Effect of Viscosity on Dynamic Evolution of Metallurgy Slag Foaming

Abstract

The viscosity is of high significance for the dynamic evolution of slag foaming in the metallurgical processes. Through physical modelling, the dynamic evolutions of slag foaming with different viscosities were analyzed by characterizing the foam structure. Furthermore, the effects of viscosity on the characteristic parameters of dynamic evolution were investigated. The experimental results indicated that the slag foaming types evolved along Partial Foaming I → Entire Foaming → Partial Foaming II with increasing superficial gas velocity, while the bubble shape of the top layers gradually transformed from the spherical to non-spherical shape. Moreover, the rise in the slag viscosity decreased the critical superficial gas velocity between the various slag foaming types by hindering the bubble shape transformation. Despite constant injection of gas, the “foam pseudo-decaying” phenomenon was observed after the liquid level of the foamed slag reached its maximum, and this phenomenon exacerbated under the higher viscosity conditions. Under the various superficial gas velocity conditions, the effects of viscosity on the characteristic parameters of the dynamic evolution presented a considerable difference. Under a lower superficial gas velocity, the viscosity had no obvious effect on the foaming and the decaying rates, whereas, under a higher superficial gas velocity, the foaming and the decaying rates decreased significantly with increasing viscosity. The foaming height reached the peak value when the slag foaming developed into the Entire Foaming, and this foaming height is basically consistent with the theoretical peak value. The superficial gas velocity required to reach this peak value decreased with increasing viscosity.

1. Introduction

Slag foaming is a common phenomenon throughout the metallurgical process, and plays an important role especially during the Basic Oxygen Furnace (BOF) steelmaking process.¹⁾ Moderate slag foaming can improve the kinetic conditions of the chemical reaction,²⁾ smelting effectiveness, and energy utilization efficiency.³⁾ However, insufficient or excessive slag foaming might reduce the smelting effectiveness and even trigger severe accidents as with splashing outside the furnace,^4,5) consequently leading to human injuries and equipment damage.⁶⁾ Therefore, the precise control of the dynamic evolution of metallurgy slag foaming is of crucial importance for the efficient and safe production.

In the metallurgical process, the foaming phenomenon of BOF slag is pretty typical. Moreover, the splashing accidents caused by the improper foaming are still the major hurdles in the development of the intelligent steelmaking. During the BOF smelting process, the constant changes in the slag physical properties including viscosity, surface tension and density would cause the extensive fluctuation of the slag foaming accordingly. The BOF slag viscosity varies usually in the range 50–1200 mPa·s or occasionally even more,⁷⁾ and is explicitly regarded as the most important factor of slag foaming among various investigated physical properties.^8,9,10) Specifically, the slag viscosity significantly affects the morphology, the motion, and the life of bubbles in the foamed slag.²¹⁾

The effect of viscosity on the slag foaming has been extensively investigated. In laboratory studies, an inert gas is usually injected into the molten slag or the simulating liquid of slag at a constant superficial gas velocity. Under this condition, the liquid height and the foam structure of slag would eventually reach the dynamic equilibrium, called the steady state of slag foaming. Fruehan et al.^{10,11,12,13,14,15)} revealed that the foaming height at steady state increased with increasing slag viscosity. Du et al.^16,17) used various types silicone oil to simulate metallurgical slag with different viscosities. The results indicated that the foaming height at steady state increased first followed by a decrease with increasing slag viscosity.^16,17) The effects of viscosity on the slag foaming presented a discrepancy between different studies. Moreover, existing studies^{10,11,12,13,14,15,16,17)} merely focused on the steady state of slag foaming while the previous researches^18,19,20) have explicitly demonstrated that the slag foaming is a complex dynamic evolution process. The viscosity plays rather different roles in the various stages of the dynamic evolution. Given the complexity of the viscosity effect, systematically studying the effect of viscosity on the dynamic evolution of slag foaming is necessary.

It is extremely difficult to observe and sample in the dynamic evolution of slag foaming through the high temperature simulation experiments. Consequently, the physical modelling was adopted to simulate the dynamic evolution of slag foaming under the various viscosity conditions in this investigation. The qualitative understanding of the effects of viscosity on dynamic evolution of slag foaming was obtained with the characteristic parameters including the foaming rate (u_foaming), the decaying rate (u_decaying) and the foaming height (Δh). The research results of this study are expected to provide a basis for the accurate control of slag foaming process.

