Flow Mechanism of Molten Steel in a Single-Strand Slab Caster Tundish Based on the Residence Time Distribution Curve and Data

Abstract

Control melt flow in tundish is very important for clean steel production. To explore the fluid flow mechanism, the RTD curves and its data, velocity vector fields and streamlines of the molten steel, and the relationship between the RTD curves and flow pattern in a single-strand slab caster tundish with a capacity of 30 tons have been investigated by both hydrodynamic and mathematical simulations. The RTD data and flow field of the original tundish have been studied in a 1:3 reduced scale hydrodynamic model. Meanwhile, the streamlines, velocity vector fields and the RTD data of the ratio of width to length (W/L) in tundish are mathematically simulated. In order to descript the flow pattern better, a new method is proposed to calculate the data of RTD curves with double peaks. The results showed that the RTD curves changed from double peaks to single peak with the increasing of the W/L in tundish. Both hydrodynamic and mathematical simulations results suggest that the W/L in tundish is the most important factor to change the flow pattern actually, that is, the short-circuiting flow disappeared with the increase of the W/H in tundish gradually. Furthermore, we have elaborated the mechanism the RTD curves change from double peaks to single peak. With increasing W/L, the wide-side walls play an important role to retard the short-circuiting flow on the inlet-outlet plane straight towards the outlet. Meanwhile, the dead region and its volume fraction are also the objects of our attention and exploration.

1. Introduction

In recent years, the continuous casting tundish has evolved into a useful reactor for liquid steel refining. As such, it plays an increasingly important role in the secondary refining process especially in the production process of clean steel¹⁾ and above its traditional role as a buffer or steel distribution vessel. Thus a modern steelmaking tundish is designed to provide maximum opportunity for carrying out various metallurgical operations such as inclusion removal, gas bubbling, alloy trimming of steel, modification of inclusions by calcium treatment, melt reheating, superheat control, thermal and particulate homogenization, collectively referred to tundish metallurgy. A great deal of tundish metallurgy operations have already been summarized in a review presented recently by Mazumdar, Guthrie²⁾ and Kinnor Chattopadhyay, Mihaiela ISAC.³⁾

Since the chemical reactor efficiency of tundish metallurgy operations is related to the nature of fluid flow, consequently the understanding of the fluid flow phenomenon is required for designing the effective flow control and tundish system. Analyzing the various residence time distribution (RTD) data is one of the most important and effective approaches to make contribution to elaborating the molten steel flow phenomena in tundish. It has been generally considered that physical and mathematical modelling is a powerful tool to simulate the molten steel flow in tundish to investigate its flow characteristics and get RTD curves with its corresponding data. Accordingly, the tremendous related studies on the method of hydrodynamic modelling,^4,5,6,7,8,9) which is based on the basis of similarity principles by using ambient temperature water to substitute for the molten steel, combined with mathematical simulation,^{10,11,12,13,14,15,16)} which is based on computational fluid dynamics (CFD) by using various sophisticated software.

Even though a lot of studies on flow behavior in tundish has been carried out using physical modeling as well as mathematical modeling, not much researches deal with RTD data and its corresponding double peak curves measured in many experiments, very little information is available on analyzing the reason why there exist double peaks. Sarbjit Singh and Satish C. Koria¹⁷⁾ have studied the RTD curves with double peaks by water model. They have found that the reason of double peaks on RTD curve is short-circuiting flow. However, both the authors have not studied in detail the relationship between single peak and double peaks on RTD curves on the basis of flow mechanism. They have not investigated the ratio of plug flow volumes versus mixed flow and dead flow volumes in the tundish while there exist a short-circuiting flow, either. Moreover, no literature have been found to describe the flow pattern with a short-circuiting flow. Therefore, it has been intended to study the mechanism of molten steel flow in a single-strand slab caster tundish with short-circuiting flow and without it.

