modelare strat fluidizat

9

Modeling

Thomas C. Ho

Lamar University, Beaumont, Texas, U.S.A.

1 INTRODUCTION

The conversion in gas–solids fluidized bed reactors hasbeen observed to vary from plug flow, to well belowmixed flow, mainly depending on reaction and fluidi-zation properties (Levenspiel, 1972). Historically, twoclasses of models have been proposed to describe theperformance of fluidized bed reactors; one is basedon a pseudohomogeneous approach and the other ona two-phase approach. The pseudohomogeneousapproach, where the existence of more than onephase is not taken into account, proposes that we usethe conventional multiphase flow models for the flui-dized bed reactors. These conventional models mayinclude ideal flow models, dispersion models, residencetime distribution models, and contact time distributionmodels. The two-phase approach, however, considersthe fluidized bed reactors to consist of at least twophases, a bubble and an emulsion, and proposes aseparate governing equation for each phase with aterm in each equation describing mass interchangebetween the two phases. Among the two-phase models,the bubbling bed model proposed by Kunii andLevenspiel (1969) and the bubble assemblage modelproposed by Kato and Wen (1969) have received themost attention. Figure 1 illustrates the developmentand evolution of these various flow models for flui-dized bed reactors.

2 PSEUDOHOMOGENEOUS MODELS

The first attempt at modeling gas–solids fluidized bedreactors employed ideal or simple one-parametermodels, i.e., plug flow, complete-mixed, dispersion,and tank-in-series models. However, the observedconversion of sometimes well below mixed flow inthe reactors could not be accounted for by these mod-els, and the approach was dropped by most research-ers. The next attempt considered residence timedistribution (RTD) model in which all the gas in thebed is considered equal in terms of gas residence time.However, since an operating fluidized bed consists of abubble phase and an emulsion phase of completelydifferent gas contacting hydrodynamics, the approachis inadequate and was also dropped.

Gilliland and Kundsen (1970) then modified theRTD models and assumed that the faster gas stayedmainly in the bubble phase and the slower in the emul-sion. Their approach was to distinguish the effect ofthe two classes of gas with different effective rateconstant. The overall conversion equation in theirproposed model is

CA

CA0

¼ð10

e�ktE dt ð1Þ

and the following equation was proposed to describethe effective rate constant, i.e.,

Copyright © 2003 by Taylor & Francis Group LLC

k ¼ k0tm ð2Þ

where m is a fitted parameter. Its value is small forshort staying gas and high for long staying gas.Inserting Eq. (2) into (1) yields

CA

CA0

¼ð10

exp �k0tmþ1

� �E dt ð3Þ

The above equation describes the conversion and isreferred to as the contact time distribution (CTD)model. Although improvement was made over theRTD models, the problem with this approach involvesobtaining a meaningful E function to use in Eq. (3)

from a measured C curve obtained at the exit of abed. Questions remain as to how the measured Ccurve represents the necessary E function in the calcu-lation due to the considerable backmixing betweenthe faster and slower gas occurring in an operatingfluidized bed.

3 TWO-PHASE MODELS

The discouraging result with the previous approachhas then led to the development of a sequence ofmodels based on the two-phase theory of fluidization

Figure 1 Development and evolution of fluidized bed models.


originally proposed by Toomey and Johnstone (1952).The two-phase theory states that all gas in excess ofthat necessary just to fluidize the bed passes throughthe bed in the form of bubbles. The term two-phasemodel, however, represents a broad range of modelswith various basic assumptions that may or may notdirectly follow the original two-phase theory. Forexample, some models consider the wakes and clouds,while others do not; some models propose the use ofsingle-size bubbles, while others allow for bubblegrowth, and some models use the two-phase flow dis-tribution following the two-phase theory, while othersneglect the percolation of gas through the emulsion. Inaddition, different models may propose different inter-phase mass transfer mechanisms.

In the following subsections, the general two-phasemodels are briefly reviewed, and then we give a moredetailed description of two of the more popular two-phase models, namely the bubbling bed modelproposed by Kunii and Levenspiel (1969) and thebubble assemblage model proposed by Kato andWen (1969).

3.1 General Two-Phase Models

In most two-phase models, a fluidized bed is consid-ered to consist of two distinct phases, i.e., a bubblephase and an emulsion phase. Each phase is repre-sented by a separate governing equation with a termin each equation describing mass interchange betweenthe two phases. A general expression of the two-phasemodels, therefore, consists of the following two equa-tions (from Wen and Fan, 1975). For the bubble phasewe have

F@CA;b

@t

� �� FDb

@2CA;b

@h2

!þ FU

@CA;b

@h

� �þ F0 CA;b � CA;e

� �þ FskCA;b ¼ 0

ð4Þ

and for the emulsion phase, we have

f@CA;e

@t

� �� fDe

@2CA;e

@h2

!þ fU

@CA;e

@h

� �þ F0 CA;e � CA;b

� �þ fskCA;e ¼ 0

ð5Þ

where the term FU represents the rise velocity of gas inthe bubble phase and the fU represents the rise velocityof gas in the emulsion phase (¼ UeÞ:

In the development of the two-phase models, mostinvestigators used a simplified form of the two-phasemodel by either assuming or estimating some of the

terms in the above two equations. Table 1 summarizesexperimental investigations of model parameters asso-ciated with the two-phase model. As indicated in Table1, most of the studies assumed steady state ð@CA;b=@t ¼0 and @CA;e=@t ¼ 0) with De ¼ 0 (plug flow in emulsionphase) or De ¼ 1 (completely mixed in emulsionphase); also Db ¼ 0 (plug flow in bubble phase) andUe ¼ Umf . The parameters investigated include gasinterchange coefficient (Fo), particle fraction in bubblephase (gb), and reaction rate constant (k). Among theinvestigations, several authors reported that the flowpatterns in the emulsion phase, whether it is assumedto be plug flow or completely mixed, do not signifi-cantly affect the model prediction (Lewis et al., 1959;Muchi, 1965). However, Chavarie and Grace (1975)reported that the two assumptions do affect their pre-dictions based on the model of Davidson and Harrison(1963).

