review by AA Kulkarni.pdf

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Bubble Formation and Bubble Rise Velocity in Gas-Liquid

Systems: A Review

Amol A. Kulkarni and Jyeshtharaj B. Joshi*

Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai-400 019, India

The formation of gas bubbles and their subsequent rise due to buoyancy are very importantfundamental phenomena that contribute significantly to the hydrodynamics in gas-liquidreactors. The rise of a bubble in dispersion can be associated with possible coalescence anddispersion followed by its disengagement from the system. The phenomenon of bubble formationdecides the primitive bubble size in the system (which latter attains an equilibrium size), whereasthe rise velocity decides the characteristic contact time between the phases which governs theinterfacial transport phenomena as well as mixing. In view of their importance, we herein presenta comprehensive review of bubble formation and bubble rise velocity in gas-liquid systems.The emphasis of this review is to illustrate the present status of the subjects under considerationand to highlight the possible future directions for further understanding of the subject. Thebubble formation at a single submerged orifice and on multipoint sieve trays in Newtonian aswell as non-Newtonian stagnant and flowing liquids is discussed in detail, which includes itsmechanism as well as the effect of several system and operating parameters on the bubble size.

The comparison of results has shown that the formulation of Gaddis and Vogelpohl22 is themost suitable for the estimation of bubble size in stagnant liquids. The special cases, such asbubble formation in reduced gravity conditions and weeping and in flowing liquids, are discussedin detail. The section on the rise of a gas bubble in liquid covers the various parameters governingbubble rise and their effect on the rise velocity. A comprehensive comparison of the variousformulations is made by validating the predictions with experimental data for Newtonian aswell as non-Newtonian liquids, published over last several decades. The results highlight thatfor the estimation of rise velocity in (i) pure Newtonian liquids, (ii) contaminated Newtonianliquids, and (iii) non-Newtonian liquids, the formulation based on the wave theory byMendelson,190 Nguyens formulation,155 and the formulation by Rodrigues,153 (last two, basedon the dimensional analysis), respectively are the most suitable. The motion of bubbles in non-Newtonian liquids and the reason behind the discontinuity in the velocity are also discussed indetail. The bubble rise is also analyzed in terms of the drag coefficient for different systemparameters and bubble sizes.

1. Introduction 5874

2. Bubble Formation in Gas-LiquidSystems

5875

2.1. Bubble Formation at SingleSubmerged Orifice

5876

2.1.1. Factors Affecting BubbleFormation

5877

2.1.1.1. Effect of LiquidProperties

5877

2.1.1.2. Effect of Gas Density 5881

2.1.1.3. Effect of OrificeConfiguration

5881

2.2. Mechanism of Bubble Formationin Newtonian Liquids

5884

2.2.1. Force Balance Approach inBottom Submerged Orifice inStagnant Liquid

5884

2.2.1.1. Kumar and Co-workers 5885

2.2.1.2. Gaddis and VogelpohlModel

5889

2.2.2. Application of Potential FlowTheory

5891

2.2.2.1. Wraith Model 5891

2.2.2.2. Marmur and RubinApproach based onEquilibrium Shape ofBubble

5892

2.2.2.3. Marmur and Rubin 5892

2.2.3. Approach based on BoundaryIntegral Method

5892

2.2.3.1. Hoopers Approach ofPotential Flow

5892

2.2.3.2. Xiao and Tans BoundaryIntegral Method

5893

2.3. Mechanism of Bubble Formation

in Top Submerged Orifice inStagnant Liquids

5893

2.3.1. Tsuge Model 5893

2.3.2. Liow Model 5894

2.4. Model for Formation ofNon-spherical Bubbles

5894

2.5. Mechanism of Bubble Formationin Flowing Liquids

5895

2.5.1. Co-current Flow 5895

2.5.2. Counter Current Flow 5896

2.5.3. Cross-Flow 5896

5873Ind. Eng. Chem. Res. 2005, 44, 5873-5931

10.1021/ie049131p CCC: $30.25 2005 American Chemical SocietyPublished on Web 05/04/2005


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2.6. Bubble Formation underReduced Gravity Condition

5898

2.6.1. Models for Bubble Formation 5898

2.6.1.1. Nonspherical BubbleFormation Model

5898

2.6.1.2. Two Stage BubbleFormation Model

5899

2.6.1.3. Mechanism of BubbleFormation in Quiescent

Liquids

5899

2.7. Bubble Formation inNon-Newtonian Liquids

5899

2.7.1. Structurally Viscous Liquids 5900

2.7.2. Viscoelastic Liquids 5900

2.8. Bubble Formation Accompaniedwith Weeping at the SubmergedOrifice

5901

2.9. Bubble Formation at PerforatedPlates and Sieve TraysSubmerged in Liquid

5902

2.9.1. Observations of BubbleFormation at Multipoint

Sparger

5902

2.9.2. Mechanism of BubbleFormation

5903

2.9.3. Models of Bubble Formation 5903

2.9.3.1. Li et al. Model 5903

2.9.3.2. Miyahara and TakahashiApproach

5904

2.9.3.3. Loimer et al. (2004) SingleBubble Approach

5904

2.9.4. Factors Affecting BubbleFormation at Sieve Plates

5904

2.9.4.1. Gas Flow Rate 5904

2.9.4.2. Pitch of Holes on SieveTrays

5905

2.10. Experimental Techniques 5905

2.10.1. Image Visualization andAnalysis

5905

2.10.2. Volume of Displaced Liquid 5905

2.11. Conclusion andRecommendations

5905

3. Bubble Rise Velocity in Liquid 5906

3.1. Factors Affecting the RiseVelocity

5907

3.1.1. Effect of Purity of Liquid onBubble Rise Velocity

5907

3.1.2. Effect of Liquid Viscosity onBubble Rise Velocity

5909

3.1.3. Effect of Temperature onBubble Rise Velocity

5910

3.1.4. Effect of External Pressureon Rise Velocity

5910

3.1.5. Effect of Initial BubbleDetachment Condition

5911

3.2. Formulations for Rise VelocityCorrelations

5911

3.2.1. Force Balance Approach 5911

3.2.2. Approach based onDimensional Analysis

5911

3.2.3. Approach through WaveAnalogy

5915

3.3. Bubble Rise Velocity inNon-Newtonian Liquids

5916

3.4. Comparison of Correlations forBubble Rise Velocity in Pureand Contaminated Liquids

5917

3.5. Drag Coefficient for Bubbles inLiquids

5918

3.5.1. Comparison of theCorrelations for Drag

Coefficient in NewtonianLiquids

5922

3.6. Bubble Trajectories 5923

3.7. Experimental Methods forMeasurement of Rise Velocity

5924

3.7.1. Nonintrusive Methods ofBubble Rise Analysis

5924

3.7.1.1. Photographic Methods 5924

3.7.1.2. PIV Measurements 5924

3.7.1.3 LDA measurements 5924

3.7.2. Intrusive Measurements 5924

3.7.2.1. Optical Probes 5924

3.7.2.2. Conductive/Capacitance

Probes

5925

3.7.3. Imaging in Opaque Systems 5925

3.7.3.1. Dynamic Imaging byNeutron Radiography forOpaque Systems

5925

3.7.3.2. X-ray Imaging 5925

3.8. Conclusion 5925

1. Introduction

Gas-liquid contacting is one of the most importantand very common operations in the chemical processindustry, petrochemical industry, and mineral process-

ing. Most commonly, it is achieved either by automationof liquid into gas in the form of drops or by bubbling(sparging) of gas into the liquid. In applications suchas absorption, distillation, and froth flotation, theinteraction of two phases occurs through bubbling of gasinto the liquid pool and the equipment is designed basedon the knowledge of the hydrodynamic parameterssuitable for desired performance. In most of suchequipment, the knowledge of the transport processesacross the gas-liquid interface is useful for the estima-tion of transfer coefficients and requires the accurateprediction of volume of discrete phase, residence timeof discrete phase, and its contribution to the mixing.Further, the physicochemical properties of liquid phase(viz. viscosity, surface tension, density, etc.) and few ofthe characteristics of the discrete phase (bubble size,bubble rise velocity, etc.) govern the hydrodynamics aswell as flow pattern in the system. For example, forliquids with low surface tension, the sizes of bubblesthat are formed for certain orifice diameters are alwayssmaller. To withstand the drag induced by the liquid,they try to maintain spherical shape, which results inan enhancement of the gas hold-up in the system. Dueto their smaller sizes, the rise velocities are slower,which results in larger residence time. In a bubblecolumn reactor, at low gas flow rates, a homogeneous

* To whom correspondence should be addressed. Tel.: 00-91-22-24145616.Fax: 00-91-22-24145614.E-mail: [email protected].

