WATER POLLUTION CONTROL RESEARCH SERIES 16080 000 7/70
OPTIMUM MECHANICAL
AERATION SYSTEMS
FOR RIVERS AND PONDS
SNVIRONMENTAL PROTECTION AGENCY WATER QUALITY OFFICE
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Reports describe the results and
progress in the control and abatement of pollution in our Nation's waters.
They provide a central source of information on the research, development,
and demonstration activities in the Water Quality Office, in the
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* about our cover
The cover illustration depicts a city in which man's activities coexist in
harmony with the natural environment. The Water Quality Control Research
Program has as its objective the development of the water quality control
technology that will make such cities possible. Previously issued reports
on the Water Quality Control Research Program include:
Report Number Title
16080 06/69 Hydraulic and Mixing Characteristics of Suction Manifolds
16080 10/69 Nutrient Removal from Enriched Waste Effluent by the
Hydroponic Culture of Cool Season Grasses
16080DRX10/69 Stratified Reservoir Currents
16080 11/69 Nutrient Removal from Cannery Wastes by Spray Irrigation
of Grassland
16080DVF07/70 Development of Phosphate-free Home Laundry Detergents
16080 10/70 Induced Hypolimnion Aeration for Water Quality Improvement
of. Power Releases
16080DWP11/70 Induced Air Mixing of Large Bodies of Polluted Water
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OPTIMUM MECHANICAL AERATION SYSTEMS
FOR RIVERS AND PONDS
by
Win. T. Hogan
P. Everett Reed
A. W. Starbird
Littleton Research and Engineering Corp.
95 Russell Street, Littleton, Massachusetts 01460
for the
ENVIRONMENTAL PROTECTION AGENCY
WATER QUALITY OFFICE
Program #16080 DOO
Contract #14-12-576
November 1970
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $1.25
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WQO Review Notice
This report has been reviewed by the Water
Quality Office and approved for publication.
Approval does not signify that the contents
necessarily reflect the views and policies
of the Water Quality Office, nor does mention
of trade names or commercial products constitute
endorsement or recommendation for use.
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ABSTRACT
The total annual cost of providing supplemental aeration of streams
and lakes by tested and untested aeration equipment is estimated.
Analytical and empirical equations are presented for the determina-
tion of operating characteristics of the various devices used to aerate
natural bodies of water. For the example stream evaluated in this
study, the most economical means of artificial aeration generally pos
sible was found to be mechanical aerators which generate a highly
turbulent white-water surface. For the example lake evaluated, the
most economical technique for the continual input of oxygen into a
lake was found to be diffused aeration using air bubbles; whereas the
most economical technique for rapid input of oxygen, operating only
while the lake is being destratified, was found to be a hybrid system
consisting of a large diameter ducted propeller which draws water
from the lake bottom and discharges it at the surface where it is aer-
ated by a mechanical aerator.
This report was submitted in fulfillment of Project Program #16080
DOO, Contract #14-12-576, under the sponsorship of the Federal
Water Quality Administration.
Key Words: Mechanical Aeration*, Aeration Efficiency*, Aeration
Devices, Aeration Methods
111
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CONTENTS
ABSTRACT iii
FOREWORD xi
I CONCLUSIONS 1
II RECOMMENDATIONS 3
III INTRODUCTION 5
IV METHODS OF INDUCING AERATION 9
V DISSOLVED OXYGEN CONCENTRATIONS IN
QUIESCENT RIVERS AND PONDS 25
VI DEVICE OPERATING CHARACTERISTICS 29
VII OPTIMUM ECONOMIC SELECTION 79
VIII ACKNOWLEDGMENTS 99
IX REFERENCES 101
X NOMENCLATURE 107
XI APPENDICES 111
A. Oxygen Capture at Air-Water Interface 111
B. Potential Flow Simulation of Sub-Surface Devices 117
C. Cost Estimate of Sub-Surface Aerators 131
v
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FIGURES
1 Diffusion of Oxygen into Water 11
2 Oxygen Transfer under Conditions where the Liquid 15
Film Controls the Mass Flow Rate
3 Comparison of Reported and Calculated Apparent 20
Liquid Film Coefficient (Adapted from Ref. 12)
4 Oxygen Transfer Rate for Various Rate Control 22
Situations
5 Annual Hydrographs for Three Stations 26
6 Schematic Sketches of Subsurface Aerators 31
7 Flow Associated with a Duct of Small Diameter Com- 33
pared with the Depth Midplane Velocity as a Function
of Radial Distance from Axis
8 Flow Associated with a Duct of Large Diameter Com- 36
pared with Midplane Velocity as a Function of
Radial Distance from Axis
9 Flow Associated with a Duct Lying on Bottom 37
Velocity on Surface in Plane of Axis as a Function
of Distance from Midplane
10 Source-Sink Arrangement to Simulate Circulation 39
Pattern Induced by a Vertical Duct and Propeller
in a Body of Water with a Horizontal Bottom
11 Source-Sink Arrangement to Simulate Circulation 41
Pattern Induced by a Vertical Duct and Propeller
in a Body of Water with a Sloping Bottom
12 Velocities with Sloping Bottom 42
13 Temperature Profiles at Various Stations in 45
Vesuvius Lake
14 Temperature Profiles at Various Stations in 45
Vesuvius Lake
VII
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15 Rise of a Bubble in Water 51
16 Measured Effect of Bubble Size and Column 57
Height on Oxygen Capture Coefficient for Water
with an Initially Low DO (~0) Level
17 Sketch of High Volume Spray Aeration Device 68
18 Comparison of Various Measured Oxygen Transfer 71
Efficiencies for White-Water Generators
B-l Distribution of Sources and Sinks to Simulate Flow 120
Induced by a Propeller in a Duct whose Width is
Small compared to the Water Depth
B-2 Distribution of Sources and Sinks to Simulate the Flow 125
Induced by a Propeller in a Vertical Duct whose Width
is Large compared to the Water Depth
B-3 Distribution of Sources and Sinks to Simulate Flow 127
Induced by a Propeller in a Horizontal Duct
C-l Sub-Surface Circulating Device - Propeller in a 134
Vertical Duct
Vlll
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TABLES
I Dissolved Oxygen Saturation Values for 6
Distilled Water, mg/liter
II Cost of Offsetting 20, 000 Ibs BOD Daily 8
III Oxygen Transfer Rate and Total Accumulated Oxygen 13
as a Function of Elapsed Time for Molecular
Diffusion
IV Oxygen Transfer Rate across a Thin Quiescent Layer 17
of Surface Stream Water
V Oxygen Transfer Rates for Turbulent Rivers 21
VI Measured Lake Destratification Time 49
VII Comparison of Destratification Efficiencies of 64
Various Studies
VIII Summary of Steady-State Field Test Data for 72
Mechanical Aerator
IX Comparison of Measured and Calculated Oxygen 74
Transfer Efficiency for Dams
X Total Power Variation with Number of 81
Sub-Surface Units
XI Oxygen Transfer Efficiencies and Power for 83
Diffused Aerators
XII Oxygen Transfer Efficiency and Power for 86
Spray Aerator and White-Water Generators
XIII Summary of Annual Cost of Aeration Systems for 91
Streams
XIV Summary of Annual Cost of Aeration Systems for 97
Lakes
IX
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FOREWORD
At the present time there is a considerable concern about nutrients,
particularly phosphates and nitrates, from properly treated sewage
effluents and from farming operations, being introduced into our
water systems. With these nutrients plant growth develops and the
water becomes fouled by the oxygen demand of this plant growth when
it decays.
On the other side of the picture, farms for growing catfish are being
developed. These fish develop most rapidly when grown in water rich
in nutrients (and also warmed). The great fishing areas of the world,
such as the Grand Banks and off the coast of Peru, are areas where the
ocean water contains many chemical nutrients which allow marine
growth that provides feed for the fish.
On the broad scale, it might be wondered why the addition of nutrients
into the water, so long as there are no poisons, shouldn't make the
stream more valuable and desirable. The answer might well be that
this would be true so long as the level of dissolved oxygen in the
stream or lake were maintained. This report is concerned with econ-
omical methods for maintaining high levels of dissolved oxygen in
quiet rivers and ponds.
In this report the annual cost of adding oxygen to streams and lakes is
estimated for a variety of existing and new aerating devices. In order
that it will be possible to determine the capacity of the machinery re-
quired for a given application, expressions are presented which des-
cribe mathematically the operating characteristics of the various aer-
ation devices. Where possible, these expressions are developed in
analytical form so that the optimum operating conditions can be made
apparent.
One of the interesting conclusions that can be drawn from these ex-
pressions is that the maximum volume of lake water that can be set
into circulation by one pump is limited to a cylindrical "cell" equal in
depth to the water depth and in diameter to a value equal to about four
times the water depth in the absence of vertical temperature gradients.
If, on the other hand, vertical temperature gradients do exist, ex-
tremely large volumes of water may be influenced by a single pump
until such time as the body of water is destratified.
A methodology for estimating the total annual cost as the sum of cap-
ital and operating cost for aeration devices is presented. The report
XI
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concludes with a section in which the total annual cost for increasing
the dissolved oxygen level of a typical stream and lake is estimated.
The economic evaluation leads to several interesting conclusions.
Included in these is the observation that diffused aeration of streams
will not be economically competitive with other methods even if the
diffused system can be designed to achieve a much higher ratio of
oxygen captured by the water to oxygen supplied than has ever been
demonstrated for comparable depths. It is also shown that the cost
of the most economical stream aerating devices can be further de-
creased by about a factor of 1. 5 if recirculation is successfully inhib-
ited. A means of accomplishing this task is suggested in the report.
Although diffused aeration is shown not to be the optimum aeration
technique for water of limited depth such as a stream, this technique
is shown to increase in its economic competitive position as the depth
increases.
XII
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I. CONCLUSIONS
1. For slow moving streams the five most economical ways to in-
crease the DO level through a given limited range in order of lowest
total annual cost are 1) venting of hydraulic turbines (if such an ar-
rangement is possible at the site in question), 2) white-water gener-
ators if recirculation can be successfully inhibited, 3) lift-drop aer-
ators if recirculation can be successfully inhibited, 4) free fall over
a dam (if such construction is possible at a given site, and 5) white-
water generators without provisions to inhibit recirculation of water
through the equipment. Of the five methods only the last one can be
applied without qualifications since methods 1 and 4 are site-depend-
ent and methods 2 and 3 depend on equipment modifications that have
not yet been proven.
2. If recirculation can be successfully inhibited in white-water
generators used in streams, their total annual cost can be expected
to decrease by approximately a factor of 1. 5.
3. The total annual cost for large diameter ducted propellers is
about a factor of 1. 4 greater than the corresponding cost for white-
water generators used in stream application without provisions for
inhibiting re circulation. However, this conclusion is based only on
the analytically estimated performance of sub-surface aerators with-
out any experimental data and the estimate could well be off by a fac-
tor comparable to 1. 4.
4. Diffused aeration of streams (depth ~ 4 feet) using air bubbles
will lead to higher annual cost than aeration by means of white-water
generators, lift-drop generators, or dams even when the diffused
aeration system has been carefully designed so that the capture coef-
ficient (f ) is considerably higher (~ . 5) than has been achieved in
practice up to the present time and the total pressure drop in the sys-
tem is maintained at a low level (~ 15 psi)0
5. The annual cost of aerating a slow moving stream by means of
diffused aeration with pure oxygen bubbles is the highest of all meth-
ods considered in this report if the capture coefficient is assumed
equal to measured laboratory values. However; since almost 90% of
the total annual cost is operating cost, the capital cost is low and this
method may be the most economical for short term application in a
given situation.
6. The volume of lake water that can be set into circulation by a
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single pump, in the absence of vertical temperature gradients, is
limited to a "cell" equal in depth to the water depth and in diameter
to a value equal to approximately four times the water depth. This
"cell" volume is not substantially influenced by the slope of the bot-
tom unless the slope is extreme (~ 45 ). If vertical temperature
gradients do exist, a considerable volume of water can be set into
circulation by a single pump until the lake is destratified and the cir-
culation "cell" is established.
7. If oxygen is to be added continuously throughout an entire lake,
diffused aeration appears to have a total annual cost which is substan-
tially lower than the cost for sub-surface aerators.
8. If oxygen is to be added to a lake only during the time required
to destratify the lake, a hybrid system, consisting of one sub-surface
circulator and a white-water generator appears to be capable of trans-
ferring oxygen far more economically than sub-surface aerators alone
or diffused aeration alone. In addition this method allows the use of
one white-water generator at several sites in a given season.
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II. RECOMMENDATIONS
1. That field tests be conducted with white-water generators that
have been modified so as to substantially inhibit re circulation. It is
recommended that the modification consist of the addition of an "L"
shaped draft tube placed so as to draw in water at some distance up-
stream from the location of the surface agitating blades.
2. That field or laboratory tests be conducted to check validity of
the predicted operating characteristics of large diameter ducted pro-
pellers as aeration devices for streams.
3. That field or laboratory tests be conducted to establish the feas
ibility of a hybrid system (consisting of a large diameter ducted pro-
peller and a white-water generator) for the rapid aeration of stratified
lakes during the destratification phase of circulation.
4. That analytical and experimental studies be conducted to est-
ablish the feasibility of using a ducted propeller to skim large quanti-
ties of cool, oxygen-rich water off a quiescent lake surface at the
time of day when the surface water temperature is a minimum and
pump it to the bottom. This technique offers the possibility of main-
taining a thermocline while increasing the DO level of hypolimnion
water.
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III. INTRODUCTION
The introduction of organic material into our rivers and ponds is a
major factor in water quality control. Conventional methods of oxid-
izing sewage use spray ponds, aeration beds (sewage sprayed over a
bed of rocks), surface agitators and bubble percolation. These are
highly efficient aeration methods and can reduce the BOD of large vol-
umes of waste in a short time and within a limited area. However,
these methods of aeration which were developed for sewage disposal
involve a high investment and operating cost.
This study is directed toward the economic improvement of the dis-
solved oxygen level of a quiescent stream or pond which is already
polluted. In this case it is not necessary to accomplish the improve-
ment in quality within a very short time nor in a very limited area,
and the simplicity and reliability of operation and the investment and
operating cost of the equipment are primary considerations.
The amount of oxygen that may be dissolved is limited by temperature
and the amounts of dissolved colloidal material present. At sea level
and at a temperature of 20°C (68 F), the saturation value of dissolved
oxygen in pure water is 9 mg/liter. The oxygen saturation value as a
function of temperature is given in Table I as adapted from Ref. 1,
Atmospheric oxygen provides the major source of dissolved oxygen
replenishment in water. It is often agreed that a minimum dissolved
oxygen content of 5 ppm is required for healthy aquatic life. If the
dissolved oxygen content of natural streams falls substantially below
this level, not only does the aquatic life deteriorate but the aerobic
bacteria that effect the decay of the organic material are replaced by
the anaerobic bacteria that generate odorous gases in the decomposi-
tion of the organic material.
The amount of oxygen dissolved in the stream water is the result of
the balance between those mechanisms which supply oxygen to the
stream; primarily, diffusion of oxygen from the atmosphere into the
water and photosyntheses, and the mechanisms which remove oxygen
from the water; namely, bio-oxidation of organic waste (including the
organic load imposed by dead algae), oxygen consumption by direct
chemical reaction, and support of aquatic life.
Dissolved oxygen content is not the only factor that defines water
quality. However, it is closely related to the odor of the water - the
most obvious and objectionable factor. Generally the introduction in-
to a stream of a pollutant having a high biological oxygen demand is
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TABLE I
Dissolved Oxygen Saturation Values
for Distilled Water, mg/liter (Ref. 1)
i empera-
ture, °C
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
14.65 14.61 14.57 14.53 14.49 14.45 14.41 14.37 14.33 14.29
14.25 14.21 14.17 14.13 14.09 14.05 14.02 13.98 13.94 13.90
13.86 13.82 13.79 13.75 13.71 13.68 13.64 13.60 13.58 13.53
13.49 13'.46 13.42 13.38 13.35 13.31 13.28 13.24 13.20 13.17
13.13 13.10 13.06 13.03 13.00 12.96 12.93 12.89 12.86 12.82
12.79 12.76 12.72 12.69 12.66 12.62 12.59 12.56 12.53 12.49
12.46 12.43 12.40 12.36 12.33 12.30 12.27 12.24 12.21 12.18
12.14 12.11 12.08 12.05 12.02 11.99 11.96 11.93 11.90 11.87
11.84 11.81 11.78 11.75 11.72 11.70 11.67 11.64 11.61 11.58
11.55 11.52 11.49 11.47 11.44 11.41 11.38 11.36 11.33 11.30
11.27 11.24 11.22 11.19 11.16 11.14 11.11 11.08 11.06 11.03
11.00 10.98 10.95 11.93 10.90 10.87 10.85 10.82 10.80 10.77
10.75 10.72 10.70 10.67 10.65 10.62 10.60 10.57 10.55 10.52
10.50 10.48 10.45 10.43 10.40 10.38 10.36 10.33 10.31 10.33
10.26 10.24 10.22 10.19 10.17 10.15 10.12 10.10 10.08 10.06
10.03 10.01 9.99 9.97 9.96 9.92 9.90 9.88 9.86 9.84
9.82 9.79 9.77 9.75 9.73 9.71 9.69 9.67 9.65 9.63
9.61 9.58 9.56 9.54 9.52 9.50 9.48 9.46 9.44 9.42
9.40 9.38 9.36 9.34 9.32 9.30 9.29 9.27 9.25 9.23
9.21 9.19 9.17 9.15 9.13 9.12 9.10 9.08 9.06 9.04
9.02 9.00 8.98 8.97 8.95 8.93 8.91 8.90 8.88 8.86
8.84 8.82 8.81 8.79 8.77 8.75 8.74 8.72 8.70 8.68
8.67 8.65 8.63 8.62 8.60 8.58 8.56 8.55 8.53 8.52
8.50 8.48 8.46 8.45 8.43 8.42 8.40 8.38 8.37 8.36
8.33 8.32 8.30 8.29 8.27 8.25 8.24 8.22 8.21 8.19
8.18 8.16 8.14 8.13 8.11 8.10 "8.08 8.07 8.06 8.04
8.02 8.01 7.99 7.98 7.96 7.95 7.93 7.92 7.90 7.89
7.87 7.86 7.84 7.83 7.81 7.80 7.78 7.77 7.75 7.74
7.72 7.71 7.69 7.68 7.66 7.65 7.64 7.62 7.61 7.59
7.58 7.56 7.55 7.54 7.52 7.51 7.49 7.48 7.47 7.45
7.44 7.42 7.41 7 40 7.38 7.37 7.35 7.34 7.32 7.31
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objectionable because it rapidly depletes the available dissolved oxy-
gen and leads to the objectionable septic condition. Likewise the in-
troduction of nutrients such as phosphates into the water is generally
considered undesirable because tney lead to aquatic growth which,
when decaying after growth, imposes a heavy organic load on tne
water that might generate septic conditions. However, marine life
feeds upon the aquatic growth and it is well known that the best fishing
grounds are those where currents rich in minerals allow growth to
provide feed for the fish. For this reason, if it is possible to control
the dissolved oxygen by artificial means, there may be benefits in tne
form of increased production of fish and marine life.
If sufficient time is available, the oxygen content of water in contact
witn air will reach its saturated or equilibrium value. Dissolved oxy-
gen values below the saturation level are an indication of the presence
of organic pollution from sewage, organic industrial waste, agricul-
tural land runoff, forest and natural land runoff or oxygen demands
exerted during respiration, and die-off of over-abundant blooms of
algae and aquatic plant life. Because dissolved oxygen levels provide
a gross appraisal 01 many factors taking place in stream water, uni-
versal use has been made of this parameter for waste treatment plant
design, stream standards for protection of aquatic iile, abatement
criteria for determining low flow augmentation requirements, and
evaluation of stream assimilative capacity in connection with permiss-
ible stream loadings.
Robert K. Davis [2] performed some economic studies of various
processes in terms of the level in pounds (BODr)' ' removed or offset
daily. A very lengthy publication on this subject was also issued by
the U. S. Army Engineer District in Baltimore [sj. The results of
these works and the works referenced in these publications show that
the most economic means of raising DO in quiet rivers and streams
are by diffused aeration and mechanical aeration.
Table II (on the following page) which shows this is based on Davis'
paper [2].
(1) The biochemical oxygen demand of polluted water is a measure of
the oxygen required to stabilize decomposable organic matter by aer-
obic bacterial action. Incubation for 5 days at 20°C (BOD^) is the
standard that is used during river basin field studies.
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TABLE II
Cost of Offsetting 20, 000 Ibs. BOD Daily [Ref. 2j
Process
Cost
Microst raining
Step Aeration
Chemical Precipitation
Powdered Adsorption
Granular Adsorption
Effluent Distribution
Diffused Aeration
Mechanical Aeration
$12,000,000
13, 000, 000
18,000,000
43, 000, 000
95,000,000
19, 000, 000
2, 200,000
1,700,000
The cost information developed for the various processes was based
on the capital costs, maintenance costs, and operating costs. The
capital costs are amortized over a 50-year period. Effluent distribu-
tion, the two carbon processes (granular and powdered adsorption),
and the combination (step aeration followed by microstraining) require
the most capital investment. (This is also the case for low-flow aug-
mentation. ) Chemical precipitation involves a low capital investment
but high operating costs. Both effluent distribution and carbon ad-
sorption have very large power requirements. It is obvious from the
cost data that the reaeration devices are the least costly of the altern-
ative processes.
Although the cost of mechanical or diffused aeration devices is far
lower than the alternatives, the total cost of numerous devices for re-
ducing the BOD in countless estuaries throughout the country runs into
the billions of dollars. This means that it is most important to devel-
op highly economical and efficient devices. It is the purpose of this
study to develop 1) the detailed criteria for design and 2) basic de-
signs of devices that will reduce the biological oxygen demand in
streams and ponds at minimum costs.