In the present work, fluid flow in a 30 tons single-strand tundish with different ratio of width to length was investigated by using physical modeling and mathematical simulation methods. In the physical modelling experiments, RTD curve of fluid flow in the tundish with original configuration was measured and flow characteristics were determined from the RTD curves. In the mathematical simulation, numerical investigations such as the commercial CFD software FLUENT 14.5 were carried out in order to get the RTD curves and data. Firstly, a physical and numerical simulation of the molten steel flow in the original tundish was carried out. The RTD curves and data of the original tundish was then investigated. Secondly, we obtained the RTD curves and data changed from single peak to double peaks under different conditions by changing the ratio of width to length in tundish (W/L). A new mathematical formulation was proposed to calculate RTD data and get the short-circuiting flow volumes, plug flow volumes, mixed flow volumes and dead flow volumes. After that, the molten steel velocity vector fields and streamlines under isothermal conditions were applied to clarify the different mechanism with changing from single peak to double peaks on RTD curves. We think that this mechanism could have a great influence on a single-strand tundish optimization design.

2. Physical Modelling

2.1. Theoretical Modeling Description

Figure 1 shows a schematic diagram of the present tundish of a single-strand slab caster having one dam, weir and a turbulence inhibitor. Figures 1(a) and 1(b) show front view and solid view of the mainly dimension of the present tundish respectively. Table 1 shows geometry of the original tundish. The molten steel is poured into the tundish through the ladle shroud (inlet of the tundish). The molten steel flow into the tundish creates enormous turbulence around the pouring point. To minimize turbulence around pouring point, the flow control devices such as dam, weir and turbulence inhibitor are used and is placed as shown in Fig. 1(b). The flow control devices are believed to guide the molten steel flow towards the outlets smoothly. The other operating parameters of tundish and the detail of casting conditions were in Table 2.

Fig. 1.

The present single-strand slab caster tundish: (a) Front view about symmetry, (b) Solid view from panoramic tundish.

Table 1. The geometry of the original tundish.

Parameters	Data
Distance from the weir to the ladle shroud (A)	913 mm
Distance between the damr and the weri (B)	640 mm
The height of the dam (C)	310 mm
Distance from the weir to the tundish bottom (D)	250 mm
Depth of the molten steel (E)	1100 mm
Width of the tundish bottom	800 mm

Table 2. The operating parameters of the present tundish.

Parameters	Data
Tundish capacity	30 t
Slab caster section size	1425 mm×220 mm
Casting speed	1.1 m/min
Inlet diameter	70 mm
Outlet diameter	70 mm
Inlet velocity, Vin	1.49 m/s
Inlet temperature	1853 K
Superheat	298 K

2.2. Design of Hydrodynamic Model

It is well-known that similarity of flow field between two models of different scales is decided by Reynolds number Re, Froude number Fr, Weber number We and so on, according to similarity principles.¹⁸⁾ The basic condition of liquid flow similarity between the model and prototype is geometrical similarity and dynamic similarity. Dynamic similarity requires that the Re number and the Fr number in the model should be equivalent to those in the prototype, respectively. However, in the simulated experiment, it is impossible to keep the condition satisfied in reduced scale modeling studies. The computational work of Sahai and Burval¹⁹⁾ and the experimental work of Singh and Koria²⁰⁾ showed that the magnitude of turbulent Reynolds number under turbulent flow range in different tundishes was very similar. Therefore, Froude number between the model tundish and the prototype one was maintained to be equivalent in this work. It indicated that the geometrical and operation parameters in the two systems can be decided by only keeping Fr equal between the model and prototype.

According to the Froude similarity criterion,   

$$F{r}_m=F{r}_p$$(1)

where m means hydrodynamic model, p means the prototype tundish.

The Froude number is defined as   

$$Fr=\frac{u^2}{gL}$$(2)

Then,   

$$\frac{u_m^2}{g{L}_m}=\frac{u_p^2}{g{L}_p}$$(3)

Thus, the main geometrical and operation parameters between model and prototype can be formulated with scale factor λ as follows:   

$$L_m/L_p=\lambda ,\mathrm{\hspace{0.17em} }\hspace{0.17em}{u}_m/u_p={\lambda }^{0.5},\mathrm{\hspace{0.17em} }\hspace{0.17em}{Q}_m/Q_p={\lambda }^{2.5},\mathrm{\hspace{0.17em} }\hspace{0.17em}{t}_m/t_p={\lambda }^{0.5}$$(4)

