MANAGEMENT AND MEASUREMENT OF "DO" IN IMPOUNDMENTS
James M. Symons, William H. Irwin, Robert M. Clark, and Gordon G. Robeck
U. S. Department of the Interior
Federal Water Pollution Control Administration
Cincinnati, Ohio 45226
September 1966
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MANAGEMENT AND MEASUREMENT OF "DO" IN IMPOUNDMENTS
James M. Symons, William H. Irwin, Robert M. Clark, and Gordon G. Robeck
This paper develops a basis for calculating the hydraulic
efficiency of any mechanical destratification device by the use of
the stability concept and a mathematical model, including a solution,
for studying the DO budget. Also, the influence of mixing on reduced
substances and DO resources in stratified impoundments is demonstrated.
INFORMATION ABSTRACT
This paper recommends a basis for the comparison of various
mechanical devices used for artificially destratifying impoundments by
calculating their hydraulic efficiency based on impoundment stability.
Three problems with this calculation are discussed to avoid misinter-
pretation, influence of size and geometry on stability, natural changes
in stability with time, and the basic hydraulic inefficiency of spring
mixing if cold water is raised. Also, a mathematical model is developed
for studying the DO budget in impoundments. A solution to the model is
presented, using specially designed DO probe equipment. Finally,
data is presented on the destratification of a 2930-ac-ft impoundment
in northern Kentucky by pumping. The hydraulic efficiency of the pump
is calculated and the influence of the mixing on the reduced substances
and the DO resources in the impoundment is shown. The most favorable
time for destratification is also discussed. A bibliography on artificial
destratification is included.
KEY WORDS
Impoundments, Stratification, Destratification, Water Quality, Stability,
Mechanical Pumping, DO Resources
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MANAGEMENT AND MEASUREMENT OF "DO" IN IMPOUNDMENTS8
1 23
James M. Symons , M.ASCE, William H. Irwin , Robert M. Clark , A.M. ASCE
4
and Gordon G. Robeck , F. ASCE
INTRODUCTION
The problem of thermal stratification in storage reservoirs is
widespread throughout the United States. Considerable attention has
been given to developing engineering methods that will prevent downstream
problems from the discharge of poor quality bottom water from impoundments,
particularly during peaking power operations. Several possible design
features that might be incorporated into dams and outlet works to
prevent the escape of poor quality water from impoundments have been
extensively reviewed by Kittrell . One of these techniques, artificial
destratification, has been researched during the past two years by
staff members of the Cincinnati Water Research Laboratory. The first
a - Presented at the ASCE National Symposium on Quality Standards for
Natural Waters, Ann Arbor, Michigan, July 19-22, 1966.
Submitted for publication in the Proceedings of the Symposium and in the
Journal Sanitary Engineering Division of the American Society of Civil Engineers,
1. In Charge, Impoundment Behavior Studies, Engineering Activities,
Research and Development, Cincinnati Water Research Laboratory, FWPCA
Cincinnati, Ohio
2. Aquatic Biologist, Impoundment Behavior Studies, Engineering Activities,
Research and Development, Cincinnati Water Research Laboratory, FWPCA
Cincinnati, Ohio.
3. Applied Mathematician, Research and Development, Cincinnati Water
Research Laboratory, FWPCA, Cincinnati, Ohio.
4. Chief, Engineering Activities, Research and Development, Cincinnati
Water Research Laboratory, FWPCA, Cincinnati, Ohio
This work was performed while the authors were members of the Division
of Water Supply and Pollution Control, Public Health Service, Department
of Health, Education, and Welfare.
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publication on this subject , discussed the theoretical aspects of
artificially breaking the thermal stratification of an impoundment,
and presented early experimental results. The second paper dealt with
a larger field trial, and extensive data were presented to demonstrate
the improvement in water quality that could be affected by pumping cold
water from the bottom of a stratified impoundment and discharging it
in the surface, thereby creating an overturn in the body of water.
Although to date (1966) the method has only been tried on
relatively small bodies of water, 3,000-4,000 ac-ft, artificial
destratification is an effective method for water quality improvement.
This technique is doubly attractive since it improves the water quality
both in the reservoir itself as well as protecting downstream quality.
This can be contrasted to some of the techniques discussed by Kittrell
that only prevent poor quality water from escaping an impoundment, and
do not prevent the formation of poor quality water in the impoundment
itself.
Since research on this process will continue, one purpose of
this article, the third in the series, is to establish a firm theoretical
basis for comparing the hydraulic efficiencies of various pieces of
equipment that might be used for artificially destratifying a body of
water. Secondly, although dissolved oxygen (DO) is not the only
quality parameter in natural waters, it certainly is an important one.
Therefore, the other purpose of this article is to examine the influence
that artificial destratification has on the DO resources in an impoundment
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from four aspects; a) the elimination of reduced substances,
b) the addition and redistribution of DO, c) when artificial
destratification should be used for quality control during a summer
season, and d) a description of a method for evaluating DO resources
in impoundments. In addition, a complete bibliography on the subject
of artificial destratification is included.
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- 4 -
THEORETICAL CONSIDERATIONS
Management of DO in Impoundments
Hydraulic Efficiency
There are four major methods currently (1966) available for
67 *
artificial destratification, mechanical pumping ' , the Air-Aqua
8 9 **10
system , compressed air, and the Aero-Hydraulics Gun . If these
four systems are to be compared, their hydraulic performance must be
judged on an equal basis, through the calculation of their hydraulic
efficiency. The hydraulic efficiency of any destratification device
is evaluated by comparing the actual energy input necessary to cause
overturn in a given body of water, to the theoretical minimum energy
required to overturn that body of water. The theoretical minimum
energy required for artificial destratification is defined as the
stability of the body of water.
The stability of stratified impoundments has been discussed by
9 11
Koberg and Ford and by Hutchinson , and can be calculated according
to the following definition - the work or the energy required to lift
the weight of the entire body of water the vertical distance between
the center of gravity of the water mass when isothermal and the mass
center of gravity when the impoundment is thermally stratified. For the
purpose of this paper the center of gravity is taken as a point anywhere
on a horizontal plane that passes through the true mass center of gravity
Figure 1 schematically illustrates the stability concept. In the
upper portion the temperature profile in a column of water, taken as
a rectangular slice from a wedge-shaped body of water, is shown at
three different months of a year. In the first column, the water is
*Product of Hinde Engineering Company, Highland Park, Illinois.
**Product of Aero-Hydraulics Corporation, Montreal, Canada.
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- 5 -
WIND
HEAT
HEAT
HEAT
ISO-
THERMAL
C.E. G.R.
x
H
Q_
UJ
Q
o
cc
UJ
liJ
I 82
FEB
JUNE
TEMPERATURE
AUG
Work of the wind
11B"
TIME
Figure 1. Schematic diagram to demonstrate stability
and work of the wind.
