Effect of Impellers’ Shape on Bubbles Dispersion Characteristics in Stirring-Injection Desulfurization System of Hot Metal

Abstract

Four types of impellers are developed to improve bubbles disintegration and dispersion characteristics in the desulfurization process of hot metal with stirring-injection method. The disintegration and dispersion behavior of bubbles are investigated by using the Euler two-phase flows approach coupled with the bubble population balance model. The bubbles transported sideward of different impellers and broken in the region near the top of impellers are well studied. Further, simulations have also been performed to understand the sensitivity of impellers types on rotation speed and surface tension. The results show that the square center impeller (impeller B) improves the flow structure of the hot metal, and makes the bubbles size much smaller than that of the other three impellers types. The blades with downward angle (impeller C and impeller D) improve the dispersion effect of the region near the impeller, and the impeller B is more sensitive to the change of the rotation speed. The best dispersion effect can be obtained by use of the impeller B, under the conditions of stirring speed with 30 rpm for 4 s switching.

1. Introduction

The hot metal desulfurization is an essential means to produce high added value steel, and the magnesium granules injection method is widely used in this process. The phenomenon of low utilization rate of desulfurizer and low desulfurization efficiency will happen in deep level desulfurization process by this method.^1,2,3) Due to the high price of magnesium, the production cost of enterprises rise. The reason for the low utilization rate of magnesium is that the bubbles emitted from the nozzle of lance are large in size, leading to the gathering of bubbles in the region around the lance and escaping rapidly, which cannot disperse in the ladle uniformly to participate in the reaction. Therefore, the key way to improve the utilization rate of magnesium is to break the large bubbles and disperse them uniformly into the ladle to participate in the reaction during the desulfurization process.

At present, in order to improve the utilization rate of magnesium and the desulfurization rate of injection method, scholars researched the bubble behavior during desulfurization;⁴⁾ studied the effect of temperature on desulfurization efficiency;⁵⁾ improved the adding mode of desulfurizer;⁶⁾ changed the nozzle structure and the spraying position of the lance;^7,8,9) but the improvement effects on the uniformity of bubbles distribution of desulfurizer are not obviously. Sigarev et al.¹⁰⁾ studied the magnesium granules injection desulfurization process through a rotating submersible tuyere, they found this type of tuyere providing high desulfurization efficiency in the process. Su et al.¹¹⁾ put forward a new method of hot metal desulfurization by bottom-blowing magnesium vapor combined with mechanical agitation, they indicated that the desulfurization with magnesium vapor proceeds easier than the desulfurization with magnesium powder relatively. However, the method of bottom-blowing magnesium vapor adds additional gasification equipment and operation system. Torres et al.¹²⁾ discussed of the effects of gas injection on the mixing process in KR mechanical stirring method, they found that the gas injection can reduce the mixing time up to 20% and increase solid dispersion up to 40%. A method had been proposed by Sano, in which the gas injection coupling mechanical stirring with an impeller was used to disintegrate bubbles and accelerate the mixing of hot metal with the magnesium bubbles. The stirring-injection system has been investigated under different stirring patterns in water model experiments.^13,14)

The power of disintegrating and dispersing bubbles comes from the impeller in the stirring-injection desulfurization process, and the mixing efficiency of desulfurizer and hot metal is also related closely to the impellers types. From the literature review, it is known that the mixing performance of impellers in gas-liquid two-phase flow needs to be investigated as very few researches have been devoted to the effects of impeller style on the gas-liquid dispersion by the numerical simulation. In addition, most of the present researches of the gas-liquid behavior are carried out in air-water system, which are different from the real high temperature and harsh working environment of the nitrogen-hot metal system. Some similarity criteria such as Weber number, Reynolds number, and Froud number etc. are unable to simultaneously meet in the lab vessel, resulting some of the essential details in the desulfurization process are overlooked. Therefore, it is necessary to analyze the bubbles behavior in the real hot metal treating processes environment.

In this work, four types of impellers which suitable for high temperature working environment are developed, the impellers are characterized by changing the center shape or setting the downward angle of the blades. The disintegration and dispersion characteristics of bubbles are studied by using four types of impellers, so as to achieve the ideal state of bubbles broken (most of bubbles sizes are less than 6 mm, and the region of gas volume fraction with 0.01–0.015 more than 60% at gas flow rate of 100–120 m³/h) by stirring-injection method. It provides a theoretical basis for further industrialization of the stirring-injection method.

2. Model Building

2.1. Physical Model Strategy

Based on the 350 tons hot metal ladle of some KR method steel plant as the prototype, simplify the ladle into the cylindrical container, contains with 1300°C hot metal, the stirring-injection impeller is inserted into the hot metal vertically conducting injection and stirring. The nozzle is located on the rotation axis under the impeller, the impeller axis center coincides with the hot metal ladle, and the stirring-injection system diagram is shown in Fig. 1, the main parameters of model are listed in Table 1.

Fig. 1.

Stirring-injection system diagram.

Table 1. Parameters of physical model.

