- Email: [email protected]
The distribution of bubbles in a gas fluidised bed
Chemical Engineering Science, 1970, Vol. 25, pp. 1587-l 593.
Pergamon Press.
Printed in Great Britain.
The distribution of bubbles in a gas fluidised bed R. MATSUNOt and P. N. ROWE Department of Chemical Engineering, University College London, London, England (Received
16 March
1970)
Abstract-Based on the observed pattern of coalescence of bubble pairs in vertical alignment in a two-dimensional bed, a theory is developed which describes how bubble size and frequency changes with height in a gas fluidised bed. It is confirmed by comparison with the reported data from 10 separate investigations which can together be described by -d,1’2(dn/dh)
= 0497 (ndgo.54)‘.64
where dB is the average bubble diameter and n the number frequency per unit cross sectional area, both at height h above the distributor.
INTRODUCTION
DURING the past decade much has been discovered about the nature and behaviour of bubbles in gas fluidised beds (e.g.[ 1,2]) and a number of models of fluidised beds as chemical reactors have been proposed [ 1,3-51 which depend on this knowledge. In each of these models it is necessary to know something of the size and distribution of the bubbles. A number of workers [6-1.51 have used various methods to measure and describe the bubble population but, apart from Kobayashi et a1.[9], there has been little attempt to rationalise the many disparate data. In this paper it is shown that a simple model based on coalescence successfully correlates the observations and a semi-empirical equation is presented. The relationship between bubble frequency and diameter By observing bubble pairs in vertical alignment in two-dimensional beds, Toei and Matsuno showed [ 161 the time needed for coalescence to occur depends only on the ratio of bubble diameters, dBl/dB2 and the initial separation, C/dBl. Their results can be expressed as, tf’resent
address:
Department
of Industrial
Chemistry,
.
(1)
We assume here that this relationship applies in normal three-dimensional beds in which there is a distribution of bubbles both in size and in position. In this case the average size of the leading bubble, dsl, can be assumed to be that of the following one, dBz, so that this ratio disappears from (1) and the suffix, 1 can be dropped. Let the number concentration of bubbles at a height h above the distributor be N. The relation between this and the number frequency of bubbles, n, crossing the horizontal plane at h is given by N = n/U,
(2)
and the average volume of bed that contains one bubble is I/N = Us/n.
(3)
If the bubbles were distributed uniformly throughout the bed, randomly or in some order, the average distance between the centres of neighbouring bubbles would be proportional to Okayama University, Okayama, Japan.
1587
R. MATSUNO
and P. N. ROWE
the cube root of this value. If on the other hand bubbles always rose along a finite number of vertical paths, the average vertical distance between the centres of neighbours, c, would be directly proportional to the volume associated with a bubble. In general we write, C z (Uj&z)“‘”
or, using(l), -d(n/Us)/dt
0~ l/t,
(2) and (4),
(5)
&‘,z (dnldh).
(13)
Equation (7) may then be written, -dP(dn/dh)
(4)
where 1 < a < 3. The rate at which bubble concentration changes as a result of coalescence will be inversely proportional to the time needed for this to occur, -dN/dt
co (1+1/2b+
=f’(
(U,,,)““d,-1)
(14)
= f”[ (dB”“/n) d,-“1 l/a
(15)
or, writing the function in an empirical form, -dB”“(dn/dh)
= K[n.dg(a-l’Z)]P.
