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Prediction of voidage fraction near bubbles in fluidized beds
Chemical Engineering Science, 1968. Vol. 23, pp. 396-397.
Pergamon Press.
Printed in Great Britain.
Prediction of voidage fraction near bubbles in fluidised beds (Received
9 October
MEASUREMENTS of changes in voidage near bubbles rising in two dimensional gas fluidised beds have been made recently by Lockett and Harrison[ 11,however the theoretical analysis which they present is in poor agreement with the results obtained and they conclude that only qualitative predictions are possible. Figure I shows group averages of the porosity data close to the vertical axis above the bubble, using the data provided in Figs. 1 and 2 of their paper, compared with various theoretical curves. Curve (I) is the theoretical result obtained from: (i) an analysis in two dimensions, equivalent to the three dimensional analysis given by Jackson[2], for the motion of particles and gas caused by a bubble in a fluidised bed, (ii) the Carman-Kozeny relationship for the dependenceon porosity of the resistance of a fixed bed to interstitial gas flow. This is the theory presented by Lockett and Hanison[ll and, separately, by Stewart[3]. As pointed out by Lockett and Harrison the discrepancy between theory and experiment is somewhat reduced by using Richardson and Zaki’s well-known fluidised bed relationship[4] in place of the Carman-Kozeny equation; the result however is still far from satisfactory- Curve (11). The bubble velocity Ub, is known to be related to the bubble radius, u, by the expression
l_Jb= k dga. In their analysis for porosity distribution Lockett and Harrison used the theoretical value of k. However, Stewart
040 hiday on verticulAxis abw &rbb
1967)
[3] showed that in predicting the gas pressure distribution determined experimentally by Reuter[S], the experimentally determined value of k gave better agreement between theory and experiment. Moreover the remaining differences could be accounted for by allowing for the viscosity of the fluidised bed. The theoretical velocity coefficient for two dimensional bubbles is O-5. Pyle and Harrison[6] recently reported measurements of the bubble velocity in two-dimensional fluidised beds and found a relationship
where a was the radius of curvature of the nose of the bubble measured from photographs of bubbles by “the best fit to a segment of arc extending some 30” on either side of the stagnation point at the nose”. However, in studies of the velocity of three dimensional bubbles and of the penetration of gas from within a two dimensional bubble into the particulate phase, Rowe[7, 81 used the radius of the circle which approximated most closely to the entire bubble boundary excluding the flattened wake portion at the bottom of the bubble. For an analysis where the movements of the particles in the whole region around a bubble are concerned, this method of measuring the radius is considered the most appropriate one. For uniformly sized Ballotini particles, and allowing for wall effects, Pyle and Harrison found as an alternative form to Eq. (1) that:
Iv\
I k- 0.5, Carman- KoJay
‘\,
In
&
044
; i
\
U
l
05, Richardsont zaki
m
.
0.7,
Iv
l
0.9,
‘\
Distance from Bubble Ccntre
(Rodii)
Fig. 1.Voidage above a bubble in a fluidised bed.
396
. -
l
U, = 16.1 Abl”
where A* is the bubble area in cm*. The photographs of two dimensional bubbles in uniformly sized 230~ Ballotini published by Rowe et af.[8] show a ratio between the height of the bubble and the diameter of the circle approximating to the bubble boundary excluding the wake, of 0.83. Using this to provide a relationship between Ab and a bubble radius of the type used by Rowe, by substitution Eq. (2) becomes: Ub = 0.65dga.
A satisfactory fit to the porosity data for sample A can be obtained by choosing k = 0.7 in the theoretical analysis -curve (III). Since Rowe et a/.[71 find that bubble rise velocity and bubble shape are dependent on the particles forming the fluidised bed, the value of 0.7 is not significantly different from the 0.65 of Eq. (3). The results for sample E cannot be fitted directly in this way. Using k = 0.9, curve (IV) is obtained which although close to the experimental results, shows a significantly different trend. Lockett and Harrison suggest three explanations of the lack of agreement between theory and experiment. (i) A number of approximations and assumptions are made in the theory, the most important of which is the elimination of variable voidage from equations where it has only a second order effect. (ii) A three dimensional component could be present in bubble motion in the two dimensional bed. (iii) Jackson’s analysis is only applicable to particles of uniform size and consequently cannot be expected to be satisfactory for a fluidised bed containing particles with a wide size range. The first two explanations do not seem to be necessary since the results for sample A may be predicted satisfactorily by inserting a more appropriate bubble velocity formula in the analysis. Collins’[9] observations with regard to a three dimensional component introduced by the front and back walls in two dimensional apparatus relate to an air/water system where the wall is wetted by the water and boundary layer effects ensure that some water will flow down all walls. In experiments with a non-wetting air/mercury system it was found that this effect could be completely avoided with carefully
