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PREDICTION OF THE MINIMUMFLUIDIZATION VELOCITY FOR FINEPARTICLES OF VARIOUS DEGREES OFCOHESIVENESSChunbao Charles Xu a & Jesse Zhu ba Department of Chemical Engineering , Lakehead University ,Thunder Bay, Ontario, Canadab Powder Technology Research Centre, Department of Chemicaland Biochemical Engineering , The University of Western Ontario ,London, Ontario, CanadaPublished online: 15 Nov 2008.

To cite this article: Chunbao Charles Xu & Jesse Zhu (2008) PREDICTION OF THE MINIMUMFLUIDIZATION VELOCITY FOR FINE PARTICLES OF VARIOUS DEGREES OF COHESIVENESS, ChemicalEngineering Communications, 196:4, 499-517, DOI: 10.1080/00986440802483855

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Prediction of the Minimum Fluidization Velocity forFine Particles of Various Degrees of Cohesiveness

CHUNBAO CHARLES XU

Department of Chemical Engineering, Lakehead University,Thunder Bay, Ontario, Canada

JESSE ZHU

Powder Technology Research Centre, Department of Chemicaland Biochemical Engineering, The University of Western Ontario,London, Ontario, Canada

Based on the well-known Ergun equation and the force balance of a particle bedunder fluidization bringing into account the interparticle forces, a new correlationfor prediction of the minimum fluidization velocity (umf) for fine particles of variousdegrees of cohesiveness has been derived. For the first time, a general correlation ofthe minimum fluidization voidage (emf) versus particle size is obtained from varioussets of experimental data. The newly derived umf correlation combined with the onefor emf proves to be superior to the traditional ones proposed by Leva (1959) andWen and Yu (1966), especially for the cases where very fine or very large particlesare employed in fluidization. The correlations of Leva (1959) and Wen and Yu(1966), both disregarding the cohesive-force effect and the effect of particle sizeon emf, result in noticeable errors in umf prediction for very fine (Geldart groupsC and C=A) and very large (Geldart group D) particles, although they worksatisfactorily for small-to-medium size particles (Geldart groups B and A). Incontrast, the prediction with the new correlation shows good agreement with theexperimental data for various types of particles ranging from Geldart group C togroup D.

Keywords Fine particles; Interparticle forces; Minimum fluidization velocity;Minimum fluidization voidage; Modeling; New correlations

Introduction

Minimum fluidization velocity (umf) is a key parameter to describe fluidization ofparticles. Minimum fluidization is defined as the state at which the weight of thewhole bed of particles begins to be completely supported by the fluidizing gas,and the pressure drop of the gas across the bed becomes constant with increasinggas velocity. The value of umf is usually determined experimentally by measuringthe pressure drop as a function of gas velocity. It can also be predicted with variouscorrelations such as the semi-empirical correlations proposed by Narsimhan (1965)and Wen and Yu (1966) based on the well-known Ergun equation (Ergun, 1952), and

Address correspondence to Jesse Zhu, Powder Technology Research Centre, Departmentof Chemical and Biochemical Engineering, The University of Western Ontario, London,Ontario, Canada, N6A 5B9. E-mail: [email protected]

Chem. Eng. Comm., 196:499517, 2009Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440802483855

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the empirical correlation of Leva (1959). All these correlations were developed basedon the data of non-cohesive particles, and the interparticle forces have not beentaken into account. Obviously, they may be not suitable for fine particles whereinthe interparticle forces are relatively strong compared with gravity.

According to Geldart (1973), very fine particles, 1030 mm or smaller in size, aregenerally classified as cohesive particles (group C) and behave quite differently fromthe aeratable particles (group A) and other larger size particles (groups B and D) influidization. Geldart group C particles are very attractive in chemical, advancedmaterial, food, and pharmaceutical industries due to the high surface area to volumeratio and other special characteristics. However, due to the presence of relativelystrong interparticle forces, group C particles tend to channel and agglomerate signifi-cantly when subject to increasing gas velocity, resulting in a very poor fluidization orcomplete de-fluidization. Although it is still a challenging task, fluidization of groupC particles can be achieved by employing mechanical=acoustic vibration or otherfluidization aids (Mori et al., 1990; Jaraiz et al., 1992; Morse, 1955; Chirone et al.,1993; Chirone and Massimilla, 1994; Russo et al., 1995; Xu and Zhu, 2006). Wankand coworkers (2001) proposed a complicated model for predicting umf of a group Cpowder (10 mm boron nitride particles) fluidized under mechanical vibration andreduced pressures, where the interparticle forces are taken into account. Several lim-itations of the model of Wank et al. (2001) may be addressed here: the calculationprocedures are very complicated, the model was proposed specifically for fluidiza-tion at reduced pressures, and no verification has been done with other sets ofdata from the literature. The main objective of the current work is thus to developa new simple correlation to predict umf for fine particles by taking into accountthe interparticle forces.

