Ind. Eng. Chem. Res. 1992,31, 1562-1568
1562
amine (in Russian). Zh. Anal. Khim. 1970a,25,943-949. Pyatnitekii, I. V.; Tabenskaya, T. V. The Effect of the Amine Nature on the Extractability of the Iron Citrate Complexes (in Russian). Zh. Anal. Khim. 1970b,25,2390-2395. Pyatnitakii, I. V.; Tabenskaya, T. V.; Sukhan, T. A. The Effect of Chemical Properties of Carboxylic Acids on the Extraction of their Iron Complexes by Triotylamine (in Russian). Zh. Anal. Khim. 1971,26,2399-2405. Pyatnitakii, I. V.; Tabenskaya, T. V.; Makarchuk, T. L. Effect of Solvents on Extraction of Citric Acid and a Citrate Complex of Iron with Tri-n-octylamine (in Russian). Zh. Anal. Khim. 1973, 28,550-564. Rabinowich, V. A.; Khavin, Z.Ya. Concise Handbook of Chemistry (in Czech); SNTL Prague, 1985;p 291. Sato, T.; Watanabe, H.; Nakamura, H. Extraction of Lactic, Tartaric, Succinic and Citric Acids by Trioctylamine (in Japanese). Bunseki Kagaku 1985,34,559-563. Sergievskii, V. V. The Influence of Hydration on Extraction Equilibria (in Russian). Ztogi Nauki Tekh. Ser.: Neorg. Khim. 1976, 5, 5-82. Shmidt, V. S. Amine Extraction (in Russian); Atomizdat: Moscow, 1980;pp 43-59. Sillen, L. G.;Martell, A. R. Stability Constants of Metal-Zon Complexes; Chemical Society: London, 1964. Tamada, J. A.; King, C. J. “Extraction of Carboxylic Acids by Amine Extractante”; Report LBL-25571;Lawrence Berkeley Laboratory: Berkeley, CA, January 1989. Tamada, J. A.; King, C. J. Extraction of Carboxylic Acids with
Amine Extractants. 2. Chemical Interactions and Interpretation of Data. Znd. Eng. Chem. Res. 1990a,29,1327-1333. Tamada, J. A.; King, C. J. Extraction of Carboxylic Acids with Amine Extractanb. 3. Effect of Temperature, Water Coextraction and Process Considerations. Znd. Eng. Chem. Res. 199Ob,29, 1933-1938. Tamada, J. A.; Kertes, A. S.; King, C. J. Extraction of Carboxylic Acids with Amine Extractants. 1. Equilibria and Law of Mass Action Modelliig. Znd. Eng. Chem. Res. 1990, 29,1319-1326. Vafiura, P.; KuEa, L. Extraction of Citric Acid by the Toluene Solutions of Trilaurylamine. Collect. Czech. Chem. Commun. 1976, 41,2857-2811. Wennersten, R. A New Method for the Purification of Citric Acid by Liquid-liquid Extraction. Proceedings of the International Solvent Extraction Conference; Liege University Press: Liege, 1980,Vol. 2, paper 80-63. Wennersten, R. The Extraction of Citric Acid from Fermentation Broth Using a Solution of Tertiary Amine. J . Chem. Technol. Biotechnol. 1983,33B,85-94. Yagodin, G. A. Liquid Extraction Fundamentals (in Russian); Khimiya: Moscow, 1981;p 27. Yang,S.-T.; White, S. A.; Hsu, S. T. Extraction of Carboxylic Acids with Tertiary and Quaternary Amines: Effect of pH. Znd. Eng. Chem. Res. 1991,30,1335-1342.
Received for review August 14,1991 Revised manuscript received February 25, 1992 Accepted March 19,1992
Axial Segregation of Multicomponent Solid Particles Suspended in Bubble Columns Toshitatsu Matsumoto,* Nobuyuki Hidaka, and Hideki Gushi Department of Chemical Engineering, Kagoshima University, Kagoshima 890, Japan
Shigeharu Morooka Department of Chemical Science and Technology, Kyushu University, Fukuoka 812, Japan
Multicomponent solid particles were suspended in vertical bubble columns of 7-cm diameter and 4.85-m height and 15-cm diameter and 2.7-m height. Axial changes in holdups of gas and solids in the column were determined by sectioning the column simultaneously with 20 horizontal plates for the 7-cm-diameter column and 11such plates for the 15-cm-diameter column. Solid particles used were sieved glass beads (118,243, and 465 pm) and mixtures thereof. A one-dimensional sedimentation-dispersion model was applied to describe the axial holdup profile of each solid component. The parameters used in the model were all correlated with experimental equations based on the results of single-componentsystems. Calculated results were in agreement with experimental ones over an entire column consisting of dense and dilute regions.
