Fluid Flow and Heat Transfer Behaviors under Non-Isothermal Conditions in a Four-Strand Tundish

Abstract

In the continuous casting process, the fluid flow of molten steel in the tundish is in a non-isothermal state. Because of the geometric shape and process parameters of a multi-strand tundish, the fluid flow behavior of each strand is quite inhomogeneous, and the difference in temperature, composition and inclusion content between each strand is great, which directly affects the quality of the steel products. In this paper, the fluid flow, heat transfer phenomena and inclusion trajectories in a four-strand tundish with and without flow-control devices (FCDs) are investigated using a water model and numerical simulation in isothermal and non-isothermal conditions. The results show that natural convection has a significant influence on the flow pattern and temperature distributions of molten steel in the tundish. Without FCDs, the average residence times of the molten steel in the tundish obtained by the isothermal water model, non-isothermal water model and non-isothermal mathematical model were 251.2 s, 263.3 s and 266.0 s, respectively, and the dead zone volumes were 21.51%, 29.26% and 28.21%, respectively. With FCDs, the average residence times of the molten steel obtained by the isothermal water model, non-isothermal water model and non-isothermal mathematical model were 293.0 s, 304.0 s and 305.2 s, respectively, and the dead zone volumes were 43.98%, 50.23% and 52.78%, respectively. The flow characteristics of the molten steel in the tundish were different between the isothermal and non-isothermal conditions. Compared with isothermal conditions, the numerical simulation results were closer to the water model results in non-isothermal conditions. The trial results showed that the fluid flow in a tundish has a non-isothermal characteristic, and the results in non-isothermal conditions can better reflect the actual fluid flow and heat transfer behaviors of molten steel in a tundish.

1. Introduction

The fluid flow behavior of molten steel is a basic phenomenon in a tundish. The fluid flow and heat transfer behaviors of molten steel in a tundish have an important effect on the cleanliness, operability and productivity of steel products. For a multi-strand tundish in particular, because of its geometric shape and process parameters, the fluid flow behavior of each strand is quite inhomogeneous and the difference in temperature, composition and inclusion content between each strand is great. The fluid flow is in a non-isothermal state, and the natural convection caused by the temperature difference in a multi-strand tundish has a significant impact on the flow field, temperature distribution and inclusion trajectories of the molten steel. Therefore, it is very important to systematically study the fluid flow and heat transfer behaviors in non-isothermal conditions.

Physical and numerical simulations of a tundish have been extensively reported. However, most previous studies have been conducted under isothermal conditions [1,2,3,4,5]. In a study on the influence of the temperature compensation caused by the external field on the flow behavior of molten steel in a tundish, the effect of temperature differences on the fluid flow was generally considered, such as in the investigation of the flow behavior using induction heating [6,7,8,9]. The effect of temperature differences on the flow behavior of molten steel cannot be ignored in the actual production process. Natural convection and forced convection together control the flow behavior of molten steel in a tundish. Many scholars have noticed this phenomenon and carried out several research works on it. Sheng et al. [10,11,12] studied the flow phenomena in non-isothermal conditions by means of physical and numerical simulations and found that the fluid flow in a single-strand tundish in non-isothermal conditions was very different from that in isothermal conditions. This research showed that the flow pattern of molten steel was jointly controlled by natural convection and forced convection. Even if the temperature difference of hot water in a tundish is only 1 K, the fluid flow state is significantly different from that of ambient-temperature water. The thermal buoyancy has a significant effect on the flow pattern and temperature distribution of molten steel in a tundish. Sun et al. [13] studied the effect of thermal buoyancy on the fluid flow, temperature distribution and residence time distribution (RTD) of molten steel in a single-strand slab tundish, and the results showed that the RTD parameters were very different in the isothermal and non-isothermal water models. Singh et al. [14] optimized the FCDs in a single-strand tundish, and the results showed that the use of a non-isothermal model is important, as it was found that even a small change in temperature played a vital role in the fluid flow inside the tundish. Alizadeh et al. [15] studied the fluid flow and mixing of molten steel in a twin-strand continuous-casting slab tundish in non-isothermal conditions. The results showed that the RTD parameters were completely different in isothermal and non-isothermal conditions. The extent of mixing in non-isothermal conditions was lower than that in isothermal conditions. Cwudzinski et al. [16,17] studied the flow phenomena of molten steel in a single-strand tundish in isothermal and non-isothermal conditions. State-of-the-art vector flow field analysis measuring systems developed by Lavision were used in the laboratory tests. In order to explain the phenomena occurring in the tundish working space, the buoyancy number (Bu) was calculated. Wen et al. [18] investigated the flow behavior in a six-strand bloom caster tundish. The results showed that, in a non-isothermal process, the fluid flow in the tundish presented a strong buoyancy pattern, which drove particles to move upwards. Cwudzinski et al. [19] simulated the RTD curves and the flow patterns in a six-strand tundish for both isothermal and non-isothermal conditions by a numerical model. Chattopadhyay et al. [20] studied the flow phenomena of molten steel for a four-strand billet tundish using water and mathematical models in non-isothermal conditions. The results showed that while the step-up conditions facilitated the flotation of inclusions because of upward buoyancy-driven flows, the step-down conditions generated catastrophic results in terms of molten metal quality. Chatterjee et al. [21,22] simulated the effect of non-isothermal conditions on melt flows in a multi-strand billet caster tundish using numerical and water models. The results showed that the fluid flow patterns changed significantly, and the temperature distribution and inclusion trajectories within the tundish were also affected due to the presence of non-isothermal conditions. Buoyancy effects were seen to dominate in regions away from the ladle shroud.

