1919
Ind. Eng. Chem. Res. 1995,34, 1919-1925
Bulk-Density Distributions of Solids in the Freeboard of a Gas-Solid Fluidized Bed Bao-Chun Shen,+L. T. Fan,* and Walter P. Walawender Department of Chemical Engineering, Durland Hall, Kansas State University, Manhattan, Kansas 66506
The freeboard region above the bubbling zone of a gas-solid fluidized bed provides the space not only for the disengagement of particles but also for additional contact and reaction between the particles and gas. The flow pattern and behavior of particles as well as their bulk-density distribution in the freeboard have a significant impact on the efficiency of fluidization. The results of numerous previous experimental studies indicate that the bulk density of solids essentially decreases exponentially as a function of the height of the freeboard. In the present work, this distribution has been obtained by first deriving the Fokker-Planck equation from the linearized equation of motion of a single particle and then transforming this Fokker-Planck equation into that for the bulk-density distribution of solids. Its simplification to the one-dimensional case readily gives rise to an exponential distribution and agrees well with the available experimental data.
Introduction Gas-solid fluidized beds have served as chemical reactors in a variety of industrial processes. Structurally, any of such fluidized beds may be roughly divided into three zones (see, e.g., Wen and Chen, 1982): the distributor zone or grid region just above the distributor in which the flow patterns of gas and solid particles (referred to as particles hereafter) are mainly determined by the type and design of air jet of the distributor, the bubbling zone above the distributor zone where the coalescence and breakage of the bubbles greatly affect the flow patterns and the voidage distribution, and the much less dense freeboard zone above the bed surface of the bubbling zone to which the particles are carried by the gas flow. The freeboard region above the bubbling zone of a fluidized bed or fluidized-bed reactor provides the space not only for the disengagement of particles but also for additional contact and reaction between the particles ejected from the bubbling zone and the gas. Particles are ejected from the bubbling zone to the freeboard primarily in two ways: ejection of the particles due to the bubble bursting or breaking a t the bed surface and the percolation of fine particles through coarse particles to the freeboard due to the driRing force, i.e., convective force, of the gas flow. The flow pattern and behavior of particles as well as the distribution of the bulk weight of these particles in unit volume, or the bulk-density distribution of solids in brief, in the freeboard have a significant impact on the efficiencies of the process as reported in numerous studies (see, e.g., Lewis et al., 1962; Horio et al., 1980). The results of these studies indicate that the bulk density of solids in the freeboard essentially decreases exponentially as a function of the distance from the surface of the bubbling zone or height of the freeboard. Nevertheless, relatively little has been done to theoretically interpret or mechanistically model the experimental data. By postulating the existence of three distinct phases in the freeboard, i.e., gas stream with completely dispersed solids, ascending agglomerates of particles,
* Correspondence should be addressed to this author. Present address: Research Department, Miles, Inc., New Martinsville, WV 26155. +
and descending agglomerates of particles, Kunii and Levenspiel (1969) have derived an expression for the solids holdup in the freeboard. Horio et al. (1980) have proposed a model for particle transport by assuming that the motion of the particles is Brownian. The objective of the present work is t o theoretically derive the expression governing the bulk-density distribution of solids in the freeboard from the dynamic equation governing the movement of a single particle by assuming Brownian motion of the particle and applying the stochastic population balance. The effect of the bulk-phase distribution of solids on the entrainment rate will be discussed.
