Measurement of Particle Velocities in Conical Spouted Beds Using an

Measurement of Particle Velocities in Conical Spouted Beds Using an Optical Fiber Probe. Martin Olazar,* Marı´a J. San Jose´, Sonia Alvarez, Albert...
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Ind. Eng. Chem. Res. 1998, 37, 4520-4527

Measurement of Particle Velocities in Conical Spouted Beds Using an Optical Fiber Probe Martin Olazar,* Marı´a J. San Jose´ , Sonia Alvarez, Alberto Morales, and Javier Bilbao Departamento de Ingenierı´a Quı´mica, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain

The vertical components of particle velocities in the spout and annular zones of conical spouted beds have been experimentally determined for different values both of contactor geometric factors (angle and air inlet diameter) and of operating conditions (particle diameter, stagnant bed height, and air velocity) by means of an optical fiber probe. Based on the effect of the operating variables on the longitudinal and radial profiles of the particle velocities, a correlation has been determined for their calculation at any position in the spout zone. The correlation proposed by Ne´meth and Pallai for the calculation of the average particle velocity at the wall in cylindrical spouted beds has been proven to be valid in conical spouted beds. Introduction Recent studies on conical spouted beds have focused on the development of fundamental hydrodynamic models to characterize their physical behavior (Olazar et al., 1992, 1993b, 1995a; San Jose´ et al., 1993, 1995). However, much of this work has focused on gas-phase flow. The present work seeks to extend the previous studies to include the solid behavior in these units. Specifically, the vertical components of the particle velocities are experimentally determined with the aid of an optical fiber probe. Given their versatility in gas and solid flow, conical spouted beds are a suitable contact method when short gas residence times are required and when a vigorous gas-solid contact is needed due to the characteristics of the solid, such as wide particle size distribution (Olazar et al., 1993a; San Jose´ et al., 1994), irregular texture (Olazar et al., 1994a), or sticky nature (Olazar et al., 1994b, 1997). As is evident from these papers, the solid flow in conical contactors is considerably different from that in cylindrical contactors. This term is applied to both spouted beds made up of a cylindrical column with a flat bottom and to conventional spouted beds, conical-cylindrical, in which most of the bed is within the cylindrical section. Consequently, in the latter case, the solid flow in the cylindrical section (most of the bed) is hardly dependent on the design of the conical section. The specific conical contactor used, with the geometric factors that influence its hydrodynamics, is shown in Figure 1. Several authors (Mathur and Gishler, 1955; Thorley et al., 1959; Mikhailik and Antanishin, 1967; Gorshtein and Mukhlenov, 1967; Lefroy and Davidson, 1969; Mathur and Epstein, 1974; Van Velzen et al., 1974; Kmiec, 1980; Suciu and Patrascu, 1978; Day et al., 1987; Kim and Cho, 1991; Roy et al., 1994) studied the particle velocity in cylindrical spouted beds and determined that the maximum velocity is reached at the axis of the spout, near the bottom of the contactor. These authors ignored the radial component of the particle velocity in * To whom correspondence should be addressed. Telephone: 34-4-4647700, ext. 2575. FAX: 34-4-4648500. Email: [email protected].

Figure 1. Geometric factors of the contactor.

the spout. Van Velzen et al. (1974) proposed the following equation for the calculation of particle velocity along the axis:

[ (

υz(0) ) [υz(0)]max 1 - 1 -

)]

z zm

2

(1)

where the maximum velocity at the axis is calculated from the operating conditions using the empirical relationship (Van Velzen et al., 1974):

F0.5G1.09 [υz(0)]max ) 40 dpD00.5H00.35Dc0.25

(2)

The longitudinal position of the maximum velocity is calculated as follows (Van Velzen et al., 1974):

zm )

F0.4G1.18 dp1.6Dc0.67

(3)

The aforementioned authors accepted a parabolic radial profile for particle velocity at any level in the spout. Epstein and Grace (1984) proposed the following equation:

[ ()]

υz ) υz(0) 1 -

10.1021/ie9800243 CCC: $15.00 © 1998 American Chemical Society Published on Web 10/07/1998

r rs

m

(4)

