Demystifying Underlying Fluid Mechanics of Gas Stirred Ladle Systems with Top Slag Layer Using Physical Modeling and Mathematical Modeling

Abstract

Effects of slag layer thickness on the fluid dynamics of liquid steel in gas-stirred ladles by bottom injection of argon was studied through water modeling experiments and numerical simulations. Mixing times increase considerably with thicker slag layers and decrease of gas flow rates. The physical properties of the system have a smaller influence on mixing time. Slag Eye Opening (SEO) area is increased under thin slag layers, increase of gas flow rates, and denser and less viscous slags. The planes close to the metal-slag interface, under the presence of thick slag layers for a given gas flow rate, are split in subregions of small velocities with different orientations making the lower fluid to come close to a stagnant condition. The presence of, either, thick or thin slag layers does not influence the axial velocity along the plume height for a fixed flow rate of gas. The SEO area follows a linear relationship with the square root of the densiometric Froude number based on the slag layer thickness.

1. Introduction

Operations of liquid steel stirring in ladles are very important to achieve clean steel products and the related literature is prolific about the different aspects involved in them. For example, dynamics of the two-phase flows in the plume has been addressed by Krishnapisharody. and Irons who established correlations to estimate the size of the eye opening and the height of the spout as function of the physical properties of the two immiscible liquids and the plume velocity.^1,2) The size of the eye increases with a lower height of the denser phase and with high gas flow rates and decreases as the upper phase thickness increases. The same authors developed correlations linking averaged velocities of the liquid and gas phases and gas volume fraction along the plume height as functions of gas flow rate and bath height.³⁾ Spot height was defined through a dimensionless variable involving gas flow rate⁴⁾ presenting a unified theory for two-phase flows dynamics in the plume.⁵⁾ Mixing time is another important issue analyzed by various authors, Iguchi et al. reported that it increases with the kinematic viscosity of liquids in bottom stirred vessels.⁶⁾ However, Mazumdar and Guthrie pointed out that the two basic forces dominating stirring are inertial and gravitational and viscosity is not involved in the criterion to scaleup operations of water modeling.⁷⁾ Mazumdar and Guthrie established expressions to estimate the plume velocity (mean rise velocity of the gas liquid mixture) as U_P = 4.4Q^2.33L^2.5R^−0.25, and mixing time (95 pct. bulk) = 25.4Q^−0.33L⁻¹.^8,9) The effects of a lighter layer (called here also as the upper or slag layer) on fluid flow and mixing times have also been objects of research starting with Computer Fluid Dynamic (CFD) simulations of multiphase systems including the denser, lighter and gas phases.^10,11,12) General findings indicate an enlargement of the eye area with increase of gas flow rate, the slag layer becomes thinner close to the eye boundary and thicker and close to the walls the ladle due to the shearing effects provided by the ascending bubbles. Low slag viscosities are suitable for metal-slag reactions like steel desulfurization but are prone to entrainment phenomena in the metal bulk.¹³⁾ All these studies report that the thickness of the slag weakens the mixing kinetics in steel; even it can be said that the thickness of the slag is more important than its physical properties in this regard.^14,15) A consequence of the affectation of slag thickness on fluid flow dynamics is the mixing time, Table 1 shows the correlations,^{15,17,18,19,20,21)} so far reported to quantify the effects of the slag layer as reported by Amaro et al.¹⁶⁾ Some observations about these correlations are appropriate to be mentioned here:

Table 1. Mixing time correlations for a denser phase with a lighter top layer.¹⁶⁾

Reference	Correlation	N	r/R	Θ	Top Layer
1. Haida et al. (17)	τm = 100$\dot{\epsilon }$−0.42	1	0, 0.5		Polystyrene balls
2. Ying et al. (18)	τm = 125$\dot{\epsilon }$−0.289	1	0
3. Yamashita et al. (19)	${\displaystyle {\tau }_m=1\mathrm{\hspace{0.17em} }910Q_g^{-0.217}{D}^{1.49}H{L}^{-1.0}{V}_L^{0.37}{\left[\frac{{\rho }_L-{\rho }_s}{{\rho }_L}\right]}^{0.243}}$	1	0		Silicon oil and pentane
4. Mazumdar-Kumar (20)	${\displaystyle {\tau }_m=60.2Q_g^{-0.33}{R}^2{H}_L^{-1.0}{h}_s^{0.6}{\left[\frac{{\sigma }_s}{{\mu }_s}\right]}^{-0.022}}$	2	0.5	180	Petroleum ether, mustard oil and benzene
5. Patil et al. (21)	${\displaystyle {\tau }_m=152Q_g^{-0.33}{R}^{2.33}{H}_L^{-1.0}{\left(\frac{h_s}{H_s}\right)}^{0.3}{V}_s^{0.033}{\left[\frac{{\rho }_L-{\rho }_s}{{\rho }_L}\right]}^{-0.044}}$	1 2	0.5 0	180	Petroleum ether, mustard oil and benzene and silicon oil
6. Khajavi-Barati et al. (15)	${\displaystyle {\tau }_m=2.33{\dot{\epsilon }}_{ms}^{-0.34}{H}_{eff}^{-1.0}}$	1	0		Kerosene and silicon oil
7. Amaro et al. (16)	${\displaystyle {\tau }_m=9.83N^{0.1025}{\dot{\epsilon }}^{-0.364}{\left(\frac{r}{R}\right)}^{-0.0051}{\left[\frac{h_s}{H_L}\right]}^{0.004}}$	1, 2 3	0.33 0.5 0.67 0.80	120 180	Engine oil blue, engine oil red and soybean oil

