Liquid fluidisation curves

Powder Technology, 60 (1990)

Liquid Fluidisation N. G. STANLEY-WOOD,

Department

(Received

E. OBATA*

University of Bradford, Bradford BD7 1DP (U.K.)

and K. AND0

of Chemical Engineering, March

61

Curves

of Chemical Engineering,

H. TAKAHASHI

Department

61 - 70

21, 1989;

Muroran Institute of Technology,

in revised form July

5, 1989)

SUMMARY

A method to calculate the particle size distribution of irregularly shaped particles from measurement of the pressure drop obtained when a liquid is initially passed through a fixed bed of powder prior to subsequent fluidisation, has been developed from the measurement of pressure drop as a function of volumetric liquid flux during transition from a fixed to a fluidised bed. The particle size distributions calculated by computer analysis of the pressure drop curve as a function of liquid fluidisation velocity, using the concepts of volume-surface mean and oversized mass fractions, were in good agreement with the particle size distributions obtained by sedimentation measurements for a range of irregular-shaped particles.

INTRODUCTION

For a particulate bed of uniform particle density, there is a range of velocities between the onset of fluidisation of the finest particles and complete fluidisation of the coarsest particle. For uniform particle density beds, segregation of particles has been shown to occur, theoretically [ 1,2] and experimentally [3 - 51 at low velocities during fluidisation. Segregation in gas-fluidised systems generally decreases with high gas velocity to approach eventually a well-mixed bed [ 6,7]. For very wide particle size distributed gas systems and also narrow and wide particle size distributed liquid systems, segregation by size is an intrinsic factor in fluidised systems [2,3 8991. *On sabbatical nology. 0032-5910/90/$3.50

from

Muroran

Institute

Muroran, Hokkaido 050 (Japan)

of Tech-

Whenever a fixed bed of powder undergoes an expansion by the passage of a liquid through the bed prior to fluidisation, segregation of particles within that bed occurs which can be related to the elutriation of certain sized particles. Conversely, as the superficial fluid velocity gradually decreases in a liquidsolid fluidisation system from that necessary to fully support the maximum diameter particle within the system -- the full supporting maximum diameter velocity - to a minimum diameter fluidisation velocity which will just fluidise the minimum diameter particles within the system, all particles in that system will have undergone size classification [ 10,111. The relationship between the full supporting maximum diameter velocity, the minimum particle size fluidisation velocity or in general terms the superficial bed velocity and the pressure drop across the bed can be regarded as having contributed to the formation of the experimental fluidisation curve. This fluidisation curve can subsequently undergo computer analysis to evaluate the sizes and distribution of particles within the fixed-fluidised system. Obata et al. initially [12,13] used a graphical method to evaluate the particle size distribution of regular glass beads fluidised by water from the observation that when a fluid passed through a packed bed the pressure drop across the bed increased as the velocity increased. When the superficial velocity reached the minimum fluidising velocity, all regular particles of minimum size were just fluidised. As the fluid velocity was increased, all regular particles were fluidised and the pressure drop was equivalent to the weight of particles per unit area. On the basis of these observations, a method of particle size distribution of regular shaped particles was developed (eqn. (1)). @ Elsevier

Sequoia/Printed

in The Netherlands

62

In this paper, however, the evaluation of the particle size distribution of particles of irregular shape has been obtained by inclusion of a shape factor into the Kozeny-Carman equation used for flow through porous media. The flow of liquid through the fluidised system must be of a magnitude to ensure low Reynolds numbers (<2) to validate the use of the Kozeny-Carman equation in this computational analysis. The particle size distribution determination, from initially a completely fluidised and then, on gradual liquid velocity reduction, to finally a fixed bed of particles, can be accomplished in situ within a process without particle removal. At the same time, the voidage and Reynolds number can be measured and calculated to justify the Kozeny-Carman equation, which is subsequently used in the particle size computations. This method of particle size analysis can be used as an adjunct to online particle size distribution measurements .as there is no requirement for irregular particle calibration inherent with laser scattering and light-obstruction techniques. The iterative solution of this deduced pressure drop superficial velocity relationship which relates, at low Reynolds number, mass frequency and particle size to pressure drop and superficial velocity when a fluidised bed is transformed into and reduced to a fixed porous bed, gave particle size distributions which were comparable to size distributions obtained by other laboratory techniques.

