ORNL-TM-3718.txt

  

MASTRR

OAK RIDGE NATIONAL LABORATORY-,
operated by -

   

» UNION CARBIDE CORPORATION w
. NUCLEAR DIVISION
| /‘} I( for the
3 U.S. ATOMIC ENERGY COMMISSION
ORNL- TM-3718
-z MASS TRANSFER BETWEEN SMALL BUBBLES AND LIQUIDS IN
. COCURRENT TURBULENT PIPELINE FLOW
.
(Thesis)
T. S. Kress
T  CISTIEITTER OF T2 nefrp v i e

" Submitted as a dissertation to the Graduate Council of The University of Tennessee in partial fulfiliment of

the 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, compileteness or
usefulness of any information, apparatus, product or process disclosed, or
represents that its use would not infringe privately owned rights,

 

 

Ny

)
ORNL~TM-3718

Contract No. W-T7L405-eng-26
MASS TRANSFER BETWEEN SMALL BUBBLES AND LIQUIDS IN
COCURRENT TURBULENT PIPELINE FLOW

T, 5. Kress

Submitted as a dissertation to the Graduate Council
of The University of Temnessee in partial fulfillment of
the requirements for the degree Doctor of Philosophy.

  
  
  
    
  
  
   

st e N O T | C E —
j‘-'i‘.ms port i , t of wor
“This t was -prepared as an accoun -
s ans;:ép?fby tigli United States Government. I::elther
'til',le"-Uni'téd States nor the United States Atomic nergg;
Commission, nogr any of their employ§e§, n?!:'p?;\ge:s
ir OF tors, or their e ees,
| their contractors, subcontractors, e sonues any
1 mskes any warranty, express or implied, s any
-Fability - bility for the accuracy,
legal -liability -or_responsi f 2 acy, com
' ness or usefulness of any information, app2 ,
‘ g}:?deuzt bf tm disclosed, or represents that its use
APRI L 1 972 woutd not infringe privately owned rights.

 

CAK RIDGE NATTONAL LABORATORY
Oak Ridge, Tennessee 37830
operated by
UNION CARBIDE CORPORATION
for the
U, 3. ATOMIC ENERGY COMMISSION
ACKNOWLEDGMENTS

This investigation was performed at the Oak Ridge National
Laboratory operated by the Union Carbide Corporation for the U. S.
Atomic Energy Commission. The author is particularly grateful for
the helpful discussions, guidance, and direction given the research
by the major advisor, Dr. J. J. Keyes, and the support of Dr.

H., W. Hoffman, Head of the Heat Transfer-Fluid Dynamics Department of
the Reactor Division.

Dunlap Scott of the Molten-Salt Reactor Program suggested the
problem and provided initial funding under the MSR Program. Dr. F. N.
Peebles, Dean of Engineering, The University of Tennessee, suggested
the use of the oxygen-glycerine-water system and carried out the original
analysis of the applicability to xenon-mclten salt systems.

The contributions of the following ORNL staff members are also
gratefully acknowledged: R. J. Kedl for his bubble generator develop-
ment work drawn upon herej; Dr. C. W. Nestor for computer solution of
the analytical model; and Frances Burkhalter for preparation of the
figures.

Special thanks are given to Margie Adair for her skillful and
cheerful preparation of the preliminary and final manuscripts, and, of

course, to my wife, Dee, for her forbearance.

ii
ABSTRACT

Liquid-phase-controlled mobile-interface mass-transfer coefficients
were measured for transfer of dissolved oxygen into small helium bubbles
in cocurrent turbulent pipeline flow for five different mixtures of
glycerine and water. These coefficients were determined by transient
response experiments in which the dissolved oxygen was measured at only
one position in a closed recirculating loop and recorded as a function
of time, Using an independent photographic determination of the inter-
facial areas, the mass-transfer coefficients were extracted from these
measured transients and determined as functions of pipe Reynolds number,
Schmidt number, bubble Sauter-mean diameter, and gravitational orienta-
tion of the flow,

Two general types of behavior were observed:

(1) Above pipe Reynolds numbers for which turbulent inertia forces
dominate over gravitational forces, horizontal and vertical flow mass-
transfer coefficients were identical and varied according to the regression

equation

Sh/Scl’ 2 = 0,34 Re®+9%¢ (dVS/D)l‘O )

The observed Reynolds number exponent agreed genefally'with other liter-
ature data for cocurrent pipeline flow but did not agree with expectation
based on equivalent power dissipation comparisons with agitated vessel
data,

(2) Below the Reynolds numbers that marked the equivalence of hor-
izontal and vertical flow coefficients, the horizontal-flow coefficients

continued to vary according to the above equation until, at low flows,

iii
iv
severe stratification of the bubbles made operation impractical. The
vertical-flow coefficients at these lower Reynolds numbers underwent a
transition to approach constant asymptotes characteristic of the bubbles
rising through the quiescent liquid. For small bubbles in the most
viscous mixture tested, both horizontal and vertical-flow coefficients
underwént this transition.,

An expression was developed for the relative importance of turbulent
inertial forces compared to gravitational forces, Fi/Fg' This ratio
served as a good criterion for establishing the pipe Reynolds numbers
above which horizontal and vertical-flow mass-transfer coefficients were
identical. In addition, it proved to be a useful linear scaling factor
for calculating the vertical-flow coefficients in the above mentioned
transition region.

A seemingly anomalous behavior was observed in data for water
(plus about 200 ppm N-butyl alcchol) which exhibited a significantly
smaller Reynolds number exponent than did data for the other fluid mix-
tures. To explain this behavior, a two-regime "turbulence interaction"
model was formulated by balancing turbulent inertial forces with drag
forces. The relationship of the drag forces toc the bubble relative-flow
Reynolds number gave rise to the two regimes with the division being at
Reb = 2, The resulting bubble mean velocities for each regime were then
substituted intb Frossling~type equations to determine the mass-transfer
behavior. The resulting Reynolds number exponent for one of the regimes
(Reb € 2) agreed well with the observed data but the predicted exponent

for the effect of the ratio of bubble mean diameter to conduit diameter,

dvs/D’ was less than that observed. The mass-transfer equations
v

v

b

particles in agitated vessels and alsc compared favorably with the

resulting from the other regime (Re, > 2) agreed well with data for
water data mentioned above,

For comparison, a second analytical model was developed based on
surface renewal concepts and an eddy diffusivity that varied with
Reynolds number, Schmidt number, bubble diameter, interfacial condi-
tion, and position away from an interface, Using a digital computer,

a tentative numerical solution was obtained which treated & dimension-
less renewal period, T,, as a parameter. This renewal period was
interpreted as being a measure of the rigidity of the interface, I, - 0
corresponding to fully mobile and T, » approximately 2.7 (in this case)

to fully rigid interfaces,
TABLE OF CONTENTS

CHAPTER
1. INTRODUCTION v & o o o o o o o o o o« o » o
II. LITERATURE REVIEW . . . . . . .
Experimental-Cocurrent Flow . . . . .

Experimental-Agitated Vessels .A. . e e

Discussion of Available Experimental Data .

Theoretical . + ¢ ¢« ¢ ¢ ¢ 4 ¢ ¢ « o o o
Surface Renewal Models . ¢« 4 o o & o &
Modeling of the Eddy Structure . .
Turbulence Interactions . . . . .7. .
Dimensional Analysis (Empiricism) . .
DisCUuSSion ¢« v v v o o « o o s o &

ITT. DESCRIPTION OF EXPERIMENT . . . ¢« « & » & &

Transient Response Technique

Apparatus . . + + . . o ; e e e e e e
Pump . . 0 0 0 0 0 0 e
Liquid Flow Measurement . . . . «
Temperature Stabilization . . . . .
Gas Flow Measurement . .

Dissolved Oxygen Measurement . . . . .
Bubble Generation . . « ¢« « o « «+ &
Bubble Separation . « « ¢ o « o « &

Test Section v v ¢« ¢« ¢ ¢« o « o o s &

vii

"PAGE

11

11
viii
CHAPTER PAGE
Bubble Surface Area Determination — Photographic
SYSTLEM + o o + « o o o « o o o o o s o o s o v o s . 38
ENG BEFECE 4 v o v o v o o o o o o o o o o w0 0w e . W7
Summary of Experimental Procedure . . . « « o o o« o o Lg
IV, EXPERIMENTAL RESULTS . & &« « ¢ 4 o o o o o s o s o o s o 55
Unadjusted Results . v ¢ ¢« ¢ o o o ¢ o o o o o o o s o 55
Equivalence of Horizontal and Vertical Flow
Mass Transfer . o o o & o o o o s o o o o o o o o o 56
Vertical Orientation Low-Flow Asymptotes . . . . . . . 60
Mass Transfer Coefficients . & o v ¢« ¢ v o o o o o & & 60
Calculating Vertical Flow Mass-Transfer Coefficients
forFi/FgLessThanl.5 C e e e e e e e e e e e . BT
Comparison with Agitated Vessels . ¢« ¢ « o ¢ o « o o & 69
Recommended Correlations . & ¢« o o+ o o o o o o o o o« o (1
V.,  THEORETICAL CONSIDERATIONS . . &+ ¢ v o 4 o o ¢ o « « o o« 6
Turbulence Interaction Model .+ v o o v o « o « « = o« &« 76
Surface Renewal Model . & v o« ¢ o 6 o o o o s o o s o 81
VI,  SUMMARY AND CONCLUSIONS + & « v o o o o o o « o s o « o o 9h
VII. RECOMMENDATIONS FOR FURTHER STUDY . v v o « « & o o o o« o 100
Experimental . . v v & ¢ o ¢ o s s o s o o o o o s s 100
Theoretical v v v v o o « « o o o o+ o o o o o o « o o » 102
ILIST OF REFERENCES .+ 4 & &« o o o o s o s o = » s & o o o o o o o o« 105
APPENDICES s e » e 6 s s s s s s 8 s ® s e s e e e o s e s s « e ¢ 111

A PHYSICAL PROPERTIES OF AQUEOUS-GLYCEROL MIXTURES . . . . . 11l
ix
CHAPTER PAGE

B DERTIVATION OF EQUATIONS FOR CONCENTRATION CHANGES

ACROSS A GAS-LIQUID CONTACTOR . v & ¢« o o o o o » & o« o 117
C INSTRUMENT APPLICATION DRAWING . & & o ¢ o o o o « o o & » 121
D INS TRU-MEN-T CALIBRATI ONS L . * . . » . . * - . * - . . . . l 2 3

E EVALUATION OF EFFECT OF OXYGEN SENSOR RESPONSE SPEED ON

THE MEASURED TRANSIENT RESPONSE OF THE SYSTEM . . . . . 129
F MASS BALANCES FOR THE SURFACE RENEWAL MODEL ., . . +» « « « 131
G ESTIMATE CF ERROR DUE TO END-EFFECT ADJUSTMENTS . . . . . 13k
H MASS TRANSFER DATA . . . & & & ¢ o &« s o s o o o o o« o «» 137

LIST OF SYMBOLS 4 & « 4 o o o o « o o o o o « s o s o « o o o o o« 163
X

LIST OF TABLES

TABLE PAGE
1. Ranges of Independent Variables Covered . . « « ¢« + + o & 3
IT. Categories of Data Correlation for Mass Transfer from

a Turbulent Liquid to Gas Bubbles . . . « +. . + « + « & 12
ITT. Physical Properties of Aqueocus-Glycerol Mixtures (25°C)
Data of Jordan, Ackerman, and Berger®® ., . ... ... 20

IV. Experimental Conditions for Runs Used to Validate

Surface Area Determination Method for Vertical

F:LOWS . . . . . » . - . . . . . . * . . . . » . . - . s L}'L"
V. Experimental Conditions for Runs Shown on Horizontal
Flow Volume Fraction Correlation . . .« « « o« =« « o .« =« 45

VI. Conditions at Which Horizontal and Vertical Flow
Mass-Transfer Coefficients Become Equal

(Lamont ! S Data)ll . * . » . . » » - - - * - . . . - . L 5:,:}
10.

11.

LIST OF FIGURES

PAGE

Photograph of the Mass Transfer FPacility « o ¢ & o o o o & 2L
Schematic Diagram of the Main Circuit of the

Experimental Apparatus . . ¢ &« & & 4 ¢ o o o 0 e o s . 25
Diagram of the Bubble Generator . . .'. s e e e s e e e s 31
Comparison of Measured Bubble Sizes with the

Distribution Function « « « « ¢« « « o ¢ o & & o o « « o« 33
Diagram of the Bubble Separator . . « « o o « o « « « » « 30
Comparison of Interfacial Areas Per Unit Volume Measured

Directly from Photographs with Those Established

Through the Distribution Function. Vertical Flow . . . 43
Correlation of Horizontal Flow Volume Fraction . . . . . . 146
Comparison of Measured and Calculated Interfacial Areas

Per Unit Volume, Horizontal Flow . . . . « « « « « « & L8
Typical Experimental Concentration Transient Iliustrating

Straight-Line Behavior on Semi-Log Coordinants . . . . 53
Typical Examples of Bubble Photographs: a. Inlet b, Exit

Vertical Flow, 37.5% Glycerine-62,5% Water, @, = 20

gpm, Qg/QE = 0.3%, D = 2 inches, and dvs = 0,023

SNCHES &+ v o o b e e e e e e e e e e e e e e e e e 5L
Mass Transfer Coefficients Versus Pipe Reynolds Number

as a Function of Bubble Sauter-Mean Diameter. Water

Plus ~200 ppm N-Butyl Alcohol, Horizontal and

Vertical Flow in a 2-inch Diameter Conduit . . . . . . 62

X1
xii
FIGURE PAGE

12, Mass Transfer Coefficients Versus Pipe Reynolds Number

as a Function of Bubble Sauter-Mean Diameter. 12,5%

Glycerine-87.5% Water, Horizontal and Vertical Flow

in a 2-inch Diameter Conduit . . v v + &« ¢« « « « . « . 63
13. Mass Transfer Coefficients Versus Pipe Reynolds Number

as a Function of Eubble Sauter-Mean Diameter. 25%

Glycerine-75% Water. Horizontal and Vertical Flow

in a 2-inch Diameter Conduilt . . . . + ¢« ¢« ¢ ¢« & o o 6L
14, Mass Transfer Coefficients Versus Pipe Reynolds Number

as a Function of Bubble Sauter-Mean Diameter. 37.5%

Glycerine-62,5% Water, Horizontal and Vertical Flow

in a 2-inch Diameter Conduit . « & .« ¢ ¢ ¢ ¢ o ¢« « o . 65
15. Mass Transfer Coefficients Versus Pipe Reynolds Number

as a Function of Bubble Sauter-Mean Diameter. 50%

Glycerine-50% Water. Horizontal and Vertical Flow

in a 2-inch Diameter Conduilt . . ¢ ¢ o ¢ « o « o & o & 66
16, Observed Types of Horizontal Flow Behavior,

dVS = 0,02 inches and D = 2 inches . ¢ & ¢ o ¢ & & o = 68
17. Equivalent Power Dissipation Comparison of Results

with Agitated Vessel Data . & ¢ & ¢« ¢« ¢ ¢ ¢ o & o o o & 70
18, Equivalent Power Dissipation Comparison of Gravity

Dominated Results with Agitated Vessel Data . . . . . . 72
19. Correlation of Horizontal Flow Data . . . « . « . + « . . Th
20, Dimensionless Variation of Eddy Diffusivity with Distance

from an Interface, Effect of Surface Condition . . . . 88
FIGURE

21,

e,

23.
2k,

25,

26,

27
28.

29,
30,
31.
32,
33.
3k,

Variation of Eddy Diffusivity with Distance from an

Interface,

xiii

Data of Sleicher . ¢« « . ¢« ¢« ¢« o o o o o o o

Numerical Results of the Surface Rénewal Model.

of a, b,

and ¢ (Exponents on Re, Sc, and d/D,

Respectively) as Functions of the Dimensionless

Period, T* . » . - - . . . o o . . - ® - . ®

Schmidt Numbers of Glycerine-Water Mixtures . . .

Henry's Law

Constant for Oxygen Solubility in

Glycerine-Water Mixtures . « « « ¢ o o ¢ « o &

Molecular Diffusion Coefficients for Oxygen in

Glycerine-Water Mixtures.,

and Berger . & o & o o s o o ¢ o o o s s e s

Densities of Glycerine-Water Mixtures . . . « « &

Viscosities

of Glycerine-Water Mixtures . ., .

Instrument Application Drawing of the Experiment

Facility
Bubble Size
Calibration
Calibration
Calibration
Calibration

Calibration

. . » . - L ] ° . . - . - . . . .

Range Produced by the Bubble Generator
of Rotameter No, 1 (100 gpm) . . . . .
of Rotameter No. 2 (40 gpm) . . . . .
of Rotameter No. 3 (8 gpm) . . . . . .
of Gas-Flow Meter at 50 psig . . . « &

of Oxygen Sensors in two Mixtures of

Glycerine and Water . . « o« ¢ o o« ¢ ¢ o ¢ o o &

Comparison of Calculated Values with

Plots

Data of Jordan, Ackerman,

PAGE

92

112

113

114
115

116

121
123
124
125
126

127

128
FIGURE

35.

36,

37.

38.

L5,

Xxiv

Unadjusted Mass Transfer Data. Water Plus ~200 ppm
N-Butyl Alcohol. Vertical Flow . . . « « « « « « &
Unadjusted Mass Transfer Data. Water Plus ~200 ppm
N-Butyl Alcohol. Horizontal Flow . . « ¢« ¢ ¢ « o« &
Unadjusted Mass Transfer Data. 12.5% Glycerine-
87.5% Water, Vertical FIOW « v « o o o o« o « & o &
Unadjusted Mass Transfer Data. 12.5% Glycerine-
87.5% Water. Horizontal FIOW . . v « « ¢ « & o o
Unadjusted Mass Transfer Data., 25% Glycerine-75%
Water., Vertical FIOW . v & o o ¢ o o o o o o o & o
Unadjusted Mass Transfer Data. 25% Glycerine-T5%
| Water., Horizontal FIlOow ¢ o o v ¢« o o o &+ o o o & &
Unadjusted Mass Transfer Data. 37.5% Glycerine-
62.5% Water. Vertical FIOW v v v o o o o o« o o o« &
Unadjusted Mass Transfer Data, 37.5% Glycerine-
62,5% Water, Horizontal FIOW « v « v o o o o « o
Unadjusted Mass Transfer Data. 50% Glycerine-50%
Water., Vertical FIOW . . o & o« ¢« « ¢ o o o &
Unadjusted Mass Transfer Data. 50% Glycerine-50%
Water., Horizontal FIlOow . « ¢ « v v ¢ o o o o o o &
Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. Water Plus ~200 ppm N-Butyl Alcohol.

Horizontal and Vertical FIOW .+ & o o o « o s o s »

PAGE

138

139

140

141

142

143

1hk

146

1h7

148
FIGURE
L6,

47.

L.

50.

51,

52.

53.

XV

Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. 12.5% Glycerine 87,5% Water. Horizontal
and Vertical FI1ow . . ¢ o ¢ « « ¢ ¢ o« o o o o o o o

Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter, ?5% Glycerine-75% Water. Horizontal
and Vertical FIOW ., o o & ¢ ¢ ¢ o o o o s o o o

Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. 37.5% Glycerine-62,5% Water. Horizontal
and Vertical Flow , . + o ¢ ¢« « « « &

Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. 50% Glycerine-50% Water. Horizontal and
Vertical FI1ow . o ¢ v v ¢« &« o o o o o « &

Mass Transfer Data Adjusted for End-Effect. Water Plus
~200 ppm N-Butyl Alcohol., Vertical Flow . . . . .

Mass Transfer Data Adjusted for End-Effect. Water Plus
~200 ppm N-Butyl Alicohol, Horizontal Flow . .

Mass Transfer Data Adjusted for End-Effect. 12,5%
Glycerine-87.5% Water, Vertical Flow . . . . « . .

Mass Transfer Data Adjusted for End-Effect, 12,5%

Glycerine-87.5% Water. Horizontal Flow . . . . . .

PAGE

149

150

151

152

153

154

155

156
FIGURE

5k,
55,
56,

57

58.

Xvi

Mass Transfer Data Adjusted for End-Effect.

Glycerine-75% Water. Vertical Flow .

- Mass Transfer Data Adjusted for End-Effect.

Glycerine-75% Water. Horizontal Flow .

Mass Transfer Data Adjusted for End-Effect.

Glycerine-62,5% Water. Horizontal Flow .

Mass Transfer Data Adjusted for End-Effect,
Glycerine-50% Water, Vertical Flow .
Mass Transfer Data Adjusted for End-Effect.

Glycerine-50% Water. Horizontal Flow .

PAGE
25%
s e e e e . . 157

25%

e 516
37.5%
e o o . 159
50%
e e e e . . 160
50%

161
CHAPTER T
INTRODUCTICN

When gas bubbles are dispersed in a continuous liquid phase,
dissolved constituents of sufficient volatility will be exchanged between
the liquid and the bubbles, effectively redistributing any concentration
imbalances that exist. Common pradtices involve contacting gas bubbles
with an agitated liquid in such a manner that a relatively large inter-
facial area is available, Techniques such as passing gas bubbles up
through a liguid column or mechanically stirring a gas-ligquid mixture in
a tank have been studied extensively and the design technology for these
is relatively firm., However, one method, cocurrent turbulent flow in a
pipeline, has not been given a great deal of attention. A review of the
literature has shown that the availlable data are insufficient to‘allow
confident determination of the mass-transfer rates in such a system.
This research, then, was undertaken to provide additional information
that will aid in determining liquid phase controlled mass-transfer rates
for cocurrent turbulent flow of small bubbles and liquids in a pipeline.

The impetus for this work was provided by the Molten Salt Breeder
Reactor (MSBR) Program of the Oak Ridge National Laboratory where recent
remarkably successful operation of a molten salt fueled nuclear reactor!
has convincingly demonstrated the feasibility of this power system. The
economic éompetitiveness of an MSBR, however, depends to a significant
extent on the breeding ratio obtainable, The production within the

liquid fuel of fission-product poisons, principally xenon-135, exerts
2
a strong influence on the neutron economy of the reactor and consequently
on the breeding ratio itself,

One method proposed for removing the xenon would require injecting
small helium bubbles into the turbulently flowing regions of the fuel-
coolant stream and allowing them to circulate with the fuel. Since such
bubbles would be deficient in xenon compared to the nearby bulk stream,
the dissolved xenon would be transferred by turbulent diffusion across
the concentration potential gradient. By continuous injection and
removal of the helium bubbles the equilibrium xenon poisoning can be
significantly reduced. Since a large amount of gas in the fuel could
influence the reactivity of the core, this system would be limited to
low volume fractions.

Peebles® showed that removal of dissolved oxygen from a given mixture
of glycerine and water by small helium bubbles could closely match the
hydrodynamic and mass-transfer conditions in an MSBR and suggested using
such a system in a similitude experiment from which the actual MSBR
behavior might be inferred. Other desirable features of such a system
include: (1) convenient variation of the Schmidt number by using differ-
ent percentages of glycerine in water, (2) operation at room temperature
using glass hardware that allows photographic measurements through an
optically clear system, and (3) easy measurement of the dissolved oxygen
content by commercially available instruments. Therefore an oxygen-
glycerine-water system was chosen for this study.

The objective of the program was to measure liquid phase controlled
axially averaged mass-transfer coefficients, k, defined by

rL

k dx
X
L .

 

k =
3

The local mass-transfer coefficients, kx’ are defined by

J=k a [Cavg —-CS] s
where J is the mass transferred from the liquid to the bubbles per unit
time per unit volume of liquid, a is the interfacial area per unit volume
of liquid, Cavg is the bulk average concentration, and CS is the inter-
facial concentration,

These coefficients need to be established as a function of Schmidt
number, Reynolds number, bubble size, conduit diameter, gravitational
orientation of the flow (vertical or horizontal), interfacial condition
(absence or presence of a surface active agent), and the volume fraction
of the bubbles. The scope of this thesis is limited to the ranges of
variables listed in Table I, below, which for the most part represent
limits of the experimental apparatus. Extensions of this program, how-
ever, are projected to include different conduit diametersland different

interfacial conditions.

