Simulation of Flow and Heat Fields in a Seven-strand Tundish with Gas Curtain for Molten Steel Continuous-Casting

Abstract

The present study involves physical and mathematical simulations to study the effect of gas bubbling curtains on the flow and temperature fields in a seven-strand tundish. In 1/3 scale tundish model experiments, RTD (Residence Time Distribution) curves were used to study the flow characteristics in the tundish with different configurations. The flow uniformity for all strands is evaluated through variance analysis of the RTD curves in different tundish cases. Besides, the tracer dispersion experiment was performed to visualize the fluid flow under the effect of the gas curtain. In order to investigate the effect of a gas curtain on the flow in the tundish, mathematical simulations were carried out mainly for two cases with or without gas bubbling curtains. The RTD curves were obtained from the species transport model. The results obtained show that gas bubbling curtains can help improve fluid flow characteristics in multi-strand tundish, and that the location of the curtains plays a key role in optimizing multi-strand tundishes configuration. Gas bubbling curtains can decrease the differences in residence times and molten steel temperatures at different outlets in multi-strand tundishes effectively. Heat transfer between different areas is promoted and the temperature field is uniformed significantly in such tundishes.

1. Introduction

With more requirements of clean steels, people pay more attention to the refining function of a continuous casting tundish. Many researchers^1,2,3,4) have significantly contributed to the improvement and optimization of melt flow controllers in tundishes. Admittedly, with traditional flow controllers, a tundish has been the best place to reduce big nonmetallic inclusions in the continuous casting process. Nevertheless, in accordance with the report from Lopez-Ramirez et al.,⁵⁾ the small inclusions do not have enough buoyancy force to float and reach to the slag layer.

Gas bubbling curtains, an approach for clean steel production, initially developed for removing small inclusions by blowing discrete bubbles from the bottom of tundish. The principle behind this device is that generating an upper flow to take inclusions toward the slag layer, and the smaller inclusions are easy to float when they are trapped by bubbles. Zhang et al.⁶⁾ simulated the water flow under gas bubbling and evaluated the effect of inclusion removal. Ramos-Banderas et al.⁷⁾ mentioned that the small rates of gas injection through a gas curtain improved the fluid flow by enhancing the plug flow volume fraction. According to the results of water-modeling experiments with argon bubbling, Zhong et al.⁸⁾ found that gas bubbling hardly influenced the minimum residence time but significantly prolonged the peak concentration time, and shortened the tail of resident time distribution (RTD) curves. Yamanaka et al.⁹⁾ claimed that owing to the use of gas curtain, a 50 pct improvement in the removal of inclusions in the range from 50 to 100 μm.

For multi-strand tundishes, gas curtain can also be used to homogenize the fluid flow characteristics. Because of its long horizontal length, the flow field in multi-strand tundishes is much more complex, especially considering the differences in residence time distributions among strands. These phenomena lead to the variations in exit temperatures from different strands and have serious negative impacts on the consistency of billet quality.

The present work, distinct from all the publications mentioned above, tends to investigate the influences of gas curtains on melt flow characteristics in a seven-strand tundish. Fluid flow in this seven-strand tundish with or without a gas curtain was performed by using water modeling and mathematical simulation methods. In physical modeling experiments, RTD curves of fluid flow with different gas curtain positions in the tundish were obtained and the optimum tundish configuration was found from the flow characteristics of strands. In mathematical simulation, the flow field and temperature field in the tundish with the optimal gas curtain position were obtained, and the results with and without gas bubbling were compared to evaluate the effect of gas curtain.

