Geoscience Reference
In-Depth Information
According to Kevin-Boltzmann statistics, the number of molecules that acquire
enough energy to break the intermolecular bonds in the water and enter the over-
lying air is related to temperature, i.e.:
n
⎛
l
⎞
“Boil off ” rate
≈
k
exp
−
(2.15)
⎜
⎟
1
kT
⎝
⎠
where
k
and
k
1
are constants and
n
is the number of molecules per unit volume.
On the other hand, the rate of capture of molecules is directly related to concentra-
tion of vapor molecules in the overlying air, i.e.:
(
)
'
(2.16)
“Capture” rate
≈−
k
1
r
e
2
where
k
2
is a constant and
r
is the fraction of water vapor molecules colliding with
the surface that are reflected without capture. If the (temperature dependent) boil
off rate exceeds the (concentration dependent) capture rate, there is net evapora-
tion and liquid water leaves the surface and enters the air as water vapor.
But what would happen if the air above the evaporating water surface was enclosed
so that evaporated molecules were not able to diffuse away higher into the atmos-
phere? Gradually the concentration of molecules in the air would rise until such time
as the rate of capture equaled the rate of boil off. There would then be no net loss of
water molecules, evaporation would cease; the air is then be said to be
saturated
.
Because net evaporation is the difference between two rates, there is a well-defined
concentration of water vapor at which the net exchange is zero. This concentration
depends on the temperature-dependent boil off rate. Were the temperature higher,
for example, the boil off rate would increase and exchange equilibrium would be
established with the saturated air having a higher concentration of water vapor.
The relationship between the saturated vapor pressure,
e
sat
, and temperature has
been defined by experiment and several empirically determined relationships
have been proposed. Here we select the following:
′
⎛
C
⎞
17.27
T
e
=
0.6108exp
kPa
(2.17)
⎜
⎟
sat
C
⎝
237.3
+
T
⎠
when the temperature is given in
C. Note that elsewhere in this text the temperature,
T
, is usually expressed in K. Here,
T
C
is used to represent temperature to emphasize
that in this empirical formula the value of temperature must be expressed in
°
C.
The gradient of the relationship between saturated vapor pressure and temperature
is often used in equations describing evaporation rate and, when used in this way,
this gradient is usually represented by
°
Δ
. Differentiating Equation (2.17) gives:
de
4098
e
Δ=
sat
=
sat
kPa °C
−
1
(2.18)
(
)
dT
C
2
C
237.3
+
T
