HomeMy WebLinkAboutEvaluation of ice problems Hydro in AK 3 of 3 1982IAHR's 8th Congress in Montreal, 1959 [(2),(3),(4)]. In January 1977,
in order to have a general survey of current ice problems in our Swedish
hydropower plants, the hydraulic laboratory sent forward a questionnaire
partly to the administrators of powerplants and partly via VAST to a
number of large power companies. As a result of this questionnaire a
lot of good information about ice problems was obtained from certain
power stations in sections of some rivers; while information from other
regions was scarce or failed to come. In order to complete this inquiry
paper a detailed questionnaire was distributed in March 1978 to power
companies that administer power stations in Lagar, Kolbacks, Dal River,
Ume River, and Lule River. The purpose of this other questionnaire was
to get -as detailed a description as possible of the ice problems in
Swedish power stations in a number of rivers from Lagar to Lule River
in the north. Importance is placed this time also on getting information
about the design of the power plants and other matters that are
important in understanding the causes of the ice problems.
Definition of Ice Terminology
Terminology within the realm of ice is not entirely unambiguous.
It also appears to vary within the country. An attempt is made below
to briefly describe some phenomena.
Ice build-up (slush, swell)
Frazil ice particles are formed on open stretches of rivers and
lakes as the water supercools. There the frazil ice particles grow in size and
form small flo? and gradually they "fasten" into numerous round
ice floes. In rivers, ice covers build up partially from the shores
(surface ice) and partially from collections of ice particles which are
2
in their different stages of development. Frazil ice particles, as
they evolve, fasten on all kinds of things that come their way. That
way anchor ice builds up and blocks the intake of a power plant by
freezing the gates.
Ice dams
Ice floes that drift with the current along an open stretch of a
river can jam against the surface of the ice edge. They can either be
stopped and build up an ice cover of loosely packed ice floes or be
sucked under the ice barrier. The maximum flow velocity for the ice
floes to build up an ice cover is about 0.6 m/sec. For collection
of slush (and thin ice floes) the maximum speed is lower and
depends on its thickness and strength. For loosely packed ice floes
the maximum flow velocity can be 0.2-0.3 m/sec. If upstream of an ice
edge, the stream velocity'is so great that ice floes get sucked under
the ice cover and get deposited under the ice, the water level rises
and the speed of the river flow is reduced. As a result ice can then
accumulate against the edge of the ice sheet causing the ice cover to
grow upstream.
Ice damming can also start as a result of ice which has run aground.
It grows then on stones and shores in such a fashion that damming occurs.
Inquiry 1 (paper 1)
The questionnaire with the requested informtion is given in Appendix
1. This information is concerned in part with ice barriers. In the
appendices 2.1-2.7 a survey of our existing Swedish powerplants over 10
MW is shown. A number of stations less than 10 MW are also reported.
3
Answers to the questionnaire concerning ice problems for each power plant
and river are reported in the appendices.
The answers that have come in concerning ice problems and ice
barriers are accounted for in a separate report (Ice problems in Swedish
hydropowerplants, part 2). The most essential problems concerning
existing ice problems are reported, besides what appears in appendices
2.1-2.7 in the section under "comments" in the following report.
Inquiry 2 (paper 2)
In March 1978 a detailed questionnaire was distributed to the
owners and administrators of hydropowerplants of Lagan, Kolbacksan,
Dalalven, Indalsalven, Umealven, and Lulealven. After a certain amount
of persuasion, information has been obtained from the aforementioned
power plants which are on selected rivers. The respondants did not
answer all the questions, which, moreover, were often not too skillfully
worded. The unanswered questions are marked with --- in the charts.
Any left out answer that applies to questions about ice problems is,
nevertheless, interpreted as a problem that does not occur. Certain
questions could be misunderstood. Moreover, it appears that people
have been less inclined to answer all the questions about power plants
if there were no ice problems occurring. A summary of the reported ice
problems is compiled in a chart form in the tables 4.1-4.15. Powerplants
from six rivers are compiled in the report. In these tables information
that is judged to•have direct interest for purposes of comparison
between different power stations, is also given. The remaining
information that was asked for in the questionnaire is not included.
2
(It can be obtained by request to the Hydropowerplant Laboratory).
It ought.to be pointed out here that time has not been granted to work
with the left out information.
Below are comments about the meanings and thoughts behind the
headings on the tables.
Power plants
Mean rate of flow and
corresponding velocity at
the upstream intake.
All power plants upstream from the
mouth of the river are considered.
Information on the mean volume of water
transported is obtained from the mean
value for a winter day and winter
night. The available information
comes from hours of different
discharge in the nearby power
stations without the regulating
possibilities. Such information
cannot, therefore, be totally
accurate. Outward bound stream
speed has been calculated from the
water mass transported and the
width and depth of the power plant.
Especially where there are no
canals, it can be assumed that the
speed of the river can be lower
than calculated (max. depth stated).
EVALUATION OF ICE PROBLEMS ASSOCIATED WITH
HYDROELECTRIC POWER GENERATION IN ALASKA:
Final Report to the State of Alaska
Department of Commerce and Economic Development
Contract 08-73-7-958/08-71-6-114
or Contract AEC81005-3
The size of the areas not
covered by ice immediately
upstream of the stations.
The heating of gates
Icing
If one supposes that freezing
occurs if the speed of the stream
falls below 0.6 m/sec., it appears
that there is an inconsistency
between the calculated speed of the
stream and data about ice -free
areas. The inconsistency can depend
upon the incorrectly calculated
speed of the stream or that the
data about the size of ice -free
surfaces is valid during the freeze-
up period.
It is not clear from the questions
in the questionnaire if the warmed
part of the total gate area is
concentrated in one (or several)
intake while gates in front of the
remaining intakes lack heating.
Information about icing occurrence
on turbine.blades (ledskenor) and
gates and whether ice gatewarmers
are switched on automatically or
manually and at what temperature
this occurs. There is information
also about production drop as a
result of icing.
6
Ice dams Presence of ice dams (description
of location and cause goes under
"other") as well as information
about production drop as a result
of reduced height of the fall.
Ice floe Problems with ice floe against the
intake. Information about causes,
point of time, necessary work, and
drop in production.
Other Added information above all about
icing problems and ice dams.
Comments
Besides the six rivers included in the paper 2, a good picture of
the state of affairs in many power stations in Gbta River, Motala River,
Mar River, and Skellefte River is obtained from the paper 2. Below
are the comments on the information received about these ten rivers.
Lagan
According to Sydkraft's description, ice problems arise mainly in
power plants where upstream is a river bed, inlet canal, or a tunnel.
Ice problems.almost never start in plants that are directly connected
with a reservoir. According to appendix 4.1, however, two out of three
power plants which have intakes in a reservoir, indicated in paper 2,
icing problems appear. The reason for this probably is that the
reservoirs in question are relatively shallow.
7
Icing is found yearly in six (seven?) out of the eight power plants
in spite.of the installed gate warmers (exception is Aby power station).
It is also stated that problems occur even when the gate warmers were
turned on before ice formation started. In three (four) of the power
plants with icing problems icing occurs -not only on gates but even on
turbine blades (ledskenorma).
The gate warmers are turned on already when the water temperature
is between 0.3 and 0.5 C. This is unusually early compared with the
practices of other stations. If problems with operating the gates
occur at +0.3 to 0.5 C, it will not probably be because of icing on the
iron of the gates. In such high water temperatures it can be assumed
that the ice problems occur when great quantities of slush block the
gate openings.
In certain canals strong anchor icing occurs at the bottom (1-1.5
m). When this anchor ice gets loose, it can contain stones (of 10-13
kg) which together with the ice can cause problems to the gates and
turbines.
In control dams the ice pressure causes damage on vertical level
gates. It should be investigated if laying out generators could help
this problem.
Sydkraft has taken different steps to reduce the ice problems. In
power stations, where the water upstream is constant, stream flow
(velocity) is reduced during the freezing period to increase the rapidity
of the ice formation upstream. Even laying out ice booms is used for
this purpose.
With variable upstream water levels, the canals must ordinarily have
open water surfaces. The great difficulty is to keep the canals ice
q
free at the start. Ice is driven out according to a certain system of
starting, stopping, and driving out the ice through the ice outlet
many times and using a certain amount of acquired experience of
opening ledskenor (blades?, guides?).
Clean up work in the waterways has also brought about as a result
a retarding effect on icing in some installations.
In four plants they have installed automatic gate cleaners for
general gate cleaning work (such as leaves, litter, ice slush), but the
cleaners have even proved to have a good effect 'against ice formation.
It is thought that the gates are given a certain vibration and that the
cleaners break the ice film on the upstream side of the gate iron and,
therefore, the handling of ice has become considerably easier.
Sydkraft maintains that without taking proper steps, it would be
scarcely possible to have power production at all times of ice
difficulties. This is especially so since nowadays and there is a lack of
personnel to solve difficult ice situations.
Gota River
To a certain degree, ice problems in G8ta River are special because
of ship traffic which has its demands for passability. On the power
plant's side, the people hope to have a fast and as complete a freeze
up as possible to avoid icing problems that otherwise appear moderately
often, especially at Lilla Edets power plant. The shipping office on
their part hopes as a principle to keep the river ice -free for shipping.
During recent years the shipping office and the power plant office have
found certain solutions that both sides can accept. The objective is
to reduce ice production in the river and this can be done by letting a
great part of the river form an ice cover. Freeze-up can be facilitated
9
by placing ice booms in strategic places. Further the power plant's
side aims at keeping a uniform and low discharge, when conditions for
freeze-up occur. This, however, is difficult since power stations in
Gbta River usually must do short term regulating.
In the river even ice dams are sometimes built up, especially on
the stretch of Lilla Edethavet. Ice dams appear, partially because of
strong drift ice formation, partially as a result of anchor ice, and
during certain years they have given cause to troubling floods.
Motala Stream
Icing has occurred only a few times in Motala and Malfors stations.
Icing has never occurred in Nykvarn.
In"Malfon and Nykvarn problems with ice floes appear occasionally.
Ice floes build up in the intake canals at nighttime if the stations
are not operating. Ice floes then break loose with the starting of
operation in the morning and travel down to the intake gates. Loss of
head appears as a result. The most difficult freeze-up appears during
Saturday and Sunday when the stations do not operate from Saturday
afternoon until Monday morning.
Klar River
Of Uddelholm's nine power stations in Klar River, there are
troublesome ice problems in Munkfors and Forshaga. In Edsforsen, Skoga
and Deje there are certain troubles with icing before freeze-up. Icing or
ice problems seldom or never occur in the remaining power stations.
In Nunkfors and Deje stations, there are trash racks that work
with icing and it seems that they make it possible to keep the operation
going on. At Deje, during part of the year, timber is laid down to
protect against icing.
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Downstream of H51jes power station there are about two ten kilometer
stretches of undeveloped (rapids). In spite of that, -the water
temperature at the station can be about +1°C in February; ice slush
and drift ice are built up along the rapids stretch. This ice gets
stored up where the water flows slowly and gives rise to damming.
Since short term regulating has been started.at Hbljes, the risk of the
problems has increased and things have been closely watched. According
to the information from the management, the ice forming is worst in
very cold weather and when discharge is low.
Kolbacksgn
Kolbacksgn is included in (Inquiry 2) Paper 2 in order to present
detailed information about a small waterway with small power stations.
Icing problems are found in five out of the eight stations that are
characterized as river power plants or that have intake canals to bring
in the water. (Gatewarmers are lacking in all power stations in Kolbackson.)
Reports on icing appear from this waterfall even before the freeze-up,
but in Trangfors icing can occur during the whole winter day and night.
Trangfors power station has a 400 m long canal that brings in water.
It is open the whole winter.
On four out of the five power stations which have icing problems
there is ice growth on both gates and ledskeneapparaten (blade apparatus?).
Smaller problems with ice floe against intake gates are reported from
two stations. The problems occur with greater discharge changes respectively
in milder weather.
Dal River
In all thirteen power stations between the confluence of the Vaster
ii
and Osterdal Rivers (Lindbyn included) and the ocean, icing has occurred
11
in the intake gates. Ice growth at both gates and ledskenor (blades?)
appears in five stations in a row from Avesta-Lillfors to Lanforsen, at
the lower part of the river.
Occasionally in Domnarvet power station icing has occurred on the
turbine blades in one of the two Kaplan turbines, this caused an
imbalance in the unit and there was some damage as a result. Ice
formation in turbine blades (in Kaplan turbines) has also been discovered
at the Lanforsen power station.
In Vaster River, icing occurs in Eldeforsen each autumn, and only
seldom in Hummelforsen. Icing in Hummelforsen happens even though
there are gatewarmers. In bsterdal River, icing occurs more seldom the
further upstream the power stations in the river are.
Ice dams occur regularly downstream of Trangslet and about 6 km
upstream of Vasa power station. Flooding caused by these ice dams
causes damage to the surrounding settlements. It is to be assumed that
in Tra�ngslet the problems are reduced after the installation of air
bubblers in the reservoir by which warm bottom water is lifted towards
the intake. During the winters 1977 and 1978, ice damming occurred in
one of the forks of the river, upstream of Untran power station. Ice
damming during these two winters was attributed to the increased water
flow of the fork of the river. It happened as a result of diversion of
water in connection of building a power station in Soderfors. The ice
dams caused the water level to rise above the dam, damaging it. In
doing so approximately 75 m3/sec water flowed on the site of Untran
power station and
lfo . intdrained river basin.
Upstream of �lvkarleby power station, bottom anchor ice can occur in
the shallow part of the river which results in the rise of the downstream
12
water level in Lanforsens power station. Problems with the ice floes
against the intake gates occur in several power stations mainly downstream
of where Vaster and Osterdal Rivers flow together.
In order to prevent ice pressure against dams and gates, current
generators are used with good results in many stations. In Avesta-Storfors
they seem to get good results by using gate cleaning machines, among other
things, to remove ice that cannot pass through the ice outlets.
The power companies agree that the most effective measures to
prevent icing are to keep low and even discharge of water during the
freeze-up period so that an ice cover is formed.
Indals River
Ice build up on intake gates occurs in Svarthalsforsen power
station (3-6 days a year). In the Stuguns and M6rsils power stations,
ice build up has occurred once since the plant started operating since
1976, and in Hammarsforsen power station once during the last 30 years.
In the rest of the power stations there are no problems with ice build up.
Downstream of Bergeforsen the river has ice every year on the shores and
bottom, decreasing the area of the river. This causes a 0.7-0.8 m drop
in the pressure head at the station.
During February -March 1972, there were serious problems with ice
slush (not ice build up) at Svarthdlsforsen. Portal cranes with ice
scoops were working in three shifts a day and night to remove the slush.
Svarthglsforsen company points out that at this time a lot of water was
lost as a result of repairs at the Krangede and Gammeldnge stations and
they speculate that ice problems in Svarthdlsforsen appeared because of
this spill. No problems, however, appeared at the Hammarforsen power
13
station that lies between Krangede and Gamnelange and Svathalforsen.
An explanation could be that the water temperature at the outlet from
Hammarforsen was lower than normal as a result of cooling in the tailrace
in Krangede and Gammelange. Ice slush build up was lighter than normal
on the relatively silty river stretch between Hammarforsen and Svathalsforsen.
The dam in Stadsforsen power station is not considered to withstand
from any ice pressure, and therefore, a canal towards the dam is kept
open. It is reported that chain saws, ice brakers and scoops of portal
cranes and also snowmobiles are used for this work. If no special
circumstances prevail in Stadsforsen, an.opening along the dam can be
maintained with current generators or floodlights which is done in
other locations.
Breaking up the ice cover together with changes in stage and ice
floe movement towards the intake occur in Midskogs, Mtlrsil, and
Svarthalsforsens power stations. Ice floe movements towards the gates
occur in some other stations in connection with freeze-up or break
up. However, these ice floes do not usually constitute any more serious
problems. As an exception, however, considerable difficulties can arise. In
M6rsils power station, thaw and strong wind in January 1973 caused a
10-15 cm thick ice on Liten Lake to break up and obstruct all of the 3
km long intake canal. The ice masses were 5-6 m deep in the dam and
caused the power production to drop for almost 24 hours.
Ume River
Icing on intake gates occurs at a few years intervals at Storrnorrfors
and Bdlforsen power stations. In Hdllforsen and Betsele, icing has
even appeared on ledskenorna (blades?). In B81forsen, Betsele and
Hallforsen power stations, icing caused a total stop of power production
in 1971, 1973, and 1975.
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At Tuggen power station ice dams appear yearly 5-10 km downstream of
the station. The downstream water level at the power station is about 2 m more
than the surface in ice -free conditions. During the day, because of water
discharge, the speed of the stream in the outflow canal is so high that
the canal does not freeze. In cold weather, great masses of ice slush
build up along the canal and along the ice -free river stretch downstream,
and come to the damming area.
There the speed of the river becomes lower and ice slush gets
accumulated under the ice. Karteringar has shown that large sections
of the river are obstructed by slush in this manner. As a result of the
increased resistance, the water level rises upstream of such an ice dam and
the stream velocity decreases and conditions for freeze-up occur. The ice
cover that gradually covers a greater part of the canal is made up of slush
and ice floes, hence, gets a moderately uneven surface. Ice floe
accumulation along the edges indicates that the stream velocity has
been very close to the limit of where the ice floes get sucked under
the ice cover.
Apart from extensive excavation work in Tuggen outlet canal, the
only possibility to reduce the risk of ice damming is to restrict
short term regulating during the time when good conditions for freeze-
up prevail.
Skellefte River
Information has been only received from Skellfte power plant.
Descriptions of ice conditions in Skellfte power station are given by
an administrator as follows.
