18-Bubble 1998 Martinez Bazan PhD Thesis

8/2/2019 18-Bubble 1998 Martinez Bazan PhD Thesis

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UNIVERSITY OF CALIFORNIA, SAN DIEGO

SPLITTING AND DISPERSION OF BUBBLES

BY TURBULENCE

A dissertation submitted in partial satisfaction of the

requirements for the degree Doctor of Philosophy

in Engineering Sciences (Aerospace Engineering)

by

Carlos Martnez

Committee in charge:

Professor Juan C. Lasheras, ChairpersonProfessor Laurence ArmiProfessor Michael BuckinghamProfessor Martin MaxeyProfessor W. Kendall MelvilleProfessor Sutanu Sarkar

Professor Forman Williams

1998


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Copyright

Carlos Martnez, 1998

All rights reserved.


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The dissertation of Carlos Martnez is approved, and it is

acceptable in quality and form for publication on microfilm:

Chair

University of California, San Diego

1998

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To my family:

my parents, sisters and Paqui.

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TABLE OF CONTENTS

Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

list of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Vita, Publications, and Fields of Study . . . . . . . . . . . . . . . . . . . . . . xvii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1A. The turbulent break up problem . . . . . . . . . . . . . . . . . . . . . . . . 1B. Background and Previous work . . . . . . . . . . . . . . . . . . . . . . . . 3

II Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10B. I mage processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13C. PDPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1. Sizing of small particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 232. Effect of Gaussian intensity profile of the incident light source . . . . . 243. Particle number density limitations . . . . . . . . . . . . . . . . . . . . 274. Measurement of size of bubbles . . . . . . . . . . . . . . . . . . . . . . 28

III Experimental Facility and Flow Conditions . . . . . . . . . . . . . . . . . . . . 32A. Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32B. F low conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

IV Break-up Frequency of Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 42A. Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42B. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1. Evolution of the volume-size bubble pdfs . . . . . . . . . . . . . . . . . 482. Rate of decay of number of bubbles of a certain class size . . . . . . . . 57

C. A phenomenological model for the bubble break-up frequency . . . . . . . 59

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V P DF of the Daughter Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B. Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71C. Comparison with other models . . . . . . . . . . . . . . . . . . . . . . . . . 76D. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

VI Characterization of the Frozen State . . . . . . . . . . . . . . . . . . . . . . . 100A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100B. Statistics of the bubble inter-arrival time . . . . . . . . . . . . . . . . . . . 101

1. The Poisson random process . . . . . . . . . . . . . . . . . . . . . . . . 1052. Effect of the Turbulent Kinetic Energy on the inter-arrival time. . . . . 106

C. Inter-arrival distance between bubbles . . . . . . . . . . . . . . . . . . . . . 1101. Effect of the Turbulent Kinetic Energy of the flow on the inter-arrival

distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113D. Dependence of the frozen bubble size pdf on . . . . . . . . . . . . . . . . 114E. Dependence of Dmax on . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

F. Dependence of the frozen bubble size pdf on the initial void fraction, . . 127

V I I C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 1

A Particle Size Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134A . D e fi n i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4B. Continuous distribution functions . . . . . . . . . . . . . . . . . . . . . . . 137

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

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LIST OF TABLES

II.1 Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 11

IV.1 Experimental Conditions. Re has been calculated based on the exit

velocity, U0, and the open section at the exit of the nozzle, DJ. Thisopen section was kept constant in the three cases of Set 3. . . . . . . . . 44

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LIST OF FIGURES

II.1 Original Image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15II.2 Processed Image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15II.3 Bubble Size Probability Density Function. . . . . . . . . . . . . . . . . . 16II.4 Particles generated from Radius= 1 pixel to Radius= 25 pixels. . . . . . 17II.5 Size of particle measured by the analysis program as function of the

size of the particle. () represents the size obtained from the mea-sured perimeter of particle and, () represents size obtained from themeasured area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

II.6 Dimensions of the probe volume . . . . . . . . . . . . . . . . . . . . . . 18II.7 Angular intensity distribution of different scattering modes of bubbles

in water. Solid line is reflection (p0), dashed line is primary refraction(p1) and dotted line is secondary refraction (p2). Figure taken fromCrowe et al., [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

II.8 Mie calculations of bubbles Phase-Size relationship for different scatter-ing angles. Figure taken from Crowe et al., [18] . . . . . . . . . . . . . . 20II.9 Phase-Diameter relation of Bubbles, = 60o. . . . . . . . . . . . . . . . 21II.10 Planar Phase Doppler Anemometry . . . . . . . . . . . . . . . . . . . . 26II.11 Standard PDPA Configuration . . . . . . . . . . . . . . . . . . . . . . . 29II.12 PDPA Configuration: Plan View . . . . . . . . . . . . . . . . . . . . . . 30II.13 Distribution of Photodetector in the Receiving Lens. D1 D2 and D3

are the three different regions of the lens capturing the scattered lightand focusing it into the respective detectors. . . . . . . . . . . . . . . . 30

III.1 Experimental Facility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33III.2 Detail of the Jet Nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

III.3 Centerline Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36III.4 Similar profiles of streamwise velocity. . . . . . . . . . . . . . . . . . . . 37III.5 Energy Spectra: Measured Spectrum, Wyngaard Corrected Spec-

trum. Lf indicates the wave number associated with the length of thefilm. X/DJ = 20, Uo = 12m/s. . . . . . . . . . . . . . . . . . . . . . . . 38

III.6 Energy Spectrum: Inertial Subrange is identified to measure the dissi-pation rate. Lf indicates the wave number associated to the length ofthe film. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

III.7 Evolution of Dissipation Rate of Turbulent Kinetic Energy, . . . . . . . 40III.8 Normalized Dissipation Rate of Turbulent Kinetic Energy, . . . . . . . 40

IV.1 Downstream evolution of the dissipation rate of TKE, . X0 4 DJindicates the virtual origin, the velocity at the exit of the nozzle isUo = 17m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

IV.2 Discretization of the flow region where the bubble break-up takes place.a) Section 1, Windows 1-5. b) Section 2, Windows 6-10. c) Section3. Windows 11-15. The downstream length of each measuring windowindicated by arrows is 7.14 mm. Experimental set 3a. Flow goes fromleft to right in each picture. . . . . . . . . . . . . . . . . . . . . . . . . . 45

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IV.3 Binary images corresponding to the flow conditions shown in figure IV.2.a) Section 1. b) Section 2. c) Section 3. . . . . . . . . . . . . . . . . . . 46

IV.4 Schematic representation of conditions used for the turbulent break-upmeasurements. Characteristic water jet Diameter, Dwj , the width andlength of all measuring windows were respectively l1 = 16.25mm and

Lw = 7.14mm. In all experiments l1/Dwj < 0.3. . . . . . . . . . . . . . 47IV.5 Evolution of the bubble V pdf, Da = 1.194mm. Experimental Set 3c. . . 49IV.6 Evidence of the existence of a frozen V pdf, which remains unchanged

in the last three measuring windows. Da = 1.194mm. ExperimentalSet 3c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

IV.7 Evolution of the flux of bubbles per window. The value refers to the to-tal flux, Nt U /Lw, obtained in each measuring window. Da = 1.194mm,experimental Set 3c. Lw is the length of the measuring window. . . . . 50

IV.8 Evolution of the bubble V pdf, Da = 0.394mm. Experimental Set 3a. . . 51IV.9 Evolution of the flux of bubbles per window. The value refers to the to-

tal flux, Nt U /Lw, obtained in each measuring window. Da = 0.394mm,

experimental Set 3a. Lw is the length of the measuring window. . . . . 51IV.10 Evolution of the bubble V pdf, Da = 0.584mm. Experimental Set 3b. . 52IV.11 Evolution of the flux of bubbles per window. The value refers to the to-

tal flux, Nt U /Lw, obtained in each measuring window. Da = 0.584mm,experimental Set 3b. Lw is the length of the measuring window. . . . . 52

IV.12 Evolution of the bubble V pdf, Da = 0.394mm. Experimental Set 1. . . 55IV.13 Evolution of the flux of bubbles per window. The value refers to the to-

tal flux, Nt U /Lw, obtained in each measuring window. Da = 0.394mm,experimental Set 1. Lw is the length of the measuring window. . . . . . 55

