Thermal Modeling of Flow in the 

San Diego Aqueduct, California, 

and Its Relation to Evaporation
Thermal Modeling of Flow in the 

San Diego Aqueduct, California, 

and its Relation to Evaporation

By HARVEY E. JOBSON

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1122

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1980
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Jobson, Harvey E.

Thermal modeling of flow in the San Diego Aqueduct, California,

and its relation to evaporation.

(U.S. Geological Survey professional paper ; 1122)

Bibliography: p. 23-24.

Supt. of Docs, no.: I 19.16:1122

1. San Diego Aqueduct, Calif.-Temperature-Mathematical models. 

2. San Diego Aqueduct, Calif.-Evaporation-Mathematical 

models. I. Title. II. Series: United States. Geological Survey. 

Professional paper ; 1122.

TC764.J62 628.1'5 79-20943

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402

Stock Number 024-001-03266-0
CONTENTS

Page

Abstract _______________________________ l

Introduction ____________________________ 1

Model development ______________________ 2

The flow model ____________________ 2

The Eulerian temperature model ______.__ 3

The Lagrangian model __________________ 7

Page

Data and model calibration ________________ 8

Data available ____________________________ 8

Model calibration ___________________ 9

Model verification and discussion ___________ 16

Summary and conclusions _________________ 22

References _____________________________ 23

ILLUSTRATIONS

Page

FIGURE l.Map showing San Diego Aqueduct showing data-collection points ________________—-———————————— 2 

2-21. Graphs showing:

2. The effect of the dispersion coefficient on the computed temperatures in the San Diego Aqueduct on August 18,

1973, in comparison with measured temperatures _____________________———-- ——————— ——— 7

3. Variation of stage and discharge for the flow transients on December 10 and 11, 1973 ——— —————— ——— — - 10

4. Comparison of wind function values for July 30, 1973, as computed with and without heat transfer between

the water and the bed _________________________________________--_———_____ 11

5. Variation of the wind function, with windspeed for the San Diego Aqueduct during July 25 through 28, 1973 __ 12

6. Variation of the wind function, with windspeed for the San Diego Aqueduct during July 29 through

August 1, 1973 _____________________________________________-_________ 12

7. Variation of the wind function, with windspeed for the San Diego Aqueduct during August 2 through 5, 1973 __ 13

8. Variation of the wind function, with windspeed for the San Diego Aqueduct during August 6 through

August 9, 1973 -________________________________________________-____——————————- — 13

9. Variation of the wind function, with windspeed for the San Diego Aqueduct during August 10 through

August 13, 1973 ______________________________________________________—— 14

10. Variation of the wind function, with windspeed for the San Diego Aqueduct during August 14 through

August 17, 1973 _____________________________________________________——— 14

11. Variation of the wind function, with windspeed for the San Diego Aqueduct during August 18 through

August 21, 1973 ___________________________________________________-_____—--- 15

12. Computed wind function values for the San Diego Aqueduct. Every other point plotted for the 28-day period

starting July 25, 1973 ____________________________________________________—— 15

13. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the calibration period ______________________--_——— — -- — —— --- 16

14. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the first verification ________________________———————————————— 17

15. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the second verification __________________ ———— -- ———— ---- —— - 19

16. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the third verification __________________________________________ ————— _—— 20

17. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the fourth verification _______________________________ ————— __ —— — 21

18. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the fifth verification ______________________—_————————————— 22

19. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the sixth verification _____________________________ ——— ___ —— _____ 23

20. Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct

during the first 4 days of the seventh verification ______________________ ——— -- — — —— - 24

21. Computed monthly average daily evaporation from the San Diego Aqueduct during the study period, July 25,

1973, to July 24, 1974 __________________________________________ — — ____—— 24

III
IV CONTENTS

TABLES

————————— Page

TABLE 1. Coefficients for use in equation 3 to determine the centerline depth in the San Diego Aqueduct — ——— _ ——— ——— __— 3

2. Summary of modeled data for the San Diego Aqueduct ____________________________————————————— 9

3. Daily values of the wind function coefficients for the San Diego Aqueduct ____—_—__ ——————— _- —————— --- 18

4. Summary of energy budget terms for individual water parcels passing through the San Diego Aqueduct _____ —————— _ 19

5. Evaporation rates for the San Diego Aqueduct during the study period, July 25, 1973, to July 24, 1974 _______ — —— __ 22

SYMBOLS

A = Cross-sectional area of the channel; Rh

Ar = Surface area of a reservoir; s

c = Specific heat of water at constant pressure;

Cj = Coefficient in equation 3; Sa

Cv = Heat storage capacity; T

dj = Coefficient in equation 3; t

DT = Time step; Ta

Dj. = Longitudinal dispersion coefficient; Tb

DX = Length, in meters of the subreach (1,299 m); T{

E — Rate of evaporation;

ea = Vapor pressure of air; T0

eik = Coefficient in equation 3;

e0 = Saturation vapor pressure of air evaluated at a tern- Tu,

perature equal to that of the water surface; U 

H(t) = Increase in heat content of the slab between time zero (7*

and t resulting from a unit increase in surface tern- V

perature at time zero; Vj 

Hh = Heat flux leaving the water caused by longwave radia- W

tion being emitted by the water; x 

He — Heat flux leaving the water as a result of the latent y

heat of vaporization; 

Hh = Heat flux conducted from the water to the air as sensi- Y,

ble heat; YM(k) 

HN = Net flux to the water caused by incoming radiation Z

from the sun and the sky; a 

Hp • = Flux of thermal energy from the bed to water; y 

HR = Heat flux added to the water by rain falling directly on A//( i)

its surface;

HT = Rate of absorption of thermal energy at the water surface per unit area; ATB 

i = Rainfall rate; ^T(jDT) 

k = Thermal diffusivity; A0 

L = Latent heat of vaporization of water; 

N = Empirical mass-transfer coefficient in the wind func- €

tion; p 

Pa = Atmospheric pressure; a 

Q(j) = Discharge at cross-section j; T 

QD = Diversion discharge;

r = Coefficient of correlation; t// 

R = Reflectivity;

Hydraulic radius;

Time-step number, before which the water temperature is assumed to have been constant; 

Sun altitude in degrees; 

Cross-sectional average water temperature; 

Time;

Air temperature; 

Arbitrary reference temperature; 

Initial temperature of a water parcel as it enters the

system; 

Final temperature of a water parcel as it passes the

downstream end of the channel; 

Wet-bulb air temperature; 

Cross-sectional average velocity; 

Shear velocity; 

Wind speed;

Volume of water in subreach j; 

Top width of the channel; 

Longitudinal coordinate; 

Distance above the insulated bottom of the slab at

which temperature ATB is computed; 

Centerline water depth in subreach j; 

Recorded centerline water depth at recorder k; 

Thickness of the bed slab; 

Constant in wind function; 

Psychrometric constant; 

Heat flux to the water from the bed during any time

step iDT to (i +DDT to result from a unit increase in

water temperature at time zero; 

Temperature rise within the slab; 

Change in water temperature to occur atjDT; 

Difference between the virtual temperature of the air

and the water surface; 

Emissivity of water; 

Density of water;

Stefan-Boltzman constant for black-body radiation; 

Travel time required for a parcel to flow through the

system; and 

Empirical wind function.

CONVERSION TABLE

Factors for converting metric units to inch-pound units are shown to four significant figures. 

Multiply metric unit By To obtain inch-pound unit

meter (m) 

kilometer (km) 

centimeter (cm) 

millimeter (mm) 

meter per second (m/s) 

cubic meter per second (m3/s) 

kilopascal (kPa)

3.281

0.6214

0.03281

0.3937

3.281

0.02832

10.00

foot (ft)

mile (mi)

foot (ft)

inch (in)

foot per second (ft/s)

cubic foot per second (ft3/s)

millibar (mb)
THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, 

CALIFORNIA, AND ITS RELATION TO EVAPORATION

By HARVEY E. JOBSON

ABSTRACT

The thermal balance of the 26-kilometer-long concrete-lined San 

Diego Aqueduct, a canal in southern California, was studied to determine the coefficients in a Dalton type evaporation formula. 

Meterologic and hydraulic variables, as well as water temperature, 

were monitored continously for a 1-year period.

A thermal model was calibrated by use of data obtained during a 

28-day period to determine the coefficients which best described the 

thermal balance of the canal. The coefficients applicable to the San 

Diego Aqueduct are similar to those commonly obtained from lake 

evaporation studies except that a greater evaporation at low 

windspeeds is indicated.

The model was verified by use of data obtained during 113 days 

(excluding the calibration data). These data verified that the derived wind function realistically represents the canal evaporation. 

An annual evaporation of 2.08 meter was computed; this amount is 

about 91 percent of the amount of water evaporated annually from 

nearby class A evaporation pans.

INTRODUCTION

Because of its large influence on many facets of the 

ecosystem, water temperature is clearly an important 

water-quality parameter. The rates of most chemical 

and biological reactions are temperature dependent, 

as are the spawning and growth cycles of most fish. 

The accuracy of any water-quality model is, therefore, 

sensitive to the accuracy of the thermal model upon 

which it must depend. The accuracy of a temperature 

model, likewise, is very dependent on the accuracy of 

the estimated exchange of energy between air and water. Finally, evaporation is a major component of the 

surface exchange process. Evaporation is, therefore, a 

key element in any water-quality model in addition to 

being a troublesome factor in computing the water 

balance of a river system.

