experimental and numerical investigation of bubble column ... · experimental and numerical...

Experimental and numerical investigation of bubblecolumn reactorsBai, W.

DOI:10.6100/IR693280

Published: 01/01/2010

Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

Citation for published version (APA):Bai, W. (2010). Experimental and numerical investigation of bubble column reactors Eindhoven: TechnischeUniversiteit Eindhoven DOI: 10.6100/IR693280

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 04. Jun. 2018

https://doi.org/10.6100/IR693280

https://research.tue.nl/en/publications/experimental-and-numerical-investigation-of-bubble-column-reactors(0a937584-dedc-4f6e-9b2e-1af83d25c458).html

Experimental and Numerical Investigation ofBubble Column Reactors

Samenstelling promotiecommissie:

prof.dr. P.J. Lemstra, chairman, Eindhoven University of Technologyprof.dr.ir. J.A.M. Kuipers, promotor Eindhoven University of Technologydr.ir. N.G. Deen, copromotor Eindhoven University of Technologyprof.dr. R.O. Fox Iowa State University, USAprof.dr.ir. B.J. Geurts Eindhoven University of Technologyprof.dr. D. Lohse University of Twenteprof.dr. R.F. Mudde Delft University of Technologyprof.dr. A.E.P. Veldman University of Groningen

Experimental and Numerical Investigation ofBubble Column Reactors

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigen

op donderdag 23 december 2010 om 16.00 uur

door

Wei Bai

geboren te Shaanxi, China

Dit proefschrift is goedgekeurd door de promotor:

prof.dr.ir. J.A.M. Kuipers

Copromotor:dr.ir. N.G. Deen

Copyright© 2010 by Wei Bai

All rights reserved. This book, or parts thereof, may not be reproduced in any formor by any means, electronic or mechanical, including photocopying, recording or anyinformation storage and retrieval system now known or to be invented, withoutwritten permission from the author.

A catalogue record is available from the Eindhoven University ofTechnology Library.

ISBN: 978-90-386-2405-1Cover design by Jie Fan.Printed by Ipskamp Drukkers B.V., Enschede, the Netherlands.

For my parents

Contents

Contents vii

1 Introduction 11.1 Bubble column reactors . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . 51.3 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . 81.4 Objectives & outline . . . . . . . . . . . . . . . . . . . . . . . . . 10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Single bubble experiment 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Bubble properties of heterogeneous bubbly flows in bubble column 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Numerical analysis of the effect of gas sparging on bubble columnhydrodynamics 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vii

Contents

4.2 Discrete bubble model . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 734.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Discrete bubble modeling of bubbly flows: Swarm effects 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2 Discrete bubble model . . . . . . . . . . . . . . . . . . . . . . . . 925.3 Drag coefficient correlations . . . . . . . . . . . . . . . . . . . . . 965.4 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 1005.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Discrete bubble modeling of bubbly flows: Implementation ofbreakup models 1176.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Discrete bubble model . . . . . . . . . . . . . . . . . . . . . . . . 1196.3 Coalescence model . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Breakup models & implementation . . . . . . . . . . . . . . . . . 1246.5 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 1316.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 Numerical investigation of gas holdup and phase mixing in bubblecolumn reactors 1497.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.2 Discrete bubble model . . . . . . . . . . . . . . . . . . . . . . . . 1517.3 Correlations of the gas holdup . . . . . . . . . . . . . . . . . . . . 1557.4 Phase mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.5 Numerical aspects of phase mixing study . . . . . . . . . . . . . 1637.6 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

viii

Contents

7.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 1667.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1747.A Runge-Kutta-Fehlberg method . . . . . . . . . . . . . . . . . . . 1787.B Interpolation methods . . . . . . . . . . . . . . . . . . . . . . . . 179References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Summary 187

Samenvatting 191

总总总结结结 195

List of publications 197

Acknowledgement 199

About the author 203

ix

Chapter

1Introduction

Bubble column reactors are commonly used in chemical, petrochemical, biochemical,pharmaceutical, metallurgical industries and so on for a variety of processes, i.e.hydrogenation, oxidation, chlorination, alkylation, chemical gas cleaning, variousbio-technological applications, etc.Bubble column reactors have advantages of ease of operation, low operating andmaintenance costs as it requires no moving parts, and compactness. Also, they havethe characteristics of high catalyst durability and excellent heat and mass transfercharacteristics. Furthermore, bubble column reactors can be adapted to specific con-figurations according to practical requirements.In spite of the simple geometry of bubble column reactors, complex hydrodynamicsand its influence on transport characteristics make it difficult to achieve reliable designand scale-up of bubble column reactors. Research on bubble column reactors coversa wide range of activities, i.e. gas holdup, bubble properties, interfacial area, flowregime, heat and mass transfer, back mixing, pressure drop, etc. Investigations on thecharacteristics of bubble column reactors include both experimental and numericalwork. Some experimental techniques and numerical methods used in multiphaseflows are briefly introduced in this chapter. Finally, the objectives and outline of thisthesis will be described.

1

1. Introduction

1.1 Bubble column reactors

In general, bubble columns are used to achieve intimate contact between acontinuous liquid and dispersed gas phase. In its most simple form, a bubblecolumn reactor consists of a vertical cylindrical vessel filled with a liquid anda gas distributor at the bottom, as sketched in Figure 1.1. Gas is spargedthrough the distributor into the column in the form of bubbles and comes incontact with the liquid.

Figure 1.1: Sketch of a simple bubble column reactor.

Bubble columns are employed in many industries, such as chemical,petrochemical, biochemical, pharmaceutical, metallurgical industries and soon. Besides a number of conventional processes, such as hydrogenation,oxidation, chlorination, alkylation, chemical gas cleaning and various bio-technological applications, bubble column reactors are also employed in theprocesses of partial oxidation of ethylene to acetaldehyde, oxidation of ac-etaldehyde to acetic acid, oxidation of p-xylene to dimethylterephthalate,

2

1.1. Bubble column reactors

Fischer-Tropsch synthesis, synthesis of methanol, synthesis of hydrocarbons,hydrolysis of phosgene, oxychlorination of ethylene to 1,2-dichlooroethane,ozonization of waste water and biological waste water treatment. In industrialapplications, bubble column reactors usually have very large dimensions. Forinstance, bubble column reactors for high-tonnage products have capacitiesof 100 − 300 m3. There are even larger bubble column reactors with capacitiesup to 3000 m3 which are employed as fermenters for protein production frommethanol. For waste water treatment, the units can be as large as 20 000 m3.

Bubble columns offer certain distinct advantages such as ease of operation,low operating and maintenance costs (as they don’t contain moving parts) andcompactness. Also, high catalyst durability can be achieved and excellentheat and mass transfer characteristics prevail. Moreover, bubble columnscan easily be adapted. For instance, the liquid phase can be operated inbatch mode, or in co-current or counter-current flow to the gas phase. Whensolid particles are suspended in the liquid, a slurry phase is formed andthe bubble column is then referred to as a slurry bubble column reactor. Inaddition, according to particular practical needs, bubble column reactors canbe modified to different forms, i.e. a cascade bubble column, a packed bubblecolumn and a multishaft bubble column, as shown in Figures 1.2(a)–1.2(c).

A simple bubble column reactor incorporating additional perforated platesforms a cascade bubble column and gas is redistributed over the perforatedplates. Consequently, the redistribution intensifies mass transfer and reducesthe fraction of large bubbles and prevents back mixing in both phases. Byusing a packing or static mixers (Figure 1.2(d)), the back mixing can be furtherreduced. Multishaft bubble column reactors prevent bulk circulation anduniform distribution of gas bubbles over the cross section can be achieved.Furthermore, the circulation can be enhanced via either an internal or externalloop, as shown in Figures 1.2(e)–1.2(f).

In spite of the simple geometry of bubble column reactors, complex hy-drodynamics and its influence on transport characteristics make it difficultto achieve reliable design and scale-up of bubble column reactors. Thereare many factors, such as column dimensions, column internals design, gasdistributor design, operating conditions, i.e. pressure and temperature, su-

3

1. Introduction

(a) Cascade bubble column (b) Packed bubble column (c) Multishaft bubble col-umn

(d) Bubble column withstatic mixers

(e) Bubble column with in-ternal loop

(f) Bubble column with ex-ternal loop

Figure 1.2: Types of bubble column reactors.

perficial gas velocity, physical properties, solid particle properties and concen-tration and so on, which influence the performance of this type of gas-liquidcontactors significantly. During the past decades, scientific interest in bubblecolumn reactors has increased considerably (Deckwer, 1992; Kantarci et al.,2005). The research on the bubble columns covers a wide range includinggas holdup (i.e. Fair et al., 1962; Shah et al., 1982; Heijnen and Van’t Riet,1984; Kawase and Moo-Young, 1987; Krishna et al., 1991; Saxena and Rao,1991; Ruzicka et al., 2001), bubble characteristics (i.e. Abuaf et al., 1978; Linand Fan, 1999; Manera et al., 2009; Guet et al., 2003; Luther et al., 2004), in-terfacial area (i.e. Kataoka et al., 1986; Tan and Ishii, 1990; Revankar andIshii, 1993; Delhaye and Bricard, 1994; Kiambi et al., 2001; Manera et al., 2009),

4

1.2. Experimental techniques

flow regime (i.e. Shah et al., 1982; Shaikh and Al-Dahhan, 2007), heat andmass transfer coefficients (i.e. Deckwer et al., 1974; Wang and Fan, 1978; Shahet al., 1982; Heijnen and Van’t Riet, 1984; Zehner, 1986; Verma, 1989; Saxenaand Rao, 1991; Avdeev et al., 1992; Lin and Fan, 1999; Merchuk et al., 1994;Schluter et al., 1995; Dudley, 1995), back mixing (i.e. Ohki and Inoue, 1970;Deckwer et al., 1974; Hikita and Kikukawa, 1974; Joshi, 1980, 1982; Heijnenand Van’t Riet, 1984; Kawase and Moo-Young, 1986; Zehner, 1986; Westerterpet al., 1987; Wachi and Nojima, 1990; Majumder, 2008), and pressure drop(i.e. Carleton et al., 1967; Gharat and Joshi, 1992; Molga and Westerterp, 1997;Majumder et al., 2006).

1.2 Experimental techniques

In order to study the characteristics of bubble column reactors, i.e. thoseintroduced above, a variety of experimental techniques have been developedand utilized. Those experimental techniques can be classified in differentways. One classification distinguishes global and local measurement tech-niques. For instance, bed expansion technique (i.e. Akita and Yoshida, 1973;Guy et al., 1986) or static pressure at different points in the column (i.e. Hikitaet al., 1980) were used to measure the overall gas holdup in bubble columns.Many types of probes, such as optical probes (i.e Abuaf et al., 1978; Cartellier,1990), resistivity probe (i.e. Herringe and Davis, 1976; Vince et al., 1981), hotfilm anemometry (i.e. Wang et al., 1984; Iskandrani and Kojasoy, 2001) andso on, were used for local gas holdup measurements.

Another classification of the experimental techniques is based on physi-cal features of the measurement and thus, the experimental techniques aredivided into two categories, that is, invasive and non-invasive measurementtechniques. For instance, those probes used for the local void fraction are cat-egorized as invasive measuring techniques. Moreover, hot film anemometrycan also be used to obtained liquid-phase characteristics, i.e. mean velocityand root-mean-square velocity.

There is a large amount of non-invasive techniques which have been uti-lized to investigate hydrodynamics of bubble column reactors. For instance,

5

1. Introduction

dynamic gas disengagement (DGD) measures the rate at which the liquidlevel or the pressure at different levels in the reactor drops after the gas flowis shut off. In such way, some characteristics, i.e. the gas holdup, bubblesize distribution, etc. can be obtained (Sriram and Mann, 1977; Daly et al.,1992; Krishna and Ellenberger, 1996). In addition, photographic techniques,tomographic techniques, radiographic techniques and so on have also beenused in study of multiphase flows. For a detailed review of the measurementtechniques we refer to Chaouki et al. (1997) and Boyer et al. (2002). Someexamples of experimental techniques used in multiphase flows are listed inTable 1.1.

Table 1.1: Experimental techniques used in multiphase flows.

Gas holdupGlobal: Bed expansion; Static pressure; Dy-namic gas disengagement (DGD); Ultrasoundattenuation technique.

Cross-sectional: Wire-mesh sensors; Electri-cal capacitance tomography (ECT); Electri-cal resistance tomography (ERT); γ-ray or X-ray computed tomography; Neutron radiog-raphy; Magnetic resonance imaging (MRI).

Local: Optical probe; Resistivity probe; Elec-trochemical probe; Hot film anemometry.

Flow regime transition Visual observation; Pressure fluctuation; Op-tical probe; Resistivity probe; Heat transferprobe; Optical transmittance probe; Acous-tic probe; Electrical capacitance tomogra-phy (ECT); Electrical resistance tomography(ERT); γ-ray or X-ray computed tomography;Neutron radiography; Computer-automatedradioactive particle tracking (CARPT); LaserDoppler anemometry (LDA); Ultrasoundcomputed tomography.

6

1.2. Experimental techniques

Bubble properties Photography; Double sensor resistivityprobe; Double optical probe; Four-point opti-cal fibre probe; Hot film anemometry; Ultra-sonic Doppler anemometer.

Interfacial area Dynamic absorption; Ultrasound attenua-tion technique; Light attenuation technique;Double sensor resistivity probe; Four-sensorconductivity probe; Four-point optical fibreprobe.

Velocity field Pitot tube; Positron emission particle tracking(PEPT); Radioactive particle tracking (RPTor CARPT); Cinematography; Laser Doppleranemometry (LDA); Particle image velocime-try (PIV); Particle tracking velocimetry (PTV);Fluorescent particle image velocimetry; Hotfilm anemometry; X-ray based particle track-ing velocimetry (XPTV); Magnetic resonanceimaging velocimetry.

Heat transfer Thermocouple; Infrared thermography.

Mass transfer Dynamic physical absorption; Chemical ab-sorption; Physical desorption; Limiting cur-rent density.

Back mixing Tracers; Computer-automated radioactiveparticle tracking (CARPT).

Pressure drop Manometry; Pitot tube.

7

1. Introduction

1.3 Computational Fluid Dynamics

Computational methods for multiphase flows have been developed duringthe past decades (Stewart and Wendroff, 1984; Crowe et al., 1996; Kuipers andvan Swaaij, 1997; Marin, 2006; Jakobsen, 2008). In general, there are two majorapproaches, that is, the Eulerian-Eulerian model and the Eulerian-Lagrangianmodel. The Eulerian-Eulerian model treats both phases as continuous phaseswhich are inter-penetrating. The Eulerian-Lagrangian model considers theliquid phase as a continuous phase, while it treats the other phases as adispersed phase in form of discrete elements, i.e. particles or bubbles. Inaddition, direct numerical simulations (DNS) that are capable of predictingthe interface as well as the flow field of the two phases are also frequentlyused in two-phase flow modelling. DNS can be used to obtain closures forforces acting on discrete elements, such as the drag, lift and virtual mass(Dijkhuizen, 2008).

For multiphase isothermal systems, the conservation equations for massand momentum in the Eulerian-Eulerian model are respectively given by:

∂

∂t(αkρk

)+ ∇ · (αkρkuk

)= Rk (1.1)

∂∂t

(αkρkuk

)+∇· (αkρkukuk

)= −αk∇p−∇· (αkτk)+Mkl+Rkuk+Sk+αkρkg (1.2)

where ρk, uk, αk andτk represent, respectively, the macroscopic density, veloc-ity, volume fraction and viscous stress tensor of the kth phase, p the pressure,Rk a source term describing mass exchange between phase k and the otherphase, Mkl the interphase momentum exchange term between phase k andphase l and Sk a momentum source term of phase k due to phase changes andexternal forces other than gravity.

The Eulerian-Eulerian model is typically used to study large-scale flowstructures and dense dispersed systems due to its lower computational de-mand compared to the Eulerian-Lagrangian model. A disadvantage of themodel is the need for appropriate closure laws for the interphase transport ofmass, momentum and heat for nonisothermal multiphase systems.

For dispersed multiphase flows, a Lagrangian description of the dispersedphase generally treats the individual elements as rigid spheres (i.e., neglecting

8

1.3. Computational Fluid Dynamics

particle deformation and internal flows). The translational motion of theparticle is governed by Newton’s second law:

ddt

(mv) = ΣF (1.3)

where m = ρV is the mass of the element. The term on the right hand side ΣFdenotes net force acting on the particle including surface and body forces, i.e.gravity, pressure, drag, lift, virtual mass, wall force, etc.

The element trajectory is then calculated as:

drdt= v (1.4)

The interfacial coupling between the phases is considered in differentways. A simple way is only to consider the impact of the local characteristicsof the continuous phase on the individual element and neglect any effectsthat the presence of the dispersed phase may have on the continuous phase.This is usually referred to as one way coupling and is only valid in systemswith a very low fraction of the dispersed phase. When the fraction of thedispersed phase is relatively high and the effects of the individual elementson the continuous phase cannot be ignored, two way coupling is required.For dense systems, four way coupling is necessary to take into account theadditional collision effects between elements.

The Eulerian-Lagrangian model has the advantage of considering the mi-croscopic transport phenomena by taking into account the direct collisionand hydrodynamic interaction between neighboring elements. Moreover, inbubble column reactors, the residence time of the gas phase in the form ofbubbles can be easily obtained with the Eulerian-Lagrangian model. Hence,it is possible to study the back mixing of the gas phase with the theory ofresidence time distribution (RTD) and then evaluate the performance of thebubble column reactor.

A disadvantage of the Eulerian-Lagrangian approach is its relatively highcomputational cost for the dispersed phase, particularly in very dense systemswhich are common in industrial multiphase chemical reactors. Even thoughefforts can be made to improve the model, i.e. improving the efficiency of the

9

1. Introduction

algorithm for dealing with collisions (Hoomans et al., 1996; Wu et al., 2010), itmay not be feasible for the Eulerian-Lagrangian model to keep track of a highnumber of discrete elements of the dispersed phase.

1.4 Objectives & outline

The objective of this thesis is to study the bubble properties, i.e. bubblevelocity, bubble size and local void fraction, in heterogeneous flow regimewith a four-point optical fibre probe and to further develop and improve theEulerian-Lagrangian model in the study of fluid dynamics, back mixing ofboth the gas and the liquid phase and performance of bubble column reactors.In the experimental work, the accuracy of a four-point optical fibre probe isinvestigated with a high speed camera in single bubble experiments. Bubbleproperties in the heterogeneous flow regime are then measured with the four-point optical probe. Effects of the superficial gas velocity and the columnheight on the bubble properties are studied. In addition, effect of the gasdistributor on the hydrodynamics and back mixing of the gas phase in bubblecolumn reactors is investigated by using the Eulerian-Lagrangian model andthe theory of residence time distribution (RTD). The swarm effects on the dragforce in bubble column are studied by comparing simulations using variousdrag coefficient correlations from literature and measurements using particleimage velocimetry (PIV). Breakup models in literature are implemented inthe Eulerian-Lagrangian model to consider the coalescence and breakup ofbubbles in turbulent flows. Finally, massless tracer particles which movewith the liquid phase are introduced into the model. Moreover, overall gasholdup and back mixing of both the gas and the liquid phases are studied andcompared with correlations in literature.The contributions to these topics are organized in chapters as follows:

Chapter 2 investigates the performance of a four-point optical fibre probefor the determination of the bubble velocity. Single bubble experiments arecarried out and five liquids with different physical properties are used. Theresults obtained from the four-point optical fibre probe are compared withthose obtained by image processing using a high speed camera. The accu-

10

Nomenclature

racy of the bubble velocity determination from the optical probe is discussedaccording to the physical properties of the liquid phase.

Chapter 3 studies the bubble properties in the heterogeneous flow regimein a square bubble column with the four-point optical fibre probe. The per-formance of the four-point optical fibre probe is further discussed. Bubblesize and specific interfacial area are estimated according to the measurementsfrom the optical probe. Furthermore, the effects of the column height and thegas flow rate on the bubble properties are investigated.

Chapter 4 discusses the effect of the gas distributor on the hydrodynamicsand back mixing of the gas phase in a square bubble column by using anEulerian-Lagrangian model. Theory of residence time distribution (RTD) isadopted to study the back mixing of the gas phase.

Chapter 5 studies the drag coefficient correlations for bubble swarms inliterature with the aid of the Eulerian-Lagrangian model. The simulationresults are compared with the measurements of particle image velocimetry(PIV).

Chapter 6 implements breakup models reported in literature in theEulerian-Lagrangian model. Furthermore, critical Weber numbers relatedto bubble breakup in turbulent flows are also considered and combined witha coalescence model. Simulation results of the Eulerian-Lagrangian model arecompared with PIV measurements. Moreover, the bubble size distributionsfrom the numerical simulations are compared with those obtained from thefour-point optical fibre probe.

Chapter 7 discusses the performance of a square bubble column. Masslesstracer particles are introduced into the Eulerian-Lagrangian model whichmove with the liquid phase in the bubble column. Gas holdup and backmixing of both the gas and liquid phases are investigated and compared withcorrelations reported in literature.

Nomenclature

F force acting on a discrete element, [N]g gravitational acceleration, [m s−2]

11

1. Introduction

m mass of a discrete element, [kg]Mkl momentum source term due to interaction between phase k and

phase l, [kg m−2 s−2]p pressure, [N m−2]r position vector, [m]Rk mass source term in phase k, [kg m−3 s−1]Sk momentum source term due to phase changes and external forces

other than gravity in phase k, [kg m−2 s−2]t time, [s]u liquid velocity, [m s−1]v velocity of a discrete element, [m s−1]

Greek letters

α volume fraction, [-]ρ density, [kg m−3]τ stress tensor, [N m−2]

References

N. Abuaf, O. C. Jones Jr., and G. A. Zimmer. Optical probe for local void frac-tion and interface velocity measurements. Review of Scientific Instruments,49(8):1090–1094, 1978.

K. Akita and F. Yoshida. Gas holdup and volumetric mass transfer coefficientin bubble columns: Effects of liquid properties. Ind. Eng. Chem. Process Des.Develop., 12(1):76–80, 1973.

A. A. Avdeev, B. F. Balunov, and V. I. Kiselev. Heat transfer in bubble layers athigh pressures. Experimental Thermal and Fluid Science, 5(6):728–735, 1992.

C. Boyer, A. -M. Duquenne, and G. Wild. Measuring techniques in gas-liquid and gas-liquid-solid reactors. Chemical Engineering Science, 57(16):3185–3215, 2002.

12

References

A. J. Carleton, R. J. Flain, J. Rennie, and F. H. H. Valentin. Some propertiesof a packed bubble column. Chemical Engineering Science, 22(12):1839–1845,1967.

A. Cartellier. Optical probes for local void fraction measurements: Charac-terization of performance. Review of Scientific Instruments, 61(2):874–886,1990.

J. Chaouki, F. Larachi, and M. P. Dudukovic. Noninvasive tomographic andvelocimetric monitoring of multiphase flows. Industrial and EngineeringChemistry Research, 36(11):4476–4503, 1997.

C. T. Crowe, T. R. Troutt, and J. J. N. Chung. Numerical models for two-phaseturbulent flows. Annual Review of Fluid Mechanics, 28:11–43, 1996.

J. G. Daly, S. A. Patel, and D. B. Bukur. Measurement of gas holdups andsauter mean bubble diameters in bubble column reactors by dynamics gasdisengagement method. Chemical Engineering Science, 47(13-14):3647–3654,1992.

W. -D. Deckwer. Bubble column reactors. Chichester [etc.] : Wiley and sons,1992.

W. -D. Deckwer, R. Burckhart, and G. Zoll. Mixing and mass transfer in tallbubble columns. Chemical Engineering Science, 29(11):2177–2188, 1974.

J. M. Delhaye and P. Bricard. Interfacial area in bubbly flow: Experimentaldata and correlations. Nuclear Engineering and Design, 151(1):65–77, 1994.

W. Dijkhuizen. Derivation closures for bubbly flows using direct numerical simu-lations. PhD thesis, Enschede, April 2008.

J. Dudley. Mass transfer in bubble columns: A comparison of correlations.Water Research, 29(4):1129–1138, 1995.

J. R. Fair, A. J. Lambright, and J. W. Andersen. Heat transfer and gas holdupin a sparged contactor. IandEC Process Design and Development, 1(1):33–36,1962.

13

1. Introduction

S. D. Gharat and J. B. Joshi. Transport phenomena in bubble colunm reactorii: Pressure drop. The Chemical Engineering Journal, 48(3):153–166, 1992.

S. Guet, R. V. Fortunati, R. F. Mudde, and G. Ooms. Bubble velocity andsize measurement with a four-point optical fiber probe. Particle and ParticleSystems Characterization, 20(3):219–230, 2003.

C. Guy, P. J. Carreau, and J. Paris. Mixing characteristics and gas hold-up of abubble column. Canadian Journal of Chemical Engineering, 64(1):23–35, 1986.

J. J. Heijnen and K. Van’t Riet. Mass transfer, mixing and heat transfer phe-nomena in low viscosity bubble column reactors. The Chemical EngineeringJournal, 28(2):B21–B42, 1984.

R. A. Herringe and M. R. Davis. Structural development of gas-liquid mixtureflows. Journal of Fluid Mechanics Digital Archive, 73(01):97–123, 1976.

H. Hikita and H. Kikukawa. Liquid-phase mixing in bubble columns: Effectof liquid properties. The Chemical Engineering Journal, 8(3):191–197, 1974.

H. Hikita, S. Asai, K. Tanigawa, K. Segawa, and M. Kitao. Gas hold-up inbubble columns. The Chemical Engineering Journal, 20(1):59–67, 1980.

B. P. B. Hoomans, J. A. M. Kuipers, W. J. Briels, and W. P. M. van Swaaij. Dis-crete particle simulation of bubble and slug formation in a two-dimensionalgas-fluidised bed: A hard-sphere approach. Chemical Engineering Science,51(1):99–118, 1996.

A. Iskandrani and G. Kojasoy. Local void fraction and velocity field descrip-tion in horizontal bubbly flow. Nuclear Engineering and Design, 204(1-3):117–128, 2001.

H. A. Jakobsen. Chemical Reactor Modeling: Multiphase Reactive Flows. Springer,1st edition, July 2008.

J. B. Joshi. Axial mixing in multiphase contactors - A unified correlation.Transactions of the Institution of Chemical Engineers, 58(3):155–165, 1980.

14

References

J. B. Joshi. Gas phase dispersion in bubble columns. The Chemical EngineeringJournal, 24(2):213–216, 1982.

N. Kantarci, F. Borak, and K. O. Ulgen. Bubble column reactors. ProcessBiochemistry, 40(7):2263–2283, 2005.

I. Kataoka, M. Ishii, and A. Serizawa. Local formulation and measurementsof interfacial area concentration in two-phase flow. International Journal ofMultiphase Flow, 12(4):505–529, 1986.

Y. Kawase and M. Moo-Young. Liquid phase mixing in bubble columns withnewtonian and non-newtonian fluids. Chemical Engineering Science, 41(8):1969–1977, 1986.

Y. Kawase and M. Moo-Young. Theoretical prediction of gas hold-up inbubble columns with newtonian and non-newtonian fluids. Industrial andEngineering Chemistry Research, 26(5):933–937, 1987.

S. L. Kiambi, A. M. Duquenne, A. Bascoul, and H. Delmas. Measurementsof local interfacial area: Application of bi-optical fibre technique. ChemicalEngineering Science, 56(21-22):6447–6453, 2001.

R. Krishna and J. Ellenberger. Gas holdup in bubble column reactors operatingin the churn-turbulent flow regime. AIChE Journal, 42(9):2627–2634, 1996.

R. Krishna, P.M. Wilkinson, and L.L. Van Dierendonck. A model for gasholdup in bubble columns incorporating the influence of gas density on flowregime transitions. Chemical Engineering Science, 46(10):2491–2496, 1991.

J. A. M. Kuipers and W. P. M. van Swaaij. Application of computational fluiddynamics to chemical reaction engineering. Reviews in Chemical Engineering,13(3):1–118, 1997.

T. -J. Lin and L. -S. Fan. Heat transfer and bubble characteristics from anozzle in high-pressure bubble columns. Chemical Engineering Science, 54(21):4853–4859, 1999.

15

1. Introduction

S. Luther, J. Rensen, and S. Guet. Bubble aspect ratio and velocity measure-ment using a four-point fiber-optical probe. Experiments in Fluids, 36(2):326–333, 2004.

