Residence Time Distribution (RTD) Applications in Continuous Casting Tundish: A Review and New Perspectives

Abstract

The continuous casting tundish is a very important metallurgical reactor in continuous casting production. The flow characteristics of tundishes are usually evaluated by residence time distribution (RTD) curves. At present, the analysis model of RTD curves still has limitations. In this study, we reviewed RTD curve analysis models of the single flow and multi-flow tundish. We compared the mixing model and modified combination model for RTD curves of single flow tundish. At the same time, multi-strand tundish flow characteristics analysis models for RTD curves were analyzed. Based on the RTD curves obtained from a tundish water experiment, the applicability of various models is discussed, providing a reference for the selection of RTD analysis models. Finally, we proposed a flow characteristics analysis of multi-strand tundish based on a cumulative time distribution curve (F-curve). The F-curve and intensity curve can be used to analyze and compare the flow characteristics of multi-strand tundishes. The modified dead zone calculation method is also more reasonable. This method provides a new perspective for the study of multi-strand tundishes or other reactor flow characteristics analysis models.

1. Introduction

The continuous casting tundish is an important connecting tool in the continuous casting process; it is the last metallurgical reactor vessel that contains refractory materials. It is vital for functions such as diversion, continuous pouring, decompression, and impurity removal [1,2]. The stimulus response method is the main method used to quantitatively investigate molten steel flow in a tundish. The flow state of molten steel in a tundish can be judged by analyzing the residence time distribution curve (RTD curve). A tracer is either instantaneously added or pulsed at the inlet, and the change in tracer concentration over time is measured at the outlet. The corresponding concentration change curve that is drawn is the RTD curve.

To obtain the tundish with the best metallurgical performance, flow control devices are usually set in the tundish, such as a turbulence controller, retaining wall, air curtain, etc. In the optimization process of the continuous casting tundish control device, the use of RTD curve analysis is commonly used to assess the flow characteristics of the tundish. Mean age theory is introduced to the tundish to characterize the mixing performance of molten steel [3]. Obviously, the accurate measure of the flow of molten steel in a tundish depends on the quality and accuracy of the model used. Sahai and Emi [4] revised the analysis model of the single flow tundish. This improved model has been widely used around the world. Their method of quantitatively studying the fluid flow characteristics of a tundish is more commonly used in single-strand casting tundishes [5,6,7,8]. However, with the development of metallurgical technology, more multi-flow tundishes are being used in production. Although it is not reasonable to apply the single-strand tundish model to the multi-strand tundish, there is no model that is specific to multi-strand tundish RTD analysis [9,10]. When employing a multi-strand tundish, the molten steel flowing through the tundish is required to have good temperature and composition consistency when entering the mold of each continuous caster, and must ensure the removal rate of inclusions in each strand. These functions make the tundish structural design more complex and diverse, and put forward higher requirements for the multi-strand tundish model.

This research compares existing single-flow tundish flow characteristics models with a multi-flow tundish flow characteristics model. The rationale of each model and their adaptability to multi-strand experimental data are assessed to provide a reference for the appropriate selection of tundish flow characteristics models. Then, we propose an analysis method based on the cumulative residence time distribution curve. This method is based on the average RTD curve measured at the outlet and analyzes the flow characteristics of the tundish by changing the cumulative RTD profile curve.

The application of RTD in single-strand and multi-strand continuous casting tundishes is investigated in detail. The volume fraction of the plug flow, mix flow, and dead zone are discussed and compared. Moreover, we present the current challenges faced by these methods, and propose ideas for improvement. The purpose of this paper is to analyze existing RTD solutions and put forward new suggestions that can be used as a reference for continuous casting tundish.

2. Residence Time Distribution Function

According to the theory of metallurgical reaction engineering, the residence time distribution function is recorded as E(t), also known as the residence time distribution density function or the exiting life distribution density function. The residence time distribution function is a continuous function. E(t) is defined as the fraction of fluid for which the residence time satisfies $t_1{<t<t}_2$ in the outlet flow.

$$\frac{\Delta N}{N}={{\displaystyle \int }}_{t_1}^{t_2}E\left(t\right)\mathit{dt}$$

(1)

E(t)dt is the fraction of the total amount of fluid entering the reactor at the same time with the residence time being t to t + dt. All fluids entering the reactor will eventually flow out as time increases. Equation (2) is the normalization condition of the distribution function.

$${{\displaystyle \int }}_0^{\infty }E\left(t\right)\mathit{dt}=1$$

(2)

The average residence time $\overline{t}$ of the fluid in the reactor is dimensionless:

$$\theta =t/\overline{t}$$

(3)

The residence time distribution function E(t), is represented by the dimensionless time θ as a variable:

$$\frac{\Delta N}{N}={{\displaystyle \int }}_{t_1}^{t_2}E\left(\theta \right)d\theta$$

(4)

Normalization condition:

$${{\displaystyle \int }}_0^{\infty }E\left(\theta \right)d\theta =1$$

(5)

where $d\theta =\overline{t}dt$,

$$E\left(\theta \right)=\overline{t}E\left(t\right)$$

(6)

For a constant volume system, assuming that the volume of the reactor is V_r and the volume flow of the system is q_v, the average residence time τ can be acquired.

$${\tau =V}_{\mathrm{r}}/q_{\mathrm{v}}$$

(7)

For a variable volume system, the mean residence time is the mathematical expectation of the distribution function E:

$$\overline{t}=\frac{{{\displaystyle \int }}_0^{\infty }\mathit{tE}\left(t\right)\mathit{dt}}{{{\displaystyle \int }}_0^{\infty }E\left(t\right)\mathit{dt}}={{\displaystyle \int }}_0^{\infty }\mathit{tE}\left(t\right)\mathit{dt}$$

