Explorations on the Multi-scale Flowstructure and Stability Condition In

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Chemical Engineering Science 62 (2007) 6978 – 6991 www.elsevier.com/locate/ces Explorations on the multi-scale flow structure and stability condition in bubble columns Ning Yang , Jianhua Chen, Hui Zhao, Wei Ge , Jinghai Li State Key Laboratory of Multi-phase Complex Systems, Institute of Process Engineering, ChineseAcademy of Sciences, P.O. Box 353, Beijing 100080, PR China Received 20 April 2007; received in revised form 19 July 2007; accepted 12 August 2007 Available online 22 August 2007 Abstract Physical understanding of heterogeneous flow structure is of crucial importance for modelling and simulation of gas–liquid systems. This article presents a review and report of recent progress in our group on exploratory application of the variational (analytical) multi-scale approach to gas–liquid systems. The work features the closure of a hydrodynamic model with the incorporation of a stability condition reflecting the compromise between the dominant mechanisms in the system. A dual-bubble-size (DBS) model is proposed to approximate the heterogeneous structure of gas–liquid systems based on a single-bubble-size (SBS) model previously established. Reasonable variation of the gas holdup and the composition of the two bubble species with operating conditions have been calculated and the regime transition can therefore be reasonably predicted for air-water system, suggesting that stability condition may provide an insightful concept to explain the general tendencies in gas–liquid systems out of their hydrodynamic complexity, and to give simple models of their overall behaviors. Of course, the diversity of the correlations for drag force and minimum bubble size and the sensitivity of the model predictions to these correlations may suggest the necessity to clarify further the essential and robust results in the current model and to reduce the uncertainties involved. 2007 Elsevier Ltd. All rights reserved. Keywords: Bubble column; Multi-scale; Stability condition; Regime transition; Flow structure; Hydrodynamics 1. Introduction Gas–liquid and gas–liquid–solid flows are widely encoun- tered in a variety of chemical and physical processes in en- gineering (Fan, 1989; Deckwer, 1992). Much work has been devoted to describing the flow behaviors and the bubble char- acteristics such as its shape, size distribution, rise velocity and wake properties in these systems. A general picture of flow regime transitions in these systems is also gradually revealed by many studies (Deckwer, 1992; Chen et al., 1994; Zahradnik and Fialova, 1996; Olmos et al., 2003; Thorat and Joshi, 2004). In bubble columns, homogeneous (bubbly flow), tran- sition and heterogeneous (churn-turbulent flow) regimes can be distinguished with the increase of gas flowrate. The mono- graphs and review articles by Fan (1989), Fan and Tsuchiya (1990), Deckwer (1992), Mudde (2005), Kantarci et al. (2005) Corresponding authors. Tel.: +86 10 62558318; fax: +86 10 62558065. E-mail addresses: [email protected] (N. Yang), [email protected] (W. Ge), [email protected] (J. Li). 0009-2509/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.08.034 and more recently by Yang et al. (2007) have served as com- prehensive accounts of the studies on the hydrodynamics of gas–liquid and gas–liquid–solid systems. These studies are crucial to the design, scaling-up and optimization of relevant processes, inspiring explorations to the physical mechanisms behind these findings. Such explorations may in turn clarify or generalize the applicability of the various correlations and criterions summarized from these findings. In this paper, we try to explain the multi-modal bubble size distribution and its variation with gas flowrate in gas–liquid systems with the so-called variational (analytical) multi-scale approach (Li and Kwauk, 2003; Li et al., 2005; Ge et al., 2007) which employs stability condition to close dynamical descriptions. As the approach has originally found application in gas–solid systems by establishing the energy-minimization multi-scale (EMMS) model (Li, 1987; Li and Kwauk, 1994; Li et al., 1999; Ge and Li, 2002), we will give a brief introduction to this model first. Previous efforts to establish a similar model in gas–liquid systems are then revisited, which have resolved the energy consumption relating to the rising of gas bubbles

Transcript of Explorations on the Multi-scale Flowstructure and Stability Condition In

Page 1: Explorations on the Multi-scale Flowstructure and Stability Condition In

Chemical Engineering Science 62 (2007) 6978–6991www.elsevier.com/locate/ces

Explorations on the multi-scale flow structure and stability condition inbubble columns

Ning Yang∗, Jianhua Chen, Hui Zhao, Wei Ge∗, Jinghai Li∗

State Key Laboratory of Multi-phase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100080, PR China

Received 20 April 2007; received in revised form 19 July 2007; accepted 12 August 2007Available online 22 August 2007

Abstract

Physical understanding of heterogeneous flow structure is of crucial importance for modelling and simulation of gas–liquid systems. Thisarticle presents a review and report of recent progress in our group on exploratory application of the variational (analytical) multi-scale approachto gas–liquid systems. The work features the closure of a hydrodynamic model with the incorporation of a stability condition reflecting thecompromise between the dominant mechanisms in the system. A dual-bubble-size (DBS) model is proposed to approximate the heterogeneousstructure of gas–liquid systems based on a single-bubble-size (SBS) model previously established. Reasonable variation of the gas holdup andthe composition of the two bubble species with operating conditions have been calculated and the regime transition can therefore be reasonablypredicted for air-water system, suggesting that stability condition may provide an insightful concept to explain the general tendencies ingas–liquid systems out of their hydrodynamic complexity, and to give simple models of their overall behaviors. Of course, the diversity ofthe correlations for drag force and minimum bubble size and the sensitivity of the model predictions to these correlations may suggest thenecessity to clarify further the essential and robust results in the current model and to reduce the uncertainties involved.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Bubble column; Multi-scale; Stability condition; Regime transition; Flow structure; Hydrodynamics