From the above research, it can be seen that most studies took a single-strand or twin-strand tundish as the research object, and there have been few studies on a multi-strand tundish in non-isothermal conditions. Because the temperature differences between each strand are great, and the fluid flow behavior of each strand is quite inhomogeneous in a multi-strand tundish, the natural convection driven by temperature differences caused by buoyancy has a greater influence on the flow behavior of molten steel. In the past, there were few studies that systematically studied fluid flow, heat transfer and inclusion trajectories in a multi-strand tundish using an isothermal water model, non-isothermal water model and non-isothermal mathematical model and compared the experimental or calculated results with the actual flow behavior of molten steel in a trial. In this paper, the fluid flow, heat transfer and inclusion trajectories in a four-strand tundish with and without FCDs are investigated by physical and mathematical simulations in isothermal and non-isothermal conditions. The effect of the isothermal or non-isothermal conditions and the inclusion/exclusion of FCDs on the average residence times, flow patterns, temperature distributions and inclusion removals of molten steel were studied and compared. The fluid flow, heat transfer and inclusion removal of molten steel in a tundish with FCDs were studied in the industrial tests. The results of physical simulation, mathematical calculation and industrial tests had been compared.

2. Water Model

2.1. Experiment Theory

The theory of the water model experiment was based on similarity theory. For geometric similarity, the ratio was 1:3 between the prototype and the model. For dynamic similarity, considering that the flow conditions and the distribution of the flow velocity had no relation with the Reynolds number when the fluid flow was in the second self-simulation area in the prototype, the Froude similarity criterion was adopted for the water model. On the other hand, natural convection had to be considered, because the tundish had a large volume and the distance between the adjacent strands was great. Therefore, the tundish similarity criterion shown in Equation (1) also had to be satisfied.

$$Tu=\frac{Gr}{{\mathrm{Re}}^2}=\frac{\beta gl\Delta T}{u^2}$$

(1)

In this equation, Gr is the Grashof number, Re is the Reynolds number, β is the coefficient of volume expansion, g is the gravitational acceleration in m²/s, l is the length in m, T is the temperature in K and u is the velocity in m/s.

The Froude criterion is shown in Equation (2):

Fr_m = Fr_p

(2)

In this equation, m is the model and p is the prototype.

Flow velocity and flow rate in the water model and the prototype can be calculated using Equations (3) and (4), respectively:

$$u_m={\lambda }^{0.5}\cdot u_p=0.57735u_p$$

(3)

$${\mathrm{Q}}_m={\lambda }^{2.5}\cdot Q_p=0.06415Q_p$$

(4)

In these equations, λ is the proportion coefficient and Q is the flow rate in m³/s.