(8)

Dispersion:

$${\sigma }^2=\frac{{{\displaystyle \int }}_0^{\infty }{(t- \overline{t})}^{2}E\left(t\right)\mathit{dt}}{{{\displaystyle \int }}_0^{\infty }E\left(t\right)\mathit{dt}}={{\displaystyle \int }}_0^{\infty }{t}^2{E\left(t\right)dt- \overline{t}}^2$$

(9)

The flow of molten steel in the tundish may be very slow or even have a local dead zone. The residence time of molten steel in the tundish is usually called the actual residence time, which is the mathematical expectation $\overline{t}$ of the obtained function E(t). The difference between the theoretical average residence time $\tau$ and the actual average residence time $\overline{t}$ is usually used as a parameter for judging the size of the dead zone. The flow characteristics of molten steel in a tundish can be obtained by analyzing the characteristics of the E curve obtained by the experiment or simulation. It is a common method to obtain the flow characteristics of molten steel in tundish by assessing the shape and characteristics of the E curve.

Table 1. The dead volume method for continuous casting tundishes.

Year	Author	Number of Strands	Dead Volume Method	Computational Model
1981	Kemeny F. [11]	Single-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}{=1- \overline{\mathsf{\theta }}}_{\mathrm{c}}$
1986	Sahai Y. [12]	Single-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}{=1- \overline{\mathsf{\theta }}}_{\mathrm{c}}$
1996	Sahai Y. [13]	Single-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{ \overline{\mathsf{\theta }}}_{\mathrm{c}}$
1997	Mazumdar Dipak [14]	Multi-Strand	C	$V_d=1-\frac{{{\displaystyle \int }}_0^{\infty }C\theta d\theta }{{{\displaystyle \int }}_0^{\infty }Cd\theta }$
2004	Vargas -Zamora [15]	Single-Strand	F	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-{\theta }_a\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}$
2005	Zheng Shuguo [16]	Multi-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{1}{N}(\frac{{\mathrm{Q}}_{1\mathrm{a}}}{{\mathrm{Q}}_1}{ \overline{\mathsf{\theta }}}_{1\mathrm{c}}+\frac{{\mathrm{Q}}_{2\mathrm{a}}}{{\mathrm{Q}}_2}{ \overline{\mathsf{\theta }}}_{2\mathrm{c}}+\dots +\frac{{\mathrm{Q}}_{\mathrm{Na}}}{{\mathrm{Q}}_N}{ \overline{\mathsf{\theta }}}_{\mathrm{Nc}})$
2005	Solorio-Díaz G. [17]	Two-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\mathsf{\theta }\frac{\mathrm{Q}}{V}$
2007	Kumar Anil [18]	Multi-Strand	C	${\mathrm{V}}_{\mathrm{d}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{\theta }_{mean}$
2008	Zhu Mingmei [19]	Multi-Strand	F	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{Q_a}{Q} \frac{\overline{t}}{\tau }$
2009	Pan Hongwei [20]	Multi-Strand	F	$\frac{V_{d,i}}{V}={{\displaystyle \int }}_2^{\infty }{E}_i\left(\theta \right)d\theta ,\infty \frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}={\displaystyle {\displaystyle \sum }_{i=1}^n}\frac{V_{d,i}}{V}$
2009	Chen Yuanqing [21]	Multi-Strand	F	$\frac{V_d}{V}=1-{{\displaystyle \int }}_0^{2}E\left(\theta \right)d\theta ·\frac{\overline{t}}{\tau }$
2009	Bensouici [22]	Single-Strand	C	$\frac{V_d}{V}=1-\left[{\displaystyle {\displaystyle \sum }_{\theta =0}^{\theta =2}}{C}_{\theta }\Delta \theta \right]·\frac{{{\displaystyle \sum }}_{\theta =0}^{\theta =2}\theta C_{\theta }}{{{\displaystyle \sum }}_{\theta =0}^{\theta =2}{C}_{\theta }}$
2010	Lei Hong [23]	Multi-Strand	C	$V_d=1-\frac{{{\displaystyle \int }}_0^{2\tau }Ctdt}{{{\displaystyle \int }}_0^{\infty }Cdt}\overline{\theta }$
2010	Cwudzinski A. [24]	Single-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{ \overline{\mathsf{\theta }}}_{\mathrm{c}},{ \overline{\mathsf{\theta }}}_{\mathrm{c}}=\frac{{{\displaystyle \sum }}_{\theta =0}^2{C}_i{\theta }_t}{{{\displaystyle \sum }}_{\theta =0}^2{C}_i}$
2010	Zhong Liang-cai [25]	Multi-Strand	C	$\frac{V_{d,i}}{V_i}=1-\frac{{\mathrm{Q}}_{\mathrm{a},\mathrm{i}}}{{\mathrm{Q}}_{\mathrm{i}}}{ \overline{\mathsf{\theta }}}_{\mathrm{c},\mathrm{i}}$
2011	Ning Ding [26]	Single-Strand	F	$V_d=1-\frac{t_a}{t_s}$
2012	Mishra [27]	Multi-Strand	C	$\frac{V_d}{V}=1-\frac{t_r}{\tau }$
2013	Sengupta A. [28]	Multi-Strand	C	$V_d=1-\frac{{{\displaystyle \sum }}_{j=0}^{j=2t_r}{C}_j{t}_j}{t_r{{\displaystyle \sum }}_{j=0}^{j=2t_r}{C}_j}$
2013	Sanchez-Ramirez R. [29]	Single-Strand	C	$\frac{V_d}{V}=1-{{\displaystyle \int }}_0^{2}E\left(\theta \right)\theta d\theta$
