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Ind. Eng. Chem. Res. 2001, 40, 420-426
Spout Geometry in Shallow Spouted Beds Marı´a J. San Jose´ ,* Martin Olazar, 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 the operating conditions (base angle, gas inlet diameter, stagnant bed height, particle diameter, and gas velocity) on spout geometry has been studied in shallow spouted beds by means of an optical fiber probe. The experimental results have been fitted to an empirical equation for calculation of the average spout diameter in shallow spouted beds. The proposed equation takes into account the effect of base angle and gas inlet diameter, which are parameters that have not been previously taken into account in the literature (for deep beds) and have a great importance in the generation of the spout in shallow spouted beds. 1. Introduction The study of the spout geometry in cylindrical spouted beds of conical base has deserved great attention in the literature1-18 because of the fact that it is a key factor for understanding the dynamics of the spouted bed. Likewise, attempts have have been made to calculate theoretically the evolution of the spout diameter.7,19-23 The experimental results are in discrepancy, although a general observation is the fact that the evolution of the spout diameter with the bed level is only important at the gas inlet zone in the bed.6-8,11-14,17 Consequently, to model the gas and solid flow in spouted beds and for the sake of simplicity, it is useful to assume that the spout diameter or radial position of the spout-annulus interface is independent of the longitudinal position along the bed.7,24-29 Usually, an average value of diameter is taken, which is estimated from semiempirical or theoretical correlations (Table 1). The correlations proposed (Table 1) are for calculation of the average spout diameter as a function of column diameter, fluid velocity, and particle size or density.3,6,7,10,13,19,20,31 Abdelrazek30 also takes into account the effect of stagnant bed height. In previous papers it has been proven that for shallow spouted beds the hydrodynamics is greatly influenced by the geometry of the contactor base, especially by the angle and by the inlet diameter.33,34 Nevertheless, the angle is not taken into account in the correlations of Table 1, which have been established from the results obtained for beds of considerable height in which the effect of the conical base is attenuated. The objective of this paper has been to analyze the effect of all of the operating conditions in shallow spouted beds and to establish a correlation for calculation of the average spout diameter, which takes into account the effect of the angle and of the inlet diameter (this latter parameter is only indirectly taken into account in the correlations of Table 1). The study has been carried out using the experimental results obtained by means of an optical fiber in a wide range of the geometric factors of the contactor-particle system and gas velocities. * To whom correspondence should be addressed. Telephone: 34-94-6015362. Fax: 34-94-4648500. E-mail: iqpsaalm@ lg.ehu.es.
Figure 1. Scheme of the equipment used and the arrangement of the optical fiber probe.
2. Experimental Section In Figure 1, a diagram of the equipment used and the arrangement of the optical fiber probe in the contactor is shown. Five contactors of poly(methyl methacrylate) have been used, which have the following dimensions (geometry defined in Figure 1): column diameter Dc, 0.152 m; base diameter Di, 0.63 m (except for γb ) 180°, for which Di ) Do); height of the conical section Hc, 0.168, 0.108, 0.078, 0.026, and 0 m; angle of the 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 have been glass spheres (density ) 2420 kg/m3) of these particle diameters: 2, 3, 4, and 5 mm. Three air velocities have been used: 1.02ums, 1.2ums, and 1.3ums. The minimum spouting velocity, ums,
10.1021/ie000340t CCC: $20.00 © 2001 American Chemical Society Published on Web 12/08/2000
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 421 Table 1. Literature Correlations for Calculation of Average Spout Diameters author al.3
Malek et Mikhailik6 Volpicelli et al.19 Abdelrazek30 glass spheres nonspherical solids Lefroy and Davidson7 McNab31 Bridgwater and Mathur20 Littman et al.32,a
Green and Bridgwater10 Wu et al.13 a
equation Ds ) (4.26 × 10-3 log Dc + 9.1 × 10-3)G0.50 Ds ) (1.67 log Dc + 2.22)(G/Fs)0.50 Ds ) (0.13 log Dc + 0.173)G0.50
(1) (2) (3)
Ds ) 0.315Dc[u/(gHo)0.50]0.33 Ds ) 0.346Dc[u/(gHo)0.50]0.50 Ds ) 1.07Dc0.75dp0.33 Ds ) 2.0G0.49Dc0.68/Fb0.41 Ds ) 0.384G0.50Dc0.75/Fb0.25
(4) (5) (6) (7) (8)
(Ho)MDs
) 0.345
[] Ds Dc
-0.384
Dc2 - Ds2 (Ho)Mri ) 0.435 + 0.020/A rc2 Ds ) (2 + Dc/2)G0.49Dc0.68/Fb0.41 Ds ) 5.61G0.433Dc0.583µ0.133/(FbFg)0.283
(9)
(10) (11) (12)
For calculation of Ds corresponding to the maximum spoutable bed height, which is calculated using eq 10.