2. Physical Modeling Experimental

A one third scale model, geometrically similar to the prototype, was established to simulate the fluid flow in the tundish. Molten steel can be replaced by water in the physical simulation because the kinematic viscosities of liquid steel and water are comparable. In general, the modified Froude number, Fr’, of the water model and prototype were kept equal, in order to ensure the dynamic similarity between them for water flow rate and gas flow rate:   

$$F{r}_{\mathrm{m}}^{\text{'}}=\frac{u_{\text{fm}}^2}{g{L}_{\text{m}}}\cdot \frac{{\rho }_{\text{fm}}}{{\rho }_{\text{lm}}-{\rho }_{\text{gm}}}=\frac{u_{\text{fp}}^2}{g{L}_{\text{p}}}\cdot \frac{{\rho }_{\text{fp}}}{{\rho }_{\text{lp}}-{\rho }_{\text{gp}}}=F{r}_{\mathrm{p}}^{\text{'}}$$(1)

where, u is the velocity of gas or liquid, m/s, ρ density of the gas or liquid, kg/m³, L characteristic length, m, and g gravity acceleration, m/s² and subscript m is for the model, p for the prototype, l for liquid phase, g for gas phase and f for gas or liquid.

According to the Froude criterions, the volume flow rate, Q_fm, in the model tundish can be calculated from the liquid steel flow rate, Q_fp, in the industrial process:   

$$Q_{\text{fm}}={\left(\frac{L_{\text{m}}}{L_{\text{p}}}\right)}^{\frac{5}{2}}\cdot {\left(\frac{{\rho }_{\text{fp}}}{{\rho }_{\text{fm}}}\right)}^{\frac{1}{2}}\cdot {\left(\frac{{\rho }_{\text{lp}}-{\rho }_{\text{gp}}}{{\rho }_{l\text{m}}-{\rho }_{\text{gm}}}\right)}^{-\frac{1}{2}}{Q}_{\text{fp}}$$(2)

The parameters of prototype and model tundish are listed in Table 1.

Table 1. Operation parameters in prototype and model tundish.

Parameter	Prototype tundish	Water model
Fluid	Molten steel	Water
Density/kg·m–3	7038	998.2
Viscosity/kg·m–1·s–1	0.0064	0.0009
Flow rate/m3·h–1	21.53	1.38
Depth of liquid/mm	800	266
Diameter of shroud/mm	70	23.3
Temperature/K	1823	298

Figure 1 shows the experimental apparatus which includes a tundish model, an argon supply system and a data acquisition system. The structure and major dimensions of the tundish prototype are shown in Fig. 2. As shown, a turbulence inhibitor and a baffle with holes are used in the tundish.

Fig. 1.

Schematic diagrams of the tundish experimental setup.

Fig. 2.

Top view of the tundish profile with key dimensions.

In the physical modeling experiment, the “stimulus-responding” method¹⁰⁾ was used to attend the RTD curves which contain the flow characteristics of corresponding strands. In the experiment, 200 ml NaCl solution (0.2 g/ml) was used as tracer. After the fluid flow in the model tundish reaches steady state, the tracer was injected into the tundish through the ladle shroud. Four conductive probes were used to measure the variation of the conductivity at each strand, and fluid flow characteristics of the strands were obtained from the RTD curves with the computer. 0.016 m³/h argon flow rate of the gas curtain was chosen on the base of trial experiments for a calm slag-metal interface.

In order to compare the flow characteristics among strands for multi-strand tundish, the calculation method of flow characteristics for single strand tundish, proposed by Sahai and Emi,¹¹⁾ was introduced in the present work by using a sub-tundish concept, that is, the n-strand tundish is consist of n sub-tundishes. The liquid volume, V_i, of a sub-tundish for strand i in n-strand tundish can be written as:   

$$V_i=\frac{q_i}{\sum _{i=1}^n{q}_i}V$$(3)

where q_i is the volumetric flow rate of strand i, m³/s, V the total liquid volume of the tundish, m³. If the flow rates of strands in n-strand tundish are identical, one can obtain V=nV_i from Eq. (3). The amount, dm_i, of tracer injected in the tundish flowing out from the strand outlet i in a period dt may be expressed as:   

$$d{m}_i=c_i(t)q_{i}dt$$(4)

where c_i(t) is the tracer concentration in outlet i. Integrating the above equation, one obtains the total amount, m_i, flowing out from the ith strand outlet:   

$$m_i={\int }_0^{+\infty }{c}_i(t)q_{i}dt$$(5)

Introducing residence time distribution density function for strand i, E_i(t), one gets:   

$${\int }_0^{+\infty }\frac{c_i(t)q_i}{m_i}dt={\int }_0^{+\infty }{E}_i(t)dt=1$$(6)

Therefore, the flow characteristics of the strand i in multi-strand tundish can be determined from the RTD curve measured at the outlet with the method in this reference.