2. Material and Methods

The BOF slag was simulated with glycerol solution. The physical properties of the BOF slag²¹⁾ and the experimental glycerol solution²²⁾ are listed in Table 1. To simulate the BOF slag with various viscosities, the glycerol solutions were prepared following the various mixture ratio of the water and the glycerol. The dynamic viscosity and the surface tension of the glycerol solution were determined using a digital viscosity meter (type: Digital Brookfield Model DV-III viscometer; USA) and GBX surface tension tester (type: 3S; France), respectively. The physical properties of slag (viscosity and surface tension) significantly affect the slag foaming.^10,17,23) Therefore, the Mo, the ratio of the viscosity force to the surface tension (Eq. (1)), was selected to measure the similarity between the BOF slag and the glycerol solution. The calculation results indicated that the Mo of the experimental glycerol solution ranged from 6.70×10⁻⁴ to 9.04×10⁻¹, which is located in the Mo range of the BOF slag 9.45×10⁻⁸ from 9.07×10⁻¹.   

$$Mo={\mu }^{4}g/{\sigma }^3\rho =v^4{\rho }^{3}g/{\sigma }^3$$(1)

where μ and v are the dynamic viscosity (Pa∙s) and the kinematic viscosity (m²/s) of liquid, respectively; ρ is the liquid density (kg/m³); σ is the liquid surface tension (N/m), and g is the gravity acceleration (m/s²).

Table 1. Physical properties of BOF slag and experimental glycerol solution.

	Temperature (°C)	Density (kg/m3)	Dynamic viscosity (mPa∙s)	Surface tension (mN/m)	Mo
BOF slag21)	1350	2800–3200	50–1200	200–600	9.45×10−8–9.07×10−1
Glycerol solution23)	(25±2)	1209–1247	68–397	62±2	6.70×10−4–9.04×10−1

A series of measures were taken to eliminate the effects of the temperature variation on the physical properties of the glycerol solution. Since the mixing process of the glycerol and the water was exothermic, the temperatures of glycerol solution were guaranteed to have been cooled to (25±2)°C before use. All experiments in this investigation were carried out at the same time every day to keep the stable ambient temperature. In addition, the glycerol solution was sealed using the plastic wrap to prevent it from absorbing the moisture in the air.

Figure 1 shows the schematic diagram of the experimental apparatus. The plexiglass vessel had a bottom length of 50 mm and a height of 1000 mm. A scale was attached to the outer wall of the plexiglass vessel, for ease of measuring the liquid height and the bubble size. The initial liquid height of the glycerol solution (volume: 250 mL) was 100 mm per experiment. Compressed air was injected into the glycerol solution through the dispersion of silicon oxide filter (pore size range: 16×10⁻⁶ to 30×10⁻⁶ m) above the gas chamber. The flow rate (Q) and pressure (P = 0.2 MPa) of the compressed air were controlled using a rotameter and a pressure stabilizing valve, respectively. The superficial gas velocity u_s is defined as the ratio of the flow rate of the compressed air to the cross-sectional area of the plexiglass vessel, as expressed in Eq. (2). A series of experiments were conducted under different viscosity (μ) and the superficial gas velocity (u_s) conditions, as listed in Table 2. The whole dynamic evolution of the glycerol solution was record using a camera (type: Nikon-3400; Japan). The videos showed the variations in the foam structure and the liquid height h with time t during the dynamic evolution of foaming. To ensure the reproducibility of the experimental results, the procedures were repeated at least thrice under the same experimental conditions.   

$$u_{\text{s}}=Q/A$$(2)

where u_s is the superficial gas velocity (m/s), Q is the flow rate of the compressed air (m³/s), and A is the cross-sectional area of the plexiglass vessel (m²).

Fig. 1.

Schematic diagram of experimental apparatus (1-Air compressor; 2-Gas tank; 3-Pressure stabilizing valve; 4-Flow meter; 5-Chamber; 6-Silicon oxide filter; 7-Flange; 8-Glycerol solution; 9-Plexiglass cylinder; 10-Video camera). (Online version in color.)

Table 2. Experimental viscosity and superficial gas velocity.

Q (mL/min) us (mm/s) μ (mPa·s)	15	30	45	60	75	90	500	1000	1500	2000	2500	3000	4000	6000	8000
68	0.10	0.20	0.30	0.40	0.50	0.60	3.33	6.67	10.00	13.33	16.67	20.00	26.67	40.00	53.33
115	0.10	0.20	0.30	0.40	0.50	0.60	3.33	6.67	10.00	13.33	16.67	20.00	–	–	–
181	0.10	0.20	0.30	0.40	0.50	0.60	3.33	6.67	10.00	13.33	–	–	–	–	–
397	0.10	0.20	0.30	0.40	0.50	0.60	3.33	6.67	10.00	–	–	–	–	–	–

3. Results and Discussion

3.1. Analysis of Dynamic Evolution of Slag Foaming with Different Viscosities

The slag foaming could be classified into three types, which have been introduced in detail in previous study.¹⁸⁾