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IV. METHODS OF INDUCING AERATION
Any mechanical aeration scheme must consist of two fundamental
steps. The oxygen must be captured by the water at some gas-water
interface and the captured oxygen must then be distributed throughout
the body of water. If the gas is air, then the rate at which oxygen is
captured at the air-water interface can be estimated by calculating
the rate at which oxygen molecules strike the air-water interface and
then multiplying this rate by a capture coefficient, that is, the ratio
of oxygen molecules that are captured by the water per unit time when
they strike to the total number that strike per unit time. The total
number of oxygen molecules that strike the interface per unit time per
unit area is shown in Appendix A (See Eq. A- 3) to be given by the ex-
pression
N = _, __ (Eq.
where N = number of molecules striking interface per second
P = partial pressure of oxygen, dynes/cm
k = Bo It zm an Constant - 138° x 10~° erg/°K
T = absolute gas temperature, K
m = mass of an oxygen molecule, grams
If the oxygen being captured by the water is supplied from the atmos-
phere with a partial pressure of 0. 21 atmosphere and a temperature
of 20°C, then the rate at which oxygen strikes the surface is found
from Eq. 1 to be 6. 30 #m/ft^sec. Of this incident flux of oxygen
molecules, some will rebound off the interface and return to the at-
mosphere and some (a fraction $) will be captured by the water at the
interface and is then available to the water below the surface if it can
reach that region by either molecular diffusion (in the absence of any
turbulence at the air-water interface) or by eddy diffusion (if sufficient
turbulence is present). The fraction of incident molecules that is cap-
tured (the capture coefficient $) is a function of the dissolved oxygen
concentration (DO) at the surface, and will be a maximum when the
surface DO is zero and will be zero when the DO at the surface cor-
responds to the saturation value. In Appendix A /3 is estimated to be
given by the expression
C C
j8 = . 74(-^- - ) (Eq. 2)
Cw
where C = concentration of DO at the surface, mg/liter
-------
C = saturation concentration of oxygen, mg/liter
s
C = concentration of water in mg/liter
w to
The rate at which oxygen is captured at the air-water interface can
now be found by multiplying Eq. 1 by Eq. 2. Thus
%
= 4.65x!0"6(C - C) #m/ft2sec (Eq. 3)
A s
where C and C are to be expressed in mg/liter
s
As an example, consider the capture of atmospheric oxygen at the air-
water interface at the instant the surface DO level is zero and the
water temperature is 20°C (Cg = 9 mg/liter). This rate is given by
Eq. 3 as 4. 1 7 x 10-5 #m/ft2sec.
The order of magnitude for the time required to saturate a newly ex-
posed water surface with an initial zero DO level can be established by
computing the time required to capture sufficient oxygen to approach
(within 99%) saturating the upper layer where dynamic capture of oxy-
gen molecules is the dominant phenomenon rather than molecular dif-
fusion, say a layer ten molecules thick. When this is done with the
use of Eq_ 3, the time is found to be 30 x 10"" sec. or 30 microsec-
onds and the total oxygen accumulated in the water during this period
is 1.25 x 10~9 #rn /ft2.
°2
In order to establish the relative importance of the two mechanisms
involved in the aeration process it is necessary to compare the sur-
face capture rate of oxygen with the rate of the molecular diffusion or
the eddy diffusion by means of which the oxygen initially captured at
the surface is transported to the water below the surface. However,
before doing this, it will be helpful to very briefly review the physics
of the gas absorption into a liquid, that is, oxygen into water.
When considering the diffusion of a gas of limited solubility, like oxy-
gen, into water, two extreme cases can be readily identified for speci-
fying the condition of the water, namely: 1) stagnant water at one end
of the spectrum and 2) extremely turbulent water with a broken
"white" surface on the other end of the spectrum. In the case of oxy-
gen being absorbed into stagnant water, the surface water when first
exposed to the atmosphere will capture oxygen at a rate given by Eq. 3;
however, once at the air-water interface the only mechanism for the
oxygen to move into the liquid is by molecular diffusion. For this case
10
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the rate of diffusion is given by Fick's law, namely:
M
O.
= D, A
2
where
L dz
M = time rate of oxygen mass transfer
L-i
DL = diffusivity coefficient for oxygen diffusing in water
A = surface area of interface
-=r = concentration gradient in direction of diffusion
Eq. 4 can be expressed in terms of C, t, DL and z only by consider-
ing the conservation of oxygen for a differential slice of the water as
shown below.
Air-water interface of Area A
dz
Fig. 1 - Diffusion of Oxygen into Water
Since oxygen is conserved, we can write for the differential control
volume:
() dz (Dq. 5)
Eq. 5 reduces to the following form when second order terms are
ne gle cte d:
(Eq. 6)
For an infinitely deep pool of water initially at a uniform dissolved
11
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oxygen content of C except at the interface where the dissolved oxy-
gen content is at the saturation values (Cs), Eq0 6 can be integrated
and the results used to evaluate the concentration gradient (SC/9z) at
the surface in Eq. 4 to obtain the mass flow rate at the surface as
given below in Eq. 7: (see Ref. 4)
(Eq. 7)
t = time elapsed from time zero (2)
In order to make comparisons between the various models to be dev-
eloped in this report, as well as with data reported in the open litera-
ture, it will be convenient to rewrite the right hand side of Eq. 7 as
the product of a coefficient and a "driving force," thus Eq. 7 can be
expressed as:
= K (C - C ) (Eq. 8)
A L s o
5
where KT = liquid film coefficient = // for this case.
L 77" t
o
If we again consider the atmosphere at 20 C to be the source of oxy-
gen, then Eq. 8 can be used to predict the rate of oxygen flow into a
semi-infinite pool of water (which is initially at zero DO) as a function
of time. A summary of the results of such a calculation are given be-
low for convenience.
(2) Although any consistent set of units may of course be used in Eq..
7, it has become the custom to speak of C in terms of mg/liter. Un
fortunately such units do not conveniently fit into Eq. 7. However-
since a mg/liter is equivalent to one part per million by weight, it
follows that one mg/liter of dissolved oxygen is equivalent to 62. 4 x
10" #rtiQ /ft . Thus a convenient form of Eq. 7 is
4_i
M /
O , /D
- = 62. 4 x 10 ° / r (C - C ) #m/ft -sec
A IT t s o
where D is in ft /sec
J_i
t is in sec
C , C are in mg/liter
12
-------
TABLE HI
Oxygen Transfer Rate and Total Accumulated Oxygen
as a Function of Elapsed Time for Molecular Diffusion
Time
1 hour
1 day
1 month
M /A, #m/ft sec
8. 55 x 10
1. 74 x 10
3. 19 x 10
-10
-10
-11
Accumulated oxygen in
water, #m/ft2 of surface
1 microsecond
1 millisecond
1 second
5. 18 x 10 5
1. 61 x 10"6
5. 18 x 10~8
-10
1. 036 x 10
3. 21 x 10"9
1. 036 x 10"7
6. 21
x
10
3. 05 x 10
1. 68 x 10
-6
-5
-4
Two observations are to be made in regard to Table III. First, when
a "new" surface is initially exposed, oxygen from the atmosphere will
tend to saturate the upper layer or interface and oxygen will be with-
drawn from this layer by molecular diffusion into the main body.
When the new surface is initially exposed to the atmosphere, the rate
at which O2 would be withdrawn from a saturated surface and zero DO
water exceeds the rate at which the atmosphere replenishes it. After
an exposure time of about 2 microseconds, the two rates become equal
and for longer exposure times the atmosphere will replenish the in-
terface oxygen at a faster rate than it will be removed by diffusion.
Thus the interface can be assumed to be at the saturation conditional
except for the short initial period of high transfer by diffusion into the
main body. Since the processes that will be of interest in the present
study will be at least of the order of microseconds, it can be safely
assumed that the interface is always saturated. Some references are
available in the literature to support this conclusion that the interface
will remain saturated by atmospheric replenishment except at the very
high diffusion rates [5j.
The second observation to be made in regard to Table III is that the
total mass of oxygen accumulated by a quiescent body of water, say a
pond in summer time, by molecular diffusion, which has its surface
renewed only once a day (for example, due only to daytime heating and
nighttime cooling) is 3. 05 x 10" 5 #mo /ft , whereas the total amount
of oxygen that could be accumulated by the same body due to mole-
cular diffusion alone could be increased to 4. 49 x 10
surface were to be replaced each second.
O
/ft if the
13
-------
Before leaving the discussion of Table III it is helpful to calculate the
time it will take for a one-foot deep pool of water to increase in DO
from a uniform value of zero to an average value of 1 mg/liter. This
can be done by combining solutions for two semi-infinite slabs as given
in Eq. 8 or by use of nondimensional solutions given in Ref. 6. When
the calculations are performed, the time is found to be 2, 4 years.
As we move from the end of the spectrum where the water is stagnant
to where the water is moving, but moving so slowly that it is in lami-
nar flow, Eq. 8 remains valid if the water is deep enough so that the
velocity profile is essentially square. As the velocity is increased or
the water is made more shallow so that the flow remains laminar but
with a pronounced velocity-depth profile, Eq. 8 must be modified.
This modification will not alter the oxygen flow rates substantially. A
stream will generally be in laminar flow if the Reynolds Number based
on the hydraulic diameter is less than 2000, that is, when
VD p
?+- < 2000 (Eq. 9)
where V = average flow velocity, ft/sec
D = hydraulic diameter, ft
H Y
= 4 times the flow cross section divided by the
wetted parameter
~ 4 times the depth for a river which is much wider
than it is deep
p - density of water, slugs/ft
jU = absolute viscosity of water, # /ft
Likewise a stream will generally be in turbulent flow if the Reynolds
Number is above 4000. Between a Reynolds Number of 2000 and 4000
the stream will be in a transition mode of flow.
As the stream velocity is increased sufficiently to just leave the lam-
inar flow regime, a thin laminar layer of water may form if the coup-
ling between the air flow in the atmosphere and the water flow is fav-
orable. If this condition exists, then the rate at which oxygen will
enter the water is fixed by the rate at which the oxygen can move
across the thin laminar layer by molecular diffusion in view of the
fact that once the oxygen has penetrated the laminar layer, it will be
transported by the turbulence eddies which is extremely more rapid
than molecular diffusion. The physical situation is depicted in Fig. 2.
The situation depicted in Fig. 2 is the well-known "liquid film" model
14
-------
Air-Water Interface
Laminar Layer 6
^~^s_ -^Vv w I *-
Talent
0 -o
Bottom
Fig. 2 - Oxygen Transfer under conditions where the
liquid film controls the mass flow rate
for diffusion of a slightly soluble gas into a liquid originally proposed
by Lewis and Whitman [?].
If the assumption is made that sufficient mixing takes place in the tur-
bulent flow zone to make the DO uniform in this region at any time,
then the rate of flow of oxygen from the atmosphere to the water is
again given by Pick's Law, Eq. 4:
= -D
O L Bz
L*
For a thin layer this equation can be simplified by replacing the con-
centration gradient, dC/3z, with AC/ Az where
Ac c - cq
= - * (Eq. 10)
where C = uniform oxygen concentration in turbulent zone
C = surface oxygen concentration, to be taken as the
saturation value
6 = thickness of laminar zone
Substituting Eq. 9 into Eq. 4 yields:
= (C.-C) (Eq. 11)
Again as in the case of Eq. 1 , it is convenient to rewrite Eq. 11 as a
product of a coefficient (the liquid film coefficient) and a "driving-
force" (the dissolved oxygen deficient), thus Eq. 11 can be expressed:
15
-------
(Eq. 12)
where K = D /6 for this case
Although it is difficult to model the quiescent layer and thereby arrive
at a value of 6, a conservative estimate (conservative in the sense that
it will predict values of 6 on the high side and the reaeration rates on
the low side) can be made by replacing the air with a smooth solid
boundary and then comparing the stream flow to a flow in a smooth
pipe the diameter of which is equal to four times the hydraulic radius.
For such a model the thickness of the laminar sublayer (6) adjacent to
the smooth walls is given by the relationship [8j:
6 = - ft (Eq. 13)
P To
where p = fluid density in slugs/ft
H - fluid viscosity in # sec/ ft
T - fluid shear stress at the walls in #,,/ft
o /
The shear stress at the wall of a pipe may in turn be expressed as a
function of the pipe friction factor, fp, for fully developed turbulent
flow and is given in the expression:
where V = flow velocity in ft/ sec
f = friction factor for smooth pipe, turbulent flow and
is a function of Reynolds Number [8j
Thus Eq. 13 can be rewritten with the aid of Eq. 14 to yield:
5D fe
=
where ReyRV =
Before calculating an example, it should be pointed out that the present
model can have validity only in or near the transition region. When
the stream turbulence is increased sufficiently beyond the transition
zone, the existence of a laminar or quiescent layer at the air-water
interface can no longer be assumed to represent the actual flow situa-
tion. Rather at sufficiently high levels of turbulence, the eddies will
16
-------
penetrate to the air-water interface and thus provide a far more rapid
mechanism for transporting surface captured oxygen away from the
interface than is provided by molecular diffusion in the present model.
For an example of an application of the present model, consider a two
foot deep stream initially at zero DO (air at 20°C) and moving at Rey-
nolds Number 2000, 4000, and 40, 000. For this case Eq. 12 and Eq.
15 yield an oxygen flow rate from the atmosphere to the water as given
in Table IV.
TABLE IV
Oxygen Transfer Rate across a Thin Quiescent
Layer of Surface Stream Water
V Surface Stream Water
Quiescent, Rey (River Velocity) M /A, #m/ft^ sec
HY ft/sec °2
2, 000
4, 000
40, 000
. 003
. 006
. 06
-1 1
4. 76 x 10
-10
1. 068 x 10
-10
7. 93 x 10
As the stream velocity is increased sufficiently, the turbulent eddies
will penetrate to the air-water interface. As a result the previous
process by which oxygen captured at the air-water interface was trans-
ported to the bulk of the water; namely, molecular diffusion across
the quiescent layer, will be replaced by the more rapid process of
eddy diffusion. To describe this process, a surface replacement
theory has been proposed and developed by several workers, most
notably by Higbie [4] and Danckwertz [9]. The replacement theory
was proposed to overcome the difficulty of accepting the existence of a
laminar or quiescent layer under conditions of strong turbulent flow in
the main body of water. In his 1951 paper, Danckwertz points out:
"The fictitious nature of the "liquid film" is prob-
ably widely suspected; nevertheless, it is constantly re-
ferred to as though it actually existed. This may be re-
garded, for many purposes, as a harmless and convenient
useage, as measured absorption rates appear to conform
to the expression:
^ = K_ (C - C )
A L s o
17
-------
where K-^, the liquid-film mass-transfer coefficient, is
constant for a given liquid and gas under given conditions.
However, if the film is in fact an unrealistic one, it may
lead to erroneous results if it is used as the basis of theor-
ies which seek to relate KL to the conditions of operation. "
In place of a laminar surface layer which always contains the same
liquid, Danckwertz assumed the existence of a laminar layer of non-
fixed identity, that is, he assumed the liquid surface layer was con-
tinually being replaced with fresh liquid (liquid from the turbulent
zone). Before this replacement theory can be applied to streams and
ponds, a method must be available for estimating the rate of surface
renewal (r) and the coefficient KL> Considerable effort has been dev-
oted to this task, notably by O'Connor and Dobbins [10, 11, 12 ],
In their 1956 paper O'Connor and Dobbins developed an expression for
the coefficient K^ and two expressions for the surface renewal, one
applicable to isotropic turbulence and one applicable to non-isotropic
turbulence. Later in 1958 O'Connor [llj pointed out that the equation
developed for the isotropic case was the more generally significant
one and that the equation for non-isotropic turbulence could be omitted
from consideration,, The equation for the liquid film coefficient given
by O'Connor and Dobbins for water deeper than . 04 cm is. (See Eq. 27,
Ref. 10):
KL =
and their expression for the surface renewal rate for isotropic turbu-
lence is (See Eq. 36, Ref. 10):
r = Jj (Eq. 17)
where u = average stream velocity
H = average stream depth
In the development of Eq. 16 the authors point out that two assump-
tions are required. First, that the concentration is uniform throughout
the depth of water and second, that the concentration does not change
with time. The first assumption will be satisfied if the river flow is
turbulent. The second assumption will be satisfied if the time re-
quired to renew the interface surface (1/r) is short compared to the
time necessary to make a substantial change in the DO concentration.
Using Eq. 16 and Eq. 17 the rate of oxygen transfer as given by the
renewal theory becomes:
18
-------
= K_ (C - C) (Eq. 18)
_L_j S
where K = v for this case
i-i H
C = bulk average DO concentration
In 1964 Dobbins [12J developed an analytical expression for the sur-
face renewal rate to be used in place of the semi-empirical express-
ion given in Eq. 17. In Dobbins' analytical expression the energy dis-
sipated per unit time per unit mass of water must be determined as it
plays a dominant role in the expression. Although the analytical ex-
pression is more general than Eq. 17 and as a result yields correla-
tions over a wider range of conditions, its use in general is hampered
by its lack of simplicity. In addition it should be noted that the analy-
tical expression and Eq. 17 yield about the same degree of agreement
between predicted and measured values of the liquid film coefficient
for natural streams but the use of the analytical expression results in
far more accurate predictions of aeration in laboratory channels as
may be seen from Fig. 3 '^) as adapted from Ref. 12.
Because of the simplicity and reasonably good agreement that Eq. 17
provides for natural waterways, we shall use this equation rather than
the more general analytical expression given by Dobbins.
It is helpful to compare the results of the previous example used in
the situation where a laminar surface layer might be assumed (low,
but turbulent Reynolds numbers) to the results given by the surface
renewal theory. Thus we again consider a two foot deep stream
(3) In making an observation on the oxygen transfer into a natural
body of water, a question arises as to the surface area to be used in
the calculation. Either the true interfacial area As can be used or the
horizontal projected interfacial area A . The two areas are related
by the following product of area and coefficient:
K A = K' A
L, s L o
where K' = apparent liquid film coefficient
Ljt
K = actual liquid film coefficient
L-t
The value of K' is plotted in Fig. 3.
19
-------
Closed Points, Data from
Laboratory Channel
Dobbins Analytical
Expression for Film Replacement
j. :o
Reported K^ , ft/day
O Open Points, Data from
" i Five Rivers
X>
100
Oj
T)
Me
a
0
U
O/Connor - Dobbins Eq. 17
? 10 100.
Reported K| , ft/day
Fig. 3 Comparison of Reported and Calculated
Apparent Liquid Film Coefficient
(Adapted from Ref. 12)
20
-------
initially at zero DO (air at 20°C) but the stream is now assumed to
move at Reynolds Numbers of 4, 000, 40, 000, 400, 000 and 4, 000, 000.
For this case the oxygen transfer rate may be found with the use of
Eq. 18. A summary of these results is given in Table V.
TABLE V
Oxygen Transfer Rates for Turbulent Rivers
u id^
Rey River Velocity
ft/ sec
4,
40,
400,
4, 000,
000
000
000
000
6
. 006
. 06
.6
. 0
2
A
5.
1.
5.
1.
#m/ft
07
60
07
60
x
x
X
X
1
1
1
1
2
0
0
0
0
sec
-9
-7
All of the oxygen transfer rates discussed so far, namely, (a) oxygen
capture at a "new" surface, (b) oxygen transport by molecular diffu-
sion across a thin laminar layer and finally (c) oxygen transport by
eddy diffusion at a surface being renewed at a finite rate, can be ex-
pressed in convenient units as was previously discussed in references
to Eq. 7, by the equation:
M -6 ,2
^ = 62. 4 x 10 K (C - C) #mn /ft sec (Eq. 19)
A L, s U2
in ft/sec for Case a
where K =
-------
-4
o
CD
CO
(M
CP
4->
rt
m
ne n
1 =
_J |_
/
I
onth
ero
_ c
7urt
-»-
Stres
/
Tra
Cas
mg/
ulenl
m
A
nsit
j b
Su
iter
/
Cas
f~*
fac^
/
2 C
.se
Cap
J
to c
ure
/
0. 01 1. 0 10 10 10 10
Reynolds Number based on Hydraulic Dia. =
10
10" 10~4 10"2 1 10
Velocity of Stream 2 ft Deep, ft/sec
10
10
10
Fig. 4 - Oxygen Transfer Rate for Various Rate Control Situations
22
-------
compared for convenience. The example used to construct Fig. 4 is
the previously used two foot deep river at zero DO and a value of C
equal to 9 mg/liter. In Fig. 4 the value of t selected for Case a is
one month.
Two significant observations can be made by inspecting Fig. 4. First,
it is noted that the rate at which oxygen is captured at the surface is
several orders of magnitude faster than the rate at which eddy diffus-
ion in a naturally flowing stream can carry the oxygen into the bulk of
the water even when the stream velocity is at its upper limit of about
10 ft/sec. Second, if turbulence is induced near the surface of a
quiescent stream in order to destroy a thin laminar layer or in order
to increase the surface renewal rates, the rate of oxygen transfer
from the atmosphere to the upper layer of water can probably be in-
creased by orders of magnitude.
In addition to the transfer of oxygen from the atmosphere to the water
at the air-water interface, it may also be considered desirable to in-
ject air or pure oxygen at the bottom of the water and let it diffuse in-
to the water as the bubble rises upward. Likewise aeration may be
accomplished by spraying the water into the air.
The engineering and economic characteristics of the various aeration
devices will be discussed in detail in the following sections.
It should be noted that the liquid film coefficient is known to depend on
the physical and chemical characteristics of the water as well as on
the flow field. In particular, the film coefficient Kj^ undergoes sig-
nificant variations with water temperature and the presence of sur-
face-active agents. These two variations are discussed briefly below.
The liquid film coefficient is known to increase with temperature.
Correlations for this variation of the form
T ? 0°C
r
where T = water temperature in C
C = a constant coefficient reported in the literature
to vary between 1. 016 to 1. 037 [l3j
If surface-active agents are present, their molecules will orient
themselves on the air-water interface and create a resistance to
molecular diffusion of oxygen across the interface. The resulting de
crease in the liquid film coefficient has been measured and can be as
high as a factor of two.
23
-------
V. DISSOLVED OXYGEN CONCENTRATION IN
QUIESCENT STREAMS AND PONDS
Both streams and ponds experience a hydrodynamic annual cycle
which may contain substantial periods of time during which conditions
are unfavorable for natural reaeration. If the oxygen demand on the
stream or pond water remains high during these same periods of low
natural reaeration, the DO level may be depressed below the level re-
quired to prevent a nuisance condition from being generated.
From Fig. 4 it is obvious that the periods of low natural reaeration
occur when the flow rate drops. The severity, length and time of year
of minimum river flow rate is highly dependent on the particular site
under study as may be seen from the sample of annual hydrographs
shown in Fig. 5 [from Ref. 14]. In general it is not unusual to find a
variation of two orders of magnitude or more between the minimum
and maximum stream or river flow rate during the year. For exam-
ple, the Mississippi River at St. Paul, Minnesota, has a minimum
flow rate of 632 cfs, a maximum of about 176, 000 cfs and an annual
average of 9, 800 cfs [15]. Although the velocity of the stream will
vary with depth as well as flow rate, stream velocities will range be-
tween 0. 01 to 10 ft/sec. From Fig. 4 it is seen that the natural oxy-
gen transfer rate also varies by about a factor of 1000 over this range
of velocities. Although the natural reaeration rate may be sufficient1
high to maintain the DO at acceptable levels during periods of modest
to high flow, it may not remain so during the period of low flow. Dur-
ing these periods artificial stream aeration could be used to supple-
ment the natural aeration in order to offset the BOD load in the stream
sufficiently to maintain the required DO level.
