Behavior of Air Bubble Screens

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CIVIL ENGINEERING STUDIES

HYDRAULIC ENGINEERING RESEARCH SERIESNO. 33

BEHAVlOR

OF

~IR BUBBLE SCREENS

By

S. TEKELI

end

W. H. C. MAXWEll

A Technical Report of Research Sponsored by the NATIONAL SCIENCEFOUNDATION

Under Grant No. ENG 76-24226

DEPARTMENT OF CIVll ENGINEERING

UNIVERSITYOF IllINOIS AT URBANA-CHAMPAIGN

URBANA, IlliNOIS

SEPTEMBER1978

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BEHAVlOR OF AIR BUBBLE SCREENS

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by

S. Tekeli and W.R.C. Maxwell

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1 ,

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A Technical Report of

Research Sponsored by the

NATIONAL SCIENCE FOUNDATION

Research Grant No. ENG 76-24226

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DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

URBANA, ILLINOIS

SEPTEMBER, 1978

26 -

G

-'l1

UILU-ENG-74-2019

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ii

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Abstract

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BEHAVIOR OF AIR BUBBLE SCREENS

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S. Tekeli and W.R.C. Maxwell Depa~tment of Civil Engineering University of Illinois at Urbana-Champaign

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Bubble screens have long been used for a variety of purposes,

rang-I

ing from control of stratification,icing and wave heights, to inducing artificial mixingdensity currents, oil spills, shoaling,for augmenting convective

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heat and mass transfer rates, aeration and water quality control~ In spite of

$uch potential, bubble screen applications have been rather limited due to lack

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of proper design information and criteria. Thus inspired, analytical and

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experimentalof the physics of bubbleinvestigationsscreens.were conducted to attain a thorough understandingDimensional and similitude considerations

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were employed to develop the necessary requirements for physical modelling of

such screens. This led to development of a set of sealing laws, into which a

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scale factor was introduced to account for distortion of similitude

require-I

ments.literatureThe scale laws were calibrated using the data availableas weIl as those collected for this program. Finallyin thea mathematical

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model, with provisions for bubble compressibility and relative mot ion in the

flow, was developed to predict the behavior and performance of both 2- and

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3-Dimensional bubble screens. Analogy with simple plumes proved to be

fruit-I

ful at various phasesAn experimentalof the above investigations.program was initiated to extend the range of the

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available data for evaluation of the sealing relationships and to determine

some of the inputs for the mathematical model. Such parameters as bubble size

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and slip veloeities were measured, along with the induced buoyancy and velocity

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distributions,data analysis technique.using hot-filmA set of preliminaryanemametry and a speciallyexperiments was followed bydeveloped digital

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iii

the final experiments, which included both a set of 3-Dimensional and a set of 2-Dimensional screens.

The major conclusions were that voids ratio distributions may be adequately represented as Gaussian, the relative spread rate for void ratios,

À, ranges from 0.45 to 0.70, and slip velocities are essentially constant in the entire flow. Moreover, scaling relationships which dep end on source strength (i.e. densimetric Froude number) and a scale factor incorporating relative pressure and relative submergence were developed. These permit transposition of physical or mathematical model results to field scale.

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iv

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Plate I Overall view of 30-ft long by 15-ft wide

by 4-ft deep test basin and associated equipment.

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Plate 11 Close-up view showing separation of personnel carriage on the left and instrument carriage on the right.

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v ACKNOWLEDGMENTS

This investigation was sponsored by the National Science

Foundation under Grant No. ENG 76-24226. Preliminary work leading to the present study was conducted as part of the research project entitled "Mechanics of Heated Surface Discharges to Rivers - Phase 11," supported by the Office of Water Research and Te chnology •

Special thanks are extended to Professor V. J. McDonald, who, in addition to providing unlimited access to the data acquisition and pro-cessing equipment used in this study, patiently spent many hours in

adapting the digital analysis programs for the minicomputer used. Finally, he critically reviewed Chapter 4.

The report is essentially the same as the doctoral thesis entitled "Air Bubble Screens" submitted by S. Tekeli. Mr. J. W. Miller played a major role in the construction of the experimental facilities. Ms. H. Dillman and Mrs. M. Johnson typed the report. The assistance of these persons is greatly appreciated.

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vi TABLE OF CONTENTS ABSTRACT PLATES I AND II ACKNOWLEDGMENTS LIST OF TABLES • • •

. ..

.

. .

LIST OF FIGURES LIST OF SYMBOLS 1. INTRODUCTION • . 1.1 Problem. • • 1.2 Objectives

2. REVIEW OF PHYSICAL PROCESSES AND RELATED LITERATURE 2.1 Generation of Bubbles •...•••

2.2 Buoyant Plumes vs. Bubble Screens ••.• 2.2.1 Turbulent Buoyant Jets and Plumes

2.2.2 Bubble Screens .

2.3 Review of Literature on Bubble Screens 2.3.1 Experimental Work •••••

2.3.2 Modelling of Bubble Screens

2.3.2.1 Physical Modelling ..••• 2.3.2.2 Analytical Modelling .

..

. . . 0. 3. THEORETICAL DEVELOPMENTS . . . • • • • • 3.1 Dimensional Considerations .•.••

3.1.1 Bubble Generation and Estimation of Terminal Velocities . . • • . . 3.1.1.1 Bubble Generation ...

3.1.1.2 Terminal Bubble Velocities ..•.•. 3.1.2 The Induced Bubble Plumes

3.2 Modelling of Bubble Plumes 3.2.1 Physical Modelling Laws 3.2.2 Mathematica! Modelling .

3.2.2.1 Zone of Established Flow (ZEF) ...• 3.2.2.2 Entrainment in Turbulent Jets

and PLumes •• • • • • • • • • • • • •

3.2.2.3 Solution of the Equations ••• 3.2.2.4 Zone of Flow Establishment (ZFE)

4.

EXPERIMENTAL PROGRAM • • • . • • • • • . • • •

4.1 ExperimentalObjectives .••••••••.

4.2 Development of the Experimental Technique • 4.2.1 Thermal Anemometry in Isothermal,

Two-Phase Flows • • • • . . • • • . 4.2.1. 1 Gene ral Review . • . • 4.2.1.2 Application in this Study •

Page ii iv v viii ix xii 1 1 3

4

5 7 7 9 10 10 14 14 16 24 24 24

25

26 26 31 31 36 36

44

47 49 51 51

52

54

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vii 4.2.2 4.2.3 4.3 4.4 4.5

Characteristics of Sensor Response .

Data Acquisition .

4.2.3.1 Averaging Period . 4.2.3.2 Heasurement Locations . Data Processing (Digita1 vs. Ana10g

Ana1ysis) .

4.2.4.1 Digitization Theory ..

4.2.4.1.1 Quantization . 4.2.4.1.2 Timing

4.2.4.2 Power Spectrum Data Reduction . . . .

4.2.5.1 Program-I: Digita1 Bubb1e Search .

4.2.5.1.1 F1uctuation Thresho1d 4.2.5.2 Program-II: Characteristic Bubb1e

Information . . . . 4.2.5.3 Program-lIl: X-Sectiona1 Profiles

Experimenta1 Procedure

Pre1iminary Experiments and their Imp1ications

Experimenta1 Apparatus ... . •

4.5.1 Reservoirs and Accessories

4.5.2 Air Supp1y and Injectors. 4.5.3 Ca1ibration System for Probes

4.5.4 Anemometer and Data Acquisition Equipment .. . . •

4.5.5 Digitization and Data Reduction Equipment

4.2.4

4.2.5

5. PRESENTATION AND DISCUSSION OF RESULTS . 5.1 Assessment of Digita1 Bubb1e Search Technique 5.2 Resu1ts of Experimenta1 Measurements

5.3 Eva1uation of Some Screen Characteristics from

Experimenta1 Data . . . . 5.4 Sealing Procedure ..•...•

5.4.1 Sealing Re1ationships . 5.4.2 Imp1ications of the Sealing Re1ations 5.5 Resu1ts of Ana1ytica1 Mode11ing

6. CONCLUSIONS AND RECOMMENDATIONS

REFERENCES

.

