The Effect of a Dissipative Ladle Shroud on Mixing in Tundish: Mathematical and Experimental Modelling

Geometry

Ladle shrouds and a small tundish

Models with a size of one-third scale of the real CLS and DLS were used for the physical and mathematical modelling. A schematic drawing of the two ladle shrouds and their dimensions are shown in Figure 1. The tube diameter of the CLS is 30 mm. The diameters of the inlet and the outlet are respectively 30 mm and 60 mm for the DLS (Figure 1(a)). There are two tundishes that are studied for modelling. A small tundish, like a water tank, is chosen to carry out the LES simulation, due to the challenges to solve a full-scaled tundish with fine grid blocks and appropriate Yplus values to achieve accurate results. The other one is a one-third scale of a real tundish, which will be described in the following section. The length, width and height of the small tundish model are chosen to be 500 mm, 250 mm and 170 mm respectively. The computational domain is meshed into ~1,750,000 hexahedral cells.

Figure 1:

(a) Schemes and dimensions of studied ladle shrouds (the unit is mm for the listed values). (b) Experimental set-up and boundary conditions of the small tundish.

For the boundary conditions, a constant inflow velocity (0.708 m/s, derived from the inlet area and flow rate of 30 L/min) profile was applied at the nozzle inlet; the outlet was set with a fixed pressure of 0 Pa to the ambient. The top free surface was simplified as free-slip boundaries with zero normal velocity. Walls were considered to be non-slip, and the near wall boundary domain was handled using the Werner-Wengle formulations [14].

A one-third scale tundish

To further the study of the effect of ladle shrouds on mixing in a tundish. A one-third scale tundish, consistent with the size scale of the ladle shrouds, was built based on industrial practices. The dimensions and structure of the tundish model are shown in Figure 2. The tundish is used in a Chinese steelmaking plant to produce slabs with a cross section of 220 mm×2000 mm. In water modelling, geometrical and dynamic Similarity criteria between the model and the prototype tundish were maintained, and the calculation procedure can be found in reference [15].

Figure 2:

Schemes and dimensions of a one-third tundish.

Mathematical modelling

Large eddy simulation (LES) is believed to be a more accurate and efficient method to get detailed information of turbulence compared with Reynolds-averaged approach and Direct Numerical Simulation [11]. LES adopts the concept of filtering and eddies are divided into large scale and sub-grid scale (SGS) categories. The filtering operation is offered by a finite-volume discretization in ANSYS-FLUENT as [16]:

where V represents the volume of the computational cell, and G(x,x′)is the filtering function. The filtered variables are denoted with an overbar. The filtered Navier-Stokes equations for incompressible fluids are as follows:

where ρ, t, p and u_i represent density, time, pressure and filtered velocities respectively. ν0 and νtare the molecule viscosity and eddy-viscosity. The subscripts i and j are the three directions in Cartesian coordinates. τij represents the unknown SGS stresses:

where ϑij represents the stress tensor resulted from molecular viscosity; the isotropic part of the SGS stresses τkk is added to the filtered static pressure term or neglected for incompressible flows [16]. Sˉijindicates the rate of strain tensor

The Wall-Adapting Local Eddy-Viscosity (WALE) model is employed to address the SGS stresses, due to its ease of dynamic procedure and suitability to complicate geometries [17]. The νt is expressed as:

where Lsis the mixing length for sub-grid scales, defined as below:

where d is the distance from the cell centre to the closest wall; the von Kármán constant κ is 0.418. Cw is chosen to be 0.325, which has been confirmed to yield satisfactory results for a wide range of flow [16].

An instantaneous tracer transfer associated with fluid flow was modelled to track the tracer dispersion by solving the following equations:

where D₀ is the molecular diffusivity of a tracer; as turbulent flow dominates the mass transfer, the Schmidt number Sc_t was chosen as one. The solutions of the species transfer and LES turbulent model are coupled together to acquire an unsteady mass distribution of the tracer.

Physical modelling and validation

A water model was established to validate the LES modelling with the same flow parameters as the small tundish. The fluid flows inside the DLS and the tundish has been verified in the previous study [11]. For the mixing, the movement of a tracer by injecting black ink from the inlet in the physical water model was recorded using a high-speed camera Hispec 5 (Figure 1(b)). The transient concentration profiles of the tracer at different times were filmed and are shown in Figure 3(a)–3(d). Correspondingly, tracer profiles of the LES are shown in Figure 3(e)–3(h). At around 0.100 s (after the injection of the tracer), the ink flows towards the left side of the first chamber (Figure 3(a) and 3(e)), and travels down along the right-side wall of the second chamber at about 0.300 s (Figure 3(b) and 3(f)). The pioneer tracer finally leaves out the shroud at about 0.600 s and tends to flow towards the right side (Figure 3(d) and 3(h)). The results of the two models match well with respect to the dispersion profiles at different time.