IV.14 Evolution of the bubble V pdf, Da = 0.394mm. Experimental Set 2. . . 56IV.15 Evolution of the flux of bubbles per window. The value refers to the to-

tal flux, Nt U /Lw, obtained in each measuring window. Da = 0.394mm,experimental Set 2. Lw is the length of the measuring window. . . . . . 56

IV.16 Downstream evolution of number of the largest class-size bubbles. Thenumber shown indicates the total number measured in each window over1,000 images corrected by the ratio of velocities between the measuringwindow and the first window measured, Nci = NiUi/U1. . . . . . . . . . 57

IV.17 Bubble break-up frequency. Experimental Set 1. . . . . . . . . . . . . . 58IV.18 Bubble break-up frequency. Experimental Set 2. . . . . . . . . . . . . . 59IV.19 Bubble break-up frequency. Experimental Set 3. . . . . . . . . . . . . . 60IV.20 Force per unit surface on the bubble. Solid line is the confining force

provided by surface tension and the broken line is the one given by the

turbulent stresses.The constant k = 6 is given in equation IV.7. . . . . 63IV.21 Evolution of the bubble break-up frequency and break-up velocity with

respect to the diameter of the bubble. = 2000 m2/s3. . . . . . . . . . . 64IV.22 Comparison of experimentally measured bubble break-up frequency with

the frequency calculated with model given in equation IV.20. a) Set 1,D0 = 2.7 mm. b) set 3b, D0 = 2.0mm. c) set 2, D0 = 1.67 = mm.Error bar indicates an estimated maximum 10% experimental error. . 66

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V.1 Difference of stresses associated to the formation of a bubble and itscomplementary, t1 = 12 (D1)2/3 6D0 and, t2 = 12 (D2)2/36D0

. In this example, the mother bubble is of size D0 = 1 mm, and

= 1000 m2 s3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72V.2 Difference of stresses associated to the formation of a bubble and its

complementary, given in equation V.10. In this particular case, themother bubble is of size D0 = 3 mm, and the dissipation rate is =1000 m2 s3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

V.3 Probability density functions of the daughter bubbles formed from thebreak-up of a mother one of size D0. a) Evolution of the pdf for variousvalues of and fixed D0 = 3mm; b) Influence of D0 on the the pf d fora fixed value of = 1000 m2 s3. . . . . . . . . . . . . . . . . . . . . . . 75

V.4 Daughter bubbles pdf predicted by model of Tsouris and Tavlarides1994. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

V.5 Daughter bubbles pdf predicted by model of Konno 1983 [36]. . . . . . 79V.6 Comparison of the daughter bubble size pdf predicted by the present

model and previously proposed models. Red line represents Tsouris andTavlarides 1994 [66], blue line represents Konno et al. 1983 [36], andblack lines represent our present model. . . . . . . . . . . . . . . . . . . 80

V.7 Right column, downstream evolution of the cumulative volume proba-bility density function. Left column, downstream evolution of the char-acteristic diameters D32 and Dv90%. . . . . . . . . . . . . . . . . . . . . 84

V.8 a) Downstream evolution of the Cumulative Volume Probability Den-sity Function. b) Downstream evolution of the Sauter Mean Diameter

Diameter, D32 =

NiD3i

NiD2iand Dv90%. Initial value of the dissipation rate

of TKE was 0 = 2, 000 m2/s3 at the air injection point, X/DJ = 15.

The solid lines represent the results obtained from the model integrat-ing equation V.32 and the solid symbols are the experimental measure-ments. Experimental Set 2. . . . . . . . . . . . . . . . . . . . . . . . . . 86


Diameter, D32 =

NiD3i

NiD2i

and Dv90%. Initial value of the dissipation rate


The solid lines represent the results obtained from the model integrat-ing equation V.32 and the solid symbols are the experimental measure-ments. Experimental Set 3a. . . . . . . . . . . . . . . . . . . . . . . . . 87

V.10 Time evolution of the characteristic break-up of a mother bubble show-

ing a binary bubble splitting mechanism. The images were taken at6,000 frames per second. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

V.11 Initial Bubble Size.a) Experimental Set 3a, Da1 = 0.394 mm, b) Experi-mental Set 3b, Da2 = 0.584 mm, c) Experimental Set 3c, Da3 = 1.194 mm. 89

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Diameter, D32 =

NiD3i

NiD2i

and Dv90%. Initial value of the dissipation rate


The solid lines represent the results obtained from the model integrat-ing equation V.32 and the solid symbols are the experimental measure-ments. Experimental Set 3c. . . . . . . . . . . . . . . . . . . . . . . . . 91

V.13 Maximum number of bubbles resulting from the break-up of a motherbubble of size D0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

V.14 Comparison of the daughter bubbles pdf resulting from a binary and atertiary splitting, = 1000 m2/s3, D0 = 2 mm . . . . . . . . . . . . . . 94

V.15 Downstream evolution of the Cumulative Volume Probability DensityFunction. The solid lines represent the results obtained using our modelassuming a tertiary breakup process model to integrate equation V.32,and the solid symbols are the experimental measurements. Experimen-tal Set 3c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

V.16 a) Downstream evolution of the Cumulative Volume Probability DensityFunction. b) Downstream evolution of the sauter Mean Diameter andDv90%.The solid lines represent the results obtained using Tsouris andTavlarides 1994 [66] model to integrate equation V.32, and the solidsymbols are the experimental measurements. Experimental Set 2. . . . 96

V.17 a) Downstream evolution of the Cumulative Volume Probability Den-sity Function (Vpdf). b) Downstream evolution of the sauter MeanDiameter and Dv90%. The solid lines represent the results obtained us-ing Konno et al. 1983 [36] model to integrate equation V.32, and thesolid symbols are the experimental measurements. Experimental Set 2. 97

VI.1 Time record of bubbles generated by a water jet. Re = 53, 000. . . . . . 102VI.2 Inter-arrival time pdf s of different classes of bubbles. Re = 53, 000.

Symbols are: 3 m < D < 20 m, 40 m < D < 60 m, 80 m < D < 100 m, 120 m < D < 200 m. . . . . . . . . . . . 103

VI.3 Pdfs of the normalized Inter-arrival time, t/tm (tm = mean inter arrival time) of different classes of bubbles. Re = 53, 000. Symbolsare: 3 m < D < 20 m, 40 m < D < 60 m, 80 m < D 200 m. . 107VI.5 Evolution of Rms/Mean Inter-arrival time with the local convective

velocity, U.The Reynolds Number of the jet has been varied from 32,000to 60,000. Symbols are: 3 m < D < 20 m, 40 m < D 200 m. Error bars are 5% of the value. . . . . . . . . . . . . . . . . . . 108

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VI.6 Evolution of the Inter-arrival time pdf of bubbles of size 40 m < D 200 m. . 111

VI.10 Evolution of the inter-arrival distance pdf of bubbles of size 40 m > Inertial range,

f(D

) ( D

)2 f or D


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If one considers an isotropic turbulent flow and introduces a lump of a second

immiscible fluid of interfacial surface tension and kinematic viscosity d, owing to

the surface tension, it will disintegrate into smaller droplets. These droplets will keep

breaking until a critical diameter, Dc, is achieved for which the straining and distortion

determined by the velocity difference, u2(D), are small and can no longer overcomethe restoring surface tension forces. Kolmogorov showed that the behavior of the droplet

can only be dependent on three dimensionless numbers: D ,d and the Weber number

We =u2(D) D

used in the present context. In the case of drops (bubbles) of diameter

of the order of the Kolmogorov scale, the three dimensionless numbers characterize the

process. However, for diameters much smaller than , viscous forces dominate over the

inertial forces and the process can be characterized by We andd . If D > > , as is the

case of interest here, the role of the viscosity of the continuous phase can be negligible

and only the Weber number characterizes the process [35]. According to Kolmogorov,

the maximum size of the stable droplet (bubble), Dmax, must correspond to a critical

Weber number, Wec, and, therefore from equations I.2 and I.3 one obtains:

Dmax = (Wec

)3/5 (

4

3)2/5 for D >> (I.4)

More recent studies [49], [5] have extended these ideas by introducing the effect of in-

termittency in equation I.4 to explain experimental results where exponents larger than2/5 have been found.

