Chemical Engineering Science 62 (2007) 70–81www.elsevier.com/locate/ces

Generalized bubbling–slugging fluidized bed reactor model

J.P. Constantineaua, J.R. Gracea,∗, C.J. Lima, G.G. Richardsb

aDepartment of Chemical and Biological Engineering, University of British Columbia, Vancouver, Canada V6T 1Z3bTeck Cominco Metals Ltd., Trail, British Columbia, Canada V1R 4L8

Available online 22 August 2006

Abstract

A generalized slugging–bubbling fluidized bed reactor model is developed which handles seamlessly the transition from bubbling to sluggingfluidization, using the concept of probabilistic averaging developed in earlier work. The ratio of the bubble diameter to the column diameteris employed to correlate the probability of each of these fluidization flow regimes. The model gives a good representation of the conversionswithin the slugging and bubbling regimes, as well as for the transition between these two regimes.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Fluidization; Chemical reactor; Modelling; Slugging fluidized beds

1. Introduction

Over the years, great success has been achieved in the de-sign and implementation of new fluidized bed processes. How-ever, the reputation of fluidized bed reactors has suffered fromsome failures which occurred even though pilot reactors oper-ated successfully. The most notable example (Geldart, 1967)was in the production of gasoline by Fischer–Tropsch synthe-sis. Pilot-scale reactors (12, 25, 50 and 195 mm diameter) wereoperated successfully with conversions exceeding 95%. Withsuch encouraging results, a complete industrial plant was builtwith a fluidized bed reactor of approximately 5 m diameter. Re-grettably, conversions of only 40–50% were obtained from theindustrial unit. After gaining experience on a 2.4-m-diameterpilot plant, the industrial unit was modified and the conversionsincreased to 65–70%. After operating for a few years, the pro-cess was uneconomic, and production ceased, highlighting thedifficulty of fluidized bed reactor scale-up.

One of the main difficulties in scale-up has been associ-ated with the use of results from pilot plants operating in theslugging fluidization regime to design large industrial reactorsoperated in the bubbling fluidzation regime. The Shell Chlori-nation Process (de Vries et al., 1972), involving catalytic oxi-dation of HCl to produce Cl2, also illustrates the difficulties of

∗ Corresponding author. Tel.: +1 604 822 3121; fax: +1 604 822 6003.E-mail address: jgrace@chml.ubc.ca (J.R. Grace).

0009-2509/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.08.011

scale-up. A 1.5-m-deep bed was required to obtain a 90%yield in a 50-mm-diameter laboratory reactor. For a 300-mm-diameter pilot-scale reactor, the same yield required a bed 2.8 mhigh. For a 3-m-diameter industrial reactor, a bed 10 m high wasrequired to reach the same yield (Werther, 1980; Wen, 1984).The strong scale-up effect is primarily due to the growth of bub-bles and their effect on the gas residence time and interphasemass transfer.

Scale-up of fluidized bed processes has received considerableattention in the literature (e.g. Volk et al., 1962; Geldart, 1967;Matsen and Tarmy, 1970; Matsen, 1985, 1996; Glicksman,1998). Similarity techniques have been developed and usedto scale fluidized beds for hydrodynamic studies (Glicksman,1998). However, reacting systems often cannot be scaled usingthese techniques, especially when the reactions are the objectof study. Different reactor models have been formulated for thebubbling and slugging fluidization regimes. No reactor modelshave been formulated to include both bubbling and sluggingfluidization flow regimes with the specific focus of addressingthe questions associated with scale-up.

Recently, a generic fluidized bed reactor model has beenformulated to span the bubbling, turbulent and fast fluidizationregimes (Abba et al., 2002, 2003). Using probabilistic aver-aging, this model successfully handles the transitions amongthese three flow regimes when the superficial gas velocity isincreased. However, it does not cover cases where transitionsfrom bubbling to slugging occurs. This article extends the

J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81 71

probabilistic approach to a new generalized slugging–bubblingfluidized bed reactor model where a single set of differentialequations describes the phenomena occurring within the bub-bles and dense-phase, and the parameters are averaged accord-ing to the probability of being in one fluidization flow regimeor the other. This model is primarily intended for group B par-ticles (Geldart, 1973) where slug flow is most commonly en-countered.

1.1. Bubbling and slugging fluidization regimes

In a bubbling fluidized bed, bubbles coalesce as they risethrough the bed causing their average diameter to increase withheight. As the bubble diameter approaches the reactor diameter,the bubbles can no longer grow, and slugging takes over. As thebubble size increases, the effect of the walls becomes greaterand one expects a smooth transition from bubbling to sluggingfluidization. Slugging may therefore be considered as a specialcase of bubbling where the bubble size is physically constrainedby the walls of the reactor.

Three conditions are necessary for slugging:

• The bed height (H) must exceed a minimum height. Baeyensand Geldart (1974) measured (HL) where bubble coalescenceis complete and stable slug spacing is achieved and proposedthe correlation:

HL = 1.3D0.175. (1)

• The superficial gas velocity (U) must exceed the minimumslugging velocity (Ums) given (Stewart and Davidson, 1967)by

Ums = Umf + 0.07√

gD (2)

Baeyens and Geldart (1974) utilized this criterion for deepbeds (Hmf > HL). For shallower beds, they proposed

Ums = Umf + 0.07√

gD + 0.16(HL − Hmf )2. (3)

• The maximum stable bubble size (De,max) must be at leastof the order of the column diameter (D).

The conditions for stable slugging clearly depend on columndiameter and bed height. A shallow bed may not experienceslugging because coalescence may be insufficient for the bub-bles to reach the bed diameter. For deeper beds, bubble growthand coalescence may be sufficient to engender slugging. Addi-tional criteria for the occurrence of slugging were summarizedby Agarwal (1987).

Baeyens and Geldart (1974) divided slugging fluidized bedsinto three zones:

• Zone I: Freely bubbling.• Zone II: Slugging with limited coalescence.• Zone III: Stable slugging with no further coalescence.