τ_m represents mixing time (s), Q_g is gas flow rate, D and H are diameter and height of the container respectively, ν_s is the kinematic viscosity of the lighter phase, ν_L is the kinematic viscosity of the denser phase, ρ_L and ρ_s represent the densities of the denser and lighter phase respectively, μ_s is the dynamic viscosity of the lighter phase, σ_s is the surface tension of the lighter phase, $\dot{\epsilon }$ is the specific potential energy input in W/ton, except for Khajavi and Barati¹⁵⁾ in W/kg, h_s is the thickness of the lighter phase, N is the number of nozzles, θ their separation angle and H_eff is and effective height including the denser and lighter phases as defined by Khajavi and Barati.¹⁵⁾ The units of all other variables are expressed in SI units. This table was complemented with the last row in the present work. R is ladle radius and r is radial coordinate.

• Correlations 1 and 2 do not involve an explicit influence of the slag thickness and explicitly express (including correlations 3–7) that the mixing times decrease with higher specific potential energy inputs.

• Correlations 3–6 agree in that the mixing time is inversely proportional to the height of the denser or lower phase, whereas correlation 7 indicates a weak influence of this variable.

• Correlations 3–5 and 7 give explicit influences of the physical properties of the lighter and denser phases tough, with different strengths. However, general speaking, the dependence of mixing time looks more dependent on the slag thickness than on the physical properties of the phases, see particularly correlations 4, 5 and 7.

• Correlations 4, 5 and 7 make explicit that the mixing time increases with thicker lighter phase layers. Correlation 6 has implicit the effect of the lighter phase layer or upper layer through the variable H_eff.

It is evident that correlations between mixing times, which are a foot print of fluid dynamics of the denser phase, are dependent on the specific experimental conditions through which they were obtained. Hence the need of an evaluation of each one of these correlations.

Steel eye opening (SEO) area is another important parameter of the process since, depending on its size, steel can be less or more contaminated by the atmospheric air. Its measurement, through infrared video-cameras, is also important to know the actual flow rates of argon as the injection through the porous plug is, most of the times, inaccurate due to the partial obstruction of the plug surface by debris of metal and slag or leaks. The operating factors affecting the size of the SEO are summarized in Table 2.^22,23,24,25) Although there are other correlations,³³⁾ they do not include direct operating parameters as those equations in Table 2 do. The observations derived from these correlations are:

Table 2. Correlations to estimate the size of the open steel eye during stirring operations in ladles.

