Effect of the Location of Tracer Addition in a Ladle on the Mixing Time through Physical and Numerical Modeling

Abstract

In the present work, the release of tracer location on the global mixing time in an agitated ladle furnace by gas bottom injection was analyzed. Then, a numerical multiphasic steel-slag-argon-air system of a prototype with a capacity of 150 tons was carried out. The simulation was validated by using a physical model with a 1/6 geometric scale using colorant, KCl dispersion measurements techniques and open slag eye opening. Four different tracer addition locations were strategically established to study the influence of tracer releasing location on chemical homogenization. From the results, it was found that the measurement of mixing times varies according to the location of the tracer addition, which to a greater extent is conditioned by the convective currents that at some extent were related to turbulent viscosity.

1. Introduction

The secondary refining process is made up of several operations which are deoxidation, desulfurization, decarburization, as well as removal of dissolved gases such as hydrogen and nitrogen. Additionally, it is possible to control the temperature and the adjustment of the composition of alloys to obtain different grades of steel.¹⁾ The above is achieved by stirring the metal bath by injecting an inert gas, commonly argon, through porous plugs located in the bottom of the metallurgical reactor that contains the steel. The gas flow injected into the system generates a column known as a plume which is made up of bubbles which contain kinetic energy. As the bubbles rise along the plume they accelerate due to the buoyant force caused by the difference that exists between the density of the steel and argon. The result of the interaction of both phases is the exchange of momentum which increases the velocity of the steel due to the dissipation of energy from the movement of the bubbles. The agitation of the liquid contained in the ladle furnace results in mixing which occurs due to convection flow, as well as turbulent and molecular diffusions.²⁾ The degree of mixing is determined by measuring the mixing time (t_mix) which represents the time necessary for a small amount of tracer added in a liquid to reach a uniform concentration in mix. The degree of mixing at 95% is usually accepted, although the degree of mixing at 99% has also been used, which is more rigorous.³⁾ The mixing phenomenon has been widely investigated and various works have been reported in literature on the main operation variables that considerably affects mixing time measurements, such as gas flow,^{4,5,6,7,8,9,10,11,12,13,14)} number and location of injections,^15,16,17,18) as well as properties¹⁹⁾ and thickness of slag.²⁰⁾ In most of the previously mentioned papers, the location of the tracer addition has minor importance within the analyzed variables and use a single addition location which, in most of the cases it is located in the slag eye opening. However, various works have been reported in literature with contrasting conclusions, such as the investigations carried out by Krishna Murthy and his colleagues^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)} in which the authors conclude that the measurements obtained of the mixing time do not depend on the location neither on the tracer amount released with which the conductivity technique is performed. On the other hand, various authors conclude that not only the location of the sensors²²⁾ affects the mixing time measurements but also the location of the tracer addition^23,24,25) is a variable that considerably affects the time required to reach a homogeneity degree of 95%. The discrepancies between authors have given rise to discussions and communications²⁶⁾ between authors with reference to the relevance of the influence that a location of the addition of the tracer has on the experimental measurements of mixing time. The last, supports the pursued objective of the present research that is to elucidate the effect that the tracer addition location has on the measurement of mixing time on a ladle furnace, four cases of study were proposed. For this, numerical simulations of a three-dimensional multiphasic system (steel, slag, argon and air) were carried out to firstly study the fluidynamics behavior in the reactor with a single gas injection configuration, and then by using the numerical non-reaction species dispersion model to compute the mixing time required to reach 95% of homogeneity. Finally, results were validated through its analogue physical modeling technique, by using KCl saturated solutions as a tracer and also colorant dispersion in water.

2. Methodology

2.1. Physical Model Design

For the validation of the numerical simulation in which the species model has to be solved, cases 1 and 2 described on Table 1 were selected. For this purpose, mixing time values were obtained using a physical model with geometric scale 1/6 of a 150 ton ladle furnace prototype, the model consists of a conical transparent acrylic cavity filled with water as the working fluid and mineral oil to simulate the upper slag layer which is shown in Fig. 1(a). The system is agitated by injecting compressed air through a nozzle located in the middle of the radius of the bottom cavity as shown in the experimental setup illustrated in Fig. 1(b). A 35 ml sample of a saturated KCl solution is released for each experiment at different coordinates as expressed in Table 1 for the studied cases. The KCl concentration in the water was monitored by electrical conductivity measurements using a fixed 9382-10D submersible electrode, located at 0.1 m from bottom and 0.01 m from ladle wall model, and HORIBA^® TDS LAQUA EC110-K conductivity meter cell-type. Measurements were captured every 5 seconds under the same conditions and sent to a PC for further processing, and to ensure the reproducibility of the results five trials were conducted for each case.

