University of Groningen Heterogeneous mass transfer models

University of Groningen

Heterogeneous mass transfer models for gas absorption in multiphase systemsBrilman, D.W.F.; Goldschmidt, M.J.V.; Versteeg, G.F.; Swaaij, W.P.M. van

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DOI:10.1016/S0009-2509(99)00491-1

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Chemical Engineering Science 55 (2000) 2793}2812

Heterogeneous mass transfer models for gas absorption in multiphasesystems

D. W. F. Brilman*, M. J. V. Goldschmidt, G. F. Versteeg, W. P. M. van SwaaijUniversity of Twente, Department of Chemical Engineering, PO Box 217, 7500 AE Enschede, The Netherlands

Received 15 March 1998; accepted 23 August 1999

Abstract

Heterogeneous, instationary 2-D and 3-D mass transfer models were developed to study the e!ect of dispersed liquid-phasedroplets near the gas}liquid interface on the local gas absorption rate. It was found among other things that droplets (or particles)in#uence local mass transfer rates over an area exceeding largely the projection of the droplets on the gas}liquid interface. Fora speci"c application particle}particle interaction was studied and could be described by a single parameter, depending only on theminimum interparticle distance. For gas absorption #ux prediction an unit cell must be de"ned. The sensitivity of the absorption #uxto the de"nition of the unit cell was investigated. Finally, a complete strategy to arrive at gas absorption #ux prediction from singleparticle simulations has been proposed. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Mass transfer; Heterogeneous model; Enhancement; Multiphase

1. Introduction

In three-phase reactors, gas}liquid}solid or gas}liquid}liquid, frequently the absorption rate of a (spar-ingly) soluble gas-phase reactant to the reaction phase israte determining (see e.g. Beenackers & van Swaaij,1993). The gas}liquid mass transfer rate may be enhancedsigni"cantly by the presence of a "nely dispersed, liquidor solid, phase present in the bulk liquid phase. This wasshown experimentally by among others Kars, Best andDrinkenburg (1979) and Alper and Deckwer (1981) forthe addition of "ne solid particles to a gas}liquid system.The addition of "ne particles caused an enhancement ofthe speci"c gas absorption rate (per unit of driving forceand interfacial area, based on the two-phase system),whereas larger particles showed almost no e!ect.

The increase of the speci"c gas absorption rate, at unitdriving force and unit interfacial area, due to the presenceof the dispersed phase can be characterized by an en-hancement factor, E. This enhancement factor is de"nedas the ratio of the absorption #ux in the presence of theparticles (which can be solid particles or liquid droplets)to the absorption #ux at the same hydrodynamic condi-tions and driving force for mass transfer without such

*Corresponding author. Tel.: 0031-534894479; fax: 0031-534894774.

particles respectively. Using this de"nition, possible ef-fects of the presence of particles on the gas}liquid inter-facial area and on local hydrodynamics are taken intoaccount. For a complete and more detailed review thereader is referred to Beenackers and van Swaaij (1993).

The enhancement of the speci"c absorption #ux due tothe presence of "ne particles has been explained by theso-called &grazing'- or &shuttle'- mechanism, see Karset al. (1979) or Alper, Wichtendahl and Deckwer (1980).According to this shuttle-mechanism particles travel fre-quently between the stagnant mass transfer zone (accord-ing to the "lm theory) at the gas}liquid interface and theliquid bulk. Due to preferential absorption of the di!us-ing gas-phase component in the dispersed phase particlesthe concentration of this gas-phase reactant in the liquidphase near the interface will be reduced, leading to anincreased absorption rate. After a certain contact time,the particle is returned to the liquid bulk where the gas-phase component is desorbed owing to the local concen-tration di!erences and the particles are regenerated. Thisshuttle mechanism requires that the dispersed phase par-ticles are smaller than the stagnant mass transfer "lmthickness, d

Faccording to the "lm theory. For gas ab-

sorption in aqueous media in an intensely agitated con-tactor, a typical value for d

pis approximately 10}20 lm,

whereas for a (laboratory) stirred cell apparatus thisvalue is typically about a few hundred microns.

0009-2509/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 4 9 1 - 1

Nomenclature

A area, m2

BCG Basic Composite Grid,c concentration, mol/m3

D di!usion coe$cient, m2/sD

Rrelative di!usion coe$cient M"D

d/D

cN,

dimensionlessd characteristic particle diameter, mdp

particle diameter, mdd

drop diameter, mE enhancement factor, dimensionlessG gas phase,I interaction parameter, dimensionlessJ mass transfer #ux, mol/m2 skL

liquid side mass transfer coe$cient, m/s¸ distance to the gas}liquid interface, mm

Rrelative solubility or distribution coe$cient,dimensionless

N number of particles, dimensionlessr radial position, dimensionlessR chemical reaction rate, mol/m3 sR

d, R

bradius of droplet and bubble, respectively, m

spc single particle cellt time, sw minimum (surface-to-surface) distance between

particles, m

x position perpendicular to gas}liquidinterface, m

y position along the gas}liquid interface, m

Greek letters

d mass transfer zone near interface, mdp

penetration depth, me$*4

fraction dispersed phase, dimensionless/ Hatta number, dimensionlessq gas}liquid contact time, s

Subscripts and superscripts

av average valueb bubblebulk at bulk liquid phase conditionscon continuous phased dropletdis dispersed phase"lm according to the "lm theorygas gas phasehet heterogeneous (model)hom homogeneous (model)max maximump physical absorption

Owing to a particle size distribution also in applica-tions where the mean particle diameter is relatively large,a signi"cant enhancement of the gas absorption rate maybe observed. This was con"rmed experimentally by Tingeand Drinkenburg (1995), who added very "ne particles toa slurry consisting already of larger ones and found thatthe enhancement of gas absorption to be similar to theenhancement of the gas absorption rate due to the addi-tion of only the same amount of "ne particles to a clearliquid. Such size distributions will certainly occur in caseof gas absorption (or solids dissolution) in a liquid}liquiddispersion. Nishikawa et al. (1994) have shown forliquid}liquid systems that the e!ect of aeration isa broadening of the droplet size distribution, i.e. more"ne droplets. This implies that especially for gas}liquid}liquid systems enhancement of gas absorptioncan be expected when, of course, the solubility of thedi!using component in the dispersed liquid phase ex-ceeds the solubility in the continuous liquid phase.

