Analogous Water Modelling of an Asymmetric Multiple Strand Tundish Using Turbulence Inhibitors and Bubble Diffusers

Abstract

The main function of the tundish is to constantly provide molten steel to each strand, for which it is important to determine the uniformity of residence times, particularly in multiple-strand tundishes. In this study, the residence time distribution, minimum and maximum concentrations, and active and dead volume fractions were determined in a 5-strands asymmetric tundish equipped with turbulence inhibitors and argon curtains. Physical modelling was used as a tool, for which a 1:3 reduced-scale analogous water model of the prototype tundish was designed and constructed. For the experimental analysis four cases were studied, the stimulus-response technique was used, in which an inert tracer (HCl) was injected as a pulse at the ladle shroud, the change in the water conductivity was measured at each of the strands, the minimum residence times and the times at which the concentration was maximum were measured and recorded, and the residence time distribution (RTD) curves were plotted; a mixed model was used to characterize the active flow volume fractions (piston and mixed) and dead volume fraction. From the analysis and discussion of the results, no uniformity was observed in any of the four cases studied for the maximum concentration time. The hexagonal turbulence inhibitor alone achieved a result very similar to that of the arrangements with argon diffuser in their characteristics and flow distribution. The use of argon diffusers is recommended to improve the cleanliness of steel.

1. Introduction

Physical modelling is a powerful tool for analyzing and designing metallurgical reactors. Its foundation lies in the similitude and analogy between the model and prototype, establishing and comparing the similarity relationships between the two flow patterns. This technique allows for the visualization of physical phenomena occurring simultaneously inside reactors, enabling the control or potential eradication of undesirable effects.

Continuous casting tundish plays a crucial role in the molten steel solidification process. Its primary functions include providing a constant supply of molten steel; supplying molds with uniform flow, temperature, and composition; reducing the inclusion count, and preventing steel contamination. By utilizing physical modelling techniques, researchers and engineers can better understand and optimize the tundish performance, ultimately improving the quality and efficiency of the continuous casting machine.

To achieve an optimal steel distribution, tundishes are typically equipped with various flow-control devices. These may include dams, weirs, deflectors, impact pads, turbulence inhibitors, and argon diffusers, as described in Table 1. Prior to conducting test campaigns to determine the internal arrangement, physical modelling must be employed. This modelling is essential to analyze and evaluate different configurations of flow control devices and their effects on steel flow patterns.^{1,2,3,4,5,6,7,8)}

Table 1. Flow Control Devices-Function and Functionality.

Flow control device (FCD)	Function	Purpose/Functionality
Dams/Weirs Baffles	Redirect steel flow	Redirect steel flow towards the free surface and/or the strands.
Turbulence inhibitors Kinetic energy dissipative ladle shrouds	Reduce turbulence at the entry zone and control the flow behaviour in the rest of tundish.	Reduce turbulence in the steel bulk and improve the distribution to the strands.
Argon diffusors	Redirect steel flow and improve steel cleanliness.	Redirect steel flow towards the free Surface and remove no metallic inclusions.
Sliding gates Stopper rods	Regulate the volumetric flow at the strands.	Maintain the metallic bath height constant in the tundish and mould.
False floor	Increase the metallic bath height over the strands.	Avoid the vortex formation and slag dragging.

Residence time distribution (RTD) measurements are crucial for determining and quantifying the operational parameters that enable the evaluation of tundish performance. RTD curve characteristics are typically determined using stimulus-response techniques, which involve the addition of an inert or colorant tracer to observe the fluid trajectory as it flows through the tundish model. By examining the minimum residence time and peak concentration time at each strand, process engineers can assess the uniformity of fluid distribution within the reactor. This information is valuable for optimizing tundish design and improving the overall process efficiency in continuous casting operations.^{9,10,11,12,13,14,15)}

Considering only the stability period without reaction and without density changes, the tundish behavior can be determined using the RTD curves to characterize the active flow volume fractions (piston and mixed) and dead volume fraction using a mixed model. Once the volume fractions are quantified, the tundish performance can be inferred. At present, there is no single mixed model for a multi-strand tundish; therefore, the one that most closely approximates the operating conditions of the real process must be chosen.^{16,17,18,19,20,21,22,23,24,25)}

Among the different flow control devices, the importance of using turbulence inhibitors is emphasized to optimize the flow characteristics of the molten steel inside the tundish. This device reduces the kinetic energy in the inlet zone and modifies the flow pattern so that it is distributed more evenly to each of the tundish outlets.^26,27,28) For the same purpose, kinetic energy-dissipative ladle shrouds have recently been proposed, showing that they can potentially control the entry flow.^{29,30,31,32,33)}

