Terminal Settling Velocity of Particle in Suspension

Abstract

In the steelmaking process, most slags and fluxes often contain a solid phase, such as CaO. The suspension in which solid phases are suspended has higher viscosity than that of a pure matrix liquid. Therefore, it is expected that the viscosity of slag containing solid phases will increase. In this study, the terminal settling velocity of particles in a suspension has been measured. The suspensions consist of a silicone oil matrix and polyethylene beads, and the settling particles are bearing balls made of stainless steel. As a result of the higher viscosity of suspension, the terminal settling velocity of the bearing ball becomes slower than that in the pure silicone oil. It was clarified that the retardation of the terminal velocity and the increase in the drag coefficient depend only on the volume fraction of the solid phase (polyethylene beads) of the suspension, and they are independent of the size of the suspended beads and the viscosity of the matrix liquid. A correlation equation for predicting the drag coefficient of particles in a suspension was proposed.

1. Introduction

Slag in converters takes the form of a complex multiphase flow. During converter operation, the slag is stirred by the gas phase owing to gas blowing and gas generation by chemical reactions, resulting in foaming. This foamed slag contains metal droplets that are entrained from the metal bath as well as undissolved slag-forming agents, forming a complex multiphase flow of gas, liquid, and solid. Proper converter operation requires accurately understanding the physical properties of the slag that has become a multiphase flow and the various physical phenomena that occur within the multiphase slag.

A suspension in which solid particles are suspended in a liquid is known to have a higher apparent viscosity than that of a single-phase liquid. Starting with Einstein’s theoretical equation for the viscosity of a dilute suspension proposed in the early 20th century, numerous equations have been proposed to date for estimating the apparent viscosity of suspensions with higher concentrations.^{1,2,3,4,5,6,7,8,9,10,11,12,13)} Einstein’s theory was derived for suspensions of infinitely dilute concentration where the interaction between particles can be ignored, and where the applicable volume fraction ϕ of the suspension particles is limited to less than approximately 0.03.¹⁾ Simha proposed an equation based on Einstein’s theory that considers the interaction between suspension particles, and is applicable to a volume fraction ϕ up to approximately 0.2.²⁾ Vand and Ford subsequently derived equations for the solid-phase volume fraction and apparent viscosity using a similar method.^3,4) Brinkman and Roscoe extended Einstein’s theory by adding a small volume of suspension particles to a dilute suspension, proposing the so called Einstein–Roscoe viscosity equation.^5,6) Additionally, Mooney analyzed the flow field when suspension particles were added using the Navier–Stokes equation, and proposed an exponential correlation equation for the solid-phase volume fraction and apparent viscosity.⁷⁾ Mooney’s equation is still relatively widely used today, and the apparent viscosity increases rapidly when the volume fraction ϕ exceeds approximately 0.3. These models of the apparent viscosity of suspensions are summarized in detail by Yagi and Ototake,⁸⁾ Pal,⁹⁾ Xia and Krueger;¹⁰⁾ please refer to these articles for details. In recent years, Saito et al. have also prepared suspensions in which solid particles are suspended in a high-viscosity liquid, and measured the apparent viscosity of the suspensions, considering the molten slags and fluxes used in the steel industry.^11,12,13) Several of these previous studies have shown that suspensions are so called non-Newtonian fluids, in which the apparent viscosity not only increases but also depends on the shear rate.

Predicting the settling velocity of particles in slag, which is a multiphase fluid, is important from the following perspective. Studies by Misra et al.¹⁴⁾ and Biswas et al.¹⁵⁾ have attempted to build a model of the refining reaction in a converter, and in both models, the residence time of metal droplets in the multiphase slag, that is, the settling velocity of the particles, played an important role. Accurately modeling various reactions in slag requires a detailed understanding of the size and settling velocity of the metal droplets and slag-forming agents present in the slag.