Although the dimensional argument proposed by Kolmogorov provides valuable

information on the maximum bubble (droplet) size, it does not give any information on

the shape of the pdf. Over the last years an important effort has been made by various

researchers to develop a model able to predict the shape of the pdfs of the droplet size

resulting from the turbulent break-up. The most successful and fundamental of these

models is based on the population-balance-equation (p-b-e), [66], which describes the

drop population history in terms of the particle properties and flow characteristics as

they interact among themselves and with the surrounding fluid [69].

p

t+x(vp) +v(Fp) =

D(R p) + Qb + Q

c + , (I.5)

where p(D, x, v, t) dD dx, dv is the probable number of particles (bubbles) in the diam-

eter range dD about D located in the spatial range dx about x with velocities in the


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range dv about v at a time t. F(D, x, v, t) = (dv/dt) is the force per unit mass on

the particle, R = (dD/dt) is the rate of change of the size D of a particle (bubble) due

to evaporation, condensation or dissolution. The rate of change of p due to break-up

or coalescence is given by Qb

and Qc

respectively and finally represents the rate of

change of the distribution function, p, caused by collisions with other particles which do

not result in coalescence. Eliminating the velocity dependency by integrating equation

I.5 over the entire velocity space, we get

n

t+x(v n) =

D(R n) + Qb + Qc , (I.6)

where Qb =

Qb dv, Qc =

Qc dv. The number of particles per unit volume and per

unit range of diameter is

n =

p dv , (I.7)

and the bar denotes an average over all velocities

v =

vp dv/

p dv , (I.8)

and

R =

R p dv/

p dv . (I.9)

Equation I.6 can just simply be expressed as

n(D, t)

t+x(v(D, t) n(D, t)) = De + Bi, (I.10)

where De and Bi respectively represent the rate of death and birth of droplets (bubbles)

due to break-up, coalescence and dissolution or evaporation. If dissolution and evapora-

tion effects are negligible, D (R n) = 0, a particle contributes to De when it disappears

after it coalesces with another one to create a bigger particle, or it breaks into smaller

ones. In the same manner, Bi, accounts for the coalescence of smaller particles and for

the break-up of bigger droplets to form smaller ones. The models used to characterize

the destruction (De) and formation (Bi) of droplets are generally based on the particle-

collision theory. Previous investigators, [66], [36] assumed that the break-up of a particle

of diameter D occurred when it collided with an eddy of size equal to or smaller than the

particle with sufficiently large energy to overcome surface tension. Similarly, coalescence

takes place when there is a collision between particles with a residence time longer than


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the time necessary to drain the film of fluid confined between the two particles.

The above birth and death terms are generally written as:

De = n(v)

0c(v, v

)hc(v, v)n(v)dv g(v)n(v) (I.11)

Bi =

v

0

c(v v, v)hc(v v, v)n(v v)n(v)dv +

v

f(v, v) m(v) g(v) n(v) dv

(I.12)

where c(v, v) is the coalescence efficiency of a collision between a particle of volume

v with another of volume v; hc(v, v)n(v) represents the collision rate of particles of

volume v with a particle of volume v; g(v) is the break-up frequency of a particle of

volume v; m(v) is the number of particles generated once a particle of volume v has been

broken, and f(v, v) is the probability density function of particles of volume v generatedfrom particles of volume v (this term is typically referred to as the probability density

function ofdaughter particles). Therefore, n(v)gc(v) = n(v)0 c(v, v

)hc(v, v)n(v)dv

is the amount of particles of volume v which die due to coalescence and g(v)n(v) is the

number of particle which die due to break-up. In the Bi equation,v0 c(v v, v)hc(v

v, v)n(vv)n(v)dv is the number of particles of volume v generated from coalescence,and

v f(v, v

)m(v)g(v)n(v)dv is the number of particles of volume v generated from

the break-up of bigger ones.

The difficulty involved in solving equation I.10 results from the need to model

the different terms of the equations, namely the coalescence efficiency, the break-up

efficiency and the pdf of the daughter droplets. Prince et al. [52] modeled the break-up

frequency, g(v), as:

gb(v) = (eddy particle collision frequency) (breakage eff iciency) (I.13)

The collision process is often described by analogy to kinetic theory of gasses whereby,

the eddy-particle collision frequency (hb) is given by:

hb(D) =

ne

Sed(u2e + u

2d)

1/2dne, (I.14)

where Sed is the collision cross-section given by Sed =(de+D)2

4 , de is the characteristic

length of the eddy which collides with a particle of diameter D and volume v = D3

6 ,

ue and ud are the characteristic turn over velocity of the eddy and the velocity of the


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particle respectively, and ne is the number density of eddies of size de. In equation I.14,

Sed and ne are two ambiguous parameters, difficult to characterize.

The breakage efficiency (b), which gives the probability of an eddy-particle

collision to result in particle break-up can be written as, [17]:

b(D) = exp

Ec

c1e

, (I.15)

where e is the average energy of an eddy, Ec is an activation energy, and c1 is a constant

of order unity which needs to be modeled. Coalescence can be modeled in the same way

as the breakage, where:

gc(v) = (particle particle collision frequency) (coalescence efficiency), (I.16)

The collision frequency between two particles of diameters Di and Dj , hc(Di, Dj ) is:

hc(Di, Dj)nj = SDiDj(u2Di

+ u2Dj )1/2nj (I.17)

where SDiDj is the collision cross-section given by SDiDj =(Di+Dj)

2

4 , Di is the diameter

of a particle, uDi is the velocity of the particle of diameter Di and ni is the number of

particles of size Di. Once the particles collide, they may stay in contact long enough

to coalesce or they may bounce back. Therefore, coalescence occurs when the residence

time is longer than the coalescence time and the efficiency can be then estimated as:

c(Di, Dj) = exp[ Tcijc2ij

], (I.18)

where Tcij is the coalescence time and ij is the residence time or contact time. Tcij can

be estimated calculating the drainage time of the fluid film formed between two disks of

diameters Di and Dj approaching at a velocity u = (u2Di

+ u2Dj )1/2.

The reliability of the model given by equations I.10-I.18 is based on the ac-

curacy of the assumed probability density function of the daughter particles, f(v, v),and the model used to account for the efficiency of both coalescence and bubble-eddy

collision. The above explained model requires the use of non-physical assumptions for

the number of eddies and the eddy-particle collision frequency and cross section which

make it questionable.

Regarding the resulting size probability density function of the daughter droplets


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8

(f(v, v)), a large number of distributions have been proposed. Some of these distribu-

tions are just based on statistical assumptions when no other information is available

[67], [17], [52]. Tsouris and Tavlarides [66] proposed a model based on surface energy

arguments. Furthermore, a combination of energy and statistical concepts have also been

used to propose more physical models, [36], [16], [41], [49]. Most of the models proposed

a daughter distribution function, which is independent of the intensity of turbulent flow

and/or the size of the mother particle, from which the daughter droplets have been

formed. All these models lack in generality and are not consistent with the experimental

observations.

The objective of this work is to obtain detailed experimental measurements

of the probability density function of the droplet sizes resulting from the break-up of

an immiscible fluid injected into a turbulent flow of known characteristics using novel

PDPA and digital image processing techniques. In order to isolate the problem, and

to prevent the additional complexity introduced by the use of turbines or any other

moving surfaces to generate the turbulence, we selected to study the turbulent break-up

by injecting air bubbles into the fully developed turbulent region along the central axis

of a high Reynolds number water jet. Through the use of Phase Doppler Techniques

(PDPA) and high speed image processing, we then measured the transient pdf as well

as the frozen pdf of the bubbles sizes resulting from the turbulent break-up over a wide

range of initial bubble sizes and turbulent conditions characterized by the turbulent ki-

netic energy (or the dissipation rate) of the underlying turbulence. These measurements

were then used to evaluate the various elements comprising the existing models such

as the break frequency g(v) and daughter droplets pdf f(v, v). The purpose of this

thesis is then to provide the fundamental knowledge of the mechanisms involved in the

turbulent break problem and their dependence on the fluids properties (surface tension,

viscosity and density ratio between the two fluids), flow properties (the dissipation rate

of turbulent kinetic energy of the underlying turbulence), and initial concentration of the

bubbles in the flow. This knowledge was then used to develop phenomenological models

based on dynamic arguments. First a model is presented for the break-up frequency

that will be used later to study the shape of the daughter pdf of the bubbles resulting

from the break of a mother one, and its dependence on the dissipation rate of turbulent


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9

kinetic energy and the size of the mother bubble. Finally, once the break process has

finished, the shape of the frozen (or unchanged) bubble size probability density function

and its dependence on the flow conditions is analyzed.


