TIME DEPENDENT MASS TRANSFER
FROM SINGLE BUBBLES
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P1935
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BIBLIOTHEEK TU Delft P 1935 2338TIME DEPENDENT MASS TRANSFER
FROM SINGLE BUBBLES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER ELEKTROTECH-NIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DONDERDAG 4 NOVEMBER 1971 TE
14.00 UUR
DOOR
JAN TOMAS LINDT
SCHEIKUNDIG INGENIEUR GEBOREN TE AMSTERDAM
/ ^ d r x?i<i5>
1971
Dit proefschrift is goedgekeurd door de promotoren DR.IR. W.J. BEEK en
r
ACKNOWLEDGEMENTS
I would like to thank my students for their contribution to this investigation.
My thanks are due to the "Laboratorium voor Fysische Technologie"
where I have been provided conditions for completing this dissertation.
Finally, I express my admiration and gratitude to my wife for her understanding and efficient help.
CONTENTS
Summary
Chapter 1 Introduction 1.1 Introduction 1.2 Scope
1.3 Structure of the thesis
Chapter 2 Experimental 2.1 Introduction
2.2 Experimental set-up and procedures 2.2.1 Hydrodynamic measurements
1 Flow visualisation experiments 2 Bubble rise velocity measurements 2.2.2 Mass transfer measurements
1 Introduction
2 Experimental principles
3 Experimental set-up and procedures 2.2.3 The reaction kinetics of carbon dioxide
absorption in buffered solutions catalyzed by arsenite ions
2.2.1+ Property measurements on the phases used 1 Gas phase 2 Liquid phase Chapter 3 Hydrodynamics 3.1 Introduction 3.2 Theory 3.2.1 Velocity of rise 3.2.2 Bubble wakes
3.3 Experimental results and discussion 3.3.1 Bubble rise velocity
3.3.2 Bubble wake visualisation
Chapter k Physical mass transfer
h,1 Introduction
k.2 Theory
1+.2.1 Mass transfer from the bubble surface to a non-separating flow
U.2.2 Wake mass transfer
U.3 Experimental results and discussion
Chapter 5 Chemical mass transfer 5.1 Introduction
5.1.1 Model chemical reaction
5.1.2 Slow and fast chemical reaction 5.1.3 Scope of this chapter
9
9
10 13 ^k1U
11» IT 19 19 20 22 25 25 25 27 29 32 3238
1*7
1*7
51*
61
61
63
68
7177
77
78
79
5.2 Theory 79 5.3 Experimental results auct discussion 81
Appendix A2.1 Physical properties of the absorptive
ooxutions 93 Appendix A3.1 Some properties of the two-dimensional
bubble wake 9^
Appendix A3.2 Estimation of changes in the captive wake
volume 95 Appendix Ak.^ Model to describe the influence of the wake
on physical mass transfer 97 Appendix Al*.2 Flow rate of the liquid passing through the
helical vortex IQl* Appendix A U . 3 Model to describe the influence of both the
wake and air desorption on physical mass
transfer IO6 Appendix A4.U Sources of possible errors in the analysis
of physical mass transfer 115 Appendix A5.1 Sources of possible errors in the analysis
of chemical mass transfer II8
References 120 Notation 123 Samenvatting 127
Summary
TIME DEPENDENT MASS TRANSFER FROM SINGLE BUBBLES
An analysis has been carried out of the physical and chemical mass transfer processes that develop when a single bubble rises in a non-degassed stagnant liquid. The system in which carbon dioxide is absorbed in an aqueous buffer solution has been used. The pseudo first order rate constant for the liquid phase chemical reaction was varied up to about lUO s by adjusting the concentration of sodium arsenite catalyst, with the absence of catalyst being used to provide the datum whereby the system could be regarded as being entirely phys-ically controlled - the reaction rate being extremely slow under these conditions (t_ << t ). In this way experimental data on physical and chemical absorption were obtained under very similar physical condi-tions. The experimental technique (described in chapter 2) provided accurate information on the instantaneous mass transfer rates from and velocities of the bubble.
The hydrodynamic analysis presented in chapter 3 is intended to provide qualitative physical insight into the periodic fluctuations which were observed in rise velocity of bubbles between 0.5 and 1.5 cm equivalent diameter. It considers questions of wake structure on which little information has been available. There is particular emphasis on the wake region adjacent to the bubble rear because of the relevance of this area to mass transfer. A captive liquid region bounded by the bubble surface and the large scale vortex system of the wake has been observed. The downstream vortex wake showed signif-icant circulation, extending in a rotating helical structure which could decay into an irregular vortex loop.
Mass transfer to the vortex wake region is postulated to tsike place by means of small scale eddies that are generated by the helical vortex within the captive liquid. Insight has been gained of the di-mension and velocity of these eddies.
A new theory has been suggested to explain the mass transfer co-efficients in the bubble wake region (chapter h). This theory has shown to be consistent with the experimental results provided for the entire range of chemical reaction rates, and it is because of this agreement that it has been possible to arrive at the conclusions concerning the captive wake. Experimental proof has also been available to confirm that the factor which accelerates mass transfer in the presence of moderately fast chemical reactions is independent of the flow field even in the complex system represented by a bubble and its associated wake.
The time dependent behaviour of the overall mass transfer coeffi-cient, which is dealt with in chapters h and 5j is explained on the
basis of the combined effects of absorption and desorption, the in-tensity of which is affected significantly by the wake structure: the thinner the captive wake, the more pronounced the effect of time dependence. The gradual development of a steady non-zero concentration of solute in the wake is found to occur but is shown to be incapable of explaining the time dependence entirely.
Chapter 1
INTRODUCTION
1.1 Introduction
Many different studies have been devoted to the analysis of drop and bubble phenomena, dealing with several fundamental aspects: formation, movement, disruption, interaction with another interface and heat smd mass transfer. Bubble behaviour, on which our attention is centred, has been observed under very different conditions: in water-like liquids, in molten metals or glass and in various non-Newtonian liquids. The size of bubbles that have been investigated has varied over a very wide range, from tiny spherical bubbles like those occuring in biological systems, up to the very large spherical cap bubbles described by Davies and Taylor (31) who calculated rather dramatically the rise velocity of a gas bubble released in a submarine explosion.
The majority of the investigations on bubble behaviour has been motivated by the relevance to industrial equipment for gas- or vapour-liquid contacting, with some interest originating from pollution problems. Although such processes usually involve bubble swarms created by a large n\jmber of nozzles or by perforations in a plate, there is some need for the study of the fundamental phenomena that occur at the surface of a single bubble in order to develop a rational understanding of industrial and biological systems.
1.2 Scope
The present work deals with the mechanism of mass transfer from a single bubble. However, although ignoring the interaction that occurs in a bubble swarm, even a single bubble presents a very complex phe-nomenon. Generally, a description of the bubble behaviour should deal with both the interior of the bubble and with the liquid phase. This contribution is, however, directed entirely towards processes taking place in the outer phase.
Our attention will be confined to a bubble rising in a stagnant Newtonian liquid of low viscosity. Of course, the term "stagnant liquid" cannot be applied near the interface of the moving bubble, but applies at some distance from the moving body. In so far as the experimental equipment has limited dimensions, the liquid is evidently not at perfect rest. Using the term "stagnant liquid" with respect to the experimental conditions, we always refer to liquid under conditions, where a single bubble rises in a relatively wide channel.
