Environmental Engineering Reference
In-Depth Information
The term coupled, associated with the calculation of AE,
means that the moisture flow partial differential equation is
solved simultaneously with the heat flow partial differential
equation. If an assumption is made regarding the relationship
between the soil temperature and the air temperature, then
the analysis for AE reduces to the solution of the moisture
partial differential equation and the solution is referred to as
an uncoupled analysis. The analysis is also referred to as an
uncoupled analysis if a closed-form, empirical relationship
is used to designate the relationship between the soil and air
temperature.
Three methods are also considered for relating the AE
to the PE: (i) the Wilson-Penman method, (Wilson, 1990),
for calculating AE (ii) the limiting function method (Wilson
et al., 1997a) and (iii) the experimental-based method (Wil-
son et al., 1997a). The three relationships between AE and
PE along with the possibility of using a coupled or uncou-
pled solution give rise to six procedures for the assessment
of AE. These calculation procedures are outlined in the fol-
lowing sections.
• Actual evaporation is calculated using the Wilson-
Penman equation, (Wilson et al., 1994).
Partial Differential Equation Governing Moisture
Flow.
Liquid water and vapor flow through an unsaturated
soil is presented in detail in Chapter 7. Equation 6.50 is a
simplified form for liquid and vapor flow and can be used
to solve for AE (Jame, 1977; Wilson, 1990; Gitirana, 2005;
M.D. Fredlund et al., 2011):
k
y
11
∂
u
w
∂y
∂
∂y
∂T
∂y
∂θ
u
∂t
ρ
i
ρ
w
∂θ
i
∂t
(6.50)
+
k
wy
+
k
y
12
+
S
sk
=
+
where
k
wy
+
k
vh
k
y
11
=
(6.51)
γ
w
k
y
12
=
k
vT
(6.52)
and
6.3.19.1 Wilson-Penman Equation for Computing AE
(Coupled Solution)
Wilson et al., (1994) proposed that the Penman (1948)
equation for PE be modified and then used to calculate
AE. The modified equation, which became known as the
Wilson-Penman equation, (Wilson et al., 1994) took into
consideration the difference in temperature and relative
humidity (and therefore vapor pressure) between the soil sur-
face and the overlying air. The difference in vapor pressure
conditions between the air and the water at ground surface
formed the basis for the development of the
soil-atmospheric
model
implemented into the SoilCover (MEND, 1993) and
SVFlux computer code (M.D. Fredlund, 2001).
The following assumptions are made when combining
moisture flow and thermal flow:
• Moisture flow and vapor flow beneath the soil surface
are governed by hydraulic head gradients, vapor pres-
sure gradients, and temperature gradients.
• Heat transfer beneath the soil surface (i.e., ground ther-
mal flux in the equation) is mainly governed by thermal
conduction. Heat transfer by thermal convection can be
included, but it can generally be neglected for most
applications.
• Soil freezing/thawing processes can be considered when
the soil temperature goes below the freezing point. For
frozen soils there must be a reduction in the hydraulic
conductivity because of the existence of ice in the pores
of the soil.
• The latent heat due to phase change, including evapora-
tion and freezing/thawing, can be considered in the heat
transfer equation beneath the soil surface.
• The soil surface temperature can be different from the
air temperature and is determined in accordance with
the heat energy balance equation (Eq. 6.8).
k
wy
=
hydraulic conductivity, m/s,
k
vh
=
water vapor conductivity by vapor diffusion within
the air phase, m/s,
k
vT
=
water vapor conductivity due to temperature gradi-
ent, m/s,
u
w
=
pore-water pressure, a component of hydraulic
head, kPa,
soil temperature,
◦
C,
T
=
t
=
time, s,
y
=
vertical direction or elevation, m,
unit weight of water, kN/m
3
,
γ
w
=
volumetric unfrozen water content in soil, m
3
/m
3
,
θ
u
=
volumetric ice content in soil, m
3
/m
3
,
θ
i
=
density of water, kg/m
3
,
ρ
w
=
density of ice, kg/m
3
, and
ρ
i
=
S
sk
=
water sink or source (i.e., water added or water
taken away),
(
m
3
/m
3
)/t
.
Partial Differential Equation Governing Heat Flow.
The heat flow partial differential equation is derived in
detail in Chapter 10. The following simplified heat flow
equations can be used to solve for AE. The heat flow
equation includes the soil freezing/thawing process and is
based on the derivations by Jame (1977), Wilson (1990),
Gitirana (2005), and M.D. Fredlund et al. (2011):
k
y
21
∂
u
w
∂y
=
C
L
f
m
i
2
∂
∂y
∂T
∂y
∂T
∂t
+
k
y
22
+
(6.53)
where
L
f
k
y
+
Lk
vh
k
y
21
=
(6.54)
γ
w
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