2. Formulation of the heat-transfer problem

The general three-dimensional (3D) heat-transfer equation that describes the temperature distribution inside a solidifying object is given by Eq. (1) according to [22]

$$
\rho c \frac{\partial T}{\partial t} = \nabla \cdot k \nabla T + \mathbf{S} \tag{1}
$$

The source term S may be considered [23] to be of the form:

$$\mathcal{S} = \mathcal{S}\_{\mathcal{C}} + \mathcal{S}\_{\mathcal{P}} \ T \tag{2}$$

Furthermore,T is the temperature, and ρ, c, and k are the density, heat capacity, and thermal conductivity, respectively. The 2D heat-transfer equation in Cartesian coordinates can be written as [22]

$$
\rho c \frac{\partial T}{\partial t} = \frac{\partial}{\partial \mathbf{x}} \left( k \frac{\partial T}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( k \frac{\partial T}{\partial \mathbf{y}} \right) + \mathbf{S} \tag{3}
$$

The boundary conditions that apply in the octagonal billet case are very similar to the ones for the rectangular billet that have been presented in detail in [24] and will not be repeated here. Nevertheless, two important boundary conditions apply in the case under study, which follow:

• Due to symmetry, the upper diagonal side (Figure 1, segment OB) of the domain of interest is supposed to be adiabatic, as well as the lower side (Figure 1, segment OK) is. In this way, the following formulation holds:

$$\begin{aligned} \frac{\partial T}{\partial s} &= 0, \quad \text{where } s \text{ is the vertical axis on the line} \\ y &= \pi \tan(\pi/8) \end{aligned} \tag{4}$$

• The primary (mold) and secondary cooling systems are the ones applied in Stomana; this is presented in detail in a previous publication [24] and will not be repeated here. However, due to the potential of the octagonal mold, a surplus of water was used in favor of an enhanced cooling behavior for the solidifying billet; this was expressed as an extra percentage of water flow deployed for the octagonal billet.

Figure 1. The salient features of an octagon and its circumscribed circle.