Experimentally, the gas absorption enhancement phe-nomenon in multiphase systems was frequently studiedand a summary of relevant studies is given in Table 1.Three types of (reaction) systems have been studied ex-perimentally; physical absorption experiments, systemswith a "rst-order reaction for the di!using component in

the continuous phase and one for a "rst-order reaction inthe dispersed phase. No experimental study concerningthe in#uence of an additional second liquid phase on theselectivity for multi-reaction systems has been found inliterature.

All presented experiments show that mass transfer andchemical reaction in mass transfer limited gas}liquidsystems can be signi"cantly enhanced by the addition ofa second dispersed liquid phase with a good solubility(m

R*10) of the solute. Enhancement factors up to 26

have been found experimentally, though usually the en-hancement factor E is within the range 1}5. Some typicalexperimental results of the enhancement factor vs. disper-sed liquid-phase hold up, as observed in G}L}L systems,are shown in Fig. 1. Note that these results are allobtained using labscale equipment.

In order to elucidate and/or describe the enhancemente!ect many theoretical models were developed. Thesecan be categorized by using the model characterizations&stationary' or &instationary' models and &(pseudo-) ho-mogeneous' models or &heterogeneous' models. For theheterogeneous models a subdivision in one-dimensional,two-dimensional and three-dimensional models can bemade, see Table 2. In Table 3 the models presented inliterature are summarized in more detail.

2794 D. W. F. Brilman et al. / Chemical Engineering Science 55 (2000) 2793}2812

Table 1Overview of experimental work on gas absorption enhancement for G}L}L systems and for some other systems

Authors System type System Apparatus Characteristics Geometric sizes

Mehra and Sharma(1985); Mehra, Panditand Sharma (1988-II)

G}L}L Absorption of isobutylene, butene-1and propylene in emulsions ofchlorobenzene in aqueous solutionsof sulphuric acid

Stirred cell /"1}15 dd"1}12 lm

e$*4

"0.02}0.30 SddT+2 lm

E"1.5}26 dp+100 lm

mR

isobutene: 16171-Butene: 2014Propylene: 396

Bruining, Joosten,Beenackers and Hofman(1986)

G}L}L Oxygen absorption into emulsionsof hexadecane in sodium sulphate

Stirred cell /+0.2 dd)10 lm

e$*4

"0.01}0.08 dp+23.5 lm

E"1}1.4m

R"11.6

Littel et al. (1994) G}L}L Physical absorption of CO2

andpropylene into toluene/wateremulsions

Stirred cell e$*4

"0.01}0.39 dd)3 lm

Laminar "lm E"1}4.1 d1,45.#%--

+300 lmm

R: CO

2"2.85 d

1,-!..&*-.+65 lm

Propylene"103Venugopal and Mehra(1994)

G}L}L Absorption of CO2

into emulsionsof aqueous sodium hydroxide in2-ethyl hexanol

Stirred cell /+6.4 dd"5}8 lm

e$*4

"0.05}0.2 dp+400 lm

E"2.4}3.8m

RCO

2"1.9

Van Ede et al. (1995) G}L}L Oxygen absorption into emulsionsof octene in sodium sulphate

Stirred cell /+0.2 dd"22 lm

e$*4

"0.05}0.5 dp+25 lm

E"1.5}3.7m

R+18

Kars et al. (1979) G}L}S Absorption of propane in a slurryof active carbon in water

Stirred cell E+1.3 dd"30}530 lm

e$*4

"0.5}5 wt% dp+20 lm

Alper et al. (1980) G}L}S Absorption of CO2

in an aqueouscarbonate-bicarbonate bu!er withcarbonic anhydrase supp. onoxirane-acrylic beads

Stirred cell /+0.32 dd"1}20 lm

e$*4

"0}0.5 wt% dp+37.5 lm

E"1.2}3.5

Pal, Sharma and Juvekar(1982)

G}L}S Oxidation of aqueous sodiumsulphite in the presence of activatedcarbon

Stirred cell /+2.7 dd"1.7 and 4.3 lm

e$*4

"0.01}2.0 wt% dp+56 lm

E"1.4}1.9Mehra et al. (1988) S}L}L Alkaline hydrolysis of solid esters

in emulsions of chlorobenzenein aqueous solution of potassiumhydroxide

Stirred cell /"0.4}1.9 d2,4vDCPB

"235 lme$*4

"0.005}0.20 dDCPB

"195 lmE"1.7}12 d

PB"120 lm

mR: PB"5170 d

d"1}12 lm

2,4-DCPB"8890 SddT+3}4 lm

DCPB"7085 dp+240 lm

Tinge, Mencke andDrinkenburg (1987)

G}L}S Absorption of a mixture of propaneand ethene in a slurry of activatedcarbon in water

Stirred cell e$*4

"0.01 dd"40}63 lm and

mR, %5)%/%

"65 500}630 lmm

R, 1301!/%"1500 d

p(100 lm

Rols, Condoret, Fonadeand Goma (1990)

G}L}L}S Oxygen absorption into emulsionsof n-dodecane and per#uoro-carbonin water, in a culture of aerobacteraerogenes

Fermentationbroth

e$*4

"0.02}0.34 dd, 40-*$4

"0.5}5 lmE"1.1}3.5 d

d,-*26*$"0.5}50 lm

mR, nv$0$%#!/%

"7.7 db"500}5000 lm

mR,1%3&-6030#.

"17Junker et al. (1989-I) G}L}L}S Oxygen absorption in

aqueous/per#uorocarbonfermentation systems

Fermentationbroth

e$*4

"0.1}0.95 dd"50}100 lm

E"5}10 db"300 lm

mR,1%3&-6030#.

"22 dp"15 lm

Van der Meer, Beenackers,Burghard, Mulder andFok et al. (1992)

G}L}L}S Oxygen absorption into emulsionsof n-octane in water, in culture ofpseud. olevorans

Fermentationbroth

e$*4

"0.05}0.11 dd"0.5 lm

E"1.1}1.6 dp"5 lm

The advantage of the homogeneous models is theirnumerical simplicity (for simple cases they can even besolved analytically) and short computation times. Forthese homogeneous models the following assumptionsare generally made.

f the dispersed phase droplets are very small with re-spect to the mass transfer "lm thickness according tothe "lm theory,

f the dispersed phase (a continuum) is homogeneouslydistributed throughout the continuous phase,

f there is no direct gas-dispersed phase contact,f transport occurs only through the continuous phase,f mass transfer resistances within the dispersed phase

are neglected.

