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Bubble frequency in gas-liquid slug flow in vertical tubes
Shorter Commumcauons
2356 PEFERENCES
[l] Scnven L E , Chem Engng SCI 1959 10 1 121 Street J R , Fncke A L and Relss L P , Znd Engng Chem Fundls 1971 10 54
]3] Rosner D E and Epstein M , Chem Engng .Sa 1972 27 69 141 Barlow E J and Langlois W E , IBM J Res Da, 1962 6 329 [Sl Szekely J and Martins G P , C’hem Engng Scr 1971 26 147
Chemrcd Ewmmw Smnce Vot 35,DP2356-2358 PcrsamonPressLtd 1980 Pnnted,n GreatBntazn
Bubble frequency in gas-liquid slug flow in vertical tubes (Recewed 31 August 1978, accepted 7 March 1980) A recent analysts showed that mterphase mass transfer m concurrent gas-hqurd slug flow m verhcal pipes depends on the frequency of the gas bubbles or the slug hquld slugs[l] The only published mformatlon on bubble frequency m slug flow IS the empulcal correlatron by Gregory and Scott[2] which LS not applicable to the problem under conslderatron since It has been derived for flow m honzontal tubes Due to the lack of mformatton on thrs parameter, and the need for it m predlctmg gas-hquld mass transfer rates, an expenmental mvesugauon was undertaken EXPJCRKMENTAL EQUIPMENT AND PROCEDURE
The expenmental equipment consisted of a verbcal glass tube 94 cm high and 0 8 cm m mslde diameter An annular inlet device was used at the bottom of the tube to introduce the gas and the hqmd mto the test section Pure carbon dloxlde. selected as the gas phase, entered through the core of the inlet device, while water, selected as the liquid phase, entered through the annulus The gas and liquid flow rates were regulated for maintaining slug flow regune m the tube If the velocity of the gas bubble IS V, (this ~111be the velocity of the hqmd slug as well), then the average bubble (or slug) frequency, Gi, can be determined from the followmg equation
average lengths of the bubbles and the slugs and then the frequency was determined through eqn (1) In using eqn (1). the bubble velocity, Vb, can be determined from the elllstmg results m the hterature[3,4] For the specific conditions existing m this work, V, was calculated using the correlations of White and Beardmore[5], because of theu apphcablhty to wade ranges of vanables RESULTS
A total of 48 expenments corresponding to dtierent values of gas and hquld velocltles were performed Superliclal velocltles of the gas and the hqmd were vmed from 17 4 to 103 4 and 5 0 @ 45 8 c-m/s respectively The expernnental results obtained for Lb and L, and the calculated values for frequency are presented m Table 1 From the data of Table 1, we see that with some exceptions, the average frequency generally decreases Hrlth the gas superficial velocity, but increases with the liquid superficial velocity Based on ths observation, the parameter (5 V&V,, was arbitranly selected as a basis for correlating the results Except for very low values of V,,, this parameter was found to be almost mdependend from the bquld super&al velocity Thus, for each V,, the average value I@( V,)/( V,,)l., was calculated and plotted against VSG as shown m Fig 1 Least mean square method was used to obtam the followmg linear correlation
I I 6-
where & and L, are the bubble and slug lengths respectively, p is the number of gas bubbles (or lrqutd slugs) along a gven section of the pipe, and bar mdlcates average quantity When the bubble vetoctty was small enough to pernut visual detection, the frequency was determmed by countmg the number of bubbles passing a certain section of the tube in a Bven time Otherwise, the test section was photographed to obtain the
VSO =0 WV,,+65 V SL cl”
when velocmes are expressed m cm/s and frequency in Using eqn (2). G was calculated for dfierent values of V, VS~.and compared Hrlth the measured quantities as shown m 2 The comparison IS satisfactory, mdlcatmg the effectiveness the correlation The correlation shows that, as a result of the change m &
V95 +65
6 4
f
-
:E
40
10
v&*
Ftg
1 Correlation
70
SC
loo
80
cm/s
of the averaged
frequency
data
110
HZ and Fig of and
2357
Shorter Commumcatlons Table YSL’
cm/s
‘SG’
5 00 5 00 5 00 5 00 5 00 5 00 5 00 5 00 10 3 10 3 10 3 10 3 10 3 10 3 10 3 10 3 15 2 15 2 15 2 15 2 12 5 15 2 15 2 15 2 25 5 25 5 25 5 25 5 25 5 25 5 25 5 25 5 35 a 35 a 35 8 35 a 35 a 35 6 35 8 35 8 45 8 45 8 45 8 45 8 45 8 45 8 45 8 45 8
Cokuhted
Comparison
of
frequm~y
the calculated measured values
1 Experrmental cm/s
17 29 41 53 65 79 91 103 17 29 41 53 65 79 91 103 17 29 41 53 65 79 91 103 17 29 41 53 65 79 91
4 5 6 7 8 0 3 4 4 5 6 7 8 0 3 4 4 5 6 7 8 0 3 4 4 5 6 7 8 0 3
T 29 41 53 65 79 91 103 17 29 41 53 65 79 91 103
t 5 6 7 8 0 3 4 4 5 6 7 8 0 3 4
data on average
4.