0

sun

u.cu;

BUQ

APi = APii + AP,i = (pressure drop in fluidised section) + (pressure drop on fixed bed)

If the minimum and maximum diameter of particles within a bed are symbolised as X, and X, respectively, then the corresponding velocities to maintain these particles in a state of minimum fluidisation can be designated U,, and U,, respectively. Pressure

drop across fixed and fluidised beds When all particles in a bed are in a state of fluidisation, the bed pressure drop across that bed (AP,) can be expressed in terms of the weight of particles present per unit area. 0

u,

As the superficial velocity within the bed is gradually decreased from a fluidisation velocity, some particles in the bed are not fully supported and therefore undergo a size classification on the reduction of velocities through the bed (Fig. 1A). At a superficial velocity, Vi, which has an intermediate value in the range U,, to U,,, particles of size x, (minimum size) to xi are fluidised while particles of size xi to x, (maximum size) remain in a fixed bed state (Fig. 1B). If the pressure drop of the fluidised and fixed bed sections are now designated aP,i and APzi, respectively, the total bed pressure drop at a velocity Vi becomes pi, which can be expressed as

THEORY

@

l

Fig. 1. Relationship between bed pressure drop and particle diameter.

= (P, - P)gM, Pp A

where A& and A are the total mass of particles and cross-sectional area of the bed, respectively.

The relationship between pressure drop and mass frequency of particles present in terms of the fluidised particles of size x, to Xi and the particles in the fixed bed of size xi to x, can then be expressed as Ap

i

=

(” -p)gMt Xifm(x) I x,

ppA

+

5(1

-

Gnf)PUi&,

emf3ppA

dX

xo

i* Mt

s %i

flYI

dx (1)

where f,(x) is the mass frequency of particle size x; M, 1:~ f,(x) dx is the total mass of all particles of size Xi to X, in a bed of cross-sectional area A, which has a surface area per unit volume Of particles Xi t0 X, Of S,_ i.

63

The relationship between Al’i and Vi has been shown by Obata et al. [12,14,15] to give the particle size distribution of regular particles within a fluidised system. The relationship between the surface area per unit volume of particles and the volumesurface mean diameter of irregular-shaped particles [16,17] can then be incorporated into the Kozeny-Carman contribution in the flow transition period from fluidised to fixed bed state.

6 3c32, i = Yhsv,

i

x0

s s x’fdx) dx x3f,(x) dx

=

5i

X0

xi

X0

s

f&x) dx

Volume-surface mean diameter If the shapes of irregular particles in the same range x, to x, are deemed to be similar and particle sizes from x, to x, are in a continuous progression, the specific surface area of all particles in the bed can be expressed as S,. n [ 171. 71

s v. n =

K,

s

6 J xn

=-

x’f,(x)

(3)

;” f,(x)

J

d;

x

xi

The volume-surface mean diameter, 232, i, in the fixed section can alternatively be defined in terms of the number of particles as

x0

71 x0

6

=

“i

i Nixi i=i

dx 532,

i =

5Mi i=i

=-

$Jixi2

x3f?l(x) dx

Xo f,(x) J X

dx

6 (Psz32, n

where x refers to the diameter of a particle of equivalent volume sphere, f,(x) a number frequency of particle size x, f,(x) is the mass frequency of particles of size x, cpsa shape factor (where cps< l.O), and 3C32,n is the volume-surface mean diameter of all the particles in the fixed and fluidised section of a bed. Within the fixed bed section (Fig. lB), however, which has particles in the size range xi to x0 only, the volume-surface mean diameter can be written in an analogous manner to eqn. (2). Thus the volume-surface mean diameter for the fixed particles in this particulate assembly section, Z32+i, becomes

i=i

i

!!!!