Fig. 1 represents these three zones in a bed at minimum sluggingvelocity as a function of height and column diameter. The two

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Zone III

Zone II

Zone I

P=1

P=0.5

P=0

D (m)

H (

m)

Fig. 1. Influence of column diameter on zones in a slugging fluidized bed,for incipient slugging, calculated using the Geldart (1972) bubble size corre-lation for porous plate distributors. Zone I: freely bubbling, Zone II: sluggingwith limited coalescence, Zone III: stable slugging with no further coales-cence, based on Baeyens and Geldart (1874). P-lines correspond to sluggingprobabilities of 0, 0.5 and 1.

dark lines were derived from an equation proposed by Baeyensand Geldart (1974). The upper line was obtained by fitting theheights observed experimentally when stable slugging was es-tablished i.e. Eq. (1). The lower dark line was obtained by cal-culating the height where the bubble diameter from the Geldart(1972) bubble size correlation is equal to half the bed diame-ter at the minimum slugging velocity, calculated from Eq. (3).Note that the minimum slugging velocity varies with columndiameter. While informative, this earlier approach adopts arbi-trary and sharp boundaries at which there are step changes inbehavior. In reality, the transitions are gradual. This paper mod-els these gradual transitions using weighting factors which de-scribe the relative probabilities of bubbling and slugging overthe height interval where the transition takes place. The thinnerdashed lines provide these weighting factors or probabilities ofslugging, as discussed below.

Other correlations can also be used to determine the heightwhere free bubbling ends and slugging takes over. For example,Darton et al. (1977) used their bubble size correlation to obtaina height condition for slugging which depends on distributordesign.

2. Modelling the transition from bubbling to slugging

Previous slugging fluidized bed reactor models have consid-ered the reactor as a simple one-zone stable slugging bed, orhave split the reactor into freely bubbling and stable sluggingzones. Currently, no model adequately accounts for the tran-sition in a slugging fluidized bed. Modelling the transition to

72 J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81

slug flow may be attempted in various ways:

• One can use two distinct models, one for bubbling and onefor slugging, applying a bubbling model up to a given height,then switching to a slug flow model. This approach has beenused to model slug flow reactors (e.g. Hovmand et al., 1971;Yates and Grégoire, 1980a, b; Yue and Zarifis, 1990; Sadakaet al., 2002). A bubbling fluidized bed reactor model is usedfor the first 1.5–2D where the bed is freely bubbling (ZoneI). Above this height, a stable slug-flow model is used (ZoneIII). However, the transition zone (Zone II) is not accountedfor. The problem with this approach is that there is a dis-continuity where the switch is made, whereas in practice thetransition occurs smoothly. Moreover, there is uncertainty inpredicting the regime transition.

• One could use the results from the two models and interpo-late or average their results. A probabilistic average, basedon the weighted sum of the results where the weights arebased on the probability of being in a given regime, wouldbetter characterize the uncertainty of the regime transition.A problem with this approach is that the interpolated resultsare not necessarily consistent with the physics of the system,especially where there are nonlinearities.

• A better method is to combine the two models into a singlemodel which approaches the two limiting models outsidethe transition range. In the transition region between thetwo flow regimes, the model probabilistically averages theparameters (not the final predictions) for each regime. Thisapproach has been used previously to model the transitionfrom bubbling to turbulent fluidization and from turbulent tofast fluidization (Thompson et al., 1999; Grace et al., 1999;Abba et al., 2002, 2003). In that case, simple Re vs Ar cor-relations were employed to describe the regime transitions(Thompson et al., 1999; Abba et al., 2003), with probabilitiesrelated to rms deviations. These probabilities then provideweighting factors for setting the values of key parameters.A somewhat similar approach was employed by Eakman etal. (1980) to model a coal gasifier subject to both bubblingand slugging. However, instead of using the probability ofbeing in a given regime, they used an arbitrary function tointerpolate the parameters in the transition interval.

Currently, the transition from bubbling to slugging fluidiza-tion is only described by the minimum slugging velocity (Ums)and the necessary conditions for slugging. The criteria for thetransition do not quantify the probability of slugging, nor dothey allow for a smooth transition from freely bubbling nearthe distributor via a smooth transition zone to stable sluggingabove.

2.1. Previous models for the bubbling and sluggingfluidization regimes

Bubbling fluidized bed reactors have received considerableattention (e.g. see Chavarie and Grace, 1975; Grace, 1986),whereas slugging fluidized bed reactors have received relativelylittle attention. Two basic models have been developed for slug-

ging fluidized beds. Neither considers the effect of the freelybubbling zone near the gas distributor. Hovmand and David-son (1968, 1971) created the first slugging fluidized bed re-actor model by extending the Orcutt et al. (1962) piston-flowfluidized bed model and deriving an interphase mass transfercoefficient expression for slugs. Raghuraman and Potter (1978)based their slug flow model on the Fryer and Potter (1972)countercurrent backmixing model for bubbling fluidized beds,and utilized the analysis of Hovmand et al. (1971) for the gasinterphase mass transfer coefficients.

The Orcutt et al. (1962), Hovmand et al. (1971) andRaghuraman and Potter (1978) models neglect reaction of thegas in contact with solids suspended in the bubbles or slugs.Grace (1984, 1986) has shown that this assumption is not validfor large effective reaction rate constants. Aoyagi and Kunii(1974) have shown experimentally that solids within bubblesmay play a significant role, even if their concentration is small.

3. Extension of the generalized model concept tobubbling/slugging

Our new steady-state fluidized bed reactor model extendsthe generalized bubbling, turbulent and fast fluidization reactormodel of Abba (2002, 2003), which accounts for variable gasdensity (due to the pressure gradient and changing compositionas one or more reactions proceed) and bulk flow in the inter-phase mass transfer using a probabilistic approach. The modeldeveloped here only considers the bubbling and slugging flowregimes. Also, the approach used to model changes in total gasmolar flow differs from that of Abba (2002).

