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Modeling the Hydrodynamics of Downers by Cluster-Based Approach Shayan Karimipour, Navid Mostoufi,* and Rahmat Sotudeh-Gharebagh Process Design and Simulation Research Center, Department of Chemical Engineering, UniVersity of Tehran, P.O. Box 11365/4563, Tehran, Iran
Cluster formation in the downers has been supported by experimental works of different investigators. A new model has been developed in this work based on the force balance over a cluster moving along the axial length of the downer. The clusters are considered to be spherical with constant diameter. Axial profiles of cluster velocity, solid holdup, and pressure in downer have been obtained by this model. Comparison between the results of the model and the experimental data found in the literature showed that the model is able to predict the experimental data satisfactorily. Also, a correlation has been proposed for equivalent diameter of clusters in the downer as a function of the superficial gas velocity and the solids circulation rate. 1. Introduction Co-current downflow circulating fluidized bed reactors, downers, are able to overcome the drawbacks encountered by the widely used riser reactors. Despite the numerous advantages of risers, they exhibit drawbacks such as severe axial backmixing of gas and solids as well as radial nonuniformity in gas velocity, particle velocity, and solids concentration. Such problems could be solved to a great extent by utilizing downers. Since the early 1990s, various researchers have investigated hydrodynamics and flow structure in the downers.1-10 However, much more works are still needed in order to adequately understand the underlying phenomena in the downers. There are three distinct flow regions along the axial length of the downer, as shown in Figure 1. In the first region, the particle velocity is lower than the gas velocity. Hence, the particles are accelerated by both the gravity and the drag force. Action of both these forces in the same direction causes the particle velocity to be increased sharply in this region. The second region begins when the particle velocity reaches the gas velocity, and the slip velocity becomes zero. Therefore, the drag and gravity forces are adversely acting on the particle, and the acceleration rate becomes much lower than that in the previous region. This region prolongs until the gravity and drag forces balance each other and the particle velocity becomes nearly constant. These three successive zones are named as the first and the second acceleration regions and the fully developed zone, respectively. Although modeling of the riser of the circulating fluidized bed could be easily found in the literature,11-15 there is little attention to the modeling of the downer. Recently, Jian and Ocone16 used the two-fluid model to obtain the radial profiles of the gas and particle flow in the downer. Li et al.17 employed the energy minimization concept to model the radial behavior of the downer. In this model, gas and particles are subject to not only mass and momentum conservation but also to minimum energy consumption. However, their results were limited to the fully developed region. Bolkan et al.18 attempted to acquire the axial profiles of particle velocity and holdup. They modified the drag coefficient, without a sound theoretical base, to provide the higher drag force and to represent the experimental data. However, higher drag force could be better realized by applying the drag on clusters instead of single particles.19 * Corresponding author, Tel.: (98-21) 6696-7797. Fax: (98-21) 6646-1024. E-mail:
[email protected].
Figure 1. Different hydrodynamic regions of the downer.