1. Introduction

Gas–liquid and gas–liquid–solid flows are widely encoun-tered in a variety of chemical and physical processes in en-gineering (Fan, 1989; Deckwer, 1992). Much work has beendevoted to describing the flow behaviors and the bubble char-acteristics such as its shape, size distribution, rise velocity andwake properties in these systems. A general picture of flowregime transitions in these systems is also gradually revealedby many studies (Deckwer, 1992; Chen et al., 1994; Zahradnikand Fialova, 1996; Olmos et al., 2003; Thorat and Joshi,2004). In bubble columns, homogeneous (bubbly flow), tran-sition and heterogeneous (churn-turbulent flow) regimes canbe distinguished with the increase of gas flowrate. The mono-graphs and review articles by Fan (1989), Fan and Tsuchiya(1990), Deckwer (1992), Mudde (2005), Kantarci et al. (2005)

∗ Corresponding authors. Tel.: +86 10 62558318; fax: +86 10 62558065.E-mail addresses: [email protected] (N. Yang),

[email protected] (W. Ge), [email protected] (J. Li).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.08.034

and more recently by Yang et al. (2007) have served as com-prehensive accounts of the studies on the hydrodynamics ofgas–liquid and gas–liquid–solid systems. These studies arecrucial to the design, scaling-up and optimization of relevantprocesses, inspiring explorations to the physical mechanismsbehind these findings. Such explorations may in turn clarifyor generalize the applicability of the various correlations andcriterions summarized from these findings.

In this paper, we try to explain the multi-modal bubble sizedistribution and its variation with gas flowrate in gas–liquidsystems with the so-called variational (analytical) multi-scaleapproach (Li and Kwauk, 2003; Li et al., 2005; Ge et al.,2007) which employs stability condition to close dynamicaldescriptions. As the approach has originally found applicationin gas–solid systems by establishing the energy-minimizationmulti-scale (EMMS) model (Li, 1987; Li and Kwauk, 1994; Liet al., 1999; Ge and Li, 2002), we will give a brief introductionto this model first. Previous efforts to establish a similar modelin gas–liquid systems are then revisited, which have resolvedthe energy consumption relating to the rising of gas bubbles

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into different portions and have proposed a stability conditionin terms of the relative dominance of these portions. Follow-ing this modelling strategy, a dual-bubble-size (DBS) model isestablished to approximately describe the heterogeneous struc-ture of gas–liquid systems instead of the previously proposedsingle-bubble-size (SBS, Zhao, 2006; Ge et al., 2007) model,and the physical implications of the results are analyzed. Lim-itations of this work and further work are also discussed.

2. The analytical multi-scale approach for gas–solidsystems

The analytical multi-scale approach to gas–solid systems wasstarted with the EMMS model. This model was aimed to givea simple but physically reasonable description of concurrent-up gas–solid flows on the macro-scale. The interactions insuch systems are resolved with respect to different scales. Theparticle-rich dense “phase” is distinguished from the fluid-richdilute “phase” by their respective voidages (�c and �f ) and flowvelocities (Udc, Udf for solids and Uc, Uf for gas). For fastfluidization, the dense “phase” occurs as discrete clusters amidthe continuous dilute “phase”, described by its volume fractionf and cluster size dcl. The flow structure within each “phase”is assumed to be homogeneous. The micro-scale interactionsof individual particles with the surrounding fluid are describedby the drag correlations. In the dilute “phase”, the drag forceon the particles is fully balanced by their gravity. In the dense“phase”, however, particle weight is only partially supported bythe gas flow in that “phase”; the rest is supported by the dragforce induced by the dilute “phase” gas flow around the parti-cle clusters. This is indeed the meso-scale interactions betweenthe characteristic “phase” structures.

Six hydrodynamic equations are found for the eight param-eters involved, Uc, Uf , Udc, Udf , �c, �f , f and dcl. They arethe momentum balance and continuity equations of the two“phases”, the pressure balance between the two “phases” andthe cluster size correlation. Apparently, these six equations arestill inadequate to construct a complete model for such hetero-geneous multi-phase systems. The remaining free variables aredetermined by the stability condition of these systems, whichreflects the compromise between the so-called dominant mech-anisms in these systems and correlates the descriptions on dif-ferent scales.

The dominant mechanisms in gas–solid systems areidentified as the tendency for the fluid to pass through the par-ticle layer with least resistance (Wst → min with Wst standingfor the volume-specific energy consumption for suspendingand transporting particles) and the tendency for the particlesto maintain least gravitational potential (� → min with �representing the local average voidage), and the stability con-dition (Nst = Wst/�p(1 − �) → min (with Nst denoting themass-specific energy consumption for suspending and trans-porting particles) reflecting the compromise between these twotendencies.

With the conservation equations and the stability condition,the model can be solved with some optimization algorithms.The typical flow regimes in circulating fluidized beds can be

well captured by this simple model. In particular, the charac-teristic state multiplicity, that is, the coexistence of a top di-lute zone with a bottom dense zone in the bed at the so-calledchoking point, is in accordance with the multiple minima ofNst found at that point. With the same expectation, this articletries to explore the possibility of understanding the multi-scalestructure in gas–liquid systems by analyzing the correspondingstability condition.

3. Modelling gas–liquid systems with SBS model

With encouraging development of the analytical multi-scaleapproach in gas–solid systems, it seems natural to explorewhether other multi-phase systems can be described in a sim-ilar manner. In fact, the similarities between gas–liquid andgas–solid systems have long been noticed (e.g., Ellenberger andKrishna, 1994). Previous works (Ge et al., 2007) have noticedthat, though gas–liquid systems lack meso-scale structures likeparticle clusters in gas–solid systems, the energy dissipationprocess during the rising, breakage and coalescence of bubblesis also characterized by a multi-scale nature. A model is thenestablished (Zhao, 2006; Ge et al., 2007) to determine a sin-gle mean diameter for the bubbles in turbulent gas–liquid flows(basically for bubble column reactors) with a stability condi-tion representing the minimization of direct energy dissipationthrough microscopic interactions. This model is briefly revis-ited in this section, from which an improved model consideringbubble size distribution is proposed in the next section.

