Destabilization of fluidized beds due to agglomeration part II: Experimental verification

Destabilization of Fluidized Beds Due to Aqqlomeration Part 11: Experimental Verification c c

GABRIEL TARDOS, DOMlNlCK MAZZONE and ROBERT PFEFFER Department of Chemical Engineering, The City College of The City University of New York, New York, New York 10031 The limiting fluidization-defluidization condition for fluidized beds of sticky particles has been studied experimentally in both low temperature beds in which defluidization occurs due to the presence of sticky fluids and in high temperature beds where agglomeration occurs due to sintering of the granules. The experimental results described herein as well as others taken from the literature are compared to the predictions of the theoretical model developed in Part I of this paper. On a 6tudiC expkrimentalement la condition limite fluidisation-defluidisation dans le cas des lits fluidisks de particules collantes dans les deux cas suivants: dans les lits B basse temperature dans lesquels la dkfluidisation se produit par suite de la presence de fluides adhesifs, et dam des lits B tempkrature 6levCe ou l’agglomkration est due du frittage des granules. On compare les rksultats experimentaux dkcrits dans le prCsent travail ainsi que ceux d’autres rapport& dans la litteratwe avec les prkvisions du modkle thCorique Ctabli dans la premiere partie de cet article.

P

art I of this paper (Tardos et al., 1985) has presented a review of theoretical and some experimental studies of agglomerating fluidized beds where aggregation of particles is caused by interparticle cohesion. In this second part, a review of some of the more recent experimental studies in the field as well as some new experimental results obtained by the authors are presented together with a comparison of these findings with the theoretical model. Basu (1982) and Basu and Sarka ( 1 983) studied agglomeration of coal-ash in an externally heated fluidized bed. Liquid petroleum gas was burned in a fluidized bed of ash particles between 0.7- 1.4 mm in diameter. The fluidizing gas flow rate was reduced slowly until defluidization occurred. The experiments were repeated at temperatures below and above the minimum sintering temperature of the ash granules in a manner similar to the experiments performed by Siegell (1976). The minimum sintering temperature was determined in a separate experiment using a dilatometer, as was proposed earlier by Gluckman et al. (1976). The experimental defluidization data were correlated by Basu (1982) by an expression of the form:

- 1

(us/~mj)

=

IJJ(T,Lo,Dp)/(l-

E)(P,

- P,).

...

(1)

where U , is the limiting gas velocity which keeps the bed ~ minimum fluidfluidized at the temperature T, u , is~ the ization velocity without agglomeration at temperature T, Lo is the bed height, D, is the particle diameter, E is the bed porosity and p., and ps are the solid and gas density, respectively. The function IJJ was written in the form:

IJJ

= K ( T - T,)fi(DP)f2(Lo). . . . . . . . . . . . . . . . . . . (2)

where T , is the minimum sintering temperature. The constant K was determined from the experiments as the slope of the experimental fluidization velocity vs. temperature, but the functions f,(D,) and f 2 ( L o )were not calculated. A more complete review of defluidization data at high temperatures was presented by Liss et al. (1983). These authors, using the experimental data of Siegell (1984), proposed an expression to compute the limiting gas velocity at high temperatures which, for small particle sizes, i.e., low Reynolds numbers, is given by 384

(U,/U,) - 1

=

a [ ( T - T,)/T,]b . . . . . . . . . . . * . . . (3)

where a and b are constants determined from experiments and T, is the minimum sintering temperature. Values for a and b were determined by curve-fitting, as shown in Figure 1. The drawback of the above equations proposed to compute the limiting velocity, Us, is that the function IJJ, in Equations (1) and (2) or the constants a and b in Equation (3) must be determined from fluidization experiments. In our companion paper (Tardos et al., 1985, Part I), a theoretical relationship for the excess gas velocity, U , - u,~,-,in a sintering fluidized bed was developed in the form: ( U s - urn!)/urnj= [3.3( 1 - ~ ) d B ” ~ d b ” ~ / ~ ~ D , ~ ’ ~ l x (psa,/+p,g”2q,)0