S. K. Majumder. Analysis of dispersion coefficient of bubble motion andvelocity characteristic factor in down and upflow bubble column reactor.Chemical Engineering Science, 63(12):3160–3170, 2008.

S. K. Majumder, G. Kundu, and D. Mukherjee. Prediction of pressure dropin a modified gas-liquid downflow bubble column. Chemical EngineeringScience, 61(12):4060–4070, 2006.

A. Manera, B. Ozar, S. Paranjape, M. Ishii, and H. M. Prasser. Comparison be-tween wire-mesh sensors and conductive needle-probes for measurementsof two-phase flow parameters. Nuclear Engineering and Design, 239(9):1718–1724, 2009. 15th International Conference on Nuclear Engineering (ICONE15).

G. B. Marin, editor. Advances in Chemical Engineering, volume 31. Elsevier Inc.,2006.

J. Merchuk, S. Ben-Zvi, and K. Niranjan. Why use bubble-column bioreactors?Trends in Biotechnology, 12(12):501–511, 1994.

E. J. Molga and K. R. Westerterp. Experimental study of a cocurrent upflowpacked bed bubble column reactor: pressure drop, holdup and interfacialarea. Chemical Engineering and Processing, 36(6):489–495, 1997.

Y. Ohki and H. Inoue. Longitudinal mixing of the liquid phase in bubblecolumns. Chemical Engineering Science, 25(1):1–16, 1970.

S. T. Revankar and M. Ishii. Theory and measurement of local interfacial areausing a four sensor probe in two-phase flow. International Journal of Heatand Mass Transfer, 36(12):2997–3007, 1993.

M. C. Ruzicka, J. Zahradnık, J. Drahos, and N. H. Thomas. Homogeneous-heterogeneous regime transition in bubble columns. Chemical EngineeringScience, 56(15):4609–4626, 2001.

16

References

S. C. Saxena and N. S. Rao. Heat transfer and gas holdup in a two-phasebubble column: Air-water system - review and new data. ExperimentalThermal and Fluid Science, 4(2):139–151, 1991.

S. Schluter, A. Steiff, and P. M. Weinspach. Heat transfer in two- and three-phase bubble column reactors with internals. Chemical Engineering andProcessing, 34(3):157–172, 1995.

Y. T. Shah, B. G. Kelkar, S. P. Godbole, and W. -D. Deckwer. Design parametersestimations for bubble column reactors. AIChE Journal, 28(3):353–379, 1982.

A. Shaikh and M. H. Al-Dahhan. A review on flow regime transition in bubblecolumns. International Journal of Chemical Reactor Engineering, 5, 2007.

K. Sriram and R. Mann. Dynamic gas disengagement: A new technique forassessing the behaviour of bubble columns. Chemical Engineering Science, 32(6):571–580, 1977.

H. B. Stewart and B. Wendroff. Two-phase flow: Models and methods. Journalof Computational Physics, 56(3):363–409, 1984.

M. J. Tan and M. Ishii. A method for measurement of local specific interfacialarea. International Journal of Multiphase Flow, 16(2):353–358, 1990.

A. K. Verma. Heat transfer mechanism in bubble columns. The ChemicalEngineering Journal, 42(3):205–208, 1989.

M. A. Vince, G. Krycuk, and R. T. Lahey Jr. Development of a radio frequencyexcited local impedance probe. Nuclear Engineering and Design, 67(1):125–136, 1981.

S. Wachi and Y. Nojima. Gas-phase dispersion in bubble columns. ChemicalEngineering Science, 45(4):901–905, 1990.

K. B. Wang and L. T. Fan. Mass transfer in bubble columns packed withmotionless mixers. Chemical Engineering Science, 33(7):945–952, 1978.

17

1. Introduction

S. K. Wang, S. J. Lee, O. C. Jones Jr., and R. T. Lahey Jr. Local void frac-tion measurement in techniques in two-phase bubbly flows using hot-filmanemometry. volume 31, pages 39–45, 1984.

K. R. Westerterp, W. P. M. van Swaaij, and A. A. C. M. Beenackers. ChemicalReactor Design and Operation. John Wiley & Sons Ltd., 2nd edition, 1987.

C. L. Wu, A. S. Berrouk, and K. Nandakumar. An efficient chained-hash-tablestrategy for collision handling in hard-sphere discrete particle modeling.Powder Technology, 197(1-2):58–67, 2010.

P. Zehner. Momentum, mass and heat transfer in bubble columns. part 2. axialblending and heat transfer. International chemical engineering, 26(1):29–35,1986.

18

Chapter

2Single bubble experiment

For the understanding of high void fraction flows in bubble columns, quantitativeinformation on bubble size, bubble velocity as well as gas void fraction is of crucialimportance. Optical probes have the advantage of low cost, simplicity of operation andeasy interpretation of the results. Moreover optical probes are well-suited to obtainbubble properties in bubbly flows at high void fraction.In this chapter, the performance of a four-point optical fibre probe was investigatedby comparing probe data with results obtained with photography. Five liquids withdifferent material properties were used in combination with air sparged from the dis-tributor at the bottom a flat bubble column.It is found that the liquid properties have a significant influence on the bubble ve-locity obtained from the optical probe. In viscous liquids, the bubble deforms and isdecelerated by the optical probe during the approach process and thus the inaccuracyof velocity determination results. In low viscosity liquids, the bubble’s wobblingbehavior also results in inaccuracy.

19

2. Single bubble experiment

2.1 Introduction

Two-phase gas-liquid flows are frequently encounted in variety of industrialprocesses, such as bubble column reactors, stirred tank reactors, waste watertreatment units and so on. Reliable reactor design and scale-up require dataon gas holdup, pressure drop, flow regime, heat and mass transfer rates andflow patterns of the phases.

Direct optical techniques i.e. particle image velocimetry (Lindken et al.,1999; Deen, 2001; Broder and Sommerfeld, 2002) which are commonly usedin gas-liquid flows, are only applicable in bubbly flows at relatively low gashold-up.

Application of optical probes as an alternative technique for measurementsin multiphase flow offers the advantages of low cost, simplicity of operationand relative straightforward interpretation of results. Optical probes can beused to measure local void fraction, bubble frequency, bubble velocity, time-average local interfacial area and mean bubble chord length. During the pastdecades, optical probes have been intensively utilized in multiphase flows(Jones and Delhaye, 1976; Boyer et al., 2002).

Multiple point probes have been proposed. Since such probes offer, inprinciple, the possibility to determine all three components of the bubble ve-locity. For instance, four-point optical probes were used to determine bubblevelocity and size (Guet et al., 2003; Luther et al., 2004).

Besides the ability of measuring bubble velocity, multiple point opticalprobes are able to recognize those bubbles with a direction of motion deviatingfrom the axial direction and thus, leading to reduced errors in axial componentof the bubble velocity.

The objective of the present study is to investigate performance of a four-point optical fibre probe with respect to the bubble velocity measurement.Furthermore, the influence of material properties of the liquid phase on theaccuracy of the velocity determination is also investigated.

For this purpose, single bubble experiments were conducted. A high speedcamera was used to record images of the approach of a bubble rising in a flatbubble column towards a four-point optical fibre probe and the subsequent

20

2.2. Experimental setup

piercing of the bubble by the tips of the probe. Meanwhile, signals generatedby the four-point optical fibre probe were recorded simultaneously. Imagestaken by the camera and signals generated by the four-point optical fibre probewere processed afterwards separately. Velocities from the image processingwere treated as reference velocities to evaluate the velocities obtained from thefour-point optical fibre probe. Five liquids with different material properties,including, viscosity, surface tension, density were used during measurements.

2.2 Experimental setup

The experimental setup mainly consists of a small flat bubble column (10 ×110 × 500 mm), an Imager Pro HS CMOS camera with 12 bit, 1024 × 1280pixel resolution, a four-point optical probe and a light source, which areschematically shown in Figure 2.1.

Figure 2.1: Sketch of experimental setup.

During the experiments, an air bubble was released from a small hole inthe center of the bottom plate with a diameter of approximately 1 mm. Therange of produced bubble sizes ranged from 3 to 5 mm.

21


The four-point optical probe was positioned at the top of the bubble col-umn, a few centimeters below the liquid surface. The signals of the four-pointoptical probe were read with a LabView program and stored onto a hard disk.

The high speed camera is mounted in front of the bubble column anda light source illuminated the bubble column from behind, thus employinga shadowgraphy technique. Shadow images of bubble and the four-pointoptical probe were recorded by the camera and stored on the computer. Theimage acquisition program is DaVis from LaVision. The applied field of viewof the camera is 700 × 1024 pixels.

The four-point optical probe has three tips of the same length that forman equivalent angle. The fourth tip is positioned in the center between theother three tips. The radial distance d = 0.5 mm and the vertical distance is∆s = 1.4 mm. The geometrical configuration of the probe tips is shown inFigure 2.2.

Figure 2.2: Geometry of four-point optical probe.

The principle of the optical probe is based on different refractive indicesof light in both the gas and the liquid phase. For instance, the signal of theoptical probe remains low when the probe is immersed in the liquid phase,whereas the signal increases to a high level in the gas phase due to totalreflection of light. With sufficient signal to noise ratio, the two phases can bedistinguished. Consequently, by using a certain separation method, the gasphase could be easily retrieved from the output signal.

Three different liquids and two aqueous mixtures with different mass frac-

22

2.3. Data processing

tion of glycerol were used during the measurements. The physical properties(at room temperature) of each of the selected liquids are listed in Table 2.1.

Table 2.1: Physical properties of employed liquids at room temperature.

Density Viscosity Surface tension[kg/m3] [mPa·s] [mN/m]

Decane 730 0.838 23.37Diethylene glycol 1120 30.2 44.7Glycerol(60%) 1153 10.7 66.9Glycerol(72%) 1187 27.6 66.5Water 998 1.00 72.75

2.3 Data processing

2.3.1 Digital image processing

The intensity images taken by the camera were processed with MatLab. Inthe image, a small region containing the bubble was cropped and then, animage segmentation method was employed within this small region to dis-tinguish the bubble contours. Image segmentation was based on the methodof Otsu (1979), which is a nonparametric and unsupervised method of auto-matic threshold selection for picture segmentation. An optimal threshold wasselected by the discriminant criterion, namely, maximizing the separability ofthe resultant classes in gray levels. A brief description of the method is givenbelow:

An image consists of L gray levels [1, 2, ..., L] and the number of pixels atgray level i are denoted by ni. In addition, the total number of pixels N isobtained by summing ni over all levels, N = n1 + n2 + ... + nL. The fraction ofpixels at level i is given by:

pi = ni/N, pi ≤ 0 (2.1)

L∑i=1

pi = 1 (2.2)

23


Suppose the pixels are dichotomized into two classes C0 and C1 (back-ground and objects, or vice versa) by a threshold at level k. C0 denotes pixelswith levels [1, ..., k], and C1 denotes pixels with levels [k + 1, ..., L]. The prob-abilities of class occurrence, ω and the class mean levels, µ, respectively aregiven by

ω0 = P(C0) =k∑

i=1

pi = ω(k) (2.3)

ω1 = P(C1) =L∑

i=k+1

pi = 1 − ω(k) (2.4)

and

µ0 =

k∑i=1

iP(i|C0) =k∑

i=1

ipi/ω0 =µ(k)ω(k)

(2.5)

µ1 =

L∑i=k+1

iP(i|C0) =L∑

i=k+1

ipi/ω1 =µT − µ(k)1 − ω(k)

(2.6)

The total mean level of the original picture is given by:

µT = µ(L) =L∑

i=1

ipi (2.7)

The between-class variance is given by:

σ2B = ω0(µ0 − µT)2 + ω1(µ1 − µT)2 = ω0ω1(µ1 − µ0)2 (2.8)

The threshold k∗ that maximizes σ2B is treated as optimal threshold to segment

the objects from the background in the image.Sezgin and Sankur (2004) compared some image thresholding techniques

and evaluated their performance quantitatively. By using several thresh-olding performance criteria, Otsu’s method performed excellent for imagesegmentation within 40 image thresholding methods. Sezgin and Sankur(2004) also reported that Otsu’s method can give satisfactory results when thenumber of pixels in each class are close to each other.

Properties of the bubble, such as area, centre of mass, perimeter, equivalentdiameter, orientation, major axis length, minor axis length and so on, can be

24


calculated through manipulating the pixels representing the bubble in theimage. After this step, the co-ordinates of the centre of mass obtained fromthe cropped region were transformed back to the coordinates system of theoriginal image.

Subsequently, the vertical component of the bubble velocity was calculatedfrom the following equation in a sequence of images:

vi =zi+1 − zi

∆t(2.9)

where z is the vertical co-ordinate of the centre of mass of the bubble and ∆tis the time difference between two subsequent images.

Three characteristic velocities were calculated separately. One is the aver-age velocity before the bubble hits the probe, which will be termed the terminalvelocity. The second characteristic velocity is the velocity at the moment ofinteraction between the bubble and the longest tip of the optical probe. Thethird one is the average velocity during the bubble-probe contact, which willbe termed the piercing velocity. These three velocities will be used as a basisfor comparison with the velocity obtained from the four-point optical probe.

2.3.2 Signal processing of optical probe

The main task of the signal processing of the optical probe data is to identifythe parts representing the bubble in the raw signal. That is, the moments ofprobe entry and exit through each bubble need to be found in each pulse. Theidea to find the relevant moments is based on different levels in the raw datasignal, i.e. the liquid level, the gas level and the noise level (Cartellier, 1992;Barrau et al., 1999; Harteveld, 2005). “Pre-signals” were observed in somebubble piercing events, i.e. as shown in Figure 2.6(a). In viscous liquids,such as glycerol solutions, the possible reason for these pre-signals is thatthe bubble surface is perpendicular as it approaches the tip. This leads todetectable reflections by the surface prior to piercing. In low viscosity liquids,wobbling behavior of bubbles with a moderate size may also result in theoccurrence of a pre-signal. Therefore, a criterion based on the noise level todetermine the entry time is debatable. Harteveld (2005) suggested that the

25


threshold should be set to 10% of the bubble plateau level. However, it is hardto remove pre-signals in this way if the signal-to-noise ratio of the probe islow.

In the present study, the entry and exit moments were determined on basisof the following equations:

Ventry = VL + 0.1(VG − VL) (2.10)

Vexit = VL − 0.1(VG − VL) (2.11)

and thus identical to the threshold used by Harteveld (2005). In some caseswhere a pre-signal occurs, additional processing was employed. The entrypoint was selected accordingly at the very beginning of the ascending rampafter the primary peak.

Figure 2.3: Signals and corresponding time differences.

Based on above steps, the moments of probe entry and exit can be deter-mined for each bubble. With such information, the time difference betweenthe upper surface of a bubble hitting the longest tip and the other three tipscan be derived as show in Figure 2.3.

The velocity of bubble was calculated from the following equation:

u =∆s∆t=

∆s13

∑∆ti, i = 1, 2, 3 (2.12)

26


where ∆s is the vertical distance between the longest tip and the other threetips, whereas ∆ti is the time difference between hitting of the longest tip andshort tip i. Due to the dynamics of a bubble during the interaction with theprobe, these three time intervals may vary from one another. This is amongstothers due to the bubble hitting the probe off-centre and irregular deformationof the upper bubble surface during the interaction. To reduce the error in thebubble velocity determination, the following selection criterion was used:∣∣∣∣∣∆ti − ∆t

∆t

∣∣∣∣∣ < β, i = 1, 2, 3 (2.13)

Mudde and Saito (2001) and Fortunati et al. (2002) studied the influence ofthis criterion on the accuracy of the bubble velocity both numerically and ex-perimentally. In Mudde’s simulations, in case 8% < β < 20% the influence onthe accuracy is negligible in comparison to other error sources. In Fortunati’sexperiment, no significant effect was observed on the average rise velocityand its standard deviation in case β < 25%. In the present single bubbleexperiments, β = 30% was adopted in order to allow more bubbles to be con-sidered. The choice of this selection criterion will be discussed in more detailin chapter 3. Moreover, the chord length of bubble can then be determinedfrom the bubble velocity u and the time span for the bubble travelling throughthe optical probe t, which will also be introduced in the following chapter.

2.3.3 Uncertainty

During image processing, the uncertainty in the instantaneous bubble veloc-ity is related to the image acquisition rate, the determination of geometricalproperties (i.e. center of mass of the bubble) and applied magnification factor.The magnification factor is obtained by taking the diameter of the probe sup-port as a reference length. The DaVis software is used to acquire images andthe time interval between sequential images is very accurate. The accuracyof geometrical properties of the bubble, such as, centre of mass of the bubble,bubble diameter and so on, mainly depends on the performance of the imagesegmentation method provided that the acquired image has good contrastbetween bubble and its background.

27


Due to inaccuracies in the manufacturing of the probe, the length of thethree short tips of the probe may not be exactly the same. The distancesbetween the longest tip and the other three short tips differ slightly. Fur-thermore, the three time differences used to calculate the bubble velocity alsovaried because of unpredictable interaction between the bubble and the four-point optical probe. These two influences were considered to analyze theuncertainty in the bubble velocity obtained from the four-point optical probe.

2.4 Results and discussion

2.4.1 Interaction between bubble and probe

Due to different physical properties of the liquids, the behavior of the bubblewhen approaching the tips of the probe differs. Particularly, the viscosity ofthe liquid influences the bubble shape significantly.

There are two main different interactions between the bubble and the four-point optical probe in the liquid. For instance, a rising bubble in a viscousliquid keeps its shape after being released from the bottom of the column andonly deforms when it gets pierced by the probe . On the contrary, however, itis hard to determine if the deformation of the bubble is due to the interactionwith the probe in liquids with low viscosity, such as decane and water. In suchliquids, bubbles start wobbling as soon as they are injected into the liquid andthe shape of the bubbles changes continuously.

A set of images depicting an air bubble rising in an aqueous mixture with72% of glycerol are shown in Figure 2.4. The bubble retains an ellipsoidalshape before encountering the probe. It can clearly be seen that the shape ofthe rising bubble changes during interaction with the tips of the optical probe.The roof of the bubble becomes flatter as it hits the probe. The retarding effectof the probe is more obvious for bubbles using in viscous liquids.

In Figure 2.5, continuous movements of a bubble rising in water arerecorded. The shape of the bubble changes continuously during the entirepath in the column and it appears that the probe has almost no effect on thebubble shape. Furthermore, the retarding effect of the probe on the bubbleis negligible. This retarding effect of liquids with different viscosity on rising

28

2.4. Results and discussion

(a) t = 0.000 s (b) t = 0.008 s (c) t = 0.016 s

Figure 2.4: Rising bubble in a glycerol solution (72%).

bubbles will be discussed in the following sections.Furthermore, the signals of the probe change when the surface of the

bubble passes through the tips of the probe and the processing programfinds the bubble according to these changes of the signals. Therefore, thewobbling behavior of bubbles in low viscous liquid may result in inaccuratedetermination of bubble velocities.

(a) t = 0.000 s (b) t = 0.006 s (c) t = 0.012 s

Figure 2.5: Rising bubble in water.

29


2.4.2 Bubble rise velocity

The signals of the probe corresponding to the above two bubble-probe inter-actions (Figure 2.4 & Figure 2.5) are plotted in Figure 2.6. The times corre-sponding to tips of the probe entering the bubble can be extracted from thesignals according to the signal processing method mentioned above. Thus,the average time difference between the central tip and the other three tipsand thus the velocity of the bubble can be determined by knowing the verticaldistance between the central tip and other tips. In the meantime, since thetrajectory of the bubble is recorded by the high speed camera, the bubblevelocity can also be calculated from the image processing method.

Figures 2.7 and 2.8 show the instantaneous positions and velocities thatwere recorded with the camera and velocity determined from the opticalprobe. Note that in the plots, the origin of coordinates is defined at the bot-tom of the central tip. Due to effects of physical properties, such as viscosity,surface tension and the size of the bubble, the trajectories of the bubble inthe low viscosity liquid are different from those of the bubble in the viscousliquid. In the low viscosity liquids, the bubble positions show more fluctua-tions compared to bubbles rising in viscous liquids because of the wobblingbehaviour, which is reflected in the corresponding velocity plots. Accordingto both figures, however, it can be found that the vertical component of thebubble position increases approximately linearly with time. Therefore, theinstantaneous velocities prior to piercing can be used to calculate the averageterminal bubble velocity in liquids of both low and high viscosity.

In viscous liquid, the ellipsoidal shape of the bubble is affected duringthe interaction with the probe. As a result, the roof of the bubble starts tobecome flat slowly. Meanwhile, the bubble is decelerated which is reflectedby the short curved part of plots of the relative position of the bubble inFigure 2.7(a). Consequently, the vertical velocity of the bubble decreases.The retarding effect of the probe in viscous liquid is obvious. The bubblevelocity from the optical probe is much smaller than the average bubblevelocity before approaching the probe, which means that the measurement ofthe bubble velocity with the optical probe may induce considerable errors inviscous liquids.

30


Tip0

Tip1

Tip2

Tip3

Signal [Voltage]

-6

-4

-2

0

Relative time [s]-0.05 0 0.05 0.1

(a) Glycerol solution (72%)

Tip0

Tip1

Tip2

Tip3

Signal [Voltage]

-6

-4

-2

0

Relative time [s]-0.02 -0.01 0 0.01 0.02 0.03 0.04

(b) Water

Figure 2.6: Example signals of the optical probe.

Relative position [m]

-0.02

-0.01

0

Time [s]0 0.05 0.1 0.15

(a) Relative position vs. time

From image

From probe

Velocity [m/s]

0.1

0.15

0.2

Relative position [m]-0.02 -0.01 0 0.01

(b) Vertical velocity vs. relative position

Figure 2.7: Air-glycerol solution (72%) system.

However, the optical probe has little retarding effect on bubbles risingin low viscosity liquid (i.e. water). It is hard to distinguish changes of theinstantaneous bubble velocity due to the presence of the probe from thosedue to the wobbling behavior of the bubble. It is also found that the bubblevelocity from the optical probe is within the range of fluctuation of the bubblevelocity.

31


Relative position [m]

-0.02

-0.01

0

Time [s]0 0.05 0.1

(a) Relative position vs. time

From image

From probe

Velocity [m/s]

0.15

0.2

0.25

Relative position [m]-0.02 -0.01 0

(b) Vertical velocity vs. relative position

Figure 2.8: Air-water system.

2.4.3 Comparison of bubble velocities

The bubble velocities obtained from both the high speed camera and theoptical probe are compared with the use of parity plots. The terminal bubblevelocities obtained from image processing are used to compare with thoseobtained from the optical probe (Figure 2.9). The terminal bubble velocityis determined by averaging the instantaneous bubble velocities before thebubble approached the optical probe. Since the field of view of the camera isfar away from the bubble injector, the bubble is believed to rise at its terminalvelocity in the region of interest. This can also be motivated from the plots ofthe bubble displacement versus time (i.e. Figure 2.7(a) & 2.8(a)). However, thepresence of the optical probe in the path of the rising bubble seems to obstructthe rising bubble. For instance, it is obvious that the bubble velocity reducesslightly when approaching the probe in Figure 2.7(b). Therefore, in orderto evaluate the performance of the optical probe, the bubble velocity at themoment of interaction with the longest tip of the optical probe is also used as abasis for comparison with results obtained from the optical probe (Figure 2.10).In addition, the velocity obtained from the four-point optical fibre probe is theone which is neither the terminal velocity nor the instantaneous velocity of thebubble at the moment that the roof of the bubble touches the longest tip of theprobe. This velocity is actually measured during the deceleration of the bubble

32


during the contact with the probe. Hence, a comparison between the velocityobtained from the probe and an average velocity during the interaction withthe probe obtained from the high speed camera is presented as well, as shownin Figure 2.11.

Diethylene glycol

Glycerol (60%)

Glycerol (72%)

Optical probe

0.05

0.1

0.15

0.2

0.25

HS camera0.05 0.1 0.15 0.2 0.25

(a) Viscous liquids

Decane

Water

Optical probe

0.1

0.15

0.2

0.25

HS camera0.1 0.15 0.2 0.25

(b) Low viscosity liquids

Figure 2.9: Parity plots of terminal bubble velocities.

The straight lines in the parity plots are identity. In Figure 2.9, it can be seenthat the presence of the four-point optical probe has a considerable retardingeffect on the bubble velocity in viscous liquids. A large viscosity of the liquidcauses deceleration of the bubble movement along the tip. Accordingly, inac-curacies arise during bubble velocity measurements using the optical probe.For instance, the average deviation between the terminal bubble velocity andvelocity measured with the probe is about 45% in diethylene glycol. In theglycerol solution, the deviation is around 33%. In the glycerol solution with60% glycerol, the deviation is about 29%.

However, the terminal bubble velocities in low viscosity liquids are muchcloser to those obtained with the camera. Meanwhile, one can find thatthere is significant scatter of the bubble velocity measured by the opticalprobe. The wobbling behavior of the air bubble in low viscosity liquidsinduces some difficulties for measuring the bubble velocity. The shape of thebubble is already changing continuously when it hits the tips. The velocitymeasurement by the probe is strongly influenced by the dynamics of the

33


bubble surface . On the contrary, the obstructing effect of the probe on therising bubble is not so significant. The average deviation between the terminalbubble velocity and the one measured with the probe is about 18% in decaneand 20% in water.

Diethylene glycol

Glycerol (60%)

Glycerol (72%)

Optical probe

0.05

0.1

0.15

0.2

0.25

HS camera0.05 0.1 0.15 0.2 0.25

(a) Viscous liquids

Decane

Water

Optical probe

0.05

0.1

0.15

0.2

0.25

HS camera0.05 0.1 0.15 0.2 0.25


Figure 2.10: Parity plots of instantaneous bubble velocities.

In Figure 2.10(a), it can be seen that points in the plot are much closerto the identity compared with the points in Figure 2.9(a). This can also beseen from the average deviations. The average deviation between the instantbubble velocity and that obtained from the optical probe in diethylene glycolreduces from 45% to 40%. In the two glycerol solutions, the average deviationalso decreases. In glycerol solution (72%), it is about 25%, whereas in theglycerol solution with 60% glycerol, the deviation is 25%. However, there isno difference in the average deviation in the two low viscosity liquids.

According to the comparison shown in Figure 2.11(a), one can find thatthe discrepancy between the velocity obtained from the optical probe andthe average velocity during the interaction determined from the high speedcamera gets smaller. For instance, the average deviation between the bubblevelocity obtained from the optical probe and the average velocity during theinteraction from image processing in diethylene glycol is 34%. In the glycerolsolutions, the average deviations for the two velocities are 18% in the glycerolsolution (60%) and 19% in the glycerol solution (72%) respectively.

34


Diethylene glycol

Glycerol (60%)

Glycerol (72%)

Optical probe

0.05

0.1

0.15

0.2

0.25

HS camera0.05 0.1 0.15 0.2 0.25

(a) Viscous liquids

Decane

Water

Optical probe

0.05

0.1

0.15

0.2

0.25

HS camera0.05 0.1 0.15 0.2 0.25


Figure 2.11: Parity plots of bubble velocities during the interaction.

2.4.4 Analysis of the probe deviation

In order to characterize the performance of the four-point optical fibre probe,the average deviations between the high speed camera and the probe arecompared in term of the Morton number (Table 2.2). Note that σ1 denotes theaverage deviation between the terminal bubble velocity and the velocity de-termined from the optical probe, σ2 represents the average deviation betweenthe instantaneous bubble velocity and that from the probe and σ3 representsthe deviation between the average velocity during the interaction from thehigh speed camera and that obtained from the optical probe.

Table 2.2: Morton numbers and average deviations.

Morton number σ1 [%] σ2 [%] σ3 [%]

Decane 5.18 × 10−10 18 18 16Diethylene glycol 8.11 × 10−5 45 40 34Glycerol (60%) 3.69 × 10−7 29 25 18Glycerol (72%) 1.64 × 10−5 33 25 19Water 2.57 × 10−11 20 20 18

Velocities measured in diethylene glycol with the probe have the highestdeviation within these five liquids. Whereas, the deviation measured in

35


decane is the lowest one. The order of magnitude of the Reynolds numberis about 103 in water and about 102 in decane. In these cases, the influenceof viscous forces is much smaller than the inertial force acting on the bubble.Therefore, viscous forces do not dominate in the range of the investigatedReynolds numbers. As a result the probe has little influence on the motion ofthe bubble in these cases. The deviation in water is slightly larger than thatin decane. This could be due to the larger surface tension of water comparedto that of decane. A large surface tension tends to prevent the bubble frombeing pierced.