3.1. Resolution of energy consumption

When inlet and outlet effects as well as wall effects are notconsidered, the energy consumption associated with the bubblesin bubble column reactors occurs on two scales. On the meso-scale, bubbles may break up under the bombarding of eddieswith characteristic sizes smaller than the target bubbles whilecontaining sufficient kinetic energy. A portion of the turbulentkinetic energy contained in the eddies is therefore convertedto surface energy and is later dissipated when the resultingdaughter bubbles merged into other bubbles. This dissipation isdenoted as Nbreak for unit mass of liquid. Even though a bubbledoes not break up or coalesce, the bubble surface may oscillatein response to the turbulence in the liquid, which producesadditional dissipation as compared with rigid solid particles ofthe same size. This surface dissipation with respect to unit massof liquid is denoted as Nsurf . The rest of the dissipation whichoccurs in the bulk of the liquid phase due to the rising bubblesis denoted as Nturb.

The total energy consumption per unit mass of liquid shouldequal the net mechanical energy fed into the system when gasinertia is negligible. Therefore, we have,

(Nsurf + Nturb) + Nbreak = Ugg. (1)

The first two portions compose the energy dissipation directlythrough microscopic interactions, whereas Nbreak represents theenergy consumption on meso-scales. The term “meso-scale”should be understood from the angle that bubbles typically

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break up at one location and coalesce at another location cover-ing a relatively “longer” temporal and spatial distance as com-pared with the viscous dissipation and surface dissipation inthe liquid and on bubble surface, i.e., Nturb and Nsurf .

Nsurf can be calculated from the difference of drag coefficientfor a bubble CDb and that of a rigid solid particle with the samediameter CDp (Zhao, 2006), that is,

Nsurf = Ugg · CDb − CDp

CDb

= Ugg · CD0,b(1 − fb)4 − CD0,p(1 − fb)

4

CD0,b(1 − fb)4

. (2)

On the other hand, Nturb can be linked with the bubble break-age rate through a kernel function. Considering that the energycontent of the colliding eddies to be greater than the corre-sponding increase of surface energy in bubble breakage and thatthe dynamic pressure of eddies to be greater than the capillarypressure of the smaller daughter bubbles, we have

Nbreak =∫ db

�min

∫ 0.5

0

�(db, �)

(1 − fb)�l + fb�g

× Pb(db, �, fBV ) · cf �d2b� · dfBV d�, (3)

where the arrival frequency of eddies �(db, �) and breakageprobability Pb(db, �, fBV ) can be obtained from the classicalstatistical theory of isotropic turbulence (Luo and Svendsen,1996; Wang et al., 2003; Kostoglou and Karabelas, 2005), asshown below.

�(db, �) = 0.923(1 − fb)nbN1/3turb

(� + db)2

�11/3, (4)

Pb(db, �, fBV ) = exp

[−max{cf �d2

b�, ���3/(3dbf1/3BV )}

�l · (�/6)�3 · (�Nturb)2/3

].

(5)

3.2. Conservation equations

For the steady state of the systems, the buoyancy force isapproximately balanced with the drag force. The conservationequation can be formulated without consideration of boundaryeffects as

fb�lg = fb

�/6 · d3b

· CDb

4d2b · 1

2�l

(Ug

fb

− Ul

1 − fb

)2

, (6)

where Ug and Ul denote the superficial gas and liquid veloci-ties, respectively, and fb is the gas volume fraction. Numerouscorrelations of drag coefficient CDb for a swarm of bubblescan be found in the literature, as reviewed by Fan and Tsuchiya(1990), Tomiyama (1998), Behzadi et al. (2004), Kulkarniand Joshi (2005) and Simonnet et al. (2007). The correlationproposed by Ishii and Zuber (1979) and Lo et al. (2000) isemployed in this model:

CDb = CD0,b(1 − fb)4, (7)

where the drag coefficient for an isolated bubble in quiescentliquid is given by

CD0,b = 4

3

gdb

U2T

�l − �g

�l

. (8)

According to Grace et al. (1976), the terminal velocity UT canbe obtained from

UT = �l

�ldb

M−0.149(J − 0.857), (9)

where

M = �4l g(�l − �g)

�2l �

3, (10)

J ={

0.94H 0.757 (2 < H �59.3),

3.42H 0.441 (H > 59.3),(11)

H = 4

3Eo · M−0.149

(�l

�ref

)−0.14

(12)

and

Eo = g(�l − �g)d2b

�. (13)

3.3. Stability condition

Based on the resolution of multi-scale interaction and energyconsumption, the stability condition is proposed as the mini-mization of the energy dissipation directly through microscopicinteractions,

Nsurf + Nturb → min . (14)

It also implies that the energy consumption induced by meso-scale structures and eventually dissipated on micro-scale needsto be maximized (Ge et al., 2007), i.e.,

Nbreak → max . (15)

From the viewpoint of gas–liquid interaction, the stability con-dition can be understood as the compromise between two dom-inant mechanisms, i.e., Nsurf reflecting the bubble oscillationdue to eddy bombardment, and Nturb involving the process ofenergy extraction from bubbles and of energy cascade fromlarger eddies to smaller eddies. According to Ge et al. (2007),lower Nsurf leads to smaller bubbles whereas lower Nturb cor-responds to larger bubbles. The joint effects of these two dom-inant mechanisms lead to an equilibrium bubble diameter (ordistribution in reality). The stability condition provides a clo-sure for the model, which includes two variables (fb, db) and aconservation equation, namely, Eq. (6). With the given super-ficial gas velocity, fb and db can be obtained by using someoptimization algorithms.