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- 6 -
isothermal, and the center of gravity along the vertical axis is
approximately 1/3 the depth of the water column. The isothermal center
of gravity does not change throughout the summer season. In the
subsequent discussion movement of the position of the center of gravity
is considered along the vertical axis only.
As the water column is heated from the surface, two different
temperature profiles, labeled "1" and "2", are shown for June and August.
Profile "1" in June, dotted, would occur if there were no wind blowing
across the surface of the water. In this condition, most of the incident
heat would be stored in the upper layers of the water, making this water
lighter, relative to the remainder of the water column. This decrease
in weight would lower the center of gravity of the stratified water
column to the position shown by CG.. . As heating continues into August,
thermal stratification becomes more intense, and the center of gravity
moves down into the water column even farther. Since the actual center
of gravity continues to lower throughout the heating season, the energy
required to mix the water column, which is evaluated by multiplying the
weight of the water column by the distance between the isothermal center
of gravity and CG1 at any given time, continually increases with time.
This is shown in the lower portion of Figure 1, by the curve labeled
"condition I."
In reality, wind is always present in any natural situation.
The influence of wind blowing across the surface of the water column
changes the temperature profile from the condition labeled "1" to the
condition labeled "2". The surface of the water column does not become as
warm, and the warm water extends farther down into the water column.
At any given time, therefore, the center of gravity of the stratified
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- 7 -
water column does not move down to the level of CG.. , but only descends
to the level shown In Figure 1 as CG2- Since wind has the effect of
preventing stratification from being as intense as without wind, the
actual energy necessary to mix the water column is considerably
reduced because of wind action. This is shown in the lower portion
of Figure 1, where the "condition 2" curve is considerably below that
of the "condition 1," curve. The distance between these two curves
•11
has been definied as "work of the wind."
In summary, at any given time, the energy that is represented
by the distance "G" in Figure 1, is the energy that would have been
required to mix the stratified water column in the absence of any
wind. Since the wind has put into the water the quantity of work
indicated by dimension "B", only the remainder, the stability, "S",
must be supplied by any mechanical device, to make the water column
isothermal. In any natural situation, the stability can be arithmetically
a
calculated . This theoretical minimum energy requirement for
destratification, when compared with the energy input of a given
mechanical device that caused destratification, defines the hydraulic
efficiency of the mechanical equipment.
The stability curve, "S" in the lower portion of Figure 1 also
shows that stability reaches a maximum in an impoundment during the
middle of the heating season, usually in late July, and then begins
to decline. The maximum stability in a body of water occurs when the
warm water has penetrated to the depth of the isothermal center of
gravity along the vertical axis. As the water warms below this depth,
and the water body approaches an isothermally warm condition, the
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distance decreases between the isothermal and the stratified centers
of gravity, and the stability declines, reaching zero when the water
body is again isothermal, but warm.
Problems with the Stability Concept
While stability is a useful method of calculating the hydraulic
efficiency of an artificial destratification device, it must be used
carefully to avoid misinterpretation. Three problems will be discussed.
The first is the effect of size and geometry on the calculation of
stability. Table 1 summarizes data from three lakes near Cincinnati,
Ohio, which are under study. Column 2 through 6 lists the pertinent
physical data for these lakes. The stability of each lake, column 7,
was calculated assuming the same temperature profile in each lake,
20°C for the upper 5 feet of water, and remainder of the water body at
4°C. As seen in column 7, the stability of these three lakes is different
Hutchinson suggested that stability should be calculated per
unit of lake surface, to eliminate differances of size and geometry.
Column 8 shows, however, this is not a reliable method of calculating
a comparable parameter. At least for these three lakes, the stability
per unit volume, column 9, is a better method of obtaining a parameter
that can be compared from lake to lake. Table 1 demonstrates that
the stability calculated for one body of water cannot be assumed to
exist in other lakes and reservoirs in the same general geographic
location. To properly evaluate the hydraulic efficiency of a mixing
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Table 1
COMPARISON OF STABILITY IN LAKES WITH
THE SAME ASSUMED TEMPERATURE PROFILE BUT DIFFERENT SIZE AND GEOMETRY
Lake
(1)
Falmouth
Bullock Pen
Boltz
Surface
ac
(2)
225
142
93
Greatest
Depth
ft
(3)
42
44
62
Avg.
Depth
ft
(4)
20.5
22.8
31.5
Vertical
Isothermal
Center
of Gravity
ft
(5)
14.232
15.059
20.867
Volume
ac-ft
(6)
4605
3237
2930
Stability
kw-hr
(7)
21.8
15.5
15.0
Stability
area
w-hr
ac
(8)
96.9
109.2
161.3
Stability
vol
w-hr
ac-ft
(9)
4.7
4.8
5.2
i f
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operation in a certain impoundment, the stability for that particular
impoundment must be calculated. Table 1 also illustrates the difficulty
in using a second nearby impoundment as a control for comparison purposes
in an artificial destratification study. Differences in size and geometry
between the two impoundments may prevent the second from being used as
an "exact" control.
Second, the continuing natural change that the stability of a given
body of water undergoes throughout a summer season is another difficulty
with its use for the calculation of hydraulic efficiency. As shown in the
lower portion of Figure 1, as stratification begins, the stability rises
from zero, reaches a maximum in late July, and declines towards zero
again, because of continued wind-mixing and nighttime cooling in August
and September. Therefore, if the stability in a given impoundment were
calculated when a mixing operating began, the natural change in stability
occurring during the time of the mixing operation, must be included in
any proper evaluation of hydraulic efficiency.
If a destratification operation should take place in the spring,
and should require a month to complete, the actual energy input during
mixing should be compared with the calculated stability at the start of
the operation, plus the natural increase in stability that would have
taken place during the month. During a fall mixing operation, the natural
decline in stability with time should be subtracted from the initial
stability, in order to properly evaluate the hydraulic efficiency of
the overturn operation.
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Third, the hydraulic inefficiency of a spring destratification
operation, in which cold vater is pumped from the bottom to the surface,
must be recognized. This is illustrated by the data in Table 2.
Here, temperature profiles were assumed for four different months
during the year and the stability calculated for each of the profiles,
(see Column 3). The energy that must be added to the lake for mixing is
low in the spring and fall, and high in mid-summer.
If destratification is to be carried out by bringing cold water
from the bottom as a heat sink to absorb heat from the upper layers
and thereby make the lake isothermal, the volume that must be moved
is the volume of cold water that is present at the start of the operation.
For this example, these volumes are shown in Column 4, Table 2.