cm
25 ii 35 1 29
9
;2 11 0 ia 8 29 7 30 8
43 1: 17 20 29 i5 169 1: 17 21 22 1
x 5 9 3 9
;: 73 11 3
11 12 17
6 1 3
:i 42 :: 13 4 13 5 13 3
s-1
frequenctes
z 9 6 1
wtth
the
frequency
cs, 22
cm
-
W* s
-1
14
20 :97 25
:: 24 f:
:: 24 207 40
f! 24 25 zZf 47 45 34 26 30 32 36 28 80 73 :: 46 El 1: 03 13 0 96 6': E94 64 18 3 17 9 13 1 11 4 1: 10 85 20 16 16 13 12
ii 9 5 9 5 7 0
1: 11
z 6
La, the bubble frequency mcreases with hqmd velocrty but decreases wtth gas velo-ctty When VsL IS Increased, LS remams almost constant but Lb decreases causmg (3 to Increase as predtctcd from_ eqn (I) On the other hand, when Vm IS increased, both Ls and Lb Increase, causmg d to decrease accordmg to eqn (1) Thus behavior can be Interpreted from a mechamcal pomt of vrew concernmg the bubble and slug formatton at various values of Vm and VsL Wtth the annular mlet devtce used tn fhts work, gas flows m a jet stream surrounded by a chmbmg hqmd film for a certam distance above the mlet The thickness of the hqmd tilm mcreases gradually unttl the grawtabona1 force on the film overcomes the drag force Induced by the gas Jet stream and the film starts to fill the entue cross sectton of the tube Thts captures part of the gas m the form of a bubble and produces a tradmg hqmd slug At large values of hqutd veloctty. one can expect the thtckenmg of the chmbmg film and the subsequent slug formatton to occur more frequently On the other hand, at targe values of gas veloctty the larger merttal force of the gas Jet Induces larger drag on the chmbmg film, delays the thtckenmg of the film and causes the slug formatton at less frequent rate It should be noted that the results presented here should be constdered as prehmmary smce they correspond only to the condtttons extstmg m thus work Work IS underway by the authors to determme the effect of the type of flulds, wider range of gas and hqutd velocttres, and dtfferent tube dtameters
2358
Shorter Commumcatlons CONCLUSION
A correlation has been obtamed for the frequency of gas bubbles m the upward slug flow of CO+vater mixtures m a 0 8 cm I d vertical pope The correlation shows that when the gas and hqmd flow rates vary from 17 4 to 103 4 and 5 0 to 45 8 cm/s respectively, the frequency mcreases Hrlth hqrud velocity but decreases with gas velocity
Va nsmg velocrty of the gas bubble, cm/s
V,
superficial
V.,,
supertic~al velocity
velocity
of the gas phase, cm/s of the hqmd phase, cm/s
Greek symbol o
gas bubble or hqmd slug frequency,
S-I
Superscnpt M VAKILOTOJJAR
-
average
K JAVDANI*
Department of Chemical Engmeenng Tehran Unrversrty of Technology Tehran, Iran
REFERENCES
[I] ValulotoJlar
NOTATION
Lb Ls
length of the gas bubble, cm length of the hqmd slug, cm
*Present bchmond.