Table I. Ranges of Independent Variables Covered

 

 

Variable Range

Schmidt Number (weight percent of glycerine) 370 - 3446

(O, 12-5, 25, 3705) 50)
Pipe Reynolds Number 8 x 10® - 1.8 x 10°
Bubble Sauter Mean Diameter 0.01 to 0,05 inches
Gas to Liquid Volumetric Flow Ratio 0.3 and 0.5 percent
Gravitational Orientation of Flow Vertical and Horizontal
Conduit Diameter 2 inches

 

The mags~transfer coefficients were extracted from measurements of
the coefficient-area products, ka, and independent photographic measure-

ments of the interfacial areas per unit volume, a. The products, ka,
L

were established by means of a unique transient response technique in
which the changes in liquid phase concentration were measured as a func-
tion of time at only one position in a closed liquid recirculating system
while helium bubbles were injected at the test section entrance and
removed richer in oxygen at the exit. The apparatus for generating
these small bubbles (with an independent control of their mean sizej and
effectively separating a high percentage from the flowing mixture had to
be developed prior to the start of this research., These are described in
Chapter III along with the photographic equipment and technique for estab-
lishing the interfacial areas.

The fesults of this study are expected to be of immediate benefit
to the MSBR Program and should also prove useful to workers in the
general chemical industry. Application may extend to such diverse areas
as general extraction of radiocactive elements from reactor effluents,
bubble lifetimes in the coolant of liquid metal fast breeder reactors,
and oxygen treatment of sewage effluents, In addition, benefi%s of a
fundamental nature may be derived in that the research concerns transfer
of a scalar in a turbulent shear flow field in which the fluid velocity
field effectively seen by the bubbles is primarily due to the turbulent
fluctuations. The characteristics of mass transfer between dispersed
bubbles and a continuous liquid phase in turbulent flow are thus seen

to be of immediate scientific and practical importance,
CHAPTER IT
LITERATURE REVIEW

A comprehensive survey was made of literature related to mass
transfer between small bubbles and liquids in cocurrent turbulent flow.
An exhaustive review of all this literature would be lengthy and some-
what pointless. Consequently, only those works that are considered
representative of the field (not necessarily of most significance) are
included in this chapter and the author intends no derogation or slight-
ing by the omission of any work, No significance should be attached to
the order in which references appear. For a fairly complete documenta-
tion of work related to this subject, the reader is referred to several

excellent review articles.2 ®

Experimental-Cocurrent Flow

There have been very few direct measurements of mass-transfer
coefficients for cocurrent turbulent flow of small gas bubbles and
liquids, perhaps because substantial special apparatus seems to be
required for thesé measurements. Recently Jepsen® measured the liguid
phase controlled product of mass-transfer coefficient, k, and inter-
facial surface area per unit volume, a, for air/water flow in horizontal
pipes with and without spiral turbulence promoters. For straight tubes
fiithout turbulence promoters he correlated his data by the equation,

kas®l 2 gV 2 (008 po-68 _ 3 47 evo.4 .
As shown in Chapter IV, Page 58, the energy dissipation per unit

volume, ¢, can be represented as
v
(o 316) = B paire (1)

Therefore, Jepsen's correlation reveals that
Sh ~ Re'*l/§ 2a

Care must be taken in interpreting the influence of Reynolds number
on k when the product, ka, is reported because the interfacial area it-
self may depend on the Reynolds number. No attempt was made by Jepsen
to separate the area from the product.

Scott and Hayduk,'© in admittedly exploratory experiments, dissolved
carbon dioxide and helium into water, ethanol, and ethylene glycol in
horizontal flow pipelines. Thus they did vary the diffusivity but, like
Jepsen, did not separate the ka product.

Their results were correlated by the equation

ka = 0.0068 v 3074 S°-51 Mo.oia &30
Dl.SB 3

 

from which may be inferred
Sh ~ Re/&*%ta .,

Lamont!! and Lamont and Scott® dissolved, in single file fashion,
relatively large CO, bubbles into water under vertical and horizontal
flow conditions. They did not vary bubble diameter or Schmidt number.
At sufficiently large Reynolds_numbers their horizontal and vertical
results became identical. The data above these Reynolds numbers were
correlated as

k ~ Re©+B2
Heuss, King, and Wilke'® studied absorption into water of ammonia

and oxygen in horizontal froth flow, The liquid phase coefficients were
T
controlling only in the oxygen runs and consequently they did not vary
the Schmidt number and their results were also obtained as the product
of ka. However, using estimates of surface area in froth flow, their
data reveal
Sh ~ Re®*® .,

Hariott'? reported mass-transfer coefficients for particles of
boric acid and benzoic acid dissolving in water flowing cocurrently in
a two-inch pipeline, A data correlation was not given but a line tan-
gent to their data at the high flow end would indicate

Sh ~ Re®*93 |

Figueiredo and Charles® measured coefficients for dissolution of
NaCl particles carried along as a "settling" suspension in water in
horizontal flow., They correlated their data with mass-transfer coeffi-
cients previously measured for transfer between a liquid and the conduit
itself. However, a line tangent to the high flow end of their data
indicates

Sh ~ Ret*® .

Experimental-Agitated Vessels

Often the data for transfer to bubbles or particles in agitated
vessels are correlated in terms of the power dissipated. Using Equation
(1) we might relate these results to what would be expected for flow in
conduits,

Calderbank and Moo-Young'® correlated data for different particles
and small bubbles dispersed in different liquids in agitated vessels,

Their equation, determined partly through dimensional analysis, is
Using Equation (1) this would give for flow in conduits
Sh = 0,082 gel/ 3 Re®+52 (2)

They also indicate that in the range of mean bubble diameters, 0.025
< dvs’g 0.1 inches, the mass-transfer coefficients increase linearly,
undergoing a transition from "small" bubble behavior where Sh ~ Sc¥ 3 to
"large" bubble behavior where S8h ~ Sc¥ 2. They conclude that this tran-
sition corresponds to a change in interfacial condition from rigid to
mobile,

Sherwood and Brian'”’ used dimensional analysis to correlate data for
particles in different agitated liquids. Their correlation graphically
related Shb/801/3 to (emd4/v3)1/3. Using Equation (1) (with ev/p = em)
and drawing a line tangent to the high power dissipation end of their
correlating curve gives

Sh -~ SC1/3 ReC* 61 (d/D)-o.le . (3)

Barker and Treybal'® correlated mass-transfer coefficients for boric
acid and benzoic acid particles dissolving in water and 45% sucrose solu-
tions with a stirrer Reynolds number, ReT, proportional to the speed of
rotation., They reported

K ~ ReTC"88 Sct/ 2 g

If the power dissipation is assumed proporticnal to the cube of the

rotation speed, then

k ~ ReC*76 Scl/:a S .
The effect of Schmidt number is not as would be inferred from the zbove
because § was reported to be essentially proportional to Scfl/z in their

experiments.
9
The preceding are representative of data available that may have
direct applicability to cocurrent flow in conduits., Some other works
that may be of indirect interest include cocurrent turbulent flow of

19-22 mass transfer

dispersed liquid drops in a continuous liquid phase,
from a turbulent liquid to a free interface,®372® and innumerable studies
of the motions of, and mass transfer from, individual bubbles or parti-
cles under steady relative flow conditions (e.g., References 26-30).

For systems in which bubbles move steadily through a fluid, some
relevant findings include the fact that, depending on bubble size and
liquid properties, the bubble motion in a gravity field may vary from
creeping flow to flow characterized by a turbulent boundary layer.
Irrespective of this, the mass-transfer correlations usually take two
basic "Frossling" forms (neglecting the constant term) depending on
whether there is a rigid interface (no slip condition) or a completely
mobile interface with internal circulation of the fluid within the
bubble (or drop). In substantial agreement with theoretical treatments,
the former data are correlated with

Sh’b ~ Rebl/ 2 g01/ 3
and the latter
Sh, ~ Re, > ® 8¢/ % =pe V2 |

Good accounts of these relative flow equations and their derivations are

given by Lochiel and Calderbank®' and by Sideman. =%
Discussion of Available Experimental Data

It is seen that there have been very few direct measurements of

mass transfer to small cocirculating bubbles in a turbulent field and
10
none that are complete in terms of all the independent variables. The
product, ka, is often not separated, because of the difficulty in estab-
lishing the interfacial area, This makes some of the available data
difficult to interpret and of limited value for application at different
conditions.

Not enough experimental information is available to assess the
influence of Schmidt number on Sherwood number although the Schmidt
nunber exponent most often appears to vary between 1/3 and 1/2 —
apparently determined by the interfacial condition (the Schmidt number
exponent may even be greater than 1/2, e.g., Reference 10).

The effects of bubble mean diameter and pipe size have received
less attention than the Schmidt number and, as yet, no systematic effect
can be confidently cited. Calderbank and Moo-Young, however, observed a
linear dependence over a limited range of bubble diameters in agitated
vessels,

The influence of Reynolds number has been the most studied. From
References 9-15, it‘yould appear that Sherwood number for gas-liquid
flow in conduits may vary with pipe Reynolds number to a power between
0.9 and 1.1 (although Lamont® found it to be 0.52). In contrast, the
effect of Reynolds number {turbulence level) in stirred vessels (Refer-
ences 16-18) would appear to yield a power between 0.6 and 0.8, This
apparent difference between agitated vessels and flow in conduits is
surprising because one would think that flowing through a closed con-
duit is Jjust another way to stir the liquid. There should be little

fundamental difference in the effect of the turbulence produced.
11

Theoretical

It is convenient to identify four different analytical approaches
designed to provide a description of mass transfer to bubbles from a
turbulent liquid that may be applicable to cocurrent flow. The author
has chosen to name these (1) Surface Renewal, (2) Turbulence Interac-
tions, (3) Modeling of the Eddy Structure, and (4) Dimensional Analysis
(Empiricism). These do not necessarily encompass all approaches and
there may be considerable overlappifig among areas (for example, a cer-
tain degree of empiricism is evident in each). There may be only an
indirect equivalence among those within a given category.

Some representative works have been categorized according to their
approach and listed in Table II. A brief discussion of each category is

given below,

Surface Renewal Models

 

This category is of considerable historical interest especially the
original contributions of Higbie®? and Danckwerts.33

The so-called surface renewal models can be envisioned by imagining
the interface as being adjacent to a semi-infinite fluid through which
turbulent eddies having uniform concentrations characteristic of the
continuous phase, periodically penetrate to "renew" the surface. The
mass transfer then depends on the rate and depth of eddy penetration and
the eddy residence time near the surface (or the distribution of eddy
ages). For a given eddy, the original models are essentially solutions

of the diffusion equation

2
L= p2s ()
Table II., Categories of Data Correlation for Mass Transfer
from a Turbulent Liquid to-Gas Bubbles

 

1., Surface Renewal

Brian and Beaverstock (40)2
Danckwerts (33)

Davies, Kilner, and Ratcliff (k1)
Gal-Or, Hauck, and Hoelscher (L2)
Gal-Or and Resnick (L43)

Harriot (LL)

Higbie (32)

King (25)

Koppel, Patel, and Holmes (L5)
Kovasy (L6)

Lamont and Scott (12)

Perlmutter (L47)

Ruckenstein (48)

Sideman (49)

Toor and Marchello (34)

3. Modeling of the Eddy
' Structure

Banerjie, Scott, and Rhodes (51)
Fortescue and Pearson (23)
Lamont (11)

2., Turbulence Interactions

Boyadzhiev and Elenkov (19)
Harriot (50)

Kozinski and King (2kL)
Levich (36)

Peebles (2)

Porter, Goren, and Wilke (20)
Sideman and Barsky (21)

4, Dimensional Analysis
(Empiricism)

Barker and Treybal (52)
Calderbank and Moo-Young (16)
Figueiredo and Charles (15)
Galloway and Sage (53)

Heuss, King, and Wilke (13)
Hughmark (54)

Middleman (38)

Scott and Hayduk (10)
Sherwood and Brian (17)

 

a
Reference number.
13

As shown by Toor and Marchello,®* the "film" model first introduced
by Whitman®® corresponds to the asymptotic solution of this equation at
long times (no surface renewal) where k would be proportional to £ and
Sherwood number would be independent of Schmidt number. The "penetra-
tion" model first introduced by Higbie®® and later extended by Danckwerts®®
corresponds to the asymptotic solution at short times where k would be
proportional to &/ 2 or Sh ~ Sc 2. Depending on the distribution of
contact times between the eddies and the surface, the transfer may take
on characteristics of either or both of the above.

King®® generalized this approach to include turbulence effects by

replacing Equation (4) with

Q/

C

[(8+u) &1,

o
oqu
e

where o is an eddy diffusivity which he arbitrarily let vary with dis-

tance from the surface as

This model approaches the same asymptote (Sh ~ Sc*/ ?) at short times but
different asymptotes at long times depending on the value of b (with b =
3, Sh~ 8c°"3%; with b = 4, Sn ~ 8c°°%9),

To establish an overall mass-transfer rate, it is necessary to
assign a frequency with which the surfaces are "renewed" {or the distri-
bution of eddy ages). The different extensions and modifications of
this model mostly involve the choice of different functions to describe
the randomness of the eddy penetrations, None of these models give
significant information as to the effect of bubble size, conduit size,

or Reynolds number. They are mechanistically unsatisfactory because the
1k

hydrodynamic effects are often ignhored or included by relating the eddy
age distribution in some way to the flow field. For example, Lamont and
Scott’® assumed that the fractional rate of surface renewal, s, (k ~'VQE3
is given by

s ~ Re J? .
There is really no clear-cut way to establish a relationship between the
rate of surface renewal and the hydrodynamics and, consequently, there
is a heavy reliance on empiricism. The original intent of these models
was to describe transfer to a surface (bubble) that has a distinct steady

flow relative to the liquid.

Modeling of the Eddy Structure

If the fluid velocity field in the vicinity of the interface could
be completely described, then the computation of transfer rates, in
principal, would be straightforward. However, at the present time,
there are no satisfactory descriptions of the details of a turbulent
veloclty field and even if such were available, the mathematical account-
ing of the differential transfer processes might become intractable.
Consequently, there have been idealizations of the eddy structure with,
admittedly, unrealistic fields and mass-transfer behavior has been conm-
puted based on these idealizations.,

Lamont's work'® provides an excellent example of this approach. He
modeled the eddy structure by considering individual eddy cells that have
a sinusoidal form at a sufficient distance away from the interface
(corresponding perhaps to an individual component of a Fourier decompo-
sition of the turbulent field). As the interface is approached, viscous

forces dampen the eddy cell velocities by an amount that depends on the
15
interfacial condition (mobile or rigid). Lamont calculated the mass-
transfer coefficient for an individual eddy cell as a function of the
daéfiing condition, fluid properties, the wave properties, and the eddy
energy. He then used a Kovasznay distribution function for the energy
spectrum and summed over a range of wave numbers to obtain the overall
coefficient, The results of this procedure were
k ~ (SC)—l/ 2 (emv)l/ 4
for a mobile interface and
k ~ (Sc)~¥3 ufiflfl4
for a rigid interface, |
Using Equation (1), these give
gh ~ Scl/ 2 Re-89
and
Sh ~ Scl/ 3 Ref- 69 ,
respectively.
The present writer feels that this approach may represent a bridge
between surface renewal models and turbulence theory and as such deserves

particular mention,

Turbulence Interactions

 

Some authors have attempted to analyze the forces and interactions
between spheres and fluid elements in a turbulent field to arrive at
equations for the fluctuating motion of the spheres. These equations
are solved to obtain a "mean" relative velocity between the bubble and
the fluid which is then substituted into a steady-flow equation (usually
of the Frossling type) to establish the mass-transfer coefficients. The

work of Levich®® is of this nature and Peebles® used this approach in
16

his document. For example, Peebles used the result of Hinze®’ for small

gas bubbles

FrE=3.hF

g
which essentially comes from an integration of the equation

dv dv
d 2 g rd? L

where W is an added mass coefficient for an accelerating spherical

bubble. The relative velocity is then

Peebles used the approximations

A/'vflz ~ V JF/2 8nd £ ~ Re”l/ B

in the above which were then substituted into Frdssling type equations
to obtain Sh ~ Re®*%® 8¢/ 2 (4/D)"* 2 for a mobile interface and Sh ~
Re®*%® get @ (a/D)™Y 2 for a rigid interface.

In a similar computation which included Stokes law to describe the
drag experienced by the bubble, Levich®® obtained for a mobile interface

Sh ~ Re¥ % gel/ 2 |

Dimensional Analysis (Empiricism)

 

some workers have chosen to postulate the physical variables that
may be controlling and have used standard dimensional analysis techniques
for ordering the experimental data, The paper by Middleman38 is a
splendid example of this approach as applied to agitated vessels. Also
for agitated vessels, Calderbank and Moo-Young'® used dimensional
anaiysis to obtain Equation (2) and Sherwood and Brian'”’ dimensionally

related shb/Sc:l/B to [emd4/v3 1 s,
17

Also included under this category is a most interesting correlation
by Figueiredo and Charles!® for a heterogeneous pipeline flow of settling
particles. They used an expression for the ratio of pressure gradient
for flow of the suspension to the pressure gradient for flow of the
liquid alone and assumed that, if altered by the ratio d/D, it could
also represent the ratio of mass-transfer coefficients for the particles
to those for transfer from the liquid to the pipe wall. They found that
they could, indeed, use this ratio to correlate their data for a settling

suspension with the data of Harriot and Hamilton.3°

Discussion

It is seen that the theoretical description of mass transfer to
bubbles in cocurrent turbulent flow has by no means been standardized.
There seems to be somewhat general agreement as to the effect of Schmidt
number, The Sherwood numbers for cases of completely rigid interfaces
with zero tangential velocity at the surface (no slip) applicable to
solid spheres, very small bubbles, and bubbles with surfactant contami-
nation in the interface are generally predicted to vary with Schmidt
number to the one-third power. Completely mobile interfaces (negligible
tangential stress with non-zero interfacial velocity) are generally pre-
dicted to yield a Sc'/ ? variation of Sh.

There is only scant and inconsistent information predicting the
effects of bubble and conduit diameter. For example, Levich predicts
no effect of d/D while Peebles predicts Sh ~ (d/D)"Y =,

There is general disagreement as to the effect of Reynolds number

as evidenced by the fact that exponents have been predicted that range
18
from 0,45 to > 1, These different exponents may not be mutually exclusive
however because an inspection of the experimental data shows disagreement
in the measured exponents also, It may be that the proper application of
these equations depends on suitable evaluation of the conditions of the

experiment.
CHAPTER IIT
DESCRIPTION OF EXPERIMENT

This experiment was designed to measure liquid phase controlled
mass-transfer coefficients for cocurrent pipeline flow of turbulent
ligquids with up to 1% volume fraction of small helium bubbles having
mean diameters from 0.0l to 0,05 inches. The liquids chosen were five
mixtures of glycerine and water (0, 12.5, 25, 37.5, and 50% by weight
of glycerine) each of which represent a different Schmidt number. The
physical properties of these mixtures, obtained from the literature,E’5
are shown graphically in Appendix A and the values used in this study

for the given mixtures are listed in Table IIT.
Transient Response Technigue

A closed recirculating system was used in which helium bubbles
were introduced (generated) at the entrance of a well-defined test
section and removed richer in oxygen at the exit, allowing only the
bubble-free liquid to recirculate,

The products of mass-transfer coefficients and interfacial areas
were measured by a transient response technique in which the system was
initially charged with dissolved oxygen. The oxygen was then progress-
ively removed by transfer to the helium bubbles while the oxygen concen-
tration was continuously monitored as a function of time at a single
position in the system.

For a test section of length, L, and cross-sectional area, A, it

can be shown (Appendix B) that the ratio of exit concentration to inlet

19
Table III. Physical Properties of Agqueous-Glycerol Mixtures (25°C)
Data of Jordan, Ackerman and Berger®®

 

 

Molecular
Glycerol Henry's Law Diffusivity of Schmidt
Content Density, p  Viscosity, u Constant, H Oxygen, & x 10° Modulus, Sc
(wt %) (1b/££>) (1b/ft ohr) (atmeliters/mole) (£t2/hr) (Dimensionless)

0 62, 43 2.15 795, 1 8.215 hi9

12.5 6L, 43 3,07 1127.6 12.865 370

25 66, 49 L, 62 1421,0 9, 261 750

37.5 67.67 6, 3k 1621, 8 L, 650 2015

50 69. 86 10.82 2011.1 4,495 3446

 

Oc
21
concentration, Ce/ci’ is a constant, K, given by

-

+ e
K = Ce/Ci = 3%71;17— ’ (5a)

where

RTQ
4 and B = kaAL(1l + v)

3, . (5b)

 

In the absence of axial smearing, each time the fluid mekes a com-
plete passage around the closed circuit (loop transit time, v, = VS/QE)
the concentration at the measuring position would (ideally) decrease
instantaneously from its value, C, to a value equal to KC. Therefore,
in actuality, the ratio, C/CO, of the concentration at any time to that
at an initial reference time {set equal to zero) would be given by

c/C_ = Exp IZ ——L—(‘Q’”f t } = Exp ;—— -————(@ns)%t } :
L L S .
Therefore a plot of @?(C/CO) versus time would be a straight line of
slope -(&EK)Qz/VS. Note that the absolute value need not be measured
because a signal that is merely proportional to the oxygen concentration
would have the same slope. If the system volumé, VS, and the liquid

volumetric flow rate, Q,, have been measured, the constant K can be

E’
extracted from the slope of the measured transient. Having a measure
also of gas volumetric flow, Qg, and the system absolute temperature, T,
and knowing R, Hy A, and L, the product, ka, can be obtained from K

through Equations (5). If an independent measure is also made of "a,"

then the mass-~transfer coefficients are fully determinable,
22

This technique was selected as being superior to a once-through
test that requires an independent measurement of the oxygen concentra-
tion at both ends of the test section for reasons illustrated by the
following comparison.

Tn a once-through system with a 37.5% mixture and conservative
values of Qg/Qz = 1%, bubble mean diameter = 0,01 inches, Reynolds
number = 6 x 10%*, and a mass-transfer coefficient of 0.7 ft/hr, a test
section length of ~100 feet would be required to obtain a concentration
change across the test section of only'

ce/ci = 0.9 .

At this level a small error in the concentration measurement would be
magnified in the determination of ka. In contrast, the same conditions
in a transient test with only a 25-~ft-long test section would give a
concentration change of C/Cofv O.1 in only about seven minutes — greatly
reducing the error magnification in ka. In return for this benefit, the
values of total system volume, Vs’ and the time coordinate, t, need also
to be measured, These, however, are parameters that can be measured
very precisely compared to the concentration measurement. Therefore,
the transient tests should result in more reliable data.

On the other hand, the concentrations in once-through tests are
measured at specific locations that bracket the region of interest and
only the transport behavior within that region is important. Whereas in
the transient tests all mass transfer occuring outside the test section
ig extranecus and represents an "end effect" contribution that must be
independently measured and accounted for in determining the "ka'" product.

This "end effect," which would include mass transfer occuring in the
23
bubble generating and separating processes, represents the most serious
disadvantage and error source in the transient measurementé. The
measurement and accounting for the "end effect” are discussed further

on page 47,
Apparatus

In constructing the main circulating systems of the experiment
exclusive use was made of stainless steel or glass hardware and all
gaskets were Teflon. This was part of careful measures taken to keep
the system free of contamination. Figure 1 is a photograph of the
facility with the test section mounted in the vertical orientation and
Figure 2 is a diagram of the main circuit portion. Figure 28, page 121
(Appendix C) is an instrument application drawing of the system which
includes an auxiliary flow circuit used for rotameter calibration and
for special tests,

The main circuit consisted of a canned rotor centrifugal pump,
three parallel rotameters, a heat exchanger, three dissolved oxygen
measuring sensors, a helium flow and metering system, a bubble generator,
the test section, a bubble separator, a photographic arrangement for
determining the bubble interfacial areas and mean diameters, and a
drain-and-fill tank equipped with scales for precise determination of
the weight percent of glycerine in the mixture., Further descriptions

of individual components are given below.

Pump
The main circulator was a 20 HP Westinghouse "100-A" canned rotor

constant speed centrifugal pump capable of delivering about 100 gpm at
 

  
  

s

Figure 1.

   

   

   

 

 

 

 

 

 

 

 

 

 

 

Photograph of the Mass Transfer Facility.
FILL AND
DRAIN TANK

 

 

 

 

_SCALES

ORNL-DWG 70-11420

~=— TEST SECTION ~ 36 ft LONG
VERTICAL ORIENTATION

 

~PHOTO
PORTS

CAMERA

     
    
 
  
     
 
  
 

FLASH

g;)LOOP PRESSURE

BUBBLE
GENERATOR

INLET

 

(LOOP PRESSURE REGULATION
FROM HELIUM BOTTLE)

BUBBLE
SEPARATOR

 

2 in. PIPING
/- =

 

 

HEAT
EXCHANGER

Vol

 

 

 

 

 

LOOP
TEMPERATURE

 

   
 

  

120 40
gpm gpm  gpm

CANNED
ROTOR
PUMP

MSBR-Mass Transfer Loop.

