EPA-68 0/2-73-02 5a
December 1973
Environmental Protection Technology Series
Hypolimnion Aeration
with Commercial Oxygen -
Vol. I - Dynamics of Bubble Plume
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. 20460
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and
Monitoring, Environmental Protection Agency, have
been grouped into five series. These five broad
categories were established to facilitate further
development and application of environmental
technology. Elimination of traditional grouping
was consciously planned to foster technology
transfer and a maximum interface in related
fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL
PROTECTION TECHNOLOGY series. This series
describes research performed to develop and
demonstrate instrumentation, equipment and
methodology to repair or prevent environmental
degradation from point and non-point sources of
pollution. This work provides the new or improved
technology required for the control and treatment
of pollution sources to meet environmental quality
standards.
-------
EPA-660/2-73-025a
^December 1973
HYPOLIMNION AERATION
WITH
COMMERCIAL OXYGEN
VOLUME I
DYNAMICS OF BUBBLE PLUME
By
R. E. Speece
Fawzi Rayyan
The University of Texas at Austin
Project 16080 FYW
Program Element 1620^5
Project Officer
Lowell E. Leach
Robert S. Kerr Environmental Research Laboratory
Ada, Oklahoma 7^820
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20^60
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price S2
-------
EPA Review Notice
This report has been reviewed by the Environmental Protection
Agency and approved for publication. Approval does not
signify that the contents necessarily reflect views and
policies of the Environmental Protection Agency, nor does
mention of trade names or commercial products constitute
endorsement or recommendation for use.
ii
-------
ABSTRACT
The characteristics of a bubble-water plume, as used in hypolimnion
aeration, were studied. The major factor introduced in the study of these
characteristics was the effect of mass transfer.
A mathematical model was developed for this case and compared with a
mathematical model which neglects the effect of mass transfer. Both the
case of slip velocity and zero slip velocity were considered and a com-
parison is presented. The model calculates the diameter of the bubble,
the diameter of the plume, the velocity of plume rise, the water flow rate,
and the momentum and energy flux for the rising plume at any level above
the diffuser. It also calculates the amount of oxygen absorbed at any
level and the increase of the dissolved oxygen concentration in the
plume for any oxygen flow rate.
The model was verified by field measurements. A description of the field
experiment and the techniques used are presented.
This report was submitted in fulfillment of Project Number 16080 FYW under
the partial sponsorship of the Office of Research and Monitoring, Environ-
mental Protection Agency.
-------
CONTENTS
Section page
I CONCLUSIONS 1
II RECOMMENDATIONS 3
III INTRODUCTION 5
IV OBJECTIVE AND SCOPE 9
V BACKGROUND 11
VI EQUIPMENT AND TECHNIQUES 31
VII MATHEMATICAL FORMULATION 43
VIII EXPERIMENTAL OBSERVATIONS 59
IX DISCUSSION 65
X ACKNOWLEDGMENTS 105
XI REFERENCES 106
XII LIST OF PUBLICATIONS 110
XIII APPENDICES 111
v
-------
FIGURES
No. Page
1 THERMAL STRATIFICATION PHENOMENON 6
2 PROPOSED MECHANISM OF BUBBLE FORMA-
TION (KUMAR (31)) 13
3 TERMINAL VELOCITY OF AIR BUBBLES IN TAP
WATER AS A FUNCTION OF BUBBLE SIZE
(HABERMAN (17)) 19
4 RELATIONSHIP BETWEEN LIQUID FILM COEFFI-
CIENT AND BUBBLE DIAMETER (MODIFICATION
OF BARNHART (31)) 23
5 RELATIONSHIP OF INITIAL AND FINAL VOLUME
OF BUBBLE FOR DIFFERENT COLUMN HEIGHTS
(DATTA (10)) 24
6 VOLUME OF BUBBLE VS. HEIGHT ABOVE OUTLET
AND CORRESPONDING VELOCITY (DATTA (10)) 24
7 POSSIBLE FLOW REGIMES WITH A BUBBLE SCREEN
IN A TWO-LAYER DENSITY SYSTEM (CEDERWALL (8)) 26
8 VELOCITY PROFILES ABOVE A SINGLE ORIFICE
(KOBUS (20)) 30
9 RATE OF SPRFJ^D OF VELOCITY PROFILES (KOBUS
(20)) 30
10 FIELD EXPERIMENTAL SET-UP 35
11 FLUOROMETER CALIBRATION CURVES 36
12 FLOW METER CALIBRATION CURVE 38
13 MAP OF TEST SITES 40
VI
-------
FIGURES
No. Page
14 DEFINITION SKETCH 45
15 THE COEFFICIENT OF ENTRAINMENT AS A
FUNCTION OF THE AIR FLOW RATE FOR TWO-
AND THREE- DIMENSIONAL AIR BUBBLE PLUMES
(CEDERWALL (8)) 51
16 AVERAGE RISING SPEED OF BUBBLE STREAM FROM
A SINGLE ORIFICE (KOBUS (20)) 54
17 DYE CONCENTRATION AT VARIOUS DEPTHS AND
DISTANCES AWAY FROM THE DIFFUSER 62
18 DYE MOVEMENT AT A DEPTH OF 75 FEET (40
FEET AWAY FROM THE DIFFUSER) 62
19 TEMPERATURE PROFILE - LAKE TRAVIS MAY 14,1972 63
20 MEASURED AND PREDICTED CENTER LINE VELO-
CITY FOR 0.75 and 2.0 L./Min. O FLOW RATE
AND INJECTION DEPTH OF 60 FEEF 67
21 MEASURED AND PREDICTED CENTER LINE VELOC-
ITY FOR 1 AND 2 L./Min. O FLOW RATE AND
INJECTION DEPTH OF 30 FEET 68
22 PREDICTION CONSIDERING THE EFFECT OF EN-
to TRAINMENT COEFFICIENT ON HALF-WIDTH OF 70-
24 PLUMS, CENTER LINE VELOCITY AND WATER FLOW 72
RATE RESPECTIVELY
25 PREDICTED BUBBLE DIAMETER AS A FUNCTION OF
DEPTH FOR 2 AND 0.2 MM. BUBBLES 73
26 COMPARISON OF PREDICTIONS OF MODEL ONE
to (0.2 and 2.0 MM. BUBBLE) AND MODEL TWO FOR 75
31 CENTER LINE VELOCITY, HALF-WIDTH OF THE PLUME ,80
WATER FLOW RATE, MOMENTUM FLUX AND KINETIC
ENERGY FLUX FOR AN OXYGEN FLOW RATE OF 1 L. /Mln.
AND INJECTION DEPTH OF 45 METERS
vii
-------
FIGURES
No. Page
32 DEFINITION SKETCH 82
33 PREDICTION CONSIDERING THE EFFECT OF
to INJECTION DEPTH ON CENTER LINE VELOCITY, 83-
37 HALF-WIDTH, WATER FLUX, MOMENTUM FLUX, 87
AND KINETIC ENERGY FLUX FOR INJECTION
DEPTHS OF 75 AND 45 METERS
38 PREDICTION CONSIDERING THE EFFECT OF FLOW
to RATE ON HALF-WIDTH OF THE PLUME, CENTER 89-
41 LINE VELOCITY, AND WATER FLOW RATE FOR 1 92
AND 3 L./MIN. AND 0.2 and 2.0 MM. BUBBLE
42 PREDICTION OF MODEL ONE (0.2 MM. BUBBLE)
to AND TWO OF THE HALF-WIDTH OF THE PLUME 94-
43 AND CENTER LINE VELOCITY CONSIDERING NON 95
ZERO SLIP VELOCITY AND COMPARED WITH PRE-
DICTIONS FOR ZERO SLIP VELOCITY
44 PREDICTION OF MODEL ONE (2.0 MM. BUBBLE)
to OF THE CENTER LINE VELOCITY AND HALF-WIDTH 96-
45 OF PLUME CONSIDERING N n ZERO SLIP VELOCITY 97
AND COMPARED TO ZERO sftp VELOCITY
46 INCREASE IN OXYGEN CONCENTRATION VS. HEIGHT
ABOVE DIFFUSER FOR A FLOW RATE OF 1 L./MIN.
AND BUBBLES OF 0.2 AND 2.0 MM. AS PREDICTED
BY MODEL ONE 98
47 RELATIONSHIP BETWEEN LIQUID FILM COEFFICIENT
AND BUBBLE DIAMETER AT 20°C 112
48 MG. OF O AS A FUNCTION OF DEPTH 113
b
49 MG. OF N AS A FUNCTION OF DEPTH 114
£t
50 FRACTION OF ORIGINAL AMOUNT OF O REMAIN-
ING AS A FUNCTION OF DEPTH 115
viii
-------
FIGURES
No. Page
51 FRACTION OF BUBBLE GAS WHICH IS O 116
52 NITROGEN/OXYGEN RATIO AS A FUNCTION OF
DEPTH 117
53 RELATIVE AMOUNT OF O IN BUBBLE NORMALIZED
TO CONSTANT RELEASE DEPTH 118
54 CSATO (Mg/mm3) VS. DEPTH
55 MG. OF CO IN BUBBLE AS A FUNCTION OF
DEPTH 120
56 CSATO VS. DEPTH 121
57 MODEL PREDICTIONS FOR CENTER LINE VELOC-
to ITY, HALF-WIDTH OF PLUME, WATER FLOW RATE 122
86 AND THE INCREASE IN D.O. FOR INJECTION DEPTHS 151
OF 25, 50, 75, 100 METERS AND FLOW RATES OF 1
AND 2 L./MIN.
IX
-------
TABLES
No.
1 MEASURED CENTER LINE VELOCITY OF RISE 60
2 MEASURED AND COMPUTED CENTER LINE
VELOCITY OF RISE 66
3 MODEL PREDICTIONS FOR 1 L./MIN. AND
DEPTH OF INJECTION OF 25 AND 50 METERS 100
4 MODEL PREDICTIONS FOR 1 L./MIN. AND
DEPTH OF INJECTION OF 75 AND 100 METERS 101
5 MODEL PREDICTIONS OF 2 L./MIN. AND
DEPTH OF INJECTION OF 25 AND 50 METERS 102
6 MODEL PREDICTION FOR 2 L./MIN. AND
DEPTH OF INJECTION OF 75 AND 100 METERS 103
7 ILLUSTRATIVE EXAMPLE 104
x
-------
SECTION I
CONCLUSIONS
1. The proposed mathematical model for the dynamics of plumes incorpo-
rating mass transfer is in good agreement with the field measurements,
2 . The effect of mass transfer on the various dynamic characteristics of
the bubble plume cannot be neglected.
3. Slip velocity as a function of diameter has a negligible effect on
plumes of small bubbles but an increasing effect with larger bubbles.
4. With larger bubbles, the bubble-water plume exhibits a higher final
center line velocity, a higher water flow rate, a higher momentum
flux, a higher kinetic energy, but a smaller half-width as compared
to smaller bubbles.
5. An increase in the oxygen flow rate from one to three 1 /min did not
lead to a noticeable corresponding increase in the water flow rate.
6. A deeper injection depth led to a lower center line velocity, momen-
tum flux, and kinetic energy flux, but to a larger plume and a higher
water flow rate, at any height above the diffuser.
7. The energy left in a plume generated by 0.2 mm diameter bubbles was
very small at the metalimnion and did not measurably cause any dis-
turbance of the stratification.
-------
SECTION II
RECOMMENDATIONS
It is recommended that the plume dynamics from a line source diffuser
be experimentally and mathematically verified. The factors which con-
trol "uncoupling" of the bubble plume at the metalimnion need to be
quantitatively defined. The entrainment coefficient,^, was taken from
the literature for this study and should be evaluated in the field. The
spatial zone of influence for a point source and line source diffuser needs
to be defined. A number of these recommendations are presently under
study in the author's laboratory. Finally, a full scale demonstration of
hypolimnion aeration with commercial oxygen is in order.
-------
SECTION III
INTRODUCTION
The phenomenon of temperature stratification of lakes and reservoirs has
been observed for several years. Thermal regimes in impoundments in
temperate climates include a period in the summer during which vertical
temperature gradients become large. Such a stratification persists until
the Fall when temperature becomes uniform throughout the impoundment
and the "Fall overturn" occurs. Stratification of the impoundment leads
to the formation of three distinct layers, as shown in Figure 1. The
epilimnion (the warmer layer on top) with almost a uniform temperature
and density throughout; the metalimnion, (the middle layer) , with a steep
temperature gradient; and finally the hypolimnion (the bottom, cold layer
of water) , with a slight temperature gradient throughout. Temperature
and density are not the only differences between these three layers . The
epilimnion has a good quality water except for the high temperature.
This of course includes a near saturation level of dissolved oxygen. The
hypolimnion, on the other hand, is a good source of cold water, but
unfortunately becomes poor water quality when the dissolved oxygen is
depleted. Depending on the depth of the impoundment and the level of
nutrients, the dissolved oxygen concentration in that layer drops to zero.
This fact limits the withdrawal of water from this nutrient-rich low-quality
layer. Literature is abundant on the water quality of hypolimnetic water
and will not be discussed here.
Several remedies for this problem have been suggested; a brief discussion
of such methods will be presented here.
Destratification
In order to avpid anaerobic conditions in the hypolimnion, the impoundment
is mixed to provide D.O. in the bottom waters. Some of the methods used
are mechanical mixing, air pumping, water pumping, etc. Regardless of
the benefits achieved by such action, certain disadvantages are also
attached to it. Such disadvantages are; the loss of the cold water resource,
the increase in productivity due to recycling nutrients and the economical
loss associated with the loss of cold water fisheries in the impoundment.
-------
EPILIMNION
DEPTH
TEMPERATURE
Fig . 1 THERMAL STRATIFICATION PHENOMENON
-------
Selective Withdrawal
This method which requires gates at various depths, besides being expen-
sive, could lead to deteriorating the water quality down-stream by re-
leasing water of high nutrient content, thus increasing its productivity.
In addition, temperature and D.C. cannot be independently regulated.
Hypolimnion Aeration
Aeration of the hypolimnion water by use of pure oxygen without des-
troying the stratification has certain engineering merits (38). Some of
the advantages of such a process are:
1. Preservation of the cold water resource;
2 . Nutrients are not recycled and are kept entrapped in the hypolimnion
away from the euphotic zone; and
3. Cold water fisheries could be sustained below the reservoir.
-------
SECTION IV
OBJECTIVE AND SCOPE
Aeration of the hypolimnion by the use of pure oxygen in the form of very
fine bubbles generates a gas-water jet and the hydro-dynamic charac-
teristics of air bubble plumes have been studied before by several inves-
tigators (1, 2, 8, 11, 14, 25, 27, 47). However, the effect of mass
transfer on the hydro-dynamics of this system has not yet been considered.
This work will consider the effect of the phenomenon of mass transfer in
conjunction with the velocity of rise, the size of the gas-water jet, water
flow, and finally the momentum and kinetic energy flux for such a rising
plume. Mathematical expressions describing this plume along with com-
puter programs for the solution of the governing equations will be developed,
This mathematical model solves for the bubble diameter, the center line
velocity of rise and the diameter of the plume at any height above the
injection point. Using these computed values, the water flow, momen-
tum flux, and kinetic energy at any level are calculated. The model
calculates the change in D.O. concentration at any level for a given
oxygen flow rate and bubble diameter.
To show the effect of the mass transfer phenomenon on the behavior of the
plume, a comparison with the case of no mass transfer will be carried out.
Consideration of zero and non zero slip velocity will also be investigated.
Field measurements to verify the model were carried out, the descrip-
tion of which is shown in Section VI.
The experimental works also included a study of the mixing and circula-
tion generated by the rising plume. Associated diffusion of this plume
was studied experimentally and described in Section VI.
-------
SECTION V
BACKGROUND
Air bubble systems have been used extensively and for a variety of pur-
poses. They have been used in industrial processes such as distillation,
evaporation, direct contact heat exchange, etc. They have also been
used for pneumatic breakwaters, prevention of ice formation and destra-
tification and quality control management of lakes and reservoirs (2 , 14,
15, 21, 23, 32) .
Numerous papers have been written on the advances in this area ranging
from fundamental studies to descriptions of several new processes and
equipment designs. The purpose of this section is to present a compre-
hensive review of those references that are related to this study. It
will cover the following topics .
1. Bubble formation
2 . Shape of bubbles
3. Path of rising bubbles
4. Velocity of rise of single bubbles and swarms of bubbles
5. Gas-liquid mass transfer
6. Induced mixing and energy consideration
Bubble Formation
Several papers have been written describing bubble formation from a
single orifice and from porous plates submerged in different liquids.
Kumar and his colleagues (6, 19, 31, 35) have reviewed the literature
in this area and published a series of articles on the basic aspects of
bubble formation from single submerged nozzles and porous plates.
Their work included bubble formation under constant flow conditions (31)
and under constant pressure (35). In their analysis they assumed that
the bubble formation takes place in two stages, the expansion stage and
the detachment stage, Figure 2 . The final volume of the bubble is the
sum of the volumes pertaining to the two stages. This in equation form is
11
-------
(i)
where: V = final volume
V = volume from first stage
ill
Q = flow rate
t = time of detachment
c
At vanishingly small flow rates, the bubble volume can be directly obtained
by equating the surface tension force with the buoyancy force. However,
at a finite flow rate, forces associated with the expansion also exert
their influences. The bubbles expand at a definite rate, thereby giving
rise to the inertial force and the viscous drag, both of which add to the
surface tension force.
The first stage is assumed to end when the downward forces are equal to
the upward forces. The quantitative expressions for various forces are;
buoyancy force = V(p, - p ) g (2)
i g
viscous drag GTT r uv (3)
e e
surface tension = rrD Y (cos 9) (4)
inertial force = (d/dt ) (Mv ) (5)
e e
3
where: V = volume of bubble in cm
3
p = density of liquid, g /cm
3
p = density of gas, g /cm
D = orifice diameter, cm
v = velocity of center of bubble in first stage,
e /
cm /sec
X = surface tension, dyn /cm
M = virtual mass of bubbles, g
12
-------
EXPANSION STAGE
DETACHMENT
STAGE
CONDITION OF
DETACHMENT
Fig. 2 PROPOSED MECHANISM OF BUBBLE
FORMATION (KUMAR (31))
13
-------
Kumar et al. (31) considered the virtual mass of the bubble to be the sum
of the mass of gas and that of 11/16 of its volume of the liquid surrounding
it.
A mathematical formulation for calculating the final volume of the bubble
is presented. The effect of various parameters, viscosity, surface tension,
liquid density, flow rate and orifice diameter on the bubble formation and
its volume is covered in these studies. Bowonder and Kumar (6) have also
presented studies on bubble formation on porous plates submerged in liquids
of various physical properties. They found that the number of effective
sites for bubble formation was much smaller than the total potentially
available sites. The number of effective sites was found to be a function
of surface tension, liquid density and gas flow rate. Pattle (29) has
studied the factors in the production of small bubbles less than 1 mm. in
diameter. Some of these are grease, low surface tension, and availabil-
ity of organic or inorganic solute.
Datta and his colleagues (10) presented a correlation of the bubble size
and the orifice diameter which is represented in the equation
R = (3ry/2pg)1/3 (6)
where: R = bubble radius
r = orifice radius
^ = surface tension of liquid
p = density of liquid
Various techniques have been used to measure the radius of the rising
bubble. They varied from photography to weighing by means of an analy-
tical balance (17, 41).
Shape of Bubble
Haberman and Morton (17) observed three types of bubble shapes. Small
bubbles are spherical, larger bubbles are ellipsoidal, and the largest bub-
bles assume a spherical cap shape. They attributed this to the surface
tension, viscous and hydrodynamic forces. For small bubbles surface ten-
sion is the governing force and tends to make the surface of the bubble as
14
-------
small as possible. For large bubbles the viscous and the hydrodynamic
forces govern and flattening of the surface occurs.
Datta (10) has also summarized the shape of the bubbles formed at cir-
cular orifices of different sizes oriented vertically upward and progres-
sively increasing in diameter:
1. For circular orifices up to 0.04 cm diameter, the bubbles are sub-
stantially spherical.
2. For orifices between 0.04 and 0.4 cm diameter, the bubbles are
spherical at the orifice, but on release rapidly assume an ellipsoidal
shape with the larger axis horizontal.
3. With orifice diameter exceeding 0.4 cm, the bubble becomes unstable.
They may assume a symmetrical saucer shape.
Path of Rising Bubbles
Haberman and Morton (17) observed three types of motion of bubbles;
a. rectilinear motion,
b. motion in a helical path, and
c. rectilinear motion with rocking .
