5098
Ind. Eng. Chem. Res. 2002, 41, 5098-5108
Effective Voidage Model of a Binary Solid-Liquid Fluidized Bed: Application to Solid Layer Inversion Suman Bhattacharya† and Binay K. Dutta* Department of Chemical Engineering, Calcutta University, 92 Acharya P. C. Road, Calcutta 700 009, India
A conceptual model based on the “effective voidage” around a “test particle” in a bidisperse solidliquid fluidized bed has been proposed and has been used to interpret and predict its phase inversion behavior. The model visualizes the test particle to be surrounded by particles of “average size”. An average size particle has a liquid envelope of average thickness which is different from the thickness of the liquid envelope surrounding the test particle. The effective voidage (eff) enjoyed by a particle of this kind has been expressed as a function of the volume fractions of individual particles in the bed. The two equations for the effective voidage (eff) for the two types of particles can be solved numerically for the local concentration pairs in the bed. The solution leads to the prediction of the “critical velocity” or the “inversion velocity” of the system. It also allows computation of the particle concentration profiles as well as the heights of different sections of the binary solid-liquid fluidized bed. The computed critical velocities for 19 systems studied by different workers are in excellent agreement with reported experimental data. The model is fully predictive and does not necessarily require any experimental bed expansion data for the prediction of the inversion velocity and other parameters. 1. Inversion Phenomenon Liquid-solid fluidization of a mixture of solid particles is involved in industrial processes such as ore beneficiation, ion exchange, fluidized-bed leaching, and crystallization. In some of these processes that aim at elutriating the lighter or the smaller particles, the phenomenon of phase inversion becomes a very important consideration. Liquid fluidization of identical particles generally produces a homogeneously expanded bed, the height of which increases with an increase in the liquid velocity. The characteristics of such a system is well represented by the Richardson-Zaki equation1
uo/ut ) n
(1)
where uo and ut are the superficial liquid velocity and the intercept velocity of the solid particle, respectively, and n is the Richardson-Zaki index for the said particle species. In most practical applications, the solid particles are not of equal size. Density variations may also occur. The expansion characteristics of a fluidized bed of such particles become more complicated. The bed voidage as well as the volumetric concentration of the different particle species varies in the axial direction of the bed; the larger or the heavier particles, as intuition would suggest, move toward the bottom of the bed and tend to form a monocomponent zone there. Similarly, the smaller or the lighter particles form the upper monocomponent zone. Between these two zones, there may exists a zone called the “transition zone” where a gradual variation in the volumetric concentration of the * Corresponding author. E-mail:
[email protected]/
[email protected]. † Present address: Oil and Natural Gas Corp. Ltd., Oil and Gas Cell (Surface), Mehsana Asset KDM Bhavan, Mehsana 384003, India.
particles of both of the species is observed in the axial direction of the bed. The height of such a transition zone generally increases with an increase in the liquid superficial velocity. The concentration profiles in a transition zone may be estimated successfully by using a diffusion-convection model. The situation becomes more complex when the smaller particles are also the denser ones. The opposing compensatory effects of the particle size and density may give rise to a very interesting phenomenon called “solid layer inversion”; the heavier but smaller particles forming the bottom layer at lower velocities may be found to form the top layer at higher velocities. This inversion of the solid layer is preceded by “complete mixing” of the bed. The velocity at which complete mixing of the bed occurs is known as the “inversion velocity” or the “critical velocity”. Another interesting feature of such a system is the formation of only two zones: a monocomponent upper zone and a uniformly mixed bottom zone consisting of both kinds of particles. The first-ever systematic study of this inversion phenomenon had been carried out by Moritomi et al.2 Moritomi et al. fluidized mixtures of glass and hollow carbon char particles with water as the fluidizing liquid. They found the critical velocity or the layer inversion velocity to depend on the bulk bed composition. The composition of the lower zone was a function only of the superficial liquid velocity regardless of the bulk bed composition. The upper layer, on the other hand, consisted of a relatively pure single component. Either glass or char could be the excess material. Inversion was achieved either by fluidizing a fixed blend of the two components at progressively increasing liquid velocity or by maintaining a fixed liquid velocity and progressively changing the bed composition by further addition of one of the solid components. The bottom zone composition remained constant throughout. Further experimental evidence of inverting systems was reported by Van Duijn and Rietema,3 Epstein and
10.1021/ie020116z CCC: $22.00 © 2002 American Chemical Society Published on Web 08/30/2002
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5099
LeClair,4 Moritomi et al.,5 Jean and Fan,6 Gibilaro et al.,7 Di Felice et al.,8 and Funamizu and Takakuwa,9 and their experimental findings corroborated the observations of Moritomi et al.2 One of the most interesting findings of these experiments is the confirmation of the existence of a uniform equilibrium composition of the bottom layer of the bed, consisting of both of the particle species and which is independent of the total amount of solid in the column. This was demonstrated conclusively by Moritomi et al.5 by mechanical sampling of the bottom layer of the fluidized beds for such inverting systems and by Di Felice et al.,8 who obtained a constant axial pressure gradient along the bed. 2. Existing Models for Solid Layer Inversion The phenomenon of bed inversion and modeling efforts for its interpretation have been briefly reviewed by Di Felice.10 One of the approaches of interpretation of the inversion phenomenon is based on the magnitude of the bed layer bulk density. Pruden and Epstein11 and Van Duijn and Rietema3 suggested that the driving force for segregation of different species of particles in a fluidized bed is the difference