EPA-660/3-75-038
JUNE 1975
Ecological Research Series
valuation of Mathematical Models
for Temperature Prediction in
Deep Reservoirs
Office of Research and Development
U.S. Environmental Protection Agency
Corvailis, Oregon 9
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This report has been reviewed by the Office of Research and
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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 *ale by the Superintendent of Document*, U.S. Government
Printing Office, Washington, D.C. 20402
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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 0075m
and a diffusion coefficient of 15 to 20 times molecular diffusion are
appropriate choices for the seven TVA 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
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TABLE OF CONTENTS
Page
List of Figures ............................ v
List of Tables ............................
Section I
Conclusions ............................. 1
Section II
Recommendations ........................... 3
Section III
Introduction ............................. 4
Section IV
Analysis of Deep Reservoir Models .................. 8
General Description of Deep Reservoir Models ............. 9
Water Resources Engineers' Model ................... 24
Principal Assumptions ....................... 25
Factors Considered and the Basic Equations ............ 25
' Direct Absorption of Solar Radiation ............... 25
Selective Withdrawal ....................... 26
Depth and Velocity Distribution of Inflow ............. 27
Internal Mixing .......................... 28
Governing Equation ........................ 29
Verification ........................... 33
Results of Test Run .................. ...... 33
MIT Model .............................. 39
Principal Assumptions ....................... 39
Factors Considered and the Basic Equations ............ 43
Varification ........ . .................. 48
Sensitivity Analysis ....................... 48
Cornell Model ............................ 48
Assumptions ............................ 48
Basic Equations .......................... 49
Verification ........................... 52
Results of Test Run ........................ 53
Problems with Deep Reservoirs Models ................. 58
Section V
Sensitivity Analysis ......................... 64
Fontana Reservoir .......................... 69
Douglas Reservoir .......................... 91
Cherokee Reservoir .......................... 109
Norris Reservoir ........................... 126
South Holston Reservoir ....................... 140
iii
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Hiwasee Reservoir 158
Fort Loudon Reservoir 176
Section VI
Analysis of Data 197
References 204
iv
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LIST OF FIGURES
Page
Figure 1 Reservoir Representation 11
Figure 2 WRE Model -- Effect of Thickness of
Horizontal Layer (July 6, 1966) 35
Figure 3 WRE Model -- Effect of Thickness of
Horizontal Layer (Sept. 1, 1966) 36
Figure 4 WRE Model -- Effect of Thickness of
Horizontal Layer (Dec. 1, 1966) 37
Figure 5 WRE Model -- Effects of Diffusion Coefficient
(July 6, 1966) 40
Figure 6 WRE Model -- Effects of Diffusion Coefficient
(Sept. 1, 1966) 41
Figure 7 WRE Model -- Effects of Diffusion Coefficient
(Dec. 1, 1966) 42
Figure 8 Cornell Model Test Run Results
July 6, 1966 55
Figure 9 Cornell Model Test Run Results
Septemper 1, 1966 56
Figure 10 Cornell Model Test Run Results
December 1, 1966 57
Figure 11 Surface Layer Schematic 61
Figure 12 Fontana Reservoif Measured Temperature
Profile 72
Figure 13 Fontana Reservoir Computed Temperature
Profile 72
Figure 14 Fontana Reservoir - Computed Outflow Temperature 76
Figure 15 Fontana Reservoir Layer Thickness Sensitivity 76
Day 75
Figure 16 Fontana Reservoir Layer Thickness Sensitivity
Day 132 77
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Figure 17 Fontana Reservoir Layer Thickness Sensitivity
Day 187 77
Figure 18 Fontana Reservoir Layer Thickness Sensitivity
Day 215 78
Figure 19 Fontana Reservoir Layer Thickness Sensitivity
Day 244 78
Figure 20 Fontana Reservoir Layer Thickness Sensitivity
Day 285 79
Figure 21 Fontana Reservoir Layer Thickness Sensitivity
Day 335 79
Figure 22 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 75 80
Figure 23 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 132 80
Figure 24 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 187 81
Figure 25 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 215 81
Figure 26 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 244 82
Figure 27 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 285 82
Figure 28 Fontana Reservoir Absorption Coefficient
Sensitivity, Day 335 83
Figure 29 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 75 83
Figure 30 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 132 84
Figure 31 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 187 84
Figure 32 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 215 85
Figure 33 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 244 85
vi
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Figure 34 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 285 86
Figure 35 Fontana Reservoir Extinction Coefficient
Sensitivity, Day 335 86
Figure 36 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 75 87
Figure 37 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 132 87
Figure 38 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 187 88
Figure 39 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 215 88
Figure 40 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 244 89
Figure 41 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 285 89
Figure 42 Fontana Reservoir Diffusion Coefficient
Sensitivity, Day 335 90
Figure 43 Douglas Reservoir Computed Temperature
Profile 93
Figure 44 Douglas Reservoir Computed Outflow
Temperature 96
Figure 45 Douglas Reservoir Layer Thickness
Sensitivity, Day 69 96
Figure 46 Douglas Reservoir Layer Thickness
Sensitivity, Day 121 97
Figure 47 Douglas Reservoir Layer Thickness
Sensitivity, Day 186 97
Figure 48 Douglas Reservoir Layer Thickness
Sensitivity, Day 218 98
Figure 49 Douglas Reservoir Layer Thickness
Sensitivity, Day 250 98
Figure 50 Douglas Reservoir Layer Thickness
Sensitivity, Day 276 99
vii
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Figure 51 Douglas Reservoir - Absorption Coefficient
Sensitivity, Day 69 99
Figure 52 Douglas Reservoir Absorption Coefficient
Sensitivity, Day 121 100
Figure 53 Douglas Reservoir Absorption Coefficient
Sensitivity, Day 186 100
Figure 54 Douglas Reservoir Absorption Coefficient
Sensitivity, Day 218 101
Figure 55 Douglas Reservoir Absorption Coefficient
Sensitivity, Day 250 101
Figure 56 Douglas Reservoir - Absorption Coefficient
Sensitivity, Day 276 102
Figure 57 Douglas Reservoir Extinction Coefficient
Sensitivity, Day 69 102
Figure 58 Douglas Reservoir Extinction Coefficient
Sensitivity, Day 121 103
Figure 59 Douglas Reservoir Extinction Coefficient
Sensitivity, Day 186 103
Figure 60 Douglas Reservoir Extinction Coefficient
Sensitivity, Day 218 104
Figure 61 Douglas Reservoir Extinction Coefficient
Sensitivity, Day 250 104
Figure 62 Douglas Reservoir Extinction Coefficient
Sensitivity, Day 276 105
Figure 63 Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 69 105
Figure 64 Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 121 106
i
Figure 65 Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 186 106
Figure 66 Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 218 107
Figure 67 Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 250 107
viii
-------
Figure 68 Douglas Reservoir Diffusion Coefficient
Sensitivity, Day 276 108
Figure 69 Cherokee Reservoir Computed Temperature
Profile 112
Figure 70 Cherokee Reservoir Computed Outflow
Temperature 113
Figure 71 Cherokee Reservoir Layer Thickness
Sensitivity, Day 66 113
Figure 72 Cherokee Reservoir Layer Thickness
Sensitivity, Day 123 114
Figure 73 Cherokee Reservoir Layer Thickness
Sensitivity, Day 186 114
Figure 74 Cherokee Reservoir Layer Thickness
Sensitivity, Day 213 115
Figure 75 Cherokee Reservoir Layer Thickness
Sensitivity, Day 248 115
Figure 76 Cherokee Reservoir Layer Thickness
Sensitivity, Day 277 116
Figure 77 Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 66 116
Figure 78 Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 123 117
Figure 79 Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 186 117
Figure 80 Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 213 118
Figure 81 Cherokee Reservoir Absorption Coefficient
Sensitivity, Day 248 118
Figure 82 Cherokee Reservoir - Absorption Coefficient
Sensitivity, Day 277 119
Figure 83 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 66 119
Figure 84 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 123 120
ix
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Figure 85 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 186 120
Figure 86 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 213 121
Figure 87 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 248 121
Figure 88 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 277 122
Figure 89 Cherokee Reservoir Extinction Coefficient
Sensitivity, Day 66 122
Figure 90 Cherokee Reservoir - Diffusion Coefficient
Sensitivity, Day 123 123
Figure 91 Cherokee Reservoir - Diffusion Coefficient
Sensitivity, Day 186 123
Figure 92 Cherokee Reservoir Diffusion Coefficient
Sensitivity, Day 213 124
Figure 93 Cherokee Reservoir Diffusion Coefficient
Sensitivity, Day 248 124
Figure 94 Cherokee Reservoir Diffusion Coefficient
Sensitivity, Day 277 125
Figure 95 Norris Reservoir Computed Temperature
Profile 127
Figure 96 Norris Reservoir Computed Outflow
Temperature 129
Figure 97 Norris Reservoir Layer Thickness
Sensitivity, Day 106 129
Figure 98 Norris Reservoir Layer Thickness
Sensitivity, Day 139 130
Figure 99 Norris Reservoir Layer Thickness
Sensitivity, Day 183 130
Figure 100 Norris Reservoir Layer thickness
Sensitivity, Day 225 131
Figure 101 Norris Reservoir Layer Thickness
Sensitivity, Day 252 131
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Figure 102 Norris Reservoir Absorption Coefficient
Sensitivity, Day 106 132
Figure 103 Norris Reservoir Absorption Coefficient
Sensitivity, Day 106 132
Figure 104 Norris Reservoir Absorption Coefficient
Sensitivity, Day 183 133
Figure 105 Norris Reservoir Absorption Coefficient
Sensitivity, Day 225 133
Figure 106 Norris Reservoir Absorption Coefficient
Sensitivity, Day 252 134
Figure 107 Norris Reservoir Extinction Coefficient
Sensitivity, Day 106 134
Figure 108 Norris Reservoir Extinction Coefficient
Sensitivity, Day 139 135
Figure 109 Norris Reservoir Extinction Coefficient
Sensitivity, Day 183 135
Figure 110 Norris Reservoir Extinction Coefficient
Sensitivity, Day 225 136
Figure 111 Norris Reservoir Extinction Coefficient
Sensitivity, Day 252 136
Figure 112 Norris Reservoir Diffusion Coefficient
Sensitivity, Day 106 137
Figure 113 Norris Reservoir Diffusion Coefficient
Sensitivity, Day 139 137
Figure 114 Norris Reservoir - Diffusion Coefficient
Sensitivity, Day 183 138
Figure 115 Norris Reservoir Diffusion Coefficient
Sensitivity, Day 225 138
