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Study of mixing in gas-fluidized beds using a DEM model
Chemical Engineering Science 56 (2001) 2859}2866
Study of mixing in gas-#uidized beds using a DEM model M. J. Rhodes , X. S. Wang *, M. Nguyen , P. Stewart, K. Li!man Department of Chemical Engineering, Monash University, Clayton, Vic. 3168, Australia Department of Pharmaceutics, Monash University, Parkville 3052, Australia Advanced Fluid Dynamics Laboratory, CSIRO DBCE, Highett 3190, Australia Received 10 May 2000; received in revised form 11 October 2000; accepted 7 November 2000
Abstract This paper examines the usefulness of a discrete element method simulation in studying solids mixing in gas-#uidized beds. At steady state, a part of the solids were tagged with a dark colour and the rest were tagged with a light colour. The state of mixing of bed solids was studied by analysing the variation of the proportions of the marked particles with time and position in the bed. This was achieved by scanning the bed using a sampling box. The variance of mixture composition based on the samples was incorporated into a mixing index which describes the degree of mixing of the particles at a particular time. This method enables assessment of the overall mixing behaviour in terms of the rate of mixing (through estimation of the time required for the mixing index to increase from zero to a certain value), together with the degree of mixing at the mixing equilibrium. Illustrative examples show that gas velocity and particle properties are important parameters in#uencing solids mixing in bubbling #uidized beds. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Solids mixing; Mixing index; DEM simulation
1. Introduction The basic mechanism of solids mixing in bubbling #uidized beds is well understood (e.g. Marsheck & Gomezplata, 1965; Rowe, Partridge, Cheney, Henwood, & Lyall, 1965; Rowe & Widmer, 1973; Nguyen, Whitehead, & Potter, 1977; Mason, 1978). This may be summarized as follows. When a bubble rises through the bed, it carries a wake of particles to the bed surface. The bubble also causes a drift of particles to be drawn up below it. A down#ow of solids exists in the region surrounding the rising bubbles, resulting an overall convection circulation of particles in the axial direction of the bed. At the same time, lateral mixing of solids occurs, which is caused partly by lateral motion of bubbles (due to interaction and coalescence of neighbouring bubbles) and partly by the lateral dispersion of particles in the bubble wake at the bed surface (due to eruption of the bubble). For more information, readers may refer to review papers such as Potter (1971), van Deemter (1985) and Baeyens and Geldart (1986). * Corresponding author. Tel.: #61-3-9905-1870; fax: #61-3-99055686. E-mail address: [email protected] (X. S. Wang).
It is still not generally possible, however, to predict the e!ect of operating parameters on the degree of mixing in a #uidized bed (Garncarek, Przybylski, Botterill, Bridgwater, & Broadbent, 1994). In fact, arguments are still raised regarding the e!ect of particle diameter and density. For example, while mechanistic models such as Yoshida and Kunii (1968) and Chiba and Kobayashi (1977) predict that the rate of mixing increases with particle diameter and density, recent experimental observations (e.g., Shen & Zhang, 1998; Lim, Gururajan, & Agarwal, 1993) appear to indicate otherwise. There is, therefore, a need to gather further evidence, preferably from a di!erent angle, to address these issues. The objective of the present work is to investigate the use of discrete element method (DEM) simulation in the study of solids mixing in gas-#uidized beds.
2. Assessment of the quality of mixing Many investigations of solids mixing in #uidized beds have been concerned with establishing the exchange coef"cient between the bubble wake and the emulsion phase. Generally speaking, a higher wake-exchange coe$cient implies a better quality of mixing. Experimental methods
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 2 4 - 8
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used in establishing the exchange coe$cient can broadly be divided into two categories: (1) injection of tracer particles at steady state, in which variation of concentration of tracer particles is monitored by specially designed probes; (2) marking part of the packed bed (e.g., using a particular colour), in which variation of concentration of the marked particles is monitored by means of external visualization. Examples which fall into the "rst category are Valenzuela and Glicksman (1984), Fan, Song, and Yutani (1986) and Shen and Zhang (1998). The uncertainty associated with this method is that the mixing process itself may be in#uenced by the injection and accumulation of the tracer particles, and also by the insertion of monitoring probes. Examples in the second category include Rowe et al. (1965), Singh, Fryer, and Potter (1972) and Lim et al. (1993). This method incurs minimum disturbance to the #ow, since no extra particles are added to the bed and since observation is made through visualization from outside the bed. The main uncertainty of this method is that particles have to be marked from a packed bed and, so, the transition from packed to #uidized bed could not be avoided. The quality of mixing can also be assessed by examining the degree of mixing of particles in the bed. So far, such studies are only limited to binary and ternary systems in which particle segregation takes place. Nienow, Rowe, and Chung (1978) and Geldart, Baeyens, Pope, and Wijer (1981) are examples of early studies in this area. More recent examples include Wang and Chou (1995) and Wu and Baeyens (1998). In these studies, the degree of solids mixing is expressed in the form of a mixing index, de"ned as x/x , where x is the concentration of the material under scrutiny at some level in the bed and x is its average concentration. This index equals 1 for a &fully mixed' bed, and 0 for a &completely segregated' bed. The value of x can be obtained experimentally by the following steps: (1) &freezing' the bed by suddenly stopping the gas supply at steady state, (2) dividing the &frozen' bed into a number of sections, (3) removing the particles in these sections, and (4) analysing the size or density distribution of the particles in various sections. Obviously, this method is not applicable to uniform particles, as these particles have the same size and density and so no segregation occurs. Investigation of the degree of mixing of uniform particles has so far been ignored by researchers in the community. The main reason is that it is di$cult to obtain information at the individual particle level, e.g., its location and velocity. On the other hand, there is interest in mixing of uniform systems. For example, we wish to know how well new feed material would mix with existing bed material. Also, if we are contacting solids with gas/liquid injection, we need to know the rate of solids turnover. Industry is also interested in how the location of the feed point and exit point a!ect the degree of mixing.
3. A mixing index based on statistical analysis Various mixing indexes are used in describing the e!ectiveness of di!erent mixers in the process industry (Poux, Fayolle, Bertrand, Bridoux, & Bousquet, 1991). Most of these indexes have been developed based on statistical analysis and especially on the de"nitions of standard deviations of a speci"ed property. These mixing indexes usually describe the closeness of a mixture to a `completely random mixturea. According to Lacey (1954), a completely random mixture is one in which the probability of "nding a particle of a constituent in every point is identical. When the components of the mixture are completely segregated, it can be assumed that any sample withdrawn from the mixture will be entirely composed of a pure component. Lacey (1943) showed that the variance of a completely segregated mixture, S , can be expressed as S "pq,
(1)
where p and q are the proportions of the two components estimated from the samples. When any sample is withdrawn from a fully randomized mixture, every particle within that sample can be considered as randomly representative of the mixture as a whole. In these conditions, the variance, S , may be calculated from 0 pq S " , 0 N
(2)
where N is the number of particles in a sample. This value is normally the minimum attainable variance within a mixture. The well-known Lacey index (Lacey, 1954) is de"ned as S !S M" , S !S 0
(3)
where S is the variance of the mixture between fully random and completely segregated mixtures. The mixing index obtained from Eq. (3) has a zero value for a completely segregated mixture and increases to unity for a fully random mixture. Due to its simplicity, the Lacey index is widely used to characterize mixers used in the process industry. This study explores its application to #uidized beds of uniform particles.
