Viscosity of Slag Suspensions with a Polar Liquid Matrix

Abstract

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.

1. Introduction

In the manufacturing of steel, the viscosity of the slag phase is one of the most important parameters for understanding and optimizing blast furnace operation and refining processes since it has a direct and critical influence on the fluid flow of slag-matrixed melts and the kinetics of slag-metal reactions. Slag viscosity is well known to be extremely sensitive to temperature, chemical composition, and the presence of secondary and dispersed phases, such as solids, gases, and other liquids. Therefore, the viscosity and related parameters, like the mass-transfer coefficient,¹⁾ diffusion constant,²⁾ etc., of slag-matrixed melts have been employed in various mathematical simulations to comprehend, control, and improve iron- and steel-making processes. Although the viscosity of molten slag has been thoroughly measured and documented,³⁾ most experiments have evaluated temperature ranges well above the liquidus temperature, where the slag is a uniform liquid. Additionally, nearly all mathematical models proposed for the viscosity of molten slag in the area of steelmaking were intended for uniformly molten slag, including the well-known Urbain,⁴⁾ Riboud,⁵⁾ Iida,⁶⁾ and Nakamoto⁷⁾ models. However, at actual operating temperatures, most slags and fluxes are not in a fully liquid state in steelmaking processes, such as hot-metal pretreatment, BOF, EAF, LF, continuous casting, and so on. They often contain solids, such as undissolved CaO, attributed to excess added refining agent, its reaction products such as 3CaO·P₂O₅, 2CaO·SiO₂, and CaS, and cuspidine in super-cooled and partially crystallized mold flux. These slags and fluxes become even more viscous than their fully liquid states, simply due to the presence of the dispersed solid phase, which can cause operational problems.

Numerous attempts have been made to investigate the effect of dispersed solid phases on the apparent viscosity of metallurgical slags. Wright et al. investigated the effect of different sizes of spinel (MgAl₂O₄) particles on the viscosity of CaO–MgO–Al₂O₃–SiO₂ slag at 1646 K and reported that the viscosity of the slag suspension increased with the addition of spinel particles; with more than 10 vol% solid particles, the slag suspension exhibited non-Newtonian behavior, namely Bingham fluid properties.⁸⁾ Wu et al. measured the apparent viscosity of silicone oil–paraffin suspensions at room temperature and studied the effect of 2CaO·SiO₂ and MgO particles on the viscosity of CaO–Al₂O₃–SiO₂–MgO slags. They found that when the volume fraction of solid particles was lower than 0.1, the suspensions still behaved as Newtonian fluids, whereas at fractions of 0.15 and higher, the suspensions showed non-Newtonian behavior.⁹⁾ Some of the authors similarly investigated the viscosity of silicone oil–polyethylene bead suspensions at room temperature^10,11) and CaO–SiO₂–R₂O (R=Li, Na, or K) slag suspensions at high temperatures,¹²⁾ revealing that the suspensions behaved as Bingham fluids when the volume fraction of the solid phase was ~30 vol% or higher. These differences in the fraction of the dispersed solid phase, at which the suspension transitions from a Newtonian to a non-Newtonian fluid, can be attributed to the size, shape, and interaction of the suspended particles with the liquid-phase matrix. In addition to experimental research, Einstein initiated theoretical investigations into the viscosity of dilute suspensions (~2 vol% solid),¹³⁾ and many subsequent studies proposed theoretical and/or empirical equations for the viscosity of suspensions.^{14,15,16,17, 18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)} Liu et al. recently reviewed these experimental and theoretical approaches to better understand the suspension viscosity of molten silicate systems and concluded with opportunities and suggestions for future work.^46,47)

One of the gaps in previous studies on the viscous characteristics of metallurgical slag suspensions is the interaction of the dispersed solid phase with the liquid matrix. Molten slag and flux are composed of various ions, including complex anions of network formers (SiO₂, Al₂O₃, P₂O₅, B₂O₃, etc.) that contain bridging and non-bridging oxygen, network modifier and charge-compensating cations, and free oxygen. The presence of these cations and anions contributes to the ionic, interfacial, and orientational polarization of the molten slag matrix, which leads to a high electrical permittivity and capacitance.⁴⁸⁾ Thus, molten slag is a highly polar liquid. Dispersed solid particles in the molten slag matrix, CaO, 3CaO·P₂O₅, 2CaO·SiO₂, CaS, cuspidine, etc., are basically ionic compounds of cations with a lower electronegativity and chalcogens or halogens with a higher electronegativity, and thus, they are also polar. Therefore, the molten slag matrix and dispersed solid particles will have strong interactions in a metallurgical slag suspension, similar to the interactions arising from the zeta potential in aqueous suspension systems. These interactions lead to repulsion between electrical double layers and can affect the viscoelastic behavior and fluid characteristics of the suspension. For instance, an electrostatically stabilized aqueous suspension of SiO₂ particles, which have a low zeta potential, has a fairly low viscosity and shows Newtonian behavior, whereas an unstable SiO₂ suspension, with a higher zeta potential, has a much higher viscosity and exhibits a high degree of shear thinning.⁴⁹⁾ However, to the best of the authors’ knowledge, there have been few experimental studies on the effect of a polar liquid matrix on the viscosity of slag suspensions in the field of pyrometallurgy.

In the present study, as a basis for evaluating the viscosity of slag suspensions with a polar liquid matrix, we measured the apparent viscosity of dispersed suspensions of spherical particles consisting of polyethylene beads in a matrix of silicone oil or aqueous glycerol at room temperature. The effects of the bead volume fraction, bead size, shear rate, and viscosity of the liquid matrix on the apparent viscosity were systematically evaluated for both systems. Then, empirical equations based on the Einstein–Roscoe equation¹⁷⁾ were proposed to accurately reproduce the viscosity of the slag suspensions on the basis of the obtained experimental values. Furthermore, we evaluated the apparent viscosity of suspensions of CaO and MgO particles dispersed in a matrix of molten CaO–Al₂O₃–SiO₂–MgO slag at 1773 K to explore the feasibility of the proposed model equations.

2. Experimental

2.1. Viscosity Measurement of Suspensions at Room Temperature

Figure 1 shows a schematic of the employed viscosity measurement apparatus, which consists of a rotational viscometer (DVII+ or DV2T, AMETEK Brookfield) and a 300-ml beaker filled with the suspension. The apparent viscosity was systematically measured for suspensions with different bead volume fractions and mean diameters, shear rates, and liquid matrix viscosities. The shear rate was calculated from the rotational speed and the dimensions of the beaker (inner diameter: 73 mm) and spindle (outer diameter: 3.2 mm) per the following equation:   

$$\gamma =\frac{2\omega }{1-{(r_i/r_o)}^2}$$(1)

where γ, ω, r_o, and r_i are the shear rate, angular rate, inner radius of the outer cylinder (beaker), and radius of the inner cylinder (spindle), respectively. The apparent viscosity of the suspension was calculated from the torque generated on the spindle by rotation. The scatter of the apparent viscosity values was approximately 10%. Further details on the viscometer have been reported elsewhere.^10,11)

Fig. 1.