Clearly, some of these assumptions are questionable. Theresults obtained with these models, however, do predict

D. W. F. Brilman et al. / Chemical Engineering Science 55 (2000) 2793}2812 2795

Fig. 1. Observed enhancement factors at increasing dispersed phase hold-up for G}L}L systems. a: Enhancement of butene-1 absorption inchlorobenzene/water emulsions (Mehra 1985, 1988). b: Enhancement of oxygen absorption in hexadecane/sodium sulphate emulsions (Bruining et al.,1986). c: Enhancement of oxygen absorption in octene/water#sulphate emulsion (Van Ede et al., 1995).

Table 2Classi"cation of models

Steady state models Instationary models

Homogeneousmodels

Mehra and Sharma(1985)

Bruining et al. (1986)

Mehra (1988)Littel et al. (1994)Van Ede et al. (1995)Nagy and Moser (1995)

Heterogeneousmodels

3-D Holstvoogd et al.(1988)

1-D Junker et al. (1990a,b)

3-D Karve and Juvekar(1990)

Vinke (1992)

Nagy (1995)Brilman (1998)

2-D Lin et al. (1999)2-D (This work)3-D (This work)

the trend of the enhancement factor with changingoperating conditions as the gas}liquid contact time, therelative solubility and the dispersed phase fraction quali-tatively reasonably well.

Since the quantitative agreement of the homogeneousmodels with the results of the experimental studies is notcompletely satisfactory, several authors even adaptedthese models to "t their experimental data better. Van

Ede, Van Houten and Beenackers (1995) assumed thedispersed phase fraction in the mass transfer zone toincrease with the distance to the gas}liquid interface froma zero fraction at the gas}liquid interface to the bulkphase holdup. Di!usion in the droplet phase was addedin this model, but this seems rather unrealistic consider-ing the discontinuous character of the dispersed phase.Littel, Versteeg and Swaaij (1994) on the contrary, as-sumed in their modeling a thin layer of the dispersedphase at the gas}liquid interface to account for the largedi!erences between the experimentally observed en-hancement factors using the stirred cell apparatus andthe ones calculated with the homogeneous model. Byusing a laminar, falling "lm apparatus Littel et al. (1994)were able to minimize the e!ect of gravity, i.e. to prohibitsettling of the dispersed phase. In the latter set of experi-ments the (homogeneous) model predictions were al-ready much better in agreement with experimentallydetermined absorption #uxes.

Heterogeneous mass transfer models, taking into ac-count the local geometry at the gas}liquid interface, willincrease the level of understanding of the mass transferenhancement phenomena at the gas}liquid interface. Forgas}liquid}liquid systems one-dimensional, instationary,heterogeneous mass transfer models for one particle inthe penetration "lm were developed by Junker, Wangand Hatton (1990a) and Nagy (1995). Their models could


Table 3Overview of mass transfer models for microphase systems

Authors Characteristics Application Model representation

Mehra and Sharma(1985)

Film model G}L}S

Stationary modelBruining et al. (1986) Penetration model G}L}L

E"J1#e(mR!1) Physical absorption

Holstvoogd et al. Unit cell approach G}L}S(1988) N ("1,2,3) particles in a row e!ect of particle

stationary model -geometryinstantaneous surface reaction

Mehra (1988) Surface renewal model G}L}Linstationary, 1st order reaction e!ect e, a

LLreaction in both phases poss.

Saraph and Mehra See Mehra (1988) G}L}S, precipitation [see Mehra (1988)](1994) (&auto-catalytic e!ect')Venugopal and See Mehra (1988) G-L-L, w/o emulsion, [see Mehra (1988)]Mehra (1994) fast reaction in the aqueous

phase

Junker et al. (1989) Penetration model G}L}L2 &plates' in series (1D) physical absorption#0th order

reactiono/w E w/o transition

Karve and Juvekar Unit cell approach G}L}S(1990) stationary model interparticle e!ect

instantaneous surface reaction 1st order reaction e!ect e, dp

equal probability for particleposition fordp/2)M(x))d!d

p/2

Rols et al. (1991) Macrosc. absorption model (kLa)

G?Lc#(k

La)

G?Ldparallel transport via directgas-dispersed phase contactcollision/break-up proposedfor G}L}L systems


Table 3 (continued)

Authors Characteristics Application Model representation

Vinke et al. (1992) Stationary "lm model for solidparticles in the "lm

G}L}Se!ect of particleadsorption capacity andadhering properties

Parallel transport covered-and uncovered surfaceparticles represented by aslab (at x"d

p/4!d

p/2)

(1D heterogeneous)

Littel et al. (1994) Penetration model G}L}Linstationary idea of microscopic phase

separation physical absorptione!ect e

*/5%3&!#%'e

"6-,

Nagy and Moser Film-penetration theory G}L}L(1995) (&pseudo'-hom. model) 0th-, 1st-order reaction

e!ect of L-L mass transfer andmass transfer inside drops

Nagy (1995) Film-penetration model G}L}L(2 parameter model) Physical absorption1D heterogeneous 0th-, 1st- order reaction

Van Ede et al. Dispersed phase holdup varies Physical absorption(1995) with distance to G/L interface 1st-order reaction

Brilman Instationary heterogeneous models:

(this work) 1-D, 2-D and 3-Df introduction forelandf e!ect geometry factors(d

p, ¸,R

b/R

d)

f particle interactionf e!ect of choice of unit cell


be solved analytically for some speci"c cases as, e.g.physical absorption. These were developed to describemass transfer (#reaction) in systems in which the dis-persed phase drops are of the same size as the penetration"lm thickness. However, especially in these cases thesphericity of the gas bubble as well as the sphericity of thedroplet may in#uence signi"cantly the mass transferrates. A numerical 1-D model, capable of handling mul-tiple particles and more complex situations was present-ed by Brilman et al. (1998).

For gas}liquid}solid systems some stationary 3-D het-erogeneous models were developed by Holstvoogd, vanSwaaij and Dierendonck (1988) and Karve and Juvekar(1990). In both studies an unit cell approach was used,containing one single particle in the cell. The results ofthe heterogeneous models by Holstvoogd et al. (1988),Karve et al. (1990) and the 1-D models of Nagy (1995)and Brilman (1998) have shown that enhancement ofgas}liquid mass transfer is dominated by the "rst par-ticles near the gas}liquid interface. Further, it was shownby Holstvoogd et al. (1988) that local geometry stronglyin#uences the enhancement factors found, indicating thata homogeneous description of the dispersed phase is notappropriate.

