- Email: [email protected]
Description of bubble shape in terms of dimensionless numbers
1495
Shorter Commumcatlons
Lkpartment of Mechanrcal Eftgoteertng Unrverstiy of Ndgena. Nsukka
P E ANAGBO
NOTATION
cqmvalcnt bubbk r&us at detachment, cut-off acceleratmn due to gravity constant volumetnc gas flow rate displacement of the bubble centre above the on&e time. to cut-off bubble volume, at detachment, at cut-off Greek
symbols PI density of hqmd 8
semi-cone
a&e
[l] Themebs N J . Tarassof P and Szekely 1, TMS-AIME I%9 245 2425 El Donald M B and Smger H , Trans Instn Chem Engrs 1959 Ftg
1 Configuratum of an envelope coatamng detachment and cut-off stages
bubbles
m the
angle hes between 190 and 22” It IS also mtcrestmg to note that both measured and denved values of the cone angle are mdependent of the ordice stz.e and the gas mjectton rate
37255 r31 Davidson J F and Schuler B 0 G , Trans Instn Chetn Engrs 1960 38 144 Wraith A E , Chem Engng Set 1971 26 1659 :; Wraith A E and Kakutam T , Chem Engng Scr 1974 29 I L M, Theoretrcal Hydrodynamics, 5th WI Mdne-Thompson Filn,p 52% Macmdlan. London 1965
lIkmcd EMlMmrrrr scum vol 35 pp 1495-1497 @ Paganon Press Ltd 1980 Prmtat m Great Bntam
Description
alo9-zsa9/a-l/oM31-1~Iy)2w0
of bubble shape in terms of dimensionless numbers
(Recerved 11 September 1979. accepted 20 September 1979)
Consldenng
a bubble which nses
m an Infinite extended hqtud spherical.. spherotdai and spherical cap bubbles Which shape the bubble assumes depends on the ratios between the forces actmg on the bubble surface These ratios can be expressed m terms of dunenslonIess numbers, of which the most common are the Reynolds-, the Eutvos-, the Froude- and the Weber-number Furthermore the fltud can be characterized by the Morton-number, defined as
three types of bubbles can be observed
+wp!x
Re4
Much effort has been made to descnbe the bubble shape by means of these dunenslonless numbers and many suggesttons can be found m bterature Naberman and Morton[l] suggested that a combmatton of the Re. We and Morton-number would be the most smtable combmation However, it IS unconvement to have two groups, m both of which the termmal velocity of the bubble plays a role Grace [2] argued that a more convement descnptlon ISgtven by the combmation of the Re-, Eii- and Morton-number From data of several mvesttgators Grace gwes a generalued graplucal correlation of these numbers, m which the three bubble shaped regons clearly could be recogmzed Spherical cap bubbles appear for m>40 and Re> 12 Spherical bubbles appear for all Reynolds numbers smaller than 12 In other regtons the Mortonnumber 1s also relevant m determmg the bubbk shape Grace’s method has the advantage that, when the bubble volume and phystcal propem of the flutd are known, the termmal velocity of the bubble also can be estunated from his graph
Nevertheless the result of Grace IS rather surpnsmg and we wonder d it IS correct It is generally accepted that mterfaclal forces tend to mmuntze the surface area of a fluid pticle, so for bubbles of sutlicrent small E&numbers, the shape wdl be sphencal From Grace’s curves It follows that under condltlons when vtscous stresses dommate the hydrodynamic forces, the bubble assumes a spherical shape, no matter how large the &number IS It would be more likely that m that re@on the bubbles wdl be ilattened We believe that Grace mlsmterpreted the results of the mvesttgators he menttoned, especmlly from those who studied bubble shapes m highly viscous hqulds It IS known from hterature[3,4] that large bubbles nsmg m hqmds wtth vtscostty > 200 CP. develop a skut which can extend to far behmd the bubbles and enclose a fhud regton approxunately completmg the sphere of which the bubble IS the upper pat-t Tadalu and MaedaIS] concluded from thev own experunental data that spherical bubbles appear tf the condition We’” * Ea”’ < 0 2 1s fulfilled, and spherical cap bubbles d Weln*Eo’“> 16 5 The disadvantage of this cntenon IS that the termmal velocity must be known In Fu 1 some of the results of the measurements of nse velocti of bubbles by Haberman and Morton[l] aud KoJuna et a/ [6] are correlated In UIIS plot a duncnsronless velocity N. defined as
Shorter Commumcatxons
14%
0.