Schematic of viscosity measuring apparatus employed for polyethylene beads dispersed silicone oil and glycerol aqueous solution suspensions at room temperature. (Online version in color.)

2.2. Suspensions with a Low-polarity Liquid Matrix

Silicone oils (KF-96, Shin-Etsu Chemical) with viscosities of 0.5, 1.0, 2.0, and 3.0 Pa·s at 24°C were employed as low-polarity liquid matrices, which have relative electrical permittivities of 2.7–2.8,⁵⁰⁾ depending on the viscosity. Polyethylene beads (LE-1080, Sumitomo Seika Chemicals) with mean diameters of 9.35, 162.5, 340.0, and 602.5 μm were selected as dispersed solid particles, which were confirmed to exhibit good sphericity and small particle distributions by scanning electron microscopy (SEM) imaging, as shown in Fig. 2. Table 1 summarizes the experimental conditions for the viscosity evaluation of the silicone oil–polyethylene bead suspensions at room temperature.

Fig. 2.

SEM microimages of polyethylene beads employed for viscosity measurements of silicone oil and glycerol aqueous solution suspensions at room temperature.

Table 1. Experimental conditions for the viscosity measurement of polyethylene beads dispersed silicone oil suspension at room temperature.

Mean diameter of beads	μm	9.35, 162.5, 340.0, or 602.5
Volume fraction of beads	vol%	0, 15, 30, 45, or 60
Viscosity of silicone oil	Pa·s	0.5, 1.0, 2.0, or 3.0
Rotating speed of spindle	rpm	10, 20, 50, 60, or 100
Shear rate	s−1	2.09, 4.18, 10.5, 12.6, or 20.9

2.3. Suspensions with a High-polarity Liquid Matrix

Glycerol (FUJIFILM Wako Pure Chemical) aqueous solutions with viscosities of 0.06, 0.22, 0.52, and 1.41 Pa·s at 20°C (80, 90, 95, and 100 mass% glycerol, respectively) were employed as high-polarity liquid matrices. The relative electrical permittivities of the aqueous glycerol solutions ranged from 41.1 to 52.3⁵¹⁾ at 20°C, depending on the concentration of water, which were roughly 15–19 times higher than those of the silicone oils. The same sizes of polyethylene beads as those used for the low-polarity matrix were employed as the dispersed solid particles. Table 2 shows the experimental conditions for the viscosity evaluation of the aqueous glycerol–polyethylene bead suspensions at room temperature. Additionally, the zeta potential of a dispersed suspension of polyethylene beads was measured to estimate the strength of the double-layer repulsion in the aqueous glycerol matrix. The zeta potential was measured by dynamic light scattering (Zetasizer Nano ZSP, Malvern Panalytical) for a dilute suspension of 9.35-μm polyethylene beads (~1 vol%) in 60 mass% aqueous glycerol (0.01 Pa·s at 20°C) as a function of pH, which was controlled by aqueous HCl or NaOH. The volume fraction of the beads and the concentration of glycerol were inconsistent with those of the viscosity measurements due to limitations on the measurable ranges. The zeta potential of the silicone oil-based suspension could not be determined because of the highly insulating property of silicone oil.

Table 2. Experimental conditions for the viscosity measurement of polyethylene beads dispersed glycerol aqueous solution suspension at room temperature.

Mean diameter of beads	μm	9.35, 162.5, 340.0, or 602.5
Volume fraction of beads	vol%	0, 15, 30, or 45
Viscosity of glycerol aqueous solution	Pa·s	0.0601, 0.219, 0.523, or 1.41
Rotating speed of spindle	rpm	10, 20, 50, 60, or 100
Shear rate	s−1	2.09, 4.18, 10.5, 12.6, or 20.9

2.4. Viscosity Measurement of CaO and MgO Particle Slag Suspensions at a High Temperature

Suspensions of CaO and MgO particles dispersed in slag were prepared by adding calcined CaO and MgO powder to CaO and MgO saturated slags, respectively. According to the phase diagram, the chemical composition of the quaternary system 53CaO-35Al₂O₃-3SiO₂-8MgO (mass%) has the eutectic temperature of lime (CaO) and periclase (MgO) at 1773 K, which suggests that CaO and MgO are not chemically soluble at the selected composition and at 1773 K. Reagent-grade CaCO₃, Al₂O₃, SiO₂, and MgO (Sigma-Aldrich Japan) powders were precisely weighed to form the given composition and thoroughly mixed in an alumina mortar. The powder batch was pre-melted in a resistance furnace using a Pt crucible at 1873 K for one hour under air, and the melt was quenched onto a water-cooled copper plate. The CaO and MgO particles for dispersion were prepared by calcination of the respective reagent powders at 1473 K for 30 min. The calcined and sieved particles of CaO and MgO were mixed with the pre-melted 53CaO-35Al₂O₃-3SiO₂-8MgO (mass%) slag before viscosity measurements to form the given compositions. Viscosity measurements were performed using the rotating crucible viscometer apparatus described in section 2.1. A Pt-20mass%Rh crucible filled with the mixed slag and CaO or MgO particles was placed in a crucible supporter in the furnace and heated to 1773 K. Then, the viscosity of the dispersed CaO or MgO slag suspension was determined at the same temperature. Further details on the viscometer have been reported elsewhere.^52,53) Table 3 shows the experimental conditions for the viscosity evaluation of the CaO and MgO particle–molten slag suspensions at a high temperature. Additionally, polished sections of the quenched slag suspensions after the viscosity measurements were observed by SEM (VE-8800, Keyence) at an accelerating voltage of 20 kV. The mean diameters of at least one hundred of the dispersed CaO and MgO particles for each experimental condition were determined using Image-J.

Table 3. Experimental conditions for the viscosity measurement of CaO or MgO particle dispersed 53CaO-35Al₂O₃-3SiO₂-8MgO (mass%) molten slag suspension at 1773 K.

Mean diameter of CaO	μm	30.4 for 10 vol%
		29.8 for 20 vol%
		28.1 for 30 vol%
Mean diameter of MgO	μm	21.5 for 10 vol%
		19.5 for 20 vol%
		19.7 for 30 vol%
Volume fraction of CaO or MgO	vol%	0, 10, 20, or 30
Viscosity of molten slag	Pa·s	0.235
Rotating speed of crucible	rpm	20, 30, or 40
Shear rate	s−1	4.71, 7.07, or 9.42

