Particle Cycle Times and Solid Circulation Rates in Conical Spouted

Article pubs.acs.org/IECR

Particle Cycle Times and Solid Circulation Rates in Conical Spouted Beds with Draft Tubes of Different Configuration H. Altzibar, G. Lopez, I. Estiati, J. Bilbao, and M. Olazar* Department of Chemical Engineering, University of the Basque Country, PO Box 64, E48080 Bilbao, Spain ABSTRACT: Particle cycle times (average, maximum, and minimum) have been measured in draft tube conical spouted beds for different geometric factors of the contactor (angle and gas inlet diameter), draft tubes (diameter, height of the entrainment zone, and width of the faces) and under different operating conditions (particle density). On the basis of the results, the effects of the type of draft tube and different factors of the contactor/draft tube/particle system have been studied, and those of greater influence have been determined. The results show that particle cycle times and solid circulation rates are highly dependent on the type of draft tube, solid density, and contactor angle. Open-sided draft tubes are the ones with the highest solid circulation rate with stable spouting performance. Therefore, this internal device is a suitable option for scaling up spouted beds with a hydrodynamic performance similar to those without tubes.

1. INTRODUCTION The spouted bed regime is an alternative contact method to fixed and fluidized beds. Different modifications of the original spouted bed (cylindrical with conical base) are proposed in the literature with the aim of improving its performance.1−4 Spouted beds with fully conical geometry combine the features of cylindrical spouted beds (such as the capacity for handling coarse particles, small pressure drop, cyclic movement of the particles, and so on) with those inherent to their geometry, such as stable operation in a wide range of gas flow rates.5−7 This versatility in the gas flow rate allows handling particles of irregular texture, fine and light particles, and those with a wide size distribution, as well as sticky solids, whose treatment is difficult using other gas−solid contact regimes.8−12 Moreover, operation can be carried out in a dilute spouted bed with short gas residence times (as low as milliseconds).8,13 There are three different zones in a conical spouted bed with a draft tube, namely, spout, annulus, and fountain. Figure 1 shows these different zones. A crucial parameter that limits the scaling up of spouted beds is the ratio between the inlet diameter and particle diameter. In fact, the inlet diameter should be no more than 20−30 times larger than the average particle diameter in order to achieve spouting status. The use of a draft tube is the usual solution to this problem. Nevertheless, solid circulation rate, particle cycle time, gas distribution, minimum spouting velocity, and operating pressure drop are influenced by the type of draft tube used. The use of a draft tube causes changes in the hydrodynamics and solid circulation rate of spouted beds.14−17 The performance of the lower conical section of the contactor is different when the draft tube is used, and largely depends on the bed geometry, draft tube diameter, length of the entrainment zone, and operating conditions.14,15,17−23 Different draft tube configurations are reported in the literature: conventional nonporous draft tubes, porous draft tubes, and open-sided draft tubes. The latter have been developed by our research group24−26 and they are especially suitable for vigorous contact. Recently, Nagashima et al.15 have © 2013 American Chemical Society

Figure 1. Zones in the conical spouted bed with a nonporous draft tube.

developed porous and nonporous draft tubes with a conicalcylindrical geometry. Particle cycle time is defined as the time the particle takes to travel from the top of the annulus downward and back again to its starting point. Since the proportion of time spent by a particle in the spout is insignificant compared with that spent in the annulus, particle cycle times can be deduced from solid flow patterns in the annulus.1,2 Knowledge of particle cycle time is very useful to ascertain the bases of the spouted bed technique. Furthermore, information on this parameter and particle trajectories is Received: Revised: Accepted: Published: 15959

May 3, 2013 September 30, 2013 October 4, 2013 October 4, 2013 dx.doi.org/10.1021/ie401412j | Ind. Eng. Chem. Res. 2013, 52, 15959−15967

Industrial & Engineering Chemistry Research

Article

urations) and without devices. Based on this information, the factors of major influence on particle cycle time and solid circulation rate are determined and their effect is quantified.

essential for spouted bed applications, given that the average cycle time regulates energy and mass transfer, and influences chemical reactions.27−29 In fact, the average particle cycle time, tc̅ , is an index of bed turbulence or vigorousness, which in spouted beds is defined as the ratio between the solid mass in the bed, W, and the solid flow rate corresponding to the spout upper surface, QH:

tc̅ =

W QH

2. MATERIALS AND METHODS The experimental unit used is described in previous papers.25,54 The study has been carried out using contactors of different angles (γ) and inlet diameters (D0) (Figure 2). These