t
I l
Water
0
turpentine
A
mlneral
FII 1 Characterlsatxon
M=2.6xlO
10
0
-9
M-1 .8x10q7 oil
M=1.5xlO
-2
100
*
castor
7
corn
011 syrup
f
transItIon from to spheroidal
of the bubble shape by means of the dunenslordess data
M-3.52 M-4910
numbers
spherzcal N. and N,,, expenmental
IO
I
0
\ \ \ 0
I
I
0
1
1
Fe 1s related to a dnnenslonless as
2 Generahzed
bubble &meter
Nb = $&glu)“’
charactenzatlon
Nb which IS defined
further are are even for spherical bubbles nsmg m hqruds havmg the M-numbers, calculated from the Hadamard [73 and relation,
N. vs l$, for spheroIdal bubbles.
I
I
10
plot to determme
bubble shape
gwen by Mendelson[9]
= EiYf2,
w&h M as parameter
In thrs plot N,. vs Ns correspondmg Rybczmsky[Lt]
-Nb
calculated
from the relation
N. vs N* for spherIcal cap bubbles, relation gwen by Davies and Taylor [lo] u, =$g
calculated
from
the
R)‘”
The cntena gven by Tadalu and Maeda, who determrne the regzons IIIwhich spherical and spherical cap bubbles occur Theexpc~entalcurvesareallslmllar, thedenvativeof log N. agamst log NS first bemg constant (equal to the value gwen by
1497
Shorter Communications Hadamard and Rybczmsky) After that. dependmg on the VIScoslty of the hqmd. tendmg more and more to the theoretlcal curve qven by Mendelson, m which U, 1s independent of the vlscosrty, and then followmg the curve gven by Davies and Taylor The minor devlatlons which occur m the mdrvidual curves are probably caused by the fact that the data were gathered from graphs pven by the mvestrgators mentioned earher The tranSitIon from spherical to spheroidal shape, as presented by the several mvestlgators IS also gven m this figure as an astensk on the different curves From these data It can be concluded that the transdion occurs d the condltlon N, * Nb < 0 5 IS fulfilled, which corresponds to Re M’” c 0 5 A strong mdrcatlon m this dvectlon 1s also that for the dtierent hqmds, the intersection between the experunental curves and the curve N, * Nb = 0 5, comcldes with the points at which the expenmental data curves begm to deviate from the relation presented by Hadamard and Rybczmsky Tadah and Maeda suggest N. * Nb = 2, which. as Fi 1 shows, fails far outside the range m which spherical bubbles were observed by other mveshgators From the F@ 1 It IS also clear that spherical bubbles cannot be expected to appear for large bubble volumes m hqurds with a large Morton-number The regon for spherical bubble shape as determmed by Grace m his graphical correlation must therefore be mcorrect In F@ 2 a generabzed plot of the relationship between N, and Ns IS gven. wluch can be used to estunate the bubble shape and termmal velocrty for smgle bubbles nsmg m mfhute extended Newtonian hqurds O& van det Gnnten Venlo. The Netherlands
J H C COPPUS
Emdhoven Umverslty of Technology Emdhoven, The Netherlands
K RIETEMA
NOTATION
equivalent diameter of the bubble, cm gravity constant. cm/set* radius of a spherical cap bubble, cm nse velocity of a bubble, cm&c vrscosity contmuous phase, gfcmlsec density continuous phase, g/cm3 surface tension, g/cm’ Eiitvos number, p g d2ia Froude number, Ub21g d. Morton number, 12 LL’II) uf
Reynolds number, p UbdJp Weber number, p U,Z~ d,lrr