In measurements with viscous liquids, the order of magnitude of theReynolds number is about 10. In this situation, viscous forces have a sig-nificant influence on the motion of the bubble. Therefore, the deviations ofbubble velocity between high speed camera and the optical probe are larger.

2.5 Conclusions

This chapter was dedicated to investigate the performance of a four-pointoptical probe with respect to the determination of bubble velocity in a flatbubble column and the influence of physical properties on the accuracy of thefour-point optical probe. For this purpose, a combined experiment was de-signed. A CMOS camera was used to verify the performance of the four-pointoptical probe. Five liquids with different physical properties were employed.

The motion of the bubble and the interaction with the optical probe wererecorded via the high speed camera. The signals of the probe were acquiredmeanwhile. Both terminal bubble velocity and instantaneous bubble velocityat the moment the bubble hits the central tip of the probe were comparedwith that from signal processing of the probe. The bubble shape remainsnearly constant before the bubble approaches the tips of the probe in viscousliquids, and the bubble only deforms when it gets pierced by the probe. Inlow viscosity liquids, however, the bubble starts wobbling and changes itsshape continuously during its rise in the column. Discrepancies of bubblevelocity between image processing and signal processing is more obvious inviscous liquids.

36

Nomenclature

Physical properties of the liquid, such as viscosity and surface tension,influence the performance of the four-point optical probe. The Reynoldsnumber and Morton number were combined to describe the extent of thediscrepancy between the bubble velocity obtained from image processingand that from the four-point optical probe. In the viscous dominant regime(low Re), the average deviation is large when the Morton number increases.When the viscosity does not dominate the motion of the bubble (high Re), theMorton number is low and the surface tension will influence the accuracy inthe velocity measurement of the probe.

In short, by applying a four-point optical fibre probe in a bubble column,inaccuracies with respect to bubble velocities cannot be avoided. Particularly,the discrepancy becomes large when the viscosity of the liquid increases.However, based on the fact of the stable behavior of bubbles rising in a viscousliquid, correction for velocity obtained from the four-point optical fibre probemay be possible.

Furthermore, in low viscosity liquid systems (i.e. high Reynolds flows),this inaccuracy is lowest. For high void fraction bubbly flow in a bubblecolumn at low liquid velocities, where PIV or LDA may not work ideally, thefour-point optical probe would also be an appropriate option.

Nomenclature

p fraction of pixels, [-]P probability, [-]∆s distance between the central tip and the other three tips, [m]∆t time difference, [s]t time of individual bubble traveling through tip of the probe, [s]u bubble velocity from four-point optical fibre probe, [m s−1]v bubble velocity from image processing, [m s−1]V voltage, [V]z vertical position of bubble, [m]

37


Greek letters

β bubble selection criterion, [-]µ class mean, [-]ω class occurence, [-]σ variance, [-]

Indices

0, 1 class, [-]B between-class, [-]T total, [-]entry probe entering bubble, [-]exit probe exiting bubble, [-]

References

E. Barrau, N. Riviere, Ch. Poupot, and A. Cartellier. Single and double opticalprobes in air-water two-phase flows: Real time signal processing and sensorperformance. International Journal of Multiphase Flow, 25(2):229–256, 1999.

C. Boyer, A. -M. Duquenne, and G. Wild. Measuring techniques in gas-liquid and gas-liquid-solid reactors. Chemical Engineering Science, 57(16):3185–3215, 2002.

D. Broder and M. Sommerfeld. An advanced LIF-PLV system for analysingthe hydrodynamics in a laboratory bubble column at higher void fractions.Experiments in Fluids, 33(6):826–837, 2002.

A. Cartellier. Simultaneous void fraction measurement, bubble velocity, andsize estimate using a single optical probe in gas-liquid two-phase flows.Review of Scientific Instruments, 63(11):5442–5453, 1992.

N. G. Deen. An experimental and computational study of fluid dynamics in gas-liquid chemical reactors. PhD thesis, Aalborg University, Denmark, 2001.

38

References

R. Fortunati, S. Guet, G. Ooms, R. V. A. Oliemans, and R. F. Mudde. Accuracyand feasibility of bubble dynamic measurement with four point optical fiberprobes. Lisbon, July 2002. 11th symposium on application of laser techniqueto fluid mechanics.

S. Guet, R. V. Fortunati, R. F. Mudde, and G. Ooms. Bubble velocity andsize measurement with a four-point optical fiber probe. Particle and ParticleSystems Characterization, 20(3):219–230, 2003.

W. K. Harteveld. Bubble columns: Structure or Stability? PhD thesis, DelftUniversity of Technology, The Netherlands, 2005.

O. C. Jones and J. -M. Delhaye. Transient and statistical measurement tech-niques for two-phase flows: A critical review. International Journal of Multi-phase Flow, 3(2):89–116, 1976.

R. Lindken, L. Gui, and W. Merzkirch. Velocity measurements in multiphaseflow by means of particle image velocimetry. Chemical Engineering andTechnology, 22(3):202–206, 1999.

S. Luther, J. Rensen, and S. Guet. Bubble aspect ratio and velocity measure-ment using a four-point fiber-optical probe. Experiments in Fluids, 36(2):326–333, 2004.

R. F. Mudde and T. Saito. Hydrodynamical similarities between bubble col-umn and bubbly pipe flow. Journal of Fluid Mechanics, 437:203–228, 2001.

N. Otsu. Threshold selection method from gray-level histograms. IEEE TransSyst Man Cybern, SMC-9(1):62–66, 1979.

M. Sezgin and B. Sankur. Survey over image thresholding techniques andquantitative performance evaluation. Journal of Electronic Imaging, 13(1):146–168, 2004.

39

Chapter

3Bubble properties of heterogeneous

bubbly flows in bubble column

This chapter focuses on the measurements of bubble properties in heterogeneous bubblyflows in a square bubble column. A four-point optical fibre probe was used for thispurpose. The accuracy and intrusive effect of the optical probe was investigated first.The results show that the optical probe underestimates bubble properties, such as,bubble velocity and local void fraction. The presence of the probe in the bubble columninfluences the local flow conditions. Particularly, when the probe is placed close to theliquid surface, this influence is more pronounced. Furthermore, two methods for thedetermination of the interfacial area were compared. The results from both methodsagree quite well at low superficial gas velocity, whereas significant discrepancies wereobtained at high superficial gas velocity. Finally, the effect of (initial) liquid heighton bubble properties was studied. No significant difference was found for the threeinvestigated (initial) liquid heights.

41

3. Bubble properties of heterogeneous bubbly flows

3.1 Introduction

Bubble column reactors are intensively utilized as multiphase contactors andreactors in chemical, petrochemical, biochemical and metallurgical industries.Typically these reactors have a rather simple geometry: a container filled withliquid where gas is introduced via a sparger at the bottom in the form ofbubbles which comes in contact with the liquid. Simple construction andlack of any mechanically operated parts are two characteristic aspects of thesereactors. Hence, little maintenance is required at low operating costs.

Bubble column reactors have excellent heat and mass transfer character-istics. In gas-liquid reactors, mass transfer between the gas phase and liquidphase constitutes an important goal of the process. The volumetric masstransfer coefficient is a key parameter in the characterization and design ofindustrial reactors. Moreover, a high heat transfer coefficient is of specialsignificance when reactions are involved with large heat effects.

Although the construction of the bubble column reactor is simple, accu-rate and successful design and scale-up of bubble column reactors requiresdetailed understanding of multiphase fluid dynamics and its impact on trans-port of mass and/or heat. Bubble characteristics, such as bubble size, voidfraction and rise velocity, have significant impact on the hydrodynamics, aswell as on heat and mass transfer coefficients in a bubble column reactor.

Optical fibre probes (Cartellier, 1990, 1992; Mudde and Saito, 2001) offer theadvantage of low cost, simplicity of setup and relative ease of interpretationof the probe signals. With a single optical fibre probe, the measurement of thelocal void fraction can be conducted. Furthermore, using a four-point opticalfibre probe, bubble characteristics, such as bubble velocity and bubble size,can also be obtained. Moreover, it is also possible to determine the specificinterfacial area using a four-point optical fibre probe.

In this chapter, bubble properties in a square bubble column were mea-sured with a four-point optical fibre probe over a relatively large range ofsuperficial gas velocities (0.005 m/s − 0.1 m/s). The flow regime changesfrom homogeneous to heterogeneous. Measurements were taken using threecolumns with different (initial) liquid heights (H/W = 3, 6 and 9).

42

3.2. Experimental setup

3.2 Experimental setup

The experiments were carried out in a square column with a cross-sectionalarea (W × D) of 0.15 × 0.15 m2. A perforated plate with 49 holes of 1 mm indiameter was used as a gas sparger at the bottom of the column. These holeswere distributed uniformly at a square pitch of 6.25 mm in the central regionof the perforated plate. Demineralized water was used as liquid phase andair as the gas phase.

A sketch of the experimental setup is shown in Figure 3.1. A four-pointoptical probe was inserted through one of the holes in the side walls. Theoptical probe could be moved in the horizontal direction to obtain measure-ments at different lateral positions (at fixed height). A differential pressuretransducer (DP) was used to measure the differential pressure in the column.

Figure 3.1: Sketch of experimental setup.

During the measurements, three different liquid heights (H/W = 3, 6and9)were investigated. The probe was moved in the middle plane horizontally.

43


For each measurement a sampling time exceeding thirty minutes was taken.

3.3 Data processing

3.3.1 Interpretation of the probe

Interpretation of signals to measure bubble velocities from the probe has beenintroduced in Chapter 2. In addition, the local void fraction can be extractedfrom the signals of the probe. It is expressed as the ratio between accumulatedtime that the probe tip is inside individual bubbles

∑t and the total sampling

time T (Cartellier, 1990):

α =

∑t

T(3.1)

As reported in the previous chapter, the bubble velocity u is determinedfrom the distance between the central tip and the other three tips and theaverage time difference between the instants that the bubble hits the centraltip and the other three tips. Once the bubble velocity u and the time span forthe bubble travelling through the optical probe t are known, it is possible tocalculate the chord length of the bubble dc.

dc = ut (3.2)

Due to the fact that many bubbles can be detected at each position withinthe sampling period, many bubble velocities and chord lengths will be ob-tained at that position for one measurement. For instance, in a small col-umn (H/W = 3), the probability density functions of bubble velocity andchord length at position x/W = 0.5, z/H = 0.75 for superficial gas velocitiesug = 0.005 m/s and ug = 0.04 m/s are respectively shown in Figure 3.2.

In order to obtain the average bubble properties at a certain position, theprobability density functions, such as f (u) and f (dc), are used to determinethe corresponding mean quantities:

u =∫ ∞

0u f (u) du (3.3)

dc =

∫ ∞

0dc f (dc) ddc (3.4)

44


f(u)

0

0.01

0.02

0.03

0.04

0.05

u [m/s]0 0.5 1 1.5

(a) Bubble velocity (ug = 0.005 m/s)

f(dc)

0

0.005

0.01

0.015

0.02

dc [m]0 0.002 0.004 0.006 0.008 0.01 0.012

(b) Chord length (ug = 0.005 m/s)

f(u)

0

0.05

0.1

0.15

0.2

u [m/s]0 1 2 3 4 5

(c) Bubble velocity (ug = 0.04 m/s)

f(dc)

0

0.05

0.1

0.15

0.2

0.25

dc [m]0 0.05 0.1

(d) Chord length (ug = 0.04 m/s)

Figure 3.2: Probability density functions.

The selection criterion has a strong effect on the fraction of “accepted“bubbles and on the accuracy of the results. As discussed in Chapter 2, theprobability density function of the bubble velocity does not change consid-erably by increasing β up to 25%. In this chapter, in order to obtain more”accepted” bubbles, a small test was done before adopting the selection crite-rion. When increasing β from 25% to 40%, the percentage of accepted bubblesincreases from about 20% to 37% accordingly. However, the resulting devia-tion of the mean bubble velocity and the mean bubble chord length is about2% and 1% respectively, which is acceptable for the present study.

During the measurements, at least 1500 bubbles selected from the criterion

45


mentioned above were used to calculate the average bubble velocity at eachlocation. Particularly, at high superficial gas velocity, more than 3000 bubbleswere selected, i.e. above 10 000 bubbles were used in case of superficial gasvelocity ug = 0.6 m/s.

3.3.2 Differential pressure transducer

By measuring the differential pressure ∆P over a certain height ∆z before andafter injecting gas into column, the integral gas holdup ε can be obtained fromthe following equation:

ε = 1 − ∆Pρlg∆z

(3.5)

It is assumed that the density of the gas phase is negligible compared to thatof the liquid phase.

3.3.3 Determination of bubble size

Bubble sizes cannot be determined from the optical probe directly since theoptical probe does not always intersect a bubble at its center. That is, achord length smaller than the largest vertical bubble dimension is typicallymeasured. In literature, some studies have been carried out to estimate theequivalent bubble size from the chord length distribution (Uga, 1972; Herringeand Davis, 1976; Thang and Davis, 1979). Based on some assumptions, suchas i) all bubbles are spherical and ii) all bubbles travel in the same directionwith the same average velocity, the equivalent bubble diameter de can bedetermined with the aid of geometrical probability analysis:

de =32

dc (3.6)

where dc is the mean chord length.Some studies also focused on inverse transformation from the chord length

distribution to bubble size distribution (Turton and Clark, 1988; Liu and Clark,1995; Liu et al., 1998) where an ellipsoidal or truncated ellipsoidal bubbleshape were under assumed. By assuming that bubbles rise vertically and

46


with a zero angle of attack, the mean horizontal diameter of a bubble withaspect ratio E (Liu and Clark, 1995) can be obtained from:

dH =32

E−1dc (3.7)

Therefore, the equivalent diameter of the bubble is:

de =32

E−2/3dc (3.8)

Once the aspect ratio of the bubble E is known, one is able to determine theequivalent diameter of the bubble. However, it may not be easy to determinethe aspect ratio in high void fraction flows. Due to the phenomena of coa-lescence and breakup of bubbles, the bubble size varies and thus, the bubbleshape varies. Hence, assumptions with respect to the shape of the bubblesmay lead to (considerable) inaccuracies in bubble size.

The local specific interfacial area is related to the local void fraction andthe local equivalent bubble size:

de =6αa

(3.9)

where α is the local void fraction.This provides an alternative means of estimating the equivalent bubble

size if the specific interfacial area is known. In the next section, we willintroduce techniques for measuring the specific interfacial area using a four-point optical fibre probe.

3.3.4 Specific interfacial area

Delhaye and Bricard (1994) pointed out that the structure of a bubbly two-phase flow can be characterized by two out of the three following parameters:the interfacial area, the void fraction, and the Sauter mean diameter. In addi-tion, in two-fluid models for two-phase flows, transport of mass, momentumand energy across the phase boundaries is represented by interfacial transferterms. In general, all interfacial transfer rate can be expressed as the productof a driving force and the inverse of a length scale L at the interface (Ishii,1975):

47


Interfacial transfer rate = 1L × driving force

The driving forces are characterized by local transport mechanisms such asmolecular and turbulent diffusion, whereas the term 1/L represents the time-average of the interfacial area per unit of volume, i.e. the specific interfacialarea a, and is related to the structure of the two-phase.

Methods of measuring local specific interfacial area have been proposedfor both a two-point and four-point probe (Kataoka et al., 1986; Tan and Ishii,1990; Revankar and Ishii, 1993; Shen et al., 2005). For a detailed study of theinterfacial area definition we refer to Ishii (1975). A brief description of themethod for a four-point probe is as follows (Tan and Ishii, 1990).

The time-average of the specific interfacial area at certain position x0 isgiven by:

a =1T

N∑j=1

1∣∣∣v0 j · n j

∣∣∣ (3.10)

where N is the number of times over the sampling time T that an interfacepasses through x0 where v0 and n represent the interface velocity and unitnormal vector of the interface respectively at x0.

Let f (x, t) = 0 represent an interface. The jth interface passes throughx0 at time t = t0 j and can be represented as where f j(x0, t0 j) is assumed tosufficiently differentiable, f j(x0, t0 j) = 0.

a =1T

N∑j=1

∣∣∣O f j(x0, t0 j)∣∣∣∣∣∣∣∣∂ f

∂t(x0, t0 j)

∣∣∣∣∣ (3.11)

Suppose that the interface passes through three adjacent positions x1, x2

and x3 at times t1, t2 and t3 respectively. That is, f j(xk, tkj) = 0, k = 1, 2, 3.If the distances sk = |xk − x0|, k = 1, 2, 3 and the time differences ∆tkj = tkj −t0 j, k = 1, 2, 3 are small in comparison with the length scale and the time scale,respectively, of the system under consideration, then each of f j(x j, tkj), k =1, 2, 3 can be written as a Taylor series expansion about x = x0 and t = t0 j:

f j(xk, tkj) = f j(x0, t0 j) + skO f j(x0, t0 j) · ξk + ∆tkj∂ f∂t

(x0, t0 j) k = 1, 2, 3 (3.12)

48


where O f j(x0, t0 j) ·ξk denotes the directional derivative of f j in the direction ofthe unit vector ξk which is parallel to the line passing through x0 and xk.

Assume the unit vector ξk, k = 1, 2, 3 is linearly independent and in termsof the Cartesian components ξkx, ξky and ξkz of ξk the direction cosines cosα j,cosβ j and cosγ j of nj can be obtained:

cosα j = −

∂ f j

∂t(x0, t0)∣∣∣O f j(x0, t0 j)

∣∣∣ A1 j

A0(3.13)

cosβ j = −

∂ f j

∂t(x0, t0)∣∣∣O f j(x0, t0 j)

∣∣∣ A2 j

A0(3.14)

and

cosγ j = −

∂ f j

∂t(x0, t0)∣∣∣O f j(x0, t0 j)

∣∣∣ A3 j

A0(3.15)

where

A0 =

∣∣∣∣∣∣∣∣∣ξ1x ξ1y ξ1z

ξ2x ξ2y ξ2z

ξ3x ξ3y ξ3z

∣∣∣∣∣∣∣∣∣ (3.16)

A1 j =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∆t1 j

s1ξ1y ξ1z

∆t2 j

s2ξ2y ξ2z

∆t3 j

s3ξ3y ξ3z

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(3.17)

A2 j =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ξ1x∆t1 j

s1ξ1z

ξ2x∆t2 j

s2ξ2z

ξ3x∆t3 j

s3ξ3z

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(3.18)

49


and

A3 j =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ξ1x ξ1y∆t1 j

s1

ξ2x ξ2y∆t2 j

s2

ξ3x ξ3y∆t3 j

s3

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(3.19)

It follows that the identity cos2α + cos2β + cos2γ = 1 and from equation 3.11that a is given by:

a =1

T|A0|

N∑j=1

√A2

1 j + A22 j + A2

3 j (3.20)

The above method makes it possible to measure the specific interfacialarea with the four-point optical fibre probe with minor (severe) assumptions.


3.4.1 Intrusive effect of the probe

There is no doubt that the optical probe, as an intrusive instrument, locallydisturbs the flow field and thus, the bubble properties obtained from theprobe. To investigate this effect and verify the feasibility of using the probe,an extra measurement was performed. The bubble properties at three differentdepths (y/D = 0.5, 0.75, and 0.875) were measured with the four-point opticalprobe. The small column (H/W = 3) was used. The optical probe was insertedthrough the side wall at nearly half height (z/H = 0.55) of the column. Twocases with different superficial gas velocities (ug = 0.02 m/s and ug = 0.1 m/s)were considered.

When the optical probe is used to obtain values for local void fractionα and mean bubble velocity u at position (x, y) in one horizontal plane, theoverall gas flow rate Q can be computed with the following formula and thus,compared with that from the flow meter to verify the accuracy of the opticalprobe.

Q =∫

Aα(x, y)u(x, y) dA (3.21)

50


where A is cross-sectional area of the column.By assuming symmetry of the flow, the total gas flow rate is obtained from

calculating the gas flow rate over half the area of the cross section and/or aquarter of the cross sectional area.

Furthermore, the surface-averaged void fraction is calculated as follows:

α =1A

∫Aα(x, y) dA (3.22)

The surface-averaged void fractions are compared with the gas holdupsobtained from the measurements using the differential pressure transducer.The comparison for both gas flow rate and gas holdup is given in Table 3.1.

Table 3.1: Comparisons of both gas flow rate and gas holdup.

Method ug = 0.02 m/s ug = 0.1 m/s

Q [m3/s] α [-] Q [m3/s] α [-]

Measured 4.49 × 10−4 0.026 2.25 × 10−3 0.11Probe, A/2 2.98 × 10−4 0.023 1.53 × 10−3 0.08Probe, A/4 3.13 × 10−4 0.024 1.58 × 10−3 0.09

It can be seen that the gas flow rates obtained from the optical probeare less than those from the flow meter. The gas flow rate from the opticalprobe differs about 30% from that obtained with the flow meter. The surface-averaged void fraction from the optical probe has a much smaller deviationcompared with results from the differential pressure transducer.

The possible reasons for the observed discrepancies may be various. Be-sides the assumption of symmetry of flows inside column, the processingmethod of the optical probe rules out many bubbles that are rising inside thecolumn at a large angle of attack and that do not hit the four tips of the probe.Furthermore, the intrusive effect of the probe is also one of the reasons. Inspite of the discrepancies, the optical fibre probe can still be used in two-phasebubbly flows to provide meaningful data.

51


3.4.2 Bubble size

The bubble size is calculated on basis of the two methods described in sec-tion 3.3.3. For the purpose of simplicity, equation 3.6 is used to calculatebubble size from the chord length distribution (method A), since the bubbleshape is difficult to determine. Furthermore, equation 3.9 is used to determinethe bubble size from the measurement of the specific interfacial area (equa-tion 3.20) (method B). Comparisons are made in a tall column (H/W = 6) withfour different superficial gas velocities (ug = 0.005, 0.04, 0.06, 0.08and0.1 m/s).The results are shown in Figure 3.3.

de [m]

0.01

0.015

0.02

x/W [-]0 0.2 0.4 0.6 0.8 1

ug =0.005m/s, Eqn. 3.6ug=0.04m/s, Eqn. 3.6 ug=0.06m/s, Eqn. 3.6 ug=0.08m/s, Eqn. 3.6 ug=0.1m/s, Eqn. 3.6

ug=0.005m/s, Eqn. 3.9ug=0.04m/s, Eqn. 3.9ug=0.06m/s, Eqn. 3.9ug=0.08m/s, Eqn. 3.9ug=0.1m/s, Eqn. 3.9

Figure 3.3: Bubble size distribution (z/H = 0.68, H/W = 6).

It can be seen that the method A agrees well with the method B at lowsuperficial gas velocity (ug = 0.005 m/s). However, discrepancies arise. Thebubble sizes from method A are much larger than those from method B. Thatis reasonable, since the shape of the bubble may no longer be spherical due tostrong coalescence of bubbles. Therefore, method B will be used for furtherdiscussions.

52


3.4.3 Effect of initial liquid height

The effect of initial liquid height on the bubble properties, such as bubblevelocity, local void fraction, interfacial area and equivalent diameter, will nowbe discussed. Measurements were taken at height z = 0.34 m in the middleplane (y/D = 0.5) for three liquid heights (H/W = 3, 6 and 9) at superficial gasvelocity ug = 0.005 m/s.

(a) Mean bubble velocity [m/s] (b) Local void fraction [-]

(c) Specific interfacial area [1/m] (d) Equivalent bubble size [m]

Figure 3.4: Distributions of bubble properties (ug = 0.005 m/s).

From Figure 3.4, one can find that there are only small differences inthe bubble properties among these three liquid heights. In Figure 3.5, bub-ble properties at higher superficial gas velocity are compared for two liquidheights (H/W = 3 and 6).

One can see that the bubble properties show little difference. Hence,

53



(c) Specific interfacial area [1/m] (d) Equivalent bubble size [m]

Figure 3.5: Distributions of bubble properties at higher superficial gas veloc-ities.

one may conclude that the initial liquid height barely influences the bubbleproperties. However, it is more obvious that the presence of the optical probein column disturbs the local flow. The optical probe is inserted through thewall at the right side of the column. The bubble properties at the right half partof the distributions, such as the mean bubble velocity, the local void fractionand the specific interfacial area, are smaller than those at the left half part ofthe distributions. The closer to the side wall the measurement point is, thelarger the difference is.

In addition, it can be seen that the profile of the mean bubble velocity atsuperficial gas velocity ug = 0.005 m/s is flat (see Figure 3.4). That is due tomeandering behavior of the bubble plume in the column. However, bubbleshave more probability to rise in the middle of the column than at the sides,which can be seen from the profiles of the local void fraction. The local

54


void fraction has a peak in the middle of the column. Furthermore, one canalso observe that the specific interfacial area has peak in the middle. Theequivalent bubble size is uniform throughout the horizontal direction in thecolumn.

As the superficial gas velocity increases, the meandering motion of thebubble plume changes to a circulation pattern. Most of the bubbles rise inthe middle of the column and some move downwards at the wall regiondue to liquid circulation. The profiles of the mean bubble velocity becomeparabolic. Moreover the mean bubble velocity, the local void fraction and thespecific interfacial area increase accordingly. For instance, the mean bubblevelocity in the center increases from about 0.43 m/s at a superficial gas velocityug = 0.005 m/s to around 0.90 m/s at a superficial gas velocity ug = 0.1 m/s.The local void fraction in the center increases about ten times (from 2.3% toaround 23%). The specific interfacial area increases from around 23 to 170.However, the equivalent bubble diameter does not increase so much. Theequivalent bubble diameter ranges from 0.0064 m at a superficial gas velocityug = 0.005 m/s while the equivalent bubble diameter becomes more or lessconstant (around 0.009 m) at higher superficial gas velocity.

3.4.4 Axial (or vertical) evolution of bubble properties

In this section, we will report the evolution of bubble properties along columnheight. Bubble properties in the center of the column (x/W = 0.5, y/D = 0.5)are considered. For the small liquid height (H/W = 3), the bubble propertiesare taken at three heights (z/H = 0.35, 0.55 and 0.75). For the relatively largeliquid height (H/W = 6), three heights (z/H = 0.38, 0.68 and 0.88) were taken.

In Figure 3.6, one can find that the bubble properties, such as the local voidfraction and the specific interfacial area, remain constant along the height atsuperficial gas velocity ug = 0.005 m/s. The mean bubble velocity and theequivalent bubble diameter possess a little variation along the vertical co-ordinate. At higher the superficial gas velocity, the mean bubble velocitydecreases slightly along the column height which may be due to the spread-ing of the bubble plume over the cross sectional area. The local void fraction

55



(c) Specific interfacial area [1/m] (d) Mean bubble size [m]

Figure 3.6: Axial (or vertical) bubble properties (H/W = 3).

and the specific interfacial area first decrease and then increase slightly. Fur-thermore, the equivalent bubble size increases a little along the height up tohalf the height of the column where the increase slows down in the upperpart of the column. One may notice that the equivalent bubble size increaseswith the superficial gas velocity up to ug = 0.06 m/s and then, decreases withthe superficial gas velocity.

The axial evolution of the bubble properties at large initial liquid height(H/W = 6) are plotted in Figure 3.7. It can be seen that the mean bubblevelocity decreases slightly along the height for all superficial gas velocities.For ug = 0.005 m/s, the local void fraction remains constant and the specificinterfacial area increases slightly. As a result, the equivalent bubble diametershows a small decrease in the upper part of the column. Beyond that super-

56

3.5. Conclusions


(c) Specific interfacial area [1/m] (d) Mean bubble size [m]

Figure 3.7: Axial (or vertical) evolution bubble properties (H/W = 6).

ficial gas velocity, the local void fraction and the specific interfacial area firstdecrease and then increase slightly. There are similar observations with thosefound in the small column. The increase of the equivalent bubble diameterslows down along the column height.

3.5 Conclusions

In the present work, bubble properties in bubble columns with different initialliquid heights have been investigated experimentally. In addition the effect ofthe superficial gas velocity has been investigated. A four-point optical fibreprobe was used to measure the relevant quantities, such as bubble velocity,local void fraction, chord length and specific interfacial area. Bubble size

57


has been determined by using two different methods. One method is basedon an inverse transform of the chord length distribution using geometricalprobability theory. The equivalent bubble diameter is then related to the meanchord length. Another method is to determine the equivalent bubble size fromthe specific interfacial area which is deduced from the data processing of thefour-point optical probe. Comparison of the two methods reveals that resultsfrom the two methods agree with each other quite well at low superficial gasvelocity. However, differences become apparent as the superficial gas velocityincreases. The former method assumes that bubbles have a spherical shape.That may be true at low superficial gas velocity due to lack of coalescence ofbubbles. The shape of bubble deviates from spherical when the bubble sizeincreases and deforms. The assumption of spherical bubbles is not valid anymore. Therefore, the latter method was used for further analysis.