The calculated mean residence times for the ith sub-tundish and the n-strand tundish, t_c,i and t_c, can be calculated by   

$$t_{\mathrm{c},i}=\frac{V_i}{q_i}=\frac{V}{\sum _{i=1}^n{q}_i}=t_{\mathrm{c}}$$(7)

The actual mean residence time, t_av,i, for outlet i is given by   

$$t_{\mathrm{a}\mathrm{v},i}=\frac{{\int }_0^{+\infty }{c}_i(t)q_{i}tdt}{{\int }_0^{+\infty }{c}_i(t)q_{i}dt}$$(8)

Or   

$$t_{\mathrm{a}\mathrm{v},i}=\frac{\sum _{j=1}^{\infty }{c}_i(t_j)t_j\mathrm{\Delta }{t}_j}{\sum _{j=1}^{\infty }{c}_i(t_j)\mathrm{\Delta }{t}_j}$$(9)

As suggested in Ref. 11), the dead volume fraction, V_d,i/V_i, for the outlet i is determined by   

$$\frac{V_{d,i}}{V_i}=1-\frac{Q_{a,i}}{Q_i}{\stackrel{-}{\theta }}_{c,i}$$(10)

where V_d,i is the dead volume, m³, V_i liquid volume in sub-tundish i, m³, Q_i and Q_a,i are flow rates of fluid, m³/s, which flows through sub-tundish i and active zone in this sub-tundish i, respectively, the term Q_a,i/Q_i is the area under the RTD curve from dimensionless time θ=0 to 2 in sub-tundish i and represents the fractional volumetric flow rate through the active region, and ${\stackrel{-}{\theta }}_{c,i}$ expresses the dimensionless meantime of the RTD curve up to the cutoff point of dimensionless time, θ=2 in sub-tundish i.

The plug flow volume fraction, V_p,i/V_i, for the outlet i is calculated with the following equation in this work   

$$\frac{V_{\text{p},i}}{V_i}=\frac{1}{2}\left(\frac{t_{\text{min,}i}+t_{\text{max,}i}}{t_{\text{c,}i}}\right)$$(11)

where t_min,i is the minimum residence time, s, t_max,i peak concentration time, s, obtained from RTD curve in sub-tundish i, and t_c,i is calculated with Eq. (7). Last, the well-mixed volume fraction, V_m,i/V_i, for the outlet i is given by   

$$\frac{V_{\text{m,}i}}{V_i}=1-\frac{V_{\text{d,}i}}{V_i}-\frac{V_{\text{p,}i}}{V_i}$$(12)

3. Mathematical Modeling Method

A three-dimensional computational domain was designed using SolidWorks 2012 and discretized into tiny hexahedral cells with ICEM CFD 12.0. For numerical simulation, the computer software FLUENT 12.0 was employed. In the computation process, one half of the tundish was taken into consideration for speeding up the calculation, considering the symmetry of the tundish. In the numerical calculation, molten steel flow was assumed to be incompressible Newtonian steady flow. As a non-isothermal process, heat loss of the tundish is regarded as constant, which can be expressed by the different heat flux values for the corresponding boundaries.