4. Use of Lacey index in studying solids mixing in 6uidized beds 4.1. The discrete element method (DEM) simulation It would be clear that in order to apply the Lacey index (Eq. (3)) to a #uidized bed, representative samples need to be taken across the bed at a given time. The DEM
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simulation of a gas-#uidized bed lends itself readily to the study of solids mixing since at any instant, we know the location and velocity of every particle in the bed. The DEM is a relatively new advance in numerical simulation of #uidized beds. In this method, the particles are traced individually by solving Newton's equations of motion, while the #uid phase is treated as a continuum (Tsuji, Kawaguchi, & Tanaka, 1993; Hoomans, Kuipers, Briels, & van Swaaij, 1996; Xu & Yu, 1997). In this study we use a DEM model, which was made available to us in the form of a computer code through our collaboration with Professor M. Horio of Tokyo University of Agriculture and Technology. Details of this model are given in Mikami, Kamiya, and Horio (1998) and are brie#y described here. For gas motion, the local averaged Navier}Stokes equations (Anderson and Jackson, 1967) were integrated by the SIMPLE method (Patankar, 1980) employing the staggered grid system. For particle motion, the Newtonian equations of motion of individual particles were integrated. The collisions between particles and between a particle and a wall were simulated using Hooke's linear springs and dash pots. The interaction between the particle and #uid phases was addressed as follows: when the void fraction was less than 0.8, the Ergun equation for packed beds was used; when the void fraction was larger than 0.8, a modi"ed equation of the #uid resistance for a single particle was used. The time step for calculating particle motions was decided based on numerical &experiments' of energy dissipation. In our study, two-dimensional beds of 0.16 m in width and 0.38 m in height were used. The thickness of the beds was the same as a single particle diameter.
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The Mikami et al. (1998) model introduced a liquid bridge interparticle force to enable simulation of wet particles. In the illustrative examples given in this paper, the amount of water in the bed was set to zero. So, we were e!ectively simulating dry particles in our study. Initially all the particles are randomly located in the bed with a randomly allotted velocity. The particles then fall freely under gravity in the absence of a gas #ow. The energy of the system dissipates in every collision between particles and between a particle and a wall due to inelastic collision. After allowing some time for energy dissipation through inelastic collisions, the bed reaches a stagnant state. The restitution coe$cient, friction coef"cient and spring constant of the particles used in this study are the same as those used by Mikami et al. (1998), i.e., 0.9, 0.3 and 800 N/m, respectively.
4.2. Illustrative examples 4.2.1. An illustration of the process of mixing Fig. 1 shows an illustrative example of the process of mixing for particles of 2650 kg/m in density and 1 mm in diameter. The super"cial gas velocity was 1.2 m/s. The bed was "rst allowed to operate for 7 s so that steadystate operation was established. The particles in the lefthalf of the bed were then tagged with a dark colour and the rest were tagged with a light colour. The time was set to zero at this moment. So, we can say that the dark and light coloured particles are completely segregated at t"0 s. As mixing proceeds, this completely segregated state gradually gives way to one in which the particles of
Fig. 1. An illustration of the process of mixing ( "2650 kg/m, d "1 mm, ;/; "1.5). Bed dimensions not to scale. N N KD
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Fig. 2. An illustration of the sampling method. (Note that the number of sampling boxes does not represent the actual number of sampling boxes used in the simulations.)
Fig. 3. E!ect of gas velocity on the Lacey index (d "1 mm, N "2650 kg/m). N
di!erent colours are partially mixed. A mixing equilibrium can be noted after about 4 s of mixing.
each sample was set at 100 (so that the level of scrutiny was "xed).
4.2.2. The sampling method By the use of DEM simulation, we can mark a part of bed particles with one colour and the rest with another colour at any time of operation. In this study, however, the particles were marked after steady-state #uidization was established. Major advantages of use of DEM for mixing studies are
4.2.3. Ewect of gas velocity In these simulations, particles of 1 mm in diameter and 2650 kg/m in density were used. The total number of particles used was 14,000. Fig. 3 shows the results for ;/; of 1.25, 1.5, 1.75 and 2.0. The x-axis represents KD time starting from solids marking, and the y-axis is the Lacey index based on Eq. (3). It shows for all the cases that as mixing proceeded, the mixing index gradually increased until an equilibrium of mixing was reached. It is interesting to note that the Lacey index at the mixing equilibrium was about the same (i.e., M+0.72) for all the gas velocities examined. This suggests that the degree of mixing achievable at this condition does not vary with gas velocity. Results shown in Fig. 3 also allow examination of the rate of mixing of the particles, through comparison of the time required for the mixing index to increase from zero to a certain value. It can be noted from Fig. 3 that the time required for the mixing index to increase from 0 (completely segregated) to 0.72 (the mixing equilibrium) decreased with increasing velocity. For example, an increase in the ;/; from 1.25 to 2.0 KD resulted in a decrease in the time from around 6 to 3 s. This would suggest that the average rate of mixing increases with increasing gas velocity for the conditions examined. It agrees with the general trend reported in the literature (e.g., Lim et al., 1993; Wu & Baeyens, 1998). We have also attempted to obtain the mixing index at ; . KD We found that the mixing equilibrium could not be reached within 15 s, suggesting that the rate of mixing was much slower under this condition. This was due to the lack of bubbles under this condition, which was evident from analysis of snapshots.