Fig. 2 represents the two phases of the model, named theL- and H-phases for low-density and high-density phases. TheL-phase represents the bubbles or slugs, while the H-phaserepresents the dense phase or emulsion. Gas enters at the bottomof the fluidized bed where it is distributed between the L- andH-phases. The gas then rises in each phase and reacts with thesolids present. Mass transfer occurs between the two phases.Axial and radial dispersion could be included within each phase.

3.1. Phase balances

As in our previous work (Thompson et al., 1999; Abba etal., 2002, 2003), the bed volume fractions of phases add up tounity, i.e.,

�L + �H = 1. (4)

Each phase volume is composed of particles and void space.The particle volume fraction (�) and gas volume fraction (�)within each phase also add up to 1, i.e. �L + �L = 1 and�H + �H = 1. The total gas molar flowrate (FT ) through thereactor equals the sum of the phase molar flowrates, i.e.

FT = FLT + FHT . (5)

J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81 73

Freeboard

Δz Δz

H-phase L-phase

FL ULUH

FT U

FH

Fig. 2. Schematic of two-phase fluidized bed model.

For ideal gases, the total concentration (CT = PT /RT). Thegaseous molar flowrates are related to the superficial gas ve-locities (U , UL and UH ) by

FT = AUCT , (6)

FHT = �H AUH CT and FLT = �LAULCT . (7)

The molar flowrates of gaseous species i in the H- and L-phasesare

FHi = �H AUH CHi and FLi = �LAULCLi . (8)

Simplifying with the aid of Eq. (5), we obtain

U = �LUL + �H UH . (9)

3.2. Gas mole balances for H- and L-phases

The differential gas mole balance over the control volumefor the low-density (L) and high-density (H) phases of Fig. 2,for a first-order irreversible reaction assuming steady-state con-ditions and neglecting axial and radial dispersion, are

− dFLi

dz+ kLH aI

(�LFHi

�H UH

− FLi

UL

)+ kr�Lvi

FLR

UL

+ kr�H �vFHR

UH

FLi

FLT

= 0, (10)

− dFHi

dz+ kLH aI

(FLi

UL

− �LFHi

�H UH

)+ kr�H vi

FHR

UH

− kr�H �vFHR

UH

FLi

FLT

= 0. (11)

The interphase mass transfer coefficient (kLH ) is defined asthe volumetric rate of transfer per unit bubble (or slug) surfacearea. The interphase mass transfer exchange area (aI ) is theinterfacial bubble area per unit bubble volume. Multiplying thetwo gives the volumetric transfer rate per unit bubble volume.Bulk interphase mass transfer is required when a change in gasvolume occurs within the H-phase. According to the two-phasefluidization theory, a flow of gas of Umf A is required for thehigh-density phase. Any excess gas enters the low-density phaseto create bubbles. In a reacting system, the requirement for thehigh-density phase is still present. However, if there is a volumechange, it must be balanced by bulk interphase transfer. Thedirection of the bulk flow and the required gas molar fractiondepends on the sign of �v. In this case, it is assumed to benegative. If �v were positive, any excess gas produced in thedense phase would go to the L-phase.

These equations are solved using an initial value solver withboundary conditions

FLi = FLi,in = �LAULCi,in (at z = 0), (12)

FHi = FHi,in = �H AUH Ci,in (at z = 0). (13)

The superficial gas velocities (UL and UH ) and the phase vol-ume fractions (�L and �H ) are calculated at a series of heightsin the bed beginning at the entrance. The superficial gas ve-locity (U ) is first calculated from the gaseous molar flowrates(FHT and FLT ). The L-phase superficial gas velocity (UL) isequal to the void velocity (Uv), calculated below. This requiresan appropriate bubble size correlation.

The low-density phase volume fraction (�L) is estimated(Clift and Grace, 1985) using

�L = U − Umf

UL

. (14)

Since the set of differential equations is solved from the distrib-utor (z = 0) to the surface of the expanded bed, the expandedbed height (H) must be known, or estimated and then calcu-lated iteratively. Since only the volume occupied by the bubblesand slugs (L-phase) contributes to the bed expansion,

H − Hmf =∫ H

0�L dz. (15)

With the appropriate choice of parameters, the model can de-scribe a bubbling fluidized bed, a purely slugging bed or abubbling–slugging bed.

3.3. Interphase mass transfer

Among the parameters in Eqs. (10) and (11), only kLH andaI require probabilistic averaging between their bubbling andslugging values. For bubbling and slugging fluidized beds, �L

and �H are usually constant. For a freely bubbling fluidized

74 J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81

Table 1Surface integral (I ) for various slug length-to-diameter ratios

L/D 0.3 0.5 1.0 2.0 3.0 4.0 5.0

I 0.13 0.21 0.39 0.71 0.98 1.24 1.48

bed, the interphase mass transfer coefficient (kLH,bubbling) andarea (aIH,bubbling) are obtained (Sit and Grace, 1981) from

kLH,bubbling = Umf

3+ 2

(Dg�mf Ub∞

�De

)1/2

, (16)

aI,bubbling = 6

De

. (17)

The interphase mass transfer coefficient (kLH ) and area (aI )for slugging fluidization can be expressed in similar form(Hovmand and Davidson, 1971), giving

kLH,slugging = Umf + 16�mf I

1 + �mf

(Dg

�

)1/2( g

D

)1/4, (18)

aI,slugging = 1

Dfs. (19)

The surface integral (I ) (see Table 1) and slug shape fac-tor (fs) are functions of the slug length-to-diameter ratio(Hovmand and Davidson, 1971) with

fs = 4V

�D3 =(

L

D

)− 0.495

(L

D

)1/2

+ 0.061. (20)

The slug length-to-diameter ratio can be obtained (Hovmand etal., 1971) by solving the quadratic equation

L

D− 0.495

(L

D

)1/2 (1 + U − Umf

0.35√

gD

)

+ 0.061 − (T /D − 0.061)(U − Umf )

0.35√

gD= 0, (21)

where T is the slug-to-slug (tail-to-nose) separation distance.The generalized interphase mass transfer coefficient and area

are calculated by weighting factors analogous to those obtainedin previous work by probabilistic averaging, i.e.

ktransition = (1 − Pslugging)kLH,bubbling + (Pslugging)kLH,slugging,

(22)

atransition = (1 − Pslugging)aI,bubbling + (Pslugging)aI,slugging.