Table 1. Numerical simulation cases.

Case	Flow (Nm3·min−1)	Location of tracer addition	Coordinates, m
			x	y	z
1	0.8	Axisymmetric bottom	0	0.25	0
2		Slag eye opening	0.7	3.2	0
3		Low recirculation zone	−1	0.25	0
4		High recirculation zone	−0.2	2.5	0

Fig. 1.

Geometrical dimensions, a) Physical scale model and b) Experimental setup.

To determine the air flow in the scale model, the modified Froude number²⁷⁾ expressed in Eq. (1) was employed.   

$${\left(\frac{Q_m}{Q_p}\right)}^2=\left(\frac{{\rho }_g}{{\rho }_a}\right)\left(\frac{{\rho }_w}{{\rho }_s}\right){\left(\frac{d_m}{d_p}\right)}^4\left(\frac{H_m}{H_p}\right).$$(1)

Where the subscripts m and p are representative of the model and the prototype, respectively, Q is the volumetric flow of the injected gas, m³·min⁻¹, d is the diameter of the nozzle, m; H is the height of liquid, m; ρ_g, ρ_a, ρ_w and ρ_s are the densities of argon, air, water, and steel respectively. From conductivity measurements, a dimensionless concentration is calculated at time t determined by Eq. (2):   

$$C_i^{\prime }=\frac{C_i-C_0}{C_{\infty }-C_0}$$(2)

Where C₀ is the initial concentration; C_i is the concentration at time t and C_∞ is the final concentration determined by the sensor.

2.2. Colorant Dispersion Technique

To validate the fluid dynamic patterns obtained from the numerical simulation, a supersaturated solution of red vegetable colorant was injected through an orifice located at the bottom of the container at a distance of 20 × 10⁻³ m from the gas injection nozzle. The dye serves as a tracer since it has the same properties as water and adopts the flow patterns once the quasi-stable state is reached approximately 15 s from the start of the air injection. Videos were captured at a speed of 20 fps with the help of a Sony^® model SLT-A37 camera with an Exmor™ CMOS sensor, which were processed to extract images every second.

2.3. Mathematical Model

2.3.1. Domain Dimensions, Boundary Conditions and Considerations

The simulation was carried out considering the geometry of a 150-ton ladle furnace, the phases and boundary conditions are shown in Fig. 2(a). The computational domain mesh consists of 809835 elements. The numerical models are solved with the help of the commercial code ANSYS FLUENT^® using the PISO algorithm²⁸⁾ for problems in transient state. The convergence criterion for all variables was established as 1 × 10⁻⁵. In the present work, the following considerations were employed:

(1) The geometry was developed in a three-dimensional Cartesian coordinate system

(2) The fluids contained in the ladle behave as Newtonian.

(3) The flow is completely turbulent.

(4) Models run under isothermal conditions.

(5) Non-slip conditions for velocity occurs on all walls.

(6) The gravity force acts only on the negative y-axis.

(7) Surface tension forces between fluids are considered.

(8) The gas injection is constant and is done through a nozzle instead of a porous plugs.

(9) Chemical reactions are not considered in the species transport model.

Fig. 2.

Computational domain of the conical ladle furnace a) boundary conditions and b) position of the sensors.

Figure 2(b) shows the arrangement of tracer sensors employed as well as their location. Additionally, two planes were established for the analysis of fluid dynamics: plane A is located at the center of the injection and divides the domain in half while plane B is perpendicular to plane A, crossing the gas injection middle in the same way, as shown in Fig. 3. Properties of the materials used in the mathematical simulation of the multiphasic system are listed in Table 2.

Fig. 3.

Arrangement of analysis planes for fluid dynamics.

Table 2. Properties of the materials used in the simulation.

