(c) Solar radiation is transmitted in the vertical direction
only and there is differential absorption of the incoming
solar radiation below the water surface.

(d) The sides and bottom of the reservoir are insulated.

(e) The density and specific heat of water are constant.

Factors Considered and Basic Equations ---

(a) Variable area with depth

(b) Direct absorption

Transmission of radiation at elevation, y, is given by

$(y) = o,(1-8) 7" Vs) (48)
where:
oy = net incident solar radiation
68 = fraction of g absorbed at the surface
n = light extinction coefficient
y, = water surface elevation
(c) Inflow ‘

(i) Inflow enters at the level at which its temperature,
or the mixed inflow temperature if entrance mixing is
allowed, matches the temperature in the reservoir.

(ii) An option to account for the travel or lag time of
inflows within the reservoir is provided.

(iii) Entrance mixing could be included by providing an
entrance mixing ratio; 100% is recommended for Fontana.

43


4---4-4-- enn gay gp --- g -- - - + - =

: H H g &
. : : . .
a---$--- fee. as00 a w---}----$3 = wonnfe---$----$---- f----}-e--}----f----}----}----f:
Zeon dt Z foe Zeon Sh 2
_ Sietee ts Lo ----}----}----}----}----$----{----}---_fSSfize_--f- 0
----}---- = gshees 2 3 e BREze2 £ 3
é cane 2 ooamay 2
H of of

4 i uu--fo---$----$----}---_fte & ns Soeeenees Senne SES Se as 22
corpo SB: . . fee
#2 gz
a7 H H a7
nan 5 Snes SnnEnSnS SnONEENE SOOIOESS SOROS SOSOnEES SO x

35 Be
BE H z=
BS | 22
SE a---}----}---- fee} edged tele-ie Fe
2 f----}---- wa--}----}--- pe F
Bs j-=--}--~-f----f----f----f----f---- f= desig 8
Fs 5; , 4 =
4 ! ; Pog

$F $y : e

7 3

40,0
4340
20,0
23,0
0

9940
40,0
$3.0
30,0

107


the stratification increases. When the surface water attains its
maximum temperature and then begins to cool, the epilimnion tends to
become more dense and unstable with respect to the lower, less dense
waters. The thermocline sinks rapidly as the epilimnion cools further
until the whole reservoir mixes or overturns and is isothermal. In
climates where the temperature falls below 4°c, two overturns may occur
per year. The reservoir then is isothermal in the early spring as well.
The cycle of stratification is very complicated. The solution of
this problem must consider all the influences from the meteorologic,
hydraulic and hydrodynamic factors and the thermal and density properties

of the water.

GENERAL DESCRIPTION OF DEEP RESERVOIR MODELS
The basic equation, 8, relating all the energy inputs to a body of

water can be solved for reservoirs, rivers and estuaries and coastal

regions.
RS &*G- Gr Aot*® WEMW W= 2 @)
Where:
Q = shortwave radiation incident to the water surface;
Q. = reflected shortwave radiation;

QQ = incoming longwave radiation from the atmosphere;

Qar = reflected longwave radiation;

bs = longwave radiation emitted by the body of water;

w = net energy brought into the body of water in inflow,
including precipitation, and accounting for outflow;

% = energy utilized by evaporation;


«=
a
2 a
n
7 2
z 12 8
“ Pat?
« BEtizs. 9
s wa--$----$----$22f8s2---439
4 BESees e
? Fhacee 2
= vernon
* ounmay
© som 5 °
2 on ----$----}----}.---}----}----}--- -- 4 2
2S .
zz 8
27 ”
at w---$----$----32
#5 =
5%
2
rae ;
22 :
ad os Sonne Soneennee ig
ard 4 as
&
2
£ 4
? ----$----3----34
*
.
2 : qe
2 w---$----$----42
S$ 5
? :
: :
: 4
z ~ *
i H
» : $2
r al
"eg 8
g & is
| ~ | ” 1s
H | ,
an
| i |
Poot | ee

210.1

=oSURFACE ELEVE

TFrPERATUE I “DEGREES ©

138

MURRTa RFSFRYUIR 197 L=-bAy 1225

MIT MUREL ®

Pigure 115


eon

— 2425 __22,0 __20)5

31003 6

22.0
32002 8

OVERLAP =
<= OUTLET

= AVERLAP
1945

<= OUTLET

© m MEASURED

1 e STAND

2 = REAL

3m RETA2
133

weSURFACE ELEVI

‘
1
1
1
1
1
1
1
meSURFACE ELEVE

119

TEMPERATURE IN DEGREES ¢
2

17400

rie)
esd

19
TENPEMATURE IY PEGREES €

Veo

MUPRTS RESERVUTA 197LeeuAYT

I
5
MUGRTS RESFRYLTK L97LOSUAYIZ25

nd
t
a 7 eee

1
T
I
| rergelcmacrec
ar rune ©
Tee
Ayr ure *

Figure 106
Pigure 105



RESEARCH REPORTING SERIES

Research reports of the Office of Research and Development,

U.S. Environmental Protection Agency, have been grouped into
five series. These five broad categories were established to
facilitate further development and application of environmental
technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in
related fields. The five series are:

Environmental Health Effects Research
Environmental Protection Technology
Ecological Research

- Environmental Monitoring
Socioeconomic Environmental Studies

This report has been assigned to the ECOLOGICAL RESEARCH STUDIES
series. This series describes research on the effects of pollution

on humans, plant and animal species, and materials. Problems are
assessed for their long- and short-term influences. Investigations
include formation, transport, and pathway studies to determine the
fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living
organisms in the aquatic, terrestrial and atmospheric environments.

EPA REVIEW NOTICE

OPWwNn—

This report has been reviewed by the Office of Research and
Development, EPA, and approved for publication. Approval does
not signify that the contents necessarily reflect the views and
policies of the Environmental Protection Agency, nor does mention
of trade names or commercial products constitute endorsement or
recommendation for use.


340

ee
20 oar
S8a05e.
PRSESS
peeere
Saungy

i%0

TEMPERATURE IN DEGREES ¢

2260 ed 2760 2965.

1945

43

ue

200

DOUGLAS REDERVUTR 1959—=0AY1121 eSURFACE FLEVE

MIT MUnEL ©

Figure 58

4---

a

<= cuTLET
1
I

140 pr 1a 1403 ito 195 2200 75 2760 2943.

cr

g
z
3
&

&

SURFACE ELEVE

OnyALAS MFSFRVUTR 195!

MIT MUL ©

Figure 59

103


time

a cad
"

phase angle

T, = average value of equilibrium temperature
over one annual cycle

ST, = one half the annual variation

K = the heat transfer coefficient at the lake

surface (BTu/£t?-day-°C)
An explicit finite difference scheme is used for numerical evaluation.
At each time step, the thermal diffusivity is evaluated from the known
temperature profile and its value is restricted to the range between
the input maximum and minimum thermal diffusivities. The variable time
increment, At, is then determined from the maximum value of the thermal

diffusivity at this step by the following equation:

2
= AZ 74
Atha 7M a al

where:
C. = a nondimensional constant, 0 <C, < 0.5
AZ = spatial mesh size
Verification

The model has not been verified explicitly in any lake. The exter-
nal parameters were chosen to correspond loosely with Cayuga Lake. The
results of the model were in qualitative agreement with the measured

values of the thermal profiles in Cayuga Lake.

52


gecenen:

sul

1740

for

1200

TERPEATURE JM DEGREES ¢

31002 An

eeSURFACE ELEVI

MUPRTS RESERVOIR LO7Le@UAVIZ25,

MUT PUTEL ©

Figure 110

20,3”

~ 2408 ___?

2
a
s
2
. ord ¢
Et 35
G2anaZe 2
w---$----4322828¢--32
Paowwee E4
poeeere
onaneey
’ i >

Os

200

5

WwW

“a8

TFRPERATURE IM MEGKEES €

307.9 4

eeSURFACE ELEVI

MQPRIS RFSFRYUIK A97LoUeY E252

“UT PUrEL ©

Pigure 111

136


is negative. The WRE model for the surface layer appears to be
incorrect. It may have contributed to the wide variation in predicted
temperatures when the layer thickness was varied.

In the Cornell model, a constant surface elevation is assumed.

63


Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

ony A HW FF WBN HB

10

11

12

13

14

15

16

LIST OF TABLES

Assumptions

Input Parameters

Factors Involved in Analysis

Model Construction

Parameters Varied in Sensitivity Analysis
Frequency Analysis of the Wind Speed
Legend for Figures 12-201

Statistical Analysis for the Predicted
Water Temperature at Outlet Level
Fontana Reservoir, 1966

Statistical Analysis for the Predicted
Surface Water Temperature Fontana
Reservoir, 1966

Statistical Analysis for the Predicted
Water Temperature at Outlet Level
Douglas Reservoir, 1969

Statistical Analysis for the Predicted
Surface Water Temperature Douglas
Reservoir, 1969

Statistical Analysis for the Predicted
Water Temperature at Outlet Level
Cherokee Reservoir, 1967

Statistical Analysis for the Predicted
Surface Water Temperature Cherokee
Reservoir, 1967

Statistical Analysis for the Predicted
Surface Water Temperature Norris
Reservoir, 1972

Statistical Analysis for the Predicted
Water Temperature at Outlet Level
South Holston Reservoir, 1953

Statistical Analysis for the Predicted
Surface Water Temperature South Holston
Reservoir, 1953

xvii

Page

14, 15

16, 17, 18, 19

20, 21

22, 23

66

70

71

73

74

94

95

110

111

128

143

144


--4 Mod a @
re! "348 1 por
1 1 | jo gt .
z4 j~-{-$ 4 | $2
—_ a-443
in i fi : z
“7 | .
z i dst —a a re ot
saa23 z 43
PELE ES la = , ;
Sea58573 2 444-24 --=- g
See Gee 5 Pom :
aagecds : d &
. tle # 4. H
af | ELBE Te PgR
- :§ (Be ]
at rom
7 > 4 coee seen Gees £1
eee | od Fe
Be i 3 4 &
FS
23 | a :
at see —_ . 2
© “ pony 4 wn
& ‘ ¥ 4 dud a a Ss 7 ‘te
rer ae 7 p 4
zo) i ; 4]
8 , 9--4-#--4 39
. 4 F v4
Po ay
ef
£ a 1 “+
P / 4
4
ar

ao

|=COMPUTED OUTFLUW TEMPERATURES®

nAYS
17h

“181

Frat Luvonu

MIT MOMEL 0

Pigure 173


1 RYT

‘© 2 OVERLAP
<= ounyit
Wi

fees

TEMPERATURE IN DEOREES

CHEAOKER RESERVOIR}

z
H
g
:
3
;

Nyy monet ©

Pigure 69

112


abe 1700 10

30

TEMPERATURE J” DEGREES ¢
CHEROKEE RESERVOIR J967eSOAYIZID FSURFACE ELEV!

Myy MODEL ©

Pigure 80

a1,

308

2605

22.0

1908

18

3240

a

TEMPERATURE [% DEGREES ¢

cor)

CHEROKEE RESERVOIR J9GTESDAYIZ0® eOBURFACE ELEVI

MJY MODEL ©

Figure 61

118


20 BH

2655

1965 “2260

<= suner
1
i

'
!
1
|
1760

1240 Perr

aS

4

a0

10040

TEMPERATURE IN DEGREES ¢

HIWASSEE RESERVOIR 1997==DAVI168

Figure 166

403,98

weSURFACE ELEVE

Uy MODEL ©

~
eo he
°
boo
; 3

anne $$ de

Tam
i
at
“T

Os!

me

1700 1905 2200 24S 2700 24

TERPERATURE IN DEGREES €
WIWASSEE RESERVOIR L9A79DAYIZ09 ==SURFACE ELEVI 46904 4

12,0 lhe

cre

in)

Figure 167

MIT MODEL ®

173


oh

|

+;

a
'
tl

= DAY! 148

= DAY! 209
= DAY! 302

0
1

2

3 = dav] 267
3 = OYE

© = OVERLAP
<= OUTLET

TEMPERATURE IN DEGREES ¢
HIWASSEE RESERVOIR 1967==NEASURED TEMPERATURE PROFILEW=

Figure 144

MIT MODEL ©

HT 4206

anne

woes

1
t
1
l

or eg cerwe

*

ra
?

TEMPERATURE IN DEGREES ¢
HIWASSEE RESERVUIR 1947==GOMPUTED TEMPERATURE PROFILER=

Figure 145

Uy HOPEL ©

162


average yearly flow is considerably greater than the volume of the
reservoir, the inflow would bring in an amount of thermal energy which
would not be negligible. The same, of course, would be true for

the energy loss due to outflow. In addition, the inflow and
withdrawal from hypolimion would provide mixing which is not
accounted for by eddy diffusion.

(iii) The eddy diffusivity depends too strongly on the wind condi-

tions above the reservoir.

PROBLEMS WITH DEEP RESERVOIRS MODELS

All of the three models evaluated use temperature for the deter-
mination of the entrance level of inflow and for evaluation of whether
an unstable condition prevails to enhance convection. Temperature is
also used for the calculation of withdrawal thickness in the MIT model.
However, the density of water is greatest at 4°c. Therefore, the
models without modification, cannot be applied to a reservoir where
the water temperature goes below 4°c. This oversight is probably due
to the fact that the temperature of the Fontana Reservoir is higher than
4°C the year round. However, water of 4°C floating on the top of the
colder water was discovered in the results from the test data supplied
with Cornell model.

The calculation of water temperature at the surface layer is an
important problem. With variable surface level, the volume of the
surface layer also changes with time. The heat transport equation des-
cribed earlier is derived for a fixed control volume with respect to

time. Therefore, directly applying the heat transport equation to the

58 -


Table 2. INPUT PARAMETERS
Input Cornell MIT Water Resources
Parameters Model Model Engineering Model
Unit Input data in Input data in can be in any units
specified units | specified units user supplies conversion
only only factors, standard units
as. indicated
Short wave No? Yes Yes! 24,8
Solar radi- (net flux) (gross flux
ation keal/m@ day i.e. no reflection)
kcal/m@ - Sec
Net Long- No Yes! #257 No .
wave kcal/m@ - day (calculated)
atmosphere .
radiation
Wind No3?10 Yes Yes®
speed m/sec m/sec
Air No? Yes Yes®
Temperature °c °c
Cloud No® Yes?»6 Yes®
Cover decimal decimal
Atmospheric No? No Yes! +28
Pressure mb
Relative No® Yes Yes?798
Humidity decimal decimal
Wet’ Bulb No No yes?+5-8
Temperature %
Dew Point No No Yes27528
Temperature %
Equilibrium | yes!2 No No
Temperature (in terms of mean

value, amplitude
and phase angle

in a sinusoidal

form)%c

16


1760

TEMPERATURE IN DEGREES ¢

FONTANA RESERVOIR 1966-~0AYi 75

2200 wes 27,0 aN!

19s:

As

12.0

50063 4

SURFACE ELEVE

Pigere 36 ty KODEL ©

15.0
30.0

45.0

60.0

27,0285

265

17.0

TENPERATURE IN DEGREES €

22.0

193

16s

95

1.0

5

3004 h

‘SeSURFACE ELEVE

FONTANA RESERVOIR 19660-0471 192

Pigore 37 wry MOnEL ©

87


Table 4 (Continued). MODEL CONSTRUCTION

Parameters Cornell MIT Water Resources
Model Model Engineering Model
Running! (400 days) 13 min. | (300 days) (300 days)
time for test for Ay = 5 ft. 1.4 min. for ay 2.6min. for Ay 2.0m
case = 2.0" at - 1 day
At 1 day 3.1 min. for Ay 1.00
2.0 min. for Ay At = 1 day
= 1.0"
dt 1 day

1. the program compiling time 1s not included

2. the densimetric Fronde number in defined as
ae
where L - reservoir length
Q - volumetric discharge through the reservoir
D - mean reservoir depth
V - reservoir volume
Py- reference density

8 ~ average density gradient -%
g - gravitational acceleration

3. Check by stability criteria, Equation (2), internally and the input At
is subdivided if necessary in that particular time step.

23


29.3

22.0

245

22.0
503eL M

1905
PeSURFACE ELEVE

10

a

Vs
TEMPERATURE IN DEGREES ©

1200
SOUTH HELSTRN 1953-=DAy1299

sr)

KIT MOMEL ®

=
<

Pigure 124

20
30,0
“30
100,0 49:

Ba

---43 ----

EL 22.
OVERLAP
<s ourelr
{
i

= MEASURED
© STAND
= DELZ)

°
ry
2
3

-4--$---

7040

2995

2740

2645

1965 22.0

1740

145

1240

mt

10°

'
100,0 ¢nenne

50202 4

woSURFAGE ELEVI

TEMPERATURE JN DEGREES ¢

148

SOUTH HOLSTON 1953~<0Ay1362

HUT MODEL ©

as

Figure 125

a0


A routine check on the second criterion was built into the program to
subdivide the time interval if the vertical velocity should become too

large.

The model was verified on Fontana Reservoir for the same nine month

period as the WRE model.

Sensitivity Analysis

The MIT model has been found to be the most satisfactory and has
been most thoroughly evaluated. The evaluation results are shown in
the Section II, Sensitivity Analyses.

CORNELL MODEL

The model was developed through an extension of a study on the
physical effects of themnal discharge into Cayuga Lake@4) _ It is a one-
dimensional model designed for deep stratified lakes. The surface ele-
vation of the lake is assumed to be constant throughout the simulation
period and the reservoir is divided into a number of horizontal layers
of equal thickness. The geometry of the reservoir is not considered.

Heat flow from inflow, outflow, and vertical advection through each
horizontal layer are not considered, nor is the differential adsorption
of incoming solar radiation. Eddy diffusivity, which is related to wind
induced turbulence and the buoyancy gradient is the primary factor of
heat transfer within the reservoir.
Assumptions

(a) Horizontal homogeneity and constant cross-section area.

48


i, 4
a. oj ~---}---
4 ha
Hl fobs ----$----}----
—--4-4--$----$2
\ “ J
: =
a---f----$----$3 0 §
a
e so: 2 .«
3 rs f 3 ‘ Ve <
Saunth z Sennst
3eazss eo H BeauZe
BestZ5 s ----}----}----}----}----3----}----3----} 22848 e---
4a secee-—--ie lg SESGS5
SEoESS =e $ Saseee
ive " ¢$ cece
teneee z
s conmay
onneey 3
H vf
wee tee tel te 3 w---$--L-$----}----$----4----}----}----}----
$e ji q
ad : H
i a H
H H H BE 7 H
H H 3
H : H a Zt $ -f---.
$----}----$----$----}----33 = 7
H H H H ing
i 2 q
2 H
i i i --a}----3 - - - $28 3 bd gd ep} ---- $---- $ ----$ ----$ ----
ta = i
Hi 2 q
{ 5
H es oc po--}----$----$----}----$----$----}----$----}----
----$----$---= w---$----$----}----$8 o£ : 7 H
e & H
H & } H . i
:
a ae Sons Sens Sneenes ie 2 2 u---}----}----}----}----$----$----$----4----}----
5 of 4
4 4 2
q
= :
H * $f ----}----}----}----4----$----$----{----
g----}----}----f----f--- pts | ge----$----
3 ; =: 2
t i t q q q
= H , H
a a+ $--Y-$--- $b d e H u-¥ fF $+ t----  -- -- t -- - -
H . <
2 < ¢ < < 3 S e 3" : 2 8 ¢ RS 2
2 a a . q a = a a s 3 s € s : Z
é 5 g s s ¢ 3 ¢ a 3 & = 2.
S Ey & Ey 8 & 3 .& } & =

27000 2S

teed

2240
27 Re

1965
eeSURFACE ELEVE

1740
erry
187

Ved
TeRPRUATYRE IY PESKFES €

9th

Vea

96>
e7aT fuels

Me
MIT PUEL ®

Figure 185



J | wb y---- ae
| ; 48
i ‘
a3 i , ~---}----f¢
H 4 :
toy yt ay .
1 | ! |
il | 2
4-4 / 3
: : a
. 4 :
. ! .
4 f
ens Sanenes See : 2 3 A ---}----$----}----}----}----fe
a o& 3
; f a § : : ; : Fy
qn
= 2
ae a z z rs
Fenn Sh 2 H 2 ar
Bezazs « i aeatee 4
o---$----}----}----}---_- fp eden u---$----$----}----}----}----}---- f 2288-2
ZebESs ¢ $ dencS5 2
BoesEE = 3 H S5beee 2
peepee ri H teaee
ounmay e H onne
;
| tout w---42
~---}----}----}----}----1----}2 2 | w---4----}----}- =
, f f . 5 8 e , . s
ge
==
H BE
| ;
wn--$----}----}----f----f---g----ta +E wu--}----$----$----}----d----}----p---- te
fog i?
agg H £
BS H
2 i
az {
5 z 4 a2 c a :
= eoikx 2
eS--$---2}----} egg edged dee te & E w---dece-$----}----}---- P---d----$ ---- fd ---- 3 ---- 32
asf 3
«< . = ©
u 2 o
2 ‘ see H
: . ‘ e a
a---}----}-- $e pepe gee beda w---}---£4.---}----p----$----p--------}----}----fs
PY :
‘
5 {
. 3 . H
; | Z { | ed f ( i,
e-,-$----$----}----}----b eld) oF canes SOs SES SUUREED SuSnEOOES SOSEENS SOSEEUINS SuOsESENS SOURS SOEEEINE 1
ce eo = 3 7 ; c
Po of, | . f
ween eene nines 4 =e eoele ew o : H
4 '
H i
f « : 4 H 4
----$-2--$----$---- fd peee deeded iis peaoof----$----}----}----}----f---- fg ---- fb ets
Soy 3
od ¢
foo 2

o

10,0
¥O,0
100,0

PEOREES F
w>SURFACE FLEVI 312e2 4
132

TEnPERATUPE 1h
MUERTS AFOFRYUTK £97Le0uP Y1135,

HAT UT eL

Pigure 103


6.

10.
1.
12.

13.

Table 2 (Continued). INPUT PARAMETERS
Only one of the three parameters is used. If more than one are presented
the last read in has the priority. But at least one of them should be
presented in input data
To be read in only when atmospheric radiation is to be calculated intern-
ally by program
If calculated by program cloud cover should be known
Meteorologic data provide at least one observation per day, constant
values such as average monthly meteorologic conditions should be avoided.
Taking account of the power plant cooling water only and in terms of heat,
p> per unit area per unit time added by power plant
Also used for friction velocity calculation externally
A constant surface area is needed for calculating 4 (see 9)
Either surface temperature (°c) or heat flux at surface (Btu/#t?-day) can
be used to replace equilibrium temperature

1

A+B sin (3h t +6)

19


o = a dimensionless constant (=0.1 for preliminary

study )

we = 1s «al «ngs
—=B, + B, sin ¢ t+ ¥), friction
i) 1 2 365 , (67)

velocity
The empirical relation of Munk and Anderson is suggested for determining

wind speeds over lakes.

- a .

R= G4) “tajz 3, Richardson Number (68)

a, =A, +A, (2 4°) +A; (CT 4)2, Coefficient
(69)

of volumetric expansion for water

where:
N = a dimensionless constant (N=2)
Ts = wind shear stress
Ay Ag sAg.By By »Cy Cy = constants
(b) Initial condition
T(z,t.) = T, (70)
(c) Boundary conditions

oT.

Gaz 79 3 (7)

OZ" 222, nm

oT. =
~(Cphy $2 -07K Mg TQ) (72)
FA a all eqs
Ty = Ty + éT, sin Ges t + 9), the equilibrium (73)
temperature
where:
tT, = temperature at water surface

51


$----

ee 7

© OUTLET

§

°
=
=
2
a
%
4
S
2
2
rs
=
e

hyd

1240

mt)

ne

ETT

Figure 168

4360 4

MIWASSEG RESERVOIR 19¢7==DAYIZ67 =sBURFACE ELEVI

MIT MODEL ®

e

2 att

Seco dh

aecoge

gees —-

Bbesee
ver
aay

a

22.0

2740

2495

17400 1965

TEMPERATURE JN DEGREES C.
HIWASSEE RESERVOIR 1967==071202 eSURFACE ELEV,

pcre)

Figure 169

46603 H

MIT MODEL ©

174


Table 20

STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

Reservoir/Year: Fort Loudon/1971

Time Period Covered: 60th - 360th Julian Day

Std. error Correlation

File Name of estimate (°c) Coefficient
STAND 2.98 0.86
DELZ 1 2.78 0.89
DELZ 2 3.09 0.85
BETA 1 2.86 0.88
BETA 2 2.88 0.88
ETA 1 2.81 0.89
ETA 2 3.38 0.81
ETA 3 -—- ----
DIFF 1 2.95 0.86
DIFF 3 2.92 0.87

180


22.0
309.9

17.0

1a
TEMPERATURE IN DEGREES €
FONTANA RESERVOIR 1966eSDAYi187 seSURFACE ELEVE

1200

Figure 38 wy none ©

I50,0.

15.0

505.9 4

2260

SeSURFACE ELEV!

88

1s
TEWPERATURE IN DEGREES ¢
FONTANA RESERVOIR 1966-=DAYiZ15

12.0

pe

1.0
MIT MODEL ©

Or
Figure 39

30,0
45.0
60.0

105.0


28

“ano

2465

22.0
30301

wa SURFACE ELEVE

Led

TEMPERATURE [M DEGREES ¢

SMUTH HOLSTIN 1953-=DAy1299

1200

MIT MODE ©

Pigure 130

= OUTLET
1
i)

1
1
!
'

1200

oo

20,0

2700 a3.