3. Results and Discussion

3.1. Rheological Behavior and Empirical Modeling of Suspensions with a Low-polarity Liquid Matrix

Figure 3 illustrates the typical measured viscosities of polyethylene beads dispersed in silicone oil (0.5 Pa·s), where the data are presented as relative viscosity (apparent viscosity divided by the viscosity of the liquid phase) as a function of the bead volume fraction. The relative viscosities increased with the bead volume fraction, regardless of the bead size, and depended on the rotational speed of the spindle above a bead fraction of 30 vol%, where lower relative viscosities were measured at higher rotational speeds, indicating shear thinning effects. As shown in Fig. 3(a), for the 9.35-μm beads, the relative viscosity gradually increased with the bead volume fraction, reaching approximately 2.1 at 15 vol%. At and above a bead fraction of 30 vol%, the relative viscosity rapidly increased with increasing solid-phase content, reaching a maximum of more than 55 at a solid-phase volume fraction of 45% and a rotational speed of 10 rpm. In other words, a solid-phase content of 45% yielded a suspension that was roughly 55 times more viscous than its pure liquid phase at a low shear rate. Figure 3(d) shows the relative viscosity of the same silicone oil (0.5 Pa·s) containing larger dispersed polyethylene beads (602.5 μm) at various rotational speeds as a function of solid-phase content. Even though the beads were much larger, the relative viscosity increased in a similar manner to the trend shown in Fig. 3(a), reaching 1.5 at a solid-phase fraction of 15 vol%. At higher solid-phase contents (i.e., 15 vol% or more), a drastic increase in relative viscosity was observed, with a value of 17 at a solid-phase fraction of 60 vol%. This increase in relative viscosity was more modest than that observed for the smaller beads. Similar trends of increasing relative viscosities were observed in the other silicone oil (1.0, 2.0, and 3.0 Pa·s) suspensions. At high solid volume fractions, the relative viscosity was strongly dependent on the rotational speed of the spindle. In contrast, this dependence was negligible at low solid-phase contents. Furthermore, for a given amount of the solid phase, the relative viscosity was higher for lower rotational speeds, which is characteristic of non-Newtonian fluids.

Fig. 3.

Typical results for viscosity measurements of polyethylene beads dispersed silicone oil suspension with liquid phase viscosity of 0.5 Pa·s, polyethylene beads size: (a) 9.35 μm, (b) 162.5 μm, (c) 340.0 μm, and (d) 602.5 μm. Numbers shown in parenthesis are shear rates.

Figure 4 shows the dependence of shear stress on the shear rate for the results shown in Fig. 3. The shear stress was estimated by multiplying the apparent viscosity by the shear rate, which was calculated from the dimensions of the viscometer and rotational speed of the spindle, in accordance with Eq. (1). For a relatively low solid-phase content (~15 vol%), the estimated shear stress increased linearly with the shear rate and eventually crossed the origin for all bead sizes, which is characteristic for Newtonian fluid behavior. On the other hand, at higher solid phase contents (30 vol% or more), the shear stress increased with the shear rate and intercepted the vertical axis; this is called yield stress τ_B and is the typical behavior of a Bingham non-Newtonian fluid. Similar variations in the shear stress, depending on the shear rate, were observed in the other silicone oil (1.0, 2.0, and 3.0 Pa·s) suspensions. Wu et al. investigated the apparent viscosity of silicone oil–paraffin suspensions at room temperature and found that when the particle fraction was lower than 0.1 (10%), the suspensions still behaved as Newtonian fluids, whereas at fractions of 0.15 (15%) and higher, the suspensions showed non-Newtonian behavior.⁹⁾ This inconsistency in the solid-phase volume fraction at which the non-Newtonian transition occurs is attributable to the size and distribution of the dispersed solid phase, as paraffin particles with 450–2000 μm diameters were employed in their work.

Fig. 4.

Variations of shear stress calculated for polyethylene beads dispersed silicone oil suspension with liquid phase viscosity of 0.5 Pa·s as a function of shear rate, polyethylene beads size: (a) 9.35 μm, (b) 162.5 μm, (c) 340.0 μm, and (d) 602.5 μm.

Various equations have been proposed for the estimation of suspension viscosity over the last century. Among them, a viscosity equation for suspensions as a function of the volume fraction of the dispersed solid phase was first announced by Einstein in 1906, as shown in Eq. (2).¹³⁾   

$$\frac{\eta }{{\eta }_L}=1+2.5\varphi$$(2)

where η, η_L, and φ are the apparent viscosity of the suspension, viscosity of the liquid phase, and volume fraction of the dispersed solid phase, respectively. However, this equation is only applicable to spherical particles within a dilute dispersed system. It was nearly half a century later that the expansion to higher volume fractions of the secondary phase gained the attention of numerous researchers.^{14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45)} Some of the authors have utilized these well-known Vand,¹⁵⁾ Mooney,¹⁶⁾ Dalla Valle,¹⁸⁾ and Yagi–Ototake¹⁹⁾ models to reproduce the viscosity data of silicone oil suspensions, which revealed that the calculated viscosities were overestimated compared with the measured values. Of these models, the Einstein–Roscoe relationship shown in Eq. (3)¹⁷⁾ is considered to be one of the most applicable.   

$$\frac{\eta }{{\eta }_L}=\frac{1}{{(1-a\varphi )}^n}$$(3)

Here, the coefficient a expresses the aggregational state of the dispersed secondary phase in the fluid matrix and increases with the agglomeration degree; thus, a can be assumed as unity. The coefficient n can be assumed to be 2.5 for a system of dispersed spherical particles.^13,15)

Some authors have already reported¹¹⁾ an empirical model for suspensions with wide ranges of dispersed solid-phase particle sizes and volume fractions, shear rates, and liquid-phase viscosities based on a modified Einstein–Roscoe equation. However, we would like to reconsider the equation presented previously because n was expressed as a function of particle size and shear rate and was not dimensionless. In the present study, the Einstein–Roscoe equation was expanded to wider ranges of solid-phase sizes and volume fractions, shear rates, and liquid-phase viscosities. It was previously suggested by Lejeune⁵⁴⁾ that n is strongly associated with the electrical, thermal, and geometric properties of the dispersed secondary phase. In addition, a can still be assumed to be unity.^13,17) Therefore, f is the only independent variable in the Einstein–Roscoe equation, and the coefficient n can be defined as a function of other experimental variables, such as the liquid-phase viscosity, shear rate, and particle size. An empirical equation for the viscosity of a suspension under various conditions can, therefore, be developed. However, as shown in Eq. (3), n should be dimensionless. Consequently, dimensionless numbers that are related to secondary-phase dispersed liquid systems were employed. The capillary (Ca) and Reynolds (Re) numbers, which represent the relative effect of viscous drag forces versus surface tension forces acting across an interface and the ratio of inertial forces to viscous forces within a fluid, respectively, were calculated based on the experimental conditions employed in the present study, and their relationship to n was estimated from the apparent viscosity of the slag suspension. Ca and Re⁵⁵⁾ can be calculated from the following equations:   

$$Ca=\frac{{\eta }_{L}V}{\sigma }=\frac{{\eta }_L\gamma d}{\sigma }$$(4)

  

$$Re=\frac{VL}{\nu }=\frac{{\left(\frac{d}{2}\right)}^2\gamma \rho }{{\eta }_L}$$(5)

where η_L, V, σ, γ, d, L, ν, and ρ are the viscosity of the liquid phase, characteristic velocity, interfacial tension, shear rate, diameter of the dispersed-phase particles, characteristic length, kinetic viscosity of the liquid phase, and density of the liquid phase,⁵⁰⁾ respectively. Here, the shear rate was calculated using Eq. (1), and the interfacial tension between the solid and liquid phases was estimated from the surface tensions of polyethylene⁵⁶⁾ and silicone oil⁵⁰⁾ on the basis of Young’s equation.⁵⁷⁾ The contact angle of polyethylene against silicone oil was assumed to be zero because of the low surface tension of the latter.⁵⁰⁾ Therefore, Ca and Re were calculated from the experimental conditions employed in the present study, and their relationship with n was estimated from the apparent viscosity of the slag suspension.