in the bed bulk densities of their monocomponent beds. The total mixing or the solid layer inversion occurs in a binary fluidized bed when the difference in bed bulk densities attains a zero value, i.e., when the monocomponent bed bulk densities of the different particle species are equal. The necessary and sufficient condition for occurrence of solid layer inversion was given by Epstein and LeClair.4 They also suggested a graphical procedure of determining the critical velocity for a binary mixture of particles with sharp and gradual transition. However, this approach does not explain the observed influence of the bulk solid composition on the inversion velocity or the observed uniform equilibrium composition of the bottom mixed layer. Gibilaro et al.7 extended the above idea and proposed a pressure drop constraint relationship for a binary fluidized system obtained by equating a modified Ergun equation for pressure drop in a mixed bed to the effective weight of the bed at a certain superficial liquid velocity. Numerical solution of the equations of his model provides all of the permissible concentration pairs that can coexist in a fluidized-bed system at a particular liquid velocity. It was further postulated from elementary stability consideration that the bottom mixed layer of the bed would contain the permissible concentration pair possessing the maximum bulk density, which need not necessarily correspond to a monocomponent zone. Unless the overall bed composition corresponds to this unique ratio, there will be an excess of one of the components, which will form a separate monocomponent zone above the mixed layer. The inversion point corresponds to the superficial liquid velocity for which the composition of the maximum bulk density mixture is the same as that of the overall bed composition. This is the first modeling approach toward predicting the bottom layer composition and is capable of explaining the dependence of critical velocity on bulk bed composition. In their latest work, Epstein and Pruden12 modified a simple bed expansion equation by Wen and Yu13 and obtained a relationship between the concentrations of the two species in a binary fluidized bed at a specific velocity. Solution of the above relationship provides all
of the permissible concentration pairs that can coexist in a fluidized-bed system at that velocity. They presented a graphical procedure, involving a plot of the bed bulk density and superficial liquid velocity, following which one can obtain the range of possible inversion velocities and the corresponding bulk densities of the mixed bed at the inversion point, depending on the overall solid composition. However, they could not use the same for calculating the inversion velocities. Patwardhan and Tien14 introduced a concept of apparent porosity or effective voidage experienced by a particle in a binary suspension, taking into account the particle-particle interaction and a modified expression of the settling velocity developed by Masliyah.15 They suggested that inversion of a bed occurred when the velocities of the particles relative to the fluid, i.e., the particle settling velocities of the two particle species, become equal. However, the predictions based on this model are not in conformity with the fact that the critical velocity increases with an increase in the volume fraction of the lighter particles. The equilibrium composition of the bottom layer also cannot be predicted using this model. Jean and Fan6 proposed a strategy for calculating the inversion velocity on the basis of “particle segregation velocity” or the convective velocity. They suggested that the velocity of segregation of the smaller but heavier particles, which they considered controlled the behavior of the bed, attains a peak value at the inversion or the critical velocity, suggesting maximum convective mixing. The model, however, provides no means to predict the equilibrium composition of the bottom mixed layer. On the basis of the force balance relationship of each particle type, Funamizu and Takakuwa9 developed a model to predict the volume fractions of particles in a mixed layer. They introduced a concept of separate hypothetical monocomponent beds for both of the species where the drag force acting on a particle component is equal to that acting on the same component in the mixed layer and presented a modified Richardson-Zaki formula for a mixed bed. The model requires experimental bed expansion data for the mixed bed for each inverting system to determine the magnitude of the exponents, which are important in characterizing particle-particle interaction in a binary fluidized system for predicting the composition of the bottom mixed layer or for predicting the inversion velocity. This model also cannot predict the thickness of different layers in an inverting system. In a later work, Funamizu and Takakuwa16 extended their model using potential energy considerations to make it capable of predicting the thicknesses of the two layers. The objective of the present work is to develop a fully predictive model capable of explaining the characteristics of an inverting system, which shows a complex behavioral pattern arising out of the opposing effects of size and density of different particle species. The model will (a) predict the inversion velocity, (b) confirm specifically the existence of a uniformly mixed zone rather than postulating the same and its position relative to the corresponding single species without setting it as a precondition that the uniformly mixed zone always forms the bottom layer, and (c) predict the equilibrium composition of the mixed layer. The model predictions on inversion or critical velocities for different inverting binary fluidized systems studied will also be compared with the observed values as well as with the
5100
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002
Figure 1. Visualization of the cell model: (a) a single particle with its liquid envelope; (b) the test particle surrounded by average particles;14 (c) an average particle surrounded by like particles; (d) a test particle surrounded by average particles as visualized in the proposed model; (e-I and e-II) limiting approach between adjacent particles as visualized in the proposed model; (f) a test particle with an effective liquid envelope as visualized in the proposed model.
predictions based on other existing models to ascertain its efficacy and standing. 3. The Proposed Model Experimental observations on different inverting systems have confirmed the existence of an equilibrium relationship between the superficial liquid velocity and the composition of the uniformly mixed bottom layer.