2445

2200
5022

1740 195
mSURFACE ELEVE

145

TEMPERATURE IY DEGREES €
151

SOUTH HALSTON 2993=-DAYIIOR

My MOREL ©

ao 48
Pigure 131

100,0


Table 12

STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL

Reservoir/Year: Cherokee/1967

Time Period Covered: 120th - 330th Julian Day

Std. error Correlation
File Name of estimate (°c) Coefficient
STAND 2.70 0.83
DELZ 1 2.99 0.80
DELZ 2 2.22 0.88
BETA 1 2.62 0.85
BETA 2 2.54 0.86
ETA 1 5.52 0.00
ETA 2 2.97 0.78
ETA 3 2.29 0.89 «
DIFF 1 1.80 0.92
DIFF 3 2.08 0.88

110


Governing equation ---
The time rate of change of thermal energy H in a control volume

of thickness Az, is:

oH. .
FE | AQ + My) + OI, Ajit Gay Gdjn cary
where subscript j indicates the increment of depth; j = 1 at

bottom surface

=
u

The heat energy stored in the control volume (kcal)

h, = Heat flow associated with inflowing water (kcal/sec)

h, = Heat flow associated with outflowing water (kcal/sec)
hy = Heat flow by advection (kcal/sec)

hg = Heat flow by diffusion (kcal/sec)
hy = Heat flow by short wave solar radiation (kcal/sec)

In terms of temperature T, Eq. 12 can be written as:

By chy an mad = 3, © TD, (05); ~P56 (Uo)y Oa)y Hy A

#05 C VSASTS Oper Var AjerTea * O5PHA; “i Pje1 Dist
Ager “in (13)
or

5 ANT ay = BYE; Cp); “B/E; (GQ)5 + 8 Aj

-cv.A.T. : =,qA:,4T: T..,-T.
+ PjCVJA;T, 541 jer fed jt -p,CD,A, ive jos

T.,3, -Ts
+ 0541 D5 atAgar jt fe js q4)

29


Table 7.

LEGEND FOR FIGURES 12-201

Fraction of

TL

Solar Radiation Radiation
Layer Absorbed at Absorption Diffusion
Thickness The Water Surface Coefficient Coefficient

AY (m) 8 nat) D(m?/day)
STAND 2.0 0.50 0.75 Dm*
DELZ 1 1.0 0.50 0.75 Dn
DELZ 2 3.0 0.50 0.75 Dm
BETA 1 2.0 0.40 0.75 Dn
BETA 2 2.0 0.20 0.75 Dn
ETA 1 2.0 0.50 0.05 Dn
ETA 2 2.0 0.50 1.40 Dn
ETA 3 2.0 0.50 0.40 Dm
DIFF 1 2.0 0.50 0.75 30 Dm
DIFF 3 2.0 0.50 0.75 100 Dm

*Dm = molecular diffusion


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

85

86

87

88

89

90

91

92

93

94

95

99

100

101

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 186

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 213

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 248

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 277

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 66
Cherokee Reservoir - Diffusion Coefficient
Sensitivity, Day 123

Cherokee Reservoir - Diffusion Coefficient
Sensitivity, Day 186

Cherokee Reservoir Diffusion Coefficient
Sensitivity, Day 213

Cherokee Reservoir Diffusion Coefficient
Sensitivity, Day 248

Cherokee Reservoir Diffusion Coefficient
Sensitivity, Day 277

Norris Reservoir Computed Temperature
Profile

Norris Reservoir Computed Outflow
Temperature

Norris Reservoir Layer Thickness
Sensitivity, Day 106

Norris Reservoir Layer Thickness
Sensitivity, Day 139

Norris Reservoir Layer Thickness
Sensitivity, Day 183

Norris Reservoir Layer Thickness
Sensitivity, Day 225

Norris Reservoir Layer Thickness
Sensitivity, Day 252

120
121
121
122

122

123
123
124
124
125
127
129
129
130
130
131

131


TABLE OF CONTENTS

List of Figures 2... . . ., ee ee ee ee v

List of Tables... ... Be ee we ewe we ee we xvii

Section I
mclusions . 2 6 ee ee 1

Section II
Recommendations... 1. ee ee ee ee 3

Section III
Introduction... fe ee es 4

Section IV

Analysis of Deep Reservoir Models... 1 1. eee ee et ee eee 8

General Description of Deep Reservoir Models. .. 1... 2. e+e eee

Water Resources Engineers' Model. ... 2... ee eee eee eens

Principal Assumptions. . 6. 6 ee ee ee et ee ee ee
Factors Considered and the Basic Equations
“Direct Absorption of Solar Radiation. ...... 2-2-2 ee eee
Selective Withdrawal . 2. 2 1 6 1 ee ee et ee ee ee
Depth and Velocity Distribution of Inflow... ....-....-.

Internal Mixing. ©... 2 ee ee ee ee ee ee
Governing Equation. ©... 1 1 eee et ee ee ee ee es
Verification... 1. ee ee ee ee ee ee ee
Results of Test Run... ee ee ee ee ee ee ee

MIT Model... 6 1 we ee ee ee ee
Principal Assumptions. . . 6 1 1 6 ee ee eee ee es
Factors Considered and the Basic Equations
Varification. 2... 1. ee ee ee ee
Sensitivity Analysis... 2... eee eee

Cornell Model 2... we ee ee
Assumptions. © 2 2. we ee ee ee ee et ee
Basic Equations. .. 1... ee ee ee ee ee ee te
Verification. ©... 1 ee ee ee ee
Results of Test RM... ww ee ee ee ee

Problems with Deep Reservoirs Models... 2... eee eet eee

Section V
Sensitivity Analysis... 6... ee ee ee es 64
Fontana Reservoir... 1... ee ee ee ee ee es 69
Douglas Reservoir. ...... i rr
Cherokee Reservoir. . 2... ee eee ee tt ee ee te te 109
Norris Reservoir... 6. 1 ee ee ee ee ee ee ee es 126
South Holston Reservoir... 1. ee eee eet ee te es 140


pat)

TFMPEPATURE 1 DEGKEES €

22D.

195

1908

307

NUPRTD KPSERVUTK L971emusyIZ5Z ©eSURFACE ELEVE

MIT FUPEL ©

Figure 106

anne

© = OVERLAP
<= OUTLET
'
1

B10 25

1945 2240 2493,

no

TELPEPATUPE IN PEGKEES

aonne.
Vad

lao

m2

492

a)

10,0

30602

seSURFACE ELEVI

MUPRTD REDFRGIA L9TLeeUAYE LOD

vir voren #

Pigure 107

134


Selective withdrawal---

G)
(ii)

(iii)

(iv)

Ww)

(vi)

Up to five outlets are allowed
Based on Debler's experimental results (10) the critical
Froude number, Fue is 0.24, therefore:

%
0.24 = 4 Po (3)
| ee

The withdrawal depth in meters, then is:

%
2 34
=2. =2. 4
a= 2.0 ae °-GoF (4)
°

where q = 2, one half of the discharge per unit width,

and w is the reservoir width

1

e= Bp oy? normalized density gradient

The velocity is uniform within the withdrawal zone

The principle of superposition is applied for the regions
in which withdrawal layers overlap

Withdrawal layers never extend through the thermocline, or
physical boundaries.

At the onset of fall cooling, when the epilimnetic region
is well-mixed and isothermal Debler's criteria do not hold '
for withdrawals from the epilimnion and Craya's approach)
is used. The flow is withdrawn from epilimion until the

discharge is larger than Craya's critical flow given by:

26


a ony

2

"
DAYS Jez.

0 « DaYy

rf

24,5 27,0 _29,8

SACOMPUTED TEMPERATURE PROFILE==

oramenees:

TRHPESATUTE IN DEGREES ¢

1240

SunTH HULSTUN 1953

ene:

2
MUy MODEL ©

Pigure 118

t

\ “
4--$--4.$----
-4¢-4--4-$----
ad e
‘ ier)
-4--}--.-

|
i

0 = MEASURED

1 KOHLER

= OVERLAP
1

eee

unt

repsesecry
t

yaaiunt

1
I
t

o
yaaa aad

rrr 27000 300.0” 33060 300.0

21900

TAS

180g

23040

seCO4PUTED OUTFLUM TEMPERATURE==

SUITH MULSTuY 1933

MAT MOMEL 1

Figure 119

145


LS

ELEVATION ( meters)

500

480)

460

440

420

FONTANA RESERVOIR, 1966
DAY: 335 ( DEC.1.)

~~ MEASURED

COUNTED AS POWER

© © © WITHOUT INFLOW/OUTFLOW

@ @ @ WITH INFLOW / OUTFLOW

PLANT COOLING WATER

Figure 10.

ite} 2. 14 16 18 20 22 24 26
TEMPERATURE (°C)

Cornell model test run results, December 1, 1966

28 30


99

Table 5.

PARAMETERS VARIED IN SENSITIVITY ANALYSIS

Units Average Low High
Values Values Values
Vertical Increments, m 2 1 3
Fraction of Radiation Absorbed
in Top Meter of Water 0.50 0.20 0.40*
Average Absorption Coefficient
of Water, m- 0.75 9.05 1.40
Vertical Diffusion Coefficient Molecular 30 Times 1000
Diffusion Molecular Times
Diffusion Molecular
Diffusion

*Not the high value


EPA-660/3-75-038
JUNE 1975

Ecological Research Series



an n of 0.05 is used. This is also verified by the statistical
analysis reported in Tables 10 and 11, where the standard error of
estimate for an n of 0.05 is twice the standard error of estimate
for the water temperature at the outlet level for the other eta
values.

In Figures 63 to 68 the effects of a variation in the diffusion
coefficient from molecular diffusion to 100 times molecular diffusion
is shown. It can be seen that 100 times molecular diffusion generally
gives poor results and only on day 121 does 30 times molecular
diffusion give predicted temperature results closer to the measured
values than does molecular diffusion. It can be seen from Table 10
and 11 that overall 30 times molecular diffusion predicts similar
temperatures to those predicted using molecular diffusion and that

using 100 times molecular diffusion gives measurably worse results.

92


Tq | | —| by
: : - : H be
a a “i | 4 ad
5 . if | i .
- : por: ot -4 r: | mi +--+ ope =
, : ! , :
r 1 is
1 q = x=
| : - 2
‘ : lo < s
—-L-$L--E | fp $-p |g} gh tg tg ge z
F eR 8 3
4 F , , Bis
5 4 Helo 4 z
: Perel 2 2. #
geahes in ee
--Lath 4p gg ~ Sesser tiie ow 4
‘Bema le bry 2 :
fi PReaee 2 8 =
; wee see = r Og
“ ° v 3 2
, : Hoe f of
1 lo 2
pe oe | eee a ee ee 4 ene
Ee s 38
é: Sc:
a: o>
a: B=
27 . 2?
z in
: s
. eu Fs
gz fg
2: &
fs Zs
Zs .&s
£3 ze
= & s Fa
cad fey 3
Zz =
om 2
8 3
3 3
3
2 2
z =
. .
Z iz
A 3
2 2
s a
= =
=
2
s
=|

166


and

= .
z 3
z z
E E

3

TEMPERATURE [N DEGREES ¢

CHEROKEE RESERVOIR 4oprenDAVIZN

TEHPERATURE 4 OEOREES
CHEROKEG RESERVOIR |9pve~oAytZLD eobURP:

MIT MODEC ©
MEY MODEE ©

Pégure 75

Phgure 74

11s


Table 15
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL

Reservoir/Year: South Holston/1953

Time Period Covered: 120th - 330th Julian Day

Std. error Correlation
File Name of estimate (°c) Coefficient
STAND 2.88 0.87
DELZ 1 2.72 0.89
DELZ 2 3.02 0.86
BETA 1 2.95 0.87
BETA 2 3.08 0.86
ETA 1 7.07 0.00
ETA 2 2.49 0.90
ETA 3 aeaiad ~oo
DIFF 1 3.27 0.83
DIFF 3 4.19 0.66

143


ABSTRACT

The deep reservoir model with one-dimensional assumptions can be
applied to a reservoir or lake where the principal variation of flow
characteristics is in the vertical direction, Among the models evaluated,
the MIT deep reservoir model appears to be most easily used and to give
results most compatible with the measured temperatures, The temperature
predicted is strongly dependent upon the magnitude of the absorption co-
efficient of water, and the diffusion coefficient. However, our sensi-
tivity analysis shows that an absorption coefficient of about 0.750
and a diffusion coefficient of 15 to 20 times molecular diffusion are
appropriate choices for the seven IVA reservoirs studied, The determina-
tion of whether or not a reservoir model depends on the Densimetric Froude
number, However, the representativeness of the result is not solely depen-
dent upon the Densimetric Froude number, By the use of a fitted curve to
the measured temperatures, it was possible to determine the maximum standard
error of estimate for the predicted outlet level temperature, 1.6°C. Temper-
atures on individual days may exceed those values and they surely are ex-
ceeded at other depths in the reservoir, These limits are suggested as the
limit of accuracy of these types of models.

This report was submitted in fulfillment of Grant R-800613 by Vanderbilt
University, Nashville, Tennessee, under the sponsorship of the Environmental

Protection Agency. Work was completed as of June, 1975.

ii


28

ino

TEMPERATUME IN DEGREES C

HIWASSEE RESERVOIR 1967=DAY;36¢ ewSURFACE ELEV!

M45

Pigure 170

434M

MJY MOEL ©

175


TEnPESATURE 1S DEGREES ¢

24745 6

seSURFACE ELEVE

seokv 1280,

FORT LOT YY 1971.

MYT MODEL ©

Figure 200

S eteosi

-4--$ =

oe

°
{goog
By &

oa?

<> ouruer

1

1 .
IT 1955 2240 aad amo -

Hed
TERPERATURE [+ PEGREES €

z
e

teed

rr

200

MaSURFACE ELEVI 2460277)

met yt3G

195

pmat LOy7y 197L

Figure 201 ary MUNEL #


--4-

4

ooo

4--4

ad

303,3
20,8.
295,35

mene as 9-4

na--¥--4-$4--4

nana peg 4 --

otis

«
ee. ne es

29055
2883

22.0 2408 22.0 2945

198

3

HD

g
FS
2
¢
=

t
;
Fr
3
3
é

DNUGLAS RESERVUIR 1909=:

MAT Monet =

Figure 43

93


Table 1. ASSUMPTIONS
Assumptions Cornel] MIT Water Respurces
Model Model Engineering Model
Horizontal Yes--one dimensional Yes--one .dimensional Yes--one dimensional
homogeneity stratification in stratification. in stratification in
vertical direction only vertical direction only vertical direction only
Primary Nonlinear interaction Differential absorption Differential absorption
mechanism between wind induced of incoming solar of incoming solar
for the turbulence and buoyancy | radiation radiation
formation of | gradient 2
thermocline
Surface one of the three is Meteorologic Input Meteorologic Input
pany specified in a sinusoidal Water surface temper- Water surface temper-
co ons orm ature calculated ature calculated
{) water surface tempera-
ture T,
ii) heat flux at surface
q
iii) equilibrium temper-
ature Te
usually TE is specified
Bottom Yes. Yes Yes
boundary
condition
(no flux)
Water budget | No No (the reservoir sur- Not directly; however
(water losses face _levation is cal- the measured daily sur-
due to evap~ culated as a function face elevations are
or_tion and of initial surface used as pool level for
gains to level and the cumulative | each simulation day.
rainfall) ‘inflow and outflow; the | The water budget impltes
‘input measured pool ele- | evaporation, rainfall
vations has never been and possible leakage
used
Advective No Yes Yes
heat (add

or subtracted)

14


L6t

Table 21
LEAST SQUARES CURVE* FIT FOR MEASURED SURFACE WATER

TEMPERATURE
No. of Data Std. Error, Correlation A B

Reservoir/Year Points of estimate (°C) Coefficient (°c) (°c) y

Fontana/1966 27 1,22 0.98 16.96 - 9.72 0.927
Douglas/1969 31 2.23 0.97 17.60 -12.82 1.180
Cherokee/1967 37 1.53 0.97 16.70 -10.08 1.006
Norris/1971 48 1.30 0.97 17.46 -11.54 1.065
South Holston/1953 12 1.79 0.98 17.01 -11.63 0.970
Hiwassee/1947 12 1.70 0.98 19.07 -10.43 0.850
Fort Loudoun/1971 12 1.50 0.99 17.88 -12.32 0.992

*Curve Type: A+ B sin (22y t + ¥)


Table 8.
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL

Reservoir/Year: Fontana/1966

Time Period Covered: 120th - 330th Julian Day

Std. error, Correlation
File Name of estimate (C) Coefficient
STAND 1.19 0.96
DELZ 1 1.25 0.96
DELZ 2 1.16 0.97
BETA 1 1.19 0.96
BETA 2 1.22 0.96
ETA 1 2.88 0.77
ETA 2 1.21 0.96
ETA 3 1.27 0.96
DIFF 1 1.15 0.97
DIFF 3 1.25 0.96

73


26706 4

oo

0 » MEASURED

t-te $tann

seSURFACE ELEVI

emp ay 206

TEHPEFATUPE 1" MEGREES €

FORT Cunpnuy 1972

MyT MOREL ©

Pigure 176

49.0

2a

273 8

22.0

1
‘
l

<= ouTLeT
1905

ine
empay1220. -=SuRFAG ELEVE

183

3
3
ed
2
Fd
&

2
PI

wooHp----
H

pe
eT Huu 1972

Ted
17 Hunet »

43

Pigure 177

roe)

20,0


(iv) In addition, the program was modified to call a subroutine TPLOT to
plot the simulated temperature profiles for chosen dates and to store
the results in the computer disc. A routine was also written to

read and plot the results of the sensitivity analysis.

One of the important parameters is the amount of evaporation. There
are a number of empirical formulas available. The MIT model uses two

different evaporation formulas.

Kohler's q, 0.000180 Wo (e, ¥e,) (84)

a

and Rohwer's Ea (0.000 308 + 0.000185W) ele, Ye_) (85)

a
where:

mass flux in kg/day-m’

Fi

density of water in kg/m?

ry =

W = windspeed in m/sec

e, = saturation vapor pressure of the air at the
temperature of water surface in mm Hg

es = saturation vapor pressure of the air at tempera-
ture Ta (air temperature) in mm Hg

y = relative humidity expressed as a fraction

The major difference in the two formulations is that Kohler's shows
no evaporation at zero windspeed while Rohwer's does indicate evapora-
tion at zero windspeed. We have found no apriori means of deciding
which is most appropriate. However, the test runs indicate Rohwer's
equation yields better results only in Fontana Reservoir; both give

about equal results in South Holston Reservoir and Kohler's shows more

68


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 148

Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 209

Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 267

Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 302

Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 364

Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 80

Hiwassee Reservoir - Extinction
Coefficient Sensitivity, Day 148

Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 209

Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 267

Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 302

Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 364

Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 80

Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 148

Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 209

Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 267

Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 302

Hiwassee Reservoir - Diffusion
Coefficient Sensitivity, Day 364

xiv

167

167

168

168

169

169

170

170

171

171

172

172

173

173

174

174

175


\ '
dng enn gett en poe ed penne penne pene a
TY , , ; ¢
H : &
4 ! i
‘ cans COVES OORInS SENS EERES GOERS GREENS SERIES GUEIEIES 54
y ;
; a
t
4 H
is Saeenns Sane SOEs SOON SONOS SEES SERS SUES SOON
;
H Fo
H
e fo 4 i
4 ! {
to; E
id H 2
2 H Seana Sennett Seanetenne Senate tetetetent Rateneieeds teneneteh Seneates tenetetenk tenenanek 07
{4 H a
q i
= H Hl e }
= 2. &
Fa Se 3h
. ----}----}----}----}----4----/----}2aes%2 a
3 H PREcES z
¢ aren
3 enwnay
ef
$e ~ wT TS
g-
#3 H
27 H H 4
5 } H *
z H H i H
fe H
ca wu--}----$----p----} ---de gedee db2
a a
5
2 H
2 : H H
3 H \ La
: ---- ----}----$---- ----$3
z ;
Z . i 2
2 ----}----}----}----}----}----}----}----d----42
= ; 4 A ;
Fd 4 H '
H
2 i ---! %
2 - -
Fs

meSuPFAcF ELEVE eS Se

Peters

189

LenseraTure bY MERNFES €
wn

Peay tu ly

eT MDNEL ©

Figure 189


The use of a horizontal thickness of one meter in the Fontana
Reservoir did simulate thermal profiles compatible with the measured
field values. However, the extrapolation of this conclusion to
other reservoirs may not be true. It is very likely that in some
reservoirs a layer thickness other than one meter is necessary in
order to satisfy the criterion mentioned above. In addition to the
encertainty of simulated results, the use of one meter instead of
two meters or a larger thickness always means longer computation
time.

The Fontana Reservoir is probably the best example for verification,
since it is very deep, 100 meters, and has useful storage of more
than one million acre-ft. It is concluded that large variations in
temperature with changes in horizontal element thickness is a serious
drawback of the WRE model.

(ii) With Variation in Diffusion Coefficient Only --- After the
coding error in the diffusion term was corrected, the 'effective’

diffusivity recommended by WRE is equivalent to:

-7 2
Do = 2.5x 10° m/sec, E<E, (45)
= -11 -0.7 2
Do = 1.5 x 10 (E) m/sec, EE, (46)
where
-1 ap

E = the stability of water, = = (47)
I

el

E, = the critical stability parameter = 1.0 x 10° Snr

which is of the same order as molecular diffusivity, 1.4 x 107? n?/sec.
Test runs using a diffusion coefficient one thousandth (1/1000) of the


27.0

oe sured
1
!

2965

10 wd creer) ino 195 22.0 2445

ad

a

TEMPERATURE IN PEOREES ¢

SOUTH HOUSTON 195300Ay1 78

520.9 4

=*SURFACE ELEVI

Figure 138 ayy monet ©

4

2g
1s
;

awa ee cure

|

----4----40
te
a
as Coens 4
5
. 7
----3----4$----§$----3%
: z
----$----$----}-- gS
s
. a
¢ .
= Se
Zernse
aeiies 4
cs cee f° T--\--2 ian
Fosse 2
terace
onamay
aod ---- $2
H
wenn genn-t----$----42
----$----4$----} -- 2
a 4 4 =
= ---}----38
-g-¢¢---- go-to :
° 4
oe
----$----$----} ----3%
<

TERPERATURE JN DEGREES ©

S242

=eSURFACE ELEVE

155

SMUTH HALSTON 1993=90AvI242

MIT MONEL *

Pigure 139


3

see seas :
---¥}o ow}. } bs

2 2 ¢ 3 8

¢ 8 ¢ $8 38

Owens wz cunwe

ie

&

5

5
a---48
Bid

$
----}3
anna
---- $2
a--- $3
3

TEMPERATURE IN DEGREES ¢

FONTAMNA RESERVOIR 19562-0471 75

500.3 4

SURFACE ELEVI

Figure 22 ry HODEL ©

s .
ee es es Cs
. gass3

ewarr a2 curwe

12040
135.0

150.0

45 0 9.5 32.0 1665 17.0 195 2240 265 27.0 2943

2.0

TEWPERATURE IN DEGREES ¢
FANTANA RESERVOIR 1986-=DAy1192 SURFACE ELEVI

50844 H

Figere 23 uy woneL ©

80


— —-,—--L-,L--L5-| --k webby pee be. Lot,-L-_|,--L_,,--L Lf, b. -| —t-
H 4 5 | : : : 5 i
i en a cae ee ee a ce
_ LHL ; __L ih EL FL tL ft mi
‘ / : ; : : |: fio did 4 4 a
? po
oso eR? LL,
ay : ‘cia ' 300 | A ; At
poo ls s 4 5 | Boe ' ; 4 120 se
[rn Sen aa b . Le. . 182-025 s
5 gS ot i —-o —frcoo pot BES ERE —-- FS
Fs Pbcse $
‘ 3 oe
. 2 4 conagy
: : : : toe t 4 i c : . j e
a ——--}----}----}----I2 3 a SS rs Ss SS SS SONS SS SNS SOS 2
4 4 ; be Ss : 4 4 q 5
| g3
gE : : :
bes w---}----}----}s
tek =
Eee ts
2 ,
zs
2 e ; ; e
Bes canals iene nO 1
RES f 4 kt
ig
. 8 ‘ :
g ' *
| oe
e ,
2p Ug
3 ;
: ‘
= ‘ ‘
3
, &
Fg
_& & | ie '
i sft :
L : :
b ‘ of kL e
z g | BIA é

44000 8

169

TENPERATURE JN DEGREES €
HIWASSEG RESERVOIR 1947e=OAYE 80 eeSURFACE ELEV!

ayT MODEL ©

Pigure 159


10 see.

----49----4----

30,0

H 7
€ i H
SANs essa
presaeenaa ina
7
©

y

“30,0

i
i
ote:
1

Se 3
23295 e--—
ghbose
ontere ai
i
|
oon
a
q

1240 165 17.0 195 22.0 2he5 27.0 2945
TEMPERATURE IN DEGREES €

mS

40

50302 4

weSURFACE ELEVI

SMUTH HOLSTON 1953-<DAvI299

MIT MOREL ®

Figure 136

S
3
2
3
é

ee mureer
!
1

2240

100,0

27,0 2905

25

1945

“M48

ideo

17460

oe no ore

ao

TEHPERATURE IM DEGREES €

5022 H

weSURPACE ELEVE

SPUTH HALSTON 1953<=DAy1362

Figure 137 ny7 MODEL ©

trnow 8

‘154


surface layer, whose volume varies with time, will cause serious
errors. To approximately account for the heat balance in the surface
layer, one should calculate the total heat content of the surface layer
at each time step. Suppose Yor Yy» ¥2 and yz are the surface elevations
at the beginning of each day ty, ty, tz, ty, etc., as shown in Figure
11. A linear variation of water surface is assumed. The mean eleva-

tion, Yp Yo and Y3 should be used at each time step, t,, t,, tz for

calculating the thickness of the surface layer water surface area and
the absorption of the solar radiation at different levels below the
water surface. The heat balance equation of the surface layer can then

be written as:

cot Dy. = cot Oy. + (COT Fin PT out out * Qy * Qy * Q)rat
with V, = v,, + Mat
where:
Vy, ° = volume in the element corresponding to the
present surface
Vs = volume of surface layer at the end of the
current time step
k = time step
e = density of water
Gin = inflow rate

Gout = outflow rate

heat transport due to diffusion

cn

59

(75)

(76)


Z0Z

Fontana
Douglas
Cherokee
Norris

South Holston
Hiwassee

Fort Loudoun

TABLE 23

DENSIMETRIC FROUDE NUMBERS FOR SOME TVA RESERVOIRS

Mean Annual Normal Maximum Reservoir Densimetric

Length of Runoff at Dam, Depth, Storage*, Surface Froude
Lake Miles 10° acft/year Feet 105 acft Area, Acres Number

29.0 2,667.7 432 1444.3 10,670 - 006

43.1 4,830.7 129 1514.1 30,600 +06

59.0 3,316.6 150 1565.4 30,200 +05

72.08% 2,956.1 196 2567.0 34,200 02

24.3 735.4 240 744.0 7,580 005

22.0 1,369.9 252 438.0 6,080 01

55.0 9,949.8 74 386.5 14,600 1,02

* At maximum controlled elevation

** Clinch River Arm Only


Table 3.