Introduction Three-phasefluidization is an important technique for contacting gas, liquid, and solid particles and is widely used in petroleum and biochemical processes. It is recognized that a three-phase fluidized bed is rather Micult to design because the phase holdups are axially distributed in the column. Many works (Cova, 1966;Imafuku et al., 1968; Farkas and Leblond, 1969;Kato et al., 1985;Smith and Ruether, 1985;Smith et al., 1986;Morooka et al., 1986; Murray and Fan, 1989)have been published, especially on the behavior of solid particles in three-phase fluidization, but most were restricted to the case of particles with the same size and density. When a mixture of solid particles is used as the solid phase, on the other hand, segregation of components naturally occurs in the bed.
* To whom correspondence should beaddressed.
The segregation pattern for liquid-solid systems has been studied phenomenologically by many researchers (Kennedy and Bretton, 1966;Al-Dibouni and Garside, 1979;Jumma and Richardson, 1983). Al-Dibouni and Garside, (1979) reported that binary mixtures with a particle diameter ratio of 2.0 were mixed uniformly throughout the bed while those with a ratio greater than 2.2 were classified. Moritomi and Chiba (1987)studied the fundamental mechanism of segregation in a liquidsolid fluidized bed consisting of hollow char particles and glass beads, the latter being heavier and smaller than the former. They predicted patterns of stratified layers from the balance between drag and buoyant forces, assuming no mixing between the stratified zones. In a gas-liquid-solid fluidized bed with multicomponent mixtures of solid particles, the segregation pattern is disturbed by the bubble flow, which causes axial mixing of liquid and solids. Then the column is roughly divided
0888-588519212631-1562$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 1563 into two regions, the lower part with a dense and nearly constant solid holdup and the upper part with a dilute solid holdup that decreases with increasing height. The boundary is often very vague at higher gas velocity, however. The axial change in solid holdup is normally described by a sedimentation-dispersion model that assumes gravitational settling of particles relative to liquid flow and turbulent dispersion of particles. Jean et al. (1989) summarized models appearing in the literature. Most of them can be applied to either the dense region or the lean region, although these regions coexist in a column. Fan et al. (1987) conducted experiments in a 10.2-cm-diameter column with coarse glass beads of 3 and 4 mm and of 3 and 6 mm in diameter in batch operation with respect to solid particles. They divided the column into three discrete regions, assuming that the concentrations of small and large particles were constant in the bottom and top regions and varied with height only in the middle region. The axial positions of the boundaries were left as arbitrary parameters which were adjusted to fit the data. Since both composition and mass concentration of particles vary continuously in the column, their axial changes must be described with unified equations that can be applied over the whole column. The model developed by Matsumoto et al. (1989) is applicable to the whole length of a threephase column, including the dense and lean regions, but the particles must be of the same size,shape, and density. Recently, Matsumoto et al. (1991) investigated the axial changes in solid holdup for two-component systems of 118 and 243 pm, 118 and 465 pm, and 243 and 465 pm. Parameters used in their model were correlated by empirical equations as functions of gas and liquid velocities, column diameter, and the properties of liquids and solids. However, experimental data were not sufficient in the bottom and top regions of the column. Extension to multicomponent systems was not confirmed. In the present study, a multicomponent mixture of glass beads is fluidized in vertical columns of 7- and 15-cm diameter. The partition of the column is increased to 20 plates, double that of the previous study (Matsumoto et al., 19911, and the axial change in solid holdup of each component in the bottom and top regions is determined in detail. The model is revised according to new data with multicomponent particles and is successfully applied over the whole column length. Experimental Apparatus and Procedure Figure 1shows a schematic diagram of the experimental apparatus. The fluidization column was made of an acrylic resin pipe with an inner diameter of 15 cm and a height of 2.7 m and was set vertically. The gas distributor was a spiral copper tube 10 mm in outer diameter, with 19 2-mm holes pointing downward, and was installed horizontally at a height of 0.24 m from the bottom. The liquid distributor was a packed bed of 15-mm-diameterglass spheres, a bronze net of 200 mesh being placed at the upper face of the packed layer. Eleven shutter plates, made of stainless steel 3 mm thick, were installed horizontally at 0.2-m intervals and were interconnected with wire rope. The fluidization column with 20 shutter plates was made of a pipe 7 cm in diameter and 4.85 m in height. The number of plates was doubled compared with the previous work (Matsumotoet d,1991). The other features were fundamentally the same as those of the 15-cm-diameter column. All the experiments were conducted batchwise with respect to the solid particles. At the top of the column, a divergent section of bronze net was used to prevent the
0 0 .Columnwall
,
l0"l
'\
-Close
I
Figure 1. Experimental apparatus (lbcm-diameter column): (1) compressor; (2) oil-mist separator; (3) orifice meter; (4) gas dietributor; (5) bronze net; (6) pump; (7) reservoir; (8)shutter plate; (9) shutter case; (10) solids withdrawal tap; (11) O-ring; (12) wire and pulley; (13) device for pulling wire.