Spherical bubbles followed either a rectilinear or helical path while
ellipsoidal and spherical cap bubbles followed all three types of motion.
They associated the type of motion of the bubble with the Reynolds number.
For a Reynolds number of approximately 300, the motion is rectilinear.
With an increase in the Reynolds number, spiraling begins and increases
in amplitude and frequency until a maximum is reached. At Reynolds num-
bers of approximately 3,000, the spiraling disappears and only rectilinear
motion with rocking is obtained.
Datta (10) reported that the spherical bubbles follow a vertical path, the
ellipsoidal bubbles follow a zigzag, while the spherical-cap shaped bub-
bles followed an irregular path.
15
-------
Haberman (17) as well as Datta (10) have observed the effect of the
liquid viscosity on the path the bubble follows. For a very viscous
liquid, the bubbles tend to preserve symmetrical shapes up to much
larger size than is the case with water and their path tends to remain
vertical.
Velocity of Rise of Bubbles in Liquid
Single Bubble
The complexity of the rise of the bubble and the many variables involved
makes it impossible to describe the bubble motion theoretically. Data
dealing with the rise of gas bubbles are scarce and often conflicting.
Most of the experimental results reported are represented by imperical
equations. In general, the equation of motion (16) of gaseous bodies is
drag force + pressure force + weight = (mass + added
mass) x acceleration (7)
This equation can be solved if the total mass of the bubble and all the
forces acting on it are known.
The drag of the bubble in liquid is in general a very complicated func-
tion of its geometry, its velocity, and the physical properties of the
medium. The shape of the bubble in turn is a complex function of the
hydrodynamics, viscous, and inertia 1 forces exerted. In addition, such
effects as the container walls and bottom and the free surface of the
liquid may also have a strong influence on the drag of a bubble.
As mentioned earlier, small bubbles can be considered and have been
observed to act as rigid spheres. The earliest work on the motion of
rigid spherical bodies was that of Stokes (40).
At large Reynolds numbers (R) and for spherical gas bubbles, Moore (24)
derived an equation for the velocity of rise which has been also arrived at
by other investigators. His conclusion was that the drag coefficient is
equal to 48/R. For non-spherical bubbles several equations are available,
such as that presented by Haberman (17).
U =1.02 (gr )1/2 (8)
16
-------
where: U = the terminal velocity
r = the equivalent radius of the bubble
e
g = acceleration due to gravity
Davies and Taylor (11) are the earliest in the field of experimentation
and analysis dealing with bubble rise velocities. They proposed the
behavior of bubbles rising through actual liquid could be approximated
by the hypothetical case of a bubble rising through an inviscid liquid
when surface tension effects are important. With the additional assump-
tion of constant pressure within the bubble, they proceeded to obtain the
velocity which satisfied Bernoulli's equation at the frontal stagnation
point. They gave the velocity of spherical-cap bubbles in an infinite
liquid as
U = 2/3 (gr )1/2 (9)
G
Maneri and Mendelson (22), using the wave theory arrived at a similar
expression for the velocity of rise between infinite parallel plates.
U - 0.486 (gr )1/2 (10)
e
Dumiterscu's analysis (13) is considered a better approximation. He
obtained the potential function for the flow in a tube and assumed a
spherical nose for the bubble. Solving simultaneously he obtained the
bubble velocity and the frontal radius of curvature. His result for the
velocity is
1 /?
U - 0.495 (gr )V (11)
e
Uno and Kintner (44) experimentally investigated the behavior of bubbles
rising in tubes. A literature review on solid spheres led them to con-
clude that the presence of the tube wall could be accounted for by the
general equation
U/U = 0(1 -d/D)n (12)
t»
where: U = rise of velocity of bubble in bounded liquid
U = rise of velocity of bubble in infinite media
CO
d = diameter of bubble (sphere)
D = diameter of tube
17
-------
By assuming that the above equation applies equally well to bubbles, they
were able to correlate their data and evaluate n and 0 to obtain
U/U = [1/0(1 -d/D)]0'765 (13)
CO
where 8 is a factor which depends on tube diameter and surface tension.
This dependence was presented in a graphical form.
Haberman and Morton (17) have carried out an extensive study on bubbles
moving in various liquids. They have noticed that below a diameter of
1 cm , bubble velocity is highly dependent upon the purity of the liquid.
An impure liquid like tap water gave results tending more towards rigid
spheres. Figure 3 summarizes their findings of the velocity of rise of
bubbles in tap and filtered water. As seen from the figure, the critical
radius at which the bubbles have maximum velocity in pure water was
found to be approximately 1.5 mm with a rise velocity of 25 cm/sec.
This radius corresponds to the largest one at which the bubble is still
spherical. They also observed that proximity of sequential bubbles might
have large effects on the rise of the bubble. One of their tests showed
that bubbles of an equivalent radius of 0.17 cm and rising through mineral
oil show an increase of 9% and 39% for spacing of 7.7 cm and 3.2 cm
respectively.
In the derivation of Stoke's equation, it is assumed that there is no slip
at the surface of a solid sphere falling through water. In the case of gas
bubbles rising through liquid there will, in general, be a finite velocity
of the liquid on the outside of the boundary envelope due to entrainment
and this will result in an increased value of the terminal velocity.
Hadamard and Rybezynski (18, 34) showed that this leads to a terminal
velocity given by
vt = 2/9 R2g (p - p-) /n [On1 + 3n)/(3rT + 2n)] (14)
where the liquid and gas being considered are isotropic media of vis-
cosities T|, n1/ respectively, p1 is the density of the gas and R is the
radius of the bubble. The expression in square brackets corrects for the
effect on velocity of internal circulation due to viscous drag, and has a
value of approximately 1.5 when r\/r\l is small and is independent of the
bubble radius. Bouissinesq (5), starting with the assumption that the
viscosities of the gas and liquid are not isotropic in the neighborhood of
the boundary envelope and there there is a surface viscosity, ri , which
S
18
-------
H
t — t
O
O
1-3
w
50
~ 1 ) ' 1 M III
T—i—r
I-TTTTJ
T T
O
0>
w
6
10
W
i i 1 i i ill i I I I I i ill 1 L
0.05 0.1
EQUIVALENT RADIUS (cm)
0.5
1.0
4.0
Fig. 3 TERMINAL VELOCITY VS. RADIUS
-------
causes a resistance to motion in the surface deduced the equation
V = 2/9 R2g(p - p')/n [(n + R/TI + R) (3iV + 3ri/3n' + 2-n)]
L S S
(15)
The expression in square brackets will change with increasing R from 1.0
to 1.5. The effect of surface viscosity would be to cause a thin boundary
layer of water to travel upwards with the bubble and in the case of small
bubbles would prevent relative motion in the boundary layer.
All the above experiments and corresponding expressions have been done
on a single bubble in a confined boundary and did not account for the
change of velocity of rise due to change in flow rate nor did they account
for the change in velocity due to a change in bubble volume caused by
mass transfer.
Swarm of Bubbles
The literature available on the motion of swarms of gas bubbles in a liquid
column is found to be rather scanty. Schmidt and his co-workers (36)
dealt with the motion of swarms of gas bubbles through liquid. Their
interest was concentrated on the motion of steam bubbles in water pipes
of steam boilers. O'Brien and Gosline (28) have given some data on the
speed of air bubbles rising in a swarm through water and two mineral oils
in a 6" tube. Nicklin (27) presented a theory on the motion of the bubbles
in a two phase flow which showed that the velocity of the bubbles consists
of a component equal to the superficial liquid velocity and a component due
to buoyancy.
His analysis included two cases: one where there is no liquid flow and
his results are summerized by
U = U =G/e A = U /(I - e) =U + G/A (16)
R o o
where: U = bubble velocity when gas is bubbled steadily
through stagnant liquid.
U = average relative velocity of the phases in two phase
H ri
flow
U = velocity of bubble due to buoyancy
G = volume flow rate of gas
20
-------
A = cross section of area of tube
e = voidage fraction = number of bubbles x volume of
bubble.
The second case was that of a finite liquid flow and his results are sum-
marized by
U = G/e A - L/(l - *)A (17)
K
where L is volume flow rate of liquid. Nicklin (27) also derived an equa-
tion describing the energy losses associated with the process.
In all of the above analyses no consideration has been given to the change
in bubble volume due to mass transfer and thus there is a change in velocity
associated with this phenomena. Since this aspect is very important and
directly associated with this study, a brief review will be given below.
Mass Transfer
The extent of change in bubble volume is dependent on the concentra-
tion of the bubble gas in the liquid, the solubility of the gas and the time
of contact, i.e. velocity, of rise and the height of the liquid column. If
the deficit is great and the gas is soluble, and the liquid column is suf-
ficiently long, all bubbles will go into solution and therefore dissolve
completely before reaching the surface. Otherwise, it will rise at such
a velocity corresponding to the size of the residual bubble. The governing
equation for mass transfer is
dm/dt = k. A(C - C ) (18)
1 s a
where: dm = the mass of gas transferred to the liquid (mg)
dt = the time of contact (sec)
k = over-all diffusion coefficient based on liquid film
resistance (mm/sec)
2
A = total interfacial area between liquid and gas (mm )
C = saturation concentration under specific partial pres-
sure of gas in contact with liquid (mg/1)
C = actual concentration (mg/1)
a
21
-------
In this equation k is a function of the bubble diameter and temperature.
Figure 4 shows clearly this dependence.
•A'inrthe above equation, which is the interfacial area, is the sum of the
surface areas of all bubbles, thus a function of bubble diameter. And
finally dt, or the contact time, is related to the velocity of rise of the
bubble which in turn is a function of the bubble diameter.
Datta and his colleagues (10) carried out an experiment to study the re-
duction of bubble volume with height due to mass transfer. Figures 5
and 6 summarize their results. Figure 5 shows the effect of column
height and Figure 6 shows the volume change and corresponding velocity
of rise. Results of observation by Pattle (29) on the terminal height of
small bubbles showed that over a range of bubble volume from 3x10 ml
to 30 x 10 ml the terminal height was roughly proportional to the volume
Induced Mixing and Energy Consideration
The use of air bubbling systems as pneumatic breakwaters and for the
elimination of ice on the surface of reservoirs, ponds, and similar
bodies of water, or for complete mixing of stratified waters has led
several investigators to study the properties of this system more thor-
oughly. Several papers (14, 21) are available on the properties of the
system when used as pneumatic breakwater. The interest in this case
is centered on the induced surface currents. The use of the system for
elimination of ice on the water surface has been used successfully but
not studied in detail. It has been suggested that bubbles transport
warmer water from the lower regions to the surface.
Other investigators have studied the use of the system for mixing
stratified bodies of water. Gay and Hagedorn (16) found that in additon
to the induced surface current, a strong bottom current toward the out-
let exists. Zieminski et al. (47) studied experimentally the properties
of this system in mixing stratified waters and found that the circulation
pattern undergoes random changes caused by the randomness of the bub-
ble motion. They also found that the mixing time decreased slowly with
dn increased air flow rate. They concluded that in decreasing the rate
df flow a critical value is reached beyond which the currents generated
are not sufficiently strong to establish a circulation of an appreciable
magnitude in the whole body of water. Their report included photographic
evidence of the induced circulation.
22
-------
r-o
CO
O
W
CO
O
W
i— i
u
I — I
CiH
PH
W
O
O
Q
O
0.5
0.4 _
0.3
0.2
0.1 _
2 468
BUBBLE DIAMETER (MM)
Fig. 4 RELATIONSHIP BETWEEN LIQUID FILM COEFFICIENT AiML- i>u BBLE DIAMETER
(MODIFICATION OF BARNHART (31))
-------
w
U
i— «
UH
o
H
<
pa
.-I
O
w
H-i
CO
03
D
03
H
1—4
Z
0.15-
0.05-
0.10
1
COLUMN HEIGHT 456 CM.
COLUMN HEIGHT 145 CM.
COLUMN HEIGHT 120 CM.
COLUMN HEIGHT 100 CM.
COLUMN HEIGHT 80 CM.
COLUMN HEIGHT 60 CM.
I 1 1
0.005
0.01
0.015
0.02
0.025
0.03
0.035
FINAL VOLUME OF BUBBLE
Fig. 5
RELATIONSHIP OF INITIAL AND FINAL VOLUME OF
BUBBLE FOR DIFFERENT COLUMN HEIGHTS (DATTA (10)}
30
. 20
U
w
CO
\
CO
2
U
"x
^-i
i—*
O
O
- 10
100
200 300 400
HEIGHT FROM NOZZLE - CM
500
Fig. 6 VOLUME OF BUBBLE VS HEIGHT ABOVE OUTLET AND
CORRESPONDING VELOCITY (DATTA (10))
24
-------
Ditmars et al. (8) hypothesized that air bubbling in a stratified body
of water could lead to an uncoupling effect between the water flow and
the air bubbles as shown in Figure 7.
One of the earliest experiments in this area was conducted by Siemes
and Weises (37) who described the mixing effect using the concept of
effective diffusivity. Their results were reported in a graphical form in
which the effective diffusivity was plotted against superficial gas velocity,
They discussed the variation of effective diffusivity with a gas flow rate
which seemed to be a linear function of gas velocity at a low gas flow
rate. The results of Tadaki and Maeda (42) showed that the effective
diffusivity D increased with gas flow rate although the liquid flow rate
had no significant effect. Reith (32) in his recent work, noted that at
very large gas flow rates the longitudinal mixing could be characterized
by a constant Peclet number (product of Prandtl and Reynolds numbers) of
3.0. The Peclet number was based on the tube diameter, gas velocity,
and effective diffusivity.
Subramanian and Chi Tien (41) studied the effective longitudinal mixing
due to single bubbles and a swarm of bubbles and described it by the
diffusion equation using the effective diffusivity. The following expres-
sion for the velocity of flow in the liquid phase in the whole system
was derived.
V = 2 X3f (19)
where: y = the bubble density
f - the frequency of bubble formation
3 = the amount of liquid carried upward by a single gas
bubble.
8 was considered to be of two components, the volume of the liquid
entrained by the boundary layer of bubbles up to the separation point and
the volume of the liquid entrained by the wake behind the bubble which
is assumed to form from the separation point. Mathematical expressions
to calculate these values are derived. Their equation of the effective
diffusivity is
D = 4.5 x 105 V2 (20)
e
and the velocity of entrained liquid is given by
25
-------
>oc
a. COMPLETE UNCOUPLING BETWEEN PLUME WATER AND
AIR-BUBBLES AT THE INTERFACE.
b. PARTIAL UNCOUPLING AT THE INTERFACE.
Fig. 7 POSSIBLE FLOW REGIMES WITH A BUBBLE SCREEN
IN A TWO-LAYER DENSITY S/STEM. (CEDERWALL (8))
26
-------
V = 0.0193758f (21)
De Nevers (12) has studied fluid circulation by bubbles in baffled and
unbaffled systems. His mathematical model is summarized by
h = W/(Plg e/2 (-dp/dl)) - 1 (22)
where: h = the height of circulation
W = the width of circulation
D = liquid density
g - the acceleration of gravity
e = the volume fraction of gas-liquid mixture occupied
by gas .
The limitation of this model is that it assumes that liquid continues to
circulate around the same closed loop while gas enters at the bottom and
leaves at the top. However, this is not the case since at the top, some
of the liquid continues to move with the gas which is leaving the circula-
tion, this other liquid must flow in somewhere to replace it. The above
model does not account for such a condition.
The convective currents which rise from heated bodies have been dis-
cussed first by Schmidt (36). He studied the behavior of convective plumes
of air above a steady point and line source of heat in a uniform, incom-
pressible atmosphere. He observed that plumes of hot air rising from small
sources tend to be confined within conical regions where the flow is
turbulent (just as in the case of forced jets) . Using this fact he discussed
the dynamics of such cases by supposing that the distribution of tempera-
ture and velocity can be found by balancing the horizontal turbulent trans-
fer of heat and momentum against the vertical transfer by convection/
allowance being made for the effect of buoyancy. Some assumptions
have to be made to connect the horizontal turbulent transfer and the mean
vertical flow before the analysis can be carried out. Schmidt (36) assumed
that there is geometrical and mechanical similarity of the process in hori-
zontal sections of the plume, and used the mixture length theories of tur-
bulence to find the complete form of velocity and temperature profiles for
both point and line sources of heat in an atmosphere at uniform tempera-
ture. His calculated results for the point source were verified by experi-
ment using small electrically heated grids of air. His experimental mea-
surements fitted quite well a normal distribution profile, exp. (-45r2/x2).
27
-------
More recently, Yih and Rouse, and Yih and Humphreys (46, 33) have
given the results of measurements of temperature and vertical velocity
above a single gas burner and above a line of gas burners in air. Those
measurements were taken at a variety of source strengths. The results
were combined in non-dimensional form to give a vertical velocity pro-
file and buoyancy profile for each of the two types of sources. For a
point source they chose the profile exp. (-96r2/x2) for the vertical
velocity and the profile exp. (-71r2/x2) for the buoyancy as giving the
best fit with the quoted experimental results.
With the exception of some comments by Batchelor (4) on convective
plumes in unstably stratified fluids, little attention has been given to
cases in which there is a density gradient in the ambient fluid or atmo-
sphere.
Taylor and his colleagues (25) were among the earliest to mathemati-
cally describe the gravitational convection from maintained and instan-
taneous sources of buoyancy in uniform and stratified fluid. The assump-
tions they followed in their derivation are:
1. The profiles of vertical velocity and buoyancy are similar, at all
heights.
2. The rate of entrainment of fluid at any height is proportional to a
characteristic velocity at that height.
3. The fluids are incompressible and do not change volume on mixing
and that local variations in density throughout the motion are small
compared to a reference density.
Their governing equations are:
d/dx (nb2U) = 2nbaU (23)
d/dx (Trb2p) = Tib2g (pQ - p) (24)
d/dxnb2U (pj - p) = 2nbaU (px - pQ) (25)
where: b = the width of the plume (m.)
U = the velocity of rise (m./sec.)
a = the entrainment coefficient relating the inflow
velocity at the edge of the plume to the vertical
velocity within the plume.
28
-------
p = the fluid density inside the plume
p = the fluid density outside the plume
p = the reference density.
The above three equations represent conservation of volume, conserva-
tion of momentum, and conservation of density deficiency.
Solution of the above equations for constant flow rate and uniform fluid
yields
b = 6a/5 x (26)
U = 5/6a (9/10aQ)1/3 x~1/3 (27)
Theoretical treatment of point source in a stratified fluid using the above
equation of conservation and assuming that the velocity and buoyancy pro-
files are normal distribution curves centered about the axis of symmetry,
led to a prediction of the final height to which a plume of light fluid will
rise in stably stratified fluid.
Kobus (20) in his analysis of the flow induced by air bubble systems and
from his experiments found that the vertical velocity profile can be repre-
sented by a Gaussian distribution curve with a linear spread in the verti-
cal. His analytical treatment included the momentum flux increase due
to the buoyancy of the air.
Figure 8 shows the observed velocity profile above a single orifice and
Figure 9 shows the rate of spread of the velocity profiles. The analyti-
cal origin was found to be dependent on the local orifice geometry and
especially to vary with the orifice elevation above the floor, and was
found for his test arrangement to be equal to 0.8 meters. He also noticed
that this value seems to increase slightly with depth.
Cederwall and Ditmars (8) in their analysis of air bubble plumes followed
the analysis of Taylor with the assumption of compressible fluid rather
than incompressible. They assumed Gaussian velocity distribution pro-
files. Their analysis included points and line sources in homogenous
ambient fluid with some attempt at the analysis in stratified fluid. They
used the experimental data of Kobus to test their model.
29
-------
H
I - 1
0
o
J
w
u
t — i
H
1.2
1.0
0.8
0.6
0.4
0.2
0
02
U
V-J
O
cc,
cu
O
O
O
Q
<
w
Cf.
C-
O
UJ
E-i
-------
SECTION VI
EQUIPMENT AND TECHNIQUES
Several laboratory techniques have been reported to be used in measuring
the velocity of rise and the induced mixing due to buoyant jets. These
techniques varied from photography to use of radio-isotopes . In this
study a dye was used as a tracer. A fluorescent dye was chosen so that
its concentration could be measured with a fluorometer. The following
sections will describe the dye selected, the fluorometer, and other
equipment used as well as the techniques employed.
Tracer
The tracer to be used should respond perfectly well to the water movement.
