Chemical Engineering Science 55 (2000) 5283}5289
Descent velocities of particle clusters at the wall of a circulating #uidized bed Peter D. Noymer*, Leon R. Glicksman Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 30 March 1999; accepted 14 January 2000
Abstract The descent velocities of clusters of particles near the wall of a circulating #uidized bed have been measured and reported by many investigators. Curiously, the measured velocities all fall within the range 0.3}2.0 m/s, regardless of particle size, material density or gas velocity. In an attempt to explain this phenomenon, a model for drag on a cluster was developed. This model neglects the interaction between the particles and the wall as well as the forced #ow of gas in the wall region, considering only the descent of a cluster in quiescent air at a terminal velocity. There are two elements to the drag force acting on a cluster in this model: one caused by #ow passing through the cluster of particles, the other caused by #ow passing around the cluster. Simpli"cation of terms in this model shows that the terminal velocity of a cluster, non-dimensionalized by the minimum #uidization velocity of the particles, can be related to the Archimedes number of the gas}solid #ow. Good agreement is obtained between this simple model and the reported measurements of cluster velocities. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The increase in research interest in #uidized beds operating in the regime of fast #uidization can be traced back to the late 1970s (Yerushalmi & Cankurt, 1979). Circulating #uidized beds (CFBs) operate in such a #ow regime, in which it is generally agreed that the gas}solid #ow takes on a core-annular structure (e.g., Yerushalmi & Cankurt, 1979; Ishii, Nakajima & Horio, 1989). This structure is characterized by a dilute concentration of upward-#owing particles in the core surrounded by a denser and more intermittent concentration of downward-#owing particles (clusters) in the annular region adjacent to the riser wall. A signi"cant amount of CFB research has been focused on understanding these gas}solid #ow patterns as a means of understanding phenomena such as heat and mass transfer (e.g., Subbarao & Basu, 1986; Brereton & Grace, 1993). For bed-to-wall heat transfer, the interactions of the particle
clusters with the wall are of particular importance, and improvements in #ow-visualization equipment have facilitated the measurement of the #ow of particles or clusters near the wall (e.g., Rhodes, Mineo & Hirama, 1992; Horio & Kuroki, 1994). One #ow characteristic that is straightforward to measure and has been reported by many investigators is the descent velocity of particle clusters near the wall of a CFB riser. Although the super"cial gas velocities in the risers vary greatly, as do the size and composition of the solid particles, the descent velocities measured are consistently between 0.3 and 2.0 m/s, with most measurements around 1.0 m/s. The fact that these measurements are nearly uniform is quite curious, given the wide variation in operating conditions. This curiosity provided the motivation for this study, in which a rather simple analysis of cluster motion results in a semi-empirical correlation that provides good agreement with the body of data available. 2. Analysis * terminal velocities of clusters 2.1. Overview and assumptions
* Corresponding author. Present address: Aradigm Corporation, 3929 Point Eden Way, Hayward, CA 94545, USA. Tel.: #1-510-2659107; fax: #1-510-265-0277. E-mail address:
[email protected] (P. D. Noymer).
The model used to predict cluster descent velocities relies on a number of simplifying assumptions which are listed below.