I
I
100
h
I
150
$ Y
20
0 2 00
300
400
500
dpi C W I Figure 2. Size distribution of glass beads used in experiments.
entrainment of particles from the column. Air and tap water at room temperature were used as the gas and liquid phases, respectively. Binary and three-component mixtures of sieved glass beads were used for the solid phase (118 and 243 pm, 118and 465 pm, and 243 and 465 pm in diameter). Glass beads with a wide size distribution (mean diameter, 550 pm) were also used for multicomponent systems. The size distributions are shown in Figure 2. After the experiments were continuously run for about 1h to reach steady state, local values of solid holdup were
1164 Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992
t
I
I
I
=243urn l&:O045mk
01
8 0 01
1
2
3
4
I
0001-
[mi
Figure 3. Experimental and calculated results of axial changes in solid and gas holdups for two-component mixtures of 243- and 465pm diameters and 118- and 243-pm diameters in a 7-cm-diameter column.
measured by a shutter method. AJl the shutter plates were closed by pulling a wire, and the feeds of gas and liquid were stopped simultaneously. By this action the column was momentarily partitioned into 12 or 21 parts. The solid particles settled on each shutter plate were withdrawn through a tap after measuring the volume of slurry in each section and were dried and separated by sieving. The local value of solid holdup of each component in the slurry phase, 4 ',was determined by weighing the particles. The solid hofiup was defied as the volume fraction of either kind of particles suspended in the unit volume of the slurry. The mean gas holdup was calculated from the tohl volume of gas in each section. The axial dispersion d i c i e n t of liquid was measured by the impulse response method. The details were published elsewhere (Matsumoto et al., 1988, 1989).
zg=0257 :01 U p s 0 285mls I UI =O029m/s Ws; 3 1 1 kg WL 5 1 5 58kg I 001
1
-
0
-
-Eg
0 01-
1
--
a 1 1
-\-
E-
l
0 OOlL
L
I
0.1 :
I
I
I
'
I
I
I
I
10.01
1 1
I w "
Experimental Results Typical results of solid holdup in the 7-cm-diameter column are plotted as a function of axial distance for the binary mixture of 243 and 465 pm and for 118 and 243 pm in Figure 3. The amount of larger particles was chosen so that larger particles formed a dense fluidized bed in the lower zone of the column. As can be seen, the holdup of the larger particles decreases simply with an increase in the axial position in the freeboard region above the dense bed, but the axial holdup distribution of the smaller particles varies widely with operating conditions. The holdup of smaller particles often shows a maximum in the 7-cm-diameter column. The solid holdup in the 15-cm-diametercolumn for the binary mixtures is shown in Figures 4-6. Axial mixing in this column is much larger than that in the 7-cm-diameter column, which makes the solid holdup flatter than in the 7-cm column. Figures 7 and 8 show the axial distributions of solid holdup for three-component systems in 7- and 15-cm-diameter columns. Figures 9 and 10 show the results with particles whose size distribution is wide, as indicated in
'4 0.00lA
Eg;0225 L!q=O.220 mls Ul :0.016 m k Ws:3.31 kg WL =10.30kg
1
1
1
2 z
I
x
0.1
'0.01
1rn1
Figure 5. Experimental and calculated results of axial changes in solid and gas holdups for a two-component mixture of 118- and 243-fim diameters in a 15-cm-diameter column.
Figures 9a and loa. The size distribution function moves in the direction of smaller size at a higher axial position, as shown in Figures 9b and lob.