Figure 116 Norris Reservoir Diffusion Coefficient
Sensitivity, Day 252 139
Figure 117 South Holston Reservoir Measured Temperature
Profile 142
Figure 118 South Holston Reservoir Computed Temperature
Profile 145
XI
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Figure 119 South Holston Reservoir Computed Outflow
Temperature 145
Figure 120 South Holston Reservoir Layer Thickness
Sensitivity, Day 78 146
Figure 121 South Holston Reservoir Layer Thickness
Sensitivity, Day 142 146
Figure 122 South Holston Reservoir Layer Thickness
Sensitivity, Day 203 147
Figure 123 South Holston Reservoir Layer Thickness
Sensitivity, Day 245 147
Figure 124 South Holston Reservoir Layer Thickness
Sensitivity, Day 299 148
Figure 125 South Holston Reservoir Layer Thickness
Sensitivity, Day 362 148
Figure 126 South Holston Reservoir Absorption
Coefficient Sensitivity, Day 78 149
Figure 127 South Holston Reservoir Absorption
Coefficient Sensitivity, Day 142 149
Figure 128 South Holston Reservoir Absorption
Coefficient Sensitivity, Day 203 150
Figure 129 South Holston Reservoir Absorption
Coefficient Sensitivity, Day 245 150
Figure 130 South Holston Reservoir Absorption
Coefficient Sensitivity, Day 299 151
Figure 131 South Holston Reservoir Absorption
Coefficient Sensitivity, Day 362 151
Figure 132 South Holston Reservoir Extinction
Coefficient Sensitivity, Day 78 152
Figure 133 South Holston Reservoir Extinction
Coefficient Sensitivity, Day 142 152
Figure 134 South Holston Reservoir Extinction
Coefficient Sensitivity, Day 203 153
Figure 135 South Holston Reservoir Extinction
Coefficient Sensitivity, Day 245 153
Figure 136 South Holston Reservoir Extinction
Coefficient Sensitivity, Day 299 154
xii
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Figure 137 South Holston Reservoir Extinction
Coefficient Sensitivity, Day 362 154
Figure 138 South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 78 155
Figure 139 South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 142 155
Figure 140 South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 203 156
Figure 141 South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 245 156
Figure 142 South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 299 157
Figure 143 South Holston Reservoir Diffusion
Coefficient Sensitivity, Day 362 157
Figure 144 Hiwassee Reservoir Measured Temperature
Profile 162
Figure 145 Hiwassee Reservoir Computed Temperature
Profile 162
Figure 146 Hiwassee Reservoir Computed Outflow
Temperature 163
Figure 147 Hiwassee Reservoir Layer Thickness
Sensitivity, Day 80 163
Figure 148 Hiwassee Reservoir Layer Thickness
Sensitivity, Day 148 164
Figure 149 Hiwassee Reservoir - Layer Thickness
Sensitivity, Day 209 164
Figure 150 Hiwassee Reservoir Layer Thickness
Sensitivity, Day 267 165
Figure 151 Hiwassee Reservoir Layer Thickness
Sensitivity, Day 302 165
Figure 152 Hiwassee Reservoir Layer Thickness
Sensitivity, Day 364 166
Figure 153 Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 80 166
xiii
-------
Figure 154 Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 148 167
Figure 155 Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 209 167
Figure 156 Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 267 168
Figure 157 Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 302 168
Figure 158 Hiwassee Reservoir Absorption
Coefficient Sensitivity, Day 364 169
Figure 159 Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 80 169
Figure 160 Hiwassee Reservoir - Extinction
Coefficient Sensitivity, Day 148 170
Figure 161 Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 209 170
Figure 162 Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 267 171
Figure 163 Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 302 171
Figure 164 Hiwassee Reservoir Extinction
Coefficient Sensitivity, Day 364 172
Figure 165 Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 80 172
Figure 166 Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 148 173
Figure 167 Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 209 173
Figure 168 Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 267 174
Figure 169 Hiwassee Reservoir Diffusion
Coefficient Sensitivity, Day 302 174
Figure 170 Hiwassee Reservoir - Diffusion
Coefficient Sensitivity, Day 364 175
xiv
-------
Figure 171
Figure 172
Figure 173
Figure 174
Figure 175
Figure 176
Figure 177
Figure 178
Figure 179
Figure 180
Figure 181
Figure 182
Figure 183
Figure 184
Figure 185
Figure 186
Figure 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
Measured Temperature
Computed Temperature
Computed Outflow
Layer Thickness
Layer Thickness
Fort Loudon Reservoir - Layer Thickness
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
Layer Thickness
Layer Thickness
Layer Thickness
Layer Thickness
Fort Loudon Reservoir 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
xv
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Figure 188 Fort Loudon Reservoir Extinction 189
Coefficient Sensitivity, Day 76
Figure 189 Fort Loudon Reservoir Extinction 189
Coefficient Sensitivity, Day 132
Figure 190 Fort Loudon Reservoir Extinction 190
Coefficient Sensitivity, Day 204
Figure 191 Fort Loudon Reservoir Extinction 190
Coefficient Sensitivity, Day 226
Figure 192 Fort Loudon Reservoir Extinction 191
Coefficient Sensitivity, Day 253
Figure 193 Fort Loudon Reservoir Extinction 191
Coefficient Sensitivity, Day 280
Figure 194 Fort Loudon Reservoir Extinction 192
Coefficient Sensitivity, Day 343
Figure 195 Fort Loudon Reservoir Diffusion 192
Coefficient Sensitivity, Day 76
Figure 196 Fort Loudon Reservoir Diffusion 193
Coefficient Sensitivity, Day 132
Figure 197 Fort Loudon Reservoir Diffusion 193
Coefficient Sensitivity, Day 204
Figure 198 Fort Loudon Reservoir Diffusion 194
Coefficient Sensitivity, Day 226
Figure 199 Fort Loudon Reservoir - Diffusion 194
Coefficient Sensitivity, Day 253
Figure 200 Fort Loudon Reservoir Diffusion 195
Coefficient Sensitivity, Day 280
Figure 201 Fort Loudon Reservoir Diffusion 195
Coefficient Sensitivity, Day 343
xvi
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LIST OF TABLES
Table 1
Table 2
Table 3
Table 4
Table 5
Table 6
Table 7
Table 8
Table 9
Table 10
Table 11
Table 12
Table 13
Table 14
Table 15
Table 16
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 Morris
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
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Table 17
Table 18
Table 19
Table 20
Table 21
Table 22
Table 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
Densiraetrie Froude Numbers for Some
TVA Reservoirs
160
161
179
180
198
199
203
xviii
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SECTION I
CONCLUSIONS
1. This study losing 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.
2. 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.
3. 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" and a diffusion coefficient of 15 to 20 times molecular
diffusion are appropriate choices for the seven TVA reservoirs
studied.
4. 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.
-------
However, the representativeness of the result is not solely dependent
upon the Densimetric Froude number.
5. 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.
6. 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.
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SECTION II
RECOMMENDATIONS
1. 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.
2. 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.
3. 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.
4. 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.
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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
i
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
4
-------
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
2
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.
-------
Therefore, the most widely known and used models, Water Resources
Engineers (Orlob) , MIT (Harleman-Huber-Ryan) , Cornell Aeronautical
^ f\ 7
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
-------
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.
-------
SECTION IV
ANALYSIS OF DEEP RESERVOIR MODELS
In temperate zones, the spring heat.ing, 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, epilimnion, 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 column at the eleva-
tion where its density is equal to that in the water column.
Thermal stratification affects not only the extent of dilution 'and,
! i
mixing of the inflow waters, but also the quality of the water in
hypolimnion. 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
8
-------
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, 1 , relating all the energy inputs to a body of
water can be solved for reservoirs, rivers and estuaries and coastal
regions.
Q Q + Q ~ Q
Where:
Qr
shortwave radiation incident to the water surface;
reflected shortwave radiation;
incoming longwave radiation from the atmosphere;
reflected longwave radiation;
longwave radiation emitted by the body of water;
net energy brought into the body of water in inflow,
including precipitation, and accounting for outflow;
energy utilized by evaporation;
-------
Q, = energy conducted from the body of water as sensible
heat;
0 = 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
-------
OUTFLOW'
ATMOSPHERIC
EXCHANGE
INFLOW'
Qln »TIn
ABSORBED SOLAR RADIATION
DIFFUSION
VERTICAL
ADVECTION
SCHEMATIC OF RESERVOIR PROBLEM
TYPICAL HORIZONTAL SLICE
FROM RESERVOIR
Figure 1. Reservoir representation
11
-------
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 Qy, 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
-------
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
i
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
-------
Table 1. ASSUMPTIONS
Assumptions
Cornell
Model
MIT
Model
Water Respurces
Engineering Model
Horizontal
homogenei ty
Yesone dimensional
stratification in
vertical direction only
Yesone .dimensional
stratification, in
vertical direction only
1
Yesone dimensional
stratification in
vertical direction only
Primary
mechanism
for the
formation of
thermocline
Nonlinear interaction
between wind induced
turbulence and buoyancy
gradient2
Differential absorption
of incoming solar
radiation
Differential absorption
of incoming solar
radiation
Surface
boundary
conditions
one of the three is
specified in a sinusoidal
form3
1) water surface tempera-
ture Ts
ii) heat flux at surface
O.S. %
lii) equilibrium temper-
ature If
usually TV is,specified
Meteorologic Input
Water surface temper-
ature calculated
Meteorologic Input
Water surface temper-
ature calculated
Bottom
boundary
condition
(no flux)
Yes
Yes
Yes
Water budget
(water losses
due to evap-
or-tion and
gains to
rainfall)
Advective
heat (add
or subtracted)
No
No
No (the reservoir sur-
face -levation 1« cal-
culated as a function
of initial surface
level and the cumulative
inflow and outflow; the
input measured pool ele-
vations has never been
used)
Yes
Not directly; however
the measured daily sur-
face elevations are
used as pool level for
each simulation day.
The water budget impltes
evaporation, rainfall
and possible leakage
Yes
14
-------
Table 1 (continued). ASSUMPTIONS
1. 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.