Another important parameter to be discussed in this work is minimum fluidiza-tion voidage (emf), an essential parameter in applying the Ergun equation to drag-force evaluation and umf prediction. This problem was circumvented in both Levas(1959) and Wen and Yus (1966) models as well as in many previous fluidizationwork in various ways, for example, (1) by simply disregarding it (Leva, 1959), (2)by assuming emf a fixed value such as 0.5 (Yates, 1996) or 0.35 (Narsimhan, 1965)for all types of particles regardless of their properties and fluidization conditions,or (3) by correlating emf with particle properties such as particle shape (Wen andYu, 1966). However, it has been found in numerous studies, such as those ofNarsimhan (1965), Wen and Yu (1966), Wilhelm and Kwauk (1948), and VanHeerden et al. (1951), that the values of emf are strongly dependent on a variety ofproperties of particles including particle size, shape, and material, as well as thegas properties. Since the size of particles is normally the primary concern for a pow-der with respect to its cohesiveness and fluidizability, to establish a correlation of emfas a function of particle size is of more significance, which is, therefore, the otherobjective of this work.

Experimental Section

The experimental data employed in this study for verification of various umf correla-tions are adopted from literature and from a previous work of the authors (Xu andZhu, 2006). Details of the experimental approaches can be found in the relevantliterature, and in Xu and Zhu (2006), whose experimental setup and proceduresare given briefly as follows. The fluidized bed column, made of Plexiglas, 100mm i.d.

500 C. C. Xu and J. Zhu

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and 500mm tall, with a porous polymer plate serving as the gas distributor, wasinstalled on a vibrating stand. Mechanical vibration was generated by a pair ofvibrators mounted opposite one another on the two sides of a steel stand and drivenby an ABB inverter with a frequency adjustable from 0 to 50Hz. By changing theunbalanced weights of the vibrator, vibration amplitude of 03mm can be obtained.The amplitude of the vibration base was precisely measured using a quartz shearICP

1

accelerometer (Dalimar Instruments Inc.) with a sensitivity of 10.09mV=m=s2

at 100Hz. Compressed air stripped of trace humidity through a fixed bed of silica gelwas used as fluidizing gas, and the flow rate was controlled by a series of rotameters(Omega Engineering Inc.) and a digital mass flow controller (Fathom Technologies,GR series). In each experimental run, the settled bed height (L0) of particles wasfixed at 0.1m. Pressure drop signals across the whole bed, DP, were measured witha differential pressure transducer (Omega PX163 series) and collected by a computer.The umf was then determined by plotting DP versus ug in loglog coordinates. A typi-cal example is shown in Figure 1 where normalized pressure drops (DP=(msg=S)) for fluidization of glass beads (65 mm) under no vibration are illustrated.

Figure 1. umf and emf determinations from the experimental data (without mechanicalvibration) for glass beads (65mm).

Minimum Fluidization Velocity for Fine Particles 501

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Here, ms denotes the weight of solids used in the experiment and S the cross-sectionalarea of the fluidized bed column. Average bed voidage (eb) in fluidization wasdetermined by measuring the bed height during the experiments. In a same manneras illustrated in Figure 1, umf and emf for a fluid catalytic cracking (FCC) catalyst(65 mm) and a series of glass beads ranging from 6 to 216 mm mean in diameter(pertaining to Geldart groups C, C=A, A, and B) in fluidization with and withoutvibration were determined. Physical properties of these particles are given inTable I. The angle of repose and bulk densities (aerated bulk density qba and tappedbulk density qbt) of the powders were determined with a Hosokawa Powder Tester(PT-N) by a standard method.

Theoretical Analysis

Theories and Existing Correlations for Prediction of umf

The pressure drop (DP) of a gas flow through a packed bed of a height L0 can beestimated from the well-known Ergun equation (Ergun 1952), which is expressed as

DPL0

drag

1501 eb2

e3b

lgug

/sdp2 1:751 eb

e3b

qgu2g

/sdp1

where ug is the superficial gas velocity of the fluidizing gas, dp is the mean particlediameter, /s is the particle sphericity, lg is the gas viscosity, qg is the gas density,and eb is the bed voidage. The first term on the right-hand side in Equation (1) repre-sents the pressure drop caused by viscous effects and is dominant at low Reynoldsnumbers, while the second term is due to inertial forces and will be dominant at highReynolds numbers.

