Settling of Particle in Foaming Slag

Abstract

During steel-manufacturing, molten slag is foamed through gas injection and gas generation reactions. However, molten iron droplets are trapped in the slag. The settling velocity of an iron droplet in the foaming slag is important because the residence time of an iron droplet in the slag is used to directly calculate the settling velocity. Previous studies have shown that the rate of settling velocity is lower than that observed in regular non-foaming slag. However, this has not been quantitatively determined. This study measured the settling velocities of particles through a foaming glycerin-water solution. A dimensionless correlation equation for particle settling velocity in the formed liquid was proposed by conducting a dimensional analysis of the experimental data. Using the obtained equation, the settling velocity of iron particles in the foaming slag was predicted. The settling velocity of iron particles was significantly affected by the volume fraction of the gas phase in the foaming slag. A threshold for the velocity was observed; the velocity of particles below this threshold was zero.

1. Introduction

Proper converter operation requires an understanding of the physical properties of foaming slag and the various physical phenomena occurring within. During steel-manufacturing, the introduction of gas by the converter results in gas generation reactions and the formation of many bubbles in the slag, resulting in a foaming state. The foaming slag is composed of complex multiphase fluids containing metal droplets and undissolved slag-forming agents. These multiphase fluids are known as foams, suspensions, and emulsions in the gas-liquid, solid-liquid, and liquid-liquid states, respectively. Their rheological properties have been previously discussed.

The apparent viscosity of a multiphase fluid, which was formed from slag, was higher than that of the single-phase fluid. In general, a rotational viscometer is used to determine the viscosity of a multiphase fluid. Sukenaga et al. and Haruki et al. fabricated suspensions using cold models, then used a rotational viscometer to determine the apparent viscosities of the simulated slags.^1,2) Studies have also determined the apparent viscosities of cold model-fabricated foams using a rotational viscometer.^3,4) The apparent viscosities of the suspensions and foams fabricated in these studies were higher than those of the single-phase fluid; additionally, the viscosities exhibited non-Newtonian properties dependent on shear rate (γ).

A high apparent viscosity value means that the relative velocity of the particles settling in the suspensions or foams are low and the duration of the residence time of the particles is extended. A previous study measured the apparent viscosity of a foam using the falling ball method.⁵⁾ Particles are dropped into the foam, and the apparent viscosity is estimated from the terminal velocity of the particles using the viscosity of the liquid, the gas phase volume fraction of the foam, and the particle density as parameters. The apparent viscosity of the foam determined using the falling ball method was higher than that of the single-phase fluid. In addition, the foam exhibited complex non-Newtonian properties.

To predict the refining reaction in slag, the size of metal droplets and their relative velocities must be known. The fabrication of a comprehensive reaction model of a converter was attempted by Misra et al.⁶⁾ Recently, Biswas et al.⁷⁾ also attempted to construct a similar refining reaction model in a converter; however, the settling velocity (residence time) of metal droplets in the foaming slag played an important role in both models. Martinsson et al.⁸⁾ measured the settling velocity of particles in foam using a cold model, resulting in the proposal of a semi-empirical model. Furthermore, Subagyo and Brooks⁹⁾ proposed an empirical formula for the settling velocity of particles in foam. According to these studies, the settling rate of particles in foam becomes significantly lower than that in a single liquid phase; additionally, some particles do not settle but are trapped within the foam.^8,9) A review by Li et al.¹⁰⁾ summarized the size of the iron particles recovered from the gas/slag/metal emulsion. Sizes ranged from several tens of micrometers to greater than 10 mm. Large iron particles are unlikely to be trapped in the single-phase slag; additionally, the interaction between the foaming single-phase slag and particles must be considered. The apparent viscosities of the foam measured using a rotational viscometer and the falling ball method were higher than that of the single-phase fluid in previous studies. However, the relationship between the apparent viscosities measured using the rotational viscometer and falling ball method are unclear.