Lakes and ponds located in regions where the water temperature can
fall below the value corresponding to maximum density (4°C) exper-
ience an annual cycle which consists of two periods of vigorous verti-
cal circulation, one in early winter when the water freezes and one in
the spring when it melts. The oxygen transfer rate during these two
periods is very high and the entire body of water is often assumed to
reach the saturated DO level during these periods [161 Between
these two periods if the body of water is of sufficient depth, it may
divide into an upper layer (the epilimnion) of warm rather turbulent
water and a lower layer (the hypolimnium) of cold relatively undisturb-
ed water. The two zones are separated at the so-called thermocline,
or the location of maximum temperature gradient. Since oxygen can
be transported across the thermocline only by molecular diffusion,
which is very slow, very little oxygen will be suppliad to the
25
-------
30
20
Chambers Cr. at Corsicana, Tex
- 958 sq ml
194445
Trinity River at Romayor, Tex
17,200 sq mi
194445
60
40
20
0
American River at Fairo
1Q?1 ^n mi
X
193
JL
wV_
7 38
A lA
(JA.
aks, Ca
jU^
f.
A
-xj
Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug.
Sept
Fig. 5 - Annual Hydrographs for Three Stations
(after Ref. 14)
26
-------
hypolimnium after the thermocline is established. On the other hand,
a shallow body of water will exhibit no such division into two layers by
a thermocline.
The depth at which a thermocline will form during the summer is de-
pendent on several factors including the wind mixing, evaporative
cooling, back radiation, and solar and atmospheric radiation. De-
pending on the relative interaction of these factors the thermocline
may be found at a depth of approximately 1 0 to 50 feet.
During mid-summer the wind velocities are at their minimum and
solar radiation is at its maximum. Under these conditions a well-
sheltered pond may become quiescent with the result that the atmos-
pheric oxygen transfer rate may decrease to values approaching that
shown as Case a in Fig. 4.
27
-------
VI. DEVICE OPERATING CHARACTERISTICS
In this section the mode of operation of existing and new aeration de-
vices is discussed and their operating characteristics are presented.
Where possible the operating characteristics are developed analytic-
ally in order to provide an insight into how these characteristics
might be optimized.
Sub-Surface Devices
Based on the fact that the rate of oxygen capture at the air-water inter
face is orders of magnitude faster than either molecular diffusion a-
cross a thin quiescent layer of thickness that might be anticipated in a
stream, or the eddy diffusion that results from the usual turbulence
present in a body of water, it can be concluded that oxygen transfer
can be induced efficiently (that is, at high values of the ratio of mass
of oxygen transferred to energy expended) by circulating the water
with a sub-surface device so that "new" or "fresh" surfaces are rap-
idly presented to the atmosphere where they can quickly capture oxy-
gen and subsequently distribute this oxygen throughout the body of
water as a result of the induced circulation. The objective of the de-
vice is thus two-fold: first to enhance the rate of surface renewal at
the air-water interface, and second to induce as large a circulation as
possible. Both objectives are to be achieved at minimum power. In
addition to the efficiency with which the device induces oxygen trans-
fer, it will also be convenient to compute the oxygen flux density for
the device, that is, the rate at which the device will cause oxygen to
flow across unit surface area per unit time.
Since it is desired to set large volumes of water into circulation with
as low an expenditure of energy as possible, the use of large diameter
slow moving propellers is indicated. That this should be the choice
may readily be seen by forming the ratio of momentum flux to kinetic
energy flux across the device, namely, [(pAV)V] / [(pAV)(V2/ 2) J and
noting that the ratio will be high for low fluid velocities and that the
mass flow can be simultaneously kept high if the area is made large.
The analytical technique employed to estimate the operating character-
istics of sub-surface devices was to simulate the device by a series of
two-dimensional potential flow functions so arranged as to satisfy the
boundary conditions at the bottom of the body of water and at the air-
water interface. Once the appropriate potential functions had been
established, they could be used to compute the flow field. Since the
flow velocity at a point will decrease exponentially to zero as the point
29
-------
is moved further away from the potentially modeled sub-surface de-
vice, the limit of the zone of induced circulation must be defined at
some low value of the velocity - in this case 1% of the centerline velo-
city was selected. Once the flow field has been calculated, the oxy-
gen transfer rate can be estimated by application of the renewal theory
as given by Eq. 18.
As a first approximation the following assumptions are made: the
bottom of the body of water is horizontal, no side walls, no density
gradients, no benefit from induced flow that persists beyond the zone
of active circulation, and no oxygen consumption processes take place
in the water. Using the above assumptions, three generalized sub-
surface aerator configurations were simulated by two-dimensional
potential flow theory. The three configurations are summarized below
and are shown schematically in Fig. 6.
Case I
This generalized configuration consists of a vertical duct through
which the water is forced to flow upward and is then allowed to
discharge somewhat below the air-water interface. After dis-
charging the water will flow away from the vertical centerline of
the duct near the bottom of the body of water. The flow was simu-
lated by infinite series of sources and sinks placed along the ver-
tical axis of the duct.
For this configuration the duct width has been made small com-
pared to the water depth so that Case I can be used to model a
large ducted propeller in a deep body of water, for example, a
ZO-foot wide duct in a 100-foot deep pool of water. It should be
pointed out that Case I can also be used to demonstrate the behav-
ior of a small width duct placed in a shallow body of water. Al-
though the last situation can be ruled impractical, it will allow
Case I to be used to quantitatively demonstrate the influence of
duct size.
Case II
Geometrically this configuration is the same as Case I; however^
in contrast with Case I the duct width for Case II has been made
large compared to the depth of the water. As a result Case II can
be used to model wide ducts in shallow water. The flow was sim-
ulated by several infinite series of sources and sinks placed along
lines parallel to the vertical axis of the duct.
30
-------
Duct Width = 0. 2H
~" Water surface
Induced
Aeration Zone'
Bottom
CASE I FLOW
Duct Width
Width of Induced
CASE II FLOW
Width of Induced
CASE IE FLOW
Fig. 6 - Schematic Sketches of Subsurface Aerators
31
-------
Case in
In this configuration water is forced to flow through a horizontal
duct and completes its circuit by passing between the outside top
of the duct and the air-water interface. The duct may be placed
on the bottom of the body of water as shown in Fig. 6 or it may
be suspended at some distance below the air-water interface.
The flow was simulated by two infinite sets of sources and sinks
placed along vertical lines passing through the duct outlet and
inlet respectively.
The duct length has been made long compared to the water depth
in order to use this case for application to shallow water such as
a stream or river.
The details of the potential flow calculations and a discussion of the
assumptions are given in Appendix B. The results of these calcula-
tions are discussed below in a brief fashion for convenience.
CASE I
The size of the circulation zone induced by the sub-surface device
shown as Case I in Fig. 6 can be established by computing the verti-
cal velocity along the horizontal line that passes through the mid-
height point of the duct. From the symmetry of the device, it is noted
that there will be no horizontal component of velocity along this line
and thus the strength of the vertical component is a measure of the
intensity of circulation at any given point. The vertical velocity along
this horizontal centerline is shown in Fig. 7.
As mentioned previously, the velocity will decrease exponentially with
distance from the duct and some arbitrary low velocity will have to be
assumed to represent the end of the circulation zone. In this case the
end of the zone was selected to be j^ 2 H for a = . 1 H and a = . 01 H
(where the velocity has decreased to about 1% of its value near the
wall of a long duct as shown in Fig. 7).
An inspection of Fig. 7 shows that although the zone of circulation de-
creases as the duct length is made shorter, the decrease is not signi-
ficant for values of "a" between 0. 01 and 0. 1. In addition it is shown
in Appendix B, based on a constant circulation zone width of 4H, that
the rate of oxygen transfer from the atmosphere to the water also de-
creases with decreasing duct length as shown by Eqa 21.
32
-------
00
H 2H
Distance from Axis of Duct
-------
/ 1/21/2
M = 50. 6 x 10" (C -C)V H #m /hr per foot of device
2 2 (For a = 0. 01H)
(Eq. 21A)
!M = 47 4 x 10 (C -C)V H #m_ /hr per foot of device
°? S 2
2 ^ (For a = 0. 1 H)
(Eq. 21B)
Jfc = 18. 8x!0"6(C -C)V H #rn /hr per foot of device
°? S 2
Z (For a = . 45 H)
(Eq. 21C)
where C = DO saturation concentration in mg/liter
s
C = DO concentration in mg/liter
V = fluid velocity in duct, ft/sec
H = water depth, ft
It can be concluded from Eq. 21 that higher oxygen transfer rates can
be achieved by having the duct extend close to :the water surface and
that the lack of any duct around the propeller would seriously degrade
the performance of the device.
The oxygen capture efficiency, TI, for this device was estimated in
Appendix B and is given by Eq. 22 below in pounds of oxygen captured
per shaft hp-hr expended.
T? = 0. 143 (C -C)H~1'2V~5/2 #m /hp-hr (For a = 0. 01) (Eq. 22A)
°2
7? = 0. 134 (C -C)H~ V"5 #m /hp-hr (For a = 0. 1) (Eq. 22B)
S °2
T?=0.053(C -C)H~1/2V~5/2 #m_ /hp-hr (For a = 0. 45) (Eq. 22C)
°2
where C , C are in mg/liter
s to
H in ft
V in ft/sec
CASE II
The vertical velocity along the horizontal line that passes through the
34
-------
duct mid-height point may again, as in Case I, be used to measure
the zone of induced circulation. This velocity is plotted in Fig. 8 as
a function of distance from the vertical centerline. Again using this
plot as a guideline the circulation zone is assumed to terminate at
_+ 5H for total width of 10 H. Based on the results of Case I, only one
value of "a" was selected, namely, a = 0. 1 H.
In a manner similar to that used for Case I, the oxygen transfer rate
and oxygen capture efficiency were estimated in Appendix B and are
given by Eq. 23 and Eq. 24 respectively:
til =(.007)(C -C)V H #m /hr per foot of device (Eq. 23)
°2 S °2
TJ=0. 657 (C -C)H~1/2V~5/Z #m /hp-hr (Eq. 24)
S 2
where C , C are in mg/liter
S
V in ft/sec
H in ft
CASE III
The zone of induced circulation for this configuration may be estimated
by computing the magnitude of the horizontal velocity component at the
air-water interface as a function of distance away from a vertical line
that passes through the mid-length point of the duct. This velocity is
shown in Fig. 9 for one value of b, namely b = 10 H. From Fig. 9
it is seen that the zone of active circulation is approximately equal to
the width of the duct. The potential flow equations developed in Ap-
pendix B indicate that the size of the circulation zone and the oxygen
transfer rate per unit length (that is, unit depth into the paper as shown
in the schematic given in Fig. 6) decrease as the duct width is de-
creased similar to the manner previously discussed for Case I. Again
the conclusion can be drawn that omission of a duct around the pro-
peller would seriously degrade the performance of the device.
The oxygen transfer rates and capture efficiencies for Case III were
estimated in Appendix B and are given below in Eq. 25 and Eq. 26 res-
pectively:
M =. 000524 (C -C)V1/2H 2 #m /hr per foot of device
? S 2 FT
L (For b = 10H, d = j)
(Eq. 25)
35
-------
IH
2H
3H 4H 5H
Distance from Axis of Duct
7H
-------
0.2b
4b 0. 6b 0.8b 1. Ob
Distance from Midplane of Duct
. 2b
1.4b
-------
1 /o c/2 H
« = 0. 589 (C -OH" V" #m /hp-hr (For b = 10H, d = )
s °2
(Eq. 26)
where C and C are in mg/liter
s
V = duct velocity in ft/sec
H = twice the water depth in ft
Although a two-dimensional potential flow model was used in order to
develop expressions for the oxygen transfer rates and capture effic-
iencies, it does not seem unreasonable at this stage of analysis to as-
sume that the performance of an actual three-dimensional device (that
is, a circular duct) can be estimated by multiplying Eqs. 21, 23, and
25 by the diameter to obtain the transfer rate per machine and to use
Eqs. 22, 24 and 26 as given.
If in addition to the induced flow a natural flow exists in the body of
water which tends to transport the water past the sub-surface aeration,
some benefit in the transfer rate and capture efficiency will be derived
from the turbulence which was initiated at the aeration station and
which will require some finite time to dissipate.
The use of a two-dimensional model to simulate an actual device
which is circular tends to overestimate the transfer rates and capture
efficiencies whereas neglecting the benefit from the turbulence that
persists beyond the aeration station tends to underestimate the same
value.
The influence of a sloping stream or pond bottom on the circulation
pattern (and hence on the oxygen transfer rates and capture efficienc-
ies) may be investigated by making suitable alterations to the potential
flow functions developed for the case of a horizontal bottom.
For an infinitely deep body of water the flow discharging from the out-
let of a vertical pipe at a distance "a" below the surface (as in Case I)
can be simulated by two sources, one a distance "a" below the inter-
face and one a distance "a" above the interface as shown in Fig. 10-a.
The source above the interface is the so-called mirror image source
used to cancel flow across the interface and thus satisfy an imposed
boundary condition. On the other hand, if the water has a finite depth,
H, the flow at the duct entrance must be simulated with a sink. In
order to satisfy the boundary condition on the bottom, a second sink
must be added a distance "a" below the bottom as shown in Fig. 10-b.
However, these two new sinks must have mirror image sinks relative
to the air-water interface and likewise the two sources must have
38
-------
-t-
Fig. 10-a
Air
^/>XV»n
Water
Bottom
Fig. 10-b
-4 Infinite Series
Bottom
Fig. 10-c
(Infinite Series
+ ?
Fig. 10 - Source-Sink Arrangement to Simulate Circulation
Pattern Induced by a Vertical Duct and Propeller
in a Body of Water with a Horizontal Bottom
39
-------
mirror image sources relative to the bottom for boundary conditions
to be satisfied. The problem of simulating flow in a vertical duct loc-
ated between the air-water interface and the bottom leads therefore to
the use of an infinite series of sources and one of sinks all placed
along a vertical line as shown in Fig. 10-c.
If the vertical line along which the sources and sinks are located is
regarded as the circumference of a circle of infinite radius, then the
same technique can be adapted for the simulation of a bottom with a
slope. This technique can be applied only to a discrete number of bot-
tom slope angles but this presents no difficulty in estimating the influ-
ence of circulation patterns. The method consists of placing the
sources and sinks along the circumference of a circle as shown schem-
atically in Fig. 11 whose radius is given by the expression
r =--,
o tan Qi
where Oi is the angle that the bottom makes with the horizontal
H is the water depth, ft
The size and intensity of the flow field created by a source and sink
located near the top and bottom of a body of water with a bottom that
slopes at an angle OL can be conveniently characterized by moving
away from the vertical centerline of the body of water along the "mid-
line" (see insert a and b on Fig. 1Z) and calculating the velocity at
right angles to the mid-line. A plot of this velocity is shown in Fig.
12 for three bottom slopes, namely, 0°, 11° and 45°.
An inspection of Fig. 12 shows that the zone of circulation is extended
to larger distances for small angles (~ 11°) with a subsequent decrease
in the mid-line velocity near the centerline of the pond as the case
must be since the source and sink strength are maintained constant
for the three cases shown in Fig. 12. When the slope angle is large
(~ 45°), the zone of circulation is decreased and the mid-line velocity
near the pond centerline increased to compensate. However, the
change in the size or intensity of the flow field is not significant even
for angles as large as 45°. Thus it may be concluded that the influ-
ence of the bottom slope on the circulation pattern can be neglected
unless the slope is very steep (~ 45°) in which case the zone of circul-
ation is decreased by the confining "side-wall" formed by the sloping
bottom.
The influence of a vertical density gradient on the potential flow pat-
terns induced by sub-surface devices is important during the period in
40
-------
Air-water interface
Of
Fig. 11 - Source-Sink Arrangement to Simulate Circulation
pattern induced by a Vertical Duct and Propeller
in a Body of Water with a Sloping Bottom
41
-------
0 \l
H 2H
Distance from Vertical Centerline Measured along Mid-line
Fig. 12 - Velocities with Sloping Bottom
3H
(M
-------
which the sub-surface aeration device is first placed in operation. As
the operating time increases any initial density gradients will be des-
troyed and the final circulation pattern will be approximated by the
potential flow fields previously discussed.
A literature survey was conducted on the influence of vertical temper-
ature gradients on potential flow patterns; however, very little mat-
erial has been found in the open literature on this topic and the task of
obtaining analytical solutions does not seem feasible at present. Al-
though it seems possible to develop a computer solution using numer-
ical techniques, the necessary effort and cost to do so appear excess-
ively high compared to the contribution that would result to the overall
program. As a result we have decided to develop approximate order-
of-magnitude expressions to deal with temperature gradient effects
and to describe the situation phenomenonologically in as much detail
as possible. These guidelines will enable us to estimate the rate at
which a temperature gradient is destroyed by the sub-surface devices
as well as the limit to the volume that will be affected.
Destratification of impoundments has received considerable attention
in recent years because of its adverse influence on the quality of water
stored below the thermocline [l?J. One of the earliest experimental
efforts to destratify a body of water was reported by Hooper; Ball and
Tanner in 1952 [18]. In their scheme water was pumped from the
hypolimnion to the surface of the lake by an on-shore centrifugal pump.
Extensive tests with mechanical pumping as a destratification tech-
nique have more recently been reported by Irwin, Symons and Robeck
[19]. Their scheme consisted of pumping cold water from near the
bottom up through a vertical duct surrounding the pump and discharg-
ing it near the air-water interface. The raft-mounted equipment of
Irwin et al introduced far less hydraulic losses than the earlier shore-
mounted equipment of Hooper et al.
When the cold hypolimnion water is discharged near the top, it mixes
with the warmer epilimnion water to produce a jet of water with a tem-
perature (and hence density) between the extreme values associated
with water originally at the very top and very bottom. As a result this
jet will descend to a water depth where the density is equal to that in
the jet. Since it is then in a neutrally buoyant configuration, the jet
will not descend any deeper but will continue to move in a horizontal
plane away from the vertical duct because of the influence of the water
being continually pumped through the duct. Such stratified flows have
been studied experimentally and analytically because of their import-
ance to water quality in impounded water and in the design of power
43
-------
plant cooling water intake structures [20,21,22]. If there is substant-
ially no mixing of the epilimnion and hypolimnion waters across the
thermocline, then there must be a drainage of cold hypolimnion water
along the bottom into the duct intake of such magnitude to equal the
rate of water addition to the epilimnion. Thus if the duct is located at
the deepest point of the body of water, these two counter flow currents
will continue (and the thermocline location will continue to drop) until
the lake is completely mixed. As the mixing nears completion, the
counter-flow currents will deform into the potential flow pattern pre-
viously calculated. In contrast to the potential flow currents, the
counter-current stratified flows will be slower but will extend to far
greater distances from the duct. Precisely how far away from the
ducts that the stratified flow will persist is difficult to assess analyti-
cally. However, from several experimental field projects it is known
that these distances are many orders of magnitude greater than the
diameter of the zone of circulation predicted by potential flow theory
for the same device operating in water with no vertical density grad-
ients. For example, in Ref. 19, the experimental destratification of
a long narrow lake, Lake Vesuvius in Ohio, is discussed. This lake
is about 16, 000 feet long and not more than 800 feet wide, with a max-
imum depth of about 30 feet and a surface of 105 acres. The pump
was located approximately 3000 feet from one end in about 30 feet of
water. For the duct diameter of 1 foot used in the experiment, poten-
tial flow theory as developed for Case I indicates a radius of about 50
feet for the zone of circulation whereas the data presented in Ref. 19
indicated that the stratified flow was felt as strongly at least 10, 000
feet away from the duct as within 50 feet (See Fig. 12) but with a time
lag of a few days.
The pumping at Lake Vesuvius was stopped after 208 hours at which
time the temperature at any given depth was almost the same at any
location on the lake. Since the pumping was discontinued, no evidence
is available from this experimental work on the transition from strat-
ified flow to a flow pattern that can be approximated by potential flow
theory.
However, Paul D, Uttermark has commented on the mechanical pump-
ing of a very similar lake in Wisconsin [23]. Uttermark pointed out
that the Wisconsin Conservation Department had been continuously
pumping a 100-acre, 28 foot deep lake with a very high volume flow
rate pump. He noted that in order to draw water from any great dis-
tance (over 100 feet) into the pump, the body of water must be strati-
fied. Once the lake was mixed, a mixing cell developed in the immed-
iate area of the pump. As Uttermark stated:
44
-------
M t» JO
After 2. 5 Days of Pumping
Fig. 13. Temperature Profiles at Various Stations
in Vesuvius Lake (from Ref. 19)
IB 20 22
TEMPERATURE. 'C
After 8. 5 Days of Pumping
Fig. 14. Temperature Profiles at Various Stations
in Vesuvius Lake (from Ref. 19)
-------
"We found that during the summer of 1967 we created a
mixing cell in the lake. We were able to pump water very
rapidly in one location but the rest of the lake was not affected
at all . . . The horizontal temperature differences from in-
side this highly mixed area to just 100 feet away were about
5°F. In the mixing cell, however, we had a very uniform tem-
perature from the surface of the lake to the bottom."
In the lake discussed by Uttermark pumping had been conducted con-
tinuously for sufficient time to pass from stratified counter-cur rent
flow through the transition phase into a region that can be approxim-
ated by potential flow. As noted by the author, the fact that a "cell"
developed is most likely the result of an equilibrium being established
between the pumping rate (which destroys stratification) and contin-
uous surface heating of the lake during the summer (which induces
stratification).
The nature of this equilibrium configuration will depend on the pump-
ing rate and the net heating of the lake. The influence of these factors
can be easily noted by considering two extreme cases. First, consid-
er a strongly stratified lake in the summer and a small pumping rate.
As cold water is discharged near the surface it mixes and settles to a
level of equal density. Since the flow rate is low, the density of the
jet will be dominated by the epilimnion water. If the pumping rate is
sufficiently low, stratified flow may continue all summer. Second,
consider a lake which is only slightly stratified (as might exist after a
few days of pumping an initially strongly stratified lake) and a very
high pumping rate. When the colder water from the bottom is dis-
charged near the top, its density can be changed only slightly because
of the relative abundance of colder water. As a result the buoyant
forces are insufficient to overcome the suction exerted at the duct in-
take and a "cell" or potential source-sink flow is established.
