Nomenclature
$\rho$	Density of continuous phase (kg·m−3)	$c_p$	Specific heat (J·kg−1·K−1)
$\overrightarrow{u}$	Velocity (m·s−1)	$L$	Latent heat (J·kg−1)
$\mu$	Viscosity (Pa·s)	$T_R$	Reference temperature (K)
$t$	Time (s)	${\rho }_p$	Density of particles (grains) (kg·m−3)
$p$	Pressure (Pa)	$d_p$	Particle diameter (μm)
$\overrightarrow{g}$	Gravity acceleration (m·s−2)	${\overrightarrow{u}}_p$	Particle velocity (m·s−1)
${\overrightarrow{S}}_u$	Momentum source term (kg·m−2·s−2)	${\overrightarrow{F}}_{\mathrm{drag}}$	Particle drag force (N)
${\beta }_T$	Thermal expansion coefficient (K−1)	${\overrightarrow{F}}_p$	Pressure gradient force (N)
$T$	Temperature (K)	${\overrightarrow{F}}_b$	Buoyancy force (N)
$\xi$	A small positive number	${\overrightarrow{F}}_{\mathrm{VM}}$	Virtual mass force (N)
$f_l$	Mass fractions of liquid phase	${\overrightarrow{F}}_s$	Saffman lift force (N)
$A_m$	Permeability coefficient (m−2)	$T_{surf}$	Temperature of billet surface (K)
$H$	Enthalpy (J·kg−1)	$T_e$	Environment temperature (K)
$k_T$	Heat conductivity (W·m−1·K−1))	$h$	Heat transfer coefficient (W·m−2·K−1)
$f_S$	Solid fraction
Subscripts
ref	Reference value	contact	Mold contact value (2000)
eff	Effective value	air	Air gap value (150)