Numerical Simulation of Residence Time Distribution (RTD) in Tundish with Channel Type Induction Heating

Abstract

Numerical simulation is an effective tool to analyze complex metallurgical multiphase flow problems especially when the high-temperature melt experiment cannot be carried out. Therefore, a three-dimensional mathematical model has been applied to investigate the fluid flow characteristic of molten steel in a tundish with channel type induction heating. The results show that the thermal buoyancy can increase the mean residence time of molten steel and decrease the dead zone fraction in the tundish with channel induction heating, although the electromagnetic force can increase the dead zone fraction, but it also increases the mixed zone fraction; the mixing of melt benefits inclusion collision-coalescence and removal. The electromagnetic force also can stir the molten steel, completely mixing it in a short time. The main flow behavior is well mixed flow; the secondary flow is plug flow in the tundish with channel-type induction heating.

Introduction

The tundish is an intermediate reactor between the ladle and the mold, used to become a multifunction vessel.^1,2,3,4 Its metallurgical operations such as inclusion removal,^{5,6,7,8,9,10,11,12} temperature compensation,^{13,14,15,16,17} alloy trimming,^1,18 and thermal and particulate homogenization are important for clean molten steel. The main currents of the presented studies concern physical experiments and numerical simulation of the melt flow phenomena and refining processes.¹⁹

For the conventional tundish, inclusion removal rate can be an indispensable evaluation target to judge the quality of molten steel, its collision-coalescence, growth, and removal process depending on the flow condition of molten steel. Research shows that fluid flow characteristics dominate the particle motion, heat transfer, and homogenization in most heteromorphic tundish. Therefore, it is necessary to analyze its fluid-dynamic behavior and thus make a suitable design or optimize the control devices (dams, baffles, weirs, turbulence inhibitors, gas curtains, etc.) to obtain the most favorable conditions to produce high-quality clean steel.^{20,21,22,23,24,25} A commonly used ideal in improving the use of internal flow space of steel of a tundish is also to combine the different flow control devices.^{20,21,22,23,24,25} Therefore, the design, operating parameters, source fields (bubble or electromagnetic force), and macroscopic transport behavior of physical systems must be optimized to ensure sufficient residence time and to produce suitable metal flow patterns.^26,27,28 Based on the above, the molten steel flow in the tundish has been widely studied by a concept residence time distribution (RTD);^29,30,31 research output in this field can be found in previous reports or literature and is methodologically diverse. RTD in physical experiments or numerical simulation is used to characterize the fluid flow state.

For the conventional tundish, control flow devices can change the melt flow effectively. Its design and optimization is relatively easy because of the vast interior space of tundish. The melt flow condition is easy to obtain through the RTD. Then, the dead zone, plug zone, and well-mixed zone in a given tundish can be readily estimated in detail. The RTD characteristics are applied to compare tundish with diverse geometric shapes, control flow equipment, and working operations. The details of flow can be obtained from a whole tundish. Generally speaking,, an appropriate tracer is injected into the fluid from the ladle shroud, and the concentration of added tracer is monitored dynamically at the exit nozzle of the tundish.¹

At present, the function of the tundish with channel induction heating is to improve the superheat and inclusion removal. Previous studies of the tundish focused on the electromagnetic, flow, temperature, and inclusion fields. It is necessary to analyze and investigate the desired characteristics for transport behavior in the tundish with channel induction heating. In the current work, an attempt was made to use numerical simulation to unveil the flow characteristics through the RTD curves occurring in a tundish with channel induction heating. Investigation of the tundish with electromagnetic force and Joule heat will be different from the optimization or comparison for conventional tundish. Three important points in this paper are: first to investigate the RTD curve of tundish with channel-type induction heating, second to investigate the flow characteristic in the tundish with Joule heat and electromagnetic force, and third to determine the tracer transport behavior in the tundish with channel-type induction heating. The numerical simulations made it possible to evaluate the macroscopic mixed flow of molten steel in the case without induction heating, Joule heat, electromagnetic force, and induction heating.

