Effect of the Injection Position on Mixing Time in a Centric Gas-stirred Ladle Water Model Assisted by a Systematic CFD Study

Abstract

Mixing time is a parameter that describes the degree of agitation in a gas-stirred ladle. The literature is not conclusive on the effect of the tracer injection position on the mixing time under equal stirring conditions. This work conducted a systematic and comprehensive study on the effect of the tracer position on the mixing time for centric injection. Results in the form of a mixing time map indicate that the diagonal formed between the inlet and the upper wall region gets the fastest mixing, and specifically in the eye of the toroid of the circulation loop is the best position to mix solute rapidly. In contrast, the dead zones at the lower near-wall part of the ladle have the poorest mixing behavior for the tracer addition. The study also tested the ladle’s axisymmetric assumption since two points were at different angular positions, but the same axial and radial points presented similar mixing time.

1. Introduction

Gas-stirred ladles are batch metallurgical reactors that refine the steel by eliminating oxygen and sulfur, promoting thermal and chemical uniformity, and eliminating non-metallic inclusions. To accelerate such refinement operations, the liquid steel is agitated by injecting argon gas from the bottom to promote turbulent fluid flow. Mixing time is a quantitative measurement of the stirring condition of a ladle. The effect of design and process variables, such as the ladle’s height and diameter, the position of plugs, the number of plugs, and the gas flow rate, among others on mixing time, has been extensively studied in the open literature through physical and mathematical models,^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)} which are excellent analysis tools, under the aggressive conditions of temperature and the opaque nature of the ladle. The excellent reviews by Mazumdar and Guthrie^20,21) and recently by Liu et al.²²⁾ present many mixing time correlations as a function of the main operating/design parameters. The mixing time correlations were mainly obtained in water physical models,^20,21,22) only a few based on industrial trials,^23,24) and even less by mathematical models.^25,26,27) Since the first decade of this century, an agreement in the scientific community indicates that the presence of the slag consumes between 8 to 20% of the stirring energy, so this phase has to be accounted for in order to study mixing. Besides, despite mixing time was reported many decades ago, including several well-recognized criteria to reach mixing based on 95 or 99% uniformity, the mixing time measurement still presents controversies: 1) The mixing time is usually measured by intrusive techniques (pH, conductivity, etc.) using solid probes that modify the flow field; 2) it gives a point of measurement that does not represent the entire volume, 3) the injection of a tracer is mainly done on the free surface on top of the position of the plug, but there are no systematic studies on the effect of the injection position on mixing time. Only a few non-systematic attempts (Oeters^28,29) and Herrera-Ortega²⁶⁾) have been published on the effect of the injection tracer position on mixing time.

In this work, a systematic and comprehensive study on the effect of the tracer injection position on the mixing time is conducted for centric single plug gas injection. The results allow us to present a map of mixing and a zone of optimum mixing resulting from a combination of bulk convection and eddy diffusion (i.e., turbulent dispersion) simultaneously acting to optimize mixing.

2. Methodology

2.1. Mathematical Modeling

The mathematical model simulates a physical acrylic water model using a 1 to 17 scale from a 200-ton industrial steelmaking ladle. The model diameter is 0.19 m, with a bath height of 0.172 m. Due to its kinematic similarity, water at 25°C is used to represent the liquid steel at 1600°C, having a kinematic viscosity of 1.0 × 10⁻⁶ m²/s and 0.97 × 10⁻⁶ m²/s, respectively. Air is injected at the bottom of the physical model to emulate argon injection. The bottom nozzle has a diameter of 7 mm, and the gas flow rate is 1.54 lpm in normal conditions. Lastly, engine oil simulates the molten slag, being 4% of the bath height. The three main phases have densities of 998.2 kg/m³, 890 kg/m³, and 1.225 kg/m³ for water, engine oil, and air, respectively.

The mathematical model simulates the solute mixing on the physical model. It was solved numerically in two stages: 1) the first consists of solving the fluid flow problem in steady state for centric gas injection, and 2) the second stage solves only the species conservation equation in transient state to simulate the solute mixing.

The following assumptions were made to simulate the fluid flow model:

1. The system is tridimensional, steady, and isothermal. Although gas is injected at the center, we solved the problem in 3D to get the solute dispersion in all the ladle’s volume since the injection of solute breaks the 2D nature due to the mass transfer problem.

2. Water, air, and engine oil’s material properties are considered constant; all fluids are assumed to be incompressible and Newtonian.

3. The air bubbles are spherical, with a constant diameter of 0.01 m.

4. An Euler-Euler approach is used to solve the multiphase fluid flow problem.

For fluid dynamics, the governing equations include the continuity and turbulent momentum equations for all phases. Turbulence is modeled using a k-ε realizable model for the main water liquid phase. The momentum exchanged between phases is accounted for via source terms in the momentum equations, like the drag force modeled using the Schiller-Naumann model,³⁰⁾ and the turbulence interaction between phases modeled using the Sato model.³¹⁾ The gas phase momentum equation contains the buoyancy source that drives the motion of the ladle. The volume fraction distribution for each phase is obtained by solving the mass conservation equation for two out of three phases, and the third phase’s volume fraction is obtained by meeting the mass balance.