Figure 5 shows the coefficient n of the Einstein–Roscoe equation calculated from the apparent viscosity of the suspension and the other experimental parameters as functions of Ca (a) and Re (b). The regression curves illustrated in Fig. 5 can be expressed as   

$$n_{silicone}=1.67C{a}^{-0.182},R^2=0.914$$(6)

  

$$n_{silicone}=0.948R{e}^{-0.098},R^2=0.774$$(7)

where R is the Pearson correlation coefficient. These equations indicate, that with a decrease in both Ca and Re, the relative viscosity of the suspension will exhibit similar exponential increases as a result of the steep decrease in n. However, according to the calculated correlation coefficients, Ca has a better correlation with n than that of Re, and thus, the relationship expressed by Eq. (6) was employed to reproduce the relative viscosity of polyethylene bead suspensions in a silicone oil matrix.

Fig. 5.

Coefficient n of the Einstein-Roscoe’s equation calculated from the apparent viscosities of polyethylene beads dispersed silicone oil suspension and the other experimental parameters as functions of (a) the Capillary number and (b) the Reynolds number estimated from the experimental conditions.

Figure 6 depicts the typical estimated relative viscosities of polyethylene beads suspended in a silicone oil matrix based on the Einstein–Roscoe equation and Eq. (6), calculated according to n and Ca from the experimental conditions. The estimated relative viscosities successfully reproduced the experimental values across the wide ranges of bead volume fractions and sizes, liquid-phase viscosities, and rotational speeds (shear rates). Here, the global delta Δ was introduced to express the relative difference between the estimated and experimental apparent viscosities:   

$$\mathrm{\Delta }=\frac{1}{N}{\sum }_{n=1}^N\left|\frac{{\eta }_n^{est}-{\eta }_n^{mea}}{{\eta }_n^{mea}}\right|\times 100(\%)$$(8)

where N, η^est, and η^mea are the number of samples, estimated apparent viscosity, and measured apparent viscosity, respectively. The global delta Δ calculated for the values estimated based on the Einstein–Roscoe equation and Eq. (6) with Ca was 7.77%. Thus, the combination of the equations proposed in the present study provides an accurate estimation of the apparent viscosity within the measurement scatter of 10%. Additionally, Δ calculated for the estimated values using the Einstein–Roscoe equation and Eq. (7) with Re was 14.0%; this lower accuracy is attributable to the lower correlation coefficient of Eq. (7).

Fig. 6.

Typical comparisons of the estimated relative viscosity values on the basis of the Einstein-Roscoe’s equation and Eq. (6) with the measured values of silicon oil suspension, (a) polyethylene beads size: 162.5 μm, viscosity of liquid phase: 0.5 Pa·s, (b) polyethylene beads size: 602.5 μm, viscosity of liquid phase: 1.0 Pa·s, (c) polyethylene beads size: 340.0 μm, viscosity of liquid phase: 2.0 Pa·s, and (d) polyethylene beads size: 9.35 μm, viscosity of liquid phase: 3.0 Pa·s.

3.2. Rheological Behavior of Suspensions with a High-polarity Liquid Matrix

Figure 7 illustrates the typical measured viscosities of polyethylene beads dispersed in aqueous glycerol (0.523 Pa·s), which are comparable to the dataset shown in Fig. 3 for 0.5 Pa·s silicone oil. The standard deviation of the viscosity data for the aqueous glycerol was found to be 8.7%, which was relatively smaller than that (18.6%) of the aqueous glycerol suspensions; thus, the variation in the viscosity of the aqueous glycerol was insignificant because of its intensive moisture-absorption characteristics. The relative viscosities increased with the bead volume fraction, regardless of the bead size, and were mostly dependent on the rotational speed of the spindle above a bead fraction of 30 vol%, which is called shear thinning. As shown in Fig. 3(a), for the 9.35-μm beads, the relative viscosity gradually increased with the bead volume fraction and reached approximately 4.7–7.7 at 15 vol%, depending on the rotational speed, which was higher than that of the silicone oil matrix. The relative viscosity rapidly increased with an increasing solid-phase content above 30 vol%, reaching a maximum of more than 45 at a solid-phase volume fraction of 30% and a rotational speed of 10 rpm, which was five times higher than that of the silicone oil matrix (9.4) under comparable experimental conditions, as shown in Fig. 3(a). Figure 7(d) shows the relative viscosity of aqueous glycerol (0.523 Pa·s) as a function of the solid-phase content of 602.5-μm beads. The relative viscosity increased similarly to the trend shown in Fig. 3(d), reaching 1.3–2.3 at a solid-phase fraction of 15 vol%, depending the rotational speed. At higher solid-phase contents of 30 vol% or more, the relative viscosity continued to increase, reaching 7.8 at a solid-phase fraction of 45 vol%, which was again higher than that of the low-polarity matrix of the silicone oil. The tendency of the relative viscosity to increase and the higher sensitivity to the increasing volume fraction of the solid phase were similarly observed in the other aqueous glycerol (0.06, 0.22, and 1.41 Pa·s) suspensions.

Fig. 7.

Typical results for viscosity measurements of polyethylene beads dispersed glycerol aqueous soulution suspension with liquid phase viscosity of 0.523 Pa·s, polyethylene beads size: (a) 9.35 μm, (b) 162.5 μm, (c) 340.0 μm, and (d) 602.5 μm. Numbers shown in parenthesis are shear rates.

Under comparable experimental conditions, the relative viscosities of the aqueous glycerol suspensions were largely higher than those of the silicon oil suspensions, which can be attributed to the difference in the interactions between the dispersed solid particles and liquid matrix, namely the repulsion between electrical double layers. Under a constant shear flow field, as employed by the rotating viscometer, this greater repulsive force can have a significant influence on the macroscopic fluid characteristics. The repulsive potential V_R between two particles in the liquid phase can be expressed as   

$$V_R=2\pi {\epsilon }_r{\epsilon }_0{\psi }_0^{2}ln\left\{1+exp(1-\kappa h)\right\}$$(9)

where ε_r, ε₀, ψ₀, κ, and h are the relative permittivity of the liquid phase, vacuum permittivity, interfacial potential, reciprocal thickness of the electrical double layer, and interparticle distance, respectively.⁵⁸⁾ In the present study, the variation in the viscosities of the three different liquid phases are discussed on the basis of the repulsive force derived from their contrasting dielectric polarizabilities. The higher-polarity aqueous glycerol exhibited relative permittivities of 41.1–52.3⁵¹⁾ at 20°C, depending on the concentration of water, which were roughly 15–19 times higher than those of the silicone oils.⁵⁰⁾ According to Eq. (9), the polyethylene particles in aqueous glycerol had a roughly 15–19 times higher repulsive potential than those dispersed in silicone oil, which is one possible reason why the glycerol suspensions were more viscous than the silicone oil suspensions. Table 4 summarizes the zeta potentials of the polyethylene beads dispersed in aqueous glycerol. The zeta potential was negative in the pH range from 2.44 to 10.94, varying from −72.9 to −44.0 mV. The pH of the as-prepared suspension was 8.54 and had the lowest-magnitude zeta potential of −44.0 mV, which resulted in a macroscopic viscosity increase in the glycerol matrix system, as shown in Fig. 7. As expressed by Eq. (9), with increasing interfacial potential ψ₀, which is closely related to the zeta potential, the repulsive potential of the dispersed particles quadratically increased, leading to an increase in the viscosity of the suspension. Unfortunately, the zeta potential of the silicone oil system was not measurable due to its insulating properties, i.e., extremely high resistivity. However, considering the smaller permittivity of silicone oil, the zeta potential of the silicone system was likely much smaller than that of the glycerol system.