FACTORS INVOLVED IN ANALYSIS

Factors Cornell MIT Water Resources
Model Model Engineering Model
Diffusivity Eddy diffusivity Molecular So called “effective
diffusion Hiffusion" accounts for
Eddy diffusivity (constant) turbulent diffusion and
>> molecular neglects konvective mixing cal-
diffusivity eg culated from temperature
turbulent Pe
diffusion profile |
c 1 c
Ay
DL = AE” for £2 E,
Where E = 3 fe stability
pf the water column, Ee
ome critical value of
tability
Stability mixing Free convection Convective Convective mixing if
process if stratification | mixing if negative temperature
is unstable negative gradient occurs
temperature T
gradient ar <0
occurs y
oT (T - temperature,
ay «<0 y - elevation,
(T - tempera- positive upward)

ture, y - ele-
vation, posi-

tive upward)

Heat transfer in

control volume

(horizontal layer)

1. Direct 1. No 1. Yes 1. Yes
absorption

2. Diffusion 2. Yes 2. Yes 2. Yes

3. Vertical 3. No 3. Yes B. Yes
advection

4. Horizontal 4. No 4. Yes H. Yes

advection

ty
a


Ce eed

my

: : be
~---}----}----}_-_- PL ------- 2
H 2
Saat ete Se A Sd 3
i . ; od
wo--d----}---p---- Pb 3

1060
2040

TEMPERATURE IN DEGREES ¢
HIWASSEE RESERVOIR 1967==DAYI267 P=SURFACE ELEV!

Pigure 150

434008

yy MODEL *

© cofo
2
----}-2-.6
Pewendes
ion
-
---- 4-8
7

ae---

—

os

= OVERLAP

°
$
g
e
i
5
a---3
z
res 54
3
fa
=
5---33
ate
:

1
(
I
i

100,0

1740
TEMPERATURE JN DEGREES C

1445

lao

408 i.) mS

20

Figure 151

44003

=eSURFACE ELEVE

HIWASSEE RESERVUTR 1967==0Ay1302

MU? MUNEL ©

165


10.

11.

12.

13.

Hanford Engineering Development Laboratory The Colheat River
Simulation Model. HEDL-TME 72-103.August, 1972.

Beard, Leo R. and Willey, R.G. An Approach to Reservoir Temperature
Analysis. Water Resources Research 6(5): October, 1970.

World Meteorological Organization, Measurement and Estimation of
Evaporation and Evapotranspiration. Geneva, Switzerland Techni-
cal Note 83, 1966,

Water Resources Engineers, Inc. Prediction of Thermal Energy

Distribution in Streams and Reservoirs. Prepared for the Department

_ of Fish and Game, State of California, Walnut Creek, California.

June, 1967.

Debler, W.R. Stratified Flow into a Line Sink. Journal of

Engineering Mechanics (ASCE)85(EM3): July, 1959.

Craya, A. Theoretical Research on the Flow of Non-Homogeneous
Fluids, LaHouille Blanche 4(1):56-64 January-February, 1949.

Huber, W.C. and Harleman, D.R.F. Laboratory and Analytical Studies
of Thermal Stratification of Reservoirs. Massachusetts Institute
of Technology, Cambridge, Massachusetts Hydrodynamics Laboratory
Technical Report No. 112 October, 1968.

Kao, T.W. The Phenomenon of Block in Stratified Flow. Journal of
Geophysical Research 70(4); February, 1965.

204


By Equations 22 and 29

at + oe = At, » (k+1)
Ka Ppa tO RK 2 I 700s G3 14,00)
-p. D4. 9 1) 0) (K) ah)
= PG Sr Sita Geta hs GY
Let
= At
$3,171 6
= p.cav.+K, , -At 33
$j,2.7 PIATING 2 “2 (33)
= At
533° %,3-2 (34)

= p(ktl) , (k+l)
F, = PY eg

(k)
j 5 -2K5 1 * 05K) 2 * 5 KG, (35)

Then Equation 31 becomes

S551 BP + 8) BOY + 85,5 Ba) = Febtenn (38)

The system is a set of implicit equations and can be solved by the
Thomas algorithm and transforms into an upper bidiagonal form. The

coefficients of this new system designated by S! ,, S! ,, S! , and
j,l? “J,2? “3,3

B are as follows:

tos j=2,3 --

Si 1705 592,35 --N (37)
1 = j=" --

St =, 521,2,3 -- N (38)
toe

51,3 = §1,3/81,25 Fy = Fy/Sy 2 (39)

32


a x --|-,--~,-4-- 4-4
: q H Et Tt 3
i t
| | ols
a oo feet tao j----}----}_---{_---fe
hoo pits, 4s
ay . | idl an
nes Sees Sens San ~---} -4.- 4 ---}4--- Ft an Sane Sees oy
3 es a é
: Bad oc) , . 4 4 if
ST 0 : i °
afb bb ele pb bie g poo} b bbb bbe
° ° 5 ° °
, | de os E : : ss
© geagzu i, © Secnat
wo--}----}---_}----}----}----$----$----s S3BSge2---$2 og wu-a}----}----}----p---_ gE bese te
Pipi 2 3 gHige 3
° Beghgs te 3 ° Boubse fs
oetuee : vetere
onnagy 3 H soamey
of .
a uo--}e---fo---}---- : poe ged pebble
$e: f f i s
. $3 : H
7 H z 7 .
es wo--}----}----}---- d wo--}o-- peepee pdb pte
i } 3 ‘
w---}e---de--- wo--}----f----}---}----}----de Be ween feee ede bee dee 2
; igi ~o-}----}----$----f 2
3 ‘ , , f s
ne Sans SE Ss SOE SOE SORE SO : -- -- 2
: : -}----$----}----}---b ts
i .
woo $----}----}--- fee fee fe y---}2 : 1 H g
, j , roy fe ae) dk pppoe
‘
----]-4--f----i, ----}----$----}---}- fds
Aoi ogo: £ i
Pel 8 1 q] : : w q
\ x
~~ §- ~---}----b.--- be
2 le 4 8 2 a
gi if | 4 g 2 8 8

295.0 5

SURFACE ELEV?

TEMPERATURE 1° DEGREES

DRUCLAS KFSERVUIK 1969"=uAyI250

MUy OPEL ©

Pigure 55


LE

500

+
i)

ELEVATION (meters)
o
3°

b
d
oO

420

FONTANA RESERVOIR, |966
DAY: 335 (DEC.1)

MEASURED
eee Az = 1.0m
poo Az = 1.5m
0 0 © @ Az= 2.0m

4 6 8 10 12 14 6 i@ 20 22 24 26 28
TEMPERATURE (°C)

Figure 4. WRE model -- Effect of Thickness of Horizontal Layer (Dec. 1)



oy :
posing tp ge
: : $
£9
1 wl a
t e
Fo | j : : s
i]
~ ' 4 a
] q___}----f----}.---$----}---- f.--- 4 ---- ft --- 43
a Hi é
= ‘ ‘ x
ee! te
ee aou-f----$----g----}----}----i2
a 8 re
a i a
=. k e :
zi 4 2 o.8 Fe
gig 2 nb }
2 Senn dh 2
ry q aeaazs ny
7 oo. w---}----$----$----f228882---32 5
e $8 : SEGGS5-—“ts 8
= 8 BEPES3 {2g
z teieee 3
3
2 cane ay 2
;
° wf
oe ' eg
= Be ieee Sena teen Genes ae 28
es : H q rg
: > : H $
Bz 3
aif qi
4 wa--}----}----$---- 2
fa H ig
5 2
a 3
Re ze
2s d___-f2¢ § 2
gee -- Gf
x >
é oy 3
2 & © 2
; 5 3 eo “ot « sfeoe to 3
ee Serereees Saeereeas - * - H 3 4 .
? a $----$----9 8 . Hoo $$ ---- 4} ---- $ ---- $ ---- ¢---- f---- be +
H eo: =
H 7 i , H a
, t
H . . .
}----$----$----4----}----} ----f epee oF : H _fe 2
H 3 : / 4 ~$3 poe $e }---- w---$----f----}----}----12 9 ¢
H t , 2 , H H ?
i H . H +
----}----}----}----$----$----f op s A os
-f----}----Is ----}----$---- ----12 8
= 3 i = 3
: , : H g H ¢
q i 7 4 3 : 3
H H Zz
} ns Sees See ee | . q v >
-- ~--- $----$ 4} fe u~u--$----$-v-_$_---$----$----$----$----$----$---- 12
« < < py a 3
* = é = 2 2 8 2 2 2 © 2 < ° ¢ © < © «
g by Py é Py ¢ * : é = € ey q = " S ¢
a $ 3 g ° 3 By g 8 < 8 ¢ : 8

188


a Ts

ot eo a a a a
a —— 3 ‘$----}----$----$----$----}---}----}.---}__- ds
aa ‘ ° e q a 1 2
et ae aes ee ee 2 an nen Sen ----f----}----$¢
os te Ca 3
Sse iasig sds a PoE 4 bg B
-e]e 23 388222 5 2
} ry J 5 ted hood ; Hose |
—--—f-~--#---_ 3 -. “4 ¥ py ---— $ ---- 3 ---- 255 SSe--- 25
q = a a a a a es Ps ET Se eee Of
Fi BsaSEs Y=
A : teseee
3 cannes
a a ee ; ot & | ae bod a FoR
4 Ef F 4 4 5 ~ ~ : q 4 =
; g: ; : q
’ . 2h , } .
an a oe zee ee ee eee Se SS ee a
24 j : if
tgs : 4 d q
ff : : i
ae?
i. =
E nl an wap s

AyY wonet ©

Pigure 88

90 senneesee

1169

122

TEMPERATURE 5M OSOREES €
CHEROKEE RESERVOIR 4907e=OAYI 06 eaSURFAGE ELEVI

MYT AUREL ©

Figure 89


Stay SHH

- St
Sha1,25je1,1 “$},3 (40)
1 : Ss. F!

Bjel _ jt ~ +1,1

j+1,2 ~ °j+1,1 “95,3 (41)

a

then Ty = RY (42)
and
i, = FS) 5 Teed »2=N1, N-2, --, 2,1 (43)
aieD = aft) +t, (eed) AE 21,2, -- N (44)

This is an implicit method of combining an explicit finite difference
and an implicit finite difference scheme. It has advantage of unconditional
stability at the cost of complexity of computation and longer computa-

tion time.

Verification

The model was verified on Fontana Reservoir for the period from
February 20, 1966 to December 31, 1966.

Results of test run ---

Since the WRE model did not furnish the test run data, a set of input
data for Fontana Reservoir was prepared in this study according to the
WRE program. Based on the sensitivity analysis of Fontana Reservoir for
the period from March 1, 1966 to December, 1966, the following effects

on temperature stratification were observed.

33


the middle or deep portion of the reservoir than the use of a larger
increment. Because there is little difference in the predicted temperatures
using either 1 or 2 meters, the use of 2 meters per layer is recommended
since this reduces computing time considerably.

The MIT model recommends a minimum of twenty layers. From the results
of this study, it is recommended that AY=2m be used unless this causes
the number of layers to be much less than twenty. In a reservoir of one
hundred meters or more, a thickness of 3 meters appears to be satisfactory.
2) Fraction of Solar Radiation Absorbed at the Water Surface, 8

The value of the surface absorption ratio, 8, is generally. assumed to
be about 0.4. Recent TVA field data suggests a value of 0.24 in Big Ridge
Lake and Fontana Reservoir. The higher the value of 8, the lower the tempera-
ture profile is likely to be, since a larger portion of the solar energy
absorbed at the water surface means less energy geing transmitted downward
into the body of water.
3) Radiation Absorption Coefficient, n

The value of n may vary with the time of year, being a function of
turbidity of the water. The values selected for analyses ranged from 0.05
for clear water, 0.40 and 0.75 for intermediate waters to 1.4 for highly
turbid water. Results on all reservoirs using 0.05 show that too much
solar energy was transmitted to too great a depth. All reservoirs tested
contained more turbidity than distilled water. Temperature profiles fol-
lowed a general pattern, being lower for higher nvalues. Because of the
temperature sensitivity to a change in n, a carefully measured value is
important to thermal simulation. For prediction of temperatures in unbuilt
reservoirs, n, between 0.75 and 1.40 can be used for the preliminary study.

199


z
id
z
2
z

247.7

FPRy Luvey" 1971 es0AYN132 ==SURFACE ELEVE

MIT MOrEL ©

Pigure 196

aaa-$
H

Wa F----

te
s
s
&
- - <
Fd
2
2.
2 ats
aecoes *
emia &t--1-\-5- en
Bhece2 £
vereas
ohnmay
2
5
w---}----f----$s
te
i
e
a g
3
— %
g
a g
3 be 2 S
: £ é
2 6¢ nS

TRAPETATURE 1S DEGREES €

267.0 #

enyhyt204 <=SURFACE FLEW!

193

Figure 197 mT MunEL ® FORT LuDTyTM 1972


4) Diffusion Coefficient, D

The MIT model authors recommend, based on their verification using
Fontana Reservoir, the use of molecular diffusion for all depths at all times,
which neglects turbulent diffusion. Other field data indicates diffusion
coefficients higher than molecular diffusion. Diffusion coefficients
are a function of density, gradient, depth, and time. Due to the approxi-
mate nature of the mathematical model and the complex interaction in
diffusion, we are unable to assign a specific diffusion coefficient.

In this study molecular diffusion coefficients 30 times, 100 times,
and 1000 times molecular diffusion were tested. The use of a coefficient
1000 times molecular diffusion always caused the model to malfunction.
This was not unexpected since the stability criteria, Equation 58, is
violated when AY = 2m and AY = 1 day are used with this diffusion
coefficient. The use of 100 times molecular diffusion also results in
most cases, in predicted temperatures different from measured tempera-
tures. The temperatures predicted with the use of 30 times molecular
diffusion and with molecular diffusion are similar. It appears that an
appropriate choice would be 15 to 20 times molecular diffusion.

5) Reservoir Classification

The criterion which is widely used for classification of a strati-
fied reservoir, is that due to Orlob‘) who introduced a densimetric
Froude number in the form

wm - 8 JE (86)

where:

nad
u

length of reservoir in meters

200


The heat transport terms due to inflow, outflow and vertical advec-

tion are:
i) Vin >0
AT + Vy = Vim “AS Tima At + AV, Cymei Ti) (80)
+ u; BayT; + At “Uy? B+ Ay: Tym At
Substituting Equations 77 and 79:
AT + Vo = Vin “A> Cima Ty) at + u; Bay (T; “Tj pat (81)
(ii) Vim <0
AT: Vy = Vin “At Tym At +u; Bay - T; + At UB ghVT jg At (82)
By Equation 77:
AT - Vo = u,Bay (T; Th) At (83)

The MIT model uses the second approximate method. The final results,
Equations 3.20 and 3.22 in Reference 4 are correct, although the con-
tinuity equation for surface layer, Equation 3.14 appears to be wrong.
Equation 3.14 implies that total inflow and outflow are always equal,
which is not always the case. The approximation has proved to be satis-
factory as determined by verification on seven TVA reservoirs when com-
pared to the results from the first approximate method which was
incorporated into the MIT model in TVA's study(17»18) |

The WRE model also uses an approximate method in which the same heat
transport and continuity equations used for internal elements are applied
to the surface layer. It is modified only when the vertical advection,

day» is positive. The quantity (@,y'T,) is then deducted from the total

surface layer heat. No modification is necessary when the advection

62


meee:
'

----$----}----

Ww sumenOZ =e

joe

cere

498.0 sene:

3s DAVE 215

e ee
H
----$----$----$----4----
: ? ?
« mone
ab mao
$----$2---4$----
4 Seas
:
H
H
H
© anene

423.0

a

40840

740 br 1240 1645 1740 195 22.0 24,5 2740 295

nd

20

39340

TEMPEPATUPE 1) DEGREES C

FATANA RFEQERYUTA 1966004F 4SUREO TEMPEKATURE PROFILES=

MET MUPEL =

Pigure 12

are

j--—

j--4—

ny

soo

a--4

aoe

330,08

3339,8

:
~ Te, 2,
at +
“ ant
al :
4
~--"3----}--2.
2 Ts
a [72
ior
venoterSe

a = rr
2 5
a z

meemeeor Hz ewe

408,0
42308

49548

ee en
|
ees eae es Seen SaeeaS Sena SEES
wee de-i-}---- fg ded
«
7
Moe
wea f--. -$----}---- pepe pee
qe
*
aw hts sags
Res fags
b Beste S55
* woes
w--d74 pote $----b--_}_--_issss
Pree
2382
* caer
e onan
Se

433,8

6 = DAYS 335
= OVERLAP

< = OUTLET

eaee

40858

WW > 32.0 1405 11.0 1945 2200 Pad 2200 2505

4d

200

393,84

TEMPERATURE IN DEGREES ¢

FONTANA RESEKYGIR 1966eeCOHUTED TEMPERATURE PRIFILE==

MAT MUPEL ©

Figure 13

72


Gg = 1-52 / gh? 2, 2h <d, for submerged (5)

outlets

and

We = 0.75 // ad, “ho for surface outlets (6)

p
where a. = depth to bottom of thermocline
h = thickness of thermocline
ro) = density above thermocline
ptAp = density below thermocline

At greater discharges water will begin to be withdrawn

from the hypolimion.

Depth and velocity distribution of inflow ---

@

Gi)

(iii)

(iv)

(vy)

The inflow is centered at the equivalent water density of
the reservoir.

The inflow layer thickness is determined from measured
inflow data by using Debler's criterion.

The distribution of inflow is proportional to the volume
of each horizontal layer within the inflow zone.

The inflow QG> is calculated from the continuity equation:

= EG 2)

where V = volume of reservoir

%

The distribution of inflow in each layer is then modified

outflow rate

by multiplying the ratio of calculated inflow rate to the

measured inflow rate.

27


ge

—--e $---=!

3

O = MEAs
Le COMPUTED

'
i
i
1

© OVERLAP

1
'
i
1

© 87000

cays

JOMPUTED OUTFLOW TEXPERATURESS

:
5
Hy
Es

UY MODEC 1

Pigure 70

TEMPERATURE JN DEGREES ¢
EHEADKEE RESERVOIR I907eSDAYE 06 SaSURRACE ELEVI 938;

May mODEC #

Pégure 71

113


2465

22.0
av

fi
i
t

1965

3

a STAND
<= OUTLET

Ino

eed

&

1200
DOUCLAS RESERVOIR L969—=0Ay1L2L

pr

MIT MUDEL ©

2

Figure. 46

20

23,0
40,0

9900
3040

2700-25

Raed

1700 1965 2240

TenPETATURE IN DEGREES C
ONYRLAD RESHRVUTR LeDge=OAVIZ8O ==SYRFACE ELEVI

120 thd
97

pt

7.0
HUY MOMEL +

or

Figure 47

pr)


+0

340

10,0

13,0
20,0

23,0

Po

a0

90,0

4240.

1965 2240 2605 270 2545

170

TPHPESATUBE IN DEGREES C

145

10 pr 12.0

5

200

20,0

29908 6

a-SuRPACE FLEVI

OAUCLAD RESFRVUIX L0b9e=DAyI121

MIT MOrEL *

Figure 64

g----

<sruner
1
'

roy 1h 1905 2240 had 2740 2%

TenPesatun:

leo

3
ee"
£8

z

299.9 6

auATIIAG weSURFACE FLEVE

OMYCLAS KFSERYUTA 10%

May MUNEL ©

Figure 65

106


© ~ HEASUR
STAND.

1

-4

—— ta

TENPERATURE I* CEGREES C

FART LOU yt 3974

weSURFACE ELEVI 24705 5

smue i226

MgT MUNEL ©

Pigure 198

oe
1

q
eee ee

3

-4--$----
r

$ e

Se
Senn Sh
perez s.
Bzkkee---
Seons 3
ghecse
renege
onnmay

49,9

eo ae

20,0

2200 Bhs 2100

1905

Wo 3? Yew Vea? iW
TENPESATURE I! PEGREES C

a>

aay

meSURFAC ELEVE 26726

wnat 4283

wth

FAT Lut!

*IT mOnEL ©

Figure 199

194


Table 13

STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

Reservoir/Year: Cherokee/1967

Time Period Covered: 60th 300th Julian Day

Std. error Correlation
File Name of estimate (C) Coefficient
STAND 2.07 0.95
DELZ 1 2.09 0.92
DELZ 2 2.11 0.95
BETA 1 2.06 0.95
BETA 2 2.01 0.96
ETA 1 1.49 0.98
ETA 2 2.25 0.95
ETA 3 1.88 0.96
DIFF 1 1.90 0.96
DIFF 3 1.72 0.97

111


TECHNICAL REPORT DATA

(Please read Instructions on the reverse before completing)

1. REPORT NO.

EPA~660/3-75-038

3. RECIPIENT'S ACCESSIONNO.

4, TITLE AND SUBTITLE

Evaluation of Mathematical Models for Temperature
Prediction in Deep Reservoirs

Is. REPORT DATE
Jume, 1975

6. PERFORMING ORGANIZATION CODE

7. AUTHOR(S)

Frank L. Parker, Barry A. Benedict, Chii-ell Tsai

8. PERFORMING ORGANIZATION REPORT NO.

9. PERFORMING ORGANIZATION NAME AND ADDRESS:

Vanderbilt University
Nashville, Tennessee 37235

10. PROGRAM ELEMENT NO,

1BA032

11. CONTRACT/GRANT NO.

R- 800613

12, SPONSORING AGENCY NAME AND ADDRESS

Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330

13, TYPE OF REPORT AND PERIOD COVERED

14, SPONSORING AGENCY CODE

16, SUPPLEMENTARY NOTES

16, ABSTRACT

compatible with the measured temperatures.

water, and the diffusion coefficient.

Froude number.

outlet level temperature, 1.6°C.

values and they surely are exceeded at other depths in the reservoir.
limits are suggested as the limit of accuracy of these types of models.

The deep reservoir model with one-dimensional assumptions can be applied to a
reservoir or lake where the principal variation of flow characteristics

is in the vertical direction. Among the models evaluated, the MIT deep
reservoir model appears to be most easily used and to give results most

The temperature predicted is
strongly dependent upon the magnitude of the absorption coefficient of
However, our sensitivity analysis
shows that an absorption coefficient of about 0.75m-1 and a diffusion
cqefficient of 15 to 20 times molecular diffusion are appropriate choices
for the seven TVA reservoirs studied. The determination of whether or not

a reservoir model depends on the Densimetric Froude number.
representativeness of the result is not solely dependent upon the Densimetric

By the use of a fitted curve to the measured temperatures, it
was possible to determine the maximum standard error of estimate for the predictef
Temperatures on individual days may exceed thes

However, the

These

or

(17. KEY WORDS AND DOCUMENT ANALYSIS

DESCRIPTORS

lb. |DENTIFIERS/OPEN ENDED TERMS |c.

COSATI Field/Group

Thermal pollution, reservoirs, mathema-
tical models, sensitivity analysis,
Tennessee Valley Authority Projects

Deep reservoir models,
Massachusetts Insittute
of Technology, Water
Resources Engineers,
Cornell Aeronautical
Laboratory

18. DISTRIBUTION STATEMENT

79. SECURITY CLASS (This Report)

21, NO. OF PAGES

20. SECURITY CLASS (This page)

22. PRICE

—
EPA Form 2220-1 (8-73)

# U.S. GOVERNMENT PRINTING OFFICE: 1975—699-073 115 REGION 10


Extirag]

© = OVERLAP
1945
UPFACE ELevt

= OUTLET

TEn*eMATUIE I) NEGREES ¢

MUPRTS AESERMUIX 1971 =udyI139

a)

w IT MOmEL #

Pigure 108

29,5

2740

2a

ee eee ae eet

1
t

1

1
310.3 4

2260

& = AVERLAP
<= OUTLET
meSURFACE FLEVE

TRnPErATUPE IY DEGREES ©

MuPKID REQFRYGTR A97Le=UAYELAS

4AT MOTEL

Figure 109



TEMPERATURE 1” DEGREES ¢

207.9

meSURFACE ELEVE

MUPRTD KFSERVUTK 197 meDAy 1252

maT PureL

Figure 116

‘ssTOP oO

139


EPA-660/3-75-038
JUNE 1975

EVALUATION OF MATHEMATICAL MODELS FOR
TEMPERATURE PREDICTION IN DEEP RESERVOIRS

By

Frank L. Parker
Barry A. Benedict
Chii-ell Tsai

Vanderbilt University
Nashville, Tennessee

Grant No. R-800613
Program Element 1BA032
ROAP 21AJH/Task 12

Project Officer

Bruce Tichenor
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330

}

NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U. S. ENVIRONMENTAL PROTECTION AGENCY

CORVALLIS, OREGON 97330

For sale by the Superintendent of Documents, U.S. Government
Printing Office, Washington, D.C. 20402


0 w MEASURED
1 a KOHLER
© OVERLAP
0 = MEASURED
1 = STAND

2 = DELZL

3 = MELzz

0a 0nenaQdewenen} Leted:

1
t
1
I
1
1
1
I
i
'
'
i
1
1
1
t

ny

mMMPUTED AUTFLUK TEYPEKATURE@*

t
t
glo
l

ays

panna

NOKETS PESESVOI® 1972

maT roret +

va dp avant varazatatar

Pigure 96

2700

= OVERLAP ===
<= curser

1

1

'
1
'
1

25

208.1 6

2240

1965

129

170
TénewaTuré 1% DEGREES ©
weSURFACE ELEVI

Ved

Ved

Murty REDFRULIK BOT LeeuAYT LOO

n>

nt nuret ©

Figure 97


one

203

22.0

----F---- 3
5
----3---- 38
---~$---- aby
—---4----35
re Dae 3
aq a

s é

a a

TEMPERATURE IN DEGREES ¢

496.8

SURFACE ELEY!

