Structure optimization of continuous casting tundish with channel - type induction heating using mathematical modeling

: There are few studies on the shape and struc - ture of the channel - type induction heating tundish on multi - physics ﬁ eld. Computational ﬂ uid dynamics has been used to study the in ﬂ uence of the structure of the tundish on the macroscopic transport behavior of the tundish with channel induction heating. The results show that increasing the depth of the molten pool is conducive to dynamic behavior of multiphase, the deeper the molten pool, the larger the active area, the longer residence time, the more inclusions removal and the higher ratio of plug to dead volume. Meanwhile, the larger the channel diameter, the more inclusions removal in the receiving chamber and channel. The channel induction heating has enough ability to increase the superheat and temperature compensate for the heat loss caused by the excessive residence time of the molten steel in the tundish. The change in the channel struc - ture is crucial to the macroscopic transport behavior of the ﬂ uid. The change in channel diameter has the greatest e ﬀ ect on the multi - physics ﬁ eld in the molten pool.


Introduction
Steel cleanliness, strict composition and suitable superheat are becoming the primary concerns of steelmakers.The tundish is basically an intermediate vessel placed between the ladle and the mold and as the last metallurgical vessel before solidifying in the continuous casting mold [1][2][3].Function of a tundish include delivery of clean molten steel to different strands at required needs.Therefore, a modern continuous casting tundish is furnished with flow control devices for carrying out various metallurgical operations, inclusion collision-coalescence, floatation, alloy trimming, induction heating, superheat control, thermal and particulate homogenization [4][5][6][7][8].Many investigations have been carried out on the abovementioned metallurgical process in the tundish by physical experiments, industrial experiment and mathematical modeling [9][10][11][12][13].However, the tundish with channel-type induction heating is a unique style of metallurgical container which is a recent novel technology equipped with heating function and electromagnetic force control units to produce improved quality casting strands and their hot rolling products.The studies and applications regarding this issue are presented with special attention for the understanding, new findings, and suggestions for the novel tundish technology for its applications and further improved steel quality.
At present, the actual industrial production of the channel-type induction heating tundish in China has not been widely promoted, and only a few steel mills are currently in trial operation.The function of the tundish is to improve the superheat and inclusion removal, and the previous studies of the tundish was focus on electromagnetic field, flow field, temperature field and inclusion field [14][15][16][17][18][19][20][21][22].But the effect on macroscopic transport behavior of the tundish structure is rarely reported [13][14][15][19][20][21][22], in order to provide the mold with molten steel whose composition and temperature meet the requirements.
Electromagnetic conditions [22], fluid flow [20], heat transfer [20], flow characteristics [21], and inclusion collision coalescence and removal process and kinetic models [17] have been studied and explored in detail in previous studies by the author.In the present work, three sets of multi-physics field were developed to show the macroscopic dynamics behavior in the tundish with channel-type induction heating, with molten pools of different depths, channel diameter and induction heating power.The fluid flow, temperature compensate, residence time distribution and inclusion spatial distribution were generated by performing the CFD simulation to analyze and investigate the desired characteristics for transport behavior in the tundish with channel induction heating.The objective of the current work is to compare and analyze the macroscopic dynamics transport behavior of multiple physical fields developed by numerical simulation.

Model development 2.1 Physical model description
Figure 1 shows the mesh, front view and top view of tundish.Simulation was performed for three important parameters that have a great influence on flow behavior for molten steel, the depth of the molten pool, diameter of the channel and induction heating power.
Table 1 shows the three sets of the tundish with channel induction heating.Case 1 to case 4 is the first group, case 5 to case 9 is the second group and case 10 to case 13 is the third group.

Assumptions
The main assumptions for the analysis of the collisioncoalescence of inclusions in molten steel are: (1) The inclusions are spherical.
(2) The effect of inclusion on the macroscopic flow morphology of molten steel is ignored.(3) The inclusion phase can be treated as continuous.(4) An inclusion is removed once it touches the slag layer of tundish, tundish wall, or channel wall.(5) The molten steel is an incompressible Newtonian fluid.(6) The flow of molten steel in the tundish is at the steady state.(7) The physical parameter of tracer is the same with molten steel in the tundish.

Maxwell's equations
In the region of molten steel, the following subset of Maxwell's equations applies: In the other region, which contains source current → J s , the equations relating the various physical quantities are constituted by the following subset of Maxwell's equations:

Fluid flow
In the case of the channel-type induction heating, the momentum conservation equations should consider the electromagnetic force → × → J B on the fluid flow:

Inclusion collision-coalescence model
The inclusion model which is verified by the industrial experiment [23] is applied to describe the inclusion collision-aggregation in the tundish: S N is related to the inclusion collision-coalescence in the molten steel [22].
3 is the characteristic inclusion radius.The constants of the − k ε two-equation turbulence model are taken as: The inclusion collision-coalescence model is applied to all types of inclusion, the main component of the particles is Al 2 O 3 , the aggregation of particles is mainly related to the density and radius size of inclusions, and therefore, the model is able to predict the spatial distribution of the vast majority of inclusions in the metallurgical vessel.
In this article, the classic combined model [23] is applied to calculate the plug zone (V pv ), the well mixed zone (V mv ) and the dead zone (V dv ) of the tundish.