Table 4. Summary of zeta-potential measurements for polyethylene beads dispersed glycerol aqueous solution suspension at room temperature.

pH	–	2.44	4.06	7.06	8.54	10.94
zeta potential	mV	−72.9	−66.5	−64.6	−44.0	−49.8

Figure 8 exhibits the dependence of the shear stress on the shear rate, corresponding to the results shown in Fig. 7. For a low solid-phase content (~15 vol%), the calculated shear stress increased linearly with the shear rate and eventually crossed the origin for all bead sizes. On the other hand, at higher solid-phase contents (30 and 45 vol%), the shear stress intercepted the vertical axis, which is the typical behavior of a Bingham non-Newtonian fluid. This dependence of the shear stress on the shear rate was similarly observed in the other aqueous glycerol (0.06, 0.22, and 1.41 Pa·s) suspensions as well as in the silicone oil systems, as shown in Fig. 4.

Fig. 8.

Variations of shear stress calculated for polyethylene beads dispersed glycerol aqueous solution suspension with liquid phase viscosity of 0.523 Pa·s as a function of shear rate, polyethylene beads size: (a) 9.35 μm, (b) 162.5 μm, (c) 340.0 μm, and (d) 602.5 μm.

Figure 9 illustrates the typical estimated relative viscosities of polyethylene beads suspended in an aqueous glycerol matrix based on the Einstein–Roscoe equation and Eq. (6), calculated with the measured data for the silicone oil system. To calculate Ca of the glycerol system, the interfacial tension between the liquid and solid phases was calculated from the surface tensions of polyethylene⁵⁶⁾ and aqueous glycerol,⁵⁹⁾ and the contact angle of polyethylene against glycerol was estimated from experimental data on the wettability of polyethylene on water and methylene iodide, which has a comparable surface tension to glycerol.⁶⁰⁾ As expected, most of the calculated relative viscosities underestimated the experimental values, and thus, the empirical equation based on a low-polarity liquid suspension failed to express the effect of repulsive forces on the viscosity of high-polarity liquid suspensions. Additionally, the global delta Δ calculated for the estimated and measured aqueous glycerol results was 22.0%.

Fig. 9.

Typical comparisons of the estimated relative viscosity values on the basis of the Einstein-Roscoe’s equation and Eq. (6) with the measured values of glycerol aqueous solution suspension, (a) polyethylene beads size: 602.5 μm, viscosity of liquid phase: 0.06 Pa·s, (b) polyethylene beads size: 0.162 μm, viscosity of liquid phase: 0.22 Pa·s, (c) polyethylene beads size: 340.0 μm, viscosity of liquid phase: 0.52 Pa·s, and (d) polyethylene beads size: 9.35 μm, viscosity of liquid phase: 1.41 Pa·s.

3.3. Rheological Behavior of the Molten Slag Matrix with Dispersed CaO and MgO Particles

Figure 10 demonstrates the microstructures of the polished sections of the quenched CaO–Al₂O₃–SiO₂–MgO slag suspensions with dispersed CaO and MgO particles after the viscosity measurements at 1773 K. The numbers shown below the micro-images are the mean diameter of CaO or MgO particles, with their standard deviations in parenthesis. As shown in Fig. 10, the mean diameters of CaO and MgO and their standard deviations were 29.4 (±9.0) μm and 20.3 (±6.3) μm, respectively. The particles were found to be spherical and well-dispersed in the slag matrix, which can be attributed to the dissolution–recrystallization reaction of dispersed particles to equalize the excess chemical potential at the particle surfaces.

Fig. 10.

SEM microimages of the polished section for quenched CaO or MgO particle dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension after viscosity measurements at 1773 K. Numbers shown below the microimages are the mean diameter of CaO or MgO particles with their standard deviations in parentheses.

Figure 11 shows the measured viscosities of the CaO–Al₂O₃–SiO₂–MgO slag suspensions with dispersed CaO and MgO particles at 1773 K as a function of CaO and MgO volume fraction, respectively, where the vertical axis is the relative viscosity calculated by dividing the apparent viscosity by the viscosity of the liquid phase, 0.235 Pa·s, at 1773 K. The relative viscosity of the molten slag suspensions increased with increasing volume fraction of CaO and MgO, which was the same as for the silicone oil and glycerol suspensions at room temperature. Surprisingly, the relative viscosity was found to be dependent on the rotational speed of the spindle, even at a solid fraction of 10 vol%, which was a lower transition fraction than those of the suspensions at room temperature, and can be attributed to interaction with much higher polar characteristics of the molten slag matrix. As shown in Fig. 11, for the dispersed CaO slag suspension, the relative viscosity first gradually increased with the volume fraction of CaO, reaching approximately 1.6–2.1 at a volume fraction of 10 vol%, depending on the rotational speed. The viscosity rapidly increased with further increasing CaO content and reached a maximum of more than 14.8 at 30 vol% CaO and a rotational speed of 20 rpm, which was a more modest increase than that observed in the glycerol suspension. As similarly shown in Fig. 12, for the dispersed MgO slag suspension, the relative viscosity increased with the volume fraction of MgO, reaching approximately 5.3–5.7 at a volume fraction of 10 vol%, depending on the rotational speed. Upon further increasing the MgO content, the viscosity drastically increased to a maximum of more than 27.4 at 30 vol% MgO and a rotational speed of 20 rpm. This increase in the relative viscosity of the MgO suspension was more pronounced than that of the CaO suspension. Magnesium is slightly more electronegative than calcium, and thus, MgO has a lower relative permittivity of 9.9 than CaO (11.95).⁶¹⁾ Therefore, the surface charge of the MgO particles should be lower than that of the CaO particles in the CaO–Al₂O₃–SiO₂–MgO molten slag, which would lead to a lower repulsive force and, thus, a lower viscosity. However, the relative viscosity of the dispersed MgO suspension increased more than in the CaO suspension, which is attributable to the smaller particle size of MgO in the molten slag. Figure 12 shows the dependence of the shear stress on the shear rate for the CaO–Al₂O₃–SiO₂–MgO slag suspensions with dispersed CaO and MgO particles at 1773 K. For a uniform liquid phase (0 vol% solid phase), the calculated shear stress increased linearly with the shear rate and eventually crossed the origin, which corresponded to Newtonian behavior. On the other hand, even at a 10 vol% solid-phase content, the shear stress intercepted the vertical axis, which is the typical behavior of a Bingham non-Newtonian fluid and was not observed in the suspension viscosities evaluated at room temperature. The shear stress of the dispersed MgO slag suspension was higher at all solid-phase contents than that of the CaO suspension, which is attributable to the higher apparent viscosity of the suspension.

