basalt from Fuji vol- cano, crystallization of thin tabular plagioclase enhanced the strong effect of crystal entanglement, whereas the changes of melt compositions did not affect the viscosity because of the increase of ferromagnesian compo- nents and the decrease of feldspathic component in the residual liquid during crystallization. The effect of the presence of crystals on the viscosity is much larger than that expected from the Einstein-Roscoe equation with Marsh's constant (η = η 0 (1 − ϕ /0.6) -2.5 ), where ϕ denotes the crystal fraction. The additional correction of Simha on the solid shape parameter slightly overestimates the increase in relative viscosity in the presence of thin platy crystals. A shear rate dependence of the viscosity was observed at lower temperature crystal-rich con- ditions.

Granite rheology: magma flow and melt migration

Granite magma rheology has long been studied by rock physicists, petrologists and experimentalists. New data on viscosity of dry and H 2 O-bearing melts and glasses in a wide range of temperatures and H 2 O content changed drastically the general view on the Arrhenian behaviour of granitic melts. Recent studies on the deformation of partially molten granite rocks at up to 50 vol% of melting show the failure of the RCMP (rheologically critical melt percentage) concept for granites. We present new direct measurements of melt migration from a combination of centrifuge experiments, falling sphere viscometry of partially molten granites assisted by a centrifugal field and viscoelastic behaviour of partially molten granites in torsion oscillation apparatus. The results suggest that the extrapolation of the Einstein–Roscoe equation for the viscosity of partially molten rocks does not give adequate results for granites with 50 vol% of melt phase. Rheology of granite magmas is very sensitive to chemical composition, and viscosity of the melt phase. Adding H 2 O to partially molten granites results in a decrease in the viscosity of the melt phase, as well as dissolution of quartz crystals. In turn, this produces more felsic and viscous melt. Thus, these factors smear out the contrast between partially and completely molten granites, and between anhydrous and H 2 O-bearing magmas at moderate H 2 O contents (1 to 2 wt%).