Ref.	Correlation		N	r/R	Systems	Constraints
22	${\displaystyle \begin{array}{l}log\left({\displaystyle \frac{A_{es}}{h_{s}H}}\right)=-0.69897+0.90032\hspace{0.17em}log\left({\displaystyle \frac{Q^2}{g{h}_s^5}}\right)\\ -0.14578{\left[log\left({\displaystyle \frac{Q^2}{g{h}_s^5}}\right)\right]}^2+0.0156{\left[log\left({\displaystyle \frac{Q^2}{g{h}_s^5}}\right)\right]}^3\end{array}}$	(1)	1	0	Mercury-oil Liquid steel-slag	Фorifice=0.5 mm ${\displaystyle 0.01\le {\displaystyle \frac{Q^2}{g{h}_s^5}}\le 10\mathrm{\hspace{0.17em} }000}$ Other diameters of the orifice give different correlations
23	${\displaystyle {\displaystyle \frac{A_e}{{\left(h_s+H\right)}^2}}=0.02\mp 0.002{\left({\displaystyle \frac{Q^2}{g{h}_s^5}}\right)}^{0.375\mp 0.0136}}$	(2)		0, 1	Mercury-oil Liquid steel-slag	It is a modification of the precedent correlation
24	${\displaystyle \begin{array}{l}{A}_e^{*}=-0.76{\left(Q^{*}\right)}^{0.4}+7.15{\left(1-{\rho }^{*}\right)}^{-1/2}{\left(Q^{*}\right)}^{0.73}{\left(h^{*}\right)}^{-1/2}\\ Q^{*}={\displaystyle \frac{Q}{g^{0.5}{H}^{2.5}}},\mathrm{\hspace{0.17em} }\hspace{0.17em}{A}_e^{*}={\displaystyle \frac{A_e}{H^2}},\mathrm{\hspace{0.17em} }\hspace{0.17em}{A}_p^{*}={\displaystyle \frac{A_p}{H^2}}=1.41{\left(Q^{*}\right)}^{0.4},\mathrm{\hspace{0.17em} }\hspace{0.17em}\\ {\rho }^{*}={\displaystyle \frac{{\rho }_s}{{\rho }_L}},\mathrm{\hspace{0.17em} }\hspace{0.17em}{h}^{*}={\displaystyle \frac{h_s}{H}}\end{array}}$	(3)	1	various	Water-paraffin, water-motor oil, CaCl2-paraffin oil, Hg-silicon oil, water-silicon oil and steel-slag	Assumed to be for general application of various systems and different orifice positions
25	${\displaystyle \begin{array}{l}{\displaystyle \frac{A_e}{h_{s}H}}=3.25{\left({\displaystyle \frac{U_p^2}{g{h}_s}}\right)}^{1.28}{\left({\displaystyle \frac{{\rho }_L}{\mathrm{\Delta }\rho }}\right)}^{0.55}{\left({\displaystyle \frac{{\nu }_s}{h_s{U}_p}}\right)}^{-0.05}\\ U_p=17.4Q^{0.244}{H}^{-0.08}{\left({\displaystyle \frac{{\rho }_g}{{\rho }_L}}\right)}^{0.0218}\end{array}}$	(4)	1	0, 0.5	Water-petroleum-ether, water-coconut oil, water-mustard oil	Applicable for ε~0.01 W/kg, 0.75 < H/D < 1.5, ${\displaystyle \nu ~{10}^{-6}{\displaystyle \frac{m^2}{s}},}$ and 0.006 < hs < 0.05 For centric position of the orifice

All correlations are given in SI units. Q is gas flow rate, h_s is slag thickness, H is the height of the denser phase, U_p is plume velocity, A_p is the plume area, A_e is the area of the eye opening, ν_s is kinematic viscosity of the lighter phase, ρ_s and ρ_L are densities of the lighter and denser phases respectively. The plume velocity was calculated using the equation of Yonezawa and Schwerdtfeger.²²⁾

• All correlations indicate that the SEO increases with the height of the denser phase and with the flow rate of the stirring phase.

• All correlations indicate a strong inverse dependence of the SEO on the thickness of the lighter or upper phase,

• Different to the mixing time, the SEO shows considerably dependence on the physical properties of the liquid phases, although with different effects, see correlations (3) and (4).

On the other hand, all steelmakers have experienced heavy slag carry-overs from the furnace to the ladle specially when the EBT (Eccentric Bottom Tapping) nozzle is worn out or the tapping sleeve of the BOF converter has cast many heats. And when this happens, either ladle-slag skimming would be necessary or prolonged processing times in the ladle furnace will be the consequences and, most of the times, those heats must be downgraded. The general perception in the plant is that the slag should be first killed and change its chemistry steadily with time to refine appropriately the liquid steel. However, this condition is more fluid flow controlled rather than chemical reaction kinetics rate controlled to attain slag conditioning and further steel refining. Hence, in the present study, the specific effects of slag thickness on mixing time, on SEO area and on underlying fluid mechanics are addressed through physical and mathematical models, defining which of the correlations presented in Tables 1 and 2 describe adequately the experimental outputs. The results of this investigation will allow us to establish practical decisions when the circumstances lead to a given ladle furnace to operate with high slag loads.

2. Experimental Setup

The experimental setup consists of a 1/3 scale ladle, made of transparent plastic, of a billet company located equipped with a bottom plug to stir steel with argon. The dimensions of the model and the positions of the plug and the slide gate are reported in Figs. 1(a) and 1(b). The ladle is filled with tap water through a top pipe until the operational scaled-height corresponding to the plant. Injection of argon is modeled using air instead, which is supplied by an air compressor and injected through an orifice in the ladle bottom, with a pressure of 2.5 kg/cm². The gas flow rate is measured through a mass flowmeter located between the compressor and the injection point. The lighter phase is food-oil and a layer of this material is conformed before starting the experiment to obtain a desired thickness. To have visualization records of the experiments, two video cameras were placed, one on the top of the bath and another one facing the wall of a flat chamber attached outside the ladle wall filled with water to avoid optical distortions from the curvature of the vessel. The camera in the top recorded the images of the bath surface without and with an oil layer. The video recordings were decomposed into images of the eye opening with a frequency of 2 s⁻¹. Quantitative measurements of these areas were performed using a previously calibrated image analyzer software,²⁷⁾ following a similar procedure as that reported by Peranandhanthan and Mazumdar.²⁵⁾

Fig. 1.

