Effect of bubble deformation on stability and mixing in bubble columns

Chemical Engineering Science 57 (2002) 3283 – 3297

www.elsevier.com/locate/ces

Eect of bubble deformation on stability and mixing in bubble columns Elizebeth Le(on-Becerril, Arnaud Cockx, Alain Lin(e ∗ Departement de Genie des Procedes Industriels, Laboratoire d’Ingenierie des Procedes de l’Environnement, Institut National des Sciences Appliquees de Toulouse, 135 Avenue de Rangueil, 31077 Toulouse Cedex, France Received 22 November 2001; accepted 10 December 2001

Abstract The paper deals with hydrodynamics in bubble columns. The objective of the paper is to study stability and mixing in a bubble column. The modeling of parameters such as stationary drag and added mass is addressed. In addition, the eect of bubble deformation in terms of eccentricity is highlighted. In a previous paper, the transition between homogeneous and heterogeneous regimes in bubble column without liquid 8ow has been shown to be driven by the deformation of the bubbles associated to drag and added mass. In the present paper, this work is generalized to bubble column with liquid 8ow and to the transition from bubble 8ow to slug 8ow in a vertical pipe. Numerical simulations of gas–liquid reactors are presented. The numerical simulations are validated in the case of gas plume after the Becker et al. data (Becker, S., Sokolichin, A., & Eigenberg, G. (1994) Gas–liquid 8ow in bubble columns and loop reactors: Part II. Comparison of detailed experiments and 8ow simulations. Chemical Engineering Science, 49 (24B), 5747–5762. The numerical simulations are ?nally applied to a bubble column. The simulations of residence time distribution coupled to transient hydrodynamics are shown to be very sensitive to the modeling of interfacial transfer of momentum from the bubbles to the liquid in terms of drag and added mass, including the eect of bubble deformation. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bubble column; Stability analysis; Added mass; Eccentricity; Two-8uid model

0. Introduction Bubble columns are mainly used in biological, chemical and petrochemical industries (Shah, Kelkar, Godbole, & Deckwer, 1982; SchlAuter, Stei, & Weinspach, 1992; Dudukovic, Larachi, & Mills, 1999). The simple design and construction of bubble columns, its ability to high mass transfer and high mixing make this reactor attractive. The high mass transfer is due to large interfacial area. The high mixing can be determined by the residence time of the liquid phase and permits control of the reaction kinetics. However, as pointed out by Delnoij, Kuipers, and van Swaaij (1999), our knowledge of the transient hydrodynamics observed in bubble columns is limited. The present paper associates a 8uid mechanics approach to a chemical engineering one. In the ?rst part of the introduction, homogeneous and heterogeneous bubbly 8ows are de?ned, respectively, in terms of steady-state and transient global hydrodynamics. Dierent ways to account for the ∗ Corresponding author. Tel.: +33-05-61-55-97-86; fax: +33-05-61-55-97-60. E-mail address: [email protected] (A. Lin(e).

transient hydrodynamics associated to large structures are brie8y reviewed. In the last part, expressions of drag and added mass coeGcients are recalled. Indeed, mixing eGciency in bubble columns is closely related to the 8ow pattern. Many experimental studies have been devoted to this problem with the objective to estimate mixing in gas–liquid reactors (see for example H(ebrard, 1995, Zahradn(Ik & Fiavol(a, 1996; Vial, 2000; Lefebvre & Guy, 1999). The two basic 8ow patterns observed in a bubble column are the homogeneous and the heterogeneous bubbly 8ows. The transition between these 8ow patterns was described by dierent authors (Chen, Reese, & Fan, 1994). The homogeneous bubbly 8ow is often characterized by uniform bubble size distribution and uniform radial pro?le of gas fraction. Indeed, such pro?les correspond to a steady-state global hydrodynamics. We consider that such a steady-state global hydrodynamics is basically the main characteristic of the homogeneous bubbly 8ow pattern. It occurs at low super?cial gas velocities with uniform injection of gas at the bottom of the column. The heterogeneous bubbly 8ow occurs at higher super?cial gas velocities. We consider that the main characteristic

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 9 9 - 9

3284

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

Table 1 Drag coeGcients for bubbles in liquid in the frame of potential 8ow

Isolated spherical bubble 48 ReD

Drag coeGcient

CD =

Reynolds number

ReD =

C dD Vrel∞

C

of the heterogeneous bubbly 8ow pattern is basically the transient global hydrodynamics: large structures are created and transported in the column, increasing eGciently the mixing in the column. This approach is coherent with previous works based on the experimental analysis of the homogeneous and heterogeneous 8ow in terms of spectral analysis of 8uctuations (DrahoNs, Zahradn(Ik, Puncoch(ar, Fiavol(a, & Bradka, 1991; Soria & De Lasa, 1992; Ranade & Utikar, 1999, Vial, 2000). Related to the transient hydrodynamics of heterogeneous bubbly 8ow, both bubble size and radial pro?les of phase fractions and velocities are non-uniform. In this case, radial pro?les of temporal average variables such as phase fractions or velocities do not give any explicit information on spatial or temporal scales of the large structures that control the mixing in the column. In the past, heterogeneous bubbly 8ow was often simpli?ed and the column was considered to be composed of a longitudinal succession of boxes, each box corresponding to a stationary circulation cell, its size being controlled by the column diameter. Computational 8uid dynamics enables simulation of transient hydrodynamics and such arbitrary simpli?cations of the bubble column seem no longer necessary. The de?nition of these two basic 8ow pattern induces the question of the transition between 8ow patterns. This transition was mainly determined from experiments: the global gas fraction in the column can be plotted versus the super?cial gas velocity classically; the plot shows dierent slopes in homogeneous and heterogeneous bubbly 8ow patterns. The super?cial gas velocity corresponding to the transition between homogeneous and heterogeneous bubbly 8ow patterns can then be estimated. We will refer to such results to validate the results of our stability analysis. The stability of bubble columns was also investigated in terms of coalescence of bubbles (McQuillan & Whalley, 1985). Indeed, we consider that coalescence can only occur when bubbles are suGciently close to coalesce. The bubble– bubble interaction will then be analysed in this paper in terms of creation of a swarm of bubbles, without bubble coalescence. The present paper is two fold: • The transition between homogeneous and heterogeneous 8ows in a bubble column will be studied analytically as well as the transition between bubble 8ow and slug 8ow in a pipe. These transitions will be shown to be controlled by the instability of the homogeneous bubbly 8ow. The eect of bubble deformation on the instability of the ho-

Non-isolated spherical bubble CD =

5=3

48 (1+G ) ReBG (1−G )