5874 Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005


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regime prevails, whereas for the same gas flow rate incoalescence-inducing liquids either a transition or aheterogeneous regime is attained, which have totallydifferent hydrodynamics than those of the homogeneousregime. One of the most important properties that helpsin finding most of the design parameters (viz., effectiveinterfacial area) is the bubble size or bubble sizedistribution. The average bubble size in a system is aneffect of the size at bubble formation and the extent ofcoalescence and dispersion (break-up), of which the latertwo are governed by the local turbulence. The discussionso far strictly applies to the system where gas is indispersed phase and volume fraction of liquid is sub-stantially higher than gas phase (such systems can beconsidered as wet foams, where bubbles are separatedby considerably thicker liquid films). At the disengage-ment zone of the column, where the bubbles leave thesystem, the energy dissipation is very high, and theliquid traps the fine bubbles, resulting in foam. In thiszone, although the gas is in the discrete phase, if thevolume of the gas is significantly higher than that ofthe liquid, i.e., gas bubbles are entrapped/separated bythin liquid films, and the phenomenon can be termedas foaming. In the majority of industrial equipment,

where the discrete presence of gas in the form of bubblesexists in bulk, at the disengagement zone, wherebubbles escape from liquid, the gas phase has significantfraction resulting in foaming. The extent of foaming isindependent of the bubble size distribution in thesystem; however, it is a combined effect of the presenceof gas bubbles and physical properties of liquid. Thus,for developing a better understanding about the role ofgas bubbles in the hydrodynamics of a system, a goodknowledge of the above-mentioned phenomena is re-quired. The subject is vast and the large amount ofliterature published over last several decades is scat-tered in different journals, books, proceedings, andreports. In view of this, taking into account the amount

of information available on these subjects individually,in this review we have decided to focus on the first twophenomena of bubble formation and bubble rise velocity.The remaining subjects of bubble coalescence, dispersion(bubble break-up) and foam break-up will be taken-upcomprehensively and separately.

Basically, the remaining part of this review containstwo sections, dedicated to the gas bubble formation ata submerged orifice/plate in liquid (Section 2) andbubble rise velocity (Section 3). In the second section ofthis review, we have discussed the bubble formationprocess over a wide range of issues, which include theeffect of physicochemical properties of a gas-liquidsystem, orifice configuration, and various mechanismsof bubble formation in liquids and proposed models, afew of the recent developments, and finally, a fewsuggestions for the direction of further investigation inthis field. In the third section, bubble rise velocity hasbeen discussed in detail, which includes relationshipbetween size, shape and rise velocity, dependence of risevelocity on system properties, various correlations forrise velocity and the drag coefficient. At the end of thissection, a critical comparison of the different correlationis discussed for various liquids and suitable correlation,which is seen to be applicable for various system isrecommended, followed by a brief review of the variousavailable experimental methods for the bubble risevelocity measurements. We conclude this review witha few recommendations with regard to the use of proper

experiments and generalized correlations, which can beused for design purpose.

2. Bubble Formation in Gas-Liquid Systems

In general, particulate systems are classified on thebasis of the state of the particle present as gas bubbles,liquid drops, and solid particles/agglomerates. The gasbubbles exist in gas-liquid, gas-solid, and gas-liquid-solid systems. The system of gas-liquid contacting

through bubbles is dynamically complex and needsattention. To pay attention to the concise field of gasbubbling in a liquid pool, it is desirable to understandthe process of bubble formation, which can be consideredas a static or quasi-static operation followed by thedynamic processes viz. coalescence, break-up, etc. Sincethe sizes of bubbles after its formation and its wakedecide the rise velocity and also the direction of rise i.e.,trajectory in the liquid, it even influences the above-mentioned dynamic processes, the overall turbulence inthe system and hence the performance of the equipmentto some extent.

The earliest studies on the formation of single bubbleand drop can be seen in Tate1 and Bashforth and

Adams.2

A significant amount of work in the area ofbubble formation at submerged orifice(s) over a widerange of design and operation parameters has appearedin the literature in last few decades (Davidson andSchuler,3,4 Tsuge et al.,5-13 Kumar and co-workers,14-17

Marmur et al.,18,19 Vogelpohl and co-workers,20-22 Tanand co-workers23-27) and the subject is interesting andimportant enough to fetch continuous attention even inthe present decade.28-41 Most of these studies can begrossly classified based on the operating conditionspertaining to the gas phase, such as the constant flow,constant pressure, and intermediate condition. In thelate 1960s, Kumar and co-workers studied the mecha-nism of bubble formation for different conditions andalso reviewed the earlier work very keenly,17 specifically,

the various methods of measuring bubble sizes experi-mentally. Later, Tsuge6 reviewed the hydrodynamics ofbubble formation from submerged orifices and discussedvarious proposed models for the mechanism of bubbleformation, while Rabiger and Vogelpohl20 have brieflydiscussed the various factors affecting bubble formation.Due to variation in the gas-liquid systems (properties),type of nozzles and operating parameters viz. gasvelocity, system pressure, etc., the observations by manyinvestigators are not concordant. This brings out a needto compare the individuals approaches, observationsand inference to develop a guideline for the forthcomingstudies in this field. Additionally, many recent develop-ments in the studies both in numerical methods usedfor the analysis of various stages in the growth processof bubble as well as the experiments conducted toinvestigate the contribution of individual forces onbubble formation need to be given attention. In view ofthis, we have reviewed the subject freshly, whichincludes most of the proposed models for bubble forma-tion and the detailed discussion of the effect of varioussystem properties on various growth phases in theprocess. This part of the review is organized as follows;the next section discusses the bubble formation at asingle submerged orifice in Newtonian and non-New-tonian liquids and it gives a qualitative descriptionabout the various modes of bubbling and the subsequenteffect on the liquid phase in its surrounding and alsothe effect of various governing parameters on the same.

Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 5875


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Prior to the description of various models of bubbleformation, since these models are based on visualizationof the phenomenon, we have discussed the variousobservations in this regard. The third subsection dis-cusses the phenomenon through force balance for themost suitable and realistically visualized bubble forma-tion process. This is followed by a comprehensivediscussion about the comparison of the experimentaldata with the predictions by various available correla-tions. We also review the multipoint bubble formationand also few subjective cases of this phenomenon (viz.reduced gravity analysis, bubble formation in the pres-ence of weeping, etc.). Finally, a brief discussion appearson a few of the experimental techniques used for theanalysis of the process of bubble formation. In theforthcoming subsections, under every parameter, thediscussion is pertaining to (i) the Newtonian liquids,unless explicitly mentioned as being about the non-Newtonian liquids and (ii) bottom submerged orifices,unless explicitly mentioned about the other ways ofsubmergence.

2.1. Bubble Formation at Single SubmergedOrifice. Bubble formation at a single submerged orificeoccurs in many modes viz. bubbling; chain bubbling,

wobbling and jetting, which depends on the orificeconfiguration, the gas velocity, gas-liquid system prop-erties, and finally magnitude of gravitational forceacting on the system. Thus, it can be a periodicphenomenon over a wide range of frequency42-44 ordeterministic process with higher numbers of degreesof freedom as an effect of jet break-up at very highvelocity36,45 which depends mainly on the liquid proper-ties. It is well-known that the modes of bubbling(mentioned above) in a given system are a strongfunction of the gas velocity and liquid depth. Muller andPrince46 have shown a typically observed regime mapfor bubbling (Figure 1A). For very low gas flow ratesand deep liquid (>100 mm), bubbling occurs singularly

only till a certain value of gas flow rate where the modechanges from single bubbling to chain bubbling. In thiscase, the bubble size is primarily decided by the orificediameter, surface tension and the density differencebetween two fluids. In the intermediate or chain bub-bling mode, bubbles are larger than the earlier mode,bubbling becomes periodic and their production rate isproportional to gas flow rate. Further, at even highergas flow rates, gas phase emerges as a continuous phaseand maintains the continuity only for a certain distancefrom the orifice mouth. Bubble formation rate is moreor less constant and bubble size increases with gas flowrate. In the jetting regime, bubble formation occursmainly due to jet break-up resulting from the instabilityof jet. Typical dynamics associated with most of thesebubbling regimes and the organized nature of flow injetting regime can be seen in Tritton and Egdell42 andKulkarni,45 respectively. Photographic images of fourregimes of bubbling are shown in Figure 1B.40 In thecase of discrete single bubble formation, the main forcesacting on a bubble in inviscid liquid are buoyancy, drag,surface tension, and gravity. In this case, the formationand detachment of bubble produces only local distur-bance in the liquid and only a volume of the order ofbubble is carried along with it as drift volume. In thecase of chain bubbling, the continuous phenomenainduces a driving force in liquid in the direction ofbubble motion and in the vicinity of the trajectories ofbubbles liquid has a positive upward velocity resulting

into a weak circulation of which intensity of circulationis directly proportional to bubble size. In the jettingregime, the strong upward motion of the jet inducesstronger recirculation in the liquid with upward motionalong the jet and downward flow away from jet. How-ever, in an infinite media, the downward velocities aredistributed over a larger cross-section. It should also benoted that bubbling regimes are strongly affected by theliquid properties (Newtonian and non-Newtonian), where

Figure 1. (A) Schematic representation of regimes of bubblegeneration (reproduced with permission of Elsevier from Muller& Prince46), (B) Modes of bubble formation at increasing gas flowrates (reproduced with permission of Elsevier from Zhang andShoji,40 Figure 2) (A) Single bubbling (q ) 1.66 mL/s, Reo ) 68);(B) pairing (q ) 5.83 mL/s, Reo ) 238); (C) double coalescence (q) 15 mL/s, Reo ) 612); and (D) triple bubble formation (q ) 25ml/s, Reo ) 1020).



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liquid depth actually represents the static head, whilebubble size/mechanism of formation is also decided bythe orifice configuration. In view of this, we criticallyanalyze the dependence of bubble size on variousgoverning parameters.