Once the fluid flow is calculated, the mixing problem is solved through the species conservation equation in transient mode.

The boundary conditions for the fluid flow problem indicate static, impermeable walls with a non-slip condition and standard wall functions for the universal velocity profiles from the wall to the turbulent core. The system has an inlet at the bottom where the gas is injected, and at the upper surface, an outlet for the gas phase is set.

For the mass transfer model, boundary conditions are zero flux of the tracer at all boundaries. The initial condition is zero concentration of solute everywhere except at the injection point, where the solute concentration is set to a value of 1 mg/liter, which was the same used in the experiments using the physical model.³²⁾

2.2. Tracer Injection Position Investigation

In order to study the effect of the injection point of solute on the mixing time, the initial point was changed all over the ladle volume. An array of injection points used in this work is shown in Fig. 1. The center plane of the ladle water model is used as a reference. Two injection points not on that plane were added: E2 has the exact coordinates as B3 but at 180° of the XY plane, and E1 is located at the same height as B3 but at 90° of plane XY. These two points were used to test if the axisymmetric assumption is valid. The 95% uniformity criterion determines the mixing time by comparing the elements’ concentration values against this criterion. When all elements in the mesh meet the criteria, the mixing time is reached.

Fig. 1. Locations of each injection point used for the simulations; the main array of points are in the XY plane. Two extra points out of the main array were added, E2 and E1. (Online version in color.)

The model was cast in the CFD software ANSYS FLUENT 2020 R2, with a mesh of over 300 k elements and a mean orthogonal quality of 0.964. The mesh has refinements near the water-engine oil-air interfaces. This grid was used after a mesh sensitivity study that warrants the grid independence of the results. The simulations were done on an Intel Xeon Gold 6130 CPU @ 2.10 GHz. It is worth mentioning that the fluid flow problem was solved in pseudo transient mode, while the mixing problem was solved in transient mode with a time step of 1 s.

The mathematical model was used to predict the fluid dynamics, and the phase distribution was extensively validated by comparing predicted flow fields against experimental measurements using Particle Image Velocimetry (PIV) for several operating conditions, which can be seen in Jardón-Pérez et al.³³⁾

The mixing problem was validated by comparing our numerical results against an experimental mixing time measured with similar conditions for injection position A1. The experimental mixing time, measured with Planar Laser-Induced Fluorescence (PLIF), is 8.65 s,³²⁾ while the simulation mixing time is 8.69 s, showing a difference of less than 1%.

3. Results and Discussion

Table 1 shows the mixing time calculated for each one of the injection positions. Figure 2 shows a contour map of the mixing times. The plot corresponds to half the longitudinal plane, where the top surface is the free surface, the bottom surface is the bottom wall and inlet, the left is the lateral wall, and the right is the symmetry axis. The region of solute injection that minimizes the mixing time is a diagonal zone from the inlet to the upper left region with less than four seconds to reach 95% solute uniformity. Below this diagonal, the mixing time increases until the maximum mixing time of 18.36 seconds is obtained when the solute is injected at the dead zone. Above the optimum diagonal, the mixing time increases, but the increment is just moderate.

Table 1. Array of mixing times in seconds as a function of the injection position. Rows are axial positions measured from the bottom of the ladle, and columns are radial positions from the center of the model.

Injection point	Radial	A	B	C	D
Axial	Position, [cm]	0.0	2.8	5.6	8.5
1	15.0	8.69	8.24	7.18	2.47
2	12.2	8.35	7.46	2.08	4.12
3	9.4	7.89	5.39	4.66	11.15
4	6.6	7.57	2.59	9.56	14.42
5	3.8	7.53	3.95	10.57	16.88
6	1.0	7.74	4.10	9.30	18.35

Fig. 2. Contours of mixing times as a function of injection point. The largest mixing times are those near the lower corner of the water model. The black area represents zones that are out of the validity of this study.

Positions E2 and E1 have mixing times of 5.78 s and 5.73 s, respectively, while on B3 (see Table 1), it is 5.39 s, showing a difference of less than 10% in the calculated mixing time. This means that the behavior of mixing time in other quadrants of the ladle is practically the same, validating the axisymmetric nature of the centric injection.