It should be noted that the oxygen transfer rate for stratified flow (for
the same DO deficit) can not be greater than for potential flow. Al-
though the flow extends over a considerably larger area for the strati-
fied case, it does so at a strata below the surface and at very low vel-
ocities such that no additional surface renewal can be anticipated at
the greater distances.
For the present study it is of interest to be able to predict the approx-
imate time required to pass from stratified flow to the potential flow
condition. This is most readily accomplished by examining the energy
required to destratify a body of water. The energy required to totally
mix a lake can be estimated as follows:
46
-------
Assume a stratified lake may be represented to two separate, homo-
geneous strata of temperatures , densities and thicknesses , T , p , a
and T , p^, b. In the stratified state the gravity center of each layer
lies in the layer midplane. The gravity center of the total stratified
system lies somewhere between, at a distance X from the bottom.
After mixing, the gravity center lies in the midplane of the lake. The
work required to mix the lake is then the change in potential between
the stratified and mixed conditions. This quantity is frequently term-
ed the "stability" of the lake and is the work required to raise the total
water weight the distance
a+b v ,
* = - X , f t (Eq. 28)
The distance X may be computed by equating to zero moments taken
about the stratified systems' gravity center.
P2a(b + |) + (P1b2)/2
. 29)
The distance between mixed and stratified centroids is then:
(ab)/2(px -p )
* - a + b > ft (Eq- 30)
The work required for mixing, W, is
W = A(a+b)gp3C , #f-ft (Eq. 31)
where p., is the density of the mixture, slugs /ft
^ 2
A is the lake surface area, ft
g is the acceleration due to gravity, 32. 2 ft/ sec
Equations 30 and 31 can be combined to yield
Ap (a+b) ab(p - p )g
Since p « p w p , this can be approximated by
1 Lt J>
W = ^-Aab(p1 -pz)g , #f-ft (Eq. 33)
In terms of lake volume, I, Eq. 33 becomes
47
-------
. -p2'« V* (Eq- 34)
3
where I is the lake volume in ft
Note that this analysis assumes the lake is of uniform depth, which is
not usually the case. Since the epilimnium thickness, a, is probably
uniform across most of the lake, an average value of b should be
us ed.
The time required to mix a lake can be estimated by equating the
product of power input to the lake and time to the total work required.
Assuming the pumping power is held constant, this condition can be
expressed as:
where DE = "de stratification efficiency" or the ratio of energy
input to the device to the energy required to shift
the center of gravity, expressed as a fraction.
P = input power, (# -ft)/ sec
t - pumping time required to destratify, sec
Since the "destratification efficiency" (DE) involves losses due to
pump efficiency, friction in the duct, and kinetic energy of the flow as
well as the induced mixing pattern, it will be dependent on the water
velocity in the duct. Lower duct velocities will give higher efficien-
cies although exit velocities must be high enough to induce sufficient
mixing near the surface to impart a buoyancy to the water from the
hypolimnium. Higher duct velocities will yield lower efficiency first
because of duct and pump losses and second, if the duct velocities are
sufficiently high, the overabundance of cold water may cause recircu-
lation of the pumped water to the duct inlet.
Based on experimental work with equipment that had a high duct veloc-
ity (8. 3 ft/ sec), Symons, Irwin and Robeck [24j reported a DE of
0. 0014 (or 0. 14%) for a 96 acre 25 foot deep (average) lake destrati-
fication (Boltz Lake in Kentucky). As the authors noted, kinetic ener-
gy losses in this equipment were high due to the excessive duct veloc-
ity and it would not seem unreasonable to anticipate values of DE at
least an order of magnitude higher for low velocity equipment.
Equations 34 and 35 may be combined to obtain an expression for the
48
-------
time required to destratify a lake, namely,
t =
I ab (pl -
2(DE)P (a+b)
sec
(Eq. 36)
To establish the order of magnitude of the time given by Eq. 36, con-
sider a 100-acre stratified lake, of average depth equal to 30 feet,
a = 15 ft, b = 15 ft, Tl = 70°F (p1 = 1. 938 slugs/ft3, TZ = 50°F (pz =
1. 940 slugs/ft3), which is to be destratified by a 1 0 hp mechanical
pump. For these conditions Eq. 36 predicts a time of 160 hours for
an assumed DE of 0. 01.
In Table VI is given a summary of the approximate destratification
times from Ref. 19 for a 16 hp pump with a flow rate of 2, 800 gpm.
TABLE VI
Measured Lake Destratification Time
Lake
Stewart Hollow
C aid well
Pine
Vesuvius
Area
(acres )
8
10
14
105
Average
Depth (ft)
15
10
7
12
Approx Time to
Destratify (hrs)
37. 5
8. 0
35. 0
208. 0
From Eq. 36 and Table VI it can be concluded that the time required
for a typical sub-surface aerator to pass through the stratified flow
mode of operation will be of the order of one week or less. During
this period the liquid film coefficient and active oxygen capture sur-
face area will not be greater than the corresponding values when the
device is operating in the potential flow mode; however, the DO deficit
near the duct outlet may be considerably higher during the initial
phase of stratified flow.
Diffused Aeration
The injection of air (or pure oxygen) at the bottom of a body of water
in order to supply oxygen for biological reactions has for some time
been applied to waste treatment plants where the BOD demand is sub-
stantial. More recently it has been applied to rivers and reservoirs
[25, 26, 27], This technique of transferring oxygen into the water con-
sists of injecting air or oxygen bubbles at the bottom of the body of
water. The bubbles rise and as they do oxygen diffuses from the
49
-------
bubble surface into the water. The diffusion process may be con-
trolled by the relatively slow molecular diffusion process if the tur-
buelnce in the water is sufficiently low or by the more rapid process
of eddy diffusion if sufficient turbulence exists.
For a single bubble rising in water the rate of oxygen transfer is
given by Eq. 8, namely
- = K (C -C) (Eq. 8)
A L/ s
where C is the uniform DO concentration at any given time
(See Eq. 7 for a discussion of the units)
In practice bubbles will be generated rapidly at a number of orifices
or diffuser plugs (porous plugs). Since the residence time of the bub-
bles in the water may be quite small, only some fraction of the oxygen
originally in the bubble may have diffused into the water before the
bubble reaches the air -water interface. As shown by Ippen and Car-
ver L28], the mass fraction of oxygen originally in the bubble which is
captured (fpn) can be found by dividing the transfer rate for all bub-
bles by the oxygen pumping rate which for air bubbles is:
K A [(C -C) x 62. 4 x 10"6]
3
where Q = volume flow rate of air, ft /sec
y - mass density of air, #m/ft
3,
A = surface area of all bubbles in the water at a
i 2
given instant, ft
C , C = dissolved oxygen concentrations , mg/liter
K - liquid film coefficient, ft/ sec
J_J
Since the surface area of all the bubbles must equal the product of the
surface area of one bubble times the number of bubbles generated per
unit time times the residence time of the bubble in the water, Eq. 37
can be expressed as the product of three nondimensional terms,
namely,
K (C -C) x62. 4x 1(T6
f -
CD T ya (Eq. 38)
50
-------
where d = bubble diameter
V = bubble terminal velocity
For a given body of water the magnitude of the depth (H) is fixed as
well as the value of the DO deficit (CS-C), with the result that one is
free to select only the size of the bubble for once this has been fixed
the value of V and K also become determined.
1 l_i
The relation between the bubble size (d) and the terminal velocity (V )
is readily found by noting that when gas is introduced near the bottom
of the water in the form of bubbles, the bubbles will experience a per-
iod of acceleration until they reach a terminal velocity as shown in
Fig. 15.
Air
"Waterbuoyant force
H
+y
drag
weight
Fig. 15 - Rise of a Bubble in Water
When the bubble has reached its terminal velocity, the net force on
the bubble will be zero, thus
,,2
w 2
= 0
(Eq. 39)
where d = diameter of the air bubble, ft
3
p = density of air, slugs/ft
a 3
D - density of water, slugs/ft
*w
g = acceleration due to gravity, 32. 2 ft/ sec
C - drag coefficient, which is defined as
DRAQ D
V = terminal velocity of bubble
51
-------
In order to solve Eq. 39 for the terminal velocity, it is necessary to
know the value of the drag coefficient CQ. The value of this coeffic-
ient is known analytically for low Reynolds number flow (Stokes Law)
and is also known experimentally over a wide range of flow conditions.
In general as the bubble diameter is increased the terminal velocity
also increases. Thus small diameter bubbles are associated with low
Reynolds numbers and large diameter bubbles with high Reynolds num-
bers. As the Reynolds number is decreased, the resident time will
increase and this by itself would result in an increase in the capture
coefficient; however, as the Reynolds number is decreased, the vis-
cous forces increase relative to the inertia forces and a thicker layer
of water can be anticipated to remain with the bubble as it rises which
by itself will decrease the capture rate since the oxygen must diffuse
across this thicker exposed layer. As a result of these considerations
it would be helpful to know the capture coefficient over a range of bub-
ble diameters and hence Reynolds numbers for application in streams
and ponds. In particular it would be helpful to have an analytical ex-
pression for the most efficient region of operation (which will be shown
to be low Reynolds number flow) to serve as a guideline for optimiza-
tion.
From experimental investigations on the terminal velocity of bubbles
rising in water, it is known L 28 J that bubbles behave like solid spheres
only up to Reynolds numbers of about 70. For Reynolds numbers up
to 1. 0, the relation between the drag coefficient (CD) and the Reynolds
number for solid spheres is known to be
24 ^ 24
(Eq. 40)
P vrr
j w T
I M
\ w
where p = density of water, slugs/ft
[I - absolute viscosity of water, # -s.ec/ft
V - kinematic viscosity of water, ft /sec
w
For the limiting case of Reynolds number = 1. 0 (bubble diameter ~
0. 12 mm), Eq. 39 can be solved for the bubble terminal velocity with
the use of Eq. 40. Thus
,, _ w
V
_
T = 18^ ' S6C (Eq. 41)
52
-------
Since the critical Reynolds number for a sphere is greater than
100, 000, the flow at these low Reynolds numbers will be laminar.
The bubble will creep through the water with a residence time of
tr = H/Vq-,. Since it is anticipated that at this very low Reynolds num-
ber the bubble will drag a layer of water with it as it rises, the appro
priate liquid film coeiticient is that given by Eq. 8, namely,
KL = V (Eq. 8)
where t = exposure time for the fluid layer
It should De pointed out that although _tne expression lor K given in
Eq. 8 will decrease with time, as 1/Vt, KL will be used only when it
is integrated over time to determine the total oxygen transfer from
the bubble. Since the integration of
t _
r dt//t
is equal to 2\/t > and since it will be somewhat more convenient to use
a constant value of K-^ which yields the correct results compared to
the exact integration procedure, the value of t to be used in Eq. 8
will be taken as (l/4)tr or (1/4)(H/VT). Thus the film coefficient can
be taken as
K^ = 2/ -^- (Eq. 42)
77
VT
Thus the liquid film coefficient, K , can be expressed by combining
Eq. 41 and Eq. 42.
2d2D g(l --^)
J-' p
w
KL = - S~n - (Eq- 43)
w
If Eq. 4.3 is used to predict the liquid film coefficient for small dia
meter bubbles, then the capture coefficient can be readily computed
by substituting Eq. 43 into Eq. 38 to obtain
100 /DT v H ' (C -C) x 62. 4x 10~6
138 / L w r s i
53
-------
An inspection of Eq. 44 clearly shows the desirability of as small a
bubble as possible in order to increase the capture coefficient. How-
ever, the rapid generation of smaller and smaller bubbles at the
diffuser plug will require progressively higher pressure drops across
the plug and hence progressively higher compression power for a
fixed mass flow rate of gas. In addition, as the number of bubbles
generated per unit time at one diffuser plug increases, there is a ten-
dency for adjacent bubbles to coalesce into large diameter bubbles.
When the flow rate is sufficiently low both Ippen et al [29] and Maier
[30] have shown that the diameter of a bubble formed at an orifice
under water is 10 to 11 times larger in diameter than the orifice. As
the gas emerges from the orifice it has a buoyancy which tends to
make it rise; however, this force is resisted by a shear force across
the orifice opening. As the gas flow rate is increased above some
threshold limit, the bubble diameter increases and the bubbles leave
in a chain-like array. Thus to produce bubbles of very small size
one must either use very small openings or provide a means to shear
the bubble off the face of the orifice before it has grown to its natural
10-11 orifice diameters. Obviously to obtain the 0. 12 mm diameters
needed to produce Reynolds numbers of 1. 0, orifice diameters of
0, 012 mm (~ . 0005") may not be practical in view of clogging difficul-
ties. However, it should be possible to approach these small bubble
sizes by use of large orifices and devices designed to increase the
shear force at the orifice face. Such work has been conducted by
many workers including Maier [30], Langelier [31] and Zieminski et
al [32].
In considering the diffusion process for bubbles, note must be taken
that Cs increases with pressure in accord with Henry's Law and hence
with depth. Thus for modes depths (< 200 ft) and air bubbles,
.21 [14.7 +
Cs ~ iCs at normal \ . 21 x 14. 7~psia (Eq' 45)
atmospheric
conditions /
where PHYD = hydrostatic pressure at depth z ft
= P Sz
The usual practice is to evaluate the gas bubble pressure and bubble
size at the mid-depth location.
54
-------
If pure oxygen is used in place of air, Eq. 37 is applicable if the fac-
tor of 0. 21 is omitted in the denominator. The remaining equations
are valid with the corresponding assumptions with the result that the
capture coefficient for pure oxygen is given by the expression
on . /D_ v H ' (C -C) x 62. 4 x l(f 6
where p = density of oxygen
°2 3
y_ = rnass density of oxygen, #m/ft
LJ
It should be noted here that although Eq. 44 and Eq. 46 differ very lit-
tle in notation form, there may exist a significant difference in the cor
rect value of the saturation concentration C to be assigned to each
equation in view of the fact that at a given temperature C increases
linearly with the partial pressure of the oxygen in accord with Henry's
Law as given by Eq. 45. For example, if the same size air and oxy-
gen bubbles are generated at some point, Cg for the pure oxygen bub-
bles will be I/. 21 times greater than for the air bubbles. As a result
the capture coefficient should be the same magnitude for air and pure
oxygen bubbles formed at the same depth and of the same size.
In order to establish the order of magnitude of the capture coefficient
for very small bubbles (Reynolds number = 1. 0, bubble diameter ~
. 12mm), it is helpful to consider the 2-foot deep body of water initial-
ly at zero DO and with C at the surface equal to 9 mg/ liter. For this
case the use of Eq. 44 indicates a capture coefficient of 0. 820. This
example indicates the possibility of high capture efficiency if the bub-
ble diameter can be made sufficiently small and it shows that a limit
exists on how small the diameter need be for a given depth. It should
be noted that in any prolonged application of diffused aeration that the
water may saturate with N^ but still have an C^ deficit with the result
that the partial pressure of O? in the bubble may decrease substant-
ially as the bubble rises and becomes almost pure N2- Under these
conditions Eq. 44 is not valid in its present form but would have to be
corrected to show how IV!Q decreases as the bubble rises due to a
falling off in the magnitude of Cg.
No data could be found in the literature relative to the capture coeffic-
ient for bubbles small enough to produce Reynolds numbers of the
order of 1. 0. Ippen and Carver [28j reported measured capture coef-
ficients as a function of depth for larger bubbles (Reynolds number
55
-------
range of approximately 300 to 900). The values reported by Ippen and
Carver are reproduced in Fig. 16 for convenience. From Fig. 16 it
is noted that for our previous example of a 2-foot deep body of water
at zero DO, an 11-fold increase in bubble diameter (from 0. 12 to
1. 32mm) results in a decrease of the calculated capture coefficient of
0. 82 to a measured value of 0. 15 or a decrease by a factor of 5. 5.
The desirability of producing small bubbles is clear.
Zieminski [32j reported capture coefficients from 0. 29 to 0. 33 for a
pilot model diffuser system specifically designed to produce small
bubbles by a shearing action at a depth of about four feet. The size of
the bubbles generated in this experimental work was not reported. For
the same depth Ippen and Carver measured a capture coefficient of
about 0. 22 for 1. 32 mm bubbles.
From Fig. 16 it is also noted that the capture coefficients for air and
pure oxygen bubbles of the same size tend to become equal only as the
water depth is increased. The authors imply that the reason for the
difference in f at low water depth is the influence of nitrogen diffus -
c o
ing from the water into the bubble at the diffuser plug, an influence
which is diminished as the column height is increased since activity at
the diffuser plug is then overshadowed by the longer main diffusion
process that takes place between the diffuser plug and the atmosphere -
water surface.
As mentioned before, increasing the flow rate through one diffuser
plug will result in the coalescence of bubbles and hence no further im-
provement in the capture coefficient. It is difficult to establish a
general criteria for the onset of this event. In order to establish a
guideline we might assume the fraction of volume below the air-water
interface occupied by air or oxygen bubbles, T] , should not exceed per-
haps a value of 0. 001. Thus the maximum air or oxygen volume flow
rate for a body of water of depth H would be
^ = 0. 001 V , ft3/ft2 sec
-t\ J-
where A = water surface area at atmosphere-water interface, ft
V = terminal velocity of bubbles, ft/sec
In order to compute the maximum mass flow rate of oxygen into the
diffuser system at low Reynolds numbers, it is necessary only to mul-
tiply Eq. 47 by . 21y for air bubbles and by yQ for pure oxygen bub-
bles and substitute Eq. 41 for VT. Thus for 2 air
56
-------
a
o
(J
Tl
-------
2
1. 167 x I0"5y d g(l - pip)
a 5X 'a 'w' ,2
, #m/ft sec
A / t w
mto W (Eq. 48)
diffuser
and likewise for pure oxygen bubbles
O,
5.56x 10-57Q d2g(l -pQ /pj
'2 ~2 " ,2
diffuser
, #m/ft sec
l/
W (Eq. 49)
When Eq. 48 or Eq. 49 is multiplied by the appropriate expression for
capture coefficient (and provided f < 1. 0), the expression for the
rate of oxygen transfer by diffused aeration is given by the following
equations for air bubbles with Reynolds numbers j< I- 0:
'/M0 \ 0. 00161 Jg 1 - p Ip )D H [(C -C) x 62. 4 x 10"6 J
cL ^V I j S
into v"w 2
water #mn /ft sec (Eq. 50)
2
where D i-s in ft /sec
H is in ft
C , C are in mg/liter
S
g is in ft/ sec
- ^2/
V is in it /sec
w
and likewise for pure oxygen bubbles:
\ 0. 00161-v/l -p_ /p )DT H [(C -C) x62. 4x 10"6]
,| 02 w L
water mo t sec
2
It is interesting to note from Eq. 50 and Eq. 51 that the oxygen trans-
fer rate per unit surface area does not depend on the bubble diameter
if the fraction of volume occupied by the bubbles is to be held at some
constant magnitude. However, the capture coefficient and hence the
oxygen capture efficiency does depend on the bubble size as will be
58
-------
discussed later.
Although the two above equations for the transfer rate of oxygen to
water from air bubbles and pure oxygen bubbles respectively appear
very similar, again it should be pointed out that the value of Cs is a
function of the partial pressure of oxygen in the bubble in accord with
Henry's Law as given by Eq. 45. As a result if air and oxygen bubbles
are generated at the same place and same size, the value of C for
s
the pure oxygen bubbles will be I/. 21 times greater than for the air
bubbles. Thus if all other factors were to remain equal, the oxygen
transfer rate for pure oxygen bubbles should be some 4. 7 times great-
er than for air bubbles generated at the same pressure and diameter.
Eq. 50 can be used to establish the order of magnitude of the maximum
oxygen transfer per unit air-water interface for air bubbles at low
Reynolds number flow. When this is done for C = 9 mg/liter and C = 0
at a depth of 2 feet, (MQ /A)max is found to be 3. 60 x lO'7 #m/ft2sec.
It is helpful to compare this value with the values given in Fig. 4.
The energy required to form a unit mass of air bubbles of uniform
diameter, D, and at a depth, H, must be determined in order to est-
ablish the overall efficiency of the diffused aeration technique. The
required energy will be composed of the energy necessary to increase
the pressure to the hydrostatic pressure at the point of release, the
pressure drop through the diffuser plug and the work required to cre-
ate new water-gas surfaces. Of the three, the last can be neglected
compared to the first two for bubbles of interest in this study. The
second work term is a function of the gas flow rate through the diffuser
plug and must go to zero as the flow rate goes to zero in the limit.
If the pressure required to force the gas through the plug and assoc-
iated piping is taken as a constant value = ^P-n > then the required ener-
gy per #m of oxygen for air bubbles based on 100% efficient adiabatic
compression for air is
w =
a
i
where P = atmospheric pressure, #7ft
P2 = ^H + Pl + APp ' Vft2
k = 1. 4
T assume 530°R
C = specific heat of air, #r-ft/#m
59
-------
W" = work to compress air per pound of O, # -ft/#m
a Z * °2
Thus
TT O Q A
W = 426,000 [(E + 1 + TT^) - U , Vft/#Tnn (Eq' 53)
a -^ i ~>-> 7 * ^2
The oxygen capture efficiency, T?, is the reciprocal of the above ex-
pression multiplied by the capture coefficient, fcQ , and compression
efficiency:
2. 34 x 10"6 (f )(|j )
co co(Eq. 54)
-286 ' o '"f
33. 9
or in the usual dimensions
4. 65 f (n )
It should be noted from Eq. 55 that it is essential to reduce the frict-
ional pressure drop in the system (AP ) to as low a value as possible
in order to achieve high oxygen transfer efficiencies. For example,
at a depth of 1 0 feet, a capture coefficient of 0. 5 and a compressor
efficiency of 80%, the capture efficiency given by Eq. 55 is 24. 2
#rriQ /hp-hr for a zero AP , and 7. 9 #ttiQ /hp-hr for a 15 psi friction
drop in the system.
If Eq. 44 is taken as the appropriate value of fcc for low Reynolds
numbers, then the oxygen capture efficiency becomes
AP
. p
644 /Dll/
l 1 i H
33.9-
-» o / V /i
.286 2 g(l
d
H ' (C -C) x 62. 4 x 10~u
W r S
-pip y
a w ' a
J
(Eq. 56)
Whipple et al [25j conducted extensive tests on a diffused aeration
system placed in a seven-foot deep pool of a river and reported values
of 77 from 0. 62 to 1.28 #m0 . hp-hr (referred to standard conditions
of zero initial DO, 20°C) and a capture coefficient, f , in the range
0. 0198 to 0. 0415. Eq. 55 yields an efficiency 1. 6 and°3. 4 #mo?/b.p-hr
60
-------
for this depth at the measured capture coefficient of 0. 0198 and 0, 0415
respectively for an assumed frictional pressure drop of zero. The
bubble size was not reported in this study but it can be assumed that
no specific design was incorporated to produce very small bubbles.