2. Heterogeneous mass transfer models

From the short overview given above it is concludedthat instationary three dimensional mass transfer modelsare desired to investigate the importance of the geometri-cal factors involved in these multiple-phase systems. It isnecessary that the models to be developed are instation-ary in order to incorporate the e!ect of &saturation' of thedispersed phase particles during the gas}liquid contacttime. Next to these e!ects of near interface geometryalso particle}particle interaction and the translation ofmodeling results to actual absorption #ux prediction willbe discussed in this work.

For the development of a three-dimensional model inwhich local geometry is taken into account the character-istic geometrical size (or -) ratios should be measured orestimated. In general, bubbles are large with respect to thedroplet size and the penetration "lm thickness. In thesecases the interface can be regarded as a plane (in this workcalled &linear' or &planar'models). For strong ionic solutions,at high power inputs in agitated systems or for a dissolv-ing solid in a liquid}liquid system this is not necessarilythe case and the bubble sphericity or solid particle sizeneeds to be taken into account. This latter e!ect is takeninto account in the so-called &angular models', presentedin this work. Therefore, a complete set of models has beendeveloped and is presented schemetically in Fig. 2.

Depending on the geometrical size ratio R"6""-%

/R$301-%5

and the relative &droplet absorption capacity' one of thesemodels will be most appropriate. Model A takes both the

Fig. 2. Set of heterogeneous, instationary mass transfer models.

Fig. 3. Gas absorption (in a multiphase system) according to thepenetration theory.

sphericity of the gas bubble and the drop into account. Ifthe bubble size is much larger than the penetration "lmthickness the sphericity of the gas}liquid interface can beneglected. For very small particles with a low capacitythe one-dimensional model B may be found convenientand su$cient accurate, whereas for somewhat larger par-ticles or particles with a higher capacity for the di!usingsolute model C is most appropriate. In theory, in all themodels described in Fig. 2 the number, position and sizeof the drops, etc. can be chosen arbitrarily.

For modeling mass transfer in multiple-phase systemsit is required to adopt a fundamental mass transfer modellike the Higbie penetration model or the "lm model. TheHigbie model is applied in the present study, thus assum-ing that a package of the bulk liquid phase, including thedispersion droplets present within the package, is trans-ported to the gas}liquid interface and remains there fora certain contact time q, before returning to the liquidphase bulk. Additionally, it is assumed that the positionand distribution of the dispersed phase droplets withinthe package remains unchanged. This is representedschematically in Fig. 3.


Fig. 4. The one-dimensional, instationary, multi-particle model.

A complication for the practical applicability of theheterogeneous models is the comparison of the modelingresults for a particular situation with experimentally ob-served absorption rates. Since, in principal, an in"nitenumber of distributions of the dispersed phase dropsthroughout the penetration "lm (liquid phase) is possible,one needs to choose either a representative &average'con"guration or use a statistical averaging technique.Since it was shown (Brilman et al., 1998) with the one-dimensional model that among others the distance of the"rst droplet to the gas}liquid interface has a major in#u-ence on the mass transfer enhancement, this aspectrequires special attention. For 2-D and 3-D models thesituation is even more complicated since particle}particleinteractions are much more complex. After discussingthe results for simulations with a single dispersed phaseparticle, e!ects of particle}particle interaction will beinvestigated using 2-D multi-particle simulations. Withthese results, e!ects of di!erent particle con"gurations ina liquid package (or unit cell) can be calculated.

2.1. The one-dimensional model

In Fig. 4 a graphical representation of the one-dimen-sional model is given. In Fig. 4 the gas phase is on the lefthand side, ¸

crepresents the continuous liquid phase and

¸$*4

the dispersed liquid-phase droplets. The parameterdp

is the penetration "lm thickness, as estimated byHigbie's penetration theory for absorption in the con-tinuous phase in the absence of the dispersed phase. Theactual penetration depth will, in general, be less thandp

due to the larger absorption capacity of the dispersedphase droplets.

The equations describing di!usion of the gas-phasesolute in the stagnant continuous liquid phase and insidethe dispersed phase (and chemical reaction if applicable)were solved numerically. The number of particles as wellas their sizes, positions and composition can be variedarbitrarily. For more details on the model the reader isreferred to Brilman et al. (1998).

2.2. Two- and three dimensional models

In Fig. 5 a typical representation of a two- or three-dimensional model is given. In this "gure the gas bubbleis surrounded by a larger number of droplets. In thisstudy both the gas bubble and the droplets are con-sidered to be spherical. The partial di!erential equations

Fig. 5. Two dimensional slice of a gas}liquid}liquid system.

describing di!usion with chemical reaction in such het-erogeneous media can only be solved numerically. Thisrequires the di!usion "eld to be covered by a computa-tional grid on which the partial di!erential equations canbe solved using a "nite di!erence approximation.

The complex geometry of the heterogeneous mediumis di$cult to describe using a single global grid. In thiswork, a composite grid, consisting of simpler componentgrids, suitably chosen to describe a particular subdomainand overlapping where they meet, is chosen to solve thisproblem. This is illustrated for the situation of one dis-persed phase droplet located near the gas}liquid inter-face. For the 2-D and 3-D linear models (type C), the basecomposite grid (BCG) used for solving the equations isrepresented by the x}y plane in Fig. 6. More details onthis overlapping grid technique are given by Chesshire andHenshaw (1994).

In Fig. 6a the geometry of the BCG for which theequations are solved is represented. The equations for the2-D models assume no variation in the z-direction, per-pendicular to the x}y plane. The 3-D single particlemodels use the same BCG. However, now the three-dimensional transport equations are solved. A large re-duction of computational e!ort is hereby achieved byusing the rotational symmetry around the line throughthe middle of the droplet, perpendicular to the gas}liquidinterface.

The composition of the composite overlapping grids ofFig. 6 is clearly demonstrated in Fig. 7. The &outer an-nulus' grid enables a numerically smooth overlap fromthe di!usion "eld around the droplet with the continuousphase di!usion "eld. The inner radius of the outer annulusgrid and the outer radius of the inner annulus grid joinprecisely and constitute the phase boundary. At thisphase boundary a user-de"ned boundary condition isimplemented, accounting for continuity of #uxes and theinstantaneous equilibrium distribution of the di!usingspecies between both phases at the interface (see Eqs. (1a)and (1b). A small square grid inside the drop is requiredto avoid very small grid sizes and a singularity at thecenter of the inner annulus.