While evaporation from lakes and reservoirs has received considerable attention, there has been almost 

no quantitative evaporation measurements from open 

channels. The purpose of this study and report is to 

develop a thermal model for open channels and to determine the applicable coefficients for a Dalton-type 

evaporation formula.

Dalton's law is often expressed as

in which .E=rate of evaporation in units of length per 

time; i//=an empirical coefficient or wind function; 

e0 — saturation vapor pressure of air evaluated at a 

temperature equal to that of the water surface; and 

ea =vapor pressure of the air. For the purposes of this 

report, the wind function is assumed to have the form

(2)

in which V=windspeed and intercept, a, as well as the 

mass-transfer coefficient, N, are assumed to be constant for the canal.

The evaluation of the coefficients a and N is difficult 

for open channels. Southern California offers an excellent average climate for the study of the coefficients in 

the wind function because the sensitivity of surface 

exchange to windspeed is large in this area of low 

humidity and high natural water temperatures. The 

San Diego Aqueduct, operated by the Metropolitan 

Water District of southern California, was chosen for 

determination of these coefficients as it has a relatively shallow depth, usually steady flow, and a simple hydraulic regime as well as being located in a hot, 

dry climate.

Meterologic data including incoming solar radiation, incoming atmospheric radiation, windspeed, 

wind direction, air temperature, wet-bulb air temperature, and rainfall were recorded at each end of the 

canal for a 1-year period beginning July 25, 1973. In 

addition, the water temperature at each end, the 

depth at five locations, and the rates of flow at the 

canal inlet and diversions were monitored. All these 

data have been presented previously (Jobson and 

Sturrock, 1976).

The interpretation of these data is accomplished 

by the use of a one-dimensional, finite-difference, 

thermal-balance model. The model, which uses both 

the Eulerian and Lagrangian reference frames, predicts the temperature at the lower end of the canal as 

a function of (1) the upstream temperature, (2) the 

rate of surface exchange due to radiation, rainfall, 

evaporation, and conduction to the air, (3) the exchange of energy between the bed and the water, and
THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

(4) longitudinal mixing within the channel. The contribution of each surface exchange and mixing process 

to the predicted temperatures is tabulated by use of a 

model written in the Lagrangian reference frame. 

This tabulation allows the coefficients in the wind 

function to be ascertained.

Data collected during 141 days were complete 

enough for use in calibrating or verifying the thermal 

model. These 141 days existed in eight groups of continuous data. Every month except January, February, 

September, and October were represented in the data 

sets.

The model was developed and calibrated using data 

obtained during the first 28 days of the study. Calibration in the sense used here implies the determination of the two coefficients in the wind function (eq. 2) 

which allow the model to best describe the thermal 

regime of the canal. The model, with the two derived 

coefficients, was then verified using the remaining 

113 days of data. The wind function coefficients found 

to apply to the San Diego Aqueduct are similar to the 

values obtained from lake-evaporation studies except 

that a larger evaporation at zero windspeed is indicated.

The following sections describe the flow and temperature models and the model calibration, including 

a discussion of a sensitivity analysis. The model verification is then presented. The report is concluded by 

presenting an estimate of the total evaporation from 

the canal throughout the year.

MODEL DEVELOPMENT

The temperature in an open channel is dependent 

on both hydraulic and meterologic variables. The flow, 

on the other hand, can usually be assumed to be independent of water temperature unless thermal stratification is significant. Flow in the San Diego 

Aqueduct was modeled independently of the temperature, and a data set was written for each time step 

which contained the average centerline water depth 

in each subreach and the discharge at each grid point. 

This data set was then used as input to the temperature model.

THE FLOW MODEL

The San Diego Aqueduct is a concrete-lined open 

canal originating near Hemet, Calif., about 120 km 

southwest of Los Angeles and flowing generally south 

for about 26 km (see fig. 1) on a slope of 0.00012. The 

canal's trapezoidal shape has a 3.66 m bottom width 

and side slopes of 1.5 to 1. At full capacity it delivers 

about 28 m3/s, but generally it flows near half capacity. During the study the flow varied from 7 to 24

117°00'

33 45'

0 5

I I I I I I

10 KILOMETERS

EXPLANATION

A Gaging station

Meteorologic station 

Evaporation pan 

Underground aqueduct 

Canal

FIGURE 1.—San Diego Aqueduct showing data-collection points.
MODEL DEVELOPMENT

m3/s. The maximum design depth is 3.05 m with a 

0.305 m freeboard.

The operation of the canal is such that the flow remains steady for several days in a row. Changes usually occur in steps. Because of equipment failures, 

shutdowns for canal maintenance, and so forth, only 

141 of the 365 days were modeled. Changes in discharge ranging from 12 to 60 percent occurred on only 

5 days of the 141 days.

Although the flow in the canal was generally 

steady, it was not uniform. Either the channel roughness or the bottom slope varied slightly with distance 

along the canal. To account for the nonuniformity of 

the flow between the five water level recorders (see 

fig. 1), longitudinal profiles of the centerline depth 

were measured under five different steady flow conditions. For computational purposes, the canal was 

schematized into 20 subreaches of equal length. Each 

subreach and the cross section at its upstream end 

were represented by the index j. The mean depth in 

each subreach was determined for each flow. An analysis of these observed depths indicated that the following expression could be used to determine the centerline water depth in each of the 20 subreaches.

+ d}

YM(4) - YM(3)

YM(k) (3)

in which Yj = centerline water depth, in meters, for 

subreach j (7=1, 2, . . . 20 with 7 = ! at the upstream 

end); Cj, dh and ej>lc are constants shown in table 1; 

YM(k)= recorded stage, in meters, at gaging station 

k, (k — l, 2, 3, 4, 5 with k = l for the gaging station in 

the farthest upstream; and Q (21)=discharge at the 

downstream end of the canal, in cubic meters per second. The constant ejik interpolates the depth from the 

two nearest gaging stations in proportion to the distance from each station. The term involving dj accounts for backwater effects and the constant Cj accounts for any vertical displacement of the bed in the 

subreach. The standard error of estimate for equation 

3, among the five flow conditions, ranged from a low of 

0.004 m for subreach 12 to a high of 0.069 m for subreach 14. The median value for all subreaches was 

0.031 m.

The water volume in each subreach was computed 

from

V, = (3.658 + 1.5Yj) Y£>X (4)

in which Vj = volume of water in subreach j, in

cubic meters; Yj = mean centerline water depth in

meters (from equation 1); and DX = length, in

TABLE 1.—Coefficients for use in equation 3 to determine the centerline depth in the San Diego Aqueduct

[All depths are expressed in meters. The value of d, is assumed zero if the absolute difference YM(4) ~ YM(3) is less than 0.04 m and the sign of dt is negative if 

YM(4) - YM<3) is negative]

Subreach

ejk

k = 2 k = 3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

-0.029

- .055

- .047

- .052

- .028

- .021

+ .024

+ .008

+ .182

+ .150

+ .139

- .019

+ .144

+ .294

+ .226

+ .202

+ .078

+ .125

+ .070

0

0

0

0

0

0

0

0

0

0

0

0

0.074

.428

.507

.519

.606

.492

.399

.158

0

1.0

.841

.689

.537

.385

.235

.081

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.159

.311

.463

.615

.765

.919

1.0

.559

.259

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.441

.741

1.0

.872

.759

.646

.533

.420

.308

.195

.082

0

0

0

0

0

0

0

0

0

0

0

0

0.128

.241

.354

.467

.580

.692

.805

.918

.500

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.500

meters, of the subreach (1,299 m). The discharge at 

each cross section was then computed from the discharge into or out of the canal and continuity using the expression

in which Q(j)= discharge at cross-section (j) and time 

iDT, V(j, t- ¥2)= volume of water in subreach./ at time 

(i — ¥2) DT; DT=time step (1 hour used in flow model); 

and QD(j)= discharge diverted from subreach./ during 

the time step. Three diversion points exist. Diversions 

near Simpson and the South End (fig. 1) were seldom 

used, and the third, downstream of Newport, is insignificant in size. The volume at time step i + Vz was determined from the average of the depth determined at 

time step i and time step i + 1.

THE EULERIAN TEMPERATURE MODEL

Applying the principle of conservation of thermal 

energy to a one-dimensional open channel, the governing equation becomes

dT

dt + U

dT

dx

d2T 

dx2

HTW 

Acp (6)

in which T=cross-sectional average water temperature; £=time; U=cross-sectional average velocity;
THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

x=longitudinal coordinate; Dx =longitudinal dispersion coefficient; #T =rate of absorption of thermal 

energy at the water surface, per unit area; W=top 

width of the channel; A = cross-sectional area of the 

channel; c=specific heat of water at constant pressure; 

p=density of water; Hp=rate at which the water absorbs heat from the bed, per unit area; and Rh =hyraulic radius. It is usually found that the crosssectional temperature variation is small for natural 

channels (Dingman and Weeks, 1970; Jones, 1965; 

and Rawson, 1970). Water temperatures were measured at four points in the vertical in the San Diego 

Aqueduct and vertical stratification was never observed. The specific heat and density are assumed to 

be constant, and the product of the dispersion 

coefficient and the area is assumed to be independent 

of x. All other varialbes, U, W, HT , A, Hp , and Rh , can 

be functions of both x and t.