i=i

Xi

where Ni is the number of particles of size Xi, and Ml the mass of particles of size xi. Since a volume-surface mean diameter is equivalent to an harmonic mean diameter on a mass basis [16,17], the volume-surface mean diameter for a range of sized particles can be defined in terms of the mass fractions of specifically sized particle [14, 151 to give For particles between the sizes x0 and Xi with a volume-surface mean diameter of $32, i at a fully supporting velocity U, at pressure API:

WI) x32.1

(54

= R&O) z32.

+

R&l)

0

-J-W,) Xl

+x0

2 For particles between x2 and x0 with a volume--surface mean diameter of 232.2 not at a fully supporting velocity and at pressure AP,: R(3~2) x32’

2 =

R(xl) 232.1

+

R(x2)

-

R(x

3c2 +x1

2

1)

(5b)

64

For particles between Xi and X0 with a volume-surface mean diameter of x32,i superficial velocity Vi :

of pressure drop across both fixed and fluidised beds (eqn. (1)) gives, when rewritten,

at a

ap

R(Xi)

%32,i =

WV,)

x2 -.

+

R&A W-I)

+

(54

R(x,)

-

R(x,-

I)

ap

2

i

=

(Pp - P)gMt[l P*

The cumulative oversized mass fraction of particle size Xi is expressed as

R(Xi) = J fm(X)do

+ 180(1-

(6)

x

which is shown diagramatically in Fig. 2 and where = 0 and R(x,,)

= 1

Measurement of size distribution from a pressure drop velocity curve Substitution of the specific surface in terms of a particle shape factor (psand the volumesurface mean diameter ~32,i (eqn. (3)), together with eqn. (6) into the initial equation

_

--R(xi)l

A

Emf)PUiMt

x

1 R(xi)

%nf3P,A(Ips2)

xi

R(x,)

(7)

i)2

Subsequently, substitution of the defined volume-surface mean diameter (Z32,i) on a mass basis (eqn. (5)) into eqn. (7) gives a quadratic equation in terms of total bed pressure drop APi, superficial velocity Vi and the cumulative oversized fraction R(xi).

xn +x,-1

?32,n-1

Emt)/JU’iMtR(Xi)

180(1-

Enh3PpA(~s~32,

For particles supported at a minimum fluidisation velocity: =

--R(xi)l

PpA

+x1 2

X32,n

(P, -~)gMt[l

(SC)

+ R(Xi) - R(xi - I)

x32,i-1

=

i

R(Xi-1) +R(xi)-RR(xi-~)

I -

x32,i-l

Xi + Xi-1

2

2

I

This deduced relationship between aPi and Vi together with the minimum fluidisation voidage emf is shown in Fig. 3 and can be obtained experimentally as the fluidisation curve. At low Reynolds number, the KozenyCarman equation, which describes the flow of fluids through packed beds of irregularly shaped particles, can be used, knowing the

0.20

‘; : 0.15 uA E 0.10 aJ 3 oE IL

005

0

Fig. 2. Relationship

j, b* h I0

I2

Diameter,

[ pm 1

I

xn

between

32,”

32.1

x

the volume-surface

(8)

I4

mean diameter

x1”. 0

X32,i and mass or number

frequency

curves.

65

wherei=1,2

,..,,

n.