Fig. 11.

Results for viscosity measurements at 1773 K of CaO or MgO dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension with liquid phase viscosity of 0.235 Pa·s. Numbers shown in parenthesis are shear rates.

Fig. 12.

Variations of shear stress calculated for CaO or MgO dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension with liquid phase viscosity of 0.235 Pa·s as a function of shear rate, (a) 29.4 μm CaO dispersed slag, and (b) 20.2 μm MgO dispersed slag.

To investigate the feasibility of the empirical model for estimating the suspension viscosity proposed in section 3.1, the variations in the relative viscosity of the CaO–Al₂O₃–SiO₂–MgO slag suspensions with dispersed CaO and MgO particles at 1773 K were calculated using the Einstein–Roscoe equation and Eq. (6). To calculate the Capillary number (Ca) of the dispersed CaO and MgO particle systems, the interfacial tension between CaO or MgO and the CaO–Al₂O₃–SiO₂–MgO slag was calculated from the surface energies of CaO and MgO,⁶²⁾ the surface tension of the molten slag,⁶³⁾ and the contact angle of CaO or MgO against the molten slag.^64,65) Here, the surface energies of CaO and MgO at 1773 K were calculated according to the values at 0 K, 895 and 1090 mJ/m² for CaO and MgO, respectively, which have a temperature coefficient of −0.16.⁶²⁾ For the surface tension of the molten slag phase, the value of the 58.8CaO-31.9Al₂O₃-9.3SiO₂ ternary slag at 1773 K, 590.3 mN/m, measured by Mukai et al. using the pendant drop method⁶³⁾ was employed as a substitute for the quaternary slag.

Figure 13 shows the estimated relative viscosities of the CaO–Al₂O₃–SiO₂–MgO slag suspensions with dispersed CaO and MgO particles at 1773 K based on the Einstein–Roscoe equation and Eq. (6). Most of the calculated values underestimated the experimental results, and thus, the empirical equation based on a low-polarity liquid suspension essentially failed to reproduce the effect of repulsive forces on the viscosity of the molten slag suspension. Additionally, the global deltas Δ calculated for the present dataset were 42.5% for the CaO suspension and 65.8% for the MgO suspension, which indicate that the repulsive forces in actual slag suspensions are much larger than those in low-polarity suspensions, like polyethylene beads dispersed in silicone oil. Therefore, it would be interesting to propose an empirical equation of viscosity for high-polarity liquid suspensions and investigate its feasibility and reproducibility for estimating the relative viscosity of slag suspensions with dispersed CaO and MgO particles at 1773 K.

Fig. 13.

Comparisons of the estimated relative viscosity values on the basis of the Einstein-Roscoe’s equation and Eq. (6) calculated from data set of silicone oil suspension with the measured values of CaO or MgO dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension, (a) 29.4 μm CaO dispersed slag, and (b) 20.2 μm MgO dispersed slag.

Here, Ca was calculated from the experimental conditions, as described in section 3.2, and the relationship with n was estimated from the apparent viscosity of the polyethylene bead–aqueous glycerol suspension. The regression curve can be expressed as   

$$n_{glycerol}=1.63C{a}^{-0.211},R^2=0.889$$(10)

Then, the variation in the relative viscosity of the CaO–Al₂O₃–SiO₂–MgO slag suspensions with dispersed CaO and MgO particles at 1773 K was calculated based on the Einstein–Roscoe equation and Eq. (10). Figure 14 demonstrates the estimation results, showing that most of the calculated values underestimated the experimental ones. Additionally, the global deltas Δ calculated for this dataset were 21.9% for the CaO suspension and 37.7% for the MgO suspension, which indicates that the estimation accuracy was improved for both the CaO and MgO suspensions. The underestimated results shown in Figs. 13 and 14 suggest that the effect of repulsive forces between the dispersed CaO and MgO particles on the viscosity of the molten slag suspensions was larger than those in the silicone oil and aqueous glycerol suspensions. Consequently, the polarizability of the studied molten slag at 1773 K is likely larger than those of the two selected liquid matrices at room temperature.

Fig. 14.

Comparisons of the estimated relative viscosity values on the basis of the Einstein-Roscoe’s equation and Eq. (10) calculated from data set of glycerol aqueous solution suspension with the measured values of CaO or MgO dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension, (a) 29.4 μm CaO dispersed slag, and (b) 20.2 μm MgO dispersed slag.

Figure 15 reveals a double logarithmic plot for the coefficient n of the Einstein-Roscoe equation calculated from the relative viscosities of the polyethylene bead-dispersed silicone oil and aqueous glycerol suspensions and the CaO- or MgO-dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension as a function of the capillary number estimated from the experimental conditions. Here, the regression line for the slag suspension illustrated in Fig. 15 can be expressed as   

$$n_{slag}=0.19C{a}^{-0.441},R^2=0.595$$(11)

Fig. 15.

Double logarithmic plot for coefficient n of the Einstein-Roscoe’s equation calculated from the relative viscosities of polyethylene beads dispersed silicone oil and aqueous glycerol suspensions and CaO or MgO dispersed CaO–Al₂O₃–SiO₂–MgO slag suspension as a function of the Capillary number estimated from the experimental conditions.

All datasets employed for the regressions expressed as Eqs. (6), (10), and (11) were plotted in a double logarithmic scale with regression lines. As can be seen in Fig. 15, after logarithmization, the coefficient n was found to monotonically and linearly decrease with an increasing logarithmic capillary number, which implies that the regression relationship calculated with the suspensions can even be applied to the slag suspension at 1773 K. Although the pre-capillary number coefficients of Eqs. (6) and (10) were found to be comparable, 1.67 and 1.63 for the silicone oil and aqueous glycerol suspensions, respectively, the slope of the regression line, i.e., the exponent of the capillary number for the aqueous glycerol suspension, had a more negative value (−0.211) compared to that (−0.182) of the silicone oil suspension. The negatively larger slope consequently resulted in a larger n, with a smaller Ca, and in a higher relative viscosity of the suspension. Therefore, the exponent of the capillary number in Eqs. (6), (10), and (11) likely reflects the strength of the dielectric polarizabilities that control the viscosity of the suspension. The exponent for the slag suspension system had the negatively largest value (−0.411), which implies that the influence of dielectric polarizabilities on the viscosity of the slag suspension was most intensive among the suspension systems investigated in the present study. For a further detailed discussion, systematic experimental and computational studies on the rheological characteristics of suspensions with a high-polarity liquid matrix, both at room and high temperatures, will be necessary.

4. Conclusions

The apparent viscosity of suspensions of dispersed spherical particles consisting of polyethylene beads in a matrix of silicone oil or aqueous glycerol at room temperature was measured. The effects of the bead volume fraction and size, shear rate, and viscosity of the liquid matrix on the suspension viscosity were systematically evaluated for both suspensions. Then, empirical equations for predicting the viscosity based on the Einstein–Roscoe equation were proposed. Furthermore, the apparent viscosity of dispersed CaO and MgO particle suspensions in a matrix of CaO–Al₂O₃–SiO₂–MgO slag at 1773 K was measured, and the feasibility of the proposed viscosity equations was investigated. The major findings of the present study can be summarized as follows:

(1) The relative viscosities of the suspensions of polyethylene beads dispersed in silicone oil and aqueous glycerol first gradually increased with increasing bead volume fraction (~15 vol%), then increased drastically at higher volume fractions (30 vol% and above).