1. Icing in Finnfors, Granfors, Krangfors, and Selfors power stations
1973-11-10 and 1973-11-11.
15
Because of low water temperature, strong winds, and low air
temperature, strong cooling of the river occured without ice formation.
We measured water temperatures to an accuracy of one -thousandth of a degree C.
Because of freezing there, and the lack of ice cover, the pressure head
dropped rapidly at the gates. So, gradually, production had to be reduced
very strongly at the power stations. Some turbines had to be stopped
because the intake gates froze totally together. In Selfors power
station, ledskenorna (blades?) in the turbines froze together so that
it became impossible to use the gear shifts. After about 40 hours,
the strong winds decreased and freeze-up on the dams of power stations
became possible (an ice cover formed). The water temperature started
to climb' up a few tenths of a degree and so the freeze-up began.
After about 44 hours, the power production was resumed to full extent.
Production drop during this period was about 630 MWH. It should
be pointed out that the above mentioned problem is very uncommon. No
information like that has ever come up before. Therefore, our actions
for above mentioned type of problem are limited to trying to keep
constant power. The most useful approach is to keep constant water
flow during the freeze-up period.
2. Ice dams downstream of Granfors power station during the winters
of 1975/76 and 1976/77.
During these winters we have had ice dams above all in downstream
Granfors power station. Because of that we got flooding in G1 turbine
pits at Granfors on January 7, 1976. The loss of pressure head on this
occasion was about 2.7 m (the height of the water head). The immediate
16
procedure was to reduce the water volume going through the station, and
to install a warning signal for high downstream water level. (The
station is far from our management center in Skelleftea.) Attempts to
break away ice dams were performed with certain success, but the problem
still remained partially during the winter 75/76 for about a month.
The drop in water height (pressure head) varied between 1.7-2.3 meters,
and the discharge was about 180-190 m3/s. We believe the best manner
to avoid the aforementioned problems, even here, is to try to keep the
discharge constant (and possibly low) during the freeze-up period, that
is to say, to keep short term regulating as low as possible.
3. Outflow stretch downstream of Kvistforsen power station.
About 1 km downstream from the central part of Skelleftea city,
ice dams often appear. In the winters of 74/75�and 75/76 the dams were
particularly big. Damming is caused by ice slush and piled up ice
blocks. The cross-section of the river is considerably smaller where
the dams start than in the upstream part of the river stretch. In
January 1975 in the city center the maximum damming was measured at 0.9
m, and in January 1976 about 1.2 m.
That the ice dams became so big in these years can, among other
things, be caused by the unusually high discharge by the power plants
during these winters.
During the winter 1975/76, damming was concentrated to quite a
short stretch, and that is why it was judged to be possible to break
away the ice. On January 8, 1976, an explosive bursting of ice was
carried out, and it resulted in an immediate decrease of the damming by
0.8 m and gradually decreased further.
17
Casualties of damming occur foremost through embankment overflows
and water seeping into the community drainage system. These require
increased pumping cost and the risks of overflowing wells. Besides,
crowding together (narrowed channels?) has caused the water velocity to
become so great increased bottom erosion- can occur.
Lule River
Apart from certain problems with ice floes against intake gates at
Letsi and Akkats power station, and a moderate risk of icing at Boden
and Laxede, serious ice problems have appeared only at Vittjdw power
station. During the first winter after the present administration took
over at Vittjaw power station they were forced to spill all of the
water through spillway which is equipped with a so called "ski jump" in
order to reduce the velocity of water downstream. This spill together
with a couple of stretches of stream open downstream of the station
produced a great amount of slush. This slush got accumulated in the
rapids and caused damming that reduced the water pressure head (fall
height) down to 2 m.
After major dredging work downstream, it appears that the
risk of ice damming is considerably smaller than earlier.
Icing at the intake gates has been a yearly recurring problem
ever since the present administration took over. As a test, gate warmers
were installed on the intake in one or two sets. No reduction of icing
or ice build up problems on the warmed intake could be established,
however. Therefore, the gatewarmers were disconnected. Neither has the
extensive cleaning work performed upstream in order to hasten freeze-
up, have substantially reduced the icing problem.
M.
INVENTORY OF ICE PROBLEMS
The nature of problems
- Ice formation on gates, ledskenor,(turbine blades) and
so on
- Ice dams, anchor ice (reduced water head, flooding)
- Ice pressure against dams and locks
- Other
The causes of problems and the measures which have been taken
- Descriptions of how often and, if possible, why the special
problems occurred
- Which measures have been taken (or should/can be taken),
examples
- Reduced water flow (the volume of water) during the freeze-
up period
- Laying out of ice boors
- Changing of water ways
- Heaters on gates and other constructions
- Laying out current generators, releasing the warmed
water or similar measures
- Other
- What effect have the measures brought about (the degree
of difficulty of the problems and frequency before and after
the steps had been taken)
Inquiry about ice booms. The purpose of laying out ice booms.
- Accumulate floating ice and thereby initiate the forming
of firm ice cover
19
Prevent drift -ice from reaching the intake constructions,
where the risk of the ice being sucked under or blocking
occurs
Direct ice to ice outlet
Other
Placing of booms (give reason for the choice of the placing of booms
please, enclose a sketch or direction)
- The surface velocity in the selected section is so low that
drift ice is gathering toward the boom
- Other
Necessary maintenance
- Recording of booms in summertime
- Exchange of the non-functioning parts
Length of life
- How long have the booms been in use
- If a boom is exchanged; give its life span and the reason to
the exchange
Expenses
- The cost of manufacturing
- The cost of placing (laying out) them, including anchoring
20
River
Stations
Owner/Administrator
u i
Indicated by X
Type of Ice Problem
Lagan
4st 10-20 MW
Sydkraft
X
Ice problems at power stations
lst 20-50 MW
10at
least sometimes each year
6st 1-10 MW
(ice dams and ice pressure
towards gates)
Mdrrumsan
4st 1-10 MW
X
The same problems as in the Lagan
Helgean
8st 1-10 MW
X
Eman
7st 1-10 M
X
Nissan
1st 10 MW
Nissastr6ms Kraft Co.
5st 1-10 MW
Atran
1st 19 MW
Papyrus Co.
1st 12 MW
"
Est 1-10 MW
Viskan
6st 1-10 MW
SAvean
4st 1-10 MW
GOta Alv
Lilla Edet 26 MW
SV
X
Serious ice problems occur every
3 or 4 years. Ice forms on the
blades so that they cannot be
maneuved and simultaneously in-
take gates freeze solid.
TrollhAttan 235 MW
SV
Varg6n 26 MW
SV
(Trollh3tte kanalverk)
SV
X
Ice dams, frazil, ice growth on
flood gates
Svartan
(Osterg8tland)
st 1-10 MW
Motala StrNyi
Motala 14 MW
SV
X
Lighter icing, once (57/58)
Malfors 21 MW
SV
X
Icing twice (in the 40's and 57/58)
ice floe towards the intake
Bergsbron 17 MW
Holmens Bruk
Nykvarn
SV
X
Ice floe towards the intake
2st 1-10 MW
River
Stations
Owner/Administrato.r
„n-de. t - t.,e
Inquiry is
Indicated by X
Type of Ice Problem
Klar3lven
H61jes 132 MW
Uddeholms Co.
X
No ice problems at the station.
Ice dams downstream.
Tasan 40 MW
Tasanskraft Co.
Skymn8s 16 MW
Uddeholms Co.
X
Never ice problems. Receives
warm water from Uvan.
Krakerum 16 MW
X
Never ice problems. Receives
warm water from Uvan.
Forshult 20 MW
X
Seldom ice problems. Receives
warm water from Uvan.
Skoga 14 MW
X
Minor risk of icing before
freezing.
Munkfors 23 MW
X
Danger every year.
Dejefors 16 MW
X
Some danger of ice problems each
year before freeze-up.
Edsforsen
X
Minor risk of icing occurs before
freeze-up.
Forshaga
X
Troublesome icing each year
(7-8 hours stop).
Svartalven
Karasen 11 MW
Bofors Co.
Atorp 10 MW
Gullspangskraft Co.
+ about 15 1-10 MW
”
Arbogaan
ca 8:1-10 MW
KolbAcksan
ca 12:1-10 MW
Vasterdalalven
Lima 13 MW
Stora Kopparberg Co.
Hummelfors 10 MW
Korsnas-Marma
Mockfj3rd 32 MW
GrAnges Kraft
X
Earlier icing on gates each
4-5 years. In 1976 damming was
increased upstream, with which
the problem will (reduce)
decrease hopefully.
River
Osterdalalven
Dalglven
(Junction of
streams
downstream)
aavlean
Jusnan
Stations Owner/Administrator
Tr&ngslet 335 MW Stropa Kopparberg Co.
Asen 26 MW 61 if
VAsa 15 MW 11 It
Blyberg 15 MW 11 It 11
Spjutmo 35 MW ° It to
Grada 24 MW It of
FSrshuvud 18 MW It 01
Lindbyn 11 MW It to It
Kvarnsveden 50 MW
Bullerforsen 18 MW
Domnarvet 16 MW
Langhag 46 MW
Skedvi 38 MW
Mansbo 11 MW
Avesta-Storfors 18 MW
Ngs <10 MW
Untra 40 MW
Lanforsen 38 MW
Alvkarleby 70 MW
About 5st 1-10 MW
Langa 160 MW
Sveg 33 MW
Byarforsen 17 MW
Krokstr5mmen 100 MW
Langstr5mmen 46 MW
Strorasstr5mmen 25 MW
Ojeforsen 26 MW
Stora Kopparberg Co.
11 11 11
11
It
of 11
Alby Klorat Co.
Avesta Jernwerks Co.
Sv
Svarthalsforsen Co.
01 it
SV
Bergvik and Ala Co.
Gullspangs Kraft Co.
11 II 11
ae is 11
11 11 11
Kema Nord Co.
Answer to the
Inquiry is
Indicated by x
x
x
x
Type of Ice Problem
Ice damming downstream at about
-30°C.
Icing on gates.
Icing on gates.
x Icing on
gates.
x Icing on
gates (see inquiry)
-78.
x Icing on
gates.
x Icing on
gates.
x Icing on
gates.
x Icing on
gates. Small problems.
x Icing on
gates. Drift ice
towards the gates.
Icing on
gates. Ice dams 76/77
& 77/78
Icing on
gates ledskenor and
turbine.
Icing on
gates and bottoms.
x Ice dams on undeveloped stretch
stream.
liver
Stations
Owner/Administrator
Answer to the
Inquiry is
Indicated by X
Type of Ice Problem
.jusnan
Lottefors 13 MW
Korsngs-Marma
D8nje 76 MW
D8nje Kraft Co.
Bergvik 18 MW
Bergvik and Ala Co.
X
No problems.
H81jebro 27 MW
It" It
X
Serious icing problems about
every 5 years in spite of gate
heating.
Ljusne Str8mmar 34 MW
It " " "
X
"
Landafors 13 MW
X
New power station (1976). No
problems during the first winter.
Ljusnefors
X
Started operating in 1976. No
experiences.
Alfta 19 MW
Voxnanskraft Co.
+ about 4 st 1-10 MW
jungan
Flasj8 20 MW
Norrlandskraft Co.
X
Minor problems.
Trangfors 72 MW
to It
" It
R8tan 58 MW
" "
X
" "
Turinge 17 MW
" it
X
ItIs
J8rnvBgsforsen 105 MW
Skad. Elverk Co.
Parteboda 35 MW
I, to "
Hermansboda 10 MW
Angefallens Kraft Co.
Ljunga 56 MW
Kema Nord
Torpshammar 120 MW
SV
Skallb8de 23 MW
Balforsens Kraft Co.
ndals3lven
Jgrpstr8mmen 118 MW
Svarthalsforsen Co.
X
Small problems. Ice floe
settles towards the gates.
M8rsil 44 MW
Krangede AB
X
Icing once. Ice dams 2 times.
Shclsjo 152 MW
Norrlandskraft Co.
X
Minor problems.
Hissmafors 60 MW
Ostersunds komun
Kattstrupefors 60 MW
Kattstrupeforsen Co.
Olden 120 MW 3
Balforsens Kraft Co.
Stensj8fallet 94 MW
Stensj8ns Kraft Co.
Kvarnfallet 17 MW
" it it
NBsaforsen 12 MW
Ostersund El Co.
Midskog 145 MW
Sv
NBrverede 62 MW
SV
iver Stations
Owner/Administrator
„nNyiet to the
Inquiry is
Indicated by X
Type of Ice Problem
ndans8lven Stugen 37 MW
SV
Krangede 240 MW
Krageede Co.
X
No problems.
Gammel8nge 72 MW
it
X
No problems.
Hammarforsen 73 MW
Balforsen Kraft Co.
X
Ice building at gate guards icing
at the intake in 1976.
Svarthalsforsen 67 MW
Svarthalsforsen Co.
X
Icing problem one week in the
autumn.
Stadsforsen 135 MW
SV
H81leforsen 140 MW
Sv
JBrkvissle 85 MW
Sv
Sillre 12 MW
SV
Bergeforsen 155 MW
SV & Balforsens Kraft Co.
igermandlven Linnvassely 70 MW
Linnvasselv Kraftlag
B1asj8n 60 MW
BlasjBns Kraft AB
Junsterforsen 40 MW
Holmens Bruk
Bagede 13 MW
It if
L8v8n 36 MW
Graningeverkens Co.
Storfinnforsen
Krangede Co.
X
Ramsele 157 MW
Krangede Co.
X
Edsele 57 MW
Balforsens Kraft Co.
X
Icing on gates yearly. Bottom
ice and drift ice towards the
intakes.
Forsse 52 MW
Graningeverkens
Hj8lta 168 MW
Norrlands Kraft Co.
X
Icing 3-5 times a year (no spill)
Solleftea 62 MW
X
Ice dams (0.5-1.0 m) downstream.
Ice pressure toward gates.
Dabbsj8 30 MW
Korsselbr8nna
Leakage in guides.
Bergvattnet 21 MW
”
Korsselbr8nna 112 MW
Balforsens Kraft Co.
Borgforsen 26 MW
Svan8 Co.
Bodum 13 MW
Fj8llsj8 13 MW
Balforsens Kraft Co.
Sil 13 MW
It u If
H811by 72 MW
Gulsele Co.
Gullsele 62 MW
" "
Degeeforsen 62 MW
Graningeverkens
Edensforsen 63 MW
19
iver
Stations
Owner/Administrator
Answer to the
Inquiry is
Indicated by X
Type of Ice Problem
Storrnorrfors 410 MW
SV
:ellefte 31v
Rebnis 64 MW
Rebnis Kraft Co.
Bastusel 108 MW
Bastusels Kraft Co.
Grytfors 32 MW
Grytforsen Co.
Gallejaur 115 MW
SV
Vargfors 70 MW
SV
Rengard 36 MW
Skelleftea Kraftverk
X
Batfors 40 MW
It It
X
Finnfors 32 MW
If
X
Severe icing on the gates on one
occasion (1973).
germanglven
Langbj8rn 92 MW
SV
Lasele 150 MW
SV
Kilforsen 275 MW
SV
NRmforsen 110 MW
SV
Moforsen 110 MW
Krangede Co.
X
Forsmo 155 MW
SV
Stalon 110 MW
SV
e Alv
Gejman 65 MW
SV
Ajaure 85 MW
SV
Gardikfors 60 MW
SV
Umluspen 95 MW
SV
Stensele 50 MW
SV
Grundfors 90 MW
SV
Rusfors 45 MW
SV
Balforsen 83 MW
Balforsens Kraft Co.
X
Icing on gates and total stop
Betsele 24 MW
in 1971, 1973 and 1975.
H911forsen 21 MW
" If it
X
X
Tuggen 105 MW
SV
Bjurfors bvre 42 MW
Bjurfors lower 78 MW
Norrlands Kraft Co.
If11 At
X
Sometimes ice dams downstream.
Harrsele 203 MW
It
X
X
Minor problems.
Pengfors 52 MW
96
X
Minor problems.
Minor problems.
River Stations
Owner/Administrator
Inquiry is
Indicated by X
Type of Ice Problem
Skellefte Xlv Granfors 39 MW
Skelleftea Kraftverk
X
Serious icing on the gates on or
occasion in 1973. Ice dams dowr
stream 75/76 and 76/77 (max 2.7
drop in pressure head).
Krangfors 58 MW
X
Serious icing on the gates in I5
Selsfors 57 MW
X
Serious icing on the gates in 1S
Even the ledskenor froze solid.
Kvistforsen 140 MW
X
Ice dams downstream station in
central Skelleftea.
Lule 31v Seitevare 220 MW
SV
Parki 20 MW
SV
Akkats 146 MW
SV
In order to sum up, it can be sa
Letsi 450 MW
SV
that ice problems mainly appear
Vietas 320 MW
SV
Laxede, VittjXry and Boden
Porjus 295 MW
SV
(icing, ice dams).
Harspranget 330 MW
SV
Ligga 160 MW
SV
Messaure 300 MW
SV
Porsi 175 MW
SV
Laxede 130 MW
SV
Vittjary 32 MW
SV
Boden 74 MW
SV
f
FOLLOW-UP QUESTIONNAIRE
1. The name of the power station and of the river.
2. (a) The owner of the station.
(b) The company responsible for the management.
3. The year when the present management took over.
4. The water head m.
5. The volume of water that goes through.
(a) Max --- m3/s
(b) The average water volume on a winter day m3/s.
(c) The average water volume on a winter night m3/s.
6. Turbines
(a)- Type
(b) How many
7. Give the type of regulating (day and night, week regulating, etc.)
and the variations of water level upstream of the power plant.
8. Type of outlet.
9. Are there current generators, air vents or similar structures
installed in front of the dams or at the outlet; describe.