(1)

Many authors have addressed the study of cycle times and solid circulation rates in conventional spouted beds.30−39 Concerning conical spouted beds, Gorshtein and Mukhlenov40 have pioneered the study of particle trajectories and solid circulation using a piezoelectric method. Subsequently, particle velocities and solid circulation rates in conical spouted beds have been monitored mainly using an optic fiber probe.41−43 These studies reveal that cycle times in conical spouted beds are significantly shorter than in cylindrical ones, especially for steep contactor angles. This is a consequence of higher particle velocities in conical beds than in cylindrical ones. The few papers dealing with the solid circulation rate in draft tube spouted beds report the influence on particle circulation of operating and geometric variables, such as draft tube diameter, particle size, height of entrainment zone, contactor angle, and gas velocity.14,15,17,19,20,44−48 Zhao et al.17,49 and Wang et al.50 found that the solid circulation rate and particle velocity in cylindrical spouted beds with a draft tube are lower than in spouted beds without this tube due to the longer trajectory of the solid in the bed when a nonporous draft tube is inserted. They also found that the height of the entrainment zone and draft tube diameter play a significant role in the solid circulation rate, i.e., solid circulation rate increases as the value of these parameters is increased. This trend with draft tube diameter has also been observed by other authors, such as Wang et al.51 Similar results have been obtained by several authors in spouted beds of different configurations. Thus, Ijichi et al.,23,44,52 operating with conventional spouted beds and nonporous tubes, found that the particle circulation rate increases as tube cone clearance, mean particle diameter, and bed mass are increased. Furthermore, the solid circulation rate is reported to increase as gas velocity is increased when both nonporous and porous tubes are used in conventional spouted beds.14,51,53 Nagashima et al.,15 operating with conventional spouted beds and different types of tubes, found that the efficiency of gas− solid contact and solid circulation rate in the bed are considerably improved by inserting a conical-cylindrical porous draft tube into the bed. To our knowledge, the solid circulation rate in a draft tube conical spouted bed reactor has only been approached by Makibar et al.,45 in order to obtain information for their 25 kg/ h pilot plant for the fast pyrolysis of biomass waste. They used a half-conical bed and visual observation to determine the cycle times of a wooden pellet in a sand bed. They found that the draft tube allows controlling the cycle time and, therefore, the average residence time of biomass particles at the desired value by changing the height of the entrainment zone. The main purpose of this paper is to gather information about particle cycle times (average, maximum, and minimum) in conical spouted beds provided with different internal devices (nonporous and open-sided draft tubes of different config-

Figure 2. Geometric factors of conical contactors.

contactors are made of polymethyl methacrylate. The dimensions of each contactor depend on the contactor angle (γ). Thus, the column diameter is the same for the different angles used, Dc, 0.36 m. The height of the conical section, Hc, is 0.60, 0.45, and 0.36 for the angles of 28, 36, and 45°, respectively. Two types of draft tubes have been used: open-sided draft tubes and nonporous draft tubes. The performance of these internal devices is very different and, consequently, their application fields are also different. The main advantage of the nonporous draft tubes is the low gas flow rate required for operation and, therefore, these tubes are an interesting alternative for improving the efficiency of the condensation section in pyrolysis processes.29,45 Moreover, the uniform solid circulation pattern achieved with nonporous draft tubes make them suitable for the coating of granular materials.55 On the other hand, open-sided draft tubes allow for an optimum gas distribution in the bed, improving heat and mass transfer in the annular region. Consequently, open-sided draft tubes are the best option for certain processes, such as drying.24,26,56 Figure 3 shows the two types of tubes used. Thus, open-sided draft tubes of different tube diameter, DT, and width of the faces (related to the aperture ratio of the tube), WH, have been used. Likewise, nonporous draft tubes of different tube diameter, DT, and height of the entrainment zone, LH, have been used. The materials used for operation are glass beads and peas. Both belong to group D of Geldart classification and they have 15960

dx.doi.org/10.1021/ie401412j | Ind. Eng. Chem. Res. 2013, 52, 15959−15967


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measuring the time required to reach the bottom of the contactor following the wall.59 All these measurements have been made operating under stable spouted bed regime, with gas velocity being 1.10 times higher than that of minimum spouting conditions in all the experiments. Accordingly all the systems studied are under the same hydrodynamic conditions (minimum particle motion). The spouting velocities referred to the inlet section range from 10 to 55 m/s.

