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Numerical Heat Transfer, Part A: Applications
An International Journal of Computation and Methodology
Volume 69, 2016 - Issue 11
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Original Articles

Investigation of the effect of magnetic field on melting of solid gallium in a bottom-heated rectangular cavity using the lattice Boltzmann method

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Pages 1263-1279 | Received 20 May 2015, Accepted 23 Jun 2015, Published online: 02 May 2016
 

ABSTRACT

A numerical study is presented for two-dimensional convection melting of solid gallium in a rectangular cavity. The bottom wall of the cavity is uniformly heated and a uniform magnetic field is applied separately in both horizontal and vertical directions. The lattice Boltzmann (LB) method considering the magnetic field force is employed to solve the governing equations. The effects of magnetic field on flow and heat transfer during melting are presented and discussed at Rayleigh number Ra = 1 × 105 and Hartmann number Ha = 0, 15, and 30. The results show that the magnetic field with an inclination angle has a significant impact on the flow and heat transfer in the melting process. For a small Hartmann number, similar melting characteristics are observed for both horizontally applied and vertically applied magnetic fields. For a high value of Hartmann number, it is found that in the earlier stage of melting process, the flow retardation effect caused by the horizontally applied magnetic field is less obvious than that caused by the vertically applied magnetic field. However, the opposite is true in the later stage.

Nomenclature

B=

magnitude of magnetic field

c=

lattice speed

cs=

sound speed of the model

C=

specific heat, J/kg × K

En=

enthalpy, J

ei=

discrete lattice velocity in direction i

Fo=

Fourier number

fl=

liquid fraction

fi=

density distribution function in direction i

=

equilibrium distribution function for density in direction i

gi=

temperature distribution function in direction i

=

equilibrium distribution function for temperature in direction i

g=

gravitational acceleration, m/s2

H=

cavity height, m

Ha=

Hartmann number

κ=

thermal conductivity, W/(m K)

L=

latent heat, J/kg

p=

pressure, Pa

q=

heat source term, J

Pr=

Prandtl number

Ra=

Rayleigh number

Ste=

Stefan number

t=

time, s

T=

temperature, K

Tm=

melting temperature, K

Tw=

temperature on hot wall, K

u=

velocity, m/s

U, V=

dimensionless velocity components

u, v=

velocity components, m/s

W=

cavity width, m

X, Y=

dimensionless coordinates

x, y=

Cartesian coordinates, m

α=

thermal diffusivity, m2/s

β=

thermal expansion coefficient, 1/K

ΔH=

latent enthalpy, J

ρ=

density, kg/m3

φ=

magnetic field orientation, °

μ=

dynamic viscosity, kg/ m s

υ=

kinetic viscosity, m2/s

σ=

electrical conductivity, A/V m

θ=

dimensionless temperature

τ=

dimensionless time

τf, τg=

dimensionless relaxation time

ω=

weight coefficient

Φ=

latent heat source term

=

additional collision term

Subscripts=
i=

direction i in a lattice

 − i=

opposite direction of direction i

l=

liquid phase

loc=

local value

max=

max value

s=

solid phase

Nomenclature

B=

magnitude of magnetic field

c=

lattice speed

cs=

sound speed of the model

C=

specific heat, J/kg × K

En=

enthalpy, J

ei=

discrete lattice velocity in direction i

Fo=

Fourier number

fl=

liquid fraction

fi=

density distribution function in direction i

=

equilibrium distribution function for density in direction i

gi=

temperature distribution function in direction i

=

equilibrium distribution function for temperature in direction i

g=

gravitational acceleration, m/s2

H=

cavity height, m

Ha=

Hartmann number

κ=

thermal conductivity, W/(m K)

L=

latent heat, J/kg

p=

pressure, Pa

q=

heat source term, J

Pr=

Prandtl number

Ra=

Rayleigh number

Ste=

Stefan number

t=

time, s

T=

temperature, K

Tm=

melting temperature, K

Tw=

temperature on hot wall, K

u=

velocity, m/s

U, V=

dimensionless velocity components

u, v=

velocity components, m/s

W=

cavity width, m

X, Y=

dimensionless coordinates

x, y=

Cartesian coordinates, m

α=

thermal diffusivity, m2/s

β=

thermal expansion coefficient, 1/K

ΔH=

latent enthalpy, J

ρ=

density, kg/m3

φ=

magnetic field orientation, °

μ=

dynamic viscosity, kg/ m s

υ=

kinetic viscosity, m2/s

σ=

electrical conductivity, A/V m

θ=

dimensionless temperature

τ=

dimensionless time

τf, τg=

dimensionless relaxation time

ω=

weight coefficient

Φ=

latent heat source term

=

additional collision term

Subscripts=
i=

direction i in a lattice

 − i=

opposite direction of direction i

l=

liquid phase

loc=

local value

max=

max value

s=

solid phase

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