The thesis concentrates on physical and chemical mass transfer from a single bubble rising in a non-degassed stagnant liquid at Reynolds numbers of practical interest. There is much evidence that the flow past a bubble can separate and form a rather large wake. There is very little known about the structure of such a wake and about the mechanism of mass transfer from the bubble surface beyond the separation ring. We will attempt to illuminate this aspect by examining the role of the wake mass transfer in the overall transfer mechanism.
The present study has not examined the influence of bulk phase properties (viscosity, surface tension, density, etc) nor has it con-sidered the influence of local changes in these properties arising from mass and heat transfer in the absorption system. Accordingly, free convection flows will not be under investigation. Interfacial phenomena such as adsorption of surface active agents, or of con-densation are also outside the present scope. The bubble phenomena which will be analyzed will not deal with heat transfer problems as '
such, except for some side effects due to the heat of solution and reaction.
In studies on mass transfer from bubbles, degassed liquids have been frequently used. In this work the bubbles were absorbed in air
saturated liquids, and thus an influence of desorption of air into the bubbles on the overall mass transfer rate has been considered. In the mass transfer system chosen the bubble volxane does not rapidly change; hence no significant radial flow is induced.
1.3 Structure of the thesis
In order to develop a better understanding of bubble mass transfer it has been necessary to develop an improved model of the flow con-ditions, particular with interest to the wake. With such a hydrodynamic basis, it has been possible to give the wake mass transfer a physical interpretation. The final step of the analysis of the mass transfer includes a verification of the model for conditions of chemical mass transfer. This approach determines the structure of the thesis.
To detect the bubble phenomena experimentally, it is necessary to measure the instantaneous hydrodynamic and mass transfer parameters since the bubble changes in size during its rise; the volume of the bubble tends to increase due to a decreasing hydrostatic head, and at the same time tends to shrink due to absorption. The method used is described in chapter 2. Besides the measurement of the instantaneous mass transfer rate and the instantaneous bubble rise velocity, the present experimental observations comprise also flow visualisation studies and measurements of physicochemical properties.
In chapter 3 the bubble rise velocity is analyzed and a basic, but qualitative description of the wake of the bubble is given.
Chapters k and 5 are very similar to each other in structure. They analyze and interpret the behaviour of the mass transfer coeffi-cient for both physical and chemical mass transfer . Instantaneous
rates of these mass transfer processes are analyzed over the entire "life" of the bubble in the experimental set-up (about 10 seconds). Different time dependent mechanisms of mass transfer will be examined, with particular attention to the bubble wake region.
Both the chapters on mass transfer (chapters it and 5) and also the chapter on hydrodynamics (chapter 3) have a simple structure of four paragraphs: introduction, theory, experimental results and discus-sion. The paragraphs marked as "theory" include the theory that has been used as a starting point, our own contributions are given in discussion paragraphs.
Chapter 2 EXPERIMENTAL
2.1 Introduction
The aim of the present study is to investigate physical and chemical absorption taking place during the rise of a bubble in a column of stagnant liquid. The system carbon dioxide-buffer solution containing a variable amount of sodium arsenite has been chosen for the present experiments; the reaction kinetics of this system are described in literature (1), (2). The kinetics are of the first order with respect to carbon dioxide. The chemical reaction rate is a function of the content of arsenite ions in the solution.
This investigation deals with a category of bubbles, the Reynolds number of which varies between 900 and 5000 (Re = d U/v , d being the equivalent bubble diameter, U the bubble rise velocity and v the liquid kinematic viscosity). These bubbles belong to the transition region between spherical and spherical cap bubble regimes. Some aspects of hydrodynamic behaviour of this bubble category are not yet clearly understood, particularly, the present knowledge on the bubble wake structure is rather limited (see chapter 3 ) . With respect to this fact we laid emphasis on flow visualisation experiments as well as on conventional bubble rise velocity measurements.
In order to analyze the mass transfer phenomena in the system concerned the following aspects are to be considered:
1. bubble rise velocity 2. shape of the bubble
3. flow pattern around the bubble
h. gas law for the gas enclosed in the bubble 5. rate of mass transfer
6. equilibrium law for the physical absorption 7. kinetics of the chemical reaction
8. physical data on the phases used.
Data with respect to the subjects 2. and 6. are available from reliable sources (3), (4), the other information has been obtained from own measurements. The description of the present experiments is divided
into four groups as indicated in Tab. 2-1. Table 2-1
Classification of the present experiments type of experiment 1 hydrodynamics description in section: 2.2.1 analysis in chapter: |
3 1
2.2 Experimental set-up and procedures
2.2.1 Hydrodynamic measurements
In the present work we carried out measurements on bubble rise velocity together with experiments on the structure of the wake behind a rising bubble. The shape of the bubble was not evaluated from the present observations but calculated from verified correlations suggested by other authors (3).
2.2.1.1 Flow visualisation experiments
Experimental work described in this paragraph was based on a photographic technique. Throughout the work we used a 6x6 cm camera HASSELBLAD and high speed negative material Kodak Tri-X 27 DIN. The visualisation experiments are summarized in Tab.2-2.
Table 2-2
Flow visualisation experiments on the bubble wake structure physical arrangement three-dimens ional bubbles two-dimensional solid models two-dimensional bubbles type of measurement direct indirect direct direct technique, tracing open tank, dye-stuff
open tank, strobo-scopic photographs of bubble rocking water tunnel, small air bubbles two-dimensional open tank, Merlite particles
The group of experiments on two-dimensional bubbles has been performed by the author recently, having been published elsewhere fSj,results of that work are given in appendix A3-1.
Flow visualisation experiments on normal three-dimensional bubbles were performed in an open perspex tank 15 x 15 x 100 cm, which is de-picted in Fig. 2-1. The bubble was introduced into the tank through a rubber stopper placed at the bottom of the perspex thimble, which was fixed in the centre of the tank base, with the aid of a graduated syringe. The air supplied was collected under a dumping cap, and at a given instant released. The experiments were carried out at room . temperature.
The first group of experiments performed in this tank was arranged to obtain some direct basic information on the flow pattern around the bubble. To get this, the streaming was visualized by means of dark blue ink. The ink was carefully supplied into the column by a syringe through the rubber stopper in the bottom of the tank to form an intensively coloured layer extending several centimeters above the
dumping cap. The bubble released from the dumping cap formed its wake in the coloured layer, and when the bubble entered clear water, the structure of the coloured wake became partly visible. Since a
rapid mass exchange between the wake and its surroundings takes place the colouring of the wake disappeared 1 perspex column rapidly. The major part of the co-2 dumping cap louring in the wake was displaced by
fresh liquid during a short time 3 stopper period due to vortex shedding, however,
h syringe a very thin layer of the dye-stuff adjacent to the bubble floor persis-ted for a relatively long time, com-pared to the vortex shedding period, before it diffused out. In spite of the fact that this sharp thin layer was visible by a naked eye, all attempts to photograph it were un-successful. However, photographs in this experimental set-up, which were Fig.2-1 Simple apparatus for ^^y.^^ within a short time after the the hydrodynamic studies bubble left the coloured region at
the bottom of the column, showed an approximate contour of the wake and to some extent also the structure of it. The photographs were taken against white background with an exposure time 1/250 s e c , using strong front lighting. The size of bubbles observed was varied between 0.5 and 1.5 cm in equivalent diameter.
We have also tried to arrange a similar experiment allowing one to stabilize colouring of the wake by generation of the colouring due to absorption of the gas enclosed in the bubble. For this pur-pose air bubbles containing ammonia and diethylamine, respectively, rising in water, which contained traces of phenolphtaline, were examined, but no significant result was obtained.
