Similarly, for Regime 1 fluidization, the coefficient is 2.17 with 959& confidence h i t s of 2.06 and 2.28. Osberg arid Charlesworth (1951) studied pas fluidization of two component systems. The experimental data were presented in a plot of the elutriation rate constant, k , to the 0.6 ~ ) o i ~ -vs. e r gas superficial J.elocity. Based on the first-order rate equation, t’lie exponent should be unity; however, it was suggested (Osberg and Charlesworth, 1951) that the driving force which determines the elut,ristion was the summation of viscous and inertial forces developed by the turbulent motion of the gas in tlie fluidized bed. In gas-fluidized beds, the occurrence of gas bubbles affects the elutriation rates (Leva and Wen, 1971).
diameter of particle diameter of column k elutriatioii rate coristant (theoretical) k’ = elutriation rate constant (experimental) 1 = length of fluidized bed Q = volume element of bed U t = terminal velocity of particle VI = superficial velocity of fluid relative t o wall V , = superficial velocity of gas TiT’(i, = weight of species i in bed TI’E(i) = weight of species i elutriated d,
DT
GREEKLETTERS pg
ps
Conclusions
The present correlatioii accounts for important parameters i i i the systeni-e.g., solids concentration, liquid superficial velocity, bed height, and physical propert’ies of part’icles and liquid. For Regimes 1 and 2 fluidization, the elutriation rate coiistaiit is defiiied by b[(V’, - L-Jl’L] n.here b has a value of 2.2 (Regime 1) and 1.5 (Regime 2 ) . The difference between t,he experimental and t,lieoreticnl elutriatioii rate coristaiits is attributed to tlie effertivt concentration of the specieq a t the surface of the bed.
= = =
= =
density of gas density of solid
Literature Cited
Hanesian, I>., Rankell, A,, Ind. Eng. Chem. Fundurnen., 7, 452 (1968). Hercules, A. S.,PIIASc Thesis, University of Toronto (1971). Jottiand, It., Chem. Eng. Sci., 3, 12 (19.54). Leva, XI., Cherri. Eng. Progr., 47, 39 (19.51). Leva, lI., ]Ten. C. Y., in “Fluidization,” ed. by J. F. Ilavidson and I>. Harrison, p 036, Academic Press, 1971. Osberg, G . I,., Charlesworth, I). I I . , Chenl. Eng. Progr., 47, 666 (1951). Schlyartz, C. E., Smith, J. >I., Ind. Eng. Cherri., 45, 1214 (1963). Thomas, W,J., et al., Brit. (:hein. Eng., 6, 170 (1061). Wen, C. Y . , Ilaijhinger, It. G., A.1.Ch.E.J., 2, 220 (1960).
Nomenclature
C
volume fraction of particles in system: concentration of bed a t any tinie Co = initial conceiitration of bed
I ~ E C P X Yfor E Dreview June 5, 1972 A4CCEPTI:DSoveniber 6, 1972
=
Work supported by the Sational Research Council of Canada.
Bed Expansion Characteristics of Annular Liquid-Fluidized Beds K. Ramamurthy and K. Subbaraju* Department of Chemical Engineering, Indian Institute of Technology, _Iladras-d6, India
Bed expansion data for annular liquid-fluidized beds have been obtained by using various sizes of glass spheres, iron balls, and lead shot fluidized by water and water-glycerol mixtures to cover wide range of variables. Generalized equations for predicting bed expansion have been obtained by extending the equation of motion of a single particle in a fluid to a multiparticle system of homogeneous fluidized bed. With the present experimental and literature data, the equation of motion tor a single particle can b e extended to a multiparticle system of homogeneous fluidized bed when the superficial velocity is modified by a factor [l-1.21 ( 1 - e)2’3]-1.56. The bed expansion characteristics in annular spaces are not different from those of tubes as long as the ratio of D,,/dp > 8.
w h e n a fluid is passed through a bed of particles a t a ve!ocity greater than minimum fluidization velocity, the bed expands and tlie particles remain in a state of suspension. The prediction of t’hisbed expansion is necessary for specifying the height of tlie fluid bed equipment. The study of the characteristics of bed expansioii has therefore attracted the attention of many investigators. Leva et al. (1948), Wilhelm arid Kwauk (1948), Lewis ct, al. (1949), Jottrand (1952), Lewis and I3owermaii (1952), 184
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1 9 7 3
Richardson and Zaki (1954), Wen and Yu (1966), and Rajagopalan and Laddha (1967) have reported in literature investigations on the bed espaiision characterist,ics of liquid-fluidized beds. The details of the eyuil)meiit, and tlie range of esperiinental variables iiivcstigated by these investigators are giveii in Table I. h riumbcr of emllirical and scmiempirical equations are suggested by these investigator.;. These equations have been develol)ed 011 the basis of either estcnding bhe cquatioiis de-
Table 1.
Investigators
Wilhelm and Kwauk
Lewis et al.
