ORNL-TM-3718 by Jon Morrow

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HELP OAK ’RIDGE

NATIONAL operated

v‘

LABORATORY.,

UNION CARBIDE CORPORATION NUCLEAR DIVISION for the

A-

U.S. ATOMIC ENERGY COMMISSION

ORNL- TM-3718

MASS TRANSFER BETWEEN SMALL BUBBLES A N D LIQUIDS IN COCURRENT TURBULENT PIPELINE FLOW (Thesis)

T. S. Kress

&the

Submitted as a dissertation to the Graduate Council of The University of Tennessee i n partial fulfillment of requirements for the degree Doctor of Philosophy.

.:.

-.- ,..

This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness M usefulness of any information, apparatus, product or process disclosed, or represents that i t s use would not infringe privately owned rights.

OFLNL-TM-3718

Contract No.

W-7405- eng-26

MASS TRANSFER BETWEEN SMALL BUBBLES AND LIQUIDS I N

COCURRENT WREYULENT PIPELINE FLOW

T. S. Kress

Submitted as a d i s s e r t a t i o n t o t h e Graduate Council o f The University of Tennessee i n p a r t i a l f u l f i l l m e n t of t h e requirements for t h e degree Doctor o f Philosophy.

prepared as an account of work United States Government. Neither nor the United States Atomic Energy any of their employees, nor any of their tpntra@or$, subcontractors, or their employees, or implied, or assumes any for the accuracy, com-

APRIL 1972

information, apparatus, or represents that its use woutd not infringe privately owned rights.

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37830 operated by UNION CARBIDE CORPORATION for t h e U. S. ATOMIC ENERGY COMMISSION

ACKNOWLEDGMENTS This i n v e s t i g a t i o n was performed a t t h e Oak Ridge National Laboratory operated by t h e Union Carbide Corporation f o r t h e U. S. Atomic Energy Commission.

The author i s p a r t i c u l a r l y g r a t e f u l f o r

t h e helpful discussions, guidance, and d i r e c t i o n given t h e research by t h e major advisor, Dr. J. J. Keyes, and t h e support of D r . H. W.

Hoffman, Head of t h e Heat Transfer-Fluid Dynamics Department of

t h e Reactor Division. Dunlap S c o t t of t h e Molten-Salt Reactor Program suggested t h e problem and provided i n i t i a l funding under t h e MSR Program,

D r . F. N.

Peebles, Dean of Engineering, The University of Tennessee, suggested t h e use of t h e oxygen-glycerine-water system and c a r r i e d out t h e o r i g i n a l a n a l y s i s of t h e a p p l i c a b i l i t y t o xenon-molten s a l t systems. The c o n t r i b u t i o n s of t h e following ORNL s t a f f members a r e a l s o gratefâ&#x20AC;&#x2122;ully acknowledged:

R. J. Ked1 f o r h i s bubble generator develop-

ment work drawn upon here; D r . C. W. Nestor f o r computer s o l u t i o n of t h e a n a l y t i c a l model; and Frances Burkhalter f o r p r e p a r a t i o n of t h e figures. S p e c i a l thanks are given t o Margie Adair f o r her s k i l l f â&#x20AC;&#x2122; u l and c h e e r f u l p r e p a r a t i o n of t h e preliminary and f i n a l manuscripts, and, of course, t o my wife, Dee, f o r her forbearance,

ABSTRACT Liquid-phase-controlled mobile-interface mass-transfer c o e f f i c i e n t s were measured for t r a n s f e r of dissolved oxygen i n t o small helium bubbles i n cocurrent t u r b u l e n t p i p e l i n e flow f o r f i v e d i f f e r e n t mixtures of glycerine and water.

These c o e f f i c i e n t s were determined by t r a n s i e n t

response experiments i n which t h e dissolved oxygen was measured a t only one p o s i t i o n i n a closed r e c i r c u l a t i n g loop and recorded a s a function

o f time.

Using an independent photographic determination of t h e i n t e r -

f a c i a l areas, t h e mass-transfer c o e f f i c i e n t s were e x t r a c t e d from t h e s e measured t r a n s i e n t s and determined as flmctions of pipe Reynolds number, Schmidt number, bubble Sauter-mean diameter, and g r a v i t a t i o n a l o r i e n t a t i o n of t h e flow.

Two general types of behavior were observed: (1) Above pipe Reynolds numbers f o r which t u r b u l e n t i n e r t i a f o r c e s

dominate over g r a v i t a t i o n a l forces, h o r i z o n t a l and v e r t i c a l flow masst r a n s f e r c o e f f i c i e n t s were i d e n t i c a l and varied according t o t h e regression e quat i on Sh/Sc’’”

0.34 Re”’94 ( dVS / D ) l ‘ ”

The observed Reynolds number exponent agreed g e n e r a l l y w i t h other l i t e r a t u r e d a t a f o r cocurrent p i p e l i n e flow but d i d not agree with expectation based on equivalent power d i s s i p a t i o n comparisons with a g i t a t e d v e s s e l data. (2)

Below t h e Reynolds numbers t h a t marked t h e equivalence of hor-

i z o n t a l and v e r t i c a l flow c o e f f i c i e n t s , t h e horizontal-flow c o e f f i c i e n t s continued t o vary according t o t h e above equation u n t i l , a t low flows, iii

iv severe s t r a t i f i c a t i o n of t h e bubbles made operation impractical.

The

v e r t i c a l - f l o w c o e f f i c i e n t s a t t h e s e lower Reynolds numbers underwent a t r a n s i t i o n t o approach constant asymptotes c h a r a c t e r i s t i c of t h e bubbles r i s i n g through t h e quiescent l i q u i d .

For small bubbles i n t h e most

viscous mixture t e s t e d , both h o r i z o n t a l and v e r t i c a l - f l o w c o e f f i c i e n t s underwent t h i s t r a n s i t i o n . An expression was developed f o r t h e r e l a t i v e importance of t u r b u l e n t i n e r t i a l f o r c e s compared t o g r a v i t a t i o n a l f o r c e s , Fi/Fg.

This r a t i o

served as a good c r i t e r i o n f o r e s t a b l i s h i n g t h e p i p e Reynolds numbers above which h o r i z o n t a l and v e r t i c a l - f l o w mass-transfer c o e f f i c i e n t s were identical.

I n addition, it proved t o be a u s e f u l l i n e a r s c a l i n g f a c t o r

f o r c a l c u l a t i n g t h e v e r t i c a l - f l o w c o e f f i c i e n t s i n t h e above mentioned t r a n s it i o n region. A seemingly anomalous behavior was observed i n d a t a f o r water

( p l u s about 200 ppm N-butyl alcohol) which exhibited a s i g n i f i c a n t l y smaller Reynolds number exponent than d i d d a t a f o r t h e o t h e r f l u i d mixtures.

To explain t h i s behavior, a two-regime ''turbulence i n t e r a c t i o n "

model was formulated by balancing t u r b u l e n t i n e r t i a l f o r c e s with drag forces.

The r e l a t i o n s h i p of t h e drag f o r c e s t o t h e bubble r e l a t i v e - f l o w

Reynolds number gave r i s e t o t h e two regimes with t h e d i v i s i o n being a t Reb = 2.

The r e s u l t i n g bubble mean v e l o c i t i e s f o r each regime were then

s u b s t i t u t e d i n t o Frgssling-type equations t o determine t h e mass-transfer behavior. (Reb

The r e s u l t i n g Reynolds number exponent f o r one of t h e regimes

2) agreed w e l l w i t h t h e observed d a t a but t h e p r e d i c t e d exponent

f o r t h e e f f e c t of t h e r a t i o of bubble mean diameter t o conduit diameter, d

/D,

w a s l e s s than t h a t observed.

The mass-transfer equations

v r e s u l t i a g from t h e o t h e r regime (Re > 2) agreed w e l l with d a t a f o r b p a r t i c l e s i n a g i t a t e d v e s s e l s and a l s o compared favorably with t h e water d a t a mentioned above.

For comparison, a second a n a l y t i c a l model was developed based on s u r f a c e renewal concepts and an eddy d i f f u s i v i t y t h a t v a r i e d w ith Reynolds number, Schmidt number, bubble diameter, i n t e r f a c i a l condit i o n , and p o s i t i o n away from an i n t e r f a c e .

Using a d i g i t a l computer,

a t e n t a t i v e numerical s o l u t i o n was obtained which t r e a t e d a dimensionl e s s renewal period, T,,

as a parameter.

This renewal pe r iod was

i n t e r p r e t e d as being a measure of t h e r i g i d i t y of t h e i n t e r f a c e , T, + 0 corresponding t o f u l l y mobile and T, + approximately 2.7 ( i n t h i s case) t o f u l l y r i g i d interfaces.

TABLE OF CONTENTS

CHAPTER I

â&#x20AC;&#x2DC;

.......................

INTRODUCTION

LITERATURE REVIEW

PAGE 1

.................... Experiment al-C ocurrent Flow . . . . . . . . . . . . . . Experimental-Agitated Vessels . . . . . . . . . . . . . Discussion of Available Experimental Data . . . . . . . Theoretical . . . . . . . . . . . . . . . . . . . . . .

...............

...........

Surface Renewal Models

Modeling of t h e Eddy S t r u c t u r e

.............. Dimensional Analysis (Empiricism) . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . 111. DESCRIPTION OF EXPERIMENT . . . . . . . . . . . . . . . . Transient Response Technique . . . . . . . . . . . . . Apparatus . . . . . . . . . . . . . . . . . . . . . . . Pump . . . . . . . . . . . . . . . . . . . . . . . . Liquid Flow Measurement . . . . . . . . . . . . . . Temperature S t a b i l i z a t i o n . . . . . . . . . . . . . Turbulence I n t e r a c t i o n s

Gas Flow Measurement

................

............ Bubble Generation . . . . . . . . . . . . . . . . . Bubble Separation . . . . . . . . . . . . . . . . . Test Section . . . . . . . . . . . . . . . . . . . . Dissolved Oxygen Measurement

vii

5 7

16 17

19 19 23 23

26 27

27 28

viii PAGE

CHAPTER Photographic Bubble Surface Area Determination .

....................... End E f f e c t . . . . . . . . . . . . . . . . . . . . . . Summary o f Experimental Procedure . . . . . . . . . . . System

EXPERIMENTAL RESULTS Unadjusted R esu lts

...................

..................

47 49 55 55

Equivalence of Horizontal and V e r t i c a l Flow

................... V e r t i c a l O r i e n t a t i o n Low-Flow Asymptotes . . . . . . . Mass T ran sfer C o e f f i c i e n t s . . . . . . . . . . . . . . Mass T ran sfer

56 60 60

C alcu latin g V e r t i c a l Flow Mass-Transfer C o e f f i c i e n t s

.............. ........... Recommended C o r r e l a t i o n s . . . . . . . . . . . . . . . V. THEOWTICAL CONSIDERATIONS . . . . . . . . . . . . . . . . Turbulence I n t e r a c t i o n Model . . . . . . . . . . . . . Surface Renewal Model . . . . . . . . . . . . . . . . . V I. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . V I 1. RECOMMENDATIONS FOR FURTHER STUDY . . . . . . . . . . . . Experiment a 1 . . . . . . . . . . . . . . . . . . . . . Theoretical . . . . . . . . . . . . . . . . . . . . . . LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . A PHYSICAL PROPERTIES OF AQUEOUS-GLYCEROL MIXTURES . . . . . Less Than 1 . 5 i 3 Comparison w ith A gita te d Vessels f o r Fi/F

94 100 100 102 105 111 111

ix W

CHAPTER B

PAGE DERIVATION OF EQUATIONS FOR CONCENTRATION CHANGES ACROSS A GAS-LIQUID CONTACTOR

.............

.............. .................

INSTRUMENT APPLICATION DRAWING

121

INSTRUMENT CALIBRATIONS

123

EVALUATION OF EFFECT OF OXYGEN SENSOR RESPONSE SPEED ON

THE MEASURED TRANSIENT RESPONSE OF THE SYSTEM

.....

....... G ESTIMATE OF ERROR DUE TO END-EFFECT ADJUSTMENTS . . . . . H MASSTRANSFERDATA . . . . . . . . . . . . . . . . . . . . LISTOFSYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . F

117

MASS BALANCES FOR THE SURFACE RENEWAL MODEL

129 131

134 137 163

LIST OF TABLES PAGE

TABLE

I. 11.

Ranges of Independent Variables Covered

...........

Physical P r o p e r t i e s of Aqueous-Glycerol Mixtures (25째C) Data of Jordan, Ackerman, and Berger"

Categories of Data C o r r e l a t i o n for Mass Transfer from

a Turbulent Liquid t o Gas Bubbles 111.

.........

........

Experimental Conditions f o r Runs Used t o Validate Surface Area Determination Method f o r V e r t i c a l

Flows..

.......................

Experimental Conditions f o r Runs Shown on Horizontal

Flow Volume Fraction C o r r e l a t i o n VI.

...........

Conditions a t Which Horizontal and V e r t i c a l Flow Mass-Transfer C o e f f i c i e n t s Become Equal

(Lamont s Data)"

...................

5 !:I

LIST OF FIGURES

PAGE

FIGURE

.........

Photograph of t h e Mass Transfer F a c i l i t y

Schematic Diagram of t h e Main C i r c u i t of t h e

................ t h e Bubble Generator . . . . . . . . . . . . .

Experimental Apparatus

Diagram of

Comparison of Measured Bubble Sizes with t h e

................. t h e Bubble Separator . . . . . . . . . . . . .

D i s t r i b u t i o n Function

Diagram of

Comparison of I n t e r f a c i a l Areas Per Unit Volume Measured

D i r e c t l y from Photographs with Those Established Through t h e D i s t r i b u t i o n Function.

... ......

V e r t i c a l Flow

C o r r e l a t i o n of Horizontal Flow Volume Fraction

Comparison of Measured and Calculated I n t e r f a c i a l Areas Per Unit Volume.

Horizontal Flow

...........

Typical Experimental Concentration Transient I l l u s t r a t i n g

....

Straight-Line Behavior on Semi-Log Coordinants

10.

Typical Examples of Bubble Photographs: a. I n l e t

b. E x i t

V e r t i c a l Flow, 37.5% Glycerine-62.5% Water, QR = 20 gpm, Q /Q g R inches

0.3%, D

= 2

inches, and dvs = 0.023

........................

11.

Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter. Plus -200 ppm N-Butyl Alcohol.

Horizontal and

V e r t i c a l Flow i n a 2-inch Diameter Conduit

Water

......

xii

PAGE

FIGURE

12.

Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number

a s a Function of Bubble Sauter-Mean Diameter. Glycerine-87.5% Water.

Horizontal and V e r t i c a l Flow

i n a 2-inch Diameter Conduit

.............

i n a 2-inch Diameter Conduit

.............

Glycerine-62.5% Water.

.............

Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter. Glycerine-50% Water.

50%

Horizontal and V e r t i c a l Flow

i n a 2-inch Diameter Conduit

.............

Observed Types of Horizontal Flow Behavior. d = 0.02 inches and D = 2 inches vs

..........

Equivalent Power Dissipation Comparison of Results with Agitated Vessel Data

18.

37.5%

Horizontal and V e r t i c a l Flow

i n a 2-inch Diameter Conduit

17.

Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter.

16.

25%

Horizontal and V e r t i c a l Flow

Glycerine-75% Water.

15.

Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter.

14.

12.5%

...............

Equivalent Power Dissipation Comparison of Gravity

...... ...........

Dominated Results with Agitated Vessel Data

19.

C o r r e l a t i o n of Horizontal Flow Data

20.

Dimensionless Variation o f Eddy D i f f u s i v i t y with Distance from an Interface.

Effect of Surface Condition

....

xiii W

FIGURE 21.

PAGE V ariatio n of Eddy D if f 'usivity with Distance from an Interface.

Comparison of Calculated Values with

Data of S l e i c h e r 22.

.... ....... .. .... ..

Numerical Results of t h e Surface Renewal Model.

Plots

of a, b, and c (Exponents on Re, Sc, and d/D, Respectively) a s Functions of t h e Dimensionless

23.

...................... Schmidt Numbers of Glycerine-Water Mixtures . . . . . . .

24.

Henry's Law Constant f o r Oxygen S o l u b i l i t y i n

Period,T,

Glycerine-Water Mixtures

25.

.. . . . .. . . . . . . . ,

..... ................. D e n s i t i e s of Glycerine-Water Mixtures . . . . . . . . . . V i s c o s i t i e s of Glycerine-Water Mixtures . . . . . . . . .

28.

Instrument Application Drawing of t h e Experiment

33.

.............. . ........ Bubble S i z e Range Produced by t h e Bubble Generator . . . . C a l i b r a t i o n of Rotameter No. 1 (100 gym) . . . . . . . . . C a l i b r a t i o n of Rotameter No. 2 (40 gym) . . . . . . . . . C a l i b r a t i o n o f Rotameter No. 3 (8 gpm) . . . . . . . . . . C a l i b r a t i o n of Gas-Flow Meter a t 50 p s i g . . . . . . . . .

34.

C a l i b r a t i o n of Oxygen Sensors i n two Mixtures of

Facility

30.

31. 32.

113

Data of Jordan, Ackerman,

and Berger

112

Molecular Diff'usion C o e f f i c i e n t s f o r Oxygen i n Glycerine-Water Mixtures.

26.

Glycerine and Water

..................

114 115

116

121

123 124 125

126 127

128

xiv PAGE

FI G U M

35.

N-Butyl Alcohol.

36.

39.

43.

Water.

44.

V e r t i c a l Flow

Horizontal Flow

143

144

37.5% Glycerine-

............ .

1L5

50% Glycerine-50%

. . . .... ..... . . . . .

Unadjusted Mass Transfer Data. Water.

45.

..............

Horizontal Flow

Unadjusted Mass Transfer Data.

142

37.5% Glycerine-

Unadjusted Mass Transfer Data.

62.5% Water.

141

25% Glycerine-75%

................

V e r t i c a l Flow

140

25% Glycerine-75%

Unadjusted Mass Transfer Data.

62.5% Water.

42.

.............

.................

Horizontal Flow

139

12.5% Glycerine-

Horizontal Flow

V e r t i c a l Flow

...........

....... .......

Unadjusted Mass Transfer Data. Water.

41.

V e r t i c a l Flow

138

12.5% Glycerine-

Unadjusted Mass Transfer Data. Water.

40.

Horizontal Flow

Unadjusted Mass Transfer Data.

87.5% Water.

............

Water Plus -200 ppm

Unadjusted Mass Transfer Data.

87.5% Water. 38.

V e r t i c a l Flow

Unadjusted Mass Transfer Data. N-Butyl Alcohol.

37.

Water Plus -200 ppm

Unadjusted Mass Transfer Data.

146

50% Glycerine-50%

...........,....

147

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter.

Water Plus -200 ppm N-Butyl Alcohol.

Horizontal and V e r t i c a l Flow

.............

1413

xv W

PAGE

FIGURE

46.

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter.

12.5% Glycerine 87.5% Water.

and V e r t i c a l Flow

Horizontal

. . . . . . . . . . . . . . . . . ..

149

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diamet e r

and V e r t i c a l Flow

48.

. Horiz ont a 1 ...................

25% Glyce r ine-75% Water

150

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter.

37.5% Glycerine-62.5% Water.

and V e r t i c a l Flow

49.

Horizontal

. . . . . . . . . . . . . . . . . ..

151

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter.

50% Glycerine-5& Water.

V e r t i c a l Flow 50.

Mass Transfer Data Adjusted f o r End-Effect.

. . . . . ..

153

V e r t i c a l Flow

......

Horizontal Flow

154

12.5%

. . . . . . . ..

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-87.5% Water.

Water Plus

Horizontal Flow

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-87.5% Water.

53.

152

Water Plus

V e r t i c a l Flow

Mass Transfer Data Adjusted for End-Effect. -200 ppm N-Butyl Alcohol.

52.

....................

-200 ppm N-Butyl Alcohol.

51.

Horizontal and

155

12.5%

........

156

xvi

FIGURE

54.

PAGE Mass Transfer Data Adjusted f o r End-Effect.

Glycerine-75% Water.

55.

Horizontal Flow

V e r t i c a l Flow

Glycerine-50% Water.

Horizontal Flow

158

37.5%

........

153

50%

..........

Mass Transfer Data Adjusted f o r End-Effect.

157

25%

.........

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-50% Water.

58.

Horizontal Flow

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-62.5% Water.

57.

..........

Mass Transfer Data Adjusted f o r End-Effect.

Glycerine-75% Water.

56.

V e r t i c a l Flow

25%

160

50%

.........

161

CHAPTER I INTRODUCTION

When gas bubbles a r e dispersed i n a continuous l i q u i d phase, dissolved c o n s t i t u e n t s of s u f f i c i e n t v o l a t i l i t y w i l l be exchanged between t h e l i q u i d and t h e bubbles, e f f e c t i v e l y r e d i s t r i b u t i n g any concentration imbalances t h a t e x i s t .

Common p r a c t i c e s involve contacting gas bubbles

with an a g i t a t e d l i q u i d i n such a manner t h a t a r e l a t i v e l y l a r g e i n t e r f a c i a l area i s available.

Techniques such as passing gas bubbles up

through a l i q u i d column or mechanically s t i r r i n g a gas-liquid mixture i n

a tank have been studied extensively and t h e design technology f o r t h e s e i s relatively f i r m .

However, one method, cocurrent t u r b u l e n t flow i n a

p i p e l i n e , has not been given a g r e a t d e a l of a t t e n t i o n .

A review of t h e

l i t e r a t u r e has shown t h a t t h e a v a i l a b l e d a t a a r e i n s u f f i c i e n t t o allow confident determination of t h e mass-transfer r a t e s i n such a system. This research, then, w a s undertaken t o provide a d d i t i o n a l information t h a t w i l l a i d i n determining l i q u i d phase c o n t r o l l e d mass-transfer r a t e s for cocurrent t u r b u l e n t flow of s m a l l bubbles and l i q u i d s i n a pipeline.

The impetus for t h i s work w a s provided by t h e Molten S a l t Breeder Reactor (MSBR) Program o f t h e Oak Ridge National Laboratory where r e c e n t remarkably successful operation of a molten s a l t f'ueled nuclear reactor' has convincingly demonstrated t h e f e a s i b i l i t y of t h i s power system.

The

economic competitiveness of an MSBR, however, depends t o a s i g n i f i c a n t e x t e n t on t h e breeding r a t i o obtainable,

The production within t h e

l i q u i d f u e l of fission-product poisons, p r i n c i p a l l y xenon-135, e x e r t s

a s t r o n g influence on t h e neutron economy of t h e r e a c t o r and consequently on t h e breeding r a t i o i t s e l f . One method proposed f o r removing t h e xenon would r e q u i r e i n j e c t i n g small helium bubbles i n t o t h e t u r b u l e n t l y flowing regions of t h e f u e l coolant stream and allowing them t o c i r c u l a t e with t h e f'uel.

Since such

bubbles would be d e f i c i e n t i n xenon compared t o t h e nearby bulk stream, t h e dissolved xenon would be t r a n s f e r r e d by t u r b u l e n t d i f f u s i o n across t h e concentration p o t e n t i a l gradient.

By continuous i n j e c t i o n and

removal of t h e helium bubbles t h e equilibrium xenon poisoning can be s i g n i f i c a n t l y reduced,

Since a l a r g e amount of gas i n t h e f u e l could

influence t h e r e a c t i v i t y of t h e core, t h i s system would be l i m i t e d t o low volume f r a c t i o n s . Peebles" showed t h a t removal o f dissolved oxygen from a given mixture of glycerine and water by small helium bubbles could c l o s e l y match t h e hydrodynamic and mass-transfer conditions i n an MSBR and suggested using such a system i n a s i m i l i t u d e experiment from which t h e a c t u a l MSBR behavior might be inferred.

Other d e s i r a b l e f e a t u r e s of such a system

include: (1)convenient v a r i a t i o n of t h e Schmidt number by using d i f f e r ent percentages of glycerine i n water, ( 2 ) operation a t room temperature using g l a s s hardware t h a t allows photographic measurements through an o p t i c a l l y c l e a r system, and ( 3 ) easy measurement of t h e dissolved oxygen content by commercially a v a i l a b l e instruments.

Therefore an oxygen-

glycerine-water system w a s chosen f o r t h i s study. The o b j e c t i v e of t h e program was t o measure l i q u i d phase c o n t r o l l e d a x i a l l y averaged mass-transfer c o e f f i c i e n t s , k, defined by f L k dx x k r to

3 The l o c a l mass-transfer c o e f f i c i e n t s , k

J = k

a [C

avg

a r e defined by

-Cs]

where J i s t h e mass t r a n s f e r r e d from t h e l i q u i d t o t h e bubbles p e r unit time p e r u n i t volume of l i q u i d , a i s t h e i n t e r f a c i a l a r e a p e r u n i t volume of l i q u i d , C

avg

i s t h e bulk average concentration, and C

i s the inter-

f ac ia 1 c onc e n t r ati on,

These c o e f f i c i e n t s need t o be e s t a b l i s h e d a s a function of Schmidt number, Reynolds number, bubble s i z e , conduit diameter, g r a v i t a t i o n a l o r i e n t a t i o n of t h e flow ( v e r t i c a l o r h o r i z o n t a l ) , i n t e r f a c i a l condition (absence or presence of a surface a c t i v e agent), and t h e volume f r a c t i o n of t h e bubbles.

The scope of t h i s t h e s i s i s l i m i t e d t o t h e ranges of

v a r i a b l e s l i s t e d i n Table I, below, which f o r t h e most p a r t represent

l i m i t s of t h e experimental apparatus.

Extensions of t h i s program, how-

ever, a r e p r o j e c t e d t o include d i f f e r e n t conduit diameters and d i f f e r e n t i n t e r f a c i a l conditions. Table I.

Ranges of Independent Variables Covered

Variable Schmidt Number (weight percent of g l y c e r i n e )

Range

370

3446

( 0 , 12.5, 25, 37.5,

Pipe Reynolds Number Bubble Sauter Mean Diameter Gas t o Liquid Volumetric Flow Ratio G r a v i t a t i o n a l Orientation of Flow C onduit Diameter

8 x io3

50)

1.8 x io6

0.01 t o 0.05 inches

0.3 and 0.5 percent V e r t i c a l and Horizontal 2 inches

The mass-transfer c o e f f i c i e n t s were e x t r a c t e d from measurements of t h e c o e f f i c i e n t - a r e a products, ka, and independent photographic measurements of t h e i n t e r f a c i a l areas p e r unit volume, a.

The products, ka,

4 were e s t a b l i s h e d by means of a unique t r a n s i e n t response technique i n which t h e changes i n l i q u i d phase concentration were measured a s a f i n c t i o n of time a t only one p o s i t i o n i n a closed l i q u i d r e c i r c u l a t i n g system while helium bubbles were i n j e c t e d a t t h e t e s t s e c t i o n entrance and removed r i c h e r i n oxygen a t t h e e x i t .

The apparatus f o r generating

t h e s e small bubbles (with an independent c o n t r o l of t h e i r mean s i z e ) and e f f e c t i v e l y s e p a r a t i n g a high percentage from t h e flowing mixture had t o be developed p r i o r t o t h e s t a r t of t h i s research.

These a r e described i n

Chapter I11 along with t h e photographic equipment and technique f o r establ i s h i n g t h e i n t e r f a c i a l areas. The r e s u l t s of t h i s study a r e expected t o be of immediate b e n e f i t t o t h e MSBR Program and should a l s o prove u s e f i l t o workers i n t h e general chemical industry.

Application may extend t o such d i v e r s e a r e a s

a s general e x t r a c t i o n of r a d i o a c t i v e elements from r e a c t o r e f f l u e n t s , bubble l i f e t i m e s i n t h e coolant of l i q u i d metal f a s t breeder r e a c t o r s , and oxygen treatment of sewage e f f l u e n t s .

I n addition, b e n e f i t s of a

fâ&#x20AC;&#x2122;undamental nature may b e derived i n t h a t t h e research concerns t r a n s f e r of a s c a l a r i n a t u r b u l e n t shear flow f i e l d i n which t h e f l u i d v e l o c i t y f i e l d e f f e c t i v e l y seen by t h e bubbles i s p r i m a r i l y due t o t h e t u r b u l e n t fluctuations.