. . . . . . . . . . . . . . . . . . . . . . (4)

where, in addition to the notations already used, dB is the bed diameter, db is the bubble size, u, is the yield strength of the sinter neck and qsis the granule surface viscosity. One has to note here that, unlike Equations (1-3), Equation (4) was determined from purely theoretical considerations and that the R.H.S. of Equation (4) does not depend explicitly on the temperature difference ( T - T,)/T, as in Equation (3),but rather depends on two physical properties, the neck yield strength a,and the surface viscosity q,, which are both temperature dependent. In fact, for a solid particle which does not deform appreciably (below the sintering temperature T s ) , the viscosity, qs,approaches infinity and the limiting velocity becomes U , = u r n ,At higher temperatures, the viscosity q, decreases and the particles stick together forming agglomerates. Thus, the definition of the minimum sintering temperature T,, as that temperature at which the fluidized granule surface viscosity suddenly decreases becomes clear. The advantage of the theoretical expression Equation (4), over the empirical equations, Equations (1-3), is that the limiting velocity Us can be determined directly, once the granule properties a, and q, are known or are measured in separate non-fluidized bed experiments. A comparison of the results obtained by Basu (1982) and Liss et al. (1983) and by the present authors is given below for both low temperature beds as well as high temperature agglomeration.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, JUNE 1985

I I

Copper o Glass

ROTAM E T E R

COMPRESSED AIR

n

STORAGE TANK

NOZZLE

a Coal Ash ( f r o m American Electric Power)

(from Institute of Gas Technology)

+ Coal Ash

TRANSDUCER

FLUIDIZED

I

Wealth Edison of

I *-

E

101 r

-.E

l

,

In

I

-1

P /

\FEE;

S I N T SUSTA""G ERED PLATE

I

Q L- ; TEMPERATURE

1

h"u + 3

Figure I - Dimensionless excess velocity vs. dimensionless temperature (after Liss et al., 1983).

Apparatus and procedure A schematic representation of the experimental apparatus is shown in Figure 2. Fluidizing air from either one of two Roots positive displacement rotary pumps is passed through an Adams Poro-Stone centrifugal air filter which eliminates oil and water from the air stream. The compressed air stream is then sent through a regulator and the total flow rate is measured by a rotameter. Air temperature and pressure are measured at the rotameter exit to correct for air density changes. From this point the air stream can be passed either through heater (1) or through heater (2), or both. Heater (1) is a Chromalox 30 kW circulation heater, which can heat 2.5 m'/min of air to a maximum of 750°C. Heater (2) is a 9 kW packed bed unit designed to heat air up to 250°C. The air stream exiting the heaters is split and enters a mixing chamber 20 cm in diameter at two opposite side openings. Subsequently, the flow enters a 15 cm diameter calming section in which the gas pressure and temperature are measured. The pressure is measured by a Validyne DP15 pressure transducer and the reading is displayed on a four channel HP Voltmeter and printed. Gas temperatures are measured by two K-type thermocouples, one of which is connected to the HP Voltmeter and its reading is printed. The other thermocouple is connected to a temperature controller, which regulates the heaters and ensures a set air temperature at the entrance of the fluidized bed. The air then passes through the distributor plate and enters the bed. The bed consists of a 15 centimeter diameter "measuring" section in which an exposed junction thermocouple measures the bed temperature which is then displayed on the HP Voltmeter and printed. Above this section a 15 cm diameter, 60 cm long Pyrex glass tube is mounted which allows for visual observation of the bed. The bed is also equipped with a spray nozzle of the Bete 40" full cone type which is

Figure 2 - Schematic of experimental apparatus to study defluidization phenomena in fluidized beds.

retractable. Spray solution stored in a steel tank is metered and is injected through the nozzle with compressed air. The sprayer is able to inject up to 90 cm3/min of spray with an air pressure of up to 300 kPa. An additional test section was designed and built to replace the transparent Pyrex tube. This section, made of stainless steel, 15 cm in diameter and 25 cm long, was mounted directly above the distributor plate and was equipped with a Validyne DP15 pressure transducer to measure the pressure drop, A P / , between two fixed points in the bed, situated at a given distance I, apart. The value, AP,, can be used to calculate the average porosity of the bed at any gas velocity since E =

1 - [AP//g(ps - p8)/l]

. .. .. .. . . .. . . . . . . .