[l] Haberman W L and Morton R K , Trans Am Sac Ctv Engrs 1956 121 227 [2] Grace J R , Tram Znstn Chem Engts 1973 51 116 [3] Angehno H , Chem Engng SKI 1966 21 541 [4] Davenport W G , Richardson F D and Bradshaw A V , Chem Engng Set 1967 22 1221 [S] Tad& T and Maeda S , Kagaku Kogaku 196125 254 [6] yoJ5m8als Akehata T and Shuar T , J Chem Engng Japan [7J Hadamard [8] Rybczynslu (91 Mendelson [lOI Davies R 1950 A200
, CR Acad Scl Pans Cmcovre 1911 152 1735 W , Buff Acad SCI 1911 A 40 H D , A ZClrEJ 1967 13 250 M and Taylor G I , Prwc Roy Sot (London) 375
J
ChamEdE&mmzas Snm Vol 3S QP1497-1499 @PcrsamonPressLtd i9w) Pnnrcd,IIGreatBnta~n
Theoretical
derivation of the mass transfer coefficient at the front of a spherical cap bubble
(Recerved 11 September 1979, accepted 20 September 1979)
Several mvestlgators [l-31 theoretically denved an expression for the mass transfer coefficient of a spherical cap bubble m relation to the equivalent diameter of the bubble and the dlffuslon coefficient of the transferred component m the bqmd All started from the dtiuslon equabon at steady state for a sphere placed m an otherwise umform flow field so that the velocity field IS axmlly symmetrical In spherical coordmates (see Fw 1) the rhffusron equation runs
The boundary
condlbons
F@ 1 Spherical coordmates
are
?>R,
@=O @=a
$0
r=Rb
e>o
c=c+
r=cO
e>o
c=o
Some of the mentioned
mvestlgators
used a too srmphfied equa-
tion, other theoretical analyses are less clear and m some cases the boundary conditions are not fully correct or incomplete In this analysis we wdl meet these ObJectins Before subst~tutmg the proper velocity dlstnbution m the tiusion equaaons, some sunphficatlons can be made by neglectmg the last term of the rgbt hand side L.evlch [S] has shown that d the Pt number (= (I,, dJD) related to the system bemg studied IS large, the distance at which mass transfer takes place, the
Shorter Commumcatlons
Lkpartment of Mechanrcal Eftgoteertng Unrverstiy of Ndgena. Nsukka
P E ANAGBO
NOTATION
cqmvalcnt bubbk r&us at detachment, cut-off acceleratmn due to gravity constant volumetnc gas flow rate displacement of the bubble centre above the on&e time. to cut-off bubble volume, at detachment, at cut-off Greek
symbols PI density of hqmd 8
semi-cone
a&e
[l] Themebs N J . Tarassof P and Szekely 1, TMS-AIME I%9 245 2425 El Donald M B and Smger H , Trans Instn Chem Engrs 1959 Ftg
1 Configuratum of an envelope coatamng detachment and cut-off stages
bubbles
m the
angle hes between 190 and 22” It IS also mtcrestmg to note that both measured and denved values of the cone angle are mdependent of the ordice stz.e and the gas mjectton rate
37255 r31 Davidson J F and Schuler B 0 G , Trans Instn Chetn Engrs 1960 38 144 Wraith A E , Chem Engng Set 1971 26 1659 :; Wraith A E and Kakutam T , Chem Engng Scr 1974 29 I L M, Theoretrcal Hydrodynamics, 5th WI Mdne-Thompson Filn,p 52% Macmdlan. London 1965
lIkmcd EMlMmrrrr scum vol 35 pp 1495-1497 @ Paganon Press Ltd 1980 Prmtat m Great Bntam
Description
alo9-zsa9/a-l/oM31-1~Iy)2w0
of bubble shape in terms of dimensionless numbers
(Recerved 11 September 1979. accepted 20 September 1979)
Consldenng
a bubble which nses