Assuming no or negligibly small cohesive force present in the particle bed,the pressure drop at umf would equate with the weight of particles per unit bedarea, i.e.,

DPL0

drag

1 emf qp qg g 2

Table I. Fine particles used in experiments of Xu=Zhu (2005)

Density (kg=m3)

Powders dpa (mm) qp qba qbt RH () AOR (deg.) Geldart group

FCC 65 1540 810 900 1.11 29.9 AGB-6 6 2500 600 1160 1.93 53.6 CGB-10 10 2500 1000 1550 1.55 49.1 CGB-39b 39 2500 1360 1510 1.11 33.5 C=AGB-65b 65 2500 1370 1510 1.11 30.4 AGB-216b 216 2500 1590 1620 1.02 20.4 B

aVolume-weighted mean diameter by laser diffraction (Malvern Mastersizer 2000).bSoda lime silica glass bead samples of spherical shape and narrow size distribution.


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where emf is the minimum fluidization bed voidage and qp is the particle density. Inthe gas-solid fluidization process, the bed maintains a state of fixed bed until thesuperficial gas velocity exceeds umf. Therefore, the pressure drop across the particlebed at the minimum fluidization velocity can still be estimated with the Ergunequation. Combining Equations (1) and (2) and then multiplying both sides by

qgd3p

1emf l2ggives

qgqp qggd3pl2g

1501 emf /2s e

3mf

qgumf dplg

! 1:75

/se3mf

qgumf dplg

!23

A dimensionless form of Equation (3) can be written as

Ar 1501 emf /2s e

3mf

Remf 1:75

/se3mf

Re2mf 4

where Ar qgqpqggd3p

l2gis the Archimedes number and Remf

qgumf dplg

is the Reynolds

number at the minimum fluidization.Wen and Yu (1966) found that the following two approximations hold true at

34% standard deviation with the experimental data for many kinds of particles overa wide range of conditions (Re 0.001 4000):

1

/se3mf

14 and 1 emf /2s e

3mf

11 5

With such simplification, Equation (4) is written as

24:5 Re2mf 1650 Remf Ar 6

which can be further rearranged to

Remf ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC21 C2Ar

q C1 7

where C1 33.7 and C2 0.0408. When the Reynolds number is very low, the vis-cous term in Equation (6) will be dominant and the inertial term becomes negligible.In such cases as with small particles, Equation (6) can be simplified to give

umf qp qggd2p

1650 lgRemf < 20 8

Another useful but completely empirical correlation was proposed by Leva (1959)for umf prediction in the case of low Reynolds number (Remf < 30), which takesthe form of

umf 7:169 104qp qg

0:94d1:82p g

q0:06g l0:88g

9

It must be noted here that the correlations of both Leva (1959) and Wen and Yu(1966) were derived based on the experimental data for particles larger than20 mm, and thus they may not be applicable to particles of smaller sizes.


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Derivation of a New Correlation to Predict umf for Fine Particlesof Various Degrees of Cohesiveness

For fine particles in the category of Geldart group C, the relative interparticle forcesin relation to the gravitational force usually become too significant to be negligible.Thus, for a settled bed of fine or cohesive particles subject to fluidization, as depictedin Figure 2, the force balance of the bed at the minimum fluidization velocity shouldbe presented as

DPL0

drag

WsSL0

r

L0

10

The term on the left-hand side of Equation (10) represents the contribution ofthe drag force due to the up-flowing gas, given by the Ergun equation inEquation (1). The first term on the right-hand side is the normal pressure causedby the weight of particles in the bed (Wsms g), which is

WsSL0

1 eb qp qg g 11

The second term on the right-hand side of Equation (10) is due to the tensile stress(r) of the particle bed resulting from the interparticle forces that must be overcomewhen the bed is fluidized. Considering a randomly packed bed of particles, the tensilestrength r can be correlated with the interparticle forces by the following equation(Molerus, 1982):

r 1 ebeb

Fcd2p

12

where Fc is the overall interparticle cohesive forces. According to Krupp (1967), theinterparticle forces could include van der Waals force and electrostatic force, as wellas interfacial forces caused by solid bridges (sintering or diffusive mixing) or liquidbridges (surface tensional and capillary) and mechanical force due to static frictional

Figure 2. Force balance for a bed of fine particles subject to fluidization.


