Application of Mean Age Theory in Multi-strand Tundish

Abstract

The mean age theory has recently been used to analyze the mixing efficiency in a single-strand tundish. The spatial distribution of the mean age was obtained by using a steady-state calculation. By applying the new theory of mean age, the computing cost was two orders of magnitude lower than when using the conventional theory of residence time distribution (RTD). This study aimed at extending the application of the mean age theory to a multi-strand tundish. Theoretical relations between the mean age and RTD were analyzed and compared for the tundish applications. A new criterion, the standard deviation of flow-weighted mean age at the multiple outlets, was applied focusing on the consistency of the flow in the multi-strand tundish. A five-strand tundish was selected as a benchmark case. The feasibility of applying the mean age theory in the multi-strand tundish was confirmed through the benchmarking results. Thus, the proposed method can be adopted as an effective tool in finding the optimal geometry of multi-strand tundish equipped with flow control devices (FCD).

1. Introduction

Considerable research efforts have been made over many decades to enhance the metallurgical performance of the tundish.^1,2,3,4) A modern tundish is designed to ensure the homogeneity of liquid steel and enhance the inclusion removal. For a billet continuous casting (CC) caster, the molten steel flow in a multi-strand tundish is more complicated and many problems can occur during the casting. The flow fields and temperature distributions among the strands are influenced by the outlet locations. A short-circuiting flow may induce a breakout in casting because of the high temperature at the outlet. On the other hand, an outlet far away from the inlet may give rise to nozzle clogging due to a higher heat loss.^5,6) In brief, the key to designing an optimum multi-strand tundish is to improve the consistency of the fluid flow and temperature distribution inside the vessel.

The residence time distribution (RTD) theory has been widely used for the analysis of flow patterns in the tundish.^7,8,9) The computational fluid dynamic (CFD) has been a long-established method to model the molten steel flow in the tundish. By applying the CFD-based RTD model, many studies have been carried out to optimize the flow control devices (FCD) in the tundish, such as dams and weirs,^10,11) turbulence inhibitors,^12,13) gas stirring,^14,15) and so forth. These works led to an improved understanding of various flow phenomena associated with tundish operations.

As a factor of the increased complexity of the CFD model, the computing time has however become a limiting factor for the application of the RTD theory in a tundish. For instance, when the tundish has a large turnover time together with a long-time tail of the RTD E-curve, the computing time is largely required to solve the transient transport equation of the tracer. In addition, to apply an advanced design optimization algorithm, a large number of case simulations are usually required. Thus, the digital design process has to be compromised by the barrier of the expensive computing costs. To address this issue, a new theory based on the steady-state spatial distribution of mean age was proposed to analyze the flow patterns in a single-strand tundish.¹⁶⁾

The mean age is defined as the average time of all materials passing through a given location in a continuous system.¹⁷⁾ The mean age is governed by a passive scalar transport equation, which can be solved by using a CFD code. With a known spatial distribution of the mean age, the mixing state in a tundish can be quantitatively defined. The mean age theory was originally developed for a steady-state incompressible flow through a closed system with one inlet and one outlet.¹⁸⁾ The present study aims at extending the application of the mean age theory to a flow system with one inlet and multiple outlets, i.e. the multi-strand tundish.

The theoretical relationship between the mean age and RTD was analyzed in both the single-strand and the multi-strand tundish. A new criterion, the standard deviation of flow-weighted mean age at the outlets, was applied focusing on the consistency of the flow in the multi-strand tundish. As a benchmark case, the mean age theory was used to characterize the flow distributions in a five-strand tundish. The applied CFD model was validated by the water model experiment. By using the Taguchi design of experiment (DOE), case studies with different FCD were carried out to test the applicability of the new theory and the new criterion in the multi-strand tundish. Through the presented results, the feasibility of applying the mean age theory in the multi-strand tundish was confirmed. The proposed design method can be adopted as an effective tool in finding the optimal geometry of the multi-strand tundish equipped with FCD.

2. Theoretical Consideration

The mean age is defined as the time duration from the melt entering the reactor (one inlet and one outlet) to its arrival at point P, as illustrated in Fig. 1(a). A small mean age value represents a quick flow through the path, and vice versa. Similarly, the time to reach the outlet from the designated point P is defined as the residual life time. The residence time, which stands for the time of the melt that stays in the reactor, is equal to the sum of the mean age and the residual life time, shown in Fig. 1(a). Specifically, when point P is located at the outlet, the mean age is equal to the residence time because the residual life time is equal to zero.

Fig. 1.

Relationship among mean age, residual life time, and residence time (a, one outlet; b, two outlets). (Online version in color.)

Similarly, the relationship between the mean age, residual life time, and residence time is shown in Fig. 1(b) for a flow system with one inlet and two outlets. The fresh melt, flowing towards outlet1 and outlet2, is marked with the red circle and blue circle, respectively. The mean age at outlet1 is lower than that at outlet2 due to the shorter flow path. Meanwhile, the residence time at outlet1 is also lower than that at outlet2.

Equation (1) is the governing equation for the mean age, which is derived from a pulse tracer input.^18,19,20,21) The mean age is a passive scalar, which is a dependent variable governed by the convection-diffusion equation. The spatial distribution of the mean age can be determined from the steady-state solution of Eq. (1):   

$$\frac{\partial u_{j}a}{\partial x_j}-\frac{\partial }{\partial x_j}\left[D_{eff}\frac{\partial a}{\partial x_j}\right]=1$$(1)

where a is the mean age of the tracer passing through a location x and D_eff is the effective diffusivity.