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 1 7 1 - 8
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Proximity of a cluster to a wall. Previous research by Lints and Glicksman (1993) indicates that clusters travel very close to the walls in CFBs, often coming as close as 100 lm to the wall. Based on arguments given later, this places clusters well within the hydrodynamic boundary layer of the gas #ow in the riser. Although the presence of the wall is likely to play a role in determining the motion of a cluster in close proximity, any e!ect of the wall is ignored for simplicity in this analysis. Implications of this assumption are discussed later. Characteristic size of a cluster. Previous research by Rhodes et al. (1992) indicates that a typical cluster in a scale-model CFB has a characteristic dimension of about 1 cm. Analysis of infrared thermography by Noymer and Glicksman (1998) con"rms this dimension, observed as typical of the lateral dimension of a cluster (i.e., the dimension across the wall normal to the direction of motion). Gas-phase boundary layer. From single-phase boundary layer theory for turbulent #ow in a pipe (White, 1991b), a boundary-layer thickness of 1.9 cm can be calculated for air at standard temperature and pressure #owing at an average velocity of 3 m/s in a duct with a diameter of 15 cm (typical of operating conditions in a scale-model CFB riser). Furthermore, work by Rashidi, Hetsroni and Banerjee (1990) shows that the presence of a signi"cant concentration of particles (as in a CFB) retards the radial development of the boundary layer, indicating that the actual thickness of the hydrodynamic boundary layer will be greater than 2 cm. Gas velocity near the downward-yowing clusters. Since a cluster is about 1 cm thick and travels as close as 100 lm to the wall of a CFB, and since the boundarylayer thickness is greater than 2 cm, it is clear that a cluster generally travels within the boundary-layer region. Ordinarily, this would mean that the opposite faces of the cluster (one facing the wall and one facing the core #ow) would see signi"cantly di!erent local gas velocities because of the velocity gradient in the boundary layer. However, the work of Rashidi et al. (1990) cited earlier indicates that the velocity gradient is very shallow in the presence of particles, implying that the local gas velocities seen by opposite faces of the cluster are likely to be similar and low. Because of this, it can be postulated that the descent velocity of a cluster should be una!ected by the average gas velocity, since the local gas velocity near the cluster will always be small. In fact, separate studies by Lim et al. (1996) and Noymer (1997) show that the cluster velocities do not vary with super"cial gas velocity for the same riser geometry, bed material and other operating conditions. As a result, this hypothesis is adopted as an assumption in this study; i.e., the gas velocity in the region where a cluster travels is negligible with respect to the descent velocity of the cluster itself. Cluster shape. Although there have been advances in the visualization of #ow in CFBs, these advances have
not been su$cient to allow for a clear de"nition of the shape of a cluster. Rhodes et al. (1992) describe clusters (viewed face on) as `swarmsa or `strandsa, depending on the shape and the operating conditions in the CFB. Lim et al. (1996) note elliptical or ellipsoidal frontal shapes. Similarly, the infrared images of Noymer and Glicksman (1998) show rounded objects traveling downward near the riser wall. Although side views of clusters do not exist, it seems reasonable from the existing information to assume that a cluster can be treated aerodynamically as a blu! body of some sort. For this analysis, the leading edge of a descending cluster is assumed to be rounded, and for simplicity, any aerodynamic analysis of a cluster will assume that the shape of a cluster can be approximated by that of a sphere. Even though the clusters may not be exactly spherical, this assumption allows for a relatively simple and concise analysis of the gas #ow around a cluster. Cluster length scale. For simplicity in this analysis, a single length scale will be used to describe a cluster. The variable D will be used to represent this length scale, where D represents the cross-sectional area of a cluster normal to its motion and D represents the volume of a cluster. Although D and D do not exactly represent the cross-sectional area and volume of a sphere, respectively, they represent the appropriate functional dependence and are su$cient for determining the semi-empirical correlation that results from this analysis. Forces acting on a cluster. The net drag force on a cluster is considered to come from both #ow through the cluster (permeable drag) and #ow around the cluster (aerodynamic drag), as depicted in Fig. 1. It is assumed that the permeable drag can be calculated using the Ergun equation (Ergun, 1952) and that the aerodynamic drag can be calculated using drag coe$cients tabulated for spheres (White, 1991a). Terminal velocities. For simplicity, only terminal velocities of clusters will be calculated in this analysis. In other words, we are considering only the period of time in which the drag force on a cluster is in equilibrium with its weight; the transient acceleration of a cluster to its terminal velocity is ignored. Although it is di$cult to determine whether the existing cluster-velocity data also correspond to this equilibrium condition, it is discussed later than this assumption over-predicts velocities at worst and one possible contribution to the error is seen in comparing some of the available data to the results of this analysis. Method of analysis. As mentioned previously, the drag force on a cluster is considered to come from two sources * from #ow around the cluster (aerodynamic drag) and from #ow through the cluster (permeable drag). In order to avoid the complexity of determining how much gas #ows through instead of around the cluster, the terminal velocity of a cluster will be found considering the two components of drag separately. In other words,
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Canceling terms in Eq. (4) results in the following expression: (1!e) k 1 o D u "o g. u #1.75 (5) Q e d e d N N Simplifying even further, let the coe$cients on the lefthand side of Eq. (5) be replaced by C and C , respectively J G (for viscous and inertial coe$cients), such that they are functions only of the void fraction of the cluster 150
Fig. 1. Schematic of simple cluster-descent model.
