112
Ind. Eng.
Chem. Fundam., Vol. 18, No. 2 ,
1979
Rise Velocity Equation for Isolated Bubbles and for Isolated Slugs in Fluidized Beds Shaflq A. Allahwala and Owen E. Potter’ Department of Chemical Engineering, Monash University, Clayton 3 168 Victoria, Australia
Measurements on bubble rise velocity in the region where transition from the bubbling regime to the slugging regime occurs are reported. Observations were made in two beds of internal diameters equal to 22.4 and 61 cm. Bubbles were injected near the bottom of incipiently fluidized beds, and the technique of change in capacitance was employed to probe bubble rise velocity. Glass beads of three different size ranges were used as the fluidizing particles; their surface mean diameters were 59, 68, and 198 pm. Employing the present data (combined with some published data) a simple equation of bubble rise velocity applicable to the full range of flow regimes in fluidized beds was formulated. Such an equation is useful in developing computer models simulating the behavior of both bubbling and slugging beds. A comparison of the simple equation with those reported in the literature for gas-liquid systems is also presented.
Introduction Among the important properties related to bubbles in fluidized beds is the bubble rise velocity. A bubble whose diameter is much smaller than the diameter of the column ( D J D < 0.1) rises with a velocity that is independent of vessel diameter. When such is the situation the bed or part thereof is said to be operating in the “bubbling regime”. However, if the relative size of a bubble is further increased its velocity becomes increasingly influenced by the walls of the column, and when the bubble diameter is almost equal to that of the column, the bubble velocity becomes independent of its volume and solely a function of the column diameter. In such an instance, the bed or part thereof is said to have entered into the “slugging regime”. In the “transition region”, where a bubble is partially under the influence of tube walls, the bubble rise velocity is a function of both the bubble and column diameters. Bubbling Regime. Many authors have measured the bubble rise velocity in isolation in both two- and threedimensional fluidized beds. Table I gives a resume of such data for three-dimensional beds only. Almost all authors have summarized their experimental observation in a form similar to the equation of Davies and Taylor (1950) ubm
= kB(gD,)’/’
(1)
It seems from the table that the value of the coefficient k B is approximately equal to 0.71-the value originally determined by Davies and Taylor (1950) for a gas-liquid system. Equation 1 now becomes U,
= 0.71g’12D,‘12
(2)
Most authors (Davidson and Harrison (1963), Potter (1971), Rowe (1971), Leva and Wen 1971, etc.) have adopted eq 2 to calculate bubble rise velocity in fluidized beds. Slugging Regime. Bubbles rising in a slugging bed are called “slugs”, but to avoid nomenclatural difficulties a common name “bubble” is used in this paper for all gas-pockets irrespective of their relative size, D,/D, except for those which could be termed fully developed slugs. Many researchers have measured the bubble rise velocity in the slugging regime, and Table I1 tabulates such information for three-dimensional beds of circular cross section. I t is seen from the table that generally the data for isolated slugs is in extremely good agreement with 0019-7874/79/1018-0112$01 .OO/O
Dumitrescu’s (1943) equation Ub
= k,(gD)’J’
(3)
with k , equal to 0.35. Transition Region. The transition from the bubbling regime to the slugging regime is slow and gradual. The present knowledge of how the transition (on a plot of U,/(gD)’J2 vs. (D,/D)’/’) from U,, to Us occurs in a fluidized bed is based on extension of observations made in gas-liquid systems (Uno and Kintner, 1956; Collins, 1965, 1967). It could be stated that bubble rise velocity in this regime may be approximated by some function of bubble diameter and tube diameter. The present paper reports on the measurements of bubble rise velocity in three-dimensional fluidized beds made primarily in the transition region. The present data (combined with some data from the literature) are then employed to develop a generalized bubble rise velocity equation which is applicable to any of the above-mentioned regimes. Experimental Apparatus Two fluidized beds were used. Both beds were of circular cross section and were basically of similar construction. Bed A. The column was 3.05 m long with an internal diameter of 22.4 cm. The distributor plate was an anodized mild steel plate on which 66 bubble caps were fitted on a triangular pitch (6 = 24.1 mm). The section of the column immediately above the distributor was a copper pipe 30.5 cm long, which was incorporated so that the plumbing work (mounting of injection tube, etc.) could