ABSTRACT
The natural convection heat transfer in a cavity filled with three layers of solid, porous medium, and free fluid is addressed. The porous medium and free fluid layers are filled with a nanofluid. The porous layer is modeled using the local thermal nonequilibrium (LTNE) model, considering the temperature difference between the solid porous matrix and the nanofluid phases. The nanofluid is modeled using the Buongiorno’s model incorporating the thermophoresis and Brownian motion effects. The governing equations are transformed into a set of nondimensional partial differential equations, and then solved using finite element method in a nonuniform grid. The effects of various nondimensional parameters are discussed. The results showed that the Brownian motion and thermophoresis effects result in significant concentration gradients of nanoparticles in the porous and free fluid layers. The increase in Rayleigh (Ra), Darcy (Da), the thermal conductivity ratios for the solid wall and solid porous matrix, i.e., Kr and Rk, enhanced the average Nusselt number. The increase in the convection interaction heat transfer parameter between the solid porous matrix and the nanofluid in the pores (H) increases the average Nusselt number in the solid porous matrix but decreases the average Nusselt number in the nanofluid phase of the porous layer.
Nomenclature
Latin symbols | = | |
C | = | nanoparticle volume fraction |
C0 | = | ambient nanoparticle volume fraction |
d | = | wall thickness (m) |
D | = | dimensionless wall thickness |
Da | = | Darcy number |
DB | = | Brownian diffusion coefficient |
DT | = | Thermophoretic diffusion coefficient |
g | = | gravitational acceleration vector (m s−2) |
hnfs | = | volumetric heat transfer coefficient between the nanofluid and solid porous matrix (W m−3 K−1) |
H | = | interface heat transfer coefficient parameter |
k | = | thermal conductivity (W m−1 K−1) |
K | = | permeability of the porous medium (m2) |
Kr | = | nanofluid to solid porous matrix thermal conductivity ratio parameter |
L | = | square cavity size (m) |
Le | = | Lewis number |
Nb | = | Brownian motion parameter |
Nr | = | buoyancy ratio parameter |
Nt | = | thermophoresis parameter |
Nu | = | local Nusselt number |
= | average Nusselt number | |
p | = | pressure (Pa) |
P | = | dimensionless pressure |
Pr | = | Prandtl number |
= | total interfacial heat flux (W m−2) | |
Qw | = | dimensionless local heat transfer through the wall |
= | dimensionless average heat transfer through the wall | |
Ra | = | Rayleigh number |
Rk | = | wall to nanofluid thermal conductivity ratio parameter |
s | = | porous layer thickness (m) |
S | = | dimensionless porous layer thickness |
Sh | = | local Sherwood number |
= | average Sherwood number | |
T | = | temperature (K) |
u, v | = | velocity components along x, y directions, respectively (m s−1) |
U, V | = | dimensionless velocity components along x, y directions, respectively |
x, y | = | Cartesian coordinates (m) |
X, Y | = | dimensionless Cartesian coordinates |
Greek symbols | = | |
α | = | effective thermal diffusivity (m2 s−1) |
β | = | thermal expansion coefficient of the fluid (K−1) |
Δ | = | difference value |
ε | = | porosity of the porous medium |
θ | = | dimensionless temperature |
μ | = | dynamic viscosity (kg m−1 s−1) |
ν | = | kinematic viscosity (m2 s−1) |
ρ | = | density (kg m−3) |
(ρc) | = | effective heat capacity (J K−1 m−3) |
τ | = | parameter defined by τ = (ρc)p/(ρc)nf |
ϕ | = | relative nanoparticle volume fraction |
Ψ | = | dimensionless stream function |
Subscripts | = | |
0 | = | ambient property |
c | = | cold |
eff | = | effective |
h | = | hot |
max | = | maximum |
nf | = | nanofluid |
nff | = | nanofluid of free fluid layer |
nfp | = | nanofluid of porous layer |
p | = | nanoparticle |
s | = | solid porous matrix |
w | = | wall |
Nomenclature
Latin symbols | = | |
C | = | nanoparticle volume fraction |
C0 | = | ambient nanoparticle volume fraction |
d | = | wall thickness (m) |
D | = | dimensionless wall thickness |
Da | = | Darcy number |
DB | = | Brownian diffusion coefficient |
DT | = | Thermophoretic diffusion coefficient |
g | = | gravitational acceleration vector (m s−2) |
hnfs | = | volumetric heat transfer coefficient between the nanofluid and solid porous matrix (W m−3 K−1) |
H | = | interface heat transfer coefficient parameter |
k | = | thermal conductivity (W m−1 K−1) |
K | = | permeability of the porous medium (m2) |
Kr | = | nanofluid to solid porous matrix thermal conductivity ratio parameter |
L | = | square cavity size (m) |
Le | = | Lewis number |
Nb | = | Brownian motion parameter |
Nr | = | buoyancy ratio parameter |
Nt | = | thermophoresis parameter |
Nu | = | local Nusselt number |
= | average Nusselt number | |
p | = | pressure (Pa) |
P | = | dimensionless pressure |
Pr | = | Prandtl number |
= | total interfacial heat flux (W m−2) | |
Qw | = | dimensionless local heat transfer through the wall |
= | dimensionless average heat transfer through the wall | |
Ra | = | Rayleigh number |
Rk | = | wall to nanofluid thermal conductivity ratio parameter |
s | = | porous layer thickness (m) |
S | = | dimensionless porous layer thickness |
Sh | = | local Sherwood number |
= | average Sherwood number | |
T | = | temperature (K) |
u, v | = | velocity components along x, y directions, respectively (m s−1) |
U, V | = | dimensionless velocity components along x, y directions, respectively |
x, y | = | Cartesian coordinates (m) |
X, Y | = | dimensionless Cartesian coordinates |
Greek symbols | = | |
α | = | effective thermal diffusivity (m2 s−1) |
β | = | thermal expansion coefficient of the fluid (K−1) |
Δ | = | difference value |
ε | = | porosity of the porous medium |
θ | = | dimensionless temperature |
μ | = | dynamic viscosity (kg m−1 s−1) |
ν | = | kinematic viscosity (m2 s−1) |
ρ | = | density (kg m−3) |
(ρc) | = | effective heat capacity (J K−1 m−3) |
τ | = | parameter defined by τ = (ρc)p/(ρc)nf |
ϕ | = | relative nanoparticle volume fraction |
Ψ | = | dimensionless stream function |
Subscripts | = | |
0 | = | ambient property |
c | = | cold |
eff | = | effective |
h | = | hot |
max | = | maximum |
nf | = | nanofluid |
nff | = | nanofluid of free fluid layer |
nfp | = | nanofluid of porous layer |
p | = | nanoparticle |
s | = | solid porous matrix |
w | = | wall |