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RR 206 Pt. II
Research Report 206 Pt. 1
THE EFFECTS OF THERMAL POLLUTION3 ON RIVER ICE CONDITIONS
Pt. U":A SIMPLIFIED METHOD OF CALCULATION
S. Lawrence Dingmanand
Andrew Assur
August 1969
DDC
O)CT 1019as
~2UU La !~
CORPS OF ENGINEERS, U.S. ARMY
COLD REGIONS RESEARCH AND ENGINEERING LABORATORYHANOVER, NEW HAMPSHIRE
THIS DOCUMENT HAS BEEN APPROVED FOR PUBLIC RELEASE
AND SALE; ITS DISTRIBUTION IS UNLIMITED.
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REPRODUCE LEGIBLY.
THE EFFECTS OF THERMAL POLLUTIONON RIVER ICE CONDITIONS
Pt. nl:A SIMPLIFIED METHOD OF CALCULATION
S. Lawrence Dingmanand
Andrew Assur
August 1969
DA TASK 1T061102B52A02
CORPS OF ENGINEERS. U.S. ARMY
COLD REGIONS RESEARCH AND ENGINEERING LABORATORYHANOVER, NEW HAMPSHIRE
THIS DOCUMENT HAS BEEN APPROVED FCR PUBLIC RELEASEAND SALE. ITS DISTRIBUTION IS UNLIMITED.
PREFACE
This report was prepared by Mr. S. Lawrence Dingman, Research Hydrologist,Research Division, and Dr. Andrew Assur, Chief Scientist, U.S. Army Cold RegionsResearch and Engineering Laboratory.
The report was published under DA Task 1T061102B52A020 Military Aspectsof Cold Regions Research.
CONTENTSPage
Introduction ........................................................... 1Basic equation ........................................................ 1Evaluating the heat-transfer coefficient .................................. 2Sample calculation ...... .......................................... 4Discussion ........................................................... 7Literature cited ....................................................... 8Appendix A: Symbols and equations used in calculating heat-loss rates ..... 9A bstract ............................................................. 11
ILLUSTRATIONSFigure
1. Total heat-loss rate ............................................. a2. Intercepts in eq 10-15 as functions of windspeed .................... 53. Slopes in eq 10-15 as functions of windspeed ....................... 5
TABLESTable
1. Comparison of ice-free reach lengths calculated by simplified method tothose calculated by method of Dingman et al ...................... 7
LBLANK PAGE
• .lI II __ II I II| II
1'4f
I, B
h _ _ __ _ _ _ __ _ _
THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS
I. A SIMPLIFIED METhOD OF CALCULATION
by
S. Lawrence Dingman and Andrew Assur
INTRODUCTION
In a previous report, Dingman et al. (1967) developed a method for calculating the tempera-ture profile of a cooling river below a source of thermal pollution and the length of ice-free reachwhich could be maintained by such a source. Computer programs were used to calculate heat-lossrates based on mean daily values of meteorological parameters and to numerically integrate a com-plicated heat-loss expression. The present paper describes a simplified approach to the sameproblem, in which heat loss is calculated as a linear function of the difference between water tem-perature and air temperature, so that the integration can be performed analytically. A simplifiedbut fairly general procedure for calculating water-air heat-loss rates on the basis of air tempera-ture, windspeed, solar radiation, and general atmospheric conditions is also presented.
BASIC EQUATION
The analysis of Dingman et al. (1967) assumed a rectangular uniform channel carrying steadyflow and subject to meteorological conditions which are constant in space and time. Heat loadwas also considered constant, and the heated water was assumed to mix instantaneously over theentire cross section (line source). A heat balance on a Eulerian volume element of the river be-low the heat source gave rise to the expression
x = Pv V, h Cp JTZ (1)Ti Q*
where
=Q T Pv (2)
and x is distance below the heat source (cm), p is mass density of water (g cm-'), vw is riverflow velocity (cm sec'"). h is river depth (cm), 7p is specific heat of water (cal g" °C"), T, iswater temperature at x (°C), Q* is heat-loss rate (cal eec" cm"-), Tat is the natural river tem-perature upstream from the heat source (OC). Qp is the heat of pollution (cal sec"), w is the riverwidth (cm), and T, is the temperature of the river water at x - 0 (OC).