3.1. Governing Equations

For liquid phase calculations, a series of governing equations including continuity, momentum, energy and standard k-ε equations were solved simultaneously in Cartesian coordinate system. These governing equations are expressed as:   

$$\frac{\partial (\rho u_i)}{\partial x_i}=0$$(13)

  

$$\frac{\partial (\rho u_i{u}_j)}{{\partial }_j}=-\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_i}\left({\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}\frac{\partial u_i}{\partial x_j}\right)+\frac{\partial }{\partial x_i}\left({\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}\frac{\partial u_j}{\partial x_i}\right)+\rho g_i$$(14)

  

$$\frac{\partial (\rho u_{i}T)}{\partial x_i}=\frac{\partial }{\partial x_j}\left(\frac{k_{\mathrm{e}\mathrm{f}\mathrm{f}}}{C_{\mathrm{p}}}\frac{\partial T}{\partial x_j}\right)$$(15)

where, ρ is the density of liquid steel, kg/m³, u the velocity of molten steel, m/s, P pressure, Pa, μ_eff effective viscosity of steel, Pa.s, which is the sum of laminar and turbulent viscosities, g gravity acceleration, m/s², k_eff represents the effective thermal conductivity, W/(m.K), which is the sum of steel laminar thermal conductivity and turbulent thermal conductivity, C_p thermal capacity of liquid steel, J/(kg.K), T is the steel temperature, K, x Cardesian space coordinates, m, and subscripts i, j are for the coordinate directions. The standard k-ε turbulence model proposed by Jones and Launder¹²⁾ are written as:   

$$\frac{\partial (\rho u_{i}k)}{\partial x_i}=\frac{\partial }{\partial x_j}\left(\frac{{\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}}{{\sigma }_k}\frac{\partial k}{\partial x_j}\right)+G_k-\rho \epsilon$$(16)

  

$$\frac{\partial (\rho u_i\epsilon )}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left({\mu }_{eff}+\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+c_1\frac{\epsilon }{k}{G}_k-c_2\frac{{\epsilon }^2}{k}\rho$$(17)

where, k is the kinetic energy of turbulence per unit mass, m²/s², ε represents for its dissipation rate, m²/s³, μ_t is turbulent viscosity of steel, Pa.s, σ_k and σ_ε are Schmidt numbers for k and ε, –, c₁ and c₂ constants, –. In these equations, G_k represents the generation of turbulence kinetic energy due to the mean velocity gradients, –, and can be expressed as:   

$$G_k={\mu }_t\frac{\partial u_j}{\partial x_i}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_i}{\partial x_j}\right)$$(18)

The turbulent-adjusted effective viscosity is calculated as follow:   

$${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mu +{\mu }_t=\mu +c_{\mu }\rho \frac{k^2}{\epsilon }$$(19)

where, μ is laminar viscosity of steel, Pa.s, c_μ empirical constant, –.

As mentioned by Launder and Spalding,¹³⁾ the other values for model constants in this study were c₁=1.44, c₂=1.92, c_μ=0.09, σ_k=1.0, σ_ε=1.3 which have been determined from experiments for fundamental turbulent flows.

Once the computation attained convergence, the species transport model was introduced to describe the tracer dispersion in tundish. The tracer was plus injected through the inlet for 1 s, and meanwhile, the mass fractions of tracer at each outlet were monitored. Thus, the RTD curves of each strand were achieved. In species transport model, the properties of tracer were set the same as those of steel for reporting the residence time distribution of tundish exactly.

3.2. Discrete Phase Simulation

For the discrete phase, the discrete phase model (DPM) was used to describe the bubbles injected in molten steel. In a Lagrangian frame of reference, the dispersed phase is tracked by solving a transport equation for each bubble as it travels through the previously obtained flow field of liquid steel, and then, the trajectory of each bubble can be tracked by DPM. The basic assumption of DPM is that the volume fraction of the discrete phase is sufficiently low.