(16)
This relates the rate of change of bubble frequency with height to the frequency and average bubble diameter at that height. It remains to determine the constants K, a and p from experimental data. CORRELATION
m (Us/dB)f-l((Ue/n)““.ds-‘)
OF
THE
DATA
Data from 10 different investigations were (6) or, since U, = dh/dt, examined [6- 151. Widely different methods of observation were used by the various workers -_[d(n/U,)/dhld, ~f-l((U&z)“a. 4-l). (7) who either measured frequency or bubble size for no one measured both together. Thus, from According to the two-phase theory and continuthe start it is necessary to use (9). It will be ity, noticed that the unknown exponent, b does not ndsb CC(U-u,,). (8) appear in the final correlating Eq. (16) but it is necessary to assume a value in order to handle If average shape is independent of absolute size the available data. The work of Everett and then b = 3 but there is some evidence that shape Rowe [ 171 has shown that this exponent varies does change consistently as bubbles grow and from material to material between the values of that b may be a little less than 3 so that generality about 2.5 and 3-O but it will here be assumed that is preserved by writing, b= 3. A total of 115 values of n or dB at a known ds 00 [ ( U - u,.f.) InI‘lb. (9) value of h were examined. Those data where dB was greater than half the bed diameter were It is generally agreed that U, COdgl/* and so we omitted as being relevant to slugging conditions may write, and inappropriate to the assumed model. From them, dB or n, (dn/dh) and ndB(a-1/2’ were d,j x d(n/Us)/dhad, x d(n/d,“2)/dh (10) calculated. For this last term, a was given successive values, 1.0, 1.1, 1.2, etc. in the range 1 < a < 3. The values of p and K were then (11) found by taking logarithms and using the method of least squares to fit a straight line. The value of (1+1/2b) a (= 1.04) that gave the minimum residual error (12) CO(U-u~,~.)‘,bn1’2b(dnldh) variance was then taken as the ‘best’ value 1588
The distribution of bubbles in agas fluidised bed
together with the related values of p and K. The resulting equation is,
led to a very similar equation and degree of correlation but ascribed what was judged to be undue weight to the three largest values with -ds”“(dn/d,+) = 0*097(t~d,~~~~)'~~~ inattention to many more low (17) corresponding value points. The method leading to Eq. (17) which is plotted together with all the original data was therefore preferred. in Fig. 1. DISCUSSION Minimising the squares of the deviations of The least squares value found for the exponent the logarithms of the data about a curve is not a was 1.04 which is not significantly different the correct procedure unless error is proportional to the absolute value of the observation which is from unity. It can be concluded from this that very rarely the case. A curve was also fitted by bubbles do indeed rise one after another along varying incrementally the values ascribed to preferred paths and the effect of increasing the p as well as to a and finding K by the method of overall gas flow rate is to increase the number in least squares using the unconverted data. This a given path rather than to create new paths.
G
3
20 cm i.d.
40 cm i.d.
10 cm i.d.
1Ocmx tOcm
4 in. i.d.
Kunii et a1.[12]
Park et a/.[ 131
Toei et&[141
Yasui ef al.[ 151
IO cm i.d.
Kobayashi ef a/.[91
Kunii et al.1 1 l]
30.8 cm id.
Geldart et a1.[81
20 cm i.d.
6 in. i.d.
Calderbank et a[.[71
Kobayashi ef a1.[10]
3 in. x 6 in.
Bed dimension
Baumgarten et a/.[61
Investigator
Porous ceramic disc (I+ in. thick)
Sintered ceramic
Canvas + perforated aluminium plate, hole 3 mm i.d. numbers 69 Canvas + perforated aluminium plate, hole 5 mm i.d. pitch 1 cm, square Porous stainless steel
Perforated plate of polyvinyl chloride thickness 5 mm hole 0.5 mm numbers 1850 Sintered vinyl chloride resin
Sintered bronze 25 p mean pore dia. Vyon porous plate
Sintered bronze 3/32 in. thick
Distributor
Glass beads
Electircally conductive coke (nearly spherical) Glass beads
Microspher’cal catal I st
Microspherical catalyst
Silica gel
Silica gel
Sand
Catalyst
Glass beads
particle used
Solid
242 p 41,75 and 175p (same fractions) 75cL
SO- 100 mesh
344 p 154 /A 86EL
15op
150 /.L
194cL
194 /I
tOOcL 83~~ 192 /A mean 128/.~
74/J
Particle size
-
2.5
1846 1.777 1.784
1.91
1.91
2.2
2.85
2.65 2.633 2.619 -
-
Particle density (g/cm”)
12.2 2.80 2.80
1.12 1.12
3.7-8.0
10~0-13 3-O-18 2.5-6.3
4.0-30.0
20.4
3-25
5.7-28.5
2.3-5.14 16-3.29 5.76-9.62 5%7.5
1.5-22
Range of fluidising gal velocity (cmlsec)
8.11
2-25
6.80 1.83 0.63
2.0
3.4
2.1
2.85
1.07 0964 3.74 1.21-140
0.733
UmJ, (cmlsec)
Table 1. List of experimental conditions
Electroresistivity probe (inside the bed) X-ray (outside the bed) Light probe (inside the bed)
Capacitance probe (inside the bed)
Light probe (inside the bed) Capacitance probe (inside the bed)
(outside the bed) Capacitance probe (outside the bed) Capacitance probe (inside the bed) and visual observation Light probe (inside the bed)
y-w
Detector or probe
2 ft
50 cm
30 cm
50 cm
30 cm
67 cm
60 cm
30cm
Maximum 3.70 ft
14.85 in.