2013	Chen D. [30]	Single-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{ \overline{\mathsf{\theta }}}_{\mathrm{c}}$
2015	Chang S. [31]	Multi-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d},\mathrm{i}}}{V_i}=1-\frac{{\mathrm{Q}}_{\mathrm{a},\mathrm{i}}}{{\mathrm{Q}}_{\mathrm{i}}}{ \overline{\mathsf{\theta }}}_{\mathrm{c},\mathrm{i}}$
2015	Lei Hong [32]	Two-Strand	C	$\mathrm{k}{\mathrm{V}}_{\mathrm{d}}=1-\frac{\overline{t}}{\tau }$
2015	Guocheng Wang [33]	Single-Strand	F	${\mathrm{V}}_{\mathrm{d}}=1-\overline{\frac{t_c}{t}}$
2015	Cui Heng [34]	Multi-Strand	F	$\frac{V_d}{V}=\left(1-{\displaystyle {\displaystyle \sum }_{i=1}^n}\left({{\displaystyle \int }}_0^{\theta }{E}_i\left(\theta \right)d\theta \right)\right)d\theta$
2016	Fei He [35]	Multi-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{ \overline{\mathsf{\theta }}}_{\mathrm{c}}$
2016	Li D. X [36]	Two-Strand	F	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-{{\displaystyle \int }}_0^2[1-F\left(\theta \right)d\theta ]$
2016	Zheng Shu-guo [37]	Multi-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{1}{N}(\frac{{\mathrm{Q}}_{1\mathrm{a}}}{{\mathrm{Q}}_1}{\overline{\mathsf{\theta }}}_{1\mathrm{c}}+\frac{{\mathrm{Q}}_{2\mathrm{a}}}{{\mathrm{Q}}_2}{ \overline{\mathsf{\theta }}}_{2\mathrm{c}}+\dots +\frac{{\mathrm{Q}}_{\mathrm{Na}}}{{\mathrm{Q}}_N}{ \overline{\mathsf{\theta }}}_{\mathrm{Nc}})$
2016	Li Dongxia [38]	Multi-Strand	F	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-{{\displaystyle \int }}_0^2\left(1-F_1-F_2-\dots -F_k\right)d\theta$
2017	Xiao-ying Wang [39]	Multi-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}=1-\frac{1}{N}(\frac{{\mathrm{Q}}_{1\mathrm{a}}}{{\mathrm{Q}}_1}\frac{\overline{t_1}}{\tau }+\frac{{\mathrm{Q}}_{2\mathrm{a}}}{{\mathrm{Q}}_2}\frac{\overline{t_2}}{\tau }+\dots +\frac{{\mathrm{Q}}_{\mathrm{Na}}}{{\mathrm{Q}}_N}\frac{\overline{t_N}}{\tau })$
2018	Braga B.M. [40]	Multi-Strand	F	$V_d=1-\frac{{{\displaystyle \int }}_0^{x}t{E}_{m}dt}{\tau {{\displaystyle \int }}_0^{\infty }{E}_{m}dt}=\left(1+x/\tau \right)e^{-\frac{x}{\tau }}$
2018	Harnsihacacha Anawat [41]	Single-Strand	C	${\mathrm{V}}_{\mathrm{d}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{\theta }_{mean}$
2020	Tkadleckova Marketa [42]	Multi-Strand	C	$\frac{V_d}{V}=1-\frac{\int c\tau d\tau }{\overline{\tau }\int cd\tau }$
2020	Sheng D.Y. [43]	Multi-Strand	F	$\frac{{\mathrm{V}}_{\mathrm{d}}}{V_t}=1-\frac{\overline{\tau }}{\tau }$
2020	Fang Qing [44]	Multi-Strand	C	$T_a=\frac{\sum t_i{c}_i\Delta t_i}{\sum c_i\Delta t_i},\frac{V_{d,i}}{V}={{\displaystyle \int }}_{2T_a}^{\infty }{C}_{i}dt,\frac{{\mathrm{V}}_{\mathrm{d}}}{\mathrm{V}}={\displaystyle {\displaystyle \sum }_{i=1}^n}\frac{V_{d,i}}{V}$
2021	Sheng D.Y. [45]	Single-Strand	C	$\frac{{\mathrm{V}}_{\mathrm{d}}}{V}=1-\frac{\overline{\tau }}{\tau }$
2022	Yadav J.N. [46]	Single-Strand	C	${\mathrm{V}}_{\mathrm{d}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{\theta }_{mean}$
2022	Yadav J.N. [46]	Single-Strand	C	${\mathrm{V}}_{\mathrm{d}}=1-\frac{{\mathrm{Q}}_{\mathrm{a}}}{\mathrm{Q}}{\theta }_{mean}$

shows the summary of using this dead volume method on continuous casting tundish in previous work. The response time and peak time of the tracer in the tundish can be directly measured, and the average residence time and dead volume fraction can be calculated by analyzing the RTD curve of the tundish. These parameters can be used to assess the performance of these models. At present, there are is no single method used to calculate the dead zone volume in the multi-strand continuous casting tundish.

3. Model of Flow Characteristics in the Tundish

3.1. Single-Strand Tundish RTD Analysis Modes

The most widely used model for the tundish is a combined model that was proposed by Sahai and Emi [13] in 1996. According to the theory, a tundish can be divided into three areas, namely the plug flow, fully mixed flow, and dead flow. A tundish can be seen as a combination of various ideal reactors. The plug and the complete mixing zone are called the active zone in the tundish. Figure 1 shows the classical RTD curve of the combined model theory, in which C is dimensionless concentration; θ is dimensionless time; θ_min is dimensionless minimum residence time; C_max is the maximum concentration of dimensionless peak; V, V_m, and V_p are the total volume of the tundish, plug, and mixing zone, respectively.

