(asif 2011) effect of volume-contraction on incipient fluidization of binary solid mixtures

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Particuology 9 (2011) 101–106

Contents lists available at ScienceDirect

Particuology

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p a r t i c

Effect of volume-contraction on incipient fluidization of binary-solid mixtures

Mohammad Asif ∗

Chemical Engineering Department, King Saud University, Box 800, Riyadh 11421, Saudi Arabia

a r t i c l e i n f o

Article history:

Received 6 August 2010

Received in revised form 11 October 2010Accepted 22 November 2010

Keywords:

Binary solid mixture

Bed void fraction

Volume-contraction

Minimum fluidization velocity

a b s t r a c t

Towards the development of a predictive model for computing the minimum fluidization velocity, the

volume-contraction phenomenon arising from the mixingof unequal solid species is accounted for in the

prediction of the bed void fraction of binary-solid mixtures at the incipient fluidization conditions. Com-parison with experimental data obtained from the literature clearly shows that significantly improved

predictions are obtained except for cases where the stratification pattern whether arising from the slow

defluidization or the difference in the densities of the two species affects the mixing of the constituent

species.

© 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of

Sciences. Published by Elsevier B.V. All rights reserved.

1. Introduction

An accurate prediction of the minimum fluidization velocity is

an important prerequisite for a successful preliminary design and

subsequent scale-up of any fluidized bed application. This issuehas been an active area of research spanning several decades. A lit-

erature review of the area yields a plethora of studies proposing

different correlations for the prediction of minimum fluidization

velocities. Despite the availability of significant literature, it is

still often difficult to accurately predict the minimum fluidization

velocity. This can often be attributed to the uncertainty associated

with the bedvoid fraction at theincipient fluidization.Even a small

error in its specification could lead to a significant error in the pre-

diction of the minimum fluidization velocity owing to the strong

dependence of the pressure drop on the bed void fraction. Widely

used correlation for predicting the pressure-drop, e.g. the Ergun

equation, contains terms that involve bed void fraction as high as

third order. Only under limiting conditionof laminar and turbulent

flows, it is possible to geta reasonable estimate of the bedvoidfrac-

tion using the well-known Richardson and Zaki correlation ( Asif,

2008).

This issue assumes added importance when two or more solid

species are present together in the same bed structure. This is due

to the fact that the mixing of unequal solid species often leads to

a contraction of the volume irrespective of whether the bed is flu-

idized or in a defluidized state (Asif, 2004b). This means that the

void fraction of the mixed bed of binary-solid is often less than

∗ Corresponding author. Tel.: +966 14676849; fax: +966 14678770.

E-mail address: [email protected]

the void fraction of the mono-component bed of either of the two

components. This phenomenon affects the pressure-drop. Ignor-

ing this aspect of the binary-solid fluidization could therefore lead

to a greater uncertainty in the specification of the minimum flu-

idization velocity. As a consequence, it is not uncommon to findseveral correlations with widely different underlying approaches

for the prediction of the minimum fluidization velocity whenever

the fluidization of two or more solid species is involved.

In spite of the important bearing of the bed void fraction on

the fluidization velocity, there is nonetheless a lack of literature

on the volume-contraction phenomenon even though its existence

at the incipient fluidization has been pointed out (Chiba, Chiba,

Nienow, & Kobayashi, 1979; Chiba, Nienow, Chiba, & Kobayashi,

1980; Formisani, 1991; Formisani, de Cristofaro, & Girimonte,

2001; Formisani, Girimonte, & Longo, 2008). Using the data avail-

able in the literature, Li, Kobayashi, Nisjimura, and Hasatani (2005)

employed the Westman equation to compute the bed voidage at

theincipient fluidization in order to predict the minimum fluidiza-

tion velocity of binary-solid mixtures. On the other hand, there

has been a growing realization of the importance of the volume-

contraction phenomenon on the hydrodynamics of binary-solid

fluidized beds, especially in connection with the prediction of the

so-called layer-inversion phenomenon. This is in view of the fact

that the mixing-induced volume-contraction phenomenon leads

eitherto an increase or a decrease in thebulk-density of themixed-

layer. The layer of higher bulk-density constitutes the lower layer

close to the distributor in conjunction with the stability considera-

tions (Asif, 2002, 2004a; Gibilaro, Di Felice, & Waldram, 1986). On

the other hand, the situation reverses for the case of the inverse

fluidization, where the layer of lower bulk density constitutes the

upper layer closer to the distributor (Asif, 2010a, 2010b; Escudié,

1674-2001/$ – see front matter © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.partic.2010.11.001

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102 M. Asif / Particuology 9 (2011) 101–106

Epstein, Grace, & Bi, 2007). Recently, there have been attempts to

modify the conventional models to predict the bed void fraction of

binary-solids subjected to liquid fluidization at the layer inversion

point (Escudié & Epstein, 2008, 2009).

