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A generalized empirical description for particle slip velocities in liquid fluidized beds
Chemical Engineering Science 54 (1999) 1045—1052
A generalized empirical description for particle slip velocities in liquid fluidized beds K.P. Galvin*, S. Pratten, G. Nguyen Tran Lam ARC Centre for Multiphase Processes, Department of Chemical Engineering, University of Newcastle, NSW 2308, Australia Received 27 July 1998; received in revised form 19 November 1998; accepted 19 November 1998
Abstract An empirical equation for calculating the slip velocity of a species in any homogeneous suspension is proposed. The Richardson and Zaki (1954, Trans. Inst. Chem. Engng, 32, 35—53) and Lockett and Al-Habbooby (1973, Trans. Inst. Chem. Engng 51, 281—292; 1974, Powder Technol., 10, 67—71) equations are special cases of the proposed equation, and arise when all species are of the same density. Therefore, the main value of the proposed equation is in describing the slip velocities of particles in suspensions containing species of different density. In this short note results from one experimental system are presented, and shown to be consistent with the model. The model is also consistent with the explanation used by Moritomi et al. (1982) to account for phase inversion in fluidized beds. The model is appealing in its simplicity, and should find favour in the design and control of process equipment. The new model predicts the strong segregation effects observed in suspensions containing particles of different density, and hence represents an immediate improvement on the Lockett and Al-Habbooby equation. Its application is expected to cover all homogeneous suspensions, in which the particles are all more dense than the suspension. At this stage its validation has been limited to low concentrations of dense particles settling through a fluidized bed of low density particles as occurs in a Teetered Bed Separator, and to phase inversion conditions in fluidized beds. It is hoped that this note might lead to a much more extensive validation of the model by workers using vastly different particle species. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Slip velocity; Liquid fluidized beds; Monocomponent fluidization data
1. Introduction A single particle of species i settles in an unbounded fluid at its terminal velocity, º . As the volume fraction RG of the species in the liquid increases, the velocity of the species relative to the liquid decreases. The complex dependence of this slip velocity, » , on the volume fracG tion of the species, , is well described by the empirical G equation of Richardson and Zaki (1954). That is, » "º (1! )LG\, (1) G RG G where n is a constant, typically 4.65 for spherical parG ticles at low Reynolds numbers, less than 0.1. For systems containing particle species which differ according to their
* Corresponding author. Present address: Department of Chemical Engineering, University of Newcastle, Callaghan, NSW 2308, Australia. Tel.: 61 2 4921 6197; Fax: 61 2 4921 6920. E-mail address: [email protected] (K.P. Galvin)
size only, the Lockett and Al-Habbooby equation (1973, 1974) is generally adequate for design purposes. For three particle species, the equation is, » "º (1! ! ! )LG\. (2) G RG G H I Although Lockett and Al-Habbooby claim that their equation is applicable to systems containing species of different density, our experience is that the equation is invalid. Lockett and Al-Habbooby used particle species which did not differ significantly in their densities. More mechanistic models have also been developed, which are concerned primarily with providing a physical understanding of hindered settling rather than an accurate description suitable for process design. Smith (1965, 1966, 1998), for example, developed a cell-model to account for differences in the local volume fraction for each species. He recognized that each species was subject to a common pressure gradient. Batchelor (1972), and Batchelor and Wen (1982) investigated the particle—particle interactions in monocomponent and multispecies
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 4 0 7 - 2
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suspensions, producing linear equations applicable to the low volume fraction limit. Experimental work concerned with suspensions containing particles of different density was generally limited to the study of equilibrium and phase inversion in liquid fluidized beds. Studies included Duijn and Rietema (1982), Moritomi et al. (1982, 1986), and Gibilaro et al. (1986). The strong segregation that develops in these systems, due to the presence of species of different density, led to the view that the buoyancy force is governed by the density of the suspension, rather than the density of the liquid. However, using an energy balance Clift et al. (1987) demonstrated that the buoyancy force is always governed by the density of the liquid. The purpose of this note is to propose a more generalized form of the Richardson and Zaki equation that is useful for all suspensions, whether they are formed using a single species, species of different size and of the same density, or by species of different size and density. In a fluidized bed consisting of a single particle species, the total pressure increases down through the bed according to, P " o gH#(1! )ogH, (3) 2 G G G where o is the density of species i, o the density of the G liquid, g the acceleration due to gravity, and H the height. Hence the total pressure gradient can be written as, dP /dH" (o !o)g#og. (4) 2 G G The pressure gradient consists of two parts, the first due to the weight of the particles in the liquid, and hence the dissipative drag force, and the second due to the hydrostatic head of the liquid. The dissipative pressure gradient is, dP/dH" (o !o)g. (5) G G If we substitute Eq. (5) into Eq. (1), a modified form of the Richardson and Zaki equation is produced. That is,
dP/dH LG\ » "º 1! . (6) G RG (o !o)g G Eq. 6 is identical to both Eqs. (1) and (2), provided the particle species are of one density. Hence, for such systems, Eq. (6) requires no new validation. For a monocomponent system the equation provides the slip velocity as a function of the pressure gradient that arises from the fluid drag. It is argued here that it does not matter how that pressure gradient is formed, whether by particles of one density, or of different densities. For a three species system, for example, Eq. (5) becomes dP/dH" (o !o)g# (o !o)g# (o !o)g. (7) G G H H I I Hence, once the pressure gradient is known or the composition of the suspension is known, the monocompo-
nent expression becomes immediately applicable for describing the slip velocity in any suspension. It should be noted that the attraction of Eq. (6) is its simplicity, and ease of use, together with its potential in process design. The equation may also be useful in process control, given that the pressure gradient is readily measured. It is probable that a more accurate equation would involve a much more complex relationship. Eq. (6) is equivalent to,
o !o LG\ G Q » "º , G RG o !o G
(8)
where o is the density of the suspension. This form of the Q equation indicates that a dimensionless density parameter can be used to describe the hindered settling. The equation also shows that the suspension density will produce a strong segregation effect when particles of different density are present. Clearly, however, this equation is invalid when the density of the species is less than the density of the suspension. It is likely, though, that such suspensions will be unstable, and generate so-called streaming effects, and lateral segregation (Batchelor and Janse van Rensburg, 1986; Davis and Gecol, 1996). These circumstances require an entirely different model. Eq. (6) can also be written in the form of Eq. (2), in terms of the volume fraction of each species. That is, » "º (1!i !i !i )LG\, G RG G G H H I I
(9)
where i "(o !o)/(o !o). H H G
(10)
Coefficients are needed for the volume fraction terms. These either enlarge the volume presence or reduce the volume presence of a given species. For example, when the velocity of a low density species is to be calculated, the volume presence of a denser species is increased by its coefficient until the effective density of the species is reduced to the value of the low density species. This adjustment is equivalent to applying a common pressure gradient to each species present at a given location.