FONTANA RESERVOIR 1966-=04¥1335

Figure 42 ty nope ©

90


aw, (T@,) + 4t, I, -(2- “a (62)
avi

a
where:
aTp = temperature rise across condenser
Ty = surface temperature
YW = pumping velocity

N
aw

length scale

When the discharge temperature is less than the surface temperature, the
effluent will remain submerged and S(z) is zero. The pumping speed is
related to pp» the heat per unit area per unit time added by the power

plant by the equation:

Wy = pp/0 “Cp “AT, (63)
where:
oCy = heat capacity per cubic foot of water

(112.32 Bru/°C-£t*)

at, = temperature difference produced by power
plant

‘T= Tg) T,) Cc) (64)
The thermal eddy diffusivity, Kp has the form given by Rossby and
Montgomery 5) ;

Ky = Kyolt * oR) (65)

where:
Kio = (Cy + Cyz)w* = the eddy diffusivity of

neutral stratification cet”, /day)

50


oF

ELEVATION ( meters)

500)

480

460

440

420

FONTANA RESERVOIR, I966
DAY: 244 ( SEPT.!)
—n" MEASURED

o o Az=0.6m

a)
ee @ @ Az=1.0m
oo o o Az=1.5m
@ @ @ © Az= 2.0m

°

Figure 3.

\2 14 16 (8 20
TEMPERATURE (°C)

22

24

26

28

WRE model -- Effect of Thickness of Horizontal Layer (Sept. 1)

30


i | f
TT —r ——- Se — —— ot Sotetet ieteteten Tetetenet 1-7
4 Jo. 4 RR. |-_-{§_4--}--4_t¢ 2
: «
5 a I . F ae 5 s
: : ! H ‘ :
Pook og) bla tay do : : -
q Sees Sees See ee eee es ee ee fb de
anne $f — $4 2
: - : , &
eos !
p_--- $d bpd pap fie g ----$---f--- fb tgs
. i FY : H a
z e
. z : : H q 6
2 : 3 Se
a . ‘ H BeeeZs *
2 --—-}----}----}----}----}----] ---} _---123ff22___ts
“ = : f £hec58 2
= : bere
5 : ertee
z i onaney
of q I. ‘ ;
oe a § ig yf fp 2
fBe --42
neg f Hl fi fi §
z& i
Bs i
3 t
eit : *
£23 “Ts
s& &
2
Sz :
BE -
BE
giz -dg
aR 2 : 4 3
=
* *
nS SOROS SES SENS SOREL SL SO SONS SORIE NNO} a----$----$-- fs
: r e .
q ta q fi qi . ‘
i ert 4 4
: “
: herr rr Sr , ' 2
oot 8.2 e ¢ e
2 wo fii tlds
S x
2 ;
&
= q
9 «
= .
§
z
- 2
a

50202 4

157

TEMPERATURE IN DEGREES ¢
SMYTH HOLSTON 1953~=0871362 SURFACE ELEVI

Figure 143° mj7 MUREL ®


: : yoo Go as
po eens 2 ~---}----}--. 2
: : ; 2 : | ie ees
oe 5 7 : . “ a :
eo--}----}---) . a---}---} ‘is
5 5 ¢ ‘ feet ite
de: : : : : : 4
‘anon,’ = ° x : :
ws & 3 ay
—s-}-g—}----}----+}---~}---- Fy ~-73}-.5-}----}----}----}----
ITs oo ‘ : A . :
met we H . = =
* ~ : : ; YBeeungt te Ff”
pe rs a a et et eer errr oe naonpe--ap—-,
tT. g :
en of
coil hc) . y : : ' ibe
30 fa. a-f----$2
“ 5 i
on Fe ‘ .
: 27 5 : :
ee a3 oF 9 —- -- + - 9
z2 4s
. : Se : 4 bs . . , :
4 ; 5 #3 B 5 a : : 4 / :
a--} eo @is w---$----}--- pp 8s} : he
: ; tas. ; : ; er ee ; : ; 8. ~---}----}----]----}g
z : é 3
é a es ees ees ee ae et So as cons meet Sy
_ 2 # Soeaneet SOSnSnIS SnOEEEnS SEEDS SEINE SEER e
; 2g : ; ---73 w---}----}----f----$3
nan 6 5 on . is . . “ST Secont nn . .
cera eres : : onelane :
; 3G Scott Satara Secret deems rere Core eee ee oad Em
$----}---- $b de bg x} Snes ES SORES SD 4 SORES SERENE SSE SORES SORES SOON ©

15.0
30,0
45.0
60.0
75.0
4.0
fos.0
15,0
30.0
45.0
60.0
15.0
90.0
fo5.0
120.0
135.0
i50,0

SeSURFACE CLEVE 50469 4
85

TERPERATURE IN DEGREES €

PONTANA RESERVOIR 19660mDAVIZ04

Figure 33 ary KODE ©


energy conducted from the body of water as sensible

aa

heat;

Qu energy carried away by the evaporated water;

Q = increase in energy stored in the body of water.

A three-dimensional analysis is so complicated that it is usually
not justified by the increased accuracy of the results. In most practi-
cal problems, one or two dimensional analyses will describe adequately
all the principal factors. The existence of horizontal isotherms, although
sometimes tilted slightly by the wind action and/or the effect of lag
time of inflow, and the much faster dispersion in the horizontal direc-
tion than in the vertical direction, ensure that the assumption of hori-
zontal homogeniety of physical properties in the model is compatible with
the prototype. Most mathematical models are based on the one-dimensional
vertical motion assumption and can predict the thermal structure of
reservoirs that are in good agreement with the measured values.

In a stratified reservoir, the body of water must be segmented into
a series of discrete horizontal elements to compute the vertical variation
of temperature. Therefore, heat flux due to vertical advection, and
diffusion between elements is added to Eq. 1 and applied to each element.
A schematic of the reservoir model considered is shown in Figure 1. For
simplicity, the elements except for the top and bottom are usually of
equal thickness. The basic heat transport equation and the continuity
equation are written for an element. At the beginning of the calculation,
the surface elevation is determined either from a measured surface eleva-
tion or calculated from measured inflow and outflow rates. The inflow

and outflow distribution in each layer is evaluated according to certain

10


Therefore, the most widely known and used models, Water Resources
Engineers (Orlob)3, MIT (Harleman-Huber-Ryan)*, Cornell Aeronautical
Laboratory (Sundaram) °, Colheat (Jaske)® and Corps of Engineers (Beard)’
have been collected and evaluated. After preliminary analysis only
three models, WRE, MIT and Cornell were further evaluated. The Colheat
Model was developed primarily to simulate the thermal properties of a
flowing water body where the vertical stratification is very weak or non-
existent. The assumption that convection is restricted only to the longi-
tudinal direction makes it unsuitable for use for deep reservoir analysis.
The Corps of Engineers' model simulates thermal properties within a
reservoir on a monthly basis. This time span is too large for the detail
required in this study.

The purposes of this study are:

1. To review each model, including analysis of available documenta-
tion and computer codes, tabulation of assumptions and factors
involved in analysis, listing of input parameters required and
the criteria for application to prototype in an explicit form.

2. To analyze the major differences between models.

3. To verify each model using numerous reservoirs' field data.

4. To perform sensitivity analysis of the most important input
parameters.

5. To recommend criteria for choosing suitable input parameters re-
quired to run the predictive models based on the field data
verification and sensitivity analysis.

It was anticipated that this study would provide clues in choosing

a suitable model, provide the necessary information about the proper use


3
%

0
1500
20.0

43.0

tor) $2.0

3 170
PURE IN DEGREES ¢

1
1
1
!
14s!

501.9

SeSURFACE ELEVE

FONTANA RESERVOIR 1966-90Ayi266

Figure 40 wey MODEL #

4300

| 5

°

stm

owe

ego otee--d----
wh ow.
ae

‘ 0d te

135.0

25

263 2700

220

---=32
by

3

ny

‘oe

--. &
ae

TEMPERATURE IN DEGREES ¢

49000

E

FONTANA RESERVOIR 191

Fégure 41 yyy MOnEL *

89


-—-1 ---

C2To 29.8

17.01% 22.0 the

Lead.

" ldo

TEMPERATURE [N DEGREES €
SOUTH HOLSTTN 195Se—eDAYI 76 emSURFACE FLEVI

320.9 M

MIT MOMEL ®

Pigure 120

29

1 és omer
1 1
1

1955 2200 2655 20

17460

TEMPERATURE IN. DEGREES ©
SMUTH HOLSTON L953——DAYELOZ w=SURFACE ELEVI-

165

1240

95

=
<

1

1

!
a0

10050

52602 4

MIT MUMEC ©

Figure 121

146


pang ennn gene . manny
a ‘a : is
: t
. : 4 , t
: fe 7 e
ns ees Soe 3 |e ~~} fleet
. § ; : 5
* i i . .
ees Sees Se 3 nas SennaER: Sanna Cees ROO: SOO SCS Se SOS OY
H { f * *
H ay
e ¢ .
$----}----}----3---- 2 g a ee ns as Ss Ss Ss Ss Sans Sees i)
aos g
H H z : , 2
; = 2 ods
2 : ead
bd se ! wzaaze «
~---}----}--- tog Sas Se) SS Se SO —--}335822---is
H i 2 3 TM siebeg "te
3 vreste
H 2 oo : Canney
; A ot | 4
oon ee a ee et feet ae. 4
w---}----f--- zo 0 : ~}----}----}. ---f_ fe
f s Be , oo] is
i i: 2
H gs i i
} BE } H
FA i H
atl i {
ee § --- - g  -- -  - ~ ra ---- wo 4-5 gf 5 8
H 222 £
H i Be 2
m i Be :
“4 i lo . /
i le ES
ens Serene Snes 235 ¢ ----f,---f--- fe p ede ge dpb te
H i ; oe _—— jr rn rn sr i roe 3
x a —
H .
5
2 | hoe ' ;
<!
—--+$----}$----}---. 2
3
. 3
. 7
2 ----}---- $-5-0© $----3----4---- 2
es | ses See ,
OO , oe b.°.Te ed.
5 q t q ,
8 -4----}--42
g
e
ee e 2
é ¢
@ ¢ gf 8

mae fez |cmrgee

iid af taf do

52402 4

weSURFACE ELEVE

TEMPERATURE IN DEGREES ¢

149

SOUTH HOLSTON 1953e=0Avt14z

Figure 127 Myy MUMEL *


ro

3 2
4 tf : 7
&
L a a--~}--- be 2
8 : 5
5 1 As
4s o
es 3 a---}----}----}--ds
= Fd
. :
q 2 s ry
‘ a “ ul a 2
oe E = s5 ” 2
, i . ; . : z i 42
H i F : 430 35
q 4, ---4 by ---. a —— $—. Bzove ey. 2
woa-da---$--- a f----f-- aba f + 2 os —}----}----}----}----}----} ----} ----} $3 sg8--_ts
: 2 8 : Pbrnse £
( = ! , veeeee
: j 3 yon onnmey
on a ' ef . :
4-$---+}---- Qe : , e
----$-2L a a3. a a--}----}----}----}----} pp te
: e 4 a Ss whom f 5
: « an gs ;
! sf
a
i 27? °
foot bos : aif : .
a---}--2-g----f_us_d---- fd dp tts St w---$----$2-- bpp ppb ds
fe8 “tt
? H : 223 z
3 4 : 2° , ,
. ° 4 ; 4 aa
- t . , 6a BE -
anna $—- $f Sn P53 ----. ee ----$--~--3----4----$ --- 3 --- 3 2
, f¢ re
j i = , 4 i ; i
z sown
5
4 2 F 4 :
p----$---- ---5$----4$----35% —---$---. 2
1 ~ *
” co a 7
qi . ° .
; : 4 4 af 4 * de | 4
q e e @ 2 e
~---}----}_---f__.---- 3 gee te oF we2$l--_-f__--g____ bgt)
.——* ° 3 fi © 6 ere <
, 4 : : ;
H - wales
Hi = .
3
---- = 4 me -$ ----$ ----4 --- -3 ----§ ----$t
. = 3 +
H
x
nae 3 ‘4
4 3

10090

sik 8

TEMPERATURE IN DEGREES €
PeSURFACE ELEVE
153

SMUTH HOLSTON 1953e=0AVI243.

MET MOMEL =

Pigure 135


Table 2 (Continued).

INPUT PARAMETERS

Input Cornel? MIT Water Resources
Parameters Model Model Engineering Model
Friction velocity [Yes (in terms of No No
mean, amplitude and
phase angle in 7
sinusoidal form!3)
ft/sec
Short wave radia- [No Yes Yes
tion extinction +
T/m in terms of extinction
coefficient depth (m)
coefficient = 6.908/
extinction depth
Diffusion coeffi- |Yes [in terms of Yes Yes, empirical deter-
cient (C, + C, Z) we] 2 mined constants A,,
1 2 m°/day 1
(upper and Tower Ags and Ag (i) dD, Ay
bound of the coeffi- for E< E
cient are also inputed) Cc 3
Ft7/day (77) D, AYE ~ for
ERE
Cc
= 140 stabiti
E= AZ stability
of the water column
m@/sec
Heat exchange Yes No No

coefficient

Btu/ft?-day-"c

1. May be calculated internally by program

2. May or may not have to be inputed

3. Wind speed, air temperature, vapor pressure, humidity, short-wave solar

radiation and cloud cover are necessary data for external calculation of

of the equilibrium temperature

4. Calculated internally both with and without reflection.

= cal.flux_ (with reflection
input, the net flux = Input flux x cal. flux (no 1

in input, the net flux cal. flux (with reflection)

1e

ection

If data are in

> if not


17 and 18 where the standard error of estimate of temperature at the
outlet level is almost 4 times the error for the standard case.

In Figures 165 to 170 are shown the effects in predicting temperatures
of varying the diffusion coefficient from molecular to 100 times
molecular diffusion. It can be seen that, in general, the use of
molecular and 30 times molecular diffusion coefficients predicts the
temperatures more closely to the measured values than does the use of
the 100 times molecular diffusion coefficient. This is also indicated

in Tables 17 and 18.

159


0

340

0

0 = MEASURED
Le STAND
2» OELZL
3s deze

OVERLAP meee.
SurLe

i

i

1

<

‘,

+e
1
'
i

110 1965 22.0 265 27.00 25

Wo He

3

10 %

us

TEMPERATURE IN DEGREES C

293,39 6

omSURFACE ELEVE

OAUGLAS RESERVUIR 196¢r=0Ay 127

mT mane, =

Figure 50

© = MEASURED

1 STAND
2 = BeraL
3 = eeTAz

'
t
t
l

gre

i
i
z

2248.

fOr) 2240 25 ane

1760

12400 Ves

no pe)

ot

3

TEMPERATURE. IN DEGREES ©
DOUGLAS KESERVOIK 1969-0VAYI 69 ~*SURFACE ELEVI

MgT wONEL ¢

Figure 51


1
T
i
: aaa
4
2943

2740

-en

eusedes
eugene.
RSSRRE
ceeesssette
gggsssskeis
Pea
228282322 |S
fucerserer
oanagnsew
bed
. fe
won $ nt 8g 9 § 9 - 9 a
=2e
—_ 2
4 ’ fo
°

----}----32

|PMEASURED TEMPERATURE PROFILER=

178

TEMPERATURE JN DEGREES €
FORT Lovonuy 1971

MIT MOREL ©

Pigure 171


TENPURATURE JH DEOREES €

CHERCKEE RESERVOIR 49O7=eDAYIR09

220109 H

URPAGE ELEYE

MET MODEC &

Figure 84

1 peorees ¢

5
2
3
:
2
5
Bo
Es
3
a?
:
z
H

32544 K

HUY MOOK ®

Figure 85

120


Table 18

STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

Reservoir/Year: Hiwassee/1947

Time Period Covered: 120th - 330th Julian Day

Std. error Correlation
File Name of estimate (C) Coefficient
STAND 1.38 0.99
DELZ 1 1.53 0.98
DELZ 2 1.34 0.99
BETA 1 1.38 0.99
BETA 2 1.37 0.99
ETA 1 1.44 0.98
ETA 2 1.40 0.99
ETA 3 aad —
DIFF 1 1.30 0.99
DIFF 3 "4.26 0.99

161


cy

ELEVATION ( meters)

500|

480

460

440

420

FONTANA RESERVOIR, 1966
DAY : 335 ( DEC.1!)

eee «= MEASURED
@@@@ STAND
©9000 Dg =10-STAND

6 8 10 $12 1416 ie 20 22 24 26 28
TEMPERATURE (°C)

Figure 7. WRE model -- Effects of Diffusion Coefficient (Dec. 1)

30


Joong tay ge

i |
4 a
1 f | |
Lis i eg dbp bp bb bp
i ae : i
2 i ig
Hy Zena dt
a go: iene leeeienes ees Seieieies eee Sais es SSS
z ' .
| | | E | ; ceangy
w---$----$----$---- $s se ° 5 anne STN SUCES (ES SULCUS SURUON SRUOURONE STRERES SEERENOnS SEER
eg ,
H H
e : ‘ H
¢ 3 op en ees es Os
f 3 :
card Pop bbb pe
ad _ ; : i : .
aa

Figure 44

0

”

110 190d 2240 240d 27.0255

1405

10 pe 1260
TEMPERATURE 1% DEGREES ¢
DAURLAS KESERYUIA L°69==04y) 69 =9SURFACE ELEVE

MYT MUREL ©

pr

Figure 45

20

20,0


heat transport due to absorption of radiation

%

Qy heat transport due to vertical advection

Another approximate method could also be derived. The surface ele-
vations are assumed to be discrete, instead of continuous, as indicated
by the solid line in the Figure 11. At each step the surface
elevation is held constant and equal to its mean value Vp Yo V3
The heat transport and continuity equations for internal elements are
the same as usual. However, the same heat transport equation can be
applied to the surface layer only by modifying the continuity equation.
The assumption of a constant surface elevation at each time interval
implies that a volume of water with a temperature equal to that of
the surface layer has been added to that layer if the surface eleva-
tion is rising, or is subtracted from the surface layer if the surface
elevation is falling. The volume of water added or subtracted is equal
to the vertical advection through the interface of the surface layer and
is the most immediate underlying layer. The continuity equation for
the surface layer is then written in the MIT model notation as:

u,BAy + Vit = u,BAy (77)

The Vin is different from Vin which when multiplied by the area be-
tween the surface layer and the immediate underlying layer is the sum of

the inflow and outflow for all elements below the surface layer, or

1 jn-1
Vin = @ -GBOIY ~ uBy)ay) ) (78)
These two terms are related to AV, as shown in Equation 79;
_AVs, 1
Vim Vin "i OK (79)

60


formula or criteria. Applying the continuity equation to each control
volume, beginning with the bottom element, the vertical advection
across the bounding surfaces can easily be found.

For a chosen period of time, At, the net change of heat content, or
the rate of heat change, in the control volume is evaluated. The heat
fluxes considered include that from inflow, outflow, vertical advection,
diffusion, and absorption of radiation energy for an internal element.
In addition to these, surface absorbed energy and surface heat losses,
due to evaporation, conduction, and longwave back radiation must also
be included in the surface layer heat balance. From the rate of heat
change, the final temperatures are obtained.

Each of the models is essentially an accounting procedure of the
energy budget over a period of time. The procedure iterates until
balance is achieved and stability criteria are satisfied, and proceeds
to the next time step.

The differences in solution method relate primarily to: a) the
handling of Q> the net energy brought into the body of water in inflow,
including precipitation, and accounting for outflow; b) the use of
directly measured or internally calculated meteorologic data, to account
for solar radiation, back radiation, conduction and evaporation (the
latter parameter includes the selection of the formula, and coefficients
from relevant meteorologic input); and c) the mathematical scheme for
numerical calculation.

The models differ in how they handle the inflow to and the outflow
from the reservoir (whether it entered or discharged at one or more

levels, how it is distributed over the vertical cross-sectional area,

12


135.0

30,0

an en er er .
3 ; ;
Pt : :
a a4 :
on

: : : :

3 .

4 "
2 3 @¢ @¢ 8
2 ¢ & & 8

emer e Hz cueme

&

e

S

3

a

: Rey

3

s

g

5
n-a-4----13
----}----}2
aa moar!
a oes <1
a aes © 1
2

~-$--- te
3 g

TEMPERATURE IN DEGREES €

FONTANA RESERVOIR 1966--08y1215

505.9 H

‘SURFACE ELEV:

Figure 18 wT MODEL &

~---+

ewan

see
2 2 8
fe ¢ 8

. 3 Wa
See eet eee FS
a an Sn £1
----}---. g

5 dd
2.
Sonu3h
S2stee_ de
wooo Begg8E---33
“gaaa338° "Is
Yerrnen 5
Saamgy

~---}----}--- te
S

coe ores ee
----}----}3
w---}----}----}s
?
a---}---- e
2 8 3“

# 8 &

TEMPERATURE IN DEGREES ¢
FONTANA RESERVOIR 1966--DAYIZ44 ==SURFACE ELEVI

Figure 19 rT MODEL ©

78


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

102

103

104

105

106

107

108

109

110

lil

112

113

114

115

116

117

118

Norris Reservoir Absorption Coefficient
Sensitivity, Day 106

Norris Reservoir Absorption Coefficient
Sensitivity, Day 106

Norris Reservoir Absorption Coefficient
Sensitivity, Day 183

Norris Reservoir Absorption Coefficient
Sensitivity, Day 225

Norris Reservoir Absorption Coefficient
Sensitivity, Day 252

Norris Reservoir Extinction Coefficient
Sensitivity, Day 106

Norris Reservoir Extinction Coefficient
Sensitivity, Day 139

Norris Reservoir Extinction Coefficient
Sensitivity, Day 183

Norris Reservoir Extinction Coefficient
Sensitivity, Day 225

Norris Reservoir Extinction Coefficient
Sensitivity, Day 252

Norris Reservoir Diffusion Coefficient
Sensitivity, Day 106

Norris Reservoir Diffusion Coefficient
Sensitivity, Day 139

Norris Reservoir - Diffusion Coefficient
Sensitivity, Day 183

Norris Reservoir Diffusion Coefficient
Sensitivity, Day 225

Norris Reservoir Diffusion Coefficient
Sensitivity, Day 252

South Holston Reservoir Measured Temperature
Profile

South Holston Reservoir Computed Temperature
Profile

xi

132

132

133

133

134

134

135

135

136

136

137

137

138

138

139

142

145


S49}9W “NOILWAS1S SOvVsuNs

Surface layer schematic

Figure 11.

61


Atmospheric radiation was calculated by the model. If measured solar
radiation is not available, the routine described in TVA Report 1419
was used for computation.
We used the original program unless we found programming or logic
errors. For this study several modifications were made:
(i) In Function 'FLXOUT(N)' the statement:
RAD = 1.13587 E  6* ( (TS + 273.16)** 4 0.937 E  5* (TAIR + 273.16)**
6* (1.0 + 0.017* CC**2)
was replaced by the following two statements
AR = 0.937 EB 5* 1.13587 E  6* (TAIR + 273.16)**6* (1.0 + 0.17*
cc#*2)
RAD = 1.13587 E  6* (TS + 273.16)** 4 AR
This is done in order to correct the error in the coefficient and
make a proper transfer of the parameter back to the main program.
(ii) In SUBROUTINE'SPEED(N)' the statement:
IF (EPSIL.LT.0.0) EPSIL = EPSIL was inserted into the program after
statement No. 15.
This is necessary only in Douglas Reservoir where a negative tempera-
ture gradient was formed. 1
(iii) In SUBROUTINE 'SPEED(N)' the cut-off gradient was put into the
routine for calculating withdrawal layer thickness when using Kao's
or Koh's formulae. The original program applied the cut-off gradient
only when the temperature gradient at the outlet was less than 0.01
and Kao's and Koh's formulae are not used in that case. This
results in withdrawal of a large amount of surface water such as

occurred in Norris Reservoir.

67


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

34

35

36

37

38

39

40

41

42

43

44

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 285

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 335

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 75

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 132

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 187

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 215

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 244

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 285

Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 335

Douglas Reservoir Computed Temperature
Profile

Douglas Reservoir Computed Outflow
Temperature

Douglas Reservoir Layer Thickness
Sensitivity, Day 69

Douglas Reservoir Layer Thickness
Sensitivity, Day 121

Douglas Reservoir Layer Thickness
Sensitivity, Day 186

Douglas Reservoir Layer Thickness
Sensitivity, Day 218

Douglas Reservoir Layer Thickness
Sensitivity, Day 250

Douglas Reservoir Layer Thickness
Sensitivity, Day 276

vii

86

86

87

87

88

88

89

89

90

93

96

96

97

97

98

98

99


STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

TABLE 11

Reservoir/Year: Douglas/1969

Time Period Covered:

60th 360 Julian Day

nantes [oe Sih | GREER
STAND 2.03 0.98
DELZ 1 2.05 0.98
DELZ 2 2.07 0.98
BETA 1 1.97 0.98
BETA 2 1.88 0.98
BIA 1 2.26 0.97
EIA 2 2.22 0.97
BIAS 2.15 0.98
DIFF 1 1.58 0.98
DIFF 3 blows up

95


where:
by = the quantity of radiation arriving at a horizontal

plane (Y meters below the water surface) in kilo-

calories

B = the fraction of radiation absorbed by the top
meter of water in the reservoir

oy = total incoming radiation in kilocalories

n = the average absorption coefficient of the water
(meterTM4), and

Y = depth below the water surface in meters

The degree of variation of the parameters is shown in Table 5. We have
tried to cover the range of values to be expected: from average flow
through times of 0.01 to 0.85 years; absorption coefficients suitable to
distilled water and to highly turbid water; diffusion coefficients from
molecular diffusion to 1000 times molecular diffusion; depth increments
from 1 to 3 meters; and fraction of radiation absorbed in the top meter
of water from 0.2 to 0.5.

Parameters not included in the sensitivity analysis were held con-
stant throughout the study. The original time step input of one day
was used, since all the meteorological and hydrological inputs were
daily averages measured on a daily basis. Inflow travel time within
reservoirs is neglected and a mixing ratio of 1.0 and four grid spaces
are used for entrance mixing. Entrance mixing is simply a way to repre-
sent the mechanisms not accounted for by the mathematical model and is

considered unsatisfactory as pointed out by the authors of the model.