2.3. Method and Criteria of Hydrodynamic Modelling

In the present work, the ratio of geometrical similarity of model tundish to the prototype was chosen to be 1:3 (λ). The experimental setup of hydrodynamic modelling is schematically illustrated in Fig. 2 which consists of a perspex glass model tundish, water supply, tracer injection system and the instrumentation to record the tracer concentration at the exit of the tundish. It is an important way to qualitatively assess fluid flow in tundish is through the study of residence time distribution (RTD). An RTD curve of the fluid flowing in the tundish can be obtained by the stimulation-response method,¹⁸⁾ in which a stimulus signal is added at the inlet and the output signal, response, is measured at the outlet for a non-ideal reactor as a black box, was applied to investigate flow filed of water in the hydrodynamic model, while the ambient temperature water was used to simulate molten steel.

Fig. 2.

Schematic illustration of hydrodynamic modelling experiment.

Blue ink was used for flow visualization and photography. In addition, sodium chloride (NaCl) was used for the measurement of residence time distribution. Before measuring, the liquid levels of the ladle and the tundish were raised to the predetermined height. Then tundish nozzles were opened. When water levels in both ladle and tundish in the hydrodynamic system were controlled stable simultaneously for more than 20 min, the saturated NaCl solution of 250 mL was used as a tracer and was injected into the water stream flowing through the ladle shroud. Meanwhile, one conductivity probe which was connected to a conductivity meter was installed below the submerged entry nozzle (SEN) of the tundish (outlet) to measure the instantaneous concentration of the tracer as a function of time. The measurement data were plotted with a recorder and input into a computer to obtain the RTD curves. All these data were further processed to calculate the fluid flow characteristics.

3. Mathematical Model of CFD

The commercial CFD (computational fluid dynamics) software FLUENT 14.5 and its pre-processor GAMBIT were used to calculate the steady state velocity field produced in the tundish due to transient turbulent flow of an incompressible single-phase fluid under isothermal condition, species transport was included to allow the derivation of the RTD curve.

3.1. Governing Equations

For the 3-dimensional steady state simulation of the fluid flow and heat transfer of the molten steel in the tundish under isothermal conditions, the following developed governing equations were involved:^21,22,23) continuity equation and momentum balance equation, Reynolds-averaged Navier–Stokes equations for incompressible Newtonian fluids, k–ε equations to simulate the turbulence, and the energy transport equation to calculate the temperature distribution.

The continuity equation is given by   

$$\frac{\partial (\rho u_i)}{\partial x_i}=0$$(5)

The Navier-Stokes equation is given by   

$$\frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}=-\frac{\partial \rho }{\partial x_i}+\frac{\partial }{\partial x_j}\left[{\mu }_{eff}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right]+\rho \beta \mathrm{\Delta }T{g}_i$$(6)

The k–ε two-equation model given by Launder and Spalding²⁴⁾ is widely used to describe turbulent kinetic energy k and its dissipation rate ε as follows   

$$\frac{\partial \left(\rho u_{i}k\right)}{\partial x_i}=\frac{\partial }{\partial x_i}\left(\frac{{\mu }_{eff}}{{\sigma }_k}\frac{\partial k}{\partial x_i}\right)+G-\rho \epsilon$$(7a)

  

$$\frac{\partial \left(\rho u_i\epsilon \right)}{\partial x_i}=\frac{\partial }{\partial x_i}\left(\frac{{\mu }_{eff}}{{\sigma }_{\epsilon }}\frac{\partial \epsilon }{\partial x_i}\right)+C_1\frac{\epsilon }{k}G-C_2\frac{{\epsilon }^2}{k}\rho$$(7b)

Concerning heat losses in the tundish, the energy equation is given by   

$$\frac{\partial \left(\rho u_{i}T\right)}{\partial x_i}=\frac{\partial }{\partial x_i}\left(K_{eff}\frac{\partial T}{\partial x_i}\right)$$(8)

The equation group of Eqs. (5), (6), (7), (8) are the main governing equations in the developed mathematical model to simulate flow field and temperature profile of molten stainless steel in the tundish. The density of molten stainless steel as a function of temperature can be calculated by²⁴⁾   

$$\rho =8\mathrm{\hspace{0.17em} }523-0.8358T$$(9)