Other diffusion experiments in which no attempt was made to efficient-
ly produce small bubbles have resulted in similar oxygen transfer effi-
ciencies of approximately 0. 5 to 1. 5 #rnQ /hp-hr for depths of 1 0 feet
and referred to zero DO level. Zieminski, Vermillion and St. Leger
[32j, under controlled laboratory conditions demonstrated a maximum
value of 7] = 5. 9 #rriQ /hp-hr at a depth of 10 feet. The energy expend-
ed included the work to drive the device used to shear the bubbles off
before they grew to the "natural" size. The size of the bubbles was not
reported.
The Penberthy Company has stated that their jet aerators have oper-
ated at 7] = 4. 85 #rriQ /hp-hr. In this device air is passed through
large diameter nozzles (~ 5/8 in), mixed with pumped water in a swirl
chamber and then the bubbles are finely divided and mixed as the air-
water mixture is discharged through an exit diffuser into the main
body of water at a depth H. The power includes both the power for the
air compressors and the water pump.
Although air or oxygen bubbles will set up circulation of water as they
rise from the bottom and as a result their influence on inducing atmos-
pheric oxygen capture at the surface similar to mechanical sub-sur-
face devices must be considered, it is known from experience with air-
lift pumps [33] that even when the bubbles are forced to flow up
through an optimum size vertical duct that the efficiency (ratio of
water power to air power) of diffused air pumping is approximately
only 50% which is substantially below the level that can be achieved
with a mechanical pump. In addition it is known that the efficiency
falls off rapidly as the duct diameter is increased beyond the optimum
size.
For example, Bernhardt [34j reported on experimental work in the
Wahnback Reservoir in Germany in which a duct was placed around
and above the diffuser plug in order to aerate the hypolimnium without
destroying the stratification. His duct was 6. 6 ft in diameter, 70 ft
long, had a measured flow rate of 67. 9 cfs of water, and air was sup-
plied by a 36. 5 kw (40 hp) compressor. If the entrance and exit losses
are assumed to each be equal to one velocity head and the frictional
losses along the duct are accounted for, then the power required to
produce this flow would be
61
-------
P = -HL rJL_ + h (!)(-! )] = . 986 hp (Eq. 57)
where m - mass flow rate, #m/sec
h = dimensionless friction factor = . 012 [8j
g = the dimensional constant 32. 2 #m-ft/# -sec
or approximately 1 hp at 100% pump efficiency. To overcome this 1 hp
requirement, he had to use a compressor input power of 49 hp.
Although the above example may be somewhat biased against diffused
aeration as a pumping device because of the depth of the Wahnback ^ ',
it does in general point out the order of magnitude argument. As a
result it will be assumed that atmospheric oxygen capture at the atmos-
phere-water interface for diffused aeration can be neglected compared
to the direct diffusion of oxygen from the rising bubbles.
Before considering the relative merits of mechanical and diffused
aeration pumping when the body of water is stratified, it will be con-
venient for a future discussion to compare the power required to pump
the water in the Wahnback Reservoir with the power of a mechanical
pump of equal capacity. Because of the very low head on such a pump
(~ . 13 ft), a propeller type pump would be the most efficient. To the
best of our knowledge, mechanical pumps with a propeller have not
been built for pumping water at these very low heads (1 ft or less).
However, from experience with marine propellers it would seem pos-
sible to construct one with an efficiency of at least 50%. If this were
the case, the power required for a mechanical (propeller type) pump
to do the same pumping as was accomplished with diffused air in the
Wahnback Reservoir would be about 2 hp. It is interesting to note
that the Aero-Hydraulic Gun (which is a high volume very low head
positive displacement pump that utilizes a very large single air bub-
ble (a foot or more in diameter) as a piston), would require approxi-
mately 10 hp to do the same pumping job [35].
(4) As the water depth is increased, mechanical pumps will display an
ever increasing advantage over diffused aeration devices in terms of
their ability to pump and circulate water in view of the fact that as the
water depth is increased, the air pressure must be increased accord-
ingly, whereas the static head on the pump remains the same and only
the frictional loss in the duct will increase, which in turn can be min-
imized by maintaining a low velocity in the duct.
62
-------
If the body of water is initially stratified, the question of the destrat-
ification efficiency of a diffused aeration system vs a mechanical
pump arises. Here it is not simply a question of which device is a
more efficient pump for the amount of water entrained in the rising
jet from the diffuser as a function of depth is an important considera-
tion. Unlike the ducted mechanical pump which must depend on mix-
ing of the cold water near the surface so as to produce a jet which
will be buoyant at some level below the surface, the diffused aeration
system will produce continual mixing from the diffuser plug upward.
Symon et al [26] reported the measured destratification efficiency
(DE) for diffused aeration, mechanical pumps and the Aero-Hydraulic
Gun. A summary given by Symon is reproduced in Table VII for con-
venience.
As noted by Symon et al the DE shown in Table VII should not be it-
self be used to conclude that diffused aeration is a more efficient tech-
nique than mechanical pumps for destratification because the pump
used in these experiments produced very high (8. 4 ft/sec) duct veloc-
ities as noted previously.
Water Spray Aeration
In this technique the water is sprayed into the air where it captures
atmospheric oxygen on the surface of the drop. The oxygen subse-
quently diffuses into the drop and eventually falls back to the main
body of water.
If, as a first approximation, it is assumed that the turbulent transport
within the drop of water is negligible, then all oxygen transport from
the drop's surface to the interior must be by molecular diffusion.
Under these conditions the oxygen transfer rate from the atmosphere
will be given by Eq. 8.
- - = K (C -C) (Eq. 8)
A L, s
where /D
K = // (See Eq. 7 for note on units)
L 771
Again since we are interested in the total oxygen accumulated by mole-
cular diffusion in the drop after a residence time in the air, the ap-
propriate "constant" K to be used is found by noting
63
-------
TABLE VII
Comparison of Destratification Efficiencies of Various Studies (from Ref. 26)
Lake
Reference
Test
Lake 1
Test
Lake 2
V e suviu s
Boltz
King Geo.
Wolf or d
Wahnbach
Cox
T-T s-i 1 1 f- . \T r
JTHJ± ±.\J W
Dates of
Mixing
6/2-6/7
6/21-6/24
7/15-7/20
8/18-8/23
1966
(5/16-5/19,
5/20-5/23)
6/10-6/15
7/ 1- 7/ 8
7/26-8/3
9/8-9/13
1966
9/3-9/17/64
8/6-9/10/65
7/13-8/17/66
4/18-4/25/62
6/9-9/9/61
6/30-7/18/66
Change in
Stability
Method Volume Uncorr (U)
acre -ft Corr. (C)
kw-hr
Diffused 2, 930
air
pump
Diffused
air 4, 600
pump
Mech. 1,260
Pump 2,930
" 18,375
Dif. Air 2,500
" 33, 740
Aefo-hyd 1, 190
Gun
27
23
33
16
7
24
35
28
8
6.
28
23.
14.
400
7.
(C)
(C)
(C)
(C)
(C)
(C)
(C)
(C)
(C)
7 (U)
(C)
8 (U)
8 (U)
(U)
9 (U)
Total Destratification Total Energy
Energy Efficiency. DE Input/Unit Volume
Input Calculated over Mixing
kw-hr entire mixing kw-hr/
period, % acre ft
1,
1,
2,
2,
3,
2,
3,
3,
2,
1,
14,
32,
2,
104,
2,
760
680
200
860
285
740
740
550
540
862
300
640
570
800
140
1.
1.
1.
0.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
5
4
5
6
2+
2
9
9
8
3
6+
4*
2
1*
6*
4*
4*
5 (avg)
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
1.
4.
1.
1.
3.
1.
2
6
6
7
0
7+
7
6
8
8
5
7+
5
9
8
0
1
8
(avg)
*These data may be somewhat in error as the change in stability was not corrected for any natural
change in stability that may have occurred during the mixing operation,
+Average for all mixings in 1966.
These data are intended as a rough guide only and are influenced by many factors: shape of lake,
time of year, meteorologic condition (temperature, sunlight, wind)
-------
JVT JVE
°o t °9 T 1/
2 = fr 2-dt = 27 -t /(C -C) (Eq. 58)
A J A 7T r s
Thus the "constant" value of K is taken as
/^iT
K = Zv7 (Ecl- 59)
L fit
const. r
The relation between the residence time (t ) and the maximum height
reached by the water above the air-water interface (h) can readily be
found in the absence of drag as
t = 2 V , sec (Eq. 60)
o
where h = spray height, ft
g = acceleration due to gravity, ft /sec
This residence time is independent of the angle of spray.
In the above discussion it has been assumed that the water leaves the
nozzle as drops which is not the actual case. In operation the water
leaves the nozzle as a sheet and then disintegrates into drops.
The bubble surface area generated per unit time is given by the ex-
pression
Q
w 2
- - , ft (Eq. 62)
where Q = "water volume flow rate through spray nozzle, ft /sec
w
d = drop diameter
The above expression can be simplified to the following:
65
-------
6Q _
A = , ft (Eq. 63)
d
The oxygen transfer rate can now be found by combining Eq. 58,
Eq. 60 and Eq. 63:
,, Q DT1/2h1/4[(C -C)x62. 4x!0"6]
2 g *
(Eq. 64)
where C , C are in mg/liter
s
For a given water flow rate through the spray nozzle Eq. 64 indicates
that the oxygen transfer rate varies inversely with the drop diameter.
In order to create the spray, energy must be invested in creating new
interfacial surfaces , imparting an initial velocity to the water and to
overcome viscous losses in the equipment,, The first of these required
energies is equal to the product of the surface tension and the created
area. It will be small compared to the second term unless the drops
are made very small (of the order of 10 molecular diameters). The
second term is readily accounted for; however, the third term de-
pends on the flow rate and specific hardware and care must be taken
to minimize this loss as it can be very high. As a first approximation
it can be assumed that the flow rates are made low enough so that the
viscous losses are negligible. Under this assumption the power re-
quired for a water volume flow rate of Q is given by the expression
w
P = Q p gh , # -ft/sec (Eq. 65)
w w w f
The oxygen transfer efficiency can now be found by dividing Eq. 64 by
Eq. 65 to obtain
1/2
14. 03 D (C -C)
T? = 3/4 ^f , #mQ /hp-hr (Eq. 66)
dh pwg 2
where C , C are in mg/liter
s 2
D is in ft / sec
J_j
Although Eq. 66 has a tendency to underestimate the transfer efficiency
because it does not account for the additional capture of oxygen that
will take place due to the turbulence created at the area where the
66
-------
spray falls back into the main body of water, it also has a tendency to
overestimate the capture efficiency because it does not include the
energy necessary to overcome viscous losses in the spray nozzle and
associated fittings.
The most common pressure nozzle for generating a spray is the "hoi
low-cone" nozzle which consists of a whirl chamber and an orifice.
The water is introduced into the whirl chamber through tangential
ducts and is thereby set into vigorous rotation. The water leaves the
whirl chamber and exits through an orifice as a conical sheet which is
subsequently broken up into drops by interacting with the air. Al-
though the drop size and size distribution are a function of the specific
hardware, it is noted in Ref. 36 that sprays of inductrial interest have
drop sizes less than 1 mm (. 039") in diameter and often less than
. 2mm.
The hollow-cone nozzle has been used at capacities up to 200 gallons/
minute for spray cooling ponds. Based on Eq. 64 and Eq. 66 a nozzle
of this capacity, with h = 10 feet, and producing drops of average dia-
meter at the low end of the range or 0. 20 mm, would induce an oxygen
transfer rate of 0. 1 7 #iriQ /hr at an oxygen capture efficiency of 0. 33
LJ
/hp-hr, and would consume 0. 5 hp not including mechanical
lo s se s.
Lueck, Blabaum, vViley and Wisniewski [3?] reported the operating
characteristics of a spray aeration unit in which the spray was pro-
duced by a propeller and nozzle as shown in Fig. 17. This machine
was specifically designed for instream aeration and had a flow rate of
250, 000 gph, a power of 19 hp, and produced a spray approximately 8
to 10 ft high and 35 ft in diameter. The authors reported an instream
oxygen transfer rate from 4. 6 to 0. 2 #rriQ /hp-hr, at an initial DO
level of less than 1 mg/liter and a temperature of about 26°C,(C =
8. 0 mg/liter).
A comparison of the performance of this spray device and the results
predicted by the analytical model used in this section can be made only
if a drop diameter is assumed. Lueck et al indicated the droplets
produced by the unit were "rather large in size" and they suggested
that the device be redesigned so that "the spray can be atomized more1.'
It is not possible to establish the approximate droplet size from this
report, but if their implication of a "coarse" spray is taken to mean
the mean between 1 and . 2 mm diameter of industrial sprays, then
the present model would predict an oxygen capture rate of 1. 13 #rriQ /
hr, a capture efficiency of 0. 12 #rnQ /hp-hr and a total power of
9. 4 hp without considering mechanical losses. As noted previously
67
-------
- wCtDLCSS DESIGN
Fit«er,o(-Ass SHSLV.
V
r
A
2.^0,000 6PH
CAt.CUUATEO
A
SPRAY
APPROX. IOFT. HIGH X 35 FT. Di.'
WATER PUMPC-T1
L.
Sec. AA
AQUA- L A7OR. M OT OK,
ie. 17 - Sketch of High Volume Spray Aeration Device (Ref. 37)
00
Sec. BE
-------
the present analytical model does not account for the capture of oxygen
that results from the turbulence induced at the area where the water
falls back into the main body of water. This oxygen transfer can be
substantial compared to the amount of oxygen transferred to the water
while it is in the air. For example, in a later section on Weirs, Dams
and Cascades it will be shown that for a carefully designed weir equal
to the spray height in the previous example (9 feet) and operating with
the same initial conditions, the measured oxygen transfer efficiency is
about 1. 2 #mQ /hp-hr.
Although the analytical model indicates that the spray aerator perform-
ance could be improved by reducing the diameter of the drops, the in-
crease in energy required to do so would probably be prohibitively
large. That the viscous power losses which have been neglected be-
come large as the drop size is decreased from 1 mm can be seen from
an estimate given in Ref. 36 in which it is noted that the viscous loss-
es encountered in atomizing 1 gram of water in 1 second to 00 001 mm
diameter amount to a power of 100 to 10, 000 hp.
Because of the high power required to rapidly create small diameter
bubbles and because diffusion rates are considerably higher in the
initial stage after surface formation, it would seem to be more econ-
omical to create new surfaces than to create a fine spray with a long
residence time. Such a step has been accomplished in a number of
commercial surface aerators such as the Bird-Simplex High Intensity
Aeration Process. This type of device is discussed in the next sec-
tion and is classified as " White Water Generators. "
vVhite Water Generators
White water generators are designed to induce transfer of oxygen from
the atmosphere to the water by rapidly exposing "new" water surfaces
as a result of energetically agitating the air-water interface. In addi-
tion, they are usually equipped with a submerged pump or draft tube
for the purpose of circulating the water. Circulation patterns estab-
lished by sub-surface pumps have been previously discussed.
Unlike spray aeration devices considered in the previous section,
white-water generaters are not designed to lift the water appreciably
above the surface with the result that the residence time of the agitated
water in the air is small. However, the amount of surface generated
per unit time is high. For a fresh surface the rate of oxygen capture
at any time is given by Eq. 8, or
69
-------
= K (C -C) (See Eq. 7 for note on units)
A I/ s
where KT
Li Tf't.
t - time elapsed from instant of surface generation
It should be noted that Eq. 8 is based on previously unexposed water
surfaces whereas the machanical aerator used in a river or stream
(with low O2 uptake) is very likely to pick up water in its surface
blades that has just been agitated and not yet transferred its oxygen
to the main body of water. In addition it is quite difficult to estimate
the active surface area produced by this device.
Because of the above considerations, it was considered not possible
to develop a reliable analytical model of the white-water generators
and as a result experimental results had to be used directly. From
Eq. 8 it is noted that the oxygen transfer rate and hence oxygen trans-
fer efficiency should increase linearly with the oxygen deficit, (C -C).
Oxygen transfer efficiencies measured in the laboratory and in the
field are shown in Fig. 18. Each of the two major sets of data shown
in Fig. 19 (University of Minnesota laboratory flow-through test and
the field test on the Passaic River) show approximately the same lin-
ear trend for the efficiency as a function of DO deficit but the slopes of
the line drawn through each set differ by about a factor of two with the
slope for the flow-through laboratory test being higher than that for
the river field test. In the case of the field test both Whipple et al [25]
and Kaplousky et al [38] have demonstrated that the oxygen transfer
efficiency increases substantially with increased river flow rates. For
example, consider the addition of oxygen to the Passaic River during
times of high flow compared to times of low flow as shown in Table VIII
from Ref. 25.
A straight line drawn through the two major sets of data shown in Fig.
18 can be expressed as Eq. 67 and Eq. 68 for the flow-through and
river test respectively.
r? = 0.46 (Cg-C) , #mQ . hp-hr (Eq. 67)
LJ
where C , C are in mg/liter
S
T? - 0. - , m-hr (Eq. 68)
70
-------
I
PH
4
O
£ 3
a
o
iH
w
W
ri
rt
^
X
O
X. X
Average Dissolved Oxygen Deficit = C (C. + C )/2, mg/liter
s A B
Chicago Canal ( High River Flow Rate), Ref. 38
Passaic River, Ref. 25
Univ. of Minn. , Flow-through Laboratory Test, Ref. 15
Aerated Lagoons, Ref. 39
C , DO above aerator; C , DO below aerator; T] = Efficiency based on Shaft HP
Fig. 18-Comparson of Various Measured Oxygen Transfer Efficiencies for White-water
Generators
-------
TABLE VIII
SUMMARY OF STEADY-STATE FIELD TEST DATA FOR
MECHANICAL, AERATOR - (From Ref. 25)
Date
8/9/6?
8/31
7/9/68
7/10
7/12
7/16
7/16
7/18
7/18
7/23
7/2L
7/26
7/29
7/30
7/31
8/1
8/6
8/7
8/7
8/9
8/9
8/22
8/27
Flow
Q
(CFS)
1620
520
159
1U7
133
111
110
13U
128
113
110
125
92
91
93
93
106
110
I0h
99
98
85
130
Water
Temp.
(°c)
23-5
22.0
2U.5
2ii.O
25.0
26.0
28.6
27.5
28.9
29. h
26.2
25. U
2U.8
23.0
2h.O
26.0
25.0
26.8
25.0
26.0
27.0
2U.9
23.0
Shaft
Power
(HP)
90.0
83. h
79.5
8U.1
71.7
80.8
78.2
82. U
77.1
76.6
75.2
71.6
lh.6
85.7
85.8
80.7
88.3
81.0
82.2
79.3
78.0
81.6
73.1
Dissolved Oxygen
(ppm)
Upstream
1.12
2.50
U. 60
3.60
U.10
3.00
3.20
2.70
2.90
2.60
1.90
1.20
2.20
3-00
2.10
2.iiO
1.50
1.70
l.UO
0.60
1.60
l.UO
1.60
Downstream
1.58
3-35
6.00
U.U5
6.U3
5.9U
5.35
U.30
U.85
5.00
U.io
U.oo
5.10
6.00
5.00
U.5o
U.50
5.20
U.70
U-50
ii.5o
U.5o
3-50
Oxygen
Added
(lbs.°2/hr.)
167. h
99.3
50.0
3U.ii
69.6
73-3
53.1
U8.2
56.1
60.9
5U.U
78.6
59.9
61.3
60.6
U3-9
71. U
86.5
77.1
86.7
63.8
59.2
55.5
IN]
)( DO samples taken 1000 ft. upstream, and 2000 ft. downstream of the aerator.
-------
Weirs, Dams and Cascades
It can be anticipated that the highly turbulent flow at the base of a weir,
dam or cascade effectively generates "new" water surfaces and hence
induces oxygen transfer from the atmosphere by surface capture. The
effectiveness of such flows in aerating the water has been documented
by several works. Gameson [40j in an extensive field study on weirs
in England showed experimentally that the increase in DO across a
free weir (that is, one in which the water falls freely and does not ad-
here to the face of the structure) was given by the expression
(C - C )
s A
(C - C ) = 1 + 0. 152a h (Eq. 69)
S X3 -L
where C = saturation DO, mg/liter
O
C = DO above weir, mg/liter
J\
C = DO below weir, mg/liter
B
a - an experimental coefficient
= 0. 85 for sewage
= 1. 00 moderately polluted water
= 1. 25 slightly polluted water
h = height of fall, ft
In the experimental study Gameson demonstrated that the head on the
weir did not influence the change in DO across the weir (at least for a
head in the range 6-13 inches) and that the majority of the oxygen
transfer took place at the splash area and not in the falling water. In
addition, the experimental work indicated that turbulent sloping chan-
nels provide less aeration than weirs for the same total head loss.
Cameron, Van Dyke and Ogden [41 J extended the field work of Game-
son with laboratory experiments on a weir where the water tempera-
ture could be controlled. They showed that Eq. 69 could be expressed
as given below so as to include the influence of temperature,
(C - C )
y-2 £_ = 1 + 0. 11 a h(l + 0. 046T) (Eq. 70)
{Cs " CB) l
where T * water temperature in C
Gannon [42 J found good agreement between the change in DO predicted
by Eq. 70 and the measured values for treated domestic waste passing
73
-------
over a 3. 1 foot weir.
If a natural drop occurs in a stream or if such a drop can be created
by a permanent structure, then no additional power would be required
to achieve the increase in DO given by Eq. 70 for flow over a weir.
If on the other hand the water must be raised, the energy to do this
may be estimated by neglecting viscous losses and kinetic energy
losses compared to the energy required to elevate the water. When
this is done, the oxygen transfer efficiency for weirs ( ' is found to be:
1 98 1 if <"
2 (Eq. 71)
where C , C are in mg/liter
S -A.
h is in ft
T is in °C
For example, if the water temperature is 20 C, CA is zero, and h = 1 0;
then T? = 1. 30 #mo /hp-hr and (C -C )/(C -C ) = 3. 64
Measurements have been made of the DO above and below a number of
dams in the northeast United States [43]. Table IX shows a compari-
son of the predicted oxygen capture efficiency as given by Eq. 71 for
weirs and measured values for flow over dams as given in Ref. 43.
