# Mathematical Model

# -D Fluid Flow and Heat Transfer Model in the Mold

(a) Assumptions

In the mathematical model, the liquid steel in mold was assumed to be a steady-state, three-dimensional, incompressible Newtonian fluid. The strand curvature, bulging, oscillation, mold taper and air gap were neglected. Besides, the heat transfer at the strand free surface was ignored.

(b) Fluid and Turbulence Model

The liquid steel flow in the liquid pool was achieved through solving the continuity and momentum equations. The popular *K-s* turbulent model [7] was chosen. The equations are:

where щ and щ is the velocity component, x_{;} and *Xj* is the spatial coordinate, p is the density of liquid steel, P is the pressure, is the effective viscosity, S_{P} is the sink term of velocity.

(c) Solidification and Heat-Transfer Model

The governing equation for heat transfer and solidification is:

where If is the effective thermal conductivity, *T* is the temperature, H is the enthalpy of steel, which is the sum of the sensible enthalpy and latent heat.

(d) Computational Domain

Due to the symmetry of fluid flow and heat transfer, only a quarter of the strand was modelled to minimize computation. The casting parameters are shown in the Table 1. Figure 1 is the mesh of the strand and the four kind of mold corner structure: right-angle, big-chamfer, multi-chamfer and fillet. The strand was meshed using about 820,000 hexahedral cells.

(e) Boundary Condition

At the inlet of SEN, the velocity-inlet boundary condition was used. The inlet velocity was calculated through the mass conservation based on the casting speed. The inlet value of turbulent kinetic energy and the rate of turbulent energy dissipation was 10 ^{5} [9], respectively. The pressure-outlet boundary condition was applied at strand exit.

The top free surface was set as zero shear stationary wall and heat insulation. At the symmetry plane, the normal velocity components and normal gradients of all other variables were assumed to be zero. At the strand surface, the heat transfer boundary condition was employed the heat flux, which decreased along the casting direction.

**Table 1 **The casting parameters of mold [8]

Operating parameters |
Values |

Mold section |
2000 mm x 260 mm |

Mold length |
800 mm |

Computer length |
2800 mm |

Inside size of SEN (submerged entry nozzle) |
80 mm x 60 mm |

Outside size of SEN |
140 mm x 140 mm |

Port size of SEN |
60 mm x 95 mm |

Port angle |
-15° |

Casting speed |
1.15 m/min |

Casting temperature |
1803 K |

**Fig. 1 ****The schematic diagram of mold and corner structure**