C "150 J
a terminal velocity will be determined "rst for the extreme case in which the gas #ows completely through the cluster, then it will be determined for the extreme case in which the gas #ows completely around the cluster. Focusing on the behavior in the two extremes will allow for clearer insight into the physical parameters governing cluster motion. 2.2. Drag resulting from permeability only The Ergun equation (Ergun, 1952) can be used to approximate the drag force on a cluster resulting from gas #ow through the cluster: *P (1!e) k (1!e) o D u , "150 u #1.75 ¸ e d e d N N
(1)
where u denotes the cluster velocity predicted when all of the #ow passes through the cluster and none of the #ow diverts around it. The Ergun equation normally depends on the sphericity of the particles as well, but this parameter can be assumed to be unity for a "rst-order approximation. Given that ¸ in Eq. (1) is the same as D, the length scale of the cluster, the drag force on the cluster is given by the pressure drop acting on the projected area of the cluster: F "150 B
(1!e) kD (1!e) o D D u . u #1.75 e d e d N N
(2)
The weight of a cluster is simply the product of its mass and the gravitational constant = "m g"o (1!e)Dg, Q
(3)
so that equating the weight of a cluster and the drag on a cluster yields the following expression for the terminal velocity: 150
(1!e) kD (1!e) o D D u u #1.75 e e d d N N
"o (1!e)Dg. Q
(4)
(1!e) e
(6a)
and 1.75 C" . G e
(6b)
Using these expressions, Eq. (5) can be re-written as k o C u #C D u "o g. (7) J d Gd Q N N Re-arranging terms and solving the quadratic polynomial in Eq. (7) for u yields: C k C k 1 4 o gd J Q N . # u " ! J # 2 C o d C o d C o G D N G D N D G (8)
Introducing the following expressions for the minimum #uidization velocity from Grace (1982): o gd Q N, u "C KD KD k
(9)
where C "0.00075, along with the following de"nition KD of the Archimedes number: (o !o )o gd o o gd D D N+ Q D N, Ar" Q k k
(10)
the expression for u in Eq. (8) can be made dimension less dividing through by u and simplifying terms to KD yield the following expression: u u H, u KD 1 C 1 J " ! 2 C C Ar G KD C 1 4 1 J # # . (11) C C Ar C C Ar G KD G KD It should be noted here that Eq. (9) is valid for gas}solid combinations with values of Ar up to about 1000, which represents conditions quite commonly found in CFBs (see Table 1). In Eq. (11), except for two constants (C and J C ) that depend on the concentration of solid particles in G the cluster, the only parameter governing the cluster
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Table 1 Measurements of cluster velocities Authors
Bed temp. (K)
Bed material
d N (lm)
o Q (kg/m)
Ar
Calculated u (cm/s) KD
Measured u (m/s)
Rhodes et al. (1992) Zhou et al. (1995) Harris et al. (1993) Ishii et al. (1989) Bader et al. (1988) Noymer and Glicksman (1998) Noymer (1997) Noymer (1997) Hartge et al. (1988) Hartge et al. (1988) Zhang et al. (1995) Yang and Gautam (1995) Wang et al. (1993) Lints (1992) Lints (1992) Wu et al. (1991) Lim et al. (1996)
300 300 300 300 300 300 300 300 300 300 1125 300 300 300 300 300 300
alum. sand FCC FCC FCC steel sand sand FCC ash n/a n/a sand sand alum. sand sand
75 213 60 60 76 69 128 182 85 120 330 250 530 80 600 170 213
2460 2640 1700 1000 1710 6980 2660 2650 1500 2600 2600 2250 2300 2660 2750 2650 2640
33 816 12 7 24 73 178 490 29 144 149 1080 11,000 44 19,000 425 816
0.54 4.6 0.24 0.14 0.38 1.2 1.6 3.2 0.42 1.5 4.7 5.2 22 0.66 31.5 3.0 4.6
0.3 1.5 0.8 0.5 0.9 1.2 1.2 1.1 1.0 1.0 0.9 0.4 2.0 1.6 1.8 1.3 1.0
Information not available; values have been estimated where necessary. Data for particles at the wall of a tube with upward gas #ow, not in a real CFB.