In thepresent work, the volume-contraction effects are incorpo-

ratedin thedescription of the voidfraction of binary-solid mixtures

at the incipient fluidization, and the improvement thus obtained

on the prediction of the minimum fluidization velocity is high-lighted. Towards this end, the available bed void fraction data (at

the incipient fluidization) reported in the literature for binary-

solid fluidized mixtures are analyzed here for volume-contraction

effects. These solid mixtures mainly differ in the size of the con-

stituent species while one of the binaries differs both in the size

as well as the density such that the smaller species is denser than

its larger counterpart. Using the well-known Westman equation

capable of accounting for the volume-contraction phenomenon in

packed structures of binary-solid mixtures, the bed void fractions

of binary-mixtures are predicted, and a comparison is made with

the experimental data here for each set the data. The minimum

fluidization velocity of the binary-solid mixture is then evaluated

using the mean solid properties in conjunction with the predicted

values of the bed void faction.

2. Predictive models

In the following, models for predicting void fractions of binary-

solidmixtures willbe firstdiscussed. Equationsfor theprediction of

the minimum fluidization velocity using the mean solid properties

will be presented next.

2.1. Prediction of bed void fraction

As a firstapproximation towards predicting thevoid fractionof a

bedcontainingtwo or more solid species, a simplearithmetic aver-

aging of their mono-component specific-volumes can be applied

to obtain the specific-volume of the mixed bed. The term specific-volume implies the volume occupied by the unit volume of the

solids in the bed environment that includes the volume occupied

by the solid and the associated void spaces. The specific-volume

additivity can be mathematically represented as:

V = X 1V 1 + (1− X 1)V 2, (1)

where V = (1−ε)−1 is the specific volume of the mixed bed of

binary-solid whereas V 1 and V 2 are the specific volumes of mono-

component beds of species 1 and species 2, respectively. And, X 1is the fluid-free volume fraction of component 1, which is chosen

here to represent the larger solid species. Commonly known as the

serial-model in the fluidization literature, Eq. (1) is often written in

terms of the bedvoidfractionas follows (Epstein, LeClair, & Pruden,

1981; Lewis & Bowerman, 1952):1

(1− ε) =

X 1(1− ε1)

+1 − X 1

(1− ε2), (1a)

which essentially implies that the total bed height is equal to

the sum of the heights of two completely segregated individ-

ual mono-component beds fluidized at the same velocity. Thus,

mixing-effects of constituent species are ignored in this approach.

As a result, no volume-contraction will be observed. It is never-

theless important to note that the volume-contraction of the bed

containing two solid species can be evaluated from the difference

between the actual height and the height computed using Eq. (1).

The greater the difference between the actual experimental values

and predictions of Eq. (1), the higher is the degree of the volume-

contraction.

Towards a more realistic prediction of the bed void fraction

however (Asif, 2001), the equation proposed by Westman (1936)

can be used:V − V 1 X 1

V 2

2+ 2G

V − V 1 X 1

V 2

V − X 1 − V 2 X 2

V 1 − 1

+

V − X 1 − V 2 X 2V 1 − 1

2 = 1. (2)The parameterG depends upon the size ratio of thetwo constituent

species comprising the bed structure. It is easy to see that setting

G = 1 in the above equation yields Eq. (1). Using a large base of

data, Yu, Standish, and McLean (1993) have proposed the following

functional form of the parameter G in the Westman equation:

1

G =

1.355r 1.566, (r ≤ 0.824)

1, (r > 0.824) (3)

where r is the size ratio (smaller to larger) of the two solid species.

Since models mentioned above require information about

the mono-component bed void fraction, i.e. V 1 and V 2, mono-

component bed void fractions at incipient fluidization are used for

the purpose. These are evaluated as:

V 1 = (1− ε1mf )−1, V 2 = (1− ε2mf )

−1, (4)

where ε1mf and ε2mf are experimentally obtained values of the bed

void fraction at incipient conditions, respectively for species 1 and

species 2.