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contact or interlocking of rough=re-entrant particle surfaces (Staniforth, 1985).Although any of these forces may dominate in particular circumstances, the forceof primary concern is the van der Waals force (Seville et al., 2000) for many powdersof practical interest. According to Krupp (1967) and by taking into account the num-ber of contact points per particle (nc 1.61eb1.48) (Jaraiz et al., 1992), the overallcohesive force per particle is

Fc 1:61e1:48bh-

8pZ20R 13

where h- is the Lifshitz-van der Waals constant, and Z0 is the distance between thetwo particles at the point of contact. R in Equation (13) is defined by

R rp1rp2rp1 rp2

14

where rp1 and rp2 are the asperity radii for the two particles. Assuming rp1 rp2 ra,substituting this into Equations (12)(14) gives

r 1:61e1:48b1 eb

eb

1

d2p

h-

16pZ20ra 15

Replacing h- with the Hamaker constant h- 4p3 AH (Krupp, 1967), one has

r 1:61e1:48b1 eb

eb

1

d2p

AHra

12Z2016

Z0 is normally assumed to be 4 1010m (Krupp and Sperling, 1966). As establishedin the well-accepted theory by Krupp (1967), Molerus (1982), and Seville et al.(2000), ra is assumed to be 0.1 mm for all particles regardless of their primary particlesize. This leads to

r 1:61e1:48b1 eb

eb

1

d2p

AH 0:1 10612 4 10102

8:38 1010e1:48b1 eb

eb

AHd2p

17

or

rL0

8:38 1010e1:48b

1 ebeb

AHL0 d2p

18

The Hamaker constant AH can be estimated as follows (Israelachvili, 1992):

AH 3

4kT

e1 e0e1 e0

2 3hve16

ffiffiffi2

p n21 n20

2

n21 n203=2

19

where k is Boltzmanns constant (k 1.381 1023 J=K), T is the absolute tempera-ture, e1 and n1 are the dielectric constant and the refractive index of the particulatematerials, e0 and n0 are the dielectric constant and refractive index of the mediumwhere the particles are present (for vacuum or air, e0 n0 1.0), h is Plancksconstant ( 6.626 1034 J s), and ve is the main electronic absorption frequencyin the UV region, which is typically around 3 1015 s1. Hence, for the particulate


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materials fluidized in air, Equation (19) may be rewritten as

AH 1:036 1023T e1 e0e1 e0

2

2:635 1019 n21 n20

2

n21 n203=2

20

For glass beads at room temperature (T 293K), e1 7.6, n1 1.5, and AHestimated by Equation (20) is 0.721 1019 J. For FCC particles, the estimated valueof AH by Equation (20) is 1.4 1019 J (e1 10.1 and n1 1.8). According toBergstrom (1997), Hamaker constants for most inorganic materials in vacuum orair are in the range of (0.51.5) 1019 J. For simplification, the average value istaken in this model, i.e., AH 1.0 1019 J.

Substituting Equations (1), (11), and (18) into Equation (10), one has thefollowing force balance for the bed at the minimum fluidization (emf and umf):

150 1 emf 2

/2s e3mf

lgumfd2p

1:75 1 emf /se

3mf

qgu2mf

dp

" #

1 emf qp qgg

8:38 109 1 emfemf

e1:48mfL0 d2p

" #21

Equation (21) can be further simplified to the following dimensionless form:

Re2mf 85:711 emf

/sRemf EG EC 22

where the two terms on the right-hand side of Equation (22), i.e., EG and EC,represent the gravitational-force effect and the cohesive-force effect, respectively,and their expressions are:

EG 0:57/se3mf Ar 23

EC 4:79 109 /se

0:52mf qgdpL0l2g

24

As shown in Equations (21)(24), minimum fluidization voidage emf is an essentialparameter for the prediction of umf. Equation (5) proposed by Wen and Yu (1966)suggests that emf is dependent only on particle shape and independent of particle size,which is not realistic according to the observations in many studies such as Wilhelmand Kwauk (1948), Van Heerden et al. (1951), and Narsimhan (1965). Even in thework of Wen and Yu (1966), for example, the data of emf for spherical particles(/s 1) are scattered in the range of 0.40.5, suggesting that other factors such asparticle size and surface roughness need to be considered in the evaluation of emffor various types of particles. For instance, Narsimhan (1965) found that emfdepended strongly on both sphericity (/s) and size (dp) of the particles. It has beencommonly accepted that the voidage of a settled bed of fine particles is related tothe interparticle attraction in such a way that the stronger the relative interparticleforce (in relation to the gravitational force), the larger the value of the settled bedvoidage. As such, the influencing factors (particle shape, size and surface roughness,etc.) for emf may be grouped into one factor, i.e., the relative interparticle force in