Bed height
0
Key
Screen+ stainless steel 0402 in. thick porosity 74 /*
Porous stainless steel Qin. thick porosity 20 p Porous ceramic +- 14in. thick porosity 55 p and IlOp
Glass beads 175 p 548 . 7.62,12.7
R. MATSUNO
and P. N. ROWE
Many observers would agree that this is a fair description. Another way of saying this is that the lateral movement of bubbles is small compared with their vertical motion. Equation (17) can be integrated, -dnldh
= 0*097n1+4dBo.3s6.
(18)
Using Eq. (9) with b = 3, -dn/n1’51 = B, (U - U,.f,)o.13 dh
It
=
(19)
in this paper, the finding that bubble frequency varies inversely with the square of bed height and the assumption that bubble volume varies as the cube of its diameter are not all compatible. Presently available data cannot resolve this problem both because size and frequency have not been measured independently and because they are not precise enough. X-ray observations should eventually rectify this and meanwhile the correlation shown in Fig. 1 gives a reasonable means of prediction.
NOTATION
WV
(h+h,)‘/051
C
B2 +
(h+ho)2(U-Um.f.)1’4
*
(21)
The constant ho is related to the bubble frequency at the distributor and thus might be expected to depend very much on its design. X-ray observation has shown (17) that nh2 = constant at a given gas flow rate using a porous plate distributor which indicates that ho is at least very small for that design. Combining this finding with (8) leads to dB = const.f(
It has been widely linearly with h at a requires b = 2 in the an improbable value.
U - U,,)
x
h21b .
(22)
reported[3] that dB varies given gas flow rate which above equation, but this is Thus, the model proposed
distance between centres of upper and lower bubble, cm diameter of bubble (suffixes 1 and 2 refer to the leading and following bubble respectively), cm integration constant, cm ho N numer of bubbles per unit volume of bed, cme3 of bubbles per unit crossn frequency sectional area of bed, numbers/cm2.sec TC dimensionless time required for coalescence t time, set tc time required for coalescence, set superficial gas velocity, cmlsec rising velocity of bubble, cmlsec u rn.f. minimum fluidisation velocity, cm/set h height from distributor, cm constants B,K
u:
REFERENCES
111DAVIDSON J. F. and HARRISON D., Fluidisedk’arricles. Cambridge University Press 1963. t21 ROWE P. N., PARTRIDGE B. A. and LYALL E., Chem. EngngSci_l964 19 97j. O., FIuidisation Engineering. Wiley 1969. r31 KUNII D. and LEVENSPIEL B. A., Trans. Instn them. Engrs 1966 44 335. [41 ROWE P. N. and PARTRIDGE P. H. and TOOR F. D.. Proc. Int. Svm~. on Fluidisatiun. D. 373. Netherlands Universitv Press 1967. 151 CALDERBANK P. K. and PIGFORD k. L.,A.I.Ch.E:JI 19606 115. ‘_ [61 BAUMGARTEN P. H., TOOR F. D. and LANCASTER R. H., Proc. Inf. Syrup. on Fluidisation, p. 652. Netherlands 171 CALDERBANK University Press 1967. PI GELDART D. and KELSEY J. R., Tripartite Chem. Engng. Conf. Symp. on Fluidisation II, p. 14, Montreal, Canada, September 1968. H., ARAI F. and CHIBA T., Kagaku Kogaku 1965 29 858. [91 KOBAYASHI H., ARAI F., TANAKA Y., SAKAGUCHI Y., SAGAWA N., SUNAGAWA T., CHIBA T. and [lOI KOBAYASHI TAKAHASHI K., Pre-print of the 6th Symp. on Chem. Reaction Engng., p. 13. Chem. Engng. Society of Japan, Nagoya, Japan, November 1966. 1592
The distribution of bubbles in a gas fluidised bed [III KUNII D., HIRAKI I. and YOSHIDA K., Pre-print of the 4th General Symp. of Chem. Engng., p. 37. Society of Japan, Tokyo, Japan, November 1965. 1121 KUNII D., INOUE E., KIKUCHI K., HIRAKI I. and YOSHIDA K., Pre-print of_ the 32ndAnnual Meeting _ c$ . Chem. Engng Society ofJapan, p. 3 16. Tokyo, Japan, April 1967. r131 PARK W. H., KUNG W. K.. CAPES C. E. and OSBERG G. L.. C&m. Enemy Sci. 1969 24 85 1. i14i TOE1 R., MATSUNO R., KOJIMA H., NAGAI Y., NAKAGAWA K. and Yu S., Mem. FUC. Engng Kyoto Univ. 1965 27 475.