Jottrand Lewis and Bowerman Richardson and Zaki
Wen and Yu Rajagopalan and Laddha
Column diam, mm
Yeor
Summary
Particles
of Experimental Studies on Bed Expansion Particle diam, mm
Particle density, g/cms
0.3-5.0 1.1Sand, Socony beads, glass 10.8 spheres, lead shot, crushed rocks Glass spheres, 0.12.361949 60.96 and Aerocat , 0.57 2.48 114.3 microspheres ... Sand 0.432 ... 1952 Glass spheres, 0.1-5.2 I . 61952 6 3 . 5 Socony 10.8 beads, lead shot 13all bearings, 0.11.061954 38.162.0 glass 6.35 11.25 spheres, lead shot, Ballotini, divinyl benzene Glass spheres, 0.2032.361966 101.6 steel balls 6.35 7.84 Glass spheres, 2.2-5.6 2.51967 3 8 . 1 steel balls, 11.71 lead shot
1948
76.2 152.4
rived for pressure drop across packed beds [Leva et al. (1948), Rajagopalan and Laddlia (196i) ] or applying dimensional analysis of parameters that influence bed expansion [Wilhelm and Kwauk (1948), Richardson and Zaki (1954)] or extending single-particle drag force equation to multiparticle system [Lewis et al. (1949), Lewis and Bowerman (1952), Wen and You (1966)]. I lowever, the applicability of these equations in wide ranges are limited, and work is still being carried out concerned with the manner in which particles are distributed in the bed and the effect on the bed expansion. Further, there is no published data on bed expansion in annular liquidfluidized beds. The present work has therefore been undertaken with the objectives to obtain extensive bed expansion data in annular liquid-fluidized beds and to develop equations for predicting bed expailsion and correlate the present as well as literature data. Theoretical Considerations
In aggregative fluidization systems, the upper boundary of the bed exhibit fluctuations due to nonhomogeneity and instability in the structure of the bed. On the other hand, in particdate fluidization, the bed is rather stable without much fluctuation. Liquid-fluidized beds may be idealized as particulate and homogeneous iii their character. 111 the case of homogeneous fluidization, particles move short distances around their average positions. The homogericoiis fluidized bed can therefore be represented as an c~nsembleof solid particles moving about points considered as nodes of imaginary lattice through whose free volume the fluid is flowing. With further simplification, the particles in t’he bed may be assumed to be in a state of stationary
Fluid den-
Fluid vixoc-
itu,
situ,
Fluidizing medium
Flow range
g/cms
CP
Air, water
0.00121.o
0.01751.o
Laminar, intermediate, turbulent
Air, water
0.0012 0.0175tol.O 1.0
Laminar, intermediate
Water Water
1. o 1.0
1.0 1.0
Laminar Laminar, intermediate, turbulent
Water, glycerin oil, bromoform, ethylene dibromide
0.812.89
1.0-113
Laminar, intermediate, turbulent
Water
1.o
1. o
Water
1. o
0.822.7
Intermediate, turbulent Turbulent
C.M.C.
suspension and uniformly distributed in the bed. Hence the forces acting on each particle may be in dynamic equilibrium. I n the case of unidimensional motion in vertical direction, the equation of motion of a single particle in a fluid can be written as:
(Acceleration force
=
gravity force
- drag force -
reaction force of apparent mass) Equation 1 can be extended to a particle belonging to a n ensemble of particles in a bed, provided a proper correction for fluid velocity is incorporated. Ranz (1952) suggested that the drag force for a fixed bed can be calculated by means of expression valid for a single sphere if the velocity is replaced by a velocity through minimum cross section of the bed. If the particles are arranged in an imaginary node of a simple cubical lattice, the velocity of the fluid in minimum cross se1:tion of the bed is given by
By averaging Equation 1 with respect to time between 0 and 7 , replacing the relative velocity, w, with (us+d - u p ) , and simplifying as suggested by Ruckenstein (1964), the resulting equation is : (3) ’Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1 973
185
Table
Liquid
Water
II. Systems and Range of Variables Investigated
Solid
Glass spheres
Water Water 60% Glycerol
Steel balls Lead shot Glass spheres
80% Glycerol
Glass spheres
Particle diam, mm
Particle density, g/cma
Porosity range
5.32 4.94 2.43 1.255 0.938 0.78 3.125 2.870 4.94 2.43 1.255 4.94 2.43 0.78
2.940 2.925 2.910 2.920 2.930 2.930 7.757 11.19 2.925 2.910 2.920 2.925 2.910 2.930
0.48-0.91 0.47-0.88 0.45-0.90 0.44-0.93 0.42-0.92 0.44-0.92 0.45-0.94 0.49-0.86 0.47-0.92 0 . 5 -0.91 0.43-0.92 0.46-0.84 0.42-0.94 0.47-0.85
Particle Reynolds number range
Viscosity, CP
340-2240 270-1818 90-750 20-250 9-140 7-100 370- 2550 370-1800 17-165 5-40 0.33-8.7 1-25 0 . 2 -7.6 0.05-0.37
0.784
0.737 0.784 8.2 8.6 9.0 28.4 28.4 28.4
2o
Galileo number
4 6 . 5 x 105 37.0 x 105 4.37 x 105 6 . 0 5 x 104 2 . 5 3 x 104 1 . 4 6 x 104 40.26 X lo5 38.29 X lo5 35.85 x 103 38.54 X lo2 486.7 30.36 X lo2 3.612 X lo2 11.98
I
PRESENT DATA; Ga