(a) it enables non-intrusive sampling; (b) we can gather information at the individual particle level at any time; (c) we can readily vary sample size; and (d) we can tag the particles after the bed is #uidized, so there is no need of going through the packed- to #uidized-bed transition. A rectangular box was used to take samples from the bed, as shown in Fig. 2. In this study, the sampling box was used to &scan' the bed while avoiding any overlapping of the samples. (Note that the particles were counted based on their centre positions.) In addition, the number of particles in each sample was kept the same. This was achieved by allowing the sampling box to expand vertically. Of course, the sampling box at the surface of the bed could contain a smaller number of particles. If the number of particles in the top boxes di!ered from the designed value by a certain margin (e.g., 10%), then this sample was disregarded. We found that this had little e!ect on the mixing index obtained, provided that a su$ciently large number of samples were taken in each snapshot. Clearly, the &scanning' approach enables the overall picture of mixing in the bed to be revealed. It is worth noting that the Lacey index (Eq. (3)) is not related to the number of particles in the bed or the number of samples, provided that a su$cient number of samples were taken. The results presented below were obtained using a sampling box of 10 mm in width. The number of particles in
4.2.4. Ewect of particle density In these simulations, the particle diameter was kept at 1 mm while the particle density was decreased from 2650
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Fig. 4. E!ect of particle density on the Lacey index (d "1 mm, N ;/; "1.5). KD
to 1590, then to 900 kg/m. The total number of particles used was again 14,000, and the simulations were conducted at ;/; of 1.5. Results for particle densities of 2650 KD and 1590 kg/m are shown in Fig. 4. The result for particle density of 900 kg/m is not included in this "gure, since it almost coincided with the result for particle density of 2650 kg/m. It can be noted that the mixing indexes at the mixing equilibrium were about 0.72 for the particle densities examined, indicating that the degree of mixing achievable in a #uidized bed was not a!ected by changes in particle density. The time required for the mixing index to increase from 0 to 0.72 increased from about 4 to 6 s when was decreased from 2650 to N 1590 kg/m. It then decreased from about 6 to 4 s when was decreased from 1590 to 900 kg/m. So, mixing N rate decreased from "2650 to 1590 kg/m, and inN creased again as was decreased to 900 kg/m. SnapN shots of the bed were examined in order to establish the causes for the results just presented. The snapshots for di!erent particle densities were taken at steady state (from t"10 s) and for the same time intervals (0.05 s). We found that as was decreased from 2650 to N 1590 kg/m, the average bubble size in the bed slightly decreased while lateral movement of bubbles became increasingly dampened. As was decreased from 1590 N to 900 kg/m, on the other hand, the average bubble size was signi"cantly decreased while lateral movement of bubbles was clearly enhanced. So, the density e!ect was the result of combined e!ects of changes in bubble size and the extent of lateral movement of bubbles. The results presented in Fig. 4 are, however, not completely consistent with experimental observations of Lim et al. (1993) and Shen and Zhang (1998). In their studies, the decreasing mixing range was not observed. This raises questions as to whether it is adequate to keep ;/; constant in investigating the density e!ect for KD particles of monosizes or narrow size distributions. Note that some researchers (e.g., Wu & Baeyens, 1998) chose to use a constant ;!; for particles of di!erent sizes. To KD
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address this problem, let's "rst look at the conditions of Shen and Zhang (1998). In order to maintain a constant ;/; of 2.0, the value of ;!; (which directly KD KD a!ected bubble size in the bed) was decreased from 0.175 to 0.1 m/s as particle density was decreased from 2600 to 1300 kg/m (d "0.5 mm). Of course, changes in the N value of ;!; depended on the value of ;/; . KD KD Similar analysis also applies to the experimental conditions of Lim et al. (1993) and also simulation conditions for Fig. 4 in this paper. We believe that the apparent inconsistency between our simulation and observations of Lim et al. (1993) and Shen and Zhang (1998) was caused by the variations in ;!; . KD We have also performed simulation &experiments' by keeping ;!; constant (at 0.4 m/s) for di!erent parKD ticle densities. The time required for the mixing index to increase from 0 to 0.72 (mixing equilibrium) was found to decrease from about 4 to 3.6, then to 3 s, respectively, as particle density was decreased from 2650 to 1590, then to 900 kg/m. Analysis of animation results shows that as particle density was decreased, bubbles grew faster, movement of bubbles became more vigorous, while the average bubble size remained relatively una!ected. Our simulation results point to the need for further experimental studies of the e!ect of particle density by keeping ;!; constant. KD 4.2.5. Ewect of particle diameter The in#uence of particle diameter was studied by increasing particle diameter from 1 to 2 mm whilst keeping the particle density at 2650 kg/m. The ;/; was again KD "xed at 1.5. The number of particles was reduced from 14,000 to 3500 so that the height of the bed was kept about the same. Results of these simulations are shown in Fig. 5. It is interesting to note that the mixing index at the mixing equilibrium decreased considerably, from about 0.72 to 0.54, when the particle diameter was increased from 1 to 2 mm. It can also be noted that the time required for the mixing index to increase from 0 to 0.54
Fig. 5. E!ect of particle diameter on the Lacey index ( "2650 kg/m, N ;/; "1.5). KD
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increased from about 2 to 5 s as particle diameter was increased from 1 to 2 mm. So, the average mixing rate would decrease with increasing particle diameter. These results are qualitatively supported by experimental observations of Lim et al. (1993) and Shen and Zhang (1998). It should be noted, however, that the initial rate of mixing (t(2 s) was not signi"cantly a!ected by particle size. Further simulations were performed to look at the e!ect of particle diameter on the basis of a constant ;!; (i.e., 0.4 m/s). We observed that results of these KD simulations qualitatively agreed with those obtained using a constant ;/; (Fig. 5). KD 5. Discussion on the proposed method 5.1. Approaches for solids marking and sampling Solids marking and sampling are obviously important issues which warrant careful consideration. Our "rst concern is regarding the timing of solids marking. Fig. 6 shows a comparison of the Lacey index obtained for two di!erent conditions under which the particles were marked: one in which the particles were marked from packed bed, and the other in which the particles were marked after steady-state #uidization was established. The main di!erence between these two cases is that a period of negligible mixing (up to about 1 s) existed in the packed-bed case. Obviously, this period of negligible mixing is associated with unlocking of the packed bed and, so, it would be strongly in#uenced by the manner by which #uidizing gas was introduced (e.g., how quickly the supply was turned on). Fig. 6 serves as an example in which the super"cial gas velocity was increased linearly to its full value within the initial 1 s of operation. Another matter concerning solids marking is by what proportion the bed should be marked with a dark colour. Experimental observations of Wu and Baeyens (1998)
using binary particles and Lim et al. (1993) using uniform particles have demonstrated that the concentration of tracer particles certainly a!ects their results. We found in a standard simulation (particle diameter: 1 mm; particle density: 2650 kg/m; ;/; : 1.5) that the resulting LacKD ey index decreased from 0.73 to 0.52 when the composition of particles with a dark colour was increased from 50 to 70%. This is expected, since the Lacey index is necessarily concentration dependent. So, as in any physical experiments, it is important that the same concentration of coloured particles be used in comparative studies of the e!ect of operating conditions. It would be advisable that half of the bed be marked with a dark colour and the other half marked with a light colour. This certainly gives rise to a relatively higher mixing index at the mixing equilibrium. It also avoids uncertainties associated with local variations in bubble properties, even at steady state (Shen & Zhang, 1998). The sampling method itself is also an important factor. In this paper we have presented a method in which the bed was &scanned'. The objective was to avoid potential problems associated with improper distribution of samples. The number of particles in each sample (i.e., sample size) re#ects the level of scrutiny. The main reason for using a constant sample size (i.e., 100) for particles of d "1 and 2 mm was that we attempted to place di!erN ent systems under the same level of scrutiny. A useful guideline in choosing an appropriate sample size would be that such a sample size should give rise to a mixing index of 0 for a completely segregated system (M ). We observed that the value of M was quite insensitive to the sample size in our system; it remained at about 0.01 as the sample size was increased from 100 to 200, then to 300 (for d "1 mm). We have also tested another approach N by which samples were taken randomly in the bed. We found, however, that large errors could be introduced due to undetected overlapping of the samples, especially those of large bubbles. 5.2. Relation to wake-exchange coezcient
Fig. 6. A comparison of the variation of Lacey index for di!erent conditions at which particles were marked (d "1 mm, "2650 kg/ N N m, ;/; "1.5). KD
The mixing index and wake-exchange coe$cient are measures of the quality of mixing from di!erent angles. The former describes the state of mixing, which, therefore, emphasizes the overall picture of mixing; the latter refers to the rate of particle exchange between the bubble wake and the emulsion phase, which, therefore, emphasizes the progress of mixing. The distribution of tracer particles in the bed can, in theory, be calculated with the knowledge of the wake-exchange coe$cient, initial concentration and location of tracer particles, together with other operating conditions. This in turn would enable calculation of the mixing index. There is, however, no simple relationship between these two measures. We would expect that a higher wake-exchange coe$cient brings about a more uniform distribution of tracer
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particles at the mixing equilibrium, resulting a higher mixing index. Our simulation results reported earlier lend qualitative support to this analysis.
