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Double-diffusive natural convection in inclined porous cavities with various aspect ratios and temperature-dependent heat source or sink

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Abstract

Laminar double-diffusive natural convective flow of a binary fluid mixture in inclined square and rectangular cavities filled with a uniform porous medium in the presence of temperature-difference dependent heat generation (source) or absorption (sink) is considered. Transverse gradients of heat and mass are applied on two opposing walls of the cavity while the other two walls are kept adiabatic and impermeable to mass transfer. The problem is put in terms of the stream function-vorticity formulation. A numerical solution based on the finite-difference methodology is obtained for relatively high Lewis numbers. Representative results illustrating the effects of the inclination angle of the cavity, buoyancy ratio, Darcy number, heat generation or absorption coefficient and the cavity aspect ratio on the contour maps of the streamline, temperature, and concentration as well as the profiles of velocity, temperature and concentration at mid-section of the cavity are reported. In addition, numerical results for the average Nusselt and Sherwood numbers are presented for various parametric conditions and discussed.

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Abbreviations

A :

enclosure aspect ratio = H/W

c :

concentration of species

c h :

high species concentration (source)

c l :

low species concentration (sink)

c p :

specific heat of the fluid

c s :

specific heat of porous medium material

C :

dimensionless species concentration = (c − c 1)/(c h − c 1) − 0.5

D :

species diffusivity

Da :

Darcy number = κ/W 2

g :

gravitational acceleration

H :

enclosure height

Le :

Lewis number = α e /D

N :

buoyancy ratio = β c (c h − c 1)/[ β T (T h − T c ) ]

\(\overline{{{\hbox{Nu}}}}\) :

average Nusselt number at heated vertical wall

p :

fluid pressure

Pr :

Prandtl number \( = v \mathord{\left/ {\vphantom {v {\alpha_{e}}}} \right. \kern-\nulldelimiterspace} {\alpha_{e}}\)

Q 0 :

heat generation or absorption coefficient

Ra :

thermal Rayleigh number = gβ T (T h − T c )W3/(α e ν)

\(\overline{{{\hbox{Sh}}}}\) :

average Sherwood number at heated vertical wall

t :

time

T :

temperature

T h :

hot wall temperature (source)

T c :

cold wall temperature (sink)

u :

horizontal velocity component

U :

dimensionless horizontal velocity component = uW/α e

v :

vertical velocity component

V :

dimensionless vertical velocity component = vW/α e

W :

enclosure width

x :

horizontal coordinate

X :

dimensionless horizontal coordinate = x/W

y :

vertical coordinate

Y :

dimensionless vertical coordinate = y/W

α:

inclination angle of the cavity

αe :

effective thermal diffusivity of the porous medium

βT :

thermal expansion coefficient

βc :

compositional expansion coefficient

ɛ:

porosity of the porous medium

ϕ:

dimensionless heat generation or absorption coefficient = Q 0 W 2/(ρ c p α e )

κ:

permeability of the porous medium

μ:

dynamic viscosity

ν:

kinematic viscosity = μ/ρ

θ:

dimensionless temperature = (T − T c )/(T h −T c ) − 0.5

ρ :

Density

ρ s :

Porous medium material density

σ:

specific heats ratio = [ɛ ρ c p +  (1 − ɛ)ρ s c s ]/(ρ c p )

τ :

dimensionless time = αe t/W 2

Ω:

vorticity

ψ:

dimensionless stream function = Ψ/α e

Ψ:

stream function

ζ:

dimensionless vorticity = Ω W2/α e

∇2 :

Laplacian operator

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Acknowledgement

The authors acknowledge and appreciate the financial support of this work by the Public Authority for Applied Education & Training under Project No. TS-05-005.

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  1. Manufacturing Engineering Department, The Public Authority for Applied Education and Training, P. O. Box 42325, Shuweikh, 70654, Kuwait

    Ali J. Chamkha & Ali Al-Mudhaf

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  1. Ali J. Chamkha
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  2. Ali Al-Mudhaf
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Correspondence to Ali J. Chamkha.

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Chamkha, A.J., Al-Mudhaf, A. Double-diffusive natural convection in inclined porous cavities with various aspect ratios and temperature-dependent heat source or sink. Heat Mass Transfer 44, 679–693 (2008). https://doi.org/10.1007/s00231-007-0299-7

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  • DOI: https://doi.org/10.1007/s00231-007-0299-7

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