For an incompressible steady-state flow in a tundish with a constant density of the fluid, the mean age has a value of zero at the inlet. Furthermore, the boundary conditions at the walls, top surface, and outlet, are represented by a zero flux. The volume integration of Eq. (1) results in the following expression:   

$$\underset{V}{\iiint }\frac{\partial u_{j}a}{\partial x_j}dV-\underset{V}{\iiint }\frac{\partial }{\partial x_j}\left[D_{eff}\frac{\partial a}{\partial x_j}\right]dV=V$$(2)

The divergence theorem can be applied to relate the flux through a closed surface to the divergence of the field of the mean age in the volume. Thereby, Eq. (2) can be rewritten as follows:   

$${∯}_{S_O}n\cdot ua\mathrm{\hspace{0.17em} }dS-{∯}_{S_I+S_O+S_W}n\cdot D_{eff}\nabla a\mathrm{\hspace{0.17em} }dS=V$$(3)

where n is the normal vector. The surface S is comprised of solid walls (S_w), one inlet (S_I), and one outlet (S_O), as shown in Fig. 1(a). By using the boundary conditions of a zero-flux of the mean age at all the surfaces, the second term of Eq. (3) vanishes.

If Eq. (3) is divided by the volumetric flow rate (Q), the following expression can be derived:   

$$a_O=\frac{{∯}_{S_O}n\cdot ua\mathrm{\hspace{0.17em} }dS}{Q}=\frac{V}{Q}=\tau$$(4)

where a_o is the flow-weighted mean age at the outlet. It is equal to the system volume divided by the volumetric flow rate (Q), which is defined as the theoretical residence time (τ) or the turnover time in the tundish.

The mathematical analysis of the flow system containing one inlet and two outlets is given below. Equation (5) can be applied to describe the system:   

$$\frac{{∯}_{S_{O1}}n\cdot ua\mathrm{\hspace{0.17em} }dS}{Q}+\frac{{∯}_{S_{O2}}n\cdot ua\mathrm{\hspace{0.17em} }dS}{Q}=\frac{V}{Q}$$(5)

For the tundish application, it is assumed that the velocities at outlet1 and outlet2 are equal because the casting speed of each strand is the same. Furthermore, Outlet1 and outlet2 (Fig. 1(b)) have the same area, which is equal to one-half of the single outlet area (Fig. 1(a)). Thereby, Eq. (5) can be written as follows:   

$$\frac{\left(a_{o1}+a_{o2}\right)}{2}=\frac{V}{Q}$$(6)

where a_o1 and a_o2 are the flow-weighted mean age at outlet1 and outlet2, respectively. The left term of Eq. (6) represents the average mean age at the two outlets.

Similarly, the average mean age of an N-strand tundish (a_t) can be calculated using the following relationship:   

$$a_t=\frac{{\sum }_{i=1}^N{a}_{oi}}{N}=\frac{V}{Q} =\tau$$(7)

where a_oi is the flow-weighted mean age at the i-th outlet, N is the number of outlets, and τ is the theoretical residence time or turnover time in a tundish.

For a multi-strand tundish, the consistency of flow among the strands should be guaranteed. Therefore, the standard deviation of flow-weighted mean age at the tundish outlets (s_n) can be used as a new criterion, as expressed by Eq. (8):   

$$s_n=\sqrt{\frac{{\sum }_{i=1}^N{\left(a_{oi}-a_t\right)}^2}{N-1}}$$(8)

It should be noted that a smaller value of s_n value indicates a higher consistency of flow among multiple strands.

3. Model Description

The numerical expressions for the RTD model and the mean age model, together with boundary conditions are described in section 3.1. In section 3.2, the validation and verification of the CFD model were given. Subsequently, to select the optimum design of the deflector hole on the U-baffle inside the tundish, the method of the Taguchi design of experiment is presented in section 3.3.

3.1. CFD Model

The assumptions made for the mathematical model of tundish flow are described below:

• The model is based on a 3D standard set of the Navier–Stokes equations.

• A non-isothermal steady-state flow is calculated.

• The density of the fluid is kept a constant.

• The realizable k-ε model is used to describe the turbulence of the fluid.

• The free surface is flat and is kept at a fixed level. The slag layer is not included.

3.1.1. Transport Equation

Equations (9), (10), (11), (12) are used to describe the continuous phase in a tundish.²²⁾   

$$\text{Continuity:}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\frac{\partial \left(\rho u_j\right)}{\partial x_j}=0$$(9)

  

$$\begin{array}{l} \text{Momentum:}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\\ \rho u_j\frac{\partial u_i}{\partial x_j}=-\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}\left[\left(\mu +{\mu }_t\right)\left\{\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right\}\right]+\rho g_i\end{array}$$(10)

  

$$\begin{array}{l}\text{Realizable}\mathrm{\hspace{0.17em} }k\text{-}\epsilon \mathrm{\hspace{0.17em} }\text{model:}\\ \frac{\partial }{\partial x_j}\left(\rho k{u}_j\right)=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+G_k+G_b-\rho \epsilon -Y_M+S_k\end{array}$$(11) ²³⁾

  

$$\begin{array}{l}\frac{\partial }{\partial x_j}\left(\rho \epsilon u_j\right)=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+\rho C_{1}S\epsilon -\rho C_2\frac{{\epsilon }^2}{k+\sqrt{v\epsilon }}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+C_{1\epsilon }\frac{\epsilon }{k}{C}_3{G}_b+S_{\epsilon }\end{array}$$(12)

where k is the turbulent kinetic energy; ε is the turbulent energy dissipation rate; µ is the molecular viscosity; μ_t is the turbulent viscosity; G_k represents the generation of turbulent kinetic energy due to the mean velocity; Y_M symbolizes the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate; υ is the kinematic viscosity; σ_k and σ_ε are the turbulent Prandtl numbers for k and ε, respectively.

Equation (1) is used as a governing equation for the mean age model. To apply the conventional RTD theory, a governing equation of tracer concentration is used in Eq. (13):   

$$\frac{\partial C}{\partial t}+\frac{\partial u_{j}C}{\partial x_j}-\frac{\partial }{\partial x_j}\left[D_{eff}\frac{\partial C}{\partial x_j}\right]=0$$(13)

where C is the tracer concentration, D_eff is the turbulent effective diffusivity, defined in Eq. (14):   

$$D_{eff}= D_m+\frac{{\nu }_t}{Sc}$$(14)

where D_m is the molecular diffusivity, ν_t is the turbulent viscosity and Sc is Schmidt number with a constant value of 0.7.

