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
Environmental Protection Agency, through inhouse research and grants and
contracts with Federal, State, and local agencies, research institutions,
and industrial organizations.

Inquiries pertaining to Water Pollution Control Research Reports should be
directed to the Head, Project Reports System, Planning and Resources
Office, Office of Research and Development, Environmental Protection Agency,
Water Quality Office, Room 1108, Washington, D. C.  20242.

 * 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

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          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

-------
  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

-------
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

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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

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    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

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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

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 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

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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

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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

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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

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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

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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

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 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

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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

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                                      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

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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

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 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

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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

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   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

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 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

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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

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                            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-
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 2.    Davis,  Robert K. , Planning a Water Quality Management Sys-
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 3.    Potomac River Basin Report  (PRB  Report), U. S. Army Engin-
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 4.    Higbie,  R. ,  The Rate of Absorption of a Pure Gas into a Still
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 5.    Bird, R. B.  , Steward,  W. E. , and  Ldghtfoot, E.  N. , Trans-
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 6.    Chemical  Engineers'  Handbook, Edited by John H. Perry, Mc-
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 7.    Lewis,  W". K. , and Whitman,  W.  C. ,  Principles  of Gas Ab-
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 8.    Sabersky and Acousta,  Fluid Mechanics,  MacMillan Co. ,  New
      York, N.  Y. ,  1963

 9.    Danckwerts,  P.  V. , Significance of Liquid Film Coefficients in
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10.    O'Connor, D.  J. ,  and Dobbins,  W.  E. ,  The Mechanism of Re-
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11.    O'Connor, D.  J. ,  The Measurements  and Calculations of Stream
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                             101

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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
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14.   Linsley, R.  K. , and Franzini, J. B. , Water Resources Engin-
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15.   Susag, R.  H. , Polta, R. D. ,  and Schroepfer, G.  J. ,  Mechan-
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16.   Hutchinson,  G. E. , A Treatise on Limnology. John Wiley and
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17.   Churchill, M. A. , Effects of Storage Impoundments on Water
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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-
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20.   Harleman, D. R.  F.  , and Huber,  W. D. , Laboratory Studies on
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      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-
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22.   Harleman, D.  R.  F.  , Mechanics of Condenser-Water Discharge
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      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
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24.   Symons,  J.  M. , Irwin, W.  H. , and Robeck,  G. G. , Impound-
      ment Water Quality Changes Caused by Mixing. , J. Sanitary
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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
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      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,
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33.   Kent.  R.  T. , Mechanical Engineer's Handbook, John Wiley and
      Sons,  New York,  N. Y. ,  pp. 2-74
                              103

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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

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                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

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 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

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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

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                              >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

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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

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              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

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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

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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

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                APPENDIX C
COST ESTIMATE OF SUB-SURFACE AERATORS
                     131

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       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

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                             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

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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

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     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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