2. 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.
3. A + B sin (||g-t + $)
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
-------
Table 2. INPUT PARAMETERS
Input
Parameters
Unit
Short wave
Solar radi-
ation
Net Long-
wave
atmosphere
radiation
Wind
speed
Air
Temperature
Cloud
Cover
Atmospheric
Pressure
Relative
Humidity
Wet BUI b
Temperature
Dew Point
Temperature
Equilibrium
Temperature
Cornel 1
Model
Input data in
specified units
only
No3
No
No3'10
No3
No3
No3
No3
No
No
Yes12
(in terms of mean
value, amplitude
and phase angle
in a sinusoidal
form)°c
MIT
Model
Input data in
specified units
only
Yes
(net flux)
kcal/m2 day
Yes1 '2;7
kcal/m2 - day
Yes
m/sec
Yes
°c
Yes2'6
decimal
No
Yes
decimal
No
No
No
Water Resources
Engineering Model
can be in any units
user supplies conversion
factors, standard units
as indicated
Yes1'2'4'8
(gross flux
i.e. no reflection)
kcal/m2 - sec
No ;
(calculated)
Yes8
m/sec
Yes8
°c
Yes8
decimal
Yes1'2'8
mb
Yes2'5'8
decimal
Yes2'5'8
°c
Yes2'5'8
°c
No
16
-------
Table 2 (Continued). INPUT PARAMETERS
Input
Parameters
Evaporation
Precipitation
Outlet elevation
Inflow rate
Inflow
Temperature
Initial
Reservoir
(i) temperature
(ii) rate of
temperature
change
Reservoir
surface
elevation
Reservoir
Geometry
.(i) Length v s.
elevation
(ii) Horizontal
cross-section
area v s. elev.
Fraction of
solar radiation
absorbed at the
water surface
Outflow rate
Cornell
Model
No
No
Yes (depth
of intake for
power plant)
ft
No9
No9
(i) Yes
(isothermal)
ac
(ii) (isor
thermal )
Yes
(constant
reservoir
depth) ft
(i) No
(ii) No11
No
No9
MIT
Model
No (Input constant
for built-in formula
to calculate heat loss)
No
Yes
m
Yes
m3/day
Yes
°c
(i) Yes (isothermal
only)
°c
(ii) (isothermal)
Yes (used for compari-
son only, the actual
value is evaluated
from continuity) m
(i) Yes
m m
(11) Yes
m^ m
Yes
(0.4 * 0.5
recommended)
Yes m3/day
Water Resources
Engineering Model
No (input evaporation
coefficient to calcu-
late heat loss)
No
Yes
m
Yes (daily avg.)
rrr/sec
Yesi (daily avg.)
°c
(i) Yes (either iso-
thermal or variable)
°c
(ii) Yes (By default use
1 x 10'9 °c/sec)
Yes (daily avg.)
m
(i) No
(11) Yes
m^ m
No
(Assume equal to
0.4 internally
for the top 0.3m
layer)
Yes (daily avg.)m /sec
17
-------
Table 2 (Continued). INPUT PARAMETERS
Input
Parameters
Friction velocity
Short wave radia-
tion extinction
coefficient
Diffusion coeffi-
cient
Heat exchange
coefficient
Cornel 1
Model
Yes (in terms of
mean, amplitude and
phase angle in a
sinusoidal formld)
ft/sec
No
Yes [in terms of
(C] + C2 Z) w*]
(upper and lower
bound of the coeffi-
cient are also inputed)
ft2/day
Yes 9
Btu/fr-day-°c
MIT
Model
No
Yes
1/m
Yes
2
m /day
No
Water Resources
Engineering Model
No
Yes
in terms of extinction
depth (m)
coefficient = 6.908/
extinction depth
Yes, empirical deter-
mined constants A, ,
A2, and A3 (i) DC A]
for E< EC A
(ii) Dc A2E 3 for
*~ c
E = - 4^- stability
of the water column
2
m /sec
No
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. If data are in
input, the net flux . Input flux x " 'f «*
in input, the net flux cal. flux (with reflection)
-------
Table 2 (Continued) . INPUT PARAMETERS
5. 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
6. To be read in only when atmospheric radiation is to be calculated intern-
ally by program
7. If calculated by program cloud cover should be known
8. Meteorologic data provide at least one observation per day, constant
values such as average monthly meteorologic conditions should be avoided.
9. Taking account of the power plant cooling water only and in terms of heat,
q , per unit area per unit time added by power plant
10. Also used for friction velocity calculation externally
11. A constant surface area is needed for calculating q (see 9)
2
12. Either surface temperature (°c) or heat flux at surface (Btu/ft -day) can
be used to replace equilibrium temperature
13. A + B sin (-1?F t + )
19
-------
Table 3. FACTORS INVOLVED IN ANALYSIS
Factors
Diffusivity
Stability mixing
process
Heat transfer in
control volume
(horizontal layer)
1. Direct
absorption
2. Diffusion
3. Vertical
advection
4. Horizontal
advection
Cornel 1
Model
Eddy diffusivity
Eddy diffusivity
» molecular
diffusivity
Free convection
if stratification
is unstable
1. No
2. Yes
3. No
4. No
MIT
Model
Molecular
diffusion
(constant)
neglects
turbulent
diffusion
Convective
mixing 1f
negative
temperature
gradient
occurs
^ - < n
ay
(T - tempera-
tare, y - ele-
vation, posi-
tive upward)
1. Yes
2. Yes
3. Yes
4. Yes
Water Resources
Engineering Model
So called "effective
diffusion" accounts for
turbulent diffusion and
convective mixing cal-
culated from temperature
jrofile
D = A, for E < Er
C 1 C
*here E = ~-ff stability
of the water column, E =
some critical value of
stability
Convective mixing if
legative temperature
gradient occurs
OT
ty <0
W J
(T - temperature,
y - elevation,
positive upward)
I. Yes
2. Yes
3. Yes
*. Yes
-------
Table 3 (Continued). FACTORS INVOLVED IN ANALYSIS
Factors
Heat transfer
across water sur-
face
1 . short wave
solar radiation
2. Net long-wave
radiation
3. Evaporation
4. Conductions
Inflow consider-
ation
1. enter at the
level of equal
temperature
2. distribution in
vertical direc-
tion
Withdrawal
consideration
1. level
2. distribution
in vertical
direction
3. limit on with-
draw zone
Cornel 1
Model
In terms of heat
flux
qs = K(TE-TJ
o t. o
K - heat exchange
coefficient
TE- equilibrium
temperature
T - water surface
temperature
1. Yes
2. No
(spread instan-
taneously into
a thin horizontal
sheet)
1. single level
outlet
2. No
(uniformly dis-
tributed at the
level of withdraw]
3. No
MIT
Model
1. Yes
2. Yes
3. Yes
4. Yes
1. Yes
2. Gaussian vel-
ocity distribu-
tion either with
or without en-
trance mixing
1. selective
withdrawal
2. withdrawal layer
thickness &
, Gaussian
' velocity dis-
tribution in
withdrawal layer
3. Yes
(withdrawal
layer never
extended over
the region with
temperature
gradient equal
or greater than
cut-off gradient)
of |£= 0.05
Water Resources
Engineering Model
1. Yes
2. Yes
3. Yes
4. Yes
1. Yes
2. uniform velocity
distribution over
the interflow
thickness deter-
mined by Debler's
critera
1. selective with-
drawal
2. withdrawal layer
thickness and
uniform velocity
distribution
over withdrawal
layer determined
by Debler's
criteria
3. Yes
withdrawal zone
always on top
or underneath
the thermocline
21
-------
Table 4. MDDEL CONSTRUCTION
Parameters
Mathematical
scheme of
approximation
Stability
criteria for
use
Applicability
criteria
Time step
(At)
Depth Interval
(Ay)
Test Case
Cornel 1
Model
Explicit finite
difference scheme
deep, stratified
turbid lake
Variable
AVrct{^ax
0< Ct < .5
KH - diffusivity
Constant
(i) maximum
100 intervals
Cayuga Lake
400 days
MIT
Model
Explicit finite
difference scheme
n At ^ , / \
(Ay)2 - * U;
u At «, -i /?\
V Ay ~ ] U;
stratified2
FD < 1 = 0.32
Constant
(1) Determine At
max. from Efil)
(ii) guided by time
step of input data
(iii) must satisfy-
stability criteria
Constant
(i) 50 intervals
or less is recom-
mended
(11) min. 20
intervals
Fontana Reservoir
3/1/66 to 12/31/66
Water Resources
Engineering Model
mplicit finite
ifference scheme
p Q (element) -At q
V V(element) '°
recommended
R < 1 always
a
2
strongly stratified
FD «0.1
Constant
(i) 1 hr..
-------
Table 4 (Continued). MODEL CONSTRUCTION
Parameters
Running
time for test
case
Cornell
Model
(400 days) 13 min.
for Ay = 5 ft.
MIT
Model
(300 days)
1.4 min. for Ay
= 2.0m
At 1 day
2.0 min. for Ay
= l.O"1
At 1 day
Water Resources
Engineering Model
(300 days)
2. 6 min. for Ay 2.0m
At - 1 day
3.1 min. for Ay 1.0m
At = 1 day
1.
2.
the program compiling time is not included
the densimetric Fronde number in defined as
-LQ /5T
m / ge
where L - reservoir length
Q - volumetric discharge through the reservoir
D - mean reservoir depth
V - reservoir volume
p - reference density
g - 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
-------
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> . 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 'effective1
diffusion, as defined by WRE. After correction of this error, the magnitude
of the 'effective1 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
-------
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
* (z) = o (1-6) e -n(zs-0.3-z) (2)
where (z) = short wave radiation at elevation z
z = elevation of water surface
3 = fraction of solar radiation absorbed at the
water surface (g=0.4 is assumed)
n = radiation extinction coefficient, m
25
-------
.(10)
Selective withdrawal
(i) Up to five outlets are allowed
(ii) Based on Debler's experimental results^UJ the critical
Froude number, FC, is 0.24, therefore:
0.24 =
(3)
The withdrawal depth in meters, then is:
d = 2.0
\%
Sp
= 2.0
feef
(4)
where q = ^, one half of the discharge per unit width,
Vi
and w is the reservoir width
e = -5^-, normalized density gradient
P0 dz
(iii) The velocity is uniform within the withdrawal zone
(iv) The principle of superposition is applied for the regions
in which withdrawal layers overlap
(v) Withdrawal layers never extend through the thermocline, or
physical boundaries.