The same tank was used for indirect measurements of the fre-quency of vortex shedding, and of the form of the wake. This has been obtained through detecting the frequency and character of bubble rocking. A rising bubble was, in this case, photographed against black background employing a stroboscopic flash at a fre-quency between 30 and ^tO cycles a second. The total exposure time was 1 second. The camera registered light reflected from the bubble upper surface, the stroscopic flash being placed above the tank, in the vertical axis of it. Thus, the photographs indicated, besides the frequency of bubble rocking, the space curve which is followed by the rising bubble due to the vorticity discharge taking place behind the bubble. This sort of experiment was carried out in the darkroom. The bubble equivalent diameter was varied between 0.5 and
— 1 > 0
-r
•=r — t x h K>®«
-Mn
•f t-?
"IP
test section model of the bubblesieve hypodermic syringe rubber ball 6 pump 7 pitot tube 8 pressure indicator 9 tank
Fig.2-2 Water tunnel
For the next group of flow visualisation experiments a two-dimensional arrangement has been set up. The bubble shape was simulated by the cross section of a metal rod, the shape of which
was calculated from Tadaki and Maeda's (3) correlations.
There were fabricated three polished stainless steel models of this sort corresponding to three-dimensional bubbles of 0.5, 1.0 and 1.5 cm in equivalent radius. These models were fixed in the test section of a water tunnel (see Fig.2-2). The mean linear velocity of water in the tunnel test section was adjusted at a value cor-responding to the rise velocity of the appropriate bubble size, i.e. 15,7, 22.1 and 27.1 cm/s. The perspex test section was 1 metre long being provided with fine wire sieves at both ends, which were built in to reduce velocity fluctuations in the velocity profile
across the test section. The equipment was built in a Streaming past the
two-dimen-sional model was traced by very small air bubbles, the size of which was controlled by adding a small amount of Teepol. Trace bubbles were generated in the pump from air present in the system. The amount of Teepol used was chosen to
pro-duce tracer bubbles of a ri se ve-locity less than 1 per cent of the mean linear velocity in the test section; the concentration of Teepol was of an order of 1O"^ per cent by weight,
The flow patterns around the solid models of bubbles were ob-tained by photographing the traces in a plane of light across the test section perpendicular to the longitudinal axis of the model;
(see Fig,2-3), Best results were obtained using the exposure time
l/60 sec. Photographs obtained showed a form of the wake similar to that observed behind a bubble in experiments using the dye-stuff tracing technique.
Photographs, showing the re-sults of the experimental methods described above, are included in chapter 3.
darkroom.
model of the bubble photographic camera slit
light source
Fig.2-3 Measurement of the two-dimensional flow pattern in the water tunnel
2,2,1.2 Bubble rise velocity measurements
The bubble rise velocity was measured simultaneously with mass transfer measurements in a single bubble column 250 cm long, see Fig.2-it. The diameter of the inner square cross section was 1^9 mm. A detailed description is given in paragraph 2.2.2.
The experiments on bubble rise velocity were arranged so that time intervals in which the carbon dioxide bubble passed 9 observa-tion points along the column after being released were measured. The bubble was released with the aid of a similar mechanism to that which was used in the open tank during flow visualisation experiments (Fig. 2-1). The stainless steel dumping cap was submerged k mm under the surface of a layer of mercury. The mercury layer was used in the system
to prevent dissolution of carbon dioxide before the bubble release, The observation points for the bubble rise velocity measurements were at levels of 22,5, 37.0, 52.0, 67.0, 137-5, I 6 T . 5 , 197-0, 227-0 and 250.0 cm above the mercury level. The relatively long distance
iM1
eö& point: 1 2 3 4 5 6 7 6 9
• n n II n
_ / _ .:
-( t - 0 )
"""
Fig, 2-lt Single bubble column
Fig, 2-5 Record of the bubble rise velocity measurement (example)
1 column 2 dumping cap 3 mercury layer k hypodermic syringe 5 thermometer 6 glass bulbs 7 pressure sensor 8 transducer
9 thermostat ing bath 10 levelling bulb
between the fourth and fifth observation points was simply due to the fact that the bubble rise was followed by eye, and the region between these two points was not accessible because a supporting construction was built at that level. For recording of visual obser-vations of the bubble rise the second marking channel of a MICROGRAPH model BD-2 recorder was used, with two rocker switches. The first switch was used for the lower part of the column, the second one for the upper part. An example of such a record is shown in Fig.2-5. The records of bubble position-time dependence were evaluated in terms of mean bubble rise velocity over the distances between ap-propriate observation points. Since the volume of the bubble was changing during its rise, the corresponding values of the bubble equivalent diameter had to be obtained from simultaneous measurements of volume changes - these experiments are described in the next section. Combining results of the measurements on the bubble rise velocity with the experimental data on changes of the bubble volume
in time, one can compile curves of the rise velocity as a function of the bubble equivalent diameter for each experimental run. The bubble rise velocity-bubble equivalent diameter curves are given in Fig,3-8.
2,2,2 Mass transfer measurements
2.2,2,1 Introduction
Experiments described in this section concern both physical and chemical absorption, since the measurement technique was iden-tical,
The experiment was set up to measure instantaneous rates of mass transfer from a pure carbon dioxide bubble during its passage through alkaline solutions. The rates of mass transfer in such a set-up are to be evaluated from changes in volume of the bubble during its rise,
Instantaneous changes in the volume of a rising bubble may be measured by direct photography, by recording volumetric changes in a system open to atmosphere, or by measuring pressure changes in sealed system.
The photographic techniques may be criticized since the evaluation of the volume and mass transfer data from projected bubble images is highly questionable.
In ^92h Ledig and Weaver (6) introduced the technique of volume measurement for studying mass transfer. The rate of gas absorption could be followed by taking shadow-graphs of the movement of a
mercury thread in a capillary sealed into the mass transfer equipment and providing the only opening to the atmosphere. Later on the tech-nique was modified by Leonard and Houghton (7), in order to follow the linear motion of the mercury by inserting a resistive platinum wire into the capillary tube and by measuring the corresponding changes of an electric current through that wire. The method can be used successfully in studying the absorption of sparingly soluble gases. For highly soluble gases like carbon dioxide in water an error will result from the viscosity of the mercury in the capillary pre-venting the meniscus from following the rapid changes in the bubble volume. A further error may be introduced, there is an imperfect contact between mercury and the platinum wire.
A pressure change technique, similar to that, which has been described by Calderbank and Loahiel (8), was chosen for the present work. This method can be considered more satisfactory for studying instantaneous transfer rates for more soluble gases, as pressure measurements of fluctuating transfer rates can be recorded with great accuracy,
2,2,2,2 Experimental principles
The equipment, which has been based on the pressure change technique, was set up as depicted in Fig,2-U, The principal part of it - the single bubble column - contained a gas space above the surface of the absorptive solution, the pressure changes of which were registered. Thus, during the bubble absorption experiments the column contained two gas volumes, the volume of the bubble V and the volume of the gas pocket at the head of the column V , For constant volume of the liquid it obviously helds that:
V^ + V^ = const. , (2-1)
The changes in the bubble volume due to a combined effect of bubble absorption and a simultaneous expansion of the bubble caused by the decreasing hydrostatic pressure within the bubble during its rise, can be detected in the gas pocket, which is, in the present arrange-ment, much larger than the bubble. To evaluate the transfer coeffi-cients for mass transfer from a bubble to the liquid from the pressure measurements in the gas pocket, one has to make some necessary
assumptions:
1, the gas in the gas pocket at the head of the single bubble column obeys the ideal gas law,
2, the concentration of the carbon dioxide at the bubble interface c can be calculated from the Henry law,
3, the gas enclosed in the bubble obeys the ideal gas law,
k. the pressure increment in the bubble due to bubble surface forces is negligible,
5. the various heat processes accompanying the bubble absorption have no significant effect on the measurements.