The temperature difference was determined by Equation (5):

Tu_m = Tu_p

(5)

The following Equation (6) was derived:

$$\Delta T_m=\frac{{\beta }_p}{{\beta }_m}\Delta T_p$$

(6)

In this equation, β_p is the coefficient of volume expansion for molten steel (1/°C) and β_m is the coefficient of volume expansion for water (1/°C).

The temperature difference of the molten steel between the injection spot of the ladle and the lowest temperature in a tundish is 15~25 °C [23], and the values of β_p and β_m are 1.16 × 10⁻⁴ and 2.79 × 10⁻⁴ [24], respectively. According to Equation (6), the temperature of hot water in non-isothermal conditions can be calculated. The temperature of hot water to simulate flow phenomena in non-isothermal conditions is defined by Equation (7).

T_{non-isothermal condition} = ΔT_m + water temperature at ambient temperature

(7)

2.2. Experimental Setup

The water model experiment apparatus schematic for the four-strand tundish is shown in Figure 1; it was 1/3 of the prototype and was made of plexiglass. Figure 2 is a schematic diagram for the tundish and FCDs. The outlets are the 1st strand to the 4th strand in sequence from left to right, as indicated in the figure.

2.3. Inclusion Behavior

In the water model, polystyrene plastic particles were used to simulate the inclusions in molten steel. Since the spherical particles form a certain wetting angle with water, and the density of the particles is lower than that of water, they could approximately simulate the spherical inclusions floating on molten steel that form a certain wetting angle with molten steel. The selection of the particle size in the water model could be calculated by Equation (8) [25].

$$\frac{R_m}{R_p}={\lambda }^{0.25}{\left[\frac{\left(1-\frac{{\rho }_p}{{\rho }_{st}}\right)}{\left(1-\frac{{\rho }_m}{{\rho }_w}\right)}\right]}^{0.5}$$

(8)

In this equation, ρ_p is the density of inclusions in molten steel, about 2500~3400 kg/m³; ρ_st is the density of molten steel, 7020 kg/m³; and ρ_w is the density of hot water, 995 kg/m³. Thus, 0.72 mm polystyrene plastic particles in water correspond to 100 μm inclusions in steel. The removal ratio is defined by Equation (9).

$$\eta =(1-Wg/We)\times 100\%$$

(9)

In this equation, We is the amount of particles injected into the domain and Wg is the amount of particles collected at outlets. Inclusion trajectories were tracked by injecting the quantitative plastic particles, observing their motions and statistically averaging the observations.

2.4. Experimental Methods

The fluid flow, heat transfer phenomena and inclusion trajectories in a four-strand tundish in isothermal and non-isothermal conditions were studied systematically. In the experiment, the inlet flow rate was equal to the exit flow rate, and the liquid level was stable. A given concentration of sodium chloride solution was added from the inlet instantaneously. The RTD curve was obtained by executing stimulus-response experiments, and the fluid flow pattern was calculated from the RTD curve. The experimental duration was two times that of the theoretical average residence time of a tundish. The full flow model described in the literature [26] was adopted for analysis. Considering the symmetry of the tundish, the data for the 1st and 2nd strands were collected.

The temperature distribution was obtained by the temperature collection system. In the experiments, the temperature of the inlet was higher than that of the outlet. To maintain the temperature difference between the inlet and outlet, a heater was used to heat up the water at the inlet. The temperature data at the outlet were measured and analyzed, that is, a temperature response curve was obtained. Considering the symmetry of the tundish, the data for the 3rd and 4th strands were collected. The temperature results were analyzed by the following equations. The average temperature variation of strand i, the maximum temperature difference between strands i and i + 1, and the average temperature difference between strands i and i + 1 could be calculated by Equations (10)–(12), respectively.

$$\overline{\Delta T_i}=\overline{T_i-T_0}=\frac{{\displaystyle \sum _{j=1}^m(T_{i,j}-T_0)}}{m}$$

(10)

$$\mathrm{Max}(\Delta T_{i,i+1})=Max|T_{i+1,j}-T_{i,j}|$$

(11)

$$\overline{|\Delta T_{i,i+1}|}=\frac{{\displaystyle \sum _{j=1}^m|T_{i+1,j}-T_{i,j}}|}{m}$$

(12)

In these equations, i is the number of the tundish outlet, T₀ is the initial temperature, j is the collection time, m is the total collection times, and $j=1,2,\cdots ,m$.

