Multiscale modeling of hydrodynamics, mass transfer and reaction in bubble column reactors

Chemical Engineering Science 56 (2001) 1067}1074

Multiscale modeling of hydrodynamics, mass transfer and reactionin bubble column reactors

Matthias Bauer, Gerhart Eigenberger*Institut fu( r Chemische Verfahrenstechnik, Universita( t Stuttgart, Bo( blinger Str. 72, D-70199 Stuttgart, Germany

Abstract

A previously derived concept of multiscale modeling of bubble column reactors has been applied to study the behavior ofa nonisothermal parallel/consecutive reaction with and without evaporation of one of the reaction products. The concept is basedupon the observation that most gas/liquid synthesis reactions are rather slow compared to the rapid local #uctuations of thegas/liquid hydrodynamics but are su$ciently fast to be considerably a!ected by the medium- and large-scale mixing behavior inindustrial scale bubble columns. A simpli"ed one-dimensional steady-state zone model for the bubble column reactor is para-meterized in its #ow and mixing parameters by the detailed, multidimensional, unsteady-state hydrodynamics and used to calculatethe respective mass, energy and bubble size balance equations for the reactor. Local bubble sizes and total mass #uxes between the gasand liquid phase are fed back from the simpli"ed reactor model into the detailed hydrodynamics until convergence between the twomodels is obtained. The results show that the important interactions between hydrodynamics and mass transfer with reaction can bewell represented by the new concept, which makes it an e$cient and comparatively simple candidate for the in detail analysis, designand scale-up of bubble column reactors. � 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Gas/liquid reactors; Bubble column; Multiscale modeling; Hydrodynamics; Reaction; Evaporation; Non-isothermal

1. Introduction

The detailed modeling of bubble column reactors re-quires accounting for interactions of hydrodynamics,mass transfer and reaction and bubble}bubble interac-tion. Substantial advances have been made in themodeling of hydrodynamics of bubble columns(Borchers, Busch, Sokolichin & Eigenberger, 1999;Sokolichin & Eigenberger, 1999; Sokolichin & Lapin,1998; Mudde & van den Akker, 1999; P#eger, Gomes,Wagner & Gilbert, 1999; Cockx, Do-Quang, Line& Roustan, 1999). It was shown (Sokolichin & Eigenber-ger, 1999) that a three-dimensional transient simulationusing an appropriate turbulence model is necessary toobtain grid independent solutions which are in goodagreement with experimental results.The detailed hydrodynamics of single or multiple

rising bubbles have been addressed in a number ofexplorative publications (Sankaranarayanan, Shan &

*Corresponding author. Tel.: #49-711-6412229; fax: #49-711-6412242.E-mail address: [email protected] (G. Eigenberger).

Kevrekidis, 1999; Delnoij, Kuipers & van Swaij, 1997)but the inclusion of detailed mass transfer and reactioninto these models is still out of sight. Mass transfer andreaction in two-phase gas}liquid #ow is therefore stillbased upon the traditional two-"lm models (Hatta,1928/1929, 1932; Whitman, 1929) or on the surface re-newal concepts (Higbie, 1935; Danckwerts, 1951).Traditionally, a "xed mean bubble size (or mass) has

been used to calculate the speci"c surface area for masstransfer in the above models. This neglects the in#uenceof bubble diameter change due to mass transfer andbubble}bubble interaction (coalescence and break-up).The proper inclusion of the in#uence of a changingbubble size distribution would require populationbalances (Hulburt & Katz, 1964) which introduce anadditional independent coordinate for each changinggas-phase species. Even for batch systems it is still anopen question of how to solve the population balanceequations e$ciently. Millies (Millies, 1996; Millies, Drew& Lahey, 1996) assumed a speci"c bubble size distribu-tion function which reduces the population balanceequation to more simple equations for the determinationof the distribution parameters but restricts the applicabil-ity to monomodal distributions. Lo (2001) suggests the

0009-2509/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 3 2 3 - 7

Fig. 1. Iteration scheme in multiscale modeling; reactor model: zone model.