The apparent viscosity of a suspension is considered to directly influence the settling velocity of suspension particles, but the relationship between the two aspects remains unclear. An increasing apparent viscosity is considered to result in a decreased settling velocity of particles, but the relationship between these two aspects is not simple for suspensions with non-Newtonian properties. Studies by Batchelor as well as Batchelor and Wen investigated the hydrodynamic effects of suspension particles on settling particles in suspensions, and theoretically found that the settling velocity of particles decreases as the solid-phase volume fraction of the suspension increases.^16,17) Meanwhile, Watanabe et al. experimentally reported that the drag coefficient of a ball falling at a high speed through a suspension decreases when the suspension is dilute.¹⁸⁾ Eguchi and Karino¹⁹⁾ compared the viscosities measured using a rotational viscometer and the falling ball method for non-Newtonian fluids, such as blood (which can be considered as a suspension of red blood cells) and polymer suspensions, and reported that the two methods were consistent with each other only for limited cases. Milliken et al. conducted the falling ball method for suspensions of spherical and rod-shaped particles.²⁰⁾ Felice and Rotondi²¹⁾ also conducted similar experiments and compared the results with those of Batchelor’s equation.^16,17) Cho et al. reported that the viscosity of non-Newtonian fluids measured using the falling ball method was not consistent with the viscosity at a steady shear rate.²²⁾ Li et al. also reported that the apparent viscosity of suspensions measured using the falling ball method exhibited complex non-Newtonian properties.²³⁾ As indicated by these previous studies, predicting the settling velocity of particles is difficult when the apparent viscosity of suspension is measured using methods, such as the rotational method.

Given the above context, the settling velocity of particles in suspensions was focused in this study. In previous studies, polyethylene beads were suspended in silicone oil to form a suspension, and the apparent viscosity was measured using the rotational method.¹³⁾ This experiment was used as a reference to create the same suspension, where a steel ball was dropped and the settling velocity was measured. In other words, apparent viscosity was measured using the rotational method and falling ball method for the same suspension. Both methods were compared and a correlation equation for the settling velocity of particles in a suspension was proposed.

2. Viscosity Measurement Using Rotational Method

The rotational method generally measures the apparent viscosity at a constant shear rate in a steady state. The method can also be applied to high-temperature systems; hence, it is often used to measure the viscosity of slags. Saito et al. measured the apparent viscosity of suspensions using the rotational method.^11,12,13) They suspended polyethylene beads in silicone oil to form a suspension, and measured the apparent viscosity of the suspension using the rotational method. They used silicone oils (Shin-Etsu Silicone, KF-96) with the kinetic viscosities of 500, 1000, 2000, and 3000 mm²/s. Table 1 lists the physical properties of the silicone oil used. The mean volume diameters of four types of polyethylene beads (Sumitomo Seika, LE-1080) are 9.35, 163, 340, and 603 µm. Scanning electron microscopy images of the beads are presented in reference,¹³⁾ which show that they are spherical and have a narrow range of particle diameters. These polyethylene beads were suspended in silicone oil to obtain suspensions with the volume fractions of 0%, 15%, 30%, 45%, and 60%. The shear rates measured using a rotational viscometer were 2.09, 4.18, 10.5, 12.6, and 20.9 s⁻¹. Please refer to reference¹³⁾ for details of the other experimental conditions.

Table 1. Properties of silicone oil.

density, ρl	970 kg/m3
kinematic viscosity, vl	500, 1000, 2000, or 3000 mm2/s
surface tension, σl	14.9, 14.8, 14.8, or 14.7 mN/m

Saito et al. measured the apparent viscosity of these suspensions, and summarized the results in the following Einstein–Roscoe dimensionless correlation equation:¹³⁾

  

$$\frac{\eta }{{\eta }_1}=\frac{1}{{\left(1-a\varphi \right)}^n},\mathrm{\hspace{0.17em} } n=1.67C{a}^{-0.182},\mathrm{\hspace{0.17em} } a=1,\mathrm{\hspace{0.17em} } Ca=\frac{{\eta }_1\gamma d_{\mathrm{p}}}{\sigma }.$$(1)

According to the results of Saito et al., the apparent density increased with increasing volume fraction, regardless of the size of the polyethylene beads or the shear rate. Additionally, except for cases where the volume fraction of the polyethylene beads is zero, the apparent viscosity depends on the shear rate (non-Newtonian fluid), with a lower shear rate resulting in a higher apparent viscosity.

