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Finally, if we assume the aerodynamic resistances for water vapour and heat to be equal
and combine Eq. (
7.12
) with the energy balance Eq. (
7.5
) (viz.,
HQ GLE
=−*
v
)
we obtain (after some serious rearrangement) the Penman equation for evaporation
from a wet surface:
ρ
c
r
(
)
p
eT e
()
−
( )
+
sat
a
a
sQ
*
G
LE
=
+
a
(7.13)
v
s
γ
s
+
γ
where we have omitted the explicit temperature dependence of
s
and
e
sat
for reasons of
clarity. Furthermore, the aerodynamic resistances for heat and water vapour have been
denoted simply by
r
a
(which is
not
equal to the aerodynamic resistance for momen-
tum, due to differences in roughness length and stability dependence). Although the
focus of the Penman equation is on evaporation, the sensible heat lux can simply be
obtained as the residual from the energy balance:
HQ GLE
=−*
v
.
The irst term on the right-hand side of Eq. (
7.13
) is called the
radiation
term
because it describes the evaporation due to energy input by radiation. The second
term is called the
aerodynamic
term because it depends explicitly on the turbulent
transport and the atmospheric moisture conditions through
e
()−
(vapour pres-
sure deicit [VPD]). Referring to the derivation of the Penman equation, the two terms
can also be interpreted as follows. The irst term is proportional to the evaporation
that is due to the fact that the temperature of the surface deviates from the air temper-
ature as a result of net energy input: a temperature contrast between surface and air
results in a contrast in
e
sat
(
T
) between surface and air. The second term is proportional
to the evaporation that would occur if the surface temperature would be equal to the
air temperature.
The Penman equation shows that even if there is not net input of energy by (
Q*-G
)
evaporation can proceed: in that case the energy required for evaporation will be
extracted from the air through a negative sensible heat lux.
e
T
a
sat
a
Question 7.3:
To circumvent the need for an iterative solution of a system of three
equations, the Penman equation uses the linearization given in Eq. (
7.11
). Determine the
error in the estimation of
e
sat
(
T
s
) due to the linearization for a situation with an air tem-
perature of 20 ºC, an aerodynamic resistance of 30 s m
-1
, and a sensible heat lux of:
a) 0 W m
-2
b) 100 W m
-2
c) 300 W m
-2
Assume an air density of 1.2 kg m
-3
. Hint: from Eq. (
7.8
) one can obtain the
real
surface
temperature for a given sensible heat lux and hence the
real
saturated vapour pressure
at the surface. Compare this with the linearization.
A number of conclusions regarding evaporation from a wet surface can be drawn,
based on this equation:
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