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* Corresponding author: E-mail: jose.rb@morelia.tecnm.mx

© 2021 The Iron and Steel Institute of Japan. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

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https://doi.org/10.2355/isijinternational.ISIJINT-2021-094

1. Introduction

The secondary refining process is made up of several operations which are deoxidation, desulfurization, decar-burization, as well as removal of dissolved gases such as hydrogen and nitrogen. Additionally, it is possible to con-trol 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 move-ment 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 diffu-sions.2) The degree of mixing is determined by measuring the mixing time (tmix) 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

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

Mario HERRERA-ORTEGA,1) José Ángel RAMOS-BANDERAS,1)* Constantin Alberto HERNÁNDEZ-BOCANEGRA1,2) and José Julián MONTES-RODRÍGUEZ3)

1) TecNM Campus Morelia, Av. Tecnológico 1500, Morelia Michoacán, 58120 México.2) Cátedras-CONACYT, Av. Insurgentes Sur 1528, CDMX, 03940 México.3) CIDESI, Av. Playa Pie de la Cuesta No. 702, Desarrollo San Pablo, Santiago de Querétaro, 76125 México.

(Received on March 20, 2021; accepted on May 6, 2021)

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 pro-totype 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 measure-ment 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.

KEY WORDS: numerical simulation; physical modeling; mixing time; tracer addition releasing.

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,4–14) number and location of injections,15–18) as well as properties19) and thickness of slag.20) In most of the previously mentioned papers, the loca-tion 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 colleagues3–21) 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 addition23–25) 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 addi-tion 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

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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 mix-ing time required to reach 95% of homogeneity. Finally, results were validated through its analogue physical model-ing technique, by using KCl saturated solutions as a tracer and also colorant dispersion in water.

2. Methodology

2.1. Physical Model DesignFor 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.

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

Q

Q

d

d

H

Hm

p

g

a

w

s

m

p

m

p

�

��

�

�� �

�

��

�

���

��

�

���

��

�

���

��

�

��

2 4��

��

. ............. (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; ρg, ρa, ρw and ρs are the densities of argon, air, water, and steel respectively. From conductivity measurements, a dimensionless concen-tration is calculated at time t determined by Eq. (2):

� ����

CC C

C Ci

i 0

0

............................... (2)

Where C0 is the initial concentration; Ci is the concentra-tion at time t and C∞ is the final concentration determined by the sensor.

2.2. Colorant Dispersion TechniqueTo 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.

Fig. 1. Geometrical dimensions, a) Physical scale model and b) Experimental setup.

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

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2.3. Mathematical Model2.3.1. Domain Dimensions, Boundary Conditions and

ConsiderationsThe simulation was carried out considering the geometry

of a 150-ton ladle furnace, the phases and boundary condi-tions are shown in Fig. 2(a). The computational domain mesh consists of 809 835 elements. The numerical models are solved with the help of the commercial code ANSYS FLUENT® using the PISO algorithm28) for problems in tran-sient 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.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.

2.3.2. Continuity EquationThe VOF model solves two or more immiscible fluids by

tracking the interfaces between the fluids through the conti-nuity equation for the volume fraction of one of the phases represented by Eq. (3).

��� � �� �� � �t

vq q q q q� � � � 0 ................... (3)

Fig. 2. Computational domain of the conical ladle furnace a) boundary conditions and b) position of the sensors.

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 7 020 0.006 –

Slag 3 500 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).

Where, for phase q: ρq is the density, kg·m −3; ∝q is the fraction of volume within the cell occupied by the phase; and vq is velocity, m·s −1. The volume fraction is defined

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by Eq. (4).

q

n

q

�� �

1

1� ................................... (4)

2.3.3. Equation of Conservation of MomentumA 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.

��� � �� �� � � �� �� � � ��� ��

��� �t

v vv p v v gT� � � � ... (5)

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

x xq q� �� ................................. (6)

2.3.4. Model of Turbulence k-εThe 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).

��� � �� �� � � � � ��

��

����

�

��

�

� � � �

tk kv k G Gt

kk b� �

�� ... (7)

��� � �� �� � � � � ��

��

����

�

��

�

�

� �� � �

tv

CkG C G C

t

k b

� � �

1 3 2 � 2

k

... (8)

Where Gk, is the generation of turbulent kinetic energy due to the average of the velocity gradients; Gb is the gen-eration of turbulent kinetic energy due to buoyancy forces; μt is the viscosity turbulent; C1ε, C2ε y C3ε, and are empirical constants whose values are 1.44, 1.92 and 1 respectively; σk and σε are the turbulent Prandtl numbers for k and ε whose values are 1.0 and 1.3 respectively.