(a) Dimensions of the model (mm). (b) Position of the nozzle.

Twenty cubic centimeters of an aqueous solution of food red-colorant was employed as tracer, injected 100 mm below the bath surface near the geometric center of the ladle. A peristaltic pump was used for extracting samples from the bath which were fed into a colorimeter cell to obtain the instantaneous concentration of the tracer. The analogical signals of the colorimeter are converted into digitals ones through a data acquisition card in a PC permitting real-time plotting of the tracer concentration vs time. Fluid flow turbulence was captured through a 10 million Hertz ultrasonic transducer immersed in the bath 20 mm below its surface and located in the ladle wall just opposite to the wall which is nearest to the injection orifice. This probe measured the horizontal velocities in the bath at this plane as well as all turbulent variables associated with the flow. Figure 2 shows a scheme of the experimental setup, here described. The physical properties of the three phases, water, oil and air as well as other details of the experimentation are shown in Table 3.³⁶⁾

Fig. 2.

Experimental set up. a) Air compressor. b) Water. c) Oil layer. d) Tracer injection. e) Flat chamber. f) Upper camera. g) Ultrasonic transducer. h) Mass flowmeter. i) Frontal camera. j) Colorimeter cell. k) Air injection.

Table 3. Experimental conditions and physical properties of the multiphase system.

Flow rates of gas	m3/s Model	5.33×10−5	1.07×10−4	2.14 ×10−4	4.28×10−4	5.50×10−4
	l/min Ladle	52	100	200	400	500

Physical properties of fluids (293 K)
	Density Kg/m3	Viscosity Pa-s	Surface tension N/m	Interfacial tension N/m
Water	1000	0.001003	0.073	0.0565
Oil	913	0.060	0.040
Air	1.24	1.8×10−5

Other features: Nozzle diameter is 6 mm, bath height is 0.90 m, scale up criterion is Fr number.

3. The Mathematical Model for Multiphase Flow

To simulate the interaction among the multiphase system, the Volume of Fluid Model (VOF) was applied.²⁸⁾ This model uses a common pressure-velocity field by solving a single set of momentum transfer equations and uses as a phase indicator, for including the presence of interfaces, the volume fraction of a phase by the solving the corresponding advection equation. The equation of the phase indicator is,   

$$\frac{\partial {\alpha }_i}{\partial t}+\left(u_m\cdot \nabla \right){\alpha }_i=0$$(1)

unit value of α_i corresponds to a cell full of fluid 1, while a zero value indicates that the cell contains no fluid 1. To avoid numerical diffusion the equation should be solved using second order explicit discretization equation in time and space, updating the indicator through the velocity field.²⁹⁾ The pressure-velocity field is simulated by resolving the continuity and Navier-Stokes equations,   

$$\nabla \cdot u_k=0$$(2)

  

$$\mathrm{k}\frac{\partial u_k}{\partial t}+u_k\cdot \nabla u_k=-\frac{1}{{\rho }_k}\nabla p_k+{\nu }_k{\nabla }^2{u}_k$$(3)

Where the u_k is the Reynolds Averaged Navier-Stokes (RANS) velocity of the turbulent flow. The interface boundary conditions or momentum jump conditions are expressed as,   

$${\displaystyle \sum }_{k=1}^2{T}_k{n}_k=2{\sigma }_I{H}_I{n}_I$$(4)

Where T_k is the totals stress interfacial tensor, n_k is the normal vector to the interfacial surface, σ_I, H_I and n_I are the surface tension, the radius of curvature and the normal vector to the interface which was simulated through the Continuous Surface Model of Brackbill.³⁰⁾ For the present case the physical properties of the multiphase system, (including the food oil³⁶⁾), are calculated as,   

$${\rho }_m={\rho }_w{\alpha }_w+{\rho }_o{\alpha }_o+{\rho }_a{\alpha }_a$$(5)

  

$${\mu }_m={\mu }_w{\alpha }_w+{\mu }_o{\alpha }_o+{\mu }_a{\alpha }_a$$(6)

The constraint for the volume fraction is,   

$${\alpha }_w+{\alpha }_o+{\alpha }_a=1$$(7)