2.1.1. Prediction of minimum fluidization velocity

For the prediction of the minimum fluidization velocity of

binary-solid mixtures, the correlation meant for determining the

minimum fluidization velocity of single solid species can be

extended to the case of the binary-solid using mean solid prop-

erties in conjunction with an appropriate description of the bed

void fraction. This consists of equating the effective bed weight

with thepressuredrop at incipient conditions.For thepredictionof

the pressure drop, any standard empirical correlation can be used.

Most notable among such correlations is the Ergun equation. Sub-

stituting the averaged values of particle properties, i.e. the mean

diameter and the mean density instead of the mono-component

properties lead to the minimum fluidization velocity of the solid

particle mixtures. This can be written as:

1.75ε−3mf

Re2mf + 150ε

−3mf

(1− εmf )Remf −Ga = 0, (5)

where theReynolds number andthe Galileo number are definedas

follows:

Remf = f U mf d

, (6)Ga =

d

3f (s − f ) g

2

. (7)

Note that the over-bar represents the mean, and the subscript ‘mf’

represents the incipient or minimum fluidization condition. Sym-

bols f and s are fluid and solid densities, respectively while other

symbols have their usual significance. The following expressions

for the mean density and the mean diameter can be used in the

above equation:

s =

n

i=

1

X isi, (8)


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M. Asif / Particuology 9 (2011) 101–106 103

Table 1

Binary-solid mixture data from the literature used in the present comparison.

Binary Size (m) Density (kg/m3) Size ratio Remark Source

Dp1 Dp2 s1 s2 Dp2/Dp1

1 335 240 2530 2530 0.716 Formisani (1991)

2 499 271 2480 2480 0.543 Formisani et al. (2008)

3 483 240 2530 2530 0.497 Formisani (1991)

4 335 153 2530 2530 0.457 Formisani (1991)5 385 163 2520 2520 0.423 Fast Chiba et al. (1979)

6 385 163 2520 2520 0.423 Slow Chiba et al. (1979)

7 483 173 2530 2530 0.358 Formisani (1991)

8 612 154 2480 2480 0.252 Formisani et al. (2008)

9 775 163 1080 2520 0.210 Fast Chiba et al. (1980)

10 775 163 1080 2520 0.210 Slow Chiba et al. (1980)

1

d=

ni=1

X iD pi

, (9)

where X i is the volume fraction of the ith solid species. Eq. (8) rep-

resents the volume-averaged density. On the other hand, Eq. (9)

represents the surface-volume mean (Sauter mean) diameter. It is

worthwhile to mention here that Eq. (8) is the only correct averag-

ing procedure for the evaluation of the mean particle density (Asif,1998) while several averaging procedures for the mean particle

diameter can be suggested (Allen, 1981). Eq. (9) is nevertheless the

most commonly used in the fluidization literature.

3. Experimental data

Data reported in the literature are utilized here. The bed void

fraction data at the incipient fluidization of binary-solid mixtures

is rather scarce in spite of a significant literature mostly report-

ing only the minimum fluidization velocity of such systems. It is

nonetheless sufficient in the present to show clearly that incor-

porating the volume-contraction effects significantly improves the

predictions of theminimumfluidizationvelocity.The dataavailable

in the literature for different systems, involving 10 binaries, andtheir sources are summarized in Table 1. The constituent species

in the first eight binaries differ only in their sizes such that the

size ratio varies from 0.25 to 0.72. Here, Binary 5 and Binary 6 of

Chiba et al.(1979) are size-differentbinarieswith same constituent

species, albeit with different defluidization behavior as mentioned

in the remark. Note that the slow defluidization process promotes

segregation of the constituent species. The obvious consequence

of the segregation is the lack of the mixing leading thereby to a

smaller degree of volume-contraction. The Binary 9 and Binary 10

of Chiba et al. (1980) are different from the rest as the constituent

solid species in this case differ in their sizes as well as densities.

Binary 9 data is based on fast defluidization while that of Binary 10

involves slow defluidization process.

