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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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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DOI: https://doi.org/10.1007/s00231-016-1811-8