*2.1. Physical Model*

According to the shape of a blast furnace trough from a steel plant in China, a physical model is established in Figure 1. During tapping, molten slag and hot metal are regarded as a mixed continuous and incompressible fluid flowing out from the tap hole, and then fall down into the main trough. The mixture fluid keeps a constant level in the main trough and around 300 mm from the upper surface in the calculation. It is separated by the skimmer, and then hot metal flows into a torpedo. Falling position of the mixture fluid trajectory (FPMFT) in the trough defines the inlet of the model. It moves from 4 m away from the origin of the coordinates in the beginning to the tap hole direction in the tapping process. According to References [5,10], Table 1 lists the physical properties of the mixture fluid and the refractory in the study. Chemical reaction between the refractory and the mixture fluid is

neglected in simulations. The diameter and the angle of the tap hole is 60 mm and 10 degrees in the simulations, respectively.

**Figure 1.** Schematic diagram of the main trough in front view: 1-1 cross-section view at the main trough; 2-2 cross-section view at the outlet.


**Table 1.** Physical properties of the mixture and the refractory.

Where temperature and viscosity are given in constant values for calculation speed, and the effect of them on the simulation will be focused on later.

#### *2.2. Mathematical Model*

#### 2.2.1. Mathematical Model of Molten Slag and Hot Metal

The governing equations of the mixture fluid include a mass conservation equation, a momentum equation based on Reynolds-averaged one, and an energy conservation equation. The fluid in the study was an incompressible Newtonian fluid and its volume expansion ratio <sup>∂</sup>*ui* <sup>∂</sup>*xi* is zero. In order to maintain the conservation of the mixture, the mass conservation equation must be met [11], as:

$$\frac{\partial \mu\_i}{\partial \mathbf{x}\_i} = \mathbf{0},\tag{1}$$

where, *xi* and *ui* express the coordinates of space points (m) and the velocity component at point *xi* of the time t coordinate (m·s−1), respectively. *<sup>x</sup>*1, *<sup>x</sup>*<sup>2</sup> and *<sup>x</sup>*<sup>3</sup> define the three directions of x, y and z, respectively.

The viscous stress tensor P and the deformation rate tensor S of Newtonian fluid have a linear and isotropic function relationship [12]. The Newtonian fluid constitutive equation is substituted into the dynamic equation to obtain the momentum conservation equation of the incompressible Newtonian fluid [11], as:

$$\frac{\partial u\_i}{\partial t} + u\_j \frac{\partial u\_i}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_i} + \nu \frac{\partial}{\partial \mathbf{x}\_j} \left(\frac{\partial u\_i}{\partial \mathbf{x}\_j}\right) \tag{2}$$

where, *p*, <sup>ρ</sup>, <sup>μ</sup> and <sup>ν</sup> express pressure of the fluid (kg·m−1·s<sup>−</sup>2), viscosity of the fluid (kg·m<sup>−</sup>3), kinetic viscosity (kg·m−1·s<sup>−</sup>1) and kinematic viscosity (m2·s<sup>−</sup>1), respectively.

In this study, the standard *k*-ε turbulence model is used in the simulation. The momentum conservation equation is a time average one to obtain the Reynolds-averaged N-S equation [13]:

$$\frac{\partial \langle \mathbf{u}\_{i} \rangle}{\partial t} + \langle \mathbf{u}\_{\dot{j}} \rangle \frac{\partial \langle \mathbf{u}\_{i} \rangle}{\partial \mathbf{x}\_{\dot{j}}} + \frac{\partial \langle \mathbf{u}\_{i}' \mathbf{u}\_{\dot{j}}' \rangle}{\partial \mathbf{x}\_{\dot{j}}} = -\frac{1}{\rho} \frac{\partial \langle p \rangle}{\partial \mathbf{x}\_{i}} + \nu \frac{\partial}{\partial \mathbf{x}\_{\dot{j}}} \Big( \frac{\partial \langle \mathbf{u}\_{i} \rangle}{\partial \mathbf{x}\_{\dot{j}}} \Big) \tag{3}$$

where, *u <sup>i</sup>* and *u <sup>j</sup>* define pulse values of the velocity (m·s−1), respectively. *ui* is time average, and *ui* = *ui* + *u i* .