4. Results and discussion

As a first step, predictions of both Eqs. (1) and (2) are presented

along with the experimental data in Figs. 1–3 f or selected bina-

ries. The overall trend of other binaries is same. The abscissa here

is the fraction of the larger species ( X 1) of the binary. Although

often termed as jetsam in the literature, the larger species, when

lighter than its smaller counterpart, could constitute the upper

layerunder certain fluidization conditions (Chibaet al., 1980). Thus,

X 1 = 0 implies a bed consisting only of smaller species, andlikewise

X 1 = 1 represents a mono-component bed of larger species. Predic-

tions of Eq. (1) are presented with solid lines in all figures, which

vary linearly with the binary composition. Noting that Eq. (1) is

based on the assumption of perfect segregation of the two com-

ponents of the binary system, any deviation from its predictions

can therefore be interpreted as the volume-change of mixing. If

the value of the bed void fraction is smaller than the one predicted

by Eq. (1), this will indicate a contraction of the volume, meaning

that the height of the mixed bed will thus be shorter than the cor-

responding height of the two segregated mono-component layers

of individual species. It is seen here that the key feature underlying

all the binaries here is the occurrence of the phenomenon of thevolume-contraction.

Experimental data as well as model predictions are shown in

Fig. 1 f or Binary 2 and Binary 8, which differ in their size ratios.

Higher degree of contraction is visible for smaller size ratio, i.e.

0.252 forBinary 8 as compared to 0.543 of Binary2. Thepredictions

of Eq. (2) show a good agreement in both cases.

Fig. 1. Comparison of experimental data with model predictions for Binary 2 and

Binary 8.


Binary 6.


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Binary 10.

Fig. 2 presents the case of binaries of Chiba et al. (1979) with

the same constituent species but different defluidization process.

It is clear from the experimental data that the lower degree of con-

tractionis observed with slow defluidization process. This is due to

the fact that as the velocity is slowly decreased, the segregation of

the mono-component layers progressively increases owing to the

difference of their bulk densities. This reflects the lack of a uniform

mixingof thetwo constituent species, andresults in a lesser degree

of the volume-contraction. On the other hand, fast defluidization

promotes mixing, leading to an enhanced degree of contraction.

At high fraction of larger component ( X 1), the degree of contrac-

tion is however comparable. This is due to the fact that a small

amount of smaller species is easily accommodated in the inter-

stices of largerspecies even duringthe slow defluidization process.

As far as the comparison with predictions is concerned, agreement

is understandably superior for Binary5 since Eq. (2) is based onthe

mixing of constituent species.

Fig. 3 highlights the case of Binary 9 and Binary 10. Constituentspecies here differ in the size as well as the density. Note that the

actualreported data,which was based onthe fractionof the heavier

component, was converted in terms of X 1 taking into account the

densitiesof constituentspecies.As faras thedifference inthe deflu-

idizationprocess is concerned, the behavior is similar to thecase of

Binaries 5 and 6. The degree of contraction in the slow defluidiza-

tion is mostly lower due to segregation tendencies, yet far more

prominent than the case of Binary 6 as the size ratio is smaller

here. At higher X 1, the bed void fraction is seen to be comparable

for Binary 9 and Binary 10 due to the filling of smaller species in

the interstices of its larger counterpart.

The difference between predictions of Eq. (2) and the experi-

mental data is quite pronounced in Fig. 3 as compared to other

binaries. Higher contraction is seen at lower X 1. Note that size ratioof Binaries 9 and 10 is comparable to Binary 8. But, in the case of

Binary 8, like any other binary, the highest degree of contraction

occurs around X 1 = 0.6. On the other hand, the highest degree of

contraction is seen to occur around 0.25 in the case of Binaries 9

and 10. This is due to the fact that the difference in the bulk densi-

ties of the two mono-component layers during the defluidization

is not significant since the larger solid species of the binary sys-

tem is lighter than its smaller counterpart. This could even lead to

the layer-inversion phenomenon as the fluid velocity is progres-

sively decreased. Thus, the magnitude of the liquid velocity fixed

just before the fluidization can generate different mixing patterns,

and therefore the bed void fraction after defluidization canchange.

It is therefore obvious that the model is not capable of describing

bed void fraction in the event of the mixing controlled by hydro-

Fig.4. Differencebetweenexperimental andpredictedvaluesas a functionof diam-

eter ratio.

dynamics.