The accuracy of the four-point optical probe was also studied. Meanbubble velocities and local void fractions at three different depths were usedto calculate area-averaged properties, such as gas flow rate and average voidfraction. The area-averaged gas flow rate was compared with that from aflow meter whereas the area-averaged void fraction was compared with thegas holdup obtained from differential pressure measurements. Discrepancieswere found for both methods. Possible reasons for the observed discrepanciesmay be due to lack of symmetry of flow inside the column, the processingmethod of the optical probe and the intrusive effect of the optical probe.

In addition, the effect of initial liquid height on the bubble properties wasinvestigated. Bubble properties were compared at three (initial) liquid heights(H/W = 3, 6 and 9) at a superficial gas velocity ug = 0.005 m/s and for twoliquid heights (H/W = 3 and 6) at higher superficial gas velocities. Resultsshow that the initial liquid height barely influences the bubble properties atthe same height. However, the axial evolution of bubble properties differsbetween the systems with different initial liquid heights. Particularly, themean bubble velocity decreases continuously along the column height atsmall liquid height while it keeps nearly constant at higher superficial gasvelocities.

58

Nomenclature

Nomenclature

a specific interfacial area, [m−1]A cross-sectional area, [m2]D column depth, [m]d bubble size, [m]dc chord length, [m]dc mean chord length, [m]de equivalent bubble size, [m]E aspect ratio of bubble, [-]f function, [-]g gravitational acceleration, [m s−2]H column height, [m]L length scale, [m]n unit normal vector of interface, [-]N number of times that an interface passes through a certain posi-

tion, [-]∆P differential pressure, [N m−2]Q volume flow rate, [m3 s−1]t time; time of individual bubble traveling through tip of the probe,

[s]T sampling time, [s]u bubble velocity, [m s−1]ug superficial gas velocity, [m s−1]v interface velocity vector, [m s−1]W column width, [m]x distance in x direction, [m]y distance in y direction, [m]z distance in z direction, [m]

Greek letters

α local void fraction, [-]

59


α surface-averaged void fraction, [-]β Bubble selection criterion, [-]ε gas holdup, [-]ρ density, [kg m−3]ξ unit vector, [-]

Indices

0 − 3 probe tipH horizontalj interfacel liquid

References

A. Cartellier. Optical probes for local void fraction measurements: Charac-terization of performance. Review of Scientific Instruments, 61(2):874–886,1990.

A. Cartellier. Simultaneous void fraction measurement, bubble velocity, andsize estimate using a single optical probe in gas-liquid two-phase flows.Review of Scientific Instruments, 63(11):5442–5453, 1992.

J. M. Delhaye and P. Bricard. Interfacial area in bubbly flow: Experimentaldata and correlations. Nuclear Engineering and Design, 151(1):65–77, 1994.

R. A. Herringe and M. R. Davis. Structural development of gas-liquid mixtureflows. Journal of Fluid Mechanics Digital Archive, 73(01):97–123, 1976.

M. Ishii. Thermo-fluid dynamic theory of two-phase flow. Collection de la Directiondes Etudes et Recherches d’Electricite de France, ISSN 0399-4198; 22. [Paris]: Eyrolles, 1975.

I. Kataoka, M. Ishii, and A. Serizawa. Local formulation and measurementsof interfacial area concentration in two-phase flow. International Journal ofMultiphase Flow, 12(4):505–529, 1986.

60

References

W. Liu and N. N. Clark. Relationships between distributions of chord lengthsand distributions of bubble sizes including their statistical parameters. In-ternational Journal of Multiphase Flow, 21(6):1073–1089, 1995.

W. Liu, N. N. Clark, and A. I. Karamavruc. Relationship between bubble sizedistributions and chord-length distribution in heterogeneously bubblingsystems. Chemical Engineering Science, 53(6):1267–1276, 1998.

R. F. Mudde and T. Saito. Hydrodynamical similarities between bubble col-umn and bubbly pipe flow. Journal of Fluid Mechanics, 437:203–228, 2001.

S. T. Revankar and M. Ishii. Theory and measurement of local interfacial areausing a four sensor probe in two-phase flow. International Journal of Heatand Mass Transfer, 36(12):2997–3007, 1993.

X. Shen, Y. Saito, K. Mishima, and H. Nakamura. Methodological improve-ment of an intrusive four-sensor probe for the multi-dimensional two-phaseflow measurement. International Journal of Multiphase Flow, 31(5):593–617,2005.

M. J. Tan and M. Ishii. A method for measurement of local specific interfacialarea. International Journal of Multiphase Flow, 16(2):353–358, 1990.

N. T. Thang and M. R. Davis. The structure of bubbly flow through venturis.International Journal of Multiphase Flow, 5(1):17–37, 1979.

N. N. Turton and R. Clark. Chord length distributions related to bubble sizedistributions in multiphase flows. International Journal of Multiphase Flow,14(4):413–424, 1988.

T. Uga. Determination of bubble-size distribution in a BWR. Nuclear Engineer-ing and Design, 22(2):252–261, 1972.

61

Chapter

4Numerical analysis of the effect of

gas sparging on bubble columnhydrodynamics

A discrete bubble model (DBM) has been used to study the effect of gas sparger prop-erties on the hydrodynamics in a bubble column. As a first step the performance ofthe model was evaluated by comparison with experimental data. Subsequently, fourdifferent perforated plates with different sparged areas were used as a gas sparger.Distributions of liquid velocity, turbulent kinetic energy and void fraction in the cen-tral plane were compared for the four different systems. Furthermore, the effect of thesparger location was also investigated. It was found that the liquid phase circulationbecomes more pronounced as the sparged area location is more distant from the centerof the bottom plate.Finally, gas phase Residence Time Distributions (RTD) were obtained from the sim-ulations. By employing standard axial dispersion model, the gas phase mixing in thebubble column was characterized. Results show that the extent of mixing increasedwhen the sparged area decreased. The axial dispersion coefficient increased as thesparged area was shifted to the edge of the bottom plate.

63

4. Effect of gas sparging on bubble column hydrodynamics

4.1 Introduction

Bubble column reactors are intensively utilized in chemical, petrochemical,biochemical and metallurgical industries. These reactors are often preferredbecause of simplicity of operation, low operating costs and ease with which theliquid residence time can be varied. This type of gas-liquid contactor has beenstudied intensively during the past decades. Parameters, such as gas holdup,gas-liquid interfacial area, interfacial mass transfer coefficients and dispersioncoefficients and heat transfer coefficient and so on, have been investigated forthe purpose of scale-up and design of bubble column reactors. Furthermore,the influence of the gas sparger on the hydrodynamical parameters has beenindicated by Shah et al. (1982).

Hebrard and Bastoul (1996) studied flow regime transition, critical gasflow rates corresponding to the transition from one regime to the other, gasholdup, bubble size distribution and axial liquid dispersion coefficients incolumns with three types of gas spargers: a perforated plate, a porous sin-tered plate and a flexible membrane disc. Their results revealed that the hy-drodynamical parameters strongly depend on the type of gas sparger used.Thorat (1998) and Veera and Joshi (1999) studied the combined effect of gassparger and height on the gas holdup by using perforated plates. They re-ported different gas holdup profiles from single point spargers and multipointspargers. Bhole et al. (2006) measured the hydrodynamics of a bubble columnwith two different gas spargers using LDA. Their results revealed that theporous plate sparger and the perforated plate sparger result in different col-umn hydrodynamics. To summarize, the effect of distributor characteristicson bubble column hydrodynamics, i.e. the details about the distributions ofliquid velocity, turbulent kinetic energy and local void fraction, is importantfor reactor design. However, the effect of sparged area of the perforated plate,such as the size of the sparged area, location of the sparged area and so on, isinteresting as well and will be examined in this chapter.

Recently, the reaction engineering community has been active in explor-ing the possibilities to utilize Computational Fluid Dynamics (CFD) in themodeling of multiphase reactors. Eulerian-Lagrangian models are one of the

64

4.2. Discrete bubble model

available Computational Fluid Dynamics tools in the field of bubble columnmodeling. Delnoij et al. (1997) used the Eulerian-Lagrangian approach tocalculate the time-dependent two-dimensional motion of small, spherical gasbubbles in a bubble column operating in the homogeneous regime. Delnoijet al. (1999) extended their two-dimensional Eulerian-Lagrangian model to athree-dimensional model. Sommerfeld et al. (2003) developed a bubble coa-lescence model on the basis of the Lagrangian approach, where bubbles aretraced through the turbulent flow field along their trajectories. Darmana et al.(2006) applied a parallel algorithm to the Eulerian-Lagrangian model em-bedding four-way coupling. Hu and Celik (2008) used Eulerian-Lagrangianapproach to study gas-liquid bubbly flow in a flat bubble column by meansof large-eddy simulation (LES) with two-way coupling. Due to additionalefforts in detailed studies of bubbly flows, the Eulerian-Lagrangian approachhas been extended recently to study liquid and gas phase mixing in bubblecolumns.

The present work studies the effect of the gas sparger on the hydrody-namics and gas phase mixing of a square bubble column using several typesof gas spargers (perforated plates) employing the Eulerian-Lagrangian ap-proach. In addition the influence of the size and the location of sparger area isinvestigated. Time-averaged quantities, such as liquid velocity, void fractionand turbulent kinetic energy, are considered. Furthermore, the residence timedistribution of the gas phase obtained from the Eulerian-Lagrangian modelis used to characterize the gas phase mixing in the bubble column in terms ofan axial dispersion model.

4.2 Discrete bubble model

Our discrete bubble model (DBM) was originally developed by Delnoij et al.(1997) and Delnoij et al. (1999). The liquid phase hydrodynamics is repre-sented by volume-averaged continuity and Navier-Stokes equations, whilethe motion of each individual bubble is tracked in a Lagrangian way.

65


4.2.1 Bubble dynamics

The motion of each individual bubble is computed from Newton’s secondlaw. The interaction with liquid phase is accounted for by an interphasemomentum transfer term. For an individual bubble, the equation of motionand the bubble trajectory equation can be respectively written as:

ρgVdvdt= ΣF (4.1)

drdt= v (4.2)

The net force acting on each individual bubble is calculated by consideringall the relevant forces. It is assumed that the net force is composed of separate,uncoupled contributions due to gravity, pressure, drag, lift, virtual mass andwall force respectively:

ΣF = FG + FP + FD + FL + FVM + FW (4.3)

The gravity force acting on a bubble in a liquid is given by:

FG = ρgVg (4.4)

The far field pressure force incorporating contributions of the Archimedesbuoyancy force, inertial forces and viscous strain is given by:

FP = −V∇P (4.5)

The drag force exerted on a bubble rising in a liquid is expressed as:

FD = −18

CDρlπd2|v − u|(v − u) (4.6)

where the drag coefficient is taken from Tomiyama et al. (1998):

CD = max[min

[ 16Re

(1 + 0.15Re0.687),48Re

],

83

EoEo + 4

](4.7)

where Re and Eo are the bubble Reynolds number and the Eotvos numberrespectively.

66


A bubble rising in a non-uniform flow field experiences a lift force due tovorticity or shear. The shear induced lift force acting on a bubble is usuallywritten as (Auton, 1987):

FL = −CLρlV(v − u) × ∇ × u (4.8)

Zhang et al. (2006) studied the influence of the lift coefficient on the bubblecolumn dynamics using a two-fluid model. It was found that the results withCL = 0.5 fit best with PIV experimental data. Therefore, CL = 0.5 is used inthe present study as well.

Accelerating bubbles experience a resistance, which is termed the virtualmass force (Auton, 1987):

FVM = −CVMρlV(Dv

Dt− Du

Dt

)(4.9)

where the D/Dt operators denote the material derivatives pertaining to therespective phase. In the present work, bubbles are assumed to have a sphericalshape and a virtual mass coefficient of CVM = 0.5 is used.

Bubbles in the vicinity of a solid wall experience a force referred to as thewall force (Tomiyama et al., 1995):

FW = −12

CWd[

1y2 −

1(L − y)2

]ρl|(v − u)·nz|2nW (4.10)

where nz and nW, respectively, are the normal unit vectors in the verticaland wall normal direction, L is the dimension of the system in the normaldirection, and y is the distance to the wall in that direction. Finally, the wallforce coefficient CW is given by:

CW =

exp(−0.933Eo + 0.179) 1 ≤ Eo ≤ 5,

0.007Eo + 0.04 5 < Eo ≤ 33.(4.11)

4.2.2 Liquid phase dynamics

The liquid phase hydrodynamics is described by a set of volume-averagedconservation equations, which consists of the continuity and Navier-Stokes

67


equations. The presence of the bubbles is reflected by the liquid phase volumefraction αl and the interphase momentum transfer rateΦ:

∂∂t

(αlρl

)+ ∇ · (αlρlu

)= 0 (4.12)

∂

∂t(αlρlu

)+ ∇ · (αlρluu

)= −αl∇p − ∇ · (αlτl) + αlρlg +Φ (4.13)

The liquid phase is assumed to be Newtonian, thus the stress tensor τl can beexpressed as:

τl = −µeff,l

[(∇u) + (∇u)T − 2

3I(∇ · u)

](4.14)

where µeff,l is the effective shear viscosity. In the present model, the effectiveviscosity is composed of two contributions, the molecular viscosity and theturbulent viscosity:

µeff,l = µL,l + µT,l (4.15)

where µT,l is the turbulent viscosity(or eddy viscosity), which is determinedfrom turbulence modeling of the liquid phase.

In the present work, a sub-grid scale model is used to simulate the turbu-lence induced by movements of bubbles. This means that the above conti-nuity and Navier-Stokes equations are resolved for the field representing the“large” eddies, while the effect of the subgrid part of the velocity representingthe ”small scales” on the resolved field is included through an eddy-viscositysubgrid-scale model.

In the present work, two eddy-viscosity models are adopted. The firsteddy-viscosity model was proposed originally by Smagorinsky (1963):

µT,l = ρl(CS∆)2|S| (4.16)

where CS is the Smagorinsky constant with a typical value of 0.1 which isalso adopted in the present work. ∆ = (Vcell)1/3 the filter width and S thecharacteristic filtered rate of strain which is calculated from:

S =√

2Si jSi j (4.17)

68


where Si j is the filtered rate of strain tensor:

Si j =12

(∂ui

∂x j+∂u j

∂xi

)(4.18)

Another eddy-viscosity model used to calculate the eddy viscosity wasproposed by Vreman (2004):

µT,l = 2.5ρlC2S

√Bβαi jαi j

(4.19)

where Bβ = β11β22 − β212 + β11β33 − β2

13 + β22β33 − β223, βi j = ∆

2mαmiαmj and

αi j = ∂u j/∂xi. ∆i is the filter width in the i direction.

4.2.3 Collision model

In the present work, a hard sphere collision model (Hoomans et al., 1996) isadopted to describe the bouncing of bubbles. It mainly consists of two parts.One is processing the sequence of collisions and another one is dealing withthe collision dynamics. The former is described in detail by Darmana et al.(2006).

A brief description of the collision model will be presented subsequently:consider a set of N bubbles with indexB = 0, 1, ...,N−1 and a set of obstacles(i.e. walls) O. For each bubble l ∈ B, a set of possible collision partners of l,N(l), can be found within all the other bubbles and obstacles.

According to Allen and Tildesley (1989), the time required for the bubblel to collide with a collision partner m ∈ N(l) can be determined from theirpresent positions and velocities:

tlm =−rlm · vlm −

√(rlm · vlm)2 − v2

lm(r2lm − (Rl + Rm)2)

v2lm

(4.20)

where rlm = rl − rm and vlm = vl − vm. Note that if rlm · vlm > 0 the bubbles aremoving away from each other and will not collide. In case of a collision witha wall the collision time follows simply from the distance to the wall and the

69


normal velocity component toward that wall, which leads for a vertical wall(i.e. xwall = 0) to the following expression:

tl,wall =rl,x − (xwall + Rl)

vl,x(4.21)

For each bubble, the minimum collision time is determined by scanningall collision partners for a possible collision. In order to prevent calculatingthe collision time for a certain bubble and its partner twice, only those bubbleswith index larger than the bubble’s index (i.e. l) and obstacles are consideredin the possible collision partnersN(l).

Once the minimum collision time for each bubble is determined, a globalminimum collision time (i.e. tab) and the corresponding collision pair (i.e. aand b) can be found. First, all bubble positions are updated to the instant ofthe collision using a simple explicit integration:

rl(t + tab) = rl(t) + vltab ∀l ∈ B (4.22)

Following the movement of all bubbles, collision pairs a and b are touchingand the collision dynamics is applied to process the collision event. The post-collision velocities of a and b can be calculated according to Hoomans et al.(1996).

4.3 Simulation details

The bubble column studied here is shown in Figure 4.1. The cross-sectionalarea of the column is 0.15 m × 0.15 m (W × D). The column is initially filledwith water to a height of 0.45 m (H). Air is used as the dispersed phase andintroduced into the column through a perforated plate at the bottom of thecolumn. The material properties of both phases are taken according to roomtemperature. The bubble column is operating at atmospheric pressure.

The effects of different configurations of gas spargers on the column hy-drodynamics are investigated. Perforated plates with 9 holes (3 × 3), 49 holes(7 × 7), 225 holes (15 × 15) and 484 holes (22 × 22) as shown in Figure 4.2 areused. All the holes have a diameter of 1 mm and are located in the centralregion of the plate with a square pitch of 6.25 mm.

70

4.3. Simulation details

Figure 4.1: Sketch of the bubble column.

Figure 4.2: Schematic representation of different perforated plates.

71


Furthermore, the effect of the position of the sparged area on the perfor-mance of the bubble column is studied by using perforated plates with 49holes. That is, besides the perforated plate with 49 holes used above, anothertwo perforated plates with the same number of holes are adopted for the pur-pose, as shown in Figure 4.3. The centers of the sparger regions are located atone-fourth, one-third and a half of the width of the column.

Figure 4.3: Schematic representation of the location of the sparged area ofperforated plates (49 holes).

A computational grid with (20×20×60) cells is adopted in the simulations.The superficial gas velocity is 0.005 m/s. The bubble diameter in the simula-tions is kept as a constant d = 0.005 m, which is obtained from experimentalobservations of a bubble column operated at the applied superficial gas ve-locity. The computational time step for the liquid phase is 1.0 × 10−3 s and thecollisions among bubbles and the movements of the bubbles are processedseveral times within each time step. The total simulation time is set as 120 s.The grid size effect was checked by comparing simulation results with thoseobtained from a finer grid (30 × 30 × 90). There was no significant differencebetween the results of these simulations.

No bubble coalescence and breakup was considered in the present work.Since the superficial gas velocity adopted in the present work is small, thehomogeneous flow regime prevails where bubble coalescence and breakupare not significant. Therefore, the bubble size is assumed to be uniform in theflow.

72



4.4.1 Comparisons with experimental data

In order to validate the discrete bubble model and the adopted closures, thesimulation results are compared with the experimental data reported by Deen(2001). Information at three different heights (z/H = 0.35, 0.55, and0.75) in thecolumn was extracted from the simulation using a distributor with 49 holes.The two eddy-viscosity models (Smagorinsky, 1963; Vreman, 2004) were usedfor the simulations.

The profiles of the time-averaged vertical velocity of the liquid phase arepresented in the Figures 4.4(a)–4.4(c). It can be seen that the results obtainedwith the eddy-viscosity model proposed by Vreman (2004) agree with theexperimental data better than those obtained with the eddy-viscosity modelof Smagorinsky (1963). This difference is much more obvious in the lowerpart of the column(i.e. z/H < 0.5), whereas in the higher part of the column,the simulation data from both two eddy-viscosity models are close to theexperimental data.

Furthermore, the profiles of turbulent kinetic energy (tke) from PIV mea-surements and simulations are also compared (Figures 4.4(d)–4.4(f)). Notethat there are only two components of the liquid velocity available from PIVmeasurements. Therefore, the reported turbulent kinetic energy obtainedfrom experimental data is calculated by assuming that the two horizontalcomponents of the liquid velocity are equal, i.e. k ≈ 1

2 (2u′2x + u′2z ). The turbu-lent kinetic energy from the simulations, however, is obtained from all threecomponents of the liquid velocity, i.e. k = 1

2 (u′2x + u′2y + u′2z ).It can be seen that the turbulent kinetic energy from the simulations is much

smaller in comparison with the experimental data at the lower height (z/H =0.35). However, the difference again becomes smaller along the column height.

Meanwhile, it can clearly be seen that the simulations with the eddy-viscosity model of Vreman (2004) are in better agreement with the experimentsthan those with the eddy-viscosity model proposed by Smagorinsky (1963).

The above comparisons demonstrate that the discrete bubble model, withsupplemented proper force closures and turbulence eddy-viscosity model,

73


PIV data

Vreman (2004)

Smagorinsky (1963)

uz [m

/s]

-0.1

0.0

0.1

0.2

0.3

x/W [-]0 0.2 0.4 0.6 0.8 1

(a) Vertical liquid velocity (z/H = 0.35)

PIV data

Vreman (2004)

Smagorinsky (1963)

uz [m

/s]

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

x/W [-]0 0.2 0.4 0.6 0.8 1

(b) Vertical liquid velocity (z/H = 0.55)

PIV data

Vreman (2004)

Smagorinsky (1963)

uz [m

/s]

-0.1

-0.05

0

0.05

0.1

0.15

0.2

x/W [-]0 0.2 0.4 0.6 0.8 1

(c) Vertical liquid velocity (z/H = 0.75)

PIV data

Vreman (2004)

Smagorinsky (1963)

tke [m

2/s

2]

0.00

0.01

0.02

0.03

x/W [-]0 0.2 0.4 0.6 0.8 1

(d) Turbulent kinetic energy (z/H = 0.35)

PIV data

Vreman (2004)

Smagorinsky (1963)

tke [m

2/s

2]

0.00

0.01

0.02

0.03

x/W [-]0 0.2 0.4 0.6 0.8 1

(e) Turbulent kinetic energy (z/H = 0.55)

PIV data

Vreman (2004)

Smagorinsky (1963)

tke [m

2/s

2]

0.00

0.01

0.02

x/W [-]0 0.2 0.4 0.6 0.8 1

(f) Turbulent kinetic energy (z/H = 0.75)

Figure 4.4: Comparisons between simulations and experimental data.

74


is suited to investigate the hydrodynamical performance of bubble columnreactors. All subsequent simulation results are based on the discrete bubblemodel utilizing the eddy-viscosity model proposed by Vreman (2004).

4.4.2 Effect of different gas sparger

This section discusses the performance of the columns with four differentperforated plates (3 × 3, 7 × 7, 15 × 15 and 22 × 22 holes). The time-averagedliquid velocity profiles in the center plane of the column (y/D = 0.5) with thefour different perforated plates are shown in Figure 4.5(a)–4.5(b).

It can be seen that the vertical liquid velocity in the column with theperforated plate of 3 × 3 has a relatively large center velocity. Since thesuperficial gas velocities of all the columns are kept the same, the columnswith less holes on the plate have larger gas inlet velocity. Therefore, the profileof vertical liquid velocity is getting flatter when the sparger area increases.

Profiles of the turbulent kinetic energy in the midplane are shown in theFigure 4.5(c)–4.5(d). It is found that the turbulent kinetic energy in the centralregion of the column with the perforated plate with 9 holes is the highest. Theturbulent kinetic energy is decreasing and becomes more flat with increasingsparged area. Furthermore, the profile of the turbulent kinetic energy alsobecomes flat with increasing height.

The distributions of the void fraction in the different systems are shownin Figure 4.5(e)–4.5(f). The column with the perforated plate (7× 7) has a veryhigh void fraction in the central region of the column (z/H = 0.35 and 0.55).Remarkably, the column with the perforated plate (3 × 3) has a lower voidfraction in that region. A reason for this can be checked by visualization of“bubbles“. Consequently, the void fraction in the column with the perforatedplate (3× 3) is lower. Furthermore, the void fraction becomes flat with heightand the void fraction in the column with more holes in the perforated plate israther flat from the beginning and does not change too much with height.

With constant superficial gas velocity, bubbles released from the plate havelower velocity with increasing number of holes. Furthermore, flows becomemore uniform when the sparged area increases. This is consistent with theabove findings. In addition, bubbles injected from small sparged area move

75


9 holes

49 holes

225 holes

484 holes

uz [m/s]

-0.1

0

0.1

0.2

0.3

x/W [-]0 0.2 0.4 0.6 0.8 1

(a) Vertical liquid velocity (z/H = 0.35)

9 holes

49 holes

225 holes

484 holes

uz [m/s]

-0.1

0

0.1

0.2

x/W [-]0 0.2 0.4 0.6 0.8 1

(b) Vertical liquid velocity (z/H = 0.55)

9 holes

49 holes

225 holes

484 holes

tke [m

2/s

2]

0

0.01

0.02

0.03

0.04

x/W [-]0 0.2 0.4 0.6 0.8 1

(c) Turbulent kinetic energy (z/H = 0.35)

9 holes

49 holes

225 holes

484 holes

tke [m

2/s

2]

0

0.01

0.02

0.03

0.04

x/W [-]0 0.2 0.4 0.6 0.8 1


9 holes

49 holes

225 holes

484 holes

αg [-]

0

0.02

0.04

0.06

0.08

x/W [-]0 0.2 0.4 0.6 0.8 1

(e) Void fraction (z/H = 0.35)

9 holes

49 holes

225 holes

484 holes

αg [-]

0

0.02

0.04

0.06

x/W [-]0 0.2 0.4 0.6 0.8 1

(f) Void fraction (z/H = 0.55)

Figure 4.5: Comparisons of different gas spargers.

76


to the sides more easily and thus, the local void fraction becomes smaller.

4.4.3 Effect of location of sparger area

The effect of different location of the same sparged area on a perforated platehas also been investigated. The locations of the sparged area on the per-forated plates are shown in Figure 4.3. For each of the configurations, thetime-averaged vertical liquid velocity, turbulent kinetic energy and local voidfraction at two different heights are plotted in Figure 4.6(a)–4.6(f).

It is quite clear that the profiles of time-averaged vertical liquid velocityare considerably different. Due to the deviation of the sparged area from thecenter of the plate, the peak of the vertical liquid velocity distribution alsomoves from the center towards the wall. Meanwhile, one can also observethat the highest vertical liquid velocity for the plate with 9 holes is slightlylarger than that of the others. This originates from the relatively intensiveliquid circulation in the bubble column.

In Figure 4.6(c)–4.6(d), the distributions of turbulent kinetic energy forthe three different systems are plotted. On the right half of the column, theturbulent kinetic energy decreases with increasing distance of the spargedarea from the center of the column. However, in the left half of the column,the distributions of the turbulent kinetic energy in the columns are somewhatmore complicated. In the lower part of the column, the turbulent kineticenergy is the highest for the W/3 case. whereas, at the intermediate height,the three turbulent kinetic energy distributions are approaching each other.

The distributions of void fraction in the three cases are also quite different(Figure 4.6(e)–4.6(f)). However, the trends follow those of the vertical liquidvelocity. That is, the peak of the void fraction distribution shifts from thecenter to the edge with increasing sparger asymmetry. Furthermore, the voidfraction is highest when the sparger is located nearest to the wall.

It can be concluded that the location of the sparged area on the perforatedplate influences the bubble column hydrodynamics significantly. It also in-fluences the gas phase mixing in the bubble column, which will be discussedin the following section.

77


W/4

W/3

W/2

uz [m/s]

-0.2

0

0.2

0.4

x/W [-]0 0.2 0.4 0.6 0.8 1

(a) Vertical velocity (z/H = 0.35)

W/4

W/3

W/2

uz [m/s]

-0.2

-0.1

0

0.1

0.2

x/W [-]0 0.2 0.4 0.6 0.8 1

(b) Vertical velocity (z/H = 0.55)

W/4

W/3

W/2

tke [m

2/s

2]

0

0.005

0.01

0.015

0.02

x/W [-]0 0.2 0.4 0.6 0.8 1

(c) Turbulent kinetic energy (z/H = 0.35)

W/4

W/3

W/2

tke [m

2/s

2]

0

0.005

0.01

0.015

0.02

0.025

x/W [-]0 0.2 0.4 0.6 0.8 1


W/4

W/3

W/2

αg [-]

0

0.02

0.04

0.06

0.08

0.1

x/W [-]0 0.2 0.4 0.6 0.8 1

(e) Void fraction (z/H = 0.35)

W/4

W/3

W/2

Void fraction [-]

0

0.01

0.02

0.03

0.04

0.05

x/W [-]0 0.2 0.4 0.6 0.8 1

(f) Void fraction (z/H = 0.55)

Figure 4.6: Comparisons of locations of gas spargers.