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relation to the gravitational force. Assuming van der Waals force as the dominantinterparticle force, the relative interparticle force in relation to the gravitationalforce, i.e., Fvan=FG, can be presented as follows:

FvanFG

AHra12Z2

0

1=6pd3pqpg

AHra12Z2

0

1=6pd3pqpg AHra

2Z20pqpg

1

d3p25

Equation (25) clearly indicates that Fvan=FG is proportional to (1=dp3). It thus

suggests that the particle size may significantly affect the settle bed voidage in sucha way that the smaller the particle size, the larger value of the settled bed voidage. Inthis regard, it would be more meaningful to develop a general correlation of emf as afunction of dp. Figure 3 shows emf as a function of dp for particles in fluidizationbased on the data from Narsimhan (1965), Van Heerden et al. (1951), and Mawatariet al. (2002) as well as from Xu and Zhu (2006). From this figure, a linear correlationof emf and dp in loglog coordinates may be obtained as

emf 0:77dp 1060:124 26

It should, however, be noted that Figure 3 (as well as Figure 5, to be presented later)contains data for very fine particles obtained with the aid of mechanical vibration.While fluidization of Geldart C particles was difficult without the aid of mechanicalvibration, mechanical vibration could also enhance fluidization of large particles(Geldart A and B group particles) through slightly decreasing both umf and emf, as

Figure 3. emf as a function of dp for fine particles.


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demonstrated by our earlier work (Xu and Zhu, 2006). As such, strictly speaking, asumf and emf are affected by vibration, the correlations for them (i.e., Equations (22)and (26)) should include parameters of vibration (frequency and amplitude). For thesake of simplification, however, the vibration parameters are not included in the cor-relations of both umf and emf. Future work is needed to improve these correlations bytaking into account the vibration parameters.

By combining Equation (22) with Equation (26), the value of Remf and hence umfcan be readily predicted, given the properties of both particles and fluidizing gas andthe initial bed height L0. For such a case when the initial bed height L0 is notprovided, its value may be estimated by the following equation given the total massof particles in the bed (ms), the cross-sectional area of the fluidization column (S),and the particle density of the powder (qp):

L0 ms

S1 emf qp27

whence emf can be estimated from dp by Equation (26).

Results and Discussion

The performance of the newly derived correlation (Equation (22)) and the correla-tions of Leva (1959) and Wen and Yu (1966) are compared in Figure 4 where thepredictions of umf from different correlations are plotted against the experimentaldata for larger particles (Geldart groups C=A, A, B, and D) fluidized withoutvibration. The properties of the particles and the raw data used in Figure 4 are alsolisted in Table II. The predictions of the three correlations show different degrees ofagreement with the experimental data, depending on the particle sizes. The correla-tions of Leva (1959) and Wen and Yu (1966) are satisfactory for groups B and Aparticles, with umf ranging roughly from 0.005 to 0.2m=s, while both of them yieldlower prediction values (or so-called under-prediction) for the group C=A particles,a category defined by Geldart (1973) for powders that behave both C-like and A-like(normally in the range of 3050 mm of particle size), and higher prediction values (orso-called over-prediction) for the group D particles. The prediction of the new cor-relation, in contrast, shows good agreement for all types of particles pertaining togroups C=A, A, B, and D. The standard deviations of the predictions from theexperimental data are 3.19, 6.50, and 0.20 for Equation (9) of Leva (1959), Equation(7) of Wen and Yu (1966), and Equation (22) in the present work, respectively.Clearly, for all the particles tested, the performance of the new correlation is muchmore satisfactory than that of the other two.

