bubble formation and bubble rise velocity in gas liquid...
TRANSCRIPT
Bubble Formation and Bubble Rise Velocity in Gas-LiquidSystems: 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 Nguyen’s 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 22. Bubble Formation in Gas-Liquid
Systems3
2.1. Bubble Formation at SingleSubmerged Orifice
4
2.1.1. Factors Affecting BubbleFormation
5
2.1.1.1. Effect of LiquidProperties
5
2.1.1.2. Effect of Gas Density 92.1.1.3. Effect of Orifice
Configuration9
2.2. Mechanism of Bubble Formationin Newtonian Liquids
12
2.2.1. Force Balance Approach inBottom Submerged Orifice inStagnant Liquid
12
2.2.1.1. Kumar and Co-workers 132.2.1.2. Gaddis and Vogelpohl
Model17
2.2.2. Application of Potential FlowTheory
19
2.2.2.1. Wraith Model 19
2.2.2.2. Marmur and RubinApproach based onEquilibrium Shape ofBubble
20
2.2.2.3. Marmur and Rubin 202.2.3. Approach based on Boundary
Integral Method20
2.2.3.1. Hooper’s Approach ofPotential Flow
20
2.2.3.2. Xiao and Tan’s BoundaryIntegral Method
21
2.3. Mechanism of Bubble Formationin Top Submerged Orifice inStagnant Liquids
21
2.3.1. Tsuge Model 212.3.2. Liow Model 22
2.4. Model for Formation ofNon-spherical Bubbles
22
2.5. Mechanism of Bubble Formationin Flowing Liquids
23
2.5.1. Co-current Flow 232.5.2. Counter Current Flow 242.5.3. Cross-Flow 24
10.1021/ie049131p CCC: $30.25 © xxxx American Chemical SocietyPAGE EST: 58.9Published on Web 00/00/0000
2.6. Bubble Formation underReduced Gravity Condition
26
2.6.1. Models for Bubble Formation 262.6.1.1. Nonspherical Bubble
Formation Model26
2.6.1.2. Two Stage BubbleFormation Model
27
2.6.1.3. Mechanism of BubbleFormation in QuiescentLiquids
27
2.7. Bubble Formation inNon-Newtonian Liquids
27
2.7.1. Structurally Viscous Liquids 282.7.2. Viscoelastic Liquids 28
2.8. Bubble Formation Accompaniedwith Weeping at the SubmergedOrifice
29
2.9. Bubble Formation at PerforatedPlates and Sieve TraysSubmerged in Liquid
30
2.9.1. Observations of BubbleFormation at MultipointSparger
30
2.9.2. Mechanism of BubbleFormation
31
2.9.3. Models of Bubble Formation 312.9.3.1. Li et al. Model 312.9.3.2. Miyahara and Takahashi
Approach32
2.9.3.3. Loimer et al. (2004) SingleBubble Approach
32
2.9.4. Factors Affecting BubbleFormation at Sieve Plates
32
2.9.4.1. Gas Flow Rate 322.9.4.2. Pitch of Holes on Sieve
Trays33
2.10. Experimental Techniques 332.10.1. Image Visualization and
Analysis33
2.10.2. Volume of Displaced Liquid 332.11. Conclusion and
Recommendations33
3. Bubble Rise Velocity in Liquid 343.1. Factors Affecting the Rise
Velocity35
3.1.1. Effect of Purity of Liquid onBubble Rise Velocity
35
3.1.2. Effect of Liquid Viscosity onBubble Rise Velocity
37
3.1.3. Effect of Temperature onBubble Rise Velocity
38
3.1.4. Effect of External Pressureon Rise Velocity
38
3.1.5. Effect of Initial BubbleDetachment Condition
39
3.2. Formulations for Rise VelocityCorrelations
39
3.2.1. Force Balance Approach 393.2.2. Approach based on
Dimensional Analysis39
3.2.3. Approach through WaveAnalogy
43
3.3. Bubble Rise Velocity inNon-Newtonian Liquids
44
3.4. Comparison of Correlations forBubble Rise Velocity in Pureand Contaminated Liquids
45
3.5. Drag Coefficient for Bubbles inLiquids
46
3.5.1. Comparison of theCorrelations for DragCoefficient in NewtonianLiquids
50
3.6. Bubble Trajectories 513.7. Experimental Methods for
Measurement of Rise Velocity52
3.7.1. Nonintrusive Methods ofBubble Rise Analysis
52
3.7.1.1. Photographic Methods 523.7.1.2. PIV Measurements 523.7.1.3 LDA measurements 52
3.7.2. Intrusive Measurements 523.7.2.1. Optical Probes 523.7.2.2. Conductive/Capacitance
Probes53
3.7.3. Imaging in Opaque Systems 533.7.3.1. Dynamic Imaging by
Neutron Radiography forOpaque Systems
53
3.7.3.2. X-ray Imaging 533.8. Conclusion 53
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].
B
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 amountof 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 contactingthrough 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 andAdams.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.
C
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 singularlyonly 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).
D
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 theliquid), 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 hasnot 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 largediameters and higher gas velocities. From the rigorousanalysis of Jamialahmadi et al.60 the size can beassigned a dependence on viscosity as ∼µ0.66.
(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 aresmall, 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 severalinvestigators (Since these are predictions, symbols correspondingto different investigators are connected by lines) and dh ) 1 mm.(Ramkrishnan et al.14: 9 Water, µ ) 0.001, σ ) 0.072, ) glycerol-aq µ ) 0.0449, σ ) 0.067, [ glycerol-aq, µ ) 0.552, σ ) 0.063),(Rabiger and Vogelpohl20, O glycerol-aq, µ ) 0.007, σ ) 0.069),(Davidson and Schuler3, ∆ glycerol-aq, µ ) 0.515, σ ) 0.063),(Terasaka & Tsuge11, 2 water, µ ) 0.012, σ ) 0.053, + Glucose, µ) 0.269, σ ) 0.08,)
E
Tab
le1.
Imp
orta
nt
Stu
die
son
the
Bu
bb
leF
orm
atio
nat
Sin
gle
Su
bm
erge
dO
rifi
cein
Liq
uid
.(T
his
tab
leco
vers
the
inve
stig
atio
ns
that
form
the
bas
isof
the
rece
nt
atte
mp
ts.W
eco
nfe
ssou
rin
abil
ity
toin
clu
de
info
rmat
ion
from
the
enti
rep
ub
lish
edli
tera
ture
onth
issu
bje
ctin
this
Tab
le)
inve
stig
ator
s
expe
rim
enta
lsy
stem
&ob
serv
atio
ns
appr
oach
inth
est
udi
espa
ram
eter
sst
udi
edan
dco
ndi
tion
sli
mit
atio
ns
con
clu
sion
&re
mar
ks
Sch
nu
rrm
ann
47B
ubb
lefo
rmat
ion
thro
ugh
apo
rou
sdi
aph
ragm
led
toob
serv
atio
nof
one
stag
ebu
bble
form
atio
n.
Pre
ssu
reva
riat
ion
over
the
diap
hra
gmw
asm
odel
edan
alyt
ical
ly.
r,µ,
σ,g
Res
ult
sw
ere
lim
ited
up
ton
onel
ectr
olyt
icso
luti
ons.
For
mat
ion
ofva
ryin
gbu
bble
ssi
zedu
eto
diap
hra
gmex
pan
sion
was
not
take
nin
toac
cou
nt.
Siz
eof
bubb
lefo
rmed
isin
depe
nde
nt
ofth
evi
scos
ity
ofli
quid
.
Eve
rsol
eet
al.48
On
est
age
bubb
lefo
rmat
ion
was
obse
rved
for
diff
eren
tor
ific
esi
zes.
Bu
bble
volu
me
was
obta
ined
byst
robo
scop
icfr
equ
ency
mea
sure
men
ts.
Hig
hga
sfl
owra
te,
µ,σ
F,P
Con
clu
sion
isge
ner
aliz
edas
sum
ing
con
stan
tbu
bble
freq
uen
cy,c
ontr
adic
tory
toth
eex
peri
men
ts.
Mec
han
ism
ofst
atic
bubb
lefo
rmat
ion
isn
otap
plic
able
.
Inpu
reli
quid
s,bu
bble
freq
uen
cyis
inde
pen
den
tof
the
gas
flow
rate
,liq
uid
prop
erti
esan
dth
eor
ific
edi
amet
er.
Kre
vele
nan
dH
ofti
jzer
49S
epar
ate
and
chai
nbu
bble
form
atio
nC
ompl
ete
wet
tin
gof
orif
ice
was
assu
med
wit
hge
ner
atio
nof
sph
eric
albu
bble
s.B
ubb
les
wer
ecl
assi
fied
assm
all,
med
ium
,an
dla
rge.
Asc
endi
ng
velo
city
was
obta
ined
from
the
gas
flow
rate
s.
Bu
bble
diam
eter
,gas
hol
d-u
p,as
cen
din
gve
loci
tyof
bubb
les,
gas
liqu
idin
terf
acia
lare
a
Gas
flow
rate
was
not
con
stan
tan
dh
ence
freq
uen
cyof
bubb
lefo
rmat
ion
was
not
the
sam
eth
rou
ghou
t.T
heo
reti
calr
esu
lts
wit
hfr
equ
ency
only
less
than
one
wer
ein
coor
din
atio
nw
ith
that
ofex
peri
men
tal.
For
sepa
rate
bubb
lefo
rmat
ion
dB
isin
depe
nde
nt
ofQ
and
prop
orti
onal
tocu
bero
otof
orif
ice
diam
eter
.F
orch
ain
bubb
lin
gd
Bis
inde
pen
den
tof
dh.
Ben
zin
gan
dM
ayer
s50D
imen
sion
alan
alys
isw
asu
sed
toco
rrel
ate
the
expe
rim
enta
lre
sult
s.
Dia
met
erof
orif
ice
(2.1
-4.
7m
m),
g,µ,
σ,F
Liq
uid
s:W
ater
,wat
er-
eth
anol
,su
gar
solu
tion
,D
ren
e-w
ater
,eth
ylen
egl
ycol
,Wes
son
oil.
Det
erm
ined
the
data
ofdr
agco
effi
cien
tfo
rve
rysm
allt
ips
wh
ich
does
not
mat
chfo
rth
em
odel
bysu
ccee
din
gin
vest
igat
ors
thou
ghth
em
eth
odof
appr
oach
seem
sto
besa
me.
For
low
con
cen
trat
ion
ofsu
rfac
tan
tsit
sef
fect
onbu
bble
size
isn
egli
gibl
e.E
ffec
tof
orif
ice
len
gth
isn
egli
gibl
eon
lyw
hen
gas
flow
rate
isre
lati
vely
smal
l.H
ugh
eset
al.51
Osc
illo
scop
ean
dst
robo
scop
ew
ere
use
dto
cou
nt
the
bubb
lefr
equ
ency
.P
roce
ssof
bubb
lefo
rmat
ion
was
assu
med
toh
ave
no
stag
es.
Dim
ensi
onal
anal
ysis
was
carr
ied
out
for
Re,
NC
and
virt
ual
mas
s.F
orce
bala
nce
was
appl
ied
ona
bubb
le.
Pre
ssu
reeq
uat
ion
sw
ere
con
side
red
and
solv
edfo
rC
D,V
T,f
ron
tala
rea
ofbu
bble
,an
dV
Bfo
rdi
ffer
ent
NC.
g,µ L
,σ,F
L,F
G,g
asve
loci
ty,l
iqu
idve
loci
ty,
vess
elco
nst
ruct
ion
,gas
inle
tco
nst
ruct
ion
,pr
essu
refl
uct
uat
ion
s(p
uls
atio
n)
atth
eti
pof
orif
ice.
Itis
dou
btfu
lwh
eth
erth
eN
Ch
asan
yin
depe
nde
nt
effe
cton
the
bubb
lefo
rmat
ion
.
Th
ech
ambe
rvo
lum
eh
asst
ron
gef
fect
ond
B.
Tip
len
gth
belo
w10
0ti
mes
tip
diam
eter
does
not
infl
uen
cebu
bble
size
.Gas
turb
ule
nce
crea
tes
non
-u
nif
orm
size
bubb
les.
VB
isin
depe
nde
nt
ofN
Cfo
rN
C<
0.8.
Lei
bson
etal
.52S
trob
osco
pic
stu
dies
wer
epe
rfor
med
.T
he
bubb
lesi
zes
wer
efo
un
dat
vari
ous
expe
rim
enta
lcon
diti
ons
and
empi
rica
lfor
mu
las
wer
ede
rive
du
sin
gfo
rmu
lafo
ror
ific
edi
sch
arge
coef
fici
ent.
Th
eto
taln
um
ber
ofbu
bble
sw
asca
lcu
late
dfr
omth
efr
equ
ency
data
for
each
size
ran
ge.
Gas
flow
rate
,(0
<R
e<
4000
0)or
ific
edi
amet
er,
and
type
ofor
ific
es
Met
hod
ofn
orm
aldi
stri
buti
ondo
esn
oth
old
accu
rate
lyfo
rve
ryh
igh
Re.
Eff
ect
ofli
quid
prop
erti
eson
bubb
lesi
zes
has
not
been
stu
died
and
hen
cere
sult
sca
nn
otbe
gen
eral
ized
.
Bu
bble
size
isre
lati
vely
un
ifor
mat
give
nR
ean
dde
pen
dsm
arke
dly
onor
ific
edi
amet
er.
F
Dav
idso
nan
dS
chu
ler3
(Vis
cou
sli
quid
s)
Str
obos
cobi
cm
eth
odw
asu
sed
for
the
expe
rim
ents
.
Equ
atio
nof
mot
ion
and
virt
ual
mas
sof
bubb
lew
asu
sed
tofi
nd
out
size
ofbu
bble
theo
reti
call
y.
g,µ,
σ,F,
gas
flow
rate
,or
ific
edi
men
sion
s.A
ssu
mpt
ion
that
bubb
leal
way
sm
oves
atit
sst
okes
velo
city
does
not
hol
dto
real
ity
and
sin
cesu
rfac
ete
nsi
onfo
rces
are
neg
lect
ed,t
he
mod
elm
ayn
otbe
vali
dfo
ral
lsiz
es.
Bu
bble
size
sw
ere
obta
ined
theo
reti
call
ym
atch
very
wel
lwit
hth
ose
ofex
peri
men
tall
y.
Dav
idso
nan
dS
chu
ler4
(In
visc
idli
quid
s)
Str
obos
cobi
cm
eth
odw
asu
sed
for
the
expe
rim
enta
lob
serv
atio
ns.
Equ
atio
nof
upw
ard
mot
ion
isco
nsi
dere
dan
dso
lved
un
der
init
ial
con
diti
ons
tofi
nd
out
crit
ical
bubb
levo
lum
e.
µ,σ,
F,ga
sfl
owra
te,
orif
ice
dim
ensi
ons
Dra
gco
effi
cien
tis
con
side
red
tobe
zero
.In
the
virt
ual
mas
sof
bubb
le,t
he
11/1
6te
rmco
mes
into
play
only
for
low
visc
osit
yli
quid
san
dth
isdi
ffer
sth
eva
lues
ofbu
bble
volu
me
calc
ula
ted
from
expe
rim
ent.
As
each
bubb
lede
tach
from
the
orif
ice
itle
aves
beh
ind
avo
lum
eof
gas
wh
ich
form
sa
nu
cleu
sfo
rth
en
ext
bubb
le.
Dro
pw
eigh
tm
eth
odca
nbe
use
dfo
rve
rylo
wfl
owra
tes
tode
term
ine
the
bubb
levo
lum
e.R
amkr
ish
nan
etal
.14S
trob
osco
bic
met
hod
was
use
dfo
rth
eex
peri
men
tal
obse
rvat
ion
s.
For
ceba
lan
cew
asdo
ne
and
virt
ual
mas
sco
nce
ptw
asu
sed
inth
efi
rst
stag
ean
dfo
rth
ese
con
dst
age
New
ton
’sse
con
dla
wof
mot
ion
was
appl
ied
toge
tth
ebu
bble
volu
me
un
der
con
stan
tfl
owco
ndi
tion
.
g,µ
(1-
552c
P),
σ(4
1.4-
71.7
Ν/m
),F
(987
-12
57kg
/m3 )
,gas
flow
rate
(1-
80cm
3 /s)
,ti
me
for
bubb
lefo
rmat
ion
,con
tact
angl
e,or
ific
edi
amet
er(1
.3-
7m
m).
Th
em
odel
can
beap
plie
don
lyw
hen
the
risi
ng
bubb
leis
not
cau
ght
up
and
coal
esce
dw
ith
the
nex
tex
pan
din
gbu
bble
.
At
extr
emel
ysm
allf
low
rate
son
lybu
oyan
cyan
dsu
rfac
ete
nsi
onfo
rces
act
onth
ebu
bble
.T
he
effe
ctof
diff
eren
tpa
ram
eter
son
volu
me
ofbu
bble
depe
nds
onth
esy
stem
un
der
con
side
rati
onan
dth
eex
peri
men
tals
etu
p.S
atya
nar
ayan
etal
.15S
trob
osco
bic
met
hod
was
use
dfo
rth
eex
peri
men
tal
obse
rvat
ion
s.
For
ceba
lan
cew
asdo
ne
wit
hou
tn
egle
ctin
gth
eef
fect
ofga
sde
nsi
tyfo
rbo
thst
ages
atco
nst
ant
pres
sure
con
diti
on.
g,µ
(60-
600c
P),
σ(3
5-72
N/m
),F L
(979
-12
46kg
/m3 )
,gas
flow
rate
(2-
250
cm3 /
s),
orif
ice
diam
eter
(0.5
-4.
05m
m).
Mod
elis
com
plic
ated
tobe
solv
edn
um
eric
ally
and
tim
e-co
nsu
min
gsi
nce
nu
mbe
rof
iter
atio
ns
tobe
carr
ied
out
isla
rge.
Th
evo
lum
esof
bubb
les
obta
ined
nu
mer
ical
lym
atch
esw
ith
that
ofth
eex
peri
men
talo
bser
vati
on.
Kh
ura
na
etal
.16In
let
pres
sure
vari
atio
n,
aver
age
flow
rate
,w
eepi
ng
rate
wer
em
easu
red
and
Str
obos
cobi
cm
eth
odw
asu
sed
for
esti
mat
ing
the
bubb
lefr
equ
ency
.
Gen
eral
ized
equ
atio
ns
ofm
otio
nan
dfo
rce
bala
nce
for
un
stea
dyst
ate
for
bubb
lefo
rmat
ion
atin
term
edia
teco
ndi
tion
sw
asst
udi
ed.
g,µ,
σ(4
0cp)
,F(9
60kg
/m3 )
,ga
sfl
owra
te(3
cm3 /
s-
20cm
3 /s)
,or
ific
edi
amet
er(3
mm
),or
ific
ech
ambe
rvo
lum
e(6
5cm
3 -60
0cm
3 ),o
rifi
cesu
bmer
gen
ce(0
.156
,0.6
6,1.
28m
).
Inth
eit
erat
ive
proc
edu
refo
rca
lcu
lati
ng
the
bubb
levo
lum
ein
stea
dof
ali
nea
r,ex
pon
enti
aldi
sch
arge
ofga
sfr
omth
ech
ambe
rsh
ould
beco
nsi
dere
d.
Con
cept
ofw
eepi
ng
tim
ean
dfo
rmat
ion
tim
eca
nbe
wel
lapp
lied
toat
inte
rmed
iate
con
diti
onto
stu
dybu
bble
form
atio
n.
Ku
mar
&K
ulo
or17
Hig
h-s
peed
phot
ogra
phy
was
use
dfo
rex
peri
men
tal
obse
rvat
ion
s.
For
ceba
lan
cew
asdo
ne
atn
egli
gibl
esu
rfac
ete
nsi
on.
g,µ,
σ,F L
,FG,g
asfl
owra
te,o
rifi
cedi
amet
er,
tim
eof
form
atio
n,
expa
nsi
onan
dde
tach
men
t,
Su
rfac
ete
nsi
onfo
rces
are
mad
en
egli
gibl
eby
usi
ng
smal
ldia
met
erca
pill
arie
s.E
xper
imen
tal
obse
rvat
ion
sw
ere
mad
eon
lyu
sin
gsm
ooth
capi
llar
ies.
Tw
ost
age
mod
elw
ith
cert
ain
con
diti
ons
deci
des
the
tim
ede
pen
den
tvo
lum
eof
the
bubb
le.
Wra
ith
53H
igh
-spe
edph
otog
raph
yw
asu
sed
for
the
expe
rim
enta
lm
easu
rem
ent
ofbu
bble
size
s.
Con
cept
ofve
loci
typo
ten
tial
was
appl
ied
for
anh
emis
pher
ical
expa
ndi
ng
bubb
leto
fin
dit
svo
lum
e.
6.3
<d
h<
19.1
mm
30<
VC
<45
cm3 .
Gas
kin
etic
ener
gy,
acou
stic
capa
cita
nce
ofth
en
ozzl
esy
stem
,gas
flow
rate
wer
eva
ried
Su
rfac
ete
nsi
onfo
rces
are
neg
lect
ed.M
odel
isap
plic
able
only
abov
ecri
tica
lgas
flow
rate
.At
hig
hga
sin
ject
ion
rate
sth
ism
odel
fail
sbe
cau
seof
coal
esce
nce
.
Intr
insi
cco
ales
cen
ceis
ten
den
cyof
atw
ost
age
mod
elfo
rth
efl
owra
tes
oth
erth
anim
med
iate
lyab
ove
crit
ical
flow
rate
.M
arm
ur
&R
ubi
n18
On
est
age
slow
bubb
lefo
rmat
ion
un
der
quas
i-st
atic
con
diti
onw
asob
serv
edvi
sual
lyan
dth
eob
serv
atio
ns
are
use
dfo
rm
odel
form
ula
tion
.
Th
eory
ofin
terf
aces
ineq
uil
ibri
um
for
abu
bble
was
appl
ied
inth
efo
rmat
ion
stag
ean
dit
str
ansi
tion
toa
dyn
amic
stat
eof
deta
chm
ent
was
deci
ded
ingr
avit
yfi
eld
usi
ng
velo
city
pote
nti
al.
Ori
fice
dim
ensi
ons,
subm
erge
nce
,con
tact
angl
e,ch
ambe
rvo
lum
e.
Eff
ect
ofva
riou
spa
ram
eter
son
dyn
amic
nat
ure
ofth
ede
tach
men
tst
age
can
not
bepr
edic
ted
byth
aton
the
quas
i-st
atic
stat
eof
bubb
lefo
rmat
ion
.
Bu
bble
sust
ain
sin
equ
ilib
riu
mat
max
imu
mra
diu
sof
the
orif
ice
and
this
radi
us
depe
nds
onth
eli
quid
prop
erti
es.
G
Tab
le1
(Con
tin
ued
)
inve
stig
ator
s
expe
rim
enta
lsy
stem
&ob
serv
atio
ns
appr
oach
inth
est
udi
espa
ram
eter
sst
udi
edan
dco
ndi
tion
sli
mit
atio
ns
con
clu
sion
&re
mar
ks
Mar
mu
r&
Ru
bin
19E
quat
ion
ofm
otio
nat
the
inte
rfac
ew
asde
velo
ped
and
toge
ther
wit
hth
eth
erm
odyn
amic
equ
atio
ns
for
gas
inth
ebu
bble
and
the
cham
ber
volu
me
belo
wit
the
inst
anta
neo
us
shap
eof
bubb
ledu
rin
gfo
rmat
ion
was
calc
ula
ted.
Ori
fice
subm
erge
nce
,g,
µ,σ,
F L,F
G,
gas
flow
rate
.
Mod
elis
rest
rict
edto
sin
gle
bubb
lefo
rmat
ion
.S
tati
cor
lift
-off
appr
oach
atth
ein
stan
tof
equ
ilib
riu
mis
inad
equ
ate.
Gas
mom
entu
mis
take
nn
egli
gibl
e.M
odel
isva
lid
for
only
very
smal
lbu
bble
s.
Inst
ant
ofde
tach
men
ti.e
.,th
ecl
osu
reof
nec
kca
nbe
dete
rmin
edth
eore
tica
lly.
Ach
arya
etal
.54S
ingl
est
age
mod
elW
ave
theo
ryw
asap
plie
dto
dete
rmin
eth
evo
lum
eof
the
bubb
lein
dyn
amic
deta
chin
gco
ndi
tion
.
Vis
cou
s,in
erti
al,i
nte
rnal
stre
sses
,sh
ear
rate
,ga
sfl
owra
te,o
rifi
cedi
amet
er.
Su
rfac
ete
nsi
onfo
rces
are
neg
lect
ed.
Inca
seof
hig
hly
visc
ous
liqu
ids
sin
cede
tach
men
tof
bubb
leis
thro
ugh
nec
kfo
rmat
ion
obse
rvat
ion
ofsi
ngl
est
age
bubb
lefo
rmat
ion
isu
nac
cept
able
.
On
lyga
sfl
owra
tean
dac
cele
rati
ondu
eto
grav
ity
deci
deth
evo
lum
eof
bubb
les
that
are
form
edin
non
-New
ton
ian
liqu
ids.