Reynolds stress tensor term, −*u i u j* , is added into Equation (3). This makes the equations disable to close and introduces a turbulence model. According to the Boussinesq hypothesis, the expression of Reynolds [13] stress is:

$$-\langle \mathbf{u}'\_i \mathbf{u}'\_j \rangle = \nu\_l \left[ \frac{\partial}{\partial \mathbf{x}\_j} \langle \mathbf{u}\_i \rangle + \frac{\partial}{\partial \mathbf{x}\_i} \mathbf{u}\_j \right] - \frac{2}{3} \delta\_{ij} \mathbf{k}\_\star \tag{4}$$

where, δ*ij* is Kronecker delta and δ*ij* = ! 1, *i* = *j* 0, *<sup>i</sup> <sup>j</sup>* . *<sup>k</sup>* is turbulent energy (m2·s<sup>−</sup>2).

Due to high velocity at the inlet, the mixture fluid in the main trough has a high Reynolds number. The standard *k*−ε turbulence model has a few empirical constants for this condition. In the standard *k*−ε model, the turbulent energy k and the turbulent dissipation rate ε are associated with the turbulence ν*t*, the formula [13] is as follows:

$$\nu\_t = \mathbb{C}\_{\mu} \frac{k^2}{\varepsilon},\tag{5}$$

where, *C*μ expresses the empirical constant and a value of 0.09 is used in the simulation.

*k* and ε are solved in an incompressible fluid using the following two equations [13]:

$$\frac{\partial(\rho k)}{\partial t} + \langle u\_i \rangle \frac{\partial(\rho k)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left[ (\mu + \frac{\nu\_t}{\sigma\_k}) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + G\_k - \rho \varepsilon\_\prime \tag{6}$$

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \langle u\_i \rangle \frac{\partial(\rho\varepsilon)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\nu\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_i} \right] + \frac{\mathbb{C}\_{1\varepsilon}\varepsilon}{k} G\_k - \mathbb{C}\_{2\varepsilon}\rho \frac{\varepsilon^2}{k},\tag{7}$$

where, *C*1ε, *C*2ε, σ*<sup>k</sup>* and σε express 1.44, 1.92, 1.0 and 1.3, respectively. *Gk* defines the increase in turbulent kinetic energy caused by the average velocity gradient and is calculated as follows:

$$\mathbf{G}\_k = \nu\_l \left( \frac{\partial \langle u\_i \rangle}{\partial u\_j} + \frac{\partial \langle u\_j \rangle}{\partial u\_i} \right) \frac{\partial \langle u\_i \rangle}{\partial x\_j} \,' \,. \tag{8}$$

The above eight equations jointly solve the velocity and the pressure of the mixture fluid region and the energy conservation equation is expressed [11] by:

$$\frac{\partial T}{\partial t} + u\_i(\frac{\partial T}{\partial \mathbf{x}\_i}) = \frac{\partial}{\partial \mathbf{x}\_i} \frac{\lambda}{\rho \mathbf{C}\_p} \frac{\partial T}{\partial \mathbf{x}\_i},\tag{9}$$

where, <sup>λ</sup> and *Cp* express the fluid heat transfer coefficient (W·m−1·K<sup>−</sup>1) and the specific heat capacity of fluid (J·m−1·s<sup>−</sup>1), respectively.

The energy conservation equation is also a time average one. Equation (9) is added to the Reynolds heat conduction term (*u i T* ) after time-average, and it becomes:

$$\frac{D\langle u\_{i}^{\prime\prime}T^{\prime}\rangle}{\partial t} = \frac{\partial}{\partial x\_{j}} \Big[ \mathcal{C}\_{T} \frac{\hbar^{2}}{\varepsilon} \frac{\partial \langle u\_{i}^{\prime\prime}T^{\prime}\rangle}{\partial x\_{j}} + a \frac{\partial \langle u\_{i}^{\prime\prime}T^{\prime}\rangle}{\partial x\_{j}} \Big] - \left( \langle u\_{i}^{\prime}u\_{j}^{\prime}\rangle \frac{\partial \langle T\rangle}{\partial x\_{j}} + \langle u\_{j}^{\prime}T^{\prime}\rangle \frac{\partial \langle u\_{i}\rangle}{\partial x\_{j}} \right) - \mathcal{C}\_{T1} \frac{\mu}{k} \langle u\_{i}^{\prime\prime}T^{\prime}\rangle - \mathcal{C}\_{T2} \frac{\partial \langle u\_{i}\rangle}{\partial x\_{j}} \langle u\_{j}^{\prime\prime}T^{\prime}\rangle,\tag{10}$$

where, *CT*, *CT*<sup>1</sup> and *CT*<sup>2</sup> are empirical coefficients. The values of them are 0.07, 3.2 and 0.5 in the simulation, respectively.