3.1.1. Under the Viscosity of 68 mPa·s

At 68 mPa·s viscosity, the liquid height is plotted against time under various conditions of superficial gas velocity in Fig. 2.¹⁸⁾ When 0.10 mm/s ≤ u_s ≤ 0.50 mm/s, only part of liquid participated to form the foam layers stacked by the spherical bubbles at the steady state (Fig. 3(a)), called Partial Foaming I. As demonstrated previously,^24,25) the structure of this spherical foam was so stable that the dynamic evolutions of Partial Foaming I was gentle. When 0.50 mm/s < u_s ≤ 16.67 mm/s, all the liquid constituted the foam layers, in which the upper bubbles transformed into the non-sphere including the deformed and polyhedron bubbles at the steady state, called Entire Foaming. When 0.50 mm/s < u_s < 10.00 mm/s, the liquid height dropped marginally following its maximum value and then fluctuated moderately near a constant in the dynamic evolution process, because the foam structure stacked by the deformed bubbles (Fig. 3(b)) was less stable than by the sphere bubbles.²⁶⁾ When 10.00 mm/s ≤ u_s ≤ 16.67 mm/s, the liquid height dropped sharply from its maximum value, and then fluctuated frequently and violently around a constant, due to the extremely poor stability of the foam structure stacked by the polyhedron bubbles (Fig. 3(c)).^10,27) When 16.67 mm/s < u_s < 53.33 mm/s, it backed to the state where just a part of the liquid participated in the formation of the foam layers though the bubbles in the top foam structure still were the polyhedron shape at the steady state, called Partial Foaming II. Its occurrence could be put down to the shorter gas residence time caused by the acceleration of bubbles rising and collapse with the larger bubble size and the violent disturbance under these extremely high values of u_s.¹⁴⁾ When u_s ≥ 53.33 mm/s, the high-speed gas flow penetrated the liquid phase, leading to the disappearance of the foam.¹⁸⁾

Fig. 2.

Variation of liquid height with time at a viscosity of 68 mPa·s during the foaming evolution (a) With the continuous gas injection; (b) Foam decaying curve following gas interruption.¹⁸⁾ (Online version in color.)

Fig. 3.

Various types of foam structures (68 mPa·s) (a) Spherical bubble; (b) Deformed bubble; (c) Polyhedral bubble. (Online version in color.)

3.1.2. Under the Viscosity of 115 mPa·s

When the viscosity increased to 115 mPa·s, the liquid height against time variation is presented under various conditions of superficial gas velocity in Fig. 4. According to the foam structure characteristics at the steady state and the change rule of h with t in Fig. 4, when 0.10 mm/s ≤ u_s ≤ 0.20 mm/s, the Partial Foaming I occurred; when 0.20 mm/s < u_s ≤ 3.33 mm/s, the slag foaming transformed into the Entire Foaming and when 3.33 mm/s < u_s < 20.00 mm/s, it developed into the Partial Foaming II.

Fig. 4.

Variation of liquid height with time at a viscosity of 115 mPa·s during the foaming evolution (a) With the continuous gas injection; (b) Foam decaying curve following gas interruption. (Online version in color.)

Figure 5 presents the foam structures against the viscosity variation. Compared the bubble size at the same superficial gas velocity and the height position, it was obvious from Figs. 5(a), 5(d) to 5(g) that there is no significant change in the bubble size with increasing viscosity under the lower superficial gas velocity, while the bubble apparently grew bigger from Figs. 5(c), 5(f) to 5(i) with increasing viscosity under the higher superficial gas velocity.

Fig. 5.

Foam structures against the viscosity variation (a)–(c) 68 mPa·s; (d)–(f) 115 mPa·s; (g)–(i) 181 mPa·s. (Online version in color.)

Notably, no polyhedral bubbles were formed with the viscosity of 115 mPa·s and 181 mPa·s as in Fig. 5 whereas they were found with the viscosity of 68 mPa·s as in Fig. 3(c). The bubble shape transformation from the sphere to the polyhedron was mainly driven by the drainage behavior. This drainage is due to the action of gravity,^28,29,30) but hindered strongly by the viscous force.³¹⁾ Galileo number (Ga) is equal to the ratio of the gravity to the viscous force, as expressed in Eq. (3). As a result, the Ga could be applied to evaluate the bubble shape transformation in this study. Obviously, the Ga decreased with increasing viscosity, indicating that the higher the viscosity, the more difficult will be the bubble shape transformation from the sphere to the polyhedron. Therefore, the increase in viscosity could hinder the bubble deformation.   

$$Ga=\frac{l^3{\rho }^2{g}^2}{\mu }$$(3)

where l is the characteristic length (m).

