Environmental Engineering Reference
In-Depth Information
The density of air can be written using the ideal gas law:
three dimensions. The two-dimensional transient air flow
equation can be written as follows:
u
a
ω
a
RT
K
¯
ρ
a
=
(9.64)
∂
v
ay
∂y
+
∂
v
ax
∂x
+
∂ρ
a
∂x
+
∂ρ
a
∂y
=
∂M
a
∂t
ρ
a
v
ax
ρ
a
v
ay
(9.70)
where:
The three-dimensional transient air flow equation can be
written as follows:
u
a
¯
=
absolute air pressure,
ω
a
=
molecular weight of air, 28.8 g/mol,
universal gas constant, 8
.
314 g/(s
2
cm
2
mol K), and
R
=
∂
v
ay
∂y
+
∂
v
ax
∂x
+
∂ρ
a
∂x
+
∂ρ
a
∂y
+
∂
v
az
∂z
+
∂ρ
a
∂z
ρ
a
v
ax
ρ
a
v
ay
ρ
a
v
az
T
K
=
absolute temperature, K.
∂M
a
∂t
The mass of air for a unit volume of soil can be written in
terms of the volume of air multiplied by the density of air:
=
(9.71)
e
e
S
a
¯
u
a
ω
a
RT
K
The air flow
M
a
is the total mass of air flowing in all
directions.
M
a
=
(9.65)
1
+
where:
9.6.6 Transient Air Flow Considering Other
Components of Air Flow
Fredlund and Gitirana (2005) presented partial differential
equations that satisfy the conservation of air mass when
three components of air flow are considered (i.e., convection,
advection, and diffusion). The flow law equations and an air
volume change constitutive equation must be combined to
satisfy the continuity for air mass.
The partial differential equation for air mass flow is formed
using three main partial derivatives for the flow in each direc-
tion. The presence of the partial derivatives is a result of
the assumption that the mass rate of air flow across a rep-
resentative elemental volume is continuously distributed in
space. Therefore, the spatial distribution of air flow rate can be
described by the partial derivative of flow for each direction.
Considering a constant referential volume
V
0
, the follow-
ing expanded form of the air flow equation is obtained under
conditions that the soil element does not change volume:
e
=
void ratio and
S
a
=
amount of voids filled with air, equal to 1
S
,
where
S
is the degree of saturation with respect to
water.
−
Darcy's law can be written for air flow in the
y-
direction:
k
ay
ρ
ma
g
∂u
a
∂y
v
ey
=
(9.66)
where:
v
ey
=
volume rate of air flow across a unit area at the
exit point under constant-pressure conditions (i.e.,
usually 101.3 kPa absolute pressure) and
ρ
ma
=
constant air density.
For a particular mass of air, the following condition exists:
k
a
g
u
w
γ
w
ρ
a
v
ay
=
ρ
ea
v
ey
(9.67)
k
ad
g
∂
∂x
∂u
a
∂x
+
∂u
a
∂x
+
∂
∂x
ρ
a
hk
w
where:
k
a
g
u
w
γ
w
+
y
k
ad
g
∂
∂y
∂u
a
∂y
∂u
a
∂y
∂
∂y
ρ
e
v
=
density of air at the exit point under constant pressure
conditions.
+
+
+
ρ
a
hk
w
k
a
g
u
w
γ
w
k
ad
g
∂
∂z
∂u
a
∂z
∂u
a
∂z
∂
∂z
+
+
+
ρ
a
hk
w
Darcy's equation can be written as
k
ay
ρ
a
g
∂u
a
∂y
W
a
RT
K
∂u
a
∂t
v
ay
=
(9.68)
=
(
1
−
S
+
hS)n
(9.72)
Substituting Darcy's law into the net air flux equation (i.e.,
Eq. 9.63) gives the following flow equation:
Equation 9.72 shows that the flow and storage of air within
a saturated-unsaturated soil is a function of the pore-air
and pore-water pressures provided the soil element remains
at constant volume. Therefore, the pore-water partial dif-
ferential equation also needs to be solved. The pore-air
conductivity function, the pore-air conductivity by diffusion
within the pore-liquid water, and the coefficient-of-water-
permeability function are related to the SWCC. The soil
property functions vary with soil suction, and therefore, the
∂
v
ay
∂y
∂ρ
a
∂y
∂M
a
∂t
ρ
a
+
v
ay
=
(9.69)
9.6.5 Two- and Three-Dimensional Transient Air Flow
The above differential equation can be expanded to form a
partial differential equation for transient air flow in two and
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