3.4. Results and discussion

Fig. 1 illustrates the variation of dimensionless energy dis-sipation with the trial value of bubble diameter. Nsurf + Nturb

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Fig. 1. Variation of dimensionless energy dissipation directly through micro-scopic interactions with trial value of bubble diameters, calculated from theSBS model of Zhao (2006). (Ug = 0.05 m/s, air–water system).

Fig. 2. Comparison between calculation from the SBS model of Zhao (2006)and experiments of Patel et al. (1989), Sada et al. (1987) and Camarasa et al.(1999): variation of bubble diameter with superficial gas velocity in air–watersystem.

reaches its minimum at a bubble diameter under given operat-ing conditions. Fig. 2 shows that the calculated average bubblediameter db decreases with the increase of superficial gas ve-locity Ug for air–water system, which is in reasonable agree-ment with experimental results for Ug < 0.1 m/s. The variationof total gas holdup calculation from the SBS model will be pre-sented in the next section in comparison with the prediction ofthe DBS model, the experiments of Camarasa et al. (1999) aswell as three other empirical correlations.

The variation of average diameters can be explained byallocating the total energy to different parts, as shown in Fig. 3.When Ug is low, Nsurf +Nturb almost equals to unity and Nbreakis much lower, suggesting that most energy is consumed to resistthe hindrance of the liquid and little is used to break the bubble,

Fig. 3. Variation of dimensionless energy consumption with superficial gasvelocity in air–water system, calculated from the SBS model of Zhao (2006).

which corresponds to the characteristics of the homogeneousregime of bubble columns. With increasing Ug , Nsurf + Nturbdecreases and Nbreak increases, indicating that more energy isconsumed in bubble breakage to decrease the average bubblediameter, corresponding to the characteristics of the heteroge-neous regime.

4. Modelling gas–liquid systems with DBS model

Although energy consumption is reasonably resolved in theSBS model, yet only the averaged bubble diameter and the over-all gas holdup can be calculated. The parameters reflecting theheterogeneous structure, however, are not taken into account.In this section we try to consider this structural heterogeneityby introducing two equivalent bubble diameters.

4.1. Structure resolution

Experiments have already revealed that the heterogeneity ofthe gas phase assumes a bimodal bubble size distribution inbubble column reactors (De Swart et al., 1996; Krishna andEllenberger, 1996; Camarasa et al., 1999; Ribeiro and Lage,2004). Accordingly, the aforementioned SBS model can be ex-tended to a DBS model involving small and large bubbles, asshown in Fig. 4. While interacting with the surrounding liquid,each bubble class can be characterized respectively by its equiv-alent bubble diameters (dS and dL), its corresponding volumefractions (fS and fL) and superficial gas velocities (Ug,S andUg,L). Correspondingly, the total energy consumption can bedecomposed into Nturb, Nbreak,S , Nbreak,L, Nsurf,S and Nsurf,L.We assume that the small and large bubbles share the sameliquid flow field and the bubbles breakup under the commoncircumstances of turbulent flow, and hence Nturb is not furtherresolved in this model. It should be pointed out that the struc-ture resolution in this preliminary study is limited to the gasphase, whereas liquid flow structure, which was described bymultiple circulation cells (Joshi and Sharma, 1979; Joshi et al.,

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Fig. 4. Schematic drawing for structure resolution of overall gas–liquid system into liquid, small bubbles, large bubbles and two interphases in the DBS model.

2002), is not considered for simplicity. Whether new dominantmechanisms will be introduced by the liquid phase is still anopen question.

4.2. Conservation equations

Fig. 4 illustrates that both small and large bubbles interactwith the surrounding liquid through the phase interfaces. Sup-pose that the slip velocities between the bubbles and the liquidare positive. Hence the force balance equations can be formu-lated for small bubbles and large bubbles, respectively, as

fS�lg = fS

�/6 · d3S

· CD,S

4d2S · 1

2�l

(Ug,S

fS

− Ul

1 − fb

)2

(16)

and

fL�lg = fL

�/6 · d3L

· CD,L

4d2L · 1

2�l

(Ug,L

fL

− Ul

1 − fb

)2

, (17)

where the drag coefficients for small and large bubbles can beobtained from Eq. (7) by modifying the drag coefficients forisolated single bubbles in quiescent liquid:

CD,S = CD0,S(1 − fb)4 (18)

and

CD,L = CD0,L(1 − fb)4. (19)

It should be noticed that the correction factor is also relatedto the bubble diameter and the bubble shapes of the small and

large bubbles. As a first approximation, we use the overall gasholdup fb to consider the effects of other bubbles on the dragcoefficient.

The mass conservation law leads to

Ug,S + Ug,L = Ug (20)

and

fS + fL = fb. (21)

Now we have a model involving six variables (dS , dL, fS , fL,Ug,S , Ug,L) and three conservation equations (Eqs. (16), (17)and (20)). It can be readily proved that for the case of uniformbubble size distribution (dS = dL = db), the above equationsof the DBS model reduce to Eq. (6) of the SBS model for anycombination of fS and fL. In what follows, we shall extendthe stability condition of the SBS model to provide a closurefor this DBS model.