The equivalent diameter of Wen and Yu is identical to the volume-surface mean diameter to ~32, n used in this paper:

Superflclal velocity ” I m/sI Fig. 3. Graphical relationship between aPi, Vi, Xi, ui, 232. i and R(Xi).

wherei=1,2

R(x,) = 1 and i i=

the superficial velocity Vi through a packed bed of known voidage, to calculate the diameter of an irregular shaped particle within a packed bed, Xi, corresponding to the superficial velocity Vi. The relationship between Vi and Xi can be expressed as Ui

Gf3

=

1w

- Glf)

(Axi

J2(Pp - Pk 18/J

(9)

The particle with the largest diameter X, within the fluidised system can be obtained from eqn. (9) by substitution of the experimentally measured fully supporting liquid velocity U,,. Similarly, the value of diameter Xi, which is smaller than X, , can be calculated from the superficial velocity Vi, which is less than U,. Particle size distribution of irregularly shaped particles To determine the particle size and particle size distribution from the superficial velocity Ui and pressure drop APi measured by the fluidisation curve, knowledge of the shape of irregular particles is necessary to evaluate the shape factor (psin eqn. (8). Since it is difficult, time-consuming and tedious to measure the profiles and then calculate the shape factors of irregular particles, the relationship of Wen and Yu [18] for fluidised particulate systems has been used. Wen and Yu reported that an equivalent particle diameter 5&, of all particles in a fluidised bed could be expressed as

n,

,...,

AR(Xi) = 1. 1

The relationship deduced by Wen and Yu between the voidage for fluidised bed and the shape factor of irregular particles can be expressed numerically as

1-Gllf = 11 (P,2e,f3

(10)

and 3

1

= 14

(11)

Wmf

The shape factor of the fluidised particles can thus be evaluated from the experimental minimum fluidisation voidage. The arithmetic mean of the shape factors calculated from eqns. (10) and (11) are used in the computer iteration analysis of eqn. (8). The oversized mass fraction of particles of size x1 was calculated by substituting into eqn. (8) the calculated values of X, and 3tI and the experimental value of API obtained from the passage of a superficial velocity U, through the particulate system. Similarly, using eqn. (9), the diameter of particle size x2 can be calculated from the experimental value of U,. This evaluated particle size x2, together with the value of x1 and R(x~) can subsequently be used (eqn. (8)) to calculate R(x~), knowing AP,. The computer iteration analysis of the fluidisation curve in terms of APi, Vi, x1, ui9 f32.i and R(Xi) using eqns. (3) and (9) can then compute a particle size distribution from the experimental fluidisation curve APi and Vi. These relationships are shown in Fig. 3 with the maximum Reynolds number,

which must be less than 2 for eqn. (9) to be valid, calculated from the relationship

(12)

EXPERIMENTAL

The water-particle-fluidised-fixed bed experimental apparatus which has been previously described for use with spherical particles [12,13] was used to determine the pressure-velocity (APi versus Vi) relationship for three powder-water systems. (i) Semi-spherical glass beads containing a 50:50 mixture of two differently sized beads in the ranges 125 - 148 pm and 74 - 88 pm respectively [ 71. (ii) Irregularly shaped crushed glass in the sieve size fraction of 62 - 500 pm.

(iii) Irregularly shaped silica sand (Toyoura, Japan) in the sieve size fraction 74 - 177 pm. The particle size distributions calculated from the iterative computer analysis of the pressure-velocity-fluidised-fixed bed relationship were compared with the particle size distributions obtained by optical microscopy using an ultra-microscopy projector (Toshiba MP30), by light attenuation using a photoscanning particle size analyser (Hitachi PAS-2) or by sedimentation using a particle size balance (Shinazu SA-2).

RESULTS AND DISCUSSION

Bimodal distribution beads

of semi-spherical

glass

The experimental results for pressure drop and superficial velocity in Table 1 were obtained from the fluidised system starting

TABLE 1 Experimental and calculated data for a mixture (50:50) ranges 125 - 148 I.trnand 74 -88 pm System data Cross-sectional area of bed Density of fluid Viscosity of fluid Initial voidage Mass of particles in bed Acceleration due to gravity