ReBG =

L dB UR (1−G )

L

Non-isolated ellipsoidal bubble CD =

48 G( ) ReB

ReD =

C dD Vrel∞

C

mogeneous bubbly 8ow is signi?cant. In fact, this analysis is based on the perturbation of an homogeneous bubbly 8ow with uniform pro?les of gas fraction. It is then restricted to the case of gas distributor such as membranes, injecting the gas uniformly over the whole section of the column. The analytical results will be validated after experiments carried out with this kind of sparger. • The computational 8uid dynamics (CFD) will be used to simulate the transient hydrodynamics induced by the large structures in gas–liquid reactors. CFD will be used in Eulerian framework. A gas plume will be studied ?rst. Both period and amplitude of the hydrodynamics induced by transient large structures associated to the oscillating gas plume will be simulated and validated after experiments. A bubble column will then be simulated. The mixing in the bubble column will be analysed. The eect of bubble deformation on mixing in a bubble column will be emphasized. This last point constitutes one of the original aspect of this paper, since in most of previous works (Delnoij et al., 1999, for example) bubbles were assumed to be spherical. The ?rst part of the study is based on the transition between homogeneous and the heterogeneous bubbly 8ows; the main problem of the stability analysis is related to the expressions of drag and added mass coeGcients. The expressions of these coeGcients are recalled. The drag and added mass coeGcients have been studied in the past. In the frame of potential 8ow theory, it is interesting to recall the expressions of both drag and added mass coeGcients in three basic cases: isolated spherical bubble, non-isolated spherical bubble and non-isolated ellipsoidal bubble (Table 1). The expression of drag coeGcient of an isolated spherical bubble in potential 8ow was given by Moore (1965) with the Reynolds number de?ned for a single bubble. For non-isolated spherical bubble, following Milne-Thomson (1968), we have derived the expression of drag coeGcient in potential 8ow (Le(on-Becerril, 2001). In this case, the Reynolds number is expressed for non-isolated bubbles. In the case of non-spherical bubbles, an ellipsoidal shape is assumed. For an oblate ellipsoid with oblatness or axis ratio (a long axis, b small axis): = a=b

(1)

then the equivalent bubble diameter of the ellipsoidal bubble is given by √ 3 dB = 2 a2 b (2a)

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

or dB =

Milne-Thomson (1968) as 2a : 1=3

(2b)

Milne-Thomson derived the expression of the drag coeGcient of the ellipsoidal bubble in potential 8ow (Table 1) with G( ) =

1 4=3 ( 2 − 1)3=2 [( 2 − 1)1=2 − (2 − 2 )sec( )] : 3 [ 2 sec( ) − ( 2 − 1)1=2 ]2 (3)

For Reynolds numbers ReB ¡ 130, following Karamanev and Nikolov (1992) and Karamanev (1994) the drag coef?cient can be expressed by the relation proposed by Turton and Levenspiel (1986) for isolated and spherical spheres as 24 CD = (0:173 ReB0:687 + 1) ReB +

0:413 1 + 16:3 ReB−1:09

if ReB ¡ 130:

(4)

For higher Reynolds numbers, ReB ¿ 130, Karamanev proposes the expression for an isolated bubble (including implicitly the deformation): CD = 0:95

if ReB ¿ 130:

(5)

In addition, the eect of a swarm of bubbles and bubble deformation can be accounted for in terms of terminal velocity. Most of the expressions including the eect of gas fraction have been written in terms of relative velocity rather than in terms of drag as UR (; ) = U∞ [1 − p2 ( )];

(6)

where U∞ is the terminal velocity for an isolated and spherical bubble with

3285

p2 =

1:43(2 + Z( )) : 3

(7)

Then for spherical bubbles, p2 = 1:43, and for an eccentricity = 1:8 it becomes p2 = 1:73. This dependency can be included in the drag coeGcient. CDeq = CD 2=3 f(; )

(8)

with f(; ) = [1 − p2 ( )]−2 :

(9)

The added mass coeGcient can also be modelled in the frame of potential theory. The expression of added mass coeGcient of an isolated spherical bubble in potential 8ow was given by Moore (1965). The expression of the added mass coeGcient of a spherical bubble as a function of the void fraction was given by Zuber (1964). Indeed, in practical applications, the bubbles are not spherical. It is then important to account for bubble eccentricity in the added mass coeGcient expression (van Wijngaarden, 1991) where Z( ) is a factor given by

Z( ) = 2

( 2 − 1)1=2 − cos−1 −1 : cos−1 −1 − ( 2 − 1)1=2 = 2

(10)

These expressions will be used in the following analysis of stability and numerical simulations, in order to highlight the sensitivity of the results of stability of bubbly 8ows and of numerical simulations of bubble plumes and bubble columns to the deformation of the bubbles. In the ?rst part of this paper, the eect of deformation of bubbles on stability of homogeneous bubbly 8ow is studied. The linear stability analysis is recalled. It is applied to determine the transition between homogeneous and heterogeneous 8ow in a bubble column with co-current gas and liquid 8ows. The stability analysis is then applied to determine the transition between bubble 8ow and slug 8ow in a vertical pipe. In the second part of the paper, the eect of deformation of bubbles on transient hydrodynamics of a gas plume and on mixing in a bubble column is studied. The numerical simulations are performed with ASTRID code. The physics of the code is brie8y recalled.The hydrodynamics of a gas plume is shown to be sensitive to added mass. The residence time distribution curves in a bubble column are shown to be very sensitive to large-scale structures induced by added mass eect including the deformation of bubbles. The second part of the study is based on the numerical simulation of hydrodynamics and mixing in a bubble column; the basic question is related to the best way to simulate the hydrodynamics. Indeed, two ways can be followed: • On one hand, one can simulate the transient transport of concentration coupled to a transient hydrodynamics, the large-scale structures being simulated and controlled by the dispersed phase 8ow. • On the other hand, one can simulate the transient transport of concentration coupled to a steady-state hydrodynamics, the mixing eect of large-scale structures being accounted for by a modi?ed “turbulent” diusivity. The ?rst way has been followed in this paper. Transient simulations of both hydrodynamics and mixing have been performed. Concerning the second way, to account for the eect of the dispersed phase on mixing, one can refer to the works of Sato, Sadatomi, and Sekogushi (1981), Biesheuvel and van Wijngaarden (1984) and Chahed (1999). In the following, the bubble dispersion will be simpli?ed. It will be assumed to be composed of only one characteristic size of bubble. In the case of non spherical bubbles, an equivalent diameter corresponding to the spherical bubble of equivalent volume will be used. The physical case of churn-turbulent regime corresponding to both small bubbles and large spherical cap bubbles (van Baten & Krishna, 2001) is out of the scope of this paper. The present paper is limited to the transition between homogeneous and heterogeneous bubbly 8ow and to the region of heterogeneous bubbly 8ow