2. Experimental methodology A fluidized bed, shown schematically in Fig. 1, was used to conduct a series of experiments in order to examine the validity of Eq. (6). The vessel was 0.173 m in diameter and 1.36 m high. A feed slurry was fed to the system continuously, producing an overflow stream, and an underflow stream was withdrawn from below. The feed slurry consisted of three particle species, PVC, glass spheres, and magnetite. Details on the monocomponent data of each of these species are given in Table 1, including fluidization results on each species to obtain º and RG
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
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n . The relatively high values of n obtained for the PVC G G and the magnetite suggest a degree of particle flocculation (Dixon, 1977). The feed entered the system and settled against the upward current of water used to fluidize the bed. The particles segregated into two main zones, with the denser magnetite and glass spheres down below, and the less dense PVC particles above. A steady flux of magnetite and glass spheres passed down through the PVC zone in order to produce the denser zone below. The system was permitted to run until steady state was reached. Samples of the suspension were withdrawn at different positions using a 60 ml syringe attached to tappings on the vessel wall. The samples were then analysed to determine the volume fraction of each species at each location. A differ-
ential pressure transducer was also used to measure the system pressure profile.
Fig. 1. Fluidized bed operating with continuous feed, separating into product and overflow streams. These systems are sometimes referred to as Teetered Bed Separators.
Fig. 2. Steady state volume fraction distribution in the fluidized bed, showing the denser phase above the base to a height of 400 mm, and the less dense phase above.
3. Results The volume fraction distribution for one experiment is shown in Fig. 2 and the overall steady state data is shown in Table 2. In the PVC zone, the magnetite and glass spheres settled at a relatively high velocity, producing the low concentrations is shown. The PVC particles did not
Table 1 Monocomponent data on species used in the feed Species
Density (kg/m)
Diameter (mm)
º (mm/s) R
Exponent n!1
PVC particles Glass spheres Magnetite
1400 2400 5000
0.125—0.180 0.063—0.075 (0.045
8.6 4.2 8.6
8.6 2.7 12.2
Table 2 Steady state experimental data — Run L3 Suspension
Feed Water Underflow Overflow
Volume fractions
Fluxes (mm/s)
PVC
Glass
Magnetite
Total Solids
0.0305 0.000 0.000 0.0229
0.0229 0.000 0.0473 0.0010
0.0301 0.000 0.0781 0.0000
0.0835 0.0000 0.1254 0.0239
1.88 1.28 0.73 2.44
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appear in the dense phase below. A peristaltic underflow pump was used to withdraw the denser particles from below, producing a steady dense phase bed height of 400 mm. In the series of four experiments reported here, the flux of magnetite fed to the system was increased from zero in the first experiment to a higher level in each experiment. This increase was achieved simply by increasing the feed pulp density via an increase in the proportion of magnetite in the feed. All other conditions remained the same, including the fluidization rate. The experimental slip velocity data reported in this study were based on volume fraction data and flux balances conducted in the steady PVC zone 610 mm above the base of the vessel. An example calculation is presented in Appendix A. It is evident in Table 3 that as the magnetite content in the feed increases, the suspension density, calculated using the corresponding pressure gradient data, rises, and the total volume fraction of the solids decreases. The experimental slip velocities of the species actually decrease, with significant reductions evident for the PVC species, and only minor changes for the other species. Table 4 provides a comparison between the experimental slip velocities and the slip velocities predicted using the Lockett and Al-Habbooby equation and the model presented in this paper. As expected, the Lockett and Al-Habbooby model predicts an increase in the slip velocity as the magnetite presence increases, because of the decrease in the volume fraction of the solids. The values predicted using the model proposed in this study, however, show remarkably good agreement with the
experimental data. These predictions were based on the suspension density data in Table 3, which were derived from the pressure gradient data. As shown in Fig. 3, the suspension density values obtained from the pressure gradient data are in excellent agreement with the suspension density data obtained from the volume fraction analysis of the samples withdrawn from the vessel.
Fig. 3. Comparison between the suspension density, obtained from the measured pressure gradient, and the suspension density obtained from an analysis of samples withdrawn from the column. The pressure gradient data were averaged from three sets taken towards the end of the experiment.
Table 3 Particle velocities relative to the water in the lower part of the PVC Phase Feed pulp density
Volume fraction
Suspension density (kg/m)
PVC (mm/s)
Glass (mm/s)
10.8 (No magnetite) 14.7 21.5 26.5
0.177 0.149 0.130 0.120
1082 1078 1105 1119
1.44 1.22 0.74 0.35
3.7 3.4 3.4 3.1
Magnetite (mm/s)
5.9 6.2 5.4
Table 4 Comparison between the particle slip velocities Experimental (mm/s)
Lockett and Al-Habbooby (mm/s)
New model (mm/s)
PVC
Glass
Magnetite
PVC
Glass
Magnetite
PVC
Glass
Magnetite
1.44 1.22 0.74 0.35
3.7 3.4 3.4 3.1
5.9 6.2 5.4
1.6 2.1 2.6 2.8
2.5 2.7 2.9 3.0
0.80 1.2 1.6 1.8
1.20 1.33 0.63 0.41
3.6 3.6 3.4 3.3
6.7 6.8 6.2 5.9
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
3.1. Phase inversion In order to further assess the new model, equilibrium data on mixed fluidized beds were obtained from the paper of Moritomi et al. (1986). The Richardson and Zaki equations for each species, and density values, were also obtained from their 1986 paper. Expressed as slip velocities, the equations produced were: 1. Char » "46.8 (1! ) \ with o "1380 kg/m and velocities in mm/s, (11) 2. Glass » "13.8(1! ) \ with o "2450 kg/m and velocities in mm/s. (12) Equilibrium data is readily calculated using the new model by recognising that the slip velocities for the two species should be identical in a mixed bed. So implementing the new model, and equating the slip velocity for each species, gives
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forms when » "» "8.9 mm/s and o "1212 kg/m. Q This solution is shown in Fig. 4, with the suspension density plotted as a function of the slip velocity for each species. Moritomi et al. (1982) used a similar condition for the mixed bed condition (see Fig. 7 in their paper), based on the proposition of Pruden and Epstein (1964), except that they equated the required superficial velocities rather than the slip velocities. It should be noted, though, that inversion is usually observed to occur over a broad range of superficial fluidization velocities. This observed range is not predicted by this model. Moritomi et al. (1982, 1986) used closely sized species in their experiments. However, the small differences in size may still have contributed to the range of inversion observed. The mixed bed equilibrium data, obtained from Fig. 6 in the paper of Moritomi et al. (1986), are shown together in Fig. 5 of this paper with the prediction based on the proposed model. The equilibrium relationship based on the present model is obtained by equating an expression for the suspension density with the critical density, o "1212 kg/m. That is, Q o "1212"1380 #2450 #1000 (1! ! ). Q (14) Rearranging gives the line in Fig. 5. That is,
o !o L\ o !o L\ Q Q » "º "º (13) R o !o R o !o Clearly, there is a unique value of o needed to solve Eq. Q (13), given that all other variables are known. Using the data in Eqs. (11) and (12), it follows that the mixed bed
"0.146!0.262 . (15) According to the above analysis, the model predicts that the inversion occurs at a given suspension density. The mixed bed composition then depends on the proportion of each species present in the system, and should
Fig. 4. Monocomponent suspension densities shown as a function of the slip velocity, for each species. The mixed bed condition is defined by the intersection of the curves. This data was obtained using Eqs. (11) and (12).