Experimental and computed data Experimental Pressure drop

of glass beads (pn = 2.487

A P

12 emf Mt g

2.734

X

X

low3kg mv3) in the size

lop3 m2

0.997x lo3 kg rnp3 0.916 x lop3 Pa s 0.424 0.400 kg 9.805 m sw2

Computed

(Pa)

Superficial velocity Vi lop4 (m s-i)

Particle diameter Xi 10m6 (m)

Volume-surface mean diameter 232.i 10m6 (m)

Mass fraction of size xi B(xi) (-)

880.0 877.0 872.0 866.0 854.0 821.0 770.0 718.0 667.0 597.0 473.0 360.0

3.70 3.40 3.10 2.80 2.50 2.20 1.90 1.60 1.30 1.00 0.70 0.50

201 193 184 175 165 155 144 132 119 105 88 74

201 197 201 192 181 168 166 167 167 154 132 124

0.00 0.08 0.06 0.10 0.18 0.47 0.51 0.50 0.49 0.60 0.83 0.92

api

Mt = 0.410 kg %,a,

= 0.072

qs = 0.883

67

with a maximum Reynolds number (eqn. (12) of Remax = (997.6)(3.7

X 10-4)(0.883)

X (2.10 X 10-4)/(9.16

X 10-4)

= 7.16 X 1O-2 < 2.0 which indicates a laminar flow of liquid around the particles and therefore justifies the use of eqn. (9).

Particle

diameter.

@%IX

a 10’

[ml

Fig, 4. Particle size distributions of bimodal 50:50 mixture of semi-spherical glass beads.

from a state of complete fluidisation and then decreasing the water velocity until the particle bed was completely settled. The minimum bed voidage emf was evaluated from the settled bed height. The results calculated from the computer iteration of eqns. (8) and (9) are also given in Table 1. The mass of particles in a fluidised bed does not contribute to the calculation of the cumulative oversize mass fraction R(IY~) when this fraction is expressed as the ratio (APj - APli)/APi. Since the pressure drop across a fluidised bed can be expressed as AF’, = (p, - p)gM,/p,A, the mass calculated from pressure drop data (M,) can be compared with experimental reality (M,). The calculated mass (M,) is M, = AP,p,A/(p,

- ,o)g = 0.410 kg

which is comparable with the experimental value of 0.400 kg. Figure 4 shows a comparison of the computer-calculated oversize fraction versus particle diameter of the bimodal glass bead mixture obtained from eqn. (8) with the size distributions obtained from microscopy and the initial geometric distribution [ 121. The apparent minimum fluidisation velocity 0, calculated from eqn. (9) was v

_ (0.424)3(0.883)2(1.15 nlO(1 - 0.424)18(9.16 x

X 10-4)2 X 10-4)

E(2.487 - 0.9976)103]9.805

lO(1- 0.424)18(9.16 = 1.20 X 10e4 m s-l

X 10-4)

Irregularly

shaped crushed

glass

Irregularly shaped particles of crushed glass were fractionated into the sieve size range of 62 - 500 pm and then fluidised in a watercrushed glass system. The pressure drop was recorded by a carbon tetrachloride manometer from the fluidisation system until, with velocity decrease, a fixed bed ensued. The system data together with the experimental and calculated data for crushed glass particles of irregular shape are shown in Table 2. A graph of the computed particle sizes and cumulative mass fraction of those sizes obtained from the fluidisation curve is shown in Fig. 5 together with the particle size distributions obtained by microscopy and photo-attenuation. The size distribution obtained by computerisation (curve 1) agrees well with the size distribution obtained by photo-attenuation. The size distribution obtained by microscopy, however, was not in accord with the computer output data. The shape factor of the crushed glass obtained from eqns. (10) and (11) gave a mean value of 0.430. The calculated value for Remax was 1.78, which is less than 2 and justifies the application of the Kozeny-Carman equation to evaluate shape factor of particles prior to the computer iteration procedure.