..

.

..

. .

.

.. .. .. .. ..

APPENDIX

1 Surface Tension and Compressibi1ity Effect .

2 Reyno1ds Number Effect

3 Procedure for Spectral Computation • 4 Experimenta1 Data

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Page 55 60 60 66

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70 72 72 72

73

76

79

83 I

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86

88

90

92

93 93 95 98

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101 103 104 104 115

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132 132 132 144 150

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164 I

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169

176

178

179 I

182

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Tab1e 3.1

4.1 I

4.2

4.3 I

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5.1

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viii LIST OF TABLES Page

Values of a and À for 2- and 3-D Jets and Plumes

45

Heat Transfer Characteristics of Air and Water .

55

Optimal Sampling for Various Percentages 65

Effect of Injector Size on Bubble Plume Behavior (3-Dimensiona1 Plume:

Q

_

0.00225 cfs STP and

a

H

=

3.54

ft) . . . 94

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Figure 2.1

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3.1

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4.2A-E4.1

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4.3 4.4

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4.5

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4.6 4.7

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_4.8

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4.9

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4.10

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4.llA-B_4.12

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4.13 4.14

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4.15

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4.164.17

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4.18 4.19

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_4.20

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ix LIST OF FIGURES Page Schematic Flow Field for a Bubble Plume • 13

Effect of À on the Buoyancy Terms • 43

A Typical Bubble Traverse . 56

Some Bubble Shapes 58

Location of the Measurement Points for Test 7 62

Effect of Averaging Time 63

Continuous, Fractional and Optimal Sampling

(Test=ll, Point=30) . •. .••. 65

Autocorrelation Functions for Test 7 67

Comparison of FS and OS: Effects of Partial

Record Length and Averaging Time • • . • • . 68 Comparison of FS and OS: Effects of Partial

Record Length and Sub record Length T • • • •

s 69

Location of Cross-Sections and Measurement Points • 71 Power Spectra for Air, Water and Mixture Phases

in a Bubble Plume •••.••.••••.••. 75

Effect of Digitization Rate on Bubble Detection . 77 Fluctuation Threshold vs the Local Air Discharge 85 Statistical Distribution of Bubble Sizes 87 Schematic Diagram of Experimental Setup • 91 Air Injectors Used in the Preliminary Experiments 96 Injectors for Point and Line Source Experiments • 97 Schematic Diagram of Velocity Calibration Setup 99 Calibration Curves for Velo city Measurements 100 Anemometer and Data Acquisition Circuitry (One

Channel) ...•... 102

Digitization (---) and Data Analysis Circuitry

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Figure 5.1 5.2 5.3 5.4 5.5 5.6A-B 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 x

General Characteristics of Flow-Bubble-Sensor Interaction

Bubble Size and Velocity versus the Traverse Time . Effect of Threshold Value E

c

Concentration Effect (when compared with Fig. 5.1) Effect of Flow Velocity (when compared with Fig. 5. 4) •••••••••••••..•. • • . • • • Probe Shape Effect

Shape of the Measurement Probes •

Velocity Distributions in a 3-D Bubb1e Plume (Test 10) • • • • • • • • . . • • • • • . • • Velocity Distributions in a 2-D Bubble P1ume

(Test 13) . • • • • • • • . • • • • • • • • • . Void Ratio Distributions in a 3-D Bubb1e Plume

(Test 10) . . . . . . . . . . . . . . . . . . . Void Ratio Distributions in a 2-D Bubb1e Plume

(Test 13) . • • . • • . • . • . • . • • . • Decay of Centerline Veloeities (A) and Void Ratios (B) •.•••••..•..••.•

Relative Turbulence Intensity Distributions in

3-D Bubble Plumes

.

. .

.

. . .

. .

.

. .

.

2-D Bubb1e Plumes

. . . .

.

. .

.

. .

.

. . . .

Bubb1e Plumes (Approximation with Exponentia1 Curves of 2nd, 4th and 6th Degrees) • • • • • • Distribution of Slip Veloeities (U ) in Bubble

Plum.es • • • • • • • • • . • • • ~ • • • • • .

Distribution of Bubble Diameters (Db) in Bubble

Plumes ... . . . . . . . . . .

Bubb1e Size Distribution for Test 10

Characteristic Bubble Sizes (Db)' Slip Veloeities (U ) and Peak Voltage Drops (E) •••.•••

s p

Sealing of Flow Widths for 3-D Bubble Screens . • . . .

J • Page 105 107 108 109 111 112 114 118 119 121 122 123 124 125

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126

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128

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129

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130 131

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135

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Figure

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5.225.21

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5.23 5.24

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5.25 5.26

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5.27

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5.28

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5.29 5.30

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5.31 5.32

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5.33

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5.34 5.35 5.36

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5.37 5.38 5.39

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xi Page

Sealing of Flow Veloeities for 3-D Bubble Screens 136

Sealing of Discharge Rates for 3-D Bubble Screens 137

Sealing of Flow Widths for 2-D Bubble Screens 138

Sealing of Flow Veloeities for 2-D Bubble Screens 139

Sealing of Discharge Rates for 2-D Bubble Screens 140

Sealing of 3-D Bubble Screens 143

Sealing of 2-D Bubble Screens 145

Sealing of Entrainment Coeffieients for 3-D Bubb1e

PLumes • • • • • • • • • • • • • • • • • • • • • • 148

Sealing of Entrainment Coeffieients for 2-D Bubb1e

P lumes . 149

Effect of Boussinesq Approximation

on Analytiea1 Modelling öf 3-D Plumes 152

Effect of Boussinesq Approximation

on Analytieal Mode1ling of 2-D Plumes 153

Comparison of Initial Conditions for 3-D Plumes

for Various Entrainment Rates • • • • 155

Comparison of Initial Conditions for 2-D Bubb1e Plumes

for Various Entrainment Rates .•.. . .•.. 156

Effect of À on Analytieal Mode11ing of 3-D Plumes 157

Effect of À on Analytical Modelling of 2-D Plumes 158

Effect of Slip Velocity U on 3-D Plume Solutions

s 159

Effect of Slip Velocity

U

on 2-D P1ume Solutions

s 160

Comparison of Numerical Solutions with the

Avail-ab1e Data (3-D P1umes) . . . .. .... 162

Comparison of Numeriea1 So1utions with the

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ai A b B

B.F.

c

d D e m E f fi F FQ[F

q]

g g' h ho

h*

H I

k.

