Ind. Eng. Chem. Res. 2004, 43, 7129-7136
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GENERAL RESEARCH On Mixing and Segregation in Binary Solid-Liquid Fluidized Beds Suman Bhattacharyya Oil and Natural Gas Corporation Ltd., Oil and Gas Cell (Surface Group), Mehsana 384003, India
Binay K. Dutta* Department of Chemical Engineering, University of Calcutta, 92, Acharya Prafulla Chandra Road, Calcutta 700009, India
A conceptual effective voidage model of a binary solid-liquid fluidized bed proposed by Bhattacharyya and Dutta (Ind. Eng. Chem. Res. 2002, 41, 5098) and successfully used to interpret the phase inversion behavior has been applied to the mixing and dispersion phenomena. A computational procedure for the determination of the axial concentration profiles of the two types of particles varying in size, density, or both has been developed. The values of the axial dispersion coefficients of the particles are evaluated simultaneously, satisfying the boundary conditions for the fluidized bed. Computed axial concentration profile and dispersion coefficients compared pretty well with available experimental data. Also, the experimental observations on the characteristics of liquid-solid fluidized beds as reported by different workers could be explained using the present model. A correlation for the dimensionless axial dispersion coefficients of the particle species was also developed. The most important feature of the model is that it relies upon the basic properties of the fluid and the solid particles and fitting of experimental axial concentration data are not required. 1. Introduction Liquid-solid fluidization of a mixture of solids is common to many industrial processes. In a fluidized bed having two types of particles there may exist a maximum of three zonessmonocomponent upper and lower zones with an intermediate “mixing or transition zone” containing both types of particles. Convection as well as dispersion of particles determines the structure of the transition zone. The particle velocity or the “convective velocity” of a particle (upi) of an individual species in a bed may be defined as the difference between the interstitial fluid velocity and the interstitial velocity required to produce a bed of same voidage when that species is fluidized on its own.
upi )
uo uo - ur i ) - uti f()
(1)
where upi, uri, and uti are the convective velocity, the particle relative velocity, and the free-fall terminal velocity respectively of a particle of species i; uo is the superficial liquid velocity; is the local bed voidage, which is defined as
)1-
∑i Ci
(2)
Ci is the volumetric concentration of species i. For particles varying in both size and density, Masliyah1 * To whom correspondence should be addressed. E-mail:
[email protected].
proposed the following equation for the convective particle velocity.
up i )
uo Fi - Fm - utini-2 Fi - Ff
(3)
where Fm ) Ff + ∑i CiFi; Ff and Fi are the densities of the fluid and the particle of species i, respectively. The difference in the convective velocities of different types of particles in the same voidage environment gives rise to the tendency of segregation in a binary fluidized bed. On the other hand, the particle dispersion in a fluidized bed leads to a tendency for mixing in the bed. The extent of segregation and mixing in a binary fluidized bed depends on the relative influence of these two opposing modes of transport, that is, dispersion and particle convection. Kennedy and Bretton2 first proposed modeling of a binary fluidized bed on the basis of a balance of convective and diffusive terms at steady state.