The best position for the tracer injection that minimizes the mixing time can be explained by analyzing the mass transfer mechanism of bulk convection and eddy diffusion that enhances the chemical homogenization. Figure 3 shows the flow patterns of the liquid phase at the middle plane. The flow pattern shows the formation of two recirculation zones adjacent to the gas plume, typical of centric injection. The dashed lines indicate the boundaries of the diagonal where the mixing times are the lowest if a tracer is injected inside these lines. The best mixing is achieved when the tracer is injected in or near the eye of the toroid. Mixing is a process that depends on the mass transfer mechanisms. Bulk convection and eddy diffusion (turbulent dispersion) are the main mass transfer mechanisms in gas-stirred ladles, as exposed by Jardón et al.³²⁾ Then, in addition to the flow patterns, Fig. 4 shows the eddy viscosity contour plots that, together with the flow patterns, explain why the diagonals are regions of the fastest mixing kinetics. The largest values of eddy viscosity are in the gas plume region and near the water-oil interface. In the case of eddy viscosity, the diagonal travels from bottom to top, increasing eddy viscosity zones. Jardón et al.³²⁾ also indicate that eddy diffusion fluxes are half the convection fluxes, but both are complementary to achieve mixing. Then, a solute injected in or near the toroids helps the solute disperse by convection, and the turbulence species diffusion helps disperse the solute all over the ladle in all directions, including the azimuthal direction.

Fig. 3. Flow pattern in the middle plane of the water model, obtained by numerical simulation. The dashed lines indicate the diagonal where the mixing times are the lowest if solute is injected in that position.

Fig. 4. Contours of water eddy viscosity in the same plane as the tracer injection points. The dashed lines indicate the diagonal where the mixing times are the lowest if solute is injected in that position.

The convective of fluid controls the mixing behavior in the toroids, and in the plume (see Fig. 3), together with the help of the lateral turbulent dispersion, develops the chemical mixing (See Fig. 4, where it can be seen that eddy diffusion is significant in the plume and the upper region of the ladle). Usually, fluid completing two circulating loops is enough to meet the mixing criterion. The mixing time curve (average concentration versus time) must show a periodic behavior of maximum and minimum values according to the circulating loops of the fluid. In every loop, the difference between the minimum and maximum is lower as chemical uniformity develops in the bath. The time between two maxima is the circulation time.^28,29)

On the other hand, in the dead zones, convection is minimal, so the eddy diffusion controls the mixing until the solute reaches the toroids. Then, both mechanisms’ synergetic contributions complete the mixing. The curve of mixing time (average concentration versus time) in a dead zone must show a steady increase in concentration until the equilibrium value is reached.^28,29)

In the diagonal region of low mixing time (see Fig. 2), we can observe that neither the most significant convection nor the most significant eddy diffusion occurs. However, the trajectory of the flow patterns (see Fig. 3) makes it possible for the tracer to reach the dead zones by convection quickly. In contrast, an adequate amount of turbulence (see Fig. 4) allows the tracer to disperse laterally while being dragged by the movement of the liquid. In this way, the synergy between the convective and turbulent diffusive mechanisms improves the mixing time.

The mixing time increases as the injection point moves away and below the diagonals because the tracer is injected directly in a dead zone where bulk convection and eddy diffusion mechanisms are present but with mitigated intensity. The largest mixing time (18.35 s) is in position D6, located in the dead zone, where mainly molecular diffusion allows the solute to slowly reach the high turbulence and convection zones. When the injection is done over the diagonal, the solute must travel all along the recirculation loop (in azimuthal direction) and then disperse downwards, thus requiring slightly longer times to mix completely.

In industrial practice, adding ferroalloys to the ladle to adjust chemical composition is mainly done on position A1, just below the slag-steel interface, and practically in the stirring gas plume (8.69 s of mixing time). The numerical results show that for a water model, the lowest mixing time is not achieved in position A1 but in position C2, located near the toroid and near the water-oil interface (2.08 s). Then, the injection of solutes through wires can be optimized by steelmaking practitioners to release solutes in the regions of highest mixing.

4. Conclusions

• The injection of solute near the recirculation vortex caused by the air injection and near the water interface close to the walls represent the lowest calculated mixing time positions (the lowest is the position C2). In contrast, the solute injection in the bottom corner of the water model, position D6, where a dead zone can be identified, shows the largest mixing time. Compared with the mixing time of the most common injection point (A1), the lowest mixing time represents a decrease of almost 70%, while the largest, a 200% increment.

• Inside the water model (ladle), there is a diagonal region where bulk convection and eddy diffusion synergize to lower the mixing times. Above the diagonal, the solute must travel all along the recirculation zone, and below the diagonal, the presence of a dead zone makes species diffusion the main mixing mechanism, thus delaying the mixing process.

• The flow and mixing time behavior is practically axisymmetric despite the turbulent flow with centric single plug injection.

Statement for Conflict of Interest

Authors state no conflict of interest.

Acknowledgments

R. Villarreal-Medina is a Ph. D. student in the “Programa de Doctorado en Ingeniería Química”, at the Universidad Nacional Autónoma de México (UNAM). He thanks CONAHCYT for receiving a doctoral fellowship (Grant Number CVU 1002868). L. E. Jardón-Pérez thanks CONAHCYT for the academic postdoctoral fellowship of the project: “Development of mathematical models for the process analysis of HVOF and HVAF thermal spray processes for biomedical implants applications”, CVU: 624968. The authors thank DGAPA-UNAM for financial support through the project grant IN 102922.

Nomenclature

k: Turbulent Kinetic Energy (m²/s²)

ε: Dissipation rate of Turbulent Kinetic Energy (m²/s³)

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