3. Particle Settling Velocity in Suspension

The settling velocity of particles in the suspension was measured. The silicone oil and polyethylene beads used in the experiment were the same as those used in the apparent viscosity measurement using the rotational method. Polyethylene beads were suspended in silicone oil to prepare suspensions with the volume fractions of 0%, 20%, 30%, and 40%. A bearing ball made of SUS304 (diameters 1, 2, 2.5, and 3.5 mm, density of 7930 kg/m³) was dropped and its movement was recorded with a video camera. The settling velocity of the particles was measured from the video. In the rotational method, the apparent viscosity is measured up to a volume fraction of 60%,¹³⁾ but in the falling ball method experiment, the volume fraction is limited to a maximum of 40%. A volume fraction of 60% for the polyethylene beads approaches the closest packing of the hard spheres. In this case, even if large particles (bearing balls) are added, the settling velocity becomes very slow, and data cannot be obtained using the falling ball method. Therefore, our experiment was limited to a relatively low volume fraction range (40% or less).

Figures 1 and 2 show examples of the measurement results. Figure 1 shows the results for a suspension of polyethylene beads in silicone oil with a kinematic viscosity of 500 mm²/s, and Fig. 2 shows the corresponding results for a kinematic viscosity of 1000 mm²/s. The figure shows that the settling velocity decreased significantly, as the volume fraction of the suspended polyethylene beads increased. Meanwhile, no apparent dependence on the size of the polyethylene beads could be observed. Similar results were obtained for the other kinematic viscosities (2000 mm²/s and 3000 mm²/s).

Fig. 1. Relationship between the terminal settling velocity and volume fraction of suspended particles. The kinematic viscosity of silicone oil is 500 mm²/s. (Online version in color.)

Fig. 2. Relationship between the terminal settling velocity and volume fraction of suspended particles. The kinematic viscosity of silicone oil is 1000 mm²/s. (Online version in color.)

Here, the settling velocity of the bearing ball was evaluated using the apparent viscosity formulated by the rotational method in Eq. (1). Assuming the Stokes region, the terminal settling velocity v_p of the particles is expressed as follows:

  

$$v_{\text{p}}=\frac{d_{\text{p}}^2\left({\rho }_{\text{p}}-{\rho }_{\text{s}}\right)g}{18{\eta }_{\text{s}}},$$(2)

where η_s and ρ_s are the apparent viscosity and apparent density of the suspension, respectively. η_s is calculated from Eq. (1). ρ_s is calculated using the volume fraction ϕ of the suspension particles:

  

$${\rho }_{\text{s}}=\left(1-\varphi \right){\rho }_1+\varphi {\rho }_{\text{b}},$$(3)

where ρ₁ and ρ_b are the densities of the silicone oil and polyethylene beads, respectively. Once the terminal velocity of the particles is determined from Eq. (2), the shear rate γ can be estimated from this value using the following equation:¹⁹⁾

  

$$\gamma =\frac{3v_{\text{p}}}{d_{\text{p}}}.$$(4)

The apparent viscosity of the suspension can be calculated from Eq. (1) using the shear rate, and the terminal velocity of the particles can be calculated from Eq. (2) using the apparent viscosity. In other words, as Eqs. (1), (2), and (4) are not closed, iterative calculations are required to find a solution.