Arguably the Richardson-Zaki equation is good for representing the same for a monocomponent bed. An equation capable of representing an equilibrium relationship in a binary solid-liquid fluidized bed, as mentioned above, will be derived here by taking into consideration the following factors: (a) the effect of the suspension density on the buoyant weight of the particles in binary fluidized systems having particles differing in densities; (b) the effect of particle-particle
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5101
interaction arising out of the difference in particle sizes that influences the flow field around a particle in a binary fluidized system. The analogue of the Richardson-Zaki equation to a multispecies suspension was proposed and used by quite a few workers.14,15
uo Fi - Fm ) ni-1 uti Fi - Ff
(2)
The above equation represents an overall voidage model of a polydisperse liquid-solid fluidized bed. It assumes inter alia that the fluid-particle hydrodynamic interaction is a function only of the overall voidage surrounding a solid particle. It fails to predict many of the characteristics and behavior of the bed including solid layer inversion. In a fluidized bed of identical particles, each particle may be visualized to be surrounded by a liquid envelope (Figure 1a) and the flow fields around the particles remain the same on average. However, because of the interaction among the particles in a binary fluidized bed differing in size and/or density, the flow field surrounding a particle becomes more complicated than that in a monocomponent bed. In a mixed bed, a “test particle” of diameter di experiences an environment in which it finds itself effectively surrounded by particles of average diameter, dav. The quantity which is very important in characterizing particle-particle interaction in a fluidized bed is the average distance between neighboring particles. At this point let us express the average particle size17 and twice the liquid envelope thickness14 surrounding a particle as
dav )
1 X2 X1 + d1φ1 d2φ2
)
d1d2φ1φ2(C1 + C2) C1d2 + C2d1
(3)
and
dav ) dav[(1 - )1/3 - 1]
(4)
respectively. Equation 4 suggested by Patwardhan and Tien14 is based on the tacit assumption that a test particle is placed at an “average distance” from the neighboring particles irrespective of its size. Thus, a test particle of diameter di is surrounded by a liquid envelope of thickness dav/2, which is the same as the envelope thickness of an average particle (Figure 1b). This appears to be an approximate picture, which is reasonable when the diameter of the test particle (di) is close to that of an average particle or, in other words, when the particle sizes are not much different. An average particle, when surrounded on all sides by like particles, maintains a liquid envelope thickness of dav/2 and the effective voidage for the average particles becomes equal to the overall voidage because of equal particle-particle interaction from all sides (Figure 1c). However, in a liquid fluidized bed of different types of particles (varying in density, size, or both), if we consider a test particle that has its identity, it finds itself surrounded by particles of all types including its own type. The corresponding distance between the test particle and a neighboring particle is not simple to determine. Here we simplify this picture by assuming
that the neighboring particles have an average diameter but the test particle has its own identity (expressed by its particle size, density, and probably the shape factor). Thus, at an overall voidage , the test particle having a liquid envelope of thickness di/2 may be considered to be surrounded on all sides by average particles with liquid envelope thickness dav/2 where each particle enjoys a voidage , if the particles are confined to their respective cells (Figure 1d). However, when the average particles surround the test particle, the flow field toward the test particle is changed, because of the difference between the liquid envelope thickness of the test particle and that of the average particles. An adjacent average particle faces the test particle only on one side, and hence the liquid envelope thickness of an average particle remains effectively the same on all sides except toward the test particle because the effect of several neighboring average particles nullifies the effect of that of the test particle. The effective voidage enjoyed by the test particle in such a case depends on the relative orientation of the test particle and the adjacent average particles. One of the limiting cases in this regard may be visualized as a situation where the average particles are stationary and the test particle is moving (Figure 1e-I). In this case the test particle enjoys a liquid envelope of thickness di/2 + dav/2, i.e., an envelope of diameter di + (di + dav). The other limiting case may be considered to be a situation where the average particles come in contact with the test particle, which is stationary. Here, the test particle enjoys a liquid envelope thickness of zero value, i.e., experiences a zero voidage (Figure 1e-II). An infinite number of such arrangements is possible between these two limiting cases. The actual picture may be closely simulated by taking the effective thickness of the liquid envelope surrounding the test particle as the average of the two limiting values, i.e., 0.5(di/2 + dav/2). Then the outer diameter of the liquid shell surrounding the test particle would be di + 0.5(di + dav) (Figure 1f). On the basis of the above argument, for particles of species i, we write
di,eff ) 0.5(di + dav)
(5)
Then the effective voidages experienced by particles of species 1 and 2 are
(
1,eff ) 1 - 1 +
[
d1
-3
)1-
{
0.5{(C1 + C2)-1/3 - 1} d1 + 1+
(
)
d2,eff d2
-3
(6a)
-3
)1-
{
0.5{(C1 + C2)-1/3 - 1} d2 + 1+
]
}
(C1 + C2)d1d2 C1d2 + C2d1
d1
2,eff ) 1 - 1 +
[
)
d1,eff
d2
]
}
(C1 + C2)d1d2 C1d2 + C2d1
-3
(6b)
Substituting the overall voidage by effective voidage in eq 2, one may get the modified Richardson-Zaki
5102
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002
equation for a binary fluidized system of particles varying in both density and size.
uo F1 - Fm ) 1,effn1-1 ut1 F1 - Ff
(7)
F2 - Fm uo ) 2,effn2-1 ut2 F2 - Ff
(8)
4. Source of Input Data for Computation
In a fluidized system where the size and density of the particles have compensatory effects on the fluidizability, it is not quite surprising if there exists an equilibrium mixture of particles at concentrations that are dependent on the fluidizing velocity. A solution of eq 6a,b exists for a specified superficial liquid velocity only when the fluidized system possesses a uniform equilibrium composition at that velocity. Therefore, simultaneous solution of the above equations provides the equilibrium volumetric concentrations C1e and C2e for a binary fluidized bed or for the section of the fluidized bed where a uniform equilibrium composition exists at the specified superficial liquid velocity, which is an essential feature of an inverting system. Hence, total mixing or inversion may occur in a fluidized bed at a specific liquid velocity, called the critical velocity or the layer inversion velocity, only when (a) a solution of eq 6a,b exists when the following condition (eq 9) is satisfied and (b) the overall composition of the bed is the same as that of the equilibrium solid components ratio obtained by solution of eq 6a,b.
ut11,eff
F n1-1 1
- Fm F2 - Fm ) ut22,effn2-1 F1 - Ff F2 - Ff
particles above the mixed layer. The model predicts a maximum of two zones for an inverting system, which is also supported by experimental observations. The model allows a quantitative prediction of the complex expansion characteristics of an inverting system.