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4521

where m varies between 1.3 and 2.2, depending on the operating conditions (m ) 2.0 for Mathur and Epstein (1974)). The radial profile determined by He et al. (1994) is more complex, since the maximum particle velocity in the spout is slightly displaced from the axis in the upper section of the bed. This displacement of the maximum velocity was theoretically predicted by Krzywanski et al. (1992), who attributed it to radial movement of the particles and interparticle collisions in the spout. The few papers dealing with particle velocities in conical spouted beds reveal peculiar characteristics of the particle velocity in the spout. Thus, Kmiec (1980) and Boulos and Waldie (1986) determined that the maximum velocities in the spouts of the conical spouted beds are higher than those corresponding to the cylindrical spouted beds and that they are reached closer to the base. Robinson and Waldie (1978) and Boulos and Waldie (1986) proved that near the interface, the particles descend relatively quickly just prior to entry in the spout. In the annular zone of the cylindrical spouted beds, several authors (Thorley et al., 1959; Suciu and Patrascu, 1978; Becker, 1961; Mathur and Epstein, 1974; Van Velzen et al., 1974; Rovero et al., 1985; Day et al., 1987; Benkrid and Caram, 1989; Roy et al., 1994) determined that in the cylindrical section of the bed, the radial profile of the particle downward velocity is almost flat, with a small velocity decrease near the contactor wall and at the interface between the annular and spout zones (Benkrid and Caram, 1989; Kim and Cho, 1991; He et al., 1994). The particle velocity has no radial component until the particle reaches the conical section, where the vertical component increases and the radial component is high. Most of the aforementioned authors assume a linear decrease of the particle velocity as the particles descend along the cylindrical section. Suciu and Patrascu (1978) proposed the following equation for the calculation of the average velocity in the annular zone, as a function of the longitudinal position:

(Hz )

(υ j a) ) 0.043 + 0.16

(5)

Roy et al. (1994) determined that the evolution of velocity with the longitudinal position is not linear but, due to the increase of bed voidage with longitudinal position (He et al., 1994), is proportional to z0.65. Experimental Section The experimental unit was described in previous papers (Olazar et al., 1992, 1993b; San Jose´ et al., 1993). The study has been carried out using contactors made of poly(methyl methacrylate). The values of the geometric factors defined in Figure 1 are as follows: column diameter, Dc ) 0.36 m; angle, γ ) 33, 36, and 45°; their corresponding height of the conical section, Hc ) 0.50, 0.45, and 0.36 m; air inlet diameter, D0 ) 0.03, 0.04, and 0.05 m. The design of the air inlet for system stability was described in detail in a previous paper (Olazar et al., 1992). The stagnant bed height was varied between 0.05 and 0.30 m. The solids used were glass spheres (density ) 2420 kg m-3) of diameters dp ) 3, 4, and 5 mm. The probe used for measurement of the vertical component of particle velocity at any position in the bed was described in a previous paper (Olazar et al., 1995b)

Figure 2. Vertical component of the particle velocity in the spout zone, at different longitudinal positions. γ ) 33°, D0 ) 0.03 m, dp ) 3 mm, H0 ) 0.18 m, u ) ums.

and consists of an encasing of stainless steel, whose maximum and minimum dimensions are 5.0 and 1.5 mm, respectively, which contains three optical fibers arranged in parallel. When the particle passes in front of the probe head, it reflects light emitted by the central fiber. The light reflected is collected in succession by the two lateral fibers and is sent to two analogical channels. From a statistical analysis, by means of the cross-correlation function (incorporated to the MATLAB 5.1 program), the signals with a statistically significant correlation coefficient (indicating that the same particles pass in front of both fibers) are accepted. These coefficients are higher than 90% in the spout zone and higher than 60% in the annular zone. From the effective distance between the two receiving fibers and the delay time between the two signals, τ (time corresponding to the maximum value of the cross-correlation function), it can be ascertained whether the displacement is upward or downward (positive or negative time delay), and the velocity of the particle along the longitudinal direction can be calculated:

υz )

de τ

(6)

The effective distance, de ) 3.3 mm, was determined on a rotary disk of known angular velocity following the procedure described by Benkrid and Caram (1989). The signals pass through a signal amplifier (-12 to +12 V). A 12-V light source sends 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 downward velocity of the particles along the wall has been measured by a technique based on video recording and image treatment, which was described in a previous paper (Olazar et al., 1996). Results Particle Velocities in the Spout. As an example of the results obtained in the spout zone by means of the optical fiber, the vertical components of the particle velocity in the spout are schematically outlined in Figure 2 for one of the experimental systems studied. The delimitation of the interface between the spout and annular zones, which is drawn in Figure 2, was carried

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Figure 3. Longitudinal profiles of the vertical component of particle velocity in the spout zone. γ ) 33°, D0 ) 0.03 m, dp ) 3 mm, H0 ) 0.18 m, u ) ums. Points, experimental results. Lines, calculated with eqs 7-9.