TABLE IX
Comparison of Measured and Calculated Oxygen
Transfer Efficiency for Dams
h (ft)
4
5
5
6
8
8
9
11
20
Temp (°C)
20. 5
20. 5
22. 5
20.5
22. 0
22. 5
27. 5
19. 0
27. 0
T?, ^Q2
77 = Mo7(measured)/Qyh
LJ
0. 95
0. 73
0. 88
1. 20
1. 35
1. 37
0. 92
0. 95
0. 74
/hp-hr
7? = Eq. 71
2. 30
2. 02
1. 98
1. 82
1. 48
1. 46
1. 25
1. 22
0. 67
(4) If the water must be pumped to raise its elevation, it can readily
be done with a propeller type fan and then allowed to splash back down
on an apron. Such a device will be referred to as a lift-drop aerator.
74
-------
Note: In Ref. 43 the initial DO is said to always be less than 2 mg/
liter and predominantly zero. In Table IX the DO is assumed to be
zero in all cases.
Although the transfer efficiency predicted for free weir fall by Eq. 71
yields results that are more than a factor of 2 higher than the mea-
sured values for the 4- and 5-foot dams, it must be noted that the
dams studied were studied in an "as found" condition which may have
been far from the optimum condition for the effective aeration on
which Eq. 71 is based.
The influence of the depth of the water at the base of the dam can read-
ily be seen by comparing two experimental programs discussed in Ref.
43. In the first program water passing over dams on the Mohawk River
and splashing into relatively deep water at the base was studied. When
the results obtained for dams up to 15 feet high were extrapolated, it
was found that a dam of about 30 feet would saturate zero DO water
passing over it. In the second study a number of dams in New England
with \vater splashing onto relatively shallow water or stone or concrete
aprons were studied. When this data was extrapolated, it was found
that a dam of only approximately 18. 5 feet was needed to saturate zero
DO water.
Although some uncertainties exist in the available data, it appears that
Eq. 71 can be used as a valid model for calculating the oxygen transfer
efficiency for weirs and dams provided that the structure is such that
the water falls free rather than adhering to the face of the structure
and provided that the water is allowed to fall on shallow water or a
masonry apron.
A comparison of Eq. 70 and Eq. 71 reveals that for the same total elev-
ation drop (h) a series of weirs (that is, a cascade) will produce a
more efficient oxygen transfer than a single weir. For example, using
water with an initial DO level of zero, a single weir with h = 9 feet,
T - 20°C, will increase the DO to 6. 32 mg/liter at an efficiency of
1. 39 #mQ /hp-hr, whereas a cascade of three 3-foot weirs would in-
crease the DO to 7. 42 mg/liter at an efficiency of 1. 63 #mQ; /hp-hr.
Hydraulic Turbine Aeration by Venting
Air may be made to flow into the exhaust end of a hydraulic power tur-
bine by simply cutting a hole through the casing and installing a control
valve provided that this section of the turbine has been designed to ope-
rate at pressures below one atmosphere. When air is allowed to enter
75
-------
the turbine, the water flow rate and power (as well as power per unit
water flow rate) decrease. The maximum oxygen transfer rates were
estimated in Ref. 44 to correspond to ADO of about 1. 8 mg/liter and
the oxygen transfer efficiency was reported to be approximately 2. 6
to 4. 1 #mo2/hp-hr.
This technique is limited to select sites. The cost of modifying ex-
isting turbines will depend considerably on the type of machine. This
technique, like diffused aeration, does offer the possibility of inject-
ing pure oxygen rather than air.
U-Tube Aeration
The U-Tube aeration method is an oxygen transfer process introduced
in the Netherlands and consists of inducing or injecting air into water
as it starts down one leg of pipe formed into a U shape. The arrange-
ment has two features which enhance the transfer of O->. First, the
LJ
water passing down the U-tube drags the air bubbles along (if the
water velocity is high enough), thus increasing the bubble residence
time. Second, as the bubbles pass down the tube, the pressure in-
creases and Cs increases according to Henry's Law.
Speece, Adams and Wooldridge [45] have presented considerable ex-
perimental data on the operation of U-tubes up to 40 feet deep and 60
inches in diameter. They showed that water could be completely sat-
urated with oxygen (at a total pressure of 1 atm) by use of these tubes
and that the oxygen transfer efficiency for water with an initial DO of
about zero could be made as high as 3. 7 #mQ /hp-hr for 40 foot deep
tubes and as high as 1. 8 #rriQ /hp-hr for 10 foot deep tubes.
Brush Aerators
Brush aeration, which consists of a rapidly rotating cylindrical brush
that projects water across the surface, have been widely used in sew-
age treatment tanks in Europe. Under the high oxygen uptake condi-
tions that exist, measured oxygen transfer rates of about 3. 0
hp-hr have been reported for low DO water [46j. It should be noted
that in the sewage treatment tanks, which are usually made about 10
to 15 feet deep and 15 to 30 feet wide, the brushes are able to estab-
lish a circular flow pattern in order to continually aerate "new" water.
However; in a natural body of water it is unlikely that any substantial
circulation pattern will be developed by the brushes because of the
lack of side walls in close proximity to the brush. As a result the
76
-------
oxygen transfer efficiency for these devices would probably decrease
considerably when they are placed in natural bodies of water.
77
-------
VII. OPTIMUM ECONOMIC SELECTION
Streams
When considering the use of supplemental aeration to prevent the dev-
elopment of critically low levels of dissolved oxygen in the most econ-
omical way, it must be noted that for any type of aeration equipment
the oxygen transfer rate and oxygen transfer efficiency depend direct-
ly on the oxygen deficit. As a result, it will be most economical to
add the oxygen when the DO has fallen to the lowest acceptable level
and in addition it will economical to increase the DO to some value
less than saturation at a given location. As the water flows away from
the aeration station its DO will again decrease if the organic load in
the water is high enough to cause bacteria to consume more oxygen
than is replenished by natural aeration. If this is the case, a second
aeration station will have to be installed when the DO again falls to its
lowest acceptable level. The process will have to be repeated until
the organic material in the water is consumed. Whipple [25] has dem-
onstrated the use of a computer simulation technique for determining
the number and spacing of supplemental aeration stations required for
a given river flow and organic loading. The lowest acceptable DO
level is usually taken as 4 mg/liter.
The economic selection will be based on the lowest combined yearly
capital cost and operating cost. The capital cost will be reduced to a
yearly value by assuming an equipment lifetime and an interest rate
for the cost of money used to purchase the equipment.
Although the optimum selection of aeration equipment will be influ-
enced by the characteristics of the particular stream, the procedure
can be demonstrated by looking at one aeration station where the DO
level has reached its minimum acceptable level of 4 mg/liter and it
is desired to increase the DO to 6 mg/liter. As an example we shall
select a 200-foot wide river, 4 feet deep, moving at a slow velocity
of 0. 05 ft/sec and at a temperature of 20°C. From Fig. 4 it is seen
that the natural aeration rate for this example is given by Case C (See
Eq. 18). The volume flow of water and the required supplemental
oxygen transfer rate for this example are 40 cfs and 18 #mQ /hr res-
pe ctively.
In searching for the economic optimum equipment, all types of aera-
tion equipment discussed in the last section will be considered in
order starting with sub-surface devices.
79
-------
Sub-surface Devices
Since the water is shallow, the most efficient of the two shallow water
configurations will be selected for evaluation, namely the large dia-
meter ducted propeller previously listed as Case II. The oxygen
transfer and transfer efficiency for Case II are given by Eq. 23 times
6H, the effective width (See Fig. 8), and Eq. 24 respectively, or
M[ = (. 007)(C -C)V1/2H1/2(6H) #rr, /hr (Eq. 72)
°2 S °2
n = 0. 657 (C -)V~5/2H~1/2 #m^ /hp-hr (Eq. 24)
°2
where C , C are in mg/liter
S
H is in ft
V is in ft/ sec
Since the effective width of a Case II device has been estimated at 1 OH
or 40 feet, a minimum of five such devices would have to placed
across the stream to aerate all the flow simultaneously. Each unit
would have to add 3. 60 #mQ /hr. However, if the number of units is
selected at too low a value, the duct velocity will have to be high in
order to achieve the necessary oxygen transfer with the result that the
transfer efficiency will be prohibitively low.
In addition to the power required to pump against the total hydraulic
head on the propeller, power will be required to overcome friction in
the bearings of the propeller. If this frictional loss is assumed to be
a constant 0. 1 hp, then the overall transfer efficiency becomes
17 T?
° p - 73>
where f\ - efficiency of device used to drive sub-surface
o
aerator shaft, assume 80%
f\ = propeller efficiency, assume 50%
Although the efficiency based on shaft power will increase as the
power per unit decreases, the frictional loss becomes more important
as the power per unit is decreased. As a result there will be an
80
-------
optimum number of units that will make the overall efficiency maxi-
mum as shown in Table X.
TABLE X
Total Power Variation with Number of Sub-surface Units
Number of
units
5
10
15
20
25
30
V
ft/ sec
7. 19
1. 79
. 79
. 44
. 29
. 20
nT
. 00387
. 120
. 785
1. 89
2. 31
2. 22
P
unit
hp
930
15
1. 5
. 48
. 31
. 27
total
hp
4650
150
22. 5
9. 6
7. 8
8. 1
From Table X the optimum number of units can be taken as 25. The
capital cost of each unit was estimated to be $1334 in Appendix C. The
The total initial or capital costs are itemized below.
Cost of Aeration Units @ $1334 each $33, 350
Cost of Hydraulic Pump and Drive 1, 000
Cost of Hydraulic Piping 1,000
Installation Cost of Aeration Units including
anchors (@ $50 each) 1, 250
Electrical Power Supply (based on 500 ft to nearest
utility wire) 1, 500
Site Preparation 2, 000
Shelter 1, OOP
Sub-Total 41,100
Contingencies and Engineering (@ 20%) 8, 220
Total Initial Cost $49,320
The estimated annual operating costs are listed below.
(5),
Electric Power (@ . 01/kw-hr, for 168 full days )
= (7. 8)(168)(24)(. 745)(. 01) 234
Maintenance @ 1% of initial cost
= (. 01)(. 8 x 49, 320) 384
7/5)Based on the work of WTiipple [25], it is assumed that the aeration
units will run for three months at 24 hours/day and for five months at
12 hours/day.
81
-------
Personnel @ 4 hours a week for 32 weeks, $10/hr
= (128)(10. 00) $ 1.Z80
Total Annual Operating Cost $ 1,898
The total annual cost can now be estimated by taking a basic interest
rate of 8%. For a life of 20 years, this results in a capital recovery
rate (interest plus amortization) of 10. 185% and a capital cost per
year of . 10185 x $49, 320 or $5, 010. Thus the total annual cost is
Capital cost per year $ 5, 010
Annual Operating Cost 1, 898
Total Annual Cost $ 6,908
Since the annual operating cost is smaller than the annual capital cost,
it is helpful to determine the total annual cost when the number of sub-
surface units is decreased. For example, if the number of units is
decreased from 25 to 15, the initial capital cost will be
Cost of Aeration Units (@ $1334) $20,000
Cost of Hydraulic Pump and Drive 900
Cost of Hydraulic Piping 600
Installation cost of Aeration Units (@ $60 each) 900
Electrical Power Supply (based on 500 ft to nearest
utility wire) 1, 500
Site Preparation 2, 000
Shelter 1, OOP
Subtotal 26,000
Contingencies and Engineering (@ 20%) 5, 360
Total initial cost $32, 169
The estimated annual operating costs are listed below:
Electric Power (@ %. 01/kw-hr, for 168 full days) 685
Maintenance (@ 1% of initial cost) 258
Personnel (@ 4 hrs/wk for 32 weeks, $10/hr) 1, 280
Total Annual Operating Cost 2,223
For an interest rate of 8% and a life expectancy of 20 years, the total
annual cost will be the sum of $3, 270 and $2, 223 or $5, 493.
It should be noted from Table X that a further decrease from 15 to 10
units will result in a substantial increase in the power and cost.
82
-------
Diffused Aeration Systems
For completeness both air bubbles and pure oxygen bubbles will be
evaluated.
Air Bubbles
In order not to possibly omit the optimum aeration equipment, three
values of oxygen transfer efficiency will be used to evaluate the econ-
omics of this method. First, the most conservative estimate would
be to select the average of the measured values reported by Whipple
et al [25], namely, 0. 85 #mQ /hp-hr at standard conditions or
0. 376 #mQ /hp-hr for the average condition of the present example.
Second, the value of 4. 85 #rriQ /hp-hr reported by the Penberthy Co.
for jet aerators at standard conditions or 2. 17 #rriQ /hp-hr for the
average condition of this example. Finally, the most optimistic of the
three, a value of 7. 8 #rriQ /hp-hr calculated by means of Eq. 55 for
standard conditions and a hypothetical system capable of producing
small enough bubbles (Reynolds number ~ 1. 0) so that the capture co-
efficient (f co) can be assumed to be as high as I/ 2 and the total
frictional pressure drop in the system as low as 15 psi. The above
value corresponds to 3. 52 #mQ /hp-hr for the conditions of the pre-
sent problem. Since the above three efficiencies do not take into ac-
count the efficiency of the device that drives the compressor, they
must be revised. The revised values of t? are tabulated below on the
assumption of an electric drive of 90% efficiency together with the
total input power to raise the DO from 4 to 6 mg/liter.
TABLE XI
Oxygen Transfer Efficiencies and Power for Diffused Aerators
o /hp-hr hp
Case 1
Case 2
Case 3
Measured in Stream
Penberthy
fco = i/2' APp= l5Psi
0. 339
1. 943
3. 17
53. 2
9. 25
5. 68
The accuracy of a cost estimate for an aeration system can be en-
hanced by experience with on-site operation. Whipple et al [25] made
cost estimates for diffused and mechanical aeration in a river with
characteristics similar to the present example in 1969. It is felt that
their estimates would be more reliable than independent ones not
83
-------
based on actual operating experience. Based on Whipple's data for an
80 hp electrically driven air blower with an initial capital cost of
$51, 000, the capital cost for Case 1, 2 and 3 would be $34, 000,
$8, 850^ and $3, 610 respectively.
The estimated annual operating costs are listed below.
Case 1 Case 2 Case 3
Eler-tric Power
(@ $. 01 kw-hr for 168 full days) $1,600 $ 277 $ 171
Maintenance
(@ 3% initial cost) 1,020 177 110
Personnel
(@ 12 hours/week, 32 weeks ) $1 0/hr 3,840 3, 840 3,840
Total Annual Operating Cost $6,460 $4,294 $4,121
If the life time of this equipment is assumed to be 10 years and the
interest rate is again taken at 8%, the capital recovery rate becomes
14. 09%. Based on the above, the total annual cost is computed below.
Case 1 Case 2 Case 3
Capital Cost per Year $4,790 $1,248 $ 509
Annual Operating Cost 6,460 5, 542 4,630
Total Annual Cost $11,250 $5,542 $4,630
Pure Oxygen Bubbles
Again in order to cover the entire possible range, two values of the
capture coefficient will be evaluated for diffusion of pure oxygen bub-
bles. First, the most conservative value will be that measured by
Ippen and Carver [29] for a depth of four feet and bubble size of
1. 50 mm, namely f = . 13 for standard conditions or 0. 0578 for the
present condition. Second, a value of f = . 50 for smaller bubbles
if they could be generated at standard conditions or . 222 for the condi-
tions in the present problem.
In order to make cost estimates, it will be assumed that the oxygen is
stored under pressure and that no additional work need be done on it to
cause it to pass through the diffuser system. Under this assumption
the capital cost will consist only of an oxygen storage tank, gas
(6) Estimated value for air blowers alone increased by 50% to account
for the fact that the energy input is by a combination of air blowers
and water pumps.
84
-------
pressure regulating system, the diffuser plugs and associated piping.
This capital cost is estimated below.
Case 1 Case 2
fco ='0578 fco ='222
Cost of O,, storage tank
(a 3-day supply) $19,900 $ 5,700
Cost of O;? pressure regulator 400 400
Cost of piping and diffuser plugs 1, 000 1, 000
Installation cost of diffuser unit 500 500
Site preparation 2, 000 2, 000
Shelter 1, OOP 1, QQQ
Sub-total 24,800 10,600
Contingencies and Engineering @ 20% 4, 960 2, 120
Total Initial Cost $29,760 $12,120
The estimated annual operating costs are listed below.
Maintenance (@ 1% of initial cost) 200 87
Personnel (@ 8 hours/week for
32 weeks, $10/hr) 2,560 2,560
Cost of pure oxygen (@ $35/ton
delivered, 3320 #m/day for
Case 1 and 863 #m/day for
Case 2, 168 full days) 21, 900 _5, 7QQ
Total Annual Operating Cost $24, 660 $ 8, 347
If it assumed that this equipment has a life-time of 20 years and the
same interest rate of 8% is again used, the capital recovery rate will
be 10. 185%. Based on this rate the total annual costs are given below.
Capital Cost per year $ 3,030 $ 1,235
Annual Operating cost 24, 660 8, 347
Total Annual Cost $27,690 $ 9,582
Water Spray Aeration and White Water Generation
In view of the consideration that for the same horsepower a spray aer-
ator and a white water aerator will have virtually the same initial cap-
ital cost as well as the same life and maintenance cost, they will differ
substantially only as a result of their different oxygen transfer effic-
iency; which is lower for the spray units that have been tested as
85
-------
previously noted.
In order to cover a wide range of operation in the search for optimum
equipment, three values of the transfer efficiency will be evaluated,
one for spray devices and two for white-water generators. The value
for spray devices will be taken as the mean of the measured range
reported by Lueck et al [37], namely, 0. 67 #mQ /hp-hr at standard
conditions or 0. 298 #mo /hp-hr for the condition's of the present ex-
ample. The two values selected for white-water generators are the
ones given by Eq. 68 and Eq. 67 (and shown in Fig. 18) for the river
test reported by Whipple et al [25] and the University of Minnesota
flow-through laboratory test [l5j respectively. These two values are
2. 16 and 4. 14 at standard conditions or 0. 96 and 1. 84 for the present
conditions respectively. Since the above efficiencies are based on
shaft power, they must be corrected to include the efficiency of the
drive unit. Based on the assumption that the drive is a directly coup-
led electric motor with an efficiency of 90%, the corrected transfer
efficiency and the total required input power is given below for the
three cases.
TABLE XII
Oxygen Transfer Efficiency and Power for Spray Aerator
and White-Water Generators
Case 1
Spray
Aerator
Case 2
River
Test
Case 3
Flow Through
Test
Overall oxygen transfer
efficiency #mo /hp-hr 0. 268 0. 865 1. 66
LA
Total required input horse-
power to increase DO from
4 to 6 mg/liter 67. 2 20. 8 10, 8
As in the case of capital cost for diffused aeration, the capital cost for
spray aerators and white-water generators will be based on Whipple's
capital cost data for white-water generators, namely. $44,000 per 75
hp electric drive unit. Based on the above the initial capital cost for
Cases 1, 2 and 3 is $39, 500, $12, 000 and $6, 350 respectively.
The estimated annual operating costs are listed below.
86
-------
Case 1 Case 2 Case 3
Flow
River
Spray Ihrough
-L est
Test
Electric Power (@ $. 01 kw-hr
for 168 full days) $2,020 $ 624 $ 324
Maintenance
(@ 3% of initial cost) 950 293 153
Personnel
(@ 4 hours/week, 32 weeks
@ $10/hr) 1,280 1,280 1, 280
Total Annual Operating Cost $4,250 $2,197 $1,757
If it is assumed that this equipment has a life-time of 10 years and the
interest rate is 8%, the capital recovery rate is 14. 09%. Based on
this rate, the total annual costs are given below.
Capital cost per year 5, 560 1, 720
Annual operating cost 4, 250 2, 197
Total Annual Cost $10,810 $ 3,917 $2,652
Weirs, Dams and Cascades
If a natural drop in the river is available for the construction of a dam,
then there will be no operating cost for energy and the cost of such
aeration reduces to the initial capital cost of construction and the an-
nual maintenance cost. On the other hand, if no natural drop in ele-
vation is available, the water can be pumped to a higher elevation and
allowed to splash down on a masonry apron or pad - a "lift-drop" aer-
ator. The capital cost of such "lift-drop" aerators will be virtually
the same as for spray and white-water generators and power will of
course have to be supplied. The total annual cost of each of these two
systems (natural drop and lift-drop) is estimated below.
Natural Drop
The height of the drop that must be available in order to increase the
DO from 4 to 6 mg/liter can be calculated from Eq. 70. When this is
done, the drop is found to be 2. 53 ft.
The initial capital costs are given below for the case where a natural
drop is available in the stream for the construction of a dam.
87
-------
Cost of dam material (concrete at $20/cu yd
for a 200' x 8' x 4' dam) $ 4, 740
Cost of forms and pouring concrete
(assumed equal to material cost) 4, 740
Excavation and site preparation
(@ $10/cu yd, 200' x 50' x 4') 14,850
Cost of temporary coffer dams 4, OOP
28,330
Contingencies and Engineering @ 20% 5, 660
Total Initial Cost $33, 990
The estimated annual operating costs are listed below.
Maintenance (@ 1% of initial cost) 283
Personnel (@ 4 hours every two weeks
@ $10/hr for 32 weeks) 640
Total Annual Operating Cost 923
If the life-time of this equipment is taken as 40 years with an interest
rate of 8%, the capital recovery rate will be . 0838= Thus the total
annual cost will be
Capital cost per year $ 2,850
Annual operating cost 923
Total annual cost $ 3, 773
Lift-Drop Aerator
If no natural drop in elevation is available, then the water can be
pumped and allowed to splash back onto an apron. The efficiency for
this operation can be estimated from Eq. 71. For a drop of 2. 53 feet,
a value of 77 = 1. 565 #rriQ /hp-hr is obtained. However, in order that
a fair comparison can be made between this type device and other de-
vices operating in a stream where some of the water will be recircul-
ated through the device before it is mixed with the main body (for ex-
ample, a comparison with white-water generators), this value must
be reduced by some factor to account for re circulation. From Eq. 67
and Eq. 68 or from Fig. 18 this factor is seen to be approximately 2. 0
for white-water generators. Since the two types of devices are simi-
lar, a factor of 2 can be assumed for this case also.
-------
For the purpose of evaluating the economics, two values of 7] will be
used to represent devices in which recirculation is and is not inhib-
ited, namely, 1. 565 and 0. 782 #m/hp-hr respectively. These two
values correspond to a total input power of 12. 8 and 25. 6 hp respect-
ively if the drive efficiency is taken as 90%.