The fluid flow pattern and the average residence time of the molten steel in the tundish were obtained by executing stimulus-response experiments. The temperature distribution was obtained by the temperature collection system. Inclusion trajectories were tracked by injecting the quantitative plastic particles and statistically averaging the observations. All experiments were repeated three times, and the measured data were averaged over the three experiments. Experimental variables between the prototype and the model are listed in

Table 1. Parameters of the tundish in the prototype and the model.

Parameter	Submergence Depth of Shroud (mm)	Depth of Molten Steel (mm)	Total Flow Rate (m3·h−1)	Flow Rate of Each Strand (m3·h−1)	Temperature Difference (K)
Prototype	120	800	15.277	3.819	25
Model	40	267	0.98	0.245	10.5

.

3. Mathematical Model

3.1. Modeling Methods

Th three-dimensional fluid flow and heat transfer of molten steel in a tundish were simulated. The flow field in the tundish was computed by solving the mass, momentum and energy conservation equations. They were solved together with the turbulence kinetic energy and dissipation rate of the turbulence kinetic energy.

Continuity:

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u_i\right)}{\partial x_i}=0$$

(13)

Momentum:

$$\frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left[{\mu }_{eff}\left[\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right]\right]+F_i$$

(14)

Effective viscosity:

$${\mu }_{eff}={\mu }_0+{\mu }_t={\mu }_0+\rho C_{\mu }{k}^2/\epsilon$$

(15)

Turbulent kinetic energy and rate of dissipation:

$$\frac{\partial }{\partial x_i}\left(\rho u_{i}K-\frac{{\mu }_{eff}}{{\sigma }_k}\cdot \frac{\partial K}{\partial x_i}\right)=G-\rho \epsilon$$

(16)

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

(17)

Constants used in the k-ε model are showed in

Table 2. The constants of the k-ε model.

Parameter	${\mathit{C}}_{\mathit{\mu }}$	${\mathit{\sigma }}_{\mathit{k}}$	${\mathit{\sigma }}_{\mathit{\epsilon }}$	${\mathit{C}}_1$	${\mathit{C}}_2$
Value	0.09	1.0	1.3	1.44	1.92

.

Energy transfer equation:

$$\frac{\partial }{\partial t}\left(\rho T\right)+\frac{\partial \left(\rho u_{i}T\right)}{\partial x_i}=\frac{\partial }{\partial x_i}\left[{\mathsf{\Gamma }}_{eff}\frac{\partial T}{\partial x_i}\right]+S_r$$

(18)

$${\mathsf{\Gamma }}_{eff}=\frac{u}{P_r}+\frac{u_t}{P_{r,t}}$$

(19)

In the above equations, $k$ is the turbulent kinetic energy in m²/s², $\rho$ is the density in kg/m, μ is the molecular viscosity of fluid in Pa·s, ${\mu }_{eff}$ is the effective viscosity of fluid in Pa·s, ${\mu }_t$ is the turbulent viscosity of fluid in Pa·s, $\epsilon$ is the dissipation rate of kinetic energy in m²/s³, T is the temperature in K, P is the pressure in Pa, t is the time in s, $\mathsf{\Gamma }$ is the diffusion coefficient and ${\mathsf{\Gamma }}_{eff}$ is the effective diffusion coefficient.

The continuity, momentum and heat-transfer equations were solved together with the k-ε equations until steady-state conditions were reached using the standard boundary conditions. The boundary conditions for momentum transfer were those of non-slipping at the solid surfaces, zero normal velocity gradients at the symmetry planes and the free surface of the liquid. The initial inlet velocity was calculated from the inlet flowrate and inlet area. The standard wall functions were required to calculate the value of a node near a solid wall. An implicit scheme with a segregated solver was applied. Particle trajectory was calculated by integrating its velocity, which was obtained by considering the drag force and gravitational force. The discrete random walk model was adopted. Particles were assumed to escape when reaching the top surface and outlets and to be reflected at solid walls. The effect of the top slag on fluid flow was ignored. During iterations, the convergence was assumed to reach that all residuals for all variables in momentum, heat, and mass transfer were less than 10⁻⁵. Since enthalpy is very sensitive to tolerance setup, the residuals of this variable were set to 10⁻⁷. The other parameters and boundary conditions of the tundish are listed in

Table 3. Parameters and boundary conditions used in modeling.