Modeling For a multicomponent mixture of solid particles in gas-liquid-solid systems,the axial change in solid holdup of each kind of component particle is analyzed with a one-dimensional dispersion model. The object of the For calculation is to obtain the steady-state value of
Ind. Eng. Chem. Res., Vol. 31, No. 6,1992 1565
' I
I
I
Eg=0.120 Ug=0.042 mls
0.1 z
01 rn
w
sa 0.01
-e"
0.001
001
0 01
,
0 001
1 1
0
1
1
0
w
1
2
z
3
[mi
0.1 0.2
-
-
W,=13.45kg
0.1 01
0
[m1
z
t 0
.,
Figure 6. Experimental and calculated resulta of axial changes in solid and gas holdups for a two-component mixture of 118- and 465-um diameters in a 15-cm-diameter column. I
'
'
I
'
I
I
'
02
I-
0 02
z =l.Srn
C
r
0.1
ln 0
1 1 - 1
5
0.1
0 0.2
0 500
o.olK\r 0.0011 0
'
'
1
I
'
\
'
I
2
3
'
'
1 4
10.01
[rnl
z
Figure 7. Experimental and calculated resulta of axial changes in solid and gas holdups for a three-component mixture of 118-, 243-, and 465-rcm diameters in a 7-cm-diameter column.
t
0.00ll 0
1
I
I
I
1
I
1
2 z
I
dDi
0.1
600 rum]
0 500 dpi
600 tum I
Figure 9. (a, top) Experimental and calculated resulta with a wide size distribution of solid particles in a 7-cm-diameter column. The solid lines are calculated axial changes in solid holdup. (b, bottom) Experimental and calculated resulta with a wide size distribution of soIid particles in a 7-cm-diameter column. The solid lines are calculated axial changes in the size distribution of particles.
multicomponent systems, however, it was impossible to find a set of appropriate values of 4pibecause the convergence of the calculation was strongly affected by the starting values. In this study, therefore, we use a set of unsteady-state material balance equations with arbitrary initial condition functions that satisfy the material balance for the total mass of i particles in the column. The steady-state solution for i particles is obtained as an asymptotic solution when the axial distribution of does not change with time. The unsteady-state material balance equation of 1 particles is given as follows:
Jo.01
[mi
Figure 8. Experimental and calculated reaulta of axial changes in solid and gas holdups for a three-component mixture of l l & , 243-, and 465-am diameters in a 15-cm-diameter column.
The initial condition is
single-component systems we used the steady-state material balance equation and began numerical iteration with an assumed value of 4 at an arbitrary z (Kato et al., 1972; Mataumoto et al., 19g9). This method was also used for two-component systems (Matsumoto et ai., 1991). For
is an arbitrary function to satisfy the following equation.
epi0
(3)
1566 Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 00 5
0.01
0 01
01 Ug
1 Imlsl
Figure 11. Correlation of El for the gas-liquid system. " L
I
02
01
1
I I 1 1 1 1 1
I
I
'
1
1
1
1
01
0 02
02
c U
e 01 .-
01
0
=s
0 A
dprum I 118 243
0
465 1300
0
C
0
" . 10
o
0.2
0
-
100
1000
Rei'
Figure 12. Correlation of Epi for a single-componentsystem.
01 0 500
600
dpl rum1
Figure 10. (a, top) Experimental and calculated results with a wide size distribution of solid particles in a 15-cm-diameter column. The solid lines are calculated axial changes in solid holdup. (b, bottom) Experimental and calculated results with a wide size distribution of solid particles in a 15-cm-diameter column. The solid lines are calculated axial changes in the size distribution of particles.
In a batch operation with respect to solid particles, the boundary conditions at the inlet and outlet are a4pi
2-0; Epi. - -a2 ---
Upi4pi
=0
(4) (5)
The time-averaged linear velocity of i particles with respect to the fixed coordinate at an axial position z is written as upi = u1- uti (6) where uti is the settling velocity of i particles in a quiescent fluid. u1is the linear velocity of liquid and is related to the superficial liquid velocity by (7) UI = U1/[(1 - e g N 1 - 4,t)l If u is negative, the particles tend to settle out and the holSup of the particles decreases with increasing axial position. The effect of interaction among randomly moving particles on upiand Epi has been well investigated for single-componentsystems (Mataumotoet al., 1988,1989). For multicomponent systems, however, the effect of particles of different classes must be considered at the same time.