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Chapter II

Experimental Techniques

II.A Introduction

Detailed measurements of two-phase flows properties are crucial to characterize

and control many natural and industrial processes. Most of the techniques used in single

phase flows are not applicable when a second phase is present in the flow due to dramatic

changes in the physical properties of the fluids. The presence of these discontinuities,

produced by the presence of the discrete phase, makes it difficult, if not impossible, to

obtain reliable measurements of the flow characteristics.

Numerous measurement techniques are available for experimental studies of

two-phase flows. These techniques can be divided into different groups depending on

their performance as shown in table II.1. In some industrial processes, where the prop-

erties of the dispersed phase are not modified by the presence of the sampling probe,

the intrusive methods have been successfully applied. For instance, the particle size dis-

tribution has been extensively measured from the mass fraction of particles in a specific

size interval using either Sieving analysis or Sedimentation methods. Since the Sedimen-

tation methods are controlled by the terminal velocity of the particles, ut =(d

) g D2

18 1,

the particle size distribution is calculated from the time dependent concentration of par-

ticles at the measuring point. Both methods are very time consuming and require a large

amount of measurements to obtain statistically reliable results.

1d is the density of the discrete phase and and are the density and viscosity of the surroundingfluid respectively.

10


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11

Property of Intrusive Non-intrusive

Particles

Size Coulter Principle Phase Doppler AnemometrySedimentation Laser Diffraction

Sieving Light ScatteringFiber Optics Probes Image Processing

Concentration Isokinetic Sampling Phase Doppler AnemometryFiber Optics Probes Laser DiffractionConductivity Probes Laser Absorption

Light Scattering

Velocity Fiber Optics Probes Phase Doppler AnemometryParticle Image VelocimetryLaser Doppler Velocimetry

Table II.1: Measurement Techniques

Measurements of the dispersed phase mass flux and concentration have been

performed in the past using isokinetic sampling. The principle of this method is based

on introducing a sampling probe in the two-phase system and extracting a representative

population of particle by suction. The particles are collected in a bag filter for an inter-

val of time and analyzed later on. In this technique, the suction velocity has to be the

same as the local velocity of the flow (isokinetic conditions). The use of optical probes

to measure concentration, velocity and size distribution has been recently increased in

the last decade, especially in connection with gas-liquid two-phase flows [15]. A common

characteristic of the techniques described above is that all necessarily disturbed the flow

because of the presence of the associated probes.

Non-intrusive techniques are necessary to study the behavior of particulates

subjected to different forces produced by the flow. Problems such as dispersion of parti-

cles in turbulent flows, atomization, etc, require measurements on-line taken in real time.

Acquisition of a sample and later analysis may modify the properties of the dispersed

phase due to the sampling procedure and the presence of the probe. To avoid theses

difficulties, optical techniques have been widely used since the development of the laser.

Among the most common on-line, optical techniques developed in the last decades, are

Flow Visualization (image processing), Particle Tracking Velocimetry (PTV) which has

evolved to a more advanced Particle Image Velocimetry (PIV), laser diffraction, laser at-


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12

tenuation, Laser Doppler Velocimetry (LDV) and Phase Doppler Anemometry (PDA).

The above optical experimental techniques can be divided in two main groups, namely,

integral methods and local methods. Integral methods, such as laser attenuation and

laser diffraction, provide time resolved, spatially averaged information of the two-phase

system along a path defined by the portion of the beam crossing through the flow. The-

ses method are extensively used in two-dimensional flows, whereas their application to

the analysis of axisymmetric flows requires deconvolution techniques to extrapolate the

local properties of the flow, thereby causing the method to be more complicated and less

reliable to apply. The time resolution of these methods is determined by their sampling

frequency. For example, laser attenuation can be as fast as the response time of the

photodiode used (> 100Khz), while sizing techniques based on diffraction are limited by

the time necessary to process the signal and fit the proper distribution to it. Typically,

it is difficult to obtain an acquisition frequency higher than 1Khz.

On the other hand, local measurement techniques allow us the determination

of the properties of the flow with a temporal and spatial resolution which depends on

the method applied. The research in the field of the multiphase flows has advanced

considerably with the development of new local, optical techniques. The capability of

some of these methods allows the simultaneous measurement of the velocity and the size

of the particles at a certain spatial location determined by the probe (sampling) vol-

ume,i.e., PDA. Other non-intrusive, optical techniques are available to obtain detailed

flow visualization and quantitative and qualitative measurements. One method which

is rapidly gaining popularity in the field of experimental fluid dynamics is the Particle

Image Velocimetry (PIV). This technique utilizes a light sheet, typically generated by

a pulsed laser with a very short life time. The light scattered by the particles moving

through a small region of the flow is instantaneously collected by an imaging device,

usually but not necessarily a CCD (Charged Coupled Device). The instantaneous ve-

locity is obtained from a cross-correlation of two consecutive images obtained from two

associated laser pulses with a known time lag, whose duration is set depending on the

velocity of the flow. A recent development of the PIV techniques is that of Pawlak and

Armi [51], who replaced the pulsed laser by a continuous laser source of light to study

the vortex dynamics on arrested flows.


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13

Inaccuracies in PIV measurements emerge from particles moving out of the

illuminated plane in three dimensional flows, resulting in incorrect correlations. In addi-

tion, although advanced correlation techniques are able to discriminate the signal coming

from the seeded particles from that of the disperse phase to develop a two-phase PIV,

no method has been successfully developed up to date to simultaneously measure the

velocity and size of the particles. To overcome these problems, holographic techniques

and techniques based on the intensity of the scattered light are under consideration.

Particle imaging methods have been commonly utilized to acquire instanta-

neous information on the particle size, shape and concentration, at a given flowfield

location. In these techniques it is especially important that the particle images are in

focus, since particles out of focus will produce blurred images, which will eventually be

translated in erroneous size measurements. Increasing the shutter speed (exposure time)

or the time duration of the illumination system as well as decreasing the aperture of

the lens to increase the depth of the field will solve these problems at the expense of

reducing the illumination. Image post-processing is used to extrapolate the size of the

particles from the images recorded. Image processing techniques have also been used

to study the transient evolution of the concentration P DF of passive scalars injected in

turbulent flows [32].

The non-intrusive techniques described above allow the measurement either of

the velocity or the size of the particles. Finally, Phase Doppler Anemometry (PDA)

enables simultaneous measurements of the size and velocity of the dispersed phase, to-

gether with the velocity of the continuous phase. The improvement of the signal pro-

cessing analysis over the last few years makes this technique one of the most appropriate

to characterize multiphase flow systems. Since both image processing and PDA are

key elements in the development of this thesis, a more through treatment of these two

techniques is now presented in the following sections.

II.B Image processing

Image processing is used to determine the spatial instantaneous properties of

the flow and the particles dispersed in it. These instantaneous images can b e obtained


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using different techniques, a pulsed source of light with a short interval of time or a

continuous source of light and a short exposure time. The light sources used during the

development of this thesis are:

Continuous laser light from a 5 Watt Argon Ion Laser. The visualizations

generated through a laser sheet gave some very good quantitative results on the effect

of the flow on the break-up process but it was not used to quantify the size and number

of bubbles produced since this technique is limited to a 2-D cut of the flow.

Pulsed diffused white light provided by a Strobe light, that synchronized witha high speed video camera was used to follow, in time, the break-up of a single bubble

under certain conditions.

Continuous diffused white light from a 1000 Watt lamp was used to studytime evolution of the bubble size pdf as described below.