Parameter	Value	Parameter	Value
Inlet temperature	1823 K	Thermal conductivity	41 w·m−1·K−1
Depth of molten steel	800 mm	Heat flux at surface	15 kw·m−2
Diameter of ladle shroud	80 mm	Heat loss from bottom wall	1.4 kw·m−2
Density of molten steel	7020 kg·m−3	Heat loss from long wall	3.2 kw·m−2
Viscosity of molten steel	0.0067 kg·m−1·s−1	Heat loss from short wall	3.8 kw·m−2
Specific heat	750 J·kg−1·K−1	Heat loss from internal walls	1.75 kw·m−2

.

Half of the tundish was chosen for the mathematical analysis. The computational domain of the tundish was divided into 500,000 fine meshes in order to obtain accurate results. Hexahedral grids were used in most areas except for the areas near the inlet and outlet of the tundish and the holes on the baffles. The mixed hexahedral and tetrahedral grids for the full-scale tundish, i.e., the prototype, were generated by GAMBIT 2.4.6 (ANSYS Inc., Pittsburgh, Pennsylvania, PA, USA), in which different mesh spacings for different portions of the tundish were applied. The mesh sensitivity was evaluated for aspect ratio and skewness, and it was found that the aspect ratio was less than 5:1 and the skewness was less than 0.80. The meshes showed strong adaptability to the model and high quality. The numerical discretization schemes were as follows: the governing equations were discretized by the finite volume method. Additionally, in the numerical solution scheme, the SIMPLE numerical algorithm was used for pressure velocity coupling and the second-order upwind scheme for momentum and scalar transport equations. The ANSYS FLUENT 15.0 (ANSYS Inc., Pittsburgh, Pennsylvania, PA, USA) was used to calculate flow and temperature field and inclusion trajectory in the tundish.

3.2. Validation

In order to validate the availability of the mathematical model, the concentration field between the physical simulation in non-isothermal conditions and the mathematical simulation were compared. Figure 3a shows the tracer dispersion in the water model with FCDs at 2, 7, 10, 17 and 22 s, respectively. Meanwhile, Figure 3b shows the results of the mathematical simulations corresponding to Figure 3a. As can be seen in the figure, at 2 s, the fluid was in the baffle; then, it reached the tundish surface from the holes of the baffle (7 s) and sunk to the bottom along the side wall of the tundish (17 s); at last, the fluid flowed to the 1st and 4th outlets (22 s). The small difference between the results was caused by the blue ink that diffused itself in the water model. As can be seen, the agreement of the dispersion profiles between the water model in non-isothermal conditions and the mathematical simulations was good. Therefore, the mathematical model was reasonable and could simulate the flow phenomena in the tundish.

4. Results and Discussion

In this paper, a total of 11 schemes of FCDs were designed to study the flow phenomena and temperature distribution of molten steel in the four-strand tundish from several perspectives, such as the position, diameter and angle of the holes in the baffle. An optimal FCD scheme was selected, as shown in Figure 2. The flow field, temperature field and inclusion motion of molten steel in a tundish in isothermal and non-isothermal conditions with and without FCDs were studied using a water model and a mathematical model.

4.1. Fluid Flow

The residence times and flow patterns of molten steel in the tundish according to the water model and the mathematical model in isothermal and non-isothermal conditions, with and without FCDs, are shown in

Table 4. The measured residence times and flow patterns of molten steel in the tundish.