m an Infinite extended hqtud spherical.. spherotdai and spherical cap bubbles Which shape the bubble assumes depends on the ratios between the forces actmg on the bubble surface These ratios can be expressed m terms of dunenslonIess numbers, of which the most common are the Reynolds-, the Eutvos-, the Froude- and the Weber-number Furthermore the fltud can be characterized by the Morton-number, defined as
three types of bubbles can be observed
+wp!x
Re4
Much effort has been made to descnbe the bubble shape by means of these dunenslonless numbers and many suggesttons can be found m bterature Naberman and Morton[l] suggested that a combmatton of the Re. We and Morton-number would be the most smtable combmation However, it IS unconvement to have two groups, m both of which the termmal velocity of the bubble plays a role Grace [2] argued that a more convement descnptlon ISgtven by the combmation of the Re-, Eii- and Morton-number From data of several mvesttgators Grace gwes a generalued graplucal correlation of these numbers, m which the three bubble shaped regons clearly could be recogmzed Spherical cap bubbles appear for m>40 and Re> 12 Spherical bubbles appear for all Reynolds numbers smaller than 12 In other regtons the Mortonnumber 1s also relevant m determmg the bubbk shape Grace’s method has the advantage that, when the bubble volume and phystcal propem of the flutd are known, the termmal velocity of the bubble also can be estunated from his graph
Nevertheless the result of Grace IS rather surpnsmg and we wonder d it IS correct It is generally accepted that mterfaclal forces tend to mmuntze the surface area of a fluid pticle, so for bubbles of sutlicrent small E&numbers, the shape wdl be sphencal From Grace’s curves It follows that under condltlons when vtscous stresses dommate the hydrodynamic forces, the bubble assumes a spherical shape, no matter how large the &number IS It would be more likely that m that re@on the bubbles wdl be ilattened We believe that Grace mlsmterpreted the results of the mvesttgators he menttoned, especmlly from those who studied bubble shapes m highly viscous hqulds It IS known from hterature[3,4] that large bubbles nsmg m hqmds wtth vtscostty > 200 CP. develop a skut which can extend to far behmd the bubbles and enclose a fhud regton approxunately completmg the sphere of which the bubble IS the upper pat-t Tadalu and MaedaIS] concluded from thev own experunental data that spherical bubbles appear tf the condition We’” * Ea”’ < 0 2 1s fulfilled, and spherical cap bubbles d Weln*Eo’“> 16 5 The disadvantage of this cntenon IS that the termmal velocity must be known In Fu 1 some of the results of the measurements of nse velocti of bubbles by Haberman and Morton[l] aud KoJuna et a/ [6] are correlated In UIIS plot a duncnsronless velocity N. defined as
Shorter Commumcatxons
14%
0.
t
I l
Water
0
turpentine
A
mlneral
FII 1 Characterlsatxon
M=2.6xlO
10
0
-9
M-1 .8x10q7 oil
M=1.5xlO
-2
100
*
castor
7
corn
011 syrup
f
transItIon from to spheroidal
of the bubble shape by means of the dunenslordess data
M-3.52 M-4910
numbers
spherzcal N. and N,,, expenmental
IO
I
0
\ \ \ 0
I
I
0
1
1
Fe 1s related to a dnnenslonless as
2 Generahzed
bubble &meter
Nb = $&glu)“’
charactenzatlon
Nb which IS defined
further are are even for spherical bubbles nsmg m hqruds havmg the M-numbers, calculated from the Hadamard [73 and relation,