According to an RTD analysis, the mean residence time can be expressed by Eq. (15).²⁴⁾   

$$\overline{\tau }=\frac{{\int }_0^{2\tau }tC\left(t\right)dt}{{\int }_0^{\infty }C\left(t\right)dt}$$(15)

The dead volume fraction (V_d/V) is calculated through using Eq. (16).   

$$\frac{V_{\mathrm{d}}}{V}=1-\frac{\overline{\tau }}{\tau }$$(16)

3.1.2. Model Domain, Boundary Conditions, and Solution

The geometric dimensions of a 35-ton five-strand tundish are illustrated in Fig. 2. Three factors of the deflector hole on the U-baffle were considered in the design optimization, including i) A (angle of deflector hole), ii) B (height of deflector hole), and iii) C (diameter of defector hole). CFD software STAR-CCM + V.15 (Siemens PLM Software, Plano, TX) was used for the flow simulations.²⁵⁾

Fig. 2.

(a) Dimensions of five-strand tundish equipped with turbulence inhibitor and U-baffle with deflector holes; (b) U-baffle with deflector holes and three design factors A, B and C; (c) turbulence inhibitor [mm]. (Online version in color.)

The volume mesh was generated with the option of trimmer and three prism layers. A base mesh size of 0.003 m was used. The average y⁺ value near the wall boundary is 1.5. A half tundish was simulated through its symmetry plane. The CFD model possesses a total of 2 million trimmer cells in the computing domain. A summary of input parameters and boundary conditions used for computational fluid dynamics (CFD) simulations is given in Table 1.

Table 1. Parameters and boundary conditions used in water model, CFD and prototype.

Water model/CFD model		Prototype
Model volume	0.2 [m3]	Capacity	35–40 [ton] (normal)
Length scale (λ)	1	Length scale (λ)	3
Water density	998 [kgm−3]	Steel density	7020 [kgm−3]
Water viscosity	0.00089 [Pa·s]	Steel viscosity	0.0062 [Pa·s]
Reference pressure	101325 [Pa]	Reference pressure	101325 [Pa]
Temperature	20 [°C]	Temperature	1550 [°C]
Inlet flow rate	0.00028 [m3 s−1]	Casting speed	0.7–1.1 [m/min] (normal)
Outlet (outflow ratio)	0.2:0.4:0.4 (Outlet 1/2/3)	Cross-section area	220 × 260 [mm2] (product)
Wall	No-slip	Liquid level	800 mm (normal)
Free surface	Free slip
Inlet (Mean age)	Zero
Inlet (RTD)	1 (t ≤ 0–2 s), 0 (t > 2 s)
Wall, surface, outlet (Mean age, RTD)	Zero-flux

The discretized equations were solved using the semi-implicit method for the pressure-linked equations (SIMPLE) algorithm. The second-order upwind scheme was applied to calculate the convective terms in the momentum equations. The solution was considered to be converged when the residuals of all solved variables were less than 1 × 10⁻⁴. The under-relaxation parameters for solving pressure, velocity, and turbulence equations were 0.3, 0.7, and 0.8, respectively. The steady-state calculations were performed to solve the transport equations of fluid flow and mean age. To apply the conventional RTD theory, the transient solver was activated to calculate the passive scalar of the tracer concentration.

3.2. Model Validation and Verification

The control of numerical uncertainties is an essential part to develop a reliable CFD model.^26,27,28) In a recent CFD benchmark exercise, the crucial aspects of CFD simulations in the tundish were addressed. They are turbulence model, meshing, boundary conditions, and discretization schemes. The proposed best practice guidelines in tundish simulation were applied in the present study.²⁹⁾

The utilization of an adequately refined mesh was an important step in achieving accuracy in CFD simulations. To investigate the effect of mesh size on the mean age distribution, two mesh sizes with 0.002 m and 0.003 m were compared. The maximum difference in the calculated mean age at the outlets is less than 0.3%. Thus, a reference mesh size of 0.003 m was used for the CFD calculations in this article. An average y⁺ value is 1.5 in the first layer of the mesh near the wall. A convergence study of prism layer thickness demonstrated only a minor impact of the prism layer thickness on the near-wall flow field resolution.

The experimental results from a 1:3 scale water model of the five-strand tundish were used to validate the CFD model.²⁵⁾ The water model was made of plexiglass. The pulse stimulus-response technique was used to obtain the RTD curves. When the flow was stabilized in the water model, a 240 mL saturated NaCl solution was quickly poured into the inlet as a tracer within 2 seconds. The change of tracer concentration was registered continuously at the outlets. The average values of three repetitions were used for data analysis. The flow pattern was observed by the intensity of the dye tracer. The parameters used in the water model were calculated by the Froude similarity criterion, listed in Table 1.

Figure 3 shows the apparatus of the tundish water model (Fig. 3(a)) and the results of the transient tracer dispersions for the tundish equipped with a U-baffle and a turbulence inhibitor (Fig. 3(b)). The tracer comes out through the deflector holes of the U-baffle and tends to flow towards the left-side wall along the top surface. This flow pattern extends the flow path, prolonging the residence time of the stream. The results of transient tracer dispersions together with RTD curves showed that CFD predictions and water model measurements kept a fairly good agreement.²⁵⁾

Fig. 3.

Transient tracer dispersion in a five-strand tundish equipped with U-baffle and turbulence inhibitor (a, water model; b, transient tracer dispersion). (Online version in color.)

3.3. Design of Experiment

Taguchi orthogonal array (OA) L9 (3 factors and 3 levels) was applied to define the design factors (Table 3).^30,31) Nine design cases were proposed with different combinations of the factor levels and displayed in Table 4. Analysis of variance (ANOVA) is applied to determine the effects of design factors in statistical analysis.³²⁾

Table 3. Controlled factors and levels.