Figure 2. Arrangement of the three optical fibers in the probe.
has been calculated with the equation of Mathur and Gishler:1
ums ) (dp/Dc)(Do/Dc)1/3[2gHo(Fs - F)/F]1/2
(13)
The technique for measurement of the spout diameter consists of an optical fiber probe and has already been used for the study of local properties of conical spouted beds.35-38 A vertical displacement device is provided for the probe (Figure 1). This device positions the probe in front of the contactor hole, at the level at which the measurement is to be carried out. The probe is manually placed in a radial position in the bed, through holes made in the contactor wall (at 20 mm intervals), as is shown in Figure 1. Grooves marked on the probe allow for setting of the radial position in the bed. The probe (Figure 2) consists of a stainless steel encasing, marked in millimeters, whose maximum and minimum external dimensions are 5 and 1.5 mm, respectively, which contains three optical fibers placed in parallel. The principle of the measurement is based on the emission of a light beam by the central fiber, which is reflected by the particles in the bed. Each time that a particle passes in front of the probe, it reflects light, which is successively collected by the two fibers placed above and below the emitting fiber. The shape of the probe avoids perturbations in the gas and solid flow. The maximum distance for light reception delimits the field range of the probe. When the light intensity is
changed, a field range of the same order as the particle size is attained, which permits a signal of maximum clarity. The effective distance between the two fibers that receive the reflected light, which is an important parameter for calculating the particle velocity, has been experimentally determined on a rotating disk of known angular velocity39 and a value of 4.3 mm was obtained. The delimitation of the interface between the spout zone and the annular zone has been carried out, as is shown in Figure 3. When the tip of the probe is in the annular zone, the corresponding signal is formed by wide peaks, because of the fact that the particles are in contact and are moving at a low velocity. When the tip of the probe reaches the spout zone, the signal registered, which is formed by narrow and pronounced peaks, is the one corresponding to particles in movement at high velocity and in contact with each other. The precise point where the change from one signal to the other happens corresponds to the radial position of the annular zone-spout zone interface. The measurements have been carried out at 20 mm intervals of the bed level and at 2.5 mm intervals for radial positions of the probe tip. In this way, the interface between the spout zone and the annular zone is delimited with an experimental error of (1.25 mm. Although the corresponding results are beyond the scope of this paper, the probe has also been used for measurement of the vertical component of particle velocity, for measurement of the local bed voidage, and for delimitation of the zones where the particles flow upward or downward in the fountain.
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Figure 3. Signals of the optical fiber probe in the annular and spout zones of the bed.
3. Results 3.1. Effect of the Operating Conditions on the Spout Geometry. When the value of each variable is changed in turn and the rest are maintained constant, the effect of the following conditions has been studied: base angle, gas inlet diameter, stagnant bed height, particle size, and gas velocity through the inlet orifice. In Figure 4, the important effect of the base angle on the geometry of the spout is shown and the evolution of the spout diameter with the bed level is plotted in Figure 5. As is observed in Figures 4 and 5, a considerable difference between the results for the angles of 30, 45, and 60° (Figure 4a-c) and for those corresponding to 120 and 180° (Figure 4d,e) is appreciated. For base angles between 30 and 60° (Figure 4a-c), the spout has a neck at an intermediate bed level, corresponding approximately to 0.10 m independently of the angle (Figure 5). The diameter of the spout at the neck, which is more than 10% greater than the inlet diameter, hardly changes with the base angle. The initial widening of the spout in the lower section of the bed and the subsequent widening in the upper section are maxima for the angle of 45° (Figure 4b). For base angles of 120 and 180°, the widening at the inlet zone is noticeably smaller than that corresponding to smaller angles and the spout does not have a neck (Figure 4d,e). It is noteworthy that the experimental systems plotted in Figure 4d,e correspond to a stagnant bed height, Ho ) 0.20 m, which is close to the maximum spoutable bed height. For lower values of Ho, the spout has a small neck when these angles are used, although it is much smaller than that corresponding to smaller base angles. When the inlet diameter is varied (Figures 6 and 7), it is observed that, as this diameter is increased, the relative neck of the spout (the ratio of the neck diameter to the inlet diameter) is less pronounced and it appears at slightly higher bed levels.