using particles of monosizes or narrow size distributions by keeping ;!; constant. KD
5.3. Accuracy of the proposed method
Notation
It needs to be stressed that the objective of this study was to provide an illustrative example of the use of DEM in studying solids mixing in #uidized beds. Accuracy of the results remains to be improved by re"ning the sampling method. Of course, the quality of the DEM model would directly in#uence the results. It is worth noting that a simpli"ed treatment was used in the original model of Mikami et al. (1998) regarding the forces acting on a single particle, i.e., Eq. (7) in their paper. In that equation, an additional term equal to the negative of product of particle volume fraction and local pressure gradient was omitted on the right-hand side. According to Horio and Rong (2000), this could cause an error of (mg);( / ) in the force at steady state, where m is the D N mass of a single particle, and are #uid and particle D N density, respectively. The actual error could be di!erent, as the gas drag component could deviate from the steady-state values. This simpli"ed treatment was also applied by Tsuji et al. (1993), as it was considered to be e!ective when #uid density was small. Simulation results such as bed pressure drop and bubble behaviour of Horio and co-workers and those of ours (e.g., Rhodes, Wang, Ngyen, Stewart, & Li!man, 2001) further con"rmed the e!ectiveness of this simpli"ed treatment. For gases of a relatively high density, e.g., under pressurized conditions, a more rigorous approach should be applied (Kaneko, Shiojima, & Horio, 1999; Rong & Horio, 2001).
d particle diameter, m N M Lacey index, de"ned in Eq. (3), dimensionless M Lacey index for a completely segregated system, dimensionless N number of particles in a sample p proportion of red particles in the samples q proportion of green particles in the samples S square root of the variance of the mixture between fully randomized and completely segregated states, dimensionless S square root of the variance of the completely seg regated states, dimensionless S square root of the variance of the fully randomized 0 states, dimensionless t time, s ; super"cial gas velocity, m/s ; minimum #uidization velocity, m/s KD
6. Conclusions The usefulness of a DEM simulation in studying solids mixing in a gas-#uidized bed has been investigated. It shows that, by using this method, the overall picture of mixing can be revealed in the form of a mixing index. It also provides information regarding the rate of mixing under speci"ed conditions. Distinct advantages of the proposed method are that the sample size can be readily varied, and that particles can be tagged after steady-state #uidization is established. Simulation results indicate that the rate of solids mixing increases with increasing gas velocity, whilst the degree of mixing achievable is una!ected by gas velocity. While the initial rate of mixing was found to be insensitive to particle size, the overall rate of mixing and the degree of mixing were found to decrease with increasing particle diameter. The e!ect of particle density was found to be not so clear-cut when mixing was measured at a constant ;/; , pointing to KD the need for further experimental studies of this e!ect
Greek letter N
particle density, kg/m
Acknowledgements The authors gratefully acknowledge the "nancial support from Australian Research Council Large Grant A89927077 and CSIRO/DBCE Grant CZ-25. They also thank Professor M. Horio and Dr. T. Mikami of Tokyo University of Agriculture and Technology, Japan, for making the original computer code available for this research, and Dr. D. Rong for very useful discussion.
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Hoomans, B. P. B., Kuipers, J. A. M., Briels, W. J., & van Swaaij, W. P. M. (1996). Discrete particle simulation of bubble and slug formation in a two-dimensional gas-#uidised bed: A hard-sphere approach. Chemical Engineering Science, 51, 99. Horio, M., & Rong, D. (2000). About the pressure gradient e!ect, Private communication. Kaneko, Y., Shiojima, T., & Horio, M. (1999). DEM simulation of #uidized beds for gas-phase ole"n polymerization. Chemical Engineering Science, 54, 5809. Lacey, P. M. C. (1943). Transactions of the Institution of Chemical Engineers, 21, 53. Lacey, P. M. C. (1954). Developments in the theory of particle mixing. Journal of Applied Chemistry, 4, 257. Lim, K. S., Gururajan, V. S., & Agarwal, P. K. (1993). Mixing of homogeneous solids in bubbling #uidized beds * theoretical modelling and experimental investigation using digital image analysis. Chemical Engineering Science, 48, 2251. Marsheck, R. M., & Gomezplata, A. (1965). Particle #ow patterns in a #uidized bed. A.I.Ch.E. Journal, 11, 167. Mason, H. (1978). Solid circulation studies in a gas solid #uid bed. Chemical Engineering Science, 33, 621. Mikami, T., Kamiya, H., & Horio, M. (1998). Numerical simulation of cohesive powder behaviour in a #uidized bed. Chemical Engineering Science, 53, 1927. Nguyen, H. V., Whitehead, A. B., & Potter, O. E. (1977). Gas backmixing, solids movement, and bubble activities in large scale #uidized beds. A.I.Ch.E. Journal, 23, 913. Nienow, A. W., Rowe, P. N., & Chung, L. Y. I. (1978). A quantitative analysis of the mixing of two segregating powders of di!erent density in a gas #uidized bed. Powder Technology, 20, 89. Patankar, S. V. (1980). Numerical heat transfer and yuid yow. New York: Hemisphere. Potter, O. E. (1971). Mixing. In J. F. Davidson & D. Harrison (Eds.), Fluidization (p. 293). London: Academic Press (Chapter 7). Poux, M., Fayolle, P., Bertrand, J., Bridoux, D., & Bousquet, J. (1991). Powder mixing: some practical rules applied to agitated systems. Powder Technology, 68, 213}234.