2.1.1. Factors Affecting Bubble Formation. Theprocess of bubble formation is governed by manyoperating parameters (i.e., gas flow rate through theorifice, mode of operation, flow/static condition of the

liquid), system properties (viz.orifice dimensions, orificechamber volume), and also the physicochemical proper-ties, such as liquid viscosity, liquid density, and natureof liquid i.e., polar or nonpolar, etc., which decide themode of bubble formation and subsequently reflects onbubble size. The main forces acting on a moving bubbleare gravity, buoyancy, drag, viscous forces, added massforce, and the lift force. In many cases, the gas -liquidproperties, orifice dimensions and the material of con-struction govern these forces. The flow rate of gasthrough the orifice and orifice dimensions mainly de-cides the bubble frequency and thus the detachmenttime, similarly orifice chamber volume decides the back-pressure on the bubble and hence bubble sizes ingeneral. In this review, the effect of gas flow rate has

not been discussed explicitly with regard to the effectof the other parameters. A chronological developmentin the understanding about this phenomenon andobservations from a few significantly important studiesthat are helping in developing a basis for the recentstudies in the subject are tabulated in Table 1. Forfurther information about the recent analyses in thisarea, readers may read the introduction and literaturereview in Nahra and Kamotani.55

2.1.1.1.Effect of Liquid Properties. (a) Viscosityof the Liquid. With the variation in the viscosity ofthe liquid, the magnitude of viscous forces exertedduring formation changes such that a stable bubblediameter is attained before its detachment. The experi-

mental observations by various investigators havebrought different views in this regard and are shownin Figure 2 and it has led us to analyze them in detail.The reported observations in such studies are contra-dicting viz. (i) the bubble sizes increase with liquidviscosity,56,16 (ii) bubble sizes are independent of liquidviscosity,57,50,51,17 and (iii) there is a very weak effect ofviscosity on size.58 In another totally different observa-tion, Siemes and Kaufmann59 have reported that forliquids of low viscosity, the bubble sizes are independentof the liquid viscosity while at higher viscosities anincrease in liquid viscosity causes an increase in bubblesize but at small flow rates. From Figure 2 and Figure5A, it can be clearly seen that viscosity affects thebubble size however its evidence diminishes for large

diameters and higher gas velocities. From the rigorousanalysis of Jamialahmadi et al.60 the size can beassigned a dependence on viscosity as 0.66.

(b) Surface Tension of the Liquid. In the case ofbubble formation at a submerged orifice, for a growingbubble, its rear surface is dragged backward along withthe liquid. In the front portion of the bubble, the surfaceis stretched and the new surface is constantly generated.In the portion close to the orifice, the bubble surface iscompressed and the liquid is pushed toward the orificeedges. As an effect, the surface forces on a bubble ariseout of the linear surface tension acting on it and it helpsa bubble to adhere to the edge of orifice,51 delaying thedetachment process. Two types of surface tension forces

act on a bubble, dynamic and static. During the initialpart of growth phase, the surface tension is dynamic asits contact angle with the orifice changes continuouslyand in the later part, it reaches to a constant contactangle approaching to static surface tension, hencesurface tension decides the orientation time/growth timefor bubbles. Although the surface tension forces are

small, they vary significantly with gas flow rate throughthe orifice61,62,63 and these forces produce sudden motionresulting in a reduced pressure at the tip of the capillaryinitiating the next bubble. This occurs mainly because,the dynamic surface effect retards the stretching of thebubble surface and makes the detachment of the liquidfilm between the bubble and orifice, faster and conse-quently, it gives very fine bubbles. In a contradictoryobservation, Davidson and Schuler4 have reported thatsurface tension affects the minimum value of absoluteresultant pressure for bubbling to occur. For smalldiameter orifices, the effect of surface tension is negli-gible at high gas flow rates, hence at the constant gasflow rate, the surface tension loses its dominance. Also,there is a noticeable influence of surface tension onbubbles formed under constant pressure conditions,through an appreciable effect on the pressure in thebubble and also to some extent it governs the flow intothe bubble. Surface tension force increases with diam-eter of the orifice and thus the orifice diameter affectsthe bubble contact/adherence time.22 Figures 2 and 3give some idea about the various experimental observa-tions made in this direction. In a recent investigation,Liow61 has reported that the surface tension forces incombination with the orifice diameter as well as thick-ness decide the bubble detachment time and hencebubble size. Also, Hsu et al.62 has reported theirobservations about the mechanism in the presence ofsurfactants, where temporal variation of dynamic sur-

Figure 2. Effect of variation in liquid phase viscosity on bubblevolume. The bubble sizes are estimated using the correlation byJamailahmadi et al.60 for the different liquids used by several

investigators (Since these are predictions, symbols correspondingto different investigators are connected by lines) and dh ) 1 mm.(Ramkrishnan et al.14: 9 Water, ) 0.001,) 0.072, ) glycerol-aq ) 0.0449, ) 0.067, [ glycerol-aq, ) 0.552, ) 0.063),(Rabiger and Vogelpohl20, O glycerol-aq, ) 0.007, ) 0.069),(Davidson and Schuler3, glycerol-aq, ) 0.515, ) 0.063),(Terasaka & Tsuge11, 2 water, ) 0.012,) 0.053, + Glucose,)0.269, ) 0.08,)



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Table1(Continued)

inves

tiga

tors

exper

imenta

l

system

&

observa

tion

s

approach

in

thestu

dies

parametersstu

died

an

dcon

ditions

lim

ita

tions

conclusion

&

remarks

Marmur

&

Ru

bin19

Equa

tionofmot

ion

atthe

interfacewas

developedan

dtoget

her

withthe

thermodynam

ic

equa

tions

forgas

inthe

bu

bblean

dthecham

ber

volume

below

itthe

instan

taneousshapeof

bu

bbleduring

forma

tion

wasca

lcu

lated

.

Ori

ficesu

bmergence

,

g,,,FL,FG,

gas

flowra

te.

Model

isrestricted

to

sing

lebu

bbleforma

tion

.

Staticor

lift

-offapproach

atthe

instan

tof

equ

ilibrium

isina

dequa

te.

Gasmomen

tum

istaken

neg

ligi

ble

.

Model

isva

lidforon

ly

verysma

llbu

bbles.

Instan

tof

detachmen

t

i.e.

,theclosureofnec

k

can

be

determ

ined

theore

tica

lly

.

Acharya

eta

l.54

Sing

lestagem

odel

Wave

theorywas

app

lied

todeterm

ine

the

volumeof

the

bu

bblein

dynam

icdetach

ing

con

dition

.

Viscous,

iner

tia

l,interna

l

stresses

,shearra

te,

gas

flowra

te,

orifice

diameter

.

Surface

tension

forcesare

neg

lected

.

Incaseof

highlyviscous

liqu

idssince

detachmen

t

ofbu

bbleisthroughnec

k

forma

tionobserva

tionof

sing

lestage

bu

bble

forma

tion

isunaccep

table

.

On

lygas

flowra

tean

d

accelera

tion

due

to

grav

ity

dec

ide

the

volumeof

bu

bbles

tha

tare

formed

innon-N

ewton

ian

liqu

ids.

Vogelpoh

l&

Ra

biger

21

Mu

ltistage

bubble

forma

tion

Forma

tionofprimary

an

dsecon

dary

bu

bbles

wasmodeled

by

force

ba

lance

g,,,FL,FG,

gas

flow

rate

,direc

tionof

flow

,

sizeofor

ifice,or

ifice

submergence

.

Model

isapp

lica

bleon

ly

injettingreg

ime.

AthigherLnec

kmot

ion

isno

longersu

fficien

tto

detach

theprimary

bu

bble

an

dbu

bblecoa

lescesw

ith

secon

dary

,w

hichisnot

pre

dictedfrom

themodel

.

Secon

dary

bu

bblesize

cannot

beca

lcu

latedfrom

the

developedmodelas

it

isan

induce

dprocess

.

Su

ctionef

fectof

the

primary

bu

bbleinduces

the

forma

tionof

secon

dary

bu

bble

.

Liqu

idmot

ion

has

strongef

fecton

the

forma

tionan

d

detachmen

tofprimary

bu

bble

.



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face tension is taken into account. The above interestingobservations show that the phenomenon still needsunderstanding in terms of the concentration polariza-tion at two opposite poles of the bubble, which willimprove our knowledge in its contribution to the de-tachment stage and hence the actual time required fordetachment. Interestingly, the effect of surface tensionforces on a clean bubble and contaminated bubble aredrastically different and needs analysis in terms ofvariation in the vorticity over different segments ofsurface during the growth and detachment.

(c) Liquid Density (GL). To attain stability, a bubble

attains a shape close to a sphere causing an earlydetachment. It is known that during the formation of abubble, its pressure energy is equal to the static headabove it, i.e., liquid density. The effect of density canbe seen from the following two observations: (I) thebubble volume decreases with increase in liquid densityand (II) it is independent of liquid density. Rise in thestatic head leads to the first observation, whereas thesecond observation should be true for very shallow liquidheads. Khurana and Kumar16 have observed that (i)when the flow rate and viscosity are small, then the firstresult is obtained, (ii) when the flow rate is large andthe viscosity is small, then for small orifice diameters,the second statement is true and finally, (iii) when theorifice diameter and viscosity are both small, again thesecond statement is found to be true.

