Page 254 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
246
Momentum equations
∂
1 ∂u
∗
u
1
i
∗
f
∂x

∂t ∗ ∗ i + u ∗ j ∂x ∗ j u ∗ i =− 1 ∂ i ∗ ( p ) − ReDa
!
∗
1.75 |V | u ∗ i J ∂ u i Gr
2 ∗
−√ √ 3/2 + ∗2 + γ i 2 T ∗ (8.23)
150 Da Re ∂x Re
i
Energy equation
!
2
∂T ∗ ∂T ∗ k ∗ ∂ T ∗
σ + u ∗ i = (8.24)
∂t ∗ ∂x i ∗ ReP r ∂x ∗2
i
In the previous equations, the parameters governing the flow and heat transfer are the
Darcy number (Da), Reynolds number (Re), Prandtl number (Pr), Grashof number (Gr),
∗
the ratio of heat capacities (σ); porosity of the medium ( ), conductivity ratio (k ), viscosity
ratio (J), and the anisotropic property ratios, for the case of an anisotropic medium. The
definitions for the scales and non-dimensional parameters are
x i u i tu a p f T − T a µ e
∗ ∗ ∗ ∗ ∗
x = ; u = ; t = ; p = ; T = ; J = ;
i
f
i
L u a L ρ f u a 2 T w − T a µ f
(ρc p ) f + (1 − )(ρc p ) s k ρ f u a L
∗
σ = ; k = ; Re = ;
(ρc p ) f k f µ f
ν f κ gβ T L 3
Pr = ; Da = 2 ; Gr = 2 (8.25)
α f L ν
f
The above scales are suitable for most forced and mixed convection problems. However,
for buoyancy-driven flows, it is convenient to handle the equations using the following
definition of the Rayleigh number (Ra), that is,
gβ T L 3
Ra = (8.26)
να
where the following different scales need to be employed in solving natural convection
problems:
u i L tα f pL 2
∗ ∗ ∗
u = ; t = 2 ; p = 2 (8.27)
i
α f L ρ f α
f
The non-dimensional governing equations for natural convection are
Continuity equation
∂u ∗
i = 0 (8.28)
∂x ∗
i
Momentum equations
1 ∂u ∗ i + u ∗ ∂ u ∗ i 1 ∂ ∗ Pru ∗ i
1
f
∂t ∗ j ∂x ∗ =− ∂x ∗ ( p ) − Da
j i
!
1.75 |V | u ∗ i JP r ∂ u i
∗
2 ∗
−√ √ 3/2 + ∗2 + γ i RaP rT ∗ (8.29)
150 Da ∂x
i