Jökull - 01.12.1968, Page 25

Water Temperature and Heat Baiance of Rivers
SIGMUNDUR FREYSTEI NSSON,
THORODDSEN AND PARTNERS, CONSULTING ENGINEERS, REYKJAVIK, ICELAND
ABSTRACT
The semi-empirical equations for viater tem-
perature i.n rivers and. the rate of heat loss
from a river surface are discussed. The heat
loss equations are used to calculate the produc-
tion of frazil ice in Thjorsa lliver, South-
wesiern Iceland, and a reasonably good agree-
rnent between calculated and measured ice pro-
duction is obtained.
INTRODUCTION
Studies on the heat balance of rivers are
limited in number and mostly connected to
practical problems at certain rivers. The pro-
blems most frequently encountered are to cal-
culate water temperatures or ice production on
the basis of hydrological and meteorological
observations. Although satisfactory agreement
between measured and calculated river temp-
eratures has been obtained in some cases, uni-
versally applicable solutions are not available
as yet.
A comprehensive study of the heat balance
of rivers with reference to ice formation was
published by Devik (1931). A review of some
of the later contributions is given by Dingman,
Weeks and Yen (1968).
In connection with the hydro-electric deve-
lopment of rivers in Iceland the forecasting of
changes in ice regime, water temperature and
frazil ice production is a matter of practical
and economical interest at certain sites. As the
different formulas available for the heat loss
from a water surface give very divergent re-
sults it was considered necessary to investigate
the subject. The equations for the water temp-
erature in rivers or canals liave also been
studied. Tliese investigations are still going on
but nevertheless a description of our approach
to the problems shoulcl be of interest to those
who are concerned with similar studies.
The gravitational or engineering system of
units is used here ancl symbols are defined
when they first appear. (kp = kilogramforce).
WATER TEMPERATURE
Here we will only consider the ice-free case
where the water temperature is everywhere
above 0° C.
An equation for the water temperature in a
turbulent stream is obtained by considering a
simply connected region, R, within the stream.
In our case the water may be assumed incom-
pressible and its properties independent of
temperature.
The heat added to R through conduction is
k grad T • dA
A
where k is the conductivity of the water, T the
instantaneous temperature, A the surface of the
region R and the surface vec.tor dA is directed
outwards from the region.
The heat added through advection is
yc T V • dA
A
where y is the specific weight and c the specific.
heat of water and v the instantaneous velocity
vector.
The lieat added through dissipation of tur-
bulent energy (frictional heating) ancl radia-
tion is omitted here. The energy from fric-
tional heating is small compared to other terms
and radiation effects will be assumed to be
confined to the surface. In fact water is prac-
tically opaque to long-wave radiation but solar
radiation may penetrate to considerable depth.
The rate of change of internal energy in-
side R is:
JÖKULL 18. ÁR 359