The dead volume fraction of the tundish is a very important parameter. According to the combined model, there are two definitions of dead zone: (1) The area in which the molten steel is in a stagnant state and does not exchange with external fluid, and (2) The area in which the fluid flow is particularly slow, resulting in a long residence time of the molten steel. Generally, the dead volume is defined as a fluid whose residence time in the container exceeds twice the theoretical residence time. Figure 2 shows the relationship between dead zones and active zones [16,47], where V_a is the volume of active area and V_d is the dead zone volume. Q_a is the volume flow of the active zone and Q_d is the volume flow of the dead zone.

In 1986, Sahai and Ahuja [12] modified their model and proposed an improved combined model. The improved combined model is as follows:

$$\frac{V_d}{V}=1-\theta$$

(10)

$$\frac{V_{\mathrm{p}}}{V}=\frac{1}{2}\left({\theta }_{\min }{+\theta }_{\mathrm{peak}}\right)$$

(11)

$$\frac{V_{\mathrm{m}}}{V}=1-\frac{V_{\mathrm{p}}}{V}-\frac{V_{\mathrm{d}}}{V}$$

(12)

where θ is dimensionless time, θ_min is the dimensionless minimum residence time, and θ_peak is dimensionless peak time.

As the original model considered the dead zone to be completely stagnant and invariable, the calculated dead zone results were often very large. Sahai and Emi [13] proposed a modified combination model based on the mixture model. This method measures the dead zone by looking at fluid that has a residence time in the tundish exceeding twice the theoretical residence time, which effectively reduces the error of the previous model in calculating the dead zone. The calculation method of each volume fraction of the modified combined model is as follows:

$$\frac{V_{\mathrm{d}}}{V}=1-\frac{Q_{\mathrm{a}}}{Q}\theta$$

(13)

$$\frac{V_{\mathrm{p}}}{V}=\frac{1}{2}\left({\theta }_{\min }{+\theta }_{\mathrm{peak}}\right)$$

(14)

$$\frac{V_{\mathrm{m}}}{V}=1-\frac{V_{\mathrm{p}}}{V}-\frac{V_{\mathrm{d}}}{V}$$

(15)

The volume of fluid with a residence time more than twice the theoretical residence time is the dead zone volume. $\frac{Q_{\mathrm{a}}}{Q}$ equals dimensionless RTD curve, with θ area between 0 and 2, as shown in Figure 3.

3.2. Multi-Strand Tundish RTD Analysis Modes

With the development of metallurgical technology, multi-flow tundishes are widely used in steel production. Although the modified combined model can accurately calculate and analyze the single-flow tundish, it is limited when analyzing the flow characteristics of the multi-flow tundish, due to the additional influences of each flow. Due to the flow characteristics of multi-flow tundishes, so far there have been no accepted models for analyzing the multi-flow tundish. A fitting model could avoid the wrong judgment of tundish flow. There have been many efforts to find a suitable method to measure the flow characteristics of the multi-strand tundish. At present, researchers often use the following methods to analyze the flow characteristics of the multi-strand tundish. We will contrast several common multi-strand tundish models, and analyze the results of the physical experiments to discuss the applicability of each model.

Model 1 assumes that each flow in the multi-strand tundish do not affect each other and deals with each flow in the multi-strand tundish as an independent single flow tundish. The model uses the same modified combined model used for the single flow tundish [47,48]. The calculation formula is as follows:

$$\frac{V_{i\mathrm{d}}}{V}=1-\frac{Q_{i\mathrm{a}}}{Q_i}{\theta }_i$$

(16)

$$\frac{V_{\mathrm{ip}}}{V}{=\theta }_{i\min }{+\theta }_{i\mathrm{peak}}$$

(17)

$$\frac{V_{i\mathrm{m}}}{V}=1-\frac{V_{i\mathrm{p}}}{V}-\frac{V_{i\mathrm{d}}}{V}(\mathrm{where} i=1~n)$$

(18)

Model 2 uses the average value of volume fraction of each flow [10], and the formula is as follows:

$$\frac{V_{\mathrm{d}}}{V}=\frac{1}{n}(\frac{V_{1\mathrm{d}}}{V}+\frac{V_{2\mathrm{d}}}{V}+\cdots +\frac{V_{\mathrm{nd}}}{V})$$

(19)

$$\frac{V_{\mathrm{p}}}{V}=\frac{1}{n}(\frac{V_{1\mathrm{p}}}{V}+\frac{V_{2\mathrm{p}}}{V}+\cdots +\frac{V_{\mathrm{np}}}{V})$$

(20)

$$\frac{V_{\mathrm{m}}}{V}=1-\frac{V_{\mathrm{p}}}{V}-\frac{V_{\mathrm{d}}}{V}$$

(21)

The processing method of this model is to superimpose and average the flow parameters of each flow in the multi-flow tundish.

Model 3 approaches the multi-strand tundish as a parallel connection of multiple single flow tundish combination models, and the overall dead zone, plug, and full mixing zone of the tundish are the sum of each flow, dead zone, plug, and full mixing zone [16,49]. The difference between these two explanations lies in the understanding of dimensionless response time. For multi-strand tundish, the response time of each strand is ${\theta }_{\mathrm{imin}}=N·\frac{V_{i\mathrm{p}}}{V}$, which is different from the understanding of response time in the single flow tundish. The formulae for Model 3 are as follows:

$$\frac{V_{\mathrm{p}}}{V}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left({\theta }_{i\min }{+\theta }_{i\mathrm{peak}}\right)$$

(22)

$$\frac{V_{\mathrm{d}}}{V}=1-\frac{1}{n}{\displaystyle \sum }_{i=1}^n\frac{Q_{i\mathrm{a}}}{Q_i}{\theta }_i$$

(23)

$$\frac{V_{\mathrm{m}}}{V}=1-\frac{V_{\mathrm{p}}}{V}-\frac{V_{\mathrm{d}}}{V}$$

(24)

Model 3 is a direct application of the modified combined model of the single flow tundish, which will certainly lead to errors in the calculations of each dead zone, plug, and full mixing zone of the multi-strand tundish.