2 THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS
Q* is the net heat-loss rate due to water-atmosphere exchanges by solar and long-wave radi-ation, convection, and evaporation, and is a complicated function of water temperature, air tem-perature, windspeed. humidity, cloud cover. and other meteorological factors. Here an approachcommon in heat-transfer problems is used. and Q* is approximated by the expression
Q* = Q 0 - q(T. - T.) (3)
where Q0 is heat loss from the surface when T, - Ta is zero (cal sec" cm"). TV, is water tem-perature. Ta is air temperature at the 2-m height (0C). and q is a conventional heat-transfer coef-ficient (cal sec" cm-' C'C). Substituting eq 3 into eq 1, and introducing an "influence length,"x* tcm). where
x* Pw v, h Cpx (4)
q
gives
x = -x* In Qo + 9(T., - Ta) (5)Qo+ q(Tj - Ta)
or
exp (-4) = Qo +q(T- Ta) (6)x Qo +q(T - Ta)
Equation 6 is a convenient non-dimensional expression for the downstream decrease of water tem-perature below a thermal pollution source. It shows a simple exponential decay with distance.
To find the total length of the ice-free reach X (cm) which would exist below a thermal pol-lution source the temperature difference Ts - Ta. where Tr is the freezing temperature of water, issubstituted for Tx - Ta in eq 5:
X x* In Qo + q(Ts - Ta) (7)QO + q(Ti - Ta)
EVALUATING THE HEAT-TRANSFER COEFFICIENT
Equation 3 states that thA heat-loss rate Q* is a linear function of Tw - Ta. The value ofthe approach outlined here obviously depends on the extent to which this is true. Many previousstudies of heat exchange between natural water bodies and the atmosphere have used expressionssimilar to eq 3 with some success. For example, the Canadian Joint Board of Engineers (1926),cited in Pruden et al. (1954), recommended
Q* = 46.4(Tw - Ta) (8)
where Q* is in cal cm" day" and Tw and T. are in 'C for a reach of the St. Lawrence River andPruden et al. (1954) developed a somewhat modified form
THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS 3
F CLEAR 'S
CLOUDY4000 -1
V011O
3000
col cmK2
doy 5I /
2000- / 7
1000,
0 20 40 60
T. -T. 0C
Figure 1. Total heat-loss rate, neglecting short-wave radia-
tion, as a function of Tw - Ta, with windspeeds at 0, 5 and 10 m sec".
Q* = 18.3.6 + 15.5 Ta + 43.1(Tw - Ta) (9)
also for a reach of the St. Lawrence River.
These equations, like many others of their type, are based on monthly averages of meteor-
ological parameters in the areas for which they were developed. The heat-transfer coefficients
and constants in these equations have "absorbed" the monthly (in these cases) averages and nor-
mal patterns of variation of all meteorologic factors in addition to air temperature which affect
heat-loss rates. Thus such equations cannot be applied with confidence outside the region for
which they were developed, nor can they be used to calculate heat-loss rates for time periods
shorter than the averaging periods used to develop them.
To provide a more general basis for calculating the constants in eq 3, the computer program
presented by Dingman et al. (1967, App. B) was used to construct graphs of total heat-loss rate
neglecting solar radiation) Q' as a function of T. - Ta over a range of windspeeds and under two
contrasting sets of general weather conditions (Fig. 1). (Appendix A lists the equations used in
the computer program in calculating the individual heat balance components.) The windspeeds v,were 0. 5. and 10 m sec" for, 1) clea,.sky with relative humidity of 50%; and 2) complete cloud
cover, cloud height less than 500 m, and relative humidity of 100%. Water temperature was taken
4 THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS
as OC. The addition of heat to the water body as solar radiation QR is not considered in evaluat-
ing the heat-transfer coefficient but, as shown later, must be taken into account in calculating Q*.