executed experiments [ IO] and a fluidised bed system where particles are only slightly retarded by the walls would be expected to be similar. Jackson’s analysis is completely general and is not specific to particles of uniform size. On the other hand it should be noted that Richardson and Zaki’s empirical relationship is based almost entirely on measurements made with narrow size ranges of particles. More likely explanations of the apparently anomalous behaviour of sample Bare as follows: (i) A relatively high velocity coefficient, could have been applicable since a bed with a wide size range of smooth particles would be expected to have a low viscosity. (ii) The indefinite bubble boundary which occurs when a wide size range of particles is used could cause a consistent underestimate of bubble radius. Bubbles with diameters of 5-10 cm appear to have been studied and errors of up to 10 per cent are quite conceivable. (iii) Accurate comparison of theoretical and practical results places heavy reliance on identification of the incipiently fluidised condition. This is difficult to establish precisely with a wide size range of particles in a fluidised bed and the porosity at incipient fluidisation for sample B could well have been 0405 instead of 040. In view of these possibilities it is difficult to decide whether or not the results for sample B were in conflict with the modified theory. It would therefore appear that Jackson’s theory, the only one which allows for porosity changes, is capable of giving a quantitative explanation of the data P. S. B. STEWART C.S.I.R.O. Division of Chemical Melbourne Australiu
Engineering
NOTATION a
A,
g k U,,
radius of curvature of the nose of a bubble or radius of circle which approximates most closely to entire bubble boundary excluding the wake area of a two dimensional bubble, cmz acceleration due to gravity coefficient in the equation for bubble velocity lJ,, = kdga bubble velocity, cm/set
REFERENCES 111 LOCKETT M. J. and Harrison D., In?. Symp. on Fluidisarion, Eindhoven, Paper 4.6.1967. PI JACKSON R., Trans. fnstn Chem. Engrs 1963 4122. I31 STEWART P. S. B., Ph.D. dissertation, University of Cambridge 1965. 1954 32 35. 141 RICHARDSON J. F. and ZAKJ. W. M.. Trans lnstn them. Ennrs .I REUTER H., Chemie-lngr-Tech: 1963 jS 98. ti; PYLE D. L. and HARRISON D., Chem. Engng Sci. 1967 22 53 1. ROWE P. N. and PARTRIDGE B. A., Trans. Instn them. ngrs 1965 43 157. t;; ROWE P. N., PARTRIDGE B. A. and LYALL E., Chem. f ngng Sci. 1964 19 973. 1965 22763. I91 COLLINS R.,J. FluidMech. [lOI STEWART P. S. B., Symposium on Two Phase Flow. Exetel. Vol. I, p. 2Y June 1965.
397 C.E.S.-H
Pergamon Press.
Printed in Great Britain.
Prediction of voidage fraction near bubbles in fluidised beds (Received
9 October
MEASUREMENTS of changes in voidage near bubbles rising in two dimensional gas fluidised beds have been made recently by Lockett and Harrison[ 11,however the theoretical analysis which they present is in poor agreement with the results obtained and they conclude that only qualitative predictions are possible. Figure I shows group averages of the porosity data close to the vertical axis above the bubble, using the data provided in Figs. 1 and 2 of their paper, compared with various theoretical curves. Curve (I) is the theoretical result obtained from: (i) an analysis in two dimensions, equivalent to the three dimensional analysis given by Jackson[2], for the motion of particles and gas caused by a bubble in a fluidised bed, (ii) the Carman-Kozeny relationship for the dependenceon porosity of the resistance of a fixed bed to interstitial gas flow. This is the theory presented by Lockett and Hanison[ll and, separately, by Stewart[3]. As pointed out by Lockett and Harrison the discrepancy between theory and experiment is somewhat reduced by using Richardson and Zaki’s well-known fluidised bed relationship[4] in place of the Carman-Kozeny equation; the result however is still far from satisfactory- Curve (11). The bubble velocity Ub, is known to be related to the bubble radius, u, by the expression
l_Jb= k dga. In their analysis for porosity distribution Lockett and Harrison used the theoretical value of k. However, Stewart
040 hiday on verticulAxis abw &rbb
1967)
[3] showed that in predicting the gas pressure distribution determined experimentally by Reuter[S], the experimentally determined value of k gave better agreement between theory and experiment. Moreover the remaining differences could be accounted for by allowing for the viscosity of the fluidised bed. The theoretical velocity coefficient for two dimensional bubbles is O-5. Pyle and Harrison[6] recently reported measurements of the bubble velocity in two-dimensional fluidised beds and found a relationship
where a was the radius of curvature of the nose of the bubble measured from photographs of bubbles by “the best fit to a segment of arc extending some 30” on either side of the stagnation point at the nose”. However, in studies of the velocity of three dimensional bubbles and of the penetration of gas from within a two dimensional bubble into the particulate phase, Rowe[7, 81 used the radius of the circle which approximated most closely to the entire bubble boundary excluding the flattened wake portion at the bottom of the bubble. For an analysis where the movements of the particles in the whole region around a bubble are concerned, this method of measuring the radius is considered the most appropriate one. For uniformly sized Ballotini particles, and allowing for wall effects, Pyle and Harrison found as an alternative form to Eq. (1) that:
Iv\
I k- 0.5, Carman- KoJay
‘\,
In
&
044
; i
\
U
l
05, Richardsont zaki
m
.