4.3. Stability condition

Conforming to the resolution of energy consumption shownin Eq. (1), the total energy consumption in the DBS model canbe formulated as

Nsurf + Nturb + Nbreak

= (Nsurf,S + Nsurf,L) + Nturb + (Nbreak,S + Nbreak,L)

= Ug · g, (22)

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N. Yang et al. / Chemical Engineering Science 62 (2007) 6978–6991 6983

where

Nsurf,S = Ug,Sg · CD,bS − CD,pS

CD,bS

= Ug,Sg · CD0,bS(1 − fb)4 − CD0,pS(1 − fb)

4

CD0,bS(1 − fb)4

, (23)

Nsurf,L = Ug,Lg · CD,bL − CD,pL

CD,bL

= Ug,Lg · CD0,bL(1 − fb)4 − CD0,pL(1 − fb)

4

CD0,bL(1 − fb)4

, (24)

Nbreak,S =∫ dS

�min

∫ 0.5

0

�(dS, fS, �)

(1 − fb)�l + fb�g

· Pb(dS, �, fBV ) · cf �d2S� · dfBV d�, (25)

Nbreak,L =∫ dL

�min

∫ 0.5

0

�(dL, fL, �)

(1 − fb)�l + fb�g

· Pb(dL, �, fBV ) · cf �d2L� · dfBV d�. (26)

The arrival frequency of eddies � and breakage probability Pb

can be obtained from the classical statistical theory of isotropicturbulence, similar to Eqs. (4) and (5), for the small and largebubble classes, respectively. Then the stability condition canbe expressed as the minimization of energy dissipation directlythrough microscopic interactions, i.e.,

Nsurf + Nturb → min . (27)

The total energy consumption can be expressed as gUg in thesame manner for both the SBS and DBS models, as shown inEq. (1). For the case of uniform bubble size distribution in theDBS model, namely, dS =dL =db, summing Eqs. (23) and (24)yields

Nsurf,SBS = Nsurf,DBS = Nsurf,S + Nsurf,L, (28)

where Nsurf,SBS and Nsurf,DBS denote the energy consumptionin the process of bubble oscillation in the SBS and the DBSmodels, respectively. In a similar manner, from Eqs. (25) and(26), we obtain

Nbreak,SBS = Nbreak,DBS = Nbreak,S + Nbreak,L, (29)

where Nbreak,SBS and Nbreak,DBS are the energy consumptionin the process of bubble breakage in the SBS and the DBSmodels, respectively. Consequently, we have

Nturb,SBS = Nturb,DBS = Nturb. (30)

The above derivation indicates that the portions of energy con-sumption in the DBS model are equivalent to those in the SBSmodel for the case of uniform size distribution.

4.4. Solution procedure

Now that the DBS model is closed by the stability condi-tion, we are in a position to obtain the six structure variables,

provided the superficial gas velocity is given. To ensure a cor-rect and complete solution to this non-linear optimization prob-lem, we have employed an ergodic global search algorithm tofind the minima among all valid combinations of dS , dL andUg,S and, at the same time, to obtain the variation of all modelvariables with these parameters. Fairly complicated landscapesof the stability criterion were found, which displays multiplelocal and global minima and strong dependence on the dragcorrelations. This dependence suggests the necessity of furtherquantification of these correlations and constraints, and on theother hand, implies the limitations of our discussions on the so-lutions of the current model. Therefore, to void any confusion,we would clarify that the results presented hereafter are onlybased on the drag correlation shown in Eqs. (7)–(13) and theminimum bubble diameter (0.55 mm) constrained by the validrange of this correlation. The behavior of the gas–liquid flowspredicted with other correlations and constraints are subject tofurther research.

Fig. 5 illustrates the global minimum point which lies in theintersection points of three profiles at Ug =0.06 m/s. Note thatthe subscripts of the bubble diameters dS and dL have onlysymbolic significance and the identification of small and largebubbles is practically dependent on the relative magnitude of thecalculated value of dS and dL. Moreover, the three dimensionaldistribution of Nsurf + Nturb should be essentially symmetricalwith respect to the z-plane of Ug,S = 0.5Ug .

Fig. 6 shows the two-dimensional contour plot of the dimen-sionless energy dissipation directly through microscopic inter-actions for the z-plane of Ug,S =0.0222 m/s which contains theglobal minimum searched from the results for all the combina-tion of the three free variables (Ug,S , dS , dL) at Ug =0.06 m/s.It is indicated that there are two local minimum points in thecontour plot and the global minimum point lies in the valleylocated in the upper-left area. However, another valley appearsin this area when Ug increases from 0.07 to 0.09 m/s, as il-lustrated in Fig. 7. Note that only the upper-left area of theoverall domain is zoomed in the two insets, indicating that theglobal minimum point is in the left valley for Ug = 0.09 m/s.The situation is further changed when Ug increases from 0.12to 0.13 m/s. Fig. 8 illustrates that for Ug less than or equal to0.12 m/s, the local minimum value in the left valley is smallerthan that in the right, that is, the global minimum point lies inthe left valley, as shown in Inset (a); whereas for Ug equal toor greater than 0.13 m/s, the position of the global minimumpoint jumps from left to the right valley, as illustrated in In-set (b). The more accurate position of this jump change of theglobal minimum point can be further located by inserting moredata points of calculation for gas velocities between 0.12 and0.13. It is found that the jump change occurs approximately atUg = 0.128 m/s.

Our calculation indicates that the model solution is sensitiveto the selection of drag coefficient correlations and this influ-ence needs further investigation. Anyway, the jump changeshown above may reflect the multiplicity of the states ingas–liquid systems. This idea has already been demonstratedin gas–solid systems by using the EMMS model. The chokingpoint, which represents the regime transition between dilute

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6984 N. Yang et al. / Chemical Engineering Science 62 (2007) 6978–6991

Fig. 5. Dimensionless energy dissipation directly through microscopic interactions as a function of the trial values of diameters for two bubble classes (dS , dL)

and the superficial gas velocity of the small bubble (Ug,S), calculated from the DBS model as described in Section 4. The position of the global minimumlies in point a intersected by the three profiles (Ug = 0.06 m/s, air–water system).