use of the MUSIG model (MUltiple SIze G�roup). In the

MUSIG model the bubble size distribution is dividedinto several size groups. With the assumption that allgroups have the same velocity the population balance isreduced to a scalar equation. Another simpli"ed alterna-tive is suggested by Kocamustafaogullari and Ishii (1995),who derive a balance equation for the interfacial areaconcentration. This balance can be rewritten to representthe mean bubble diameter or the number density ofbubbles (Millies, 1996).For a proper description of bubble column reactors it

would be necessary to combine the above-mentionedsubmodels for hydrodynamics, mass transfer with reac-tion and bubble}bubble interaction into one global re-actor model. While it is fairly straightforward to specifythe system of coupled model equations, a detailedsimultaneous solution is presently out of reach becausea simultaneous solution would have to cover time andlength scales over more than six orders of magnitude(from the surface layer reaction over the bubble}bubble interactions up to the reactor scale transportequations).In Bauer and Eigenberger (1999) we presented a multi-

scale modeling concept which allows to combine detailedmodels on di!erent scales in an iterative process witha reactor model of medium complexity. This concept isbased upon the main physical interactions between thedi!erent bubble column sub-models which are con-sidered with the required degree of detail. In the previouspaper we presented the concept, validated the reactormodel of medium complexity and gave an example fora simple, isothermal reaction with substantial reactivegas consumption. It is the purpose of this contribution toextend the model to the nonisothermal, multi-reactioncase, to include the in#uence of liquid evaporation intothe rising bubbles and to demonstrate the usefulness ofthe concept for the study of design options like di!erentsparger con"gurations for large-scale bubble columnreactors.

2. Multiscale modeling

Bubble column reactors with multi-step reactions haveso far only been simulated with "xed and strongly simpli-"ed #ow patterns. This is appropriate if a stable #owstructure prevails like in bubble column loop reactorswith internal or external recirculation and if the changeof the mean bubble size and the local gas content can beneglected. Typical counterexamples are `emptya bubblecolumns (without draft tubes). They are characterized bystrongly #uctuating vortical #ow patterns which dependon the column design, the speci"c details of the gassparger location and on the reactive gas consumption orevaporation. None of the traditional hydrodynamic sim-pli"cations like the cell models (Joshi & Sharma, 1979;Zehner, 1988) or the circulation #ow models (Ueyma&Miauchi, 1979) are able to describe the in#uence of the#ow pattern on the reaction in these cases with thenecessary degree of detail.The appropriate three-dimensional unsteady-state

hydrodynamic models (Borchers et al., 1999) on the otherhand, demand such a high computer power for theirsolution that it seems not feasible to extend them withdetailed reaction kinetics. But in many cases there is alsono need to do so since the rather slow reaction kinetics ofmost industrial gas/liquid synthesis reactions donot require the "ne time and space resolution of thehydrodynamics. What is needed instead is a reasonablerepresentation of the axial convection and the lateralmixing patterns. Such a representation has been incorp-orated into our simpli"ed reactor model. For the bubblecolumn it consists of a 1D dispersion model with severalup- and down#ow zones and lateral mixing between thezones. Contrary to conventional dispersion models theaxial dispersion and the lateral cross #ows of liquid aretaken from long time averages of the detailed unsteady-state hydrodynamic model. This leads to axially varyingin- and out#ows between the zones (see Fig. 1, middle).Liquid continuity balances determine the axial convec-tion in each zone. Mass, energy and bubble number

1068 M. Bauer, G. Eigenberger / Chemical Engineering Science 56 (2001) 1067}1074

Fig. 2. Initial results (without accounting for reaction and mass trans-fer) of the hydrodynamic model: left: gas holdup, right: streamlines.

density equations are then formulated and solved for thedi!erent zones of the simpli"ed reactor model (see Bauerand Eigenberger (1999) for further details). The hydro-dynamics are primarily a!ected by the locally varyingmass #uxes between the gas and the liquid phase and bythe locally varying bubble size (both due to mass transferwith reaction).This information is transferred back from the

steady-state reactor model to the detailed unsteady-state hydrodynamics in an iteration loop until conver-gence between the two model levels has been obtained(Fig. 1).For the following examples we will use a two-zone

reactor model with up#ow in the reactor center anddown#ow at the walls, and lateral mixing as explainedabove (Fig. 1, middle). The detailed unsteady-statehydrodynamic model will be calculated for a 2D rectan-gular grid with adjusted constant viscosity to account forthe turbulent viscosity increase. These simpli"cations donot change the general #uctuating vortical character ofthe hydrodynamics but allow for a much faster solution.For a quantitative comparison with experimental resultsor for realistic design studies a three-dimensional un-steady-state hydrodynamic model with proper turbu-lencemodeling like in Sokolichin and Eigenberger (1999),Borchers et al. (1999) would be required. Such work ispresently in progress.