Di
dCi ) Ciupi dz
(4)
Di is the dispersion coefficient or the diffusivity of a particle of species i. The above model has been used by many researchers to study the mixing and segregation phenomena in a liquid-fluidized bed and also for the determination of particle concentration profiles along the bed (AlDebouni and Garside,3 Juma and Richardson,4 Gibilaro et al.,5 and Dutta et al.6). The bed voidage, , is probably the single most important factor to determine the characteristics of a
10.1021/ie030681h CCC: $27.50 © 2004 American Chemical Society Published on Web 09/25/2004
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binary solid fluidized bed. All the existing models for predicting the particle concentration profiles in binary solid-liquid fluidized beds are based on the overall voidage model, and consider the fluid-particle hydrodynamic interaction to be governed solely by the overall voidage surrounding a particle, regardless of the relative contribution of each solid species. This is likely to introduce an error in the estimation of the characteristics of such systems, especially when the particle diameters of different species are not in a close range. Moreover, most of the existing models require experimental data fitting for prediction of the particle concentration profiles or dispersion coefficients. In the present work a new theoretical framework of a binary liquid fluidized bed has been suggested on the basis of an “effective voidage model”. It has three basic objectives: to explore the suitability of the “effective voidage model” (Bhattacharyya and Dutta7). to interpret the mixing and segregation phenomena in a binary liquid-solid fluidized bed; second, to suggest an easyto-use computational procedure requiring only the knowledge of operating conditions (superficial liquid velocity and overall bed composition), basic properties of the liquid (density and viscosity) and of the particles (diameters, densities, freefall terminal velocities, and Richardson-Zaki indices) for predicting different system characteristics such as concentration profiles, dispersion coefficients, etc.; and, finally, to obtain a correlation for particle dispersion coefficients in terms of relevant dimensionless groups. 2. The Proposed Model In a previous communication (Bhattacharyya and Dutta7), we proposed a “cell model” of a binary liquidsolid fluidized bed and used it successfully for prediction of the “bed inversion phenomenon”. Expressions for the effective voidage enjoyed by particles of each kind were developed on the basis of possible orientations of a “test particle” and the surrounding “average particles”. In this work we use the same model for prediction of mixing and segregation behavior of a binary mixture of particles. The effective porosities enjoyed by the two types of particles are given by (see Bhattacharyya and Dutta7)
(
1,eff ) 1 - 1 +
[
d1
-3
)1-
{
0.5{(C1 + C2)1/3 - 1} d1 + 1+
(
)
d2,eff d2
-3
) )
(6a)
dC2 uo F2 - Fm D2 ) C2 - (ut22,eff)n2-2 dz F2 - Ff
(6b)
For equal density particles differing in size only the above equations reduce to
Di
(
)
dCi uo ) Ci - uti(i,eff)ni-2 ; dz
i ) 1, 2
(7)
Equations 6a and 6b are the mass balance equations in a binary fluidized bed corrected for the effects of suspension density on buoyant weights of the particles and the effect of particle-particle interaction arising out of the difference in sizes of the different particle species. Solution of the equations provides the axial concentration profiles for different particle species in the fluidized bed. 3. Computational Procedure The mass balance equations (6a) and (6b) may be recast in the following form,
uo F - Fm 2-2 2 - ut2n2,eff C2 F2 - Ff dC2 ) D* dC1 C1 uo F - Fm 2-2 1 - ut1n1,eff F1 - Ff
(8)
where D* is the diffusivity ratio. Solution of the above equation provides all the permissible concentration pairs, which may coexist in a fluidized bed in the specified superficial velocity. A boundary condition for eq 8 may be obtained from the limiting case of a monocomponent bed.
For C1 ) 0, C2 ) Co2
(9)
Because of the indeterminacy at this boundary, application of L’Hospital rule provides the initial gradient required for the solution.
(
dC2 | ) D* dC1 C1)0
uo - ut1(1 - Co2)n1-1
) )
F1 - Ff F2 - Ff (10) F - Ff o 2 1 - C2 F1 - Ff
Co2ut2(1 - Co2)n2-1 n2 - 1 +
(5a)
(
For equal density particles, eq 10 takes the form
-3
)1-
{
0.5{(C1 + C2)1/3 - 1} d2 + 1+
]
}
(C1 + C2)d1d2 C1d2 + C2d1
d1
2,eff ) 1 - 1 +
[
)
d1,eff
( (
dC1 uo F1 - Fm D1 ) C1 - ut1(1,eff)n1-2 dz F1 - Ff
d2
]
}
(C1 + C2)d1d2 C1d2 + C2d1
-3
(5b)
Substituting the overall voidage by the effective voidage in the expressions for the convective velocity, eq 3, the differential mass balance eq 4 takes the form
o n2-1
o
C2ut2n2(1 - C2) dC2 |C1)0 ) D* dC1 u - u (1 - Co )n1 o
t1
(11)
2
In the present case, unlike that in the paper of Gibilaro et al.,5 D* is also a parameter to be evaluated. The permissible concentration pairs are obtained by solving eq 8 in conjunction with eq 10 by an iterative technique based on the fourth order Runge-Kutta method. Starting at C1 ) 0 (and C2 ) Co2), iteration
Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004 7131 Table 1. Physical Properties of the Particles and Fluids solid particle ref.