The force balance equations of each bubble show as follow:   

$$\frac{d{u}_{\mathrm{p}}}{dt}=F_{\mathrm{D}}(u_i-u_{\mathrm{p}})+\frac{g({\rho }_{\mathrm{p}}-\rho )}{{\rho }_{\mathrm{p}}}$$(20)

where, u_p is velocity of bubbles, m/s, ρ_p density of gas, kg/m³, and F_D drag, s^–1. F_D(u_i–u_p) expresses the drag force per unit bubble mass and F_D can be expressed as:   

$$F_{\mathrm{D}}=\frac{18\mu }{{\rho }_{\mathrm{p}}{d}_{\mathrm{p}}^2}\frac{C_{\mathrm{D}}Re}{24}$$(21)

here, C_D is drag coefficient, –, d_p bubble diameter, m, and Re relative Reynolds number. Re can be defined as:   

$$Re=\frac{\rho d_{\mathrm{p}}|u_{\mathrm{p}}-u_i|}{\mu }$$(22)

In the gas bubbling process, each bubble is treated as a smooth sphere. So, according to the spherical drag law, the drag coefficient, C_D, for a bubble can be computed by the following equation:   

$$C_{\mathrm{D}}=a_1+\frac{a_2}{Re}+\frac{a_3}{Re}$$(23)

where a₁, a₂, and a₃ are empirical constants which apply to some ranges of Reynolds number.

By default, DPM just considers impacts of the continuous phase on the discrete phase. However, in this study, the interaction with continuous phase option was enabled for coupling the interactions of discrete phase with the subsequent continuous phase, including the energy and momentum transformer between them. This coupling calculation is achieved by alternately solving the discrete and continuous phase equations until the residuals of both two phases converge to a specified range. Therefore, the Two-Way turbulence coupling method is used in order to perform the turbulent eddies formed by particles.

3.3. Boundary Conditions

No slip conditions were applied to all the solid walls in the tundish, which implies all velocity components were zero at these walls. For the near-wall treatment, the standard wall functions proposed by Launder and Spalding¹³⁾ were used to incorporate the influence from turbulence flow. At the free surface, the zero shear stress boundary condition was set. Symmetry boundary condition was applied to the symmetry plane by setting zero normal velocity and gradients of all variables for that plane. Inflow of the liquid steel from the shroud was simulated by defining a velocity inlet condition, with a velocity of 1.55 m/s, which is perpendicular to the inlet flat. The inlet parameter of turbulent kinetic energy and dissipation rate were calculated as follow:   

$$k=\frac{3}{2}{(u_{\mathrm{a}\mathrm{v}\mathrm{g}}I)}^2$$(24)

  

$$\epsilon =c_{\mu }^{3/4}\frac{k^{3/2}}{l}$$(25)

Where, I and l represent the turbulence intensity and turbulence length scale and were assigned I=0.039 and l=0.0049 m, respectively. In view of the equality of each outlet flow in industrial process, outflow boundary conditions were applied to all the outlets. For heat transfer computation, the values of heat flux recommended by Chakraborty and Sahai¹⁴⁾ through the bottom, lateral wall, frontal walls and top surface were 1.4, 3.2, 3.8 and 15 kW/m² respectively. The temperature of molten steel from the shroud was 1823 K.

In the discrete phase model, the bubble size was assumed as constant during its rising process in tundish and the interaction between bubbles was ignored in the tundish. Argon was simplified to be injected uniformly through the surface of argon bubbling strip, with an initial velocity of 0.18 m/s. For discrete phase properties, the bubble diameter was set as 2 mm and the total volume flow rate was 0.25 m³/h. In order to determine the fate of the particles at boundary, the reflect boundary condition was applied to the solid wall in the tundish. Those bubbles which float to a top surface and flow to the outlets were considered to have escaped.

3.4. Solution Procedure

The computational domain was meshed to 1080000 hexahedral cells in a non-uniform grid system. In the numerical solution scheme, the Semi-Implicit Method for Pressure Linked Equation algorithm was used for pressure velocity coupling and a second order upwind scheme was adopted for convective term in the momentum equations. The solution was judged to be converged, when the normalized residuals of all the variables were less than 10^–4. Initially, the flow and temperature fields were calculated in steady state. Once obtaining the converged results, the numerical calculation was changed to unsteady state to solve the discrete phase equations with a time step for 0.001 s. The value of the under-relaxation factors were kept as defaults during the steady state period, while in the gas blowing process, these values were reduced by 10–20% to guarantee the convergence in complex multi-phase computations.