To do so, its specific gravity, viscosity, and other physical characteris-
tics should match those of the water it is dispersed into. It should be
easy to handle, cheap, non-toxic to human beings and harmless to aquatic
biota, and above all should be easily detected quantitatively. Rhodamine
B, a fluorescent tracer of a commercial organic pigment, is an excellent
tracer. It is available commercially in powder form or in a 40% by weight
solution in acetic acid at a cost of approximately $5.00 per pound. It
is quite soluble in water and is also readily soluble in methanol. This
property makes it possible to adjust the solution density to match that of
the water in which it is to be dispersed. It is non-toxic if used at a low
concentration. The FDA has established as a pro tern allowance limit for
Rhodamine B in drugs and proprietary products a continuing ingestion rate
not to exceed 0.75 mg per day.
This tracer is easily detected by its fluorescence which reaches a maximum
at 570 m/i. To avoid any interference with naturally occuring substances
which usually peak at a shorter wave length, selection of the primary and
secondary filter is very important.
Pritchard and Carpenter (30) found that Rhodamine B is not seriously af-
fected by bacterial action and that it decomposes only slowly on exposure
to light. Its fluorescence remains constant over a pH range of 4 - 10.
They tested the adsorption properties of this dye on living matter and
found that a sample containing 0.4 ppb of Rhodamine B and a large algae
population showed no decrease in fluorescence after four days. However,
this dye is very sensitive to temperature. Pritchard and Carpenter found
that the fluorescence of this dye decreases with increasing temperature
at the rate of 2 - 3% per degree centigrade.
Feuerstein and Selleck (15) found the response to temperature change to be:
31
-------
F = F ent (28)
o
where: F = the fluorescence reading in units
F = the corresponding reading at 0°C
n = constant = -0.027
t = the sample temperature (°C)
Therefore, for accurate work, the temperature of the samples must be
taken and correction applied.
Fluorometer
Rhodamine B can be easily detected quantitatively with a fluorometer.
The Turner Model III flourometer is commonly used and this was used
in this study. The fluorometer is basically an optical bridge which is
analogous to the accurate wheatstone bridge used in measuring electrical
resistance (43). The optical bridge measures the difference between light
emitted by the sample and that from a calibrated rear light path. A single
photomultiplier surrounded by a mechanical light interrupter sees light
alternately from the sample and the rear light path. The photomultiplier
output is alternating current, permitting a drift-free AC amplifier to be
used for the first electronic stages. The second stage is a phase-sen-
sitive detector whose output is either positive or negative, depending
on whether there is an excess of light in the forward (sample) or rear
light path respectively. Output of the phase detector drives a servo
amplifier which is in turn connected to a servo motor. The servo motor
drives the light cam (and the fluorescence dial) until equal amounts of
light reach the photomultiplier from the sample and the rear light path.
The quantity of light required in the rear path to balance that from the
sample is indicated by the fluorescence dial.
The fluorometer may be equipped with a variety of sample doors . Two
types were used in this study; namely the discrete sample cuvette door
and the continuous sample door. It also has four operating ranges and
different levels of sensitivity.
Fundamentally, fluorescence is the emission of radiation from a molecule
or atom following absorption of radiation. From an analytical standpoint,
the intensity of fluorescent light emitted by a sample under constant in-
put light intensity is directly proportional to the concentration of the
fluorescent compound. Many investigators have agreed with the manu-
facturer that at low concentration of the fluorescent material, the response
of the fluorometer is linear with the concentration of tracer. The operating
and service manual (1964) lists three reasons for deviation from linearity:
32
-------
1. Extremely high concentration of the fluorescent material, leading
to self-absorption of light. This concentration quenching effect
frequently occurs at concentrations above about one ppm and should
be checked.
2. High concentration of a material in the reagents which absorbs either
the exciting ultraviolet light, or the emitted light.
3. Non-linearity in a chemical reaction which is used to convert the
unknown to a fluorescent material, or non-linear recoveries.
None of these reasons were applicable in this study.
Feuerstein and Selleck (Is) reported that the response of Rhodamine B is
not linear in the Turner Model III fluorometer, but has the relationship:
C = KFn (29)
where: C = the concentration
F = the fluorescence reading
K,n = constants
Wilson's (45) findings agreed with the findings of Feuerstein and Selleck,
as well as this study.
Diff users
Two types of diffusers were used and tested.
1. Porous PVC diffuser with a bubble of approximately 2 mm in diameter
2. Porous ceramic diffuser which produces bubbles in the range of 0.2 -
0.3 mm in diameter and a maximum of 0.5 mm. For best results and
to minimize coalescence this diffuser should be horizontal. This
was achieved by strapping the diffuser to a metal rack and balancing
it by four ropes tied to the oxygen line.
Other Equipments
1. Gear type, small size, model 7012 positive displacement pump with
a capacity of approximately 42 ounces of water per minute. This
pump was used for the sample collection.
2. Submersible pump.
33
-------
3. Buoys (floats) shown in Figure 10. They are made of four foot by
four foot by two foot deep plywood and are filled with stryofoam.
They house two 55 gallon drums to accomodate four oxygen cylinders,
a pressure regulator, and a flow meter.
4. Flow meter
5 . Boat and motor
Calibration of Equipment
A laboratory study was conducted to calibrate the fluorometer and to
determine the characteristics of the fluorescence phenomena of Rhodamine
B in the aquatic environment of the experiment.
A ten-liter sample of lake water was collected and'left to come to room
temperature (21°C). Rhodamine B solutions of various concentrations
were made to use in constructing a standard calibration curve. Using
those solutions, corresponding fluorescence readings on the four scales
were recorded. These fluorescence readings are plotted against the
concentration of Rhodamine B in Figure 11. The fluorescence was found
to be non-linear with concentration. This agreed with the results of
Feuerstein and Selleck (15), and Wilson (45).
The following equations express the concentration-fluorescence relation-
ships on the four sensitivity ranges.
Ix C = 1.95 (F)0-89 (30)
3x C = 0.55 (F)0'92 (31)
n Q9
lOx C = 0.195 (F)u'^ (32)
n Qc;
30x C = 0.048 F(F)U'y:5 (33)
where: C = concentration of Rhodamine B in micrograms/liter
F = fluorescence reading at 21°C
No attempt was made to make temperature corrections, since all readings
were done at room temperature, which was 21 ± 0.5°C. Calibration of
the instrument as well as later readings were made in a dim room, to
avoid light leakage which would affect the fluorescence readings.
Experimental Method
General Procedure Outline
1. Set the oxygen flow at a predetermined flow rate and wait for a
34
-------
PRESSURE
REGULATOR
OXYGEN
CYLINDERS
it
I I
SAMPLER
DYE
OXYGEN HOSE
P.V.C.
DIFFUSER
Fig. 10 FIELD EXPERIMENTAL SET-UP
35
-------
100
w
i—I
.-)
CO
O
O
o:
0
O
i— <
H
<
OS
w
U
S
O
U
10
PRIMARY FILTER:
SECONDARY FILTER:
TEMPERATURE:
546
590
20.8°C
10
FLUOROMETER READING
100
Fig. 11 FLUOROMETER CALIBRATION CURVES
36
-------
minimum of half an hour to establish the flow field and have steady
state conditions .
2 . Inject dye .
3. Sample at the center of the plume at known intervals and depths for
determining the center line velocity.
4. Sample at known intervals and depths at known distances from the
diffuser, (point sample or continuous sample), for determining mixing
patterns .
5. Read collected samples on the fluorometer and reduce data.
Setting the Flow
As indicated in the experimental set-up, the oxygen cylinders adapted
with pressure regulators and flow meters described earlier were used.
The following equations and the calibration curve shown in Figure 12
were used in setting the flow meter at the desired rate.
°
(34)
q'
where:
°
°
' =
q
,9
" G
Injection of Dye
q°GP/760 (35)
standard air flow as read from graph
standard gas flow in same units
gas flowing at P but volume reduced to measurement
at standard condition
viscosity of gas in centipoises at standard condition
absolute pressure in mm of Hg
Method I
Forty to 50 ml of 40% by weight of Rhodamine B is diluted with lake water
to approximately \ liter volume and poured into the dye container. The
outlet of the submersible pump is then connected to the dye container and
power is turned on. This forces the dye down the hose which is approxi-
mately five inches above the diffuser. Enough water was pumped to insure
that the dye had discharged. This is considered to be equal to the volume
of the dye hose.
37
-------
00
o
\~f
a
Q
-------
Method II
The dye solution is made into a two-liter bottle with an outlet near the
bottom. This is connected to a small gear pump to force the dye into
a stiff plastic hose five to ten inches above the diffuser.
Sampling
The most critical part of the sampling was determining the instant that the
dye reached the diffuser. To determine this instant, a sampling stiff
plastic hose was attached to the dye hose and ended one inch beyond it.
The other end of this hose was connected to one of the small gear pumps.
The dye was injected with close monitoring of the stiff plastic hose that
was connected to the gear pump. As soon as the dye was detected in
the hose (i.e. at the surface) the pump was turned off (to avoid spilling
of dye on the surface which would lead to erroneous results) and the time
was recorded. With the known pump flow rate and the volume of the hose,
the lag time was calculated and the instant the dye hit the diffuser was
set. For velocity measurements three or four stiff plastic hoses varying
in length were suspended in the center of the plume (or as closely as
possible) and connected to the gear pumps. Samples of ten to 20 ml
volume are collected into BOD bottles from all sampling hoses and at
recorded intervals of time. The time interval between each sample and
the next was ten seconds reduced to five seconds around the time the dye
was expected to hit that sampling line. The determination of this time
was based on a trial run with sampling at intervals of 15 - 20 seconds.
For measurements of the circulation pattern and the dye diffusion, samples
were collected using stiff plastic hoses and the gear pumps. The samples
were collected at known depths and distances away from the diffuser at
recorded time intervals .
Test Sites
The test sites were located in Lake Travis , the largest of the Highland
Lakes of Central Texas. Lake Travis is impounded by Mansfield Dam,
which was constructed by the United States Bureau of Reclamation in
1934. It serves for flood control, irrigation, and power generation. The
lake is 65 miles long with 270 miles of shoreline and a maximum depth
of 225 feet. The surface is 42,000 acres and the capacity is two million
acre feet.
Figure 13 shows the three test sites where the experiments were conducted
39
-------
Fig. 13 MAP OF TEST SITES (SCALE 1:62500)
40
-------
Difficulties Encountered
This study was carried out in three separate sites on Lake Travis in Texas.
The first of these sites utilized the set-up shown in Figure 10 and the use
of the boat. The boat with sampling instruments used to be tied to the
buoy on the side of the diffuser. However, surface currents induced by
wind and passing motor boats caused oscillation and rotating of the boat
and the buoy which in turn caused movement of the diffuser. This move-
ment along with surface surges caused an unstable plume to exist. The
movement of the plume, as would be expected, did not follow the movement
of the buoy nor the diffuser and sampling lines attached to it for measure-
ment of the center line velocity. Thus, regardless of the many runs made,
no reliable results were obtained. The alternative was to conduct this
part of the study on a more stable base and thus the second and third areas
were chosen.
The quantity and concentration of dye used along with the instability of
the plume limited the number of runs carried out. The dye concentration
was still detectable in the plume vinicity after two hours and at different
levels of concentration which made it difficult to establish a base line
and start a second run. The reason for this was attributed to the random
motion of the bubble plume, for the above mentioned causes, such that
no definite circulation patterns were defined.
Another problem encountered later on at the second and third sites was
the entrapment of bubbles in the velocity measurement sampling lines.
The quantity of those air bubbles was sometimes of such a magnitude
that discontinuity of flow in the sampling lines was brought about. Re-
starting the flow in them, although timed, caused in most instances missing
the right interval for sampling at that depth. Such a condition used to
happen most often in the timing hose, the closest to the diffuser, used
for setting the zero time for sampling. Although the hose end was turned
upward away from the diffuser, bubbles were sucked into it and caused
the discontinuity of flow. Such an incident caused the failure of the
whole run, due to lack of knowledge of the lag period.
41
-------
SECTION VII
MATHEMATICAL FORMULATION
Detailed analysis of an air bubble plume is virtually impossible due to
the many variables associated with it. Taylor and his colleagues (25) ,
as mentioned earlier, presented the first thorough description of the plume,
Cederwall (8) adopted the techniques of Taylor with the assumption of
compressible fluid rather than incompressible fluid assumed by Taylor.
However, neither of the above authors nor other investigators have at-
tempted to consider an important variable, which is the mass transfer
and the associated changes in the plume's characteristic parameters;
i.e. velocity or rise, radius of the plume, and energy.
Most bubble plume mathematical formulations have been made in con-
junction with high intensity air injection rates where gas transfer was
minor. The concept of hypolimnion aeration with commercial oxygen of
necessity incorporates significantly lower oxygen injection rates and
essentially complete absorption. Therefore, mass transfer must be
incorporated into the plume hydrodynamics.
This work will adopt the techniques of Taylor and Cederwall (25, 8) and
the following assumptions.
1. All bubbles produced are of the same volume and no coalescence
takes place.
2. At any cross section, change in volume of each bubble due to mass
transfer is the same regardless of its location in the plume.
3. No mass transfer occurs between the mathematical source and the
actual source.
4. The number of bubbles produced per unit time is constant.
5. The rate of entrainment is constant, not proportional to center line
velocity.
6. Vertical velocity and buoyancy force in a horizontal cross-section
are of similar form at all heights and assumed Gaussian.
7. Water density is constant (no density gradient) .
43
-------
8. The bubbles are compressible and their expansion is isothermal.
Governing Equations
Zero Slip Velocity
The governing equations for a three-dimensional symmetrical bubble
plume in ambient fluid are:
Rate of water discharge = Q = / VdA (36)
r 2
Water momentum flux = M = /.p V dA (37)
•'Am
Air-water buoyancy flux = B = /.. (o - p ) VdA (38)
J A w m
Water kinetic energy (K.E.) flux =jt(p V )/2dA (39)
where: V = local plume velocity
o = ambient water density, (surrounding the plume)
w
p = air-water mixture density
m
A = cross -sectional area
Since no measurements of the velocity or density distribution across
the plume were made, the assumption used by other investigators (8,
25, 20) i.e. Gaussian will be used here.
_2/h2
V = V e ' (40)
c
where: V = center line velocity of the plume
c
r = lateral co-ordinate
b = nominal half -width of the plume related to the
deviation of the velocity distribution by 2o = /2 b
44
-------
SOURCE
OF
BUBBLES
NTRAINIv^ENT
MATHEMATICAL ORIGIN
Fig. 14 DEFINITION SKETCH
45
-------
X = a non-dimensional parameter representing the ratio
of sideway spread between the velocity and the
density profile.
Solution of Equations 36 through 39 using the velocity and density dis-
tribution assumed above leads to
~ -r2/b2 „
rdr
water flow = Q = f 2nVrdr = 2rr f V e'
r, r, O O
•2 2 /, 2 -
C —r / b 2
= 2ir / V e rdr = rrV b (42)
Jo c
/2
p V dA
m
2/2 2 ?
= 2rrp f (V e 7 ) rdr = nV p b /2 (43)
m / c cm
o
Making the Boussinesq assumption that density difference may be ne-
glected except in buoyancy terms, p =* p in Equation 43; thus
m w
M = nV 2p b2/2 (44)
c w
The buoyancy equation for a zero slip velocity is
00 _ _
B = f 2iT(V) (p - p ) rdr = 2 V Ap Te~r /b . e
J w m c Km J
2/u2 2 „_. 2
-r /b -r /(Xb)
o
(45 a)
B = irV Ap (X2b2/l + X2) (45 b)
c m
The kinetic energy flux is
2/2
3 (46a)
o
= n Vc Pw / (e"f /b )3 rdr = nVc Pwb2/6 (46 b)
o
In all the above equations V , Ap , and b (the radius) are the unknowns,
c m
However, considering a small segment of the plume of a thickness dx,
46
-------
the water flux is
dQ = 2jrb dx(V__) (47)
ri
where V is the horizontal flow velocity related to the vertical center
line velocity by a constant such that V = aV where a is constant and
equal to the entrainment coefficient. Thus equation 47 becomes
dQ/dx = 2nb(aV ) (48)
c
The driving force of the plume is the buoyancy and the momentum flux is
CO
dm/dx = /2rr(p -p )grdr
J w m
o
2 2
dm/dx - rrb gAp X (49)
m
The buoyancy at any level x above the diffuser is
B = q (o - p ) H /(H + H - x) (50 a)
o w a oo
where: H = piezometric head equivalent to the atmospheric
o
pressure
H = depth above diffuser
q = air flow rate at atmospheric pressure
since p « p
a
B = q (p ) H /(H + H - x) (50 b)
o w o o
equating equations 45 and 50 gives
Ap (X2bV(l + X2) = q (p ) H /(H + H - x) (51)
c m o w o o
and equating equations 49 and 44 yields
d/dx(rrV 2p b2/2) = nb2gAp \
c w ni
47
-------
or
2 2
d/dx (V b ) = 2b2g (Ap /p ) X (52)
c m w
which upon rearrangement yields
Ap /p = d/dx (V2 b2) /2bV2 (53)
m w c
but from 51
IT
Ap /p - q („ ° )/nV xV/(l+X2) (54)
m w o H + H - x c
equation 53 and 54 yields
LJ
d/dx (V 2b2) /2b2g\2 = q (
c ' ' y'v ^o v H +H-X
2
(H +H-x)__c Ti(H +H-x)V
also from 42 and 48
(ss,
dx
d/dx (V b2) = 2baV (56)
c c
In equations 55 and 56, assuming that values could be assigned for a
and \ the only two unknowns are b and V . However, the above two
equations cannot be solved in a close form but rather a step - by - step
integration must be used to solve them for the unknowns.
The first step is to reduce the equations in the following form.
48
-------
- 2a - gqH(l+X) (57)
n (H +H-x)bV
o c
0 _ 2av (58)
dx TT(H +H-x) b2V 2 -77
o c b
The term q in equations 57 and 58 is the gas flow rate which is equiva-
lent to the number of bubbles of known volume produced per unit time, i.e.
q = nrTD3/6 (59)
where: n = number of bubbles produced per unit time
D = diameter of bubbles produced
However, the diameter due to mass transfer is a function of x, (D ) .
Thus substituting this equality in equations 57 and 58 yields
„ 3
,( . rm D _
-SpL = 2a - g ( — r-5-) H (1 + \) (60)
dx _ 6 _ o _
n(H +H-x) bV 3
o c
_ ^y v -^ i •!••* vx * A. i r\ TT / c "M
— O « OC V ^ O JL /
dx 22~T~
TT(H +H-x) b V
o c
Cederwall and Ditmars (8) normalized equations 57 and 58 and used
numerical integration to solve for the unknowns. Solution of equations
60 and 61 was also by numerical integration. However, the introduc-
tion of the new variable D and calculating its changes with depth for
v
each step used in the numerical integration was found best to be done
by setting up a computer program to solve first for the diameter at each
step and supply that to equations 60 and 61 to solve for the other two
variables.
Before proceeding in describing this computer program, a comment on
some of the assumptions and determing the values of a and X seems to
be necessary.
49
-------
Bubble size is unifontL
The diffusers used in this study which were described earlier and ac-
cording to manufacturers' claims and visual observation do seem to pro-
duce uniform initial bubbles and almost no coalescence. This assump-
tion was necessary to simplify the solution and any other assumption
would have been impossible to verify.
Bubble volume due to mass transfer is reduced
It is true that the bubbles at the side of the plume will be reduced in
volume at a faster rate than those at the center and any distribution like
Gaussian could be adopted, but again the verification of this is very
difficult, if not impossible. However, considering the fact that there is
hardly any variation in deficit across the plume, this assumption is
acceptable.
No mass transfer^occurs between the mathematical origin
and the actual source
The mathematical origin which is a function of the properties of the dif-
fuser is considered here a constant = 1.0 meter. This increase in depth
is incorporated in the solution of the differential equations. However,
the injection depth is considered to be the actual diffuser depth and mass
transfer calculation is based on that assumption. Thus no mass transfer
was considered to take place in that region.
The number of bubbles produced per unit time is constant
Control of the flow rate and equal distribution of flow across the dif-
fuser makes this assumption acceptable.
The rate of entrainment is constant
There is no way to verify this assumption or even to measure it under
the experimental conditions on the lake. The coefficient used in this
analysis is that derived by Cederwall (8) as illustrated in Figure 15,
based on the experimental observation of Kobus (20).