Although the solids concentration in the downer is lower than that in the riser,8 particle clustering is still the predominant phenomenon, resulting in slip velocities several times greater than the terminal velocity of a single particle.3 Existence of clusters has been also confirmed by researchers who employed imaging techniques for investigating the flow pattern in the downers.4,5,9 Nevertheless, characteristics of the clusters in the downers are still not completely investigated, and it is necessary to be further studied. A new model has been developed in this work on the basis of the force balance over the clusters moving along the axial length of the downer. Axial profiles of cluster velocity, solids holdup, and pressure drop in the downer could be obtained using the model, accordingly. 2. Modeling In a recent work, Bolkan et al.18 adopted the force balance equation for one-dimensional motion of single particles in the downer (particle-based approach, PBA). However, based on various experimental observations of clusters in downer, it has been proven that clusters are dominated in the downer and must be taken into account in any modeling efforts. Therefore, in the present study, the model is developed on the basis the forces
10.1021/ie060026k CCC: $33.50 © 2006 American Chemical Society Published on Web 09/20/2006
Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7205 Table 1. Correlations of Effective Drag Force Coefficient20 CD,0 )
24 0.413 (1 + 0.173Recl0.657) + Recl 1 + 16.300Re
CD ) fCD,0 f ) -m m ) 3.02Ar0.22Ret-0.33
Once the velocity profile of clusters in the downer is determined from eq 5, the corresponding solids holdup could be calculated from
-1.09 cl
s ) 1 - g )
() dcl ds
0.40
exerted on a cluster (cluster-based approach, CBA), neglecting the effect of single particles. If the clusters are assumed to be spheres of constant diameter, the force balance equation would be as follows:
|
|(
)
dVcl 1 U0 U0 FclVcl ) Fg - Vcl - Vcl AclCD + (Fcl - Fg)Vclg dt 2 g g (1) The drag force term as expressed in eq 1 is valid in both first and second acceleration zones (see Figure 1) regardless of the direction of the relative movement of clusters. It is worth mentioning that the void fraction in this model should be in fact calculated on the basis of the volume of bed minus the volume of clusters. However, it is substituted by the volume fraction of gas (the volume of bed minus the volume of solids) in eq 1 since both these void fractions are very close to 1 and such a substitution does not introduce significant error in the results. Substituting Vcl and Acl in eq 1 by
π Vcl ) dcl3 6
(2)
π Acl ) dcl2 4
(3)
dVcl dVcl ) Vcl dt dz
(4)
eq 1 could be rearranged as
|(
)
U0 dVcl 3FgCD U0 g - Vcl - Vcl + (F - Fg) (5) ) dz 4dclFclVcl g g FclVcl cl The effective drag coefficient, CD, in eq 5 could be estimated from the correlation of Mostoufi and Chaouki20 as given in Table 1. In the above equation, the porosity is determined from the solids mass balance equation
Gs ) Fs(1 - g)Vcl
(6)
in which the solids velocity is in fact considered to be the same as cluster velocity since all solids are assumed to move as clusters. At the top, the particles are fluidized at the minimum fluidization condition before entering the downer.21 Therefore, the initial value of the cluster velocity could be obtained from the solids mass balance, and eq 5 should be solved subject to the following initial condition:
Vcl|z)0 )
(8)
The momentum balance over the downer could be used to obtain the profile of the pressure drop along the downer. Such a momentum balance could be expressed as follows:
dp dp dp dp ) dz dz head dz acceleration dz friction
( )
( )
( )
(9)
where
() dp dz
(dpdz)
head
) Fssg + Fggg
) FssVcl
acceleration
(10)
()
dVcl U0 d U0 + F g g dz g dz g
(11)
The friction pressure drop comprises two terms, i.e., gas-wall and particle-wall frictions:
(dpdz)
) friction
(dpdz)
+ gas-wall
(dpdz)
particle-wall
(12)
These pressure losses are defined by the Fanning equation as 2
dp 1 U0 ) f g Fg dz gas-wall 2D g
( ) (dpdz)
cluster-wall
1 ) fs FssVcl2 2D
(13) (14)
where the gas-wall friction factor, fg, could be obtained from the Blasius formula,22
and considering the equality
|
Gs FsVcl
Gs Fs(1 - mf)
(7)
fg )
0.316 Reg e 105 Reg0.25
(15)
and the cluster-wall friction factor could be estimated using the correlation of Kanno and Saito:23
fs )
0.057 (gD)1/2 2Vcl
(16)