The proposed model is fully predictive in nature and requires only the knowledge of the operating conditions (superficial liquid velocity and overall feed composition) and basic properties of the liquid (density and viscosity) and of the particles (diameter, density, freefall terminal velocity, or the “intercept velocity”, the Richardson-Zaki index) and does not require any experimental data fitting. The values of the parameters mentioned above, for different systems studied, were extracted from the published literatures. However, because of the nonavailability of monocomponent bed expansion data or the intercept velocities (ut) of the individual particles, for the systems studied by Gibilaro et al.7 and for some of the systems studied by Moritomi et al.2, the intercept velocity of a particle species was obtained first by calculating the terminal velocity uT using the correlation for CD cited by Clift et al.18
CD )
(10)
and then correcting the same for use in the RichardsonZaki correlation1
log uT ) log ut +
(9)
Unless the overall composition matches exactly to the solid component ratio obtained from the solution of eq 6a,b, there will be an excess of one of the solid components that will form a monocomponent layer. The predictions of the model, discussed in a later section, indicate an increase in the critical velocity for a fluidized system with an increase in the volume fraction of the lighter particles. Hence, for an inverting system at a velocity below the inversion velocity, there will be an excess of the lighter component, which will form a monocomponent layer. The bulk density of the uniformly mixed layer is always higher in this case than that for the monocomponent layer of the lighter particle. This suggests, as per stability criteria, the formation of a monocomponent layer of lighter particles above the mixed layer. For velocities above the critical velocity, there will be an excess of the heavier particle species that will form a monocomponent layer. In this study it is not set as a precondition that the excess material of the heavier particle species will form the upper monocomponent layer. The difference in the suspension densities of the uniformly mixed layer and the monocomponent layer formed by the heavier material will decide their relative arrangement as per the bed stability criteria. This makes the model more flexible toward predicting system characteristics of a binary fluidized system in which a uniform mixed layer exists. For all of the systems studied, the predictions of the model, however, always indicated a higher bulk density of the mixed zone than that for the monocomponent zone at higher velocities than the critical velocity, thereby suggesting an upper monocomponent layer of denser
24 (1 + 0.14ReT0.7) ReT
dp Dc
(11)
where Dc is the diameter of the fluidizing column and dp is the particle diameter. Richardson-Zaki index n for a particle of a species, whenever not available in the literature, was calculated using the most widely used Richardson-Zaki1 correlation, which is given by
(
)
dp n ) 4.45 + 18 ReT-0.1 Dc
1 < ReT < 200
(12)
Physical properties of fluid and particles for different systems studied here for the purpose of predicting system characteristics are listed in Table 1. 5. Results and Discussion 5.1. Layer Inversion Velocity or the Critical Velocity. Predictions based on the present model, obtained by numerical solution of eq 6a,b, of the layer inversion velocity or critical velocity for 19 different inverting systems are presented in Table 2 along with the experimental observations and predictions based on other models. The predictions based on the present model have been found to agree with the experimental observations of Moritomi et al.2 and of others, that the critical velocity is dependent on the bulk bed composition and increases with an increase in the volume fraction of the lighter but larger particles. The same is apparent from eq 6a,b because the effective voidage experienced by particles of both species becomes higher because of an increase in the effective voidage experienced by average particles. The above increase in the volume fraction of the lighter particles also leads to an
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5103 Table 1. Physical Properties of the Particles and Fluids for the Systems Studied in the Present Worka solid particle properties
fluid properties
ref
particle
Fp, kg/m3
dp, mm
Jean and Fan.6
AC GB GB163 GB214 HC775 HC460 HC385 zirconia copper glass ballotini
1509 2510 2450 2450 1500 1500 1500 3800 8800 2730
0.778 0.193 0.163 0.214 0.775 0.46 0.385 0.7 0.135 2.28
nickel ballotini Ni-coated glass alundum C500
8900 2910 4500 3950 1386
0.456 1.079 0.542 0.645 0.448
0.707 0.899 0.481 0.607 2.428
4.75 5.13 4.84 5.36 3.65
G130
2476
0.14
1.3028
3.87
Moritomi et al.2
Gibilaro et al.7 Epstein and Le Clair4
Funamizu and Takakuwa9 a
uT, m/s × 102 5.44 2.05 2.4926 3.0514 2.4213 15.7123 5.04347
ut, m/s × 102
n
3.7 1.75 1.531 2.4681 5.1286 2.9875 2.3789 15.7123 5.04347 2.64
3.851 4.251 3.175 3.829 3.466 3.544 3.6705 2.78 3.67 4.59
F f, kg/m3
µ, N‚s/m2
water
1000
1.0 × 10-3
water
1000
1.0 × 10-3
water
1000
1.0 × 10-3
polyethylene glycol aqueous solution
1070
0.14
water
1000
1.0 × 10-3
fluid
C, AC: activated carbon. G, GB: glass beads. HC: hollow char.