shaped silica sand Irregularly shaped particles of silica sand were sieve fractionated into sizes in the range 74 - 177 pm prior to measurement of the fluidisation curve. Experimental data from the fluidisation curve, together with the computer analysis output data, are shown in Table 3. The mean Wen and Yu shape factor calculated from the Irregularly

68 TABLE 2 Experimental and calculated data for irregularly shaped particles of crushed glass (pp = 2.500 the size range 65 - 500 pm System data Cross-sectional area of bed Density of fluid Viscosity of fluid Initial voidage Mass of particles in bed

A P l-l

lo3 kg me3) in

3.589 X lop3 m* 0.997 x lo3 kg me3 0.837 X 10m3 Pa s 0.566 0.300 kg

%lf

Mt

Experimental and computed data Experimental

x

Computed

2)

Superficial velocity Ui 10M4 (m s-r)

Particle diameter Xi 1O-6 (m)

Volume-surface mean diameter X3*,i lop6 (m)

Mass fraction of size Xi E(xi) (-)

494.9 494.0 488.0 478.0 465.5 442.0 391.0 305.8 176.4 72.5

4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.20

771 727 680 630 575 514 445 364 257 163

771 749 712 715 704 602 556 508 461 429

0.00 0.03 0.16 0.15 0.17 0.40 0.59 0.78 0.94 1.00

Mt = 0.301 kg Remax = 1.78

cps= 0.403

Pressure drop

-o-

This

work

-+-Microscopy -,a-.

Photo-

0 0

6

4

2 Particle

diameter

,

0%~

~10'

8 [m

I

Fig. 5. Particle size distributions of irregularly shaped crushed glass.

minimum fluidisation voidage and eqns. (10) and (11) was 0.525. Figure 6 shows the size distributions obtained by computer, microscopy and sedimentation (curves 1, 2 and 3, respectively). The particle size distribution obtained by computer analysis is in good agreement with the particle size distribution obtained by

microscopy. The discrepancy between the size distribution observed by microscopy, based on projected area, to that obtained by computer or sedimentation is to be expected because of the departure of irregular particles from sphericity. Such differences are normally expected between distributions obtained by different methods. The projected area diameter measured by microscopy is non-comparable to the diameters evaluated by fluidisation to maintain a particle supported in a fluid stream. This method of particle size determination which measures particle sizes from fluidisation curves is based on the use of the KozenyCarman equation and the force balance on a fluidised particle. A similar technique of analysis may be applicable at higher Reynolds numbers by use of the Ergun equation. CONCLUSIONS

Irregular particle size distributions can be measured from the fluidisation curve which relates pressure drop and superficial

69 3

TABLE

Experimental and calculated the size range 74 - 177 pm

data for irregularly

System data Cross-sectional area of bed Density of fluid Viscosity of fluid Initial voidage Mass of particles in bed

Experimental Experimental Pressure

and computed

shaped

particles

of silica sand (pu = 2.647

A P

3.589 0.997

I-1

0.865 0.523 0.230

%If

Mt

X X X

x 10-a

kg m-3) in

10m3 m2 lo3 kg m-3 10e3 Pa s

kg

data Computed

drop

Superficial Ui lo-4 (m s-l)

api (Pa)

velocity

Particle diamter Xi 10e6

Volume-surface mean diameter Z3.2, i 10m6

(m)

Mass fraction

of size xi

R(xi) (-)

(m) 397 358 324 286 241 216 187 153 108

1.35 1.10 0.90 0.70 0.50 0.40 0.30 0.20 0.10

386.9 381.4 375.2 366.4 350.8 336.1 286.3 208.5 107.5

0.00 0.14 0.08 0.16 0.21 0.30 0.82 0.88 1.00

397 377 414 350 324 288 226 221 204

Mt = 0.227 kg Re,,, = 0.32

-

1.0

;

0.8

1 \

z ; .t

0.6

e

0.4

--*--Microscopy \

\.

--•--- Settling \\ \

balance

=:

systems (bimodal mixture of semi-spherical glass beads, irregularly shaped crushed glass and silica sand) in good agreement with distributions obtained from sedimentation measurements.