3. Modeling of non-isothermal bubble column reactors

Subsequently, a #at bubble column reactor of 0.5 mwidth and 5 m height will be considered. Unless other-wise speci"ed the liquid feed and the gas sparger coverthe center half of the bottom area of the reactor where thesparger produces bubbles of 4 mm diameter. The productoutlet is at the top of the reactor.An exothermic parallel-consecutive reaction

A�#B�PC��, �H�

"!90 kJ/mol, r�"k

�c�c�,

A�#C�PD�, �H�

"!90 kJ/mol, r�"k

�c�c�,

was chosen for all of the following simulations. Thissequence of overall reactions is typical for partial oxida-tions or hydrogenations with C being the desired prod-uct. For the subsequent simulations the second step ofthe reaction was assumed to be more temperaturesensitive.

3.1. Example 1: exothermic reaction without evaporation

3.1.1. Initial results or frozen yow xeldA common practice for the simulation of bubble col-

umn reactors, loop reactors or stirred tank reactors is to"rst simulate the hydrodynamics for the non-reacting#ow conditions. The obtained #ow "eld is then used for

the simulation of reaction and mass transfer, neglectingany feedback of the reaction on the hydrodynamics(`frozen #ow "eld assumptiona). Snapshots of such initialresults of the detailed hydrodynamics are shown in Fig.2 (gas holdup left, streamlines right). Due to the strongcentral aeration the lower part of the column up toa height of 2 m is very well mixed. Further upwards thegas is dispersed more uniformly over the whole columncross-section and larger circulation cells can be observed.Parameterizing the reactor model with these initialhydrodynamic results yield the vertical liquid velocitiesshown in Fig. 3 top, left. The high liquid up- and down-#ow velocities at the lower part of the reactor drop toa minimum at a height of about 2.2 m. This is an indica-tion of a large vortex in that region even in the longtime-averaged results. A second long time-averaged vor-tex can be observed in the upper part of the columnalthough the up- and down#ow velocities are here con-siderably reduced.This division of the column into a two-tanks-in-series

behavior is also obvious both in the concentration andthe temperature pro"les (Fig. 3 bottom), all of whichshow a plateau in the lower and upper part of the col-umn. At the bottom of the column the gas holdup in thetwo sections, around the sparger (}�}) and near wall(}£}), are rapidly approaching each other and afterabout 1 m height the holdup only deviates slightly be-tween the up- and down#ow zones. Both holdup andbubble diameter decreases with increasing height due tothe reactive gas consumption. The mixing of the gasphase between the zones occurs mainly laterally. There isalmost no back#ow of gas in the top part of the reactor

M. Bauer, G. Eigenberger / Chemical Engineering Science 56 (2001) 1067}1074 1069

Fig. 3. Results from the reactor model, using initial hydrodynamics: top: liquid velocity, gas velocity, holdup, bubble diameter; bottom: weight fractionC and D, temperature. � close to the column wall £ column center.

Fig. 4. Converged results for example 1 of the hydrodynamic model:accounting for reaction and mass transfer left: gas holdup, right:streamlines.

because the slip velocity is almost always higher than thedownward wall liquid velocity.

3.1.2. Converged result using the adapted yow xeldThe feedback of reaction and mass transfer on the

hydrodynamics is taken into account by using the abovedescribed concept of multiscale modeling (Section 2). Thebubble diameter and the total gas balance source termare fed back into the detailed hydrodynamic model ateach height and in each zone. These iterations convergedafter four iteration steps.In Fig. 4, it can be seen that the gas holdup now

decreases to almost zero at the top of the reactor. It issigni"cantly less dispersed over the cross section of thereactor and only a narrow bubble hose rises from thesparger.This lateral di!erence in the gas holdup causes a

stronger driving force for the axial #ow. The up- anddown#ow velocities are now considerably higher and novortex can be observed in the long time-averaged #ow"elds anymore. A good overall mixing in the entire col-umn is the consequence.The overall mixing changes the reactor behavior from