increase in the density correction terms in said equations. Computed results also indicate an increase in the critical velocity with the increase in diameter of the smaller but heavier particle species and the diameter of the lighter but larger particle species, with the solid component ratio remaining constant. The reverse is true when the diameter of the lighter but larger particle species is increased. These predictions are also in agreement with the experimental observations of Moritomi et al.2 Predictions of critical velocities based on the proposed model are in excellent agreement with the experimental observations on most of the inverting systems with the wide range of density and diameter ratios studied (Table 2). The only significant deviation of the predicted result (28%) has been observed for the fluidized system of 1.079 mm ballotini and 0.645 mm alundum particles in poly(ethylene glycol) studied by Epstein and LeClair.4 However, this particular system showed a fuzzy, as opposed to a sharp, inversion involving a large velocity range over which a large mixed layer existed. The above deviation may be partly a result of the effect of dispersion that has not been taken into account in the present work to avoid complicating the problem and partly a result of the nonuniformity of the alundum particles used in the experiments. In most cases the deviation has been found to be within (10%. The average deviation of the predicted values from the experimental results remains within 7%. Model predictions of Jean and Fan6 fair well for fluidized systems of binary particle mixtures with higher particle diameter and density ratios when the experimental bed voidage is known and show a sharp transition during inversion. When applied to all of the 19 systems considered in this work, the critical velocity predicted by their model shows an average deviation of about 20% and a maximum deviation of 119% from the experimental values. Gibilaro’s model7 works better; the maximum deviation remains within (22%, and the average deviation is about 12%. The model of Patwardhan and Tien,14 though capable of accounting for the effect of bulk bed composition on the critical velocity, is unable to show that the critical velocity increases with an increase in the volume fractions of the lighter particles. This may be a result of the assumption in their
cell model, as discussed earlier, that the thickness of the liquid envelope surrounding the test particle is the same as that for an average particle. This leads to the overprediction of the effective voidage enjoyed by the smaller particles and the reverse for the larger ones, thereby introducing inaccuracy in the estimation of the particle relative velocities or the settling velocities. The model of Epstein and LeClair,4 on the other hand, predicts the critical velocity to be independent of the bulk bed composition (Table 2), which is in conflict with the experimental observations of Moritomi et al.,2 Jean and Fan,6 Gibilaro et al.,7 and others. This model, based on the equality of the monocomponent bed bulk densities of the different species at the inversion point, visualizes complete segregation of the bed before and after the total mixing and suggests a singular inversion velocity for an inverting system which is independent of the bulk bed composition. The model also neglects the effect of particle-particle interaction in a fluidized bed, which is dependent on the volume fraction of different species in the bed and also on their sizes. This approach is also restricted to the situation when the particle flow regimes for the different particle species are identical. Equation 2, based on the overall voidage model and the model of Masliyah,15 in principle, is expected to be able to predict the inversion velocity and the equilibrium composition of the bottom mixed layer in an inverting system. However, its predictions are found to be far away from the observed values, and in some cases the model fails to predict any inversion velocity (Table 2). Predictions based on the model of Funamizu and Takakuwa9 are dependent on experimental results and therefore are not considered for comparative study in the present work. The model developed in the present work has been found to perform much better than the existing models described above for predicting the critical or inversion velocities in most of the cases studied. Figure 2 presents a comparison of the values of the inversion velocity as predicted by the models of Jean and Fan,6 Gibilaro et al.,7 Patwardhan and Tien,14 and Epstein and Pruden12 and by the proposed model against the observed values reported by these researchers. 5.2. Composition of the Bottom Mixed Layer and Heights of Different Zones Formed. The predicted
HC775-GB163-water HC775-GB163-water HC775-GB163-water HC775-GB163-water HC775-GB214-water HC385-GB163-water HC460-GB163-water glass ballotini- nickelpolyethylene glycol ballotini-Ni-coated glasspolyethylene glycol ballotini-alundumpolyethylene glycol zirconia-copper-water
C500-G130-water
Moritomi et.al.2
Funamizu and Takakuwa16
Gibilaro et
al.7
Epstein and Le Clair4
AC778-GB193-water
system
Jean and Fan6
ref
0.639
1.67
3.2
0.261
0.359
0.536
1.99
5.185
0.212
0.345
0.337
(F1 - Ff)/ (F2 - Ff)
4.75 4.75 4.75 4.75 3.62 2.36 2.82 5
4.03
d1/d2
1.564 1.787 0.78 0.83 0.89 0.93
0.33 0.5 0.6
0.18
0.223