65


A | *
ies | ee to ee So 8 Fite ieee &
: : Ring é : g
: q i He | Poyda ib
a . ha oe . aL: i an u 1 2
| A es a on ue Seen en ~ R 4 as
—f | 2 wa--f----}----}---- anes eee ere Ct
yy ~~ ro
i | te | | © ‘ .
a : oF ---}2
bopopoio a oe ee ee Ge i iF
: : °
. ‘ a a f Fy ; a a a 3 oe
: = Beeses *
---- _ ~}---- a " a }----}----$----}----  2aEEZe---12
- ee s Caiet imate Sennen SESSS5 g
3 2 gassed &
‘ = crrete
2 ounmay
bo 4 5 5 ee ee ee : fp} bpd de
q “fe os oon =
4 af ; 583 : :
Paes
; g&
4 fo BE H
4 4 fo BE H
: ”
w---$----}--- pb sit ens Conan Sanne Snes SEE 4
, — ios Soccer Seceieten Sonia Seen Seite Ta rier Senet Seine Seine Soe £
: : Y : PSs ,
' i ; . ’ 4 Z£
; 4 4 ! e5 3
on : ‘ 4 . \ ek : : ---+$2
---}----}_---}----}----}---_J_---L__}___-te Fs uueepee-- peepee ged --- $----g---- 3
t 4 4 : - BFS 3 i { i i f j q 3
A i e 2
4 p : A fi +
fT 7 a
2 ----}----$---- fete pe eps pocor ct
= z 7 q
i : : : B
. .
2 & feed 4 4 : __le
2 fsss-g .
. :
= :
‘ 8
$
be 7 2
fhe OF
i
| 3 el BR] @ ¢
¢
8; 3 B 3 g
H

44600 8

eeSURFACE ELEV!

172

TEMPERATURE JN DEGREES ©

MIWASSEE RESERVOIR 1947e=DAYI BO

NBT MODEL ©

Figure 165


Douglas Reservoir
Figures 43 and 44 show the computed temperature data for Douglas

Reservoir for 1969 at various depths and the computed outflow
temperature. These can be compared with the measured data as shown
in Figures 45 through 50. It can be seen from the Figures and

as shown in Tables 10 and 11 that the predicted results, standard errors
of estimate of 2.19 and 2.0° for outlet water temperatures and
surface water temperature, respectively, are not as good as they were
for Fontana Reservoir. In Figures 45 to 50 it can also be seen that,
the variation in the horizontal increments from 1 to 3 meters

makes a random difference in the predicted results, with 3 meters
giving better results on day 121 and 1 meter giving better results
on day 186. Overall, as shown in the statistical analysis in

Tables 10 and 11, the change in thickness in the horizontal

segments makes little difference. In Figures 51 to 56 the effect of
a change in 8, the fraction of solar radiation absorbed at the water
surface, from 0.20 to 0.5 has a negligible effect on the predicted
temperature. As shown in the discussion on the variation of the
horizontal segments, the predicted values vary randomly both greater
and less than the measured values.’ Over the whole year, however,
the variation in the predicted values do not differ markedly from
the standard errors of estimate as shown in Tables 10 and 11.

In Figures 57 to 62, the effect of a variation inn, the radia-
tion absorption coefficient from 0.05 to 1.40 is shown. While there
are slight differences from each other in the predicted temperatures
when n of 0.75 and 1.40 are used, there is a very large difference
from the other predicted temperature and the measured temperature when

91


prcccprscprs pepe
2

a---}----feewedsT bbb te
g
2 Es

eis Ses

3 2.9
~---}----}---etS_ S3F_}--_fe
3 BBE 2
3 4

ao ter
w---}----}--- $F. a---}---- bits
=

| a
a---}----}--- feat Es z
e 3

of 3 zg

H es
----}----}----}----f- bb 3
----}----}----}---- bbb eb te

°3
----}----}----}----} db ape
2 &
----$----#.---}----}----}-- BBE
e@ 8 @ 8 2&2 8&8 8 8 @ 288
8 s ¢ 8 § £§ gg fF SF &

Poxewede sew wee

navy

FONTAN KESFRYUIN 4966==CO4PUTED NUTFLLY TEXPERATUKE==

MIT RUPEL

Figure 14

120,0 sene=:

IEASURED .

onK

{33,0

22.0

ane

%

165 17.0

120

3

TO

405

TEMPERATURE IN DEGREES ¢

weSURFACE ELEVE

FONTANA RESERVOIR 1966—=0AY1 75

Figure 15 uty MODEL *

76


H
i
gi

eo

H
2

$ s
ry 7
Seonns
22535
aegas
gha8e
teen
cane

12.0 1645 17.0 1965 2200 Eye ame

15

720

TEMPERATURE IN DEGREES ©
MIWASSEE RESERVOIR 1967==UAYIL6@ -=SURFACE ELEV!

Pigure 148

463.9 4

MIT MODEL ©

nob plese penny rman
4 : g
: g
f : yl is
: “
TT £
&
aa}
TT s
a

foo.

Sena dh
geaszs te
~---}----}---- ] 235382 ds
Sege85-—"fe
Bogase [se

rraeie

coamey
gz
----4-- 2
eres eens ee ees oes
e---}----}----}----4----432
bd a

2
ve
° S
= 2
.

~t2
=
4
a---f----i--}
+
‘ °
; 3

TEMPERATURE IN PEGREES ©

Pigure 149

weSURFACE ELEVE 46346 m1

HIWASSEE RESERVUIK 1947==D871209

WbT MOREL &

164


0 - HEASURI

1

TY

40369 8

N DEORERE €

u
RESERVOIR LOOTeCOAYILOR ee BURFACE ELEVI

=
Fi
2
=
i

MAT MOORE ©

Figure 156

TEMPERATURE IN DEGREES ©
HIWASSEG RESERVOIR J9g7@=DAYIZ09 -ewSURFACE BLEVI 46206

py MODEL ©

Figure 155

167


(iv) Inflow velocity profile is approximated by a

Gaussian distribution, at elevation y

(orvintt}
20, 2
U, ty) Ui x(t) (49)
where:
Oi ax (t) = maximum value of the inflow velocity at

time t,

is determined from:

Y

Q(t) = 30) U;@) dy (50)
Yb

where:

GQ) = total inflow

Ys = surface elevation

Yy = bottom elevation

B(y) = width of the reservoir at elevation y

Yjn(t) = elevation of inflow

Oo; = inflow standard deviation

(d) Outflow

(i) multiple outlets
(ii) outflows are centered at the outlet with a Gaussian

velocity distribution:

44


Ov
ELEVATION (meters)

500

480)

460 1
e

i e

e

440
¢ FONTANA RESERVOIR, 1966
,, DAY : 187 ( JULY 6)
e
° a MEASURED
® e@ @ @ @e STAND

420 © 0 O @ De=I0-STAND

10) 2 4 6 8 lo =12 14 16 18 20 22 24 26 28 30
TEMPERATURE (°C)

Figure 5. WRE model -- Effects of Diffusion Coefficient (July 6)


However, the representativeness of the result is not solely dependent

upon the Densimetric Froude number.

Perhaps, the most important water temperatures are those at the
surface and in the withdrawal layer. By the use of a fitted

curve to the measured temperatures at these elevations, it was
possible to determine the maximum standard error of estimate for
the predicted surface temperature, 1.2°C, and of the predicted
outlet level temperature, 1.6°C. Temperature variations on
individual days may exceed these values and they surely are exceeded
at other depths in the reservoir. These limits are suggested as

the limit of accuracy of these types of models.

In a reservoir where the temperature goes below 4°C, the density
instead of temperature, should be used for the determination of the
entrance level of inflow, the thickness of withdrawal layer and the
condition for the convection to occur. The models evaluated, MIT,
WRE and Cornell must be modified before they can be applied to such

conditions.


Tea" Tea
Equation 15 becomes:

p;cAv.t. +K, , Ts ,#K. oP.+K. ,f.., = P.+Q.
hak he Ae PRE i De Ns Pes Mae 5°

where
DA,
+ p.c td
Kj ja 7 7 05° ae
DA, D. Ay
— i +141.
Kj 27 05° ae * jaa az G1 *%,x)

De WA;

= +1 j+1
Kj 37 - jae ae
Taylor series of 2nd order

" At?

T(t+At) = T(t) + T(t)At + T (t) >
and by definition of derivative
mw _ i T - T
T(t) = real Etta) T(t)
for small At
T(teat) = T(t) + Tet + T(trat) s

or use superscript k to indicate time step

T&D). 7 5 pO Bt, FOC) oe. OO , pO) ae

where a) = 709 + T i) s

a9)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)


Se

ELEVATION (meters)

500

480

460r

440

420

FONTANA RESERVOIR , 1966
DAY: 187 (JULY 6)

ome «= MEASURED

@? © © 0 o Az= 0.6m
° e © © @ Az=1.0m
& oo o o Azel.dm
° 0 0 0 @ Az=2.0m

2 4 6 8 ite) 2. 14 \6 18 20 22 24 26 28
TEMPERATURE (°C)

Figure 2. WRE model -- Effect of Thickness of Horizontal Layer (July 6)

30


SECTION V
SENSITIVITY ANALYSIS

Based on results from the test runs on Fontana Reservoir, the three
models were evaluated. Only the MIT model was chosen for further
sensitivity analysis of input parameters on seven TVA reservoirs. The
Cornell model, though it has some attractive theoretical features,
lacks means of inputing inflow, outflow, and the variation of surface
elevation. The model, therefore, does not approximate the most typical
cases. The running time for the model is considerably greater than
for the other two models. In addition, the test run results are poor
compared to the measured field data.

The Water Resources Engineers' Model has also been found to be
inappropirate due to the complex structure of the program, the longer
computer run times for the program and because of inadequate documenta-
tion of the model. In addition, a drastic change in the predicted
temperature in response to variation in the layer thickness was observed.

The MIT model is, in our opinion, the most easily used. We have
applied the model to seven TVA reservoirs, having widely different flow
through times, volumes and depths. We have varied the height of the
vertical increments, Ay, the fraction of radiation absorbed in the top
meter of water in the reservoir, 8, and the average absorption coeffi-
cient of water, n, and the vertical diffusion coefficient. Transmission ,

of the radiation below the water surface is related to 8 and n as shown

nN
by 6, 1 8)®

in Equation 48.

(48)

64


NORRIS RESERVOIR

Figures 95 and 96 show the computed reservoir temperatures and the
computed outflow temperatures respectively, at Norris Reservoir for 1971.
These can be compared with the measured temperatures as shown in Figures
97 to 101. It can be seen from the Figures and in Table 14 that the
predicted temperatures have larger standard errors of estimate than do
the predicted temperatues for Fontana Reservoir.

In Figures 97 to 102 it can be seen that the variation in thickness
of the horizontal segments from 1 to 3 meters makes minor differences
though the use of the 3 meter segment does lead to a larger error as
shown in Table 14.

It can also be seen from Figures 102 to 106 that a variation from
0.2 to 0.5 in 8, the fraction of solar radiation absorbed at the water
surface, makes little difference in the predicted temperatures. This is
varified in Table 14, where the standard errors of estimate of tempera-
ture are essentially the same for all three fs tested.

It can be seen from Figures 107 to 111 that a change from 0.05 to
1.40 per meter in n, the radiation absorption coefficient, predicts the
temperature poorly when the absorption coefficient of 0.05 is used. The
error is as great as 12°C in some instances. This is verified in
Table 14, where the standard error of estimate of the temperature is
19°C greater with the use of the lowest absorption coefficient.

In Figures 112 to 116, the effect of varying the diffusion coefficient
from molecular to 100 times molecular diffusion is shown. It can be seen
that in general the use of 30 times molecular diffusion yields the best
prediction of temperature with the use of 100 times molecular yielding

the worst.

126


4 f J i ; A a ES
o--}----}----} ede ----43 nt ns ene eens Cone: Se 3
~~--$----}---- 3s ¢ a: a See e
. ; wes 7 3
a = 2 7 s
{ ° 3 . .
: bY 4 4 & . doo? $6 4 4 pk 4 e
a won fe---f----fo---f---f---42 ----}----}----}----}----f----}----}----f----}- 2
° g 1 : ”
° 2: *
a wo--f----f----f-5--}----f----}2 FE ----}----}----$---- wane feo eed ----f dt
: Pa i 3 :
z= q a ’ ¥ q aa te
= j----$ 4 : Zé — a - --}----}---$-- 3g
" < v
i 3 2
.
— — = #
= 8
— =
4 it g
boy ; boy 2
---- #----f----$~---3-~-- ¢--Y-$---- ee ee 2
cy 2 2 3 8 2 8
: 5 = PS = Ps
gs ¢€ &s & & 28

316.9 6

119

TENPERATURE 1% DEGREES ¢
CHEAMKEE RESERVOIR J9$Te=DAY! 66° e=TURFACE FLEE

myy HOEC &

Figure 83


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

Fort Loudon Reservoir
Profile

Fort Loudon Reservoir
Profile

Fort Loudon Reservoir
Temperature

Fort Loudon Reservoir
Sensitivity, Day 76

Fort Loudon Reservoir
Sensitivity, Day 132

Fort Loudon Reservoir
Sensitivity, Day 204

Fort Loudon Reservoir
Sensitivity, Day 226

Fort Loudon Reservoir
Sensitivity, Day 253

Fort Loudon Reservoir
Sensitivity, Day 280

Fort Loudon Reservoir
Sensitivity, Day 343

Fort Loudon Reservoir

Measured Temperature

Computed Temperature

Computed Outflow

Layer Thickness

Layer Thickness

- Layer Thickness

Layer Thickness

Layer Thickness

Layer Thickness

Layer Thickness

Absorption.

Coefficient Sensitivity, Day 76

Fort Loudon Reservoir

Absorption

Coefficient Sensitivity, Day 132

Fort Loudon Reservoir Absorption
Coefficient Sensitivity, Day 204

Fort Loudon Reservoir - Absorption
Coefficient Sensitivity, Day 226

Fort Loudon Reservoir Absorption
Coefficient Sensitivity, Day 253

Fort Loudon Reservoir Absorption
Coefficient Sensitivity, Day 280

Fort Loudon Reservoir - Absorption
Coefficient Sensitivity, Day 343

178

181

181

182

182

183

183

184

184

185

185

186

186

187

187

188

188


----F--__.

v,
0

ERATURE JX DEGREES €

ww
CHEAOKER RESERVOIR 1967e~DAYI219

220K

IRFACE ELEVE

MjY MODEC ©

Figure 85

TEMPERATURE 1" DEGREES €

arya N

CHEROKEE RESERVOIR J9OTeRDAYIZOG @=SURFAGE BLEVE

My ManEL ©

Pigure $7

121


‘effective' diffusivity give almost identical results as runs using
the 'effective' diffusivity (Equations 45 and 46). The comparison
of the measured temperature profiles with those calculated with the
diffusion coefficients listed in Equations 45 and 46, marked STAND,
and diffusion coefficients 10 times as large are shown in Figures 5
through 7. For the early summer, Figure 5; late summer, Figure 6; and
winter, isothermal, Figure 7; both diffusion coefficients give similar

results and closely approximate the measured values.

MIT MODEL
The details of the development and testing of the thermal strati-
fication model are shown in Reference 12. The current version has some
modifications in the numerical scheme, selective withdrawal, etc., as
pointed out in the "Foreward'' of Reference 4. Most of the assumptions,
factors considered in analysis and input data are similar to WRE's. The
major differences between these two models are in the numerical scheme
and the handling of inflows and outflows. The computer program is clear
and easily followed. The inclusion of the test data on Fontana Reservoir
in the report provided useful guidance to new users.
Principal Assumpti: -os
(a) Thermal gradients exist in the vertical direction only, i.e.,
horizontal isotherms.
(b) The diffusion coefficient (molecular) is constant at all
depths and at all times; mixing due to unstable density

profile accounts for convection in the epilimion.

39


Hiwasee Reservoir... 6 1 ee ee ee ee ee ee 158

Fort Loudon Reservoir... 1. eee ee ee ee ee ee 176
Section VI

Analysis of Data... 1... ee ee 197
References . 2 6 1 6 ee ee ee ee te ee 204

iv


HIWASSEE RESERVOIR

Figures 144 and 145 show the computed and measured temperature
profiles for Hiwassee Reservoir for 1947. It can be seen that the
computed temperatures do compare, in general, reasonably well with the
measured temperatures. This can also be seen in Tables 17 and 18 where
the standard errors of estimate are quite small, 1.00 and 1.4° for the
outlet and surface water temperatures, respectively. The computed
outflow temperature is compared with the measured outflow temperature in
Figure’ 146 and is found to predict the temperatures very closely as is
shown in Table 17. In Figures 147 to 152, is shown the effect of the
variation in the thickness of the horizontal segments from 1 to 3 meters.
The change causes only minor differences in the predicted temperature.
This is verified in Tables 17 and 18 where the standard errors of
estimate are only slightly different from those calculated using a
2 meter segment.

In Figures 153 to 158 the effect of the variation of 8, the fraction
of the solar radiation absorbed at the water surface, from 0.2 to 0.5
on the predicted temperature is shown to be negligible. This is verified
in Tables 17 and 18, where the standard errors of estimate are shown
to be similar to those obtained with the standard 8 of 0.5.

In Figures 159 to 164 is shown the effect of variation in n, the
radiation absorption coefficient, from 0.05 to 1.40 per meter. It can
be seen that the use of an absorption coefficient as low as 0.05 per
meter gives very different results on day 209 in Figure 161 and similar

results during most of the rest of the year. This is verified in Tables

158


0 w MEASURED
STAND
= OVERLAP.

Le

— nO 1965.

TEMPERATURE JN DEGREES ¢

Eros

Te.

8d

24602

weSURFAGE ELEV!

esvAyt 76

MT MODEL ® PRT LuNDMUY 3972

Figure 174

0 » MEASURED

Pad.

22.9

1945

TEMPEPATURE IN NEGREES

26767 6

wr SURFACE ELEVE

eA yH132

FORT LOWDAZe 197)

MIT MOREL ©

Pigure 175

182


14.

15.

16.

17,

18.

19.

Sundaram, T.R. Easterbrook, C.C,, Piech, K.R. and Rudinger, G.
An Investigation of the Physical Effects of Thermal Discharges

into Cayuga Lake. Cornell Aeronautical Laboratory. Buffalo, New
York Report VI-2616-0-2. November, 1969.

Rossey, C.C. and Montgomery, B.R. The Layer of Frictional
Influence in Wind and Ocean Currents, Physical Oceanography
3(3):101. 1935.

Munk, W.H. and Anderson, E.R. Notes on the Theory of the
Thermocline, Journal of Marine Research 1:276. 1948.

Tennessee Valley Authority Water Temperature Prediction Model for
Deep Reservoirs. Water Resource Management Methods Staff.
Technical Report No. A-2, October, 1973.

Tennessee Valley Authority. Temperature Predictions for TVA Reser-
voirs Graphical Presentations, Water Resources Management Methods

Staff, Report A-6 December, 1973.

Tennessee Valley Authority. Heat and Mass Transfer Between a Water
Surface and the Atmosphere. Water Resources Research Laboratory
Report No. 14. April, 1972.

205


H | . wenn gene gen
wenn penne gage —- 4 n,
7 i
- yo | 1 . i . | i at
1 i i
1 7 1 1 1 4 js a By ,
tos i Pa a as o3
a a ee at a i : 4g
: TPG : ~ !
: . , , \ | foldl § 5 : s
- , i j 5 a---}----}----f-- ds
we-a}----}----$----f----f 4-4 fd 2
ae 1 * 4 .
°. . / . { :
4 eos _ e
wonngannofon-f---}----}----}----f----f--------32 8 a 43
7 = ‘ o
H H . = bo omk fos
H « . BY Benhes «
~---}-s--$----}----$----f a a ee oe eee eet ee et on on + ey
2 3 . Pi ns
“ i : veers
2 he eunaay
° ' of i ! :

. wh ;
wf Zp pdf Ze, es Jn g-—~—$_—- bf dtd
, 4 7 Oh! Sg 3 : j 5

; Bz F
A . :
et aif sh, «
----}--9-g----$----}---- fo. a, ----$----}2 asp} dd eee
uf doma nt 3
To H See : a
Hi ° f q q 4 : q z: B 4 :
he , gg .
----g----JS_-_ Pg Fe w---$----}----$----g---- dite
ar re . i 4 4 s
i : Pon :
s : : * a ~---}----}---- ts
a ~---}----}2 — 2
a oi a “I * 1 ol .
: 5 oad :
t 2 # ° ; : $ 2 z
ns eee ----$----}¢# ----}----}----F-00-fe-02}----f---- }----f----}----fe
ino 3 s ; . , =
: . : :
, = :
. gy : *
~---}----$---- I-42 og - _. - %
g i H ; , :
q q é q :
a a |---| 3.
2 S q 2 le
£2 2 zs} 8
\ i | I i
(4 | |

51106 8

eoSURFACE ELEVE

147

TEMPERATURE JN DEGREES ¢
SOUTH MDOLSTON 49S3e=-DAYIZES

Figure 129 MY MUREL ©


9s

ELEVATION ( meters )

500

480

460

440

420

FONTANA RESERVOIR, 1966
DAY: 244 (SEPT. |.)

—_— MEASURED

° ° © WITHOUT INFLOW/OUTFLOW

Figure 9.

TEMPERATURE (°C)

, WITH INFLOW / OUTFLOW oe
COUNTED AS POWER
PLANT COOLING WATER

fe} 2 4 6 8 lo «12 4 16 18 20 22 24 26 28 30

Cornell model test run results, September 1, 1966


ATMOSPHERIC
OUTFLOW: EXCHANGE INFLOW:

%+ To | i Qin sTin

ASSOREED SOLAR RADIATION
DIFFUSION

aan es Ee Salk

VERTICAL,
ADVECTION

SCHEMATIC OF RESERVOIR PROBLEM

TYPICAL HORIZONTAL SLICE
FROM  RESERVCIR

Figure 1. Reservoir representation

11


of the diffusion coefficient with respect to depth and temperature
gradient, the conversion will be very complicated. Since the model has
not provided any method to account for vertical distortion and no
available scheme is applicable to this situation, the values directly
calculated are used for evaluation.

The profiles for typical times of the year are shown in Figures 8
through 10 for runs without inflow/outflow and with inflow/outflow con-
verted to that corresponding to power plant cooling water. Both runs
used a diffusion coefficient of 7.98 x 10? £t2/day as in the Cornell
example. As shown in Figure 8 for July 6th, the predicted values for
both conditions are reasonable approximations of the measured values
except at the surface where the difference is 6c. In Figure 9, for
September Ist, the predicted values vary greatly from the measured
values, in some instances as much as 6°c. In Figure 10, December 1st,
the predicted values are in better agreement with the measured values
though differences as great as 4°C are noted. The predicted values for
the inflow/outflow case are in much better agreement with the measured
results than is the case without inflow/outflow. When higher diffusion
coefficients were used, the time limit of 15 minutes was exceeded.

The differences between computed and measured temperatures may be due
to:

(i) distortion of the vertical scale of reservoir caused by use of

a constant surface elevation and cross-section area.
(ii) neglect of inflow and outflow other than cooling water. Since

in most of the reservoirs suitable for analysis of these models,


ah see

'
t
t
1

eee

an

2
=, <
Souns
Benne
S2555
Srane
Poses
coors
onnn

< OUTLET

INO

TENPEPATURE |W TECKEES €

at

2a!

20

198

Ved

1240

B22 6

=ASURFACE FLEVI

SUPRID REDFRYUIK L9TLemusyI139.

HIT Mure =

Figure 98

g----

Aowen3-0.

os

2905

Pas 2700

2240

TW WD yar

a

“no

eye

TERPESATURE 1” DEGREES ©

91003 4

=eSURFAGE ELEVE

PATS WESFRYLIK LOTLeouAYE ABS

“yr eunet ®

Pigure 99

130


U, (y) = inflow velocity at elevation y

Uy (y) = outflow velocity at elevation y

Ty; = inflow temperature

A(y) = area at elevation y

t = time

a = molecular diffusivity

¢(y) = transmission of radiation at elevation y

and the continuity equation can be written as

F VONAW)) = BY) U,0)-U,0)) (57)

The isothermal profile at the beginning of the Spring provided the
initial condition and the two boundary conditions are given by the no
heat flux through the reservoir bottom and the balance of heat input
at the water surface.

The mathematical model used is an explicit finite difference scheme.

The selection of layer thickness, Ay, is restricted by the stability

criteria:
dt <k

> ay ~ (58)
ve st (59)
where:

D = diffusion coefficient

At = time increment

v = vertical advection velocity

Ay = depth increment

47


SECTION III
INTRODUCTION

Water quality is one of the major considerations in water resources
planning, and water temperature is a key factor in determining water
quality. Due to the influence of temperature on the physical and chemi-
cal properties of water and on the aquatic life within a water body, a
reasonable prediction of the temporal and spatial variation of the
thermal structure within the reservoir is essential for successful
management. This has become even more evident with passage of PL 92-500,
Federal Water Pollution Control Act Amendments of 1972, where thermal
water quality standards are specifically included in the term "water
quality standards" (Section 303-h).

Relief from these standards may be obtained under Section 316 if the
petitioner can show that the effluent limitations are "more stringent
than necessary to assure the projection and propagation of a balanced,
indigenous population of shellfish, fish and wildlife in and on the
body of water into which the discharge is to be made ...".

With the increase in the size of power production units and plants and
the trend to nuclear power, the prediction of the thermal structure in
receiving waters is even more so a necessity. Consequently, a large ;
number of mathematical models have been constructed but most have had only
very limited field verification. All of the models have certain
features in common and it has, therefore, become more important to

verify the models' field results and obtain the range and mean of the


2. distribution in
vertical direc-
tion

(spread instan-
taneously into

a thin horizontal
sheet)

ocity distribu-
tion either with
or without en-
trance mixing

Table 3 (Continued). FACTORS INVOLVED IN ANALYSIS
Factors Cornel] MIT Water Resources
Model Model Engineering Model
Heat transfer In terms of heat
across water sur- flux
face ag = K(Tg-T,)
1. short wave where 1. Yes H. Yes
solar radiation
2. Net long-wave K- feat exchange 2. Yes 2. Yes
radiation Toe coud libri
3. Evaporation temperature 3. Yes 3. Yes
4. Conductions T,- water surface 4. Yes 4. Yes
temperature
Inflow consider-
ation
1. enter at the 1. Yes 1. Yes 1. Yes
level of equal +
temperature 2. No 2. Gaussian vel- —/2. uniform velocity

distribution over
the interflow
thickness deter-
mined by Debler's
critera

Withdrawal
consideration

1. level

2. distribution
in vertical
direction

3. limit on with-
draw zone

. Single level
outlet

No

(uniformly dis-
tributed at the
level of withdraw )}

3. No

selective
withdrawal
withdrawal layer
thickness &
Gaussian
velocity dis-
tribution in
withdrawal layer
Yes

(withdrawal
layer never
extended over
the region with
temperature
gradient equal
or greater than
cut-off gradient}

of zg = 0.05

n

3.

selective with-
drawal
withdrawal layer
thickness and
uniform velocity
distribution
over withdrawal
layer determined
by Debler's
criteria

Yes

withdrawal zone
always on top

or underneath
the thermocline

21


In Figures 138 to 143 are shown the effects of varying the
diffusion coefficient from molecular to 100 times molecular diffusion on the
temperatures. It can be seen that, in general, the use of the molecular
diffusion coefficient predicts the temperatures most closely and that
the use of 100 times molecular diffusion predicts the temperature
most poorly as indicated in Tables 15 and 16.