2.1.1.2. Effect of Gas Density (GG). The gas-phasedensity can be increased by increasing the pressure oreven by using a higher molecular weight gas. As a resultof increased gas density, the difference between thedensities of two phases goes down resulting into thesmaller sized bubbles and also reduction in the buoy-ancy force. The gas density comes into picture in termsof the added mass force. Interestingly, its contributionto the added mass coefficient is not very significant sincethe total density of the fluid which affects the motionof a single bubble can be given as F ) FG + (11/16)FL (11/16)FL. For low gas density, at low gas flow rates, thesurface tension force is dominant and the detachment

is delayed. This reasoning is invalid for very smalldiameter orifices, as the drag force is the main resis-tance for bubble detachment and surface forces arenegligible in comparison. Idogawa63 (air, He and H2) andWilkinson64 (He, N2, Ar, CO2, SF6) have studied theeffect of gas density on bubble sizes. They have reportedthat for a large diameter orifice, the detachment isfaster in the case of higher density bubbles, whereasfor very small diameter orifice, the gas density hasnegligible influence on the detachment. He has alsoshown that an increase in gas density for large chambervolumes results in an increase in gas momentum, a

higher pressure drop at the orifice, and an increasedrate of bubble necking, which finally lead to a smallerbubble at formation. Figure 4 depicts the quantitativeanalysis of these observations. This effect can also beseen by increasing system temperature, provided thedifference between the molecular weights of gas andliquid are noticeable.

2.1.1.3. Effect of Orifice Configuration. The bubbleformation in a pool of liquid takes place simply byinjecting gas through the capillary or orifice or dia-phragm or puncture in a membrane. Orifice submer-gence, orifice chamber volume, orifice diameter, type ofthe orifice, orifice material, etc. are important orifice-related parameters in the bubble formation process.Here, we have discussed their dependence in brief andhave critically compared them for reaching an optimumconfiguration for the desired bubbling operation.

(a) Orifice Construction and Type of the Orifice.Many variations in the gas inlet design are possible, butthe circular-cylindrical vertical tipped orifice is the mostcommonly used in these studies. In the case of a sharp-edged orifice operating in a gas-liquid system, thestream filaments of the orifice get converged to aminimum section downstream from the orifice and thendiverge. An orifice of this type has a square cross-sectiontoward upstream edge. The upstream edge of a round-edged orifice is bevelled to minimize the constriction ofthe stream filaments at the throat of the orifice. In asharp-edged orifice, the pressure downstream is re-

Figure 3. Effect of variation in surface tension of liquid on bubble

volume. (+ water 0.073 N/m, n-Propanol 0.0238N/m, Ethanol0.0228 N/m, OMethanol 0.0227 N/m, 0 i-Propanol 0.0217 N/m).

Figure 4. Effect of gas density on bubble size from single orifice.Helium: ([ 0.1 Mpa, 2 0.6 MPa, b 2.1 MPa), Nitrogen: (] 0.1

MPa, 0.3 MPa, O2.1 MPa). For the inset box, abscissa is thegas density in gm/cm3 and ordinate is the bubble size in mm.



10/59

duced, while a constant pressure is maintained up-stream from the orifice. In the case of a long thincapillary, there is generally sufficiently high flow re-sistance in the tube such that the pressure gradient inthe tube prevents events in the gas chamber upstreamof the tube from interacting with the events at thebubble forming point. In this case, the size of thechamber has a negligible effect on the bubble formationmechanism and the bubble volume is a function of orificeradius given byVb ) 2Ro/(Fg) known as Tates law.Leibson et al.52 have reported that the behavior of athick plate orifice is similar to the sharp edged orifice.

(b) Orifice Diameter (dh).Flow through an orificeis proportional to its cross-sectional area and the extentof growth of a bubble, i.e., its volume depends greatlyon the orifice diameter (i.e., inner diameter, unlessexplicitly mentioned) as the surface tension variescontinuously over the rim of orifice. Generally, it isassumed that the distance over which surface tensionvaries is O(dh) and thus, depending upon the gas flowrate, the bubble volume increases until it is detached.It is always observed that the effect of diameter in smallorifices is negligible, while for large diameter orifices,the bubble volume increases with flow rate.5 For the

smallest orifice diameter when the inverse of Webernumber is equal to the sine of the angle between theinterface the relation almost exactly represents the firststage of formation and the deviation occurs at largerorifice diameters. Further, the external diameter of theorifice also decides the maximum possible bubble vol-ume as the contact angle of bubble varies with thethickness affecting the detachment period and hence thebubble sizes.61 Figure 5 shows the experimental obser-vations on effect of orifice diameter on bubble size underconstant flow as well as constant pressure conditions.

In the case of PAA solutions (Figure 5C), the bubblevolume has a very weak positive dependence on theorifice diameter for the gas flow rates (QG


11/59

equal to the initial pressure PC(t). Thus, the inflow ofgas to the bubble (QC) during bubble formation can belarger than the gas flow rate to the chamber and willbe independent of the flow to chamber but will dependon the value of the initial pressure and the pressuredrop across the orifice.65,16,5,66 In the case of larger orificediameters at higher flow rates, an increase in chambervolume decreases the bubble frequency as a result oflarge bubbles generated at the orifice. However, forlower gas flow rates with the same orifice-chamberconfiguration, bubble frequency is always lower and isindependent of orifice diameter. In the case of the topsubmerged orifices, the bubble size is almost indepen-dent of the chamber volume since the back pressureeffect is nominal. In air-water system, using sharpedged square shaped orifice and over a wide range ofchamber volume, Antoniadis et al.66 observed that, evenat low gas flow rates, the number of bubbles in a groupand their volume increased with an increase in chambervolume, whereas at higher gas flow rates, single bubbleswere generated with the same effect of increased VC.For small orifices, an increase in VC causes a muchsmaller pressure drop in the chamber and as a result,at higher volumes, the bubble detachment becomes

unstable, producing many bubbles rapidly in succession.Also, for the sameVCbut with varied chamber diameter,the number of bubbles formed in groups is the same.In the case of larger orifice diameters at higher flowrates, increase in chamber volume decreases the bubbleformation frequency as a result of large bubbles gener-ated at the orifice. These results are certainly helpfulin deciding the range of flow rates to be maintained forgetting desired bubble sizes for a given orifice diameterandVC; however, it would be helpful to understand theeffect of chamber geometry on bubble size. A detailedaccount of the efforts in this direction are given inHughes et al.,51Antoniadis et al.,66 McCann & Prince,67

Marmur & Rubin,18,19 Kumar & Kuloor,17 Park et al.,68

and Wilkinson.64

A typical pressure variation cycle in a gas chamberduring bubble formation is studied by Kupferberg andJameson,65 Khurana and Kumar,16 Hsu et al.62 Theiranalysis supports the observations by Tsuge and Hibino5

where, in a system with wide variation in the gas-liquidproperties, the bubble volume was found to increasewith chamber volume and gas flow rate only for certain

capacitance number NC (4VCgFL)/(dh2Ph), however,

bubble volume was found to be independent of gas flowrate beyond (NC > 25). In another studies, Davidson andAmick69 have observed that, the bubble volume wasindependent ofNCforNC < 0.8, while at higher gas flowrates, critical value of NC drops down to (0.2 VC),which supported the results by Hughes et al.51 and is

contradictory to the above results. In the case of the topsubmerged orifices, the bubble size is almost indepen-dent of the chamber volume since the back-pressureeffect is nominal.70

In non-Newtonian liquids at large flow rates, bubblevolume increases with increasing chamber volume,mainly due to the fact that accumulation of the gas fedinto the gas chamber results in a higher chamberpressure (than the hydrostatic pressure) at the orificeand helps to increase the volume of the bubble. How-ever, at low gas flow rates and large chamber volumes,the time period without bubble growth (waiting time)is required so that the bubble volume depends onchamber volume. On the other hand, for the case of high

gas flow rates the waiting time is short enough so thatthe effect of the chamber volume on bubble volume islow.

(d) Orifice Submergence. An orifice can be sub-merged in three ways: top submergence, bottom sub-mergence, and side submergence (Figure 6). The firstkind of submergence is very common in the metallurgi-cal industry, while the later two are usually found inthe chemical process industry where the aim is uniformdispersion of gas. In the case of bottom submergence,the viscous forces, surface tension forces, pressure,gravitational forces, and inertia are counter-balancedby the buoyancy and gas flow rate through the orifice.Kumar and co-workers16,14,15 have shown that thebubble volume reduces exponentially with increase inorifice submergence. This rate of reduction of bubblevolume is higher for larger chamber in the wide rangeof orifice submergence. Under constant flow and con-stant pressure conditions, submergence has a negligibleeffect on bubble volume. For the top submerged orifice,the pressure required to initiate bubble formation ishigher and once the bubble goes past its maximumbubble pressure, the chamber volume rapidly depres-surizes leading to rapid growth and detachment. Incomparison with the bottom submerged orifices, here,the surface tension forces and viscous forces are of less

importance and friction in the capillary is dominant. Ithas been reported that the growth rate of bubbles inthis case is comparably small, and bubble sizes (traversediameter) are larger than the earlier case,57,71 andbubble size depends up on the thickness of the orifice61

as many stable contact angles exist for the same orifice.Thus, depending up on the thickness of the orifice, thenumber of stable contact angles changes and gives arange of possible bubble sizes. The detachment stage isbetter controlled by the orifice diameter. Finally, sidesubmergence is very rare, but many times it is used inthe air-loop lift reactors and small bubbles are formedas liquid is in cross-flow mode resulting in an earlydetachment. Recently, Iliadis et al.72 have studied theeffect of orifice submergence on bubble formation ex-perimentally, using four orifices (of diameters 1.15, 2.1,3.25, and 4.35 mm). Experiments were conducted in therange of gas flaw rates 0.75-56.7 mL/s, chambervolumes 150-7000 mL and orifice submergence 0.1-1.5 m. Results show that in the region of single bubbleformation the bubble size increases with the orificesubmergence. In the region of group formation ofbubbles, the individual bubble size is reported to beindependent of the orifice submergence, ranging from0.1 to 0.5 mL for the smallest orifice and from 0.2 to 1mL for the other ones. No effect of orifice submergencewas observed on weeping.