The velocity and buoyancy force distribution at all levels is similar
Difficulties encountered in running the experiment made it impossible
to verify this assumption. However, Kobus (20) did measure the velocity
50
-------
d
H
O
H
2
r-J
h— I
u
w
O
O
D4
0. 16
0. 12
0.08
0.04
1 - 2-D
2 = 3-D
10
THE AIR FLOW RATE q
o
10 3 m3/s; 3-D
10~2 m2/s; 2-D
Fig . 15 THE COEFFICIENT OF ENTRAINMENT AS A FUNCTION OF
THE AIR FLOW RATE FOR TWO- AND THREE-DIMENSIONAL
AIR BUBBLE PLUMES (CEDERWALL (8))
51
-------
distribution in a small scale lab experiment and found it to be Gaussian
Other investigators (8, 15, 33) also used this assumption.
The ambient water condition is uniform
A study of a typical temperature profile of a deep stratified reservoir,
Figure 1 , and the almost constant temperature below the metalimnion makes
this assumption acceptable.
The bubbles are compressible
Expansion of rising air bubbles is neither adiabatic nor truly isothermal.
However, for the sake of simplicity the assumption of isothermal con-
ditions is adopted.
The spreading ratio (X) is constant
Cederwall et al. (8) , in a photographical recording of air bubble plumes
found that the lateral spreading of the air bubbles is slow relative to the
expansion of the plume. They concluded that the X is fairly constant
throughout the rise of the plume and of a magnitude of about 0.2 . This
value was used in this work. Other values of 0.1 and 0.3 were also
tested by Cederwall and in this work, but showed practically no dif-
ference in results .
Slip Velocity Consideration
The previous mathematical formulation of the governing equations of the
water-bubble plume, considered the velocity of the bubbles relative to
the surrounding water (slip velocity) to be zero. This section will incor-
porate the slip velocity and present the differential equations governing
this consideration. Essentially the governing equations are the same
except in equation 45 a, the conservation of buoyancy. The velocity
term is of two components; local plume velocity (V) , and the velocity of
the rising bubbles relative to the water (V, ) . This consideration leads to
b
B = f 2n(V+V ). (o - o ) rdr (62)
J D w m
+ 2 <63>
which is equal to the buoyancy flux at any level (equation 50) , thus
52
-------
rrV Ao X2b2 + rrV Ao X2b2 = q D HO (64)
cm — b m ° m 1
1+X o
continuing the solution in the fashion presented earlier for zero slip
gives the final two equations to be solved;
d(Vcb2) = 2av b (65)
c
dx
2gHo(mrD3/6) (66)
dx ^(H +H-x) (Vc+Vb)
which yield
db _ , _3
—: = 2a - g (nnp ) H
(67)
nnD
- 2g (
c
nV 2 b(H +H-x) (Vc + V, )
c o 2 "
l+\
3
dvc = 2^ 6 ) H0 - 2aVc (68)
nV b2(H +H-x) (V + V )
c o c b
Equations 67 and 68 cannot be solved in a closed form and thus a nu
merical integration has to be used to solve for the unknowns.
Cederwall and Ditmars (8) in their solution assumed that at x - O
V and reduced the equations accordingly. They also used the value of
Vb = constant =0.3 meters per second based on Kobus1 (20) observation
reproduced in Figure 16. The values presented by Kobus are a function
of the air flow rate. It also shows that the orifice diameter is not a fac-
tor in the velocity of rise. Haberman and Morton (17) observed that
proximity could affect the speed of the rising bubbles. But they did not
pursue this fact and no measurement was taken.
53
-------
w
W
,-J
CO
CD
D
CQ
IX,
O
Q
w
w
cu
CO
O
ce
2.0
1.5
1.0
0.5
0
2000
4000
6000
8000
AIR SUPPLY (cm /s)
Fig. 16 AVERAGE RISING SPEED OF
BUBBLE STREAM FROM A
SINGLE ORIFICE. (KOBUS (20))
54
-------
Since no measurement of the velocity of rise of the bubble column rela-
tive to the water as a function of the bubble diameter, which is an im-
portant factor, was made in this study, the results of the measurement
of Haberman and Morton (17), although for single bubbles, will be used
here. These results are presented in Figure 3 and approximated by the
equations
Vfe = 24.4 x D in cm/sec for 0.0 < D < 0.72 mm.
V = 17.6 + (D - 0.72) for 0.72
-------
and other parameters before it calls the next sub-program RKAM. With
the values of the variables being calculated and stored it solves for the
water flow rate, momentum, kinetic energy flux and increase in D.O.
Sub-prog ram RKAM
This sub-program, which is a general routine for solving up to ten simul-
taneous first - order differential equations, is written by the computation
center at U.T. at Austin, but was modified to suit the new conditions.
This program uses the Runge - Kutta procedure which is a self-starting
single step method, and the Adams - Maulton procedure which is a multi-
step method which requires a self-starting method (Runge - Kutta) together
with a predictor formula (Adams - Bashforth). The program carries out
error analysis and doubles or halves the size of the step if necessary to
keep the results within certain bounds of error set by the user. However,
this program was forced to use only the indicated step value with no
halving or doubling of the step size so as to match the step used in HYPO.
This program calls the next sub-program F.
Sub-program F
This is a listing of the differential equations to be solved.
Sub-program VELB (DYAM)
This sub-program is added in case of slip velocity considerations and
equations 67 and 68 will replace 60 and 61 in sub-program F.
Thus the program HYPO solves for the diameter at each step and sup-
plies HYDRO with those values to solve the two differential equations.
The final output is either in the form of listing or a plot by calling a
system routine (PLOTT). This routine won't be described here nor will
listing of it be included in the listing of the program in Appendix. How-
ever, a write-up on that routine is available from the computation center
at U.T. at Austin.
To start the integration process of equations 60 and 61, initial values
of the two dependent variables for a corresponding value of the indepen-
dent variable should be specified. Solution of the simple plume equations
was used in predicting the initial velocity of rise.
56
-------
V
25 gq H (1+A )
o o
24 a TT(H +H)
O
1/3
x
-1/3
(69)
where x corresponds to the height above the mathematical source. This
value, which is a function of the properties of the source used, was
found by Kobus (20) to be 0.8 meter. In this study this depth was con-
sidered to be equal to 1.0 meter, and thus the velocity was calculated
accordingly. The corresponding radius of the plume was considered to
be equal to the actual source.
Integration Step Length
To carry out the numerical integration of the two differential equations ,
a step size should be indicated. As mentioned earlier, the program can
vary the step size to keep the error within a set limit. However, this
was not possible in considering the mass transfer, so the step size was
taken to be fixed at a value of 0.1 meter. The single step error associated
with this step size was of the order of 10"10 - 10~13 which is a negligible
error. Larger steps could be used at a larger error. However, for small
bubble size, e.g. 0.2 mm. , the step should be as small as possible to
account for the time of contact. Thus it is up to the user to indicate the
step size required.
57
-------
SECTION VIII
EXPERIMENTAL OBSERVATIONS
As mentioned previously, the general purpose of this investigation was
to measure the velocity of rise of the buoyant plume, and the circulation
pattern generated by such a plume through measurement of the dispersion
of the Rhodamine B dye.
Velocity of Rise
Measurement of the velocity of rise of the water-bubble column was taken
by the procedure outlined earlier. In spite of the frequent runs made
(over 50 runs) , only a few meaningful results were obtained. The run was
considered successful if the lag time was established accurately and
the results obtained were duplicated in at least two more runs under the
same conditions . Table 1 shows the experimental results obtained for
the velocity of rise using the P.V.C. (2 mm bubbles) at three flow rates
and different injection depths. Two measurements at 1 .0 and 2 .0 1/min
at a depth of injection of 30 feet using the ceramic diffuser (0.2 mm
bubbles) yielded a velocity of rise of 0.042 and 0.06 m/sec at 20 feet
above the diffuser. The latter reported observations were duplicated once
Other results are not reported as they were not possible to duplicate and
were considered unreliable.
General
At quiescent conditions and at gas flows of 1.0 and 2 .0 1/min the bubble
plume showed good stability and was observed at the surface directly
above the diffuser. Measurements of the water-bubble column radius were
not possible under the conditions of the experiment. The bubble column
at shallow injections (low mass transfer) showed quite a spread and was
symmetrical. However, at higher injection depths, the observed diame-
ter of the bubble plume at the surface was much less. Bubble size asso-
ciated with deep injections was at least an order of magnitude less than
those for shallow injection depths. Many bubbles from the P.V.C. dif-
fuser (2 mm) did surface from an injection depth of 90 feet, and a flow
59
-------
TABLE 1
Measured Center Line Velocity of Rise
Using P.V.C. Diffuser (2 mm Bubbles)
Flow Rate
1 / min
Injection
Depth
Distance of
Rise (ft)
Measured Velocity
cm /sec
0.75
0.75
2.0
2.0
2.0
1.0
2.0
60
60
60
60
60
30
30
20
60
20
40
60
30
30
10.4
9.3
15.4
11.4
10.8
12.2
17.8
60
-------
rate of 2 1/min but at a much smaller volume than when generated at the
diffuser. Most of the bubbles from the ceramic diffuser were absorbed
before reaching the surface. Only a very minor number of larger bubbles
surfaced. These large bubbles were attributed in part to coalescence due
to the presence of the bands strapped around the diffuser and the balancing
lines (Figure 10) , and in part attributed to not having the diffuser per-
fectly horizontal. No measurement of bubble diameters were made.
Induced Mixing and Circulation
In this work, an attempt was made to study experimentally the induced
mixing and circulation. The experimental procedure is covered in Sec-
tion VI.
The sampling was carried out at several points away from the diffuser.
The farthest was 110 feet away. The dye was detectable at that distance
from the diffuser more than two hours from injection time. Typical results
are shown in Figure 17.
Since almost no movement exists in the hypolimnion, one could reason
that the only way that the dye reached that distance is the induced mix-
ing and turbulence generated by the rising plume. In an attempt to mea-
sure the diffusion coefficient for such a case, the continuous sampling
gate was used on the fluorometer and continuous sampling 40 feet away
from the diffuser was conducted. A plot, for this observation, of con-
centration vs. time at a sampling depth of 75 feet and an injection depth
of 90 feet is shown in Figure 18. During the sampling period, tests for
concentration at other depths were carried out. The concentration of dye
from the surface to a depth of 55 - 60 feet was almost zero and increased
suddenly beyond that level. This observation and consideration of the
temperature profile (Figure 19) for the lake during that period, substantiate
the complete uncoupling hypothesized by Cederwall et al. (8) and shown
graphically in Figure 7.
Another thing that was noticed and is shown in Figure 18 was that after
going down, the dye concentration started going up again and stayed al-
most at the last level for over half an hour before starting to go down
again. This impulse-like behavior was so often observed that the idea of
calculating a diffusion coefficient for this system was dropped. How-
ever, a listing of the collected data on the induced mixing and circula-
tion is included in the Appendix (Part A) .
61
-------
0 -
10
20
DEPTH
(ft) 30
40
50
60
FLOW RATE = 1 l./min,
1 30' AWAY
2 50' AWA/
3 100' AWAY
SCALE: 30x
10 20 30 40
FLUORESCENCE UNITS
50
60
Fig. 17 DYE CONCENTRATION AT VARIOUS DEPTHS AND DISTANCES
AWAY FROM THE DIFFUSER.
30
co 25
H "
P 20
w
U
w 15
O
co 1 _
u 10
O
FLOW RATE =2 l./min.
DEPTH OF INJECTION = 90 ft,
SCALE (3Ox)
_L
JL
J_
95
115 135 155 175
TIME (min.) FROM INJECTION
195
Fig. 18 DYE MOVEMENT AT A DEPTH OF 75 FEET (40 FEET
AWAY FROM THE DIFFUSER) .
62
-------
/5
en
H
PH
OH
W
P
1'JO
125
is:
175
30
40
J I L
50 60 70
TEMPERATURE (°F)
80
90
Fig. 19 TEMPERATURE PROFILE OF LAKE TRAVIS - MAY 14, 1972
-------
SECTION IX
DISCUSSION
In the following ais . jssion the model with mass transfer will be referred
to as Model Cris and the model with no mass transfer will be referred to
as Model Twc ..
Comparison With Experimental Results
Center line velocity was the only variable measured experimentally.
Thus comparison with the model prediction will be only for this variable.
Table 2 lists the field observations and the model prediction with mass
transfer as compared with the no mass transfer case. Graphical compari-
son is also shown in Figures 20 and 21. This prediction is based on
zero slip velocity and an entrainment coefficient of 0.03. As can be seen
from the table and the graphical representation, the prediction of the
model with mass transfer is in very good agreement with the field obser-
vation. For an oxygen flow rate of 0.75 1/min at an injection depth of
60 feet, the model prediction was within 5.8% and 9.7% of the measured
velocity at two sampling depths. Moreover, for a flow rate of 2 1/min
at an injection depth of 60 feet, the model prediction was within 0% to
7.5% of the measured velocity at three different levels. The model pre-
diction for a flow rate of 1. 0 1/min at a depth of 30 feet is within 13%
of the measured velocity, but was within 2 .5% for a flow rate of 2 1/min
at the same injection depth.
Since consideration of slip velocity means less entrainment, the model
prediction with mass transfer and non-zero slip velocity fit the field data
at an entrainment coefficient of approximately 0.02.
Thus the model is in excellent agreement with the field observation at the
three oxygen flow rates used as demonstrated above.
Effect of Entrainment Coefficient
To demonstrate the effect of the value of the entrainment coefficient on
the prediction of the dynamic characteristics of the plume, three different
65
-------
TABLE 2
Measured and Computed Velocity of Rise for a 2 .0 mm Bubble Plume
Flow Rate
1 /min
0.75
0.75
2.0
2.0
2.0
1.0
2.0
Injection
Depth
60
60
60
60
60
30
30
Distance
of Rise
20
60
20
40
60
30
30
Measured
Velocity
cm /sec
10.4
9.3
15.4
11.4
10.8
12.2
17.8
computed*
Velocity
with mass
transfer
cm /sec
11.0
8.4
15.3
12.2
11.6
13.8
17.36
computed*
Velocity
without mass
transfer
cm /sec
13.0
11.96
18.04
16.5
16.5
16.2
20.45
*Computed velocities considering zero slip and an entrainment
coefficient of 0.03
66
-------
O
w
o
o
)—t
w
>
w
w
r __.
\Z
W
.3,
.3U
.11
.!<
O - i'v sasureJ velocity
a = U.03
2 . , . 75 =• Oxygen Flov Rate
L/Min
0
16
18
Fig. 20
2 4 :5 8 10 12 14
HEIGHT ABOVE DIFFUSER (M)
MEASURED AND PREDICTED CENTER LINE VELOCITY FOR 0.75 AND 2.0
L/iVIN OXYGEN FLOW RATE AND AN INJECTION DEPTH OF 60 FEET (2MM BUBBLE)
-------
CD
CO
.40
.35
£ ,30
8
w
O
.25
,20
-> .15
o;
w
H
.10
.05
0
0
1
I
T
T
T
O = Measured velocity
a = .03
2 ., 1. = Oxygen Flow Rate
L/Min
1.
I
I
I
8
234567
HEIGHT ABOVE DIFFUSER (M)
Fig. 21 MEASURED AND PREDICTED CENTER LINE VELOCITY FOR 1.0 AND 2.0 L/MIN
AND AN INJECTION DEPTH OF 30 FEET {2.0 MM BUBBLE )
-------
values of 0. 01, 0.03, and 0.05 for a constant oxygen flow rate were
used. For an oxygen flow rate of 1 1/min, the results are shown graph-
ically in Figures 22 and 24, for the half-width of the plume, velocity
of rise, and water flow, respectively. The initial velocity was calcu-
lated based on Equation 69 and the initial half-width was assumed equal
to 0.1 meter, and the initial bubble diameter was equal to 0.2 mm. A
higher entrainment coefficient led to a lower velocity (Figure 23) , but to
a larger plume (Figure 22) , and higher water flow rate, (Figure 24) .
Thus the choice of the entrainment coefficient is very critical and more
studies need to be made to evaluate this constant for more reliable pre-
diction.
Comparison of the Two Models
Illustration of the prediction of Model One and comparison with the
prediction of Model Two was found to be best done by an example.
Assume an impoundment 150 feet (46 m) deep with a metalimnion approxi-
mately 50 feet (15m) below the surface. Using a 0.1 meter radius dif-
fuser that produces 0.2 mm bubbles and an oxygen flow rate of 1.0 1/min,
it is required to know the velocity, width, water flow rate, momentum
flux, and kinetic energy flux at the metalimnion for an injection depth
of 45 m. Compare the results to the case of no mass transfer. Compare
the results if a 2.0 mm bubble diffuser was used instead of a 0.2 mm bubble
one. Assume an entrainment coefficient of 0.03 for all cases and zero
slip velocity. The models' predictions are shown graphically in Figure
25 through Figure 31 .
Figure 25 shows that a 0.2 mm bubble was absorbed after rising a dis-
tance of 6.5 m while the 2 . 0 mm bubble had a residual size at the
metalimnion of .87 mm. Cf course, the model with no mass transfer did
not account for such a condition. After the bubbles disappeared the two
differential equations took the form
db/dx - 2 a (70)
dV /dx - -2a (V ,/b) (71)
c c
69
-------
^
w
ol
!-IH
O
E-i
Q
0.01, 0.03, 0.05
30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 22 HALF WIDTH OF PLUME (M) MS HEIGHT ABOVE DIFFUSER (M) FOR AN OXYGEN
FLOW RATE OF I I/MIX AND AN INJECTION DEPTH OF 75 M AND AN ENTRAINMENT
COEFFICIENT OF 0.01, 0.03, AND O.OG
-------
u
H
CO
H
i — i
U
o
,._]
w
w
[-H
^
w
O
,10
0
10
50
Fig. 23
20 30 40
HEIGHT ABOVE DL-FUSER (M)
CENTER LINE VELOCITY (M/SEC) VS HEIGHT ABOVE DIFFUSER (M) FOR AN
OXYGEN FLOW RATE OF 1 L/MIN AND AN INJECTION DEPTH OF 75 M AND AN
ENTRAINMENT COEFFICIENT OF 0.01, 0.03, AND 0.05
-------
vj
0.01, 0.03, 0.05 =a
30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 24 WATER FLOW VS HEIGHT ABOVE DIFFUSER (M) FOR AN OXYGEN FLOW RATE OF
1 I/MIM AMD AN INJECTION DEPTH OF 75M AND AN ENTRAINMENT COEFFICIENT
OF 0.01, 0.03, AND 0.05
-------
CO
W
hJ
CQ
CQ
E>
CD
fM
O
w
H
W
IS
P
2.0
1.5
1.0
.5
Fig. 25
0.2/2.0 = bubble diameter (mm)
(0.2)
111
0
10
15
25
30
35
40
20
DEPTH (M)
PREDICTED DIAMETER OF BUBBLE (MM) AS A FUNCTION OF DEPTH (M)
FOR 2.0 AND 0.2 MM BUBBLES AT AN INJECTION DEPTH OF 45 M
45
-------
and the plume continued to rise driven by its momentum. The effect of
the vanished buoyancy caused a faster deceleration of the 0.2 mm bub-
ble plume compared to the 2 .0 mm bubble or the no mass transfer case,
Figure 26. The deceleration of the plume was considered by Albertson
et a 1. (1) to be coupled with an acceleration of the surrounding fluid and
an increase of the half-width of the plume causing the total flow rate of
the flow past successive sections to increase with distance above the
outlet. Such an increase in the diameter of the plume and the increased
flow rate of the plume of the 0.2 mm bubble which continued to expand
as a direct function of height (Equation 70) would be larger than the
radius of the plume due to the 2 .0 mm bubble and the no mass transfer
case. Regardless of the larger radius, the water flow which is a func-
tion of the radius to the second power and the center line velocity, the
water flow rate of the 0.2 mm bubble was the smallest of the two cases,
while the no mass transfer was the largest, Figure 28. This demonstrates
the effect of the high reduction in the velocity of rise of the plume for
small bubbles. This effect was demonstrated also in the momentum flux
of the plume, Figure 29. The momentum flux of the 0.2 mm bubble in-
creased to the point where the bubbles were absorbed and then continued
at a constant rate as shown to a larger scale in Figure 30. However,
the momentum flux due to the 2 .0 mm bubble plume continued to increase
with height but still at a lower rate than the no mass transfer case. This
was also demonstrated in the kinetic energy flux which is a function of
the velocity to the third power. This led to a decrease in the kinetic
energy flux with height for the 0.2 mm bubble plume and a slight increase
for the 2 .0 mm bubble but caused a high increase in the no mass transfer
case as shown in, Figure 31. Thus Model One can predict the point at
which buoyancy has completely vanished (as in the case of 0.2 mm bubbles)
and shows its effect on the characteristics of the plume. This effect is
well demonstrated in the low center line velocity of the plume compared
to the no mass transfer case. Although this work assumed constant water
density in the hypolimnion, the effect of the mass transfer was very
significant. However, if we consider a very small density gradient and
couple it with the effect brought about by the mass transfer, and con-
sider again the center line velocity of the plume predicted by Model
One, one can assume that the plume could have come to a stop before
reaching the metalimnion.