141


Ty

500

480

460

ELEVATION (meters )

>
b
f°]

FONTANA RESERVOIR, !966
DAY : 244 ( SEPT. |)

MEASURED

© © © @ STAND
© 0 0 © Dg=l0+STAND

6 8 lo (2 14 16 18 20 22 24 26 28

TEMPERATURE (°C)

Figure 6. WRE model -- Effects of Diffusion Coefficient (Sept. 1)

30


of each model and lessen the difficulties one may encounter in running
such programs, to the extent that a new investigator in the field will
not be tempted to build his own program, but would further detail and

verify the existing programs.


Table 14

STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

Reservoir/Year: Norris/1972

Time Period Covered: 60th - 360th Julian Day

Std. error Correlation
File Name of estimate (°c) Coefficient
STAND 2.31 0.94
DELZ 1 2.29 0.94
DELZ 2 3.16 0.92
BETA 1 2.32 0.94
BETA 2 2.28 0.94
ETA 1 3.21 0.89
ETA 2 2.36 0.93
ETA 3 2.39 0.94
DIFF 1 2.46 0.93
DIFF 3 2.64 0.92

128


Table 9.

STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

Reservoir/Year: Fontana/1966

Time Period Covered: 60th - 360th Julian Day

Std. error Correlation
File Name of estimate (°c) Coefficient
STAND 1.74 0.95
DELZ 1 1.75 0.95
DELZ 2 1.70 0.96
BETA 1 1.74 0.95
BETA 2 1.70 0.96
ETA 1 2.20 0.92
ETA 2 1.74 0.95
ETA 3 1.75 0.95
DIFF 1 1.76 0.95
DIFF 3 1.81 0.95

74


----4---4y---- 4 - -
* dq ‘
co : “ . a
a ee — 3 (---- 3 —--43--~- 4---—.
‘ ‘ tt “a : .
& Hy
| | ’ :
woot} e- gl 4--. 3 woe $---- $f} eee
<
ioe 2
° . . . * cf 4 ot a a . . . ” .
, : x
. :
q e os
wn--$--e-}$---- $----}----}----}----}----}----$----}3 2 nas SS SEDNNNOS SOTES SONEETEE SunOS Suen
| = e
‘ “ 4 . a ‘ te Ts
4 i a Sernst
« 4 = . A aeeigs
a---$--%-$---- $----}----$-5-- #---i2 oy j---- }---- $----}----}---- pga Eb ge——-
2 2 4 noose
bo . 2 J ecreee
* i; 2 1, aanay
°o i Iv 7 °
~---$--T§$__-- eb lao —
a q q bh so Sg
at g8
a: i
: ea
H 4 H
et TM az? °
a---$--2-9----$.---$ ---- fe a---$5 74 ----}---~$---- $.---} ----}----$ ----$ ----$----3----
= eu os
age “
° 4 4 f 4 4
ot , | Sez i
= é
fone f a 2
w---$---- 3 2 tg ----32 5 € weg ---- 3 ----
oom ok a a ir q +
° ts z
5
4 . + 2
: ‘ &
----4---~§ -- +. a 4 or] q —
ee
a Boe 4 B ay
} .
a a q 4 ob z
e 8
w---$----$---- a ns ns as Sy mee gana fg ----
e 68
F | 4 2 4 4 :
¢ # =
| f ' f q z
a a {-$ ——— 4 ~---4|---_-$% ----4¢ ----$----3----
: ¢
mt I i :
: q 7
z
—--F---$--_# j--§-- Fg eg FB $3 fs ----f----$----
g a S iS. 2 e °
s s e ry o s <
2 & Es S & a i

1905 2200 2655 2700 2903
S16 4

weSURFACE ELEVE

1200 Ved 1100
TEMPERATURE IN DEGREES €

SOUTH HOLSTON 1993=—DAYI2Z45

ry

NBT MOMEL ©

ad

Pigure 142

- bed

10050


Internal mixing ---

@
Gi)

(iii)

Free convection occurs if stratification is unstable.
The heat transport, H, due to diffusion can be written as:
H= pc D(z,t) - STGst) (8)

The general properties of diffusion coefficient D are as

follows:

(a) It is usually greatest near the surface but declines
rapidly with depth and attains the minimum at thermo-
cline.

(b) In the hypolimnion, it increases with depth in an
erratic manner, reaching a maximum at about mid-depth,
thereafter decreases as the bottom is approached.

The form of the diffusion coefficient used is:

D = A; (constant), E< E, (9)

D = AES, E>, (10)

where E = stability of the water colum

E=-i 2 (11)

E iC critical stability parameter

The following values are suggested:

Ay = 2.5 x 1074 m/sec
-§ 183

Ag =1.5x10°m /sec

Az = -0.7

E,=1.0x 10° wt

28


favorable results in the rest of the reservoirs tested. Whichever
evaporation formula more nearly predicted the measured values was sub-
sequently used in all of the sensitivity analyses for that particular
reservoir. The frequency analysis of wind speeds shown in Table 6, seems
to indicate that Kohler's formula yields better results in the reservoir
where the wind speeds are mostly higher than 2 mph.

The results of the sensitivity analysis are shown in Figures 12-201.

The legend for these figures is shown in Table 7.

FONTANA RESERVOIR

Figure 12 shows the measured Fontana Reservoir Temperature data
with depth for selected days. Figure 13 shows the computed temperature
data for the same days using the corrected MIT model. As can be seen
from the figures and from Tables 8 and 9, the predicted and measured
values for the outlet water temperature and the surface water tempera-
ture have standard errors of estimate of 1.2 and 1.7 degrees C respec-
tively. Such good agreement might have been expected, since the model
coefficients were adjusted to fit the measured values of this reservoir
for this year. We were interested in how well the model would fit the
measured temperatures for other reservoirs and other years.

The closeness of the estimate of the outflow temperature using the
Rohwer evaporation formula to the measured outflow temperature is shown
on Figure 14. The effects of the variation of the layer thickness,
fraction of solar radiation absorption at the water surface, radiation
absorption coefficient, and the diffusion coefficient for selected days

are shown on Figures 15-21, 22-28, 29-35 and 36-42 respectively. It

69


Table 17

STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL

Reservoir/Year: Hiwassee/1947

Time Period Covered: 120th - 330th Julian Day

Std. error Correlation
File Name of estimate (°c) Coefficient
STAND 1.01 0.98
DELZ 1 0.96 0.98
DELZ 2 1.03 0.98
BETA 1 1.03 0.98
BETA 2 1.06 0.98
ETA 1 3.95 -0.79
ETA 2 1.08 0.98
ETA 3 ---- ---
DIFF 1 1.06 0.98
DIFF 3 1.44 0.97

160


SECTION VI

ANALYSIS OF DATA

The surface temperature, being related to the equilibrium tempera-
ture, can be well represented by a simple sinusoidal relationship. A
sinusoidal type function was least square fitted to the measured
surface water temperatures as shown in Table 21. The water temperature
at the outlet level cannot be fitted as easily, because it is dependent
upon several variables, e.g. time (day of the year), depth of outlet,
outflow rate and variation of thermocline depth, etc. However, if the
time period considered was shortened, such as using only the data between
early summer and winter instead of the full year, a least square fit of
a third degree polynomial can be fitted reasonably well. The results
of a third degree polynomial fitted to the measured water temperatures
at the outlet level are shown in Table 22. The average standard error
of estimate for the curves fitted are 1.2°C for the measured surface
temperature and 1.6°C for the measured water temperature at outlet
level, respectively for seven TVA reservoirs. The predicted temperatures
were then evaluated for the standard error of estimate with respect to
the aforementioned fitted least square curves.

In general, the sensitivity analysis of the input parameters has
shown that for:

1) Vertical Increments or Horizontal Layer Thickness, AY

The effects of variation of layer thickness on the temperature profile
are different from one reservoir to another. The general trend is that the
use of a smaller AY, 1 meter, yields temperature profiles a little lower in

196


26707 4

© = MEASURED

eeSuRFACE ELEVE

os08y1132

TRHPEPATURE IN DEGREES C

Prat Lunwryt 1972

HUT wOneL =

Pigure 182

----$4---

(2955

ano

thd

2240
26700

© » MEASURED
€ + ouTLET
'
1
1965
SURFACE ELEVE

1
i
i
1
i
1
1
i

1740

186 -

Vad

29,0

TERPEPATUPE 1% DEGREES ©

eet

&
>

wd
Fear Lou

m0
NET wane ©

or

oe

Figure 163

33.0
49,0
4300


normalized density gradient = t $ (55)

o
"

g = gravitational acceleration

If the temperature gradient at the outlet is smaller than the value

specified above, the withdrawal layer is restricted by the thermocline.

The built-in cut-off gradient is set at 0.05 °c/m.

(e)

(£)

(iv) The velocities from each outlet are superimposed on
one another.

Variable water surface elevation

The surface level is calculated from the initial surface
level and the cumulative inflow and outflow. The measured
elevations are used as reference only. The reservoir is
schematized into a series of horizontal elements with con-
stant thickness, Ay, except the bottom element, which is
half as thick, and the surface element, which varies
between 0.25Ay and 1.25Ay to account for the variation in
the surface elevation.

Governing Equations

The heat transport equation applied to each horizontal layer
has the following form:

Fe ROT wy AO) OD = cate BAMIIOD)- — (56)

AGT VOAWITON + xy U_OBOIT,-U,O)BOITOD)

where:
T(y) = temperature at elevation y
V(y) = vertical velocity at elevation y

46


OL

Table 6.

FREQUENCY ANALYSIS OF THE WIND SPEED

1 2 3 4 5 6 7 8 9 10 11
Wind Speed 0.0 2.0 4.0 6.0 8.0 | 10.0 12.0 16.0 | 16.0 £8.0
>20.0
(MPH) 2.0 4.0 6.0 8.0 10.0 | 12.0 14.0°' | 16.0 | 18.0 20.0
No 18.0] 117.0 | 100.0 | 39.0 18.0 | 10.0 3.0 1.0 0.0 0.0 0.0
Fontana
% 5.9] 38.2 32.7 | 12.7 5.9 3.3 1.0 0.3 0.0 0.0 0.0
Douglas No 4.0{ 162.0 | 128.0 | 46.0 20.0 3.0 2.0 0.0 0.0 0.0 0.0
% 1.1] 44.4 35.1 | 12.6 5.5 0.8 0.5 0.0 0.0 0.0 0.0
Cherokee| N° 3.0} 49.0 | 112.0 {103.0 48.0 |} 31.0 13.0 2.0 3.0 1.0 0.0
a 0.8{ 13.4 30.7 | 28.2 13.2 8.5 3.6 0.5 0.8 0.3 0.0
No 0.0] 31.0 | 128.0 | 87.0 58.0 {| 31.0 16.0 9.0 2.0 3.0 0.0
Norris
4’ 0.0 8.5 35.1 | 23.8 15.9 8.5 4.4 2.5 0.5 0.8 0.0
South No 22.0] 91.0 94.0 | 48.0 33.0 | 12.0 2.0 4.0 0.0 0.0 0.0
Holston
4’ 7.2] 29.7 30.7 | 15.7 10.8 3.9 0.7 1.3 0.0 0.0 0.0
Ht No 1.0] 20.0 | 128.0 | 91.0 71.0 | 33.0 8.0 7.0 2.0 3.0 1.0
wassee|
a 0.3 5.5 35.1 | 24.9 19.5 9.0 2.2 1.9 0.5 0.8 0.3
Fort No 0.0; 31.0 | 128.0 | 87.0 58.0 | 31.0 16.0 9.0 2.0 3.0 0.0
Loudoun —
% 0.0 8.5 35.1 | 23.8 15.9 8.5 4.4 2.5 0.5 0.8 0.0



ie a ee sO

&
: t
1 q : 2
palye ea. 3
ave eo:
w---}----f---#ee--}.---f---fdee-p----p ie g Sanenens SnOSEEnS SEEN SOEEEEEE SORES OEE SOS SOOO SOS SO
” ~ ' . ‘
ease z 2
if é 2
wae-$----}---- bed pd z--$2 -o--}----f----f fd
2 fs 8 ?
: é z :
H q of
w---}----}---- n---f----f----f----f----12 g See oe oe oo
, + ze
H i i H af
bo---f----} fff feted ft od w---}----}----f----} fb pp
sé
£3 7
= « 7
. . :
; -- ~---}---- a--be--e pee tee-ie FT :
5 ,
; « co foe
a---}----f----}----} ----}----}---}----}2s Sf} gpg pd
; : -.
: , . i f q : | .
i : 2
W---f---+-$---- $----¥---5$-i--f---- §----. aon FE an Fe Fe Fe Fo

rr
8.

ror 17.0 193 22.0 Ene 210-2903
=eSURFACE FLEVE

TEMPERATURE I” PEOREES ©

10 pr 120
DAUELAS KESERVUTH 2989==0871 89,

MAT MUDEL >

pry

Figure 63

30,0.


Table 19
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: Fort Loudon/1971

Time Period Covered: 120th - 330th Julian Day

Std. error Correlation
File Name of estimate (C) Coefficient
STAND 1.53 0.92
DELZ 1 1.56 0.92
DELZ 2 1.37 0.94
BETA 1 1.54 0.92
BETA 2 1.58 0.92
ETA 1 2.05 0.86
ETA 2 1.43 0.93
ETA 3 — ~~
DIFF 1 1.59 0.92
DIFF 3 1.69 0.90

179


a eee ae

? euieee ienemeien or
: | ods
¢----}$----4---- $----$----}---- et oe od
" a
; j
a *
ssipe gd.
web --- Fhe 0--}---- $$ ---- p+ $ ---- $5 8
a 3
ae ala tala : 24
° Senn dt
monn geno $---- $----$----$----}----$---- ie
Sbe5ee
oaangy
----}----{----}----}.---}.---B--- bpd e
4 , | ie
----}----}----}----}_---f-- te bd
. 4 4 M 4 ”
!
—--~ $---- §----$----3--_- ----4----§---- 3
2 38 g 2 ¢ °¢& (8 4 ¢
* ¢ 8s & 8 2, gd 8

29303 0

seSURFACE ELEVE

108

TENPEPATURE 1° DEGREES ©

DNYELAS RFSFAVUIK L959~=uAYIZT®

AQT MUPEL ©

Figure 68


Table 1 (continued). ASSUMPTIONS

The reservoir system can be represented by more than one segment, with
thermal simulation carried downstream segment by segment, to achieve a

quasi two dimensional solution from a series of one dimensional solutions.
The assumption that the bulk of incoming solar radiation is absorbed with-
in a small layer near the surface is implicit in the governing equations.
on
A+B sin gg t +o)
where A - mean value
B - amplitude

t - time in day, t = o corresponding to the time when reservoir
temperature profile is isothermal

@ - phase angle

15


SOUTH HOLSTON RESERVOIR

Figures 117 and 118 show the computed and measured temperature
profiles for South Holston Reservoir for 1953. It can be seen that the
computed temperatures do compare, in general, reasonably well with the
measured temperatures. However, some of the predicted temperatures are
quite different from the measured values and account for the fairly
large standard errors of estimate for outlet level and surface water
temperatures, as shown in Tables 15 and 16, 2.4° and 2.9° respectively.
The computed outflow temperature using the Kohler evaporation formula is
shown in.Figure 119. In Figures 120 and 125 it is shown that the effect
of the variation in thickness of the horizontal segments from 1 to 3
meters makes only minor differences in the predicted temperatures. This
is verified in Tables 15 and 16 where the standard errors of estimate
are similar to those predicted by using the standard thickness of 2
meters.

In Figures 126 to 131 the effect of the variation of 8, the fraction
of solar radiation abosrbed at the water surface, from 0.2 to 0.5 on the
predicted temperature is shown to be negligible. This is verified in
Tables 15 and 16 where the standard errors of estimate are shown to
be similar to those obtained for the standard 8 of 0.5.

In Figures 132 to 137 is shown the effect of variation in n, the
radiation absorption coefficient, from 0.05 to 1.40. It can be seen
that the use of the 0.05 coefficient predicts temperatures quite different
from the measured values and quite different from the predicted values |
using other absorption coefficients. This is verified in Tables 15 and
16, where the standard errors of estimate are one half and twice those

predicted with the standard absorption coefficient, 0.75.

140


2905

2740

245

2240

1945

1760

aed
TEMPERATURE IN DEGREES C

7
weespnppp a a Ss eS SE SEE SS SOOO OES Sana
fm
i
b }---}----}-- ppp
, z
22. zg 2
=e e Ve Ss
PTs
32 4 Senn st
ee ore es . a 4 4 4 ZENNEe
~}----}---- } SSF
#8! a S2gz85
: eoaBs2
te 5 : teeetr
a g
° z H conney
t 4 3 i
oe ee res © ----$----$----4----4----4----4$---+--}--~--$----}----
8
, j é . ,
:
i : 8 H
- . H H
H fz of i
fo of $-- $2 w---}----f----} pd
isk |
ie bd
iS gp S|
Peed
é 3
i 8
: FA
# }-_--f wo--t----}---- fed edd
3 i { ,
3
3 | ; : ‘ H
g. i
24
Fs

3 #
= 2 ---- 4
g 2
5
=
_. eo 2
IS i nt
TE $ og i
Sto lo :
.| 2 a
is
a lo
=

446.0 8

weSURFAGCE ELEVI

40 mS 12.0
HIWASSEE KESEAYUIK 4967==DAy! 80

AUT MOneL ©

Pigure 147


Figure 68

Figure 69

Figure 70

Figure 71

Figure 72

Figure 73

Figure 74

Figure 75

Figure 76

Figure 77

Figure 78

Figure 79

Figure 80

Figure 81

Figure 82

Figure 83

Figure 84

Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 276

Cherokee Reservoir Computed Temperature
Profile

Cherokee Reservoir Computed Outflow
Temperature

Cherokee Reservoir Layer Thickness
Sensitivity, Day 66

Cherokee Reservoir Layer Thickness
Sensitivity, Day 123

Cherokee Reservoir Layer Thickness
Sensitivity, Day 186

Cherokee Reservoir Layer Thickness
Sensitivity, Day 213

Cherokee Reservoir Layer Thickness
Sensitivity, Day 248

Cherokee Reservoir Layer Thickness
Sensitivity, Day 277

Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 66

Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 123

Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 186

Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 213

Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 248

Cherokee Reservoir - Absorption Coefficient
Sensitivity, Day 277

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 66

Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 123

ix

108

112

113

113

114

114

115

115

116

116

117

117

118

118

ng

119

120


=

26706 8

© » MEASURED

Ls STAND.
ooSURFACE ELEVI

=e0Ay1204

TEMPERATURE IN DEGREES ¢

FORT Luvoruy 1971

Figure 190 yyy mune, ®

23,0

20,0 eas
—. Bao

----#----

2

é

a

35,0

2405 2760
26S

22.0

1949

meSuRFACE ELE VE

DErREES C
south yi220

190

Lead

TRaPe*ATUPE 1”

FT hg taut 97h

wit “oren ©

Pigure 191



: : oo 2 ----$----$----}---_-} --- pf} a
: £ zate - 5 z
t 4 ‘ : Y a 4 4 = Fr “te d - f 4 5 : a
: . . g 27
ay q ----}----}----}¢ 2 p____ $2. .-F----$---_-$_---_-$_----}----} e
eea-p----$---- pe |---| : a a oot f----3 Saat pana} $——-$----43
‘ ‘ — 7 eft
, oS 4. . q 2.4
pout bp pb eeeBgB de ~---}----}____} 282828 te
Fhssas f2 = Pissee {f
4 8 = vaeere
: 3 ounnee
7-7. p--—$____f____f____F__-_§----}----}----Te . a-g-}----}----f----f----t2
¥-F-f---- a fs Bs i m
: ‘ H gz: «de ode : *
Ze 4 ane eo
2% : 2 ftw fee °
ras w---$----§ ---- 43 ----$ 2g 8b 3 tg it
rans a
fs ‘ 1
fe . |
ze --9-_----}-__ te
ad q a
g ~~ qs
3 3 3 ; F ; ie F
= }---- $---- $----$---- (~--- $----$--2.

MIT MODEL =

Piguce 90

325,04
123

TEMPERATURE IN DEGREES ¢
CHERTKEE RESERVUIR J9O7eDAYIL00 enSURFACE ELEVE

NYT MODE ©

Figure 91


ta
3.
ce
wee
cat
;
;
ads

0 @ MEASURED
Le STAND

2 OBLzL
3m DELI?
= OVERLAP
<> OUTLET

eones: —
cre tt) 1645

“Tak

hae

TFRPEPATURE [~ TEGREES C

310. 4

meSURFACF ELEVE

SWOKTS KESERUTH L9TLeeubytZ23

ayy rureL ©

Figure 100

o n
pave

2 rs
= =
Sound
Pe tetetd
Brie
PRESS
eeeer
onan

-- 2
ec
ae
- 3
2
. ry
a
48
iy
4
#---3%
3 e
.
v

-- 2
:
4
H
2

TenPETATURE f° PERREES C

207.9

rok 29719=u6 71292 ==SURFAGE FLEVI

MUcKTS KESER

MIT MUPEL ©

Figure 101

131


7 203

2740

2445

20801 4

ae geel

weSURFACE FLEVE

wonneen32:
Prarie,
TENPERATURE 1" DEGREES €

MURATS KESFRYUTK LOTLO=OAYILOG

NIT HUnEL =

Figure 12

2eeh _270 2

2200
ata2 4

= OVERLAP
190d

MEASURED

2+ STAND

2 = O1FFL

32 njer3

<= OUTLET
weSURFACE ELEWE

i
1
1
'

137

TERPERATUPE IN DEGREES ¢

PuRKTS RESEAYUTR L97L==0AY1139

wT Pure ©

Figure 113



DAs De aAsar

= Vi .
T PET Qt mee Tha, (5S t Ojea® a Ta

+ Pj41°

where

P.
j

s

T

Ty

qy

oI

D. As as)
jt j+l
ta "}43/2
Bs (a7), 8: P . + hia. 16
PsC(Ty)5(47)5  P5CCTQ)5 (9Q)5 + BUA (16)
BSCVSALTS Pyar Vy ag Ajay Tia (17)

water temperature (°c)

inflow temperature (Pc)
inflow rate (m?/sec) into control volume
outflow temperature (Pc)
outflow rate (m /sec) from control volume

net insolation heat flux per unit area (kcal/m?-sec)
diffusion coefficient (m/sec)

vertical advection velocity (m/sec)

thickness of control volume (m)

average density of water (kg/m*)

density of water (kg/m*)

Aj-a2 volume of jth control volume (m°)

vertical axis, positive upward (m)

cross-section area at depth step j (m?)

(Aj*Ay41)/2 (18)

mean water temperature of jth control volume eC c)

specific heat of water (= 1 kcal/kg-°c)

30


e
*
~ é
®
¢ ----$2
2 a
Fa
z i
. 2. n
‘ #---ts
8 5 S
3 +2 £
= :
2 v
3
of 2
3 eg ee te te tern tes
geo deed =
eg i di :
: Ea
: z? :
----}----$-4-- ---4 2) i ns Sas SOE SOE SEEDS SRS SOC SEE SOE ©
ES f
5 |
fe j
[ne es Soe ee | ais i a---$a---$---- $2
3 j---- gi -f--_f__--- pee fe 3
. se |
. ee 2
a oe lk aod 3: ‘
oe a o 1 ' 3 1 " 1
: , » 2- 3 2 *
gee bp en | eee ae 4 2 a--a$----}----}----}----}----42
: mop $n---$----} ---- ;
A : 1 es 8 ss-ee
. oi ’ a 4 7 7 “ om : 7
. egi. eo joe
: q q 2 %
w---$--- -$- ge y -$----}--. a 2 Totes $i ---$----4$----42
q | h . ; ie ns Canes SanOnnnen 52a nk Seen Oo i ig
f 4 z f
w---$----$---- Pf Fa _---}_---}----}----}----}----}----}----{----4----12
j i
£ 1 '
. , . , , £ H
‘ H f z
$----$----$-¥--$-- E ~$-¥--F----
5 & 2 < 2 2 Pg 8 e 2
z a 2 8 J s s s

Zhen

meSURFACE ELEVE

espAyt To

192 -

TRPESATUPE LY DEGREES C
FORT tunueye 1972

MIT MUnEL

Figure 195


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

SL

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

Douglas Reservoir - Absorption Coefficient
Sensitivity, Day 69

Douglas Reservoir Absorption Coefficient
Sensitivity, Day 121

Douglas Reservoir Absorption Coefficient
Sensitivity, Day 186

Douglas Reservoir Absorption Coefficient
Sensitivity, Day 218

Douglas Reservoir Absorption Coefficient
Sensitivity, Day 250

Douglas Reservoir - Absorption Coefficient
Sensitivity, Day 276

Douglas Reservoir Extinction Coefficient
Sensitivity, Day 69

Douglas Reservoir Extinction Coefficient
Sensitivity, Day 121

Douglas Reservoir Extinction Coefficient
Sensitivity, Day 186

Douglas Reservoir Extinction Coefficient
Sensitivity, Day 218

Douglas Reservoir Extinction Coefficient
Sensitivity, Day 250

Douglas Reservoir Extinction Coefficient
Sensitivity, Day 276

Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 69

Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 121

Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 186

Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 218

Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 250

viii

99
100
100
101
101
102
102
103
103
104
104
105
105
106
106"
107

107


A brief description of each model evaluated is given in the fol-

lowing sections.