Figure 3 shows the obtained results. The horizontal axis shows the measured settling velocity, and the vertical axis shows the settling velocity calculated from Eqs. (1), (2), and (4). The straight line in the graph indicates consistency between the measured and calculated values. The figure shows that, in cases where the volume fraction of the polyethylene beads is zero, that is, when the silicone oil is a single phase, the measured and calculated values are consistent with each other. This indicates that the viscosity measurement results from the rotational method and falling ball method are consistent with each other. Silicone oil is a Newtonian fluid. The previous literature has reported that the results of viscosity measurements using the rotational method and falling ball method are consistent with each other for Newtonian fluids in the Stokes region.¹⁹⁾

Fig. 3. Relationship between the observed terminal velocity and the terminal velocity estimated using Eqs. (1) and (2). (Online version in color.)

Meanwhile, cases where the volume fraction of the polyethylene beads is nonzero result in the values shifting above the solid 1:1 line. In other words, the measured settling velocity is lower than the calculated settling velocity. This indicates that the presence of polyethylene beads inhibits particle settling. The degree of this inhibition also depends solely on the volume fraction, with a larger volume fraction considerably inhibiting settling. Studies by Batchelor as well as Batchelor and Wen derived a relational equation between the solid-phase volume fraction of the suspension and the settling velocity of the particles, where they reported that the settling velocity of the particles decreased as the solid-phase volume fraction increased.^16,17)

Although it is the same as above, the discrepancy between the calculated and measured values in Fig. 3 indicates that the apparent viscosity obtained using the rotational method is not consistent with the apparent viscosity evaluated using the falling ball method. Assuming the Stokes region, the apparent viscosity η_s of the suspension obtained using the falling ball method can be calculated by transforming Eq. (2) into the following equation:

  

$${\eta }_{\text{s}}=\frac{d_{\text{p}}^2\left({\rho }_{\text{p}}-{\rho }_{\text{s}}\right)g}{18v_{\text{p}}}.$$(5)

Figure 4 shows a graph with the apparent viscosity calculated using Saito’s correlation equation Eq. (1) on the horizontal axis and the apparent viscosity evaluated using the falling ball method Eq. (5) on the vertical axis. In cases where the solid-phase volume fraction ϕ=0%, which signifies a Newtonian fluid, the two values are consistent with each other, but in cases where the solid-phase volume fraction ϕ is not 0%, which signifies a non-Newtonian fluid, the two values diverge, with the apparent viscosity evaluated using the falling ball method being higher. The apparent viscosity obtained using the rotational method is the viscosity that occurs when the shear rate is constant and in a steady state. Meanwhile, the falling ball method involves a ball falling into a stationary suspension; hence, the shear rate is high around the ball and decreases further away from the ball, and thus, the shear rate has a spatial distribution. Furthermore, the liquid and beads in the stationary state deform as the ball approaches, and return to a stationary state as the ball moves away. In other words, the liquid and beads follow a nonstationary and transient history. The different measurement conditions for the two methods are considered to result in different apparent viscosities, particularly in the case of non-Newtonian fluids.

Fig. 4. Comparison of apparent viscosity measured using a rotational viscometer and the falling ball method. (Online version in color.)

4. Drag Coefficient of Particle in Suspension

The drag coefficient of a particle settling at a constant velocity v_p in a stationary fluid can be calculated by the following equation:

  

$$C_{\text{D}}=\frac{\frac{\pi }{6}{d}_{\text{p}}^{3}g\left({\rho }_{\text{p}}-{\rho }_{\text{s}}\right)}{\frac{\pi }{4}{d}_{\text{p}}^2\cdot \frac{1}{2}{\rho }_{\text{s}}{v}_{\text{p}}^2}.$$(6)

This is the drag coefficient calculated from the measured value v_p. Meanwhile, in the Stokes region, the drag coefficient is theoretically evaluated using the following equation:

  

$$C_{\text{D}}=\frac{Re}{24}=\frac{1}{24}\cdot \frac{{\eta }_{\text{s}}}{d_{\text{p}}{\rho }_{\text{s}}{v}_{\text{p}}}.$$(7)

Figures 5, 6, 7, 8 show graphs with the Re number corresponding to the particle settling velocity on the horizontal axis and the drag coefficient on the vertical axis. The graphs correspond to the solid-phase volume fractions of ϕ=0%, ϕ=20%, ϕ=30%, and ϕ=40%, respectively.