Predictions of umf by using different correlations for fluidization of glass beadsof 5 to 300 mm average particle size are comparatively illustrated in Figure 5 againstthe experimental data of Mawatari et al. (2002) and Xu and Zhu (2006) with andwithout vibration. Similarly, all three correlations work satisfactorily well for largerparticles (Geldart groups B and A). Nevertheless, both correlations of Leva (1959)and Wen and Yu (1966) result in noticeable errors when applyied to the finer parti-cles (groups C and C=A) with an average size less than about 30 mm, both yieldingmuch smaller prediction values (under-prediction) than the experimental data. Incontrast, much better predictions are obtained using either Equation (22) or theErgun equation for all sizes of the particles tested (Geldart C, A, and B groups),as clearly shown in the figure. Equation (22) was derived directly from the Ergun


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equation, and the two correlations differ only in whether the cohesive-force effect(represented by the term of EC as given in Equation (25)) is taken into account.As expected and clearly shown in Figure 5, for fluidization of larger particles, wherethe interparticle forces are negligibly small in comparison with the gravitational forceof the particles, both correlations yield very similar predictions. For particles smallerthan 30 mm (essentially, Geldart group C particles), the predictions using Equation(22) are apparently better than those with the Ergun equation. It can be reasonablyaccounted for by the fact that the magnitude of the interparticle forces in relation tothe gravitational force of the particles increases as the particle size decreases. Itshould be noted that in both predictions (using Equation (22) and the Ergun equa-tion) for umf, emf was calculated from Equation (26). The results shown in Figure 5suggest that by using the correlation of Equation (22) or the Ergun equation,

Figure 4. Comparison of experimental and predicted values of umf using different correlations forlarger particles (Geldart groups C=A, A, B, and D). No vibration was applied in the experiments.


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Table

II.Comparisonofexperim

entalandpredictedvalues

ofumfusingdifferentcorrelationsforlarger

particles(G

eldart

groupsC=A,

A,B,andD)

umfpredicted(cm=s)

Particles

dp(mm)

qp(kg=m

3)

/s(

)umfaexperi.(cm=s)

Leva

Wen=Yu

Thiswork

XuandZhu(2006)

FCC

catalyst

65

1540

1.0

0.35

0.24

0.21

0.41

Glass

beads

39

2500

1.0

0.30

0.15

0.12

0.31

Glass

beads

65

2500

1.0

0.68

0.38

0.34

0.67

Glass

beads

216

2500

1.0

4.30

3.40

3.76

4.20

Mawatariet

al.(2002)

Glass

beads

60

2500

1.0

0.40

0.33

0.29

0.59

Glass

beads

100

2500

1.0

0.80

0.84

0.81

1.29

Wilhelm

andKwauk(1948)

Glass

beads

286

2492

1.0

5.49

5.69

6.62

6.37

Glass

beads

396

2492

1.0

9.14

10.27

12.66

10.31

Glass

beads

5182

2492

1.0

143.3

1100

2160

98.50

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Sea

sand

518

2639

1.0

10.06

17.60

22.86

15.80

Sea

sand

549

2639

1.0

17.98

19.53

25.63

17.16

Sea

sand

4572

2639

1.0

36.58

926.0

1780

96.29

Soconybeads

1006

1603

1.0

57.91

36.80

52.22

23.57

Soconybeads

3353

1603

1.0

76.20

329.0

581.3

62.28

Leadshot

1280

10792

1.0

103.63

344.0

570.7

113.07

VanHeerden

etal.(1951)

Carborundum

82

3180

0.92

1.13

0.73

0.69

1.03

Carborundum

94

3180

0.95

1.48

0.94

0.91

1.35

Carborundum

95

3180

0.94

1.50

0.96

0.93

1.34

Carborundum

117

3180

0.83

2.19

1.40

1.40

1.44

Carborundum

162

3180

0.90

4.56

2.53

2.69

2.78

Carborundum

192

3180

0.99

5.82

3.44

3.78

4.36

Carborundum

225

3180

1.0

7.09

4.60

5.19

5.65

Ironoxide

92

5180

0.88

2.43

1.43

1.41

1.82

aFluidizationin

ambientair(293K)withoutvibration;settledbed

height:L00.070.2m.

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combined with Equation (26) for emf, more satisfactory predictions of umf areobtained than those by the correlation of Wen and Yu (1966), which was alsoderived from the Ergun equation. Although derived from the same Ergun equation,the correlations of Equation (22) and Wen and Yu (1966) yield much differentpredictions for umf, mainly due to different correlations of emf used: the emf usedby Wen and Yu (1966) was correlated with particle shape as given by Equation(5), while a new correlation of emf versus particle size as given by Equation (26)was developed and used in the present work. The above discussion may thus suggestthat the development of Equation (26) for emf is an important contribution of thiswork, in addition to the new correlation of Equation (22) for the prediction of umftaking into account the interparticle forces.