Although the volume of cold water that must be pumped in the
spring is high, the work requirements are not, since the elevation
12
head through which the water must be lifted, based on density difference ,
is much smaller in the spring than in the summer or fall. Practically,
however, mixing equipment cannot take advantage of this low elevation
head in the spring and so the actual energy input at any time is
directly proportional to the volume of water that must be moved,
Column 4, Table 2. Therefore, although theoretical energy requirements
an
are low in the spring,/hydraulically inefficient operation results
because of practical considerations. The improvement in hydraulic
efficiency as the summer season progresses is again shown in Column 5,
Table 2 where the volume required to mix the lake is calculated per
unit of original stability.
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- 12 -
Table 2
VARIATION IN PUMPING REQUIREMENTS TO DESTRATIFY BULLOCK PEN LAKE AT
DIFFERENT SEASONS
Month
(1)
April
Assumed
Temperature
Profile
(2)
0-5 ft - 20°C
Stability
kw-hr
(3)
13.8
Cold Water
Volume
Required
to mix
Lake ac-ft
(4)
2560
Cold Water
Volume Required
to mix the
Lake per Unit
Stability
ac-ft /kw-hr
(5)
180
June
August
September
5-45 ft - 9°C
0-15 ft - 23°C
15-45 ft - 9°C
0-25 ft - 25°C
25-45 ft - 10°C
0-40 ft - 25°C
40-45 ft - 10°C
32.9
28.2
2.5
1440
640
35
43
23
15
Note: Data calculated for Bullock Pen Lake, see Table 1. for physical data.
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- 13 -
The practical inefficiency of destratification in the spring could
be overcome by pumping the smaller volume of warm water down into the
colder layers. The theoretical energy requirements would be the same,
since pumping cold water up involves moving a large volume through a
small elevation head while pumping warm water down would result in
moving a small volume through a larger elevation head. As noted before,
however, most mechanical equipment cannot take advantage of the elevation
head differences, therefore the actual input energy is directly proportional
to the water volume moved. This would make pumping warm water down in
the spring a hydraulically more effecient operation.
This variation in pumping requirements with changing seasons
does not eliminate the usefulness of the stability concept for calculating
hydraulic efficiency, but the discussion above should indicate that
a low calculated hydraulic efficiency may occur during a spring destratification
operation and should also indicate that different season comparisons
are unfair.
Another practical consideration for any mixing operation is the
change in hydraulic efficiency as the mixing nears completion. In
9
Figure 2 from the work of Koberg and Ford , the rate of change in
stability per unit of time decreases as the stability approaches zero.
There are at least three reasons for this. First, since so much of the
water in most lakes is in the upper portions, a very slight temperature
gradient can produce some stability. Practically, complete elimination
of stability, particularly in the spring may be impossible. Second,
as mixing nears completion the density of the body of water is nearly
uniform. Under these conditions some of the water that has been raised
to the surface may sink to the bottom and be repumped. This would lead
to excessive energy inputs to complete the job.
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- 14 -
Finally, as a body of water is cooled to near its final
temperature, there is a smaller decrease in stability per unit volume
of water pumped. This is caused by the non-linear change in water
12
density with changing temperature. In the example shown in Table 2,
for the April data, the first 155 ac-ft of bottom water pumped up
decreases the stability 2.3 kw-hrs, while the last 155 ac-ft of water
mixed only eliminates 0.8 kw-hrs of stability. The higher the
average (final) temperature of the water, the less pronounced this
effect would be, as the density change of water tends to be linear
over small temperature ranges. Keeping a continuous record of
stability during a destratification operation is recommended to avoid
excessive energy inputs in the late stages of the mixing.
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MIXING TIME, hours
Figure 2. Decrease in stability with application of compressed
a ir (Reference 9).
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Measurement of DO Budget in Impoundments
Mathematical Model
One common mathematical model used to describe the change
of DO concentrations with time in the euphotic zone of impoundments,
where sedimentation is not an important factor, is shown in Equation (1) ,
13
(see O'Connell and Thomas .)
*- k (L -y) - k (D) + (R-P) (1)
cl
where D = DO deficit at any time
t = time
k- = Deoxygenation rate constant
L = Total organic biochemical oxygen demand
a
y = Biochemical oxygen demand satisfied at any time
(y-L [1-exp flc.t)]}
a. J-
k0 = Atmospheric rearation rate constant
R = Rate of oxygen demand by the algal population
P = Rate of oxygen produced by the algal population
Integration yields Equation (2).
l. [expC-^t) - exp(-kzt)]+ - [l-exp(-k2t)] + DjexpC-k^) ] (2)
where D = Initial DO deficit.
a
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- 17 -
14
Although Hull's suggestion of measuring all the DO sources
and sinks in a given impoundment to evaluate the DO budget is valid,
this is often time-consuming and evaluates the budget only under one
set of environmental conditions. If a reliable model can be developed,
the changes in the constants, as related to changing sunlight, wind,
algal counts, and nutrient level, will permit general conclusions to
be drawn, permit predictions to be made, and permit comparisons of
one impoundment to another.
Through the years, research efforts have attempted to devise
methods to evaluate and understand the physical meaning of the five
constants (k-, k_, L , R, and P) in Equation (2). While excessive
J. f- Q-
concern over the precise meaning of these constants has, with some
14 15
justification, been criticized by Hull and Ettinger , this does not
negate the utility of the overall model. The important feature of
any mathematical model is whether or not its results accurately
reproduce what is occurring in nature. The model shown in Equation (2)
may not be an exact representation of the complete DO budget in an
impoundment, but, as will be shown in Figure 10, the change in "D"
with time derived from the solution of the model, and the change in
the dissolved oxygen deficit with time in the natural environment are
similar. Therefore, the model is acceptable for further use.
Evaluation of the Model by the DO Probe Method
There are many different methods of studying the DO budget in
natural waters, for example, Odum - diurnal cycle method, Hull
19
- "light" and "dark" - bottle method, Goldman and Carter - carbon-14
20
method, and Verduin - changes in C02 concentration in
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- 18 -
unconfined water method. Each method has advantages and drawbacks.
The method developed in this study, called the DO probe method, using
special equipment has three distinct advantages; (a) the inaccuracies
in bottle experiments are minimized, (b) DO is measured directly
and (c) the model, Equation (2), may be solved from data gathered.
Future study will undoubtedly reveal further improvements in this
approach, but this is a significant advance.
Details of the design and operation of this equipment, and a
discussion of its improvements over the bottle technique are contained
in reference 21, and shall be noted only briefly here. The apparatus
consists of three DO probes that will record the changes in DO
concentration with time under three different conditions. One probe
is placed in a black container that prevents the admission of either
sunlight or air. Under these conditions the constants "k2", and "P"
are zero in Equation (1). Integrating Equation (1) under these
conditions yields Equation (3).