Model	FCDs	Conditions	$\overline{{\mathit{t}}_1}/\mathbf{s}$	$\overline{{\mathit{t}}_2}/\mathbf{s}$	$\overline{\mathit{t}}/\mathbf{s}$	${\mathit{V}}_{\mathit{p}}/\mathit{V}$	${\mathit{V}}_{\mathit{d}}/\mathit{V}$	${\mathit{V}}_{\mathbf{s}}/\mathit{V}$
Water	no	isothermal	241.1	261.3	251.2	0.2151	0.3592	0.0396
	yes	isothermal	285.1	300.8	293.0	0.4398	0.1951	0.0000
	no	non-isothermal	255.1	271.4	263.3	0.2926	0.3242	0.0000
	yes	non-isothermal	298.3	309.7	304.0	0.5023	0.1513	0.0000
Mathematical	no	non-isothermal	258.9	273.1	266.0	0.2821	0.3126	0.0000
	yes	non-isothermal	299.1	311.3	305.2	0.5278	0.1461	0.0000

. The RTD curves from the water model and the mathematical model in isothermal and non-isothermal conditions with FCDs are shown in Figure 4. The velocity field from the mathematical model with and without FCDs is shown in Figure 5.

In this table, $\overline{t_i}$ is the average residence time for strand I in s; $\overline{t}$ is the average residence time for the whole tundish in s; and $V_p/V$, $V_d/V$ and $V_{\mathrm{s}}/V$ are the piston flow volume, the dead volume and the short-circuit flow volume, respectively.

In the isothermal water model without FCDs, a short current flow obviously occurred near the second and third outlets; the inclusions, therefore, spent little time floating out to the top surface and tended to flow away from the outlets, and meanwhile, the dead volume reached 36%. In the non-isothermal water model, however, a short current flow was not found, and the dead volume reached 32%. The mathematical model results showed that there was no short circuit flow in the tundish, and the dead volume reached 31%.

In the isothermal water model with FCDs, the previous short current flow was eliminated, and the vigorous turbulence was controlled in the inlet zone; the flow pattern was therefore improved. Meanwhile, compared to the model with no FCDs, the average residence time of the molten steel was increased by 16.7% from 251.2 s to 293.0 s. The difference in the average residence time between the first and second outlets decreased by 22.3% from 20.2 s to 15.7 s, which shows that the fluid flow between the first and second outlets was more uniform. The dead zone volume decreased by 45.7% from 35.92% to 19.51%. The piston flow volume increased by 104.5% from 21.51% to 43.98%. This shows that the flow pattern in the tundish was more reasonable.

The results of the water model experiment in non-isothermal conditions were consistent with those in isothermal conditions. There was no short circuit flow in the tundish. Compared to the model with no FCDs, the average residence time increased by 15.5% from 263.3 s to 304.0 s. The difference in the average residence time between the first and second outlets decreased by 30.1% from 16.3 s to 11.4 s. The dead zone volume decreased by 53.3% from 32.42% to 15.13%. The piston flow volume increased by 71.7% from 29.26% to 50.23%.

The results of the mathematical calculations were also consistent with the results of the water model experiment. Compared to the model with no FCDs, the average residence time increased by 14.7% from 266.0 s to 305.2 s. The difference in the average residence time between the first and second outlets decreased by 14.1% from 14.2 s to 12.2 s. The dead zone volume decreased by 53.3% from 31.26% to 14.61%. The piston flow volume increased by 87.1% from 28.21% to 52.78%.

The results of isothermal and non-isothermal water model without FCDs were compared with the results of the mathematical model. The average residence times were 251.2 s, 263.3 s and 266.0 s, respectively. Compared with the numerical simulation results, the differences of the isothermal and non-isothermal experiments were 14.8 s and 2.7 s, respectively. The difference in the average residence time between the first and second outlets were 20.2 s, 16.3 s and 14.2 s, respectively. Compared with the numerical simulation results, the differences of isothermal and non-isothermal experiments were 6.0 s and 2.1 s, respectively. The dead zone volumes were 35.92%, 32.42% and 31.26%, and the differences were 4.66% and 1.16%, respectively. The piston flow volumes were 21.51%, 29.26% and 28.21%, and the differences were 6.7% and 1.05%, respectively.