a two-tanks-in-series behavior to a single-stirred-tank.As a consequence the reactor performance worsens andthe concentration of the desired product drops. For com-parison the results from the initial `frozena #ow "eld areshown in Fig. 5 with dashed thin lines.The temperature at the bottom of the column is now

higher than before, the temperature-sensitive second re-action is therefore signi"cantly faster and the concentra-tion of D rises. In the upper most part of the column the

temperature is lower than before but since the concentra-tion of A in the liquid is low in the upper part of thereactor consequences are negligible. As a result of thechanging hydrodynamics the selectivity from A toC dropped from initially 86 to 71%.For the gas phase a di!erence between the initial

calculation and the converged calculation can also be


Fig. 5. Results from the reactor model example 1, using converged hydrodynamics (*) in comparison with the initial hydrodynamics (- - -). Top: liquidvelocity, gas velocity, holdup, bubble diameter; bottom: temperature and weight fraction C and D. (}�}) close to the column wall, (}£}) columncenter.

seen. Due to the strong liquid circulation over the totalcolumn height the total gas holdup has somewhat de-creased. A bigger di!erence between the diameters of therising bubbles in the column center and the back carriedbubbles in the downward wall #ow develops due todi!erences in the mean bubble residence time, since thelateral mixing is somewhat reduced. Nevertheless, lateralmixing has an important in#uence on the mean bubblesize in the down#ow region, since it causes the observedsize increase with decreasing height. At the bottom of thecolumn the down carried bubbles of about 3 mm dia-meter mix with the fresh bubbles from the sparger (4 mm)which results in a mean bubble diameter of 3.5 mm forthe rising bubbles just behind the sparger. At the top ofthe column the mean bubble size is bigger than for thefrozen #ow "eld simulation since the rising time has beensubstantially reduced.

3.2. Example 2: diwerent sparger conxguration

The drawbacks of the "rst example resulted from thestrong liquid circulation loop which extended over thetotal height of the column and resulted in a stirred tankbehavior of the total reactor. It is the aim of the secondexample to show how multiple spargers could be used totransform the #ow pattern back into a tanks-in-seriesbehavior without increasing the total gas #ow rate. Asshown in Fig. 6, left, nine spargers have been introducedin the column. The "rst sparger is located at the very

bottom of the column and feeds 1% of the total gas feed.The other eight spargers feed the rest of the gas consecu-tively in central and peripheral positions as indicated. Allspargers are assumed not to obstruct the liquid #ow.They produce bubbles of 4 mm diameter.Vortices are developing (Fig. 6) which increase the

lateral mixing and decrease the vertical velocities inthe column. Several maxima of the gas holdup canbe observed along the column height at the respectivesparging positions (Fig. 7). The gas holdup decreasesbetween the spargers but overall it increases slightlytowards the top of the reactor, and contrary to theprevious examples the gas is not totally consumed at thecolumn top.Fig. 7 gives a comparison of the previous results of

example 1 (thin dashed lines) with the results of example2 (thick solid lines). One striking result is that the vertical(up- and down#ow) velocities decreased from almost1 m/s to below 0.15 m/s. The small undulations in thevelocity pro"les indicate the vortices between the spargerpositions. The reduced backmixing causes lower temper-atures at the column bottom where the fresh gas A nowreacts slower but more selectively towards the desiredproduct C. With increasing height the temperatureincreases but stays well below the previous example. Ascan be concluded from the gas holp-up and the bubblediameter pro"les, the exit conversion of A is considerablyreduced but the yield of C is almost the same as in theprevious example at a substantially increased selectivity


Fig. 6. Left: Sketch of the column with sparger locations for example 2. Middle: Snapshots of the converged column hydrodynamics with gas holdup(three left "gures) and liquid velocity streamlines (three right "gures). Right: Long time-averaged hydrodynamics, left: gas holdup, right liquidvelocities.

Fig. 7. Comparison of example 1 with converged hydrodynamics (dashed lines) and example 2 (solid lines). Top: liquid velocities, gas holdup, bubblediameter; bottom: temperature and weight fraction C and D. (}�}) close to the column wall, (}£}) column center.

(see the di!erence in the side product D pro"les). It isobvious, that the current operation conditions could befurther improved but this was not the intention of thisexample. It should only demonstrate the degree of detailand physical insight which can be expected and handledwith the modeling concept presented.