3286

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

close to the transition. The eect of bubble deformation and local phase fraction, expressing bubble–bubble interactions, on drag and added mass is emphasized. This is coherent with the conclusion of Ranade and Utikar (1999) who considered that the bubble–bubble interactions play a much stronger role than coalescence in determining the dynamic characteristics of bubble columns. The main characteristic of bubble column being related to so-called coherent structures (van den Akker, 1998), the objective of this paper is to highlight the physical parameters that may control the prediction of such coherent structures in bubble columns. 1. Eect of deformation of bubbles on bubble ow transition

In the liquid phase, Stuhmiller (1977) and Prosperetti and Jones (1984) have proposed to express the dierence of pressure as follows: PL − PLi = % L (UG − UL )2 ;

(14)

with % = 0:25. The third term of the r.h.s. of Eq. (12) expresses interfacial transfer of momentum. It includes dierent contributions: the classical drag force, induced by pressure and viscous stresses around the bubbles in steady-state 8ow and the added mass force, induced by pressure distribution around the bubbles in accelerated 8ow: i i MGi = MG1 + MG2 :

(15)

The drag force is classically modelled as

1.1. Stability analysis The transition between homogeneous and heterogeneous bubbly 8ow in a bubble column can be expressed in terms of instability of the homogeneous bubbly 8ow. The homogeneous bubbly 8ow is perturbed. A linear analysis evaluates the conditions in which the perturbation is damped (stability) or ampli?ed (instability). The bubbly 8ow is known to be stable as long as the slowest dynamic wave celerity is smaller than the kinematic one (Wallis, 1969). The present work is based on previous studies of bubble 8ow stability (Biesheuvel & Gorissen, 1990; Pauchon & Banerjee, 1986, 1988). The two-8uid model equations are space averaged over the section of the column. The basic equations express mass and momentum balances in each phase (k = G in the gas and k = L in the liquid): @k @(k Uk ) + = 0; (11) @t @z 
@k k Uk2 @Pk @k @k k Uk + = −k − (Pk − Pki ) @t @z @z @z + Mki + Tkw − k k g +

@ (k $tk ) @z

(12)

with k being the density of each phase, k the volume fraction of each phase, Uk the velocity of each phase, Pk the pressure in each phase, Mki the interfacial transfer of momentum, Tkw the wall transfer of momentum, g the gravity and $tk the longitudinal turbulent diusion of momentum.

i MG1 = −ai 21 L CD |UR |UR

(16)

with ai being the projected interfacial area of the bubbles per unit volume, CD the drag coeGcient and UR the relative velocity between the phases. Indeed, the additional velocity UD expresses a drift velocity induced by the gas dispersion. This velocity has been modelled by UD = −

D @ :  @z

(17)

In bubble columns, with low liquid 8ow rate, the diusivity D cannot be modelled by reference to turbulent shear 8ow in the liquid. In fact, the gas phase induces a “pseudo-turbulence”. The diusivity will then be related to a length scale and a velocity scale characteristic of the bubble motion. Following Batchelor (1988) and Biesheuvel and Gorissen (1990) it becomes: D () = rVR ()H ()1=2 ;

(18)

where r is the bubble radius and H () is a function de?ned by Batchelor (1988) as     H () = 1− where cp = 0:62: (19) cp cp The added mass force can be modelled by the expression given by Voinov (1973) i i ML2 = −MG2

= − L CA 

1.2. Closure relations



@ @ (UG − UL ) + @t @z



UG2 U2 − L 2 2

 : (20)

In the gas phase, we can assume that the pressure at the interface in the gas bubbles is very close to the averaged pressure in the gas. Then, one can write

The fourth term on the r.h.s. of Eq. (12) represents the wall transfer momentum. It is negligible for the gas phase if we assume that it is the dispersed one. Then

PG − PGi = 0:

TGw = 0:

(13)

(21)

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

However, for the liquid phase this term can be expressed as in single phase 8ow as follows: 4 L fLw UL2 ; 2DH

(22)

where DH is the hydraulic diameter and fLw is the wall friction factor. In the liquid phase, Nigmatulin (1979) and Biesheuvel and van Wijngaarden (1984) have proposed a model for the Reynolds stress due to bubble-induced turbulence. The term is given in detail in Lahey (1990). For one-dimensional 8ow, it reduces to $tL; z = −k L (UG − UL )2

with k = 1=5:

10

jL (cm s−1)

TLw = −

3287

1

0 1 (a)

3 jG (cm s−1)

5

3

5

10

(23)

Eqs. (11) and (12) can be written in each phase and given the modelling of closure terms, Eq. (12) can be developed in the gas phase and in the liquid phase. Equating the equations of momentum, the pressure gradient term can be eliminated. After linearization, one obtains a wave equation for the perturbation of gas fraction  : A

2 

2 

2 





@ @  @ @ @  +G +B +E 2 +F = 0: @t 2 @z@t @z @t @z

(24)

The celerities of the dynamic waves are expressed as    2 B B 1 E c± = (26) −4 ± 2 A A2 A

1

0 1 (b)

Eq. (24) can be expressed in terms of wave equation in relation to the void fraction: 

@ @ @ @ @ @ + c+ + c−  + + c = 0: $e @t @z @t @z @t @z (25)

jG (cm s−1)

Fig. 1. Flow maps in bubble column with liquid 8ow rate: analytical derivation of the transition between homogeneous and heterogeneous 8ow pattern for 2 oblateness: (a) 1.1 and (b) 2.

of a driver to changes in local gradients of concentration (of cars). If brakes are bad, then the dynamic wave velocity is decreased. If it is decreased below the continuity wave velocity the result is a  pile up  involving large numbers of cars”. Then the instability is ampli?ed. 1.4. Bubble :ow transition in column and pipe

and the celerity of the kinematic wave is given by c = G=F;

jL (cm s−1)

1.3. Wave equation

(27)

where A, B, E, F and G are given in Appendix A. The stability analysis has been compared to previous works of Biesheuvel and Gorissen (1990) and Pauchon and Banerjee (1986, 1988). The dierences between these approaches have been explained (Le(on-Becerril & Lin(e, 2001). These models have been uni?ed and applied to bubble columns. The homogeneous bubbly 8ow is stable as long as c− ¡ c ¡ c+ . The transition is shown to occur when c − = c . A pedagogic illustration of the above relation is given by Lighthill and Whitham (1955) who are cited in Wallis (1969). The text of Wallis is recalled: “continuity wave theory may be used to describe unsteady 8ow in a stable stream of traGc : : : We can see qualitatively what occurs by considering the dynamic waves that are the result of the response