(5)

where ps is the density of the solid particle, pn is the density of the fluidizing gas, and g is the acceleration due to gravity. The test section is also equipped with ceramic heating elements which were regulated to keep the bed wall temperature equal to the gas temperature and thus minimized horizontal temperature gradients in the bed. All temperature, pressure and flow rate readings are also recorded and displayed on a Digital Equipment Minc 1 1 data acquisition system so that defluidization data are readily obtained directly from the computer. During a typical low temperature (liquid coating) experiment, the total amount of fluid sprayed by the nozzle onto the bed is metered .and pressure drop data at a given gas velocity is recorded until defluidization occurs. Defluidization is observed when the pressure drop through the bed decreases suddenly because particles are no longer sustained by the gas stream. When the data were compared with the theoretical predictions, it was assumed that the coating (high viscosity fluid) was evenly distributed in the bed. The premise for this assumption was met by adding the sticky coating over a long period of time to ensure that a thorough

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, JUNE 1985

385

4.00

1

2-14 mesh. The material properties of the particles and the gas were: p s = 1400 kg/m3, pn = 1.2 kg/m3, p,,, = 1200 kg/m3, d, = 0.0762 m, W , = 0.4 kg and E = 0.4. Using these properties the experimental data could be correlated as seen in Figure 3 by the expression

TCC Catalyst

A 6p

c

E

I

.14cm

repression line to f i t all experimental data.

m

0.00

I

I

0

5

(Us-

15

10

25

20

Wrn

~ 3 / 2d 3 / 2 4112 m p FJ

Figure 3 - Dimensionless excess velocity vs. dimensionless liquid content of the bed. TABLE1 Dimensionless Excess Velocity as a Function of Average Bed Temperature for Two Sizes of Polyethylene Particles

uM)/urnf =

0.098( W r n / p r n D p 3 / 2 d g 3 / 2.~. ”. 2. )(6)

whereas the theoretical result (Equation (I@, in Part 1 of this paper (Tardos et a]., 1985)), assuming that the critical agglomerate size dagES d,, yields a coefficient (quantity outside the brackets) of 0.040. Taking into account the large number of assumptions made in order to obtain the theoretical results, the fit of the experimental data is quite good. The main reason why the theory underpredicts the value of the gas velocity Usis that the aggregate strength is in fact larger than that initially assumed which takes into account only surface tension. In reality, the force separating the particles in the agglomerate has to overcome not only the pressure due to the surface tension, but also the viscous flow of the material around the contact point. This effect was not taken into account in the theoretical considerations. Defluidization experiments at high temperature

Low temperature (coating) experiment

The sintering experiments were performed using two sizes of polyethylene particles of diameter Dp = 0.32 cm and Dp= 0.07 cm. The minimum fluidization velocity, umf, at room temperature was determined from the intersection of two linear least-squares fitted lines corresponding to the pressure drop in the fixed bed and the fluidized bed region, respectively. The experiment was repeated at moderate temperatures between 25°C and 90°C to see if there was any agglomeration effects at temperatures below the observed minimum sintering temperature, T,. It was noted that the adhesive forces between particles at these temperatures were not strong enough to have an effect on the minimum fluidization velocity and the fluidized bed could be operated with no concern for the possibility of defluidization. Experiments at higher temperatures were performed to obtain the minimum fluidization velocity under conditions where agglomeration would occur. In these experiments, the bed was kept in the fluidized state at a given temperature T, while the gas velocity was reduced to the point of the limiting velocity, Us and only brought below this value for a short time in order to get an accurate value of Us.Table 1 shows the dimenversus temperature sionless excess velocity (Us- urnf)/urnf for the two sizes of particles used in the experiments, above the minimum sintering temperature of the material. The two fundamental physical quantities required for predicting defluidization velocities as determined by Equation (4) are the surface viscosity, qs,and the yield strength, us. Surface viscosity measurements were performed as described in Tardos et al. (1984), while the yield strength of the sinter neck was estimated using adhesion theory which assumes that van der Waals forces are the dominant (shortrange) attractive forces within particle-particle bonds once they are formed. This assumption is correct for weak particle-particle bonds. The strength of a particle-particle bond is given by Krupp (1967) as

Some typical results usig a coating material (Ace pMstic coating #3100, Ace Glass, Vineland, New Jersey) with a surface tension of yrn = 0.039 N/m were obtained using TCC (thermal cracking catalyst) beads in the size range from

us = fc0/81~~Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . (7) where h o is the Lifshitz-van der Waals constant, which depends on the material involved and Z, is the separation distance of the granules. Values of the Hamaker constant, A,

us- ulqf

us- urn/

Temperature

urn/

umf

(“C)