LIST

.-:

cps = 0.525

OF SYMBOLS

0.2

0 0

0

Fig. 6. Particle silica sand.

1

2

Partacle

diameter,

size distributions

4

3 0,x

~10’

I ml

of irregularly

shaped

velocity within a fluidised system. With particles in the size range Xi to x, in a fixed bed, the volume-surface mean diameter Z32,i within that fixed bed can be expressed as a function of the oversize fraction R(xi) of the particle size 3ti. Computer iteration analysis of the fluidisation curve gave particle size distributions for three irregularly shaped particulate

fn(x)

F4 N

g

i$) Remax &

u

cross-sectional area of bed, m2 mass frequency of particle size x, m-l number frequency of particle size X, m-l acceleration of gravity, m ss2 mass of particles, kg number of particles bed pressure drop, Pa pressure drop of fluidised bed, Pa pressure drop across fixed beds, Pa cumulative oversize fraction of particle size x Reynolds number, p,U,(~,x,)/~ specific surface area, m-l superficial liquid velocity, m s-l

70

u?l X

&, ?32. i

Gnf

P

P PP A

apparent minimum fluidising velocity, m s-l particle diameter, m equivalent diameter of Wen and Yu, m volume-surface mean diameter from xi tox,,m voidage at the incipient fluidisation viscosity of liquid, kg m-l s-l density of liquid, kg me3 density of particle, kg mP3 shape factor from Wen and Yu

Subscripts C calculated i ith 0 largest nth or smallest t” total REFERENCES J. L.-P. Chen, Chem. Eng. Commun., 9 (1981) 303. J. F. Richardson, in J. F. Davidson and D. Harrison (eds.), Fluidisation, Academic Press, London and New York, 1971, pp. 25 - 64. J. L.-P. Chen and D. L. Keairns, Can. J. Chem. Eng.,

53 (1975)

395.

4 J. F. Richardson and W. N. Zaki, Trans. Znstit. Chem. Engrs., 32 (1954) 35. 5 A. W. Nienow, P. N. Rowe and T. Chiba, AZChE Symp. Ser., No. 176, 74 (1978) 45. 6 R. R. Cranfield, AZChE Symp. Ser., No. 176, 74 (1978) 54. 7 L. T. Fan and Y. Chang, Can. J. Chem. Eng., 57 (1979) 88. 8 D. Geldart, J. Balyens, D. J. Pope and P. Van de Wijer, Powder Technol., 30 (1981) 195. 9 R. di Felice, L. G. Gibilaro, S. P. Waldram and P. U. Foscolo, Chem. Eng. Sci., 42 (1987) 639. 10 R. P. Vaid and P. Sen Gupta, Can. J. Chem. Eng.,

56 (1978)

292.

11 M. R. Al-Dibouni and J. Garside, Trans. Chem.

Engrs.,

(1986)

619.

57 (1979)

Inst.

94.

12 E. Obata, H. Watanabe and N. Endo, J. Chem. Eng. Japan, 15 (1982) 23. 13 E. Obata and H. Watanabe, Encyclopedia of Fluid Mechanics, Vol. 4, Gulf Publishing, Huston, 1986, p. 221. 14 E. Obata, H. Watanabe, K. Mukaida, M. Akiyoshi and K. Ando, Kagaku Kougaku Ronbunshu, 12 15 E. Obata, H. Takahashi, M. Akiyoshi, K. Ando and H. Watanabe, Kagaku Kougaku Ronbunshu, 14 (1988) 103. 16 N. Stanley-Wood, Enlargement and Compaction of Particulate Solids, Butterworth, London, 1982. 17 G. Herdan, Small Particle Statistics, Butterworth, London, 2nd edn., 1960. 18 C. Y. Wen and Y. H. Yu, Chem. Eng. Progr. Symp. Series, 62 (1966) 100.