Page 253 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
where V is the velocity vector in the field. By substituting Equation 8.17 into Equation 8.14,
we obtain
2 ! 2 2 ! 245
ρ f ∂u 2 ∂ u 1 u 2 ∂ u 2 1 ∂ µ e ∂ u 2 ∂ u 2
+ + =− (p f ) + +
∂t ∂x 1 ∂x 2 ∂x 2 ∂x 2 ∂x 2
1 2
µ f u 2 1.75 ρ f |V|
+(ρ ref − ρ f )g − − √ √ u 2 (8.18)
κ 150 κ 3/2
Similarly, other momentum components can also be derived, and the final form of the
governing equations for incompressible flow through a porous medium in dimensional form
can be given, using indicial notation, as
Continuity
∂u i
= 0 (8.19)
∂x i
Momentum
2
ρ f ∂u i ∂ u i u j 1 ∂ µ e ∂ u i
+ =− (p f ) +
∂t ∂x j ∂x i ∂x 2
i
µ f u i 1.75 ρ f |V|
+(ρ ref − ρ f )gγ i − − √ √ 3/2 u i (8.20)
κ 150 κ
The previous equation can be simplified by substituting Equation 8.19 into
Equation 8.20. The energy conservation equation is also derived in a similar manner. The
final form of the energy equation is
Energy
!
2
∂T ∂T ∂ T

(ρc p ) f + (1 − )(ρc p ) s + (ρc p ) f u i = k (8.21)
∂t ∂x i ∂x 2
i
In the above equation, t is the time, c p is the specific heat, γ i is a unit vector in the
gravity direction, T is the temperature and k is the equivalent thermal conductivity. The
subscripts f and s stand for the fluid and solid phases respectively.
It should be noted that the permeability and thermal conductivity values can be direc-
tional, in which case they are tensors.

8.2.1 Non-dimensional scales
The non-dimensional form of the equations simplifies most of the calculations. The fol-
lowing final form of the non-dimensional equations may be obtained by suitable scaling.

Continuity equation
∂u ∗
i = 0 (8.22)
∂x ∗
i