3. Results and Discussion To examine the results of the proposed model, four series of experimental data, reported by various researchers, were used. Solving the model needs a correlation for the cluster diameter which does not exist in the literature. Hence, the experimental data of Bolkan et al.18 were used to establish such a correlation. The data of Liu et al.,8 Cao and Weinstein,24 and Zhang and Zhu7 were then used for validation of the model. In all cases, eq 5 was solved with the assumption that the porosity of the cluster is at the minimum fluidization condition. 3.1. Cluster Size. It has been shown that the size of the clusters is increased with an increase in the solids circulation rate and/or a decrease in the superficial gas velocity in the downer.9 This could be attributed to the fact that the higher the solid concentration is, the easier the larger clusters would be formed. With an increasing the number of particles in the cluster, the effective drag force that each particle sustains is reduced, which results an increase in the cluster velocity.4 However, a
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Table 2. Operating Condition Used for Estimating the Cluster Diameter18 case
particle
dp (µm)
Fp (kg/m3)
Gs (kg/m2s)
U0 (m/s)
1 2 3 4 5 6 7 8 9
FCC FCC FCC FCC FCC FCC FCC FCC FCC
67 67 67 67 67 67 67 67 67
1500 1500 1500 1500 1500 1500 1500 1500 1500
49 49 50 101 101 102 194 208 205
3.7 7.3 10.1 3.7 7.2 10.2 3.7 7.2 10.2
further increase in the velocity of the cluster leads to a decrease in the cluster size due to dissipation and eruption of clusters. On the other hand, increasing the solid circulating rate leads to an increase in the concentration of solids, followed by formation of larger clusters. In the absence of a correlation to estimate the cluster size, experimental data of velocity and solids holdup of Bolkan et al.18 have been used in this work to estimate the equivalent cluster diameter at different operating conditions given in Table 2. For each set of data, eq 5 was solved by guessing different values of dcl. The cluster size at the specific operating condition was set to the value that best fits the experimental data. A sample result of such a sensitivity analysis is shown in Figure 2 for the cluster size. As can be seen in this figure, increasing the cluster size raises the cluster velocity by increasing the weight of the cluster. The cluster size at which the model best fits the experimental data is considered as the cluster diameter in the corresponding operating condition. Cluster sizes in the downer were determined by this method at different superficial velocities and solids circulation rates. On the basis of the cluster size provided by the above method, the following correlation for the cluster diameter as a function of operating conditions was developed:
dcl ) 19.33 + 0.03Gs - 1.87U0 dp
Figure 2. Effect of cluster velocity on model predictions (Gs ) 49 kg/ (m2‚s), U0 ) 7.3 m/s).
(17)
with a Pearson correlation coefficient of 0.962. The standard deviation of the constants of the correlation were estimated to be 3.98, 8.78 × 10-5 (m2‚s)/kg and 0.05 s/m, respectively. A comparison between the results of the model presented in this work with the experimental data is made in Figure 3 in terms of the relative cluster diameter. As it could be seen in this parity plot, there is a good agreement between the cluster diameters calculated from eq 17 and those obtained from the fitting of eq 5 to the experimental data. Comparison between the model predictions and the experimental data of solids velocity of Bolkan et al.18 is presented in Figure 4a-c for various solids velocity and Figure 5a-c for solids holdup along the downer. The cluster size in the model was determined from eq 17. It could be concluded from these figures that the cluster size might be a key parameter in estimating the hydrodynamic variables in the downer and its proper estimation could directly improve the model predictions. A correlation similar to eq 17 was also proposed by Bolkan et al.18 for estimating the cluster size. However, their approach for evaluating the cluster size is quite different from the present study. Bolkan et al.18 proposed a modified drag coefficient and determined the cluster size based on the solids velocity in the fully developed zone. It was shown in the present study that there is no need to modify the drag coefficient artificially while suitable determination of the hydrodynamic status of the downer is possible by taking into account the movement of the clusters
Figure 3. Comparison between cluster sizes obtained from sensitivity analysis and predicted from the correlation.