The transition between homogeneous and heterogeneous 8ow in a bubble column has already been presented (Le(on-Becerril & Lin(e, 2001), in the case of a column without liquid 8ow. In the present paper, the transition in a bubble column with liquid 8ow is analysed. The experimental data have been carried out by Moustiri (2000) in a column of 15 cm i.d., with air and tap water, for super?cial liquid velocities between 0.62 and 2:16 cm s−1 and super?cial gas velocity up to 6 cm s−1 . The transition between homogeneous and heterogeneous bubbly 8ow was observed to occur for super?cial gas velocities between 4 and 5 cm s−1 . In addition, the in8uence of the super?cial liquid velocity was shown to be weak. Such a result is classic (Reilly, Scott, de Brujin, Jain, & Piskorz, 1986). The stability analysis is applied to this column. For each couple of gas and liquid velocities, the transition has been determined in terms of instability of the homogeneous 8ow. The resulting 8ow maps are plotted on Figs. 1a and b,

3288

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

2. Numerical simulation of hydrodynamics and mixing 2.1. CFD code

Fig. 2. Flow map in gas–liquid 8ow in a pipe after Taitel et al. (1980): analytical derivation of the transition between bubble 8ow and slug 8ow for 2 oblateness: (a) 1.1 and (b) 2.

respectively, for oblatness values of 1.1 and 2. The 8ow map coordinates are the super?cial velocities of the gas and the liquid. The transition between homogeneous and heterogeneous 8ow pattern corresponds to the curve derived from linear stability analysis. The ?rst case corresponds to almost spherical bubbles. In this case, the transition occurs at a gas velocity of 5 cm s−1 . The second case corresponds to ellipsoidal bubbles. In this case, the transition occurs at a gas velocity of 3 cm s−1 . This result shows the ability of the stability analysis to determine the 8ow transition between homogeneous and heterogeneous bubbly 8ow. In addition, it highlights the sensitivity of the result to the eccentricity of the bubble which is included in drag and added mass modelling. The stability analysis is then applied to gas–liquid 8ow in a vertical pipe. The data have been presented by Taitel, Barnea, and Dukler (1980). The 8uids are air and tap water. The pipe is 5 cm i.d. The 8ow map after the authors is plotted on Fig. 2. The 8ow map coordinates are the super?cial velocities of the gas and the liquid. The transition between bubble 8ow and slug 8ow regimes corresponds to the curve obtained after stability analysis. Calculations of stability have been derived for two eccentricity values of 1.1 and 2. Once again, the ability of the stability analysis to explain the 8ow transition between bubbly 8ow and slug 8ow is shown. In addition, the sensitivity of the result to the eccentricity is shown to be larger for small liquid velocities. The sensitivity of the transition to the eccentricity decreases with increasing liquid velocity. In the following cases (gas plume and bubble column), the liquid super?cial velocity being equal to zero, the eect of eccentricity will be emphasized.

Detailed presentations of ASTRID code have been published (Simonin and Viollet, 1988; Simonin, 1990). Their model is based on continuity and momentum balance equations expressed in two-phase 8ows. The interfacial momentum transfer is modelled for spherical shape and uniform size dispersed phase. It accounts for drag and added mass forces. The expressions aforementioned (Table 2) have been introduced in the code in order to account for bubble deformation on drag and added mass. The turbulence model in the continuous phase is a two-equation (k; ,) model of turbulence adapted to dispersed two-phase 8ow. The development of the two-8uid model is now classical. Consequently, the basic equations driving two-phase 8ow hydrodynamics will be brie8y recalled. The mass balance writes as @k k + ∇k k Uk = mk ; @t

(28)

where k is the phase retention, index k stands for gas (G) or liquid (L), k (kg m−3 ) is the phase density and Uk (m s−1 ) is the instantaneous and local statistical averaged phase velocity. On the right-hand side appears a new term (mk = k (ui − uk )nki 0i ) represents the statistical averaged interfacial mass transfer. The momentum equation writes as @ (k k Uk ) + ∇k k Uk Uk @t =k k Bk − ∇k Pk + ∇k ($k − k uk uk ) + uk mk + Mk (29) the gravitational acceleration, with Bk (m s−2 ) Pk (kg m−1 s−2 ) the pressure of phase k; uk uk (m2 s−2 ) the phasic Reynolds stress tensor and Mk (kg m−2 s−2 ) the interfacial transfer of momentum due to pressure and viscous stress distributions over the gas–liquid interface (Mk = −pk nik 0i + $k nik 0i ). Eq. (2) can be rewritten as follows: @ (k k Uk ) + ∇ · k k Uk Uk @t =k k g − k ∇PL + ∇ · k ($k − k uk uk ) + uk mk + Ik ; (30) Table 2 Added mass coeGcients for bubbles in liquid in the frame of potential 8ow Added mass coeGcient Isolated spherical bubble Non-isolated spherical bubble Non-isolated ellipsoidal bubble

CA = 12 CA () = 12 1+2 1− CA (; ) = 12 Z( )(1 + (1 + 2=Z( )))

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

where Ik represents the interfacial term of momentum transfer. This term accounts for drag, added mass and turbulent pressure. It can be expressed as (Mudde and Simonin, 1999) 1

L CD U r Vr 2
 @Ur  u ] +  G L C A + UG :∇Ur + ∇[G L CA uG r @t

IG = −IL = ap

 u :∇ : − L uG G L

(31)

The ?rst term on the right-hand side of the previous equation accounts for drag. The second and third terms account for added mass eects. The last term accounts for turbulent pressure (Bel Fdhila & Simonin, 1992). Since liquid motion is controlled by the gas 8ow, and primarily by buoyancy eects, we will focus on drag and added mass mechanism. In the momentum equation of the two-8uid model, the interfacial transfer of momentum related to drag is expressed as IDG = −IDL = ap 12 L CD Ur Vr :

(32)

In this drag term, appear the relative velocity Vr and the drag coeGcient CD . In the case of spherical bubbles, the projected interfacial area ap writes as ap =

3 G ; 4 rb

(33)

where rb is the bubble radius; hence IDG = G L

3 CD Ur V r : 8 rb

(34)

In practice, the bubble is distorted. Hence, rb is the radius of a sphere of equivalent volume than the distorted bubble. It is then possible to introduce the oblatness of the bubble in the expression of the projected area. For sake of simplicity, the eect of bubble deformation and non isolated bubbles (in terms of local gas fraction) are included in the expression of the drag coeGcient and equivalent bubble radius is used. CDeq = CD 2=3 f(G )

(35)

with IDG = G L

3 CDeq U r Vr 8 rbeq

(36)

In addition, the relative velocity calculated in ASTRID code is not equal to the dierence of velocity of the two phases. It is de?ned as Vr = UG − UL − [uL ]:

(37)

The relative velocity Vr is the statistical average of the local instantaneous relative velocity between each bubble and the surrounding ?lm. In other words, it represents the statistical average of relative velocity of the liquid seen by the gas.