In addition, argon diffusers have been developed with the primary aim of removing inclusions and transporting them from the metal bath to the free surface so that they can be absorbed by the slag. However, for the solid particles to adhere to the argon bubble, the bubble size must be very small, less than 5 microns.^34,35,36) Therefore, its main effect is on the flow pattern, promoting a better flow distribution along the tundish, modifying it to move close to the metal-slag interface, and thus redirecting the inclusions towards the slag.^{34,35,36,37,38,39,40)}

Despite all the previous studies, it is important to determine the uniformity of the minimum arrival times at each strand and the time at which the concentration of the tracer is maximum, which can be measured directly in the tundish model, and its performance is evaluated using these parameters. Therefore, the purpose of the present study was to quantitatively determine the distribution of the minimum arrival times, peak concentration time, and active and dead volume fractions in a 5-strands asymmetric delta tundish equipped with turbulence inhibitors and argon diffusers.

2. Experimental Setup and Methodology

Analogous water model of the prototype tundish was designed and constructed, together with the flow control devices at a 1:3 reduced-scale. The model design was base in the Froude similarity criteria establishing the relationship among the inertial to gravitational forces expresses in the Froud number; it has been demonstrated that can be satisfy at any scale to study the behavior of two analogous flow patterns, the molten steel flow at 1600°C and tap water at room temperature.³⁾ The model was equipped with turbulence inhibitors and stone air diffusers, in accordance with the scaled geometric dimensions of the prototype, as shown in Fig. 1.

Fig. 1. a) Schematic diagram and b) Photograph of the asymmetric model including hexagonal and square turbulence inhibitors, argon diffuser and stopper rods (scale 1:3). (Online version in color.)

To analyze the main function of the tundish, it is important to determine the uniformity of the fluid arrival time for each strand, and the time with the tracer concentration is the maximum. An inert tracer was injected as a pulse in the model entry shroud, and the change in the water conductivity was measured at each individual model outlet strand nozzle. The residence times were plotted in a “C” type curve to quantify the active and dead volume fractions, which are important parameters for evaluating tundish performance.

Four cases were studied: a) Case 1 was equipped with a simple square turbulence inhibitor with a flat bottom, false floor, stopper rods, and closed air diffusers; b) Case 2 was equipped with a hexagonal turbulence inhibitor with a corrugated bottom, false floor, stopper rods, and closed air diffusers; c) Case 3 was equipped with a simple square turbulence inhibitor with a flat bottom, false floor, stopper rods, and open air diffusers; and d) Case 4 was equipped with a hexagonal turbulence inhibitor with a corrugated bottom, false floor, stopper rods, and open air diffusers.

For the experimental analysis of the different cases, the stimulus-response technique was used, in which a dissolved red vegetal coloring and an inert tracer (HCl) were injected as a pulse at the ladle shroud, the change in color of the water was filmed for flow pattern visualization, and the water conductivity change was measured at each strand. The minimum residence times and times at which the concentration was maximum were measured and recorded, and the residence time distribution (RTD) curves were plotted. Table 2 the modelling parameters for a casting speed 2.1 m/min in the prototype.

Table 2. Water modelling parameters.

Model parameters	Quantity
Total volume (L)	123
Entry ladle shroud transversal area (m2)	0.00053
Entry flow (L/s)	0.2873
Entry nozzle transversal area (m2)	0.00011
Outlet flow per strand (L/s)	0.05747
Time interval between measurements (s)	1
Mean residence time (s)	428
Amount of Inert tracer HCl (mL)	20

Mixed models are very useful for quantifying the fluid element dispersion or mixing, which is obtained from the response of the tracer addition, and the concentration measurements are plotted in an RTD curve. By applying an appropriate mixing model, the proportion of the non-ideal volume fraction can be determined, for the present work the combined mixed model was chosen.¹⁶⁾

2.1. The Combined Mixed Model Applied for Multistrand Tundishes

Fluid flow characterization in multistrand tundishes can be performed by calculating the volume fractions of ideal flows present inside the reactor as the fluid passes through, for which concentration measurements are needed from each outlet, and then a mixed model must be applied.

Assuming that during the stable period, the outlet volumetric flow rate (q_N) is equal in each one of the outlets; therefore, the entry volumetric flow rate (Q_T) is divided by the number of the strands (N). As expressed in Eq. (1).