1 K L xii

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LIST OF SYMBOLS

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numerical coefficients (i.e.in entrainment function)

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area; anemometer calibration constant

nominal plume width

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b/H; anemometer calibration constant; general variabIe (as in

Section 4.2.5.3.); bandwidth (spectral)

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buoyancy force

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concentration of bubbles or voids ratio

injector diameter

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diameter

mean error defined by Eq. 4.4

1 . f . ( 2 (K-l) /2)

vo tage; entralnment unctlon

=

on

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frequency

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functions

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densimetric Froude number

2 5 1/5 source strength

=

(Q /g'H ) a

[=

(q;/g'H3

)1/3]

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acceleration of gravity

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effective g (= ~pg/pw)

depth of surface current

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atmospheric pressure as head of water

H/ho

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depth of injector submergence

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constant characterizing similarity of shear stresses

numerical constants

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=0 indicates 2-Dimensionality whereas K=l indicates 3-D

circumference of injector; length of 2-D injector

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N q Q p (r ,z) s u

u

u'

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t T

T

.

1. v

v

V m

(V,U)

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x,y

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xiii

number of (--~) (i.e.bubbles); number of bits in an AID

converter

dis charge ratel unit length discharge rate

scale factor; pressure

terms used to simplify the equations in the mathematical model

2-Dimensional (including axisymmetric) coordinate system located at the injector

z/H

turbulent fluctuation of U velocity

r.m.s. value of u

fictitious velocity _(g,2 Qa)l/S (=(g' qa)1/3)

terminal bubble velocity temperature; time

length of time (i.e.averaging time) integral time scale

turbulent fluctuation of V

nondimensional velocity, Um/UQ; volume maximum velocity in surface currents velocities in the (r,z) system

general variables

location of the centerline w.r.t. (X ,Y ) for velocity o 0

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(x ,Y ) o 0 (l IJ. y À u p o T Subscripts : a b c e f g i L m max min o p q r xiv

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reference coordinates for cross-sectional measurements

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entrainment coefficient

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( )w-( )

specific weight

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spread rate of voids ratio relative to velocity profiles

dynamie viscosity

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density

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surface tension coefficientj characterizes bubble size distribution; acceptable error in defining averaging times

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time lag in autocorrelation computations

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refers to air; atmospheric conditions; average

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refers to bubbles

refers to concentration of buoyancy; cutoff (frequency);

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critical

denotes entrainment

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denotes fluid

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denotes gaps

denotes the i th member in a group

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refers to local conditions (FL)

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refers to centerline values; modeIs; and denotes mean or maximum

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denotes maximum denotes minimum

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refers to souree conditions

denotes peak (voltage drop); partial; prototype

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refers to 2-D plumes

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refers to r coordinate of (r,z)

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xv

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R refers to 3-D plumes denote ratio

Q

s denotes slip (veloeity); sensor; subreeord; sample

tm denotes true mean

u denotes u or U

w denotes water

z refers to loeal values (at z or (r, z)

Superseripts:

powers of scale factor P

powers of FQ( or Fq) in sealing relationships

denotes "from the virtual souree"; denotes "w.r.t. mixture volume" (for m'.llticomponentmixtures)

temporal rnean of turbulent quantities

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1 1. INTRODUCTION 1.1 Prob1em

Air re1eased at some dep th in a water body breaks into bubb1es of various sizes. As a resu1t of their inherent buoyancy, the bubb1es rise in

the water, dragging the surrounding water partieles a10ng. Thus induced, the bubb1y water stream moves upward, and in the process entrains more water and spreads laterally. Upon reaching the surface, the flow transforms into sur-face currents, which radiate away from the center of the vertica1 stream. Such a flow is of ten referred to either as a bubb1e screen or an air curtain

(Baines and Hami1ton, 1959; Larsen, 1960; Bulson, 1961; Abraham and Burgh, 1964; Kurihara, 1965; Kobus, 1968; McAna11y, 1973). Bubb1e screens have been used to augment convective heat and mass transfer rates in various appli-cations (Bergles, 1969; Kenning and Kao, 1972); to induce artificia1 mixing for control of stratification and water qua1ity in 1akes and reservoirs (Ford, 1963; Bernhardt, 1967; Leach et al., 1970; Zieminski and Whittemore 1970; lto, 1972; Stefan et al., 1976); as pneumatic barriers against density currents (Larsen, 1960; Abraham and Burgh, 1964; Rahm and Sjoberg, 1965; Lucas, 1969; Bruun, 1975); in pneumatic breakwater app1ications

(Evans, 1954; Tay10r, 1955; Straub et al., 1959; Brevik, 1976); in con-tro11ing shoa1ing in estuaries and harbors (Simmons, 1967; Bruun, 1975); in preventing icing in navigationa1 waterways (Proc. of the Symposium on Air Bubb1ing, 1961; Berge, 1965; Communicator, 1978). Recent1y it has found new use in containment of oi1 spilIs during un10ading operations (Kobus, 1975; Sea Techno10gy, 1975), and in providing instream aeration, especia11y

in navigab1e waterways (Water Research in Action, 1976). In almost all cases, the important factor has been the induced currents: either

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the vertical stream, as in prevention of shoaling and stratification, or the surface currents, as in the case of pneumatic breakwaters or oil spill containment. Such applications as the control of density currents, icing in waterways and water quality depend on both type of currents. The presence of bubbles has not been instrumental except in some aeration work.

It was during the exploration of the feasibility of another possible application area, that of enhancing dilution capabilities at wastewater dis-posal sites by bubble screens, that the subject of the present study

emerged.

In spite of the enormous potentialof bubble screens, their appli-cation has been severely limited due to lack of proper design information and criteria. The main obstacle in this area has been the difficulty in

establishing general model-prototype relationships. This has forced potential users to perform prototype experiments for their specific purposes. Even those suffered as much from complications due to field conditions as the inability to measure such important parameters as the induced local buoyancy, possibly due to inadequacy of instrumentation. Furthermore, these problems have hindered comparison of the technique with alternate means such as jet induced mixing (Iamandi and Rouse, 1969; Ditmars, 1970).

Initial attempts for generalization of the available limited data have relied mainly on analogy with simple plurne analysis (Taylor, 1955). Later, it was realized that air bubble plumes differ from simple plumes because of compressibility of the bubbles and their relative motion with respect to the surrounding water, which depends on the size of the bubbles involved (Kobus, 1970). Even incorporation of these factors into the formu-lations feIl short of achieving the desired end result (Cederwall and

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Ditmars, 1970; Brevik, 1977), since the mode1s we re high1y empirica1 and based on sparse data.

1.2 Objectives

The promise of vast app1icabi1ity potential, together with the know1edge that bubb1ing systems are easi1y installed and require re1ative1y sma11 additiona1 costs for their operation (Proc. of the Symp. on Air

Bub-bling, 1961) strongly suggest the desirability of further investigations

for a more thorough understanding of the physics of bubb1e screens and deve10pment of the appropriate sealing 1aws. Specifica11y the objectives for this investigation wi11 be:

a) to critica11y review avai1ab1e experimenta1 and ana1ytic works to determine the areas and reasons for their shortcomings; and if needed,

b) to investigate experimenta11y in the laboratory the effect of the relevant flow and geometrie parameters

c) to deve10p the appropriate transformations for applications of the resu1ts to prototype situations

d) to genera1ize the resu1ts for various app1ications, a math-ematica1 model wi11 be deve10ped to predict the induced flow behavior

e) to eva1uate any empirica1 re1ations or coefficients which may be needed for the model from the avai1ab1e data

f) to verify the performance of the model

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4

2. REVIEW OF PHYSICAL PROCESSES AND RELATED LITERATURE

Energy and/or mass exchange between two fluids may be achieved by providing direct contact between the two media, as by passing one through the other (a gas through a liquid), or mixing one with the other (gas into gas or liquid into liquid).