Dynamics of Motion of a Single Particle When a solid particle is thrown from the bubbling zone into the freeboard by the gas flow during fluidization, it is subjected t o a combination of various forces. Consequently, it will move around inside the freeboard and its velocity and position in the freeboard are governed by the equation of motion. Equation of Motion. The dynamic equation for the movement of a single spherical solid particle in the gas phase can be written as (see, e.g., Ruckenstein, 1964; Houghton, 1966)
(e, + xeg>v,dt dv - qlu - Vln-l(U
- v)
+
where u and v are the gas and particle velocities, respectively, and a, is the drag coefficient. The left-hand side of eq 1 represents the product of the apparent mass of a particle, (ep xeg)VP,and its acceleration, dvldt. The term (ep xeg)Vpindicates that for a particle of mass QpVpto have an acceleration of dvldt, it is necessary to accelerate the virtual mass, i.e., the associated mass of surrounding gas, x.pgVp,where x is the dimensionless coefficient of virtual mass; this coefficient signifies the ratio of the volume of the accelerated gas to that of the particle. The first term on the right-hand side of eq 1 represents the nth power drag law of the resistant force by gas to the acceleration of the particle. In the case of a
0888-588519512634-1919$09.00/00 1995 American Chemical Society
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1920 Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995
single particle in an infinite fluid, the drag coefficient, a,, is a function only of the physical properties of the particle and gas, whereas in a fluidized bed, a, may depend on the void fraction of the system, E , as well. The second term is the buoyant force experienced by the particle. The third term represents the reaction of the gas t o the movement of the particle; more specifically, it signifies the pressure gradients arising from the accelerated gas displaced by the particle together with the virtual mass of the displaced gas. The last two terms on the right-hand side are the random acceleration force arising from the collisions of the particle with other particles and the Basset force; the Basset force introduced here accounts for the effect of deviation of the flow pattern around the particle from that at steady state; it is negligible if the particle size is relatively small and the fluid density is low compared to that of the particle. In principle, the exact solution of eq 1, subject to the appropriate initial and boundary conditions, leads to the particle's instantaneous velocity and position in the freeboard. Consequently, the solid-phase density distribution and the rate of carry-over of particles from the freeboard may be evaluated by tracing the movements of all the particles in the freeboard. Nevertheless, it is extremely difficult, if not impossible, to obtain the exact expressions for the terms in eq 1,such as q(t)and B(t), corresponding to the effects of the random contact collisions of the particle with other particles and the Basset force, respectively. In fact, instead of a continuous function, q(t)may be best represented by a discrete function to account for the randomness and the discontinuity of the particles' collisions. These unknown terms, therefore, hinder the exact prediction of the velocity and position of the particle inside the freeboard in solving eq 1. On the other hand, by accepting the fact that the particle's movement cannot be exactly predicted, the unknown terms in eq 1 may be treated collectively as the fluctuating term, i.e., a source of noise and randomness. Consequently, eq 1 is considered as a stochastic differential equation. As such, we are able to predict the possibility or probability of the particle to be at a certain point at a certain time, from which the solid-phase density distribution is obtained. The drag term in eq 1 is usually nonlinear in terms of the solid velocity, v; it is linear only when n = 1and a, = 3npd,, corresponding to creeping flow. When the particle velocity is small compared to the gas velocity and the radial gas velocity profile does not vary significantly in the freeboard, 1u in the first term on the right-hand side of eq 1 may be approximated as Un-',consequently, this term can be represented by a linear expression with respect to v as where E ( t ) is the error from approximation, and U is the superficial gas velocity. By letting U* = u - Uk (3) Equation 1 can be rewritten as a stochastic differential equation dvldt = -Pv
+ F +A
where
qv-l =
(e, + xe,)V,
(4)
A=
(e, + xe,)V,
[qu"-'u* + ( 1 + &,V, dt du +
The first term on the right-hand side, -PV,represents the dynamic friction experienced by the particle; the second term, F, the convective force of the surrounding flowing gas to the particle minus the gravity force; and the third term, A, the term characterizing the randomness and fluctuations arising from the interactions between the particle and the surrounding turbulent gas flow and those between the particle and other particles as well as the errors from approximations. By followingUhlenbeck and Ornstein (19301, the third term on the right-hand side of eq 4,A(t),is assumed to be close to a white noise (without memory). Naturally, we assume that it possesses the following properties. 1. The ensemble average ofA(t) at any time t is zero, i.e.,
(Ai)= 0
(8)
and so is the temporal average of A(t). 2. The values of A(t)at different times tl and t 2 are correlated only when It2 - tll is very small. More explicitly, = 4&-t1), i = 1, 2, 3 (9) where 4 ( t 2 - t l ) is a function with a very sharp maximum at t 2 - tl = 0. Consequently, the integration of & ( t 2 - t l ) over all possible time differences, zi's, is finite, i.e., (Ai(t&+(tl))
and (A,(t,)A,(tJ)= 0, ij = 1, 2, 3; i * j
(11)
Probability Distribution of a Particle's Movement. From the stochastic differential equation governing the moving dynamics of a solid particle, eq 4, the probability distribution of the particle's movement in the freeboard can be determined. Let flr,t;ro,tO)be the probability density function of the particle being at position r in the freeboard a t time t , given that it was a t the position, ro, a t time to. In other words, f(r,t;ro,to)AVis the probability or possibility of the particle being at a position within a volume element of AV(r) in the freeboard a t time t , given that it was at the position, 1 0 , at time to. It is assumed that the movement of the particle in the freeboard is Markovian, i.e., its future depends only on the present information of the particle and is independent of its past history. Let At denote an interval of time much shorter than the intervals during which the velocity of the particle changes by appreciable amounts and q(r- Ar;Ar)be the transition probability density function of the particle moving from (r - Ar) to r during time interval (t,t+At), then, the equation governing the evolution of f(r,t;ro,tO) is (see, e.g., Chandrasekhar, 1943) f(r,t+At;ro,to) = Sf(r-Ar,t;ro,to) q(r-Ar,Ar) d(Ar) (12)
Ind. Eng. Chem. Res., Vol. 34, No. 5 , 1995 1921 Expanding flr,t+At;ro,to), flr-Ar,t;rO,tO), and q(rAr;Ar) in this equation into Taylor series yields (Appendix A)
(16)
(hxi),( h i 2 ) , and (
h i h j ) can be
evaluated by integrating the stochastic differential equation of motion, eq 4, whose ith component can be written as
dv.
