Local Bed Voidage in Spouted Beds - American Chemical Society

in the spout, annulus, and both the fountain core and periphery of spouted beds, ... Correlations are proposed for calculation of the local bed voidag...
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Ind. Eng. Chem. Res. 2001, 40, 427-433

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Local Bed Voidage in Spouted Beds Martin Olazar,* Marı´a J. San Jose´ , Miguel A. Izquierdo, Sonia Alvarez, and Javier Bilbao Departamento de Ingenierı´a Quı´mica, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain

The effect of operating conditions (base angle, gas inlet diameter, stagnant bed height, particle diameter, and gas velocity) on longitudinal and radial profiles of bed voidage has been studied in the spout, annulus, and both the fountain core and periphery of spouted beds, by means of an optical fiber probe. Correlations are proposed for calculation of the local bed voidage in spout and annular zones, along the spout axis and in proximity to the contactor wall. Introduction Progress in gas and solid flow modeling in spouted beds is being made simultaneously with the knowledge of local bed properties. Consequently, the results of these models are very very sensitive to small variations of the local bed voidage, especially in the spout zone, where slight differences in bed voidage measurement give way to important errors when mass, energy, and momentum balances are solved. Consequently, great attention has been given to the spout bed voidage in the literature since the development of spouted bed technology,1 and different authors have developed theoretical models to predict the spout bed voidage.2-4 Likewise, numerous experimental studies on the spout bed voidage have been carried out.5-11 For the sake of simplicity, bed voidage in the annular zone of the spouted bed has usually been taken as constant and equal to the bed voidage corresponding to the loose bed,1,5,7,12,13 which is in turn taken as equal to the bed voidage of minimum fluidization with a minimum value of 0.42 for equally sized spherical particles.7,14 Nevertheless, the differences in the experimental measurements obtained in different positions of the annular zone are significant, especially in the upper half of the bed15 and when the gas velocity is higher than that for minimum spouting.10,11 Experimental measurements of the bed voidage in the fountain have been carried out because the fountain is especially important in granulation and coating processes.16,17 Besides, when the spouted bed is to be used in any application, it must be taken into account that the solid contained in the fountain is an important fraction of the total solid inventory (up to 9% in deep beds and even higher in shallow spouted beds) and, consequently, this zone cannot be ignored. The values of the experimental results for fountain bed voidage are greater in shallow beds than in deep beds.18 Moreover, if two hydrodynamically distinct regions are considered in the fountain (Figure 1), the voidage in the fountain core (where the particles and fluid move concurrently upward) is lower than the voidage in the fountain periphery (where particles returning to the top of the annulus countercurrently encounter fluid rising from the top of the annulus). This voidage is lowered as the * To whom correspondence should be addressed. Telephone: 34-94-6012527. Fax: 34-94-4648500. E-mail: [email protected].

Figure 1. Zones in the spouted bed.

level in the fountain core is increased and increases as the gas flow velocity is increased.11 In this paper bed voidage has been experimentally determined in the three zones of the spouted bed by means of an optical fiber probe. The measurements have been carried out under different operating conditions: base angle, gas inlet diameter, stagnant bed height, particle diameter, and gas velocity. The results have been fitted to correlations, which will be useful for modeling the solid flow throughout the whole bed. Experimental Section The unit used (Figure 2) has been described in detail in previous papers.19,20 Five contactors of poly(methyl methacrylate) have been used, which have the following dimensions: column diameter Dc, 0.152 m; base diameter Di, 0.06 m; height of the conical section Hc, 0.168, 0.108, 0.078, 0.026, and 0 m; their corresponding angles

10.1021/ie0003741 CCC: $20.00 © 2001 American Chemical Society Published on Web 12/08/2000

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through a signal amplifier (-12 + 12 V). A 12 V light source transmits light to the emitting fiber, and a filter controls the intensity of the beam. An analogical/digital interface sends the data to the computer for processing. The intensity of the light reflected by the particles that pass in front of the fiber depends on the type or composition of the particle, on its size or size distribution, and on the bed voidage. For this reason a calibration has been carried out for this solid so that the local bed voidage has been related to the probe signals, either in the spout zone or in the annular zone. The calibration procedure has been previously detailed.19 The average relative error of the measurement, which has been repeated three times at each point, is 4%. On the other hand, it has been proven that the position of the probe in the bed does not affect the resulting calibration curve. Results