Fig. 6. Transformation of the two-dimensional plane on which the mass balances for the 2-D and 3-D linear, heterogeneous models are solved into thethree-dimensional situation that these equations represent. (a) Geometry of a grid on which the model equations are solved, (b) 3-D geometry asdescribed by the equations of the 2-D linear model, (c) 3-D geometry described by the equations of the 3-D linear model.

Fig. 7. Construction of the basic composite (overlapping) grid from the component grids.


The model equations for the 2-D and 3-D models arelisted in Table 4. In all these equation R

#0/(or R

$*4) is the

source term due to chemical reaction. This term causescoupling of the mass balance of solute A to the massbalances of other component in case of multi-componentreactions. The mass transfer resistance in the gas phase isneglected in this work.

The angular models (type A) di!er from the linearmodels (type C) only via the background-grid. For theA-type models the background is chosen to be a fraction(+1/6) of an annular grid. The equations describingdi!usion with chemical reaction for the other componentgrids remain unchanged.

From Table 4c it can be noted that boundary condi-tions at overlapping component grids do not need to bespeci"ed. The continuity of the concentration pro"le andthe conservation of mass is accounted for by the interpo-lation and overlap algorithms within the Overture soft-ware package (Los Alamos Nat. Lab., USA). At thedroplet surface the inner and outer annulus componentgrids do not overlap, but join precisely. Here, the bound-ary conditions need to be speci"ed to ensure the continu-ity of #uxes and to account for the distribution of thesolute between both liquid phases. The equilibrium at theinterface is assumed to be instantaneously fast.

r"Rd

C$*4

"mRC

#0/, (1a)

C!D$*4

LC$*4

Lr"!D

#0/

LC#0/

Lr Dr/Rd

. (1b)

The result of the models described above consists ofa concentration pro"le for the di!using component(s) inthe di!usion "eld (at every gridpoint) for both the con-tinuous and the dispersed phase. From the model thetime-dependent speci"c rate of absorption per unit timeand unit of gas}liquid interface, u(t) in [mol/m2 s], andthe average speci"c rate of absorption over the gas}liquidcontact time, u

!7(q) in [mol/m2 s], are obtained at every

y-position at the gas}liquid interface. The local en-hancement factor E(y, t) is de"ned by the ratio ofthese #uxes to their equivalent for gas absorption underidentical conditions without the presence of a dispersedphase:

E(y, t)"u(y, t)

SD

cpt

, (2a)

E!7

(y, q)"u!7

(y, q)

2SD

cpq

. (2b)

The enhancement factors E(y) mentioned refer always tothe contact time-averaged enhancement factor E

!7(y,q),

unless mentioned otherwise. In the simulations the

depth of the di!usion "eld was taken 1.5 times the pene-tration depth of the di!using gas-phase component

(d1%/

"2JpDcq), thus su$ciently large to assure that the

concentration pro"le does not reach to the end of thedi!usion "eld. The model was validated against analyti-cal solutions for physical absorption and for absorptionaccompanied by a homogeneous "rst-order chemical re-action in the continuous phase for situations withoutparticles.

From these test cases it is concluded that the concen-trations and #uxes calculated on overlapping grids areusually calculated within 1% deviation. Convergencyorders in time- and space stepsize are consistent with theEuler explicit "nite di!erence formula used to approxim-ate the partial di!erential equations on the componentgrids. Extrapolation to in"nitely small step sizes yieldeddeviations from the analytical solutions of less than0.03% in the test cases.

3. Results of simulations with one dispersedphase particle

For the single particle simulations values for the vari-ous physico-chemical parameters were taken from theexperimental study of Littel et al. (1994) were chosen asdefault values, see Table 5. A typical single particle modelsimulation result is given in Fig. 8. The enhancementfactor not only varies with the position along thegas}liquid interface, but also changes signi"cantly withtime (see e.g. Fig. 9c). A maximum is observed due tocounteracting e!ects. The initial increase of E(y, t) withtime is due to the fact that the propagating concentrationpro"le needs to reach the droplet, before its in#uence willbecome noticeable.

However, as time propagates, the droplet will get&saturated' with the di!using solute. The local decreaseof its concentration in the continuous phase near thegas}liquid interface will diminish, and with this the localenhancement factors will decrease, ultimately leading toE(y, t)"1 at every y-position. Another important aspectwhich can be recognized from Fig. 8 is the interfacial areaover which the in#uence of the mass transfer enhance-ment is noticeable. In this work, it is proposed to call thisthe &foreland' of the droplet.

For the single droplet case the in#uence of severalprocess parameters was studied, and especially the dis-tance of the front of the particle to the gas}liquid inter-face, the droplet diameter and the relative solubilitym

Rwere varied. Examples of results are given in Fig. 9a}e

for 2-D simulations. The default parameter values usedare given in Table 5.

Fig. 9c shows clearly that the width of the forelandchanges in time and will depend on droplet diameter,relative solubility and the continuous phase di!usion


Table 4Mass balances and boundary conditions for the 2-D and 3-D heterogeneous mass transfer models

(a) 2-D models; Instationary mass balances for di!using gas phase component A

2D models Instationary mass balances

Continuous phase (rectangular&Background')

Lc#0/

(x, y, t)

Lt"D

#0/AL2c

#0/(x, y, t)

Lx2#

L2c#0/

(x, y, t)

Ly2 B#R#0/

Continuous phase! (angular&Background')

Lc#0/

(rb, h

b, t)

Lt"D

#0/A1

rb

LLr

bArb

Lc#0/

(rb, h

b, t)

Lrb

B#1

r2b

L2c#0/

(rb, h

b, t)

Lh2b

B#R#0/

Continuous phase around dispersedphase droplet" (&outer annulus')

Lc#0/

(rd, h

d, t)

Lt"D

#0/A1

rd

LLr

dArd

Lc#0/

(rd, h

d, t)

Lrd

B#1

r2d

L2c#0/

(rd, h

d, t)

Lh2d

B#R#0/

Dispersed phase" (&inner annulus') Lc$*4

(rd, h

d, t)

Lt"D

$*4A1

rd

LLr

dArd

Lc$*4

(rd, h

d, t)

Lrd

B#1

r2d

L2c$*4

(rd, h

d, t)

Lh2d

B#R$*4

Dispersed phase" (&inner square') Lc$*4

(x, y, t)

Lt"D

$*4AL2c

$*4(x, y, t)

Lx2#

L2c$*4

(x, y, t)

