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Variation in shape with size of bubbles in fluidised beds
Shorter Communications This is also the fraction of bed cross-sectional area occupied by bubbles, A,/A. AB is that area through which the upward particle flow of Or6 QB is assumed to occur. The average upward particle velocity then is given by
QB=A(U-U,,,,).
(11)
In which case (10) reduces to U,, = 0.6 Q.q/AB = O-6 U,
(5) tc = h,,,,/0*6 (U-U,)
and since there is no net particle movement out of the bed &OWN= O-6Qsl(A -AB)
(6) The average particle circulation time around the bed will then be
= hAIO.6 QB.
(7)
(8)
or
h= hti/(l--&)
(12)
In this form it is seen that the model breaks down at such high gas velocities that Eq. (12) becomes negative. The most likely purpose in calculating an average particle circulation time is to compare it with the reaction rate to test, in cases where the particle composition is changing, whether the bed can be assumed perfectly mixed (Refs. [2 and 31). For this purpose little more than an order of magnitude estimation is required when the simplifying assumptions in this analysis are likely to be reasonable. Department of Chemical Engineering University College London WCl, England
This requires knowledge of the expanded bed height, h, which can be exchanged for the estimated bubble velocity since the bubble hold-up,
Qds=A(h-hm,)
l-v].
A AB h hw n
from Eq. (2). Thus,
(10)
It should be noted that Ah,,,, is the volume of the bed at minimum fluidisation conditions and according to the two-phase
P. N. ROWE
NOTATION cross-sectional area of the bed, mp cross-sectional area occupied by bubbles, m* bed height, m bed height at U,, m number of bubbles per second crossing a horizontal plane, set-’ volumetric bubble flow, m3/sec residence time of a bubble in the bed, set average particle circulation time, set superficial gas velocity, m/set bubble rising velocity, m/set minimum fluidisation velocity, m/set average particle velocities up and down, m/set bubble volume, m3 fraction of bed occupied by bubbles
REFERENCES [I] ROWE P. N. et al., Trans. lnstn Chem. Engrs (1965) 43T271. [2] ROWE P. N., Chem. Engng Progr. (1964) 60 75. [3] KUNII D. and LEVENSPIEL O., Fluidization Engineering. Wiley, New York 1962. Chemical
Engineering
Science, 1973, Vol.
28,~~.
980-981.
Pergmon
Press.
Printed in Great Britain
Variation in shape with size of bubbles in fluidised beds (Received 28 July 1972) IT WAS recently reported (Ref. [l]) that X-ray pictures of bubbles in fluidised beds show a systematic change of shape with size. Relatively undisturbed bubbles are spherical but contain a particle wake that cuts off the lower part of the sphere. With increasing size the wake becomes proportionately greater so that larger bubbles appear flatter than smaller ones. It was proposed (in Ref. [l]) that volume varies as diameter raised to the power 2f and this reconciled measured bubble diameters with the known volumetric
980
bubble flow. It was subsequently confirmed by detailed measurements on a sample of rather more than a hundred bubbles. The approximate relation I’, = O+5ds5’2
(1)
was a convenient form for tbe purpose then intended but leaves the empirical coefficient with uncomfortable dimensions. This note reports more extensive measurements and the search for a more satisfactory relationship than Eq. (1).
Shorter Communications A total of 1867 bubbles were examined as X-ray photographs taken in beds of alumina, powdered glass, carbon and Ballotini fluidised by air. They varied in diameter between about 1 and 16 cm. Observations were made at 24 different combinations of flow rate, bed height and powdered material, an average sample of about 80 bubbles for each condition. The experimental details are exactly as reported previously (Ref. [l]). Every bubble in the sample was measured except those that were obviously on the verge of coalescing. The photograph was projected slightly larger than natural size on to a screen and the boundary of each bubble was traced by hand using a stylus linked to a PCD diaital data recorder- which registered -a series of boundary co-ordinates on paper tape. From these data the bubble diameter and volume were calculated assuming a shape of the general form shown in Fig. 1 and that the bubble is a body of rotation about an axis in the direction of motion.
can be fitted with an almost identical residual variance. This again makes unsatisfactory predictions outside the experimental range both at large diameters and as dB + 0 for hydrodynamic considerations would suggest a spherical bubble as Re + 0. A more acceptable form of equation is V,/Vs = e-“+%
(4)
which fits the data with a standard deviation & 0.25 which is a barely signiticant increase over that about Eq. (2) at the 1 per cent probability level. These equations together with the originally proposed form are all shown in Fig. 2. Equation (4) is the recommended
I
7
'i
Direction of motion
O
t
I
2
I
4
I
I
6
6
Bubblsdia.