Material/Interface	Density kg·m−3	Viscosity kg·m−1·s−1	Surface tension N·m−1
Steel	7020	0.006	–
Slag	3500	0.2664	–
Argon	1.6228	2.125×10−5	–
Air	1.225	1.7894×10−5	–
Water	998.2	0.001003	–
Oil	889	0.1589	–
Steel-Argon	**	**	1.8229)
Slag-Argon	**	**	0.5829)
Steel-Slag	**	**	1.15
Steel-Air	**	**	1.82
Water-Oil	**	**	0.04
Water-Air	**	**	0.072
Air-oil	**	**	0.021

**  Calculated by Eq. (6).

2.3.2. Continuity Equation

The VOF model solves two or more immiscible fluids by tracking the interfaces between the fluids through the continuity equation for the volume fraction of one of the phases represented by Eq. (3).   

$$\frac{\partial }{\partial t}\left({\alpha }_q{\rho }_q\right)+\nabla \cdot \left({\alpha }_q{\rho }_q{\overrightarrow{v}}_q\right)=0$$(3)

Where, for phase q: ρ_q is the density, kg·m⁻³; ∝_q is the fraction of volume within the cell occupied by the phase; and ${\overrightarrow{v}}_q$ is velocity, m·s⁻¹. The volume fraction is defined by Eq. (4).   

$${\displaystyle \sum }_{q=1}^n{\alpha }_q=1$$(4)

2.3.3. Equation of Conservation of Momentum

A single equation of momentum is solved along the domain, the resulting velocity field is shared between the present phases. Equation (5) is dependent on the volume fractions of all phases through the properties ρ and μ.   

$$\frac{\partial }{\partial t}\left(\rho \overrightarrow{v}\right)+\nabla \cdot \left(\rho \overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+\nabla \cdot \left[\mu \left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^T\right)\right]+\rho \overrightarrow{g}$$(5)

Where ρ is the density, kg·m⁻³; μ is the viscosity, Pa·s; ∇p is the pressure gradient, Pa; and $\overrightarrow{g}$ is the acceleration of gravity, m·s⁻¹. The properties of the mixture, such as density, viscosity, etc. are calculated for the volume fraction and the properties of each phase x_q through Eq. (6).³⁰⁾   

$$x=\sum {\alpha }_q{x}_q$$(6)

2.3.4. Model of Turbulence k-ε

The turbulence phenomenon is simulated by means of the standard k-ε model,³¹⁾ which solves two equations for the turbulent kinetic energy, k, and the rate of dissipation of turbulent kinetic energy, ε, which are expressed by Eqs. (7) and (8).   

$$\frac{\partial }{\partial t}\left(\rho k\right)+\nabla \cdot \left(\rho k\overrightarrow{v}\right)=\nabla \cdot \left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\nabla k\right]+G_k+G_b-\epsilon \rho$$(7)

  

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\rho \epsilon \right)+\nabla \cdot \left(\rho \epsilon \overrightarrow{v}\right)=\nabla \cdot \left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\nabla \epsilon \right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+C_{1\epsilon }\frac{\epsilon }{k}\left(G_k+C_{3\epsilon }{G}_b\right)-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k} \end{array}$$(8)

Where G_k, is the generation of turbulent kinetic energy due to the average of the velocity gradients; G_b is the generation of turbulent kinetic energy due to buoyancy forces; μ_t is the viscosity turbulent; C_1ε, C_2ε y C_3ε, and are empirical constants whose values are 1.44, 1.92 and 1 respectively; σ_k and σ_ε are the turbulent Prandtl numbers for k and ε whose values are 1.0 and 1.3 respectively.

2.3.5. Species Transport Equation

To simulate the tracer dispersion, the species transport model was employed which solves Eq. (9).   

$$\frac{\partial }{\partial t}\left({\alpha }_q^i{\rho }_q^i{Y}_i^q\right)+\nabla \cdot \left({\alpha }_q^i{\rho }_q^i{\overrightarrow{v}}_q^i{Y}_i^q\right)=-\nabla \cdot {\overrightarrow{J}}_i$$(9)

Where ${\alpha }_q^i$, ${\rho }_q^i$ and ${\overrightarrow{v}}_q^i$, represent the fraction of volume, density and velocity of phase q that contains species i; $Y_i^q$ is the mass fraction of species i contained in phase q; ${\overrightarrow{J}}_i$ is the diffusive flow of species i.³⁰⁾