Fig. 5. Relationship between the Reynolds number and drag coefficient. Volume fraction of solid phase, ϕ=0%. (Online version in color.)

Fig. 6. Relationship between the Reynolds number and drag coefficient. Volume fraction of solid phase, ϕ=20%. (Online version in color.)

Fig. 7. Relationship between the Reynolds number and drag coefficient. Volume fraction of solid phase, ϕ=30%. (Online version in color.)

Fig. 8. Relationship between the Reynolds number and drag coefficient. Volume fraction of solid phase, ϕ=40%. (Online version in color.)

Cases where the solid-phase volume fraction is zero (Fig. 5) signify that the fluid is not a suspension but a single phase of silicone oil. In such cases, the apparent viscosity in Eq. (1) is the viscosity of the silicone oil, and the measured value is consistent with the Stokes equation in Eq. (7). A solid-phase volume fraction of ϕ=20% (Fig. 6) indicates that a measured drag coefficient is larger than that of the Stokes equation. This corresponds to the decrease in the settling velocity of the particles in the suspension. Although there is some variation, the rate of increase in the drag coefficients are almost equivalent regardless of the viscosity of the silicone oil or the size of the polyethylene beads. Furthermore, at larger solid-phase volume fraction values of ϕ=30% or ϕ=40% (Figs. 7 and 8), the drag coefficient deviates more from Stokes’ equation, the rate of increase in the drag coefficient does not depend on the viscosity of the oil or the size of the beads. In other words, the drag coefficient increases solely depending on the solid-phase volume fraction.

The following correlation equation is used to express the dependence of the drag coefficient on the solid-phase volume fraction. This function is used because (a) the drag coefficient always increases, (b) the increase in drag coefficient depends only on the solid-phase volume fraction ϕ, and (c) the drag coefficient C_D=C_D0 at the solid-phase volume fraction ϕ=0%:

  

$$C_{\text{D}}=C_{\text{D}0}\left(1.0+\alpha {\varphi }^{\beta }\right),$$(8)

where C_D0 is the drag coefficient when the solid-phase volume fraction is zero. The measured data were fitted to this equation, and the coefficients α and β were determined, yielding the following equation:

  

$$C_{\text{D}}=C_{\text{D}0}\left(1+13.8\hspace{0.17em}{\varphi }^{1.50}\right).$$(9)

Figure 9 shows a comparison between the drag coefficient obtained from the measurement and that obtained from Eq. (9). The two values are consistent with each other over a wide range. The correlation coefficient is R=0.987.

Fig. 9. Comparison between the observed and estimated drag coefficients.

In the Stokes region, the drag coefficient is inversely proportional to the terminal settling velocity. In other words, this becomes

  

$$\frac{v}{v_0}=\frac{1}{1+13.8\hspace{0.17em}{\varphi }^{1.50}},$$(10)

where v₀ is the terminal settling velocity at ϕ=0%. Figure 10 shows Eq. (10) plotted against the solid-phase volume fraction. The box plots in the figure are measured values, and the outliers in the figure have upper and lower whisker lengths that are 1.5 times the interquartile range.

Fig. 10. Change in the terminal velocity with the volume fraction of suspension particles. (Online version in color.)

