Page 165 - Bird R.B. Transport phenomena

Problems 149
in which e, the porosity, is the ratio of pore volume to total volume, and к is the permeability of
the porous medium. The velocity v 0 in these equations is the superficial velocity, which is de-
fined as the volume rate of flow through a unit cross-sectional area of the solid plus fluid, av-
eraged over a small region of space—small with respect to the macroscopic dimensions in the
flow system, but large with respect to the pore size. The density and pressure are averaged
over a region available to flow that is large with respect to the pore size. Equation 4C.3-2 was
proposed empirically to describe the slow seepage of fluids through granular media.
When Eqs. 4C.3-1 and 2 are combined we get

(4C.3-3)

for constant viscosity and permeability. This equation and the equation of state describe the
motion of a fluid in a porous medium. For most purposes we may write the equation of state as
m
p = Po p e^ (4C.3-4)
in which p is the fluid density at unit pressure, and the following parameters have been given: 6
0
1. Incompressible liquids m = 0 /3 = 0
2. Compressible liquids m = 0 /3 Ф 0
3. Isothermal expansion of gases /3 = 0 m-\
4. Adiabatic expansion of gases /3 = 0 m = C /C = \/y
v
p
Show that Eqs. 4C.3-3 and 4 can be combined and simplified for these four categories to give
(for gases it is customary to neglect the gravity terms since they are small compared with the
pressure terms):
Casel. V p = 0 (4С.З-5)
2

Case 2. ( ^ ) Ц- = V p - (V • p /3g) (4C.3-6)
2
2
Case 3. ( ^ ^ j ^ = V p 2 (4C.3-7)
2
Case 4. /(m + \)e^ \ ^ ^
/m
= у 2 ( 1 +т)/т (4Q 3 g )
Note that Case 1 leads to Laplace's equation, Case 2 without the gravity term leads to the heat-
conduction or diffusion equation, and Cases 3 and 4 lead to nonlinear equations. 7

4C.4 Radial flow through a porous medium (Fig. 4C.4). A fluid flows through a porous cylindri-
cal shell with inner and outer radii R } and R , respectively. At these surfaces, the pressures
2
are known to be p and p , respectively. The length of the cylindrical shell is h.
2
x
Porous medium Fluid

w = mass
rate of flow
] J Fig.4C4. Radial flow
\ \ I through a porous
Pressure p } Pressure p 2 medium.

в
М. Muskat, Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill (1937).
7 For the boundary condition at a porous surface that bounds a moving fluid, see G. S. Beavers and
D. D. Joseph, J. Fluid Mech., 30,197-207 (1967) and G. S. Beavers, E. M. Sparrow, and B. A. Masha, AIChE
Journal, 20, 596-597 (1974).