78


4.4.4 Comparisons of residence time distributions

The concept of residence time distributions (RTD), indicated by E(t), is a veryimportant concept in the analysis of chemical reactors. The idea of RTD’s hasbeen introduced by Danckwerts (1953).

Residence time distribution theory produces valuable insight with respectto consequences of non-ideal flow for the macro-mixing in process equip-ment. The distribution of residence times provides considerable informationabout homogeneous, isothermal reactions. For single, first-order reactions,knowledge of the residence time distribution allows the yield to be calculatedexactly, even in flow systems of arbitrary complexity. For other reaction or-ders, it is usually possible to calculate tight limits, within which the yield mustlie. Even if the system is nonisothermal and heterogeneous, knowledge of theresidence time distribution provides substantial insight regarding the flowprocesses occurring within it (Levenspiel, 1999; Nauman, 2002; Scott Fogler,2005).

The residence time distribution E(t) is defined in such way that the areaunder the curve is unity: ∫ ∞

0E(t) dt = 1 (4.23)

The mean residence time tm and the variance σ2 of the residence timedistribution are calculated from:

tm =

∫ ∞

0tE(t) dt (4.24)

σ2 =

∫ ∞

0(t − tm)2E(t) dt (4.25)

The axial dispersion model has been used extensively to characterize phasemixing in chemical reactors. The residence time distribution can be used toevaluate the dispersion coefficient Da in the axial dispersion model. Accordingto the axial dispersion model, the mass balance of a species in an unsteady-state is given by (Levenspiel, 1999):

∂C∂t= Da

∂2C∂z2 −U

∂C∂z

(4.26)

79


where the velocity U represents the mean velocity (for the gas phase U =Ug/εg).

In a so-called closed-closed system, a Peclet number Pe can be calculated(Levenspiel, 1999) by using tm and σ2 determined from RTD data:

σ2θ =σ2

t2m=

2Pe− 2

Pe2 (1 − e−Pe) (4.27)

where σθ is the dimensionless variance in the residence time. The dispersioncoefficient Da can now be determined from the Peclet number:

Da =UHPe

(4.28)

In the discrete bubble model, the determination of the residence time of thegas phase is straightforward. Each bubble traveling in the reactor is trackedexactly by the discrete bubble model. Therefore, the residence time of eachbubble can be determined from the DBM simulations directly and thus theresidence time distribution of the gas phase can be obtained.

The resulting residence time distributions of the columns with the fourdifferent perforated plates are shown in Figure 4.7.

9 holes

49 holes

225 holes

484 holes

E(t) [1/s]

0

0.01

0.02

0.03

0.04

0.05

t [s]0 1 2 3 4 5 6 7

Figure 4.7: Bubble residence time distributions

One can find that the residence time distributions obtained from the DBMsimulations agree well with characteristics of typical residence time distribu-

80


tion of the bubble phase reported by Molerus (1986). That is, (i) even thefastest bubbles rise with a finite absolute velocity, i.e. the RTD shows a mini-mum time; (ii) a steep slope of the RTD is observed for short residence times;(iii) partial recirculation of small bubbles results in a long tail of the RTD.

The mean residence time, variance, Peclet number and dispersion coeffi-cient in each column have been calculated from the residence time distribu-tions shown in Figure 4.7 and are listed in Table 4.1.

According to the Table 4.1, the gas phase dispersion coefficient Da ofthe column decreases with decreasing sparger area. This implies that thedegree of backmixing of the gas phase in the bubble column is increasingwith decreasing sparger area.

Table 4.1: Gas holdup and calculations from RTDs of the columns.

Perforated plate 3 × 3 7 × 7 15 × 15 22 × 22W4

W3

W2

ε [-] 0.016 0.014 0.015 0.016 0.020 0.024tm [s] 1.37 1.23 1.31 1.35 1.61 1.92σ2 [s2] 0.34 0.25 0.25 0.25 0.32 0.18σ2θ [-] 0.18 0.17 0.15 0.14 0.12 0.05

Pe [-] 9.93 13.50 11.00 12.64 15.13 39.93Da [m2/s] 0.014 0.015 0.012 0.011 0.008 0.002

In addition to the comparisons of the columns with different sparger area,the mixing in the columns with the same sparger area but with different loca-tions on the bottom plate is also studied. The perforated plates with 49 holesas shown in Figure 4.3 are used for this purpose. The residence time distri-butions of these three cases are shown in Figure 4.8. The relevant parameterscharacterizing the residence time distribution and the phase mixing in thecolumns are calculated and listed in Table 4.1.

The residence time distributions of the three columns have the same vari-ance but the mean residence time decreases as the location of the sparged areadeviates more from the center of the bottom plate. The axial dispersion coef-ficient, however, increases meanwhile, which means that the extent of mixing

81


W/4

W/3

W/2

E(t) [1/s]

0

0.02

0.04

0.06

t [s]0 1 2 3 4 5 6 7

Figure 4.8: Gas phase residence time distributions (49 holes)

becomes more intensive. That is quite reasonable because the flow field in thecolumn with a asymmetrical gas sparger is much more irregular than that ina column with a symmetrical gas sparger.

4.5 Conclusions

A discrete bubble model (DBM) has been utilized to investigate the perfor-mance of a laboratory scale bubble column with a square cross-section. Firstly,the model including its closures has been validated by comparing the simula-tion results with experimental data. Two eddy-viscosity models were adoptedfor the comparison purpose. Computed time-averaged profiles of vertical liq-uid velocity and turbulent kinetic energy of the liquid phase at three differentheights (z/H = 0.35, 0.55 and 0.75) were compared with PIV data. The resultsshow that the eddy-viscosity model proposed by Vreman (2004) performsbetter than the model proposed by Smagorinsky (1963). Therefore, the for-mer eddy-viscosity model was adopted for further investigations. Severalcolumns with different perforated plates were simulated to study the effectof the gas sparger on the bubble column hydrodynamics. It was found thatthe distributions of liquid velocity, turbulent kinetic energy and void fraction

82

Nomenclature

become uniform as the sparger area increases.Furthermore, the effect of the sparger location was investigated. The liquid

phase circulation becomes more pronounced as the sparger is moved towardsthe wall.

The gas phase residence time distributions of the columns were also ob-tained from the discrete bubble model and were used to characterize themacro-mixing of the gas phase in the column in terms of an axial dispersionmodel. It is found that the extent of mixing increases when the sparger areadecreases. For instance, the dispersion coefficient of the column with the per-forated plate of 3 × 3 is seven times larger than that of the column with theperforated plate of 22 × 22. Furthermore, the effect of the sparger location onthe phase mixing is also investigated by comparing columns with three dif-ferent sparger locations. In terms of the gas phase residence time distribution,the mean residence time decreases as the sparger area deviates more from thecenter of the plate. The variance of the distribution, however, does not changewith the location of the sparger area. That is, the width of the distributionremains the same. It also turns out that the location of the sparged area onthe perforated plate influences the phase mixing. The dispersion coefficientincreases as the sparger is moved towards the wall.

The present work focused on a small scale bubble column operating atlow superficial gas velocity without consideration of bubble coalescence andbreakup. However, bubbly flows in large scale reactors at high superficial gasvelocity are more common in industrial applications. Therefore, discrete bub-ble modeling with bubble coalescence and breakup model is necessary for thesimulation of large scale bubble column reactors operating at superficial gasvelocity. Future work will focus on these aspects. In addition to experimentalstudies, discrete bubble simulation offers us an alternative method to studyphase macro-mixing in bubble column reactors.

Nomenclature

B bubblesC model coefficient, [-]; concentration of a species, [kg m−3]

83


d bubble diameter, [m]D column depth, [m]Da dispersion coefficient, [m2 s−1]

Eo Eotvos number, Eo =(ρl − ρg)gd2

σ, [-]

E(t) residence time distribution, [s−1]F force vector, [N]g gravitational acceleration, [m s−2]H column height, [m]I unit tensor, [-]n unit normal vector, [-]N collision partnersO obstaclesp pressure, [N m−2]Pe Peclet number, [-]r position vector, [m]R bubble radius, [m]

Re bubble Reynolds number, Re =ρl|v − u|dµl

, [-]

S characteristic filtered strain rate, [s−1]t time, [s]u liquid velocity vector, [m s−1]U mean bubble velocity, [m s−1]v bubble velocity vector, [m s−1]V bubble volume, [m3]W column width, [m]y distance to the wall, [m]

Greek letters

α void fraction, [-]∆ subgrid length scale, [m]ε gas holdup, [-]µ viscosity, [kg m−1 s−1]

84

References

Φ volume averaged momentum transfer due to interphase forces,[kg m−2 s−2]

ρ density, [kg m−3]σ surface tension, [N m−1]σ2 variance, [s2]τ stress tensor, [N m−2]

Indices

cell computational cellD drageff effectiveg gas phaseG gravityi i directionj j directionl liquid phaseL lift; molecular viscositym mean; bubble index of possible collision partnerP pressureS subgridT turbulentVM virtual massW wallz vertical direction

References

M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford Univer-sity Press, USA, June 1989.

T. R. Auton. The lift force on a spherical body in a rotational flow. J. FluidMech., 197:241–257, 1987.

85


M. R. Bhole, S. Roy, and J. B. Joshi. Laser Doppler anemometer measurementsin bubble column: Effect of sparger. Industrial and Engineering ChemistryResearch, 45(26):9201–9207, 2006.

P. V. Danckwerts. Continuous flow systems : Distribution of residence times.Chemical Engineering Science, 2(1):1–13, February 1953.

D. Darmana, N. G. Deen, and J. A. M. Kuipers. Parallelization of an Euler-Lagrange model using mixed domain decomposition and a mirror domaintechnique: Application to dispersed gas-liquid two-phase flow. Journal ofComputational Physics, 220(1):216–248, 2006.


E. Delnoij, F. A. Lammers, J. A. M. Kuipers, and W. P. M. van Swaaij. Dynamicsimulation of dispersed gas-liquid two-phase flow using a discrete bubblemodel. Chemical Engineering Science, 52(9):1429–1458, 1997.

E. Delnoij, J. A. M. Kuipers, and W. P. M. van Swaaij. A three-dimensionalCFD model for gas-liquid bubble columns. Chemical Engineering Science, 54(13-14):2217–2226, 1999.

G. Hebrard and D. Bastoul. Influence of the gas sparger on the hydrodynamicbehaviour of bubble columns. Chemical Engineering Research and Design, 74(3):406–414, 1996.


G. Hu and I. Celik. Eulerian-Lagrangian based large-eddy simulation of apartially aerated flat bubble column. Chemical Engineering Science, 63(1):253–271, 2008.

O. Levenspiel. Chemical Reaction Engineering. John Wiley & Sons, 3rd edition,1999.

86

References

O. Molerus. Modelling of residence time distributions of the gas phase in bub-ble columns in the liquid circulation regime. Chemical Engineering Science,41(10):2693–2698, 1986.

E. B. Nauman. Chemical Reactor Design, Optimization, and Scaleup. McGraw-Hill, 2002.

H. Scott Fogler. Elements of Chemical Reaction Engineering. Prentice Hall PTR,4th edition, August 2005.


J. Smagorinsky. General circulation experiment with the primitive equation.Monthly Weather Review, 91:99–165, 1963.

M. Sommerfeld, E. Bourloutski, and D. Broder. Euler/Lagrange calculationsof bubbly flows with consideration of bubble coalescence. Canadian Journalof Chemical Engineering, 81(3-4):508–518, 2003.

B. N. Thorat. Effect of sparger design and height to diameter ratio on fractionalgas hold-up in bubble columns. Chemical Engineering Research and Design,76(A7):823–834, 1998.

A. Tomiyama, T. Matsuoka, T. Fukuda, and T. Sakaguchi. A simple numericalmethod for solving an incompressible two-fluid model in a general curvi-linear coordinate system. Advances in Multiphase Flow, pages 241–252,Amsterdam, 1995. Society of Petroleum Engineers Inc.

A. Tomiyama, I. Kataoka, I. Zun, and T. Sakaguchi. Drag coefficients of singlebubbles under normal and micro gravity conditions. JSME internationaljournal. Ser. B, Fluids and thermal engineering, 41(2):472–479, 1998.

U. P. Veera and J. B. Joshi. Measurement of gas hold-up profiles by gamma raytomography: Effect of sparger design and height of dispersion in bubblecolumns. Chemical Engineering Research and Design, 77(4):303–317, 1999.

87


A. W. Vreman. An eddy-viscosity subgrid-scale model for turbulent shearflow: Algebraic theory and applications. Physics of Fluids, 16(10):3670–3681,2004.

D. Zhang, N. G. Deen, and J. A. M. Kuipers. Numerical simulation of the dy-namic flow behavior in a bubble column: A study of closures for turbulenceand interface forces. Chemical Engineering Science, 61(23):7593–7608, 2006.

88

Chapter

5Discrete bubble modeling ofbubbly flows: Swarm effects

The performance of several drag correlations reported in literature for bubble swarmshas been investigated. A discrete bubble model (DBM) based on the Eulerian-Lagrangian approach was adopted for this purpose. Numerical simulations for asquare bubble column were performed and the results were compared with PIV mea-surements. The drag model reported by Lima Neto et al. (2008) predicts the verticalliquid velocity and the relative velocity better than the other drag models at a superfi-cial gas velocity ug = 0.0024 m/s. As the superficial gas velocity increases, however,Lima Neto’s drag model and Wen & Yu’s model tend to overestimate the relativevelocity between the two phases. Among the other models, Rusche’s model gives abetter prediction of the vertical liquid velocity in the lower part of the bubble column.Furthermore, most of the models can predict the liquid velocity in higher parts of thecolumn well. However, there are still some aspects that need to be considered andimproved to advance the accurate simulation of bubbly flows at high void fraction.

89

5. Discrete bubble modeling of bubbly flows: Swarm effects

5.1 Introduction

Two-phase gas-liquid flows are encountered in several large scale processes,in the chemical, biochemical, metallurgical and petrochemical industries. De-tailed knowledge on hydrodynamics of two-phase gas-liquid flows is essentialfor design and scale-up of chemical reactors. Computational Fluid Dynamics(CFD) offers the possibility to investigate hydrodynamics of these complexsystems and reduce the experimental effort required for design and scale-upof related process equipments. During the past decades, numerical simulationof multiphase flows has received considerable attention (Jakobsen, 2008).

Eulerian-Lagrangian modeling, as one of the two major numerical ap-proaches, has been adopted to study bubbly flows in recent years (Trapp,1993; Lapin, 1994; Delnoij et al., 1997, 1999; Sommerfeld et al., 2003; Darmanaet al., 2006; Hu and Celik, 2008).

For numerical simulation of bubbly flows, reliable closures are required torepresent the interfacial momentum transfer rate (i.e. the effective drag actingon bubbles). A number of theoretical and experimental studies have beenconducted to evaluate the drag coefficient of single bubbles rising in quies-cent liquids (Clift et al., 1978; Zun and Groselj, 1996; Tomiyama et al., 1998).However, industrial processes are generally operated at high void fractions.Ishii and Zuber (1979) and Ishii and Hibiki (2005) have developed constitu-tive relations for the drag force and the relative velocity in bubbly, dropletand particulate flows for a wide range of dispersed volume fraction. Theyidentified four flow regimes for bubbles: viscous regime, distorted regime,churn-turbulent regime and slug regime. According to them, the dispersedphase fraction has a considerable influence on the drag coefficient of the dis-persed phase. Holland and Bragg (1995) reported that the relative velocitybetween the dispersed phase and the continuous phase is expressed as theproduct of the terminal velocity of a single dispersed element and a correctionfactor (1 − αg)n−1. For bubble columns, the expression shows that the relativevelocity of a bubble rising in a swarm is lower than the terminal velocity ofan isolated bubble in the liquid. It follows that the presence of neighboringbubbles increases the drag on a bubble. Moreover, the value of n can be ap-

90

5.1. Introduction

proximated as n = 2 in practice. Garnier et al. (2002) measured the relativevelocity between the gas and the liquid phase in an air-water system at highvoid fractions (0−0.3) using a double optical probe and hot-film anemometry.They found that the average relative velocity decreases when the void frac-tion increases. Their measurements implied that the ratio between the dragcoefficient of a bubble in a swarm and that of an isolated bubble in an infiniteliquid can be expressed as (1 − Cα1/3

g )−2. where C is a constant and aboutunity. Rusche (2002) reviewed the correlations to determine the drag forcein dispersed two-phase flows at low and high void fractions and proposedhis own correlation. By comparing those correlations with experimental datain literature, Rusche found that the new correlation gives the best predictionfor the terminal velocity of bubbles rising in swarms. Meanwhile, he alsopointed out that the correlation of Wen and Yu (1966) can give reasonableresults, although it was originally developed for fluid-particle systems. Thecorrelations of Rusche have been implemented in a two-fluid model and im-provement of the quality of the predictions was reported (Behzadi et al., 2004).By means of the volume of fluid method, Bertola et al. (2004) simulated themotion of a bubble swarm. They showed that different bubbles of differentsize possess a different trend of relative velocity versus gas holdup. Simonnetet al. (2007) studied the relative velocity in a swarm of bubbles using LaserDoppler Velocimetry (LDV) and a double optical probe. They found thatbelow a critical value of the local void fraction, the relative velocity decreasesdue to hindrance effects when the void fraction increases. Beyond this crit-ical value, the relative velocity increases with the local void fraction. Theyreported that the presence of surfactants gives a totally different trend com-pared to that for pure water. Finally, they proposed a new correlation for thedrag coefficient embedding Jamialahmadi et al. (1994)’s correlation for predic-tion of terminal rise velocity of single bubbles. In their numerical simulationsGentric et al. (2008) reproduced the characteristic evolution of the gas holdupagainst superficial gas velocity and captured some characteristic features ofdifferent flow regimes by using this new drag closure. Lima Neto et al. (2008)investigated air-water bubbly jets in a large water tank experimentally. Theyfound that the relative velocities exceeded the terminal velocities of isolated

91


bubbles given by Clift et al. (1978) and proposed a new correlation for thedrag coefficient as function of the bubble Reynolds number.

In the present work, Eulerian-Lagrangian modelling of bubbly flows in asquare column is investigated with emphasis on the performance of differentclosures for drag.


Our discrete bubble model (DBM) was originally developed by Delnoij et al.(1997, 1999) and is based on volume-averaged continuity and Navier-Stokesequations for the liquid phase, while the motion of each individual bubble istracked in a Lagrangian fashion taking into account bubble-bubble encoun-ters.


The motion of each individual bubble is computed from Newton’s secondlaw. For an individual bubble, the equation of motion and bubble trajectoryequation can respectively be written as:

ρgVdvdt= ΣF (5.1)

drdt= v (5.2)




FG = ρgVg (5.4)

92



FP = −V∇P (5.5)


FD = −18

CDρlπd2|v − u|(v − u) (5.6)

where CD represents the drag coefficient (see section 5.3).A bubble rising in a non-uniform liquid flow field experiences a lift force

due to vorticity or shear in the flow field. The shear induced lift force actingon a bubble is usually written as (Auton, 1987):

FL = −CLρlV(v − u) × ∇ × u (5.7)

The lift coefficient is calculated according to Tomiyama et al. (2002):

CL =

min[0.288tanh(0.121Re), f (EoH)] EoH < 4,

f (EoH) 4 ≤ EoH ≤ 10,

−0.29 EoH > 10.

(5.8)

wheref (EoH) = 0.00105EoH

3 − 0.0159EoH2 + 0.474 (5.9)

EoH is the Eotvos number defined by using the maximum horizontal dimen-sion of a bubble:

EoH =(ρl − ρg)gd2

H

σ(5.10)

The maximum horizontal diameter of the bubble is obtained from thebubble aspect ratio E according to Wellek et al. (1966):

E =dV

dH=

11 + 0.163Eo0.757 (5.11)

where dV is the maximum vertical diameter of the bubble and Eo is the Eotvos

number, Eo =(ρl − ρg)gd2

σ.

93


The relation between the above two diameters and the diameter of thebubble d in the discrete bubble modeling is as follows:

d = (dVd2H)1/3 (5.12)

Accelerating bubbles experience a resistance, which is described as thevirtual mass force (Auton, 1987):

FVM = −CVMρlV(Dv

Dt− Du

Dt

)(5.13)

where the D/Dt operators denote the substantiative derivatives pertaining tothe respective phases. In the present work, bubbles are assumed to have aspherical shape and a virtual mass coefficient of 0.5 is used.


FW = −12

CWd[

1y2 −

1(L − y)2

]ρl|(v − u)·nz|2nW (5.14)

where nz and nW, respectively, are the normal unit vectors in the vertical andwall normal direction, L is the dimension of the system in the wall normaldirection, and y is the distance to the wall in that direction. Finally, the wallforce coefficient CW is given by:

CW =

exp(−0.933Eo + 0.179) 1 ≤ Eo ≤ 5,

0.007Eo + 0.04 5 < Eo ≤ 33.(5.15)


The liquid phase hydrodynamics is described by a set of volume-averagedconservation equations for mass and momentum. The presence of the bub-bles is reflected by the liquid phase volume fraction αl and the interphasemomentum transfer rateΦ:

∂∂t

(αlρl

)+ ∇ · (αlρlu

)= 0 (5.16)

∂∂t

(αlρlu

)+ ∇ · (αlρluu

)= −αl∇p − ∇ · (αlτl) + αlρlg +Φ (5.17)

94



τl = −µeff,l

[((∇u) + (∇u)T − 2

3I(∇ · u)

](5.18)

where µeff,l is the effective viscosity. In the present model, the effective viscos-ity is composed of two contributions, the molecular viscosity and the turbulentviscosity:



In the present work, a sub-grid scale model is used to simulate the tur-bulence induced by bubble movement. This means that the conservationequations account for “large eddies”, while the effect of the “subgrid“ eddiesare accounted for through an eddy-viscosity model.

The model proposed by Vreman (2004) was used to calculate the eddyviscosity:

µT,l = 2.5ρlC2S

√Bβαi jαi j

(5.20)

where Bβ = β11β22 − β212 + β11β33 − β2

13 + β22β33 − β223, βi j = ∆

2mαmiαmj and

αi j = ∂u j/∂xi. ∆i is the filter width in the i direction.


In the present paper, a hard sphere collision model (Hoomans et al., 1996)is adopted to describe the (possible) bouncing of bubbles. It consists of twomain parts. One part is processing the sequence of collisions and another partis dealing with the collision dynamics. The former is described in detail byDarmana et al. (2006). More details are given in Chapter 4 of this thesis.

95


5.3 Drag coefficient correlations

5.3.1 Wen & Yu’s model

Although the correlation of Wen and Yu (1966) was proposed for gas-solid sys-tems, we used this model here for the purpose of comparison. The associatedexpression for the drag coefficient is give by:

CD =

24Res

(1 + 0.15Re0.687s )(1 − αg)−1.65 Res < 1000

0.44(1 − αg)−1.65 Res ≥ 1000(5.21)

where the bubble Reynolds number Res is based on the superficial velocity:

Res =(1 − αg)ρl|v − u|d

µl(5.22)

5.3.2 Ishii & Zuber’s model

According to Ishii and Zuber (1979) and Ishii and Hibiki (2005), for the dragcorrelation of bubbles four flow regimes should be distinguished: the viscousregime, the distorted flow regime, the churn-turbulent flow regime and theslug regime.

The drag coefficient correlations are given as follows:

CD =

24Res

(1 + 0.1Re0.75s ) Viscous regime

23

√Eo

[1 + 17.67(1 − αg)1.3

18.67(1 − αg)1.5

]2

Distorted regime

83

(1 − αg)2 Churn-turbulent regime

9.8(1 − αg)3 Slug regime

(5.23)

During the numerical simulations, the flow regime is distinguished ac-cording to Morud and Hjertager (1996). That is, the flow is in the distortedregime for αg ≤ 0.3 and in the churn-turbulent regime for 0.3 < αg < 0.7.

96

5.3. Drag coefficient correlations

5.3.3 Holland’s model

According to Holland and Bragg (1995), the drag coefficient of a bubble in abubble swarm can be expressed as follows :

CD = CD∞(1 − αg)−2 (5.24)

where CD∞ is the drag coefficient of an isolated bubble.

5.3.4 Tomiyama’s single bubble model

Tomiyama et al. (1998) developed a simple but accurate drag coefficient modelfor single bubbles. The model consists of three correlations, respectivelycorresponding to pure, slightly contaminated, and contaminated gas-liquidsystems. For the purpose of comparison of the drag coefficient for the casewith bubble swarms, the correlation for pure gas-liquid systems is adopted:

CD∞ = max[min

[ 16Re

(1 + 0.15Re0.687),48Re

],

83

EoEo + 4

](5.25)

where Re is the bubble Reynolds number, Re =ρl|v − u|dµl

.

5.3.5 Rusche’s model

According to Rusche (2002), the drag coefficient can be expressed as the prod-uct of the drag coefficient for an isolated bubble in an infinite stagnant liquidand a correction factor:

CD = CD∞ f (αg) (5.26)

where f (αg) represents the effect arising from the presence of other bubbles:

f (αg) = exp(3.64αg) + α0.864g (5.27)

5.3.6 Simonnet’s model

Simonnet et al. (2007) gave the following relation for air-pure water systems:

CD = CD∞(1 − αg)

(1 − αg)25 +

(4.8

αg

1 − αg

)25−2/25

(5.28)

97


where CD∞ is the drag coefficient of single bubbles in infinite liquids, whichis deduced from Jamialahmadi et al. (1994)’s correlation:

V∞ =V1V2√V2

1 + V22

(5.29)

where

V1 =118∆ρ

µlgd2 3µl + 3µb

2µl + 3µb(5.30)

and

V2 =

√2σ

d(ρl + ρg)+

gd2

(5.31)

The drag coefficient for a single bubble in infinite liquids CD∞ is thendetermined as follows:

CD∞ =43

(ρl − ρg)gd

V2∞ρl

(5.32)

Due to the fact that Simonnet’s model only provides the drag coefficientcorrelation in the range of the local void fraction between 0 and 0.3, we usethe drag coefficient at 0.3 when the local void fraction exceeds 0.3.

5.3.7 Lima Neto’s model

Lima Neto et al. (2008) studied air-water bubbly jets in a large water tank andproposed the following correlation for the drag coefficient in bubble swarms:

CD = 0.0828lnRe − 0.403 450 < Re < 10000 (5.33)

where Re is the bubble Reynolds number.Since Lima Neto et al. (2008) only gave the drag coefficient correlation

within the range of Re from 450 to 10, 000, the drag coefficient at Re below 450is determined from the correlation from Tomiyama et al. (1998).

5.3.8 Comparison of the drag coefficient models

In order to compare the above models for the drag coefficient in an air-watersystem, we here assume that the bubble size is equal to 0.005 m and addition-ally that the relative velocity between the gas phase and the liquid phase is

98

5.3. Drag coefficient correlations

0.2 m/s. And thus, the drag coefficient corresponding to each model can becalculated as function of the local void fraction, which is shown in Figure 5.1.Note that the drag coefficients from those models independent of the localvoid fraction, such as Tomiyama’s model and Lima Neto’s model, are treatedas constants in the plot. Moreover, for those models without explicit specifi-cation of the correlation for single bubble, i.e. Holland’s model and Rusche’smodel, the drag coefficient CD∞ is calculated according to the correlation ofTomiyama et al. (1998).

Wen & Yu (1966)

Ishii & Zuber (1979)

Holland (1995)

Tomiyama (1998)

Rusche (2002)

Simonnet (2007)

Lima Neto (2008)

CD [-]

0

2

4

6

8

10

αg [-]0 0.2 0.4 0.6 0.8

Figure 5.1: Comparison of the drag coefficient models at a relative velocity of0.2 m/s.