The c h a r a c t e r i s t i c s of mass t r a n s f e r between dispersed

bubbles and a continuous l i q u i d phase i n t u r b u l e n t flow a r e t h u s seen t o be of immediate s c i e n t i f i c and p r a c t i c a l importance.

CHAPTER I1

LITERATURE REVIEW A comprehensive survey was made of l i t e r a t u r e r e l a t e d t o mass

t r a n s f e r between small bubbles and l i q u i d s i n cocurrent t u r b u l e n t flow. An exhaustive review o f a l l t h i s l i t e r a t u r e would be lengthy and somewhat p o i n t l e s s .

Consequently, only those works t h a t a r e considered

r e p r e s e n t a t i v e of t h e f i e l d (not n e c e s s a r i l y of most s i g n i f i c a n c e ) a r e included i n t h i s chapter and t h e author intends no derogation or s l i g h t i n g by t h e omission of any work.

N o s i g n i f i c a n c e should be attached t o

t h e order i n which references appear.

For a f a i r l y complete documenta-

t i o n o f work r e l a t e d t o t h i s subject, t h e reader i s r e f e r r e d t o s e v e r a l e x c e l l e n t review a r t i c l e s .

3-8

Experiment al-C ocurrent Flow There have been very few d i r e c t measurements o f mass-transfer c o e f f i c i e n t s f o r cocurrent t u r b u l e n t flow of small gas bubbles and l i q u i d s , perhaps because s u b s t a n t i a l s p e c i a l apparatus seems t o be required f o r t h e s e measurements.

Recently Jepseng measured t h e l i q u i d

phase c o n t r o l l e d product of mass-transfer c o e f f i c i e n t , k, and i n t e r f a c i a l surface a r e a p e r u n i t volume, a, f o r a i r / w a t e r flow i n h o r i z o n t a l p i p e s w i t h and without s p i r a l turbulence promoters.

For s t r a i g h t tubes

without turbulence promoters he c o r r e l a t e d h i s data by t h e equation,

A s shown i n Chapter I V , page 58, t h e energy d i s s i p a t i o n per unit

volume,

can be represented a s

Therefore, Jepsen’s c o r r e l a t i o n reveals t h a t

Care must be taken i n i n t e r p r e t i n g t h e influence of Reynolds number on k when t h e product, ka, i s reported because t h e i n t e r f a c i a l area itNo attempt was made by Jepsen

s e l f may depend on t h e Reynolds number. t o s e p a r a t e t h e a r e a from t h e product. S c o t t and Hayduk,”

i n admittedly exploratory experiments, dissolved

carbon dioxide and helium i n t o water, ethanol, and ethylene g l y c o l i n h o r i z o n t a l flow pipelines.

Thus they d i d vary t h e diff’usivity but, l i k e

Jepsen, d i d not s e p a r a t e t h e ka product. Their r e s u l t s were c o r r e l a t e d by t h e equation ka = 0.0068

~ Po.o8 *

A9°90.39 ~ ~ 7

~1.88

from which may be i n f e r r e d

S h - Re/&090’61a Lamont’’

and Lamont and Scott’“ dissolved, i n s i n g l e f i l e fashion,

r e l a t i v e l y l a r g e CO, bubbles i n t o water under v e r t i c a l and h o r i z o n t a l flow conditions.

They d i d not vary bubble diameter or Schmidt number.

A t s u f f i c i e n t l y l a r g e Reynolds numbers t h e i r h o r i z o n t a l and v e r t i c a l

r e s u l t s became i d e n t i c a l .

The data above t h e s e Reynolds numbers were

correlated as

ReoeS2

Heuss, King, and Wilke13 studied absorption i n t o water of ammonia and oxygen i n h o r i z o n t a l f r o t h flow.

The l i q u i d phase c o e f f i c i e n t s were

7 c o n t r o l l i n g only i n t h e oxygen runs and consequently t h e y d i d not vary t h e Schmidt number and t h e i r r e s u l t s were a l s o obtained a s t h e product of ka.

However, using estimates of surface a r e a i n f r o t h flow, t h e i r

data reveal

S h - Re0”

Hariott14 reported mass-transfer c o e f f i c i e n t s f o r p a r t i c l e s of b o r i c acid and benzoic acid d i s s o l v i n g i n water flowing cocurrently i n a two-inch pipeline.

A d a t a c o r r e l a t i o n was not given but a l i n e t a n -

gent t o t h e i r d a t a a t t h e high flow end would i n d i c a t e Sh

- Reoog3 .

Figueiredo and Charles16 measured c o e f f i c i e n t s f o r d i s s o l u t i o n of NaCl p a r t i c l e s c a r r i e d along a s a “ s e t t l i n g ” suspension i n water i n h o r i z o n t a l flow.

They c o r r e l a t e d t h e i r d a t a with mass-transfer c o e f f i -

c i e n t s previously measured f o r t r a n s f e r between a l i q u i d and t h e conduit itself.

However, a l i n e tangent t o t h e high flow end of t h e i r data

indicates S h - Re’’’

Experimental-Agitated Vessels Often t h e data f o r t r a n s f e r t o bubbles or p a r t i c l e s i n a g i t a t e d v e s s e l s a r e c o r r e l a t e d i n terms of t h e power dissipated.

Using Equation

(1)we might r e l a t e t h e s e r e s u l t s t o what would be expected f o r f l o w i n

conduits. Calderbank and Moo-Youn&

correlated data f o r different p a r t i c l e s

and small bubbles dispersed i n d i f f e r e n t l i q u i d s i n a g i t a t e d vessels. Their equation, determined p a r t l y through dimensional analysis,. i s C

-J

Using Equation (1) t h i s would give f o r flow i n conduits

They a l s o i n d i c a t e t h a t i n t h e range of mean bubble diameters, 0.025 N

dvs -s 0.1 inches, t h e mass-transfer c o e f f i c i e n t s i n c r e a s e l i n e a r l y ,

undergoing a t r a n s i t i o n from “small” bubble behavior where Sh “ l a r g e ” bubble behavior where Sh

Scl/

’.

- Scl’

They conclude t h a t t h i s t r a n -

s i t i o n corresponds t o a change i n i n t e r f a c i a l condition from r i g i d t o mobile. Shemood and Brian17 used dimensional a n a l y s i s t o c o r r e l a t e d a t a for p a r t i c l e s i n different agitated liquids. r e l a t e d S%/SC’/~ t o ( e d4/v3)’13.

Their c o r r e l a t i o n g r a p h i c a l l y

Using Equation (1) (with e V / p = c m )

and drawing a l i n e tangent t o t h e high power d i s s i p a t i o n end of t h e i r c o r r e l a t i n g curve gives Sh

Sc113

Barker and Treyball’

(d/D)-’”“

(3)

c o r r e l a t e d mass-transfer c o e f f i c i e n t s f o r b o r i c

a c i d and benzoic a c i d p a r t i c l e s dissolving i n water and 45% sucrose solut i o n s with a s t i r r e r Reynolds number, ReT, p r o p o r t i o n a l t o t h e speed of rotation.

They reported k

- ReTSae3

Scl/

If t h e power d i s s i p a t i o n i s assumed proportional t o t h e cube of t h e

r o t a t i o n speed, then k-

Scl’”

The e f f e c t of Schmidt number i s not as would be i n f e r r e d from t h e above because 69 was reported t o be e s s e n t i a l l y proportional t o Sc-l’ experiments.

in their

9 Y

The preceding a r e r e p r e s e n t a t i v e of d a t a a v a i l a b l e t h a t may have d i r e c t a p p l i c a b i l i t y t o cocurrent flow i n conduits.

Some o t h e r works

t h a t may be of i n d i r e c t i n t e r e s t include cocurrent turbulent flow of dispersed l i q u i d drops i n a continuous l i q u i d p h a ~ e , ' ~ - "mass ~ transfer from a t u r b u l e n t l i q u i d t o a f r e e i n t e r f a ~ e , " ~ - and " ~ innumerable s t u d i e s of t h e motions of, and mass t r a n s f e r from, i n d i v i d u a l bubbles or p a r t i c l e s under steady r e l a t i v e flow conditions (e. g., References 26-30). For systems i n which bubbles move s t e a d i l y through a f l u i d , some r e l e v a n t f i n d i n g s include t h e f a c t t h a t , depending on bubble s i z e and l i q u i d p r o p e r t i e s , t h e bubble motion i n a g r a v i t y f i e l d may vary from creeping flow t o flow characterized by a t u r b u l e n t boundary layer. I r r e s p e c t i v e of t h i s , t h e mass-transfer c o r r e l a t i o n s u s u a l l y t a k e two b a s i c " F r k s l i n g " forms (neglecting t h e constant term) depending on whether t h e r e i s a r i g i d i n t e r f a c e (no s l i p condition) o r a completely mobile i n t e r f a c e with i n t e r n a l c i r c u l a t i o n of t h e f l u i d within t h e bubble (or drop).

I n s u b s t a n t i a l agreement with t h e o r e t i c a l treatments,

t h e former d a t a a r e c o r r e l a t e d with S%

- R eb

Sc"

and t h e l a t t e r

Good accounts of t h e s e r e l a t i v e flow equations and t h e i r d e r i v a t i o n s a r e given by Lochiel and Calderbank31 and by Sideman. 2 2 Discussion of Available Experimental Data It i s seen t h a t t h e r e have been very few d i r e c t measurements of

mass t r a n s f e r t o small c o c i r c u l a t i n g bubbles i n a t u r b u l e n t f i e l d and

none t h a t a r e complete i n terms of a l l t h e independent variables.

The

product, ka, i s o f t e n not separated, because of t h e d i f f i c u l t y i n establ i s h i n g t h e i n t e r f a c i a l area.

This makes some of t h e a v a i l a b l e data

d i f f i c u l t t o i n t e r p r e t and of l i m i t e d value f o r a p p l i c a t i o n a t d i f f e r e n t conditions. Not enough experimental information i s a v a i l a b l e t o a s s e s s t h e i n f l u e n c e of Schmidt number on Sherwood number although t h e Schmidt number exponent most o f t e n appears t o vary between 1/3 and 1/2 apparently determined by t h e i n t e r f a c i a l condition ( t h e Schmidt number exponent may even be g r e a t e r than 1/2, e. g., Reference 10). The e f f e c t s of bubble mean diameter and p i p e s i z e have received l e s s a t t e n t i o n than t h e Schmidt number and, a s y e t , no systematic e f f e c t can be c o n f i d e n t l y c i t e d .

Calderbank and Moo-Young, however, observed a

l i n e a r dependence over a l i m i t e d range of bubble diameters i n a g i t a t e d ves s e l s

The influence of Reynolds number has been t h e most studied.

From

References 9-15, it would appear t h a t Sherwood number f o r g a s - l i q u i d flow i n conduits may vary with p i p e Reynolds number t o a power between

0.9 and 1.1 (although Lamontâ&#x20AC;&#x2122;â&#x20AC;&#x2122;

found it t o be 0.52).

I n contrast, t h e

e f f e c t of Reynolds number (turbulence l e v e l ) i n s t i r r e d v e s s e l s (References 16-18) would appear t o y i e l d a power between 0.6 and 0.8.

This

apparent d i f f e r e n c e between a g i t a t e d v e s s e l s and flow i n conduits i s s u r p r i s i n g because one would t h i n k t h a t flowing through a closed cond u i t i s j u s t another way t o s t i r t h e l i q u i d .

There should be l i t t l e

fundamental d i f f e r e n c e i n t h e e f f e c t of t h e turbulence produced.

Theoretical

It i s convenient t o i d e n t i f y f o u r d i f f e r e n t a n a l y t i c a l approaches designed t o provide a d e s c r i p t i o n of mass t r a n s f e r t o bubbles from a t u r b u l e n t l i q u i d t h a t may b e applicable t o cocurrent flow.

The author

has chosen t o name t h e s e (1)Surface Renewal, ( 2 ) Turbulence I n t e r a c t i o n s , (3) Modeling of t h e Eddy Structure, and (4) Dimensional Analysis (Empiricism).

These do not n e c e s s a r i l y encompass a l l approaches and

t h e r e may be considerable overlapping among areas ( f o r example, a cert a i n degree of empiricism i s evident i n each).

There may be only an

i n d i r e c t equivalence among those within a given category. Some r e p r e s e n t a t i v e works have been categorized according t o t h e i r approach and l i s t e d i n Table 11. A b r i e f discussion of each category i s given below. Surface Renewal Models This category i s of considerable h i s t o r i c a l i n t e r e s t e s p e c i a l l y t h e o r i g i n a l c o n t r i b u t i o n s of Higbie3" and Danckwerts.

The so-called surface renewal models can be envisioned by imagining t h e i n t e r f a c e as being adjacent t o a s e m i - i n f i n i t e f l u i d through which t u r b u l e n t eddies having uniform concentrations c h a r a c t e r i s t i c of t h e continuous phase, p e r i o d i c a l l y p e n e t r a t e t o "renew" t h e surface.

The

mass t r a n s f e r then depends on t h e r a t e and depth of eddy p e n e t r a t i o n and t h e eddy residence time near t h e surface ( o r t h e d i s t r i b u t i o n o f eddy ages).

For a given eddy, t h e o r i g i n a l models a r e e s s e n t i a l l y s o l u t i o n s

o f t h e diff'usion equation

-ac- - 8 -a2c

3Y"

Table 11.

Categories of Data C o r r e l a t i o n for Mass Transfer from a Turbulent Liquid t o . Gas Bubbles

Brian and Beaverstock (40)" Danckwerts (33) Davies, Kilner, and R a t c l i f f (41) G a l - O r , Hauck, and Hoelscher ( 4 2 ) Gal-Or and Resnick (43) Harriot (44) Higbie (32) King (25) Koppel, P a t e l , and Holmes (45) Kovasy (46) Lamont and S c o t t ( 1 2 ) Perlmutter (47) Ruckenstein (48) Sideman (49) Toor and Marchello (34)

3. Modeling of t h e Eddy Structure Banerjie, S c o t t , and Rhodes ( 5 1 ) Fortescue and Pearson (23) Lamont (11) a

Turbulence I n t e r a c t i o n s

Boyadzhiev and Elenkov (19) Harriot (50) Kozinski and King (24) Levich (36) Peebles ( 2 ) Porter, Goren, and Wilke ( 2 0 ) Sideman and Barsky (21)

Dimensional Analysis (Empiricism)

Barker and Treybal ( 5 2 ) Calderbank and Moo-Young (16) Figueiredo and Charles (15) Galloway and Sage (53) Heuss, King, and Wilke (13) Hughmark (54) Middleman (38) S c o t t and Hayduk (10) Sherwood and Brian (17)

Reference number.

A s shown by Toor and M a r c h e l 1 0 , ~ t~h e " f i l m "

model f i r s t introduced

by Whitman36 corresponds t o t h e asymptotic s o l u t i o n of t h i s equation a t long times (no surface renewal) where k would be proportional t o B and Sherwood number would be independent of Schmidt number.

The "penetra-

t i o n " model f i r s t introduced by Higbie3" and l a t e r extended by Danckwerts"" corresponds t o t h e asymptotic s o l u t i o n a t s h o r t times where k would be proportional t o

dsl'

o r Sh

Sc"

Depending on t h e d i s t r i b u t i o n of

contact times between t h e eddies and t h e surface, t h e t r a n s f e r may t a k e on c h a r a c t e r i s t i c s of e i t h e r or both of t h e above. King26 generalized t h i s approach t o include turbulence e f f e c t s by replacing Equation ( 4 ) with

where

IJ-

i s an eddy d i f f u s i v i t y which he a r b i t r a r i l y l e t vary with d i s -

tance from t h e surface a s b PeNY

This model approaches t h e same asymptote ( S h - Scl'")

a t s h o r t times but

d i f f e r e n t asymptotes a t long times depending on t h e v a l u e of b ( w i t h b

3, Sh

S C " ' ~ with ~; b =

4, Sh

- S C ~ '>. " ~

To e s t a b l i s h an o v e r a l l mass-transfer r a t e , it i s necessary t o a s s i g n a frequency with which t h e surfaces a r e "renewed" ( o r t h e d i s t r i bution of eddy ages).

The d i f f e r e n t extensions and modifications o f

t h i s model mostly involve t h e choice of d i f f e r e n t functions t o describe t h e randomness of t h e eddy penetrations.

None of t h e s e models give

s i g n i f i c a n t information a s t o t h e e f f e c t of bubble s i z e , conduit s i z e ,

o r Reynolds number.

They a r e mechanistically u n s a t i s f a c t o r y because t h e

hydrodynamic e f f e c t s a r e o f t e n ignored o r included by r e l a t i n g t h e eddy age d i s t r i b u t i o n i n some way t o t h e flow f i e l d .

For example, Lamont and

Scott’” assumed t h a t t h e f r a c t i o n a l r a t e of s u r f a c e renewal, s, ( k

i s given by s

dz)

dF .

There i s r e a l l y no clear-cut way t o e s t a b l i s h a r e l a t i o n s h i p between t h e r a t e of surface renewal and t h e hydrodynamics and, consequently, t h e r e

i s a heavy r e l i a n c e on empiricism.

The o r i g i n a l i n t e n t of t h e s e models

w a s t o describe t r a n s f e r t o a surface (bubble) t h a t has a d i s t i n c t steady

flow r e l a t i v e t o t h e l i q u i d . Modeling of t h e Eddy S t r u c t u r e If t h e f l u i d v e l o c i t y f i e l d i n t h e v i c i n i t y of t h e i n t e r f a c e could

be completely described, then t h e computation of t r a n s f e r r a t e s , i n p r i n c i p a l , would be straightforward.

However, a t t h e present time,

t h e r e a r e no s a t i s f a c t o r y d e s c r i p t i o n s of t h e d e t a i l s of a t u r b u l e n t v e l o c i t y f i e l d and even i f such were available, t h e mathematical accounting of t h e d i f f e r e n t i a l t r a n s f e r processes might become i n t r a c t a b l e . Consequently, t h e r e have been i d e a l i z a t i o n s of t h e eddy s t r u c t u r e with, admittedly, u n r e a l i s t i c f i e l d s and mass-transfer behavior has been computed based on t h e s e i d e a l i z a t i o n s . Lamont’s work”

provides an excellent example of t h i s approach.

modeled t h e eddy s t r u c t u r e by considering i n d i v i d u a l eddy c e l l s t h a t have a s i n u s o i d a l form a t a s u f f i c i e n t d i s t a n c e away from t h e i n t e r f a c e (corresponding perhaps t o an i n d i v i d u a l component of a Fourier decompos i t i o n of t h e t u r b u l e n t f i e l d ) .

A s t h e i n t e r f a c e i s approached, viscous

f o r c e s dampen t h e eddy c e l l v e l o c i t i e s by an amount t h a t depends on t h e

15 i n t e r f a c i a l condition (mobile o r r i g i d ) .

Lamont c a l c u l a t e d t h e mass-

t r a n s f e r c o e f f i c i e n t f o r an i n d i v i d u a l eddy c e l l as a m c t i o n of t h e damping condition, f l u i d p r o p e r t i e s , t h e wave p r o p e r t i e s , and t h e eddy energy.

He then used a Kovasznay d i s t r i b u t i o n f’unction f o r t h e energy

spectrum and summed over a range of wave numbers t o obtain t h e o v e r a l l coefficient.

The r e s u l t s of t h i s procedure were k-

(SC)

(gmv)‘’

(SC)-~

(cmV)’/

f o r a mobile i n t e r f a c e and k

for a rigid interface. Using Equation (l), t h e s e give Sh

Scl/

Reo’ 69

Sc’/

Reoo6’

and Sh

respectively. The present w r i t e r f e e l s t h a t t h i s approach may represent a bridge between s u r f a c e renewal models and turbulence theory and a s such deserves p a r t i c u l a r mention. Turbulence I n t e r a c t i o n s Some authors have attempted t o analyze t h e f o r c e s and i n t e r a c t i o n s between spheres and f l u i d elements i n a t u r b u l e n t f i e l d t o a r r i v e a t equations f o r t h e f l u c t u a t i n g motion of t h e spheres.

These equations

a r e solved t o obtain a “mean” r e l a t i v e v e l o c i t y between t h e bubble and t h e f l u i d .which i s then s u b s t i t u t e d i n t o a steady-flow equation ( u s u a l l y of t h e FrGssling type) t o e s t a b l i s h t h e mass-transfer c o e f f i c i e n t s . ~ t h i s nature and Peebles2 used t h i s approach i n work of L e ~ i c hi s~ of

The

16 h i s document.

For example, Peebles used t h e r e s u l t of H i n ~ ef o~r ~small

gas bubbles

which e s s e n t i a l l y comes from an i n t e g r a t i o n of t h e equation

where W i s an added mass c o e f f i c i e n t f o r an a c c e l e r a t i n g s p h e r i c a l bubble.

The r e l a t i v e v e l o c i t y i s then

Peebles used t h e approximations

i n t h e above which were then s u b s t i t u t e d i n t o Frijssling type equations t o o b t a i n Sh

Reo.

(d/D)-”

Sell

ReooG6Scl’ a (d/D)-l/ a

for a mobile i n t e r f a c e and Sh

f o r a r i g i d interface.

I n a similar computation which included Stokes law t o describe t h e drag experienced by t h e bubble, L e ~ i c hobtained ~ ~ f o r a mobile i n t e r f a c e Sh

Re3/

Scl’

Dimensional Analysis (Empiricism) Some workers have chosen t o p o s t u l a t e t h e physical v a r i a b l e s t h a t may be c o n t r o l l i n g and have used standard dimensional a n a l y s i s techniques

for ordering t h e experimental data.

The paper by Middleman38 i s a

splendid example of t h i s approach a s applied t o a g i t a t e d vessels. f o r a g i t a t e d v e s s e l s , Calderbank and Moo-Young’‘

Also

used dimensional

a n a l y s i s t o obtain Equation ( 2 ) and Sherwood and Brian17 dimensionally r e l a t e d S%/Scl’

t o [c d4/v3I1’ m

’.

17 Also included under t h i s category i s a most i n t e r e s t i n g c o r r e l a t i o n by Figueiredo and Charles15 f o r a heterogeneous p i p e l i n e flow of s e t t l i n g They used an expression f o r t h e r a t i o of pressure gradient

particles.

f o r flow of t h e susFension t o t h e pressure gradient f o r flow of t h e l i q u i d alone and assumed t h a t , i f a l t e r e d by t h e r a t i o d/D, it could a l s o represent t h e r a t i o of mass-transfer c o e f f i c i e n t s f o r t h e p a r t i c l e s t o those f o r t r a n s f e r from t h e l i q u i d t o t h e pipe w a l l .

They found t h a t

t h e y could, indeed, use t h i s r a t i o t o c o r r e l a t e t h e i r d a t a f o r a s e t t l i n g suspension with t h e d a t a of Harriot and Hamilton. 39 Discussion It i s seen t h a t t h e t h e o r e t i c a l d e s c r i p t i o n of mass t r a n s f e r t o

bubbles i n cocurrent t u r b u l e n t flow has by no means been standardized. There seems t o be somewhat general agreement as t o t h e e f f e c t of Schmidt number.

The Sherwood numbers f o r cases of com-pletely r i g i d i n t e r f a c e s

w i t h zero t a n g e n t i a l v e l o c i t y a t t h e surface (no s l i p ) applicable t o

s o l i d spheres, very s m a l l bubbles, and bubbles with s u r f a c t a n t contamination i n t h e i n t e r f a c e are generally predicted t o vary with Schmidt number t o t h e one-third power.

Completely mobile i n t e r f a c e s ( n e g l i g i b l e

t a n g e n t i a l s t r e s s with non-zero i n t e r f a c i a l v e l o c i t y ) a r e g e n e r a l l y predicted t o yield a

~ va~ r i a t i â&#x20AC;&#x2122;o n o~f

Sh.

There i s only scant and i n c o n s i s t e n t information p r e d i c t i n g t h e e f f e c t s of bubble and conduit diameter.

For example, Levich p r e d i c t s

no e f f e c t of d/D while Peebles p r e d i c t s S h -

(d/D)-l/â&#x20AC;?.

There i s general disagreement a s t o t h e e f f e c t of Reynolds number a s evidenced by t h e f a c t t h a t exponents have been p r e d i c t e d t h a t range

18 from 0.45 t o > 1. These d i f f e r e n t exponents may not be mutually exclusive however because an inspection of t h e experimental d a t a shows disagreement i n t h e measured exponents also.

It may b e t h a t t h e proper a p p l i c a t i o n of

t h e s e equations depends on s u i t a b l e evaluation of t h e conditions of t h e experiment.

CHAPTER I11 D E S C R I P T I O N OF EXPERIMENT This experiment w a s designed t o measure l i q u i d phase c o n t r o l l e d mass-transfer c o e f f i c i e n t s f o r cocurrent p i p e l i n e flow of t u r b u l e n t l i q u i d s with up t o 1% volume f r a c t i o n of small helium bubbles having mean diameters f i o m 0.01 t o 0.05 inches.

The l i q u i d s chosen were f i v e

mixtures o f glycerine and water (0, 12.5, 25,

37.5,

and 50% by weight

of glycerine) each of which represent a d i f f e r e n t Schmidt number.

The

p h y s i c a l p r o p e r t i e s of t h e s e mixtures, obtained from t h e l i t e r a t u r e , 55 a r e shown g r a p h i c a l l y i n Appendix A and t h e values used i n t h i s study f o r t h e given mixtures a r e l i s t e d i n Table 111. Transient Response Technique A closed r e c i r c u l a t i n g system w a s used i n which helium bubbles

were introduced (generated) a t t h e entrance of a well-defined t e s t s e c t i o n and removed r i c h e r i n oxygen a t t h e e x i t , allowing only t h e bubble-free l i q u i d t o r e c i r c u l a t e .

The products of mass-transfer c o e f f i c i e n t s and i n t e r f a c i a l areas were measured by a t r a n s i e n t response technique i n which t h e system was i n i t i a l l y charged w i t h dissolved oxygen.

The oxygen was then progress-

i v e l y removed by t r a n s f e r t o t h e helium bubbles while t h e oxygen concent r a t i o n w a s continuously monitored as a function of time a t a s i n g l e p o s i t i o n i n t h e system.

For a t e s t s e c t i o n of length, L, and c r o s s - s e c t i o n a l area, A, it can be shown (Appendix B) t h a t t h e r a t i o of e x i t concentration t o i n l e t

Physical Properties of Aqueous-Glycerol Data of Jordan, Ackerman and Berger"

Table III.

Glycerol Content Cd %>

(

Density, (lb/ft3)

Viscosity, (lb/ft*hr)

Henry's Law Constant, H (atm*liters/mole)

Mixtures

(25째C)

Molecular Diffusivity of Oxygen, d9 x 10' (ft"/hr)

62.43

2.15

795.4

12.5

64.43

3.07

1127.6

12.865

370

66.49

4.62

1421.0

9.261

750

37.5

67.67

6.34

1621.8

4.650

2015

69.86

10. a2

2011.1

4.495

3446

a. 215

Schmidt Modulus, SC (Dimensionless

419

21 W

concentration, C /C i s a constant, K, given by e i'

K E C / C . = y l++ey e 1

where RTQ a

y=-

and

kaAL(1

HQg

I n t h e absence of a x i a l smearing, each time t h e f l u i d makes a comp l e t e passage around t h e closed c i r c u i t (loop t r a n s i t time, T,

= Vs/QR)

t h e concentration a t t h e measuring p o s i t i o n would ( i d e a l l y ) decrease instantaneously from i t s value, C, t o a value equal t o KC. i n a c t u a l i t y , t h e r a t i o , C/Co,

Therefore,

of t h e concentration a t any time t o t h a t

a t an i n i t i a l reference time ( s e t equal t o zero) would be given by

C/Co

Exp

[-I

Therefore a p l o t of h ( C / C o ) slope - ( h K ) Q

a/V s'

Exp

V S

versus time would be a straight l i n e of

Note t h a t t h e absolute value need not be measured

because a s i g n a l t h a t i s merely proportional t o t h e oxygen concentration would have t h e same slope. volumetric flow r a t e , Q

If t h e system volume, V s J and t h e l i q u i d

a' have been measured, t h e constant K can be

e x t r a c t e d from t h e slope of t h e measured t r a n s i e n t .

Having a measure

a l s o of gas volumetric flow, Q and t h e system a b s o l u t e temperature, T, g' and knowing R, H, A, and L, t h e product, ka, can b e obtained from K through Equations ( 5 ) .