Figure 4. Effect of the base angle on the spout geometry. Dc ) 0.15 m, Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums. (a) γb ) 30°, (b) γb ) 45°, (c) γb ) 60°, (d) γb ) 120°, and (e) γb ) 180°.
Figure 5. Effect of the base angle on the evolution of the spout diameter with the bed level. Do ) 0.03 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums.
As is observed in Figures 8 and 9, the spout narrows less and the neck appears at a higher bed level as the stagnant bed height is increased. For Ho ) 0.30 m, there is a very slight neck and the only change in the spout diameter takes place at the inlet zone. This result is evidence of the little importance of the neck in beds where Ho/Do > 10. As the particle diameter is increased (Figures 10 and 11), the initial widening of the spout is more important, whereas the neck of the spout is less pronounced and appears at higher bed levels.
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 423
Figure 6. Effect of the gas inlet diameter on the spout geometry. γb ) 45°, Dc ) 0.15 m, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums. (a) Do ) 0.03 m, (b) Do ) 0.04 m, and (c) Do ) 0.05 m.
Figure 8. Effect of the stagnant bed height on the spout geometry. γb ) 45°, Dc ) 0.15 m, Do ) 0.03 m, dp ) 4 mm, and u ) 1.02ums. (a) Ho ) 0.15 m, (b) Ho ) 0.20 m, (c) Ho ) 0.25 m, and (d) Ho ) 0.30 m.
Figure 7. Effect of the gas inlet diameter on the evolution of the spout diameter with the bed level. γb ) 45°, Ho ) 0.20 m, dp ) 4 mm, and u ) 1.02ums.
The effect of increasing the gas velocity above that corresponding to minimum spouting (Figures 12 and 13) is similar to that aforementioned for the particle diameter increase. Thus, the widening of the spout at the inlet zone is more pronounced, whereas the neck is less pronounced and appears at higher bed levels. 3.2. Average Spout Diameter. As has been shown in the previous section, the size of the neck as well as its position changes with the following two operating conditions: contactor geometry and solid characteristics. When these results are compared with those of the literature for conventional beds, in the shallow beds studied in this paper the spout shape changes along most of the bed. This phenomenon, which is usually treated in a very simplified way for deep beds by assuming a slight widening of the spout at the inlet zone and a constant diameter at higher bed levels, is very important in shallow spouted beds because it affects a large fraction of the bed volume and has an influence on gas and solid flow. The geometry of the spout in shallow spouted beds is very sensitive to operating conditions, especially to the geometric factors of the contactor inlet (base angle and
Figure 9. Effect of the stagnant bed height on the evolution of the spout diameter with the bed level. γb ) 45°, Do ) 0.03 m, dp ) 4 mm, and u ) 1.02ums.
inlet diameter), which are not taken into account in Table 1, and this explains the fact that the correlations shown in Table 1 do not satisfactorily estimate the experimental results of the average spout diameter determined in our systems. Equation 7 of McNab31 is the one that best predicts the average spout diameter values. Nevertheless, for certain systems this equation underpredicts the experimental values. Thus, the deviations for relatively large contactor inlet diameters (Do ) 0.05 m) are over 30%, which is due to the fact that McNab used Do/Dc ratios lower than those of our experimental systems. On the other hand, for near-flat contactor base angles the spout diameters calculated with eq 7 deviate positively from the experimental ones. Thus, the calculated values are between 15 and 20% higher than the experimental ones (Table 2). Consequently, a new equation based on that of McNab31 (eq 7) is proposed by introducing two more
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Figure 10. Effect of the particle diameter on the spout geometry. γb ) 45°, Dc ) 0.15 m, Do ) 0.03 m, Ho ) 0.20 m, and u ) 1.02ums. (a) dp ) 3 mm, (b) dp ) 4 mm, and (c) dp ) 5 mm.
Figure 12. Effect of the gas velocity on the spout geometry. γb ) 45°, Dc ) 0.15 m, Do ) 0.03 m, Ho ) 0.20 m, and dp ) 4 mm. (a) u ) 1.02ums, (b) u ) 1.2ums, and (c) u ) 1.3ums.