Figure 4. Radial profiles of the vertical component of the particle velocity in the spout zone. γ ) 33°, D0 ) 0.03 m, dp ) 3 mm, H0 ) 0.18 m, u ) ums. Points, experimental results. Lines, calculated with eqs 7-9.

out in a previous paper (Olazar et al., 1995b) by means of the optical probe. The spout shape is qualitatively similar for all the experimental systems studied. At first, it widens in a very pronounced way near the contactor inlet and then there is a neck, and finally the spout widens again up to the bed surface. It is noteworthy that the average diameter of the spout is much greater than the contactor inlet diameter. The values of the vertical component of the particle velocity plotted in Figure 2 are shown in more detail in Figures 3 and 4, where the longitudinal and radial profiles, respectively, are plotted. In these figures, points are experimental results and lines are calculated with the correlations proposed further on. In Figure 3, it is observed that at a given radial position, the vertical component of the particle velocity reaches a maximum value at a given longitudinal position near the bottom. As the radial position is higher, this maximum becomes less pronounced and is located at a higher longitudinal position. In Figure 4, it is observed that the vertical component of the particle velocity has a maximum value at the axis for all the longitudinal positions. Besides, the higher the longitudinal position along the spout, the less pronounced this maximum. The radial and longitudinal profiles of velocity in the spout change with the geometry of the spout and, consequently, with the operating conditions that influ-

ence this geometry. With the aim of analyzing in detail the effect of the operating conditions on the particle velocity, the velocity results obtained for several radial positions along the axis and near the bottom will be subsequently studied. These results are representative of the shape of the velocity profiles in the spout. In Figure 5, the effect both of the geometric factors of the contactor (angle and inlet diameter) and of the operating conditions (particle diameter, stagnant bed height, and air velocity) on the longitudinal profile of the vertical component of the particle velocity along the spout axis is analyzed. These results show that the maximum particle velocity is reached at a position between 0.02 and 0.03 m from the air inlet, and this position is not significantly affected by the operating variables. However, at higher axial locations, the particle velocity decreases linearly with distance. Of the variables studied, Figure 5, the stagnant bed height appears to have the most significant effect on the maximum particle velocity. Particle size also has a great influence (although smaller than that of the stagnant bed height). The other variables have a smaller influence. When the effect of the operating variables is analyzed, stagnant bed height is the one of greater influence, Figure 5d. Thus, the higher the stagnant bed height, the higher the particle velocity along the spout axis. This is followed by particle size, which also has a great influence, Figure 5c. As the particle size is smaller, the particle velocity along the spout axis increases noticeably. The influence of the remaining variables is smaller. When the contactor angle, Figure 5a, and the air velocity (over that corresponding to the minimum for spouting) are increased, Figure 5e, the particle velocity along the spout axis increases slightly, whereas as the air inlet diameter is increased, the particle velocity decreases, Figure 5b. The radial profile of the vertical component of particle velocity at the base of the spout, at z ) 0.025 m, is analyzed in Figure 6. Points are experimental results, and lines are calculated with the correlations proposed further on. The variables of greater influence are, in order of influence, stagnant bed height, Figure 6d, particle size, Figure 6c, and air inlet diameter, Figure 6b (in this case due to the modification of the spout geometry induced by a change in inlet diameter). The influence of the contactor angle, Figure 6a, and of air velocity, Figure 6e, are small. With the aim of having an equation valid for calculation of the particle local velocities, several mathematical expressions were tried. The best fit was obtained with an expression similar to that proposed by Epstein and Grace (1984), i.e., eq 4, but taking into account that, in conical spouted beds, the radial profile of the vertical component of velocity changes with the longitudinal position and, consequently, the exponent (named mz) will also change with bed level. The proposed expression is