(23)

3.4. Void velocities and weighting factors forbubbling/slugging

For the bubbling fluidization regime, the isolated bubble risevelocity is

Ub∞ = 0.71√

gDe. (24)

For freely bubbling beds, the bubble rise velocity (Ub) is givenby

Ub = Ub∞ + (U − Umf ). (25)

At minimum fluidization, Eq. (25) predicts a non-zero bubblerise velocity. As in previous work (Thompson et al., 1999) theexpression for bubble rise velocity is therefore modified so thatUb → 0 as U → Umf :

Ub = (U − Umf )

(1 + 0.71

U

√gDe

). (26)

This change makes negligible difference in the overall predic-tions, but facilitates the use of the probabilities as weightingfactors below.

Since the velocity of bubbles depends on their size, the effec-tive bubble size (De) must be calculated as a function of height.Numerous correlations are available (e.g. Mori and Wen, 1975;Darton et al., 1977; Horio and Nonaka, 1987). Since particlesof Geldart Group A behave differently from Group B particles,especially with respect to the maximum bubble size, the bubblesize correlation must be chosen carefully.

The velocity of a single slug (Us∞) is given by

Us∞ = 0.35√

gD. (27)

For a continuously slugging bed, the slug velocity (Us) is

Us = Us∞ + (U − Umf ). (28)

In order to bridge the bubbling and slugging fluidizationregimes, the void (bubble or slug) rise velocities (Uv∞) aredescribed in dimensionless form

Fr = Uv∞√gD

(subscript � = b or s) (29)

Eqs. (24) and (27) may then be written as

Bubbling Fr = 0.71�, (30)

Slugging Fr = 0.35, (31)

where � is the ratio of square root of diameter,√

De/D.Allahwala and Potter (1979) measured isolated bubble and

slug rise velocities in a gas–solid fluidized bed and correlatedthe results by

Fr = 0.35 tanh1/1.8[3.6(�)1.8]. (32)

Instead of using this relatively complex equation to calculatethe velocities and evaluate the probabilities, a simpler fittingapproach is employed here. Fig. 3 presents the velocities in anair–water system. The transition between the limits of singleisolated bubbles and single isolated slugs given by Eqs. (30)and (31) is smooth and occurs for � between about 0.38 and0.7 (Hovmand and Davidson, 1971). The behavior at each endof the transition interval clearly approaches the two limitingcases, with the slopes equal to each of the limiting values.

We require an interpolation between the two limits whichfits the experimental data and gives the proper limiting slopes.

J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81 75

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.319 0.714

No wall effect Transition Slug

0.493

Fr = 0.35

Fr =

0.7

1ζ

ζ

Fr

Fig. 3. Bubble rising velocity in water in the presence of wall effects.Experimental points from Hovmand and Davidson (1971).

A fit in the intermediate region (between pure bubbling andpure slugging) was obtained by a transformation of variables.The resulting piecewise fit is

Fr = 0.71� (��0.319), (33)

Fr = 0.2265 + 0.2805X − 0.1904X2 + 0.0334X3

(0.319 < � < 0.714), (34)

Fr = 0.35 (��0.714), (35)

where

X = � − 0.319

0.395. (36)

This piecewise fit is shown in Fig. 3.Outside the transition interval, the weighting assigned to

slugging is taken as 0 for ��0.319 and 1 for ��0.714.In the transition interval, the slugging weighting factor in-creases smoothly from 0 to 1. A reasonable estimate ofin the intermediate region, 0.319 < � < 0.714, was obtainedby considering the growing deviation of the correspondingvoid velocity (Eq. (34)) from the corresponding wall-effect-independent bubble rise velocity (Eq. (33)) in the lower partof the transition interval, and then the shrinking differencebetween the void velocity and the slug flow (Eq. (35)) limitin the upper section of the transition interval. The condition(� = 0.35/0.71 = 0.493) when the two limiting velocities areequal defines the point where the weighting factors for slug-ging and bubbling are equal, i.e., Pbubbling =Pslugging =0.5. Theweighting factors are then assigned (see Constantineau, 2004

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Psl

uggi

ng

0.493

ζ

Fig. 4. Probability of slugging as a function of � = √De/D.

for details) as

Pslugging = 0 (��0.319), (37)

Pslugging = 2.794X2 − 0.491X3 (0.319 < ��0.493), (38)

Pslugging = −0.813 + 4.116X − 2.794X2 + 0.491X3

(0.493 < � < 0.714), (39)

Pslugging = 1 (��0.714), (40)

The gas velocity is assumed to be low enough that we are onlyconsidering bubbling and slugging. Hence

Pbubbling = 1 − Pslugging. (41)

Fig. 4 presents the resulting weighting factor for sluggingas a function of �. This procedure allows a smooth transi-tion between pure bubbling (Pbubbling = 1) and pure slugging(Pslugging = 1) as � increases from 0.319 to 0.714. While thisapproach is conceptually similar to the probabilistic transitionsin the generalized fluidized bed reactor model (Abba et al.,2002, 2003), in the current case, the transition takes place overheight as � grows due to coalescence, whereas in the bub-bling/turbulent/fast fluidization transitions of the generalizedfluidized bed reactor model, the transitions are taken to be onlyfunctions of the superficial gas velocity.

