Environmental Engineering Reference
In-Depth Information
D
ay
=
D
a
), and the air flow equation becomes
D
a
∂
2
u
a
∂x
2
∂
2
u
a
∂y
2
∂D
a
∂x
∂D
a
∂y
∂u
a
∂x
+
∂u
a
∂y
=
+
+
0 (9.59)
where:
D
a
=
air coefficient of transmission in the
x
- and
y
-
directions.
The partial differential equations for steady-state air flow
bear a similarity in form to those previously presented for
water flow.
9.6.3 Three-Dimensional Steady-State Air Flow
The three-dimensional steady-state partial differential equa-
tion for air flow requires that a third dimension be added to
the previously presented two-dimensional air flow equation:
D
a
∂
2
u
a
∂x
2
Figure 9.9
Two-dimensional steady-state air flow through an
unsaturated soil element.
∂
2
u
a
∂y
2
∂
2
u
a
∂z
2
∂D
a
∂x
∂D
a
∂y
∂u
a
∂x
∂u
a
∂y
Substituting Fick's law for the mass rates
J
ax
and
J
ay
gives the following nonlinear partial differential equation:
+
+
+
+
∂D
a
∂z
∂u
a
∂z
D
ax
(u
a
−
D
ay
(u
a
−
+
=
0
(9.60)
∂
∂x
u
w
)
∂u
a
∂x
∂
∂y
u
w
)
∂u
a
∂y
+
=
0
The three-dimensional geometry must be input to the com-
puter and a numerical technique must be used to solve
three-dimensional problems.
(9.57)
where:
D
ax
(u
a
−
u
w
)
=
air coefficient of transmission as a func-
tion of matric suction and
9.6.4 One-Dimensional Transient Air Flow
The derivation for the differential equation for transient air
flow is derived using Darcy's flow law for air. The flow
of air in and out of a referential element must take density
changes into consideration. Both the density of air and the
velocity take the form of a field representation across the soil
element. The velocity of air flow multiplied by the density
of air is equal to the mass of air flow:
ρ
a
+
∂u
a
/∂x
=
pore-air pressure gradient in the
x
-di-
rection.
Let us write
D
ax
(u
a
−
u
w
)
and
D
ay
(u
a
−
u
w
)
simply as
D
ax
and
D
ay
, respectively. The coefficient of transmission
D
ax
is related to the air coefficient of permeability
k
ax
by
the gravitational acceleration (i.e.,
D
ax
=
k
ax
/g
). Expanding
Eq. 9.57 results in the following flow equation:
ν
ay
+
dy
dx dz
∂ν
ay
∂y
∂ρ
a
∂y
∂M
a
∂t
(9.61)
−
ρ
a
ν
ay
dx dz
=
∂D
ay
∂y
∂
2
u
a
∂x
2
∂
2
u
a
∂y
2
∂D
ax
∂x
∂u
a
∂x
∂u
a
∂y
D
ax
D
ay
+
+
+
=
0
where:
(9.58)
ρ
a
=
density of air,
where:
v
ay
=
velocity of air flow in the
y
-direction, and
M
a
=
mass of air in the element.
∂D
ax
/∂x
=
change in air coefficient of transmission in
the
x
-direction.
The net air flux in the
y-
direction is obtained by multiply-
ing out the terms in the above equation:
ρ
a
dx dy dz
Spatial variations in the coefficients of transmission are
accounted for by the third and fourth terms in Eq. 9.58.
These two terms produce the nonlinearity in the flow equation.
Equation 9.58 describes the pore-air pressure distribution in
the
x
-
y
plane of the soil mass during two-dimensional steady-
state air flow.
For the
heterogeneous
,
isotropic
case, the coefficients of
transmission in the
x
- and
y
-directions are equal (i.e.,
D
ax
=
∂ν
ay
∂y
∂ρ
a
∂y
∂M
a
∂t
+
ν
ay
=
(9.62)
Equation 9.62 can be written as follows since a unit vol-
ume is being considered:
∂ν
a
∂y
∂ρ
a
∂y
∂M
a
∂t
ρ
a
+
ν
ay
=
(9.63)
Search WWH ::
Custom Search