Factor	Unit	Level 1	Level 2	Level 3
A (angle of deflector hole on the U-baffle)	[°]	6	8	10
B (distance of deflector hole to tundish bottom)	[mm]	46.5	62	77.5
C (diameter of deflector hole)	[mm]	7.5	10	12.5

Table 4. Taguchi L9 OA, mean age at outlets and signal-to-noise (S/N) ratios.

Case	A	B	C	Outlet1 [s]	Outlet2 [s]	Outlet3 [s]	Std [s]	S/N
C1	1	1	1	748.1	764.3	789.8	21.0	−26.5
C2	1	2	2	767.8	773.9	770.5	3.1	−9.7
C3	1	3	3	803.6	783	744.7	29.9	−29.5
C4	2	1	2	778.2	781.3	758.3	12.5	−21.9
C5	2	2	3	799.2	781.7	748.2	25.9	−28.3
C6	2	3	1	751.3	768	784.8	16.8	−24.5
C7	3	1	3	801.3	785.9	742.8	30.3	−29.6
C8	3	2	1	719.9	787.8	780.5	37.3	−31.4
C9	3	3	2	760.6	785.9	764.3	13.7	−22.7

Minitab V.18 (Minitab, LLC, State College, Pennsylvania, PA) software was used for the DOE analysis.³³⁾ Taguchi DOE analysis uses a classical signal-to-noise (S/N) ratio as a measure for deciding the optimal circumstances. In this study, the design criterion is the standard deviation of flow-weighted mean age at the multiple outlets of the tundish. The smaller-is-better S/N ratio is selected for the optimization, which is expressed by Eq. (17):   

$$S/N=-10log\left(\frac{1}{n}\sum _{i=1}^n{y}_i^2\right)$$(17)

Where y is the performance characteristic value, and n is the observation repeat number.

4. Results

Section 4 consists of three parts. Section 4.1 presents a detailed flow analysis when using the RTD model and the mean age model. Then, the results through Taguchi DOE analysis are given in section 4.2 to suggest the most influential parameters in designing the deflector hole of the U-baffle in the tundish. Lastly, a detailed comparative analysis was carried out in section 4.3 based on the modelling results from the two different models.

4.1. Flow Analysis

Two cases with different configurations were comparatively studied in the five-strand tundish. They are i) Case B1, a bare tundish and ii) Case B2, a tundish with FCD - a U-baffle with deflector holes and a turbulence inhibitor (UB + TI).

4.1.1. Flow Pattern

Figure 4 shows the calculated streamlines for Cases B1 and B2. To visualize the data, the clipping range of velocities is set in the range 0–0.2 ms⁻¹. The recirculation loops are formed in the inlet chamber (Fig. 4(a)). In the bare tundish, the entering flow moves along the bottom and spreads quickly to the outlet regions. When the tundish was equipped with U-baffle and turbulence inhibitor, the entering flow was reoriented (Fig. 4(b)). The appearance of a turbulence inhibitor provided more surface-directed flow, which can improve the inclusions removal and reduce the shear stress on the U-baffle. When the flow was controlled by the U-baffle, the incoming stream could only pass through the deflector holes located in the front and side wall. The streamlines show that the high velocities are created near the deflector holes. The flow velocities in the center of the tundish increased due to the existence of the U-baffle, which improves the mixing in the tundish.

Fig. 4.

Calculated streamlines in tundish (a, Case B1-bare; b, Case B2-UB+TI). (Online version in color.)

4.1.2. Mean Age at Section Planes

Figure 5 displays the contour plots of the mean age on the different section planes through the outlets of the water model. The clipping range of mean age is set in the range 0–900 s. The plots clearly illustrate the change of the mean age distribution from low (blue) to high (red), which also describes the location of dead zones where the mean ages are high (red). The lowest mean age region is located near the inlet, due to the young material feed. Also, a short-circuiting path can be identified by the smaller mean age. As shown in Fig. 5(a), there is a short-circuiting path from the inlet to outlet1. Figure 5(b) shows a lower mean age zone in the inlet chamber, due to the presence of U-baffle and turbulence inhibitor. This indicates a strong mixing in the inlet chamber. In the outlet chamber of Case B2, the mean age is more evenly distributed compared to the bare tundish (Case B1). The jet effect caused by deflector holes is observed on the section plane of outlet 2 where the mean age is smaller, as indicated by the arrow.

Fig. 5.

Contour plots of mean age [s] on the different outlet section planes of water model (a, Case B1-bare; b, Case B2-UB+TI). (Online version in color.)

4.1.3. RTD Measurement

Figure 6 shows the measured conductivity curves at the three outlets of the water model. As shown in Fig. 6(a), for outlet1 in the bare tundish, the measured breakthrough and peak conductivity occur at a relatively earlier time, 4 s and 36 s, respectively. A sharp rising in the tracer concentration indicates a short-circuiting flow path which is undesirable for the mixing in the tundish. Double peaks are shown at outlet2, indicating the presence of a considerable large dead volume. The measured dead volume fraction of three strands is 77% (outlet1), 57% (outlet2) and 58% (outlet3), respectively (Table 2). The measured conductivity curves of Case B2 (UB+TI) are displayed in Fig. 6(b). As listed in Table 2, the measured dead volume fraction of the three strands decrease to 8.3% (outlet1), 8.4% (outlet2) and 7.3% (outlet3), respectively. The mean residence time of outlet1 was prolonged from 186 s to 708 s owing to the installation of U-baffle and turbulence inhibitor. The flow control devices lead to a more uniform flow distribution in the tundish and minimize the dead volume variance among the outlets. It should be mentioned that the measurement uncertainties can affect the results. The possible sources of the measurement uncertainties include the mass flow rate, the amount of tracer injected, the injection rate and the conductivity measurements. An example is that the tracer injected through the inlet takes about several seconds and the numeric predicted breakthrough time at oulet1 is only 4 s in the bare tundish. Therefore, the relative error can become large.