Vog
elpo
hl&
Rab
iger
21M
ult
ista
gebu
bble
form
atio
nF
orm
atio
nof
prim
ary
and
seco
nda
rybu
bble
sw
asm
odel
edby
forc
eba
lan
ce
g,µ,
σ,F L
,FG,g
asfl
owra
te,d
irec
tion
offl
ow,
size
ofor
ific
e,or
ific
esu
bmer
gen
ce.
Mod
elis
appl
icab
leon
lyin
jett
ing
regi
me.
At
hig
her
µ Ln
eck
mot
ion
isn
olo
nge
rsu
ffic
ien
tto
deta
chth
epr
imar
ybu
bble
and
bubb
leco
ales
ces
wit
hse
con
dary
,wh
ich
isn
otpr
edic
ted
from
the
mod
el.
Sec
onda
rybu
bble
size
can
not
beca
lcu
late
dfr
omth
ede
velo
ped
mod
elas
itis
anin
duce
dpr
oces
s.
Su
ctio
nef
fect
ofth
epr
imar
ybu
bble
indu
ces
the
form
atio
nof
seco
nda
rybu
bble
.L
iqu
idm
otio
nh
asst
ron
gef
fect
onth
efo
rmat
ion
and
deta
chm
ent
ofpr
imar
ybu
bble
.
H
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 bubbleattains 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, ahigher 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 bubblevolume. (+ water 0.073 N/m, ∆ n-Propanol 0.0238N/m, × Ethanol0.0228 N/m, O Methanol 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: (] 0.1MPa, ∆ 0.3 MPa, O 2.1 MPa). For the inset box, abscissa is thegas density in gm/cm3 and ordinate is the bubble size in mm.
I
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 by Vb ) 2πRoσ/(∆Fg) known as Tate’s 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 thesmallest 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 < 2 × 10-6
m3/s) and above this value, the dependence is verystrong. For small diameter orifice, the forces of attach-ment are very strong and also the contact angles havewide variation. At low gas flow rates where turbulenceis lower, for smaller orifice diameters, the waiting timeis longer, whereas for a high gas flow rate the detach-ment is faster.
(c) Orifice Chamber Volume (VC). It is known that,the mechanism of bubble formation depends greatly onthe mode of gas supply and hence on the chambervolume. Thus, for a very small chamber, the in-flowinggas to the chamber will be approximately equal to thegas flow into the bubble. This condition is known as theconstant flow operation. On the other hand, for largechamber volumes, as a result of damped pressurefluctuations, the in-flowing gas to the bubble is not thesame as into the chamber. Thus, the pressure forcesthrough the chamber volume, which are dynamic innature, affect the bubble formation in various ways.Satyanarayan15 has observed that the bubble volumeincreases with chamber volume by almost the sameamount at all flow rates. However, at low orificesubmergence, the effect becomes more pronounced. Theconstant pressure conditions can be said to exist whena further increase in chamber volume does not vary thebubble volume where the discharge resistance is small.Thus, in smaller chambers, initially, the pressure in thebubble and in the chamber will increase beyond theinitial pressure as an effect of the inflow of gas and the
inertia of liquid. This results in the growth of the bubbleand thus suppresses the effect of surface tension forces,which tends the bubble to grow faster than the rate atwhich gas is being supplied to the chamber volumedeveloping a negative pressure gradient. As a result ofthis, the chamber pressure PC(t) decreases while, thebubble pressure PB(t) would decrease further. From athermodynamic viewpoint, the rate at which these twopressures decrease (as a result of velocity fluctuations)depends on the system pressure, chamber volume andheat capacity ratio.62 In the case of a larger chamber,the pressure in the orifice PO(t) will remain virtually
Figure 5. Effect of orifice diameter on bubble volume. (A) Filledsymbols correspond to the bubble sizes obtained for orifice ofdiameter 3.67 mm and empty symbols correspond to the bubblesfrom orifice of diameter 5.94 mm, for different liquid viscosities,(B) effect of orifice diameter on bubble size in water, (C) effect oforifice diameter on bubble size in different viscous liquids (PAA)9 µ ) 1.09 cP, 0 µ ) 6.19 cP.
J
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 becomesunstable, producing many bubbles rapidly in succession.Also, for the same VC but 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 diameterand VC; 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.,51 Antoniadis 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 certaincapacitance number NC ≈ (4VCgFL)/(πdh
2Ph), 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 of NC for NC < 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 iscontradictory 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 lessimportance 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.
K
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 constructionof 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 byvarious 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 rangeof 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, Davidson’s 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
L
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 rate Q, the bubble expands at a definite rate givingrise to inertial and viscous drag, which adds to surfacetension. 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 ) 6πµvere
surface tension forces ) πDγcosθ,
inertial force ) Q2(FG + 1116
FL) V-0.66
12π ( 3
4π0.66) (1)
ve ) (dre/dte) ) Q/4πre2 (2)
dMve
dte) M
dve
dte+ ve
dMdte
dve
dte) - [Q2
6π( 3
4π2/3)] V-5/3, dMdte
) (1116
F1)Q (3)
dMve
dte) Q2 (11
16F1)V-2/3/12π ( 3
4π)2/3(4)
M
Table 3. Various Correlations for Bubble Sizes
investigatorsmodel proposed through
experimental observations method of approach
assumptionsused in
formulationa diameter/volume of bubble
Eversole et al.48 One stage model wasobserved for different sizeorifices.
Bubble volume was obtained bystroboscopic frequencymeasurements.
1,2,3, 4, 5, 6,7, 20, 27a, 9a10, 39, 40, 48
dB ) 81.18 σPσg
Krevelen andHoftijzer49
Separate and chain bubbleformation
Bubbles were classified as small,medium, and large. Ascendingvelocity was obtained from flow rateof gas. The same was studied forthree different regimes of gas flow.
1, 9a, 6, 7, 8,32, 13, 27a,31, 48
dB )6dh
g∆F
Benzing andMayers50
Static and dynamic region ofbubble formation processwere considered.
Dimensional analysis was usedto correlate results.
1, 5, 6, 7, 8,9a, 10, 11,13, 16, 20, 46
dB ) 1.82 dσ0.25
gdh2FL
Hugheset al.51
Process of bubble formationwas considered to be acontinuous one without anystages in it.
Dimensional analysis was carriedout for Re, Nσ, virtual mass.Force balance was applied on abubble. Pressure equations wereconsidered and solved for dragcoefficient, bubble velocity,frontal area of bubble, andbubble volume this was verifiedfor different NC.
1, 5, 6, 7, 8,9a, 10, 11,13, 20, 46
VB ) 1.82πgdpσg∆F
Leibsonet al.52
Standard deviation andprobability methods wereused to find the range of sizesof bubbles. Effect of orificesdimensions and gas flow rateon dB was studied usingstroboscopy.
The bubble diameters were foundat various experimentalconditions and empiricalformulas were derived usingformula for orifice dischargecoefficient.
1,5,6,7,8,9a,20, 39,
dB ) 0.19Do0.48 NRe
0.32
Davidson andSchuler3
(Viscousliquids)
Single stage model wasstudied for constant flow andconstant pressure conditions.
Equation of motion and virtualmass of bubble was used to findout size of bubble theoretically.
27a, 1, 8, 10,11, 12, 13,20, 32, 19, 48
V ) 1.772 G6/5
g3/5
Davidson andSchuler4
(Inviscidliquids)
Single stage model inunsteady state was studiedfor constant flow andconstant pressure wasconsidered.
Equation for upward motion isconsidered and solved underinitial conditions to find outcritical bubble volume.Assumptions used for modelformulation are different withrespect to gas flow rates.
27a, 1, 8, 10,11, 12, 13,20, 32, 19,42, 2, 48
R ) 16g11 [t2
4+
Vot2Q
-V0
2
2Q2ln(Qt + V0
V0)]
V0 ) 4πr02/3
Kumar & co-workers14-17
Two-stage model. Force balance was carried outassuming negligible surfaceforce.
27b, 1, 5, 19,10, 48, 43,32, 34, 20,
VF ) (4π2 )0.25(15µQ
2F1g )0.75
Wraith53 Two-stage model. Concept of velocity potential wasapplied for a hemisphericalexpanding bubble to find its volume.
6, 7, 8, 27b,13, 44, 5, 16,45, 48
V ) 1.09Q6/5 g-3/5
Park et al.68 Model is based on the materialbalance and force balance usingTate’s law.
1, 6, 7, 8, 46,48 Vb )
2πRoσ∆Fg
Acharyaet al.54
Single stage model Wave theory was applied todetermine the volume of thebubble in dynamic detachingcondition.
1,5,6,7,8,9a,27a, 46, 44 Vb ) 0.976 (Q2
g )3/5
N
Rabiger andVogelpohl 20
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 ) x33σdh
FL+ x(3σdh
FL)2
+KVG
2 dh
g
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 )8πσrH
F1g+ 2 G2
g ( 34πVf
2)0.33
Tsuge andHibino5
Two-stage model Bubble volume is obtained byforce balance including virtualmass 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 )
M′X - [4FgQ2t2
πDo2
- 0.5CDF1(dXdt )2
πr2t2](F1 - Fg)Vbgt
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 ) [(6DoσFg ) + 81νV
πg+ (135V2
4π2g )5/3]0.25
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 ) 6.9 ( σF1
)0.5ug
0.44
Sada74 Single stage model Model is entirely based on theexperimentally observed bubbledimansions.
27c, 9, 14,17, 20, 23,34, 48
dB ) (a2 b)1/3
a: maximum bubble diameterb: minimum bubble diameter
Tsuge et al.8 Continuous bubble growthmodel at high systempressure
Bubble surface is divided into anumber of axisymmetricelements 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 )πdhσ∆Fg
Marshallet al.76
Two-stage model Potential flow theory was appliedwith Bernoulli’s 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 ) 0.48 Ro0.286(Q1/πRo
2
U1)0.36
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 ) πd03/12
Pamperin &Rath78
Two-stage quasi-staticunsteady shape bubbleformation under reducedgravity.
Force balance including virtualmass was used to find thedetachment stage
27a, 27b, 35,34, 37, 25,38, 20, 9a
dB ) D0A x(FL - FG)(We/(We - 8))
A ) x 4Re
+ 0.124xRe
Re )2vBD0
νf
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 ) 4
11gt2 + 8
11kQπro
2t
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 )
πd0σFLg
O
Making a force balance and letting V ) 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 Newton’s second law of motion along withforce balance as
where v′ 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 (θ) 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
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VE5/3 ) 11
192π(3/4π)2/3gQ2 + 3
2(3/4π)1/3g×
µF1
QVE1/3 + πDσ
gF1VE
2/3 (5)
M dv′dt
+ v′ dMdt
) [(VE + Qt)FLg - 6πµr(v + ve) -
Qt2(11/16FL)
12π[3(VE + Qt)/4π]2/3- πDγcosθ] (6)
(Dv/dT) + A(v/T) ) B - GT-4/3 - ET-5/3 - CT-1
(7)
A ) 1 +6π(3/4π)0.33(1.25)VE
0.33µ
Q(11/16FL), B ) 16g
11Q,
C ) 16πDγcosθ11QFL
(8)
E ) Q12π(3/4π)0.66
, and G ) 24µ11(3/4π)0.33FL
rE ) B2Q(A + 1)
(VF2 - VE
2) - ( CAQ) (VF - VE) -
3G2Q A-1/3
(VF2/3 - VE
2/3) (9)
P
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 for M(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 bubblesize. 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)
12π(3/4π)2/3Vt2/3 [3Vt
Qt
dte+ Qt
2] +
πDγcosθ + 3µQt
2(3/4π)0.33Vt
1/3 (10)
d(Mv′)dt
) Vt∆Fg - 6µπr′v - πDγcosθ (11)
Mdvdt
+ vdMdt
) Vt∆Fg - 6πµr(v +ve) - πγDcosθ -
(Fg + 1116
F1)(3Vt
dQt
dt+ Qt
2)12π(3/4π)2/3Vt
2/3(12)
Mdvdt
+ vdMdt
+ Mdve
dt+ ve
dMdt
)
Vt∆Fg - 6πµr(v + ve) - πγDcosθ (13)
M ) Vt(Fg + 1116
Fl) and dMdt
) Qt(Fg + 1116
Fl) (14)
ddt
(Mv′) ) Vt∆Fg - 6µπrv′ - πDγcosθ (15)
Vt ø dvdt
+ Qtøv ) rVt∆Fg - 6πµ(v + ve) - πγdhcosθ -
ø(3Vt
dQt
dt+ Qt
2)12π(3/4π)0.66Vt
2/3(16)
ø ) (FG + 1116
FL)
Figure 7. Model for spherical bubble formation at constant flowconditions (reproduced with permission of Elsevier from Gaddisand Vogelpohl22).
Q
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 1/4th
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 to Fm+ Fp ) Fs. Having an inverse dependence of We on 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, µ) 0.552 cP, (2) Ramkrishnan et al.14: 9 dh ) 0.00367 m, µ ) 0.045cP, (3) Davidson and Schuler4: O aq. glycerol dh ) 0.00067m, µ )1.05cP, (4) Rabiger270: water 0 dh ) 0.002 m, µ ) 0.001 cP, (5)Morgenstern and Mersmann93: Glucose, ∆ dh ) 0.0002 m, µ )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 Leibson’s formulation: The line numbers correspond tothe predictions for individual data sets as numbered above. (C)Effect of orifice diameter on bubble size (Terasaka & Tsuge11)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 ) π6
d3(FL - FG)g, Fp ) π4
dh2(pG - pL),
Fm ) π4
dh2FGwG
2
Fs ) πdhσ, Fd ) π4
d2CD
FLwL2
2, and Fi ) (FGV - FLV1)a
(17)
Fb + Fm ) Fs + Fd + Fi (18)
d3 ) S + L/d + T/d2 (19)
S )6dhσ(4 - We)
4∆Fg, L ) 81ηV
π∆Fg,
T ) (1354π2
+27FG
π2FL) FLV2
∆Fg, We )
16FGV2
π2dh3σ
(20)
dB ) [(6dhσFLg )4/3
+ (81νVπg ) + (135V2
4π2g )5/3]1/4
(21)
R
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 Leibson’s 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 mechanismfor 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 ‘r’takes form as
at constant flow rate, the growth rate can be given as φ) Q/(4πr2), 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 the‘hemispherical 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
Pθ has 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 . πa2 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).
PFL
+ 0.5 r4
R4 (drdt)2
) F(t) (23)
Φ ) r2
Rdrdt
, q ) r2
R2drdt
(24)
dΦdt
) r2
R(d2rdt2) + 2 r
R(drdt)2
) 0 (25)
da/dt ) Q/(2πa2)
Pθ )FLQ2
8π2a4+ Po - (FLga)cosθ (26)
Pe
FL+ Q2
8π2a4) 0 (27)
FR -FLQ2
8πa2- πaPo + 2
3πa3FLg ) 0 (28)
FR - 2πa2∫0π/2Pθcosθsinθdθ ) 0 (29)
ah ) 0.453 Q2/3g-1/5 and Vh ) 0.194Q6/5 g-3/5
(30)
VB ) 1.09 Q6/5 g-3/5 (31)PFL
+ 0.5q2 - ∂Φ∂t
) F(t) (22)
S
Interestingly, the bubble volume is shown to have noeffect of gas-liquid system’s 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 Rubin18 Approach based onEquilibrium Shape of Bubble. Equilibrium shape ofa bubble generated at a submerged orifice, detachment,delayed release of bubbles and finally multiple bubbleformation 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. Thebubbles’ 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) Thebubble 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 ofthe 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 ofinviscid 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 Greens’s 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 methodhas 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. Hooper’s 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 thesurrounding liquid to be inviscid. The velocity was rela-ted with the gradient of stream function as u ) ∆φ and
Bernoulli’s equation was applied for the liquid in motionin terms of velocity potential.
As
hence
∂Φ∂t
+ 12|u2| +
(P1 + FLgz)FL
) C(t) (32)
zf∞, P1 + FLgzf0
C ) 0
T
Writing the equation in Lagrangian form for conven-ience yields
For the compressible fluid the continuity equation canbe written in Laplace form as Λ2 φ ) 0
Since there is no flow through the orifice pate at z )0, dφ/dz ) 0.
At the interface the normal velocity of the liquid andof the bubble are equal and surface tension results inthe pressure jump. Hence on S, dφ/dn ) vs and P1 ) Pb- σk.
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 FCare approximately constant and equals PE and FE.
(ii) Expansion of bubble is isothermal.The equations were solved under the boundary condi-
tions, which relate P1 to Pb and the gas equations fromwhich Pb can be calculated are substituted into equa-tions of motion. Initial conditions for the values of Pband PC and the initial position of bubble surface andthe value of φ 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 conditionsand 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 Tan’s 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 usedLaplace’s 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) Laplace’s 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 Green’s 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 Euler’s 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 ofavailable computational power.
2.3. Mechanism of Bubble Formation in TopSubmerged 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
DΦDt
) 12|u2| - 1
FL(P1 + FLgz) (33)
H ) k(πro2) x(PC - Pb)/FC
U
where the initial conditions are r ) di/2 and dr/dt ) Qg/πdi
2. The detachment stage then begins when the forcebalance is hold vertically.
where VB2 is the volume of the bubble less hatched bythat of orifice which gives the virtual mass (M1′) wasassumed for an ascending spherical bubble parallel tothe wall as 0.5FL VB1. In this case, every time as thevolume of the bubble increases, the volume hatchedalong with it also increases, and thus the virtual massof the bubble changes with time toward the detachmentstage. Now from the balance of forces the earlierequation to the equation of motion in the detachmentstage gets modified and in the same way M2′ alsochanges. This equation can be solved for initial condi-tions to get the volume VB and dB at detachment stageas
(B) Nonwetted Nozzles. This condition has beenobserved in the molten liquids and the equations result-ing from the force balance and equation of motion aresame as the earlier case except the initial conditions.Hence, the equations for calculating volume and thediameter remain same, but in both the cases a discrep-ancy remains between the calculated and the experi-mentally observed values of diameter of the bubble. Thiswas because (i) Fractional gas hold up affects the overallliquid density if liquid is a molten metal. (ii) As the heattransfer between gas bubble and the liquid metal wasnot considered in this model, the bubble size will bepredicted to be smaller by this model than by theapproach which considers the interfacial heat transfer.(iii) Constant flow condition is not actually fulfilled.Thus, balancing the forces is of much reliability ascompared to the other methods to calculate the volumeof the bubble.
2.3.2. Liow Model. The mechanism of the bubbleformation at a top submerged orifice having a lengthlonger than that of the normal orifices was experimen-tally studied and modeled by Liow.61 In the case of topsubmerged orifice, the orifice contact angle varies whilethe bubble grows, but the bubble size mainly dependson the orifice diameter rather than the contact angle.In view of this, Liow61 has taken the Laplace equationof the interface in equilibrium with gravity as
using few dimensionless parameters, the bubble volumewas obtained by solving
where
by subjecting it to initial conditions: φ ) 0, z* ) 0, R*) 0, dφ/ds* - La/2 at s* ) 0 using MATHEMATICA.The variation in the maximum bubble pressure for arange of Laplace number (La) and Bond number (Bo)was obtained which showed that although the dimen-sionless pressure increases with Bo, the actual valuesof the pressure for a given system fall with increasinglance diameter. As mentioned by many investigators,the maximum bubble size occurs when φ ) P at thelance. The bubble maximum diameter is shown toincrease faster than the bubble heights with increasinglance diameter leading to flatter bubbles.
His analysis shows that the mechanism of bubbleformation in the case of top submerged orifice is a threestep process. In the first step, a spherical cap is formedat the lance opening which reaches to a hemisphericalshape and the point of attachment moves away fromthe orifice diameter. Thus, the bubble size is governedby the lance inner diameter. Further, due to localimbalance, bubble prefers to grow outward at one sideof the lance and begins to slip past the outer diameter.It clings to the lance before it becomes unstable leadingto detachment. Here the intermediate portion betweenthe inner and outer diameters of the lance governs themaximum stable bubble size. This is because, for a givensystem, there are many stable contact angles and thelarger the thickness, the more time is taken to achievethe maximum bubble size. Once the maximum bubblepressure is attained, the bubble formation process iscontrolled by the outer diameter of the lance. Thus, thelarger the lance diameter, the better is the control onthe detachment stage. The typical sequence of bubbleformation at top submerged orifice is shown in Figure9B. The entire process of detachment is dynamic sinceonce the maximum pressure of the bubble at the innerdiameter has been reached, the bubble is in a quasi-steady state, which continually moves toward a newequilibrium position until outer lance diameter is unableto keep the growing bubble stable. When compared withthe experimental data, the model showed a slightdifference in the predictions because the increase inbubble volume due to excess chamber pressure adds tothe volume of the bubble at the detachment stage, whichwas not taken into account in the model.
2.4. Model for Formation of Non-spherical Bub-bles. In most of the realistic cases, the bubbles that areformed at the orifice are nonspherical. The reasons aremainly the surrounding flows. A few attempts appearin the literature11,19,23,84 to understand the mechanismof nonspherical bubble formation. In this subsection, wehave discussed the model by Terasaka and Tsuge,11
while the other attempts are discussed elsewhere inappropriate subsections.
Nonspherical Bubble formation under constant flowconditions is studied by Terasaka and Tsuge,11 and amodel based on the following assumptions was devel-oped: (i) The gas flow rate into the bubble through anorifice or a nozzle is constant. (ii) Liquid surroundingthe bubble is at rest. (iii) Bubble shape is symmetricalabout the vertical axis. (iv) Bubble motion is not affectedby the presence of other bubbles.
Q )dVB
dt)
d(4πr3/3)dt
) constant (34)
FLVB2g ) 0.5CDFLπr2(drdt)2
+d(M′1dr/dt)
dt+
4FGQG2
πDi2
+
πDiσ (35)
VB ) 4π r3
3- π
dh2r2
(36)
d2z/dr2
[1 + (dz/dr)2]3/2+ dz/dr
r[1 + (dz/dr)2]1/2- F* gz
γ) - P*
γ(37)
dV*ds*
) r*2dz*ds*
(38)
dr*ds*
) cosφ, dz*ds*
) sin φ and dφ
ds*)
La + Bo z* - sin φ
r*
V
The bubble surface was divided into ‘n’ elements, andfor every element, two equations of motion, in radialand vertical directions, are solved to give its radial andaxial velocities. Then, the positions of the elements aredetermined. Pressure change in the bubble can beobtained by the application of energy balance and theideal gas law for adiabatic expansion of gas bubble as
Further, equivalent radii (RE) are found out using two-dimensional symmetric elements, and R′ is the radiusof the elemental circle, then
Pressure balance on the gas liquid interface is takenon any element ‘j’ at which bubble radius is R, then amodified Rayleigh equation (the force balance) was usedas
Now, introducing the equation of motion of a risingbubble, which is again a force balance, we get
To solve these two differential equations, a procedurewas followed to get the volume of the bubble in boththe stages of bubble formation as follows: (i) Determinethe initial bubble pressure. (ii) Calculate PB at any time‘t’ using the first equation. (iii) At any element on thebubble surface, evaluate R and R′ from the geometryand calculate RE by the second equation. (iv) Evaluatethe radial acceleration and velocity of the bubble surfaceby solving the equation of pressure balance between theinside and the outside the bubble surface using thirdequation. (v) Evaluate the vertical acceleration andvelocity of the bubble by solving the equation of motionof the rising bubble by fourth equation. (vi) Convergenceof these calculations occurs on closing of the calculatedbubble neck. This novel procedure to evaluate non-spherical bubble formation deals with the sequenceright from the beginning up to the detachment. Asmentioned previously, other attempts in this directionare explained elsewhere in this Review.
2.5. Mechanism of Bubble Formation in FlowingLiquids. Sieve plates in distillation columns, orificeson the moving blades of gas inducing impellers, spargersin up-flow and down flow bubble columns, air-liftreactors, fermentation systems and the ring sparger inthe stirred tank are a few examples of the bubbleformation where liquid is also in motion. In general,these situations are complicated, and the mechanismof the bubble formation is slightly different than in thestagnant liquid. The motion of liquid phase has an
important effect on bubble formation. In the co-currentand the cross-current flows, the drag force developedby the liquid flow results in an early detachment of thebubble, and hence, the bubble sizes are smaller. As aneffect, these small-sized bubbles, being more in number,result in a dispersion with narrow bubble size distribu-tion and high void fraction. Hence, in addition to theparameters, which affect the bubble sizes in stagnantliquid, here, the liquid velocity and direction of flow arealso important. In this section, we have criticallydiscussed the effect of liquid flow on mechanisms ofbubble formation, various existing models and a com-parison in their approaches toward development of amodel has been given based on their results. On thebasis of the direction of the flow with respect to the gasflow, this subsection is divided into three parts as co-current flow, counter current flow, and cross-flow.