In all cases it is evident that the curves of Q' vs Tw - Ta are close to linear over the range
0 < (T, - Ta) < 50 and that the effect of complete low cloud cover is, as expected, to lower heat-
loss rates. Regression techniques were used to approximate the curves of Figure 1. with the fol-
lowing results:
Q' . 105.18 + 35.08(Tw- T,); v% = 0 m sec" (10)
Clear Q'= 220.90 + 56.27(Tw - T,); Va = 5 In sec" (11)
Q , = 336.62 + 77.47(Tw - Ts); va = lom sec" (12)
Q' = -72.85 + 37.10(Tw - Ta); Va m 0 m sec" (13)
Cloudy QV -27.44 + 59.99(T, - Ta), v, = 5 In sece' (14)
Q' - 17.97 + 82.88(Tw - T,); Va -10 In see" (15)
where Q' is in cal cm"7 day".
In eq 10-15 the values of the slopes give the heat-transfer coefficient q which is effectiveat a given windspeed. Since incoming solar radiation lowers the beat-loss rate by the amount QR,this value must be subtracted from the intercept to give the value of Q0 for a particular case. Thevalues of the slopes and intercepts in eq 10-15 increase with increasing va for each set of generalmeteorological conditions. Figures 2 and 3 show that these relationships are very close to linearand can be approximated by
Clear Ia0 105.18+ 23.14 % (16)
1 #o - 3 5.08 + 4.24 va (17)
Cloudy a I -72.85 + 9.08 v, (18)
1 9l 37.10 + 4.58 va (19)
where a is the intercept and IS is the slope, and the subscripts 0 and 1 refer to clear and cloudyconditions, respectively.
SAMPLB CALCULATION
The following sample calculation of the length of an ice-free reach is carried out for com-parison with results from the computer program presented by Dingman et al. (1967, App. A). Apower plant which discharges a beat of pollution (Q.) of 5 x 10 cal sec" into a river of uniformwidth (w - 300 m), depth (h = 1.75 m). and velocity (vw = 0.75 In sec-') is considered. Assuming
complete mixing at the pollution site, this heat input causes a temperature rise of
Qp
PVw hCP W
(5) (10) 1.o3)(I g cm")(75 cm see")(175 cm)(r3a Il14] cm)(1 cal g" 0 C") 1
THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONOITIONS 5
20 T 350
I
o -300
a, -7285-9 08o//
-20
25 C1250cal Cf- doy a.,105 8+23 14v. CO Cr,?dy '
-40 /200,/
-60 1500
-80 I 000 4 8 12
Figure 2. Intercepts in eq 10-15 as functions of windspeed.
100
80 /
,3710 +4 58 v.//(CLOLUbyl
/
60 ///I
ct cm', day C/ / ,35 08,4 24 ao -/col c z da "C //•//(CL E ARýl
40// 2
20r
0 4 12
lo m sec
Figure 3. Slopes in eq 10-15 as functions of windspeed.
6 THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS
above the natural water temperature of OC. Weather conditions above the river are: cloud cover= 0: relative humidity = 50%; windspeed = 2 m sec'-; air temperature - -25C; heat gain due tosolar radiation (QR) = 85 cal cml day".
Using eq 15
a0 = 105.18 + 23.14(2) = 151.46 cal cm-' day" - Q1
and
Q= Q' - QR - 66.46 cal cm-2 day".
Equation 17 gives
Po 35.08 + 4.24(2) = 43.56 cal cm-2 day" 0C'. = q.
For the calculation of x* it is convenient to change the units of q to cal cm"2 sec"1 CC-,:
q = (43.56 cal cm-' day" "C-C)(11.157][10• day sec")
( C5.04)(10') cal e"ra sec" °C".
Then
x* (1)(75)(175)(1) (2.604)(10') cm.(5.04) (IT"')
The units of Qo are now similarly changed:
Q0 - (66.46)(1.157)(l0") - (7.69)(10"') cal cm"- sec".
Substituting the values of x*. Q0. q. Ts, T1, and Ta Into eq 7
X = - (2.604)(10) In (7.69)(10') + (5.04)(10"4)(25)(7.69)(10") + (5.04)(10"')(26.3)
X = -(2.604)(10') In (7.69)(10") + (1.26)(10")
(7.69) (10") + (1.33) (l0")
X = -(2.604)(10'1)ln 0.9474
X = (1.409)(10')cm - 14.09 km.