2.3.5. Species Transport EquationTo simulate the tracer dispersion, the species transport

model was employed which solves Eq. (9).

��� � �� �� � � �� �t

Y v Y Jqi

qi

iq

qi

qi

qiiq

i� � � �

........... (9)

Where αqi , ρqi and vqi , represent the fraction of volume,

density and velocity of phase q that contains species i; Yiq is the mass fraction of species i contained in phase q; Ji 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).

Ck

Ct jt

j

k�

��1 1 ........................... (10)

Where Cjt is the concentration measured by sensor j at

time t; k, is the number of sensors considered, in the pres-ent 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 StructureIn 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.

3.2. Validation of ResultsThe flow dynamics technique by means of colorant dis-

persion 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 stream-lines in plane A of the numerical model and the physical simulation is shown in Fig. 5.

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

Fig. 4. Steel streamlines for steel-slag system colored by velocity magnitude, a) Plane A and b) Plane B. (Online version in color.)

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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 redi-rected 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 recircu-lation 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.

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 axi-symmetric 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 domi-nant 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 sen-sors 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 rela-tive 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 TimeFigure 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 homoge-nizes 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,

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.)

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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)

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 resolu-tion of cases 3 and 4.

3.4. Mixing Dependence on Turbulent ViscosityIn previous works the turbulent kinetic energy,33) the

effective viscosity34) and the angular momentum35) associ-ated to tracer releasing location in the mixing phenomenon

Fig. 6. Mixing time obtained for case 1 a) experimental and b) numerical simulation; Case 2 c) experimental and d) numerical simulation.

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.)

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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 tur-bulent 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 ana-lyzed. 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 influ-ence 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.

3.5. Slag Eye OpeningAs 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. How-ever, the aperture is slightly smaller for the physical simu-lation 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.

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 sat-isfactory 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 recircula-tion 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 consider-able 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

Fig. 8. Mixing time obtained through numerical and physical simulation for all cases.

Fig. 9. Turbulent viscosity contours, a) Plane A and b) Plane B. (Online version in color.)

Fig. 10. Slag eye opening obtained from a) Physical model and b) Numerical simulation. (Online version in color.)

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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.

AcknowledgmentsThe 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.

NomenclatureSymbol Description C0, C∞, Ci: Initial, final and time t concentrations. C1ε, C2ε, C3ε: Turbulence model constants. Ct : Average mixing degree at time t, Dimension-

less. ′Ci : Average dimensionless concentration, Dimen-

sionless. Cj

t : Mixing degree of sensor j at time t, Dimen-sionless.

dm: Physical model diameter, m dp: Full scale prototype diameter, m

g : Gravitational acceleration, m·s −2

Gb: Generation of turbulence kinetic energy due to buoyancy.

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

Hm: Physical model bath height, m Hp: Full scale prototype bath height, m K: Turbulent kinetic energy, J·kg −1

p: Pressure, Pa Qm: Physical model gas flow, Nm3·min −1

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

Ji : Diffusion flux of species i, kg m −2 s

v : Velocity, m·s −1

Yiq: 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 μt: Turbulent viscosity, Pa·s ρ: Density, kg·m −3

ρa: Air density, kg·m −3

ρg: Argon density, kg·m −3

ρs: Steel density, kg·m −3

ρw: Water density, kg·m −3

σ : Surface tension, N·m −1

σk: Turbulent Prandtl number for k, Dimension-less.

σε: Turbulent Prandtl number for ε, Dimension-less.

τmix: Mixing time, s.

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