2.5.1. Co-Current Flow. In co-current flow for thebottom submerged orifice or the sieve plates, thedetachment of the bubble occurs at an earlier stage ascompared to the other two flows because, the resultantforce due to liquid flow and the buoyancy is moredominant. Many investigators have modeled this phe-nomenon by force balance at the orifice, and here, webriefly discuss a few of these models. Chuang andGoldschmidt85 have developed one stage model basedon the force balance at the orifice (0.0136 and 0.0481mm i.d.) under constant gas flow conditions. Theirexperimental observations show that the bubble vol-umes do not depend on the gas flow rate for liquid flowvelocities higher than 0.303 m/s in the experimentalconditions. Considering all the forces explained in themodel by Kumar86 and also the momentum applied dueto the liquid motion, the force balance can be given as
where VL and fB are the liquid flow velocity and thebubble formation frequency. This equation can be solvedwith suitable boundary condition at the orifice to getthe final bubble volume. The predictions from this modeldeviate from the experimental observations mainly dueto the fact that, for such small sized orifices, the bubblesizes are also small and hence the buoyancy hasnegligible contribution, while inertial forces and theliquid flow were dominant. Hence to overcome thisdifference, later, few investigators have introduced thetwo-stage model in bubble formation for the samesituation. Sada et al.87 studied the phenomenon usingstroboscopic methods and observed that, at any gas flowrate, the bubble size decreased with increasing super-ficial liquid velocity. A simple force balance approachwas followed to develop a relationship between bubblevolume and flow regimes using Froude number. Thismodel was further modified by Takahashi et al.88 byconsidering the process to be of two stages and allequations are solved using boundary conditions depend-ing upon the liquid flow direction. Since the contributionduring detachment was also taken into account theresults obtained after modification were found to matchvery well with the experimental bubble sizes.
Since the experimental observations clearly indicatethat the bubble shapes under the liquid flow condition
dPB
dt)
PB
VB(QG -
dVB
dt ) (39)
1RE
) 12 (1R + 1
R′) (40)
PB - PH ) FL[Rd2R2
dt2+ 3
2(dRdt )2] + 2 σ
RE+
4µLdRR
dt -
12
FG(4QGcosθ
πdi2 ) (41)
ddt(M′dz
dt) )
(FL - FG)VBg - 12CDFL(dz
dt)2 πdM2
4+ 4FG
QG2
πdi
2 (42)0.5 d
dt (mj dsdt) )
πFLdB3
6+
CDπFL(dB2 - di
2)4 (FL
2 (VL -dB fB
6 ) - πdiσ)(43)
W
deviated from the spherical ones, Terasaka et al.13
developed a model for nonspherical bubble formation inco-currently upward flowing liquid. Initially, equivalentradius of bubble is obtained by dividing the surface ofnonspherical bubble in various elements. Further, pres-sure balance is taken over the surface by applyingBernoulli’s equation as
where PB, PH, F, σ, µ, and VL are the pressure in bubble,static liquid pressure at any element, liquid density,surface tension, viscosity, and liquid velocity, respec-tively. In the second stage, when the growth of thebubble is included and the center of the bubble isconsidered to go away from the orifice surface in theform of equation of motion as
where mj , VB, DM, Di, QO are virtual mass, bubblevolume, maximum horizontal bubble diameter, orificeinner diameter, and gas flow rate through an orificerespectively and CD is the drag coefficient. Further,since the elements on the bubble surface undergo motionto follow the changes in shape of bubble, under theassumption of axisymmetric expansion, the mean ex-pansion velocity of each element was defined in termsof the reference angle between the normal axis to bubblesurface at the edge of the nozzle and bubble volume.This expansion velocity along with the liquid velocityand the apparent rise velocity together give the absoluterise velocity. Since VC and PC fluctuations have notice-able effects on the bubble size, the pressure fluctuationsin chamber are taken into account along with thepressure drop across the nozzle (which depends on thetype of nozzle) to obtain an overall pressure balance inthe system. The volume of the bubble was obtained bysolving all the pressure balance equations simulta-neously using finite difference method. The results fromnumerical solution for the effect of the liquid velocity,gas velocity on bubble volume and shape were comparedwith the experimental observations and the model wasfound to give excellent match for both the spherical aswell as the nonspherical bubbles.
2.5.2. Counter-Current Flow. The bubble formationat an orifice in counter current flow of the liquid is acommon phenomenon in distillation operations and alsoin the counter current bubble column operations. In thiscase, for a bubble to form, the pressure drop across theorifice should be much larger to overcome the hydro-static pressure force and the liquid force acting down-ward together. Also the shape of the bubble no longerremains spherical, as it expands against the liquid forceand it starts to be an ellipsoid so that the resistance tothe opposing forces is minimized. The theoretical modelfor this type of bubble formation is similar to that ofthe co-current except that sign of forces will changesince the direction of liquid flow acting on bubble surfaceis changed. For the quantified difference in variousliquid flow conditions, readers may refer to Takahashiet al.88
2.5.3. Cross-Flow. As mentioned previously, themotion of liquid in the traverse horizontal direction isvery common in the case of flow-over sieve plates,distillation trays, air-loop reactor and in a spargedstirred tank where the liquid flow is largely due toimpeller motion. In the case of relative liquid motion,two effects are observed prominently. (i) The drag forceby the liquid flow results in early bubble detachmentand hence smaller bubbles resulting into the enhance-ment of gas-liquid interfacial area and improvedboundary layer transport causing higher kLa. (ii) Bubblecoalescence is prevented because of liquid motion, andthus bubble sizes are controlled. Maier89 was among thefirst to demonstrate that the shear force experiencedby a growing bubble in a flowing liquids causes its earlydetachment and it is at a maximum when the liquidhas an exact cross-flow with respect to the nozzle axis.Stich and Barr90 studied this phenomenon experimen-tally and proposed an empirical correlation for the finalbubble size. Sullivan and Hardy,91 systematically ana-lyzed the phenomena through a semiempirical modelbased on expansion, vertical displacement, horizontaldisplacement of the spherical shaped bubbles and finallycorrelated the bubble volumes to Re and Fr. Otherstudies in this most realistic case are by Kawase andUlbrecht,92 Morgenstern and Mersmann,93 Wace et al.,94
Marshall et al.,76 Forrester and Rielly81 and the mostrecent being Zhang and Tan.27 Most of these studies arebased on the spherical shapes of bubbles, whereas arecent analysis by Tan et al.23 discusses the complexcase of the nonspherical bubble formation in liquidcross-flow. Here we discuss a few of these modelsdeveloped specifically for the case of liquid cross-flow.
Model by Marshall et al.76 for bubble formation in thecross-flow is based on many assumptions. Althoughthese assumptions are far from reality, they may beaccepted for gross simplification of physical reality todevelop a relevant model. The actual assumptions canbe seen from the assumption numbers given in Table 3as listed in Table 2. In the studies, they tried totransform the experimental results into the model bypotential flow theory. In the case of the uniform streamcondition, bubble slides over the orifice plate and itresembles the situation like a flow past sphere in theliquid. Since the knowledge of the velocity potentialenables the liquid pressure on the surface of the sphereto be determined from the unsteady form of Bernoulli’sequation, we get
where P is the pressure on the sphere surface and P∞ isthe hydrostatic pressure of the undisturbed liquid. Sinceby the first assumption liquid pressure on the bubblesurface is uniform, it can be given as
The radius of the bubble can be calculated using thenormal procedure of applying the equation of motion forthe system under consideration and considering thevirtual mass of the system. The virtual mass coefficientwas obtained from the potential flow solution by con-sidering the kinetic energy of the liquid and then theLagrangian equation of motion was considered. To get
∆PF
)P - P∞
F) ∂Φ
∂t- 0.5|VL|2 (46)
∆pF
)P - P∞
F) 1
4πR2× ∫0
2x ∫0
x ∆PF
R2 sin R d R d ψ
(47)
PB - PH ) FL[R d2Rdt2
+ 32 (dR
dt )2+
VL2
2 ] + 2σR
+4µL
RdRdt(44)
ddt [mj dz
dt] ) (FL - FG)VB - πdiσ -
12
CDFL[dzdt]2 πdM
2
4+
4FGQo2
πdi2
(45)
X
the Reynolds number for the system, the drag coefficientwas calculated. Knowing the liquid and bubble velocitiesand the orifice radius, bubble radius can be easilycalculated.
where VG is the superficial gas velocity given by VG )Qi/πRo
2. Further modification yields
Thus, knowing the average bubble rise velocity andother parameters, bubble size can be estimated. Theseinvestigators have reported that the chamber pressurefluctuations do not significantly affect the bubble sizein cross-flow situation. Although the authors couldobserve that the initial stages of bubble formation andalso the detachment by neck closure both are dynamical,it was not considered while formulating the model.
Sullivan and Hardy68 assumed that bubble formationhas three sequential steps and the first two steps arethe expansion and translation of bubble along the liquidflow while the third is that of detachment from theposition of adherence. To model this condition (i) theforce balance was taken in the vertical direction, (ii)Newton’s law of motion was applied assuming the forcesto be in horizontal direction, and finally, (iii) theequations are solved at specific boundary conditions toget the volume of the bubble. Further, Takahashi et al.88
modified the two-stage model by considering that theformation occurs at an inclination of angle θ to thevertical line. Their force balance gave the bubble volumein terms of very low and very high velocities as
and a′ ) 1, b ) 0.242 are obtained experimentally.Tsuge et al.95 studied this phenomenon under the
varying gas pressure conditions in the chamber usinggeneralized Bernoulli’s equation in terms of velocitypotential. The pressure equation was solved numericallyto get a simplified inequality
which gives the minimum distance (S) between thecenter of the bubble and orifice at the final condition ofdetachment. Ui is the velocity of the growing surface attime t.
Recently, Tan et al.23 proposed a model for nonspheri-cal bubble formation in cross-flow condition by interfa-cial element approach. Initially, force balance is takenover each surface element to obtain a set of differentialequations of motion in cylindrical coordinates as
where r and z are the radial coordinate from the axis ofthe bubble and the axial coordinate from the orificehorizontal level, respectively, ∆P is the pressure differ-ence between the bubble and the liquid pressure at each
element, â is the angle defined as â ) tan-1(∂z/∂r)andmj ) (0.6875FL + FG)VB is the virtual mass. For thesolution of this problem, equations of mass balance atbubble, chamber and the orifice are required.
(i) The equation for the mass balance on the chamberis given as
(ii) The pressure equation on the chamber is given as
(iii) The equation for gas in the bubble and the chamberis given as
where, VB is the bubble volume, γ is the adiabatic gasconstant, and aO is the cross-sectional area of the orifice.
(iv) Finally, the orifice equation is given as follows
where C is the orifice constant. The inclination in theangle of a growing bubble due to liquid cross-flow (VL)was obtained as ω ) tan-1(VLVB/QRB), where RB is theequivalent radius of nonspherical bubble. Virtual bubblecoordinate system along with the stationary axisym-metry was used to find the stagnation point θ, whichindicates the outermost point of attack. Equations 53-56 were solved simultaneously for discerning the de-velopment of bubble growth in liquid cross-flow condi-tion. The results from the simulations showed that thebubble growth begins immediately in a flowing liquidwhen compared with the stagnant liquid, and they aresmaller. The comparison of the predictions with theexperiments by Marshall96 showed a reasonable match.The simulations showed a difference in the mechanismat higher flow rates where the bubbles stick to the plateof orifice during its growth and thus delay the detach-ment of simulated bubbles.
In a different situation of bubble formation ac-companied with weeping at the submerged orifice witha liquid cross-flow, Zhang and Tan27 have used potentialtheory to describe the pressure field around a bubble.The bubble growth is assumed to be the same as in thecase of stagnant liquid followed by a displacement androtation before it gets detached. The liquid head isassumed to be high and has no effect on weeping andbubbling, which are supposed to be occurring alter-nately. Their differential equation for the camber fluc-tuations is same as in the quiescent liquid, while theorifice equation is given in terms of the growth rate andpressure difference (Miyahara and Takahashi97) as
The generalized Bernoulli equation is used for thedetermination of liquid pressure at different positions(r, θ) and procedure from Tsuge et al.98 is followed toderive the velocity potential, which can be used for the
RB ) 0.48Ro0.826(VG/VL)0.36 (48)
UB ) 0.375CD (VL - Ub)
2
mj R(49)
VB ) V1(1 - eú) + V2eú, where ú ) -a′/U1
b (50)
S g [R + (1 - 0.02U1)Di] [1 - (CDF1 Ui2 R2/2diσ)]0.5
(51)
r∆Pdr - σd(r sin â) ) ddt
(Uzmj )
r∆Pdz + σ[d(rcosâ) - dzsin â] ) d
dt(Urmj ) (52)
VC
dFC
dt) FaQ - FCq (53)
VC
dPC
dt) γ(PaQ - PCq) (54)
VB
dPB
dt+ γPb
dVB
dt) γ PCq + (γ - 1)
FCq3
2aO2
(55)
q ) CπRO2[PC - PB
FC]0.5
(56)
dVB
dt) πRo
2x2/FGCG(PC - Pa)0.5 (57)
Y
estimation of local absolute velocity. The forces actingon a bubble in the vertical and transverse directions aremodeled separately under the assumption of no slip. Thebubble detachment criteria is based on the angleachieved during rotation of bubble such that if thedistance from the bubble center to the orifice center isx, then a bubble would detach if x g R + (1-0.02VL)dh.In addition to this, the condition of weeping is dependenton the instantaneous pressure fluctuations in thechamber and hence the dynamics of this pressurefluctuation can be modeled from the gas flow equationin terms of differential pressure followed by the integra-tion over the weeping period. Their results are seen tomatch very well with their own experiments and thepressure cycles are analogous to the published litera-ture. An interesting observation is the adaptation ofsequential process of growth, displacement and rotation,which may not be realistic, as in reality, these processesoverlap to some extent.
More recently, Loubiere et al.32 studied the impact ofthe liquid cross-flow on the bubble generation at theflexible orifice made out of a rubber membrane throughforce balance in two different directions and also bystroboscopic method. Their observations have clearlyproved that the liquid velocity has a strong impact onbubble formation such that (i) the bubbles of signifi-cantly smaller sizes are generated at higher formationfrequencies and (ii) the bubble coalescence and thelikelihood of breakage during their ascent in the system.The following important observations are reported: (i)during its growth, the bubble moves downstream andis flattened due to effect of the liquid motion; (ii) exceptfor the first moments, the bubble inclination angleremains roughly constant during the bubble growth; itdepends only on the liquid velocity and can be calcu-lated, (iii) as under quiescent liquid conditions, thebubble volume varies linearly with the growth time, (iv)the bubble formation at a flexible orifice is a continuousphenomenon, (v) as under quiescent liquid conditions,the bubble spreads over the orifice surface during itsgrowth; the bubble advancing contact angle is largerthan the receding one, (vi) drag force due to the liquidcross-flow is responsible for the bubble detachment andnot the buoyancy force.
The above discussion and force balance based modelsclearly indicate that, although this approach is simple,it is not very easy to incorporate all of the dynamicalfeatures of the phenomenon for the accurate predictionof bubble size. However, the use of surface disintegra-tion followed by the application of finite elements orpotential flow based approaches seems to be robust evenfor the case of nonspherical bubbles.
2.6. Bubble Formation under Reduced GravityCondition. During the process of bubble formation, thesurface forces due to pressure difference between thegas in the bubble and the liquid and the line forces aredue to surface tension. For a static interface, these forcesare in equilibrium but in the process of dynamicformation, the resultant of these forces is equal to therate of change in the liquid momentum assuming thegas phase momentum is negligible. As it can be clearlyseen from the earlier discussion in this review, theprocess of bubble formation is grossly made up of twostages, the expansion/growth of bubble (when surfaceforces are dominant) and the detachment (when buoy-ancy force is dominant and hence important). For verysmall bubbles, buoyancy is very small and the larger
the bubble, the greater is the buoyancy resulting innonideal liquid flow in its vicinity, which even inducesa liquid motion in its vicinity and thus helps in sever-ance of the neck. Hence, in order to understand theexact contribution of the buoyant forces in the processof bubble formation, it is required to physically elimi-nate the forces acting in the downward direction. Inview of this, recently, many studies are being carriedout55,78,79,99-103 under the condition of reduced gravity.Additionally, the effect of individual component forcesin the force balance equations by changing eitherviscosity or surface tension or flow rate of gas or liquidindividually can also be seen when gravity has no roleto play in the overall mechanism. These types ofanalyses are important since the gas liquid systems arecommonly observed in propulsion, power generation,storage and life supporting systems in the space.
To understand the bubble formation in the reducedgravity condition, in one of the first attempts, Pamperinand Rath78 observed only two regimes of bubble forma-tion i.e., with and without bubble detachment. Theydefined the detachment criteria on the basis of value ofWe. Their stroboscopic observations showed that thebubbles formed at all the gas flow rates were sphericalin shape. Their force balance was based on the assump-tion of immediate detachment of bubble from the orifice.Their observations showed that the bubble sizes in theabsence of buoyancy are directly proportional to theorifice diameter to the power S, such that 0.33 < S < 1.Also, the bubble that already detached least affects thebubbles getting formed at the orifice due to the smalllevel of turbulence developed in the liquid due to smallsizes of bubbles, thus eliminating the idea of primaryand secondary bubbles.
2.6.1. Models for Bubble Formation. 2.6.1.1. Non-spherical Bubble Formation Model (Tsuge et al.10).The phenomenon of bubble formation, where the liquidflow was considered as an external force was carefullyanalyzed by Tsuge et al.9,10 under the hypothesis thatthe liquid flow causes the bubbles to take a definiteshape. The procedure followed for the theoretical de-velopment of model was based on the concept of non-spherical bubble formation model by dividing the bubblesurface into several elements. The force balance in thereduced gravity condition can be given as
where mj is the virtual mass of the bubble, and dM isthe maximum vertical bubble diameter of the bubble.Using Reynolds number it is easy to obtain the velocityof the element on bubble surface which gives thehorizontal distance covered by the bubble in certain timeand this in turn gives the radius of the bubble at anytime if the bubble contact angle with the orifice isknown. This can be solved numerically only if initialconditions are known. They observed that when theliquid flow was co-current the bubble growth rateincreases with an increase in the gas flow rate. Underconstant flow condition the bubble volume increases inproportion to the bubble growth time. In the case ofcounter current flow the bubble growth rate increaseswith increasing gas flow rate and the bubble grows
ddt (mj dz
dt) ) ∆FVB - 0.5CDFL(dzdt)2 ×
π(dM2 - dh
2)4
+4FGQo|Qo|
πdh2
(58)
Z
under constant flow conditions. During the experiments,it was observed that the bubble did not grow sym-metrically about the horizontal axis so that the bubblesturn away and detach from the nozzle earlier. In crosscurrent flow, bubbles are formed at regular intervals.Bubble shape changes from spherical to nonsphericaldue to the bubble movement and expansion by theflowing liquid. But the model developed by them couldnot give results in coordination with that of the experi-mental results. It was also observed that bubble volumewas the smallest among all due to regular detachmentof bubbles from the nozzle.
2.6.1.2. Two Stage Bubble Formation Model.Buyevich and Webbon79 have discussed the effect ofgravity on bubble detachment through a two stagebubble formation model. The model assumes the bubbleto maintain spherical shape throughout the process. Thesimulations from their model showed that the durationof expansion in normal gravity condition is independentof the gas flow rate whereas, the final bubble volume isroughly proportional to it. However, in reduced gravityconditions, expansion time is inversely proportional tothe gas flow rate and expansion volume is independentof it indicating the static regime of bubble formation,where the bubble size is only a function of orifice radiusand the density difference. In the case of constantpressure condition, the authors have reported a differentkind of mechanism of bubble formation with modifica-tions in the dependencies of bubble size on orifice radiusand gas density. Their results to understand the effectof higher and lower gravity have shown that the bubblevolume continuously increases at low gravity with slowlengthening in the neck. This happens mainly becauseat low gravity, the force due to injection of gas intobubble is the only force which pushes the bubble upwardand the momentum arising out of it is weak andcompensated by the inertial forces arising out of in-creasing bubble volumes. At high gravity, the bubblevolume increases with very rapid lengthening of neckleading to instant detachment. The most importantfeature of this work is a careful, universal analysis offorces experienced by the growing bubble. Since thereis a discrepancy in the conditions at which the model isformulated and analysis of experimental results, themodel is still not validated by experimental results butthe interpretations from the results are physicallyrelevant to the process of bubble formation. Althoughthe model seems to have taken care of all the possibleconditions, the classification of gravity level is not clearas they mention low, reduced and high gravity in thepaper and in the absence of experimental data atreduced gravity, the comparison was done only undernormal gravity.
2.6.1.3. Mechanism of Bubble Formation in Qui-escent Liquids. In another similar investigation, Tsugeet al.9 analyzed the bubble formation mechanism undermicrogravity conditions. Similar to the earlier modeldeveloped by Terasaka and Tsuge104 for nonsphericalbubbles assuming the behavior of gas in the chamberis to be polytropic and the formulation and solutionprocedures are similar to their earlier work. The ex-perimental system, which was used for the validationof model consisted of three liquids of different properties,and the nozzle was top submerged. They found out thatvolume of bubble is proportional to the inverse ofgravitational acceleration for a small gas flow rate whenother conditions are the same. The experimental obser-
vations showed that the bubble growth rate increasewith the gas flow rate and the bubbles do not detachfrom the nozzles during the 10 s under low gravity inlow gas flow rates. They estimated bubble volume as
Bubble formation time increases with a decrease in thegas flow rate and gravity level resulting in increase inbubble volume when bubble detaches from the nozzle.Similarly, when the ratio of gravitational accelerationto the terrestrial gravitational acceleration is equal to0.001, bubbles do not detach from the nozzle underconstant flow conditions. The predictions from theirmodel are seen to be very good on comparison with theirexperimental results and that by Pamperin and Rath.78
To adequately disperse bubbles in liquids for masstransfer or chemical reaction processes at relatively lowgas flow rates under reduced gravity, it becomes neces-sary to force bubbles to get detached from nozzles byexternal forces such as liquid flows. Such an analysishas been carried out by Tsuge et al.10 The same modelas above was also used for external forces acting onbubble in the form of liquid flow in reduced gravitycondition. The investigators studied the variation inbubble formation mechanism under the conditions of co-current, counter-current and cross-current flow of liquid.In the co-current flow situation, the predicted bubbledetachment volumes were found to increase with thegas flow rate. They observed that in the case of constantflow conditions, in the counter-current liquid flow situ-ation the bubble growth rate increased with liquid flowrate and the growth was asymmetric. In crosscurrentflows, bubbles were formed regularly with irregularshapes, and bubble volumes were the smallest amongall the flow conditions due to early detachment of thesame.
Similar analysis was done by Nahara and Kamota-mi103 who observed that the effect of liquid flow is morewhen the experiments are carried out in reduced gravityconditions. Their investigation involved the analysisbased on force balance, where the main forces that weretaken into account were the buoyancy, surface tension,momentum flux of the gas, liquid drag in two orthogonaldirections, liquid inertia in two orthogonal directions,and shear-induced lift force due to the liquid cross-flow.At normal gravity conditions, the bubble formation wasseen to follow a two stage mechanism, and the liquidvelocity was seen to have a low effect on the detachment.Under reduced gravity conditions, in the absence of anyneck the formation mechanism is reported to have asingle stage with relatively higher bubble sizes even inthe presence of liquid cross-flow. In a recent compre-hensive attempt by these authors,103 a two-dimensionalmodel was developed on the basis of global force balanceon bubbles evolving from a wall orifice with liquid cross-flow. The analysis is complete in many respects andshows a good comparison match with the experimentaldata at reduced gravity conditions.
2.7. Bubble Formation in Non-Newtonian Liq-uid. Similar to the case of bubble formation in Newto-nian liquids, bubbles formed in the non-Newtonianliquids follow certain mechanism before they actuallydetach from the source of generation. The mechanismof generation of a bubble is almost the same until thestage of expansion, but during its detachment, the neck
VB )πdhσFLg
(59)
AA
motion is mainly governed by the rheology of the liquidand the effect of the neck formation induced on theliquid motion. Non-Newtonian liquids that are consid-ered here are structurally viscous liquids105 and vis-coelastic liquids. The characteristic feature of bubblesin non-Newtonian liquids is their larger size than in aNewtonian liquid having the same apparent viscosity3.The factors that govern the phenomenon are viscosity,viscoelasticity, gas flow rate, surface tension, chambervolume, the orifice diameter and the complex hydrody-namics of these fluids.
Apart from the parameters discussed in Section (2.1.2)of Newtonian liquids, two more parameters: concentra-tion of liquid and viscous stresses are very muchimportant in non-Newtonian liquids and are inter-related. If the concentration of the viscoelastic liquidincreases, then the viscoelastic stresses become largeand affect the bubble growth time (defined as the timefrom the beginning of bubble formation to the detach-ment of the bubble) by elongating it, and thus thevolume of the detached bubble increases. The contribu-tion of the viscous stresses and extensional stressescomes into picture when a bubble undergoes deforma-tion due to the internal stresses. These stresses alsoaffect the growth of the bubble through the character-istic stretch time, which is much smaller than therelaxation time for viscous liquids. At very high con-centrations of the viscoelastic liquid the rheologicalbehavior makes the system complicated to predict thebubble behavior during its formation.
2.7.1. Structurally Viscous Liquids. The bubbleformation in these liquids is a multistage process andthe diameter of the primary bubble steadily increaseswith augmenting flow rate of gas. The rheologicalbehavior of the liquid exerts a decisive influence on thesecondary bubble formation, and thus, the dispersionof the primary bubble is favored by the induced liquidmotion and non-Newtonian flow behavior. Thus, morethorough dispersion of the primary bubbles and earlierinception of the jet regime are the two characteristicfeatures of the process of bubble formation in non-Newtonian liquids. Coalescence of two successive bubblesduring/immediately after the detachment limited byhigher value of viscosity is reached. During the forma-tion of primary bubbles, vigorous motion of the liquidcauses deformation of the upper contour of the bubblesince stronger inertial forces act on the bubble duringthe first stage of formation, consequently larger bubblesare formed.