[ ()]

υz ) υz(0) 1 -

r rs

mz

(7)

By nonlinear regression fitting of the experimental results (Box, 1965) to eq 7, the parameter mz takes values between 1.0 and 3.0. In the upper section of the spout, mz takes the value 1.0 for all the experimental

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4523

Figure 5. Effect of the geometric factors of the contactor (a, contactor angle; b, inlet diameter) and of the operating conditions (c, particle diameter; d, stagnant bed height; e, air velocity) on the longitudinal profiles of the vertical component of the particle velocity along the spout axis.

systems studied, while the maximum value of mz (mz ) m0) is reached at a distance between z ) 0.020 and z ) 0.030 m from the contactor inlet. It has been proven that mz decreases from its maximum value, m0, with the longitudinal position, in all the experimental systems, according to the following expression:

mz ) 1 + (m0 - 1) exp[-100(z - 0.025)2]

(8)

In eq 8, m0 is a function of both the geometric factors of the contactor and of the operating variables. By studying these variables, grouped into the conventional dimensionless moduli used in the hydrodynamic study of conical spouted beds (Olazar et al., 1992), the following expression is obtained:

() ( ) ( )

m0 ) 1.41

dp Di

-0.31

H0 Di

0.77

u ums

0.80

(

γ0.50 exp

)

-1.87D0 Di (9)

The fitting of the experimental results of the vertical component of the velocity, υz, to eqs 7-9 was carried

out by the complex method of nonlinear regression (Box, 1965) and gave a global regression coefficient of r2 ) 0.90, with a maximum relative error of 12%. The adequacy of the fitting is shown in Figures 4 and 6, where points are experimental results and lines are calculated with eqs 7-9. The validity of eq 7 for the prediction of the scarce experimental data published in the literature for conical spouted beds was analyzed. It is noteworthy that these data correspond to particular experimental systems. Thus, Kmiec (1978) used shallow beds, and consequently, only a qualitative assessment may be made about the validity of eq 7, which provides values of the same order, although slightly smaller than the experimental ones of Kmiec (1980). The results of Waldie and Wilkinson (1986) are average values for the spout section at several levels, and they are lower than those calculated by means of eq 7. Particle Velocity in the Annular Zone. The results of Figures 7 and 8, which correspond to one of the experimental systems studied, are an example of the general shape of the longitudinal profiles, Figure

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Figure 6. Effect of the geometric factors of the contactor (a, contactor angle; b, inlet diameter) and of the operating conditions (c, particle diameter; d, stagnant bed height; e, air velocity) on the radial profiles of the vertical component of the particle velocity at the base of the spout, at z ) 0.025 m. Points, experimental results. Lines, calculated with eqs 7-9. Table 1. Equations Proposed in the Literature for Calculation of the Particle Velocity at the Wall authors Thorley et al. (1959) Becker (1961) Shigeo (1965) Matsen (1968) Abdelrazek (1969) Ne´meth and Pallai (1970)

equation W)υ j wAa(1 - a)F ) K(D0/Dc)-0.25 (Us/Ums)1.23(H0/Dc)1.0 [(υ j w)2/gH0]0.4 [Ua/(Ums)M]2 ) B(z/HM)b where at z/HM < 0.25, B ) 0.055, and b ) 1; at z/HM > 0.25, B ) 0.21, and b ) 2 υ j w/Ums ) 7.6 × 10-3(gDc/Ums2)-0.4(D0/Dc)-0.7(Us/Ums)1.7 υ j w/(υ j w)ms ) Us/Ums log(υ j w/Ums ) ) (3.42 × 10-4)Re(H0/Dc) - 1.543 υ j w ) (υ j w)M(H0/HM)1/3 υ j w ) (υ j w)ms(Us/Ums)3/2

7, and of the radial profiles, Figure 8, of the vertical component of the particle velocity along the annular zone. In these figures, points are experimental results and lines are traced along the points in order to follow the tendency of the results. In Figure 7, it is observed that given the conical geometry of the contactor, the closer the particles to the bottom of the contactor, the higher their acceleration and, consequently, the maximum component of the downward velocity is reached near the bottom and close to the interface between the spout and annular zone. At the bottom of the contactor, the main component of the particle velocity is the radial one.