The results of isothermal and non-isothermal water model with FCDs were compared with the results of the mathematical model. The average residence times were 293.0 s, 304.0 s and 305.2 s, and the differences were 12.2 s and 1.2 s, respectively. The differences in the average residence time between the first and second outlets were 15.7 s, 11.4 s and 12.2 s, respectively. Compared with the numerical simulation results, the differences of the isothermal and non-isothermal experiments were 3.5 s and 0.8 s, respectively. The dead zone volumes were 19.51%, 15.13% and 14.61%, and the differences were 4.9% and 0.52%, respectively. The piston flow volumes were 43.98%, 50.23% and 52.78%, and the differences were 8.8% and 2.55%, respectively.

From Table 2 and Figure 4 and Figure 5, the following conclusions about the fluid flow can be drawn: (1) The effect of the natural convection of the molten steel in a multi-strand tundish on the fluid flow should be considered. For the current tundish, the section was φ220 mm², the casting speed was 2.3 m/min, and the average velocity was 0.067 m/s. It has been reported that if the average flow velocity is less than 0.14 m/s, the natural convection cannot be ignored [25]. The results of the theoretical calculations, water model experiment and numerical simulation showed that the natural convection driven by the temperature difference caused by buoyancy had a great influence on the flow behavior of molten steel. (2) The flow characteristics of the molten steel in the tundish were different between isothermal and non-isothermal conditions. A short-circuit flow appeared in the isothermal water model experiment without FCDs; however, there was no short-circuit flow in the non-isothermal water model experiment and the numerical simulation. (3) The RTD parameters were very different in the isothermal and non-isothermal water models. (4) The performance with FCDs was better than without FCDs. The average residence time increased, the fluid flow was more uniform and the flow pattern was more reasonable. (5) Compared with the absence of FCDs, non-isothermal conditions had less influence on the residence time and flow pattern of the molten steel with FCDs in the tundish. (6) The numerical simulation results were closer to the experimental results in the non-isothermal water model.

4.2. Heat Transfer

The dimensionless temperature versus time of the molten steel at the outlet of the tundish is shown in Figure 6, and the analyzed results are shown in

Table 5. Temperature measurement in non-isothermal conditions.

Case	$\mathbf{Max}(\mathbf{\Delta }{\mathit{T}}_{3,4})/\mathbf{K}$	$\overline{\mathbf{\Delta }{\mathit{T}}_{3,4}}/\mathbf{K}$	$\overline{{\mathit{T}}_3-{\mathit{T}}_0}/\mathbf{K}$	$\overline{{\mathit{T}}_4-{\mathit{T}}_0}/\mathbf{K}$
Without FCDs	2.2	0.43	6.64	6.25
With FCDs	1.7	0.28	6.89	6.52

. The dimensionless temperature is the temperature data measured at the outlet divided by the temperature at the inlet of the tundish. Without FCDs, the fluid directly reached the third outlet and entered the mold without full heat exchange with the rest of the molten steel in the tundish; thus, there was a strong temperature fluctuation at the third outlet area, as shown in Figure 6a. It is shown that the flow pattern in the tundish was complex, and the temperature distribution was not uniform. With the optimized FCDs in the non-isothermal water model, in opposition to the isothermal water model experiments, the fluid reached the fourth outlet first and then traveled back to the third outlet due to the density difference induced by the temperature difference. The difference between the third and fourth outlets became smaller as time passed, as shown in Figure 6b. It should also be noted that the temperature fluctuation at the third outlet was much smoother than that in the tundish without FCDs. The above observation indicates that the temperature distribution in the tundish with FCDs was more uniform than that without FCDs.

Compared to the model with no FCDs, the maximum and average temperature differences between the third and fourth outlets decreased from 2.2 K and 0.43 K to 1.7 K and 0.28 K with FCDs, respectively. This indicates that the temperature distribution in the tundish was more uniform with FCDs. The average temperature changes of the third and fourth outlets increased from 6.64 K and 6.25 K to 6.89 K and 6.52 K, respectively. The values of the average temperature changes of each strand were larger, which shows that the temperature rose quickly and the mixing extent of the molten steel was better. The temperature fluctuations at the third outlet were not as strong as those without FCDs. As more time passed, the temperature fluctuations gradually disappeared. The average temperature differences were smaller, and the temperature distribution was more uniform.

Figure 7 shows the calculated temperature distribution in the tundish. The maximum temperature difference in the tundish without FCDs was 13.5 K, and it was 10.5 K with FCDs. The temperature distribution was more homogenous with FCDs.

