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Lattice Boltzmann technique for heat transport phenomena coupled with melting process

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Abstract

In this work, the heat transport phenomena coupled with melting process are studied by using the enthalpy-based lattice Boltzmann method (LBM). The proposed model is a modified version of thermal LB model, where could avoid iteration steps and ensures high accuracy. The Bhatnagar–Gross–Krook (BGK) approximation with a D1Q2 lattice was used to determine the temperature field for one-dimensional melting by conduction and multi-distribution functions (MDF) with D2Q9 lattice was used to determine the density, velocity and temperature fields for two-dimensional melting by natural convection. Different boundary conditions including Dirichlet, adiabatic and bounce-back boundary conditions were used. The influence of increasing Rayleigh number (from 103 to 105) on temperature distribution and melting process is studied. The obtained results show that a good agreement with the analytical solution for melting by conduction case and with the benchmark solution for melting by convection.

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Abbreviations

c :

Lattice speed

c p :

Specific heat at constant pressure

c s :

Sound speed of the lattice

e i :

Discrete lattice velocity in direction i

En :

Total enthalpy

En l :

Total enthalpy of the liquid phase

En s :

Total enthalpy of the solid phase

f l :

Volume fraction of liquid

f i :

Particle distribution function in direction i for velocity field

\(f_{i}^{eq}\) :

Equilibrium distribution function in direction i for velocity field

F :

Buoyancy force

g :

Acceleration due to gravity

g i :

Particle distribution function in direction i for temperature field

\(g_{i}^{eq}\) :

Equilibrium distribution function in direction i for temperature field

l :

Appropriate length scale

L f :

Latent heat of phase change

Nu :

Nusselt number

p :

Pressure

Pr:

Prandtl number (ν/α)

Ra :

Rayleigh number (gβΔTl 3/να)

Ste :

Stefan number (c p ΔT/L f )

t :

Time

T :

Temperature

T 0 :

Initial temperature

T b :

Temperature of the left wall

T m :

Melting temperature

u :

Velocity

Δt :

Lattice time step

Δx :

Lattice space

α :

Thermal diffusivity

β :

Volume expansivity

ε :

Small expansion parameter

κ :

Thermal conductivity

ν :

Kinematic viscosity

ω i :

Weight coefficient in direction i

Ω:

Collision operator

τ, τ f , τ g :

Relaxation time

*:

Dimensionless symbols

k :

Iteration

n :

Time step

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Aswan University, Aswan, 81528, Egypt

    A. M. Ibrahem, M. F. El-Amin & A. A. Mohammadein

  2. Effat University, Jeddah, 21478, Kingdom of Saudi Arabia

    M. F. El-Amin

  3. Department of Mechanical Engineering, University of Akron, Akron, OH, 44325, USA

    Rama Subba Reddy Gorla

Authors
  1. A. M. Ibrahem
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  2. M. F. El-Amin
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  3. A. A. Mohammadein
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  4. Rama Subba Reddy Gorla
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Correspondence to Rama Subba Reddy Gorla.

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Ibrahem, A.M., El-Amin, M.F., Mohammadein, A.A. et al. Lattice Boltzmann technique for heat transport phenomena coupled with melting process. Heat Mass Transfer 53, 213–221 (2017). https://doi.org/10.1007/s00231-016-1811-8

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