In the case of bubble formation in moving non-Newtonian liquid, the main effect is on the detachmentstage. For structurally viscous liquids at low gas veloci-ties the primary bubble diameter does not show anynoticeable effect on the change in gas velocities, whilethe formation of the secondary bubbles is mainly dueto the rearrangement of the flow profile at low velocityas an effect of the primary bubble. The neck formed inthe second stage is longer than that in the Newtonianliquids and it induces a more pronounced motion of theliquid after severance of the bubble than would be thecase in a liquid at rest. Consequently, primary bubblesare more effectively dispersed. As the liquid velocityincreases the forced motion of the liquid dominates overthe rearrangement of the flow profile, and the neckbecomes shorter. Because of the altered rheologicalbehavior in the viscous liquids very high velocities ofgas are needed for introducing in jetting regime.
If the motion of the liquid is counter-current to thebubble motion, then the most noticeable effects are onthe detachment motion of the bubble and on the liquidmotion induced by neck. Slower detachment of thebubble is observed along with a longer neck as comparedto that of in the stagnant liquids. Severance of thebubble results in the more vigorous liquid motioninduced by neck causing more effective disintegrationof the primary bubbles and decrease in the secondarybubble diameter at low liquid velocities. Further, theassociated increase in the gas content in the channelno longer permits direct action of the liquid flow on theprocess of the primary bubble formation, and hence, theprimary bubbles again become smaller while the disin-tegrating effect of the induced liquid motion becomesless pronounced. Many times at higher viscosity of theliquid maximum of the primary bubble diameter andminimum of the secondary bubble diameter are notobserved but the diameter of the previous one remainsconstant with augmenting liquid velocity and disinte-grating action of the induced liquid motion steadilydecreases.
2.7.2. Viscoelastic Liquids. The flow media, pos-sessing time dependent flow behavior are termed as theviscoelastic liquids and are characterized by viscousbehavior and the deformational elasticity. The stressstate of the viscoelastic liquid at a given time dependson the previously observed deformation in it. The effectof the formation of the primary bubble is similar to thatin the case of the viscous liquids. After the severance ofthe neck from the capillary orifice, the elements of theliquid are strongly deformed in the direction of motionas a result of the subsequent acceleration of the neckend. This causes a pronounced increase in the elonga-tion viscosity and results into elongational flow in theliquid in the wake. The resistance due to deformationof the liquid elements dominates over the shear viscosityand is perpendicular to the direction of the motion. Thisrheological behavior causes the subsequent bubble tobe elongated as a result of the stronger suction, and eventhe detachment process is decisively affected. At thestart of the secondary bubble formation the effect of theprimary bubble still continues. After the induction ofliquid motion by the neck, liquid penetrates into theprimary bubble and draws a second, which is smallerin size. With increase in the pressure in the primarybubble the neck decelerates and the liquid tends torecover the energy expended resulting in dispersion ofthe liquid elements. Consequently, the bubbles coalescebecause of the increase in the pressure predeterminedby the bubble shape, since this pressure cannot beovercome. The decreasing deformation in the neck iscompensated by the augmenting deformation of theliquid elements in the annular flow. As compared to theviscous liquids here, no deviation in diameter of theprimary bubbles takes place. Much longer bubbles arecreated in the viscoelastic liquids at equilibrium and thisbubble shape persists after their detachment becauseof the properties of the liquid.
Ghosh and Ulbrecht106 introduced a three-stage bubbleformation model with bubble growth, elongation and awaiting stage. The first two stages correspond to thetotal formation time of a gas bubble and the last stageis responsible for the lapse time during which pressurebuilds up in the chamber to form an interface at theorifice. The elongation stage ends when the neck of thebubble breaks off and the bubble detaches. The system
AB
was studied under constant gas flow condition. Thetheoretical analysis was based on a few assumptionsand force balance was taken including terms expressedby L’ecuyer and Murthy107 for pseudoplastic and vis-coelastic liquid
The effect of gas motion is found out from streamfunction corresponding to the circulatory motion of gasinside the bubble, and the pressure distribution due tothe gas momentum at the interface. The velocity po-tential around the bubble is obtained from the summa-tion of different potentials as mentioned in the case ofthe Newtonian liquids. Since besides the radial expan-sion of the bubble, vertical translation also occurs, thenet upward force due to the vertical component of theliquid pressure becomes positive. To find out the dimen-sions of bubble its surface is divided in ‘N’ grid points(Figure 10). As the bubble starts growing, the newcoordinate positions of the grid points on the bubblesurface are computed by employing a finite differenceprocedure for moving boundary. The integral methodand truncated cone method are used to estimate thebubble volume, which yielded a good comparison withthe experimental results. Also in the case of polymericliquids, the bubble shape is elongated to its vertical axiseven at lower gas flow rates due to complicated stressfield around the axisymmetric body in a viscoelasticliquid.
Terasaka and Tsuge12 studied the formation of bubblesat single orifice in viscoelastic liquid viz. PAA andproposed a model for nonspherical bubble formation. Inthe model, the bubble surface is divided into a numberof elements as in the earlier case, and the flow of gas isconsidered to be constant. Two equations of motion inthe radial and vertical directions are simultaneouslysolved to calculate the radial and vertical velocities foreach element on the bubble surface and the positionsof elements are estimated. The convergence was sub-jected toward the detachment of the bubble. The modelis based on the assumptions that the strain in a liquiddoes not exist at all before bubble growth and bubblegrows symmetrically about the vertical axis on the
center of orifice. The concept of relaxation time of thebubble was used and it was found out that at high gasflow rates the relaxation time was smaller than thegrowth time except for the initial period of bubble for-mation. But in the case of low flow rates, the elasticityinfluences the bubble growth as well as the initial periodbecause of the temporary decreasing bubble volume atthat point. Ghosh and Ulbrecht106 followed the two-parameter version of the Oldroyd model for bubbleformation in viscoelastic liquids, which is convenient tomeasure the dynamic elasticity and viscosity. Fromtheir observations, Terasaka and Tsuge12 have sug-gested that for polymeric liquids other than PAA, it isbetter to follow Maxwell’s rheological model rather thanOldroyd’s model. For the details about bubble growthin viscous Newtonian and non-Newtonian liquids, read-ers may refer to Favelukis and Albalak.108
2.8. Bubble Formation Accompanied with Weep-ing at the Submerged Orifice. In the operation ofdistillation and absorption using sieve trays, weepingthrough the holes is a significant and most commonlyobserved problem affecting the performance of theoverall operation of the process equipment. In view ofthis, it is required to understand the mechanism ofbubble formation while weeping occurs. Weeping ratecan be measured either physically by collecting samplesof liquid accumulated in the chamber from time to timeor by measuring the pressure fluctuations in the cham-ber, which represents the transient pressure differencebetween the liquid at the orifice and the gas chamber.McCann and Prince109 were the first to study thisproblem systematically, by following a potential theorybased approach to predict the liquid pressure distribu-tion in the wake of bubbles. The understanding devel-oped by these investigators based on the experimentalobservations is generally used as a basis for developingweeping models; however, the two main assumptionsof sphericity and negligible effect of the external orificediameter on bubble volume deviate the predictions.Kupferberg and Jameson65 observed that at low gas flowrates, the process of bubble formation follows dumpingin the orifice-chamber region and dumping occurs justbefore the weep point is reached. In the process ofdumping, the pressure drop across the liquid above andthe orifice chamber is just sufficient to maintain a liquidlevel above the sparger and thus, any value below itstarts weeping. During weeping bubbles are generatedvery discretely and only large bubbles are formed.Recently, Zhang and Tan24 have studied this problemvery carefully using the first law of thermodynamics andthe Bernoulli’s equation across the orifice mouth andthe potential theory for understanding the surroundingliquid flow. Their predictions showed that wake pres-sure exerts a back-pressure on the orifice such thatwhile it goes away from the orifice, chamber pressuregoes down. The predicted weeping velocity for variousgas flow rates show small deviation from the experi-ments mainly because; the model assumes that thebubble attains its terminal rise velocity immediatelyafter its detachment. Also, the weeping rate is greatlyaffected by the orifice diameter as well as the chambervolume and hence an optimum combination of the twoshould be reached to eliminate the weeping condition.Miyahara et al.110 have also studied the phenomena ingreater detail. Byakova et al.33 have reported experi-mental results of the surface phenomena effect onbubble formation from a single orifice (1 mm diameter)
Figure 10. Bubble surface divided in 'N' grid points at gas-liquidinterface. (reproduced with permission of Elsevier from Ghosh andUlbrecht106).
PB + PG - PL ) ∆pσ +∆pµ (60)
AC
submerged in water with extremely small gas flow rate(2 cm3/min). Bubble formation was studied for a widerange of contact angles (68°-110°) at liquid-orifice plate-gas interface using key geometrical parameters of abubble: volume (VB), surface area (S), radius at the tip(R0) and the dimension of bubble periphery at the base(dh). The formation was observed to occur in fourstages: (1) nucleation period, (2) under critical growth,(3) critical growth, and (4) necking. It was determinedthat bubble volume essentially depends on wettability.With severing wetting conditions (e.g., equilibriumcontact angle increases from 68° to 110°) the bubble sizewas reported to increase at the same gas flow rate.
2.9. Bubble Formation at Perforated Plates andSieve Trays Submerged in Liquid. In the case ofbubble formation under submerged multi-orifice sys-tems such as sieve plates and perforated plates althoughthe applicability is wide, very less attention has beengiven mainly because, a number of bubbles are gener-ated simultaneously and the system is much morecomplex than the single submerged orifice. In such acase, bubbles are not spherical because (i) the pressuredrop across each hole is not the same resulting in thestratification of the liquid in its vicinity causing distor-tion of shapes, and (ii) during its expansion, bubbleadheres to the plate surface deviating its shape fromthe spherical ones. Also the formation of a bubble is ahindered process and the hindrance needs to be includedin the models in terms of number of bubbles formedsimultaneously, pitch of holes and distance betweensieve plates and the hole-gas velocity.
Mechanism of bubble formation is same as that fromsingle orifice but the bubble generation from each orificeor perforation occurs at different time and also its modeof detachment, and hence, the overall mechanism ofbubble formation on sieve trays is different from a singleorifice. Generally, bubble formation under sieve platesis considered to have two different mechanisms. In thefirst mechanism, as in the case of single orifice, bubblesare generated under low gas velocities such that everysingle bubble is detached individually while in thesecond, bubbles are formed as a consequence of jetbreak-up, where a fine dispersion is seen. Most of thetime, in operating bubble columns, the gas is suppliedfor achieving jetting regime, which leads to bubbleformation due to jet break-up. In such a case instead ofthe bubble sizes, the bubble size distributions arepredicted and used in design of the system. As men-tioned previously, at low gas velocities, individualbubbles are formed at each orifice on a sieve platesparger. Most of the bubbles are of the same size. Withincrease in the superficial gas velocity, depending uponthe chamber volume, a pressure drop profile is gener-ated inside the chamber, and it achieves a dynamicnature. As a result, the formation of bubbles at the sieveplate occurs dynamically and varies in space. This leadsto turbulence in the sparger zone. At even higher gasvelocities, each hole acts as an independent orifice andleads to jetting. The state of the different holes on thesparger at any instant can be in the range of blockedorifice (time gap between two bubbling/jetting cycles)to completely open orifice (jetting). However, dependingupon the chamber volume, static liquid height above thesparger and the gas velocity, the duration for thisphenomena and its dynamics can vary. On the otherhand, with a porous plate with very fine holes (20 µ <dh < 200 µ), the pressure drop across the sparger is
extremely high and this leads to equal bubbling fromall the orifices in time. Subsequently, continuous pro-duction of the tiny bubbles results in froth formationabove the sparger with very narrow bubble size distri-bution in the sparger region. This happens at relativelyhigher gas velocities, while at lower gas velocities,depending upon the properties of the liquid, bubblescoalesce right above the sparger and thus result innormal bubbly flow. The important difference betweenthe two spargers is the formation of froth (poroussparger) and jetting (sieve plate). Here, we have com-piled the significant observations of many investigators.
2.9.1. Observations of Bubble Formation at Multi-Point Sparger. Houghton et al.111 were the first toestimate the bubble sizes produced from porous platesin different liquids using the hold-up and gas flow rate.Mersmann112 was among the first few investigators toexperimentally study the bubble formation on sieveplates and perforated plates and outline the differencefrom the single orifice mechanism. Later in due coursemany investigators (Kumar and Kuloor,17 Miyahara andco-workers,113-115 Koide,116,117 and Schwarzer et al.118)made attempts to propose various mechanisms of bubbleformation under submerged sieve trays and perforatedplates including the effect of various physical param-eters that affect the bubble formation. Until now, all ofthem have tried to validate their theoretical model withexperimental observations, and thus, we have a numberof correlations, which give bubble diameter or volumewith certain limitations.
As we shift from single orifice to multi-orifice condi-tion the complexity in the system increases and theexperimentally observed bubble dimensions deviatefrom that of theoretical. McCann and Prince119 devel-oped a bubble interaction model to predict bubblefrequencies placed in a line of five orifices with 2.3 cmspacing. They have reported that bubble frequencydepends on the total chamber volume however; theinteraction model did not appear to be applicable toother orifice configurations. As discussed previously, forthe case of a single orifice bubbling, the effect ofchamber volume pressure fluctuations is significant ondeciding the bubbling frequency as well as the bubblevolume to some extent, with an increase in orificenumber this effect decreases and finally vanishes whenthe orifice number is more than 15.120 Many authorshave tried to calculate the bubble diameter in terms ofSauter mean diameter, equivalent spherical diameterand volumetric mean diameter. Mersmann120 has rec-ommended that for bubble formation at sieve plates fora complete flow, Weber number We > 2. Miyahara andHayashino115 analyzed the process considering non-spherical shape of bubble and volumetric mean diameterwas found out theoretically assuming logarithmic bubblesize distribution. Li et al.121 studied the same phenom-enon using a narrow slot by producing bubbles simul-taneously from the closely spaced slots. Southern andWraith122 and Li and Harris123 have shown that bubblesare formed continuously at a series of spontaneoussources distributed on a slot length and each source actslike a self-contained, self-regulating nozzle. The numberof sources on slot and size of bubbles they produce varywith gas flow rate and the width of the slot opening andmany small bubbles are generated at low flow rate.Ruzicka et al.124,125 investigated bubble formation at twoorifices and identified two types of bubbling modesthrough the analysis of pressure fluctuations in the gas
AD
chamber. On increasing the orifice number to 13, moretypes of bubbling modes were identified based onvarious plate configurations.
Since the forces acting on a bubble in multipointspargers or a multi-orifice configuration are slightlydifferent from the single orifice, to understand thiscomplex phenomenon, it is required to compare theavailable theoretical models for its mechanism.
2.9.2. Mechanism of Bubble Formation on Multi-Orifice Systems. As mentioned previously, bubble for-mation on multi-orifice systems takes place in twomodes, chain bubbling and the dispersion due to jetbreakage. The geometry of the hole affects the shape ofresulting bubble by virtue of contact angle. In themajority of literature, Mersmann’s analogy120 of suc-cessive hole occupation in sieve trays is followed toderive a theoretical model and in most of the efforts,the results obtained from single hole orifice can beextended towards sieve plates. According to this anal-ogy, gas first passes through a single hole and then itgoes to the next until the first hole attains jet regime.In the bubbling regime, bubbles are of a constant size.Before studying the various available mechanisms, itis essential to note that in a bubble column, fourdifferent regimes viz. bubbling, froth, cellular foam, andjetting are observed as a result of bubble generation atsubmerged sieve and perforated plates at various gasflow rates.
Similar to that of the single submerged orifice, themechanism here can also be understood through theforce balance, which includes surface tension force,pressure force, viscous force, drag force, gravity and twomore features in the column i.e., liquid circulation andliquid turbulence influence the bubble formation. Onporous plates with small holes, small bubbles are formedand gas momentum is too small to cause any gas densityinfluence on the bubble formation size. Average bubblesize depends on the bubble size at the sparger hole. Themechanism of bubble formation is considered to takeplace in two or more stages. Neck formation at the trayhole takes place in very small span and especially inthis case, due to high gas velocity, the momentum isimparted by primary bubbles onto secondary and thesurrounding liquid motion is high. This momentum isnot only because of the effect of earlier bubble but alsobecause of liquid motion due to bubbles generating fromsurrounding holes. As a result of these forces, shape ofbubble under formation no longer remains spherical andchanges to ellipsoidal and then to spherical cap (de-pending on the liquid properties) as it goes away fromsieve tray holes and experiences coalescence/dispersion.In the process of bubble formation by jet breaking,36 thelevel at which bubbles are formed is given by averagejet penetration depth. For a bubble generating from anupward facing orifice, its center of gravity rises andresults in reduction in the pressure on the downstreamside of the orifice. Since the pressure drops across theorifice, it further causes pressure variation inside form-ing bubbles and thus a decrease in the surface tensionpressure can be seen.
The observations by Klug and Vogelpohl126,127 supportthe Mersmann criteria for generation of bubbles in jetregime while the mechanism of bubble formation onsieve trays is found similar to that at single submergedorifice. At high Weber numbers, secondary bubbles aresmaller than those generated at the single hole plateand bubbles are generated through all the holes, though
they do not approach the jet regime. In the case of sieveplates having the closest distance between holes, noindividual bubbles are formed. Even before primarybubbles form, they coalesce to larger agglomerateswhich detach from the plate and again disintegrate dueto shear stress of the surrounding liquid. This motionof liquid above the sieve plate is a considerable resis-tance in forming primary bubbles, which also followsMersmann’s criteria.
Miyahara and Tanaka114 observed that all the bubblesascend individually without coalescence and all that getdispersed have uniform size from a sintered mesh plate.Miyahara and Hayashino’s115 observation for perforatedplates shows that bubble coalescence takes place aftercertain time immediately after formation yielding a log-normal probability distribution of the bubble sizes. Inanother extensive study on bubble formation on perfo-rated plates Bowonder and Kumar128 suggested themechanism based on an assumption that the numberof effective sites (active holes at any instant) such thatthe bubbles formed cover the whole area as if fromsingle nozzles when arranged in close packing. Thus,their analysis accepts the same mechanism as for singleorifice. Wraith53 has considered it as a two-stage phe-nomenon considering the line of contact between bubbleand the edge of slot.
2.9.3. Models of Bubble Formation in Multi-Orifice Configuration. Under the crowded sourceconditions that are likely to apply at high flow rates,lateral interaction between growing bubbles will impedeformation and a model based upon spherical growthmay not be adequate. Effectively, all of the models arethe same as for single orifice except some times it isrepresented in terms of size distribution.
2.9.3.1. Li et al. Model121. According to Rayleigh-Taylor instability criteria,129 horizontal interface be-tween two stratified fluid layers at rest is generallyobserved to be unstable in a gravity or acceleration fieldif the density of the upper fluid is larger than that ofthe lower fluid. When gas flows vertically through anarrow, slot-shaped nozzle submerged in liquid, the gasliquid interface within the slot can be consideredcylindrical. The meniscus is unstable because gasunderlies liquid, and its radius of curvature is likely tovary with gas pressure. On this basis, the interface isassumed to get disturbed by the presence of peaks ornodes distributed along the slot at the most dangerouswavelength (λd) based on acceleration field developedin the vicinity of nozzle mouth which results in thegeneration of bubble. This dangerous wavelength canbe given by
For the present case, the analogous dimensionless formcan be given as follows
where λd/ is the dimensionless dangerous wavelength
and R* ) RoxFLQ/σ is the dimensionless radius ofcurvature. Model of bubble formation considering eachhole as independent source on the same slot is proposedand at low gas flow rates, the bubbles generated by
λd ) 2πx 3σg(F1 - F2)
(61)
λd/ ) 2.067 + 0.3108x3R*1.434
1 + 0.3108R*1.4342π
x1 + (1/R*2)(62)
AE
capillary action are represented by force balance as
where FS effective surface tension force at the nozzledefined in terms of the total length (L) of contact at theorifice mouth. At large gas flow rates force balanceyields
where KB is the constant that depends on the bubbledetachment condition. From the independent analysis,they have estimated the values of λ for different slotspacings (w ) L/N) with N sources and correlated it withthe system We as
And further, these values are used for the calculationof bubble volume using the superficial gas velocity (VG)as
This formulation is later modified130 to achieve thebubble sizes directly from the superficial gas velocityand the properties of the gas-liquid system and givenas
2.9.3.2. Miyahara and Takahashi (1986)113 Ap-proach. According to this model, bubble volume at theend of detachment remains the same for all the bubblesgenerated from all holes on a porous plate. Referring tothis assumption, these investigators calculated thevolume of individual bubble based on experimentallyobserved number of bubbles that are generated due tosupply of certain volume of gas and then by forcebalance on a single bubble, diameter of individualbubble (dB) was given as
For plate geometries, equation for dB changes owing tochange in shape of bubble. Similarly, for the case of sievetrays, Biddulph and Thomas131 have used just thebuoyancy and surface tension forces and employed thedetachment point condition to predict the bubble sizesas
To consider the actual size distribution, the diameter ofgenerated bubble was considered to be Sauter mean dia-meter neglecting viscous drag force acting on bubble.115,119
which is reasonably close to Tate’s law (dB ) 1.8
x3dhσ/Fg).2.9.3.3. Loimer et al. (2004) Single Bubble Ap-
proach.132 In a recent work, Loimer et al.132 have
investigated the bubble formation over porous platesand sieve plates. Their theoretical analysis is based onthe assumption that every bubble forms singularlywithout any hindrance due to surrounding bubbles andused Tate’s law for the estimation of bubble size underquasi-steady regime as well as the dynamic regime. Forsingle bubble formation in the dynamic regime, theyhave used the relative variation between the surfacetension force and the pressure force to estimate the gasflow rate (QSI) for the transition from the quasi-steadyto dynamic regime as QSI ) π(0.66/g2k3)0.17(dhσ/F)0.83,where k is the added mass force. For the case of wettedsieve plates, they estimated the minimum flow rate forthe bubble formation at more than one orifice as Qmin
) 2π xσdh3/FG and the number of active orifices is
estimated in terms of the ratio of total gas volume fluxto the Qmin. For a densely packed sieve plate, theiranalysis yield the bubble sizes
where VG is the superficial gas velocity. On the non-wetted sieve plates, the analysis also includes thedynamic regime. Their analysis assumes that below thesieve plate, no spatial pressure variations exist andhence the formation is a random process. However, thisassumption is contradictory to the visual observa-tions,133 with Mersmann’s criteria and also the CFDresults by Dhotre et al.134 The approach can be consid-ered valid for porous plates which offer very highpressure drop across them yielding practically negligiblepressure gradient below the plate. However, the pres-sure variation below a sieve plate is not only spatial butalso has temporal dynamics associated with it, whichmakes the process complicated. Moreover, the visualobservations indicate that the dynamics are well orga-nized and follow systematic patterns, which can makethe mathematical analysis simpler.
For the wetted porous plates, the analysis assumes apressure field variation below the plate and a charac-teristic radius (RC) is identified for the simultaneousbubble formation at two locations separated by a locusof circle of RC. Since the formulation for Qmin is basedon Darcy’s law, it is independent of pore size and thebubble size is given as
Similar analysis is also done for a nonwetted porousplate. The analysis has shown a reasonably good matchbetween the experimental and predicted results. How-ever, the random distribution of the bubbling positionsmainly depends on the pore/hole size and their distribu-tion and the observations need not be valid for largesystems, due to the dynamics of bubble formation atlarge holes and the associated liquid flow in its vicinity.Nevertheless, the approach is simple and may be testedfor large systems as well.
2.9.4. Factors Affecting Bubble Formation atSieve Plates. 2.9.4.1 Gas Flow Rate. Similar to thesingle submerged orifice, for sieve trays bubble sizeincreases with increase in gas velocity,114 and theyascend individually without coalescence and get dis-persed with uniform sizes. According to Miyahara andHaga,135 at low gas flow rates, bubbles ascend individu-ally, whereas at some higher flow rate formation of thesecondary bubble is affected by the primary one because
VB ) FS/(FL - FG)g (63)
VB ) KBQ6/5/g3/5 (64)
λw
) 17.6(FL/FG)0.16We-0.25 (65)
VB ) 0.01(VGwλ)1.37 (66)
VB ) 26.2((VGw)2
g )(FL
FG)0.192
(Fr4/3We)-3/10 (67)
dB ) (∑6VB/πn)1/3 (68)
dB,Max ) (6sin θ)1/3[ dhσ
(FL - FG)g]1/3
(69)
d30 ) 2.9[Fg/σdh]-1/3 (70)
R ) 1.881VG2 /g (71)
R ) x(8πσk/3x3VGµG) (72)
AF
bubbles from two holes ascend in close proximity witheach other. At low pitch to hole-diameter ratio, gasvelocity has no effect on bubble formation. Kumar andKuloor17 have reported above observations for sievetrays. At small flow rates, bubble size increases withhole-diameter and bubbles are formed at certain fre-quency and this has a unimodal distribution in size.With an increase in flow rates, relative influence ofsurface tension goes down resulting in diminishing ofdifference between bubble volumes obtained from dif-ferent pore diameters and bubbles of nonuniform sizesare formed (bimodal distribution). At medium flow rates,single bubble formation takes place due to coalescencebefore detachment (multimodal distribution). At veryhigh rates, the effect of surface tension is totallyeliminated and a continuous linear blanket of gas formsat surface along the slot mouth and local getting andsubsequent break-up takes place; also, bubbles ascendin groups and thus formation of the new bubble has toface the resistance due to turbulence by liquid in itsvicinity. When spacing between sources is larger thanthe free bubble diameter the sources are discrete andindependent and the performance is least affected atthe orifice mouth. Li et al.121 have also shown that evenfor a system having several bubbling sources, the bubblesize increased with the gas flow rate; however, thenature of variation differed with the orifice configura-tion.