3289

The statistical average of the turbulent 8uctuating liquid velocity in the liquid is equal to zero, but the statistical average of the turbulent 8uctuating liquid velocity seen by the gas is not equal to zero. Therefore on the right-hand side of the previous equation, the two ?rst terms account for the total relative mean velocity and an additional third term appears as a drift velocity due to the correlation between instantaneous distribution of bubbles and large-scale liquid motions (Simonin & Viollet, 1988). In order to perform a residence time distribution experiment from a numerical point of view, it is necessary to recall the equation governing the specie transport phenomena. The general transport equation of the concentration of species in two-phase 8ow writes @k Ck + ∇ · k Ck Uk = k Sk − ∇ · k (Jk + ck uk ) @t + ck m k + L k ;

(38)

where Lk represents the interfacial transfer of concentration between the two phases: Lk = −Jk nik 0i

(39)

and Jk represents the molecular diusion modelled by Fick’s law JK = −DK ∇ · CK . In the following, the interfacial transfer of concentration between the two phases will not be simulated. This set of equations and closure relations are solved in ASTRID code and dierent simulations of gas plume and bubble column will be discussed below. 2.2. Transient hydrodynamics of a gas plume In a ?rst step, ASTRID code has been applied to the data of Becker, Sokolichin, and Eigenberg (1994) and present results can be compared to the works of Sokolichin and Eigenberg (1994, 1999) and Sokolichin, Eigenberg, Lapin, and LAubbert (1997). The pilot is a rectangular one, 2 m high, 0:5 m long and 0:08 m wide (Fig. 3a). The height of tap water is 1:5 m, the gas sparging is 40 mm diameter. It is not centered at the bottom of the column but located 0:15 m from the left side of the column. The gas 8ow rate is 1:6 l min−1 , the bubble diameter is equal to 3 mm. The gas plume is shown to oscillate (Fig. 3b). The axial component of the liquid velocity was measured in two locations (point A: 35, 40, 900 mm will be only refered in this paper). Experimental results (Figs. 3c and d) show periodic motion with a period of 40 s and an amplitude of 0:2 m s−1 . This case is often used to test numerical simulations. In the present paper, the numerical simulations will be validated in this case. Then, a bubble column will be simulated in the next part, without experimental reference. The mesh of the geometry is shown on Fig. 4. The simulation is 3D. The mesh contains 26877 nodes (37×17×51).

3290

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

Fig. 3. Gas plume experiment after Becker et al. (1994): (a) pilot, (b) gas plume, (c) liquid velocity at point A, (d) liquid velocity at point B.

0.50 m

0.08 m

Gas sparger 1.5 m

Numerical probes k j i z y x

Gas Fig. 4. Mesh of the gas plume experiment.

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

3291

Fig. 5. Spatial distributions of liquid velocity and gas fraction at dierent time steps: (a) with constant drag coeGcient and without added mass (b) with constant drag coeGcient and constant added mass coeGcient.

Three simulations have been performed: • with constant drag coeGcient and without added mass; • with constant drag coeGcient and constant added mass coeGcient; • with drag and added mass coeGcient functions of both local gas fraction and bubble eccentricity. In the ?rst case (with constant drag coeGcient and without added mass), the gas plume does not oscillate. Spatial distributions of gas fraction and liquid velocity ?eld are plotted on Fig. 5a at dierent time steps. The axial component of the liquid velocity at point A versus time is plotted on Fig. 6a. The 8uctuations are very small and no periodic motion is detected. Such a simulation is then unable to catch the physical characteristic of the transient 8ow. In the second case (with constant drag coeGcient and constant added mass coeGcient), the results obtained by Mudde and Simonin (1999) are con?rmed. Spatial distributions of gas fraction and liquid velocity ?eld are plotted on Fig. 5b at dierent times. The axial component of the liquid velocity at point A versus time is plotted on Fig. 6b. The liquid velocity at point A shows periodic motion with a period of 35 s and an amplitude between 0.2 and 0:25 m s−1 . Added mass is then shown to play a key role in the modeling of transient hydrodynamics.

Since the deformation of the bubbles has been shown to be signi?cant in the prediction of the transition between homogeneous and heterogeneous bubbly 8ows, this parameter is introduced in the simulation of the gas plume. Then, in the third case (with drag and added mass coeGcient functions of both local gas fraction and bubble eccentricity), the axial component of the liquid velocity at point A versus time is plotted in Fig. 6c. The liquid velocity at point A shows periodic motion with a period of 40 s and an amplitude between 0.25 and 0:30 m s−1 . In this case, the improvement of the simulations related to the deformation of the bubble is not obvious. The main result of this part is the key role of added mass on the transient behaviour of the gas plume. In addition, this part validates the present simulations by reference to Mudde and Simonin (1999) work. Delnoij et al. (1999) also simulated oscillating bubble plumes with an Eulerian–Lagrangian approach, including added mass in their model. 2.3. Hydrodynamics and mixing of a bubble column The gas plume has been simulated to validate the numerical approach. Indeed, the hydrodynamics induced by a gas plume looks simple compared to the more complex

3292

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

UZL (m s-1)

1 0.5 0 -0.5 0

50

100

150

200

250

t (s)

(a)

UZL (m s-1)

1 0.5 0 -0.5 0

50

100

150

200

250

t (s)

(b)

UZL (m s-1)

1

0.5

0

-0.5 0 (c)

50

100

150

200

250

t (s)

Fig. 6. Simulated axial component of the liquid velocity at point A versus time: (a) with constant drag coeGcient and without added mass; (b) with constant drag coeGcient and constant added mass coeGcient; (c) with drag and added mass coeGcient functions of both local gas fraction and bubble eccentricity.