(2) Under comparable measurement conditions, the relative viscosities of the aqueous glycerol suspensions were higher than those of the silicone oil suspensions, which is attributable to the larger repulsive potential of the former.

(3) In both the silicone oil and aqueous glycerol suspensions, the increase in relative viscosity was more pronounced with smaller beads than with larger beads.

(4) For most measurement combinations, at a 30 vol% solid-phase content, the relative viscosities of the room-temperature suspensions were dependent on the rotational speed of the viscometer and the shear rate; the dependency on the rotational speed was stronger at higher solid volume fractions.

(5) For a given amount of beads, the relative viscosities were higher for lower rotational speeds, which is a typical characteristic of Bingham non-Newtonian fluids according to the dependence of the shear stress on the shear rate.

(6) The proposed viscosity model based on the Einstein–Roscoe equation and the capillary number Ca can reproduce the viscosity of silicone oil suspensions but underestimates the viscosity of aqueous glycerol suspensions due to repulsive forces in the high-polarity liquid matrix.

(7) Suspensions of CaO and MgO dispersed in CaO–Al₂O₃–SiO₂–MgO molten slag were obtained by adding calcined CaO or MgO particles to pre-melted slag to achieve CaO or MgO saturation. The trend of an increasing relative viscosity of the CaO– and MgO–molten slag suspensions was similar to that of the room-temperature suspensions, revealing Bingham non-Newtonian behavior.

(8) The proposed viscosity models based on the Einstein–Roscoe equation and Ca using the experimental measurements from the silicone oil and aqueous glycerol suspensions underestimated the viscosity of the molten slag suspensions, which is due to the repulsive forces in high-polarity liquid matrices. These calculation results suggest that the dielectric polarizability of the studied molten slag at 1773 K was likely the largest among the suspension systems investigated in the present study.

The findings listed above can initiate consideration of the effects of the polarizability of molten slag matrices and dispersed solids on macroscopic fluid characteristics for a better understanding of iron- and steelmaking processes. For this purpose, systematic experimental and computational studies on the rheological characteristics of suspensions with a high-polarity liquid matrix, both at room and high temperatures, will be necessary.

Acknowledgements

One of the authors (NS) wishes to thank KAKENHI, Grant-in-Aid for Scientific Research (B), Japan Society for the Promotion of Science (JSPS), Project ID: 18H01762, and the team members of Visualization of Slag for the Better Understanding of Multi-phase Melts Flow, the Iron and Steel Institute of Japan (ISIJ).