Parameter	Hot metal system value	Water model value
Diameter of vessel (m)	4	0.44
Height of vessel (m)	5	0.5
Height of level (m)	4.3	0.37
Distance from impeller bottom to ladle bottom (m)	0.4	0.07
Nozzle diameter (m)	0.02	0.002
Distance from nozzle to vessel bottom (m)	0.35	0.06
Surface tension (N/m)	1.8	0.072
Liquid density (kg/m3)	7800	1000
Gas density (kg/m3 )	0.23	1.2
Liquid viscosity (Pa·s)	0.0063	0.001
Gas viscosity (Pa·s)	5.3 × 10−5	1.8 × 10−5
Gas flow rate (Nm3h−1)	100	1.0

Four simple impellers coated with refractory materials have been developed to meet harsh conditions of high temperature and corrosion. The four types of impellers and dimensions are shown in Fig. 2. The flow state of the fluid on the back of the blades would be improved by setting a special-shaped structure at the center of the blades (the impeller B with square structure, the impeller C with disc structure, and the impeller D with twist structure, respectively). The hot metal diffusion ability around the blades would be increased by setting the downward angle of the blades (the impeller C and the impeller D) and the hypotenuse of the blades (the impeller D), so as to improve the disintegrating bubbles ability of impellers in the region near the impellers. The disintegration and dispersion bubbles effect of the three new types of impellers (impeller B, C, and D) are compared with the traditional impeller type A.

Fig. 2.

Diagram of impellers. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D.

Due to the limitation of wear and corrosion of the impeller and ladle in hot metal, the high-speed rotation mode cannot be adopted in stirring-injection process. However, the bubbles cannot be broken when the rotation speed is less than the “gas flooding speed”. In this view, the variable speed stirring mode is adopted, and the bubbles disintegration and dispersion by using the power generated with the instantaneous speed change in variable speed mode. The variable speed mode is conducted with clockwise-anticlockwise rotation as shown in Fig. 3. The switch period of impeller is set as T, and the impeller rotates at angular velocities of N and -N in the former T/2 and the later T/2, respectively. Because the surface tension of hot metal is high, so the bubbles can be broken with a small speed change. In present work, a speed change mode of less than 30 rpm for 4 s switching is used.

Fig. 3.

Variable speed rotation diagram.

In order to analyze the effect of different impellers types on the bubble disintegration and dispersion behavior, the desulfurization process is simplified as follows,

(1) The influence of slag is ignored and only the disintegration and dispersion characteristics of bubbles are considered.

(2) The temperature drop of hot metal is ignored and it is regarded as an isothermal process.

(3) The effects of desulfurization reaction and mass transfer are not considered.

(4) The magnesium granular are vaporized and mixed with the carrier gas before passing through the nozzle of the stirring-injection impeller, and using the nitrogen instead of the mixing gas.

2.2. Mathematical Modeling

The gas-liquid two phases are coupled in the process of bubbles disintegration and dispersion in hot metal ladle, and the multiple size group equation of bubble population balance model coupling with Euler two-phase flow model are developed to deal with the two-phase flows in which the dispersed phase features a large variation in its characteristic sizes.

2.2.1. Euler Multiphase Hydrodynamic Equations

The equations for the liquid and bubbles are derived based on the assumption that both phases could be described as the continua. Therefore, the whole system is considered as the “Euler model”. The mass conservation equation and momentum conservation equation are shown as follow,   

$$\frac{\partial ({\alpha }_q{\rho }_q)}{\partial t}+\nabla \cdot ({\alpha }_q{\rho }_q{\mathit{u}}_{\mathit{q}})=0,$$(1)

  

$$\begin{array}{l}\frac{\partial ({\alpha }_q{\rho }_q{\mathit{u}}_q)}{\partial t}+\nabla \cdot ({\alpha }_q{\rho }_q{\mathit{u}}_q{\mathit{u}}_q)=\\ -{\alpha }_q\nabla p+\nabla \cdot \left({\alpha }_q{\mu }_{eff}(\nabla {\mathit{u}}_q+{(\nabla {\mathit{u}}_q)}^{\text{T}})\right)+{\alpha }_q{\rho }_q\mathit{g}+{\mathit{F}}_q,\end{array}$$(2)

where, ρ_q, u_q, p, μ_eff, g, and F_q denote density, averaged velocity of liquid phase (q = l) and gas phase (q = g), pressure, effective viscosity, gravitational acceleration, and inter-phase momentum exchange item, respectively.

The effective viscosity μ_eff is included in μ_eff,l and μ_eff,g. The effective viscosity of the liquid phase μ_eff,l can be expressed as,¹⁵⁾   

$${\mu }_{eff\text{,}l}={\mu }_l+{\mu }_t+{\mu }_{Bl}\text{,}$$(3)

where μ_l is the molecular viscosity; ${\mu }_t=C_{\mu }{\rho }_l\frac{k^2}{\epsilon }$ is the turbulence viscosity; and μ_Bl = ρC_μ,Blα_gd$\left|{\mathit{u}}_l-{\mathit{u}}_g\right|$ is an extra term due to bubble induced turbulence;¹⁶⁾ C_μ and C_μ,Bl are the empirical constant, and C_μ,Bl = 0.6;¹⁶⁾ d is the bubble diameter, and d is calculated by population balance model and coupled with Euler model; k is the turbulent kinetic energy; ε is its energy dissipation rate.