In order to further verify the correlations, a set of discrete particle model (DPM)simulation data of Ye et al. (2005) for particles (qp 1495 kg=m3) of an average sizeranging from 40 to 180 mm (pertaining to Geldart groups A and B) is adopted. Thesimulation results of umf of Ye et al. (2005) and the prediction results using variouscorrelations are presented against particle size in Figure 6. The parameters used inthe DEM simulation are: number of particles simulated 36000, L0 0.00368m, q-p 1495 kg=m3, qg 1.2 kg=m3, lg 1.8 105 Pa.s, and/s 1.0. It should be notedthat the initial bed height (L0) employed for the simulation, limited by the number ofparticles used in the DEM model, is impractically small ( 0.00368m). Since L0 is acritical parameter for umf prediction with Equation (22) where the term of cohesive-force effect EC (given by Equation (24)) is strongly dependent on the value of L0, theauthors have done calculations based on two different values of L0: one is the originalvalue (0.00368m) used by Ye et al. (2005) in the simulation and the other is a more

Figure 5. umf vs. dp for glass beads fluidized in ambient air at room temperature andL0 0.070.1m; experimental data obtained with and without mechanical vibration andpredictions with different correlations.


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practical value (0.1m). As apparent in Figure 6, the increase in the initial bed heightleads to a decrease in umf for all types of particles, which can be accounted for byEquation (24) where a smaller EC (or a reduced cohesive-force effect) is predictedat a larger L0. Figure 6 also shows that the predictions of both Levas and Wen=Yuss correlations yield significant deviation from the simulation results for the two large-size powders (group B), 125 and 180 mm, although acceptable agreement is obtainedfor group A particles. The same as shown in Figures 4 and 5, the new correlation ofEquation (22) with L0 0.1m yields satisfactory prediction compared with the simu-lation results for all types of particles employed (groups A and B).

Figures 46 all reveal that correlations of both Leva (1959) and Wen and Yu(1966) yield noticeable errors in prediction of umf for either very fine (groups C=Aand C) or very large (such as group D) particles. Possible reasons are discussed asfollows. For fine and cohesive particles subject to fluidization, as shown inFigure 2, the drag force due to the up-flowing gas is partially balanced by the tensilestrength resulting from the interparticle forces, which would accordingly lead to anincrease in umf due to the addition of the term of cohesive-force effect, EC, to theterm of gravitational-force effect, EG, in Equation (22). As such, a lower value ofumf would be predicted if the cohesive-force effect were not taken into account.The relative importance of EC compared with EG may be revealed by the ratioEG=EC, given in Table III for various types of particles. As shown in Table III,

Figure 6. Comparison of umf predictions using different correlations with the 3-D discreteparticle model (DPM) simulation results by Ye et al. (2005). Conditions for the simulationin Ye et al. (2005) are: no vibration, L0 0.00368m, qp 1495 kg=m3, qg 1.20 kg=m3,lg 1.8 105 Pa.s, T 293K, and /s 1.0.


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the value of EG is more than two orders of magnitude larger than that of EC forparticles of Geldart groups A, B, and D, while the ratio is within less than two ordersof magnitude or even in the same order of magnitude for the Gedart groups C=A andC fine powders. Therefore, for fine or cohesive particles (groups C and C=A pow-ders) where the interparticle forces are relatively significant, the cohesive-force effectshould be taken into account in prediction of umf. Any correlations disregarding thecohesive-force effect, such as ones by Leva (1959) and Wen and Yu (1966), might besafe for large particles (Geldart groups A, B, and D), whereas they will yield notice-able errors (under-prediction) in prediction of umf for fine or cohesive particles(groups C and C=A), as revealed in Figures 46. On the other hand, the reasonfor the over-prediction of umf by using both Levas and Wen=Yus correlations forvery large particles (e.g., Geldart group D powder) may be due to the improperevaluation of emf. In terms of the Ergun equation, Equation (1), a smaller emf mayresult in a larger drag force under the same superficial gas velocity in fluidization,

Table III. Ratio of cohesive force-effect to gravitational-force effect, EG=EC, forvarious types of particles

ParticlesGeldartgroup dp (mm)

qp(kg=m3) /s () L0 (m) EG=ECa

Xu and Zhu (2006)Glass beads C 6 2500 1.0 0.1 3.2Glass beads C 10 2500 1.0 0.1 7.6FCC catalyst A 65 1540 1.0 0.1 110Glass beads C=A 39 2500 1.0 0.1 75Glass beads A 65 2500 1.0 0.1 179Glass beads B 216 2500 1.0 0.1 1.37 103