From Figure 5.1, one can see that the drag coefficient obtained from LimaNeto’s model is very small compared to the others and about 8.5 times smallerthan that from Tomiyama’s single bubble model. Based on Wen & Yu’s model,the drag coefficient increases monotonically and the slope becomes larger withincreasing local void fraction. Furthermore, the drag coefficient obtainedfrom Ishii & Zuber’s model first increases slightly up to αg = 0.3 and thendecreases. The drag coefficient increases considerably with αg fraction inHolland’s and Rusche’s models. On the contrary, the drag coefficient obtainedfrom Simonnet’s model first slightly increases with αg. Subsequently the dragcoefficient decreases rapidly beyond the local void fraction of 0.15. When

99


the local void fraction approaches 0.3, the drag coefficient obtained fromSimonnet’s model approaches the value from Lima Neto’s model.


The bubble column studied here is shown in Figure 5.2. The cross-sectionalarea of the column is 0.15 m × 0.15 m (W × D). The column is initially filledwith water to a height of 0.45 m (H). Air is used as the dispersed phase andintroduced into the column through a perforated plate with 49 holes (7× 7) atthe bottom of the column. The material properties of both phase are taken atroom temperature (20 °C) and atmospheric pressure.

A computational grid with (20×20×60) cells is adopted in the simulations.Four superficial gas velocities, i.e. ug = 0.0024 m/s, 0.0049 m/s, 0.0073 m/s and0.0097 m/s are used. The bubble diameter in the simulations is assumed tobe a constant, 0.005 m. Note that this assumption may not be correct sincethe bubble size may vary in the applied range of superficial gas velocities dueto coalescence and breakup of bubbles. For the purpose of comparison ofdifferent drag coefficient correlations for bubbles rising in a swarm, however,a uniform bubble size is used here. Moreover, the time step in the simulationfor the liquid phase is set as 1.0 × 10−3 s and the collisions among bubbles andthe movements of the bubbles are processed multiple times within each timestep. The total simulation time is 150 s.


5.5.1 Instantaneous gas holdup and liquid velocity

The gas holdup in the bubble column was logged at every 0.04 s. An exampleof the history of the overall gas holdup obtained with Rusche’s drag model isshown in Figure 5.3(a) for ug = 0.0024 m/s.

It can be seen that the gas holdup first increases rapidly since the sim-ulation starts and reaches a maximum value very soon. After reaching themaximum value, the gas holdup decreases in a very short time and then,starts fluctuating. The increase of the gas holdup reflects that the bubbles

100



are continuously injected into the bubble column and start rising in the col-umn. At the moment when the peak maximum is reached, some bubbleshave already reached the level of the liquid surface and most of them escapefrom the column. Hence, there is a rapid decrease in the curve right after thepeak. Once the circulation flow pattern has developed, some bubbles at thetop of the column may be trapped by the circulating liquid and remain in thecolumn. Some of these bubbles may leave the column soon, whereas othersmay take a little more time to escape. These effects produce the fluctuationof the gas holdup in the curve. In addition, the instantaneous vertical liquid

101


ε [-]

0

0.002

0.004

0.006

0.008

0.01

t [s]0 50 100 150

(a) Instantaneous gas holdup [-]

uz [m/s]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

t [s]0 50 100 150

(b) Instantaneous liquid velocity [m/s]

Figure 5.3: Instantaneous quantities vs simulation time.

velocity at position (x/W = 0.5, y/D = 0.5, z/H = 0.96) is also plotted versustime (Figure 5.3(b)). The obtained trend is similar to that obtained for the gasholdup. However, unlike the gas holdup, the liquid velocity shows more fluc-tuations at the beginning of the simulation. The averaging of hydrodynamicquantities, i.e. velocities of both phases, is initiated 30 seconds after the startof the simulation.

5.5.2 Comparisons of different drag models

In this section, the simulation results will be compared with PIV measure-ments of Deen (2001). The comparison is divided into four parts according tothe superficial gas velocity. For each superficial gas velocity, average quan-tities, such as vertical liquid velocity and relative velocity, at three differentheights (z/H = 0.3, 0.5 and 0.7) in the central plane (y/D = 0.5) of the bubblecolumn will be compared using the drag closures reported in section 5.3.

ug = 0.0024 m/s

Profiles of the average vertical liquid velocity and relative velocity betweenthe gas phase and the liquid phase at three heights are shown in Figure 5.4and compared with the PIV measurements at corresponding heights. The

102


results reveal that Lima Neto’s drag correlation predicts the vertical liquidvelocity better than all other correlations in the central region of the column.However, the model underestimates the vertical liquid velocity in the regionbetween the center and the wall, particularly in the right part of the column(x/W > 0.5). Among the other five drag coefficient correlations, Wen & Yu’smodel exhibits better performance for the prediction of the vertical liquidvelocity compared to the other models. Simonnet’s model has the largestdeviation in the central region in lower parts of the column (z/H ≤ 0.5). Ishii& Zuber’s correlation and Tomiyama’s single bubble correlation show similarperformance regarding the prediction of the vertical liquid velocity in thecentral region. Rusche’s model combined with Tomiyama’s single bubblecorrelation shows moderate performance of all the six models. The similarbehavior is also found with Holland’s model combined with Tomiyama’ssingle bubble correlation.

In addition, by comparing the relative velocity between the two phasesobtained from both the PIV measurements and the simulations, one can findthat the relative velocities between the gas phase and the liquid phase ob-tained with Wen & Yu’s model and Lima Neto’s model for the drag coefficientcorrelations are higher than those with the other four correlations. This isdue to the fact that the drag forces calculated from these two correlations aresmaller than the others. Furthermore, the other four correlations producesimilar profiles along the x direction at the three heights. In the lower partsof the column, i.e. z/H = 0.3, Lima Neto’s model overestimates the relativevelocity in the central region. Along the height, however, this model canpredict the relative velocity quite well. This also holds for Wen & Yu’s model.

ug = 0.0049 m/s

In Figure 5.5, the simulation results and PIV measurements at ug = 0.0049 m/sare compared with each other. It can be seen that all the models again overesti-mate the vertical liquid velocity in the central region at low part of the column(z/H ≤ 0.5). Ishii & Zuber’s model and Tomiyama’s model for single bubblesexhibit similar trends and produce the largest deviation from the PIV data in

103


(a) Vertical liquid velocity (z/H = 0.3) (b) Relative velocity (z/H = 0.3)

(c) Vertical liquid velocity (z/H = 0.5) (d) Relative velocity (z/H = 0.5)

(e) Vertical liquid velocity (z/H = 0.7) (f) Relative velocity (z/H = 0.7)

Figure 5.4: Comparisons at superficial gas velocity ug = 0.0024 m/s.

104


that region. Moreover, the other four models yield better predictions withrespect to the vertical liquid velocity. At increased heights, i.e. z/H = 0.7,one can find that Wen & Yu’s model and Simonnet’s model perform well.Furthermore, Rusche’s model also predicts the vertical liquid velocity quitewell in the central region. However, the profile of the vertical liquid velocityaccording to Rusche’s drag model seems to underestimate the velocity in theleft part of the column (x/W < 0.5) and slightly overestimates the velocityin the right part (x/W > 0.5). On the contrary, Holland’s model seems canpredict the vertical liquid velocity in the left part of the column well, whilethe model underestimates the liquid velocity in the right part. In addition, itis worth to mention that Simonnet’s drag model can predict the vertical liquidvelocity away from the central region very well in the entire bubble column.

Comparisons for the relative velocity are presented in Figure 5.5(b), Fig-ure 5.5(d) and Figure 5.5(f). It is quite clear that Lima Neto’s drag modeloverestimates the relative velocity at all three heights. Wen & Yu’s model canpredict the relative velocity well in the central region of the lower part of thecolumn (z/H ≤ 0.5) compared with the other models. However, it overesti-mates the relative velocity at the height z/H = 0.7. Furthermore, Simonnet’sdrag model, Ishii & Zuber’s model as well as Tomiyama’s model are able topredict the relative velocity quite well. Finally Rusche’s drag model underes-timates the relative velocity between the two phases slightly, while Holland’smodel has better estimation.

ug = 0.0073 m/s

When the superficial gas velocity increases up to 0.0073 m/s, it can be seenthat Tomiyama’s model for isolated bubbles produces large deviations for thevertical liquid velocities in the central region of the column. However, themodel predicts the vertical liquid velocity well in the upper part of the column.It is also clear that all the other models again overestimate the vertical liquidvelocity in the central region in the lower part of the column (z/H ≤ 0.5). Themodels due to Ishii & Zuber, Holland, Rusche and Lima Neto yield betterprediction with respect to the vertical liquid velocities. In the upper part ofthe column (z/H = 0.7), most of the models can predict the vertical liquid

105






106

5.6. Conclusions

velocity quite well except that the models of Wen & Yu, Holland, Simonnetand Lima Neto underestimate the vertical liquid velocity in either the left orthe right part of the column.

By comparing the profiles of the relative velocity from numerical sim-ulations with PIV measurements, it can been seen that Wen & Yu’s modeland Lima Neto’s model clearly increasingly overestimate the relative velocitybetween gas phase and liquid phase along the column height. In addition,the other models predict the relative velocity quite well in the central regioncompared to the wall region. However, one can find that these models agreewith PIV measurements at height z/H = 0.7 very nicely.

ug = 0.0097 m/s

At a superficial gas velocity ug = 0.0097 m/s, one can find that the verticalliquid velocity predicted with all the drag models is much larger than thePIV measurements in the central region in the lower part of the column. Wen& Yu’s model are much closer to the PIV data than the others. Moreover,Rusche’s model and Lima Neto’s model show moderate performance. At theheight z/H = 0.7, however, Wen & Yu’s model apparently underestimates thevertical liquid velocity in the central region of the column. In addition theresults from Holland’s model and Rusche’s model also show slightly differentprofiles compared with the PIV data. The others agree with the data quitewell.

When looking at the comparisons of the relative velocity between the twophases, one can see that Wen & Yu’s model and Lima Neto’s drag modeloverestimate the relative velocity at all three heights in the bubble column.The other models show similar trends for the relative velocity along the xdirection, particularly at the height z/H = 0.7. However, the profile of therelative velocity in the upper part of the column is reproduced.

5.6 Conclusions

In this chapter, the performance of several drag coefficient correlations pro-posed in literature has been tested by performing detailed Euler-Lagrange

107






108

5.6. Conclusions





109


simulations. In our study, we focused on the swarm effect with respect tothe effective drag experienced by a bubble rising in a swarm. The simulationresults were compared with PIV measurements to evaluate these correlations.Three heights in the bubble column were considered: z/H = 0.3, 0.5 and 0.7.The results show that Lima Neto’s drag model and Wen & Yu’s drag modelgive better prediction of both the vertical liquid velocity and the relative ve-locity between the gas phase and the liquid phase in the entire column at lowsuperficial gas velocity (i.e. ug = 0.0024 m/s). As the superficial gas velocityincreases, however, Lima Neto’s model tends to overestimate the relative ve-locity considerably even though it still can predict the vertical liquid velocitywell. Wen & Yu’s drag model shows a large deviation with the prediction ofboth the vertical liquid velocity and the relative velocity at high superficialgas velocity (i.e. ug > 0.0049 m/s. Among the other models, Rusche’s modeland Holland’s model predict the vertical liquid velocity in the lower part ofthe column better than the other three models at the moderate superficial gasvelocity (i.e. 0.0049 m/s ≤ ug ≤ 0.0073 m/s). At ug = 0.0097 m/s, however,Rusche’s model gives a better performance with the prediction of the verticalliquid velocity in the lower part of the column. In the higher part of the col-umn, these five drag models have a similar performance with the predictionof the vertical liquid velocity particularly at high superficial gas velocity. Inaddition, the five models also have a similar trend on predicting the relativevelocity between the gas phase and the liquid phase. Meanwhile, it can beseen that Rusche’s drag model predicts the relative velocity better at highsuperficial gas velocity.

The present work has compared the performance of different drag closureswith the prediction of hydrodynamics of bubbly flows. We found that LimaNeto’s drag model and Wen & Yu’s model have a better performance at lowsuperficial gas velocity. As the superficial gas velocity increases, these twomodels tend to overestimate the relative velocity between the gas phase andthe liquid phase. It can also be seen that Rusche’s model can predict the hy-drodynamics of the bubbly flows better compared to the other models at highsuperficial gas velocity. However, there is also need for further improvementof the drag coefficient correlation with swarm effects. Some factors important

110

Nomenclature

in bubbly flows as well need to be considered, i.e. the effect of bubble swarmson the lift force. Meanwhile, coalescence and breakup of bubbles should alsobe taken into account in the modelling in order to simulate the bubbly flowsreasonably at high void fraction flows.

Nomenclature

C model coefficient, [-]CS Smagorinsky constant, [-]d bubble diameter, [m]D column depth, [m]E bubble aspect ratio, [-]


σ, [-]

F force vector, [N]f functiong gravity acceleration, [m s−2]H column height, [m]; horizontal directionI unit tensor, [-]n unit normal vector, [-]p pressure, [N m−2]r position vector, [m]R bubble radius, [m]


, [-]

t time, [s]u liquid velocity vector, [m s−1]v bubble velocity vector, [m s−1]V volume, m3; terminal velocity, [m s−1]W column width, [m]

Greek letters

α void fraction, [-]

111


∆ subgrid length scale, [m]ε gas holdup, [-]µ viscosity, [kg m−1 s−1]Φ volume averaged momentum transfer due to interphase forces,

[kg m−2 s−2]ρ density, [kg m−3]σ surface tension, [N m−1]τ stress tensor, [N m−2]

Indices

b bubbleD drageff effectiveg gas phaseG gravityH horizontal directioni i directionj j directionl liquid phaseL lift; molecular viscositym mixtureP pressures superficial velocityT turbulenceV vertical directionVM virtual massW wall∞ infinite medium

112

References

Acknowledgement

The authors would like to gratefully acknowledge Dr. Lima Neto for sharinghis experimental results.

References


A. Behzadi, R. I. Issa, and H. Rusche. Modelling of dispersed bubble anddroplet flow at high phase fractions. Chemical Engineering Science, 59(4):759–770, 2004.

F. Bertola, G. Baldi, D. Marchisio, and M. Vanni. Momentum transfer in aswarm of bubbles: estimates from fluid-dynamic simulations. ChemicalEngineering Science, 59(22-23):5209–5215, 2004. ISCRE18.

R. Clift, J. R. Grace, and M. E. Weber. Bubbles, Drops, and Particles. New York[etc.] ; London : Academic Press, 1978.





113


C. Garnier, M. Lance, and J. L. Marie. Measurement of local flow characteristicsin buoyancy-driven bubbly flow at high void fraction. Experimental Thermaland Fluid Science, 26(6-7):811–815, 2002.

M. Gentric, C. Olmos, E. Midoux, and N. Simonnet. Cfd simulation of the flowfield in a bubble column reactor: Importance of the drag force formulationto describe regime transitions. Chemical Engineering and Processing: ProcessIntensification, 47(9-10):1726–1737, 2008.

F. A. Holland and R. Bragg. Fluid Flow for Chemical Engineers. Elsevier Ltd.,second edition, 1995.


M. Ishii and T. Hibiki. Thermo-fluid dynamics of two-phase flow. Springer, 2005.

M. Ishii and N. Zuber. Drag coefficient and relative velocity in bubbly, dropletor particulate flows. AIChE Journal, 25(5):843–855, 1979.


M. Jamialahmadi, C. Branch, and H. Muller-Steinhagen. Terminal bubblerise velocity in liquids. Chemical Engineering Research and Design, 72(A1):119–122, 1994.

A. Lubbert A. Lapin. Numerical simulation of the dynamics of two-phasegas-liquid flows in bubble columns. Chemical Engineering Science, 49(21):3661–3674, 1994.

I. E. Lima Neto, D. Z. Zhu, and N. Rajaratnam. Bubbly jets in stagnant water.International Journal of Multiphase Flow, 34(12):1130–1141, 2008.

K. E. Morud and B. H. Hjertager. LDA measurements and CFD modellingof gas-liquid flow in a stirred vessel. Chemical Engineering Science, 51(2):233–249, 1996.

114

References

H. Rusche. Computational fluid dynamics of dispersed two-phase flows at high phasefractions. PhD thesis, Imperial College, London, 2002.

M. Simonnet, C. Gentric, E. Olmos, and N. Midoux. Experimental determi-nation of the drag coefficient in a swarm of bubbles. Chemical EngineeringScience, 62(3):858–866, 2007.




A. Tomiyama, H. Tamai, I. Zun, and S. Hosokawa. Transverse migration ofsingle bubbles in simple shear flows. Chemical Engineering Science, 57(11):1849–1858, 2002.

J. A. Trapp. A discrete particle model for bubble-slug two-phase flows. Journalof Computational Physics, 107(2):367–377, 1993.


R. M. Wellek, A. K. Agrawal, and A. H. P. Skelland. Shape of liquid dropsmoving in liquid media. AIChE Journal, 12(5):854–862, 1966.

C. Y. Wen and Y. H. Yu. Mechanics of fluidization. Chemical EngineeringProgress Symposium Series, 62:100–111, 1966.

115


I. Zun and J. Groselj. The structure of bubble non-equilibrium movement infree-rise and agitated-rise conditions. Nuclear Engineering and Design, 163(1-2):99–115, 1996.

116

Chapter

6Discrete bubble modeling of

bubbly flows: Implementation ofbreakup models

Breakup models developed in literature were implemented in our model, which isbased on an Eulerian-Lagrangian approach. Moreover, the critical Weber number forbubble breakup utilized by many authors in turbulent flows was also incorporated inthe model. Only binary breakage is considered in this work. For the models utilizingthe critical Weber number, two different daughter size distributions, namely bell shapeand a U shape were used.First the implementation was verified by simulating bubbly flows in a square bubblecolumn. The simulated breakup frequency and daughter size distribution were com-pared with those obtained from the models. Subsequently, the simulation results werecompared with detailed PIV measurements. Finally, the predicted bubble size distri-butions were compared with chord length distributions obtained from measurementswith a four-point optical fibre probe.

117

6. Implementation of breakup models

6.1 Introduction

In two-phase gas-liquid flows, the properties of the dispersed phase are veryimportant (i.e. interface topology). The complex interface topology and itsdynamics (due to coalescence and breakup) poses a considerable difficulty inthe study of bubbly flows. Hence, mechanisms of coalescence and breakupare of special significance for the investigation of two-phase gas-liquid flows.

The deformation and breakup of a bubble can be described with a sim-ple static force balance, which introduces the ratio between the force thatcauses the deformation and the surface tension that tends to counteract thedeformation (Hinze, 1955). Depending on the type of flow, the cause of thedeformation varies. No matter what the nature of the deformation is, how-ever, the breakup occurs when the ratio exceeds a critical value. The ratiois expressed as a critical Weber number. Sevik and Park (1973) studied thecritical Weber number for air bubbles in a high Reynolds number water jet(Kolev, 2007). Walter and Blanch (1986) proposed an expression for the maxi-mum stable bubble size in solutions and conducted a comprehensive study onthe effect of the presence of surfactants on bubble breakup. Risso and Fabre(1998) analyzed the breakup mechanism of a bubble in turbulent flows undermicrogravity conditions and estimated the critical Weber number. Qian et al.(2006) studied the critical Weber number of bubbles in homogeneous turbu-lence under zero gravity conditions using the lattice Boltzmann method. Adetailed review of the mechanisms of deformation and breakup of drops andbubbles has been given by Risso (2000).

In recent years, CFD has emerged as a powerful tool to study multiphaseflow phenomena and assess their impact on the performance of multiphasechemical reactors (Jakobsen, 2008). In particular population balance mod-elling has received considerable attention in the past decade to account forthe size distribution of the dispersed phase. In order to close the populationbalance equations, i.e. through expressions for the birth rate and the deathrate due to breakup of bubbles, many efforts on the mathematical formulationof coalescence and breakup models in turbulent flows have been made sincethen. Based on the turbulent nature of liquid-liquid dispersion, a phenomeno-

118


logical model was developed to describe drop breakup by Coulaloglou andTavlarides (1977). The basic premise is that an oscillating deformed drop willbreak if the turbulent kinetic energy transmitted to the drop by turbulent ed-dies exceeds the drop surface energy. The breakup frequency of a drop of sized was defined as:

Ω(d) =(

1Breakup time

) (fraction of drops breaking

)(6.1)

Other formulations of breakup mechanisms in turbulent flows have beenpresented by Mihail and Straja (1986), Lee et al. (1987), Prince and Blanch(1990), Tsouris and Tavlarides (1994), Luo and Svendsen (1996), Sathyagalet al. (1996), Kostoglou et al. (1997), Martınez-Bazan et al. (1999a), Martınez-Bazan et al. (1999b), Hagesaether et al. (2002), Lehr et al. (2002), Wang et al.(2003) and Kostoglou and Karabelas (2005). Moreover, detailed reviews aboutbreakage models can be found in Lasheras et al. (2002) and Liao and Lucas(2009).

Population balance modelling coupled to the two-fluid approach has beenfrequently used to study the multiphase flows (Lehr and Mewes, 2001; Olmoset al., 2001; Chen et al., 2005; Wang et al., 2006; Bannari et al., 2008). Howeverfor the Eulerian-Lagrangian approach, there are barely studies on hydrody-namics of two-phase gas-liquid flows with breakup models integrated intothe model.

In the present work, an attempt has been made to integrate two breakupmodels from literature into an Eulerian-Lagrangian (EL) framework. In addi-tion, an expression for the critical Weber number for bubble breakup reportedin literature has been adopted. The simulation results were compared withPIV measurements to evaluate the performance of the EL model incorporatingcoalescence and breakup.


Our discrete bubble model (DBM) was originally developed by Delnoij et al.(1997, 1999). The liquid phase is described by volume-averaged continuity

119


and Navier-Stokes equations, while the motion of each individual bubble istracked in a Lagrangian fashion.


The motion of each individual bubble is computed from Newton’s secondlaw. For an individual bubble, the equation of motion can be written as:

ρgVdvdt= ΣF (6.2)

drdt= v (6.3)




FG = ρgVg (6.5)


FP = −V∇P (6.6)


FD = −18

CDρlπd2|v − u|(v − u) (6.7)

The drag coefficient CD is determined by taking swarm effect into account.According to Rusche (2002), the drag coefficient in a swarm of bubble can beexpressed as the product of the drag coefficient for an isolated bubble risingin an infinite quiescent liquid and a correction coefficient:

CD = CD∞ f (αg) (6.8)

120


where f (αg) represents the effect arising from the presence of other bubbles:

f (αg) = exp(3.64αg) + α0.864g (6.9)

The drag coefficient for an isolated bubble CD∞ is calculated according toTomiyama et al. (1998):

CD∞ = max[min

[ 16Re

(1 + 0.15Re0.687),48Re

],

83

EoEo + 4

](6.10)

A bubble rising in a non-uniform liquid flow field experiences a lift forcedue to vorticity or shear in the flow field. The shear induced lift force actingon a bubble is usually written as (Auton, 1987):

FL = −CLρlV(v − u) × ∇ × u (6.11)

The lift coefficient is calculated according to Tomiyama et al. (2002):

CL =


f (EoH) 4 ≤ EoH ≤ 10,

−0.29 EoH > 10.

(6.12)

where

f (EoH) = 0.00105EoH3 − 0.0159EoH

2 + 0.474 (6.13)

EoH is the Eotvos number defined by using the maximum horizontal dimen-sion of a bubble:


H

σ(6.14)


E =dV

dH=

11 + 0.163Eo0.757 (6.15)



σ.

121



d = (dVd2H)1/3 (6.16)


FVM = −CVMρlV(Dv

Dt− Du

Dt

)(6.17)

where the D/Dt operators denote the substantiative derivatives pertaining tothe respective phase. In the present work, bubbles are assumed to have aspherical shape and a virtual mass coefficient of 0.5 is used.


FW = −12

CWd[

1y2 −

1(L − y)2

]ρl|(v − u)·nz|2nW (6.18)


CW =

exp(−0.933Eo + 0.179) 1 ≤ Eo ≤ 5,

0.007Eo + 0.04 5 < Eo ≤ 33.(6.19)


The presence of the bubbles is reflected by the liquid phase volume fractionαl and the interphase momentum transfer rate Φ in the volume-averagedconservation equations:

∂

∂t(αlρl

)+ ∇ · (αlρlu

)= 0 (6.20)

∂∂t

(αlρlu

)+ ∇ · (αlρluu

)= −αl∇p − ∇ · (αlτl) + αlρlg +Φ (6.21)

122

6.3. Coalescence model


τl = −µeff,l

[((∇u) + (∇u)T − 2

3I(∇ · u)

](6.22)



where µT,l is the turbulent viscosity (or eddy viscosity), which is determinedfrom turbulence modeling of the liquid phase.


In the present paper, a hard sphere collision model (Hoomans et al., 1996)is adopted to describe the bouncing of bubbles. It consists of two mainparts; i) processing the sequence of collisions and ii) dealing with the collisiondynamics. The former is described in detail by Darmana et al. (2006). Thisapproach is also used in this work.

6.3 Coalescence model

Bubble coalescence is considered by comparing the film drainage time andthe contact time between two bubbles (Prince and Blanch, 1990). The filmdrainage time can be obtained with a simple expression:

ti j =

√Ri jρl

16σln

h0

h f(6.24)

where h0 is the initial film thickness which is assumed to be 1 × 10−4 m for anair-water system. The critical film thickness where rupture occurs is given as1 × 10−8 m. The equivalent bubble radius for two different sized bubbles isgiven by Hofman and Chesters (1982):

Ri j =12

(1Ri+

1R j

)−1

(6.25)

123


The contact time is dependent on the bubble size and the turbulent intensity.An estimate of the contact time in turbulent flows is given as:

τi j =R2/3

i j

ε1/3(6.26)

where ε is the turbulent energy dissipation rate.Sommerfeld et al. (2003) provided an estimate of the contact time by as-

suming that it is proportional to a deformation distance divided by the normalcomponent of the relative velocity:

τi j =CCRi j

|vni − vnj|(6.27)

where CC is a model constant.The properties of the new bubble in case of coalescence are obtained from

conservation of mass and momentum.

6.4 Breakup models & implementation

6.4.1 Breakup models

Breakup frequency

According to Martınez-Bazan et al. (1999a), the basic premise for a bubble tobreak is that its surface has to deform, and enough energy must be providedby the turbulent stresses of the surrounding liquid. They postulated that thebreakup frequency is proportional to the difference between the turbulentstresses 1

2ρlβ(εd)2/3 and the surface pressure 6σ/d. In other words, the largerthe difference is, the larger the probability that the bubble will break in acertain time. On the other hand, the breakup frequency should decrease tozero in case this difference vanishes. Thus, the breakup frequency is estimatedas:

Ω(d) =0.25

d

√β(εd)2/3 − 12σ

ρld(6.28)

where the constant β = 8.2 was given by Batchelor (1953) and the constant0.25 was found experimentally by Martınez-Bazan et al. (1999b).

124

6.4. Breakup models & implementation

The dependence of the breakup frequency on the bubble diameter is shownin Figure 6.1(a). The breakup frequency possesses a maximum as the bubblediameter increases and after reaching the maximum, the breakup frequencydecreases monotonically with increasing bubble size.

ε=0.5 m2/s3

ε=1.0 m2/s3

ε=2.0 m2/s3

Ω(d) [1/s]

0

5

10

15

20

25

d [m]0 0.02 0.04 0.06 0.08 0.1

(a) Martınez-Bazan et al. (1999a)

ε=0.5 m2/s3

ε=1.0 m2/s3

ε=2.0 m2/s3

Ω(d) [1/s]

0

5,000

1e+04

1.5e+04

2e+04

d [m]0 0.02 0.04 0.06 0.08 0.1

(b) Lehr et al. (2002)

Figure 6.1: Breakup frequency vs bubble diameter.

Lehr et al. (2002) assumed that the breakup of a bubble is determinedby the balance between the interfacial force acting at the bubble surface andthe inertial force of an impinging eddy. The interfacial force depends on theshape of the bubble and on the size of daughter bubbles. They computedthe breakup probability based on the criterion that the kinetic energy of theeddy exceeds a critical energy which is obtained from the force balance. Thebreakup frequency is then expressed as:

Ω(d) = 0.5d5/3ε19/15ρ7/5

l

σ7/5exp

−√

2σ9/5

d3ρ9/5l ε

6/5

(6.29)

As shown in the Figure 6.1(b), the breakup frequency increases monoton-ically with increasing bubble diameter and there is no maximum, which isdifferent from that of Martınez-Bazan et al. (1999a).