be done easily. The remaining section of the column comprised three identical glass tubes each 91.4 cm long. T o measure bubble rise velocity, bubbles were injected only in the center of the column at 15.2 cm above the distributor. Injection tubes of two different diameters were employed: 2.8 and 1.3 cm. Bed B. The column was approximately 2 m long with an internal diameter of 61 cm. The distributor was made of a mild steel plate on which 430 bubble caps were fitted on a triangular pattern (6 = 25.4 mm). The segment of the column immediately above the distributor was made of a stainless steel pipe. The remainder of the column consisted of a Quickfit glass tube 1 m long and a Perspex 8 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
Table I. Summary of Data on Bubble Rise Velocity in Isolation in Three-Dimensional Beds“ authors bed size, cm particle size measuring technique Davidson et aJ. (1959)b
round 7.6 square 15.2
Harrison and Leung (1962)b Rowe and Partridge (1962) Toei et al. (1966)b
square 61.0
Park et al. (1969)c
round 10.0
Rowe and Matsuno (197 1 I Donsi et al. (1972)
rectangular 29.5 X 14.4 round 35.0
round 14.0 7.6 10.0
kS= 150 pm & = 400 p m
d, = 1 7 0 pm S, 60-150 B.S. Mesh dGs = 50 pm GS, 80-100Y. GS, 16-24f S, 80-120e S, 35-48#
PVC, 80-1001: d , = 344 pm d , = 1 5 4 pm d , = 86 pm d G s , 300-400 pm d,,,
170-350 pm
113
given relation
capacitance probe
ubm =
o.79g1’2vb’i‘
capacitance probe dissection of the bed X-ray photography and capacitance probe
ubm =
o.71g1/2vb1/5
ubm =
o,74gl/zvb1’6
electroresistivity probe X-ray photography photography
ubm =
o.70g1/2vb1’5
ub =
0.67g1’2D,1’2
ub = 0.717gl’ZVb1’S‘75
ub = 0.77[g(Db/Z)]
I/’
These authors particle diameter; GS = glass spheres; S = sand ac: alumina catalyst; c = coke; ss = swede seeds. used wall correction factor m from Uno and Kintner (1956). Measured bubble rise velocity in a freely bubbling bed and determined the value of the coefficient h~ using - urn*) = Ubm. a d =
(u
Table 11. Summary of Measurements of Slug Rise Velocity in Three-Dimensional Fluidized Beds of Circular Cross Section
authors
bed diameter, cm
Yasui and Johanson (1958)
10.2 15.2
Lanneau ( 1 9 6 0 ) Kadlec et al. ( 1961)b Angelino et al. (1964) Pyle and Stewart (1964) Ormiston e t al. (1965) Kehoe and Davidson (1970) Hovmand et al. ( 1 9 7 1 ) Thiel et’al. (1971)
7.6 5.1 2.5 18.0
2.5 5.7 14.0 2.5 5.1 10.2 46.0 5.1
system continuously slugging
particles
kA
glass beads crushed rock catalyst magnesite coal catalyst
0.35
1.0 1.0
70
0.41a
1.0
100
0.33 0.33 0.35
-
0.35
-
0.36 0.37 0.38
2.8 1.6 5.8
0.35
1.0
0.35
1.0
0.35
1.2
continuously slugging continuously sand slugging isolated glass beads slugs (used data of Angelino et al. (1964)) isolated slugs and continuously slugging continuously slugging continuously slugging continuously slugging
catalyst
230
42
sand quartz glass beads sand sand
49-275 68-275 125, 145 74 and 125
glass beads catalyst cat.-alumina
3,9} -
}
-
a Slightly higher value is thought t o originate from measurements at relatively lower levels in the bed (effect of coalescence). Obtained from Stewart and Davidson (1967).
pipe 85 cm long. The mounting of the injection tube was slightly different from that in apparatus A: the injection tube was made to pass through the windbox and distributor plate into the center of the column. Bubbles were injected 2.5 cm above the distributor to conserve the column height. Injection tubes of diameters equal to 2.4 and 1.5 cm were used. A piece of sintered glass was glued onto the open end of each injection tube. In each apparatus, bubbles were injected by opening and closing a solenoid valve between a pressure vessel and the tube. The combined volumes of the pressure vessels and immediate pipe work were 39.9 and 40.0 L in apparatus A and B, respectively. The pressure gauges used were pretested precision gauges. The opening and closing times of the solenoid valves employed were determined in the laboratory at different operating
pressures. The valves were selected for quick operational response. The valves were monitored by an electronic timer, which was connected to a Beckman frequency counter that kept count of the number of bubbles injected. A compressed air supply (100 psig) was used to supply fluidizing fluid for the beds after it had passed through a dryer and a pressure regulator. The regulator decreased the air pressure from 100 to 20 psig. The moisture content of the air after it had passed through the dryer was measured. The air was found to have a relative humidity of 25 to 30%. In each apparatus, the gas flow to the bed was controlled and metered using a rotameter assembly, which comprised four variable area rotameters of different sizes. Beds were operated a t atmospheric pressure. A more detailed description of the experimental equipment is given elsewhere (Allahwala, 1975).