Geometrical Model

Figure 1 and Table I show the more specific details of structure, size, and operating parameters of the tundish with channel-type induction heating.^11,13 The channel plays two important key roles in the tundish. The first function is to heat the molten steel by Joule heat, which is caused by induced current. The second function is to stir the molten steel by electromagnetic force, which is caused by the interaction between the inducted current field and magnetic field.¹⁴

Fig. 1

Schematic of tundish with channel-type induction heating (all dimensions are in mm).

Table I Operating, geometry, and physical parameters of computational models

Mathematic Models

The electromagnetic field, fluid flow, heat transfer, and tracer transport in the tundish can be described as the following equations.^11,13

Magnetic field constitutive equation:

$$ \nabla \left( {\frac{1}{\mu }\nabla \times \vec{A}} \right) + \sigma \left( {\frac{{\partial \vec{A}}}{\partial t} + \nabla \varphi } \right) = 0\quad {\text{in molten steel region}} $$

(1)

$$ \nabla \times \left( {\frac{1}{\mu }\nabla \times \vec{A}} \right) = \mathop {\vec{J}}\nolimits_{{\text{s}}} \quad {\text{in other regions}} $$

(2)

$$ \nabla \cdot \vec{A} = 0 $$

(3)

Continuity equation:

$$ \frac{{\partial \rho_{f} }}{\partial t} + \nabla \cdot (\rho_{f} \vec{u}_{f} ) = 0 $$

(4)

Momentum equation:

$$ \begin{aligned} \frac{{\partial \left( {\rho_{f} \vec{u}_{f} } \right)}}{\partial t} + \nabla \cdot \left( {\rho_{f} \mathop{u}\limits^{\rightharpoonup} _{f} \mathop{u}\limits^{\rightharpoonup} _{f} } \right) &= - \nabla P + \rho_{f} \mathop{g}\limits^{\rightharpoonup} + \beta (T_{0} - T)\rho_{f} \vec{g} \, \\ & \quad + \nabla \cdot \left[ {\mu_{{{\text{eff}}}} \left( {\nabla \mathop{u}\limits^{\rightharpoonup} _{f} + \left( {\nabla \mathop{u}\limits^{\rightharpoonup} _{f} } \right)^{T} } \right)} \right] + \vec{J} \times \vec{B} \\ \end{aligned} $$

(5)

Turbulent kinetic energy and rate of dissipation:

$$ \frac{{\partial (\rho_{f} \kappa )}}{\partial t} + \frac{{\partial \left( {\rho_{f} \kappa \vec{u}_{f} } \right)}}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{u_{t} }}{{\sigma_{k} }}} \right)\frac{\partial \kappa }{{\partial x_{j} }}} \right] + G_{k} - \rho_{f} \varepsilon $$

(6)

$$ \frac{{\partial (\rho_{f} \varepsilon )}}{\partial t} + \frac{{\partial (\rho_{f} \varepsilon u_{j} )}}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{u_{t} }}{{\sigma_{\varepsilon } }}} \right)\frac{\partial \varepsilon }{{\partial x_{j} }}} \right] + \frac{{C_{1\varepsilon } \varepsilon }}{\kappa }G_{\kappa } - C_{2\varepsilon } \rho_{f} \frac{{\varepsilon^{2} }}{\kappa } $$

(7)

Energy conservation equation:

$$ \frac{{\partial \left( {\rho c_{P} T} \right)}}{\partial t} + \nabla \cdot \left( {\rho_{f} T\mathop{u}\limits^{\rightharpoonup} } \right) = \nabla \cdot \left( {\lambda \nabla T} \right) + \sigma J^{2} $$

(8)

Tracer transport equation:

$$ \frac{{\partial \left( {\rho_{f} C} \right)}}{\partial t} + \nabla \cdot \left( {\rho_{f} \mathop{u}\limits^{\rightharpoonup} _{f} C} \right) = \nabla \cdot \left( {D_{{{\text{eff}}}} .\nabla C} \right) $$