The discussion in the foregoing was mainly limited to a qual-

itative comparison between the experimental data and model

predictions. For a quantitative evaluation, the percent differencewas computed for each individual binaryusingthe followingequa-

tion:

Difference (%) =1

N − 2

N −1i=2

εi(experimental)− εi(predicted)εi(experimental)

×100. (10)

The end points, i.e. X 1 =0 and X 1 = 1 corresponding to i =1 and i = N

were excluded in the calculation of the average error as experi-

mental values were used for V 1 and V 2 as mentioned in Eq. (4).

The results of all binaries are summarized in Fig. 4. Eq. (1) repre-

sents the difference between experimental values and predictions

of Eq. (1), and therefore an indication of the degree of volume-

contraction. It is clear that the bed volume-contraction dependsupon the size difference of the two constituent species. As the dif-

ference in thesize of twospeciesdecreases, thevolume-contraction

also decreases. The same is true for two cases involvingslow deflu-

idization that are shown with open circles. The curve represented

byEq. (2) shows the difference between the model predictions and

experimental data. It is clear here that the agreement is good as

the difference in most cases is less than 3% except for the cases

of Binary 9 and Binary 10 due to reasons explained earlier. It is

worthwhile to mention here that the mean error of all binaries is

8.5% for Eq. (1) and 2.4% for Eq. (2) when Binaries 9 and 10 are

excluded. It can therefore be concluded at this stage that Eq. (2)

is capable of describing the volume-contraction behavior at the

incipient fluidization conditions.

It is important to integrate the predictive capability of theproposed model in order to improve the prediction of the min-

imum fluidization velocity of the binary-solid mixtures. In this

connection, the first difficulty was faced with respect to the mini-

mum fluidization velocity of the mono-component species. Using

the reported diameter in conjunction with experimental bed void

fraction yielded a substantial difference in the prediction of the

minimum fluidization velocity. In some cases, it was found to be

over 40%, e.g. 0.163mm and 0.335 mm glass bead of Formisani

(1991) and 0.775 hollow char of Chiba et al. (1980). In order to

address this issue, the effective diameter was computed using the

experimentally reported minimum fluidization velocity with the

help of Eq.(5) f or eachindividual species. These values are reported

in Table 2. Note that the difference between the reported andeffec-

tive diameter is seen to be as high as 20% (species 1 of Binaries 9


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5. Conclusions

It is clear here that the volume-contraction effects prevail dur-

ing the incipient fluidization of binary-solid mixture. This mainly

depends upon the difference in the diameter of the constituent

species. Smaller size ratios yield larger contraction of the bed. The

Westman equation used here can correctly describe the volume-

contraction behavior so long as the hydrodynamics-controlledsegregation tendencies, whether arising from the slow defluidiza-

tion or the difference in the densities of the two species, do

not affect the mixing of two unequal species. Accounting for the

volume-contraction effects in the prediction of the minimum flu-

idization velocity significantly improves the prediction.

During the comparison of the minimum fluidization velocities,

it is clear that the effective diameter needs to be used for any

meaningful comparison instead of the reported or the measured

diameter of the solid sample. Such a difference is often attributed

to the sphericity of particles in the literature. In the present case

however,the particlesize distribution appears to be responsible for

thesignificantlyloweredvalues of the effectiveparticle diameteras

smaller particlefraction, owing to their larger surface area,controls

the pressure dropand consequently theincipient fluidization of the

mono-component solid samples. In this connection, it is important

to note that the value of theeffective diameter could vary as signif-

icantly as those of non-spherical particle depending upon the flow

regime (Asif, 2009). Therefore, such effective diameter should not

be universally be used for specifying other hydrodynamic charac-

teristics of the solid samples.

Acknowledgement

This work was supported by the SABIC grant (Project ENG-30-

34), King Saud University, Riyadh.

Notations

De effective particle diameter (mm)Dp particle diameter (reported) (mm)

G parameter G of Westman equation (Eq. (2))

Ga Galileo number

r size ratio (smaller to larger)U mf minimum fluidization velocity (m/s)

V overall specific volume=

total bed volumetotal solids volume

V i mono-component specific volume of ith species

X 1 fluid-free volume fraction of particle species 1

Greek symbols

ε overall bed void fraction

εi void fraction of mono-component bed of ith particle

species

fluid viscosity (Pa s)

f fluid density (kg/m3)

s solid density (kg/m3)

Subscript

1 larger component

2 smaller component

mf minimum fluidization

References

Allen, T. (1981). Particle size measurement . London: Chapman and Hall. (Chapter 4).Asif, M. (1998). Generalized Richardson–Zaki correlation for liquid flu-

idization of binary solids. Chemical Engineering & Technology, 21,77–82.