2.9.4.2. Pitch of Holes on Sieve Trays. For smallpitch on the sieve trays, bubbles get coalesced to largerones before it detaches. Smallest bubbles are formed atsieve plates with the closest hole-spacing of 6 mm asthe momentum transfer becomes intensified with in-creasing proximity of bubbles in swarm. At high viscosi-ties, smallest bubbles are formed at sieve plates withthe largest spacing. Generally, large hole-spacing isrequired in order to suppress the increased coalescencein the sparger region. In dispersion zone, arrangementof bubbles is not affected by hole-spacing. Varying hole-spacing leads to different bubble sizes, and it affects theascending velocity of bubbles. Decrease in pitch belowless than bubble diameter stretches a bubble in verticaldirection,100 and consequently shape changes to anoblate shape, the pressure inside the bubble keepsreducing and finally produce a larger gas flow ratethrough the orifice and thus a larger detachment bubblevolume. In a systematic analysis, Li et al.121 have shownthat with increase in the distance between two holes,larger bubble size is achieved even at smaller gas flowrates, while at high gas flow rates (gas flow rate perbubble source > 0.7 mL/s), the bubble sizes are notsignificantly affected by the pitch. When estimated forthe various slot sizes they have used, the nozzle veloci-ties were seen to be closer to the transition regionbetween the steady jet and pulsating jet depending onthe depth of orifices. Since very little has been studiedand known about the bubble formation in sieve platsand porous plates, it is required to understand thebubble formation process to have a control on theprocess by creating only required interfacial area andby reducing back mixing.
In a bubble column reactor, the regime of operationi.e., homogeneous, heterogeneous and transition arebasically governed by the bubble sizes and their slipvelocities. In many of the cases, the homogeneousregime prevails at low gas velocities, whereas at higherVG heterogeneous regime is attained. It is possible that,
at low VG, transition or heterogeneous regime can beattained by using a sieve plate having either large holediameter124,125 or very small pitch. This way, it ispossible to control the regime of operation to someextent by making proper choice of the sparger design.
2.10. Experimental Techniques. Motion of bubblesin liquids can be observed experimentally either byvisualization or by taking photographs at high speed.This is possible only in the case of transparent or semi-transparent liquids. Here, we discuss the above twomethods in brief followed by their pros and cons.Further, we also discuss the possible experimentalmethods of visualization, which would be possible usingadvanced flow visualization techniques. We have alsotabulated the methods used by various investigators forthis analysis in Table 1.
2.10.1. Image Visualization/Analysis. Most com-monly used methods for the experimental observationof bubble formation and its rise is image visualizationthrough stroboscopy or high-speed photography andX-ray.136 The Stroboscopic method and high-speed pho-tography is reliable even for jetting regime as thenumber of bubbles formed and their sizes can bemeasured from the photographs. However, this methodis limited by its applicability in transparent and semi-transparent liquids in near wall region. This limitationcan be overcome by the use of X-ray as it can even beused in opaque systems but at higher cost of measure-ment and includes the danger of exposure to hazardousrays. The details of these techniques are explained inthe third section of this Review.
2.10.2. Volume of Displaced Liquid. In this method,bubbles are continuously generated in a pool of liquidat certain known frequency (usually measured throughvisual observation) and the increase in liquid volumeis measured. From the knowledge of the frequency ofbubbling (under the assumption of constancy in size)bubble size is estimated. This method is suitable evenfor the opaque medium, provided the frequency ofbubbling should be measured using some invasive/noninvasive methods for the recording of event on anoscilloscope. This method is limited for the generationof single bubbles and cannot be used for the cases, whereprimary and secondary bubbling or even bubblingthrough jet break-up occurs. Further, for very smallsized bubbles, it is required to measure the exact changein liquid volume, which can make significantly largeerrors in results.
2.11. Conclusion and Recommendations. Bubbleformation at a single orifice has been given sufficientattention and various research groups have developedthis concept very well. Published literature shows thatapproaches for studying the bubble formation can beclassified mainly as, (i) the experimental analysis bystroboscopic method, (ii) modeling the process throughsimple force balance/dimensional analysis, and finally,(iii) solving the equations of motion and continuity bypotential flow approach for various conditions. A few ofthe recent studies illustrate that a combination of anyof the above two is very useful.
Single stage bubble formation model holds well onlyfor a few cases, viz. in the case of liquid flow where thebubble detachment is very fast and the volume occupiedby bubble neck is negligible. The results from the single-stage bubble formation model do not compare very wellwith the experiments at high gas velocity and in non-Newtonian liquids since the contribution of bubble neck
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to the overall bubble volume is not taken into account.Since the stroboscopic pictures have shown that theprocess is not a single stage process, it is needed todivide it in various stages and model it subsequently.The predicted results can be matched with the experi-mental ones only when the process of formation, growthand detachment are studied separately by applying theequations of motion to the system in consideration.Further, for a multi-stage consideration, it can be clearlyseen that the predictions are improved over a widerange of operating and system parameters. This isbecause, the process of bubble growth is being taken intoaccount very carefully and even the elongation of neckat low gas flow rates is also considered. The results ofmodels based on the potential flow theory18,102 lookpromising since the bubble growth at the orifice can beobserved very neatly in time domain. Still, the forcebalance approach is simple, reliable and not computa-tionally intensive as for the potential flow approach.
In most of the proposed models, bubbles are assumedto be spherical, which is not true always. For high gasflow rates in a pool of viscous liquid with large orificediameters, the bubbles generated are either ellipsoidsor hemispheres and pressure inside the bubble will bedifferent at all the surface elements. Also, for non-spherical bubbles formed in the case of continuous phaseflows, the detachment condition and the nonidealitiesin the shape also need to be considered. Similarly, tounderstand the phenomenon at high temperatures,where the gas does not remain ideal and becomescompressible, further modifications need to be made.With increased computational power, it seems possibleto release such assumptions from the existing modelsand formulate a generalized approach. The approach ofdividing the interface in various small parts and thenapplying the force balance on each surface can becombined with the potential flow method to meet theabove requirement with appropriate inclusion of thethermodynamic laws.
As mentioned earlier, when the liquid is in motion,the bubble detachment occurs at an early stage andhence an additional term of the virtual mass force istaken into account. The results obtained even by thepotential flow approach show little deviation from theexperiments, which focuses on the empirical nature ofthe virtual mass coefficient and brings out the require-ment of modifying it.
Various investigators have shown that the chambervolume and the orifice shape are important designparameters affecting the bubble sizes. The first param-eter needs to be studied more methodically since theeffect of back pressure has rarely been taken intoaccount while modeling the process. This is mainlybecause, at higher flow rates, the primary bubble hasnoticeable effect on the secondary bubble volume andalso in multipoint sparger, the hole-to-hole interactionand variation in the pressure drop across neighboringholes is mainly governed by the pressure fluctuationsin the chamber. Also, in the case of column type ofreactors, the near sparger region shows a complex flowpattern, which is greatly affected by the chamberdynamics and can be discerned by characterizing thebubble formation using advanced measurement tech-niques137 and by using fast cameras to capture requiredfeatures.
In practice, the single point spargers have limitedapplications and multipoint spargers are used com-
monly. The process of bubble formation at the multi-point spargers has not been studied thoroughly andneeds to be understood by applying the first principals.The bubbling at the multipoint sparger is accompaniedwith weeping. Since the type of liquid and orificematerial decide the wettability, conditions and hencethe possibility of weeping and bubble formation in thepresence of weeping24,109-111 from multipoint spargersneeds to be studied. Also, the bubble formation atdifferent temperature-pressure conditions and for liq-uids of various types needs to be studied since many ofthe industrial operations run at nonambient conditionswith different liquids with various properties. Most ofthe industrial columns are operated in jetting regime,where the effect of chamber volume is negligible it isrequired to improve upon the design of trays to mini-mize weeping.
In the published literature, the focus is on the effectsof various experimental conditions on the bubble sizesgenerated from the orifice, which shows discrepanciesin the observations. It is a requirement to compare themon a common basis and suggest the scope for furtherwork to reach to a conclusion or a generalized approachto study this phenomenon.
New approach of employing neural network-param-eter estimation technique toward this problem by Jami-alahmadi et al.60 looks very interesting. However, effortsin reducing the deviation from the experimental dataneed to be made and more attempts toward increasinglycomplex systems (liquid flow, bubble swarm, extremeoperating conditions, etc.) are required.
Approaching the complex problem of formation ofbubbles in the presence of hindrance seems difficult atthis stage. An attempt can be made by formulating amodel, which takes into account the effect of presenceof surrounding bubbles, associated liquid motion andthe various additional forces acting on it. The localbubble number density, bubble size distribution, thevariations in the local liquid velocity in the vicinity ofeach hole, the dynamics of formation process, unequalpressure distribution over the orifice can be used toformulate a population balance based model along withNavier-Stokes equations for both the phases may bethe next step in solving this problem.
Since the discrepancy in the various experimentalobservations is critical in some cases, it would be nowmore realistic to develop the further understanding ofthis phenomenon through rigorous CFD, which will helpin visualizing the contribution of individual forces andparameters over a wide range of systems to a betterextent.
3. Bubble Rise Velocity in Liquid
After the detachment of a bubble from the orifice, thebuoyant forces cause its rise in the liquid. The dynamicsassociated with rise are mainly due to temporal varia-tion in bubble characteristics or even the variation inthe system properties. The phenomenon is very simpleand observed very commonly in many industrial gas-liquid contactors viz. absorbers, flotation tanks, bubblecolumns, stirred gas-liquid hydrogenation reactors,viscbreaking, etc. Bubble rise in liquid is a classicalexample of flow past immersed bodies and showsdifferent behavior under the conditions of with andwithout slip at the boundary. Rise velocity (generallytermed as the terminal rise velocity in stagnant liquidsand slip velocity in the case of moving liquids) of gas
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bubbles in liquid dispersions is one of the parameters,which decide the gas phase residence time and hencethe contact time for the interfacial transport andsubsequently contributes to the performance of theequipment. In the applications of flotation, bubble risevelocity decides the operation time138 and for enhancingthe performance of the equipment, bubble size and risevelocity are important controlling parameters. Thephenomenon of bubble rise in liquid in different condi-tions, i.e., quiescent and bubble swarm is under inves-tigation over many decades. In the early years ofattempting this problem,139 it was assumed that thefluid surrounding the bubble has zero viscosity, andthere is no slip at the boundary which simplifies thecase to a greater extent when compared with the caseof real fluids, where the complexity is higher. In the caseof a system where fluid is real, gas bubbles experiencea slip and the presence of viscous and inertial forcesmakes the analysis complicated and understandingmeager.
The regimes of bubble motion in liquid are grosslyclassified as Stokes regime, Hadamard regime, Levichregime, and Taylor regime.140 All of these regimes havespecific conditions for a bubble motion conditional totheir size, shape and nature of interface, i.e., free orrigid. The phenomenon is very interesting as thecharacteristics of a rising bubble (i.e., its size, risevelocity, trajectory, etc.) show dynamic behavior fromsystem to system, making it very tedious to develop ageneral understanding for all the cases and also todevelop a generalized correlation for rise velocity in allabove regimes for different gas-liquid systems. Thesubject of bubble motion in liquid has been veryinteresting for the physicists (Auton,141 Eames & Hunt,142
Magnaudet & Eames,143 Mei & Klausner144) and thisproblem is generally attempted through the force bal-ance. The main forces, which act on a bubble during itsmotion in liquid are gravity, buoyancy, drag, surfacetension, viscous forces, added mass force, history force(arising out of the unequal distribution of vorticity), andfinally lift force (either due to baroclinically developedpressure gradients or unsteadiness of the flow). Thecontribution of individual forces varies from the case tocase and most of the earlier work involves the develop-ment of suitable formulation for the same. However, inthis section of the review, we have focused our attentionmainly on the rise of bubbles in liquids and the effectof various physicochemical parameters on the risevelocity. Many investigators (Astarita,145 Davis andAcrivos,146 Haque et al.,147 Clift et al.,148 Abou-el-Hassan,149,150 Chhabra,151,152 Rodrigue153,154) have workedextensively in various gas-liquid systems and devel-oped several correlation for a specific range of condi-tions. Since the chronological development in this areais not very important (pertaining to the development ofgeneralized correlation) the attempts in this directioncan be grossly classified on the basis of the nature ofliquid phase (Newtonian, non-Newtonian, aqueous,organic etc.). Only the recent attempts by Nguyen155 andRodrigue153,154,156,157 on developing generalized correla-tion for various liquid systems can be considered to beapplicable over a wide range; however, still manyimportant deviations from the regularity need to beincorporated. In view of this, here we have made anattempt to review the published literature on bubble risevelocity and have compared the existing correlation forthe experimental data by various investigators for
recommending the most suitable correlation applicableover a wide range of parameters. In addition to this,we have also compared all these observations andidentified the possible reason behind every result andthe physical significance of the particular experimentalobservation is also discussed in detail. This part of thereview is arranged as follows: In the next subsection,we discuss the various parameters governing the phe-nomenon of bubble rise and effect of them on bubblerise velocity. Further, a qualitative description of bubbletrajectories and different modes of oscillation is given.In the fourth subsection, we have reviewed the observa-tions by various investigators on bubble rise velocity,and their analogy for development of correlation for thesame is discussed and finally recommendations aregiven for the use of the same.
3.1. Factors Affecting the Rise Velocity. The riseof a bubble in liquid is a function of several parametersviz. bubble characteristics (size and shape), propertiesof gas-liquid systems (density, viscosity, surface ten-sion, concentration of solute, density difference betweengas and liquid), liquid motion (direction), and operatingconditions (temperature, pressure, gravity). Here, wehave discussed a few of these in regard to the impor-tance given to them in many studies.
3.1.1. Effect of Purity of Liquid (Surface Ten-sion) on Bubble Rise Velocity. As mentioned earlier,the regime of bubble motion in liquid shows the effectof various parameters on rise velocity and thus varyingthe physicochemical properties of liquid the regime ofoperation can be manipulated. In the case of Newtonianinviscid liquids, the rise velocity depends on bubble size,temperature, extent of contamination, and pressure inthe system. The viscous forces have no noticeableinfluence on the rise velocity and only the gravity,buoyancy, and drag matter. Typical observations in suchsystems are shown in Figure 11A. It can be clearly seenthat at ambient conditions the plot of rise velocity versussize in clean (pure) liquid, the velocity increases withsize and attains a peak followed by a decrease in velocityover a very small range of bubble size after which itattains a weakly positive dependence on bubble size.Depending upon the type of contamination, i.e., surfac-tant, electrolyte and their concentration, the rise veloc-ity characteristics of a bubble change significantly. Inthe case of the electrolyte solution, the effect is similarto that of surfactants; however, the additional electro-kinetic forces repel them further causing existence ofsmaller bubbles rising with still lower velocity.
The mechanism by which surfactants modify thevelocity field in the vicinity of a fluid-fluid interfacewas first described by Frumkin & Levich158 and Levich,159
who proposed that surfactants tend to be advected alongthe bubble surface and accumulate in the rear region.Since the surface tension gradient produced by anygradient of the surface concentration of surfactant mustbe balanced by a jump of the shear stress across theinterface (Marangoni effect), this mechanism impliesthat a shear-free boundary condition is no longerimposed in the liquid at the gas-liquid interface, andthis leads to an increase in the drag force. This is knownas stagnant cap hypothesis and has been used success-fully for the estimation of bubble rise velocity and alsoused for the prediction of drag coefficient with respectto extent of surface contamination.
Although the effect of surfactants/contaminants (maybe even polymers) on bubble rise velocity is known for
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a while, very few detailed investigations exist where theeffects of surfactants are studied. Since bubble size andits characteristics form an important component ofdesign of a multiphase system operated in industriesand in reality the industrial liquids are not always pure/clean, the laboratory studies with such liquids areimportant. Surfactants mainly modify the surface of abubble resulting in change in its characteristics in bothNewtonian and non-Newtonian liquids in a differentmanner. In the case of Newtonian contaminated liquids,if the contamination is due to surface active agents, asbubble rises, the bubble surface is dragged backwardalong with the liquid. The surfactants, specifically, areknown to bridge the gap between solid rigid bodies andclean bubbles by immobilizing the bubble surface thusmodifying the velocity field. In the front portion of thebubble, the surface is stretched and the new surface isconstantly generated. As a result, surface tensiongradients are generated resulting in increase in dragand reduction in the mobility of the gas-liquid interface.Thus, the internal circulation is reduced and also thetendency to coalesce with other bubbles. As a result offormation of rigid interface and enhancement in drag,the rise velocity is lesser than the clean bubble of thesame size. For the case of liquid drops, the criticaldiameter increases with increasing surfactant concen-tration.160 However, the effect is significant dependingupon the solubility of the surfactant. Kopf-Sill andHomsey161 have used Hele-Shaw cell for analyzing theeffect of surfactants and have used the capillary numberand Bond number for identifying six different shapes.
Studies by De Kee et al.162 have shown that thereduction in surface tension (by addition of 0.042 mass%of sorbitan monolaurate) in the solution of 1 mass%polyacrylamide (AP-30) in 50/50 water/glycerine mixtureresults in the decrease of bubble rise velocity of smallbubbles without any discontinuity. In another attempt,Rodrigue et al.156 have studied the velocity of singlebubble in Newtonian (three different concentrations ofglycerine in distilled water) and non-Newtonian (solu-tions of CMC, gellan gum, poly(ethylene oxide) andPAA) fluids containing impurities (by adding severalconcentrations of SDS). They have reported that withincrease in surfactant concentration the bubble risevelocity decreases till certain critical bubble volume,after which the bubble velocities are not affected by thesurfactant concentration. The critical volume is a func-tion of the solvent composition for their system. Radiusof bubble at the critical volume can be estimated as rC
) xσ/g∆F (Bond and Newton163). Thus, the observedreduction in the rise velocity is a function of thereduction in surface tension as well as higher liquidviscosity, which cumulatively lead to lowering of inter-facial motion and hence the rise velocity. According totheir observation, the effect of surfactants is noticeableonly at relatively larger bubble volumes (beyond Stokesregime). For larger bubbles, which move at highervelocity, although more area is exposed to the fluid,diffusion rates of the surfactant do not allow for asignificant immobilization and hence the effect of sur-factants is negligible.
For the case of non-Newtonian liquids, the effect ofsurface tension on bubble rise velocity is similar to theearlier case, however, the observations for viscoelasticliquids are significantly different. In viscoelastic liquidssuch as solutions of PAA, bubble experiences a jump inthe liquid velocity. The role of surface tension in thisphenomenon is important as the jump is a result of thetransformation of the rigid interface to a flexible one,and the extent of rigidity of the interface is decided bythe surface tension to some extent. Several investigatorshave studied this phenomenon and proposed variouspossible explanations based on their observations. Oneof the possible explanations comes from the surfacetension model by Zana and Leal,164 where the existenceof a bubble is proposed only under its surface beingeither totally covered or totally free of surfactants andalso the elasticity plays an important role. However, theliquid viscosity is equally important and needs carefulanalysis. The remaining analysis of this phenomenonis explained in detail later at appropriate position inthis Review.
For the case of power-law non-Newtonian liquids(CMC), Tzounakos et al.165 have studied the effect ofsurfactants (SDS) in detail through the analysis of thedrag curve. According to their investigation, the motionof bubble in a shear thinning liquid can be explainedon the basis of two equations. For rigid bubbles havingimmobile surface
and
while for the bubbles having mobile surface, since the
Figure 11. (A) Typical trends in rise velocity with bubble sizefor pure and contaminated liquids (reproduced with permissionfrom Clift et al.148). (B) Effect of initial bubble deformation onterminal rise velocity in Distilled water. (reproduced with permis-sion from Tomiyama185).
CD ) 24Re
(1 + 0.173 Re0.657) where Re ) FLdbnVT
2-n/m
for Re < 135 (73a)
CD ) 24/Re for Re > 135 (73b)
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shape deviations are possible at larger bubble volume,the correlation comes out to be
and
where m is the consistency index. Thus, there exists adifference in the respective drag curves and the additionof surfactant to the solution. Thus, for a given solutionof fixed concentration, with increase in the bubblevolume, the drag curve experiences a smooth transitionfrom the curve explained by eq 74a to the curveexplained by eq 74b. Details of this effect are quantifiedlater.
Recently quantitative experiments performed withcarefully controlled contamination have allowed thetime evolution of the rise velocity of the bubble to becorrelated to the physicochemical properties and theconcentration of surfactant in the surrounding liq-uid.166,167 On carrying out the experiments with bubblesrising in water contaminated with several surfactants,it was observed that below a critical bulk concentration,the final rise velocity is insensitive to the presence ofthe surfactant, whereas the rise velocity decreasesabruptly to the value corresponding to a solid sphereabove the critical bulk concentration. The critical con-centration is reported to depend on the nature of thesurfactant and also the bubble diameter. This alsoshowed that a higher concentration is required to coverlarger bubbles. Very recently, in a systematic experi-mental investigation by Ybert and Di Meglio,168 bubbleswere released preloaded with the surfactant, the mea-sured rise velocity as a function of distance traveledindicated that after the initial stage of acceleration fromrest, the instantaneous rise velocity of a given bubbledepends only on the total amount of surfactant adsorbedon the bubble surface. It was also shown that thesurfactants are not only absorbed onto the bubblesurface, but may also be desorbed, which result in risein the bubble velocity owing to remobilization of theinterface finally reaching a constant equilibrium valueof the surfactant concentration. This observation rulesout the proposition by Zana and Leal164 that no inter-mediate stages of surfactant presence on bubble surfacecan exist.
In an elegant approach, Alves et al.169 have shownthat with increase in the extent of contamination overa bubble surface, the drag coefficient of a 1.5 mmdiameter bubble increases gradually and reaches aplateau after an exposure for about 960 s, which is evenseen to be consistent for other bubble sizes. The experi-mental analysis by these authors in terms of the risevelocity of bubbles for a range of contamination level isconsistent when compared with the classical bubble risevelocity plot (Figure 11A).
3.1.2. Effect of Liquid Viscosity on Bubble RiseVelocity. In Newtonian viscous liquids160 (viz. aqueoussolutions of glycerol) modes of bubble rise show differentbehavior than the inviscid liquids as well as some ofthe non-Newtonian liquids. Rodrigue et al.156 havereported an almost linear relationship between risevelocity and the bubble volume dependent on the liquid
viscosity of the aqueous solutions of glycerine at differ-ent concentrations. Further, the bubble rise velocity isalso reported to decrease with viscosity with the pos-sibility of reduction of the interfacial motion due toviscous forces.
For the case of different non-Newtonian liquids, therelationship between the bubble size and rise velocityshows different nature depending upon the nature ofviscosity. In certain viscoelastic liquids (viz. aqueoussolutions of PAA), terminal rise velocity (VT) exhibits adiscontinuity at certain critical value of bubble sizewhile in other shear thinning liquids (viz. solutions ofdifferent polysaccharide, CMC)167 such an observationis never reported. Leal et al.,173 Astarita and Apuzzo,140
Acharya et al.,55 and Rodrigue173 have investigated thisphenomenon in detail. It has been proposed that thejump in the velocity is an effect of the transition fromthe Stokes regime to the Hadamard regime where theviscoelasticity shows its dominance. These authors havecarried out their analysis for various viscoelastic liquidsof different concentrations and found that the transitionsize is a function of the concentration of the solute.Further, their analysis has shown that the effect ofelastic forces prohibits any regime transition and alsothe contribution of elastic forces is independent of thepresence or absence of no slip condition. However, theobservations of Acharya et al.174 have shown that in thecreeping flow regime (i.e. the size below the transitionsize) elasticity has no effect on the rise velocity, whichthey have shown through CD-Re relationship. Theseobservations are concordant with the earlier inferenceand clearly indicate that the transition in rise velocityfollows from a deviation in shape from sphere. Also, thetransition from Stokes to Hadamard regime is gradualin the case of Newtonian purely viscous shear thinningliquids, and not sudden as in the case of viscoelasticliquids. Figure 12 shows the regime transition in thecase of various viscoelastic liquids. In contrast to theabove observation, Carreau et al.,175 Margaritis et al.,167
Dekee et al.176 have observed no discontinuity in therespective solutions and their observations are sup-ported even in the case of dissolution of a bubble inliquid. To verify the existence of this transition even in
CD ) 16Re
(1 + 0.173 Re0.657) where Re )FLdb
nVT2-n
mEn
for Re < 60 (74a)
CD ) 16/Re for Re > 60 (74b)
Figure 12. Typical experimental data showing the regimetransition during the motion of a bubble in non-Newtonian liquids.Several such transition plots for various liquids may be seen inTzounakos et al.,165 Rodrigue et al.157,173
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liquids with very low surface tension Belmonte,177 hasstudied the effect of micellar concentration on themodified viscosity of a cetltrimethylammonium bromidesolution, and the overall effect of modification of theproperties on bubble rise velocity. This attempt isdifferent from the earlier as the surface tension wasreduced to form micelles, which reduces the surfacemobility. They have observed change in bubble shapeat the transitional bubble size that results in disconti-nuity in the velocity. The details of this phenomenonand the possible reasons behind it are discussed sepa-rately.