(D,= 0.07 cm)

(D,= 0.32 cm)

-

0.18

108 110 111 115 120

1.61

-

2.14 4.30

0.20 0.32 0.50

-

NOTE:Minimum sintering temperature T, = 106°C.

mixing of the particles occurred during the experiment. A typical high temperature (sintering) experiment is performed as follows: the air flow is started to fluidize a bed of particles at approximately 2.5 to 3 times the minimum fluidization velocity. The temperature controller is set to the desired air temperature to be maintained throughout the run, and the heaters are energized. The system is allowed to reach thermal equilibrium for about 30 min. The experiment is then started by decreasing the gas velocity in equal increments. The rotameter reading along with the pressure and temperature at the rotameter are recorded. The temperature and pressure of the gas under the distributor plate as well as the temperature and pressure in the bed itself are all displayed and printed. The gas velocity is decreased in set increments repeatedly until the bed defluidizes. The gas velocity is further decreased in the fixed bed state so that six to eight data points can be recorded. The velocity is then increased in set increments so that the bed is refluidized. Pressure drop through the bed versus gas velocity data at a preset gas (bed) temperature are automatically recorded and stored in the computer.

386

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, JUNE 1985

5.00

TABLE2 Effect of Temperature on the Breakage Strength of Agglomerates in Streams of Hot Air

4.00

2

Air velocity Mass Temperature Run no. (m/s) (kg) X lo3 ("C)

3.00

3 \ I

L

E

'

->l

2.00

1.00

0.00 0.00

0.05

0.10

0.15

0.20

0.25

(T-Ts)/Ts

Figure 4 - Dimensionlessexcess velocity vs. dimensionlesstemperature for a bed of polyethylene particles.

A- I A-2 A-3 A-4 A-5 A-6 A-7 B- 1 8-2 B-3 B -4 B-5 B-6

6.82 8.99 9.68 10.46 11.16 11.97 5.31 6.82 8.99 9.68 10.46 11.16 10.46

0.941 0.779 0.820 0.877 0.486 1.181 1.120 0.860 0.760 0.977 0.659 1.099 1.159

Critical drag force, uos (kg/m2)

165 160 161 160 156 149 167 156 155 I52 146 I48 153

0.641 1.075 0.979 1.075 1.494 2.404 0.510 1.002 1.102 1.425 2.219 1.935 1.313

NOTE: A, agglomerate polypropylene particles produced at 164°C; 5.00

B, agglomerate polypropylene particles produced at 160°C.

--Polyethylene 0

o

4.00

2,

L

2

-

3.00

6, = 0.07crn 6, = 0.32crn = 0.6nrn

\

L

E

l

-y

-

2.00 0

1 .oo

0

n nn 0.00

0

I

I

I

1

0.05

0 .lo

0.15

0.20

0.25

( T -Ts) / T,

Figure 5 - Dimensionlessexcess velocity vs. dimensionlesstemperature for two sizes of polyethylene particles.

for different materials which is related to the Lifshitz constant by the relation: A = 3 h w / 4 ~. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . (8) can be found in Visser (1972). The value for polyethylene Joules (0.624eV). The only is given as A = 1.0 X quantity undetermined in the calculation of u, using Equation (7) is the separation distance, Zo. This was left as a free parameter in the comparison of the experimental data with the theoretical model. Figure 4 shows the comparison between experimentally determined defluidization velocities as compared to the theoretical model (using Equation (4)) versus temperature. These results are for the polyethylene granules of average size equal to 0.07 cm and for a bubble diameter db taken equal to one quarter of the bed diameter. The choice of Zo plays an important role in determining the fit between experiment and theory, and as seen from Figure 4 an average value of Zo equal to 0.6 nm yields a good fit of the dimensionless excess velocity ( U , - umr)/umfversus temperature as compared to the experimental data. Figure 5 presents defluidization velocities versus temperature for both samples of polyethylene using a separation distance of 0.6 nm.