2.3.6. Measurement of Mixing Time in the Mathematical Model

To measure the mixing time, the methodology proposed by Villela et al.³²⁾ was employed in which the average normalized concentration is determined for a system with multiple sensors in time t, which is determined by Eq. (10).   

$${\overline{C}}_t=\frac{1}{k}{\sum }_{j=1}^k{C}_j^t$$(10)

Where $C_j^t$ is the concentration measured by sensor j at time t; k, is the number of sensors considered, in the present work there are 15 sensors. The 95% degree of mixing criterion was employed, which means that the measurement of normalized concentration is within a variation range of ±5% of the measurements average.

3. Results and Discussion

3.1. Fluid Dynamics Structure

In Fig. 4, contour lines colored by velocity magnitude are observed for both analysis planes, in which the presence of a single dominant recirculation is more clearly observed in Fig. 4(a), which is located on the axis of symmetry in the area near the slag layer, as well as a smaller recirculation in the upper part near the argon gas plume and close the interface steel-slag. This is because once the bubbles rise due to the buoyant forces, they impact the slag layer and change their direction towards the walls while the slag layer deforms to create the slag eye opening. Once the flow meets the domain walls, it begins to descend, slowing down as it approaches the bottom of the ladle. On the other hand, in Fig. 4(b) two recirculations are observed, which gives three-dimensional toroid-shape behavior of the flow generated by the column of bubbles, these two recirculations extend on both sides of the gas injection. Gas plume behavior is dependent of the bubbles properties as they ascend to the top slag layer as reported by Villela et al.³¹⁾ in a previous work.

Fig. 4.

Steel streamlines for steel-slag system colored by velocity magnitude, a) Plane A and b) Plane B. (Online version in color.)

3.2. Validation of Results

The flow dynamics technique by means of colorant dispersion was used to qualitatively analyze at different times the fluid dynamic behavior once the quasi-stable state was reached. The comparison between the analysis of streamlines in plane A of the numerical model and the physical simulation is shown in Fig. 5.

Fig. 5.

Colorant dispersion at a) 4 s, b) 6 s, c) 8 s, d) 10 s and e) streamlines of the numerical simulation on quasi-stable state conditions. (Online version in color.)

At the beginning of the experiment, once the vegetable colorant is injected, it immediately adopts the direction of the air flow that is introduced through the nozzle as observed in Fig. 5(a). Due to the buoyancy forces, the flow of air bubbles is directed towards the oil layer in an upward movement towards ①. Later in Fig. 5(b) it is observed that after 6 seconds the flow separates, a part of the flow is redirected towards the nearest wall while the other advances in the opposite direction under the oil layer towards ②. After 8 seconds, two incipient recirculations are observed at ① and ② because the flow impinges the cylindrical wall and is directed towards the bottom of the cavity as shown in Fig. 5(c). On the other hand, in Fig. 5(d) it is observed that as the flow descends it loses velocity and once it reaches the bottom it goes back towards the plume at ③ thus creating a recirculation circuit. There is good agreement between the experiment and the numerical simulation since in Fig. 5(e) the recirculation at the ❷ is in charge of transporting the tracer to the majority of the volume occupied with water. The recirculatory flow rises again at ❸ in both analysis and finally a small secondary recirculation is observed at ❶ by both techniques. Qualitatively and employing these last results, the tracer addition location could be determined for cases 3 and 4, which corresponds to a low and high recirculation zone respectively. The quantitative validation of the numerical model was carried out by means of mixing time measurements for cases 1 and 2 as shown in Figs. 6(a)–6(b) and 6(c)–6(d), respectively.

Fig. 6.

Mixing time obtained for case 1 a) experimental and b) numerical simulation; Case 2 c) experimental and d) numerical simulation.