This study focused on the falling velocities of particles in foam. The apparent viscosity of the foam and terminal settling velocity of the particles in the foam can be determined using the falling ball method. Measuring the apparent viscosity of a foam using the falling ball method is also interpreted as an assessment of the terminal settling velocity of particles.⁵⁾ This study also aimed to examine the relationship between the apparent viscosity determined using the falling ball method and that determined using the rotational method. Dimensional analysis was performed, and the results were summarized as a dimensionless correlation equation on the settling velocity of particles in foam.

2. Falling Ball Methodology

The falling ball method was performed using a cold model. To simulate the settling of particles in the foaming slag, a gas-liquid mixed phase fluid comprising an aqueous glycerol solution system was prepared¹¹⁾ using the gas generated from the chemical reaction in the simulated slag. The reaction between sodium bicarbonate (NaHCO₃) and oxalic acid (C₂H₂O₄) was used to generate gas based on the following equation.

  

$${\text{C}}_2{\text{H}}_2{\text{O}}_4+2{\text{NaHCO}}_3\to {\text{Na}}_2{\text{C}}_2{\text{O}}_4+{\text{H}}_2\text{O}+{\text{CO}}_2\mathrm{\hspace{0.17em} }\text{(g)}$$(1)

Carbon dioxide gas was generated by the reaction between sodium bicarbonate and oxalic acid in an aqueous solution of glycerol in advance and mixing them in an experimental vessel. The experimental equipment is made of acrylic board with a thickness of 3 mm, and its internal dimensions are 25 mm × 44 mm and the height is 300 mm. Balls of different densities (stainless steel, titanium, glass) with a diameter of 2 mm were added to the top of a vessel filled with foam, then the settling velocities were measured. The detailed experimental methodology is described in a previous study.⁵⁾

The summarized experimental conditions are listed in Table 1. The experiment was performed using the following parameters: the concentration (x) of the aqueous glycerol solution, the volume fraction (ϕ) of the gas phase of the foam, and the density (ρ_p) of the falling particles. The average bubble diameter of the foam (d_f) was calculated by analyzing the images of the foam. The d_f value varied from 1.19 mm to 1.80 mm depending on the concentration of the aqueous glycerol solution and the volume fraction of the gas phase of the foam (Table 2). Because the diameter of the falling ball was 2 mm, the bubble diameter of the foam is equal to or lower than that value.

Table 1. Experimental conditions.

particle diameter, dp	2 mm
density of particle, ρp	2500, 4510, and 7930 kg⁄m3
glycerol mass fraction, x	0.2, 0.35, 0.5, 0.65, and 0.8
density of glycerol solution, ρl	1051, 1090, 1129, 1168, and 1208 kg⁄m3
viscosity of glycerol solution, μl	1.76, 3.06, 6.00, 15.2, and 60.1 mPa s
surface tension of glycerol solution, σl	71.7, 70.4, 69.3, 68.3, and 67.4 mN⁄m
volume fraction of gas, ϕ	0.3, 0.5, and 0.7
bubble diameter of foam, df	1.19–1.80 mm

Table 2. Mean diameter of the foam bubble (unit: mm).

＼		glycerol mass fraction, x
		0.2	0.35	0.5	0.65	0.8
Volume	0.3	1.19	1.31	1.43	1.43	1.43
fraction	0.5	1.31	1.43	1.55	1.59	1.64
of gas, ϕ	0.7	1.54	1.59	1.64	1.72	1.80

Using the falling ball method, the apparent viscosity of the foam was evaluated by determining the terminal settling velocity of the particles. Assuming Stokes’ region, the terminal settling velocity of the particles (v_p) is

  

$$v_{\text{p}}=\frac{d_{\text{p}}^2({\rho }_{\text{p}}-{\rho }_{\text{f}})g}{18{\mu }_{\text{f}}}$$(2)

where, ρ_f and μ_f are the apparent viscosity and apparent density of the foam, respectively. Using ϕ, ρ_f and μ_f are calculated as follows

  

$${\rho }_{\text{f}}=(1-\varphi ){\rho }_{\text{l}}+\varphi {\rho }_{\text{b}}\approx (1-\varphi ){\rho }_{\text{l}}$$(3)

  

$${\mu }_{\text{f}}=\frac{d_{\text{p}}^2({\rho }_{\text{p}}-{\rho }_{\text{f}})g}{18v_{\text{p}}}$$(4)