Figure 11. Effect of the particle diameter on the evolution of the spout diameter with the bed level. γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, and u ) 1.02ums.
Figure 13. Effect of the gas velocity on the evolution of the spout diameter with the bed level. γb ) 45°, Do ) 0.03 m, Ho ) 0.20 m, and dp ) 4 mm.
dimensionless moduli in his equation, Do/Dc and γb, which, as was aforementioned, characterize the hydrodynamics of shallow spouted beds. For all of the experimental systems studied, the average spout diameter, Ds, has been calculated based on the volume occupied by the spout, Vs, which has been calculated from the data of evolution of the spout radius, rs, with the bed level, z, by means of the following equation:
Table 2. Experimental Values and Those Calculated Using Equations 7 and 15 of the Average Spout Diameter for the Different Experimental Systems
rs
Vs )
H
π
∑0 ∑0 ∆z × 2πr] ) 4(Ds)2H [∆r
(14)
The equation obtained by fitting the experimental results by nonlinear regression is the following:
(
)( )
G0.49Dc0.68 Do Ds ) 6.6 Dc F 0.41 b
0.76
γb-0.15
(15)
with a regression coefficient of r2 ) 0.95 and a maximum error of 5%. The interactions are not significant at a 95% confidence interval.
γb, deg
Do, m
dp, mm
Ho, m
Ds , m u/ums calcd (eq 15) calcd (eq 7)
exptl
30 45 45 45 45 45 45 45 45 45 45 60 120 180
0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05 0.03 0.03 0.03 0.03
4 3 4 4 4 4 4 4 4 4 5 4 4 4
0.20 0.20 0.15 0.20 0.20 0.20 0.25 0.30 0.20 0.20 0.20 0.20 0.20 0.20
1.02 1.02 1.02 1.02 1.20 1.30 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02
0.041 0.036 0.037 0.040 0.043 0.044 0.045 0.048 0.049 0.059 0.043 0.037 0.036 0.035
0.039 0.038 0.037 0.040 0.044 0.045 0.042 0.045 0.051 0.062 0.043 0.039 0.036 0.034
0.038 0.036 0.037 0.040 0.044 0.045 0.042 0.045 0.041 0.042 0.043 0.041 0.042 0.042
As an example of the suitability of the experimental results for eq 15, the results obtained with eq 15 are compared in Table 2 with the experimental values. It must be pointed out that eq 15 is reduced to eq 7 of McNab31 for a ratio of Do/Dc ) 1/5 and for contactor
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 425
base angles between 30 and 60°. For these values the group 6.6(Do/Dc)0.76γb-0.15 takes a value near 2.0. On the other hand, an alternative method of using eq 15 for calculation of the average spout diameter in cylindrical spouted beds with flat or near-flat base (120° < γb < 180°) is proposed in this paper. In this contactor it has been observed that when the maximum spoutable bed height or slightly lower heights are used (Figure 4d,e), the cross section of the spout is almost uniform with the bed level and the average spout diameter is slightly greater than the contactor inlet diameter. In virtue of this result, the average spout diameter corresponding to the maximum spoutable height, (Ds)M, has been taken as a reference for the calculation of the average spout diameter in any operating conditions. Moreover, because the average spout diameter depends on the air velocity, the modulus relating the air velocity to the minimum spouting velocity corresponding to the maximum spoutable height, (ums)M, has also been adopted as a variable. When the experimental results of the average spout diameter are fitted by nonlinear regression, the following correlation has been obtained (regression coefficient, r2 ) 0.89, with an average relative error lower than 8%):
Ds ) 1.00(Ds)M[u/(ums)M]0.58
(16)
In eq 16, the average spout diameter corresponding to the maximum spoutable bed height, (Ds)M, is calculated by using the maximum spoutable bed height, (Ho)M, in the equation proposed by Littman et al.:40
(Ho)M(Ds)M Dc2 - (Ds)M2
) 0.345
( ) (Ds)M Dc
-0.384
(17)
It must be noted that Littman et al.40 proposed eq 17 to be applied the opposite way to that used here, that is, for calculation of the maximum spoutable bed height. For calculation of (Ds)M by means of eq 17, the experimental values of (Ho)M have been used. These satisfy the equation of Morgan et al.41 and, consequently, this equation is valid for calculation of (Ho)M:
(Ho)MDo 2
Dc
) 0.101