Fig. 6.

Measured conductivity at the outlets of tundish water model. (a) Case B1-bare; (b) Case B2-UB+TI. (Online version in color.)

Table 2. Data analysis of CFD predicted mean age and measured RTD curves.

Case		τ a)	Mean age (CFD)				RTD (water model)
			MA b)	STD (MA) c)	MMA d)	Diff. (τ) e)	RTD	Vd/V f)	STD (RTD)
B1(bare)	Outlet1	802	630.0	124	801.6	0.1%	186	77%	90
	Outlet2		850.5				347	57%
	Outlet3		838.6				335	58%
B2(UB+TI)	Outlet1	772	785.9	18.2	771.6	0.1%	708	8.3%	4.9
	Outlet2		782.6				707	8.4%
	Outlet3		753.2				716	7.3%

^a)  τ: turnover time [s]

^b)  MA: mean age [s]

^c)  STD: standard deviation [s]

^d)  MMA: Mass-flow-averaged mean age = 0.2*mean age(outlet1)+0.4* mean age (outlet2)+0.4* mean age (outlet3) [s]

^e)  Diff(τ) = (1- mass-flow-averaged mean age/turnover time)*100%

^f)  V_d/V = (1-RTD/turnover time)*100%

4.1.4. Mean Age at Outlets

Figure 7 displays the contour plots of mean age at the three outlets of the water model (zoomed-in view) for Case B1 (bare) and Case B2 (UB+TI). For the bare tundish (Case B1), the mean age at outlet1(607.5 s–650.9 s) is much lower than that at outlet2 (826.3 s–868.4 s) and outlet3 (834.5 s–841.4 s), indicating the presence of a short-circuiting low path. This is consistent with the RTD measurements. The incoming flow jet is probably split into two main streams. One part of the stream flows along the tundish bottom to reach the nearest outlet1 and outlet2. However, this partial stream seldom reaches outlet 3 due to the long flow path and the low kinetic energy of the stream. Another partial stream forms a circulation flow after hitting the tundish bottom and then moves towards the top surface and flows back to reach outlet3, outlet2 and, then outlet1. The existence of different outgoing stream paths at outlet1 and outlet2 can explain why the variation of the mean age at outlet1 (43.4 s) and outlet2 (42.1 s) are higher than that at outlet3 (6.9 s). When comparing the variation of mean age at outlets from the RTD measurements in Fig. 6, the RTD variation at outlet2 is 1196 s (from 4 s to 1200 s) while the mean age variation at outlet 2 is only 42.1 s (from 826.3 s to 868.4 s). This can be explained by the fact that the mean age distributions were solved using steady-state conditions, while the RTD curves recorded the temporal distribution for materials passing through the outlets.

Fig. 7.

Contour of mean age (s) at the three outlets (zoomed-in view) for Case B1 (bare) and Case B2 (UB+TI). (Online version in color.)

The variation of mean age at three outlets for Case B2 UB+TI decreases (outlet1:27 s; outlet2: 21 s; outlet3: 1 s) as compared to Case B1. The equipped flow control devices in the tundish improved the consistency of flow among the three strands. As shown in Fig. 7, it suggests that the main stream firstly reaches outlet3 (753 s–754 s), then outlet2 (768 s–789 s), and lastly outlet1 (773 s–800 s). This agrees well with the formed big recirculation flow in the outlet chamber, as illustrated in Fig. 4(b).

Table 2 lists the data analysis of the CFD predicted mean age distribution and the measured RTD curves at the three outlets. According to the definition, the flow-weighted mean age at the outlet should be equal to the turnover time in a flow system with a closed inlet and outlets. The average flow-weighted mean age at the outlets is 0.1% away from the turnover times for both Case B1 and Case B2. These data show an excellent mass balance in the CFD calculations.

4.1.5. Data Analysis

The turnover time (τ) of Case B1 and Case B2 is 802 s and 772 s, respectively. Installation of U-baffle and turbulence inhibitor decreases the fluid volume in the tundish, which leads to a decreased turnover time (Case B2). The standard deviation of the mean age at three outlets decreases from 124 s (Case B1) to 18.2 s (Case B2). The result reveals that the flow control devices (UB+TI) can improve the flow consistency in the tundish. The standard deviation for the measured RTD data decreases from 90 s (Case B1) to 4.9 s (Case B2), while keeping the same tendency as the predicted mean age. A discrepancy between the mean age model and the RTD model is observed. The calculated mean age value at outlet3 (753.2 s) in Case B2 is lower than at outlet2 (782.6 s), while the RTD value at outlet3 (716 s) is slightly higher than at outlet2 (707 s). The possible sources of the discrepancy are i) the experimental uncertainties of the water model and ii) the numerical uncertainties of the CFD model. In the water model experiment, the measurement uncertainties are influenced by the amount of tracer injected, the injection rate and the conductivity measurements. A typical example is that the denser NaCl tracer may disturb the flow pattern to a limited extent.³⁴⁾ A sink slow can be formed due to the high density of the tracer compared to that of water. This phenomenon was not modelled in the CFD calculations of the mean age. In section 3.2, the normal sources of numerical uncertainties are discussed, including the turbulence model, meshing, and discretization schemes. There is another important parameter, the turbulent Schmidt number, which needs to be justified. It represents the ratio of the turbulent eddy viscosity and the turbulent mass diffusivity. A constant value (Sc = 0.7) was used to solve the tracer concentration equation (Eq. (13)) since it is set as default in the CFD software. In author’s opinion, this value needs to be carefully investigated in future work since it may have a significant effect on the predictions.

4.2. DOE Analysis

Taguchi L9 OA (3 factors and 3 levels) was applied to define the design cases. In Table 3, three factors with high (level 3), medium (level 2), and low (level 1) values were considered in DOE. Level 3 and level 1 are plus/minus 25% from level 2, respectively. Nine CFD cases were proposed with different combinations of the factor levels, listed in Table 4.