4 = -Bui
dt
+ Fi +Ai,
i = 1,2,3
(18)
Solving this stochastic differential equation and taking the means over the resultant expression, subject to the assumptions in eqs 8-11 and the assumption that BAt >> 1, lead to (see, e.g., Uhlenbeck and Ornstein, 1930; Houghton, 1966; also see Appendix B)
Fi
(hi) = - At
(19)
B
and
(bibj) = o(At), i #
j
(21)
conditions, leads t o the probability density distribution of the particle’s position in the freeboard. Density Distributions and Entrainments of Particles The freeboard contains numerous moving particles whose dynamics and probability density functions are governed by eqs 1 and 23, respectively. Equation 23 may be transformed into the governing equation for these particles’ density distribution in the freeboard, which is the primary concern of this work. Usually, the particles ejected from the bubbling zone into the freeboard are disperse. In other words, they contain the whole spectrum of particle sizes present in the bubbling zone. The simplest case is that the sizes of the particles in the freeboard are uniform. Density Distribution of Particles with Uniform Size. Let g(r,t) dV(r) be the number of particles in dV(r), where dV(r) denotes an infinitesimal volume in the vicinity of r in the freeboard. Then, a population balance of the particles in dV(r) at time t yields g(r,t>dV(r) =
where Vf is the total volume of the freeboard, NO,the total number of particles in the freeboard at the initial time, to, each with an initial probability distribution of p(ro,to),A, the area of the freeboard entrance, i.e., the area separating the freeboard from the bubbling zone of the fluidized bed, and w(rA,z) dA, the flow rate in terms of the number of particles entering the freeboard through dA, an infinitesimal area of the freeboard entrance. The first term on the right-hand side of eq 25 signifies the number of those particles in the volume, dV(r), at time t, which were originally in the freeboard at time to. The second term represents the number of those particles in the volume, dV(r), at time t, which have entered the freeboard from the bubbling zone during the time interval, (to$). Hence, the bulk density of solids or the weight of the particles in unit volume of the freeboard, Q(r,t), may be defined as
where Di is the effective particle diffusivity or mixing coefficient; it is defined as
D.=-
Ti
i = 19 2 9 3
2p2’
(22)
Note that this mixing coefficient is allowed t o be anisotropic since DI,Dz,and D3 can be different. Substituting eqs 18-21 into eq 13 gives rise to
(26) Differentiating this equation with respect to time and position and in the light of eqs 23and 25, we obtain the following equation for the bulk density of solids in the freeboard, Q (Appendix C).