10. Type of water intake.
(a) Open canal
(b) River power plant (special intake canal missing)
(c) Intake in connection of reservoir (directly or via a tunnel)
11. The dimensions of the intake canal.
(a) length-m
(b) breadth-m
(c) depth
12. The dimensions of the river upstream of the power plant (river
power plant).
(a) breadth-m
(b) depth-m
13. Water intake
(a) Now many intake openings
(b) The breadth and height of the intake openings
14. Gates: built in
(a) Yes
(b) No
15. Gates leaning out from vertical plane (0* for vertical gates).
16. The dimensions of vertical racks
(a) diameter--mm
(b) approximate separation--mm
17. Are there mechanical gate cleaners.
(a) Yes
(b) No
18. The heating of gates.
(a) Lacking
(b) The fraction of total gate area which is heated (example 2/3)
(c) The entire gate area is warmed
19. The type of warming the gates have
(a) Induction
(b) Circulation of warm water
(c) Other
20. The power (electric) on the gates
(a) Total --kw
(b) Per facing surface
21. Temperature and observing the ice formation. Reading slush term (0 meter)
(kvicksilver term) ------------ times a day.
22. Re"
a� temperatures recorded automatically.
(a) Yes
(b) No
23. If the account of recorded temperatures is missing, how often is
the temperature observed when there is a risk of icing. --------
times a day.
24. Is there installed a meter over the gates for measuring the loss
of the fall.
(a) yes
(b)- no
25. Are cables (chains?), cords etc., used for detecting the beginning
of icing.
(a) yes
(b) no
26. At what temperature are the gate warmers switched on. ------ °C
27. Are the gatewarmers switched on manually or automatically.
(a) manually
(b) automatically
28. During winter, the areas which are not covered by ice, directly
upstream of the station (including the intake canal), cracks (rifts?)
(a) length ----- m
(b) breadth ---- m -
29. Stretches of rapids (streams?) upstream.
(a) distance from the station -------- m
(b) the cracks length ---------------- m
(c) the cracks breadth---------------- m
30. Stretches of rapids (stream?) downstream (incoming) minor rivers
upstream of the station.
31. Minor river ------- m3/s
32. Temperature in relation to the main river.
(a) same
(b) colder
(c) warmer
33. Measures to hasten freeze-up upstream of the station. Laying out
4
ice booms.
(a) yes
(b) no
34. Reduction of water flow during the freeze-up.
(a) yes
(b) no
35. Is the surface of water constant upstream during freeze-up.
(a) yes
(b) no
36. Other
37. Occurrence of icing on gates.
(a) yes
(b) no
38. Tracks or turbine blades
(a) yes
(b) no
39. Give the time (morning, daytime, evening, night) and the type of
weather (temperature, precipitation, the direction of wind and
the wind velocity, etc.) when ice forming usually happens. Also
inform in what direction the intake canal is (example North -South).
40. How often does the icing occur.
41. Estimate the average production drop per year during the last
10 years ----------- kWh/a year
42. Has icing occurred in spite of that the gate warming was switched
on before ice forming started.
(a) yes
(b) no
43. Occurrence of ice dams. Where do the ice dams originate (give
likely causes).
44. Consequences of ice dams (reduced height of fall, flooding, etc.).
45. How often do ice dams appear.
46. Estimate the average drop in production during the last ten years
----=-------- kWh/a year
47. Ice floes against intake canals. Interferences in running (extra
work input) ----------- man hours/a year
48. When do the ice dams appear.
49. Is there an ice outlet.
50. Does the ice outlet (isutskov) work.
(a) yes
(b) no
(c) partly
51. How is ice removed if the ice outlet does.not function.
52. Estimate the average drop in production per year during the last
10 years (because of the decreased waterflow)-------- kWh/a year.
53. Other ice problems (ice pressure against dams, ice on gates and
gate folds, etc.).
54. Give the effective methods of fighting against ice.
55. More information (the rest) (for example: details of the form
of the station; details which have importance on the occurring ice
problems).
EXPLANATION FOR TABLE COLUMNS
1. Type of intake.
2. Average water flow (m3/s) and corresponding velocity (m/s)
upstream of the intake (winter day).
3. Average water flow (m3/s) and corresponding velocity (m/s)
upstream of the intake (winter night).
4. Size of the areas which are not covered by ice and are directly
upstream of the station (m2).
5. The heating of gates. The total warmed gate area, (m2).
6. The heating of gates. Total power (Kw).
7. (°C) temeperature at which gate warming is switched on (a) auto-
matically or (m) manually.
8. Icing. Tracks (?)
9. Icing. Gates.
10. Icing. Drop in production (MWh/a year).
11. Ice dams.
12. Ice dams. Production drop (MWh/a year).
13. Ice floes against the gates. Reason, point of time.
14. Ice floes against the gates. Work in man hours.
15. Ice floes against the gates. Production drop (MWh/a year).
16. Other.
L. aG,.n
Ka. Se. or
� KOjjab,
KArA
Majenp on 6
,ajcni. rs N
'i rar jd
Aby
1
Intake in
1450 m canal
475 m canal
1000 m canal
300 m canal
Intake in
River power
Intake in
reservoir
reservoir
plant
reservoir
2
Max Vol. 180
Max Vol. 155
Max Vol. 126
Max vol. 91
Max vol. 63
Max vol. 65
60--0.1-0.2
Max Vol.16
3
Max vel. 0.8
Max vel. 0.4
40--0.1
4
250 x 100
----------
475 x 12
1000 x 22
300 x 25
25 x 15
16 x 5
vary
5
lacking
total
total
total
total
total
total
lacking
6
----------
-----------
200
200
about 200
200
7
----------
(m)
(m) 0.5
(m) 0.5
(m) 0.5
(m) 0.3
8
Yes(?)
Yes
Yes
No
Yes
No
No
No
9
----------
Yes
Yes
Yes
Yes
Yes
No
Yes
10
----------
----------
-----------
-----------
-----------
-----------
0
-----------
11
----------
----------
No
-----------
No
No
No
Yes
12
----------
----------
-----------
-----------
-----------
-----------
-----------
-----------
13
----------
----------
-----------
-----------
-----------
-----------
-----------
-----------
14
----------
----------
-----------
900
100
100
-----------
0
15
----------
----------
-----------
-----------
-----------
-----------
-----------
-----
16
Icing has
Icing has
Icing about
Icing about
Icing about
Icing 2-3
happened
occurred
2 times a
2 times a
2 times a
times a year
in spite
in spite
year (has
year (has
year (has
of gate
of the
occurred
occurred
occurred
warming.
warming of
in spite
in spite
in spite
the gates.
of the
of the
of the
warming
warming
warming
of the
of the
of the
gates).
gates).
gates).
stationen
1
River power
250 m canal
River power
River power
32 m canal
station
plant
plant
2
17--0.1-0.2
17--0.4
17--0.1-0.2
32--0.4
2--<O.l
3
17--0.1-0.2
17--0.4
17--0.1-0.2
32--0.4
2--<O.1
4
100 x 50
1000 x 50
40 x 30
Up to the
-------
power station
(about 400 m)
5
lacking
lacking
lacking
lacking
lacking
6
7
8 Yes No
9 Yes Yes
10-------- ----------
11 No No
12
13 --------
14-------- ----------
15 -------- ----------
16 Icing occurs Icing occurs
often before often before
freeze-up. freeze-up.
No Yes No
No Yes No
--------- 500
No At gates ----------
----------- ----------
--------- Mild winter ----------
--------- 50 ----------
--------- ----------- ----------
Icing when
weather changes,
northerly wind
and -100C or
colder.
u IL r a a
400 m canal River power River power
plant plant
16 17--0.2 17--0.1-0.2
16 17--0.2 17--0.1-0.2
400 m long 100 x 30 150 x 150
---------- lacking lacking
---------- Yes No
---------- Yes No
---------- ----------
---------- No No
---------- ----------
---------- ----------
Icing occurs Often icing
day and night before freeze -
when the up evening
-nertir wind and night.
blows from
the northl'
I"'-* of nvrMeasf
(4 times
during
Jan., Feb.,
and March
1978).
250 m canal
300 m canal
1000 m canal
Canal
150 m canal
325--0.8
300--0.4
300--0.8
Max Vol.180
83--0.3
100--0.3
200--0.3
200--0.5
83--0.3
250 x 30
2000 x 100
1500 x 80
500 x 100
850 x 110 m
the canal can
freeze-up totally.
1/2
2/5
1/2
lacking
lacking
700
400
1500 KVA
(a)0.04
(m) 0.01
(a) 0.005
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
--------
---------
Little
700
2
Yes (anchor
Yes
Yes
Yes
No
ice)
Ice cover breaks
up because of
storm or
variations of
water level
.10
Ice forming against dams
and gates. Icing on gates
as a rule each winter be-
fore freeze-up in SW -wind
and at colder temp. than
-20C (canal SN). Great
problems if ice cover
breaks up and get into
the intake canal).
During
freeze-up.
Icing about 1 week a year
in the evenings and at
times in negative
temperatures and north-
westerly wind (canal EW).
Appears
Total of 200-300 hours of
extra work a year because
of icing problems in the
evenings and at nighttime
in NW wind (canal in NW
direction).
--------- When there
is a change
in water use
in Storforse
50
1
Icing 0-8 Icing on the afternoons
times a and nights about 2
year when times a year in the
there is a westerly wind (the
westerly canal is in E-W direc-
wind and tion) and when it is
-8°C (or colder than -10°C.
colder).
U U'
1 93 m canal
100 m canal
River power
River power
River power
River power
River power
plant
plant
plant
plant
plant
2 210--0.4
125--1.0
300--0.2
300--0.2
210--0.2
210--0.1-0.2
210--0.1-0.2
3 210--0.4
125--1.0
240--0.1
240--0.1-0.2
210--0.2
210--0.1-0.2
210-4.1-0.2
4 400 x 100 m2
150 x 25-50
Minimum
Minimum
100 x 50
150 x 70
0 x 0
from 70 m up-
20 x 20-30
20 x 20-30
stream of the
station
5 lacking lacking lacking
6
7
8 No
9 Yes
10 10
11 No
12---------
13 When there
are changes
in water
level.
No
Yes
2
In spring,
in fast
changes in
water volume
that goes
through.
No
Yes
No
lacking lacking lacking lacking
No No
No No
Yes Yes
Yes Yes
--------- 0
--------- 0
No No
--------- No
--------- 0
-------------------
14 100
--------
0
0
-------------------
15 12
--------
0
0
0----------
16 Ice in gates Icing once a
Strong icing
Strong icing
Icing 2 times
Icing once in Icing once in
and gate year.
2 times in
2 times in
in 10 years
every 7 years 3 years.
guards. Icing
30 years
30 years
at 10°C or
at -10% or
about once
under north-
under north-
colder as
colder as well
a year.
erly wind
erly wind
well as in
as in wind and
(canal is
(canal is
wind and
precipitation.
NW/SE).
NW/SE).
precipitation.
Li..db.yn
ocKfjaro
fluforsen
huuuneltursen
rorshuvuo Lima
River power
River power plant
250 m canal
River power plant
River power River power
plant
plant plant
60--<0.1
60--<0.1
25--0.2
25--<0.1
210--0.1-0.2 20--<0.1
60--<0.1
60--<0.1
25--0.2
25--<0.1
210--0.1-0.2 20--<0.1
0 x 0
0 x 0
-----------
-----------
400 x 40 0 x 0
lacking
lacking
lacking
total
lacking lacking
150 .KVA
(m)
No
No
No
No
No No
Yes
No
Yes
Yes
Yes No
About100
---------
12
-----------
-------------------
---------
---------
-----------
No No
---------
---------
-----------
0 0
---------
At the time of ice -breakup.
At the time of ice -breakup.
-----------
----------- --------
---------
---------
-----------
0
0 0
---------
---------
-----------
-----------
0 0
The last time
No icing after a rise in
'Icing each autumn. At the
Icing extremely seldom
Icing about
icing occur-
the height of the dam since
time of icing, the station
but it has happened in
once in every
red 1975.
a stretch of rapids about
is stopped overnight so
spite of the fact that
3 years at
1 km upstream of the
that (freezing can start)
the gate warmers have
-10°C and colder
station "disappeared"
freeze-up can happen,
been installed.
temperatures as
well as in wind
(gale) and snow-
fall. Ice on
gates and gate
guards.
SH,u..,Oo
�1}..e,
1
River power
River power
River power
River power plant
plant
plant
plant
2
160--0.2
80--<0.1
80--<0.1
80--<0.1
3
160--0.2
50--<0.1
50--<0.1
50--<0.1
4
--------
0 x 0
0 x 0
0 x 0
5
lacking
lacking
lacking
lacking
6
7
8
No
No
No
No
9
Yes
No
No
Yes
10
15
0
0
6
11
No
No
No
6 km upstream
12
0
0
0
0
13
---------
----------
-----------
No
14
0
0
0
0
15
---------
----------
-----------
-----------
16
Icing each
Icing once in every 5 years
3rd year.
(can occur at any time day
or night).
Intake in
reservoir
80
50
0 x 0
lacking
irung.dc
Intake in reservoir
150
0
5 m long
lacking
No
No
No
No
0
0
No---------
----------
0 , .
----------
0
---------
0
----------
0
Prior to the installation of air
outlet very deep in the reservoir
(air outlet lifts up the warm
bottom water to the intake) ice
dams appeared downstream of the
power station at very low
temperatures (-30°C) they can
still appear.
uergetorsen Sillre • JBrkvissle H81leforsen Stadsforsen Svarthalsforsen Hammarforsen
1 River power Intake in
plant reservoir
(300 m long
canal)
2 450-500--0.5 Max volumcr 8 250
3 120--0.1-0.2 Max vel. 0.2 100
4 0 x 0-------------------
5 lacking lacking lacking
6
7
8 No-------------------
9 No --------- ----------
10-------------------
11 Yes --------- ----------
12 500---------
13 Sometimes in ---------
spring.
14 ---------
15 ---------
16 Ice dams down-
stream reduce
the height of
fall by about
0.7-0.8 m.
---------- ---------
At freeze -up ----------
--at breakup
River power Intake in
plant reservoir
Intake in River power
reservoir plant
450 425 Max vol4me'525
350 250 Max vel. 0.2
0 x 0 0 x 0-----------
lacking----------- lacking
No No No
No No Yes
----------- 0
---------- No-----------
0
----------- In the atmos-
pheric distur-
bances in the
system.
-------v-- 0
Ice in gate In order to Icing in the
guards and eliminate ice mornings 3-6
on sills. pressure days a year.
against dam,
a lead is
kept open next
to the dam.
River power plant
Max volyme' 460
Max vel. 0.3
85 x 15
lacking
No
No
T,
If,
Icing on the intake gates
in January 1976. Previously
icing has not occurred for
about 30 years.
Gammel3nge
Krangede
Stugun
NBverede
Midskog
M8rsil
1 River power
River power
Intake in
Intake in
Intake in
Intake in
plant
plant (intake
reservoir
reservoir
reservoir
reservoir
in reservoir)
(a river
(river power
via 100 m
power plant)
plant)
long canal
2 400--0.1
400--0.1
Max vol. 600
Max vol. 600
Max vol; 600
150--0.2
3 360--0.1
360--0.1
Max vel. 0.2
Max vel. 0.1
Max vel. 0.1
60--0.1
4--------
--------
40 x 0-2
40 x 0-2
50 x 2
50 x 25
5 lacking lacking
6
7
8 No No
9 No No
10
it No No
12 0 0
13------------------
JBrpstr8mmen Olden
River power Intake in
plant reservoir
180--0.2 Max vol. 34
160--0.2
1000-1500 x 10 x 6
80-300
----------------- lacking lacking --------- lacking
-----------------
-----------------
-------- No No No No
-------- No No Yes No
-------- --------- ---------
-------- --------- ---------- Yes (seldom) No
-------- --------- ----------
In connection In connection Late bA When discharge Very seldom
of break-up. of break-up. up and9on- is increased.
section of
switching off.
14 0 0
---------
15 0 0
---------
16 The size of
Icing once
Icing only
the reservoir
since 1A56'
once 4449
(deep) makes
since the
since the
the water tem-production
production
perature at
started.i)
started.
the intake
M 5 G
1949
stay above
+0.04°C.
No
No
No
Storrnorrfors
Pengfor's
Harrsele
Bjurfors
nedre (lower)
Bjurfors
llvre (upper)
Tuggen
HAllforsen
1
2500 m canal
•Intake in
Intake in
Intake in
Intake in
Intake in
Intake in
reservoir
reservoir
reservoir
reservoir
reservoir
reservoir
2
210--0.5
284--0.1
284--<0.1
284--<0.1
284--0.1
330--<0.1
Max volume 300
3
120--0.3
86--<0.1
86
86
86
165
Max vel. 0.1
4
0 x 0
--------
---------
---------
--------
0 x 0
-----------
5
lacking
lacking
lacking
lacking
lacking
lacking
lacking
6
7
8
No
No
No
No
No
No
Yes
9
Yes
No
No
No
No
No
Yes
10
-------
10
11
-------
--------
---------
---------
--------
Each year
------------
downstream.
12
-------
--------
--------
3000
------------
13
Before freez-
--------
---------
---------
--------
ing on canal.