D = Lfl [l-exp(-k1t)] + R(t) + Dfl (3)
A second DO probe is in a clear plastic container that admits
sunlight, but not air. Under these conditions, Equation (1) can be
integrated with "k" = 0. This integration yields Equation (4).
D = L [l-exp(-k t)] + (R-P)(t) + D (4)
a 1 a
The third DO probe is unconfined, therefore the change in DO
deficit with time is represented by Equation (1), which when
integrated yields Equation (2).
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- 19 -
Evaluation of the five constants in the model is accomplished
by solving Equation (3) for "In " t "R", and "La" using DO vs. time
data obtained from the DO probe in the dark container. These constants
are then entered in Equation (4) and "P" is evaluated through the use
of the DO vs. time data taken from the light container DO probe.
Finally, Equation (2) is solved from the DO data taken with the probe
that is unconfined, and the previously developed constants.
For those interested in the details of the solution, an example
is included in the Appendix. The application of this method, using
21
data from the equipment developed for this purpose , will be demonstrated
in the results section of this paper.
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- 20 -
EQUIPMENT AND METHODS
Management of DO in Impoundments
The equipment and methods used in this study have been explained
In detail in references 7 and 21 and shall be briefly summarized
here. Artificial destratification was studied from August 6, to
September 10, 1965 in Boltz Lake, a 2,930 ac-ft impoundment near
Cincinnati, Ohio. Destratification was accomplished by pumping
cold water from within 2 feet of the bottom and discharging it in
the surface with a 13 ac-ft/day capacity, raft-mounted pump, driven by
a 15.5 kw gasoline engine. Analysis of temperature, and DO, sulfide,
manganese, and total COD concentrations at various depths, weekly
from May 14 to October 29, 1965 demonstrated the changes in stability
and DO resources that occurred in this impoundment before, during,
and after the pumping operation. A nearby, unmixed lake, Bullock
Pen, served as the control. Physical details of both lakes are
summarized in Table 1.
Measurement of DO Budget in Impoundments
The ability of the special DO probe equipment to produce data
necessary for the solution of Equations (2), (3), and (4) was demonstrated
by making studies in a 750-gal. tank filled with artificial impoundment
water. The simulated impoundment water was created by mixing some
settled sewage, nitrogen, phosphorus, sodium bicarbonate, and algal
seed with Cincinnati, Ohio tap water. This mixture was stirred 15
minutes each hour and illuminated 14 hours each day until an observable
algal growth developed. The day of the test, some sewage was added
as a fresh source of organic material. The three DO probes, with
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- 21 -
open trap doors on the containers were lowered into the tank, the
triggers were tripped, and the test begun. The DO concentration
data was obtained for an 8-hour period, under illumination, using
the recording and timing circuitry detailed in reference 21. Before
the test began, the DO probes were calibrated and adjusted against
the Winkter method for DO analysis. The temperature of the test
was fairly constant at 20°C.
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- 22 -
RESULTS
Management of DO in Impoundments
Hydraulic Efficiency
The approximate magnitude of the work of the wind in the control
lake, Bullock Pen, during the 1965 season is shown in Figure 3. The
"G" energy was approximated by assuming that all of the heat incident
to the lake was concentrated in the upper 5 feet of the water. The
"G" energy is much higher than the stability, the lower curve,
indicating that, at least for this impoundment, the wind has applied
much more energy for mixing than would mechanical equipment.
Figure 4 compares the stability pattern for the test and the
control lakes. The control lake, Bullock Pen, shows the typical
stability pattern, reaching a maximum stability of 45 kw-hrs in
mid-July. At this time the calculated vertical distance between the
isothermal and actual centers of gravity was 0.17 inches. The test
lake, Boltz Lake, also reaches its maximum stability about the same
time, but the calculated stability for Boltz Lake is somewhat above
that for the control throughout the unmixed period because of
differences in size and geometry. In spite of this slight
inconsistency, Bullock Pen Lake was assumed to be a proper control
for this operation.
Artificial destratification of Boltz Lake was started on
August 6, 1965 when the pumping equipment became available. As shown
in Figure 4, through the 5 weeks of pumping, the stability in
Boltz Lake dropped more sharply than the natural decline of the control.
After the 5 weeks of pumping, the calculated stability in the test
-------
200
Work of the wind- B
STABILITY- "S"
MAY
JUNE
JULY AUG
1965
SEPT
OCT
Isi
I
Figure 3. Estimation of the work of the wind in Bullock Pen
Lake (Control lake).
-------
50
40
. 30
CD
<
20
TEST LAKE
CONTROL LAKE
NET A "S"
18 kw-hrs
i
N>
JULY AUG.
1965
APR. MAY JUNE
Figure 4. Stability in test and control lakes.
SEPT
OCT
NOV.
-------
- 25 -
lake was 18 kw-hr lower than the control at the same time. Therefore,
the hydraulic efficiency of this particular piece of mixing equipment,
calculated according to Equation (5) was a rather low 18/14,300 = 0.13%.
Net change in stability „ . ... ... J
Total energy input = «ydraulic efficiency (5)
Elimination of Reduced Substances
In Figure 5, the total quantity of sulfide, in the control lake,
calculated as the equivalent oxygen demand rises throughout the summer,
reaching a peak in mid-September. After this time, the natural
overturn of the lake provides DO throughout the depth, thus oxidizing
and eliminating the sulfides. The test lake rises similarly to the
control prior to the artificial destratification operation. As
pumping provided DO throughout the depth of the test lake, the sulfides
were oxidized and their quantity lowered, as shown in the dotted portion
of the test lake curve. Assuming that the test lake would have risen
to a level approximately equal to that of the control lake on September 10,
the net change in the sulfide content caused by pumping was 13,000
pounds, as the oxygen equivalent. The oxygen equivalent of the sulfide
was calculated as twice its concentration, since two moles of oxygen
are required to oxidize one mole of sulfide.
A similar, but smaller, change is shown in the manganese oxygen
demand in Figure 6. For this analysis, all of the manganese measured
in both lakes was taken to be soluble, divalent manganese and the oxygen
equivalent of the manganese was calculated as 0.29 of its concentration
since one-half mole of oxygen is required to oxidize one mole of
manganous manganese. The net change in the manganese oxygen equivalent
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- 26 -
between the control and test lakes at the end of the pumping operation,
September 10, was 1,300 pounds of oxygen equivalent. As with the
sulfides, the oxidation of the manganese in the test lake was
accomplished by the destratification providing DO to the lower
level of the lake.