[151 YASUIG.andJOHANSON L.N.,A.I.Ch.E.JI 19584445. [16] TOE1 R. and MATSUNO R., Proc. Int. Symp. on Fluidisation, p. 271. Netherlands University Press 1967. 1171 EVERETT D. J. and ROWE P. N., X-Ray Studies of Fluidised Beds, University College London 1970. (TO be published.) Resume- Une theorie decrivant le changement de la frequence et de la grosseur des bulles en fonction de la hauteur, dans une couche de gaz fluidisee, est developpee sur le modele observe de la coalescence de paires de bulles en alignement vertical dans une couche bi-dimensionde. Ce fait est confirm6 par la comparaison avec les donntes provenant de 10 investigations &par&es, qui, ensemble, peuvent &tre d&rites par: = 0 7097 (nd Bos.4)1.64
--d,“*(dn/dh)
dans laquelle dB est le diametre moyen dune bulle et n la frequence en nombre par section transversale unitaire, a la hauteur h au-dessus du distributeur. Zusammenfassung-Auf Grund des beobachteten
Bildes der Koaleszenz von Blasenpaaren in Vertikalausrichtung in einem zweidimensionalen Bett, wird eine Theorie entwickelt, die darlegt wie sich die Blasengrosse und die Frequenz in einem gas-durchwirbeltem Bette mit der Hohe Indern. Sie wird bestltigt durch Vergleich mit den aus 10 Einzeluntersuchungen erhaltenen Daten, die zusammen durch den Ausdruck -d,“2(dn/dh)
= 0,097 (ndE0~54)‘.64
beschrieben werden kiinnen, worin dB der durchschnittliche Blasendurchmesser ist und n die Zahlenfrequenz pro Querschnittsfhacheneinheit, beide in einer HGhe h oberhalb des Verteilers.
1593
Pergamon Press.
Printed in Great Britain.
The distribution of bubbles in a gas fluidised bed R. MATSUNOt and P. N. ROWE Department of Chemical Engineering, University College London, London, England (Received
16 March
1970)
Abstract-Based on the observed pattern of coalescence of bubble pairs in vertical alignment in a two-dimensional bed, a theory is developed which describes how bubble size and frequency changes with height in a gas fluidised bed. It is confirmed by comparison with the reported data from 10 separate investigations which can together be described by -d,1’2(dn/dh)
= 0497 (ndgo.54)‘.64
where dB is the average bubble diameter and n the number frequency per unit cross sectional area, both at height h above the distributor.
INTRODUCTION
DURING the past decade much has been discovered about the nature and behaviour of bubbles in gas fluidised beds (e.g.[ 1,2]) and a number of models of fluidised beds as chemical reactors have been proposed [ 1,3-51 which depend on this knowledge. In each of these models it is necessary to know something of the size and distribution of the bubbles. A number of workers [6-1.51 have used various methods to measure and describe the bubble population but, apart from Kobayashi et a1.[9], there has been little attempt to rationalise the many disparate data. In this paper it is shown that a simple model based on coalescence successfully correlates the observations and a semi-empirical equation is presented. The relationship between bubble frequency and diameter By observing bubble pairs in vertical alignment in two-dimensional beds, Toei and Matsuno showed [ 161 the time needed for coalescence to occur depends only on the ratio of bubble diameters, dBl/dB2 and the initial separation, C/dBl. Their results can be expressed as, tf’resent
address:
Department
of Industrial
Chemistry,
.
(1)
We assume here that this relationship applies in normal three-dimensional beds in which there is a distribution of bubbles both in size and in position. In this case the average size of the leading bubble, dsl, can be assumed to be that of the following one, dBz, so that this ratio disappears from (1) and the suffix, 1 can be dropped. Let the number concentration of bubbles at a height h above the distributor be N. The relation between this and the number frequency of bubbles, n, crossing the horizontal plane at h is given by N = n/U,
(2)
and the average volume of bed that contains one bubble is I/N = Us/n.