(vi) At the onset of fall cooling, when the epilimnetic region
is well-mixed and isothermal Dabler's criteria do not hold
for withdrawals from the epilimnion and Craya's approach^ ^
is used. The flow is withdrawn from epilimnion until the
discharge is larger than Craya's critical flow given by:
26
-------
qc = 1.52 /gli HE, 2h
-------
Internal mixing
(i) Free convection occurs if stratification is unstable.
(ii) The heat transport, H, due to diffusion can be written as:
H = pc D(z,t) . rcg'fl (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.
(iii) The form of the diffusion coefficient used is:
D = Ai (constant), E < EC (9)
D = A2EA3, E>Ec (10)
where E = stability of the water column
E = critical stability parameter
The following values are suggested:
Ai = 2.5 x 10" 4 m2/sec
A2 = 1.5 x 10"8 n/'Vsec
A3 = -0.7
Ec = 1.0 x 10"6 m"1
28
-------
Governing equation
The time rate of change of thermal energy H in a control volume
of thickness Az, is:
3H.
where subscript j indicates the increment of depth; j = 1 at
bottom surface
H = The heat energy stored in the control volume (kcal)
hj = Heat flow associated with inflowing water (kcal/sec)
h = Heat flow associated with outflowing water (kcal/sec)
h = Heat flow by advection (kcal/sec)
h^ = Heat flow by diffusion (kcal/sec)
h = Heat flow by short wave solar radiation (kcal/sec)
Sri
In terms of temperature T, Eq. 12 can be written as:
3T. ,
p. CA. Az _l±i = p. c (TI) . (qlD . -p.c CT0) . (q^ + h. Aj
9T
A .Al
*J*1 - (13)
or
Pj
p CD.A-
J J J Az
C14)
29
-------
D.A. D.A.
+ P-;
/I
where
Pj = PjcCWqI}J Pj'CVj Cq^+hft C16)
T = vater temperature (°c)
T,. = inflow temperature ( c)
q., = inflow rate (m3/sec) into control volume
T = outflow temperature (°c)
a = outflow rate (m /sec) from control volume
h' = net insolation heat flux per unit area (kcal/m2-sec)
D = diffusion coefficient (m2/sec)
v = vertical advection velocity (m/sec)
Az = thickness of control volume (m)
p = average density of water (kg/m3)
p = density of water (kg/m3)
AV.J = A-'Az volume of jth control volume (m3)
z = vertical axis, positive upward (m)
A- = cross -section area at depth step j Cm2)
(18)
f = mean water temperature of jth control volume (°c)
c = specific heat of water (= 1 kcal/kg-°c)
30
-------
since
TJ-% = TJ-1 (19)
(20)
(21)
Equation 15 becomes:
where
D.A.
Kj,2
T ftl - lim TCt+At) - T(t)
1 ttj " - At ^"
Taylor series o£ 2nd order
T(t+At) = T(t) + T(t)At + T (t) Q- (26)
Zi
and by definition of derivative
for small At
T(t+At) = T(t) + T(t)-| + T(t+At) ^ (28)
or use superscript k to indicate time step
= T(k) + t(k) At + j(fcfl) At = a(k) + ^.(k+1) At (29)
2 ^ Z
where a = T + (30)
31
-------
By Equations 22 and 29
f (35)
Then Equation 31 becomes
" (36)
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 Si ,, Si 7, Si _ and
J j-1- 3 >* 3 >*>
F! are as follows:
Sj,l = °' j=2'3 " N
Sj>2 = 1, j=l,2,3 -- N (38)
Sl,3 = S1,3/S1,2; Fl = F1/S1,2 (39)
32
-------
C40)
, ,'Sj,3 C41)
then TN = F^ (42)
and
' * = N'lj N"2> "' 2'1 C43)
.(k+1) -At f j=1>2> . N (44)
j J &
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
-------
Ci) 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 wanning period, Spring and early Summer,
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
-------
500
480
Q)
460
O
i
UJ
440
420
FONTANA RESERVOIR, 1966
DAY'- 187 (JULY 6)
MEASURED
O O o O A z = 0.6m
Az = I.Om
Q Q Q Q Az = 1.5 m
o o o A z = 2.0m
8 10 12 14 16 18 20 22 24 26 28 30
TEMPERATURE (°C)
Figure 2. WRE model -- Effect of Thickness of Horizontal Layer (July 6)
-------
500
480
tt)
0)
E
-" 460
i
UJ
UJ
440
420
FONTANA RESERVOIR, 1966
DAY: 244 (SEPT.n
MEASURED
O O o O Az = 0.6 m
Az = 1.0 m
o o D a Az = 1.5m
o o o o Az = 2.0m
8 10 12 14 16 18
TEMPERATURE (°C)
20 22 24 26 28 30
Figure 3. WE model -- Effect of Thickness of Horizontal Layer (Sept. 1)
-------
500
480
o>
E
460
§
UJ
UJ44Q
420
o
FONTANA RE
DAY: 335 (
D D a a
o o o o
ISERVOIR, 1966
DEC. 1)
MEASURED
Az = I.Om
Az
Az
= 1.5m
= 2.0m
8 10 12 14 16 18 20
TEMPERATURE (°C)
22
24 26 28 30
Figure 4. WRE model -- Effect of Thickness of Horizontal Layer (Dec. 1)
-------
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
oncertainty 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:
DC = 2.5 x 10"7 m2/sec, EEC (46)
where
E = the stability of water, = |£ (47)
(J (3 £*
-6 1 '
E = the critical stability parameter = 1.0 x 10 m
which is of the same order as molecular diffusivity, 1.4 x 10"7 m2/sec.
Test runs using a diffusion coefficient one thousandth (1/1000) of the
38
-------
'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 Assumptions
(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 epilimnion.
39
-------
500
480
vt
w
V
^5
E
r 460
UJ
_l
UJ
440
420
FONTANA RESERVOIR, 1966
DAY 187 ( JULY 6)
_ MEASURED
STAND
O O O O Dc = 10-STAND
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
TEMPERATURE (°C)
Figure 5. WRE model -- Effects of Diffusion Coefficient (July 6)
-------
500
480
M
w
0>
0>
460
§
UJ
_i
UJ
440
420
FONTANA RESERVOIR, 1966
DAY : 244 ( SEPT. I)
MEASURED
STAND
O O O O Dc = 10 STAND
8 10 12 14 16 18 20 22 24 26 28 30
TEMPERATURE (°C)
Figure 6. WRE model -- Effects of Diffusion Coefficient (Sept. 1)
-------
500
480
(A
k.
0)
*-
0>
~ 460
O
i
LM
^440
420
FONTANA RESERVOIR, 1966
DAY: 335( DEC.i)
__^ MEASURED
STAND
O OO O Dc = 10-STAND
8 10 12 14 16 18 20 22
TEMPERATURE (°C)
24 26 28 30
Figure 7. WRE model -- Effects of Diffusion Coefficient (Dec. 1)
-------
(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
0(l-3) e"n frs'tf (48)
where:
<|> = net incident solar radiation
B = fraction of absorbed at the surface
n = light extinction coefficient
y = water surface elevation
o
Cc) 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
-------
(iv) Inflow velocity profile is approximated by a
Gaussian distribution, at elevation y
*i2
U. (y) = U, me (49)
1 ^-maxW
where:
i (t) = maximum value of the inflow velocity at
IftcLX
time t,
is determined from:
rYe
(S
lift) =) B(y) Ujl
'
^Y) dy (50)
where:
Qi(t) = total inflow
y = surface elevation
y, = bottom elevation
B(y) = width of the reservoir at elevation y
y. (t) = elevation of inflow
a. = inflow standard deviation
(d) Outflow
(i) multiple outlets
(ii) outflows are centered at the outlet with a Gaussian
velocity distribution:
44
-------
u°Cy) =
where:
U = maximum velocity or velocity at y=y
max
y t = elevation of centerline o£ outlet
a0 = the outflow standard deviation calculated on
the basis that 95% of the outflow comes from
the calculated withdrawal layer or:
-
-------
e = normalized density gradient = -£ (55)
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.
(iv) The velocities from each outlet are superimposed on
one another.
(e) 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.
(f) Governing Equations
The heat transport equation applied to each horizontal layer
has the following form:
(56)
CV60Atjr)T(jr)) + ^j (Ui(y)B(y)Ti-U0(y)B(y)T(y))
where :
T(y) = temperature at elevation y
V(y) = vertical velocity at elevation y
46
-------
l^Cy) = inflow velocity at elevation y
UQ(y) = outflow velocity at elevation y
T. = inflow temperature
A(y) = area at elevation y
t = time
a = molecular diffusivity
(j>(y) = transmission of radiation at elevation y
and the continuity equation can be written as
-£ (V(y)A(y)) = B(y) (Ui(y)-Uo(y)) (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:
%
(59)
where :
D = diffusion coefficient
At = time increment
V = vertical advection velocity
Ay = depth increment
47
-------
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.