114
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 Table 111. Physical Properties of Solids Used in the Experiments
4 1
CAPACITANCE PROBE
m
specific surface mean diameter, pm
U,f, cm/s
5-105 5-125 15-265
59 68 198
0.89 1.08 3.74
size range, glass beads
GB. 1 GB.2 GB. 3
for connecting both probes to the same proximity meter was to avoid the installation of a grounded plate between the probes. OUTLET
& -. & -
w
S O L E N O I D VALVF REGULATING VPLVE ON/OFF VALVE
Figure 1. Schematic diagram of apparatus A (all dimensions in cm).
n
:AS
$
W
-
ON/OFF VALVE
&
-
SOLENOIO VALVE
Figure 2. Schematic diagram of apparatus B (all dimensions in cm).
Measurement of Capacitance In apparatus A, as manifested in Figure 1, two capacitance probes identical in construction were used. A probe was made of two aluminum strips bent in the form of a semicircle. The two probes were positioned around the fluidized bed column a t a known vertical distance from each other. Each probe was connected to a Fielden Proximity Meter, metering capacitance, which in turn fed an output signal to an ultraviolet oscillograph. The electrical field from one probe was stopped from interfering with the field of another by placing a grounded plate around the bed projecting outwards halfway between the probes. In apparatus B when two capacitance probes were positioned around the bed, bubbles of a somewhat smaller size went past the probe levels undetected. To overcome this problem the probes were placed inside the fluidized bed. A probe consisted of two aluminum strips bent in the shape of an arc. These two plates were positioned in the bed at the same vertical distance from the distributor such that the concave surfaces faced each other and the column axis. This was achieved as shown in Figure 2. The construction was such that a probe could easily be moved up or down vertically in the column. Both probes were connected to the same proximity meter which in turn sent output signals to the ultraviolet oscillograph. The reason
Solids Used Glass beads approximately spherical in shape of three different average sizes were used during the experiments. The size analysis was performed from photographs of samples of a powder under a microscope. Results of the particle size analysis are tabulated in Table 111. Analysis of duplicate samples of solids showed insignificant difference. The incipient fluidization velocity, Umhof each sample was determined experimentally in bed A by plotting the variation of pressure drop across the bed against increasing fluidizing gas velocity. Urn,was taken as the gas velocity a t the point where the straight lines in the fluidized and fixed regions intersected with each other. The incipient fluidization velocities of the solids used are also tabulated in Table 111. Experimental Procedure Both beds A and B were used. Bubbles of different sizes were injected in the center of the beds maintained a t or near incipient fluidization conditions. This condition in a fluidized bed was attained as follows: a bed was vigorously fluidized for a few minutes. Then by cutting the gas supply slowly, the formation of bubbles in the bed was made to cease while maintaining the bed in the fluidized state. The range of gas rates permitting maintenance of the bed in this state was narrow; a slight increase in the gas supply resulted in a continuous formation of bubbles and a slight decrease resulted in partial defluidization of the bed. It was nearly impossible to avoid the formation of an occasional small bubble in the bed and a t the same time maintain its fluidized state. Different sized bubbles were produced in the beds by varying the pressure in the pressure vessels. The pressure in the vessels was varied from 16 to 40 psig. The injection time, the length of time during which the solenoid valve remained open, was kept in all cases less than 0.05 s. The size of an injected bubble was calculated by determining the number of injections that produced a pressure drop of 2 psig in a pressure vessel. It was then assumed that the bubbles produced were of equal size and that there was no leakage of gas from the bubble, as it was formed, to the dense phase or vice versa. X-ray observations (Rowe and Partridge, 1965;Rowe and Matsuno, 1971) show that the volume of an injected bubble produced could be less than, equal to, or greater than the volume of the intruded gas depending on the main gas flow rate in the bed. In view of the results of Rowe and Matsuno (1971), it is believed that in the present experiments the superficial gas velocity in the beds was kept close to the value where an injected bubble has a volume nearly equal to the volume of the gas injected. Rise velocity of the injected bubbles was measured by positioning two capacitance probes a t a known vertical distance from each other. In apparatus A, the two probes
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
were positioned at three different vertical separations; most results were acquired using a probe separation of 37 cm, while some results were obtained with probes 25 cm and 50 cm vertically apart. As the variation in the results acquired applying different probe separations was insignificant, all measurements in bed B were made using only one separation distance. They were positioned a t 50 cm from each other. Observations of bubble rise velocity were made a t 30 and 25 cm above the bubble injection point in bed A and bed B, respectively. This was done to allow the injected bubbles to attain their natural rise velocity. Donsi et al. (1972) reported that the unsteady state after bubble injection ends about 10 cm above the immersed nozzle.