If an independent measure i s a l s o made of "a,"

then t h e mass-transfer c o e f f i c i e n t s a r e f u l l y determinable.

This technique was s e l e c t e d a s being superior t o a once-through t e s t t h a t r e q u i r e s an independent measurement of t h e oxygen concentrat i o n a t both ends of t h e t e s t s e c t i o n f o r reasons i l l u s t r a t e d by t h e following comparison. I n a once-through system with a 37.5% mixture and conservative values of Q /Q = 1%, bubble mean diameter = 0.01 inches, Reynolds g R number = 6 x lo4, and a mass-transfer c o e f f i c i e n t of 0.7 f%/hr, a t e s t s e c t i o n l e n g t h of -100 f e e t would be required t o o b t a i n a concentration change across t h e t e s t s e c t i o n of only Ce/Ci

0.9

A t t h i s l e v e l a small error i n t h e concentration measurement would be

magnified i n t h e determination o f ka.

I n c o n t r a s t , t h e same conditions

i n a t r a n s i e n t t e s t w i t h only a 25-ft-long t e s t s e c t i o n would give a concentration change of C/Co

0.1 i n only about seven minutes

reducing t h e e r r o r magnification i n ka.

- greatly

In return for t h i s benefit, the

values of t o t a l system volume, Vs, and t h e time coordinate, t, need a l s o t o be measured.

These, however, a r e parameters t h a t can be measured

very p r e c i s e l y compared t o t h e concentration measurement.

Therefore,

t h e t r a n s i e n t t e s t s should r e s u l t i n more r e l i a b l e data. On t h e other hand, t h e concentrations i n once-through t e s t s a r e measured a t s p e c i f i c l o c a t i o n s t h a t bracket t h e region of i n t e r e s t and only t h e t r a n s p o r t behavior within t h a t region i s important.

Whereas i n

t h e t r a n s i e n t t e s t s a l l mass t r a n s f e r occuring outside t h e t e s t s e c t i o n

i s extraneous and represents an "end e f f e c t " c o n t r i b u t i o n t h a t must be independently measured and accounted f o r i n determining t h e "ka" product. This "end e f f e c t , " which would include mass t r a n s f e r occuring i n t h e

23 W

bubble generating and separating processes, represents t h e most s e r i o u s disadvantage and e r r o r source i n t h e t r a n s i e n t measurements.

The

measurement and accounting f o r t h e "end e f f e c t " a r e discussed f u r t h e r on page 47. Apparatus I n c o n s t r u c t i n g t h e main c i r c u l a t i n g systems of t h e experiment exclusive use was made of s t a i n l e s s s t e e l or g l a s s hardware and a l l gaskets were Teflon.

This was p a r t of c a r e f u l measures taken t o keep

t h e system f r e e of contamination.

Figure 1 i s a photograph of t h e

f a c i l i t y with t h e t e s t s e c t i o n mounted i n t h e v e r t i c a l o r i e n t a t i o n and Figure 2 i s a diagram of t h e main c i r c u i t portion.

Figure 28, page 1 2 1

(Appendix C ) i s an instrument a p p l i c a t i o n drawing o f t h e system which includes an a u x i l i a r y flow c i r c u i t used f o r rotameter c a l i b r a t i o n and

for s p e c i a l t e s t s . The main c i r c u i t consisted of a canned r o t o r c e n t r i f i g a l pump, t h r e e p a r a l l e l rotameters, a heat exchanger, t h r e e dissolved oxygen measuring sensors, a helium flow and metering system, a bubble generator, t h e t e s t section, a bubble separator, a photographic arrangement f o r determining t h e bubble i n t e r f a c i a l areas and mean diameters, and a d r a i n - a n d - f i l l tank equipped with s c a l e s f o r p r e c i s e determination of t h e weight percent of glycerine i n t h e mixture.

Further d e s c r i p t i o n s

of i n d i v i d u a l components a r e given below.

pump The main c i r c u l a t o r was a 20 HP Westinghouse "100-A"

canned r o t o r

constant speed c e n t r i f u g a l pump capable of d e l i v e r i n g about 100 gpm a t

Figure 1.

Photograph of t h e Mass Transfer Facility.

ORNL-OWG

?O-11420

TEST SECTION -36 ft VERTICAL ORIENTATION

(LOOP

PRESSURE FROM HELIUM

REGULATION BOTTLE) \lLET GAS

BUBBLE SEPARATOR i

2 In._PIPING

-’

HEAT EXCHANGER

FILL DRAIN -

AND TANK

TEMPERATURE > He:OR’O,

CA-&Z0 ROTOR PUMP

MSBR-Mass

Figure

Transfer

Loop.

Schematic Diagram of the Main Circuit Apparatus.

of the Experimental

LONG

26 about 180 f e e t of head.

The motor cannings, housing, and impeller were

s t a i n l e s s s t e e l and t h e bearings were g r a p h i t a r - l u b r i c a t e d t h e loop f l u i d .

s o l e l y by

An a u x i l i a r y c i r c u i t required t o cool t h e pump motor

windings c i r c u l a t e d transformer o i l through t h e windings and through an e x t e r n a l c i r c u i t containing an a u x i l i a r y o i l pump, a f i l t e r , and a small water-cooled heat exchanger.

The pump was s a f e t y instrumented t o t u r n

off on l o s s of pressure i n t h e o i l c i r c u i t or on high temperature of t h e motor housing. Liquid Flow Measurement The l i q u i d flow r a t e was c o n t r o l l e d by t h r e e p a r a l l e l s t a i n l e s s s t e e l globe valves downstream of t h e pump a t t h e entrances t o t h e r o t a meters.

Three p a r a l l e l rotameters of d i f f e r e n t c a p a c i t i e s (100, 40,

and 8 gpm) were used f o r measuring l i q u i d volumetric flow r a t e s .

judicious use of t h e rotameter s c a l e s , p a r a l l e l rotameters provide g r e a t e r p r e c i s i o n when measurements a r e required over a wide flow range. I n each experiment, however, some flow was allowed t o go through each rotameter t o prevent having regions t h a t might "lag" t h e r e s t of t h e

loop during t h e t r a n s i e n t t e s t s and thereby become concentration "capacitance" volumes. Because of t h e l a r g e d i f f e r e n c e s i n v i s c o s i t i e s over t h e range of glycerine-water mixtures used, t h e rotameters were c a l i b r a t e d , i n place, f o r both water and a 50% mixture,

These c a l i b r a t i o n s were obtained by

t h e use of two i d e n t i c a l 6-inch-diameter,

6-feet-long g l a s s tanks i n

t h e a u x i l i a r y c i r c u i t valved together i n such a way t h a t , while one was being f i l l e d , t h e other was being drained.

Changing t h e p o s i t i o n of one

l e v e r reversed t h e process before t h e l i q u i d could s p i l l over t h e top.

27 W

The time required t o f i l l ( o r empty) a known volume of e i t h e r of t h e s e tanks was measured over t h e e n t i r e range of each rotameter.

These c a l -

i b r a t i o n s a r e given i n Appendix D. Since t h e r e was only a small d i f f e r e n c e i n t h e c a l i b r a t i o n between 0 t o 50% glycerine, t h e flow f o r in-between mixtures was determined by

l i n e a r l y i n t e r p o l a t i n g between t h e two curves according t o t h e v i s c o s i t y .

Temperature S t a b i l i z a t i o n The f l u i d temperature was measured a t t h e i n l e t and e x i t of t h e t e s t s e c t i o n by standard s t a i n l e s s s t e e l sheathed chromel-alumel thermocouples immersed i n t h e f l u i d .

The f r i c t i o n and pump heat were removed and t h e

t e s t s e c t i o n temperature held a t 25OC f o r a l l t e s t s by a s t a i n l e s s - s t e e l , water-cooled,

shell-and-tube heat exchanger.

Gas Flow Measurement Helium f o r generation of t h e bubbles was obtained from standard commercial c y l i n d e r s metered through a pressure r e g u l a t o r , a s a f e t y r e l i e f valve, and a flow c o n t r o l needle valve.

The r a t e of helium flow

was determined by measuring both t h e e x i t pressure and t h e p r e s s u r e drop across a 6-foot l e n g t h of tubing of about 1/16-inch i n t e r n a l diameter. These measurements were made with a Bourden type pressure gage and a w a t e r - f i l l e d U-tube manometer, respectively. C a l i b r a t i o n a t atmospheric conditions was obtained p r i o r t o operat i o n by comparing with readings from a w e t - t e s t meter timed with a s t o p watch.

The c a l i b r a t i o n a t 50 p s i g e x i t pressure (normal operating con-

d i t i o n ) i s given i n Figure 33, page 127 (Appendix D).

The c a l i b r a t i o n

28 and t h e l e a k t i g h t n e s s of t h i s system were checked p e r i o d i c a l l y over t h e course of t h e experimental program. Dissolved Oxygen Measurement Two i d e n t i c a l commercially a v a i l a b l e â&#x20AC;&#x153;Polarographicâ&#x20AC;? type i n s t r u -

ments were used t o measure t h e dissolved oxygen concentration (Magna Oxymeter Model 1070, Magna Corporation-Instrument Division, 11808 South Bloomfield Avenue, Santa Fe Springs, C a l i f o r n i a ) .

Two were used s o

t h a t an automatic continuous check was provided by comparing t h e readings of one with t h e other.

It w a s f e l t u n l i k e l y t h a t both would use up

t h e i r e l e c t r o l y t e or f a i l simultaneously.

These instruments used polar-

ographic type sensors i n s e r t e d i n t o t h e flowing l i q u i d through penetrat i o n s i n t e e s provided f o r t h a t purpose.

E l e c t r i c a l s i g n a l s produced

by t h e sensors were f e d through recording adaptors and t h e r e s u l t i n g m i l l i v o l t s i g n a l s recorded on a Brown Multipoint recorder having a measured c h a r t speed of 1.18 inches/sec. Each sensor assembly consisted of an e l e c t r o l y t i c c e l l made up of a cathode, anode, and an e l e c t r o l y t e mounted i n a p l a s t i c c y l i n d r i c a l housing.

The end o f t h e housing, containing t h e c e l l , w a s encased i n

a t h i n oxygen-permeable Teflon membrane which a l s o acted t o contain t h e electrolyte.

The dissolved oxygen i s e l e c t r o l y t i c a l l y reduced a t t h e

cathode causing a current t o flow through t h e system from cathode t o anode.

The magnitude of t h i s current i s p r o p o r t i o n a l t o t h e oxygen

concentration i f s u f f i c i e n t l i q u i d v e l o c i t y e x i s t s (-2 f t / s e c ) t o prevent concentration p o l a r i z a t i o n a t t h e membrane.

29 The response times for t h e s e instruments a r e g r e a t e r than

30 seconds.

90% i n

An a n a l y s i s showing t h a t t h i s response produces an accept-

a b l e e r r o r i n t h e t r a n s i e n t t e s t s i s given i n Appendix E. Since t h e t r a n s i e n t response technique used i n t h e s e t e s t s r e q u i r e s a s i g n a l t h a t i s merely p r o p o r t i o n a l t o t h e oxygen concentration, an absolute c a l i b r a t i o n of t h e s e instruments was not necessary.

Neverthe-

l e s s c a l i b r a t i o n t e s t s were made for two d i f f e r e n t mixtures of glycerine and water by bubbling a i r through t h e mixtures a t d i f f e r e n t pressures

u n t i l t h e y became saturated.

Knowing t h e s o l u b i l i t y of oxygen i n t h e

mixtures, t h e meter reading could be s e t on t h e c a l c u l a t e d concentration f o r an i n i t i a l "set-point" pressure and subsequent readings a t d i f f e r e n t p r e s s u r e s compared with c a l c u l a t e d values (assuming a Henry's Law r e l a tionship).

C a l i b r a t i o n s obtained i n t h i s manner a r e shown on Figure 34,

page 128 (Appendix D ) which includes readings made with a t h i r d i n s t r u ment s i m i l a r t o t h e Magna instruments but made by a d i f f e r e n t company. The response speed of t h i s t h i r d sensor proved t o be slow compared t o t h e Magna sensors and consequently it was used only as an independent monitor on t h e o p e r a b i l i t y of t h e Magna sensors throughout t h e s e experiment s. Bubble Generation S p e c i a l apparatus was required t h a t could generate a d i s p e r s i o n of small bubbles whose mean s i z e could be c o n t r o l l e d and v a r i e d over t h e range 0.01 t o 0.05 inches independently of t h e p a r t i c u l a r l i q u i d mixture being used and of t h e flow r a t e s of gas and l i q u i d .

Two devices con-

sidered and discarded as belng inadequate were (1) a f i n e p o r o s i t y f r i t t e d g l a s s d i s c through which t h e gas was blown i n t o t h e l i q u i d , and

30 ( 2 ) two p a r a l l e l s t a i n l e s s s t e e l d i s c s , a r o t o r and a s t a t o r , each

equipped with intermingling blades.

1 '

The gas-liquid mixture flowed

between t h e blades and t h e gas w a s broken i n t o f i n e bubbles by t h e shearing action. The bubble generator designed and developed f o r t h i s p r o j e c t i s shown diagrammatically on Figure 3.

The l i q u i d flowed through a con-

verging diverging nozzle with a 1-inch-diameter t h r o a t and a 2-inchdiameter entrance and e x i t .

The s e c t i o n downstream of t h e t h r o a t

diverged a t an angle of about 1 2 degrees.

A c e n t r a l "plumb-bob"

probe of maximum c r o s s - s e c t i o n a l diameter of -0.812

shaped

inches was movable

and could be c e n t r a l l y positioned anywhere i n t h e diverging s e c t i o n including t h e t h r o a t and e x i t . c a r r i e d t h e gas i n t o the system.

T h i s probe w a s supported by a tube which

The tube, i n t u r n , was supported by a

"Swagelok" f i t t i n g p e n e t r a t i n g a flange on t h e end of t h e s t r a i g h t l e g of a t e e connected t o t h e nozzle entrance.

Four small p o s i t i o n i n g rods

near t h e t h r o a t centered t h e probe within t h e nozzle and helped support

it.

They also acted as holders f o r a s e c t i o n of "honey-comb" s t r a i g h t e n -

i n g vanes used t o minimize t h e l i q u i d swirl induced by t h e r i g h t angle t u r n a t t h e t e e entrance t o t h e nozzle. Gas entered t h e l i q u i d through 48 holes (1/64-inch-diameter)

around

t h e probe periphery a t i t s maximum thickness and e x i t e d a s a s e r i e s of p a r a l l e l plumes which were broken i n t o i n d i v i d u a l bubbles by t h e turbul e n c e i n t h e diverging s e c t i o n of t h e nozzle.

The mean bubble s i z e f o r

a given flow and mixture was c o n t r o l l e d by t h e p o s i t i o n of t h e probe within t h e nozzle

( t h e c l o s e r t h e probe was t o t h e t h r o a t t h e smaller

t h e mean bubble s i z e produced).

ORNL-DWG

STRAIGHTENING

Figure 3.

POSITION AND SUPPORT RODS

Diagram of t h e Bubble Generator,

71-8047

Bubbles generated by t h i s device were found t o follow c l o s e l y a s i z e d i s t r i b u t i o n function proposed by Bayens‘“

and previously used t o

d e s c r i b e d r o p l e t s i z e s produced i n spray nozzles, The f’unction, defined as f ( 6)d6

= that

f r a c t i o n of t h e t o t a l number

of bubbles t h a t have diameters, 6, l y i n g i n t h e range 6 f 1/2 d6, i s given by f(6) =

(C?/IT)~’’

6“ Exp ( 4 6 ” )

(6)

i n which

[4JGN/6@IV3

This function has been normalized so t h a t m

f(6)d6 = 1

An i n d i c a t i o n of t h e s u i t a b i l i t y of t h i s d i s t r i b u t i o n function i s given i n Figure

4 where

measured cumulative s i z e d i s t r i b u t i o n s f o r

bubble populations produced by t h e bubble generator a r e compared with t h e d i s t r i b u t i o n s c a l c u l a t e d from t h e f i n e t i o n a t d i f f e r e n t l i q u i d flows and d i f f e r e n t r a t i o s of gas t o l i q u i d flows.

The measured d i s t r i -

butions were obtained by painstakingly s c a l i n g t h e s i z e s of a s u f f i c i e n t number of bubbles d i r e c t l y o f f photographs taken of t h e bubble swarms a t each condition.

These measured areas should be accurate within about

10%. The range of mean bubble s i z e s capable of being produced by t h i s bubble generator were measured a t a constant gas-to-liquid volumetric of 0.3% a t d i f f e r e n t l i q u i d flow r a t e s , d i f f e r e n t flow r a t i o , Q /Q g A’ mixtures of glycerine and water, and d i f f e r e n t probe p o s i t i o n s . The r e s u l t s a r e shown on Figure 29, page 123 (Appendix D ) .

The mean

99 9 99 8

71-7999

ORNL-DWG

99 5 99 98 95

2I-

W _I

0 [L

a 20 10

5 2 1 0 04

0 02

005

0 2

d , BUBBLE D I A M E T E R ( i n )

Figure 4.

Comparison of Measured Bubble Sizes with the Distribution Function.

34 diameter used throughout t h i s r e p o r t i s t h e "Sauter" mean defined by W

6" f ( 6 ) d 6

which i s t h e volume-to-surface weighted mean commonly used i n masst r a n s f e r operations

Bubble Se-Daration Since t h i s p r o j e c t uses t h e t r a n s i e n t mode of t e s t i n g , bubbles t h a t r e c i r c u l a t e and e x t r a c t dissolved oxygen from t h e l i q u i d i n regions outs i d e t h e t e s t s e c t i o n c o n s t i t u t e an e r r o r source i n t h e measurements. Consequently a high degree of separation i s d e s i r a b l e f o r t h i s method of t e s t i n g .

Some techniques considered were (1) g r a v i t a t i o n a l separa-

t i o n i n a tank, ( 2 ) c e n t r i f u g a l separation through t h e use of vanes t o induce a s t r o n g vortex, and (3) s e p a r a t i o n by flowing through a porous metal which might a c t a s a p h y s i c a l b a r r i e r t o t h e bubbles.

Each of

t h e s e had shortcomings t h a t prevented t h e i r use i n t h i s p r o j e c t .

For

example, with g r a v i t a t i o n a l separation t h e tank s i z e required for t h e viscous mixtures was ponderously large.

This i n c r e a s e s t h e system

volume r e s u l t i n g i n a "sluggish" loop and an accompanying increase i n t h e measurement e r r o r . With c e n t r i f b g a l separation t h e r e were problems i n s t a b i l i z i n g t h e gaseous core of t h e vortex over a wide range of operating conditions.

I n addition, l a r g e by-pass of bubbles ( i n e f f i c i e n t separation) w a s observed and t h e r e was t o o much l i q u i d carryover through t h e gas removal duct

35 The porous physical b a r r i e r s t e s t e d required l a r g e f r o n t a l areas

or had p r o h i b i t i v e p r e s s u r e drops, and t h e bubbles were observed t o regularly penetrate these barriers. A s a t i s f a c t o r y s e p a r a t o r w a s f i n a l l y developed t h a t combined fea-

t u r e s of each of t h e above. Figure 5.

A diagram of t h i s s e p a r a t o r i s shown on

The liquid-bubble mixtures entered t h e bottom of a 6-inch-

diameter pipe.

A s e r i e s of P l e x i g l a s vanes j u s t beyond t h i s entrance

c r e a t e d a s w i r l flow within t h e tank which tended t o f o r c e t h e bubbles t o t h e middle.

The spinning mixture flowed upward i n t o a converging

cone-shaped region with s i d e s of 500-mesh s t a i n l e s s s t e e l screen.

When

wetted by t h e l i q u i d , t h e screen acted a s a physical b a r r i e r t o t h e bubbles but allowed t h e l i q u i d t o pass through.

The l i q u i d e x i t e d from

t h e s e p a r a t o r while t h e bubbles continued t o r i s e through t h e t r u n c a t e d end of t h e c o n i c a l screen t o an i n t e r f a c e where t h e gas was vented through a small e x i t l i n e .

The system pressure l e v e l was a l s o con-

t r o l l e d a t t h i s i n t e r f a c e by providing an a u x i l i a r y sweep of helium through t h e e x i t l i n e . Good s e p a r a t i o n w a s achieved with t h i s apparatus over t h e t e s t cond i t i o n s of t h i s t h e s i s .

No bubbles could b e detected i n photographs

taken downstream of t h e separator.

However, with t h e use of a l i g h t

beam, some bubbles t h a t appeared t o be smaller than t h e screen mesh s i z e could be detected v i s u a l l y .

A f t e r passing through t h e pump and

e n t e r i n g a higher pressure region t h e s e bubbles apparently went i n t o s o l u t i o n because they could no longer be v i s u a l l y detected downstream of t h a t region.

If indeed t h e y d i d go i n t o s o l u t i o n along w i t h t h e i r

small amount of e x t r a c t e d oxygen, they would have h a r d l y c o n s t i t u t e d

T--;GAs O R N L-DWG

SYSTEM PRESSURE CONTROL --c

71- 80 4 6

OUT

L I Q U I D OUT

5 0 0 MESH CONICALLY SHAPED STAINLESS S T E E L SCREEN

[SWIRL

Figure 5.

GENERATOR

-)--

Diagram o f t h e Bubble Separator.

LIQUID AND GAS

37 W

a s i g n i f i c a n t e r r o r i n t h e mass-transfer measurements.

Nevertheless,

s e v e r a l "special" t e s t s were made i n which about 10% of t h e normal gas flow w a s purposely introduced downstream of t h e s e p a r a t o r and allowed t o recirculate.

The measured r a t e s of change i n loop concentration

under t h e s e conditions were always l e s s than 3% of t h e normal r a t e and t h e e f f e c t of t h e apparently much smaller amounts of by-pass t h e r e f o r e were f e l t t o be acceptable. This s e p a r a t o r was t h e major f a c t o r i n l i m i t i n g t h e ranges of Reynolds numbers t h a t could b e obtained i n t h i s system.

For a given

mixture, a s flow was increased a flow r a t e was eventually reached a t which t h e r e was an observed "breakthrough" of many l a r g e bubbles t h a t would continue t o r e c i r c u l a t e .

A t t h i s l e v e l o f flow it was necessary

t o terminate t h e t e s t s with t h e p a r t i c u l a r mixture.

I n a d d i t i o n t o t h e flow l i m i t i n g aspect of t h e separator, an unexpected l a r g e amount of mass t r a n s f e r occurred t h e r e - p r o b a b l y due t o t h e energy d i s s i p a t i o n o f t h e s w i r l and t h e r e l a t i v e l y l a r g e amount of contact time between t h e l i q u i d and gas.

Consequently a l a r g e r than

a n t i c i p a t e d "end e f f e c t " r e s u l t e d t h a t had t o be accounted for i n d e t e r mining t h e mass-transfer c o e f f i c i e n t s applicable t o t h e t e s t s e c t i o n only.

This c o r r e c t i o n r e s u l t e d i n decreased r e l i a b i l i t y of t h e r e s u l t s .

Test Section The t e s t s e c t i o n was considered as t h a t p o r t i o n of conduit between t h e bubble generator e x i t and t h e entrance of an elbow leading i n t o t h e s e p a r a t o r entrance pipe ( s e e Figure 1, page 24).

It consisted of f i v e

s e c t i o n s of 2-inch-diameter conduit flanged together with Teflon gaskets. A s encountered i n t h e d i r e c t i o n of flow t h e s e were a 4-foot-long

section

38 of g l a s s pipe, a 10-foot-long s e c t i o n of g l a s s pipe, a 6 1/2-foot-long s e c t i o n of s t a i n l e s s s t e e l "long-radius" U-bend, another 10-foot-long s e c t i o n o f g l a s s pipe, and a 5-foot-long s e c t i o n of g l a s s pipe, f o r a t o t a l of 35 1 / 2 f e e t of length.

The t e s t s e c t i o n and bubble generator

were connected t o t h e r e s t of t h e loop piping through t h e bubble genera t o r t e e a t t h e entrance and an elbow a t t h e e x i t which served as p i v o t p o i n t s t o permit t h e t e s t s e c t i o n t o be mounted i n any o r i e n t a t i o n from horizontal t o vertical. Bubble Surface Area Determination

- Photographic

System

The mean s i z e s and i n t e r f a c i a l areas p e r u n i t volume of t h e bubble dispersions were determined photographically using a Polaroid camera and two Strobolume f l a s h u n i t s .

To reduce d i s t o r t i o n t h e photographs were

taken through rectangular g l a s s p o r t s f i t t e d around t h e c y l i n d r i c a l g l a s s conduit and f i l l e d with a l i q u i d having t h e same index of r e f r a c t i o n a s t h e glass.

The p o r t f o r ' ' i n l e t " p i c t u r e s was l o c a t e d about one

f o o t downstream from t h e bubble generator e x i t and t h e " e x i t " p o r t was l o c a t e d about two f e e t upstream from t h e t e s t s e c t i o n e x i t . The Polaroid camera was equipped with a s p e c i a l l y made t e l e s c o p i c l e n s t h a t permitted taking photographs i n good focus across t h e e n t i r e c r o s s s e c t i o n of t h e conduit.

The camera was semi-permanently mounted

onto t h e f a c i l i t y s t r u c t u r e i n such a manner t h a t photographs could be taken a t t h e " i n l e t " p o r t and then t h e camera pivoted f o r t a k i n g a subsequent p i c t u r e through t h e " e x i t " port.

For v e r t i c a l o r i e n t a t i o n of

t h e t e s t section, photographs were taken d i r e c t l y through t h e ports.

39 For h o r i z o n t a l t e s t s , t h e camera remained i n i t s " v e r t i c a l o r i e n t a t i o n " p o s i t i o n and t h e photographs were taken through high q u a l i t y f r o n t surf a c e mirrors. With t h e camera focused along t h e a x i s of t h e conduit, bubbles

c l o s e r t o t h e camera appear l a r g e r and those f u r t h e r away sppear smaller.

To determine t h e magnitude of t h i s p o s s i b l e e r r o r source, small wires of known diameter were mounted i n s i d e t h e conduit across t h e cross section. Photographs obtained a f t e r focusing on t h e c e n t r a l wire indicated l e s s t h a n one percent maximum e r r o r i n t h e apparent diameter reading. The Strobolume f l a s h u n i t s (one f o r each p o r t ) produced p i c t u r e s of b e s t c o n t r a s t when mounted t o provide d i f f u s e back l i g h t i n g i n which t h e l i g h t s were aimed d i r e c t l y i n t o t h e camera l e n s from behind t h e photo ports.

Semi-opaque "milky" P l e x i g l a s s h e e t s between t h e l i g h t s and t h e

photo p o r t s served as t h e l i g h t diff'users. Bubble diameters could have been scaled d i r e c t l y o f f t h e photographs

for each run and used t o e s t a b l i s h t h e i n t e r f a c i a l areas and mean diamet e r s j u s t a s w a s done t o v a l i d a t e t h e bubble s i z e d i s t r i b u t i o n f'unction. However, this proved to be such an onerous and time-consuming procedure

t h a t it would have been p r o h i b i t i v e due t o t h e l a r g e number of experimental runs and need f o r a t l e a s t two photographs for each run. quently, t h e following use was made of t h e d i s t r i b u t i o n f'unction. The i n t e r f a c i a l a r e a per u n i t volume i s defined as W

N n6" f ( 6 ) d6 0

and t h e bubble volume f r a c t i o n i s given by @

f(6) d6 0

Conse-

40 Recalling t h e d e f i n i t i o n of t h e Sauter mean diameter, Equation

(7),it

i s seen from t h e above t h a t , regardless of t h e form of t h e d i s t r i b u t i o n function, t h e i n t e r f a c i a l area p e r u n i t volume can b e expressed a s a

E-

6m d vs

(9)

(6), Equation (8)

For t h e d i s t r i b u t i o n function of Equation

may be i n t e -

grated t o give

3 / 4

Therefore, by measuring t h e volume f r a c t i o n , @,

it was only necessary

t o count t h e number of bubbles per u n i t volume from t h e photographs and Equation (9) w a s then used t o

use Equation (10) t o e s t a b l i s h t h e areas.