The calculation of the effective gas viscosity μ_eff,g is based on the effective liquid viscosity as proposed by Jakobsen et al.,¹⁷⁾   

$${\mu }_{eff,g}={\mu }_{eff,l}\frac{{\rho }_g}{{\rho }_l}.$$(4)

The turbulent kinetic energy k and its energy dissipation rate ε are calculated from their governing equations,¹⁸⁾   

$$\begin{array}{l}\frac{\partial }{\partial t}({\alpha }_l{\rho }_{l}k)+\nabla \cdot \left({\alpha }_l{\rho }_l{\mathit{u}}_{l}k\right)=\\ \nabla \cdot \left({\alpha }_l\frac{{\mu }_{t,l}}{{\sigma }_k}\nabla k\right)+{\alpha }_l{G}_k-{\alpha }_l{\rho }_l\epsilon +{\alpha }_l{\rho }_{l}k,\end{array}$$(5)

  

$$\begin{array}{l}\frac{\partial }{\partial t}({\alpha }_l\rho {}_l\epsilon )+\nabla \cdot ({\alpha }_l{\rho }_l{\mathit{u}}_l\epsilon )=\\ \nabla \cdot \left({\alpha }_l\frac{{\mu }_{t,\mathrm{\hspace{0.17em} }l}}{{\sigma }_{\epsilon }}\nabla \epsilon \right)+{\alpha }_l\frac{\epsilon }{k}(C_{1\epsilon }{G}_k-C_{2\epsilon }{\rho }_l\epsilon )+{\alpha }_l{\rho }_l\epsilon ,\end{array}$$(6)

where the standard values are used for the turbulence parameters, C_1ε = 1.44, C_2ε = 1.92, C_μ = 0.09, σ_k = 1.0, σ_ε = 1.3, G_k is the turbulent kinetic energy induced by the additional bubbles, which is modeled following Elghobashi et al.¹⁹⁾ work.

The interfacial momentum transfer term between two phases F_q has a dominant effect in the multi-phase momentum equations. The total interfacial forces considered in the present work can be categorized into the drag force F_D, the lift force F_L, the virtual mass force F_VM, and the turbulence dispersion force F_TD. The interfacial force between the two phases F_q is given as follows,   

$${\mathit{F}}_q={\mathit{F}}_{lg}=-{\mathit{F}}_{gl}={\mathit{F}}_D+{\mathit{F}}_L+{\mathit{F}}_{VM}+{\mathit{F}}_{TD}.$$(7)

The drag force F_D is given as follows,¹⁸⁾   

$${\mathit{F}}_D=\frac{3}{4}\frac{{\mu }_l{C}_D{Re}_b}{{\rho }_g{d}^2}\left|{\mathit{u}}_l-{\mathit{u}}_g\right|({\mathit{u}}_l-{\mathit{u}}_g),$$(8)

where C_D is the drag coefficient, and Re_b is the bubble’ Reynolds number. Although the small bubbles shape in hot metal under ideal conditions is circular, but the bubbles size is different and their profile is variable under the alternating direction rotation mode in the stirring-injection vessel. So, the C_D can be written as follow based on the Tomiyama et al.,²⁰⁾   

$$C_D=\mathrm{M}\mathrm{a}\mathrm{x}\left\{\mathrm{M}\mathrm{i}\mathrm{n}\left[\frac{16}{R{e}_b}(1+0.15{Re}_b^{0.687}),\frac{48}{{Re}_b}\right],\frac{8}{3}\frac{\mathrm{E}{\mathrm{o}}_b}{(\mathrm{E}{\mathrm{o}}_b+4)}\right\},$$(9)

where ${Re}_b=\frac{{\rho }_{l}d\left|{\mathit{u}}_l-{\mathit{u}}_g\right|}{{\mu }_l}$, and $\mathrm{E}{\mathrm{o}}_b=\frac{\mathit{g}({\rho }_l-{\rho }_g)d^2}{\sigma }$.

The lift force F_L mainly originates from the velocity gradients in the shear flow, which is calculated as follow,   

$${\mathit{F}}_L=C_L\frac{{\rho }_l}{{\rho }_g}({\mathit{u}}_g-{\mathit{u}}_l)\times (\nabla \times {\mathit{u}}_l),$$(10)

where C_L is the lift force coefficient, the sign of C_L depends on the bubble size, in present work C_L is given as follow based on the Tomiyama et al.,²¹⁾   

$$\begin{array}{l}{C}_L=\\ \{\begin{array}{l}min\left[0.288tanh(0.121{Re}_b),f(\mathrm{E}{\mathrm{o}}_b)\right],\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{E}{\mathrm{o}}_b<4\\ f(\mathrm{E}{\mathrm{o}}_b),\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }4\le \mathrm{E}{\mathrm{o}}_b<10,\\ -0.27,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{E}{\mathrm{o}}_b\ge 10\end{array}\end{array}$$(11)

where $f(\mathrm{E}{\mathrm{o}}_b)=0.00105\mathrm{E}{\mathrm{o}}_b^3-0.0159\mathrm{E}{\mathrm{o}}_b^2-0.0204\mathrm{E}{\mathrm{o}}_b+0.474$.