  

$$q_N=Q_T/N$$(1)

The total mass of the injected tracer is:

  

$$M_T={\displaystyle \sum }_1^N{m}_i$$(2)

The amount of injected tracer leaving for any of the outlets (N) in a given period Δt, as Δt → 0 and Δm_i → 0, can be expressed by Eq. (3):

  

$$d{m}_i=c_i(t)q_{N}dt$$(3)

where c_i(t) is the tracer concentration at time t. Similarly, the fraction of tracer leaving for the outlets (i) during time period dt is given by

  

$$\frac{d{m}_i}{M_T}=\frac{c_i(t)q_{N}dt}{M_T}$$(4)

By integrating Eq. (4) from the initial limit t = 0 to t = ∞ and applying the summation from j = 1 to N the following expression is obtained:

  

$${\displaystyle \sum }_{j=1}^N\left(\frac{m_i}{M_T}\right)={\displaystyle \sum }_{j=1}^N\left(\frac{{{\displaystyle \int }}_0^{\infty }{c}_i(t)q_{N}dt}{M_T}\right)$$(5)

Substituting Eqs. (1) and (2) and rearranging, the next equation represents the general function of the residence time distribution for multiple strands, which is basically the mean individual RTD function and can be expressed as follows:

  

$${\displaystyle \sum }_{j=1}^N\left(\frac{{{\displaystyle \int }}_0^{\infty }{c}_i(t)Q_{T}dt}{M_{T}N}\right)=1$$(6)

Therefore, for a multiple-strand tundish, an RTD curve can be obtained from the data of the individual strands. Because Q_T and M_T are constant in the reactor, C_i(t) is directly proportional to concentration (c). The dimensionless concentration for the pulse input of tracer is C = c/(q/V); where q is quantity of tracer, and q/V is the average concentration of the tracer when it dissolves in the fluid volume, V, in the vessel; thus, the C-curve starts at a dimensionless concentration.³⁾ The following equations are obtained:

  

$$\frac{1}{N}{\displaystyle \sum }_{j=1}^N\left({\int }_0^{\infty }{C}_i(t)dt\right)=1$$(7)

where the area under the individual overall curve is equal to unity.

  

$${\int }_0^{\infty }Cdt=1$$(8)

The combined mixed model of multiple strands considers that the total molten steel flow rate entering the tundish is divided into two regions as a function of the flow characteristic: an active region and a passive region.¹⁶⁾ This, in turn, gives rise to the fractions of the total volume of the reactor that characterize the steel flow of the tundish, as shown in Fig. 2.

Fig. 2. Diagram of the combined mixing model for multiple strands.

For this model, the characteristic dead or inactive volume fraction and active region were obtained from the overall RTD curve as follows:

  

$$\frac{V_D}{V}=1- \frac{Q_A}{Q}\overline{{\theta }_C}$$(9)

Where:

  

$$\overline{{\theta }_C}=\frac{{{\displaystyle \sum }}_{\theta =0}^2{C}_i{\theta }_i}{{{\displaystyle \sum }}_{\theta =0}^2{C}_i}$$(10)

And θ is the dimensionless time, which is an indication of the fractional residence, obtained by diving any time by the average residence time ($\theta =t/\overline{t}$).³⁾

The active region fraction is.

  

$$\frac{Q_A}{Q}={\displaystyle \sum }_{\theta =0}^2{C}_i\mathrm{\Delta }\theta$$(11)

The combined mixing model considers that after a dimensionless time greater than 2 on the C curve, the fluid movement is very slow and can be assumed to be dead flow. Additionally, the model predicts the relationship between the piston volume fraction and the mixed volume fraction inside the tundish in a simple manner.

3. Results and Discussion

3.1. Analysis of the Minimum Residence Time (θ_min) and Concentration Peak Time Measured (θ_cmax) per Strand

The analysis of the results starts by showing the images of flow behavior taken from the video recordings at different significant moments as the red ink tracer moves inside the water model, which are shown consecutively along the section. Then, the minimum residence time (θ_min) and concentration peak time measured (θ_cmax) are shown graphically for each individual strand: first, for cases 1 and 2, where the air diffusors are off, and second, for the others two cases 3 and 4, where the air diffusors are on.