To estimate the parameters in eq 1 from the results of single-component systems, the following assumptions are made. Gas Holdup. The local gas holdup for a gas-liquidsolid system with single-component particles is little dependent on particle size and is calculated as E , = U,(l - R)/(0.29[1 + 2.54pp85] + CIUg(l- R)) (8) where R = e,U,/[(l - e,)U,]. The constant c1 in eq 8 was determined to be 1.8 for DT = 0.07 m and 2.4 for DT = 0.15 m. Equation 8 was verified in the range of U,= 0.01-0.3 m d , U,= 0-0.15 ms-l, 4, = 0-0.3, and d, < 465 pm. Axial Dispersion Coefficient of Solid Particles. The axial dispersion coefficient of liquid for the air-water system in the case of a spiral tube used as gas distributor is expressed by
for DT = 0.07 m: k = 0.3[1 + 4U,/(1 - e,)] for DT = 0.15 m: k = 0.3 This equation can be used over a range U,= 0.01-0.3 ms-l and Ul= 0 . 1 5 m d as shown in Figure 11,including the data published previously (Matsumoto et al., 1988,1989, 1991). The axial dispersion of i particles, Epi,is related to El, as shown by Matsumoto et al. (1989). = 1 - 0.01(~e~*)2/3
(10) for Rei* = dPi[a(1 - R)(g&/2)]"2/v < 500. Figure 12 shows the correjfationof EPi. Since the effect of particle size on E, is not large, it was assumed that there was no difference in Epi among the
Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 1567 components. The axial dispersion coefficient of i particles in the multicomponent system is expressed by the following equation, which takes account of the volume fraction of each component.
f
t
-I
I
Ug =0.045m/s
N Ep
= CEpi4pi/4pt i=l
(11)
Relative Velocity. The relative velocity between single-componenti particles and liquid velocities is expressed as uti = fiUti (12) The terminal velocity of an isolated i particle falling in a liquid, uti, is expressed by Stokes', Allen's, and Newton's laws. For simplification of the analysis, the following equation is adopted for the whole region with a maximum error of about 2% (Matsumoto et al., 1989) Gai w4i (13) Y [M4j5 ( C U ~ / ~ . O ) ~ / ' ] ~ / ~
-- -
+
where Gai = d~3g(p,/pl - 1)/9.When a single particle falls through a gas-liqud-olid flow, however, eq 13 should be corrected as follows: utidpi GaiSi (14) Y [184/5+ (G~i{i/3.0)~/']'/~ where
-- -
Equation 14 was obtained experimentally by Matsumoto et aL (1989). All the phenomenologicalconstants appearing in eqs 14 and 15 can be unequivocally calculated by knowing tg,R, and Cui. The voidage function in eq 12 for single-component systems is given as follows: (16) fi = (1- $ J p p - l The value of ni for liquid-solid systems with single-component particles is given by Garside and Al-Dibouni (1977). According to Matsumoto et al. (19891, on the other hand, ni for three-phase systems is correlated by the following equation: (ni - 2)/(5 - ni) = ~.OGU;'/~ (17) Equation 17 gives a slightly smaller value than that pred i d by Garside and Al-Dibouni for liquid-solid systems. For multicomponent particle systems in a three-phase flow, the relative velocity between liquid and i-particle velocities is affected by its own holdup as well as that of others. Matsumoto et al. (1991) developed equations of fi for two-component systems (i = L and S). for S particles: fs = (1- 4 p-~4,~)"~-'(1 - 4 p ~ ) n ~ - n(18) s = (1 - 4ps- (bpL)"W - 4pS)ns-n~ (19) where (1- 4pL)n~-n~ and (1- 4 S ) n s - n ~are introduced to express the interaction between and L particles, and both are unity in the case of single-component systems. The values of ns and nLin eqs 18 and 19 are given by eq 17. This model is extended to multicomponent systems as follows:
for L particles:
fL
8
1
04
I
I
I
06 1
08
1
4,
Figure 13. Relative velocity, ut,between u, and up for a singlecomponent system. The solid and broken lines are calculated with and without the factor [l - (&J&J3]1/3, respectively.
The factor [ l - (4pt/4P33]1/3 in eq 20 was newly introduced to obtain a better fit with the data. Figure 13 compares results with and without the factor, and r # was ~ ~ ~obtained as 0.55 in this study. A few other expressions for fi were tried. Equation 20 gave the best result.
Numerical Calculation Equations 1-5 are rewritten in dimensionless forms by introducing the following dimensionless variables. t* = tUl/H, Z* = z/H, uPi* = Up,/U1, EPi* = Epi/UIH (21) Then the equations were rewritten in finite-difference forms by using upwind-difference approximation. Grid sizes of &* = 0.025 and At* = U,/H were used for the present computation. The convergence criterion at each - 4;)/4p' < 0.01. The point of the grid was given as superscript j is the number of numerical iterations. Thus a set of asymptotic steady-state solutions for solid holdup was obtained for each component of solid particles.