Fig. 5. Comparison between the new model and literature data on phase inversion in fluidized beds (Moritomi et al., 1986).
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Fig. 6. Monocomponent suspension density versus the superficial velocity, for each species. The data was produced using Eqs. (11) and (12), modified to give the superficial velocities, and hence are similar to the data in Fig. 7 of Moritomi et al. (1982).
correspond to the relationship given by Eq. (15). This linear relationship is in reasonable agreement with the mixed bed data shown in Fig. (5). The magnitudes of the errors are comparable to those obtained using the more complex models of Moritomi et al. (1986). Indeed, simply predicting the existence of a range of mixed bed equilibrium volume fractions is a reasonable result. It is worth considering the process of phase inversion by using a plot of the monocomponent suspension density as a function of the superficial fluidization velocity. This data, shown in Fig. 6, is similar to Fig. 7 in the paper of Moritomi et al. (1982). The data, here, was obtained using Eqs. (11) and (12), corrected to give superficial velocities. Hence the exponent n rather than n !1 was G G used in Eqs. (11) and (12). Also, the density values shown with Eqs. (11) and (12) were used. At low superficial velocities, Moritomi et al. (1982, 1986) observed a monocomponent zone of char above a monocomponent zone of glass spheres. The model predicts the first mixed bed condition to occur when the glass zone has a monocomponent volume fraction of 0.146. At this stage, the density of the glass zone must be 1212 kg/m. As shown in Fig. 6, this occurs when the density of the char zone is 1175 kg/m, and the superficial velocity is 7.6 mm/s. In this case the char can enter the glass zone, but only generate an infinitesimal concentration. (Given that the slip velocity in the glass zone is 8.9 mm/s, the superficial velocity must be 8.9 (1!0.146)"7.6 mm/s.) This velocity is significantly higher than the minimum velocity of 5 mm/s obtained by Moritomi et al. (1982, 1986) because the present model predicts a starting concentration of 0.146 which is too low, and predicts a slip velocity which
is too high. Of course, as noted earlier, part of this discrepancy may be due to the existence of a finite range of particle sizes for each species in the experiments. In the study by Moritomi et al. (1982), a mixed bed containing 100 g of glass and 50 g of char was formed. Using the corrected densities in their 1986 paper the ratio, / , will be 1.12. So, using Eq. 15, the new model predicts a fully mixed bed with "0.105 and
"0.118, and hence a total volume fraction of 0.223. The model also predicts a slip velocity of 8.9 mm/s, and hence a required superficial velocity of 8.9 (1—0.223) "6.9 mm/s. According to Fig. 6, the monocomponent char zone has a density of 1180 kg/m and the monocomponent glass zone a density of 1235 kg/m. Presumably, particles near the interface between the char and glass zones gradually create a mixed interface density of 1212 kg/m. This condition then propagates down through the glass zone. So, this model predicts a superficial inversion velocity of 6.9 mm/s, which is lower than the experimental value of 8.4 mm/s shown in Fig. 3 of the paper by Moritomi et al. (1982). It is evident that, despite its simplicity, the proposed model offers a reasonable description of particle slip velocities. However, the model is entirely empirical, and hence its potential must be limited. For example, the model predicts that the fully mixed bed is produced at º"6.9 mm/s. However, to reach such a state, the lower phase must pass through a state involving a high glass concentration ( "0.146) and a low char concentration ( &0). At this point the superficial velocity is 7.6 mm/s, a value slightly higher than that predicted for the fully mixed bed. So arguably the model predicts that to achieve the fully mixed state, it is necessary to initially raise the fluidization velocity to a value higher than 6.9 mm/s to allow the char to move into the glass phase. Once some char is in the glass phase, the fully mixed bed could then be produced by lowering the superficial velocity back to º"6.9 mm/s to achieve the mixed bed state. We have seen no evidence to support this more detailed mechanistic prediction, and hence suggest some caution in using the model. Experiments involving identical particles within a given species would be needed in order to investigate this finer detail. Finally, it is noted that an exceedingly broad range of circumstances, involving particles of different size and density can arise in theory. However, in practical engineering problems the range of circumstances encountered is arguably much more narrow. For example, in batch fluidization of a binary system of particles, the range of conditions capable of producing a mixed bed equilibrium condition is relatively narrow. Moritomi et al. (1986) provides a window of conditions, showing for example that large low-density particles mix with small high density particles. When the particles are of comparable size, the particles should also be of comparable density. Of course, in a Teetered Bed Separator, a broader range of
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
circumstances can arise, because the system is run as a continuous separator. However, even in a continuous separator homogeneous operation involves conditions which do not extend all that far beyond the window provided by Moritomi et al. (1986). It was seen in this study that relatively small high density particles of magnetite settled through a zone of relatively large low density particles of PVC. Consider now the case of a high concentration of nearly neutrally buoyant particles, containing a very low concentration of much denser particles. The model proposed in this study will predict that the dense particles will settle at a velocity close in value to the infinite dilution terminal value. If these dense particles are very much smaller than the low-density particles, such a result is reasonable. The small dense particles will, on average, be many particle diameters from the large low-density particles, and will not be affected significantly by them. This circumstance is similar to that which occurred in this study, and that which occurs in mixed beds. However, if the dense particles are much larger than the low-density particles, they will experience a reduction in velocity. This result is not predicted by the model. The suspension of low-density particles effectively produces a more viscous liquid. A similar resistance will also arise if the two particle species are also of similar size. It is argued, however, that these circumstances are more the exception than the rule. In a continuous separator the low-density particles would simply move upwards through the vessel with the liquid, and would therefore not co-exist with the dense particles at all. If, however, the system is fluidized with a liquid containing these lowdensity particles, then clearly these cases could arise. However, the sensible approach here is to simply ignore the presence of the low-density particles and treat the fluidizing medium as a more viscous liquid. The terminal velocity of the dense particles would then be modified accordingly. In this way, the model can readily be extended to cover these more unusual cases.