The generation rate of turbulence energy G and effective viscosity μ_eff appearing in the k-ε two-equation model in Eqs. (7a) and (7b), can be calculates as follows   

$$G={\mu }_t\frac{\partial u_j}{\partial x_i}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\hspace{1em}{\mu }_{eff}=\mu +{\mu }_t=\mu +C_{\mu }\rho \frac{k^2}{\epsilon }$$(10)

The effective molecular and turbulent thermal conductivity of molten steel K_eff defined in Eq. (8) can be calculated by   

$$K_{eff}=\frac{\mu }{{\sigma }_T}+\frac{{\mu }_t}{{\sigma }_{t,T}}$$(11)

The five constants²⁴⁾ appearing in the k-ε two-equation model defined in Eqs. (7a) and (7b) and two constants defined in Eq. (7)²⁴⁾ take the values as follows:   

$$\begin{array}{c}{C}_{\text{1}}=\text{1}\text{.43, }{C}_{\text{2}}=\text{1}\text{.92, }{C}_{\mu }=\text{0}\text{.09, }{\sigma }_k=\text{1}\text{.0,}\\ {\sigma }_{\epsilon }=\text{1}\text{.3, }{\sigma }_T=\text{1}\text{.0, }{\sigma }_{t,T}=\text{0}\text{.9}\text{.}\end{array}$$

3.2. Boundary Conditions and Assumption

To ensure that the accuracy of these evaluations is reliable enough for design decisions to be based upon, care has to be taken with the theoretical modeling accuracy and assumptions.

(1) Molten steel is treated as three-dimension steady incompressible viscous fluid, and the flow in the tundish is turbulence with a turbulent intensity of 10%.The physical properties of the molten steel such as density, viscosity, specific heat and thermal conductivity are constant value. They are showing in Table 3.

Table 3. Physical properties of the molten steel at 1853 K.

Parameters	Data
Density	7000.0 kg/m3
Viscosity	5.3×10–3 kg/(m·s)
Specific Heat	822 J/(kg·K)
Thermal conductivity	30 W/(m·K)
Mass Diffusion Coefficients of Tracer	1.1×10–8 m2/s

(2) The surface fluctuation of molten steel in the tundish is ignored and the influence of slag layer and secondary oxidation of molten steel on flow field in the tundish are also neglected. Slag-molten steel surface is treated as free surface.

(3) Any solid wall in the tundish is considered as no slip stationary wall and the standard wall function was used to incorporate the variation due to turbulence. A zero shear stress boundary condition was applied for the free surface of the present tundish. A zero gradient was applied for the tracer concentration on the walls, free surface and outlets for the tracer dispersion.

(4) The heat loss of the tundish is caused by heat radiation and conduction. The values of heat transfer coefficient through slag layer, side walls, bottom wall, dam and weir of the present tundish are shown in Table 4 respectively.

Table 4. The values of heat transfer coefficient.

Parameters	Heat transfer coefficient [W⋅m−2⋅k−1]
Slag layer	10
Side walls	4
Bottom wall	4
Dam and weir	Adianatic

(5) Velocity inlet and pressure outlet boundary conditions were applied at inlet and outlet of the tundish respectively. Inlet velocity is 1.49 m/s, vertically downward. Pressure boundary condition of 1 atm was fixed at the outlets of the tundish.

As mentioned above, the tundish grid is automatically generated for all the design iterations using GAMBIT, the pre-processor of FLUENT. A fully unstructured grid is generated using tetrahedron and hexahedron elements. As representative, the mesh of the starting design configuration for original tundish is shown in Fig. 3 and consists of approximately 565973 cells. The model tundish was fabricated to have complete geometrical similarity with the full-scale tundish system.

Fig. 3.

Three-dimensional view of unstructured mesh and boundary conditions.