An

example for the latter would be disposal of pollutants into the atmosphere and water bodies. As an example of the former, one may cite the subject of this investigation, bubble screens, in which the potential energy of a gas is transferred to the surrounding liquide In the following,

the review will focus on this case, air and water being used for the gaseous and liquid phases, respectively. Even with this restrietion the subject matter is vast and the investigations involved are numerous. Therefore, only the appropriate generalities and the pertinent details will be mentioned.

When air is injected through a submerged device, such as a nozzle, orifice or even a simple port, into water, the air stream disintegrates into bubbles of various sizes, thus producing a large contact area between the two phases. The transfer process may occur in either or both the generation stage

or the rising stage of the bubbles. There has been interest in the fluid dynamie (gross and/or detailed) as weIl as the transfer aspects of each stage. Examples may be found in Ecuyer and.Murthy (1965), who studied the heat trans-fer from the surrounding liquid to the bubbles generated during the formation

stage, in Speece et al. (1973), who, for purposes of aeration, were concerned

with the flow induced in a bubble screen as weIl as the mass transfer occurring

between the bubbles and the surrounding liquid, and in Haberman and Morton

(156) and Baker and Chao (1963), who investigated the details in and around

individual bubbles. The primary concern has been for estimation of the shape

and size of the bubbles generated, the velocity of rise of the individual or

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5

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2.1 GenerationGeneral discussionsof Bubb1es of the process of bubb1e formation and

inf1uenc-I

ing factors have been presented by Ecuyer and Murthy (1965), Wallis (1969),

and Speece and Rayyan (1973). Here on1y those aspects which may help for

I

understanding of the characteristics of bubb1e screens wi11 be out1ined. The

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variables that p1ay a role in bubb1e formation include the gas injection rate, the gas and 1iquid properties, and the size and shape of the injector and the gas supply 1ine. Their influences emerge in the form of f1uid dynamic and interfacia1 forces due to injection momentum, inertia of the disp1aced 1iquid,

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buoyancy, drag on bubb1es during rise and interfacia1 tension.

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The injection 1ine may be such as to maintain a gas flow at either a "constant rate" or "constant pressure". The bubb1e formation and gas flow are uncoup1ed in the former case, but not in the latter case.

Depending on the gas flow rate,

Q ,

two main regimes of bubb1e a

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formation have been identified (Everso1e et al., 1941; Krevelen and Hoftijzer,

I

1950; Benzing and Myers, 1955; Davidson and Amick, 1956; Hayes et al., 1959). At very low flow rates (Q < (Q) - 1 cc/s), the terminal bubb1e volume Vb

a - a c

I

can be determined by equating the buoyancy and surface tension forces:

I

(2.1)

in which L

=

circumference of the injector, pand p

=

density of the 1iquid

w

a

I

and gas respective1y, and cr

=

the surface tension coefficient. This regime is

referred to as the static regime to distinguish it from the dynamic regime,

I

which occurs when

Q

is increased beyond

(Q)

.

The frequency of bubb1e

pro-a a c

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ductdependsion,on1y on the buoyancyf, has been found to be constant, whi1e the resu1ting bubb1eand 1iquid inertia forces in 1iquids of lowsize Db

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kinematic viscosity, such as water:

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6

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Q2 1/5 Db = k(_j!)

s

(2.2)

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in which g = acceleration of gravity and values of k = 1.38 and 1.49 were found

I

by Davidson and SchuIer (1960), and Krevelen and Hoftijzer (1950) respectively.

For separation of the two regimes, Walters and Davidson (1962)

I

reconnnended:

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(Qa)c ~ 4.95!s d5 (2.3)

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in which d

=

injector diameter. For both regimes, Q may be expressed as a a

combination of the bubble volume and the production frequency:

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Q = V ·f

a b (2.4)

In the statie regime, since

V

b

=

const., f becomes proportional to Qa. Thus both regimes allow for the periodic formation of bubbles, whose sizes are

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unique for the given Q . a

At still higher flow rates, Ecuyer and Murthy distinguished a third

I

regime, which they refer to as the turbulent regime. Leibson et al.'s (1956)

I

photographic study showed a discontinuous jet of gas, which disintegrates about 3-4 inches above the injector to produce a swarm of bubbles with

nonuni-I

form sizes. In this regime, both the bubble size as weIl as the format ion

frequency were found to vary randomly. Leibson et al. (1956), and Silberman

I

(1957) were able to approximate the different bubble sizes by a log-normal

I

probability function. Silberman (1957) predicted the maximum bubble size by a relationship of the form of Eq. 2.2 with k

=

1.41. This relationship is

still independent of the injector, and the air and 1iquid properties. To

I

insure the occurrence of this regime, Si1berman (1957) requires:

y Q2 3/5 d < 0.47 ~ (_j!) max o g

I

(2.5)

I

(24)

I

7

Generally the presence of both induced or forced liquid motion around the injector was noted to reduce the bubble size. In the turbulent regime, the reduction was proportional to the 1/2 power of the relative liquid velocity

(Silberman, 1957).

Although the above observations are clearly valid for the "constant flow ratelt, the same conclusion cannot be made.without hesitation for the "constant pressure" cases.

2.2 Buoyant Plumes vs. Bubble Screens

2.2.1 Turbulent Buoyant Jets and Plumes

When one of two miscible fluids, to be identified as an effluent, is discharged through a submerged outlet into the other, to be called the ambient fluid, the resulting flow behavior would be similar whether they are gases or liquids. The momentum inherent in the effluent, due to its discharge rate, causes mixing with the ambient fluid. Due to density differences between the discharged and ambient fluids, the effluent may possess buoyancy forces in addition to its initial momentum. The relative influence of each is

characterized by a densimetric Froude number, which represents the ratio of the initial momentum to the initial buoyancy force. When momentum is the dominant factor, as in the case of thermal and sewage discharges, the resulting flow is referred to as a buoyant jet. If, as in the case of smoke stack effluents, buoyancy dominates, the flow would be known as a plume. At the two extremes, one would have either a simple jet when the jet is nonbuoyant, or a simple plume when the plume has no initial momentum. The distinction between buoyant jets and plumes is not clear-cut. It has been observed that what starts out to be a buoyant jet, would eventually behave more like a plume

(25)

8

since, as the initia1 velocity excess is dep1eted, the buoyancy effect takes over (Kotsovinos and List, 1977; Wright, 1977).

To provide a basis for comparison with bubb1e p1umes, the flow field for turbulent buoyant jets and p1umes wi11 be brief1y described.

Both a submerged vertica1 turbulent buoyant jet and a p1ume have three distinct zones. For the sake of simp1icity both wi11 be referred to as a jet.

In what is known as the zone of flow establishment (ZFE), the jet enters the ambient f1uid. The shear generated between the jet and the

surrounding f1uid induces a progressive1y growing mixing zone that spreads both toward the center1ine and into the ambient f1uid. The point at which the inner boundary of the mixing zone reaches the center1ine marks the end of this zone. The region 1ying inside the inner boundaries has a uniform velocity and

con-centration, equa1 to the outfa11 va1ue and has usua11y been referred to as the potentia1 core (Albertson et al., 1950; Abramovich, 1963). The core leng th for

concentrations of heat and mass is about 10% shorter than that for velocity, which is about 4-7 outfa11 diameters long (Forstall and Gaylord, 1955; Kiser, 1963). Except for simp1e p1umes, this zone is dominated by the initia1 momentum.