(3)
If the bubbles were distributed uniformly throughout the bed, randomly or in some order, the average distance between the centres of neighbouring bubbles would be proportional to Okayama University, Okayama, Japan.
1587
R. MATSUNO
and P. N. ROWE
the cube root of this value. If on the other hand bubbles always rose along a finite number of vertical paths, the average vertical distance between the centres of neighbours, c, would be directly proportional to the volume associated with a bubble. In general we write, C z (Uj&z)“‘”
or, using(l), -d(n/Us)/dt
0~ l/t,
(2) and (4),
(5)
&‘,z (dnldh).
(13)
Equation (7) may then be written, -dP(dn/dh)
(4)
where 1 < a < 3. The rate at which bubble concentration changes as a result of coalescence will be inversely proportional to the time needed for this to occur, -dN/dt
co (1+1/2b+
=f’(
(U,,,)““d,-1)
(14)
= f”[ (dB”“/n) d,-“1 l/a
(15)
or, writing the function in an empirical form, -dB”“(dn/dh)
= K[n.dg(a-l’Z)]P.
(16)
This relates the rate of change of bubble frequency with height to the frequency and average bubble diameter at that height. It remains to determine the constants K, a and p from experimental data. CORRELATION
m (Us/dB)f-l((Ue/n)““.ds-‘)
OF
THE
DATA
Data from 10 different investigations were (6) or, since U, = dh/dt, examined [6- 151. Widely different methods of observation were used by the various workers -_[d(n/U,)/dhld, ~f-l((U&z)“a. 4-l). (7) who either measured frequency or bubble size for no one measured both together. Thus, from According to the two-phase theory and continuthe start it is necessary to use (9). It will be ity, noticed that the unknown exponent, b does not ndsb CC(U-u,,). (8) appear in the final correlating Eq. (16) but it is necessary to assume a value in order to handle If average shape is independent of absolute size the available data. The work of Everett and then b = 3 but there is some evidence that shape Rowe [ 171 has shown that this exponent varies does change consistently as bubbles grow and from material to material between the values of that b may be a little less than 3 so that generality about 2.5 and 3-O but it will here be assumed that is preserved by writing, b= 3. A total of 115 values of n or dB at a known ds 00 [ ( U - u,.f.) InI‘lb. (9) value of h were examined. Those data where dB was greater than half the bed diameter were It is generally agreed that U, COdgl/* and so we omitted as being relevant to slugging conditions may write, and inappropriate to the assumed model. From them, dB or n, (dn/dh) and ndB(a-1/2’ were d,j x d(n/Us)/dhad, x d(n/d,“2)/dh (10) calculated. For this last term, a was given successive values, 1.0, 1.1, 1.2, etc. in the range 1 < a < 3. The values of p and K were then (11) found by taking logarithms and using the method of least squares to fit a straight line. The value of (1+1/2b) a (= 1.04) that gave the minimum residual error (12) CO(U-u~,~.)‘,bn1’2b(dnldh) variance was then taken as the ‘best’ value 1588
The distribution of bubbles in agas fluidised bed
together with the related values of p and K. The resulting equation is,
led to a very similar equation and degree of correlation but ascribed what was judged to be undue weight to the three largest values with -ds”“(dn/d,+) = 0*097(t~d,~~~~)'~~~ inattention to many more low (17) corresponding value points. The method leading to Eq. (17) which is plotted together with all the original data was therefore preferred. in Fig. 1. DISCUSSION Minimising the squares of the deviations of The least squares value found for the exponent the logarithms of the data about a curve is not a was 1.04 which is not significantly different the correct procedure unless error is proportional to the absolute value of the observation which is from unity. It can be concluded from this that very rarely the case. A curve was also fitted by bubbles do indeed rise one after another along varying incrementally the values ascribed to preferred paths and the effect of increasing the p as well as to a and finding K by the method of overall gas flow rate is to increase the number in least squares using the unconverted data. This a given path rather than to create new paths.
G
3
20 cm i.d.
40 cm i.d.
10 cm i.d.
1Ocmx tOcm
4 in. i.d.
Kunii et a1.[12]
Park et a/.[ 131
Toei et&[141
Yasui ef al.[ 151
IO cm i.d.
Kobayashi ef a/.[91
Kunii et al.1 1 l]
30.8 cm id.
Geldart et a1.[81
20 cm i.d.