Figure 2. Schematic Diagram of the Main Circult of the Experimental
Apparatus.

G2
26

about 180 feet of head, The motor cannings, housing, and impeller were
stainless steel and the bearings were graphitar — lubricated solely by
the loop fluid. An auxiliary circuit required to cool the pump motor
windings circulated transformer oil through the windings and through an
external circuit containing an auxiliary oil pump, a filter, and a small
water-cooled heat exchanger. The pump was safety instrumented to turn
off on loss of pressure in the oil circuit or on high temperature of

the motor housing.

Liguid Flow Measurement

The liquid flow rate was controlled by three parallel stainless
steel globe valves downstream of the pump at the entrances to the rota-
meters. Three parallel rotameters of different capacities (100, Lo,
and 8 gpm) were used for measuring liquid volumetric flow rates. By
judicious use of the rotameter scales, parallel rotameters provide
greater precision when measurements are required over a wide flow range.
In each experiment, however, some flow was allowed to go through each
rotameter to prevent having regions that might "lag" the rest of the
loop during the transient tests and thereby become concentration
"capacitance" volumes.

Because of the large differences in viscosities over the range of
glycerine-water mixtures used, the rotameters were calibrated, in place,
for both water and a 50% mixture. These calibrations were obtained by
the use of two identical 6-inch-diameter, 6-feet-long glass tanks in
the auxiliary circuit valved together in such a way that, while one was
being fillied, the other was being drained. Changing the position of one

lever reversed the pfocess before the liquid could spill cver the top.
27
The time required to fill (or empty) a known volume of either of these
tanks was measured over the entire range of each rotameter. These cal-
ibrations are given in Appendix D.
Since there was only a small.difference in the‘calibration between
0 to 50% glycerine, the flow for in-between mixtures was determined by

linearly interpolating between the two curves according to the viscosity.

Temperature Stabilization

The fluid temperature was measured at the inlet and exit of the test
section by standard stainless steel sheathed chromel-alumel thermocouples
immersed in the fluid., The friction and pump heat were removed and the
test section temperature held at 25°C for all tests by a stainless-steel,

water-cooled, shell-and-tube heat éxchanger.

Gas Flow Measurement

Helium for generation of the bubbles was obtained from standard
commercial cylinders metered through a pressure regulator, a safety
relief valve, and a flow control needle valve. The rate of helium flow
was determined by measuring both the exit pressure and the pressure drop
across a 6-foot length of tubing of about l/l6-inch internal diameter.
These‘measurements were made with a Bourden type pressure gage and a
water-filled U-tube manometer, respectively.

Calibration at atmospheric conditions was obtained prior to opera-
tion by comparing with readings from a wet-test meter timed with a stop
watch., The calibration at 50 psig exit pressure (normal operating con-

dition) is given in Figure 33, page;lET(Appendix D), The calibration
28
and the leak tightness of this system were checked periodically over

the course of the experimental program.

Dissolved Oxygen Measurement

Two identical commercially available "Polarographic" type instru-
ments were used to measure the dissolved oxygen concentration (Magna
Oxymeter Model 1070, Magna Corporation-Instrument Division, 11808 South
Bloomfield Avenue, Santa Fe Springs, California). Two were used so
that an automatic continuocus check was provided by comparing the readings
of one with the other. It was felt unlikely that both would use up
their electrolyte or fail simultaneously. These Instruments used polar-
ographic type sensors inserted into the flowing liquid through penetra-
tions in tees provided for that purpose. Electrical signals produced
by the sensors were fed through recording adaptors and the resulting
millivolt signals recorded on a Brown Multipoint recorder having a
measured chart speed of 1.18 inches/sec,

Bach sensor assembly consisted of an electrolytic cell made up of
a cathode, anode, and an electrolyte mounted in a plastic cylindrical
housing., The end of the housing, containing the cell, was encased in
a thin oxygen-permeable Teflon membrane which also acted to contain the
electrolyte. The dissolved oxygen is electrolytically reduced at the
cathode causing a current to flow through the system from cathode to
anode., The magnitude of this current is proportional to the oxygen
concentration if sufficient liquid velocity exists (~2 ft/sec) to pre-

vent concentration polarization at the membrane,
29

The response times for these instruments are greater than 0% in
30 seconds. An analysis showing that this response produces an accept-
able error in the transient tests is given in Appendix E.

Since the transient response technigque used in these tests requires
a signal that is merely proportional to the oxygen concentration, an
absolute calibration of these instruments was not necessary. Neverthe-
less calibration tests were made for two different mixtures of glycerine
and water by bubbling air through the mixtures at different pressures
until they became saturated. Knowing the solubility of oxygen in the
mixtures, the meter reading could be set on the calculated concentration
for an initial "set-point" pressure and subsequent readings at different
pressures compared with calculated values (assuming a Henry's Law rela-
tionship). Calibrations obtained in this manner are shown on Figure 34,
page 128 (Appendix D) which includes readings made with a third instru-
ment similar to the Magna instruments but made by a different company.
The response speed of this third sensor proved to be slow compared to
the Magna sensors and consequently it was used only as an independent
monitor on the operability of the Magna sensors throughout these experi-

ments,

Bubble Generation

 

Special apparatus was required that could generate a dispersion of
small bubbles whose mean size could be controlled énd varied over the
range 0,01 to 0.05 inches independently of the particular liquid mixture
beifig used and of the flow rates of gas and liquid. Two devices con-
sidered and discarded as being inadequate were (1) a fine porosity

fritted glass disc through which the gas was blown into the liquid, and
30
(2) two parallel stainless steel discs, a rotor and a stator, each
equipped with intermingling blades. The gas-liquid mixture flowed
between the blades and the gas was broken into fine bubbles by the
shearing action.

The bubble generator designed and developed for this project 1is
shown diagrammatically on Figure 3. The liquid flowed through & con-
verging diverging nozzle with a l-inch-diameter throat and a 2-inch-
diameter entrance and exit. The section downstream of the throat
diverged at an angle of about 12 degrees. A central "plumb-bob" shaped
probe of maximum cross-sectional diameter of ~0.812 inches was movable
and could be centrally positioned anywhere in the diverging section
including the throat and exit. This probe was supported by a tube which
carried the gas into the system. The tube, in turn, was supported by a
"Swagelok" fitting penetrating a flange on the end of the straight leg
of a tee connected to the nozzle entrance, Four small positioning rods
near the throat centered the probe within the nozzle and helped support
it. They also acted as holders for a section of "honey-comb" straighten-
ing vaneg used to minimize the liquid swirl induced by the right angle
turn at the tee entrance to the nozzle,

Gas entered the liquid through 48 holes (1/64-inch-diameter) around
the probe periphery at its maximum thickness and exited as a series of
parallel plumes which were broken intc individual bubbles by the turbu-
lence in the diverging section of the nozzle. The mean bubble gize for
a given flow and mixture was controlled by the position of the probe
within the nozzle (the closer the probe was to the throat the smaller

the mean bubble size produced).
31

ORNL-DWG 71-8017

 
   
 
  

2

   

SITION AND
SUPPORT RODS

THROAT 4in.

     
    

LIQUID

FLOW STRAIGHTVEAN!NG

NE Q.812 in.

   

48 HOLES, 1/64 in. DIAM

GAS FLOW ~— = -we—

41/2———-{

Figure 3. Diagram of the Bubble Generator.
32

Bubbles generated by this device were found to follow closely a
size distribution function proposed by Bayens56 and previously used to
describe droplet sizes produced in spray nozzles.

The function, defined as £(6§)d8 = that fraction of the total number
of bubbles that have diameters, &, lying in the range & * 1/2 dé, is
given by

£(8) = b (/MY 2 82 Exp (-6?) (6)
in which |
o = [h/w/68]¥ 2 |

This function has been normalized so that

o
[ f(s8)as =1 .
o

An indication of the suitability of this distribution function is
given in Figure U4 where measured cumulative size distributions for
bubble populations produced by the bubble generator are compared
with the distributions calculated from the function at different liquid
flows and different ratios of gas to liquid flows. The measured distri-
butions were obtained by painstakingly scaling the sizes of a sufficient
number of bubbles directly off photographs taken of the bubble swarms at
each condition., These measured areas should be accurate within about
10%.

The range of mean bubble sizes capable of being produced by this
bubble generator were measured at a constant gas-to-ligquid volumetric
flow ratio, Qg/Qz’ of 0.3% at different liquid flow rates, different

mixtures of glycerine and water, and different probe positions.

The results are shown on Figure 29, page 123 (Appendix D). The mean
PERCENT LESS THAN d

33

ORNL-—-DWG 74-7999

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

99.9 v
( | | | T | {
998 N BUBBLE DIAMETER —
00 5 |l v CUMULATIVE DISTRIBUTIONS
' / A { CALCULATED® MEASURED ]
99 w1 A —
v / P p a x10”" SYMBOL Q_ (gpm) Qg/Q_ (%)
98 VH—ar—tF 1 00789 o 20 0.5 —
/ '. 2 01667 ® 40 0.3
a5 fall | 3 04359 A 80 05 ]
o ¥ é A’/& e 4 04981 A 60 03
90 oY 4 / 5 0.8325 7 40 oA —
// 7/ ® 6 1.9024 v 60 04
L
80 fi o — n
v }{ / /2 CALCULATED FROM ASSUMED DISTRIBUTION FUNCTION
-0 / /
‘ ;//’ @ ;1)
; / o / Nt/2 2 L
20 7 /] 1/ o £(d) = 4(%) 4" EXP (-ad?)
// /© B
7 ./-"' N 2/3 1
fi/ ./ (/ WHERE o = [——-——4 5 ]
20 T CP |
10 b /Ff N = THE NUMBER OF BUBBLES PER UNIT VOLUME
7 —
A —0F @ = THE VOLUME FRACTION OF BUBBLES —
5 E j/f - - g
si?zimf_dm L o
2 v
f
0.01 002 0.05 o1 0.2 0.5 1 2

d, BUBBLE DIAMETER (in)

Figure k4, Comparison of Measured Bubble Sizes with the Distribution
Function.
3k
diameter used throughout this report is the "Sauter" mean defined by

- 83 £(8)ds
., 2o D IOT .

Vs @
jo 52 £(8)as

which is the volume-to-surface weighted mean commonly used in mass-

transfer operations.

Bubble Separation

 

Since this project uses the transient mode of testing, bubbles that
recirculate and extract dissolved oxygen from the liquid in regions out-
side the test section constitute an error source in the measurements.
Consequently a high degree of separation is desirable for this method
of testing. Some technigues considered were (1) gravitational separa-
tion in a tank, (2) centrifugal separation through the use of vanes to
induce a strong vortex, and (3) separation by flowing through a porous
metal which might act as a physical barrier to the bubbles; Each of
these had shortcomings that prevented their use in this project. For
example, with gravitational separation the tank size required for the
viscous mixtures was pondercusly large. This increases the system
volume resulting in a "sluggish'" loop and an accompanying increase in
the measurement error.

With centrifugal separation there were problems in stabilizing the
gaseous core of the vortex over a wide range of operating conditions.

In addition, large by-pass of bubbles (inefficient separation) was
observed and there was too much liquid carryover through the gas removal

duct.
35

The porous physical barriers tested required large frontal areas
or had prohibitive pressure drops, and the bubbles were observed to
regularly penetrate these barriers,

A satisfactory separator was finally developed that combined fea-
tures of each of the above. A diagram of this separator is shown on
Figure 5. The liquid-bubble mixtures entered the bottom of a 6-inch-
diameter pipe. A series of plexiglas vanes Jjust beyond this entrance
created a swirl flow within the tank which tended to force the bubbles
to the middle. The spinning mixture flowed upward into a converging
cone-shaped region with sides of 500-mesh stainiess steel screen, When
wetted by the liguid, the screen acted as a physical barrier to the
bubbles but allowed the liquid to pass through. The liquid exited from
the separator while the bubbles continued to rise through the truncated
end of the conical screen to an interface where the gas was vented
through a small exit line. The system pressure level was also con-
trolled at this interface by providing an auxiliary sweep of helium
through the exit line,

Good separation was achieved with this apparatus over the test con-
ditions of this thesis., No bubbles could be detected in photographs
teken downstream of the separator. However, with the use of a light
beam, some bubbles that appeared to be smaller than the screen mesh
size could be detected visually. After passing through the pump and
entering a higher pressure region these bubbles apparently went into
solution because they could no longer be visually detected downstream
of that region. If indeed they did go into solution along with their

small amount of extracted oxygen, they would have hardly constituted
36

ORNL—-DWG 71-8016

SYSTEM PRESSURE CONTROL =~ - = — -== GAS OUT

 

 

i

~— |~

 

 

2' g—- LIQUID OUT
i

500 MESH CONICALLY SHAPED
6 1/2 STAINLESS STEEL SCREEN

 

 

— e ——

 

 

 

SWIRL GENERATOR

 

 

|

IN
2 —__ AND GAS

 

S —

Figure 5. Diagram of the Bubble Separator.
37
a gignificant error in the mass-transfer measurements. Nevertheless,
several "special" tests were made in which about 10% of the normal gas
flow was purposely introduced downstream of the separator and allowed
to recirculate. The measured rates of change in loop concentration
under these conditions were always less than 3% of the normal rate and
the effect of the apparently much smaller amounts of by-pass therefore
were felt to be acceptable,

This separator was the major factor in limiting the ranges of
Reynolds numbers that could be obtained in this system. For a given
mixture, as flow was increased a flow rate was eventually reached at
which there was an observed '"breskthrough'" of many large bubbles that
would continue to recirculate. At this level of flow it was necessary
to terminate the tests with the particular mixture,

In addition to the flow limiting aspect of the separator, an
unexpected large amount of mass transfer occurred there — probably due
to the energy dissipation of the swirl and the relatively large amount
of contact time between the liquid and gas. Consequently a larger than
anticipated "end effect” resulted that had to be accounted for in deter-
mining the mass-transfer coefficients applicable to the test section

only. This correction resulted in decreased reliability of the results.

Test Section

The test section was considered as that portion of conduit between
the bubble generator exit and the entrance of an elbow ieading into the
separator entrance pipe (see Figure 1, page 24). It consisted of five
sections of 2-inch-diameter conduit flanged together with Teflon gaskets.

As encountered in the direction of flow these were a L4-foot-long section
38

of glass pipe, a 1l0-foot-long section of glass pipe, a 6 l/2-foot-long
section of stainless steel "long-radius" U-bend, another 10-foot-long
section of glass pipe, and a 5-foot-long section of glass pipe, for a
total of 35 1/2 feet of length. The test section and bubble generator
were connected to the rest of the loop piping through the bubble gener-
ator tee at the entrance and an elbow at the exit which served as pivot
points to permit the test section to be mounted in any orientation from

horizontal to vertical.
Bubble Surface Area Determination — Photographic System

The mean sizes and interfacial areas per unit volume of the bubble
dispersions were determined photographically using a Polarold camera and
two Strobolume flash units. To reduce distortion the photographs were
taken through rectangular glass ports fitted around the cylindrical
glass conduit and filled with a liquid having the same index of refrac-
tion as the glass. The port for "inlet" pictures was located about one
foot downstream from the bubble generator exit and the "exit" port was
located about two feet upstream from the test section exit.

The Polaroid camera was equipped with a specially made telescopic
lens that permitted taking photographs in good focus across the entire
cross section of the conduit. The camera was semi—permanently mounted
onto the facility structure in such a manner that photographs could be
taken at the "inlet" port and then the camera pivoted for taking a sub-
sequent picture through the "exit" port. For vertical orientation of

the test section, photographs were taken directly through the ports,
39
For horizontal tests, the camera remained in its "vertical orientation”
position and the photographs were taken through high quality front sur-
face mirrors.

With the camera focused along the axis of the conduit, bubbles
closer to the camera appear larger and those further away appear smaller.
To determine the magnitude of this possible error source, small wires of
known diameter were mounted inside the conduit across the cross section,
Photographs obtained after focusing on the central wire indicated less
than one percent maximum error in the apparent diameter reading.

The Strobolume flash units (one for each port) produced pictures
of best contrast when mounted to provide diffuse back lighting in which
the lights were aimed directly into the camera lens from behind the photo
ports., Semi-opaque "milky" plexiglas sheets between the lights and the
photo ports served as the light diffusers.

Bubble diameters could have been scaled directly off the photographs
for each run and used to establish the interfacial areas and mean diame-
ters Jjust as was done to validate the bubble size distribution function.
However, this proved to be such an onerous and time-consuming procedure
that it would have been prohibitive due to the large number of experi-
mental runs and need for at least two photographs for each run, Conse-
quently, the following use was made of the distribution function,

The interfacial area per unit volume is defined as

ag ” N 82 £(8) as (8)
O

and the bubble volume fraction is given by

6 = | T8 a6y g
=] = -
Lo
Recalling the definition of the Sauter mean diameter, Equation (7), it
is seen from the above that, regardless of the form of the distribution

function, the interfacilal area per unit volume can be expressed as

. (9)

Q—-|O\
o1

a
Vs

For the distribution function of Equation (6), Equation (8) may be inte-

grated to give

a8 =

ny

N\ 2
<€E> 31\11/3 3% 3 = L, po NV 3 ¥ 3 | ' (10)

Therefore, by measuring the volume fraction, ¢, it was only necessary
to count the number of bubbles per unit volume from the photographs and
use Equation (10) to establish the areas. Equation (9) was then used to
determine the mean bubble diameters. Counting the number of bubbles in
a representative area of the photographs was a considerably easier task
than measuring the actual sizes of each bubble, However, it was then
necegsary to have an independent determination of the volume fraction
occupied by the bubbles,

Hughmarks4 presented a volume fraction correlation that graphically

related a flow parameter, X, defined from
pQ | |
;‘é"=1"9—<-%> (11)
g'g e

= (Re)V ® (Fr)¥ e /vy *

to the parameter

3
il

where
B
vaa,/e,+a) .
For p >> D2 Equation (11) reduces to

3 =X Qg/QJz . (12)
bl
Hughmark's correlation for X at sufficiently large Z is nearly flat
with X changing from 0.7 to 0.9 over a 10-fold change in Z. For the
conditions of the experiments in this report, X was considered to be
constant at an average value of 0,73.
When volume fractions were measured in the vertical flow tests,
it was found that

3 = 0.73 Qg/%

gave a good measure of the mean value for a given test but that the
volume fractions were sometimes considerably smaller than this in the
riser leg of the test section and, at the same time, comparably larger
in the downcomer. It was apparent that this difference was due to
buoyancy driven relative flow between the bubbles and the liquid.,
Separate volume fractions were therefore determined for each leg based
on a mass balance, This mass balance between the riser and downcomer

sectionsg in a constant area conduit takes the form

() = (W),
Letting
Vr =V + Vb
and
Vd = V‘—'Vb
then
vV + Vb
@d/@r = Nd/Nr = v——_'_—vb- . (13)

The bubble terminal velocity, Vb, depends on the bubble Reynolds

number, Re (= Vbdvs/v).
Lo

If Reb < 2, then Stokes law results in

2 -
. a® &lp — o )
b~ 184 *
If Re, > 2, then V. is determined from a balance between the drag

b b
2 2 3
force [(Cde b/EgC)(nd VS/M)] and buoyancy (pmd vsg/gc6) to be

i/ =
e [o0e
= = A ’
b 3 Cd ]
wherelthe drag coefficient, Cd’ is given by
_ 0.8
Cd = 18.5/Reb .
It was further assumed that the average of the riser and downcomer

volume fractions could be calculated by

@r + @d )
~—=0.730a/q, . (1)

Then with iterations to establish dvs’ Vb’ and Reb, Equations (13) and
(14) were solved to determine the individual leg vertical flow volume
fractions, and Equation (10) was used to establish the interfacial areas
per unit volume, The averages were used to extract the mass-transfer
coefficients from the ka products.

As a further indication of the accuracy of the distribution function
and the validity of this technique for establishing the vertical flow
surface areas, Figure 0 compares some surface areas determined as out-
lined above with the areas measured directly from the photographs. The
experimental conditions for the run numbers identifying each point are
listed in Table IV.

In horizontal flows the volume fractions were the same in each leg

but stratification of the bubbles near the top of the conduit, especially
ORNL—DWG 71-—8000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ac, CALCULATED INTERFACIAL AREA PER UNIT VOLUME (in~h

 

 

 

 

 

 

 

 

 

 

 

 

2.0
VERTICAL FLOW IN 2-in. DIAMETER CONDUIT
19 = 4, = INTERFACIAL AREA MEASURED
DIRECTLY FROM PHOTOGRAPHS
18 |- '
a, = 4.22 N3 $2/3 (FROM DISTRIBUTION
. FUNCTION)
WHERE N = NUMBER OF BUBBLES PER UNIT
16 |- VOLUME TAKEN FROM PHOTOGRAPHS
® = VOLUME FRACTION OCCUPIED BY
5 | THE BUBBLES
THE NUMBERS IDENTIFYING THE DATA POINTS ARE
1.4 |— RUNS FOR WHICH CONDITIONS ARE LISTED
ELSEWHERE
3 L, +20%
. ]_ /
/
.2 /
1
0217
e 53
J/~20%
0.9 ,
08 ’
0.7
0.6
05 ‘
0.4 /
/ o /7
0.3 o1 f 27
(£
02 |—85 100
01 (=
76
0
0 0.2 0.4 06 0.8 1.0 $.2 1.4

am, MEASURED INTERFACIAL AREA PER UNIT VOLUME (in7h

Figure 6, Comparison of Interfacial Areas Per Unit Volume Measured
Directly from Photographs with Those Established Through
the Distribution Function. Vertical Flow.
iy

Table IV. Experimental Conditions for Runs Used to Validate
Surface Area Determination Method for Vertical Flows

 

 

Mixture
Run No Qfl (gpm) Qg/QE (%) (% glycerine)
71 20 0.5 0
73 20 0.1 0
76 20 0.1 0
83 4o 0.5 0
85 Lo 0.1 0
87 Lo 0.3 0
91 60 0.1 0
92 60 0.5 0
93 60 0.3 0
100 80 C.1 0
10k 80 0.5 0
119 20 0.5 50
130 L0 0.5 50
142 10 0.5 50
155 50 0.3 50
162 20 0.5 37.5
165 30 0.5 37.5
171 Lo 0.5 7.5
198 4o 0.5 37.5
213 20 0.5 37.5
217 4o 0.5 37.5

 
L5

at low flows, invalidated the use of Equation (14). It was found

possible however to correlate the horizontal flow volume fractions at

Qg/Qz = 0.3% with the ratio, v/vb, of the axial liquid velocity to the

bubble terminal velocity in the liquid.

Figure 7 with the identification of the randomly selected runs given

in Table V,

This correlation is shown in

Table V, Experimental Conditions for Runs Shown on
Horizontal Flow Volume Fraction Correlation

 

 

. Mixture
Run No. %Y (gpm) % (inches) (% glycerine)
376 35 0.033 50
390 50 0.028 50
382 L0 0.059 50
389 50 0.024 50
391 30 0. 037 50
365 60 0.026 0
355 20 0.049 0
370 30 0,01k 50
368 70 0.01k 0
400 30 0.066 37.5
Lok 35 0.061 37.5
ele 55 0.026 37.5
Lo7 60 0.030 37.5

 

For V/Vi) less than 30, a least squares line,

§ = 0.0018 + 0.021/(V/V.)

was used while for V/Vb greater than 30 a constant value,

¢ = 0,0025

2

(15)

was used. Severe stratification prevented experimentation at values of

V/Vb less than about 3.
BUBBLE VOLUME FRACTION

&,

0.010

0.009

0.008

0.007

0.006

0.005

0.004

©.003

0.002

0.001

Figure 7.

L6

ORNL-DWG 71-8001

 

VOLUME FRACTION CORRELATION
HORIZONTAL FLOW IN A 2-in. DIAMETER CONDUIT
Qq/QL = 0.3%

- MEASURED FROM PHOTOGRAPHS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V/Vp = 0.0188 Q_(u/p)°°1%/a,s"27
WHERE Q_ = gpm
@ = Iby/ft-hr
3 p = Ibm/ft3
dys = inches
|
355
| @400
‘ i
® = 0.0018 + 0.021/(V/Vy) LEAST
SQUARE LINE FOR DATA WITH V/Vp <30
404 @
382
376 @427
\OA365 |
\42 389 370
— @O e O D) e N
O 390 368
391
GLYCERINE
(%)
0O 50
375 NUMBERS IDENTIFYING DATA POINTS
® : ARE RUNS LISTED ELSEWHERE
A 0 '
i
0 10 20 30 40 50 60

RATIO OF AXIAL TO TERMINAL VELOCITY (V/Vp)

Correlation of Horizontal Flow Volume Fraction.