WATER RESOURCES ENGINEERS' MODEL
The WRE model has been developed by Water Resources Engineers, Inc.,

through a series of studies for various agencies 3,9) It was the
first comprehensive model proposed for predicting the thermal structure
in reservoirs. The computer program of WRE model used for this study is
the modified EPA version. Although this model has many versions and is
widely used, most users indicate that an intensive effort is needed to
have this model run properly. This is partially due to the difficulty
of acquiring full documentation and partially to some computer coding
problems. Several errors were detected during this study. The most
serious one is the concept of ‘effective’ diffusion, based on the numeri-
cal evaluation of the eddy conductivity coefficient from the measured
temperature profile in a reservoir. By comparing the rederived heat
transport equation, as shown later in this section, with the WRE's
computer program, a mistake in the computer code related to the diffusion
term was discovered. The density of water, p, appears to have been
omitted from the diffusion term. Since the MKS system is used, the actual
diffusion is approximately one thousandth (1/1000) of the 'effective'
diffusion, as defined by WRE. After correction of this error, the magnitude
of the 'effective' diffusion coefficient is then of the same order of
magnitude as the molecular diffusion coefficient.

Other difficulties in using the WRE model were related to the numerical

scheme for evaluation of the heat flow term at the top layer. This

24


--4-

<= runer
i
'

1260

ino 1965 22.0 2663 27,0 2953.

Wed

pr

TEMPERATURE IM DEGREES ¢

DOULAS RESERVUIK 29

weSURFACE ELEN

mvAYIZ18

AT MOnEL =
!
i
i

Figure 60

Lo

9540

40,0

peer es
fnnbee
renee

9900

10 Dry li 140d 17.0 1968 22.0 hed 2760-2948.
TENPERATURE 1” DEGREES €

pr)

20

20,0

295.0 6

=*SURFACE ELEVE

DAYELAS RESERVITR 1¢59-evAyIZS0

qT MOREL ©

Figure 61

104


Table 22
LEAST SQUARES CURVE* FIT FOR MEASURED WATER TEMPERATURE

AT OUTLET LEVEL

Std. Time Period
No. of.Data Error of Correlation of Fitted Curve
Reservoir/Year Points Estimate (°c) Coefficient (Julian Days)

Fontana/1966 22 0.96 0.97 103 341
Douglas/1969 27 1.10 0.98 69 352
Cherokee/1967 32 1.06 0.96 117-332
Norris/

South Holston/1953 9 1.09 0.96 113 (362
Hiwassee/1947 10 1.50 0.96 80 364
Fort Loudoun/1971 10 1.56 0.92 76 343

* Third Degree Polynomial

198


SECTION IT
RECOMMENDATIONS

Though it is possible to further refine the mathematical models
used in temperature predictions for deep reservoirs, it appears
that at this time it would be advisable for operational purposes
to utilize the existing models, such as the MIT model, which
have been most thoroughly verified.

Though an analysis of the reservoir temperatures has been made
for a series of the TVA reservoirs, no analysis has yet been
made for the temperature distribution of a single reservoir over
a period of years to determine the proper coefficients to be
used in the model. This should be done.

An intensive thermal analysis of reservoirs in the southeast
section of the United States has been made. Such an analysis
needs to be made for other sections of the country to determine
if the coefficients most suitable for the southeastern section
of the United States are also most suitable for the other sections.
For a better theoretical and emperical understanding of thermal
regimes in reservoirs, further detailed laboratory and field
studies on the effects of multiple inflows on the assumption of
horizontal homogeneity need to be carried out. In addition,
further field studies on withdrawals is required to determine
the layers affected and the internal currents induced by the

intermittent releases.


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

188

189

190

191

192

193

194

195

196

197

198

199

200

201

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 76

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 132

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 204

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 226

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 253

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 280

Fort Loudon Reservoir Extinction
Coefficient Sensitivity, Day 343

Fort Loudon Reservoir Diffusion
Coefficient Sensitivity, Day 76

Fort Loudon Reservoir Diffusion
Coefficient Sensitivity, Day 132

Fort Loudon Reservoir Diffusion
Coefficient Sensitivity, Day 204

Fort Loudon Reservoir Diffusion
Coefficient Sensitivity, Day 226

Fort Loudon Reservoir - Diffusion
Coefficient Sensitivity, Day 253

Fort Loudon Reservoir Diffusion
Coefficient Sensitivity, Day 280

Fort Loudon Reservoir Diffusion
Coefficient Sensitivity, Day 343

189

189

190

190

191

191

192

192

193

193

194

194

195

195


Figure 1
Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11
Figure 12

Figure 13

Figure 14
Figure 15

Figure 16

LIST OF FIGURES

Reservoir Representation

WRE Model -- Effect of Thickness of
Horizontal Layer (July 6, 1966)

WRE Model -- Effect of Thickness of
Horizontal Layer (Sept. 1, 1966)

WRE Model -- Effect of Thickness of
Horizontal Layer (Dec. 1, 1966)

WRE Model -- Effects of Diffusion Coefficient
(July 6, 1966)

WRE Model -- Effects of Diffusion Coefficient
(Sept. 1, 1966)

WRE Model -- Effects of Diffusion Coefficient
(Dec. 1, 1966)

Cornell Model Test Run Results
July 6, 1966

Cornell Model Test Run Results
Septemper 1, 1966

Cornell Model Test Run Results
December 1, 1966

Surface Layer Schematic

Fontana Reservoir Measured Temperature
Profile

Fontana Reservoir Computed Temperature
Profile

Fontana Reservoir - Computed Outflow Temperature

Fontana Reservoir Layer Thickness Sensitivity
Day 75

Fontana Reservoir Layer Thickness Sensitivity
Day 132

Page

11

35

36

37

40

41

42

55

56

57
61

72

72
76
76

77


5 OUTLET

200

Ine

3
=

TEWPERATURE IN DEBREES ¢
CHEROKEG RESERVOIR L9GTrWDAYIZTT coSURFAGE ELEVI

aa

Kir MODEC ©

Figure 76

1% DEOREES ¢

TenPaRATy
CHEROKEE RESERVOIR 49¢TERDAYI 66

Neen

EeBURPAGE ELEVE

HUY MODEC ©

Figure 77

LG

i


TABLE 10

STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL

Reservoir/Year: Douglas/1969
Time period covered: 120th - 330th Julian Day

Fite Nane of estinate (°c) coetficient
STAND 2.11 0.95
DELZ 1 2.23 0.94
DELZ 2 2.07 0.95
BETA 1 2.11 0.94
BETA 2 2.15 0.95
ETA 1 5.73 0.28
ETA 2 2.28 0.94
ETA 3 2.12 0.94
DIFF 1 2.09 0.94
DIFF 3 4,97 0.63

94


Results of Test Run
A set of test data was prepared from Fontana Reservoir field data

to evaluate the model. Several runs were made for different input
conditions. Problems about the selection of input parameters were
evident during the preparation of the data and during the running of
the program. It became obvious that long computer times were required.

In the numerical example given by Cornell, the friction velocity,
w*, is taken to be the surface current velocity at 0.1 ft/sec. However,
calculations according to Munk and Anderson's empirical relationship
yield w* at about one tenth of the value given in the example. The
resultant eddy diffusivity, Kyo, is 4.96 x 10° £t?/day compared to 7.98
x 10” £t2/day used by the Cornell Aeronautical Laboratory. The eddy
diffusivity affects both the thermal diffusivity and time step, and
therefore the simulated thermal structure in reservoir. The current
velocity at the surface in Fontana Reservoir is not available. If the
suggested relation is used, the eddy diffusivity is of order of 5.0 x
10° £t2/day and the computer time for a 300 day run will be about 90
minutes. Therefore, a maximum time limit of 15 minutes was set for all
runs. Other parameters defined loosely are the dimensionless constant
o,in Equation 65,the length scale,'a',in Equation 62,and the maximum
and minimm thermal diffusivities.

In a lake of variable cross-sectional area with depth, the assump-
tions used in this model, constant cross-sectional area and horizontal
homogeneity, imply a distortion of the vertical scale. Hence, the
computed temperature profile has to be converted in some way before
it can be compared with the measured value. Due to the nonlinearity

53


~ 44-3 4--- monn tend= 4
ag ed
| | tg
; ;
: : e
a---}---- g
‘ a
4
w---$---- +
é
. . 3
°
ae 4 ¢
ab anne 4
2 Fa 2 i
5d Fe 2) ote
23 : Atanes a
#3 2 3 ---}----}----{----}2223%¢---i2
eerie mys) = 7 wee $$ 8 g223e5. 2
Aa ~ e q H PResze 2
en : ennmey
7 i
Be wen }----$----$----$----$----4$----4----4----
z 8 ; , , ,
g 3 i
a : H
©¢ A H H
2? hon H
2 | $e g----}----f----
2 i H
z i
Se H
= 5 H
Rr wee _$_---f----}----}----3----4----$----
3 1.
2 : 4
2
a a
2 4
ze ----$----}----
i
,
F;
5
3
ty
g _--}----}----}4
H
g H
H
z H
- ,
H
be
i$ $----$----32
° ° Pa Pg
5 °° 2 co
& 2 g

meSueFACE ELEVE 267e3 8

moony ize

TenPERATURE IN PECREES €
191

pray LuNoty" 1978

MAT HOTEL ©

Pigure 193


SECTION IV
ANALYSIS OF DEEP RESERVOIR MODELS

In temperate zones, the spring heating, primarily by the absorption
of the solar and atmospheric radiation, tends to warm up the waters
closest to the surface. However, surface cooling, due to back radiation,
evaporation and conduction, and wind-induced turbulence will cause mixing
whenever the density gradient is too shallow and too weak to maintain a
stable condition. During this period the temperature distribution is
only weakly stratified. The heat in the surface layers is transported
slowly down to the deep water primarily by advection. As solar heating
continues, the temperature of the upper region, epilimmion, increases,
while the lower region, hypolimnion, remains cool and relatively undis-
turbed. A zone in between the two regions in which the temperature
gradient is the largest is called the thermocline. This steep density
gradient tends to inhibit the transfer of heat and momentum between the
warm upper layer, and the underlying cooler waters. The tributary
inflows, which tend to be warmer than the lower reservoir waters during
the summer season, mix and enter the reservoir water colum at the eleva-

tion where its density is equal to that in the water colum.

Thermal stratification affects not only the extent of dilution ‘and
Py

mixing of the inflow waters, but also the quality of the water in
hypolimion. The development of deficits in the hypolimnetic dissolved
oxygen concentration usually follows the establishment of thermal

stratification. After its formation, the thermocline moves downward as


Bena genne pene
i
js ns rs See
~---}----}----$----1-
----$----4----.3----
H ; .
823850
sESRes
i sesesie__ta
[eens oes ees See eer gEgee--49
Z2n55--is
B2S2322
rereaa?
oanagey

----$----$----3----3----

127

NYT, weCP OUTER TEMPERATINE PROFILER®

TEL CE ATUPE [7 PERKFES €
WOT IS Peden WBF

aT rune

Figure 95


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

South Holston Reservoir
Temperature

South Holston Reservoir
Sensitivity, Day 78

South Holston Reservoir
Sensitivity, Day 142

South Holston Reservoir
Sensitivity, Day 203

South Holston Reservoir
Sensitivity, Day 245

South Holston Reservoir
Sensitivity, Day 299

South Holston Reservoir
Sensitivity, Day 362

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

South Holston Reservoir
Coefficient Sensitivity,

Computed Outflow
Layer Thickness
Layer Thickness
Layer Thickness
Layer Thickness
Layer Thickness
Layer Thickness
Absorption

Day 78

Absorption
Day 142

Absorption
Day 203

Absorption
Day 245

Absorption.
Day 299

Absorption
Day 362

Extinction
Day 78

Extinction
Day 142

Extinction

Day 203
Extinction
Day 245
Extinction
Day 299

xii

145

146

146

147

147

148

148

149

149

150

150

151

151

152

152

153

153

154


FCAMSS SAMA AS

2300

Saiatet Saneenel Suneneny Salen ueienet Rein net Sonne SaeteD ET £3
°

2

9a

TEMPERATURE IN DEGREES ¢
CHEKKEE RESERYUIR L987ES0AYI2TT weSURFAGE ELEV!
125

maT MOREE &

Figure 9%


———

43600 4

TEMPERATURE [N DEGREES €

MIWASSEG RESERVOIR L967e=DAYIZ67 eeSURFACE ELEV!

NET MODEL ©

Figure 156

nc

22.0
44009 4

1740

1s
168

TEMPERATURE JN DEGREES ¢

HIWASSEE RESERVOIR 19¢7e@DAYI90Z weSURFACE ELEV!

1240

af
MIT MODEL =

or
Yigure 157



and the entrance mixing); and in determining whether or not there should
be a time delay in the water inflow input to take into account time of
flow in the reservoir; in determining whether the horizontal segments
should not also be divided into fast flowing and slower flowing sectors.

The physical aspects of solar radiation, back radiation, conduction
and evaporation, etc. do not, of course, change. However, not all of the
measured meteorologic data are available or even feasible or economic to
measure on a daily basis. Some of them must be evaluated from relevant
input data (for example, the evaporation calculated from wind speed, rela-
tive humidity or dew point temperature etc.). The models may differ in
the formulas used; in the coefficients chosen; in the mechanism of
testing for stability in the horizontal slices and thereby initiating the
thermocline; and in the mechanisms determining the heat transfer in the
vertical directions.

The models also differ in the thickness of the top and bottom layers,
and the time increments iterated.

Tables 1-4 illustrate the major differences between models as
determined by reading the documentation and the computer program listings
and test runs on the data furnished with the model and/or test runs on
data prepared from Fontana Reservoir if original model test data is not
available. The assumptions made are shown in Table 1; the input parame-
ters in Table 2; the factors involved in the analysis in Table 3; and the
model construction in Table 4. Some difficulty (due to program mistakes
or inadequate information provided in the model) was experienced in getting
some of the programs to run and in obtaining full documentation on each
model.

13


i
=o wang mete
Ty j <
| Dobe &
: {pd ite
Ss : =
i | 4 . f aE
4 Y
. ~---$2---4----}----4----]2
z é
«
2
f e 3 e
aa--}a---$---- bbe 2g wf fb be
“ a : . §
5 e
4 A . A z , : ‘ , ge .
: 2 ° 8 Senndt
« “ ! ezaae0 a
a a ae as 2 eas SOR SE ROS SS SES SOs SOE Sf fee oy
i ' g 3 : H $5bh28 g
« j : teres
5
= onnmay
, ° i 7 ‘
sb | pis tool fe
---}_-- cane — 2
: is 8s 4 4 fs
2&8 .
: =:
4 #2
Fy
4 2?
ow q nme “
w---$--2-F_--- bb nent ---- a---$----$--- pe puts
i fe8 2
L ‘ f 37, | 4 i
: 2 t : : Hl
8 kara j H
, oe : BE
a as =e : ego---f---_-$--- dese
Fi ee jpn ge gee 3
mo i ot ' . of re " a q - ot Mee
oat z
, 5
a u “ a al a H
= + 7 2 =$----}$----$----3$----$%
---$----$----}-- ba ~---$---}---2}_. 3
° . : °
b ; ‘ , : ; de |
a ° z : co-fe-o2h----$----}__--- dds
Fs -- =! - 2
; 2 : ,
A i » "os eo otee
, i = q ,
. © : “
a---$---- fe 8 woof} fd df debts
: 8 ‘ ;
It | i '
§ wi i af : :
3 iF. f i q q
z
Zee. ~~ $4 —
| 4 <
i $
1

veomalea fog jeer anad '
| ian

1 S1keo 4

weSURFAGE ELEVE

150

TEMPERATURE IN DEGREES ¢

SMYTH HPLSTAN L95S==0AvIZ45

Figure 129 MjT MONEL ©


me

ms

2200

40306 MH

19605

TF

120

TEMPERATURE IN DEGREES €

MIMASSEE RESERVOIR 1967e=DAYILO®

Figure 160

«
ees
2 3
z 2
: a
2 . =
42) ge Ey
ge at 2
EP
as es Sn Se eS SO SO DD
SS5585-—-42 og
Ssstse {= §
cS . =
. F
:
F ?

MIT MODEL ©

5

1700

TEMPERATURE. IN DEGREES ¢
HIWASSEE RESERVOIR 1967e=DAYI209 ==SURFACE ELEVI

1655

4240

igure 161

MJY MOREL ©

170


15.0

30.0

45.0

oe
°
°
°
*
$

£

wee

Quart wz curwe

- $3
: : : B if
a---$----}--- byte
‘ A ¥ a ns
: : : : F : we
wn-- $e }---- peddle
sgezz5___ fn
qoo-d----f----f---- fp eam de
5} : : deekhas fe
: : Peete 5
: ‘Posen gv

4s
----13
w---$2
----48
g

Pd
a---43
88

3

TENPERATURE IN DEGREES ¢

SURFACE ELEVI 30169

FONTANA RESERVOIR 1956-~DAyIZ06

Figure 26 ayy KONEL ©

wnee

15.0

3040

4500

ooo
°

©
moot ae

°
we
¢
3
ewars

105.0

120.0

133.0

2965

2605

1700 193 22.0 2760
TEMPERATURE IN DEGREES C
FONTANA RESERVOIR 1966-=DAy1285 =-SURFACE ELEVE

12.0

45

150.0

49603 4

Fégure 27 uty wove #

82


cy aaa eee 3
= 5 A 1 4 4 ; ; HR
fe a---}--_ pts
¥3 : 1 Rid
ans eee Sees fy
= $
«x ‘ ‘ ©
t 2
e i e 2
2 2 a-fubit:
ry Pr da
| og "ig os ob &
,  § eo
a ne a eee SR 3 | ee
2 3 =T om Bisse f= 3
. ree
7 f * ado . . 3
a , Bo ; q | hoo boc hoot
2 Se ----3----f ----} ----f ----4---- —-Fe$---- --2-$---- z a :
i? po. oe
nee - ak
3if a---}----}----}- pd pp ep 3%
= Be 738
1 gs ; f q T t ' f Be
£3] i ' ; : | : tee
Fa _— : . ‘ ----}----}e # =
ans a 4 : of a . a be oa arr g
£ : z
2 o¢ : $ f : $f
e & . a §
2 a en ens ee Snes een | ----}2E
. . , q q :
re a a ‘ Pepe ple
rr ner er : | +4 ff a8
— a Sa : ~---}----}---- bee bpd bb
; ; ‘ & s
----$----#----} fds es a a ns ns ans a nS ae Oe
2 2 ec 2 2 2 2 e e oe ce J e e ° c e . . ° cy °”
* ¢ ey a Py s s = s “ - s o S a y 5 s os =
£ s #€ 28 #§ & & & 8 = # s 8 € 8&8 & & 2

124


can be seen in Figures 15-21, that the variation in the thickness of the
horizontal layers from 1 meter to 3 meters had virtually no effect on
the predicted temperatures. The standard errors of estimate for the
temperature when varying the depth confirm this.

It can be seen in Figures 22-28 that the variation from 0.2 to 0.5
in the fraction of the solar radiation absorbed at the surface made
little difference in the predicted temperature. This is also confirmed
in Tables 8 and 9.

It can be seen in Figures 29-35 that the variation of the radiation
absorption coefficient from 0.05 to 1.40 made a great difference but
the variation from 1.40 to 0-75 made little difference. The use of
a coefficient of 0.05, however, made a large difference between the
predicted and measured temperatures. This is also confirmed in Tables 8
and 9.

It can be seen in Figures 30-42 that a variation in the diffusion
coefficient from molecular diffusion to 100 times molecular diffusion
resulted in a great difference between the predicted temperature and
the measured temperature for the greater diffusion coefficient. This
difference was most pronounced during the warmer part of the year as

shown in Figures 38-40 and as indicated in Tables 8 and 9.

75


5.

REFERENCES

Shirazi, Mostafa A. and Davis, Lorin R. Workbook of Thermal Plume
Prediction, Volume 1: Submerged Discharge. National Environmental
Research Center, Corvallis, Oregon. U.S. Environmental Protection

Agency Technology Series EPA-R2-72-005a.August, 1972.

Shirazi, Mostafa A. and Davis, Lorin R. Workbook of Thermal Plume
Prediction, Volume 2; Surface Discharge. National Environmental
Research Center, Corvallis, Oregon. U.S. Environmental Protection

Agency Technology Series EPA-~R2-72-005b. 1974.

Water Resources Engineers, Inc. Mathematical Models for the Predic-
tion of Thermal Energy Changes in Impoundments. Federal Water
Quality Administration, Washington, D.C. Water Pollution Control
Research Series 16130EXT12/69. December, 1969.

Ryan, Patrick J. and Harleman, Donald R.F. Prediction of the Annual
Cycle of Temperature Changes in a Stratified Lake or Reservoir:
Massachusetts Institute of Technology, Cambridge, Mass.
Mathematical Model and User's Manual, MIT Hydrodynamics Laboratory
Report No. 137. April, 1971.

Sundaram, T.R. Rehm, R.G., Rudinger, G. and Merritt, G.E. A Study
of Some Problems on the Physical Aspects of Thermal Pollution
Report VI-2790-A-1, Cornell Aeronautical Laboratory, Buffalo, New

York. June, 1970.

203


somewhat larger at the outlet than with the standard absorption
coefficient of 0.75 per meter.

In Figures 195 to 201 are shown the effects in predicting temperature
of varying the diffusion coefficient from molecular to 100 times
molecular diffusion. It can be seen that the variation of the diffusion
coefficient makes little difference in the predicted temperatures. This
is verified in Tables 19 and 20 where the standard errors of estimate

are all similar.

177


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

South Holston Reservoir Extinction
Coefficient Sensitivity, Day 362

South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 78

South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 142

South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 203

South Holston Reservoir Diffusion
Coefficient: Sensitivity, Day 245

South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 299

South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 362

Hiwassee Reservoir Measured Temperature
Profile

Hiwassee Reservoir Computed Temperature
Profile

Hiwassee Reservoir Computed Outflow
Temperature

Hiwassee Reservoir Layer Thickness
Sensitivity, Day 80

Hiwassee Reservoir Layer Thickness
Sensitivity, Day 148

Hiwassee Reservoir - Layer Thickness
Sensitivity, Day 209

Hiwassee Reservoir Layer Thickness
Sensitivity, Day 267

Hiwassee Reservoir Layer Thickness
Sensitivity, Day 302

Hiwassee Reservoir Layer Thickness
Sensitivity, Day 364

Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 80

xili

154

155

155

156

156

157

157

162

162

163

163

164

164

165

165

166

166


ng eg an png ng ng a pn tg nn eg ennn gn no eee
: : ; : : ‘le ; : ; :
: : 5 : ‘ ye i : :
wo - ten § i $--- $$ --- ae amet cient 4 Facets ete oo: a
: _ @o$ ¢.
° é &
°8 _ . # ° Zoayst
84d : ; 3 3 n--"$----}----}----p----f---- be
foe i : : 2 4 . Penhes3 77
q 3 . : ‘ peewee
3 ? ° onaney
. . 5, a ens 22). ‘| 2
$ fas -a--fogf-}----}----f---- ppb
shoe of A . ‘ ‘ : a x
of BE ote H
-$----}---- pee a : 2 :
----}-_- io eens Saenens tee ne SS SO SOS SS Se
: : : Be 4 Hoo / .
: . : 5 4,28 : : H
a en ge , : : i :
; : : : ; 4228 an oe w---}---- fp
. a . = q ‘ a 4 . ‘
. : : i : :
a oe ----f---f----f dis E i; nr es Sees Se
H H { °
; z wane ponnn penne foo png} --- p= p---f--b
i 5 : ; . .
feos dee bo : eealece de
a fonnn fanaa fone a $---- f----f----f----g---E wa-- feed} bb
: 5 : : 2
z
ne ene OEE £1 y.
$ + i a---$----$---- $e bbb
2 oq 3 q 0638 s 3 ¢ 2 8 @ 2 8 8
= a g e cy es ra s a s s S
& 8 3S g $3 $ ok $ 3 g $68

26.3 27.0 298

17.0 1965 22.0
505.9 4

165

TEMPERATURE IN OECREES ¢
FONTANA RESERVOIR L9SG-=DAYIZIS ==SURFACE ELEVI

7.0 Corres
81

405
Pégure 25 yy NOMEL =


45508

rhe

15
maSURFACE ELEVE

an

Wn

tees
TEMPERATURE [N DEGREES ¢

HIWASSEE RESERVOIR 19467 0=0AVIZ07

MIT MODEL ®

Pigure 162

eo) ee Sa

sn8 = OVERLAI
<= OUTLET
!
I
WS 22.0 245

1740

TEMPERATURE JN ODBGREES ¢
HIWASSEE RESERVOIR 196Te=DAYIIOZ enSURFACE ELEVI 44603 4

171

15

AUT MODEL =

s
<

Figure 163



!
t
{
!
i
l

|

!

shesss sees,

!

1
ezerleseseses

200
40a

S0SURFACE ELEVE

1740

168

TEMPERATURE IN DEGREES €

12.0

FONTANA RESERVOIR 1966e~DAyi265

po

1.0

Figure 36 ry HODEL +

0
13,0
13.0
60,0
75.0

AG ote

Savtesse

90,0
103.0

2200

2760

265

1760

“168
TAWPERATURE IN DEGREES ¢

as ne g

oo et

none $3

~ a
2 2 8
s a ¢
# 8 &

86

§
i
i

Figure 35 wrt MODEL ©


po:

19.3... 22,0 ___24,5 27,0 29,5

ee ees ee

2houd

i
1

0 MEASURED
“DVERL

om OUTLET,

nat
weSURFAGE ELEVI

1
1
i

say e363

TARPESATURE IN DEGREES €

197,

2 0—81

MIT MOMEL ® FEAT LOUDN YE

Figure 180

----$----

22.0
2hoed n

1955

9 OUTLET
meSURPACE ELEVE

10
DEGREES ¢
snvart 70
185

14>

TENPERATURE I

Vaed

et)
Figure 161 myy Mune, © FRAT Luuurut 197)



FORT LOUDON RESERVOIR

Figures 171 and 172 show the computed and measured temperature
profiles for Fort Loudon Reservoir for 1971. It can be seen that the
computed temperatures predict the water temperature at the outlet
reasonably satisfactorily. The good fit of the predicted outlet water
temperature is shown in Figure 173. These results are statistically
verified in Tables 19 and 20 which show standard errors of estimate of
1.5°C and 3.0°C for the outlet and surface waters, respectively.