For Case 1, in Figs. 3(a)–3(c), the addition of the red ink tracer impacts the bottom of the turbulence inhibitor almost instantaneously and redirects the flow towards the free surface, as shown in Fig. 3(a). Once it reached the free surface, the red ink advanced towards the lateral walls above the false floor and close to the back wall. It also moves to the front where the center strands are located, as shown in Fig. 3(b). As shown in Fig. 3(c), the flow continued to move toward strand 5. For case 2, Figs. 3(d)–3(f) show the red ink tracer addition which impacts the corrugated bottom of the turbulence inhibitor instantaneously and redirect the flow towards the free surface and also the flow coming out from the lateral orifices moves towards to the back wall of the model as show in Fig. 3(d), once it leaves the turbulence inhibitor is observed that the red ink advances towards the laterals walls above the false floor and close to the back wall and for the free surface towards the frontal and lateral wall as show to Fig. 3(e), the flow continue to move to strand 5 as shown in Fig. 3(f).

Fig. 3. Visualization of the ink tracer for Case 1 (a–c) and Case 2 (d–f) in a frontal view and upper view at different times. (Online version in color.)

The quantification of the times for the first tracer appearance (θ_min) and when the concentration was maximum (θ_cmax) is shown in Figs. 4(a)–4(b). In Case 1, the first recorded time was for central strands 2 and 3. Then, the water flow moved towards the lateral wall, the tracer arrived at strands 1 and 4, and took three times the first recording (θ_min) to reach strand 5, as shown in Fig. 4(a). In contrast, for Case 2, the tracer arrives first at strands 1 and 4; then, the water flow moves towards the frontal wall and the tracer arrives at central strands 2 and 3, and takes more than three times the first recording to reach strand 5, as shown in Fig. 4(b). For the maximum concentration time, the results in both cases show the same tendency; that is, the times recorded are first for central strands 2 and 3 in case 1, and for case 2, are strands 1 and 4, as shown in Figs. 4(a)–4(b). In Fig. 4(b) show that the maximum concentration times are more uniform compare with the Fig. 4(a), this means a better flow distribution to the 5 strands, mainly due to the effect of the lateral orifices of the hexagonal turbulence inhibitor; this also reduces the minimum time to reach strand five with respect to the square turbulence inhibitor.

Fig. 4. Dimensionless graph of the measured minimum residence time (θ_min) and the maximum tracer concentration time (θ_cmax) per strand Case 1 (a) and Case 2 (b). (Online version in color.)

The next analysis considers when the air diffusers in the model operate. Figures 5(a)–5(c) shows the red ink tracer injection for Case 3, where can be observed that it impacts the flat bottom almost instantaneously and redirects the flow to the free surface more slowly due to the effect of the curtains as shown in Fig. 5(a). Once it reaches the free surface, it is observed that the red ink advances towards the lateral walls above the false floor and close to the back wall; then, the ink tracer rises at the air curtains, and another part of the flow goes to the front wall where strands 2 and 3 are located, as shown in Fig. 5(b). Because of the effect of the air curtains, the flow continues to advance towards the lateral walls at a higher velocity, so it takes less time to reach the strands after the curtains, strands 1 and 4; the flow advances close to the free surface and then descends. Once strand 4 was reached, the ink tracer continued slowly to strand 5 and moved along the bottom of the model, as shown in Fig. 5(c). For case 4, Figs. 5(d)–5(f) shows the red ink tracer addition, which instantaneously impacts the corrugated bottom of the turbulence inhibitor and redirects the flow towards the free surface. The flow coming out at a high velocity from the lateral orifices moves towards the air curtains, as shown in Fig. 5(d), where the ink tracer rises at the air curtains and another part of the flow goes to the front wall where strands 2 and 3 are located. Because of the effect of the air curtains, Fig. 5(e) shows a flow behavior very similar to that observed in Case 3, where the ink tracer reaches strands 1 and 4 at a lower measurement time. The ink tracer continued slowly to strand 5 and moved along the bottom of the model, as shown in Fig. 5(f).

Fig. 5. Visualization of the ink tracer for Case 3 (a–c) and Case 4 (d–f) in a frontal view and upper view at different times. (Online version in color.)

The quantification of the times for the appearance of the first ink tracer (θ_min) and when the concentration is maximum (θ_cmax) is shown in Figs. 6(a)–6(b). The effect of the air curtains is that, as the flow advanced at a higher velocity, the time to reach the strands after the curtains was reduced, and an even first appearance time (θ_min) was observed for all strands. In contrast, the maximum concentration (θ_cmax) increased considerably for strands after the curtains, which can be attributed to the agitation of the bubbles in this zone.

Fig. 6. Dimensionless graph of the measured minimum residence time (θ_min) and the maximum tracer concentration time (θ_cmax) per strand Case 3 (a) and Case 4 (b). (Online version in color.)