Figure 3:

Tracer dispersion profiles at different time: (a) 0.104 s, (b) 0.304 s, (c) 0.477 s, and (d) 0.598 s in the physical modelling; (e) 0.100 s, (f) 0.300 s, (g) 0.470 s and (h) 0.610 s in the LES modelling.

Stimulus-response experiments were conducted using the tracer to obtain RTD curves by injecting 50 ml saturated KCl solution from the inlet in the one-third scale tundish. The electric conductivity was recorded at the outlet with an interval of 1.0 s. The characterization of the dead volume, plug volume and mixed volume have been well developed for single-strand tundishes in reference [18].

Flow and mixing inside the ladle shrouds

The mass transfer is dominated by the fluid flow inside the ladle shroud with high turbulence (Reynolds number, ~20,000). It is reasonable to elucidate mixing from the point view of fluid flow. Vortex (or eddy) is one of the typical attributes of turbulent flow, which implies the rotation of fluid elements as they travel in a fluid field. A measurement of rotation, vorticity, is defined as the curl of the velocity vector (ζ=∇×ui). The interaction of different scale eddies, dissipation in particular, strongly contributes the intermixing of fluid elements [19], which can be characterized by the rate of turbulence energy dissipation (ε). The relationship between the vorticity and the dissipation rate is illustrated as [11]:

Thus, the existence and magnitude of vortices, along with the basic notion of flow velocity, are selected as the main features of turbulence to facilitate the discussion of mixing inside the ladle shrouds in the succeeding sections.

A typical comparison of the flow profiles inside the CLS and the DLS is shown in Figure 4. The directions of velocity vectors in the CLS seem to be consistent and flow straight down along the ladle shroud (Figure 4(a)), similar to the plug flow. Smaller velocity vectors are observed along the wall boundaries and virtually no vortex can be visualized on the longitudinal (XZ) plane in the enlarged view. On the transversal plane (XY), the velocity is quite small in the X and Y directions, less than 0.03 m/s. Only weak vortices are created along the wall due to the boundary effect. In the DLS, the velocity decreases in the chambers and increases in the straight sections; many vortices are formed inside the shroud, especially inside the chambers. The main stream swings inside the shroud, and vortices are intensified at the enlarged sections of the chamber in Figure 4(b). Meanwhile, an anticlockwise vortex is formed at the corner of the chamber, which can drive the intermixing of neighbour fluid elements in longitudinal direction.

Figure 4:

Typical velocity vector profiles, with enlarged transversal and longitudinal section of the (a) CLS and (b) DLS at the central XZ plane.

To clearly reveal the fluid flow in the DLS, velocity vector and contour profiles are also displayed in the transversal sections of the three chambers and the outlet in Figure 5. The colour indicates the overall velocity magnitude in the XYZ directions; the arrow is just the velocity vector in the XY plane. Tangential velocity vectors can be observed and numerous vortices are formed on the selected planes which are mainly caused by the sudden change of the velocity direction from the divergent part into the convergent part of the chamber. A variety of time-dependent vortices is formed and can intermix with the longitudinal vortices in Figure 4(b). The velocity in XY plane reaches nearly 0.2 m/s, which is much higher than that of CLS. This means the mass transfer is radially intensified inside the DLS. The vortices observed in the selected sections imply that the fluid elements are easy to intermix with each other both in the longitudinal and radial directions inside the DLS. This can also be verified by the ink dispersion profiles filmed by a high-speed camera in Figure 6. At 0.1728 s (after the injection of ink in another tracer test), the stream flows down mainly towards the left side and in prone to turn anticlockwise to the right side. At 0.2065 s and 0.2445 s, the anticlockwise vortex is fully developed. With the effect of the flow and diffusion, the ink finally gets mixed inside the first chamber.

Figure 5:

Velocity vector and contour profiles in the transversal sections of the three chambers and the outlet.

Figure 6:

Ink dispersion in the first chamber filmed by a high-speed camera.

The above discussion indicates that the design of three chambers has revolutionized the flow structure and mixing inside the ladle shroud. The reinforced turbulence enhances the movement and intermixing of fluid elements in all directions, rather than solely flowing straight down with less interaction. In other words, the fluid premixing ahead of the tundish is improved inside the ladle shroud, being potentially changing the mixing inside the tundish as well.