3. RESULTS AND DISCUSSION Runs have been carried out in conical spouted beds of different geometry according to a factorial design of experiments. The purpose of this design has been to study the effect on particle cycle times of the following: the insertion of a draft tube, draft tube configuration, and the different factors of the contactor and particle system. Given that different geometric factors are analyzed in each tube configuration, three sets of runs have been carried out, each one corresponding to a different type of tube. Tables 1, 2, and 3 show the factors and their levels when open-sided tubes and nonporous tubes are used and when no Table 1. Factors and Their Levels for Open-Sided Tube Systems factors

levels

γ (deg) D0 (m) WH (m) DT (m) ρs (kg/m3)

28 0.04 0.01 0.04 1230

36 0.05 0.018 0.05 2420

45 0.025

Table 2. Factors and Their Levels for Nonporous Tube Systems factors

levels

γ (deg) D0 (m) LH (m) DT (m) ρs (kg/m3)

Figure 3. Scheme of open-sided and nonporous draft tubes.

a similar particle diameter (4 mm glass beads and 3.4 mm peas). However, the density of the glass beads is approximately two times higher (2420 kg m−3) than that of the peas (1230 kg m−3). This is the case in processes we are currently studying, such as biomass pyrolysis,57,58 in which the biomass particles and char particles have a similar size but very different density. The stagnant bed height is around 0.27 m, which is also that corresponding to our 25 kg/h pyrolysis pilot plant. Particle cycle times has been measured by monitoring a marked (painted) particle of the same material, with visual observation in the fountain through the transparent wall and from the top of the bed. This method has shown to be very accurate and reliable for coarse particles such as those used in the present study. Thus, the painted particle is deposited on the surface of the annulus zone and the time the particle takes to reappear in the fountain, i.e., to complete a cycle, is measured successively. After a significant number of measurements have been made, an average cycle time is determined. The minimum cycle time is determined as the lowest value of the cycle times measured, and the maximum cycle time is measured by dropping the particle onto the bed adjacent to the wall and

28 0.04 0.07 0.04 1230

36 0.05 0.15 0.05 2420

45

Table 3. Factors and Their Levels for Systems without a Draft Tube factors

levels

γ (deg) D0 (m) ρs (kg/m3)

28 0.04 1230

36 0.05 2420

45

draft tube is used, respectively. To avoid instability problems, the runs with draft tube have been performed with the gas inlet diameter (D0) equal to that of the tube (DT). Experimental runs have been carried out by combining all these contactor geometries and draft tube configurations, i.e., approximately 70 runs involving nonporous draft tubes, 90 with open-sided tubes, and 35 without a draft tube. Particle cycle times (minimum, maximum, and average) have been determined in each of the runs whose operating conditions have been chosen according to the design of the experiments. Thus, at least fifty cycle times have been measured in each system in order to have a representative sample for 15961



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that contactor angle (γ) is the factor of greatest influence on the three cycle times analyzed (average, maximum, and minimum), followed by factors related to the draft tube (WH and LH) and density of the solid (ρs). The effect of gas inlet diameter (D0) is less significant. The quantitative effect of the factors analyzed on the parameters studied may be observed by plotting the values for the response vs the factor levels, i.e., main effect plots. Figure 5

statistical inference. To visualize the distribution of the random cycle times measured, histograms are plotted for each one and the average cycle time is obtained. Figure 4 shows the distribution of cycle times for a given system provided with an open-sided draft tube.

Figure 4. Cycle time distribution for a given system. Experimental conditions: γ 36°; WH, 0.018 m; DT, 0.04 m; D0, 0.04 m; material, glass beads; H0, 0.27 m.

Figure 5. Influence of contactor angle on the average cycle time for the three systems studied. Experimental conditions: γ 28, 36, and 45°; WH, 0.01 m; LH, 0.07 m;. DT, 0.04 m; D0, 0.04 m; material, peas; H0, 0.27 m.