--------
------------
14
10
--------
---------
---------
--------
--------
------------
15
--------
---------
---------
--------
--------
------------
16
Icing once
every 3-5
Icing has
High velocity
Icing 5 times
years
(cold, no
occurred once
in the drain-
since,4AJW,
precipitation,
20 years ago.
age canal
s#xlow' the plant
calm, canal
causes strong
started operation.
ice free).
ice production
1 Nf964
and thereby
ice dams down-
stream (up till
2 m each year).
aetsele 0alforsen Rusfors Grundfors Stensele Umtuspen
1 Intake in Intake in Intake in reservoir
reservoir reservoir
2 Max vol. 300 Max vol. 300 160--<O.1
3 Max vel. 0.1 Max vel.<0.1 100
4---------- ---------- 0 x 0
5 lacking lacking lacking
6
7
8 Yes No No
9 Yes No No
10 12 ----------------------
11 -------------------- no (see notes)
12 --------------------------------
13--------------------------------
14 --------------------
1s---------- ----------
16 Icing 4 times Icing 3 times Swell (surge?) has been
sinee1965, rinse-1958, caused by leaking gates.
since the since the Anchor ice has occurred
plant plant 3-4 km downstream of the
started started station with accompanying
operation. operation. lossAof water heat?
in 19F.S In l959
Intake in Intake in
reservoir reservoir
220--<0.1 220--<0.1
120---------
0 x 0 0 x 0
lacking lacking
Intake in reservoir via
300 m long canal
250--0-0.2
110--<0.1
40 x 40
lacking
No No No
No No No
---------- No No
---------- ----------
---------- Sometimes in Sometimes in spring.
spring.
Sometimes swell
forming in
leaking gates.
Gardikfors
Ajaure
Gejman
1 Intake in reservoir
2 150--<O.1
3 150--<O.1
4 5 x 15
5 lacking
6
7
8 No
9 No
10
11 No
12--------------
13
14
15
16 Swell (surge) of ice because of
leaking gates.
No
No
V1•
-------------------
Swell (surge) of ice because of
leaking gates.
Intake in reservoir
23--<O.l
23--<O.1
-------------------
lacking
No
No
No
Boden
Vittjgry
Laxede
Porsi
1
Intake in reservoir
River powerplant
River powerplant
Intake in reservoir
2
450--0.1-0.2
450
450--0.3
450--<0.1
3
450--0.1-0.2
450
450--0.3
350--<0.1
4
0 x 0
----------------
0 x 0
0 x 0
5
lacking
lacking
lacking
lacking
6
7
8
No
No
No
No
9
Yes
Yes
Yes
No
10
21
----------------
1
11
No
Yes
-----------------
No
12
---------------
----------------
13
---------------
Ice floe can be sucked
Do not exist
-----------------
under towards the gates.
14
---------------
----------------
0
0
15
---------------
----------------
-----------------
16
Icing 2 times since
Risks of considerable ice
Icing 1 time since the
1971 when the plant
damming downstream after
plant started operating
started operations
extensive dredging.
in 1962. (Evening strong
Considerable icing problems
snowfall and-10°C),Ipe
on intake gates come up
damming has stopped
each year before freeze-
after building of
up period.
Vittj3ry power station.
Letsi
1 Intake in reservoir via
100 in long canal.
2 330--1.2
3 90--0.3
4 5 x 30
5 lacking
6
7
8
No
9
No
10
------------
11
No
12
13
At the break-up
14
2
15
------------
16
Akkats
Randi
Parki
Seitevare
Messaure
Intake in reservoir(?)
2100 m canal
Intake in
Intake in
Intake in reservoir
480 m tunnel
reservoir(7)
reservoir
via 75 m long canal.
(tunnel &
via 210 in
canal).
long canal
330--<0,1
330--1,1
150
110--<0.1
350
0
0
150
50
195
5 x 20
2000 x 20
20 x 20
0 x 0
15 x 20
(in midwinter
the canal freezes
up).
lacking
lacking
lacking
lacking
lacking
(being con-
structed)
140
No
No
No
No
No
-------------
No
No
No
No
No
No
No
No
No
------------- Does not Does not Does not
exist. exist. exist.
2-----------
In 1974-02-25 the station(sin
cing in 1976
lost out because ice floesce the plant
stopped the intake. tarted operating,
Ligga
Harspranget
Porous
1
Intake in reservoir
Intake in reservoir
Intake in reservoir
2
380
350--<0.1
380
3
175
205--<0.1
240
4
30 x 40
40 x 30
30 x 40
5
lacking
lacking
lacking
6
7
8
No
No
No
9
No
No
No
10
11
No
No
No
12
13
Does not exist.
Does not exist.
Does not exist.
14
15
16
V1etas
Intake in reservoir (tunnels)
350
0
50 x 50 at two tunnel intakes
lacking
APPENDIX 4
LENGTH OF THE OPEN -WATER REACH
BELOW A DAM OR RESERVOIR:
Report to the State of Alaska,
Department of Commerce Economic Development
by
J. P. Gosink
Geophysical Institute
University of Alaska
Fairbanks, Alaska 99701
May 7, 1984
ABSTRACT
The prediction of the open water length downstream from a dam is an
essential safety concern for hydroelectric development in Alaska. This
information provides the position of the ice front and determines the
stability of that ice front during changes in atmospheric conditions
and/or changes in discharge from the dam. During very cold weather (lair <
-25°C) the open water reach will be the site of severe ice fog, causing
icing on structures, visibility problems, and adversely affecting nearby
residents. In addition, the open water reach may eliminate a traditional
winter crossing route for man and animals.
In this report we examine different approaches for the prediction of
the open water length; they are compared for simplicity, for generality
and for accuracy. Formulae for direct application of certain of the
models are given in tabular and/or graphical format. Several simple -to -
use analytic formulae are given for steady-state and transient boundary
conditions. The impact of various complications, including lateral
temperature gradients, effects of side streams, water clarity and braided
channels, which characterize realistic conditions in Alaskan rivers but
which unfortunately are not included in the simpler formulae, are discussed
and methods are suggested for the quantitative analyses of these problems.
Finally, a finite difference computer program of the transient river
temperature distribution for the single channel, constant discharge case
is given.
INTRODUCTION
Hydroelectric development in Alaska is proceeding at an accelerating
pace. The recently completed hydroelectric projects at Solomon Gulch,
Green Lake and Tyee Lake will provide 48.5 megawatts to cities in
southeastern Alaska. Other projects either under construction or
recommended for construction, including the Susitna site, can provide
some 1800 megawatts to the state.
The creation of new reservoirs or the deepening of existing lakes
and reservoirs can drastically alter the thermal regime in the lake basin
and in the downstream river. Water released from the dam will be warmer
in the winter and colder in the summer than under pre -construction
conditions. In Arctic and sub -arctic regions the temperature of the
outfall water during winter is a critical parameter controlling the length
of the open water reach downstream from the dam and the position of the
leading ice edge. There is a great deal of concern regarding the length
of the open water reach since the released water vapor will cause icing
on nearby structures and equipment, and will produce thick ice fog during
periods of extremely cold temperatures. In addition traditional winter
crossing routes for man and animals would be eliminated by the open water
reach.
Several different methods have been used to determine the length of
the open water reach. In general these methods could be classified as
either statistical or semi -empirical. The first class uses data acquired
for many years at a particular site to establish a curve or set of limits
for the length of the open water reach as a function of meteorological
and discharge parameters. Two examples of this procedure are the analyses
of Goryunov and Perzhinskiy (1967) and of Gotlib and Gorina (1974). Only
e:
one of the examples of the statistical method shows actual comparisons
of the predictions with measured open water lengths. The statistical
method is useful only at sites where there exists a long data base for
analysis. In addition, the predictions are no longer valid when the
hydrology of the reservoir basin and river system are appreciably altered,
as for example, by deepening of the reservoir.
The other class of semi -empirical methods for finding the open water
length is analytical in the sense that some attempt is made to model the
basic physics of the problem. These models vary in the assumptions made,
but in general, they utilize a semi -empirical heat balance for the open
water reach. A major shortcoming of all the models considered in this
report is that none takes the dynamics of the ice cover into consideration;
that is, all of the models are primarily thermodynamic. This approach is
suitable as long as the ambient conditions (discharge and meteorology)
are relatively stable, so that changes in the ice conditions occur rela-
tively slowly. These models are not applicable for example, during a
sudden thaw or with a sudden drastic change in discharge.
For stable winter conditions, the analytic models yield reasonably
accurate predictions of open water length. Both steady state and transient
models are available, and the steady state assumption allows a particularly
simple closed form solution to be written for the open water length. In
the present report we introduce a closed form solution to the transient
problem which is exact whenever the air temperature and/or short wave
radiation can be expressed as a sum of sinusoidal terms of arbitrary
frequencies - a fairly common case. All closed form solutions are based
on the assumption of uniform river hydrology, i.e., constant width,
depth, velocity, discharge and specifically, no braided channels or
3
stream inflow. If variations in these parameters are to be included, a
finite difference or finite element solution of the governing equations
is necessary. An example of this type of finite difference model is
given by Ashton (1979). His model allows arbitrary variations in air
temperature, and changes in river width and depth and may be modified to
improve the surface heat transfer expression or to include the thermal
effects of inflowing streams.
The purpose of this report is to summarize and assess models for the
prediction of the open water length downstream from a dam in arctic and
sub -arctic conditions. We include two statistical models to demonstrate
their use and the required data. The primary emphasis is on analytical
models which are of general applicability. We explain the derivation of
the governing equations and differences in the surface heat transfer
expression. Using comparisons with data from sub -arctic rivers, we
demonstrate that the Dingman and Assur (1969) version of the "Russian
winter equation" for linearized heat exchange provides the best estimate
of surface flux. The simple closed form solutions of the heat balance
equations are presented for both steady state conditions and for sinusoi-
dally varying air temperatures. These closed form solutions are useful
estimates of the open water length when there are no side streams entering
the river, and little variation in river width and depth. Finally, for
more general applications, we present a finite difference model based on
the Ashton (1979) model, which may easily be extended to include heat
flux from side streams and heat exchange by the Dingman, Weeks and Yen
(1967) formulae. Other complications including water clarity and transverse
mixing are discussed quantitatively, and recommendations are made for
Alaskan applications.
0
Classification of Models to Determine the
Lead Area Downstream of a Power Station
There are two distinct types of models which predict the open lead
area downstream of a power station. The first is the totally statistical
technique suggested by Goryunov and Perzhinskiy (1967) and by Gotlib and
Gorina (1974). All the remaining models discussed in this paper may be
classified as semi -empirical models. The models to be discussed are
listed for reference in Table 1.
Statistical Techniques
1) Gotlib and Gorina (1974)
Gotlib and Gorina present a graph (Figure 1) which represents the
length of the open lead downstream from the Bratsk hydroelectric plant
under cold winter conditions (air temp.: Dec. and Jan., - 290C). D is
outflow discharge in m3/sec, and L is open water length in km. These
curves represent the minimum length of the lead; a maximum length of
30-48 km is suggested for warm -winter conditions. Each curve is associated
with a specific water outflow temperature at the dam ranging from 1.0°C
to 3.0°C with increments of O.M. Apparently winter discharge tempera-
tures at the Bratsk hydroelectric site always vary between 1.0°C and
3.0°C.
From Figure 1, it is evident that the length of the open lead varies
directly with reservoir discharge and with the temperature of the outflow
water (Tw). No details are given by Gotlib and Gorina (1974) regarding
their analysis; furthermore, no comparison with data is given. Although
the length of the open water reach increases with the magnitude of the
warm discharge and with the temperature of the discharge, neither increase
G
j
U
M
3
0
v
m
i
b
L
U
N
C
Y
24
+1.0 1.4 1.8 2.2 2.6 3.0
u lu Zu 30 510 SO
Length of open water, L (km)
Figure 1. (from Gotlib and Gorina, 1974) Open water length downstream
from the Bratsk hydroelectric plant vs. discharge from the
dam. The curves represent lines of constant outfall temperature
ranging from 1.0°C to 3.0°C in increments of 0.2°C.
P
-100
-200
-4700
- 900
-500
-COO
V
-700
-800
-000
=1000
.I
i-t p "`T T
1 2 8 y S 0
I�
^)
r
72 14 76 78 ZO 2Z Z � 2e ZB JO Z 9 5'
April P May
Figure 2. (from Gotlib and Gorina, 1974) Date of the onset of break-up
downstream from the Bratsk hydroelectric plant vs. sum of the
daily air temperatures during March and April. The curves 1
througi 6 represent the following conditions: 1) discharge =
3500 m /sec and downstream ice thickness before break-up = 0.6 m;
2) 3000 m3/sec and 0.6 m; 3) 3500 m3/sec and 0.0 m; 4) 3000 m3/sec
and 0.3 m; 5) 3500 m3/sec and 1.0 m; 6) 3000 m3/sec and 1.0 m.
is linear. This implies that extrapolation beyond the range of the curves
is impossible, since the systematic variation of the open water length
with these parameters is not provided, nor is any indication of open water
length for varying winter air temperatures.
Gotlib and Gorina (1974) also provide graphical information con-
cerning the date of the initiation of ice edge recession (i.e., date of
onset of break up) as a function of positive degree days, reservoir
discharge, and existing ice thickness (see Figure 2). Parameters for the
six curves, discharge and initial ice thickness, are given in the caption.
Figure 2 demonstrates that for constant ice thickness, break-up of the
ice edge begins 3-4 days earlier when the discharge D = 3500 m3/sec than
when D = 3000 m3/sec. Furthermore, for constant discharge, say 0 = 3500
m3/sec, ice edge recession is delayed 4 days for every 0.2 meter of
ambient ice thickness above 0.6 meter.
2) Goryunov and Perzhinskiy
Goryunov and Perzhinskiy (1967) present an empirical curve (see Figure 3)
to represent the relation between the length of the open water lead, L,
and the sum of the negative degree days. The formula suggested by Goryunov
and Perzhinskiy (1967) is:
L (in km) = 5.5 • 106 (E-Tair)-2 (1)
This formula is applicable to the Lower Volga downstream of the
Volgograd Reservoir. Discharge rates and meteorological conditions are
not given; thus direct comparison with analytic techniques cannot be
made. The data points represent the open lead length for a particular
winter as a function of total negative degrees. It is not clear whether
this length is a minimum or a winter average. It would be interesting to
see whether discharge rates from the Volgograd Reservoir varied during
0
u
ua
c
z
m
w
ct P.O
=roo
wnF -s
Z w O
Nv 3
O i
onsp o
nk a
w c
n o
s -- o
m <
rc z ri
oaa
0. (p (D C
w m a
< <
rt o -v
- (D
a+ � z
ci O N
n) a J
ro 0. N
o m
<(-
m ru
a < ko
o rn
8 V
O
ct z O
w -a
N
7c < E
m
o 0 -1
= c
0. g
m
E o o
-ht
r rt
�C
m
-,a
m a
ut h
� 1
<
!D
� O
Dam of hydrostadon�
Volgograd
Krasnoarmeysk
�•"� o o
G
Svetlyy Yar
�� o
o
�
Kamennyy Yar
p 00
/
e
o
•
s
p
Y
:�
Chernyy Yar o/
q p p
1
1
1
•1
�^
r
r
tvD
400
toO
Nikol'skoyc
• q o4 G
°i
O
w
8
t
l
i
l
w
t
t
p
M
�►
14
Ycnotaycvka pond
Seroglazovka c
G
N. Lebyazh'ye
rv.
Astrakhan' o
x^
9
the years of observation, since the analytical models discussed in the
following sections all predict a linear increase of open water length with
discharge. The data from the Lower Volga suggest that the least variabi-
lity in open water length occurs during the warmer periods (smaller
degree days). Furthermore, local hydrologic effects would appear to play
a major role in establishing ice coverage at Yenotayevka pond; although
never stated explicitly, it is reasonable to assume that the pond is a
wide river reach with slow water velocities.
Finally, it should be noted that the general appearance of the L vs.
-lair curve found by statistical methods by the Russian investigators is
similar to the theoretical curve predicted by the semi -empirical models.
The latter models suggest a relationship of the form L a In [1+Tw/-Tairl
for steady-state conditions, and this logarithmic function approaches
L - 1/-Tair when Tw << -Tair-
Statistical models can provide useful guidelines at existing sites
where a good data base already exists. They have no predictive value at
the site for any major alterations in the reservoir -river system or for
weather extremes. They are not useful as predictive tools for the planning
of new projects or expansions. These statistical models yield the following
qualitative information on open water length: open water length decreases
with negative degree days and with decreasing dam water outflow temperature,
and increases with reservoir discharge. With respect to the timing of
ice cover break-up, the statistical models suggest accelerated break-up
with increased discharge and thinner initial ice thickness.
10
Analytical Techniques
Another approach to finding the open water length involves analysis
of the basic physics or thermodynamics of the flow. Consider the thermal
balance on a slab of fluid:
UT + a(UT)/ax ox]
Rate of change of = Heat - Heat + heat flux + Other
heat in the slab convected in convected out through top heat sources
(1) pcp a(hboxT)/at = pcphbUT-pcp(hbUT+a(hbUT)/ax ax) + Q box + E S
or
(2) pcp [a(hbT)/at + a(UhbT)/ax] = Qb + ES'
where p is water density, cp is specific heat, h is river depth, b is
river width, T is water temperature, t is time, U is average streamwise
velocity, x is streamwise distance, Q is net surface heat exchange [W/m2],
and E S' represents the sum of other heat sources including side stream
inflow and longitudinal heat diffusion.
Initially we will consider only rivers with constant discharge (Uhb = D =
constant), constant width, depth and velocity, and zero stream inflow.