4. Results and Discussion

4.1. Physical Modeling

A previous study¹⁵⁾ conducted by one of the authors and his research group, the results had shown that, under the “no gas” bubbling condition, the optimal tundish configuration consists of a baffle with two holes on both sides, two dams and a square turbulence inhibitor, as optimized from water modeling experiments, but a big difference in flow characteristics between strand 4# and the other strand was found. In the present study, the influence of the combination of the square turbulence inhibitor and the baffle with two gas curtains on flow characteristics in the seven-strand tundish was investigated. In the physical experiments, more than 20 cases were performed, in order to optimize the tundish configuration, the gas flow rates and position of gas curtain. Here, only four tundish arrangements and their flow characteristics are given, as shown in Fig. 3 and Table 2.

Fig. 3.

Tundish cases without or with gas bubbling curtain at different positions.

Table 2. Flow characteristics for different tundish configurations.

Case	Strand	tmin,i, s	tmax,i, s	tav.i, s	Vp,i/Vi, %	Vd,i/Vi, %	Vm,i/Vi, %
1	1	40.5	190.0	389.2	23.0	22.2	54.8
	2	38.0	175.5	395.5	21.4	20.9	57.7
	3	43.0	206.5	404.6	24.9	19.2	55.9
	4	75.5	326.0	420.4	40.2	15.9	43.9
2	1	48.5	173.5	396.8	22.2	20.7	57.1
	2	31.5	168.5	370.4	20.0	25.9	54.1
	3	38.0	214.5	376.7	25.2	24.7	50.1
	4	34.5	245.0	398.3	28.0	20.3	51.7
3	1	45.0	186.5	391.2	23.2	21.7	55.1
	2	36.5	144.5	375.0	18.1	25.0	56.9
	3	42.5	177.0	383.2	22.0	23.3	54.7
	4	60.5	266.0	406.1	32.7	18.8	48.5
4	1	41.5	238.5	412.2	28.0	17.5	54.5
	2	42.5	239.0	412.6	28.2	17.4	54.4
	3	39.0	240.5	407.5	27.9	18.5	53.6
	4	39.9	182.0	393.6	21.2	21.2	57.6

As can be seen from Table 2, the residence times, t_min, t_max and t_av of strand 4 in case 1 where there were no gas curtains are the longest and its plug volume is the largest, while its dead volume the lowest. It is known from the results that the flow characteristics between strand 4 and the other three strands are large. In tundish cases 2 and 4, the difference in flow characteristics among strands becomes small due to the gas curtains at suitable positions. Comparing the flow characteristics in these two cases, case 4 has larger plug flow volume fraction and lower dead volume fraction and its minimum residence times among the four strands are more uniform. Therefore, case 4 is considered to be the optimum one. In tundish case 3, the difference in flow characteristics among these four strands is still large even though gas bubbling curtains were applied. Because the gas bubbling curtains in case 3 are near to the baffle, the results obtained in case 3 are certainly similar to those in case 1.

Figure 4 presents the RTD curves from tundish case 1. In this tundish case, there are significant differences in RTD curves between strand 4 and the other stands. The RTD curve of strand 4 obviously moves rightward at a distance along the time coordinate direction. Interestingly, the peak times show the same characteristic as the minimum residence times. In case 1, the fluid first flows to the two end wall areas through the holes in the baffle, and then turns back to the outlet 4# located at the middle of the tundish. Therefore, more time is needed for the fluid to reach the outlet 4# and the residence times, such as, t_min, t_max and t_av of strand 4 in case 1 are the longest and its RTD curve move toward right side, as shown in Fig. 4. As a result, the flow characteristics of strand 4 are greatly different from those of the other 3 strands.

Fig. 4.

Experimental RTD curves for case 1.

Figure 5 shows the RTD curves in tundish case 4. It is obvious that the RTD curves among the four strands in this case are much more uniform each other. This means that the flow characteristics among theses strands in tundish case 4 are almost the same, as given in Table 2.

Fig. 5.

Experimental RTD curves for case 4.