The virtual mass force accounts for the additional work performed by bubbles in accelerating the liquid surrounding the bubbles, which is given by,   

$${\mathit{F}}_{VM}=\frac{{\rho }_l}{{\rho }_g}{C}_{VM}\frac{D}{Dt}({\mathit{u}}_l-{\mathit{u}}_g),$$(12)

where the virtual mass force coefficient C_VM is taken to be 0.5 for bubbles.²²⁾

The turbulent dispersion force results in the additional dispersion of phases from high volume fraction regions to low volume fraction regions due to turbulent fluctuations, which is given based on the Favre-averaged drag model,²³⁾   

$${\mathit{F}}_{TD}^{}=C_D{C}_{TD}\frac{{\mu }_{t,g}}{{\sigma }_{t,g}}\left(\frac{\nabla \cdot {\alpha }_l}{{\alpha }_l}-\frac{\nabla \cdot {\alpha }_g}{{\alpha }_g}\right),$$(13)

where C_TD and σ_t,g are the turbulent dispersion coefficient and the turbulent Schmidt number of the gas phase, respectively. By default, the turbulent dispersion coefficient C_TD = 1 and the turbulent Schmidt number σ_t,g = 0.9 are adopted.²³⁾

2.2.2. Bubble Population Balance Model

To calculate the nonuniform bubble size distribution, the PBM (Population Balance Model) by the discrete method is employed. The coalescence of bubbles is formulated to the random collision driven by turbulence and wake entrainment, and the breakage of bubbles is formulated through the impact of turbulent eddies. The Prince aggregation model and the Luo breakage model are both included. The effect of bubble diameter on flow field is considered by transferring the bubble Sauter mean diameter data between the PBM and the Euler momentum equation. The Sauter mean diameter is defined as,²⁴⁾   

$$d_{32}=\frac{\sum _{i=1}^{N}n{d}_i^3}{\sum _{i=1}^{N}n{d}_i^2}.$$(14)

The PBM can be expressed as follows,²⁵⁾   

$$\begin{array}{l}\frac{\partial n(d,t)}{\partial t}+\nabla ({\mathit{u}}_{g}n(d,t))=\frac{1}{2}{\int }_0^d{a}_c(d^{\prime },d^{″},t)n(d^{\prime },t)n(d^{″},t)\mathrm{d}{d}^{\prime }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{\mathrm{d}}^{\infty }n(d^{\prime },t)\cdot a_b(d,t)\cdot b_b(d,d^{\prime },t)\mathrm{d}{d}^{\prime }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\int }_0^{\infty }{a}_c(d^{\prime },d^{″},t)n(d^{\prime },t)n(d^{″},t)\mathrm{d}{d}^{\prime }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-n(d^{\prime },t)a_b(d,t),\end{array}$$(15)

where n(d,t) is defined as the bubble number density function; u_g is the velocity of the bubbles, which can be calculated using the Eulerian-Eulerian approach; a_b(d,t) and a_c(d′,d″,t) are breakage rate function of bubbles and coalescence rate function of bubbles for size d′ and size d″, respectively; b_b(d,d′,t) is the daughter bubble size distribution form size d to size d′. To solve Eq. (15), reliable models of a_b(d,t), b_b(d,d′,t), and a_c(d′,d″,t) are needed for PBM closure.

(1) Coalescence Model

Prince and Blanch modeled bubble coalescence by considering bubble collisions due to turbulence and wake entrainment. It is assumed that collisions from these various mechanisms are cumulative, i.e.,²⁶⁾   

$$a_c(d^{\prime },d^{″},t)=\left[{\omega }_T(d^{\prime },d^{″},t)+{\omega }_l(d^{\prime },d^{″},t)\right]P(d^{\prime },d^{″},t),$$(16)

the turbulent collision rate ω_T(d′,d″,t) is expressed as,   

$${\omega }_T(d^{\prime },d^{″},t)=0.089\pi {\epsilon }^{1/3}{({d^{\prime }}_{}+{d^{″}}_{})}^2{({d^{\prime }}_{}^{2/3}+{d^{″}}_{}^{2/3})}^{1/2},$$(17)

the buoyancy-driven collision rate ω_l(d′,d″,t) is calculated as,   

$${\omega }_l(d^{\prime },d^{″},t)=\frac{\pi }{4}{(d^{\prime }+d^{″})}^2\left|u^{\prime }-u^{″}\right|,$$(18)

where u′ and u″ are bubble rise velocity of d′ and d″, respectively. The bubble rise velocity is written as u =$\sqrt{2.14\sigma /{\rho }_{l}d+0.505gd}$.