Ly2 B#R$*4

(b) 3-D models; Instationary mass balances for di!using gas phase component A

3D models Instationary mass balances

Continuous phase (rectangular&Background')

Lc#0/

(x, y, t)

Lt"D

#0/AL2c

#0/(x, y, t)

Lx2#

1

y

LLyAy

Lc#0/

(x, y, t)

Ly BB#R#0/

Continuous phase! (angular&Background')

Lc#0/

(rb, h

b, t)

Lt"D

#0/A1

r2b

LLr

bAr2b

Lc#0/

(rb, h

b, t)

Lrb

B#1

r2b

sin hb

LLc

#0/Asin h

b

L2c#0/

(rb, h

b, t)

Lh2d

BB#R#0/

Continuous phase arounddispersed phase droplet"(&outer annulus')

Lc#0/

(rd, h

d, t)

Lt"D

#0/A1

r2d

LLr

dAr2d

Lc#0/

(rd, h

d, t)

Lrd

B#1

r2d

sin hd

LLh

dAsin h

d

Lc#0/

(rd, h

d, t)

Lhd

BB#R#0/

Dispersed phase" (&inner annulus')Lc

$*4(rd, h

d, t)

Lt"D

$*4A1

r2d

LLr

dAr2d

Lc$*4

(rd, h

d, t)

Lrd

B#1

r2d

sin hd

LLc

$*4Asin h

d

L2c$*4

(rd, h

d, t)

Lh2d

BB#R$*4

Dispersed phase" (&inner square')Lc

$*4(rd, h

d, t)

Lt"D

$*4A1

r2d

LLr

dAr2d

Lc$*4

(rd, h

d, t)

Lrd

B#1

r2d

sin hd

LLc

$*4Asin h

d

L2c$*4

(rd, h

d, t)

Lh2d

BB#R$*4

(c) 2-D and 3-D models; Initial and boundary conditions#

2D#3D models Boundary conditions

Continuous phase (rectangular IC t"0 x*0 c#0/

"0 (or c#0/

"c#0/,"6-,

)&Background') BC t'0 x"0 c

#0/"c

'!4M

glxPR c

#0/P0 (or c

#0/Pc

#0/,"6-,)

Continuous phase (angular IC t"0 r"*R

"c#0/

"0 (or c#0/

"c#0/,"6-,

)&Background') BC t'0 r

""R

"c#0/

"c'!4

mGL

r"R c

#0/P(or c

#0/Pc

#0/, "6-,)

Continuous phase around IC t"0 rd)R

dc#0/

"0 (or c#0/

"c#0/,"6-,

)droplet (&outer annulus') BC t'0 r

d"R

dc$*4

"c#0/

mR

J#0/

"J$*4

Dispersed phase IC t"0 rd)R

dc$*4

"0 (or c$*4

"c$*4,"6-,

)(&inner annulus') BC t'0 r

d"R

dc$*4

"c#0/

mR

J#0/

"J$*4

Dispersed phase (&inner square') IC t"0 x*0 c$*4

"0

!The center of the coordinate system is placed at the center of the droplet."The center of the coordinate system is placed at the center of the gas bubble.#No boundary conditions are required at the overlap-boundaries: &outer annulus'/&Background' and &inner square'/&inner annulus'. The boundary

condition at the droplet surface rd"R

dis de"ned in Eqs. (1a) and (1b)


Fig. 8. Typical simulation results: a concentration "eld around the droplet and the local enhancement factors, evaluated at the gas}liquid interface.

Table 5Input parameters

Independent parameter values Depending parameter values

kL

1.16]10~4 (m/s) q 0.117332 (s)D

c1.24]10~9 (m2/s) d

p42.76]10~6 (m)

Dd

2.30]10~9 (m2/s) DR

1.85 (!)m

R103 (!)

dp

3]10~6 (m)!

!Littel et al. (1994) mentioned that `the dispersed phase droplets weresmaller than 3 lm.a

coe$cient (not shown in the "gures). The distance of thedroplet to the gas}liquid interface seems to have nosigni"cant in#uence on the foreland area (Fig. 9a), how-ever, the enhancement factors obtained increase stronglywith decreasing distance of the droplet to the gas}liquidinterface. This latter result was also found for the 1-Dmodel (see Brilman et al., 1998).

The importance of the radius of the foreland wasfurther analyzed with the aid of Figs. 10a and b. In Fig. 10athe local enhancement factors E(y, q) at di!erent radialpositions from the projected center of the particle on thegas}liquid interface as well as the corresponding inter-facial area are represented for a 3-D simulation. In Fig.10b these are combined to yield the relative contributionto the additional #ux due to the presence of the dropletfor every radial position. Note that for a 2-D calculationa similar representation can be constructed, but in thesecases the di!erences in fractional surface &area' (line pieces)of the gas}liquid interface, will be less pronounced.

In Fig. 10b it can be recognized that only about 1/3 ofthe additional gas-phase component absorbed is ac-counted for in the area covered by the projection of thedroplet on the gas}liquid interface, and thus 2/3 of theadditional absorption occurs in the foreland outside thisprojection area.

The total amount of gas-phase component which willbe absorbed in addition to the amount absorbed by an


Fig. 9. Some typical (2-D) single particle simulation results showing the e!ects of several parameters (default values are given in Table 5 and default¸ value"1.7 mm). a. E!ect of distance to gas}liquid interface (¸). b. Variation of relative solubility m

R. c. E(y) at di!erent contact times 0)t)4q. d.

Variation of droplet diameter. e. E(t) for 0)t)4q for 5 y-positions.

equivalent liquid-phase package without a droplet can becalculated by integrating the additional #ux (which isequal to: J

1)) (E(y, t)!1)) over the foreland area. It was

found that the total amount of gas-phase componentabsorbed in the dispersed phase droplet exceeds the addi-tional amount absorbed through the gas}liquid interface.In other words, the droplet acts as a sink for the di!usinggas-phase component due to the local distribution of thegas-phase component.

4. Particle}particle interaction

Another important feature is that the presence ofother particles in the vicinity of the particle consideredmay a!ect its in#uence on the gas absorption. In thissection it is investigated how the presence of additionalparticles in#uences the overall absorption #ux. In thesesimulations the default physico-chemical parametervalues are taken equal to those presented in Table 5,


Fig. 10. Analysis of the contribution to mass transfer enhancement atdi!erent radial positions. a. Local enhancement factors at di!erenty-positions. b. Contribution to mass transfer enhancement at di!erentradial positions.