The total COD content of the test and control lake was also
calculated for each week during the study period. These data were
more scattered than those shown in Figures 5 and 6, however, and no
significant reduction in the total COD content in the test lake could
be noted during the pumping period. This is probably because the
degradable portion of the total COD was small, so even though the
addition of DO to the lake enhanced the biodegradation of this portion
of the organic material, the total COD measurement probably was too
insensitive to detect any biodegradation.
In summary, the artificial destratification operations on
Boltz Lake, from August 6 through September 10, 1965 supplied a total
of 14,300 pounds of DO, which satisfied an equivalent quantity of
inorganic oxygen demand.
Addition and Redistribution of DO
The decline in the total quantity of DO in both lakes through
the early part of the summer season, because of continuing oxygen
demand in the water, is shown in Figure 7. When pumping began in
the test lake, the quantity of DO rose, but at this same time, because
of continued wind-mixing, additional DO was appearing in the control lake.
-------
20
3
O
16
UJ
Q
Z
LU
O
>-
X
UJ
Q
L
_J
I
12
8
o
CONTROL LAKE
i
!
APR
MAY
JUNE
JULY AUG.
1965
SEPT.
OCT.
NOV.
Figure 5. Total sulfides as oxygen equivalent in test and control lakes,
-------
-Q
in
3
O
e
Ill
•>
§2
CONTROL LAKE
o
PUMPING
' -'
00
APR.
MAY
JUNE
JULY AUG.
1965
SEPT.
OCT.
NOV.
Figure 6. Total manganese as oxygen equivalent in test and control lakes.
-------
- 29 -
Based on these data, although 14,300 pounds of DO were supplied to the
test lake (Figures 5 and 6) the pumping capacity of the equipment was
not sufficient to add the desired quantity of DO above the inorganic
oxygen demand. The oxygen demand caused by the natural fall overturn
in late September is also shown in Figure 7. This occurred even in
the test lake, which restratified slightly after the pumping operation
stopped.
Although the desired quantities of DO beyond the inorganic oxygen
demand were not added to Boltz Lake, Figure 8 shows the redistribution
of the existing DO resources caused by the pumping operation. Here,
DO, absent below 20 feet before pumping began, was present, although
in low concentrations, at all depths after pumping. Since the goal
of any destratification operation is to provide reasonably high levels
of DO at all depths, Figure 8 indicates further the undercapacity of
this particular piece of equipment. It does show in general, however,
that a favorable redistribution of DO resources is accomplished by
artificial destratification.
Optimum Time for Destratification
Figures 5, 6, and 7 show a general deterioration in water quality
from May to August. The total quantity of DO is lowered in the lakes,
and the quantity of sulfide and manganese rises. While the mixing
operation improved the water quality markedly in August, the goal of
managing the DO resources in an impoundment through artificial destratification
is to maintain quality throughout the summer season. The Boltz Lake
studies were planned to begin in spring, in an effort to prevent
any water quality deterioration, but the lake was actually mixed in
August because the mixing equipment was not ready for operation.
Early season mixing is currently (1966) under study.
-------
CONTROL LAKE
' .
I
APR.
MAY
JUNE
JULY AUG.
1965
SEPT.
OCT.
NOV.
Figure 7. Total quantity of dissolved oxygen in test and control lakes.
-------
- 31 -
BEFORE PUMPING
10
20
JE 30
o.
LL)
Q
40
501
60
AFTER 38 PUMPING DAYS
I
1
I
I 2345678
DISSOLVED OXYGEN, mg/l
Figure 8. Redistribution of dissolved oxygen
resources in Boltz Lake during
artificial destratification.
-------
- 32 -
Spring mixing has some hydraulic drawbacks: (a) the practical
inefficiency of raising cold water when the volume of cold water is high
(see Table 2) and (b) a naturally increasing stability with time at this
season of the year, (see the lower portion of Figure 1 and Figure 4).
Quality protection, not hydraulic efficiency, is the goal of this
technique, however, and so some hydraulic sacrifices should be made
in an attempt to prevent the quality deterioration that naturally occurs
from the beginning of the heating season through the time of maximum
stability in late July. Some of this inefficiency could be overcome by
pumping the warm water layer down into the water mass in the spring.
Measurement of DO Budget by the DO Probe Method
The data plotted in Figure 9 was obtained with the special equipment
21
previously described. Some important water quality parameters of the
artificial impoundment water used for the test are noted on Figure 9.
This figure shows the excellent data that may be obtained using this
equipment, note the extremely expanded ordinate scale. Approximate
curves showing the trends have been placed by eye through the data points.
The solutions to Equations (2), (3), and (4) are taken from the
deficit data taken from these curves. Table 3 summarizes the data used for
the solution of the equations.
Using these data and the method in Appendix I the solution for
Equation (2) is:
D = WQ.Q^5 V {exp[(-0.15)(t)] - exp[(-0)(t) ]} + ^~^ {l-exp[ (-0) (t) ]
Da{exp[(-0)(t)]} (6)
-------
3.4O —
I
TEST CONDITIONS
pH - 7.9
ALKALINITY-132 mg/l AS CaC03
COD-l5mg/l
BODS - 0.8 mg/l
VSS-5mg/l
ALGAL COUNT-3xlOVml
TEMPERATURE-2C°C
ILLUMINATED
DARK DISSOLVED OXYGEN
PROBE DATA-USED FOR
EQUATION (3)
V.LIGHT DISSOLVED OXYGEN PROBE
DATA-USED FOR EQUATION (4)
UNCONFINED DISSOLVED OXYGEN
PROBE DATA-USED FOR EQUATION (2)
i •
•
B
TIME,hours
Figure 9. Data from special dissolved oxygen probe equipment.
-------
- 34 -
TABLE 3
DATA FROM FIGURE 9 FOR THE SOLUTION OF THE MATHEMATICAL MODEL
DO Deficit - mg/1
Time
Hours
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Dark Probe
2.82
2.88
2.94
2.99
3.03
3.07
3.11
3.14
3.18
3.21
3.24
3.27
3.30
3.33
3.36
3.39
3.41
Light Probe
2.77
2.82
2.86
2.90
2.94
2.98
3.01
3.04
3.08
3.10
3.13
3.16
3.19
3.21
3.23
3.25
3.27
Unconfined Probe
2.74
2.78
2.81
2.84
2.87
2.90
2.92
2.95
2.97
2.99
3.00
3.01
3.01
3.01
3.00
2.99
2.98
-------
- 35 -
Figure 10 compares the line of best fit from Equation (6) with the
deficit data obtained with the unconfined DO probe. The fairly
close comparison of the data generated from the mathematical model
and the data taken with the DO probe shows the general reliability of
the model. The low values of k 2 and (R-p) are in agreement with the
low turbulence and low algal population in the test media. These data
show that the "DO probe method," using this special equipment, and
the mathematical model represented by Equation (2), is a useful method
for studying the DO budget in impoundments.