hydrodynamics in a bubble column. Consequently, the last part of the present paper focuses on hydrodynamics and mixing in a bubble column. The goal of this study is to highlight the role of bubble deformation and bubble–bubble interaction on mixing. A rectangular column is simulated (Fig. 7). The height is 1 m. The base is rectangular (0:2 m × 0:1 m). The gas is injected uniformly at the bottom of the column. There is no liquid 8ow rate. The bubble diameter is ?xed to 3:5 mm. In order to emphasize the eect of bubble deformation, an excessive oblatness of 3.5 is ?xed. The super?cial gas velocity is 1:25 cm s−1 . It corresponds to the transition between homogeneous and heterogeneous bubbly 8ows in the column. The mesh is 40 × 20 × 100, resulting in 80,000 nodes. In the simulations, transient hydrodynamics is simulated. Previous authors (Delnoij et al., 1999) pointed out that time averaged 8ow pattern in a bubble column does not resemble the 8ow ?eld prevailing in bubble columns. They suggested that experiments in gas–liquid bubble column should be con-

ducted using pseudo-instantaneous measurements. Their results underline the importance of dynamic modelling of bubble columns. In this scope, the present transient simulations of the hydrodynamics will be used to analyse the transient tracer response in the bubble column. Given the local and instantaneous simulation of a tracer injected in the column, the transient tracer response in the liquid phase will be analysed in terms of one dimensional axial dispersion model. A specie is injected as a Dirac function, at the bottom of the column and measured at dierent sections up to the top. The main point of the simulations is that transient transport of the specie is simulated with a transient hydrodynamics. Numerical results obtained with transient transport of a specie simulated with a steady-state hydrodynamics will not be presented here since such simulations are uneGcient to reproduce residence time distribution (Le(on-Becerril, 2001). In such simulations, the eGcient mixing induced by the transient structures is underpredicted, since these transient structures are ?ltered by steady-state simulations and classical turbulent eddy diusivity remains unable to reproduce the eect of these coherent structures driven by the gas dispersion. Three kinds of simulation have been performed: • with constant drag coeGcient and constant added mass coeGcient; • with drag and added mass coeGcient functions of local gas fraction; • with drag and added mass coeGcient functions of both local gas fraction and bubble eccentricity. In each case, the simulated concentration versus time has been plotted at two locations, 0:75 m and at 0:95 m, respectively, in Figs. 8a and b. The concentration is injected at the bottom of the column as a Dirac pulse. The numerical diffusion can be estimated (Cockx, 1997): Dnum = XtXx=5, it corresponds to a value of 10−6 m2 s−1 and can be neglected. It is clear that the residence time distribution is shown to be sensitive to the modeling of interfacial transfer in terms of drag and added mass. Accounting for both local gas fraction and eccentricity of the bubbles does modify signi?cantly the mixing in the bubble column. We can try to evaluate the global mixing in the bubble column in terms of axial dispersion. Given the simpli?ed equation: @C @2 C = EZL 2 : @t @z

(40)

An analytical solution was derived by Inoue and Kafarov (1973): C(z + ; t) C0  2 2  ∞
 n 6 (cos n6z + ) exp − 2 EZL t ; =1+2 HC n=1

E=

(41)

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

3293

0.1 m Numerical probe (0.052, 0.05, 0.93) 0.20 m

1.0 m

k j i

z y x

Uniform gas inlet

Fig. 7. Mesh and size of the bubble column.

where z + is a non-dimensional ordinate in the column where concentration is (z + = z=HC ), t is the time after injection, EZL is the axial dispersion coeGcient, C0 is the space average concentration, C(z + ; t) is the local and instantaneous concentration. The response to the pulse is compared to the theoretical expression by adjusting the two parameters EZL and HC . The numerical simulations of concentration plotted on Fig. 9 show that: • The axial dispersion determined at the outlet of the column is signi?cantly increased when accounting for local gas fraction and bubble eccentricity: EZL = 0:001 m2 s−1 for constant drag and added mass coeGcients, EZL = 0:0035 m2 s−1 for drag and added mass coeGcients depending on local gas fraction and EZL = 0:005 m2 s−1 for drag and added mass coeGcients depending both on local gas fraction and bubble eccentricity, • the axial dispersion coeGcient seems not to be constant in the column, since the dimensionless curves plotted at two locations are dierent. The present simulations tend to show that the axial dispersion is larger at the bottom of the column than in the upper part. This result may not be physical and has to be revisited by experiments. Numerically it may be related to larger transient structures simulated in the lower part of the column than in the upper part. Indeed, the free surface is simpli?ed in the simulations: simulations do not account for the deformation of the free surface and the transient large structures may be numerically damped in this region. Similar simulations

have carried out by van Baten and Krishna (2001) in a churn-turbulent regime; in this case axial dispersion is mainly induced by spherical cap bubbles. The authors do not mention that axial dispersion varies along the column. The main result of these simulations is that the: • transient hydrodynamics related to large structures is ampli?ed by accounting for local gas fraction and bubble eccentricity in the modeling of drag and added mass coeGcients; • mixing expressed in terms of axial dispersion coeGcient is signi?cantly increased by accounting for local gas fraction and bubble eccentricity in the modeling of drag and added mass coeGcients. 3. Conclusion The transitions between homogeneous and heterogeneous 8ows in bubble column as well as the transition between bubbly 8ow and slug 8ow in a vertical pipe have been shown to be controlled by the instability of the homogeneous bubbly 8ow. The eect of bubble deformation on the instability of the homogeneous bubbly 8ow has been highlighted. In a gas plume, computational 8uid dynamics has been used to simulate the transient large structures. The period and amplitude of the hydrodynamics induced by such transient large structures have been simulated and validated after experiments, in the case of a gas plume. Added mass is a key parameter to reproduce these transient structures.

3294

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

1.5

1

0.5

0 0

50

100 t (s)

(a) 1.5

1

0.5

0 0

50

100 t (s)

(b) CA= ½,

CA = CA (αG),

CA = CA (αG,χ),

Fig. 8. Simulated concentration versus time at two locations: (a) 0:75 m and (b) 0:95 m, for dierent modelling of added mass coeGcient: with constant drag coeGcient and constant added mass coeGcient, with drag and added mass coeGcient functions of local gas fraction, with drag and added mass coeGcient functions of both local gas fraction and bubble eccentricity.