The computer-calculated value of X for this case is 13.18 km. Other calculations by this simpli-fied method also differed by less than 10% from the computer-calculated values (see Table I).The simplified method tends to slightly overestimate ice-free reach lengths when the initialpolluted water temperature is above about 1C. This is as expected, since the heat-loss ratescalculated by this method are based on a water temperature of OC, whereas in the actualsituation (and in the computer calculation), heat-loss rates are somewhat higher becausewater temperature is greater than OC throughout the open reach.
THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS 7
Table 1. Comparison of ice-free reach lengths calculated bysimplified method with those calculated by method of Dingman et &1. (1967).
Trial Trial Trial Trial1 8 3 4
Qp (cal se"') 5X 10e a 0° 3 top 8X leW (m) 300 60 200 100h (im) 1.75 0.6 5 3
Vw (mesec") (075 0.6 1 1.5Tna (CC) 0 0 0 0
QCL (cal 'mn2 day-)t 98.55 98.55 270.31 270.31C f 0 0 10 10H (m)t 500 500ea (mb)t 0.403 1.43 0.189 4.21va (m sec-) 2 6 4 8
Ta (*C) -25 -10 -40 -5
X (kin)(Dingman et al., 1967) 13.18 15.44 6.21 186.6
X (kin)(Simplified method) 14.09 15.97 6.16 202.5
t See Appendix A for definitions.
DISCUSSION
Using eq 10-19, calculations can nowbe made of heat-loss rates, temperature profiles below athermal pollution source in a cooling river, and the length of ice-free reach maintained in a river bya thermal pollution source, under fairly general conditions and without recourse to a computer.Equations 10-15. and hence eq 16-19, are derived considering that Tw= 0C. This is true only atthe downstream end of an ice-free reach, but little error is introduced by this assumption unlessTw - Ta is small relative to Ti. Use of heat-loss equations other than those listed in Appendix Awould change the constants in eq 10-19; however, the equations in Appendix A have proved ade-quate in several field tests, and Dingman et al. (1967) found good agreement between observedlengths of ice-free reaches and those calculated using the equations in the one case they tested.
8 THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONS
LITSRATUhE CITED
Anderson, E.R. (1954) "Energy budget studies." In Water-loss Investigations: Lake Mef.ner studies. Technical Report. U.S. Geological Survey, Professional Paper 269. p 71-118.
Dingman, S.L.; Weeks, W.F.. and Yen, Y.C. (1967) The effects of thermal pollution on riverice conditions - 1: A general method of calculation. U.S. Army Cold Regions Researchand Engineering Laboratory Research Report 206, Pt. I (AD 688205).
List. R.J. (Ed.) (1963) Smithsonian meteorological tables. Washington, D.C.: SmithsonianInstitution, 527 p.
Pruden, F.W.; Wardlaw, R.L.: Baxter, D.C.. and Orr. J.L. (1954) A study of wintertime heatlorses from a water surface and of heat conservation and heat addition to combat iceformation in the St. Lawrence River. National Research Council of Canada, Report no.MD-42. 56 p.
Rimsha, V.A. and Donchenko, R.V. (1957) The investigation of heat loss from free watersurfaces in wintertime. Leningrad Gosudarstvennyi Gidrologicbeskii Institut, Trudy,
vol. 65. p. 54-83 (text in Russian).
9
APPENDIX A: SYMBOLS AND EQUATIONS USED INCALCULATING HEAT-LOSS RATES
Symbols
Symbol Meaning Units
C Cloud cover dimensionless (0< C .< 10)
H Cloud height m
Qa Heat added to river by long-wave radiation cal cm-2 day-"
Qar Long-wave radiation reflected at river surface cal cm-2 day-'
% Net heat lost from river as long-wave radiation cal cm-' day"-
Qbs Long-wave back radiation from river sirface cal cm' day"-
QC L Incoming clear-sky solar radiation cal cm"2 day"-
QE Heat lost from river by evaporation cal cm-' day"-
QH Heat lost from river by convection to atmosphere cal cm-2 day"'
QR Heat added to river by solar radiation cal cm2 day-1
QR1 Solar radiation incident at river surface cal cm-2 day-I
QRR Solar radiation reflected at river surface cal cm-2 day"'
Ta Air temperature (2 m height) 0C
Tw Water surface temperature 0C
a (See eq A8)
b (See eq A9)ea Vapor pressure of air (2 m height) mb
eew Saturation vapor pressure at temperature Tw mb
kn (See eq A13)
0 Stefan-Boltzmann constant (1.171X10") cal cm'- day-' OK'
Equations
(See Dingman et al., 1967 for discussion.)