(9)

where $\mathop{A}\limits^{\rightharpoonup}$ is the magnetic vector potential, $\phi$ is the electric scalar potential, $\sigma$ is the electrical conductivity, $\vec{J}_{s}$ is the source current, $\mu$ is the magnetic conductivity, $\mu_{{{\text{eff}}}}$ is the effective viscosity, $\rho_{f}$ is the fluid density, $\vec{u}_{f}$ is the fluid velocity, P is the pressure, $\vec{g}$ is the gravitational accelerate, $\kappa$ is the turbulence kinetic energy, $\varepsilon$ is the turbulent kinetic energy rate, $\sigma_{\kappa }$ and $\sigma_{\varepsilon }$ represent the Schmidt number for $\kappa$ and $\varepsilon$, $G_{k}$ is the generation rate of turbulence energy, $C_{p}$ is the heat capacity, T is the temperature, and C is the mass fraction of the injected tracer.

The most widely used analysis models of the RTD (residence time distribution) curve are the classic combined model and the Sahai modified model at present.³² By comparing the two models, the classic combined model is applied to calculate the plug, the well-mixed, and the dead zones of the tundish.

The classic combined model is applied to calculate the plug zone fraction ($V_{{{\text{pv}}}}$), the well-mixed zone fraction ($V_{{{\text{mv}}}}$), and the dead zone fraction ($V_{{{\text{dv}}}}$) of the tundish. The C curve is applied to obtain the volume fraction; the calculation formula is

$$ V_{{{\text{dv}}}} = 1 - \frac{{t_{{{\text{av}}}} }}{{t_{s} }} $$

(10)

$$ V_{{{\text{pv}}}} = \frac{{t_{\min } + t_{\max } }}{{2t_{0} }} $$

(11)

$$ V_{{{\text{mv}}}} = 1 - V_{{{\text{dv}}}} - V_{{{\text{pv}}}} $$

(12)

where the $t_{{{\text{av}}}}$ is the mean residence time, $t_{0}$ is the theoretical residence time, $t_{\min }$ is the minimum residence time, $t_{\max }$ is the peak concentration time, $V_{{{\text{mv}}}}$ is the volume fraction of well-mixed zone, $V_{{{\text{dv}}}}$ is the volume fraction of dead zone, and $V_{{{\text{pv}}}}$ is the volume fraction of plug zone.

Model Assumptions

• (1) The iron core is isotropic, and its electromagnetic properties are constant.

• (2) The effect of fluid flow on the electromagnetic field is too weak to be neglected.

• (3) The fluid flow in the tundish is steady and incompressible turbulent flow.

• (4) There is no chemical reaction in the tundish.

• (5) The tracer has the same physical parameter as the fluid in the tundish.

• (6) The free surface is flat in the tundish.

Boundary Conditions

Electromagnetic field and flux parallel boundary conditions are imposed on the exterior surface of the computational domain. The low field, with four types of boundaries, encloses the domain: the inlet, outlet, free surface, and solid wall. The symmetry condition is applied at the free surface, and the wall function method is applied near the wall. For the temperature field, there are also four types of boundaries, the inlet, outlet, free surface, and tundish wall. Table II shows the heat flux at the free surface and the tundish walls. For the tracer transfer, a zero gradient or flux boundary condition was applied for the tracer concentration on the walls, free surface, and outlets for the tracer dispersion equation.

Table II Heat losses from the tundish

Numerical Solution and Convergence Criteria

The governing equations are solved by the commercial software package CFX and ANSYS-EMAG module (version 11.0, ANSYS, Pittsburgh, PA, USA), which is based on the finite volume method and finite element method. The calculation domain is covered by about 300,000 hexahedral grids. The fluid flow model and the heat transport model are coupled to obtain the flow field and the temperature field, based on the known flow field; the tracer transport equation is solved to obtain the RTD curve. During iteration, the convergence was assumed to reach a point where all the normalized residuals are < 10⁻⁵.