Equation 6 is an Eulerian expression; it is a description of the variation of temperature with respect 

to a fixed coordinate system. This equation was solved 

numerically by the finite-difference technique using 

an implicit six-point forward difference scheme. The 

method of Stone and Brian (1963) was modified by 

giving the spatial derivative evaluated at the new 

time step a weight of 0.55 and the derivative 

evaluated at the old time step a weight of 0.45. This 

modification is similar to the method proposed by 

Fread (1974) for a four-point scheme. A 20-minute 

time step was selected for use in the temperature 

model because it was an even multiple of the data recording frequency (10 minutes) and because of 

economic reasons. Once the time step was determined, 

the distance step was selected such that the average 

value of U DT/DX was about 1.0 in order to optimize 

the accuracy of the numerical scheme. Using the same 

cross sections as in the flow model, the canal length 

was divided into 20 subreaches (21 grid points).

The last two terms in equation 6 were evaluated by 

use of meterologic and hydraulic data; the data were 

evaluated at a time midway between the time steps. If 

the water temperature is required in evaluating these 

terms, however, the old value at the cross section was 

used. It can be easily demonstrated, by use of numerical experiments, that use of the old temperature 

values has little effect on the accuracy of the solution 

provided DT W/A [d(HT/cp)/dT] < 0.2. For a water 

depth greater than 50 cm, this restriction reduced to 

d(HT/cp)/dT < 700 cm/day. Jobson (1973) has shown 

that this criteria will be met except under very exceptional conditions. The value of d(HTlcp)ldT will be 

recognized as the kinematic surface exchange coefficient.

The next to the last term in equation 6 represents 

the rate at which the water absorbs energy from the

atmosphere. The San Diego Aqueduct contains several 

inverted siphons to carry the water under roadways or 

drainageways. When a subreach contained a siphon 

the effective surface area of the subreach was reduced 

to account for the fact that no surface exchange can 

occur unless there is a free surface. Siphons were assumed to have no effect on the velocity or volume of 

water in the subreach.

For purposes of this report, the surface exchange 

will be expressed as the sum

=HN —Hi, +HR ~He — (7)

in which HN =net heat flux to the water caused by incoming radiation from the sun and the sky; Hb =heat 

flux leaving the water caused by longwave radiation 

being emitted by the water; HR = heat flux added to the 

water by rain falling directly on its surface; He =heat 

flux leaving the water as a result of the latent heat of 

vaporization of the evaporated water; and Hh =heat 

flux conducted from the water to the air as sensible 

heat.

The net incoming radiation is composed of four 

components: The solar radiation incident to the water 

surface less the reflected component plus the incoming atmospheric radiation less its reflected component. 

The solar radiation incident to an exposed horizontal 

surface was measured directly at each end of the 

canal. During times when the entire water surface 

was exposed to the sun, the percentage of the incident 

solar radiation to be reflected was determined by use 

of the expression:

= 1.18 Sa (8)

in which R=reflectivity and Sa =sun altitude in degrees (Anderson, 1954). Equation 8 was developed for 

clear sky conditions, but it is used here under all 

weather conditions. Cloud cover, which was fairly 

rare, appears to have a fairly small effect on the 

reflectivity for sun altitudes above 30°. For low sun 

altitudes, the water surface was shaded by high banks 

on either side of the canal. Shading was accounted for 

by increasing the reflectivity.

For the purposes of determining the shading due to 

the banks, the canal was assumed to flow directly 

south (fig. 1) and the top of the bank was assumed to 

be 16.38 m from the centerline of the channel. The 

average bank height was found to be 8.11 m, measured relative to the invert of the canal. For each time 

step, the location of the sun (azimuth and elevation) 

was determined. Knowing the sun's azimuth, an altitude for complete shading of the water surface and 

for complete exposure were determined. If the water
MODEL DEVELOPMENT

surface was completely shaded, the value of reflectivity was assumed to be 0.9. In other words it was assumed that the diffuse component of solar radiation 

(about 10 percent) would be available even in the 

shade. If the water surface was completely exposed, 

equation 8 was used to determine the reflectivity. The 

reflectivity was assumed to vary directly with the 

sun's altitude ranging from a value of 0.9 for complete shading to the value given by equation 8 for 

complete exposure.

The amount of incoming atmospheric radiation was 

measured directly. Unfortunately, this measurement 

proved to be very difficult, and reliable values are not 

available for all the 141 days. In some cases, the incoming atmospheric radiation was estimated by use of 

a method proposed by Koberg (1964). Three percent of 

the incoming atmospheric radiation was assumed to 

be reflected (Anderson, 1954).

The longwave radiation emitted by the water surface was computed using the Stefan-Boltzman law for 

blackbody radiation.

(9)

in which e=emissivity for water (0.97); <r=StefanBoltzman constant for black-body radiation; and 

273.16 converts to the kelvins when the water temperature, T, is given in degrees Celsius.

The energy added by rainfall, which was a very rare 

occurrence, was determined by

HR =cpi(Tr-Tb ) (10)

in which i=rainfall rate; Tw = wet-bulb air temperature; and Tb is an arbitrary reference temperature 

which is taken to be zero.

The other terms in equation 7 are related to the 

rate of evaporation, E. The thermal energy utilized by 

evaporation is expressed as

He =pL$(e0 -ea ) (ID

in which L= latent heat of vaporization of water.

Heat exchange by conduction has received relatively little attention because its magnitude is usually 

small in comparison to the evaporation heat exchange. Assuming that eddy diffusivities of heat and 

mass are equal, which leads directly to the Bowen 

ratio concept, the conduction term can be expressed as

(12)

in which -y=the psychrometric constant (0.0061 Pa );

Pa =atmospheric pressure in kilopascals; and Ta =air 

temperature that should be measured at the same 

elevation as the vapor pressure.

The last term in equation 6 represents the thermal 

flux at the bed of the canal. In the past, attempts have 

been made to model the bed conduction term; the heat 

flux has been estimated to be the product of the thermal conductivity and the temperature gradient within 

the bed. Measurements of bed temperatures are 

difficult to take so are seldom available. In addition, 

bed conditions are seldom uniform. Even though the 

bed conduction term has been shown to be significant 

(Brown, 1969; Pluhowski, 1970) for shallow depths, it 

is usually ignored.

The concrete and earth under the San Diego 

Aqueduct were approximated as an infinitely thick 

conducting medium; the thermal properties of which 

were estimated. The thermal conductivity of flowing 

water is much greater, because of turbulence, than 

that of the concrete and soil; so the surface temperature of the bed can be assumed to equal water temperature.

Mathematical expressions for the temperature distribution and heat fluxes within a semi-infinite 

medium which result from an arbitrary temporal 

variation in surface temperature are relatively simple 

(Carlslaw and Jaeger, 1959, p. 64). Unfortunately, 

these expressions converge slowly and their use in a 

numerical model would be expensive. On the other 

hand, if the bed were considered to be a slab, insulated on the bottom and of an arbitrary thickness, Z, 

the equations are still fairly simple but converge 

much faster. If the temporal variations in surface 

temperature are cyclical, the heat fluxes determined 

by the semi-infinite and finite thickness slab equations become indistinguishable as the slab thickness 

increases. In fact, assuming a diurnal water temperature swing of 5°C and thermal properties for concrete, 

the surface heat fluxes for a slab only 25 cm thick are 

within 7 percent of the values for a semi-infinite 

medium.

The heat exchange between the water and the bed 

was, therefore, estimated by considering the bed to be 

a homogenous slab, insulated on the lower face and 

with a top-surface temperature equal to that of the 

overlying water. The heat flux into or out of the bed 

was then determined as a function of the past history 

of the water temperature. Only the thermal diffusivity and heat-storage capacity of the bed needed to be 

assumed. A slab thickness of 25 cm was assumed; this 

thickness is believed to be sufficiently thick to give 

the desired accuracy.

The temperature distribution within a slab, initially at constant temperature, for which the surface 

is subjected to a unit increase in temperature at time
THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

zero is given by (Carslaw and Jaeger, 1959, p. 102)

ATB = 1 - -

-

"-!)"

i 77- n = o 2n + 1

exp [-/K2n + l)2 77- 2 £/4Z2l cos [(2n + 1) 7ry/2Z] (13)

in which ATB = the temperature rise within the slab; 

k = the thermal diffusivity; Z = thickness of the bed 

slab; and y = the distance above the insulated bottom 

of the slab at which temperature (ATB ) is computed. 

The increase in the heat content of the slab can be 

evaluated at any time by multiplying equation 13 by 

the heat-storage capacity, then integrating over the 

total thickness

H(t) = C,Z ( 

exp [—k(2n

(-1)"

7T-2 n = 0 (2/1 + I)2

l)277-2£/4Z2] sin ,[(2/i 7T-/2]) (14)

in which H(t) = the increase in heat content of the 

slab between time 0 and t resulting from the unit increase in surface temperature at time zero; and Cv 

the heat-storage capacity of the slab which is the 

product of the density and specific heat. Of course this 

heat must have been provided from the overlying 

water.