The measured values for ϕ=0%, 20%, 30%, and 40% correspond to Figs. 5, 6, 7, 8, respectively. Theoretically, v/v₀ should be exactly 1 at ϕ=0%, but the measured values were scattered in the range of 0.87 to 1.03 (median of 0.94). However, this is the result of consolidating all the measured values in Fig. 5 into one point, and Fig. 5 shows that the measured values were consistent with the theoretical values over a wide range of Re numbers. In the case of suspensions (ϕ=20%, 30%, 40%), the particles do not always settle straight at a constant speed, but rather settle with irregular changes in speed or by changing direction in a zigzag pattern. Reflecting this tendency, the measured values had considerable variability, but the settling velocity decreased as the solid-phase volume fraction increased.

Additionally, the data from the previous literature are plotted for comparison in Fig. 10. Li et al.²³⁾ suspended polystyrene beads in silicone oil to form a suspension, and SUS304 bearing balls were allowed to settle like the present study. Meanwhile, the experiment by Eguchi and Karino¹⁹⁾ involved dropping polystyrene particles into blood. In the case of blood, the suspension particles correspond to red blood cells, and their solid-phase volume fraction is approximately 45%. The data in both studies show the same tendency as Eq. (9), that is, the settling velocity of the particles decreases as the solid-phase volume fraction of the suspension particles increases. With the exception of the small solid-phase volume fraction results in the data by Li et al., all the data in the studies quantitatively were consistent with Eq. (9).

Figure 10 shows that, for example, when the solid-phase volume fraction is 10%, the terminal settling velocity is predicted to be further 0.7 times lower (v/v₀=0.70) than the terminal settling velocity calculated from the apparent viscosity obtained using Saito’s correlation equation Eq. (1). Similarly, when the solid-phase volume fraction is 20%, the terminal settling velocity is predicted to be 0.45 times lower, indicating that a larger solid-phase volume fraction results in a lower terminal settling velocity. This result is considered to be useful for predicting the settling velocity of particles in suspended slags and fluxes. Conversely, the results of this study can be used to estimate the apparent viscosity obtained via the rotational method using the apparent viscosity measured via the falling ball method.

5. Conclusions

In this study, the falling velocity of particles in solid particle suspensions was experimentally investigated and the following conclusions were obtained.

(1) In cases where solid particles were not suspended, that is, a single-phase silicone oil, which is a Newtonian fluid, the viscosity values measured using the rotational method were consistent with the viscosity values measured using the falling ball method.

(2) In cases where solid particles were suspended, that is, non-Newtonian fluids, the apparent viscosity values in the suspension were higher than that of a single-phase liquid.

(3) The apparent viscosity of the suspension measured using the falling ball method was higher than that measured using the rotational method, and the settling velocity of the particles was overestimated when the viscosity measured using the rotational method was used.

(4) The increase in the drag coefficient of particle in the suspension was determined solely by the volume fraction, not by the size of the suspension particles. We proposed the following correlation equation for predicting the drag coefficient of particle in the suspension:

  

$$C_{\text{D}}=C_{\text{D}0}\left(1+13.8\hspace{0.17em}{\varphi }^{1.50}\right).$$

Acknowledgments

This study was supported by the Iron and Steel Institute of Japan “Slag visualization for understanding flow of multiphase melts” research group. We would like to express our gratitude to this group here.

Symbols Used

a[–]: constant in Einstein–Roscoe equation

d[m]: diameter

g[m/s²]: gravitational acceleration

n[–]: exponent in Einstein–Roscoe equation

v[m/s]: velocity

Dimensionless numbers

C_D[–]: drag coefficient of particles

C_D0[–]: drag coefficient of particles at ϕ=0

Re[–]: Reynolds number, $Re=\frac{d_{\text{p}}{v}_p{\rho }_{\text{s}}}{{\eta }_{\text{s}}}$

Greek letters

γ[1/s]: shear rate

η[Pa s]: viscosity

σ[N/m]: interfacial tension

ϕ[–]: solid-phase volume fraction

Subscripts

b: bead (suspension particle, polyethylene composition)

l: liquid (silicone oil)

p: particle (settling particle, SUS304 bearing balls)

s: suspension

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