3.1.3. Under the Viscosity of 181 mPa·s

When the viscosity increased to 181 mPa∙s, the liquid height is plotted against time under various conditions of superficial gas velocity in Fig. 6. According to the foam structure characteristics at the steady state and the change rule of h with t shown in Fig. 6, when the u_s was 0.10 mm/s, the Partial Foaming I occurred (Fig. 5(g)); when 0.10 mm/s < u_s ≤ 0.50 mm/s, the slag foaming changed into the Entire Foaming (Fig. 5(h)) and when 0.50 mm/s < u_s < 13.33 mm/s, developed into the Partial Foaming II (Fig. 5(i)).

Fig. 6.

Variation of liquid height with time at a viscosity of 181 mPa·s during the foaming evolution (a) With the continuous gas injection; (b) Foam decaying curve following gas interruption. (Online version in color.)

The superficial gas velocity ranges of Partial Foaming I, Entire Foaming, and Partial Foaming II are listed in Table 3. The slag foaming evolved along the Partial Foaming I→Entire Foaming→Partial Foaming II with increasing superficial gas velocity. Simultaneously, the bubble shape transformed from the spherical to the non-spherical shape including the deformed and the polyhedron bubbles in the top foam layers.

Table 3. Superficial gas velocity ranges of the various types with different viscosity conditions.

Types Viscosity	Partial Foaming I	Entire foaming	Partial Foaming II
68 mPa∙s	0.10 mm/s≤us≤0.50 mm/s	0.50 mm/s<us≤16.67 mm/s	16.67 mm/s<us<53.33 mm/s
115 mPa∙s	0.10 mm/s≤us≤0.20 mm/s	0.20 mm/s<us≤3.33 mm/s	3.33 mm/s<us<20.00 mm/s
181 mPa∙s	us=0.10 mm/s	0.10 mm/s<us≤0.50 mm/s	0.50 mm/s<us<13.33 mm/s

The classification of foaming type was believed to depend to a large degree on slag viscosity.³²⁾ The required conditions for Partial Foaming I, Entire Foaming, and Partial Foaming II are illustrated in Fig. 7. The increase of the viscosity could decrease the critical superficial gas velocity between the various slag foaming types. Since the rise in the viscosity could prevent the foam structure from transforming into the polyhedron with the highest collapse rate, the lower superficial gas velocity was required to reach the critical gas fraction for the foam structure transformation under higher viscosity, thereby triggering the evolution of slag foaming types.

Fig. 7.

Required conditions region for Partial Foaming I, Entire Foaming and Partial Foaming II (× point-experimental critical superficial gas velocity). (Online version in color.)

In generally, the foam will decay after the gas interruption. During the foam decaying, the foam layer thickness would decrease due to the gas escape from the foam structure caused by the bubble collapse. It is worth noting that, even if the gas was continuously injected into the liquid at a constant rate, the foam decaying in various degrees was observed after the liquid height reached to its maximum during the dynamic evolution of Entire Foaming and Partial Foaming II, as presented in Figs. 2(a), 4(a) and 6(a). This phenomenon was defined as “foam pseudo-decaying” in this investigation. Under a viscosity of 68 mPa·s, the “foam pseudo-decaying” phenomenon turned up in the manner of a sudden foam collapse and an instantaneous drop of the liquid height (the grey area in Fig. 2(a)). In contrast, the rate of foam collapse in the “foam pseudo-decaying” process decreased significantly under the higher viscosity of 115 mPa·s and 181 mPa·s, accompanied by the continuous reduction in the liquid height over a span of time (Figs. 4(a) and 6(a), respectively).

In essence, the reduction of liquid height in the “foam pseudo-decaying” meant that the gas escape at the top was greater than the gas supply at the bottom. During the initial period of the gas supply, the number of bubbles was relatively small and the newborn bubbles still remained within their lifetime. Without gas escaping, a large number of bubbles gradually accumulated in this period. The liquid height kept rising with time during the initial period. When the accumulated bubbles collapsed at large scale so that the rate of gas escape exceeded the rate of gas supply, the “foam pseudo-decaying” phenomenon evolved.^31,33) With increasing viscosity, the “foam pseudo-decaying” behave differently between the sudden foam collapse and the slow foam collapse, because the increase in the viscosity prevented the bubble from deforming, consequently bringing with the more stable foam structure. Therefore, the main reason for the “foam pseudo-decaying” phenomenon was the large-scale and intensive collapse of the accumulated bubbles in the initial period.^31,33)

With the consumption of the accumulated bubbles in the initial period, the gas escape rate reduced to be equal to the gas supply rate again. Subsequently, the liquid height would be the dynamic equilibrium (the steady state) without further reduction, marking the end of the “foam pseudo-decaying” phenomenon.