Verification
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
f!41
physical effects of thermal discharge into Cayuga Lake*- '. 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
-------
(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:
J "
or
where:
T
t
z
KJJ
w
z.,z,
zm
S(z)
= temperature (°C)
= time (day)
= depth below the water surface (ft)
1 2
= thermal diffusivity (ft /day)
= the specified pumping velocity (ft/day)
= the specified intake and discharge depths (ft)
= the depth of the lake (ft)
= the explicit thermal discharge or heat input
term (°C/day)
49
-------
S + ATp Tsl -CZ-Z (62)
-" e
where:
ATp = temperature rise across condenser
T = surface temperature
w = pumping velocity
z = 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 CL , the heat per unit area per unit time added by the power
plant by the equation:
WP = W'0? 'ATP (63)
where:
p-Cp = heat capacity per cubic foot of water
(112.32 BTU/°C-ft3)
AT = temperature difference produced by power
plant
ATp = T(zd) T(Z;L) (°C) (64)
The thermal eddy diffusivity, K., has the form given by Rossby and
Montgomery1- ':
KJJ = K^0(l + CTR^"1 (65)
where:
= (C-j^ + C2z)w* = the eddy diffusivity of
neutral stratification (ft /day)
50
-------
a = a dimensionless constant (=0.1 for preliminary
study)
W* = T<5 ?TT
-= = B, + B? sin G£r t + ₯), friction
P i L ttb
velocity
The empirical relation of Munk and Anderson is suggested for determining
wind speeds over lakes.
cz N-1 3T
R. = ( STT) a z TT~ ' Richa-rdson Number (68)
ay = A1 + A2 (T 4°) + A3 (T 4)2, Coefficient
of volumetric expansion for water
where:
N = a dimensionless constant (N=2)
T = wind shear stress
o
\.9,A,,B, ,B2,C, ,C2 = constants
(b) Initial condition
T(z,tQ) = TQ (70)
(c) Boundary conditions
(S___ = 0 i
?TT
T,, = TQ + 5TQ sin (2£r t + <}>), the equilibrium
i; e e juj r73")
temperature
where:
T = temperature at water surface
51
-------
t = time
= phase angle
T = average value of equilibrium temperature
over one annual cycle
STe = one half the annual variation
K = the heat transfer coefficient at the lake
surface (BTU/ft3-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 (74)
where:
C. = a nondimensional constant, 0 <(L < 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
-------
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
3 2
resultant eddy diffusivity, IC.Q, is 4.96 x 10 ft /day compared to 7.98
2 2
x 10 ft /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 ft2/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
a.in Equation 651the length scale,'a1,in Equation 62,and the maximum
and minimum 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
-------
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
2 2
used a diffusion coefficient of 7.98 x 10 ft /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 6°C. In Figure 9, for
September 1st, 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,
54
-------
500
en
01
FONTANA RESERVOIR , 1966
DAY: is? (JULY 6)
MEASURED
WITHOUT INFLOW/OUTFLOW
WITH INFLOW/OUTFLOW
COUNTED AS POWER
PLANT COOLING WATER
10 12 14 16 18 20
TEMPERATURE (°C)
Figure 8. Cornell model test run results, July 6, 1966
-------
en
ON
500
FONTANA RESERVOIR, 1966
DAY : 244 ( SEPT. I.)
480
0)
E
460
UJ
_l
UJ
440
420
MEASURED
O O WITHOUT INFLOW/OUTFLOW
WITH INFLOW / OUTFLOW
COUNTED AS POWER
PLANT COOLING WATER
O
8 10 12 14 16 18 20 22 24 26 28 30
TEMPERATURE(°C)
Figure 9. Cornell model test run results, September 1, 1966
-------
en
500
480
0)
E
460
UJ
_l
UJ
440
420
FONTANA RESERVOIR, 1966
DAY: 335 ( DEC. i.)
^ MEASURED
000 WITHOUT INFLOW/OUTFLOW
WITH INFLOW / OUTFLOW
COUNTED AS POWER
PLANT COOLING WATER
0 2 4 6 8 10
12 14 16 18 20
TEMPERATURE (°C)
22 24 26 28 30
Figure 10. Cornell model test run results, December 1, 1966
-------
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 hypolimnion 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 M3DELS
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
-------
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 y , y-^, y and y, are the surface elevations
at the beginning of each day tp t2, t^, t4, etc., as shown in Figure
11. A linear variation of water surface is assumed. The mean eleva-
tion, y, , y^, and y,, should be used at each time step, t, , t-, t, 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:
(75)
with Vs = VSQ + |]:At (76)
where:
V = volume in the element corresponding to the
present surface
V = volume of surface layer at the end of the
current time step
k = time step
p = density of water
q. = inflow rate
^m
a = outflow rate
Q, = heat transport due to diffusion
59
-------
Q, = heat transport due to absorption of radiation
Q = 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 y,, y^, y,.
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:
utBAy + VjjjA = uQBAy (77)
The V! is different from V- 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 J«-l
jm A i = ^ o
These two terms are related to AV as shown in Equation 79:
_ AVs 1
~ A
60
-------
(ft
I
co
TIME.days
Figure 11. Surface layer schematic
61
-------
The heat transport terms die to inflow, outflow and vertical advec-
tion are:
« Vjm>0
AT Vs » Vjm A . T.m_± At + AV£ (Tja_. T.J (80)
+ u- BAyT. At -u B Ay T.- At
i ' i o ' jm
Substituting Equations 77 and 79:
AT ' Vs - Vjm ' A (Tjm-i - V At * ui B^ (Ti 'VAt
(ii) Vjm<0
AT Vs = V.jm - A Tjm At + Ui BAy - T. - At -u^AyT^At (82)
By Equation 77:
AT Vs = uiBAy(Ti -T^) 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 * ' '.
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,
qav, is positive. The quantity (
-------
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
-------
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, 3, 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 $ and n as shown
in Equation 48. / , ,.. 0^&~^
rf0 (1 3) (48)
64
-------
where:
o = total incoming radiation in kilocalories
TI = the average absorption coefficient of the water
(meter ), 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-
i
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
-------
Table 5.
PARAMETERS VARIED IN SENSITIVITY ANALYSIS
Units
Average
Values
Low
Values
High
Values
ON
ON
Vertical Increments, m
Fraction of Radiation Absorbed
in Top Meter of Water
Average Absorption Coefficient
of Water, m~l
Vertical Diffusion Coefficient
0.50
0.75
Molecular
Diffusion
0.20
p.05
30 Times
Molecular
Diffusion
0.40*
1.40
1000
Times
Molecular
Diffusion
*Not the high value
-------
Atmospheric radiation was calculated by the model. If measured solar
radiation is not available, the routine described in TVA Report 14
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 E 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. i
(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
-------
(iv) In addition, the program was modified to call a subroutine TPI£T 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 E = 0.000180 Wp (ec YeJ
in o 9.
and Rohwer's ^ = (0.000 308 + 0.000185W)
where:
(84)
(85)
E_ = mass flux in kg/day-m
p = density of water in kg/m
W = windspeed in m/sec
e = saturation vapor pressure of the air at the
temperature of water surface in mm Hg
e0 = saturation vapor pressure of the air at tempera-
cL
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
-------
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
I
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 6.
Wind Speed
(MPH)
Fontana
Douglas
Cherokee
Norris
South
Holston
Hiwassee
Fort
Loudoxm
No
%
No
%
'No
%
No
%
No
%
No
%
No
%
1
OvO
2.0
18.0
5.9
4.0
1.1
3.0
0.8
0.0
0.0
22.0
7.2
1.0
0.3
0.0
^~0.0
2
2.0
4.0
117.0
38.2
162.0
44.4
49.0
13.4
31.0
8.5
91.0
29.7
20.0
5.5
31.0
8.5
3
4.0
6.0
100.0
32.7
128.0
35.1
112.0
30.7
128.0
35.1
94.0
30.7
128.0
35.1
128.0
35.1
4
6.0
8.0
39.0
12.7
46.0
12.6
103.0
28.2
87.0
23.8
48.0
15.7
91.0
24.9
87.0
23.8
5
8.0
10.0
18 '.0
5.9
20.0
5.5
48.0
13.2
58.0
15.9
33.0
10.8
71.0
19.5
58.0
15.9
6
10.0
12.0
10.0
3.3
3.0
0.8
31.0
8.5
31.0
8.5
12.0
3.9
33.0
9.0
31.0
8.5
7
12.0
14.0
3.0
1.0
2.0
0.5
13.0
3.6
16.0
4.4
2.0
0.7
8.0
2.2
16.0
4.4
8
16.0
16.0
1.0
0.3
0.0
0.0
2.0
0.5
9.0
2.5
4.0
1.3
7.0
1.9
9.0
2.5
9
16.0
18.0
0.0
0.0
0.0
0.0
3.0
0.8
2.0
0.5
0.0
0.0
2.0
0.5
2.0
0.5
10
18.0
20.0
0.0
0.0
0.0
0.0
1.0
0.3
3.0
0.8
0.0
0.0
3.0
0.8
3.0
0.8
11
> nr\ r\
? zu . u
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.3
0.0
0.0
-------
Table 7.
LEGEND FOR FIGURES 12-201
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Layer
Thickness
AY(m)
2.0
1.0
3.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
Fraction of
Solar Radiation
Absorbed at
The Water Surface
3
0.50
0.50
0.50
0.40
0.20
0.50
0.50
0.50
0.50
0.50
Radiation
Absorption
Coefficient
0.75
0.75
0.75
0.75
0.75
0.05
1.40
0.40
0.75
0.75
Diffusion
Coefficient
D(m2/day)
Dm*
Dm
Dm
Dm
Dm
Dm
Dm
Dm
30 Dm
100 Dm
*Dm «= molecular diffusion
-------
543,0 *-
1
!
1
1
1
1
1
1
1
1
1
1
1
1
i
i
i
i
1
1
i
i
i
i
i
i
i
i
i
1
1
1
1
1
t
1
1
1
1
1
1 0
1
01
1
1 1
1
1
1 1
1
1
1
1
0 1 2
21
1 3 *
*l * *
* 516
4 1
5 1
6 1
1
1
1
1
1
1 1
1
1 1
1 1
1 1
1 I
1 1
1 1
1 1
I 1
i 1
161)2
1112
H 1
11 2 3
1 1 3
12 1
7 1 3
> 131 4
1 1
13 1
1 1 4
1 1
1 4
\ 1 1
16 4 1
4 4 15 5 55
5 15 1
1
1
1
1
1
1
1
1
Z
3
3
4
4
5
0 BA>
OA1
OA>
OA1
oyf
< oyl
> ?
3
4
4
l\ T3
1 I32
1 1'7
'1 215
J ?44
'1 285
'1 3*9
LET
2 2
4
4
2
1
"
.
1
1
1
J
1
\
~,
1
1
2,9 4,5 7.0 9,5 12.0 14.5 17.0 19.5 22.0 24.5 27.0 29,3
Figure 12 HJT HQnf-L *
TFMPE°ATJPE 1-1 DEGREES c
F1.IA KFiFRVUM 19661F4SUKEO TE"PEKATUKE
g
I
f
y
A
T
0
N
N
H
g
I
E
*
1
1
X
I.
1
1
1
1
1
1
49.1 »a +BT r,..-j
1
1
1
1
I
1
1
1
4$S,« **,.,-.-
1
J
1
1
*sne f,
1
1
1<
1
439,8 *c........