Figure 2. Scheme of the equipment used and the arrangement of the optical fiber probe.

of contactor base γb, 30, 45, 60, 120, and 180°; gas inlet diameter Do, 0.03, 0.04, and 0.05 m; stagnant bed height Ho, between 0.05 and 0.35 m. The solids studied are glass spheres (density ) 2420 kg/m3) of particle diameters 2, 3, 4, and 5 mm. Three gas linear velocities have been used: 1.02ums, 1.2ums, 1.3ums. The values of minimum spouting velocity corresponding to each system have been determined experimentally, and it has been proven that they fit the equation of Mathur and Gishler1 with an average relative error of lower than 15%:

ums ) (dp/Dc)(Do/Dc)1/3[2gHo(Fs - F)/F]1/2

(1)

The optical fiber probe and the experimental procedure for voidage measurement have been described in detail in previous papers in which the local bed voidage19 and the vertical component of the particle velocity20 have been studied in conical spouted beds. The probe used for voidage measurement consists of a stainless steel encasing with maximum and minimum dimensions of 3.0 and 1.5 mm, respectively, containing three optical fibers in parallel. When a particle passes near the head of the probe, it reflects the light emitted by the central fiber. The reflected light is collected in succession by the two fibers located at the extremes, between which there is an effective distance of 4.3 mm. This distance has been determined on a rotary disk of known angular velocity. The probe section reduces to a minimum perturbation in the solid flow (both upward and downward). The greatest precision is obtained with a range field equal to the particle diameter. The intensity of light emission or sensitivity is regulated by means of a powermeter, allowing for the range to be changed between 1 and 6 mm and, consequently, operated with beds of different particle sizes within this range. The frequency of the light used is 50 Hz. The light signal is collected by photodiodes and converted into voltage (0-100 mV). The signals pass

Bed Voidage along the Spout Axis. From analysis of the results of longitudinal evolution of the bed voidage at different radial positions, it is concluded that the more pronounced change in voidage takes place at the axis of the contactor. The effect of operating conditions on this parameter, (0), has been studied, and the results are shown in Figure 3, where each plot corresponds to the analysis of the effect of each operating condition. In Figure 3a it is observed that an increase in the bed voidage along the axis is produced as the base angle of the contactor is increased from 30° to 45°. For angles greater than 45°, the decrease in (0) attenuates as the angle is increased. (0) is constant in the range between 120° and 180°. As the gas inlet diameter is increased in the range between 0.03 and 0.05 m, the bed voidage increases in all of the longitudinal positions along the spout axis (Figure 3b). The same result is obtained when the stagnant bed height is increased in the range between 0.15 and 0.30 m (Figure 3c), when the particle diameter is increased between 3 and 5 mm (Figure 3d), and when the ratio of the inlet gas velocity to the minimum spouting velocity is increased in the range between 1.02 and 1.3 (Figure 3e). In view of these results, it is concluded that the factor of greater influence on the bed voidage along the axis of the contactor is the total height of the bed (height of the developed bed including the fountain), HT. Likewise, the evolution of (0) with longitudinal position follows a sigmoidal shape. Consequently, the experimental results of (0) have been fitted to the following general expression:

(0) ) [1 + E(z/HT)a]b

(2)

Taking a and b as constant, the following values have been obtained: a ) 3; b ) -0.22. E is a parameter whose value is between 15 and 50 depending on the operating conditions. When the conventional dimensionless moduli for the design of spouted beds are taken into account, the following expression is obtained for this parameter:

E ) 7.37

() () () ( ) dp Do

-0.14

H Dc

0.85

Do Dc

-0.39

u ums

-0.76

γb0.62

(3)

The points in Figure 3 correspond to the experimental results and the lines to the values calculated using eq