3.3. Numerical Solution Procedure

The set of governing equations were discretized using the finite volume technique in a computational domain and solved with the help of above boundary conditions using the commercial CFD software FLUENT 14.5. An implicit scheme with segregated solver was applied. In the numerical solution scheme Semi Implicit Method for Pressure Linked Equation (SIMPLE) algorithm²⁵⁾ was used for pressure velocity coupling in the momentum equation. Second order up wind scheme was used for discretization of convective term in the governing equations to provide higher order accuracy. The convergence criterions for scaled residuals were set to be less than 10^–3 except for concentration which is set to ≤ 10^–5. Figure 4 shows the numerical scheme used to determine RTD curve in the present work. Once the flow field was converged to steady state, the problem defined module of the FLUENT solver was changed to unsteady state for solving the transient tracer dispersion with appropriate initial and boundary conditions. Consequently, at t=0, a small volume element was assumed to be filled with the tracer at the inlet plane and subsequently, at t>0, the tracer concentration was predicted computationally for various instants of time at the outlet plane of the tundish to obtain the RTD curve.

Fig. 4.

The numerical scheme applied to predict RTD curve.

4. Results and Discussions

4.1. RTD Curves Change with the Increase of the Ratio of Width to Length

Figure 5 shows the experimental RTD curves measured by water modelling. Figure 6 shows the mathematical measured RTD curves of the original tundish (width = 800 mm) which are plotted between the molar concentration and time. It can be seen that both the experimental and mathematical RTD curves show a double peaks obviously. This is a reminder that double peaks is not an accident, even not an experimental error. It reflects the influence of tundish structure on flow pattern of the molten steel.

Fig. 5.

RTD curves of the original tundish (W/L=0.20) by water modelling.

Fig. 6.

RTD curves of the original tundish (W/L=0.20) by mathematical modelling.

Actually, appearance of double peaks in the RTD curves indicates that short-circuiting flow phenomena appeared in the tundish fluid flow system. Fluid flow in the region of short-circuiting, they have no enough residence time and spaces to mix and react with other fluids. It means that the effect of the tundish metallurgy is failed. Thus, short-circuiting of the fluid is an undesirable feature in both tundish designs and operations.

It was found that the width of the tundish can control the short-circuiting flow in the tundish fluid flow system. All other parameters such as bath height, inlet volumetric flow rate, inlet-exit distance, exit flow control device (stopper rod or nozzle), slag cover, submergence depth of the ladle shroud and perpendicular or inclined tundish walls have produced the flow pattern according to the tundish width. They have no influence on the shape of the RTD curve. Therefore, under the conditions of other technological parameters unchanged, we obtained the different RTD curves and data by changing the ratio of (W/L) in tundish in the present experimental conditions. The ratio of (W/L) in tundish are set as Table 5.

Table 5. The different ratio of width to length in tundish.

Case	The width of Tundish	The ratio of width to length (W/L)
T1	700 mm	0.18
T2	850 mm	0.22
T3	900 mm	0.24
T4	950 mm	0.26
T5	1000 mm	0.28
T6	1050 mm	0.30

The results of measured RTD curves from the different ratio of width to length in tundish in mathematical modelling are shown in Fig. 7. It can be observed clearly that the RTD curve shows a very sharp peak which indicates very poor mixing when the ratio of (W/L) in tundish in the present experimental conditions is 0.18. It is because the fluid travel straight towards the outlets once they impinge on the tundish bottom, without getting any chance to mix with the surrounding fluids. This affects negatively the performance of this tundish as a reactor. Then with the increasing of the ratio of (W/L) in tundish in the present experimental conditions, the shape of RTD curves change from double peaks to single peak gradually. When the ratio of (W/L) in tundish between 0.26 and 0.28, the short-circuiting flow disappeared. When the ratio of (W/L) in tundish is 0.30, the RTD curve shows a single peak completely. Meanwhile, the steepness of the different RTD curves is becoming more and more slowly with the increasing of the ratio of (W/L) in tundish in the present experimental conditions.

Fig. 7.

Typical RTD curves for the different ratio of width to length in tundish.

When the W/L is made from 0.28 to 0.30, the peak is significantly smaller. Actually, the available region is divided into two regions: the plug flow region and the well-mixed flow region. The combination of the plug flow region and the well-mixed flow region jointly influence the shape of RTD curves. The peak concentration is depend on the plug flow region and the well-mixed flow region affect the steepness of the RTD curve. The more plug flow region, the less peak concentration. The more well-mixed flow region, the shape of a flat RTD curve (exponential curve). Meanwhile as shown in Fig. 7, the lowest RTD curves (T6), it reflect the molten steel flow characteristics and behaviors in the tundish, which eliminated the short-circuiting flow to a great extent.