Model 4 is a method based on the overall RTD curve. The characteristic RTD curve is derived from the RTD curve of each flow, and then processed with the combined model [50]. The approach is as follows:

$${{\displaystyle \int }}_0^{\infty }E\left(t\right)\mathit{dt}={{\displaystyle \int }}_0^{\infty }{E}_1\left(t\right)\mathit{dt}+{{\displaystyle \int }}_0^{\infty }{E}_2\left(t\right)\mathit{dt}+\cdots +{{\displaystyle \int }}_0^{\infty }{E}_n\left(t\right)\mathit{dt}=1$$

(25)

$$\frac{V_{\mathrm{d}}}{V}=1-\frac{Q_{\mathrm{a}}}{Q}\theta$$

(26)

$$\frac{V_{\mathrm{p}}}{V}{=\theta }_{\min }$$

(27)

$$\frac{V_{\mathrm{m}}}{V}=1-\frac{V_{\mathrm{p}}}{V}-\frac{V_{\mathrm{d}}}{V}$$

(28)

Model 5 calculates the overall average residence time based on the RTD curve of each flow. Because this method cannot use the combined model, it can only use the mixed model for subsequent processing [51]. The calculating method is as follows:

$${ \overline{t}}_{\mathrm{c}}=\frac{{{\displaystyle \sum }}_{t=0}^{2 \overline{t}}{c}_{\mathrm{a}{v}_i}{t}_i}{{{\displaystyle \sum }}_{t=0}^{2 \overline{t}}{\mathrm{c}}_{\mathrm{a}{v}_i}}, (i=1~N)$$

(29)

$$\frac{V_{\mathrm{d}}}{V}{=1- \overline{\theta }}_{\mathrm{c}}$$

(30)

$$\frac{V_{\mathrm{p}}}{V}{=\overline{\theta }}_{\min }$$

(31)

$$\frac{V_{\mathrm{m}}}{V}=1-\frac{V_{\mathrm{p}}}{V}-\frac{V_{\mathrm{d}}}{V}$$

(32)

Model 6 takes the outflow parameters of each flow as the weight value to obtain the overall residence time distribution (RTD) curve of the multi-strand tundish. The overall RTD curve of the multi-strand tundish is then investigated using the classical analysis model [20,40]. The minimum response time is defined as the time when the concentration reaches 1% of the peak concentration. The formula is as follows:

$$\mathrm{C}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\mathrm{c}}_i$$

(33)

Model 7 uses statistical theory to put forward the concepts of active zone influence factor. The dead zone influence factor (${\alpha }_i$) is defined as the proportion of the ith flow of tundish from the dimensionless RTD curve between the area of 0~2 and each flow in the tundish. The dead zone influence factor (${\beta }_i$) is defined as the proportion of the ith flow of the tundish dimensionless RTD curve. The area from 2 to ∞ is on the dimensionless RTD curve of each flow in the tundish, from the ratio of the total area of 2~∞ [52], where $Q_i{=Q}_{\mathrm{a},i}{+Q}_{\mathrm{d},i}$

$${\alpha }_i=\frac{Q_{\mathrm{a},i}/Q_i}{Q_{\mathrm{a},1}/Q_1{+Q}_{\mathrm{a},2}/Q_2{+\dots +Q}_{\mathrm{a},n}/Q_n}$$

(34)

$${\beta }_i=\frac{Q_{\mathrm{d},i}/Q_i}{Q_{\mathrm{d},1}/Q_1{+Q}_{\mathrm{d},2}/Q_2{+\dots +Q}_{\mathrm{d},n}/Q_n}$$

(35)

Model 8 analyzes the flow characteristics of tundish based on the F-curve. The model is calculated using F-curves made from experimental data. Combined with the F-curve, the area enclosed by the curve after θ > 2 and F = 1 is considered to be the fluid volume in the tundish whose residence time is longer than twice the theoretical residence time, that is, the dead zone [38]. The formula for calculating the dead zone ratio using this method is:

$$\frac{V_{\mathrm{d}}}{\mathrm{V}}={{\displaystyle \int }}_0^{\infty }(1-F)d\theta =1-{{\displaystyle \int }}_0^2(1-F)d\theta$$

(36)

For the multi-strand tundish, the calculation method is as follows:

$$\frac{V_{\mathrm{d}}}{V}={{\displaystyle \int }}_0^{\infty }(1-F_1-F_2-\dots F_{\mathrm{k}}\left)d\theta =1-{{\displaystyle \int }}_0^2\right(1-F_1-F_2-\dots F_{\mathrm{k}})d\theta$$

(37)

3.3. Application for Multi-Strand Tundish RTD Analysis Models

The flow of molten steel in the tundish is due to viscous incompressible flow under the action of gravity, and the hydraulic model system satisfies geometric and dynamic similarities. The hydraulic simulation experiment of the continuous casting tundish meets the second self-modeling zone under turbulent conditions. The water model experiment was carried out according to the similarity ratio of the model to the prototype being 0.5. During the experiment, the liquid level in the tundish was raised to the experimental working liquid level, and then a large ladle and tundish nozzles were opened. After the liquid level of the tundish became stable, NaCl aqueous solution was added to the long nozzle of the tundish by pulse. We started timing and measuring data at the same time the tracer was added, and continuously measured NaCl concentration in the tundish through the data acquisition system. The tracer concentration at the outlet obtained by stimulus response experiments were consistent with hydraulic simulation experiments in the literature [41]. The eight models listed above are relatively common for analyzing the flow characteristics of multi-stand tundishes at present. To determine the practicability of these models, we analyzed these eight models by comparing them with results from the physical experiment. A comparative study was carried out to compare and analyze the advantages and disadvantages of the various models, and to find out the most suitable model for the multi-strand tundish flow characteristics in this experiment. Figure 4 shows the tracer concentration distribution in a six-strand tundish in the water modeling experiment. The RTD curves of the water modeling experiment is shown in Figure 5.