Discussion The solid lines in Figures 3-8 are calculated from the present model and are in agreement with the data. Figures 9 and 10 reveal that the particle-size distributions change with the axial coordinate z. The particle-size distribution is divided into 10 sections, and the material balances, eq 1, for each section are solved simultaneously. The experimental size distributions are well described by the model proposed. Jean et al. (1989) derived a mass balance equation for multicomponent systems and defined the linear velocity of i particles as
Equation 22 is identical to eq 6 used in the present study. Jean et al. (1989) further assumed upi = 0 for the batch system with respect to the solids and concluded that the model could not be applied to the batch system. However, the flux of the solid phase consists of convective flow, up$ and diffusive flow, -E,(&$,/dz), and upiis not necessarii);: zero in the column even without the feed of particles. In fast fluidized beds for gas-solid systems, uti increases with increasing solid holdup, which is a function of the feed of solids (Yerushalmi and Avidan, 1985). Then upican be negative in the lower part of the bed even if the gas velocity is much higher than the settling velocity of each particle, uti. In gas-liquid-solid fluidized beds, however, uti normally decreases with increasing solid holdup, as indicated in eqs 12 and 16. Therefore, the axial distribution of solid holdup disappears when the gas velocity becomes larger than uti.
1568 Ind. Eng. Chem. Res., Vol. 31, No. 6,1992
When the present model is applied to the system with a solids feed, the boundary conditions eqs 4 and 5 must be changed to (23)
Greek Letters e, = gas holdup 1;. = correction factor defined by eq 15 v = kinematic viscosity of liquid, mz/s pI = density of liquid, kg/m3 pp = density of solid particles, kg/m3 qbpi = holdup of ith particles in slurry
= total solid holdup,
As indicated by Jean et al. (1989),different authors use different definitions. The validity of a model should be judged on the basis of its ability to describe complex phenomena with appropriate values of the parameters used in the model. It is very desirable that all the parameters be correlated in experimental equations, as is done in the present study. With increasingcolumn diameter, the axial dispersion of particles increases in accordance with eqs 8-10. Then the axial distribution of solid holdup becomes legs notable in a larger column, and the segregation pattern is less dependent on upi,which is very sensitive to the experimental conditions. Equation 1, on the other hand, gives a proper value of Epifor every group of particles. The model developed by Murray and Fan (1989)emphasizes the entrainment of particles by bubble wake. Fundamentally, their model is related to eq l by replacing the mass flus due to bubble wakes and the wake-emulsion exchange rate with Epi and uti. Conclusion In a gas-liquid-solid fluidized bed containing a two- or three-component mixture of solid particles smaller than 465 pm, the axial distribution in solid holdup of each component was well expressed by eq 1. When particles with a broad size distribution were suspended, the size distribution at an arbitrary height was estimated. The calculation based on the model proposed was in agreement with the data. The parameters used in the model were correlated by eqs 8-17 and 20 as functions of particle size, mass of the particles in the column, and liquid and gas velocities. Nomenclature
DT = diameter of column, m d = diameter of ith particle, m
8 = axial dispersion coefficient of liquid, mz/s
E , = axial dispersion coefficient of ith particles, mz/s Fpi= feed flux of ith particles in inlet and outlet, m3/(m2.s) f = voidage function in eq 12 g = gravitational acceleration, m/s2 Cui = modified Galilei number of ith particles defined by dpi3gb /PI - U / V 2
H = heigkt of column, m
k = correction factor defined by eq 9 ni = exponent in eq 16 R = dimensionless parameter defined by c,Ul/(l - c,)U Rei* = Fbynolda number defied by d,&(l- R)(gDT/2)J1h/v U = superficial velocity of gas, m/s L( = superficial velocity of liquid, m/s ul = linear velocity of liquid, m/s u, = linear velocity of ith particles with respect to fixed coordinate, m/s Val= superficial velocity of slurry, m/s uti = relative velocity between uIand upi,m/s uti = terminal velocity of a single ith particle, m/s Wi = mass of ith particles in column, kg z = axial distance from gas distributor, m
Subscripts i = component of particles S = small-size particles L = large-size particles M = middle-size particles
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Receiued for reuiew November 20, 1991 Accepted March 3, 1992