4.2. The Characterization of Flow Zone by RTD Curves

In several studies,^26,27) the modified mixed flow model was used by many researchers to characterize melt flow in continuous casting tundish systems. Schematics of the modified mixed flow models is shown in Fig. 8. The volume fraction of three zones in a tundish can be calculated by Eqs. (12), (13), (14), (15), (16), (17), (18). However the specific situation, including short-circuiting flow phenomena in the reactors, was no defined.

Fig. 8.

Schematics of the modified mixed flow models.

According to the modified mixing model of flow,^2,28) fluid flow in a tundish can be divided into three zones as dispersed plug zone, well-mixed zone and dead zone. The shape of the curve is of interest since it correlates with the ratio of plug flow volumes versus mixed flow and dead flow volumes in the tundish. It also included the minimum residence time, peak concentration residence time and the mean residence time. It is desirable to have high a plug flow volume compared to the mixed flow and dead flow volumes when mixing in the tundish is to be kept to a minimum. The volume fraction of three zones in a tundish can be calculated by following equations.

Theoretical mean residence time:   

$$t_a=\frac{V}{Q}$$(12)

V—Volume of tundish

Q—Volumetric flow rate of the molten steel

Actual mean residence time:   

$$\overline{t}=\frac{\sum _{i=1}^{N}C(t_i)\cdot t_i}{\sum _{i=1}^{N}C(t_i)}\hspace{1em}\hspace{1em}{t}_i\in \left[0,+\infty \right]$$(13)

The 2 times the theoretical mean residence time:   

$${\overline{t}}_c=\frac{\sum _{i=1}^{N}C(t_i)\cdot t_i}{\sum _{i=1}^{N}C(t_i)}\hspace{1em}\hspace{1em}{t}_i\in \left[0,2t_a\right]$$(14)

The plug flow residence time:   

$$t_p=(t_{min}+t_{peak})/2$$(15)

The volume fraction of dead region:   

$$V_d=1-{\overline{t}}_c/\overline{t}$$(16)

The volume fraction of plug flow:   

$$V_p=t_p/t_a$$(17)

The volume fraction of well-mixed flow:   

$$V_m=1-V_p-V_d$$(18)

So far, the majority of research is based on the single peak of RTD curves and data. The above calculational method is not suitable for the double peaks of RTD curves and data. Therefore, to study the calculational method which including short-circuiting flow is not only important theoretical value, but also important practical significance, especially in the tundish design.

However, we can not define the short-circuiting flow like define the dead zone (the region which fluid flow slowly in the tundish and stay for longer than two times the mean residence time). Therefore, to compared with traditional method, which the tundish can be divided into three regions, it can be divided the overall fluid flow region into two regions in tundish, namely the available region and dead region. This method elaborate the phenomenon of the fluid flow in tundish which include the short-circuiting flow more accurately.

Actually, as shown in Fig. 7, double peaks, the first peak is stand for short-circuiting flow. Similar to the method for calculating the plug flow volume fraction, the short-circuiting flow volume fraction is related to t_min and t_peak for the first peak on RTD curve. Schematic diagram of the modified mixed flow models with short-circuiting flow is shown in Fig. 9. The short-circuiting flow exists together with the plug flow parallelly.

Fig. 9.

Schematic diagram of the modified mixed flow models with short-circuiting.

According to the new mathematical formulation, the RTD characterization results is shown in Table 6, Figs. 10 and 11 respectively. The results of the typical RTD curves for the ratio of (W/L) in tundish in the present experimental conditions show that volume fraction of dead zone V_d is about 11.76%, and the minimum breakthrough time t_min is 130 s in Case Original with the corresponding t_peak is about 294 s. According to sum of order based on the ratio of (W/L) in tundish in the present experimental conditions, the optimal integrated tundish structural parameters can be found in Case T1, Case T2, Case T3, Case T4, Case T5 and Case T6, in which t_min can increase by 7.69%, 11.54%, 17.69%, 23.08% and 38.46%, t_peak increase by 9.52%, 17.68%, 30.61%, 43.54%, 57.14%, while V_d can decrease by 4.76%, 8.42%, 18.48%, 21.09% and 30.44%, respectively, compared with those in Case Original. Obviously, the currently applied tundish with structural parameters shown in Fig. 1 does not behave perfectly for flow field of molten steel.