Fig. 6. Two-dimensional contour plot of (Nsurf +Nturb)/(gUg) for the z-planeof Ug,S = 0.0222 m/s extracted from Fig. 5. The global minimum point liesin the upper-left valley (Ug = 0.06 m/s, air–water system).

transport and fast fluidization, is captured by the jump changebetween two stable solutions of the EMMS model. Consid-ering the analogy between gas–liquid and gas–solid systems,we expect that the jump change found in this study mayalso supply a physical explanation of the regime transition ingas–liquid systems, although it needs further verification underappropriate drag coefficient correlations and minimum bubblediameter constraints.

4.5. Results and discussion

Fig. 9 compares the prediction of total gas holdup withthe DBS and SBS models and the experiments of Camarasaet al. (1999) as well as other empirical correlations. The predic-tion from the DBS model is fairly consistent with experimentalresults when Ug is less than 0.07 m/s, and therefore more ac-curate than the prediction from the SBS model. Compared tothe SBS model, the improvement of model prediction is con-sidered to result from the introduction of structure resolution inthe DBS model. It can also be noticed that the transition pointat Ug =0.04 m/s where the slope of the curve begins to change,can be correctly captured by the model, presumably reflectingthe regime transition from homogeneous to transition regimes.

For Ug greater than 0.07 m/s, the total gas holdup first in-creases to a maximum approximately at 0.128 m/s and then fallsdown to a smaller value at 0.129 m/s. Although the decrease ofholdup is not quantitatively consistent with the shoulder area ofthe experimental results cited in Fig. 9, the jump change mayreflect the flow regime transition from the homogenous andtransition regimes to the fully developed heterogeneous regime.The jump change of the total gas holdup can be understood asa switch of the relative magnitude between the two local min-ima, as illustrated in Fig. 8. The global minimum point there-fore jumps from the left to the right so that the small bubblediameter changes to a larger value and thereby gives rise to thejump change of other structure parameters.

The model prediction of gas holdup for small and large bub-bles is compared with experimental measurements of Camarasa

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Fig. 7. Zoomed contour plot of (Nsurf + Nturb)/(gUg) for the upper-left area of the overall domain of dS and dL. Inset (a) indicates that only one valleyoccurs for Ug = 0.07 m/s. Inset (b) shows the existence of two valleys for Ug = 0.09 m/s. (a) Ug = 0.07 m/s, Ug,S = 0.0252 m/s and (b) Ug = 0.09 m/s,Ug,S = 0.0315 m/s.

Fig. 8. Jump change of the global minimum point between the two valleys: the global minimum (solid line) lies at the left valley in Inset (a) at Ug = 0.12 m/s;whereas it moves to the right valley in Inset (b) at Ug = 0.13 m/s. The dash line represents another local minimum. (a) Ug = 0.12 m/s, Ug,S = 0.0408 m/s;and (b) Ug = 0.13 m/s, Ug,S = 0.0377 m/s.

et al. (1999) for the case of multiple orifice nozzle in Fig. 10.Apparently, the model prediction is comparable to the experi-mental results. Moreover, the jump change of total gas holdupresults from the abrupt decrease of small bubble holdup.

The variation trend of the calculated results for rise veloc-ities of large bubbles (Ug,L/fL), as shown in Fig. 11, agreesreasonably with the experiments of Camarasa et al. (1999), in-dicating that the rise velocity of large bubbles increases withincreasing Ug . Krishna et al. (1999) proposed a correlation forthe rise velocity of large bubbles with the large bubble diame-ter, the column diameter, the superficial gas velocity and the gasvelocity at the regime transition point, the last of which can beobtained from experimental correlation of Reilly et al. (1994).For the small bubbles, experimental results in Fig. 11 indicatea slight decrease of rise velocity with increasing Ug , whereasthe rise velocity calculated from the DBS model (Ug,S/fS)

increases to higher value when the regime transition happens(Ug > 0.128 m/s), which seems to be related to the unrealisticdecrease of small bubble holdup illustrated in Fig. 10.

Fig. 12 shows that the calculated number density of largebubble increases with increasing Ug , whereas the number den-sity of small bubbles first increases gradually to form a ridge,and then falls down dramatically at the end of the ridge, whichcorresponds to the jump change in Figs. 9 and 10 where theholdup of small bubble decreases dramatically at this point.The small bubble diameter shifts to a larger value, as illustratedin Inset (b) of Fig. 8. Therefore, the joint effect of holdup anddiameter for the small bubble leads to the jump change of itsnumber density.

To show the influence of superficial gas velocity on bubblesize distribution, the calculated relative frequency (the num-ber density of each bubble classes divided by the total number

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Fig. 9. Comparison between calculation from the DBS and the SBS mod-els and correlations of Bach and Pilhofer (1978); Hikita et al. (1980) andHughmark (1967) as well as experiments of Camarasa et al. (1999): variationof total gas holdup with superficial gas velocity for air–water system.

Fig. 10. Comparison between calculation from the DBS model and experi-ments of Camarasa et al. (1999): variation of the holdup of small and largebubbles with superficial gas velocity.

density of small and large bubbles) is illustrated in Fig. 13. It isindicated that the small bubble diameter ranges from 1 to 3 mmand the large bubble diameter is in the range of 7–21 mm, whichis in reasonable agreement with literature reports (Clift et al.,1978; De Swart et al., 1996; Krishna et al., 2000). The numberdensity of large bubbles is generally far less than that of smallbubbles due to its large volume, and the bimodal size distri-bution is more distinct with increasing Ug . Similar trend canalso be observed in Fig. 14 which illustrates the gas holdup ofcorresponding bubble diameters at different gas velocities. Thejump change illustrated by the arrows in Figs. 13–14 shows thatwhen Ug increases from 0.12 to 0.13 m/s, both the small bubblediameter and the large bubble relative frequency increases dra-matically and the small bubble holdup decreases, which maysuggest that gas tends to flow out of the columns in the formof large bubbles at higher gas velocity.