Furthermore, variance analysis for RTD curves in the tundish is applied to evaluate the difference between all the RTD curves and the average RTD curve of the tundish quantitatively. First, the mean concentration out of the tundish is obtained with Eq. (26) (here n=4 for the 7-strand tundish where 4 strands were measured) and the difference in concentration between each strand and the tundish, [E_i(θ) – E(θ)]², is calculated. In order to consider the elapsed measurement time of the difference, it is multiplied by a time interval. Then, the variance of the tundish, σ², is worked out with Eq. (27). This tundish variance represents the total dispersion degree of all the measured RTD curves relative to the average RTD curve of the tundish. The results of the variance analysis are listed in Table 3 for different tundish cases. As shown in Table 3, the tundish variance in case 4 is 0.00715, only one twelfth of that in case 1, and less than the other two cases. It means that the flow characteristic of each strand is the most uniform in case 4. This result is consistent with the results of previous discussions.   

$$\overline{E(\theta )}=\frac{1}{n}\sum _{i=1}^n{E}_i(\theta )$$(26)

  

$${\sigma }^2=\sum _{\theta =0}^{{\theta }_{\mathrm{e}\mathrm{n}\mathrm{d}}}\sum _{i=1}^n\{{[E_i\left(\theta \right)-\overline{E\left(\theta \right)}]}^2\times \mathrm{\Delta }\theta \}$$(27)

Table 3. Tundish variance of RTD curves in each case.

Case	Variance
1	0.08825
2	0.03342
3	0.05648
4	0.00715

The fluid flow in tundish case 1 and 4 at different time are displayed with color tracer and captured by a digital camera, in order to understand the effect of gas bubbling curtain on getting uniform flow characteristics in the tundish. The photos of the fluid flow at different time in case 1 and 4 are given in Figs. 6 and 7, respectively.

Fig. 6.

Red tracer dispersion in tundish case 1 without gas curtain at different times.

Fig. 7.

Red tracer dispersion in tundish case 4 with gas curtain at different time.

Figure 6 shows the tracer depression in the tundish without gas curtain. After coming out from the diversion hole in the baffle, the tracer flows to the end wall along the surface. In this process, part of tracer falls to the outlet 2# due to its attraction effect. The other goes straight to the end wall. Once reaching the end wall, it turns back and flows to the symmetry plane along the bottom of the tundish. Evidently, the path from inlet to outlet 4# is the longest among all the outlets, that is, the residence time of outlet 4# is the longest among the outlets.

In Fig. 7, the fluid firstly flows through the holes in baffle. Once reaching the gas blowing region, the fluid flew up to the free surface and then is divided into two parts on the surface; one flows across the bubble curtain and the other turns back to the middle of the tundish owing to the effect of bubble flowing. As a result, the minimum residence time of outlet 4 reduces from 75.5 s in case 1 to 39.9 s in case 4, a drop of 47%. The difference in minimum residence time among the 4 strands in case 4 is as low as 3.5 s and it is less than that in case 1 by 34 s. For the same reason, the differences in mean residence time and peak time between the 4 strands in case 4 are declined apparently. Therefore, thanks to the effect of two strategically placed gas curtains, the fluid flow characteristics of the tundish are significantly improved and trended uniform among the strands.

4.2. Mathematical Simulation

Figures 8 and 9 present the RTD curves obtained in physical modeling and mathematical simulation in tundish case 1 and 4, respectively. As shown in Figs. 8 and 9, dimensionless RTD curves predicted by numerical simulation are in good agreement with those from the water experiments, which means that the results from the mathematical simulation are reasonable.

Fig. 8.

Comparison between computational and experimental RTD curves in case 1.

Fig. 9.

Comparison between computational and experimental RTD curves in case 4.