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Figure 3. Effect of operating conditions on the longitudinal profiles of the bed voidage along the axis of the spout. Points, experimental results. Lines, calculated with eq 2. (a) Effect of the base angle. Experimental system: Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, u ) 1.02ums. (b) Effect of the inlet diameter. Experimental system: γb ) 45°, Ho ) 0.20 m, dp ) 4 mm, u ) 1.02ums. (c) Effect of the stagnant bed height. Experimental system: γb ) 45°, Do ) 0.03 m, dp ) 4 mm, u ) 1.02ums. (d) Effect of the particle diameter. Experimental system: γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, u ) 1.02ums. (e) Effect of the air velocity. Experimental system: γb ) 45°, Ho ) 0.20 m, Do ) 0.03 m, dp ) 4 mm.

2. The fit of the experimental results of (0) to the calculated ones has a regression coefficient of r2 ) 0.95 and a maximum relative error of 7%. Bed Voidage at the Wall. The local bed voidage decreases radially from the axis to the proximity of the contactor wall. The minimum value, termed the bed voidage at the wall, (w), corresponds to the value measured at a distance from the wall of approximately one particle diameter. In Figure 4 the effect of operating conditions on longitudinal profiles of bed voidage at the wall is analyzed. In all of the experimental systems, bed voidage at the wall decreases from the base to the surface of the bed, where the value is almost equal to the bed voidage of the loose bed, o, except for γb ) 180° (flat base) and γb ) 120°(nearly flat base). In this case, near the base of the contactor, there is a slight increase in the bed voidage with longitudinal position, which is

a consequence of the existence of a dead zone at the base of the contactor. In Figure 4a it is observed that bed voidage at the wall reaches its maximum value for an angle of the base of γb ) 45°. Likewise, it reaches a maximum value for an intermediate stagnant bed height, Ho ) 0.20 m (Figure 4c). At bed levels near the surface of the bed, voidage is almost independent of Ho. The bed voidage gradually decreases as the gas inlet diameter (Figure 4b), particle diameter (Figure 4d), and gas velocity (Figure 4e) are increased. The experimental results have been fitted by nonlinear regression to the following expression:

(

(w) ) o 1 +

H - z 0.25 H

)

(4)

The fitting has a regression coefficient of r2 ) 0.98 and a maximum relative error of 5%. In Figure 4, the

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Figure 4. Effect of operating conditions on the longitudinal profiles of the bed voidage at the wall. Points, experimental results. Lines, calculated with eq 4. (a) Effect of the base angle. Experimental system: Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, u ) 1.02ums. (b) Effect of the inlet diameter. Experimental system: γb ) 45°, Ho ) 0.20 m, dp ) 4 mm, u ) 1.02ums. (c) Effect of the stagnant bed height. Experimental system: γb ) 45°, Do ) 0.03 m, dp ) 4 mm, u ) 1.02ums. (d) Effect of the particle diameter. Experimental system: γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, u ) 1.02ums. (e) Effect of the air velocity. Experimental system: γb ) 45°, Ho ) 0.20 m, Do ) 0.03 m, dp ) 4 mm.

points are the experimental results and the lines have been calculated using eq 4. When this equation was applied in order to calculate the data in Figure 4c, the experimentally determined effect of the stagnant bed height, Ho, on the operation bed height, H, was taken into account. Likewise, in Figure 4d the effect of the particle diameter, dp, on the loose bed voidage, o, has been taken into account. Radial Profile of Bed Voidage in Spout and Annular Zones. Taking into account that the radial profile of the bed voidage at each bed level depends mainly on the bed voidage at the axis, on the bed voidage at the wall, and on the radial position of the inflection point (spout radius), the bed voidage data have been fitted by nonlinear regression to the equation

(r) )

(0) - (w) + (w) r - 0.90rs 1 + exp 72.0rs2.50

(

)

(5)

where (0) and (w) are calculated using eqs 2 and 4, respectively. The values of the spout radius have been experimentally determined in a previous paper in which the evolution of the radius with longitudinal position was proven to depend on the operating conditions.21 The regression coefficient is r2 ) 0.92, and the maximum relative error is 11%. The adequacy of the fitting is shown in Figure 5, which corresponds to an experimental system taken as an example and where the curves correspond to bed voidage values calculated using eq 5 and the points are the experimental values at different bed levels. Although the information of bed voidage at any given point is needed for a rigorous design of spouted beds, the knowledge of the average bed voidage at each bed level in annular and spout zones is interesting for a simplified design. As an example, the evolution of these average voidages with bed level is plotted in Figure 6 for the same experimental system as that shown in Figure 5. The average bed voidage in the

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Figure 5. Radial profiles of the bed voidage. Experimental values (points) and calculated using eq 5 (lines). Experimental system: γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, u ) 1.02ums.