Equation 6 is solved by use of a finite-difference approximation that advance in time by discrete steps of 

duration at DT. The heat flux from the slab to the water bH(i) during any time interval iDT to (i + l)DT 

which results from a unit increase in water temperature at time zero, can be computed as

= H(iDT) -H [d + 1) DT]. (15)

The &H(i)'s describe the time variation of the response of the system to a unit change in water temperature.

Water temperature fluctuations can be approximated by a series of step changes, and the heat flux is 

linear with respect to temperature; so the superposition principle is used to determine the heat flux from 

the bed to the water for any temperature history by 

use of the equation

Hp (iDT) = ^T(kDT) - k) (16)

in which Hp(iDT) is the heat flux to the water from 

the bed during the time iDT to (i + 1)DT; &T(kDT) is 

the change in water temperature which occurred at

kDT (k ^ i); A/? is given by equation 15; and the water temperature is assumed to have been constant for 

times before t = (i - 72)DT. Equation 16 is solved for 

each cross section and each time step in a temperature model. The water temperature was assumed to 

have been constant before the model started (AT1 = 0 

for k < 0), and the bed conduction term was limited 

to a 24-hour memory (s = i - 72, since DT = 20 minutes).

Thermal diffusivity and heat-storage capacity of 

the bed material are the only additional parameters 

which are needed to include the bed conduction term 

in a thermal model. Heat-storage capacities are relatively easy to estimate. A thermal diffusivity of 0.01 

cm2/s was found to work best for the concrete-lined 

canal.

No water was added to the canal, and the diversions 

had no effect on the temperature of the water remaining in the canal except insofar as they affected the 

depth and velocity. The effect of the diversions are accounted for in the flow model, and no special precautions were needed in the temperature model.

The flow and temperature models were coupled in 

that the output of the flow model was used as input to 

the temperature model. Hydraulic data at times midway between the time steps of the temperature model 

were obtained by straight-line interpolation between 

the values generated by the flow model.

The dispersion coefficient in equation 6 represents 

the combined results of several physical processes 

(Jobson and Yotsukura, 1973), and the estimation of 

its value is difficult. Fortunately, the predicted temperatures are not very sensitive to the assumed value 

of the dispersion coefficient. Figure 2 is presented to 

illustrate this insensitivity. The temperature measured at the downstream end of the San Diego 

Aqueduct on August 18, 1973, is shown in the figure, 

as well as the predicted temperatures obtained by solving equation 6 with two different values of the dispersion coefficient, Dx . One value of Dx was assumed 

to be equal to 7,500 Rh U., where t/* is the shear velocity and the other assumed 1)^=250 Rh U*. Changing 

the dispersion coefficient by a factor of 30 changed the 

predicted temperature at any time by less than 0.08 

Celsius degree. The vertical scale, in figure 2, has 

been expanded during part of the day to illustrate 

that the value of 7500 Rh U* appears to overly smear 

small temperature variations. The large dispersion 

coefficient was observed in the Missouri River by Yotsukura, Fischer, and Sayre (1970), and the small 

value represents the approximate mean of all the 

measured values summarized by Fischer (1973). A 

value of Dx/Rh U*=25Q is used throughout this report 

and the value of Rhwas evaluated for the section near 

the center of the canal reach.
MODEL DEVELOPMENT

EXPLANATION

Observed

Computed Dx /Rh U* = 250 

Computed Dx /Rh U* = 7500 UJ

TIME OF DAY, IN HOURS

FIGURE 2.—The effect of the dispersion coefficient on the computed temperatures in the San Diego Aqueduct

on August 18, 1973, in comparison with measured temperatures.

THE LAGRANGIAN MODEL

Equation 6 is an Eulerian equation, meaning that 

it is a description of the variation of temperature with 

respect to a fixed coordinate system. Another description, the Lagrangian, considers the variation of the 

temperature of a given parcel of fluid or fluid lump, as 

the parcel moves through the system. In the Lagrangian framework, one conceptually follows an individual fluid parcel while keeping track of the factors 

which tend to change its temperature. Applying the 

thermal continuity equation to a unit mass of fluid, 

the Lagrangian equivalent of equation 6 is

dt

d2T 

dx2

HTW

Acp (17)

Integrating both sides of equation 17 over the traveltime, one obtains

Acp

in which T0 = final temperature of the water parcel as 

it passes the downstream end of the channel reach; 

T{ — initial temperature of the water parcel as it enters at the upstream end of the reach; and T = traveltime required for the parcel to be convected through

the system. Expanding the right-hand side of equation 18 yields

d2T

dt

( T

J o

(HN — HI, + HR)

W 

Acp dt +

W {L(e0 - ej + yL(T - Tj} -- dt. (19)

Equation 19 is extremely useful to a modeler because 

it allows the contribution of each exchange process to 

be isolated and evaluated. For example, integrating 

the first term in the second integral would determine 

the net temperature rise of a parcel due to absorbed 

atmospheric and solar radiation. Comparison of this 

value to the total temperature change (T0 - TJ then 

is a measure of the sensitivity of the predicted temperatures to errors in measurements of solar or atmospheric radiation.

One of the main purposes of this report is to determine the coefficients in the wind function which are 

applicable to the San Diego Aqueduct. Equation 19 

provides a powerful tool for investigating the wind 

function. To make use of this tool we must assume 

that the wind function, i//, applicable to a particular 

water parcel, is constant. Solving for the wind function by factoring it out of the integral gives
THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

T Ti ~ rTo dt

T {L(e0 - ea)

- TJ W 

Ac

(20)

dt

The wind function in equation 20 is computed as a 

residual in that it is the average value which must be 

used during the traveltime, T, to force the energy budget of the Lagrangian parcel to balance. Any measurement error or any physical process which is not 

correctly modeled will manifest itself as an error, or 

scatter, in the value of the wind function computed by 

use of equation 20. Nevertheless, if measurements 

are carefully made on a well-selected site, the results 

can be consistent and useful. _ 

Before equation 20 can be solved, the variation of 

water temperature with time and distance must be 

known or assumed. The resulting wind function value, 

however, is not very sensitive to this water temperature distribution. Equation 6 was used, before the application of equation 20, to estimate the internal 

water temperature distribution. In the solution to 

equation 6, the wind function was assumed to be related to the windspeed as shown in equation 2. The 

final result was obtained by use of an iterative process 

in which (a) the coefficients in equation 2 were assumed; (b) equation 6 was solved for the 28-day calibration period; (c) equation 20 was solved for the wind 

function at 1,890 equally spaced times during the calibration period; (d) the wind function values were 

plotted against the windspeeds representative of the 

time of passage of the water parcel and new estimates 

of the coefficients in equation 2 obtained; and (e) if the 

new estimates were significantly different from the 

previous estimates, the process was repeated. Equations 6 and 20 were solved using numerical techniques and the process converged very rapidly. Each 

term in equation 20 was calculated separately so that 

the relative contribution of each physical process to 

the total temperature changes of any particular fluid 

parcel could be determined.

DATA AND MODEL CALIBRATION

DATA AVAILABLE

Meterologic data including incoming solar radiation, incoming atmospheric radiation, windspeed, 

wind direction, air temperature, and wet-bulb air 

temperature, were continuously recorded at each end 

of the canal for a 1-year period beginning July 25, 

1973. In addition, the water temperature at each end, 

the depth at five locations, and the flow rate were continuously monitored.

The water temperature and all meteorologic data, 

except rainfall which was a rare occurrence, were recorded every 10 minutes in digital form on magnetic 

tape. The five stages were recorded continuously on 

analog charts. The charts were digitized to give 

hourly stages. During times of steady flow, the discharges were furnished by the Metropolitan Water 

District from current-meter measurements and (or) 

gate ratings. After February 28, 1974, a continuous 

analog record of discharge was available from a sonic 

flow meter. Before February 29, 1974, unsteady flow 

rates were computed from Manning's equation using 

the farthest upstream stage measurement. The 

roughness coefficient applicable in the Manning equation was determined during the periods of steady flow.

For further information concerning the site layout 

or the data collection effort, the reader is referred to 

Jobson and Sturrock (1976).

Because of equipment malfunctions and times when 

there was no flow in the canal, the data for all 365 

days of the study were not complete. A total of 141 

days were judged to have sufficient data to calibrate 

or verify the thermal model. These 141 days existed in 

eight groups of continuous data. The length and dates 

for each group are shown in table 2. The smallest 

group contained 4 consecutive days and the largest 

group 45 days. To be considered acceptable for analysis, certain criterion had to be met. These were:

1. The water temperature must be available at 

both ends of the canal.

2. Flow data must be reasonably complete.

3. All meterologic parameters, except atmospheric radiation, must be available at 

either the Cottonwood (upstream) or the 

Skinner (downstream) station.

Even during the 141 days, many time periods of 

short duration existed for which the data were not 

complete. ,For example, there were frequent gaps in 

the wet-bulb temperature record because the wick of 

the wet-bulb probe dried out for short periods of time. 