70
b
This horizontal flow volume fraction correlation in conJunction
with Equation (10) was used to establish the horizontal flow interfacial
areas per unit volume. An indication of the adequacy of this procedure
is given in Figure 8 in which calculated and measured areas are compared

for the runs identified in Table V,

End Effect

In the transient response mode of operation all mass transfer
occuring outside the test section (principally in the bubble separator
and generator) must be independently measured and accounted for in
establishing the ka products applicable only to the test section.

"End-effect" measurements were made after all other scheduled tests
were completed by moving the bubble generator to a position at the test
section exit which allowed the bubbles to flow directly from the genera-
tor into the separator — effectively by-passing the test section. All

tests were then repeated duplicating as nearly as possible the original

 

conditions. With the end-effect response‘so measured, the correction
was determined as follows,

Consider three regions of mass transfer in series representing the
bubble generator (Region 1), the test section (Region 2), and the bubble
separator (Region 3). The original measurements, indicated here by &
subscript "I," determined the ratio, Kps of the outlet to inlet concen-
‘tration across all three regions. Therefore

KI=K1 Kg Ks 3

where K,, K5, and K5 are the outlet and iniet concentration ratios across

the individual regions.
1.9
1.8
1.7
‘l-'—: 1.6
£
15
=
_ 1.4
o
>
- 1.3
=
- 942
a
5
1.1
<J
o
T 10
I
< 09
Q
N
a 0.8
Lo
|_
< 0.7
O
Eo06
I
3
3 0.5
-
I
© 04
o
0.3
0.2
01
O
Figure 8.

L3

ORNL-DWG 74—8002

 

HORIZONTAL FLOW I[N 2-in. DIAMETER CONDUIT

am = MEASURED DIRECTLY FROM
i PHOTOGRAPHS
ac= 4.22 N3 &2/3 (FROM DISTRIBUTION
FUNCTION)
WHERE N = NUMBER OF BUBBLES PER UNIT
i VOLUME FROM PHOTOGRAPHS

® = VOLUME FRACTION OCCUPIED BY
- THE BUBBLES

RUN NUMBERS SHOWN ARE IDENTIFIED
"ELSEWHERE

L

b

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘+20°/o
/3700
/I
yi
7
} —2 0% ——]
/ 7
///
7
///
/
O 0.2 0.4 0.6 0.8 1.0 {.2
dm MEASURED INTERFACIAL AREA PER UNIT VOLUME (in~"

Comparison of Measured and Calculated Interfacial Areas

Per Unit Volume, Horizontal Flow.
ho
The second series of tests, sfibscripted "IT," with only the bubble
generator and separator regions entering into the mass transfer, deter-
mined the ratio

KII = Kl Ka .

Consequently the desired ratio, K5, across the test section only, was
determined from

K =Kg = KI/KII .

An estimate of the error involved in this procedure is given in Appendix

G.
Sfimmary of Experimental Procedure

The mode of experimentation was transient with the independent
variables being Schmidt number (depending on percent glycerine in
glycerine-water mixtures), Reynolds number (liquid flow), bubble mean
diameter (controlled by bubble generator probe position), and test sec-
tion orientation (vertical or horizontal), Other parameters that were
held constant fdr most of these tests include the test section conduit
diameter (D = 2 inches), the ratio of gas to liquid volumetric flow
(Qg/Qfl = 0.3%) and the fluid temperature (25°C).

It was found that the only effect of volume fraction up to 1% was
in the highly predictable change in surface area, No significant
difference was detected in the mass-transfer coefficients themselves
which are on a unit area basis. Consequently with the exception of
some of the early runs most of the experiments were performed at a con-
venient volume fraction of 0.3%. In addition it was found that for the

distilled water runs (no glycerine) the rapid agglomeration of the
50
bubbles at the flows obtainable prevented meaningful interpretation of
the data. Consequently all water tests were performed with the addition
of about 200 ppm of normal butyl alcohol which effectively inhibited the
agglomeration but may have resulted in a different surface condition
compared to the other mixtures. The addition of this same amount of
N-butyl alcohol to the glycerine-water mixtures made no significant
difference,

For a given liquid mixture and orientation of the test section, the
procedure followed to obtain a series of data is outlined in detail
below:

1. The loop was first purged repeatedly with distilled water to
remove residual liquid from previous experiments and the system allowed
to dry by blowing air through it overnight.

2., The mixture of glycerine and water to be used was precisely
made up in the weigh tank and then thoroughly mixed by vigorous stirring
produced by pumping the liguid from the bottom of the tank back into the
top.

3. The loop was filled using a small auxiliary pump and the system
operating pressure was set at a nominal 4O psia by helium pressure over
the interface in the bubble separator.

4, ILiquid flow was established by energizing the main loop circu-
lator and the flow was set at the desired level by throttling through
all three rotameters.

5. The system was charged with oxygen to about seven or eight
parts per million by passing oxygen bubbles through the bubble generator,

the test section, and the bubble separator. The system was allowed to
51
run a sufficient time after the oxygen flow had been terminated to
insure that the concentration readings were steady.

6. The bubble generator probe position was set to obtain the
first desired mean bubble diameter for the given test conditions.

7. The helium filow, having been preset to give Qg/QE = 0.3% at
the given liguid flow, was turned on initiating the transient experiment
which was usually allowed to continue for 10 to 15 minutes.

8. The oxygen concentration was continuously recorded and data
sheet loggings were made of liquid flow through each rotameter, test
section inlet pressure, test section pressure drop, helium pressure
at the capillary tube exit, pressure drop across the capillary tube,
loop temperature, bubble generator probe position, and atmospheric
pressure.

9. About midway through the transient for each test, a Polaroid
picture of the bubbles was made through one cf the photo ports (entrance
or exit) and then the camera was pivoted and a picture taken through the
other photo port.

10. For the given liquid flow, the bubble generator probe position
was varied to produce different mean diameters. Five values were desired
and usually obtained, For each position the above procedure (5-9) was
repeated. Occasionally to produce extra large bubbles, the gas was
introduced through the test-section inlet pressure tap — bypassing the
bubble generator itself.

11, The liquid flow was varied over the desired range and the above

procedure (5-10) was repeated for each flow setting.
52

Typical transient data of.&zc/co versus time taken directly from
the oxygen concentration recording chart is shown on Figure 9 which
illustrates the constancy of the slope (#%zKI QE/VS)'

The system volume had been previously measured to be ~2,52 ft3 by
filling the system completely with water which was then collected and
weighed. Using this and the measured values of Qz, the constants KI
were determined from the slopes of the curves.

After all vertical and horizontal tests were completed, "end effects”
were measured by moving the bubble generator to the test section exit and
repeating each experiment with the original conditions duplicated as
nearly as possible.,

1"

The values of K_.. were then calculated from the slopes of the "end

IT

effect" curves and K's were calculated from
K = K/Kpp o
The products, ka, were extracted from K through Equations (5),

The bubble photographs were analyzed to obtain the interfacial
areas per unit volume and the mean diameters, Typical examples of an
inlet and exit photograph are shown on Figure 10. The outlined regions
were used as the sample populations for counting the number of bubbles
per unit wvolume, N,

The applicable volume fraction correlation [either Egquations (13),
(14), or (15)] was used to determine ¥ and Equations (10) and (9) were
used to calculate the interfacial areas per unit volume and the mean
bubble diameters, respectively. Finally, the averages of the inlet and

exit areas were used to extract k from the ka products.
OXYGEN CONCENTRATION/OXYGEN CONCENTRATION

23

ORNL-DWG 71-8015

 

 

\ TYPICAL EXPERIMENTAL DATA
HORIZONTAL FLOW IN 2-in. DIAMETER CONDUIT

FLUID =12.5% GLYCERINE + 87.5% WATER
QL = 85 (gpm)
Qg/QL= 0.3%

\ BUBBLE MEAN DIAMETER, dys = 0.015 in.

 

 

 

 

0.5

 

0, (C/Cy)

N

 

AT t =
|

 

0.2 \

 

 

 

 

 

 

 

 

 

01
O 1 2 3 4 5 6 7

t, EXPERIMENT TIME (min)

Figure 9, Typical Experimental Concentration Transient TIllustrating
Straight-ILine Behavior on Semi-Log Coordinants.
 

5k

.PHOTO 1854-71

 

 

   

(5)

Figure 10. Typical Examples of Bubble Photographs: a. Inlet b. Exit.
Vertical Flow, 37.5% Glycerine-62,5% Water, Q, = 20 gpm,
Qg/Qg = 0.3%, D = 2 inches, and dyg = 0.023 inches.

 
CHAPTER IV
EXPERTMENTAL RESULTS

The experimentally measured mass-transfer transients initially were
converted into pseudo mass-transfer coefficients without any adjustment
being made for mass-~-transfer occurring outside the test section. The
results thus obtained are not the true mass-transfer coefficients
because significant mass transfer occurred in the bubble generating and
separating equipment, Nevertheless, considerable information can be
- gathered from this "unadjusted" data because of its presumed greater
precision. The true mass-transfer coefficients with extraneous mass-

transfer effects accounted for are presented later in this Chapter.
Unadjusted Results

The "unadjusted" mass-transfer coefficients determined as outlined
in Chapter III as functions of bubble mean diameter, Reynolds number,
orientation of the test section, and Schmidt number are given in Appendix
H (Figures 35-44, pages 138-147),

The "raw' data which consists of recorder charts of oxygen concen-
tration versus time, innumerable photographs of bubble populations, and
log book records of flows, probe settings, temperature, pressure and
other conditions are on file in the Heat Transfer-Fluid Dynamics Depart-
ment, Reactor Division of the Oak Ridge National Laboratory and are
available upon request.

It is instructive to consider the crossplots (Figures 45-49, pages
148-152), Similar to Lamont's'' results, the horizontal and the vertical

55
56
flow values were identical above sufficiently high Reynolds numbers. As
flows were decreased below these Reynolds numbers, however, the vertical
flow coefficients were larger than the horizontal flow coefficients and
seemed to asymptotically approach constant values. The horizontal flow
data, on the other hand, continued along straight line variations (on
log-loglcoordinates) until either the flow was too low to prevent con-
centration polarization at the oxygen sensors or in some cases severe
stratification of the bubbles prevented further testing., The pertinent
results to be considered, based on these unadjusted data, are the values
of the Reynolds number at which vertical énd horizontal flow results
become identical and the apparent asymptotes approached by the vertical
flow coefficients at low flows., Mass transfer occurring outside the
test section should not affect either of these and their values should be
the same as for the data presented later that represents the true mass-

transfer coefficients.

Fquivalence of Horizontal and Vertical Flow Mass Transfer

 

It seems evident that gravitational forces (buoyancy) tend to
establish a steady relative flow between the bubbles and the liguid if
the bubbles are free to move in the vertical direction {as they would
be in vertical orientations of the test sections) and are not restricted
by physical boundaries (as they would be in horizontal orientations).
The bubbles are also acted upon by inertial forces generated by the
turbulent motions within the liquid. These turbulent inertial forces
are randomly directed and thus tend, on the average, to counteract the
gravitational forces. Therefore it would be reasonable to assume that

if the magnitudes of the turbulent inertial forces, Fi’ were known
57
compared to the gravitational forces, Fg’ then their ratio, Fi/Fg’ would
be a measure of the relative importance of these forces in establishing
the mass-transfer coefficients. At sufficiently high values of Fi/Fg
one would expect the turbulent forces to dominate and there should then
be no detectable difference in the horizontal and vertical flow results,
The gravitational force on a bubble of diameter, d, is the weight

of the displaced fluid
md>
R TR (16)
The turbulent inertial force exerted on a bubble essentially
traveling at the local fluid velocity in a turbulent liquid is not so
easily determined., Consequently, use was made of dimensional arguments.
In a turbulent fluid the mean variation in velocity, AV, over a

distance, \, (greater than the microscale) is given dimensionally by

. /3
;E:?tg 1

where €, is the power dissipation per unit volume, The 1/3 power on A

 

agrees with the result of Hinze (Reference 37) for the variation in tur-
bulent intensity required to result in the Kolmogoroff spectrum law,
Similarly, the period, 6, for such velocity variations is given dimen-

sionally by
3

e ii”' A.g 1/
e

Following Levich,86 it is postulated that the mean acceleration al

undergone by a fluid element of size, X, is

e g8 \¥3
0 =&V [ Xc z @
A dt 0 :
£

 
58
A spherical fluid element with this mean acceleration must have

experienced a "mean" force given by

3 ¥s
sz NFWK ('VC> A

It is further postulated that a bubble of diameter, d, in the turbulent

 

ligquid will be subjected to the same mean forces as those exerted on a
fluid element of the same size. Therefore the mean turbulent inertial

force on the bubble is given by

3

.~-%— Vc. /a3, (17)

Dividing by Equation (16) the ratio of inertial forces to gravitational

 

forces is given by

 

' % 3
F./F [ / &/ 3g (18)
e\ ’
For flow in conduits, the power dissipation per unit volume can be

expressed as

=y 9P
e, =V 5 (Reit/Do) £ dx

and the pressure gradient can be determined from the Blasius relationship,

aP £ oV® 2 3 2
= = D——P—ggc = (f p3/2g_ D® p) Re

Using the friction factor for smooth tubes,
f = 0.316/(Re)¥ ¢ ,

the power dissipation per unit volume is

e = <——O'316\—-——“8 Rell/ %, (19)

v Egc / p2D4
59
Substitution into Equation (18) and replacing the bubble diameter
by the Sauter mean gives

-~

! 3 11/ 4 12/:3
F/F NJO.316}.L Re /d1/3 g . (20)
i'Tg | Vs

L 2% D*

Since Equation (20) was established on dimensional grounds, there
exists a proportionality constant of unknown magnitude. To establish
the value that the ratio should have to serve as a criterion for deter-
mining when horizontal and vertical flow mass-transfer coefficients
become identical, use was made of the data of Lamont gathered from his

report as listed in Table VI below.

Table VI. Conditions at Which Horizontal and Vertical Flow
Mass Transfer Coefficients Become Equal (Lamont's Data)l?

 

 

Case I Case IT
Conduit Diameter, D (inches) 5/16 5/8
Reynolds Modulus, Re 10% 3 x 10*%
Liquid Viscosity, u (centipoise) 0.89 0.89
Liquid Density, o (g/cm®) 1.0 1.0
Bubble Diameter, d (inches) ~5/32 ~5/32

 

Substitution of the data of Case I into Equation (20) gives
Fi/Fg = 1,5

As a check the data of Case II are compared,

; 11/ 6 N 8/ 3
(Fi/Fg)I _ ."I! 104 \ / / 5/16 \\i 0. 82
F/F ) \ 3 ¢ 10, -

For the present investigation, the loci of points for Fi/Fg = 1,5

as calculated from Equation (20) are shown on Figures 45-149, pages 148-

152,
60
It is seen that the ratio Fi/Fg seems to be a good predictor for

the equivalence of the horizontal and vertical results.

Vertical Orientation Low-Flow Asymptotes

 

As liquid flow is reduced, the gravitational forces become more and
more dominant over the turbulent inertial forces. Consequently, at low
flows, the vertical flow mass-transfer coefficients approach the values
that would be expected for the bubbles rising through a quiescent liquid.

The conditions of mass transfer for bubbles rising through a column
of liquid have been extensively studied (e.g., References 26-30).

Resnick and Gal-Or®” have recommended for surfactant-free systems

-
[Spg (Y2 1-38
k =0,109 | =2  ——— . 7
a ? l_ Moo (1 — g% 3)V/ 2 Vs

They caution that this equation may give values slightly higher than the
observed data in particular for lower concentrations of glycerol in
water-glycerol systems.

In the present investigation, the volume fraction is low so that

the above equation was approximated as

k, = 0.109 [8pg/ul” ? Ja__ (21)

Vs
and used to determine the "calculated asymptotes” for the vertical flow

results as indicated on the various data plots,
Mass-Transfer Coefficients

With the end-effect accounted for as outlined in Chapter I1III, the
mass-transfer coefficients measured in this investigation are given in

Figures 50-58, pages 153-161 {Appendix G). The more revealing crossplots
61
of mass-transfer coefficients versus Reynolds number are shown in
Figures 11-15 which contain regression lines fitted to the horizontal
flow data and calculated lines for the vertical flow cases. Vertical
flow data are not shown for the 37.5% mixture because the end effect
adjustments were not satisfactory. Excessive vibration of the bubble
generation probe that occurred during the 37.5% experiments was elimi-
nated by redesign of the probe before the horizontal data were obtained.
Time did not permit a reorientation of the system to the vertical posi-
tion to repeat the runs,

From these figures it is seen that the horizontal flow data for
water (plus N-butyl alcohol) apparently have a lesser slope than that
for the glycerine-water mixtures. Therefore a regression equation was
determined for the water runs alone and a separate regression equation
was determined for the combined data for the 12,5, 25, and 37.5%
glycerine mixtures., A third behavior was observed for the 50% glycerine
mixture (Figure 15). It is seen that all the data for this mixture were
obtained at Reynolds numbers less than that required for Fi/Fg = 1.5,
However, instead of a steady march of the horizontal flow data down a
straight line as observed for the other mixtures, the small bubble
horizontal flow mass-transfer coefficients tended to behave like those
for vertical flows, This behavior implies that, if the liquid is viscous
enough, small bubbles apparently can establish steady relative flow con-
ditions in their rise across the conduit cross section, In these runs,
the pipe wall apparently did not significantly inhibit the bubble rise
rate during transit through the test section and, evidently, the bubbles

behaved exactly as if they were rising through a vertical conduit.
MASS TRANSFER COEFFICIENT (ft/hr}

k,

62

ORNL-DWG 71-7989

/CALCULATED ASYMPTOTES

VERTICAL FLOW
> (USING Fi/Fg AS SCALE FACTOR)

LT

WATER + ~200 ppm N-BUTYL ALCOHOL LOCUS OF Fi/Fg=15 ~

SCHMIDT NO. = 449 . 1 1

O
1 HORIZONTAL FLOW REGRESSION
BUBBLE EQUATION

MEAN

DIAMETER HORIZONTAL VERTICAL (Sh = 19436 Re

079 (913)0,85)
(in) FLOW FLOW D

 

C.015 ®
0.02 u
0.03 A
004 v

 

0.2
103 2

n

104 2 5 105 2 5 108
PIPE REYNOLDS NC., Re = VD/v

Figure 11. Mass Transfer Coefficients Versus Pipe Reynolds Number
as a Function of Bubble Sauter-Mean Diameter. Water
Plus ~200 ppm N-Butyl Alcohol., Horizontal and Vertical
Flow in a 2-inch Diameter Conduit.
MASS TRANSFER COEFFICIENT (ft/hr)

k:

63

ORNL-DWG 7%-7990

 

 

10
CALCULATED ASYMPTOTES
5
VERTICAL FLOW
(USING F,/F, AS SCALE FACTOR)
2
OCUS OF F/Fy = 1.5
1
12.5% GLYCERINE
SCHMIDT NG. = 370 FROM HORIZONTAL FLOW REGRESSION
EQUATION
05 BUBBLE
' MEAN 22 o071 (dusy! O7
DIAMETER HORIZONTAL VERTICAL (Sh = 000538 Re ™™™ Sc™ (“6“) )
(in.) FLOW FLOW
0015 ® '
002 »
003 A
02 004 v
o1
103 4 2 5 10° 2 5 10%

2 5 10
‘ PIPE REYNOLDS NO, Re = VD/v

Figure 12, Mass Transfer Coefficients Versus Pipe Reynolds Number
as a Function of Bubble Sauter-Mean Diameter. 12.5%
Glycerine-87.5% Water. Horizontal and Vertical Flow
in a 2-inch Diameter Conduit.
MASS TRANSFER COEFFICIENT (ft/hr)

K,

64

ORNL—DWG 71-7991

 

 

10
ALCULATED ASYMPTOTES
5
2
VERTICAL FLOW
{USING Fi/fg AS SCALE FACTOR)
LOCUS OF
1 FisrFg=15
25% GLYCERINE O
SCHMIDT NO. = 750
0.5 BUBBLE FROM HORIZONTAL FLOW
MEAN REGRESSION EQUATICON
DIAMETER HORIZONTAL VERTICAL o
(in) FLOW FLOW Sh = 000538 Re' 22 5cO 7! (fifl
0.015 ® O
002 » 0
0.2 0.03 A A
.04 v vV
04

03 2 5 104 2 5 10° 2
PIPE REYNOLDS NO., Re = VD/u

Figure 13. 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.
k, MAS5S TRANSFER COEFFICIENT (ft/nr)

ORNL—-DWG 71-7992

CALCULATED ASYMPTOTES LOCUS OF Fi/Fq =15

VERTICAL FLOW FROM HORIZONTAL FLOW
(USING Fi/Fg AS SCALE FACTOR REGRESSION EQUATION
I | .

37.5% GLYCERINE

c A g A0
SCHMIDT NO. = 2015 sh : 000538 Re'2? 57" (_y;)

RIZONTAL FLOW DATA) 0

BUBBLE
MEAN
IAMETER

{in.}

10
5
2
i
0.5 HO
D
0.02
0.0
103
Figure

 

0.015
0.02
0.03
0.04

2 5 104 2 5 10° 2
PIPE REYNOLDS NO., Re = VD/v

14, Mass Transfer Coefficients Versus Pipe Reynolds Number
as a Function of Bubble Sauter-Mean Diameter. 37.5%
Glycerine-62.5% Water. Horizontal and Vertical Flow
in a 2-inch Diameter Conduit.
k, MASS TRANSFER COQEFFICIENT (ft/hr)

66

ORNL-DWG 71-7993

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
50% GLYCERINE
SCHMIDT NO = 3446
BUBBLE '
3 MEAN
DIAMETER HORIZONTAL VERTICAL
(in) FLOW FLOW
— 0.015 ® O
D ASYMPTOT .
CALCULATED ASYMPTOTES doe in) | 005 . 0
f e ] 0.04 0.03 A A
2 ——
—— =4 joos y '1',.. 0.04 v v
——
~_ T ”0915‘] ~ b 1/,
1 ~ ‘Lol' 8 Z -
NG \}\‘Q ’
g \0_ FROM REGRESSION EQUATION FITTING ONLY
™ THE 12.5, 25, AND 37.5% GLYCERINE DATA,
107
0.5 d
N Sh = 0.00538 Re'?22 5c07! (—5%
| T T 0
CALCULATED LINE LOCUS OF Fi/Fg =15
FOR dys = 0.015 |
USING Fi/Fg=15
AS SCALE FACTOR
[ . ot N N
02 (USING Fi/Fg=1C AS ]
SCALE FACTOR)
0.1
10? 2 5 104 2 5 105 2 5 106

PIPE REYNOLDS NO., = VD/v

Figure 15. Mass Transfer Coefficients Versus Pipe Reynolds Number
as a Function of Bubble Sauter-Mean Diameter. 50%
Glycerine-50% Water. Horizontal and Vertical Flow
in a 2-inch Diameter Conduit.
67

These three kinds of observed horizontal flow behavior are further
illustrated on Figure 16 for 0,02-in. mean diameter bubbles. The
regression slope of 0,94 for the glycerine-water mixtures agrees gener-
ally with the literature as discussed in Chapter II and the slope of
0.52 for the water plus N-butyl alcchol is, coincidentally, exactly what
Lamont found, However, the combined regression slope (0.79) for all the
water data which includes the other bfibble mean diameters was greater

than the wvalue for the 0.02-in. bubbles by themselves.