Figure 6. Average bed voidage at different levels in the spout and annular zones. Experimental system: γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, u ) 1.02ums.

spout has been calculated by means of the following expression:

For the spout zone: dr ∫0r 2πr πr 2

(6)

dr ∫rr π(r 2πr 2 - r 2)

(7)

js )

s

s

For the annular zone: ja )

w

s

w

s

In Figure 6 it is observed that the average bed voidage in the spout first decreases from unity (for z ) 0) to 0.93 in proximity to the contactor base. This decrease is due to an important solid cross-flow into the spout near the contactor inlet. Subsequently, the bed voidage in the spout goes on to decrease with the bed level, first (down to approximately z ) 0.06 m) in a very attenuated way and then in a more pronounced way near the region where the neck is produced (at the end of the conical section). In the cylindrical section, the decrease in voidage is much more moderate, until a minimum value of 0.46 is reached in the upper surface of the bed. These results of the average bed voidage in the spout do not follow the trend described by Day et al. in cylindrical contactors of flat base whereby the average bed voidage in the spout is unity up to the level in which the evolution of pressure drop peaks (at z/H ) 0.20).7 The radial average bed voidage in the annular zone (Figure 6) starts with a value of 0.49 and decreases, initially in a more pronounced way, in the conical

Figure 7. Radial bed voidage profiles in the fountain core. γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums.

section of the bed, whereas the decrease in the cylindrical section is linear until a value of 0.39 is reached in the upper surface level. The effects of the base angle and the stagnant bed height are noteworthy. The bed voidage in the upper surface of the spout is greater for the contactor of 45° base angle ( ) 0.46) than for the contactor of γb ) 120° ( ) 0.41). It is also observed that the average bed voidage, either in the spout zone or in the annular zone, increases as the stagnant bed height is increased. Fountain Bed Voidage. Taking into account the geometry of the fountain (Figure 1), the properties must be studied in the core of the fountain (where the particles move upward) and in the periphery (where the particles return to the top of the annulus). These zones have been delimited in a previous paper by means of an optical fiber probe,22 for all of the experimental systems studied in this paper. In the axis of the fountain, the bed voidage decreases following the same trend observed in the axis of the spout. Thus, as has been proven from the experimental results, eq 2 is suitable for calculation of voidage along the axis of the bed, from its base to the top of the fountain. It is noteworthy that, contrary to what is usually assumed,18 voidage values at the top of the fountain are not as low as those of the loose bed. The radial profiles of the bed voidage in the fountain core and periphery are considerably different. Whereas in the core the voidage values and the effect of operating conditions on the radial voidage profiles are qualitatively similar to those previously observed for the spout zone, in the periphery the results are different from those of the annular zone of the bed. Thus, bed voidage in the fountain periphery is noticeably higher than that in the annular zone and it slightly changes with the operating conditions. As an example, in Figures 7 and 8 the radial profiles of voidage in the fountain core and periphery, respectively, are shown for an experimental system. In Figure 7, it is observed that the bed voidage at any level in the fountain decreases from the axis to the coreperiphery interface. At any radial position, the bed voidage decreases smoothly with height. These results are in agreement with those obtained by He et al. for a base angle of 60°.11 In the downflow zone (Figure 8), the bed voidage increases in a very pronounced way with the radius at positions near the core-periphery interface, until it reaches a high and almost constant value. This voidage follows a slightly decreasing trend toward the outside of the fountain. This peculiarity, namely, that the

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the bed voidage in the fountain core become very pronounced. Conclusions

Figure 8. Radial bed voidage profiles in the periphery (solid downflow zone) of the fountain. γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums.