(e) Orifice Contact Angle. It is known that, thesurface tension force affects even the bubble shapes and

Figure 6. Schematics of various ways of orifice submergence. (A)top submergence, (B) bottom submergence, (C) side submergence.



12/59

the effect of surface tension can be taken into accountby including the angle of inclination of the interface atthe triple point of solid-liquid-gas while estimating thesurface forces. This inclination angle depends on thecontact angle and type of orifice (sharp-edged or curved).For a sharp-edged orifice, the angle of contact is ap-proximately the same as the angle of inclination,whereas for a curved orifice, there is a large differencebetween the two because of the local conditions i.e.,radius of bubble curvature, thickness of the orifice, thegas pressure inside the bubble neck and the localcontamination. Details of the effect of contact angle onbubble size can be found in Marmur and Rubin. 18

(f) Orifice Orientation.In the cases where the axisof the orifice is inclined with respect to the vertical, theformation, growth, and expansion of a bubble takesplace in vertical direction. Hence, during force balance,the vertical component of the surface tension force isalways considered. Kumar and Kuloor17 have discussedthis aspect in detail. This particular aspect is importantfor modeling the bubble formation over the holes inblades of a gas inducing impeller system.

(g) Material of Construction of the Orifice.Surface forces depend on the material of construction

of orifice and hence decide bubble size. As discussedearlier, the contact angle between the gas-liquid andsolid is one of the parameters determining the detach-ment time, which mainly depends on the wettingproperties of material of construction. Ponter and Su-rati73 have reviewed the effect of material of construc-tion and have recommended the suitable material basedon the hydrophilic/hydrophobic nature of liquid (wetta-bility) and polarity. In another investigation underconstant gas flow conditions, Sada74 has reported that,when water is in the liquid phase, for carbon nozzles,bubbles contact the inner surface while, for Teflonnozzle, bubbles were formed at the outside surface ofthe nozzle owing to nonwettability of the surface. Thus,

the bubble size changes depending upon the wettabilityproperty of the nozzle, for the same inner diameter andvaried outer diameter of the orifice. Hence, for obtainingsmaller size bubbles, the orifice material should bewettable for the liquid under consideration. Hughes etal.51 have shown that for the case of glass and brassorifices of the same diameter, the variation in bubblesizes with respect to gas flow rate is negligible. Wraith53

has given the variation in the hemispherical radius ofa bubble versus gas flow rate from orifice of differentmaterials of construction, where it is shown that for thesame gas flow rate, the bubble radius is always greaterin Perspex orifice than in brass orifice.

The most important messages from the above discus-sion are the very diverse experimental observations by

various investigators, which have appeared in therespective correlation for bubble size. Hence, it is ofutmost importance to identify a correlation, which wouldbe more generalized (i.e., it should show a good agree-ment with the experimental observations over widerange of parameters).

2.2. Mechanisms of Bubble Formation in New-tonian Liquids. Bubble formation in a gas-liquidsystem is a sequential process, and several approacheshave been followed to understand the mechanism ofbubble formation. In general, we know that when gasis passed through the capillary or porous material, asan effect of certain forces acting on the system the gasphase cannot come as a continuous stream up to a finite

Reynolds number, whereas only at very high gas flowrates continuous gas jet may impinge from the orifice/pores. The discontinuity in the streamlined gas resultsinto the formation of small packets of fluid, which weterm, bubbles. The models proposed for understandingthe mechanism of bubble formation can be grosslyclassified based on the number of stages considered i.e.,one stage model, two-stage model, and multistagemodel. Although the modes of bubble formation totallydepend on the system properties described earlier, thenumber of stages is a function of the gas flow rate andthe orifice dimensions. When gas starts flowing throughthe capillary, initially for a while, it passes at constantflow condition raising the pressure inside the bubblelinearly. The rapidly varying volume of the bubble atthe tip induces oscillations in the liquid in the immedi-ate vicinity of orifice at a frequency equal to that ofbubble formation. The sudden motion resulting from thedetachment of a bubble may produce a reduced pressureat the tip of the capillary and help to initiate formationof the next bubble. The length of neck varies with thegas flow rate. Rabiger and Vogelpohl21 have discussedthe variation in neck length in detail. In the majorityof studies, various assumptions are made for conven-

ience as well as for simplicity in the model. A list ofassumptions made for the development of these modelsis given in Table 2. A summary of the models considered,correlation developed along with the limitations of eachmodel and formulation for bubble size/volume are givenin Table 3. These correlations are either complicated,require iterative procedure for solving or very simplebut applicable for only a narrow range of conditions. Onthe gross level, the approaches for all these studies canbe classified as (i) basic force balance, (ii) dimensionalanalysis, and finally (iii) use of finite difference or finiteelement methods. The details of models based on forcebalance (as a simple case) and the comparison of itspredictions with experimental data over a wide range

of parameters are discussed below.2.2.1. Force Balance Approach in Bottom Sub-merged Orifice in Stagnant Liquid. The expansionof bubble followed by its detachment is governed by thedominance of different forces at different instants clearlyindicates that the process of bubble formation is not asingle stage process and hence, Davidsons one stagemodel cannot be accepted as universal, and it is requiredto analyze the two stages, i.e., expansion and thedetachment of bubble separately. In view of this, Sa-tyanarayan and Kumar15 proposed the concept of two-stage bubble formation for the first time. During thefirst stage, the bubble expands while its base remainsattached to the tip of the orifice whereas in the detach-ment stage the bubble base moves away from the tip,while bubble itself would be in contact with the orificethrough a neck. According to Siemes and Kaufmann,59

for the inviscid liquids, in the two-stage mechanism, theprocess is divided in the following two stages: (i) thefirst stage is presumed to be exactly analogous to theformation of a bubble when gas flow rate approacheszero and (ii) the second stage begins when the buoyancyforces exactly balance the surface tension forces and thebubble proceeds to detach itself. However, in the caseof viscous liquids, it has been assumed that the volumeof bubble does not depend on the gas flow rate. Thisapproach has resulted in faulty estimation of detach-ment time during which, the bubble even gets inflated.In two-stage mechanism, the effect of drag force during



13/59

initial expansion (growth) has been neglected and in noway, the reliable detachment time could be estimated.Hence, Satyanarayan and Kumar15 came out of areasonably acceptable model, which can be used forexplaining the process of bubble formation. Accordingto them, the final volume is the sum of the volumespertaining to two stages. Here, we discuss some of themodels based on this approach.

2.2.1.1. Kumar and Co-workers.14,15. At finite gasflow rateQ, the bubble expands at a definite rate givingrise to inertial and viscous drag, which adds to surface

tension. The first stage is assumed to end when theforces in opposite direction become equal. The forcesconsidered in this approach are as follows

and virtual mass of the bubble is given as M) (11/16FL)Q te.

The uppermost point of the bubble is assumed to movewith a velocity equal to the rate of change of the bubblediameter and hence the average bubble growth velocityis the velocity of its center, which is equal to the rate ofchange of bubble radius, i.e.

and balancing the corresponding momentum gives

where

The values of dve/dte and dM/dte are obtained ondifferentiating two Equations in Davidson & Schuler4

and on simplification gives

Table 2. List of Assumptions Made for Modeling the Process of Bubble Formation

1. The bubble is spherical throughout.2. The drag coefficient is inversely proportional to the instantaneous Reynolds number of the bubble.3. The bubble velocity is proportional to the flow rate with no allowance for change in bubble cross-section during growth.4. The frontal area for the drag term is constant at its final value.5. The carried mass of liquid is constant during bubble growth.6. Diameter of liquid column is so large that the wall does not influence ascending bubbles.7. Liquid column does not contain any obstacles.8. Liquid in the column is not circulated except by the action of the bubbles themselves or circulation of liquid is negligible.9. a) Complete wetting of orifice, b) incomplete wetting of orifice.10. The motion of the bubble is not affected by the presence of another bubble immediately above it.11. The momentum of the in-flowing gas is negligible.12. The bubble is at all instants moving at the Stokes velocity appropriate to its size.13. Liquid is infinite in comparison to bubble volume.14. The liquid is inviscid.15. The gas injection rate is constant and the gas incompressible.16. Gas density is neglected.17. The bubble is a volume of revolution around the axis of the orifice.18. The interface is acted upon by pressure difference between gas and liquid and surface forces.19. Added mass coefficient of inviscid fluid is taken constant.20. The gas is ideal.21. The gas in the bubble, as well as the gas in the chamber flows and expands adiabatically.22. Pressure difference exists across the orifice, which determines the rate of gas flow into the bubble.23. The pressure within the bubble is uniform and the same holds for the pressure in the chamber underneath the orifice.24. Bubble detachment occurs when the neck that forms narrows to zero at one of its points.25. Volumetric gas flow is constant26. The flow rate of gas flowing into the bubble through the nozzle is constant during bubble formation,

27. The bubble formation is a a) one, b) two and b) three stage process.28. The pressure in the bubble is uniform as that in the chamber.29. A pressure difference exists across the orifice and it determines the rate of gas flow into the bubble.30. Flow of gas in the chamber is an adiabatic process. In the orifice and bubble it is isothermal.31. The gas liquid interface is acted on by a pressure difference between the gas and the liquid and by surface tension forces.32. The interfacial surface tension is constant and uniform.33. In the case of the liquid cross-flow, the flow is isothermal, uniform, inviscid and irrotational.34. There is no energy exchange or mass transfer across the interface.35. At reduced gravity condition, buoyancy forces are negligible when compared with liquid drag.36. The volume contribution by neck is negligible as compared to the spherical portion of the bubble.37. Completion of the growth cycle occurs when the gas neck severs.38. For constant flow condition, the bubble growth rate is equal to the gas flow rate entering the system.39. Formation frequency is constant.40. Bubble growth occurs at constant rate.41. Flow around a bubble is irrotational and unseparated.42. First stage in the formation process ends when the upward and downward forces equate.43. Surface tension has not effect on bubble size.