(10) (11) (12) (13) (14) (15) (16)

The radial profiles of the vertical component of the particle velocity, Figure 8, show the existence of a maximum value at a radial position, which shifts toward higher radial positions as the longitudinal position in the bed is higher. The average values of the particle velocity in the spout and annular zones are plotted in Figure 9 against the longitudinal position. As is observed, the average upward particle velocity in the spout decreases almost linearly with the longitudinal position. The average downward velocity along the annular zone has a slight acceleration in the upper section of the bed and increases linearly toward the bottom of the bed.

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4525 Table 2. Comparison of the Experimental Values of the Particle Velocity at the Wall with Those Calculated Using Equation 16 Proposed by Nemeth and Pallai (1970) γ, deg 33

Figure 7. Longitudinal profiles of the vertical component of the particle velocity in the annular zone. γ ) 33°, H0 ) 0.18 m, dp ) 4 mm, D0 ) 0.03 m, u ) ums.

Figure 8. Radial profiles of the vertical component of the particle velocity in the annular zone. γ ) 33°, H0 ) 0.18 m, dp ) 4 mm, D0 ) 0.03 m, u ) ums.

Figure 9. Longitudinal profiles of average particle velocities in the spout and annular zones. γ ) 33°, D0 ) 0.03 m, dp ) 4 mm, H0 ) 0.18 m, u ) ums.

Particle Velocity at the Wall. The different correlations proposed in the literature for calculation of the particle downward velocity along the wall in cylindrical spouted beds are set out in Table 1. Mathur and Epstein (1974) drew attention to the limitations on the application of these correlations, as their validity was restricted to the range of experimental conditions in which they were determined. Several authors (Rovero et al., 1985; Boulos and Waldie, 1986; Randelman et al., 1987; Day et al., 1987; Benkrid and Caram, 1989; Roy et al., 1994; He et al., 1994) proved that the particle velocity at the wall is severely affected by wall friction

D0, m 0.03

dp, m

H0, m

0.004

0.18

0.005

0.18

0.04

0.004

0.18

0.05

0.004

0.18

36

0.03

0.004

0.18

45

0.03

0.004

0.18

υ j w, m s-1 u/ums 1.02 1.20 1.30 1.02 1.20 1.30 1.02 1.20 1.30 1.02 1.20 1.30 1.02 1.20 1.30 1.02 1.20 1.30

exptl 10-2

3.1 × 3.8 × 10-2 4.2 × 10-2 3.7 × 10-2 4.1 × 10-2 5.0 × 10-2 2.8 × 10-2 3.4 × 10-2 4.0 × 10-2 2.2 × 10-2 2.7 × 10-2 3.0 × 10-2 3.7 × 10-2 3.9 × 10-2 5.2 × 10-2 4.5 × 10-2 5.2 × 10-2 5.8 × 10-2

eq 16 4.1 × 10-2 4.6 × 10-2 4.9 × 10-2 5.5 × 10-2 3.7 × 10-2 4.2 × 10-2 3.0 × 10-2 3.3 × 10-2 4.9 × 10-2 5.5 × 10-2 5.9 × 10-2 6.6 × 10-2

and that it is consequently a poor representation of the particle velocity in the annular zone. In Figure 10, the effect of the operating variables on the longitudinal profile of the particle velocity along the wall (velocity measured in the wall direction) is analyzed. It is observed that the particle velocity at the wall increases as the particles descend. At first, the increase is exponential and then the velocity increases linearly to a maximum value near the bottom of the bed. The particle velocity at the wall is higher as the value of the following variables is increased: contactor angle, Figure 10a; particle size, Figure 10c; stagnant bed height, Figure 10d; and air velocity, Figure 10e. On the other hand, the particle velocity at the wall decreases as the contactor inlet diameter is increased, Figure 10b. The adequacy of the equations of Table 1 for predicting the experimental values of this study has been analyzed. It must be pointed out that eq 11 (Becker, 1961) and eq 15 (Ne´meth and Pallai, 1970) are not applicable to conical spouted beds, as these equations take into account the maximum spoutable bed height, HM. In conical spouted beds, there is no maximum spoutable bed height (Olazar et al., 1993a,b). On the other hand, in conical spouted beds, the variation of the particle velocity at the wall with the longitudinal position is great, Figure 10, and for this reason, the values fitted to the equation in Table 1 are average values between the surface and the bottom of the bed. It has been proven that eq 16 proposed by Ne´meth and Pallai (1970) suitably fits the experimental results, Table 2, with a regression coefficient r2 ) 0.98 and a relative error lower than 3%. The fitting of eq 12 proposed by Shigeo (1965) is also acceptable, with a regression coefficient r2 ) 0.87 and a relative error lower than 14%. The fitting of the other correlations is poor. In these calculations, the value of Dc for conical beds has been taken as the arithmetic mean between the base diameter, Di, and the upper diameter of the stagnant bed height, Db. Conclusions A strong dependency of the geometric factors of the contactor and of the operating conditions has been found on the vertical component of particle velocity in the spout and annular zones of conical spouted beds. The