in the downer whose existence has been already proved experimentally. As a result, the cluster sizes obtained in the present work could be considered to be more realistic than that proposed in the literature. 3.2. Model Validation. Results of the model were compared with experimental data in terms of solids holdup, solids velocity, and pressure drop along the downer height in Figure 6a-c, respectively. Corresponding operating conditions of the experimental data are given in Table 3. Results of the CBA model and the solids velocity reported by Zhang et al.7 are shown in Figure 6a. This figure illustrates that the CBA results are in close agreement with the experimental data of the solids velocity in the downer. Results of the PBA model are also included in this figure. As could be seen in this figure, the PBA model presents a very short acceleration zone height, while this zone is considerably longer according to the experimental data. The PBA model also underpredicts the solids velocity, which
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Figure 4. Comparison between the model results and the experimental data18 for the cluster velocity along the downer height: (a, top) Gs ) 49 kg/(m2‚s), (b, middle) Gs ) 101 kg/(m2‚s), and (c, bottom) Gs ) 200 kg/ (m2‚s).
indicates that the drag force exerted on the solids is not being estimated accurately in the PBA model. In other words, the solids gain momentum slower and move faster than if the solids were moving as isolated particles. Such drawbacks of the PBA
Figure 5. Comparison between the model results and the experimental data18 for the solids holdup along the downer height: (a, top) U0 ) 3.7 m/s, (b, middle) U0 ) 7.2 m/s, and (c, bottom) U0 ) 10.2 m/s.
model further confirm that the solids, indeed, move in the form of clusters rather than isolated particles. In the CBA model,
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Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 Table 3. Operating Conditions Used for the Model Validation case
particle
dp (µm)
Fp (kg/m3)
U0 (m/s)
bed diam (m)
Ref
1 2 3 4
FCC FCC FCC FCC
70 82 67 67
1300 1480 1500 1500
5.44 2.9 5.7 8.06
0.025 0.127 0.1 0.1
8 24 7 7
pressure gradient in the downer consists of three components. The static head causes the pressure to increase along the downer, whereas acceleration and frictional effects result in the pressure loss. Figure 6c demonstrates that there is a negative pressure gradient at the entrance of the downer (i.e., acceleration zone) which is due to high values of particles acceleration in this region. With decreasing the rate of acceleration, the static head becomes dominant and leads to the positive values of pressure gradient. The pressure gradient eventually reaches a constant value, indicating the complete flow development (fully developed zone). Good agreement between the results of the model and the experimental data confirms that the same effective drag coefficient, which is used in other forms of fluidized systems, is also applicable to the down-flow streams and modification of drag coefficient as suggested by Bolkan et al.18 is not necessary. 3.3. Effect of Acceleration on Pressure Drop. The pressure drop across the length of the fluidized beds is caused by static head, acceleration, and gas-to-pipe and solid-to-pipe friction. The friction is proved to have a little effect on the total pressure drop and is neglected in this part. Although, in many experimental attempts the pressure drop has been considered to be made up of only a static head (from which the average voidage of the section is evaluated), the influence of solids acceleration cannot be neglected in the acceleration zones. Figure 7 demonstrates the effect of particle acceleration on the total pressure drop in the downer. As can be seen in this figure, in the initial zones of the downer, which are the first and second acceleration zones, particle acceleration has a remarkable effect on the total pressure drop and cannot be neglected. Moving forward along the downer diminishes the influence of particle acceleration until the fully developed region in which the pressure drops is only due to the static head of the gas and solids, increasing the cluster velocity and reducing the length of the acceleration region. This comparison indicates that, only in the fully developed region and at high velocities, the solid concentration could be calculated by measuring the pressure drop
Figure 6. Comparison between the model results and the experimental data18 in terms of (a, top) solids velocity along the downer height, (b, middle) solids holdup in various solids circulating rates, and (c, bottom) pressure drop gradient along the downer height.
the clusters are larger species than the particles, thus, gaining momentum faster, and could reach higher velocities. Experimental solids holdups reported by Liu et al.8 and Cao and Weinstein24 at various solids circulation rates are compared with the model predictions in Figure 6b. As can be seen from this figure, the suggested model is capable of predicting these experimental data satisfactorily. Figure 6c provides a comparison between the experimental data of Bolkan et al.18 and the results of the model in terms of pressure gradient along the downer. According to eq 9, the
Figure 7. Effect of considering the acceleration term in the pressure drop equation.18
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gradient. Both the static head and acceleration should be considered in the acceleration regions, especially in the beginning of the downer.