Fp,
particle
Juma and Richardson4
glass
Dutta et al.6
R.D. Felice12
kg/m3
dp, mm
ut, m/s ×
1.9 2.98 3.7 2.14 3.87
16 26.3 32.9 32.3 48.6
2960
3.5 3.29 3.21 2.49 2.18
fluid
Ft, kg/m3
µ, Nsm-2
paraffin oil
873
10.6 × 10-3
water
1000
1 × 10-3
3000 2500 1950
0.55 0.55 0.92
8.45 6.02 4.59
3.175 3.285 3.28
water
1000
1 × 10-3
lead glass copper zirconia
2900 8800 3800
0.425 0.085 0.7
7.2768 2.4812 15.5444
3.19 4.14 2.83
water
1000
1 × 10-3
C2 ) 0; C1 ) Co1
(12)
A unique value of D* is obtained when the above boundary condition is satisfied in totality. In the calculations, the maximum allowable value for C2 at this boundary have been taken to be of the order of 10-6. The increments in C1 during iteration of eq 8 have been taken to be about 0.0005 to maximize the accuracy in calculation. A locus of permissible concentration pairs in the form of a fifth-order polynomial equation is obtained afterward, which provides flexibility during computation of the concentration profiles. Once C2 is found as a function of C1 and the value of D* is known, the solution of eqs 6a and 6b becomes easier, provided the bed height L is known. Here, the bed height for a particular system was estimated using the serial model of Epstein et al.8
V1 uo A 1ut1
V2 uo A 1ut2
[ ()] [ ()] 1/n1
+
1/n2
(13)
The next step is the solution of the mass balance equations (11a) and (11b) subject to the mass balance constraints,
∫0L Ci dz;
Vi ) A
n
Chalcopyrite glass coke
continues until the following second boundary condition is reached.
L ) Ho1 + Ho2 )
fluid 102
i ) 1, 2
(14)
To start with, the concentration pair (C1 ) Co1, C2 ) 0) is assigned to the location z ) 0. Taking a reasonable guess value of D1, eq 6a is integrated forward until the distance L is reached and if required the zero location is shifted to the next concentration pairs and the process repeated successively until the mass balance constraint (eq 14) is satisfied. If both the mass balance constraints are satisfied simultaneously, we obtain the required concentration profile for both the species as well as the value of the dispersion coefficient D1. The dispersion coefficient of the second species, D2, can be calculated from the chosen value of D*. In the case of the mass balance constraint for i ) 2 not being satisfied, the assumed value of D1 needs adjustment and the abovementioned procedure is repeated until both the mass balance constraints are satisfied. However, eq 6b may also be integrated for the concentration profiles. To obtain accuracy, the increment in z was kept less than 1.5 mm for tall beds and lesser for relatively small beds. Less than 0.5% deviation in V1 and V2 was allowed.
Accuracy of calculation has been found to be highly dependent on increments in z. No maximum or “hook” has been observed in the predicted concentration profiles, unlike in the predictions of Kennedy and Bretton2 or Al Debouni and Garside.3 The input parameters for the numerical solution include the operating conditions (superficial liquid velocity, overall feed composition) and basic properties of the fluid (density and viscosity) and of the particles (diameter, density, free-fall terminal velocity, and Richardson-Zaki indices). No experimental data fitting is necessary. The values of the parameters mentioned above, for different systems studied, were extracted from published literature. In a few cases, particle parameters such as ut and n were not available and were estimated by using standard correlations described in Bhattacharyya and Dutta.7 Physical properties of the fluid and the particles for different systems studied here for the purpose of predicting system characteristics are listed in Table 1. Minimum fluidization velocity, if not available in the literature, was calculated by using correlation of Wen and Yu.9 4. Results and Discussions 4.1. Permissible Concentration Pairs. Figures 1 (particles of the same density) and 2 (particles of different densities) show typical plots of permissible concentration pairs in a fluidized bed. The computed ratios of the dispersion coefficients as well as one set of available experimental concentration pair data are also shown in the figures. For a mixture having particles of different sizes only, the concentration pair map is nearly linear.
Figure 1. Predicted solid concentration relationship for the fluidized system 3 mm (1) and 2 mm (2) glass particles in paraffin oil (b experimental data for uo ) 0.109 m/s and D* ) 1.391, Juma and Richardson4).
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Figure 2. Predicted solid concentration mapping for the fluidized system of 0.425 mm lead glass (1) and 0.085 mm copper (2) particles in water; experimental data from Di Felice et al.12
Figure 4. Comparison of the computed and experimental (Juma and Richardson4) concentration profiles in a mixture of 3 mm (1) and 2 mm (2) glass particles in paraffin oil. Experimental concentration: b total; 4 species 1; 3 species 2. Model prediction: ‚‚‚ Juma and Richardson;4 s present model.