1
1
1
1
1
1
1
1
1
1
1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 ! 1
1 1 1
« +»HW^««nwV<»*^0«VI«+0*B»«» I
101 1*1
00 | 11*1'
01 I 116
00 1 11 1 1 *
U 1 11 1 1 22
0111 1 «
0 111 | 2 2 *
0 11 1 216
__,0 -11 .*__._2-2--* - -J
Oil 12 1*3
0 11 1 |22 1 * 33
Oil 21 1 63
0* 1 22 | 1 S »
p-»0 l-»*-+«-2H»«+«»»« -*3. «»-6
1 1 22 I 3 J 66
* 12 I 3 16
22 1 » 06
* 21 1 » * 1
...*-22p-*~ 3-3*- 66-* 4 -- 4-
*2 13 1 641
23 31 4 46* 15
*3 4 1 51 * 1
»* 51 66 1
61 6 1 I
61 | 1
61 | 1
1 1 1
! 1 1
1 1 1
1 1 1
1 1 1
1
|
i '
1
j
|
1
J.
|
1 j
1 1 1
1 1
1 2
2 21 *
2 I A j
2 1 3*3
33 *
1 1 5
33 1 *
3 1 i
3........* .-_+<)_4
1 *
1 64 f
1 4 5
44 *
-..-..-*.+ J.._
44 1 13
4 1 5
4 4 US
!
-M~-. -*-..«.»_
5 1
» 1
1
1
2
2
..- -_3
1 4
1 1*
4
44
44
4
*
.........
._.._-««
1 0 f>AY| 75
1 1 OAYI 132
1 2 9«Y| 187
1 3 "AVI 213
1 ' HAYI 285
1 » DAYI 335
i OVERLAP
1 < OUTLET
2
2
4 3
.*«......
-_«-.__
.........
1 1
1 I
i 4.
1 1
i 1
1 1
1 1
-1 J.
1 1
L
1 1
1 1
1 1
i 1
J. 4.
1 1
i 1
1 1
1 1
1 1
1 1
I 1
1 1
1 |
. _| ,*
1 1
1 1
1 I
1 1
1 1
1 1
1 1
1 |
1 1
1 1
1 1
1 1
1 1
1 1
ZtO 4,3 7.0 ».» 12.0 14.5 17.0 19,5 22.0 ?4.5 27.0 29,9
Figure 13 M|T Hur>El * FONTANA KFJFK'AJIR 1966CnHPOTfcn TEMPERATURE PR1F1LE--
72
-------
Table 8.
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: Fontana/1966
Time Period Covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate ( C)
1.19
1.25
1.16
1.19
1.22
2.88
1.21
1.27
1.15
1.25
\
Correlation
Coefficient
0.96
0.96
0.97
0.96
0.96
0.77
0.96
0.96
0.97
0.96
73
-------
Table 9.
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reservoir/Year: Fontana/1966
Time Period Covered: 60th - 360th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate ( G)
1'.74
1.75
1.70
1.74
1.70
2.20
1.74
1.75
1.76
1.81
Correlation
Coefficient
0.95
0.95
0.96
0.95
0.96
0.92
0.95
0.95
0.95
0.95
74
-------
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
-------
iT,9
J5.0
ZZ,5
Z0,0
IT, 3
15.0
It. 9
10,0
T,S
5,0
t.3
..r»...'
0 0
iniiiiii
0
0
1U1111U
0
0
1
1 IIUU
u
0
11111
1111
0
r>
U
U
111
1
1
1
1
01
111
1
10
1
1
0 . HE
1 - ftO
- Oy
0
0
11 11
10
0
SUREO
HER
HUP
0
1
1*1* 0
1
0
1
1*
UO
110
Figure 14 HjT
"»V4
INA KFSFK"dlK t<>66--C
.0 4
43.0
90.0
2
<
.0 4
l
...
,i
0
*
J
.....
7
2<
*0
.........
.0 9
.3 U
....
.0 U
3 IT
o ME
1 ST
2 Of
3 »t
< ou
.0 19
»...*....*
ISURID .
IND
S -
TL6T
.
.9 Z2
....
. -.1
.
.0 U
....
i IT
1
"
1
.
1
0 J9.J
Figure 15 H,T MQDEL ,
TFMPERATUKE IN DECREES C
FONTANA RESERVOIR 1966OAVI T3 .-SURFACE ELEVl 300,1 N
76
-------
19.0
»0.0
49.0
60.0
73.0
90.0
109.0
120.0
119.0
190.0 .
2.
*-
<
_
--?-»*
i
i
i
i
i
i *
1 0
1 «3
21
21 3
1
21
213
01
1
i
i
i
... i
i
i
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21
0
*9
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01 0
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> OE
.;.« ov
< ou
sumo
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n
12
RLA» «
LIT
- ..
.
^ » <
0 4.3 7.0 9*.5 12.0 14.3 17.0 19.3 12.0 24.3 17
**«£*
0 2t.
Figure 16 HIT MOOEL *
TIM»E«ATU«E IN DEGREES C
FONTANA RESERVOIR 1966OAyll32 SUHF1CE ELEVI 901.
.0
19.0
10.0
49.0
D
I 60.0
P
T
H
I 73.0
N
H
E
T 90.0
E
R
109.0
120.0
139.0
190.0
2
.-._»-.-
<
21
Zl 1
1
*
*
» 0
1
9
0
0
21
-*l-»---
3
. ° 12
0 *
1
1
0 1
*
3
0
.0* 1
_
_
2
010
1
_
o - MEASURED
1 - STAND
2 - OELll
3 - OEIZ2
< - OUTLET
1
1
0 0
"
)^.0^
)
.
-
-^
"""" "\
...;.,..
---i-i-i.
>. -». -*^ ^ i2.^ ^ I7i() WiJ J2_0 M;5 Ht0 Mj,
TEHpE>ATUHE IN OESREES C
Figure 17 M,T HOOEl; * FONTAXA RESSRVOIR 1956.-DAYJH7 SURFACE ELEVI 309.9 N
77
-------
.0
15.0
30.0
45.0
40.0
75.0
90.0
105.0
120.0
135.0
150.0
2
<
0 4.
.
....
5 7
0
* 3
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.
0
213
1 3
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'
0 *
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3
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0
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< ou
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.
.
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22
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tl 1
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0
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0
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.
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1
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T
I
1
1
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. . . 1
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1
0 9'.5 12.0 14'.5 17.0 19.5 22.0 24.5 S7.0 2*. 3
Figure 18 H,T HOOElI
TEH'ERATUIE IN DEGKEES C
FDNTAMt RESERVOIR 19»6--0tf1215 ^-SURFACE ELIVl 505.9 H
15.0
30.0
45.0
D
E 60.0
t
T
H
I 75.0
N
M
E
T 90.0
E
R
105.0
120.0
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14A-A j
... -_!---
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21
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2*
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.-.-.1-1-
0
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213
31 1.1.
i .1.1.4
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0
12
12
*
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o > MEASURED
1 - STAND
2 - OELZl
. 3 . OELZ2
* - OVERLAP --!-<
< - OUTLET
1
1
!
0
0
..
"
"
....»..,
.........
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1
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.
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2.0 4.5 7.0
Fljur. 19
H,T
9.5 12.0 1».5 " 17.0 19^5 22^0 24^5
TEMPERATURE IN DEGREES C
FONTANA RESERVOIR 1966--DAYI244 --SURFACE ELEVI 501.9 N
17.0 29,3
78
-------
13.0
45(o
90.0
tzo.o
_
0
*J
*
... .....
.........
2 01
0 2 1
* ...
0
3
0 **
0
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0 Mi
1 JT
i oe
. 3 w
< ou
SURiD
NO
Zl
12
LIT
.........<
1 . .-
m^~+mmmm
....^..*.
14.5 1T.O 19'. 3 21.0 2«'.5
TEMPE«»TURJ IN DECREES C
Figure 20 HIT HOieC * FONTAXA RfSERVDIR If »6.'OAylZ65 .'SURFACE eiEVI *»*.3 N
1T.O 2».»
.0
15.0
JO.O
45.0
D
E 60.0
P
T
H
I 75.0
N
H
E
T »0.0
e
R
105.0
120.0
135.0
ISO. A 4
»«....«...
<
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t»-...^.-.
.
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.........
.
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.
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1 - ST
2 - OE
3 - OE
< - DU
.........
...-.-«.
SUREO -
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S .
LET
.....IM*
.
i,:ii-
,-,-...
.
-....-«
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J
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TEMl>E
-------
75.0
to.o
105.0
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IM.H.J
<
^^.
fe«r--s-
*3
ft*
*
.
___.-
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Ftjure 22 M,T mnfL « FC1NTAHA RESERVOIR 1966--OAYI 75 i-SURFACE ELEVl 500.3 M
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Figure 23 H,T HOBEI;
TEMpEI>ATURE IX DEGREES C
FnNTANA RFSERVOIR 1966 DAyl 132 SURFACE ELEVI 50«,4 M
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Figure 26 H,T HQnEL , FnNTAIA RESERVOIR l«66DAyl244 "SURFACE ELEVI 901.9 N
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Figure 27 M,T HDOE,_ » FPNTANA RESERVOIR 1966DAyl2S5 SURFACE ELEVI 494.3 M
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Figure 28 MT MODEL * FONTANA RESERVOIR 1966OAyUJJ SURFACE ELEVI 4«».« H
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Figure 30
TfHPEHATUHE IN DECRIES C
HIT MODEL * FONTANA RESERVOIR 19&6-.D4Y1132 -.SURFACE ELEVl 501.4 M
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Ptgur. 31 H,T HonEL
TeHl>E«ATU«E IN DEGREES C
FONTANA RESERVOIR 19&6--D4yl197 ..SURFACE ELCV 509.9 M
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32 H,T HonEL . FHNT4N4 MSIRvOIR H66OAVI213 i-SUXFACC CLIVI 303.9 H
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Figure 33 HJT MDOE|; » poNTANA RESERVOIR 1966OAVI244 i-SURFACE ELIVI 501.9
85
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Figure 34 H,T HOO|I;
T(H»E»*TUH6 IN DECRIES C
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Fl,ur« 35
TtH'EAATURE IN DECREES C
FONTANA RESERVOIR 1966«>OAyll35 (-.SURFACE Iklyi 49*.S M
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Figure 36 M,T
TEMPERATURE IN OECME5 C
FONT4MA MSERVOIR 1*»«~.04YT TS i^SURFACI EllVl JOOtJ
.a <
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Figure 37 M,T HDnE|; t
TEHFERATURE IN gEGREES C
FONTANA RESERVOIR 1966.-04YIU2 '^.SURFACE EllVl 90*.4 M
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TBH'E«ATg«E IN oESREES C
MIT MODEL * FONTANA RESEKVOIR l*66DAVi215 ^'SURFACE ELEVI 505.9 M
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Figure 40 HIT MODEL' * FONTANA RESERVOIR 19t6..0Ayi244 V.SURFACE EllVl" 301.9 M
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««ur« 41 H]T
TEMPERATURE IN OeORKS C
FONTANA RtSERVOIR 1966--OAYI2U i-SUKFACE tLEVI 494.3 N
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Figure 42 H|T MaoE|; »
TIMPEIATURE IN OECKIES C
FnNTAMA RtSIRvalR l«66"0tnl>! ^SURFACE ELIVI »»».« H
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Douglas Reservoir
Figures 43 and 44 show the computed temperature data for Douglas
Reservoir for 1969 at various depths and the confuted 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 2.1° 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 3, 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
-------
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
-------
' .