Once the average cycle time tc̅ has been obtained and, given that the bed mass in each experiment is known, the solid circulation rate at the surface of the bed (ascending in the spout and descending in the annulus) is obtained by dividing the total mass by the average cycle time. Accordingly, the solid circulation rate is defined as the solid mass that is ascending through the spout and descending through the annulus per time unit. The knowledge of this parameter is essential to determine the amount of solids that can be treated, and consequently, the volume of the contactor. To ascertain the influence of the different factors on cycle time and, therefore, solid circulation rate, an analysis of variance (ANOVA) has been carried out for each type of draft tube by means of a standard statistical package (SPSS 12.0). The factors of greatest influence on cycle time are shown in Table 4 according to their significance order. The results show that the average cycle time is influenced by both the type of draft tube used and the geometrical factors of the contactor and material. The three systems studied confirm

shows the change in the average cycle time under given experimental conditions caused by the factor of greatest influence (contactor angle) in the three types of systems analyzed. As shown in Figure 5, as the angle of the contactor is greater, the average cycle time for the three systems is longer. This is explained by the higher amount of solid in the bed for the same stagnant bed height (which is 70% higher in the 45° contactor than in the 28° contactor) and by the longer particle trajectories (those descending near the wall) in the contactor with the higher angle. Therefore, operation with low contactor angles has a positive effect on solid circulation rate. Thus, higher contactor wall slope enhances solid circulation due to the increase in particle descending velocity in the annulus. These results are consistent with those obtained by Olazar et al.41 and Ijichi et al.23,44,52 in conical or conical-cylindrical spouted beds without a draft tube. A comparison of the systems studied shows that those without a tube are the ones with the shortest cycle times, followed by those with open-sided tubes, whereas the longest cycle times are those corresponding to a nonporous draft tube, with the increase in the latter being more than proportionate to the angle. Although this trend is general for all the systems analyzed, the differences attenuate when the longer entrainment height is used. Thus, when a 0.15 m entrainment zone height is used instead of 0.07 m shown in Figure 5, the differences between the cycle times corresponding to nonporous draft tubes and open-sided ones are smaller. Figure 6 shows the effect of gas inlet diameter on the average cycle time for the three different systems studied. As observed, an increase in gas inlet diameter (and draft tube diameter at the same time) causes an increase in the solid

Table 4. Significance Order of Factors and Interactions for All Systems and Materials (95% Confidence Interval)

tc̅

tcmax

tcmin

system

influence order

open-sided nonporous without tube open-sided nonporous without tube open-sided nonporous without tube

γ > WH > ρs > D0 > γ × ρs L H > γ > D 0 > L H× D 0 γ γ > ρs> WH > γ × ρs > D0 γ > LH > ρs > D0 > γ × ρs γ > ρs > γ × ρs > γ × D0 γ > ρs > WH > γ × ρs LH > γ > LH × D0 > D0 γ 15962



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Figure 8. Influence of the width of the faces of open-sided tubes on the average, maximum, and minimum cycle times. Experimental conditions: γ 36; WH, 0.01, 0.018, and 0.025 m; DT, 0.04 m; D0, 0.04 m; material, peas; H0, 0.27 m.

Figure 6. Influence of gas inlet diameter on the average cycle time for the three systems studied. Experimental conditions: γ 45°; WH, 0.018 m; LH, 0.07 m; DT, 0.04 and 0.05 m; D0, 0.04 and 0.05 m; material, glass beads; H0, 0.27 m.

circulation rate and, consequently, a reduction in the average cycle time. In fact, a higher cross-sectional area of the spout caused by a greater inlet diameter gives way to an increase in gas flow rate in this zone and, therefore, to an increase in the solid flow rate ascending in the spout. Figure 7 shows the change in the average cycle time caused by particle density for the three types of systems. As observed,

Figure 9. Influence of the height of the entrainment zone of nonporous tubes on the average, maximum, and minimum cycle times. Experimental conditions: γ 36; LH, 0.07 and 0.15 m;. DT, 0.04 m; D0, 0.04 m; material, peas; H0, 0.27 m.