Where the sub-indexes w, o and a hold for water, oil and air, respectively. The k-ε model^31,32) was used to simulate the turbulence of the flow combined with Eqs. (2) and (3) to obtain the pressure-velocity field which is employed to update the advection equation of the indicator. The model, which is based in the turbulent viscosity hypothesis,^31,32) requires the solution of two other equations for the turbulent kinetic energy and its dissipation rate,   

$$\begin{array}{l}{\displaystyle \frac{\partial \left({\rho }_{m}k\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_{m}k{u}_i\right)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\mu }_m+{\displaystyle \frac{{\mu }_m^t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}+G_k+G_b-{\rho }_m\epsilon \end{array}$$(8)

  

$$\begin{array}{l}{\displaystyle \frac{\partial \left({\rho }_m\epsilon \right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_m\epsilon \right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\mu }_m+{\displaystyle \frac{{\mu }_m^t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial }{\partial x_j}}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}\left(G_k+C_3\epsilon G_b\right)-C_{2\epsilon }{\rho }_m{\displaystyle \frac{{\epsilon }^2}{k}}\end{array}$$(9)

G_k is the generation of kinetic energy due to the interaction between the gradients of the mean velocity and the Reynolds stresses (energy extracted from the mean flow):   

$$G_k=-{\rho }_m{u}_i^{\prime }{u}_j^{\prime }\frac{\partial u_k}{\partial x_j}$$(10)

And G_b is the energy generated by buoyancy forces and is given by Eq. (11):   

$$G_b=-g_i\frac{{\mu }_m^t}{P{r}_t}\frac{\partial {\rho }_m}{\partial x_i}$$(11)

The scalars k and ε are used to calculate the turbulent viscosity through Eq. (12),   

$${\mu }_m^t={\rho }_m{C}_{\mu }\frac{k^2}{\epsilon }$$(12)

Where C_μ=0.09, C_1ε=1.44, C_2ε=1.92, σ_k=1.0, σ_ε=1.3, C_3ε=1.0.

4. Numerical Solution

The solution of the governing equations with boundary conditions & all source terms are obtained through the commercial package ANSYS.³³⁾ In all solid surfaces a no-slip boundary condition is applied, and the wall log-law is used to link the outer grid with the computational elements in the boundary layer. In the nozzle, an entry gas-velocity boundary condition was applied and in the top surface of the bath a pressure one was applied. The calculation domain was divided by polyhedral 1525000 cells and the calculations were conducted under transient conditions. A criterion for convergence was fixed when all residuals for the dependent variables add up to less than 10⁻⁴.

5. Results and Discussion

5.1. Mixing Times

Figure 3(a) show the experimental results for the mixing times without and with the presence of the upper layer, naturally, as the stirring energy, increases these times decrease. Figure 3(b) shows the corresponding results for the SEO (area of the spot eye opening), here, it is clear the tendency of smaller areas with thicker upper layers and a decrease of the mixing time with higher flow rates of the stirring gas. There is always useful, for simple engineering calculations, to find correlations, based on macroscopic models, to estimate mixing times using basic process parameters. Table 4 shows some of these correlations for bath systems without an upper layer, including that of Mazumdar and Guthrie.^8,9) Hence, the experimental results obtained here are plotted in Fig. 4(a) against the calculated ones, under the conditions of the present experiments, using the correlations found in Table 4. As seen, correlations reported by Mazumdar-Guthrie^7,8,9) and Haida et al.¹⁷⁾ yield very good agreement with the measured times. However, the presence of an upper layer inevitable leads to more complex flows and thereby the use of correlations to predict mixing times is not as straightforward as Fig. 4(a) shows for baths without the upper layer. Indeed, Fig. 4(b) shows a plot of the experimental times obtained in this work against those calculated using all the correlations presented in Table 1. As seen, there is a widespread distribution of the data and only the correlation of Patil et al.²¹⁾ provides the better agreement with the experimental measurements. These results highlight the important effect of the thickness and the physical properties of the upper phase.

Fig. 3.

Effects of gas flow rate on: a) Mixing time. b) Eye slag area.

Table 4. Mixing time correlations for baths without an upper layer.