Rhodes, M. J., Wang, X. S., Ngyen, M., Stewart, P., & Li!man, K. (2001). Use of discrete element method simulation in studying #uidization characteristics: In#uence of interparticle force. Chemical Engineering Science, in press. Rong, D., & Horio, M. (2001). Behaviour of particles and bubbles around immersed tubes in a #uidized bed at high temperature and pressure: a DEM simulation. International Journal of Multiphase Flow, in press. Rowe, P. N., & Widmer, A. J. (1973). Estimation of solids circulation rate in a bubbling #uidised bed. Chemical Engineering Science, 28, 979. Rowe, P. N., Partridge, B. A., Cheney, A. G., Henwood, G. A., & Lyall, E. (1965). The mechanisms of solids mixing in #uidised beds. Transactions of the Institution of Chemical Engineers, 43, T271. Shen, L. H., & Zhang, M. Y. (1998). E!ect of particle size on solids mixing in bubbling #uidized beds. Powder Technology, 97(2), 170. Singh, B., Fryer, C., & Potter, O. E. (1972). Solids motion caused by a bubble in a #uidized bed. Powder Technology, 6, 239. Tsuji, Y., Kawaguchi, T., & Tanaka, T. (1993). Discrete particle simulation of a #uidized bed. Powder Technology, 77, 79. Valenzuela, J. A., & Glicksman, L. R. (1984). An experimental study of solids mixing in a freely bubbling two-dimensional #uidized bed. Powder Technology, 38, 63. van Deemter, J. J. (1985). Mixing. In J. F. Davidson, R. Clift, & D. Harrison (Eds.), Fluidization (p. 331). London: Academic Press. Wang, R. C., & Chou, C. C. (1995). Particle mixing segregation in gas}solid #uidized bed of ternary mixtures. Canadian Journal of Chemical Engineering, 73, 793. Wu, S. Y., & Baeyens, J. (1998). Segregation by size di!erence in gas #uidized beds. Powder Technology, 98(2), 139. Xu, B. H., & Yu, A. B. (1997). Numerical simulation of the gas-particle #ow in a #uidized bed by combining discrete particle method with computational #uid dynamics. Chemical Engineering Science, 52, 2785. Yoshida, K., & Kunii, D. (1968). Stimulus and response of gas concentration in bubbling #uidized beds. Journal of Chemical Engineering of Japan, 1, 11.
Study of mixing in gas-#uidized beds using a DEM model M. J. Rhodes , X. S. Wang *, M. Nguyen , P. Stewart, K. Li!man Department of Chemical Engineering, Monash University, Clayton, Vic. 3168, Australia Department of Pharmaceutics, Monash University, Parkville 3052, Australia Advanced Fluid Dynamics Laboratory, CSIRO DBCE, Highett 3190, Australia Received 10 May 2000; received in revised form 11 October 2000; accepted 7 November 2000
Abstract This paper examines the usefulness of a discrete element method simulation in studying solids mixing in gas-#uidized beds. At steady state, a part of the solids were tagged with a dark colour and the rest were tagged with a light colour. The state of mixing of bed solids was studied by analysing the variation of the proportions of the marked particles with time and position in the bed. This was achieved by scanning the bed using a sampling box. The variance of mixture composition based on the samples was incorporated into a mixing index which describes the degree of mixing of the particles at a particular time. This method enables assessment of the overall mixing behaviour in terms of the rate of mixing (through estimation of the time required for the mixing index to increase from zero to a certain value), together with the degree of mixing at the mixing equilibrium. Illustrative examples show that gas velocity and particle properties are important parameters in#uencing solids mixing in bubbling #uidized beds. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Solids mixing; Mixing index; DEM simulation
1. Introduction The basic mechanism of solids mixing in bubbling #uidized beds is well understood (e.g. Marsheck & Gomezplata, 1965; Rowe, Partridge, Cheney, Henwood, & Lyall, 1965; Rowe & Widmer, 1973; Nguyen, Whitehead, & Potter, 1977; Mason, 1978). This may be summarized as follows. When a bubble rises through the bed, it carries a wake of particles to the bed surface. The bubble also causes a drift of particles to be drawn up below it. A down#ow of solids exists in the region surrounding the rising bubbles, resulting an overall convection circulation of particles in the axial direction of the bed. At the same time, lateral mixing of solids occurs, which is caused partly by lateral motion of bubbles (due to interaction and coalescence of neighbouring bubbles) and partly by the lateral dispersion of particles in the bubble wake at the bed surface (due to eruption of the bubble). For more information, readers may refer to review papers such as Potter (1971), van Deemter (1985) and Baeyens and Geldart (1986). * Corresponding author. Tel.: #61-3-9905-1870; fax: #61-3-99055686. E-mail address: [email protected] (X. S. Wang).
It is still not generally possible, however, to predict the e!ect of operating parameters on the degree of mixing in a #uidized bed (Garncarek, Przybylski, Botterill, Bridgwater, & Broadbent, 1994). In fact, arguments are still raised regarding the e!ect of particle diameter and density. For example, while mechanistic models such as Yoshida and Kunii (1968) and Chiba and Kobayashi (1977) predict that the rate of mixing increases with particle diameter and density, recent experimental observations (e.g., Shen & Zhang, 1998; Lim, Gururajan, & Agarwal, 1993) appear to indicate otherwise. There is, therefore, a need to gather further evidence, preferably from a di!erent angle, to address these issues. The objective of the present work is to investigate the use of discrete element method (DEM) simulation in the study of solids mixing in gas-#uidized beds.