Models 1 and 2 directly apply the single-strand tundish combined model to the multi-strand tundish, solving each strand separately and then averaging the results. The basis of these two models is that each strand in the multi-strand tundish is irrelevant to the others. However, the flow areas in the multi-strand tundish are interconnected, and there are no obvious boundaries. Therefore, parameter analysis methods in the RTD curve of the single flow tundish are not suitable for the parameter analysis of each flow in the multi-strand tundish. Therefore, the results obtained by Models 1 and 2 are inconsistent with reality.

Table 2. Volume fraction results by Models 1 to 8.

Area	Analysis of Model
	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Model 7	Model 8
$V_{\mathrm{d}}$	0.364	0.351	0.350	0.450	0.325	0.214	0.334	0.190
$V_{\mathrm{p}}$	0.099	0.111	0.116	0.049	0.103	0.070	0.115	0.103
$V_{\mathrm{m}}$	0.537	0.538	0.534	0.502	0.572	0.716	0.534	0.707

shows the results of Models 1 to 8. It can be seen from the table that the corresponding dead volume fraction, plug volume fraction, and overall integral of the mixing zone obtained by different multi-flow tundish models are also different according to the experimental data.

Models 2 and 3 both measure the experimental data by means of the average value. The flow analysis model of a single-flow tundish cannot be directly used to analyze the flow characteristics of each flow in a multi-flow tundish. Although Model 3 has similarities to Model 2, its interpretation of the dimensionless minimum residence time of multi-strand tundish is more reasonable. In Model 3, the multi-flow tundish is regarded as the parallel connection of multiple single-flow tundish models. The number of flows of the tundish should be considered in relation to the dimensionless minimum residence time of each flow and volume fraction of the mixing zone, rather than simply averaging each strand. At the same time, the sum of the dead zone and plug of a single strand in the multi-strand tundish is less than 1, and it is meaningless to directly calculate the dead zone for every strand. The dead volume fraction of the experiment is 0.350 by Model 3.

Model 4 obtains the overall RTD curve based on the RTD curve of each strand. It can be seen from Table 2 that the volume fraction of the plug of the experiment is 0.049. As Model 4 lacks a definition for response time, the model considers that the earliest response time for each flow is the response time of the multi-flow tundish system. However, each flow of the multi-flow tundish affects one other and the response times are not the same. It is unreasonable to simply use the first response of each flow as the response time of the system. Figure 6 shows the characteristic RTD curve of the model. The overall RTD curve of the six-strand tundish in this experiment fluctuates violently when it reaches the peak value, which is clearly different from the description of the response time in the combined model, that is, the stagnation time and the peak value times are not equal. The dead volume fraction of the experiment calculated by Model 4 is 0.450. The dead zone calculation is too large.

Model 5 gives an overall average residence time from a formulaic point of view and uses a mixed model to solve it. It can be seen from Table 2 that the volume fraction of the dead zone is 0.325 according to Model 5. Compared with Models 3 and 4, the calculation for the dead volume fraction of Model 5 is clearly decreased. As Model 5 directly gives the formula for the average residence time, the ratio between the volume fraction of the active area and total volume cannot be calculated, and the modified combination model cannot be used. Meanwhile, the response time of Model 5 is defined as the average value of each response time. Averaging the response times of different outlets ignores fluctuations in each outlet. There is also a certain error in the calculation results by Model 5.

Model 6 proposes an overall RTD curve method for the tundish to calculate the flow characteristics of multi-strand tundish, defining 1% of the peak concentration as the response time, and the average residence time of the overall curve as the weighted average of the total amount of tracers in each flow. The model has a clear definition for the response time, which is beneficial for a reasonable interpretation of the response time in the multi-flow tundish. It can be seen from Table 2 that the volume fractions of the plug obtained by Model 6 is 0.070. The volume fraction of the plug is more reasonable. The dead volume fraction of Model 6 is 0.214. Compared with the previous models, the calculation size of the dead zone is smaller and the volume fraction is more reasonable. At the same time, Model 6 uses all the data of the RTD curve, so the average residence time calculated by the overall RTD curve can effectively characterize the basic characteristics of the RTD curve. The average residence time can also be used as a tundish flow consistency. Model 6 in the literature [53] has also effectively overcome situations where the dead zone has a negative volume. In general, Model 6 is a more comprehensive model for multi-strand tundish analysis.

Model 7 proposes the concept of an active area influence factor and dead area influence factor from a statistical point of view. Table 2 shows the dead volume fraction, plug volume fraction, and total mixing zone volume fraction of the experiment by Model 7. The sum of the three volume fractions calculated by Model 7 is less than 1, and Model 7 also lacks a response time definition. The calculated dead zone volume is also large.