Reference	Mixing Time	Energy	Units of energy
1. Haida et al.17)	τm = 100$\dot{\epsilon }$−0.42	$\dot{\epsilon }=\frac{{\rho }_{L}gQ{H}_L}{{\rho }_L\pi R^2{H}_L}$	W/ton
2. Ying et al.18)	τm = 125$\dot{\epsilon }$−0.289	$\dot{\epsilon }=\frac{{\rho }_{L}gQ{H}_L}{{\rho }_L\pi R^2{H}_L}$	W/ton
3. Mazumdar and Guthrie.9)	${\displaystyle {\tau }_m=37{\epsilon }_m^{-0.33}{H}_L^{-1}{R}^{1.66}}$	$\dot{\epsilon }=\frac{{\rho }_{L}gQ{H}_L}{{\rho }_L\pi R^2{H}_L}$	W/kg
4. Iguchi et al.6)	${\displaystyle {\tau }_m=1\mathrm{\hspace{0.17em} }200Q_g^{-0.47}{D}^{1.97}{H}_L^{-1}{v}_L^{0.47}}$

ρ_L liquid density, g is gravity constant, Q_g is gas flow rate, H_L is bath height, R ladle radius, D is ladle diameter, v_L kinematic viscosity of the liquid. $\dot{\epsilon }$ is the stirring energy per mass unit.

Fig. 4.

Experimental vs numerical mixing times. a) Bath without an upper layer. b) Bath with an upper layer.

5.2. Areas of the Slag Eye Opening (SEO) and Velocities Near the Bath Surface

Figure 5 shows a typical behavior of SEO area with time found in the experiments for an upper phase thickness of 0.02 m. There, clearly is seen the stochastic variations of area but in the long trend this area has a defined average value for each flow rate of gas. Different to the other three flow rates, the increase of the area in the last two highest flow rates seems to be smaller and this is explained by the larger buildup of oil, opposing resistance to widening effect of the eye area, in the opposite ladle wall to the position of the injecting nozzle. A Fast Fourier Transforms analysis of these data yielded small characteristic frequencies (0.5–1 s⁻¹) without any apparent relation with the frequency of the bubbling in the ladle, similarly to the findings of Krishnapisharody and Irons.²⁾ Figures 6(a)–6(d) show the SEO photos of a bath with an oil layer 0.02 m thick stirred with different flow rates of gas together with images of the corresponding simulations using this multiphase model. At a glance, there is a very good agreement not only with the size of the SEO but even with its shape between numerical predictions and the experimental images. The quantitative test of this apparent agreement is presented in Fig. 7(a), for a bath with an oil layer 0.02 m thick, where the numerical simulations are compared with the experimental results. There is a very good agreement between numerical results and the measured SEO areas. Given the very good agreement between predicted and experimental results it can be claimed that the CFD approach can be considered as a useful tool to understand this three-phase flow system and to test the reliability of other correlations. Besides, simpler engineering tools to calculate the SEO are evaluated here by plotting all correlations presented in Table 2, except for correlation (1) whose results lay out of the plot, in the same Fig. 7(a). As seen, Peranandhanthan-Mazumdar´s correlation yields the closest match with the present experimental data, while others predict very different magnitudes of the dimensionless SEO area. However, at higher flow rates of gas the predictions made by this correlation fall apart from the experimental measurements probably because it was derived from experiments using a centric injection nozzle. A similar comment is applicable to correlation (3) in Table 2, and correlations (1) and (2) yield definitively very different results. Therefore, according to the correlation of Peranandhanthan-Mazumdar´s the thickness and the physical properties of the upper layer have important influences on the SEO area and this corroborated by the present results.

Fig. 5.

Effects of gas flow rate for an upper layer thickness of 0.02 m on the variations of the eye-opening area with time.

Fig. 6.

Comparison between experimental and numerical views of the slag eye opening area for an oil thickness 0.02 m. (a) flow 5.33 × 10⁻⁵m³/s. (b) 1.07 × 10⁻⁴m³/s. (c) 2.14 × 10⁻⁴m³/s. (d) 4.28 × 10⁻⁴m³/s. Red color is the upper phase (oil), blue color is the lower phase (water).

Fig. 7.

Slag eye opening area. a) Test of correlations obtained experimentally and numerical results of the present work. b) Relation with the densiometric Froude number for different systems.

Indeed, when the experimental dimensionless SEO area is plotted against the square root of the densiometric Froude number²⁾ given by,   

$$F{r}_D=\frac{U_P^2}{g{h}_s}\frac{{\rho }_g}{\mathrm{\Delta }\rho }$$(13)

where the velocity of the plume is calculated with the expression suggested by Castello-Branco and Schwerdtfeger²⁶⁾   

$$U_P=17.4Q^{0.244}{h}_s^{0.244}{\left(\frac{{\rho }_g}{{\rho }_L}\right)}^{0.0218}$$(14)

a straight line is obtained as seen in Fig. 7(b). In the same figure there are data for different systems of upper and lower liquids, laying, all of them, in straight lines with slopes that depend of the corresponding square root of the densiometric Froude number of each system as was reported by Krishnapisharody and Irons.²⁾