Table 6. The flow quantification results derived from the RTD curves.

The ratio of width to length (W/L)	The peak concentration
0.18	0.4117
0.20	0.3515
0.22	0.3244
0.24	0.3044
0.26	0.2876
0.28	0.2768
0.30	0.0536

Fig. 10.

The characteristic time t_min and t_peak.

Fig. 11.

Overall RTD parameters and the associated volume fractions.

From the above drawing, we can see clearly that, due to the existence of short-circuiting flow, it has negative influence on the overall flow pattern of the molten steel in tundish. As shown in Fig. 10, the ratio of (W/L) in tundish in the present experimental conditions increases successively from 0.18 to 0.30, the minimum residence time (t_min) increases from 113 s to 180 s, the peak concentration residence time (t_peak) increases continuously from 251 s to 462 s. With the increasing of the ratio of (W/L) in tundish in the present experimental conditions, the fraction of available volume increased gradually, the fraction of dead volume decreased. And further calculate the ratio of V_a to V_d, they are 7.93, 8.29, 9.43, 9.76 and 11.22, respectively. The ratio of V_a to V_d is also an important standard to reflect the effect of tundish metallurgy. It can be concluded that, to the present single-strand tundish, the ratio of V_a to V_d increase with the increasing the ratio of (W/L) in tundish in the present experimental conditions, this is the result we expect for. Meanwhile, as shown in Table 6, with increasing the ratio of (W/L) in tundish, the peak value of molar concentration are decreased by 7.71%, 13.40%, 18.18%, 21.25% and 84.75%. When the ratio of width to length is 0.30, the RTD curve becomes smooth and the peak value of molar concentration decreased obviously, which indicate that the extent of mixing in tundish is becoming more and more better and the residence time of the molten steel in the tundish is longer with the increasing of the ratio of (W/L) in tundish in the present experimental conditions.

It would be specially mentioned that the ratio of (W/L) in tundish is a key factor to control the flow field of molten steel. With increasing the ratio of (W/L) in tundish, some important criteria such as V_a, V_d, the ratio of V_a to V_d, t_min, t_peak and the peak value of molar concentration present a regular change pattern. To some extent, this regular change pattern may provide guidelines and plays an instructional role for practical single-strand tundish optimization design.

4.3. Effect of the Ratio of Width to Length on Velocity Fields under Nonisothermal Condition

Figure 13 shows the calculated velocity vectors in the symmetrical plane and its corresponding 3-D streamlines in the tundish. And the photographed streamlines in the original tundish in the hydrodynamic modelling by injecting blue ink as color tracer is shown in Fig. 12. As shown in Figs. 12 and 13, the streamlines of both molten steel and ambient temperature water in the tundish show that fluid flow is highly turbulent near the inlet region. From the inlet region the tracer disperses in all the directions. It maybe seen that a small portion of the molten steel is advancing on the inlet-outlet plane straight towards the outlet. This tracer front on exiting the tundish gives rise to first peak in Fig. 7. Then the rest of the molten steel continues according to the flow field in the tundish. The RTD curve shows another peak value in concentration at sometime after the time for the appearance of the first peak. After the second peak, the tracer concentration decays exponentially with time. Therefore, we can conclude that the first peak is closely related to the short-circuiting flow.

Fig. 12.

Photograph showing the tracer dispersion.

Fig. 13.

The velocity vectors and its corresponding 3-D streamlines.

By increasing the ratio of (W/L) in tundish in the present experimental conditions, the wide-side walls retard the movement of the aforementioned portion of the molten steel which advance on the inlet-outlet plane straight towards the outlet and at a critical width the width-side walls force the portion of the molten steel towards the middle of the tundish where it mixes with the remaining liquid. Thus, straight movement of the portion of the molten steel and the short-circuiting contained in it are eliminated as a consequence that RTD curve shows a single peak.