A number of theoretical studies on the two-phasemodel were also carried out and reported in the litera-ture. They are listed in Table 2. The ones proposed byDavidson and Harrison (1963) and Partridge andRowe (1966) are briefly reviewed below.

3.1.1 Model of Davidson and Harrison (1963)

One of the representative two-phase models is the oneproposed by Davidson and Harrison (1963). Thismodel follows the two-phase theory of Toomey andJohnstone (1952) and has the following assumptions:

1. All gas flow in excess of that required for inci-pient fluidization passes through the bed asbubbles.

2. Bubbles are of uniform size throughout thebed.

3. Reaction takes place only in the emulsionphase with first-order kinetics.

4. Interphase mass transfer occurs by a combinedprocess of molecular diffusion and gasthroughflow.

5. Emulsion phase (dense phase) is either per-fectly mixed (DPPM) or in plug flow (DPPF).

Note that the key parameter in their model is theequivalent bubble diameter, which was assumed to bea constant.

Chavarie and Grace (1975) reviewed the model andstated that the DPPM model is too conservative in itsestimate of overall conversion and that the concentra-tion profiles are in no way similar to their experimentalcounterparts. Regarding the DPPF model, they statedthat while the DPPF model offers a fair dense phase


Table 1 Experimental Investigation of Two-Phase Mode Parameters

Authors Model Experimental conditions Experimental results

Shen & Johnstone

(1955)

�b ¼ 0

Ue ¼ Umf

De ¼ 0 or 1

decomposition of nitrous oxide

dt ¼ 11:4 cm

Lmf ¼ 26 � 32 cm

dp ¼ 60 � 200 mesh

parameter Fo

k ¼ 0:06 � 0:05 (1/s)

Mathis & Watson

(1956)

De ¼ 0

Ue ¼ Umf

decomposition of cumene

dt ¼ 5 � 10:2 cm

Lmf ¼ 10 � 31 cm

dp ¼ 100 � 200 mesh

parameter: Fo, �bk ¼ 0:64 (1/s)

Lewis et al.

(1959)

Ue ¼ 0

De ¼ 0 or 1hydrogenation of ethylene

dt ¼ 5:2 cm

Lmf ¼ 11 � 53 cm

dp ¼ 0:001 � 0:003 cm

parameter: Fo, �bk ¼ 1:1 � 15:6 (1/s)

�b ¼ 0:05 � 0:18, F ¼ 0:4 � 0:8

Gomezplata & Shuster

(1960)

�b ¼ 0

Ue ¼ Umf

De ¼ 0

decomposition of cumene

dt ¼ 5 � 10:2 cm

Lmf ¼ 3:8 � 20 cm

dp ¼ 100 � 200 mesh

parameter: Fo, �bk ¼ 0:75 (1/s)

Massimila & Johnstone

(1961)

�b ¼ 0

Ue ¼ Umf

De ¼ 0

oxidation of NH3

dt ¼ 11:4 cm

Lmf ¼ 26 � 54 cm

dp ¼ 100 � 325 mesh

parameter: Fo

k ¼ 0:071 (1/s)

Orcutt et al.

(1962)

�b ¼ 0

Ue ¼ 0

decomposition of ozone

dt ¼ 10 � 15 cm

Lmf ¼ 30 � 60 cm

dp ¼ 0:001 � 0:003 cm

k ¼ 0:1 � 3:0 (1/s)

Kobayashi et al.

(1966a)

Ue ¼ Umf

De ¼ 0


dt ¼ 8:3 cm

Lmf ¼ 10 � 100 cm

dp ¼ 60 � 80 mesh

parameter: �bk ¼ 0:1 � 0:8 1/s

�b ¼ 15 ðL=Lmf � 1Þ

Kobayashi et al.

(1966b)

Ue ¼ Umf

De ¼ 0


dt ¼ 20 cm

Lmf ¼ 10 � 100 cm

dp ¼ 60 � 80 mesh

parameter: �bk ¼ 0:2 � 3:5 1/s

�b ¼ 0:1 � 0:3

Kato

(1967)

Ue ¼ Umf

De ¼ 0

packed fluidized bed

hydrogenation of ethylene

dt ¼ 8:7 cm

Lmf ¼ 10 � 30 cm

dp ¼ 100 � 200 mesh,

dp ¼ 1 � 3 cm

parameters: �bk ¼ 1:1 � 3:3 1/s

�b ¼ 0:35 � 0:45

Kobayashi et al.

(1967)

De ¼ 0

Ue ¼ Umf

residence-time curve

gas: air

tracer: He

particle: silica, gel

dt ¼ 8:4 cm

Fo ¼ 11=db

De Grout

(1967)

De ¼ 0

Ue ¼ Umf

residence time curve

gas: air

tracer: He

particle: silica

dt ¼ 10 � 150 cm

HK ¼ 0:67 d0:25t L0:5

where L ¼ bed height (m)

HK ¼ U=Fo; U[m/s]

Kato et al.