3.5. Transition to slugging

According to the equations presented in the preceding sec-tion, slug flow is dominant for ��0.714, i.e for De �0.51D,while wall effects are negligible for ��0.319, i.e., for

76 J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81

De �0.1D. Combining a correlation for bubble diameter withthe definition of �, i.e., � = √

De/D, and Eqs. (37)–(41) al-lows one to predict lines of constant probability of slugging forfluidized beds operated in the bubbling and slug flow regimesas plotted in Fig. 1. Since Baeyens and Geldart (1974) usedthe Geldart (1972) bubble size correlation, the same correlationhas been employed in this figure to calculate the probability ofslugging. Note that the superficial gas velocity in Fig. 1 varieswith column diameter. The location of the line with a proba-bility of slugging of 1 is virtually equal to that calculated byBaeyens and Geldart (1974) for the transition from freely bub-bling to slugging with limited coalescence. An important fea-ture of this figure is that the transition range of the probabilities(i.e. 0 < Pslugging < 1) falls within Zone I, closer to bubbling forPslugging �0.5 and approaching slug flow for Pslugging > 0.5.

4. Reactor model predictions

With the weighting factors for the two flow regimes now de-fined, the generalized bubbling–slugging fluidized bed reactormodel proceeds in an analogous manner to the generalized bub-bling turbulent model (Thompson et al., 1999). The probabili-ties are used as weighting factors to adjust and interpolate suchfactors as the interphase mass transfer coefficients in the range(0.319 < � < 0.714) between pure bubbling and pure slugging.

4.1. Comparison with previous models

Previous models in the literature have assumed either purebubbling or pure slugging in any given zone. Fig. 5 presentsconversions calculated from the Hovmand et al. (1971) slug-ging bed model, the Orcutt et al. (1962) mixed-flow and plug-flow models, the Grace (1984) two-phase bubbling bed reactormodel and our generalized slugging–bubbling model (GSBM)for pure slugging and pure bubbling with constant average bub-ble size and variable bubble size for a first order chemical re-action, with the operating conditions as given in Table 2. Theconstant average bubble size case is considered because boththe Orcutt and Grace models use an average bubble size. Sinceprevious models do not consider changes in gas volumetric flowdue to variations in hydrostatic pressure and total molar flow,the comparisons here are for constant total volumetric and mo-lar flow.

Table 2Model parameters used to compare the generalized bubbling slugging modelto earlier slugging and bubbling models

Figs. 5, 6 and 7 Fig. 8

U 0.5 m/s 0.5 m/sUmf 0.02 m/s 0.02 m/sH 1.0 m CalculatedHmf Calculated 1.0 m�mf 0.45 0.45Gas diffusivity: Dg 1.0 × 10−4 m2/s 1.0 × 10−4 m2/sBubble correlation Mori and Wen (1975) Mori and Wen (1975)Initial bubble size: Deo 0.01 m 0.01 m

In general, the predictions for the bubbling fluidized bedmodels are higher than for the slugging models. Note that thefluidized bed models are applied without any allowance forslugging, even when the bubble size exceeds the reactor diame-ter. For this reason, the predictions for small column diametersmust be considered with caution.

When the weighting factor for slugging is 1, the results ofthe Hovmand et al. (1971) slugging model and the GSBMmodel are indistinguishable in Fig. 5. The GSBM thereforeessentially reduces to the Hovmand et al. slugging modelin the ��0.714, Pslugging = 1 limit. When considering purebubbling behavior (Pbubbling = 1), the GSBM is seen to cal-culate similar conversions to the bubbling bed models con-sidered, though with small differences due to variations inassumptions.

4.2. Influence of effective reaction rate constant

The influence of the effective reaction rate constant, kr , onthe gas conversion is presented in Fig. 6 for different reactordiameters with some solids in the low-density phase, �L =0.005 and neglecting any solids within the bubbles or low-density phase (i.e., �L=0). The figure presents the results of thecomplete generalized slugging–bubbling model, as well as thepredictions when the probabilities of bubbling and slugging areequal to one (entirely slugging or bubbling). When Pslugging =1, no bubbling region is considered prior to slugging. WhenPbubbling =1, the model either calculates a given bubble size foreach height or uses a constant average bubble size calculatedat 40% of the expanded bed height (z = 0.4H ). The modelparameters and conditions are again as given in Table 2.

For slugging beds (Figs. 6a and b), bubbling bed models tendto overpredict the conversions. For small kr , the generalizedslugging–bubbling fluidized bed model predicts conversionssimilar to those for pure slugging (where Pslugging=1) since theconversion is primarily reaction-controlled. For higher kr , thegeneralized slugging–bubbling fluidized bed model predictionsdiverge from the pure slugging predictions due to the increas-ing importance of interphase mass transfer from the bubblesand slugs to the dense phase. For such conditions, the bubblingregion at the base of the bed before the onset of slugging is ofcritical importance to the conversion. The presence of particleswithin the bubbles (�L) significantly augments the predictedconversions for large kr by compensating for limited interphasemass transfer.

For large vessels where slugging does not occur (Figs. 6eand f), the GSBM model predicts the same conversions as forpure bubbling (Pbubbling =1). The use of an average bubble sizeresults in reduced gas conversions for large reaction rate con-stants. For large kr , mass transfer from the bubbles to the densephase is rate-controlling. There is then a significant beneficialeffect on interphase mass transfer from the small bubbles nearthe distributor, which disappears when a single average bub-ble size is adopted. As for slugging beds, accounting for thepresence of particles within the bubbles (�L) is significant forlarge kr . For fluidized beds of intermediate sizes (Figs. 6cand d), the GSBM model predictions diverge slightly from the

J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81 77

10-1 1000

0.1

0.2

0.3

0.4

0.5

Hovmand

GSBM, PSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.1

0.2

0.3

0.4

0.5

Orcutt, Plug FlowOrcutt, MixedGSBM, PBubbling=1, Constant DeGrace 2-phaseGSBM, PBubbling=1, Variable De

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.2

0.4

0.6

0.8

1

HovmandGSBM, PSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.2

0.4

0.6

0.8

1

Orcutt, Plug FlowOrcutt, MixedGSBM, PBubbling=1, Constant DeGrace 2-phaseGSBM, PBubbling=1, Variable De

Bed Diameter (m)

Con

vers

ion

(-)

Slug flow models, kr = 10 s-1

Slug flow models, kr = 1 s-1

Bubbling bed models, kr = 10 s-1

Bubbling bed models, kr = 1 s-1

Fig. 5. Comparison of the GSBM model to the Hovmand et al. (1971) slugging model and to the Orcutt et al. (1962) and Grace (1984) bubbling bed models.For conditions, see Table 2.

predictions for pure bubbling due to the effect of the walls asbubbles grow with height, causing Pslugging to increase, andhence the interphase mass transfer to decrease.