address Stauffer CA 94804. U S A
Ch.?mKa/ lznm.mw Peqtamn Press Lid
Chenucal
Co,
1200 S 47th
St
,
M , M S Thesis, Tehran Umversrty of Technology. Tehran. Iran 1978 121 Greg0ryG.A andScottD S,AIChEJ 1%9f5933 [3] Govler G W and Auz K , The Flow of Complex Mrxtures m hpe Van Nostrand Remhold. New York 1972 [4] Walk G B , One I)rmenslonal Two-Phase Flow McGrawHdl. New York I%9 [51 Wlute E T and Beardmore R H , Chem Engng Scr 1%2 17 351
scunce Vol 33 pp 2338-2360 I9Rl Pnntcd q Great Bntam
Application of Gale&in technique to diffusion problems in tubular reactor (Recerved 20 November 1979, accepted 4 March 1980) The Taylor’s dlsperslon theory[l] has been used by many works to predict chemical converstion m a lammar flow tubular reactor[2-71 Kulkarm and Vasudeva[8] have revlewed the earher work and have offered an emplrtcal expresslon for the effective dlsperston coefklent which turns out to be dependent on reactIon rate constant The vahdrty of reacrlon dependent dlsperslon coefficient for the unsteady state lammar reactor has been expenmentally estabhshed by Nlgam and Vasudeva[9] In a separate study they have shown{101 that the apphcabddy of the Fan and Hwang[ 111 analysts to the lirst order reaction m lammar flow of power law fhnds depend on the values of a, 7 and n They have suggested that the Fan and Hwang analysis holds m the case of reactmg systems for the values 0 5 zz n 52 5 provided P >O I5 and ~3> I 0 They have also developed empIrIcal expresstons for effective dlsperslon coefliclent which turns out to be dependent on the reactlon rate constant and permit the use of axial dlsperslon model with a much greater accuracy over a wide range of parameter values of practical Interest The purpose of this commumcatlon IS to present an analytlcal solution for the effective dlsperslon coefficient for a homogeneous first order ureverslble chemtcal reactlon takmg place m the bulk of a non-Newtoman hqmd by usmg Galerkm techmque [ 12,131
$$=Oat
Y=Oand
Y=l
(2b)
Galerkm technique has been used to solve eqns (I) and (2) Under this techmque, let C’,,, denotes the m th order approxlmatlon for C, and Cn =oo+oJI(Y)+ozfu)+
+anfm(Y)
(3)
, m) are the first m functions f,(Y), (J = I, 2,3, of an mfimte -iequeice v,(Y)}. (J = 1,2,3 ) such that each function f,(Y) IS a twice contmuously dlfferentlable function of Y on - I & Y 5 I and satlslies the boundary condltlons (2) Also, these functions form a lmearly Independent set of functions on -I
The functions
L(C,)f,(Y)YdY=Oforj=1,2,3,
,m
(4)
where
ANALYSIS
The steady-state two dlmenslonal convecelve dlffuslon equatlon m lammar flow of power law flmds In circular ducts for a movmg coordmate system with reference to axls 1s grven m Ref [ 1I] (eqn 31) Takmg mto account the homogeneous first order reactIon, this equation becomes
>-/3C The relevant
boundary
-&(2-(n+3)Y”+‘}
condttlons C=latX=O
$j=O
(I)
are Pa)
--&(2-(n+3)Y”+‘}$$
=0
This system of equations has resulted due to orthogonahty L(C,,,) to each f,(Y) over the cross-sectlon of the tube calculate a0 the foltowmg condltlons IS used
(5) of TO
I
L(C,,,)YdY=O Thus the Important feature of this techmque sequence of functions u,(Y)} satrsfymg the
(6) 1s to select the mentloned con-
2356 PEFERENCES
[l] Scnven L E , Chem Engng SCI 1959 10 1 121 Street J R , Fncke A L and Relss L P , Znd Engng Chem Fundls 1971 10 54
]3] Rosner D E and Epstein M , Chem Engng .Sa 1972 27 69 141 Barlow E J and Langlois W E , IBM J Res Da, 1962 6 329 [Sl Szekely J and Martins G P , C’hem Engng Scr 1971 26 147
Chemrcd Ewmmw Smnce Vot 35,DP2356-2358 PcrsamonPressLtd 1980 Pnnted,n GreatBntazn