I
I
10
12 d,
I
14
I
I
16
16
I-
20
cm
Fig. 2. The equations compared. Fig. 1. Typical bubble shape. Contrary to what was concluded from a very much smaller sample (Ref. [2]), there is no systematic change in shape with change of material amongst the four examined. There is, not surprisingly, considerable scatter in a plot of bubble diameter against volume but from so many data a trend such as expressed by Eq. (1) is obvious. It is most convenient to compare the bubble volume with that of a sphere of the same diameter and by the method of least squares the data lead to V,/Vs = 1 - O+I5Od,
(2)
with a standard deviation for a single observation of f 0.20. Although this describes data within the range examined it is obviously unsuitable at larger diameters and the equation V,/Vs = 0.86-O.O18Od,
(3)
one and, if it is close to the correct form, should predict the most probable volume ratio with an error of order & 1 per cent rather than something like a50 per cent which is all that is possible to estimate for an individual bubble. For most purposes the average over a relatively long period of time is the only quantity of interest. All the equations predict this fairly well within the experimental range but (4) is most likely to be reasonable outside it.
ds V, Vs
NOTATIGN bubble diameter, cm bubble space volume, cm3 sphere volume (7r/6)ds3,cm3
Department of Chemical Engineering University College London W.C.I., England
REFERENCES [l] ROWE P. N. and EVERETT D. J., Trans. Inst. Chem. Engrs (1972) 5042; 49; 55. [2] ROWE P. N. and PARTRIDGE B. A., Trans. Inst. Chem. Engrs (1965) 43T157.
981
P. N. ROWE A. J. WIDMER
QB=A(U-U,,,,).
(11)
In which case (10) reduces to U,, = 0.6 Q.q/AB = O-6 U,
(5) tc = h,,,,/0*6 (U-U,)
and since there is no net particle movement out of the bed &OWN= O-6Qsl(A -AB)
(6) The average particle circulation time around the bed will then be
= hAIO.6 QB.
(7)
(8)
or
h= hti/(l--&)
(12)
In this form it is seen that the model breaks down at such high gas velocities that Eq. (12) becomes negative. The most likely purpose in calculating an average particle circulation time is to compare it with the reaction rate to test, in cases where the particle composition is changing, whether the bed can be assumed perfectly mixed (Refs. [2 and 31). For this purpose little more than an order of magnitude estimation is required when the simplifying assumptions in this analysis are likely to be reasonable. Department of Chemical Engineering University College London WCl, England
This requires knowledge of the expanded bed height, h, which can be exchanged for the estimated bubble velocity since the bubble hold-up,
Qds=A(h-hm,)
l-v].
A AB h hw n
from Eq. (2). Thus,
(10)
It should be noted that Ah,,,, is the volume of the bed at minimum fluidisation conditions and according to the two-phase
P. N. ROWE
NOTATION cross-sectional area of the bed, mp cross-sectional area occupied by bubbles, m* bed height, m bed height at U,, m number of bubbles per second crossing a horizontal plane, set-’ volumetric bubble flow, m3/sec residence time of a bubble in the bed, set average particle circulation time, set superficial gas velocity, m/set bubble rising velocity, m/set minimum fluidisation velocity, m/set average particle velocities up and down, m/set bubble volume, m3 fraction of bed occupied by bubbles
REFERENCES [I] ROWE P. N. et al., Trans. lnstn Chem. Engrs (1965) 43T271. [2] ROWE P. N., Chem. Engng Progr. (1964) 60 75. [3] KUNII D. and LEVENSPIEL O., Fluidization Engineering. Wiley, New York 1962. Chemical
Engineering
Science, 1973, Vol.