Effect on Metalimnion
Regardless of the argument presented above that the plume might come to
a stop before reaching the metalimnion, this work will consider it to
74
-------
.25
O
W
CO
u
o
i
w
.20
.15
W
S .10
H
O
.05
T
T
NT = no mass transfer
i2.0, 0.2 = bubble diameter (mm) v;ith mass transfer
25
30
Fig. 25
0 5 10 15 20
HEIGHT ABOVE DIFFUSER (M)
CENTER LINE VELOCITY VS MEIGIii ABOVE DIFFUSER PREDICTED BY MODEL
ONE AND MODEL TWO FOR AN INJECTION DEPTH OF 45 M AND AN OXYGEN
FLOW RATE OF 1 L / iWIN
-------
2.0
1.6
rc
E-i
p
0.8
0.4
0
T
T
T
T
NT = no mass transfer
0.2, 2.0 = bubble diameter (mm) with mass transfer
(0.2)
I
0
10
25
30
15 20
HEIGHT ABOVE DIFFUSER (M)
Fig. 27 HALF WIDTH OF PLUME VS HEIGHT AT'''VE DIFFUSER PREDICTED BY MODEL
ONE (0.2 AND 2.0MM BUBBLE PLUM.) AND MODELTWO FOR AN INJECTION
DEPTH OF 45 M AND AN OXYGEN FLOW RATE OF 1 L/MIN
-------
u
w .24
CQ
£
U
o
v-1
w
f-H
NT = no mass transfer
0.2, 2.0 = bubble diameter (mm) with mass transfer
.08
0
JU^_^_B__^_an>^^^i^^____Ba_>_
10 15 20
HEIGHT ABOVE DIFFUSER (M)
Fig. 28 WATER FLOW VS HEIGHT ABOVE DIFFUSER PREDICTED BY MODEL ONE (0.2
AND 2.0MM BUBBLE PLUME) AND MODEL TWO FOR AN INJECTION DEPTH
OF 45 M AND AN OXYGEN FLOW RATE OF 1 L/MIN
-------
00
I I I I
NT = no mass transfer
0.2, 2.0 = bubble diameter (mm) with mass transfer
0
I 1 I I
10 15 20
HEIGHT ABOVE DIFFUSER (M)
Fig. 29 MOMENTUM FLUX VS HEIGHT ABOVE DIFFUSER PREDICTED BY MODEL ONE
(0.2 AND 2.0 MM BUBBLE PLUME) AND MODEL TWO FOR AN INJECTION
DEPTH OF 45 M AND AN OXYGEN FLOW RATE OF 1 L/MIN
-------
U
Q)
in
\
6
i
en
X
D
.-1
w
S
o
1.15 _
1.10
1.05
1.0
.95
.90
.85
80
0.2 = BUBBLE DIAMETER (mm) WITH MASS TRANSFER
(0.2)
I
I
I
I
10
25
30
15 20
HEIGHT ABOVE DUFFUSER (M)
Fig. 30 MOMENTUM FLUX VS HEIGHT ABOVE DIFFUSER PREDICTED BY MODEL ONE
i OR A 0.2 MM BUBBLE PLUME AT A DEPTH OF INJECTION OF 45 M AND FLOW
RATE OF 1 L/MIN
-------
00
o
U
w
X
w
S
w
o
I— (
H
W
•3
.2
.1
NT = no mass transfer
0.2, 2.0 = bubble diameter (mm) with mass transfer
25
30
Fig. 31
0 5 10 15 20
HEIGHT ABOVE DIFFUSER (M)
KINETIC ENERGY FLUX VS HEIGHT ABOVE DIFFUSER PREDICTED BY MODEL
ONE (0.2 AND 2.0 MM BUBBLE PLUME) AND MODEL TWO FOR AN INJECTION
DEPTH OF 45 M AND AN OXYGEN FLOW RATE OF 1 L/MIN
-------
continue to rise and show that its effect on the metalimnion is negligi-
ble.
As the plume rises it will expand and entrain more and more water, causing
it to decelerate. This rising water jet will raise the layer above it or
penetrate through it depending on the energy left in it. When this plume
hits the metalimnion, the difference in density coupled with a low energy
could bring this plume to a stop. This phenomenon was shown to exist
as indicated in Section VIII - Induced Mixing and Circulation,. The con-
figuration for this condition is shown in Figure 32 , where the plume will
cause a rise in the metalimnion (Ah) . Analysis of this condition on a
three-dimensional basis is rather complicated. However, a one-dimen-
sional analysis could lead to a rough measure of this rise (Ah):
p V 2 - Ah(o - o ) g (72)
1C -L J_
Using this equation, the plume of the example in Section IX - Compari-
son of the Two Models, Model One for which the velocity at the metalimnion
is 1.4 cm/sec will cause a h = 1.06 cm fora temperature difference of
20°F. Cn the other hand, Model Two will cause a h = 37 cm for the same
condition.
Effect of Injection Depth
To demonstrate the effect of the injection depth on the various charac-
teristics of the plume, the model was run for an injection depth of 45
meters and an injection depth of 75 meters for an impoundment with a
metalimnion 15 meters below the surface. The diffuser was assumed to
be 0.1 meter in radius and to produce 0.2 mm bubbles. The flow rate was
considered 1 1/min and the entrainment coefficient 0.03. The results
are shown in Figure 33 through Figure 37. The initial velocity was cal-
culated according to Equation 69 and the final velocity at the metalimnion
is shown in Figure 33. The increase in depth will lead to an increase in
the half-width of the plume, Figure 34, and in the water flow rate, Figure
35. However, the effect of the smaller initial velocity and the earlier
disappearance of bubbles for a 75 meter injection depth caused the momen-
tum flux to be less than that for a 45 meter injection depth, Figure 36.
The kinetic energy flux, Figure 37, shows also the effect of the above
mentioned factors.
81
-------
Pig. 32 DKK1NJLTIQN SKETCH
82
-------
oo
OJ
u
w
U
o
v—I
w
w
w
H
£
W
O
10
15
20
25 30
.25
.20
.15
.10
.05
0
45,75 = injection depth (m)
I
I
10
50
60
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 33 CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR INJECTION DEPTHS
OF 75 AND 45 METERS AS PREDICTED BY MODEL ONE
-------
0 5 10 15 20 25 30
00
4.0
3.0
O 2.0
JC
EH
Q
1.0
45,75 = injection depth (m)
T—I—Fl
0
10
50
Fig. 34
20 30 40
HEIGHT ABOVE DIFFUSER (M)
HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR INJECTION DEPTHS
OF 75 AND 45 METERS AS PREDICTED BY MODEL ONE
-------
oo
Cn
55, 75 = injection depth (m)
30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 35 WATER FLOW VS ^EIGLT ABOVE DIFFUSER FOR INJECTION
75 AND 45 METERS AS PREDICTED BY MODEL ONE
50
60
or
-------
CO
en
1.0
u
w
oo
X
H
o
.75
.50
.25
0
45,75 = injection depth (m)
10
15 20 25 30
i—i—r
(45)
(75)
o
10
50
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 36 MOMENTUM FLUX VS HEIGHT ABOVE DIFFUSER FOR INJECTION DEPTHS OF
75 AND 45 METERS AS PREDICTED BY MODEL ONE
60
-------
10
15
20
25
30
CO
oo
•K
*
o
w
I
I
X
f
i-l
w
£
w
U
1—4
H
W
.06
.05
45,75 = injection depth (m)
.03
.02
.01
0
0
10
50
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 37 KINETIC ENERGY FLUX VS HEIGHT ABOVE DIFFUSER FOR INJECTION DEPTHS
OF 75 AND 45 METERS AS PREDICTED BY MODEL ONE
60
-------
Effect of Flow Rate
The effect of varying the flow rate on the characteristics of the plume
will be shown here for 1 and 3 1/min and an injection depth of 75 meters
with a rise of 60 meters. Since the two flows are rather close in magni-
tude and the uncertainty of the value of the entrainment coefficient, a
single value was used for both cases, namely 0.03. To demonstrate the
effect of bubble size again, the output from 0.2 and 2.0 mm bubbles was
considered. The initial velocity was calculated based on Equation 69
and the initial radius of the plume was assumed 0.1 meter for all cases.
Consideration of Equation 57 leads to the conclusion that change in the
radius of the plume with height is essentially a function of the entrain-
ment coefficient (8). Since the same value of entrainment coefficient
was used for the two cases, Figure 38 shows the half-width of the plume
to be approximately the same for 1 and 3 1/min for 0.2 and 2 .0 mm bubbles
The complete absorption of the 0.2 mm bubble after a short rise made the
expansion of the plume a function only of the entrainment coefficient re-
gardless of the flow rate. The 2.0 mm bubble continued to rise causing
a smaller plume diameter than the 0.2 mm bubble, with a rather unnotice-
able effect due to the higher flow rate.
The effect of the initial velocity of the two cases and with 0.2 and 2.0
mm bubbles is shown in Figures 39, 40. Those two variables led to the
water flow rates shown in Figure 41 which indicate a higher flow rate due
to 3 1/min than to 1 1/min for 0.2 and 2.0 mm bubbles. The flow rate due
to the 2.0 mm bubble was higher in both cases than the 0.2 mm bubble.
Increasing the air flow rate three-fold did not cause a similar increase
in the water flow rate. For the 0.2 mm bubble the increase in water flow
rate was 35% while for the 2.0 mm bubble the increase was 42%.
Prediction Considering Slip Velocity
The speed of rise of a swarm of bubbles relative to the surrounding water
has not been measured in this work and the literature is very conflicting on
this aspect. The mathematical model with mass transfer treated it as
a function of the diameter utilizing the available data in the literature
(Section VII) . However, the model with no mass transfer used constant
slip velocity considered to be equal to 0.25 m/sec.
88
-------
CO
to
w
CL,
Pn
O
H
Q
ffi
T
1,3 = flow rate (1/min)
0.2, 2.0 = bubble diameter (mm)
(0.2)
(1,3)
0
10
50
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 38 HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR FLOW RATES OF
1 AND 3 L/MIN AND BUBBLES OF 0.2 AND 2.0 MM IN DIAMETER AT AK
INJECTION DEPTH OF 75 METERS
60
-------
CO
o
O
w
O
O
w
53
M
H
£
w
O
1,3 = flow rate (1/min)
0,2 = bubble diameter (mm)
,05 —
30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 39 CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR FLOW RATES OF
1 AND 3 L/MIN AND BUBBLE DIAMETER OF 0.2 MM AT AN INJECTION
DEPTH OF 75 METERS
-------
,25
O
W
O
c
W
Pi
w
W
u
.20
.15
.10
.05
0
T
1,3 = flow rate (1/min)
2.0 = bubble diameter (mm)
(1,2.0)
I
I
I
0
10
50
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 40 CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR FLOW RATES OF
1 AND 3 L/MIN AND BUBBLE DIAMETER OF 2 .0 MM AT AN INJECTION
DEPTH OF 75 METERS
60
-------
to
.70
.60
U
w
40 .50
s
QQ
O .40
I
cx;
w
.30
.20
.10
0
1,3= flow rate (1/min)
0.2, 2.0= bubble diameter (mm)
0
( 3,2
I
10
50
60
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 41 WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR FLOW RATES OF 1 AND 3 L/MIN
AND BUBBLES OF 0.2 AND 2.0 MM IN DIAMETER AT AN INJECTION DEPTH OF
75 METERS
-------
Figure 42 through 45 show the output of the two models and compared it
to the output considering zero slip velocity. A 0.2 mm bubble has a low
initial slip velocity (4 cm/sec) which continues to decrease as a function
of the residual diameter until it reaches zero. This consideration caused
the center line velocity of the plume to remain the same as that for zero
slip velocity. However, the constant slip velocity of 0.25 m/sec used
in Model Two caused a large effect on the center line velocity. The half-
width of the plume remained the same for Model One while it increased
with Model Two. However, the effect of the slip velocity on a 2.0 mm
bubble plume is well pronounced in Figure 44 and 45. The initial slip
velocity for a 2 .0 mm bubble was 23 cm/sec and decreased in relation
to the residual diameter. The effect of this high initial slip velocity
is well demonstrated in Figure 44, and led to a larger plume as shown
in Figure 45.
Although the 2.0 mm bubble plume has approximately the same initial
slip velocity used for the no mass transfer case, the reduction in velocity
of rise and the increase in the half-width of the plume was lesser than
the no mass transfer case which demonstrated the effect of the reduction
in bubble diameter and thus the slip velocity.
Other Model Predictions
The mass transfer model, in addition to solving for residual bubble dia-
meter, velocity, radius, etc., solves for other important parameters.
The program HYPO calculates for such important variables as the fraction
of oxygen left, the per cent of oxygen in off gas, the oxygen to nitrogen
ratio and others. Samples of such outputs are shown in the Appendix.
For a certain injection rate of oxygen and with the calculated fraction of
oxygen left, the amount of oxygen absorbed and thus the increase in
the D.O. level in the plume is calculated. The fraction of oxygen absorbed
calculated by HYPO is passed on to sub-program HYDRO which calculates
the water flux caused by a known flow rate of oxygen. The increase in
dissolved oxygen of the water can then be calculated. An example of
such an output is shown in Figure 46 for 0.2 and 2.0 mm bubble plumes
for an injection depth of 45 meters and an oxygen flow rate of 1. 0 1/min.
Initially, as the water flux in the plume is small and the relative amount
of oxygen absorption is high, there is a positive increase in D.O. Sub-
sequently, as the water flux rate increases and the oxygen absorption
decreases, the increase in D.C. declines toward zero.
93
-------
2.0
1.5
0,
O 1.0
Q
H—i
£
-------
CD
en
O
o
w
I—I
1-1
w
H
W
O
.25
o -20
w
.15
,10
.05
NT = no mass transfer
z = zero slip velocity
s = non zero slip velocity
0.2 = bubble diameter (mm)
I
0
10
25
30
Fig. 43
15 20
HEIGHT ABOVE DIFFUSER (M)
CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER AS PREDICTED BY MODEL
ONE (2MM BUBBLE) AND MODEL TWO CONSIDERING NON ZERO SLIP VELOCITY
AND COMPARED TO ZERO SLIP VELOCITY
-------
CD
0
W
o
o
w
O
.25
.20
.15
w
§ .10
.05
2.0 = bubble diameter
z = zero slip velocity
s = non zero slip velocity
I
I
0
25
30
Fig. 44
5 10 15 20
HEIGHT ABOVE DIFFUSER (M)
CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER AS PREDICTED BY MODEL
ONE (2.0 MM BUBBLE) AND MODEL TWO CONSIDERING NON ZERO SLIP VELOCITY
AND COMPARED TO ZERO SLIP VELOCITY
-------
2.0
to
•vj
W
S
3
OH
P-.
o
Q
i—i
<;
ffi
1.0
0.5
0
z = zero slip velocity
s = non zero slip velocity
2.0 = bubble diameter (mm)
I
I
0
10
25
30
15 20
HEIGHT ABOVE DIFFUSER (M)
Fig. 45 HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR A 2.0 MM BUBBLE
PLUME AS PREDICTED BY MODEL ONE CONSIDERING ZERO SLIP VELOCITY
AND COMPARED TO NON ZERO SLIP VELOCITY
-------
L.2
ID
O3
i.o-
0.8
O
u
2
w
0.6
w 0.4
CO
U
S3
0.2
0
0
0.2, 2.0 = bubble diameter
(mm)
25
5 10 15 20
HEIGHT ABOVE DIFFUSER (M)
Fig 46 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF I L/MIN AND BUBBLKS OF 0.2 A*!D 2.0 VIM AT A'\r """
DEPTH OF 45M
30
-------
The complete absorbance of the 0.2 mm bubbles and the lower water
flow leads to a higher increase in D.O. in the plume than that caused
by the 2 .0 mm bubble .
General Guide
The purpose of this section is to present the prediction of the model for
various conditions of injection depth, bubble diameters and oxygen flow
rates to be used as a general guide for field work. The given prediction
is for injection depths of 25, 50, 75, and 100 meters and oxygen flow
rates of 1 and 2 1/min for diffusers that produce 0.2, 0.5, and 1,0 mm
bubbles. In all cases the metalimnion is assumed at a depth of 15 meters
below the surface. The prediction for the center line velocity, half-
width of the plume, water flow and increase in D.C. are tabulated in
Tables 3 through 6 at the quarter points of the distance from the diffuser
to the metalimnion (x) . The prediction is also shown graphically in the
Appendix (Part C) .
The following symbols are used in Tables 3 through 7;
BD = Bubble Diameter (mm)
h - Distance above diffuser (M)
X = Distance between diffuser and metalimnion (m)
V = Center line velocity (M/sec)
c
b = Half width of plume (M)
WQ = Water flux (M /sec)
ADO = Increase in plume dissolved oxygen (mg/1)
Illustrative Example of Use of Tables
To illustrate the use of those tables and graphs, let us assume a lake
of approximately 90 meters in depth which is to be aerated using commer-
cial oxygen. We would like to know the velocity, width, water flow and
increase in D.O. of the plume if we used a 0.5 mm bubble diffuser at
a depth of 85 meters and an oxygen flow of 2 1/min. From the tables and
graphs for a depth of 75 and 100 meters the above mentioned variables can
be obtained at any height above the diffuser. With these predictions we
can approximate the behavior of this plume at 85 meters injection depth.
This interpolation is shown in Table 7. Similar tables could be constructed
for a 0.2 or 1.0 mm diffuser. Knowing these characteristics we can decide
on the best diffuser to use, for a certain flow rate, to yield the highest
increase in D.C. with the least disturbance to the stratification.
99
-------
TABLE: 3
Model Predictions for 1. L/Min
and Depths of 25 and 50 Meters.
Oxygen Flow = 1 L/Min
25 M
X=10
BD H
0
X
4
••> f
3X
4
X
0
X
4
O.S f
3X
4
X
0
X
4
1.0 f
3X
4
X
V
c
0.261
0.135
0.089
0.064
0. 049
0.261
0.15
0.117
0.099
0.085
0.261
0.152
0.124
0.108
0.093
b
0.1
0.226
0.361
0.51
0.66
0.1
0.216
0.323
0.43
0.54
0.1
0.214
0.32
0.42
0.52
WQ
x!02s
.82
2.2
3.66
5.2
6.7
.82
2.2
3.86
5.76
7.9
.82
2.2
3.9
5.9
8.1
AD.o.
xlO •
0.0
8.7
6.4
4.6
3.5
0.0
4.1
3.9
3.2
2.7
0.0
3.1
3.0
2.6
2.2
V
c
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
22
045
024
016
012
22
072
042
0276
02
22
08
055
042
035
50 M
X=35
b
0.1
.59
1.11
1.64
2.2
0.1
0.5
0.95
1.46
1.99
0.1
0.48
0.86
1.26
1.7
WQ
xlQ2 •
0.
4.
9.
13.
18.
0.
5.
11.
18.
25.
0.
5.
12.
21.
30.
69
9
4
7
2
69
5
9
5
2
69
7
9
3
5
AD.o.
xlO»
0.
4.
2.
1.
1.
0.
3.
1.
1.
0.
0.
2.
1.
1.
0.
0
8
5
7
3
0
6
97
3
94
0
8
7
1
77
100
-------
TABLE: 4
Model Predictions for 1. L/Min
and Depths of 75 and 100 Meters
Oxygen Flow = 1 L/Min
75 M
X=60
BD H
0
X
4
0. 2 —
3X
4
X
0
X
4
0.5 f
2
3X
4
X
0
X
4
,0 f
3X
4
X
V
" c
0.196
0. 025
0. 013
. 0086
. 0065
0. 196
. 041
0. 02
0.013
0. 0098
0. 196
0.052
0.03
0. 02
0. 014
L
0.
0.
1.
2
3.
0.
0.
1.
2.
3.
0.
0.
1.