Moreover, the “foam pseudo-decaying” phenomenon was exacerbated under the higher viscosity conditions. For example, the liquid height decreased drastically even approaching to the initial level with the viscosity of 397 mPa·s, as shown in Fig. 8. As the viscosity increased, the bubble accumulation became more serious in the initial period, because of the time extension of the bubble rising motion and the bubble life. Consequently, the “foam pseudo-decaying” phenomenon turned more prominent.

Fig. 8.

Variation of liquid height with time at a viscosity of 397 mPa·s during the foaming evolution (a) With the continuous gas injection; (b) Foam decaying curve following gas interruption. (Online version in color.)

In summary, based on the foam structure characteristics at the steady state and the change rule of h with t, the slag foaming was classified into three types. With increasing superficial gas velocity, the slag foaming types evolved along Partial Foaming I → Entire Foaming → Partial Foaming II while the bubbles shape gradually transformed from the spherical to the non-spherical shape in the foam structure of the top layers. With increasing viscosity, the critical superficial gas velocity between the various slag foaming types was reduced by hindering the bubble shape deformation. The “foam pseudo-decaying” phenomenon exacerbated under the higher viscosity conditions.

3.2. Effects of Viscosity on Characteristic Parameters of the Dynamic Evolution of Slag Foaming

The dynamic evolution of slag foaming could be divided into following stages: foaming stage→equilibrium stage→foam decaying stage (Fig. 9). To quantitatively describe the effects of viscosity on the dynamic evolution of slag foaming, the foaming rate (u_foaming), the decaying rate (u_decaying) and the foaming height (Δh) were defined as the characteristic parameters, which were calculated by Eqs. (4), (6) and (7), respectively.   

$$\mathrm{\Delta }h=h_{max}-h_0$$(4)

where Δh is the foaming height (m), h_max is the maximum liquid height in the foaming stage under a certain superficial gas velocity (m), and h₀ is the initial liquid height (m).

Fig. 9.

Schematic diagram of the dynamic evolution of slag foaming. (Online version in color.)

The average time $\overline{t}$ of the foaming stage could be denoted:   

$$\overline{t}=\frac{\underset{t_0}{\overset{t_1}{\int }}(h-h_0)dt}{\mathrm{\Delta }h}$$(5)

where t₀ and t₁ are the beginning and end moment of the foaming stage (s), respectively

Furthermore, the foam volume produced per unit time was regarded as u_foaming, as expressed by Eq. (6):   

$$u_{\text{foam}}=\frac{\mathrm{\Delta }h}{\overline{t}}A=\frac{{(\mathrm{\Delta }h)}^{2}A}{\underset{t_0}{\overset{t_1}{\int }}(h-h_0)dt}$$(6)

where u_foaming is the foaming rate (m³/s), and A is the cross-sectional area of the plexiglass vessel (m²).   

$$u_{\text{decaying}}=\frac{{(h^{\theta }-h_0)}^2\cdot A}{\left[\underset{t_2}{\overset{t_3}{\int }}(h-h_0)dt\right]}$$(7)

where u_decaying is the decaying rate (m³/s), h^θ is the liquid height at the start of foam decaying (m), and t₂ and t₃ are the time points at the start and end of the foam decay stage (s), respectively.

Figure 10 presents the foaming rate at various viscosity values. Under a lower superficial gas velocity (0.10 mm/s ≤ u_s ≤ 0.60 mm/s), the viscosity had almost no obvious effect on the foaming rate, because the bubble size remained basically unchanged with increasing viscosity, which could be demonstrated by comparing Figs. 5(a), 5(d) and 5(g). Under a higher superficial gas velocity (0.60 mm/s < u_s < 53.33 mm/s), the foaming rate decreased with increasing viscosity, due to the growing size of bubbles from Figs. 5(c), 5(f) to 5(i). These larger size bubbles would lead to the higher rate of gas escape during the foaming stage, because of the acceleration of bubble rising and collapse. Consequently, the foaming rate decreased with increasing of viscosity at higher superficial gas velocity.

Fig. 10.

Foaming rate u_foaming under various viscosity conditions. (Online version in color.)

Figure 11 presents the decaying rate under various viscosity conditions. Under a lower superficial gas velocity (0.10 mm/s ≤ u_s ≤ 0.60 mm/s), the decaying rate only changed slightly with increasing viscosity. Under a higher superficial gas velocity (0.60 mm/s < u_s < 53.33 mm/s), the decaying rate slowed down with increasing viscosity. Combining with the foam structure shown in Fig. 5, the decaying rate of the foam stacked by the spherical bubbles (Figs. 5(a), 5(d) and 5(g)) was almost unaffected by viscosity, while that of the foam formed by the non-spherical bubbles (Figs. 5(b), 5(c), 5(e), 5(f), 5(h) and 5(i)) was reduced with increasing viscosity, attributing to different foam decaying mechanisms. For the foam stacked by the spherical bubbles, the gas diffusion among bubbles was the dominant cause for decay, and the effect of viscosity on the gas diffusion rate was negligible. However, for the foam stacked by the non-spherical bubbles, the main driving force to decay was the drainage behavior,^{34,35,36,37,38,39)} which retarded with increasing of viscosity.^38,40,41,42) In addition, these results also indicated that the decaying rate of the foam stacked by the spherical bubbles was much lower than that of the foam stacked by the non-spherical bubbles, again proving that the stability of the foam stacked by the spherical bubbles was indeed higher than that of the foam stacked by the non-spherical bubbles.