3.1.3. Effect of Temperature on Bubble RiseVelocity. It is well-known that for ideal gases, atconstant pressure, with increase in temperature, thevolume of gas increases and the average density goesdown. But, generally, to maintain the liquid state of theliquid phase reactant the condition of high temperatureis accompanied with high pressure. There are very fewstudies made on the effect of temperature on the risevelocity of bubble. At ambient conditions, the risevelocity follows certain trend with bubble size. Twodifferent cases may be considered: rise of a gas bubblein insoluble liquid and rise of vapor bubble in liquid atrelatively low temperature. Both the cases are common,while only the second case is studied very systematicallyand the complexity of accompanied condensation makesit more complicated. In the first case of rise of a bubbleof ambient condition in heated liquid, the heat transferacross the interface causes increase in the bubblevolume. Also, at higher temperature the density differ-ence between two fluids goes down, the buoyancyexperienced by the bubble decreases causing the sub-sequent reduction in the rise velocity. Thus, the coun-teracting effects are dominant in different ranges oftemperature and in certain temperature range risevelocity is a strong function of size. This particular caseis important but still remains unattended in manyrespects and very meager information is available.Leifer et al.178 have studied the effect on small bubblesof 0.1 to 3.5 mm diameter. Under the assumption of theno slip at the surface and no internal circulation, thekinematic viscosity, and the densities are expressed asa function of temperature in clean water as
where Fw ) 1 + 8.124 × 10-5T - 1.018 × 10-5T2 + 1.146× 10-7T3 - 9.246 × 10-10T4 + 2.986 × 10-12T5 whichcan be substituted in Stokes equation VT ) (2grB
2)/9νto estimate the bubble rise velocity. Few of the mostimportant observations by these authors include: (i) Theoscillatory nature of bubble for varying temperatureranges (0-40 °C). (ii) For the bubble sizes below 0.3 mm,the rise velocity increased with temperature, (iii) Forbubble sizes greater than 0.3 mm (∼0.67 mm), at lowertemperatures, the observations followed the abovetrend, (iv) above 25 °C, the rise velocity decreased withtemperature (Figure 13). For larger sizes, the risevelocity is shown to have a weak negative dependenceon temperature, whereas on the other hand, the oscil-lation amplitude was found to increase positively. Inreality, the situation is quite different in terms ofcontamination of liquid, nonisothermal operations, widebubble size distribution, coalescence-break-up phenom-enon, and so forth, and hence, it is required to quanti-
tatively understand the dynamics of bubble motion insuch systems.
Since liquid density (FL) is inversely proportional tothe temperature, the possible effect of liquid density onbubble rise velocity can be realized from the aboveexplanation. Higher liquid density results in increasedbuoyant forces which lead to faster bubble rise. How-ever, this effect is not significant when compared toother physical properties of the liquid that that affectthe bubble rise velocity.
3.1.4. Effect of External Pressure on Rise Veloc-ity. Although most of the laboratory scale investigationson bubble dynamics are carried out at ambient condi-tions, the operating conditions in industrial systems aresignificantly different from the ambient ones. Forexample, at elevated pressure, the hydrodynamic condi-tions in the column type of reactors are very muchdifferent as the increased outside pressure affects thebubble dynamics, the mode of bubble formation andhence the overall flow pattern. To be specific, as a resultof rise in the outside pressure, bubbles are formed onlyafter a sufficient minimum gas velocity is reached(which is higher than at ambient conditions), while theirsizes are relatively smaller. This is mainly because, towithstand the outside pressure, the bubbles prefer tohave as smaller size as possible and hence their risevelocities are lesser than that at the ambient condition.Due to the elevated external pressure, pressure headon bubble is very high resulting in a slow rise. Anotherreason behind the existence of smaller bubbles is theincreased bubble break-up frequency with increase insize. Letzel179 has systematically analyzed the effect ofgas density on the rise velocity of large bubbles. In asimilar attempt, under the assumption of homogeneousnature of the suspension, Luo et al.180 have studied theeffect of external pressure on bubble rise velocity in agas-liquid-solid fluidized bed (Newtonian viscous sys-tem). It can be clearly seen from Figure 14 that, withincrease in pressure, for the same solid loading, the risevelocity had a manifold decrease at two temperatureconditions. Further, the increased pressure was found
Figure 13. Experimental data on the effect of temperaturevariation on bubble rise velocity. [ dh ) 0.375 mm, 0 dh ) 0.677mm, ∆ dh ) 1.003 mm, ] dh ) 2.0 mm, 9 dh ) 3.462 mm.
ν ) 0.01Fw
e[-24.71 +4209/T + 0.04527T - 3.37 × 10-5T2] (75)
AL
to have low effect for higher temperatures, mainlybecause, the difference between the vapor pressure ofliquid and external pressure goes down.
3.1.5. Effect of Initial Bubble Detachment Con-dition. In Figure 11A, the plot of bubble terminal risevelocity vs size is shown for an inviscid liquid. Asmentioned in published literature (Clift et al.,148 Fanand Tsuchiya181), the upper bound of the VT data isconsidered to correspond to the bubbles in pure liquidswhile in the presence of trace amount of surfactants,drastic increase in the drag, causes the reduction invelocity leading to a scatter in the data below the curvefor pure liquid. In a recent analysis, Tomiyama etal.182,183 and Wu and Gharib184 have reported that themain reason behind this scatter is not the extent ofcontamination but the difference in the way of bubblerelease from the orifice i.e., the initial condition beforedetachment. According to their analysis, for low initialshape deformation, bubble rise occurs at lower velocitywith its trajectory being either a rectilinear or zigzagpath. For large initial shape deformation terminalvelocity is high and produces wide scatter dependingon the different possible release conditions and defor-mations. This observation is depicted in Figure 11B,which is obtained from the measurements at 170 mmabove the nozzle tip. Recently Tomiyama185 has shownthat the Figure 11B is comprised of data points thatcan be bounded with three different drag curves. Basi-cally, for bubble having constant equivalent diameterincrease in the aspect ratio is shown to reduce VT. Also,for constant aspect ratio, VT was found to decrease withincreasing equivalent bubble diameter. It is howeverrequired to check the validity of such observation withthe bubble shape map (Figure 15) proposed by Grace186
on the basis of dimensionless numbers.3.2. Formulations for Rise Velocity Correlations.
In view of the important contribution of rise velocity indeciding the overall hydrodynamics of the system, it isrequired to know the bubble rise velocity in a system ofknown physicochemical properties and also possibly thebubble size. Since the rise velocity shows varying naturewith the system properties, many investigators havedeveloped empirical, semiempirical and theoretical for-
mulation for the rise velocity, which many times remainsubjective to a narrow range of governing parameters.We begin with the fundamental approach based onsimple force balance and then move to empirical andsemiempirical ones.
3.2.1. Force Balance Approach. One of the earliestinvestigations in the bubble rise have shown that forsmall spherical bubbles, the drag force can be calculatedand combined with the buoyancy force to yield Stokes’law as
under the assumption that the bubbles do not have anyinternal circulation and no slip exists at the boundary.Peebles and Garber187 have done extensive experimentalmeasurements of the drag coefficient analysis of bubblesin liquids with wide range of physical properties. Theircorrelation is given in Table 4 and the comparison ofthe predictions with experiments is shown in Figure 16.
The authors have obtained different correlations fordifferent size regions, which bring out the need fordeveloping a generalized correlation, which would workindependent of the region. Davies and Taylor198 havecorrelated the bubble rise velocity and the radius ofcurvature in an innovative way for the case of sphericalcap bubbles (6.9 mm < Rb < 36 mm). They assumedthat the pressure over the front of the cap shaped bubbleis the same as on a completely spherical bubble and therise velocity was independent of the liquid properties.They proved that the rise velocity VT ≈ (RC)0.5.
3.2.2. Approach based on Dimensional Analysis.The main forces, which govern the phenomenon of
Figure 14. Experimental data on the effect of external pressureon bubble rise velocity (Luo et al.180) for different liquid viscosities.
Figure 15. Bubble shape map. (reproduced with permission fromFan and Tsuchiya181).
VSt )2g(FL - FG)Rb
2
9µ(76)
AM
Table 4. Correlations for Bubble Rise Velocity in Newtonian and Non-Newtonian Liquids
investigator systemphysicochemical
properties correlation remarks
Stokes188 Pure gases- clean liquidsVT )
2gFrB2
9µ
Simple in formulation, however applicable onlyfor the cases, where slip at interface as wellas internal circulation in bubble is negligible.
Haberman &Morton189
Air-cold, hot water, mineral oil,varsol, turpentine, methanol,corn syrup-in water, glycerin-water, ethanol-water, olive oil
782 < F < 1480 kg/m3
0.02 < σ < 0.72 N/m0.52 < µ < 18000 cP
VT ) 1.02xgrBAlthough simple in formulation, not applicable
to most of the systems, except where effect ofphysical properties is negligible.
Mandelson190 Published data of Habbermanand Morton (1953)
782 < F < 1480 kg/m3
0.02 < σ < 0.72 N/m0.52 < µ < 18000 cP
VT ) x σrBFL
+ grB
Suits well for medium sized bubbles (>2 mm) inmost of the pure liquids. Modification is requiredfor including very small bubbles andcontaminated liquids.
Peebles andGarber187
Measured data for:Air-water, acetic acid, ethylether, oil, ethyl acetate-oil,aniline, acetone, methanol,butanol, benzene, toluene,nitro benzene, pyridine,2-propanol
626 < F < 1071 kg/m3
0.016 < σ < 0.72 N/m0.22 < µ < 31 cP
VT1 ) 2gFLrB2/9µ, Re < 2 Applicable for certain ranges of Re and Mo.
VT2 )0.33g0.76FL
0.52rB1.28
10µ0.52, 2 < Re < 4.02Mo-0.214
VT3 ) 1.35x σgrBFL
, 4.02Mo-0.214 < Re < 3.10Mo-0.25
VT4 ) 1.18(σg2
FL), 3.10Mo-0.25 < Re
Harmathy191V∞ ) 1.53[g∆Fσ
FL2 ]1/4
, moderately distorted ellipsoids Applicable for bubbles with Re > 500.
V∞ ) λxg∆FDFL
for cylindrical slugs
Astarita andApuzzo140
Air-viscous liquids(Separan AP30, J100 in water) Hadamard region VT )
126VB0.66
ν
Levich region VT )42VB
0.66
ν
Known correlations are categorized for differentregions in terms of shapes and bubble sizes.Since it is similar to Stoke’s formulation,applicability is limited.
Taylor region VT ) 23(gR′)0.5
R′ - Radius of curvatureAngelino192 Air-Viscous liquids VT ) K(4rB
3)m/3where K ) 25/(1 + 0.33Mo0.29);and m ) 0.167(1+0.34Mo0.24)
Equation suits well for spherical cap shaped bubblesin various liquids204. Needs modification in termsof verification against complete force balance.
Dumitrescu193 Air-PVA V∞ ) 0.35x2grbApplicable for slugs with clean interface and potential
flow of liquid in its vicinity.Lehrer194 Published data of
Habberman andMorton (1953)
782 < F < 1480 kg/m3
0.02< σ < 0.72 N/m0.52 < µ < 18000 cP
VT ) x 3σFdB
+dBg∆F
2F
Modification of waveanalogy. Fits well forexperimental in a better way thanwave theory.
Clift et al.148 Published data Air-Newtonianliquids over wide range ofproperties
782 < F < 1410 kg/m3
0.015 < σ < 0.72 N/m0.72 < µ < 29000 cP
VT ) ( µFLG)Mo-0.149(J - 0.857) Equation is suitable for pureliquids over
wide range of properties. Deviation isvisible for spherical cap shaped bubbles.J ) 0.94 H0.747 for 2 < H < 59.3
J ) 3.42 H0.441 for H>59.3
H ) - 43EoMo-0.149( µ
µw)-0.14
Abou-elHassan150
Published data Air-Newtonianliquids over wide range ofproperties
835 < F < 1039 kg/m3
0.015 < σ < 0.072 N/m0.233 < µ <59 cP
V ) 0.75(log F)2
V - Velocity numberF - Flow number
Excellent match with experimental datain terms of Velocity and Flow number.
AN
bubble motion in a pool of liquid, are buoyancy, gravity,viscosity, drag, interfacial tension, and the memory force(arising through the gradient of vorticity on the surfaceof a bubble). Since few of the forces are formulated usingthe simple physicochemical parameters, the simplestapproach for correlating the bubble size and rise velocityis through the use of density of gas, density of liquid,viscosity of liquid, surface tension, gravity, equivalentbubble diameter, and obtain a relation between dragcoefficient and Reynolds number. Several authors haveused this approach of using the above parameters fordimensional analysis of the phenomenon and expressthe dependence of bubble rise on various other param-eters. Among one of the initial attempts, Abou-el-hassan149 has used the force ratios of buoyancy toinertial, viscosity and interfacial tension respectivelyand expressed as
He obtained two dimensionless numbers, flow number(F) and velocity number (V)
The range of physicochemical parameters considered byhim is given in Table 4. The correlation proposed byhim, which fits very well for the data in the range ofstudies is
Further, he has considered various flow regimes byvarying the index of flow regime and modified theearlier correlation as
where n′ is the index of flow regime. Further, for non-Newtonian fluids, he replaced the viscosity by a correc-tion term based on the consistency (k) and pseudoplas-ticity index (n) as
which results into
These correlations are found to be independent of flowregimes and the bubble shapes and are applicable forthe range of Reynolds numbers from Stokes region toNewton’s law region. Recently, this methodology wasadapted by Rodrigue157 to develop a general correlationfor the rise velocity of single gas bubbles in Newtonianliquids.
Recently, Tomiyama185 and Rodrigue157 have proposedexcellent correlations based on the dimensional analysis.For the first, the approach is mainly oriented to capturethe bubble shape variation in the Newtonian liquids andit gives excellent way of understanding the trends inrise velocity for different liquid properties and fordifferent bubble shapes over a wide range of possibledistortions. The correlation by Rodrigue157 extends theapproach of Abou-El-Hassan150 and derives the bubblerise velocity in terms of velocity number and flowN
icke
ns
etal
.195
Pu
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data
Air
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Dat
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Pee
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Gar
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9)62
6<
F<
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Vsp
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ir-
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3
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imat
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[g(∆F)d5/3/(u(Fµσ)1/3)] (77)
F ) [gd8/3(∆F)F2/3/µ4/3σ1/3] (78)
V ) [VTd2/3F2/3/µ1/3σ1/3] (79)
V ) 0.75(log F)2 (80)
VT ∝ (g∆F)n′µ(1-4n′)/3σ(1-n′)/3F(2n′-2)/3d(8n′-2)/3 (81)
µ ) kF(n)(u/d)n-1 (82)
VT ∝ (g∆F)1/n[kF(n)]1/nd(n+1)/n (83)
AO
AP
number. According to his correlation
where
3.2.3. Approach through Wave Analogy. In aninnovative attempt, Mendelson190 proposed wave theory
for prediction of bubble rise velocity. In his analogy, heassumed bubbles interfacial disturbances for whichtheir dynamics are similar to those of waves on an idealfluid and can be correlated in terms of the fluidproperties and bubble size (r) for the terminal risevelocity of a bubble as
The results have shown very good agreement with theexperiments except for very small bubbles (<0.5 mm),
Figure 16. Comparison of predictions by various correlations with published experimental data. (Abbreviations for the differentauthors: Abou-el-Hassan149 [VA], Davies and Taylor198 [VDT], Haberman & Morton207 [VHM], Mandelson190 [VM], Nickens and Yannitell195
[VNS], Nguyen155 [VN], Rodrigue153 [VR], Jamialahmadi et al.196 [VJ]). Experimental data is shown in symbols and lines indicate thepredictions. For the data by Raymonds & Rosent209 (N) for Glycerine solution of different viscosities (cP), the numbers in the bracket ( )indicate the predictions while symbols indicate the experimentally measured rise velocities in the respective viscous solutions. (1) +1250, (2) 2 1245, (3) O 1230, (4) ] 1222, (5) 0 1205, (6) ∆ 1190, (7) × 1172, (8) b 1150.
V ) F12[(1 + 1.31 × 10-5Mo11/20F73/33)21/176
(1 + 0.02F10/11)10/11 ]V ) VT(F2db
2
σµ )1/3
and F ) g[F5db8
σµ4 ]1/3
(84) VT ) x σrF
+ gr (85)
AQ
where the surfaces are rigid and as no interfacialdisturbances are present no waves are formed. Figure16 shows the comparison of the predictions from waveanalogy for different experimental data. The theory wasmodified by Lahrer194 who argued that during rise,potential energy associated with bubble is converted intokinetic energy, followed by its dissipation in the wake,which on incorporation in the wave analogy resulted in
Mendelson’s wave theory was extended by Maneri199 fortwo special cases of rise of a planar bubble and the risein a rectangular duct. The final equations for the bubblerise velocity follow the wave analogy but their ap-plicability is restricted toward the limited cases, whichare not common in practice. The comparison of thevarious experimental data sets over a wide range ofliquid properties is shown in Figure 16.
Jamialahmadi et al.196 has used the wave analogy andmodified the correlation by Hadamard,144 i.e., no slipat the boundary for the rise velocity of bubbles for a widesize and shape range. The correlation is given in Table4 and it is worth mentioning that for the case of pureliquids, correlation by Jamialahmadi et al.196 yieldsexcellent results when compared with the experimentaldata for various pure liquids (Figure 16).
In addition to the few known approaches of under-standing the variation in terminal rise velocity withbubble size in various liquid that are discussed earlier,one of the ways is through the variation in the dragcoefficient (CD) with Re. This particular approach andthe corresponding developments are discussed sepa-rately.
3.3. Bubble Rise Velocity in Non-NewtonianLiquids. Rheological behavior of non-Newtonian liquidsis complex and the motion of bubbles in these liquidshas different characteristics than observed in Newto-nian liquids. As mentioned earlier, the jump in risevelocity and early transition in the shapes are few ofthe peculiar characteristics of bubbles in a special classof non-Newtonian liquids. Depending upon the type ofliquid, the rheological properties change and hence thecritical bubble size. Chhabra200 has reviewed the pub-lished literature in detail, whereas Dekee et al.176 haslisted the transition bubble volumes for various viscousNewtonian and non-Newtonian liquids. However, it isrequired to compare the different methods of investiga-tion and results from different theoretical studies. Mostof these studies are carried out to develop a correlationbetween the CD and Re. The complexity in the estima-tion of CD as well as Re is due to the time dependentviscosity as well as its effect on the bubble shape. Non-Newtonian liquids mainly considered in most of thestudies can be categorized in two classes, shear thinningliquids and structural viscous liquids. In shear thinningliquids, the apparent viscosity (i.e., ratio of shear stressby strain rate) decreases with increasing shear rate.Prediction of the thinning characteristics is solely afunction of the time dependent shear rate variation andcan be written as a power-law model in terms of theconsistency index (m) and flow index (n), where for thesekind of liquids, n < 1. Typically, the shear stress (τ) canbe given a function of shear rate (γ) as τ ) mγn.Depending upon the nature of flow and the operatingconditions, the formulation for viscosity takes differentforms. Before starting with various models, for the case
of viscous Newtonian and power-law non-Newtonianliquids it is important to note that, in these cases, themodified viscosity i.e., µ ) m(U/D)n-1 is used instead ofmolecular viscosity. With the earlier explained reasonsbehind the jump in rise velocity in Section 3.2, here wehave given stress on the attempts that are made topossibly explain this phenomenon theoretically andsome of the experimental observations by variousinvestigators.
Recently, Rodrigue et al.173 has developed criteria forjump conditions based on the drag analysis. Accordingto their analysis, the drag coefficient in an incompress-ible fluid can be given as
where Re is for the non-Newtonian fluids. According totheir analysis, for meeting the jump condition, a bubblehas to follow a condition, R ) CaDe/Ma ∼ 1, where Ca) η0VT/σ is the capillary number, De ) N(γ)/2τ is theDeborah number (with N ) aγb represents the powerlaw relation for primary normal stresses) and Ma ) ∆σ/(τR) is the Marangoni number with ∆σ being thedifference in the surface tension of the pure solvent, andthe solution and R is the equivalent bubble radius. Here,R represents the balance between different forces actingon bubble and it clearly indicates that it is required toreach a particular condition where all the parameterslead to R ) 1 causes a jump in the bubble velocity. Thus,at the rear stagnation point, the local stresses are largeand cause high curvature and local deformation. Gen-eration of high normal stresses in this region cause thecontaminant molecules to get stretched and induce achange in the fluid properties. Thus, the action of thenormal forces in the vicinity of the bubble result inremoving the contaminant molecules from the bubblesurface causing a sudden change in the interfacialconditions, which finally may lead to the jump in risevelocity. Further, since on the rear stagnation point theunidirectional flow is continuous it may lead to removalof contaminant molecules thus helping the jump condi-tion. However, this process need not be restricted forcertain bubble size but needs to be verified for smallerbubbles for longer time, which may show similarfeatures depending upon the extent of surface contami-nation.200
We begin with an industrial process where the natureof liquid has utmost importance in controlling the flowregime. In fermentation processes, the rheological be-havior of the liquid changes from Newtonian in thebeginning to non-Newtonian at the end of the reactions.Although very less attention has been paid in thisinteresting phenomenon, recent studies by Margaritiset al.171 on the effect of such a transition on the risevelocity of single bubble are encouraging in terms ofexperimental observations. The important feature beingthe dynamics associated with the rise velocity, it isindeed required to follow the dynamics of viscosity,which would decide the nature of CD vs Re curve. Sincethe flow index of the liquids used here is in the rangeof 0.63 and 0.68, these authors have followed thesuggestion of estimating the drag coefficient usingHadamard-Rybczynsky202 correlation by Miyahara andYamanaka.203 Their observations in CMC, xanthangum, Na-alginate, dextran and pullulan showed that forconstant shear rate, the shear stress decreases with
VT ) ( 3σ2Fr
+ rg∆FF )0.5
(86)
CD ) 16Re[2n-13(n-1)/21 + 7n - 5n2
n(n + 2) ] (87)
AR
concentration and can be correlated by power law.Further, they have also reported that for same concen-tration of the above six liquids, the mean rise velocitywas a strong function of the K and n, and for the samefluid, the drag coefficient plot did not show muchvariation for different concentration values. For thesolutions of pullulan they observed different nature ofthe drag curve for bubbles of different shapes. Unlikeother investigators, these studies did not show any jumpin the velocity with change in the regime of bubblemotion. The generalized correlation proposed by theseauthors clearly shows deviation from the experimentaldata where the drag coefficient is proposed to remainconstant after Re > 135. Further, the conclusion basedon the experimental drag curve that most of the bubbleswould be in Stokes region in Newtonian liquids is notacceptable as the spread in the data clearly shows thediscrepancy and demands modification in the correlationin terms of the dynamic nature of viscosity. Rodrigue154
has followed the similar approach for the prediction ofdrag coefficient in various power law non-Newtonianliquids as in Newtonian liquids. Since the approachadopted here is the same as discussed earlier, thedimensional analysis has suppressed the actual infor-mation to a great extent and error in the predictionsmakes the model unacceptable. However, the modifica-tion in the formulation based on the variable Mortonnumber has shown that the following correlation forvelocity and bubble diameter at the minimum in theCD-Re curve hold very well.
The above discussion evidently shows a requirement forthe development of correlation based on the dynamicsof bubble shapes and their rise velocity.
Few other attempts include Barnett et al.,204 Haqueet al.,205 Margaritis et al.,171 Miyahara and Yamanaka203
who studied the rise velocity in highly viscous Newto-nian and non-Newtonian liquids. Their analysis mainlyfocuses on the shape variation in bubble at variousranges of physicochemical properties and it indicatesthat irrespective of type of liquid, the rise velocityincreases with bubble size, however, it brings out theneed to incorporate the extent of deformation andassociated oscillation frequency in the model. Dekee etal.176 have shown that for the viscous Newtonian liquids,the rise velocity is a weak function of the bubble volume,while for non-Newtonian liquids, it has a linear depen-dence but the velocities in the earlier case are muchhigher than the later. Their observations are correlatedand given in Table 4. Their experimental observationsclearly show that with addition of a shear thinningliquid (1% PAA) in a viscous Newtonian solution (glyc-erol-water), the rise velocity decreases drastically.Figure 17A shows the comparison of predictions fromthe correlations by Miyahara and Takahashi206 for pureliquids, where different correlation are used for differentranges of Re.