Figure 3. Predicted concentration profiles for a fluidized mixture of 3 mm (1) and 2 mm (2) glass particles in paraffin oil.
Comparison with experimental C1-C2 data is reasonably good. It should, however, be noted that the accuracy of the terminal velocity, ut, and of the Richardson-Zaki index, n, greatly determines how closely the experimental C1-C2 data match the computed results. 4.2. Axial Concentration Profiles. Figure 3 represents a typical computed concentration profile for a mixture of glass particles (2 and 3 mm) fluidized by paraffin oil showing segregated and transition regions. Concentration profiles of the particles in such a mixture computed by using the present model are compared with the experimental data of Juma and Richardson4 and also with the model predictions presented by the same authors in Figure 4. The same for a mixture of 4 and 2 mm glass particles is shown in Figure 5. While the comparison is satisfactory for the former mixture, it is much better for the latter one. This indicates that the effective voidage model is more effective in predicting the characteristics of systems if the particle diameter ration is larger. It is to be noted that the concentration profiles computed by Juma and Richardson4 were based on direct experimental data fitting, whereas the predictions of the present local voidage model do not involve any such data fitting at all. Further computation of concentration profiles in paraffin-fluidized systems of such mixtures of glass particles has also been carried out at a few other superficial liquid velocities (Table 2). Average deviation of predicted values from the experimental observations remain within 4% for the paraffin oil fluidized systems. Predicted and experimental concentration profiles for fluidization of a mixture of 0.55
Figure 5. Comparison of the computed and experimental (Juma and Richardson4) concentration profiles in a mixture of 4 mm (1) and 2 mm (2) glass particles in paraffin oil. Experimental concentration: b total; 4 species 1; 3 species 2. Model prediction: ‚‚‚ Juma and Richardson;4 s present model.
mm glass particles and 0.92 mm coke particles (Dutta et al.6) are shown in Figure 6. 4.3. Axial Dispersion Coefficients and the Effect of Different Parameters Thereon. The computed particle dispersion coefficients in different fluidized systems are listed in Table 2. The effects of different system parameters on the axial dispersion coefficient observed in this work and reported by other workers are discussed below. 4.3.1. Effects of Superficial Liquid Velocity and Voidage. Juma and Richardson4 reported the variation of the dispersion coefficients of glass beads with increasing superficial liquid velocity and the bed voidage when fluidized with paraffin oil. They obtained the values of dispersion coefficients by fitting experimental axial concentration profile data of the particles in their empirical model. Values of the quantity have been computed by using the present model at the experimental conditions mentioned by these workers. Figures 7 and 8 show that the experimentally determined and computed values of the dispersion coefficients for mixtures of glass beads are in excellent agreement. In a mixture of 2 and 3 mm particles (diameter ratio ) 0.6675) the dispersion coefficient increases with the liquid velocity as well as with the bed voidage. But the same for 2 and 4 mm beads
Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004 7133 Table 2. Particle Dispersion Coefficients and Their Ratios Estimated Values present model ref.
Juma and Richardson4
Dutta et al.6
uo m/s‚102
3 mm (1) and 2 mm (2) glass particles in paraffin oil
published data
AV
D1 m2/s‚104
D2 m2/s‚104
D*
D1 m2/s‚104
D2 m2/s‚104
3.7 7.3 10.9 12.5
0.6 0.739 0.832 0.87
9 15.5 26.3 40
6.62856 11.2966 18.9073 28.4091
1.35776 1.3721 1.391 1.408
9.4 15.8 25.9 39.2
6.2 11.1 18.9 29.2
4 mm (1) and 2 mm (2) glass particles in paraffin oil
3.7 7.3 9.3 11 12.5
0.589 0.71 0.758 0.8 0.83
11 24.5 27.7 28 27
7.29951 15.9256 17.7792 17.6745 16.711
1.50695 1.5384 1.558 1.5842 1.6157
9.8 25 28
7.9 18.6 18.8
4 mm (1) and 2 mm (2) particles in water
18.2 25.4
0.727 0.83
16 20
11.8125 15.2335
1.3545 1.3129
15.9 21.2
12.4 20.7
system
0.55 mm chalcopyrite (1) and 0.55 mm glass (2) particles in water
2.16
0.695
41
35.9495
1.14048
36
43
0.55 mm glass (1) and 0.55 mm coke (2) particles in water
1.91
0.724
26
25.1844
1.0324
17
26.5
(diameter ratio ) 0.5) in a mixture increases with the liquid velocity, reaches a maximum, and then decreases
Figure 6. Comparison of the computed and experimental (Dutta et al.6) concentration profiles in a mixture of 0.5 mm glass (1) and 0.92 mm coke (2) particles in water. Experimental concentration: b total; 4 species 1; 3 species 2. Model prediction: ‚‚‚ Dutta et al.;4 s present model.