In case 1, Fig. 6(a) illustrates the mixing time obtained by the conductimetry technique in which the results of 5 trials and their average are plotted. The dispersion among measurements at the first seconds is due to the fact that the tracer is added through a hole in the center of the bottom of the cavity and it is observed that the tracer is dispersed in different directions in each experiment before adopting the recirculation streamlines. However, as the domain occupied by tracer becomes homogeneous, the error decreases until reaching a 95% of mixing degree. The above behavior does not interfere with the overall mixing time which is 106 s. Figure 6(b) shows the normalized concentration variation of the tracer species for the multi-sensor system in the numerical simulation. Sensor 15 is located right in the axisymmetric location of tracer addition reaching extremely and obviously high values in relation to the other sensors and it was not plotted to better appreciate the time in which the mixing degree of 95% is reached, which corresponds to 109 s. On the other hand, for case 2, Fig. 6(c) shows in the same way the measurements of the 5 trials and the average of them, in which the addition of tracer is carried out at the opening of the oil layer. A lower dispersion in comparison to case 1 is observed in the first seconds of experimental data, which is due to the fact that the addition at the slag eye opening is carried out right in the flow generated by the rising bubbles and the tracer immediately adopts the dominant recirculation flow pattern quickly transporting itself to the rest of the domain occupied by free-tracer water. It is observed that for the 5 experiments the mixing time has a variation of less than 5% after around 98 seconds. Figure 6(d) shows the normalized concentration variation for case 2 of the numerical simulation. It can be seen that in the sensors located in the H3 plane they register the lowest values since they are located in the area furthest from the tracer addition, unlike sensors 1 and 3 that are close to the slag eye opening. Sensors 2, 4, and 13 record the highest values relative to the other sensors and are omitted from the graph to better visualize the other normalized concentration curves.

3.3. Tracer Dispersion Analysis for Mixing Time

Figure 7 shows normalized concentration contours for cases 1 and 2 at different times of the tracer dispersion. For case 1, Figs. 7(a)–7(f), it is observed how the tracer species adopts the flow pattern moving towards the gas inlet and heads in an upward direction towards that of the slag, subsequently integrating into dominant recirculation. It can be seen that although the variation in concentration is within the range of ±5% at 109 seconds, there are areas of greater homogeneity. And for case 2, Figs. 7(g)–7(l), it can be highlighted that as the tracer species rapidly adopts the recirculation pattern and the concentration rapidly homogenizes in the central region of the domain. However, the area where a slight recirculation is created near the wall closest to the gas injection, a zone of slow movement is generated compared to the rest of the domain and it is the zone where takes the longest to reach the mixing criteria. For the last, it is clear that bubbles plume acts with a virtual wall-type effect that inhibits the mixing phenomena as observed in last times for case 2, such phenomenon was already described elsewhere.⁶⁾

Fig. 7.

Normalized concentration contours of KCl tracer for the cases 1 and 2 at different times, a) and g) 0 s, b) and h) 25 s, c) and i) 50 s, d) and j) 75 s, e) and k) 100 s, f) and l) 125 s. (Online version in color.)

The mixing times obtained for the numerical modeling in the four proposed cases as well as the validation of cases 1 and 2 through measurements obtained with the physical model are shown in Fig. 8. According to the results obtained through both models, it can be seen that there is a strong agreement in the results within an acceptable margin of error. From the validation of cases 1 and 2 it is concluded that the numerical model used is acceptable for the resolution of cases 3 and 4.

Fig. 8.

Mixing time obtained through numerical and physical simulation for all cases.

3.4. Mixing Dependence on Turbulent Viscosity

In previous works the turbulent kinetic energy,³³⁾ the effective viscosity³⁴⁾ and the angular momentum³⁵⁾ associated to tracer releasing location in the mixing phenomenon have been analyzed. However, turbulent viscosity can be a parameter to locate the optimum release tracer zones that reduce the total mixing time. Since in this work it will be applied that the zones of high turbulent viscosity agree with zones of large recirculation favoring the mass transfer due eddy or turbulent diffusion which occurs due to the dissipation of the turbulent kinetic energy of the system.³⁶⁾ Figure 9 shows the turbulent viscosity contours for both analysis planes in which it can be seen that in the center of the domain adjacent to the gas plume there is a greater amount of turbulent viscosity. In the species model the turbulent viscosity plays a preponderant role in the transport of the tracer species since in the mathematical formulation the turbulent viscosity is directly proportional to the mass diffusion of the tracer species. It can be said based on the results of Fig. 9 that the tracer species will diffuse to a greater extent to the center of the domain coinciding with the location of the dominant recirculation previously analyzed. It is also observed that for the region of the domain in the wall closest to the gas injection, the intensity in the turbulent viscosity contours is lower, thus explaining why this region is the last to reach the mixing criterion observed in Fig. 8 of normalized concentration contours. The influence of the area in which the tracer is added may be due to a greater extent since there are regions where mixing is less intense and consequently releasing the tracer in these areas causes the time in which the variation in the concentration of tracer increases.