103,3
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Figure 43
M,T
TfMPE«»TuBE ]N DECREES C
t«6»Cdl'MTtD T6«PE*»TURe KlOFUE
93
-------
TABLE 10
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: Douglas/1969
Time period covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°c)
2.11
2.23
2.07
2.11
2.15
5.73
2.28
2.12
2.09
4.97
Correlation
coefficient
0.95
0.94
0.95
0.94
0.95
0.28
0.94
0.94
0.94
0.63
94
-------
TABLE 11
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reservoir/Year: Douglas/1969
Time Period Covered: 60th 360 Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate ( C)
2.03
2.05
2.07
1.97
1.88
2.26
2.22
2.13
1.98
blows up
Correlation
Coefficient
0.98
0.98
0.98
0.98
0.98
0.97
0.97
0.98
0.98
95
-------
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Figure 44
M,T
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Figure 45
HJT
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106904 Y I <« SU»fACE ELEVI
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-------
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T!HPE'ATUȣ |N DEGREES C
Flgure-46 H,T MUPEL UOU6LA5 «SER"l)IR 1969--1)A»I121 Su«f*CB HEVI J9y.« M
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Plgure 47
ttnu'!t«»
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97
-------
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Figure 48 H)T MUOEl »
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TEM'E>ATutE (N DECREES c
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Figure 45 HjT HOPEL
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TEHPE°ATURE IN DEGREES C
ODUPHS RESERVUIR I960 OAYI250 SURfACE ELEVI 2»9.« H
22.0 24.5 27.0 Z'.J
98
-------
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PlgUM 51 H,T BQOEL
7EnPt«ATu»E.JN DEGREES c
ESERVQIH 1969--UATI 6« '-SURFACE ELEVI
-------
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figure 32 H[T
r.i 12.0 14.J 17.0 If.9 12.0
TEn»e«ATU«E IN DEfiREEJ C
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Figure 54
M)T
12,0 14,9 17.B 19.5 22.Q
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Figure 55
I'l BtRMES C
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Figure 56
HJT
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Figure 58 NJT HuneL ,
TEMPERATURE 1" DEGREES C
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Figure 39 HJT HU«H. -
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108
-------
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 3, 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
-------
Table 12
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: Cherokee/1967
Time Period Covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
2.70
2.99
2.22
2.62
2.54
5.52
2.97
2.29
1.80
2.08
Correlation
Coefficient
0.83
0.80
0.88
0.85
0.86
0.00
0.78
0.89 .
0.92
0.88
110
-------
Table i-3
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reservoir/Year: Cherokee/1967
Time Period Covered: 60th 300th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
2.07
2.09
2.11
2.06
2.01
1.49
2.25
1.88
1.90
1.72
Correlation
Coefficient
0.95
0.92
0.95
0.95
0.96
0.98
0.95
0.96
0.96
0.97
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125
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NORRIS RESERVOIR
Figures 95 and 96 show the computed reservoir temperatures and the
computed outflow temperatures respectively, at Morris 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 g, 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 3s 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
1°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
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127
-------
Table 14
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reserveir/Year: Norris/1972
Time Period Covered: 60th - 360th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
2.31
2.29
3.16
2.32
2.28
3.21
2.36
2.39
2.46
2.64
Correlation
Coefficient
0.94
0.94
0.92
0.94
0.94
0.89
0.93
0.94
0.93
0.92
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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 B 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
-------
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
-------
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142
-------
Table 15
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: South Holston/1953
Time Period Covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
2.88
2.72
3.02
2.95
3.08
7.07
2.49
3.21
4.19
Correlation
Coefficient
0.87
0.89
0.86
0.87
0.86
0.00
0.90
0.83
0.66
143
-------
Table 16
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reservoir/Year: South Holston/1953
Time Period Covered: 60th - 360th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
2.44
2.53
3.09
2.32
2.29
1.05
2.82
2.18
1.75
Correlation
Coefficient
0.96
0.96
0.94
0.96
0.96
0.99
0.95
0.97
0.98
144
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Figure 139 MJT
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Figure 141 HJT MongL > SOUTH HBLSTDrf 1953«BA»I2*» SURFACE ELEVI 511,6 M
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Figure 143 MJT
TEH'EtATURE (H OE8RKS C
i953--o*ri3»z SURFACE ELEVI $02.2 H
157
-------
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, 1.0° 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 3, 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 3 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
-------
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
-------
Table 17
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: Hiwassee/1947
Time Period Covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
1.01
0.96
1.03
1.03
1.06
3.95
1.08
1.06
1.44
Correlation
Coefficient
0.98
0.98
0.98
0.98
0.98
0.79
0.98
0.98
0.97
160
-------
Table 18
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reservoir/Year: Hiwassee/1947
Time Period Covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
1.38
1.53
1.34
1.38
1.37
1.44
1.40
1.30
' 1.26
Correlation
Coefficient
0.99
0.98
0.99
0.99
0.99
0.98
0.99
0.99
0.99
161
-------
490,6
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1 DAVI 148
DAVI 209
OAVl 267
OAVI 364
OVERLAP
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2.0 4,9 7.0 9,9 12.0 U,9 17.0 19.3 22.6 24,9 17.0 29,9
TEMPERATURE IN DEGREES t,
Figure 144
«|T HOOEL « HIWASSES RESERVOIR i947--HEASu*Ei> TEMPERATURE PROFILE
490,6 4
410.6
470,4
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Figure 145
N DECREES C
H(T HQI>EL HIUASSEE KFSERVUIR 1947COMPUTED TEMPERATURE PROFILER-
162
-------
J'.o
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OAVS
Figure 146
H(T HQQEL 1 HIHASSEE RESERVOIR 19*7 'COMPUTED OUTFLOW TEMPERATURE
1 01 2 i\
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TFM'E«ATU»E »N PEP-KEtS C
Figure 147^ ^^ ^ H,WASSEfc RESEK-'ulK 19*7--OAYI 80 -SURFACE ELEVI 4*6,0 H
163
-------
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figure 148
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42
ERtAP ---
JL.T
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figure 161
N DECREES C
HJHASSBE RESERVUIR 19«T.-DAYI209 -nSURFACE ELEVI 463,4 M
170
-------
1
!
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1
1
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Figure 162
TlH"E«ATU«E I* DEBRCH 6
MIT MODEL HJWAISBE RESERVOIR i9»T«-oAYt2»T ,>
nevi *s*,i
,
1
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.««!.» *?"?*;- ':J:9"li-=~* *-' Jj.o n.s 17.0 19,5 22.0 **.S tT.U M.S
Figure 163
HIT MODEL
JN 06«R*E»
MStRVQIR 19»T«-OAYI102
fLtVI *46.» H
171
-------
1 I 'I 1 0
1 I 1 1 0
1 -[ | * ! 0
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l<
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Figure 164
HIT MODEL
TEMPERATURE ]N DEOREES C
H|WASm RBSERVOIR 1**T..OAVI3A* *«IU*FACE BLEYI 414.1 n
1
1
1
1
l<
1
1
1
1
01
0!
01
I
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- p»l
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l/t
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100,0^--,.---.- =-- =»- --*- - l?-0 ,, MiQ -., 27l0 29fS
Figure 165
H|T MODEL »
TBhPERATURE JN DECREES C
HjHASSEt RESfRVOIR 194T*-OAYI "0 SURFACE ELEVI 446,0 M
172
-------
lo,e «.;-..*..-
20,0 *»rri
_ _ !<
j.:-_:__..
1
L _.
70,0 *.?.-;--.
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1
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o - ^«TJ
Figure 166
HJT HOCEL
TEMPERATURE |N DECREES C
HIWASSEE RESERVOIR 19*7--OAVII*B ..-SURFACE euvi
1
1 1
1
j
I 1
1 . 1 . -
l« 1
1
! i
i i
i i
1 1 o
I i 02
_ j. j
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1 I
1 i
1 1
I 1
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1 2
3
0
3
a
01 2
3
0
0 I
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3
0
0 1 2
0 1 2
I
2 S
3
o HE/
1 STJ
z DIP
3 DIP
< DUI
1 02
1 0 2
0
3
3
i
SURED
NO
Fl
FS
RIAP -"-
l«T
u
12 J.
3
l
>..... ..].!*........*
01 1
13 01 1
0 1 1
1 1
1 1
1 1
1
1 1
1
1 . 1
1 I
1 1
.'--..'
1 .
T
i
T~
1
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J ... i
i
i i
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i i
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i i
Flgur.
