Numerical Modeling of the Fluid Flow in Continuous Casting Tundish with Different Control Devices

and Applied Analysis 3 and a pair of baffles (Case III). The cross-section of slab is 1600mm × 200mm and the casting speed is 1.2m/min, so the corresponding flow rate is 0.768m/min. 2. Mathematic Formulation 2.1. Fundamental Equations. The liquid steel flow in the continuous casting tundish can be considered to be threedimensional, Newtonian, and incompressible turbulence. A layer of slag at the top of liquid is neglected and the melt surface is assumed to be flat.Themathematical model, which is used to simulate the melt flow inside the tundish as well as the chemical mixing process of the tracer injected by a pulse in the incoming stream,was formulated based on the solution of the three-dimensional Navier-Stokes equations, the mass transfer equation, and two equations for the k-εmodel chosen to represent turbulent viscosity. Continuity equation:


Introduction
In the steel production route, continuous casting of liquid steel is the most important process.Basically, the tundish in the continuous casting is an intermediate vessel between the ladle and the mold to distribute and supply liquid steel to different molds at an approximately constant rate (Figure 1).In recent years, tundish has become more of a continuous reactor than merely an intermediate buffer to distribute and supply liquid steel with an approximately constant rate [1].Various metallurgical operations, such as alloy trimming of steel, thermal and chemical homogenization, inclusion separation and floatation, are carried out in steel making tundishes.The time available for these operations is very short, and therefore, it is necessary to have a good understanding of fluid flow behavior in the tundish.Generally, two research methods, physical modeling and mathematical simulation [2][3][4][5], are used for the tundish configurations optimization and flow control devices, such as weirs, dams, baffles with inclined holes, and turbulence inhibitors (TI), have been widely used to increase residence times and plug flow volume of liquid steel [6][7][8].
Since detailed knowledge of the molten steel flow is a prerequisite to any effective flow-control optimization, significant efforts have been made by researchers to investigate fluid flow phenomena in tundish systems.Estimation of the various residence time distribution (RTD) parameters via the pulse tracer addition technique has been widely used to study the fluid flow patterns in tundish system [9][10][11].In such studies, a tracer (e.g., dye, acid, or salt) is injected through the incoming water stream and its concentration at the exit is recorded as a function of time.The plot of the exit concentration against time is known as the RTD curve.The RTD of the fluid in a tundish is analyzed to characterize the flow which, normally, includes the determination of the extent of mixing (plug and mixed volumes) and the dead volume in the tundish.And it has been generally considered that the mathematical model is able to simulate RTD phenomena realistically [12][13][14].
In the present, work fluid flow in a 30 t tundish with different flow control devices was investigated by mathematical models.In each case of study, flow characteristics, velocity patterns, RTD curves, and inclusion distribution were obtained.The objective in this work was to study the effects of the flow control devices on the fluid flow pattern and RTD curves in a T-type two-strand tundish of a bloom caster.
Figure 2 shows the physical dimensions of this problem.Design of the baffles and the turbulence inhibitor as well as the positions of the TI and baffles inside the tundish is also indicated in Figure 2. Three types of the tundishes were studied as follows: the bare tundish (Case I), a tundish with a pair of baffles (Case II), and a tundish with a TI  and a pair of baffles (Case III).The cross-section of slab is 1600 mm × 200 mm and the casting speed is 1.2 m/min, so the corresponding flow rate is 0.768 m 3 /min.