Counting t h e number of bubbles i n

determine t h e mean bubble diameters.

a r e p r e s e n t a t i v e a r e a of t h e photographs was a considerably e a s i e r t a s k t h a n measuring t h e a c t u a l s i z e s of each bubble.

However, it was then

necessary t o have an independent determination of t h e volume f r a c t i o n occupied by t h e bubbles. HughmarkS4 presented a volume f r a c t i o n c o r r e l a t i o n t h a t graphic a l l y r e l a t e d a flow parameter, X, defined from

xm) t o t h e parameter Z

(Re)'/

(Fr)l/8/Y''

where

Y For p >> p

' Equation

Q ~ / ( Q-t-~ Q ~ }

(11)reduces t o @ = X Q / Q g R

41 Hughmark's c o r r e l a t i o n f o r X a t s u f f i c i e n t l y l a r g e Z i s n e a r l y f l a t with X changing from

0.7 t o 0.9 over a 10-fold change i n Z.

For t h e

conditions of t h e experiments i n t h i s r e p o r t , X was considered t o b e constant a t an average value of 0.73. When volume f r a c t i o n s were measured i n t h e v e r t i c a l flow t e s t s ,

it was found t h a t @ =

O.73

QgIQA

gave a good measure of t h e mean value f o r a given t e s t but t h a t t h e volume f r a c t i o n s were sometimes considerably smaller than t h i s i n t h e r i s e r l e g of t h e t e s t s e c t i o n and, a t t h e same time, comparably l a r g e r i n t h e downcomer.

It was apparent t h a t t h i s d i f f e r e n c e was due t o

buoyancy driven r e l a t i v e flow between t h e bubbles and t h e l i q u i d . Separate volume f r a c t i o n s were t h e r e f o r e determined f o r each l e g based on a mass balance.

This mass balance between t h e r i s e r and downcomer

s e c t i o n s i n a constant a r e a conduit t a k e s t h e form

Letting

vr = v + vb and

vd = v -

then

@ /@ E N / N d r d r

v+ =

The bubble t e r m i n a l v e l o c i t y , Vb, number, Re

V d /Y). b vs

V-Vb

depends on t h e bubble Reynolds

42 If Reb < 2, then Stokes l a w r e s u l t s i n

If Re > 2, then V i s determined from a balance between t h e drag b b

1 and

~) f o r c e [ ( C , P V " ~ / ~ ~(nd2,,/4)

where t h e drag c o e f f i c i e n t , Cd, C

buoyancy ( p1-rd~~,g/g,6)t o be

i s given by 18.5/Re:'6

It w a s f'urther assumed t h a t t h e average of t h e r i s e r and downcomer

volume f r a c t i o n s could be c a l c u l a t e d by

mr +

0.73 Qg/Qa

2 Then with i t e r a t i o n s t o e s t a b l i s h dvs, Vb,

and Reb, Equations (13) and

(14) were solved t o determine t h e i n d i v i d u a l l e g v e r t i c a l flow volume f r a c t i o n s , and Equation (10) was used t o e s t a b l i s h t h e i n t e r f a c i a l a r e a s p e r u n i t volume.

The averages were used t o e x t r a c t t h e mass-transfer

c o e f f i c i e n t s from t h e ka products. A s a f'urther i n d i c a t i o n of t h e accuracy of t h e d i s t r i b u t i o n f'unction

and t h e v a l i d i t y of t h i s technique f o r e s t a b l i s h i n g t h e v e r t i c a l flow surface areas, Figure 6 com-pares some surface areas determined as outl i n e d above with t h e a r e a s measured d i r e c t l y from t h e photographs.

The

experimental conditions f o r t h e run numbers i d e n t i f y i n g each p o i n t a r e l i s t e d i n Table I V . I n h o r i z o n t a l flows t h e volume f r a c t i o n s were t h e same i n each l e g but s t r a t i f i c a t i o n of t h e bubbles near t h e t o p of t h e conduit, e s p e c i a l l y

43 ORNL-DWG

71-8000

2 .o

1.9 1.8

1.7

- a, = 4.22 FUNCTION) WHERE N = NUMBER

1.6

I .

V E R T I C A L FLOW I N 2-in. D I A M E T E R C O N D U I T = INTERFACIAL AREA MEASURED D I R E C T L Y F R O M PHOTOGRAPHS

- a,

O F B U B B L E S PER U N I T V O L U M E T A K E N FROM PHOTOGRAPHS

@ = V O L U M E F R A C T I O N OCCUPIED BY

w 3

1.5

1.4

-1

> t-

(FROM DISTRIBUTION

THE BUBBLES THE N U M B E R S I D E N T I F Y I N G T H E DATA POINTS ARE R U N S FOR WHICH C O N D I T I O N S A R E L I S T E D ELSEWHERE

1.3

CK w

a a w

1.2

1.1

a 1.0

a -

0 Q

0.9

0 8

0 W

0.7

I-

0.6

3 0 -1 a

0.5

,”

0.4

0.3 0.2

0.1 0

0.2

0.4

0.6

0.8

1 .O

1.2

a m , M E A S U R E D I N T E R F A C I A L A R E A P E R U N I T VOLUME

Figure 6.

1.4

(in:’)

Coqarison of Interfacial Areas Per Unit Volume Measured Directly from Photographs with Those Established Through the Distribution Function. Vertical Flow.

Table IV. Experimental Conditions f o r Runs Used t o Validate Surface Area Determination Method f o r V e r t i c a l Flows

Q ~ I(%IQ ~

Mixture

(% g l y c e r i n e )

Run No.

QR ( a m )

71 73 76

20 20 20

0.5

0.1 0.1

40 40 40

0.5 0.3

85 87 91

0.1

0.5

0.3

80 80

0.1 0.5 0.5 0.5

100 104

40 10 50 20

30 l+O

0 0 0 0 0 0 0 0 0

0.5 0.3 0.5 0.5 0.5

0.5 0.5

0.5

50 50

50 50 37.5 37.5 37.5 37.5 37.5 d

17.5

45 W

a t low flows, i n v a l i d a t e d t h e use of Equation

(14).

It was found

p o s s i b l e however t o c o r r e l a t e t h e h o r i z o n t a l flow volume f r a c t i o n s a t

/Q = 0.3% with t h e r a t i o , V/Vb, of t h e a x i a l l i q u i d v e l o c i t y t o t h e g R bubble terminal v e l o c i t y i n t h e l i q u i d . This c o r r e l a t i o n i s shown i n Q

Figure

7 w i t h t h e i d e n t i f i c a t i o n of t h e randomly s e l e c t e d runs given

i n Table V.

Table V. Experimental Conditions for Runs Shown on Horizontal Flow Volume Fraction C o r r e l a t i o n

376 390 382 389 391 365 355 370 168

404 422 427

55 60

For V/Vb

0.037 0.026 0.049 0.014

0.014

70 30 35

0.066 0.061 0.026

was used while f o r

0.030

v/vb

0.0018 + 0.021/(v/vb)

g r e a t e r than $ =

V/V

37.5 37.5 37.5 37.5

l e s s than 30, a l e a s t squares l i n e ,

w a s used.

50 50 50 0 0 50 0

0.024

50 30 60 30 30

Goo

50 50

0.033 0.028 0.059

30 a constant value,

0.0025

Severe s t r a t i f i c a t i o n prevented experimentation a t values of

l e s s than about 3.

46 ORNL-DWG

71-8001

0.010 V O L U M E F R A C T I O N COR R E L AT1 0 N H O R I Z O N T A L F L O W IN A 2 - i n . D I A M E T E R C O N D U I T Q ~ / Q L=

0.3%

@ MEASURED FROM PHOTOGRAPHS

0.009

v/vb

0.0188 QL(/.dp)0'5'5/d"s'.27

WHERE

Q L = gpm = Ib,/ft.hr p = Ibm/ft3 dvs = inches

0.008

0.007 2

0 l0

a a:

0.006

w E

= 0.0018 t O . O Z l / ( V / V b ) L E A S T SQUARE L I N E FOR DATA W I T H v/vb e 3 0

3 -I

0.005

> w -I

m m 3 m

0.004

6 0.003

0 .oo2 GLYCERINE (O/O)

0.001

37.5

N U M B E R S I D E N T I F Y I N G DATA P O I N T S

0 I

R A T I O OF A X I A L TO T E R M I N A L V E L O C I T Y ( v / v b )

Figure 7.

Correlation of Horizontal Flow Volume Fraction.

47 W

This h o r i z o n t a l flow volume f r a c t i o n c o r r e l a t i o n i n conjunction with Equation (10) was used t o e s t a b l i s h t h e h o r i z o n t a l flow i n t e r f a c i a l areas p e r u n i t volume.

An i n d i c a t i o n of t h e adequacy of t h i s procedure

i s given i n Figure 8 i n which c a l c u l a t e d and measured areas a r e compared

f o r t h e runs i d e n t i f i e d i n Table V. End Effect I n t h e t r a n s i e n t response mode of operation a l l mass t r a n s f e r occuring outside t h e t e s t s e c t i o n ( p r i n c i p a l l y i n t h e bubble separator and generator) must be independently measured and accounted f o r i n e s t a b l i s h i n g t h e ka products applicable only t o t h e t e s t section. "End-effect" measurements were made a f t e r a l l other scheduled t e s t s were completed by moving t h e bubble generator t o a p o s i t i o n a t t h e t e s t s e c t i o n e x i t which allowed t h e bubbles t o flow d i r e c t l y from t h e generat o r i n t o t h e separator

- e f f e c t i v e l y by-passing

t h e t e s t section.

All

t e s t s were then repeated d u p l i c a t i n g a s n e a r l y as p o s s i b l e t h e o r i g i n a l conditions.

With t h e end-effect response s o measured, t h e c o r r e c t i o n

was determined as follows. Consider t h r e e regions of mass t r a n s f e r i n s e r i e s representing t h e bubble generator (Region l), t h e t e s t s e c t i o n (Region 2 ) , and t h e bubble s e p a r a t o r (Region 3).

The o r i g i n a l measurements, i n d i c a t e d h e r e by a

s u b s c r i p t lrI,tIr determined t h e r a t i o , K t r a t i o n across a l l t h r e e regions.

where K,, W

K,,

of t h e o u t l e t t o i n l e t concen-

Therefore

and K3 a r e t h e o u t l e t and i n l e t concentration r a t i o s across

t h e i n d i v i d u a l regions.

48 ORNL-DWG

7.1-8002

1.9 HORIZONTAL F L O W IN 2-117. DIAMETER C O N D U I T

1.8

a,=

1.7 c -i

MEASURED DIRECTLY FROM PHOTOGRAPHS

a c = 4 . 2 2 N 1 I 3 O 2 l 3 (FROM DISTRIBUTION FUN C T I ON)

1.6

.-l i

WHERE

N = NUMBER OF B U B B L E S PER U N I T

VOLUME FROM PHOTOGRAPHS

1.5

3 3

= VOLUME FRACTION OCCUPIED BY 1.4

THE BUBBLES

1.3

RUN NUMBERS SHOWN ARE I D E N T I F I E D ELSEWHERE

1.2

> I-

z a

t20%

a a W ir

1.1

t .o

a -

0.9

0.8

W I-

z -

0.7

n W

0.6

3 0 -I

a u

0.5

0.4

u 0

0.3 0.2

0.1

0 0 ,a,

Figure 8.

0 2

0.4

0.6

0 8

1.0

1.2

MEASURED I N T E R F A C I A L AREA PER U N I T VOLUME (in.-')

Comparison of Measured and Calculated I n t e r f a c i a l Areas Horizontal Flow. Per U n i t VoJume.

49 The second s e r i e s of t e s t s , subscripted "11," with only t h e bubble

generator and s e p a r a t o r regions e n t e r i n g i n t o t h e mass t r a n s f e r , d e t e r mined t h e r a t i o

KII

Consequently t h e d e s i r e d r a t i o , K,,

K1 K3

across t h e t e s t s e c t i o n only, was

determined from K - K a = K /K

I I1

An estimate of t h e e r r o r involved i n t h i s procedure i s given i n Appendix G.

Summary of Experimental Procedure The mode of experimentation was t r a n s i e n t with t h e independent v a r i a b l e s being Schmidt number (depending on percent glycerine i n glycerine-water mixtures), Reynolds number ( l i q u i d flow), bubble mean diameter ( c o n t r o l l e d by bubble generator probe p o s i t i o n ) , and t e s t seet i o n orientation ( v e r t i c a l o r horizontal).

Other parameters t h a t were

held constant f o r most of t h e s e t e s t s include t h e t e s t s e c t i o n conduit diameter ( D = 2 i n c h e s ) , t h e r a t i o of gas t o l i q u i d volumetric flow ( Q /Q = 0.3%) and t h e f l u i d temperature (25OC). g a It was found t h a t t h e only e f f e c t of volume f r a c t i o n up t o 1% was

i n t h e highly p r e d i c t a b l e change i n s u r f a c e area.

No significant

d i f f e r e n c e was detected i n t h e mass-transfer c o e f f i c i e n t s themselves which a r e on a u n i t area basis.

Consequently with t h e exception of

some of t h e e a r l y runs most of t h e experiments were performed a t a convenient volume f r a c t i o n of 0.3%.

I n a d d i t i o n it was found t h a t f o r t h e

d i s t i l l e d water runs (no g l y c e r i n e ) t h e r a p i d agglomeration o f t h e

50 bubbles a t t h e flows obtainable prevented meaningful i n t e r p r e t a t i o n of t h e data.

Consequently a l l water t e s t s were performed with t h e addition

of about 200 ppm of normal b u t y l alcohol which e f f e c t i v e l y i n h i b i t e d t h e agglomeration but may have r e s u l t e d i n a d i f f e r e n t s u r f a c e condition compared t o t h e o t h e r mixtures.

The addition of t h i s same amount of

N-butyl a l c o h o l t o t h e glycerine-water mixtures made no s i g n i f i c a n t difference.

For a given l i q u i d mixture and o r i e n t a t i o n of t h e t e s t s e c t i o n , t h e procedure followed t o o b t a i n a s e r i e s of d a t a i s o u t l i n e d i n d e t a i l below:

1. The loop was f i r s t purged repeatedly with d i s t i l l e d water t o remove r e s i d u a l l i q u i d from previous experiments and t h e system allowed

t o dry by blowing a i r through it overnight. 2.

The mixture of g l y c e r i n e and water t o be used was p r e c i s e l y

made up i n t h e weigh tank and t h e n thoroughly mixed by vigorous s t i r r i n g produced by pumping t h e l i q u i d from t h e bottom of t h e tank back i n t o t h e top.

The loop was f i l l e d using a small a u x i l i a r y pump and t h e system

operating pressure was s e t a t a nominal

40 p s i a

by helium p r e s s u r e over

t h e i n t e r f a c e i n t h e bubble separator.

Liquid flow w a s e s t a b l i s h e d by energizing t h e main loop c i r c u -

l a t o r and t h e flow was s e t a t t h e desired l e v e l by t h r o t t l i n g through a l l t h r e e rotameters.

The system was charged with oxygen t o about seven or e i g h t

p a r t s per million by passing oxygen bubbles through t h e bubble generator, t h e t e s t section, and t h e bubble separator.

The system was allowed t o

51 run a s u f f i c i e n t time a f t e r t h e oxygen flow had been terminated t o i n s u r e t h a t t h e concentration readings were steady.

The bubble generator probe p o s i t i o n was s e t t o obtain t h e

f i r s t d e s i r e d mean bubble diameter f o r t h e given t e s t conditions. The helium flow, having been p r e s e t t o give Q /Q = 0.3% a t g J t h e given l i q u i d flow, was turned on i n i t i a t i n g t h e t r a n s i e n t experiment

which was u s u a l l y allowed t o continue for 10 t o 15 minutes.

8. The oxygen concentration was continuously recorded and d a t a sheet loggings were made of l i q u i d flow through each rotameter, t e s t s e c t i o n i n l e t pressure, t e s t s e c t i o n pressure drop, helium p r e s s u r e a t t h e c a p i l l a r y tube e x i t , pressure drop across t h e c a p i l l a r y tube, loop temperature, bubble generator probe p o s i t i o n , and atmospheric

p r e s sure.

About midway through t h e t r a n s i e n t for each t e s t , a Polaroid

p i c t u r e of t h e bubbles was made through one of t h e photo p o r t s (entrance

or e x i t ) and t h e n t h e camera was pivoted and a p i c t u r e taken through t h e o t h e r photo port. 10.

For the given l i q u i d flow, t h e bubble generator probe p o s i t i o n

was v a r i e d t o produce d i f f e r e n t mean diameters. and u s u a l l y obtained. repeated.

Five values were d e s i r e d

For each p o s i t i o n t h e above procedure (5-9) was

Occasionally t o produce e x t r a l a r g e bubbles, t h e gas was

introduced through t h e t e s t - s e c t i o n i n l e t p r e s s u r e t a p

- bypassing

the

bubble generator i t s e l f .

11. The l i q u i d flow was v a r i e d over t h e desired range and t h e above procedure (5-10)was repeated f o r each flow s e t t i n g .

Typical t r a n s i e n t data of h C / C o versus time taken d i r e c t l y from t h e oxygen concentration recording chart i s shown on Figure 9 which i l l u s t r a t e s t h e constancy of t h e slope (+h KI QA/Vs). The system volume had been previously measured t o be -2.52

f t 3 by

f i l l i n g t h e system completely w i t h water which was then c o l l e c t e d and weighed.

Using t h i s and t h e measured values of Q

t h e constants K

were determined from t h e slopes of t h e curves.

After a l l v e r t i c a l and h o r i z o n t a l t e s t s were completed, "end e f f e c t s " were measured by moving t h e bubble generator t o t h e t e s t s e c t i o n e x i t and repeating each experiment with t h e o r i g i n a l conditions duplicated as n e a r l y a s possible. The values of K

I1 were then c a l c u l a t e d from t h e slopes of t h e "end e f f e c t " curves and K's were c a l c u l a t e d from K

K ~ / K ~ ~

The products, ka, were e x t r a c t e d from K through Equations (5). The bubble photographs were analyzed t o o b t a i n t h e i n t e r f a c i a l areas per u n i t volume and t h e mean diameters.

Typical examples of an

i n l e t and e x i t photograph a r e shown on Figure 10.

The o u t l i n e d regions

were used a s t h e sample populations f o r counting t h e number of bubbles p e r u n i t volume, N. The applicable volume f r a c t i o n c o r r e l a t i o n [ e i t h e r Equations (13),

(14), o r (15)]

was used t o determine

and Equations (10) and (9) were

used t o c a l c u l a t e t h e i n t e r f a c i a l areas per u n i t volume and t h e mean bubble diameters, r e s p e c t i v e l y .

F i n a l l y , t h e averages of t h e i n l e t and

e x i t areas were used t o e x t r a c t k from t h e ka products.

OR N L- D WG 7 1 - 8045

1.0

I-

4 IL I-

2 W

T Y P I C A L EXPERIMENTAL DATA HORIZONTAL FLOW IN 2 - i n DIAMETER CONDUIT FLUID = 12 5% GLYCERINE 8 7 5 % WATER QL = 8 5 ( g p m ) Q,/QL= 03% B U B B L E M E A N DIAMETER, d Y S = 0 0 1 5 in

0.5

2 0 V

0 -

( 3 \

e 6 z

0 -

U I -

I-

Z W

0 2 0

0.2

2 W (3

> X

0.1 0

1, E X P E R I M E N T T I M E (min)

Figure

Typical Experimental Concentration Transient I l l u s t r a t i n g Straight-Line Behavior on Semi-Log Coordinants.

PHOTO 1854-71

Figure 10.

Typical Examples of Bubble Photographs: a. Inlet b. Exit. Vertica3 Flow, 37.5% Glycerine-62.5$ Water, QJ = 20 gpm, Qg/Qk = 0.35, D = 2 inches, and hS = 0.023 inches.

CHAPTER I V EXPEXIMENTAL RESULTS The experimentally measured mass-transfer t r a n s i e n t s i n i t i a l l y were converted i n t o pseudo mass-transfer c o e f f i c i e n t s without any adjustment being made f o r mass-transfer occurring outside t h e t e s t section.

The

r e s u l t s t h u s obtained a r e not t h e t r u e mass-transfer c o e f f i c i e n t s because s i g n i f i c a n t mass t r a n s f e r occurred i n t h e bubble generating and s e p a r a t i n g equipment.

Nevertheless, considerable information can be

gathered from t h i s "unadjusted" d a t a because of i t s presumed g r e a t e r precision.

The t r u e mass-transfer c o e f f i c i e n t s with extraneous mass-

t r a n s f e r e f f e c t s accounted f o r a r e presented l a t e r i n t h i s Chapter. Unadjusted Result s The "unadjusted" mass-transfer c o e f f i c i e n t s determined as o u t l i n e d i n Chapter I11 as functions of bubble mean diameter, Reynolds number, o r i e n t a t i o n of t h e t e s t section, and Schmidt number a r e given i n Appendix

H (Figures 35-44, pages 138-147). The "raw" data which c o n s i s t s of recorder c h a r t s of oxygen concent r a t i o n versus time, innumerable photographs of bubble populations, and l o g book records of flows, probe s e t t i n g s , temperature, pressure and

o t h e r conditions a r e on f i l e i n t h e Heat Transfer-Fluid Dynamics Department, Reactor Division of t h e Oak Ridge National Laboratory and a r e a v a i l a b l e upon request. It i s i n s t r u c t i v e t o consider t h e c r o s s p l o t s (Figures 45-49, pages

148-152).

Similar t o Lamont s l l r e s u l t s , t h e h o r i z o n t a l and t h e v e r t i c a l

56 flow values were i d e n t i c a l above s u f f i c i e n t l y high Reynolds numbers.

flows were decreased below t h e s e Reynolds numbers, however, t h e v e r t i c a l flow c o e f f i c i e n t s were l a r g e r than t h e h o r i z o n t a l flow c o e f f i c i e n t s and seemed t o asymptotically approach constant values.

The h o r i z o n t a l flow

data, on t h e o t h e r hand, continued along s t r a i g h t l i n e v a r i a t i o n s (on log-log'coordinates) u n t i l e i t h e r t h e flow w a s t o o low t o prevent conc e n t r a t i o n p o l a r i z a t i o n a t t h e oxygen sensors o r i n some cases severe The p e r t i n e n t

s t r a t i f i c a t i o n o f t h e bubbles prevented f'urther t e s t i n g .

r e s u l t s t o be considered, based on t h e s e unadjusted data, a r e t h e values of t h e Reynolds number a t which v e r t i c a l and h o r i z o n t a l flow r e s u l t s become i d e n t i c a l and t h e apparent asymptotes approached by t h e v e r t i c a l flow c o e f f i c i e n t s a t low flows.

Mass t r a n s f e r occurring o u t s i d e t h e

t e s t s e c t i o n should not a f f e c t e i t h e r of t h e s e and t h e i r values should be t h e same as f o r t h e d a t a presented l a t e r t h a t r e p r e s e n t s t h e t r u e masstransfer coefficients. Equivalence of Horizontal and V e r t i c a l Flow Mass Transfer It seems evident t h a t g r a v i t a t i o n a l f o r c e s (buoyancy) t e n d t o

e s t a b l i s h a steady r e l a t i v e flow between t h e bubbles and t h e l i q u i d i f t h e bubbles a r e f r e e t o move i n t h e v e r t i c a l d i r e c t i o n ( a s they would be i n v e r t i c a l o r i e n t a t i o n s of t h e t e s t s e c t i o n s ) and a r e not r e s t r i c t e d by p h y s i c a l boundaries ( a s t h e y would be i n h o r i z o n t a l o r i e n t a t i o n s ) . The bubbles are a l s o acted upon by i n e r t i a l f o r c e s generated by t h e t u r b u l e n t motions within t h e l i q u i d .

These t u r b u l e n t i n e r t i a l f o r c e s

a r e randomly d i r e c t e d and thus tend, on t h e average, t o counteract t h e g r a v i t a t i o n a l forces.

Therefore it would be reasonable t o assume t h a t

i f t h e magnitudes of t h e t u r b u l e n t i n e r t i a l forces, Fi,

were known

57 then t h e i r r a t i o , F./F would compared t o t h e g r a v i t a t i o n a l f o r c e s , F 1 g' g' be a measure of t h e r e l a t i v e importance of t h e s e f o r c e s i n e s t a b l i s h i n g t h e mass-transfer c o e f f i c i e n t s .

A t s u f f i c i e n t l y high values of Fi/F

g one would ex-pect t h e t u r b u l e n t forces t o dominate and t h e r e should then

be no d e t e c t a b l e d i f f e r e n c e i n t h e h o r i z o n t a l and v e r t i c a l flow r e s u l t s . The g r a v i t a t i o n a l f o r c e on a bubble of diameter, d, i s t h e weight of t h e displaced f l u i d

The t u r b u l e n t i n e r t i 1 f o r c e exerted on

bubble e s s e n t i a l l y

t r a v e l i n g a t t h e l o c a l f l u i d v e l o c i t y i n a t u r b u l e n t l i q u i d i s not s o e a s i l y determined.

Consequently, use was made of dimensional arguments.

I n a t u r b u l e n t f l u i d t h e mean v a r i a t i o n i n v e l o c i t y , AV, over a distance, 1, ( g r e a t e r than t h e microscale) i s given dimensionally by

where ev i s t h e power d i s s i p a t i o n p e r u n i t volume.

The

1/3 power on h

agrees with t h e r e s u l t of Hinze (Reference 37) f o r t h e v a r i a t i o n i n turbulent i n t e n s i t y required t o r e s u l t i n t h e Kolmogoroff spectrum law. Similarly, t h e period, 8, f o r such v e l o c i t y v a r i a t i o n s i s given dimens i o n a l l y by

Following L e ~ i c h , ~it " i s p o s t u l a t e d t h a t t h e mean a c c e l e r a t i o n a undergone by a f l u i d element of s i z e , h , i s

58 A s p h e r i c a l f l u i d element with t h i s mean a c c e l e r a t i o n must have

experienced a “mean” f o r c e given by

It i s f’urther p o s t u l a t e d t h a t a bubble of diameter, d, i n t h e t u r b u l e n t

l i q u i d w i l l be subjected t o t h e same mean f o r c e s a s those exerted on a f l u i d element of t h e same size.

Therefore t h e mean t u r b u l e n t i n e r t i a l

f o r c e on t h e bubble i s given by

Dividing by Equation (16) t h e r a t i o of i n e r t i a l f o r c e s t o g r a v i t a t i o n a l f o r c e s i s given by

For flow i n conduits, t h e power d i s s i p a t i o n p e r u n i t volume can be expressed as

and t h e pressure gradient can be determined from t h e Blasius r e l a t i o n s h i p , dP f = dx D

V” 2gc

= ( f p2/2gc D3 p ) Re”

Using t h e f r i c t i o n f a c t o r f o r smooth tubes, f = O.316/(Re)l1

t h e power d i s s i p a t i o n p e r u n i t volume i s

59 S u b s t i t u t i o n i n t o Equation (18) and replacing t h e bubble diameter

by t h e S a u t e r mean gives P

Since Equation ( 2 0 ) w a s e s t a b l i s h e d on dimensional grounds, t h e r e e x i s t s a p r o p o r t i o n a l i t y constant of unknown magnitude.

To e s t a b l i s h

t h e value t h a t t h e r a t i o should have t o serve a s a c r i t e r i o n f o r d e t e r mining when h o r i z o n t a l and v e r t i c a l flow mass-transfer c o e f f i c i e n t s become i d e n t i c a l , use w a s made of t h e d a t a of Lamont gathered from h i s r e p o r t as l i s t e d i n Table V I below.

Table V I . Conditions a t Which Horizontal and V e r t i c a l Flow Mass Transfer C o e f f i c i e n t s Become Equal (Lamont â&#x20AC;&#x2DC; s Data)â&#x20AC;?

Case I

Case I1

Conduit Diameter, D (inches)

5/16

Reynolds Modulus, Re

io4

Liquid Viscosity, p ( c e n t i p o i s e )

0.89

5/8 3 x io4 0.89

Liquid Density, p (g/cm3)

1.0

Bubble Diameter, d ( i n c h e s )

-5132

4/32

S u b s t i t u t i o n of t h e d a t a o f Case I i n t o Equation ( 2 0 ) gives

F . / F = 1.5 = g A s a check t h e data of Case I1 a r e compared,

x 104

For t h e p r e s e n t i n v e s t i g a t i o n , t h e l o c i of p o i n t s f o r F./F = 1.5 1 g a s c a l c u l a t e d from Equation (20) a r e shown on Figures 45-49, pages 148W

152.