Fig. 11. Two particle con"gurations.

only the number of particles and their position isvaried.

Interaction between the enhancing e!ect of particlesoccurs when the total e!ect of the particles di!ers fromthe sum of the individual contributions of the separateparticles if no other particles were present. This interac-tion is studied for several particle con"gurations, usingthe two-dimensional model. First, interactions betweentwo particles were considered as schemetically shown inFig. 11.

Some results of these simulations are shown in Fig. 12.In these "gures the calculated local enhancement curvefor the two particle con"guration as well as the indi-vidual contributions of the particles respectively and thesummation of the their contributions is presented. Forparticles located directly behind each other (with respectto the interface, see Fig. 12c) it is found that the particleslocated directly behind the "rst one contribute margin-ally to the mass transfer enhancement. This is not onlydue to the larger distance to the interface for the secondparticle, but mainly to the shielding e!ect by the "rstparticle.

As it may be expected, the interaction (in fact thedeviation between the two particle simulation and thesum of both single particle contributions) increases withdecreasing distance between the particles considered. Forthree particle simulations similar results were found, seeFigs. 13a}d.

The interaction may be quanti"ed by de"ning an inter-action factor I as the ratio of the actual enhancementfactor to the summation of the contributions of the par-ticles separately;

I(y)"+N

i/1E(y)!(N!1)

EN1!35*#-%4

(y). (3)

The value found for this interaction factor will vary withthe y-position along the gas}liquid interface. For thesimulation case of Table 5, the interaction factor I de-creases strongly (and thus interaction becomes increas-ingly important) at interparticle distances below 5 lm. Atinterparticle distances exceeding 5 lm the particle}particle interaction may be neglected (deviations lessthan 3%). For other sets of physico-chemical parametersthe mentioned threshold value of 5 lm will be di!erent,e.g. for larger values of D

#0/the threshold value will

be higher.Since the number of multi-particle con"gurations

possible is in"nite, it would be advantageous if theenhancement factors for a speci"c multi-particle con-"guration could be described on beforehand from theresults for single particle simulations. It was foundthat the calculated local enhancement factors for thetwo- and three-particle curves can be described satisfac-tory accurate well using the following correlation, inwhich all binary particle}particle interactions areaccounted for:

EN1!35*#-%4

(y)"N+i/1

Ei(y)!

(N!1)

1/2N(N!1)

]AN~1+i/1

N+

j/i`1iEj

(Ei(y)E

j(y))C(wi?j )B. (4)

In this correlation the only remaining ("t)-parameter isC, which is a function of the interparticle distance(surface to surface) *w

i?j. For the parameter set of


Fig. 12. a. 2-D simulations for two particles an interparticle distance of 17 lm. b. 2-D simulations for two particles an interparticle distance of 2 lm. c.2-D simulations for two particles directly behind each other. d. 2-D simulations for a 2 lm and a 3 lm particle located at the same distance to the G}Linterface.

Table 5 the correlation of C with w is found to be:

w)5 lm C"0.1#(5.0 ) 10~6!wi?j

)0.05/wi?j

(wi?j

in [lm])

w'5 lm C"0.1

For the considered set of parameter values, this particleinteraction description was tested for a 5 particle con-"guration. The results are given in Fig. 14. Fromthese results it is concluded that the E(y, q) pro"le ispredicted reasonably accurate from the single particlesimulations.

5. On the prediction of absorption 6uxes

For calculating the overall gas absorption #uxes intoemulsions or liquid}liquid dispersions using a heterogen-eous model all the possible particle con"gurations at theinterfaces must be taken into account and the enhance-ment factors for these situations should be calculated.Alternatively, an unit cell may be de"ned in which theparticle con"guration is such that the absorption #ux forthis unit cell represents the average absorption #ux for

the particular system and all interactions are correctedfor. Holstvoogd et al. (1988) arbitrarily choose to placeone particle in the center of a unit, cubic, cell (see Table3). Karve and Juvekar (1990) took a cylindrical unit cellwith one particle (see Table 3). However, this unit cell,implemented with a symmetry boundary at the cylin-drical wall overestimates the dispersed phase holdupsurrounding this unit cell, owing to the cylindrical ge-ometry. The same holds for the work of Lin, Zhou andXu (1999).

Single particle calculations have shown that especiallythose particles located most closely to the interface con-tribute to the mass transfer enhancement. In multipar-ticle simulations the mass transfer enhancement dependson the position of the particles with respect to each otherand to the gas}liquid interface. The interaction algo-rithm, taking the binary particle}particle interactionpairs into account, was found to describe reasonably wellthe local enhancement factors for the 2-D cases con-sidered. With this, the e!ect of interaction of di!erentparticle con"gurations can now be evaluated. Since it isimpossible to evaluate every possible con"guration, thesensitivity of the average absorption #ux in a multi-particle cell to the particle con"guration is studied. From


Fig. 13. a. Three identical particles at equal distance from the gas}liquid interface and an interparticle distance of 2 lm. b. Three particles, at equaldistance from the G}L interface. Interparticle distance: 7 lm. c. Three particles, the outer particles are located more close to the gas}liquid interface. d.Three particles, the middle one is located closer to the gas}liquid interface.

Fig. 14. Five particle simulation; testing the interaction correlation.

this it might be ultimately possible to arrive at a represen-tative unit cell con"guration.

For this purpose a multi-particle cell was de"ned, with12 particles present in a certain area in combination withvarious combinations. The area was chosen such that theholdup (by surface for the 2-D model) was equal to 3.4%.The area d

p]*> was divided (in 6 di!erent ways) in

Fig. 15. Some particle con"gurations used in the sensitivity study.

12 equally sized rectangular single particle cells, which,on the average, will contain one single particle each(so-called &single particle cells'). Examples of these sub-divisions are given in Fig. 15.

For these con"gurations one single particle cell locatedat the gas}liquid interface was considered and the localenhancement curves E(¸, y, q) were integrated and aver-aged over all possible particle positions within the cell, onthe centerline perpendicular to the gas}liquid interface,to yield EM

L(y, q). To allow analytical integration, the

E(¸, y, q) curves were "tted (without signi"cant lossin accuracy) using a Cauchy distribution function


Fig. 17. Procedure for calculating the average enhancement factor from heterogeneous, single particle models Mspc"single particle cellN.

Fig. 16. Shifted half-a-cell position aside to avoid the maximum shield-ing e!ect.