Fig. 11. Comparison between calculation from the DBS model and experi-ments of Camarasa et al. (1999): variation of rise velocity of small and largebubbles with superficial gas velocity.

Fig. 12. Variation of the number density of small and large bubbles calculatedfrom the DBS model.

It should be pointed out that the variation of the gas holdupand the bubble size distribution is also influenced by otherimportant factors, like sparger types and column sizes, whichare, however, not taken into account in our current model. Asa preliminary study, we do not expect close coincidence ofcalculation with experimental reports; rather, the DBS modelonly serves as a conceptual model to study the general trend ofvariation of structure parameters in gas–liquid systems throughthe analysis of the compromise between dominant mechanisms,which is expressed as the stability condition in the model.

Fig. 15 compares the variation of the dimensionless energydissipation directly through microscopic interactions calculatedfrom the SBS and DBS models, respectively. Nsurf + Nturbcalculated from the DBS model is much lower than that fromthe SBS model. This can be explained by the resolution ofstructure of gas phase in the DBS model, which reflects the

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Fig. 13. Bubble size distribution at different superficial gas velocities. Thetwo arrows show the jump change of small bubble diameter and of largebubble number density at from Ug = 0.12 m/s to Ug = 0.13 m/s.

Fig. 14. Gas holdup of corresponding bubble diameters at different gasvelocities. The two arrows show the jump change of small bubble diameterand of small bubble holdup at Ug = 0.12 m/s to Ug = 0.13 m/s.

physical situation better than the SBS model. Since the totalenergy is equal for the two models, this difference shown inFig. 15 corresponds to that of Nbreak shown in Fig. 16. Thedifference suggests that the system consumes more energy forbubble breakup and coalescence in the case of non-uniformbubble size distribution reflected in the DBS model. For both themodels, Nbreak increases with increasing Ug . Fig. 17 comparesthe variation of Nbreak for small and large bubbles. Apparently,almost all Nbreak is consumed on the large bubble breakage andthe small bubble consumes little.

4.6. Exploring the compromise between dominant mechanisms

The stability conditions shown in Eqs. (14) and (27) try toreflect the joint effects of two dominant mechanisms, that is,

Fig. 15. Comparison of the dimensionless energy dissipation directly throughmicroscopic interactions calculated from the SBS and DBS models.

Fig. 16. Comparison of the dimensionless energy consumption for bubblebreakage calculated from the SBS and DBS models.

the compromise between Nsurf → min and Nturb → min.According to Ge et al. (2007), smaller bubbles dominated bysurface tension tend to achieve lower Nsurf , whereas largerbubbles interacting with turbulent flow tend to achieve lowerNturb. The joint effects of these two dominant mechanisms pro-duce an equilibrium bubble diameter. This has been verifiedby performing CFD simulation incorporating with a popula-tion balance model (PBM) (Zhao, 2006; Ge et al., 2007) inwhich the coalescence model of Prince and Blanch (1990) andthe breakup model of Luo and Svendsen (1996) are employed.The number density for each bubble size classes can be ob-tained by solving the population balance equations and thenthe average bubble diameter at local points can be calculated.It can be seen from Figs. 18(b) and (c) that larger bubbles ex-ist at some local points (e.g., point B) whereas smaller bubblesdistribute at other points (e.g., point A), implying the differ-ence of dominant mechanisms between these different points,

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Fig. 17. Comparison of the dimensionless energy consumption in bubblebreakage for the small and large bubbles.

Fig. 18. Relationship between bubble size distribution and energy consumption in air–water system (Ug=0.048 m/s, Ul =0 m/s, �l =998 kg/m3, �g=1.2 kg/m3,� = 0.072 N/m, �l = 0.001 Pa s, from Zhao, 2006; Ge et al., 2007).

i.e., Nturb → min for the former whereas Nsurf → min for thelatter. The statistics for the global system shown in Fig. 18(d) in-dicates the bi-modal bubble size distribution, supplying furtherevidence of the alternative dominance of the two mechanisms.The dependence of the energy dissipation directly through mi-croscopic interactions Nsurf + Nturb on the average bubble di-ameters for points A and B and for the global system can beseen in Fig. 18(e). While points A and B lie on the left andright slopes of the curve, respectively, the global mean bubblediameter locates almost in the valley, which corresponds to theminimum of Nsurf +Nturb and in this way the proposed stabilitycondition seems reasonable.

5. Further extension

We find that some model prediction, like the position andthe number of minimum points, is influenced by the drag coef-ficient correlation. In literature, there is no general agreement

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on the form of the drag coefficient correlation. Therefore, theinfluence of different drag coefficient correlations needs furtherinvestigation. On the other hand, in gas–liquid systems, inertiaand inertia-induced forces like the virtual-mass force and theshear-induced lift may affect the flow structure significantly(Magnaudet and Eames, 2000). The force balance equation usedin the study, i.e, Eqs. (16) and (17), considers only the effectsof buoyancy and drag force for simplicity. The effects of inertiashould be taken into account in further study.