0.99 1.232 0.91 0.7 0.84 0.917 0.983 1.25 1.216 1.133 0.445
expt
0.2831 0.4968 0.2
0.493
0.618
0.48 0.85 0.16 0.164 0.37 0.495 0.578 0.495 0.495 0.495 0.74
vol fraction of lighter particles, X1
0.88 0.89 0.91
1.6816 1.741 0.87
0.23
0.222
1.035 1.222 0.939 0.762 0.7935 0.8218 0.845 1.416 1.166 1.0213 0.435
present work
6.02 0 -2.15
7.52 -2.57 11.54
0.711 0.761 0.798
2.1445 2.375 0.681
0.3947
0.3252
-0.45 27.78
0.973 1.235 0.86 0.691 0.748 0.796 0.831 1.243 1.036 0.918 0.4397
Jean and Fan6
4.5 -0.81 3.18 8.86 -5.53 -10.4 -14 13.28 -4.11 -9.86 -2.25
% dev
-14.3 -14.5 -14.2
37.11 32.9 -12.7
119.3
1.65 1.87
0.186
0.18
1.05 1.5 0.82 0.596 0.695 0.745 0.794 1.43 1.234 1.069 0.36
-1.7 0.24 5.49 -1.28 -11 -13.2 -15.5 -0.56 -14.8 -19 -1.19 45.83
Gibilaro et al.7
% dev
5.5 4.64
3.33
-19.3
6 21.75 9.89 14.86 -17.3 -18.8 -19.2 14.4 1.48 -5.65 -19.1
% dev
uc (critical velocity), m/s × 102
1.262 1.333 1.368 0.842 0.836 0.811 0.785 1.46 1.244 1.1
Patwardhan and Tien14
Table 2. Comparison of Critical Velocities for Solid Layer Inversion between Experimental Results and Predictions Based on Different Models
0.248
0.497
1.196 1.196 1.196 0.889 0.889 0.889 0.889
Epstein and Pruden12
eq 2 0.342 0.08 fails
5104 Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5105 Table 3. Equilibrium Composition of the Bottom Mixed Layer at Different Superficial Liquid Velocities along with the Suspension Densities of the Mixed Layer and the Monocomponent Upper Layer for a Water-Fluidized Bed System of 0.778 mm Activited Carbon (0.3075 kg) and 0.193 mm Glass Beads (0.5533 kg) X1e
uo, m/s × 102
C1e
C2e
e
Fm, kg/m3 × 103
Fb1, kg/m3 × 103
0.050 000 0.200 000 0.300 000 0.400 000 0.480 000 0.550 000 0.650 000 0.750 000 0.800 000 0.850 000 0.950 000 0.980 000
0.916 000 0.950 000 0.974 500 1.005 000 1.034 500 1.062 000 1.110 000 1.160 000 1.191 000 1.221 500 1.290 000 1.300 000
0.007 263 0.030 375 0.047 186 0.605 067 0.080 335 0.094 600 0.116 443 0.141 300 0.155 600 0.171 264 0.209 000 0.225 400
0.138 000 0.121 500 0.110 500 0.097 600 0.087 030 0.077 400 0.062 700 0.047 100 0.038 900 0.030 223 0.011 000 0.004 600
0.854 737 0.848 125 0.842 314 0.837 333 0.832 635 0.828 000 0.820 857 0.811 600 0.805 500 0.798 513 0.780 000 0.770 000
1.212 100 1.198 900 1.190 900 1.180 500 1.172 300 1.165 000 1.153 900 1.143 000 1.137 900 1.132 800 1.123 000 1.121 700
1.154 400 1.146 200 1.136 600 1.114 200
Fb2, kg/m3 × 103
1.047 900 1.047 800 1.044 800 1.042 400 1.040 100 1.034 900 1.034 200
Figure 2. Comparison of observed critical velocities with the predictions of various models including the present one.
equilibrium composition of the bottom mixed layer at different superficial liquid velocities for an inverting binary water-fluidized bed of a mixture of glass beads and carbon particles (AC778-GB193-water) having 48 vol % of the lighter particles6 is presented in Table 3. Layer inversion takes place at a water velocity at which the equilibrium solid component ratio equals that in the feed (i.e., a well-mixed bed). In the above case, as per prediction of the present model, the inversion occurs at a liquid velocity of 1.0345 × 10-2 m/s when the equilibrium volume fraction of the lighter particles (X1e ) 0.48) equals that in the feed. The observed inversion velocity in this case is 0.99 × 10-2 m/s. The above table also incorporates the suspension density of the uniformly mixed layer and that of the monocomponent upper layer. In all of the cases, the suspension density of the uniformly mixed layer has been found to be more than that of the monocomponent upper layer. This suggests, as mentioned earlier, a monocomponent layer formed by the excess material of one of the solid species above the mixed layer. The lighter particles form the upper monocomponent layer at velocities below the critical one, while the heavier ones form the upper monocomponent layer above the critical velocity. Jean and Fan6 presented experimentally measured heights of different zones of the fluidized bed but did not make any attempt to apply their model to predict these quantities. However, as Figure 3 shows, the proposed model is capable of predicting these quantities. The bottom or mixed zone heights and the total heights, represented by hbo and hT in the figure, converge to a single point at the critical or layer inversion velocity and
Figure 3. Comparison of the observed heights of different layers of a bed with the values predicted by the proposed model [system: 0.778 mm diameter active carbon particles (0.3075 kg)-0.193 mm glass beads (0.5533 kg)-water]: (a) observed heights (Jean and Fan6); (b) predicted values.
diverge rapidly thereafter. The bottom height reaches a maximum and the upper monocomponent zone height reaches a zero value at the inversion velocity, indicating total mixing of the bed. Beyond the velocity range for which the equilibrium uniform composition exists in the bottom layer, the top and bottom zones are monocomponent in nature. The denser but smaller size glass particles form the bottom monocomponent zone at lower velocities and the upper monocomponent zone at higher velocities beyond the range mentioned above, and the reverse is the case for the lighter but larger particles. Our computed results (not presented here) show that the proposed model satisfactorily predicts the bed height data of all of the systems studied by Jean and Fan.