In the near field, close to the injection point, since the bubbles are not spheri-

cal, it was not possible to use PDA techniques which require the particle to be spherical

and the evolution of the bubbles size probability density function was characterized by

means of analyzing images. The images were taken by illuminating the flow with a dif-

fused white light and capturing the attenuated light by a Sony XC-77R CCD camera

placed in front of the source of light at a short exposure time, 1/80000 seconds. The

768(H) x 493 (V) image was recorded through a 640 x 480 frame grabber on a computer

for later processing. Each image was divided into windows of equal size for later process-

ing. Each window consisted of a rectangular section of approximately 7.14 x 16.25 mm2

area of the flow and was digitalized with a resolution of 200 x 455 pixels. The images

stored on the hard drive of the computer were analyzed as follows:2

The original image, shown in figure II.1, was imported to subtract the back-ground level before applying a threshold. Once the background was subtracted, a thresh-

old, close to a white level to eliminate random noise, was applied in order to detect the

particles contained in the image.

The portion of the images enclosed in each window was then made binary,so that, pixels with an intensity value lower than the threshold were assigned a zero

2To describe the image processing methodology more clearly, a case where the bubbles are nearlyspherical after pinching off from the air needle was selected as shown in figure II.1. In this particularexample the image was divided into four independent windows.


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Figure II.1: Original Image.

Figure II.2: Processed Image.


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16

0

2

4

6

8

10

0.5 1 1.5 2

Section 1Section 2Section 3Section 4

PDF(D

)

Equivalent Diameter (mm)

Figure II.3: Bubble Size Probability Density Function.

value and pixels with an intensity value higher than the threshold were given a value of

one, detecting in that way any particle contained in the image as shown in figure II.2.

From the binary images, the area occupied by each particle was easily determined by

counting the number of enclosed pixels. The equivalent diameterof the particle was then

calculated assuming a spherical shape, according to Dp = 2

A .

In the following development, particles that were touching the edges of the win-

dows were rejected to avoid sizing errors. Similarly, particles whose areas were less than3 pixels ( 100 m) were also rejected, thereby limiting the resolution of the systemto particles larger than 100 m. By processing a large number of images as described

above and calculating the size of the particles detected in the pictures, it was possible

to compute the bubble size probability density function at several downstream locations

in the flow, corresponding to each section or window, as shown in figure II.3.

To analyze the accuracy of the image processing routine used, a set of par-

ticle diameters ranging from 71 m to 2 mm was computationally generated as shown

in figure II.4. The diameter obtained from measuring the area and also from an alter-

native technique based on measurements of the perimeter of the contour of particle are

represented as a function of their real diameter in figure II.5, along with their associated

errors. As can be seen, the error of the diameter calculated by the processing program,

shown in figure II.5 remains always smaller than 15% in both cases and drops to less


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17

Figure II.4: Particles generated from Radius= 1 pixel to Radius= 25 pixels.

than 5% for particles of radius larger than 5 pixels (around 350m) diameter. Sincethe measurement technique based on the effective area is considerably more accurate,

resulting in error below 2% for particles of diameter larger than 280 m, this method

was selected to process the results presented in this thesis.

0

5

1 0

1 5

2 0

2 5

0 5 1 0 1 5 2 0 25

Rm

(pix

els)

R (pixels)

-15

-10

-5

0

5

10

15

0 5 10 15 20 25

Error(%

)

Radius (pixels)

Figure II.5: Size of particle measured by the analysis program as function of the size ofthe particle. () represents the size obtained from the measured perimeter of particle

and, () represents size obtained from the measured area.

II.C PDPA

Phase Doppler Anemometry (PDA) is a non-intrusive, optical technique that

processes the light scattered by a spherical particle as it crosses through a probe volume


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18

formed at the intersection of two monochromatic, coherent laser beams, as shown in

figure II.6. The PDA used for this work was a Phase Doppler Particle Analyzer (PDPA)

manufactured by Aerometrics. The multi-line beam light coming from a 2 Watts Innova

70C Coherent Argon Ion laser was driven into the Fiber Drive (FBD240) of the PDPA

system. The incoming beam was directed through an acousto-optics modulator (Bragg

Cell) which split the beam into two beams of the same intensity, one with the same

frequency as that of the incoming beam and the other (shifted beam) with a frequency

40 MHz higher. The two beams were directed into a color dispersion prism that produced

four monochromatic laser beams, two blue beams of wavelength 488 nm and two green

ones of wavelength 514 nm. Each individual beam was then picked off and directed

to a different exit of the Fiber Drive depending on its wavelength and frequency shift.

Therefore, at the exit of the Fiber Drive there were two unshifted beams (green and blue)

and two shifted beams (also green and blue), all coupled into a Coupler Assembly. A

Probe Volume

Y

X

dy

dx

dz

Figure II.6: Dimensions of the probe volume

10m cable containing 4 single mode, polarization preserving optical fibers connected the

Fiber Drive to the XMT204 transmitter. The transmitter had a 50 mm clear aperture

and, provided with a 500 mm focal length, focused two sets of orthogonal beams into a

probe volume. The two green beams, aligned in the plane of the mean direction of the

flow (vertical), were used to measure the axial component of the velocity and size of the

bubbles. The two blue beams, perpendicular to the flow, provided a measurement of the


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19

Figure II.7: Angular intensity distribution of different scattering modes of bubbles inwater. Solid line is reflection (p0), dashed line is primary refraction (p1) and dotted lineis secondary refraction (p2). Figure taken from Crowe et al., [18]

radial component of the velocity. The beam separation at the exit of the transmitter lens

was 20 mm and the beam diameter 0.7 mm. The corresponding crossing angle at the

probe volume was /2 = 1.14

o

, and the focused beam waist dw = 4f/dl 467 m,where f is the focal length of the transmitter lens, is the wavelength of the laser

beam and dl represents the beam diameter at the exit of the transmitter. For the above

configuration, indicated in figure II.6, the probe volume located at the focal point of the

lens used in the transmitter is an ellipsoid of the following dimensions:

dx =dw

cos(/2)= 467 m,

dz = dw = 467 m,

dy =dw

sin(/2)= 20 mm,

The light scattered by particles crossing the probe volume is composed of

diffraction, reflection and refraction [68]. The diffraction component of the scattered light

can be minimized when detected from an angle, , far from the forward direction. For

the case of air bubbles in water, whose relative index of refraction is m = na/nw = 0.75,


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Figure II.8: Mie calculations of bubbles Phase-Size relationship for different scatteringangles. Figure taken from Crowe et al., [18]

the most important scattering mode is reflection when acquiring the scattered light at

an angle in the range 50o and 80o, as described in detail in [18]. Bachalo, [3], showed

that the diameter of the particle is proportional to the phase shift between the incident

and scattered light when one of the scattering modes is dominant. Although, as seen

in figure II.7, the optimum scattering angle to capture the reflective light seems to be

between 70o and 80o, Mie calculations included for completeness in figure II.8, show that

the relationship between the phase shift and the diameter of the bubble remains linear

for scattering angles as small as 55o.

In the case of experimental facilities where there is an interface between two

different media separated by a wall, as is the case of a water tank, it is very impor-

tant that the receiver be installed perpendicular to the wall. In order to have accuratemeasurements, as described in the following sections, and to satisfy at the same time

the restrictions imposed by the experimental facility, we chose to work with the 60o

scattered light, an angle that guarantees a linear relationship phase-diameter, as shown

in figure II.9. Therefore, the RCV208 receiver unit was then placed at an angle of 60o

from the plane formed by the set of the green beams (vertical plane) and in the plane of


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21

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0

Phase 12Phase 13

Phase

(degrees)

D i a m e t e r (m )

Figure II.9: Phase-Diameter relation of Bubbles, = 60o.

incidence of the two blue beams. The scattered light by a bubble crossing through the

probe volume is acquired by the 72-mm clear aperture receiver with 500-mm lens, which

provides a higher quality signal with a higher Signal to Noise Ratio (SNR) than smaller

aperture receivers such as that documented in previous work by Kiger [33]. The signal

is focused through an internal 250 mm lens into 150 m spatial filter, thereby reducing

the length of the probe volume to dzef f = 150 500/250/sin(60o) = 346 m.After being color filtered, the light detected from three, well-defined, different

regions of the receiving lens is sent to four photodetectors. Three of the photodetec-tors capture the green component of the scattered light collected on the three distinct

regions. Subsequent processing enables the calculation of one component of the veloc-

ity (streamwise component) and also of the size of the bubble ([10], [3]). Similarly the

fourth photodetector captures the blue component of the light to determine the radial

component of the velocity, as in a LDV system.