slowly. Thus, the two mixtures show different behaviors. A larger value of the dispersion coefficient for a mixture is qualitatively indicative of mixing, and a smaller value means segregation or classification. Thus, segregation expectedly plays a more dominant role in a fluidized mixture of 2 and 4 mm particle mixture which is expected from physical considerations. It may be noted that, for a mixture of 2 and 3 mm particles, the depth of the transition region is larger than that of the other mixture under identical conditions. This is also indicative of dominance of mixing over segregation. Moreover, the maximum of the dispersion coefficient occurs at a higher liquid velocity (and therefore at a higher voidage) for the larger particles (4 mm) compared to that of the smaller particles (2 mm) in a mixture. This observation is in agreement with that of Kang et al.10 Conputed values of the dispersion coefficient of the particles of the above mixtures have been presented in Table 2 for a larger range of liquid velocity than used by Juma and Richardson.4 4.3.2. Effect of Liquid Viscosity. Particle dispersion coefficients are found to decrease with a decrease in the liquid viscosity as observed by Juma and Richardson4
Figure 7. Dependence of particle dispersion coefficient on voidage in the transition region. Estimated values (from Juma and Richardson4): 2 4 mm and 9 2 mm glass particles in their mixture; b 3 mm and 1 2 mm glass particles in their mixture. Prediction of the present model: 4 4 mm and 0 2 mm glass particles in their mixture; O 3 mm and 3 2 mm glass particles in their mixture.
Figure 8. Dependence of particle dispersion coefficient on superficial liquid velocity. Estimated values (from Juma and Richardson4): 2 4 mm and 9 2 mm glass particles in their mixture; b 3 mm and 1 2 mm glass particles in their mixture. Prediction of the present model: 4 4 mm and 0 2 mm glass particles in their mixture; O 3 mm and 3 2 mm glass particles in their mixture.
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Figure 9. Predicted bulk density of the bed on volumetric concentration of the larger particles for water-fluidized mixtures of 0.425 mm lead glass and 0.085 mm copper particles.
for the fluidized system of 2 and 4 mm glass particles. In a water-fluidized bed, the particle dispersion coefficients are less than that in paraffin oil. 4.3.3. Effect of Particle Diameter and Density. For fluidized systems having particles varying only in size, the dispersion coefficient of the larger particle species is always greater than that of the smaller one at the same liquid velocity. This observation conforms to that of Juma and Richardson,4 Al Debouni and Garside,3 and Asif and Peterson.11 However, Kennedy and Bretton2 suggested a different trend. Also the dispersion coefficient of a species becomes larger if the size of the coparticles is bigger. This phenomenon may be attributed to the greater effective voidage experienced by particles of a species in the presence of larger coparticles as suggested by the cell model of Bhattacharyya and Dutta.6 However, for a fluidized system of particles varying both in size and in density, the larger particle species may not always acquire a higher dispersion coefficient than the smaller one. The combined effect of size and density of the two particle species decide the relative magnitudes of the dispersion coefficients at a certain superficial liquid velocity. For example, in the water fluidized system of 0.55 mm glass and 0.92 mm coke (Figure 6) the effect of density and size more or less compensate each other and dispersion coefficients are nearly equal at the given velocity, but the higher density glass particles have slightly higher dispersion coefficients. 4.4. The Bed Density. Allowable concentration pairs and associated bed bulk densities are of great help in interpreting qualitatively complex behaviors of systems with particles varying both in size and in density. The maximum bulk density, Fb, in a binary system of particles differing only in size always corresponds to a monocomponent bed consisting of the larger particles. This implies that the larger particles will have a tendency to settle at the bottom to form the bottom monocomponent zone there. But the same may not be true for systems having particles varying both in size and density. The plots of bed bulk density Fb vs larger particle concentration C1, obtained from the predicted solid concentration relation for the water-fluidized system of 0.425 mm lead glass and 0.085 mm copper particles at different superficial velocities, are represented in Figure 9. Plot a in Figure 9 (uo ) 0.012) represents the situation where the maximum bulk density corresponds to the monocomponent copper zone,