2. Assessment of the quality of mixing Many investigations of solids mixing in #uidized beds have been concerned with establishing the exchange coef"cient between the bubble wake and the emulsion phase. Generally speaking, a higher wake-exchange coe$cient implies a better quality of mixing. Experimental methods
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 2 4 - 8
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used in establishing the exchange coe$cient can broadly be divided into two categories: (1) injection of tracer particles at steady state, in which variation of concentration of tracer particles is monitored by specially designed probes; (2) marking part of the packed bed (e.g., using a particular colour), in which variation of concentration of the marked particles is monitored by means of external visualization. Examples which fall into the "rst category are Valenzuela and Glicksman (1984), Fan, Song, and Yutani (1986) and Shen and Zhang (1998). The uncertainty associated with this method is that the mixing process itself may be in#uenced by the injection and accumulation of the tracer particles, and also by the insertion of monitoring probes. Examples in the second category include Rowe et al. (1965), Singh, Fryer, and Potter (1972) and Lim et al. (1993). This method incurs minimum disturbance to the #ow, since no extra particles are added to the bed and since observation is made through visualization from outside the bed. The main uncertainty of this method is that particles have to be marked from a packed bed and, so, the transition from packed to #uidized bed could not be avoided. The quality of mixing can also be assessed by examining the degree of mixing of particles in the bed. So far, such studies are only limited to binary and ternary systems in which particle segregation takes place. Nienow, Rowe, and Chung (1978) and Geldart, Baeyens, Pope, and Wijer (1981) are examples of early studies in this area. More recent examples include Wang and Chou (1995) and Wu and Baeyens (1998). In these studies, the degree of solids mixing is expressed in the form of a mixing index, de"ned as x/x , where x is the concentration of the material under scrutiny at some level in the bed and x is its average concentration. This index equals 1 for a &fully mixed' bed, and 0 for a &completely segregated' bed. The value of x can be obtained experimentally by the following steps: (1) &freezing' the bed by suddenly stopping the gas supply at steady state, (2) dividing the &frozen' bed into a number of sections, (3) removing the particles in these sections, and (4) analysing the size or density distribution of the particles in various sections. Obviously, this method is not applicable to uniform particles, as these particles have the same size and density and so no segregation occurs. Investigation of the degree of mixing of uniform particles has so far been ignored by researchers in the community. The main reason is that it is di$cult to obtain information at the individual particle level, e.g., its location and velocity. On the other hand, there is interest in mixing of uniform systems. For example, we wish to know how well new feed material would mix with existing bed material. Also, if we are contacting solids with gas/liquid injection, we need to know the rate of solids turnover. Industry is also interested in how the location of the feed point and exit point a!ect the degree of mixing.
3. A mixing index based on statistical analysis Various mixing indexes are used in describing the e!ectiveness of di!erent mixers in the process industry (Poux, Fayolle, Bertrand, Bridoux, & Bousquet, 1991). Most of these indexes have been developed based on statistical analysis and especially on the de"nitions of standard deviations of a speci"ed property. These mixing indexes usually describe the closeness of a mixture to a `completely random mixturea. According to Lacey (1954), a completely random mixture is one in which the probability of "nding a particle of a constituent in every point is identical. When the components of the mixture are completely segregated, it can be assumed that any sample withdrawn from the mixture will be entirely composed of a pure component. Lacey (1943) showed that the variance of a completely segregated mixture, S , can be expressed as S "pq,
(1)
where p and q are the proportions of the two components estimated from the samples. When any sample is withdrawn from a fully randomized mixture, every particle within that sample can be considered as randomly representative of the mixture as a whole. In these conditions, the variance, S , may be calculated from 0 pq S " , 0 N
(2)
where N is the number of particles in a sample. This value is normally the minimum attainable variance within a mixture. The well-known Lacey index (Lacey, 1954) is de"ned as S !S M" , S !S 0
(3)
where S is the variance of the mixture between fully random and completely segregated mixtures. The mixing index obtained from Eq. (3) has a zero value for a completely segregated mixture and increases to unity for a fully random mixture. Due to its simplicity, the Lacey index is widely used to characterize mixers used in the process industry. This study explores its application to #uidized beds of uniform particles.
4. Use of Lacey index in studying solids mixing in 6uidized beds 4.1. The discrete element method (DEM) simulation It would be clear that in order to apply the Lacey index (Eq. (3)) to a #uidized bed, representative samples need to be taken across the bed at a given time. The DEM
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simulation of a gas-#uidized bed lends itself readily to the study of solids mixing since at any instant, we know the location and velocity of every particle in the bed. The DEM is a relatively new advance in numerical simulation of #uidized beds. In this method, the particles are traced individually by solving Newton's equations of motion, while the #uid phase is treated as a continuum (Tsuji, Kawaguchi, & Tanaka, 1993; Hoomans, Kuipers, Briels, & van Swaaij, 1996; Xu & Yu, 1997). In this study we use a DEM model, which was made available to us in the form of a computer code through our collaboration with Professor M. Horio of Tokyo University of Agriculture and Technology. Details of this model are given in Mikami, Kamiya, and Horio (1998) and are brie#y described here. For gas motion, the local averaged Navier}Stokes equations (Anderson and Jackson, 1967) were integrated by the SIMPLE method (Patankar, 1980) employing the staggered grid system. For particle motion, the Newtonian equations of motion of individual particles were integrated. The collisions between particles and between a particle and a wall were simulated using Hooke's linear springs and dash pots. The interaction between the particle and #uid phases was addressed as follows: when the void fraction was less than 0.8, the Ergun equation for packed beds was used; when the void fraction was larger than 0.8, a modi"ed equation of the #uid resistance for a single particle was used. The time step for calculating particle motions was decided based on numerical &experiments' of energy dissipation. In our study, two-dimensional beds of 0.16 m in width and 0.38 m in height were used. The thickness of the beds was the same as a single particle diameter.
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The Mikami et al. (1998) model introduced a liquid bridge interparticle force to enable simulation of wet particles. In the illustrative examples given in this paper, the amount of water in the bed was set to zero. So, we were e!ectively simulating dry particles in our study. Initially all the particles are randomly located in the bed with a randomly allotted velocity. The particles then fall freely under gravity in the absence of a gas #ow. The energy of the system dissipates in every collision between particles and between a particle and a wall due to inelastic collision. After allowing some time for energy dissipation through inelastic collisions, the bed reaches a stagnant state. The restitution coe$cient, friction coef"cient and spring constant of the particles used in this study are the same as those used by Mikami et al. (1998), i.e., 0.9, 0.3 and 800 N/m, respectively.
4.2. Illustrative examples 4.2.1. An illustration of the process of mixing Fig. 1 shows an illustrative example of the process of mixing for particles of 2650 kg/m in density and 1 mm in diameter. The super"cial gas velocity was 1.2 m/s. The bed was "rst allowed to operate for 7 s so that steadystate operation was established. The particles in the lefthalf of the bed were then tagged with a dark colour and the rest were tagged with a light colour. The time was set to zero at this moment. So, we can say that the dark and light coloured particles are completely segregated at t"0 s. As mixing proceeds, this completely segregated state gradually gives way to one in which the particles of
Fig. 1. An illustration of the process of mixing ( "2650 kg/m, d "1 mm, ;/; "1.5). Bed dimensions not to scale. N N KD
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Fig. 2. An illustration of the sampling method. (Note that the number of sampling boxes does not represent the actual number of sampling boxes used in the simulations.)
Fig. 3. E!ect of gas velocity on the Lacey index (d "1 mm, N "2650 kg/m). N
di!erent colours are partially mixed. A mixing equilibrium can be noted after about 4 s of mixing.
each sample was set at 100 (so that the level of scrutiny was "xed).