Bubble frequency in gas-liquid slug flow in vertical tubes (Recewed 31 August 1978, accepted 7 March 1980) A recent analysts showed that mterphase mass transfer m concurrent gas-hqurd slug flow m verhcal pipes depends on the frequency of the gas bubbles or the slug hquld slugs[l] The only published mformatlon on bubble frequency m slug flow IS the empulcal correlatron by Gregory and Scott[2] which LS not applicable to the problem under conslderatron since It has been derived for flow m honzontal tubes Due to the lack of mformatton on thrs parameter, and the need for it m predlctmg gas-hquld mass transfer rates, an expenmental mvesugauon was undertaken EXPJCRKMENTAL EQUIPMENT AND PROCEDURE
The expenmental equipment consisted of a verbcal glass tube 94 cm high and 0 8 cm m mslde diameter An annular inlet device was used at the bottom of the tube to introduce the gas and the hqmd mto the test section Pure carbon dloxlde. selected as the gas phase, entered through the core of the inlet device, while water, selected as the liquid phase, entered through the annulus The gas and liquid flow rates were regulated for maintaining slug flow regune m the tube If the velocity of the gas bubble IS V, (this ~111be the velocity of the hqmd slug as well), then the average bubble (or slug) frequency, Gi, can be determined from the followmg equation
average lengths of the bubbles and the slugs and then the frequency was determined through eqn (1) In using eqn (1). the bubble velocity, Vb, can be determined from the elllstmg results m the hterature[3,4] For the specific conditions existing m this work, V, was calculated using the correlations of White and Beardmore[5], because of theu apphcablhty to wade ranges of vanables RESULTS
A total of 48 expenments corresponding to dtierent values of gas and hquld velocltles were performed Superliclal velocltles of the gas and the hqmd were vmed from 17 4 to 103 4 and 5 0 @ 45 8 c-m/s respectively The expernnental results obtained for Lb and L, and the calculated values for frequency are presented m Table 1 From the data of Table 1, we see that with some exceptions, the average frequency generally decreases Hrlth the gas superficial velocity, but increases with the liquid superficial velocity Based on ths observation, the parameter (5 V&V,, was arbitranly selected as a basis for correlating the results Except for very low values of V,,, this parameter was found to be almost mdependend from the bquld super&al velocity Thus, for each V,, the average value I@( V,)/( V,,)l., was calculated and plotted against VSG as shown m Fig 1 Least mean square method was used to obtam the followmg linear correlation
I I 6-
where & and L, are the bubble and slug lengths respectively, p is the number of gas bubbles (or lrqutd slugs) along a gven section of the pipe, and bar mdlcates average quantity When the bubble vetoctty was small enough to pernut visual detection, the frequency was determmed by countmg the number of bubbles passing a certain section of the tube in a Bven time Otherwise, the test section was photographed to obtain the
VSO =0 WV,,+65 V SL cl”
when velocmes are expressed m cm/s and frequency in Using eqn (2). G was calculated for dfierent values of V, VS~.and compared Hrlth the measured quantities as shown m 2 The comparison IS satisfactory, mdlcatmg the effectiveness the correlation The correlation shows that, as a result of the change m &
V95 +65
6 4
f
-
:E
40
10
v&*
Ftg
1 Correlation
70
SC
loo
80
cm/s
of the averaged
frequency
data
110
HZ and Fig of and
2357
Shorter Commumcatlons Table YSL’
cm/s
‘SG’
5 00 5 00 5 00 5 00 5 00 5 00 5 00 5 00 10 3 10 3 10 3 10 3 10 3 10 3 10 3 10 3 15 2 15 2 15 2 15 2 12 5 15 2 15 2 15 2 25 5 25 5 25 5 25 5 25 5 25 5 25 5 25 5 35 a 35 a 35 8 35 a 35 a 35 6 35 8 35 8 45 8 45 8 45 8 45 8 45 8 45 8 45 8 45 8
Cokuhted
Comparison
of
frequm~y
the calculated measured values
1 Experrmental cm/s
17 29 41 53 65 79 91 103 17 29 41 53 65 79 91 103 17 29 41 53 65 79 91 103 17 29 41 53 65 79 91
4 5 6 7 8 0 3 4 4 5 6 7 8 0 3 4 4 5 6 7 8 0 3 4 4 5 6 7 8 0 3
T 29 41 53 65 79 91 103 17 29 41 53 65 79 91 103
t 5 6 7 8 0 3 4 4 5 6 7 8 0 3 4
data on average
4.