The second of the three zones is called the zone of estab1ished flow (ZEF), and it starts with the end of the potentia1 core and continues to

spread 1inear1y into the ambient f1uid unti1 the initia1 momentum and buoyancy excesses are depleted. The flow pattern is independent of the outfa11

characteristics, becoming sole1y dependent on the 10ca1 velocity, buoyancy and ambient flow conditions. Even though the velocity and concentration profiles are common1y assumed to be simi1ar from the start, measurements in nonbuoyant jets have shown that the jets do not become tru1y se1f-preserving unti1 some

(26)

I

9

I

Gaussian distributions70 diameters downstreamhave been used for the simi1arityof the outfa11 (Wygnanski and Fied1er,functions.1969). aften

I

The third zone, referred to as the far field, is reached when the

velocity excess of the jet is depleted. The ambient characteristics dominate

I

the flow behavior. In a 1iquid, a buoyant jet may reach the surface before

I

dep1etingsurface currents.its velocityIn a stratifiedexcess and thus, it may spread horizonta11yambient f1uid, this zone may be tota11ycreating

I

submerged (Abraham, 1972).

I

2.2.2 Bubb1e ScreensThe bubb1e screen has of ten been referred to as a bubb1e p1ume,

con-I

sidering the simi1arities it displays with respect to buoyant jets and p1umes. At the injector, the bubb1e screen possesses both momentum and buoyancy. The

I

re1ative effects of each may be determined by defining the relevant densimetric

I

Froude number, F:

I

Pa U2 • Pa

u

2 F = -:-=----:---:- - - -(p -P )gd - p gd w a w (2.6)

in which U = injection velocity. Using a typica1 set of Kobus's (1968) data:

I

1 1 F

=

800 • 980 2 (150)

=

0 14 (0.2) •

The ratio wou1d drop even further just above the injector. Thus, the bubb1e screen can be c1assified as a p1ume. Indeed bubb1e screens exhibit th~ same gross structure as plumes. Such a conc1usion wou1d be we1come since it wou1d

I

provide access to the findings of the vast buoyant jet and p1ume 1iterature.

I

As a consequenceUnfortunate1y a c10ser observationof the immiscibi1ityprohibitsof air and water, Kobusone from further genera1ization.(1970) notes the

I

fo11owing differences:

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10

1) In ZFE, the flow structure is quite different than that described for plumes because of the way in which bubbles are generated

(Section 2.1).

2) The finite bubble size and the resulting bubble mot ion relative to the liquid, known as slip velocity must be accounted for.

3) The slip velocity, and in turn the spread rate of the plume are found to depend on the air discharge rate.

4) Effects due to bubble compressibility must be considered. The significance of these differences will become apparent as the relevant experimental and analytical investigations are reviewed.

2.3 Review of Literature on Bubble Screens

A general review of the experimental and analytical work done up to 1961 was presented by Orsoni (1961) along with a historical account of the use of pneumatic breakwaters. Simmons (1967) provided a discussion of the possible application areas, whereas a symposium in 1961 brought out the

experience related to practice (Proceedings of the Symposium on Air Bubbling, 1961).

2.3.1 Experimental Work

Most of the studies mentioned in the Introduction as weIl as a number of investigations in Russia, Germany, Japan and U.S.A. (all cited in Orsoni, 1961) present accounts of actual applications, and model and full-scale experiments. Those conducted before 1960 were mostly concerned with the breakwater application and were directed toward collection of such design information as the type of waves that could be damped and the amount of air injection required to achieve the desired attenuation. It was Evans

(28)

I

11

attenuation process. Owen (1942) conducted laboratory tests at a 1-1 scale to deterrnine the best orifice size and shape, and direction of discharge for producing the finest bubbles and the largest currents.

A set of the experiments conducted in Germany at water depths of 22.2, 33.3, 85.0 and 240 cm, was aimed at determination of scale effects in bubble screens; however no scale influence was noticed (reported by Hansen, 1955). Straub et al. (1959), during comparison of efficiency of pneumatic and hydraulic breakwaters, arrived at the same conclusion in models with depth of 1.0 and 4.5 ft. The reason for this apparent failure will be pointed out in Section 3.1.2 and 3.2.1.

Since 1960, the scope of experimental work has been extended to all phases of application. With few exceptions measurements were aimed at the specific purpose at hand; thus the results are difficult to generalize, and the improvement of understanding of the dynamics of the induced currents has been minimal.

The first attempt to investigate the dynamics of a bubble screen was made by Baines and Hamilton (1959). Using single orifices of various sizes, they observed and measured veloeities in both the vertical and the surface currents in a 8' x 8' x 6' deep tank. Based on their scanty measure-ments, they observed the following trends: for any given discharge rate,

the flow generated was independent of the orifice size, but was quite un-stabIe due to random oscillations. The maximum vertical velocity stayed constant with height, while the flow width increased with the square root of the height. The velocity profiles looked similar.

Af ter having found the flow, induced in the laboratory by a

multi-port diffuser, to be independent of orifice size and spacing, and that current meters operate satisfactorily in the bubble screen, Bulson (1961)

(29)

I

12

I

conducted large-scale measurements in a 912 ft long, 102 ft wide and 36

I

ft deep dock. Af ter checking the lateral and longitudinal distributions of

surface currents, one set of measurements were aimed at determination of the

I

surface current's depth and its maximum velocity (which was estimated to be

at a distance equal to the flow depth from the axis) as a function of

I

diffuser submergence, port size and injection ràte. The results were

pre-I

sented as: V m

₌

_{1.46 (1}

₊

_{g_)-1/3}

h o (2.7)

I

(gq )

1/3

a

I

H ho H 0.32

H

ln(l

+

h)

o (2.8)

I

h

- =

in which h

=

atmospheric pressure as head of water, H and q

=

injection

o a

I

depth and rate respectively, V

=

maximum surface velocity and h

=

depth of

m

the surface current (Fig. 2.1). For a water depth of 34 ft and

Q

a 1 cfs

I

per ft, the variation of the center line velocity with depth as weIl as four

cross-sectional profi1es were measured and presented; however the point

I

measurements for profi1es are too few to be used for quantitative purposes.

The most extensive measurements we re undertaken by Kobus (1968) to

I

study so1e1y the vertical portion of both 2 and 3 dimensional screens.

°

l

I

Measurements consisted of center 1ine veloeities, 3 to 6 cross-sectional

velocity profiles and average density measurements using radio-isotopes.

The velocity profiles obtained by current meters, were approximated by

I

Gaussian distributions in the least-square sense, and thereby, the local

plume widths were deduced. The profiles were also integrated to determine

the local fluid discharge rates. The widths and local densities were used,

I

along with the injection rate, to compute an average bubble rise velocity.

I

(30)

I

13 Zone of Estab-lished Flow(ZEF) r2 = exp(-

2)

b c H

\ I '

\

,

\ \

.

,

\I'

\ 1

'<ë-1

Virtual Souree

Fig. 2.1 Schematic Flow Field for a Bubble Plume

V m U m 2 r exp(- -) b2 u

(31)

I

14

I

This is a rather crude way to determine the bubble veloeities, since the

bubbles occupy only a limited portion of the flow area. Also this procedure

I

provides no information on the individual bubble characteristics. The 3

I

dimensional screen had a 4.5 m water depth and the di.scharge rate ranged

from 400 to 6200 cm3/sec. For the 2 dimensional case, water depth was 2.0

and 4.3 meters. The dis charge varied between 30 and 100 cm3/s/cm.