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by the liquid in order to support the weight of all the particle species. Implicit in the model is the view that it is not important how the pressure gradient arises. A pressure gradient simply exists, which in turn governs the hindered settling of each species in virtually the same way as it does under monocomponent conditions.
Appendix A. Calculation of slip velocities for Run L3 In order to calculate the slip velocities for Run L3, refer to the steady state data in Table 2. Note that any error in the mass balance is most likely in the feed data and the overflow rate. The fluidization water rate is considered accurate. The total underflow flux of water is 0.73(1— 0.1254)"0.64 mm/s, and hence the upward flux of water through the PVC phase is 1.28—0.64"0.64 mm/s. At a position 610 mm from the base, the total volume fraction of solids is 0.1295, as shown in Fig. 2, and hence the velocity of the upward moving water relative to the vessel is 0.64/(1—0.1295)"0.74 mm/s here. Given that the PVC velocity relative to the vessel is zero, the PVC slip velocity is 0.74 mm/s. The underflow flux of the glass spheres is 0.0473;0.73"0.0345 mm/s. The volume fraction of the glass in Fig. 2, 610 mm from the base, is 0.013, and hence the velocity of the glass relative to the vessel is 0.0345/0.013"2.66 mm/s. So, the slip velocity of the glass spheres is 2.66#0.74"3.4 mm/s. Similarly for the magnetite, with a volume fraction of 0.0104, the slip velocity is 0.0781;0.73/0.0104#0.74"6.2 mm/s.
Acknowledgements The authors acknowledge the financial support of Rio Tinto Research and Technology Development, Novatech, and the Australian Coal Association Research Program (ACARP). Further support from the ARC Centre for Multiphase Processes is also gratefully acknowledged.
4. Conclusions It is proposed that the slip velocity of a species in a multispecies suspension, involving particles of different size and density, can be predicted using the following empirical equation, based on the Richardson and Zaki equation:
dP/dH LG\ . »"º 1! G RG (o !o)g G The only data required is the monocomponent fluidization data for each species. The pressure gradient term is common to all particle species, and is a consequence of the dissipative processes, that is, the drag force produced
Notation g H n G P P 2 º º RG » G
acceleration due to gravity, ms\ height down the column, m exponent pressure due to weight of particles in the liquid, Pa total pressure due to weight of particles and liquid, Pa superficial fluidization velocity, m s\ terminal velocity of species i at infinite dilution, m s\ slip velocity of species, i ms\
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i o o G o H o I o Q
G
H
I
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
dimensionless density as defined by Eq. 10 density of liquid, kg m\ density of species i, kgm\ density of species j, kg m\ density of species k, kg m\ density of suspension, kg m\ volume fraction of species i volume fraction of species j volume fraction of species k
References Batchelor, G.K. (1972). Sedimentation in a dilute suspension of spheres, part 2. J. Fluid Mech., 52, 245—268. Batchelor, G.K., & Wen, C.S. (1982). Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. numerical results. J. Fluid Mech., 124, 495—528. Batchelor, G.K., & Janse van Rensburg, R.W. (1986). Structure formation in bidisperse Sedimentation. J. Fluid Mech., 166, 379—407. Clift, R., Seville, J.P.K., Moore, S.C., & Chavarie, C. (1987). Comments on buoyancy in fluidised beds. Chem Engng Sci., 42(1), 191—194. Davis, R.H., & Gecol, H. (1996). Classification of concentrated suspensions using inclined settlers. Int. J. Multiphase Flow, 22(3), 563—574.