Figures 174 to 180 show the effect of the variation in the
thickness of the horizontal segments from 1 to 3 meters. The change
causes only minor differences in the predicted temperatures. This is
verified-in Tables 19 and 20 where the standard errors of estimate
are only slightly different from those calculated using a 2 meter
segment.

In Figures 181 to 187 the effect of the variation of 6, the fraction
of the solar radiation absorbed at the water surface, from 0.2 to 0.5
is shown to be negligible. This is verified in Tables 19 and 20 where
the standard errors of estimate are shown to be similar to those obtained
with the standard 8 of 0.5.

In Figures 188 to 194 is shown the effect of variation in n, the
radiation absorption coefficient, from 0.05 to 1.40 per meter. It can
be seen that the use of an absorption coefficient as low at 0.05 predicts
the temperature poorly on day 132. On the other days the differences
between the temperatures predicted using 0.05 and the other coefficients
are relatively small. This is also verified in Tables 19 and 20 where,

the differences in the standard errors of estimate of temperatures are

176


j 4 denn pecee pees
4 z
eo: e
2 8 w---}----}----}----}----$----f----f----}----f----43
‘ FI Ege of
----f---- to3 ns Ss SOE SONOS SONNE SEES SOUS SOOO YD ee
: = ¢ sritze  [s
: i vertet
! 3 oonngy
4 : q ev! e
foes ieee ee ees ee eee ae ees nes nee (0 0 Tn Ses See Sel ad a Sa
fg ‘ f f
f Bz
~--- a---$----} eee. sit woe nae Sans SS SOS SOS SOE ©
age £
a , Bs , q
; be BZ q , e
noon conn finnegonnngon--d----4 3 FOE Sa Se a a a es Steer ee od
; ae 8 : i ‘ , , h 3
i 3
, 3 a 4 .
| 3 ts
.
| Bo pcppecic be bbb be
E;
‘cn fel sao 8
Prop ry a. i
i ' z

=7SUSFAGE FLEVE

TEMPERATURE [% DEGREES ¢

102

DOUGLAS KESERVUIR L999—=0Ay 69

MIT MOMEL ©

Figure 57


(b) Lake is deep and isothermal during the springtime.

(c) The lake is turbid and the incoming solar radiation is ab-
sorbed within a small layer near the surface.

(d) Eddy diffusivity accounts for all heat transfer within the
lake except for the heat added by the power plant and pumping.

(e) The annual equilibrium temperature and wind speed over the
lake can be expressed in a sinusoidal form.
Basic Equations

(a) Governing equations

The change in temperature with depth when the plant discharge

surfaces is:
MD. bg) y, MEBs 6c sor sgr e — )
STE) = 38 (MEd) + s(2) ocaxey or 24 <28%q (61)
where:

T = temperature (°c)

t = time (day)

Zz = depth below the water surface (ft)

!

Ky = thermal diffusivity (£t?/day)

“ = the specified pumping velocity (ft/day)

24.24 = the specified intake and discharge depths (ft)

Zn = the depth of the lake (ft)

S(z) = the explicit thermal discharge or heat input

term (°C/day)

49


ss
ELEVATION (meters)

o-
500
480
460
FONTANA RESERVOIR, 1966
440 DAY: 187 (JULY 6)
ee «69 MEASURED
© © WITHOUT INFLOW /oUTFLOW | 4
© = @ WITH INFLOW / OUTFLOW
420f COUNTED AS POWER
PLANT COOLING WATER

Oo 2 4 6 8 lo 12 14 6 18 20 22 24 26 28 30
TEMPERATURE (°C)

Figure 8. Cornell model test run results, July 6, 1966


°
q
=
Pry
2 Ey
g
& --
: de E e
zg : rs
= 5
: i Sr,
33 : ‘ Sonush
--+333 i 3g5hze «
Py 3 —4i-d--4 nae n gH $$ 2255S e~-- 4 8
2g 3 oe 1 Segas5 e
= i 2o8see a
. =
s j : tunece
nat gf q onnmay
et |
2 i a
Pa { 4 af oe
oF
© 3
z 7
aa y
2 a
s
z i
Par H
PPG ie
z- w--- Hdd tg
3 4 : : f x
3 , 4
z Hau- $e ~-- $----4$----$----44
z e
* ql : :
. d
4 eon-$-~-- $----$----4$----4¢2
= B : q -
g fi 4
5 ----4----4---- 3
H y Hy :
f . .
7 a
. 3
= ¢
3 &

MeSURFACE ELEVE 26765 01

ss0AyI280

184

TFHPETATURE IN DEGREES ¢

Figure 179 yy #OMEL ® FART LUT 1972


Table

Table

Table

Table

Table

Table

Table

17

18

19

20

21

22

23

Statistical Analysis for the Predicted
Water Temperature at Outlet Level -
Hiwassee Reservoir, 1947

Statistical Analysis for the Predicted
Surface Water Temperature Hiwassee
Reservoir, 1947

Statistical Analysis for the Predicted
Water Temperature at Outlet Level
Fort Loudon Reservoir, 1971

Statistical Analysis for the Predicted
Surface Water Temperature - Fort Loudon
Reservoir, 1971

Least Squares Curve Fit for Measured
Surface Water Temperature

Least Squares Curve Fit for Measured
Water Temperature at Outlet Level

Densimetric Froude Numbers for Some
TVA Reservoirs

xviii

160

161

179

180

198

199

203


Table 16

STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE

Reservoir/Year: South Holston/1953

Time Period Covered: 60th - 360th Julian Day

Std. error Correlation

File Name of estimate (°c) Coefficient
STAND 2.44 0.96
DELZ 1 2.53 0.96
DELZ 2 3.09 0.94
BETA 1 2.32 0.96
BETA 2 2.29 0.96
ETA 1 1.05 0.99
ETA 2 2.82 0.95
ETA 3 — ----
DIFF 1 2.18 0.97
DIFF 3 1.75 0.98

144


weg any ag ng ag 9p ag gn — *
: : y ; : : y : qs : A : : : : 4*
: ¥ : a Sons Ss SE Soe Sa S| in Ss Cs Ss Ses Se Gees : : 4
: ‘ ; ‘ : ‘ ‘ . We a . ‘ ‘ . : : ’ &
a a a at ae i 0 2 ent een Deen Sennen Sena, Sa: SO 3
° 2
e ¢
----}-~--}----}---- ; 2 : ; nninno-f-—-43 8
z 3 5
----$----}----}----}----}----}----}---- s a---$----$----}----}---- fp de Sw
: : : : 2 : 4 : . ri cy 3
: 4 : : 5 5 : 4 Aa :
: . : ~j--~-f2 3 7 _ ect
wena }---u}----}----}----}---- ff. fey wane fonenp----f----f---- fff tt de 2
soa Bae
. & > : : Ss
z gs
: de zd : pa
“35 - "s
5 ’ ge
“e see : Fa : Zs
a . abe “ . ¥ tok A o “ . : aoe
2. Sooo seé ee ere — ees
w---}-~--}----}-.2-}----}----}----f----f -i¢ 2 & w---}- ~}----f----4---~f_---] dts Bg
eu 1 a we = a “ “ as 4
: . ° ; = i . : :
. feo. : ; 5 z ; : : : : i
. . = =
----}----}----}----$---«$----$----$---- Fa e-nn---- p----f---- ppd ee -ped--eteeads 2
°
. : dee
. .
° ‘ “8. z
----$----}----}----f.--5f---- pp de nia d~---$----$----$----}---- fff pede
, 2 i «8 2
/ Heo: : : S : . . . ra
osed ae z . z
wana fone n pent ---- fa f= ff fb i ----}----}----}----}----}----f----f----}.---f----i?
: ‘ : 2
= z
H y 1 e y e
----$----}----}--¥_}----}----4----f----}---- bets w---}----$----b eb debts
2 8 g ¢ 3 ba 2 5 3 s e 3 3 8 3 g bd 8 2 3
s 5 3 Py 3 Py Ps § é Py g s é 3
2 gs 38 $$ #8 ¢€ § g€ § & 2 ¢ $$ $s 8 ¢ 8 g§ § §

ewert wz xurwe owark az curwe

83


necessary coefficients than to further elaborate on the details of the
models themselves. For practical field applications, this is especially
true. For research purposes some of the phenomena should be disaggre-
gated and be more intensively studied but for purposes of meeting the
legal requirements of Section 316, determination with greater confidence
of the coefficients of the aggregated phenomena is more important.

Therefore, a series of comparative studies of mathematical models
of typical thermal systems have been commissioned by EPA. These include
a comparative study of the local thermal structure of submerged discharges
into large bodies of water, of the local thermal structure of surface
discharges into large bodies of water”, and this work on the thermal
structure of large, deep bodies of water (i.e., reservoirs or lakes).

The distinction between a deep reservoir or lake and a shallow, run
of the river reservoir is the maintenance of horizontal isotherms and a
strong stratification during summer. Also deep reservoirs usually have
a low annual through-flow to volume ratio.

Though there are many mathematical models available today for
calculation of temperature rises in reservoirs, few of them have been so
thoroughly explicated that they can be used easily except by a specialist
in the field. Though many of the models have been developed under EPA
contracts, there is no detailed evaluation and comparison of the various
models. Therefore, it is not possible to know beforehand what informa-
tion is required for the computer program evaluation, which factors are
considered in the models, the sensitivity of the models to changes in the
variables nor how the results would differ from one another if the

various models were used.


(i) With Variation in Thickness of Horizontal Layer, Az, Only ---
WRE's report states that calculation with a constant thickness of
one meter most closely matches the measured values of temperature.
However, no details of the test runs are given. The results of

our study show that the thickness of horizontal layer affects the
thermal simulation greatly. This is undesirable. In general, our
results indicate that in the warming period, Spring and early Sumer,
the larger the horizontal layer thickness the higher the temperature
profile. A typical example can be seen in Figure 2. The temperature
profile of Az = 2m is 2°C to 3°C higher than that of Az = 1m and the
profile for Az = 1.5m falls in between.

In the early fall, as the water body loses heat, the temperature
profile predicted with the larger horizontal element followed the
trend but at a more rapid rate at the water surface than at inter-
mediate depths. Calculation with the larger vertical increments
predicts a higher temperature than with the smaller vertical incre-
ments as indicated in Figure 3. During the period that the reservoir
is isothermal, the profiles predicted are very similar for all thick-
nesses as shown in Figure 4.

The use of Az = 0.6m probably violates the criterion that within
the time period simulated the through flow must be less than the volume
in that element. The temperature profile became isothermal in Fall as
shown in Figure 3. Since no remedy is provided within the model and
no warning is given if the situation exists, all simulation results should

be examined very carefully.

34


0

aeee

15.0

eeee

Se eee eet ee ee

Ye
Ps :
‘toe
Seen ed et et
. ehevonson
yoy : ‘
= 8 2 $ 2 8g
= 3 Ff gg g &

ewers =z xorue

i98.0

1%

TEMPERATURE IN DEGREES ¢

5 17.0028

24

220

195

12.0 16s

De

<

A909

woSURFACE ELEVi

FONTANA RESERVOIR 1966-=DAy 1205

Figure 20 uty MODEC #

15.0

20.0

°
°
ee eer eee oe
: des
coed oe
¥

= 2 8
3 8 8

eweer =z curwe

as Sas ae £1
~---}----} ds
1 5 8
----}----}----}2
Py
f .
enya
=~} 388522 te
#22
#5a858
BS earierran
onnnay
w---}-- fife
S
----}--- pote
w--~}----}---- fe
=
---}-- Lhe g
= eg g 3
gs 8 2 8

TEMPERATURE IN DEGREES €

2
?

SURFACE ELEVS

FONTANA RESERVOIR 1966--04y1335

Figure 21 wq7 HODEL ©

79


Table 2 (Continued).

INPUT PARAMETERS

Input Cornell MIT Water Resources
Parameters Model Model Engineering Model
Evaporation No No (Input constant No (input evaporation

for built-in formula coefficient to calcu-
to calculate heat loss) | late heat loss)
Precipitation No No No
Qutlet elevation | Yes (depth Yes Yes
of intake for| m TM
power plant)
ft
Inflow rate No? Yes Ygs (daily avg.)
m3/day [sec
Inflow No? Yes Yes (daily avg.)
Temperature % %
Initial (i) Yes (i) Yes (isothermal (i) Yes (either iso-
Reservoir isothermal ) gnly) , feral or variable)
{3} temperature
ii) rate of G i) (iso- Gi) (isothermal ) (in Yes {ay default use
temperature thermal) x 1077 °c/sec)
change.
Reservoir Yes Yes (used for compari- |Yes (daily avg.}
surface (constant son only, the actual m
elevation reservoir value is evaluated
‘ depth) ft from continuity) m
Reservoir (i) No (i) Yes (i) No
Geometry u mn om
(i) Length v s. (ii) No” (ij) Yes (ij) Yes
elevation m m m om
(ii) Horizontal
cross-section
area v s. elev.
Fraction of No Yes No
solar radiation (0.4 = 0.5 (Assume equal to
absorbed at the recommended) 0.4 internally
water surface for the top 0.3m
layer)
9 3 3
Outflow rate No Yes m/day Yes (daily avg.) m/sec

17


a4

- moe 4
: °
bg.
ot Hee nw
oe ote e

- --49— .
F . q

| J

= ¥ ¥ _ Wl be
3 |—_g_j__ $3

| P f ; rps

: i” , tb
ad on 5
ce ers Serene eee é

g
3
3
. oo
i
5 *
zis
fy a
.
.
$2
i 5
J—-- = 3S
jn--- $2
3

m0

TEMPERATURE IN DEGREES €

520.9 4

SOUTH HOUSTON 1953-<DAy1 78 e~SURFACE ELEV!

Figure 132 37 MOnEL ©

+O » MEASURED

H
{
'

< © ourber
I
t

90,0

3

140 cry 1200 Load 170 5 2260 2405 2710
TEMPERATURE IN DEGREES ¢

4nd

100,

S202 H

SOUTH HOLST"N 1953—=DAyI142 —=SURFACE ELEVE

Figure 133 jy HOMEL ©

152


fy

"
ai

)

8R0
3

s 5
* = . a a - a = —
= 8 me
Es =
3 abe
tes ep FR WPF eb e
a 2 : : ;---"$3-— ; ; —---}----}____fe
3 3 tt is

we

TENPERATURE. I I
CHEAOKEG RESERVOIR L96TeRDAYILE® eSBURPAGE ELEVE

ATURE JW DEGREES ©

CHEROKEE RESERVOIR 49OVeSDAYILED

180

=

eg oye
£ 4
Bs a 4 a a

* a a as a aes Ses 1)

Hn

117

MY MQOEE ®

Figure 79


SECTION I
CONCLUSIONS

This study using field data indicates that the deep reservoir model
with one-dimensional assumptions can be applied to a reservoir or
lake where the principal variation of flow characteristics is in
the vertical direction. The factors considered in the heat trans-
port equation must include the heat gained or lost through the
water surface, heat transported by inflow, outflow and vertical

advection, and the mixing mechanism due to diffusion and convection.

Among the models evaluated, the MIT deep reservoir model appears to
be most easily used and to give results most compatible with the

measured temperatures.

The temperature predicted is strongly dependent upon the magnitude
of the absorption coefficient of water, and the diffusion coefficient.
However, our sensitivity analysis shows that the value of about

0.75m"2

and a diffusion coefficient of 15 to 20 times molecular
diffusion are appropriate choices for the seven TVA reservoirs

studied.

The Densimetric Froude number is an important parameter to indicate
the degree of stratification of a body of water. The determination
of whether or not a reservoir or lake is suitable for the application

of a deep reservoir model depends on the Densimetric Froude number.


Uo) =U e () (51)
° 0 max (t)
where:
vonax = maximum velocity or velocity at YYout
Yout = elevation of centerline of outlet
% = the outflow standard deviation calculated on
the basis that 95% of the outflow comes from
the calculated withdrawal layer or:
> _ 6/2
° 196
with US evaluated from (52)
max
Vs
Qo Ct) 4 B(V)U, Ody (53)
Db
where:

Q(t) is total outflow at the specified outlet

(iii) Withdrawal thickness calculated from modified Kao's

a3) if the temperature gradient, at at the outlet

equation ay

is greater than or equal to 0.01 c/n.

8 = 4.8 ae (54)
where:

6 = thickness of withdrawal layer

q = outflow rate per unit width

4S


volumetric discharge through the reservoir in m/sec

mean reservoir depth in meters

< G8 4
u

= reservoir volume in cubic meters

o
u

average normalized density gradient in reservoir

qo? my

f

g = gravitational constant (= 9.8 m/sec’)

substituting the average values, we have

L
IF, = s20¢ @ (87)

For IF, < t , the reservoir is considered strongly stratified.

The densimetric Froude numbers for the seven reservoirs tested are
listed in Table 23. Except for Fort Loudoun Reservoir, the densimetric
Froude numbers are much smaller than 1/Il. Based on the Froude number
criterion, all but one are considered to be strongly stratified.

When we compare the computed temperatures with measured temperatures,
we note that an adequate simulation by the deep reservoir model for a
particular reservoir does not solely depend on its densimetric Froude
number.

The densimetric Froude numbers of the seven reservoirs do not
extend over the whole range of interest. Therefore, no critical value
can be established with regard to the applicability of the model.
However, it appears that for a reservoir with large depth, low densimetric
Froude number and small variation in surface elevation, such as Fontana
Reservoir, small differences in predicted and measured temperatures
result from the application of the MIT deep reservoir model. The greater

the deviation from deep reservoir conditions, the less accurate the

calculated temperatures will be.
201


2000

vy

CHEROKEE RESERVOIR 1907E~OAYISRD -EQSUAPACE

Mby MOET #

Figure 72

FH
x
z
:

:
5
$
2

CHEROKEE RESERVOIR 4967 e=DA

MU? MDOEE ®

Figure 73

114


a

=

~---$--c0

enna

eggRe--~

aeeES
reser
canna

27,0295

2405

¢
8

1700 rt

1n5
TEMPEPATURE IN DEGREES ¢

12.0

08

9

2nT

ONYOLAS RFSERVUTR 196%~-04yIZ1b -=SURFACE ELEVE

MYT MUDEL *

Figure 68

7
Sek

900
$300

<= ouner
1
I

12.0

30,0,

14ed 1m 1905 22.0 Er 2740 -2%3

TENPEPATURE IN DEGREES ¢

pe

2.0

emSURFACE ELEVI 29540 4

AY 1250

VOUCLAS RESERVUIR,

AUT Monet =

Figure 69

98


denends

o

923

30

a---$----

Hoyos «
w--t}----f----fe 6 ¢
B a.* ee
igo
Secszt oi
—---}383222---i2 —
fcbkea Is
veenes
.

ae-p-4--

24a 8709 2949

170

143
TEMPERATURE JN DERKEES €

OMYSLAS RESERVUIK 1969=vAyI121 -=SURFACE ELEVI

MIT MOREL ©

1
evry

Figure 52

2
asec:
S5ee
tere

peer

= OVERLAP
<> cuner
t
t

5
i
i
1

270

25

te pr 1200 Vad ine 1965 22,0 Baad

405

TERPERATURE IN DEGREES €

299.9 4

SURFACE ELEVE

£
3
z
2
Ea
S

AUT MOMEL ©

Figure 53

100


Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

Fontana Reservoir Layer Thickness Sensitivity

Day 187

Fontana Reservoir Layer Thickness Sensitivity

Day 215

Fontana Reservoir Layer Thickness Sensitivity

Day 244

Fontana Reservoir Layer Thickness Sensitivity

Day 285

Fontana Reservoir Layer Thickness Sensitivity

Day 335

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 75

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 132

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 187

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 215

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 244

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 285

Fontana Reservoir Absorption Coefficient
Sensitivity, Day 335

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 75

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 132

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 187

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 215

Fontana Reservoir Extinction Coefficient
Sensitivity, Day 244

vi

77

78

78

79

79

80

80

81

81

82

82

83

83

84

84

85

85


B fee ate wed ane:
wooo $4}
2 ¢ 8
¢ oe8 2
ewere oz runwe

Tos.0

27.00 28

2240

165 1760

12.0

TEMPERATURE IN DEGREES ¢

FONTANA RESERVOIR 1!

Sones H

SeSURFACE ELEVI

onDhy 1132

Figure 16 HIT MODEL ©

17.0 1963 2260 as 27.0 38S

15

TEMPERATURE IN DEGREES
FONTANA RESERVOIR 1986~=0Ayi207 ==SURFACE ELevE

pe fer}

1.0

or

fon gmp pen gig ge pn gp
g---}—--}---}----f pp
23 Senuss
~2a-f—--}----}----]----}_ pe
rs 3 PESsSES
~ : . : Weerrer
3 : cnwngy
te:
----}----fee_} ppp
, : 1} ita : : :
. Te
:
Tate
ast
: Thee, :
: Oa sedecendes
ea--fe fp
. ¥
a es eS OS et OO SS a ES SE
2 ¢ 2 8 $ $8 © $ 2 8
2 gg 8 ¢ #£ & $$ g &

260

130,0 +

509.9 4

Fégure 17 ry HOOEL *

77


Table 4. MODEL CONSTRUCTION
Paramets C 11 MIT Water Resources
aramevers Nodel Model Engineering Model
Mathematical Explicit finite Explicit finite Implicit finite
scheme of difference scheme difference scheme ifference scheme
approximation
Stability .
At recommended
y by S Vivveee (2) R, <1 always
Applicability deep, stratified | stratified” strongly stratified”
criteria turbid lake Fy < i. 0.32 Fp 0.1
Time ster Variable Constant Constant
2 (i) Determine At
Ate Se es. max. from E,(1) Gy), hr. <At< day
(44) guided by time|{44) guided by time
o< Cy < 45 step of input data step of input
Ky > diffusivity (iii) must satisfy. ata
stability criteria
bepte Interval Constant Constant Constant
Ay) (4) maximum (4) 50 intervals |(i) 1 meter
100 intervals or less is recom- recommended
mended be
(ii) min. 20 (44) max.number of steps
intervals is 200 i

Test Case

Cayuga Lake
400 days

Fontana Reservoir
3/1/66 to 12/31/66

Fontana Reservoir
3/1/66 to 12/31/66

22


— on ee a a ao jn--- 43
: f 4
z 3
a
. . |
-
] e
7 a
‘ , 3
. n ragers.
; S8ze2s.
wcececdl ta
waa $e--- bed bbb ssssssBe ts
: ze525585-13
23338322 $5
. coamguey
i .
--f----}----f pbb pds
J ‘ , f : , §
H
i
H H *
----}----}----}--- w---f---- be ds
| “
- s °
: 3
a 2
3
bath
; ate k
eahlse fe
ne w2phiSe__fe
an 2
i “a 0 ,
i :
4
a-4-$ 4. ----b---}y
7 ;
q a 3
J alt s
d 3
a / 3

TraPEVaTUME IM DEGREES ©

SWUTH MLSTOW 19939—4EASUSED TEMPERATURE PROFILES=

MUT MOTEL *

Figure 117

sTOp® 0

142


CHEROKEE RESERVOIR

Figures 69 and 70 show the computed reservoir temperatures and the
computed outflow temperature at Cherokee Reservoir respectively, for
1967. These can be compared with the measured data as shown in Figures
71 to 76. It can be seen from the Figures and as shown in Table 12 and
13 that the predicted temperatures at the outlet and at the surface
have larger standard errors of estimate, 2.7° and 2.1° respectively,
than the results for Fontana Reservoir. It can also be seen from
Figures 71 to 76 that the variation in thickness of the horizontal
segments from 1 to 3 meters makes minor differences in the predicted
results with the measured temperature than does the 2 meter thickness.
This is also evident in Tables 12 and 13.

It can be seen from Figures 77 to 82 that a variation in 6, the
fraction of solar radiation absorbed at the water surface, from 0.2 to
0.5 makes little difference in the predicted temperatures. This is also
evident from Tables 12 and 13.

In Figures 83 to 88 the effect of a change in n, the radiation
absorption coefficient, from 0.05 to 1.40 is shown. It is also shown
that the use of a sorption coefficient of 0.05 predicts the temperature
very poorly and that in general the value of 0.4 gives the best predic-
tions. This is verified in Tables 12 and 13 where the standard error
of estimate is 2.3° and 1.9°C for the outlet temperature and the surface
water temperature, respectively.

In Figures 89 to 94, the effects of varying the diffusion coefficient
from molecular to 100 times molecular diffusion are shown.

109


Tos.0

’ F Ts
wenn genes
panna n= 48
2 .
Beanaze__ in
GESSSSES Fs
Sp oeepartoe +
onmmg ay
aoe be
§
w---}----}e

20.0

i350

avs

o5

1
1
'
1

1.0

Or

150.0

TEMPERATURE IN DEGREES ¢

FONTANA RESERVOIR 1986-~DAvi192

5084

nASURFACE ELEVI

Figure 30 ry moneL *

$----

15.0

30.0

we
.
, z eve

eee

a---3----$-2--$----

3,
60.0
75.0

wee dee

90.0
508.0

120.0

133.0

2
Be

150.0

2903

27.0

45 0 De 1240 165 17.0 1965 22.0

20

TEMPERATURE IN DEGREES ¢
FONTANA RESERVDIR 1986==DAyI187 SURFACE ELEVI

509.9

Pégure 31 ayy MOREL ©

84


general problem will be discussed in detail in a separate section.

We were unable to acquire the original WRE test run data. There-

fore, Fontana Reservoir data used in the evaluation were prepared from

TVA measured field data.

Principal Assumptions

(a) There is horizontal homogeneity, i.e., stratification is in

the vertical direction only.

(b) Effective diffusion accounts for the heat transfer due to

wind mixing turbulent motion and reservoir instability,

(c) There is differential absorption of incoming solar radiation.

(d) There is no flux, of volume or heat, through the reservoir

bottom or sides except that due to inflow and outflow.

Factors Considered and the Basic Equations

Since the heat transport equations used in the computer program

are different than those indicated in WRE report, the heat transport

equations were rederived. The set of implicit equations were solved

by the Thomas Alogrithm, and are shown below.

Direct absorption of solar radiation ---

# (2) = 6, (1-8) © M5052) @)
where o, = net short wave radiation at water surface
¢(z) = short wave radiation at elevation z
Z, = elevation of water surface
8 = fraction of solar radiation absorbed at the
water surface (8=0.4 is assumed)
n = radiation extinction coefficient, wt

25