I

In a lake, Speece and Rayyan (1973) traeed Rhodamine B dye to

I

measure the centerline veloeities induced by pure oxygen injection. The

injections were made at depths of 30 and 60 ft for discharge rates of 1.0

I

(0.75 for the 60 ft injection) and 2.0 liters/min. In each case, the

veloc-I

ities could be determined at on1y a few e1evations. Even then the reliabi1ity

of the results is dubious due to entrapment of bubbles by the dye sampling

I

line.

I

2.3.2 Modelling of Bubb1e Screens

Lack of proper model1ing 1aws, physica1 and analytical, for bubb1-

I

ing systems has long 1imited investigation of such systems to prototype

I

situations, and thus, their potential application. Efforts along this 1ine,

although not few, had on1y 1imited succes~. These wi1l be reviewed in the

I

following sections.

I

2.3.2.1 Physica1 Model1ing

For estimation of the air discharge rates needed for prototype

I

application from model results, attention was focused on determination of

the appropriate scale transformations. Considering the requirement of Froude

I

similitude for the modelling of waves, Schijf (1940) formulated a

dimension-less combination of the air discharge rate required for attenuation of a wave

I

(32)

I

15

I

of given height and length, and the water depth. The necessary coefficient

I

was evaluated from the four sets of German scale experiments; but the

dis-I

charge predictionsLaurie, 1955). Similar transformations were deduced elsewhereturned out to be unnecessarily high (Hansen , 1955;(Straub et

I

al., 1959). Discussions of such failures centered around the need for consideration of such effects as viscosity and turbulence in addition to

I

Froude scaling (Laurie, 1955).

I

vertical plume, was done by KuriharaA more orderly investigation(1965).of the bubble screen, includingGeneration of the surfacethe

I

currents and the wave attenuation mechanism were recognized as independent phenomena. Furthermore, the role of such factors as compressibility of

I

bubbles and their slip velocity were considered in a dimensional analysis and

I

yielded (for a 2-D screen):

I

H , h

o

(2.9)

in which band U characteristic width and velocity of the induced current,

I

respectively and U

=

terminal bubble velocity. It was realized that for

t

large bubbles (or Ut) weaker currents (lower U and b) would be induced. At

the time, the dependence of Ut on qa (Chapter 3) was not noticed. On the

other hand, this was confirmed by the analytical developments to be

dis-I

I

cussed in the next section. Specification of the combination of the

I

remaining terms requires further investigation.

I

expected to achieve similarity in the bubble screensSatisfaction of the above conditions, in their reduced form, was(Kurihara, 1965).

I

Implications and feasibility of this situation were discussed by Orsoni

I

(33)

16

(1961) at great length. For complete similitude, operation of a model under subatmospheric conditions and/or use of a heavier ambient liquid was recom-mended.

The difficulty of satisfying the similitude requirements has been associated with the uncertainty of scaling bubble induced effects, which closely depend on the bubble size (Abraham and Burgh, 1965). Manyattempted to avoid this by scaling only some characteristic velocity of the induced currents (McAnally, 1973). In a study of bubbly flows in siphons and vortex chambers, Boughton (1959) matched the ratio of model-to-prototype terminal bubble velocities with the ratio of flow velocities. To achieve this without bubble scaling, kerosene bubbles were used in the model. The first technique ignores kinematic similarity in favor of dynamic similarity (induced buoyancy force), whereas the second sacrificed dynamic similarity for kinematic

similarity.

The above discussions point to the need for some orderly investi-gation in this area.

2.3.2.2 Analytical Modelling

Earlier efforts along this line were of semi-empirical nature and

attempted to predict the maximum velocity and the thickness of the surface

currents. Taylor (1955) assumed the air bubbles to be small enough to follow

the liquid motion. Then analogy with Schmidt's (1941) theory for heat plumes

from heated bodies, led to (given the form of Eqs. 2.7 and 2.8)

kl

=

1.9

and

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I

17

I

Kurihara (1965) arrived at similar results:

I

1.994 (1

+

H/h )-1/3 o h k2 = 0.30 (1

+

_Q_) In (1

+ ~ )

H

0

Experimentally the numerical constant in k2 was found to be 0.21. The form

of these relationships have been used to present the experimental results.

Measurements of Gordier (1959) on jets of air-water mixture

dis-charging into a current indicated that the mixture cannot be treated as a

homogeneous fluid.

Attempts to predict the local plume velocities and widths in the

vertical portion of bubble screens were made by Baines and Hamilton (1959),

Kobus (1968), Cederwall and Ditmars (1970), Speece and Rayyan (1973), and

Brevik (1977). Of these, only the last four produced fully developed

modeIs.

Based on the theory of turbulent buoyant jet and plumes, the usual

assumptions (Morton et al., 1956) of a) stagnant, incompressible and

homo-geneous ambient fluid; b) steady and fully turbulent flow; c) Boussinesq

approximation for small density variations d) boundary layer approximations;

e) approximation of velocity and buoyancy profiles by Gaussian distributions

and similarity requirement in the flow direction; and f) existence of a

virtual source below the real one were adopted. To account for

compress-ibility effects, the bubbles were allowed to respond isothermally to changing

ambient hydrostatic pressure. The effect of relative bubble motion is

incorporated by allowing the induced buoyancy to move with the bubbles. The differences among the models and the accuracy of their predictions will be

pointed out in the following portions of this section.

(35)

I

18

I

Kobus (1968) developed a highly empirical theory. As a result of the

11

similarity assumption, the flow width was taken to grow linearly, the growth

ra te represented by an empirical spread coefficient. Also using experi-

I

mentally deduced bubble velocities (Section 2.3.1), the local buoyancy force,

B.F., was formulated as:

I

Qa z )

B.F.

=

-y h -- In (1 - h

+

H

w 0 Ub 0 (2.10)

I

This, when introduced into the momentum flux equation, yields an expression

I

for the centerline velocity. The width and the velocity expressions were

I

solved explicitly by the experimental virtua1 source depths, spread rates

and the bubb1e velocities. The results show close agreement with the

mea-I

surements except in the initial region and the surface zone. Besides the

empirical information required, the model was criticized by Ditmars and

I

Cederwall (1974) for the form of its buoyancy function which led to the poor

Af ter incorporating the above approximations, Cederwall and Ditmars

I

agreement at the source.

(1970, 1974) obtained the following equations for the fluxes of volume,

rnornentumand buoyancy in the plurne (Fig. 2.1):

s.,

[(nb2)j U ] dz rn K 2(nb) a U TIl (2.11)

I

d 2' 2 2' ~p -- [eb )J U ]

=

(2À b)J ~ dz rn g p

~...,

• ~p 2 2 j h (nb2)J __!!! U =«l+À )IÀ) • 0 LK-l pw m 1 + (1+À2)j U

[u

h + H-z Qa s m 0 (2.12)

I

(2.13)

I

(36)

I

19

I

in which U and ~p

=

loca1 center1ine velocity and density deficiency

m m

respective1y whi1e b

=

flow width at some e1evation z above the source;

I

u

s

=

slip velocity due to the relative bubble motion; À

=

ratio of relative

spread of the lateral density and velocity distributions; Cl

=

entrainment

I

coefficient; and j and K are re1ated by:

I

j (K

+

1) /2

I

Here K

o

and 1 designates 2 and 3 dimensional plumes, respectively.

To determine the appropriate initial conditions, the analogy with thermal plumes was extended further by assuming that the flows behave

similarly between the virtual source and the end of ZFE. Thus in this region,

I

z=O

and

U

»U.