3.3. Example 3: exothermic reaction with evaporation

Many reactions take place where one or more of theproducts evaporate. In some cases the entire productstream exits the reactor as vapor. Also evaporation isoften desired for an internal cooling of the reactor.


Fig. 8. Comparison of examples 1 and 3 (with evaporation of C). Top: liquid velocity, gas holdup, bubble diameter; bottom: temperature and weightfraction C and D. � close to the column wall, £ column center.

In the last example the desired productC is assumed toevaporate. The whole system runs close to the boilingpoint of C, the heat of evaporation is then constant(�H

�"9 kJ/mol

�). The broken lines in Fig. 8 result

from the simulation without evaporation (see Fig. 5) andthe solid lines are converged results from the simulationassuming evaporation of the product C.Now, the gas holdup has two maxima, one at the

bottom of the reactor where the pure gas A enters andone at the top of the reactor where the gas consistsmostly of the evaporated product C. The gas is very welldispersed over the cross section of the bubble columnafter about 1 m height and the di!erence in the gasholdup between the center and wall is almost negligible.Since this di!erence is the driving force for the liquidcirculation the total liquid circulation is considerablyreduced if compared with example 1. Only one big vortexcan be observed close to the sparger where lateral di!er-ences of the gas holdup still exist. This hydrodynamicbehavior corresponds to several tanks in series. At thebottom of the column the big vortex creates a "rst mixingsection. The division into further mixing sections alongthe column axis are indicated by the oscillations of thevertical liquid velocities in the long time-averaged results(Fig. 8 top, left). The tanks-in-series behavior is alsoindicated in the stepwise increase of the temperature andthe liquid concentrations of C and D along the columnheight.GasA is almost totally consumed (not shown in Fig. 8)

which causes higher temperatures inspite of the evapo-ration of C. The system reaches almost the same outlet

temperature as in example 1 without evaporation. Thegreat advantage here is that the temperature at the bot-tom of the column where gas A is available in largeamounts is lower which causes the second reaction to beslow. Gas A is being consumed with increasing heightand the temperature rises. The rising temperature hasnow a smaller e!ect on the decomposition of C becauseof the lower concentrations of A.A second aspect contributes to the very low concentra-

tions of D and therefore to good selectivities in thisscenario: Since component C is evaporating it iswithdrawn from further decomposition in the secondreaction. Snapshots of the gas holdup and the liquidvelocity streamlines are given in Fig. 9. It should bementioned that for a gas holdup of more than ten percentthe validity of the detailed hydrodynamics model used isquestionable. The results of the last example shouldtherefore be taken with some caution.

4. Conclusion and summary

For the modeling of interactions between hydrodyn-amics, mass transfer and reaction in bubble columnreactors a new concept for multiscale modeling wasapplied.It is based upon the observation that the often rather

slow reaction kinetics of gas/liquid synthesis reactionsneed not be solved at the same detailed time and spacescales as the unsteady-state hydrodynamics. This allowedto solve the mass, energy and bubble number density


Fig. 9. Snapshots of converged column hydrodynamics for example3 (with evaporation of C): Left: gas holdup, right: liquid velocitystreamline.

balances of the bubble column reactor using a simpli"edsteady-state reactor model. The examples presentedshow that it was essential that this simpli"ed reactormodel has to be parameterized with the detailedhydrodynamic model and that the local bubble sizeas well as the local source terms of the overall gasbalance are fed back to the hydrodynamic model inan iterative way. The frequently applied assumptionof `frozen hydrodynamicsa (from an initialhydrodynamic simulation) was shown to yield resultswhich are substantially di!erent from the converged iter-ative solution.The potential and the feasibility of the new concept

was demonstrated via three di!erent examples for a par-allel/consecutive reaction which include the in#uence ofevaporation of one of the products as well as the in#u-ence of location and design of the gas spargers on thereactor behavior. In both cases the intended change ofthe liquid #ow pattern from one big circulation over thewhole column height to a cascade of well-mixed circula-tion cells resulted in a substantial improvement of thereactor behavior. Based upon these results it can beconcluded that the concept presented forms a reasonablebasis for a rational design and scale-up of gas/liquidbubble column reactors.

Acknowledgements

Support of this work through Deutsche Forschungs-gemeinschaft is gratefully acknowledged.

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Multiscale modeling of hydrodynamics, mass transfer and reaction in bubble column reactors