It i s seen t h a t t h e r a t i o F./F 1

seems t o be a good p r e d i c t o r f o r

t h e equivalence of t h e h o r i z o n t a l and v e r t i c a l r e s u l t s . V e r t i c a l Orientation Low-Flow Asymptotes A s l i q u i d flow i s reduced, t h e g r a v i t a t i o n a l f o r c e s become more and

more dominant over t h e t u r b u l e n t i n e r t i a l forces.

Consequently, a t low

flows, t h e v e r t i c a l flow mass-transfer c o e f f i c i e n t s approach t h e values t h a t would be expected f o r t h e bubbles r i s i n g through a quiescent l i q u i d . The conditions of mass t r a n s f e r f o r bubbles r i s i n g through a column of l i q u i d have been extensively studied (e. g., References

26-30).

Resnick and Gal-0rS7 have recommended f o r s u r f a c t a n t - f r e e systems

They caution t h a t t h i s equation may give values s l i g h t l y higher than t h e observed data i n p a r t i c u l a r f o r lower concentrations of g l y c e r o l i n water-glycerol systems. I n t h e present i n v e s t i g a t i o n , t h e volume f r a c t i o n i s low so t h a t t h e above equation was approximated a s

and used t o determine t h e "calculated asymptotes" f o r t h e v e r t i c a l flow r e s u l t s a s i n d i c a t e d on t h e various d a t a p l o t s . Mas s-Transf e r C o e f f i c i e n t s With t h e end-effect accounted f o r as o u t l i n e d i n Chapter 111, t h e mass-transfer c o e f f i c i e n t s measured i n t h i s i n v e s t i g a t i o n a r e given i n Figures

50-58, pages 153-161 (Appendix

G).

The more revealing c r o s s p l o t s

61 of mass-transfer c o e f f i c i e n t s versus Reynolds number a r e shown i n Figures 11-15 which contain regression l i n e s f i t t e d t o t h e h o r i z o n t a l flow d a t a and c a l c u l a t e d l i n e s f o r t h e v e r t i c a l flow cases.

Vertical

flow d a t a a r e not shown f o r t h e 37.5% mixture because t h e end e f f e c t adjustments were not s a t i s f a c t o r y .

Excessive v i b r a t i o n of t h e bubble

generation probe t h a t occurred during t h e 37.5% experiments was eliminated by redesign of t h e probe before t h e h o r i z o n t a l d a t a were obtained. Time did not permit a r e o r i e n t a t i o n of t h e system t o t h e v e r t i c a l posit i o n t o repeat t h e runs. From t h e s e f i g u r e s i t i s seen t h a t t h e h o r i z o n t a l flow data f o r water ( p l u s N-butyl alcohol) apparently have a l e s s e r slope than t h a t f o r t h e glycerine-water mixtures.

Therefore a regression equation was

determined f o r t h e water runs alone and a s e p a r a t e regression equation was determined f o r t h e combined d a t a f o r t h e 12.5, g l y c e r i n e mixtures. mixture (Figure 15).

25, and 37.5%

A t h i r d behavior was observed f o r t h e 50% glycerine

It i s seen t h a t a l l t h e d a t a f o r t h i s mixture were

obtained a t Reynolds numbers l e s s t h a n t h a t required f o r Fi/F

1.5.

g However, i n s t e a d of a steady march of t h e h o r i z o n t a l flow d a t a down a s t r a i g h t l i n e a s observed f o r t h e o t h e r mixtures, t h e small bubble h o r i z o n t a l flow mass-transfer c o e f f i c i e n t s tended t o behave l i k e those f o r v e r t i c a l flows.

This behavior implies t h a t , i f t h e l i q u i d i s viscous

enough, small bubbles apparently can e s t a b l i s h steady r e l a t i v e flow cond i t i o n s i n t h e i r r i s e across t h e conduit cross section.

I n t h e s e runs,

t h e pipe w a l l apparently did not s i g n i f i c a n t l y i n h i b i t t h e bubble r i s e r a t e during t r a n s i t through t h e t e s t s e c t i o n and, evidently, t h e bubbles behaved e x a c t l y a s i f t h e y were r i s i n g through a v e r t i c a l conduit.

OR NL- DWG 7 1-7989

103

HORIZONTAL

VERTICAL

1 o4

105

I I Ill![

P I P E R E Y N O L D S N O , R e =_ V D / V

Figure 11. Mass as a Plus Flow

Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number Function of Bubble Sauter-Mean Diameter. Water -200 ppm N-Butyl Alcohol. Horizontal and V e r t i c a l i n a 2-inch Diameter Conduit.

106

ORNL-DWG

77-7990

5 ,--

c \

0 LL LL

0 I+

LL v)

a n

DIAMETER (in 1

0 0 A

003 004

LL 2

(Sh

H O R I Z O N T A L VERTICAL FLOW FLOW

000538 Re"'

Sco71

(%)'

07)

to4

1o5

PIPE REYNOLDS NO, Re E VD/v

Figure 12.

Mass T ran sf e r C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter. 12.5% Glycerine-87.5% Water. Horizontal and V e r t i c a l Flow i n a 2-inch Diameter Conduit.

TO6

ORNL-DWG

0 015 .

0 02

003

103

10" PIPE R E Y N O L D S

Figure 13.

71-7991

2 N O , Re E

105

VD/u

Mass Transfer Coefficients Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter. 25% Glycerine-75% Water. Horizontal and Vertical Flow in a 2-inch Diameter Conduit.

ORNL-DWG

7t-7992

0 5

0 02

0 04

103

104

PIPE REYNOLDS N O , Re

Figure

14.

105

VD/v

Mass T ran sfer C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter. 37.5% Glycerine-62.5% Water. Horizontal and V e r t i c a l Flow i n a 2-inch Diameter Conduit.

O R N L - D W G 7.1-7993

0 5 m

m a 2

0 2

io3

io4

105

PIPE R E Y N O L 3 S N O , E V D / V

Figure 15.

Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter. 50% Glycerine-53% Water. Horizontal and V e r t i c a l Flow i n a 2-inch Diameter Conduit.

106

67 These t h r e e kinds of observed h o r i z o n t a l flow behavior a r e fâ&#x20AC;&#x2122;urther i l l u s t r a t e d on Figure 16 f o r 0.02-in.

mean diameter bubbles.

The

regression slope of 0.94 f o r t h e glycerine-water mixtures agrees genera l l y with t h e l i t e r a t u r e a s discussed i n Chapter I1 and t h e slope of

0.52 f o r t h e water p l u s N-butyl alcohol i s , coincidentally, e x a c t l y what Lamont found,

However, t h e combined regression slope (0.79) f o r a l l t h e

water d a t a which includes t h e o t h e r bubble mean diameters was g r e a t e r than t h e value f o r t h e 0.02-in.

bubbles by themselves.

Calculating V e r t i c a l Flow Mass-Transfer C o e f f i c i e n t s f o r Fi/Fg Less Than 1 . 5 Since t h e r a t i o of t u r b u l e n t i n e r t i a l f o r c e s t o g r a v i t a t i o n a l f o r c e s i s seen t o be a good p r e d i c t o r of t h e Reynolds number a t which h o r i z o n t a l and v e r t i c a l flow mass-transfer c o e f f i c i e n t s become i d e n t i c a l ,

it i s proposed t h a t t h e varying r a t i o might a l s o serve a s a s c a l i n g f a c t o r a t a l l Reynolds numbers t o determine t h e r e l a t i v e importance of t h e p u r e l y t u r b u l e n t c o e f f i c i e n t s (Fi/F

c o e f f i c i e n t s ( v e r t i c a l flow asymptotes).

> 1 . 5 ) and t h e r e l a t i v e flow T h a t i s , i f t h e values a r e

known f o r t h e s t r a i g h t l i n e v a r i a t i o n a t higher Reynolds numbers where v e r t i c a l and h o r i z o n t a l c o e f f i c i e n t s a r e equal along w i t h t h e v e r t i c a l flow asymptotes, it i s proposed t h a t t h e intermediate v e r t i c a l flow mass-transfer c o e f f i c i e n t s can be c a l c u l a t e d by using F./F a s a l i n e a r 1 g s c a l i n g f a c t o r between t h e two.

Since F./F = 1.5 appears t o mark t h e 1 g

Reynolds numbers a t which t u r b u l e n t i n e r t i a l f o r c e s dominate over gravit a t i o n a l forces, t h e a c t u a l r a t i o o f f o r c e s a t t h a t condition a r e assumed t o be of t h e order of 10 t o 1 f o r g r a v i t a t i o n a l f o r c e s t o begin

ORNL-DWG

Figure 16.

1 o4 2 5 P I P E REYNOLDS N O , R e Z VD/V

105

Observed Types of Horizontal Flow Behavior, d = 0.02 inches and D = 2 inches. VS

71-7995

69 W

Consequently 10 (Fi/F )/1.5 was chosen a s an approg p r i a t e l i n e a r s c a l i n g f a c t o r and t h e v e r t i c a l flow mass-transfer c o e f f i t o be negligible.

c i e n t s were calculated from

i n which k

i s t h e c a l c u l a t e d asymptote given by Equation ( 2 1 ) and

% is

t h e value a t t h e given Reynolds number t h a t would be obtained by extendi n g t h e s t r a i g h t - l i n e v a r i a t i o n of t h e h o r i z o n t a l flow data. Using separate regression l i n e s f o r

5,t h e v e r t i c a l

flow mass-

t r a n s f e r c o e f f i c i e n t s c a l c u l a t e d from Equation (22) a r e compared with t h e d a t a on Figures 11-15, pages 62-66.

Except f o r t h e

5% mixture data,

Equation (22) provides a r e l a t i v e l y good d e s c r i p t i o n of t h e data. Comparison with Agitated Vessels A comparison of t h e h o r i z o n t a l flow data with t h a t of Sherwood and

Brian17 for p a r t i c u l a t e s i n a g i t a t e d v e s s e l s i s shown on Figure 17. Sherwood and Brianâ&#x20AC;&#x2122;s coordinates a r e used by converting c m through Equation (19) f o r flow i n conduits.

ev/p)

It i s seen t h a t , although

t h e r e l a t i v e magnitudes of t h e c o e f f i c i e n t s a r e comparable on an equival e n t power d i s s i p a t i o n b a s i s , t h e r e i s a Schmidt number separation of t h i s d a t a i n d i c a t i n g mobile i n t e r f a c i a l behavior.

I n agreement with t h e

f i n d i n g s of o t h e r i n v e s t i g a t i o n s reported i n Chapter 11, t h e v a r i a t i o n with Reynolds number f o r flow i n conduits i s much s t e e p e r than would have been expected from t h e a g i t a t e d v e s s e l data. A p o s s i b l e explanation f o r t h i s d i f f e r e n c e i n slope observed between

a g i t a t e d v e s s e l s and flow i n conduits may l i e i n t h e r e l a t i v e importance W

o f t h e g r a v i t a t i o n a l forces.

For example, t h e d a t a of t h i s research f o r

100

0.5

H O R I Z O N T A L FLOW I N 2 - l n

D I A M E T E R CONDUIT

0.2

0.1 1

Figure

lo2

17. Equivalent Power D i s s i p a t i o n Comparison of R e s u l t s with Agitated Vessel Data.

io3

71 W

small bubbles i n a 50% mixture of glycerine and water were obviously s t r o n g l y g r a v i t a t i o n a l l y dominated as evidenced by t h e e q u a l i t y of t h e h o r i z o n t a l and v e r t i c a l flow c o e f f i c i e n t s even a t very low Reynolds numbers.

A comparison of t h e s e "gravity-influenced" d a t a with Sherwood

and B r i a n ' s c o r r e l a t i o n shown on Figure 18 i n d i c a t e s a remarkable s i m i larity.

It may be t h a t g r a v i t a t i o n a l forces a r e g e n e r a l l y l e s s important

f o r flow i n conduits than f o r flow i n a g i t a t e d v e s s e l s where t h e r e may be a g r e a t e r degree of anisotropy.

Recommended Correlations A r e g r e s s i o n l i n e through a l l t h e h o r i z o n t a l flow d a t a except t h e

water and t h e 50% mixtures has a Schmidt number exponent of 0.71 using the literature"

values of 8.

These values of 8 (Figure 25, page 114,

Appendix A ) f i r s t increase with a d d i t i o n of glycerol, reach a maximum a t about 12.5% glycerol, and then decrease.

This behavior represents a

s t r i k i n g departure from t h e Stokes-Einstein behavior u s u a l l y observed f o r aqueous mixtures.

If, i n s t e a d of using t h e s e values f o r 8, a smooth

monotonically decreasing l i n e i s drawn through t h e f i r s t , f o u r t h , and f i f t h data p o i n t s of Figure 25 and t h e values of 8 taken from t h a t l i n e ,

a r e g r e s s i o n a n a l y s i s y i e l d s a Schmidt number exponent of 0.58

- not

much d i f f e r e n t than t h e value of 0.5 expected f o r mobile i n t e r f a c e s . A r e g r e s s i o n a n a l y s i s of a l l t h e h o r i z o n t a l d a t a f o r t h e glycerine-

water mixtures (except f o r t h e 5%

mixture) using t h e o r i g i n a l values of

a9 (Table 111, page 20) and forcing t h e Schmidt number t o have an exponent

of 1 / 2 r e s u l t s i n t h e equation, Sh =

0.34

S C " ~( dVS / D ) l "

(23)

ORNL-DWG

71-7997

100

0.5

0.2

0.1 1

102

Figure 18. Equivalent Power Dissipation Comparison of Gravity Dominated Results with Agitated Vessel Data.

1 o3

73 W

with a standard deviation i n Q r (Sh/Scl’ ”) of determination o f 0.86.

i s shown i n Figure

0.19 and an index of

The comparison of t h e d a t a with t h i s equation

19.

Since a Schmidt number exponent of 1/2 i s expected on t h e o r e t i c a l grounds and s i n c e t h e r e i s l i t t l e l o s s of p r e c i s i o n by using t h i s exponent, it i s recommended f o r design purposes t h a t t h e h o r i z o n t a l flow mass -t rans f e r eo e f f i c i e n t s as long a s V/V

be c a l c u l a t e d from Equation (23)

i s g r e a t e r than about 3.

Operation below V/V

not recommended because of severe s t r a t i f i c a t i o n .

3 is

Equation ( 2 3 ) can

a l s o be used t o c a l c u l a t e t h e v e r t i c a l flow c o e f f i c i e n t s , kv, as long a s Fi/Fg,

a s determined by Equation (20), i s g r e a t e r than 1.5.

Other-

wise, Equation ( 2 2 ) i s recommended f o r t h e v e r t i c a l flow c o e f f i c i e n t s with t h e asymptotic values,

ka 7

t o be c a l c u l a t e d from Equation (21).

A s evidenced by t h e observed high Schmidt number exponent, t h e s e

recommendations a r e f o r contamination f r e e systems only.

For a con-

taminated system with r i g i d i n t e r f a c i a l conditions, t h e Schmidt number exponent i s expected t o be l / 3 and t h e c o e f f i c i e n t multiplying t h e equation should also be d i f f e r e n t .

I n t h e absence of supporting experi-

mental data, a t e n t a t i v e c o r r e l a t i o n f o r r i g i d i n t e r f a c i a l conditions might be i n f e r r e d from Equation ( 2 3 ) t o be Sh = 0.25 Re0’”* S C ’ ’ ~ ( d

/D)’*”

The c o e f f i c i e n t , 0.25, w a s obtained by multiplying 0.34 [ t h e c o e f f i c i e n t of Equation ( 2 3 ) ] by t h e r a t i o of rigid-to-mobile c o e f f i c i e n t s of equations a p p l i c a b l e t o bubbles moving s t e a d i l y through a liquid.31

similar transformation of Equation ( 2 1 ) would be required t o obtain t h e W

r i g i d - i n t e r f a c e values of t h e v e r t i c a l flow asymptotes.

The above

Figure

19. Correlation of Horizontal Flow Data.

75 equation f o r r i g i d i n t e r f a c e s should be used w i t h caution a s it has not been v a l i d a t e d by experimental data. observed l i n e a r v a r i a t i o n w i t h ( d

t i o n from rigid-to-mobile

I n addition t h e experimentally

/D) may have been caused by a t r a n s i -

i n t e r f a c i a l condition.

For s t r i c t l y r i g i d

i n t e r f a c e s no such t r a n s i t i o n would be expected t o occur and t h e exponent on (dvs/D) might then be l e s s than 1.0.

CHAPTER V

THEORETICAL CONSIDERATIONS

Two d i f f e r e n t viewpoints were considered t o describe mass t r a n s f e r between small bubbles and l i q u i d s i n cocurrent t u r b u l e n t flow.

In the

f i r s t , a turbulence i n t e r a c t i o n approach, t h e bubbles were considered t o be subjected t o turbulence f o r c e s which impart random motions r e s u l t i n g i n "mean" r e l a t i v e v e l o c i t i e s between t h e bubbles and t h e f l u i d .

These

"mean" v e l o c i t i e s were then considered as "steady" ( a l b e i t multi-direct i o n a l ) and as d i c t a t i n g t h e mass-transfer behavior. I n t h e second, a surface renewal approach, t h e bubbles were viewed a s being associated with a s p h e r i c a l s h e l l of l i q u i d f o r an i n d e f i n i t e time during which mass exchange t a k e s place by t u r b u l e n t d i f f u s i o n . This i n d e f i n i t e time was assumed t o be r e l a t e d t o t h e bubble s i z e and t h e average r e l a t i v e v e l o c i t y between t h e bubble and t h e l i q u i d . Turbulence I n t e r a c t i o n Model A small bubble suspended i n a t u r b u l e n t f i e l d w i l l be subjected t o

random i n e r t i a l forces created by t h e t u r b u l e n t f l u c t u a t i o n s .

Under t h e

a c t i o n of a given force, i f s u f f i c i e n t l y p e r s i s t e n t , t h e bubble may achieve i t s terminal v e l o c i t y and move a t a steady pace through t h e l i q u i d before being r e d i r e c t e d by another f o r c e encounter within t h e random f i e l d .

If t h e "average" value representing t h e bubble r e l a t i v e

v e l o c i t y i n such a t u r b u l e n t f i e l d could be determined, then a convenient formulation would be t o use t h a t v e l o c i t y t o determine an average bubble

77 W

Reynolds number and s t a y within t h e confines of t h e well-established r e l a t i v e - f l o w FrBssling-type equations t o determine t h e mass-transfer coefficients. The movement of t h e bubbles through t h e l i q u i d w i l l be r e s i s t e d p r i m a r i l y by viscous s t r e s s e s .

The drag f o r c e on a sphere moving

s t e a d i l y through a l i q u i d i s o f t e n expressed i n terms of a drag c o e f f i c i e n t , Cd, by t h e equation, CdApv2b

Cdn pa

2gC

F =

8gc

i n which t h e drag c o e f f i c i e n t i s i t s e l f a f'unction of t h e bubble

( = vb dp/p).

Reynolds number, R%

In r e l a t i v e flows, however, t h e drag

coefficient-Reynolds number c o r r e l a t i o n depends on t h e p a r t i c u l a r Reynolds number range.

Frequently, two regimes of flow a r e i d e n t i f i e d

with t h e d i v i s i o n occurring a t Re

z 2.

Common c o r r e l a t i o n s f o r t h e

drag c o e f f i c i e n t s i n t h e s e two regimes a r e given below.

For Re

5 2,

For 2 < Reb

24/Reb and Fd = 3rrp2 Reb/gco

( 24-a )

200,

C d = 18.5/Re0*6 and Fd = 18.5rrp2 Re:'*/8gcp b

(24-b)

I n Chapter I V , an expression was developed f o r t h e i n e r t i a l f o r c e s experienced by a bubble i n a t u r b u l e n t f l u i d , 2

1(d/D)"l3 (Re)"/ Pg-L

It might be reasonable t o determine "mean" bubble v e l o c i t i e s from a balance between t h e i n e r t i a l f o r c e s and t h e drag forces for l a t e r subW

s t i t u t i o n i n t o t h e Friissling equations.

If it i s p o s t u l a t e d t h a t t h e

78 above two r e l a t i v e flow regimes a l s o e x i s t f o r bubbles i n a t u r b u l e n t

f i e l d , then two d i f f e r e n t s e t s of equations describing t h e mass t r a n s f e r

w i l l result.

Since t h e i n e r t i a l f o r c e s depend on t h e bubble s i z e , a

d i s p e r s i o n of bubbles with a d i s t r i b u t i o n o f s i z e s may have bubbles i n e i t h e r or both regimes simultaneously and t h e mass-transfer behavior may be described by e i t h e r s e t of equations or t a k e on c h a r a c t e r i s t i c s of a combination of t h e two.

The mass-transfer equations r e s u l t i n g for

t h e two separate regimes a r e discussed below. Regime-1: Reb

5 2

If t h e bubble motion were predominantly governed by t h e regime,

Re r; 2, t h e drag forces would be given by Equation (24-a). b

A balance

between t h e i n e r t i a l and drag forces, F.1 = Fd’ would then give f o r t h e bubble Reynolds number Reb

- (d/D)”/

Re’’’

(26)

By t h i s formulation, t h e bubble r e l a t i v e flow Reynolds number depends only on t h e r a t i o , d/D,

and on t h e pipe Reynolds number which,

f o r a given bubble s i z e , e s t a b l i s h t h e turbulence l e v e l .

The Sherwood

number f o r mass t r a n s f e r can t h e r e f o r e be determined a s a f u n c t i o n of t h e s e v a r i a b l e s by s u b s t i t u t i o n of Equation (26) i n t o mass-transfer equations t h a t have been e s t a b l i s h e d as applicable t o a sphere moving through a l i q u i d .

These a r e t h e Frgssling-type equations which, f o r

l a r g e Schmidt numbers, u s u a l l y take t h e forms S%

- Reb’/

Scl’ a

79 W

f o r mobile and r i g i d i n t e r f a c e s , respectively. Sh

(D/d) Sh,,

Making t h e conversion,

and s u b s t i t u t i n g Equation (26) gives for t h e mobile and

r i g i d i n t e r f a c e pipe Sherwood numbers applicable t o cocurrent t u r b u l e n t flow, Sh

Scl’

“

Re”””

(27)

(d/D)’/

and Sh

Re0””

(d/D)’/

( 28)

r e spect i v e ly. Consequently, i n t h i s regime, t h e pipe Reynolds number exponent i s

0.92.

For comparison, t h e experimentally determined value for t h e

water-glycerine mixtures i n t h i s i n v e s t i g a t i o n was 0.94. bubble diameter dependence, (d/D)l/

mentally determined l i n e a r v a r i a t i o n .

The t h e o r e t i c a l

however i s l e s s than t h e experiCalderbank and Moo-Young’

point

out t h a t t h e l i n e a r v a r i a t i o n they observed f o r bubbles i n t h i s s i z e range probably r e s u l t e d from a t r a n s i t i o n from r i g i d t o mobile i n t e r f a c i a l conditions because small bubbles tend t o u n i v e r s a l l y behave a s r i g i d spheres while l a r g e r bubbles r e q u i r e t h e presence of s u f f i c i e n t s u r f a c e a c t i v e i n g r e d i e n t s t o immobilize t h e i r surface.

If such a t r a n s i t i o n i s t h e reason f o r t h e l i n e a r v a r i a t i o n i n t h i s

i n s t a n c e a l s o , then t h e e f f e c t of conduit diameter w i l l be d i f f e r e n t from t h a t implied i n Equation ( 2 3 ) which d i d not include a c t u a l v a r i a t i o n s i n conduit diameter.

Consequently, a n t i c i p a t e d f u t u r e experiments

with v a r i a t i o n s i n t h e conduit diameter should h e l p c l a r i f y t h e influence of bubble mean diameter.

I n addition, exploratory experiments i n t h i s

study i n d i c a t e d t h a t t h e l i n e a r v a r i a t i o n d i d not continue up t o l a r g e r bubble s i z e s and may, therefore, be l i m i t e d t o t h e r e l a t i v e l y narrow mean

80 diameter range of approximately 0.01 t o 0.05 inches.

A t l a r g e r diameters,

t h e dependence tended t o l e s s e n u n t i l above mean diameters of about 0.08 inches where t h e Sherwood number appeared t o decrease with i n c r e a s i n g bubble diameter.

Since t h e bubble generator was not g e n e r a l l y capable

of producing l a r g e r bubbles, f u r t h e r i n v e s t i g a t i o n of t h e bubble s i z e influence was not p o s s i b l e i n t h i s experiment. Regime-2: Reb > 2 I f t h e bubble motions were predominantly i n t h e regime, Reb > 2, t h e

drag f o r c e s would be given by Equation (24-b).

The balance, F.1 = Fd'

would t h e n give

The r e l a t i v e - f l o w bubble Reynolds number i n t h i s regime s t i l l depends on t h e v a r i a b l e s t h a t e s t a b l i s h t h e turbulence l e v e l but t h a t dependence i s d i f f e r e n t from t h a t of Regime 1. When s u b s t i t u t e d i n t o t h e F r g s s l i n g equations for mobile and r i g i d i n t e r f a c e s , t h e r e s u l t s a r e Sh

(d/D)-g'd

Scl'

4. 2

and Shr e s p e ct i v e l y

Re"*66

(d/D)-o*d4.2

(30)

For t h i s regime t h e Reynolds number exponent i s 0.66.

Consequently,

i f bubbles i n cocurrent t u r b u l e n t flow experience d i f f e r e n t flow regimes

s i m i l a r t o bubbles i n r e l a t i v e flow, a t r a n s i t i o n would be expected a t higher pipe Reynolds numbers i n which t h e Reynolds number exponent would tend t o become smaller.

I n t h e present experiments, t h e data f o r water

( p l u s -200 ppm N-butyl alcohol) with no glycerine added was obtained a t

a1 t h e highest range of Reynolds numbers covered.

The experimentally

measured Reynolds number exponent f o r t h e water runs was lower than f o r t h e glycerine-water mixtures and compared favorably with t h e above results.

I n addition, Equation ( 3 0 ) compares q u i t e well w i t h d a t a f o r

p a r t i c l e s i n a g i t a t e d v e s s e l s [for example s e e Equation (3)].

It i s f e l t t h a t t h e p o s s i b l e existence of d i f f e r e n t flow regimes even i n cocurrent t u r b u l e n t flows i s an important concept t h a t , i f f'urther developed, could help explain some of t h e apparent discrepancies i n t h e l i t e r a t u r e data.

For example, t h i s may explain t h e d i f f e r e n t

slopes observed i n t h i s study and may be t h e reason f o r observed d i f f e r ences between mass t r a n s f e r i n a g i t a t e d v e s s e l s and i n conduits.

It i s

more l i k e l y , however, t h a t t h e l a t t e r d i f f e r e n c e i s due t o g r e a t e r g r a v i t a t i o n a l influence i n a g i t a t e d vessels. Surface Renewal Model I n t h i s a n a l y s i s each bubble i s considered t o be surrounded by, and exchanging mass with, a s p h e r i c a l s h e l l of t u r b u l e n t l i q u i d i n which t h e turbulence i s isotropic. A mass balance (Appendix F) i n a s p h e r i c a l d i f f e r e n t i a l element of

f l u i d r e s u l t s i n t h e equation

Making Reynolds assumptions,

ur and time averaging gives

=u'

I n t u r b u l e n t s c a l a r t r a n s f e r , t h e "Reynolds" term p 'C

', i s

often

assumed t o be expressible with an eddy diff'usivity, E, defined by

However, it i s more convenient h e r e t o use a r e c e n t eddy v i s c o s i t y d e f i n i t i o n by P h i l l i p s , 6 8 f o r which an analogous d e f i n i t i o n f o r an eddy d i f f u s i v i t y i n s p h e r i c a l coordinates would be

-d ( r 2 u cI ) / dr

= p e d- - ( r 2 dsC)

(33)

Using t h i s d e f i n i t i o n , Equation ( 3 2 ) i s expressed more simply as

The view i s now t o be taken t h a t , on t h e average, a bubble remains associated with a s p h e r i c a l s h e l l of l i q u i d f o r some i n d e f i n i t e time a f t e r which i t s s u r f a c e i s completely "renewed"

- that

i s , associated

with an e n t i r e l y d i f f e r e n t s p h e r i c a l s h e l l of l i q u i d t h a t has an i n i t i a l uniform concentration c h a r a c t e r i s t i c of t h e bulk f l u i d .