Then the governing equation simplifies to:
(3) pcph[aT/at + U aT/ax] = Q
11
All of the analytical models use a simplified form of equation (1),
usually equation (3). Q, the surface heat transfer expression, is
determined from semi -empirical models for radiative, turbulent, latent
and bottom heat exchange; the formulae for Q vary substantially between
the different models, and the complexity of the formulae for Q determines
whether or not a closed form solution is available for T(x,t). The
expressions for Q take three forms: Q is a function of atmospheric
parameters only, Q is linearly proportional to the air and water temperature
a
difference, and Q is a complex function of water temperature and atmospheric
parameters. Details of the second and third types of expression will be
given in subsequent sections. In the following section we discuss models
for open water length based on all three types of expressions for Q. We
have listed the models in the order of increasing mathematical complexity,
i.e., increasing complexity of the functional form of Q. In a subsequent
section predictions of the models and field data will be compared to
assess their realiability; finally guidelines will be offered for the
selection of an appropriate model for a given application.
Asvall (1972)
Asvall (1972) greatly simplifies equation (1) by assuming steady
state conditions, constant discharge, river depth and width, and a surface
heat transfer expression for Q which depends only on atmospheric conditions.
Asvall suggests using the net surface heat loss expression for Q from
Devik (1964); this is given in equation 35 of this report and will be
discussed subsequently. Since Q is assumed to be a constant (a known
function of air temperature and wind velocity), equation (1) may be
integrated directly to become,
12
(4) PcpUhTo/IQI = L
where L is the open water length and To is the outflow temperature at the
dam. Another way of determining this simple formula for L is by equating
the net heat into the river at the dam (= PcpUhbTo) and the net heat
lost over the open water area (=QLb). However equation 4 fails to take
into consideration the fact that the surface exchange Q is a function of
water temperature, time and river location.
In order to incorporate in a simple way the variation of Q with
water temperature, various linearized expressions have been determined
for the surface heat exchange. A particularly useful linearization
formula expresses Q as a linear function of water temperature,
(5) Q=A+BT
When an expression of the form of equation (5) is assumed for Q,
simple steady state solutions of equation (1) exist, and yield more
reliable estimates of the open water length. Consequently, there is no
real advantage in using constant values of Q, and a real physical advantage
in including water temperature dependency in the formulation of Q.
Dingman and Assur (1969)
Dingman and Assur (1969) introduce a simple steady state analysis
for open water length. The major simplification comes from the linearization
of the surface heat transfer expression Q, given previously in equation (5)
and written in a more general form below as,
(6) Q = -Qo - K(T-Tair)
13
The linear coefficients, Qo and K, were obtained by analysis of the
empirical expressions of Dingman at al. (1967) for net long wave sensible
and latent heat flux; net measured short wave radiation should be added
to the expression. The coefficients are given as functions of wind
velocity and cloud cover in Table 2. With this linearization, the steady
state heat balance from equation (3) becomes
(7) pcphU dT/dx = -Qo -K(T-Tair)
and this equation has the closed form solution,
(8) T = To -CQo/K + To - Tair][1-exp(-Kx/pcpUh)]
where To = T(x=o) is the average well -mixed temperature at the outfall.
The temperature of the water decreases exponentially with distance from
the outfall, and approaches a theoretical equilibrium temperature (Te =
Tair - Qo/K) at x = In actuality the temperature decreases to DOC at
the leading edge of the ice; beyond this distance the expression for
surface heat exchange is no longer valid, the water temperature remains
DOC, and heat loss through the ice cover implies ice growth. The position
of the zero isotherm, L, can be found by setting T = 0°C in equation (8):
(9) L = (pcpUh/K) In [1 + KTo/(Qo-K Tair)]
Note that the coefficients for atmospheric heat transfer, Qo and
K, are simple functions of wind velocity and air temperature; clearly
equation (9) is a very easy -to -use formula for open water length. However,
the assumptions required for this derivation should be kept in mind.
These include: 1) uniform and constant river discharge, width and depth,
2) constant air tempratures and wind velocity, 3) no inflowing streams,
14
4) no heat flux from or to the river bottom, 5) applicability of the
linearized surface heat transfer expressions. Where these assunptions
are violated, an appropriate strategy might be the use of equation (9) as
a first estimate of the open water depth, with subsequent analysis of the
effects of other parameters. Quantitative discussion of some of these
parameters follows in a subsequent section.
Paily, Magagno and Kennedy (1974)
Paily et al. (1974) solve the following version of equation (2),
(10) aT/at + U aT/ax = Q/pc ph + E 32T/3x2
with a linearized heat exchange expression for Q similar to the expression
used by Dingman and Assur (1969), but involving different values of the
linearization coefficients K and Qo. In Paily et al. (1974) the coeffi-
cients are given in tabular form rather than as functions of wind speed
and cloud cover; the coefficients are presented in Table 3 of the present
report. The additional term, Ea2T/ax2, represents streamwise diffusion
of heat. Obviously diffusion is a much less effective mechanism for heat
transport in a river than is convection. Nevertheless it is included in
this model for completeness and to demonstrate the relative effect of
longitudinal diffusion of heat. For steady state cases, Paily et al.
(1974) give a closed form solution of equation (10);
(11) T = To - [(Qo/K) + To - Tair]C1-exp{(Ux/2E)(1- 1+4KE/pcphU,2)' 11
This solution was devised earlier by Daily and Harleman (1966). It is
important to note that in the limit as E approaches zero, the argument of
the exponential term goes to-Kx/pcpUh, exactly as predicted by the
Dingman-Assur (1969) model (see equation 8); this can be seen either by
15
Table 3. Values of Qo and K from Paily et al. (1974)
i
I I
I Heat ex -
Base
I change co -
Wind
I I
heat
I efficient, K
I
velocity
I 1
exchange
I in Watts
I
in miles
I I
rate, Qo
I per square
Air tem- I
per hour
I Relative I
in Watts
I meter
perature, I
(meters
I humidity, I
per square
per degree
in degrees I
per
I as a per- I
meter
I Celsius
Celsius I
second) I
centage I
(1) i
(2) i
(3) i
(4)
i (5)
-1.0
11.0
70.0
16.25
31.40
(4.95)
-3.0
11.0
70.0
65.35
32.50
(4.95)
-5.0
1170.0
70.0
114.67
33.58
(4.95)
-10.0
11.0
70.0
239.39
36.22
(4.95)
-15.0
11.0
70.0
366.96
38.77
(4.95)
-18.0
11.0
70.0
445.27
40.28
(4.95)
-5.0
0.0
70.0
23.04
16.67
(0.0)
-5.0
3.7
70.0
53.59
23.30
(1.65)
-5.0
7.4
70.0
84.13
27.94
(3.30)
-5.0
11.0
70.0
114.67
33.58
(4.95)
-5.0
14.7
70.0
145.22
39.21
(6.60)
-5.0
18.4
70.0
175.76
44.85
(8.25)
-5.0
11.0
10.0
171.79
34.25
(4.95)
-5.0
11.0
30.0
152.75
34.02
(4.95)
-5.0
11.0
50.0
133.71
69.80
(4.95)
-5.0
11.0
70.0
114.67
33.58
(4.95)
-5.0
11.0
90.0
95.64
33.35
(4.95)
-5.0
11.0
100.0
86.12
33.24
(4.95)
-for
a ues valid
range of
water temperature
between 0 C and C; va
ues o other
meteorological
variables
are: barometric pressure = 996.0 mb; cloud
height = 3,275
ft (1,000 m);
cloud cover
= .6; and visibility = 1.87 miles (3 km).
applying L'Hopital's rule or by expanding the square root with the binomial
expansion. The latter procedure yields the following series for the
argument,
(12) (Ux/2E) 1-2(KE/pcphU2) + 2 (KE/pcphU2)2 - 4(KE/pcphU2)3 + ... }
This series shows that the diffusion term lessens the longitudinal
temperature decrease, producing a slightly longer open water length. The
tempering effect of the diffusion term can also be seen directly from
` equation (10) when it is noticed that the second derivative term is
Positive definite in these problems.
A closed form expression for the open water length can be written as
follows,
(13) L = (pc pUh/K)(1/2 + 1/4 + KE/pcphU12)ln[l + KTo/(Qo-K Tair)]
Here the effect of the diffusion term on the open water length is
immediately apparent. Clearly when E << 4pcphU2/K there is very little
increase in open water length. Numerical estimates of this increment for
typical Alaskan conditions will be given in a subsequent section. Note
that if E is small, then differences between values of L calculated by
the Paily et al. (1974) formulae and the Dingman and Assur (1969) formulae
will depend primarily upon the linearization coefficients, K and Qo, in
the respective formulae.
Paily et al. (1974) also provide a closed form solution for the
transient case of equation (10) for linearized surface heat exchange and
particular initial boundary conditions. However the specific initial and
boundary conditions assumed by these authors are not appropriate for the
temperature regime for water released from a dam. Paily et al. (1974)
17
are interested in the temperature regime in a flowing river with a heat
source at x = o in which the entire river including upstream (x < o) is
subject to atmospheric heat transfer. That is to say, the water arriving
at x = o from upstream is changing temperature due to atmospheric forcing.
The application of the Paily et al. (1974) model is to temperature
prediction in a river with a thermal effluent injected at x = o. Therefore
they assume that the boundary temprature T(x = o, t) is not constant, but
instead, equals the sum of the inflow temperature To plus the transient
river response to uniform atmospheric heat transfer. This boundary
condition is not appropriate for the water released from a dam. Water
released from the dam is at a constant temperature since this water comes
from depth below the ice cover, and reservoir water under the ice cover
has very little if any diurnal temperature variation. During breakup or
during intense wind mixing, or when alternative outlets from the dam are
used, the released water temperature will vary, but the released water
temperature cannot be predicted from a simple river temperature model.
It is essential that a reliable reservoir temperature model be used to
define the outflow temperature. In the present analysis we consider the
outflow temperature as given either through measurements or by prediction
from a reservoir model.
Other analytic solutions
An interesting and useful analytic solution can be found for the
problem of the transient response of the river to periodically varying
meteorological conditions. The meteorological condition may represent
diurnal variation in air temperature and/or short wave radiation, or
alternately, seasonal climatic variation. The formal statement of the
problem consists of the governing equation (equation 3) with the atmospheric
heat transfer expression as follows,
I
(14) Q = - Qo - K(T-Tair) + Q sin wt
with the initial conditions,
(15) T(x,o) = To - [(Qo/K) + To - Tair][1-exp(-Kx/pcpUh)]
Note that this initial condition has the expected behavior at x = o,
i.e., T(x = o, t) = To, the constant outflow temperature. Furthermore,
the initial condition is actually the steady state temperature for the
case when Q = 0. The solution then defines the transient river response
to sinusoidal atmospheric forcing. The analytic solution to this problem
is,
(16) T = To - [(Qo/K) + To - Tair]11-exp(-kx/Uh)1
+ AT {sin(wt-a) - exp(-kx/Uh) sin (wt-wx/U-g)}
where k = K/pcp
AT = Q/pcp k2+w2h2
and a = sin-1 [wh/ k2+w2h2]
The form of the solution highlights the role played by the periodic air
boundary conditions. If Q =_ 0, or a constant air temperature is assumed,
the solution reduces to the steady-state case. When Q + 0, the periodic
nature of the temperature distribution in the river becomes evident. The
river temperature lags the air temperature by the phase angle a. This
Phase lag is directly proportional to river depth and inversely proportional
to the surface heat loss coefficient, matching the intuitive expectation
for river temperature adjustment to air temperature variation. That is,
shallow rivers (h + 0) cool faster than deeper rivers with the same
discharge, and rapid heat transfer (k >> 0), which occurs for example
with high winds, is characterized by rapid temperature adjustment. An
19
estimate of the typical diurnal adjustment time lag for winter conditions
may be found by assuming reasonable values for K, w and h: w = 27r/(24.3600)
s-1, h = 3 m, and k = 7 • 10-6 m/sec. This value of k corresponds to a
coefficient of Tair in the Dingman-Assur formula (Table 2) equal to 30
W/m2. These parameters suggest a daily time lag between air and water of
about 5.8 hours during winter conditions. A similar estimate can be made
for a seasonal time lag when an annual period is assumed for the air tem-
perature; this estimate suggests a time lag of about 5 days.
A still more general transient solution may be found for the case
where the atmospheric heat transfer can be represented by a sum of
periodically varying terms of arbitrary frequency and magnitude. This
boundary condition may represent the combination of diurnal and seasonal
variation in air temperature, and in short wave radiation or other
parameters, or it may represent a complex transient surface heat flux
determined from measured values by harmonic analysis. For this general
case the heat transfer expression is,
N
(17) Q = -Qo - K (T-Tair) + E Qi sin (wit+oi)
i=1
and the initial conditions are given by equation (15). The analytic
solution is,
(18) T = To - C(Qo/K) + To - Tair][1 - exp(-kx/Uh)]
N
+ E oTi{sin(wit+oi-si)-exp(-kx/Uh)sin(wit+oi-wix/U-si)}
i=1
where oTi = Qi/Pco k2+wih2
and Bi = sin-1 Cwih/42+w?h2]
20
Each phase si can be calculated independently, and each phase lag is
directly related to the period of the respective heat flux fluctuation.
The amplitude of the periodic temperature waves in the river, oTi is
inversely proportional to the forcing frequency; i.e., short period fluctu-
ations in air temperature are hardly felt in the river and longer period
fluctuations are strongly impressed upon river temperature. In all cases,
the amplitude of periodic temperature waves in the river is inversely
related to river depth, and if river depth is very small, that amplitude
approaches the amplitude of air temperature variation (Qi/K).
Finally, a slightly more general transient solution may be found for
the case where the atmospheric heat transfer varies in a known way as a
Polynomial function of river distance. This boundary condition may represent
a spatially varying air temprature because of lapse rate, weather pattern,
or systematic change in radiative heating. For this general case the heat
transfer expression is,
N — M
(19) Q = -Qo - K(T - Tair) + E Qi sin(wit+(Di) + E gixi
i=1 i=1
where qi represent the known longitudinal variation, and the initial
conditions are given by equation (15). The analytic solution is given by
equation (18) plus a linear summation from the longitudinal variation:
(20) T = To - C(Qo/K) + To - Tair][1-exp (-kx/Uh)]
N
+ E oTi{sin(wit+Oi-Si)-exp(-kx/Uh)sin(wit+oi-wix/U-si)}
i=1
M
+ E gixi+l/(i+1)PcpUh
i=1
21
This solution is the most general closed form expression for the
temperature regime downstream from a dam when the surface heat transfer has
been linearized. Allowable functional forms for the surface heat transfer
(equation 19) can be quite general including differing periodicity of air
temperature and radiation, as well as combinations of diurnal, seasonal and
episodic events and arbitrary persistent longitudinal variation.
Hone of the transient analytic solutions for temperature (equations
16, 18 or 20) can be directly inverted to determine open water length since
the equations are transcendental. However the temperature regime can be
easily calculated as a function of x and t, and, for a particular time, the
open water length determined directly.
It is important to remember the limitations of all the analytic models.
First, none of the analytic thermal models include latent heat exchange
with an ice cover and are therefore only useful for river temperatures
above or equal to 0°C. They can be directly applied only in uniform river
stretches, i.e., with no variation in river width, depth and velocity and
no inflowing streams. The allowable heat transfer functions, although
reasonably general, are based on linearized analysis of higher order surface
heat transfer expressions, and the appropriateness of the linearizations
must be considered. In the following section we shall consider semi -empirical
formulae for surface heat expressions, and discuss some assumptions involved
in the linearization of these formulae. Measurements of open water length
in typical Alaskan conditions will be compared with predictions from the
different linearization expressions.
22
Heat Transfer Expressions
Dingman, Weeks and Yen (1967) provide a very extensive analysis of
the mechanisms of heat transfer to a flowing stream. These authors consider
the following eight heat transfer terms:
(21) Q=QR+Qg+QE+QH+QS+QG+QGW+QF
where QR short wave radiative flux
QB net long wave exchange with the atmosphere
QE evaporative heat exchange
QH sensible or turbulent heat flux
QS heat lost by influx of snow
QG heat added by geothermal transfer
QGW heat added by ground water
QF heat added by friction from stream bottom
The expressions for each of these terms are given in Table 2. Dingman et
al. (1967) were particularly interested in the selection of appropriate
expressions for QE and QH in arctic and sub -arctic conditions. They compared
the formulae of Kohler (1954) and of Rimsha and Donchenko (1957) to cold
region data and determined that the "Russian winter equation" as given by
Rimsha and Donchenko (1957) was the more accurate of the two formulae. We
have included both the Kohler (1954) and the Rimsha and Donchenko (1957)
formulae for QE and QH in Table 2 for comparison.
More recent formulae for water -atmospheric heat transfer have been
given by the Tennessee Valley Authority (1972), Hicks (1972), Pond et al.
(1974), and Holmgren and Weller (1968); however, the first three of these
were devised primarily for temperate regions, and all four were devised for
deep water. McFadden (1974) presented a comprehensive comparison of heat
23
transfer formula with measurements for arctic conditions; the reader is
referred to that report for details of the comparisons. In this report we
shall not attempt to compare in detail the formulae for heat transfer
mechanisms given by each author. Instead we shall make recommendations for
both the full empirical formulae and for the linearized versions of these
formulae based on our calculations and those of Dingman et al. (1967) and
of McFadden (1974). In all these discussions the units of heat flux, Q, are
W/m2.
QR: Short wave radiation
Short wave radiation is always positive and represents a relatively
small component of the heat budget of Alaskan rivers in winter. McFadden
(1974) cites several references which report the daily flux of short wave
radiation near 65° latitude to be less than 5 W/m2 in December. Wendler
(1980) gives the average measured short wave radiative flux as less than
5 W/m2 during November, December and January. This contrasts with lower
latitudes where the short wave radiation is often the dominant mode of heat
transfer to a water surface (e.g., see Fischer et al. 1979). Because of
the reliability and simplicity of short wave radiometer systems, it is
recommended that short wave radiation be measured directly at the site, and
the measured values used in the calculations for open water length. In the
linearization formulae, QR can be added directly to the heat flux terms.
