Environmental Engineering Reference
In-Depth Information
10.4.4 Heat Flow Equations Including Vapor Flow
There can also be vapor flow associated with a thermal gra-
dient; however, vapor flow is closely associated with flow in
the pore-air and pore-water phases. The components related
to vapor flow are mentioned below, but further details related
to the theory and solution to practical engineering problems
are mentioned under the topic of coupling liquid and vapor
flow through unsaturated soils.
Fredlund and Gitirana (2005) presented the vapor flow in
terms of two components: one due to vapor diffusion within
the air phase and a second due to advection within the free
pore-air. The velocity of vapor flow can be written as follows:
the above equation: the thermal conductivity function, the
vapor conductivity function (diffusion), and the vapor advec-
tion function. These soil property functions can vary with
the stress state (e.g., soil suction), and therefore, the partial
differential equations are nonlinear.
10.4.5 Temperature and Heat Flux Boundary
Conditions
There are two common thermal boundary conditions asso-
ciated with the solution of heat flow problems: Dirichlet-
type boundary conditions (i.e., the primary variable specified
is temperature
T
) and Neumann-type boundary conditions
(i.e., the derivative of the primary variable or the heat flux
q
h
). The Neumann heat flux can be specified as zero when
the boundary is perfectly insulated. The boundary condi-
tion can also be considered as a zero heat flux bound-
ary under conditions of symmetry for the material being
modeled (e.g., a two-dimensional section out of a laterally
extending continuum).
Most geotechnical engineering problems involve exposure
to the ground surface where weather conditions are rapidly
and continuously changing. The nighttime and morning peri-
ods are generally cool relative to the midday temperatures.
Also, the temperatures from one day to the next can change
significantly, as can the conditions from one month to the
next. In other words, the temperature boundary condition at
ground surface is in an unsteady or transient state.
Some geotechnical engineering problems can be mod-
eled using average daily temperature values. However, other
engineering problems may require a more rigorous represen-
tation of the thermal boundary conditions (e.g., a mathemati-
cal function approximating changes in temperature through-
out the day). Weather stations commonly collect data and
report the information on a daily basis (Fig. 10.13). Quite
often the average daily temperature is recorded along with
values for the maximum and minimum temperatures for the
day. Temperature data can also be collected on an hourly
basis but the amount of data becomes excessive.
Temperature as a Dirichlet-type boundary condition can
readily be applied at the ground surface. The temperature
may be assumed to vary according to a particular pattern
during the day. For example, temperature may take the form
of a sine function with the peak temperature near noontime.
It is also possible to use measured hourly temperatures as
input for numerical modeling purposes.
Temperature at a weather station is generally recorded at
1 or 2 m above ground surface. It is the ground surface tem-
perature, however, that forms the more accurate boundary
condition for modeling geotechnical engineering problems.
The soil temperature at the ground surface may not be the
same as the air temperature above the ground surface. The
difference in temperature between the air and the soil can
be taken into consideration when performing coupled heat
and moisture flow analyses for the calculation of actual
evaporation at ground surface. Wilson (1990) suggested a
k
vd
γ
w
k
vd
γ
w
∂u
w
∂y
+
u
w
∂T
∂y
v
y
=
v
v
y
v
v
y
+
=−
T
+
273
.
15
k
va
γ
a
∂u
a
∂y
−
(10.29)
where:
v
y
=
total velocity of vapor flow, m/s,
v
v
y
=
velocity of vapor flow by diffusion through the air
phase, m/s,
v
v
y
=
velocity of vapor flow by advection in the air phase,
m/s,
k
vd
=
pore-water vapor conductivity by vapor diffusion
within
the
air
phase;
which
can
be
written
as
ω
v
u
air
D
v
∗
ρ
w
v
γ
w
,m/s
ρ
w
R(T
+
273
.
15
)
k
va
=
pore-water vapor conductivity by advection within
ρ
v
ρ
a
the free pore-air, which can be written as
γ
a
D
a
∗
ρ
w
,m
/
s
unit weight of water, kN/m
3
,
γ
w
=
density of the vapor, kg/m
3
,
ρ
v
=
density of water, kg/m
3
,
ρ
w
=
density of air, kg/m
3
,
ρ
a
=
u
air
v
=
vapor pressure, kPa,
u
a
=
pore-air pressure, kPa,
u
w
=
pore-water pressure, kPa,
temperature,
◦
C,
T
=
ω
v
=
molecular weight of water vapor, 18.016 kg/kmol,
R
=
universal gas constant, 8.314 J/(mol.K),
D
v
∗
=
molecular diffusivity of vapor through soil, m
2
/s, and
D
a
∗
=
(
1-
S)nD
v
ω
v
/
RT
,kg
·
m/kN
·
s.
The above equation shows that the flow and storage of heat
within a saturated-unsaturated soil can be a function of three
primary variables:
u
w
,
u
a
, and
T
. Volume changes associated
with the relative phases of the soil are not explicitly pre-
sented as primary variables; however, volume-mass changes
affect thermal conductivity and volumetric heat capacity. Sev-
eral unsaturated soil property functions can be identified in
Search WWH ::
Custom Search