Because the model requires a complete data set for 

operation, the observed data were copied on a temporary data file and all missing or questionable data were 

filled in or replaced before the model was run. If a 

parameter was missing for only an hour or so, the 

missing values were interpolated. If a longer period of 

record was missing, the missing values were assumed 

to be equal to the values observed at the other station.
DATA AND MODEL CALIBRATION 

TABLE 2.—Summary of modeled data for the San Diego Aqueduct

[RMS = root mean square]

of

data

1

2

3

4

5

6

7

8

Beginning

Date

07-25-73

11-28-73

12-19-73

03-07-74

03-13-74

04-18-74

05-08-74

07-05-74

Hour

1440

1720

1000

2220

920

1020

1520

1040

Ending

Date

08-21-73

12-12-73

12-24-73

03-10-74

03-21-74

05-01-74

06-22-74

07-23-74

Hour

0400

1440

1140

1200

2400

1240

1420

1420

Length 

of

hours

638

334

122

62

207

314

1079

436

Error in 

computed 

temperature (°C)

Mean

+0.01

- .08

- .16

+ .01

- .04

- .07

- .16

- .13

RMS

0.14

.23

.22

.12

.14

.22

.26

.23

Mean daily flow rate, in 

cubic meter per second

Mean

13.03

6.65

11.10

23.58

19.36

13.95

15.98

20.34

Standard

deviation

0.41

1.58

.69

.56

5.30

1.07

1.35

.20

Vapor pressures and wet-bulb temperatures were 

handled with special care. Vapor pressures were computed from the observed wet- and dry-bulb temperatures at both stations before any missing records were 

filled. Then the vapor pressure was treated as if it 

were a measured parameter so that in no case was the 

vapor pressure computed from wet- and dry-bulb temperatures obtained at different locations.

It was observed that the diurnal range in atmospheric radiation differed depending upon the sensor 

type (Eppley or Gier-Dunkle) 1 used in its detection. 

Although daily mean values appeared consistent, the 

observed diurnal variation appeared to be too large for 

both sensor types (Jobson and Sturrock, 1976). It was 

assumed, therefore, that the incoming atmospheric 

radiation did not vary during a day, and the daily average value was used. During the calibration period 

July 25, 1973, to August 21, 1973, the atmospheric 

radiation was observed at both ends of the canal by 

use of Eppley pyrgeometers. Between November 28, 

1973, and March 10, 1974, only the Eppley pyrgeometer at Skinner was used. After March 13, 1974, the 

incoming atmospheric radiation component was estimated by use of the procedure outlined by Koberg 

(1964).

MODEL CALIBRATION

No flow model calibration, as such, was involved in 

this study. As stated previously, the flow was unsteady for only 5 of the 141 days, and the stage was 

continuously monitored at five points along the canal 

length. The mean and standard deviation between the 

daily average ^flow rates for each data group is shown 

in table 2. Knowing the flow rate into the canal, as 

well as the canal depth as a function of time and distance (from eq. 3), the discharge at any point and time 

was uniquely determined from continuity considerations (eq. 5). In order to illustrate the response of the

'The use of brand names in this report is for identification purposes only and does not 

imply endorsement by the U.S. Geological Survey.

system to a typical transient, the observed stages as 

well as the discharge values at the inlet and outlet are 

shown in figure 3 for transients which occurred on December 10-11, 1973. The Cotton wood, Simpson, Newport, South End, and Skinner gages are 0.3, 9.1, 13.5, 

25.0, and 26.0 km downstream of the canal entrance 

respectively. The Cottonwood discharge is computed 

from the observed stage at Cottonwood, and the Skinner discharge is determined from equation 5.

Before October 1, 1973, the roughness of the canal 

was observed to be constant with a value of 0.0175 but 

after October 1, the observed roughness appeared 

somewhat erratic (Jobson and Sturrock, 1976, p. 69). 

The results of the thermal model obtained for the second period, November 28 to December 12, 1973, 

seemed to be more consistent when the roughness was 

assumed to be 0.0175 than when it was assumed to be 

0.0151 as published by Jobson and Sturrock (1976). 

Therefore, the results for this period were obtained by 

assuming the Manning's roughness coefficient to be 

_0.0175.

The modeled and observed temperatures on the 5 

days when unsteady flow occurred were critically reviewed, but no systematic error that appeared to be 

related to inadequate flow model results could be detected. A sensitivity analysis, conducted in some preliminary runs during the calibration period, indicated 

that neither the RMS (root-mean-square) error in predicted temperatures nor the derived coefficients in the 

wind function were extremely sensitive to discharge 

errors. A one percent increase in computed discharge 

increased the intercept coefficient, a, in equation 2 by 

about 4 percent and decreased the mass-transfer coefficient, N, by about 4 percent.

It was also determined by a sensitivity analysis of 

data input for the calibration period that the intercept 

was not very sensitive to errors in the measured solar 

radiation or the assumed thermal diffusivity of the 

bed, but that it was fairly sensitive to errors in the observed atmospheric radiation. A one percent increase
10 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

Cottonwood

y

Simpson

DC

LUhLU

South End\

Skinner. —-

Newport^

15

10

O 

o

LU 

V)

DC 

LU 

Q_ 

C/)

DC 

LU

ho

CO

o 

z 

- 5

LU 

O 

DC

X 

O

Cottonwood\ /

Skinner

0000 0600 1200 1800 

DECEMBER 10, 1973

0000 0600 1200 1800 

DECEMBER 11, 1973

2400

FIGURE 3.—Variation of stage and discharge for the flow transients on December 10 and 11, 1973.

in the values of solar radiation, atmospheric radiation, and thermal diffusivity changed the intercept by 

+0.4, +3, and 0.1 percent, respectively. The fitted 

mass-transfer coefficient, on the other hand, was 

somewhat sensitive to errors in solar radiation but not 

to errors in atmospheric radiation or bed diffusivity.

One percent increases in solar radiation, atmospheric 

radiation, and thermal diffusivity changed the masstransfer coefficient by +2, +0.6, and -0.02 percent, respectively.

In summary, the computed coefficients in the wind 

function were sensitive to certain measured parame-
DATA AND MODEL CALIBRATION 11

ters such as flow rate and radiation, but measurement 

errors were not unreasonably amplified.

The RMS error in computed temperatures was not 

very sensitive to assumed value of thermal diffusivity 

of the bed as long as it was within the range of 0.008 

cm2/s to 0.015 cm2/s. Outside this range, the RMS 

error increased rapidly. By comparison, the range of 

thermal diffusivities observed in concrete dams on the 

Tennessee River was from 0.0061 cm2/s for Bull Run 

Dam to 0.0141 cm2/s for Norris Dam (Troxell and 

others, 1956). If the assumed thermal diffusivity was 

outside the indicated range, the computed values of 

the wind function did not appear to be linearly related 

to windspeed. Figure 4 illustrates the effect of thermal 

diffusivity on the wind function values computed for 

July 30, 1973. With the diffusivity set equal to zero, 

the wind function values for the day define a loop 

rather than a straight line. Assuming diffusivities 

larger than 0.015 cm2/s also created a loop, but the 

loop progressed counterclockwise with increasing 

time.

Equation 20 was solved for the wind function at

I,890 equally spaced times during the 28-day calibration period. The average windspeed, during the 

transit of each parcel, was also tabulated, and the 

wind function values were plotted as a function of 

windspeed. The results are shown in figures 5 through

II. Also shown in each illustration is a straight line of 

best fit for the day as well as the correlation 

coefficient.

A composite of all of the results shown in figures 

5-11 is shown in figure 12 along with the equation

i//=3.02+1.13 V (21)

in which the wind function is given in millimeters per 

day per kilopascal when the windspeed, V, is given in 

meters per second. Equation 21 was derived by use of 

a regression analysis on all 1,890 values of windspeed 

and wind function during the calibration period, and 

it was used in equation 6 for all days in the analysis 

procedure. Equation 21 is considered to be the principal result of this study.

An idea of the total variability of the wind function 

can be obtained from figure 12. As can be seen from

10

o

CO

<

Q. 

O

cc

LU 

Q.

Q 

CC

CO 

DC 

LU

H 

LU

= 4

o 

o

D

LL 

Q

EXPLANATION

O Thermal diffusivity = 0.01 cm2/s 

A Thermal diffusivity = 0

Hour of water exit

WINDSPEED, IN METERS PER SECOND

FIGURE 4.—Comparison of wind function values for July 30, 1973, as computed with and without heat transfer between the water and

the bed.
12 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

12

10

a. 4

oc 2

m 0

ii 12

10

JULY 25, 1973

0 = 2.5 + 1.2P

JULY 26, 1973

r = 0.98

^ = 2.8 + 1.3K

4

2 -

JULY 27, 1973

r = 0.87

ifr = 3.6+0.8K

JULY 28, 1973

r =0.95

<l> = 2.9 + 1.2K

3 40 1 

WINDSPEED, IN METERS PER SECOND

FIGURE 5.—Variation of the wind function, with windspeed for the San Diego Aqueduct during July 25 through 28, 1973.

12

10

5 8

a. 4

>-

Q 

oc 2

LLJ 

O. 