If short wave radiation measurements are not available, then the
following estimation procedure modified from Dingman et al. (1967) is recommended:
(21) QR = 0.892 QRI + 1.397 • 10-4 QR12 [W/m2]
and
24
QRI = QCL (0.96 - 0.61C)
[W/m2]
where C is cloud cover in tenths (e.g., complete cloud cover implies C = 1.0)
alnd QCL is incoming short wave radiation for a cloudless sky. QCL may be
found for various latitudes as functions of season in tablular and graphical
form (see TVA, 1972 and Bolsenga, 1964).
A distinctive feature of the short wave radiation fl ux is the fact
that it is not completely absorbed at the water surface; it penetrates to
some depth depending upon the water clarity and turbidity. The short wave
flux available at a depth y is usually assumed to follow Bouger's Law for
absorption:
(22) QR (Y) = QR (Y=o) exp (-nY)
where n is an extinction coefficient ranging from about 0.2 m-1 for very
clear water,to 4.0 m-1 for turbid water. This implies that in a very clear
shallow stream with depth equal to 1 meter, only 20% of the short wave
radiation is absorbed by the water column, and the remaining 80% penetrates
into the river bottom. At night some of this stored heat flux is released
into the water column, implying an increase in geothermal heat flux QG
which lags the short wave flux. In sediment laden streams Ti may be even
larger than 4.0 m-1, and therefore virtually all short wave radiation is
absorbed in the topmost meter of the water column. In order to reliably
model the bottom flux it would be necessary to couple the river temperature
model to a ground thermal model. However, due to the fact that QR in late
fall is only a minor component in the thermal budgets for high latitude
rivers, it is uaualiy possible to ignore heat absorption in the river
bottom. In any case the main effect of bottom heat absorption on river
temperatures would be a lag in the diurnal temperature maximum of the river
25
or a slower decrease in river temperature in the evening. Quantitatively
QR will represent less than 5% of the overall river heat budget from late
fall through early spring; therefore the lagged release of heat from bottom
sediments may equal 4% of the river heat budget in clear streams. In this
report we will not propose a mathematical model which couples predictions
of the ground thermal regime to predictions of water temperatures. For
rivers deeper than 2 meters and in sediment laden streams, we recommend
assuming that QR is entirely absorbed by the river, unless it is critical
at the particular site to determine the diurnal variation in water tempera-
ture. For clear shallower streams we recommend that an experimental study
be undertaken to determine the diurnal lag in river temperature due to
gradual release of stored radiative heat in the river bottom.
QB: Net long wave radiation exchange with the atmosphere
Net long wave exchange with the atmosphere consists of the outgoing
long wave radiation emitted from the water surface QW plus the net incoming
radiation from the atmosphere Qp:
(23) QB = -QW + Qq
The net long wave exchange may be measured directly at the site. If these
measurements are not available, then the long wave exchange may be estimated
by semi -empirical formulae relating QB to water and air temperatures. The
radiation from the water surface is modeled by the Stefan formula,
(24) QW = eWa(T + 273)4
where eW is the emissivity of water (=0.97), a is the Stefan-Boltzman
constant (5.67 • 10-8 W/m2K4) and T is the surface water temperature in °C.
This formula is widely accepted in the literature and is recommended here.
26
There has been some speculation that at the time of ice formation a thin
supercooled layer of water may exist on the river surface. While this
assumption may be valid in quiescent ponds, it has been shown to be unfounded
in turbulent rivers (Osterkamp et al., 1983). For rivers with mean velocity
greater than about 0.6 m/sec the surface water temperature may be assumed
to be equal to the mean river temperature.
Usually the atmospheric radiation can be modeled by a Stefan formula
(25) QA = f(e,C,H,a) v (Tair + 273)4
where f is a function of air vapor pressure (e), cloud cover (C), cloud
height (H), absorptivity of the water surface (a) and Tair is the air
temperature in °C at a specific height, usually 10 meters. McFadden (1974)
has discussed various expressions for f in some detail, and proposed a
complex formula especially for cooling ponds which includes an additional
dependency on the cooling pond shape factor. McFadden (1974) also compared
long wave radiation data at a site with ice fog with the predictions of
long wave radiation determined by the formulas of Brunt (1944), Angstrom
(1920), Elsasser (1942) and Andersen (1952), and used a correlation technique
to modify these formula and thereby improve the agreement with the data.
We recommend McFadden's (1974) modified version of the Andersen (1952)
formula. The Anderson equation (1952), both in the original format and in
the modified version, exhibited the minimum standard error of all those
investigated. The Andersen equation (1952) was also adopted in the Dingman,
Weeks and Yen (1967) river thermal model. The modified version of the
Andersen equation is:
(26) QA = E.814 + .11C exp (-.19H)
+ ea (.0054 - .000594 C exp (-.197H))] a (Tair + 273)4
27
where H is cloud height in km and ea is vapor pressure of the air in mb.
QE: Evaporative heat flux QH: Turbulent heat flux
Semi -empirical formulae for turbulent heat flux are usually written in
the form,
(27) QH = (A + Bw)(T-lair)
where w is wind speed, and A and B are empirically determined parameters.
There is an extensive core of literature related to the determination of A
and B (e.g., Friehe and Schmitts, 1976; Kohler, 1954; Rimsha and Donchenko,
1957; TVA, 1972; Hicks, 1972; Kays, 1966). The form of the equation models
the intensification of convective or turbulent heat transfer by strong
winds and increased temperature difference between the air and water. In
addition, the parameter A assures upward heat transfer from a water surface
which is warmer than the air even when the wind velocity is small. This
situation frequently occurs in interior Alaska where air temperatures 300
below water temperatures may exist with no wind. Under these conditions
the air is buoyantly unstable, and strong vertical motion in the form of
thermal plumes or buoyant convective cells may develop, facilitating surface
heat transfer.
Evaporative heat loss QE occurs when there isa net upward transport
of vapor from the water surface; the heat loss is the product of the specific
heat of the vapor and the evaporation rate. There is extensive literature
on evaporative heat loss (e.g., Hicks, 1972 and 1975; TVA, 1972; Friehe and
Schmitt, 1976; Anderson, 1954; Pasquill, 1949; Rimsha and Donchenko, 1957;
Devik, 1964). It is usually assumed to be linearly proportional to the air -
water specific humidity difference and is modeled by equations of the form,
(28) QE = (C + Dw) (e - eair)
a3
where w is wind speed, C and D are empirically determined parameters, a is
the saturated vapor pressure of air at the temperature of the water, and
eair is the vapor pressure of the air at a specific height, usually 10
meters. It should be noticed that the transfer of water vapor or any gas
across the water surface is a complex problem and the subject of intense
recent research (e.g., see Brutsaert and Jirka, 1984).
Dingman et al. (1967) and McFadden (1974) reviewed several models for
QH and QE, and compared the predictions of these models with data from
arctic conditions. Both concluded that the Rimsha-Donchenko (1957) formulae
for QH and QE more accurately predicted turbulent and evaporative heat
exchange in arctic conditions than did other models under consideration.
The Rimsha-Donchenko formulae are given here:
(29) QH = [3.87 + 0.17 (T - Tair) + 1.89 w ] (T - Tair)
(30) QE = [6.04 + 0.264 (T - Tair) + 2.94 w ] (e - Bair)
where QH and QE are in W/m2, w is wind speed in m/sec, T is water temperature,
e is saturated vapor pressure at T, Tair is air temperature at 2 meters, and
eair is vapor pressure at 2 meters.
QS, QG, QGW and QF
For the four types of heat transfer, QS, QE, QGW and QF we follow the
recommendations of Dingman et al. (1967).
Latent heat exchange from snow QS is proportional to the snow accumulation
rate A:
(31) Qs = cA [G + Ci (T - Tair)]
where A is given in g/cm day, a is the latent heat of fusion of ice in
cal/g, Ci is the heat capacity of ice in cal/g °C, and c is a dimension
29
conversion constant c = 0.484 [W/m2 : cal/cm2 day]. If snow accumulation
rate is not available, then A may be estimated as a function of visibility
by an expression of the form (Mellor, 1964):
(32) A = 7.85 v"2.375
where v is visibility in km. For consistency the river discharge should
be increased by A times the river width, although the net change in discharge
would be very small.
QG is the geothermal heat flux below the river plus heat released from
bottom sediments and must be determined from local data. The geothermal
flux is expected to be small except possibly in areas of high geothermal
flux (Osterkamp, Kawasaki and Gosink, 1983). As discussed earlier some of
the daily short wave radiation QR may penetrate through the river and be
absorbed into the river bottom. This stored heat may then be released
later in the day, thus delaying the diurnal river temperature decrease.
Accurate knowledge of this effect can only be established by analysis which
couples temperature distribution in the river with temperature distribution
in the bottom sediments. The effect will not be significant (< 4% of total
heat flux) for rivers deeper than 1 meter with extinction coefficient
greater than about 0.2 m-1. If it is essential to determine the diurnal
temperature regime in a very shallow and clear stream, then a more complex
coupled analysis of river and sediment temperature is necessary.
As a general rule, where there are no indications of high geothermal
heat flux, where the river is deeper than about 2 m, and where the short
wave extinction coefficient is greater than 0.2 m-1, the total geothermal
flux QG may be considered negligible.
IN
QGW is the heat added by flow of ground water and smaller streams into
the river. In order to model this heat flux, information is needed on both
the ground water recharge or stream discharge and the temperature of the
inflowing water. Note that QGW affects both the right hand and left hand
sides of equation (3), by changing the heat input and the river discharge
respectively. If stream inflow and temperature measurements are known,
these may be incorporated into the model by relatively small changes in the
finite difference form of equation (3).
QF is the heat added to the river due to friction of the water flowing
over the river bottom. It is generally assumed that the decrease in
potential energy in the river as it flows downhill is compensated for by
the frictional drag at the bottom; subsequently, the drag creates turbulent
eddies which, through the turbulent energy cascade, ultimately cause viscous
heating. The major problem with this assumption is the neglect of the wall
(river bottom) temperature, since if the river bottom is colder than the
bulk river temperature, frictional heating will be directed downward into
the sediment (Schlichting, 1968). Therefore, the model for frictional
heating suggested here and in Dingman et al. (1967) or Starosolszky (1970)
should be considered an upper limit to heat flux by frictional heating of
the river.
The relation between bottom shear stress and the change in potential
energy of a volume of water is given by standard hydraulic theory (Henderson,
1966). The shear stress at the river bottom is,
(33) Tw = PwghS [kg/m-secZ]
where pw is water density in kg/m3, g is the gravitational constant in
m/secZ, h is river depth in m, and S is the slope of the water surface.
Then the heat flux generated by this stress is (Ince and Ashe, 1964),
31
(34) QF = U Tw = PwgUhS
For steep rivers both U and S may be high, suggesting that frictional
heating may be a significant fraction of the total heat transfer Q. For
example, for h = 3m, U = 2m/sec and S = 10-3, QF = 60 W/m2, and for water
at 0°C, the long wave radiative flux QW = 305 W/m2. Even if it is assumed
that half the frictional heating is directed upward into the water, QF
represents at least 10% of the long wave radiation and therefore should be
included in the total budget. It should be noted that Dingman et al. (1967)
suggest that QF is insignificant while Starosolszky (1970) recommends that
QF be included in the heat budget. QF is relatively easy to estimate for a
given river reach, and its magnitude may be included in the governing
equation (equation 3) as an additive constant, posing no real complication
to the solution of the governing equation. We suggest including QF when
the river slope is greater than about 10-4
Linearization formulae
The long wave radiation from the water surface (equation 24) and the
turbulent heat flux (equation 29) depend nonlinearly upon water temperature;
due to this fact an analytic solution of equation 3 is generally not
available. However, it is possible to solve equation 3 when all heat fluxes
are expressed as linear combinations of water temperature and other
parameters, as demonstrated by the solutions given in equations 4, 8, 11,
16, 18 and 20. Therefore, several authors have determined linearized forms
of several terms in the heat budget, specifically Q8, QH and QE. It is
assumed that since the remaining heat flux terms, QR, QS, QG. QGW and QF
are not dependent upon water temperature, their cumulative effect is
equivalent to an additive constant in Q, i.e., they are simply added to the
linearization constant Qo in equation 6:
32
(6) Q = QR + QS + QG + QGW + Qp - Qo - K (T - Tair)
Using regression techniques, Dingman and Assur (1969) determined the
following expressions for Qo and K:
Qo
50.93
+ 11.21 w
(clear
sky)
(35)
-35.28
+ 4.40 w
(cloudy
sky)
K =
16.99
+ 2.05 w
(clear
sky)
17.97
+ 2.22 w
(cloudy
sky)
where w is wind speed in m/sec (the height of the anemometer was not given),
and the units of Qo are W/m2 and of K, W/m2-°C. These expressions are
linearizations of the Dingman et al. (1967) heat flux formulae for long
wave radiation, and turbulent and latent heat flux (see equations 24, 26,
29 and 30).
Other linearization expressions include formulae derived specifically
for a reach of the St. Lawrence River by Pruden et al. (1954):
QB + QH + QE = - 88.91 - 7.5 Tair - 20.87 (T - Tair)
and the formulae given by Asvall (1972) and adapted from Devik (1964):
(36)
136.05
+ 2.09 w
C
= 0.0
Qo = 77.38
+ 2.09 w
C
= 0.5
23.00
+ 2.09 w
C
= 1.0
12.59
+ 1.63 w
C
= 0.0
K' = 9.44
+ 2.41 w
C
= 0.5
10.92
+ 2.05 w
C
= 1.0
where C is cloud cover and K' multiplies - Tair (°C) instead of T - Tair
(°C) as in equation (6). 33
Paily et al. (1974) also determined empirical fits to the Dingman et
al. (1967) formulae for QB, QH and QE by a least squares polynomial approxi-
mation technique. Values of Qo and K determined by Paily et al..(1974) are
given in tabular form in Table 3. These coefficients differ from the set
given by Dingman and Assur (1969) and there are two reasons for the differ-
ences: 1) values of Qo and K from equations (35) are not dependent on air
temperature while the Paily et al. (1974) coefficients are, and 2) the
Paily et al. (1974) coefficients were selected as best fits over the range
of air temperatures -180C < Tair < DOC, while the coefficients in equation
(35) were selected as best fits over the range of air temperatures
-50°C < Tair < 0°C. This latter effect becomes critical for application to
Alaskan rivers. Although the Paily et al. (1974) expressions for Qo and K
are reliable within their range of applicability, they deviate from the
complex Dingman et al. (1967) formulae when air temperatures are substan-
tially below -18°C.
Since only discrete values of Qo and K are given in the Paily et al.
(1974) report, we have determined the following interpolation formulae
which agree with their tabular values with a maximum deviation of 1.7% and
an average deviation less than 0.5%.
(37) K = 14.795 + 3.45 w - 1.11 • 10-2 q + .540ITairl - 1.12 • 10-31Tair12
Qo=-32.796 + 18.513 w -.952q + (24.290-K)ITairl + 4.016 • 10-21Tair12
+ 2.696 • 10-51Tair14
where w is wind speed in m/sec, q is humidity in % (100. is saturated), and
Tair is air temperature in *C. Note that K must be calculated first, since
it is used in the evaluation of Qo.
34
The linearized heat transfer from equations 35, 36 or 37 represents QS
the sum of long wave radiative exchange plus evaporative and turbulent heat
flux, as given by the approximate expression:
QS = Qw - QA + QH + QE = Qo + K (Tw - Tair)
Both left and right hand sides are functions of Tw the water temperature.
The agreement between the different linearization formulae and the "exact"
formulae may be tested for an appropriate range of river temperature and
atmospheric conditions. We shall plot both sides of the expression for the
range of values, 1.0 z Tw < 4.0, with the terms Qw, QA, QH and QE calculated
from equations 24, 26, 29 and 30 respectively and Qo and K from equations
35, 36 and 37. In Figure 4, we assume zero wind velocity and clear sky or
zero relative humidity. The "exact" values of QS (as given by equations
24, 26, 29 and 30) are shown for air temperatures Tair = {0, -10, -20,
-30, -40 } by the vertical braces. (Note that in some cases the vertical
braces have been shifted slightly left or right for clarity). Since the
Devik (1964) formula (equation 36) is not a function of water temperature,
only a single value of QS may be plotted at each air temperature. The
Dingman and Assur (1969) expressions or equation 35 yield the range of QS
denoted on Figure 4 by the solid bar; and the Paily et al. (1974) expressions
or equation 37 yield the range of QS denoted by the open bar. There is a
clear tendency for equation 37 to diverge from the exact solution, becoming
less accurate as the air temperature decreases below -20°C. Equation 36
(from Devik (1964)) also diverges from the exact solution with decreasing
air temperature. The Dingman and Assur (1969) expression or equation 35
provides the best overall estimate of the exact solution.
35
1200
1000
LEE
600
400
mil]
I "exact" solution
Dingman and Assur (1969)
p Paily et al. (1974)
x Devik (1964)
5
Tair
x
Figure 4. Comparison of complete heat flux equations with linearized
approximations for zero wind velocity.
W.
In Figure 5, we assume a wind velocity of 10 m/sec and clear sky or
zero relative humidity. Again the Dingman and Assur (1969) expression or
equation 35 provides the best overall agreement with the exact solution,
keeping pace with the intense heat transfer associated with high wind -low air
temperature. The Devik (1964) expression consistently underestimates the
heat transfer rate, and the Paily et al. (1974) expression diverges from
the exact solution beyond about -15°C. The Dingman and Assur (1969) ex-
pressions or equation 35 are significantly more accurate than the others
at low air temperatures.