Therefore, the apparent viscosity of the foam can be evaluated from the terminal velocity of the particles. Figure 1 shows the relationship between the concentration of the aqueous glycerol solution and the terminal settling rate of the measured particles at a specified volume fraction (0.7) of the gas phase of the foam. The rate of the settling velocity decreases as the glycerol concentration and the viscosity of the liquid increases. However, the rate of the settling velocity increases as the mass of the particles increases. Figure 2 shows the relative viscosity (apparent viscosity of the foam divided by the viscosity of the liquid) at a gas volume fraction of 0.7. Increasing the viscosity of the liquid increases the relative viscosity up to 100-fold. However, the relative viscosity increases by approximately 20-fold at the highest glycerol concentration (0.8), which shows that the relationship was not monotonic. In addition, the relative viscosity generally decreased as the density of the particles decreased.

Fig. 1. Experimental results: relationship between the glycerol mass fraction of the liquid and terminal velocity. The gas fraction of the foam was 0.7. (Online version in color.)

Fig. 2. Experimental results: relationship between the glycerol mass fraction of the liquid and the relative viscosity of the foam. The gas fraction of the foam was 0.7. (Online version in color.)

3. Apparent Viscosity of the Foam Using the Rotational and Falling Ball Methods

Yamashita et al.³⁾ used a cold model to measure the apparent viscosity of the foaming fluid using a rotational viscometer. According to their results, the ratio of the apparent viscosity of the foam to the viscosity of the liquid was summarized as a function of ϕ using the Einstein-Roscoe equation as follows.

  

$$\frac{{\mu }_{\text{f}}}{{\mu }_{\text{l}}}=\frac{1}{{(1-a\varphi )}^n}$$(5)

where

  

$$n=0.148\mathrm{\hspace{0.17em} }C{a}^{-0.371},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }Ca=\frac{{\mu }_{\text{l}}\gamma d_{\text{b}}}{{\sigma }_{\text{l}}}$$(6)

The shear rate γ of a rotational viscometer can be expressed as

  

$$\gamma =\frac{2\omega }{1-(r_{\text{i}}/r_{\text{o}})}$$(7)

using the inner cylindrical radius (r_i), the outer cylindrical radius (r_o), and the rotational angular velocity (ω). The shape factor (a) in Eq. (5) is a constant (value of 1) when the bubble is spherical. Figure 3 shows the relationship between the γ and apparent viscosity of the foam when the glycerol concentration is x=0.8. Based on Eq. (5), the apparent viscosity increases as ϕ of the foam increases or γ decreases.

Fig. 3. Relationship between the shear rate and apparent foam viscosity. (Online version in color.)

The formula that estimates the viscosity using the rotational method can be applied to the foam produced by the experiment of the falling ball method. The γ value of the flow field generated by the particles can be estimated as

  

$$\gamma =3v_{\text{p}}/d_{\text{p}}$$(8)

where v_p represents the falling velocity of the particle.¹²⁾ By sequentially substituting the v_p value obtained from the measurements of the falling ball method into Eqs. (8), (6) and (5), the viscosity of the foam estimated via the rotational method can be evaluated. Figure 4 shows a diagram of the foam viscosity obtained in this way. The horizontal and vertical axes of Fig. 4 represent the apparent viscosities calculated using Eq. (4) (falling ball method) and Eq. (5), respectively. The slopes of both lines were similar; however, their magnitudes differed. The viscosity evaluated via the falling ball method was approximately 20-fold higher than that of the rotational method. This difference in magnitude was likely due to the volume fraction of the gas phase. Therefore, the terminal velocity of the particles in the foam is overestimated when it is estimated using the apparent viscosity obtained by the rotational method. The experimental values obtained by Martinsson et al.^8,13) were also plotted in Fig. 4. The slopes of the values obtained from those studies were similar to those of this study; additionally, the viscosity determined using the falling ball method was higher than that determined by the rotational method.

Fig. 4. Foam viscosities estimated using the falling ball method and a rotation viscometer. (Online version in color.)