It can be seen from the above analysis that the influence of the temperature in the tundish system, that is, the influence of natural convection on the fluid flow, cannot be ignored. It is well known that natural convection induces an upwards flow, and this is very important, especially when a tundish has a great volume and a long flow space.

4.3. Inclusion Trajectories

The influence of FCDs on the inclusion removal in the tundish is shown in

Table 6. Effect on inclusion removal ratio of molten steel in tundish.

Case	Isothermal Model without FCDs	Isothermal Model with FCDs	Non-Isothermal Model without FCDs	Non-Isothermal Model with FCDs	Mathematical Model without FCDs	Mathematical Model with FCDs
Inclusion removal ratio, %	90.2	95.3	92.8	97.3	92.0	98.0

, which indicates that the removal of particles to the top surface was greatly improved by the FCDs. Around 95.3–98.0% of the particles were removed with FCDs, which is about 4.5–6.0% higher than without FCDs. The calculated inclusion trajectories are shown in Figure 8. With FCDs, because of the upward flow pattern generated, more and more particles moved to the top surface, and only a few inclusions reached the fourth outlet. Without FCDs, because of the short current flow, the particles quickly reached the bottom of the tundish and moved directly to the third outlet, and many also reached the fourth outlet.

Without FCDs, the inclusion removal ratios of the isothermal water model, non-isothermal water model and mathematical model were 90.2%, 92.8% and 92.0%, respectively. Compared with the mathematical model results, the differences of the isothermal and non-isothermal water models were 1.8% and 0.8%, respectively. With FCDs, the inclusion removal ratios were 95.3%, 97.3% and 98.0%, and the differences were 2.7% and 0.7%, respectively.

The inclusion removal ratio of the non-isothermal water model was higher, and it was closer to the numerical simulation results. The molten steel flowed upward due to the natural convection effect in the non-isothermal water model, while the fluid flowed downward and directly reached the bottom of the tundish in the isothermal water model. This upward flow pattern surely favors inclusions removal, because inclusions are able to reach the top surface more quickly.

5. Trial Results

When FCDs were used in the trial, the results showed that the maximum temperature difference between adjacent strands was within 5 °C. The total amount of non-metallic inclusions in the steel decreased by 41.1% on average. For the non-metallic inclusions larger than 100 μm, the removal ratio could reach 100%. The trial results were closer to those of the non-isothermal water model and numerical simulation. The trial results showed that the fluid flow in the tundish had non-isothermal characteristics, and the research results in non-isothermal conditions could better reflect the actual fluid flow and heat transfer behaviors of molten steel in a tundish.

6. Conclusions

In this paper, the fluid flow, heat transfer and inclusion trajectories in a four-strand tundish were investigated by physical and mathematical simulations in isothermal and non-isothermal conditions. The following conclusions can be drawn.

(1)

The natural convection driven by temperature differences has a significant influence on the flow pattern and temperature distributions of molten steel in a tundish.

(2)

Without FCDs, the average residence times of the molten steel in the tundish obtained by the isothermal water model, non-isothermal water model and non-isothermal mathematical model were 251.2 s, 263.3 s and 266.0 s, respectively, and the dead zone volumes were 21.51%, 29.26% and 28.21%, respectively. With FCDs, the average residence times of the molten steel obtained by the isothermal water model, non-isothermal water model and non-isothermal mathematical model were 293.0 s, 304.0 s and 305.2 s, respectively, and the dead zone volumes were 43.98%, 50.23% and 52.78%, respectively. The flow characteristics of the molten steel in the tundish were different between isothermal and non-isothermal conditions. The RTD parameters were very different in the isothermal and the non-isothermal water models.

(3)

The trial results showed that the maximum temperature difference between adjacent strands was within 5 °C. For the non-metallic inclusions larger than 100 μm, the removal ratio could reach 100%. The fluid flow in the tundish had non-isothermal characteristics, and the trial results were closer to those of the non-isothermal water model and numerical simulation.

(4)

The research results in non-isothermal conditions could better reflect the actual fluid flow and heat transfer behaviors of molten steel in a tundish.