Assuming the capital cost for the Lift-Drop device is the same as for
white-water generators, the capital cost will be $7, 500 and $15, 000
respectively for the two values of TJ.
The estimated annual operating costs are listed below for the two
cases.
Case I Case II
77= 1. 565 n = 0. 782
Electric Power (@ $0. 01/kw-hr for
168 full days) $ 385 $ 769
Maintenance (@ 3% of initial cost) 180 360
Personnel (24 hrs/wk, 32 weeks,
@$10/hr) l, 280 1,280
Total Annual Operating Cost 1, 845 2, 409
If this equipment is assumed to have a life-time of 1 0 years and the
interest rate is 8%, then the annual capital cost will be $1, 055 and
$2, 110 for the two cases. The annual cost is given below.
Capital Cost per year 1, 055
Annual Operating Cost 1, 845
Total Annual Cost $2,900 $4,519
Hydraulic Turbine Aeration be Venting
The initial capital cost of providing a vent for a new turbine at the de-
sign stage is insignificant. However, the cost of providing a vent in
existing equipment may run into several thousand dollars. In addition
to the vent, an air flow control must be provided.
If the initial capital cost is taken at $1000 to cover the control system
and $3000 for a vent in existing equipment, and if the equipment life is
assumed to be 20 years, then the annual capital cost will be $406 for an
interest rate of 8%.
The total loss in electric power produced by the turbulence because of
air venting may be estimated by taking the mean value of T? reported
89
-------
Lakes
When the supplemental aeration of lakes is to be evaluated for optimum
economics, the same procedure that was used for streams can be em-
ployed. However, two additional factors must be considered. First,
whether the lake is initially stratified or not; second, the time span
in which the DO level must be improved from its initial value to the
final desired value.
It is helpful to recall a few previously discussed points concerning the
flow patterns induced in lakes. If the lake is stratified with high DO
water in the epilimnion and low DO water in the hypolimnion and no
appreciable organic load remains in the hypolimnion, then it might
suffice to simply mix the water vertically. It has been shown that this
can be done from a single location with one pump.
If on the other hand it is necessary to continue to add O? after the
water has been mixed vertically, the influence of even the most effect-
ive circulator (the large diameter ducted propellers) can only circul-
ate the water within a cylindrical "cell" equal in depth to the water
depth and in diameter to about four times the depth for a deep lake.
The economic selection of equipment will be demonstrated by an ex-
ample in which a 100-foot deep stratified lake with a thermocline at a
depth of 20 feet, a DO level in the epilimnion of 6 mg/liter and a DO
in the hypolimnion of 4 mg/liter is to be raised to a DO level of 6 mg/
liter everywhere. Thus the amount of oxygen that must be added per
acre of surface is 435 #mQ If the time in which the DO increase
must take place is assumea to be three months, then this corresponds
to 0, 198 #rriQ /hr per acre. If the lake surface area is taken at 100
acres, then the total oxygen flow rate will be 19. 8 #mQ /hr.
L*
A sub-surface aerator, a diffused aeration system with air bubbles,
and a hybrid system consisting of a white-water generator and a duct-
ed propeller will be evaluated
Sub-Surface Aerators
Since the water is deep, the ducted propeller with diameter = . 2H pre-
viously listed as Case I will be selected as the sub-surface device.
The oxygen transfer and transfer efficiency for Case I are given by Eq.
21B times 4H and Eq. 22B respectively, or
92
-------
tii = 47. 4xlO"6(C -C) V1/2H (4H) #m /hr (Eq. 21B)
T? = 0. 134 (C -C) H~1/2V~5/2 #rn /hp-hr (Eq. 22B)
°2
where C , C are in nig/liter
s
H is in ft
V is in ft/ sec
The last equation, however, must be corrected for drive efficiency,
propeller efficiency and friction loss in the bearings similar to the
manner used to develop Eq. 73. Thus the expression for overall effic
iency becomes
1
. 134(C -C)V"5/
0. 1 hpl
2-1/2 K4
2
(Eq. 74)
\vhere T\ - efficiency of device used to drive sub-surface
aeration shaft, assume 80%
T\ - propeller efficiency, assume 50%
Since the effective diameter of a Case I unit has been estimated at 4H
ft, a minimum of approximately 34 units will be required for contin-
uous aeration (that is, to continue to circulate all the water after de-
stratification). At this number of units, each unit would have to add
0. 582 #mo /hr. For this transfer rate the duct velocity and oxygen
transfer efficiency are found to be 0. 77 ft/sec and 0. 04 #rriQ /hp-hr
respectively. The total power required is thus equal to 19. 87- 04 or
494 hp.
The initial capital cost and annual operating costs are given below.
Initial Capital Cost
Cost of aeration units @ $2844/ea, See App. C)
Cost of hydraulic pump and drive
Cost of hydraulic piping (based on
2, 090' x 2, 090' surface area of
7x2, 090 @ $1. 00/ft)
Installation cost of aeration units
Including anchor (@ $100/ea)
$ 96,500
20, 000
14,630
3, 400
93
-------
Electric power supply (based on 500'
to nearest utility wire $ 1, 500
Site preparation 2, 000
Shelter 1. OOP
119, 030
Contingencies and Engineering @ 20% 23. 806
Total Initial Cost $142,836
Annual Operating Cost
Electric power (@ $0. 01/kw-hr, 91. 5 full days) 8, 900
Maintenance (@ 1% of initial cost
= (. 01)(. 8 x 142,836) 1, 190
Personnel (@ 4 hrs/wk, 12 weeks, $10/hr) 480
Total Annual Operating Cost $ 9, 760
If the interest rate is taken at 8% and the life-time of the equipment at
20 years, then the total annual cost will be the sum of $14, 500 and
$9, 760, or $24,260.
Diffused Aeration - Air Bubbles
In view of the fact that no comprehensive experimental data could be
found relative to the oxygen transfer efficiency for water depths of the
order of 100 feet, it will be necessary to estimate the capture coeffic-
ient and subsequently the efficiency from Eq. 55, or
7] =
4.65f n
co cornp
+ 1
#m /hp-hr (Eq. 55)
°2
From Fig. 16 and the previous discussion of this data, the value of f
will approach unity and may be assumed to be unity without introducing
serious error. If the frictional pressure drop is assumed to be 15 psi,
the compressor efficiency is assumed to be 80% and the electric drive
efficiency is taken at 90%, then the overall transfer efficiency will be
2. 56 #mQ /hp-hr. The total required power will be 7. 75 hp.
The initial capital cost and the annual operating cost are estimated
be low.
94
-------
Initial Capital Cost
Cost of compressor and drive $ 4, 950
Cost of air pipes (spaced 200 feet apart for a
2, 090' x 2, 090' surface
= 23, 000 ft at $1. 00/ft) 23, 000
Installation cost of piping (@ $1. 00/ft) 23, 000
Electric power supply (based on 500 feet
to nearest utility wire) 1 500
Site preparation 2, 000
Shelter 1, OOP
55, 450
Contingencies and Engineering (@ 20%) 11, 090
Total Initial Cost $66, 540
Annual Operating Cost
Electric power (@ $0. 01/kw-hr for 91. 5 full days) 127
Maintenance (@ 1% of initial cost) 554
Personnel (@ 4 hrs/wk for 12 weeks, $10/hr) 480
Total Annual Operating Cost $1, 161
If the interest rate is taken at 8% and the life-time of the equipment at
20 years, the total annual cost will be the sum of $6, 770 and $1, 161
or $7, 931.
Hybrid System
It is interesting to compare the above results for diffused aeration and
sub-surface aeration with a hybrid system consisting of one sub-sur-
face device used to produce stratified counter-current flow and a white
water generator placed at the outlet of the sub-surface device and
operated only during the time interval in which the lake is being de-
stratified. In order to estimate the cost, one of the 34 sub-surface
units will be selected. Since each of the 34 units had a horsepower
input of 14. 5 hp, the time to destratify the lake is found to be 313. 2
hours from Eq. 36 if a destratification efficiency (DE) of 2. 5% is as-
sumed for an epilimnion temperature of 70° F and a hypolimnion temp-
erature of 50°F. The white-water generator must therefore induce
a transfer of 139 #mQ /nr ^or 313. 2 hours in order to increase the
initial lake water from a DO level of 4 to 6 mg/liter. Since the water
that goes through the white -water generator will not be circulated
through it again until it is well mixed because of the stratified flow,
the appropriate value of 77 is taken as that measured in the University
95
-------
of Minnesota flow-through test, namely, 4. 14 #mo /hp-hr at stand-
ard conditions or 1. 84 for the present conditions based on shaft power.
If the drive is assumed to be electric at 90% efficiency, the total input
power for the white-water generator would be 139 [. 9(1. 89)] or 83. 7
hp. Based on Whipple's data the initial capital cost of the white-water
generator installation alone would be $49, 100. In addition to this
capital cost the sub-surface initial cost, as given below, must be in-
cluded.
Cost of one sub-surface device $ 2,844
Cost of hydraulic pump and drive 2, 000
Cost of hydraulic piping (based on 1/2 x 2, 090
at $1. 00/ft) 1, 045
Installation cost of one unit 500
6,389
Contingencies and Engineering (@ 20%) 1, 278
Sub-surface total 7, 667
White-water generator total 49, 100
Total Initial Capital Cost of Hybrid System $56, 767
The annual operating costs are estimated below:
Electric Power (@ $0. 01/kw-hr for 156. 6 hrs
(83. 7 + 14. 5)(156. 6)(. 745)(. 01) 114
Maintenance (@ 3% of initial cost) 1, 370
Personnel (16 hrs/cycle for 1 cycle, @ $lO/hr) 160
Total Annual Operating Cost $1, 644
If the interest rate is taken at 8% and the life-time of the equipment
is taken at 20 years, then the total annual cost will be the sum of
$5, 760 and $1,644, or $7,404.
In view of the fact that the white-water generator represents a major
portion of the capital cost and since this equipment can be made rela-
tively portable, it may be desirable to design the hybrid system with a
permanent sub-surface device but with a portable white-water gener-
ator that could be moved easily from one lake to another, thus reduc-
ing the total annual cost per lake.
A summary of the cost for each of the three systems for supplemental
aeration of the 100-acre lake is given in Table XIV.
96
-------
TABLE XIV
Summary of Annual Cost of Aeration Systems for Lakes
Type of Aerator
Comments
Total Annual
Cost
Hybrid, Sub-Surface
plus White-Water
Generator
Sub-Surface
Aerators
Diffused Aeration
System in operation only
during the 31 3° 2 hours re-
quired to destratify the
lake
Parallel pipes spaced 200 ft
apart, with an assumed cap-
ture coefficient of unity, and
an assumed frictional press-
ure drop of 14. 7 psi. Sys-
tem in continuous operation
for 3 months to achieve re-
quired ADO of 2 mg/liter
34 units in continuous
operation for 3 months to
achieve required iiDO of
2 mg/liter
7, 404
7, 931
24, 260
In regard to the question of whether destratification is desirable or
not in a given lake, it should be noted that for the example lake used
in this section, the hybrid system would destratify the lake in about
two weeks (based on an assumed DE of 2. 5%), the sub-surface aera-
tion would destratify the lake in less than one day at the same value of
DE, and the diffused aeration system would require about four weeks
to destratify the lake if it operated at the same value of DE.
97
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VIII. ACKNOWLEDGMENTS
The authors would like to acknowledge the advice and guidance of
Dr. William R. Duffer of the Kerr Water Laboratory of the Federal
Water Quality Administration in Ada, Oklahoma, who served as the
Technical Monitor of this study.
We would also like to acknowledge the assistance of Dr. R. W. Pat-
terson who assisted in the flow studies.
99
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IX. REFERENCES
1. Eckenfelder, W. W. , Industrial Water Pollution Control, Mc-
Graw Hill Cook Co. , New York, N. Y. 1966
2. Davis, Robert K. , Planning a Water Quality Management Sys-
tem: The Case of the Potomac Estuary, published in "Water
Research", Kneese and Smith, Editors, Johns Hopkins Press,
1966
3. Potomac River Basin Report (PRB Report), U. S. Army Engin-
eer District, Basin Studies Branch, Baltimore, Feb. 1963
4. Higbie, R. , The Rate of Absorption of a Pure Gas into a Still
Liquid During Short Periods of Exposure, Trans. Am. Inst.
Chem. Engrs, 31, 1935, pg. 365
5. Bird, R. B. , Steward, W. E. , and Ldghtfoot, E. N. , Trans-
port Phenomena, John Wiley and Sons, Inc. , New York, N. Y. ,
1966, pg. 522
6. Chemical Engineers' Handbook, Edited by John H. Perry, Mc-
Graw Hill Book Co. , New York, N. Y. , 1963, pp. 14-20, 21
7. Lewis, W". K. , and Whitman, W. C. , Principles of Gas Ab-
sorption, Industrial and Engineering Chemistry, V. 17, 1924,
pg. 1215
8. Sabersky and Acousta, Fluid Mechanics, MacMillan Co. , New
York, N. Y. , 1963
9. Danckwerts, P. V. , Significance of Liquid Film Coefficients in
Gas Absorption, Industrial and Engineering Chemistry, V. 43,
June 1951
10. O'Connor, D. J. , and Dobbins, W. E. , The Mechanism of Re-
aeration in Natural Streams, J. Sanitary Engrg. , ASCE, Dec.
1956, Pg. 641
11. O'Connor, D. J. , The Measurements and Calculations of Stream
Reaeration Ratio, Proc. Seminar in Oxygen Relationships in
Streams, U. S. Public Health Service, Robert A. Taft Sanitary
Engrg. Ctr. , Cincinnati, Ohio, March 1958
101
-------
12. Dobbins, W. E. , BOD and Oxygen Relationships in Streams,
J. Sanitary Engrg. , ASCE, June 1964
13. O'Connor, D. J. , St. John, J. P. , and DiToro, D. M. , Water
Quality Analysis of the Delaware River Estuary, J. Sanitary
Engrg. , ASCE, December 1968
14. Linsley, R. K. , and Franzini, J. B. , Water Resources Engin-
eering, McGraw Hill Book Co. , New York, N. Y. , 1964
15. Susag, R. H. , Polta, R. D. , and Schroepfer, G. J. , Mechan-
ical Surface Aeration of Receiving Waters, WPCF Journal, V.
38, No. 1, January 1966
16. Hutchinson, G. E. , A Treatise on Limnology. John Wiley and
Sons, New York, N. Y. , 1957
17. Churchill, M. A. , Effects of Storage Impoundments on Water
Quality, J. Sanitary Engrg. , ASCE, 1957
18. Hooper, F. F. , Ball, R. C. , and Tanner, H. A., An Experi-
ment in the Artificial Circulation of a Small Michigan Lake,
Trans. Am. Fisheries Soc. , 1952, pp. 82-222
19. Irwin, W. H. , Symons, J. M. , and Robeck, G. G. , Impound-
ment Destratification by Mechanical Pumping, J. Sanitary
Engrg, ASCE, December 1966
20. Harleman, D. R. F. , and Huber, W. D. , Laboratory Studies on
Thermal Stratification in Reservoirs, Proc. of the Specialty
Conference on Current Research into the Effects of Reservoirs
on Water Quality, Sponsored by ASCE, Vanderbilt Univ. , 1968
21. Brooks, N. , Koh, C. Y. , Selective Withdrawal from Density-
Stratified Reservoirs, Proc. of the Specialty Conference on
Current Research into the Effects of Reservoirs on Water Qual-
ity, Sponsored by ASCE, Vanderbilt Univ. , 1968
22. Harleman, D. R. F. , Mechanics of Condenser-Water Discharge
from Thermal-Power Plants, Proc. of National Symposium on
Engineering Aspects of Thermal Pollution, Vanderbilt Univ. ,
August 1968
23. Uttermack, P. D. , Discussion on "Control of Reservoir Water
102
-------
Quality by Engineering Methods, Proc. of Specialty Confer-
ence on Effects of Reservoirs on Water Quality, sponsored by
ASCE, Vanderbilt Univ. , January 1968
24. Symons, J. M. , Irwin, W. H. , and Robeck, G. G. , Impound-
ment Water Quality Changes Caused by Mixing. , J. Sanitary
Engrg. , ASCE, SA2, April 1967
25. Whipple, W. Jr. , Hunter, J. V. , Davidson, B. , Dettman, R. ,
and Yu, S. , Instream Aeration of Polluted Rivers, Water Re-
search Institute, Rutgers Univ. , New Brunswick, N. J. , August
1969
26. Symons, J. M. , Irwin, W. H. , Robinson, E. L. , and Robeck,
G. G. , Impoundment Destratification for Raw Water Quality
Control Using Either Mechanical or Diffused-Air Pumping, J.
AWWA, October 1967
27. Leach, L. E. , Duffer, W. R. , and Harlin, C. C. , Pilot Study
of Dynamics of Reservoir Destratification, FWPCA - Robert S.
Kerr Water Research Center, Ada, Oklahoma, 1968
28. Ippen, A. T. , and Carver, C. D. , Basic Factors of Oxygen
Transfer in Aeration Systems, Sewage and Industrial Waste,
July 1954
29. Ippen, A. T. , Campbell, L. G. , and Carver, C. D. , The Deter-
mination of Oxygen Absorption in Aeration Processes, Tech.
Rept. 7, MIT Hydrodynamics Lab. , Cambridge, Mass. , 1952
30. Maier, C. G. , The Ferric Sulf ate-Sulfur ic Acid Process, Bui.
Bu. of Mines No. 260, U. S. Commerce Dept. , Washington,
D. C., 1927
31. Langelier, W. F. , The Theory and Practice of Aeration, J.
AWWA, January 1932
32. Zieminski, S. A., Vermillion, F. J. , and St. Ledger, B. G. ,
Aeration Design and Development, Sewage and Industrial Waste,
October 1958
33. Kent. R. T. , Mechanical Engineer's Handbook, John Wiley and
Sons, New York, N. Y. , pp. 2-74
103
-------
34. Bernhardt, H. , Aeration of Wahnbach Reservoir Without
Changing the Temperature ^Profile, J. AWWA, August 1967
35. Aero-Hydraulic Technique as Applied for De stratification of
Impounded Water, Aero-Hydraulic Corp. , Montreal, Canada,
March 1962
36. Perry, R. H. , Chilton, C. H. , and Kirkpatrick, S. D. , Chem-
ical Engineers' Handbook, McGraw Hill Book Co. , New York,
N. Y. , 1963, pp. 18-60
37. Lueck, B. F. , Blabaum, C. J. , Wiley, A. J. , and Wisniewski,
T. F. , Evaluation of the Spray Type "Aqua-Lator" for River
Reaeration, Wisconsin Committee on Water Pollution, Bui.
No. WP-109, Madison, Wisconsin, March 1964
38. Kaplovsky, A. J. , Walters, W. R. , andSosewitz, B. , Artifi-
cial Aeration of Canals in Chicago, J. WPCF, V. 36, No. 4,
1964.
39. McKinney, R. E. , and Benjes, H. H. , Evaluation of Two Aer-
ated Lagoons, J. Sanitary Engrg. , ASCE, December 1965
40. Gameson, A. L. H. , Weirs and the Aeration of Rivers, J.
Inst. Water Engrg. , 11, 1957, pg. 477
41. Gameson, A. L. H. , Van Dyke, K. G. , Ogden, C. G. , The
Effect of Temperature on Aeration at Weirs, Water and Waste
Engineering, 1958, pg. 489
42. Gannon, J. J. , Aeration at Waste Treatment Plant Outfall
Structures, Water and Wastes Engineering, 1967, pg. 62
43. Artificial Reaeration of Receiving Waters, National Council of
the Paper Industry for Air and Stream Improvement, Tech. Bui.
No. 229, Tufts Univ. , Medford, Mass. , August 1969
44. Wiley, A. J., Lueck, B. F. , Scott, R. H. , and Wisniewski, T.
F. , Turbine Aeration of Streams, J. SPCF, I960, pg. 186
45. Speece, R. E. , Adams, J. C. , and Wooldridge, C. B. , U-Tube
Operating Characteristics, Presented at the 23rd Annual Purdue
Industrial Waste Conference, Purdue Univ. , Lafayette, Indiana,
May 1968
104
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46. Von der Emde, W. , Advances in Water Pollution Research,
Vol. II, Pergamon Press, New York, N. Y. , 1964
105
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X. NOMENCLATURE
a a given or fixed distance, ft
a an empirical coefficient
2
A area, ft
A' apparent surface area, ft
b a given or fixed distance, ft
C concentration,mass per unit volume, see text for units
C a constant
C specific heat, # -ft/#m
P f
C drag coefficient
d width or diameter, ft
2
D diffusion coefficient, ft /sec
J_*
D hydraulic diameter, ft
ri Y
DE destratification efficiency, fraction
f oxygen capture coefficient
co &
f pipe friction factor
P
F force, #
2
g acceleration of gravity, 32 ft/sec
g dimensional constant = 32. 2 #m-ft/# -sec
h height, ft
H water depth, ft
I Volume, ft
k Boltzmann constant or ratio of specific heats
K liquid film coefficient, ft/sec
L-i
K' apparent liquid film coefficient, ft/sec
LJ
m mass of a molecule, grams
m mass flow rate, #m/sec
M mass flow rate, #m/sec
N number of molecules striking unit surface per unit time
p gas pressure or partial pressure, psi or psf
107
-------
P power, # -ft/sec or hp
* 3
Q volume flow rate, ft /sec
r surface renewal rate, I/sec
r radius along which potential functions are located, ft
t time
T temperature, F
u average stream velocity, ft/sec
V velocity, ft/sec
W work or energy, #f-ft
x distance along x axis, ft
z distance along z axis, ft
Oi angle between horizontal and water bottom or spray direction,
degrees
|8 surface capture coefficient
y mass density, #m/ft
6 film thickness, ft
distance between centroids, ft
77 efficiency factor
fj, absolute viscosity, # -sec/ft
2
V kinematic voscosity, ft /sec
3
p mass density, slugs/ft
T wall shear stress, ft/sec
o
#, force expressed in pounds
#m mass expressed in pounds
Subscripts
a air
A above
B below
Comp Compression
108
-------
Subscripts (cont. )
N nitrogen
Lt
O oxygen
Lj
p frictional
r residence
s saturation
t total or overall
T terminal
w water
109
-------
APPENDIX A
OXYGEN CAPTURE AT AIR- WATER INTERFACE
111
-------
OXYGEN CAPTURE AT AIR- WATER INTERFACE
If the air in the vicinity of the water surface is assumed to have a
Maxwellian equilibrium velocity distribution, then the number of oxy-
gen molecules from the air that strike unit area of the interface per
unit time can be found by integrating the velocity distribution with
respect to orientation and speed. When this operation is carried out,
the number of oxygen molecules that leave the air and strike the inter-
face is given by the expression
(Eq.A-l,
where n = number density of oxygen molecules, no/cm
k = Boltzmann constant, 1. 380 x 10 erg/ K
T = absolute gas temperature, K (assume 20 C)
m = mass of oxygen molecules, grams
Since air will behave essentially as a perfect gas at modest pressures
and temperatures, the perfect gas equation of state can be introduced
into Eq. A 1, namely,
P = nkt (Eq. A-2)
where P = partial pressure of oxygen in air, assume
2 to be 0. 2 1 atmospheres
When Eq. A-2 is substituted in Eq. A-l, the expression for the num-
ber of oxygen molecules that strike unit area of water surface, per unit
time becomes
p
//2TT m k T
The mass of oxygen that leaves the air and strikes the water per unit
surface area per unit time is readily found by multiplying Eq. A-3 by
the mass of an oxygen molecule. When this is done and the resulting
expression evaluated for a temperature of 20°C and a partial oxygen
pressure of 0. 21 atm, the rate of oxygen flow to the water surface is
found to be
M = 6. 30 #m /ft sec (Eq. A-4)
°2 2
113
-------
Some of the molecules that strike the surface will be captured by the
water and some will rebound back into the air. Given time (and in the
absence of diffusion of oxygen from the interface to the main body of
water), the surface will come to a dynamic equilibrium where the flux
of oxygen molecules leaving the water and returning to the air will
equal the flux of oxygen molecules leaving the air and striking the
water. This dynamic equilibrium may be characterized by means of
the apparent fraction of the oxygen molecules that leave the air, strike
the water surface and are captured by the water. This fraction of in-
cident molecules which is captured by the water (/3) is a function of the
degree to which the liquid interface is saturated with oxygen. As the
dissolved oxygen level at the interface approaches saturation, /Smust
approach zero. The magnitude of ft may be approximated by equating
it to the ratio of volume available to oxygen molecules in the liquid
interface region to the total volume of the region at the time the mole-
cules strike the surface. This ratio can be expressed as
(Eq. A-5)
where m = mass of a molecule
A. N. = Avogardro number
D = molecular collision diameter
C = saturation concentration of DO, me/liter
s &
C = concentration of DO at the interface, mg/liter
C - concentration density of water, mg/liter
When Eq. A-5 is evaluated, the expression for /3 reduces to
j8 = 0. 74 x 10~6(C -C) (Eq. A-6)
S
Eq. A-6 and Eq. A-4 can now be combined to determine the mass of
oxygen captured per unit time per unit surface area, namely3
MQ =6.300 #m02/ft2sec (Eq. A-7)
Z(captured)
or
O 2
1 *y
(m^Q)U.N.