Bubbles in real systems have a continuous size distribution,and therefore the DBS model can be further extended to accom-modate multiple bubble sizes. Suppose that we have m classesof bubbles, each with a volume fraction of fi , a diameter of di

and a corresponding superficial gas velocity of Ui . For the ithclass, the conservation equations can be written as

fi�lg = fi

�6 · d3i

· CDi

4d2i · 1

2�l

(Ug,i

fi

− Ul

1 − fb

)2

, (31)

where∑Ug,i = Ug (32)

and∑fi = fb. (33)

By the same token, the stability condition can be expressed asthe minimization of energy dissipation directly through micro-scopic interactions for all classes of bubbles, namely,∑

i

(Nsurf + Nturb) → min . (34)

The model involves 3m variables and m + 1 equations, andthe stability condition may serve as a closure for the model tocalculate the number density of each bubble class.

The stability condition may also change with the scale of thesystem when a certain level is reached. A good example canbe found in the nano-gas–liquid flow enclosed in a segment ofstraight pipe with periodic boundaries in the axial direction anddriven by a constant bulk force field, as investigated by Ge et al.(2007). It was found that extreme tendencies still take effectand the dominant mechanisms have been identified as the min-imization of viscous dissipation and the interfacial potential. Itwould be interesting to see how a more generalized stabilitycondition can be proposed for different gas–liquid systems.

6. Conclusions and prospects

Gas–solid and gas–liquid systems are characterized byheterogeneous structures and multi-scale behaviors. The com-plexity of these systems can be understood from the angle ofstability condition and the compromise of different dominantmechanisms in the systems. This study shows that the strategyof establishing and utilizing a stability condition can possi-bly be applied to gas–liquid systems also. The prediction ofstructural parameters from the single-bubble-size (SBS) andthe dual-bubble-size (DBS) models for gas–liquid systems

based on this strategy has shown reasonable agreement withexperimental measurements and correlations. The variation ofthe structural parameters and the different portions of energyconsumption can qualitatively reflect the evolution tendenciesof the structures in these systems, and the two gas velocitiesat which regime transition occurs in air-water system can bereasonably predicted with the DBS model. Moreover, similarstability condition has been found for nano-gas–liquid flows,suggesting that the strategy in seeking stability condition, aspresented in this study, may be of general relevance and henceof significance to the fundamentals of multi-phase flow.

We anticipate further extension of the gas–liquid modelsbased on stability conditions. First, as the models based on thisstrategy embody information of the multi-scale interaction ofeddies and bubbles, and can reflect the compromise betweendominant mechanisms, the models could be further developedto provide closures for the interphase momentum transfer forCFD models to predict more detailed flow structure. Second,the model established in this study is still at its preliminarystage. Some specific aspects of the theoretical predictions inthe current model are still dependent on the phenomenologicalinputs to the model, such as the drag coefficient, minimum bub-ble diameters and bubble oscillation characters. These inputsneed further quantification in both experimental measurementsand micro-scale simulations, and correspondingly, the model issubject to further improvement, extension or even integrationof additional dominant mechanisms. At this preliminary stage,we expect no more than these conceptual results due to thelimitation of knowledge on the complexity of gas–liquid multi-scale interactions. However, with the future works discussed,it would be promising to develop a relatively simple modelfor describing the multi-scale behavior in practical gas–liquidsystems with a reasonable accuracy.

Notation

cf coefficient of surface area increase, cf = f2/3BV +

(1 − fBV )2/3 − 1, dimensionlessCDb drag coefficient for a bubble in a swarm, dimen-

sionlessCD0,b drag coefficient for a bubble in a quiescent liquid,

dimensionlessCDp drag coefficient for a particle in multi-particle sys-

tems, dimensionlessCD0,p drag coefficient for a particle in a quiescent fluid,

dimensionlessdb bubble diameter, mdL bubble diameter of large bubbles, mdmin minimum bubble diameter, mmdS bubble diameter of small bubbles, mDT column diameter, mEo Eotvos number, dimensionlessfb volume fraction of gas phase, dimensionlessfL volume fraction of large bubbles, dimensionless

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fS volume fraction of small bubbles, dimensionlessfBV breakup ratio of daughter bubble to its mother bub-

ble, dimensionlessg gravitational acceleration, m/s2

Mo Morton number, dimensionlessnb number density of bubbles, 1/m3

Nbreak rate of energy consumption due to bubble breakageper unit mass, m2/s3

Nsurf rate of energy dissipation due to bubble oscillationper unit mass, m2/s3

Nturb rate of energy dissipation in turbulent liquid phaseper unit mass, m2/s3

Nst rate of energy dissipation for suspending and trans-porting particles per unit mass, m2/s3

Pb bubble breakup probability, dimensionlessUg superficial gas velocity, m/sUg,L superficial gas velocity for large bubbles, m/sUg,S superficial gas velocity for small bubbles, m/sUl superficial liquid velocity, m/sWst rate of energy dissipation for suspending and trans-

porting particles per unit volume, m2/s3

Greek letters

� voidage, dimensionless� character size of eddy, m� viscosity, Pa s� density, kg/m3

� surface tension, N/m� collision frequency, 1/s

Subscripts

DBS double-bubble-sizeg gasl liquidL large bubblep particleS small bubbleSBS single-bubble-size

Acknowledgments

The authors would like to express their appreciation to Prof.Mooson Kwauk for his encouragement and valuable sugges-tions to this work. Thanks are also extended to Dr. Yushan Zhuof Tsinghua University for his help on trying the method ofsimulated annealing, to Dr. Wei Wang for some literature sur-vey, to Mr. Limin Wang and Mr. Guangzheng Zhou for theirhelp on data visualization and to Ms. Bona Lu and Mr. FeiguoChen for analyzing the solutions of a non-linear model dur-ing the exploration. The long term supports from the NationalNatural Science Foundation of China, in particular, Grant nos.20406022, 20221603, 20490201 and 20336040, as well as thesupport from the hi-tech research and development programof China under Grant 2006AA030202, are gratefully acknowl-edged.

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