As illustrated in Fig. 12, the streamlines of the molten steel in tundish shows that the turbulence inhibitor and sidewall of the tundish lead molten steel to disperse and flow upwards and then come to the main zone via the weir-dam arrangement, while some remaining molten steel mix with the new molten steel in the turbulence inhibitor zone, and then flow to the main zone of the tundish. In the main zone, there is a strong recirculation between dam and outlet. From the perspective of metallurgical kinetics, the strong recirculation provides enough power to promote inclusion’s collision, agglomeration and flotation. The weir-dam combination has the flow guide function for molten steel: a weir can make all molten steel flow horizontally along the tundish bottom, and a dam can lead some molten steel flow to SEN of the tundish directly and some other flow to the upper region in the main zone. With the increasing of the ratio of width to length, there are more and longer streamlines of molten steel in the overall tundish, it can make longer residence time and longer flow distance for the molten steel, and lead more molten steel to mix better and better.

In addition, it is almost certain that the more and longer streamlines zone, the less dead volume. As shown schematically in Fig. 14, there may exist dead regions on the downstream side of the dams and weirs,or near the end wall.

Fig. 14.

A schematic diagram of the fluid flow in the tundish.

5. Conclusions

The fluid flow phenomena and mechanism of the molten steel in an asymmetrical single-strand tundish with capacity of 30 tons have been investigated in a 1:3 hydrodynamic model combined with a mathematical modelling. From the present study, the main conclusions can be summarized as follows:

(1) A new mathematical formulation is proposed to analyze the RTD curves with double peaks which indicate a short-circuiting flow in the tundish. Actually, the short-circuiting flow exists together with the plug flow parallelly.

(2) The ratio of width to length (W/L) in tundish in the present experimental conditions is a key factor to control the short-circuiting flow in the tundish fluid flow system. With increasing the ratio of (W/L) in tundish, the wide-side walls play an important roles to retard the short-circuiting flow on the inlet-outlet plane straight towards the outlet.

(3) To some extent, the location, the length and relative distribution density of 3-D streamlines reflects the influence of tundish structure on flow pattern of the molten steel. We can find the dead volume by analyzing the 3-D streamlines of the molten steel in tundish. The more and longer streamlines zone, the less dead volume.

Nomenclature

C(t_i): Molar concentration of tracer measured at individual time t_i (kmol/m³)

C₁,C₂,C_μ: Constants in the k-ε two-equation turbulence model (—)

Fr: Froude number (—)

g: Acceleration of gravity (9.8 m/s²)

G: Generation rate of turbulence kinetic energy (—)

k: Turbulent kinetic energy (m²/s²)

K_eff: Effective thermal conductivity (kg/(m·s))

L: Characteristic length (m)

p: Pressure (kg/(m·s²))

Q: Volumetric flow rate of the molten steel (m³/h)

Re: Reynolds number (—)

t: Time (s)

t_a: Theoretical mean residence time (s)

t: Actual mean residence time (s)

t_c: The 2 times the theoretical mean residence time (s)

t_min: the minimum residence time (s)

t_peak: time taken to attain peak concentration of tracer at outlet (s)

t_p: The plug flow residence time (s)

T: Temperature (K)

u: Flow velocity (m/s)

V: Volume of tundish (m³)

V_a: Volume fraction of available region (—)

V_d: Volume fraction of dead region (—)

V_p: Volume fraction of plug flow (—)

V_m: Volume fraction of well-mixed flow (—)

x_i,x_j: i-th or j-th Cartesian space coordinate (m)

Greek symbols

β: Coefficient of volumetric thermal expansion of molten steel (K^–1)

ε: Dissipation rate of turbulent kinetic energy (m²/s³)

λ: Geometrical scale factor (—)

μ: Molecular viscosity (kg/(m·s))

μ_eff: Effective viscosity (kg/(m·s))

μ_t: Turbulent viscosity (kg/(m·s))

ρ: Density of molten steel (kg/m³)

σ_k,σ_ε: Schmidt number for k and ε (—)

σ_T,σ_t,T: Laminar and turbulent Prandtl number (—)

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