CO

LU 0 

| 12

I 10

JULY 29, 1973

r =0.95

^ = 3.2 + 1.1K

JULY 30, 1973

r = 0.97

* = 3.1 + l.Of

JULY 31, 1973

r ={).97

<!, = 2.4 + 1.2K

AUGUST 1, 1973

r =0.92

0 = 2.4 + 1.1K

3 40 1 

WINDSPEED, IN METERS PER SECOND

FIGURE 6.—Variation of the wind function, with windspeed for the San Diego Aqueduct during July 29 through August 1, 1973.
DATA AND MODEL CALIBRATION 13

3 40 1 

WINDSPEED, IN METERS PER SECOND

FIGURE 7.—Variation of the wind function, with windspeed for the San Diego Aqueduct during August 2 through August 5, 1973. 

12

i 10 AUGUSTS, 1973 AUGUST 9, 1973

r =0.98

i// = 3.2 + 1.1K

3 40 1 

WINDSPEED, IN METERS PER SECOND

FIGURE 8.—Variation of the wind function, with windspeed for the San Diego Aqueduct during August 6 through August 9, 1973.
14 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

12

10

Q. 4

a: 2

12

10

AUGUST 10, 1973

r = 0.94

AUGUST 12, 1973

r =0.92

^ = 3.2 + 1.2V

AUGUST 11, 1973

r = 0.95

0 = 3.5 + 1.1K

AUGUST 13, 1973

r =0.61

^ = 3.5 +0.9K

3 40 1 

WINDSPEED, IN METERS PER SECOND

FIGURE 9.—Variation of the wind function, with windspeed for the San Diego Aqueduct during August 10 through August 13, 1973.

3 40 1 

WINDSPEED. IN METERS PER SECOND

FIGURE 10.—Variation of the wind function, with windspeed for the San Diego Aqueduct during August 14 through August 17, 1973.
DATA AND MODEL CALIBRATION 15

AUGUST 19, 1973

r =0.93

<li =3.1 + 1.0K

3 40 1 

WINDSPEED, IN METERS PER SECOND

FIGURE 11.—Variation of the wind function, with windspeed for the San Diego Aqueduct during August 18 through August 21, 1973.

1 2

WINDSPEED, IN METERS PER SECOND

FIGURE 12.—Computed wind function values for the San Diego Aqueduct. Every other point plotted for the 28-day period starting

July 25, 1973.
16 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

figures 5 through 11, the data for any particular day 

generally defines a reasonably straight line, but the 

mass-transfer coefficient and intercept varies from 

day to day. Ignoring the results for 2 of the 26 complete days, the intercept varies from 1.7 to 3.9 and the 

mass-transfer coefficient varies from 0.5 to 1.6. 

Nevertheless, equation 21 seems to be fairly well defined by the data and indeed it is entirely possible 

that the intercept and mass-transfer coefficient are 

somewhat dependent on wind direction or other 

meteorologic factors which vary from day to day.

With the wind function defined by equation 21, 

equation 6 was capable of predicting the downstream 

temperature during the 28-day calibration period with 

a RMS error of 0.14°C. A sample of the predicted and 

measured temperatures is shown in figure 13.

MODEL VERIFICATION AND DISCUSSION

The ability of equation 21 to accurately define evaporation from the San Diego Aqueduct was verified by

solving equation 6, wherein the wind function is defined by equation 21, for the remaining 113 days of 

data. These days were scattered throughout the year 

and included days in the months of November, December, March, April, May, June, and July. Visual 

verification with typical results is illustrated in 

figures 14 through 20 which contain the observed and 

modeled temperatures during the first 4 full days of 

each of the seven groups of verification data. The 

mean and the RMS error in the predicted temperatures for each group are shown in table 2. The overall 

RMS error during the 113 days of verification is 

0.23°C.

Errors tend to be smaller during the winter; this 

should be expected because of the smaller diurnal 

range. The observed temperatures during March 1974 

(figs. 16 and 17) appear somewhat erratic possibly because of instrumentation errors. The temperature recorder at the lower end of the canal definitely malfunctioned soon after March 21, 1974. Overall, the

29

28

27

26

i——r

CO

B

CO

LU 

LJJ 

CC 

O 

LU

LU

CC

D

<

CC 

Uj 

Q_ 

S 

UJ

24

23

EXPLANATION

Modeled 

Observed 

Observed upstream

/\

26 27 28 

JULY 1973, IN DAYS

29

FIGURE 13.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct during the

first 4 days of the calibration period.
MODEL VERIFICATION AND DISCUSSION 17

CO

c/j

LU

o

LU 

LU 

OC

LU 

Q

OC

D

<

DC 

LU

a.

17

16

15

14

13

12

11

EXPLANATION

Modeled 

Observed 

Observed upstream

\

NOV. 30 DEC. 1 DEC. 2 DEC. 3

DAY IN 1973

FIGURE 14.- -Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct during the

first 4 days of the first verification.

results of the verification appear to be very good.

The results illustrated in figures 14-20 are believed 

to verify that equation 21 predicts representative 

evaporation rates during all seasons of the year. On 

the other hand, the results shown in figures 5 through 

11 suggest that the wind function coefficients in equation 21 vary from day to day. In order to determine if 

these coefficients varied in any systematic manner 

throughout the year, equations 6 and 20 were solved 

simultaneously for the seven verification periods in 

the same manner as for the calibration period. Plots 

similar to those shown in figures 5-11 were inspected, 

and the daily values of the wind function coefficients 

were determined. Table 3 contains a tabulation of 

the results for all days in which the correlation 

coefficients for the linear regression exceeded 0.7. 

Fewer results are shown for the cooler months. During this time, evaporation became a small factor in 

the energy budget so that random errors in the other 

components were amplified. Although significant scatter in the daily mass transfer coefficients exist, no

seasonal pattern is apparent. The intercept values, on 

the other hand, appear to increase between March and 

July. Since the intercept values are fairly sensitive to 

errors in the atmospheric radiation term and no actual measurements of this term was available after 

March 13, 1974, little emphasis is placed on the apparent trend.

Figures 13 through 20 demonstrate that equation 

21 provides accurate predictions of water temperature 

when incorporated in a thermal model. But before the 

accuracy of equation 21 can truly be verified, it must 

be demonstrated that the accuracy of the predicted 

temperatures is reasonably sensitive to the evaporation rate. The diurnal variation in water temperature 

at the canal entrance is very small as can be seen in 

figures 13 through 20. The San Diego Aqueduct is fed 

by a pipeline which passes under the San Jacinto 

Mountains just upstream of the canal entrance. The 

ability of the model to start with a relatively constant 

inflow temperature and to accurately predict the 

diurnal swings at the downstream end implies that
18

TABLE 3.— Daily

Date

07-26-73

07-27-73

07-28-73

07-29-73

07-30-73

07-31-73

08-01-73

08-03-73

08-04-73

08-05-73

08-06-73

08-07-73

08-08-73

08-09-73

08-10-73

08-11-73

08-12-73

08-14-73

08-15-73

08-16-73

08-17-73

08-18-73

08-19-73

08-20-73

11-30-73

12-11-73

12-22-73

03-18-74

03-20-74

03-21-74

04-19-74

04-20-74

04-22-74

04-23-74

04-24-74

04-25-74

04-26-74

04-27-74

04-28-74

04-29-74

05-10-74

05-11-74

05-12-74

05-13-74

05-16-74

05-17-74

05-18-74

THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT,

values of the wind function coefficients for the 

San Diego Aqueduct

Intercept (a)

[mm/d-kPa]

2.80

3.60

2.90

3.20

3.10

2.40

2.40

1.70

1.70

2.10

2.00

2.60

3.40

3.20

3.20

3.60

3.20

3.90

3.20

2.80

3.10

3.60

3.10

.60

2.60

2.50 -.80

-1.10

-1.70

-1.50

-2.20

1.10

1.60

1.40

-2.00

.80

.50

.30

.80

1.60

-3.90

-1.70

.40

-3.60

-1.00 -.70

.60

Mass transfer

(N)

[mm/d'kPa-m/s]

1.30

.80

1.20

1.10

1.00

1.20

1.10

1.50

1.60

1.20

1.50

2.00

1.60

1.10

1.20

1.10

1.20

.60

1.10

1.60

1.40

.60

1.00

2.50

1.10

1.80

2.60

6.00

2.50

5.00

3.60

2.50

2.50

1.70

3.00

1.60

2.00

2.40

2.10

1.20

4.60

3.80

2.30

3.50

2.30

2.40

1.70

Correlation

coefficient

0.98

.87

.95

.95

.97

.97

.92

.93

.93

.94

.81

.76

.85

.98

.94

.95

.92

.82

.78

.85

.91

.90

.93

.92

.72

.88

.84

.88

.74

.81

.80

.73

.86

.82

.91

.95

.86

.89

.92

.72

.92

.95

.84

.91

.92

.94

.94

TABLE 3

Date

05-19-74

05-20-74

05-21-74

05-22-74

05-23-74

05-24-74

05-26-74

05-27-74

05-28-74

05-29-74

05-31-74

06-01-74

06-02-74

06-03-74

06-04-74

06-05-74

06-06-74

06-07-74

06-08-74

06-09-74

06-10-74

06-11-74

06-12-74

06-13-74

06-14-74

06-15-74

06-16-74

06-17-74

06-18-74

06-19-74

06-20-74

06-21-74

07-06-74

07-07-74

07-08-74

07-09-74

07-10-74

07-11-74

07-12-74

07-13-74

07-16-74

07-17-74

07-18-74

07-20-74

07-21-74

07-22-74

CALIFORNIA

— Daily values of the wind function coefficients for the 

San Diego Aqueduct — Continued

Intercept (

[mm/d-kP

1.80

1.20

1.40

1.60

1.90

1.80

2.60

3.10

1.20

1.80

.70

1.30

2.30

1.90

2.00

2.00

2.40

.10

1.00

2.40

2.30

1.90

1.50

2.40

2.50

2.50

2.70

2.40

2.10

2.30

1.90

2.30

2.80

2.00

1.50

2.40

.80

1.90

2.60

2.70

2.20

2.40

3.30

1.70

1.90

1.70

Mass transfer

a) (N}

a] [mm/d-kPa-m/s]