The foam, which must be analyzed, rotates in a steady state during the rotational method. Therefore, the volume fraction of the gas phase can be distributed inside, and outside of the rotation. In addition, the rotational method measures viscosity under constant γ spatially and temporally. Consideration of the non-uniformity of the foam associated with the rotation was not required because the particles were dropped into a uniform foam in the falling ball method. In addition, the spatial and temporal distribution of γ around the particles was not constant because the particles exhibited relative motion based on the stationary state of the foam. Therefore, the rotational and falling ball methods measured viscosity at different states.

When the bubble diameter of the foam is approximately the same or higher than that of the falling particles (d_b≥d_p), the foam cannot be considered a uniform fluid, and the particles are significantly affected by the structure of the foam. Therefore, (1) the particles repel the foam bubble, (2) fall along the structure of the foam in a zigzag pattern, or (3) fall linearly while degrading the structure of the foam.^5,13) In either case, the settling particles need to do extra work compared to when they are in a uniform fluid. Under these influences, the terminal settling velocity, which estimated from the apparent viscosity measured by the rotation method, is overestimated.

Eguchi and Karino compared the viscosities measured using a rotational viscometer and the falling ball method for non-Newtonian fluids such as blood and polymer suspensions. The study showed that both values were similar in a limited number of cases.¹²⁾ Cho et al.¹⁴⁾ showed that the viscosities of non-Newtonian fluids measured using the falling ball method were not similar to the viscosity at the steady γ. Evaluating the settling velocity of particles in non-Newtonian fluids using the apparent viscosity measured by the rotational method is not possible (Fig. 4 and previous studies).

4. Dimensionless Correlation Equation

Two dimensionless numbers—the Reynolds and Archimedes numbers—can be used when assessing the settling/floating of particles in a stationary fluid at terminal velocity. Multiplying both sides of Eq. (4) of Stokes’ law by d_pρ_f⁄μ_f results in

  

$$\frac{d_{\text{p}}{v}_{\text{p}}{\rho }_{\text{f}}}{{\mu }_{\text{f}}}=\frac{1}{18}\cdot \frac{d_{\text{p}}^2{\rho }_{\text{f}}({\rho }_{\text{p}}-{\rho }_{\text{f}})g}{{\mu }_{\text{f}}^2}$$(9)

therefore,

  

$$Re=\frac{1}{18}\cdot Ar$$(10)

is obtained, where Re denotes the particle Reynolds number corresponding to the terminal settling velocity, and Ar represents the Archimedes number—a dimensionless number that expresses the ratio of buoyancy to viscous force in a gravitational field. Haider and Levenspiel corrected this equation to facilitate its application outside of Stokes region. The study proposed the following dimensionless correlation equation.¹⁵⁾

  

$$Re={\left(\frac{18}{Ar}+\frac{2.412}{4A{r}^{1/2}}\right)}^{-1}$$(11)

The term of 2.412⁄(4Ar^1⁄2) in Eq. (11) represents the difference from the Stokes law of Eq. (10). Equation (11) is the same as Eq. (10) of Stokes law when Ar≪1.

Figure 5 shows a plot of the experimental terminal velocity (falling ball method) on the Re-Ar diagram. The density and viscosity used in the calculation of dimensionless numbers are those evaluated by Eqs. (3) and (5), respectively. The settling rate of the particles in the foam were significantly lower than that of the terminal velocity of a single phase. Experimental results exhibited a tendency for deviation from the Haider and Levenspiel equation as the volume fraction of the gas phase of the foam increased. However, based on the results of a previous study by Martinsson et al. (gas phase volume fraction ϕ=0.73–0.85), this deviation was not only due to the changes in the volume fraction of the gas phase. The deviation from the Haider and Levenspiel equation (single-phase flow) was mainly influenced by the volume fraction of the gas phase; however, factors, such as the surface tension, should also be considered.

Fig. 5. Re-Ar diagram. (Online version in color.)

For particles that settle in a stationary fluid at terminal velocity, a dimensionless correlation equation is constructed using the following equation.