I
115
1
HASTER PLOT
.. ..
.. ,
Results One of the main objectives of the study was to formulate a simple equation which could be employed in computer models to calculate bubble rise velocity in all three regimes of fluidization. T o develop such a relation empirically, it was essential to have experimental data on bubble rise velocity in the region (0.3 < D,/D < 0.9) where adequate experimental data are not available in the literature. Rise velocity, u b , of 465 bubbles was measured in beds of glass beads of three different average particle sizes. The dimensionless equivalent diameter, D,/D, of injected bubbles was varied from 0.9 to 0.5 in bed A and from 0.5 t o 0.3 in bed B. In order to obtain a simple empirical equation applicable to all regimes, the following data were annexed to the present data. In the range 0 < D,/D < 0.7 the bubble rise velocity ub, of different sized bubbles was calculated using the equation u b , = 0.7(gD,)1/zand was fed to the computer as data. In the range 0.06 < D,/D < 0.32 results of Harrison and Leung (1962) were used. Results of the present study obtained in beds of glass beads of dp equal to 198,68, and 59 pm are plotted in Figure 3. Results of Harrison and Leung (1962) are also plotted in the figure. Several equations were fitted to the experimental data, using the method of nonlinear regression analysis developed from Marquardt (1963) by Coulter (1970).
+ Pz
(4)
y = P(l - e-px)
(5)
y = Ple-Px
+
y = Pl(l - P2e-PlX (1 - Pz)e-PZX)
(6)
y = PI tanh1/P{P2xPi
(8)
where x and y, except in eq 6 were (D,/D)’i2 and ub/ (gD)’12 (Froude number based upon the tube diameter), respectively. In eq 6, x and y were ub/ub- and (D,/D), respectively. In the above equations p , pl, p z , P, PI,and P2 were the parameters to be determined. It was found that among the above equations, eq 8 best represents the trend of the data and gives the least value of the sum of squares of residuals. Using eq 8 the line of best fit obtained is shown in Figure 4. A complete list of the data is given elsewhere (Allahwala, 1975). The least-square estimates of parameters P1,P,, and p for the three different particle sizes were calculated. The line of best fit shown in Figure 3 is given by the following relation
- ub-
v@
- 0.35 tanh1/1.8[ 3.6
(4m)1’8] (9)
’0 2
t
. 2
’
.-
S I N G L E DATA P O I N T
0-
D U P L I C A T E DATA P O I N T
0 -
TRIPLICATE
e-
H A R R I S O N & LELiElG
.
.
.
3 3
3.2
0.4
i
( O R MORE)
. 0.5
DATA P O I N T
. 0.6
PRESENT EXPERINENTS
.
.
.
0.7
0.8
0 9
’ O
‘
’
Figure 3. Combined experimental data collected using three different average sized glass beads, and the curve fitted to this data using eq
a.
.
.
1 8
2 2
. 0 )
. 3 1
. c 5
. 3 6
.
0 7
2 6
I ?
‘ a
Figure 4. Comparison of eq 9 with other equations available in the literature: -, eq 9; - -, Uno and Kintner (1956)expression: U,/ Ub, = { [ l / b ] [ l - (De/D)]}0.765, with b = 0.9; ----,Collins(1967) eauation; -..-, u b , = 0.71(gD,)’/2. Also the variation of &/ub,, as predicted by eq 9, with ( D , / D ) ” 2 .
