Page 249 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
2
Where u i are the seepage velocity components, κ (m ) is the permeability of the
medium, µ is the dynamic viscosity of the fluid, p is the pressure and x i are the coordinate
axes. For two-dimensional flow, we can rewrite the velocity components as 241
κ ∂p
u 1 =−
µ ∂x 1
κ ∂p
u 2 =− (8.2)
µ ∂x 2
It is interesting to note that the above equation is very similar to Ohm’s law for the flow
of electricity, Fourier’s law of heat conduction and Fick’s law for mass diffusion. However,
simple relations such as Darcy’s law are not always applicable, and further modifications
or extensions are necessary in order to accurately predict the flow field in porous media.
Several years after the introduction of Darcy’s law, two major additions to the model
have extended its use in many engineering disciplines including chemical, mechanical and
civil engineering. The first extension was due to Forchheimer (Forchheimer 1901), and this
modification accounted for moderate and high Reynolds number effects with the addition
of a nonlinear term in the Darcy equation. A relationship for the drag force was introduced
by Forchheimer, Figure 8.1, as
D p = au i + bu 2 i (8.3)

D p

Solid particle

Flow direction

Figure 8.1 Drag force on a porous medium grain