at where
F3=
%u“ - (e, - eg)VPg (e, + x e p ,
(24)
This is the Fokker-Planck equation governing the particle’s movement in the freeboard. Solving this equation, subject to the appropriate initial and boundary
-
P
+
h 3
c-
a2(DiQ>
1a(F3Q)
aQ
i
(27) k i 2
Solving this equation, subject to the appropriate initial and boundary conditions, leads t o the solid-phase density distribution in the freeboard. For a cylindrical freeboard under the steady-state conditions, eq 27 becomes (28)
1922 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995
Let us consider a special case in which the diameter of the fluidized bed is relatively large so that the wall effect is insignificant and in which the distributor is well designed so that there is no significant angular dependence of the bulk density of solids, 8. For such a case, 8 may be considered as a function only of the distance from the entrance to the freeboard, i.e., from the surface of the bubbling zone, z. Thus, eq 28 reduces t o
(29) with the boundary conditions z=o, z = 00,
P=Po
(30)
P=O
where
(31) Subject to the boundary conditions in eq 30, eq 29 can be separately solved for the following three cases. 1. If U, < U , i.e., F3 > 0, the steady-state distribution of the bulk density of solids cannot exist in the freeboard. In other words, the size of the particles ejected from the fluidized bed is small enough that they are eventually carried out of the freeboard. 2. If U = Ue, i.e., F3 = 0, the bulk density of solids reaches the critical axial distribution, i.e., it is uniform throughout the freeboard. 3. If U, > U , i.e., F3 0, there exists a steady-state axial distribution of the bulk density of particles; it is given by
P
= Po exP[
u" - u," u"-lD2
z]
This expression indicates that the bulk density of solids in the freeboard falls off exponentially from the entrance of the freeboard. The existence of the stable exponential distributions may be due to the turbulent and fluctuating gas flow and the gas-particle and particle-particle interactions in the freeboard. Ue, upon which the criterion for attainment of a steady-state bulk-density distribution of solids is based, may be referred t o as the entrainment velocity of the particles in the freeboard. Density Distributions and Entrainments of Disperse Particles. When the bubbling zone in a fluidized bed contains particles with multiple sizes, the particles ejected from this zone into the freeboard also contain the whole spectrum of sizes present in the bed. We may divide the size distribution of the particles in the freeboard into narrow intervals and assume that the density distribution of the particles in each size interval can be independently evaluated. Then, for the particles whose entrainment velocities, Uei, are less than the superficial gas velocity, U , no stable exponential density distributions can be established; the particles will be continuously entrained from the fluidized bed, and their entrainment rates are independent of the height of the freeboard. For the particles whose entrainment velocities are greater than U, their density distributions in the freeboard can be evaluated from eq 32. In other words, the particles in the ith size interval have the axial density distribution of
Pi= Qoi exp
[
1'
u" - uein u"-'D,,
(33)
Obviously, this expression indicates that if Uei > U , the bulk density of the particles in the ith size interval decreases with the height in the freeboard. The entrainment rate of the ith size particles depends on the height of the freeboard, the higher the freeboard, the less the entrainment rate. Eventually, the transport disengaging height (TDHi) for the ith size particles is reached above which their entrainment rate does not change appreciably. The overall transport disengaging height of the freeboard (TDH) may be considered as that of the smallest particles with U, =- U. According to the procedure proposed by Zenz and Weil (1958) and also by Gugnoni and Zenz (1980), the size distribution of particles is divided into narrow intervals, and only the small particles within the intervals for which ut < U are considered to be entrained. From the discussion in the preceding paragraph, however, it is likely that the particles ejected from the bubbling zone into the freeboard due to the bursting of bubbles or percolation contain the whole spectrum of particle sizes present in the bed, of which the smaller particles will be transported out of the freeboard. The bulk-density distributions of the larger particles in the freeboard, for which ut > U , may follow exponential distributions, as indicated in eq 35. In other words, some of these larger particles will be suspended in the freeboard by the turbulent gas flow once they are ejected from the bubbling zone into the freeboard by the bursting of bubbles. Consequently, the larger particles can also be entrained from the freeboard. Nevertheless, their entrainment rate decreases with an increase in the height of the freeboard. The entrainment rates of larger particles could become significant when the height of the freeboard is considerably shorter than TDH. Model Validation The expression in eq 32 for the bulk density of solids in the freeboard is in accord with the expression derived by Kunii and Levenspiel (1969) and also with the experimental results of Lewis et al. (1962) and Horio et al. (1980). The axial mixing coefficient, D,, is recovered by fitting eq 32 to the available experimental data. This requires a general expression for the drag coefficient, a,,.Usually, a, is a function of the physical properties of the solid particles and gas as well as the void fraction of the system. For the case of entrainment, however, the voidage in the freeboard is most likely close to one; hence, its effect on a,,may be negligible. Consequently, a,,can be expressed in terms of the conventionally defined friction factor, f , as (see, e.g., Bird et al., 1960)
a, = Thus,
(34)
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1923 Table 1. Mixing Coefficients Evaluated by Comparing Eq 32 with Data of Lewis et al. (1962) u (WS) D,(ft2/s) u (WS) D,(fi2/s) 1.5 1.7
81.68 88.10
1.9 2.1
119.48 173.25
by Lewis et al. (1962),and the errors induced by the approximations in deriving eq 32. Concluding Remarks
Symbol v
1.7, e x p . 1.9, exp. 2.1, exp. present model
A
10
0
1.5, e x p .