Model 8 derives the calculation method of dead volume based on the cumulative time distribution F-curve and uses the classical combined model to calculate the plug and total mixing zone. Figure 7 is one of the sample F-curves. It can be seen from the figure that the F-curve is getting close to 1 as time increases. It can be seen from Table 2 that the dead volume fraction of Model 8 is 0.190. This dead zone calculation is the smallest among the eight models. Model 8 considers the area of the upper part of the F-curve θ > 2 as the volume of fluid in the tundish with a residence time greater than twice the residence time. The dead volume fraction obtained by Model 8 is the smallest.

Table 2 shows the calculation results of the dead zone and plug volume fractions for the eight models. It can be seen from the table that the calculation result of the dead volume fraction fluctuates, the minimum and maximum values of dead volume fraction are 0.19 and 0.45, respectively. This shows that there are obvious differences in the sizes of the dead volume fractions obtained by different models. Considering the actual situation of the flow in the tundish, the volume fraction of dead zone should be less than 40%. The dead volume fraction of the tundish in this experiment was larger according to Models 3, 4, 5, and 7. The calculation results obtained by Models 6 and 8 were relatively close, and the dead zone of the tundish calculated by Model 8 was relatively small. The response time defined by Model 6 was also reasonable. The model has better adaptability to the experimental data, and the obtained volume fraction of each region was also more reasonable.

4. F-Curve as an Analysis Model for Multi-Flow Tundish Flow Characteristics

The F-curve is an exponential response curve, which is defined as the fraction of the outlet strand that has a residence time in the reactor less than t. It can also be interpreted as adding a certain concentration of tracer to the reactor continuously within a certain period, the tracer flowing out from the outlet after flowing through the reactor, and measuring the total concentration of the tracer flowing at the outlet over time. The change curve is called the F-curve.

The relationship between F(t) curve and E(t) can be expressed as:

$$F\left(t\right)={{\displaystyle \int }}_0^{t}E\left(t\right)\mathit{dt}$$

(38)

Expressed as dimensionless time:

$$F\left(\theta \right)={{\displaystyle \int }}_0^{\theta }E\left(\theta \right)d\theta$$

(39)

Figure 8 shows the F-curve of each flow and the total F-curve of the original data of this experiment. The continuous casting tundish is a non-ideal reactor, and the curve of the cumulative flow tracer concentration increases with time. The curve fluctuates, which is not as smooth as the full mixed flow reactor. We can intuitively see the proportion of tracer flowing out from each strand of the multi-strand tundish through the F-curve. The value of this proportion reflects the contribution of the strand to the entire multi-flow tundish and is an important parameter reflecting the flow characteristics of the multi-flow tundish.

4.1. Modification of Dead Zone Calculation Method

According to the definition of Sahai and Emi [13], a dead zone is an area where the residence time of the fluid flow is more than twice the average residence time. In the combined model, the effective region consisting of plug flow and total mixed flow is parallel to the dead zone. However, some tracers may also stay in the plug for more than twice the average residence time due to diffusion or convection as the tracer passes through a well-mixed area. It will cause errors in the calculation results of the dead volume fraction. Therefore, we should take this error into account when calculating the dead volume using the combined model.

According to the assumptions of the combined model, the volume fraction of the well-mixed region in the reactor should be equal to the multiplicative inverse of the dimensionless maximum concentration C_max. However, this calculation method is rarely used by researchers because the calculated dead zone is generally large, which is inconsistent with the observations in the experiment. In both the traditional and the Sahai method, the volume for uniform mixing is calculated by formula. Su [54] considered that the well-mixed region of the tundish could be divided into two zones in the modification of the dead zone calculation model: an equivalent well-mixed region and a perfect well-mixed zone. In an actual tundish, well-stirred regions can be considered fully mixed regions, such as the impingement zone of the jet and each side of the air curtain. If the fluid in the dead zone is considered completely stagnant, the volume fraction of the dead zone mixing area is:

$$\frac{V_{\mathrm{md}}}{V_{\mathrm{mp}}}=\frac{1}{C_{\max }}$$

(40)

$$\frac{V_{\mathrm{md}}}{V}=\frac{V_{\mathrm{md}}}{V_{\mathrm{mp}}}\frac{V_{\mathrm{mp}}}{V}=\frac{1}{C_{\max }}{(1-\theta }_{\mathrm{mC}})$$

(41)

where V_md and V_mp are the volumes of the dead zone and the fully mixed zone, respectively, V is the volume of fluid in the tundish, and θ_mC is the dimensionless mean residence time of the RTD curve at θ = 2 in a well-mixed reactor. The tundish mixing area is shown in Figure 9 [54].

The well-mixed region in the combined model can be divided into the combination of the effective well-mixed region (Region 1) and the ineffective mixing region (Region 2). In Region 2 of the invalid mixing zone, it is regarded as a dead zone. Subtracting $\frac{1}{C_{\max }}{(1-\theta }_{\mathrm{mC}})$ from the calculated dead volume to calculate the dead volume is more accurate.

4.2. Model Application

4.2.1. F-Curve and Overall Intensity Curve

The F-curve reflects the change in the concentration of tracer flowing out of each outlet of the tundish, and the F-curve gets infinitely close to 1 over time from the cumulative time distribution curve. Figure 10 shows the F-curve of each strand of the tundish. It can be seen from Figure 10 that the F value of each strand of the multi-flow tundish is less than 1, and these values reflect the ratio of the tracer flow from each strand. The volume flow rate of each strand in the tundish is usually the same. However, the tracer flows out from each strand in different ratios after entering the tundish. These ratios reflect the degree of influence each strand has on the flow characteristics in the tundish. As the position of the guide hole on the tundish baffle wall faces strand no. 4 and 5, they have the largest outflow of tracer. Strand no. 2 is far from the main stream. Therefore, the position of strand no. 6 is the lowest on the F-curve. Although the distance between other outlets and inlets are not the same, the positions of other outlets also change due to the influence of the flow field. The multi-strand tundish requires better consistency in each strand, which is shown in the F-curve as the consistency of the F-curve of each strand and the consistency of the F value. The F-curve is suitable for analysis of the flow characteristics of the multi-flow tundish, and can reflect the flow differences of each strand more intuitively.