A brief description of the measurements of velocities and size of particles will

be presented in the following paragraphs. For particles moving at velocities much slower

than the speed of light (U c), the velocity component perpendicular to the bisectorof the two incident beams is given by,

fD =U

f, (II.1)

where U is the velocity of the particle, f = /2sin(/2) is the fringe spacing of the

interference pattern formed by the intersection of the two laser beams, crossing at angle


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22

at the probe volume as seen in figure II.6, and fD is the frequency Doppler detected

by the photodetector. The phase shift between the Doppler signals detected by two

different detectors provides a measure of the size, D, of the moving droplet (bubble), as

described in [3]:

ab =2D

(m,p,geometry) . (II.2)

The parameter (m, p, geometry) depends on the ratio of index of refraction of the two

media, m, the scattering mode (p = 0 reflection, p = 1 first refraction, p = 2 second

refraction, . . . ) and the geometrical configuration of the receiver. A two-photodetector

system enables only the measurements of phase shifts between zero and 2. Therefore a

three-detector system is commonly used, providing two phase differences, 12 and 13,

from the comparison of the measurements of pairs of detectors having different separa-

tions. An example of such measurements is shown in figure II.9. The first approximation

of the diameter is obtained from the phase shift of two detectors (12). A more refined

measure of the diameter of the particle is calculated from the shift of a second pair of

detectors (13). Comparison between the diameter of the particle extrapolated from

(12) and that from (13) gives a final measure of the size of the droplet if both values

are within a certain tolerance or the measure is rejected otherwise.

The processor used is a RSA (Real time Signal Analyzer), younger brother of

the DSA (Doppler Signal Analyzer) whose operation has been very well documented

elsewhere [33]. A summary of the similarities and differences between the DSA and the

newer RSA processors is described below. The signal from the photodetector is sent to

the analog processor and processed as described in [33]. The quadrature downmixing

performed by the RSA processor is also sampled using 1-bit sampling to accelerate the

processing time. As described by Ingrahain et al, [31], the error increases for a certain

range of SNR (Signal to Noise Ratio) of the burst. Both processors have a burst detec-

tor in the analog and frequency domain (FTBD) to be able to validate signal with SNR

lower than 0dB while rejecting possible coherent noise. The main difference between the

DSA and the RSA processor is that, while the DSA is basically a sampler and relies on

a computer to do all the data processing, the RSA performs a large number of DFTs

( Direct Fourier Transforms) at once. Therefore, depending on the burst length, the

number of samples used to perform the DFT and compute the Doppler frequency is


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optimized in real time, resulting in greater accuracy and resolution in every burst. The

RSA processor normally performs four DFT in velocity measurements and up to fifteen

DFT in sizing measurements for each burst, providing an averaged value if the variance

of the collected data is within a certain tolerance. The algorithm uses the four greatest

number of samples obtained while the DSA uses a fixed number of samples chosen by

the user to perform the DFT. Unlike the DSA, in the RSA all the processing is done by

the processor and sent to the computer for display afterwards.

Although PDA has been accepted as one of the most reliable techniques to

simultaneously measure the size and velocity of particles in turbulent flows, it is not

an error-free technique and it could produce misleading results when improperly used.

The following sections summarize some of the inherent problems faced when using Phase

Doppler Anemometry and suggestions to overcome such problems. Special attention has

been given to obtaining reliable measurements on air-water systems

II.C.1 Sizing of small particles

The principle of geometrical optics used to compute the light scattered by par-

ticles and its applications in this thesis is based on the assumption that the particle

diameter is larger than the wavelength of the laser light source . This approach implies

that no interference of the scattering modes is considered, [68], and also that one of

the modes is the dominant component of the light scattered. The applicability of the

principle of geometric optics is restricted to relatively large particles of diameter, D,

such that 1 = D/, while for smaller particles diffraction becomes an importantcomponent of the scattered light, thereby invalidating the present method. For instance,

for the Argon ion laser source used in this study of = 514 nm, the minimum particle

size that can be accurately measured is approximately 2 m. Since the diameter of the

smallest bubbles measured in our experiments is 10 m, the linear relationship betweenthe phase shift and the bubble diameter is, therefore, adequately satisfied for the bubble

diameter range present in this work.

Furthermore, Bachalo and Sankar, [4], have shown that the interference be-

tween the reflecting and refracting modes gives rise to oscillations in the phase-diameter

calibration curve of the PDPA. Those oscillations can be minimized and even eliminated


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24

by choosing a proper PDPA configuration. Sankar et al., [59], using PSL (polystyrene

latex) fine particles of diameter smaller than 10 m, diluted in water, of relative index of

refraction m = 1.20, demonstrated that the uncertainty of the measurements decreases

from

1 m to

0.4 m when the mean collective angle is increased from 20o to 70o.

Calibration curves were generated by Sankar and Bachalo, [57], solving the

Lorenz-Mie theory and performing spatial integration of the light scattered over the

three different collection areas occupied by the three detectors, and therefore simulating

the light detected by the three distinct regions of the collecting lens. They showed that

the strong oscillations appearing in the calibration curves for certain configurations were

the result of erroneous calculations based on calculating the phase at single points on

the receiver surface instead of integrating along the entire surface. The receiver lens

readily removes these oscillations by performing a spatial integration of the scattered

light collected in the selected area.

II.C.2 Effect of Gaussian intensity profile of the incident light source

As described in the previous section the correct application of the PDPA is

based on using intensity of the dominant scattering mode to calculate the appropriate

phase-diameter calibration curve. However, since the incident beams have a gaussian

intensity profile, depending on the particle trajectory across the probe volume and on

the probe volume size, an incorrect, unexpected scattering mode may become dominant

and lead to erroneous diameter measurements, as described in references [57], and [58].

Sankar and Bachalo [57] analyzed the effect of the particle trajectory by using a geomet-

rical optics approach and taking into account the nonuniform illumination experienced

by the particles as they move through the measuring volume. They showed that wa-

ter drops of diameter comparable to that of the focused beam were reported as smaller

particles when they crossed the probe volume through its outer part, farther away fromthe receiver lens. Similarly, very large particles were erroneously reported as even larger

ones. They showed how errors were minimized and even eliminated by choosing the

proper optical configuration and also by increasing the ratio of beam diameter to maxi-

mum particle size. In particular they proved that increasing the beam intersection angle

changed the phase relationship between reflected and refracted modes, and effectively


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25

overcame the nonuniform illumination effect. For instance, by increasing the beam cross-

ing angle from 1.8o to 5.4o, the sizing error was decreased from 50% to 10% for particles

crossing through the edge of the measuring volume as presented in [58].

In the case of bubbles for a collecting angle of 60 o used in this work, reflec-

tion is the dominant mode. When a bubble crosses the probe volume through the edge

closer to the receiver, the refracted light emerges from the inner portion of the volume,

where the intensity of the beam is higher due to its gaussian profile. Similarly, the re-

flected component comes from the outer portion of the volume where the light intensity

is lower. Therefore, the refracted component becomes dominant in this case. Since the

phase-diameter correlation has been calculated assuming a reflected dominant mode, the

associated measurements would be erroneous. These errors caused by inhomogeneous

illumination are relatively small for small bubbles and also when the size of the probe

volume is large compared with that of the particles.

Grehan et al.,[26] studied the trajectory-ambiguity effect in air bubbles moving

in water configurations, dominated by reflection (refractive index of 0.75). They showed

that trajectory ambiguity overestimates the bubble diameter, in contrast with the case

of water droplets moving in air where it underestimates their size, with the effect being

less pronounced in the case of bubbles. As a solution that produces the smallest trajec-

tory effects, they proposed an optical configuration of the receiver unit of 70o off-axis,

resulting in errors smaller than 10% in all case.

After reviewing the published literature, the ambiguity error in our experimen-

tal facility was minimized by combining the following three methods:

Optimizing the crossing angle of the incident laser beams and using an optical config-uration of 60o off-axis, similar to the one proposed by Grehan, [26].

Using two set of photodetectors to perform phase validation. The diameter is mea-sured from the phase shift of two detectors,

12. This value is compared to the diameter

measurement obtained from the phase shift of another pair of detector, 13. The mea-

surement is validated if the values of both diameters agree. This method is not completely

effective by itself as reported by Bachalo [2]. A combination of this method with the one

described in the following point has been proved to reduce the measurement errors.