The liquid side mass transfer coefficient k is given by definition as:
-|f= V^^'-'^J ' (2-2)
where dN/dt is the mass flux across the interface, A the interfacial area, and c the solute concentration in the bulk of the liquid. Assuming the concentration of the gas in the bulk c to be negli-gible compared to the interfacial concentration c and using the Henry law, the concentration difference of the right hand side of Eq.(2-2) can be expressed as:c* - c = P /He , (2-3)
CO ^
P_ being the total pressure in the bubble, and He the Henry constant. The value of He for the system carbon dioxide-water at 25 C was cor-rected for the solutions used after Krevelen and Hoftyzer (9). The total pressure in the bubble P is a sum of the pressure in the gas pocket P and the hydrostatic pressure:
The influence of desorption from the non-degassed liquid will be considered later on (chapters I4 and 5 ) .
Pg = Pp + pgh , (2-U) p is the density of the liquid, g the acceleration due to gravity,
and h the height of the liquid column above the bubble. Since we have assumed an ideal behaviour of the gas in the gas pocket, we can write that:
PpVp = const. , (2-5) at any instant, as the temperature of the gas in the gas pocket does
not change in time, and there is no mass exchange between this gas space and the liquid. These conditions were fulfilled to a reasonable degree during the present experiments; the gas pocket was filled with air saturated with water vapour, the liquid was saturated with air. The system was thermally isolated,
The bubble volume and the pressure in it obey the ideal gas law. Thus, for any instant of the bubble rise one can write:
PgVg = NRT . (2-6) Combining Eq,(2-2), Eq.(2-6) and using the Henry law expressed in
Eq,(2-3) one obtains:
dV V dP
V RT M t P^ dt ^ • ^'^ "
B
Before relating Eq.(2-7) to measurable quantities it is convenient to introduce a pressure quantity p defined as: '
P = Pp - Pp, , (2-8) where P is the pressure inside the air pocket before injecting the
bubble into the column, P was equal to the atmospheric pressure in the present experiments. Now, Eq,(2-7) may be easily re-written substituting the bubble volume and pressure terms and their deriva-tives in terms of the initial pressure and volume of the air pocket P and V , the liquid density p, the bubble rise velocity U, and the time derivative cf the pressure term p, dp/dt. These measur-able quantities can be employed as follows:
(for p « P p ^ ) V„ = V„^ - ^ , (2-9a) and ^B = d t P - P = V P P ° P ° Ppo ^Po " p o ^B = Ppo ^ P d t + Pgh (2-9b) (2-10)
Pg = PpQ + pgh , (2-10a) the time derivative is obtained from Eq,(2-10), which yields:
dPn A
^ = f - p g U , (2-lOb) Expressions (2-9a), (2-9b), (2-10a) and (2-lOb) substituted into
Eq,(2-7) allow one to calculate the product k A from measured values of the bubble rise velocity U and the pressure p-time curves. The final relation for evaluating the experimental data is:
k A = — — f E (ocU - ^ ) - ^ } (2-11) V RT P„ ^ P„ + pgh ^P^" dt' dt ^ • ^'^ " '
Po Po
Eq,(2-1l) may be simplified for a case when pgU >> dp/dt, the sim-plified expression takes a form:
k A = S£ ! P ^ { Pg"P _ ^ } . (2-12)
V RT P^ ^ pgh + P„ dt ' ^'^ ''"
Po Po
This simplification seems to be permissible for conditions kept during the experiments performed as the ratio pgU/(d£/dt) was large, considering mean values of the bubble rise velocity U, and of the time derivative of the overpressure term dp/dt. This ratio, depending
on the ratio V /V , estimated as:
^ ^ . 1 0 0 , (2-13)
t being the total time of the bubble rise; H the length of the single bubble column, which was 250 cm; Ap the maximum pressure difference in the air pocket, 2,8.10^ dyne/cm^,
2,2,2.3 Experimental set-up and procedures
Inspecting formulae (2-11) and (2-12), respectively, it is evident that besides equipment constants V , P , H , the tempera-ture T and the density of the liquid p, there are two functions to be measured to determine the product k A:
1. a change of the pressure in the gas pocket, p, 2. the bubble rise velocity, U.
Measurements of the bubble rise velocity were described in section 2.2.1.2; measurements of the pressure changes in the gas pocket are a subject of this paragraph.
As mentioned above the measurements were carried out in a perspex column (see Fig,2-1+) 250 cm high, having a square cross-section IU9 X IU9 mm. The walls were made of material 11 mm thick. The
column was equipped with a jacket, giving a channel 10 mm wide through which water was circulated at 25 C.
In the bottom of the column a perspex thimble was fixed con-taining a mechanism used to form bubbles (see section 2.2.1), Besides that there was an outlet cock in the bottom end plate.
The top of the column was covered by a perspex plate 11 ram thick with a perspex tube screwed in. This tube was 135 mm long, having an inner diameter of 27 mm. The volume of this extension tube formed a part of the air pocket, this extension tube was connected with the pressure sensor by tubing. Between the pressure sensor and the exten-sion tube there was one of two inter-changeable volume capacities (75 and 500 cm^, respectively) to get a suitable volume for the air pocket. The column top plate also contained a thermometer socket, and a cock connected with a levelling bulb,
Pressure variations occuring in the air pocket were scanned by a membrane capacity sensor connected to a transducer with linear voltage output in an overpressure range 0 - 2,8 x 10^ dyne/cm^. In order to apply this pressure range to experiments with all bubble sizes, the two different additional air volume capacities (see above) were used alternately. For registration of the transducer output the first channel of the recorder was used,
Since the measurements were very sensitive to temperature va-riations, the whole column was additionally isolated with a layer of foam polyurethane 20 mm thick with openings to allow measurement of the bubble rise velocity by sight, as described in section 2,2,1.
The single bubble column filled with the absorptive solution was brought to a temperature of 25 C by circulating water in the thermostat jacket for three hours before starting the experiment, When the temperature of 25 C was reached one had to stop circulating through the jacket since pressure variations in the thermostat system would disturb the mass transfer measurements. Closing the inlet to the jacket and opening the top of it to the atmosphere, an equilib-rium between pressure forces, acting on the walls of the column from both absorptive liquid and thermostating liquid sides, was soon reached. After mechanical and thermal equilibrium was reached the experiments could start. The temperature during the experiments was kept to 2 5 + 0 , 1 °C,
The carbon dioxide bubbles were supplied to the column with the aid of a gas hypodermic syringe which had been carefully made; see Fig,2-U. The syringe contained a small amount of mercury to fill in the syringe "dead" space. After the syringe needle was pulled out from the injection point at the bottom of the thimble, the pressure change corresponding to the volume of the bubble supplied under the dumping cap could be read from the record.
When the bubble was released, it first expanded rising through the column of mercury, and the absorption began after the bubble entered the absorptive solution. Because of the absorption, carbon dioxide bubbles decreased in size during the rise. All pressure (and
hence volume) changes were recorded by the technique described above, A typical example of such a record is shown in Fig,2-6. After the bubble reached the level of the top of the liquid in the column, the
space of the air pocket was opened to the atmos-phere , and purged with the aid of a leveling bulb. The pressure sen-sor was calibrated before and after each series of experiments and no sig-nificant changes in performance were observ-ed.