whereas plot b (uo ) 0.015) and plot c (uo ) 0.017 m/s) illustrate cases where the maximum bulk density corresponds to a blend of two solid components. Whether the smaller or the larger particle species will form the upper monocomponent zone solely depends on the overall ratio of the solid components in the bed. The range of experimental solid concentration data Di Felice et al.12 for these two higher velocities suggest that the monocomponent copper zone having a lower bulk density forms the upper zone and the bottom layer consists of a blend of both lead glass and copper particles. The above-mentioned behavioral pattern suggests an inversion of the bed between the two lower velocities and this is in agreement with the experimental observations of Di Felice et al.12 4.5. A Correlation for the Dispersion Coefficient. A knowledge of particle dispersion coefficients is of great importance in design or scale-up of a liquid-solid fluidized bed. The data and information derived from this study and the particle dispersion coefficients estimated in this work using the effective voidage model as well as those reported in the literature (Table 2) have been utilized here for developing a correlation that allows estimation of the particle dispersion coefficients. Due to lack of sufficient information on particle dispersion coefficients in fluidized beds containing particles differing both in size and density, the proposed correlation is restricted to fluidized systems in which the particles differ in size only. Dorgelo et al.13 proposed the following correlation for predicting the axial dispersion coefficient in a liquid fluidized bed.
Di ) 0.1uo2
(15)
Kang et al.10 presented a correlation that shows the dependence of the particle dispersion coefficient on the liquid superficial velocity and the minimum fluidization velocity.
Di ) 2.97 × 10-3(uo + umf)0.802
(16)
Both the correlations suggest a strong dependence of the dispersion coefficient in a monodispersed fluidized bed on the liquid superficial velocity. But these do not show the effects of other relevant parameters on the dispersion coefficients. For example eq 15 wrongly assumes the dispersion coefficient to be independent of the particle size. Moreover, both the correlations unrealistically suggest substantial particle mixing even when the liquid superficial velocity is very near the minimum fluidization velocity. Equation 15 also fails to take into account the effect of viscosity on the dispersion coefficient. It is always desirable to have a correlation in terms of dimensionless parameters. Asif and Petersen11 first proposed a correlation for particle dispersion coefficients in terms of dimensionless groups
Fr′ Pe
(
) K1
)
uo - umf ut
K2
(17)
where
Fr′ )
Ff uo2 u od ; Pe ) Fp - Ff gd D
(18)
Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004 7135
K1 ) 7.9 ( 1.1 and K2 ) 2.141 ( 0.054. The correlation coefficient is 0.951 with a 16.4% standard error of prediction. The correlation accounts for important particle and fluid properties but fails to predict the effect of neighboring particles of the other species on the dispersion coefficient of the particles of a given type. For example, it suggests the same dispersion coefficient for a particle species when it is fluidized separately with particles of different diameters. The above correlation is, in fact, suitable only for predicting “self-diffusivities” of the particles in a monodisperse fluidized bed rather than for predicting particle diffusivities or dispersion coefficients in a binary mixture. Considering the different parameters that may influence the particle dispersion coefficients, the following dimensionless groups are selected.