The 95% confidence contours for the empirical constants were computed. They showed that in eq 8 the true values of parameters P1,P,, and p lie in the ranges 0.35 f 0.01, 3.6 f 2.1, and 1.8 f 0.8, respectively. The above equation was not derived on a theoretical basis, but has some resemblance to the equation for bubble rise velocity predicted by the wave theory (Mendleson, 1967). The transition from the bubbling to the slugging starts when (D,/D)1/2approximately equals 0.3 and is complete
116
Ind. Eng. Chem. Fundarn., Vol. 18, No. 2, 1979
when (D,/D)1/2approximately equals 0.9. In other words, when (De/D)'i2 I0.3-where the slope of the curve is approximately equal to 0.61-the equation for ub,, and when (De/D)'/2> 0.9, the equation for Us,could be applied safely to determine bubble and slug rise velocities, respectively. The scatter evident in Figure 3 is greater than that expected from the errors involved in the measurements. Most experimental results of bubble rise velocity reported in the literature (Rowe and Partridge, 1962, 1965; Kobayashi et al., 1966; Toei et al., 1966; Park et al., 1969; Rowe and Matsuno, 1971; Donsi et al., 1972; Chiba e t al., 1973) show similar scatter. Rowe and Partridge (1962), and Rowe and Matsuno (1971) from X-ray cine films of bubbles rising in fluidized beds observed a considerable variation in the velocity of the same bubble without any apparent change in conditions. From this, it appears that rise velocity of a bubble in a fluidized bed is not strictly constant but varies around a mean value. In the present experiments, in order to obtain a reliable estimate of bubble rise velocity, measurements were continued until collection of additional data made an insignificant difference to the results. A comparison of the predictions of eq 9 with those of other equations of bubble rise velocity (Collins, 1967; Uno and Kintner, 1956) is made in Figure 4. It is seen that Collins' (1967) equation predicts a slightly higher bubble rise velocity (on average about 3.5%) compared to that predicted by eq 9 in the range 0.3 < (D,/D)1/2< 0.7. The Froude number, Ub/ was calculated using the equation for u b , and Uno and Kintner's wall correction factor and was found to be in good agreement with that predicted by eq 9 a t different values of (D,/D)''2. This is shown in Figure 4. Also plotted in Figure 4 is the variation of u b / u b , with ( D e / D ) 1 / 2as predicted by eq 9.
Nomenclature d = diameter of a particle d = average diameter of a sample of particle 8 = diameter of a fluidized bed column Db = diameter of a spherical cap bubble De = equivalent bubble diameter: diameter of a sphere having the same volume as that of the bubble g = acceleration due to gravity GB = measured bubble flow rate in a fluidized bed k A = constant in equation us= ks(gD)1'2+ kA(U - umf) kB = dimensionless constant in the bubble rise velocity equation (see eq 1) k, = dimensionless constant in the slug rise velocity expression (see eq 3) p , pl, p 2 = constants in different equations P , P1,P2 = constants in different equations U , = bubble rise velocity in a freely-bubbling fluidized bed ub = rise velocity of a bubble U = superficial gas velocity in a fluidized bed ub, = rise velocity of a bubble as calculated using eq 2 Urn,= incipient fluidization velocity Us = rise velocity of a fully developed slug as calculated using eq 3 6 = spacing between the centers of bubble caps on the distributor plate Literature Cited Allahwala, S. A,, Ph.D. Thesis, Department of Chemical Engineering, Monash University, Australia, 1975. Allahwaia, S.A., Potter, 0. E., Paper E5.7, Symposium on Fluidization, CHISA Conference, Prague, Sept 1972. Angelino, H., Charzat, C., Williams, R., Chem. Eng. Sci., 19, 289 (1964). Chiba, T., Terashima, K., Kobayashi, H., J . Chem. Eng. Jpn., 8 , 79 (1973). Collins, R., J . Fluid Mech., 22, 763 (1965). Collins, R., J . Fluid Mech., 28,97 (1967). Couiter, P. R., Report CHER-70-2, School of Engineering, Monash University, Victoria, Australia, 1970. Davidson, J. F., Harrison, D., "Fluidized Particles", Cambridge University Press,
1963. Duxbury, H. A,, Trans. Inst. Cbem. Davidson, J. F.. Paul, R. C., Smith, M. J. S., €no. (London\. 37. T323 (1959). Davies, M., Taylor, 'Sir Geoffrey,'Proc. R. SOC.London, Ser. A , 200, 375
d.