velocity in this analysis is the Archimedes number. Given that a typical volumetric concentration of particles in a cluster is about 10% (Lints & Glicksman, 1993), an `order-of-magnitudea expression for the dimensionless cluster velocity can be obtained:
10 3;10 1 10 u H" ! # # . 2 Ar Ar Ar
(12)
In the limit when Ar100, the right-hand term under the square-root dominates in a CFB, a typical value for Archimedes number might be Ar*100. Using this simpli"cation, Eq. (12) becomes 900 u H+ . (Ar
(13)
This is an expression for the terminal velocity of a cluster that depends on Ar and u , assuming that the only drag KD force acting on the cluster is that due to the air #owing through the cluster. 2.3. Drag resulting from aerodynamics only Assuming that all of the #ow goes around the cluster, the drag force is given by F " c o u D, (14) B B D where c is the drag coe$cient for #ow around a sphere, B and u represents the velocity of the cluster when ana lyzed for aerodynamic drag only. Equating the expression in Eq. (14) to the weight of a cluster as given in
Eq. (3) results in a force balance that can be re-arranged to provide the following solution for u : 2 o Q (1!e)gD. u " c o (15) B D As before, the expression in Eq. (15) can be made dimensionless by dividing through by the expression for u in KD Eq. (9) and re-arranging terms:
u 1 2 D (1!e). u H, " u (16) c d KD CKD (Ar B N Performing an `order-of-magnitudea analysis similar to that used to derive Eq. (12) from Eq. (11), several of the assumptions that have already been stated can be used. For example: the volumetric fraction of solids in a cluster is about 10% (Lints & Glicksman, 1993); a typical cluster length scale is about 1 cm (Rhodes et al., 1992); a typical particle size in a CFB is about 100 lm, and the drag coe$cient is on the order of unity for a sphere with a Reynolds number of about 1000 (based on a diameter of 1 cm, a velocity of 1 m/s and a kinematic viscosity of 10\ m/s) (White, 1991a). Again, each of these numbers is good within an order of magnitude, but the assumptions result in the following simpli"cation to Eq. (16): 6000 u H" . (Ar
(17)
The functional dependence of cluster velocity on Archimedes number is exactly the same as that given in Eq. (13), and the constants of proportionality are of similar magnitude. In other words, whether considering
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the drag forces caused by #ow through a cluster or #ow around a cluster, the result di!ers only by a constant of proportionality.