4.2.2. The sampling method By the use of DEM simulation, we can mark a part of bed particles with one colour and the rest with another colour at any time of operation. In this study, however, the particles were marked after steady-state #uidization was established. Major advantages of use of DEM for mixing studies are
4.2.3. Ewect of gas velocity In these simulations, particles of 1 mm in diameter and 2650 kg/m in density were used. The total number of particles used was 14,000. Fig. 3 shows the results for ;/; of 1.25, 1.5, 1.75 and 2.0. The x-axis represents KD time starting from solids marking, and the y-axis is the Lacey index based on Eq. (3). It shows for all the cases that as mixing proceeded, the mixing index gradually increased until an equilibrium of mixing was reached. It is interesting to note that the Lacey index at the mixing equilibrium was about the same (i.e., M+0.72) for all the gas velocities examined. This suggests that the degree of mixing achievable at this condition does not vary with gas velocity. Results shown in Fig. 3 also allow examination of the rate of mixing of the particles, through comparison of the time required for the mixing index to increase from zero to a certain value. It can be noted from Fig. 3 that the time required for the mixing index to increase from 0 (completely segregated) to 0.72 (the mixing equilibrium) decreased with increasing velocity. For example, an increase in the ;/; from 1.25 to 2.0 KD resulted in a decrease in the time from around 6 to 3 s. This would suggest that the average rate of mixing increases with increasing gas velocity for the conditions examined. It agrees with the general trend reported in the literature (e.g., Lim et al., 1993; Wu & Baeyens, 1998). We have also attempted to obtain the mixing index at ; . KD We found that the mixing equilibrium could not be reached within 15 s, suggesting that the rate of mixing was much slower under this condition. This was due to the lack of bubbles under this condition, which was evident from analysis of snapshots.
(a) it enables non-intrusive sampling; (b) we can gather information at the individual particle level at any time; (c) we can readily vary sample size; and (d) we can tag the particles after the bed is #uidized, so there is no need of going through the packed- to #uidized-bed transition. A rectangular box was used to take samples from the bed, as shown in Fig. 2. In this study, the sampling box was used to &scan' the bed while avoiding any overlapping of the samples. (Note that the particles were counted based on their centre positions.) In addition, the number of particles in each sample was kept the same. This was achieved by allowing the sampling box to expand vertically. Of course, the sampling box at the surface of the bed could contain a smaller number of particles. If the number of particles in the top boxes di!ered from the designed value by a certain margin (e.g., 10%), then this sample was disregarded. We found that this had little e!ect on the mixing index obtained, provided that a su$ciently large number of samples were taken in each snapshot. Clearly, the &scanning' approach enables the overall picture of mixing in the bed to be revealed. It is worth noting that the Lacey index (Eq. (3)) is not related to the number of particles in the bed or the number of samples, provided that a su$cient number of samples were taken. The results presented below were obtained using a sampling box of 10 mm in width. The number of particles in
4.2.4. Ewect of particle density In these simulations, the particle diameter was kept at 1 mm while the particle density was decreased from 2650
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Fig. 4. E!ect of particle density on the Lacey index (d "1 mm, N ;/; "1.5). KD
to 1590, then to 900 kg/m. The total number of particles used was again 14,000, and the simulations were conducted at ;/; of 1.5. Results for particle densities of 2650 KD and 1590 kg/m are shown in Fig. 4. The result for particle density of 900 kg/m is not included in this "gure, since it almost coincided with the result for particle density of 2650 kg/m. It can be noted that the mixing indexes at the mixing equilibrium were about 0.72 for the particle densities examined, indicating that the degree of mixing achievable in a #uidized bed was not a!ected by changes in particle density. The time required for the mixing index to increase from 0 to 0.72 increased from about 4 to 6 s when was decreased from 2650 to N 1590 kg/m. It then decreased from about 6 to 4 s when was decreased from 1590 to 900 kg/m. So, mixing N rate decreased from "2650 to 1590 kg/m, and inN creased again as was decreased to 900 kg/m. SnapN shots of the bed were examined in order to establish the causes for the results just presented. The snapshots for di!erent particle densities were taken at steady state (from t"10 s) and for the same time intervals (0.05 s). We found that as was decreased from 2650 to N 1590 kg/m, the average bubble size in the bed slightly decreased while lateral movement of bubbles became increasingly dampened. As was decreased from 1590 N to 900 kg/m, on the other hand, the average bubble size was signi"cantly decreased while lateral movement of bubbles was clearly enhanced. So, the density e!ect was the result of combined e!ects of changes in bubble size and the extent of lateral movement of bubbles. The results presented in Fig. 4 are, however, not completely consistent with experimental observations of Lim et al. (1993) and Shen and Zhang (1998). In their studies, the decreasing mixing range was not observed. This raises questions as to whether it is adequate to keep ;/; constant in investigating the density e!ect for KD particles of monosizes or narrow size distributions. Note that some researchers (e.g., Wu & Baeyens, 1998) chose to use a constant ;!; for particles of di!erent sizes. To KD
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address this problem, let's "rst look at the conditions of Shen and Zhang (1998). In order to maintain a constant ;/; of 2.0, the value of ;!; (which directly KD KD a!ected bubble size in the bed) was decreased from 0.175 to 0.1 m/s as particle density was decreased from 2600 to 1300 kg/m (d "0.5 mm). Of course, changes in the N value of ;!; depended on the value of ;/; . KD KD Similar analysis also applies to the experimental conditions of Lim et al. (1993) and also simulation conditions for Fig. 4 in this paper. We believe that the apparent inconsistency between our simulation and observations of Lim et al. (1993) and Shen and Zhang (1998) was caused by the variations in ;!; . KD We have also performed simulation &experiments' by keeping ;!; constant (at 0.4 m/s) for di!erent parKD ticle densities. The time required for the mixing index to increase from 0 to 0.72 (mixing equilibrium) was found to decrease from about 4 to 3.6, then to 3 s, respectively, as particle density was decreased from 2650 to 1590, then to 900 kg/m. Analysis of animation results shows that as particle density was decreased, bubbles grew faster, movement of bubbles became more vigorous, while the average bubble size remained relatively una!ected. Our simulation results point to the need for further experimental studies of the e!ect of particle density by keeping ;!; constant. KD 4.2.5. Ewect of particle diameter The in#uence of particle diameter was studied by increasing particle diameter from 1 to 2 mm whilst keeping the particle density at 2650 kg/m. The ;/; was again KD "xed at 1.5. The number of particles was reduced from 14,000 to 3500 so that the height of the bed was kept about the same. Results of these simulations are shown in Fig. 5. It is interesting to note that the mixing index at the mixing equilibrium decreased considerably, from about 0.72 to 0.54, when the particle diameter was increased from 1 to 2 mm. It can also be noted that the time required for the mixing index to increase from 0 to 0.54
Fig. 5. E!ect of particle diameter on the Lacey index ( "2650 kg/m, N ;/; "1.5). KD