3.4. Comparison of Correlation for Bubble RiseVelocity in Pure and Contaminated Liquids.Stokes188 derived the simplest and the earliest knowncorrelation resulting from force balance on very smallbubbles in a quiescent liquid (eq 76), however, theapplicability of this equation is indeed limited for thevery small sized bubbles mainly in contaminated liquid,
where the internal circulation is almost minimal. Theabove statement is found true for the polymeric liquids(Newtonian) where for very small sized bubbles theviscous forces are strong enough to resist any surfacemotion as well as internal circulation and result in arigid surface. The comparison of the experimental datafor very small bubbles in a pure liquid system is foundnever to match with the predictions by the Stokesformulation as the rise velocities are always higher,while for the contaminated liquids with low viscosity,the deviation between predictions and experimentaldata is noticeable which diminishes with increase in theliquid viscosity. It shows an inherent inability to showa good match with most of the experimental data, exceptfor the viscous non-Newtonian liquids below criticalbubble size. In the next attempt in this directionHabberman and Morton,207 (will be referred here-onward as HM) proposed rise velocity only as a functionof bubble size under the assumption that the velocityis independent of the physical properties of liquid.Although the resulting correlation is simple, its limitedapplicability may be only for the data by few investiga-tors17,51,52,58 who have observed negligible effect ofviscosity on size and Davidson and Schuler4 who proposeno effect of surface tension specifically for the case ofconstant gas flow rate system. A poor agreement wasobserved on comparison of the predictions from thecorrelation by HM with the experimental data forseveral pure liquids (Figure 16). Density differencebetween the two phases being the most importantparameter for rise of a bubble (which is not taken intoaccount), the validity of this correlation is suspicious.Its applicability is restricted for very small bubbles incontaminated liquid, where (i) irrespective of viscositybubble surface is rigid, (ii) Eo is also very small, and(iii) it does not depend on viscosity. As a result, althoughMo varies over a wide range bubble rise velocity is
VTm ) 1.37(gσF )1/4
and CDm ) (39.9mFg)1/(n+1)
(VTm)n/(n+1)
(88)
Figure 17. Typical drag curves for bubble motion in liquid. (CDvs. Re for different liquids) (reproduced with permission fromPeelis and Garber187).
AS
simply a function of its size. However, a large numberof other studies have shown a noticeable effect ofviscosity and surface forces on size. We prefer tomention that, the discussion in this subsection is basedon the results available in the literature and comparisonwe have shown in Figure 16.
For pure Newtonian liquids, the correlation by Davisand Taylor198 is applicable only for large size sphericalcap bubbles which for the available experimental dataof Devenport et al.,208 shows a very good match in low-density liquids. Correlation by Clift et al.148 gives goodpredictions for the pure liquids as it even takes intoaccount the shape variation in a bubble through the useof dimensionless numbers (viz. Ta), however, the methodof estimation includes numerous iterations to reach thefinal value of velocity for each size. It is generallypreferred to follow simpler formulation for designpurpose. Formulation by Karamanev197 is an extensionto that by Clift et al.148 which is generally applicableunder the assumption of rigid surface irrespective of itsshape and this procedure also requires tedious iterativecalculations. The interesting part is that this formula-tion is bubble shape dependent and hence can be usedfor the correct estimation of the drag coefficient over awide range of Re, however, it is required to quantify thedifference in the drag coefficient arising out of a rigidand a free surface. In a recent attempt Rodrigue153 hascorrelated most of the published data on pure Newto-nian liquids. It is to be noted from the method ofanalysis that the dimensional analysis loses the sensi-tivity for few of the parameters whereby the effect ofphysical properties cannot be understood from thevariation in velocity with size. The comparison of thepredictions by Rodrigue’s153 formulation with the ex-perimental data of Reymonds et al.209 for bubble rise inaqueous glycerol solutions shows excellent match forliquids with relatively low viscosity, while for higherviscosities, predictions deviate by about 15%.
Correlation by Nguyen155 for contaminated liquidsstands well for the Newtonian liquids. The correlationsuits very well for a range of bubble sizes and followsthe curve for contaminated liquids given in Clift et al.148
The available data for such studies is from the flotationof minerals and the correlation is found to give excellentmatch for such data. However, the lack of data incontaminated liquids of different properties is thebottleneck for the thorough validation of this correlationover a range of parameters.
Wave analogy based correlation by Mendelson190 isfound to be suitable for various pure liquids only afterthe region of very small bubbles where the rise velocityincreases with size and reaches maximum followed bya decrease in the velocity. The correlation is not suitablefor the bubbles in viscous systems viz. glycerol solution,corn syrups, except for the glycerin solutions of variousconcentrations and few of the oils. However, in lowviscosity oils and other organic liquids the comparisonis reasonable but not completely acceptable. Indeed, theexperimental data would also get corrupted to someextent due to the possible small trace of impurity/contamination present in the liquid or even in the gasphase. But the other modified forms of the wave analogyby various investigators have shown some success forpure liquids. The predictions from correlation by Ja-mailamhadi196 show an excellent agreement with theexperimental data for liquids over a wide range ofphysical properties (i) low viscosity organic liquids (i.e.
Aniline, nitrobenzene, etc.), (ii) low-density viscous oils(olive oil, caster oil, mineral oil, hydrocarbon oil, cottonseed oil, turpentine, etc.), (iii) viscous Newtonian liquids(aqueous solutions of glycerin and glycerol, varsol, etc.).For the case of viscous solutions of Glycerol in water,the correlation is found to agree with the experimentaldata discretely indicating that, the correlation shouldfollow a smooth variation in bubble shape provided thesurface tension should also vary with viscosity. In mostof the cases, the surface of bubbles is rigid approachingthe state of contaminated Newtonian liquids with higherviscosity either due to dominant surface forces or viscousforces. The correlation also gives excellent match withthe data for pure water in cold as well as ambientconditions.
Trends obtained from the geometry based correlationby Davis and Taylor198 are restricted for spherical capsin various systems, but the deviation of the predictionsshows that even for spherical cap shapes, since theshape depends on the physical properties of liquid, itwould be unsafe to use a geometry based correlation.Similar analysis also applies for Habberman and Mor-ton’s correlation.210 Correlation by Abou-al-hassan149
successfully correlates the flow number (F) and velocitynumber (V) estimated for a range of experimental data,however, in the form of dimensionless numbers, thedenominator of the dimensionless numbers is very highas compared to the numerator and thus sensitivity tothe variation with respect to various parameters is lost.This correlation may be used to check the validity ofthe experimental data by checking the nature of varia-tion in terms of V and F for Newtonian liquids. Here,we can conclude that correlation by Jamialahmadi etal.196 can be used for various pure liquids however thecorrelation needs to be modified for its applicability evenfor contaminated liquids, for which one by Nguyen155
can be used.The results from the correlation, when compared with
the experiments, clearly show that the region of risevelocity being independent of the size does not show anymatch and hence it is required to improve-upon thecorrelation for that particular region.
3.5. Drag Coefficient for Bubbles in Liquids.Since it is not always possible to obtain the rise velocitydirectly, its variants viz. drag coefficient (CD), Re areestimated and a correlation is developed for understand-ing the relationship between size and velocity. Mostcommonly, the relation between bubble rise velocity andthe size is seen through the variation in CD with Re,provided, for nonspherical bubbles the effect of physi-cochemical properties of gas-liquid system is incorpo-rated into the correlation. The most important effect ofviscosity on the displacement of a flow past body is toproduce a drag force which slows down its relativemotion in the surrounding fluid. Conceptually, dragcoefficient is analogous to Fanning friction factor211 butthe formulation is complex as it is a result of both, theform drag and skin drag. In the past decade, Karamanevand Nikolov212 experimentally showed that the trajec-tory of free rising solid particles in Newtonian fluids isdifferent from the free falling particles, until then it wasassumed that the free rise of buoyant particles shouldobey the laws of free settling of heavy particles. Theyalso reported that the terminal rise velocity of free risingparticles is much lesser than the falling ones. For thedetailed analysis of this phenomenon, reader may referto Karamanev,213 where it is shown that the free rise
AT
characteristics of light rigid particles are similar to thoseof bubbles. Before discussing more onto the drag coef-ficient of bubbles, here we briefly discuss the subject ofthe drag force.
Production of drag force that tends to slow the relativemotion of a body is one of the most important effects ofviscosity on the displacement of a body in a fluid.Although the theory is well characterized for the lowRe regime, a complete understanding of bluffed bodiesmoving at high Re needs more careful analysis. For thecontaminated bubble surfaces, the low Re theory is wellapplicable. This is mainly because of the zero shearstress for the tangential component of the liquid velocitywhich allows the liquid to slip along the surface of thebubble leading to unseparated flow in its wake andchanges the mechanism of the vorticity production onthe surface of the bubble.214 The drag force (FD) deter-mined from a balance between the rate of work doneby the drag force and the viscous dissipation within thefluid interior, which leads to its formulation as
where CD is the proportionality constant and for low Re,it can be given as
In reality, this correlation215,159 changes depending onthe range of Re and also the physicochemical propertiesof the liquid. Importantly, Kang and Leal216 have shownthat the pressure variation over a bubble surfaceaccounts for one-third of the total drag in eq 90, whilethe rest of it is provided by the normal viscous stress.The formulation in eq 90 is further corrected by Moore217
by taking into account the boundary layer thickness atdifferent surface regions of a bubble. Several numericalattempts in this direction can be seen in the litera-ture.218-226 It is known that CD decreases linearly withincrease in Re. The most general formulation for CD canbe given as
i.e., for the case of passage of gas bubble into liquid,value of the term in bracket is negligible and correlationtends to CD ) 16/Re, whereas for the case of solid-liquidsystem, CD ≈ 24/Re. thus the value of CD at a particularRe are bounded in this range. However, the experimen-tally obtained CD values have shown different trendsin variation with Re, which has encouraged manyinvestigators to propose various formulation for CD.Here, we have reviewed many such attempts and havecompared their results to recommend the most suitableformulation applicable over a wide range of parameters.For the detailed discussion of the drag force in a bubblyflow, reader may refer to Magnaudet and Eams143 andMiyahara and Takahashi.206
The values of CD depend on the system’s physico-chemical properties and the bubble dimensions. Beforediscussing the dependence of the drag coefficient ondifferent parameters, here we summarize three distinctregions that show a different dependence on the bubblerise velocity. (1) A first region (viscosity-dominated), forvery low Reynolds number, in which bubbles are spheri-
cal, viscosity forces dominate the terminal motion andterminal velocity increases with diameter. (2) An inter-mediate region (surface-tension-dominated), in whichsurface tension and inertia forces determine the termi-nal velocity. Bubbles are no more spherical in this regionand terminal velocity may increase or remain constantor decrease with equivalent diameter. According to Cliftet al.,148 at least for air-water systems this regime holdsfor about 0.25 < Eo < 40, however the boundaries(especially the lower one) are somewhat arbitrary. (3)A last region (inertia-dominated), for high Eo, in whichthe bubbles are spherical-cap or bullet-shaped and themotion is dominated by the inertia forces. Velocityincreases with equivalent diameter in this regime.
In continuation to the above explanation, for verysmall sized bubbles, (Re < 2) Stokes law is obeyed,whereas for Re > 2, departure from Stokes law issignificant and correlation for the drag coefficient showsrelatively weak dependence on Re. However, withdeparture from Stoke’s regime bubble shape also changesand also with transition from the clean to contaminatedliquids, the extent of internal circulation also changeswhich requires the surface properties also to be incor-porated into the correlation.227 The nature of bubblerising in pure liquids and the contaminated liquids arecompletely different and the same is with its risevelocity. For example, air bubble behaves like solidspheres only until a certain critical Re, beyond which,it changes its shape and a slip will be associated withits motion. Rosenberg227 has made an attempt toidentify critical Re (Recrit) and found that Recrit for thedistilled and contaminated water are 40 and 130,respectively. In the case of the rigid spherical bubbles,internal circulation is absent and this condition can evenbe achieved if the liquid is contaminated with surfac-tants resulting into constant drag coefficient. Manyinvestigators have formulated several correlation for theprediction of drag coefficient in Newtonian148,187,206,223
and non-Newtonian151,153,173,228 liquids which are sum-marized in Table 5 and few are discussed here.
For the case of contaminated Newtonian liquids,Karamanev213 has formulated the drag coefficient as
and CD ) 0.95 for Re > 135, where the value of Ret fora spherical particle in power-law fluid can be given asRet ) dp
n V t2-nF/K, which for nonspherical axisymmet-
ric particles becomes Ret ) dhn V t
2-nF/K, where dh is thediameter of the horizontal projection of bubble.
Kamaranev197 has given the rise velocity of singlebubble in quiescent liquids in terms of the drag curveunder the assumption that the difference between thedrag coefficients given in the literature and that of thesolid particles is only due to the use of equivalentdiameters as measure of the bubble size. Further, fortaking into account the departure from spherical shapes,the surface area was determined using the projecteddiameter of the horizontal plane and the modifiedequation for drag coefficient appears as
The range of parameters used for checking the validity
FD )CDFVT
2A2g
(89)
CD ) 48/Re (90)
CD ) 24Re
2 + 3(µD/µC)
3 + 3(µD/µC)(91)
CD ) 24Ret
(1 + 0.173Ret0.657) + 0.413
1 + 16300Ret-1.09
for Re < 135 (92)
CD )4g∆FdB
3
3Fdh2 VT
2(93)
AU
Table 5. Correlations for Drag Coefficient (CD)
investigatorsystems under consideration/
bubble shapes correlation applicability
Levich215 Single cleanbubble in water
CD ) 48/Re Applicable for the systems, with free surface and energytransferred to liquid is directly dissipated into internal energy.
Peebles &Garber187
Single bubble risein water, 2-propanol,ethyl ether, pyridine,nitrobenzene, aniline,acetone, n-butanol,methanol, benzene,toluene, ethyl acetate,cotton seed oil,glacial acetic acid
CD ) 24Re-1 Bubbles with Re < 2 rise rectilinearly. Applicable for the regionwhere CD decreases with Re.
CD ) 18.7 Re-0.68 ) 8rg3V∞
2Re > 2, till the range where CD abruptly starts increasing with Re.
CD ) 0.0275( gµ4
Fσ3g3)Re4, Re > 4.02Mo-0.21 Region, where CD increases with Re and depends on the Mo to anoticeable extent.
CD ) 0.82( gµ4
Fσ3g3)ReCD is independent of Re, dominance of physicochemical properties
of liquid is more than Re.
Moore217 Clean sphericalbubbles CD ) 48
Re(1 - 2.21Re0.5) + O(Re-11/6) Bubble is associated with a thin boundary layer and wake.
The formulation takes into account the dissipation occurringin the above two regions.and
CD ) 5.66Mo0.2We-3/5f4/5
We ) 4x-4/3(x3 + x - 2)[x2 sec-1 x-(x2 - 1)1/2]2
(x2 - 1)3 x is the deformation factor or the aspect ratio in thecase of ellipsoidal bubbles.
f ) 0.4x4/3[(x2 - 1)0.5 - (2 - x2)sec-1x
x2 sec-1 x - (x2 - 1)0.5 ]Taylor and Acrivos223 Clean spherical
bubbles CD ) 16Re
+ 2 Limited applicability mainly in contaminated liquids withnegligible internal circulation.
Takahashi et al.228 1.19e-7 < Mo < 1.04e2 CD′ ) 1 + 16Re′
Applicable for Mo > 10-7, where CD decreases with Re.
CD′ ) 13Re′-0.7 Mo < 10-7, Re > 10, where CD increases with Re.Bhaga and Weber229 Bubbles in viscous fluids CD ) [2.42 + (16/R)0.9]1/0.9 Excellent match for various viscous liquids for Mo > 0.004.Miyahara &
Takahashi 206Mo > 10-7 CD′ ) 0.03(Re′)1.5Mo0.3 Applicable also for nonspherical bubbles in the range,
where Takahashi’s correlation cannot be applied.Prior knowledge of the shape factor is required.
Turton andLevenspiel230
Spherical bubbles incontaminated liquids. CD ) 24
Re(1 + 0.173Re0.657) + 0.413
1 + 16300Re-1.09Applicable for rigid bubbles with Re < 130.
Delnoij et al.231 Spherical bubbles in water. CD ) 24Re
(1 + 0.15Re0.687) for Re < 1000 Applicable for a bubble swarm with
CD ) 0.44 for Re > 1000Ford & Loth232 Ellipsoidal clean bubble CD ) e[2.1785ln(Re) - 15.054] Formulation is yielded from the measured terminal velocity for
successively increasing bubble diameters. Applicability islimited for the case where CD increases with Re.
Kendoush233 Bubble swarm inNewtonian liquid
CD ) 48/Re(1 - R)2,R: void fraction
For known bubble size, its velocity can be predicted atvarious gas void fractions in bubble swarm.
Rodrigue153 Viscous fluids CD ) 16Re[(12 + 32Θ + 1
2x1 + 128Θ)1/3
+ Applicable for a wide range of bubble sizes rising throughvarious viscous liquids.
(12 + 32Θ - 12x1 + 128Θ)1/3
+
0.036(1283 )1/9
Re8/9 Mo1/9]4/9
where Θ ) (0.018)3(23)1/3Re8/3Mo1/3
Margaritis etal. 171
Applicable fordeformed bubbles CD ) 16
Re(1 + 0.173Re0.657) + 0.413
1 + 16300Re-1.09Suitable for bubbles in viscous non-Newtonian liquids.
Modification is necessary to incorporate the region,specific to correlation by Miyahara and Takahashi81
AV
of this equation is given in Table 5. From his experi-mental studies and the observations based on the solidbubbles’ motion, it has been concluded that the internalrecirculation has no detectable effect of the dynamicsof the bubble rise in contaminated liquids. The semi-analytical correlation for the rise velocity from aboveobservations is given as
where ‘a’ and ‘b’ are empirical constants. Further, sincethe correlation takes into account the variation in aspectratios, it can be transformed into various other correla-tions specific to particular bubble shape. Tadaki number(Ta) can have various ranges pertaining to differentbubble shapes and hence the coefficients ‘a’ and ‘b’change accordingly. Since the Ta and CD both dependon the bubble rise velocity, the equation becomescomplex and solution procedure to converge at a par-ticular value is tedious.
Since in the above correlation, the rise velocity isdependent on the bubble drag coefficient and theprocedure for the estimation of drag coefficient isnumerically tedious. In view of this, Nguyen155 hasdeveloped simplified approach for the prediction ofterminal rise velocities in contaminated liquids throughthe use of Archimedes number (Ar ) D3δ2g/µ2) andLyashchenko number (Ly ) V∞
3δ/g/µ). The correlationfor CD follows the improvement by expressing it in termsof Mo and Re as
and used the same for expressing modified Ar and Ly.Since the bubble shape also reflects into the values ofRe, different correlations were developed for variousranges of Re, i.e., different shapes (Figure 15). For therange below Re < 130, the correlation is a modifiedversion of Stokes’ equation.
It even takes into account the deviation in terminal risevelocity with rigidity of bubble. For higher Re, it wasassumed that terminal rise velocity is independent ofRe and the correlation was modified in terms of onlyAr and Mo as
where the drag coefficient was considered to be constantin that range as taken by Karamanev197 for rising solidspheres. Since the formulation is specific for the con-taminated liquids, the assumption of a rigid interfacesupports the formulation. However, for the liquidscontaminated with electrolytes or polymeric dilute solu-tions, the nature of the drag curve is different, sincethe bubble shapes deviate from the spherical ones, andhence, checking the validity of this equation againstsuch experimental data is required. The distinct featureof this correlation is its ability to predict the transitionpoint in the Re (∼130), where the shape deviates fromspherical and also the equation can be reduced to
Mei
&K
lau
sner
144
Cle
ansp
her
ical
bubb
les
CD
)16 R
e{1+
[8 Re
+1 2(1
+3.
315R
e-0.
5 ) ]-1 }
Val
idfo
ral
lRe.
Sad
hal
and
Joh
nso
n23
4
(198
3)
Cle
anbu
bble
risi
ng
inco
nta
min
ated
liqu
idC
D)
CD
,rig
id-
CD
,mob
ile
2π(2
φ+
sinφ
-2s
in(2
φ)
-0.
33si
n(3
φ))
+C
D,m
obil
e
Use
fulf
oran
alyz
ing
the
effe
ctof
con
tam
inat
ion
onth
eri
seve
loci
tyof
sph
eric
alca
psh
aped
bubb
les.
wh
ere
φis
cap
angl
ede
pen
den
ton
the
tru
nca
tion
poin
t.T
omiy
ama18
5N
ewto
nia
nli
quid
s
CD
)
{8 3E
o(γ
E)2/
3 (1-
E2 )-
1E
o+
16E
4/3F
-2
for
E<
1
6γ-
1/3
for
E)
1
8 3γ1/
3 Eo
(γE
)2/3 (E
2-
1)-
1E
o-
16E
4/3
G-
2fo
rE
>1
App
lica
ble
for
pure
and
con
tam
inat
edbu
bble
sof
allp
ossi
ble
shap
esin
New
ton
ian
liqu
ids.
wh
ere
E)
b/a,
γis
the
dist
orti
onfa
ctor
F)
cos-
1E
-Ex 1
-E
2
1-
E2
and
G)
Ex E
2-
1-
tan
h-
1 (E-
1 xE
2-
1)
E2
-1
Rod
rigu
e157
New
ton
ian
liqu
ids
CD
)16 R
e(1
+0.
02Y
10/1
1 )10/1
1
X21
/176
Su
itab
lefo
ral
lNew
ton
ian
invi
scid
and
visc
ous
liqu
ids.
Itgi
ves
anex
cell
ent
fit
even
for
the
regi
on,w
her
eth
eef
fect
ofsu
rfac
ete
nsi
onis
take
nov
erby
the
visc
ous
forc
esth
atm
obil
ize
the
bubb
lesu
rfac
e.S
how
sex
cell
ent
com
pari
son
for
the
data
for
wat
er,d
iffe
ren
tco
nce
ntr
atio
ns
ofgl
ycer
inin
wat
er,e
than
ol,m
ercu
ry,a
nd
mol
ten
silv
er.
wh
ere
X)
(1+
1.31
×10
-5 M
o11/2
0 Y10
/11 )
Y)
[3 4CD
Re2
Mo ]8/
9
VT ) 40.3V1/6de
dh) 40.3V1/6aTab (94)
CD ) 8a2Dg3VT
2Mo0.46bRe2b (95)
VT ) VSt{1 + Ar/96(1 + 0.079Ar0.749)0.755}-1
(96)
VT ) x3gµδ {4a2Mo0.46b
3CD}1/(2-2b)
Ar(2b+1)/(2-2b) (97)
AW
specific expressions suitable for specific conditions, viz.for very small bubbles, rise velocity can be expressedas a simple function of bubble size and gravity. For avery critical analysis of the drag coefficient based onthe comparison of the predictions from various proposedcorrelation and the experimental values, we recommendthe reader to refer to Marco et al.235 Typical drag curvesfrom the literature are shown in Figure 17. The differentcorrelations for drag coefficient by several investigators(discussed here and also given in Table 5) when puttogether on the same plot could not clearly bring outtheir distinct features (due to several overlaps) andhence here we prefer to discuss the nature of differentdrag curves from the various correlations and their rageof validity.
An empirical drag law (given in Table 5) was proposedby Mei and Klausner225 is considered to yield accuratedrag coefficient at almost all Re. In a recent attempt,Tomiyama et al.,236 have developed a generalized cor-relation with four coefficients
and it gives an excellent match with the whole dragcurve. The range of the coefficients C1 ≈ 16-24, C2 ≈48 - ∞, C3 ) 0, C4 ) 2 were found suitable to cover thewhole range of Re for different gas-liquid systems. Wehave discussed the comparison of the predictions fromthis formulation for different cases in subsection 3.6.
On the basis of the approach by Abou-al-hasan,149
where V and F are used for characterizing the risevelocity, recently Rodrigue153,156 has developed excellentmethod in correlating bubble rise velocity with govern-ing parameters for various liquids. He has used dimen-sional analysis to modify the flow number and velocitynumber and proposed generalized correlation for bubblerise velocity in pure liquids. The results from the modelhave shown that it gives a very good match with theexperimental values for a wide range of data. The risevelocity is shown to follow three different relationshipsVT ∝ µ-1d2, VT ∝ µ0d0, VT ∝ g7/16d1/2 in laminar, turbulentand intermediate regimes, respectively, which clearlyshows its applicability for pure and clean liquids only.For the case of inviscid and viscous Newtonian liquids,the correlation by Rodrigue157 gives excellent compari-son the data experimental for water, different concen-trations of glycerin in water, ethanol, mercury andmolten silver. The single correlation is capable of evenfollowing the trends of the Re, where the drag coefficientgoes through a minimum and then increases to attaina plateau. However, in this particular region, thecorrelation seems to under predict the drag coefficientwith decreasing Mo, and the error is seen to increasewith lowering Mo value.
Non-Newtonian Liquids. In general, for the non-Newtonian liquids, the drag coefficient is given as
where Re ) FVT2-ndn/m, XE ) µd/m(V∞/d)n-1}, and X is
the drag correction factor. Although the values of ‘m’and ‘n’ are the properties of liquid and X for various
fluids can be expressed in different forms. Chhabra200
has listed known formulations for X. The value of X isunity for Newtonian liquids and increases for shearthinning liquids.
