3-D Mathematical Modeling of the Effect of Shunt Plate Hole Distribution and Magnetic Field on Transport Phenomena Within DC Casting of Magnesium Alloy Slab

Abstract

Due to the nonaxisymmetric feature of slabs, the macrotransport phenomena during direct chill (DC) casting are more complex than the macrotransport phenomena of round billets and are tightly associated with their dimensions, width-thickness ratio and even the structure of the shunt plate used to feed the melt to the slab edge. In the present research, aiming to find the desirable structure of the shunt plate and clarify how the form of the magnetic field (MF) affects the transport phenomena, a 3-D coupled mathematical model was set up and used to study the fluid flow, heat transfer and solidification characteristics during electromagnetic DC casting of a magnesium alloy slab with dimensions of 1200 mm (width) × 400 mm (thickness). The effects of the hole distribution of the shunt plate under MF or no MF and different forms of MF on the corresponding physical fields were studied systematically. The predicted results, including the Lorentz force distribution, macro-flow and temperature fields, cooling curve and sump depth, were presented and compared. The results indicated that reducing the holes facing the wide surface was helpful for obtaining desirable variations, such as more homogeneous melt flow and temperature fields, shallower sump depth and faster cooling rate. Using a specific out-of-phase pulsed magnetic field also promoted the above variations more than other forms of magnetic field.

Appendix

Nomenclature
Latin symbols
A	mushy zone constant (kg/(m3 s))	X, Y, Z	wide direction, thickness direction, vertical direction
$\vec{B}$	magnetic flux density vector (T)
C	specific heat (J/(g K))	x, y, z	coordinate value at wide direction, thickness direction, vertical direction
C eff	equivalent specified heat (J/(g K))
D	extra dissipation term in k-equation
d	secondary dendrite arm spacing (m)
$\vec{E}$	electric field intensity vector (V/m)	Greek symbols
E	extra generation term in ε-equation	μ 0	magnetic permeability
$\vec{f}$	electromagnetic force (N/m3)	ε 0	dielectric constant
fl, fs	liquid fraction, solid fraction	ρ e	electric charge density (C/m2)
f1, f1, fμ	constants used in low Reynolds k-ε	σ	electrical conductivity (S/m)
fk, fcr	constants related to mushy zone	μ	laminar viscosity (Pa s)
$\vec{F}$	mean electromagnetic force (N/m3)	μ T	turbulent viscosity (Pa s)
FX, FY, FZ	Time-averaged electromagnetic force components in three directions (N/m3)	ρ	density (kg/m3)
		β	volume expansion coefficient (1/K)
$\vec{g}$	acceleration of gravity (m/s2)	λ	thermal conductivity (W/(m K))
h	heat transfer coefficient (W/(m2 K))	ε	turbulent kinetic energy dissipation rate (m2/s3)
$\vec{J}$	induced current density vector (A/m2)	Φ	generalized dependent variable
k	turbulent kinetic energy (m2/s2)	ΓΦ	diffusion coefficient associated with Φ
L	latent heat (kJ/kg)	S Φ	Source term associated with Φ
P	pressure (Pa)
P K	production term in turbulent kinetic energy equation
		Subscripts
Q	joule heat (W/m3)	air	Air
Q ave	mean joule heat (W/m3)	cast	Cast
Re	Reynolds number	con	contact
r e	hydraulic radius of the nozzle (m)	en	environment
R T	turbulent Reynolds number based on the turbulent quantities	film	Film
		i, j, k	coordinate direction
S T	turbulence intensity	inlet	inlet
T	temperature (K)	l	liquid
T P	period of the varying current (s)	oulet	oulet
t	time (s)	pour	pouring
U	velocity at wide direction (m/s)	ref	reference
V	velocity at thickness direction (m/s)	s	solid
W	velocity at vertical direction (m/s)