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increased from about 2 to 5 s as particle diameter was increased from 1 to 2 mm. So, the average mixing rate would decrease with increasing particle diameter. These results are qualitatively supported by experimental observations of Lim et al. (1993) and Shen and Zhang (1998). It should be noted, however, that the initial rate of mixing (t(2 s) was not signi"cantly a!ected by particle size. Further simulations were performed to look at the e!ect of particle diameter on the basis of a constant ;!; (i.e., 0.4 m/s). We observed that results of these KD simulations qualitatively agreed with those obtained using a constant ;/; (Fig. 5). KD 5. Discussion on the proposed method 5.1. Approaches for solids marking and sampling Solids marking and sampling are obviously important issues which warrant careful consideration. Our "rst concern is regarding the timing of solids marking. Fig. 6 shows a comparison of the Lacey index obtained for two di!erent conditions under which the particles were marked: one in which the particles were marked from packed bed, and the other in which the particles were marked after steady-state #uidization was established. The main di!erence between these two cases is that a period of negligible mixing (up to about 1 s) existed in the packed-bed case. Obviously, this period of negligible mixing is associated with unlocking of the packed bed and, so, it would be strongly in#uenced by the manner by which #uidizing gas was introduced (e.g., how quickly the supply was turned on). Fig. 6 serves as an example in which the super"cial gas velocity was increased linearly to its full value within the initial 1 s of operation. Another matter concerning solids marking is by what proportion the bed should be marked with a dark colour. Experimental observations of Wu and Baeyens (1998)
using binary particles and Lim et al. (1993) using uniform particles have demonstrated that the concentration of tracer particles certainly a!ects their results. We found in a standard simulation (particle diameter: 1 mm; particle density: 2650 kg/m; ;/; : 1.5) that the resulting LacKD ey index decreased from 0.73 to 0.52 when the composition of particles with a dark colour was increased from 50 to 70%. This is expected, since the Lacey index is necessarily concentration dependent. So, as in any physical experiments, it is important that the same concentration of coloured particles be used in comparative studies of the e!ect of operating conditions. It would be advisable that half of the bed be marked with a dark colour and the other half marked with a light colour. This certainly gives rise to a relatively higher mixing index at the mixing equilibrium. It also avoids uncertainties associated with local variations in bubble properties, even at steady state (Shen & Zhang, 1998). The sampling method itself is also an important factor. In this paper we have presented a method in which the bed was &scanned'. The objective was to avoid potential problems associated with improper distribution of samples. The number of particles in each sample (i.e., sample size) re#ects the level of scrutiny. The main reason for using a constant sample size (i.e., 100) for particles of d "1 and 2 mm was that we attempted to place di!erN ent systems under the same level of scrutiny. A useful guideline in choosing an appropriate sample size would be that such a sample size should give rise to a mixing index of 0 for a completely segregated system (M ). We observed that the value of M was quite insensitive to the sample size in our system; it remained at about 0.01 as the sample size was increased from 100 to 200, then to 300 (for d "1 mm). We have also tested another approach N by which samples were taken randomly in the bed. We found, however, that large errors could be introduced due to undetected overlapping of the samples, especially those of large bubbles. 5.2. Relation to wake-exchange coezcient
Fig. 6. A comparison of the variation of Lacey index for di!erent conditions at which particles were marked (d "1 mm, "2650 kg/ N N m, ;/; "1.5). KD
The mixing index and wake-exchange coe$cient are measures of the quality of mixing from di!erent angles. The former describes the state of mixing, which, therefore, emphasizes the overall picture of mixing; the latter refers to the rate of particle exchange between the bubble wake and the emulsion phase, which, therefore, emphasizes the progress of mixing. The distribution of tracer particles in the bed can, in theory, be calculated with the knowledge of the wake-exchange coe$cient, initial concentration and location of tracer particles, together with other operating conditions. This in turn would enable calculation of the mixing index. There is, however, no simple relationship between these two measures. We would expect that a higher wake-exchange coe$cient brings about a more uniform distribution of tracer
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particles at the mixing equilibrium, resulting a higher mixing index. Our simulation results reported earlier lend qualitative support to this analysis.
using particles of monosizes or narrow size distributions by keeping ;!; constant. KD
5.3. Accuracy of the proposed method
Notation
It needs to be stressed that the objective of this study was to provide an illustrative example of the use of DEM in studying solids mixing in #uidized beds. Accuracy of the results remains to be improved by re"ning the sampling method. Of course, the quality of the DEM model would directly in#uence the results. It is worth noting that a simpli"ed treatment was used in the original model of Mikami et al. (1998) regarding the forces acting on a single particle, i.e., Eq. (7) in their paper. In that equation, an additional term equal to the negative of product of particle volume fraction and local pressure gradient was omitted on the right-hand side. According to Horio and Rong (2000), this could cause an error of (mg);( / ) in the force at steady state, where m is the D N mass of a single particle, and are #uid and particle D N density, respectively. The actual error could be di!erent, as the gas drag component could deviate from the steady-state values. This simpli"ed treatment was also applied by Tsuji et al. (1993), as it was considered to be e!ective when #uid density was small. Simulation results such as bed pressure drop and bubble behaviour of Horio and co-workers and those of ours (e.g., Rhodes, Wang, Ngyen, Stewart, & Li!man, 2001) further con"rmed the e!ectiveness of this simpli"ed treatment. For gases of a relatively high density, e.g., under pressurized conditions, a more rigorous approach should be applied (Kaneko, Shiojima, & Horio, 1999; Rong & Horio, 2001).
d particle diameter, m N M Lacey index, de"ned in Eq. (3), dimensionless M Lacey index for a completely segregated system, dimensionless N number of particles in a sample p proportion of red particles in the samples q proportion of green particles in the samples S square root of the variance of the mixture between fully randomized and completely segregated states, dimensionless S square root of the variance of the completely seg regated states, dimensionless S square root of the variance of the fully randomized 0 states, dimensionless t time, s ; super"cial gas velocity, m/s ; minimum #uidization velocity, m/s KD
6. Conclusions The usefulness of a DEM simulation in studying solids mixing in a gas-#uidized bed has been investigated. It shows that, by using this method, the overall picture of mixing can be revealed in the form of a mixing index. It also provides information regarding the rate of mixing under speci"ed conditions. Distinct advantages of the proposed method are that the sample size can be readily varied, and that particles can be tagged after steady-state #uidization is established. Simulation results indicate that the rate of solids mixing increases with increasing gas velocity, whilst the degree of mixing achievable is una!ected by gas velocity. While the initial rate of mixing was found to be insensitive to particle size, the overall rate of mixing and the degree of mixing were found to decrease with increasing particle diameter. The e!ect of particle density was found to be not so clear-cut when mixing was measured at a constant ;/; , pointing to KD the need for further experimental studies of this e!ect
Greek letter N
particle density, kg/m
Acknowledgements The authors gratefully acknowledge the "nancial support from Australian Research Council Large Grant A89927077 and CSIRO/DBCE Grant CZ-25. They also thank Professor M. Horio and Dr. T. Mikami of Tokyo University of Agriculture and Technology, Japan, for making the original computer code available for this research, and Dr. D. Rong for very useful discussion.
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