There are 95 sets of experiments concern-ing eight solutions each with a range of about 12 bubble sizes. bubbte r«linq in^p »» Lima ^o^ each of these sets
a number of measurements (at least four) were done Fig.2-6 Pressure-time record (example) ^o^" ^ particular bubble
size until satisfactory reproducibility
in the measured volume changes was achieved. The initial bubble equi-valent diameter was varied from 0.7 to 1.5 cm, however, because of the bubble dissolution a range from 0.3 to 1.5 cm can be considered.
For reading of the injected bubble volume the difference between the levels A and C was used (inspect Fig,2-6), Evaluating the absorp-tion-expansion curve E one has to take into account the expansion of the bubble during its rise through the mercury layer. The corresponding pressure increase is illustrated by the "cut" D; this point was taken as a starting point for evaluating of the pressure curve. The time required for the passage of the bubble through the mercury was neg-lected, From each curve nine equidistant pressure values were read, Results in terms of a product of the mass transfer coefficient and the bubble surface area were obtained from the pressure change records using Eq,(2-12), It is convenient to base the mass transfer coefficient on the equivalent sphere surface area i,e, surface area of a sphere having the same volume. The results obtained are given below in chapters h and 5. B K A . > ^ \ V "
'H,
Injectiori* of l.r>€ bubbtcx C M1
labsora peak due t o the passage of Ihe bubble Ihrough Lhe m e r c u r y2.2.3 The reaction kinetics of carbon dioxide absorption in buffered solutions catalyzed by arsenite ions
The aim of these measurements was to verify the chemical reaction kinetics which have been described in literature (1), (2). The present experiments were performed in a wetted wall column, principal parts of which have been designed by De Waal (W).
This sort of apparatus has been extensively used for determina-tion rate constants for both physical and chemical absorpdetermina-tion, and the experimental principles have been described in many references
e.g. (V, (10), (TV.
The falling film, through which the absorption took place, was formed on the outside of a l6 mm outer diameter glass tube. The liquid passed up the inside of the tube and was distributed around its circum-ference by a ca-p fitted over the top of it. The liquid flowed from the inside of the tube, through five 3 mm holes drilled around its circum-ference, into two small annular chambers, and from there it flowed through the annular gap formed between the cap and the tube, finally falling freely down the outside of the tube - forming a laminar film. The length of this film, where mass transfer takes place, was 25 cm. The falling film was removed from the column by means of a channeled teflon collar, having been built after a suggestion of Danakwerts (1).
Carbon dioxide was supplied through a constant temperature bath and saturator into the reactor. The gas space of the reactor was
connected tp a gas burette into which a movable soap film was introduced. Before each experiment the film reactor was purged with carbon dioxide. The rate of absorption was measured with eight absorptive solutions of different arsenite content. The measurements consisted of the determination of the absorbed volume of carbon dioxide per unit time. The volume absorbed was followed with the aid of the soap-film flow meter.
All the absorptive solutions had approximately the same value ' of the carbonation ratio (= {C03"}/{HC03'}), ionic strength, and pH; these values were O.96, 0.9** gion/litre and 9-7, respectively. The concentrations of arsenite in the absorptive solutions were: 0.000, 0,139, 0.288, 0.392, O.U90, 0.691, 0.71+0 and 0.900 mole/litre.
During the measurements the whole system was kept at 25 C. For the results see chapter 6,
2,2,U Property measurements on the phases used
2,2.1+. 1 Gas phase
Generation of the carbon dioxide
Carbon dioxide was obtained free from the vapour of organic compounds, which might contaminate the bubble interface using a simple glass apparatus to generate the gas by decomposition of potassium
carbonate with dilute sulphuric acid. The carbon dioxide produced was of purity higher than 99.6 per cent by volume. The carbon dioxide was carefully washed by bubbling through a sodium bicarbonate solution at 25 C, thus being purified from traces of other acid gases, and satura-ted with water vapour. The tubing of the apparatus terminasatura-ted with a rubber stopper, the carbon dioxide was drawn off with the aid of a hypodermic needle.
Analysis of the carbon dioxide
The quality of the carbon dioxide used was checked up by
Soholander's (12) analyzer of respiratory gases. For the present
gas pipcllc
A reaction chamber B compensaton chamber C drop indicator D tube connecting the
reaction chamber and the reservoir of the absorptive solution E reservoir of the absorbent F micrometer G levelling bulb H syringe needle I rubber seal J mercury drop K thermostating bath
purposes one had to slightly modify the original apparatus, which was more suitable for dilute mixtures of carbon dioxide. The original modi-fication had two absorption chambers for both oxygen and carbon dioxide, The analyzer used in the present study (see Fig.2-7) was provided with a chamber for absorption of carbon dioxide only.
The gas sample was drawn off from the carbon production line and transferred to the analyzer using a special gas pipette. The gas pipette was made of a glass capillary 30 cm long, 3 mm internal diameter. At one end the capillary was cemented a syringe needle. The gas pipette was closed by a mercury droplet about 3 mm long. Before filling the pipette with a carbon dioxide sample the mercury drop was at the bottom of the capillary. During draw-off the mercury drop gently moved up. The appa-ratus is prepared such that the micrometer is turned completely to the left, so that the mercury level reaches the bottom of the compensation chamber B, A drop C of an solution inert to CO2 floates on the mercury; it will be used both as a seal and as an indicator,
With the needle of the pipette inserted through the cock above the compensation chamber B up to a position underneath the drop C the gas sample is sucked into the reaction chamber A by lowering the mercury level with aid of the micrometer,
Consequently the absorbing liquid is tilted into the reaction cham-ber from reservoir E and the position of the drop C is kept stationary between the chambers A and B during the absorption by adjustment of the mercury level. Volumes were read in terms of micrometer divisions. The present modification allowed one to supply very small amount of the ab-sorptive solution to the reaction chamber, this was achieved by choosing the most suitable inclination angle of the tube D. The capillary
con-necting the reaction and compensation chambers was longer than in a commercial Scholander analyzer, A detailed description of the apparatus and procedure is given in Soholander's (12) original work. The whole analyzer was placed in a thermostating bath at 25 C.
2.2.1*. 2 Liquid phase
Preparation of the solutions
The appropriate amount of technical grade arsenous oxide was dissolved in a small amount of hot concentrated sodium hydroxide solution. After filtration the solution was diluted by destined water to 50 liters. When the solution was cooled down to room tem-perature pH was measured. By adding a small amount of hydrochloric acid pH of the solution was adjusted to a value 9-9, corresponding to a buffer solution of a carbonation ratio ({C03"}/{HC03'} = O.96). Finally, a calculated amount of sodium carbonate and sodium bicar-bonate of an buffer ratio of 0,96 was added to maintain the ionic strength 0.91+. A solution for physical absorption of carbon dioxide in a buffer had the same carbonation ratio and ionic strength, it was
Chemical analysis and pH measurements on the absorptive solutions
The determination of the arsenite concentration in the absorp-tive solutions was carried out by iodometric titration.
The sample to be analyzed was diluted with distilled water 1:10. The pH of the sample was brought to a slightly alkaline level of about 8. Afterwards, an excess of 0.1N iodine solution was added to the arsenite solution. The surplus of iodine was back-titrated by a neutral 0.1N solution of arsenite in the presence of starch.