Results and Discussion

Grid Independence

The grid system consists of 196,156, 248,300, 300,318, 324,848, and 350,580 grids, which are applied to ensure the accuracy and reliability of the numerical result. Table III gives the related analysis result of test point K. From the mesh 196,156 to 350,580, the variation of velocity value is on the third digit (the maximum related error is only 1.12%). Therefore, the 300,318 mesh can be applied to obtain the RTD curves.

Table III Analysis result of velocity for different grid numbers

Model Validation

Figure 2 and Table IV give the RTD curve and the fluid flow characteristic in the water model and simulation calculation. The measured values of volume fraction are 33.72%, 9.48%, and 56.8%, and the predicted results are 33.05%, 9.55%, and 57.40%. The related errors are only 2.03%, 0.75%, and 1.78%, respectively, between the predicted and measured results; the relative error is < 5%, and the predicted result agrees with the experimental data of the water model. The difference between the predicted curve and the experimental curve comes from several factors. (1) In the experiment, the tracer has solute buoyancy in water, but the mathematical model ignores the solute. (2) The fluid flow in the water model is turbulent, which is anisotropic, while in the mathematical model, the fluid flow is isotropic fluid. (3) The oscillating free surface in the water model has an impact on curves, while it is assumed to be flat in the mathematical model.

Fig. 2

RTD curve in the tundish for water model and numerical simulation.

Table IV Analysis results in the tundish for water model and numerical simulation

Analysis of RTD

Electromagnetic force and Joule heat are two key characteristics of a tundish with channel type induction heating. Therefore, the influence of electromagnetic force and Joule heat on the macroscopic mixed flow of fluid needs to be deeply studied.

To determine the details about the effect of the source field (electromagnetic field and Joule heat field) on RTD (residence time distribution) in a tundish with channel induction heating, the calculation is divided into four cases, shown in Table V. Case 1 is the tundish without channel induction heating. Case 2 needs to solve the energy equation with Joule heat. Case 3 needs to solve the momentum equation with electromagnetic force, and the Case 4 needs to solve the energy equation with Joule and the momentum equation with electromagnetic force.

Table V Four cases in the numerical simulation

To describe the spatial distribution of fluid flow and tracer concentration in the tundish clearly, some characteristic cross sections need to be defined, as shown in Fig. 3. Section A-A and section B-B pass through the center axis of the channel.

Fig. 3

Characteristic sections in the tundish.

Figure 4 gives the RTD curve in the tundish with channel induction heating under four conditions (without induction heating, Joule heat, electromagnetic force, and induction heating). The curve shows some interesting features. (1) In the case without induction heating and Joule heat, the double peak phenomena appeared in the RTD curve. This is due to the obvious presence of the double-vortex phenomenon in the receiving chamber, as shown in the Fig. 5a. (2) In the case of electromagnetic force and channel induction heating, the double-peak phenomena do not appear in the RTD curve. This is because there is a big and a small recirculation zone in the receiving chamber, but the smaller counterclockwise vortex has little influence on tracer transport at the lower left corner, and the larger counterclockwise recirculation zone leads to a homogeneous mixing of the tracer, as shown in Fig. 5b. The two RTD curves basically coincide in shape; this is because the electromagnetic force has a larger driving force than Joule heat. As a result, the Joule heat can be ignored under the strong electromagnetic force. (3) The sequence of the peak concentration from small to large for the different cases is: Case 4 $\to$ Case 3 $\to$ Case 1 $\to$ Case 2. This is because the electromagnetic force increases the mixed effect of molten steel in the whole tundish; meanwhile, the rapid flow of molten steel in the tundish leads to premature peak concentration. In the case of Joule heat, due to the high temperature, molten steel flows from the channel to receiving chamber; the fluid flows upward and extends the flow trajectory of fluid in the receiving chamber. Therefore, the residence time of molten steel increases greatly in the receiving chamber.

Fig. 4

RTD curves in the tundish for four cases.

Fig. 5

Outlet cross section of tundish in the receiving chamber.