The mean ages at three outlets for the selected cases were calculated from the CFD model. The standard deviation of mean age among the three outlets and their corresponding S/N ratios are listed in Table 4. The standard deviation of the mean age varied from 3.1 s (Case C2) to 37.3 s (Case C8). The response table for the design factors A (angle of deflector hole), B (height of deflector hole), and C (diameter of deflector hole) was created in an integrated manner, and the results are given in Table 5. A greater S/N value corresponds to a better performance. The contribution order of the design factors is the following: C > A > B. From the analysis of the S/N ratio (Fig. 8), the optimum levels of the design factors are determined as A1B2C2 (A = 6°, B = 62 mm, and C = 10 mm).

Table 5. Response table for signal-to-noise ratios of the standard deviation of mean age at outlets.

Level	A	B	C
1	−21.89	−26.00	−27.45
2	−24.89	−23.14	−18.12
3	−27.92	−25.57	−29.14
Delta	6.03	2.87	11.02
Rank	2	3	1

Fig. 8.

Mean of S/N ratio for each factor at levels 1–3. (Online version in color.)

ANOVA was chosen to verify the results from signal-to-noise ratio regarding the significance of the design factors on design response (the standard deviation of flow-weighted mean age at three different outlets), which is C > A > B. As listed in Table 6, the term DF represents the degree of freedom. The adjusted sums of squares (Adj SS) indicate the relative importance of each factor. The adjusted mean squares (Adj MS) measure how much variation a term explains. The factor with the biggest coefficient and the biggest sum of squares has the greatest impact. The results of ANOVA show that factors A, B, and C influenced the response values by 17.2%, 0.6%, and 65.9%, respectively. The contribution rate of error was 16.3%, which comprises different error parts, such as the external sources and the calculation errors. Factor C (diameter of deflector hole) had the highest percentile (65.9%) contribution to the response. The significance of the design factors is determined as C > A > B based on the F-value, which is consistent with the results from the Taguchi analysis.

Table 6. ANOVA table for design analysis.

Source	DF	Adj SS	Adj MS	Contribution	F-Value
A	2	158.644	79.322	17.2%	1.06
B	2	5.929	2.965	0.6%	0.04
C	2	606.955	303.478	65.9%	4.07
Error	2	149.258	74.629	16.3%
Total	8	920.787		100%

A confirmation test is usually necessary to validate the conclusion drawn during the DOE analysis phase. From the Taguchi analysis of the S/N ratio (Fig. 8), the optimum levels of the design factors are determined as A1B2 C2, which is the same as the design of Case C2 in the Taguchi L9 OA. Therefore, no additional simulation case is needed as the confirmation test. Since the mean age theory is relatively new in the tundish applications, a detailed analysis between the mean age model and the conventional RTD model was carried out. Two cases of the lowest (Case C2) and the highest (Case C8) standard deviation of mean age at the outlets were selected for the comparison.

4.3. Detailed Analysis (Mean Age vs. RTD)

Figure 9 displays the calculated RTD curves at the three outlets of Case C2 and Case C8. The difference in RTD curves between the three outlets is smaller in Case C2. The RTD curve at outlet1 of Case C8 is apart from the other two curves. The calculated dead volume fraction of the strand to outlet1 in Case C8 is 11.1%, which is much higher than for the other two strands (2.6% and 3.4%).

Fig. 9.

Displays the calculated RTD curves at the three outlets of (a) Case C2 and (b) Case C8. (Online version in color.)

Figure 10 shows the contour plots of mean age at the three outlets of Case C2 and Case C8. The mean age at outlet1 (703.5 s–740.9 s) of Case C8 is lower than for the other two outlets (outlet2: 755.6 s–805.3 s; outlet3: 779.9 s–781.5 s), indicating a shorter residence of the melt in the tundish, which can cause a higher dead volume fraction. Furthermore, the deviation of mean ages at three outlets of Case C8 (101.8 s, from 703.5 s to 805.3 s) is bigger than that of Case C2 (34.6 s, from 754.7 s to 789.3 s). This is consistent with the observation of RTD curves in Fig. 9, where the RTD curve at outlet1 of Case C8 is apart from the other two curves. The calculated mean age has a tighter distribution compared to the calculated RTD curves.

Fig. 10.

Contour of mean age (s) at the three outlets (zoomed-in view) for Case C2 and Case C8. (Online version in color.)

Table 7 lists the data analysis results of the mean age distribution and the RTD curves of Case C2 and Case C8. The flow-weighted mean age at the outlets is 0.1% away from the turnover times for both Case C2 and Case C8. The geometric parameters of the baffle hole lead to different mean age distribution and RTD in the two cases. The standard deviation of mean age at three outlets increases from 3.1 s (Case C2) to 37.3 s (Case C8). The result reveals that the deflector hole on the U-baffle has a significant effect on the flow distribution in the tundish. The standard deviation of the calculated mean residence-time increases from 3.2 s (Case B1) to 36.5 s (Case B2), which is similar to the standard deviation of the calculated mean age (deviations: <3.3%). It is interesting to note that both the mean age and RTD can identify the small deviation between the outlets and draw the same conclusion, for example, the deviation between outlet2 and outlet3 in Case C2.

Table 7. Data analysis of mean age distribution and RTD curves calculated by CFD.