44. Liquid viscosity has not effect on bubble size.45. Bubble is symmetric about the axis.46. Liquid is stationary.

buoyancy force ) V(FL - FG)g,

viscous drag ) 6vere

surface tension forces ) Dcos,

inertial force ) Q2(FG + 1116FL)V-0.66

12 ( 340.66) (1)

ve ) (dre/dte) ) Q/4re2 (2)

dMvedte

) M

dvedte

+ vedMdte

dvedte

) - [Q2

6( 342/3)]V

-5/3,dMdte

) (1116 F1)Q (3)

dMvedte

) Q2 (1116F1)V-2/3/12(34)

2/3(4)



14/59


15/59

Rabiger andVogelpohl20

Multistage bubble formation Formation of primary andsecondary bubbles was modeledby force balance

45, 20, 15,11, 27c, 6, 7,8, 18, 40, 32,34, 22, 23, 48

dB

Rice &Lakhani75

Two-stage model For an elastic hole from rubbersheet bubble volume of bubblewas found out applying smalland large deflection theories

27a, 1, 8,10, 11, 12,13, 20, 32,19, 42, 2, 48

Vb

Tsuge andHibino5

Two-stage model Bubble volume is obtained byforce balance including virtual

mass of bubble. It is given interms of dimensionless groups.

27b, 1, 5, 6,7, 8, 9a, 16,

19, 20, 25,28, 32, 34,37, 21, 23, 48

Vb

Gaddis andVogelpohl22

Multistage model Bubble detachment diameter isdeveloped by force balance interms of a few dimensionlessnumbers.

27b, 5,6,7, 8,20, 15, 1, 45,28, 32, 34,42, 48

dB

Tsuge et al.7 Two-stage model for bubbleformed at downward facingorifice submerged in liquid.

Model is obtained by forcebalance, equation of motion ofbubble for both wetted as well asnonwetted orifices for constantflow condition

26, 1, 10, 42,14, 27b, 5,19, 40, 24,20, 5, 6, 7,9a, 9b, 48

dB

Sada74 Single stage model Model is entirely based on theexperimentally observed bubbledimansions.

27c, 9, 14,17, 20, 23,34, 48

dBab

Tsuge et al.8 Continuous bubble growthmodel at high system

pressure

Bubble surface is divided into anumber of axisymmetric

elements which are characterizedradii of curvature. Underpolytropic high-pressure systemModified Rayleigh equation andequation of motion are solved byfinite differences.

15, 20, 25, 6,7, 8, 9, 34,

17, 47, 10, 48

VB

Marshallet al.76

Two-stage model Potential flow theory was appliedwith Bernoullis equation andequation of continuity to getvolume of bubble formed undercross-flowing liquid.

23, 22, 21,20, 11, 32,33, 34, 35, 1,27b, 36, 37, 19

Rb

Wilkinson &vanDierendonck77

Nonspherical two-stagebubble formation model

Thermodynamic equation of gasflow along with the equation ofmotion was solved.

20, 21, 27b,44, 46, 25

VB

Pamperin &Rath78

Two-stage quasi-staticunsteady shape bubbleformation under reduced

gravity.

Force balance including virtualmass was used to find thedetachment stage

27a, 27b, 35,34, 37, 25,38, 20, 9a

dB

A)

Re

Buyevichet al.79

Two-stage spherical bubblemodel

Force balance neglecting thegravitational force was carriedout for getting bubble size

5,6,7,8,9a,35, 36, 46, 44 RB

Tsuge et al.10 Two-stage nonsphericalbubble formation inquiescent liquid.

Equation of motion and modifiedRayleigh equation were used tosimplify the formula for bubblevolume at reduced gravity.

2, 4, 5, 6, 7,8, 9a, 27b, 46 VB


16/59

Making a force balance and lettingV) VE at the endof the first stage yields

When the first two terms on the right side are neglectedthe equation reduces to the one used for evaluatingstatic bubble volume.

In the second stage of detachment, since upwardforces are dominant than the downward forces, thebubble accelerates. The bubble is assumed to detachwhen its base has covered a distance equal to the radiusof the force balanced bubble. It is assumed that, therising bubble is not caught up and coalesced with thenext expanding bubble. Further, the bubble motion canbe given by Newtons second law of motion along withforce balance as

wherev is the velocity of the center of the bubble i.e.,v ) v + (dr/dt).

Above expression in terms of two velocity componentsv and dr/dt divides the drag into two terms and ondividing the simplified equation by (11F1/16) (VE + Qt)we yield

where

for boundary condition of at x ) rE, T) VF, it gives

Thus, the radius of a force-balanced bubble can be easilyobtained using the two-stage model. Though the for-mulation of model is simple, information about thecontact angle () is required.

For the case of intermediate region between constantpressure and constant flow rate, Khurana and Kumar16

have formulated a two-stage model. Considering thebubble formation at constant flow rate, a two-stageT

able3(Continued)

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VE5/3

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192(3/4)2/3gQ2 +

3

2(3/4)1/3g

F1QVE

1/3+D

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2/3 (5)

Mdv

dt

+ vdM

dt

)

[(VE + Qt)FLg - 6r(v + ve) -

Qt2(11/16FL)

12[3(VE + Qt)/4]2/3

- Dcos] (6)

(Dv/dT) + A(v/T) ) B - GT-4/3 - ET-5/3 - CT-1(7)

A ) 1 +6(3/4)0.33(1.25)VE

0.33

Q(11/16FL) ,B )

16g11Q

,

C )16Dcos

11QFL(8)

E ) Q

12(3/4)0.66, andG )

24

11(3/4)0.33FL

rE ) B

2Q(A + 1)(VF

2- VE

2 ) - ( CAQ)(VF - VE) -3G

2Q A-1/3(VF

2/3- VE

2/3) (9)



17/59

model can be generalized and the resulting force balanceyields.

The first term on right-hand side of (eq 10) is theexpansion force d(Mve)/dte, which on substitution in theforce balance equation gives the generalized equationfor the first stage. For the second stage of the detach-ment, inertial force is obtained by equating the inertialforce to the net upward force acting on the bubble.Velocity of the bubble center can be given as (v ) v +ve) and the force balance can be given as

substitution for v in above equation gives

where

substituting forM(dve/dt) + ve(dM/dt) in above equationyields

But for using these general equations it is necessary toexpress flow rate as a function of time and also thechange in radius in the second stage was also includedto yield

where

This model takes into account the detachment stage,all the viscous forces and the pressure gradient in theorifice-chamber as an effect of submergence and theweeping time as an effect of transient pressure variationin the chamber volume at the stage of bubble detach-ment. This helps in correcting the chamber pressure

term and increases the accuracy in predicting the bubble

size. Additionally, the model can also be used for findingthe bubble sizes in extreme conditions of constant flowor pressure. However, the number of parameters con-sidered for calculations are more as compared to theearlier models. Since the calculation procedure for thesize is tedious, here we have discussed another simplermodel which can be used for the estimation of bubblesize and comparison with experiments.

