Influence of the shape, thermal radiation, and variable properties on the turbulent buoyancy-driven airflow inside cavities with Trombe wall geometry

ABSTRACT

The effects of the radiative heat transfer and the influence of air variable properties on the natural convection flows in cavities with Trombe wall geometry are numerically investigated. The influence of considering surface-radiative effects on the differences reached for the Nusselt number and the mass flow rate obtained with several heating intensities is studied. The effects of the shape change consisting of varying slope angle of the heated wall is also analyzed. The influence of the thermal radiation on the appearance of the burnout phenomenon becomes particularly relevant. Important changes in the thermal and dynamic flow patterns are observed for high enough values of the wall slope angle.

Nomenclature

b	=	width of the vents, m (Figure 1)
bc	=	vertical channel wall-to-wall spacing, m (Figure 1c)
bw	=	width of the inside solid wall, m (Figure 1c)
cp	=	specific heat at constant pressure, J/kg·K
Ds	=	radiative stopping distance, m
g	=	gravitational acceleration, m/s2
GrH	=	Grashof number for isothermal cases,
GrH	=	Grashof number for heat flux cases,
H	=	height of the cavity and the heated wall (Figure 1), m
hy	=	local heat transfer coefficient,, W/m2 K
I	=	turbulence intensity
J	=	radiosity (W/m2)
k	=	turbulent kinetic energy, Eq. (16), m2/s2
ks	=	coefficient of radiative scattering, 1/m
L, Lc	=	length of the cavity (Figure 1), characteristic length, m
M	=	dimensionless mass flow rate,
m	=	mass flow rate, kg/s · m (two-dimensional)
n	=	coordinate perpendicular to wall, m
NuH	=	average Nusselt number based on H, Eqs. (8) and (9)
Nuy	=	local Nusselt number, hyH/κ
P	=	average reduced pressure, N/m2
PT	=	total-average reduced pressure, N/m2
p	=	pressure, N/m2
Pr	=	Prandtl number, μcp/κ
q	=	wall heat flux (convective), W/m2
qr	=	boundary heat flux (radiative), W/m2
R	=	constant of air, R = 287 J/kg · K
RaH	=	Rayleigh number based on H, (GrH)(Pr)
Sij	=	mean strain tensor, 1/s
T,T′	=	average and turbulent temperatures, K
	=	average turbulent heat flux, K·m/s
	=	average and turbulent components of velocity, respectively, m/s
	=	turbulent stress, m2/s2
uτ	=	friction velocity, , m/s
V	=	dimensionless average velocity, V = UH/α
x,y	=	Cartesian coordinates (Figure 1), m
y1	=	wall to first grid point distance, m
y+	=
α	=	thermal diffusivity, κ/ρcp, m2/s
αr	=	coefficient of radiative absorption, 1/m
β	=	coefficient of thermal expansion, 1/T∞, 1/K
γ	=	wall slope angle (Figure 1c)
δT	=	thickness of the thermal boundary layer, m
ϵ	=	coefficient of surface radiation emissivity
κ	=	thermal conductivity, W/m·K
Λ	=	heating intensity, Eqs. (2) and (6) for UWT and UHF heating, respectively
μ	=	viscosity, kg/m · s
ν	=	kinematic viscosity, μ/ρ, m2/s
θ	=	dimensionless temperature difference,
ρ	=	density, kg/m3
σ	=	Stefan–Boltzmann constant, σ = 5.6678 × 10−8 W/m2K
τw	=	wall shear stress, N/m2
ω	=	specific dissipation rate of k, 1/s
Subscripts	=
B	=	constant properties and Boussinesq approximation
r	=	radiative
t	=	turbulent
w	=	wall
∞	=	ambient or reference conditions
Superscripts	=
−	=	averaged value
Abbreviations	=
max,min	=	maximum/minimum value
UHF,UWT	=	Uniform Heat Flux/Uniform Wall Temperature

Nomenclature

b	=	width of the vents, m (Figure 1)
bc	=	vertical channel wall-to-wall spacing, m (Figure 1c)
bw	=	width of the inside solid wall, m (Figure 1c)
cp	=	specific heat at constant pressure, J/kg·K
Ds	=	radiative stopping distance, m
g	=	gravitational acceleration, m/s2
GrH	=	Grashof number for isothermal cases,
GrH	=	Grashof number for heat flux cases,
H	=	height of the cavity and the heated wall (Figure 1), m
hy	=	local heat transfer coefficient,, W/m2 K
I	=	turbulence intensity
J	=	radiosity (W/m2)
k	=	turbulent kinetic energy, Eq. (16), m2/s2
ks	=	coefficient of radiative scattering, 1/m
L, Lc	=	length of the cavity (Figure 1), characteristic length, m
M	=	dimensionless mass flow rate,
m	=	mass flow rate, kg/s · m (two-dimensional)
n	=	coordinate perpendicular to wall, m
NuH	=	average Nusselt number based on H, Eqs. (8) and (9)
Nuy	=	local Nusselt number, hyH/κ
P	=	average reduced pressure, N/m2
PT	=	total-average reduced pressure, N/m2
p	=	pressure, N/m2
Pr	=	Prandtl number, μcp/κ
q	=	wall heat flux (convective), W/m2
qr	=	boundary heat flux (radiative), W/m2
R	=	constant of air, R = 287 J/kg · K
RaH	=	Rayleigh number based on H, (GrH)(Pr)
Sij	=	mean strain tensor, 1/s
T,T′	=	average and turbulent temperatures, K
	=	average turbulent heat flux, K·m/s
	=	average and turbulent components of velocity, respectively, m/s
	=	turbulent stress, m2/s2
uτ	=	friction velocity, , m/s
V	=	dimensionless average velocity, V = UH/α
x,y	=	Cartesian coordinates (Figure 1), m
y1	=	wall to first grid point distance, m
y+	=
α	=	thermal diffusivity, κ/ρcp, m2/s
αr	=	coefficient of radiative absorption, 1/m
β	=	coefficient of thermal expansion, 1/T∞, 1/K
γ	=	wall slope angle (Figure 1c)
δT	=	thickness of the thermal boundary layer, m
ϵ	=	coefficient of surface radiation emissivity
κ	=	thermal conductivity, W/m·K
Λ	=	heating intensity, Eqs. (2) and (6) for UWT and UHF heating, respectively
μ	=	viscosity, kg/m · s
ν	=	kinematic viscosity, μ/ρ, m2/s
θ	=	dimensionless temperature difference,
ρ	=	density, kg/m3
σ	=	Stefan–Boltzmann constant, σ = 5.6678 × 10−8 W/m2K
τw	=	wall shear stress, N/m2
ω	=	specific dissipation rate of k, 1/s
Subscripts	=
B	=	constant properties and Boussinesq approximation
r	=	radiative
t	=	turbulent
w	=	wall
∞	=	ambient or reference conditions
Superscripts	=
−	=	averaged value
Abbreviations	=
max,min	=	maximum/minimum value
UHF,UWT	=	Uniform Heat Flux/Uniform Wall Temperature