TEMPERATURE IN OERREES C
19VT*«OAYI209 SURFACE ELEVI »»3.* H
173
-------
1 1 1 1
1 1 1 1
1 1
1 |
* ' 1 u 1 1
1 1 1
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, .-..-i-U-,-!---;.--,,
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~~~~ ~~ 1 - - - . [ - - 1 ....
v
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3
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it I) I 1
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I z OIFFI i "I I
i 3 BIFFS i i i
i < OUTUT i i i
i ill
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i . i " . . . I
k
k
IAO_ . *i» 7.0 9-.S 12.0 !*,» 17,0 19. S 12.0 14,5 ZT.O Z»,5
Figure 168
M(T HQOiL
TEM'ERATURE |N DECRIES
HINASSEE RESERVOIR l»*T--OAY 1267
UiVl «9«,t M
* ~K"Z - STI * * ----- -..- ---, - . . f
i i
i , .... i
i
i
i
i r i
i i
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I 1 0
i i 20"
1 12 0
1 I 2
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1
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1 1 1
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3
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0
0
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3 BIFFS
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0
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ft
t f-
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- f
*«» «i» T,0 9,9 12.0 1*,S 17,0 19,5 22.0 H.t 27,0 j»-,!T
,. ... TEMPERATURE »N DEBREES C
HIT MODEL HIHASSEE RESERVOIR
o«rii°2 ^-SURFACE ELEVI
174
-------
J
J
1
1
1
1
1
_ 1
1
1
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,
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Figure 170
MIT
TenM«*TU»E |N DECRIES C
HJHASSEE K§«*VOU i»«T--o»ri»M *
ELEVI «»,i H
175
-------
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 3, 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 $ of 0.5.
In Figures 188 to 194 is shown the effect of variation in ri, 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
i
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
-------
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
-------
T4.O t
.
.
-» «0
IJ'lO
2
* * *
^
- . -
' -«
.
0 4
.
1
.
.
S 7
0
e
0
0 9
....
.
6
6
6
6
6
» 12
.
_
1
0 14
1
1
1
> 17
_
. .
-
I 1
1
0 DA
1 DA1
t DA
3 t»A
s BA
» DA
0V
< nu
0 19
L ... _
_
.. . ..,..
i
l\ T6
n 112
n 2
-------
Table 19
STATISTICAL ANALYSIS FOR THE PREDICTED
WATER TEMPERATURE AT OUTLET LEVEL
Reservoir/Year: Fort Loudon/1971
Time Period Covered: 120th - 330th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std. error
of estimate (°C)
1.53
1.56
1.37
1.54
1.58
2.05
1.43
1.59
1.69
Correlation
Coefficient
0.92
0.92
0.94
0.92
0.92
0.86
0.93
0.92
0.90
179
-------
Table 20
STATISTICAL ANALYSIS FOR THE PREDICTED
SURFACE WATER TEMPERATURE
Reservoir/Year: Fort Loudon/1971
Time Period Covered: 60th - 360th Julian Day
File Name
STAND
DELZ 1
DELZ 2
BETA 1
BETA 2
ETA 1
ETA 2
ETA 3
DIFF 1
DIFF 3
Std . error
of estimate (°C)
2.98
2.78
3.09
2.86
2.88
2.81
3.38
2.95
2.92
Correlation
Coefficient
0.86
0.89
0.85
0.88
0.88
0.89
0.81
0.86
0.87
180
-------
'"ft ^^^-^^
1 1
1
l_ _-
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20 ^9 TiO 0* ? 12iO 1^*9 17*0 l?i" ?2tO ?4i? 17 §0 J*i
Figure 172
M(JnEL .
FDRT
TEMl>E«ATu>E I" PEOKCES C
1971 r-cn«(PUT60 TEHRtRATURE
II11
1 | 1 I
1 ! ' 1
-I | 1 1
1 1 i >
(III
j-jj.-- 1 ' I _ 1 1 I
1 i i i 11
,.| | .. 1 1 11
*0'BT" " * i n
. + ||0
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1 I 10 0 ._!_..
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S 2*0.o" 270.0 . 300.0 ' 3JO,
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Figure 173 MJT
FORT
PAYS
LU"ur>U" 1«71 .-COMPUTED OUTFLUW TiHpfRATUHE.
181
-------
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Figure 198 HJT p.unEl . pnRT LUl'ii' « )»71
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Figure 199 MJT
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194
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-------
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
-------
Table 21
LEAST SQUARES CURVE* FIT FOR MEASURED SURFACE WATER
TEMPERATURE
ID
Reservoir/Year
Fontana/1966
Douglas/1969
Cherokee/1967
Norris/1971
South Holston/1953
Hiwassee/1947
Fort Loudoun/1971
No. of Data
Points
27
31
37
48
12
12
12
Std. Error
of estimate( C)
1.
2.
1.
1.
1.
1.
1.
22
23
53
30
79
70
50
Correlation A
Coefficient (°C)
0.
0.
0.
0.
0.
0.
0.
98
97
97
97
98
98
99
16.
17.
16.
17.
17.
19.
17.
96
60
70
46
01
07
88
B
- 9.
-12.
-10.
-11.
-11.
-10.
-12.
:)
72
82
08
54
63
43
32
»
0.
1.
1.
1.
0.
0.
0.
927
180
006
065
970
850
992
*Curve Type: A + B sin (J^, t + T)
-------
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
Douglas/1969
Cherokee/1967
22
27
32
0.96
1.10
1.06
0.97
0.98
0.96
103
69
117
341
352
332
Morris/
South Holston/1953
Hiwassee/1947
Fort Loudoun/1971
9
10
10
1.09
1.50
1.56
0.96 113 362
0.96 80 364
0.92 76 343
* Third Degree Polynomial
198
-------
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, 3
The value of the surface absorption ratio, 3, 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 3, 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,!!
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
-------
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 Or lob * ' who introduced a densimetric
Froude number in the form
Uv. = =£ I (86)
where:
L = length of reservoir in meters
200
-------
Q - volumetric discharge through the reservoir in. ra3/sec
D = mean reservoir depth in meters
V = reservoir volume in cubic meters
e = average normalized density gradient in reservoir
(1(T6 m'1)
g = gravitational constant (=9.8 m/sec2)
substituting the average values, we have
IFD ' 320I5 $ (87)
For IFD < ^ , 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/n. 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
-------
TABLE 23 DENSIMETRIC FROUDE NUMBERS FOR SOME TVA RESERVOIRS
Mean Annual Normal Maximum Reservoir Densimetric
Fontana
Douglas
Cherokee
Norris
South Holston
Hiwassee
Fort Loudoun
Length of
Lake Miles
29.0
43.1
59.0
72.0**
24.3
22.0
55.0
Runoff at Dam,
10^ acft/year
2,667.7
4,830.7
3,316.6
2,956.1
735.4
1,369.9
9,949.8
Depth,
Feet
432
129
150
196
240
252
74
Storage* ,
103 acft
1444.3
1514.1
1565.4
2567.0
744.0
438.0
386.5
Surface
Area, Acres
10,670
30,600
30,200
34,200
7,580
6,080
14,600
Froude
Number
.006
.06
.05
.02
.005
.01
1.02
* At maximum controlled elevation
** Clinch River Arm Only
-------
REFERENCES
1, 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.
2. 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.
3. 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.
4. 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.
5, 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 VT-2790-A-I.Cornell Aeronautical Laboratory, Buffalo, New
York.June, 1970.
203
-------
6. Hanford Engineering Development Laboratory The Colheat River
Simulation Model. HEDL-TME 72-103*August, 1972.
7. Beard, Leo R. and Willey, R,G, An Approach to Reservoir Temperature
Analysis. Water Resources Research 6(5): October, 1970.
8. World Meteorological Organization, Measurement and Estimation of
Evaporation and Evapotranspiration. Geneva, Switzerland Techni-
cal Note 83. 1966,
9. 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.
10, Debler, W.R. Stratified Flow into a Line Sink* Journal of
Engineering Mechanics (ASCE)85(EM3): July, 1959.
11. Craya, A. Theoretical Research on the Flow of Non-Homogeneous
Fluids, LaHouille Blanche 4(1):56-64 January-February, 1949.
12. 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.
13. Kao, T.W. The Phenomenon of Block in Stratified Flow. Journal .of
Geophysical Research 70(4); February, 1965.
204
-------
14. 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 VT-2616-0-2. November, 1969.
15. Rossey, C.C. and Montgomery, B.R. The Layer of Frictional
Influence in Wind and Ocean Currents» Physical Oceanography
3C3):101. 1935.
16. Munk, W.H. and Anderson, E.R. Notes on the Theory of the
Thermocline. Journal of Marine Research 1:276. 1948.
17. Tennessee Valley Authority Water Temperature Prediction Model for
Deep Reservoirs. Water Resource Management Methods Staff,
Technical Report No. A-2, October, 1973.
18. Tennessee Valley Authority. Temperature Predictions for TVA Reser-
voirs Graphical Presentations. Water Resources Management Methods
Staff, Report A-6 December, 1973.
19. Tennessee Valley Authority, Heat and Mass Transfer Between a Water
Surface and the Atmosphere. Water Resources Research Laboratory
Report No. 14. April, 1972.
205
-------
TECHNICAL REPORT DATA
II lease read Instructions on the reverse before completing)
1. REPORT NO.
EPA-660/3-75-038
2.
3. RECIPIENT'S ACCESSIONING.
4. TITLE AND SUBTITLE
Evaluation of Mathematical Models for Temperature
Prediction in Deep Reservoirs
5. REPORT DATE
6. PERFORMING ORGANIZATION CODE
1Q7CJ
7.'AUTHOR(S)
Frank L. Parker, Barry A. Benedict, Chii-ell Tsai
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORG \NIZATION 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
13. TYPE OF REPORT AND PERIOD COVERED
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330
14. SPONSORING AGENCY CODE
16. SUPPLEMENTARY NOTES
16. 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 MET 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 an absorption coefficient of about 0.75m~l and a diffusion
coefficient 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. However, the
representativeness of the result is not solely dependent 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 predicte
outlet level temperature, 1.6°C. Temperatures on individual days may exceed thes
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.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
Thermal pollution, reservoirs, mathema-
tical models, sensitivity analysis,
Tennessee Valley Authority Projects
b.lDENTIFIERS/OPEN ENDED TERMS
Deep reservoir models,
Massachusetts Insittute
of Technology, Water
Resources Engineers,
Cornell Aeronautical
Laboratory
:. COSATI Field/Group
18. DISTRIBUTION STATEMENT
19. SECURITY CLASS (This Report)
20. SECURITY CLASS (Thispage)
22. PRICE
EPA Form 2220-1 (9-73)
U.S. GOVERNMENT PRINTING OFFICE: 1975-699-073 I\S REGION 10
------- |