4526 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998

stagnant bed height and particles size have a great influence on the velocity profile in the spout. The maximum value of the upward velocity in the spout zone is reached at the axis of the spout and near the contactor inlet (at z between 0.02 and 0.03 m). For longitudinal positions further away from the bottom, the maximum velocity also corresponds to the spout axis. For the mathematical modeling of the solid flow pattern in the spout zone, the adequacy of eqs 7-9 has been proven in a wide range of experimental conditions. These equations, together with others determined in a previous paper (San Jose´ et al., 1998) for calculation of local bed voidages in the spout and annular zones, will be used in subsequent papers for modeling the solid flow pattern and defining solid trajectories throughout the entire bed. The downward velocity of the particles in the annular zone has a maximum value near the bottom of the contactor and at a radial position close to the interface between the spout and annular zone. As the bed level is higher, the maximum velocity corresponds to a radial position further away from the interface. Equation 16, proposed by Ne´meth and Pallai (1970), has been proven to be suitable for calculation of the average particle velocity along the contactor wall. Acknowledgment This work was carried out with the financial backing of the Department of Education, University and Research of the Government of the Basque Country (Project No. PI94/33), and of the Ministery of Education and Culture of the Spanish Government (Project DGICYT No. PB94-1359). Nomenclature

Figure 10. Effect of the geometric factors of the contactor (a, contactor angle; b, inlet diameter) and of the operating conditions (c, particle diameter; d, stagnant bed height; e, air velocity) on the longitudinal profiles of particle velocity at the wall.

Aa ) cross section of the annular zone, m2 B, b ) parameters in eq 11 Db, Dc, Di, D0 ) upper diameter of the stagnant bed and diameters of the column, of the bed bottom, and of the inlet, respectively, m de ) effective distance between the two receiving fibers, mm dp ) particle diameter, mm G ) gas mass flow rate per unit of column cross section, kg m-2 s-1 H, Hc, H0 ) heights of the developed bed, of the conical section, and of the stagnant bed, respectively, m HM ) maximum spoutable bed height, m m ) exponent in eq 4 m0 ) parameter defined by eq 9 mz ) exponent in eq 7, defined by eq 8 r ) radial distance from the axis of the conical contactor, m Re ) superficial particle Reynolds number rs ) spout radius at level z, m u, ums ) velocity and minimum spouting velocity of the gas, m s-1 Ua, Ums, (Ums)M ) superficial gas velocity in the annular zone, minimum spouting velocity, and minimum spouting velocity at the maximum spoutable bed height, respectively, m s-1 W ) solids circulation rate, kg s-1 z ) longitudinal distance from the bottom of the conical contactor, m zm ) longitudinal position of the maximum particle velocity, m

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4527 Greek Letters a ) bed voidage in the annular zone γ ) contactor angle, rad F ) density of the solid, kg m-3 τ ) delay time between two signals, s υw, υ j w, (υw)M, (υw)ms ) particle velocity parallel to the wall, average particle velocity, and particle velocities corresponding to the maximum spoutable bed height and minimum spouting velocity, respectively, m s-1 υ z, υ j z ) component of particle velocity in the z direction and its average value at a given level, m s-1 υz(0), [υz(0)]max ) component of particle velocity in the z direction at the axis and its maximum value, m s-1 υ j a ) average particle velocity in the annular zone, m s-1

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Received for review January 14, 1998 Revised manuscript received August 10, 1998 Accepted August 11, 1998 IE9800243