-------
U 3
LL
LU
Q
* E
o c
Q
LU
I
SOLUTION FROM EQUATION (6)-
\
UNCONFINED PROBE DATA FROM FIGURE 9
;
I
I
1
A A A
\
0 I 234567
TIME, hours
Figure 10. Comparison of Equation (6) with the unconfined probe data.
U)
••'
8
-------
- 37 -
FUTURE PLANS
Continued study and research are necessary to completely
define the potential of artificial destratification for water quality
control. Comparison of different destratification methods, spring
mixing, and the mixing of larger bodies of water, must be accomplished.
The first two items are currently (1966) under study, a diffused air
system was used, starting in the spring of 1966, for the destratification
of Boltz Lake. This system has the disadvantage that it cannot be
adapted to mixing warm water down in the spring. Eventually, other
destratification methods must be tried, and water quality control by
this technique must be demonstrated in a large storage reservoir.
Finally, the DO probe equipment must be used in the field to gain
experience with the influence of various natural environmental factors
on the constants in the mathematical model, and to determine whether
good correlations between the model and data taken may be obtained
under a variety of conditions.
-------
- 38 -
SUMMARY AND CONCLUSIONS
Management of DO in_ Impoundments
Hydraulic Efficiency
The data presented here, and in reference 6 and 7 have indicated,
that artificial destratification is an effective method of impoundment
water quality improvement, although to date (1966) it has only been
researched on a relatively small scale. Any piece of mechanical
artificial destratification equipment should be as hydraulically
efficient as possible, and this can be evaluated by calculating the
stability of the impoundment to be mixed. The calculated hydraulic
efficiency, "the net change in stability/the total energy input,"
is a good method of comparing one piece of destratification equipment
with another, or evaluating various designs of a given piece of
equipment. The equipment used in this study had a relatively poor
hydraulic efficiency, 0.1370, probably because of poor hydraulic design.
Oxygenation Efficiency
Effective water quality improvement is even more important
than a high hydraulic efficiency, and although the hydraulic efficiency
of this pumping equipment was relatively low, the pump did supply
14,300 pounds of oxygen to the test impoundment with an expenditure
of 14,300 kw-hr of input energy. The calculated oxygenation efficiency
of this equipment was 1 pound of oxygen/kw-hr. This is an acceptable
value, but this equipment was still undersized for Boltz Lake, as
only enough oxygen was provided to satisfy the inorganic oxygen
demand, and very little DO was actually added to the water. In
spite of this, the DO resources in the impoundment were favorably
redistributed, and at least some DO was present at all levels of the
-------
- 39 -
lake after the mixing operation.
Best Time for Destratification
The major advantage of artificial destratification over the
devices for preventing escape of poor quality water from impoundments ,
is that mixing creates a two-fold benefit, protecting downstream users
as well as improving the quality of the water in the impoundment
itself. In order to obtain these benefits, an impoundment should
be destratified in the spring, to prevent the formation of bad
quality water, rather than in mid-summer to improve poor quality water.
Spring mixing is necessary, in spite of the practical hydraulic
inefficiency of raising cold water at this season of the year, and
the necessity to occasionally remix during the summer season.
These hydraulic sacrifices must be made in order to maintain good
quality water throughout the heating season.
Measurement of DO Budget
To properly evaluate the DO budget in impoundments, an adequate
mathematical model and a method of solving the model are needed.
The model presented in Equation (2),
Mk.)
2 1
[l-exp(-k2t)] + DjexpC-k^) ] (2)
21
is adequate, and the special three DO probe system is effective for
determining the five constants. Use of this tool will permit further
study of the DO budget and will permit comparison of the DO budget
from one impoundment to another.
-------
- 40 -
Conclusions
Although research on this project is continuing, the conclusions
that may be drawn at this time are:
1. Engineering methods are available that will prevent the
escape of poor quality bottom water from impoundments.
2. Artificial destratification is a better method of impoundment
water quality control since it has the double benefit of
preventing downstream damage and improving water quality in
the impoundment itself.
3. The proper use of the stability concept for the calculation
of the hydraulic efficiency of a mechanical mixing device is
a useful method of comparing various pieces of equipment and
improving equipment design.
4. In this study, reduced materials, sulfides and manganese,
were eliminated from the test lake by the artificial destratification
operation.
5. The equipment used in this study was insufficient in capacity
to add the desired dissolved oxygen beyond the inorganic oxygen
demand, but the available DO resources in the test lake were
favorably redistributed by the pumping so that DO was present at
all levels.
6. In spite of the practical hydraulic inefficiency of raising
large volumes of cold water, spring mixing is better than mid-
summer destratification because early mixing will prevent any
bottom water quality deterioration.
-------
- 41 -
7. Equation (2)
: iexp^-knt) - exp(-k t)]+ (~1 [ l-exp(-k t)] +D [exp(-k t) ] (2)
K._"'*•-, L *- K2
is a satisfactory mathematical model for representing the DO
budget in impounded waters.
8. The special three DO probe equipment designed for this
study is capable of supplying the data necessary for solving
the DO model.
-------
- 42 -
ACKNOWLEDGMENTS
The authors wish to thank Lavrence Kamphake, George Holtzer, and
Richard Shibiya for the laboratory chemical analyses, Glenn R. Gruber
for help with the field work, and Thomas A. Entzminger of the Ohio
River Basin Project who supplied the computer program for the solution
of the Table 3 data, yielding Equation (6). In addition, acknowledgment
is made of the cooperation of the personnel of the State of Kentucky
Department of Fish and Wildlife Resources.
-------
- 43 -
REFERENCES
5. Kittrell, F.W., "Effects of Impoundments on Dissolved
Oxygen Resources," Sewage and Industrial Wastes. Vol. 31, 1959
p. 1065-1078.
6. Irwin, W.H., Symons, J.M., and Robeck, G.G., "Impoundment
Destratrlfication by Mechanical Pumping," Presented at ASCE National
Symposium on Sanitary Engineering Research, Development, and Design
at University Park, Pa., July 27-30, 1965. In Press.
7. Symons, J.M., Irwin, W.H., and Robeck, G.G., "Impoundment
Water Quality Changes Changes Caused by Mixing," Presented at the
ASCE Water Resources Engineering Conference, Denver, Colorado,
May 16-20, 1966. In Press.
8. Ogborn, C.M., "Aeration System Keeps Water Tasting Fresh,"
Public Works Magazine. Vol. 97, April 1966, p. 84-86.
9. Koberg, G.E. and Ford, M.E., Jr., "Elimination of Thermal
Stratification in Reservoirs and the Resulting Benefits," Geological
Survey Water Supply Paper 1809-M, U.S. Government Printing Office,
Washington, D.C., 1965, 28 pp.