Determination of the density, viscosity and surface tension of the solutions
The material properties concerned were measured with the aid of conventional methods as shown in Table 2-3.
Table 2-3
Measurements on physical properties of the solutions property
density viscosity surface tension
technique used specific gravity bottle
capillary viscosimeter 1 ring method
Results of the measurements of the physical properties are summarized in appendix A2.1.
Chapter 3
HYDRODYNAMICS
The present experiments include hydrodynamic and mass transfer measurements on bubbles of the equivalent bubble diameter d between 0.6 and 1.3 cmi the equivalent bubble diameter is defined as
d = (6VT,/TT) , V_ being the volume of the bubble. Hydrodynamic be-haviour in this region of bubble sizes is not yet clearly understood, particularly, with respect to the structure of the wake behind a bubble rising in a liquid. The region of bubble sizes investigated in the present study lies between the spherical and spherical cap bubble regimes, for which a large amount of both theoretical and experimental information is available. Considering the region of bubble sizes investigated in the present work as intermediate between the two regimes we treat bubble dynamics over a very wide range of Reynolds numbers; Re = d U/v, U is the bubble rise velocity and V the kinematic viscosity of the liquid. A special interest is devoted to bubble wake phenomena because of their importance- to mass transfer.
3.1 Introduction
Observing a gas bubble introduced into a stagnant column of a liquid one can determine the terminal velocity, the bubble shape and trajectory during its rise. Knowing the bubble volume, the other properties in terms of which one can attempt to interpret the
experimental data are: the densities of the gas and liquid, p'and p; the viscosities, v' and v; the interfacial tension, a; and the
acceleration due to gravity, g. Usually, the number of properties to be considered can be reduced if it is assumed that the motion of the enclosed gas has a negligible effect on the flow. For air or carbon dioxide bubbles rising in water, the ratios of pressure and viscous stress forces exerted by the gas on the interface to the same sort of forces acting from the liquid aide are p'/p and p'v'/pv, respectively, which are of the order of 10 and 10~ . The remaining physical parameters can be combined to fonn the dimensionless ratio M (= gp /po ) , y is the dynamic viscosity. In the course of the present work M = 10 , and the following discussion is restricted to Newtonian liquids of constant, low M-values. For such bubbles
Rosenberg (13) has given the following summary of their shape and motion character:
Table 3-1
Bubble behaviour in Newtonian liquids of low M-values
class 1 2 3 lt+ 5+ 6+
7
equivalent radius r <0.0i* cm e 0.0lt<rg<0.062 cm 0.062<r <0.077 cm 0.077^r <0.2it cm 0.2lt<r <0.35 cm 0.35<r <0,88 cm r^>0,88 cm Reynolds number Re<70 70<Re<lt00 lt00<Re<500 500<Re<1100 1100<Re<l600 l600<Re<5000 Re>5000 descriptionspherical bubbles, recti-linear path, Cjj as for rigid spheres
sphere, rectilinear path, C less than solid spheres oblate spheroid,
rectilinear motion oblate spheroid, helical motion
irregular oblate spheroid, almost rectilinear motion transition from oblate spheroid to spherical cap, almost rectilinear motion spherical cap,
rectilinear motion The bubble deformation has a significant effect on the rise velocity. Usually, the form of a bubble interface is discussed in terms of Weber number. We = d U p/a. The Weber number measures the ratio of the hydrodynamic pressure forces to the surface tension forces which are maintaining the shape of the bubble. Aybers and Tapuau (14) presented an extensive experimental study, including a summary on the relationship between bubble shape and We, Their rough classification is given by Tab, 3-2, In contrast to the
observation by Rosenberg (13), Aybers and Tapuau (14) have reported
the first distortion of a spherical bubble can occur with an equivalent radius equal to 0,05 cm,
The shape of class 1 bubbles is determined by the condition that the surface energy is a minimum, so that the bubble is spherical, Sy et al. (IS) proved that the bubble will remain
spherical as long as the inertial term in the Navier-Stokes equation is negligible,
It is these regions which have been studied in our experimental programme. This symbol "+" will be used hereafter in the same sense,
Table 3-2
Shape of the bubbles for different Weber numbers class cf. Tab.3-1 ^'2 1
2,3,V
3\b*7
Weber number 0<We<0.62 0.62<We<3.70 3.70<We<5.35 We>5.35 eccentricity E = a/b E = 1 1<E<2 2<E<i+shape of the bubble
spherical
e.nipsoidal, no 1 surface oscillation
ellipsoidal, increasing! surface oscillation distorted bubble and 1 spherical cap bubble |
Moore (16) showed that for slightly distorted bubbles, class 2 and 3 bubbles, which are oblate spheroids, the eccentricity E can be expressed as:
1 +
W
We (3-1)As We increases from 0.62 to 5.35 bubbles in water adopt an
ellipsoidal shape and the stability of its surface decreases (14),
At large Reynolds numbers bubbles, indicated as class 7 in Tab,3-1, have a spherical cap shape, which is determined by the pressure forces, Moore (17), and Rippin and Davidson (18) explained this shape as the result of flow separation,
Tadaki and Maeda (3) correlated the shape
ratio i,e, a ratio of equivalent spheri-cal diameter to the major axis of the bubble 2r /a (see Fig,3-l), with the Re and M groups for a large set of experi-mental data. The authors have presented four approximate empirical formulae ig-noring the flattening of class 3 bubbles, and considering bubbles of classes k and 5 to be geometrically similar. Their results re-written for bubbles rising in water are shown in Table 3-3.
ellipsoidal bubble
spherical cap bubble
Fig. 3-1 Symbols in the bubble description
Table 3-3
Shape o f t h e b u b b l e s r i s i n g i n w a t e r ( c o r r e l a t i o n s a f t e r Tadaki and "4aeda (3) ) c l a s s c f . Tab. 3-1 1 , 2 , 3
k\-,^
ê
7
R e y n o l d s number Re<550 550<Re<l650 l650<Re<U520 Re>U520 shape r a t i o 2r / a = 12r / a = 3.06Re-°-^^H
2 r > = 5 - 7 8 R e - ° - 2 « 2r / a = 0 . 6 2 e 1 3.2 Theory 3.2.1 Velocity of riseThe conditions of motion of a gas bubble in a liquid are de-termined by three factors: the value of the Reynolds number, the shape of the bubble and the interface conditions.
The interface of a gas bubble may realize any condition between the two extreme cases of a free and of a rigid interface. A free interface is such that there is no tangential stress at it if the gas is assumed to be inviscid. In a case of a rigid interface there is no relative velocity between liquid elements on the interface and the centre of gravity of the bubble.
Table 3-1* shows the theoretical solutions which are reviewed below.