Dixon, D.C. (1977). Comments on hindered settling theories. Powder ¹echnol., 17, 147—149. Duijn, G.V., & Rietema, K. (1982). Segregation of liquid-fluidized solids. Chem. Engng Sci., 37(5), 727—733. Gibilaro, L.G., Felice, D.I., Waldram, S.P., & Foscolo, P.U. (1986). A predictive model for the equilibrium composition and inversion of binary-solid liquid fluidised beds. Chem. Engng Sci., 41(2), 379—387. Lockett, M.J., & Al-Habbooby, H.M. (1973). Differential settling by size of two particle species in a liquid. ¹rans. Inst. Chem. Engng., 51, 281—292. Lockett, M.J., & Al-Habbooby, H.M. (1974). Relative particle velocities in two-species settling. Powder ¹echnol., 10, 67—71. Moritomi, H., Iwase, T., & Chiba, T. (1982). A comprehensive interpretation of solid layer inversion in liquid fluidized beds. Chem. Engng Sci., 37(12), 1751—1757. Moritomi, H., Yamagishi, T., & Chiba, T. (1986). Prediction of complete mixing of liquid-fluidized binary solid particles. Chem. Engng Sci., 41(2), 297—305. Pruden, B.B., Epstein, N. (1964). Stratification by size in particulate fluidization and in hindered settling. Chem. Engng Sci., 19, 696—700. Richardson, J.F., & Zaki, W.N. (1954). Sedimentation and Fluidization: Part I. ¹rans. Inst. Chem. Engng, 32, 35—53. Smith, T.N. (1965). The differential sedimentation of particles of two different species. ¹rans. Inst. Chem. Engng, 43, T69—73. Smith, T.N. (1966). The Sedimentation of particles having a dispersion of sizes. ¹rans. Inst. Chem. Engng, 44, T153—157. Smith, T.N. (1998). A model of settling velocity. Chem. Engng Sci., 53(2), 315—323
A generalized empirical description for particle slip velocities in liquid fluidized beds K.P. Galvin*, S. Pratten, G. Nguyen Tran Lam ARC Centre for Multiphase Processes, Department of Chemical Engineering, University of Newcastle, NSW 2308, Australia Received 27 July 1998; received in revised form 19 November 1998; accepted 19 November 1998
Abstract An empirical equation for calculating the slip velocity of a species in any homogeneous suspension is proposed. The Richardson and Zaki (1954, Trans. Inst. Chem. Engng, 32, 35—53) and Lockett and Al-Habbooby (1973, Trans. Inst. Chem. Engng 51, 281—292; 1974, Powder Technol., 10, 67—71) equations are special cases of the proposed equation, and arise when all species are of the same density. Therefore, the main value of the proposed equation is in describing the slip velocities of particles in suspensions containing species of different density. In this short note results from one experimental system are presented, and shown to be consistent with the model. The model is also consistent with the explanation used by Moritomi et al. (1982) to account for phase inversion in fluidized beds. The model is appealing in its simplicity, and should find favour in the design and control of process equipment. The new model predicts the strong segregation effects observed in suspensions containing particles of different density, and hence represents an immediate improvement on the Lockett and Al-Habbooby equation. Its application is expected to cover all homogeneous suspensions, in which the particles are all more dense than the suspension. At this stage its validation has been limited to low concentrations of dense particles settling through a fluidized bed of low density particles as occurs in a Teetered Bed Separator, and to phase inversion conditions in fluidized beds. It is hoped that this note might lead to a much more extensive validation of the model by workers using vastly different particle species. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Slip velocity; Liquid fluidized beds; Monocomponent fluidization data
1. Introduction A single particle of species i settles in an unbounded fluid at its terminal velocity, º . As the volume fraction RG of the species in the liquid increases, the velocity of the species relative to the liquid decreases. The complex dependence of this slip velocity, » , on the volume fracG tion of the species, , is well described by the empirical G equation of Richardson and Zaki (1954). That is, » "º (1! )LG\, (1) G RG G where n is a constant, typically 4.65 for spherical parG ticles at low Reynolds numbers, less than 0.1. For systems containing particle species which differ according to their
* Corresponding author. Present address: Department of Chemical Engineering, University of Newcastle, Callaghan, NSW 2308, Australia. Tel.: 61 2 4921 6197; Fax: 61 2 4921 6920. E-mail address: [email protected] (K.P. Galvin)
size only, the Lockett and Al-Habbooby equation (1973, 1974) is generally adequate for design purposes. For three particle species, the equation is, » "º (1! ! ! )LG\. (2) G RG G H I Although Lockett and Al-Habbooby claim that their equation is applicable to systems containing species of different density, our experience is that the equation is invalid. Lockett and Al-Habbooby used particle species which did not differ significantly in their densities. More mechanistic models have also been developed, which are concerned primarily with providing a physical understanding of hindered settling rather than an accurate description suitable for process design. Smith (1965, 1966, 1998), for example, developed a cell-model to account for differences in the local volume fraction for each species. He recognized that each species was subject to a common pressure gradient. Batchelor (1972), and Batchelor and Wen (1982) investigated the particle—particle interactions in monocomponent and multispecies
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 4 0 7 - 2
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suspensions, producing linear equations applicable to the low volume fraction limit. Experimental work concerned with suspensions containing particles of different density was generally limited to the study of equilibrium and phase inversion in liquid fluidized beds. Studies included Duijn and Rietema (1982), Moritomi et al. (1982, 1986), and Gibilaro et al. (1986). The strong segregation that develops in these systems, due to the presence of species of different density, led to the view that the buoyancy force is governed by the density of the suspension, rather than the density of the liquid. However, using an energy balance Clift et al. (1987) demonstrated that the buoyancy force is always governed by the density of the liquid. The purpose of this note is to propose a more generalized form of the Richardson and Zaki equation that is useful for all suspensions, whether they are formed using a single species, species of different size and of the same density, or by species of different size and density. In a fluidized bed consisting of a single particle species, the total pressure increases down through the bed according to, P " o gH#(1! )ogH, (3) 2 G G G where o is the density of species i, o the density of the G liquid, g the acceleration due to gravity, and H the height. Hence the total pressure gradient can be written as, dP /dH" (o !o)g#og. (4) 2 G G The pressure gradient consists of two parts, the first due to the weight of the particles in the liquid, and hence the dissipative drag force, and the second due to the hydrostatic head of the liquid. The dissipative pressure gradient is, dP/dH" (o !o)g. (5) G G If we substitute Eq. (5) into Eq. (1), a modified form of the Richardson and Zaki equation is produced. That is,
dP/dH LG\ » "º 1! . (6) G RG (o !o)g G Eq. 6 is identical to both Eqs. (1) and (2), provided the particle species are of one density. Hence, for such systems, Eq. (6) requires no new validation. For a monocomponent system the equation provides the slip velocity as a function of the pressure gradient that arises from the fluid drag. It is argued here that it does not matter how that pressure gradient is formed, whether by particles of one density, or of different densities. For a three species system, for example, Eq. (5) becomes dP/dH" (o !o)g# (o !o)g# (o !o)g. (7) G G H H I I Hence, once the pressure gradient is known or the composition of the suspension is known, the monocompo-
nent expression becomes immediately applicable for describing the slip velocity in any suspension. It should be noted that the attraction of Eq. (6) is its simplicity, and ease of use, together with its potential in process design. The equation may also be useful in process control, given that the pressure gradient is readily measured. It is probable that a more accurate equation would involve a much more complex relationship. Eq. (6) is equivalent to,
o !o LG\ G Q » "º , G RG o !o G
(8)
where o is the density of the suspension. This form of the Q equation indicates that a dimensionless density parameter can be used to describe the hindered settling. The equation also shows that the suspension density will produce a strong segregation effect when particles of different density are present. Clearly, however, this equation is invalid when the density of the species is less than the density of the suspension. It is likely, though, that such suspensions will be unstable, and generate so-called streaming effects, and lateral segregation (Batchelor and Janse van Rensburg, 1986; Davis and Gecol, 1996). These circumstances require an entirely different model. Eq. (6) can also be written in the form of Eq. (2), in terms of the volume fraction of each species. That is, » "º (1!i !i !i )LG\, G RG G G H H I I
(9)
where i "(o !o)/(o !o). H H G
(10)
Coefficients are needed for the volume fraction terms. These either enlarge the volume presence or reduce the volume presence of a given species. For example, when the velocity of a low density species is to be calculated, the volume presence of a denser species is increased by its coefficient until the effective density of the species is reduced to the value of the low density species. This adjustment is equivalent to applying a common pressure gradient to each species present at a given location.