Figure 9. Radial profiles of the bed voidage in the fountain core. γb ) 120°, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums.

The results of the bed voidage in the different zones of the spouted bed for the base angle of 60° are similar to those obtained by He et al.,10 for this angle. The bed voidage in spouted beds has a maximum value at the inlet region of the contactor, where it is almost unity and decreases with radial and longitudinal positions. Equation 5, as proposed in this paper, is interesting for gas and solid flow modeling because it faithfully describes the radial variation of the bed voidage at any bed level once three properties of the bed are known: the bed voidage along the axis of the contactor, the bed voidage in proximity to the contactor wall, and the spout radius. The latter must be determined experimentally. Correlations have been determined for predicting the bed voidage along the axis of the spout (eq 2) and the bed voidage in proximity to the contactor wall (eq 4). Equation 3 takes into account the important effect on the bed voidage along the axis of the spout of the following operating variables: base angle, gas inlet diameter, stagnant bed height, particle diameter, and gas velocity. The study of the bed voidage in both the fountain core and periphery has shown that there are considerable differences between these zones. Thus, whereas the bed voidage in the core or particle ascent zone follows a trend similar to that in the spout zone, in the periphery or particle downflow zone of the fountain bed, voidage is noticeably higher and almost independent of operating conditions. Given the present empirism in the modeling of the solid flow pattern in spouted beds, an important improvement in the results is expected by incorporating the correlations proposed in this paper, which allow for reducing more than 50% of the relative error introduced in the determination of the local solid flow by using the simplification of assuming radially uniform bed voidage. Acknowledgment

Figure 10. Radial profiles of the bed voidage in the fountain core. γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.3ums.

downflow zone is almost empty, is more pronounced as the air velocity is increased. It is also observed that the bed voidage increases with the bed level except at radial positions near the interface, where the opposite is true. The base angle of the contactor has a great influence on the bed voidage in the fountain core. When the results in Figure 7 (γb ) 45°) are compared with those in Figure 9 (γb ) 120°), it is observed that, as the base angle is increased, the bed voidage decreases at any position in the fountain core. Moreover, the radial profile of the bed voidage is more uniform, with values within a narrow range between 0.38 and 0.50. Another operating condition of great influence on the bed voidage in the fountain core is the gas velocity. When the results in Figure 7 (u ) 1.02ums) are compared with those in Figure 10 (u ) 1.3ums), it is observed that, as the gas velocity is increased, the radial profiles of

This work was carried out with the financial support of the Ministry of Education and Culture of the Spanish Government (Project QUI98-1105) and of the Government of the Basque Country (Project EX-1998-136). Nomenclature a, b ) parameters defined in eq 2 Db, Dc, Di, Do ) upper diameter of the stagnant bed height and diameters of the column, of the contactor base, and of the inlet, respectively, m dp ) particle diameter, m E ) parameter defined in eq 3 g ) gravity constant, m s-2 H, Hc, Hf, Ho, HT ) height of the developed bed, of the conical section, of the fountain, and of the stagnant bed and the total height of the bed (H + Hf), m r, z ) radial and longitudinal coordinates, m rs, rw ) spout radius and radial position of the wall, m u, ums, ut ) gas velocity, minimum spouting velocity and terminal velocity, respectively, m s-1 zf ) longitudinal coordinate in the fountain, measured from its base, m

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 433 Greek Letters , (0), (w), o ) bed voidage and bed voidage at the axis of the spout, in the proximity of the contactor wall, and of the loose bed, respectively ja, js ) average bed voidage in the annular and spout zones γb ) angle of the conical base of the contactor, rad F, Fs ) gas and solid densities, kg m-3