Table 3-lt
Mathematical models on bubble dynamics revieved in § 3.2 surface c o n d i t i o n s , range of Re r i g i d i n t e r f a c e R e « 1 free i n t e r f a c e Re<<1 free i n t e r f a c e R e » 1 bubble shape | sphere Stokes Eq.{3-3) Oseen Eq.(3-'>) Hadaaard Eq.(3-6) Rybczynski Eq.(3-6) Boussinesq Eq.(3-7) Levich Eq.(3-9) Moore Eq.(3-10) spheroid
-Moore Eq.(3-12) * Mendelson Eq.(3-l't) spherical cap 1
-Davies-Taylor Eq.(3-15) 1 Mendelson Eq.{3-llt) Collins Eq.(3-17) Rippin-Davidson Eq.(3-18)
Spherical bubbles (classes 1 and 2 of Table 3-1)
Low Reynolds (lumber regimes
Stokes solution
The physical model is based on three assumptions: that the bubble is spherical, the interface is rigid, and the liquid is in creeping flow. These three conditions are to be satisfied provided the bubble is small enough,
The terminal velocity is limited by viscous drag and the equations of motion can, in principle, be solved rigorously. Thus for a very' small Reynolds number the approximate solution of the Navier-Stokes equations was obtained by retaining only viscous terms. The drag force was calculated to be:
D = ÖTTprU , (3-2)
and this force balanced by the buoyancy force results in the rise velocity of the bubble being:
0 = 1 ^ . (3-3)
neglecting the density of the gas enclosed in the bubble.Effects which arise when the Reynolds number is small but not wholly negligible have been treated by techniques which attempt to ap-proximate the inertial terms in the Navier-Stokes equations,
Oseen (19) obtained an approximate solution for flow past a sphere which has given the drag force as:
D = 6iTprU { 1 + "I Re + 0(Re2) } , (3-1+)
A higher order approximation has been worked out by Proudman and Pearson (20). Numerical methods for treating streaming flows past spheres and circular cylinders in the Reynolds number range inter-mediate between the creeping flow and boundary layer flows were
reviewed by Jensen (21).
Hadamard-Rybczynski solution
The bubble corresponding to this conception is also situated in a creeping flow field. The bubble is spherical, but in
contra-distinction to the Stokes model, the interface is free,
Solving the equations of motion past an internally circulating fluid sphere Hadamard (22) and Rybazynski (23) have independently
g i v e n t h e d r a g a s :
D = ItTTprU , ( 3 - 5 ) and t h e b u b b l e v e l o c i t y a s :
U = l ^ ' . ( 3 - 6 ) 3 V
Much controversy has centered around the question as to whether the assumptions of the spherical shape and the free interface can be realized simultaneously. Water, and diluted aqueous solution, have shown conspicuous interfacial phenomena, which make doubtful the hypothesis of free interface even for larger bubbles,
Both Hadamard (22) and Rybazynski (23) considered that the tan-gential velocity components and the normal and shear stresses at both sides of the interface were equal, Boussinesq (24) included changes in interfacial stresses resulting from motion. It is due to the combined effect of the usual surface tension and a "dynamic" incre-ment, e. The latter varies over the surface of the sphere. Boussinesq expressed the drag as:
D = 67rMrU ^ - ± ^ . (3-7)
e + 3ur
For small Re or large e, Boussinesq's expression approaches Stokes' law. The other limit is the Hadamard-Rybazynski solution. However, there is no experimental evidence that such "dynamic" surface viscos-ity exists, as postulated by Boussinesq.
High Reynolds number regimes
Levich-Moore and Chao's solutions
The bubble considered has a spherical form and moves under invis-cid flow conditions. The interface is free. These assumptions are only hypothetical; it is impossible to produce a spherical bubble of high Reynolds number in the gravitational field. In real liquids a bubble large enough to make the Reynolds number large has a shape which is severely distorted from spherical.
Leviah (25) postulated that the tangential stress vanished at the bubble-liquid interface. With this hypothesis, he attempted to integrate the equation of motion without consideration of the internal circulation. The solution of the problem was obtained by evaluating the viscous energy dissipation rate for the flow field constructed according to potential flow theory and by equating this to the potential
e n e r g y l o s s r a t e a s t h e b u b b l e r i s e s .
The d r a g f o r c e e v a l u a t e d a s t h e t o t a l d i s s i p a t i v e f o r c e e x e r t e d on t h e b u b b l e i s :
D = 12TTurU , ( 3 - 8 )
provided that the dissipation in the wake is much smaller than the dissipation in the boundary layer. This leads to a value for the velocity of rise:
U ^ f ' • (3-9) Moore's (26) analysis supported Leviah's concept that the flow
is essentially irrotational and that the drag could be calculated from the dissipation in the irrotational field. Moore (26) consid-ered the rotational flow to be confined to a thin boundary layer on the surface of the bubble which separated at the rear stagnation point to form a thin rotational wake of negligible dissipation energy rate. Calculating the dissipation, the drag force has been expressed by:
5
D = 12TTMrU { 1 - -^^ + 0(Re ) } . (3-10)
> ^
From a comparison of Leviah's and Moore's solutions it appears that the latter is closer to experimental results.
The problem of the rise of a spherical bubble has been also examined with the aid of a perturbation method. Chao (27) has assumed that on either side of the interface there exist boundary layers and that flow separation is insignificant. Evaluating the external velocity field Chao (27) computed the drag force from the surface stresses with the boundary condition for the continuity of shear stress at the interface as:
D = 8TTyrU { 1 - - 2 ^ ^ } . (3-11)
* ^
However, it has been pointed out by Moore (26) that Chao's conclusion, that pressure forces are unimportant in the boundary layer, is in error; the incorrect result of Chao is in agreement with Moore's (17) earlier theory, which was based on the same erroneous concept.
S p h e r o i d a l b u b b l e s ( c l a s s e s 3,1*,5,6 of Table 3-1) Moore's s o l u t i o n
For s l i g h t l y d i s t o r t e d b u b b l e s of f r e e i n t e r f a c e , which may be assumed t o o b l a t e s p h e r o i d s , i n an i n v i s c i d flow f i e l d ( R e > > l ) , Moore
(16) has e s t a b l i s h e d a t h e o r y l e a d i n g t o ;
1 _ i
D = 12TiprUG(E) { 1 + H(E)Re"^ + 0(Re ^) 1 , (3-12)
where E i s t h e e c c e n t r i c i t y , i , e . t h e r a t i o of t h e c r o s s - s t r e a m a x i s t o t h e p a r a l l e l - s t r e a m a x i s of t h e o b l a t e e l l i p s o i d . Moore (16) has g i v e n t h e f u n c t i o n G ( E ) by:
G ( E ) = 1 E 3 ( E 2 - 1)2 | ( E 2 . 1)2 _ (2 _ ^^^ . . . - ^ / [ E ^ S C C - ^ E - ( E ^ - I ) ^ ]
( 3 - 1 3 )
Omitting the second and third term in the parentheses in Eq.(3-12) the formula for drag is asymptotically correct as Re ->• "> , which is the analogous case to Levioh's solution for a spherical bubble. Applying the boundary layer and wake dissipation analysis Moore
gives a table of calculated values of H ( E ) . The equation (3-12) is in fair quantitative agreement with Haberman and Morton's (28) data for bubbles of the bubble classes 2,3,1* of Tab.3-1.
Mendelson's small amplitude theory
For liquids of low viscosity Mendelson (29) presented a solution deduced from an assumed similarity between the behaviour of rising bubbles and that of surface waves propagated over deep water. The analogy contains the assumptions of free bubble interface and poten-tial flow past the bubble.
The wavelength was interpreted in terms of the bubble dimensions as A = 2irr . Then the equation for waves for small wavelength compa-red to the depth of the liquid given by Lamb (30) can be re-written by:
U = - 2 - + gr . (3-11*) pr ^ e
e
The first term corresponds to a surface tension dominated regime, which is of theoretical interest only, and the second one to the regime governed by the buoyancy force. With the aid of Mendelson's
about 0.6 cm rising in water have a lower terminal velocity than those of any other size greater than 0.12 cm. Mendelson's solution is a good approximation of bubble velocities for bubbles classified in groups 3,1*,5,6 of Tab.3-1.