The coalescence efficiency is calculated as,²⁷⁾   

$$p(d^{\prime },d^{″},t)=exp\left(-\frac{t_{d^{\prime }{d}^{″}}}{{\tau }_{d^{\prime }{d}^{″}}}\right)=exp\left(-\frac{r_{d^{\prime }{d}^{″}}^{5/6}{\rho }_l^{1/2}{\epsilon }^{1/3}}{4{\sigma }^{1/2}}ln\frac{h_o}{h_f}\right),$$(19)

where the equivalent radius r_d′d″ is calculated as r_d′d″ =$\frac{1}{2}\frac{r_{d^{\prime }}{r}_{d^{″}}}{r_{d^{\prime }}+r_{d^{″}}}$.²⁸⁾ The initial liquid film thickness h₀ and the critical film thickness h_f are 10⁻³ and 10⁻⁶ m according to the work of Wang.²⁹⁾

(2) Breakup Model

Luo and svendsen proposed the bubble breakup model based on random turbulence collision theory. The equations can be expressed by,³⁰⁾   

$$a_b(d,t)={\int }_0^{1/2}b(f_v|d)\mathrm{\hspace{0.17em} }\mathrm{d}{f}_v,$$(20)

  

$$b_b(d,d^{\prime },t)=\frac{2b(f_v|d)}{{\int }_0^{1}b(f_v|d)\mathrm{\hspace{0.17em} }\mathrm{d}{f}_v},$$(21)

where b(f_v|d) is the breakup rate of the bubble with size d and the breakup fraction f_v, which is calculated as,   

$$\begin{array}{l}b(f_v|d)=0.923f(1-{\alpha }_g){\left(\frac{\epsilon }{d^2}\right)}^{1/3}\\ {\int }_{{\xi }_{min}}^1\frac{{(1+\xi )}^2}{{\xi }^{11/3}}\cdot exp\left(-\frac{12C_f\sigma }{\beta {\rho }_l{\epsilon }^{2/3}{d}^{5/3}{\xi }^{11/3}}\right)\mathrm{d}\xi ,\end{array}$$(22)

where ε is the energy dissipation rate; ξ is the ratio crushing of bubble and $\xi =\frac{\lambda }{d}$; σ is the surface tension; f is the breakage calibration factor, and 0.5 is adapted in present work;¹³⁾ β is model parameter, 2.0 is adapted based on the work of Liu.³¹⁾

2.2.3. Numerical Details and Boundary Conditions

The commercial software Fluent 2020 is used to calculate the bubbles disintegration and dispersion process in this work. Euler two-fluid model is adopted by the implicit separation method. The slippage grid technique is used to simulate the flow field. In the population balance model, the bubbles are grouped into two dispersed phases which travel with two individual flow fields. The first dispersed phase is assigned for bubbles ranging from 10 to 50 mm in the region around of nozzle according to Sun et al.,³²⁾ Sano et al.,³³⁾ and Iguchi et al.³⁴⁾ research, while the second dispersed phase is dedicated for bubbles ranging from 0 to 10 mm. The number of bubble bins is specified 10 of each dispersed phase for balancing the computational cost and the accuracy for predicting the bubble size distribution. The range of bubble size is determined by the analysis of the experimental results and the research of Liu et al.³¹⁾ The phase coupled SIMPLE scheme is used for the pressure-velocity coupling of the two-phase flow, and the momentum equation is discretized by the second-order upwind scheme. The top surface of ladle is set as degassing boundary condition, and the nozzle is set as the velocity inlet for gas; the stirring mode realized by UDF (User Defined Function); and the ladle wall, impeller wall including shaft wall are set as no slip wall with the standard wall function. A physical time scale of 0.005 s is adopted for all the unsteady simulations. The scaled residuals for all variables are less than 1 × 10⁻⁴ as convergence condition.

3. Model Validation

In order to validate the accuracy of the results calculated by the applied mathematical model, a water model of the stirring-injection system is established, which is modified from that used in the previous work.¹³⁾ As shown in Fig. 1, a cylindrical vessel contained with a stirring-injection lance. The sizes of water model and other parameters adopted in the water model are listed in Table 1. The nitrogen is injected into the shaft of the impeller and jets out from the nozzle at the bottom of the impeller. The impeller type employed with the impeller C, and the size is 0.1 times of the impeller C as shown in Fig. 2(c). The stirring mode is the clockwise-anticlockwise rotation pattern, and the stirring speed is 120 rpm for 2 s switching. Once the liquid surface and air injection reach a steady state, the distribution of gas volume and bubbles sizes are visually captured by a high-speed camera. Then, the location and size of bubbles are measured through the image analysis software of ImageJ. And the bubble size distribution and the gas volume fraction are attained from the experiment. Figure 4 shows the comparison of gas volume distribution and bubble size distribution of the experiment and simulation results, the monitor points are located at the radius direction and the height are 0.22 m, 0.28 m, and 0.34 m away from the bottom of ladle. It can be seen that the shape of the gas volume fraction in experiment and simulation are consistent, the bubbles size distribution is consistent, and the maximum error between the calculated values of bubbles size and those in experiment are less than 10%, which indicating that the calculated results in this work are reliable.

