Abstract

Combined heat and mass transfer process by natural convection along a vertical wavy surface in a thermal and mass stratified fluid saturated porous enclosure has been numerically analysed. Finite element method has been used and the influence of varying flow, heat and mass transfer governing parameters has been reported. Presence of thermal and mass stratification terms reduces the Nusselt number and Sherwood number values in all the cases. The flow circulation pattern which is anti-clockwise when the species buoyancy forces are opposing the thermal buoyancy forces, gets clockwise when the forces are aiding. When the two buoyancy forces are equal and opposing, a multi-cellular pattern with alternating circulation orientation manifests. Several other interesting features such as thermal and mass boundary layers, thermal plumes, secondary circulation zones, flow intensification etc. are noticed in the flow, temperature and concentration fields with varying flow, heat and mass transfer governing parameters.

Keywords: Double diffusion; Natural convection; Finite element method; Porous enclosure; Thermal and mass stratification; Wavy surface

Nomenclature

a

amplitude of the wavy wall

B

Buoyancy ratio (=β_c(c_w − c_∞,0)/β_t(t_w − t_∞,0))

c

dimensional species concentration

C

non-dimensional species concentration (=(c − c_∞,x)(c_w − c_∞,0))

D

mass diffusivity

e

typical element in finite element formulation

g

gravitational acceleration

k

thermal conductivity

K

permeability of the porous medium

L

length or the mean width of the porous cavity

Le

Lewis number (=α/D)

n

outward drawn unit normal to the wavy surface

N

number of waves considered per unit length

N_i

quadratic interpolation function

Nu

Nusselt number

QH_x

cumulative heat flux

QM_x

cumulative mass flux

Ra

Rayleigh number (=Kgβ_tL(t_w − t_∞,0)/αν)

s_c

dimensional mass stratification parameter (=dc_∞,x/dx)

s_t

dimensional thermal stratification parameter (=dt_∞,x/dx)

S_C

non-dimensional mass stratification parameter (=1/(c_w − c_∞,0)*(dc_∞,x/dX))

S_T

non-dimensional thermal stratification parameter (=1/(t_w − t_∞,0)*(dt_∞,x/dX))

Sh

Sherwood number

S(ξ)

arc length of the wavy wall

t

dimensional temperature

T

non-dimensional temperature (=(t − t_∞,x)/(t_w − t_∞,0))

u, v

dimensional velocity components in x and y directions

U, V

non-dimensional velocity components in X and Y directions (U = u/V_c,V = v/V_c)

V_c

convective velocity (=gβK(t_w − t_∞,0)/ν)

w

weight function used in the finite element formulation

x, y

dimensional cartesian coordinates

X, Y

non-dimensional cartesian coordinates (X = x/L, Y = y/L)

almost less than to

almost greater than to

Greek symbols

α

thermal diffusivity

β_c

concentration expansion coefficient (=−(1/ρ)(∂ρ/∂c)_P,t)

β

thermal expansion coefficient (=−(1/ρ)(∂ρ/∂t)_P,c)

phase of the wavy surface

ρ

fluid density

Ψ

non-dimensional stream function (U = ∂Ψ/∂Y, V = −∂Ψ/∂X)

ν

fluid kinematic viscosity

ξ

arc length variable

Ω

domain considered in the problem

Γ

boundary of the domain

Subscripts

0, ∞

ambient points

P

pressure

w

evaluated at the wall temperature

x

evaluated at point x

Article Outline

Nomenclature

1. Introduction

2. Mathematical model

3. Finite element formulation

4. Results and discussion

4.1. Influence of wave amplitude (a)

4.2. Influence of wave number (N)

4.3. Influence of the phase () of the wavy surface

4.4. Influence of Rayleigh number (Ra)

4.5. Influence of buoyancy ratio (B)

4.6. Influence of Lewis number (Le)

4.7. Influence of stratification terms (S_T and S_C)

5. Conclusion

References