28,~~.
980-981.
Pergmon
Press.
Printed in Great Britain
Variation in shape with size of bubbles in fluidised beds (Received 28 July 1972) IT WAS recently reported (Ref. [l]) that X-ray pictures of bubbles in fluidised beds show a systematic change of shape with size. Relatively undisturbed bubbles are spherical but contain a particle wake that cuts off the lower part of the sphere. With increasing size the wake becomes proportionately greater so that larger bubbles appear flatter than smaller ones. It was proposed (in Ref. [l]) that volume varies as diameter raised to the power 2f and this reconciled measured bubble diameters with the known volumetric
980
bubble flow. It was subsequently confirmed by detailed measurements on a sample of rather more than a hundred bubbles. The approximate relation I’, = O+5ds5’2
(1)
was a convenient form for tbe purpose then intended but leaves the empirical coefficient with uncomfortable dimensions. This note reports more extensive measurements and the search for a more satisfactory relationship than Eq. (1).
Shorter Communications A total of 1867 bubbles were examined as X-ray photographs taken in beds of alumina, powdered glass, carbon and Ballotini fluidised by air. They varied in diameter between about 1 and 16 cm. Observations were made at 24 different combinations of flow rate, bed height and powdered material, an average sample of about 80 bubbles for each condition. The experimental details are exactly as reported previously (Ref. [l]). Every bubble in the sample was measured except those that were obviously on the verge of coalescing. The photograph was projected slightly larger than natural size on to a screen and the boundary of each bubble was traced by hand using a stylus linked to a PCD diaital data recorder- which registered -a series of boundary co-ordinates on paper tape. From these data the bubble diameter and volume were calculated assuming a shape of the general form shown in Fig. 1 and that the bubble is a body of rotation about an axis in the direction of motion.
can be fitted with an almost identical residual variance. This again makes unsatisfactory predictions outside the experimental range both at large diameters and as dB + 0 for hydrodynamic considerations would suggest a spherical bubble as Re + 0. A more acceptable form of equation is V,/Vs = e-“+%
(4)
which fits the data with a standard deviation & 0.25 which is a barely signiticant increase over that about Eq. (2) at the 1 per cent probability level. These equations together with the originally proposed form are all shown in Fig. 2. Equation (4) is the recommended
I
7
'i
Direction of motion
O
t
I
2
I
4
I
I
6
6
Bubblsdia.
I
I
10
12 d,
I
14
I
I
16
16
I-
20
cm
Fig. 2. The equations compared. Fig. 1. Typical bubble shape. Contrary to what was concluded from a very much smaller sample (Ref. [2]), there is no systematic change in shape with change of material amongst the four examined. There is, not surprisingly, considerable scatter in a plot of bubble diameter against volume but from so many data a trend such as expressed by Eq. (1) is obvious. It is most convenient to compare the bubble volume with that of a sphere of the same diameter and by the method of least squares the data lead to V,/Vs = 1 - O+I5Od,
(2)
with a standard deviation for a single observation of f 0.20. Although this describes data within the range examined it is obviously unsuitable at larger diameters and the equation V,/Vs = 0.86-O.O18Od,
(3)
one and, if it is close to the correct form, should predict the most probable volume ratio with an error of order & 1 per cent rather than something like a50 per cent which is all that is possible to estimate for an individual bubble. For most purposes the average over a relatively long period of time is the only quantity of interest. All the equations predict this fairly well within the experimental range but (4) is most likely to be reasonable outside it.
ds V, Vs
NOTATIGN bubble diameter, cm bubble space volume, cm3 sphere volume (7r/6)ds3,cm3
Department of Chemical Engineering University College London W.C.I., England
REFERENCES [l] ROWE P. N. and EVERETT D. J., Trans. Inst. Chem. Engrs (1972) 5042; 49; 55. [2] ROWE P. N. and PARTRIDGE B. A., Trans. Inst. Chem. Engrs (1965) 43T157.
981
P. N. ROWE A. J. WIDMER