Fig. 11.

Decaying rate u_decaying under various viscosity conditions. (Online version in color.)

Figure 12 presents the foaming height at various viscosity values. With rising viscosity, the foaming height increased under a lower superficial gas velocity (0.10 mm/s ≤ u_s ≤ 0.40 mm/s), whereas, the foaming height decreased under a higher superficial gas velocity (6.67 mm/s < u_s < 53.33 mm/s). These results revealed that the effect of viscosity on the foaming height varied with the change in superficial gas velocity, explaining the discrepancy in the literature results.^{10,11,12,13,14,15,16,17)} Under a certain superficial gas velocity, the effect of viscosity on the foaming height conformed to the following rules. With increasing of viscosity, the foaming height of Partial Foaming I increased, there was no apparent variation in the foaming height of Entire Foaming, and the foaming height of Partial Foaming II decreased significantly. When the multiple slag foaming types presented under a certain superficial gas velocity, the foaming height of Entire Foaming was higher than that of Partial Foaming I /Partial Foaming II.

Fig. 12.

Foaming height Δh under various viscosity conditions. (Online version in color.)

The foaming height always reached nearly the same peak (about 4.30 times of the initial liquid height) when the slag foaming developed into the Entire Foaming under various viscosity conditions. This reflects that the maximum gas fraction of the liquid-gas multiphase system with various viscosities was almost equal. The peak of foaming height has a huge guiding significance for the regulation of the BOF slag foaming. Therefore, this peak value of foaming height would be theoretically deduced, as discussed below.

Since the local gas fraction φ usually varied with the location and time, an average gas fraction $\overline{\varphi }$ should be defined for a particular foaming system. The relationship between the foaming height and the average gas fraction was expressed by Eq. (8).⁴³⁾ Based on the maximum gas fraction of the foam stacked by the spherical bubbles of 0.74, the foaming height of the foam stacked by the spherical bubbles (Partial Foaming I) should be no more than 2.86 times of the initial liquid height.   

$$\begin{array}{l}\overline{\varphi }\text{(}t\text{)}=V_g\text{(}t\text{)/}V\text{(}t\text{)}\\ \mathrm{\hspace{1em}\hspace{1em}}=\left[V\text{(}t\text{)}-V_l\text{(}t\text{)}\right]\text{/}V\text{(}t\text{)}\approx \left[h\text{(}t\text{)}-h_0\right]/h\text{(}t\text{)}\approx \mathrm{\Delta }h\text{(}t\text{)}/h\text{(}t\text{)}\end{array}$$(8)

where $\overline{\varphi }(t)$ is the average gas fraction at time t, V_g(t), and V_l(t) are the volume of the gas and liquid at time t, respectively (m³), and h(t) and Δh(t) are the liquid height and the foaming height at time t, respectively (m).

For Entire Foaming, the bottom and top of the foam layers were stacked by the spherical bubbles and the polyhedral bubbles, respectively. It could be assumed that the gas fraction at the bottom and top foam layers were 0.74 and 1, respectively.^31,32,44) Based on some boundary conditions, the literature³²⁾ finally calculated that the average gas fraction of Entire Foaming $\overline{\varphi }(t)$ should be 0.82. Correspondingly, the peak foaming height is 4.55 times of the initial liquid height in theory according to Eq. (8), basically consistent with the above-mentioned experimental value (4.30 times the initial liquid height) in Fig. 12.

Although the peaks of foaming height were approximately equal under various viscosity conditions, the superficial gas velocity required to reach this peak was reduced obviously with increasing viscosity (Fig. 12).

4. Conclusions

In conclusion, the slag foaming was investigated by physical modeling according to the similarity principle via Morton number. The dynamic evolution of slag foaming under various viscosity conditions was analyzed in combination with the foam structure characteristics. The effects of viscosity on the characteristic parameters of dynamic evolution of slag foaming were quantitatively described, including the foaming rate, the decaying rate and the foaming height.

(1) The slag foaming types evolved along Partial Foaming I→Entire Foaming→Partial Foaming II with increasing superficial gas velocity. Correspondingly, the bubble shape gradually transformed from the spherical to non-spherical shape including deformed and polyhedron shape bubbles in the foam structure of the top layers. The rise of viscosity decreased the critical superficial gas velocity between the various slag foaming types by hindering the bubble shape transformation.