Figure 10 shows the velocity vectors at vertical outlet plane and path lines of liquid steel in case 1. At first, the liquid steel flow down to the turbulence inhibitor from the shroud. The turbulence inhibitor provides an upward motion which makes the liquid steel go back to the free surface. Then, the liquid steel goes through an upward inclined hole in the baffle and flows to the end wall along the surface. After hitting the end wall, it turns back and flows to the symmetry plane along the bottom of the tundish and finally reaches the outlet 4#. It can be concluded that in case 1, more time is needed for the fluid to reach the outlet 4#, which causes lower temperature of liquid steel near outlet 4#.

Fig. 10.

Flow pattern of molten steel in a half section of a tundish; case 1.

Figure 11 presents the velocity field and streamlines of molten steel in case 4. It can be seen from the figure that when the liquid steel reaches the bubble curtain region, it is divided into two parts. One flows through the bubble region to the end wall, and then turns back to the bubble curtain. The other is driven back to outlet 4# along the liquid surface, under the impact of the circular flowing caused by the gas curtain. So, the residence times of outlet 4# are reduced effectively, which can raise the molten steel temperature near outlet 4# and makes the flow characteristics of outlet 4# become closer to those of the other three outlets. It should be noted from the velocity fields at vertical plane along outlets in Figs. 10 and 11, that the velocity vectors in the tundish case 4 are greater than those in case 1 due to the gas bubbling, especially in the area near the gas bubbling curtain.

Fig. 11.

Flow pattern of molten steel in a half section of a tundish; case 4.

Computed temperature contour plots at the vertical outlet plane in the two tundish cases are shown in Fig. 12. In case 1, the temperature gradient of molten steel is great, especially near the top surface. The temperature of molten steel on the top surface at outlet 4# is as low as 1812.00 K, which is 11 K lower than that of the incoming stream (1823.00 K). However, the temperature in the upper region between strand 2 and strand 3 can reach over 1820 K. As shown in Fig. 12, the temperatures in outlet 1#, 2#, 3# and 4# are 1818.55 K, 1819.75 K, 1819.27 K and 1816.36 K, respectively. It is worth noticing that the maximum temperature difference among the 4 strands is up to 3.39 K.

Fig. 12.

Comparison of temperature fields between case 1 and case 4.

However, the temperature distribution in the tundish for case 4 is more homogeneous, compared with that in case 1. This is mainly because the gas bubbling creates a float force to drive molten steel to the top surface, which promotes the heat transfer between the upper and lower tundish. In case 4, the outlet temperature is in the range of 1817.64 K to 1818.47 K, with the maximum difference of 1.17 K among the 4 outlets. The steel temperature on the top surface at outlet 4# reaches 1815.6 K, which is in favor of preventing outlet 4# clogging due to too low a steel temperature.

5. Conclusions

In present work, the numerical simulations and the water model experiments are performed in a seven-strand tundish with or without gas curtains. The effects of gas curtains on fluid flow characteristics, velocity fields and temperature fields among four strands in the tundish with different tundish configurations are studied. The main conclusions are listed as follow.

(1) In case 1, under the control of baffle with diversion holes, the liquid steel flow forms a big circular flowing on either side of the tundish. It leads to too long residence time for outlet 4#, resulting in a notable difference in flow characteristics between strand 4 and the other strands in the tundish.

(2) The uniformity of the fluid flow characteristics in the tundish are significantly improved by using gas bubbling curtains. Some liquid steel heads back to the outlet 4# as it meets the gas bubbling curtains due to the two recirculation zones at both sides of the curtain. Owing to the use of gas curtain, the differences of residence times among the 4 strands are reduced greatly.

(3) Gas bubbling forms recirculation flows to promote the heat transfer between different areas in the tundish. The temperature field in the tundish is homogenized significantly, as well as the molten steel temperature for the 4 outlets.

(4) The position of gas bubbling curtain plays a key role in optimizing multi-strand tundishes configuration. An appropriate gas bubbling curtain location can improve fluid flow characteristics and promote homogenize temperature with in liquid steel contained in multi-strand tundish, effectively.

Acknowledgments

Financial support from the National Natural Science Foundation of China (Project No. 61333006) and from the National Undergraduate Innovative Experiment Program (Project No. 110136) in this key research project is gratefully appreciated.

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