Calculating Vertical Flow Mass-Transfer Coefficients

 

for Fi/Fg Less Than 1.5

 

Since the ratio of turbulent inertial forces to gravitational
forces is seen to be a good predictor of the Reynolds number at which
horizontal and vertical flow mass-transfer coefficients become identical,
it is proposed that the varying ratio might alsoc serve as a scaling
factor at all Reynolds numbers to determine the relative importance of
the purely turbulent coefficients (Fi/Fg > 1,5) and the relative flow
coefficients (vertical flow asymptotes). That is, if the values are
known for the straight line variation at higher Reynolds numbers where
vertical and horizontal coefficients are equal along with the vertical
flow asymptotes, it is proposed that the intermediate vertical flow
mass-transfer coefficients can be calculated by using Fi/Fg as a linear
scaling factor between the two. Since Fi/Fg = 1.5 appears to mark the
Reynolds numbers at which turbulent inertial forces dominate over gravi-
tational forces, the actual ratio of forces at that condition are

assumed to be of the order of 10 to 1 for gravitational forces to begin
Sh/sc!/2

68

ORNL-DWG 71-7995

 

103
GLYCERINE SCHMIDT
(%) NO.
50 3446
¢ 37.5 2015
5 o5 250 SLOPE OF 0.52
12.5 370 (WATER + 200 ppm
0 419 N-BUTYL ALCOHOL
= kD/%
= ,u./p75
2 Re = VDp/u

8UBBLE DIAMETER, dys = 0.02 in.
HORIZONTAL FLOW IN 2-in. CONDUIT

 

102
CALCULATED 5
e (50% GLYCERINET SLOPE OF 0.94
° ObOo (375, 25, AND 12.5%
. GLYCERINE)
®
2
101

103 2 5 104 2 5 10% 2 5
PIPE REYNOLDS NO, Re = VD/v

Figure 16. Observed Types of Horizontal Flow Behavior.

dvs = 0,02 inches and D = 2 inches.
69
to be negligible. Consequently 10 (Fi/Fg)/l.5 was chosen as an appro-
priate linear scaling factor and the vertical flow mass-transfer coeffi-

cients were calculated from

 

 

10(F,/F )/1.5 7
kK =k < vk | o | (22)
v~ al|l+ 10(F./F )/L.5 1+ 10(F,/F_)/1.5 | ?
. i g i g
in which ka is the calculated asymptote given by Equation (21) and K, is
the value at the given Reynolds number that would be obtained by extend-
ing the straight-line variation of the horizontal flow data.
Using separate regression lines for kh, the vertical flow mass-
transfer coefficients calculated from Equation (22) are compared with
the data on Figures 11-15, pages 62-66. Except for the 50% mixture data,

Equation (22) provides a relatively good description of the data.

Comparison with Agitated Vessels

 

A comparison of the horizontal flow data with that of Sherwood and
Brian'’ for particulates in agitated vessels is shown on Figure 17.
Sherwood and Brian's coordinates are used by converting e_ (= ev/p)
through Equation (19) for flow in conduits. It is seen that, although
the relative magnitudes of the coefficients are comparable on an equiva-
lent power dissipation basis, there is a Schmidt number separation of
this data indicating mobile interfacial behavior. In agreement with the
findings of other investigations reported in Chapter II, the variation
with Reynolds number for flow in conduits is much steeper than would
have been expected from the agitated vessel data.

A possible explanation for this difference in slope observed between
agitated vessels and flow in conduits may lie in the relative importance

of the gravitational forces. For example, the data of this research for
70

 

 

100
BUBBLE"
50 Dlnfill\zlléfiER SCHMIDT NUMBER, Sc
(in.) 419 370 750 2013
0.015 o
0.02 o
20 0025 o
0.03 @
10 0.035 ®
% GLYCERINE — O
5
]
~
‘—U
<
= SHERWOQOD AINDIBRMN
w
2 Sc = 518
1
0.5
HORIZONTAL FLOW IN 2-in. DIAMETER CONDUIT
0.2
o1
/3
4
“mdvs.
y3
Figure 17.

with Agitated Vessel Data,

ORNL-DWG 71-7996

 

Equivalent Power Dissipation Comparison of Results
71
small bubbles in a 50% mixture of glycerine and water were obviously
strongly gravitationally dominated as evidenced by the equality of the
horizontal and vertical flow coefficients even at very low Reynolds
numbers. A comparison of these "gravity-influenced" data with Sherwood

and Brian's correlation shown on Figure 18 indicates a remarkable simi-

larity. It may be that gravitational forces are generally less important
for flow in conduits than for flow in agitated vessels where there may be

a greater degree of anisotropy.

Recommended Correlations

 

A regression line through all the horizontal flow data except the
water and the 50% mixtures has a Schmidt number exponent of 0.71 using
the literature®® values of 8. These valueé of § (Figure 25, page 11k,
Appendix A) first increase with addition of glycerol, reach a maximum
at ebout 12,5% glycerol, and then decrease. This behavior represents a
striking departure from the Stokes-Einstein behavior usually observed
for aqueous mixtures. If, instead of using these values for 8, a smooth
monotonically decreasing line is drawn through the first, fourth, and
fifth data points of Figure 25 and the values of 8§ taken from that line,
a regression analysis yilelds a Schmidt number exponent of 0.58 — not
much different than the value of 0.5 expected for mobile interfaces.

A regression analysis of all the horizontal data for the glycerine-
water mixtures (except for the 50% mixture) using the original values of
0 (Table III, page 20) and forcing the Schmidt number to have an exponent

of 1/2 results in the eqguation,

Sk = 0.3)4 Reo.faéc Scl/z (dVS/D)l.o , (23)
70

ORNL~DWG 71-7997
100

50

20

10

'SHERWOOD AND BRIAN
SCHMIDT NO. = 3600

O

HORIZONTAL OR VERTICAL FLOW IN
2 2—in. CONDUIT
SCHMIDT NO. = 3446

50% GLYCERINE

Sh/sc!/3

REYNOLDS
NO.

05 12205

15272
18306
20342
22374

 

01
2 3
1 2 5 10 2 5 10 2 5 10

(€m d?s/vsf/3

Figure 18. Equivalent Power Dissipation Comparison of Gravity
Dominated Resultgs with Agitated Vessel Data.
73
with a standard deviation in fr (8h/Sc'/ ?) of 0.19 and an index of
determination of 0.86, The comparison of the data with this equation
is shown in Figure 10,

Since a Schmidt number exponent of 1/2 is expected on theoretical
grounds and since there is little loss of precision by using this
exponent, it is recommended for design purposes that the horizontal
flow mass-transfer coefficients, k , be calculated from Equation {23)
as long as V/Vb is greater than about 3. Operation below V/Vb = 3 is
not recommended because of severe stratification. Equation (23) can
also be used to calculate the vertical flow coefficients, kv, as long
as Fi/Fg’ as determined by Equation (20), is greater than 1.5. Other-
wise, Equation (22) is recommended for the vertical flow coefficients
with the asymptotic values, ka, to be calculated from Equation (21).

As evidenced by the observed high Schmidt number exponent, these
recommendations are for contamination free systems only., For a con-
taminated system with rigid interfacial conditions, the Schmidt number
exponent is expected to be 1/3 and the coefficient multiplying the
equation should also be different., In the absence of supporting experi-
mental data, a tentative correlation for rigid interfacial conditions
might be inferred from Equation (23) to be

Sh = 0,25 Re®:%% gel/ @ (dVS/D)l'O .

The coefficient, 0.25, was obtained by multiplying 0.34 [the coeffi-
cient of Equation (23)] by the ratio of rigid-to-mobile coefficients of
equations applicable to bubbles moving steadily through a liquid.®! A
similar transformation of Equation (21) would be required to obtain the

rigid-interface values of the vertical flow asymptotes. The above
sn /[s¢"2 (dys/D)]

109

HORIZONTAL FLOW

Th

ORNL~-DWG 71-79%94

REGRESSICN LiNE FORCED TQ FIT sct@

Yo

dVS

BUBBLE

MEAN SCHMIDT NO., Sc

DIAMETER

(in.) 370 750 2013 419

C.015

0.02

0.025

0.03

0.035
GLYCERINE — 12.5

Figure 19,

o
>
<
O
>

| d
- 0335 Re®%% 5c'? (%)

1
1
1
r
|
‘

 

10 2 5 10% 2 5

PIPE REYNOL_DS NO.,, Re = VD/v

Correlation of Horizontal Flow Data.
75
equation for rigid interfaces should be used with caution as it has not
been validated by experimental data, In addition the experimentally
observed linear variation with (dVS/D)‘may have been caused by a transi-
tion from rigid-to-mobile interfacial condition. For strictly rigid
interfaces no such transition would be expected to occur and the exponent

on (dvs/D) might then be less than 1.0,
CHAPTER V
THEORETICAL CONSIDERATIONS

Two different viewpoints were considered to describe mass transfer
between small bubbles and liquids in cocurrent turbulent flow. In the
first, a turbulence interaction approach, the bubbles were considered to
be subjected to turbulence forces which impart random motions resulting
in "mean" relative velocities between the bubbles and the fluid. These
"mean" velocities were then considered as "steady" (albeit multi-direc-
tional) and as dictating the mass-transfer behavior.

In the second, a surface renewal approach, the bubbles were viewed
as being associated with a spherical shell of liquid for an indefinite
time during which mass exchange takes place by turbulent diffusion.
This indefinite time was assumed to be related to the bubble size and

the average relative velocity between the bubble and the liquid,
Turbulence Interaction Model

A small bubble suspended in a turbulent field will be subjected to
random inertial forces created by the turbulent fluctuations. Under the
action of & given force, if sufficiently persistent, the bubble may
achieve its terminal velocity and move at a steady pace through the
liquid before being redirected by another force encounter within the
random field, If the "average" value representing the bubble relative
velocity in such a turbulent field could be determined, then a convenient

formulation would be to use that velocity to determine an average bubble

76
T

Reynolds number and stay within the confines of the well-established
relative-flow Frossling-type equations to determine the mass-transfer
coefficients.

The movement of the bubbles through the liquid will be resisted
primarily by viscous stresses. The drag force on a sphere moving
steadily through a liquid is often expressed in terms of a drag coeffi-

cient, Cd’ by the equation,

2 2 paZ
_CdApVb_CdTTp Reb

F = ’
d EgC —Bgc 0

in which the drag coefficient is itself a function of the bubble
Reynolds number, Reb (= vy, dp/u). In relative flows, however, the drag
coefficient-Reynolds number correlation depends on the particular
Reynolds number range. Frequently, two regimes of flow are identified
with the division occurring at Reb ~ 2, Common correlations for the

drag coefficients in these two regimes are given below.

For Re, 5 2,

b
— — 2 -,
Cy= EM/Reb and F, = 3mu Reb/gco . (2hea)
For 2 < Reb < 200,
_ Oeb6 _ 2 1.4 -
Cq = 18.5/Reb and F, = 18.5mu® Rey /Bgcp . (24-b)

In Chapter IV, an expression was developed for the inertial forces

experienced by a bubble in a turbulent fluid,

] |
L 8/ 3 11/ 6 5
F, 5 (a/D) (Re) ) (25)

It might be reasonable to determine "mean" bubble velocities from a

balance between the inertial forces and the drag forces for later sub-

stitution into the Frossling equations. If it is postulated that the
78
above two relative flow regimes also exist for bubbles in a turbulent
field, then two different sets of equations describing the mass transfer
will result. Since the inertial forces depend on the bubble size, a
dispersion of bubbles with a distribution of sizes may have bubbles in
either or both regimes simultanecusly and the mass~transfer behavior
may be described by either set of eguations or take on characteristics
of a combination of the two. The mass-transfer equations resulting for

the two separate regimes are discussed below.

Regime-1: Rey < 2

 

If the bubble motion were predominantly governed by the regime,
Re, < 2, the drag forces would be given by Equation (2k-a), A balance
between the inertial and drag forces, Fi = Fd’ would then give for the

bubble Reynolds number

Re, ~ (4/D)8’ @ Retl/ & | (26)

By this formulation, the bubble relative flow Reynolds number
depends only on the ratio, d/D, and on the pipe Reynolds number which,
for a given bubble size, establish the turbulence level, The Sherwood
number for mass transfer can therefore be determined as a function of
these variables by substitution of Equation (26) into mass-transfer
equations that have been established as applicable to a sphere moving
through a liquid. These are the Frossling-type equations which, for
large Schmidt numbers, usually take the forms

~ 1/ 2 1/ 2
Shb Reb Sc

and

Sh-b ~ Rebl/ 2 Scl/ 3
79

for mobile and rigid interfaces, respectively. Making the conversion,
sh = (D/d) Sh, , and substituting Equation (26) gives for the mobile and

rigid interface pipe Sherwood numbers applicable to cocurrent turbulent

flow,

Sh ~ Sl 2 Re®°22 (d/D)l/S (27)
and

Sh ~ Sc'/ ® Re®®2 (4/D)V 2 , (28)
respectively.

Consequently, in this regime, the pipe Reynolds number exponent is
0.92. For comparison, the experimentally determined value for the
water-glycerine mixtures in this investigation was 0,94, The theoretical
bubble diameter dependence, (d/D)Y 2, however is less than the experi-
mentally determined linear variation, Calderbank and Moo-Young* ® point
out that the linear variation they observed(for bubbles in this size
range probably resulted from a transition from rigid to mobile inter-
facial conditions because small bubbles tend to universally behave as
rigid spheres while larger bubbles require the presence of sufficient
surface active ingredients to immobilize their surface,

If such a transition is the reasdn for the linear variation in this
instance also, then the effect of conduit diameter will be different
from that implied in Equation (23) which did not include actual varia-
tions in conduit diameter. Consequently, anticipated future experiments
with variations in the conduit diameter should help clarify the influence
of bubble mean diameter. In addition, exploratory experiments in this
study indicated that the linear variation did not continue up to larger

bubble sizes and may, therefore, be limitéd to the relatively narrow mean
80
diameter range of approximately C.01l to 0.05 inches. At larger diameters,
the dependence tended to lessen until above mean diameters of about 0.08
inches where the Sherwood number appeared to decrease with increasing
bubble diameter., Since the bubble generator was not generally capable
of producing larger bubbles, further investigation of the bubble size

influence was not possible in this experiment.

Regime-2: Reyp > 2

 

If the bubble motions were predominantly in the regime, Re,_ > 2, the

b

drag forces would be given by Equation (24-b). The balance, F, = Fg

would then give

Re, ~ (/D)3 ++2 Re*/ o5 (29)

The relative-flow bubble Reynolds number in this regime still depends
on the variables that establish the turbulence level but that dependence
is different from that of Regime 1. When substituted into the Frossling
equations for mobile and rigid interfaces, the results are

ah ~ SCL/E ReC* 86 (d/D)—o.a/4.2
and

Sh ~ qel/ 3 Rel-86 (d/D)—O.a/é.E , (30)
respectively.

For this regime the Reynolds number exponent is 0,66. Consequently,
if bubbles in cocurrent turbulent flow experience different flow regimes
similar to bubbles in relative flow, a transition Qould be expected at
higher pipe Reynolds numbers in which the Reynolds number exponent would
tend to become smaller, In the present experiments, the data for water

(plus ~200 ppm N-butyl alcohol) with no glycerine added was obtained at
81
the highest range of Reynolds numbers covered. The experimentally
measured Reynolds number exponent for the water runs was lower than for
“the glycerine-water mixtures and compared favorably with the above
results. In addition, Equation (30) compares quite well with data for
particles in agitated vessels [for example see Equation (3)7.

It is felt that the possible existence of different flow regimes
even in cocurrent turbulent flows is an important concept that, if
further developed, could help explain some of the apparent discrepancies
in the literature data. For example, this may explain the different
slopes observed in this study and may be the reason for observed differ-
ences between mass transfer in agitated vessels and in conduits, It is
more likely, however, that the latter difference is due to greater

gravitational influence in agitated vessels,

Surface Renewal Model

In this analysis each bubble is considered to be surrounded by, and
exchanging mass with, a spherical shell of turbulent liquid in which the
turbulence is isotropic,

A mass balance (Appendix F) in a spherical differential element of

fluid results in the eguation

A _gl3%C, 23 |, 123
a_t—sl:arg+r-a-1_':]+rgar(r UI‘C) . (31)

Making Reynolds assumptions,

and time averaging gives
82

dC d%C 2 aC 1 3 s A7
.fi=s[;+;—r}+—za(r MC) . (32)

H

In turbulent scalar transfer, the "Reynolds" term p’'C’, is often

assumed to be expressible with an eddy diffusivity, E, defined by

However, it is more convenient here to use a recent eddy viscosity defi-
nition by Phillips,58 for which an analogous definition for an eddy

diffusivity in spherical coordinates would be

 

d 5 dC
(r dr)

d 2 F
= (PP u'c) = = : (33)

Using this definition, Equation (32) is expressed more simply as

- (8+1) [fi+£§€] . (34)

Q| Q/
2

or?

The view is now to be taken that, on the average, a bubble remains
associated with a spherical shell of liquid for some indefinite time
after which its surface is completely "renewed" — that is, associated
with an entirely different spherical shell of liquid that has an initisal
uniform concentration characteristic of the bulk fluid., It is felt that
the times of asscciation between the bubble and a given region of liquid
should be related to the magnitude of the turbulent inertial forces or
alternatively to the mean relative velocity between the bubble and the
liquid as established by the balance of the inertial and the viscous
resisting forces,

Therefore a nondimensional time for comparison purposes is proposed
to be

tv
b, = —2

* d *
83
Using this definition along with the following additional definitions

of dimensionless quantities

C, = E/co

r, = r/d
Reb = vbdp/p
Re = VDp/p
Sc =p/pd ,

Equation (34) can be expressed in nondimensional form as

 

3, (L +u/8) [ 3%, , 2 Cy
ot Re, Sc o1, 2 r, Ory j .

Assuming the bubble motion is predominantly in Stokes' regime, Equation

(26) can be used to estimate Re,  and substituted into the above equation

 

to give
3¢, (1 +u/9) To%c, 5 9, ]
3t, | s (35)
*  Cy8c(d/D)¥ @ RV ® | 3r,®  x OTx

where C, 1s a proportionality constant of unknown magnitude but assumed
to be of the order ~1072, A similar equation can be developed for
Regime-2 of the previous model by using Equation (29) for Re,. Logical
boundary conditions for Equation (35) would be

l. Cy (o, r,) =1,

2. ¢, (t>o0, 1/2) = 0, and

3C,

— = 1/ 3
3. 33 =0at o, 1/2 3

»
The third boundary condition above arises from equating the volume

fraction with the ratio of bubble volume to equivalent sphere volume.
Bl

A solution of Equation (35) would give C, as a function of r, and

 

t,. If a radial average, 6;, is defined as
r
*e 2
— _ Il/g Cyx (r*, t*) Ty~ dry
C-X- (t.)(-) = 3

e
2
Il/g Ty~ Ory
then the Sherwood number as a function of time can be expressed as

ac,
.é-;;_- II'_X_ = 1/2

sh (t,) = —-zigifii:———- .

If a bubble is assumed to remain associated with a fluid element for

some unspecified time, T then the average Sherwood number for that

)

period 1s

T-)(—
", sho(t,) dt,

Sh = T . (36)

 

The above analysis is similar to normal surface renewal models in
that the dimensionless time period T, is analogous to a surface age.
There is no real basis for being able to relate T, to the flow hydrody-
namics or the surface conditions; however, it could be treated as a
parameter and the mass-transfer coefficients determined as a function
of this parameter. "Surface age" distributions could then be established
from the experimental data or specified arbitrarily just as they have
been in other surface renewal models. For example, one common assump-
tion has been that the surface is "renewed" each time the bubble travels
(relative to the fluid) a distance egual to its diameter. With the
formulation used here, this assumption would be particularly convenient

because then T* =1,
85
Equation (35) along with its boundary conditions is considered as
a8 surface renewal model. For a solution, a function,

He/"g (Re, sc, d/D: r*) »

- must first be established to describe the variation in eddy diffusivity.
In arriving at his eddy viscosity definition, Phillips®® used a
Fourier decomposition of the turbulent field and, by an elegant analysis,
determined the contributions to the local eddy viscosity due to each

component "wave" making up the field.

Through a paraliel analysis for mass transfer, it is inferred here
that the individual component contributions to the eddy diffusivity are
proportional to the energy of the transverse velocity fluctuations and
inversely proportional to their wave number,

Me,n ™ u?/n - (37)

Defining f(n)dn as the fraction of eddies that have wave numbers in

the range n * 1/2 dn, and summing the contributions over all wave numbers

gives

po ~ [, o= f(man . (38)

&

If XKolmolgoroff's energy spectrum is used, the distribution function
defined above can be assumed to be inversely proportional to the wave
number,
£(n) ~ l/n ’
and Equation (38) becomes
My N'fn (Ei/ne) dn . (39)
To assess the effect of the interface, use was made of Lamont's'?

analysis in which he idealized each component as & sinusoidal viscous
86

"eddy cell" in which the velocities are damped by viscous stresses as
an interface is approached. His analysis gave for a spacial average

(parallel to the interface),

-1/ 3

s

u ~ E (¥)

1

where y is a coordinate defined as y = r —-d/2 and E(y) is a damping
factor depending on the interfacial condition. Lamont's solution of
the viscous "eddy-cell" equation gave for a rigid interface,

§r = [0.294 ny sinh ny + 0.388 sinh ny —0.388 ny cosh ny] ,

and for a mobile interface,

§m = [0.366 sinh ny —0.089 ny cosh ny]

In addition, it is assumed here that only the range of eddy sizes smaller
than, or equal to, the bubble diameter interact with a bubble to produce
eddy transfer to the bubble itself and that each of these eddies is
effectively damped only if it is within a distance from the interface
equal to the wave gize. The eddy sizes assumed present range from a
minimum given by the Kolmolgoroff microscale for pipe flow,

N = D/Rell/ 16
min

to an arbitrary maximum of one-half the pipe diameter,

Mooy = D/2

Consequently, using Equation (39), the ratio of eddy diffusivity effec-
tive to the bubble at a position y to the eddy diffusivity existing away
from the interface, pe/po, is calculated from the following relations:

1. For fl/y > fl/d,
87

fl/y fl/d
. In/k n~® 3 gn + J‘fl/ n~® 3 g2 gn
_ min Y )-F
= =7 . (L4oa)
max o-s/s g

ffl/k

D

 

=

min

2. For m/y < n/d (no damping),

 

fl/d
I n—8/ 3 dn
He fl/Kmin
== =7 . (Lob)
”O j‘ max n_s/ a an
/.
min

A numerical integration of Equation (40) with Mgy = & 1s shown on
Figure 20 for both mobile and rigid damping.

The actual relative eddy diffusivity variation calculated from
Equation (40) will not approach unity in midstream as in Figure 20
because the integration of the numerator is to include only eddies up
to the size of the bubble diameter whereas the denominator is to be
integrated over all wave sizes in the field.

Comparing the mobile and rigid interface curves on Figure 20 indi-
cates that the two conditions would result in very little difference in
mass~-transfer behavior for an essentially passive bubble being acted
upon simultaneously by many eddies — a result not too displeasing
intuitively., A significant difference in behavior then, by this
formulation, must come about by assigning a longer renewal period, T,
to rigid interfaces than to mobile interfaces.

The variation of pe/fi required for a solution to Equation (35) can

be obtained from the product

IJ‘e “O
o) () “*”

if the values for eddy diffusivity in midstream, Hos are known.
e Mo
o o
®  ©

06

0.5

0.4

0.3

0.2

0.1

88

ORNL-DWG 74-8013

 

I I l | o | I

 

 

PLOT OF
/N
/Yy T/ Amax
[ N—8/3 + N—8/3 62 dN
T dN fwy
Ho 7/ X max

- l; N—8/3 dN

~ FOR A RIGID INTERFACE

L AND & = [0.366 sinh NY — 0.089 NY cosh NY:I
FOR A MOBILE INTERFACE

|

!

WHERE £R = [0.294 NY sinh NY + 0.388 sinh NY — 0.388 NY cosh NY:,

—

 

 

 

 

 

 

 

 

y
— MOBILE /4?

/

 

// RIGID

4

 

/

 

 

 

Figure 20,

 

 

 

 

 

 

 

 

A

 

 

 

 

 

 

O 01 02 03 04 05 06 07 08 09

Y/ Xmax

Condition.

Dimensionless Variation of Eddy Diffusivity with
Distance from an Interface, Effect of Surface
89
For the standard definition of eddy diffusivity, Groenhof®® gives

a correlation applicable to the midsection of a pipe,

E = 0,04 J%Wgc7p D . (42)

Letting 7= f p V2/8gC and f = 0,316/Re? * for smooth tubing, then E
from Equation (42) is given by
E/v = 0,04 J/F/8 Re = 0.04 ,/0.316/8 Re™/ & . (L43)
Phillip's definition of eddy viscosity reduces to the standard
definition in the midsection of a pipe. Consequently, it is acceptable

to convert Equation (43) to

po/ag = 0.04 ,/0.316/8 Sc Re /8 (L)

which along with Equation (41) and Equations (40) fully determine a
function

uo/,@ (Re, Sc, d/D, r,)

for use in solving Equation (35).