A bubble column has then been simulated. In bubble columns, computational 8uid dynamics has been used to simulate the transient large structures that characterize the heterogeneous bubbly 8ow pattern close to the transition between homogeneous and heterogeneous bubbly 8ows. The mixing in the bubble column has been analysed in terms of axial dispersion. The eect of bubble deformation on transient mixing in the bubble column related to these large scale structures has been emphasized, the axial dispersion being multiplied by 5 when local gas fraction and bubble eccentricity are introduced in the modeling of drag and added mass coeGcients. At this stage of the study, only trends can be proposed. Experiments are needed to con?rm these trends. Dierent perspectives can be addressed: • Experimental work is needed to evaluate the characteristics of transient large-scale structures in bubble column in

order to validate the trends exhibited by numerical simulations in the last part of this paper. • It should be interesting to account for dierent bubble size in heterogeneous bubbly 8ow by coupling population balance model of bubble to the local two-8uid model. This should be important in heterogeneous bubbly 8ows with non-uniform bubble population at the sparger device or with spherical caps resulting from coalescence. Experimental work is also needed in this area. Such simulations could be helpful to predict the distributions of bubble size within the column. • Transient simulations have been performed to highlight the increase of mixing due to eccentricity of bubbles in terms of drag and added mass. Indeed, another way could have been followed to account for mixing induced by the dispersed phase in terms of modi?ed eddy viscosity for example. Experimental work is also needed in this area.

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

3295

CA= ½,

C/Co (-)

1

0.5

0 0

1

2

(a)

3 t*=π2EZL t/HC2

4

5

CA = CA (αG)

C/Co (-)

1

0.5

0 0

1

2

(b)

3 t*=π2EZL t/HC2

4

5 CA = CA (αG,χ)

C/Co (-)

1

0.5

0 0

1

(c)

2

3 t*=π2EZL t/HC2

4

5

Theoretical model: Numerical simulation:

z=0.75 m,

z=0.95 m

Fig. 9. Non-dimensional simulated concentration versus non dimensional time, for dierent modeling of added mass coeGcient: (a) with constant drag coeGcient and constant added mass coeGcient; (b) with drag and added mass coeGcient functions of local gas fraction; (c) with drag and added mass coeGcient functions of both local gas fraction and bubble eccentricity; at two locations ( ) 0:75 m and ( ) 0:95 m. Comparison to the model of Inoue and Kafarov.

Appendix A The variables introduced in the stability analysis are listed below  L A = − G − CA L − ; (1 − )





(UG − UL ) L −2 B = 2UL (1 − )  H ( G + L CA ) −% + − k

L  L − G − (1 − )



G (1 − )

L

3296

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297

 UG UL − L (UG − UL ); +  (1 − )   UL UG + [ − 2H ( G + L CA ) E = (UG − UL )  (1 − ) 

− 2 L CA

+ 2 L % + 22 L k] −

 UL2 − G UG2 (1 − )

+ (UG − UL )2  L  % H ( G + L CA ) (−1 + 2)  − + k   (1 − ) ×  (1 − )H  ( +  CL )  G L A    + HCA − k −% +

L   2 1 ai L CD (UG − UL )D UG UL2 −  L CA + + ;  (1 − ) 2 (1 − ) F =−

ai L CD (UG − UL ) 4 L fLw UL − ; DC (1 − ) (1 − )2

G=−

2 L fLw UL2 ai L CD (UG − UL ) − DC (1 − )2 (1 − )2

×(UL + (1 − )(UG − UL ) + (1 − )VR ): References Batchelor, G. K. (1988). A new theory of the instability of a uniform 8uidized bed. Journal of Fluid Mechanics, 193, 75–110. Becker, S., Sokolichin, A., & Eigenberg, G. (1994). Gas–liquid 8ow in bubble columns and loop reactors: Part II. Comparison of detailed experiments and 8ow simulations Chemical Engineering Science, 49(24B), 5747–5762. Bel Fdhila, R., & Simonin, O. (1992). Eulerian prediction of a turbulent bubbly 8owdownstream of a sudden expansion. Sixth workshop on two-phase 8ow predictions, 30 March–2 April, Erlangen, Germany. Biesheuvel, A., & Gorissen, W. C. M. (1990). Void fraction disturbances in a uniform bubbly 8uid. International Journal Multiphase Flow, 16(2), 211–231. Biesheuvel, A., & van Wijngaarden, L. (1984). Two-phase 8ow equation for dilute suspension of gas bubbles in liquid. Journal of Fluid Mechanics, 148, 301–318. Chahed, J. (1999). Forces interfaciales et turbulence dans les e coulements a@ bulles: modelisation et e tudes de cas de references. ThYese de Doctorat es-Sciences, Ecole Nationale d’Ing(enieurs de Tunis, Tunisie. Chen, R. C., Reese, J., & Fan, L-S. (1994). Flow structure in a 3-D bubble column and three-phase 8uidized bed. A.I.Ch.E. Journal, 40, 1093–1104. Cockx, A. (1997). Modelisation de contacteurs gaz–liquide: Application de la Mecanique des Fluides Numeriques aux airlifts. ThYese de Doctorat, I.N.S.A. Toulouse, France. Delnoij, E., Kuipers, J. A. M., & van Swaaij, W. P. M. (1999). A three-dimensional CFD model for gas–liquid 8ow bubble columns. Chemical Engineering Science, 54, 2217–2261. DrahoNs, J., Zahradn(Ik, J., Puncoch(ar, M., Fiavol(a, M., & Bradka, F. (1991). Eect of operating conditions on the characteristics of pressure 8uctuations in a bubble column. Chemical Engineering and Processing, 29, 107–115. Dudukovic, M. P., Larachi, F., & Mills, P. (1999). Multiphase reactors— revisited. Chemical Engineering Science, 54, 1975–1995.