It i s f e l t t h a t

t h e times of a s s o c i a t i o n between t h e bubble and a given region of l i q u i d should be r e l a t e d t o t h e magnitude of t h e t u r b u l e n t i n e r t i a l f o r c e s o r a l t e r n a t i v e l y t o t h e mean r e l a t i v e v e l o c i t y between t h e bubble and t h e l i q u i d as e s t a b l i s h e d by t h e balance of t h e i n e r t i a l and t h e viscous r e s i s t i n g f o r c es

Therefore a nondimensional time f o r comparison purposes i s proposed t o be

tvb d

83 Using t h i s d e f i n i t i o n along with t h e following a d d i t i o n a l d e f i n i t i o n s of dimensionless q u a n t i t i e s

Equation (34) can b e expressed i n nondimensional form a s

Assuming t h e bubble motion i s predominantly i n Stokesâ&#x20AC;&#x2122; regime, Equation

(26) can be used t o estimate R% and s u b s t i t u t e d i n t o t h e above equation t o give

where C , i s a p r o p o r t i o n a l i t y constant of unknown magnitude but assumed t o be of t h e order -lo-â&#x20AC;?.

A s i m i l a r equation can b e developed f o r

Regime-2 of t h e previous model by using Equation ( 2 9 ) for R%.

Logical

boundary conditions f o r Equation (35) would be

1. C, 2.

(0,

r,) = 1,

(t >

1 / 2 ) = 0, and

The t h i r d boundary condition above a r i s e s from equating t h e volume f r a c t i o n w i t h t h e r a t i o of bubble volume t o equivalent sphere volume.

84 A s o l u t i o n of Equation

t,.

(35) would give

If a r a d i a l average, C*,

as a f'unction of r, and

i s defined a s

c* (t,) =

then t h e Sherwood number as a function o f time can be expressed a s

If a bubble i s assumed t o remain associated with a f l u i d element f o r

then t h e average Sherwood number f o r t h a t

some unspecified time, T,, period i s

Sh =

rn I*

Sh (t,) dt, T*

The above a n a l y s i s i s s i m i l a r t o normal surface renewal models i n t h a t t h e dimensionless time period T, i s analogous t o a surface age. There i s no r e a l b a s i s f o r being able t o r e l a t e T,

t o t h e flow hydrody-

namics o r t h e surface conditions; however, it could be t r e a t e d as a parameter and t h e mass-transfer c o e f f i c i e n t s determined a s a function of t h i s parameter.

"Surface age" d i s t r i b u t i o n s could then be e s t a b l i s h e d

from t h e experimental d a t a or s p e c i f i e d a r b i t r a r i l y j u s t a s t h e y have been i n o t h e r surface renewal models.

For example, one common assump-

t i o n has been t h a t t h e surface i s "renewed" each time t h e bubble t r a v e l s ( r e l a t i v e t o t h e f l u i d ) a distance equal t o i t s diameter.

With t h e

formulation used here, t h i s assumption would be p a r t i c u l a r l y convenient because t h e n T,

= 1.

85 Equation (35) along with i t s boundary conditions i s considered a s a surface renewal model.

For a solution, a function,

must f i r s t be e s t a b l i s h e d t o describe t h e v a r i a t i o n i n eddy d i f f u s i v i t y . I n a r r i v i n g a t h i s eddy v i s c o s i t y d e f i n i t i o n , P h i l l i p s 6 8 used a Fourier decomposition of t h e t u r b u l e n t f i e l d and, by an elegant a n a l y s i s , determined t h e c o n t r i b u t i o n s t o t h e l o c a l eddy v i s c o s i t y due t o each component "wave" making up t h e f i e l d . Through a p a r a l l e l a n a l y s i s f o r mass t r a n s f e r , it i s i n f e r r e d h e r e t h a t t h e i n d i v i d u a l component contributions t o t h e eddy d i f f u s i v i t y a r e p r o p o r t i o n a l t o t h e energy of t h e t r a n s v e r s e v e l o c i t y f l u c t u a t i o n s and i n v e r s e l y p r o p o r t i o n a l t o t h e i r wave number,

Defining f ( n ) d n a s t h e f r a c t i o n of eddies t h a t have wave numbers i n t h e range n

1 / 2 dn, and summing t h e c o n t r i b u t i o n s over a l l wave numbers

gives

If Kolmolgoroff's energy spectrum i s used, t h e d i s t r i b u t i o n function

defined above can be assumed t o be i n v e r s e l y p r o p o r t i o n a l t o t h e wave nuuiber, f(n>

- Jn

(?/n2)

l/n

and Equation (38) becomes 'e

To a s s e s s t h e e f f e c t of t h e i n t e r f a c e , use was made of

(39)

Lament's''

a n a l y s i s i n which he i d e a l i z e d each component a s a s i n u s o i d a l viscous

86 "eddy c e l l " i n which t h e v e l o c i t i e s a r e damped by viscous s t r e s s e s a s

H i s a n a l y s i s gave f o r a s p a c i a l average

an i n t e r f a c e i s approached. (parallel t o the interface),

- d/2

and t ( y ) i s a damping

f a c t o r depending on t h e i n t e r f a c i a l condition.

Lamont's s o l u t i o n of

where y i s a coordinate defined a s y

t h e viscous "eddy-cell" equation gave f o r a r i g i d i n t e r f a c e ,

[0.294 ny s i n h ny + 0.388 s i n h ny 4.388 ny cosh ny]

and f o r a mobile i n t e r f a c e ,

EO. 366

s i n h ny -0.089 ny cosh ny]

I n addition, it i s assumed here t h a t only t h e range of eddy s i z e s smaller than, o r equal t o , t h e bubble diameter i n t e r a c t with a bubble t o produce eddy t r a n s f e r t o t h e bubble i t s e l f and t h a t each of t h e s e eddies i s e f f e c t i v e l y damped only i f it i s w i t h i n a d i s t a n c e from t h e i n t e r f a c e equal t o t h e wave size.

The eddy s i z e s assumed present range from a

minimum given by t h e Kolmolgoroff microscale f o r p i p e flow,

t o an a r b i t r a r y maximum of one-half t h e pipe diameter, 'max

D/2

Consequently, using Equation (39), t h e r a t i o of eddy d i f f u s i v i t y e f f e c t i v e t o t h e bubble a t a p o s i t i o n y t o t h e eddy d i f f ' u s i v i t y e x i s t i n g away from t h e i n t e r f a c e , p /po, e 1.

For rr/y > v/d,

i s c a l c u l a t e d from t h e following r e l a t i o n s :

-'e - PO

A numerical i n t e g r a t i o n of Equation (40) with Amax

= d i s shown on

Figure 20 f o r both mobile and r i g i d damping. The a c t u a l r e l a t i v e eddy d i f f b s i v i t y v a r i a t i o n c a l c u l a t e d from Equation (40) w i l l not approach u n i t y i n midstream a s i n Figure 20 because t h e i n t e g r a t i o n of t h e numerator i s t o include only eddies up t o t h e s i z e of t h e bubble diameter whereas t h e denominator i s t o be i n t e g r a t e d over a l l wave s i z e s i n t h e f i e l d . Comparing t h e mobile and r i g i d i n t e r f a c e curves on Figure 20 i n d i c a t e s t h a t t h e two conditions would r e s u l t i n very l i t t l e d i f f e r e n c e i n mass-transfer behavior f o r an e s s e n t i a l l y passive bubble being acted upon simultaneously by many eddies intuitively.

-a

r e s u l t not t o o d i s p l e a s i n g

A s i g n i f i c a n t d i f f e r e n c e i n behavior then, by t h i s

formulation, must come about by assigning a longer renewal period, T,, t o r i g i d i n t e r f a c e s than t o mobile i n t e r f a c e s . The v a r i a t i o n of I.,/& required f o r a s o l u t i o n t o Equation (35) can be obtained from t h e product

i f t h e values f o r eddy d i f f ' u s i v i t y i n midstream, 1-1

, are

knm.

88 0 RNL-DW G 74- 8013

PLOT OF

NHERE

-AND

CR =

&,,=

b . 2 9 4 NY sinh NY t 0.388 sinh NY - 0.388 NY cosh NY] FOR A RIGID INTERFACE [0.366 sinh NY - 0.089 NY cosh NY]

FOR A MOBILE INTERFACE 1.o 0

0.9

< 0.8 0.7 0.6

0.5 0.4 0.3

0.2 0.1

0 0

0.1 0.2 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Y/Xrnax

Figure 20.

Dimensionless Variation of Eddy D i f f u s i v i t y with Distance from an Interface. E f f e c t of Surface Condition.

89 W

For t h e standard d e f i n i t i o n of eddy d i f f u s i v i t y , Groenhof5' gives a c o r r e l a t i o n applicable t o t h e midsection of a pipe, E = 0.04+/,t Letting

T~ =

f p v2/8gC and f = O.316/Re1I4

from Equation

(42) i s

for smooth tubing, then E

given by

E/v = 0.04 h/fTg Re = 0.04

8d-

Re7/

(43)

P h i l l i p ' s d e f i n i t i o n of eddy v i s c o s i t y reduces t o t h e standard d e f i n i t i o n i n t h e midsection of a pipe.

Consequently, it i s acceptable

t o convert Equation (43) t o Po/&!! = 0.04

which along with Equation

m Sc

7 / 8

(44)

(41) and Equations (40) f u l l y determine a

f o r use i n s o l v i n g Equation (35).

It i s r e a l i z e d t h a t P h i l l i p ' s a n a l y s i s f o r eddy v i s c o s i t y i s not s t r i c t l y a p p l i c a b l e near an i n t e r f a c e nor i s t h e "eddy-cell" i d e a l i z a t i o n a r e a l i s t i c p i c t u r e of t h e turbulence.

Nevertheless, t h e v a r i a t i o n i n

eddy d i f f u s i v i t y based on t h e s e concepts was determined through Equations

(43) and (40).

It i s f e l t t h a t t h e behavior of a pseudo-turbulence such

a s t h i s may be s i m i l a r t o a r e a l t u r b u l e n t f i e l d i n t h a t t h e e s s e n t i a l f e a t u r e s a r e r e t a i n e d and t h e t r e n d s p r e d i c t e d i n t h i s manner may be useful.

For example, f o r t h e condition of t u r b u l e n t t r a n s f e r t o a con-

d u i t w a l l i t s e l f t h e r e have been measurements o f t h e standard eddy diffusivity distributions.

Therefore, a comparison was made i n Figure

2 1 of eddy d i f f u s i v i t i e s c a l c u l a t e d i n t h e above manner with S l e i c h e r ' s

ORNL-DWG

71-8014

320

300 280

p 0 = 0 . 0 2 8 &Re

260

AND A m a x ASSUMED = 0.5 ro = PIPE RADIUS

240 2 20

200 180 a

' a,

160

140

120 I00

80 60

O /

20 0 0

0.2

0.4

0.6

0.8

1 .o

Y/r,

Figure 21.

Variation of Eddy D i f f u s i v i t y with Distance from an I n t e r f a c e . Comparison of Calculated Values with Data of Sleicher.

91 data.60

For t h i s a p p l i c a t i o n of t r a n s f e r t o a conduit, t h e value of d

i n Equation (boa) ( t h e maximum eddy s i z e i n t h i s case) was a r b i t r a r i l y s e t equal t o 1 / 2 of t h e pipe radius, ro, and t h e c o e f f i c i e n t i n Equation

(43) was adjusted s l i g h t l y t o r e q u i r e pe/v t o coincide e x a c t l y with S l e i c h e r ' s value i n t h e pipe midsection a t Re = 14,500.

Considering t h e

d i f f e r e n c e i n t h e eddy d i f f u s i v i t y d e f i n i t i o n s , t h e comparison i s favora b l e and it appears t h a t use of a pseudo-turbulence i d e a l i z a t i o n such as t h i s may provide a unique means of p r e d i c t i n g eddy v i s c o s i t y and eddy diffusivity variations.

Since t h e determination of eddy d i f f u s i v i t i e s

and t h e i r v a r i a t i o n was not t h e primary concern of t h i s t h e s i s , f u r t h e r

development of t h e s e concepts was not considered. Equations (35), (36), ( b o ) , and

(44), which

represent t h e p r e s e n t

s u r f a c e renewal model were programmed on a d i g i t a l computer and numeric a l s o l u t i o n s obtained using T,

as a parameter.

Time did not permit a

complete evaluation of t h i s computer program and t h e r e s u l t s can only be presented here as t e n t a t i v e .

Figure 22 i l l u s t r a t e s t h e values of

t h e exponents obtained f o r an equation of t h e form S%

as a f u n c t i o n of T,.

- Re"

Scb (d/D)'

The value of T,

f o r which t h e Schmidt number

exponent was 1/3 (corresponding t o r i g i d i n t e r f a c e s ) was approximately

2.7.

A t t h i s value of T,,

t h e s o l u t i o n f o r t h e time-averaged p i p e

Sherwood number was e s s e n t i a l l y independent of t h e bubble diameter and v a r i e d according t o S h - Reoo8' scl' The computer results as

(45)

approached zero appeared t o approach t h e

c l a s s i c a l p e n e t r a t i o n s o l u t i o n of Equation (35) obtained f o r IJ-

= 0,

ORNL-DWG

7 1 - 1 0753

0.5

0.2

0.1 1

100

Figure 22. Numerical Results of the Surface Renewal Model. Plots of a, b, and c (Exponents on Re, Sc, and d/D, Respectively) as Functions of the Dimensionless Period, T , .

93 Sh

JSc

(d/D)8/

Re”’

(d/D)

or Sh

Scl/

Re0”

which i s i d e n t i c a l t o Equation (27).

( d/D)

Consequently, i f t h e s u r f a c e

renewal period, T*, i s i n t e r p r e t e d a s being a measure of t h e r i g i d i t y of t h e i n t e r f a c e , T,

T, -+ -2.7

-t

0 being c h a r a c t e r i s t i c of mobile i n t e r f a c e s and

( i n t h i s case) being c h a r a c t e r i s t i c of r i g i d i n t e r f a c e s , t h e n

t h i s s u r f a c e renewal model may be useful. Neither t h i s model nor t h e preceding turbulence i n t e r a c t i o n model s a t i s f a c t o r i l y p r e d i c t t h e observed v a r i a t i o n of pipe Sherwood number w i t h bubble diameter f o r t h i s range of bubble s i z e s .

I n d i r e c t support

i s t h e r e f o r e provided f o r t h e supposition t h a t t h e observed l i n e a r

v a r i a t i o n may b e t h e r e s u l t of a t r a n s i t i o n from r i g i d t o mobile behavior.

CHAPTER V I SUMMARY AND CONCLUSIONS

Transient response experiments were performed using f i v e d i f f e r e n t mixtures of g l y c e r i n e and water.

Liquid-phase-controlled mass-transfer

c o e f f i c i e n t s were determined f o r t r a n s f e r of dissolved oxygen i n t o small helium bubbles i n cocurrent t u r b u l e n t gas-liquid flow.

These c o e f f i -

c i e n t s were e s t a b l i s h e d a s functions of Reynolds number, Schmidt number, bubble mean diameter, and g r a v i t a t i o n a l o r i e n t a t i o n of t h e flow. An a n a l y t i c a l expression was obtained f o r t h e r e l a t i v e importance of t u r b u l e n t i n e r t i a l f o r c e s compared with g r a v i t a t i o n a l f o r c e s , Fi/Fg.

For conditions i n which t h i s r a t i o was g r e a t e r than -1.5,

the variation

i n t h e observed mass-transfer c o e f f i c i e n t s with Reynolds numbers was l i n e a r on log-log coordinates with i d e n t i c a l behavior f o r h o r i z o n t a l Below F . / F = 1.5, t h e h o r i z o n t a l c o e f f i c i e n t v a r i 1 g a t i o n continued t o b e " l i n e a r " u n t i l t h e r a t i o of l i q u i d a x i a l v e l o c i t y and v e r t i c a l flows.

t o bubble terminal v e l o c i t y , V/Vb,

decreased t o about 3, where severe

s t r a t i f i c a t i o n made operation impractical.

The v e r t i c a l flow c o e f f i -

c i e n t s underwent a t r a n s i t i o n from t h e " l i n e a r " v a r i a t i o n and approached constant asymptotes c h a r a c t e r i s t i c of bubbles r i s i n g through a quiescent The v a r i a b l e r a t i o of F . / F proved t o be a u s e f u l l i n e a r s c a l i n g 1 g f a c t o r f o r describing t h e v e r t i c a l flow c o e f f i c i e n t s i n t h i s t r a n s i t i o n

liquid.

region f o r which Equation ( 2 2 ) i s t h e recommended c o r r e l a t i o n . The Schmidt number exponent f o r t h e s t r a i g h t - l i n e p o r t i o n s of t h e d a t a was observed t o b e g r e a t e r than 1/2 based on p h y s i c a l p r o p e r t y data f o r 8 which may be suspect.

F i t t i n g t h e d a t a with a Schmidt number

95 W

exponent of 1/2 r e s u l t e d i n only s l i g h t l y l e s s p r e c i s i o n than f o r t h e case i n which t h e a c t u a l regression exponent was used, and a d e f i n i t i v e choice could not be made between t h e two.

Based on theo-

r e t i c a l expectations, a Schmidt number exponent of 1 / 2 would seem t o be appropriate, and consequently, t h e recommended c o r r e l a t i o n i s Equation ( 23). The v a r i a t i o n i n mass-transfer c o e f f i c i e n t with bubble mean diameter over t h e range covered was observed t o be l i n e a r i n agreement with t h e f i n d i n g s of Calderbank and Moo-Young''

f o r a g i t a t e d vessels.

Some pre-

l i m i n a r y runs made with bubble mean diameters o u t s i d e t h e range of t h i s r e p o r t i n d i c a t e d t h a t t h e l i n e a r v a r i a t i o n does not continue b u t t h a t t h e c o e f f i c i e n t s l e v e l o f f a t both smaller and l a r g e r diameters. Furthermore t h e c o e f f i c i e n t s t e n t a t i v e l y appear t o decrease slowly w i t h i n c r e a s i n g mean diameters above about 0.08 inches. Consistent w i t h findings of o t h e r i n v e s t i g a t i o n s , t h e Reynolds number exponent was s i g n i f i c a n t l y g r e a t e r than expected based on a g i t a t e d v e s s e l d a t a comgared on an equivalent power d i s s i p a t i o n b a s i s .

One

explanation i s t h a t there may e x i s t g r e a t e r g r a v i t a t i o n a l influence i n a g i t a t e d vessels.

Another i s t h e p o s t u l a t e d existence of d i f f e r e n t

bubble r e l a t i v e flow regimes. A seemingly anomalous behavior was observed f o r t h e Reynolds number

v a r i a t i o n i n t h a t t h e d a t a f o r water ( p l u s about 200 ppm N-butyl alcohol) exhibited s i g n i f i c a n t l y smaller Reynolds number exponents and a c o r r e s pondingly smaller exponent f o r t h e r a t i o , (d/D), than t h a t f o r t h e glycerine-water mixtures.

There may have been a d i f f e r e n c e i n t h e

i n t e r f a c i a l conditions ( t h e a d d i t i o n of t h e s u r f a c t a n t c r e a t e s a " r i g i d " W

96 i n t e r f a c e while t h e glycerine-water mixtures apparently g e n e r a l l y had "mobile" i n t e r f a c i a l behavior).

However, under steady r e l a t i v e flow

conditions t h i s would r e s u l t i n no d i f f e r e n c e i n t h e Reynolds number exponent.

Consequently, it was p o s t u l a t e d t h a t t h i s d i f f e r e n c e r e s u l t e d

from t h e p o s s i b l e e x i s t e n c e of d i f f e r e n t bubble r e l a t i v e flow regimes. I n support of t h e above contention, a two-regime "turbulence i n t e r a c t i o n " model was formulated by balancing t u r b u l e n t i n e r t i a l f o r c e s w i t h drag forces t h a t depend on t h e bubble r e l a t i v e flow Reynolds num-

ber.

The r e s u l t i n g mean bubble v e l o c i t i e s were s u b s t i t u t e d i n t o

"Fr8ssling" equations t o determine t h e mass-transfer behavior.

The

r e s u l t i n g Reynolds number exponent f o r one regime (Reb < 2) agreed v e r y w e l l w i t h t h e experimental value f o r t h e glycerine-water mixtures and t h a t f o r t h e o t h e r regime (Re

> 2) compared favorably with t h e water

d a t a and with a g i t a t e d v e s s e l d a t a on an equivalent power d i s s i p a t i o n basis. The dependence of Sherwood number on t h e bubble-to-conduit diameter r a t i o , d/D, p r e d i c t e d by t h e i n t e r a c t i o n model did not agree with t h e observed l i n e a r v a r i a t i o n .

Calderbank and Moo-Youn$

pointed out t h a t

t h e l i n e a r v a r i a t i o n t h e y observed i n a g i t a t e d v e s s e l s f o r bubbles of t h i s s i z e range probably r e s u l t e d from a t r a n s i t i o n from ''small'' bubble

t o "large" bubble behavior.

Such a t r a n s i t i o n could a l s o explain t h e

present observations, however, t h e r e was no s a t i s f a c t o r y means f o r v a l i dating this.

For comparison, a second a n a l y t i c a l model was developed based on surface renewal concepts which could a l s o include d i f f e r e n t flow regimes. This model incorporated an eddy d i f f u s i v i t y t h a t v a r i e d with Reynolds

97 number, Schmidt number, bubble diameter, i n t e r f a c i a l condition, and p o s i t i o n away from t h e i n t e r f a c e .

The v a r i a t i o n of eddy diff'usivity was

e s t a b l i s h e d by using a pseudo-turbulent model i n which t h e turbulence was simulated by superposed viscous eddy-cells damped by t h e bubble i n t e r f a c e i n a manner determined by Lamont." The s u r f a c e renewal model assumed t h a t t h e "renewal" period f o r t h e bubbles was r e l a t e d t o t h e bubble "mean" v e l o c i t y r e s u l t i n g from a balance between t u r b u l e n t i n e r t i a l f o r c e s and viscous r e s i s t i n g f o r c e s , t h u s allowing t h e c a s t i n g of t h e equations i n t o nondimensional form with t h e pipe Reynolds n u d e r , t h e Schmidt number, and d/D as parameters.

closed s o l u t i o n of t h e equations was not obtained but a t e n t a t i v e numeri c a l s o l u t i o n employing a d i g i t a l computer i n d i c a t e d t h a t , i n t h e limit of small dimensionless renewal period, T,

- interpreted

as representing

mobile i n t e r f a c i a l behavior, t h e c l a s s i c a l p e n e t r a t i o n s o l u t i o n of t h i s p a r t i c u l a r form of t h e d i f f i s i o n equation r e s u l t e d . A s T,

approached a value of approximately 2.7 ( i n t h i s case), t h e

computer s o l u t i o n was independent of (d/D) and r e s u l t e d i n a Schmidt number exponent of -l/3.

Therefore, t h i s value of T , w a s i n t e r p r e t e d

a s r e p r e s e n t i n g r i g i d i n t e r f a c i a l behavior. E x p l i c i t r e s u l t s based'on t h e models described above along with a l i s t i n g o f t h e more s i g n i f i c a n t observations of t h i s study a r e given below: 1. Bubbles generated i n a t u r b u l e n t f i e l d a r e w e l l characterized

by t h e d i s t r i b u t i o n function f (6) =

4 ($/IT)''

6" Exp (-6")

where

[4 ,& N / 6 @ J V 3

98 2.

The average volume f r a c t i o n s occupied by gas i n bubbly flow

a r e approximated by Hughmark's c o r r e l a t i o n 6 4 only a t higher flows.

h o r i z o n t a l flow, when t h e r a t i o of a x i a l v e l o c i t y of t h e l i q u i d t o t h e bubble terminal v e l o c i t y i s below -25,

Hughmark's c o r r e l a t i o n p r e d i c t s

volume f r a c t i o n s lower than those observed.

I n v e r t i c a l flow, while

t h e volume f r a c t i o n s a r e higher i n downcomer l e g s than i n r i s e r legs, t h e y can be e s t a b l i s h e d by using Hughmark's c o r r e l a t i o n f o r t h e mean and accounting f o r t h e buoyant r e l a t i v e v e l o c i t y of t h e bubbles i n each leg.

A t low t u r b u l e n t flows s t r a t i f i c a t i o n of t h e bubbles i n h o r i -

z o n t a l conduits prevented operation f o r r a t i o s of a x i a l v e l o c i t y t o bubble t e r m i n a l v e l o c i t y below -3.

Even a t Reynolds numbers w e l l i n t o t h e t u r b u l e n t regime, h o r i -

z o n t a l and v e r t i c a l flow mass-transfer c o e f f i c i e n t s d i f f e r .

The Reynolds

numbers above which t h e y become equivalent a r e marked by t h e dominance of t u r b u l e n t i n e r t i a l f o r c e s over g r a v i t a t i o n a l forces.

A s Reynolds numbers a r e reduced, v e r t i c a l flow mass-transfer

c o e f f i c i e n t s approach asymptotes c h a r a c t e r i s t i c of bubbles r i s i n g through a quiescent l i q u i d .

The r a t i o of t u r b u l e n t i n e r t i a l f o r c e s t o g r a v i t a -

t i o n a l f o r c e s serves a s a usef'ul l i n e a r s c a l i n g f a c t o r f o r estimating t h e mass-transfer c o e f f i c i e n t s a t t h e s e lower Reynolds numbers.

The e f f e c t of Reynolds number on Sherwood number f o r flow i n

conduits i s not as would b e expected based on comparison with a g i t a t e d v e s s e l d a t a on an equivalent power d i s s i p a t i o n b a s i s .

For example, t h e

observed turbulence-dominated d a t a are c o r r e l a t e d by Sh/Scl/

0.34 Reoog4 (d/D)l * O

(23)

99 W

whereas one o b t a i n s from t h e a g i t a t e d v e s s e l data of Calderbank and

Moo-Youn$

f o r small bubbles Sh/Scl/

= 0.082 Reom6'

(2)

and of Sherwood and Brian17 f o r p a r t i c u l a t e s

Sh/Scl'

(d/D)-"'l"

(3)

I n t h i s t h e s i s t h e two-regime turbulence i n t e r a c t i o n model and t h e s u r f a c e renewal model e x h i b i t i d e n t i c a l r e s u l t s f o r mobile i n t e r f a c e s i n t h e "Stokes" regime (Reb Sh

5 2),

Reo *'

Scl/

( d/D)l/

which compares w e l l with t h e observations represented by Equation (23). I n t h e second regime (Re > 2 ) , t h e turbulence i n t e r a c t i o n model b for rigid interfaces results i n S h - sc1/3

Re0*66

(d/D)-O*2/4*2

and t h e r i g i d i n t e r f a c e i n t e r p r e t a t i o n of t h e surface renewal model gives Sh

Scl/

as compared, for example, with Equations ( 2 ) and (3).

The observed l i n e a r v a r i a t i o n of Sherwood number with bubble

diameter was not predicted t h e o r e t i c a l l y . Calderbank and Moo-Young,"

Consequently, following

it i s conjectured t h a t t h i s v a r i a t i o n r e -

s u l t s from a t r a n s i t i o n from r i g i d ( s m a l l bubbles) t o mobile ( l a r g e bubbles) i n t e r f a c i a l behavior f o r t h i s s i z e range.

Data of t h i s study t h a t were obviously g r a v i t a t i o n a l l y i n f l u -

enced compare favorably with d a t a for p a r t i c u l a t e s i n a g i t a t e d v e s s e l s , giving r i s e t o t h e speculation t h a t g r a v i t a t i o n a l forces may b e more i n f l u e n t i a l i n a g i t a t e d v e s s e l s where t h e r e may e x i s t a g r e a t e r degree of anisotropy compared with flow i n conduits.

CWTER VI1 RECOMMENDATIONS FOR FURTHER STUDY

Experimental Time did not permit a complete i n v e s t i g a t i o n of t h e e f f e c t s of a l l t h e independent v a r i a b l e s .

Consequently, p r o j e c t i o n s of t h i s study i n t o

t h e f u t u r e include experiments involving v a r i a t i o n s of t h e conduit diame t e r and t h e i n t e r f a c i a l condition.