3. Comparison of analysis to existing data The preceding analysis suggests that the terminal velocity of a cluster, when non-dimensionalized by the minimum #uidization velocity of the particles, can be correlated to the square-root of the Archimedes number of the gas}solid system. In order to test this analysis, a large number of cluster-velocity measurements have been collected from the available literature on CFBs (Ishii et al., 1989; Rhodes et al., 1992; Noymer & Glicksman, 1998; Lim et al., 1996; Noymer, 1997; Zhou, Grace, Brereton & Lim, 1995; Harris, Davidson & Xue, 1993; Bader, Findlay & Knowlton, 1988; Hartge, Rensner & Werther, 1988; Zhang, Johnsson & Leckner, 1995; Yang & Gautam, 1995; Wang, Lin, Zhu, Liu & Saxena, 1993; Lints, 1992; Wu, Lim, Grace & Brereton, 1991). These measurements and the conditions under which they were made are summarized in Table 1; all the data were obtained under atmospheric-pressure conditions. The data presented in Table 1 are plotted in Fig. 2 in the form (u )/(u ) vs. Ar. In addition to these data, a line KD representing the following function is also plotted in Fig. 2: u 1000 " . u KD (Ar
(18)
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By inspection, this line provides a reasonably good curve "t to the data. The fact that the constant of proportionality is 1000 bodes well for the preceding analysis, since the constants found in the two extremes were of that order of magnitude. All of the data appear to follow the trend line fairly well. Only the data of Rhodes et al. (1992), Zhang et al. (1995), and Yang and Gautam (1995) deviate appreciably, and for each of these, the simple correlation appears to over-predict the cluster velocity. There is nothing that stands out in any of those measurements that indicates a reason for the deviation of our model from the measurements. One possible explanation is that the measured velocities do not represent the terminal velocities of the clusters and are therefore lower than what has been predicted. However, it seems that for the relatively small amount of e!ort expended, a fairly good correlation for cluster velocity has been established.
4. Discussion 4.1. Comparison to previous correlations Although this analysis is not exact, the data indicate that the actual behavior might very well be some combination of the behavior in the two extremes. Since the empirical constant of 1000 falls between the derived constants for drag resulting from #ow through the cluster and #ow around the cluster, it seems plausible that the total drag is a result of some combination of the two. In fact, these results agree with a correlation recently published by Gri$th and Louge (1998). In their paper, they
Fig. 2. Cluster velocities * measurements vs. model predictions.
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curve "t the following function to a set of data similar to ours: u "36(gd . (19) N Our analysis allows for the consideration of more variables than simply the gravitational acceleration and the particle size; speci"cally, it allows for the consideration of variations in solid or gas density as well. Using Eq. (9) to rewrite Eq. (18) in dimensional terms yields the following equation:
o Q gd . u "0.75 N o (20) D So, our analysis retains the same functional dependence on g and d as that of Gri$th and Louge, while also N predicting a dependence on o and o . Since the Q D solid}gas density ratio ranges from 1000 to 6000 in the data considered herein, the coe$cient resulting from the product of 0.75 and the square-root of the density ratio ranges from 24 to 58. In other words, the analysis of Gri$th and Louge is something of a nominal result, while our analysis predicts additional variations in cluster velocity due to variations in solid or gas density.
Given the simplicity of this model and the data available, it seems that the present results are more than satisfactory.
5. Conclusions A new and simpli"ed model has been developed to predict the terminal descent velocities of particle clusters traveling near the wall of a circulating #uidized bed. This model assumes that the mechanisms for drag on a cluster come from gas #owing through the cluster and gas #owing around the cluster. The interaction between the cluster and the wall is neglected, as is the opposing upward gas velocity, which can be shown to be negligible in the wall region. This model results in an expression for cluster velocity that is dependent only on the physical properties of the solid particles and the gas * the Archimedes number and the minimum #uidization velocity. Good agreement is found in comparing this model to existing data, but the level of simpli"cation implied by the assumptions warrants further investigation in order to truly understand the nature of the gas}solid #ow near the wall of a CFB riser.