Comparison with Data: Example 1
Studies of ice -free reaches downstream from a warm discharge seldom
contain complete meteorological and hydrological conditions. For example,
Carlson et al. (1978) do not report air temperature, wind velocity, cloud
cover or radiation data. However, this information is sometimes available
from local weather records. The information should be acquired from weather
stations as close as possible to the study site to minimize errors in the
determination of heat loss and whenever possible, at the study site.
Carlson et al. (1978) specify that the data were recorded at the MUS Power
Plant in Fairbanks during December 1971. Thus, referring to Fairbanks
meteorological reports for this period, it is possible to calculate heat
loss with the different linearization models, and then to compare calculated
and measured open water areas. In particular we wish to compare the
linearization formula of Dingman and Assur (1969) (equation 35), Asvall
(1972) (equation 36), and Paily et al. (1972) (equation 37) and the analytic
solutions for river temperature as given by equations 4, 9 and 13.
For the month of December 1971, discharge rates for the Chena River and
the MUS Power Plant are approximately 80o ft3/sec and 25 ft3/sec respectively.
37
1600
1600
1400
1000
L
d
t-
x
U-
u..
AIR
400
200
;act" solution
igman and Assur (196
ly et al. (1974)
,ik (1964)
Figure 5. Comparison of complete heat flux equations with linearized
approximations for wind velocity = 10 m/s.
If complete mixing near the discharge is assumed, the effective temperature
rise becomes:
To = 10° (25/800) = 0.31°C
when the effluent temperature is 10°C.
The MUS Power Plant uses two different types of discharge. The first
and more conventional mode of discharge is the subsurface diffuser. When
this technique is employed, there is considerable turbulent mixing near the
diffuser. Consequently, mixing may be assumed to be complete, and the one-
dimensional assumption implicit in the models is appropriate. Measurements
of the open water length in the Chena when the subsurface diffuser was in
use in December 1971 indicated an ice -free area of 15 acres (Carlson et
al., 1978).
Measurements of the open water length were also made when the second
type of discharge, the surface dispersion field, was in use. In this mode,
the effluent enters the stream at the surface through a series of pipes
with little turbulent mixing. Hence, the dispersion field operates as a
surface spreading scheme. Heat transfer is rapid, since heat loss is pro-
portional to the temperature differences between the water and the air. As
expected, the surface dispersion scheme produces smaller ice -free area; in
December 1971; average areas of 8 acres were measured. The surface dispersion
field is characterized by strong vertical and lateral temprature gradients.
The existence of steep temperature gradients invalidates the assumptions
implicit in the one-dimensional models, indicating that comparisons of
prediction schemes with existing data are appropriate for only the subsurface
diffuser.
Fairbanks weather data for the month of December 1971 was compiled by
the Environmental Data Service, National Oceanic and Atmospheric Administration.
39
Average temperature for the month was -21°C; wind speed, 3.7 mph = 1.65
m/sec; cloud cover, 0.7. Air temperature was about normal for December,
and cloud cover, heavier than normal. We have used the above values of the
mean air temperature, wind speed, and cloud cover in each of the linearization
formulae (equations 35, 36 and 37) and have determined the parameters Qo
and K listed in Table 4. For equations 35 and 36, interpolation between
cloud covers of 5 and 10 was required. Equation 37 contains no functional
dependence on cloud cover but is dependent upon humidity, which, for this
test case was assumed to be 10%. There is a surprising lack of agreement
of the calculated values of Qo and K between the different models, particularly
between the Dingman and Assur (1969) and the Paily et al. (1974) formulae
which both represent linearizations of the Dingman et al. (1967) formulae.
However it is encouraging to note that the open water areas predicted by
these two expressions are in good agreement and bracket the measured open
water length of 15 acres. The Paily formula is somewhat sensitive to the
selection of humidity, and when a humidity of 90% is assumed, the predicted
open water area is 16.6 acres or identical with the Dingman and Assur (1969)
prediciton. We have determined the open water area for the Paily (1974)
model using E = 4.51 m2/sec which is the value recommended by Paily et al.
(1974) and E = 0 to test the sensitivity of the longitudinal diffusion
term; as expected the difference is negligible. It should be noted that
4.51 m2/sec is about twice the value of the longitudinal dispersion coefficient
calculated by using the Fischer et al. (1978) expression for dispersion
coefficient. As previously suggested, longitudinal diffusion of heat
becomes important only for slow rivers, in particular for conditions in
which the ratio KE/pcpA is about 0.1 or greater (see equation 13). For
this example when E = 4.51, the ratio is 2 • 10-5. Whenever the ratio
M
Table 4. Calculations for Example 1
(measured open water area = 15 acres)
Model
Equations for
length Q K
Q (W/mZ)
equation 6
Q (W/m2)
K (W/m2 °C)
Area
E (m2/sec) 104 m2 jAcres
Asvall (1972)
4
36
-348.22
59.08
13.77
- 8.71
21.51
Dingman and
-447.46 +
Weeks (1969)
9
35
21.25 T
1.214
21.25
- 6.73
16.62
Paily et al.
-521.27 +
(1974)
13
37
31.22 T
-134.35
31.22
4.51 5.76
14.24
Paily et al.
-521.27 +
(1974)
13
37
31.22 T
-134.35
31.22
0 5.76
14.24
KE/PcpA is less than 0.1, we recommend the simpler Dingman and Assur
(1969) formula for open water length (equation 9) over the Paily et al.
(1974) formula (equation 13).
Comparison with Data: Example 2
Data from W.A.C. Bennett Dam on the Peace River in British Columbia
can also be used to compare the accuracy of the various models. Measurements
of open water length downstream from the dam are available for the winters
of 73/74, 74/75, 75/76, 76/77 and 77/78 (British Columbia Hydro and Power
Authority, personal communication), with the length varying between 60 and
203 miles during these years. However, it is difficult to apply the
theoretical models for open water length directly to the Peace River data
for several reasons related to the assumptions implicit in the models: 1)
the models (equations 4, 9 and 13) are all steady state cases, implying
both steady discharge and meteological parameters; 2) the Peace River
meanders in the region of interest and a typical river width is difficult
to determine; 3) the closest meteorological data come from Fort St. John
about 15 miles downstream from the dam, and meteorological data from this
location often disagree substantially with data from the next downstream
source, Peace River some 60 miles from the dam.
Nevertheless it is useful to determine rough estimates of the open
water length for the five winters by using "mean" meteorological and hydrau-
lic parameters at the site. Discharge and average outflow temperature are
known (British Columbia Hydro and Power Authority, personal communication).
We assume a constant river width of 200 meters. A mean air temperature is
probably the most subjective choice since it is not clear whether the period
of averaging should include the entire winter or a specific period proceeding
the time of the minimum open water length. We have chosen a degree day
method to determine the mean air temperature. Using the measured air
42
temperatures from Fort St. John assembled by British Columbia Hydro we
divide the maximum accumulated degree days by the number of degree days;
these "mean" air temperatures are listed in column 4 of Table 5. The
minimum measured open water length and the measured average winter discharge
(Nov -Feb) were provided by British Columbia Hydro. We have determined mean
winter wind velocity and cloud cover from the Meteorological Data for Canada.
Humidity was not available, but we have assumed a constant 10% throughout
the winter which may be slightly high considering the cold air temperatures
4
at the site. Using this combination of averaged meteorological and hydrological
data, we determined Qo and K according to equations 35, 36, and 37 and
applied the heat loss coefficients to the appropriate models of open water
length, i.e., we used equations 36 and 4 to determine open water length
according to Asvall (1972); equations 35 and 9 according to Dingman and
Assur (1969); and equations 37 and 13 according to Paily et al. (1974) with
E set equal to 0.0. These calculated open water lengths appear in columns
8, 9 and 10 of Table 5. Clearly the Asvall (1972) formulae consistently
overpredicts open water length. The Dingman and Assur (1969) and the Paily
et al. (1974) formulae are in substantial agreement, as should be expected
considering that both heat loss formulae are linearizations of the earlier
Dingman et al. (1967) equations. It appears that equation 35 is in better
agreement with the data than equation 37, with the former yielding an aver-
age deviation from measured open water length of 13 miles and the latter,
an average deviation of 15 miles. However, considering the assumptions
employed in determining an "average" air temperature, wind speed, discharge
etc., the difference is not significant. It is worth noting that the
Asvall (1972) formulae overpredicts open water length both in this example
and in the earlier example, and that equations 35 and 37 predict the same
trend in open water length as is found in the measured open water length.
43
TABLE 5 - Example 2s Peace River
Measured Open
Measured
Mean Air
Water Length
Discharge
Temperature
Year
(miles)
(m3/sec)
(°C)
73/74
60
1201.3
-14.7
74/75
103
1581.5
-9.7
75/76
98
1213.4
-10.1
76/77
203
1572.3
-5.0
77/78
102
1725.2
-12.4
A
Mean Wind
Velocity
(m/sec)
Cloud
Cover
Humidity
M
Calculated Open Length (miles)
with Heat Loss Expression From
Eq. 35 36 37
Dingman & Assur Asvall Paily
4.01
.70
10
72.7
90.6
72.0
5.22
.62
10
120.3
142.9
123.5
4.91
.70
10
94.0
112.9
94.5
5.71
.63
10
190.9
214.8
185.2
3.9
.62
10
120.5
146.5
120.8
s
Based on the two examples for the Chena River and the Peace River, we
recommend either the Dingman and Assur (1969) heat loss expressions (equation
35) or the Paily et al. (1974) expression (equation 37) when a simplified
version of surface heat transfer is to be used and when the air temperature
is warmer than -19°C. Since the Paily et al. (1974) formulae have not been
tested below about -21°C, and since they were derived explicitly for
temperatures greater than -19°C, we suggest using equation 35 exclusively
whenever air temperatures below -190C are possible.
Finite difference methods
In the foregoing sections of this report, we have primarily examined
steady state and analytic models for the temperature regime in a river.
These are important tools for environmental assessment for known meteorological
forcing. That is, for design purposes when only the large scale hydrologic
conditions and climatic variability are known, the analytic models provide
useful estimates of the expected open water length. However for operational
purposes on a day to day basis, a finite difference on finite element model
is needed to simulate the site specific variations in river hydrology, and
variations in discharge and meteorology.
General finite difference models for arbitrary surface heat loss have
been given by Dingman, Weeks and Yen (1967), by Ashton (1979), and more
complicated models for coupled hydrodynamic and thermodynamic analysis have
been given by Chaudry et al. (1983) and Bowles et al. (1977). The first
model is for steady state conditions and therefore, except for allowing the
non -linear surface heat transfer expressions (see equations 26 and 29),
offers no real advantage over the analytic models when reliable linearization
formulae are used (e.g., equations 35 and 37). The coupled hydrologic
thermal models of Chaudry et al. (1983) and Bowles et al. (1977) represent
45
a very sophisticated approach to the analysis of river temperature. However,
at this time we do not recommend this level of modeling for application to
Alaskan rivers due to the scarcity of the necessary hydrological data. It
should be noted that in order to simulate continually varying discharge it
is necessary to use the coupled hydrologic thermal models; for gradually
changing discharge the thermal models described in this report should give
reasonable estimates of the water length. Furthermore, if a coupled
hydrologic thermal model were to be used it is essential that the surface
heat transfer expressions be based on formulae appropriate for arctic
conditions as discussed earlier. It would be necessary to change the
thermal portion of the model to follow the suggestions given earlier for
surface heat transfer. The finite difference model from Ashton (1979)
allows daily variations in meteorology and local variations in river width
and mean velocity. Variations in discharge, both from changes at the dam
and from stream inflow downstream are not included in the Ashton (1979) mo-
del; ice dynamics are also not included. However, the Ashton (1979) model
provides a useful framework for the study of transient effects, and is
easily modified to include a variety of site specific adaptations. A copy
of the Ashton (1979) computer model is included in Appendix A. In the
present section, we shall briefly describe the model, its limitations and
assumptions and discuss refinements which could be included.
The Ashton (1979) model is a numerical solution of equation 2 in which
the width b, depth h and mean velocity U are allowed to vary with downstream
distance x; Q the surface heat exchange is calculated according to a
simplified air temperature -wind velocity formula, and no other heat transfer
terms S' are included. The river discharge (D = Uhb) is assumed to remain
constant over the calculation period. The simulations are done in a
M
Lagrangian reference frame, following a fluid parcel downstream; therefore
downstream distance steps are set internally depending on local velocity.
The inclusion of local river geometry in terms of variable U, h and b
is clearly an improvement over analytic models in which these terms are
held constant. This feature is important for the heat balance since the
net heat loss is directly proportional to river width b. presently, Ashton
(1979) assumes that the river hydrology h(x), b(x) and D are known, and
calculates U(x) locally assuming a rectangular basin. In principle, any
measured river width and depth distribution, including the total river
width in a braided section of the river, may be used as data. The extension
of the model for alternate basin geometry (e.g., trapezoidal or multi -channel)
is straightforward requiring only the inclusion of a flag variable to
define basin geometry at each subreach (alter statement 40 in the model to
define area discharge relation and statements 10 and 20 to define the basin
geometry flag).
In its present form, the model features a simplified expression for Q
which is calculated daily based on mean air temperature and wind velocity.
For application to Alaskan rivers we recommend using the linearized
expressions from Dingman and Assur (1969) (see equation 35). These
expressions are only slightly more complicated than those in the Ashton
model, and programming changes to the model would be minimal (alter statement
87).
If small streams enter the main channel, they will increase river
discharge and alter the thermal balance. In principle, this effect can be
handled by solution of equation (1) in which the other heat sources are the
known stream input in terms of stream water and ice discharge and water
temperature. The solution procedure will "step downstream", and a new
47
increased discharge calculated for the next calculation reach. As a
practical matter, information on small stream discharge, temperature and
particularly ice content, is usually not available. In addition, if the
stream inflow is at a different temperature from the river, it will also be
at a different density, and subsequently will not mix instantaneously with
the main flow. However, a reasonably good literature exists regarding
theoretical and field examinations of transverse mixing in rivers, and a rough
estimate can be made of the distance required for complete transverse mixing.
If this distance is substantially°less than the estimated open water length
(from equation 9), then the thermal effects of inflowing streams can be
simulated by adding discrete heat and mass sources to the governing equations
at the appropriate locations. If the mixing distance is of the same order
as the open water distance, then a two-dimensional model involving downstream
convection of heat and cross -stream diffusion of heat must be used. An
example of such a two-dimensional model is given by Ashton (1979) and is
listed in Appendix B.
The determination of whether a two-dimensional model is required hinges
on the estimate for transverse mixing length Lt. Transverse mixing for
open channel flow is determined by the transverse mixing coefficient et
where et = chU* and h is depth, U* is friction velocity and c is a scale
constant (Fischer et al., 1978). Ashton (1979) assumes c - 0.2, but a more
recent compilation of typical values suggests c = 0.6 is more appropriate
for the winding rivers characteristic of Alaska. Diffusion theory predicts
that a passive tracer will diffuse as (time)1/2:
/N
where a2 is the variance of the diffusion and a is the appropriate
diffusion coefficient. Fischer et al. (1979) suggest that a reasonable
criterion for substantially complete transverse mixing is when the tracer
is diffused to within 5% of its mean value everywhere on the cross-section.
Assuming a Gaussian distribution for the tracer, this occurs when o = 0.5b
where b is river width. The time required for this to occur following a
fluid parcel is Tt m a2/et = 0.25 b2/et, and the downstream distance
travelled is Lt = TtU = .25b2U/et = 0.25 b2U/(.6U*h). Since a reasonable
approximate value for U* - 0.1U, we have
(38) Lt = 4b2/h
As a rule of thumb, the river and side stream inflow are well mixed at
the distance Lt. If this distance is the same order of magnitude as L the
estimated open water length from equation 9, then a two-dimensional model
is necessary. If on the other hand Lt < 0.1 L, a one-dimensional model is
acceptable.
Assuming that Lt << L for all small streams entering the main river, a
procedure could be devised to alter the one-dimensional Ashton model (1979)
to include these additional thermal sources. The simplest way to do this
appears to be: first, make_ discharge a variable (alter statements 8, 12,
15, 30, 40) in particular defining the subreach velocity by the reach
characteristics (statement 40 becomes U(J) = DISCH(I)/(SB(I)* SD(I))), and
second, define a new variable giving the temperature increment from the
small stream and insert it where the subreach characteristics are defined,
say after statement 46. It would be of the form TINC(J) = TINFL(I)*(DISCH(I)
-DISCH(I-1))/DISCH(I), and TINFL is temperature of the stream water. This
would have the effect of adding the additional heat only where the stream
enters and weighting it proportional to the stream discharge. Finally, the
third step would require that TINC(J) be added to the local temperature, by
altering statement 80 to read TWOUT(J) = TWOUT(J) + DELTW + TINC(J). A few
additional alterations would be required to change format statements, and
to zero unaffected TINC(J), etc.
A discussion of the Ashton (1979) finite difference model is incomplete
without reference to the two modes of thermal equilibrium used in the model.
The first mode states that length of the open water reach is coincidental
with the position of the zero degree (°C) water temperature. This is the
mode that has been assumed throughout this report, and is implicit in the
analytic solutions (see equations 9 and 13). The Ashton model uses this
definition (Tw(L) = 0°C) to define L when the ice cover is newly forming or
melting.