2,
3.
xlS -
1
96
87
76
67
1
83
7
6
5
1
76
5
35
24
0.
7.
13.
20,
27.
0.
8.
18.
28.
38.
0.
9.
21.
34.
47.
6
2
9
6
4
6
8
6
3
6
4
5
6
0
£D. o.
0.0
3.3
1.7
1.4
.86
0.0
2. 6
1.27
0.83
. 62
0.0
2. 2
1. 1
0. 68
0. 5
V
c
0.179
0. 016
0.0033
. 0056
. 0042
0.179
0.026
0.013
0.008
.006
0.179
.036
0.018
0.012
. 003
100 M
X-85
b
0, 1
1.33
2. 6
3. 88
5.16
0. 1
1.2
2.47
3.74
5.02
0.1
1.07
2.26
3. 5
4. S
WQ
xlO2*
0.
9
17.
26.
35.
0.
11.
24.
36.
49.
0.
12.
29.
45.
61.
56
2
9
7
5
56
7
3
8
4
56
9
4
.3
^D. o.
xlO =
0. 0
2. 6
1.3
0.88
0.6
0.0
2.0
.97
0. 64
0.4
0.0
1.74
0.8
0.52
0. ?8
101
-------
TABLE: 5
Model Predictions for 2. L/Min
and Depths of 25 and 50 Meters.
Oxygen Flow = 2. L/Ml*i
25 M
X=10
BD H
0
X
4
«•> f
3X
4
X
0
X
4
0.5 f
3X
4
X
0
X
4
,0 f
3X
4
X
V
c
0.33
0.17
0.11
0.08
.062
0.33
0.19
0.147
0.124
0.108
0.33
0.192
0.156
0.136
0.123
b
0.1
0.226
0.36
0.51
0.66
0.1
0.22
0.32
0.43
0.54
0.1
0.21
0.32
0.42
0.51
WQ
x!02 =
1.03
2.7
4.6
6.5
8.5
1.03
2.8
4.9
7.25
9.9
1.03
2.8
4.9
7.4
10.2
AD.O.
xlO*
0.
13.
10.
7.
5.
0.
6.
6.
5.
4.
0.
4.
4.
4.
3.
0
7
0
2
6
0
5
2
2
3
0
9
7
1
4
V
c
0.
0.
0.
•
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
276
057
03
021
016
276
09
053
035
026
276
10
072
053
044
b
0.1
0.58
1.11
1.64
2.2
0.1
0.49
0.95
1.5
1.98
0.1
0.476
0.82
1.3
1.7
50 M
X=35
WQ
x!02s
0.37
6.2
11.8
17.3
22.8
0.87
7.0
15.
23.3
31.7
0.87
7.2
15.1
26.8
38.4
/.D. o.
xlO*
0.
7.
4.
2.
2.
0.
5.
3.
2.
1.
0.
4.
2.
1.
1.
0
6
7
1
0
7
1
5
0
5
3
7
22
102
-------
TABLE: 6
Model Predictions for 2. L/Min
and Depths of 75 and 100 Meters.
Oxygen Flow = 2. L/Min
75 M
X=60
BD H
0
X
4
.., f
3X
4
X
0
X
4
0.5 f
_3X
4
X
0
X
4
,0 f
3X
4
X
V
c
0.246
.03
0.016
0.011
0. 008
0.246
0. 052
0.025
0.017
0.012
0. 246
0. 065
0. 038
0.025
0. 013
b
0.1
0.96
1.9
2.8
3.7
0.1
0. 83
1-71
2.6
3.5
0. 1
0.76
1.51
2.35
3. 24
WQ
x!02r
0.
9.
17.
25.
34.
0.
11.
23.
35.
47.
0.
11.
27.
43.
60.
77
5
9
4
77
1
4
6
9
77
8
1
,6
.4
103
£D. o.
xlOs
0.0
5.2
2.7
1.8
1.4
0.0
4.2
2.
1.3
0.98
0.0
3.5
1.7
1.1
0.8
V
c
0.
0.
0.
0.
•
0.
0.
0.
0.
o.
o.
0.
0.
0.
0.
226
021
Oil
0071
0053
226
033
016
Oil
0078
226
045
023
015
Oil
100 M
X^85
b
0.1
1. 33
2. 6
3.88
5.2
0.1
1.2
2. 5
3.7
5.02
0. 1
1.07
2.26
3.53
4. 81
WQ
x!02 =
0.7
11.5
22.6
33.6
44.7
0.71
14. 8
30. 6
46.4
62.2
0.71
16.3
36.6
57.2
77.9
,'D. o.
xlOs
0.
4.
2.
1.
1.
0.
3.
1.
1.
0.
0.
2.
1.
0.
0.
0
1
1
4
1
0
2
5
76
0
8
3
8
6
-------
7A.1LS 7
Illustrative £xa;nple
Vc at
Yc at
Vc at
Vc at
b at
b at
b at
b at
V/Q at
WQ at
WQ at
¥Q at
AE.O.
AD.C.
AE.C.
AD.O.
x/4
x/2
3x/4
X
x/4
x/2
3x/4
X
x/4
x/2
3x/4
X
at x/4
at x/2
at 3x/4
at x
75 Meters
.052
.025
.017
.012
.33
1.71
2.G
3.5
11.1 x 10~2
23.4 x 10~2
35.6 x 10"2
47.9 x 10~2
4.2 x 1CT1
2 x 10~1
1.3 x 10~1
.98 x 10"1
Interpolated
33 Meters Lif-
1000 y.eters fuser depths
.033 .042
.016 .021
.011 .014
..0078 .01
1.2 1.0
2.5 2.1
3.7 3.15
5.02 4.25
14.6 x 10~2 13 x 10~2
30.-: x 10~2 27 x 10~2
46.4 x 10'2 41 x 10~2
62.2 x 10~2 55 x 10~2
3.2 x 10~1 3.7 x lO'1
1 .5 x 10~1 1 .7 x 10-1
1 x 1C'1 1 .15 x 10"1
.76 x 10~1 .SB x 10-1
104
-------
SECTION X
ACKNOWLEDGMENTS
This study was sponsored by the Office of Research and Monitoring,
Environmental Protection Agency. Grateful appreciation is extended
to Richard Hiller, who was the initial Project Officer and Lowell
Leach, who subsequently served as project officer for the major portion
of the grant. These men took a genuine interest in the project and
whole heartedly supported it. Appreciation is also extended to Dr.
Curtis C. Harlin, Jr. for his support of this project as well as the
general area of river and impoundment aeration.
The University of Texas at Austin staff of Environmental Health Engineering
and Dr. Gus Fruh in particular are gratefully acknowledged for their
consultation and encouragement. The staff and facilities of the Center
for Research in Water Resources are much appreciated.
105
-------
SECTION XI
REFERENCES
1. Albertson, M .L. , Dai, Y.B. , Jensen, R.A. and Rouse, H. ,
Diffusion of Submerged Jets", Trans. A.S.C.E.. V. 115, 1950,
p. 639.
2. Baines, W.D., Hamilton, G.F., "On the Flow of Water Induced
by a Rising Column of Air Bubbles" , International Association for
Hydraulic Research, 8th Congress, Montreal, August 24-29, 1959.
3. Barnhart, Edwin L., "Transfer of Oxygen in Aqueous Solutions",
proceedings of the American Society of Civil Engineers, Sanitary
Eng. Div. , June 1969.
4. Batchelor, G.K., 1954 Quart. J.R. Met. Soc. 80, p. 339.
5. Bouissinesq, Comptes Rendues, 1912, 153, pp. 983, 1035,
1040, 1124.
6. Bowonder, B., Kumar, R., "Studies in Bubble Formation-IV Bubble
Formation at Porous Discs", Chem. Engr. Science, 25:25, 1970.
7. Brasher, P., Contribution to Compressed Air Magazine, April 1907.
8. Cederwall, K., and Ditmars, J.D., "Analysis of Air-Bubble Plumes",
Report #KH-R-24, Sept. , 1970, California Institute of Tech.
9. Collins, R.J. , Fluid Mech. , 22, 763, 1965.
10. Datta, R.L., Napier, D.H., Newitt, D.M., "The Properties and
Behaviors of Gas Bubbles Formed at Circular Orifice" , Trans.
Instn. Chem. Engrs., 28: 14, 1950.
11. Davies, R.M., Taylor, G.I., "The Mechanics of Large Bubbles
Rising Through Extended Liquids and Through Liquids in Tubes,"
Proc., Royal Soc. of London, Vol. 200, Series A, 1950, p. 395.
12. DeNevers , N. , "Bubble Driven Fluid Circulation", AIChE Journal,
Vol. 14, No. 2, March 1968.
106
-------
13. Dumitrescu, D.T., Z. Angew, Math. Mech. 23, 139, 1943.
14. Evans, J.T., "Pneumatic and Similar Breakwaters" , Proc . Roy.
Soc. (London), Series A, Vol. 231, 1935, p. 457.
15. Feuerstein, D.L., and R.E. Selleck (1963), "Fluorescent Tracers
for Dispersion Measurements" , J. of San. Engr. Div. , ASCE, 89,
SA 4, August.
16. Gay, F., and Hagedorn, Z., "Forced Convection in Stratified Fluid
by Air Injection, " M .1 .T. Thesis , Jan. 1962.
17. Haberman, W.L. , Morton, R.K., "An Experimental Study of Bubbles
Moving in Liquids" , Trans . Amer. Soc. Civil Engr., 121:227, 1956.
18. Hadamard, Comptes Rendues, 1911, 152, 1735, and 1912, 154, 109.
19. Khurana, A.K., Kumar R., "Studies in Bubble Formation III" , Chem.
Engr. Sci. , 24: 1711, 1969.
20. Kobus, H .£., Analysis of the Flow Induced by Air Bubble Systems',
Coast. Eng. Conf. Vol. II, London, 1968.
21. Kurihara, M., "Pneumatic Breakwater, Section I, II, and III" ,
Translated by K. Horikawa, Univ. of California, Institute of
Engineering Research, Wave Research Lab. , Series 104, Issues
4, 5, and 6, 1954.
22. Maneri, C.C., Mendelson, H.D., "The Rise Velocity of Bubbles
in Tubes and Rectangular Channels as Predicted by Wave Theory",
AIChE Journal, Vol. 14, No. 2, page 295, 1968.
23. Mathers , W.G ., Winter, E.E., "Principles and Operation of an
Air-Operated Mixer-Settler. The Can. J. Chem. Engr., 37:99, 1959
24. Moore, D.W. , "The Velocity of Rise of Distorted Gas Bubbles in
a Liquid of Small Viscosity," J. Fluid Mech. , 23 (4): 749, 11963.
25. Morton B.R., Taylor, G.I., and Turner, J.S., "Turbulent Gravi-
tational Convection From Maintained and Instantaneous Sources,"
Proc. Roy. Soc., Vol. A236, (1936).
107
-------
26. Murphy, D., Clark, D.S., Lentz, C.P., "Aeration in Tower
Type Fermenters" , The Can. J. ofChem. Engr., 37:157, 1959.
27. Nicklin, D.J. , "Two-Phase Bubble Flow" , Chem. Eng. Science,
Vo. 17, pp. 693-702, 1962.
28. O'Brien and Gosline, Ind. Eng.Chem., 1935, 27, 1436.
29. Pattle, R.E. , "Factors in the Production of Small Bubbles" , Trans.
Instn. Chem. Engrs., 28; 14, 1950.
30. Pritchard, D.W., and J.H. Carpenter, (1960), "Measurement
of Turbulent Diffusion in Estuarine and Inshore Waters." Bull.
Int. Assoc. Sci. Hydrol. No. 20.
31. Ramakrishnan, S., Kumar, R., Kuloor, N.R., "Studies in Bubble
Formation I; Bubble Formation Under Constant Flow Conditions,"
Chem. Engg. Sci., 24, 731, 1969.
32. Reith, T. , Renken, S., Israel, B.A., "Gas Hold-up and Axi 1
Mixing in the Fluid Phase of Bubble Columns." Chem. Engr.
Sci., 23, 619, 1968.
33. Rouse, H., Yih, C.S., and Humphreys, H.W. , "Gravitation
Convection from Boundary Source" , 1952, Tellus, 4, 201.
34. Rybczynski, Bull. Acad. Sci. Gracovie, 1911, 1, 40.
35. Satyanarayanan, A., Kumar, R., Kuloor, N.R., "Studies in
Bubble Formation II: Bubble Formation Under Constant Pressure
Conditions," Chem. Engg. Sci., 24: 749, 1969.
36. Schmidt, W. , 1941, Z. Angew. Math. Mech. 21, 265, 351.
37. Siemes, W. , Weises, W., "Mixing of Fluid in Narrow Blow Columns"
Chem. Eng. Tech., 29, 727, 1957.
38. Speece, R.E. , "Hypolimnion Aeration" , Journal American Water Works
Association, Vol. 63, No. 2, January 1971.
39. Speece, R.E., "The Use of Pure Oxygen in River and Impoundment
Aeration." Presented at the 24th Purdue Industrial Waste Conference,
May 8, 1969.
108
-------
40. Stokes, G.G., Mathematical and Physical Papers, Cambridge
University Press, 1880, Cambridge, U.K.
41. Subramanian, R., and Chi Tien, "Longitudinal Mixing in Liquid
Columns Due to Bubble Motion", Department of Chem. Eng .
and Metallurgy, Syracuse University, Syracuse, N.Y., Research
Report #70-2
42 . Tadaki, T. , Maeda, S. , Chem. Eng. Japan, 28: 270, 1964.
43. Turner, G.K., Associates. "Cperating and Service Manual, Model
III Fluorometer."
44. Uno, Seiji, and Kintner, R.C ., AIChE Journal, 2, 3, Sept. 1956.
45. Wilson, J.R. , and Masch F.D. , "Field Investigation of Mixing
and Dispersion in a Deep Reservoir" , Hydraulic Eng. Lab. ,
University of Texas, Austin Texas , Tech. Report HYD 10-6701,
June 1967.
46. Yih, C.S. , 1951, Proc. 1st Nat. Cong. Appl. Mech. , p. 941.
47. Zieminski, S.A., and Whittemore, R.C., "Induced Air Mixing
of Large Bodies of Polluted Water." EPA Program #16080 DWP. ,
1970.
109
-------
SECTION XII
LIST OF PUBLICATIONS
One publication, to date, has resulted from this project. It is
entitled "Alternative Considerations in the Oxygenation of Reservoir
Discharges and Rivers" by R. E. Speece, Fawzi Rayyan, George
Murphee. It is a publication in the Conference Proceedings -
Applications of Commerical Oxygen the Water and Wastewater
Systems University of Texas Press (In Press).
No patents resulted from this study.
110
-------
SECTION XIII
APPENDICES
111
-------
o
w
CO
H
g
W
W
O
u
D
i — i
t»
a
.50
0
8
246
BUBBLE DIAMETER (MM)
Fig. 47 RELATIONSHIP BETWEEN LIQUID FILM COEFFICIENT AND BUBBLE DIAMETER
AT 20° CENTIGRADE
-------
3.0
2.5
2.0
bubble diameter = 2.0 mm
0
10
15
30
40
20 25
DEPTH (M)
Fig. 48 MO OF OXYGEN AS * FUNCTION OF DEPTH FOR AN INJECTION DEPTH
OF 45 M AND A BUBBLE DIAMETER OF 2 .0 MM
45
-------
-
O
O
.10
.08
.06
.04
.02
0
bubble diameter = 2.0mm
10
15
30
35
40
?ig. 49
20 25
DEPTH (M)
MG OF N2 AS A FUNCTION OF DEPTH FOR AN INJECTION DEPTH OF 45M
AND A BUBBLE DIAMETER OF 2.0 MM
-------
Cn
III!
Duoble oian.eter = 2 . •„ :;,n~.
Fig. 50
XA',/110:
DI:;FC;
20 25 30 35 40
DEPTH
Or Qr.rAi-.hl, AWOUMT OF O2 REMAINING AS A FUNCTION OF
10?. A:; n'jEr 'no'1" DEPTH OF 45M A:JD A BUBBLE DIAMETER
OF 2.0MM
-------
1.0
CM
00.8
CO
i—i
U
coO.6
c'
CO
QQ
CO
O
0.4
H
U
§0.2
bubble cu
^r = 2.0
I
I
I
I
0
5
i:
15
30
35
10
20 25
LEPTH (M)
Fig.51 FRACTION OF BUBBLE GAS WHICH IS O2 AS A FUNCTIOI- OF DEPTH FOR
AN INJECTION DEPTH OF 45M AND A BUBBLE DIAMETER OF 2.0MM
-------
9
8
O
h—4
I 6
W c
W
I I
bubble diameter = 2 .Omm
10
15
30
35
40
Fig. 52
20 25
DEPTH (M)
NITROGEN/OXYGEN RATIO AS A FUNCTION OF DEPTH FOR AN INJECTION
DEPTH OF 45 M AND A BUBBLE DIAMETER OF 2 .0 MM
-------
00
1.0
0.8
O
0.6
O
0.4
0.2
0
bubble diameter = 2.0 mm
0
0.5
1.0
3.5
I
r
4,0
1.5 2.0 2.5 3.0
NORMALIZED RELEASE DEPTH
Fig.53 RELATIVE AMOUNT OF O2 IN BUBBLE VS NORMALIZED RELEASE DEPTH FOR
AN INJECTION DEPTH OF 45M AND A BUBBLE DIAMETER
OF 2.0 MM
4.5
-------
W
H
W
U
O
CQ
O
.00023
.00020
,00017
,00014
I I I I
bubble diameter = 2.0 mrn
,00011
_
2T5
DEPTH (M)
Fig. 54 CSATO VS DEPTH FOR AN INJECTION DEPTH OF 45 M AND A BUBBLE DIAMETER
OF 2.0 MM
-------
U
w
U
O
.25
.20
.15
g
g .10
a
H
w nc
U -05
0
0
T
2 , . 5 ,1. = Bubble diameter
(mm)
1
I
I
10
Fig. 55
2468
HEIGHT ABOVE DIFFUSER (M)
CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF
1 L/MIN AND BUBBLE DIAMETER OF 0.2,0.5 AND 1.0 iMM AND AN
INJECTION DEPTH OF 25 M
-------
0.7
1 ~
w
a.
O
K
H
P
ffi
0.6
0.5
0.4
0.3
0.2
0.1
. 2 ,. 5,1 .= Bubble diameter
(mm)
Fig. 56
0
I
I
10
2468
HEIGHT ABOVE DIFFUSER (M)
HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF
1 L/MIN AND BUBBLE DIAMETER OF 0.2,0.5, AND 1.0 MM AND AK
INJECTION DEPTH OF 25 M
-------
u
W
^
CO
O
I
.-I
W
T
T
.08
.2,.5,1.= Bubble diameter
(mm)
.06
(1.0)
.04
.02
0
I
I
0 2 4 6 8 10
HEIGHT ABOVE DIFFUSER (M)
Fig. 57 WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF 1 L/MIN
AND BUBBLE DIA METER OF 0.2, 0.5, AND 1.0 MM AND AN
INJECTION DEPTH OF 25M
-------
1.0
to
OJ
o
u
£
o
u
H
p
w
PC:
O
.6
O
S -4
i—i
H
.2
0
I I I •
.2,.5,1.= Bubble diameter (mm)
LO
2468
HEIGHT ABOVE DIFFUSER (M)
Fig. 58 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE
OF U 1,/MIN AND RUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM AND.AN
INJECTION DEPTH OF 25 M
-------
CO
U
w
in
8
W
55
i—i
^
W
W
U
.20
.15
10
05
2, .5, 1.= Bubble diameter
(mm)
I
I
I
I
I
0
30
5 10 15 20 25
HEIGHT ABOVE DIFFUSER (M)
Fig. 59 CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF
1 L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5, AND 1.0 MM AND
AN INJECTION DEPTH OF 50 M
-------
en
w
2
ED
I-H
cu
O
H
P
-------
to
CD
u
«
^
S
CO
t>
U
W
I
30
.25
.20
.15
.10
.05
0
0
I I I
.2, .5, 1.= Bubble diameter
(1.0)
I
I
I
I
I
10
I
30
r
15 20 25
HEIGHT ABOVE DIFFUSER (M)
Fig. 61 WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF L L/MIN
AND BUBBLE DIAMETER OF 0.2, 0.5, AND 1.0 MM AND AN INJECTION
DEPTH OF 50 M
-------
o
o
s
o
o
5
H
X
o
H
w
DC:
O
.2, .5,1.= Bubble diameter (mm)
0.2
0
10 15 20 25
HEIGHT ABOVE DIFFUSER (M)
35
Fig. 62 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 1. L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5, AND 1.0 MM
AND AN INJECTION DEPTH OF 50 M
-------
CO
u
w
8
w
S3
w
H
S
W
O
02 —
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 63 CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF
1 L/MIN AND BUBb^E DIAMETER OF 0.2, 0.5, AND 1.0 MM AND AN
INJECTION DEPTH OF 75 M
-------
.50
Ni
.40
O
w
^ .30
PQ
u
O -20
PH
W
I
.10
0
I
.2, .5, 1.- Bubble diameter
(i.O)
0
10
50
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 64 WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF 1 L/MIN
AND BUBBLE DIAMETER OF 0.2, 0.5, AND 1.0 MM AND AN
INJECTION DEPTH OF 75 M
-------
3.5
3.0
2 2.5
PL,
g 2.0
E-i
Q
<: 1.5
i-J
<
'j;
1.0
0.5
T
T
.2, .5, 1.= Bubble diameter (mm)
I
0
10
50
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 65 HALF WIDTH OF PLUME VS HEIGHT ABOVE DUFFUSER FOR A FLOW RATE
OF 1 L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 75 M
60
-------
1.2
Bubble diameter
(:nm)
20 30 40
"EIGHT ABOVE DIFFUSER (M)
Fig. 56 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 1. L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 A*TD 1.0 MM
AND AN INJECTION DEPTH OF 75 M.