5.3. Fluid Flow Structure

The measured instantaneous horizontal-velocities in the subsurface of the meniscus are plotted in Figs. 8(a)–8(d), for a bath without the upper layer, and with layers 0.02, 0.03, and 0.4 m in thickness, respectively. Without the upper layer, Fig. 8(a), there are highly fluctuating-stochastic velocities close to the position of the transducer. The negative values indicate that the liquid flows toward the transducer and the positives that the liquid flows toward the position of the injecting nozzle. Hence the flow near the ladle wall consists of forth and back flow directions and as the distance comes closer to the two-phase plume the fluctuations of velocity increase because of the ascending bubbles bursting through the bath surface. The presence of the upper layer, Figs. 8(b), 8(c) and 8(d), mitigates the velocity fluctuations near the wall where there is the transducer. Therefore, the turbulence in this region is strongly diminished with increases of the thickness of the upper layer. With thick upper layers, the only region left with high turbulence is just the SEO area where the bubbles travel toward the bath surface bursting there. The continuous line in those plots correspond to the averaged measured velocities for a given flow time, and the interrupted line corresponds to the velocity profile by the VOF model. There is a qualitative agreement between the measured average velocities and those predicted by the model. The reason behind the lack of a better agreement is that it is not possible to match the times between the measurements and the mathematical model to make a viz to viz comparison. Besides, the VOF model calculates an averaged-in-time velocity of the three phases as only a single set of Navier-Stokes equations is solved, and this is certainly another big approximation to the actual dynamics of the flow. Given the circumstances adduced here, it can be said that the model predicts acceptably well also the experimental velocities.

Fig. 8.

Measured vs simulated velocities 20 mm below the metal-slag interface for different thicknesses of the upper phase layer at a fixed gas flow rate of 1.07 × 10⁻⁴m³/s. a) 0 m. b) 0.01 m. c) 0.02 m. d) 0.04 m. Continuous line, measured velocities, interrupted line, numerical velocities with the VOF model, dotted line, averaged experimental velocities.

Figures 9(a)–9(d) show the numerical results in a vertical plane passing through the axis of the injection nozzle, for a bath without an upper layer and with upper layer with thicknesses of 0.01, 0,02 and 0.04 m, respectively. The plume develops following a conical shape and in the bath surface the liquid flows horizontally dragging the upper phase. As the upper layer thickness increases, the long recirculating flow assisting the mixing process decreases until a condition that it is almost suppressed with a layer thickness of 0.04 m. The recirculation flow in the left side, between the ladle wall and the plume, is intensified due to the dissipation of energy by the presence of the layer in the right side. A most dramatic illustration of the effect of thick upper layers can be seen through the streamlines of the flow shown in Figs. 10(a)–10(d), for a bath without an upper layer and with upper layer thicknesses of 0.01, 0,02 and 0.04 m, respectively. Without an upper layer, Fig. 10(a), the streamlines indicate that practically all the volume of the lower phase is stirred by the plume. However, as the thickness of the layer grows, the motion of the lower phase is suppressed as seen in Figs. 10(b)–10(d). The suppressing of the flow in the lower phase explains the longer mixing times found in the present work when the upper layer thickness is as large as 0.04 m. Figures 11(a)–11(d), show the streamlines in a horizontal plane located 20 mm below the interface for a bath without an upper layer and with upper layer thicknesses of 0.01, 0,02 and 0.04 m respectively, just the same plane where the ultrasound transducer is located. When there is not an upper layer, Fig. 11(a), the streamlines are distributed from the spout regularly throughout the plane. The presence of the upper layer develops the formation of recirculating flows at both sides of the location of the transducer. With thicker layers, Figs. 11(b) and 11(c), the recirculation becomes slower but the volume of liquid included in it grows and with the thickest layer, Fig. 11(d), the flow field is split in many regions of secondary flows with different directions and small liquid velocities making the flow practically stagnant.

Fig. 9.

Simulated velocity fields of the liquid phase for a gas flow rate of 2.14×10⁻⁴ m³/s and different thicknesses of the oil layer. a) 0 m. b) 0.01 m. c) 0.02 m. d) 0.04 m.

Fig. 10.

Numerical streamlines of the liquid lower phase for different thicknesses of the upper layer. (a) Oil thickness 0 m. (b) Oil thickness 0.01 m. (c) Oil thickness 0.02 m. (d) Oil thickness 0.04 m.

Fig. 11.

Streamlines of the flow for different thicknesses of the upper layer for a gas flow rate of 2.14×10⁻⁴ m³/s. a) 0 m. b) 0.01 m. c) 0.02 m. d) 0.04 m.