(1967)

Ue ¼ 0

De ¼ 0:68U �Umf

Umf

� �dp"p

residence-time curve

gas: air, H2, N2

tracer: H2, C2H4, C3H5

particle: silica-alumina, glass

dt ¼ 10 cm, dp ¼ 1 � 3 cm

Fo ¼ 5 � 3 1/s for

U=Umf ¼ 2 � 30

M ¼ 0:4 � 0:2 1/s for

U=Umf ¼ 2 � 30


Table 2 Theoretical Study of the Two-Phase Model

Authors Parameter assumed Method �b or F0 Remarks

Van & Deemter

(1961)

�b ¼ 0

Ue ¼ 0

De ¼ Ds

A steady state analysis of gas

backmixing and residence time

curve and first-order reaction

by two-phase model

Hk ¼ FoL

U

Hk ¼ 0:5 � 2:5; �b ¼ 0

Parameter Fo is not

related to the bubble

movement in the bed

Davidson & Harrison

(1963)

De ¼ 0 or 1�b ¼ 0

Estimation of conversions for

a first-order reactionFo ¼ 5:85D1=2g1=4

d5=4b

þ 4:5Umf

db

Parameter db model

does not account for

bubble growth in the

bed

Muchi

(1965)

Ue ¼ Umf

0 < De <1A study of effect of Fo, �b, De

Ue on conversion of a first-

order reaction

No relation between

bubble movement and

parameter

Mamuro & Muchi

(1965)

Ue ¼ Umf

�b ¼ 0

Analysis of a first-order reaction

based on the two-phase cell

model

Fo=� ¼ 0:05� ¼ shape factor of bubble

Kobayashi & Arai

(1965)

Ue ¼ 0

De ¼ 0

A study of the effect of k, �b,De, and Fo on conversion of a

first-order reaction

Parameters �b, Fo, De

are not related to the

bubble movement

Partridge & Rowe

(1966)

De ¼ 0

�b 6¼ 0

Consider bubble-cloud phase

reaction in bubble-cloud

phase with non-first-order

kinetics

Shc ¼ 2þ 0:69Sc0:33Re0:5c

Shc ¼Kg;cdc

D

Allow the variation

of bubble population

with height

Van & Deemter

(1967)

De ¼ 0 Analysis of backmixing

residence time curve of tracer

gas and the first-order

reactions

Fo ¼ 0:4 � 1:2 (1/s) Parameters �b, Fo, Ue,

are not related to the

bubble growth in the

bed


concentration profile, predicted dense phase concentra-tions are far too high and too close to the bubble phaseconcentrations. They concluded that the mass transferrate in the model is too high.

3.1.2 Model of Partridge and Rowe (1966)

Another representative two-phase model is the oneproposed by Partridge and Rowe (1966). In thismodel, the two-phase theory of Toomey andJohnstone (1952) is still used to estimate the visiblegas flow, as in the model of Davidson and Harrison(1963). However, this model considers the gas inter-change to occur at the cloud-emulsion interphase,i.e., the bubble and the cloud phase are considered tobe well-mixed, the result being called bubble-cloudphase. The model thus interprets the flow distributionin terms of the bubble-cloud phase and the emulsionphase. With the inclusion of the clouds, the model alsoallows reactions to take place in the bubble-cloudphase. The rate of interphase mass transfer proposedin the model, however, considers the diffusive mechan-ism only (i.e., without throughflow) and is much lowerthan that used in the model of Davidson and Harrison(1963).

Chavarie and Grace (1975) also reviewed the modeland found that the model has a tendency to overesti-mate the cloud size which deprives the concentrationprofiles and related features obtained from the modelof any physical meaning, owing to the inherent physi-cal incompatibility. A similar problem regarding themodel was also reported by Ellis et al. (1968).

3.1.3 Modifications and Applications

In recent years, several modified versions of the two-phase model were proposed for modeling fluidized bedreactors. They include a model proposed by Werther(1980) for catalytic oxidation of ammonia, in which themass transfer process is expressed in terms of filmtheory, as described in Danckwerts (1970); a modelproposed by Werther and Schoessler (1986) for cataly-tic reactions; a model proposed by Borodulya et al.(1995) for the combustion of low-grade fuels; amodel proposed by Arnaldos et al. (1998) for vacuumdrying; and a model proposed by Srinivasan et al.(1998) for combustion of gases. The modificationsinclude the consideration of axial mass transfer profile,the inclusion of a wake phase in addition to the bubbleand emulsion phases, and the consideration of thegrowth of bubbles in the bubble phase.

3.2 Bubbling Bed Model

The bubbling bed model proposed by Kunii andLevenspiel (1969) can be considered a modified versionof the two-phase model where, in addition to the bub-ble and the emulsion phases, a cloud-wake phase isalso considered. The model represents a group of mod-els often referred to as backmixing or dense phase flowreversal models (see also Van Deemter, 1961; Lathamet al., 1968; Fryer and Potter, 1972). A key differencebetween this model and the rest of the two-phase mod-els is that the interphase mass transfer considers twodistinct resistances, one from the bubble phase to thecloud-wake phase, and the other from the cloud-wakephase to the emulsion phase.

The derivation of the model involves the followingbackground theory and observations reported byDavidson and Harrison (1963) and Rowe andPartridge (1962), i.e.,

1. Bubble gas stays with the bubble, recirculatingvery much like smoke rising and only penetrat-ing a small distance into the emulsion. Thiszone of penetration is called the cloud since itenvelops the rising bubble.

2. All related quantities such as the velocity of therise, the cloud thickness, and the recirculationrate, are simple functions of the size of risingbubble.

3. Each bubble of gas drags a substantial wake ofsolids up the bed.

3.2.1 Derivation of the Model

Based on the above observations, the bubbling bedmodel assumes that

1. Bubbles are of one size and are evenly distrib-uted in the bed.

2. The flow of gas in the vicinity of rising bubblesfollows the Davidson model (Davidson andHarrison, 1963).

3. Each bubble drags along with it a wake ofsolids, creating a circulation of solids in thebed, with upflow behind bubbles and down-load in the rest of the emulsion.

