Page 250 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
242
which is balanced by the pressure force as follows:
2 ∂p
au i + bu =− (8.4)
i
∂x i
In the above equation, the first term on the left-hand side is, in essence, similar to the
linear drag term introduced by Darcy, and the second term is the nonlinear drag term. The
parameters a and b are determined by empirical relations and one such correlation was
given by Ergun (Ergun 1952), that is,
2
(1 − ) µ f
a = 150 (8.5)
3 d p 2
and
(1 − ) ρ f
b = 1.75 (8.6)
3 d p
It should be noted, however, that other suitable correlations may also be employed for
different ranges of the bed porosity, , to obtain the non-Darcian flow behaviour inside a
porous medium. In the above equations, d p is the solid particle size in a porous medium,
and ρ f is the fluid density. The above solid matrix drag relation can also be expressed in
terms of the medium permeability κ by defining
3 2
d p
κ = (8.7)
150(1 − ) 2
The flow relationship, given by Equation 8.4, can be rewritten in terms of permea-
bility as
µ f u i 1.75 ρ f |V| ∂p
+ √ √ u i =− (8.8)
κ 150 κ 3/2 ∂x i
Although the above equation gives an accurate solution at higher Reynolds numbers,
it is not accurate enough to solve flow in highly porous and confined media. In order to
deal with the viscous and higher porosity effects, Brinkman introduced an extension to
the Darcy model in 1947, which included a second-order viscous term with an equivalent
viscosity for the porous medium (Brinkman 1947). The viscous extension, as given by
Brinkman, can be written as (Figure 8.2)
2
∂p ∂ u i
au i =− + µ e (8.9)
∂x i ∂x 2
i
where µ e is the equivalent viscosity of the porous medium. This modification takes into
account the no-slip conditions that exist on the confining walls (Tong and Subramanian
1985).
The Darcy model and the extensions discussed above have been widely used in the past.
However, a generalized model, incorporating the flow regimes covered by both Darcy’s
model and its extension, will have several advantages (Hsu and Cheng 1990; Nithiarasu
et al. 1997, 2002; Vafai and Tien 1981; Whitaker 1961). One of these is that the gener-
alized flow model approaches the standard incompressible Navier–Stokes equations when