Boundary conditions
Electromagnetic field: flux parallel boundary condition is imposed on the exterior surface of the computational domain which is discretized by using 500,000 nonuniform tetrahedral grids [22].
Flow field: four types of boundaries enclose the domain: the inlet, the outlet, free surface and the solid wall.The wallfunction method is applied near the wall [20,21].
Temperature field: here also there are four types of boundaries, the inlet, the outlet, free surface and the tundish wall [20][21][22][23].
Tracer field: the tracer concentration is considered to be impervious for tundish wall [21], the other boundary conditions of inclusion can be found in Table 2.
Inclusion field: two mechanisms for inclusion to move toward the free surface and the tundish wall: diffusion and convection [17].
The other boundary conditions of inclusion can be found in Table 3.

Grid system, convergence criteria and numerical solution
The tundish domain is discretized by using a nonuniform grid system which encloses the computational domain.
The number of grids is about 300,000.ANSYS CFX software (version 11.0, ANSYS, Pittsburgh, PA, USA, 2008) is applied to solve these partial differential equations in order to obtain the multi-physics field, the convective discrete scheme adopts first-order upwind style, the convergence criteria is that the normalized residual for variables should be less than 10 −5 .
The calculation procedure can be described as follows: (1) Finite element method is applied to solve the Maxwell's equation.
(2) A Fortran program is applied to do the interpolation of electromagnetic body force per unit and Joule heat power density from ANSYS to CFX. (3) Finite volume method is applied to solve the mass/ momentum/energy equations, tracer transport equation and the k − ε turbulence equation.(4) Finite volume method is applied to solve the inclusion mass conservation and number conservation equations.
Structure optimization of continuous casting tundish  463 3 Results and discussion

Model validation
The mathematical model of flow field, temperature field, electromagnetic force, inclusion field and solute field has been verified in previous papers [17,[20][21][22], hence, no need to elaborate here.

Comparison of performance of tundish at different molten pool depth
The plant has operated at different molten pool depth.It was expected that the results can be altered due to the effect of change in the molten pool depth.
Table 4 shows some noticeable information: (1) The effective volume increases gradually with the increase in the molten pool depth of receiving chamber within a limited range, from 0.875 to 0.884, an increment of 0.9%, the reason for this can be attributed to the long mean residence time of molten steel, from 1812.48 to 2011.71 s.Meanwhile, long mean residence time is conducive to decrease dead volume fraction, from 12.49 to 11.58%, a decrease of 0.91% and increase the high ratio of plug to dead volume, from 0.400 to 0.415, an increase of 1.5%.Therefore, the investigations have shown that if the molten pool depth of receiving chamber increased, then the hydrodynamic conditions in the tundish are significantly improved.(2) The longer mean residence time of molten steel could lead to a temperature decrease, comparing case 1 and case 4, only a decrease of 0.5℃ due to the constant Joule heat, the heat loss mainly comes from the free surface, side wall and channel wall of tundish.If there is no external heat source, the temperature drop will be increased for the tundish with large capacity.Therefore, the channel induction heating is essential to produce the high-quality steel in continuous casting tundish.To sum up, the thermal effect can effectively compensate for the heat loss caused by the excessive residence time of the molten steel in the tundish.
The inclusion removal rate and characteristic inclusion radius at outlet can be seen from Figure 2. The result shows that a deeper molten pool depth can lead to a slight improvement in inclusions removal, from 42.0 to  43.9%, and the size of the inclusions slightly increased from 3.64 to 3.85 µm, due to the longer mean residence time for large capacity tundish that can be seen in Table 4. Therefore, increasing the depth of the molten pool can increase the inclusions removal and the collisions probability of inclusions.And if the heat loss is less than guaranteed, extending the residence time of molten steel as much as possible is beneficial for continuous casting.