2. Experimental methodology A fluidized bed, shown schematically in Fig. 1, was used to conduct a series of experiments in order to examine the validity of Eq. (6). The vessel was 0.173 m in diameter and 1.36 m high. A feed slurry was fed to the system continuously, producing an overflow stream, and an underflow stream was withdrawn from below. The feed slurry consisted of three particle species, PVC, glass spheres, and magnetite. Details on the monocomponent data of each of these species are given in Table 1, including fluidization results on each species to obtain º and RG
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n . The relatively high values of n obtained for the PVC G G and the magnetite suggest a degree of particle flocculation (Dixon, 1977). The feed entered the system and settled against the upward current of water used to fluidize the bed. The particles segregated into two main zones, with the denser magnetite and glass spheres down below, and the less dense PVC particles above. A steady flux of magnetite and glass spheres passed down through the PVC zone in order to produce the denser zone below. The system was permitted to run until steady state was reached. Samples of the suspension were withdrawn at different positions using a 60 ml syringe attached to tappings on the vessel wall. The samples were then analysed to determine the volume fraction of each species at each location. A differ-
ential pressure transducer was also used to measure the system pressure profile.
Fig. 1. Fluidized bed operating with continuous feed, separating into product and overflow streams. These systems are sometimes referred to as Teetered Bed Separators.
Fig. 2. Steady state volume fraction distribution in the fluidized bed, showing the denser phase above the base to a height of 400 mm, and the less dense phase above.
3. Results The volume fraction distribution for one experiment is shown in Fig. 2 and the overall steady state data is shown in Table 2. In the PVC zone, the magnetite and glass spheres settled at a relatively high velocity, producing the low concentrations is shown. The PVC particles did not
Table 1 Monocomponent data on species used in the feed Species
Density (kg/m)
Diameter (mm)
º (mm/s) R
Exponent n!1
PVC particles Glass spheres Magnetite
1400 2400 5000
0.125—0.180 0.063—0.075 (0.045
8.6 4.2 8.6
8.6 2.7 12.2
Table 2 Steady state experimental data — Run L3 Suspension
Feed Water Underflow Overflow
Volume fractions
Fluxes (mm/s)
PVC
Glass
Magnetite
Total Solids
0.0305 0.000 0.000 0.0229
0.0229 0.000 0.0473 0.0010
0.0301 0.000 0.0781 0.0000
0.0835 0.0000 0.1254 0.0239
1.88 1.28 0.73 2.44
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appear in the dense phase below. A peristaltic underflow pump was used to withdraw the denser particles from below, producing a steady dense phase bed height of 400 mm. In the series of four experiments reported here, the flux of magnetite fed to the system was increased from zero in the first experiment to a higher level in each experiment. This increase was achieved simply by increasing the feed pulp density via an increase in the proportion of magnetite in the feed. All other conditions remained the same, including the fluidization rate. The experimental slip velocity data reported in this study were based on volume fraction data and flux balances conducted in the steady PVC zone 610 mm above the base of the vessel. An example calculation is presented in Appendix A. It is evident in Table 3 that as the magnetite content in the feed increases, the suspension density, calculated using the corresponding pressure gradient data, rises, and the total volume fraction of the solids decreases. The experimental slip velocities of the species actually decrease, with significant reductions evident for the PVC species, and only minor changes for the other species. Table 4 provides a comparison between the experimental slip velocities and the slip velocities predicted using the Lockett and Al-Habbooby equation and the model presented in this paper. As expected, the Lockett and Al-Habbooby model predicts an increase in the slip velocity as the magnetite presence increases, because of the decrease in the volume fraction of the solids. The values predicted using the model proposed in this study, however, show remarkably good agreement with the
experimental data. These predictions were based on the suspension density data in Table 3, which were derived from the pressure gradient data. As shown in Fig. 3, the suspension density values obtained from the pressure gradient data are in excellent agreement with the suspension density data obtained from the volume fraction analysis of the samples withdrawn from the vessel.
Fig. 3. Comparison between the suspension density, obtained from the measured pressure gradient, and the suspension density obtained from an analysis of samples withdrawn from the column. The pressure gradient data were averaged from three sets taken towards the end of the experiment.
Table 3 Particle velocities relative to the water in the lower part of the PVC Phase Feed pulp density
Volume fraction
Suspension density (kg/m)
PVC (mm/s)
Glass (mm/s)
10.8 (No magnetite) 14.7 21.5 26.5
0.177 0.149 0.130 0.120
1082 1078 1105 1119
1.44 1.22 0.74 0.35
3.7 3.4 3.4 3.1
Magnetite (mm/s)
5.9 6.2 5.4
Table 4 Comparison between the particle slip velocities Experimental (mm/s)
Lockett and Al-Habbooby (mm/s)
New model (mm/s)
PVC
Glass
Magnetite
PVC
Glass
Magnetite
PVC
Glass
Magnetite
1.44 1.22 0.74 0.35
3.7 3.4 3.4 3.1
5.9 6.2 5.4
1.6 2.1 2.6 2.8
2.5 2.7 2.9 3.0
0.80 1.2 1.6 1.8
1.20 1.33 0.63 0.41
3.6 3.6 3.4 3.3
6.7 6.8 6.2 5.9
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
3.1. Phase inversion In order to further assess the new model, equilibrium data on mixed fluidized beds were obtained from the paper of Moritomi et al. (1986). The Richardson and Zaki equations for each species, and density values, were also obtained from their 1986 paper. Expressed as slip velocities, the equations produced were: 1. Char » "46.8 (1! ) \ with o "1380 kg/m and velocities in mm/s, (11) 2. Glass » "13.8(1! ) \ with o "2450 kg/m and velocities in mm/s. (12) Equilibrium data is readily calculated using the new model by recognising that the slip velocities for the two species should be identical in a mixed bed. So implementing the new model, and equating the slip velocity for each species, gives
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forms when » "» "8.9 mm/s and o "1212 kg/m. Q This solution is shown in Fig. 4, with the suspension density plotted as a function of the slip velocity for each species. Moritomi et al. (1982) used a similar condition for the mixed bed condition (see Fig. 7 in their paper), based on the proposition of Pruden and Epstein (1964), except that they equated the required superficial velocities rather than the slip velocities. It should be noted, though, that inversion is usually observed to occur over a broad range of superficial fluidization velocities. This observed range is not predicted by this model. Moritomi et al. (1982, 1986) used closely sized species in their experiments. However, the small differences in size may still have contributed to the range of inversion observed. The mixed bed equilibrium data, obtained from Fig. 6 in the paper of Moritomi et al. (1986), are shown together in Fig. 5 of this paper with the prediction based on the proposed model. The equilibrium relationship based on the present model is obtained by equating an expression for the suspension density with the critical density, o "1212 kg/m. That is, Q o "1212"1380 #2450 #1000 (1! ! ). Q (14) Rearranging gives the line in Fig. 5. That is,
o !o L\ o !o L\ Q Q » "º "º (13) R o !o R o !o Clearly, there is a unique value of o needed to solve Eq. Q (13), given that all other variables are known. Using the data in Eqs. (11) and (12), it follows that the mixed bed
"0.146!0.262 . (15) According to the above analysis, the model predicts that the inversion occurs at a given suspension density. The mixed bed composition then depends on the proportion of each species present in the system, and should
Fig. 4. Monocomponent suspension densities shown as a function of the slip velocity, for each species. The mixed bed condition is defined by the intersection of the curves. This data was obtained using Eqs. (11) and (12).