Case		τ	Mean age (CFD)				RTD (water model)
			MA b)	STD (MA) c)	MMA d)	Diff. (τ) e)	RTD	Vd/V f)	STD (RTD)
C2	Outlet1	772	767.8	3.1	771.3	0.1%	739	4.2%	3.2
	Outlet2		773.9				745	3.5%
	Outlet3		770.5				744	3.6%
C8	Outlet1	772	719.9	37.3	771.3	0.1%	686	11.1%	36.5
	Outlet2		787.8				752	2.6%
	Outlet3		780.5				746	3.4%

^a)  τ: turnover time [s]

^b)  MA: mean age [s]

^c)  STD: standard deviation [s]

^d)  MMA: Mass-flow-averaged mean age = 0.2*mean age(outlet1)+0.4* mean age (outlet2)+0.4* mean age (outlet3) [s]

^e)  Diff(τ) = (1- mass-flow-averaged mean age/turnover time)*100%

^f)  V_d/V = (1-RTD/turnover time)*100%

5. Discussions

The mean age model provides a complete three-dimensional distribution of mean age inside the multi-strand tundish, which can easily identify the size and location of the undesired regions such as dead zones or short-circuiting flow paths. Also, from the variance results of the mean age at multiple outlets, the consistency of flow in the tundish can be quantitatively evaluated.

Compared to the conventional RTD method, the mean age model requires only a computing time of 0.6 hours (Case C2) using an engineering workstation with 12-core and 64GB of RAM, while a transient RTD solution requires 60 hours on the same computer. The computing resource effectivity is significantly improved by using the proposed new method.

When solving a time-dependent equation, the numerical error introduced at a time step can propagate to the final solution. In this regard, the use of the mean age model can provide more accurate results compared to when using the RTD model, since the time integration has been done analytically during the derivation of the governing equation. The numerical error from the transient solution can therefore be eliminated by using the mean age model. However, it is also worth noting that the mean age model has a limitation since it only provides a steady-state solution using the averaged information in the temporal domain. Thus, if it is more interesting to study transient phenomena, the RTD model has the advantage over the mean age model.

6. Conclusions

A new method was developed to quantitatively characterize the flow consistency in the multi-strand tundish. This method was based on the recently developed mean age theory for the mixing study in the single-strand tundish. Theoretical analysis was carried out to extend the application of the mean age theory, from the single-strand tundish to the multi-strand tundish. Case studies were carried out to test the applicability of the new method in a five-strand tundish. CFD model and Taguchi DOE were applied to optimize the flow control devices. The main conclusions can be drawn as shown below:

• The computing time using the new mean age model in the five-strand tundish is two orders of magnitude faster compared to the conventional model of RTD. An accurate mass balance (deviations: 0.1%) in CFD simulations was observed by using the mean age model.

• The standard deviation of the flow-weighted mean age at the multiple outlets decreases with the installation of a U-baffle and a turbulence inhibitor. The overall comparison between the simulation and the experiment is satisfactorily close (deviations: < 3.3%).

• Three design factors were considered in the Taguchi DOE analysis: i) A, deflector hole’s angle, ii) B, deflector hole’s height, and iii) C, deflector hole’s diameter. The calculated standard deviation of mean age at the three outlets varies from 3.06 s (Case C2) to 37.27 s (Case C8). The optimum levels of the design factors were proposed as A1B2C2 (A = 6°, B = 62 mm, and C = 10 mm). The results of ANOVA revealed that the significance of the design factors is as follows: C (65.9%) > A (17.2%) > B (0.6%).

• The conclusions of the design analysis are almost identical when comparing the mean age model with the RTD model. The mean age model shows a significant advantage that a complete three-dimensional picture of the mixing states inside the tundish can be visualized. Undesired flow regions such as dead zones or short-circuiting flow paths can be easily identified.

• The coupling of CFD and DOE analysis allows the exploration of tundish designs on a larger scale. The new mean age method can significantly improve the efficiency and accuracy during the scientific computing. The proposed method in this article can be widely used in the design of metallurgical reactors.

Acknowledgements

The author would like to acknowledge the Swedish Foundation for Strategic Research (SSF) for their financial support via Strategic Mobility Program (2019). The author would like to thank Professor Qiang Yue (Anhui University of Technology) for his valuable contribution to the water model test data for the model validation. The help of Professor Pär Jönsson (KTH Royal Institute of Technology) with the preparation of the manuscript is gratefully acknowledged.

Funding

Open access funding is provided by Royal Institute of Technology.

Conflict of Interests

The corresponding author states that there is no conflict of interest.

Nomenclature and Abbreviations

a: Mean age [s]

a_o: Flow-weighted mean age at outlet [s]

a_oi: Flow-weighted mean age at the ith outlet [s]

a_t: Total average mean age [s]

C: Tracer concentration

D_eff: Effective diffusivity [m² s⁻¹]

D_m: Molecular diffusivity [m²]

g: Gravity [m s⁻²]

G_k: Generation of k

k: Turbulent kinetic energy [m² s⁻²]

N: Number of outlets

Q: Volumetric flow rate [m³ s⁻¹]

Sc: Schmidt number

S_n: Standard deviation of mean age

Sε: Source term of ε

S_k: Source term of k

u: Velocity of the flow field [m s⁻¹]

V: System volume [m³]

V_d/V: Dead volume fraction

y: Performance characteristic value

τ: Theoretical residence time (turnover time) [s]

υ: Kinematic viscosity [m² s⁻¹]

μ: Molecular viscosity [Pa s]

μ_t: Turbulent viscosity [Pa s]

ρ: Density [kg m⁻³]

ε: Turbulent energy dissipation rate [m² s⁻³]