2.2.1.2. Gaddis and Vogelpohl Model.22 A simplemodel for bubble formation in quiescent liquids underconstant flow condition was proposed by Gaddis andVogelpohl.22 Comparison between theoretical predic-tions and experimental measurements made by largenumber of investigators has shown that the model is,valid over a wide range of viscosity and gas flow rates,

and hence highly reliable. The prediction of the detach-ment volume is complicated, but it was obtained byreleasing the constraints of spherical bubble shape andcylindrical neck. Further, equation of motion was ap-plied for the resulting geometries of hemisphere and theconical neck, which were not taken into considerationby earlier investigators (Davidson,3,4 Kumar and co-workers,14-17 Marmur and Rubin18,19), etc. But theapproach to solve the equations in this way is lengthyand rarely used in practice. In view of this, the modelunder discussion22 suggested two different approaches;(i) the equation of motion was applied to the neck atthe moment of detachment and then with the help of afew assumptions simplified equations were obtained,which were later solved analytically, and (ii) in thesecond approach total volume of the bubble just beforedetachment was calculated as the sum of the volume ofthe bubble at the end of detachment stage and thevolume of gas in the neck during detachment. Thevolume of the bubble at the end of the detachment stagewas found out by force balance. Since the secondapproach carries a number of errors in including theneck volume correctly the first is always used in theinvestigation for the dynamics of bubble formation.Considering the following conditions again (i) constantvolumetric gas flow, (ii) Liquid is quiescent and hasinfinite intensity in every direction, and (iii) Bubble isspherical in geometry and various forces (with referenceto Figure 7) can be defined as

Vt(FL - FG)g )(FG + 11/16FL)

12(3/4)2/3Vt2/3 [3Vt

Qt

dte+ Qt

2] +Dcos + 3

Qt

2(3/4)0.33Vt

1/3 (10)

d(Mv)

dt ) VtFg - 6rv - Dcos (11)

M

dv

dt + v

dM

dt ) Vt

Fg - 6r(v +ve) - Dcos -

(Fg + 1116F1)(3VtdQtdt

+ Qt2)

12(3/4)2/3Vt2/3

(12)

Mdvdt

+ vdMdt

+ Mdvedt

+ vedMdt

)

VtFg - 6r(v + ve) - Dcos (13)

M ) Vt(Fg + 1116Fl)anddM

dt ) Qt(Fg + 1116Fl) (14)

ddt

(Mv) ) VtFg - 6rv - Dcos (15)

Vtdvdt

+ Qtv ) rVtFg - 6(v + ve) - dhcos -

(3Vt

dQt

dt + Qt2

)12(3/4)0.66Vt2/3 (16)

) (FG + 1116FL)

Figure 7. Model for spherical bubble formation at constant flowconditions (reproduced with permission of Elsevier from Gaddisand Vogelpohl22).



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Different investigators have tried to solve the equationsresulting from the force balance using wide range ofdrag coefficient values for different systems. The resultsshowed that the maximum neck length is always 1/4 th

of d and displacement of the bubble center from thenozzle at the moment of detachment is 3/4th of d.Balancing these forces as

Can be given entirely in terms of bubble size (d) as

where

Then three different conditions can be applied to controlthe detachment diameter to yield a generalized equationfor dB as

Since in the transition to the jetting regime, drag andthe inertial force due to liquid acceleration differ greatlycompared with those calculated by the given formulas,the validity of above equation can be considered onlyup to the point of transition to the jetting regime. A jetis formed only when (i) the force due to gas momentumexceeds the surface tension force and (ii) surface tensionis not capable of forming the spherical or quasi-sphericalbubble. This occurs at a critical Weber number whichis equal to 4, and thus the force balance changes toFm+Fp ) Fs. Having an inverse dependence ofWeon holediameter, the results clearly show that for achieving acritical We, the nozzle velocities should increase withincreasing hole diameter. The predictions from this

Figure 8. Comparison of the predictions with experiments forvarious liquids. See details at right.

Figure 8 (continued). In A and B, symbols indicate the ex-perimental data of (1) Ramkrishnan et al.14: b dh ) 0.00367 m,) 0.552 cP, (2) Ramkrishnan et al.14: 9 dh ) 0.00367 m, ) 0.045cP, (3) Davidson and Schuler4: O aq. glyceroldh ) 0.00067m, )1.05cP, (4) Rabiger270: water 0 d h ) 0.002 m, ) 0.001 cP, (5)Morgenstern and Mersmann93: Glucose, dh ) 0.0002 m, )0.268 cP. (A) Predictions based on Gaddis and Vogelpohl22 forcebalance based model: The line numbers correspond to the predic-tions for individual data sets as numbered above. (B) Predictionsbased on Leibsons formulation: The line numbers correspond tothe predictions for individual data sets as numbered above. (C)Effect of orifice diameter on bubble size (Terasaka & Tsuge 11)bubbles generated in water ([dh ) 0.0003 m, ] dh ) 0.0004 m, dh ) 0.00198 m, dh ) 0.003 m) and in 68% glycerol solu-tion (*dh ) 0.0003 m, ] dh ) 0.0004 m, dh ) 0.00198 m,dh ) 0.003 m).

Fb )

6d3(FL - FG)g,Fp )

4dh

2(pG - pL),

Fm )

4dh

2FGwG

2

Fs ) dh,Fd )

4d2CD

FLwL2

2 , andFi ) (FGV - FLV1)a

(17)

Fb + Fm ) Fs + Fd + Fi (18)

d3 ) S + L/d + T/d2 (19)

S )6dh(4 - We)

4Fg ,L )

81VFg

,

T ) (13542 +27FG

2FL)

FLV2

Fg,We )

16FGV2

2dh3

(20)

dB ) [(6dh

FLg)

4/3

+ (81Vg ) + (135V2

42g)5/3

]1/4

(21)



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correlation show a very good match with the experi-mental data (Figure 8A), whereas the same data it canbe seen to have less agreement for Leibsons correlation,

indicating the importance of surface tension in theformulation of dB. Typical simulation result for thesequence of bubble formation in bottom submergedorifice can be seen in Figure 9A.11

In addition to the above two simple ways, severalother methods of bubble formation analysis exist. Theseinclude the analysis based on finite elements methods,the boundary integral methods and the use of Laplaceintegral equations. Here we discuss a few other inves-tigations which are important in the process of under-standing the subject of bubble formation.

2.2.2. Application of Potential Flow Theory.2.2.2.1. Wraith Model (1971).53 For the first time,Wraith53 showed that the velocity potential can besuccessfully used in developing a two stage mechanism

for bubble formation and it could be used to calculatethe volume of the bubble. Here, two-stage model hasbeen considered with an additional concept of theexpanding envelope. The model is based on the followingassumptions: (i) The liquid is inviscid and of largeextent, (ii) Surface tension may be neglected, (iii) Thegas injection rate is constant and the gas incompress-ible, (iv) Gas density is negligible, and (v) the bubblesurface is spherical.

Initially applying the pressure equation to a gasbubble formed in the liquid

which for a stationary expanding sphere of radius rtakes form as

at constant flow rate, the growth rate can be given as ) Q/(4r2), which on substitution in the pressureequation at R > r yields

Assuming that the pressure at infinity is negligible, inthe above equation, we have F(t) ) 0. The visualobservations can be used to take into account thehemispherical bubble growth with a as the radius ofsphere.

Then

Again using the pressure equation for the hemisphericalsurface at R ) a.

Equating the surface forces and the reactive forces inthe liquid gives

Phas two components as the hydrostatic pressure andthe pressure due to expansion. Since there is no expan-sion at the plate surface

Substituting in the force balance equation and integrat-ing over the hemispherical solid angle

if FR . a2 Po, the bubble is in equilibrium, and itremains in contact with the plate. This gives themaximum radius attainable by the bubble and themaximum hemispherical bubble volume as

respectively. In comparison to the earlier attempts, thismodel is distinct as it takes into account the growth ofbubble accompanied with the change in shape fromspherical to hemispherical during expansion beforedetachment. The resulting bubble volume is given as

Figure 9. (A) Sequence of simulated bubble shapes during itsgrowth at constant gas flow from a bottom submerged orifice (Rh) 1.52 mm,G ) 1.74 10-6 m3/s) (reproduced with permission ofElsevier from Terasaka and Tsuge11). (B) Sequence of bubbleformation from a top submerged orifice (reproduced with permis-sion of Elsevier from Liow61).

P

FL+ 0.5

r4

R4(drdt)

2) F(t) (23)

)r2

R

drdt

,q ) r2

R2drdt

(24)

ddt

)r2

R(d2r

dt2) + 2r

R(drdt)2

) 0 (25)

da/dt ) Q/(2a2)

P )FLQ

2

82a4+ Po - (FLga)cos (26)

Pe

FL+

Q2

82a4) 0 (27)

FR -FLQ

2

8a2- aPo +

23a3FLg ) 0 (28)

FR - 2a20

/2Pcossind ) 0 (29)

ah ) 0.453Q2/3g

-1/5 andVh ) 0.194Q6/5 g

-3/5

(30)

VB ) 1.09Q6/5 g

-3/5 (31)P

FL+ 0.5q2 -

t ) F(t) (22)



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Interestingly, the bubble volume is shown to have noeffect of gas-liquid systems physicochemical propertiesas well as the orifice properties, which reduces itspossible use for realistic estimations of bubble volumesfor a wide range of gas-liquid systems.

2.2.2.2. Marmur and Rubin18Approach based onEquilibrium Shape of Bubble.Equilibrium shape ofa bubble generated at a submerged orifice, detachment,delayed release of bubbles and finally multiple bubble

formation can be studied based on the Laplace equationand its further simplification. Marmur and Rubin18

studied the effect of chamber volume on bubble forma-tion by taking into account the detailed force balanceacross the interface, where the interface was discretizedin several elements. Their approach for understandingthe effect of chamber volume on bubble size assumesthat the gas flow rate is maintained such that thebubble remains attached to the orifice and the bubbleformation process is quasi-static.