0
0
U (ft/!
o
20
30
50
40
60
70
80
H e i g h t of Freeboard (in.)
Figure 1. Bulk-density distributions of solids in the freeboard with the column diameter of 0.75in., the bubbling-zone depth of 4 in., and with fluidized particles of glass spheres (Lewis et al., 1962). 2
2.3ln~$;
for
Re
for
2 < Re
for
5x
~ - 2 ~ ~ 3 1 5
I0.1;
n=1
I5
x
lo2; 1 < n < 2
lo2 < R e ; n = 2
The exponential expression of eq 32 is compared with the bulk densities of solids in the freeboard experimentally measured by Lewis et al. (1962). The superficial gas velocity in their entrainment experiments ranges from 1.5 to 2.1 fvs, corresponding to the Reynolds number ranging from 244 t o 343. Thus, 1 < n < 2 and a, can be calculated from the middle expression in eq 35. As depicted in Figure 1,the model prediction agrees well with the experimental results. The recovered values of the mixing coefficient, D,, reflecting the intensity of fluctuations arising from the gas-particle and particle-particle interactions, are given in Table 1. It is worth noting that D,increases with an increase in the superficial gas velocity. This may be due to the fact that as the gas velocity increases, the turbulence of the gas flow is intensified; this is accompanied by the magnification in amplitude of the fluctuations of particle motion. The recovered values of D, in Table 1 are significantly higher than those in the bubbling zone of the fluidized bed obtained from some other experimental studies (Kunii and Levenspiel, 1991). This is probably attributable to the turbulent nature of the freeboard, the high superficial gas velocities in the experiments
A stochastic differential equation, i.e., the FokkerPlanck equation, has been derived in this work from the linearized dynamic equation governing the movement of a single particle. By applying the stochastic population balance, this stochastic differential equation has been transformed into that for the bulk-density distribution of solids. Upon simplification and solution, this equation has yielded the exponential distribution of the solid-phase density along the height of the freeboard, which is in good accord with the available experimental data. It has been revealed that particles with all size intervals can possibly be entrained out once they are ejected from the bubbling zone into the freeboard by the bursting of bubbles or percolation. The present model can be applied readily to liquidsolid and possibly to three-phase fluidized beds. The expressions for the bulk-density distributions of solids in a liquid-solid fluidized bed may be derived through a procedure parallel to that of the present work, while for a three-phase fluidized bed, the dynamic equation of motion needs to be modified to estimate the bulkdensity distributions of solids. Acknowledgment
This is contribution no. 94-300-5from the Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506. The authors also thank Dr. M. Horio for his valuable suggestions. Nomenclature A: area of the freeboard entrance Bl(t): Basset force d,: particle diameter Di: effective particle diffusivity or mixing coefficient E(t): error from linearization f: friction factor flr,t;ro,to): probability of the particle being at position r at time t, given that it was at position r o at time to g: acceleration of gravity g(r,t) dV(r): number of particles in dV(r) No: total number of particles in the freeboard initially p(r0,to): initial probability distribution of the particle ql(t): random acceleration force arising from particles’ collisions Re: Reynolds number t : time u: gas velocity U superficial gas velocity U,: entrainment velocity of particles in the freeboard v: particle velocity Vf: total volume of the freeboard V,: particle size Greek Letters a,: drag coefficient @r,t):bulk density of solids eg: gas density
1924 Ind. Eng. Chem. Res., Vol. 34, No. 5,1995
ep: particle density
dui
a function with a very sharp maximum at t = 0 x: ratio of the volume of the accelerated gas to that of the particle q(r-Ar;Ar):transition probability density function of the particle a(rA,'A,t): flow rate in terms of number of particles through unit area of the freeboard entrance #&):
-dt_ - -Pui
+ Fi +A,,
i = 1, 2 , 3
(B1)
Integration of this equation gives rise to h, = x 1. - x lo. 1
Appendix A. Derivation of Eq 13 The equation governing the evolution of fTr,t;rO,tO), eq 12 in the text, is
flr,t+At;ro,to) = Jflr-Ar,t;ro,to)q(r-Ar,Ar) d(Ar) (Al) The terms, fTr,t+At;ro,to),flr-Ar,t;ro,to),and ly(rAr;Ar),in this equation can be expanded into Taylor series, respectively, as
Taking the means over this expression, subject t o the assumption in eq 8 and the assumption that ,8At >> 1, leads t o ( h i )
Fi
= -At
P
Squaring both sides of eq B2 and taking the means over the resultant expression, subject to the assumptions in eqs 9 and 10 and noting that pAt > 1, give
flr-Ar,t;ro,to)= flr,t;ro,to) -
(hi2) = W,At
af + e-&, i
(B4)
where Di is the effective particle diffusivity or mixing coefficient and is defined as
h i
Similarly, evaluating Lsxihxj from eq B2, taking the means to the resultant expression, subject to the assumption in eq 11, and noting that /?At >> 1 yield (hihi) = o(At), i # j
By substituting eqs A2-A4 into eq A1 and noting that
flr,t;ro,to) = J-Tflr,t;ro,to) q(r,Ar>d(Ar) (-45) we have
(B6)
Equations B3, B4, B6, and B5 are eqs 19, 20, 21, and 22 in the text, respectively.