Spherical cap bubbles (class 7 of Table 3-1) Davies and Taylor's solution
The authors assumed potential flow around the nose of the bubble and supported this assumption with experimental evidence. The bubble under discussion has a free interface.
The pressure distribution over the upper surface of the sphe-rical cap is considered to be the same as that calculated over a complete sphere in inviscid flow. Combining the equation of the pressure distribution along the bubble upper surface with a condi-tion of constant pressure through the bubble interior the authors obtained that:
U = I /iR^ , (3-15)
where R is the curvature radius of the spherical cap. c • •
In order to get the relationship between the bubble rise veloc-ity and the equivalent radius of the bubble the angle of the sphe-rical sector 6] corresponding to the sphesphe-rical cap has to be known. Davies and Taylor (31) presented a value of 9i of about 100 , Aatarita and Apuzzo (32) indicated the same value by experiments. For Bi = 100° Eq.(3-15) may be re-written as:
u = 1,01 /gF" , (3-i6) Moore (17) assumed a spherical cap bubble followed by an
in-finite ajci-symmetric wake of stagnant liquid, and from simple considerations the author deduced as a value for the angle Sj 78 .
Collins (33) satisfied the requirement of constant pressure at the upper bubble surface to a higher order than Davies and Taylor (31) and found that:
u = 0.988 / ^ . (3-17)
Movement of spherical cap bubbles may be approximated well using the formulae (3-l6), (3-17) and Mendelson's small amplitude theory, Eq.(3-ll*), although these theories neglect the effect of the wake behind a bubble,
A model based on wake properties has been described by Rippin and Davidson (18) for spherical cap bubbles. They postulated the existence of a wake formed by an infinite column of stagnant liquid
below the spherical cap bubble which was separated from the rest of the flow by a surface of discontinuity, the properties of which are a constant dynamic pressure and a constant velocity. They computed this free streamline and found that the velocity of rise was given by:
U = 1,20 (3-18)
Comparing experimental data with Eq.(3-l8), Rippin and Davidson pre-dicted values of the bubble rise velocity of about 20 per cent higher. They ascribe this discrepancy to an influence of turbulence inside the wake,
3.l2^2_ Bubble_wakes
The previous paragraph has dealt with the flow around the nose of a bubble up to the separation ring; it is evident that there is a great deal of information available. The flow at the rear of a bubble has been less understood, and the published information on it is, so far, mostly of qualitative nature, giving no systematic classi-fication over a range of Reynolds numbers,
Magarvey and Bishop (34) observed and classified a related pheno-menon, wakes behind drops falling through quiescent water, and noted the approximate ranges of Reynolds numbers corresponding to the tran-sition from one wake configuration to the next. Their classification IS given in Table 3-5.
Table 3-5
The wake of the drops falling in water for different Re numbers
(after Magarvey and Bishop (34) )
wake class I II III IV V VI Reynolds number 0<Re<210 210<Re<270 270<Re<290 290<Re<l*10 290<Re<700 700<Re<2500 nature of trail single trail | double trail
double trail with waves | procession of vortex loops | double row of vortex rings |
In the following there are two categories of wakes distinguished: a) steady wakes,
b) unsteady wakes.
A steady wake is one in which the rate of vorticity generation is balanced by the rate of continuous transfer to the wake surroundings, with no vorticity accumulation in the wake. In contrast an unsteady wake is characterized by the spontaneous discharge of vorticity after the wake becomes unstable due to vorticity accumulated in it.
Steady wakes
As Magarvey and Bishop (34) indicated in Table 3-5 for the class I wake the configuration behind a freely falling drop and a rigidly held solid sphere were identical. Inspecting Tab.3-1 one can expect similar behaviour for bubbles smaller than O.Ol* cm in equivalent radius rising in water. There has been a lot of controversy upon behaviour of small spherical bubbles, Leviah (25) stated that the apparent "solidification", and therefore the existence of a consid-erable wake behind small bubbles, was related to the effect of trace amounts of surface active materials present in liquid media unless they were purified extremely carefully. One can understand that visualisation techniques using a dye-stuff (49) are not able to answer the question, and no attempt has been made to verify existence of the wake using other visualisation procedures. Since the results of experimenters on the drag coefficients of small bubbles indicate a rigid interface we will follow the discussion by Goldstein (36)
on steady wakes behind a solid bluff body which are, in principle, similar to wakes behind spheres at moderate Re (compare Fig,3-2).
The first stage in the development of a wake behind a solifl bluff body appears in the broadening out of the streamlines (Fig.3-2a). As the Reynolds number is increased, the streamlines widen out further and form a closed region behind the obstacle. Within this region there is an inflow along the axis of the wake and a flow in the general direction of motion in the outer portions (Fig.3-2b); in the two-dimensional flow around a cylinder, this circulatory motion constitutes a vortex pair. As the Reynolds number increases the vortices become more and more elongated in the direction of flow. Above some critical Reynolds number the vortices become unstable and wavy (Fig.3-2c) and ultima-tely leave the obstacle and move downstream (Fig.3-2d). The critical value of Re is between 30 and 90.
According to Batahelor (37) a similar sequence of changes as Re is increased from values near unity occurs in the flow past most other bodies. The region of closed streamlines behind a sphere which is formed at about Re = 25 contains a standing ring vortex, which corresponds to the two-dimensional counterpart depicted in Fig.3-2b. Again there is instability of the flow pattern above a critical value
a, very low Reynolds numbers, cylinder and sphere
b, steady twin-vortex stage cylinder: 3 < Re < 1*0 sphere: 25 < Re < 130
intermediate range cylinder: 30 < Re < 90 sphere: 130 < Re < 500
d, periodic wake (vortex street) 80 < Re < 300
for cylinder only
Fig,3-2 Development of the wake behind a two-dimensional cylinder and a sphere
of the Reynolds number, and it has been found that the ring vortex first begins to oscillate gently at about Re = 130, This is a con-siderably higher value than that for the two-dimensional flow past
a circular cylinder.
The class II wakes were also classified as steady, the configura-tion is characterized by a double vortex trail and an intricate spiral-ling structure immediately behind the drop. There is a continuous back-flow which impinges a restricted area of the back surface of the drop and escapes as one of the spiralled layers which form the double vortex trail, this situation is depicted in Fig.(3-3a).
Periodic wakes
The class III wake is a development from the class II wake in for-mation and structure. However, the vorticity generation rate is increased
with increasing Re and the balance between generation and dispersion rates is main-tained by small periodic vorticity errup-tions. These appear as waves spaced equal-ly on the double trail, see Fig.3-3b.
In contrast to this behaviour of drops, bubbles rising at corresponding Re move rectilinearly. This indicates a re-duced slip of the drop interface due to the higher viscosity of the inner phase. An instability of the bubble wake analo-gous to class III drop weike may be ex-pected at higher Reynolds numbers.
(è>
(d)^
IÖ, e. Karman model Fig.3-2 (cont.)Two-dimensional; solid bodies
In order to illuminate the problem of the establishment and character of periodic wakes, the better understood phenomena of two-dimensional periodic wakes behind long bluff bodies will be discussed. Figures 3-2a - 3-2d represent the major identifiable regimes of flow around circular cylinder over a Reynolds number range up to 300.
At a critical Reynolds number be-tween 30-90, the vortex pair which is formed behind the obstacle at the begin-ning of the motion, after becoming more and more elongated in the direction of flow, takes up an asymmetrical position and The asymmetrical arrangement alters the then moves away from the body.
