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An Improved Lattice Boltzmann Model for Convection Melting in the Existence of an Inhomogeneous Magnetic Field

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Abstract

An improved single-relaxation-time lattice Boltzmann model of the melting with natural convection in a cavity submitted to an inhomogeneous magnetic field is developed and comparatively investigated in this paper, which can not only reduce the numeric diffusion at the phase boundary obviously but also recover the macroscopic energy equations correctly. Three numerical benchmark cases are performed, including the half-space conduction melting, convection-dominated melting, and natural convection under an inhomogeneous magnetic field. Impacts of the dimensionless magnetic force parameter and the magnetic field inclination angle on the melting process are presented in relation to the average Nusselt number, liquid fraction, temperature profile, melting front and pressure distribution. The results show that the numerical predictions based on the current model agree with analytical solutions and present better accuracy than those reported in previous studies. Like the buoyancy force, the magnetic force also has a vital influence on the natural convection development in the melting process. Compared with the case without a magnetic field, the magnetic field angle can be adjusted to enhance and suppress the melting process by employing the coupling effect of the magnetic force and gravity. Moreover, there exists an optimal magnetic field inclination angle for maximizing the melting heat transfer performance. Besides, increasing the dimensionless magnetic parameter expands the gain of melting enhancement while expanding the range of inclination angles that enhances melting performance.

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Abbreviations

a m :

Magnetic acceleration(m/s2)

c :

Lattice speed (m/s)

c 0 :

Constant

c p :

Specific heat at constant pressure (kJ/(kg K))

c s :

Sound speed (m/s)

e i :

Discrete lattice velocity in direction i (m/s)

En :

Total enthalpy (kJ/kg)

En 0 :

Total enthalpy at the beginning (kJ/kg)

En 1 :

Total enthalpy at the end (kJ/kg)

f i :

Density distribution function in direction i (kg/m3)

f i eq :

Equilibrium distribution function of density in direction i (kg/m3)

f l :

Liquid volume fraction

F :

Body force per unit mass (N/kg)

F i :

Discrete body force in direction i (kg/(m3 s))

F m :

Magnetic force per unit mass (N/kg)

Fo :

Fourier number, Fo = αt/L2

g :

Acceleration due to gravity (m/s2)

g i :

Temperature distribution function in direction i (K)

g i eq :

Equilibrium temperature distribution function in direction i (K)

H :

Height of the domain or characteristic length (m)

H m :

Magnetic field intensity

k :

Thermal conductivity coefficient (W/(m K))

L :

Weight of the domain or characteristic length (m)

L a :

Latent heat of phase change(kJ/kg)

Nu :

Nusselt number

Nu avr :

Average Nusselt number

p :

Pressure (Pa)

P :

Dimensionless pressure

Pr :

Prandtl number

r :

Space position (m)

Ra :

Rayleigh number

Sr i :

Discrete source term (W/m3)

Ste :

Stefan number

t :

Time (s)

T :

Temperature (K)

T c :

Temperature of cold wall (K)

T h :

Temperature of hot wall (K)

T init :

Initial temperature (K)

T m :

Melting temperature (K)

u :

Velocity (m/s)

u x :

X velocity (m/s)

u y :

Y velocity (m/s)

v :

Kinematic coefficient of viscosity (m2/s)

w i :

Weight coefficient

x, y :

Cartesian coordinates (m)

α :

Thermal diffusivity (m2/s)

β :

Coefficient of thermal expansion (1/K)

γ :

Dimensionless magnetic force

Δx :

Lattice space (m)

Δt :

Time step (s)

θ m :

Inclination angle

λ :

Melting coefficient

μ 0 :

Vacuum permeability

υ :

Kinematic viscosity (m2/s)

ξ :

A constant parameter

ρ :

Density (kg/m3)

τ f , τ T :

Dimensionless relaxation time

χ m :

Magnetic susceptibility

s :

Solid phase

l :

Liquid phase

i :

Direction i in a lattice

PCM:

Phase change material

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 51876184) and the Science Fund of Nanjing Institute of Technology (No. CXY201924 and No. CKJA201604).

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Authors and Affiliations

  1. School of Energy and Power Engineering, Nanjing Institute of Technology, Nanjing, 211167, PR China

    Xianqi Cao & Dongyan Gao

  2. School of Energy and Environment, Southeast University, Jiangsu 210096, Nanjing, PR China

    Yongping Huang

  3. College of Electrical, Energy and Power Engineering, Yangzhou University, Yangzhou, 225127, PR China

    Xiangdong Liu

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  1. Xianqi Cao
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  2. Dongyan Gao
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  4. Xiangdong Liu
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Correspondence to Xiangdong Liu.

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Cao, X., Gao, D., Huang, Y. et al. An Improved Lattice Boltzmann Model for Convection Melting in the Existence of an Inhomogeneous Magnetic Field. Microgravity Sci. Technol. 33, 56 (2021). https://doi.org/10.1007/s12217-021-09903-6

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