Mixing in Tundishes

Stimulus-respond technique was utilized to examine the mixing phenomena by analysing RTD curves. The analysis of RTD characteristics is deemed as an effective approach to evaluate the plug volume, dead volume and mixed volume. The comparison in different scaled tundishes could be helpful to investigate the effect of ladle shroud on different sized tundishes as well as the inner and outer outlets in multi-strand tundishes.

In the small tundish

The RTD curve profiles of the studied cases are given in Figure 7 and the analysed results are shown in Table 1 for the small tundish. The time and concentration of tracer are both dimensionless to facilitate the discussion in Figure 7. It can be seen that the RTD curve of the DLS is much smoother than that of the CLS, with low peak concentration and relatively long breakthrough time. When the CLS is replaced by the DLS, the plug volume is dramatically increased from 3.53 % to 11.53 %, and the dead volume is reduced from 28.00 % to 10.79 % respectively. A longer breakthrough time (1.5 s and 4.9 s for the CLS and DLS, respectively) is contributed. These are attributed to the swinging jet flow and damping effect of the DLS to mix with more fluid around the jet, rather than directly impacting the tundish bottom or quickly flowing out from the tundish outlet [12]. Although the outlet of the CLS is featured with higher velocity than that of the DLS, the tundish under the DLS is allowed with larger active volume (the sum of plug and mixing volume [18]). The high turbulent energy of the CLS jet is partially consumed by impacting on the tundish bottom and fluctuating the free surface, rather than by activating the fluid flow inside the tundish.

Figure 7:

RTD curves of different ladle shrouds in the small tundish.

Table 1:

RTD characteristics of studied cases in two types of tundishes.

		Plug volume Vp (%)	Dead volume Vd(%)	Vp/Vd	Breakthrough time (s)
Small tundish	CLS	3.53	28.00	0.13	1.5
	DLS	11.53	10.79	1.07	4.9
One-third tundish	CLS	16.47	23.78	0.69	41.5
	DLS	23.61	14.32	1.65	59.5

In the one-third scale tundish

The one-third scale tundish matches the size of the ladle shrouds, and it is expected to perform more closely to the real tundish. Meanwhile, all-scale turbulence information is included in the physical modelling; no time averaging or filtering is involved to simplify the fluid flow. Hence, physical modelling is considered to be a direct method to study the mixing in tundish [18]. The achieved RTD curves are shown in Figure 8. Microscopic fluctuation of the electric conductivity can be seen along the RTD curves. The breakthrough time is 59.5 s (in Table 1) with the DLS, longer than that with the CLS (41.5 s), bringing about an increase of plug volume 14.67 % to 23.61 %. Meanwhile, the values of V_p/V_d are augmented with the DLS in the two tundishes. Thus, the dissipation effect of the DLS is effective in the two tundishes to benefit the flow characteristics, potentially improving cleanness and diminishing the differences of temperature and composition for the molten steel in practice. It is also noted that the increase of the V_p/V_d is 723 % (from 0.13 to 1.07) in the small tundish, and is 139 % (from 0.69 to 1.65) in the one-third scale tundish, when the CLS is replaced with the DLS. Thus, the DLS seems to perform more effectively in improving the fluid mixing in the smaller volume tundish than in the larger volume tundish due to its increased volume weight (volume of ladle shroud/ volume of tundish) to control the fluid flow. The comparison of mixing in these two single-strand tundishes with different sized can be also extended to the mixing in multi-strand tundishes, such as four or six-strand tundish. The outlets of the small and the one-third scaled tundishes correspond to the inner outlet and outer outlet (using the position of ladle shroud as reference) in a multi-strand tundish. The DLS can especially benefit the flow characteristics and mixing for the flow of the inner outlet (s), which is consistent to the study of Morales-Higa et al. [12].

Figure 8:

RTD curves of different ladle shrouds in the one-third scale tundish.

In addition, grade transition frequently happens in a single casting sequence, which involves intermixing of the pre-existing steel and the newly introduced steel from the ladle shroud inside a tundish [20]. It is desirable to minimize the transition length (or tonnage) of hybrid chemistry product, as it commonly fails to meet the required composition of any grade. A perfect plug flow in a tundish would obtain an ideal steel grade transition without any intermixing. Fortuanately, the adoption of the DLS is able to contribute a larger plug volume (as just discussed above, and also applicable in transition operation), which means the DLS can promote a more plug-like flow pattern inside the tundish. In this case, a shorter transition length can be expected to reduce the tonnage of hybrid product.

It should be pointed out that the present study only deals with the fluid flow and mass transfer in reduced scale of ladle shrouds and tundish models. Water is used as fluid to replace molten steel to simplify the validation and discussion, which is a common alternative to study steelmaking practices. Future attention should be paid on having a more practical insight into the ladle shrouds.