Using Intensity Validation to reject particles which pass through the edge of the probe


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X

Y

Z

Flow Direction

Plane ZOY

O

Plane XOY

Receiver

D4

D3

D2

D1Planar PDA: D3, D4

Modified Standard

PDA : D1, D2

Figure II.10: Planar Phase Doppler Anemometry

volume producing wrong measurements. A reported large bubble with a very low inten-

sity could be a erroneous measure of a small bubble crossing through the edge of the

probe volume.

Although it has not been used in the present work, it is very instructive to

describe one of the most recent systems developed to eliminate trajectory errors. The

Dual Phase Doppler Anemometry (DPDA) developed by Tropea et al. [65], consists of

a set of four photodetectors integrated in a single receiver unit combining a Planar PDA

(PPDA) with an Modified Standard PDA (MSPDA) configuration, the DPDA is shown

in figure II.10. Two of the detectors are located in the plane of the intersecting laser

beam containing the axis of the main direction of the flow. Although both the reflective

and refractive modes of the scattered light are collected, their contributions come with

time-delay and the refractive component has a higher amplitude. Therefore, the signal

from the refractive mode can be distinguished from that of the reflective one so that siz-

ing error can be eliminated. The other two detectors are positioned in a matter similar

to that of a standard PDA although the beam intersection plane is perpendicular to the

main direction of the flow in contrast to standard PDA where the main direction of the

flow is parallel to the beam intersection plane. Combination of the size measured by the

two configurations produces a trajectory error-free measurement validating only particles


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27

for which the size obtained using the MSPDA agrees with the one determined using the

PPDA within a certain tolerance. In addition, this system has been used to measure par-

ticle index of refraction combining the phase measurements detected from reflection and

refraction in SPDA configuration [50]. The diameter of the particle is determined from

the reflective mode since it is independent of the particle index of refraction. Knowing

the diameter and the optical parameters, the particle index of refraction can be obtained

from the phase of the refractive mode.

II.C.3 Particle number density limitations

The Phase Doppler method requires that only one particle be detected in the

probe volume at a time to avoid overlapping of doppler signal from different particles.

If two particles of different size are crossing the measuring volume at the same time, the

signal processed to calculate the diameter will have contributions from both particles

and must be rejected. The size of the probe volume has to be defined, for a given

concentration of particles, to minimize the probability of having more than one particle

in the sampling volume at the same time.

If the probability of finding n particles in the probe volume follows a Poisson

distribution given by:

Pn = N

n

eN

n! , (II.3)

where N = cpVm is the number of particles found in the probe volume, Vm, and cp is the

concentration of particles in the flow. The relative probability of finding two particles is:

P2 =P2P1

=N

2. (II.4)

For a maximum coincidence tolerance of 5%, P2 = 0.05, the maximum number of par-

ticles has to be N= 0.1 and, therefore, the optimal concentration of particle in the flow

must be:

cpopt =0.1

Vm. (II.5)

Considering the worst case of the experiments reported in this thesis where the flow rate

of air injected is Qa = 70ml/min, if a characteristic volume mean diameter of the bubble


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28

distribution is D30 = 200 m, the number of bubbles generated per unit time is:

N =Qa6

D330= 27 104 particles

second. (II.6)

At twenty jet diameters, where the bubble size distribution is measured, the amount of

liquid entrained by the water jet can be calculated by Qe 0.32(XD )Qo [54]. For thecase of the minimum flow rate utilized, Qo = 63 106m3/s, the total flow rate at 20

diameters is Q = 441 106m3/s and, therefore, the mean concentration of particles in

the flow is NQ = 0.61109particles/m3. On the other hand, and for a characteristic size of

the probe volume ofVm = 75.5

1012m3, the optimal concentration of bubbles obtained

from equation II.5 is cpopt = 1.33109particles/m3. Therefore cpopt NQ in all the cases

tested and the probability of finding more than one particle in the probe volume is kept

smaller than 5% having a very low rejection rate in our measurements. Additionally,

more recent developments in signal processing have been developed to measure the signal

at extremely high rates using fourier transform. With the new processors, included in

our system, it has been possible to measure the size of particles even when more than one

are present in the sampling volume by analyzing different portions of signal, providing

that the burst emitted by two different particles do not overlap. In all the experiments

reported here, the Doppler signal has been visualized and controlled on an Oscilloscope

in order to obtain a single burst and good visibility to guarantee reliable results.

II.C.4 Measurement of size of bubbles

The PDA technique has been successfully used to size water droplets in air for

many years. Not much work has been dedicated to light particles like air bubbles in water

using PDA. Until now, most of the data related to bubble size, velocity and concentra-

tion has been obtained using high speed imaging techniques, a technique which is very

limited and time consuming. Brena de la Rosa et al., [10], developed a theoretical and

experimental study of the characterization of bubbles using light scattering interferome-

try. This study was devoted to develop calibration curves for different configurations of


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29

X

Y

Z

Flow Direction

OBeam 1

Beam 2

= 60o

D1

D2D3

X

Z' in plane YZ

Figure II.11: Standard PDPA Configuration

the Phase Doppler Particle Analyzer (PDPA) in order to find an optimal configuration.

The configuration proposed by Brena de la Rosa et al. was used in a posterior study

by Brena de la Rosa et al., [9], to analyze the morphology of spheroidal bubbles using

light scattering interferometry. Since the reflective component of the light scattered by

air bubbles in water is dominant at some scattering angles (Figure II.7), the differenceof phase between the incident light and the reflected light is directly proportional to the

spherical scatterer as shown by Bachalo and Houser, [3], and plotted in Figure II.8.

Theoretical studies show that scattering collecting angles higher than 55o can

be used to measure the size of bubbles using a PDPA, [18], although, some experimental

studies often use a configuration with a receiver unit placed at 70o, [26] and 75o [11].

Most of the above work is related to size of bubbles of the order millimeter, among

experimental measurements concerning microbubbles of diameter from a few microns to

several hundred microns are those by Rightley, [55], and by Grehan at al. [26]. The for-

mer compares the measurements obtained with a PDPA with those done using a Malvern

diffraction system resulting in a good agreement when there is coincidence between the

probe volume formed by the two sets of laser beams (blue and green) used by the PDPA.

The later shows that the trajectory errors, discussed earlier, are eliminated when mea-


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30

Laser Beams

TransmitterReceiver

= 60o

Z

Y

Figure II.12: PDPA Configuration: Plan View

D1 D2 D3X

Z'

Figure II.13: Distribution of Photodetector in the Receiving Lens. D1 D2 and D3 arethe three different regions of the lens capturing the scattered light and focusing it intothe respective detectors.

suring bubbles at a receiving angle of 70o. Therefore, after an extensive review of the

bibliography and analysis of the most common problems associated with the measure-

ment of the size of bubbles (described in the previous sections) the configuration chosen

for this study was the one sketched in figures II.11 and II.12 (plan view) and whosedescription is presented in the following paragraph.

The water tank was designed in an hexagonal cross-section in order to have

both the transmitter and the receiver perpendicular to walls and avoid any undesired

scattering effect due to the presence of the interface, respecting at the same time the

optimal optical configuration described above. The vertical plane YX, containing the


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31

main direction of the flow, was formed by the intersection of the two green beams, as

seen in figure II.11. The receiver unit was placed on the horizontal plane YZ at an

angle = 60o off-axis from the transmitter axis. The scattered light collected on three

different areas of the receiver lens was sent to three photodetectors for later processing

as described earlier. The three areas were parallel to each other as shown in figure II.13

and, must be parallel to the plane of the fringes formed in the probe volume (which lay

in the YZ plane) to obtain a linear calibration curve between the phase shift and the size

of the bubble. Finally, the value of the receiver focal length was corrected using Snells

law to take into account the path of the scattered light changes due to the changes in

the index of refraction of the two propagating media.