Fig. 4.

Comparison diagram of experiment and numerical simulation. (a) Experimental photograph; (b) Gas volume distribution; (c) Bubbles diameter distribution. (Online version in color.)

4. Results and Discussion

4.1. Effect of Impellers Types on Flow Structure

Understanding the fluid flow is of great importance for predicting and controlling the bubbles disintegration and dispersion actions in stirring-injection system. In the variable speed stirring mode, the impeller provides the energy for the flow of bubbles and hot metal, which carries on the complex momentum exchange and relative motion with the bubbles and hot metal. Strong turbulent disturbances and hydrodynamic interactions are the principal physical mechanisms in the stirring-injection system, which gradually result in bubbles deformation and disintegration when the restoring interfacial forces are exceeded.

The large vortexes of the hot metal are generated around the blades in the low rotation speed with variable speed stirring mode, and the vortexes are changed slightly with time. Different impellers types form different vortexes structures, and the turbulent kinetic energy generated in the vortex will affect the size and the distribution of bubbles. Figure 5 shows the vortexes generated at the moment when the impellers switched from the short sides to the long sides of the blades at the speed of 30 rpm for 4 s switching. It can be seen that the interface of vortex 1 and vortex 2 is located at the height of 0.7 m with the stirring impeller center, because vortex 1 and vortex 2 rotating in opposite directions, the surrounding fluid is aroused to collide with the side wall and forms vortex 3. The combined action of these vortexes leads to the disintegration and dispersion of bubbles. The size of vortex 1 and vortex 2 of impeller B is similar and the symmetry degree of upper and lower vortexes is the highest, and the vortexes 3 generated by impeller B are the largest and symmetric, as a result, the impeller B has a larger dynamic to breaking bubbles. Vortexes 1 of impeller C and D are closer to the side wall, while vortexes 2 are less aligned with it, and vortexes 3 excited by impeller C and D are smaller than that of the impeller B. The integrity of vortex 2 and vortex 3 of impeller A are the worst, and the dynamic of the broken bubbles is the lowest.

Fig. 5.

Instantaneous vortex distributions of different impellers. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D. (Online version in color.)

Figures 6 and 7 show the distribution of horizontal and vertical velocities of hot metal in the vertical profile at the corresponding time and the operating conditions same with Fig. 5. It can be seen from Fig. 6 that the velocity of hot metal around the impellers changes dramatically, and there are three layers of varying wildly velocity distributions in the opposite direction, these three layers of velocity have different position and shapes with different impellers types. Through the water model experiment and simulation observation, it is found that most of bubbles are thrown out after rising to the top end face of the blades. Therefore, there are few bubbles broken in the boundary region between velocity 1 zone and velocity 2 zone except for impeller D, so this region effecting on bubble disintegration and dispersion behavior is relatively small except for the impeller D. However, part of bubbles are thrown out from velocity 1 zone and disintegrated preliminarily with impeller D, because the location of velocity 1 region is higher than that of other impellers types. Most of the bubbles are broken in the boundary region between velocity 2 and velocity 3 zone. It can be seen that the velocity 3 zone of impeller A and B showed an upward trend, while the velocity 3 zone of impeller C and D tended to the side wall. The velocity 2 zone and velocity 3 zone of impeller B are connected closely and the contact area is the largest, followed that of with impeller C, and the minimum of that for impeller A and D. This shear action of upper and lower velocities layers can provide greater energy to the region, causing the bubbles to disintegrate and disperse. The velocity changes in the middle and upper parts of the ladle are less violent, and the ability to disintegrate bubbles is poor.

Fig. 6.

Horizontal velocity distribution of different impellers types. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D. (Online version in color.)

Fig. 7.

Vertical velocity distribution with different impellers types. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D. (Online version in color.)

As can be seen from Fig. 7, the interface with drastic velocity changes appears near the impellers. The dashed lines in the Fig. 7 are the interface area of velocity 2 and velocity 3 in opposite directions which the location is same as Fig. 6. It can be seen that those regions are not only the junction areas with opposite horizontal velocities direction, but also the junction areas with opposite vertical velocities direction, which confirm that the bubbles are broken most violently here. The impeller B and D have large contact surfaces in those regions, where the upward and downward velocity change dramatically, which plays an important role in bubble breakage. The bubbles of impeller D are subjected to the larger shear action of upward and downward speeds on this region, which are superimposed with the horizontal force, so the effect of bubbles breakage is better than that of impeller A. The zones where the velocity changes violently in the upward and downward directions of impeller A and impeller C are concentrated on the above of the blades end, the shear action of the upward and downward are not as stronger as that of the impeller B and D, and the ability of breaking bubbles in this direction is lower compared with that of the impeller B and D. In conclusion, from the perspective of flow structure, the impeller B has the strongest disintegrate bubble capacity, and the impeller A is the weakest one. In addition, the areas of upward velocity distribution in upper part of the ladle are large, and the gas volume fraction in this area is also large (see Fig. 8 for gas volume distribution). At this time, the bubbles drive the hot metal upward under the action of buoyancy.