4. The emulsion stays at minimum fluidizing con-ditions; thus the relative velocity of gas andsolid remains unchanged.

With the above assumptions, material balances forsolids and for gas give in turn


ðUpflow of solids with bubble)

¼ ðDownflow of solids in emulsionÞ ð6ÞðTotal throughflow of gas)

¼ ðUpflow in bubbleÞ þ ðUpflow in emulsionÞð7Þ

Letting

ubr ¼ 0:711ðgdbÞ0:5 ð8Þthe above material balances give (1) The rise velocity ofbubbles, clouds, and wakes, ub:

ub ¼ U �Umf þ ubr ¼ U �Umf þ 0:711ðgdbÞ0:5ð9Þ

(2) the bed fraction in bubbles, d:

d ¼ U � ½1� d� ad�Umfð Þub

ffi U �Umf

ubð10Þ

(3) the bed fraction in clouds, b:

b ¼ 3dðUmf="mf Þubr � ðUmf="mf Þ

ð11Þ

(4) the bed fraction in wakes, o:

o ¼ ad ð12Þ(5) the bed fraction in downflowing emulsion includingclouds, �oo:

�oo ¼ 1� d� ad ð13Þ(6) the downflow velocity of emulsion solids, us:

us ¼adub

1� d� adð14Þ

(7) the rise velocity of emulsion gas, ue:

ue ¼Umf

"mf

� us ð15Þ

Using Davidson’s theoretical expression for bubble-cloud circulation and the Higbie (1935) theory forcloud-emulsion diffusion, the interchange of gasbetween bubble and cloud is then found to be

Kbc ¼ 4:5Umf

dbþ 5:85

D0:5g0:25

d1:25b

ð16Þ

and between cloud and emulsion

Kce ¼ 6:78ð"mfDubÞ

d3b

� �0:5

ð17Þ

The above expressions indicate that if "mf , �, Umf , andU are known or measured, then all flow properties and

regional volumes can be determined in terms of oneparameter, the bubble size. The application of thismodel to chemical conversion is described below.

3.2.2 Model Expression for First-Order Kinetics

For a first-order catalytic reaction occurring in a gas–solid fluidized bed with "A ¼ 0, the rate equation maybe expressed as

�rA;s ¼1

Vs

dNA

dt

� �¼ kCA ð18Þ

If the bed is assumed to be operated at a fairly high gasflow rate with vigorous bubbling of large rising bub-bles, then both the gas flow in the emulsion and thecloud volume become so small that we can ignore thethroughflow of gas in these regions. Consequently, asan approximation, flow through the bed occurs only inthe bubble phase. The disappearance of A in risingbubble phase, therefore, can be formulated as (seeFig. 2):

ðDisapparance from bubble phaseÞ ¼ ðReaction in

bubble)+(Transfer to cloud and wakeÞðTransfer to cloud and wake)=(Reaction in cloud

cloud and wake)+(Transfer to emulsion)

ðTransfer to emulsion)=(Reaction in emulsion)

ð19ÞIn symbols, the above expressions become

�rA;b ¼ � 1

Vb

dNA

dt

� �¼ gbkCA;bþ

KbcðCA;b � CA;cÞKbcðCA;b � CA;cÞ ¼ gckCA;c þ KceðCA;c � CA;eÞKceðCA;c � CA;eÞ ¼ gekCA;e ð20Þ

Experiment has shown that

gb ¼ 0:001 � 0:01 ð21Þand

a ¼ 0:25 � 1:0 ð22ÞAlso, by the material balance expressions, Eqs. 6 to 15,it can be shown that

gc ¼ ð1� "mf Þ 3

Umf"mf

�ubr �

umf

"mf

� �þ a

26643775 ð23Þ


and

ge ¼ð1� "mf Þð1� dÞ

d� gb � gc ð24Þ

On eliminating all intermediate concentration in Eqs.20a, b, and c, we find that

�rA;b ¼ gbkþ 1

1

� �CA ð25Þ

where

1 ¼1

Kbc

þ 1

2ð26Þ

2 ¼ gckþ 1

3ð27Þ

and

3 ¼1

Kce

þ 1

ðgekÞð28Þ

Inserting for plug flow of gas through the bed yieldsthe desired performance expression, or

lnCA0

CA

� �¼ gbkþ 1

1

� �Lfluidized

ubð29Þ

where approximately

Lfluidized

ub¼ 1� "packed

1� "mf

Lpacked

ubrð30Þ

Note that since the bubble size is the only quantity thatgoverns all the rate quantities with the exception of k,

the performance of a fluidized bed is a function of thebubble size. For small bubbles, the results from thebubbling bed model may range between a plug flowmodel and a mixed flow model; for large bubbles, how-ever, they may be well below those predicted by amixed flow model.

3.2.3 Examples and Model Applications

Examples illustrating the model and additional discus-sions on the model are found in Levenspiel (1972) andKunii and Levenspiel (1991). Applications of themodel were reported in several recent studies, includingscale-up studies of catalytic reactors by Botton (1983)and Dutta and Suciu (1989), a gasification study ofcoal carried out by Matusi et al. (1983), and a studyof fast fluidized bed reactors by Kunii and Levenspiel(1998).

3.3 Bubble Assemblage Model

Since the development of the bubbling bed model, var-ious other hydrodynamic models have been proposedusing other combinations of assumptions such aschanging bubble size with height in the bed, negligiblebubble-cloud resistance, negligible cloud-emulsionresistance, and nonspherical bubbles. Among them,the bubble assemblage model, proposed by Kato andWen (1969), considers changing bubble size with heightin the bed. The model has the following assumptions:

Figure 2 Bubbling bed model. (From Kunii and Levenspiel, 1969.)


1. A fluidized bed may be represented by n com-partments in a series. The height of each com-partment is equal to the size of each bubble atthe corresponding bed height.

2. Each compartment is considered to consist of abubble phase and an emulsion phase. The gasflows through the bubble phase, and the emul-sion phase is considered to be completelymixed within the phase.

3. The void space within the emulsion phase isconsidered to be equal to that of the bed atthe incipient fluidizing conditions. The upwardvelocity of the gas in the emulsion phase is Ue.

4. The bubble phase is assumed to consist ofspherical bubbles surrounded by sphericalclouds. The voidage within the cloud isassumed to be the same as that in the emulsionphase, and the diameter of the bubbles andthat of clouds is given by Davidson (1961) as

dcdb

� �3

¼ ubr þ 2ðUmf="mf Þubr � ðUmf="mf Þ

for ubr �Umf

"mf

ð31Þ

where

ubr ¼ 0:711ðgdbÞ0:5 ð32ÞNote that the calculation proposed above would not beapplicable for large particles where ubr may be less thanUmf="mf .