Fg ) gas density (kg/m3) Fs ) solids density (kg/m3) Literature Cited
4. Conclusions On the basis of numerous pieces of evidence about the existence of clusters in downers and the shortcoming of particlebased models, a model was developed for describing the hydrodynamics of the downer of a circulating fluidized bed. The clusters were assumed to be rigid spheres, having porosity equal to voidage at minimum fluidization. The main equation of the model is Newton’s second law written for the forces acting on a cluster. In the absence of reliable experimental data in the literature on the cluster size in the downer, a sensitivity analysis was performed to investigate the effect of the cluster size and estimate the equivalent cluster diameter. On the basis of such an analysis, a correlation has been proposed for estimating the cluster diameter as a function of operating conditions. Comparison of the model results and the experimental data showed that there is a good agreement between the model and the data. It was also found that the effective drag coefficient which is used in the modeling of the other forms of fluidized beds is also applicable to the down-flow streams and there is no need to modify such a drag coefficient, as proposed by other researchers. It was also shown that the estimation of the solids holdup based on only pressure drop gradient (i.e., neglecting the particle acceleration term) would lead to erroneous results in the acceleration regions. Nomenclature Acl ) cross-sectional area of cluster (m2) Ar ) Archimedes number (ds3Fg(Fs - Fg)g/µ2) CD ) effective drag coefficient CD,o ) standard drag coefficient dcl ) cluster diameter (m) dp ) particle diameter (m) D ) downer diameter (m) f ) drag coefficient correction factor fs ) solid-wall friction factor fg ) gas-wall friction factor g ) acceleration of gravity (m/s2) Gs ) solids flux (kg/m2s) m ) exponent in correction factor of drag coefficient in Table 1 p ) pressure (Pa) Reg ) gas Reynolds number (DU0Fg/µg) Recl ) cluster Reynolds number (dclUclFg/µg) Ret ) terminal Reynolds number of cluster (dclUtFg/µg) t ) time (s) U0 ) superficial gas velocity (m/s) Vcl ) cluster velocity (m/s) Vcl ) cluster volume (m3) z ) downer height from top (m) Greek Letters g ) voidage s ) solids holdup mf ) voidage at minimum fluidization Fcl ) cluster density (kg/m3)
(1) Herbert, P. M.; Gauthier T. A.; Briens, C. L.; Bergougnou, M. A. Application of Fiber Optic Reflection Probes to the Measurement of Local Particle Velocity and Concentration in Gas-Solid Flow. Powder Technol. 1994, 80, 243. (2) Herbert, P. M.; Gauthier, T. A.; Briens, C. L.; Bergougnou, M. A. Flow Study of a 0.05 M Diameter Downflow Circulating Fluidized Bed. Powder Technol. 1998, 96, 255. (3) Tuzla, K.; Sharma, A. K.; Chen, J. C.; Schiewe, T.; Wirth, K. E.; Molerus, O. Transient Dynamic of Solid Concentration in Downer Fluidized Bed. Powder Technol. 1998, 100, 166. (4) Krol, S.; Pekediz, A.; de Lasa, H. Particle Clustering in Down Flow Reactors. Powder Technol. 