With these simplifications, the continuity and momentum

m s

I

equations can be solved to get:

For 3-D case: b

= &.

Cl z'

5 (2.l4a)

I

2 1 1/3

gQ

1/3 U = [25 l+À _.1.. ] (---1!) m 24 n 2 l+H/h Cl 0

z'

(2.l4b)

I

For 2-D case: b = -2a

z'

;;

(2.l5a)

I

i

]

1/3 (gQ )1/3 Cl(l+ ~) a h o (2.l5b)

I

where z' is the distance above the virtual source. If the compressibility of

bubbles had been neglected,

Q

(or q ) being constant, the (l+H/h ) term

a a 0

would equal 1 in the above expressions. Then these results would be equivalent to those obtained for pure plumes (Morton et al., 1956). The width and centerline velocity were computed for the given virtual depth, and

I

with these, the full equations were numerically integrated to find the

(37)

I

20

I

step-by-step variation of U , b and C. The va1ues of a, À and U needed was

m m . s

estimated as described below.

I

The authors generated an a vs.

Q

re1ationship using Eqs. 2.14a and a

2.15a a10ng with Kobus's data for p1ume widths. a was found to vary between

I

0.044 and 0.08, approaching 0.08 for large

Q

in 3-D plumes, whereas for the

. a

2-D case a

=

0.085 to 0.115. Due to the 1imited range of

Q

avai1ab1e in the

a

I

second case, the asymptotic va1ue of a cou1d on1y be obtained from simp1e p1ume

I

studies. U was estimated as 0.3 mIs from Haberman and Morton's (1954) data.

s

I

Based on the observations of the bubb1e screens, À was expected to be 1ess than

1.0. Therefore, 0.1, 0.2 and 0.3 were tried, and finding no apparent

sensitiv-I

ity to changes in À it was set to 0.2. This is quite low when compared with

Chuang and Go1dschmidt's (1970) À 0.665, obtained in measurements invo1ving

I

the spread of bubb1es in a vertica1 simp1e jet. Comparison of the model's

pre-I

dictions with Kobus's (1968) data was found to be satisfactory by the authors.

However this conclusion will be criticized at a later point.

I

Speece and Rayyan (1973) adopted Cederwall and Ditmar's (1970) model

with modifications to study the mass transfer between the bubb1es and the sur-

I

rounding water. The injected air was assumed to break into bubbles of one

I

size, Db' which was assumed to vary with the mass transfer rate. Thus the air

Q

=

N

.!

D3

a 6 bz

I

discharge was set to be:

in which N = number of bubbles (N =

Qa/(i

D~», and Dbz is the local bubble size.

During the numerical solution, computation of Dbz at each new step was followed

with the usual computations. The predictions for center1ine velocity were

com-I

I

parabIe with their field data when a

=

0.03, U

=

0.0, and À

=

0.2. Solutions s

to test the sensitivity of the model for certain parameters have yielded

I

certain interesting results:

I

(38)

I

21

I

correspondinga) As a is increased from 0.01 to 0.05, banddecrease in U •

Q

increase with a m

I

b) Due to mass transfer, bubbles experienced a rapid size reduc-tion in the vicinity of the injector. Bubbles of 0.2 mm were completely

I

absorbed, while the 2.0 mm ones were reduced to 0.9

mm

for an injection depth

I

of 4.5 m. c) For U

=

0.0 and a

=

0.03, the results for the no-mass transfer

s

I

case (NT) indueed higher U

,Q,

momentum and kinetie energy flux than those

m

with mass transfer (MT), the smallest bubbles (0.2 mm vs 2.0

mm)

produeing

I

the lowest values. The trend was reversed for the plume width.

I

both the MT and NT.d) ConsiderationHowever,of slip velocityin case of MT, the effect disappearedled to higher (lower) b(U ) withif Dbm

I

was small (0.2 mm).

The effect of mass transfer was further investigated in the

labor-I

I

atory by Speece and Hurfee (1973). Considering the transfer of O2 along with

that of N2 and 002, they have found negligible variations in Db for

sub-mergences of 8 m or less, the variation being smaller for the smaller bubbles

I

(0.2-0.5 mm vs. 1.0-2.0 mm). However, as the submergence was inereased, the bubble sizes were reduced, even to the point of complete absorption.

I

To consider the 2-D plume, Brevik (1977) replaced the volumetrie

flux, in Cederwall and Ditmars' (1970) model, with the kinetie flux, Dut

to obtain a closed system of equations, had to assume the similarity of the

I

shear stresses.a. At the virtual source both dUm/dz' and b we re equatedThus a constant eharacterizing this similarity,to zero.I, replacesThese

I

initial conditions, when eombined with predietions of À, U and I, provide s

the data and conditions needed for numeri cal solution of the model. For

I

what appeared to be sensible estimates of À, U and I, eomparison of the

s

(39)

I

22 I

solutions with Kobus (1968) and Bu1son's (1961) data showed velocity

I

discrepancies, which were greater with Bu1son's than with Kobus's. At this point, it seems appropriate to quote Brevik (1977)'s conc1usion regarding

I

the discrepancies:

"This circumstance indicates one of the fol1owing two things: Either the present theory is incapab1e to give an accurate description of the p1ume, or systematic errors are contained in the observed data. The general suggestion resu1ting from app1ication of the theory to the three examp1es [referring to both Kobus and Bulsons data] be10w is that measuring difficu1ties in the two-dimension~l air curtain have produced too low center 1ine ve1ocities. It is of interest to note that the same sug-gestion was made by Cederwa11 and Ditmars. These authors found that reasonab1e input va1ues for Ure1 [Us) and entrainment co-efficient a would lead to somewhat 1arger center 1ine ve10cities than those Kobus observed. This feature a1so provides another demonstration of the dua1ity of the two theories."

I

There are several points raised in the above quotation which de-

I

serve criticism. The dua1ity cited to back up the model under consideration

I

is rea11y not there, since the continuity and kinetic energy flux equations

are interchangeab1e, and preference of one over the other wi11 not introduce

I

any new insight (Morton, 1971). Cederwa11 and Ditmars' (1970) suggestion of

too low center1ine ve10cities due to integration of the measurements over a

I

5 minute period shou1d have been va1id for the point source injection as

I

weIl. The osci11ations associated with the finite tank size in Kobus (1968)

shou1d not have been a factor in Bu1son's (1961) test; however, the magnitude

I

of the discrepancies disputes this hypothesis. Speece and Rayyan's (1973)

I

sensitivity ana1ysis indicates that the modeIs, if used with va1ues of a or I predicted to approximate the p1ume width (velocity), wou1d overestimate

I

the p1ume velocity (width).

Before finding fault with either the experimenta1 data or the

I

theory, it seems appropriate to seek verification that the estimates for

I

parameters like U , À and a (or I) are reasonable. Then the va1idity of

s

(40)

I

23 I

certain assumptions made in the theory development must be investigated.assumptions of constancy of U in the flow field, and the approximation ofThe

s

I

density deficiency distributions are two which will require experimental back-up.

I

The Boussinesq approximat ion , which is considered to be valid only

I

up to density variations of 2 to 2.5%, came under scrutiny when variationsup to 15% were observed at as far as z/H

=

0.2 from the souree. Therefore,of

I

reformulation of the theory in such a way as to remedy this situation becomes highly desirabIe.

I

Another questionable area involves the initial conditions used.

I

The analogy assumed in the initial zone is physicallyif an air stream is injected into water, there will be no entrainment bet-unacceptable, since,

I

ween air and water due to the extreme density difference as weIl as the

immiscibility of the fluids (Batchelor, 1954; Morton et al., 1956).