  

$$Re=f(Ar,\mathrm{\hspace{0.17em} }La,\mathrm{\hspace{0.17em} }\varphi )$$(12)

Where, La is the Laplace number, which is a dimensionless number expressing the ratio of viscous force to the product of inertial force and surface tension. Martinson showed that the falling velocity of particles tends to decrease as the surface tension of the foam increased.⁸⁾ Therefore, the terminal velocity of the particles (Re) may be correlated with a dimensionless number that includes surface tension. Several possible dimensionless numbers, such as the capillary and Weber numbers, were considered; however, the Laplace number, which exhibited the strongest correlation, was adopted in this study.

In this study, the rate of the settling velocity of the particles generally decreased with increasing foam ϕ and La number. Therefore, the following form was adopted as a function form of the dimensionless correlation equation.

  

$$Re={\left(\frac{18}{Ar}+\frac{2.412}{4\mathrm{\hspace{0.17em} }A{r}^{1/2}}\right)}^{-1}\frac{1}{1.0+\alpha \mathrm{\hspace{0.17em} }L{a}^{\beta }{\varphi }^{\gamma }}$$(13)

When the foam ϕ or La number is zero, Eq. (13) reduces to Eq. (11) (Haider and Levenspiel equation) for single-phase flow. By fitting the experimental data, coefficients α, β, and γ of Eq. (13) were determined as follows.

  

$$Re={\left(\frac{18}{Ar}+\frac{2.412}{4\mathrm{\hspace{0.17em} }A{r}^{1/2}}\right)}^{-1}\frac{1}{1.0+18.6\mathrm{\hspace{0.17em} }L{a}^{0.226}{\varphi }^{5.26}}$$(14)

The dependence of the La number and ϕ were exponents of 0.226 and 5.26, respectively (Eq. 14). These values were mainly dependent on ϕ.

Figure 6 shows the settling velocity of the particles evaluated using the dimensionless correlation equation with an experimental value. The experimental data and the correlation equation are consistent over a wide range, the difference between the experimental data and the correlation equation reaches a maximum of 5-fold. As a reference, the experimental data of Martinsson et al.^8,13) were also plotted (Fig. 6). The trends were similar to those observed in this study. The apparent viscosity of the foaming slag determined using the rotational method depended on γ (Eq. (5)); therefore, the settling velocity of the particles, and the apparent viscosity increased as the settling rate of the particles decreased. Additionally, the apparent viscosity decreased as the rate of the particles increased. Because the settling velocity of particles has such a dependency on the apparent viscosity, even a slight difference is greatly fed back, resulting in a significant difference in settling velocity. Therefore, the variation in the data (Fig. 6) was assumed to have occurred.

Fig. 6. The terminal settling velocity of the experiments and the correlation equation. (Online version in color.)

5. Estimating the Settling of Iron Particles in Foaming Slag

Using the proposed dimensionless correlation equation, the settling velocity of iron particles in the foaming slag was estimated. The physical properties used in the calculation¹⁶⁾ are shown in Table 3. Equations (5), (6), and (8) were used to estimate the apparent viscosity of the foam. Therefore, the viscosity of the foam depended on the falling velocity of the particles; simultaneously, the falling velocity depended on foam viscosity. Therefore, the equations are not closed, and an iterative method must be used. The diameter of the target iron particles was set between 10 µm to 10 mm, and the calculation was performed at a foam ϕ of 0.6, 0.7, 0.8 and 0.9.

Table 3. Properties used for estimation.¹⁶⁾

particle diameter, dp	10 μm–10 mm
density of particle, ρp	7580 kg⁄m3
density of molten slag, ρl	3500 kg⁄m3
viscosity of molten slag, μl	0.1 Pa s
surface tension of molten slag, σl	0.470 N⁄m
bubble diameter of foam, df	5 mm
volume fraction of gas, ϕ	0.6, 0.7, 0.8, and 0.9

Figure 7 shows the calculated results. The single-phase slag is represented by the line at ϕ=0. The rate of the settling velocity of the particles decreases when the slag foams. The higher the ϕ, the lower the rate of the settling velocity. A threshold in the particle diameter was observed. Below this threshold, the rate of the settling velocity of the particles significantly decreases until it reaches a value of zero. The apparent viscosity of the foaming slag determined using the rotational method is dependent on γ; additionally, the apparent viscosity increased as the rate of the particles decreased (Eq. 5). This feedback occurs when the particle diameter decreases to a value below that of the threshold and the rate of the settling velocity of the particles decreases to zero. The trapping of particles in the foaming fluid and the reduction of the rate of the settling velocity to zero are often observed. The presence of a threshold likely corresponds to this trap phenomenon.