When an ice cover is already present, an alternative criterion for ice
edge position is adopted in the Ashton (1979) model which is referred to as
the equilibrium criterion. The equilibrium criterion is derived from the
heat balance equation through the ice cover:
(39) (Tm-Tair)/(n/ki + 1/hia) - hiw (Tw - Tm) = Pi a do/dt
where n is ice thickness, Tm is the melting point (Tm = 0°C), Tair is air
temperature, Tw is water temperature, ki is thermal conductivity of the ice,
Pi is ice density, a is the heat of fusion and hia and hiw are the
ice/air and ice/water heat transfer coefficients respectively. This equation
in turn is derived from the energy balance at the water/ice interface:
(40) �i - �wi = Pi a do/dt
50
where �i is the heat flux by conduction through the ice and Owi is the
heat flux from the water to the ice. It is assumed that �i = pia where
pia is the heat flux from the top surface of the ice to the atmosphere.
The equilibrium criterion for the leading edge of the ice is determined
from the condition that n = do/dt = 0 in equation (39). This condition
then defines the equilibrium temperature of the water at the leading edge:
(41) Twe = -hia/hiw Tair
Clearly TWe is not in general equal to 0°C; in fact, TWe < 0 whenever Tair
> 0 and TWe > 0 whenever Tair < 0. The first condition is clearly meaningless
and therefore in the model the equilibrium criterion is inoperative whenever
Tair > 0; the more standard zero isotherm criterion is adopted for the
position of the ice edge if Tair > 0. The equilibrium criterion is used in
the model only when a presently existly ice edge is growing or decreasing
in length and the air temperature is less than zero; under all other
conditions including the first formation of the ice, the zero isotherm
criterion is used.
There are several basic problems associated with the use of the
equilibrium criterion. This criterion is determined from equation 40 with
the additional assumption that the conductive heat transfer through the ice
exactly balances an expression for ice/atmosphere heat transfer. It should
be noted that: 1) equation 40 neglects the possibility of surface melt,
defining all melting on the water/ice interface; 2) the expression used for
conductive heat transfer across the ice is the steady state linear formula
(�i =-kiTs/n where Ts is the top surface temperature of the ice) which
is not realistic during a period of ice growth or decay; 3) the expression
used for the ice/atmosphere heat transfer pia is a simple linearization
51
formula (Oia = hia (Ts - Tair)) and thus Oia effectively ignores effects
of melt puddles and short wave radiative exchange; 4) equating pia to
conductive heat transfer 01 is a questionable assumption, particularly
when the ice is wet and Ts is close to DOC while Tair « 0 °C; and finally,
5) there are no data available which would indicate that the equilibrium
criterion is actually an improvement on the zero isotherm criterion.
The zero isotherm criterion may be implemented as the only criterion
by the following program modification. Between statements 105 and 106 add
the statement,
IF (ETA(J).GT.O. .AND. TWOUT(J).GT.O.) ETA(J) = 0.
Finally, note a correction to the Ashton model; statements 103 and 104
should be reversed.
52
Conclusions
This report has reviewed several approaches to the problem of the
determination of the length of open water downstream from a dam or thermal
source in winter.
Open water lengths have been predicted by several Russian studies by
statistical approaches based on local data, and are appropriate only for
particular locations. More general types of analyses for rivers were
introduced by Asvall (1972), Dingman, Weeks and Yen (1967), Paily et al.
(1974) and Harleman (1972). These analyses are based on semiempirical
formulae for the rate of heat transfer from an open water surface to the
atmosphere by evaporation, radiation and sensible heat transfer, and
possibly including infiltration of ground water and frictional effects.
The heat transfer expressions are applied to the one-dimensional equation
for conservation of thermal energy in the river, yielding solutions which
predict temperature in the river. Since the formulae for radiative heat
transfer are non -linear functions of water temperature, in general numerical
methods must be used to determine temperature distributions in the river.
However, there are several "linearized" versions of the surface heat
transfer expressions, including those by Paily et al. (1974) and by
Dingman and Assur (1969). The application of these linear heat transfer
expressions greatly simplifies the mathematics involved in the determination
of river temperatures, and in fact, allows closed form analytic solutions
to be found for a limited number of boundary conditions. The most obvious
of these analytic solutions is the steady-state case, given by Dingman and
Assur (1969) and defined in this report by equation 9.
53
We compared the steady-state solution with measured ice -free area in
the Chena River and in the Peace River for three linearizations of the
surface heat transfer expressions: Dingman and Assur (1969); Paily et al.
(1974); and Asvall (1972). The linearizations given by Dingman and Assur
(1969) and by Paily et al. (1974) were based on the "Russian winter
equation" of Rimsha-Donchenko (1957) and produced the best agreement with
the data. However, since the Paily et al. (1974) linearization formulae
were derived primarily for air temperatures greater than -19°C, the
Dingman and Assur (1969) formulae given by equation 35 are recommended.
Paily et al. (1974) found an additional analytic solution, the
transient response of an intially uniform river temperature distribution
to a given temperature increment at x = o, with constant air temperature
and solar radiation. In equation 20 of this report we introduce a new
analytic solution, the transient response of river temperature to periodic
air temperature and/or solar radiation. The latter analytic solution
provides information on the phase lag between atmospheric forcing and
river response, indicating that this lag increases with increasing river
depth and decreases with surface heat loss rate and is independent of
river width. This closed form solution also includes the effects of
spatially varying air temperature and therefore, provides a general model
for temperature prediction in rivers with uniform flow and uniform cross -
sectional area. Another important use of this transient analytic solution
is for comparison with numerical models. Since the analytic solution is
exact, it provides a reliable gauge for the accuracy of finite difference
or finite element models, thus providing confidence in the applicability
of these models.
54
General finite difference models for arbitrary surface heat loss and
changing river basin geometry have been given by Dingman et al. (1967),
Ashton (1979), Chaudry et al. (1983) and Bowles et al. (1977). For
general applicability in Alaskan rivers where the hydrological data base
is sparse, we recommend the Ashton (1979) finite difference model for
river temperature analysis. The model predicts the transient response
of water temperature for constant discharge, spatially varying cross -
sectional area, and temporally varying air temperature and discharge
temperature. We have discussed several refinements to Ashton model
including arbitrary (non -rectangular) cross -sectional area, the implementation
of the Dingman and Assur (1969) heat transfer expressions, heat flux from
small streams and an alternative criterion for leading ice edge position.
Any of these modifications may be rather simply applied to the Ashton
(1979) model.
None of the models discussed in this report are applicable when
river discharge is changing drastically. In this case, ice movement and
ice front position is a mechanical -hydrodynamic problem, only slightly
affected by thermal changes.' At this time there is no reliable theore-
tical or numerical model available for ice front behavior with rapid
changes in discharge. For clear strategic reasons, field measurements
of these events are rare. In this report we have reviewed the thermo-
dynamic models which are appropriate only for gradually changing discharge
when the ice conditions and water temperatures are controlled by the
discharge temperature and the local meteorology. Comparisons with data
indicate that under these conditions, the appropriate thermodynamic
models yield realistic estimates of open water length.
55
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Laboratory Research Report 206, Part II, Hanover, N.H.
Dingman, S., W. Weeks and Y. C. Yen, 1967. The Effects of Thermal Pollution
in River Ice Conditions -I: A General Method of Calculation, U.S. Army
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Harvard Meteorological Study No. 6, Harvard Univ. Blue Hill Met. Observatory.
F:
Fischer, H., J. Imberger, E. J. List, R. Koh and N. Brooks, 1979. Mixing
in Inland and Coastal Waters, Academic Press, N.Y.
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face fluxes of sensible heat and moisture by the bulk aerodynamic
formulas, J. of Physical Oceanography, 61 801-809.
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No.- 4.
Gotlib and Gorina, 1974. Gidrotekhnicheske, Stroitel`stvo, No. 11.
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ambri ge, ass.
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Coefficients Over Water, Boundary Layer Meteorology, 8, pp. 515-524.
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Over Water Bodies of Different Sizes, Boundary Layer Meteorology, 3,
pp. 201-213. —
Holmgren, B. and G. Weller, 1974. Local Radiation Fluxes Over Open and
Freezing Leads in the Polar Ice Pack, AIDJEX Bulletin, 27, pp. 149-166.
Ince, S. and G. Ashe, 1964. Observations on the Winter Temperature Structure
of the St Lawrence River, Proceedings of the Eastern Snow Conference,
p. 1-13.
Kays, W. M., 1966. Convective Heat and Mass Transfer, McGraw Hill, New York.
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Lake Hefner Studies, U.S. Geological Survey Professional Paper 269,
PP• -
McFadden, T., 1974. Suppression of Ice Fog from Power Plant Cooling Ponds,
Ph.D. Dissertation, University of Alaska, Fairbanks, Alaska.
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Osterkamp, T. E., R. E. Gilfilian, J. P. Gosink and C. S. Benson, 1983.
Water Temperature Measurements in Turbulent Streams During Periods
of Frazil Ice Formation in Rivers, Annals of Glaciology, 4,
pp. 209-215. —
Osterkamp, T. E., K. Kawasaki and J. P. Gosink, 1983. Shallow Magnetic
Induction Meausrements for Delineating Near -Surface Hot Groundwater
Sources in Alaskan Geothermal Areas, J. Energy Res. Tech., 105, pp.
156-161.
57
Paily, P., E. Macagno and J. F. Kennedy, 1974. Winter -Regime Thermal
Response of Heated Streams, ASCE Hydraulics Division, HY4, pp. 531-550.
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1052, pp. 116-140.
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Coefficients for Sensible Heat and Moisture Fluxes, Boundary -Layer
Meteorology, 6, pp. 333-339.
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Report No. MD42, National Research Council of Canada, Ottawa, Canada.
4
Rimsha, V. A. and Donchenko, R. V., 1957. The Investigation of Heat Loss
From Free Water Surfaces in Winter Time. Tr. Leningrad Gas. Gidrol.
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0
Table 1 - Models of open water area
1) Statistical Models
Investigator Relationship for open water area
Gotlib and Gorina (1974) 1) Graphical relationship between length
Godrotekhnicheskoe of open water (L), reservoir discharge
Stroitel'stvo, No. 11 (D) and temperature of discharge (Tw)
L = f (D, Tw) for cold winters
2) For warm -winter conditions, one specific
L = f(D)
Goryunov and Perzkinskiy
(1967) Soviet Hydrology;
Selected Papers, Issue No. 4
2) Semi -Empirical Models
Investigator
Asvall (1972)
Proc. of Banff Symposia
on the role of snow and
ice in hydrology
3) Graphs for transient response of location
of ice edge under warming conditions
L (t)/L (t=o) = f(D,ni,E-lair) where
ni = ice thickness, Tair = air temp.
(Note: This graph is not well labeled.
The location of the ice edge is not de-
fined clearly. Only qualitative infor-
mation regarding the effects of variation
in D and ni may be discerned).
L = 5.5 • 106 (E - Tair)-2
L gives the transient location of ice
edge since t - Tair sums over all negative
degree days
Surface Heat
Loss Definition
Tabular values heat loss
(Q), as a function of
cloud cover (C), wind
velocity (w) and air
temp. (Tair); Q = f (C,
w, Tair)• Q can be
formulated,
Q = ao + al Tair where ao,
al = f (C,w)
Relationship for
Open Water Area
Area (Lb) times sur-
face loss (Q) equals
heat input from reser-
voir
LbQ = Ro
L = Ro/Qb
E'�1
Surface Heat
Relationship for
Investigator
Loss Definition
Open Water Area
Paily, Magagno, Kennedy
Q = -K (T-TE)
Solves the one-
(1974), Jrnl. of
where K is a surface exchange
dimensional partial
Hydraulics Div., ASCE
coefficient
diff. equation for
T is local water temperature
conservation of
TE is "equilibrium temp".,
thermal energy. It
water temp. at which there is
is assumed that
no exchange of heat across
T = T(x,t) only, and
the water surface with the
the equation is
atmosphere.
integrated over a
cross -sectional area.
Since a linear re-
ation is assumed
for tie equation
may be solved analy-
tically, yielding an
expression T = T (x,t)
Dingman, Weeks, Yen
Q = non -linear function of
Numerically integrates
(1967), CRREL Res.
T, Tair, w, e, D, S where
the one-dimensional,
Rpt. 206
new variables are
steady-state partial
e = evaporation pressures
differential equation for
D = discharge
conservation of thermal
S = river slope or
energy assuming negli-
Q = non -linear function of
gible longitudinal
T, Tair. w, D, S
diffusion.
Dingman, Assur (1969)
Q = Q'o + K (T - Tair)
Closed form solution of
CRREL Res. Rpt. 206
Part II
Q'o and K from regression
one-dimensional steady -
analysis of the non -linear
state ordinary differ -
function in Dingman et al.
ential equation (linear)
(1967).
for conservation of thermal
energy.
Table 2 - Surface Heat Transfer Definitions
Investigator Surface Heat Transfer Expressions in W/m2
Asvall (1972) Cloud cover = 0.0, Q = 136.05 + 2.09 w +(12.59 + 1.63 w)ITa
= 0.5, Q = 77.38 + 2.09 w +(9.44 + 2.41 w)IT r
1 0, Q = 23.00 + 2.09 w + (10.92 + 2.05 w)T}a
where Q = [W/m�], w = [m/sec] Tair = ['C]
Paily, Macagno, Graphs of a and n for
Kennedy (1974)
Q =
eT+n
where e =
e (Tair,
w, R.H)
n =
n (Tair,
w, R.H)
and R.H.
= relative
humidity
It is assumed
that
barometric pressure
= 99.6 mb
cloud height
= 3,275 ft
cloud cover
= 6
visibility
= 1.87 miles
Dingman, Weeks, Yen and QR - QB - QE - QH - QS + QG + QGW + QF
QR is heat from short wave radiation
QB net loss of heat by exchange of long -wave rad. w. atmos.
QE heat loss due to evaporation
QH sensible heat loss
QS heat lost by influx of snow
QG heat added by flow of geothermal heat
QGW heat added by flow of ground water
QF heat added by friction on stream bottom
QR = QRI QRR = incoming -reflected short wave radiation
and QRI = QCL 1.17 + .30 (1-C)]
QCL is incoming short wave radiation
C is cloudiness in to the 2
QRR = .052 QRI - 3.28 • 10-� QRI
'QB = Qa - Qar - Qbs
Qa = long wave radiation from atmosphere
Qar = .03 Qa = reflected incoming long wave radiation
Qbs = long, wave radiation from water surface
61
Qa = (a + bea) a Tair4
a = .36 + .12 C exp [-1.92 • 10-4 Z]
b = 2.8 • 10-3 - 26.1 • 10-4 C exp [-1.97 10-4Z]
ea = vapor pressure of air (mb)
Z = cloud height (m)
Qar = .03 Qa o
Qbs = •97 a Tw
QE and QH were estimated by two approaches.
Kohler formulae:
QE = (1.52 + 3.55 w) (eSW - ea)
QH = (.92 + 2.16w) (Tw - Tair)
w wind velocity at 2 m.
Rimsha and Donchenko formula
QE = (1.56 kN + 2.94w) (eSW - ea)
QH = (kN + 1.89w)(Tw - Tair)
kN = 3.87 + .17 (Tw - Tair)
eSW = saturation vapor pressure (mb)
Qs = A [a + Ci (Tw - Tair)]
A is snow ar�cpglation rate
A . 7.85 V-
V is visibility in km
Ci heat capacity of ice
a latent heat of ice
QG by local measurements of geothermal gradient
QGW by local measurements of ground water flows
QF = OyS/Jb
D is river discharge [m3/sec]
y is weight density of water = pwg = [kg/m2 sec21
S is water surface slope
b is river width
Dingman, Assur (1969) Clear (C=O.) Q = 50.93 + 11.21w + (16.99 + 2.05w) (Tw-Tair)
Cloudy (C=1.0) Q = -35.28 + 4.40w + (17.97 + 2.22w) (Tw-Tair
w = wind velocity
Tw = local water temperature, Tw (x)
Tair = ambient air temperature
62
APPENDIX 5
REPORTS
Osterkamp, T. E., J. P. Gosink, K. Kawasaki and G. Penn, "Annotated biblio-
graphy listing sources of information for small hydropower users in cold
climates", An Interim Report to the Alaska Energy Center under Contract
No. AEC81005-3, December 1981.
Osterkamp, T. E., J. P. Gosink, K. Kawasaki and G. Penn, "Survey of manu-
facturers of hydroelectric equipment", An Interim Report to the Alaska
Energy Center under Contract No. AEC81-005-3, December, 1981.
Osterkamp, T. E. and J. P. Gosink, Ice cover development on interior Alaska
Streams, Report No. UAGR-293, Geophysical Institute, University of
Alaska, Fairbanks, Alaska, December, 1982.
PAPERS PRESENTED AT MEETINGS
Gosink, J. P. and T. E. Osterkamp, Hydraulic resistance generated by frazil
ice formation, Paper presented at the Workshop on Hydraulic Resistance
of River Ice, Burlington, Ontario, Canada, September, 1980.
Osterkamp, T. E., R. E. Gilfilian, J. P. Gosink and C. S. Benson, Water
temperature measurements in turbulent streams during periods of frazil
ice formation. Paper presented at the Second Symposium on Applied
Glaciology, Hanover, NH, August 23-27, 1982.
Gosink, J. P. and T. E. Osterkamp, Measurements and analyses of velocity
profiles and frazil ice crystal rise velocities during periods of frazil
ice formation on rivers. Paper presented at the Second Symposium on
Applied Glaciology, Hanover, NH, August 23-27, 1982.
Gosink, J. P. and T. E. Osterkamp, Preliminary evaluation of hydroelectric
power generation in cold climates, Paper presented at the 33rd Alaska
Science Conference, AAAS, September 1982, Fairbanks, Alaska.
Gosink, J. P., T. E. Osterkamp and P. A. Hoffman, Modeling of ice covered
lakes. Poster session presented at Frontiers in Hydrology Speciality
Conference of ASCE, M.I.T., Cambridge, Massachusetts, August 1983.