10. Bryan, J.G., "Physical Control of Water Quality," The
Journal. British Waterworks Association. Vol. XLVI, No. 395, August 1964,
p. 546.
11. Hutchinson, G.E., A Treatise on Limnology. John Wiley &
Sons, Inc., New York, New York, 1957, Vol. 1.
12. Symons, J.M., Weibel, S.R., and Robeck, G.G., "Impoundment
Influences on Water Quality," Journal American Water Works Association,
Vol. 57, 1, Jan. 1965, pp. 51-75.
13. O'Connell, R.L., and Thomas, N.A., "Effect of Benthic Algae
on Stream Dissolved Oxygen," Journal of the Sanitary Engineering
Division, ASCE, Vol. 91, SA3, Proc. Paper 4345, June 1965, pp. 1-16.
14. Hull, C.H.J., Discussion of "Effect of Benthic Algae on
Stream Dissolved Oxygen" by Richard L. O'Connell and Nelson A. Thomas,
Journal of the Sanitary Engineering Division, ASCE, Vol. 92, SA1,
Proc. Paper 4637, Feb. 1966, pp. 306-313.
15. Ettinger, M.B., "How to Plan an Inconsequential Research
Project," Journal of the Sanitary Engineering Division. ASCE, Vol. 91,
SA4, Proc. Paper 4437, August, 1965, pp. 19-22.
16. Odum, H.T., "Primary Production in Flowing Waters,"
Limnology and Oceanography, Vol. 1, 1956, pp. 102-117.
17. Hull, C.H.J., "Oxygenation of Baltimore Harbor by Planktonic
Algae," Journal Water Pollution Control Federation, Vol. 35, 1963,
pp. 587-606.
-------
- 44 -
18. Hull, C.H.J., "Photosynthetic Oxygenation of a Polluted
Estuary," Advances in Water Pollution Research, Proceedings of the 1st
International Conference, held in London, United Kingdom, in
September, 1962, Pergamon Press, London, U.K., 1964, Vol. 3, pp. 347-374.
19. Goldman, C.R, and Carter, R.A., "An Investigation by
Rapid Carbon-14 Bioassay of Factors Affecting the Cultural Eutrophication
of Lake Tahoe, California-Nevada," Journal Water Pollution Control
Federation. Vol. 37, 1965, pp. 1044-1059.
20. Verduin, J., "Energy Fixation and Utilization by Natural
Communities in Western Lake Erie," Ecology, Vol. 37, 1956, pp. 40-50.
21. Symons, J.M., Discussion of "Effect of Benthic Algae on
Stream Dissolved Oxygen," By Richard L. O'Connell and Nelson A. Thomas,
Journal of the Sanitary Engineering Division, ASCE, Vol. 92, SA1,
Proc. Paper 4637, Feb. 1966, pp. 301-306.
22. Willers, F.A., Practical Analysis. Dover Publications,
New York, New York (1948).
23. Whittaker, E.T. and Robinson, G., The Calculus of Observations.
D. Van Nostrand Company, Princeton, New Jersey (1926).
-------
- 45 -
APPENDIX I
Solution of Mathematical Model with Assumed Data
The assumed constants for this example are: k.. = 0.10/hr,
k0 = 0.05/hr, R = 0.10/hr, P = 0.30/hr, L = 3.00 mg/1, D = 0.47 mg/1.
2 a a
Using the appropriate constants and solving Equations (2), (3), and
(4) yields the deficit data shown in Table 4, and plotted in Figure 11.
For simplicity, the units of the constants will not be used.
Several approaches to the solution of this problem were
attempted. Various forms of exponential approximation or Prony's
22 23
method , and a modification of Newton Raphson's method were
tried without success. Much of the difficulty in using these techniques
was caused by an insufficient number of significant figures and error
caused by rounding off. The most productive technique was simple
iteration, which will be developed using the data in Table 4. The
solution presented was developed using a Honeywell 400 computer.
Considering the simulated dark probe data and solving Equation (3)
at t = 7.5 and t = 8.0 yields,
2.81 = L { 1-exp fck.O.S)]] + R (7.5) + 0.47 (7)
3. L
2.92 = L (1-exp [-k (8.0) ]} + R (8.0)+ 0.47 (8)
^ 4-
Solving Equations (7) and (8) for R and L yields,
fl
2.45 (1-exp [-^(7.5)]} - 2.34 fl-exp [-^(8.0)]} (g)
R * 8.0 {1-exp [-1^(7.5)1} - 7.5 {1-exp [-^(7.5)]}
8.0 (2.34) - 7.5 (2.45J . ^_
a 8.0 {1-exp [-1^(7.5) ]} -7.5 {1-exp [-^(8.0)]} (10)
-------
- 46 -
TABLE 4
DO DEFICIT DATA FROM ASSUMED CONSTANTS
D6 Deficit - mg/1
Time
hrs
(1)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
kl
D
a
Eq. (3)
Simulated
Dark Probe
(2)
0.47
0.67
0.87
1.04
1.21
1.38
1.55
1.72
1.86
2.00
2.14
2.28
2.42
2.56
2.67
2.81
2.92
= 0.10, k =0
« 0.47 mg/1
Eq. (4)
Simulated
Light Probe
(3)
0.47
0.52
0.57
0.59
0.61
0.63
0.65
0.67
0.66
0.65
0.64
0,63
0.62
0.61
0.57
0.56
0.52
k =0.10, k =0
Eq. (2)
Simulated
Unconfined Probe
(4)
0.47
0.52
0.56
0.59
0.61
0.63
0.65
0.66
0.66
0.66
0.65
0.64
0.62
0.60
0.58
0.55
0.52
k^O.10, k2=0.05
P * 0, R » 0.10 P=0.30, R=0.10 P=0.30, R=0.10
L - 3.00 L =3.00 L =3.00
a . a a
-------
3.0
CJ>
E
2.0
O
u,
LU
Q
z
LU
X
O
Q
UJ
O
GO
CO
Q
1.0
I I
ASSUMED CONSTANTS IN THE MODEL
K, - 0.10
K2 - 0.05
La - 3.00 mg/l
R - 0.10
P * 0.30
Da» 0.47 mg/l
SIMULATED DARK DISSOLVED
OXYGEN PROBE DATA
(EQUATION 3)
SIMULATED UNCONFINED DISSOLVED
OXYGEN PROBE DATA
(EQUATION 2)
i
i
SIMULATED LIGHT DISSOLVED
OXYGEN PROBE DATA
(EQUATION 4)
I
I
I
I
I
I
•
TIME, hours
Figure 11. Assumed dissolved oxygen data to verify solution of dissolved
oxygen model.