5. The total volume of the gas bubbles within thebed may be expressed as (L� Lmf ÞS.

6. Gas interchange takes place between the twophases. The overall mass interchange coeffi-cient per unit volume of gas bubbles is given by

Fd ¼ Fo þ K 0M ð33Þ7. The bubbles are considered to grow continu-

ously while passing through the bed until theyreach the maximum stable size or reach thediameter of the bed column. The maximumstable bubble size, db;t, can be calculated by(Harrison et al., 1961)

db;t ¼ut

0:711

�2 1

gð34Þ

8. The bed is assumed to be operating under iso-thermal conditions since the effective thermaldiffusivity and the heat transfer coefficient arelarge.

3.3.1 Key Equations in the Bubble AssemblageModel

In addition to the above assumptions, the model hasthe following key equations to estimate variousbubbling properties:

1. Bubble size, based on Cooke et al. (1968):

db ¼ 0:14rpdpU

Umf

� �hþ do ð35Þ

where

do ¼ 0:0256ðU �Umf Þ=ðnopÞ½ �0:4

g0:2ð36Þ

2. Bubble velocity, following Davidson andHarrison (1963):

ub ¼ ðU �Umf Þ þ ubr ¼ ðU �Umf Þþ 0:711ðg dbÞ0:5

ð37Þ

3. Bed expansion, based on Assumption 5 andEqs. 35 through 37,

L� Lmf

Lmf

¼ U �Umf

0:711ðg db;aÞ0:5ð38Þ

where db;a is the average bubble diameter ofthe bed given by

db;a ¼ 0:14rpdpU

Umf

Lmf

2þ do ð39Þ

4. Voidage of the bed, "(a) For h Lmf ,

1� " ¼ Lmf

L1� "mfð Þ ð40Þ

(b) for Lmf h Lmf þ 2ðL� Lmf Þ,

1� " ¼ Lmf

Lð1� "mf Þ

� 0:5Lmf ð1� "mf Þðh� Lmf Þ

2LðL� Lmf Þð41Þ

5. Superficial gas velocity in emulsion phase, Ue

Ue

Umf

¼ 1� "mfa0yub

Umf ð1� y� a 0yÞ ð42Þ

where

y ¼ L� Lmf

Lð43Þ

and a 0 is the ratio of the volume of emulsiontransported upward behind a bubble (volume


of wake) to the volume of a bubble. The valueof a 0 is approximately 0:2 � 0:3 according tothe experimental study of Rowe and Partridge(1965). Therefore under normal experimentalconditions, Eq. (42) yields Ue=Umf ¼ 0:5for U=Umf ¼ 3, and Ue=Umf ¼ 0 forU=Umf ¼ 5 � 6. However, in the model, Ue

was assumed to be zero based on the experi-mental findings of Latham (1968) and theargument presented by Kunii and Levenspiel(1969).

6. Interchange coefficient, Fd, based on Eq. (33)Since no experimental data are available forthe particle interchange rate, M, or the adsorp-tion equilibrium constant for the reacting gason particle surfaces, K 0, the model neglects gasinterchange due to adsorbed gas on interchan-ging particles. Equation (33) therefore can bereduced to

Fd ¼ Fo ð44Þwhere the following equation based on theexperimental work of Kobayashi et al. (1967)was proposed to describe Fo:

Fo ¼ 0:11

dbð45Þ

3.3.2 Calculation Procedure Based on BubbleAssemblage Model

Let the height of the nth compartment be hn, wheren ¼ 1; 2; 3, to n (see Fig. 3). Based on an arithmeticaverage of the bubble size, the height of the initialcompartment immediately above the distributorbecomes

�h1 ¼do þ ðc�h1 þ doÞ

2ð46Þ

or

�h1 ¼2do2� c

ð47Þ

where

c ¼ 0:14rpdpU

Umf

ð48Þ

which is a proportionality constant relating the bubblediameter for a given operating condition. The height ofthe second compartment then becomes

�h2 ¼ 2do2þ c

ð2� cÞ2 ð49Þ

and that of nth compartment becomes

�hn ¼ 2doð2þ cÞn�1

ð2� cÞn ð50Þ

The number of bubbles in the nth compartmentbecomes

N ¼ 6Sð"� "mf Þpð�hnÞ2ð1� "mf Þ

ð51Þ

The volume of cloud in the nth compartment can becomputed from Eq. (31) as

Vcn ¼Npð�hnÞ3

6

3ðUmf="mf Þubr �Umf="mf

ð52Þ

where

ubr ¼ 0:711 gð�hnÞ½ �0:5 ð53ÞThe total volume of the bubble phase (bubble andcloud) and that of the emulsion phase in the nth com-partment are, respectively,

Vbn ¼ Npð�hnÞ36

ubr þ 2ðUmf="mf Þubr � ðUmf="mf Þ

ð54Þ

and

Ven ¼ S�hn � Vbn ð55ÞThe distance from the distributor to the nth compart-ment is then

hn ¼ �h1 þ�h2 þ�h3 þ þ�hn ð56ÞThe gas interchange coefficient based on unit volumeof bubble phase (bubble and cloud) can be shown as

F 0on ¼ Fon

ubr � ðUmf="mf Þubr þ 2ðUmf="mf Þ

ð57Þ

Hence, the material balance for the gaseous reactantaround the nth compartment becomes, for the bubblephase,

ðSUCA;bÞn�1 ¼ F 0onVb CA;b � CA;e

� �� n

þ ðrA;cVcÞn þ ðSUCA;bÞnð58Þ

and for the emulsion phase,

F 0onVb CA;b � CA;e

� �� n¼ rA;eVe

� �n

ð59Þwhere rA;c and rA;e are the reaction rates per unitvolume of the cloud and the emulsion phase, respec-tively. Note that for the first-order reaction, the expres-sions for rA;c and rA;e are rA;c ¼ krCA;b andrA;e ¼ krCA;e, respectively.