References

• 1)   N.  Kikuchi,  A.  Matsui and  Y.  Uchida: Tetsu-to-Hagané, 105 (2019), 610 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-157
• 2)   K. C.  Mills: Slag Atlas, 2nd Edition, Verlag Stahleisen GmbH, Düsseldorf, (1995), 541.
• 3)  e.g.  K. C.  Mills: Slag Atlas, 2nd Edition, Verlag Stahleisen GmbH, Düsseldorf, (1995), 349.
• 4)   G.  Urbain,  Y.  Bottinga and  P.  Richet: Geochim. Cosmochim. Acta, 46 (1982), 1061. https://doi.org/10.1016/0016-7037(82)90059-X
• 5)   P.  Riboud,  Y.  Roux,  L.  Lucas and  H.  Gaye: Fachber. Huttenprax. Metallweiterverarb., 19 (1981), 859.
• 6)   T.  Iida,  H.  Sakai,  Y.  Kita and  K.  Murakami: High Temp. Mater. Process., 19 (2000), 153. https://doi.org/10.1515/HTMP.2000.19.3-4.153
• 7)   M.  Nakamoto,  J.  Lee and  T.  Tanaka: ISIJ Int., 45 (2005), 651. https://doi.org/10.2355/isijinternational.45.651
• 8)   S.  Wright,  L.  Zhang,  S.  Sun and  S.  Jahanshahi: Metall. Mater. Trans. B, 31 (2000), 97. https://doi.org/10.1007/s11663-000-0134-8
• 9)   L.  Wu,  M.  Ek,  M.  Song and  D.  Sichen: Steel Res. Int., 82 (2011), 388. https://doi.org/10.1002/srin.201000207
• 10)   S.  Sukenaga,  S.  Haruki,  Y.  Yamaoka,  N.  Saito and  K.  Nakashima: Tetsu-to-Hagané, 95 (2009), 807 (in Japanese). https://doi.org/10.2355/tetsutohagane.95.807
• 11)   S.  Haruki,  S.  Sukenaga,  N.  Saito and  K.  Nakashima: High Temp. Mater. Process., 30 (2011), 405. https://doi.org/10.1515/htmp.2011.062
• 12)   N.  Saito,  S.  Yoshimura,  S.  Haruki,  Y.  Yamaoka,  S.  Sukenaga and  K.  Nakashima: Tetsu-to-Hagané, 95 (2009), 282 (in Japanese). https://doi.org/10.2355/tetsutohagane.95.282
• 13)   A.  Einstein: Ann. Phys., 324 (1906), 289.
• 14)   H.  Eilers: Kolloid-Z., 97 (1941), 313. https://doi.org/10.1007/BF01503023
• 15)   V.  Vand: J. Phys. Chem., 52 (1948), 277. https://doi.org/10.1021/j150458a001
• 16)   M.  Mooney: J. Colloid Sci., 6 (1951), 162. https://doi.org/10.1016/0095-8522(51)90036-0
• 17)   R.  Roscoe: Br. J. Appl. Phys., 3 (1952), 267.
• 18)   C.  Orr, Jr. and  J. M.  Dalla Valle: Chem. Eng. Prog. Symp., 50 (1954), 29.
• 19)   S.  Yagi and  T.  Ototake: J. Chem. Soc. Jpn., 61 (1956), 1404 (in Japanese).
• 20)   D. G.  Thomas: J. Colloid Sci., 20 (1965), 267. https://doi.org/10.1016/0095-8522(65)90016-4
• 21)   M. M.  Cross: J. Colloid Sci., 20 (1965), 417. https://doi.org/10.1016/0095-8522(65)90022-X
• 22)   N.  Frankel and  A.  Acrivos: Chem. Eng. Sci., 22 (1967), 847. https://doi.org/10.1016/0009-2509(67)80149-0
• 23)   R.  Farris: Trans. Soc. Rheol., 12 (1968), 281. https://doi.org/10.1122/1.549109
• 24)   J.  Chong,  E.  Christiansen and  A.  Baer: J. Appl. Polym. Sci., 15 (1971), 2007. https://doi.org/10.1002/app.1971.070150818
• 25)   D.  Quemada: Rheol. Acta., 17 (1978), 632. https://doi.org/10.1007/BF01522036
• 26)   F.  Ferrini,  D.  Ercolani,  B.  De Cindio,  L.  Nicodemo,  L.  Nicolais and  S.  Ranaudo: Rheol. Acta., 18 (1979), 289. https://doi.org/10.1007/BF01542776
• 27)   T.  Dabak and  O.  Yucel: Rheol. Acta., 25 (1986), 527. https://doi.org/10.1007/BF01774404
• 28)   M.  Sengun and  R.  Probstein: Rheol. Acta., 28 (1989), 394.
• 29)   T.  Kattamis and  T.  Piccone: Mater. Sci. Eng. A, 131 (1991), 265. https://doi.org/10.1016/0921-5093(91)90403-A
• 30)   M.  Hirai,  K.  Takebayashi,  Y.  Yoshikawa and  R.  Yamaguchi: ISIJ Int., 33 (1993), 405. https://doi.org/10.2355/isijinternational.33.405
• 31)   A.  Nzihou,  L.  Attias,  P.  Sharrock and  A.  Ricard: Powder Technol., 99 (1998), 60. https://doi.org/10.1016/S0032-5910(98)00091-6
• 32)   O.  Melnik and  R.  Sparks: Nature, 402 (1999), 37. https://doi.org/10.1038/46950
• 33)   D. M.  Liu: J. Mater. Sci., 35 (2000), 5503. https://doi.org/10.1023/A:1004885432221
• 34)   B.  Bournonville and  A.  Nzihou: Powder Technol., 128 (2002), 148. https://doi.org/10.1016/S0032-5910(02)00192-4
• 35)   A.  Costa,  O.  Melnik and  R.  Sparks: Earth Planet. Sci. Lett., 260 (2007), 137. https://doi.org/10.1016/j.epsl.2007.05.024
• 36)   R.  Champallier,  M.  Bystricky and  L.  Arbaret: Earth Planet. Sci. Lett., 267 (2008), 571. https://doi.org/10.1016/j.epsl.2007.11.065
• 37)   H.  Ishibashi: J. Volcanol. Geotherm. Res., 181 (2009), 78. https://doi.org/10.1016/j.jvolgeores.2009.01.004
• 38)   B.  Cordonnier,  K. U.  Hess,  Y.  Lavallee and  D. B.  Dingwell: Earth Planet. Sci. Lett., 279 (2009), 263. https://doi.org/10.1016/j.epsl.2009.01.014
• 39)   C. I.  Mendoza and  I.  Santamaria-Holek: J. Chem. Phys., 130 (2009), 044904. https://doi.org/10.1063/1.3063120
• 40)   P.  Senapati,  B.  Mishra and  A.  Parida: Powder Technol., 197 (2010), 1. https://doi.org/10.1016/j.powtec.2009.07.005
• 41)   H.  Brouwers: Phys. Rev. E., 81 (2010), 051402. https://doi.org/10.1103/PhysRevE.81.051402
• 42)   B. A.  Horri,  P.  Ranganathan,  C.  Selomulya and  H.  Wang: Chem. Eng. Sci., 66 (2011), 2798. https://doi.org/10.1016/j.ces.2011.03.040
• 43)   A.  Vona,  C.  Romano,  D.  Dingwell and  D.  Giordano: Geochim. Cosmochim. Acta, 75 (2011), 3214. https://doi.org/10.1016/j.gca.2011.03.031
• 44)   F.  Boyer,  É.  Guazzelli and  O.  Pouliquen: Phys. Rev. Lett., 107 (2011), 188301. https://doi.org/10.1103/PhysRevLett.107.188301
• 45)   R.  Blissett and  N.  Rowson: Fuel, 111 (2013), 555. https://doi.org/10.1016/j.fuel.2013.03.003
• 46)   Z.  Liu,  L.  Pandelaers,  B.  Blanpain and  M.  Guo: ISIJ Int., 57 (2017), 1895. https://doi.org/10.2355/isijinternational.ISIJINT-2017-288
• 47)   Z.  Liu,  L.  Pandelaers,  B.  Blanpain and  M.  Guo: Metall. Mater. Trans. B, 49 (2018), 2469. https://doi.org/10.1007/s11663-018-1374-9
• 48)   N.  Saito,  K.  Kusada,  S.  Sukenaga,  Y.  Ohta and  K.  Nakashima: ISIJ Int., 52 (2012), 2123. https://doi.org/10.2355/isijinternational.52.2123
• 49)   M. N.  Rahaman: Ceramic Processing, Taylor & Francis, Boca Raton, FL, (2007), 183.
• 50)  Shin-Etsu Chemical Co., Ltd.: Silicone Fluid KF-96 Performance Test Results, https://www.shinetsusilicone-global.com/catalog/pdf/kf96_e.pdf, (accessed 2020-06-24).
• 51)   P. S.  Albright: J. Am. Chem. Soc., 59 (1937), 2098. https://doi.org/10.1021/ja01290a005
• 52)   N.  Saito,  N.  Hori,  K.  Nakashima and  K.  Mori: Metall. Mater. Trans. B, 34 (2003), 509. https://doi.org/10.1007/s11663-003-0018-9
• 53)   S.  Sukenaga,  N.  Saito,  K.  Kawakami and  K.  Nakashima: ISIJ Int., 46 (2006), 352. https://doi.org/10.2355/isijinternational.46.352
• 54)   A. M.  Lejeune and  P.  Richet: J. Geophys. Res.: Solid Earth, 100 (1995), 4215. https://doi.org/10.1029/94JB02985
• 55)   A. L.  Graham and  R. B.  Bird: Ind. Eng. Chem. Fundam., 23 (1984), 406. https://doi.org/10.1021/i100016a005
• 56)   S.  Wu: J. Colloid Interface Sci., 31 (1969), 153. https://doi.org/10.1016/0021-9797(69)90321-X
• 57)   T.  Young: Philos. Trans. R. Soc. Lond., 95 (1805), 65.
• 58)   A.  Kitahara: Kaimen Colloid Kagaku No Kiso (Fundamentals of Interfacial and Colloid Science), Kodansha, Tokyo, (1994), 105 (in Japanese).
• 59)   R. W.  Gallant: Hydrocarb. Process., 46 (1967), 201.
• 60)   A.  Baszkin and  L.  Ter-Minassian-Saraga: J. Colloid Interface Sci., 43 (1973), 190. https://doi.org/10.1016/0021-9797(73)90364-0
• 61)   M. A.  Subramanian,  R. D.  Shannon,  B. H. T.  Chai,  M. M.  Abraham and  M. C.  Wintersgill: Phys. Chem. Miner., 16 (1989), 741. https://doi.org/10.1007/BF00209695
• 62)   D. T.  Lively and  P.  Murray: J. Am. Ceram. Soc., 39 (1956), 363. https://doi.org/10.1111/j.1151-2916.1956.tb15606.x
• 63)   K.  Mukai and  T.  Ishikawa: J. Jpn. Inst. Met., 45 (1981), 147 (in Japanese). https://doi.org/10.2320/jinstmet1952.45.2_147
• 64)   Y.  Taniguchi and  S.  Seetharaman: Metall. Mater. Trans. B, 43 (2012), 587. https://doi.org/10.1007/s11663-012-9639-1
• 65)   J.  Park,  K.  Lee,  J.  Pak and  Y.  Chung: ISIJ Int., 54 (2014), 2059. https://doi.org/10.2355/isijinternational.54.2059