125


Daughter size distribution

By assuming that a bubble only breaks into two bubbles, Martınez-Bazan et al.(1999b) postulated that the probability of the formation of a pair of bubblesof sizes d1 and d2 from a mother bubble of size d, P(d∗), is weighted by theproduct of two surplus stresses which are associated with the formation of thetwo daughter bubbles. Hence, the daughter size distribution can be writtenas:

β(d∗) =P(d∗)∫ 1

0 P(d∗) dd∗(6.30)

and

P(d∗) =(d∗2/3 − 12σ

8.2ρld5/3ε2/3

) [(1 − d∗3)2/9 − 12σ

8.2ρld5/3ε2/3

](6.31)

where d∗ = d1/d. Note that the dimensionless diameter d∗ can easily beconverted to a dimensionless volume V∗: V∗ = d∗3. Hence, for the sake ofclarity, only the dimensionless volume V∗ is considered.

The daughter size distribution according to Martınez-Bazan et al. (1999b)is plotted in Figure 6.2(a). One can see that the daughter size distribution hasa bell shape. In other words, it is more likely for a mother bubble to break intotwo bubbles with equal volume. Moreover the distribution becomes wider asthe mother bubble size increases.

Lehr et al. (2002) described the daughter size distribution as a lognormaldistribution which gives:

β(V∗) =

exp

−94

ln22/5d1ρ

3/5l ε

2/5

σ3/5

2

V∗√π

1 + erf

32

ln

21/15dρ3/5l ε

2/5

σ3/5

(6.32)

where V∗ = V1/V.As shown in Figure 6.2(b), unequal-size breakage is more likely according

to the above daughter size distribution. And the probability of unequal-sizedbreakage increases rapidly with the mother bubble size.

126

6.4. Breakup models & implementation

d=0.005 m

d=0.007 m

d=0.01 m

β(V

*)

0

1

2

3

V* [-]0 0.2 0.4 0.6 0.8 1

(a) Martınez-Bazan et al. (1999b)

d=0.005 m

d=0.007 m

d=0.01 m

β(V

*)

0

5

10

15

20

25

V* [-]0 0.2 0.4 0.6 0.8 1


Figure 6.2: Daughter size distribution.

6.4.2 Critical Weber number

When a bubble moves in a turbulent flow, the turbulent eddies deform itssurface and may eventually cause its breakup. According to Hinze (1955),only the eddies with length scale smaller than the bubble size are capable tobreak the bubble while larger eddies merely transport the bubble. This leadsto a quantitative description of turbulence induced bubble breakup. Theturbulent shear stress imposed by the continuous phase deforms the bubbleand breaks it if it overcomes the surface tension force. The relevant Webernumber is thus defined as:

We =ρlu′2dσ

(6.33)

By assuming that the turbulence is isotropic at the bubble scale and the bubblediameter belongs to the inertial turbulent subrange, the Weber number isexpressed as a function of the dissipation rate ε (Risso, 2000):

We =2ρlε2/3d5/3

σ(6.34)

For the breakup of a gas jet in a turbulent liquid stream, the critical Webernumber is given as We = 2.48 (Kolev, 2007). By investigating the behaviorof bubbles of different diameters under microgravity conditions, Risso andFabre (1998) found that the critical Weber number ranges between 2.7 and

127


7.8. They also obtained a value of the critical Weber number close to 4.5 byanalyzing how the mean deformation is related to the mean turbulence in-tensity. Hence, they suggested that a minimal Weber number of about 5 isnecessary for breakup when the turbulence is the only cause of deformation.Qian et al. (2006) performed lattice Boltzmann simulations for the deforma-tion and breakup of bubbles in homogeneous turbulence under zero gravityconditions. The minimum Weber number for bubble breakup was found tobe about 3.0.

The beta distribution can be used to describe the daughter bubble sizewhen a critical Weber number is involved to determine bubble breakup in theturbulent flows. The beta distribution is formulated as follows:

f (V∗) =Γ(a + b)Γ(a)Γ(b)

V∗a−1(1 − V∗)b−1 (6.35)

where Γ is the gamma function and a and b shape parameters.

a=b=0.5

a=b=2.0

f(V*)

0

0.5

1

1.5

2

2.5

V* [-]0 0.2 0.4 0.6 0.8 1

Figure 6.3: Beta distributions.

In order to obtain different daughter size distributions, two different setsof shape parameters of the beta distribution, a = b = 0.5 and a = b = 2.0,were used. Hence, two different shapes of the beta distributions, such asbell shape and U shape, can be obtained, as shown in Figure 6.3. By using

128


different daughter size distributions, one can get equal-sized breakage of amother bubble with either the highest probability or the lowest probability.

6.4.3 Implementation

The above breakup models provide quantitative information on dynamicsof bubble breakage in turbulent flows. For instance, the breakup frequencyis related to the time interval in which the bubble breakage occurs. Hence,one may find a way to integrate the bubble breakage into the discrete bubblemodel (DBM) by taking the breakup time into account.

In the discrete bubble model, each bubble is tracked individually in theturbulent flow field. The bubble properties, such as bubble velocity, bubblediameter and bubble position in the flow, and local flow properties, i.e. dis-sipation rate of turbulence kinetic energy, are known. Hence, it is possibleto determine the breakup time for each bubble at its position in the flow. Ifthe breakup time of a bubble is smaller than the computational time step, thebubble breaks immediately within that time step. On the other hand, if thecalculated breakup time of the bubble exceeds the computational time step,the program starts counting the time since that instant. The computationmoves to the next time step and once the counted time exceeds the breakuptime, the bubble breaks. In this approach, the history of turbulent eddiescolliding with the bubble is considered.

The resulting daughter bubble sizes are determined according to the aboveintroduced daughter size distributions. The velocities of the two daughterbubbles are assumed to be the same as that for the mother bubble. One of thetwo daughter bubbles is located at the same position as the mother bubblewhile the other daughter bubble is located around that daughter bubble ran-domly. After breakage, each bubble in the turbulent flow field is then trackedby considering the breakage criterion again.


The bubble column studied here is shown in Figure 6.4. The cross-sectionalarea of the column is 0.15 m × 0.15 m (W × D). The column is initially filled

129


with water to a height of 0.45 m (H). Air is used as the dispersed phase andintroduced into the column through a perforated plate with 49 holes (7× 7) atthe bottom of the column. The material properties of both phases are taken atroom temperature and atmospheric pressure.


A computational grid with (20×20×60) cells is adopted in the simulations.Three superficial gas velocities, i.e. 0.0049, 0.0073 and 0.0097 m/s were used.The initial (i.e. at the distributor) bubble diameter in the simulations is 0.005 m,which is consistent with experimental observations in a bubble column atthe applied superficial gas velocity. The computational time step for the

130


liquid phase is set at 1.0 × 10−3 s and the collisions among bubbles and themovements of the bubbles are processed multiple times within each timestep. The total simulation time is 150 s.

The coalescence model and two breakup models described above are usedin the simulations. Moreover, two critical Weber numbers, i.e. We = 2.48 andWe = 5.0, are adopted for the purpose of comparison.

The sub-grid scale model is required to represent the turbulent flow in-side the bubble column. This means that the continuity and Navier-Stokesequations are resolved for the “large” eddies, while the ”small scales” of theresolved filed are included through an eddy-viscosity subgrid-scale model.

The eddy-viscosity model proposed by Vreman (2004) was used to calcu-late the eddy viscosity:

µT,l = 2.5ρlC2S

√Bβαi jαi j

(6.36)

where Bβ = β11β22 − β212 + β11β33 − β2

13 + β22β33 − β223, βi j = ∆

2mαmiαmj and

αi j = ∂u j/∂xi. ∆i is the filter width in the i direction.In the large eddy simulations, however, the dissipation rate of turbulent

kinetic energy cannot be obtained directly. Moreover, the dissipation rate onlyfrom the resolved scales is negligible compared with that from the subgridscales. By assuming a local equilibrium between turbulent kinetic energyproduction and dissipation, the dissipation rate can be estimated as (Jimenezet al., 2001; Hartmann et al., 2004; Delafosse et al., 2009):

ε =(µL,l + µT,l)

2ρl

(∂ui

∂x j+∂u j

∂xi

)2

(6.37)


6.6.1 Validation of the implementation

Before discussing the performance of different breakup models with respectto prediction of bubbly flow characteristics, it is necessary to first verify theimplementation of the models, such as, breakup frequency and daughter sizedistribution. In Figure 6.5, the breakup frequency of each breaking bubble

131


with size d at a certain dissipation rate in a numerical simulation is comparedwith the breakup frequency curve calculated from the corresponding breakupmodel at the same dissipation rate. Three different energy dissipation rates,ε = 0.5, 1.0 and 2.0 m2/s3, are chosen for the comparison. One can clearly seethat the breakup frequencies obtained from the numerical simulation agreevery well with the curve calculated from the corresponding breakup model.This indicates that the breakup frequency models have been implementedcorrectly.

Model (ε=0.5 m2/s3)



DBM (ε=0.5 m2/s3)

DBM (ε=1.0 m2/s3)

DBM (ε=2.0 m2/s3)

Ω(d) [1/s]

0

10

20

30

40

d [m]0 0.005 0.01 0.015 0.02

(a) Martınez-Bazan et al. (1999a)




DBM (ε=0.5 m2/s3)

DBM (ε=1.0 m2/s3)

DBM (ε=2.0 m2/s3)

Ω(d) [1/s]

0

200

400

600

d [m]0 0.005 0.01 0.015


Figure 6.5: Verification of implementation for the breakup frequency.

Moreover, the daughter size distribution obtained from the numericalsimulations is also compared with the probability distributions of bubblesizes obtained from the breakup model (see Figure 6.6). It can be seen that thebubble size distribution obtained from the numerical simulations reproducesthe trend of the corresponding model very well. In addition for the caseswith critical Weber numbers, the bell-shaped and U-shaped daughter sizedistributions are also reproduced well.

6.6.2 Comparison of different breakup models

In order to evaluate the performance of the above breakup models and thecritical Weber numbers for bubble breakup, the coalescence model in combi-nation with the hydrodynamic model, averaged quantities, such as vertical

132


Model (d=0.005 m) DBM (d=0.005 m)

Model (d=0.007 m) DBM (d=0.007 m)

Model (d=0.01 m) DBM (d=0.01 m)

β(V

*)

0

1

2

3

V* [-]0 0.2 0.4 0.6 0.8 1

(a) Martınez-Bazan et al. (1999b)

Model (d=0.005 m)

Model (d=0.007 m)

Model (d=0.01 m)

DBM (d=0.005 m)

DBM (d=0.007 m)

DBM (d=0.01 m)

β(V

*)

0

5

10

15

20

25

V* [-]0 0.2 0.4 0.6 0.8 1


β(V

*)

0

0.02

0.04

0.06

0.08

V* [-]0 0.2 0.4 0.6 0.8 1

(c) Bell-shaped distribution

β(V

*)

0.05

0.1

0.15

V* [-]0 1

(d) U-shaped distribution

Figure 6.6: Verification of implementation for the daughter size distribution.

liquid velocity and relative velocity between the gas phase and the liquidphase, will be considered. Comparisons are carried out at three heights inthe column (z/H = 0.3, 0.5 and 0.7). Moreover, simulations assuming uni-form bubble size with and without considering swarm effects are included forthe purpose of reference. In the following figures, the symbols representingdifferent conditions of simulations are kept the same.

ug = 0.0049 m/s

The average vertical liquid velocity and the local void fraction obtained fromthe simulations with superficial gas velocity ug = 0.0049 m/s are plotted in

133


Figure 6.7. It can be seen that the value of the critical Weber number influencesthe vertical liquid velocity considerably, particularly in the central region ofthe bubble column. Assuming the same daughter size distribution, i.e. thebell-shaped distribution, the vertical liquid velocities predicted with We = 5.0are higher than those predicted with We = 2.48. Furthermore, assuming thesame critical Weber number, i.e. We = 2.48, but with different daughter sizedistributions, i.e. bell shape and U shape, the difference between the predictedand experimental vertical liquid velocities in the central region is getting largeralong column height. Martınez-Bazan’s breakup model produces similar per-formance regarding the prediction of the vertical liquid velocity assuming thecritical Weber number We = 2.48 and a bell-shaped daughter size distribu-tion. Lehr’s breakup model produces a relatively large deviation. For thecase of uniform bubble size with a single bubble drag closure also a largediscrepancy with the experiments is obtained. However, it can be seen that atthe relatively low superficial gas velocity, i.e. ug = 0.0049 m/s, application ofa uniform bubble size considering the swarm effects on the drag force givesbetter predictions of the vertical liquid velocity in the central region of thecolumn.

Comparison of local void fraction reveals that the local void fractionspredicted with the critical Weber number, We = 5.0, in the center is largerthan those predicted with other models. The predicted local void fractionsassuming an uniform bubble size considering the swarm effect and criticalWeber number, We = 2.48, with bell-shaped daughter size distribution areclose to each other.

ug = 0.0073 m/s

In Figure 6.8, profiles of the vertical liquid velocity and the local void fractionat three heights in the bubble column are compared. The simulation with acritical Weber number, We = 5.0, again shows the largest discrepancy with re-spect to the prediction of the vertical liquid velocity in the center. In the centralregion, Martınez-Bazan’s model and Lehr’s model have similar performancecompared to the case with the uniform size and single bubble drag closure inthe lower part of the column (z/H ≤ 0.5). Moreover, the models with a critical

134


(a) Vertical liquid velocity (z/H = 0.3) (b) Local void fraction (z/H = 0.3)

(c) Vertical liquid velocity (z/H = 0.5) (d) Local void fraction (z/H = 0.5)

(e) Vertical liquid velocity (z/H = 0.7) (f) Local void fraction (z/H = 0.7)


135


Weber number, We = 2.48, predict better the liquid velocity in the lower partof the column. However, they underestimate the liquid velocity in the higherparts of the column. The model with uniform bubble size considering theswarm effect predicts the liquid velocity quite well in the bubble column.

Profiles of the local void fraction show that the model with the criticalWeber number, We = 5.0, gives the highest void fraction in the central regionof the column. Furthermore, the local void fractions obtained from the modelswith the critical Weber number, We = 2.48, are close to those predicted withthe model with uniform bubble size considering the swarm effect in the lowerpart of the column (z/H ≤ 0.5). However, they start to deviate from each otheralong the column height.

ug = 0.0097 m/s

At large superficial gas velocity, ug = 0.0097 m/s, profiles of the vertical liquidvelocity and the local void fraction obtained from the above models (Fig-ure 6.9) show some differences compared with low superficial gas velocity.It can be seen that the model with the critical Weber number, We = 5.0, canpredict the vertical liquid velocity quite well in the central region of the bubblecolumn. On the contrary, the model with critical Weber number, We = 2.48and bell-shaped daughter size distribution and the model with uniform bub-ble size considering swarm effects produce large deviations in the lower partof the column. However, the former predicts the profile of the vertical liquidvelocity in the higher part of the column (i.e. z/H = 0.7) well. The model withuniform bubble size and single bubble drag closure has the largest deviationhowever in the higher part of the column this model performs quite well.Martınez-Bazan’s model, Lehr’s model and the model with the critical Webernumber, We = 2.48 and U-shaped daughter size distribution show similarperformance in predicting the liquid velocity in the lower part of the bubblecolumn (z/H ≤ 0.5). With increasing column height, however, these modelsshow a worse performance.

In addition, one can see that the local void fractions obtained from themodel with uniform bubble size are slightly higher than those obtained fromthe other models in the central region. In the lower part of the column, the

136






137


model with uniform bubble size and swarm effects predicts higher values ofthe local void fraction than the others in the central region. The model thatuses the critical Weber number We = 2.48 and the U-shaped daughter sizedistribution has similar performance as Martınez-Bazan’s model. Along thecolumn height, differences between the models are getting more pronounced.





138


6.6.3 Bubble size distribution

In order to evaluate the performance of the breakup models in combina-tion with a coalescence model with respect to the prediction of bubblesize, the bubble size distributions at two positions in the bubble column(z/H = 0.55, x/W = 0.5 and z/H = 0.75, x/W = 0.5) are compared with thoseobtained from a four-point optical fibre probe at corresponding positions. InChapter 3, we introduced Uga (1972) and other authors’ work on the deter-mination of the bubble size from a measured chord length distribution. Forinstance, by assuming that all bubbles are spherical and travel in the same di-rection with the same average velocity, Uga (1972) determined the equivalentbubble diameter de from the mean chord length as:

de =

∫ ∞

0d f ′(d) dd =

32

dc =32

∫ ∞

0dc f (dc) ddc (6.38)

where dc is the chord length.We hereby assume that the probability density function of the bubble size

is the same as that of the chord length, that is, f ′(d) = f (dc). Therefore, thebubble size can be estimated as d = 1.5dc roughly and the bubble size distribu-tion obtained from the four-point optical fibre probe can then be plotted in theFigure 6.10. The estimated bubble size distributions are stretched to the rightside from the chord length distributions in the plots. The bubble size distri-butions also show that the bubble size d = 0.005 m has the highest probabilityin the bubble column approximately. Moreover, due to the geometry of thefour-point optical fibre probe, those bubbles with sizes or the chord lengthspierced by the probe smaller than 0.0015 m can not be detected. Hence, thereare no data points for those bubbles in the plots.

Subsequently, the bubble size distributions determined from the simula-tion results are then compared with those estimated from the chord lengthdistributions using the measurements with the four-point optical fibre probe,as shown in the Figure 6.11.

When comparing the distributions at larges bubble size, it is found thatthe bubble size distributions predicted by the above models have very similartrends with those obtained from the optical probe except for Martınez-Bazan’s

139


Chord length

Bubble diameter

Distribution [-]

0

0.1

0.2

0.3

0.4

0.5

Dimension [m]0 0.005 0.01 0.015 0.02

(a) z/H = 0.55, x/W = 0.5

Chord length

Bubble diameter

Distribution [-]

0

0.1

0.2

0.3

0.4

Dimension [m]0 0.005 0.01 0.015 0.02

(b) z/H = 0.75, x/W = 0.5

Figure 6.10: Distributions of the chord length and the bubble size.

model. At the central part of the column, the bubble size distribution pre-dicted by Martınez-Bazan’s model has higher probability than others at aboutd = 0.007 m. At the higher part of the column, the distributions obtainedfrom the numerical simulations are slightly shifted to the larger bubble sizecompared with that determined from the optical probe. That suggests thatthe implemented coalescence and breakup models into the discrete bubblemodel are able to predict the bubble size distribution reasonably well. Fur-thermore, the model that uses a U-shaped daughter size distribution has ahigh probability to obtain some very small bubbles, which suggests that theimplementation of the U-shaped daughter size distribution is successful.

6.7 Conclusions

In this work, coalescence and breakup models have been combined with adetailed Euler-Lagrange (EL) model incorporating four-way coupling. Theimplementation of the breakup models in the EL model is based on the con-cept of breakup frequency (breakup time). By considering the breakup timefor bubbles in turbulent flows and the history of bubble and turbulent eddiesencounter, bubble breakup is incorporated into the discrete bubble model.Only binary breakup is considered in the breakup models. The daughter size

140

6.7. Conclusions

0.000 0.005 0.010 0.015 0.020 0.0250.0

0.1

0.2

0.3

0.4

0.5

Bu

bb

le s

ize

dis

trib

utio

n

d [m]

We=2.48, bell shape

We=2.48, U shape

We=5.0, bell shape

Optical probe

(a) z/H = 0.55, x/W = 0.5

0.000 0.005 0.010 0.015 0.020 0.0250.0

0.1

0.2

0.3

0.4

0.5

Bu

bb

le s

ize

dis

trib

utio

n

d [m]

Martinez-Bazan, 1999

Lehr, 2002

Optical probe

(b) z/H = 0.55, x/W = 0.5

0.000 0.005 0.010 0.015 0.0200.0

0.1

0.2

0.3

0.4

Bu

bb

le s

ize

dis

trib

utio

n

d [m]

We=2.48, bell shape

We=2.48, U shape

We=5.0, bell shape

Optical probe

(c) z/H = 0.75, x/W = 0.5

0.000 0.005 0.010 0.015 0.0200.0

0.1

0.2

0.3

0.4

Bu

bb

le s

ize

dis

trib

utio

n

d [m]

Martinez-Bazan, 1999

Lehr, 2002

Optical probe

(d) z/H = 0.75, x/W = 0.5

Figure 6.11: Bubble size distribution.

distribution after bubble breakage is determined according to models avail-able in literature. In addition, critical Weber numbers for bubble breakup inturbulent flow reported in literature have been adopted as well. Two differentdaughter size distributions, i.e. bell shaped and U shaped distribution, wereused in this case.

The correct incorporation of the breakup models is first verified by com-paring the breakup frequency and the daughter size distribution obtainedfrom both numerical simulations and the underlying breakup models. Theresults show that the implementation of the breakup models was correct.

141


Furthermore, the comparison of the simulated and experimental average ver-tical liquid velocity revealed that the simulations considering coalescenceand breakup of bubbles show quite different results when compared with themodel without considering coalescence and breakup. For instance, the modelthat uses the critical Weber number We = 2.48 gives better predictions on thevertical liquid velocity in the lower part of the bubble column. Meanwhile,it is also found that the model with uniform bubble size and swarm effectperform better at low superficial gas velocity (ug = 0.0049 m/s).

In addition, the bubble size distributions obtained from the numericalsimulation were compared with those estimated from the chord length dis-tributions with a four-point optical fibre probe. The results suggest that theincorporation of both coalescence and breakup models in the discrete bubblemodel produces reasonable results with respect to the evolution of bubblesize in bubbly flows.

Nomenclature

a shape parameter of beta distribution, [-]b shape parameter of beta distribution, [-]C model coefficient, [-]d bubble diameter, [m]D column depth, [m]


σ, [-]

f function, [-]F force vector, [N]g gravity acceleration, [m s−2]h0 initial film thickness, [m]h1 critical film thickness, [m]H column height, [m]I unit tensor, [-]n unit normal vector, [-]p pressure, [N m−2]

142

Nomenclature

r position vector, [m]R bubble radius, [m]


, [-]

t time, [s]u liquid velocity vector, [m s−1]v bubble velocity vector, [m s−1]V bubble volume, [m3]V∗ relative bubble volume, [-]W column width, [m]We Weber number, [-]

Greek letters

α void fraction, [-]β daughter size distribution, [-]ε turbulent kinetic energy dissipation rate, [m2 s−3]Γ gamma distribution, [-]µ viscosity, [kg m−1 s−1]Φ volume averaged momentum transfer due to interphase forces,

[kg m−2 s−2]Ω breakup frequency, [s−1]ρ density, [kg m−3]σ surface tension, [N m−1]τ stress tensor, [N m−2]; contact time, [s]

Indices

∆ subgrid length scale, [m]D drageff effectiveg gas phaseG gravityH horizontal direction

143


i bubble indexj bubble indexl liquid phaseL lift; molecular viscositym mixtureP pressureT turbulenceVM virtual massW wall∞ infinite medium

References


R. Bannari, F. Kerdouss, B. Selma, A. Bannari, and P. Proulx. Three-dimensional mathematical modeling of dispersed two-phase flow usingclass method of population balance in bubble columns. Computers andChemical Engineering, 32(12):3224–3237, 2008.

G. K. Batchelor. The theory of homogeneous turbulence. Cambridge UniversityPress, 1953.

P. Chen, M. P. Dudukovic, and J. Sanyal. Three-dimensional simulation ofbubble column flows with bubble coalescence and breakup. AIChE Journal,51(3):696–712, 2005.

C. A. Coulaloglou and L. L. Tavlarides. Description of interaction processesin agitated liquid-liquid dispersions. Chemical Engineering Science, 32(11):1289–1297, 1977.


144

References

A. Delafosse, J. Morchain, P. Guiraud, and A. Line. Trailing vortices generatedby a Rushton turbine: Assessment of URANS and large Eddy simulations.Chemical Engineering Research and Design, 87(4):401–411, 2009.



L. Hagesaether, H. A. Jakobsen, and H. F. Svendsen. A model for turbulentbinary breakup of dispersed fluid particles. Chemical Engineering Science, 57(16):3251–3267, 2002.

H. Hartmann, J. J. Derksen, C. Montavon, J. Pearson, I. S. Hamill, and H. E. A.van den Akker. Assessment of large eddy and RANS stirred tank simu-lations by means of LDA. Chemical Engineering Science, 59(12):2419–2432,2004.

J. O. Hinze. Fundamentals of the hydrodynamic mechanism of splitting indispersion processes. AIChE Journal, 1(3):289–295, 1955.

A. K. Hofman and G. Chesters. Bubble coalescence in pure liquids. AppliedScientific Research, 38(1):353–361, 1982.



C. Jimenez, F. Ducros, B. Cuenot, and B. Bedat. Subgrid scale variance anddissipation of a scalar field in large eddy simulations. Physics of Fluids, 13(6):1748–1754, 2001.

145


N. I. Kolev. Multiphase Flow Dynamics 2: Thermal and Mechanical Interactions.Springer, 3rd edition, 2007.

M. Kostoglou and A. J. Karabelas. Toward a unified framework for the deriva-tion of breakage functions based on the statistical theory of turbulence.Chemical Engineering Science, 60(23):6584–6595, 2005.

M. Kostoglou, S. Dovas, and A. J. Karabelas. On the steady-state size distri-bution of dispersions in breakage processes. Chemical Engineering Science,52(8):1285–1299, 1997.

J. C. Lasheras, C. Eastwood, C. Martınez-Bazan, and J. L. Montaes. A reviewof statistical models for the break-up an immiscible fluid immersed into afully developed turbulent flow. International Journal of Multiphase Flow, 28(2):247–278, 2002.

C. H. Lee, L. E. Erickson, and L. A. Glasgow. Bubble breakup and coalescencein turbulent gas-liquid dispersions. Chemical Engineering Communications,59(1-6):65–84, 1987.

F. Lehr and D. Mewes. A transport equation for the interfacial area densityapplied to bubble columns. Chemical Engineering Science, 56(3):1159–1166,2001.

F. Lehr, M. Millies, and D. Mewes. Bubble-size distributions and flow fieldsin bubble columns. AIChE Journal, 48(11):2426–2443, 2002.

Y. Liao and D. Lucas. A literature review of theoretical models for drop andbubble breakup in turbulent dispersions. Chemical Engineering Science, 64(15):3389–3406, 2009.

H. Luo and H. F. Svendsen. Theoretical model for drop and bubble breakupin turbulent dispersions. AIChE Journal, 42(5):1225–1233, 1996.

C. Martınez-Bazan, J. L. Montanes, and J. C. Lasheras. On the breakup of anair bubble injected into a fully developed turbulent flow. Part 1. Breakupfrequency. Journal of Fluid Mechanics, 401:157–182, 1999a.

146

References

C. Martınez-Bazan, J. L. Montanes, and J. C. Lasheras. On the breakup of anair bubble injected into a fully developed turbulent flow. Part 2. Size PDFof the resulting daughter bubbles. Journal of Fluid Mechanics, 401:183–207,1999b.

R. Mihail and S. Straja. A theoretical model concerning bubble size distribu-tions. The Chemical Engineering Journal, 33(2):71–77, 1986.

E. Olmos, C. Gentric, Ch. Vial, G. Wild, and N. Midoux. Numerical simulationof multiphase flow in bubble column reactors. Influence of bubble coales-cence and break-up. Chemical Engineering Science, 56(21-22):6359–6365, 2001.

M. J. Prince and H. W. Blanch. Bubble coalescence and break-up in air-spargedbubble columns. AIChE Journal, 36(10):1485–1499, 1990.

D. Qian, J. B. McLaughlin, K. Sankaranarayanan, S. Sundaresan, and K. Kon-tomaris. Simulation of bubble breakup dynamics in homogeneous turbu-lance. Chemical Engineering Communications, 193(8):1038–1063, 2006.

F. Risso. Mechanisms of deformation and breakup of drops and bubbles.Multiphase Science and Technology, 12(1):1–50, 2000.

F. Risso and J. Fabre. Oscillations and breakup of a bubble immersed in aturbulent field. Journal of Fluid Mechanics, 372:323–355, 1998.


A. N. Sathyagal, D. Ramkrishna, and G. Narsimhan. Droplet breakage instirred dispersions. breakage functions from experimental drop-size distri-butions. Chemical Engineering Science, 51(9):1377–1391, 1996.


147





C. Tsouris and L. L. Tavlarides. Breakage and coalescence models for dropsin turbulent dispersions. AIChE Journal, 40(3):395–406, 1994.

T. Uga. Determination of bubble-size distribution in a BWR. Nuclear Engineer-ing and Design, 22(2):252–261, 1972.


J. F. Walter and H. W. Blanch. Bubble break-up in gas-liquid bioreactors:Break-up in turbulent flows. The Chemical Engineering Journal, 32(1):B7–B17,1986.