If one chooses a regime-specific model (either pure bubblingof pure slugging) for cases where there are zones of Psluggingbetween 0 and 1, there can be substantial errors. For reactionswith large values of the effective reaction rate constant (kr ), themodel should not use an average bubble size to characterizethe entire reactor and should include the effect of solids withinthe bubbles. The generalized slugging–bubbling fluidized bedmodel allows the modelling of fluidized systems where a transi-tion from bubbling to slugging occurs within the bed, or wherethere are appreciable wall effects.

4.3. Effect of reactor diameter

The predicted effect of scale-up of a reactor for different re-actor diameters and different rate constants is shown graphi-cally in Fig. 7. The GSBM model is seen to approach the slug-ging limit for small columns and then to approach the bubblingbed limit with increasing column diameter, giving a smoothtransition between these limits in the intervening region. Forthe conditions considered, the GSBM model predictions de-

part from pure slugging for reactor diameters larger than about0.1 m. The departure from pure slugging at the higher reactionrate constant results from the bubbling region near the distrib-utor. In the calculations leading to Fig. 7, as shown in Table 2,the expanded bed depth, H, was fixed, consistent with the in-dustrial equipment related to the modelling, for which therewas a weir, limiting the bed expansion. In Fig. 8, on the otherhand, as in most comparisons in the literature, the bed inven-tory, and hence Hmf is fixed. In the latter case, we observe theusual overall trend of decreasing conversion with increasingcolumn diameter. In both cases the model smoothly interpolatesbetween the two limiting cases of pure slugging and pure bub-bling as D increases. The minimum in the conversion predictedin Fig. 8 for D ≈ 0.3 m is not expected, and it is not knownwhether there are any experimental results which would confirmsuch a trend.

In the predictions of Figs. 5–8, the bubble diameters havebeen based on the correlation of Mori and Wen (1975). Otherbubble diameter correlations would result in somewhat differentpredictions. For the conditions studied, the predictions of theGSBM model converge to pure bubbling at column diametersclose to 1 m. This is the minimum reactor diameter for thesystem to be modelled as pure bubbling. Note that results from

78 J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81

10-2 100 1020

0.2

0.4

0.6

0.8

1

PSlugging=1PBubbling=1, Constant DeGSBMPBubbling=1, Variable De

Con

vers

ion

(-)

10-2 100 1020

0.2

0.4

0.6

0.8

1

PSlugging=1PBubbling=1, Constant DeGSBMPBubbling=1, Variable De

Con

vers

ion

(-)

10-2 1020

0.2

0.4

0.6

0.8

1

PSlugging=1PBubbling=1, Constant DeGSBMPBubbling=1, Variable De

Con

vers

ion

(-)

10-2 100 1020

0.2

0.4

0.6

0.8

1

PSlugging=1PBubbling=1, Constant DeGSBMPBubbling=1, Variable De

Con

vers

ion

(-)

10-2 100 1020

0.2

0.4

0.6

0.8

1

PSlugging=1PBubbling=1, Constant DeGSBMPBubbling=1, Variable De

Con

vers

ion

(-)

10-2 100 1020

0.2

0.4

0.6

0.8

1

PSlugging=1PBubbling=1, Constant DeGSBMPBubbling=1, Variable De

Con

vers

ion

(-)

D = 0.1 m φL = 0

D = 0.1 m

φL = 0.005

D = 0.5 m φL = 0

D = 0.5 m

φL = 0.005

D = 1 m

φL = 0 D = 1 mφL = 0.005

kr, s-1

kr, s-1 kr, s

-1

kr, s-1

kr, s-1kr, s

-1

100

(a) (b)

(c) (d)

(e) (f)

Fig. 6. Comparison of conversions predicted by GSBM model and its limiting models as a function of reaction rate constant and column diameter. Foroperating conditions see Table 2.

small and large industrial fluidized beds must be compared withcare. In the transition region, the generalized model predictsconversions intermediate between those for pure bubbling andfor pure slugging.

5. Discussion

For reactions where interphase mass transfer is rate-limiting,the size of bubbles has a critical effect on the conversion. The

model developed here has the same structure as the genericfluidized bed reactor model formulated to span the bubbling,turbulent and fast fluidization regimes (Abba et al., 2002,2003). To construct a model useful for both diameter andvelocity scale-up, one must account for the transition fromslugging to turbulent fluidization. Since the superficial gasvelocity where the transition to the turbulent regime occurs(Uc) varies with bed diameter (Bi et al., 2000), a generalizedslugging–bubbling–turbulent regime probability model which

J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81 79

10-1 1000

0.1

0.2

0.3

0.4

0.5

0.6

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.1

0.2

0.3

0.4

0.5

0.6

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.2

0.4

0.6

0.8

1

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.2

0.4

0.6

0.8

1

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

kr = 1 s-1, φL = 0 kr = 1 s-1, φL = 0.005

kr = 10 s-1,φL = 0 kr = 10 s-1,φL = 0.005

Fig. 7. Comparison of the conversions calculated using GSBM model and its limiting cases as a function of bed diameter for different gas reaction rateconstants with constant expanded bed depth. For conditions, see Table 2.

10-1 1000

0.1

0.2

0.3

0.4

0.5

0.6

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.1

0.2

0.3

0.4

0.5

0.6

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.2

0.4

0.6

0.8

1

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

10-1 1000

0.2

0.4

0.6

0.8

1

PBubbling=1, Variable De

GSBMPSlugging=1

Bed Diameter (m)

Con

vers

ion

(-)

kr = 10 s-1, φL = 0 kr = 10 s-1, φL = 0.005

kr = 1 s-1, φL = 0.005 kr = 1 s-1, φL = 0

Fig. 8. Comparison of the conversions calculated using GSBM model and its limiting cases as a function of bed diameter for different gas reaction rateconstants with constant bed depth at minimum fluidization. For conditions, see Table 2.

