2.2.2. Mathematical Model of Refractory

Heat transfer between the mixture fluid and the refractory is coupled to each other. For the refractory heat transfer calculations, only the Fourier equation [14] is solved:

$$\frac{\partial \langle T \rangle}{\partial t} = \frac{\partial}{\partial \mathbf{x}\_i} \frac{\lambda}{\rho \mathbb{C}\_p} \frac{\partial \langle T \rangle}{\partial \mathbf{x}\_i},\tag{11}$$

where, <sup>λ</sup> defines the solid heat transfer coefficient (W m−1·K<sup>−</sup>1). *Cp* expresses the specific heat of the refractory (Al2O3–SiC–C) and is 0.628 kJ·kg−1·K<sup>−</sup>1.

#### *2.3. Boundary Conditions (cf. Figure 2)*

(1) Inlet boundary conditions. Due to the decrease of the pressure in the furnace during tapping, mass flow of the mixture flow from the tap hole decreases and FPMFT moves to the tap hole direction. Therefore, boundary condition at the inlet is velocity type and it can change in the direction and in the magnitude at the same time. A parabolic Equation (12) is used to define the velocity. Its maximum magnitude is 6.635 m·s−<sup>1</sup> and is estimated from the FPMFT at the beginning of the tapping. Thermal and pressure boundary conditions at the inlet are constants of 1773 K and zero gradient, respectively.

$$y = x \cdot \tan \alpha - \frac{\mathcal{S}}{2u\_0^2 \cos^2 \alpha} \cdot x^2,\tag{12}$$

where, α is the inclined angle of the tap hole (◦), *u*<sup>0</sup> defines the velocity of the mixture fluid stream (m·s<sup>−</sup>1) and changes with the time, x and y are the coordinates of FPMFT (m, m) and g expresses the acceleration of gravity (m·s<sup>−</sup>2).


**Figure 2.** Computational grid: (**a**) the grids at the inlet and the upper wall, (**b**) the grids at the outlet.