3.3.3 Computational Procedure for Conversion

The computational procedure for conversion and con-centration profile in a fluidized bed reactor is givenbelow. The following operating conditions are needed,i.e., particle size (dp), particle density (rp), minimumfluidization velocity (Umf), gas superficial velocity (U),distributor arrangement (no), column diameter (dt),incipient bed height (Lmf), reaction rate constant (kr),and order of reaction. It should be noted that themodel requires no adjustable parameters.

First, Eq. (38) is used to calculate the expanded bedheight, L. Next, Eq. (50) is used to compute the size ofthe nth compartment. Using Eqs. (51) through (55), thevolumes of the cloud, the bubble phase, and the emul-

sion phase for the nth compartment are then calculated.The nth compartment concentrations, CA;bn and CA;en,are computed from (CA;bÞn�1 and (CA;eÞn�1 using Eqs.(58) and (59). The calculations are repeated from thedistributor until the bed height equivalent to Lmf isreached. For bed height above Lmf , the voidage isadjusted by Eq. (41), andVcn,Vbn, andVen are obtainedusing the same procedure as that shown for the heightsmaller than Lmf . The calculation is repeated until thebed height reaches Lmf þ 2ðL� Lmf).

3.3.4 Discussion of the Bubble Assemblage Model

The computation using the bubble assemblage modelindicates that for most of the experimental conditions

Figure 3 Main features of bubble assemblage model. (From Kato and Wen, 1969.)


tested, the number of compartments is usually greaterthan 10. This means, in terms of the flow pattern, thatthe gas passing through the bubble phase is close toplugflow. In the emulsion phase, because Ue ¼ 0 isused, the flow pattern is essentially considered as adead space interchanging gas with the bubble phasesimilar to that in the bubbling bed model. The flowpattern in the emulsion phase, however, does notexpect to affect significantly the reactor behavioraccording to Lewis (1959) and Muchi (1965). It isworth pointing out that when the reaction is slow,any model, either a plugflow, a complete mixing, abubbling bed, or a bubble assemblage model will rep-resent the experimental data well. However, whenthe reaction is fast, a correct flow model is needed torepresent the data.

3.3.5 Examples and Applications

More detailed discussion regarding the model perfor-mance is given by Wen and Fan (1975) and Mori andWen (1975). The literature-reported applications ofthis model include a combustion study of coal withlimestone injection by Horio and Wen (1975), a coalgasification study by Mori et al. (1983), a study oncatalytic oxidation of benzene by Jaffres et al. (1983),a catalytic ammoxidation of propylent by Stergiou andLaguerie (1983), a silane decomposition study by Li etal. (1989), a catalytic oxidation of methane by Mleczkoet al. (1992), and a study on chlorination of rutile byZhou and Sohn (1996).

3.4 Comparison of Models

In an attempt to compare the effectiveness of varioustwo-phase models, Chavarie and Grace (1975) carriedout a study of the catalytic decomposition of ozone ina two-dimensional fluidized bed where the reactor per-formance and the ozone concentration profiles in bothphases were measured. They compared the experimen-tal data with those predicted from various two-phasemodels reviewed above and reported that

1. The model of Davidson and Harrison (1963),which assumes perfect mixing in the densephase (DPPM), underestimates seriously theoverall conversion for the reaction studied.While the counterpart model that assumes pis-ton flow in the dense phase (DPPF) gives muchbetter predictions of overall conversion, stillthe predicted concentration profiles in the indi-vidual phases are in poor agreement with theobserved profiles.

2. The model of Partridge and Rowe (1966)makes allowance for variable bubble sizesand velocities and for the presence of clouds.Unfortunately, for the conditions of theirwork, the overestimation of visible bubbleflow by the two-phase theory of Toomey andJohnstone (1952) led to incompatibilitybetween predicted cloud areas and the totalbed cross section. This mechanical incompat-ibility prevented direct application of thesemodels to the reaction data obtained in theirwork.

3. The bubbling bed model of Kunii andLevenspiel (1969) provides the best overallrepresentation of the experimental data inthis study. Predicted bubble phase profilestend gently to traverse the measured profiles,while dense phase profiles are in reasonableagreement over most of the bed depth.Overall conversions are well predicted. Thesuccess of the model can be mainly attributableto (1) the moderate global interphase masstransfer, (2) the negligible percolation rate inthe dense phase, (3) the occurrence of reactionwithin the clouds and wakes assumed by themodel, and (4) the use of average bubble prop-erties to simulate the entire bed.

4. The Kato and Wen (1969) bubble assemblagemodel, though better suited to represent com-plex hydrodynamics due to allowance for vari-able bubble properties, fails to account forobserved end effects in the reactor. While thismodel was found to give the best fit for thebubble phase profile, dense phase profiles andoutlet reactant concentrations were seriouslyoverpredicted.

It is worth pointing out that a general observationmade by Chavarie and Grace (1975) was that none ofthe models (tested correctly) accounted for the consid-erable end effects observed at both the inlet and theoutlet, i.e., near the distributor (grid region) and in thespace above the bed surface (freeboard region).

4 MULTIPLE-REGION MODELS

The grid region near the distributor and the freeboardspace above the bed surface have bed hydrodynamicssignificantly different from the main body of the bub-bling bed reactor. A number of authors have observedthe abnormally high rate of reaction near the distribu-tor, apparently due to the very high rate of interphase


mass transfer. Similarly, many observations have beenmade of temperature increase in the freeboard region,indicating additional reaction in the region. A realisticapproach for modeling a fluidized bed reactor wouldtherefore require the consideration of three consecutiveregions in the bed: the grid region near the bottom, thebed region in the middle, and the freeboard regionabove the bed surface.