4.2. Limitations of this analysis Notation There are some limitations to this analysis that must be addressed. First, it is impossible to determine whether the #ow actually goes through or around the cluster without analyzing the problem in greater detail. Therefore, the method in this analysis cannot be taken as an absolute explanation for the phenomenon of similar cluster velocities even though the results of this analysis appear to adequately predict cluster velocities. Second, in simplifying Eq. (12) to Eq. (13), it was assumed that Ar100. However, the expression for u introduced in KD Eq. (9) is best for Ar)1000, so there is some inconsistency in the underlying assumptions. Third, the data points from Rhodes et al. (1992), Zhang et al. (1995), and Yang and Gautam (1995) deviate from the model by 100% or more (the scale in Fig. 2 is logarithmic, so the proximity of the points to the line can be misleading). However, the discrepancies between their data and the model is such that the model over-predicts the cluster velocities, indicating the possibility that the data reported from these three sources might not actually be terminal velocities. Finally, it is known that the clusters travel very close to the wall in a CFB, so that ignoring the e!ect of the wall on the #ow "eld around the cluster is physically incorrect * the wall should have some in#uence on the drag coe$cient. For these reasons, a more precise analysis of the #ow on a cluster near a wall is required, and for this level of complexity, computational models are required (e.g., Noymer & Glicksman, 1999).
Ar c B C G C KD C J d N D F B g ¸ m P u u u u KD =
Archimedes number drag coe$cient constant from inertial term in Ergun equation constant from approximate expression for umf constant from viscous term in Ergun equation mean particle diameter cluster length scale drag force gravitational acceleration length cluster mass pressure cluster terminal velocity considering only #ow through a cluster cluster terminal velocity considering only #ow around a cluster cluster velocity minimum #uidization velocity cluster weight
Greek letters e o D o Q k
volumetric void fraction of particles in a cluster gas density solid material density gas dynamic viscosity
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Acknowledgements This work was conducted under a grant from the National Science Foundation. References Bader, R., Findlay, J., & Knowlton, T. M. (1988). Gas/solid #ow patterns in a 30.5-cm-diameter circulating #uidized bed. In P. Basu, & J. F. Large, Circulating yuidized bed technology II (pp. 123}137). Oxford: Pergamon Press. Brereton, C. M. H., & Grace, J. R. (1993). Microstructural aspects of the behavior of circulating #uidized beds. Chemical Engineering Science, 48, 2565}2572. Ergun, S. (1952). Fluid #ow through packed columns. Chemical Engineering Progress, 48, 89}94. Grace, J. R. (1982). Fluidized bed hydrodynamics. In G. Hetsroni, Handbook of multiphase systems. Washington, DC: Hemisphere (Chapter 8). Gri$th, A. E., & Louge, M. Y. (1998). The scaling of cluster velocity at the wall of circulating #uidized bed risers. Chemical Engineering Science, 53, 2475}2477. Harris, B. J., Davidson, J. F., & Xue, Y. (1993). Axial and radial variation of #ow in circulating #uidized bed risers. In A. Avidan, Proceedings of the 4th International Conference on Circulating Fluidized Bed (pp. 103}110), Somerset, PA. Hartge, E.-U., Rensner, D., & Werther, J. (1988). Solids concentration and velocity patterns in circulating #uidized beds. In P. Basu, & J. F. Large, Circulating yuidized bed technology II (pp. 165}180). Oxford: Pergamon Press. Horio, M., & Kuroki, H. (1994). Three-dimensional #ow visualization of dilutely dispersed solids in bubbling and circulating #uidized beds. Chemical Engineering Science, 49, 2413}2421. Ishii, H., Nakajima, T., & Horio, M. (1989). The clustering annular #ow model of circulating #uidized beds. Journal of Chemical Engineering of Japan, 22, 484}490. Lim, K. S., Zhou, J., Finley, C., Grace, J. R., Lim, C. J., & Brereton, C. M. H. (1996). Cluster descending velocity at the wall of circulating #uidized bed risers. Proceedings of the xfth international conference on circulating yuidized beds. Beijing, Peoples's Republic of China. Lints, M. (1992). Particle-to-wall heat transfer in circulating #uidized beds. Sc.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
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