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Chapter III

Experimental Facility and Flow

Conditions

The turbulent break-up problem has been extensively studied in various ex-

perimental configurations. Most of the experiments have been done in stirred tanks,

commonly used within the chemical engineering community. Their experimental results

have produced very useful data, mainly for their practical applications, but they have

been very difficult to understand due to the complicated type of turbulence produced in

this type of experiments. The flow is highly anisotropic consisting of strong shear regions

near the blades where the break-up mainly takes place. Although several correlations

have been developed to account for this effect, knowledge of the turbulent characteristics

is still very poor. To study the particles break-up into a homogeneous, isotropic and

nearly in equilibrium turbulent flow, a more controlled and much better understood type

of flow was chosen: an axisymmetric turbulent water jet. This type of flow, free of solid

boundaries, does not have the interaction of the particles with walls or any turbulence

generator, i.e, the blades used in the stirred tanks, which makes it more suitable to

produce a better understanding of the physical problem.

III.A Experimental Set-up

The experimental facility, shown in figure III.1, consisted of a submerged water

jet where air was injected at a given position along its centerline. In order to maximize

32


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33

10 mm

710 mm

Transmitter

Receiver

1.8 m

10 mm

600

PDPA

PDPA

Transmitter Receiver

Figure III.1: Experimental Facility.

the accuracy of the phase Doppler and other optical measurements, the tank in which

the jet discharged was designed with an hexagonal cross-section. The water jet nozzle

was located at the bottom and the jet discharged vertically upwards into the tank. In

order to minimize the recirculating flow produced in the tank by the entrainment of the

high momentum water jet, the water was allowed to overflow from the top of the tank

through a set of gutters placed on each side. Uniform velocity was achieved at the exit

of the nozzle in the submerged water jet by the use of two perforated plates located

upstream of a high-contraction-ratio (250:1) nozzle, as seen in figure III.2. Although

different nozzle exit diameters could be used, in all the experiments reported here the

nozzle exit diameter was 3.1 mm. The jet Reynolds number, Re = U0DJ/, based on

the velocity at the exit of the water jet, U0, on the diameter of the nozzle DJ and on


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34

the kinematic viscosity of the water, , could be systematically varied up to 105. Air

was injected coaxially on a selected downstream location at the axis of the submerged

turbulent water jet through a small hypodermic needle. To avoid any undesirable vibra-

tion effects at the air injection point, the needle was supported at the crossing points

with perforated plates. The bubble injection point, which determines the value of the

130mm

40mm

50 mm

PerforatedPlate

Water Input Water Input

Needle

3.1 mm

Figure III.2: Detail of the Jet Nozzle.

turbulent kinetic energy of the underlying turbulence where the bubble break-up pro-

cess starts, could be varied along the axis of the water jet from 10 to 50 jet diametersby moving the needle vertically. All the experiments presented in this work, however,

corresponded to injection points located between 10 and 25 jet diameters downstream

from the nozzle exit section. These positions were several diameters downstream from

the end of the potential core region of the jet in all Reynolds number cases studied and

the turbulence was fully developed in the scales of interest to our problem.


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35

The water flow rate, Qw, could be varied from 6.13 105 m3s1 to 2.3 104 m3s1 providing, for the 3.1 mm diameter nozzle, a range of exit velocities, Uo,

from 8 m/s to 30 m/s. The flow rate of air, Qa, could be increased from 5.8 108 m3s1

to 1.25

106 m3s1, resulting in the initial void fraction of air at the point of injection

= Qa/Qw, ranging from 2.5 104 to 2.0 102.

From the one-dimensional spectrum of the fluctuating component of the axial

velocity, and considering the turbulence to be locally homogeneous and isotropic, the

longitudinal integral scale, Lx, the dissipation rate of turbulent kinetic energy, , longi-

tudinal Taylor microscale, t, and the Reynolds number based in the Taylor scale, Rt ,

can be estimated as:

Lx =

E11(k1 = 0)

2 u2 ,

= 15

0

k21E11(k1)dk1,

1

2t=

30 u2 ,

Rt =u t

, (III.1)

where u

=

u2 is the rms of the streamwise component of the fluctuating veloc-

ity, and E11 is the one-dimensional spectrum of energy [29, 25, 23]. For example,

for a local velocity value of 6.4 m/s, measured at X/DJ = 20, the fluctuating veloc-

ity is approximately u = 1.6 m/s and 1000 m2/s3. Under this conditions theKolmogorov scale is = (3/)1/4 6 m, the Taylor microscale, estimated fromequation III.1 is t = 277 m and the corresponding Reynolds number is Rt = 443.

These values are consistent with those calculated using the correlations proposed by

[23], t = 0.88 (Re)1/2 X 240 m.

III.B Flow conditions

The axisymmetric jet has been extensively studied experimentally, analytically

as well as computationally during the past years. Important information has been col-

lected in the reviews by Monin [43], Hinze [29], and Townsend [64]. The work of Wyg-

nanski and Fiedler [70] has become an standard reference for quantitative descriptions of


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36

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60

Uo/UcBu=4.08, xo=3.78

Uo

/Uc

X/D

Figure III.3: Centerline Velocity.

the mean velocity profiles and turbulent properties of jets. The centerline axial velocity

for a self-preserving jet is given by [30]:

U0

Uc=

1

Bu(

X

DJ X0

DJ) , (III.2)

where U0 is the velocity at the exit of the nozzle, Uc is the centerline velocity at a distance

downstream X/DJ, Bu is an empirical constant which determines the rate of decaying

of the axial velocity and X0 is the virtual origin. The values of Bu and X0 may depend

on the exit conditions as pointed out by Hussein et al.[30].

The measured centerline velocity has been plotted in figure III.3 as a function

of the axial position, X/DJ. The axial centerline velocity, Uc, has been normalized

with the exit velocity, U0 and the downstream coordinate by the diameter of the nozzle,DJ. The experimental measurements have been compared with the equation III.2. A

downstream rate of decaying of the centerline velocity with a constant Bu = 4.08 and

a virtual origin, defined as the distance between the origin and the interception of the

straight line with the x-axis, X0/DJ = 3.78 has been found. These results are in a

good agreement with the constants presented by previous investigators [70],[30]. The


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37

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

6D

10D

15D

20D

30D

U/U

c

r /x

Figure III.4: Similar profiles of streamwise velocity.

centerline velocity velocity follows the characteristic 1/X decaying law of round jets for

downstream distances X/DJ > 10. The mean axial velocity normalized by the centerline

velocity, U/Uc, is represented versus the non-dimensional radial distance, r/x, in figure

III.4. The mean velocity profiles collapse at downstream distances of X/DJ > 10,

giving a radial position for which the axial velocity has decayed to 50% of its value at

the centerline, (r/x)1/2 0.1.Since the mean rate of dissipation, , will be necessary in the following chapters

to characterize the bubbles break-up, it was measured for several flow conditions at

different downstream positions along the axis of the jet to study its evolution. The energy

spectrum, shown in figure III.5, was calculated from hot film measurements applying

the Taylor hypothesis to convert from the temporal domain (frequency) to the spatialdomain (wave number), k = 2 f / U . Notice that the length scale associated with

the length of the film, Lf, falls within the inertial subrange of the energy spectrum.

Therefore, the spectrum drops off at a wave number slightly larger than that associated

with the length of the probe and, it was not possible to resolve smaller scales of the

flow. In order to overcome this problem, a first attempt to correct the measured one-


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38

10-9

10-8

10-7

10-6

10-5

0.0001

0.001

0.01

1 10 100 1000 104 105

Energy Spectrum

Corrected Energy

E(m3/s2)

k=2f/U (1/m)

-5/3

Lw

Figure III.5: Energy Spectra: Measured Spectrum, Wyngaard Corrected Spectrum.Lf indicates the wave number associated with the length of the film. X/DJ = 20,Uo = 12m/s.

dimensional spectra was done by using Wyngaards correction, [71], represented in figure

III.5. Although Wyngaards correction has been successfully used in many cases, the

values of obtained using equation III.1 with E11 being the corrected spectra were much

smaller than expected. An alternative procedure, described below, was applied.

To calculate the dissipation rate of turbulent kinetic energy, , an alternative

method was applied. Since the inertial subrange can still be identified in the measured

energy spectra as shown by the -5/3 line in figure III.5, it can be used to infer the

dissipation rate, . At high enough Reynolds numbers, the inertial range obeys the

fo

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18-Bubble 1998 Martinez Bazan PhD Thesis