1.40

1.80

1.50

1.30

1.00

1.40

3.60

1.60

1.60

1.10

.90

.90

.60

.90

.80

.80

.80

1.00

.90

.40

.60

1.00

1.30

.90

.60

.40

.80

.70

.80

.80

1.00

.80

.70

.80

1.20

.80

1.50

.90

.50

.60

.80

.50

.60

1.10

.80

2.00

Correlation

coefficient

0.88

.90

.79

.83

.84

.71

.86

.97

.94

.92

.91

.95

.86

.88

.90

.94

.94

.87

.93

.73

.93

.88

.90

.95

.89

.85

.93

.97

.97

.89

.94

.97

.97

.97

.90

.95

.91

.82

.90

.82

.92

.90

.84

.89

.94

.78

the surface exchange, including evaporation, is being 

accurately modeled.

The study site was selected such that it would be in 

a climate where a large part of the total surface exchange is due to evaporation. Equation 19 provided 

the means of evaluating the contribution of each process to the energy budget of an individual water parcel passing through the canal. In order to illustrate 

the relative contributions of each process, the mean 

and standard deviation of each major term in equation 

19 are tabulated in table 4 for each group of data. It is 

seen that evaporation He , as well as the net radiation, 

HN -Hb , are major components of the energy budget. 

Conduction to the air or the bed are small components. Comparing the RMS error in the predicted 

temperature (table 2) to the temperature change due

to evaporation (table 4), it is seen, that except perhaps 

for a short period in March when there may have been 

instrument malfunction, the evaporation must have 

been modeled fairly well. It is concluded, therefore, 

that the evaporation from the San Diego Aqueduct is 

realistically represented by the wind function of equation 21.

In order to compare equation 21 with existing wind 

functions, a simple seventh-root velocity law was used 

to convert windspeeds for all equations to a common 

reference height of 4 m, which was the approximate 

height of the anemometers used in this study. No correction was made for the measurement height of the 

vapor pressure, and it was assumed that the effect of 

averaging data over periods of time of 1 day or less 

had no effect on the coefficients in the wind function.
MODEL VERIFICATION AND DISCUSSION 

TABLE 4.—Summary of energy budget terms for individual water parcels passing through the San Diego Aqueduct

19

Group

1

2

3

4

5

6

7

8

Beginning 

date

07-25-73

11-28-73

12-19-73

03-07-74

03-13-74

04-18-74

05-08-74

07-05-74

Mean and standard deviation of temperature increase, 

in degrees Celsius, to result from the indicated cause 

[Standard deviations in parentheses]

Observed

temperature

increase

-0.21

(0.86)

-0.54

(0.73)

-0.33

(0.55)

-0.22

(0.44)

+ 0.12

(0.62)

+0.01

(1.00)

-0.08

(0.98)

-0.06

(0.86)

Incoming

radiation

2.78

(1.00)

3.12

(0.88)

1.94

(0.56)

1.20

(0.40)

1.58

(0.69)

2.53

(1.14)

2.52

(1.11)

2.27

(1.04)

Back

radiation

-1.95

(0.05)

-2.80

(0.33)

-1.82

(0.02)

-1.06

(0.02)

-1.31

(0.29)

-1.84

(0.04)

-1.82

(0.22)

-1.57

(0.03)

Evaporation

-1.00

(0.35)

-0.71

(0.22)

-0.37

(0.15)

-0.25

(0.05)

-0.12

(0.25)

-0.58

(0.29)

-0.68

(0.31)

-0.72

(0.35)

Conduction

to the air

0.0

(0.16)

-0.13

(0.21)

-0.08

(0.11)

-0.14

(0.05)

-0.02

(0.19)

-0.08

(0.13)

-0.08

(0.13)

0.0

(0.13)

Conduction

to the bed

0.00

(0.09)

0.01

(0.12)

0.01

(0.06)

0.00

(0.03)

0.00

(0.05)

0.01

(0.11)

0.01

(0.09)

0.00

(0.08)

17

16

C/3

D

C/3

B 15

C/3 

LU 

LU

tr

LU 

Q

LU

tr

D

<

tr LU0.

14

13

12

11

——'———————I————I——r— 

EXPLANATION

— — — — - — — Modeled

—————— Observed

— — ——— Observed upstream

20 21 22 

DECEMBER 1973, IN DAYS

23

FIGURE 15.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct during the

first 4 days of the second verification.

Harbeck (1962) summarized evaporation studies inputs and having surface areas ranging from 4xl03 

made on reservoirs having little or no artificial heat to nearly 1.2xl08m2 . He proposed the relation
20 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

2.43

(22)

in which Ar is surface area of the reservoir, in meters 

squared; and V is the windspeed at the center of the 

reservoir, in meters per second. The major difference 

between equation 21 and 22 is that Harbeck assumes 

no evaporation will occur unless a measurable 

windspeed exists. This is probably a reasonable assumption if the windspeed is measured near the center of a large lake, but it is probably unrealistic 

for computation of evaporation from streams. Near 

streams and rivers the rough terrain limits windspeeds to values less than the stall speeds of most 

anemometers during a fairly larger percentage of the 

time. In addition the flow velocity generates a relative 

motion between the water and air even at zero 

windspeeds. For a surface area of 150 m2 , (the average width of the canal was about 9 m) equations 21

and 22 intersect at a windspeed of about 4 m/s.

Above an artificially heated water surface, free convection may significantly increase the rate of evaporation. For artificially heated reservoirs, Ryan and Stolzenbach (1972) present the formula

i|/=0.95(A0)I/3 +0.91V (23)

in which A0 is the difference between the virtual temperature of the air and the water surface. The virtual 

temperature is the temperature at which dry air 

would have the same density as the moist air mixture 

assuming constant pressure. They suggest that the 

value of A0 should be taken as zero for water bodies 

with no thermal loading. The mass-transfer coefficients (slope) in equations 21 and 23 differ by 19 percent.

A commonly used formula was developed by Meyer 

in 1942 and published by Linsley, Kohler, and 

Paulhus (1958).

17

16

I———I———I———I———I———I———I———I———I———I———I———I———I———I———I———I———T

14

CO

D

CO

O

CO

LU 

LLJ 

CC.

LU 

DC

QC 10 LU lv5

CL

2 

LU

12

11

EXPLANATION

Modeled 

Observed 

Observed upstream

\

/ \

L I i i i i i i

8 9 

MARCH 1974. IN DAYS

10

FIGURE 16.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct during the

third verification.
MODEL VERIFICATION AND DISCUSSION 21

^=2.7+0.66 V. (24)

Although the mass-transfer coefficient is smaller, the 

intercept in formula 24 is only 11 percent less than 

that for equation 21.

The Tennessee Valley Authority (1972), as well as 

Ryan and Stolzenbach (1972), have given excellent 

summaries of many of the existing empirical wind 

function formulas. Equation 21 is, however, believed 

to be the only equation ever developed from the thermal balance of an open channel. In summary, the 

wind function for the San Diego Aqueduct (eq. 21) indicates a larger evaporation at low windspeeds than is 

indicated by most lake evaporation formulas, but the 

mass-transfer coefficient is within the range of values 

which have been reported. For the average windspeed 

at the San Diego Aqueduct, about 1.5 m/s, equation 21 

indicates a slightly greater evaporation rate than 

would be indicated by most lake evaporation formulas.

In order to estimate the evaporative water loss from

the San Diego Aqueduct, equations 1 and 21 were 

solved using daily averaged values of windspeeds, 

water temperature, and vapor pressure for each end of 

the canal (Jobson and Sturrock, 1976). The evaporative rate at each end of the canal was computed for 

each day, and the resulting monthly averaged evaporation rates were determined. The monthly average 

results are shown in figure 21 as well as in table 5. 

Because of missing data, mainly the vapor pressures 

of the air, a considerable amount of interpolation was 

required to estimate monthly evaporation rates. 

Using days when the daily average vapor pressure 

was available at both ends of the canal, a monthly 

value of the ratio of these vapor pressures was established. Missing vapor pressures were interpolated 

using these ratios as well as the monthly mean values 

and any other factors considered to be significant. 

Missing water temperatures were interpolated by paying special attention to the seasonal variation of the 

mean difference in the water temperatures at the ends 

of the canal. Very few windspeed data were missing.

18

17

D

C/5

_1

W 

O

(/> 

LLJ 

LLI 

DC 

O 

LLJ

LLI 

CC

D

LLI

I15

14

13

12

EXPLANATION

Modeled 

Observed 

Observed upstream

14 15 16 

MARCH 1974, IN DAYS

17

FIGURE 17.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego Aqueduct during the

first 4 days of the fourth verification.
22 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

16 20 21 

APRIL 1974, IN DAYS

FIGURE 18.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego

Aqueduct during the first 4 days of the fifth verification.