Figure 8 shows that an increase in the width of the faces (aperture ratio of the tube is decreased) increases the three particle cycle times, especially the maximum cycle time. This trend is explained by the lower aperture ratio when the width of the faces is increased, which causes fewer particles to enter the spout through the open side of the tube and, therefore, the average particle trajectory is longer and solid circulation is lower. It should be noted that cycle times decrease monotonically in the three cases as the width of the faces is decreased. In fact, a value of zero for the width of the faces means operation without tube and the cycle times correspond to this situation. As shown in Figure 9, when the entrainment zone is higher the three average cycle times are lower and more uniform (smaller differences). This is explained by a larger fraction of particles entering the spout when the entrainment zone is higher, which gives way to higher solid circulation rate and lower cycle times.16,17,45,49,51,53,60,61 Furthermore, the air flow rate required for stable spouting is also higher as the entrainment zone is longer. Consequently, more air diverts from the spout into the annulus at the bottom of the bed,

Figure 7. Influence of particle density on the average cycle times. Experimental conditions: γ 36; WH, 0.018 m; LH, 0.07 m; DT, 0.04 m; D0, 0.04 m; material, peas and glass beads; H0, 0.27 m.

solid density has a significant role only when open-sided draft tubes are used but not with nonporous tubes or without internal device. An increase in solid density with open-sided draft tubes enhances solid circulation and, therefore, reduces the cycle time. An explanation lies in the stability provided by the open-sided draft tube. Thus, solid circulation with this internal device is high and uniform, whereas big bubbles and even slugging may appear when coarse particles are used without tube. Figures 8 and 9 show the change in the average, maximum, and minimum particle cycle times caused by the width of the faces in the open-sided tubes, WH, and by the height of the entrainment zone in the nonporous tubes, LH, respectively. 15963



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porous draft tube is used instead of a nonporous one in conventional spouted beds. As with open-sided tubes, porous tubes allow annulus aeration as gas diverts from the spout into the annulus along the whole length of the spout. Furthermore, the values of the average cycle time for opensided tubes are closer to those without a tube than to those with nonporous tubes. This is explained by the rather high aperture ratios of the open-sided tubes used (42, 57, and 78%). In fact, the higher value of aperture ratio used (78%) provides cycle times similar to those without a tube, with the advantage of greater stability. To visualize the ranges between the maximum and minimum average cycle times for given operating strategies, they have been plotted in Figures 11−13 grouped according to different

thereby contributing to a more vigorous solid circulation in the whole bed. Figures 8 and 9 also show that the values of the maximum cycle time are approximately 3-fold higher than the values of the minimum cycle time. The former correspond to the trajectory in the annulus following the wall of the contactor, whereas the latter corresponds to the trajectory along the external surface of the draft tube. In fact, several authors measured the maximum cycle times (particle velocity at the wall) and proved that the maximum cycle time is severely affected by wall friction.30,32,37,38,62−64 It should also be noted that, in view of the slope of the lines in Figures 8 and 9, the width of the faces and entrainment zone height have a greater impact on the maximum cycle time than on the minimum and average cycle times. To gain an overall view of the effect caused by the use or not of different draft tubes, Figure 10 shows the average values of the cycle times for the three types of systems studied under the whole range of conditions studied.

Figure 11. Average cycle time ranges for the three systems studied.

operating alternatives. Figure 11 shows the maximum and minimum values for the systems grouped according to the type of draft tube or to the operation without a tube. As shown in Figure 11, the average cycle time range is very different depending on the system used. Unlike conventional spouted beds, the widest range corresponds to the nonporous draft tube systems, whereas the open-sided draft tube systems and those without draft tube have much narrower ranges, with the latter ones having the narrowest ranges and shortest cycle times. The fact that nonporous tube systems have the widest ranges evidence that, under the experimental conditions used in this work, the factors related to the nonporous tube have a greater influence on the average cycle time than those related to the open-sided draft tube. Thus, the particles in a conical spouted bed with a nonporous draft tube follow very different trajectories from the surface to the bottom. Thus, the trajectories of the particles that descend near the wall are much longer and their velocity is lower than those that descend near the draft tube. The difference in the length of the trajectories is not significant in conventional cylindrical beds, but it is significant in conical spouted beds, especially for high stagnant beds and wide angles. When conical spouted beds are fitted with an open-sided draft tube, the solids may enter the spout at any level, and this means cycle times are more uniform. Furthermore, the higher flow rate required for spouting gives way to a more vigorous bed, which contributes to shortening cycles times and rendering them more uniform. When there is no draft tube, the air flow

Figure 10. Overall average cycle times (tc) for the three systems studied.