Tt is realized that Phillip's analysis for eddy viscosity is not
strictly applicable near an interface nor is the "eddy-cell" idealization
a realistic picture of the turbulence. WNevertheless, the variation in
eddy diffusivity based on these concepts was determined through Equations
(43) and (40)., It is felt that the behavior of a pseudo-turbulence such
as this may be similar to a real turbulent field in that the essential
features are retained and the trends predicted in this manner may be
useful. For example, for the condition of turbulent transfer to a con-
duit wall itself there have been measurements of the standard eddy
diffusivity distributions. Therefore, a comparison was made in Figure

21 of eddy diffusivities calculated in the above manner with Sleicher's
ORNL—-DWG 71-8014

 

 

 

 

2
320 XCALCULATED FROM CURVE OF
300 MHe/Ho VERSUS y/Amax (RIGID INTERFACE)
WITH:
280
p, = 0.028 f/2 Re
260 AND A\mgx ASSUMED = 0.5,
r, = PIPE RADIUS
240

 

 

 

' :
DATA OF SLEICHER

 

220 |[O@ CALCULATED| o o__¢o
®

 

 

 

 

VALUES™
200
Re = 80,300
180
~ 160
@O
1 /
140

 

 

120 //
o
100

 

o |/

 

 

o L/

 

 

 

 

——Re = 14

 

 

 

 

 

 

 

20 o
M
O

 

 

0 0.2 0.4 0.6 0.8 1.0
y/rg

Figure 21, Variation of Eddy Diffusivity with Distance from an
Interface. Comparison of Calculated Values with
Data of Sleicher,
91
data,®° For this application of transfer to a conduit, the value of a
in Equation (40a) (the maximum eddy size in this case) was arbitrarily
set equal to 1/2 of the pipe radius, s and the coefficient in Equation
(43) was adjusted slightly to require ue/v to coincide exactly with
Sleicher's value in the pipe midsection at Re = lh,SOO. Cofisidering the
difference in the eddy diffusivity definitions, the comparison is favor-
able and it appears that use of a pseudo-turbulence idealization such as
this may provide a unique @eans of predicting eddy viscosity and eddy
diffusivity variations. Since the determination of eddy diffusivities
and their variation was not the primary concern of this thesis, further
development of these concepts was not considered.

Equations (35), (36), (40), and (uh),-which represent the present
surface renewal model were programmed on a digital computer and numeri-
cal solutions obtained using T, as a parameter. Time did not permit a
complete evaluation of this computer program and the results can only
be presented here as tentative. Figure 22 illustrates the values of
the exponents obtained for an equation of the form

Sh, ~ Re® scP (a/D)°
as a function of T,. The value of T, for which the Schmidt number
exponent was 1/3 (corresponding to rigid interfaces) was approximately
2.7. At this value of T,, the solution for the time-averaged pipe
Sherwood number was essentially independent of the bubble diameter and
varied according to

Sh ~ Re®*%% get/ 2 (45)
The computer results as T, approached zero appeared to approach the

classical penetration solution of Equation (35) obtained for pe/ 8 =0,
10

a, b, ¢

0.5

02

0.1

Figure 22.

 

92

ORNL-DWG 71-10753

SURFACE RENEWAL MODEL
PLOT OF a, b AND ¢ VERSUS Ty
FOR A SOLUTION OF THE FORM

Sh, ~ Re® sc” (d/D)°
VARIABLE VALUE HELD CONSTANT

S¢ = 3000

Re d/D = 0.01

Re 5 x 103

{
{

ASYMPTOTIC SOLUTION AT T, =
OTIC SOLUTION « =0 e = 5 x 107

Sc 3000

2 5 10 20 50 100

Ty

Numerical Results of the Surface Renewal Model., Plots
of a, b, and ¢ (Exponents on Re, Sc, and d/D, Respectively)
as Functions of the Dimensionless Period, T,.
93

 

Sh ~ ./8c (a/D)¥ 2 Re*'Y © / (a/D)
or

Sh ~ qpl/ 2 Reg.ez (d/D)l/B
which is identical to Equation (27). Consequently, if the surface
renewal period, T, , is interpreted as being a measure of the rigidity
of the interface, T, + O being characteristic of mobile interfaces and
T, + ~2,7 (in this case) being characteristic of rigid interfaces, then
this surface renewal model may be useful,

Neither this model nor the preceding turbulence interaction model
satisfactorily predict the observed variation of pipe Sherwood number
with bubble diameter for this range of bubble sizes. Indirect support
is therefore provided for the supposition that the observed linear
variation may be the result of a transition from rigid to mobile

behavior.
CHAPTER VI
SUMMARY AND CONCLUSIONS

Transient response experiments were performed using five different
mixtures of glycerine and water. Liquid-phase-controlled mass-transfer
coefficients were determined for transfer of dissolved oxygen into small
helium bubbles in cocurrent turbulent gas-liquid flow. These coeffi-
cients were established as functions of Reynolds number, Schmidt number,
bubble mean diameter, and gravitational orientation of the flow.

An analytical expression was obtained for the relative importance
of turbulent inertial forces compared with gravitational forces, Fi/Fg'
For conditions in which this ratio was greater than ~1.,5, the variation
in the observed mass-transfer coefficients with Reynolds numbers was
linear on log-log coordinates with identical behavior for horizontal
and vertical flows. Below Fi/Fg = 1.5, the horizontal coefficient vari-
ation continued to be "linear" until the ratio of liquid axial velocity
to bubble terminal wvelocity, V/Vb, decreased to about 3, where severe
stratification made operation impractical., The vertical flow coeffi-
cients underwent a transition from the "linear" variation and approached
constant asymptotes characteristic of bubbles rising through a quiescent
liguid. The variable ratio of Fi/Fg proved to be a useful linear scaling
factor for describing the vertical flow coefficients in this transition
region for which Equation (22) is the recommended correlation.

The Schmidt number exponent for the straight-line portions of the
data was observed to be greater than 1/2 based on physical property
data for & which may be suspect. Fitting the data with a Schmidt number

oL
B
exponent of 1/2 resulted in only slightly less precision than for
the case in which the actual regression exponent was used, and a
definitive choice could not be made between the two. Based on theo-
retical expectations, a Schmidt number exponent of 1/2 would seem to
be appropriate, and consequently, the recommended correlation is
Equation (23).

The variation in mass-transfer coefficient with bubble mean diameter
over the range covered was observed to be linear in agreement with the
findings of Calderbank and Moo-Young'® for agitated vessels. Some pre-
liminary runs made with bubble mean diameters outside the range of this
report indicated that the linear variation does not continue but that
the coefficients level off at both smaller and larger diameters.
Furthermore the coefficients tentatively appear to decrease slowly with
increasing mean diameters above about 0.08 inches.

Consistent with findings of other investigations, the Reynolds
number exponent was significantly greater than expected based on agitated
vessel data compared on an equivalent power dissipation basis, One
explanation is that there may exist greater gravitational influence in
agitated vessels. Another is the postulated existence of different
bubble relative flow regimes.

A seemingly anomalous behavior was observed for the Reynolds number
variation in that the data for water (plus about 200 ppm N-butyl alcohol)
exhibited significantly smaller Reynolds number exponents and a corres-
pondingly smaller exponent for the ratio, (d/D), than that for the
glycerine-water mixtures., There may have been a difference in the

interfacial conditions (the addition of the surfactant creates a "rigid"
96
interface while the glycerine-water mixtures apparently generally had
"mobile" interfacial behavior). However, under steady relative flow
conditions this would result in no difference in the Reynolds number
exponent, Consequently, it was postulated that this difference resulted
from the possible existence of different bubble relative flow regimes.

In support of the above contention, a two-regime "turbulence
interaction" model was formulated by balancing turbulent inertial forces
with drag forces that depend on the bubble relative flow Reynolds num-
ber. The resulting mean bubble velocities were substituted into
"Frossling”" equations to determine the mass-transfer behavior. The
resulting Reynolds number exponent for one regime (Reb < 2) agreed very
well with the experimental value for the glycerine-water mixtures and
that for the other regime (Reb > 2) compared favorably with the water
data and with agitated vessel data on an equivalent power dissipation
basis.

The dependence of Sherwood number on the bubble-to-conduit diameter
ratio, d/D, predicted by the interaction model did not agree with the
observed linear variation. Calderbank and Moo-Young'® pointed out that
the linear variation they observed in agitated vessels for bubbles of
this size range probably resulted from a transition from "small" bubble
to "large" bubble behavior. Such a transition could also explain the
present observations, however, there was no satisfactory means for vali-
dating this.

For comparison, a second analytical model was developed based on
surface renewal concepts which could also include different flow regimes.

This model incorporated an eddy diffusivity that varied with Reynolds
o7
number, Schmidt nunber, bubble diameter, interfacial condition, and
position away from the interface. The variation of eddy diffusivity was
established by using a pseudo-turbulent model in which the turbulence
was simulated by superposed viscous eddy-cells damped by the bubble
interface in a menner determined by Lamont,!!?

The surface renewal model assumed that the "renewal" period for the
bubbles was related to the bubble "mean" velocity resulting from a
balance between turbulent inertial forces and viscous resisting forces,
thus allowing the casting of the equations into nondimensional form with
the pipe Reynolds number, the Schmidt number, and d/D as parameters. A
closed solution of the equations was not obtained but a tentative numer-
ical solution employing a digital computer indicated that, in the limit
of small dimensionless renewal period, T, — interpreted as representing
mobile interfacial behavior, the classical penetration solution of this
particular form of the diffusion equation resulted.

As T, approached a value of approximately 2,7 (in this case), the
computer solution was independent of‘(d/D) and resulted in a Schmidt
number exponent of.ul/3. Therefore, this value of T, was interpreted
as representing rigid interfacial behavior.

Explicit results based 'on the models described above along with a
listing of the more significant observations of this study are given
below:

1. Bubbles generated in a turbulent field are well characterized
by the distribution function

£(8) = 4 (o®/mMV 2 82 Exp (—a8®) ,

where

Q
I

= [b /7 N/687% 2
98

2, The average volume fractions occupied by gas in bubbly flow
are approximated by Hughmark's correlation®% only at higher flows. In
horizontal flow, when the ratio of axial velocity of the liquid to the
bubble terminal velocity is below ~25, Hughmark's correlation predicts
volume fractions lower than those observed, In vertical flow, while
the volume fractions are higher in downcomer legs than in riser legs,
they can be established by using Hughmark's correlation for the mean
and accounting for the buoyant relative velocity of the bubbles in each
leg.

3. At low turbulent flows stratification of the bubbles in hori-
zontal conduits prevented operation for ratios of axial velocity to
bubble terminal velocity below ~3.

4, Even at Reynolds numbers well into the turbulent regime, hori-
zontal and vertical flow mass-transfer coefficients differ. The Reynolds
numbers above which they become equivalent are marked by the dominance
of turbulent inertial forces over gravitational forces.

5. As Reynolds numbers are reduced, vertical flow mass-transfer
coefficients approach asymptotes characteristic of bubbles rising through
a quiescent liquid, The ratio of turbulent inertial forces to gravita-
tional forces serves as a useful linear scaling factor for estimating
the mass-transfer coefficients at these lower Reynolds numbers.

6. The effect of Reynolds number on Sherwood number for flow in
conduits is not as would be expected based on comparison with agitated
vessel dafia on an equivalent power dissipation basis, For example, the
observed turbulence-dominated data are correlated by

Sh/scl/ 2 = 0,34 Re®*®% (g/D)1*° (23)
29
whereas one obtains from the agitated vessel data of Calderbank and
Moo-Young' ® for small bubbles
Sh/Sct/ ® = 0,082 Re®-%° (2)
and of Sherwood and Brian'” for particulates
Sh/Schs ~ ReC* 61 (d/D)—o.lz . (3)
In thisrthesis the two-regime turbulence interaction model and the
surface renewal model exhibit identical results for mobile interfaces

in the "Stokes" regime (Re, < 2),

b
Sh ~ ScV/ 2 Re®?2 (4/D)Y 2

which compares well with the observations represented by Equation (23).

In the second regime (Re. > 2), the turbulence interaction model

b

for rigid interfaces results in
Sh ~ Scl/ 8 Re0-86 (d/D)‘o'a’4'2
and the rigid interface interpretation of the surface renewal model
gives
Sh ~ Scl/ 3 Re©-85
as compared, for example, with Equations (2) and (3).

7. The observed linear wvariation of Sherwood number with bubble
diameter was not predicted theoretically. Consequently, following
Calderbank and Moo-Young,'® it is conjectured that this variation re-
sults from a transition from rigid (small bubbles) to mobile (large
bubbles) interfacial behavior for this size range.

8. Data of this study that were obviously gravitationally influ-
eficed compare favorably with data for particulates in agitated vessels,
giving rise to the speculation that gravitational forcegs may be more
influential in agitated vessels where there may exist a greater degree

of anisotropy compared with flow in conduits.
CHAPTER VII
RECOMMENDATIONS FOR FURTHER STUDY

Experimental

Time did not permit a complete investigation of the effects of all
the independent variables, Consequently, projections of this study into
the future include experiments involving variations of the conduit diam-
eter and the interfacial condition. It is anticipated that these studies
will help clarify the role of d/D, in particular with regard to the
observed linear variation of mass-transfer coefficient with bubble diam-
eter. These projected studies will also attempt to extend the ranges of
variables covered through improvements in the bubble generating and
separating equipment., It is hoped that these improvements will reduce
the magnitude of the "end-effect" and thereby provide greater precision
to the data. Parenthetically, the high rates of mass transfer observed
in the bubble separator may qualify it for further investigation as a
possible efficient in-line gas-liquid contactor.

For practical purposes it is recommended that mass-transfer rates
also be measured in regions of flow discontinuities such as elbows, tees,
valves, venturis, and abrupt pipe size changes. An objective of these
"discontinuity" studies would be to test Calderbank and Moo-Young's
hypothesis that mass-transfer rates can be universally correlated with
the power dissipation rates.

As a direct extension of the work of this thesis, others might con-
sider use of different fluids to provide a more definitive variation of

the Schmidt number and of the interfacial condition. The studies could

100
101
have the additional objective of demonstrating that surface tension is
not an influential variable other than for its effect on the mobility
of the interface., Experiments designed to look at the actual small-
scale movements of bubbles in cocurrent turbulent flow and the eddy
structure very close to the interface would help guide further theoret-
ical descriptions and may help validate the dimensionally determined
expression for the average turbulent inertial forces,

One contention of the present work, the possible existence of
different flow regimes yielding different Reynolds number exponents,
should be further tested. A substantially widened range of Reyneclds
number for a given bubble size in a viscous fluid might uncover a tran-
sition from one regime to another.

In practical applications, the interfacial area available for mass
transfer is equally as important as the mass-transfer coefficient.
Therefore, for systems in which relatively long term recirculation of
the bubbles is anticipated, the bubble dynamic behavior becomes of
interest. For example, more information is needed on bubble breakup
and coalescence which tend to establish an equilibrium bubble size in
a turbulent field. More important perhaps, is the effect of bubbles
passing through regions with large changes in pressure (e.g., across a
pump) where they may go into solution and, as the pressure is again
reduced, renucleate and grow in size. The effects on mean bubble sizes
and the interfacial areas available under such conditions are not well
known and this particular aspect of bubble behavior could provide a

fruitful field for further research,
102

Theoretical

Two extreme viewpoints were taken in this report in which bubbles
were considered as being either essentially passive in a turbulent field
with the mass-transfer behavior being governed by the "sweeping" of the
surface with random eddies or, alternatively, as moving through the tur-
bulent liquid and establishing a boundary-layer type of behavior.

The "surface renewal" model developed in this report was only ten-
tatively evaluated. TFurther development of the model is anticipated and
additional solutions should demonstrate the technique by which surface

renewal concepts can be applied to cocurrent turbulent flow.,

A complete mechanistic description of mass transfer between bubbles
and liquids in cocurrent turbulent flow would presumably include the
transient effects of a developing boundary layer as a bubble is acceler-
ated in first one direction and then the other by random inertial forces,
Superimposed on this would be the effects of the surrounding eddy struc-
ture and the characteristics of the eddy penetrations through the
developing boundary layer. Further efforts to theoretically describe
these simultaneous effects should be considered with possible solutions
on a digital computer.

The use of pseudo-turbulent fields (e.g., an eddy-cell structure)
to determine the transport rates and to establish such properties as an
eddy diffusivity should provide useful insights into the actual behavior
in real fluids and should help predict data trends. For example, the
miltiple boundary layer structure established by Busse®! for the vector

field that maximizes momentum transport in a shear flow strongly resembles
103

an artificial eddy-cell structure. Starting with such a structure, one
could work "backwards" to calculate eddy viscosities (for example) as a

function of position away from a solid boundary.
LIST OF REFERENCES

1. Oak Ridge National Laboratory, Molten Salt Reactor Program Semi-
annual Progress Report for Period Ending February 29, 1968, USAEC
Report ORNL-425L4, August (1968).

2, Peebles, F. N., "Removal of Xenon-135 from Circulating Fuel Salt
of the MSBR by Mass Transfer to Helium Bubbles," USAEC Report
ORNL-TM-2245, Oak Ridge National Laboratory, July (1968).

3.  Harriott, P., "A Review of Mass Transfer to Interfaces,” The Can.
Journal of Chem. Eng., April (1962).

L,  Calderbank, P. H., "Gas Absorption from Bubbles-Review Series No.
3., The Chemical Engineer, 209-233, October (1967).

 

5. Gal-Or, B., G. E. Klinzing, and L. L. Tavlarides, "Bubble and Drop
Phenomena,” Ind. and Eng, Chem., 61(2): 21, February (1969).

6. Tavlarides, L. L., et al., "Bubble and Drop Phenomena,"” Ind. and
Eng. Chem., 62(11): 6, November (1970).

7. Regan, T. M. and A. Gomezplata, "Mass Transfer,” Ind. and Eng.
Chem,, 69(2): 41, February (1970).

8. Regan, T. M. and A, Gomezplata, "Mass Transfer,” Ind. and Eng.
Chem., 62(12): 140, December (1970).

9. Jepsen, J. C., "Mass Transfer in Two-Phase Flow in Horizontal
Pipelines," AIChE Journal, 16(5): 705, September (1970).

10. Scott, D. S. and W. Hayduk, "Gas Absorption in Horizontal Cocurrent
Bubble Flow," Can, J. Chem. Eng., Lh: 130 (1966).

11. Lamont, J. C., "Gas Absorption in Cocurrent Turbulent Bubble Flow,"
PhD Thesis, The University of British Columbia, August (1966).

12, Lamont, J. C. and D. S. Scott, "Mass Transfer from Bubbles in
Cocurrent Flow," Can. J. Chem. Eng., 201-208, August (1966).

13. Heuss, J. M., C. J. King, and C. R, Wilke, "Gas-Liquid Mass Transfer
in Cocurrent Froth Flow," AIChE Journal, 11(5): 866, September (1965).

14, Harriott, P., "Mass Transfer to Particles: Part II, Suspensed in a
Pipeline," AIChE Journal, 8(1): 101, March (1962).

15« Figueiredo, O. and M. E, Charles, "Pipeline Processing: Mass

Transfer in the Horizontal Pipeline Flow of Solid-Liquid Mixtures,"
The Can. J. of Chem, Eng., 45: 12, February (1967).

105
106

16. Calderbank, P. H. and M. B, Moo-Young, "The Continuous Phase Heat
and Mass-Transfer Properties of Dispersions," Chem, Eng. Sci.,

16: 37 (1961).

17. Sherwood, T. K. and P. L. T. Brian, "Heat and Mass Transfer to
Particles Suspended in Agitated Liquids,” U. S. Dept. of Interior
Research and Development Progress Report No. 334, April (1968).

18, Barker, J. J. and R, E. Treybal, "Mass Transfer Coefficients for
Solids Suspended in Agltated Liquids," AIChE Journal, 6(2): 289-.295,
June (1960).

19. Boyadzhiev, L. and D. Elenkov, "On the Mechanism of Liquid-Liquid
Mags Transfer in a Turbulent Flow Field," Chem. Eng. Sci., 21: 955
(1966).

20. Porter, J. W., S. L. Goren, and C. R, Wilke, "Direct Contact Heat
Transfer Between Immiscible Liquids in Turbulent Pipe Flow," AIChE
Journal, 14(1): 151, January (1968).

21, Sideman, S. and Z. Barsky, "Turbulence Effect on Direct-Contact
Heat Transfer with Change of Phase,” AIChE Journal, 11(3): 539,

May (1965).

22, Sideman, S., "Direct Contact Heat Transfer Between Immiscible
Liquids," Advances in Chemical Engineering, Vol. 6, pp. 207-286,
Academic Press, New York (1966),

 

23, Fortescue, G. E, and J. R. A. Pearson, "On Gas Absorption into a
Turbulent Liquid," Chem. Eng. Seci., 22: 1163-1176 (1967).

24, Kozinski, A. A, and C. J. King, "The Influence of Diffusivity on
Liquid Phase Mass Transfer to the Free Interface in a Stirred
Vessel, AIChE Journal, 12(1): 109-110, January (1966),

25. King, C. J., "Turbulent Liquid Phase Mass Transfer at a Free Gas-
Liquid Interface," Ind, Eng. Chem. Fund., 5(1): 1-8, February (1966).

26, Peebles, F. N. and H. J, Garber, "Studies on the Motion of Gas
Bubbles in Liquids, Chem. Eng. Progr., 49(2): 88-97, February (1953).

27. Ruckenstein, E., "On Mass Transfer in the Continuous Phase from
Spherical Bubbles or Drops," Chem. Eng. Sci., 19: 131-146 (1964),

28. Griffith, R. M., "Mass Transfer from Drops and Bubbles," Chem. Eng.
Sci., 12: 198-213 (1960).

29, Redfield, J. A, and G. Houghton, "Mass Transfer and Drag Coefficients
for Single Bubbles at Reynolds Numbers of 0.02-5000," Chem. Eng. Sci.,
20: 131-139 (1965).
30.

31,

32.

33.

3k,

35.
36.

37.

38,

39.

Lo,

L1,

Lo,

b3,

L,

107

Chao, B, T., "Motion of Shperical Gas Bubbles in a Viscous Liquid
at Large Reynolds Numbers," Phys. Fluids, 5(1): 69-79, January
(1962).

Lochiel, A, C. and P. H. Calderbank, "Mass Transfer in the Contin-
uous Phase Around Axisymmetric Bodies of Revolution,'" Chem. Engr.
Sci., 19: 475-4BL, Pergamon Press (1964).

Higbie, R., "The Rate of Absorption of a Pure Gas Into a Still

Liquid During Short Periods of Exposure,’ Presented at American
Institute of Chemical Engineers Meeting, Wilmington, Delaware,

May 13-15 (1935).

Danckwerts, P. V., "Significance of Liquid-Film Coefficients in
Gas Absorption,” Ind. Eng. Chem., Engineering and Process Develop-

ment, 43(6): 1460-1467, June (1951).

Toor, H. L. and J. M, Marchello, "Film-Penetration Model for Mass
and Heat Transfer," AIChE Journal, 4{(1): 97-101, March (1958).

Whitman, W. G., Chem. and Met. Eng,, 29: 147 (1923).

Levich, V. G., Physicochemical Hydrodynamics, Prentice-Hall, Inc.,
Englewoods Cliffs, N. J. (1962).

 

Hinze, J, O., Turbulence, McGraw-Hill Book Co., Inc,, New York

(1959).

Middleman, S., "Mass Transfer from Particles in Agitated Systems:
Application of the Kolmogoroff Theory," AIChE Journal, 11(4):
750-761, July (1965).

Harriott, P. and R. M, Hamilton, "Solid-Liquid Mass Transfer in
Turbulent Pipe Flow," Chem, Eng. Sci., 20: 1073-1078 (1965).

Brian, P. L. T. and M. C. Beaverstock, "Gas Absorption by a Two-
Step Chemical Reaction," Chem, Eng. Sci., 20: 47-56 (1965).

Davies, J. T., A. A. Kilner, and G. A. Ratcliff, "The Effect of
Diffusivities and Surface Films on Rates of Gas Absorption,"

Chem, Eng. Sci., 19: 583-590 (1964).

Gal-Or, B., J. P. Hauck, and H., E. Hoelscher, "A Transient Response
Method for a Simple Evaluation of Mass Transfer in Liquids and Dis-
persions,”" Intern. J. Heat and Mass Transfer, 10: 1559-1570 (1967).

Gal-Or, B. and W. Resnick, "Mass Transfer from Gas Bubbles in an
Agitated Vessel with and without Simultaneous Chemical Reaction,”

Chem, Eng. Sci., 19: 653-663 (1964),

Harriott, P., "A Random Eddy Modification of the Penetration
Theory," Chem. Eng. Sci., 17: 1h49-15L (1962),
L5,

Le,

e

ho.

50,

51.

52,

5Lk,

5.

56.

57.

58.