-------
to
U
w
U
o
£
w
PC:
w
H
£
W
U
T
T
.15
.2, .5, 1.= Bubble diameter
(mm)
.10
.05
0
Fig. 67
I
I
10
20
70
30 40 50 60
HEIGHT ABOVE DIFFUSER (M)
CENTEP LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 1 L/MIN BUBBLE DIAMETER OF 0.2, 0.5, AND 1.0 MM
AND AN INJECTION DEPTH OF 100 M
80
-------
oo
oo
W
S
£
J
cu
PM
O
K
H
P
1 = Bubble diameter (mm)
40 50 60
HEIGHT ABOVE DIFFUSER (M)
HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 1 L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND
1.0 MM AND AN INJECTION DEPTH OF 100 M
80
-------
CO
O
W
n
CO
D
U
w
H
0.6
0.5
0.4
O 0.3
0.2
0.1
0
.2, .5, 1.= Bubble diameter
I
10
20
I —
70
30 40 50 60
HEIGHT ABOVE DIFFUSER (M)
Fig.69 WATEil FLOW VS HEIGHT ABOVE DUFFUSER FOR A FLOW RATE 'OF 1
L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 100 M
80
-------
CO
en
I I I
2, .5,1.= Bubble diameter (mm)
0 . ( ( .
30 40 50 60
HEIGHT ABOVE DIFFUSER (M)
Fig. 70 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 1. L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 100 M
-------
cr>
O
w
.30
.25
8 .
w
w
H
20
.15
s -10
.05
.2, .5,1.= Bubble diameter
(mm)
2.0
4.0 6.0 8.0 10.0
HEIGHT ABOVE DIFFUSER (M)
Fig. 71 CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE
OF 2 L/MIN AND BUBBLE DIAMETER OF 0.2, 0,5 AND 1.0 MM
AND AN INJECTION DEPTH OF 25 M
-------
co
0.7
0.6
§
H 0.5
i-l
. 4
Q
£0.3
u,
(-»
-------
0.12 pfZI |
.2, .5, l.= Bubble diameter (mm)
0.10
OJ
00
u
w
CQ
£
U
£
O
Pi
W
0.08
0.06
0.04
0.02
.2)
I
2.0
Fig. 73
4.0 6.0 8.0
HEIGHT ABOVE DIFFUSER (M)
WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF 2
L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 25 M
10.0
-------
CO
UD
o
2
O
U
S
w
g
W
CO
-------
.25
u
w
.20
O .15
w
w
H
55
w
O
10
.05
0
2, .5, 1.= Bubble diameter
(mm)
I
I
10
30
Fig. 75
15 20 25
HEIGHT ABOVE DIFFUSER (M)
CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE
OF 2 L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 50M
35
-------
2.0
w
CL,
PH
o
1.5
H 1.0
Q
i-l
.50
— .2, .5, 1.= Babble diameter (r;'T;)
10
30
Fig. 76
15 20 25
HEIGHT ABOVE DIFFUoER (M)
HALF WIDTL OF PLUME V,.> HEIGHT ABOVE DIFFUSER FOR A FLOW RAVE
OF 2 L/.VJrJ AND BUBBLE DIAMETER OF 0.2 , 0 .5 AND 1.0 Iv.M
AND AN INJECTION DEPTH OF 50 M
-------
CO
o
W
2
CQ
O
I
H
,40
.35
.30
.25
.20
.15
.10
.05
.2, .5, l.= Bubble diameter (mm)
I
I
5 10 15 20 25 30
HEIGHT ABOVE DIFFUSER (M)
Fig. 77 WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF 2 L/MIN
AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM AND AN INJECTION
DEPTH OF 50 M
-------
OO
o
£
o
o
s
w
1
w
CO
S
«
o
2, .5, 1.= Bubble diameter
(mm)
0.2 —i
0
15 20 25
HEIGHT ABOVE DIFFU3ER (M)
Fig. 78 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 2/L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 50M
-------
.25
Ji.
U
w
CO
U
o
w
OS
w
H
£
W
U
2, .5 ,1.= Bubble diameter
(mm)
.20
.15
.10
.05
1
I
0
Fig. 79
10
20
50
30 40
HEIGHT ABOVE DIFFUSER (M)
CENTER LINE VELOCITY VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE
OF 2 L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0
MM AND AN INJECTION DEPTH OF 75 M
-------
Cn
W
OH
PH
O
Q
PH
3.0
2.0
1.0
0
,2, .5, 1.= Bubble diameter (rnni)
I
0
10
50
20 30 40
HEIGHT ABOVE DIFFUSER
Fig. 80 HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR .A FLOW RATE
OF 2 L/MIN AND BUBBLE DIAMETER OF 0 .2 , 0.5, AND 1.0 MM
AND AN INJECTION DEPTH OF 75 M
-------
a>
0.6 __
0.5
u
w
§ 0.4 —
ca
£3
(J
£ 0.3
O
W
0.2
.2, .5, 1.= Bubble diameter (mm)
0.1 —
0
^^^^^^^^^^^^^^^^^^^^^^^^^^^
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 81 WATER FLOW VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE OF 2
L/MIN AND BUBBLE DIAMETER OF 0.2 , 0.5 AND 1.0 MM
AND AN INJECTION DEPTH OF 75 M
-------
1.8
>£>.
2, .5, !•= Bubble Diameter
mm)
20 30 40
HEIGHT ABOVE DIFFUSER (M)
Fig. 82 INCREASE IN OXYGEN CONG. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 2. L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM AND
AN INJECTION DEPTH OF 75M
-------
00
O
w
u
O
w
oi
w
H
Z
w
U
.20
.15
10
05
T
T
T
.2, .5, 1.= Bubble diameter
(mm)
t- I
10
20
70
30 40 50 50
HEIGHT ABOVE DIFFUSER
Fig. 83 CENTER LINE VELOCIT ; VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 2 L/MIN AND BUBBLE DIAMETER OF 0.2 , 0.5 AND 1.0
MM AND AN INJECTION DEPTH OF 100M
80
-------
w
o
f-T-i
H-;
H
Q
ffi
,2, .5, 1.= Rubble diameter (mm)
I -
0
Fig. 84
10
20
70
80
30 40 50 60
• EEIGKT ABOVE DIFFUSER (IV)
HALF WIDTH OF PLUME VS HEIGHT ABOVE DIFFUSER FOR A FLOW RATE
OF 2 L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0
MM AND AN INJECTION DEPTH OF 100 M
-------
.80
.64
U
w
.48
On
O
o
I-J
w
H
-------
2.0
O
0
2
O
o
s
w
1
w
CO I
< •
W )
&, •
U J
1.5
1.0
0.5
.2 , .5, l.= Bubble diameter
(mm)
40 50 60 70
HEIGHT ABOVE DIFFUSER (M)
Fig. 86 INCREASE IN OXYGEN CONC. VS HEIGHT ABOVE DIFFUSER FOR A FLOW
RATE OF 2/L/MIN AND BUBBLE DIAMETER OF 0.2, 0.5 AND 1.0 MM AND
AN INJECTION DEPTH OF 100 M
-------
FIELD DATA
ON
INDUCED MIXING AND CIRCULATION
152
-------
Field Data
on
Induced Mixing; and Circulation
Date; 8-10-1971
Diffuser Depth: 100 feet
Flow Rate: 2 l./min. Type of Diffuser: P.V.C.
Depth
(feet)
90
80
70
60
50
40
30
20
10
Sampling Location in
Feet from Diffuser
24'
F.R. Temp.°F.
2 65 1)
1 70
2.5 74 2 )
11 79
3 79.5
7.5 79.7
8.5 80 3)
39 30
2 80
4)
Remarks
F.R. = Fluores-
cence Reading
All readings on
30x scale unless
indicated other-
wise.
Reading taken 24
hrs. from injec-
tion time.
Approximate samp-
ling interval is
2-3 min.
153
-------
Date; 8-11-1971
Diffuser Depth: 100 ft.
Flow Rate: 2
l./fflin.
Type of Diffuser: P.V.C.
Sampling Location in
Depth
(feet)
90
80
70
60
50
40
30
20
10
2.0'
F.R.
4.5
2.5
0
0.5
1
0.5
0
0.5
0
Feet
12'
F.R.
6
0.5
0.5
0.5
0.5
0.2
0.4
0.5
0.5
from Diffuser
24'
F.R.
3
1 .5
0
0
0
0
0.5
0.5
0
Remarks
12'
Temp.°F.
65 Data collected
70 48 hours from
74 injection time.
79
79.5
79.7
80
80
80
154
-------
Date; 8-26-1972
Diffuser Depth: 100 ft.
Plow Rate: 2
l./min,
Type of Diffuser: P.V.C.
Depth
(feet)
90
80
70
60
50
40
30
20
10
Sampling Location in
2'
P.R.
3
7
2.5
3
-z.
^
5
7
1.5
3.5
Feet
12'
F.R.
5
1
3.5
6
5
5
7
2
5
from Diffuser
24'
P.R.
2.5
3
7
2
9
2
16
1
Remarks
Temp.°P.
70 Data Collected
75 24 hours after
78 injection.
79
79
79.5
81.5
8?
83.5
155
-------
Date: 9-16-1971
Diffuser Depth: 60'
Flow Rate: 0
.75 l./min.
Type
of Diffuser: P.V.C.
Sampling Location in
Depth
(feet)
60
50
40
30
.20
10
25'L
P.R.
10
8
4
1 1
8.5
3.5
12'L
P.R.
55(3x)
37
18
6
7
2
Feet
10'R
F.R.
70
20
8
8
2
1
from Diffuser
30'R
F.R.
58
3
13
7
5
6
50 'R
F.R.
24
17
60
8
4
3
100'R
F.R.
7
11
4
5.3
6
5
2'R
Temp.°F
80
80.2
80.5
80.7
81.1
81 .2
Remarks:
1 ) Lf R, = to the left and to the right of diffuser
respectively.
2) F.R. = Fluorescence reading
3) Sampling started two hours after injection.
156
-------
Date; 9-27-1971 Diffuser Depth; 60'
Flow Rate: 0,
,75
Type of Diffuser: P.V.C.
Sampling Location
Septh 25 'L
20'R
40 'R
In Feet from Diffuser
80 'R
100'R 0
30 'R
Temp.
OT^
(feet)
60
50
40 80
30 26
20 6
10 2
14
6
2
2
2
12
14
2
4
1
1
1
9
1
2
1
3
0
19
4 61
1 83
0.5 50
1 15
1 6
22
32
3
2
1
1
77.7
78
78
78
78
78
Remarks: Sampling started two hours from dye injection.
157
-------
Date; 10-1-1971 Diffuser Depth: 60 ft
Flow Rate:
.75 1.
/min
Type
of
Diffuser:
P.V.C.
Depth
(feet)
60
50
40
30
20
10
Sampling
25'L
4
1
0
0
0
6
15
61
10
0
0
Location in
20'R
0.5
6
0
2
0
0
Feet from Diffuser
40'
0
0
0
1
0
0
'L 80'L
1
0
0
1
0
0
10'L
0
1
0
1
0
0
Temp.
°.P
78
78
78.2
78.2
78.2
78.3
Remark s:
Sampling started two hours after dye infection.
158
-------
Date: 10-13-1971 Diffuser Depth: 60 ft.
Flow Rate: 2 l./min.
Type of Diffuser
: P.V.C.
Depth
(feet)
60
50
40
30
20
10
Sampling
25L 20R
20
17
27 16
39 16
65 20
32(1 Ox) 55
Location
40R
13
12
12
12
10
15
in Feet
70R
15
0
8
9
17
9
from Diffuser
100R
8
8
8
1 1
5
85
Temp.°F.
76
76
76
76
76
76
Remarks:
Sampling started 60 minutes from injection time.
159
-------
Date; 10-13-1971 Diffuser Depth: 60 ft.
Flow Rate: 2 l./mln. Type of
Diffuser:
P.V
.C.
Sampling Location In Feet from Diffuser
Depth
(feet)
60
50
40
30
20
10
20'L 40'R
10 30 60 120 40 80
min. rain. min. min. min. min.
39 1 3
32 12
3 60 27 8 30 12
(3x)
6 75 39 31 27 12
5 30 65 67 10
^2 10 15
(10x)
20'R
70
min.
20
17
16
16
20
(3x)
55
140
min.
6
4
5
4
27
10
0
130
min.
10
5
7
6
10
6
Temp.
°F.
76
76
76
76
76
76
Remarks:
Time (min.) from time of dye injection.
(-x) = Fluorescence scale
160
-------
Date; 10-18-1971 Diffuser Depth; 60 ft.
Flow Rate: 2 l./min.
Type of Diffuser:
P.V.C.
Sampling Location in Feet from Diffuser
Depth
(feet)
60
50
40
30
20
10
25'L
16 31 45
min. min. min.
8 35 32
Ox)
6 28 27
(10x)
7 45 30
15 78 25
40 'R
60 25 45
min. min. min.
7 5
6
24 8 4
5
12 5 6
41 8
60
min.
3
3
2
3
3
7
20R
67
min.
36
23
23
20
14
70R
75
min.
20
17
13
10
13
6
160R
82
min.
5
4
10
4
1
Temp.
°F.
75.8
75.8
75.8
75.8
75.8
75.8
Remarks:
Indicated time (min.) from injection of dye .
L,R = To the left or right of diffuser.
161
-------
Date; 10-24-1971
Diffuser Depth:
Flow Rate
: 1 l./min.
Type of Diffuser: Oerarclc
Sampling Location in
Depth
(feet)
60
50
40
30
20
10
Feet
25'L
10 20
1
0 1
1.5 64
(3x)
1 95
40 75
(10x) (1x)
from Diffuser
20 ' R 50 ' R
1 5 52 45
17 2
15 12 1
499
375
0
Temp.
74.9
74.9
74.9
74.9
74.9
74.9
162
-------
Typical
Date: 9-5-1972
Plow Rate: 2 l./mln
Start Up G0 Flow at
Distance Away
(feet)
12
12
12
12
12
12
12
7
7
7
•-7
f
V
1
V
7
7
7
7
7
7
7
•
1 2 : 30
Time
1 :55
1 :57
• :59
2:00
2:02
2:04
2:05
2:07
2:09
2:12
2:15
2:16.5
2:13
2:19
2:18.5
2:20.5
2:23
2:26
2:27.5
2:29
Depth, of Diffuser
Type of Diffuser:
Dye In;]ection at
Depth
75
'•O
60
50
40
50
20
80
70
60
50
40
50
20
80
70
60
50
60
70
: 90 ft..
P . V . C .
1 :45
?.R.
15
10
•n
1
2
\
2
40*
25*
2
2
O
t_
1 .5
2
1C*
35*
2
2
1 5*
12
163
cont.
-------
Distance Away Time Depth j\R,
(Feet)
7
7
7
7
7
7
2: 51
2:33
2:34
2:37
2:39
2:40
cO
50
3C
70
n
50
2
1 .5
-,ote: All readings from zero to •. 0 were beu/een cne and two,
•»• went down to 2 after 15 seconds t:ien started cin:. up a ain
164
-------
Typical
Late: 14-
?low Rate
Start Up
Depth of
5-1972
: 2 l./min.
00 Plow: 2:15
Sampling: 75 ft.
Diffuser Depth:
Type of Diffuser
Distance from Di
Dye Injection at
90 ft.
: P . V . G .
ffuser: 40 '
3:22
Continuous Sampling Gate (30x)
Time
3:40
3:50
3:55
4:05
4:08
4:1 1
4:12
4:13
4: 14
4:15
4:1C
4:16
4:17
4: 19
4:20
4:24
F.R.
3
12
14
13
17
14
20
22
20
16
10
1 3
10
1 1
5
8
•Jlme
4:27
4:29
4:^0
4: vl
4:33
4: -36
4: -:3
4:41
4:42
4:44
4:48
4:52
4:53
5:01
5:07
5: 10
?.H.
11
1 3
14
8
13
1 6
21
20
22
U;
1 ;
20
24
20
20
4:
7
5:
18
4:26 10 b:40 1 -.
i:ote: Measurements at deptns for surface to ::.0' were between
1 and ' ?.R. (Fluorescence Readin;'). Temp, profile is shown
in Figure 5-3.
165
-------
PROGRAM LISTING
AND
USER'S GUIDE
166
-------
PROGRAM LISTING
and
USER'S QUIDS
The listing included hereafter is for Model One
(mass transfer) with slip velocity, However, the equations
for zero slip velocity consideration are also listed.
To run the program as it stands one need first (if
not at U.T.) to delete those statements for the PLOTT sub-
routine and 'include his plotting sub-routine or add a print
statement to get a listing of the output. However, a listing
of PLOTT sub-routine is available from U.T. Computation
Center. The user needs to do the following:
1) Set the value of the depth of injection in program HYPO-
DEPTH 1 - in meters.
2) Set the bubble diameter - DIAM 1 in mm.
3) Set the value of 1 in HYDRO equal to DEPTH 1 + 1 .0 in
meters.
4) Set the value of Omega in HYDRO to whatever level needed.
( Omega & Z) in meters.
5) set BD = DIAJVI 1 .
6) Set PR - the oxygen flow rate in m. /sec.
7) Set the value of Alpha = initial value of x in meters.
8) Set YJ(1) = initial half-width of plume in meters.
9) Set YJ(2) = center line velocity in m./sec.
To run the program with zero slip condition insert
the corresponding equations, DP(M,1), DF(M,2) in place of the
167
-------
listed equations and delete Function V2L3 (DIAM).
To run the program with no mass transfer, start the
program with HYDRO.
This program has been run on CDC 6600 and 6400.
168
-------
LIST OF ABBREVIATIONS
********************
DIAM1 a DIAMETER OF BUBBLE CMMJ
DEPTH1 B DEPTH OF INJECTION(M)
ZMGO cMG OF 02
ZMGN«MG OF N2
ZMGC«MG OF C02
T02L*»FRACTION OF ORIGINAL AMOUNT OF 02 REMAINING
RNTO « NITROGEN/OXYGEN
XMGC E RELATIVE AMOUNT OF 02 IN BUBBLE NORMALIZED TO CONSTANT
RELEASE DEPTH
CSTO»CSAT OF 02 (MG/L)
CSTC sCSAT OF C02 CMG/L)
PC02 ^RELATIVE AMOUNT OF 02 IN BUBBLE NORMALIZED TO CONSTANT
RELEASE DEPTH
T02A sFRACTION OF 02 ABSORBED
KLOZ BOVER ALL MASS TRANSFER COEFFICIENT (025
BD "BUBBLE DIAMETER (MM) «DIAM1
HO «ATM PRESSURE CM)
Z * DEPTH OF INJECTION + 1.