cm
25 ii 35 1 29
9
;2 11 0 ia 8 29 7 30 8
43 1: 17 20 29 i5 169 1: 17 21 22 1
x 5 9 3 9
;: 73 11 3
11 12 17
6 1 3
:i 42 :: 13 4 13 5 13 3
s-1
frequenctes
z 9 6 1
wtth
the
frequency
cs, 22
cm
-
W* s
-1
14
20 :97 25
:: 24 f:
:: 24 207 40
f! 24 25 zZf 47 45 34 26 30 32 36 28 80 73 :: 46 El 1: 03 13 0 96 6': E94 64 18 3 17 9 13 1 11 4 1: 10 85 20 16 16 13 12
ii 9 5 9 5 7 0
1: 11
z 6
La, the bubble frequency mcreases with hqmd velocrty but decreases wtth gas velo-ctty When VsL IS Increased, LS remams almost constant but Lb decreases causmg (3 to Increase as predtctcd from_ eqn (I) On the other hand, when Vm IS increased, both Ls and Lb Increase, causmg d to decrease accordmg to eqn (1) Thus behavior can be Interpreted from a mechamcal pomt of vrew concernmg the bubble and slug formatton at various values of Vm and VsL Wtth the annular mlet devtce used tn fhts work, gas flows m a jet stream surrounded by a chmbmg hqmd film for a certam distance above the mlet The thickness of the hqmd tilm mcreases gradually unttl the grawtabona1 force on the film overcomes the drag force Induced by the gas Jet stream and the film starts to fill the entue cross sectton of the tube Thts captures part of the gas m the form of a bubble and produces a tradmg hqmd slug At large values of hqutd veloctty. one can expect the thtckenmg of the chmbmg film and the subsequent slug formatton to occur more frequently On the other hand, at targe values of gas veloctty the larger merttal force of the gas Jet Induces larger drag on the chmbmg film, delays the thtckenmg of the film and causes the slug formatton at less frequent rate It should be noted that the results presented here should be constdered as prehmmary smce they correspond only to the condtttons extstmg m thus work Work IS underway by the authors to determme the effect of the type of flulds, wider range of gas and hqutd velocttres, and dtfferent tube dtameters
2358
Shorter Commumcatlons CONCLUSION
A correlation has been obtamed for the frequency of gas bubbles m the upward slug flow of CO+vater mixtures m a 0 8 cm I d vertical pope The correlation shows that when the gas and hqmd flow rates vary from 17 4 to 103 4 and 5 0 to 45 8 cm/s respectively, the frequency mcreases Hrlth hqrud velocity but decreases with gas velocity
Va nsmg velocrty of the gas bubble, cm/s
V,
superficial
V.,,
supertic~al velocity
velocity
of the gas phase, cm/s of the hqmd phase, cm/s
Greek symbol o
gas bubble or hqmd slug frequency,
S-I
Superscnpt M VAKILOTOJJAR
-
average
K JAVDANI*
Department of Chemical Engmeenng Tehran Unrversrty of Technology Tehran, Iran
REFERENCES
[I] ValulotoJlar
NOTATION
Lb Ls
length of the gas bubble, cm length of the hqmd slug, cm
*Present bchmond.