Appendix C. Derivation of Eq 27 Equation 27 in the text is derived here in the system of rectangular coordinates. Substituting eq 25 into eq 26 in the text leads to
Differentiating this equation with respect to time yields
(hihj) = J: mhihjq(r,Ar)d(Ar)
(A101
Equation A6 is eq 13 in the text.
Appendix B. Derivation of Eqs 19-22 The ith component of the stochastic differential equation of motion, eq 4, can be written as
Since r is a point within the freeboard, and rA is a point at the entrance of the freeboard, the probability for a particle t o be at two different positions at the same time is zero, i.e.,
flr,t;rA,t) =0
(C3)
By substituting this expression into eq C2, we have
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1925 solids in the freeboard,
ag
-=--at B
Multiplying both sides of eq C 1 by F3 and differentiating the resultant expression with respect to x3 yield
Similarly, multiplying both sides of eq C 1 by Di,i = 1-3, and differentiating the resultant expressions twice with respect to xi give rise t o
e.
a(~,g)+ ca2coig) ax3
i
hi2
(C8)
This is eq 27 in the text.
Literature Cited Bird, R. B.; Steward, W. E.; Lightfoot, E. N. Tranport Phenomena; John Wiley & Sons: New York, 1960;pp 181-200. Chandrasekhar, S. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 1943,15,1-89. Gugnoni, R. I.; Zenz, F. A. Particle Entrainment from Bubbling Fluidized Beds. In Fluidization ZZ& Grace, J. R., Matsen, J. M., Eds.; Plenum: New York, 1980;pp 501-508. Horio, H.; Taki, A.; Hsieh, Y. S.; Muchi, I. Elutriation and Particle Transport through the Freeboard of a Gas-Solid Fluidized Bed. In Fluidization ZZ& Grace, J. R., Matsen, J. M., Eds.; Plenum: New York, 1980;pp 509-518. Houghton, G.; Particle and Fluid Diffusion in Homogeneous Fluidization. Znd. Eng. Chem. Fundam. 1966,5,153-164. Kunii, D.;Levenspiel, 0. Entrainment and Elutriation from Fluidized Beds. J . Chem. Eng. Jpn. 1969,2,84-88. Kunii, D.; Levenspiel, 0. Fluidization Engineering; ButtenvorthHeinemann: Boston, 1991;pp 213-233. Lewis, W. K.; Gilliland, E. R.; Lang, P. M. Entrainment from Fluidized Beds. Chem. Eng. Prog. Symp. Ser. 1962,58(38),6578. Ruckenstein, E. Homogeneous Fluidization. Znd. Eng. Chem. Fundam. 1964,3,260-268. Uhlenbeck, G. E.; Ornstein, L. S. On the Theory of the Brownian Motion. Phys. Rev. 1930,36, 823-841. Wen, C. Y.; Chen, L. H. Fluidized Bed Freeboard Phenomena: Entrainment and Elutriation. AZChE J. 1982,28,117-128. Zenz, F. A.; Weil, N. A. A Theoretical-Empirical Approach to the Mechanism of Particle Entrainment from Fluidized Beds. AZChE J. 1968,4,472-479.
Received for review January 24, 1995 Accepted February 9,1995 @
IE940349T
Substituting eq 23 in the text into this expression gives rise t o the following equation for the bulk density of
Abstract published in Advance ACS Abstracts, April 1, 1995. @