59,

108

Koppel, L. B., R. D. Patel, and J. T. Holmes, "Statistical Models
for Surface Renewal in Heat and Mass Transfer,” AIChE Journal,

12(5): 9k1-955, September (1966),

Kovdsy, K., "Different Types of Distribution Functions to Describe

a Random Eddy Surface Renewal Model," Chem. Eng. Sci., 23: 90-91
(1968).

Perlmutter, D. D., "Surface-Renewal Models in Mass Transfer," Chem,
Eng. Sci., 16: 287-296 (1961).

Ruckenstein, E., "A Generalized Penetration Theory for Unsteady
Convective Mass Transfer," Chem, Eng. Sci., 23: 363-371 (1968).

Sideman, S., "The Equivalence of the Penetration and Potential Flow
Theories,”" Ind. Eng. Chem., 58(2): 54-58, February (1966).

Harriott, P., '"Mass Transfer to Particles: Part I. Suspended in
Agitated Tanks," AIChE Journal, 8(1): 93-101, March (1962).

Banerjee, S., D. S. Scott, and E. Rhodes, "Mass Transfer to Falling
Wavy Liquid Films in Turbulent Fiow," Ind. Eng. Chem, Fund., 7(1):
P2-26, February (1968),

Barker, J. J. and R. E. Treybal, "Mass Transfer Coefficients for
Solids Suspended in Agitated Liquids," AIChE Journal, 6(2): 289-295,
June (1960),

Galloway, T. R. and B. H. Sage, "Thermal and Material Transport
from Spheres,'" Intern. J. Heat Mass Transfer, 10: 1195-1210 (1967).

Hughmark, G, A,, "Holdup and Mass Transfer in Bubble Columns,"
Ind. Eng. Chem., Process Design and Development, 6(2): 218-220,
April (1967).

Jordan, J., E. Ackerman, and R. L. Berger, "Polarographic Diffusion
Coefficients of Oxygen Defined by Activity Gradients in Viscous
Media," J. Am. Chem. Soc., 78: 2979, July (1956).

Bayens, C., PhD Thesis, The Johns Hopkins University, Baltimore,
Maryland (1967).

Resnick, W. and B. Gal-Or, "Gas-Liquid Dispersions,' Advances in
Chemical Engineering, Academic Press, New York, Vol. 7, pp. 295-395
(1963).

 

Phillips, O. M., "The Maintenance of Reynolds Stress in Turbulent
Shear Flow," J. Fluid Mech., 27(1): 131-144 (1967).

Groenhof, H, C., "Eddy Diffusion in the Central Region of Turbulent
Flows in Pipes and Between Parallel Plates," Chem, Eng. Sci., 25:
1005-1014 (1970).
60.

61.

109

Sleicher, Jr., C. A., "Experimental Velocity and Temperature Pro-
files for Air in Turbulent Pipe Flow," Transactions of the ASME,
80: 693-704, April (1958).

Busse, F. H., "Bounds for Turbulent Shear Flow,” J. Fluid Mech.,
h1s 219-240 (1970).
APPENDIX A

PHYSICAL PROPERTIES OF AQUEOUS-CLYCEROL MIXTURES

111
112

ORNL-DWG 71-8003

106

SCHMIDT NUMBERS CALCULATED
FROM DATA OF JORDAN, ACKERMAN
AND BERGER - 25°C

5
2
109
N MTX
GLYCERINE AND WATER
5
P
S
N
Il 2
o
o)
o
=
4
. 10
o
=
T
&
5
2
103
5
2

 

0 20 40 60 80 100
GLYCERINE (%)

Figure 23, Schmidt Numbers of Glycerine-Water Mixtures.
P/C (atm~liters/mole x 1072)

HENRY'S LAW CONSTANT, H

113

ORNL—-DWG 71—8004

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4
l | [ [
SOLUBILITY OF OXYGEN IN MIXTURES
OF GLYCERINE AND WATER - 25°C
3
2
,'---_’/
1 e
C
DATA OF JORDAN, ACKERMAN
AND BERGER
0

 

 

0 10 20 30 40 50 60 70 80
GLYCERINE (%)

Figure 2L, Henry's Law Constant for Oxygen Solubility in Glycerine-
Water Mixtures.

90
11k

ORNL-DWG 71-8005

 

DIFFUSION OF OXYGEN THROUGH
DIFFERENT MIXTURES OF GLYCERINE
AND WATER AT 25°C

/C\

 

 

 

%, MOLECULAR DIFFUSION COEFFICIENT (cm@/sec x 10°)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

90

S
O\
\O\
\\
N
O-O“O
DATA OF JORDAN, ACKERMAN
AND BERGER
10 20 30 40 50 60 70 80
GLYCERINE (%)
Figure 25. Molecular Diffusion Coefficients for Oxygen in Glycerine-

Water Mixtures, Data of Jordan, Ackerman, and Berger.
p, DENSITY (g/cc)

115

ORNL-DWG 71-8006

 

DENSITIES OF GLYCERINE AND
WATER MIXTURES - 25°C

‘o/fig

v

 

1.200

1.100

 

y
¥

/O

 

 

 

 

 

 

 

’b\
> @ O

O
>

 

 

JORDAN, ACKERMAN AND
BERGER

HANDBOOK OF CHEMISTRY
AND PHYSICS

CHEMICAL ENGINEERING
HANDBOOK

BEILSTEIN'S HANDBOOK OF
ORGANIC CHEMISTRY

| |

 

 

1.000 E?//

20 40 60

80 100 120

GLYCERINE (%)

Figure 26, Densities of Gly

cerine-Water Mixtures.

140
p, VISCOSITY (CP)

109

116

ORNL-DWG 71-8007

VISCOSITY OF GLYCERINE-WATER
MIXTURES AT 25°C

DATA OF JORDAN, ACKERMAN
AND BERGER

10 20 30 40 50 60
' ' GLYCERINE (%)

Figure 27, Viscosities of Glycerine-Water Mixtures.

 

70
APPENDIX B

DERIVATION OF EQUATIONS FOR CONCENTRATION CHANGES

ACROSS A GAS-LIQUID CONTACTOR

Consider the cocurrent flow of a gas and a liquid in a constant
area pipeline of cross section AC and length L., In a differential
element of length df, a dissolved constituent of concentration C in

the ligquid is transferred into the gas as shown below.

 

 

e ' ; _— -_—
Q,C - I[_ . ] +Q, (C + 4ac)
- v ; — -
, d
et L 0% (€« Cg)
< eee df e

A mass balance for the dissolved constituent gives

Q,dC = —ka A dd (C — CS) (B-1)

il

c C - -2
dicg +ka A a4 (C cs) , (B-2)
where C is the liquid phase average concentration,'fig is the gas phase
concentration, and CS is the concentration existing at the gas-liguid

interface,

Dividing Equetion (B-2) by Equation (B-1) gives

ac Q
dC g
Integrating Equation (B-3) and letting Eé - O when C = Cy gives
G, = (a /e )(c; = C) . (B-4)

If the interfacial concentration is assumed to be at "equilibrium"

and the solubility of the dissolved constituent is expressible by Henry's

117
118
Law, then

HC = ¢ RT . (B-5)

Q
cs=§i (?”i)(ci—fi) . (B-6)

Equation (B-6) can be substituted into Equation (B-1) to obtain

Q, &€ = —xa A, 4 [T = (RI/H)(Q,/Q)(c; —T)]

i

~ka A_ 44 C (1 + RTQE/HQg) — (RTQz/HQg) c.l . (-7

Expanding and dividing by dez gives

 

= — kaAc'YCi By
mPC T T G -

where B’ = ka A, (1 + Y)/Qfl and y = (RT/H)(Qfl/Qg).

 

/
Use of the integration factor eB 4 permits the following solution
= Y \ B4
0 = <l.+-y‘/ci + (const.) e . (B-9)

At 4 = 0, C = Ci’ therefore the constant of integration is
(const.) = Ci/(l +v)

and

 

5=< Y \C.+(—-]—'——\.e"eifi’c
i i

l+y/ \ 1+,
Therefore the ratio of the exit (4 = L) to inlet concentration, Ce/Ci,
is

1 _B'L
€ ’

 

Ce/Ci=l+y+1+y

or defining B = B'L,

Y+ e B
Ce/ci Tl + vy 2
119
where

ka AC L (1 +v)

 

v = (RT/H)(Qfi/Qg) and B = T,
121

APPENDIX C
INSTRUMENT APPLICATION DRAWING

| R Ne [H0497-QF-001-D-0 |

T REVISIONS| | [ T T T T 17
R

 

 

 

 

 

 

 
  
     

 
    
            
    
 
   

 

 

 

 

    
  

 

 

    
    
 
    
   

 

 

 

 

 

 

  
   

  

 

 

 

 

 

 

 

 

 

 

  
 
   

 

  
      

 

 

 

 

 

 

 

 

  
 
  
 
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     
 

 

   

 

 
  

 

       
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

APFPD. APPD. ASPD. APPD
-— - - — ——————— - — U
N
: [ : X
A TR- 1, ; ’ rerofoTsTorde o1 oo 3
| PT6 >~ l | ) SEDESSR NI | : ettt
‘ _ )RR ANEMOMETER
' ; S S LS B s j mmm e .
g ' \ | ¢ gvn¥oen oy ' w '
RNy ey oo : ™ STROBELUME
3 : N & KRRRRARIRE HOT Frm u SELLIME.
i : N _ : 9 | PROBE 25 LA e
| i
] 0.k = e T e
9 g9 12
§ ¥R ¥ 9 5 = SAFETY SHIELDING i
3 & E ;
Q - TN . \\ - 2
S | e 1 : TR-1, ;
3 T AR i < T
XX N L | R o
38 L Pl & :
32 ] ; @ v gt ¥
Q vl B i forerermterdy o770 &
| v | : BRI T
TN ¥ B 8 . T ARG SR Q
83 ; ; 0-/100F51 f‘?-' ?_-g—i-i-‘ TN S
) - 5 _ - ¢ '
Ifx | : R S S I ¢
§ N g 3 0 by boodge E o
11 OXYCEN Wl o, . CALIBRATION ; I bedo—m -4
FY¥QY W TANKS b r-=-—2 Y
SATURATION, ST N I ! I " .
4 TANK NN s b N el o (0 pgAN¥ Mo TTTme-- : ! : , | |
AANK N o N POLAROI O '
5 Y oM\ Aogh
| gl kT acrivarep CAMERA @ @ @— &
Nady BY CAMERA
3k LENS OPENING. ! ! ! ! !
4 T =X F oy ey . e P e b - i ( ! U Aad
! l
< ‘3 : S o~ FLASH CONTROLS! : i ! | 28/
w Qg o TURN ON & THME ! ! ! ! |
| v R 2O ¢ bLenTH OF FLASHL _ A AN A A
TILY _ B COOLING - oY o @ ?
i Tuohk swe| |} N 00 6.8 M. DRAIN He0 b o - 8 p t:';‘ y 8 ‘;’]" -
} 5 EXPANSION e <Ap.5-58 | < < = 92 <
JOINTS 3 2 @'— @ % :
\‘-.:‘\ P $ i - ‘(AoeM' 7 |
N x ¥ P01 AROCRAPHIC I
% ) O METER PROBES |
g ! “ PHOTO ~PORT # 1
| 0-40 L
E o | i fM R
3 qQ T
< 3 BUBBLE |
| OXYGEN—fl‘j 3+ ;: - GENERATOR | STROBELUME SAFETY SHIELDING
= : R R | LAMP #/
] DISPERSER. ’ I BT Lo
i
4 DRAIN é #vv7
) -0 1€ Hi 1B —— | pewvEnT
- 41k HV-#ID
Hy-DI% W ~ — —_— - —_
- N x g
Q _ Hv-0/5 6 % § ;
2 [
¥ . i1 | *
i z 2 5
x T
| D g MR g
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4 0-200 PSS
z .
£
L] B 9
23 < FCC4
z ELEMENTARY WIRING DIAGRAM QF-002-0
e —~ECCH
0-1060% |8 REFERENCE DRAWINGS DWG. NO.
4 0-3000 PST BRAWH TATE | DRAWING CHECK GATE | APFROVED BATE
; ECKEITH 1227 oo iicassen
. OESIGNED DATE | SENIGN CHECK DATE | COLLAN. AFRD_ DATE
| sicnr . | WK, 1970 '
i{l 7 w coonomator | W, KREWSCON
| ":. AIR SUPPL Y PROJECT TITLE
i MSBR MASS TRANSFER LOOP
Vi e
% P Pz \ INSTRUMENT
HELIIM | ‘ COOLING O/L HELIIN HELIUM OXYGEN
SupPLY_ | | PUMP PWAY— —— ® 7N SUPPLY SURPPLY SUPPLY APPLICATION DIAGRAM
CYLINDERT3 8/ ! CYLINDER* 2 CYLINDER #t cyenoER #y |
L — —{PwR. T RECCA- | INSTRUMENTATION AND CONTROLS DIVISION
Y _ — — - — _ — - _ _ _ | OAX RIDGE NATIONAL LABORATORY
— — — _ > . —_ e —— - — — - UNLESS STHLRwisx UNION CARBIDE CORP.
I | | 1 E | I EE%E?:. E:';i ,j 5. K’//uM—-A—,/ A Hitrc s, IW%, She o
No. i“:'-‘::o’":':_"l ario I DatE i oRAWN I ArFD. l Arrml arrD. Ann,—l REVISIONS . 'C‘L-'NONE I_IO497—QF_OO |_D_O

 

 

 

 

 

 

Figure 28. Instrument Application Drawing of the Experiment Facility.
dys, BUBBLE MEAN DIAMETER (in)

123

APPENDIX D

INSTRUMENT CALIBRATIONS

ORNL-DWG 71-7998

 

 

1 T L r 1 r
BUBBLE GENERATOR CHARACTERISTICS

LIQUID FLOW, Q_ (gpm)—~20 30 35 40 45 50 55 60 65 70 75 80 85 90
SYMBOL—~©C ® A &4 V ¥ 0 © & & 4 & V¥V V

 

 

 

010 || Lt ! || sese GENERATOR (SCHEMATIC)
"~ | 6As To LIQUID VOLUMETRIC FLOW .
! PROBE SHOWN AT 4 1/2-in.
RATIO, Qg/QL = 0.3% POSITION (FULLY WITHDRAWN)

 

 

0.09 FLIQUID = INDICATED MIXTURE OF <=FLOW DIRECTION (LIQUID}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

GLYCERINE AND WATER Y a-—(GAS)
oop LOAS = HELIUM . .
A <7 { % Yo 37.5% 50%
0% GLYCERINE 12.5% 25% QL (gFIH'n)
0.07 | 30y QL (gpm)
' a_ (gpm) \, 30
35 Q_ (gpm) \ a5
0.06 — N
20 \ N\ N \
N\, 5 X LN A
005 === 5 % 40 \ \
\ 50 \ N
AN \ a5 A\ A
0.04 _50\\\ h 50 VN ~ 45
A
N 559, \ 50 50N
\ A P> v\ ~N \ v
A\ NGO 55 N\ 4\ A

 

 

 

60
003 :
653\
e,

 

 

 

\ —656 Ny \ o5 \\fi\, NN 554
0, v
\\ ; \ 75-?"&- \ ?o'°x\\ ,&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

?O-a..._\ L. i ; "
0.02 |80=ggi ™o A)‘, —( Sa ¥, A= ~ QA |
OBV | [T ~ o
RNy d .l
0.01
0
3 5 7 9 3 5 7 9 3 5 7 o 3 5 7 9 3 5 7 9

PROBE POSITION (in)

Figure 29. Bubble Size Range Produced by the Bubble Generator.

 
ROTAMETER SCALE

12k

ORNL-DWG T1-8008

 

 

 

 

 

 

 

100
WATER (0% GLYCERINE)
S0
80
" <
60
50% GLYCERINE
| 1
UPSCALE
DOWNSCALE
50

 

o \

 

 

|
-
.18/

 

 

 

 

 

CALIBRATION OF ROTAMETER 1

FOR TWO MIXTURES OF GLYCERINE

AND WATER

0 | |
0 20 40 60 80 - 100

FLOW (gpm)

 

 

 

 

Figure 30. Calibration of Rotameter No. 1 (100 gpm).
125

ORNL—DWG 74—8009

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 /
o0 //
®
80 /
50% GLYCERINE /
70
UPSCALE /
DOWNSCALE&
/ WATER {0% GLYCERINE)
60
/
S @
2 /
50
b /
bl
-]
[
QO
7 o\
40
30
./
20 [l
10
CALIBRATION OF ROTAMETER 2
FOR TWO MIXTURES OF GLYCERINE
AND WATER
o | | |
0 10 20 30 40 50
FLOW (gpm)
Figure 31. Calibration of Rotameter No. 2 (40 gpm).
SCALE READING (%)

126

ORNL-DWG 71—-80!0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 .:O
20 /
@
80 50% GLYCERINE
 UPSCALE V
DOWNSCALE&
70 /
60 @
WATER (0% GLYCERINE)
50
2 / /
@
30 //
20 &4
{0
CALIBRATION OF ROTAMETER 3
FOR TWO MIXTURES OF GLYCERINE
AND WATER
i | |
0 2 4 6 8 10
FLOW (gpm)

Figure 32. Calibration of Rotameter No. 3 (8 gpm).

 
AP, PRESSURE DROP ACROSS CAPILLARY TUBE (inches of water)

127

ORNL-DWG 71-8011

 

CALIBRATION OF CAPILLARY-TUBE FLOWMETER
AT A PRESSURE OF 50 psig

ATMOSPHERIC PRESSURE, Pg = 29.41 in. HG
ATMOSPHERIC TEMPERATURE, To = 27.6°C

 

HELIUM FLOW MEASURED WITH
A WET-TEST METER

 

 

 

 

 

 

 

~

‘./

 

 

 

 

 

 

 

0

0.02 C.04 0.06 0.08 0.10
HELIUM FLOW (SCfM)

Figure 33. Calibration of Gas-Flow Meter at 50 psig.

0.12
DISSOLVED OXYGEN READING (ppm)

40

38

36

34

32

30

28

26

24

22

20

128

ORNL-DWG 71-8018

 

®
A

POLAROGRAPHIC MEASUREMENT OF DISSOLVED
OXYGEN CONCENTRATION IN TWO MIXTURES
OF GLYCERINE AND WATER

O MAGNA CORPORATION PROBE f
MAGNA CORPORATICON PROBE 2
BECKMAN CORPORATION PROBE

 

 

 

 

/

 

 

 

 

 

 

 

oL

 

WATER (0% GLYCERINE) CURVE CALCULATED
FOR HENRY'S LAW CONSTANT (H = P/C) EQUAL
TO 1.74 psi AIR/ppm DISSOLVED OXYGEN
TAKEN FROM PERRY, "CHEMICAL ENGINEERS
HANDBOOK"

 

 

 

 

/|

 

3/

 

 

 

 

/

 

 

A/}?.S% GLYCERINE CURVE CALCULATED
/7~ FOR H = 355 psi AIR/ppm DISSOLVED

 

 

W

ACKERMAN, AND BERGER

 

© OXYGEN TAKEN FROM DATA OF JORDAN,

 

W

 

/

l//////
N\

 

e

 

W

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3k,

20

30

AlR

40 50 60 70 80
PRESSURE (psiq)

Calibration of Oxygen Sensors in two Mixtures of
Glycerine and Water.

S0
APPENDIX E

EVALUATION OF EFFECT OF OXYGEN SENSOR RESPONSE SPEED

ON THE MEASURED TRANSIENT RESPONSE OF THE SYSTEM

Instrument responses are typically exponential in nature. Thus,
if the sensor reading is defined as Cr’ and the actual loop concentra-
tion as 6'(both functions of time) it is safe to assume an instrument
response equation of the form

dC
T

=k, (C-c) , (E-1)

where Kr is an instrument response coefficient.
The transient response of the loop itself is given by an equation

of the form

Therefore, Equation (E-1) can be expressed as

-z-z-ll +kK.C, =K, Coe-KLt . (E-2)
Integration of Equation (E-2) with the initial condition Cr = Co at
t = 0 gives
cr/co = [1 - Kr/(Kr - %)) ey [Kr/(Kr — k)] e Xpt
- A (E-3)

The manufacturers stated response time for the Magha oxygen sensors
is 90% in 30 seconds., This response results in a value of

K =4.61 .
r

The maximum observed rate of change of oxygen concentration in the
transient experiments corresponded to

129
130

KL = 0.75

(On the average, the experiment transients resulted in KL < 0.3.) There~
fore, for this case, A = =0.19, B = 1.19 and

cr/co = 1,19e7°*78% — g 19e %81t | (E-k)

An examination of Equation (E-U4) shows that as time progresses the
second term becomes negligible compared to the first, and the measured
slope approaches the actual transient slope of 0,75, For example, the
measured slope for this "worse" case is 0.7k after only one minute of
transient compared to the real value of 0.75. Therefore, to further
minimize this possible error, the slopes of the measured transients were
taken only from the final six minutes of the curve permitting an initial

n

"response adjustment" time of several minutes. The error due to the

instrument response, then, is assumed to be negligible,
APPENDIX F
MASS BALANCES FOR THE SURFACE RENEWAL MODEL

Consider a differential region in a spherical shell of fluid

surrounding a bubble as shown below,

 

Mass balances for the concentration, C, of a dissolved constituent

within the liquid are obtained as follows:

 

Convection
in: U C bmr®
T
3(U c)
out: Um (r + dr)® EU C +-—7§——— dr
3(U c)
net convection = (out-in) = [4mr? “5“_ dr + 8mrdr (U C)! (F-1)
Diffusion
ins O Lmr® oC
ar
out: 8 bw (r + dr)® [ BC + é——-d ]
o r 3p?
r
X A A 5 8 C ac
net diffusion = (out-in) = |8 bmr dr + 8 8mrdr 3> (F-2)
ar® |

131
132

Storage
-LI-TT a a3 oC
net loss = —- [(r + dr) r3] %
N2 o
= |=brr® dr 5t | (F-3)

 

Summing the contributions (F-1) through (F-3) gives

ot

a(Urc) 5
+hrr? dr | ——+=(UC)} =0
r T
Dividing by Umr® dr gives

2 3(r°U_C)
53_(1:39 é_.g.+§..a_q +.1'......_.....__r...... . (F_)_@)
ot arg r or rg

2
orr? dr £ 4§ bmr® dr [_a_ng%g_g]

Meking the Reynolds assumptions

substituting into Equation (F-4), expanding and collecting terms gives

~ ’ 2~ 2n ! ~ ’
_a_c_+ac=&[ac+ac+_2_ac+§§_c_]
r or r

 

3r

 

ot ot 372 3r2 T
d0(u’C) |, d3(u’c’) . 2, = P oyt
$ommml S 2 (u'C) + = (u'c?) . (F-5)

The time average of a quantity, C, is defined as

jtta ¢ dt
1

in which the time interval, (t; — t,), is long enough for the time
average of the fluctuating quantities in Reynolds assumption to be
zero but short compared to the transient changes in C. Therefore, =

time average of Equation (F-5) gives
(F-6)
APPENDIX G
ESTIMATE OF ERROR DUE TO END-EFFECT ADJUSTMENTS

The measured ratios of exit-to-inlet concentration, K, across a
gas-liquid contacter were extracted from the measured slope, S, of the

log-concentration versus time data by the relation

-8V /Q
K = e s" Al

The error involved in measuring the various quantities used to establish

K are estimated to be

AS
'S_" -~ O. Ol s
oy
£~ 0.03

S
and

N
L. 0.03 .
9

Consequently, the error in K can be estimated from

K - K

 

 

LK max min
K K ’
where
(8 —AS)(VS -—AVS)
Kpax = Bxp | — (Qz + AQE)
and
(8 + M8)(V_ + AVS)
Kpin = B¥P |~ g —Acs,zj :
PRy ]
135
The minimum ratio measured for K was ~0.9, therefore the maximum estimate
of the error is

.99) .01) (1.03)
K O.gclaz%%ggg%fi _.0,9(L—g%54§7%§
K

0.9

 

~ 0,02

In Chapter III, the ratio, Ky, applicable to the test section above

was calculated from
Ko = KI/KII ,

where KI was the measured ratio across the bubble generator, the test

section, and the bubble separator together, and KII was that across

just the bubble generator and bubble separator. Therefore, the maximum

instrument-precision induced error in Kg is estimated to be

 

 

K3 Kz,max u'Ka,min
Kz ~ Kz
1l + 0,02 1 -0,02
KI/KII<1—O.OE> KI/KII<1 +o.02> .
~ K /K < lOO .
T 11

In establishing KIland KII in separate tests, the inability to
exactly duplicate conditions results in an error greater than the above.
An estimate of the maximum magnitude of this error can be had by examin-
ing the data for the 75% water-25% glycerine mixture (Figure 13, page 63).
Before the end-effect adjustment, the calculated vertical and horizontal
flow mass-trénsfer coefficients for the 0.02-inch mean diameter bubbles
were essentially identical. However, after the adjustment they differed
by ~25%, It is felt that this difference mostly arises from the inabil-

ity to exactly recreate the vertical flow conditions as a result of

alterations made in the bubble generator between the original test and
136
the end-effect test, Consequently the horizontal flow data are consid-
ered the more "exact" although they should still reflect the ~10% error

estimated due to measurement precision,
APPENDIX H

MASS TRANSFER DATA

137
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

38
WATER + ~200 ppm N-BUTYL ALCOHOL
VERTICAL FLOW IN 2-in. CONDUIT

3 b

® [scHmioT No. u 419 Q/QL (%) Qi {gpm)  Re

34 0.3 0.5
O ® 20 35,583

32 O N 35 62,269
A A 40 71,165

30
¢ ¢ 50 88,955

>8 Vv v 60 106,748
A ¢ 65 115,642

26 v v 80 142,381
A A 100 177,913

24

22 100

20

18

l‘ ‘,/15’65
,80
16 }VZ pad
14
’,/' ',r35
' VW & de0 49 50,!]//,20
VeV Q‘A// Q00
10 AN AT L= e
s //
8 A a’
/ 0O
6 SR
~ |

4

2

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
dys, BUBBLE MEAN DIAMETER (in.)
Figure 35. Unadjusted Mass Transfer Data., Water Plus 200 ppm

138

ORNL-DWG 71-7965

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N-Butyl Alcohol. Vertical Flow.