6 in. i.d.
Calderbank et a[.[71
Kobayashi ef a1.[10]
3 in. x 6 in.
Bed dimension
Baumgarten et a/.[61
Investigator
Porous ceramic disc (I+ in. thick)
Sintered ceramic
Canvas + perforated aluminium plate, hole 3 mm i.d. numbers 69 Canvas + perforated aluminium plate, hole 5 mm i.d. pitch 1 cm, square Porous stainless steel
Perforated plate of polyvinyl chloride thickness 5 mm hole 0.5 mm numbers 1850 Sintered vinyl chloride resin
Sintered bronze 25 p mean pore dia. Vyon porous plate
Sintered bronze 3/32 in. thick
Distributor
Glass beads
Electircally conductive coke (nearly spherical) Glass beads
Microspher’cal catal I st
Microspherical catalyst
Silica gel
Silica gel
Sand
Catalyst
Glass beads
particle used
Solid
242 p 41,75 and 175p (same fractions) 75cL
SO- 100 mesh
344 p 154 /A 86EL
15op
150 /.L
194cL
194 /I
tOOcL 83~~ 192 /A mean 128/.~
74/J
Particle size
-
2.5
1846 1.777 1.784
1.91
1.91
2.2
2.85
2.65 2.633 2.619 -
-
Particle density (g/cm”)
12.2 2.80 2.80
1.12 1.12
3.7-8.0
10~0-13 3-O-18 2.5-6.3
4.0-30.0
20.4
3-25
5.7-28.5
2.3-5.14 16-3.29 5.76-9.62 5%7.5
1.5-22
Range of fluidising gal velocity (cmlsec)
8.11
2-25
6.80 1.83 0.63
2.0
3.4
2.1
2.85
1.07 0964 3.74 1.21-140
0.733
UmJ, (cmlsec)
Table 1. List of experimental conditions
Electroresistivity probe (inside the bed) X-ray (outside the bed) Light probe (inside the bed)
Capacitance probe (inside the bed)
Light probe (inside the bed) Capacitance probe (inside the bed)
(outside the bed) Capacitance probe (outside the bed) Capacitance probe (inside the bed) and visual observation Light probe (inside the bed)
y-w
Detector or probe
2 ft
50 cm
30 cm
50 cm
30 cm
67 cm
60 cm
30cm
Maximum 3.70 ft
14.85 in.
Bed height
0
Key
Screen+ stainless steel 0402 in. thick porosity 74 /*
Porous stainless steel Qin. thick porosity 20 p Porous ceramic +- 14in. thick porosity 55 p and IlOp
Glass beads 175 p 548 . 7.62,12.7
R. MATSUNO
and P. N. ROWE
Many observers would agree that this is a fair description. Another way of saying this is that the lateral movement of bubbles is small compared with their vertical motion. Equation (17) can be integrated, -dnldh
= 0*097n1+4dBo.3s6.
(18)
Using Eq. (9) with b = 3, -dn/n1’51 = B, (U - U,.f,)o.13 dh
It
=
(19)
in this paper, the finding that bubble frequency varies inversely with the square of bed height and the assumption that bubble volume varies as the cube of its diameter are not all compatible. Presently available data cannot resolve this problem both because size and frequency have not been measured independently and because they are not precise enough. X-ray observations should eventually rectify this and meanwhile the correlation shown in Fig. 1 gives a reasonable means of prediction.
NOTATION
WV
(h+h,)‘/051
C
B2 +
(h+ho)2(U-Um.f.)1’4
*
(21)
The constant ho is related to the bubble frequency at the distributor and thus might be expected to depend very much on its design. X-ray observation has shown (17) that nh2 = constant at a given gas flow rate using a porous plate distributor which indicates that ho is at least very small for that design. Combining this finding with (8) leads to dB = const.f(
It has been widely linearly with h at a requires b = 2 in the an improbable value.
U - U,,)
x
h21b .