5106
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002
Table 4. Equilibrium Composition of the Bottom Mixed Layer of Different Superficial Liquid Velocities along with the Suspension Densities of the Mixed Layer and the Monocomponent Upper Layer for a Water-Fluidized Bed System of 0.775 mm Hollow Char (0.03 kg) and 0.163 mm Glass Beads (0.1 kg) X1e
uo, m/s × 102
C1e
C2e
e
Fm, kg/m3 × 103
Fb1, kg/m3 × 103
0.050 000 0 0.164 000 0 0.200 000 0 0.300 000 0 0.370 000 0 0.400 000 0 0.495 000 0 0.578 000 0 0.700 000 0 0.850 000 0 0.950 000 0
0.751 000 0 0.762 000 0 0.766 600 0 0.781 000 0 0.793 500 0 0.799 000 0 0.821 800 0 0.845 000 0 0.891 000 0 0.965 000 0 0.027 000 0
0.010 326 3 0.035 587 6 0.044 097 5 0.069 192 9 0.088 154 0 0.096 667 0 0.125 269 0 0.152 581 0 0.197 797 0 0.265 993 0 0.326 420 0
0.196 200 0 0.181 410 0 0.176 390 0 0.161 450 0 0.150 100 0 0.145 000 0 0.127 800 0 0.111 400 0 0.084 770 0 0.046 940 0 0.017 180 0
0.793 473 7 0.783 002 4 0.779 512 5 0.769 357 1 0.761 746 0 0.758 333 0 0.746 510 0 0.736 019 0 0.717 433 0 0.687 067 0 0.656 400 0
1.289 600 0 1.280 800 0 1.277 800 0 1.268 700 0 1.261 700 0 1.258 600 0 1.247 500 0 1.237 800 0 1.221 800 0 1.201 000 0 1.188 100 0
1.211 600 0 1.206 700 0 1.203 000 0 1.178 200 0
Figure 4. Comparison of the observed heights of different layers of a bed with the values predicted by the proposed model [system: 0.775 mm diameter hollow char particles (0.03 kg)-0.163 mm glass beads (0.1 kg)-water]: (a) observed heights (Moritomi et al.2); (b) predicted values.
The experimental results of Moritomi et al.2 on different zone heights for the water-fluidized bed system of 0.775 mm hollow char (0.03 kg) and 0.163 mm glass beads (0.1 kg) are presented in Figure 4, and also shown are the predictions based on the present model. Table 4 represents the equilibrium composition of the bottom mixed layer at different superficial liquid velocities for the system studied. The inversion of the bed is predicted at a liquid velocity of 0.7935 × 10-2 m/s, versus the experimental value of 0.84 × 10-2 m/s. The monocomponent bed heights of the glass particles for the same amount as exists in the feed (hGB°) are found in Figure 4a to be larger than the bottom mixed heights (hbo) before the layer inversion point, and the
Fb2, kg/m3 × 103
1.232 400 0 1.084 900 0 1.082 600 0 1.076 500 0 1.066 700 0 1.058 700 0
two merge at a point close to the layer inversion point. The reverse is found to be true for the results reported by Jean and Fan.6 Moreover, the total bed height (hT) and hGB° are about the same after the layer inversion point in Moritomi et al.’s work, whereas hT is considerably higher than hGB° in Jean and Fan’s work. Predictions based on the present model support the experimental observations of Jean and Fan. This discrepancy may be due to the nonuniformity of particles used in the work of Moritomi et al.2 and may also be due to the uneven flow distribution or difference in flow patterns arising out of the use of a perforated distributor plate in Moritomi et al.’s work versus the more efficient porous plates used by Jean and Fan. The deviations observed in different model predictions on inversion velocities from the experimental observations of Moritomi et al. may also be a result of the factors mentioned above. Figure 5 shows the effect of bulk bed composition on the pre-inversion overall and post-inversion mixed bed heights. Critical velocity, as mentioned before, is found to increase with an increase in the volume fraction of the lighter particles. The minor deviations in the predicted bed heights from the observed values may be attributed mainly to the nonuniformity of the particles used and to some extent to the mixing at the phase boundary. The discontinuity in the gradient of the mixed bed height beyond the inversion velocity corresponds to the point at which the bottom zone becomes a monocomponent zone consisting of lighter particles. Apart from the model of Gibilaro et al.,7 the present model is the only one that is capable of predicting the equilibrium composition of the uniformly mixed layer and the thickness of different layers in an inverting binary fluidized bed without the help of any experimental data. However, the present model is the only model that is fully predictive in nature in its true sense for a binary fluidized system of particles varying in both size and density. It was postulated in the model proposed by Gibilaro et al.,7 from elementary stability consideration, that the bottom mixed layer of the bed would contain the permissible concentration pair possessing the maximum bulk density. The model cannot specifically predict the existence of a uniformly mixed layer in a binary fluidized bed. It is not always necessary that the uniformly mixed layer consisting of both of the particles will have a maximum bulk density. A situation may arise when a uniformly mixed layer exists in a bed, the maximum bulk density corresponds to a monocomponent layer, and the said model will fail in such a situation. The model developed in the present work, as discussed earlier, is more flexible in this regard.
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5107
axial concentration profiles of both of the solid species. Interestingly, the model need not rely upon experimentally measured values of the terminal velocities of the particles or the values of the Richardson-Zaki index. Comparison of the model prediction with experimental data for 19 systems reported by various workers as well as with the predictions of the existing models establishes the effectiveness of the proposed model. The average deviation of the predicted inversion velocities remains within 7% of the experimental values. The proposed cell model has also been successfully used to predict the dispersion phenomenon in a binary solidliquid fluidized bed and will be reported in another paper. Notation
Figure 5. Effect of bulk bed composition on the critical velocity [system: 0.775 mm hollow char-0.163 mm glass beads-water]: (a) experimental results of Moritomi et al.;2 (b) values predicted by the proposed model.
