Local thermal nonequilibrium conjugate natural convection heat transfer of nanofluids in a cavity partially filled with porous media using Buongiorno’s model

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., K_r and R_k, 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