The annual evaporation rate for the canal was estimated to be 2.08 m. This number is significantly 

larger than the estimated lake evaporation for the region of 1.32 m (Kohler and others, 1959) and slightly 

greater than the regional estimate for evaporation 

from a class A evaporation pan of 1.78 m. On the 

other hand, the average pan evaporation during the 

study, from class A evaporation pans maintained at 

each end of the canal by the Metropolitan Water District, is shown, in table 5 along with the computed

TABLE 5.—Evaporation rates for the San Diego Aqueduct during the 

study period, July 25, 1973, to July 24, 1974

Evaporation rate 

in millimeters per day

Month

January

February

March

April _ _

May

June

July1 _________

August

September

October

November

December _

Total ____

Canal

____ 1.99

___ 2.19

_ 2.47

__ 5.19

____ 6.62

____ 9.31

____ 9.58

____ 7.96

____ 7.89

____ 7.41

__ 4.48

_— 3.39

NOTE: — 1,000 m3 = 0.811 acre feet. 

'Represents composite of 1973 and

Class A 

pan

2.57 

4.30 

3.45 

6.90 

6.97 

9.65 

10.65 

9.80

6.72 

3.35

2.74

1974 data.

Pan 

coefficient

0.77 

.51 

.72 

.75 

.95 

.96 

.90 

.81

1.10 

1.34 

1.24

Water loss by 

the Canal, in 

thousands of 

cubic meters

15.4 

4.3 

19.8 

43.7 

55.3 

78.2 

81.6 

59.7 

57.6 

55.8 

31.0 

22.4

.___ 524.8

canal evaporation and a monthly pan coefficient. Assuming a pan evaporation of 8.3 mm/d during September, the annual pan loss was observed to be 2.29 m 

giving an annual observed pan coefficient of 0.91.

Comparing the computed canal evaporation with 

the observed pan evaporation values, one again 

reaches the conclusion that evaporation rate from the 

San Diego Aqueduct is a little greater than reservoir 

evaporation for the same area.

The evaporation rates estimated above were converted to water use by multiplying the average evaporation rate for each day by the effective surface area 

of the canal for that day. The surface area was computed from the mean daily stage values in conjunction 

with the shape of the canal and the data contained in 

table 1. The monthly water losses are also shown in 

table 5. The water loss for February is very low because the canal was empty during much of the month. 

Flow was continuous for all other months. The July 

value represents the loss during July 26-31, 1973, 

plus the loss during July 1-22, 1974, prorated to 31 

days. Even during June, July, and August, the 

evaporative loss represents less than 0.2 percent of 

the total flow.

SUMMARY AND CONCLUSIONS

A one-dimensional temperature model has been developed by use of flow and meteorologic data collected
SUMMARY AND CONCLUSIONS 23

25

24

22

21

20

19

I ' I ' I ' I 

EXPLANATION

Modeled 

Observed 

Observed upstream

10 11 12 

MAY 1974, IN DAYS

13

FIGURE 19.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego

Aqueduct during the first 4 days of the sixth verification.

on the San Diego Aqueduct in southern California. 

Analyzing the problem from both the Eulerian and 

Lagrangian point of view, it was possible to determine 

the two coefficients in a Dalton-type evaporation formula (eq. 21) by balancing the energy budget of the 

canal. The model was calibrated, that is, the 

coefficients were determined, using 28 days of data 

and verified with 113 different days of data.

During all seasons of the year, equation 21 appears 

to realistically represent the evaporation from the 

canal. Equation 21 is believed to be the first wind 

function ever derived from the thermal balance of an 

open channel. The derived wind function for the San 

Diego Aqueduct indicates a larger evaporation at low 

windspeeds than would be indicated by most lake 

evaporation formulas, but the mass-transfer coefficient is within the range of values commonly reported. An annual canal evaporation of 2.08 m is indicated; this figure is about 91 percent of the amount of 

water evaporated annually from nearby class A evaporation pans.

REFERENCES

Anderson, E. R., 1954, Energy budget studies, in Water-loss investigations: Lake Heftier studies, Technical Report: U.S. Geological Survey Professional Paper 269, p. 71-120.

Brown, G. W., 1969, Predicting temperatures of small streams:

Water Resources Research, v. 5, no. 1, p. 68-75.

Carslaw, H. S., and Jaeger, J. C,, 1959, Conduction of heat in solids 

(2d ed.): New York, Oxford University Press, p. 104.

Dingman, S. L. and Weeks, W. F., 1970, Temperature and ice distribution in the North Saskatchewan River below the Edmonton generating plant: Cold Regions Research and Engineering 

Laboratory, Hanover, N.H., Special Report 152, 31 p.

Fischer, Hugo B., 1973, Longitudinal dispersion and turbulent mixing in open channel flow: Annual Review of Fluid Mechanics, p. 

59-78.

Fread, D. L., 1974, Numerical properties of implicit four-point finite 

difference equations on unsteady flow: National Oceanic and 

Atmospheric Administrative Technical Memorandum NWS 

HYDRO-18, 38 p.

Harbeck, G. E., Jr., 1962, A practical field technique for measuring 

reservoir evaporation utilizing mass-transfer theory: U.S. 

Geological Survey Professional Paper 272-E, p. E101-E105.

Jobson, Harvey E., 1973, The dissipation of excess heat from water 

systems: American Society of Civil Engineers, Journal of the 

Power Division, v. 99, no. P01, p. 89-103.

Jobson, Harvey E., and Sturrock, A. M., Jr., 1976, Comprehensive 

monitoring of meteorology, hydraulics and thermal regime of 

the San Diego Aqueduct, California: U.S. Geological Survey 

Open-File Report 76-628, 102 p.

Jobson, Harvey E. and Yotsukura, Nobuhiro, 1973, Mechanics of 

heat transfer in nonstratified open-channel flows, in Shen, 

Hsieh Wen, ed., Environmental impact on rivers (River 

mechanics III): Fort Coiling, Colo., Hsieh Wen Shen, 67 p.

Jones, E. J., 1965, Temperature in California streams; part I, 

evaluation of thermograph records: U.S. Geological Survey 

open-file report, 31 p.

Koberg, Gordon, E., 1964, Methods to compute long-wave radiation 

from the atmosphere and reflected solar radiation from a water
24 THERMAL MODELING OF FLOW IN THE SAN DIEGO AQUEDUCT, CALIFORNIA

28

27

D 

CO

W O OR^"

CO

111 

111 

cc 

O

HI

2 25

cc

D

<

cc 24

111 *-~

23

22

EXPLANATION

Modeled 

Observed 

Observed upstream

789 

JULY 1974. IN DAYS

FIGURE 20.—Comparison of the modeled and observed temperatures at the downstream end of the San Diego

Aqueduct during the first 4 days of the seventh verification.

Linsley, R. K., Jr., Kohler, M. A., and Paulhus, J. L. H., 1958, Hydrology for engineers: New York, McGraw-Hill, 98 p.

Plukowski, E. J., 1970, Urbanization and its effect on the temperature of the streams on Long Island, New York: U.S. Geological 

Survey Professional Paper 627-D, 110 p.

Rawson, J., 1970, Reconnaissance of water temperature of selected 

streams in southeastern Texas: Texas Water Board, Austin, 

Tex., Report 105, 12 p.

Ryan, P. J., and Stolzenbach, K. D., 1972, Environmental heat 

transfer, in Engineering aspects of heat disposal from power 

generation: Mass. Institute of Technology, Cambridge, Ralph 

M. Parsons Laboratory for Water Resources and Hydrodynamics, summer session, 75 p.

Stone, H. L., and Brian, P. L. T., 1963, Numerical solution ot convective transport problems: American Institute of Chemical 

Engineers Journal, v. 9, no. 5., p. 681-688.

Tennessee Valley Authority, 1972, Heat and mass transfer between 

a water surface and the atmosphere: Water Resources Research 

Laboratory Report 14, Norris, Tenn., 147 p.

Troxell, George Earl, and Davis, Harmer E., 1956, Composition and 

Properties of Concrete: New York, McGraw-Hill, 334 p.

Yotsukura, N. B., Fischer, H. B., and Sayre, W. W., 1970, Measurement of mixing characteristics of the Mississippi River between Sioux City, Iowa and Plattsmouth, Nebraska: U.S. 

Geological Survey Water-Supply Paper 1899-G, 29 p.

z> 10

*<

UJ Q 

I— p

< cc 8

o:£

Z CO fi

0 CC b

h-£

2S! 4

0 -

Q. _l

< _i 2

Si

n

—

—

—

-

—

-

—

-

—

—

—

-

—

-

—

-

M M J J 

MONTH

O N

FIGURE 21.—Computed monthly average daily evaporation from 

the San Diego Aqueduct during the study period, July 25, 1973, 

to July 24, 1974.

surface: U.S. Geological Survey Professional Paper 272-F, p. 

F107-F136.

Kohler, M. A., Nordenson, T. J., and Baker, D. R., 1959, Evaporation maps of the United States: U.S. Weather Bureau Technical 

Paper 37, Washington, D.C., 13 p.