Environmental Engineering Reference
In-Depth Information
due to convection. Air flow by conduction is assumed to be
negligible because air acts as an insulator. Air flow resulting
from radiation was discussed in Chapter 6 for the assessment
of actual evaporative flux for climatic boundary conditions.
Air flow by diffusion (e.g., air dissolving in water) and air
flow by advection (e.g., air swept along by water flow) are
briefly discussed.
Air in an unsaturated soil can exist either as a continu-
ous phase or in the form of occluded air bubbles (Fig. 9.1).
The air phase generally becomes continuous as the degree
of saturation becomes less than about 85% (Corey, 1957).
The flow of free air through an unsaturated soil commences
as the air phase becomes continuous. When the degree of
saturation is above about 90%, the air phase generally exists
as occluded air bubbles and air flow is reduced to the dif-
fusion of air through the pore-water (Matyas, 1967) as well
as dissolved air flow by advection.
The flow of air through a soil can be caused by factors
such as variations in barometric pressure, rainwater infil-
tration that compresses the air in the pores of the soil,
temperature changes, and externally applied loads. There are
also some situations where human impact results in changes
in air pressure and subsequent convective air flow.
the rate of mass transfer of a diffusing substance across a
unit area is proportional to the concentration gradient of the
diffusing substance. It is reasonable to apply Fick's law for
the flow of free air through an unsaturated soil, the flow of
water vapor in response to a vapor pressure gradient, and
the diffusion of air through fluids such as water.
Air has a low density and as a result the driving potential
for flow can be taken as the air pressure gradient. The mass
of air flow,
m
a
(as opposed to the volume of air flow), can be
written using Fick's (1855) constitutive flow relationship:
du
a
dx
m
ax
=−
D
ax
(9.1)
du
a
dy
m
ay
=−
D
ay
(9.2)
du
a
dz
m
az
=−
D
az
(9.3)
where:
m
ax
,m
ay
,m
az
=
mass flow rate in the
x-
,
y-
, and
z
-
directions, respectively, and
D
ax
,D
ay
,D
az
=
air diffusivity in the
x-
,
y-
and
z-
directions, respectively.
9.2.1 Driving Potential for Air
The flow of free air is governed by a concentration, den-
sity, or pressure gradient when the air phase is continuous.
The elevation gradient is negligible. The pressure gradient is
most commonly considered as the only driving potential for
the air phase. Both Fick's law and Darcy's law can be used
to describe the flow of air through a porous medium. Fick's
law may appear to be more fundamentally correct, but it
is also useful to be able to apply a Darcy-type formulation
with its associated air coefficient of permeability.
The air flow law can also be written as a velocity of air
flow,
v
a
,
similar to Darcy's (1856) law, thereby taking on
the following form (Blight, 1971):
k
ax
du
a
dx
v
ax
=−
(9.4)
du
a
dy
v
ay
=−
k
ay
(9.5)
du
a
dz
v
az
=−
k
az
(9.6)
9.3 FICK'S LAW AND DARCY'S LAW
FOR AIR FLOW
The air coefficient of permeability
k
a
(and air diffusivity)
is a mathematical function in the sense that the transmission
of air varies with the amount of air in the soil. The amount
of air can be described using the SWCC. Figure 9.2a shows
a SWCC for sand and Fig. 9.2b illustrates the form of the
air permeability function (Ba-Te et al., 2005). The air per-
meability function takes on an inverse form to that of the
water permeability function.
The magnitude of the air coefficient of permeability tends
toward the diffusion of air through water when soil suc-
tion becomes less than the air-entry value of the soil. The
air coefficient of permeability increases by several orders of
magnitude as suction is increased once the suction in the soil
exceeds the air-entry value. The low viscosity of air means
that air can flow through a soil with much greater ease than
water flow through a soil of equal porosity.
The void space of a porous medium (i.e., porosity of the
soil) can be used as the reference. There is one extreme sit-
uation where the void space is filled with water and another
Fick's (1855) law is often used to describe the diffusion of
gases through liquids. A modified form of Fick's law can
be applied to the air flow process. Fick's first law states that
Flow systems common
to unsaturated soils
Continuous air phase
(two-phase flow)
Air diffusion
through water
Water
Air
Occluded air bubbles
(compressible
pore fluid flow)
Figure 9.1
Subdivision of air flow systems in unsaturated soil.
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