Fig. 8.

Bubbles size and volume fraction distribution at the monitor points of different impellers. (Online version in color.)

4.2. Effect of Impellers Types on Bubbles Dispersion

The disintegration and adequate dispersion of bubbles are critical factors for the efficiency of desulfurization process when the stirring-injection is employed. The bubbles size distribution and volume distribution in each position did not change when the injection and stirring reached a steady state. The effects of impellers types on the gas volume distribution and the bubbles diameter distribution are shown in Fig. 8, where the monitor points are located at the radius direction and the height are increased from 1.4 m to 3.8 m away from the bottom of ladle, and the rotation speed is 30 rpm for 4 s switching. The results show that due to the effect of buoyancy force, most of the bubbles are dispersed at the region of upper impeller, the bubbles are rarely found at the bottle of the ladle except for around the nozzle, and the bubbles size distribution with different impellers types are different.

It can be seen from Fig. 8 that the most of bubbles are located at the central area of the ladle, and there is less gas within 0.5 m from the side wall of the ladle. So, all of the impellers do not have enough power to distribute the bubbles to the bottom and the region close to the wall of the ladle evenly when the rotating speed is low. When the monitor points are lower than 2.6 m, the bubbles size of the impeller B is smaller than that of other impellers, and the uniformity of gas volume fraction distribution is weaker than that of other impellers. Those are because although the impeller B has a larger capacity to breakup bubbles, the trend of bubbles distribution to the side wall is small, which can be verified by the distribution of velocity 3 in Fig. 6. The gas volume distribution of impeller D is better than that of other impellers types, it is due to the hypotenuse and downward inclination of the blades, which pushed the hot metal towards the wall, then made the bubbles dispersed more evenly. With the increase of the height of the monitor points, the bubbles size of each impellers increases slowly, which is due to the coalescence of bubbles and the decrease of static pressure. The distribution of gas volume fraction becomes gradually evenly, and correlation with the impellers types becomes smaller with the increase of the height of the monitor points. The region where the bubbles size changes dramatically above the impeller will be discussed in detail at the next section.

4.3. Effect of Impellers on Bubbles Breakage Surrounding Them

In present work, the gas flow rate of 100 m³ h⁻¹ is adopted according to Sun et al.³²⁾ research, which is very high considering the small internal lance diameter of 20 mm, thus a gas jet would form at front of the lance nozzle, and the bubbles plume is formed before rising upwards. When the plume reaches to the region around the rotating impeller, most of bubbles gather on the back of the blades, while the direction of rotation changes, the bubbles at behind of the blades are thrown out, and broken into small bubbles, when they rise to an area where the speed changes dramatically, the bubbles are broken again by vortexes formed with stronger shear forces of the hot metal.

Figure 9 shows the state of the bubbles been thrown out at the moment when the short sides of the blades change direction to the long sides of the blades for different impellers types under the rotation speed of 30 rpm for 4 s switching, and the horizontal and vertical velocity changes at the corresponding time in the region of drastic velocity changes. It can be seen that different types of impellers carry and throw out different volumes of gas on the back of the blades, and the gas is thrown out from the top of the blades of impeller A and B, while a small amount of gas is thrown out from the middle of the blades of impeller C, and impeller D throws out much more gas at middle of the blades than that of the impeller C. Due to different impeller structures, the instantaneous velocity changes are obviously different. The bubbles dispersion will be affected by the flow patterns developed by the impeller, as shown in Fig. 9(b), the velocity changes of impeller B and D are more obvious than that of impeller A and C. As a result, the position of the thrown gas is changed, and the volume of the thrown gas is different, so do the heights and bubble sizes of the regions with dramatic velocity change (as can be seen from Fig. 10).

Fig. 9.

The instantaneous tendency of impellers change speed. (a) Transient gas volume change; (b) Transient velocity change in the severe crushing region. (Online version in color.)

Fig. 10.

Bubbles size distribution in the severe crushing region. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D. (Online version in color.)

The bubbles are broken into different sizes after being thrown out, when rising to the strong shear layer where the velocity changes dramatically, the bubbles are broken violently. The bubbles size distribution is shown in Fig. 10 with the rotation speed of 30 rpm for 4 s switching, and the gas volume fraction distribution is shown in Fig. 11. Since the layer with drastic velocity change is relatively thin, the gas volume fraction distribution in this layer does not change very much, so the value of the entering region is used to represent the gas volume distribution of the changed layer. It can be seen that the initial size and gas volume distribution of different impellers types are different in the region where the velocity changes dramatically. The maximum initial bubble size reaches 8 mm of impeller A, 6.7 mm in impeller B, and the maximum initial bubble size of impeller C and D are between of them. After bigger bubbles being broken in this layer, most of the bubbles size decreases to less than 4 mm. The gas volume fraction distribution of the region with dramatic velocity change are more uniform with the impeller C and D than that of impeller A and B due to the downward inclination of the blades of impeller C and D.