Mawatari et al. (2002)Glass beads C 6 2500 1.0 0.065 2.1Glass beads C 10 2500 1.0 0.065 4.9Glass beads C 20 2500 1.0 0.065 16Glass beads C=A 30 2500 1.0 0.065 31Glass beads A 60 2500 1.0 0.065 101Glass beads A 100 2500 1.0 0.065 241

Wilhelm and Kwauk (1948)Glass beads B 286 2492 1.0 0.1 2.19 103Glass beads D 5182 2492 1.0 0.1 2.95 105Sea sand B=D 518 2639 1.0 0.1 6.33 103Socony beads D 1006 1603 1.0 0.1 1.18 104Socony beads D 3353 1603 1.0 0.1 9.07 l04Lead shot D 1280 10792 1.0 0.1 1.20 105

Van Heerden et al. (1951)Carborundum B 82 3180 0.92 0.1 337Carborundum B 117 3180 0.83 0.1 615Carborundum B 225 3180 1.0 0.1 1.86 103Iron oxide B 92 5180 0.88 0.1 667

aFluidization carried out in ambient air at room temperature without vibration.


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thus leading to a lower umf value. As discussed in the previous section and in Figure 3,the minimum fluidization voidage emf decreases with increasing particle size. Shouldthis trend of emf not be taken into consideration, such as in correlations of Leva(1959) and Wen and Yu (1966), larger values of umf would be predicted (i.e., over-prediction) for group D particles.

Conclusions

A new correlation to predict the minimum fluidization velocity (umf) of fine particleshas been developed from the well-known Ergun equation, based on the force balanceof a settled bed of fine solids subject to fluidization, taking into account the interpar-ticle forces. A correlation between minimum fluidization voidage (emf) and particlesize is also obtained for the first time from the experimental data. Compared withtwo commonly used correlations for umf of Leva (1959) and Wen and Yu (1966),the newly developed correlation, combined with the emf correlation, gives better pre-dictions (in better agreement with the experimental data) for various types of parti-cles pertaining to Geldart groups C, C=A, A, B, and D. Both correlations of Leva(1959) and Wen and Yu (1966) yield noticeable errors in the prediction of umf foreither very fine (groups C=A and C) or very large (such as group D) particles. Thisstudy suggests that (1) for fine or cohesive particles (groups C and C=A powders)where the interparticle forces are relatively significant, the cohesive-force effectshould be taken into account in the prediction of umf, and (2) for the effect of particlesize on emf, and hence umf, needs to be considered.

Acknowledgments

The authors are grateful to the Ontario Research and Development Challenge Fundfor supporting this study, through a Research Chair Program awarded to Jesse Zhu.

Nomenclature

A amplitude of vibration, mAH Hamaker constant, JAOR angle of repose, deg.Ar Archimedes numberdp mean particle size, mEG gravitational-force effectEC cohesive-force effectFc overall interparticle cohesive force, NFG gravitational force of a particle, 1=6pdp3qpg, NFvan van der Waals force, Ng gravity acceleration, 9.81m=s2h Plancks constant, 6.626 1034 J sk Boltzmann constant 1.381 1023 J=KL0 settled bed height, mms mass of solids in the bed, kgn0 refractive index of mediumn1 refractive index of the solidsnc number of contacting points per particle


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DP pressure drop across the bed, Parp1; rp2 asperity radii for two particles, mra asperity radius for particle, mR parameter used in Fvan calculationRe Reynolds numberRemf Reynolds number at umfRH Hausner ratio (qbt=qba)S cross-sectional area of the fluidization column, m2

T absolute temperature, Kug superficial gas velocity, cm=sumf minimum fluidization velocity, cm=sWs weight of particle bed (ms g), NZ0 distance between two particles, m

Greek Letters

eb average bed voidageemf minimum fluidization voidagee0 dielectric constant of mediume1 dielectric constant of the solidslg gas viscosity, kg=m=sve main electronic absorption frequency in the UV region, 3 1015 s1qba aerated bulk density, kg=m

3

qbt tapped bulk density, kg=m3

qp particle density, kg=m3

qg gas density, kg=m3

r tensile strength of the particulate bed, Pa/s particle sphericityh- Lifshitz-van der Waals constant, eV

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