address Stauffer CA 94804. U S A
Ch.?mKa/ lznm.mw Peqtamn Press Lid
Chenucal
Co,
1200 S 47th
St
,
M , M S Thesis, Tehran Umversrty of Technology. Tehran. Iran 1978 121 Greg0ryG.A andScottD S,AIChEJ 1%9f5933 [3] Govler G W and Auz K , The Flow of Complex Mrxtures m hpe Van Nostrand Remhold. New York 1972 [4] Walk G B , One I)rmenslonal Two-Phase Flow McGrawHdl. New York I%9 [51 Wlute E T and Beardmore R H , Chem Engng Scr 1%2 17 351
scunce Vol 33 pp 2338-2360 I9Rl Pnntcd q Great Bntam
Application of Gale&in technique to diffusion problems in tubular reactor (Recerved 20 November 1979, accepted 4 March 1980) The Taylor’s dlsperslon theory[l] has been used by many works to predict chemical converstion m a lammar flow tubular reactor[2-71 Kulkarm and Vasudeva[8] have revlewed the earher work and have offered an emplrtcal expresslon for the effective dlsperston coefklent which turns out to be dependent on reactIon rate constant The vahdrty of reacrlon dependent dlsperslon coefficient for the unsteady state lammar reactor has been expenmentally estabhshed by Nlgam and Vasudeva[9] In a separate study they have shown{101 that the apphcabddy of the Fan and Hwang[ 111 analysts to the lirst order reaction m lammar flow of power law fhnds depend on the values of a, 7 and n They have suggested that the Fan and Hwang analysis holds m the case of reactmg systems for the values 0 5 zz n 52 5 provided P >O I5 and ~3> I 0 They have also developed empIrIcal expresstons for effective dlsperslon coefliclent which turns out to be dependent on the reactlon rate constant and permit the use of axial dlsperslon model with a much greater accuracy over a wide range of parameter values of practical Interest The purpose of this commumcatlon IS to present an analytlcal solution for the effective dlsperslon coefficient for a homogeneous first order ureverslble chemtcal reactlon takmg place m the bulk of a non-Newtoman hqmd by usmg Galerkm techmque [ 12,131
$$=Oat
Y=Oand
Y=l
(2b)
Galerkm technique has been used to solve eqns (I) and (2) Under this techmque, let C’,,, denotes the m th order approxlmatlon for C, and Cn =oo+oJI(Y)+ozfu)+
+anfm(Y)
(3)
, m) are the first m functions f,(Y), (J = I, 2,3, of an mfimte -iequeice v,(Y)}. (J = 1,2,3 ) such that each function f,(Y) IS a twice contmuously dlfferentlable function of Y on - I & Y 5 I and satlslies the boundary condltlons (2) Also, these functions form a lmearly Independent set of functions on -I
The functions
L(C,)f,(Y)YdY=Oforj=1,2,3,
,m
(4)
where
ANALYSIS
The steady-state two dlmenslonal convecelve dlffuslon equatlon m lammar flow of power law flmds In circular ducts for a movmg coordmate system with reference to axls 1s grven m Ref [ 1I] (eqn 31) Takmg mto account the homogeneous first order reactIon, this equation becomes
>-/3C The relevant
boundary
-&(2-(n+3)Y”+‘}
condttlons C=latX=O
$j=O
(I)
are Pa)
--&(2-(n+3)Y”+‘}$$
=0
This system of equations has resulted due to orthogonahty L(C,,,) to each f,(Y) over the cross-sectlon of the tube calculate a0 the foltowmg condltlons IS used
(5) of TO
I
L(C,,,)YdY=O Thus the Important feature of this techmque sequence of functions u,(Y)} satrsfymg the
(6) 1s to select the mentloned con-