0.7,
Iv
l
0.9,
‘\
Distance from Bubble Ccntre
(Rodii)
Fig. 1.Voidage above a bubble in a fluidised bed.
396
. -
l
U, = 16.1 Abl”
where A* is the bubble area in cm*. The photographs of two dimensional bubbles in uniformly sized 230~ Ballotini published by Rowe et af.[8] show a ratio between the height of the bubble and the diameter of the circle approximating to the bubble boundary excluding the wake, of 0.83. Using this to provide a relationship between Ab and a bubble radius of the type used by Rowe, by substitution Eq. (2) becomes: Ub = 0.65dga.
A satisfactory fit to the porosity data for sample A can be obtained by choosing k = 0.7 in the theoretical analysis -curve (III). Since Rowe et a/.[71 find that bubble rise velocity and bubble shape are dependent on the particles forming the fluidised bed, the value of 0.7 is not significantly different from the 0.65 of Eq. (3). The results for sample E cannot be fitted directly in this way. Using k = 0.9, curve (IV) is obtained which although close to the experimental results, shows a significantly different trend. Lockett and Harrison suggest three explanations of the lack of agreement between theory and experiment. (i) A number of approximations and assumptions are made in the theory, the most important of which is the elimination of variable voidage from equations where it has only a second order effect. (ii) A three dimensional component could be present in bubble motion in the two dimensional bed. (iii) Jackson’s analysis is only applicable to particles of uniform size and consequently cannot be expected to be satisfactory for a fluidised bed containing particles with a wide size range. The first two explanations do not seem to be necessary since the results for sample A may be predicted satisfactorily by inserting a more appropriate bubble velocity formula in the analysis. Collins’[9] observations with regard to a three dimensional component introduced by the front and back walls in two dimensional apparatus relate to an air/water system where the wall is wetted by the water and boundary layer effects ensure that some water will flow down all walls. In experiments with a non-wetting air/mercury system it was found that this effect could be completely avoided with carefully
executed experiments [ IO] and a fluidised bed system where particles are only slightly retarded by the walls would be expected to be similar. Jackson’s analysis is completely general and is not specific to particles of uniform size. On the other hand it should be noted that Richardson and Zaki’s empirical relationship is based almost entirely on measurements made with narrow size ranges of particles. More likely explanations of the apparently anomalous behaviour of sample Bare as follows: (i) A relatively high velocity coefficient, could have been applicable since a bed with a wide size range of smooth particles would be expected to have a low viscosity. (ii) The indefinite bubble boundary which occurs when a wide size range of particles is used could cause a consistent underestimate of bubble radius. Bubbles with diameters of 5-10 cm appear to have been studied and errors of up to 10 per cent are quite conceivable. (iii) Accurate comparison of theoretical and practical results places heavy reliance on identification of the incipiently fluidised condition. This is difficult to establish precisely with a wide size range of particles in a fluidised bed and the porosity at incipient fluidisation for sample B could well have been 0405 instead of 040. In view of these possibilities it is difficult to decide whether or not the results for sample B were in conflict with the modified theory. It would therefore appear that Jackson’s theory, the only one which allows for porosity changes, is capable of giving a quantitative explanation of the data P. S. B. STEWART C.S.I.R.O. Division of Chemical Melbourne Australiu
Engineering
NOTATION a
A,
g k U,,
radius of curvature of the nose of a bubble or radius of circle which approximates most closely to entire bubble boundary excluding the wake area of a two dimensional bubble, cmz acceleration due to gravity coefficient in the equation for bubble velocity lJ,, = kdga bubble velocity, cm/set
REFERENCES 111 LOCKETT M. J. and Harrison D., In?. Symp. on Fluidisarion, Eindhoven, Paper 4.6.1967. PI JACKSON R., Trans. fnstn Chem. Engrs 1963 4122. I31 STEWART P. S. B., Ph.D. dissertation, University of Cambridge 1965. 1954 32 35. 141 RICHARDSON J. F. and ZAKJ. W. M.. Trans lnstn them. Ennrs .I REUTER H., Chemie-lngr-Tech: 1963 jS 98. ti; PYLE D. L. and HARRISON D., Chem. Engng Sci. 1967 22 53 1. ROWE P. N. and PARTRIDGE B. A., Trans. Instn them. ngrs 1965 43 157. t;; ROWE P. N., PARTRIDGE B. A. and LYALL E., Chem. f ngng Sci. 1964 19 973. 1965 22763. I91 COLLINS R.,J. FluidMech. [lOI STEWART P. S. B., Symposium on Two Phase Flow. Exetel. Vol. I, p. 2Y June 1965.
397 C.E.S.-H