σ_k: Turbulent Prandtl number for k

σ_ε: Turbulent Prandtl number for ε

λ: Length scale

Adj SS: Adjusted Sums of Squares

Adj MS: Adjusted Mean Squares

ANOVA: Analysis of Variance

CAD: Computational Aided Design

CC: Continuous Casting

CFD: Computational Fluid Dynamics

DF: Degree of Freedom

Diff: Difference

DOE: Design of Experiment

FCD: Flow Control Device

MMA: Mass-flow-averaged Mean Age

OA: Orthogonal Array

RTD: Residence-time distribution

S: Surface

STD: Standard Deviation

S/N: Signal-to-noise ratio

References

• 1)   J.  Szekely and  O.  Ilegbusi: The Physical and Mathematical Modeling of Tundish Operations, Springer-Verlag, New York, (1989), 1.
• 2)   D.  Mazumdar and  R. I. L.  Guthrie: ISIJ Int., 39 (1999), 524. https://doi.org/10.2355/isijinternational.39.524
• 3)   Y.  Sahai: Metall. Mater. Trans. B, 47 (2016), 2095. https://doi.org/10.1007/s11663-016-0648-3
• 4)   K.  Chattopadhyay,  M.  Isac and  R. I. L.  Guthrie: ISIJ Int., 51 (2011), 759. https://doi.org/10.2355/isijinternational.51.759
• 5)   I. V.  Samarasekera and  J. K.  Brimacombe: Can. Metall. Q., 38 (1999), 347. https://doi.org/10.1179/cmq.1999.38.5.347
• 6)   R.  Agarwal,  M.  Singh,  R.  Kumar,  B.  Ghosh and  S.  Pathak: World J. Mech., 9 (2019), 29. https://doi.org/10.4236/wjm.2019.92003
• 7)   L.  Neves and  R. P.  Tavares: Ironmaking Steelmaking, 44 (2017), 559. https://doi.org/10.1080/03019233.2016.1222122
• 8)   C.  Damle and  Y.  Sahai: ISIJ Int., 35 (1995), 163. https://doi.org/10.2355/isijinternational.35.163
• 9)   H.  Tang,  J.  Liu,  K.  Wang,  H.  Xiao,  A.  Li and  J.  Zhang: Acta Metall. Sin., 57 (2021), 1229. https://doi.org/10.11900/0412.1961.2021.00046
• 10)   M.  Bensouici,  A.  Bellaouar and  K.  Talbi: J. Iron Steel Res. Int., 16 (2009), 22. https://doi.org/10.1016/S1006-706X(09)60022-4
• 11)   K. J.  Craig,  D. J.  de Kock,  K. W.  Makgata and  G. J.  de Wet: ISIJ Int., 41 (2001), 1194. https://doi.org/10.2355/isijinternational.41.1194
• 12)   A.  Cwudziński: Steel Res. Int., 81 (2010), 123. https://doi.org/10.1002/srin.200900060
• 13)   P. K.  Jha and  S. K.  Dash: Int. J. Numer. Methods Heat Fluid Flow, 14 (2004), 953. https://doi.org/10.1108/09615530410544283
• 14)   L. C.  Zhong,  L. Y.  Li,  B.  Wang,  L.  Zhang,  L. X.  Zhu and  Q. F.  Zhang: Ironmaking Steelmaking, 35 (2008), 436. https://doi.org/10.1179/174328108X318365
• 15)   D.  Chen,  X.  Xie,  M.  Long,  M.  Zhang,  L.  Zhang and  Q.  Liao: Metall. Mater. Trans. B, 45 (2014), 392. https://doi.org/10.1007/s11663-013-9941-6
• 16)   D. Y.  Sheng: Metall. Mater. Trans. B, 53 (2022), 2735. https://doi.org/10.1007/s11663-022-02563-w
• 17)   P. V.  Danckwerts: Chem. Eng. Sci., 8 (1958), 93. https://doi.org/10.1016/0009-2509(58)80040-8
• 18)   D. B.  Spalding: Chem. Eng. Sci., 9 (1958), 74. https://doi.org/10.1016/0009-2509(58)87010-4
• 19)   M.  Liu: Can. J. Chem. Eng., 89 (2011), 1018. https://doi.org/10.1002/cjce.20563
• 20)   M.  Sandberg: Build. Environ., 16 (1981), 123. https://doi.org/10.1016/0360-1323(81)90028-7
• 21)   D.  Russ and  R.  Berson: Chem. Eng. Sci., 141 (2016), 1. https://doi.org/10.1016/j.ces.2015.10.030
• 22)   S. V.  Patankar: Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, (1980), 16.
• 23)   T. H.  Shih,  W. W.  Liou,  A.  Shabbir,  Z.  Yang and  J.  Zhu: Comput. Fluids, 24 (1995), 227. https://doi.org/10.1016/0045-7930(94)00032-T
• 24)   D. Y.  Sheng and  Q.  Yue: Metals, 10 (2020), 1084. https://doi.org/10.3390/met10081084
• 25)  STAR-CCM + User Guide Ver. 15.04, Siemens Digital Industries Software, Munich, (2019).
• 26)   P. J.  Roache: J. Fluids Eng., 138 (2016), 101205. https://doi.org/10.1115/1.4033979
• 27)   M.  Casey and  T.  Wintergerste: ERCOFTAC Best Practice Guidelines, ERCOFTAC Special Interest Group on “Quality and Trust in Industrial CFD”, ERCOFTAC, Bushey, (2000), 18.
• 28)   H.-J.  Odenthal,  M.  Javurek and  M.  Kirschen: Steel Res. Int., 80 (2009), 264. https://doi.org/10.2374/SRI08SP163
• 29)   D. Y.  Sheng: Materials, 14 (2021), 5453. https://doi.org/10.3390/ma14185453
• 30)   G.  Taguchi,  S.  Chowdhury and  Y.  Wu: Taguchi’s Quality Engineering Handbook, John Wiley and Sons, Inc., Hoboken, NJ, (2005), 56.
• 31)   G.  Taguchi: Proc. Int. Conf. on Quality Control, (Tokyo), (1978), 1.
• 32)   R. A.  Fisher: Statistical Methods for Research Workers, Oliver and Boyd, Edinburgh, (1925), 25.
• 33)  Minitab, Inc.: Website of Minitab, (2017), https://www.minitab.com, (accessed 2017-06-07).
• 34)   C.  Chen,  L. T. I.  Jonsson,  A.  Tilliander,  G. G.  Cheng and  P. G.  Jönsson: Metall. Mater. Trans. B, 46 (2015), 169. https://doi.org/10.1007/s11663-014-0190-0