Fig. 11.

Gas volume fraction distribution in the severe crushing region. (Online version in color.)

4.4. Sensitivity of Impellers Types to Rotation Speed

It is very important to research the relationship between bubbles disintegration and rotational speed for reducing energy consumption and improving the life of stirring-injection impeller. In order to compare the sensitivity of the impellers types to the change of rotational speed, the bubbles disintegration and dispersion are researched at 10 rpm, 20 rpm and 30 rpm for 4 s switching, respectively. The bubbles distribution in most areas above the impeller is stable relatively. Therefore, the bubbles size distribution and gas volume distribution at the monitor points with a height of 3.4 m from bottle of ladle are used to represent the effects of different rotation speeds on bubbles disintegration and dispersion.

Figure 12 shows the change of bubbles size and gas volume distribution for four impellers types. It can be seen that the bubbles size and volume distribution of the four impellers types show the same trend. The larger the speed is, the smaller the bubbles size is, and the more uniform the gas volume distribution is. When the rotating speed is 10 rpm for 4 s switching, the bubbles are concentrated on the areas around the impellers shafts and the sizes of the bubbles are large, so the effect of bubble disintegration and dispersion is poor. The bubbles size distribution and gas volume distribution are improved obviously when the speed is increased to 20 rpm for 4 s switching. When the rotational speed changed from 20 rpm to 30 rpm for 4 s switching, the average bubbles size of impeller A decreased from 6.4 mm to 3.9 mm, and the change rate of gas volume fraction distribution is 11.9%; the average bubbles size of impeller B decreased from 3.56 mm to 2.12 mm, and the change rate of gas volume distribution is 25%; the average bubbles size of impeller C decreased from 4.35 mm to 3.65 mm, and the change rate of gas volume fraction distribution is 9.2%; the average bubbles size of impeller D decreased from 4.21 mm to 3 mm, and the change rate of gas volume fraction distribution is 1.1%, respectively. So, the change rate of bubbles size and gas volume fraction of impeller B are the largest with the increase of rotation speed.

Fig. 12.

Bubbles size and gas volume distribution of four impellers types at different rotational speeds. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D. (Online version in color.)

4.5. Sensitivity of Impellers Types to Surface Tension

The surface tension of hot metal varies greatly with the composition and temperature change. According to the research of Su et al.,³⁵⁾ the surface tension of hot metal ranges from 1.4 N/m to 2.4 N/m. The surface tension of hot metal-nitrogen is much higher than that of water-nitrogen model, so a smaller impact force can break up the bubbles. It is clear that the surface tension is an important parameter affecting the size and distribution of bubbles in the stirring-injection ladle. The bubbles size distribution and gas volume distribution at a height of 3.4 m from bottle of ladle are used to show the effect of different surface tension on bubbles disintegration and dispersion, as shown in Fig. 13. It can be seen that the impeller type has little effect on the distribution of gas volume fraction. With the increase of surface tension, the change rate of the average bubbles size is in the order of impeller D, impeller C, impeller B, and impeller A , respectively.

Fig. 13.

Bubbles size and gas volume fraction distribution of four impellers types at different surface tension. (a) Impeller A; (b) Impeller B; (c) Impeller C; (d) Impeller D. (Online version in color.)

5. Conclusion

In this work, four kinds of stirring-injection impellers which suitable for high temperature work environment are developed to improve the bubbles disintegration and dispersion. The influence of the structure of impeller on bubbles breakage at the surrounding of the impeller region and the severe crushing region are investigated. The following conclusions were drawn,

(1) The volume of bubbles wrapped on the back of blades of four impellers is different, and the positions of bubbles thrown out are also different, the vortex around blades is the main factor causing bubble breakage. So, the special-shaped impeller’ center and the downward angle of the blades can promote the disintegration and dispersion of the bubbles.

(2) The square center impeller type (impeller B) is the best impeller type to develop the bubbles to the ideal state of magnesium, the average size of bubbles in most area is less than 3 mm, and the bubble volume fraction reached to 0.005– 0.01 in most region except for the bottom and the side of the ladle at the condition of the rotation speed is 30 rpm for 4 s switching. The downward angle of the blades (impeller C and impeller D) can improve the bubble volume fraction dispersion effect effectively in the region around the impeller.

(3) In variable speed mode, the bubbles size decreases obviously and the gas volume distribution becomes more uniform with the stirring speed increasing, but too large speed will lead the impeller and ladle to wear.

(4) A better bubbles disintegration and dispersion effect can be obtained by the rotating speed of 30 rpm for 4 s switching with impeller B.

Acknowledgement

The present work was supported by Education Department of Liaoning Province of China (No. LJ2019JL016).

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