Since eq 92 over predicts the drag coefficient for non-Newtonian shear thinning liquids, in an extensiveexperimental analysis, Margaritis et al.171 have studiedthe drag coefficient variation for bubbles over a widerange of Re in different non-Newtonian polysaccharidesolutions and proposed the following correlation
and CD ) 0.95 for Re > 60, where Re is estimated usingthe equivalent bubble diameter. This correlation isshown to give very good match with the experimentaldata, except in the region of transition and the reasonis attributed to the shape changes. However, it isimportant to notice that the region where the drag curvechanges its direction, it goes though a minimum and itmainly depends on the local fluid phase physical prop-erties and hence the local turbulence. Thus, change inshape needs to be taken into account correctly to yield agood understanding of this region, which demands sub-stantial modification of above correlation. This require-ment is noticeable from Figure 12 of Margaritis et al.171
Recently, Rodrigue et al.173 has developed a correla-tion for the bubbles in non-Newtonian viscoelasticliquids. According to their analysis, the drag coefficientin an incompressible fluid can be given as and
depending upon the bubble surface characteristics. X(n)is a correction function dependent on the power-lawindex n. Thus, when the surface is rigid, X(n) can beused, which represents a correlation dependent on thepower law index (n), while for the case of mobile bubblesurface, Y(n) can be used, which characterizes the visco-elastic nature of the fluid. They have shown that in atypical plot of CD vs Re, for viscoelastic liquids, at lowRe, the variation of CD follows eq 101 while, on the jumpof velocity, the nature of surface changes and it followsthe properties of viscoelastic liquid. Thus, before thejump, the drag coefficient can be given using eq 101,while after the jump, it can be correctly given througheq 102. For the case of viscous liquids Rodrigue154 hasmade an attempt to take into account the bubble shapesby expressing CD in terms of Re and Mo. The correlationis given in Table 5. The predictions from it show excel-lent comparison for the experimental data by Peeblesand Garber187 and Moore237 over a wide range of Mo (1.93× 10-1 < Mo < 7.71 × 10-12) for various viscous fluids.
3.5.1. Comparison of the Correlation for DragCoefficient for Newtonian Liquids. From the vari-ous data and comparison available in the literature, wehave made the following observations. The selection ofthe correlation should be made on two bases: ap-plicability for a certain range of Re and variation in theshape which reflect upon the nature of drag curve.Correlations given by Miyahara and Takahashi206 are
CD ) max {min[C1
Re(1 + 0.15Re0.687),
C2
Re(1 - C3Re-0.5)], 8
3Eo
Eo + 2C4} (98)
CD ) 24Re
2 + 3XE
3 + 3XEX (99)
CD ) 24Re
(1 + 0.173Re0.657) + 0.4131 + 16300Re-1.09
for Re < 60 (100)
CD ) 16Re
X(n) (101)
CD ) 16Re
Y(n), where Y(n) )
[2n-13(n-1)/2(1 + 7n - 5n2)/n(n + 2)] (102)
AX
sufficiently good to track the experimental drag curvein different ranges however the change in shape (whichshould be known a priory) is incorporated by modifyingthe values on abscissa and ordinate through the ratioof equivalent diameter to major axis. The results of thevarious gas-liquid systems have shown that datamatches very well for the modified CD and Re. For thecase, where Mo is sufficiently high (high viscosity andlow surface tension), drag curve follows curve close tothat of a rigid bubble. Correlation by Turton andLevelspiel230 can be used with some modificationstoward changing viscosity (for the case of shear thinningliquids). These set of correlation are even found to giveexcellent match for the experimental data (Figure 17B)by Margaritis et al.,171 except for the region, where dragcoefficient increases with Re, where correlation byMiyahara and Takahashi206 shows excellent agreement.Correlation by Mei and Kalusner144 suits well for verysmall bubbles. The plots of comparison of the predictionswith the experimental ones show that the drag coef-ficient cannot be predicted simply from a single correla-tion and various correlations are applicable for differentshape regimes. Among all, the recent correlation byRodrigue157 seems to be excellent for the Newtonianliquids (the extent of variation in the CD in the region,where the shape changes show an effect should beverified with the correlation by Miyahara and Taka-hashi206) properties change, while for the non-Newto-nian liquids, (i) correlation by Margaritis et al.171 issuitable for shear thinning power law fluids, and (ii) forthe viscoelastic liquids, the correlation by Rodrigue etal.173 can be used provided, the correction functionshould be changed appropriately to account for thevariation in the surface properties of bubble.
3.6. Bubble Trajectories. As mentioned in theintroductory part, the rise velocity is associated withcertain dynamics. So far, we have discussed only themagnitude part of it, while directional component isequally important to understand this vectorial quantity.General aspects of the rise of single medium-sizedbubbles (between 1 and 10 mm diameter) produced froman underwater nozzle have been well documented.148 Adetailed observation is discussed in de Vries et al.,239
where they have experimentally investigated the motionof gas bubbles in highly purified water. In the beginningof bubble size range (<2 mm), smaller bubbles undergoaxisymmetric shape oscillations. At a certain height,depending on the conditions and hence the size (>2mm), bubbles establish an approximately ellipsoidalshape. Feng and Leal240 have discussed the variouspossible bubble trajectories in different shape regimes.A single bubble can follow a zigzag path (Re ≈ 600),coordinated with vortex shedding behind the bubble.However, under the same experimental conditions, thebubble might also follow a spiral trajectory withoutvortex shedding.241 The sideways motion of bubbles maycontribute significantly to the turbulence generated bybubbles in mixing situations. Furthermore, the overalllength of the trajectory influences the residence timeof the bubble in the liquid. This is very important forgas absorption. Generally, the dynamics of bubble riseare nonlinear and the extent of nonlinearity increaseswith bubble size. In a bubble swarm, the trajectoriesare strongly dominated by the surrounding liquid flowand associated dynamics is more complex. The observa-tions have shown that bubble trajectories have aprimary and secondary structure. The primary structure
probably represents the greatest kinetic energy, i.e., thesideways oscillation within the spiral part of the trajec-tory.242 The secondary structure is the large-scale shapeof the trajectory. This is a single path up to a criticalheight, where it bifurcates into a pair of (or several)intertwined spirals. For smaller bubbles (2.10 ( 0.05mm) in stagnant liquid, a stable, curved trajectory isreported up to a critical height, which here is about 17bubble diameters, after which several spiral trajectoriesmay begin. They are also reported to oscillate withinthe trajectories, but this is less noticeable than similaroscillations made by the larger bubbles. The cross-flowcauses the bubbles to spiral immediately and spread oftrajectories is proportional to the cross-flow velocity. Forbubbles of sizes 4.3 ( 0.2 mm in diameter, without anyliquid cross-flow, Yoshida and Manasseh241 have foundthat bubbles follow a perfectly repeatable trajectory withslight curvature (mainly due to lift forces and vorticitygradients developed on the surface of a bubble) up to acritical height about 45 mm (10 bubble diameters) abovethe orifice. At the critical height, axisymmetric oscilla-tions are found to become complex associated withnonaxisymmetric surface distortion, followed by a spiralpath. In fact, their observation has shown that thetrajectory bifurcates into two spirals at the criticalheight, beyond which their amplitude increases withheight with sideways oscillations to certain extent. Inthe cross-flow situation, the critical height is loweredand multiple trajectories (spirals) can be seen. Theshape of the bubble at the critical height may be whatdetermines the trajectory it will follow. For the case ofviscous Newtonian and non-Newtonian liquids, irre-spective of the size of the bubble, the trajectories arerectilinear,242 mainly because of the large viscous drag.In the simulations of bubble dynamics, Blanko243 andTakagi and Matsumoto244 have performed detailedaxisymmetric computations of transient evolution of adeforming bubble rising in a stagnant liquid. Blanko243
has reported that these oscillations occur when the
capillary time x8Fr3σ is larger than the gravitationaltime x2r/g, which supports the experimental observa-tions related to the path instability of a rising bubble.The relationship between the wake structure andtrajectory of clean bubbles (helical or zigzag) has beenclarified by the flow visualizations of Lunde & Per-kins.245 They have showed that the helical path isassociated with a steady wake made of two vortexthreads, whereas the zigzag path is observed whenhairpin vortices are shed in the wake. Thus, thereshould possibly exist a close correspondence between thewake structure and the bubble trajectory. Similarexperiments with expanded-polystyrene spheroids alsoshowed zigzag motion with relatively larger angulardisplacements from the vertical, which shows that pathinstability does not require shape variations or for liquidto slip along the interface. These experiments appearto indicate that the path instability is controlled by themode by which the bubble sheds its vorticity, and thismode itself depends on the amount of vorticity producedon the bubble surface. Thus, bubble deformability in aninviscid flow is not able to produce path instability,although wake dynamics seem sufficient to qualitativelyexplain most of the observations. However, the contri-bution of the bubble deformation on the trajectory orthe mode is not negligible as the wake dynamics areinherently governed by bubble shape. However, detailsof the bubble trajectories are not well understood;
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moreover, the nature of the trajectories when a streamof bubbles is produced deserves further investigation.For further reading on this subject, readers are recom-mended to refer to Fan and Tsuchiya181 (specifically onthe effect of wake structure and dynamics on the bubblemotion), Wu and Gharib184 and Feng and Leal.240
3.7. Experimental Methods for Measurement ofRise Velocity. The most commonly used methods forthe measurement of rise velocity include high-speedphotography and use of intrusive and nonintrusivemethods. For detailed information about these differenttechniques the readers may refer to the review byAdrian.246 Here, we give a brief account of the variousexperimental techniques that are employed for themeasurement of bubble rise velocity and associatedparameters.
3.7.1. Nonintrusive Methods of Bubble RiseAnalysis. 3.7.1.1. Photographic Methods. Thismethod is one of the most commonly used experimentaltechnique for the measurement of bubble size and risevelocity in transparent systems. Most often, high-speedphotography (500-2000 fps) is used for the rise velocitymeasurements and for tracking the rise trajectories(Willert and Gharib,247 Lunde and Perkins,245 Hsieh etal.,248 Lu and Zang,249 Wu and Gharib,184 Tomiyama etal.,183 Takemura and Magnaudet250). The optimumchoice is made based on the possible coalescing tendencyof bubble and speed of photography. This methodologysuits very well for the case of two-dimensional transpar-ent systems, and even for 3-D systems (only when singleor very few bubbles are rising in stagnant/flowingliquid). Further, for opaque systems and turbid liquids,this method is not suitable. Although the spatial resolu-tion achieved by these techniques is excellent, itsreliability goes down with increase in number of movingparticles, nature of curvature of the system and alsowith the extent of turbulence. Trajectory analysis usingcombination of mirrors is possible with a single camera.Such a procedure and details are given by de Vries etal.239 In many cases, to analyze the oscillations of a gasbubble, bubble motion is recorded using a rotatingmirror camera and acoustic waves radiated by thebubble are monitored using a needle hydrophone.251 Itis also possible to carry out the analysis using laserinduced fluorescence that helps in tracking the bubbletrajectory based on the wake movement as well as themovement of bubble interface from the fluorescenceemitted by the photosensitive dye.
With the existing techniques, it is difficult to analyzethe effect of gradual contamination on bubble charac-teristics. To be able to follow bubble dissolution for muchlonger periods of time, Vasconcelos et al.252 and Alveset al.169 have adapted a technique where individualbubbles are kept stationary in a downward liquid flow.The liquid flow is in a closed circuit, allowing continuouscleaning of the liquid. By keeping the individual bubblesstationary in a downward liquid flow, it is possible tosimultaneously (i) follow mass transfer to/from a singlebubble as it inevitably gets contaminated; (ii) follow itsshape as it increases its sphericity; and (iii) periodicallymeasure its terminal velocity along the process ofcontamination. The experimental setup allows a bubbleto be visually followed for an indefinite period time asit gathers contaminant. Periodic measurement of ter-minal velocity as a function of bubble age can be easilyachieved just by stopping the pump and allowing thebubble to ascend along the tube. This technique can
even be used for analyzing the gradual changes in thewake dynamics.
3.7.1.2. PIV Measurements. Many studies of thebubble motion in liquid shows that the size, shape, andposition of the moving bubbles are unsteady and thesechanges are aperiodic. Such an unsteady-state analysisof the bubble motion is possible through the use ofdigital image processing techniques (image filtering,edge detection, thresholding and binary images, etc.).One such method is the use of particle image velocime-ter (PIV) for the measurement of time varying bubblecharacteristics.253,254 This technique has the capabilityto track detailed characteristics of individual bubblesmoving through a turbulent flow field, e.g., size, shape,velocity, and acceleration, and simultaneously to mea-sure the instantaneous fluid velocity field on a two-dimensional plane. Typical system configurations andmethod of data analysis can be seen in Raffel et al.255
Important features of this technique are the ability tocapture bubble trajectories for long periods of time andthe use of bubble images which appear as two fine pointimages for each instant, from which centroid and di-ameter can be deduced. For the details of using PIV orparticle tracking velocimetry (PTV), readers may referto the extensive work by Murai256 and Yamamoto.257
3.7.1.3. LDA Measurements. The method of em-ploying photographic technique to quantify the risevelocity and departure frequency of bubbles requires asignificant time investment to provide accurate time-averaged results. It is possible to use LDA to quantifythese parameters. On the passage of air bubbles throughthe measurement volume the data acquisition unitreceives a time gap, frequencies of which can beestimated directly from the signal. Recent attempts byKulkarni et al.258 have shown that LDA can be success-fully used in forward scatter mode for the exact mea-surement of bubble rise/slip velocity even in a bubbleswarm without any hindrance to the flow. The advan-tage of a system with a facility to measure three velocitycomponents simultaneously can be used to get themagnitude as well as the direction of bubble motion. Thefiber optic based LDA makes the system simple but theapplicability is limited due to the measurements inbackscatter mode. Recently, Terasaka et al.259 havedeveloped a novel sensing system using three coupledlaser sensors, which is capable of yielding measurementof 3-dimensional shape and behavior of single bubblein liquid in real time.
3.7.2. Intrusive Measurements. 3.7.2.1. OpticalProbes. For intrusive techniques, the performancemust be rated based on the quality of data and hin-drance caused to the local flow in their presence. Theprobes based on the principal of utilizing the differencein optical reflectivity of the liquid and gas phases areknown as optical probes and are known for theirexcellent de-wetting property. In the presence of bubbles,they produce a pulse of voltage which due to nonwet-tability gives accurate time for the presence of bubbles.Most commonly, a pair of probes is used under theassumption that the rise is rectilinear. The method ofestimation of velocity is simple and just requires inputin terms of time required for bubble for interceptionbetween two probe tips with known separation distance.However, for the systems where bubble passage is notrectilinear but moves in different directions, more thantwo probe tips with known distance of separation arerequired. Saberi et al.260 have given a detailed account
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of the previous attempts in this direction and have madesystematic analysis of the experimental data using thismethod for a bubble column. Similar analysis is alsocarried out by Chabot et al.,261 where the spherical bulbfiber optic probes are applied for bubble characteriza-tion. The principle of operation of these optical fiberprobes is based on the difference in refractive indicesbetween the gas and the liquid phases. The investigatorshave verified the method through simultaneous photo-graphic measurements.
3.7.2.2. Conductive/Capacitance Probes. Theprobes based on the principal of utilizing the differencein electrical conductivity between the liquid and gasphases are widely used for the measurement of bubblevelocities and sizes to some extent.262,180 Conductiveprobes have wetting characteristic and hence theirapplicability is restricted for the bubbling systems withhigher gas fraction. The discussion for optical probesregarding the direction of bubble motion and numberof probe tips required also applies for the capacitanceprobes. Kuramoto et al.263 have used high-speed pho-tography accompanied with hot-wire probe for themeasurement of detailed flow structure and propertiesof dispersed as well as continuous phase transportphenomena in a 2D column.
3.7.3. Imaging in Opaque Systems. 3.7.3.1. Dynam-ic Imaging by Neutron Radiography for OpaqueSystems. For opaque test systems, it is possible tocharacterize bubble motion using neutron flux method(Mishima et al.,264 Hibiki et al.,265 Saito et al.266). Thepenetrated neutron flux conveys the information of two-phase flow in the test section, since the incident neutronflux is attenuated when the neutron beam passesthrough the test section. Generally, a neutron source isfollowed by a converter having high sensitivity andquick light decay characteristics, and for the imagingsystem, silicon intensifier target (SIT) tube camera fordynamic imaging or a high-speed video camera for highframe-rate imaging is used. The bubble images are otherparameters are analyzed on the basis of different imagebrightness with gas single-phase, liquid single-phase,and two-phase flow based on the scaling method.
3.7.3.2. X-ray Imaging. X-rays have the advantagethat they are not affected by the various refractionindices of the multiphase system and penetrate themultiphase flow in undistorted straight lines. Thetechnique falls within a broad array, including gamma,beta, and neutron radiations, but as a general rule,quantitative measurements have remained local (i.e.,narrow beams), and mostly steady state. Hence, incontrast to optical methods, both of these X-ray-basedmethods are independent of the void fraction and areapplicable to opaque liquids. Recently, Seeger et al.267
have developed a methodology that extracts the infor-mation concerning continuous or discreet phase isextracted using X-ray-based flow visualization andX-ray-based particle tracking velocimetry (PTV). Hei-ndel268 has used flash X-ray radiography (FXR) tovisualize rising air bubbles in fiber suspension. The FXRtechnique allows for bubble visualization in the bulksuspension and the images of bubbles reveal that thebubble rise characteristics and patterns are highlydependent on suspension properties. Theofanous et al.269
have given detailed information about the techniquesbased on X-ray radiography.
Controlled Experiments. In most of the experi-ments where the terminal bubble rise velocities are
measured, the observation is made based on the equiva-lent bubble diameter, its possible deformation (throughsome a priory known rules or photographic images) andthe rise velocity. However, the important factor ofdetachment condition or bubble release condition israrely observed. In the available literature that accountsfor this kind of an analysis, the generation of bubble ismade in different manners. Typically, orifices of differ-ent diameter are used for analyzing the release condi-tion or using an orifice of fixed dimensions the bubbleis generated either by direct injection or by injectionassisted with transportation of bubble using the samefluid as in the tank. Using the former method thedeformation can be enhanced by using larger orificediameters. The later method is considered to reduce theextent of deformation. Details of this method can be seenin Wu and Gharib184 and Tomiyama.185
3.8. Conclusions. Experimental methods for themeasurement of bubble rise velocity include photogra-phy, which is useful for the analysis in stagnant liquidsor even for the dispersed two-phase flows, the reliabilityof the results in the second case is limited by bubblefrequency. Recent attempts to measure the rise velocitybased on the use of intrusive or nonintrusive anemom-eters are encouraging and can be used over a wide rangeof operating conditions provided the noise part of thedata should be carefully eliminated. Further, extensivedata should be collected for the rise of bubbles instagnant and flowing liquids of various properties tounderstand the sensitivity of rise velocity to differentphysical properties and their combined effects.
Formulation for bubble rise velocity should be basedon the complete force balance on a rising bubble. Inmany cases, the forces, which are taken into accountare always taken either for convenience in simplificationof the formula or based on the understanding andcompared with the experiments. However, the minutedetails viz. vorticity on the surface of a bubble (whichvaries over the surface itself) has an important contri-bution in deciding the shape and hence the rise velocityof a bubble, are bypassed. The approach based on thepotential flow or the solution of equation of motion andcontinuity would give better predictions than the em-pirical correlation. Results from the approach based onthe dimensionless numbers are acceptable to certainrange of system parameters however the attempts tomake it generalized or robust result in the loss ofsensitivity toward few parameters.
Wave-based analogy of bubble rise and its anomaliesshow a good comparison with the experimental condi-tions for a wide range of parameters. However, theyshow poor agreement for the contaminated liquids inthe range of bubbles of dB < 5 mm. For the operations,where, deformed, large size bubbles exist the correlationbased on wave analogy190 may even be used for theprediction of rise velocity. Nevertheless, the correlationneeds modification in terms of the effect of externalpressure and temperature on the rise of a bubble.
Acknowledgment
Authors acknowledge the permission from (i) Elsevierfor the reproduction of Figures 1, 2 7, 9, and 10, (ii)Professor R. Clift and Elsevier for Figure 11A, (iii)Professor Tomiyama for Figure 11B, (iv) Professor L.S. Fan for Figure 15, and (v) AIChE for Figure 17.(Sources of the individual Figures are mentioned in theFigure caption) Amol Kulkarni gratefully acknowledges
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(i) the financial support from Professor M. M. SharmaEndowment (University of Mumbai) during this workand also (ii) the Alexander von Humboldt Foundation(Bonn) and the Max-Planck-Institute for Dynamics ofComplex Technical Systems, Magdeburg for the finan-cial support and facilities, respectively during the revi-sion of this paper.
Nomenclature
a ) radius of bubble (eq 25-27), bubble acceleration at theend of bubble detachment (eq 17), constant in Eq. (95,97), maximum bubble diameter (Table 3, Sada74)
ao ) cross-sectional area of the orifice eq 55 (m2)Ar ) Archimedes number (-)b ) constant in Eq. (95, 97) (-), minimum bubble diameter
(Table 3, Sada74)Ca ) Capillary number (-)CD ) drag coefficient (-)CDm ) minimum drag coefficient in CD-Re plot (-)d ) bubble diameter (m)d30 ) sauter mean diameterdB, ) bubble diameterde ) equivalent diameter of bubble (m)dh ) hole diameter on sieve plates, horizontal projection
of a bubble (eq 92) (m)dm ) maximum horizontal bubble diameterDe ) Deborah number (-)Di ) orifice inner diameterE) distortion factor (Table 5, Tomiyama185)F ) Flow number (-)Fb, fb ) buoyancy force, bubble frequencyFd ) drag forceFi ) inertial forceFm ) gas momentum forceFp ) pressure forceFR ) Froude numberFs ) surface tension force (Nm/s)g ) acceleration due to gravity (m/s2)H ) orifice coefficient (-)QG, G ) gas flow rate (m3/s)k ) added mass force (N)K ) pulsation parameterL ) parameter in eq 20, length of contact/perimeter of
orifice (m)La ) Laplace number (-)Ly ) Lyashchenko number (-)mj , M ) virtual mass of the bubblem ) flow indexMa ) Marangoni number (-)Mo ) Morton number (-)n ) number of holes on sieve plate, consistency indexn′ ) the index of flow regime (eq 81)N ) number of sources (-)NC ) capacitance number (-)P, Pb ) Pressure in the bubble (N/m2)PB, Ph ) pressure with in the bubble and the static head
(N/m2)PC, PB ) pressures in the gas chamber and in the bubble
(N/m2)PG ) gas pressure (N/m2)PO ) hydrostatic pressure at the orifice plate (N/m2)Q ) gas flow rate at any one of sources on a slot (m3/s)Qt ) volumetric flow rate at any instant 't’ (m3/s)Qmin ) minimum gas flow rate required for bubbling (m3/s)QSI ) gas flow rate for transition from quasi-steady state
to dynamic regime (m3/s)r,R ) radius (m)rC ) radius of bubble at critical bubble volume (m)rO ) orifice radius (m)R′ ) radius of elemental circle (eq 40)
RC ) radius of curvature, characteristic radius (m)RE ) equivalent radii (eq 40)Re ) Reynolds number (-),re, rE ) radius of bubble reaching to the end of first stage/
expansion stage (m)Recrit ) critical Reynolds number (-)Ro ) orifice radius (m)s ) distance of center of bubble from point of gas supply (m)S ) bubble surfaceSD ) perpendicular distance between bubble center and
orificeTa ) Tadaki number (-)T, Tc, Ti ) temperature (°C)te ) bubble expansion time (s)Ub ) bubble rise velocity for spherical cap shaped bubbles
(m/s)Ui ) velocity of the surface element (m/s) (eq 51)V ) velocity number (-), bubble volume (eq 1), bubble rise
velocity (m/s)VB ) bubble volume at any instant (m3)VC ) chamber volume (m3)Ve ) velocity relating to first stage (m/s)VE ) volume of the force balance bubble (m3)VF ) final volume of a bubble (m3)VL ) mean velocity of liquid (m/s)Vs ) slip velocity (m/s)VT ) terminal rise velocity (m/s)VTm ) terminal rise velocity at minimum in CD-Re plot (m/s)w ) slot spacing (eq 65, 66)We ) Weber number (-)X ) drag correction factor (-)X(n) ) correction function dependent on the power-law index
n (eq 101)z ) Eulerian axial coordinate
Greek Letters
R ) orifice coefficient (-), void fraction (-)â ) bubble growth angle (eq 52)∆F ) density difference between gas and liquidφ ) cap angle dependent on the truncation point (degrees)φ ) velocity potentialγ ) shear rate (1/s)λ ) wavelength (m)µ ) viscosity (Pas)θ ) contact angle (degrees)F ) density (kg/m3)σ ) surface tension (Nm/s)τ ) shear stress (m2/s2)v )kinematic viscosity of the liquidø ) virtual mass coefficient (-)ú ) pulsation function
Subscript
0 ) initial stageb, B ) bubblee ) indicates the first stage of formationG ) gash ) hemispherical geometry, orifice/holei ) time unitL ) liquidO ) orificeq ) angular coordinate on the hemispherical surface (eq
26, 29)w ) water
Superscript
* ) dimensionless quantities
BB
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Received for review September 9, 2004Revised manuscript received February 9, 2005
Accepted February 11, 2005
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