Fig. 7. Estimation of the settling terminal velocity of iron particles in foaming slag. (Online version in color.)

In general, before the slag is discharged, it is allowed to settle and the remaining iron particles are recovered via sedimentation after blowing in the converter. Assuming a slag thickness and settling time of 5 m and 300 sec, respectively, the velocity required for all particles at the top of the slag to settle is 16.7 mm/s. For a foaming slag with a high ϕ (ϕ=0.9), all particles larger than 7 mm will settle and all particles smaller than 3 mm will be trapped (Fig. 7). For a foaming slag with a low ϕ (ϕ=0.6), all particles larger than 2 mm will settle and all particles smaller than 65 µm will be trapped. Because the settling velocity depends on ϕ, the ϕ must be reduced to increase the settling velocity.

Li et al.¹⁰⁾ showed that the size of the small and large iron particles recovered from emulsified slag were tens of micrometers and millimeters (1–10 mm), respectively. These include those recovered from the slag during operation and those recovered from the solution that escaped the converter, which were not necessarily the same as the iron particles in the slag during settling. However, the order of particle size was consistent with the predicted value in Fig. 7.

6. Conclusion

In this study, the falling velocity of particles in a foaming fluid was experimentally investigated, and the following conclusions were obtained.

(1) The apparent viscosity of the foam was significantly higher than that of only the liquid.

(2) The apparent viscosity of the foam measured by the falling ball method was higher than that measured by the rotational method; therefore, evaluating the settling velocity of the particles using the viscosity measured by the rotational method overestimates the terminal velocity.

(3) The following dimensionless correlation equation for predicting the terminal settling velocity of particles in the foam was developed.

  

$$Re={\left(\frac{18}{Ar}+\frac{2.412}{4\mathrm{\hspace{0.17em} }A{r}^{1/2}}\right)}^{-1}\frac{1}{1.0+18.6\mathrm{\hspace{0.17em} }L{a}^{0.226}{\varphi }^{5.26}}$$

(4) When the terminal settling velocity of iron particles in the foaming slag was evaluated using the obtained dimensionless correlation equation, its rate was lower than that of the single-phase slag. A particle diameter threshold was observed, and the iron particles below that threshold exhibited a settling velocity of zero and were trapped in the foam.

Acknowledgements

This research was supported by The Iron and Steel Institute of Japan’s “Visualization of Slag for Better Understanding of Multi-phase Melts Flow” study group. We would like to express our gratitude by writing here.

Symbols Used

a [−]: constant in the Einstein-Roscoe Equation

d [m]: diameter

g [m⁄s²]: gravitational acceleration

n [−]: exponent in the Einstein-Roscoe Equation

v [m⁄s]: velocity

x [−]: glycerol mass concentration

Dimensionless Numbers

Ar [−]: Archimedes number, $Ar=\frac{d_{\text{p}}^{3}g{\rho }_{\text{f}}({\rho }_{\text{p}}-{\rho }_{\text{f}})}{{\mu }_{\text{f}}^2}$

Ca [−]: capillary number, $Ca=\frac{{\mu }_{\text{l}}\gamma d_{\text{b}}}{{\sigma }_{\text{l}}}$

La [−]: Laplace number, $La=\frac{d_{\text{p}}{\sigma }_l{\rho }_{\text{f}}}{{\mu }_{\text{f}}^2}$

Re [−]: Reynolds number, $Re=\frac{d_{\text{p}}{v}_p{\rho }_{\text{f}}}{{\mu }_{\text{f}}}$

Greek Letters

γ [1/s]: shear rate

μ [Pa s]: viscosity

σ [N⁄m]: surface tension

ϕ [−]: gas phase volume fraction

Subscripts

b: bubble

f: foam

l: liquid

p: particle

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