80 J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81

accounts for superficial gas velocity, bed diameter and height,as well as gas and particle properties, must first be developedbefore adding the slugging fluidization flow regime into thegeneric fluidized bed reactor model. Once a consistent descrip-tion of the probabilities is available, adding the slugging regimeto the generic fluidized bed reactor model should be straight-forward.

6. Conclusion

A new model is proposed to predict the performance of flu-idized bed reactors operated wholly or partially in the slug flowregime. This model simulates the transition from bubbling toslugging within a given reactor as operating variables such assuperficial gas velocity, bed inventory and height change. Itinterpolates smoothly between the limits where the bed oper-ates entirely in the bubbling bed flow regime and the slug flowregime as height increases within a given reactor, or as the col-umn diameter is varied.

Notation

A bed cross-sectional area, m2

Ad distributor area per orifice, m2

aI interphase mass transfer exchange area, m−1

CHi molar concentration of species i in H-phase,mol/m3

Ci,in inlet molar concentration of species i, mol/m3

CLi molar concentration of species i in L-phase,mol/m3

CT total gaseous molar concentration(CT = PT /RT), mol/m3

D reactor diameter, mDe effective bubble diameter, mDe,0 initial effective bubble diameter, mDe,∞ maximum bubble diameter due to coalescence

and growth, mDg gas diffusivity, m2/sFHi molar gas flowrate of species i in H-phase,

mol/sFHi,in inlet gas molar flowrate of species i in H-phase,

mol/sFHT total gas molar flowrate in H-phase, mol/sFLi molar gas flowrate of species i in

L-phase, mol/sFLi,in inlet gas molar flowrate of species i in L-phase,

mol/sFLT total gas molar flowrate in L-phase, mol/sFr Froude number = U�∞/

√gD, dimensionless

FT total gas molar flowrate in reactor, mol/sfs slug shape factor, m3/m3

g acceleration due to gravity, m/s2

H expanded bed height, m

HL limiting bed height where coalescence is com-plete and stable slug spacing achieved, m

Hmf bed height at minimum fluidization, mI slug surface integral function,dimensionlesskLH interphase mass transfer coefficient, m/skr reaction rate constant, s−1

L slug length, mPbubbling weighting factor analogous to probability of

bubbling,dimensionlessPslugging weighting factor analogous to probability of

slugging,dimensionlessPT total reactor pressure, PaR universal gas constant, 8.314 J/(mol K)T slug-to-slug (tail-to-nose) distance, mU reactor superficial gas velocity, m/sUb bubble rise velocity, m/sUb∞ isolated bubble rise velocity, m/sUH H-phase superficial gas velocity, m/sUL L-phase superficial gas velocity, m/sUmf superficial gas velocity at minimum fluidization,

m/sUms superficial gas velocity at minimum slugging,

m/sUs slug velocity, m/sUs,∞ slug velocity of a single slug in a bed at

minimum fluidization, m/sU•

t terminal velocity of spherical particles ofdiameter 2.7dp, m/s

Uv void (bubble or slug) rise velocity, m/sUv∞ free void (bubble or slug) velocity, m/sz height coordinate, m

Greek letters

�v difference in gas total stoichiometric coefficientdue to reaction, dimensionless

�z finite height in bed (control volume), m�H H-phase gas volume fraction, m3/m3

�L L-phase gas volume fraction, m3/m3

�mf bed voidage at incipient fluidization, m3/m3

� square root of bubble-to-column diameterratio = √

De/D, dimensionless�I Stoichiometric coefficient of reactant i, dimen-

sionless�H H-phase solids volume fraction, m3/m3

�L L-phase solids volume fraction, m3/m3

�H H-phase volume fraction, m3/m3

�L L-phase volume fraction, m3/m3

Acknowledgments

The authors thank Teck Cominco Metals Ltd. and the Nat-ural Science and Engineering Research Council of Canada(NSERC) for financial contribution and the Science Councilof British Columbia for supporting J. Pierre Constantineauthrough a GREAT Scholarship.

J.P. Constantineau et al. / Chemical Engineering Science 62 (2007) 70–81 81

References

Abba, I.A., Grace, J.R., Bi, H.T., 2002. Variable-gas-density fluidized bedreactor model for catalytic processes. Chemical Engineering Science 57,4797–4807.

Abba, I.A., Grace, J.R., Bi, H.T., Thompson, M.L., 2003. Spanning theflow regimes: generic fluidized-bed reactor model. A.I.Ch.E. Journal 49,1838–1848.

Agarwal, P.K., 1987. Effect of bed diameter on bubble growth and incipientslugging in gas fluidised beds. Chemical Engineering Research and Design65, 345–354.

Allahwala, S.A., Potter, O.E., 1979. Rise velocity equation for isolatedbubbles and for isolated slugs in fluidized beds. Industrial and EngineeringChemistry Fundamentals 18, 112–116.

Aoyagi, M., Kunii, D., 1974. Importance of dispersed solids in bubblesfor exothermic reactions in fluidized beds. Chemical EngineeringCommunications 1, 191–197.

Baeyens, J., Geldart, D., 1974. An investigation into slugging fluidized beds.Chemical Engineering Science 29, 255–265.

Bi, H.T., Ellis, N., Abba, I.A., Grace, J.R., 2000. A state-of-the-art review ofgas–solid turbulent fluidization. Chemical Engineering Science 55, 1–37.

Chavarie, C., Grace, J.R., 1975. Performance analysis of a fluidized bedreactor. Industrial and Engineering Chemistry Fundamentals 14, 75–91.

Clift, R., Grace, J.R., 1985. Continuous bubbling and slugging. In: Davidson,J.F., Clift, R., Harrison, D. (Eds.), Fluidization, second ed. Academic Press,London, pp. 73–132.

Constantineau, J.P., 2004. Fluidized bed roasting of zinc sulfide concentrate:factors affecting the particle size distribution, Ph.D. Thesis, University ofBritish, Columbia.