(2) Despite keeping constantly injecting gas, the “foam pseudo-decaying” phenomenon was observed after the liquid level reaching its maximum value in the dynamic evolution process, and this phenomenon exacerbated at the higher viscosity, because of the large-scale and intensive collapse of the accumulated bubbles in the initial period.

(3) Under a lower superficial gas velocity (0.10 mm/s ≤ u_s ≤ 0.60 mm/s), there was no obvious change in the foaming and decaying rates with various viscosities; whereas, under a higher superficial gas velocity (0.60 mm/s < u_s < 53.33 mm/s), the foaming and decaying rates decreased significantly with increasing of viscosity.

(4) The foaming height reached to its peak value when the slag foaming developed into the Entire Foaming, and this value was approximately 4.55 times the initial liquid height in theory, basically consistent with the experimental value (about 4.30 times the initial liquid height). The superficial gas velocity required to reach this peak value decreased with increasing viscosity.

Acknowledgments

Financial support to this project is provided by National Key R & D Program of China, China, Grant No. 2017YFC0805100; National Natural Science Foundation of China, China, Grant No. 51774087; Liaoning Provincial Natural Science Foundation of China, China, Grant No. 2019-MS-123 and Fundamental Research Funds for the Central Universities, China, Grant No. N180725008.

References

• 1)   C.  Kattenbelt,  E.  Spelbos,  P.  Mink and  B.  Roffel: Steel Res. Int., 79 (2008), 821. https://doi.org/10.1002/srin.200806205
• 2)   G.  Qiu,  C.  Shan,  X.  Zhang and  X.  Lv: Ironmaking Steelmaking, 44 (2017), 246. https://doi.org/10.1080/03019233.2016.1210360
• 3)   Y.  Zhang and  R. J.  Fruehan: Metall. Mater. Trans. B, 26 (1995), 813. https://doi.org/10.1007/BF02651728
• 4)   S. M.  Jung and  R. J.  Fruehan: ISIJ Int., 40 (2000), 348. https://doi.org/10.2355/isijinternational.40.348
• 5)   B.  Deo,  B. A.  Overbosch,  B.  Snoeijer,  D.  Das and  K.  Srinivas: Trans. Indian Inst. Met., 66 (2013), 543. https://doi.org/10.1007/s12666-013-0306-2
• 6)   J.  Ruuska,  A.  Sorsa,  S.  Ollila and  K.  Leiviskä: IFAC-PapersOnLine, 48 (2015), 171. https://doi.org/10.1016/j.ifacol.2015.10.098
• 7)   A. A.  Kozhukhov: Steel Transl., 44 (2014), 136. https://doi.org/10.3103/S0967091214020107
• 8)   R.  Jiang and  R. J.  Fruehan: Metall. Trans. B, 22 (1991), 481. https://doi.org/10.1007/BF02654286
• 9)   S.  Barella,  A.  Gruttadauria,  C.  Mapelli and  D.  Mombelli: Ironmaking Steelmaking, 39 (2012), 463. https://doi.org/10.1179/1743281212Y.0000000009
• 10)   Y.  Zhang and  R. J.  Fruehan: Metall. Mater. Trans. B, 26 (1995), 803. https://doi.org/10.1007/BF02651727
• 11)   K.  Ito and  R. J.  Fruehan: Metall. Trans. B, 20 (1989), 509. https://doi.org/10.1007/BF02654600
• 12)   S. S.  Ghag,  P. C.  Hayes and  H. G.  Lee: ISIJ Int., 38 (1998), 1201. https://doi.org/10.2355/isijinternational.38.1201
• 13)   S. S.  Ghag,  P. C.  Hayes and  H. G.  Lee: ISIJ Int., 38 (1998), 1216. https://doi.org/10.2355/isijinternational.38.1216
• 14)   L.  Pilon,  A. G.  Fedorov and  R.  Viskanta: J. Colloid Interface Sci., 242 (2001), 425. https://doi.org/10.1006/jcis.2001.7802
• 15)   D.  Lotun and  L.  Pilon: ISIJ Int., 45 (2005), 835. https://doi.org/10.2355/isijinternational.45.835
• 16)   M. Y.  Zhu and  D.  Sichen: Steel Res., 71 (2000), 76. https://doi.org/10.1002/srin.200005693