Another important issue to deal with is the evaluation of how much the presence of the upper layer would affect the velocities of the plume and its geometry or shape. To answer this question, the axial velocities simulated along the plume height are plotted in Fig. 12, as seen here, there is no evidence that the presence of an upper layer affects the velocities along the plume. With or without the upper layer the plume yields large velocities close to the injecting nozzle decreasing abruptly, until about 0.2 m from the ladle bottom, maintaining later an approximately constant velocity until very close to the spout where it decreases abruptly again to zero as has been reported by other authors.^{1,2,3,4,5,35)} Regarding the plume diameter, Fig. 13(a) shows the simulated plume diameter along the plume, at low flow rates of gas, this parameter increases linearly, without apparent effect of the upper layer, resembling to conical shapes. At high flow rates of gas, the plume diameter observes a waved behavior ascribed more to the turbulent effects rather than to the presence of the upper layer. The relations between the simulated and the experimental SEO's and the simulated plume diameter are shown in Fig. 13(b), finding a linear tendency and with minor differences between experimental and simulated SEO's diameters. Figures 14(a)–14(d) show, for the four cases being discussed here, 3D views of the interaction of the three phases. It is worthy to mention that, according to the physical and simulated results, for a given flow rate of gas the entrainment of the upper phase becomes more frequent and the size of the droplets increase with thicker upper layers, (see Figs. 14(c) and 14(d)). A further observation is that thick upper layers produce droplets that are transported further down the bath. Then, in summary, thick slags are more prone to entrainment phenomena of large droplets into the metal bulk in a more frequent basis and having long residence times before floating back.

Fig. 12.

Velocity along the plume axis with a gas flow rate of 1.07 × 10⁻⁴m³/s for different thicknesses of the upper layer.

Fig. 13.

(a) Numerical relation between plume diameter and bath height for different flow rates of gas and an upper layer thickness of 0.02 m. b) Relation between calculated slag eye diameter and plume diameter at bath height.

Fig. 14.

Two-phase plumes and slag eye opening areas calculated numerically with the VOF model for different upper layer thicknesses at a fixed gas flow rate of 1.07 × 10⁻⁴m³/s. a) 0 m. b) 0.01 m. c) 0.02 m. d) 0.04 m.

The operation of a ladle with a thick slag layer will have poor performance for bath homogenization and floatation of inclusions, requiring larger stirring energies at expenses of entrainment of slag particles, affecting steel cleanliness. If slag skimming facilities are not available, the only viable solution is to prolong the standstill time of the ladle aiming at the floatation of inclusions. However, someone must be aware of the possibility of losing the casting sequence. According with the present results, a reasonable slag thickness which permits efficient bath homogenization and refining capacity, would be about 2.2–2.5% of the bath height. Finally, these results confirm that stirring of liquid steel in ladles with argon bubbling is inefficient with bad performance for mixing and general refining purposes. Future breakthroughs in this process will be the generalized employment of magnetic stirring instead of pneumatic stirring. This approach will require large initial investment costs but with better economical expectations in the long run due to the decrease of refractory consumption, processing time, better metallurgical performance and improved control, free from the irregular working patterns of porous plugs.

6. Conclusions

Effects of slag layer thickness on the fluid dynamics of liquid steel in bottom purged ladles was studied using physical and mathematical modelling. The conclusions derived from this study are as follows:

(1) Mixing times increase considerably by thicker slag layers and decrease of gas flow rates. The physical properties of the system have a smaller influence on mixing time. Correlation (5) in Table 1 is recommended to estimate mixing times in ladles with a slag layer. For systems without a slag layer or thin layers (about 1% of the bath height), correlations (1) and (3) in Table 4 are recommended.

(2) Slag eye opening (SEO) area is increased by thin slag layers, increase of gas flow rates, and denser and less viscous slags. Correlation (4) in Table 2 is recommended for the estimation of this operating parameter. It is also possible to use Fig. 7(b) and the square root of the densiometric Froude number for steel-slag systems.

(3) The SEO dimensionless area follows a linear relationship with the square root of the densiometric Froude number based on the slag thickness.

(4) The presence of, either, thick or thin slag layers does not influence the axial velocity along the plume height for a fixed flow rate of gas. The diameter of the plume is affected by the presence of slag layers only through an upper layer of the bath close to the meniscus.

(5) The numerical VOF model yields numerical data that agree with the experimental observations of SEO area and liquid velocities close to the metal-slag interface. The present mathematical approach can be used as a standard to test other empirical correlations required for engineering calculations.

Acknowledgements

The authors give the thanks to Instituto Politecnico Nacional and the University of Toronto for supporting this work. Thanks, are also given to the Consejo Nacional de Ciencia y Tecnologia of Mexico for its continuous support.

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