Fig. 5. Comparison between the new model and literature data on phase inversion in fluidized beds (Moritomi et al., 1986).
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Fig. 6. Monocomponent suspension density versus the superficial velocity, for each species. The data was produced using Eqs. (11) and (12), modified to give the superficial velocities, and hence are similar to the data in Fig. 7 of Moritomi et al. (1982).
correspond to the relationship given by Eq. (15). This linear relationship is in reasonable agreement with the mixed bed data shown in Fig. (5). The magnitudes of the errors are comparable to those obtained using the more complex models of Moritomi et al. (1986). Indeed, simply predicting the existence of a range of mixed bed equilibrium volume fractions is a reasonable result. It is worth considering the process of phase inversion by using a plot of the monocomponent suspension density as a function of the superficial fluidization velocity. This data, shown in Fig. 6, is similar to Fig. 7 in the paper of Moritomi et al. (1982). The data, here, was obtained using Eqs. (11) and (12), corrected to give superficial velocities. Hence the exponent n rather than n !1 was G G used in Eqs. (11) and (12). Also, the density values shown with Eqs. (11) and (12) were used. At low superficial velocities, Moritomi et al. (1982, 1986) observed a monocomponent zone of char above a monocomponent zone of glass spheres. The model predicts the first mixed bed condition to occur when the glass zone has a monocomponent volume fraction of 0.146. At this stage, the density of the glass zone must be 1212 kg/m. As shown in Fig. 6, this occurs when the density of the char zone is 1175 kg/m, and the superficial velocity is 7.6 mm/s. In this case the char can enter the glass zone, but only generate an infinitesimal concentration. (Given that the slip velocity in the glass zone is 8.9 mm/s, the superficial velocity must be 8.9 (1!0.146)"7.6 mm/s.) This velocity is significantly higher than the minimum velocity of 5 mm/s obtained by Moritomi et al. (1982, 1986) because the present model predicts a starting concentration of 0.146 which is too low, and predicts a slip velocity which
is too high. Of course, as noted earlier, part of this discrepancy may be due to the existence of a finite range of particle sizes for each species in the experiments. In the study by Moritomi et al. (1982), a mixed bed containing 100 g of glass and 50 g of char was formed. Using the corrected densities in their 1986 paper the ratio, / , will be 1.12. So, using Eq. 15, the new model predicts a fully mixed bed with "0.105 and
"0.118, and hence a total volume fraction of 0.223. The model also predicts a slip velocity of 8.9 mm/s, and hence a required superficial velocity of 8.9 (1—0.223) "6.9 mm/s. According to Fig. 6, the monocomponent char zone has a density of 1180 kg/m and the monocomponent glass zone a density of 1235 kg/m. Presumably, particles near the interface between the char and glass zones gradually create a mixed interface density of 1212 kg/m. This condition then propagates down through the glass zone. So, this model predicts a superficial inversion velocity of 6.9 mm/s, which is lower than the experimental value of 8.4 mm/s shown in Fig. 3 of the paper by Moritomi et al. (1982). It is evident that, despite its simplicity, the proposed model offers a reasonable description of particle slip velocities. However, the model is entirely empirical, and hence its potential must be limited. For example, the model predicts that the fully mixed bed is produced at º"6.9 mm/s. However, to reach such a state, the lower phase must pass through a state involving a high glass concentration ( "0.146) and a low char concentration ( &0). At this point the superficial velocity is 7.6 mm/s, a value slightly higher than that predicted for the fully mixed bed. So arguably the model predicts that to achieve the fully mixed state, it is necessary to initially raise the fluidization velocity to a value higher than 6.9 mm/s to allow the char to move into the glass phase. Once some char is in the glass phase, the fully mixed bed could then be produced by lowering the superficial velocity back to º"6.9 mm/s to achieve the mixed bed state. We have seen no evidence to support this more detailed mechanistic prediction, and hence suggest some caution in using the model. Experiments involving identical particles within a given species would be needed in order to investigate this finer detail. Finally, it is noted that an exceedingly broad range of circumstances, involving particles of different size and density can arise in theory. However, in practical engineering problems the range of circumstances encountered is arguably much more narrow. For example, in batch fluidization of a binary system of particles, the range of conditions capable of producing a mixed bed equilibrium condition is relatively narrow. Moritomi et al. (1986) provides a window of conditions, showing for example that large low-density particles mix with small high density particles. When the particles are of comparable size, the particles should also be of comparable density. Of course, in a Teetered Bed Separator, a broader range of
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
circumstances can arise, because the system is run as a continuous separator. However, even in a continuous separator homogeneous operation involves conditions which do not extend all that far beyond the window provided by Moritomi et al. (1986). It was seen in this study that relatively small high density particles of magnetite settled through a zone of relatively large low density particles of PVC. Consider now the case of a high concentration of nearly neutrally buoyant particles, containing a very low concentration of much denser particles. The model proposed in this study will predict that the dense particles will settle at a velocity close in value to the infinite dilution terminal value. If these dense particles are very much smaller than the low-density particles, such a result is reasonable. The small dense particles will, on average, be many particle diameters from the large low-density particles, and will not be affected significantly by them. This circumstance is similar to that which occurred in this study, and that which occurs in mixed beds. However, if the dense particles are much larger than the low-density particles, they will experience a reduction in velocity. This result is not predicted by the model. The suspension of low-density particles effectively produces a more viscous liquid. A similar resistance will also arise if the two particle species are also of similar size. It is argued, however, that these circumstances are more the exception than the rule. In a continuous separator the low-density particles would simply move upwards through the vessel with the liquid, and would therefore not co-exist with the dense particles at all. If, however, the system is fluidized with a liquid containing these lowdensity particles, then clearly these cases could arise. However, the sensible approach here is to simply ignore the presence of the low-density particles and treat the fluidizing medium as a more viscous liquid. The terminal velocity of the dense particles would then be modified accordingly. In this way, the model can readily be extended to cover these more unusual cases.