• 17)   L. S.  Wu,  G. J.  Albertsson and  D.  Sichen: Ironmaking Steelmaking, 37 (2010), 612. https://doi.org/10.1179/030192310X12690127076550
• 18)   R. F.  Wang,  B.  Zhang,  C. J.  Liu and  M. F.  Jiang: Exp. Therm. Fluid Sci., 113 (2020), 110041. https://doi.org/10.1016/j.expthermflusci.2020.110041
• 19)   A.  Kapilashrami,  M.  Görnerup,  A. K.  Lahiri and  S.  Seetharaman: Metall. Mater. Trans. B, 37 (2006), 109. https://doi.org/10.1007/s11663-006-0090-z
• 20)   R. D.  Morales,  H.  Rodríguez-Hernández,  A.  Vargas-Zamora and  A. N.  Conejo: Ironmaking Steelmaking, 29 (2002), 445. https://doi.org/10.1179/030192302225004629
• 21)   R. J.  Fruehan: The Making, Shaping and Treating of Steel: Steelmaking and Refining Volume, The AISE Steel Foundation, Pittsburgh, PA, (1998), 101.
• 22)   K.  Takamura,  H.  Fischer and  N. R.  Morrow: J. Pet. Sci. Eng., 98–99 (2012), 50. https://doi.org/10.1016/j.petrol.2012.09.003
• 23)   D.  Exerowa and  P. M.  Kruglyakov: Foam and Foam Films: Theory, Experiment, Application, Elsevier, Amsterdam, (1997), 4.
• 24)   K.  Wu,  W.  Qian,  S. J.  Chu,  Q.  Niu and  H. W.  Luo: ISIJ Int., 40 (2000), 954. https://doi.org/10.2355/isijinternational.40.954
• 25)   C.  Nexhip,  S.  Sun and  S.  Jahanshahi: Int. Mater. Rev., 49 (2004), 286. https://www.tandfonline.com/doi/abs/10.1179/095066004225021945
• 26)   T. X.  Zhu,  K. S.  Coley and  G. A.  Irons: Metall. Mater. Trans. B, 43 (2012), 751. https://doi.org/10.1007/s11663-012-9654-2
• 27)   R. A. M.  de Almeida,  D.  Vieira,  W. V.  Bielefeldt and  A. C. F.  Vilela: Mat. Res., 20 (2017), 474. https://doi.org/10.1590/1980-5373-MR-2016-0059
• 28)   D. D.  Joseph: J. Fluids Eng., 119 (1997), 497.
• 29)   J. T.  Davies: Interfacial Phenomena, Elsevier, Amsterdam, (2012), 398.
• 30)   P.  Walstra: Foams: Physics, Chemistry and Structure, Springer, London, (1989), 2.
• 31)   J. A.  Attia,  S.  Kholi and  L.  Pilon: Colloids Surf. A, 436 (2013), 1000. https://doi.org/10.1016/j.colsurfa.2013.08.025
• 32)   L.  Pilon,  A. G.  Fedorov and  R.  Viskanta: Chem. Eng. Sci., 57 (2002), 977. https://doi.org/10.1016/S0009-2509(02)00010-6
• 33)   S.  Hartland and  A. D.  Barber: Trans. Inst. Chem. Eng., 52 (1974), 43.
• 34)   V.  Gergely and  T. W.  Clyne: Acta Mater., 52 (2004), 3047. https://doi.org/10.1016/j.actamat.2004.03.007
• 35)   M.  Tong,  K.  Cole and  S. J.  Neethling: Colloids Surf. A, 382 (2011), 42. https://doi.org/10.1016/j.colsurfa.2010.11.007
• 36)   J. L.  Wang,  A. V.  Nguyen and  S.  Farrokhpay: Adv. Colloid Interface Sci., 228 (2016), 55. https://doi.org/10.1016/j.cis.2015.11.009
• 37)   R.  Lemlich: Ind. Eng. Chem. Fundam., 17 (1978), 89. https://doi.org/10.1021/i160066a003
• 38)   P.  Stevenson: Chem. Eng. Sci., 61 (2006), 4503. https://doi.org/10.1016/j.ces.2006.02.026
• 39)   A. V.  Nguyen: J. Colloid Interface Sci., 249 (2002), 194. https://doi.org/10.1006/jcis.2001.8176
• 40)   J. J.  Bikerman: Sep. Sci., 7 (1972), 647. https://doi.org/10.1080/00372367208057973
• 41)   J. J.  Bikerman: Foams, Springer Science & Business Media, New York, (2013), 159.
• 42)   A.  Bhakta and  E.  Ruckenstein: Adv. Colloid Interface Sci., 70 (1997), 1. https://doi.org/10.1016/S0001-8686(97)00031-6
• 43)   J.  Martinsson,  B.  Glaser and  D.  Sichen: Ironmaking Steelmaking, 46 (2019), 777. https://doi.org/10.1080/03019233.2017.1410950
• 44)   J.  Wang,  A. V.  Nguyen and  S.  Farrokhpay: Adv. Colloid Interface Sci., 228 (2016), 55. https://doi.org/10.1016/j.cis.2015.11.009