H(ebrard, G. (1995). Etude de l’in:uence du distributeur de gaz sur l’hydrodynamique et le transfert de mati@ere gaz–liquide des colonnes a@ bulles. ThYese de Doctorat, I.N.S.A. Toulouse, France. Inoue, & Kafarov (1973). Method for determining the coeGcient of longitudinal mixing for the continuous phase under the conditions of a non 8ow-through system in high rate extractors. Moscow, Theoreticheskie Osnovy Khimicheskoi Tekhnologii, 7(4), 550–556 (J.I. Mendeleev Institute of Chemical Engineering; in Russian). Karamanev, D. G. (1994). Rise of gas bubbles in Quiescent liquids. A.I.Ch.E. Journal, 40(8), 1418–1421. Karamanev, D. G., & Nikolov, L. N. (1992). Free rising spheres do not obey Newton’s law for free settling. A.I.Ch.E. Journal, 38(11), 1843–1846. Lahey Jr., R. T. (1990). Void wave propagation phenomena in two-phase 8ow (Kern Award Lecture). A.I.Ch.E. Journal, 37(1), 123–135. Lefebvre, S., & Guy, C. (1999). Characterization of bubble columns with local measurements. Chemical Engineering Science, 54, 4895–4902. Le(on-Becerril, E. (2001). Analyse de stabilite et simulation numerique des colonnes a@ bulles. ThYese de Doctorat, I.N.S.A. Toulouse, France. Le(on-Becerril, E., & Lin(e, A. (2001). Stability analysis of a bubble column. Chemical Engineering Science, to appear. Lighthill, M. J., & Whitham, G. B. (1955). On kinematic waves. II. A theory of traGc 8ow on long crowded roads. Proceedings of the Royal Society, A229, 317–345. McQuillan, K. W., & Whalley, P. B. (1985). Flow patterns in vertical two-phase 8ow. International Journal Multiphase Flow, 11(2), 161– 175. Milne-Thomson, L. M. (1968). Theoretical hydrodynamics (5th ed.). New York: Macmillan. Moore, D. W. (1965). The velocity rise of distorted gas bubbles in a liquid of small viscosity. Journal of Fluid Mechanics, 23, 749–766. Moustiri, S. (2000). Hydrodynamique des colonnes a bulles fonctionnant a@ co-courant de gaz et de liquide: EDet hydrodynamique produit par la presence d’un garnissage speciEque. ThYese de Doctorat, I.N.S.A. Toulouse, France. Mudde, R. F., & Simonin, O. (1999). Two and three-dimensional simulations of a bubble plume using a two-8uid model. Chemical Engineering Science, 54, 5061–5069. Nigmatulin, R. I. (1979). Spatial averaging in the mechanics of heterogeneous and dispersed systems. International Journal Multiphase Flow, 5, 353–385. Pauchon, C., & Banerjee, S. (1986). Interphase momentum interaction eects in the averaged multi?eld model. Part I: Void propagation in bubbly 8ows International Journal Multiphase Flow, 12(4), 559–573. Pauchon, C., & Banerjee, S. (1988). Interphase momentum interaction eects in the averaged multi?eld model. Part II: Kinematic waves and interfacial drag in bubbly 8ows International Journal Multiphase Flow, 14(3), 253–264. Prosperetti, A., & Jones, A. V. (1984). Pressure forces in disperse two-phase 8ow. International Journal Multiphase Flow, 10(4), 425–440. Ranade, V. V., & Utikar, R. P. (1999). Dynamic gas–liquid 8ows in bubble column reactors. Chemical Engineering Sciences, 54, 5237–5243. Reilly, I. G., Scott, D. S., de Brujin, T. J., Jain, A., & Piskorz, J. (1986). A correlation for gas holdup in turbulent coalescing bubble columns. The Canadian Journal of Chemical Engineering, 64, 705–717. Sato, Y., Sadatomi, L., & Sekogushi, K. (1981). Momentum and heat transfer in two-phase bubbly 8ow. International Journal of Multiphase Flow, 7, 167–190. SchlAuter, S., Stei, A., & Weinspach, P. M. (1992). Modelling and simulation of bubble column reactors. Chemical Engineering and Processing, 31, 97–117. Shah, Y. T., Kelkar, B. G., Godbole, S. P., & Deckwer, W. D. (1982). Design parameters estimations for bubble column reactors. A.I.Ch.E. Journal, 28(3), 353–379.

E. Leon-Becerril et al. / Chemical Engineering Science 57 (2002) 3283–3297 Simonin, O. (1990). Eulerian formulation for particle dispersion in two-phase 8ows. Fifth workshop on two-phase :ow predictions, 19 –22 March, Erlangen, R.F.A. Simonin, O., & Viollet, P.L. (1988). On the computation of turbulent two-phase 8ows in Eulerian formulation. Turbulent Two-Phase System, Euromech 234, Toulouse, France. Sokolichin, A., & Eigenberg, G. (1994). Gas–liquid 8ow in bubble columns and loop reactors: Part I. Detailed modelling and numerical simulation Chemical Engineering Science, 49(24B), 5735–5746. Sokolichin, A., & Eigenberg, G. (1999). Applicability of the standard k–, turbulence model to the dynamic simulation of bubble columns: Part I. Detailed numerical simulations Chemical Engineering Science, 54, 2273–2284. Sokolichin, A., Eigenberg, G., Lapin, A., & LAubbert, A. (1997). Dynamic numerical simulation of gas–liquid two-phase 8ows Euler=Euler versus Euler=Lagrange. Chemical Engineering Science, 52(4), 611–626. Soria, A., & De Lasa, H. (1992). Kinematic waves ad 8ow patterns in bubble columns and three-phase 8uidized beds. Chemical Engineering Science, 47(13=14), 3403–3410. Stuhmiller, J. H. (1977). The in8uence of interfacial pressure forces on the character of two-phase 8ow model equations. International Journal Multiphase Flow, 3, 551–560. Taitel, Y., Barnea, D., & Dukler, A. E. (1980). Modelling 8ow pattern transitions for steady upward gas–liquid 8ow in vertical tubes. A.I.Ch.E. Journal, 26(3), 345–354. Turton, R., & Levenspiel, O. (1986). A short note on drag correlation for spheres. Powder Technology, 47, 83.

3297

van den Akker, H. E. A. (1998). Coherent structures in multiphase 8ows. Powder Technology, 100, 123–136. van Baten, J. M., & Krishna, R. (2001). Eulerian simulations for determination of the axial dispersion of liquid and gas phases in bubble columns operating in the churn-turbulent regime. Chemical Engineering Science, 56, 503–512. van Wijngaarden, L. (1991). Bubble deformation in bubbly liquids and its eect on the stability of voidage waves. In T. Miloh (Ed.), Mathematical approaches in hydrodynamics. (pp. 38– 49). Philadelphia, PA: SIAM. Vial, C. (2000). Apport de la mecanique des :uides a@ l’etude des contacteurs gaz–liquide experience et simulation numerique. ThYese de Doctorat, I.N.P.L., France. Voinov, O. V. (1973). Force acting on a sphere in an homogeneous 8ow of an ideal incompressible 8uid. Journal of Applied Technical Physics, 14, 592–594. Wallis, G. B. (1969). One-dimensional two-phase :ow. New York: McGraw-Hill. Zahradn(Ik, J., & Fiavol(a, M. (1996). The eect of bubbling regime on gas and liquid phase mixing in bubble column reactors. Chemical Engineering Science, 51(10), 2491–2500. Zuber, (1964). The eect of bubbling regime on gas and liquid phase mixing in bubble column reactors. Chemical Engineering Science, 51(10), 2491–2500.