It i s a n t i c i p a t e d t h a t t h e s e s t u d i e s

w i l l help c l a r i f ' y t h e r o l e of d/D, i n p a r t i c u l a r with regard t o t h e

observed l i n e a r v a r i a t i o n of mass-transfer c o e f f i c i e n t with bubble diameter.

These p r o j e c t e d s t u d i e s will a l s o attempt t o extend t h e ranges of

v a r i a b l e s covered through improvements i n t h e bubble generating and s e p a r a t i n g equipment.

It i s hoped t h a t t h e s e improvements w i l l reduce

t h e magnitude of t h e "end-effect" and thereby provide g r e a t e r p r e c i s i o n t o t h e data.

P a r e n t h e t i c a l l y , t h e high r a t e s of mass t r a n s f e r observed

i n t h e bubble s e p a r a t o r may q u a l i f y it f o r f i r t h e r i n v e s t i g a t i o n as a p o s s i b l e e f f i c i e n t i n - l i n e gas-liquid contactor. For p r a c t i c a l purposes it i s recommended t h a t mass-transfer r a t e s

a l s o be measured i n regions of flow d i s c o n t i n u i t i e s such a s elbows, t e e s , valves, v e n t u r i s , and abrupt pipe s i z e changes.

An o b j e c t i v e o f t h e s e

"discontinuity" s t u d i e s would be t o t e s t Calderbank and Moo-Young' s hypothesis t h a t mass-transfer r a t e s can be u n i v e r s a l l y c o r r e l a t e d with t h e power d i s s i p a t i o n r a t e s . A s a d i r e c t extension of t h e work of t h i s t h e s i s , o t h e r s might con-

s i d e r use of d i f f e r e n t f l u i d s t o provide a more d e f i n i t i v e v a r i a t i o n of t h e Schmidt number and of t h e i n t e r f a c i a l condition.

100

The s t u d i e s could

101 W

have t h e a d d i t i o n a l o b j e c t i v e of demonstrating t h a t s u r f a c e t e n s i o n i s not an i n f l u e n t i a l v a r i a b l e other than f o r i t s e f f e c t on t h e mobility of t h e i n t e r f a c e .

Experiments designed t o look a t t h e a c t u a l small-

s c a l e movements of bubbles i n cocurrent t u r b u l e n t flow and t h e eddy s t r u c t u r e very c l o s e t o t h e i n t e r f a c e would help guide f u r t h e r t h e o r e t i c a l d e s c r i p t i o n s and may h e l p v a l i d a t e t h e dimensionally determined expression f o r t h e average t u r b u l e n t i n e r t i a l forces. One contention of t h e present work, t h e p o s s i b l e e x i s t e n c e of d i f f e r e n t flow regimes y i e l d i n g d i f f e r e n t Reynolds number exponents, should be f u r t h e r t e s t e d .

A s u b s t a n t i a l l y widened range of Reynolds

number f o r a given bubble s i z e i n a viscous f l u i d might uncover a t r a n s i t i o n from one regime t o another. I n p r a c t i c a l a p p l i c a t i o n s , t h e i n t e r f a c i a l a r e a a v a i l a b l e f o r mass t r a n s f e r i s equally a s important a s t h e mass-transfer c o e f f i c i e n t . Therefore, f o r systems i n which r e l a t i v e l y long term r e c i r c u l a t i o n of t h e bubbles i s a n t i c i p a t e d , t h e bubble dynamic behavior becomes of interest.

For example, more information i s needed on bubble breakup

and coalescence w h i c h tend t o e s t a b l i s h an equilibrium bubble s i z e i n a turbulent field.

More important perhaps, i s t h e e f f e c t of bubbles

passing through regions with l a r g e changes i n pressure ( e , g.,

across a

pump) where they may go i n t o s o l u t i o n and, a s t h e pressure i s again reduced, r e n u c l e a t e and grow i n s i z e .

The e f f e c t s on mean bubble s i z e s

and t h e i n t e r f a c i a l a r e a s a v a i l a b l e under such conditions a r e not w e l l known and t h i s p a r t i c u l a r aspect of bubble behavior could provide a

f r u i t f u l f i e l d f o r f u r t h e r research.

102 Theoretical Two extreme viewpoints were taken i n t h i s r e p o r t i n which bubbles

were considered a s being e i t h e r e s s e n t i a l l y passive i n a t u r b u l e n t f i e l d with t h e mass-transfer behavior being governed by t h e "sweeping" of t h e surface with random eddies or, a l t e r n a t i v e l y , a s moving through t h e t u r bulent l i q u i d and e s t a b l i s h i n g a boundary-layer type of behavior. The "surface renewal" model developed i n t h i s r e p o r t was only t e n t a t i v e l y evaluated.

Further development of t h e model i s a n t i c i p a t e d and

a d d i t i o n a l s o l u t i o n s should demonstrate t h e technique by which s u r f a c e renewal concepts can be applied t o cocurrent t u r b u l e n t flow. A complete mechanistic d e s c r i p t i o n of mass t r a n s f e r between bubbles

and l i q u i d s i n cocurrent t u r b u l e n t flow would presumably include t h e t r a n s i e n t e f f e c t s of a developing boundary l a y e r a s a bubble i s a c c e l e r a t e d i n f i r s t one d i r e c t i o n and t h e n t h e o t h e r by random i n e r t i a l forces. Superimposed on t h i s would be t h e e f f e c t s of t h e surrounding eddy s t r u c t u r e and t h e c h a r a c t e r i s t i c s of t h e eddy p e n e t r a t i o n s through t h e developing boundary layer.

Further e f f o r t s t o t h e o r e t i c a l l y d e s c r i b e

t h e s e simultaneous e f f e c t s should be considered w i t h p o s s i b l e s o l u t i o n s

on a d i g i t a l c o q u t e r . The use of pseudo-turbulent f i e l d s (e. g., an eddy-cell s t r u c t u r e ) t o determine t h e t r a n s p o r t r a t e s and t o e s t a b l i s h such p r o p e r t i e s a s an eddy d i f f ' u s i v i t y should provide u s e f u l i n s i g h t s i n t o t h e a c t u a l behavior i n r e a l f l u i d s and should help p r e d i c t data trends.

For example, t h e

multiple boundary l a y e r s t r u c t u r e e s t a b l i s h e d by Busse'l

for t h e v e c t o r

f i e l d t h a t maximizes momentum t r a n s p o r t i n a shear flow s t r o n g l y resembles

103 an a r t i f i c i a l eddy-cell s t r u c t u r e .

S t a r t i n g with such a s t r u c t u r e , one

could work tlbackwardsll t o c a l c u l a t e eddy v i s c o s i t i e s (for example) as a m e t i o n of p o s i t i o n away from a s o l i d boundary.

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Gal-Or, B., G. E. Klinzing, and L. L. Tavlarides, "Bubble and Drop Phenomena," Ind. and Eng. Chem., 61( 2) : 21, February (1969).

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Lamont, J. C. and D. S. S c o t t , "Mass Transfer from Bubbles i n Cocurrent Flow," Can. J. Chem. Eng., 201-208,August (1966).

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H a r r i o t t , P., "Mass Tr a nsf e r t o P a r t i c l e s : P a r t 11. P ip elin e, AIChE Jo u r na l, 8(1): 101, March (1962).

Figueiredo, 0. and M. E. Charles, "Pipe line Processing: Mass T ran sfer i n t h e Horizontal P i p e l i n e Flow of Solid-Liquid Mixtures, The Can. J. of Chem. Eng., 45: 12, February (1967).

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16: 37 (1961). 17

Sherwood, T. K. and P. L. T. Brian, "Heat and Mass Transfer t o P a r t i c l e s Suspended i n Agitated Liquids," U. S. Dept. of I n t e r i o r Research and Development Progress Report No. 334, A p r i l (1968).

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Barker, J. J. and R. E. Treybal, "Mass Transfer C o e f f i c i e n t s f o r S o l i d s Suspended i n Agitated Liquids," AIChE Journal, 6( 2) : 289-295, June (1960).

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P o r t e r , J. W., S. L. Goren, and C. R. Wilke, "Direct Contact Heat Transfer Between Immiscible Liquids i n Turbulent Pipe Flow, A I C h E Journal, 14(1): 151, January (1968).

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Sideman, S. and Z. Barsky, "Turbulence E f f e c t on Direct-Contact Heat Transfer w i t h Change of Phase," A I C h E Journal, ll(3): 539, May (1965).

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Sideman, S., "Direct Contact Heat Transfer Between Immiscible Liquids," Advances i n Chemical Engineering, Vol. 6, pp. 207-286, Academic Press, New York (1966).

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Fortescue, G. E. and J. R. A. Pearson, "On Gas Absorption i n t o a Turbulent Liquid, Chem. Eng. Sei. , 22: 1163-1176 (1967).

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Kozinski, A. A. and C. J. King, "The Influence of D i f f u s i v i t y on Liquid Phase Mass Transfer t o t h e Free I n t e r f a c e i n a S t i r r e d Vessel, AIChE Journal, 1 2 ( 1): 109-110, January (1966).

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Peebles, F. N. and H. J. Garber, "Studies on t h e Motion of Gas Bubbles i n Liquids, Chem. Eng. Progr., 49(2) : 88-97, February (1953).

27.

Ruckenstein, E., "On Mass Transfer i n t h e Continuous Phase from Spherical Bubbles o r Drops," Chem. Eng. Sei., 19: 131-146 (1964).

2a.

G r i f f i t h , R. M., "Mass Transfer from Drops and Bubbles," Chem. Eng. S e i . , 1 2 : 198-213 (1960).

29.

Redfield, J. A. and G. Houghton, "Mass Transfer and Drag C o e f f i c i e n t s f o r Single Bubbles a t Reynolds Numbers of 0.02-5000," Chem. Eng. Sei., 20: 131-139 (1965).

30.

Chao, B. T., "Motion of Shperical Gas Bubbles i n a Viscous Liquid a t Large Reynolds Numbers, Phys. Fluids, 5 (1): 69-79, January

(1962) 31.

Lochiel, A. C. and P. H. Calderbank, "Mass Transfer i n t h e Continuous Phase Around Axisymmetric Bodies of Revolution," Chem. Engr. S'c i 9 19: 475-484, Pergamon Press (1964).

32* Higbie, R.,

"The Rate of Absorption of a Pure Gas I n t o a S t i l l Liquid During Short Periods of Exposure," Presented a t American I n s t i t u t e of Chemical Engineers Meeting, Wilmington, Delaware, May 13-15 (1935).

33.

Danckwerts, P. V. "Significance of Liquid-Film C o e f f i c i e n t s i n Gas Absorption," Ind. Eng. Chem., Engineering and Process Development, 43 (6) : 1460-1467, June (1951).

34.

Toor, H. L. and J. M. Marchello, "Film-Penetration Model f o r Mass and Heat Transfer, A I C h E Journal, 4( 1): 97-101, March (1958).

35.

Whitman, W. G.,

36.

Levich, V. G., Physicochemical Hydrodynamics, Prentice-Hall, Englewoods C l i f f s , N. J. (1962).

37.

Hinze, J. O., Turbulence, McGraw-Hill Book Co.,

Chem. and Met. Eng.,

29: 147 (1923).

Inc.

, New

Inc.

York

(1959)

38.

Middleman, S., "Mass Transfer from P a r t i c l e s i n Agitated Systems: Application of t h e Kolmogoroff Theory, I t A I C h E Journal, 11(4) : 750-761, J u l y (1965).

39.

H a r r i o t t , P. and R. M. Hamilton, "Solid-Liquid Mass Transfer i n Turbulent Pipe Flow," Chem. Eng. Sci., 20: 1073-1078 (1965).

40.

Brian, P. L. T. and M. C. Beaverstock, "Gas Absorption by a TwoStep Chemical Reaction,'' Chem. Eng. Sci., 20: 47-56 (1965).

41.

Davies, J. T., A. A. Kilner, and G. A. R a t c l i f f , "The E f f e c t of D i f f u s i v i t i e s and Surface Films on Rates of Gas Absorption," Chem. Eng. Sci., 19: 583-590 (1.964).

42.

G a l - O r , B., J. P. Hauck, and H. E. Hoelscher, "A Transient Response Method f o r a Simple Evaluation of Mass Transfer i n Liquids and D i s persions," I n t e r n . J. Heat and Mass Transfer, 10: 1559-1570 (1967).

43.

Gal-Or, B. and W. Resnick, "Mass Transfer from Gas Bubbles i n an Agitated Vessel with and without Simultaneous Chemical Reaction," Chem. Eng. S e i . , 19: 653-663 (1964).

44.

H a r r i o t t , P., "A Random Eddy Modification of t h e Penetration Theory," Chem. Eng. Sci., 17: 149-154 (1962).

108

45.

Koppel, L. B., R. D. P a t e l , and J. T. Holmes, " S t a t i s t i c a l Models f o r Surface Renewal i n Heat and Mass Transfer," AIChE Journal, 1 2 ( 5 ) : 941-955, September (1966).

46.

Kov8sy, K., "Different Types o f D i s t r i b u t i o n Functions t o Describe a Random Eddy Surface Renewal Model," Chem. Ebg. Sci., 23: 9 - 9 1

( 1968)

47.

Perlmutter, D. D., "Surface-Renewal Models i n Mass Transfer," Chem. Eng. s c i . , 16: 287-296 (1961).

48.

Ruckenstein, E., "A Generalized Penetration Theory f o r Unsteady Convective Mass Transfer," Chem. Eng. Sci., 23: 363-371 (1968).

49.

Sideman, S., "The Equivalence of t h e Penetration and P o t e n t i a l Flow Theories, Ind. Eng. Chem., 58( 2) : 54-58, February (1966).

50.

H a r r i o t t , P., "Mass Transfer t o P a r t i c l e s : P a r t I. Suspended i n Agitated Tanks," A I C h E Journal, 8(1): 93-101, March (1962).

51. Banerjee, S., D. S. S c o t t , and E. Rhodes, "Mass Transfer t o F a l l i n g Wavy Liquid Films i n Turbulent Flow," Ind. Eng. Chem. Fund., 22-26, February (1968).

7(l) :

52.

B a r k e r , ' J . J. and R. E. Treybal, "Mass Transfer C o e f f i c i e n t s f o r S o l i d s Suspended i n Agitated Liquids," AIChE Journal, 6(2): 289-295, June (1960).

53.

Galloway, T. R. and B. H. Sage, "Thermal and M a t e r i a l Transport from Spheres," Intern. J. Heat Mass Transfer, 10: 1195-1210 (1967).

54.

Hughmark, G. A., "Holdup and Mass Transfer i n Bubble Columns," Ind. Eng. Chem., Process Design and Development, 6 ( 2 ) : 218-220, A p r i l (1967).

55.

Jordan, J . , E. Ackerman, and R. L. Berger, "Polarographic Diffusion C o e f f i c i e n t s of Oxygen Defined by A c t i v i t y Gradients i n Viscous Media," J. Am. Chem. Soc., 78: 2979, J u l y (1956).

56.

Bayens, C . , PhD Thesis, The Johns Hopkins University, Baltimore, Maryland (1967).

57.

Resnick, W. and B. Gal-Or, "Gas-Liquid Dispersions, Advances i n Chemical Engineering, Academic Press, New York, Vol. 7, pp. 295-395 ( 1968)

58.

P h i l l i p s , 0. M., "The Maintenance of Reynolds S t r e s s i n Turbulent Shear Flow," J. Fluid Mech., 2 7 ( l ) : 131-144 (1967).

59.

Groenhof, H. C . , "Eddy Diffusion i n t h e C e n t r a l Region of Turbulent Flows i n Pipes and Between P a r a l l e l P l a t e s , " Chem. Eng. S c i . , 25: 1005-1014 (1970).

60.

Sleicher, Jr., C. A., "Experimental Velocity and Temperature Prof i l e s f o r A i r i n Turbulent Pipe Flow," Transactions of t h e ASME, 80: 693-704, A p r i l (1958).

61.

Busse, F. H.,

"Bounds f o r Turbulent Shear Flow," J. Fluid Mech.,

41: 2l9-d-O (1970).

APPENDIX A

PHYSICAL PROPERTIES OF AQUEOUS -CLYCEROL MIXTURES

111

112 ORNL-DWG

71-8003

IO6

105

I 04

403

5 I

2 0

GLYCERINE

Figure 23.

100

(O/o)

Schmidt Numbers of Glycerine-Water Mixtures.

â&#x20AC;&#x2DC;3

71-8004

ORNL-DWG

S O L U B I L I T Y OF OXYGEN I N M I X T U R E S OF GLYCERINE A N D WATER - 25OC

DATA OF JORDAN, A C K E R M A N A N D BERGER

GLYCERINE

Figure

24.

Henry's Law Constant f o r Oxygen S o l u b i l i t y i n GlycerineWater Mixtures.

114

ORNL-DWG

71-8005

D I F F U S I O N OF OXYGEN THROUGH D I F F E R E N T M I X T U R E S OF GLYCERINE A N D W A T E R AT 25OC

' \

7 / /

\ \

.\O

D A T A O F JORDAN, A C K E R M A N AND BERGER

G L Y C E R I N E (%)

Figure 25.

Molecular Diffusion C o e f f i c i e n t s f o r Oxygen i n GlycerineWater Mixtures. Data of Jordan, Ackerman, and Berger.

115

ORNL-DWG

71-8006

D E N S I T I E S OF G L Y C E R I N E A N D WATER M I X T U R E S - 25OC

1.200

0 0 \

0 Y

I-

w n

1 .io0

JORDAN, A C K E R M A N AND BERGER HANDBOOK OF CHEMISTRY AND PHYSICS

CHEMICAL ENGINEERING HANDBOOK

B E I L S T E I N ' SH A N D B O O K O F ORGAN I C C H E M I S T R Y

1.000

60 GLYCERINE

Figure

I 100

(O/o)

26. Densities of Glycerine-Water Mixtures.

I 120

140

116

ORNL-DWG

71-8007

ACKERMAN

10'

-a -

c_ v)

0 V

100

5 0

30 GLYCERINE

(%I

Figure 27. Viscosities of Glycerine-Water Mixtures.

APPENDIX B DERIVATION OF EQUATIONS FOR CONCENTRATION CHANGES

ACROSS A GAS-LIQUID CONTACTOR Consider t h e cocurrent flow of a gas and a l i q u i d i n a constant a r e a p i p e l i n e of c r o s s s e c t i o n A

and length L.

In a differential

element o f length dR, a dissolved c o n s t i t u e n t of concentration

t h e l i q u i d i s t r a n s f e r r e d i n t o t h e gas a s shown below.

A mass balance f o r t h e dissolved c o n s t i t u e n t gives

Q adE = +a Q dc

where

Ac dR

+ka Ac dR

(c - C s )

(B-1)

(c - C s ) , -

i s t h e gas phase g i s t h e concentration e x i s t i n g a t t h e gas-liquid

i s t h e l i q u i d phase average concentration, C

concentration, and C

interface. Dividing Equation (B-2) by Equation (B-1) gives

&a A - =--. dC

03-31

I n t e g r a t i n g Equation (B-3) and l e t t i n g

= 0

when

= Ci

gives

If t h e i n t e r f a c i a l concentration i s assumed t o be a t "equilibrium"

and t h e s o l u b i l i t y of t h e dissolved c o n s t i t u e n t i s e x p r e s s i b l e by Henry's

118 Law, then HCs =

(B-5)

S u b s t i t u t i n g Equation (B-4) i n t o Equation (B-5) gives

c =E s

Equation (B-6)

( $ ) ( C i 4 )

can be s u b s t i t u t e d i n t o Equation (B-1) t o o b t a i n

Ac dR

[F - (RT/H)(Q R/Q g ) ( C i - C ) ]

Expanding and dividing by Q dR gives

where B

‘ = ka

Ac (1+ y)/QR and y

(RT/H) (QR/Qg)

Use of t h e i n t e g r a t i o n f a c t o r eB’’ permits t h e following s o l u t i o n

= 0,

= C

i’

t h e r e f o r e t h e constant of i n t e g r a t i o n i s

and

Therefore t h e r a t i o of t h e e x i t is

or defining B

R’L,

= L ) t o i n l e t concentration, Ce/Ci,

119 W

where

121

APPENDIX C INSTRUMENT APPLICATION DRAWING

;;-:>-g -1 , itI I

Li.. HV-UlU

ORPlN

HVNl4

HV-AIS C

E ?! 0

z ELEMENTARY WIRING DIAGRAM PS

QF-002-0 DWG. NO.

REFERENCE DRAWINGS DlTl

0111"

E.C.KEITH C.c.,a*mm

I-Z-71 DLTl

O".w,*a

E"1S.

. 1 . 0

A.P"O"11)

,&/KIIYII( C*.CI

ol.

EOLL,.

1970 r o o m m w i i o m J.W. KREWSCjN

.--o

-*,

J.W.K.

AJR SUPPLY

."D,.CT

oA,

Tllll

MSBR MASS TRANSFER INSTRUMENT APPLICATION DIAGRAM

Figure 28. Instrument Application Drawing of the Experiment Facility.

.,vm

LOOP

123 APPESSDIX D INSTRUMENT CALIBRATIONS

ORNL-DWG 71-7998 I

BUBBLE GENERATOR LIQUID FLOW, QL (gpm)-20 ~YMBOL-O

0 IO

GAS TO LIQUID VOLUMETRIC FLOW RATIO, Q Q / Q = 0.3% LIQUID = IND~CATED MIXTURE OF GLYCERINE AND WATER GAS = HELIUM O%GLYCERlNE

0 09

-5 W

30 35 40 45 50 55 60 65 70 75 80 85 90 A A v v a o e o A A v

1 I

0.08

-----(GAS)

007

c W

006

005

n z W

I 004 m

m 3

003

0 02

0 01 0 3

I 5

PROBE POSITION (in.)

Figure 29.

Bubble S i z e Range Produced by t h e Bubble Generator.

124 ORNL-DWG

71-8008

100

WATER ( 0 % GLYCERINE)

10 CALIBRATION OF ROTAMETER 1 FOR T W O M I X T U R E S OF GLYCERINE AND W A T E R

0 0

100

FLOW ( g p m )

Figure 30.

C a l i b r a t i o n of Rotameter No. 1 (100 g-pm).

125 ORNL-DWG 71-8009

100

-8 c

W _J

10 C A L I B R A T I O N O F ROTAMETER 2 FOR TWO M I X T U R E S OF GLYCERINE A N D WATER

0 0

Figure

30 FLOW ( g P m)

31. C a l i b r a t i o n of Rotameter No.

(40 g p m ) .

126 ORNL-DWG

71-8010

100

-8 c

5 50

rr W -I

Q V

10 C A L I B R A T I O N OF R O T A M E T E R 3 FOR T W O M I X T U R E S OF G L Y C E R I N E AND WATER

0 0

Figure 32.

4 6 FLOW (gpm)

C a l i b r a t i o n of Rotameter No. 3 (8 gpm).

ORNL-DWG

74-804 1

C A L IB R AT I 0 N 0 F C A P I L L A R Y - T U B E FLOW MET E R AT A PRESSURE OF 5 0 p s i g

0 Ill

L 0

-.C

m 3

I-

> U a _I

-J -

a a 0

0 rK 0

a 0 rK

n w rK 3

cn cn w

a' Q

0.02

0.04

0.06

0.08

0.10

H E L I U M FLOW ( S C f M )

Figure 33.

C a l i b r a t i o n of Gas-Flow Meter a t 50 psig.

0.12

ORNL-DWG

71-8018

40 38 36

OXYGEN CONC OF GLYCERINE

0 0

MAGNA CORPORATION PROBE i MAGNA CORPORATION PROBE 2

34 32 30

-E

FOR HENRY'S LAW CONSTANT (H P CI) EQUAL TO '1.74 p s i A I R / p p r n DISSOLVED OXYGEN T A K E N FROM P E R R Y , " C H E M I C A L ENGINEERS

n Q

0 t. X 0

22 20 18

7 5 % GLYCERINE CURVE CALCULATED FOR H = 3 5 5 p s i A I R / p p m DISSOLVED

n W

?! cn !

n 12

40 8 6 4

0 20

AIR PRESSURE ( p s i g )

Figure 34.

C a l i b r a t i o n o f Oxygen Sensors i n two Mixtures of Glycerine and Water.

APPENDIX E EVALUATION OF EFFECT OF OXYGEN SENSOR RESPONSE SPEED ON THE MEASURED TRANSIENT RESPONSE OF THE SYSTEM

Instrument responses a r e t y p i c a l l y exponential i n nature. i f t h e sensor reading i s defined a s Cr,

t i o n as

Thus,

and t h e a c t u a l loop concentra-

(both functions of time) it i s s a f e t o assume an instrument

response equation of t h e form

where Kr i s an instrument response c o e f f i c i e n t . The t r a n s i e n t response o f t h e loop i t s e l f i s given by an equation

Therefore, Equation (E-1) can be expressed a s dC

r + Kr dt

C r = Kr Coe 4 Lt

Integration of Equation (E-2) with the initial condition C

= 0

(E-2)

gives

=Ae

4 t 4 t r +Be L

(E-3)

The manufacturers s t a t e d response time f o r t h e Magna oxygen sensors

90%

i n 30 seconds.

This response r e s u l t s i n a value of K

4.61

The maximum observed r a t e of change of oxygen concentration i n t h e t r a n s i e n t experiments corresponded t o

129

0.75

. % < 0.3.)

(On t h e average, t h e experiment t r a n s i e n t s r e s u l t e d i n

There-

fore, f o r t h i s case, A = 4 . 1 9 , B = 1.19 and C /Co = l.19e-0*76t

An examination of Equation

- 0.19e-4*61t

(E-4)

(E-4) shows t h a t a s time progresses t h e

second term becomes n e g l i g i b l e compared t o t h e f i r s t , and t h e measured slope approaches t h e a c t u a l t r a n s i e n t slope of 0.75.

For example, t h e

measured slope f o r t h i s "worse'' case i s 0.74 a r t e r only one minute of t r a n s i e n t compared t o t h e r e a l value of 0.75.

Therefore, t o f b r t h e r

minimize t h i s p o s s i b l e e r r o r , t h e slopes of t h e measured t r a n s i e n t s were taken only from t h e f i n a l s i x minutes of t h e curve p e r m i t t i n g an i n i t i a l "response adjustment" time of s e v e r a l minutes.

The e r r o r due t o t h e

instrument response, then, i s assumed t o be n e g l i g i b l e .