As observed, the experimental systems with the lower average cycle time (higher solid circulation rate) are those without a draft tube.17 Open-sided tubes have intermediate average cycle times, and nonporous ones are the systems with the highest values (between double and triple those without a tube). The decrease in cycle time (increase in solid circulation rate) is closely related to the vigorousness of the system, which is higher in systems without a tube, given that they require a much higher air flow rate for spouting. Nevertheless, instability is greater without a tube, and this is a serious drawback for scaling up. An open-sided draft tube stabilizes the bed, although the cycle time is slightly higher. Nonporous tubes give way to the highest values of average cycle time due to the poor vigorousness of the system caused by the small amount of solid cross-flow from the annulus into the spout (only in the entrainment zone) and the small air fraction diverting into the annulus, given that most of the air fraction rises through the spout.14,21,51 Consequently, the air flow rate required for spouting is much smaller, the amount of solids circulating in the contactor is lower, and the gas−solid contact is poorer, but this system is more stable than any other one.14,25 As happens with open-sided tubes, Nagashima et al.15 and Wang et al.50 found that the circulation rate increases when a 15964



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contactor angle (γ), the factors of each type of tube (WH, LH), gas inlet diameter (D0), and solid density (ρs). The experimental systems with the lowest cycle times and highest solid circulation rates are those without draft tubes, but they have more stability problems and their minimum spouting velocity and operating pressure drop are higher. Open-sided draft tubes register intermediate particle cycle times and solid circulation rates, and nonporous ones are the systems with the highest particle cycle times and lowest solid circulation rates. Furthermore, open-sided tube systems have better gas−solid contact than nonporous tube systems. The system without a draft tube is adequate at bench-scale, but draft tubes are necessary on a larger scale. Draft tubes give stability to the system. Furthermore, the desired cycle times of the particles can be obtained for the conical spouted bed by choosing the suitable configuration of both the bed and the draft tube. The width of the faces of the open-sided tubes and the height of the entrainment zone of the nonporous tubes are two effective factors for optimizing the solid circulation rate in draft tube conical spouted beds. Consequently, the solid circulation rate can be easily controlled by aerating the bed and adjusting the amount of solids entering the spout through the width of the faces and the height of the entrainment zone, thereby giving the design flexibility.

rate required for spouting is even higher, and although the bed has greater instability problems, cycle times are more uniform. Consequently, the desired particle cycle times can be obtained for a conical spouted bed by choosing the suitable configuration of the bed and draft tube. Figures 12 and 13 show the cycle time ranges for the different levels of the factors studied in the two types of tubes, i.e., width of the faces and height of entrainment zone, respectively.

■

Figure 12. Average cycle time ranges when open-sided tubes of different face width are used.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +34 946392527. Fax: +34 946393500. E-mail: martin. [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been carried out with the financial support of the Ministry of Science and Education of the Spanish Government (Project CTQ2010-16133) and of the Basque Government (Project IT748-13).

■ Figure 13. Average cycle time ranges when nonporous tubes of different entrainment height are used.

Figure 13 shows that the variation of the factor (entrainment height) in a nonporous draft tube involves a great variation in the spouted bed operating regime. A similar effect is noted with the width of the faces in open-sided tubes (Figure 12), although the difference is not so significant. It should be noted that a decrease in the width of the faces contributes mainly to decreasing the maximum average cycle time, but the minimum is approximately the same independent of the width.

4. CONCLUSIONS The results obtained based on an experimental design show that particle cycle times (average, maximum, and minimum) and solid circulation rates are influenced by both the type of draft tube used and its geometry. For all systems, the parameters influencing particle cycle times are (according to a decreasing order of significance):

NOTATION DC = column diameter, L Di = contactor base diameter, L D0 = gas inlet diameter, L DT = diameter of the draft tube, L H0 = stagnant bed height, L HC = height of the conical section, L LH = height of the entrainment zone of the nonporous draft tube, L tc̅ = average cycle time, t tcmax = maximum cycle time, t tcmin = minimum cycle time, t W = bed weight, M WH = width of the face of the open-sided draft tube, L QH = solid circulation mass flow rate, M t−1

Greek Letters

■

γ = cone angle, rad ρs = density of the particle, ML−3

REFERENCES

(1) Mathur, K.; Epstein, N. Spouted Beds; Academic Press: New York, 1974.

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