Figure 11 shows the overall intensity curves of the experiment results, because the intensity curves are related to the E curve, and the total amount of tracer flowing out of each tundish strand is less than 1. Therefore, to avoid errors, the overall calculation method can be used in the multi-strand tundish intensity curve. It can be seen from the figure that there are obvious peaks, indicating that there is a dead zone in the system. From the definition of the intensity function, after the tracer is added in the three cases, the escape fraction of the tracer quickly reaches a peak value after a short response time. The tracer quickly flows out from the outlet after being added to the tundish. This indicates that there is a short-circuit flow in the tundish. The remaining tracer continues to flow in the tundish due to diffusion and convection, and the later it flows out of the outlet, the longer the residence time. Therefore, the fluctuation of the intensity curve and the difference between the curves of each scheme can be used to judge the short-circuit flow and flow consistency of the tundish. The ideal tundish flow should have good consistency in the flow of each strand, and the fraction of tracer flowing out of the outlet over time should also be consistent.

4.2.2. The Dead Volume Fraction

The F-curve and strength curve allow us to judge the flow consistency of each strand of the multi-flow tundish, the existence of a dead zone, and short-circuit flow from a graphical point of view. These zones can be used as a reference for making a judgment on the most optimal model for multi-strand tundishes. In terms of model adaptability and rationality, the overall RTD curve method of Model 6 is preferable. The process of calculating F-curves and the proportion of tracer flowing out of each strand is similar to this model. These can provide a reference for the optimization of the tundish. Therefore, we suggest using the cumulative residence time distribution curve to analyze the flow characteristics of the multi-flow tundish. Using our experimental data, the cumulative residence time distribution curve of each strand was obtained. Each flow F-curve is at the position of twice the residence time, $F_{\mathrm{i},\mathsf{\theta }}=2$, which represents the tracer in the tundish after twice the residence time. According to the modified combinatorial model, the dead zone is defined as the volume of fluid in the tundish that has a residence time of more than twice the residence time, so we can calculate the dead zone by subtracting the total tracer at the θ = 2 position for each strand. Then, the contribution of the fully mixed region of the dead zone to the dead zone ($\frac{1}{C_{\max }}{(1-\theta }_{\mathrm{mC}})$) is subtracted as a correction to the calculation result of the dead zone. For the dimensionless concentration processing method in this work, the size of $\frac{1}{C_{\max }}$ is related to the length of the experimental data measurement time. At the same time, $F_{i,\theta =2}$ reflects the difference of the proportion of each strand, so the variance can be used as the basis for judging the consistency level of each strand. The dead zone calculation formula can be expressed as:

$$\frac{V_{\mathrm{d}}}{V}=1-{{\displaystyle \sum }}_{i=1}^n{F}_{i,\theta =2}-\frac{1}{C_{\max }}{(1-\theta }_{\mathrm{mC}})$$

(42)

Consistency criteria for each strand:

$$S=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left({\mathrm{F}}_{i,\theta =2}-\overline{F_{\theta =2}}\right)}^2}{n}}$$

(43)

In this experiment, the data recording time was 2.5 times the residence time during the experiment. The dead volume fraction of the experiment was 17% as shown in

Table 3. Dead volume fraction.

Parameters	Value
$F_{i,\theta =2}$	0.817
$\frac{1}{C_{\max }}{(1-\theta }_{\mathrm{mC}})$	0.012
$V_{\mathrm{d}}$	0.17

according to Equation (42). When using the experimental data to make the cumulative residence time distribution curve, it was approximated that the total amount of tracer flowing at 2.5 times residence time is 1. However, it can be seen from the moving trend of the F-curve that the F-curve is close to 1. A short experimental recording time has an impact on the results of data processing. From the processing results in this paper, the experimental duration of 2.5 times the residence time is obviously short, which will bring certain errors to the data processing results. Therefore, it is recommended to extend the experimental time as much as possible in the process of experimental recording to ensure the correctness of experimental data processing.

The F-curve and the intensity function can be used to analyze and compare the flow characteristics of the tundish, and the data processing results of the F-curve can also be used to calculate the dead zone of the tundish. The further optimized dead zone calculation method can effectively reduce the dead zone calculation results. At the same time, to ensure the accuracy of the experimental data processing results, it is recommended to extend the experimental time as much as possible.

5. Conclusions

In this paper, the rationality of several existing single-flow tundish flow characteristic analysis models and multi-flow tundish flow characteristic analysis models are compared and analyzed. The applicability of various models to this experiment is discussed. The following main conclusions can be drawn.

Based on analyzing and summarizing previous research works, this paper proposes a multi-flow tundish flow characteristic analysis method based on the cumulative residence time distribution curve.

The practical application shows that the F-curve and the strength curve can be used to analyze and compare the flow characteristics of the multi-flow tundish, and the modified dead zone calculation method is more reasonable.

The dead zone correction calculation of F-curve is based on the mixed model, and there are certain requirements for the experimental recording time, so there is still a certain error. This method can provide a feasible idea for the study of flow characteristics of multi-flow tundish.

The minimum and maximum values of dead volume fraction are 0.19 and 0.45 with different models. This shows that there are obvious differences in the size of the dead volume fraction obtained by different analysis models. The Model 6 and new model have better adaptability to the experimental data, and the obtained volume fraction of each region is also more reasonable.