Mathematic Formulation
2.1.Fundamental Equations.The liquid steel flow in the continuous casting tundish can be considered to be threedimensional, Newtonian, and incompressible turbulence.A layer of slag at the top of liquid is neglected and the melt surface is assumed to be flat.The mathematical model, which is used to simulate the melt flow inside the tundish as well as the chemical mixing process of the tracer injected by a pulse in the incoming stream, was formulated based on the solution of the three-dimensional Navier-Stokes equations, the mass transfer equation, and two equations for the k- model chosen to represent turbulent viscosity.Continuity equation: Momentum equation: where  , are the time-averaged fluid velocities in the ith and jth directions, respectively,  is the liquid density,  is the pressure in the fluid,  eff is the effective turbulent viscosity,   is the gravitational acceleration in the ith direction,   ,   are the spatial coordinates in the ith and jth directions, respectively, and ,  denote the three directions in the global Cartesian coordinate system.Effective viscosity  eff is the sum of laminar viscosity  and turbulent viscosity   : The following equation describes the turbulent kinetic energy: The following equation describes the dissipation rate of turbulence energy: where  is the generation of turbulence kinetic energy caused by the mean velocity gradients: The values for the constants in this - model  1 ,  2 ,   ,   , and   are 1.44, 1.92, 0.09, 1.00, and 1.30, respectively [15].
Mass transfer equation: In the mass transfer equations, C is the tracer concentration. eff =   +   is the effective mass transfer diffusivity, which is the summation of molecular and turbulent diffusivities, respectively.The turbulent diffusivity D  is related to the turbulent viscosity   by which means that the turbulent Schmidt number is equal to one.

Boundary and Initial
Conditions.Due to symmetry, only a half tundish from the symmetric central-longitudinal plane was chosen for this mathematical simulation.On the top surface of the bath and in symmetry planes, the fluxes of all variables were set equal to zero.No slip conditions were applied to all solid surfaces of the tundish including baffles, and interior walls of the tundish and standard wall functions were applied.The tundish exit is computationally treated as a plane, at which flow occurs at an ambient pressure.
The vertical velocity profiles of the liquid steel at the inlet of the tundish were assumed to be uniform through the cross-sections and the other two velocity components were assumed to be zero.The values of k and  at the inlet were calculated from the inlet average velocity through the following equations: The initial condition for ( 5) is that the tracer concentration is zero except in the ladle nozzle from the injection point to the nozzle tip.

Geometry and Numerical Methods. A control volume-
based technique was used to convert the nonlinear governing equations to algebraic equations that can be solved numerically using the 3D mesh (Figure 3).A dense mesh was employed near the wall of the tundish.SIMPLE algorithm [16] was applied for the pressure-velocity coupling.A segregated and implicit solver was used to solve the governing equations and the second-order accuracy upwind differencing scheme was adopted to improve the accuracy of the solution.The open source Computational Fluid Dynamics software OpenFOAM [17] was used to compute the velocity and concentration fields.OpenFOAM is an open source code which is designed for continuum mechanics applications and has attracted much attention recently.It is an object oriented  4) were solved together with their boundary conditions until the steady state of the fluid flow was reached.Then the velocity field was employed to solve (7) for the tracer concentration under unsteady state conditions.Here, the tracer was treated as a passive scalar; that is, the presence of the tracer does not affect the melt flow.A criterion for convergence was set to be less than 10 −5 on all variables and computations were carried out until the relative sum of residuals on all variables fell below the stipulated value.

Model Validation.
In this section, the fluid flow model was validated against the experimental data [18] in the water model for a single inlet, single outlet tundish in which the tracer concentrations with time at the outlet were measured.The bath height was kept at 260 mm in the experiment and the same height was used for the model accordingly.Figure 4 shows the temporal variation of the tracer concentration (nondimensional) with nondimensional time and its comparison with the experiment.It can be seen that the overall characteristics of the RTD curve predicted by the present  of the velocity at various transverse positions in the symmetry plane of the bare tundish ( = 0, −1.066 ≤  ≤ 0.324,  = 0.1).According to the results, the grid size schemes of 63 × 79 × 40 and 76 × 96 × 48 have almost the same velocity magnitude distribution, which demonstrates that the grid size scheme of 63 × 79 × 40 is accurate enough for the present numerical calculation.