-------
- 48 -
To solve for k , Equation (3) can be reformulated as follows:
D - L [1-exp (-k.t)] + R(t) + D (3)
a I a
D
-D -R(t) = L [1-exp (-k.t)]
a a i
(11)
D-D -R(t)
1 1 -exp (-ktt) (12)
D-D -R(t)
l-[ 1 1 - exp (-kxt) (13)
a
D-D -R(t)
D-D -R(t)
-t L(t) » - '"I
a
D-D -R(t)
Using [ !L ] as K, Equation (15) may be written
Jj
a
In a value may be obtained for k. at each
3 i.
set of values fer D and t using the dark probe data of Table 4 and
2
Equation (16). The average value for k- is the new value, k . Then
2 2
Equations (9) and (10) are resolved for L and R . The process is
3
repeated until kn - k and the solution has converged. A tolerance of
0.005 is placed on k- for convergenge, that is when k is within
0.005 of k. the iteration stops.
-------
- 49
Rearranging Equation (3) into a)
, , _ r
L + R L
_
a l-exP(-klt)
a least squares technique gives the best value of L and R, using the
3
final value of k. .
To illustrate, starting with k. = 0,50 and L = 0.76799 and
JL 9
1 2
R = 0.21201 from Equations (9) and (10), the solution for k^ would
be as shown in Table 5. Discarding any values for which 1-K is zero
or negative, k1 = 0.33426. Continuing in this manner until k = k^
1 o
kl" = 0.11. The least squares solution of Equation(16$with k.^ = Q.l\
yields R - 0.10 and L = 2.72.
In the next step, using k, = 0.11 and L = 2.72 from the
X 3
previous solution, Equation (4) is solved for (R-P) at each time
except t=0 using the light probe data in Table 4. This yields 16
values for (R-P) that are averaged to yield the final value of (R-P) .
For these data (R-P) = -0.19. P can then be evaluated by difference,
if required.
In solving Equation (2) a slightly different approach is used.
Here, equally spaced abscissa values are used, then Equation (2)
is reformulated as follows,
D - C1[exp(-k1t)] + C2[exp (-k^) ] + Q (17)
where
2 Vkl k2 a
(19)
-P) (20)
k2
-------
- 50 -
TABLE 5
DATA FOR SOLUTION OF EQUATION (16)
t
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
R(t)
0.0
0.105
0.212
0.315
0.424
0.525
0.630
0.735
0.840
0.945
1.050
1.155
1.260
1.365
1.470
1.575
1.680
D-D
a
0.00
0.20
0.40
0.57
0.74
0.91
1.08
1.25
1.39
1.53
1.67
1.81
1.95
2.09
2.20
2.34
2.45
D-D -R(t)
r a i . ,,
1 L J K
a
0.0000
0.12239
0.24478
0.32811
0.41144
0.49477
0.57810
0.66142
0.70569
0.74995
0.79422
0.83848
0.88275
0.92701
0.93222
0.97648
0.98168
ln,l-K . k
t 1
-
-0.26111
-0.28075
-0.26511
-0.26504
-0.27309
-0.28766
-0.30943
-0.30578
-0.30802
-0.31619
-0.33148
-0.35724
-0.40269
-0.38449
-0.50000
-0.50000
-------
. 51 -
Solving Equation (17) at (t , D ) for C [exp(-k t )] yields,
J_ JL £» ff i
C2[exp (-k2t1^ * Di"Ci ^exp ("k1t1)J ~Q (21)
The solution for C-[exp(-k0t )] at (t0,D0) is
C2[exp (-k2t2)] = E>2-C [exp(-k1t2)] - Q (22)
Dividing Equation (21) by Equation (22) yields,
exp [-k.(t -t )] - -~j;
Repeating this process at(t2>D2)and(t3,D3)yields,
D -C [exp(-k t )] -Q
exp [-kCt-t)] - .^t ,] .-, <»)
Since equally spaced abscissa values are used, (t- -t2) = ^t2~t
the right hand side of Equations (23) and (24) may be equated,
yielding
D1-C1[exp(-k t )]-Q D -C [exp(-k t ) ]-Q
1111 ] (27)
(28)
(29)
and ZtDD-CD)] (30)
-------
- 52 -
Using values of k , L , and (R-P) derived from the solution of
J- 3
Equations (3) and (4), the quadratic Equation (26) may be solved
for its positive root, as follows.
The coefficients for k~ and the constant in Equation (26)
may be determined from each three successive data points, for
example (t=0.5, 1.0, 1.5), (t=1.0, 1.5, 2.0) and so on. This
yields 14 polynomial equations in the form of Equation (26). The
fourteen sets of coefficients and constants are then averaged to
yield an "average" Equation (26) which is then solved for k_ as the
only unknown, which in this example = 0.09. A summary of all of the
solutions for the constants in Equation (2) is shown in Table 6.
TABLE 6
COMPARISON OF SOLUTIONS FOR CONSTANTS IN EQUATION (2)
R
Assumed 0.10 0.05
Dark Probe 0.11
Light Probe
Unconfined - 0.09
Probe
0.10
0.10
(R-P)
-0.20
-0.19
3.00
2.72
-------
- 53 -
To demonstrate the adequacy of the model, Equation (2), and
the method of solution, deficit data was generated using the
constants of Table 6, in Equation (2). This curve is
plotted in Figure 12 and is compared to the simulated unconfined DO
data from Column 4 in Table 4. The fit curve is low as time progresses,
probably because k« is too large. Fit curves calculated for the dark-
and light-probe systems are much better, as ky *-s not included.
The solution is very sensitive to the final value of k-.
The computational procedure was repeated using a tolerance of 0.001
as a measure of convergence for k1. Under these circumstances k
is reduced 0.03 to 0.08. This slight reduction has a large impact
on the remaining calculations. One method of reducing this instability
might be to evaluate Equations (9) and (10) at each group of two
times (0.5, 1.0; 1,0, 1.5; and so forth) instead of just at
t = 7.5 and 8.0 to obtain an "average" L and R before calculating
a
2
k . In spite of these minor difficulties the usefulness of this
equipment and method for studying the DO budget in impoundments has
been demonstrated.
-------
M
o
u.
UJ
Q
z
UJ
CD
§
Q
UJ
O
CO
1.5
1.0
0.5
0.0
SIMULATED DATA-COL. 4-TABLE 4
I
SOLUTION TO EQUATION (2)
USING CONSTANTS
TABLE 6
I
4 5
TIME, hours
8
! 'I
I
Figure 12. Comparison of the solution of Equation (2) with the simulated data.
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- 55 -
APPENDIX II
BIBLIOGRAPHY
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- 56 -
15. Yount, J.L., Biologist, Florida State Board of Health,
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11
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29. Burns, J.M., Jr., "Reservoir 'Turn-over' Improves Water
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