(1950).
Conclusion (1) Equation 9 predicts the rise velocity of isolated bubbles in fluidized beds, irrespective of the ratio of bubble diameter, De, to vessel diameter, D. Equations of Uno and Kintner (1956) and Collins (1967) are equally validated for fluidized bed systems. (2) Assuming that in fluidized beds bubble flow rate, GB, is given by the two-phase theory of Toomey and Johnstone (1952), eq 9 could be extended to bubbles rising in freely bubbling beds. An analogous equation to that given by Nicklin and Davidson (1962) for gas-liquid systems could then be written as
UA = 0.35@
tanh1/'.8[ 3.6
((m)'"I + ( U - Umf) (10)
(3) The Collins (1967) predictions require lengthy computations and Uno and Kintner's (1956) correction fails a t De = D. Predictions using eq 9 involve simple calculation and are useful in computer models (Allahwala and Potter, 1972) simulating the bubble behavior in fluidized beds when slugging conditions are approached.
Acknowledgment One of the authors (S.A.A.) would like to thank Monash University for the financial assistance received, during the course of study, in the form of a Monash Graduate scholarship.
Donsi, G., Massimilla, L., Crescitelli, S.,Volpicelli, G., Powder Techno/.,8. 21 7
(1972). Dumitrescu, D. T., Angew. Math. Mech., 23, 139 (1943). Harrison, D., Leung, L. S.,Trans. Inst. Chem. Eng. (London),40, 146 (1962). Hovmand, S.,Freedman, W., Davidson, J. F., Trans. Inst. Cbem. Eng. (London),
49, 149 (1971). Kadlec, R. H., Williams, B., Rudd, D. F., 54th Nati. Mtg. Amer. Inst. Chem. Engrs.,
1961. Kehoe, P. W. K., Davidson, J. F., Aust. Acad. Sci., Chemeca ' 70, 97 (1970). Kobayashi, H., Arai, F.,Chiba, T., Kagaku Kogaku(Abridged Ed.) 4, 147 (1966). Lanneau, K. P., Trans. Inst. Cbem. Eng. (London). 38. 125 (1960). Leva, M., Wen C. Y., "Fluidization". J. F. Davidson and D. Harrison, Ed., Academic Press, London, Chapter 14, 1971. Marquardt, D. W., J . SOC. Indust. Appi. Math., 11. 431 (1963). Mendleson, H. D.. AIChE J . , 13, 250 (1967). Nickiin, D. J., Davidson, J. F., Cbem. Eng. Sci.. 17,693 (1962). Ormiston. R. M., Mitchell, F. R. G., Davidson, J. F.. Trans. Inst. Chem. Eng. (London),43,T209 (1965). Park, W. H., Kang, W. K., Capes, C. E., Osberg, G. L., Chem. Eng. Sci., 24, 851 (1969). ---, Potter, 0. E., "Fluidization", J. F. Davidson and D. Harrison, Ed., Academic Press, London, Chapter 7, 1971. Pyle. D L., Stewart, P. S. B., Chem. Eng. Sci., 19, 842 (1964). Rowe, P. N., "Fluidization", J. F. Davidson and D. Hanison, Ed., Academic Press, London, Chapter 4, 1971. Rowe, P. N.,Matsuno, R., Chem. Eng. Sci., 28,923 (1971). Rowe, P. N., Partrige, B. A,. "Interaction Between Fluids and Particles", Inst. Chem. Engrs., London, p 135, 1962. Rowe, P. N., Parb'ige, B. A,, Trans. Inst. Chem. Eng.(London),43, TI57 (1965). Stewart, P. S. B., Davidson, J. F., Powder Techno/., I, 81 (1967). Thkiel, W. J., uhherr, P. H. T., Potter, 0. E.. "Proceedings, 4th Australian Conference on Hydraulics and Fluid Mechanics", Melbourne, Australia, 1971. Toei, R., Matsuno, R., Kojima, H., Nagai, Y., Nakagawa, K.. Yu, S.,Kagaku Kogaku (Abridged Ed.), 4, 142 (1966). Toomey, R. D., Johnstone, H. F., Chem. Eng. Prog.. 48, 220 (1952). Uno, S.. Kintner, R. C., AIChE J., 2, 420 (1956). Yasui, G..Johanson, L. N.. AIChE J . , 4,445 (1958). \
Received for review December 27, 1977 Accepted January 12, 1979
