Jökull - 01.12.1968, Page 31

Evaporation and convection
The transfer of water vapour and sensible
heat frorn a water surface to the atmosphere
is analogous to the heat transfer in a turbulent
river, provided the wind velocity is substantial.
fn a thin layer near the surface the transfer
is effected by molecular diffusion but higher
up in the wind turbulent diffusion is prevail-
ing. The diffusion equations are analogous to
(3) or (4) but instead of cT there we have now:
for convection: cp 0, where cp is the speci-
fic heat of air at constant pressure and 0
the potential temperature;
for evaporation: q, the specific humidity.
The use of 0 and q, rather than other expres-
sions for the temperature and humidity, is
preferred because they are conservative with
respect to dry-adiabatic motions (Haltiner and
Martin 1957).
Taking the x-axis in the wind direction and
neglecting lateral diffusion the diffusion equa-
tion for evaporation for the steady state be-
comes
3(vq) J_a_ /K J3(vq) \
3x 3z \ E 3z /
(21)
where y is the specific weight of air.
The eddy-diffusion coefficients for evapora-
tion, Ke, and convection, Kn, are functions of
the height above the ground, wind velocity, the
stability of the air etc. The coefficients are of
the same order of magnitude as the coefficient
for transfer of momentum, the eddy viscosity,
Kj,. From the phenomenological turbulence
theories an expression for KM can be obtained.
(corresponding to (7)) but KE and KH can not
be derived in a corresponding way. The simp-
lest hypothesis is to assume
Ke = KH = Km.
With this and certain other assumptions Snt-
ton (1953) has solved the equation (21) for an
area which is infinite across wind and finite
down wind and the wind direction is per-
pendicular to the area. He uses a power law
for the wind profile and an expression for KM
which is derived on basis of statistical theory
and mixing length theory. Sutton’s solution can
be written
E = constant ■ (qw — qa) vz° ,'s x0°-88 (22)
where E is the total rate of evaporation per
unit of cross wind length, x0 is the down wind
dimension of the area, vz is the wind velocity
at height z, qw is the specific humidity at the
water surface and qa is the specific humidity
of the air. It should by noted that E is not
directly proportional to the area of the water
surface. The relation (22) is in good agreement
with laboratory experiments and observations
in the field (Sutton 1953). The relation (22)
is valid for a smooth surface but Sutton states
that a virtually identical expression is obtained
for a rough surface. It is to be expected that
a relation similar to (22) is valid for evapora-
tion from rivers. In practical cases the wind
direction is however more or less variable and
besides it would be complicated to include the
river width in computations.
The assumption of the equality of the dif-
fusion coefficients needs a closer consideration.
According to Sutton (1953) KH > KB = KM in
unstable conditions and KH = KB > KM in
stable conditions. When we are concerned with
heat loss from open water during frost it is
thus possible that the ratio KE/KH is a func-
tion of the difference Tw — Ta, where Tw and
Ta are the temperatures of the water and the
air. Undoubtedly the wind structure is also
significant. Smith (1951) has shown that con-
tinuous records of the wind direction can be
used as a basis for classification of the turbul-
ence of the air. It seems worth-while to in-
vestigate if there is a correlation between KE/
ICH and Smith’s types of turbulence as con-
tinuous records of wind direction are readily
obtainable.
For comparison of the rate of lieat loss by
evaporation and convection and to arrive at
some approximate formulas we shall study the
simplest case: the steady state over infinite
water surface, assuming that KB = KH = KM.
In this case the rate of heat loss per unit area
by evaporation is
= <*»>
JÖKULL 18. ÁR 365