From Figure 5, one can clearly see that, using the same separation distance, the model is able to predict defluidization velocities for both relatively small granules as well as larger granules. It is important to observe that the value found for the separation distance Zo = 0.6 nm is very close to the widely accepted value of Z, = 0.4 nm (Rumpf, 1977), thus attesting to the soundness of the proposed model. Additional defluidization experiments were performed with two different sizes of polypropylene granules of diameter, Dp = 0.2 cm and Dp = 0.32 cm, respectively. To eliminate the use of the van der Waals force for estimating the agglomerate strength, which is a somewhat arbitrary choice and which only holds for very weak agglomerates, experiments were performed on the polypropylene agglomerates formed in the fluidized bed. The measurements were performed by a group at Chuo University in Japan (Sekiguchi and Tohata, 1982) using the following procedure. A weighed agglomerate is mounted on the head of a needle shaped sample holder, which is connected to a microbalance. The microbalance is kept in equilibrium while a stream of hot air is passed around the agglomerate at steadily increasing velocities until the agglomerate fails. From these experiments the critical drag force, uos,at which agglomerate breakup occurs is obtained. Table 2 shows the critical drag force as a function of air stream temperature. The yield strength of the agglomerate a,can be obtained by using the equation: U, = [ ~ / ( 1- ~ ) ] ( l / ~ ~ ) ( D ~ / b ). ~ . .u.o. .s . . . . . . . . (9) and values so calculated were used in the model instead of using the van der Waals estimation for the strength of the polypropylene agglomerates. Figure 6 shows the comparison between experiment and theory, for a slugging bed (db = d B )and a neck radius, b, taken as 3.5 percent of the average particle diameter. This value gave the best agreement with the experimental data.

Comparison with other experimental data Figure 7 shows a comparison of the theoretical and experimental data from all of the works discussed above. The

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, JUNE 1985

387

0.80

E

Polypropylene

0.60

I

-Dp =

0.2cm

0

Dp =

0.32cm

----

Correlation by L I m et al. ( 1 9 8 3 ) of data by Slegell ( 1 9 7 6 ) Theoretical results of Tardos et 01. ( 1 9 8 5 )

9

101 L

-

0

L

Dp = O.6Smm Dp = 1.20mm

aE

\

L

=E

0.40

I

Baiu(1982)

Polyet hylene

100 T

I

Dp = 0.7mm

r

-

YI

E

3

\ ) .

0.20

E

-

I

-s I

I

0.10

0.15

I

0.00

0.05

0.00

16' r

Po I y p l _ o p y e

Dp = 3.2mm

/ 0.20

( T - T S ) / T, I

I

Figure 6 - Dimensionless excess velocity vs. dimensionless temperature for beds of polypropylene particles.

101

E

----

Correlation by Lisa et al. (1983) of data by Siegell ( 1 9 7 6 ) Theoretlcal results o f Tardoi et al. ( 1 9 8 5 )

P

/

x

Dp Polyethylene =0 . 7 m m K : $

Dp = 3.2mm

-== .='

/"\

Polypropylene

Dp = 2.0mm

Figure 7 - Comparison of theoretical and experimental results.

dimensionless excess velocity (Us- umf)/umfis plotted here as a function of the temperature difference ( T - T,)/T,. As seen in the figure the dependency of U, - umfwith temperature is similar in all cases but appears to depend on, at least, one more parameter. Inspecting Equation (4)the most likely additional parameter is the granule size, Dp.In Figure 8, the same data were replotted against the quantity [ ( T - Tx)/Ts].(Dp/dB)-3'2 and as seen the fit between the different data is much better, especially at the higher temperatures. At low temperatures one cannot expect an empirical relationship of the form given by Equation (3) to be valid whereas the computed curves based on Equation (4) appear to show the correct trend.

[(T-Ts)/T,]

(sintering) effects. For the injection of liquid material in the bed, the limiting velocity, U,, was seen to be dependent on the dimensionless amount of liquid added as well as measurable bed and fluid properties. For agglomeration due to temperature effects, the theoretical model requires knowledge of two fundamental physical quantities, the surface viscosity, q,, and the sinter neck yield strength, us,both as functions of temperature. Surface viscosity measurements can be performed using a dilatometer and the yield strength can either be estimated using adhesion theory or calculated from experimental data. It was demonstrated that the minimum sintering temperature T, occurs at the temperature where the surface viscosity undergoes a significant change in magnitude over a very small temperature range. At temperatures below T,, destabilization due to agglomeration is of no concern, but above this temperature higher gas velocities, as predicted by the model, are necessary to maintain stable fluidization.