Sc )
2 µf Re2 uo uo ; Fr ) ) ; ; FfD Ar dpg ut i,eff
(19)
where Fr ) two-phase Froude number, Sc ) Schmidt number, Re ) Reynolds number, and Ar ) Archemedes number d2pF2f g/µ2f . For the purpose of correlating the particle dispersion coefficients in fluidized beds, data covering a reasonably broad spectrum of fluid and particle properties is considered in this work. The measurement techniques used by different workers are different. The dispersion coefficients for particles for the fluidized systems studied in the work of Juma and Richardson4 have been estimated with the help of the model presented in this work. Kang et al.10 employed a pressure relaxation method from the histogram of pressure fluctuation in the fluidized bed for qualifying the particle mixing characteristics. Carlos and Richardson14 used a tracer particle detection technique for particle movement in a fluidized bed for evaluating the particle dispersion coefficients whereas Dorgello et al.13 measured the particle dispersion coefficient in a liquid-fluidized bed applying a random walk method. Al Debouni and Garside3 calculated the axial dispersion coefficients of particles in a binary fluidized bed by direct sampling and data fitting in their model. A wide range database used in this work for the purpose of correlating axial dispersion coefficients with important relevant parameters is obtained by different measurement techniques. Therefore, the correlation is expected to be free from any error inherent to the specific measurement techniques. A log-log plot of Fr‚Sc1/2‚ i,eff versus uo - umf/ ut - uo was found to be the most appropriate to correlate the experimental axial dispersion coefficient values (Figure 10). Accordingly, the best correlation is found to be in the following form,
Fr‚Sc1/2‚i,eff ) K1
(
)
uo - umf ut - u o
K2
(20)
where K1 ) 0.02438 and K2 )0.9209. The correlation coefficient is 0.955. The database of the present correlation covers the following range of dimensionless variables:
6.55 e Re e 982.98; 0.0377 e Fr e 3.0731 0.46 e i,eff (or ) e 0.907
Figure 10. Correlation for the particle dispersion coefficient.
The correlation predicts a rise in the axial dispersion coefficient with increasing superficial liquid velocity since a larger liquid velocity increases the turbulence in the bed. It also increases with increasing viscosity of the fluidizing liquid. It may be noted that the terminal settling velocity of a particle decreases with increasing liquid viscosity and the bed voidage decreases. This results in less segregation and a larger dispersion coefficient. 5. Concluding Remarks The effective voidage model of a fluidized bed developed by Bhattacharyya and Dutta7 has been found to be applicable for interpretation and prediction of mixing and segregation behavior of a particle mixture in a fluidized bed. The computed axial concentration profiles and dispersion coefficients closely match the experimental data available in the literature. The effects of various system parameters on the dispersion coefficients of the species in a binary mixture have been analyzed in light of the proposed model. A correlation for the axial dispersion coefficient has been developed in terms of relevant dimensionless groups. The model used in the present work is fully predictive in nature, and no experimental data fitting is necessary for calculating the axial concentration profiles and dispersion coefficients of the particles. Acknowledgment Use of library and other facilities available at the Indian Institute of Chemical Engineers, Dr. H. L. Roy Building, Calcutta 700032, is gratefully acknowledged. Notations C ) volumetric concentration of solid particles, kg/m3 D ) dispersion coefficient, m2/s D* ) ratio of particle dispersion coefficients ()D1/D2) d ) diameter of solid particles, m d,eff ) characteristic linear dimension of the liquid space surrounding a particle as visualized in the “cell model”, m Fr ) Froude number, uo2/dg H ) height of the expended bed, m L ) length of the fluidized section, m n ) Richardson-Zaki index Re ) particle Reynolds number Sc ) Schmidt number upi ) convective particle velocity, m/s uri ) particle relative velocity, m/s ut ) free-fall terminal velocity, m/s Vi ) total volume of the ith particles in the bed, m3
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z ) axial distance along the bed, m uo ) superficial liquid velocity, m/s A ) area, m2 Greek Letters F ) density, kg/m3 Ff ) density of the fluid, kg/m3 ) porosity of the bed Fm ) density of a mixture, kg/m3 µ ) viscosity, kg/m‚s Subscripts i ) ith particle 1,2 ) particles of type 1 and 2 p ) particle eff ) effective f ) fluid
Literature Cited (1) Masliyah, J. H. Hundred settling in a multi-species particle system. Chem. Eng. Sci. 1979, 34, 1166. (2) Kennedy, S. C.; Bretton, R. H. Axial dispersion of spheres fluidized with liquids. AIChE J. 1966, 12, 24. (3) Al-Dibouni, M. R.; Garside, J. Particle mixing and classification in liquid fluidized beds. Trans. Inst. Chem. Eng. 1977, 57, 94. (4) Juma, A. K.A; Richardson, J. F. Segregation and mixing in liquid fluidized beds. Chem. Eng. Sci. 1983, 38, 955. (5) Gibilaro, L. G.; Hossain, I.; Waldram, S. P. On the Kennedy and Bretton model for mixing and segregation in liquid fluidized beds. Chem. Eng. Sci. 1985, 40, 2333-38
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Received for review August 20, 2003 Revised manuscript received July 1, 2004 Accepted July 1, 2004 IE030681H