N. vs l$, for spheroIdal bubbles.
I
I
10
plot to determme
bubble shape
gwen by Mendelson[9]
= EiYf2,
w&h M as parameter
In thrs plot N,. vs Ns correspondmg Rybczmsky[Lt]
-Nb
calculated
from the relation
N. vs N* for spherIcal cap bubbles, relation gwen by Davies and Taylor [lo] u, =$g
calculated
from
the
R)‘”
The cntena gven by Tadalu and Maeda, who determrne the regzons IIIwhich spherical and spherical cap bubbles occur Theexpc~entalcurvesareallslmllar, thedenvativeof log N. agamst log NS first bemg constant (equal to the value gwen by
1497
Shorter Communications Hadamard and Rybczmsky) After that. dependmg on the VIScoslty of the hqmd. tendmg more and more to the theoretlcal curve qven by Mendelson, m which U, 1s independent of the vlscosrty, and then followmg the curve gven by Davies and Taylor The minor devlatlons which occur m the mdrvidual curves are probably caused by the fact that the data were gathered from graphs pven by the mvestrgators mentioned earher The tranSitIon from spherical to spheroidal shape, as presented by the several mvestlgators IS also gven m this figure as an astensk on the different curves From these data It can be concluded that the transdion occurs d the condltlon N, * Nb < 0 5 IS fulfilled, which corresponds to Re M’” c 0 5 A strong mdrcatlon m this dvectlon 1s also that for the dtierent hqmds, the intersection between the experunental curves and the curve N, * Nb = 0 5, comcldes with the points at which the expenmental data curves begm to deviate from the relation presented by Hadamard and Rybczmsky Tadah and Maeda suggest N. * Nb = 2, which. as Fi 1 shows, fails far outside the range m which spherical bubbles were observed by other mveshgators From the F@ 1 It IS also clear that spherical bubbles cannot be expected to appear for large bubble volumes m hqurds with a large Morton-number The regon for spherical bubble shape as determmed by Grace m his graphical correlation must therefore be mcorrect In F@ 2 a generabzed plot of the relationship between N, and Ns IS gven. wluch can be used to estunate the bubble shape and termmal velocrty for smgle bubbles nsmg m mfhute extended Newtonian hqurds O& van det Gnnten Venlo. The Netherlands
J H C COPPUS
Emdhoven Umverslty of Technology Emdhoven, The Netherlands
K RIETEMA
NOTATION
equivalent diameter of the bubble, cm gravity constant. cm/set* radius of a spherical cap bubble, cm nse velocity of a bubble, cm&c vrscosity contmuous phase, gfcmlsec density continuous phase, g/cm3 surface tension, g/cm’ Eiitvos number, p g d2ia Froude number, Ub21g d. Morton number, 12 LL’II) uf
Reynolds number, p UbdJp Weber number, p U,Z~ d,lrr
[l] Haberman W L and Morton R K , Trans Am Sac Ctv Engrs 1956 121 227 [2] Grace J R , Tram Znstn Chem Engts 1973 51 116 [3] Angehno H , Chem Engng SKI 1966 21 541 [4] Davenport W G , Richardson F D and Bradshaw A V , Chem Engng Set 1967 22 1221 [S] Tad& T and Maeda S , Kagaku Kogaku 196125 254 [6] yoJ5m8als Akehata T and Shuar T , J Chem Engng Japan [7J Hadamard [8] Rybczynslu (91 Mendelson [lOI Davies R 1950 A200
, CR Acad Scl Pans Cmcovre 1911 152 1735 W , Buff Acad SCI 1911 A 40 H D , A ZClrEJ 1967 13 250 M and Taylor G I , Prwc Roy Sot (London) 375
J
ChamEdE&mmzas Snm Vol 3S QP1497-1499 @PcrsamonPressLtd i9w) Pnnrcd,IIGreatBnta~n
Theoretical
derivation of the mass transfer coefficient at the front of a spherical cap bubble
(Recerved 11 September 1979, accepted 20 September 1979)
Several mvestlgators [l-31 theoretically denved an expression for the mass transfer coefficient of a spherical cap bubble m relation to the equivalent diameter of the bubble and the dlffuslon coefficient of the transferred component m the bqmd All started from the dtiuslon equabon at steady state for a sphere placed m an otherwise umform flow field so that the velocity field IS axmlly symmetrical In spherical coordmates (see Fw 1) the rhffusron equation runs
The boundary
condlbons
F@ 1 Spherical coordmates
are
?>R,
@=O @=a
$0
r=Rb
e>o
c=c+
r=cO
e>o
c=o
Some of the mentioned
mvestlgators
used a too srmphfied equa-
tion, other theoretical analyses are less clear and m some cases the boundary conditions are not fully correct or incomplete In this analysis we wdl meet these ObJectins Before subst~tutmg the proper velocity dlstnbution m the tiusion equaaons, some sunphficatlons can be made by neglectmg the last term of the rgbt hand side L.evlch [S] has shown that d the Pt number (= (I,, dJD) related to the system bemg studied IS large, the distance at which mass transfer takes place, the