4.1 Models for the Grid Region

The grid region plays an important role in determiningthe reaction conversion of fluidized bed reactors, espe-cially for fast reactions where the mass transfer opera-tion is the controlling mechanism. Experimentalstudies have indicated that changing from one distri-butor to another, all other conditions remaining fixed,can cause major changes in conversion (Cooke et al.,1968; Behie and Kehoe, 1973; Bauer and Werther,1981). It is generally observed that, in the grid region,additional mass transfer can take place owing to theconvective flow of gas through the interphase of theforming bubbles. The flow of gas through the formingbubbles into the dense phase, and then returning to thebubble phase higher in the bed, represents a netexchange between the two phases. The experimentalwork of Behie and Kehoe (1973) indicated that themass transfer coefficient in the grid region, kje, can be40 to 60 times that in the bubble region, i.e., kbe.

Behie and Kehoe (1973) and Grace and De Lasa(1978) proposed similar sets of equations to describefluidized bed reactors considering both the grid andbed regions. The model of Grace and De Lasa (1978)contains the following three equations: (1) for the jetphase in the grid region (0 h J),

UdCA;j

dh

� �þ kjeajðCAj � CA;eÞ ¼ 0 ð60Þ

(2) for the bubble phase in the bed region (J h L),

b 0UdCA;b

dh

� �þ kbeab CA;b � CA;e

� � ¼ 0 ð61Þ

and (3) for the emulsion phase in both the grid and bedregions (0 h L),

Uð1� b 0ÞðCA;e � CA;jJÞ þðJ0

kjeajðCA;e � CA;jÞdh

þðLJ

kbeabðCA;e � CA;bÞ dhþ krCA;eLmf ¼ 0

ð62Þ

It should be noted that the difference between thismodel and the corresponding model for the bed regiononly is the replacement of Eq. (61) by Eq. (60) in thegrid region (0 h Lj). The model therefore predictsa much higher conversion in the grid region becausethe mass transfer coefficient in the region can be 40 to60 times greater than in the bed region, as reported byBehie and Kehoe (1973). Among the grid region stu-dies, Ho et al. (1987) reported dynamic simulationresults of a shallow jetting fluidized bed coal combus-tor using the grid region model.

In another attempt, Sit and Grace (1986) measuredtime-averaged concentrations of methane in the entryregion of beds of 120 to 310 mm particles contained in a152 mm column with a central orifice of diameter 6.4mm and auxiliary tracer-free gas. The following equa-tions were proposed for the particle–gas mass transferin this region:

Vb

dCA;j

dt

� �¼ QorCA;or þ kbe1Sex;bCA;e

� Qor þ kbe1Sex;b

� �CA;j

ð63Þ

where Qor represents the convective mechanism definedas

Qor ¼ uorp4

�d2or ð64Þ

and kbe1 represents the diffusive mechanism that can bepredicted by the penetration theory expression, i.e.,

kbe1 ¼4D"mf

ptf

� �0:5

ð65Þ

Their results indicated that the convective mechanismhas a greater effect on the grid region mass transfer.Since bubbles are also smaller near the grid, they con-cluded that favorable gas–solid contacting occurs inthis region primarily due to convective outflow fol-lowed by recapture. Note that Eq. (63) can be usedto replace Eq. (60) for the grid region modeling.

4.2 Models for the Freeboard Region

When bubbles burst and release their gases at the sur-face of a fluidized bed, it is generally observed thatparticles are ejected into the freeboard space abovethe surface. These particles can originate either fromthe dense phase just ahead of the bubble at the momentof eruption (Do et al., 1972) or from those solids thattravel in the wake of the rising bubbles (Basov et al.,1969; Leva and Wen, 1971; George and Grace, 1978).


As for studies of freeboard region hydrodynamicsand reactions, Yates and Rowe (1977) developed asimple model of a catalytic reaction based on theassumption that the freeboard contained perfectlymixed, equally dispersed particles derived from bubblewakes. The fraction of wake particles ejected, f 0, was amodel parameter. The governing equation of themodel was proposed as

� dCA;cell

dt¼ kgAp

Vcell

CAh � CAp

� � ð66Þ

where Vcell can be determined by

Vcell ¼3Vp

f 0ð1� "mf ÞU �Umf

U � utð67Þ

and kg can be evaluated from the expression of Roweet al. (1965), i.e.,

Sh ¼ kgdp

D ¼ 2þ 0:69 Sc0:33Re0:5t ð68Þ

with the Schmidt number, Sc, defined as

Sc ¼ mrgD

ð69Þ

and the terminal Reynolds number, Ret, defined as

Ret ¼utdprg

mð70Þ

The model equations were solved for various combina-tions of the parameters, and it was reported that inmany cases freeboard reactions lead to greater conver-sion than that achieved by the fluid bed itself. Thedetails of the results are also reviewed by Yates (1983).

5 CONCLUDING REMARKS

Numerous theoretical and experimental studies havebeen carried out in the past five decades in an attemptto model gas–solids fluidized bed reactors. However,the modeling of such reactors remains an art ratherthan a science. Models that work well for certain reac-tion processes may not work for others. In otherwords, there has not been any single model universallyapplicable to all processes carried out in such reactors.However, all these modeling attempts do generatevaluable insights regarding reactor behavior, whichleads to improved design and operation.Nevertheless, additional studies are needed to furtherour knowledge in the modeling of such reactors.

NOTATION

Ap = surface of freeboard particles appearing in Eq.

(66), m2

ab = transfer area of bubbles to emulsion per unit

volume of reactor, m2=m3

aj = transfer area of jets to emulsion per unit

volume of reactor, m2=m3

CA = concentration of A, kg mol/m3

CAh = concentration of A at periphery of gas cell

appearing in Eq. (66), kg mol/m3

CAo = initial concentration of A appearing in Eqs. (1)

and (3), kg mol/m3

CAp = concentration of A at particle surface

appearing in Eq. (66), kg mol/m3

CA;b = concentration of A in bubble phase, kg mol/

m3

CA;bn = concentration of A in bubble phase at nth

compartment, kg mol/m3

CA;c = concentration of A in cloud phase, kg mol/m3

CA;cell = concentration of A in gas cell appearing in Eq.