Darton, R.C., LaNauze, R.D., Davidson, J.F., Harrison, D., 1977. Bubblegrowth due to coalescence in fluidized beds. Transactions of the Institutionof Chemical Engineers 55, 274–280.

de Vries, R.F., van Swaaij, W.P.M., Mantovani, C., Heijkoop, A., 1972.Design criteria and performance of the commercial reactor for the shellchlorine process. Proceedings of the Fifth European Second InternationalSymposium on Chemical Reaction Engineering. Elsevier, Amsterdam,pp. B9:59–69.

Eakman, J.M., Wesselhoft, R.D., Dunkleman, J.J., Vadovic, C.J., 1980. Gasifieroperation and modeling in the Exxon catalytic coal gasification process.Coal Processing Technology 6, 146–158.

Fryer, C., Potter, O.E., 1972. Countercurrent backmixing model for fluidizedbed catalytic reactors. Applicability of simplified solutions. Industrial andEngineering Chemistry Fundamentals 11, 338–344.

Geldart, D., 1967. The fluidized bed as a chemical reactor: a critical reviewof the first 25 years. Chemistry and Industry 1474–1481.

Geldart, D., 1972. The effect of particle size and size distribution on thebehavior of gas-fluidised beds. Powder Technology 6, 201–215.

Geldart, D., 1973. Types of gas fluidization. Powder Technology 7, 285–292.Glicksman, L.R., 1998. Fluidized bed scale-up. In: Yang, W.-C. (Ed.),

Fluidization, Solids Handling, and Processing: Industrial Applications.Noyes, Westwood, NJ, pp. 1–110.

Grace, J.R., 1984. Generalized models for isothermal fluidized bed reactors.In: Doraiswamy, L.K. (Ed.), Recent Advances in the Engineering Analysisof Chemically Reacting Systems. Wiley Eastern, New Delhi, pp. 237–255.

Grace, J.R., 1986. Fluid beds as chemical reactors. In: Geldart, D. (Ed.), GasFluidization Technology. Wiley, New York, pp. 285–339.

Grace, J.R., Abba, I.A., Bi, H.T., Thompson, M.L., 1999. Fluidized bedcatalytic reactor modeling across the flow regimes. Canadian Journal ofChemical Engineering 77, 305–311.

Horio, M., Nonaka, A., 1987. A generalized bubble diameter correlation forgas–solid fluidized beds. A.I.Ch.E. Journal 33, 1865–1872.

Hovmand, S., Davidson, J.F., 1968. Chemical conversion in a sluggingfluidised bed. Transactions of the Institution of Chemical Engineers 46,190–203.

Hovmand, S., Davidson, J.F., 1971. Pilot plant and laboratory scale fluidizedreactors at high gas velocities: the relevance of slug flow. In: Davidson,J.F., Harrison, D. (Eds.), Fluidization, first ed. Academic Press, Londonand New York, pp. 193–259.

Hovmand, S., Freedman, W., Davidson, J.F., 1971. Chemical reaction ina pilot-scale fluidized bed. Transactions of the Institution of ChemicalEngineers 49, 149–162.

Matsen, J.M., 1985. Fluidized beds. In: Bisio, A., Kabel, R.L. (Eds.), Scale-up of Chemical Processes. Wiley, New York, pp. 347–405.

Matsen, J.M., 1996. Scale-up of fluidized bed processes: principle andpractice. Powder Technology 88, 237–244.

Matsen, J.M., Tarmy, B.L., 1970. Scale-up of laboratory fluid-bed data: thesignificance of slug flow. A.I.Ch.E. Symposium Series 66 (101), 1–7.

Mori, S., Wen, C.Y., 1975. Estimation of bubble diameter in gaseous fluidizedbeds. A.I.Ch.E. Journal 21, 109–115.

Orcutt, J.C., Davidson, J.F., Pigford, R.L., 1962. Reaction time distributionsin fluidized catalytic reactors. Chemical Engineering Progress SymposiumSeries 58 (38), 1–15.

Raghuraman, J., Potter, O.E., 1978. Countercurrent backmixing model forslugging fluidized-bed reactors. A.I.Ch.E. Journal 24, 698–704.

Sadaka, S.S., Ghaly, A.E., Sabbah, M.A., 2002. Two phase biomass air–steamgasification model for fluidized bed reactors. Biomass and Bioenergy 22,439–487.

Sit, S.P., Grace, J.R., 1981. Effect of bubble interaction on interphase masstransfer in gas fluidized beds. Chemical Engineering Science 36, 327–335.

Stewart, P.S.B., Davidson, J.F., 1967. Slug flow in fluidised beds. PowderTechnology 1, 61–80.

Thompson, M.L., Bi, H.T., Grace, J.R., 1999. A generalized bubbling/turbulent fluidized-bed reactor model. Chemical Engineering Science 54,2175–2185.

Volk, W., Johnson, C.A., Stotler, H.H., 1962. Effect of reactor internals onquality of fluidization. Chemical Engineering Progress 58 (3), 44–47.

Wen, C.Y., 1984. Flow regimes and flow models for fluidized bed reactors.In: Doraiswamy, L.K. (Ed.), Recent Advances in the Engineering Analysisof Chemically Reacting Systems. Wiley Eastern, New Delhi, pp. 256–290.

Werther, J., 1980. Mathematical modeling of fluidized bed reactors.International Chemical Engineering 20, 529–541.

Yates, J.G., Grégoire, J.-Y., 1980a. An experimental test of slugging-bedreactor models. In: Grace, J.R., Matsen, J.M. (Eds.), Fluidization. PlenumPress, New York, pp. 581–588.

Yates, J.G., Grégroire, J.-Y., 1980b. Catalytic oxidation of o-xylene in aslugging fluidized bed. Chemical Engineering Science 35, 380–385.

Yue, P.L., Zarifis, N., 1990. Catalytic dehydration of ethanol in a fluidizedbed reactor. A.I.Ch.E. Symposium Series 86 (276), 119–125.