3.2. Analysis of the Residences Time Distribution (RTD) per Strand

The RTD or “C” curves are obtained from the chemical tracer concentration measured at the strand outlets and the time that a fluid element takes from the entrance to each individual strand. Figure 7(a) shows the curves for Case 1, where each one has a different height and shape, and the first-time appearance is also different. This implies that the flow distribution is not uniform, which is attributed to the asymmetric geometry of the tundish. Consequently, the RTD curves are not uniform; comparing each strand, it can be seen that the central strands present less dispersion because they are closest to the entrance point. It is important to observe that the curve for strand 2 presents a slight double peak that occurs before reaching its maximum concentration owing to the presence of a counterflow coming from the side wall. The RTD curves for strands 1 and 4 are similar because the flow falls down before reaching these strands, having a higher dispersion than strand 3 with the lowest dispersion, being the furthest strand 5 the one that shows a greater dispersion because it has more time and bulk to disperse. Figure 7(b) shows the RTD curves for Case 2, which shows that there continues to be an uneven distribution of residence times; however, the flow patterns are different because of the complex geometry of the hexagonal turbulence inhibitor, as shown previously. Strand 1 shows the lowest dispersion owing to the stream exiting with high velocity from the orifices in the sidewalls of the turbulence inhibitor, and one of the streams reaches the closest sidewall of the tundish and reaches its maximum concentration time very quickly. This does not occur for strand 4 because there is no resistance when the streams move over the false floor and when they fall to the strand meets with a counter-flow coming from the furthest side wall. This phenomenon does not affect strand 5, which also shows a higher dispersion because it has more time and bulk to disperse; strands 2 and 3 have better dispersion because they are closest to the entry point.

Fig. 7. RTD curves per strand a) square turbulence inhibitor (CASE 1) and b) hexagonal turbulence inhibitor (CASE 2). (Online version in color.)

An analysis of the RTD curves for Cases 3 and 4 when the air diffusers are open is presented. Figure 8(a) shows the curves for Case 3, and it is observed that they have better uniformity in the times at which the first appearance is recorded, as shown previously. For central strands 2 and 3, the curves show similar heights and lower dispersion because the flow falls down and moves towards the front wall; for the strands behind curtains 1 and 4, a similar dispersion occurs because the flow when it hits the curtains is redirected to the free surface and the side walls; and for strand 5, which also shows a higher dispersion because it has more time and bulk to disperse. Figure 8(b) shows the curves, and it is observed that central strands 2 and 3 have similar heights and lower dispersion because the flow falls down and moves towards the front wall. Strand 1 presents a dispersion similar to that of the central strands because the flow is redirected to the free surface and side wall, and then falls down to this strand. Strands 4 and 5 have a higher dispersion because the flow is redirected by the curtain to the further side wall and encounters a counter-flow coming from that wall, spreading the flow further.

Fig. 8. RTD curves per strand a) square turbulence inhibitor with argon diffusers (CASE 3) and b) hexagonal turbulence inhibitor with argon diffusers (CASE 4). (Online version in color.)

For the overall analysis and quantification of the volume fraction parameters of the tundish total volume for the studied cases, the overall RTD curves were obtained from the measured tracer concentration data at each strand for different times, as shown in Figs. 9(a)–9(b), in which it can be seen that the height and shape of the curves are similar, which means that the overall residence time and bulk flow mixing are practically the same. Case 1 had a higher dispersion than the other cases owing to the counter-flow effect discussed above, which resulted in a slight double peak at strand 2. For case 2, there is a lower degree of dispersion owing to the effect of the hexagonal turbulence inhibitor, which distributes the flow more evenly in the model and eliminates the counter-flow; thus, a slight step is no longer present in the overall RTD curve, as shown in Fig. 9(a). For Cases 3 and 4, it is observed that both cases have a similar dispersion and very similar shapes owing to the effect of the curtains that distribute the flow more uniformly along the model, as shown in Fig. 9(b).

Fig. 9. Overall RTD curves a) with turbulence inhibitors and b) with turbulence inhibitors with argon diffusers. (Online version in color.)