(66), kg mol/m3

CA;e = concentration of A in emulsion phase, kg mol/

m3

CA;en = concentration of A in emulsion phase at nth

compartment, kg mol/m3

CA;j = concentration of A in jet phase, kg mol/m3

CAjJ = concentration of A at the tip of jets, kg mol/

m3

CA;or = concentration of A at the distributor orifice,

kg mol/m3

Db = axial dispersion coefficient of reactant in

bubble phase, m2/s

De = axial dispersion coefficient of reactant in

emulsion phase, m2/s

db = bubble diameter, m

db;a = average bubble diameter, m

db;t = maximum stable bubble diameter, m

dc = diameter of cloud, m

d0 = initial bubble diameter at the distributor, m

dor = diameter of distributor orifice, m

dp = particle diameter, m

dt = diameter of bed column, m

E = probability density function

F = volumetric fraction of gas in the bubble phase

Fd = overall gas interchange coefficient per unit

volume of gas bubble, 1/s

Fo = gas interchange coefficient per unit volume of

gas bubble, 1/s

Fon = gas interchange coefficient at the nth

compartment per unit bubble volume, 1/s

F 0on = Fon per unit volume of bubble phase (bubble

and cloud), 1/s

Fs = volume fraction of solids in bubble phase

f = volumetric fraction of gas in the emulsion

phase


f 0 = fraction of wake particles ejected appearing in

Eq. (66)

fs = volumetric fraction of solids in emulsion phase

g = gravitational acceleration, m/s2

h = distance from the distributor, m

h1 = distance between the distributor and the top of

the first compartment, m

�h1 = length of the first compartment, m

hn = distance between the distributor and the top of

the nth compartment, m

�hn = length of the nth compartment, m

J = height of jet phase, m

K 0 = adsorption equilibrium constant

Kbc = gas interchange coefficient between the bubble

and cloud phases, 1/s

Kce = gas interchange coefficient between the cloud

and emulsion phases, 1/s

k = reaction rate constant, 1/s

kbe = bubble to emulsion mass transfer coefficient,

m/s

kbe1 = bubble to emulsion mass transfer coefficient

during bubble formation, m/s

kg = mass transfer coefficient, m/s

kje = jet to emulsion mass transfer coefficient, m/s

k0 = reaction rate constant defined in Eq. (2), 1/s

kr = first-order reaction rate constant based on

volume of emulsion or cloud, 1/s

L = bed height, m

Lfluidized = bed height under fluidization condition, m

Lmf = bed height at minimum fluidization, m

Lpacked = bed height under packed condition, m

M = Solid interchange coefficient between bubble

and emulsion phases per unit volume of

bubble, 1/s

m = fitted parameter appearing in Eqs. (2) and (3)

N = number of bubbles in nth compartment

NA = moles of A, kg mol

n0 = number of distributor holes

Qor = gas flow through orifice, m3/s

rA;b = rate of reaction per unit bubble volume, kg

mol/(m3 s)

rA;c = rate of reaction per unit cloud volume, kg

mol/(m3 s)

rA;e = rate of reaction per unit emulsion volume, kg

mol/(m3 s)

rA;s = rate of reaction per unit solid volume, kg mol/

(m3 s)

Ret = Reynolds number of particle at terminal

velocity

S = cross-sectional area of the bed, m2

Sex;b = surface area of bubbles, m2

Sc = Schmidt number, defined as Sc ¼ m=ðrgDÞSh = Sherwood number, defined as Sh ¼ kgdp=Dt = time, s

tf = bubble formation time, s

U = superficial gas velocity, m/s

Ue = superficial gas velocity in the emulsion phase,

m/s

Umf = minimum fluidization velocity, m/s

ub = rise velocity of bubbles in a fluidized bed, m/s

ubr = rise velocity of a single bubble in a fluidized

bed, m/s

ue = rise velocity of emulsion gas, m/s

uor = velocity of gas through orifice, m/s

us = downflow velocity of emulsion solids, m/s

ut = terminal velocity of particle, m/s

Vb = volume of bubble phase, m3

Vbn = volume of bubble phase at the nth

compartment, m3

Vc = volume of cloud phase, m3

Vcell = volume of a freeboard gas cell expressed in Eq.

(67), m3

Vcn = volume of cloud phase at the nth

compartment, m3

Ve = volume of emulsion phase, m3

Ven = volume of emulsion phase at the nth

compartment, m3

Vp = volume of freeboard particles appearing in Eq.

(67), m3

Vs = volume of solids, m3

a = ratio of cloud volume to bubble volume

a 0 = ratio of wake volume to bubble volume

b = bed fraction in clouds

b 0 = fraction of gas flow in the bubble or jet phase

gb = volume fraction of solids in bubbles in a

fluidized bed

gc = volume fraction of solids in clouds in a

fluidized bed, -

ge = volume fraction of solids in emulsion phase in

a fluidized bed,

D = diffusivity of reactant gas, m2/s

d = bed fraction in bubbles

" = void fraction

"A = fractional volume change on complete

conversion of A

"mf = void fraction at minimum fluidization

"packed = void fraction under packed bed condition

y = bed expansion fraction defined in Eq. (43)

p = constant, � 3:1416rg = gas density, kg/m3

rp = density of particle, kg/m3

m = gas viscosity, kg/(m s)

1 = expression defined in Eq. (26)



o = bed fraction in wake, appearing in Eq. (12)�oo = bed fraction in downflowing emulsion,

appearing in Eq. (13)

c = expression defined in Eq. (48)


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modelare strat fluidizat

Documents

dispersion models

conventional models

series models

twoclasses of models

slower gas

faster gas

staying gas

operating uidized bed