[ ]
Do + 0.12 umf Dc ut
(18)
It is noteworthy that, following the procedure proposed and using eqs 16-18, the effect on the average spout diameter of the variables whose influence on the spout geometry has been proven is taken into account. These variables are contactor inlet diameter, stagnant bed height, particle size, and ratio of the air velocity to the minimum spouting velocity. The effect of the base angle is not taken into account because it has a minor effect when γb > 120°. The validity of eqs 16-18 is shown in Table 3, in which the experimental and calculated values of the average spout diameter are compared. In any case, eq 16 is of less general application than eq 15, because the latter takes into account the contactor base angle. 4. Conclusions For deep spouted beds, the variation of the spout diameter with the bed level can be treated in a simpli-
Table 3. Experimental Values and Those Calculated Using Equations 16-18 of the Average Spout Diameter for the Different Experimental Systems γb, deg
Do, m
dp, mm
Ho, m
u/ums
D s, m calcd exptl
120 120 120 120 120 120 120 120 180 180 180 180 180 180 180 180
0.03 0.03 0.03 0.03 0.03 0.04 0.05 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05 0.03
3 4 4 4 4 4 4 5 3 4 4 4 4 4 4 5
0.20 0.15 0.15 0.15 0.20 0.20 0.15 0.15 0.20 0.15 0.15 0.15 0.20 0.20 0.15 0.15
1.02 1.02 1.20 1.30 1.02 1.02 1.02 1.02 1.02 1.02 1.20 1.30 1.02 1.02 1.02 1.02
0.032 0.033 0.037 0.039 0.036 0.046 0.052 0.036 0.031 0.032 0.035 0.037 0.034 0.045 0.051 0.035
0.033 0.032 0.037 0.040 0.036 0.047 0.053 0.036 0.032 0.031 0.036 0.038 0.035 0.046 0.051 0.034
fied way by introducing a slight widening of the spout at the lower section of the bed. On the other hand, for shallow spouted beds, it is a relatively more important phenomenon because it affects a great fraction of the bed volume and has an influence on gas and solid flow. In shallow spouted beds, it has been proven that both the size of the neck and its longitudinal position change with the two operating conditions: contactor geometry and solid characteristics. For calculation of the average spout diameter, the correlation proposed in this paper (eq 15) is suitable and its contribution to the correlation proposed by McNab31 is the consideration of the effect of the base angle and the contactor inlet diameter. This equation is valid in a wide range of experimental conditions: 30° < γb < 180°; 1/5 < Do/Dc < 1/3; 0.05 < Ho < 0.35 m; 2 < dp < 5 mm; 1.02 < u/ums < 1.3. Its application is restricted to ambient conditions of pressure and temperature. Equation 15 satisfies the equation proposed by McNab31 for a ratio of Do/Dc ) 1/5 and contactor base angles between 30 and 60°. Likewise, when the base angle is between 120 and 180°, eq 16 together with the equations of Littman et al.40 and of Morgan et al.41 can be used for calculation of the average spout diameter. Acknowledgment 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 ) parameter in eq 10, FumfuT/(Fs - F)gDo Db, Dc, Di, Do ) diameter of the upper level of the stagnant bed, of the column, of the bed base, and of the inlet, respectively, m Ds, (Ds)M ) average spout diameter and average spout diameter for the maximum spoutable bed height, m dp ) particle diameter, m G ) mass flow rate referring to the inlet section, kg m-2 s-1 g ) gravity constant, m s-2 H, Ho, (Ho)M ) height of the developed bed and of the stagnant bed and maximum spoutable height, m r, z ) radial and longitudinal coordinates, m
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u, umf, ums, (ums)M, ut ) velocity of gas, of minimum fluidization, and of minimum spouting, for the maximum spoutable bed height, and terminal velocity, respectively, m s-1 rc, ri, rs ) radius of the column, of the contactor base, and of the spout, respectively, m Vs ) volume occupied by the spout, m3 Greek Letters γb ) contactor-included base angle, rad µ ) viscosity, kg m-1 s-1 F, Fb, Fs ) density of the gas, of the static bed, and of the solid, respectively, kg m-3
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Received for review March 21, 2000 Accepted October 3, 2000 IE000340T