>M4 H 0M w'
LJ
114
-------
^O = 4. 65xlO~6(C -C)
2 (captured) « C) o sec (Eq. A-8)
where C C are in mg/liter
115
-------
APPENDIX B
POTENTIAL FLOW SIMULATION OF
SUB-SURFACE DEVICES
117
-------
POTENTIAL FLOW SIMULATION OF
SUB-SURFACE DEVICES
The flow induced by a sub-surface device may be approximated by a
family of potential flow functions. Once the flow field has been deter-
mined, the rate of surface renewal can be estimated and hence the
rate of oxygen transfer from the atmosphere to the water can be esti-
mated.
The flow patterns for various device configurations (See Fig. 6 for a
schematic of the three flow cases) will first be computed based on the
assumption that the water is of uniform density, the bottom is horizon-
tal and the side walls are infinitely far away. The influence of a verti-
cal temperature and a sloping bottom and side walls is discussed in
the body of the report. Each of the three flow cases is developed sep-
arately below.
Case 1 FLOW
The flow pattern induced by a propeller and a vertical duct, for a duct
width which is small compared to the water depth, can be approximat-
ed by a combination of an infinite series of sources and an infinite
series of sinks placed along the vertical centerline of the duct as
shown in Fig. B-l. The sources and sinks are so placed that no flow
takes place across the bottom or across the air-water interface.
Hence these two boundaries are treated as flat plates.
The complex potential function for all the sources, F , can be ex-
pressed in the x, y cartesian coordinate system (or z complex coordi-
nate system) as
F = L n {- JJn [z - in(2H-a)] - ^~ An [z - in(2H+a)J]
4- n=0 277 277
+ £,{- 4n [z + in(2H-a)J - ~- 4n[z + in(2H+a)]} (Eq. B-l)
n= 1 27T £7r
where Q = constant strength of the sources, ft3 per ft length of
device
Eq. B-l can be expressed in closed form as a hyperbolic function,
namely
B-2)
In a similar fashion the complex potential function for all the sinks,
119
-------
2a
H
2a
H
1
1 Air a~~T
a
00
a
* 4
'DUC
'
r
Wa
t
Water iLJ
[
ter Depth
a.
Bottom a '
Width
Propeller and '
o _
Za
Duct ,
i
L .
' Y
^ 1
r ~u~ '
m J^.
\
t
L .
i
l-^xvJ
t
»
f
r J
r Air
^ Water
I
Bottom
1
i
_
Potential Flow
Functions
Fig. B-l - Distribution of Sources and Sinks to Simulate Flow
induced by a Propeller in a Duct whose Width is
Small compared to the Water Depth
120
-------
F , can be expressed in the x, y cartesian coordinate system (or z
complex coordinate system) as
Q
. , 7T(z+iH)
-
(Eq. B-3)
\vhere Q = constant strength of the sinks, ft /sec per
ft length of device
The complex function for the source-sink flow between the air-water
interface and the bottom can now be found by adding Eq. B-2 and Eq.
B- 3 to obtain
Q . r . . 77 z 1 Q r . 77 z
' = ~^n Lsinh - - J - £n Lsinh . . J
(2H-a) 277
. .
(2H+a)
Q « r i, 77(z-iH)l . Q , r . , 7T(z+iH) -,
7T -^n Lsinh J + ~ &n Lsinh . J
2TT (2H-a) 277 (2H+a)
(Eq. B-4)
The velocity in the x-direction, u, and the velocity in the y-direction,
v, can now be determined at any point by noting that the following rela-
tion exists between F, u and v:
= - u + iv
(Eq. B-5)
From Eq. B-5 and Eq. B-4, the velocity components are found to be
Q
2H
Q
v = -
4H
(Eq. B-6)
(cont. )
121
-------
*4H
. 77(y+H-a)
Q -
cot ZH
r 7r(y+H-a)-i2
7T(y+H+a)
-L 2H
a 2H LC°LU2
r , 7TX -|2
+ [coth ]
7T(y+H+a) r
-<- 2H L^LL
HJ
TTv -i ?
(/ -A. ^ <-»
1- B-7)
[cot
The magnitude of the circulation zone can now be determined by plot-
ting v along the horizontal line that passes through the mid-height
point (See Fig. 7). From the symmetry u will be zero along this line
and hence v serves as a measure of the intensity of circulation at a
given point.
Once a duct width has been selected, the volume flow rate through the
duct per unit length of device [= (width)(v)] can be related to the
strength of the individual sources and sinks per unit length of device
(Q) by estimating what fraction of the water that emanates from the
source near the top of the water and passes down to the sink at the bot-
tom by taking a path outside of the duct walls, and hence which frac-
tion passes down to the sink at the bottom by taking a path inside the
duct walls. This estimate was made by plotting the vertical compon-
ent of velocity at various depths as a function of horizontal distance
from the vertical centerline. For a = 0. 01H these two ratios were
estimated at 0. 174 and 0. 826 respectively, whereas for a = 0. 1 OH and
a = 0. 45H, the corresponding ratios were o. 236, 0. 764 and 0. 700,
0. 300 respectively.
The liquid film coefficient Kj__ and hence the oxygen transfer rate from
the atmosphere to the water can now be estimated from the flow field.
The liquid film coefficient for this type flow is given by Eq. 16 as
KL =
where D = diffusion coefficient for O in water,
2. 65 x 10-8 ft2/sec 2
r = surface renewal rate
The surface renewal rate can be approximated by means of Eq. 17 as
r = { )
Ay upper water layer
122
-------
If the right hand side of Eq. B-9 is evaluated by taking the difference
in the x-direction velocity at the surface and at mid-depth, then the
liquid film coefficient at some distance x from the duct centerline be-
comes
/ U r
T, L . surface .
KL = ^DL( R/2 )
Since the zone of circulation is shown to be - 2H < x < -f 2H in the re-
port, the oxygen transfer rate per unit length of device is given by the
expres sion
(E,. B-H)
where u is given by Eq. B-6 when y is made equal to zero.
su. rici CG
When Eq. B-ll is evaluated, the final expression for the transfer rate
of oxygen for Case I Flow becomes
MQ = C1(10~6)(C -C)V1/2H #m/hr per ft length of (Eq. B-12)
2 device
where C , C are in mg/liter
s
V = duct velocity for the assumption that the duct
width is equal to 0. 2H
C = 50. 6 for a = 0. 01H
C = 47. 4 for a = 0. 1H
C = 18. 8 for a = 0. 45H
H = water depth, ft
In order to estimate the energy that must be supplied to the propeller
to produce a flow with velocity V in the duct, it is noted that in steady
operation the required energy will be equal to the energy dissipated in
frictional heating as the water enters the duct at the bottom, moves
through the duct, exits from the duct at the top and the loss of kinetic
energy as the water flow in the external circuit from the top to the
bottom of the duct. Since it is anticipated to operate the device at
velocities of a. few ft/ sec or less, the entrance losses and losses along
the duct can be neglected compared to the exit loss and kinetic energy
loss in the external circuit. These two losses together at most should
equal the kinetic energy of fluid in the duct just before it passes through
123
-------
the exit of the duct. Assuming the required energy is equal to a max-
imum value of one kinetic energy head, the oxygen transfer efficiency
can be found by dividing Eq. B-12 by the required power, namely
P = £1 (Eq. B-13)
where Q = volume flow rate through the duct,
= . 2HV per ft length of device
When Eq. 12 is divided by Eq. 13, the oxygen transfer efficiency is
found to be
C(C"C)
H
where C = 0. 143 for a = 0. 01
L*
C = 0. 134 for a = 0. 10
C = 0. 053 for a = 0. 45
L+
Case II FLOW
The flow pattern induced by a propeller and a vertical duct, for a duct
width which'is wide compared to the water depth, can be approximated
by a combination of several infinite series of sources and several in-
finite series of sinks placed along lines parallel to the vertical center-
line of the duct as shown in Fig. B-2. The sources and sinks are so
placed that no flow takes place across the bottom or across the air-
water interface. As a first approximation eleven infinite series of
sources and eleven infinite series of sinks were selected to character-
ize the flow as shown in Fig. B-2.
The complex potential for the infinite series of sources and sinks
along any one vertical line parallel to the duct centerline is given by
Eq. B-4. Thus the components of velocity (u and v) at any point are
given by adding the contribution of each of the eleven series together.
The contribution of each individual series is given by Eq. B-6 for the
x-direction component of velocity and by Eq. B-7 for the y-direction
component of velocity.
Proceeding in the same manner as for Case I Flows, the surface
124
-------
>uct -L Width
Propeller and Duct
H
2a
Water Depth
-y-
Bottom
H
H
+ t- !--«< t--t--f--+ -r
t- + + -t- ~- -t- -t- -«-- +
2a
2a
Potential Flow Functions
Fig. B-2 Distribution of Sources and Sinks to Simulate the Flow
Induced by a Propeller in a Vertical Duct whose Width
is Large compared to the Water Depth
125
-------
renewal rate can be approximated from Eq. B-9 as
surface /Tr, ,-. -, r\
r _ (Eq. B-15)
a
Since in the text of the report the zone of circulation is shown to be
- 5H < x < + 5H for a duct width of six times the water depth, the oxy-
gen transfer rate per unit length of device is given by the expression
x = +5H / u
M =J yo ( SUraaCe)(Cs-C)dx (Eq. B-16)
2 x = -5H
When Eq. B-16 is evaluated for the total contribution of the eleven in-
finite series, the final expression for oxygen transfer becomes
ill = 7000xlO~6(C -C)V1/2Hly'2 #m/hr per ft length (Eq. B-17)
2 of device
where C , C are in mg/liter
s
V = duct velocity, ft/sec
H = water depth, ft
If Eq. B-17 is divided by Eq. B-13, the expression for the oxygen
transfer efficiency becomes
0. 657 (C -C)
T? = - 1/2 S - #tn /hp-hr (Eq. B-18)
H 2
Case III FLOW
The flow pattern induced by a horizontal ducted propeller for a duct
height which is small compared to the water depth can be approximated
by the combination of an infinite series of sources and an infinite ser-
ies of sinks. The sources are placed along a vertical line that passes
through the duct outlet and the sinks are placed along a parallel verti-
cal line that passes through the duct inlet as shown in Fig. B-3.
The complex potential function for the infinite series of sources with
respect to the x, y cartesian coordinate system (or the z complex
coordinate system) is given by the expression
126
-------
Water Depth
_~-iL^-
H
H
H
f
H
| d
Height
|-<-b=duct length*-]
Propeller and Duct
Bottom
X
H/2 = water depth
- Bottom
Image of Air -
Water Interface
Potential Flow Functions
Fig. B-3 - Distribution of Sources and Sinks to Simulate Flow
Induced by a Propeller in a Horizontal Duct
127
-------
F = S f- JJnCz-inH]} + £ . {- ^- ^n [z+inH]} (Eq. B-19)
+ n=0 277 n= 1 277
Eq. B-19 can be expressed in closed form by means of a hyperbolic
function, thus
Q r . , 77 Z -,
F, = - Jin Lsinh J
T 277 ri
(Eq. B-20)
In the same manner the complex potential function for the infinite
series of sinks can be expressed in closed form as
v t r u
F = An Lsinh
277 H
(Eq. B-21)
where b = length of duct, ft
The total complex potential function for the two infinite series is
given by adding Eq. B-20 and Eq. B-21, namely
F =-^T ^nCsinh ff J + 4n[sinh ^~
277 H 277 H
(Eq. B-22)
The velocity components (a and v) can now be found at any point with
the aid of Eq. B-5, thus
%
r 77V -i 2 77x 77x ;
Lcot f-J coth + coth- i
ri ri rl
u =
v =
2H
-Q
2H
[cot^]2 + [coth^]2 j
Q
2H
*. ff
C0tl
1 Q
' 2H
r--t ^y i2 i r.-tl- 77(x+b)n2 f
L cut J "1 Lcutii J {
ri ri ]
y , 77y r 77x -, 2
I C0t H [cothl^ I
" V ~i r -,7/x~i^
I H" 1 c o til * "" J
H H
r_t ^7 ..t 77y r..t1 77(x+b) n2
cot cut *: Lcotn J
ri ri ri
r 77y -,2 p 77(x+b) ^2
H H
(Eq. B-23)
(Eq. B-24
128
-------
Since it is anticipated that this configuration might be applied to a
stream, the situation requires b > H. For this case the x-direction
component of velocity as given by Eq. B-23 does not vary much with
distance in the y-direction for a given value of x between x=0 and x=-b.
As a result the velocity-depth profile will be similar to that found in a
natural stream and hence the surface renewal rate can be estimated
from Eq, B-25 (See. Eq. 36, Ref, 10).
0. lu
ave
r =
0.
where u = average of x-direction velocity over the depth
at a given location
Since the zone of circulation is shown to correspond approximately to
the duct width for b > H, the oxygen transfer rate per unit length of
device is given by the expression
x=-b Iu"
M = I VD ( suriace ) (C -C) dx (Eq. B-26)
°2 x=0 L f -d
When Eq. B-26 is evaluated with the use of Eq. B-23 for d = H/4, the
expression for the oxygen transfer rate per unit length of device be-
comes
M = C (10~6)(C -C)V1/2H1/2 #m/hr per ft length (Eq. B-27)
\*s J. S ,. .. .
2 of device
where C , C are in mg/liter
s
V = duct velocity, ft/sec
H = twice the water depth, ft
C = 524 for b = 10H, d = . 5H
C = 262 for b = 5H, d = . 5H
2
The oxygen transfer efficiency can again be estimated by assuming the
required power input must be equivalent to the dissipation of one veloc'
ity head. Thus the transfer efficiency, T\, can be found by dividing
Eq. B-27 by Eq. B-13 to obtain
129
-------
C,(C -C)
- *mo
H v 2
where C = 0.589 forb=10H, d = . 5H
LJ
C = 0. 2 14 for b = 5H, d = . 5H
130
-------
APPENDIX C
COST ESTIMATE OF SUB-SURFACE AERATORS
131
-------
COST ESTIMATE OF SUB-SURFACE AERATORS
The capital cost for Case I and Case II devices are estimated in this
Appendix.
CASE II DEVICE
Fig. C-l shows a sketch of a sub-surface device designed for a
stream depth of four feet. The cost estimate for this device is given
below.
Material Cost
Part
Top Ring
Bottom Ring
Side Channels
Propeller Braces
Side Wall
Fan (fabricated)
Hydraulic Motor
Total Material Cost
Fabrication Cost
Total Unit Cost
Material
Alum. 6" x .5"
Alum. 6" x . 5"
Alum. 4" x . 180"
Alum. 4" x . 180"
Fiberglass 4' x 75'
x . 25"
Fiberglass
-
Weight
208#
208#
150#
38#
Area=300 ft
o
@ $. so/fr
-
-
Cost
$ 208
208
150
38
I =10
J. _J \J
200
80
$1, 034
300
$1, 344
CASE I DEVICE
Case I differs from CaseII primarily in the fact that the duct length
is long in the Case I device and short in the Casell device. Case I
flow could be created by the same unit shown in Fig. C-l with the add-
ition of a long duct held up by buoyant material in the form of a number
of collars. If the cost of this extension is taken at $0. 25/ft , then the
total cost of a Case I unit for a 100-foot deep lake would be $1, 334 +
(.25)(96)(77 x 20) or $2844.
133
-------
24' D.
Side
Channels
ZO per unit
Top
Ring
AL - 6"xO. 4"
Side
Wall
Fiberglass
Propeller Braces
4 per unit
Bottom
Ring
AL - 6"xO. 4"
Propeller, Fiberglass
Fig. C-l - Sub-Surface Circulating Device
Vertical Duct
- Propeller in a
134
-------
BIBLIOGRAPHIC:
Hogan, W.T., Reed, F. E. and Star-
bird, A. W. , Mechanical Aeration Sys-
tems for Rivers and Ponds, Littleton
Research & Engineering Corp. , Final
Rept. FvVQA Contract 14-12-576, 11/70
ABSTRACT
A study of methods of increasing the
dissolved oxygen in rivers and ponds.
Analytical and empirical relations est-
ablish operating characteristics of test-
ed and untested aerating devices. From
estimates of cost and efficiency, the
most economical methods of aerating
rivers and ponds are determined.
ACCESSION NO.
KEY WORDS
Mechanical
Aeration
Aeration
Efficiency
Aeration
Rivers
Ponds
Economic
Prediction
Hydraulic
Engineering
Water Quality
Control
BIBLIOGRAPHIC:
Hogan, W.T., Reed, F. E. and Star-
bird, A. W- , Mechanical Aeration Sys-
tems for Rivers and Ponds, Littleton
Research & Engineering Corp. , Final
Rept. FvVQA Contract 14-12-576, 11/70
ABSTRACT
A study of methods of increasing the
dissolved oxygen in rivers and ponds.
Analytical and empirical relations est-
ablish operating characteristics of test-
ed and untested aerating devices. From
estimates of cost and efficiency, the
most economical methods of aerating
rivers and ponds are determined.
ACCESSION NO.
KEY WORDS
Mechanical
Aeration
Aeration
Efficiency
Aeration
Rivers
Ponds
Economic
Prediction
Hydraulic
Engineering
Water Quality
Control
BIBLIOGRAPHIC:
Hogan, W.T., Reed, F. E. and Star-
bird, A. vV. , Mechanical Aeration Sys-
tems for Rivers and Ponds, Littleton
Research & Engineering Corp. , Final
Rept. FvVQA Contract 14-12-576, 11/70
ABSTRACT
A study of methods of increasing the
dissolved oxygen in rivers and ponds.
Analytical and empirical relations est-
ablish operating characteristics of test-
ed and untested aerating devices. From
estimates of cost and efficiency, the
most economical methods of aerating
rivers and ponds are determined.
ACCESSION NO.
KEY WORDS
Mechanical
Aeration
Aeration
Efficiency
Aeration
R i v e r s
Ponds
Economic
Prediction
Hydraulic
Engineering
Water Quality
Control
-------
Access/on /Vum&er
Subject Field & Group
04A, 05F
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organ iza (ion
Littleton Research and Engineering Corp.
Littleton, Massachusetts 01460
Title
MECHANICAL AERATION S YSTEMS FOR RIVERS AND PONDS,
I Q Authors)
Hogan ,
Reed, F
C-J- -«"U4 -« J
William T.
. Everett
A 1 !_ 4- TAT
16
21
Project Designation
16080D0007/70
Note
22
Citation
23
Descriptors (Starred First)
Aeration, * Rivers , * Ponds , * Economic Prediction, * Hydraulic Engineering, *
Air Entrainment, Bubbles, Dissolved Oxygen, Water Circulation, Mixing,
Oxygenation, Water Quality Control, *
25
Identifiers (Starred First)
Mechanical Aeration, * Aeration Efficiency,* Aeration Devices, Aeration Methods
27
Abstract
The total annual cost of providing supplemental aeration of streams
and lakes by tested and untested aeration equipment is estimated.
Analytical and empirical equations are presented for the determina-
tion of operating characteristics of the various devices used to aerate
natural bodies of water. For the example stream evaluated in this
study, the most economical means of artificial aeration generally pos-
sible was found to be mechanical aerators which generate a highly
turbulent white -water surface. For the example lake evaluated, the
most economical technique for the continual input of oxygen into a
lake was found to be diffused aeration using air bubbles; whereas the
most economical technique for rapid input of oxygen, operating only
while the lake is being destratified, was found to be a hybrid system
consisting of a large diameter ducted propeller which draws water
from the lake bottom and discharges it at the surface where it is aer-
ated by a mechanical aerator.
Abstractor
Wm. T. Hogan
Instituti
Littleton Research and Engineering Corp.
WR:10Z (REV JULY 1969)
WRSI C
SEND TO: WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U S. DEPARTMENT OF THE INTERIOR
WASHINGTON. D. C 20240
* GPo: 1969-359=339
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