 
26

249

20

18

k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

139

ORNL—-DWG 71-7966

 

 

 

 

 

 

 

 

 

WATER + ~200 ppm N-BUTYL ALCOHOL
[HORIZONTAL FLOW IN 2-in. CONDUIT Q, (gpm) re
Qg/QL = 0.3% _ —

5o [ SCHMIDT NO. = 419 O 50 88,955 _

® 60 106,748
A 70 124,537 |
A 80 142,331
v 90 160,119 _
50
. P
80
14 //
1 70
A /
12 P4 60
A /
/AS o

 

 

 

'
48’0'—__0_0‘0"‘---50

 

 

 

 

 

 

 

 

 

 

 

 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
d,,, BUBBLE MEAN DIAMETER (in.)

Figure 36. Unadjusted Mass Transfer Data. Water Plus ~200 Ppm
N-Butyl Alcohol. Horizontal Flow.
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

18

16

14

12

10

140

ORNL-DWG 71-7967

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12.5% GLYCERINE
Qqg/QL = 0.3%
VERTICAL FLOW IN 2-in. CONDUIT
FSCHMIDT NO. = 370 W1 THOUT™ WITH® Q. (gpm) Re —
O ® 15 19,288
0 B 20 25,718
A A 35 45,006
O ¢ 50 64,294
v v 65 83,583
*ADDITION OF ~200 ppm N-BUTYL ALCOHOL -
65
/ |
50
v// /35
\ O/ ¢ b H,20
O
\
N5
P
-~
B
0 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08
dys, BUBBLE MEAN DIAMETER (in.)
Figure 37. Unadjusted Mass Transfer Data. 12.5% Glycerine-87.5%

Water.

Vertical Flow.
14

12

k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

141

ORNL—-DWG 71—7968

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HORIZONTAL FLOW IN 2—in. CONDUIT
12.5% GLYCERINE Q_ (gpm) Re
SCHMIDT NO. = 370 O 35 45,006
- o
Qg/QL = 0.3% o 50 64.294
A 65 83,583
A 75 96,442
v 85 109,302
85 75
A
10 /
/ 65
8 ’
A"
A’ ._,-1-50 ‘
6 % o 0— i
— |
®
a , / 35
0 0.0 0.02. 0.03 0.04 0.05 0.06 0.07 0.08

dys, BUBBLE MEAN DIAMETER (in)

Figure 38. Unadjusted Mass Transfer Data. 12.5% Glycerine-87.5%
Water. Horizontal Flow.
UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

k,

1k2

ORNL-DWG 74-7969

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
25% GLYCERINE
Qg/QL = 0.3%
VERTICAL FLOW IN 2—in. CONDUIT ,
9 | SCHMIDT NO. = 750 )
/
8 V.
2
6
5
4
3
2 /o QL (gpm) Re
20 O ® 20 17,636
O n 30 26,454
1 A A 40 35,272
O ¢ 50 44,090
Vv v 60 52,908
o TADDITION( OF ~ 200 ppm N-BUTYL ALCOHOL

 

 

 

 

 

0 0.01 0.02 003 0.04 0.05
dys, BUBBLE MEAN DIAMETER {in.)

.06

0.07

0.08

Figure 39. Unadjusted Mass Transfer Data. 25% Glycerine-75% Water.

Vertical Flow.
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

143

ORNL-DWG 71-7970

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
7
6
5
4
3
> O 40 35,272
® 45 39,681
A 50 44,090
9 56: 48,498 25%, GLYCERINE
57,317
{ - ’ 1 Qq/QL = 0.32 —
v 76 66,135 97/t
SCHMIDT NO. = 750
HORIZONTAL FLOW IN 2-in
CONDUIT
0 _ { | {
0 0.01 002 0.03 0.04 0.05 0.06 0.07
dys, BUBBLE MEAN DIAMETER (in.)
Figure L40. Unadjusted Mass Transfer Data., 25% Glycerine-75% Water.

Horizontal Flow.
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

14k

ORNL-DWG 71-—79714

 

 

WITHOUT™®  WITH* QL (gpm) Re
o O o 20 13,079
o » 30 19,619
A A 40 26,159
v v 45 29,429

—*ADDITION OF ~200 ppm N-BUTYL ALCOHOL

 

 

/45

 

 

 

 

/

/A

Y
prd

// /’/Qr

 

40

30
é |
20
s Or

 

 

 

A

7
/

~

 

 

 

 

 

 

 

 

 

 

 

|
0
I/.

-

e -
37.5% GLYCERINE
SCHMIDT NO. = 2015
Og/QL = 0.3%
VERTICAL FLOW IN 2-in. CONDUIT

0 0.04 0.02 0.03 0.04 0.05 0.06 0.07
dys, BUBBLE MEAN DIAMETER (in.)
Figure L1, Unadjusted Mass Transfer Data. 37.5% Glycerine-62,5%

Water.

Vertical Flow.

 

0.08
UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

K,

10

145

CRNL—DWG 71-7972

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.08

l I
QL (gpm) Re
O 30 19,619
@ 35 22,889 —
A 40 26,159
A 45 29,429
v 50 32,699
v 55 35,968 ——
a 60 39,238
N 7C 45,777
60
/55
Y
_, 50
45
35
P e— i-.___ i
_‘0'30
37.5% GLYCERINE
SCHMIDT NO. = 2015
—— Qg/QL = 0.3%
HORIZONTAL FLOW IN 2-in. CONDUIT
O 0.01 0.02 0.04 0.05 C.06 0.07
dvs, BUBBLE MEAN DIAMETER (in.)
Figure 42, Unadjusted Mass Transfer Data. 37.5% Glycerine-62.5%

Water.

Horizontal Flow.
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

146

ORNL-DWG 71-7973

 

]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

WITHOUT®  WITH*® oL (gpm) Qg/QL (%) Re
— O o 10 0.5 4,068
0 n 20 0.5 8,137
A A 30 0.5 12,205
O ¢ 40 0.5 16,274
v v 50 0.3* 20,342
*ADDITION OF ~200 ppm N-BUTYL ALCOHOL
tEXCEPT WHERE NOTED
'/50
ro.5°/o
| |
o 30
] /v,ro.s /o /A//.zo
4
Z A
/g/fi 8 / a
A %& 10 _]
T o7
@
- rfff” L~ ~
S -~
/
u 5 +
50% GLYCERINE
SCHMIDT NO. = 3446
VERTICAL FLOW IN 2-in. CONDUIT
| l |
0.04 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
dyvs, BUBBLE MEAN DIAMETER (in.)
Figure 43. Unadjusted Mass Transfer Data. 50% Glycerine-50%

Water.

Vertical Flow.
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

147

ORNL—DWG 71{-7974

 

 

 

 

 

Q. (gpm) Re
O 30 12,205
° 35 14,238
A 40 16,274
A 45 18,306
Vv 50 20,342
v 55 22,374

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

@ —— 35
—\m
30

| 50% GLYCERINE

SCHMIDT NO. = 3446

Qg/QL = 0.3%

HORIZONTAL FLOW IN 2-in. CONDUIT

i { | ]

O 0.014 0.02 0.03 0.04 0.05 0.06 0.07

dvs, BUBBLE MEAN DIAMETER (in.)

Figure 44, Unadjusted Mass Transfer Data. 50% Glycerine-50%
Water, Horizontal Flow,
UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

K,

148

ORNL-DWG 71-7984

 

 

LOCUS OF
FI/FQ = 1

71

/ HORIZONTAL
VERTICAL FLOW

FLCW

5 104 2 5 105 2 3
PIPE REYNOLDS NO., Re = VD/»

100
WATER + ~200 ppm N~BUTYL ALCOHOL
SCHMIDT NO. = 419
BUBBLE
50 MEAN
DIAMETER HORIZONTAL VERTICAL
" (in) FLOW FLOW
0.015
0.02
0.03
20 0.04
10
CALCULATED
ASYMPTOTES
5
>
1
103 2
Figure b5,

Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. Water Plus ~200 ppm N-Butyl Alcochol.,
Horizontal and Vertical Flow.
UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

K,

100

50

20

149

 

—

n—— —

VERTICAL/
FLOW

2

PIPE REYNOLDS NO., Re = VD/v

12 5% GLYCERINE
SCHMIDT NO. = 370
BUBBLE
ME AN
DIAMETER HORIZONTAL VERTICAL
(in) FLOW FLOW
0.015 ° O
0.02 N a
0.03 A A
0.04 v v
CALCULATED
ASYMPTOTES
103 2 5 104
Figure 46,

ORNL-DWG 71-7985

OCUS OF

Y
"/

\—HORIZONTAL

FLOW

 

105

Unadjusted Mass Transfer Coefficients Versus Pipe

Reynolds Number as a Function of Bubble Sauter-Mean

Diameter.

12.5% Glycerine-87.5% Water.

and Vertical Flow.

Horizontal
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/hr)

50

20

150

ORNL-DWG 74-7986

25% GLYCERINE
SCHMIDT NO. = 750

BUBBLE
MEAN
DIAMETER
{in))

! |

HORIZONTAL VERTICAL
FLOW FLOW

 

©.015
0.02
0.03
0.04

LOCUS OF Fi/Fg = 1.5

ALCULATED

ASYM

 

PTOTES

t
HORIZONTAL FLOW

VERTICAL FLOW

103 2 5 104 2 5 105

PIPE REYNOLDS NO., Re = VD/v

Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. 25% Glycerine-75% Water. Horizontal and
Vertical Flow.
UNADJUSTED MASS TRANSFER COQEFFICIENT {ft/hr)

k,

151

ORNL—DWG 71-7987

100
37.5% GLYCERINE

SCHMIDT NO. = 2015

o
o

BUBBLE
ME AN

DIAMETER HORIZONTAL VERTICAL
(in.) FLOW FLOW

0.015
0.02
0.03
0.04

™
o

LOCUS OF Fi/Fg = 15

o

|82

/
' o1

C T f!f
—— __CALCULATED —
/ASYMPTOTES - l’..’
| HORIZONTAL
A FLOW

|

VERTICAL
FLOW

 

103 2 5 104 2 5 10°
' PIPE REYNOLDS NO., Re = vD/»

Figure 48. Unadjusted Mass Transfer Coefficients Versus Pipe
Reynolds Number as a Function of Bubble Sauter-Mean
Diameter. 37.5% Glycerine-62,5% Water. Horizontal
and Vertical Flow.
k, UNADJUSTED MASS TRANSFER COEFFICIENT (ft/nr)

50

no
O

o

w

N

152

50% GLYCERINE

SCHMIDT NO. = 3446
|

 

 

ORNL-DWG 71—7988

5 103

BUBBLE
MEAN
DIAMETER HORIZONTAL VERTICAL
(in.) FLOW FLOW
0.015 O
0.02 0
0.03 A
0.04 v
LOCUS OF Fi/Fq = 1.5
A4S
A
CALCULATED V/fi—-fi'
ASYMPTOTES 1%
Ua®
-0
- o
103 2 5 104 2
PIPE REYNOLDS NO., Re = VD/v
Figure 49, Unadjusted Mass Transfer Coefficients Versus Pipe

Reynolds Number as a Function of Bubble Sauter-Mean

Diameter., 50% Glycerine-50% Water.
Vertical Flow.

Horizontal and
k, MASS TRANSFER COEFFICIENT (ft/hr)

153

ORNL—DWG 71-—7975

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

QL (gpm) Re

O 20 35 583

P 35 62,269
o A 50 88,955

A 70 124,537

v 80 142 328

v 100 177,913
5

/20, 35
6
° /
4
e A =7 50, 70
3 J
100/
2 / A , _ WATER + ~200 ppm N-BUTYL ALCOHOL
8O// SCHMIDT NO. = 419
A Qg/OL = 0.3%
70/{ VERTICAL FLOW IN 2—in. CONDUIT
1 504
20, 35
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

dys, BUBBLE MEAN DIAMETER (in.)

Figure 50. Mass Transfer Data Adjusted for End-Effect. Water Plus
~200 ppm N-Butyl Alcohol, Vertical Flow. '
154

ORNL-DWG

7T1-7976

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7 o
QL (gpm) Re
S o 50 88,955
_ ® 60 106,746
5 A 70 124,537
5 La 80 142,328
E v 90 160,119
5‘5 e = == | EAST SQUARES LINES PASSING THROUGH
= ORIGIN FOR DIAMETERS UP TO 0.035
w 4
o 70,80
o ?
/ 60
1t p /
i 90
2 / & 50
3 A\ A7
s’
- g 5
<
p= ‘f///
xfl / /
/ WATER + ~200 ppm N-BUTYL
1 ALCOHOL |
/’ SCHMIDT NO. = 419
Qg/QL = 0.3%
HORIZONTAL FLOW IN 2-in. CONDUIT
0 i | l
0 0 .01 0.02 0.03 0.04 0.05 0.06
dys, BUBBLE MEAN DIAMETER (in.)
Figure 51, Mass Transfer Data Adjustéd for End-Effect. Water Plus

~200 ppm N-Butyl Alcohol.

Horizontal Flow.
k, MASS TRANSFER COEFFICIENT (ft/hr)

155

ORNL—DWG 71-7977

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dvs, BUBBLE MEAN DIAMETER (in.)

Figure 52. Mass Transfer Data Adjusted for End-Effect. 12.5%
Glycerine-87.5% Water. Vertical Flow,

9
QL (gpm) Re
O 20 25,718
8 L © 35 45,006
A 50 64,294
A 65 83,583
.
20
6
5 / 35,50
o
/‘65
A
o
A 7
A”A
A
12.5% GLYCERINE
SCHMIDT NO. = 370
Qq/QL = 03%
VERTICAL FLOW IN 2-in. CONDUIT
| | | I
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
K,

MASS TRANSFER COEFFICIENT (ft/hr)

156

ORNL—-DWG 71-7978

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
12.5% GLYCERINE
SCHMIDT NO. = 370
Qg/QL = 0.3%
2 HORIZONTAL FLOWAIN 2—in. CONDUIT
QL.(gpm) BQ
O 35 45,006
6 —e 50 64,294 —
A 65 83,583
A 75 96,442
v 85 {09,302
5 — —_— ]
—-— | FAST SQUARES LINES PASSING THROUGH
ORIGIN FOR DIAMETERS UP TO 0.035 in.
1
//85
4
3
’.—-
2 35
o~
/
i
O
O 0.01 0.02 0.03 0.04 0.05 0.06

dys, BUBBLE MEAN DIAMETER (in.)

Figure 53. Mass Transfer Data Adjusted for End-Effect. 12.5%
Glycerine-87.5% Water. Horizontal Flow.
k, MASS TRANSFER COEFFICIENT (ft/hr)

157

ORNL—DWG 71-7979

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dvs, BUBBLE MEAN DIAMETER (in.)

7 ; |
25% GLYCERINE
SCHMIDT NO. = 750
s | Qa/Q = 03%
VERTICAL FLOW IN 2-in. CONDUIT
OL (gpm) Re
O 20 17,636
5 ® 30 26,454
A 40 35 272
A 50 44,090
| v 60 52,908
4 | 207
O 0
30
/
3 ®
60 y N
2 / //
Y/
j/d o’
Az
”
1 |
/ x M_.
|
0 i
0 0.01 002 0.03 0.04 0.05 0.06

Figure 54. Mass Transfer Data Adjusted for End-Effect. 25%
Glycerine-75% Water. Vertical Flow.
158

ORNL—-DWG 71{—7980

 

 

 

 

 

 

 

k, MASS TRANSFER COEFFICIENT (ft/hr)

 

 

 

 

 

 

 

 

 

 

 

7
25% GLYCERINE
SCHMIDT NO. = 750
Qg/Q-{_ = 0.3%
6 HORIZONTAL FLOW IN 2—in. CONDUIT
| | |
Q_ (gpm) Re
o 40 35272
5 o 45 39,681
A 50 44,090
A 55 48,498
v 65 57,317
v 75 66,135
4 }— ' ]
LEAST SQUARES LINES PASSING THROUGH
ORIGIN FOR DIAMETERS UP TO 0035 in.
3
2
1
O
O Q.01 0.02 003 0.04 0.05 0.06

dvs, BUBBLE MEAN DIAMETER (in.)

Figure 55. Mass Transfer Data Adjusted for End-Effect. 259
Glycerine-75% Water. Horizontal Flow.
159

ORNL—DWG 71—7981

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

S [37.5% GLYCERINE
SCHMIDT NO. = 2015
Qg/QL = 0.3%
HORIZONTAL FLOW IN 2-in. CONDUIT
- 5
= | |
< QL {gpm) Re
}_ O 35 22.889
zZ ., |e 40 - 26,159
o A 45 29,429
i A 50 32,699
il
S v 55 35,968
x ° [V 60 39,238
§ D 70 45,777
2 55
<g
T oo
w ” o 3.5
w
<{
=
~
X
0
0.0 0.02 0.03 0.04 0.05
dys, BUBBLE MEAN DIAMETER (in.)
Figure 56, Mass Transfer Data Adjusted for End-Effect.

Glycerine-62,5% Water.

Horizontal Flow.

0.06

37. 5%
MASS TRANSFER COEFFICIENT (ft/hr)

k,

160

ORNL—DWG 71—-7982

 

 

 

 

 

 

 

 

 

 

 

 

 

5
50% GLYCERINE
SCHMIDT NO. = 3446
VERTICAL FLOW IN 2-in. CONDUIT
4 B I
QL (gpm) Re
O 20 8,137
3 o 30 12,205 20,30
A 40 16,274 ]
A 50 20,342
2 "’,f(ffli””””
® Y -A———"'A"""'4015O
a—"
4
O A
0.01 0.02 0.03 0.04 0.05 0.06 0.07

dvs, BUBBLE MEAN DIAMETER (in.)

Figure 57. Mass Transfer Data Adjusted for End-Effect, 50%
Glycerine-50% Water. Vertical Flow.
k, MASS TRANSFER COEFFICIENT (ft/hr)

Y

W

N

-

161

ORNL—DWG 71-7983

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dys, BUBBLE MEAN DIAMETER (in)

50% GLYCERINE
SCHMIDT NO. = 3446
HORIZONTAL FLOW IN 2-in. CONDUIT
i |
— Q_ (gpm) Re
O 30 12,205
® 40 16,272
| A 45 18,306
A 50 20,342
v 55 22,374
45 155
] ¢—40,50
-a‘--——‘\\o
30
0 0.0 0.02 0.03 0.04 0.05 0.06

Figure 58, Mass Transfer Data Adjusted for End-Effect. 50%

Glycerine-50% Water.

Horizontal Flow.

0.07
£(8)

f(n)

LIST OF SYMBOLS

Bubble interfacial area per unit volume

Mean acceleration of a fluid element of size A in a turbulent
field

Conduit cross sectional area

Bubble projected cross sectional area

Local concentration of a diésolved constituent in a liquid

Time averaged component of C

Turbulent fluctuating component of C

Bulk-average concentration of a dissolved constituent in a liquid
Drag coefficient for a bubble moving through a liquid

Gas~liquid contactor inlet value of Ca

Vg

Gas—liquid contactor exit value of Cavg

Initial value of C
avg
Interfacial value of C
Bubble diameter
Sauter-mean diameter of a bubble dispersion
o0 [=-] .
[=] 8°e(8)as / [ 82£(s)as]
o 0
Conduit diameter
Molecular diffusion coefficient
Eddy viscosity
Blasius friction coefficient
Bubble size distribution function

Frequency distribution function for turbulent eddies of wave

number n

163
164
Drag force on a bubble moving through a liquid
Mean inertial force on a bubble due to turbulent fluctuations
Gravitational force on a bubble (buoyancy)
Gravitational acceleration
Dimensional proportionality constant relating force to the
product of mass and acceleration
Solubility constant in Henry's Law relations
Mass transfer per unit time per unit volume of liquid
Local mass-transfer coefficient
Axially averaged mass-transfer coefficient
Horizontal flow values of k
Vertical flow values of k
Low flow asymptotic value of kv
Ratio of test section exit-to-inlet concentration, Ce/Ci
Loop response coefficient
Oxygen sensor response coefficient
Test section length
Mass of a fluid element
Wave number of a turbulence component
Number of bubbles per unit volume of liquid
Local pressure in the conduit
Volumetric flow rate of gas bubbles
Volumetric flow rate of liguid
Universal gas constant
Radial coordinate

Fractional rate of surface renewal
AV

165
Time coordinate
Absolute temperature
Radially directed velocity in spherical coordinates
Fluctuating component of Ur
Contribution to u’ of eddies of wave number n
Ligquid axial velocity
Bubble terminal velocity within a liquid in a gravity field
Mean fluctuating velocity of a bubble in a turbulent fluid
Mean fluctuating velocity of a fluid element in a turbulent field
Relative mean fluctuating velocity between a bubble and the liquid
Bubble total velocity in the riser leg of a vertical test section
Bubble total velocity in the downcomer leg of a vertical test
section |
Mean variation in velocity over a given distance in a turbulent
field
Volume of the closed recirculating experiment system
Added mass coefficient for an accelerating spherical bubble
Axial coordinate
A flow parameter used by Hughmark in correlating volume fractions
Ratio of liquid-to-total volumetric flow [Q .0/ (Q ,* Qg)]
A flow parameter used by Hughmark in correlating volume fractions

(E Rel/6 Fr1/8 / Yl/ 4)

Greek Symbols

o

8

Y

Parameter in bubble size distribution function
Gas-liquid contacter parameter [= ka AL (1 + y)/sz

Gas-liquid contacter parameter [= RTQI/HQg]
166
Bubble diameter used in distribution function (same as d)
Energy dissipation per unit mass in a turbulent liguid
Energy dissipation per unit volume in a turbulent ligquid
Distance scale in a turbulent ligquid
Minimim eddy size in a turbulent liquid
Maximum eddy size in a turbulent ligquid

Liquid viscosity

'Bddy diffusivity

Contribution to Mo from turbulent component of wave number n
Undamped eddy diffusivity away from an interface

Kinematic viscosity (= p/p)

Interfacial darping function for wviscous eddy cells

Rigid interface form of §

Mobile interface form of £

Liquid densgity

Wall shear stress

Bubble volume fraction

Bubble volume fraction in the downcomer leg of a vertical test
section

Bubble volume fraction in the riser leg of a vertical test

section

Dimensionless Quantities

 

Cx

Cx

Fr

Pe

Dimensionless concentration (EVCO)
Radial average of C_
Froude number (= V?/gD)

Bubble Peclet number (= Reb Sc)
*e

Re

Re

Re

167
Dimensionless radial coordinate (= r/d)
Dimensionless radius of spherical shell of liquid surrounding a
bubble [= 1/2 &/ 2]
Pipe Reynolds number (= VDp/p)
Bubble Reynolds number (= vbdp/u)
Stirrer Reynolds number defined as the product of the stirrer
rotation speed, square of the stirrer diameter, and = divided
by the kinematic viscosity
Schmidt number (= p/ph)
Pipe Sherwood number (= kD/8)
Time average of Sh
Bubble Sherwood number (= kd/®)
Dimensionless time coordinate (= tvb/d)

Period for surface renewal
OO FR P

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