Table VI gives the flow characteristic of molten steel for the different types of cases. It shows that:

• (1) The sequence of the mean residence time of molten steel from small to large for the different cases is: Case 4 $\to$ Case 3 $\to$ Case 1 $\to$ Case 2 (1812.1 s < 1813.7 s < 1850.1 s < 1904.9 s). Case 2 demonstrates a longer mean residence time compared to Case 1, 3 and 4; this is because there are two recirculation zones in the discharging chamber. One small recirculation is on the left side and another large recirculation is in the middle area, and there is a large recirculation zone in the lower region in the receiving chamber. The three recirculation zones and velocity of molten steel with no speed difference from the tundish without channel induction heating lead to a longer mean residence time, as shown in Fig. 6a and b. In the case of electromagnetic force and channel induction heating, the residence time of molten steel is shorter. This is because that the velocity of molten steel with electromagnetic force is much greater than that without electromagnetic force. The molten steel flows out of the outlet quickly, as shown in Fig. 6c and d. Therefore, according to the above, the electromagnetic force will shorten the residence time of molten steel in the tundish with channel induction heating.

• (2) The sequence of the dead zone fraction from small to large for the different cases are: Case 2 $\to$ Case 1 $\to$ Case 3 = Case 4 (8.0% < 10.6% < 12.4% = 12.4%). Case 3 and Case 4 have longer mean residence time compared to Cases 1 and 2; this is because the electromagnetic force leads to an increasing velocity of molten steel and short residence time of molten steel, and the dead zone is determined by the residence time. On the contrary, the Joule heat leads to a longer residence time of molten steel, resulting in an absolutely natural smaller dead zone.

• (3) The sequence of the plug zone fraction from small to large for the different cases are: Case 2 $\to$ Case 1 $\to$ Case 3 = Case 4 (60.0% < 61.8% < 62.6% = 62.6%). Case 3 and Case 4 have longer mean residence time compared to Cases 1 and 2. This is because the magnitude of electromagnetic force is on the order of ${10}^{{5}}$ N/m³, which is greater than the gravity of molten steel ${7} \times {10}^{{4}}$ N/m³, and promotes better stirring of the molten steel by electromagnetic force.¹¹ Although the Joule heat extends the flow trajectory of molten steel, it decreases the recirculation zone and the mixing effect in the receiving chamber, as shown in Fig. 6b.

• (4) The sequence of the minimum residence time of tracer from small to large for the different cases is: Case 4 = Case 3 $\to$ Case 1 $\to$ Case 2 (44 s = 44 s < 76 s < 131 s). The minimum residence time of Case 2 is the largest compared to Cases 1, 3 and 4. The peak concentration of tracer from small to large for different cases are: Case 4 $\to$ Case 3 $\to$ Case 1 $\to$ Case 2 (990 s < 991 s < 1065 s < 1194 s). Two physical quantities conform to the same rule; the electromagnetic force and Joule heat show a different effect on molten steel. Both sources can change the flow behavior, but the former mainly focuses on stirring and the latter mainly on temperature compensation.

Table VI Fluid flow characteristics in the tundish with channel induction heating for different type of cases

Fig. 6

Velocity field in the tundish at A-A section.

Conclusion

The following conclusions are summarized as follows:

• (1) The predicted RTD curve agrees well with the water model.

• (2) The mean residence times of molten steel of 1904.9 s in the tundish with Joule heat are 54.8 s, 91.2 s, and 92.8 s greater than the mean residence time of 1850.1 s, 1813.7 s, and 1812.1 s in the tundish without induction heating, with electromagnetic force and induction heating, respectively.

• (3) The thermal buoyancy can increase the mean residence time of molten steel and decrease the dead zone fraction in the tundish with channel type.

• (4) The electromagnetic force increases the dead zone fraction, but it increases the mixed zone fraction. The mixing of molten steel benefits inclusion collision-coalescence and growth.

• (5) The main flow behavior is well mixed flow; the secondary flow is plug flow in the tundish with channel type induction heating.