2.2.2.3. Marmur and Rubin.19 In bubble formationprocess, the second stage of detachment is a result ofthe dominance of the buoyancy over the other forces,which was considered in all previous models. Buoyancyis dominant if the bubbles are of large sizes. The

bubbles lift-off process from the orifice due to buoyancywas put as a misleading approach by Marmur andRubin19 who proposed the process of detachment as adynamic process caused by the inward radial motion ofthe liquid at the orifice. This motion, which narrows theneck of the bubble up to detachment, takes finite time,during which the bubble continues to grow. Hence,according to them the static approach to lift-off at theinstant of equilibrium is inadequate. Since the wholeanalysis is highly involved, here we have restrictedourselves to explain the approach and important find-ings in a qualitative manner. To solve the underlyingproblem, they have solved the equation of motionnumerically under the following assumptions: (i) The

bubble is a volume of revolution around the axis of theorifice. (ii) The gas liquid interface is acted upon bypressure difference between the gas and the liquid andby surface tension forces. (iii) Inertial mass is assignedto the bubble interface, which equals the instantaneousmass of liquid being accelerated. Added mass of inviscidfluid exists for every bubble. (iv) The volume of liquidaround the bubble is very large as compared to thebubble volume. (v) The gas in the bubble, as well as thegas in the chamber flows and expands adiabatically. (vi)Pressure differences across the orifice determine the gasflow rate into the bubble. (vii) The pressure within thebubble is uniform and the same holds for the pressurein the chamber underneath the orifice. (viii) The mo-mentum of the gas is negligible. (ix) The formation of

the bubble is unaffected by the presence of the otherbubbles. (x) Bubble detachment occurs when the necknarrows to zero at one of its points.

Approximate equation of motion for the gas-liquidinterface was used. It was considered that the surfaceforce is due to pressure differences between the gas inthe bubble and the liquid and the line forces are due tosurface tension. In the dynamic process of bubbleformation the resultant of these forces is equal to therate of change in the liquid momentum, assuming thatthe gas momentum is negligible. In the analysis ofgrowing bubble surface, a systematic approach wasdeveloped for taking into account the position of eachsurface element so that a converging solution of equa-

tions was achieved. Under the assumption that themotion of interface was transient with respect to anypoint on the bubble surface, concept of added mass wasused to solve the Navier-Stokes equation by theLagrangian approach. The equations were solved usingthe boundary conditions of symmetry at the top and thatat the three-phase point at the edge of the orifice. Inaddition, two more boundary conditions of restrictingthe bubble growth with respect to orifice and no slipwere used. To neglect the effect of buoyancy, bubbleswere assumed to be of small size. To correlate thesolution of equation of motion and bubble volume, anapproach through thermodynamic equation of gas phasewas used.

A set of 11 equations was solved using finite differ-ence. It was observed that the solution could convergevery well for values of added mass coefficient of 0.85and orifice coefficient 0.65. To maintain the forcebalance across the interface, the computations weresubjected to adoption of bubble shape during its growth.The model was found to be sensitive to orifice diameterat higher flow rates, as it resulted in deviating bubblesizes. Their results were found to support the conclusionas deduced by Davidson and Schuler4 in the case of

inviscid fluid.2.2.3. Approach based on Boundary IntegralMethod. In general, boundary value problems withinterfaces exhibit relatively large surface-to-volumeratio, and in this case, the boundary element method isexpensive compared to domain techniques such as thefinite element method. Boundary element methods arevery important for solving boundary value problems inthe PDEs. Many boundary value problems of PDEs canbe reduced into boundary integral equations by thenatural boundary reduction. The boundary integralmethod is based on Greenss formula, which on refor-mulation of the potential problem as the solution of aFredholm integral equation helps in reducing the di-mension of the problem by one. Although the method

has been used in several different areas of research, veryfew investigators have used this approach to evaluatethe problem of bubble formation. Below, we discuss thetwo important contributions made in the subject underconsideration.

2.2.3.1. Hoopers Approach of Potential Flow.82

The boundary element method has been applied for thestudy of the bubble formation, which also helps inmathematically understanding the mechanism of de-tachment of the bubble using the equations of motionfor the liquid. It has been assumed that volume of liquidis large so that the effect of side walls is negligible andthe depth is much greater than the largest size of thebubble. Hooper82 applied this technique assuming the

surrounding liquid to be inviscid. The velocity was rela-ted with the gradient of stream function asu ) and

Bernoullis equation was applied for the liquid in motionin terms of velocity potential.

As

hence

t +

12|u2| +

(P1 + FLgz)

FL) C(t) (32)

zf,P1 + FLgzf0

C ) 0



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Writing the equation in Lagrangian form for conven-ience yields

For the compressible fluid the continuity equation canbe written in Laplace form as 2 ) 0

Since there is no flow through the orifice pate at z )

0, d/dz ) 0.At the interface the normal velocity of the liquid andof the bubble are equal and surface tension results inthe pressure jump. Hence onS, d/dn ) vsand P1 ) Pb-k.

For a closed system with radius rO the orifice coef-ficient was given as

Above equations were solved simultaneously to take intoaccount the pressure correction at each time step. Thefollowing assumptions were made while solving theequations:

(i) For very small chamber volume, the gas flowsdirectly into bubble at the entrance pressure,(ii) When chamber volume is very large, PC and FC

are approximately constant and equals PE and FE.(ii) Expansion of bubble is isothermal.The equations were solved under the boundary condi-

tions, which relateP1to Pband the gas equations fromwhich Pb can be calculated are substituted into equa-tions of motion. Initial conditions for the values ofPband PC and the initial position of bubble surface andthe value of on bubble surface are needed. A system-atic approach has been used for analyzing the bubblegrowth in time, and the results are validated with thepublished literature. However, the processing time forthe calculations greatly depends on the initial conditions

and boundary conditions. Second, since the model doesnot take into account the effect of wake pressure, theeffect of the primary bubble on the secondary bubble isneglected. Pinczewski83 has also followed a similarapproach for describing bubble formation at a singlesubmerged orifice.

2.2.3.2. Xiao and Tans Boundary IntegralMethod.28 In a very recent attempt, Xiao and Tan28

have proposed an improved boundary integral methodwhich is an extension to the above-discussed analysisby Hooper.82 Under the assumption of constant flow ratecondition, the viscosity of liquid is considered to benegligible and the flow is irrotational. They have usedLaplaces equation to describe the velocity potential andBernaulli integral is applied between the liquid side ofthe bubble surface and a point in the liquid. Theirsystematic approach includes the following steps andequations: (i) Laplaces equation, (ii) thermodynamicequations for the gas flow, (iii) curvature of bubblesurface from the analytical planar geometry, (iv) volu-metric growth rate of bubble, (v) estimation of thenormal velocity through Greens integral formula for apiecewise smooth surface, (vi) reduction of dimensionby assuming axisymmetry of bubbles, where the surfaceis divided in several elements and an isoparametriclinear approximation is used for the surfaces andfunctions through proper discretization, and (vii) systemof images used for satisfying the zero-normal-velocitycondition at the rigid boundary. (vii) The surface velocity

specifications are completed by defining the tangentialvelocity over the surface through cubic spline interpola-tion over the surface, and finally, (viii) solution of theseset of equations is carried out in time domain while aniterative trapezium rule with Eulers method is used forupdating the time and space coordinates of a growingbubble. The approach is systematic and has a fewimportant advantages over the method by Hooper,82

however the estimated bubble sizes are noticeablyhigher when compared with the experimentally ob-tained bubble sizes of Kupferberg and Jameson65 (Fig-ure 4 from Xiao and Tan28). Also, the simulated bubblesequences do not show the existence of bubble neck,which in reality is known to exist from several experi-mental observations in the literature.

Importantly, these approaches based on detailedmodeling of the development of bubble surface form animportant direction for further research in systemswhere the experimental analysis is always not feasible(viz. bubbling in plasma, reduced gravity conditions,bubbling in opaque liquids such as mercury, etc.). Themost important component is the validity of the ap-proach through comparison with the cold flow experi-ments, which should be satisfied to optimize the use of

available computational power.2.3. Mechanism of Bubble Formation in Top

Submerged Orifice in Stagnant Liquids. The ad-vantages of quick withdrawal of orifice in the event offailure of gas supply and less clogging of the orificefacilitates easy control of operation and hence makesthe top submerged orifice very commonly used in themetallurgical industry. In earlier studies, Datta et al. 42

have shown that the diameter of a bubble formed at thetip of the top submerged orifice is larger than that ofthe bottom submerged orifice of the same diameter.However, the diameter of the bubble is not a strongfunction of the way of submergence but of the diameterof the orifice. In this subsection, we discuss the modelsexplaining the mechanism of bubble formation at thetop submerged orifice in brief.

2.3.1. Tsuge Model6. The two-stage model for bubbleformation under constant flow condition was used forthe prediction of the size of the bubbles formed. Themathematical model that was used to validate theexperimental data was based on the following assump-tions: (i) The flow rate of gas into the bubble throughthe nozzle is constant during bubble formation. (ii) Thebubble maintains spherical shape during its growth.(iii) Bubble motion is not affected by the presence ofother bubbles. (iv) The bubble formation consists of twostages.

In this case, the first step is similar to earlierdescribed expansion stages while during the detachmentstage the buoyancy force is balanced by the other forcesand the bubble continues to grow while lifting upvertically but the gas is still fed through the nozzle. Thedetachment stage comes to an end when the base of thebubble detaches from the nozzle tip and the bubbleseparates. According to their approach, wettability ofthe nozzle can be considered as the basic criteria forstudy and thus the wetted and nonwetted nozzles wereconsidered.

(A) Wetted Nozzle Condition. Bubble formationprocess in the case of liquids such as water and organicliquids (in a wetted nozzle) takes place in two stages.In the expansion stage constant flow condition is writtenas

DDt

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