Further-I

more the initial values of b and U , computed as Cederwall and Ditmars

m

I

(1970, 74) described, yield too low buoyancy va lues at the injector area. Besides, for prediction purposes, the presence of a virtual source

intro-I

duces the additional_{there are little guidelines except for the knowledge}requirement of predicting the virtual source,_{that it should be below}for which

I

the physical source (Kotsovinos, 1976).

I

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24

3. TIIEORETICAL DEVELOPMENTS

In this chapter, dimensional and analytical considerations will be employed to understand, model and predict the behavior of both 2 and

3-Dimensional bubble plumes, generated in an otherwise homogeneous stagnant

ambient fluid. Any temperature variations between the injected air and the

ambient fluid will be ignored. Considerable attention will be devoted to

investigation of the feasibility of physical modelling of the induced flow

pattern. Various aspects of the concepts and formulations discussed in the

previous chapter will be modified and/or used as guidelines for further

developments.

Of the three zones in the flow field only the zones of flow

estab-lishment (ZFE) and established flow (ZEF) will be considered in detail. The

belief, that once the flow behavior in the ZEF is known, the developments such

as that of Taylor (1955), Bulson (1961), and Kobus (1975) wi11 adequately

describe the situation in the surface zone, has been the motivation for the

above decision.

3.1 Dimensional Considerations

In complex flow situations, understanding of the relevant parameters,

their orderly investigation, interpretation of the investigation resu1ts, and

transformation of these results to prototype applications would require

deter-mination of the important parameters and the pertinent sealing laws. In what

follows, dimensional analysis will be employed to aid in the selection of the

main parameters and then, the appropriate modelling laws will be established.

3.1.1 Bubble Generation and Estimation of Terminal Veloeities

The ana1ysis wil1 be carried out in stages with the hope that a

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25 I

3.1.1.1 Bubb1e GenerationThe first question that wi11 be posed is that of estimating the

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size of the bubb1es (Db) generated when air,is injected through a submerged

opening of diameter d into a liquide The resu1ting bubb1e size is in general

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a function of the injector geometry, density Pa' and vo1wnetric flux, Qa of

the air as we11 as the properties of the surroUnding 1iquid: density P ,

w

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viscosity 1-1, surface tension coefficient cr, and final1y, the acceleration of

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gravity g. Thus:

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Db

=

f(d, Qa' Pa' Pw' u , c , g) (3.1)

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in which Pa and g can be rep1aced by the effective buoyancy ~Ya since

~Y = (p - p )g a w a (3.2)

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Then Db

=

f1(d,

Q ,

P , cr,~Y , 1-1) a w a (3.3) or in dimension1ess form:

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Q2_a _PwQ_a -~Y__';=--

']:;cl)

a d5 Pw

Availab1e experimenta1 and ana1ytica1 studies (Wallis, 1969) indicate that

2'

d

(3.4)

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the above re1ationship, at low Q ,reduces to:

a

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for d>(_2:__) 1/2 s:::l 0.11

~y

a

(3.5)

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At higher Q , the surface tension and the injector size effects become a

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neg1igib1e (Wallis, 1969). When the surrounding 1iquid is of low kinematic viscosity, 1ike water, the re1ationship simp1ifies to the fo11owing:

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26

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Db

=

k )1/5 _(3.6)

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This particular form was arrived at by experimental and analytical

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considerations under various conditions (Section 2.1).

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3.1.1.2 Terminal Bubble Velocities

The terminal velocities, Ut which the generated bubbles will attain in a stagmant fluid can be expressed as:

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U

=

f(Db, ~y , p , cr, ~, C)

t a

w

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Here

C

is a void ratio or a concentration parameter. In situations where the

concentration of bubbles is low enough for interference among bubbles to be

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negligible: Pw UtDb u or cr ~Ya fl (-cr 2 Db P ~Y Db3 wa) 2 ~

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(3.7)

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This is too complex to be used for practical purposes; fortunately it has been

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possible to divide the full range into regions where only part of the above influences are dominant (Wallis, 1974).

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3.1.2 The Induced Bubble Plumes

When an injector, say a point source, discharges compressed air at a

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rate Q , the induced flow properties, such as the axia1 f1uid velocity U , the

a m

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p1ume width b or simply the dis charge Q at sorne elevation z above the source

can be expressed, keeping the previous sections in mind, as (Fig. 2.1):

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Q(z)

=

f(z, H, h , Db' U , crb' C, ~Y , P , u)

o t a w (3.8)

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where crb is a parameter characterizing bubb1e size distribution. Here

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27

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atmospheric head h is inc1uded to account for the variation of the bubb1e o

volume, thus effective buoyancy ~y , due to compressibi1ity of bubb1es. A1so

a

eoneentration of the bubb1es, C, may be interpreted as the loea1 void ratio.

Surfaee tension is not ine1uded sinee, as observed earlier, it affeets only

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Db and Ut for the bubb1es and not the f1uid flow. This is discussed further in Appendix 1 al.ong wi th the compressibili ty effect.

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Eq.

3.8

can be simp1ified by noting that the bubb1es affect the flow

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main1y (or on1y) through the buoyancy flux that they induce. Thus Db' Ut and

C,

whieh essentia11y determine Q , ean be rep1aeed in Eq.

3.8

by Q :

a a

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Q(z)

=

fez, H, h , Q , ~, ~y , p )

o a a w (3.9)

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App1ying Buekingham's IT-Theorem and choosing H, Q ,p as the

a w

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repeating variables, one ean obtain:

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(3.10)

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PwQa . Pw pwQ(z)

When ~ 1S combined with Q(z)/Qa to yie1d ~H ' it can be rec-ognized as a 1iquid Reyno1ds Number. Jet and p1ume studies indicate 1itt1e

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_{dependence of the flow characteristics} _{on this when it is high enough to insure}

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a turbulent flow strueture. This is shown to be satisfied in Appendix 2.

~y

Thus , letting g ' = ~ be the effeetive acce1eration of gravity, Pw

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~

Q

_a ~ f (z1

H'h'

H

0

(3.11)

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The last term expressesof densimetric Froude number.the source strength, and can be reeongized as a form

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28

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Qaz h o h

+

H-z Qa o

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Due to compressibi1ity of the air bubb1es, the air discharges vary with depth of submergence, that is, assuming isothermal conditions:

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or Qa = Qaz 1

+

!!__ (1 - ~)

h

H

o (3.12)

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Since ~ may be rep1aced in any expression by: o [1

+

!!__ (1 - ~)] -1 H ho H

-=----=---

_h o

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(1 - ~)

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it fo11ows that Eq. 3.11 may be expressed as

Qz ~ f (- , H' 2 Qa 2 H z Qa [1

+ -

(1 - -)], --)

=

0 h_o H

_g

'H5

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Combining this expression with Eq. 3.12 wil1 yie1d, after some further manipu1ation: H

.Q_Ç&

= f (.!, [1

+

h(l Q 3 H 0 az

Q2

- .!)], ~) H g'H5

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(3.13)

.9J&

= f(~, [1

+ ~

(1 - ~)], qaz 0 q2 ~) g'H3 (3.14)

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For a 2-Dimensional injection source, the appropriate re1ationship would be:

Even as such the re1ationship is still too ambiguous to be used. It

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with q as the air discharge rate (at STP) per unit 1ength of the source. a

was hoped that, since

Q

a1ready embodies the compressibi1ity effect, az