The quantified parameters of the flow volume fractions of the overall RTD curves using the combined mixed model for a multi-strand nonsymmetric tundish determined the characteristic performance of the reactor using different arrangements and geometries of turbulence inhibitors and argon curtains. Table 3 shows the calculated values of the overall active volume (mixed plus piston) and dead flow volume, in addition to the dimensionless averaging time that indicates how the bulk flow behaves in the model, where it is observed that Cases 3 and 4 present similar values of dimensionless averaging time, as well as of dead and active flow volume fractions, which indicates that both have a very similar overall fluid dynamic behavior due to the effect of the argon curtains, which contribute to improve flow distribution. Owing to the effect of the orifices in the side walls of the hexagonal turbulence inhibitor, Case 2 also presents values similar to those of Cases 3 and 4, showing by itself a good flow distribution in the model. This is in contrast to Case 1, where it was observed that the results were different in each of the volume fractions and in the dimensionless average time owing to the effect of the square turbulence inhibitor, which takes longer to distribute the flow and does not distribute it uniformly in the model. From the results obtained and shown in Figs. 9(a)–9(b) and Table 3, the hexagonal turbulence inhibitor achieved by itself a very similar result to that of the argon diffuser-operated arrangements, both in its characteristics and flow distribution.

Table 3. Overall volume fractions of the tundish bulk flow.

Studied Cases Internal Arrangements	Passive flow volume fraction	Active flow volume fraction	Piston flow volume fraction	Mixed flow volume fraction	Mean residence time
Square turbulence inhibitor and close argon diffusers	0.2	0.80	0.27	0.53	0.88
hexagonal turbulence inhibitor and close argon diffusers	0.25	0.75	0.17	0.58	0.83
Square turbulence inhibitor and open-argon diffusers	0.28	0.72	0.16	0.56	0.81
hexagonal turbulence inhibitor and open-argon diffusers	0.28	0.72	0.15	0.57	0.81

4. Conclusions

The following conclusions can be drawn from the analysis and discussion of the results:

(a) By contrasting the use of turbulence inhibitors, the hexagonal geometry achieves a better uniformity in the flow distribution, in the arrival times at each strand and the time where the concentration is maximum reducing both at the more distant strand this is mainly attributed to the sidewall orifices.

(b) The use of argon diffusers promotes more uniformity and a reduction in arrival times at each strand, which is shorter when the hexagonal turbulence inhibitor is used. At the maximum concentration, no uniformity was observed in any of the studied cases, and this time was longer for the hexagonal turbulence inhibitor than for the square geometry.

(c) The residence time distribution curves do not show uniformity when turbulence inhibitors are used without any other device. However, the flow patterns were different when a hexagonal turbulence inhibitor was used. When turbulence inhibitors were combined with argon diffusers, the residence time distribution curves exhibited better uniformity.

(d) The overall passive volume fraction of the total tundish volume shows an increase when the turbulence inhibitors are combined with argon diffusers, this is due to the fact that at the strands after the location of the argon diffusers the average residence time increases.

(e) The hexagonal turbulence inhibitor itself shows results similar those two of the arrays operating with argon diffusers. The use of the hexagonal turbulence inhibitor with or without argon diffusers provided a good flow distribution to the five strands.

Further Remarks

Defining the two main technological options of the proposed arrangements, according to the operational needs of the shop, the use of a hexagonal turbulence inhibitor with or without argon diffusers is recommended, as it provides efficient flow distribution to all five strands. The use of argon diffusers is recommended only if a greater inclusion removal is required.

Statement for Conflict of Interest

Authors declare that is not any conflicts of interest related to this research.

Acknowledgments

The authors would like to thank CONAHCYT for the financial support provided for graduate studies. To the institutes TecNM-ITM for funding the research project and SNII for the support provided to the research group.

Nomenclature

Variables Concepts Units

C_i Measured tracer concentration (M)

C_j Measured tracer concentration in the strand (M)

C Output concentration (M)

C_θ Dimensionless output concentration (-)

M Amount total tracer (mL)

m_i Amount of tracer per strand (Ω)

N Number of strands (-)

q_N Volumetric flow per strands (m³/s)

Q_T Entry volumetric flow rate (m³/s)

Q_A Active flow (m³/s)

Q_D Dead flow (m³/s)

t_i Measure time (s)

V Tundish fluid volume m³

$\frac{V_D}{V}$ Dead flow volume fraction (-)

$\frac{V_m}{V}$ Mixed flow volume fraction (-)

$\frac{V_p}{V}$ Piston flow volume fraction (-)

$\overline{t}$ Average residence time (s)

θ Dimensionless time (-)

θ_cmáx Dimensionless maximum concentration time (-)

θ_min Dimensionless minimum time (-)

$\overline{{\theta }_C}$ Dimensionless mean time (-)

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