E(¸, y, q)"1#A/(1#(y/B)2) in which the parametersA and B depend only on ¸.

EL(y, q)"

:L/d41#~1@2dpL/0

E(¸, y, q)d¸:L/d41#~1@2dpL/0

d¸. (5)

In the next step, the interaction between particles inadjacent cells was taken into account via Eq. (4), usingthe ¸-averaged enhancement relations from Eq. (5). By

doing so, it is assumed that the interaction of the EL(y, q)

curves for the separate particles is a good approachto averaging over all possible con"gurations, each withits own interaction characteristics. This assumptionwas supported by test calculations with three adjacentcells. In each of these cells 20 particle positions on thecenterline were chosen. The average E(y, q) relation forthese 20]20]20 situations was found to be equal to theE(y, q) relation obtained by the interaction of the three

EL(y, q) relations. In another test the assumption of locat-

ing all particles at the centerline of their respective single


Table 6E!ect of single particle cell con"guration

Con"guration E!7,#%--

(3 lm)E!7,#%--

(2 lm)

1]12 1.73 2.14

2]6 1.71 2.09

3]4 1.72 2.05

4]3 1.76 2.10

6]2 1.77 2.10

12]1 1.88 2.15

particle cell was tested by allowing 45 positions (3 lines]15 positions per line) per cell, again for three adjacentcells. All tests showed that these assumptions cause littleerror ((5%).

Single particle simulations have shown that dropletslocated at ¸'1/3d

phave no signi"cant in#uence on the

#ux enhancement, see also Brilman et al. (1998). There-fore, for those con"gurations where the depth of thesingle particle cell exceeds this value, the cells behind thecell adjacent to the gas}liquid interface were neglected.However, this is not the case for con"gurations like inFig. 15d, where also the second row of single particle cellswill contribute to the mass transfer enhancement. How-ever, a maximum shielding e!ect is obtained if the par-ticles are located directly behind one another, as will beclear from Fig. 12c. To avoid this, the second row isshifted half-a-cell position aside. This is represented inFig. 16. When N rows should be taken into account, eachrow is shifted (1/N)th part aside. After this, the contribu-tion of the second row of cells, taking into account theinteraction terms, was also included. The 2-D singleparticle results (for 3 lm and for 2 lm droplets) were usedto evaluate the di!erent single particle cell con"gura-tions, see Table 6.

From these results it is concluded that the con"gura-tion of the unit cell does not in#uence signi"cantly theoverall enhancement factor calculated, if the con"gura-tion of the single particle cell is chosen not too extreme.In Fig. 17 the complete procedure for #ux predictionfrom single particle simulations using particle-particleinteraction is represented.

This procedure is used to evaluate the average en-hancement factor using the 3-D simulations for the para-meter set of Table 5, representing the measurements ofLittel et al. (1994). Using these 3-D single particle simula-tions and the strategy proposed in Fig. 17 the results ofFig. 18 were obtained.

Fig. 18. Comparison of results obtained with the 1-D and 3-D in-stationary heterogeneous models with a homogeneous model and ex-perimental data by Littel et al. (1994).

It can be seen from the results of the 3-D heterogen-eous model that inclusion of the particle interaction leadsto a more logarithmic shape of the E}e curve, which isalso predicted by the homogeneous models and observedin other experimental studies (see Fig. 1). This was notfound for the 1-D model, because in the latter modelparticle interaction is negligible (all particles lie behindone another) and an increase in the dispersed phase hold-up only leads to a smaller distancer of the "rst particle tothe gas}liquid interface. Although the results of the muchsimpler homogeneous model are reasonably good in thiscase, it must be realized that the physical nature of thatmodel is not correct, and this might lead to unreliablemodelpredictions in more complicated situations.

The simulation results of the 3-D heterogeneous modelfor the data by Littel et al. (1994) show an underpredic-tion of the mass transfer enhancement when assuminga droplet size of 3 lm. For a slightly smaller droplet sizea better description would have been obtained (see alsoTable 6). The average droplet size in the experimentsmight have been less than the 3 lm used in the simula-tions. However, since the exact average dropsize (and sizedistribution) is not known, no further validation ispossible.

The 3-D simulation data presented in Fig. 18 werecalculated for di!erent numbers of rows and for di!erentaspect ratios of the unit cell. The latter e!ect was negli-gible, but the number of rows had a minor in#uence((5%) at higher holdups. This is believed to be causedby the limitations of the "t of the simulation data at smalldistances to the gas}liquid interface. More single particlesimulations under these conditions could resolve thisin#uence.

6. Discussion and conclusions

Heterogeneous mass transfer models have been de-veloped to study the e!ect of particles near the gas}liquid


interface on the gas-absorption rate. Using the developedmodels the relative importance of particle capacity para-meters and particle position are now clear from singleparticle simulations. Particle interaction was studied ex-tensively for the 2-D models, and to a lesser extent for the3-D models. An interaction algorithm valid for the speci-"c set of parameters considered (Table 5) was identi"edand gave good results in predicting the absorption #uxesfor multi-particle simulations. However, for other sets ofphysico-chemical parameters the (coe$cients used in the)algorithm should be determined again by "tting anotherset of simulations (multi particle simulations or 2-Dsingle particle simulations using symmetry at the bound-aries of the single particle cell). Taking all particle inter-actions into account, the shape of the single particle cell(the size is already determined by the holdup) can bechosen rather arbitrarily. A procedure for predicting theabsorption #ux for gas absorption in heterogeneous me-dia has been proposed in Fig. 17.

From the results of Fig. 18 for the variation of theenhancement factor at increasing holdup, it is clear thatparticle}particle interaction should be taken into ac-count in the heterogeneous models in order to be able todescribe the enhancement factor leveling o! at higherholdups. Note, that at higher holdups also other hy-drodynamic parameters (like the characteristic surfacerenewal time) may have changed in the experiments andeven direct gas-dispersed liquid phase contact may be-come important.

The results obtained using the models developed andthe strategy proposed seem to follow the general experi-mental trends and described reasonably well the data byLittel et al. (1994). However, accurate comparison isdi$cult since no accurate data on dispersed phase drop-let size and distribution were reported. Further valida-tion of the model is desired and awaiting new, accuratelyde"ned experiments.

Acknowledgements

The authors wish to acknowledge W.D. Henshaw(Los Alamos Nat. Lab., USA) for providing the OvertureOverlapping Grid software and S. Maas for her contribu-tion to the development, and application of the models.

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University of Groningen Heterogeneous mass transfer models