Solar radastion
0R = QR "QRR (Al)
where
QRI = QcL(0.35 + 0.061(10 - C)A (List, 1963) (A2)
(• = 0"I0 8 QuI - (6.766)(10") QLj (Dingman et al., 1967) (A3)
10 APPENDIX A
LAg-wave radiation
=B Qb. " Q, ÷ Qr (A4)
where
Q.= 0.970o TW (AS)
,= (0.68 + 0.036 \/), T,4 (clear sky) (Anderson. 1954) (A6)
Q= (a + bea)o Ta4 (cloudy sky) (Anderson, 1954) (A7)
a = 0.740 + 0.025C exp R(-1.92)(10")HI (Anderson. 1954) (A8)
b = (4.9)(10") - (5.4)(10")C exp 1(- 1.97)(10")¶H] (Anderson, 1954) (A9)
Qr - 0.03 Qa. (AlO)
Convection and evaporation
QH = (k:, + 3.9 va)(Tw - Ta) (Rimsha and Donchenko, 1957) (All)
QE = (1.56kn + 6.08va) (esw- ea) (Rimsha and Donchenko, 1957) (A12)
where
kn - 8.0 + 0.35(Tw - Ta) (Rimsha and Donchenko, 1957). (A13)
UinclassifiedSerudti Ciasuiftcation
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U.S. Army Cold Regions Research and Engineering Laboratory Unclassif ledHanover 2A. gROUPNew Hampshire 0M755
S. REPORT TITLE
THE EFFECTS OF THERMAL POLLUTION ON RIVER ICE CONDITIONSIL. A SIMPLIFIED METHOD OF CALCULATION
4. DESCRIPTIVE NOTES 1Tlrp .fi pt M~ et fa. ist"das)
S. AV THNOISI (jPIkt in.. olid. t.Iinje. last sNeH)
S. Lawrence Dingman and Andrew Assur
6. REPORT DATE 786 TOTAL NO. OP PAGES 71L No. OP REps
August 1969 11 5W5 CONTRACT OR GRANT NO. W. ORISINATOfrI REPORT NWM21fIS)5
B.PROJECT NO. Research Report 206, Pt. 11
DA Task 1T061102B52A02 S6. *THER REPORT NO421 (AW 04baresa Amm moFU besweSlfem
IS. O4STRIOUITION STATEMENT
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I$. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
U.S. Army Cold Regions Research and EngineeringLaboratory
_____________________________ Hanover, New Hampshire 03755
In part I of this study. an expression relating distance x below a thermal pollution source to water temperature T,under certain conditions was developed:
x = C f d TwQ'
where C is a constant. T, and T5 are the water temperatures at x = 0 and x, respectively, and Q* is the heat-lossrate. Q*was represented therein by a complicated expression requiring numerical integration of the above relation.Here, the more complete equation is shown to be well approximated by Q* = Qo + 9(Tw - Ti). where Ta is air tem-perature. Qo is heat-loss rate when TS, - Ta= 0. and q is a heat-transfer coefficient. it is further shown that for agiven set of general meteorologic conditions, q is a linear function of windapeed va and go is a linear function of vaand net incoming solar radiation QR. Thus by specifying general meteoroogc Conditions Va, QR, li T - Ta oncan evaluate the simple linear relation for Q*. and the expression relating x and Tw can he easily determined byanalytical integration. This relation takes the formi of an exponential-decay curve. Results using these approxima-tions are within 10% of those calculated by numerical integration for the cases tested.
IAL KEY WORDS
Stream Pollution Thermal power plantsThermal pollution Waterways (transportation)
.61= 10473 ::: AN6.ICNUn lff