(22)
reported[3] that dB varies given gas flow rate which above equation, but this is Thus, the model proposed
distance between centres of upper and lower bubble, cm diameter of bubble (suffixes 1 and 2 refer to the leading and following bubble respectively), cm integration constant, cm ho N numer of bubbles per unit volume of bed, cme3 of bubbles per unit crossn frequency sectional area of bed, numbers/cm2.sec TC dimensionless time required for coalescence t time, set tc time required for coalescence, set superficial gas velocity, cmlsec rising velocity of bubble, cmlsec u rn.f. minimum fluidisation velocity, cm/set h height from distributor, cm constants B,K
u:
REFERENCES
111DAVIDSON J. F. and HARRISON D., Fluidisedk’arricles. Cambridge University Press 1963. t21 ROWE P. N., PARTRIDGE B. A. and LYALL E., Chem. EngngSci_l964 19 97j. O., FIuidisation Engineering. Wiley 1969. r31 KUNII D. and LEVENSPIEL B. A., Trans. Instn them. Engrs 1966 44 335. [41 ROWE P. N. and PARTRIDGE P. H. and TOOR F. D.. Proc. Int. Svm~. on Fluidisatiun. D. 373. Netherlands Universitv Press 1967. 151 CALDERBANK P. K. and PIGFORD k. L.,A.I.Ch.E:JI 19606 115. ‘_ [61 BAUMGARTEN P. H., TOOR F. D. and LANCASTER R. H., Proc. Inf. Syrup. on Fluidisation, p. 652. Netherlands 171 CALDERBANK University Press 1967. PI GELDART D. and KELSEY J. R., Tripartite Chem. Engng. Conf. Symp. on Fluidisation II, p. 14, Montreal, Canada, September 1968. H., ARAI F. and CHIBA T., Kagaku Kogaku 1965 29 858. [91 KOBAYASHI H., ARAI F., TANAKA Y., SAKAGUCHI Y., SAGAWA N., SUNAGAWA T., CHIBA T. and [lOI KOBAYASHI TAKAHASHI K., Pre-print of the 6th Symp. on Chem. Reaction Engng., p. 13. Chem. Engng. Society of Japan, Nagoya, Japan, November 1966. 1592
The distribution of bubbles in a gas fluidised bed [III KUNII D., HIRAKI I. and YOSHIDA K., Pre-print of the 4th General Symp. of Chem. Engng., p. 37. Society of Japan, Tokyo, Japan, November 1965. 1121 KUNII D., INOUE E., KIKUCHI K., HIRAKI I. and YOSHIDA K., Pre-print of_ the 32ndAnnual Meeting _ c$ . Chem. Engng Society ofJapan, p. 3 16. Tokyo, Japan, April 1967. r131 PARK W. H., KUNG W. K.. CAPES C. E. and OSBERG G. L.. C&m. Enemy Sci. 1969 24 85 1. i14i TOE1 R., MATSUNO R., KOJIMA H., NAGAI Y., NAKAGAWA K. and Yu S., Mem. FUC. Engng Kyoto Univ. 1965 27 475.
[151 YASUIG.andJOHANSON L.N.,A.I.Ch.E.JI 19584445. [16] TOE1 R. and MATSUNO R., Proc. Int. Symp. on Fluidisation, p. 271. Netherlands University Press 1967. 1171 EVERETT D. J. and ROWE P. N., X-Ray Studies of Fluidised Beds, University College London 1970. (TO be published.) Resume- Une theorie decrivant le changement de la frequence et de la grosseur des bulles en fonction de la hauteur, dans une couche de gaz fluidisee, est developpee sur le modele observe de la coalescence de paires de bulles en alignement vertical dans une couche bi-dimensionde. Ce fait est confirm6 par la comparaison avec les donntes provenant de 10 investigations &par&es, qui, ensemble, peuvent &tre d&rites par: = 0 7097 (nd Bos.4)1.64
--d,“*(dn/dh)
dans laquelle dB est le diametre moyen dune bulle et n la frequence en nombre par section transversale unitaire, a la hauteur h au-dessus du distributeur. Zusammenfassung-Auf Grund des beobachteten
Bildes der Koaleszenz von Blasenpaaren in Vertikalausrichtung in einem zweidimensionalen Bett, wird eine Theorie entwickelt, die darlegt wie sich die Blasengrosse und die Frequenz in einem gas-durchwirbeltem Bette mit der Hohe Indern. Sie wird bestltigt durch Vergleich mit den aus 10 Einzeluntersuchungen erhaltenen Daten, die zusammen durch den Ausdruck -d,“2(dn/dh)
= 0,097 (ndE0~54)‘.64
beschrieben werden kiinnen, worin dB der durchschnittliche Blasendurchmesser ist und n die Zahlenfrequenz pro Querschnittsfhacheneinheit, beide in einer HGhe h oberhalb des Verteilers.
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