Page 252 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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fractional area a f may vary from location to location on the macro-length scale ‘L’ of the
physical problem owing to the variation of the bed porosity.
The porosity, , of the medium is defined as CONVECTION IN POROUS MEDIA
void volume a f x 1 x 2
= = = a f (8.10)
total volume x 1 x 2
Now, the mass balance of an arbitrary control volume, as shown in Figure 8.3, gives
(refer to Chapter 7)
∂ρ f ∂(ρ f u 1f ) ∂(ρ f u 2f )
+ + = 0 (8.11)
∂t ∂x 1 ∂x 2
where the subscript ‘f’ stands for fluid, ρ is the density and u 1 and u 2 are the velocity com-
ponents in the x 1 and x 2 directions respectively. The volume averaged velocity components
may be defined as (Nield and Bejan 1992),

(8.12)
u 1 = u 1f u 2 = u 2f
Equation 8.11 can be simplified for an incompressible flow (constant density) as
follows:
∂u 1 ∂u 2
+ = 0 (8.13)
∂x 1 ∂x 2
Similarly, the equation for momentum balance can be derived. For instance, in the x 2
direction, the momentum balance gives
2 !
ρ f ∂u 2 ∂ u 1 u 2 ∂ u 2
+ + =
∂t ∂x 1 ∂x 2
!
2
2
1 ∂ µ e ∂ u 2 ∂ u 2
− (p f ) + 2 + 2 + (ρ ref − ρ f )g − D x 2 (8.14)
∂x 2 ∂x ∂x
1 2
where µ e is the equivalent viscosity, p f the fluid pressure, g the acceleration due to gravity
is the matrix drag per unit volume of the porous medium. The particle drag can
and D x 2
be expressed in the following form, as discussed in Section 8.1:
D p = aV + bV 2 (8.15)
for a one-dimensional flow with velocity V . For two-dimensional flow, the drag in the x 2
direction is given as
2 1/2
2
= au 2 + b(u + u ) (8.16)
D x 2 1 2 u 2
by resolving the vertical drag expression along the x 2 direction. In the present formulation,
Ergun’s correlation for the constants a and b, given in Equations 8.5 and 8.6, will be used.
can be written as
Now, the solid matrix drag component D x 2
µ f u 2 1.75 ρ f |V|
= + √ √ (8.17)
D x 2 u 2
κ 150 κ 3/2