Comparison of performance of tundish at different diameters of channel
Table 5 shows the fluid flow characteristics obtained by performing the RTD analysis.The flow characteristics of the fluid do not show regular pattern due to the small change in tundish volume.Therefore, the inclusion field could act as an assessment criterion to evaluate the effect of channel diameter on the fluid flow, heat transfer and inclusion spatial distribution in the tundish with channeltype induction heating.
Figure 3 shows some interesting regularity phenomena which is characterized as follows.(1) In the receiving chamber, the larger the channel diameter, the more the inclusion removal, from 10.74 to 26.22%, an increase of 15.48%, a small change in diameter can lead to a great inclusion removal.The reason for this can be attributed to the fluid flow characteristics in the tundish, as shown in Figure 4.It can be seen that the larger the channel diameter, the greater the macro mixing in the tundish.When d = 100 mm, there is recirculation zone in the receiving chamber and shows a gentle clockwise flow phenomena.As the diameter increases, the flow characteristics change, when d = 150 mm, there are two small recirculation zones in the receiving chamber, the two small recirculation zones can increase the collision chance of particles.When d = 200 mm, the recirculation zone becomes larger compared to that when d = 100 mm, it is beneficial to mix for molten steel.When d = 250 mm, an enormous recirculation zone is formed in the receiving chamber which leads to inclusion collision-coalescence and grow up.When d = 300 mm, a clockwise flow around the recirculation zone with a very regular and long streamline will increase the inclusion  Structure optimization of continuous casting tundish  465 collision chance and the inclusion removal by slag layer and tundish wall.
(2) In the channel, the order of particle removal is from least to most: Case 5 (9.28%) < Case 6 (14.90%) < Case 7 (23.57%)< Case 8 (24.95%) < Case 9 (25.18%),Case 9 has the best inclusion removal compared to other cases.Therefore, the larger the channel diameter, the more the inclusion removal, from 9.28 to 25.18%, an increase by 15.9%.The reason for this can be attributed to flow characteristic in the channel as shown in Figure 4.
With the increase in the channel diameter, the flow from chaos to orderly, and the backflow phenomena occurs in the channel, it is beneficial for inclusion collision-aggregation and adsorption by channel wall.From the above data, it can be inferred that when the channel diameter comes to a certain extent, the inclusion removal has reached a limited.Therefore, it is crucial to choose a suitable channel diameter for tundish with channel type.(3) In the discharging chamber, the order of particle removal is from least to most: Case 9 (2.21%) < Case 8 (3.16%) < Case 7 (5.6%)< Case 6 (11.17%) < Case 5 (15.04%), the larger the channel diameter, the lesser the inclusion removal, from 15.04 to 2.21%, this is because the inclusion collision-aggregation and grow up occurs in the receiving chamber and channel for large size channel diameter of tundish due to the superior flow behavior; however, the inclusion collision-aggregation and grow up occurs in the discharging chamber for small size diameter of tundish channel induction heating, therefore the channel is one of the important place to remove inclusions.(4) The larger the channel diameter, the smaller the characteristic inclusion radius at outlet of tundish and larger the characteristic inclusion radius in channel outlet due to the complex fluid behavior.

Comparison of performance of tundish at different induction heating powers
Figure 5 shows that the greater the induction heating power, the more the inclusion removal in the tundish, due to the strong electromagnetic force.The inclusion removal rate increases from 19.7 to 62.7%, an increase by 43%, but the excessive electromagnetic force can cause negative effect, leading to a discontinuous flow phenomena in the tundish.Therefore, in the actual industrial production, the appropriate induction heating power should be selected reasonably according to the production needs.The order of the characteristic inclusion radius is from small to large: Case 5 (3.16 µm) < Case 6 (3.59 µm) < Case 7 (3.77µm) < Case 8 (3.89 µm), Case 8 has the largest characteristic inclusion radius compared to Case 5 to Case 7, this is because the higher induction heating power lead to a great electromagnetic force and a great turbulent kinetic energy, so the small inclusions have more chance to collision-coalescence and float to slag layer.

Conclusion
The mathematical model was developed to investigate the optimum structure for multi-physics field in the present work.The following conclusions were presented from the study: (1) The deeper the molten pool depth, the more effective the volume, the longer the residence time of molten steel, the more the inclusion removal rate and higher the ratio of plug to dead volume.
(2) The larger the channel diameter, the greater the macro mixing effect, the greater the inclusion removal rate and the smaller the characteristic inclusion radius at outlet of tundish.(3) Receiving chamber and channel facilitate inclusion removal, and the lager channel is beneficial for inclusion grow up in the receiving chamber.(4) The greater the induction heating power, the higher the temperature compensate, the more the inclusion removal rate and the smaller the characteristic inclusion radius at outlet of tundish.

Figure 1 :
Figure 1: Schematic of tundish with channel-type induction heating (all lengths are in mm).

Figure 2 :
Figure 2: Inclusion removal rate and characteristic inclusion radius of tundish with different molten pool depths.

Figure 3 :
Figure 3: The inclusion removal rate in chambers and characteristic inclusion radius distribution at different positions for different tundishes.
the injected tracer C p heat capacity D 0 molecular diffusion coefficient → E electric field strength F C c transport flux of inclusion volume concentration F N c Transport flux of inclusion number density F C d diffusion flux of inclusion volume concentration F N d diffusion flux of inclusion

Figure 5 :
Figure 5: Inclusion removal rate and characteristic inclusion radius of tundish with different induction heating.

Table 1 :
Simulation performed for following cases

Table 2 :
Boundary condition for tracer transport

Table 4 :
Analysis results for the residence time distribution (RTD) curve and temperature value for cases Note: V 1 represents the volume of receiving chamber and V 3 represents the volume of discharging chamber.

Table 5 :
Analysis results for the RTD curve for case 5 to case 9