Literature Cited (1) Mathur, K. B.; Gishler, P. E. A Technique for Contacting Gases with Coarse Solid Particles. AIChE J. 1955, 1, 157. (2) Lefroy, G. A.; Davidson, J. F. The Mechanics of Spouted Beds. Trans. Inst. Chem. Eng. 1969, 47, 120. (3) Morgan, M. H., III; Day, J. Y.; Littman, H. Spout Voidage Distribution, Stability and Particle Circulation Rates in Spouted Beds of Coarse Particles. I. Theory. Chem. Eng. Sci. 1985, 40, 1367. (4) Krzywanski, R. S.; Epstein, N.; Bowen, B. D. MultiDimensional Model of a Spouted Bed. Can. J. Chem. Eng. 1992, 70, 858. (5) Mathur, K. B.; Epstein, N. Spouted Beds; Academic Press: New York, 1974. (6) Grbavcic, Z. B.; Vukovic, D. V.; Zdanski, F. K.; Littman, H. Fluid Flow Pattern, Minimum Spouting Velocity and Pressure Drop in Spouted Beds. Can. J. Chem. Eng. 1976, 54, 33. (7) Day, J. Y.; Morgan, M. H., III; Littman, H. Measurements of Spout Voidage Distributions, Particle Velocities and Particle Circulation Rates in Spouted Beds of Coarse Particles. II. Experimental Verification. Chem. Eng. Sci. 1987, 42, 1461. (8) Chandnani, P. P.; Epstein, N. Spoutability and Spout Destabilization of Fine Particles with a Gas. In Fluidization V; Ostergaard, K., Sorensen, A., Eds.; Engineering Foundation: New York, 1986; p 233. (9) Wu, S. W. M.; Lim, C. J.; Epstein, N. Hydrodynamics of Spouted Beds at Elevated Temperatures. Chem. Eng. Commun. 1987, 62, 261. (10) Cai, P.; Dong, X. R.; Jin, Y.; Yu, Z. Q. A New Technique for Determining the Hydrodynamic Characteristics of Spouted Beds. Can. J. Chem. Eng. 1992, 70, 835.

(11) He, Y. L.; Qin, S. Z.; Lim, C. J.; Grace, J. R. Particle Velocity Profiles and Solid Flow Patterns in Spouted Beds. Can. J. Chem. Eng. 1994, 72, 561. (12) Matsen, J. M. Void Fraction Variation in the Spouted Bed Annulus. Ind. Eng. Chem. Process Des. Dev. 1968, 7, 159. (13) Epstein, N. Void Fraction Variation in the Spouted Bed Annulus. Ind. Eng. Chem. Process Des. Dev. 1968, 7, 157. (14) Epstein, N.; Grace, J. R. Spouting of Particulate Solids. In Handbook of Powder Science and Technology; Otten, L., Fayed, M. E., Eds.; Van Nostrand Reinhold: New York, 1984; Chapter 11. (15) Heertjes, P. M.; Khoe, G. K. Advances in Spouted Bed Technology. Chem.-Ing.-Tech. 1980, 52, 333. (16) Robinson, T.; Waldie, B. Dependency of Growth on Granule Size in a Spouted Bed Granulator. Trans. Inst. Chem. Eng. 1979, 57, 121. (17) Waldie, B. Initial Stage of Coating and Granulation in Spouted Beds. Inst. Chem. Eng. Symp. Ser. 1981, 63, 1. (18) Grace, J. R.; Mathur, K. B. Height and Structure of the Fountain Region above Spouted Beds. Can. J. Chem. Eng. 1978, 56, 533. (19) San Jose´, M. J.; Olazar, M.; Alvarez, S.; Bilbao, J. Local Bed Voidage in Conical Spouted Beds. Ind. Eng. Chem. Res. 1998, 37, 2553. (20) Olazar, M.; San Jose´, M. J.; Alvarez, S.; Morales, A.; Bilbao, J. Measurement of Particle Velocities in Conical Spouted Beds Using an Optical Fibre Probe. Ind. Eng. Chem. Res. 1998, 37, 4520. (21) San Jose´, M. J.; Olazar, M.; Izquierdo, M. A.; Bilbao, J. Spout Geometry in Shallow Spouted Beds. Ind. Eng. Chem. Res. 2000, in press. (22) Izquierdo, M. A. Bed Characterization and Trajectories of the Solid in Cylindrical Spouted Beds. Ph.D. Thesis, University of the Basque Country, Bilbao, Spain, 1998.

Received for review March 30, 2000 Revised manuscript received September 26, 2000 Accepted October 4, 2000 IE0003741