Fig. 9.

Turbulent viscosity contours, a) Plane A and b) Plane B. (Online version in color.)

3.5. Slag Eye Opening

As well as the fluidynamics and the mixing time, the opening of the slag layer was simulated by means of a numerical and physical modeling and results are shown in Fig. 10. In this figure, it is observed that the opening shape of the slag layer is very similar by both techniques. However, the aperture is slightly smaller for the physical simulation in Fig. 10(a) the percentage of the exposed area is 26.5% while for the case of the numerical simulation in Fig. 10(b) the percentage of exposed area is 28.6%. In general, a good concordance is observed between the physical model and the multiphasic numerical model employed.

Fig. 10.

Slag eye opening obtained from a) Physical model and b) Numerical simulation. (Online version in color.)

4. Conclusions

A multiphasic numerical simulation of a ladle furnace was developed to study the influence of tracer location on the mixing time, results were validated by a scaled physical model and the main conclusions can be drawn as follows:

(1) The numerical simulation was validated in a satisfactory way by the techniques of dispersion of colorant, qualitative and quantitative measurement of mixing times and also by computing slag eye opening through a physical scale model.

(2) The fluid dynamic structure is made up of a single dominant recirculation, owing to the off-centered bottom injection of argon gas, however, there is a small recirculation close the top slag layer between the gas plume and the nearest ladle wall that in a three-dimensional viewpoint a toroid would be formed by streamlines.

(3) The location of tracer addition has a considerable influence on the mixing time measurements, this due to combined effects derived of fluid dynamics structure, such as to the turbulent viscosity, virtual wall-type effects between encountered flows and low-movement zones that as a whole affect the transport of the tracer species. This brings as a consequence some areas with different concentrations within the domain, even when the global mixing criterion is reached.

Acknowledgments

The authors want to acknowledge to the TecNM-ITM, CATEDRAS-CONACyT, CIDESI, CONACyT and SNI for the permanent support to the academic groups of Modeling of Metallurgical Processes and Thermofluids.

Nomenclature

Symbol Description

C₀, C_∞, C_i: Initial, final and time t concentrations.

C_1ε, C_2ε, C_3ε: Turbulence model constants.

${\overline{C}}_t$: Average mixing degree at time t, Dimensionless.

$C_i^{\prime }$: Average dimensionless concentration, Dimensionless.

$C_j^t$: Mixing degree of sensor j at time t, Dimensionless.

d_m: Physical model diameter, m

d_p: Full scale prototype diameter, m

$\overrightarrow{g}$: Gravitational acceleration, m·s⁻²

G_b: Generation of turbulence kinetic energy due to buoyancy.

G_k: Generation of turbulence kinetic energy due to mean velocity gradients.

H_m: Physical model bath height, m

H_p: Full scale prototype bath height, m

K: Turbulent kinetic energy, J·kg⁻¹

p: Pressure, Pa

Q_m: Physical model gas flow, Nm³·min⁻¹

Q_p: Full scale prototype flow, Nm³·min⁻¹

${\overrightarrow{J}}_i$: Diffusion flux of species i, kg m⁻² s

$\overrightarrow{v}$: Velocity, m·s⁻¹

$Y_i^q$: Mass fraction of species i in phase q.

Greek symbols

α: Volume fraction, Dimensionless.

ε: Dissipation rate of turbulent kinetic energy, m²·s⁻³

μ: Molecular viscosity, Pa:s

μ_t: Turbulent viscosity, Pa·s

ρ: Density, kg·m⁻³

ρ_a: Air density, kg·m⁻³

ρ_g: Argon density, kg·m⁻³

ρ_s: Steel density, kg·m⁻³

ρ_w: Water density, kg·m⁻³

σ: Surface tension, N·m⁻¹

σ_k: Turbulent Prandtl number for k, Dimensionless.

σ_ε: Turbulent Prandtl number for ε, Dimensionless.

τ_mix: Mixing time, s.

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