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by the liquid in order to support the weight of all the particle species. Implicit in the model is the view that it is not important how the pressure gradient arises. A pressure gradient simply exists, which in turn governs the hindered settling of each species in virtually the same way as it does under monocomponent conditions.
Appendix A. Calculation of slip velocities for Run L3 In order to calculate the slip velocities for Run L3, refer to the steady state data in Table 2. Note that any error in the mass balance is most likely in the feed data and the overflow rate. The fluidization water rate is considered accurate. The total underflow flux of water is 0.73(1— 0.1254)"0.64 mm/s, and hence the upward flux of water through the PVC phase is 1.28—0.64"0.64 mm/s. At a position 610 mm from the base, the total volume fraction of solids is 0.1295, as shown in Fig. 2, and hence the velocity of the upward moving water relative to the vessel is 0.64/(1—0.1295)"0.74 mm/s here. Given that the PVC velocity relative to the vessel is zero, the PVC slip velocity is 0.74 mm/s. The underflow flux of the glass spheres is 0.0473;0.73"0.0345 mm/s. The volume fraction of the glass in Fig. 2, 610 mm from the base, is 0.013, and hence the velocity of the glass relative to the vessel is 0.0345/0.013"2.66 mm/s. So, the slip velocity of the glass spheres is 2.66#0.74"3.4 mm/s. Similarly for the magnetite, with a volume fraction of 0.0104, the slip velocity is 0.0781;0.73/0.0104#0.74"6.2 mm/s.
Acknowledgements The authors acknowledge the financial support of Rio Tinto Research and Technology Development, Novatech, and the Australian Coal Association Research Program (ACARP). Further support from the ARC Centre for Multiphase Processes is also gratefully acknowledged.
4. Conclusions It is proposed that the slip velocity of a species in a multispecies suspension, involving particles of different size and density, can be predicted using the following empirical equation, based on the Richardson and Zaki equation:
dP/dH LG\ . »"º 1! G RG (o !o)g G The only data required is the monocomponent fluidization data for each species. The pressure gradient term is common to all particle species, and is a consequence of the dissipative processes, that is, the drag force produced
Notation g H n G P P 2 º º RG » G
acceleration due to gravity, ms\ height down the column, m exponent pressure due to weight of particles in the liquid, Pa total pressure due to weight of particles and liquid, Pa superficial fluidization velocity, m s\ terminal velocity of species i at infinite dilution, m s\ slip velocity of species, i ms\
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i o o G o H o I o Q
G
H
I
K.P. Galvin et al. /Chemical Engineering Science 54 (1999) 1045—1052
dimensionless density as defined by Eq. 10 density of liquid, kg m\ density of species i, kgm\ density of species j, kg m\ density of species k, kg m\ density of suspension, kg m\ volume fraction of species i volume fraction of species j volume fraction of species k
References Batchelor, G.K. (1972). Sedimentation in a dilute suspension of spheres, part 2. J. Fluid Mech., 52, 245—268. Batchelor, G.K., & Wen, C.S. (1982). Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. numerical results. J. Fluid Mech., 124, 495—528. Batchelor, G.K., & Janse van Rensburg, R.W. (1986). Structure formation in bidisperse Sedimentation. J. Fluid Mech., 166, 379—407. Clift, R., Seville, J.P.K., Moore, S.C., & Chavarie, C. (1987). Comments on buoyancy in fluidised beds. Chem Engng Sci., 42(1), 191—194. Davis, R.H., & Gecol, H. (1996). Classification of concentrated suspensions using inclined settlers. Int. J. Multiphase Flow, 22(3), 563—574.
Dixon, D.C. (1977). Comments on hindered settling theories. Powder ¹echnol., 17, 147—149. Duijn, G.V., & Rietema, K. (1982). Segregation of liquid-fluidized solids. Chem. Engng Sci., 37(5), 727—733. Gibilaro, L.G., Felice, D.I., Waldram, S.P., & Foscolo, P.U. (1986). A predictive model for the equilibrium composition and inversion of binary-solid liquid fluidised beds. Chem. Engng Sci., 41(2), 379—387. Lockett, M.J., & Al-Habbooby, H.M. (1973). Differential settling by size of two particle species in a liquid. ¹rans. Inst. Chem. Engng., 51, 281—292. Lockett, M.J., & Al-Habbooby, H.M. (1974). Relative particle velocities in two-species settling. Powder ¹echnol., 10, 67—71. Moritomi, H., Iwase, T., & Chiba, T. (1982). A comprehensive interpretation of solid layer inversion in liquid fluidized beds. Chem. Engng Sci., 37(12), 1751—1757. Moritomi, H., Yamagishi, T., & Chiba, T. (1986). Prediction of complete mixing of liquid-fluidized binary solid particles. Chem. Engng Sci., 41(2), 297—305. Pruden, B.B., Epstein, N. (1964). Stratification by size in particulate fluidization and in hindered settling. Chem. Engng Sci., 19, 696—700. Richardson, J.F., & Zaki, W.N. (1954). Sedimentation and Fluidization: Part I. ¹rans. Inst. Chem. Engng, 32, 35—53. Smith, T.N. (1965). The differential sedimentation of particles of two different species. ¹rans. Inst. Chem. Engng, 43, T69—73. Smith, T.N. (1966). The Sedimentation of particles having a dispersion of sizes. ¹rans. Inst. Chem. Engng, 44, T153—157. Smith, T.N. (1998). A model of settling velocity. Chem. Engng Sci., 53(2), 315—323