Asif, M. (2001). Expansionbehavior of a binarysolid–liquidfluidized bedwith largeparticle size difference. Chemical Engineering & Technology, 24, 1019–1024.

Asif, M. (2002). Predicting binary-solid fluidized bed behavior using averagingapproaches. Powder Technology, 127 , 226–238.

Asif, M. (2004a). Thecompletesegregation model fora liquidfluidized bed: Formu-lation and related issues. Powder Technology, 140, 21–29.

Asif,M. (2004b). Volume-contraction behavior of binary-solid liquidfluidized beds.Powder Technology, 145, 113–122.

Asif, M. (2008). Bed void fraction at minimum fluidization conditions for limitingcases. Journal of Chemical Engineering of Japan, 41, 161–164.

Asif, M. (2009). Flow dynamic characterization of cylindrical particles. Particle & Particle Systems Characterization, 26, 210–219.

Asif, M. (2010a). Stratification pattern in inverse binary-solid liquid-fluidized beds.International Journal Mineral Processing , 97 , 79–84.

Asif, M. (2010b). Prediction of layer-inversion behavior of down-flow binary

solid–liquid fluidized beds. Chemical Engineering & Technology, 33,475–480.Chiba,S., Chiba,T., Nienow, A.W., & Kobayashi, H.(1979). Theminimumfluidization

velocity, bed expansion and pressure-drop profile of binary particle mixtures.Powder Technology, 22, 255–269.

Chiba, S., Nienow, A. W., Chiba, T., & Kobayashi, H. (1980). Fluidized binary mix-tures in which the denser component may be flotsam. Powder Technology, 26,1–10.

Epstein, N., LeClair, B. P., & Pruden, B. B. (1981). Liquid fluidization of binaryparticle mixtures—I. Overall bed expansion. Chemical Engineering Science, 36 ,1803–1809.

Escudié, R., & Epstein, N. (2008). Property-averaging applied to determination of volume contraction in binary-solid liquid-fluidized beds. Canadian Journal of Chemical Engineering , 86, 261–266.

Escudié, R., & Epstein, N. (2009). Voidage at the layer inversion point in binary-solid liquid-fluidized beds. Industrial & Engineering Chemistry Research, 35,182–184.

Escudié, R., Epstein, N., Grace, J. R., & Bi, H. T. (2007). Layer inversion and bed con-traction in down-flow binary-solid liquid-fluidized beds. Canadian Journal of

Chemical Engineering , 85, 25–35.Formisani, B. (1991). Packing and fluidization properties of binary–mixtures of

spherical particles. Powder Technology, 66, 259–264.Formisani, B., de Cristofaro, G., & Girimonte, R. (2001). A fundamental approach to

the phenomenologyof fluidizationof sizesegregating binary mixturesof solids.Chemical Engineering Science, 56, 109–119.

Formisani, B., Girimonte, R., & Longo, T. (2008). The fluidization process of binarymixturesof solids:Developmentof theapproachbasedon thefluidizationveloc-ity interval. Powder Technology, 185, 97–108.

Gibilaro,L. G.,Di Felice, R.,& Waldram,S. P.(1986).A predictivemodelfor theequilib-rium composition and inversion of binary-solid liquid fluidized beds. ChemicalEngineering Science, 41, 379–387.

Lewis, E. W., & Bowerman, E. W. (1952). Fluidization of solid particles in liquids.Chemical Engineering Progress, 48, 603–608.

Li, Z., Kobayashi, N., Nisjimura, A., & Hasatani, M. (2005). A method to predict theminimum fluidization velocity of binary mixtures based on particle packingproperties. Chemical Engineering Communications, 192, 918–932.

Westman, A. E. R. (1936). Thepacking of particles: Empiricalequations forinterme-diate diameter ratios. Journal of the American Ceramic Society, 19, 127–129.

Yu, A. B., Standish, N., & McLean, A. (1993). Porosity calculation of binary mix-ture of nonspherical particles. Journal of the American Ceramic Society, 76,2813–2816.

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