Effects of the Control Devices on the RTD Curve.
Figure 6 shows the comparison of the RTD for the three studied cases.The curve for Case I is characterized by two peak values of concentrations, one soon after the tracer injection and then the second.After then, the tracer concentration decreases continuously with time.This is because a portion of the liquid moves on the horizontal plane straight towards the exit [19].
The existence of the two peak values of concentration may  suggest that the flow field in the tundish is short circuited and that is undesirable in tundish fluid system.Thus the bare tundish is associated with considerable short-circuiting and large dead volumes, which are potentially detrimental to the floatation of nonmetallic inclusions.As seen from Figure 6, the RTD parameters in Case II and III are improved considerably and only one peak value appeared.
From the RTD curves, minimum residence time,  min , peak concentration time,  max , and mean residence time,  av , can be obtained for each case.Considering that there is fluid exchange between the fluids in the dead zone and in the active zone, the flow model proposed by Sahai and Emi [20] was employed in this work to calculate dead volume fraction, the fractions of plug flow and well-mixed volumes.
Table 1 shows a summary of the flow characteristics in the tundish with different flow control devices.Each volume fraction of flow for the studied cases is shown in Figure 7.It can be seen that the associated dead volume is significantly high for the bare tundish.The application of baffles brings down the dead volume from 54% to 24% and it is further decreased to 16% with the TI+baffle configuration as is clearly mentioned in Table 1.The better mixing in the tundish occurs corresponding to higher well-mixed along with lower dead volume and therefore it is also expected for better temperature homogenization.At the same time, higher plug volume with minimum dead region shows the better inclusion removal in the tundish.region to make any alloy additions in the tundish.And along with the higher convective and diffusive mixing in this region, the alloy will get uniformly distributed in the rest of the tundish, which results in homogeneity during the casting.

Analysis of Velocity Fields.
Figure 9(a) through Figure 9(c) shows the predicted turbulence kinetic energy contours along the top surface of one half of the tundish for Cases I, II, and III, respectively.The high turbulence kinetic energy is associated with the liquid steel top surface instability and the entrainment of the slag.As seen from Figure 9(a), the highest value of the turbulence kinetic energy is found in the region just above the inlet stream and it is 6.5 − 5 m 2 /s 2 per kg of the liquid.With a pair of baffles, the highest value is lowered to 5.0 − 5 m 2 /s 2 per kg of the liquid (Figure 9(b)).For Case III as shown in Figure 9(c), the turbulence kinetic energy is greatly decreased and the highest value is 2.8 − 5 m 2 /s 2 per kg of the liquid.

Conclusions
A numerical study was conducted to investigate the fluid flow and residence time distribution of the T-type two-strand continuous casting tundish with and without flow control devices.The following conclusions have been drawn from the present study.
(i) The application of a pair of baffles eliminates the short circuiting phenomena in the tundish and brings down the dead volume from 54% to 24%.
(ii) The employment of a TI and a pair of baffles is more effective to increase the plug fraction than a bare tundish or a tundish with just a pair of baffles, and it was found to be an optimum configuration of the two-strand tundish in the present study.(iii) With a TI, the velocity vectors directed toward the tundish floor are smaller and the turbulence kinetic energy of the liquid steel top surface is lower than the other two arrangements, which means a more stable slag-metal interface.

Figure 1 :
Figure 1: Schematic diagram of a continuous casting setup.

Figure 2 :
Figure 2: Physical dimensions of the tundish and the control devices.

Figure 3 :
Figure 3: 3D computational mesh employed in the present model.

Figure 4 :
Figure 4: Comparison of the predicted and measured RTD curves.

Figure 5 :Figure 6 :
Figure 5: Grid independence check of solutions in a bare tundish.

Figure 7 :
Figure 7: Volume fraction of flow for the studied cases.

Figure 8 :Figure 9 :
Figure 8: Predicted metal flow field in tundish for all cases: (a) Case I, (b) Case II, and (c) Case III.

Table 1 :
RTD parameters in the tundish system.