2000, 108, 6. (5) Nova, S.; Krol, S.; de Lasa, H. Particle Velocity and Particle Clustering in Down-Flow Reactors. Powder Technol. 2004, 148, 172. (6) Zhang, H.; Zhu, J. X.; Bergougnou, M. A. Hydrodynamics in Downflow Fluidized Beds (1): Solids Concentration Profiles and Pressure Gradient Distributions. Chem. Eng. Sci. 1999, 54, 5461. (7) Zhang, H.; Zhu, J. X. Hydrodynamics in Downflow Fluidized Beds (2): Particle Velocity and Solids Flux Profiles. Chem. Eng. Sci. 2000, 55, 4367. (8) Liu, W.; Liu, K. B.; Zhu, J. X.; Beeckmans, J. M. Characterization of High-Density Gas-Solids Downward Fluidized Flow. Powder Technol. 2001, 115, 27. (9) Lu., X.; Li, S.; Du, L.; Yao, J.; Lin, W.; Li, H. Flow Structures in The Downer Circulating Fluidized Bed. Chem. Eng. J. 2005, 112, 23. (10) Ball, J. S.; Zhu, J. X. A Preliminary Study into the Local Solids Fluxes in a Downer Reactor. Powder Technol. 2001, 114, 96. (11) Zhou, H.; Flamant, G.; Gauthier, D.; Lu, J. Lagrangian Approach for Simulating the Gas-Particle Flow Structure in a Circulating Fluidized Bed Riser. Int. J. Multiphase Flow 2002, 28, 1801. (12) Das, A. K.; Baudrez, E.; Marin, G. B.; Heynderickx, G. J. ThreeDimensional Simulation of a Fluid Catalytic Cracking Riser Reactor. Ind. Eng. Chem. Res. 2003, 42, 2602. (13) Bollas, G. M.; Vasalos, I. A.; Lappas, A. A.; Iatridis, D. Modeling Small-Diameter FCC Riser Reactors. A Hydrodynamic and Kinetic Approach. Ind. Eng. Chem. Res. 2002, 41, 5410. (14) Vandewalle, Ch.; Baeyens, J.; Claerbout, A. Hydrodynamic Model for the Gas Flow in Circulating Fluidized Bed Reactors. Ind. Eng. Chem. Res. 2002, 41, 5983. (15) Huilin, L.; Gidaspow, D. Hydrodynamic Simulations of Gas-Solid Flow in a Riser. Ind. Eng. Chem. Res. 2003, 42, 2390. (16) Jian, H.; Ocone, R. Modeling the Hydrodynamics of Gas-Solid Suspension in Downers. Powder Technol. 2003, 138, 73. (17) Li, S.; Lin, W.; Yao, J. Modelling the Hydrodynamics of the Fully Developed Region in a Downer Reactor. Powder Technol. 2004, 145, 73. (18) Bolkan, Y.; Berruti, F.; Zhu, J.; Milne, B. Modeling Circulating Fluidized Bed Downers. Powder Technol. 2003, 132, 85. (19) Sabbaghan, H.; Sotudeh-Gharebagh, R.; Mostoufi, N. Modeling the Acceleration Zone in the Riser of Circulating Fluidized Beds. Powder Technol. 2004, 142, 129. (20) Mostoufi, N.; Chaouki, J. Prediction of Effective Drag Coefficient in Fluidized Beds. Chem. Eng. Sci. 1999, 54, 851. (21) Johnston, P. M.; de Lasa, H. I.; Zhu, J. Axial Flow Structure in the Entrance Region of a Downer Fluidized Bed-Effects of The Distributor Design. Chem. Eng. Sci. 1999, 54, 2161. (22) Fox, R. W.; McDonald, A. T.; Pritchard, P. J. Introduction to Fluid Mechanics, 6th ed.; Wiley: New York, 2003. (23) Kanno, H.; Saito, S. Pneumatic Conveying of Solid through Straight Pipes. J. Chem. Eng. Jpn. 1969, 2, 211. (24) Cao C.; Weinstein, H. Characterization of Downflowing High Velocity Fluidized Beds. AIChE J. 2000, 46, 515.
ReceiVed for reView January 9, 2006 ReVised manuscript receiVed August 20, 2006 Accepted August 21, 2006 IE060026K