APPJiYDIX F

MASS BALANCES FOR THE SURFACE RFNEWAL MODEL

Consider a d i f f e r e n t i a l region i n a s p h e r i c a l s h e l l of f l u i d surrounding a bubble as shown below,

Mass balances f o r t h e concentration, C, of a dissolved c o n s t i t u e n t within t h e l i q u i d a r e obtained a s follows: Convection in: out:

ur c

4nr2

4rr (r

dr)” [Ur C

a (urc) dr

a (urc 1

4m2 7d r

n e t convection = ( o u t - i n ) =

in:

out:

k~ 4nr2

8mdr ( U C ) ’

ac ar

417( r +

dr)’ -

net d i f f i s i o n = (out-in) =

[ .-

ac + a2c d r ] ar

ar2

.. -..

a 2C d r + 4nr2 -

r9 81-rrdr ac

Storage net l o s s =

lcn - [(r + 3

ac - r3] at

dr)3

Summing t h e contributions (F-1) through (F-3) gives

U L

2 +(urc) r

Dividing by

4m2 d r

= 0

gives

a(r2u rc)

at Making t h e Reynolds assumptions

c ur

=i?+C‘ = u t

s u b s t i t u t i n g i n t o Equation (F-h), expanding and c o l l e c t i n g terms gives

ai? + -ac‘ =a at

[-a r 2

a2i? + -a2c’ +--

ai?

r ar

ar2

+ a aro + aa r o + r 3(u?)

ar 2

f r (u’c’)

(F-5)

The time average of a quantity, C, i s defined as

Jt-‘’ C

i n which t h e time i n t e r v a l , ( t a- t l ) , i s long enough f o r t h e , t i m e average of t h e f l u c t u a t i n g q u a n t i t i e s i n Reynolds assumption t o be

z e r o but s h o r t compared t o t h e t r a n s i e n t changes i n time average of Equation (F-5) gives

Therefore, a

APPENDIX G ESTIMATE OF ERROR DUE TO END-EFFECT ADJTJSTMENTS

The measured r a t i o s of e x i t - t o - i n l e t concentration, K, across a gas-liquid c o n t a c t e r were e x t r a c t e d from t h e measured slope, S, of t h e log-concentration versus time d a t a by t h e r e l a t i o n

The e r r o r involved i n measuring t h e various q u a n t i t i e s used t o e s t a b l i s h K a r e estimated t o be

A5 - 0.01 , S

and

nQ, - N

0.03

Consequently, t h e e r r o r i n K can be estimated from DK - N K

max

Kmin

where

and =

Kmin

Exp

[-(s +

&)(V

(Q, - AQ,)

+ nvs)

135 The minimum r a t i o measured f o r K was -0.9,

t h e r e f o r e t h e maximum estimate

of t h e e r r o r i s

I n Chapter 111, t h e r a t i o , K2, applicable t o t h e t e s t s e c t i o n above was c a l c u l a t e d from

where K

was t h e measured r a t i o across t h e bubble generator, t h e t e s t

section, and t h e bubble s e p a r a t o r together, and KII j u s t t h e bubble generator and bubble separator.

was t h a t across

Therefore, t h e maximum

instrument-precision induced e r r o r i n KB i s estimated t o be &a

Ka, min

K2,max

- N

1 + 0.02 K I / K I I (1 0.02

In establishing K

) - .KI/KII(

and KII

1 0.02 1 + 0.02

)< 10% .

i n separate t e s t s , t h e i n a b i l i t y t o

e x a c t l y d u p l i c a t e conditions r e s u l t s i n an e r r o r g r e a t e r than t h e above. An estimate of t h e maximum magnitude of t h i s e r r o r can be had by examini n g t h e d a t a f o r t h e 75% water-25% glycerine mixture (Figure 13, page 63). Before t h e end-effect adjustment, t h e c a l c u l a t e d v e r t i c a l and h o r i z o n t a l flow mass-transfer c o e f f i c i e n t s f o r t h e 0.02-inch mean diameter bubbles were e s s e n t i a l l y i d e n t i c a l . by-25%.

However, a f t e r t h e adjustment t h e y d i f f e r e d

It i s f e l t t h a t t h i s d i f f e r e n c e mostly a r i s e s from t h e i n a b i l -

i t y t o e x a c t l y r e c r e a t e t h e v e r t i c a l flow conditions a s a r e s u l t of a l t e r a t i o n s made i n t h e bubble generator between t h e o r i g i n a l t e s t and

136 t h e end-effect t e s t .

Consequently t h e h o r i z o n t a l flow d a t a a r e consid-

ered t h e more "exact" although they should s t i l l r e f l e c t t h e -10% error estimated due t o measurement precision.

APPENDIX H

MASS TRANSFER DATA

137

ORN L-DWG

7 1-7965

WATER t -200 ppm N-BUTYL ALCOHOL VERTICAL FLOW IN 2-in. CONDUIT SCHMIDT NO. 4f9

Re -

(gpm)

0.3

0.5

0 U

35,583

62,269

71,165

88,955

106,748 1 15,642

+h A+

142,381

100

177,913

a :

24 0 V

Iv, 3 3

3 Y

0.01

0.02

0.03 d,,,

Figure 35.

0 04

0.05

0.06

0.07

0.08

BUBBLE MEAN DIAMETER (in.)

Unadjusted Mass Transfer Data. Water Plus ,200 N-Butyl Alcohol. V e r t i c a l Flow.

ppm

139

ORNL-DWG 7 1 - 7 9 6 6

WATER t -200 ppm N - B U T Y L A L C O H O L H O R I Z O N T A L FLOW IN 2-in. CONDUIT

Q,/QL

= 0.3%

S C H M I D T NO.

0.01

= 419

0.02 d,,,

Figure 36.

0.03

I (gpm)

0.04

0.05

Re -

88,955

0.06

B U B B L E M E A N DIAMETER (in.)

Unadjusted Mass Transfer Data. Water Plus -200 pprn N-Butyl Alcohol. Horizontal Flow.

0.07

140

O R N L - DWG 7 1-7967

12.5 O/o G L Y CE R I NE &/QL = 0.3% VERTICAL FLOW IN 2-in. CONDUIT SCHMIDT NO. 370

w I THOUT*

w I TH"

0 h

0 v

v L

I-

*ADDITION

OF - 2 0 0

QI (gpm)

Re -

19,288

20 35

25,718 45,006

50 65

64,294 83,583

ppm N-BUTYL ALCOHOL

LL LL W

0 0

z a

+ v, v,

n w 1 - 6 v, 3 7

a z 2

' I

0 0

0.01

0.02

0.03

0.04

0.06

0.05

0.07

0.08

d u s t B U B B L E MEAN DIAMETER (in.)

Figure 37.

Unadjusted Mass Tr a nsf e r Data. Water. Vertical Flow.

12.5% Glycerine-87.5%

141

ORNL-DWG

iORIZONTAL FLOW IN 2-in 2 5% GLYCERINE iCHMIDT NO = 370 2 g / Q ~= 0 3%

CONDUIT

0 0

A A

c L

I-

71-7968

Re -

(gpm)

35 50 65 75 85

45,006 64,294 83,583 96,442 109,302

0 LL LL

:10 0

11: W LL

I-

cn cn

a 2

n W

0 ’ 4

a z

3 1

0 0

0.ot

0.02.

0.03

0.04

0.05

0.06

0.07

dvs, BUBBLE M E A N DIAMETER (in.)

Figure 38.

Unadjusted Mass Transfer Data. Water. Horizontal Flow.

12.5% Glycerine-87.5%

0.08

142 v

71-7969

ORNL-DWG

10 25% GLYCERINE Q ~ / Q L = 0.3% VERTICAL FLOW I N 2-in. SCHMIDT NO. = 750

CONDUIT

8 c L

1 .c

I - 7 z

LL Lc

w 0

W LL

2 a

CT I-

cn cn 2 4 n W

Iv)

2 3 Q 2 3 1

WITHOUT*

w I TH*

0 U

17,636

I A

26,454

0 v

*ADDITION

OF - 2 0 0

(gpm)

35,272

4 4,090

52,908

ppm N-BUTYL ALCOHOL I

0.01

0.02

0.03

0.04

Re -

0.05

0.06

0.07

0.08

d v s , BUBBLE MEAN DIAMETER (in.)

Figure 39.

Unadjusted Mass Transfer Data. Vertical Flow.

25% Glycerine-75$,Water.

143

ORNL-DWG

71 - 7 9 7 0

7 h

r \ 4-

v Y-

1-6 2

w -

u LL LL

LT W IL

rn 2 a

I-

m m

a 2

D W

QL ( g p d

rn 3 7

a a

-0 0

A -v

Figure 40.

40 45 50 55 65 75

Re 35,272

39,681

44,090 48,498 57,317 66,135

25O/o GLYCERINE Q g / Q ~= 0 32 SCHMIDT NO = 750 HORIZONTAL FLOW I N 2-117 CONDUIT I

Unadjusted Mass Transfer Data. Horizontal Flow.

25% Glycerine-75% Water.

144

ORNL-DWG

7 1-7971

8 WITHOUT"

0 D

7 c

QL ( w m )

40 45

WITH*

Re -

13,079 19,619 26,159 29,429

-*ADDITION

I - 6

OF -200

ppm N-BUTYL ALCOHOL

0 LL

W LL

a g 4 [L

I-

cn cn

n W

I-

cn 3

0 ' 2

2 3

37.5% GLY CE R I N E SCHMIDT NO. = 2015

Q g / 0 ~= 0.3%

VERTICAL FLOW IN 2 - i n . CONDUIT

0 0

0.01

0.02

0.03

dvS, BUBBLE

Figure

41.

0.04 0.05 0.06 MEAN DIAMETER ( i n . )

Unadjusted Mass Transfer Data. Water. V e r t i c a l Flow.

0.07

37.5% Glycerine-62.5%

0.08

145 W

OR N L-DW G 7 i-7 9 7 2

30 35 40 45 50

19,6! 9 22,889

55 60

35,968 ___ 39,238 45,777

1 D 1

L 7

1 I

r \

QL (gpm)

26,159 29,429 32,699

c v .c

I - 7

uLL LL

8 6

cn $ 5 CK I-

cn cn a I 4 n w F

z 3 x

2 37.5"10 GLYCERINE SCHMIDT NO. = 2015 - Q g / Q ~ = 0.3%

HORIZONTAL FLOW I N 2 - i n .

CONDUIT

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

dvs, BUBBLE MEAN DIAMETER (in.)

Figure

42.

Unadjusted Mass Tr a nsf e r Data. Water. Horizontal Flow.

37.5% Glycerine-62.5$

0.08

146

ORNL-DWG

WITH*

Q L (gpm)

Q ~ / Q L (%)

10 20 30 40

0.5 0.5 0.5 0.5

4,068 8,137 12,205

0.3+

20,342

2 7

4 v

*ADDITION OF -200 ppm N-BUTYL +EXCEPT WHERE NOTED

Re -

WITHOUT*

c L

9 LL

71-7973

16,274

ALCOHOL

LL W

u E

m z a

LT I-

m 4 m

2 n W

+ - 3 m 3 -3

a a

= 2

c;9--A&

C '

j O % GLYCERINE SCHMIDT NO. = 3 4 4 6 d E R T I C A L F L O W IN 2-1n

I I

CONDUIT

0 0.04

0.02

0.03

d,,,

Figure 43.

0.04 0.05 0.06 0.07 B U B B L E MEAN DIAMETER (in.)

Unadjusted Mass Transfer Data. Water. V e r t i c a l Flow.

0.08

50% Glycerine-50%

0.09

OR N L- DWG 7 1 - 79 74

n A

v I

- 50%

Qg/QL

12,205 14,238 16,274 18,306 20,342 22,374

' I

= 3446

= 0.3%

HORIZONTAL

0.01

F L O W I N 2-in.

0.02 dvs,

Figure

30 35 40 45 50 55

Re -

GLYCERINE

S C H M I D T NO.

(gpm)

44,

0.03

CONDUIT

0.04

0.05

0.06

BUBBLE M E A N D I A M E T E R ( i n . )

Unadjusted Mass Transfer Data. Water. Horizontal Flow.

5% Glycerine-50%

0.07

ORNL-DWG

100

-1

-WATER -

MEAN DIAMETER ' (in.)

Li Li

419

I I I I I

pprn N - B U T Y L

I-

NO. =

0.015 0.02 0.03 0.04

ALCOHOL

I I II

- BUBBLE

. I -

+ -200

SCHMIDT

71-7984

HORIZONTAL FLOW

VERTICAL

OCUS OF

-/CALCULATED ASYMPTOTES -

I-

v, v,

= n w

I-

v, 3 3

n Q

z 3

1 5

1 103

io4

PIPE REYNOLDS

Figure 45.

NO.,Re

105

= VD/u

Unadjusted Mass Tr a nsf e r C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter. Water Plus -200 ppm N-Butyl Alcohol, Horizontal and V e r t i c a l Flow.

ORNL-DWG

74-7985

100

-2 5 0 \

DIAMETER

HORIZONTAL

0.015

0 I

VERTICAL

v Y-

0.04

+LOCUS

cn Z

a IO K

cn cn

0 W

3 -3

a z

io3

5 PIPE

Figure 46.

1 o4

1o5

R E Y N O L D S NO.. R e = V D / v

Unadjusted Mass Tr a nsf e r C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter. 12.5% Glycerine-87.5% Water. Horizontal and V e r t i c a l Flow.

150

ORNL-DWG

7f-7986

25% GLYCERINE

- SCHMIDT NO = 7 5 0

L \

v e

t2 20

BUBBLE MEAN DIAMETER

LL LL W

0 OI

(In

HORIZONTAL FLOW

VERTICAL FLOW

0.015 0 02

0.03 004

LOCU'S OF F i / F g I

I l l 1

l.5 1

L/l#

LL v)

z a

a: t-

m In

I D W

tIn 3

n 3

a z

' l-H,ORl;ONTAL 1 -

-cL--

FLOW

VERTICAL FLOW

103

t 04

105

P I P E REYNOLDS NO., Re Z VD/w

Figure 47.

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter. 25% Glycerine-75% Water. Horizontal and V e r t i c a l Flow.

151

ORNL-DWG

71-7987

100

50 c L

MEAN DIAMETER

\ c â&#x20AC;&#x2122;c Y

(in.)

0.015 0.02 0 03 0.04

2 LL LL

HORIZONTAL FLOW

VERTICAL FLOW

cc w

LL v)

z f0 a

111

I c3

3 -3

z 3 Y

1 103

104

P I P E R E Y N O L D S NO., R e

Figure 48.

VD/v

Unadjusted Mass Tr a nsf e r C o e f f i c i e n t s Versus Pipe Reynolds Number a s a Function of Bubble Sauter-Mean Diameter. 37.5% Glycerine-62.5% Water. Horizontal and V e r t i c a l Flow.

io5

ORNL-DWG

71-7988

I I 5 0 % GLYCERINE S C H M I D T NO. = 3 4 4 6 BUBBLE MEAN DIAMETER (in.)

HORIZONTAL FLOW

VERTICAL FLOW ~

0 0 0 0

015 02 03 04

103

- ~ - - - -

104

P I P E R E Y N O L D S NO., R e

Figure 49.

Z VD/v

Unadjusted Mass Transfer C o e f f i c i e n t s Versus Pipe Reynolds Number as a Function of Bubble Sauter-Mean Diameter. 5% Glycerine-5% Water. Horizontal and V e r t i c a l Flow.

153 ORNL-DWG

QL ( g p m )

0 0

20 35

35 7 83 62,269

88,955

124,537

142,328

I00

177,913

v v

7 1-7975

r20, 35

-i - WATER

50, 70

-200 ppm N - B U T Y L ALCOHOL = 4 19 Q g / Q ~= 0.3'10

SCHMIDT NO.

V E R T I C A L FLOW I N 2 - i n .

0.01

0.02 dvs,

Figure 50.

0.03 BUBBLE

0.04

CONDUIT

0.05

0.06

0.07

M E A N D I A M E T E R (in.)

Mass Transfer Data Adjusted f o r End-Effect. -200 ppm N-Butyl Alcohol. V e r t i c a l Flow.

Water Plus

ORNL-DWG

0 0 A

50 60 70

88,955 106,746 124,537

-A V

142,328

160,119

---

Figure 51.

71-7976

LEAST SQUARES L I N E S PASSING THROUGH ORIGIN FOR D I A M E T E R S UP T O 0.035

0.01

0.02

0.03

0.04 0.05 dvS, B U B B L E MEAN DIAMETER ( i n . )

Mass Transfer Data Adjusted for End-Effect. Horizontal Flow.

,200 ppm N-Butyl Alcohol.

0.06

Water Plus

155 ORNL- DWG 71 -797 7

QL (gpm)

Re -

20 35 50 65

25,7 18 45,006 64,294 83,583

-a

SCHMIDT NO. = 3 7 0 Q ~ / Q L = 0.3% VERTICAL FLOW I N 2 - i n . CONDUIT

dvs, BUBBLE

Figure 52.

MEAN DIAMETER (in.)

Mass Transfer Data Adjusted f o r End-Effect.

Glycerine-87.5% Water.

Vertical Flow.

12.5%

ORNL-DWG

71-7978

12.5% G L Y C ER I N f SCHMIDT NO. = 370 Q,/QL = 0.3% HORIZONTAL FLOW I N 2-in. CONDUIT

0 -0 A

-v

QL (gpm)

Re -

35 50 65 75 85

45,006 64,294 83,583 96,442 109,302

--- L E A S T SQUARES

LINES PASSING THROUGH ORIGIN FOR DIAMETERS UP TO 0.035 in.

0.01

0.02

0.03

0.04

0.05

0.06

dvS, B U B B L E MEAN DIAMETER (in.)

Figure 53.

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-87.5% Water. Horizontal Flow.

12.5%

157

ORN L-DWG

74-7979

1 25% GLYCERINE SCHMIDT NO. = 750 Q ~ / Q L= 0.3% VERTICAL FLOW I N 2-in. CONDUIT

0 0

n A V

0.01

0.02

0.03

Q, ( g p m ) 20 30 40 50 60

0.04

Re 17,636 26,454 35,272 44,090 52,908

0.05

0.06

dvs, B U B B L E M E A N DIAMETER ( i n . )

Figure 54.

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-75% Water. V e r t i c a l Flow.

25%

158

ORNL-DWG

71-7980

25% GLYCERINE SCHMIDT NO. = 750 Q ~ / Q L= 0.3%

40 45 50 55 65 75

A A

(9Pm)

35 F 72 39,681 44,090 48,498 57,317 66,l3 5

LEAST SQUARES LINES PASSING THROUGH ORIGIN FOR D I A M E T E R S UP TO 0.035 in.

0.01 dvs,

Figure 55.

0.02

0.03

0.04

I 0.05

0.06

B U B B L E M E A N D I A M E T E R (in.)

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-75% Water. Horizontal Flow.

25%

1 59

ORNL-DWG

SCHMIDT NO. = 2015 ) ~ / Q L= 0.3% iORIZONTAL FLOW I N 2-in.

(4pm) 35 40

2 9,429 32,699

35,968

-'I

39,238

45,7 77

0.02

0.03

-e

CONDUIT

71-7981

22,889 26,159

0.01

0.04

0.05

0.06

d V S , BUBBLE MEAN DIAMETER (in.)

Figure 56.

Mass Transfer Data Adjusted for End-Effect. Glycerine-62.5% Water. Horizontal Flow.

37.54'0

ORNL-DWG

5 0 % GLYCERINE SCHMIDT NO. = 3 4 4 6 V E R T I C A L FLOW I N 2-in.

0.01 Figure 57.

QL (gpm)

Re -

71-7982

CONDUIT

0.03 0.04 0.05 0.06 dvs, B U B B L E MEAN DIAMETER (in.)

0.02

Mass Transfer Data Adjusted f o r End-Effect. Glycerine-5% Water. V e r t i c a l Flow.

0.07

50%

161

ORNL-DWG

5 0 % GLYCERINE SCHMIDT NO. = 3 4 4 6 HORIZONTAL FLOW IN 2-in.

a -A

A V

CONDUIT

30 40 45 50

12,205 16,272 18,306 20,342 22,374

I 0

71-7983

I 0.01

0.02

0.04

0.03

0.05

0.06

d v s , B U B B L E MEAN DIAMETER (in.)

Figure 58.

Mass Transfer Data Adjusted f o r End-Effect. Hori zont a 1 Flow. Gly c er i n e - 5% Water

50%

0.07

LIST OF SYMBOLS

Bubble i n t e r f a c i a l a r e a per unit volume

a a

Mean a c c e l e r a t i o n of a f l u i d element of s i z e h i n a t u r b u l e n t field Conduit c r o s s s e c t i o n a l a r e a

Bubble p r o j e c t e d c r o s s s e c t i o n a l a r e a C

-C

Local concentration of a dissolved c o n s t i t u e n t i n a l i q u i d Time averaged component of C

C’

Turbulent f l u c t u a t i n g component of C

Bulk-average concentration of a dissolved c o n s t i t u e n t i n a l i q u i d

avg

Drag c o e f f i c i e n t f o r a bubble moving through a l i q u i d

‘d

‘i

Gas-liquid contactor i n l e t value of C Gas-liquid contactor e x i t value of C

avg

I n i t i a l value of C

avg I n t e r f a c i a l value of C

Bubble diameter

d vs

Sauter-mean diameter of a bubble dispersion m

63f(6)d6

s2f(s)d6] 0

Conduit diameter Molecular d i f f u s i o n c o e f f i c i e n t E

Eddy v i s c o s i t y Blasius f r i c t i o n co e f f i c i e n t Bubble s i z e d i s t r i b u t i o n f’unction Frequency d i s t r i b u t i o n f’unction for t u r b u l e n t eddies of wave number n

164 Fd Fi

Drag f o r c e on a bubble moving through a l i q u i d Mean i n e r t i a l f o r c e on a bubble due t o t u r b u l e n t f l u c t u a t i o n s G r a v i t a t i o n a l f o r c e on a bubble (buoyancy) Gravitational acceleration

Dimensional p r o p o r t i o n a l i t y constant r e l a t i n g f o r c e t o t h e product of mass and a c c e l e r a t i o n

S o l u b i l i t y constant i n Henry's Law r e l a t i o n s

Mass t r a n s f e r p e r u n i t time p e r u n i t volume of l i q u i d

Local mass-transfer c o e f f i c i e n t

A x i a l l y averaged mass-transfer c o e f f i c i e n t

Horizontal flow values of k

kv ka

V e r t i c a l flow values of k Low f l o w asymptotic value of k

Ratio of t e s t s e c t i o n e x i t - t o - i n l e t concentration, C /C e i Loop response c o e f f i c i e n t

Oxygen sensor response c o e f f i c i e n t

Test s e c t i o n length

Mass of a f l u i d element

Wave number of a turbulence component

Number of bubbles per u n i t volume of l i q u i d

Local p r e s s u r e i n t h e conduit

Volumetric flow r a t e of gas bubbles

Volumetric flow r a t e of l i q u i d

Universal gas constant

Radial coordinate

F r a c t i o n a l r a t e of surface renewal

Time coordinate

Absolute temperature

r I

Radially d i r e c t e d v e l o c i t y i n s p h e r i c a l coordinates F l u c t u a t i n g component of U

Contribution t o u ' of eddies of wave number n

Liquid a x i a l v e l o c i t y

Bubble terminal v e l o c i t y within a l i q u i d i n a g r a v i t y f i e l d

b' V

Mean f l u c t u a t i n g v e l o c i t y of a bubble i n a t u r b u l e n t f l u i d Mean f l u c t u a t i n g v e l o c i t y of a f l u i d element i n a t u r b u l e n t f i e l d

r' d'

Relative mean f l u c t u a t i n g v e l o c i t y between a bubble and t h e l i q u i d Bubble t o t a l v e l o c i t y i n t h e r i s e r l e g of a v e r t i c a l t e s t s e c t i o n Bubble t o t a l v e l o c i t y i n t h e downcomer l e g of a v e r t i c a l t e s t section

Mean v a r i a t i o n i n v e l o c i t y over a given d i s t a n c e i n a t u r b u l e n t field

Volume of t h e closed r e c i r c u l a t i n g experiment system

Added mass c o e f f i c i e n t f o r an a c c e l e r a t i n g s p h e r i c a l bubble

Axial coordinate

A flow parameter used by Hughmark i n c o r r e l a t i n g volume f r a c t i o n s

Ratio of l i q u i d - t o - t o t a l volumetric flow [Q,/(Q,

A flow parameter used by Hughmark i n c o r r e l a t i n g volume f r a c t i o n s

a,)]

Greek Symbols

Parameter i n bubble s i z e d i s t r i b u t i o n f'unction

Gas-liquid c o n t a c t e r parameter [s ka AL (1 + y)/Q ]

Gas-liquid contacter parameter [e RTQ

166 6 c â&#x201A;Ź

Bubble diameter used i n d i s t r i b u t i o n f'unction (same as d) m v

'min 'max I-1

'e 'e, n I-10

Energy d i s s i p a t i o n per u n i t mass i n a t u r b u l e n t l i q u i d Energy d i s s i p a t i o n p e r u n i t volume i n a t u r b u l e n t l i q u i d Distance s c a l e i n a t u r b u l e n t l i q u i d Minimum eddy s i z e i n a t u r b u l e n t l i q u i d Maximum eddy s i z e i n a t u r b u l e n t l i q u i d Liquid v i s c o s i t y Eddy d i f f ' u s i v i t y Contribution t o p

from t u r b u l e n t component of wave number n

Undamped eddy d i f f ' u s i v i t y away from an i n t e r f a c e

Kinematic v i s c o s i t y ( z p / p )

I n t e r f a c i a l damping function f o r viscous eddy c e l l s

5, 'm P 7W

m 'd

Rigid i n t e r f a c e form of 5 Mobile i n t e r f a c e form of 5 Liquid d e n s i t y Wall shear s t r e s s Bubble volume f r a c t i o n Bubble volume f r a c t i o n i n t h e downcomer l e g of a v e r t i c a l t e s t section

Bubble volume f r a c t i o n i n t h e r i s e r l e g of a v e r t i c a l t e s t section

Dimensi onle s s Q u a n t i t i e s

Dimens i o n l e s s cone ent r a t i o n (?!/Co)

Radial average of C,

Froude number

Peb

V2/gD)

Bubble Peclet number

(E Reb

Sc)

167 r* r*e

Dimensionless r a d i a l coordinate

( 3

r/d)

Dimensionless radius of s p h e r i c a l s h e l l of l i q u i d surrounding a bubble

1/2

@â&#x20AC;&#x2122;â&#x20AC;&#x2122; ]

Pipe Reynolds number

Bubble Reynolds number

S t i r r e r Reynolds number defined as t h e product of t h e s t i r r e r

VDp/p)

vbdp/p)

r o t a t i o n speed, square of t h e s t i r r e r diameter, and by t h e kinematic v i s c o s i t y sc

Schmidt number

Pipe Sherwood number

p/pB) (E

kD/B)

Time average of Sh

Bubble Sherwood number

Dimensionless time coordinate

Period f o r s u r f a c e renewal

kd/B) (E

tvb/d)

I-I

divided

ORNL- TM-3718 INTERNAL DISTRIBUTION 1. 2.

3. 4. 5. 6. 7. 8. 9. 10.

11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23,.

24. 25.

26. 27. 28.

29. 30.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

41. 42.

43. 44. 45. 46. 47. 48. 49.

L. J. C. H. S. E. R. E. H. R. S. D. C. H. C. E. J. L. W. J.

F. J. J. S. W. D. J. D. L. M. A.

Alexander Anderson Baes Bauman Beall S. B e t t i s Blumberg G. Bohlmann I. Bowers B. Briggs Cantor W. Cardwell J. C laffey D. Cockran W. C o l l i n s L. Compere W. Cooke T. Corbin B. C o t t r e l l L. Crowley L. C u l l e r H. DeVan R. DiStefano J. D i t t o P. E ath erly M. Eissenberg R. Engel E. Ferguson M. F e r r i s H. Fontana P. Fraas G. L. F. F. E.

J. H.

Frye

L. C. F u l l e r W. K. Furlong C. H. Gabbard W. R. Gambill W. R. Grimes A. G. G r i n d e l l R. H. Guymon R. L. Hamner T. H. Harley W. 0. Harms P. N. Haubenreich R. E. Helms P. G. Herndon D. M. Hewett H. W. Hoffman W. R. Huntley H. Inouye

50.

51. 52.

53. 54. 55.

56. 57. 58. 59-63. 64. 65. 66. 67. 68. 69. 70

71. 72. 73.

74 75 76. 77 78 79.

80. 81. 82.

J. K. Jones (K-25) W. H. Jordan P. R. Kasten R. J. Ked1 J. J. Keyes, Jr. S. S. K r e s l i s 0. H. Klepper J. W. Koger A. I. Krakoviak T. S. Kress J. A. Lane Kermit Laughon, AEC-OSR A. W. Longest M. I. Lundin R. N. Lyon H. G. MacPherson R. E. MacPherson C. L. Matthews, AEC-OSR H. E. McCoy H. C. McCurdy D. L. McElroy H. A. McLain L. E. McNeese J. R. McWherter A. P. Malinauskas A. S. Meyer A. J. M i l l e r W. R. Mixon R. L. Moore J. W. Michel F. H. N e i l 1

83 84. 85. E. L. Nicholson 86. T. S. Noggle 87. P. P a t r i a r c a

88.

A. M. Perry H. B. Pipe r 90. D. M. Richardson 91. R. C. Robertson M. W. Rosenthal 92- 93 94. H. M. Roth, AEC-OR0 95 W. K. S a r t o r y 96 Dunlap S c o t t 97 R. L. Senn 98 J. H. Sha f f e r 99. Myrtleen Sheldon 100. J. D. Sheppard 101. M. D. Silverman 102. M. J. Skinner 103. G. M. Slaughter

89.

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Norton Haberman, AEC-Washington Milton Shaw, AEC-Washington Division of Technical Information Extension (DTIE) Laboratory and University Division, OR0 Director, Division of Reactor Licensing Director, Division of Reactor Standards Executive Secretary, Advisory Committee on Reactor Safeguards