Acknowledgement This research was supported by National Science Foundation grants CPE79-5054 and CPE82- 13062. Additional support from the International Fine Particle Research Institute (IFPRI) is also greatly appreciated. The authors also wish to thank Mr. Dov Firnberg for constructing the experimental set-up and for running some of the experiments.

Nomenclature A b dog

db

DP

A model has been developed which predicts limiting gas velocities necessary to maintain a potentially agglomerating fluidized bed in uniform fluidization for two different situations, one in which a liquid is injected in the bed and the other where particles become sticky due to high temperature 388

(Dp/d,)-3'2

Figure 8 - Comparison of experimental data and theoretical calculations using a modified temperature scale.

ds

Conclusions

X

g fio

Lo 1, AP, T T,

Hamaker constant radius of the bonding neck, m diameter of agglomerate, m diameter of bubble, m diameter of the bed, m particle diameter, m = acceleration due to gravity, m/sz = Lifshitz-van der Waals constant, J = bed height at rest, m = distance between two fixed points in the bed, m = pressure drop between two fixed points in the bed, Pa = temperature, "C = minimum sintering temperature, "C

= = = = = =

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 63, JUNE 1985

hf

Us

U,r WS

Wm

Z”

minimum fluidization velocity, m/s limiting velocity, m/s excess velocity, m/s mass of solid in bed, kg = mass of moisture in bed, kg = separation distance, nm

= = urn,= =

Greek letters E

ll* P8

Pm

$

= coating liquid surface tension, N/m = porosity, dimensionless = surface viscosity, kg/m.s

= gas density, kg/m’ = density of coating liquid, kg/m3 = density of solids in bed, kg/m3 = shape factor, dimensionless

uos

= maximum tensile strength, N/m2 = yield strength of neck, N/mZ = strength of a sintered agglomerate, N/m2

4J

= function defined by Equation (2), dimensionless

0 0

u,

References Basu, P., “A Study of Agglomeration of Coal-Ash in Fluidized Beds”, Can. J. Chem. Eng. 60, 791 (1982). Basu, P. and A. Sarka, ‘.‘Agglomerationof Coal Ash in Fluidized Beds”, Fuel, 62, 924 (1983). Gluckman, M. J . , J. Yerushalmi and A. M. Squires, “Defluidization Characteristics of ‘Sticky or Agglomerating Beds”, in

Fluidization Technology, D. L. Keairns (Ed.), Vol. 11, 395 (1976). Krupp, H., “Particle Adhesion, Theory and Experiment”, Adv. Coll. Interf. Sci. 1, 1 1 1 (1967). Liss, B., T. R. Blake, A. M. Squires and R. Bryson, “Incipient Defluidization of Sinterable Solids”, Fourth International Conference on Fluidization, Japan, June (1983). Rumpf, H., “Particle Adhesion”, in “Agglomeration, 1977”, K. V. S. Sastry (Ed.), AIME, 97 (1977). Sekiguchi, I. and H. Tohata, “Effect of Temperature on the Agglomerate Strength of Polymeric Particles: Part 2”, 1982 Annual IFPRI Report, Chuo University, Japan (1982). Siegell, J. H., “Defluidization Phenomena in Fluidized Beds of Sticky Particles at High Temperatures”, Ph.D. Dissertation, The City University of New York (1976). Siegell, J. H., “High-Temperature Defluidization”, Powder Tech. 38, 13 (1984). Tardos, G . , D. Mazzone and R. Pfeffer, “Measurement of surface viscosities using a dilatometer”, Can. J. Chem. Eng. 62, 884 (1984). Tardos, G., D. Mazzone and R. Pfeffer, “Destabilization of Fluidized Beds due to Agglomeration Part 1: Theoretical Model”, Can. J . Chem. Eng. 63, 377 (1985). Visser, J., “On Hamaker Constants: A Comparison Between Hamaker Constants and Lifshitz-Van der Waals Constants”, Adv. Coll. Interf. Sci. 3, 331 (1972). Manuscript received July 7, 1983; revised manuscript received October 18, 1984; accepted for publication November 14, 1984.

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