FR »02 FLOW RATE (CUB, M./SEC.3
Q fe NUMBER OF BUBBLES PRODUCED /SEC,
VEL "CENTER LINE VELOCITY OF PLUME
RAD« HALF WIDTH OF PLUME
WO = WATER FLUX
WM = MOMENTUM FLUX
KE a KINETIC ENERGY FLUX
02T » INCREASE IN D.O,
169
-------
C
C
C
c
C
c
c
c
PROGRAM HYPO (INPUT, OUTPUT)
THIS PROGRAM CALCULATES THE TRANSFER OF OXYGEN OUT OF AND THE
TRA\SFER DF NITROGEN INTO A BUBBLE INJECTED AT THE BOTTOM OF
A RESERVOIR --- INITIALLY THE BUBBLE is PURE OXYGEN
TrE UMTS FOR THE DEPTH AND THE DIAMETER ARE METERS AND
RESPECTIVELY
REAL
REAL
REAL
REAL
REAL
REiL
REAL
RfciL
REAL
KGCZ
HGCZ1
KLOZ
MGOZl
MGOZ
MGNZ1
COMMON /!/ TEMP
DI HENS ION DPTH(100fl)fDYAM<10C»0),ZMGOU0r0),ZMGN(1000)
DIMENSION T02L(ie>00),PC02U0&P>,RNTOn00n,DPTNUe00)
DIMENSION X«GO ( 1 000), Z*GC ( 1 000 ) ,CSTOU 000) ,CSTC( 1000)
AREA(13,5l),YSCALE(51),T02A(1000)
NFHAXsl
NF»i
MY«0
MX*0
MH B 1
XLIK-EB0.0
LL = 1
NtS = 36
MCL s 80
NDMAX&100
NCTP *0
TEHP=20.
DIAH B a.e
DO i I s l, 100
DIAM s DIAM+,1
ZMGOCI)=
-------
TO » (TEMP*(.01M*T£Mp«l,6088)+69.528>/1000000.
TN = (TE*P*(.01ie9*TEMP-.87a6i)+30.5335/1000000.
TC = CTEMP*(1.2043*TEMP-106,231«U3319.657)/1000000.
T K 273./C273.+TEMP3
PI = 3,1416
CNATO « 3.0*28,3
CMTN «i 15.5E-6
CwATC=0,00
BICM02 a 1.0
BICNN2 a 0.0
BICNC02 = :.-BICM02-BICNN2
DEPTH1 a a5.0
PRINT 13
DEPTH = OEPTH1
OIAM1S2.
DIAM cOIAMl
STEP «0.1
PRINT 23, I, DEPTH, DIAM, TEMP
PRINT i«
KTKNT a DEPTH/2.0
IF (KTKNT ,EQ. 0) KTKNT a 1
NCTP»0
DEEP1 s DEPTH
DEPTH cDEPTHl
DIAM aDIAMl
ATM = 1.+DEPTH/10.3632
VOLBB CC1./6.5*PI*CDIAM**3))M00.
MGOZ s VOLB*ATM*BICN02*T*l.a28SE-3
MGOZ1 « MGOZ
MGNZ « VOL8*ATM*BICNN2*T*l,2500E-3
MGNZI » MGNZ
MGCZ » VOLB*ATM*BICNC02*T*l,9642E-3
MGCZi s MGCZ
VOLOZ = VDL8*BICN02
VOLNZ = VOL0*BICNN2
VOLCZ = VOLB*BICNC02
PCOZ » BICN02
PCCZSBICNC02
PCNZ*BICNM2
CSATO s ATM*PCOZ*TO
CSATC = ATM*PCCZ*TC
VOL1B «VOLB/100.
IF (DIAM ,GT. 2.0) GO TO 3
ARE »PI*(DIAM**2)*100.
GO TO a
E • ,125*DIAM-.25
A*EXP(AUOG(3,*VOLlB/(a.*PI*SORTCl.-E**2)))/3.)
ARE = C(PI*A**2) *(Z.+(l.-E**2)*ALOG(Cl.+E)/(lt»E))/E))*100.
CONTINUE
IF(NCTP.LT.NOMAX)GO TO 5
KTKKT * 2*KTKNT
171
-------
GO TO 2
5 CONTINUE
IFCDIAM.GE. 0.fl ,AND. DIAH .LE, .72) VEL"2«.0*DIAM*0.01
IF(D!AH .GE. .72.AND.DIAM .LE. 5,) VEL"( 17.6+C ( OIAM«0,72)*1000. )
l/(30.+15b.*CDIAM-0.72)))*0.0i
3fl CONTINUE
STEP B0.1
C
7 00 9 KT » 1» KTKNT
NCTP sNCTP+J
OLEFT « MGOZ/HGOZi
02*B«1.-OLEFT
PC02 a VOLOZ/VOLB
PCNZ « VOLNZ/VOL6
PCC2 a VOLCZ/VOL8
RNZOZ » MGNZ/MGOZ
OPTH(NCTP)aDEPTH
DVAM(NCTP)*DIAM
T02LCNCTP)»OLEFT
T02A(NCTP)=02AB
ZHGO(NCTP)«MGOZ
CSTO(NCTP)«C8ATO
CSTCtNCTP)»CSATC
PC02(NCTP)»PCOZ
RNTO(NCTP)«RNZOZ
XMGO(NCTP)»MGOZ/MGOZ1
OPTN(NCTP)«OEPTH/DEEP1
ZMGCCNCTP)«MGCZ
CSATO * ATM*PCOZ*TO
CSATW a ATM*PCNZ*TN
CSATC » ATM*PCCZ*TC
DMOZ « KLOZ(OIAM)*ARE*(CSATO-CWATO)*(STEP/VEL)
MG02 * MGOZ-OMOZ
IF (MGOZ .LE, 0.0) GO TO 10
KLNZ s ,89*KLOZ(OIAH)
-------
IF (VOL18.ST. ii. 1555) D I AM=EXP ( ALOG ( 6 ,*VCLl B/PI) /3. )
IF CDIAM ,GT. 2.?) GO TO 6
ARE =PI*(DIAM**2)*100,
GC TO 9
E = «l25*DIA*-,25
AKE =
-------
CALL HYDRO(DYAM,T02A)
C *****************
13 FORMAT (2X*LABFl*,7X*RELEASE DEPTH*,10X*DIAMETER*,i0X*TEMPERATUR
1E*3
14 FORMAT <*!*)
15 FORMAT £3<»X*MG OF 02 AS A FUNCTION OF DEPTH IN METERS*)
1*. FP»*ATC33X*MG OF N2 AS A FUNCTION OF DEPTH IN METERS*)
17 FCRMAT(2?X*FHACTION OF ORIGINAL AMOUNT OF 02 REMAINING AS A FUNCTI
ION OF DEPTH IN METERS*)
18 FORMAT (3PX*DIAMETER,IN MILLIMETER,OFBUBBLE AS A FUNCTION OP DEPTH
1*)
1
2« FORMAT (20X *RELATIVE AMOUNT OF 02 IN BUBBLE NORMALIZED TO CONST**.
IT RELfcASE DEPTH*)
26 FORMAT (3feX* MGOF C02 IN BUBBLE AS AFUNCTION OF DEPTH(M.)*)
31 FORMAT(3!»X*CSATC VS. DEPTW(M)*)
32 FORMAT C30X*CSATO (MILLIGRAMS/CUBIC MILLIMETER) VS. DEPTH*)
END
174 '
-------
FUNCTION! KLOZ fDIAM)
REAL KLOZ
COMMON /!/ TEMP
IF (DIAM .GT. 0,0) 60 TO 1
KLOZ«0.013
RETURN
T a 1.028**(TEMP-20.)
IF (OIAM ,GT. 2.2) 60 TO 2
KLOZ«0. 0134-DI AM* (0,04088 + 0. 0962*0 T AM )*T
RETURN
IF COIAM .GT, 2.5) GO TO 3
KLOZ B ,5555*T
REtURN
KLOZ » (400,/(a32.*OIAM«360.))*T
RETURN
END
175
-------
SUBROUTINE HYDRO CDYAM,T02A)
DIMENSION YJ(1B)
DIMENSION T02A (1 000) ,02T t 1000) ,DYAM( 1000)
DIMENSION AREAUS,51),YSCAL£(51)
COMMON G, Q, HO. 91, S, Z, C
DIMENSION RAD c 1000 ), VELH000) » «Q ( 1000 )>WM{ 1000), KE( 1000)
COMMON/CSAVE/3AVEC2000)
DOUBLE SAVE
COMMON/ XS A V/DEPTC 1000)
REAL KE
EXTERNAL F
NDATA*301
NDMAX*}01
ISYMBL*1H1
XL1NE*0.0
YLINEB0.0
MX80
NCL*80
NLSB36
ALPHA*!.
G»9.815
BD«2(
H0»10.362
OMECA«3l,
Z*A6.
s«e.2
C«0.03
HS0.1
YJ(l)e0.1
YJ(2)s0.2233
HMlN«1.0E»6
IPR*0
HMAX»1.0
CALL R« AM (F,YJ, ALPHA, OMEGA rH,IPR,HMIN,HMAX,EM!N, EM AX, DYAM)
176
-------
Ks2*MDATA-l
II =1
DO 30K 1=1, K, 2
RACCII3 *SAVE(I3
VELCIIJs SAVEtI+1)
II =11+1
DO S20 I«1,NDATA
*cm=PI*(VELCI33*CRAD(I5)**2
w ( I ) = ( PI *CRADm**23*(VELm**23* 1000.3/2.
KE(I3 =
-------
SUBROUTINE RKAM IF, YJ, ALPHA, OMEGA, H, IPR, HMlN, HMAX, E*IN, EM
1AX,DYAM)
C*
c*
C*
c*
C*
C*
C*
C*
C*
C*
C*
C*
C*
C*
ADAMS-MOULTON PROCEDURE WITH RUNGE-KUTTA STARTER ********
F -• USER-SUPPLIED SYSTEM
ALPHA -- INITIAL VALUES OF INDEPENDENT VARIALBE
EXIT POINT SINGLE-STEP ERROR
OMEGA -- TERMINAL (EXIT) VALUE OF INDEPENDENT VARIABLE
M — INITIAL (EXIT) VALUE OF STEP SIZE
IPR — PRINTING INDEX(£VERY IPP-TH STEP PRINTED)
ERROR ANALYSIS PARAMETERS
HHlN --
EMAX --
SET HMIN
SET EMIN
SET EMAX
STEP SIZE
MAXIMUM STEP SIZE
MINIMUM SINGLE-STEP ERROR
MAXIMUM SINGLE-STEP ERROR
s HMAX FCR NO ERROR ANALYSIS
= 3. FOR NO DOUBLING OF STEP SIZE
= A LARGE VALUE FOR NO HALVING OF STEP SIZE
DIMENSION
1(4,4),
DIMENSION
X(7), Y(7,1P), YJ(IB), 0(7,12), DF(7,10),
BP(0), BC(4), PHK10), YS(7,10)
DYAMC100B)
Y3
xsm, *(u), B
DOUBLE SAVE
COMMON/ XS A V/DEPT( 1000)
DOUBLE PRECISION Y, PHI,
SAVECi)s0.1
IF (EMJM .GT. EMAX .OR. HMlN .GT, HMAX) 20,10
10 IF (HMlN .EO. HMAX) GO TO 30
IF (HKIN .GT, H .0*. HMAX .LT, H) 20,30
20 J'RI.N'T 520
PSINT 5t0, HMlN, HMAX, EMlN, EMAX
P?INT 550, ALPHA, OMEGA, M
RETURN
; ....... COEFFICIENTS
KK B
A(2)
A CO)
6(2,
B(3,
BU,
10 =
FCT
4
= A(3) «
s i .
i) • B{3,
1) s 9{«,
3) » i.
a
= 19./270.
,5
2) * .5
i) * Bca,
2)
BP(1)
178
-------
BP(2) s 37./24.
BP(3) a -59./24.
BP(4) B 5S./24,
BC(l) » 1./24.
BC(2) « -S./24,
BCC3) a 19./24.
BC(4) a 9.X24.
C ....... RKAM ENTRY POINT
CALL F (ALPHA,YJ,1,OF,N,DYAM(II))
IF JIPR) 40,70,40
40 PRINT 530
IF (HMIN eEO, HMAX) 60,50
50 PRINT 540, HMIN, HMAX, EMIN, EMAX
60 P^INT 550, ALPHA, OMEGA, H
PRINT S60, {YJ(I>, I « 1, N)
70 ITC » 1
1QM1 = IO1
1QP1 « IQ*1
IQP s 2*IQ-i
ISTP = a
SIGN a 1.
IF (H .LT. 0.) SIGN s -1.
x(i) a ALPHA
DEP7U5 =ALPHA
00 60 I s 1, N
80 Y<1, I) s YJ(I)
90 MM = i
IFLG a 0
100 KCOUNT » 0
110 M = MM
MM a M+l
IF (MM .GT. IGP) MM = 1
X(MM) a XCM3tH
OEPTCII) cX(MM)
IF (ABSCOMEGA-X(MM)) ,LE. SIGN*H) 120,150
120 IF (A8SCOM£GA-X(MM)) §EO, SIGfJ*H) 140,130
130 H s OMEGA-X(M)
I FUG a 0
X(MM) a X(M)+H
OEPTCII)sX(MM)
140 ISTP s 1
150 XJ = X(M5
00 160 I * I, N
160 YJ(I) » Y(M, I)
IF (IFLG ,NC, 0) GO TO 270
C ....... RUNGE-KUTTA PROCEDURE
DO 220 K s 1, KK
IF (K ,£Q. 1) GO TO 200
XJ s XCM)+H*A(K)
179
-------
DO '.70 I a 1, N
170 PHI(I) a 0,000
KM! s K-l
DO 1*0 I » 1, N
00 I8fc J s i, KMi
180 PHICX) » PHI(I3*H*B(K, J)*DCJ, I)
19? YJ(I) » VCM, D+PHICI)
200 CAUL F(XJ,YJ,K»D,N,DYAM(II)J
IF (K ,K-Eg i) GO TO 220
DO 2i0 I * i* N
210 Or' CM, i: - 0(1, I)
220 CONTINUE
00 230 I * i, N
230 PHlCX) a e,«500
:vO 250 I s <, H
JO 2^0 K a ; , KK
2«0 r»Hl»Y(MM,I)
JJaJJ+1
250 CONTINUE
IF (MM .EQ. 10) IFLG « 2
IF (!TC-ITC/!PR*IPR ,NE. 0) GO TO 260
^Si.\V 57i3f XCMM), H
P^INT 560, (Y(MM, I), I « 1, fJ)
260 *TC » ITC*1
If (1ST? ,EQ, i) GO TO 510
GO TO 110
; _...^.. i^OfeMS PREDICTOR-CORRECTOR PROCEDURE
270 C^LL F(XJ,YJ,M,OF,N,DYAM(II))
DO 280 i B 1, N
280 PHia) a 0.0DB
CO 290 :•: = 1* 10
J t= K4-KCOUNT
IF ^J w^Y, IQP) J = J-IQP
00 ?
-------
SINGLE-STEP ERROR
IF CHMIN ,EQ, HMAX) GO TO 500
OLTMX « 0.
DO 340 i • i, N
VMM » YCMM, I)
DtT a ABSCYMM-YJCI5)
IF CDLT ,LE. DLTMX) GO TO 340
OLTMX a OLT
IDLT s I
340 TEST B DLTMX/YCMM, IDLT)
IF QUOTIENT OVERFLOW 350,360
350 SSE a ABS(FCT*DLTMX)
GO TO 370
360 SSE a ABS (FCT*TEST)
370 IF (ITOITC/IPR*IPR ,NE, 0) GO TO 380
PRINT 590, XCMM), H, SSE
PRINT 580, CYCMM, I), I « 1, N)
380 ITC c ITC+1
C ...-.-• ERROR ANALYSIS
IF CEMIN ,LT. SSE .AND. SSE ,LT. EMAX) 60 TO 500
IF CSSE ,GT, EMAX) 390,420
390 H a H/2.
IF (SIGN*H .LT. HMIN) 490,400
400 IF (IFLG .EG. 2) GO TO 90
M * M-i
IF CM ,EQ. 0) M s IQP
X(l) « XCM)
DO 410 I * i, N
410 Yd, I) * YCM, I)
GO TO 90
420 IF CISTP .EQ, 1) GO TO 510
H * 2.*H
IF (SIGN*H .GT. HMAX) 490,430
430 IF CIFLG .EQ. 2) GO TO 90
L * 0
DO 470 K « 1, IQ
J = IQP1-K
M * MM-L
IF (M .LE. 0) 440,450
440 M = M+IQP
L « 0
MM » M
450 IF CX(M) ,LT, ALPHA) GO TO 90
XS(J3 » X(M)
DO 460 I •
1* N
D(J, I) - DFCM, I)
460
470
YSCJ, I) a
L = L + 2
DO 480 K *
YCM, I)
1, IQ
XCK) c XSCK)
181
-------
DO U8P I s 1, N
DF(K, I) s D(K, I)
Y(K, I) = YS(K, I)
MV r JO
IFLG s i
GO TO 180
PRIM 622, H
PRINT 5U3, HHIM, UMAX, EMJN, EMAX
GO TO 513
50? IF (ISTP .EC, 1) GO TO 510
IFLG = 1
KCOUMT s xCOUNT+1
IF (KCOUNT .EC. IQP) KCOUNT » 0
GO TO 110
c .—. RKi^suB EXIT POINT
51? 0MEG4 s XJ
IF (HHlN .ME. HMAX) ALPHA a SSE
RETURN
52e FORMAT c *0 Rx*1" -- INITIAL PARAMETERS INCORRECT*)
53? FORMAT c*e •- R**M ••*>
540 FCRPAT (*B HMlNs*E15.8,2X,*HMAXB*E15.8,2X
1*E15.8)
553 FORMAT (* ALPWA=*E15.8,2X,*0«EGAa*E15.8,2X,*Ha*El5.8)
562 FORMAT (* VJ AR»AV * /(2X,7C2X,E15.8)))
572 FORM*T (*0 X a *E15.8,5X,*H s *E15,85
FORMAT (* YJ ARRAY * /(2X,7(2*,015.8))}
FORMAT C*0 X s *El5.8,5X,*w a *E15,8,5X,*SSE * *E15.8)
FORMAT (*0 RKAM -- STEP SIZE OUT OF BOUNDS -- Hs*E15.8)
END
182
-------
SUBROUTINE F(X,YJ,M,DF,N,DYAM)
J G, Q, HO, PI, S, Z, C
OY DF(7, I?), YJU0)
9 ) /6 . 5 / ( PI * Y J ( 2) * C Y J ( 1J **2
)))-C2.*C*YJ(2)/YJ(D)
RETURN
END
183
-------
FUNCTION VELB(OYAM)
IFCDVAM.GE. 0.72 .AMD. OYAM .(.E. 5.)GO TO 10
VELB *2«.a*DYAM*0.01
RETURN
10 VEIB«U7,64
RETURN
END
184 OU.S. GOVERNMENT PRINTING OFFICE: 1574 54o-3i4, 208 1-3
-------
SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Re
tfa.
W
TVk-
Hypolimnion Aeration with Commercial Oxygen -
Vol. I - Dynamics of Bubble Plume
.Speece, R.E.; Rayyan, F; Murfee, G.
The University of Texas at Austin
Austin, Texas 78712
5. R ittD
6
•7. P<~ 'cTmi" Ozgz ition
16080 FYW
13
JPerioa ^'
-nd
I?.
Environmental Protection Agency, report number
EPA-660/2-73-Q25a, December 1973.
I-.,. .4BJ(/Lct This study deals with a proposed scheme for restoration and maintenance of
dissolved oxygen in the hypolimnion of stratified impoundments without disturbing the
stratification. Commercial oxygen is an economical alternative to air as an oxygen sourc
for hypolimnion aeration. In addition, it possesses many other advantages, the main one
being avoidance of nitrogen gas supersaturation and its related toxicity to fish.
Laboratory and lake studies were conducted to demonstrate the gas transfer
dynamics of a bubble plume generated by the injection of pure oxygen bubbles within the
hypolimnion. The experimental results verified the practicality of the original concept -
hypolimnion aeration which preserves stratification. Efficient oxygen absorption was
achieved within the hypolimnion.
Mathematical models were formulated and calibrated by experimental data.
The calibrated model was then used to predict the oxygen transfer and hydrodynamic
characteristics of the bubble plume for various oxygen injection rates and injection depths
The sensitivity of the model to the various input parameters was shown.
J7a. Descriptors
Water Quality Modeling, Hypolimnion Aeration, Stratification, Water
Quality Control
I7h. Identifiers
Lake Travis (Tex.)
CCWKR Field S- Group
IS
19. S 'jrityf -ss.
(Report)
20, Seciui .y Class.
21. : . of
Pages
2t. Price
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON. O. C. 2O24O
R. E. Speece
I Iilsr,,u
------- |