In the present work the effects of dispersion are not taken into account because doing so would have made the problem more intractable. Dispersion, nonetheless, may have a significant effect on bed inversion. Bed inversion is possible over a very limited range of values of particle density and size under conditions where the effects of dispersion might be expected to be large compared to those attributable to the difference in bed densities. Sphericity of particles is an important parameter necessary for the calculation of the average particle diameter of a bidisperse system. However, as a matter of fact, values of this parameter were not reported in the experimental works cited here and used for comparison with our computed results. We have assumed the sphericity to be unity for all particles. This may be one of the reasons behind the deviation of the computed results from the observed values. 6. Concluding Remarks The proposed model of a bidisperse solid-liquid fluidized bed provides a simple, useful, and effective technique of predicting the phase inversion phenomenon. The model predicts the inversion velocity, the heights of the mixed and the monocomponent zones of an inverting system, and the nondispersive equilibrium
C ) volumetric concentration of solid particles in a binary fluidized bed C° ) volumetric concentration of a solid particle species in a monocomponent fluidized bed or in a monocomponent layer CD ) drag coefficient, for a single particle system d, dp ) diameter of solid particles, mm dav ) average particle diameter in a binary fluidized bed, mm d ) characteristic linear dimension of the liquid space surrounding a particle as visualized in a cell model, mm Dc ) internal diameter of the fluidizing column, mm h° ) height of the monocomponent bed formed by a solid particle species, m hbo, hup, hT ) heights of the bottom layer, upper monocomponent layer, and total bed, respectively, in an inverting fluidized bed, m n ) Richardson-Zaki index ReT ) terminal Reynolds number, Ffuodp/µf uc ) critical or layer inversion velocity, m/s uo ) superficial liquid velocity, m/s uT ) freefall terminal velocity of a particle, m/s ut ) intercept velocity, i.e., the extrapolated liquid velocity as the bed voidage approaches unity obtained generally from the log(uo) vs log() plot based on the monocomponent bed expansion data, m/s W ) weight, kg X ) volume fraction of the solid particle Greek Letters ) fractional voidage or porosity µf ) viscosity of the fluid, Ν‚s/m2 Fb ) bulk density of a monocomponent bed, kg/m3 Fi ) particle density of species i, kg/m3 Ff ) density of a fluid, kg/m3 Fm ) suspension density of a binary fluidized bed, kg/m3 φ ) sphericity of a particle Subscripts 1 ) larger but lighter particle species 2 ) smaller but heavier particle species av ) average c ) at critical condition e ) at equilibrium condition eff ) effective value f ) fluidizing medium i ) particle type GB ) glass bead HC ) hollow char AC ) activated carbon
5108
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002
M ) moving S ) stationary
Literature Cited (1) Richardson, J. F.; Zaki, W. N. Sedimentation and fluidization, Part 1. Trans. Inst. Chem. Eng. 1954, 32, 35. (2) Moritomi, H.; Iwase, T.; Chiba, T. A comprehensive interpretation of solid layer inversion in liquid-fluidized beds. Chem. Eng. Sci. 1982, 37 (12), 1751. (3) Van Duijn, G.; Rietema, K. Segregation of liquid-fluidized solids. Chem. Eng. Sci. 1982, 37, 727. (4) Epstein, N.; LeClair, B. P. Liquid fluidization of binary particle mixtures - II. Bed inversion. Chem. Eng. Sci. 1985, 40, 1517. (5) Moritomi, H.; Yamagishi, T.; Chiba, T. Prediction of complete mixing of liquid-fluidized binary solid particles. Chem. Eng. Sci. 1986, 41, 297. (6) Jean, R.-H.; Fan, L. S. On the criteria of solid layer inversion in a liquid-solid fluidized bed containing a binary mixture of particles. Chem. Eng. Sci. 1986, 41, 2811. (7) Gibilaro, L. G.; Di Felice, R.; Waldram, S. P. A predictive model for the equilibrium composition and inversion of binary solid-liquid fluidized beds. Chem. Eng. Sci. 1986, 41 (2), 379. (8) Di Felice, R.; Gibilaro, L. G.; Foscolo. P. U. On the inversion of binary solid-liquid fluidized beds. Chem. Eng. Sci. 1988, 43, 979. (9) Funamizu, N.; Takakuwa, T. An improved Richardson-Zaki formula for computing mixed layer composition in binary liquidsolid fluidized beds. Chem. Eng. Sci. 1995, 50, 3025.
(10) Di Felice, R. Hydrodynamics of liquid fluidization. Chem. Eng. Sci. 1995, 50, 1213. (11) Pruden, B. B.; Epstein, N. Stratetification by size in particulate fluidization. Chem. Eng. Sci. 1964, 14, 696. (12) Epstein, N.; Pruden, B. B. Liquid fluidization of binary particle mixturessIII. Stratification by size and related topics. Chem. Eng. Sci. 1999, 54, 401. (13) Wen, C. Y.; Yu, Y. H. Mechanics of fluidization. Chem. Eng. Prog., Symp. Ser. 1966, 62, 100. (14) Patwardhan, V. S.; Tien, C. Sedimentation and liquid fluidization of solid particles of different sizes and densities. Chem. Eng. Sci. 1985, 40, 1051. (15) Masliyah, J. H. Hindered settling in a multiple-species particle system. Chem. Eng. Sci. 1979, 34, 1166. (16) Funamizu, N.; Takakuwa, T. A minimal potential energy model for predicting stratefication pattern in binary and ternary solid-liquid fluidized beds. Chem. Eng. Sci. 1996, 51, 341. (17) Carman, P. C. Flow of Gases Through Porous Media; Butterworth Scientific Publications: London, 1956. (18) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and particles; Academic Press: New York, 1978.
Received for review February 8, 2002 Revised manuscript received June 20, 2002 Accepted June 20, 2002 IE020116Z