T. Wang, J. Wang, and Y. Jin. A novel theoretical breakup kernel function forbubbles/droplets in a turbulent flow. Chemical Engineering Science, 58(20):4629–4637, 2003.

T. Wang, J. Wang, and Y. Jin. A CFD-PBM coupled model for gas-liquid flows.AIChE Journal, 52(1):125–140, 2006.


148

Chapter

7Numerical investigation of gas

holdup and phase mixing inbubble column reactors

A discrete bubble model (DBM) has been used to study the overall gas holdup and thephase mixing in bubble column reactors. The Eulerian-Lagrangian approach has theadvantage that it is possible to study the overall gas holdup and the gas phase mixingin a direct way. Furthermore, by introducing tracer particles, also the liquid phasemixing can be studied in a Lagrangian manner.Comparisons suggest that the overall gas holdups obtained from the discrete bubblemodel agree very well with the correlations of Kumar et al. (1976), Heijnen andVan’t Riet (1984) and Ruzicka et al. (2001) within the applied range of the superficialgas velocity. The gas phase dispersion coefficients from the simulation results agreepretty well with Wachi and Nojima (1990)’s correlation, which is derived based onthe recirculation theory. In addition, the turbulent diffusion coefficient calculatedfrom the tracer particles velocities are very close to those from the literature withinthe applied range of the superficial gas velocity.

149

7. Gas holdup and phase mixing in bubble column reactors

7.1 Introduction

Bubble column reactors are frequently utilized in the chemical, petrochemical,biochemical and metallurgical industries. A bubble column reactor is basicallya liquid filled vessel with a gas distributor at the bottom. The gas is spargedthrough the distributor in the form of bubbles and comes into contact with theliquid. The resulting buoyancy driven flow creates strong liquid recirculation.Bubble column reactors have the advantage that little maintenance and lowoperating costs are required due to lack of moving parts and compactness.Moreover, they have excellent heat and mass transfer characteristics.

There are some important parameters, such as gas holdup, gas-liquidinterfacial area, interfacial mass transfer coefficient, dispersion coefficient andheat transfer coefficient and so on, which are essential to characterize, scale upand design the bubble column reactors. For instance, the gas holdup gives thevolume fraction of the phases present in the reactor. On the other hand, thegas holdup combined with knowledge of the mean bubble diameter allowsdetermination of the specific interfacial area and the related volumetric masstransfer rates between the gas and liquid phase. Moreover, the gas holdup isusually used to identify the flow regime in bubble column reactors. Duringthe past decades, extensive studies on the gas holdup have been carried outin several contexts, such as flow regime identification and factors that mayinfluence the gas holdup in bubble column reactors, such as gas flow rate,liquid flow rate, geometry of the bubble column, operating conditions (suchas pressure and temperature), physical properties of both phases, sparger typeand so on. The related work has been reviewed by many authors (Shah et al.,1982; Heijnen and Van’t Riet, 1984; Deckwer and Schumpe, 1993; Kantarciet al., 2005; Ribeiro Jr. and Lage, 2005; Shaikh and Al-Dahhan, 2007; Joshiet al., 2008).

Dispersion coefficients are of significance in chemical reactors as well.There are two main model reactors and related assumptions regarding pre-vailing state of mixing, ideally mixed flow and plug flow in either singlephase or multiphase chemical reactors (Westerterp et al., 1987; Levenspiel,1999; Scott Fogler, 2005). In the ideally mixed flow system, the composition of

150


the reaction mixture is uniform and is typically used to describe either batchreactors or continuously mixed tank reactors. On the other hand, mixing ordiffusion of the reacting mixture does not occur in the flow direction in plugflow reactors. Real chemical reactors, however, may neither exhibit perfectmixed nor plug-flow behavior. For this reason, axial dispersion models, havebeen developed to describe the deviations from these simplified model reac-tors. In addition, the concept of residence time distribution (RTD) has provena powerful tool to diagnose and characterize non-ideal reactors since Danck-werts (1953). In bubble column reactors, investigations on the mixing of boththe gas and liquid phase have been reviewed by several authors, such as Joshi(1980, 1982); Shah et al. (1982); Heijnen and Van’t Riet (1984) and Deckwerand Schumpe (1993).

Computational Fluid Dynamics (CFD) has gained considerable interest inrecent years in the field of chemical engineering (Jakobsen, 2008). In additionto experimental routes, it provides an alternative way to study and character-ize chemical reactors in detail. On the other hand, chemical reactor modelingis also crucial for scaleup and design of chemical reactors.

In this work we will focus on the prediction of gas holdup and mixing ofboth the gas and liquid phase using the Eulerian-Lagrangian approach. Thesimulation results will be compared with corresponding correlations obtainedfrom literature.


The discrete bubble model (DBM) used in this study was originally developedby Delnoij et al. (1997) and Delnoij et al. (1999). In this model, the liquidphase hydrodynamics is represented by the volume-averaged continuity andNavier-Stokes equations while the motion of the each individual bubble istracked in a Lagrangian way.


The motion of each individual bubble is computed with Newton’s secondlaw. The liquid phase contributions are taken into account by the interphase

151


momentum transfer experienced by each individual bubble. For an individualbubble, the equations can be written as:

ρgVdvdt= ΣF (7.1)

drdt= v (7.2)

The net force acting on each individual bubble is calculated by consideringall contributing forces. It is assumed that the net force is composed of separate,uncoupled contributions due to gravity, pressure, drag, lift, virtual mass andwall force respectively:



FG = ρgVg (7.4)


FP = −V∇P (7.5)


FD = −18

CDρlπd2|v − u|(v − u) (7.6)

where the drag coefficient CD is determined according to Rusche (2002) whichtakes swarm effects into account:

CD = CD∞[exp(3.64αg) + α0.864

g

](7.7)

where the drag coefficient of a single bubble CD∞ is taken from Tomiyamaet al. (1998):

CD∞ = max[min

[ 16Re

(1 + 0.15Re0.687),48Re

],

83

EoEo + 4

](7.8)

152


where Re is the bubble Reynolds number, Re =ρl|v − u|dµl

.

A bubble rising in a non-uniform liquid flow field experiences a lift forcedue to vorticity or shear in the flow field. The shear induced lift force actingon a bubble is usually written as (Auton, 1987):

FL = −CLρlV(v − u) × ∇ × u (7.9)

where the lift coefficient is calculated according to Tomiyama et al. (2002):

CL =


f (EoH) 4 ≤ EoH ≤ 10,

−0.29 EoH > 10.

(7.10)

wheref (EoH) = 0.00105Eo3

H − 0.0159Eo2H − 0.0204EoH + 0.474 (7.11)

The modified Eotvos number, EoH is defined by using the maximum horizon-tal dimension of a bubble as a characteristic length as follows:


H

σ(7.12)


E =dV

dH=

11 + 0.163Eo0.757 (7.13)



σ.


d = (dVd2H)1/3 (7.14)


FVM = −CVMρlV(Dv

Dt− Du

Dt

)(7.15)

153


where the D/Dt operators denote the substantiative derivatives pertaining tothe respective phases. In the present work, bubbles are assumed to have aspherical shape and a virtual mass coefficient of 0.5 is used.


FW = −12

CWd[

1y2 −

1(L − y)2

]ρl |(v − u)·nz|2 nW (7.16)


CW =

exp(−0.933Eo + 0.179) 1 ≤ Eo ≤ 5,

0.007Eo + 0.04 5 < Eo ≤ 33.(7.17)


The liquid phase hydrodynamics are described by a set of volume-averagedequations, which consists of the continuity and the Navier-Stokes equations.The presence of the bubbles is reflected by the local volume fraction of theliquid phase αl and the interphase momentum transfer rateΦ:

∂∂t

(αlρl

)+ ∇ · (αlρlu

)= 0 (7.18)

∂∂t

(αlρlu

)+ ∇ · (αlρluu

)= −αl∇p − ∇ · (αlτl) + αlρlg +Φ (7.19)


τl = −µeff,l

[((∇u) + (∇u)T − 2

3I(∇ · u)

](7.20)



154

7.3. Correlations of the gas holdup


In the present work, large eddy simulation (LES) is used to simulate theturbulence induced by the rising bubbles. Therefore, the above continuity andNavier-Stokes equations are resolved for the field representing the “large”eddies, while the effect of the subgrid part of the velocity representing the”small scales” is included through an eddy-viscosity subgrid-scale model.The eddy-viscosity model proposed by Vreman (2004) is used to calculate theeddy viscosity:

µT,l = 2.5ρlC2S

√Bβαi jαi j

(7.22)

where αi j = ∂u j/∂xi, βi j = ∆2mαmiαmj and Bβ = β11β22−β2

12+β11β33−β213+β22β33−

β223. ∆i is the filter width in the i direction.


In this work, a hard sphere collision model developed by Hoomans et al.(1996) is adopted to describe the bubble-bubble interaction. It mainly consistsof two parts. The first part is processing the sequence of collisions and thesecond part is dealing with the collision dynamics. A detailed descriptionabout the model is referred to Darmana (2006) and Hoomans (2000).

7.3 Correlations of the gas holdup

In bubble column reactors, the gas holdup is an important parameter whichcan be used to characterize the hydrodynamics. It is defined as the overallvolume fraction of the gas phase present in the form of bubbles. The gasholdup mainly depends on the superficial gas velocity. It can be used toestimate the interfacial area and is thus required to estimate the volumetricmass transfer rates between the gas and liquid phase.

Fair et al. (1962) carried out measurements of the gas holdup and heattransfer in two commercial scale bubble columns with an air-water system.The diameters of the columns were 0.54 m and 1.26 m respectively. By com-

155


paring with results obtained in those columns with smaller diameter, theyfound that the gas holdup varies directly with the superficial gas velocity andis less in the column with a large diameter than in the column with a smalldiameter. However, when the diameter is larger than 0.54 m, the gas holdupis no longer influenced by the column diameter. The experimental data repro-duced from Fair et al. (1962), a linear fitted curve and the gas holdup curvefrom Shulman (1950) for a bubble column with a diameter of 0.12 m are shownin Figure 7.1.

Φ=0.54 m

Φ=1.26 m

Fitted curve

Φ=0.12 m

Gas holdup [-]

0

0.1

0.2

0.3

ug [m/s]0 0.02 0.04 0.06

Figure 7.1: Gas holdup curve (reproduced from) Fair et al. (1962).

Kato and Nishiwaki (1972) proposed the following correlation for the gasholdup in an air-water system (Shah et al., 1982):

ε =2.51ug

0.78 + βu0.8g (1 − eγ)

(7.23)

whereβ = 4.5 − 3.5 − 2.548φ1.8 (7.24)

andγ = 717u1.8

g /β (7.25)

156

7.3. Correlations of the gas holdup

Kumar et al. (1976) correlated 382 gas holdup data points (ε < 0.35andug <

0.15 m/s) with the following equation:

ε = 0.728u∗ − 0.485u∗2 + 0.0975u∗3 (7.26)

where u∗ is defined as a dimensionless gas velocity given by:

u∗ = ug

ρ2l

σ∆ρg

0.25

(7.27)

According to the circulation theory and an assumption for the single bub-ble rise velocity, Heijnen and Van’t Riet (1984) proposed the following expres-sion for the homogeneous flow regime:

ε = 4ug (7.28)

which is consistent with the experimental observations by Krishna et al. (1991).Based on the concept of a characteristic turbulent kinematic viscosity

in bubble columns, Kawase and Moo-Young (1987) developed a theoreticalmodel for the gas holdup with Newtonian and non-Newtonian fluids. ForNewtonian fluids, the equation is given as:

ε = 1.07

u2g

φg

1/3

(7.29)

Based on hydrodynamic coupling between both phases, Ruzicka et al.(2001) developed a simple physical model for the homogeneous to heteroge-neous regime transition in a bubble column. For the homogeneous regime,the formula for the gas holdup is given by:

ε =ug + uT −

√B

2(1 + a)uT(7.30)

whereB = u2

g + u2T − 2(2a + 1)uguT (7.31)

For a bubble column with dimensions (H = 1 m, φ = 0.15 m), the two param-eters, uT and a, are 0.22 m/s and 0.35 respectively according to experimentaldata.

157


7.4 Phase mixing

There are several factors that may induce longitudinal mixing in a bubblecolumn, such as turbulent eddies, a nonuniform velocity distribution over thecross section of the reactor and molecular diffusion. In most of the practicalcases, the former two factors dominate the molecular diffusion (Westerterpet al., 1987).

7.4.1 Axial dispersion model

Axial dispersion models are often used to characterize the extent of phase mix-ing of chemical reactors. It assumes a diffusion-like process superimposed onplug flow. The unsteady state one-dimensional convection diffusion equationis given by Levenspiel (1999):

∂C∂t= D∂2C∂z2 −

∂(UC)∂z

(7.32)

where C represents the species concentration, D the axial dispersion coefficientand U a characteristic velocity that depends on the phase. For instance for thegas phase, the characteristic velocity is defined as U = ug/ε.

The axial dispersion coefficient D is directly related to the process of mixingin reactors. For instance, a large D means rapid mixing while a small D impliesslow mixing, whereas D = 0 corresponds to plug flow conditions.

When the boundary conditions are known, i.e. open-open boundary con-ditions, the above equation can be solved analytically. From a pulse tracerexperiment, the dispersion coefficient can then be obtained from the tracercurve at the outlet of the reactor, that is, from the mean and variance of theresidence time distribution of the tracer.

7.4.2 Residence time distribution

Since Danckwerts (1953) introduced the concept of residence time distribution,it has been used as an important tool in the analysis of chemical reactorperformance.

158

7.4. Phase mixing

The residence time distribution E(t) normally has the property that thearea under the curve is unity: ∫ ∞

0E(t) dt = 1 (7.33)

In addition, the mean residence time tm and the variance σ2 of the residencetime distribution can be calculated as follows:

tm =

∫ ∞

0tE(t) dt (7.34)

σ2 =

∫ ∞

0(t − tm)2E(t) dt (7.35)

Consequently the axial dispersion coefficient can be obtained with themean residence time tm and the variance σ2. For instance, the Peclet numberPe can be calculated in a closed-closed system (Levenspiel, 1999):

σ2θ =σ2

t2m=

2Pe− 2

Pe2 (1 − e−Pe) (7.36)

The dispersion coefficient D can hence be determined from the Pecletnumber:

D =UHPe

(7.37)

where H is the height of the system.

7.4.3 Lagrangian description of the mixing

In a Lagrangian view, the diffusion coefficient can be evaluated from thechange in mean-square displacement of fluid elements z2 with time t (Brodkey,1972):

D =12

dz2

dt(7.38)

where the mean-square displacement z2 is calculated as:

z2 =1N

N∑i=1

(zi(t) − zi(0)) − 1N

N∑i=1

(zi(t) − zi(0))

2

(7.39)

159


where N represents the number of fluid elements.In turbulent flows, a turbulent diffusion coefficient can be expressed as

(Brodkey, 1972):DT = v′2TL (7.40)

where v′ is the root-mean-square value of the fluctuating part of the instanta-neous fluid velocity and the Lagrangian time scale TL is defined as:

TL =

∫ ∞

0RL(τ) dτ (7.41)

where

RL(τ) =v(t)v(t + τ)

v′2(7.42)

7.4.4 Correlations for the gas phase dispersion coefficient

For the axial dispersion coefficient of the gas phase in a bubble column, Dibounand Schugerl (1967) suggested (Westerterp et al., 1987):

Bo =urφ

Dg= 0.2 (7.43)

where Bo is Bodenstein number and ur is the relative velocity between the gasand the liquid phase.

Men’shchikov and Aerov (1967) studied the gas phase axial mixing in abubble column (φ = 0.3 m) with the superficial gas velocity varying in therange of 0.008-0.1 m/s and proposed the following correlation (Joshi, 1982):

Dg = 1.47u0.72g (7.44)

Towell and Ackermann (1972) correlated their own experimental data andthose from the literature with a correlation as follows (Joshi, 1982):

Dg = 19.7φ2ug (7.45)

The superficial gas velocity applied for the correlation ranges from 0.009 m/sto 0.13 m/s, whereas the diameter of the bubble column ranges from 0.09 m to1 m.

160

7.4. Phase mixing

Joshi (1982) reviewed six experimental investigations for the gas phasedispersion in bubble columns. Based on this, he proposed the followingcorrelation which covers all the experimental data presented in the six exper-imental studies:

Dg = 110φ2u2

g

ε(7.46)

Heijnen and Van’t Riet (1984) suggested that the dispersion coefficient isconstant in the homogeneous flow regime according to experimental data inthe literature. They gave the following dispersion coefficient correlation inthe heterogeneous flow regime for the air-water system:

Dg = 78(ugφ)1.5 (7.47)

However, they also mentioned that the transition from the homogeneousto heterogeneous flow regime is difficult to distinguish due to geometricparameters, such as sparger location and uniformity of gas distribution.

On the basis of the recirculation theory for a bubble column in the turbulentflow regime, Wachi and Nojima (1990) derived the following equation for theaxial dispersion coefficient of the gas phase:

Dg = 20φ3/2ug (7.48)

They also measured the gas phase dispersion coefficients in two bubblecolumns (φ = 0.2 m and 0.5 m) using pulse experiments at superficial gasvelocities in the range of 0.029-0.456 m/s. Comparison between the theoreti-cal correlation and the measurements revealed that the derived correlation isable to reflect the dependencies of the axial dispersion of the gas phase on thecolumn diameter and the superficial gas velocity.

7.4.5 Correlations for the liquid phase dispersion coefficient

Ohki and Inoue (1970) measured the axial liquid phase dispersion coefficientin batch type bubble columns of three different diameters (φ = 0.04 m, 0.08 mand 0.16 m) and established a correlation as follows:

Dl = 0.30φ2u1.2g + 170δ (7.49)

161


where δ is hole diameter of the perforated plate. Note that the units in theequation are non-SI units. That is, the unit of dispersion coefficient is cm2/s,superficial gas velocity has unit of cm/s and the hole diameter is in cm.Furthermore, the range of superficial gas velocity in his study ranges 0.02 m/sand 0.25 m/s.

Towell and Ackermann (1972) developed a correlation for dispersion coef-ficient as a function of the column diameter and the gas velocity (Majumder,2008):

Dl = 1.23φ1.5u0.5g (7.50)

The diameters of the bubble columns used in this study are 0.04 m and 1.07 m,while the superficial gas velocity ranged from 0.016 to 0.13 m/s and 0.009 to0.034 m/s respectively (Joshi, 1982).

By measuring the dispersion coefficient in two tall bubble columns with(φ = 0.15 m and 0.2 m), Deckwer et al. (1974) correlated his own experimentaldata with those in literature and proposed an expression which is similar tothe correlation proposed by Towll and Ackermann (1972):

Dl = 0.678φ1.4u0.3g (7.51)

The superficial gas velocity used in the two bubble columns ranged from0.004 to 0.13 m/s and 0.01 to 0.14 m/s respectively.

Hikita and Kikukawa (1974) studied the liquid dispersion coefficient in twobubble columns and presented a correlation for liquid dispersion coefficient:

Dl = (0.15 + 0.69u0.77g )φ1.25 (7.52)

Baird and Rice (1975) adopted dimensional analysis to derive the disper-sion coefficient based on isotropic turbulence model:

Dl = 0.35φ4/3(ugg)1/3 (7.53)

Even though the turbulence in a bubble column is not necessarily isotropic,the correlation using an isotropic turbulence model as the basis gives a di-mensionally consistent and simple equation.

Zehner (1986) proposed a correlation based on a vortex cell model:

Dl = 0.368φ4/3(ugg)1/3 (7.54)

162

7.5. Numerical aspects of phase mixing study

where the constant 0.368 is an adjustable parameter.Heijnen and Van’t Riet (1984) derived the liquid dispersion coefficient

from either the liquid circulation velocity or the Peclet number according toJoshi (1980). They found out that the two correlations are in good agreementwith each other except for the constant C. The correlations have the followinggeneral form:

Dl = Cφ4/3(ugg)1/3 (7.55)

where C equals either 0.33 or 0.36.By relating the axial dispersion coefficient to the longitudinal displacement

of a fluid element during a certain period, Kawase and Moo-Young (1986)derived the following correlation valid for Newtonian liquids:

Dl = 0.343φ4/3(ugg)1/3 (7.56)

Note that most of the above correlations that are either obtained fromexperiments or derived theoretically in the literature show proportional de-pendencies of the liquid phase axial dispersion coefficient on the diameter ofthe bubble column and the superficial gas velocity. The only difference is thatthe powers of these two parameters and the constant vary slightly from eachother.

7.5 Numerical aspects of phase mixing study

7.5.1 Residence time distribution of the gas phase

In Eulerian-Lagrangian modeling of bubbly flows, each bubble is trackedindividually according to Newton’s second law. Hence, the residence timeof each bubble in the system can be determined directly from the simulationand from this information, the axial dispersion coefficient can be determinedaccording to the axial dispersion model.

7.5.2 Tracer particles of the liquid phase

Massless tracer particles have been used to visualize unsteady flows in Com-putational Fluid Dynamics (CFD) (Hin and Post, 1993; Lane, 1997; Bauer et al.,

163


2002). In the present work, tracer particles are used to study the mixing of theliquid phase.

For the liquid phase, the Navier-Stokes equations are solved on an Euleriangrid and the liquid velocities are obtained on the faces of each computationalcell. The velocity of a tracer particle v at a certain moment t can then beinterpolated according to its position r(t) in the grid. Accordingly, after a time∆t, the new position of the tracer particle can be calculated as:

r(t + ∆t) = r(t) +∫ t+∆t

tv(r(t), t) dt (7.57)

Equation 7.57 can be evaluated using a numerical integration scheme.Lane (1997) has reported that the first-order integration scheme could lead toerroneous results and suggested a higher-order integration to be used. Hence,the Runge-Kutta-Fehlberg method is used in the present simulations. Detailsof this method are given in appendix 7.A.

Yeung and Pope (1988) have shown that the accuracy of the tracer particletrajectories depends on the accuracy of the interpolation scheme used to cal-culate the tracer particle velocities rather than on the time-stepping method.In the present work, two interpolation methods, i.e. trilinear interpolationand tricubic interpolation (Lekien and Marsden, 2005), are compared witheach other. More information on these interpolation methods is given inappendix 7.B.


The bubble column studied here is shown in Figure 7.2. The cross-sectionalarea of the column is 0.15 m × 0.15 m (W × D). The column is initially filledwith water to a height of 0.6 m (H). Air is used as the dispersed phase andintroduced into the column through a perforated plate at the bottom of thecolumn. The material properties of both phases are taken according to theirroom temperature values. The bubble column is operating under atmosphericpressure.

A perforated plate with 576 holes (24 × 24) is used as gas distributor. Allthe holes have a diameter of 1 mm and are positioned in the central region of

164


Figure 7.2: Sketch of the square bubble column.

the plate with a square pitch of 6.25 mm.A grid with (20×20×80) cells is adopted in the simulations. Five superficial

gas velocities (ug = 0.005, 0.01, 0.015, 0.02 and 0.025 m/s) are adopted. Thediameter of the bubbles in the simulations is kept as a constant, 0.004 m. Dueto the fact that the perforated plate adopted has many holes and bubblesinjected from each hole have very low velocities, the coalescence and breakupof bubbles can be neglected. The time step for the liquid phase equations is setto 1.0 × 10−3 s and for the bubbles it is set to 5.0 × 10−5 s. The total simulationtime is 150 s.

There are two sets of tracer particles being used in the simulations. The

165


two sets of tracer particles are released uniformly over the cross-section of thebubble column. The set of tracer particles released from the bottom of thebubble column is denoted as “Tracer0” and the other set of tracer particlesreleased from the top of the bubble column is denoted as “Tracer1”. Moreover,both sets of tracer particles are released simultaneously at the simulation timet = 30 s at which the flow in the bubble column is fully developed.


7.7.1 Gas holdup

Determination of the gas holdup

The instantaneous gas holdup is recorded at every 0.04 s during the sim-ulation. Examples of gas holdup histories at a superficial gas velocityug = 0.005 m/s and ug = 0.025 m/s are plotted in Figure 7.3.

ε [-]

0

0.01

0.02

0.03

t [s]0 50 100 150

(a) ug = 0.005 m/s

ε [-]

0

0.05

0.1

0.15

t [s]0 50 100 150

(b) ug = 0.025 m/s

Figure 7.3: Instantaneous gas holdup [-].

The plots show that the gas holdup increases monotonically at the begin-ning of the simulation. After a short time, the gas holdup reaches a peakand then starts to fluctuate during the simulation. Bubbles are injected intothe bubble column continuously and rise in the vertical direction, which isreflected in the increasing initial part of the curves. Once the bubbles reach the

166


top of the bubble column, most of them escape from the column directly. Thisleads to the small decrease of the gas holdup after the first peak. As soon as aliquid circulation pattern in the bubble column has developed, some bubblesmay be trapped in the circulating liquid and remain in the column for a longertime. Hence, fluctuations in the gas holdup curves result.

In order to obtain the overall gas holdup in the bubble column for eachsuperficial gas velocity, the instantaneous gas holdups are averaged after 30seconds, in a similar fashion as described in Chapter 5. The resulting overallgas holdup for each superficial gas velocity is listed in Table 7.1. From thistable it can be seen that the standard deviation of the averaged gas holdup σis quite small compared to the overall gas holdup. Furthermore, within theadopted range of the superficial gas velocity, the overall gas holdup is relatedto the superficial gas velocity with a multiplication factor of 4 approximately.

Table 7.1: Overall gas holdup.

ug [m/s] 0.005 0.01 0.015 0.02 0.025

ε [-] 0.0234 0.0471 0.0671 0.0916 0.1119σ [-] 0.0004 0.0017 0.0040 0.0056 0.0060

Comparison with literature correlations

In Figure 7.4, the overall gas holdups obtained from the simulations are com-pared with the literature correlations introduced earlier.

It is seen that the simulation results agree with the correlations of Kumaret al. (1976), Heijnen and Van’t Riet (1984) and Ruzicka et al. (2001) quite well.These results indicate that the overall gas holdup increases nearly linearlywith the superficial gas velocity with a factor of 4 in the adopted range ofthe superficial gas velocity. It can also be seen that the gas holdups obtainedfrom the correlation of Shulman (1950) are higher than those obtained fromothers, which may be due to the diameter of the bubble column as reportedby Fair et al. (1962). In addition, the correlations of Fair et al. (1962) and Katoand Nishiwaki (1972) are close to each other when the superficial gas velocity

167


Shulman (1950)

Fair (1962)

Kato (1972)

Kumar (1976)

Kawase (1987)

Heijnen (1984)

Ruzicka (2001)

DBM

ε [-]

0

0.05

0.1

0.15

0.2

0.25

ug [m/s]0 0.005 0.01 0.015 0.02 0.025 0.03

Figure 7.4: Comparison of the computed gas holdup with literature correla-tions.

ug < 0.025 m/s. As the superficial gas velocity increases, the gas holdupobtained according to Kato and Nishiwaki (1972) increases faster than thatobtained from Fair et al. (1962). Moreover, Kawase and Moo-Young (1987)’scorrelation possesses a more flat trend.

7.7.2 Gas phase dispersion

Residence time distribution

The residence times of all individual bubbles in the bubble column arerecorded during the simulation and from this data set, the residence timedistribution is determined. In Figure 7.5, the residence time distribution foreach superficial gas velocity is shown. The plot shows a single peak in theresidence time distribution, which means that most of the bubbles in the bub-ble column exit the column at the corresponding time. Moreover, the long tailof the curve indicates that some bubbles remain in the column for longer res-idence times due to reasons mentioned before. As the superficial gas velocityincreases, it is found that the peak of the residence time distribution shifts tosmaller residence time. That can be explained by the fact that at higher gas

168


loading a stronger liquid circulation is created leading to a faster transport ofthe bubbles. In addition, the distribution becomes wider at higher superficialgas velocity.

ug=0.005 m/s

ug=0.01 m/s

ug=0.015 m/s

ug=0.02 m/s

ug=0.025 m/s

E(t) [1/s]

0

0.02

0.04

0.06

t [s]0 2 4 6 8 10

Figure 7.5: Residence time distributions of the gas phase.

Relevant moments of the residence time distributions, such as the meanresidence time tm and the variance σ2 are determined from the curves. Byusing the solution for a closed-closed system, the Peclet number and thegas phase dispersion coefficients can be determined. The computed resultsare shown in Table 7.2. From this table, it can be seen that the variance σ2

and the dimensionless variance σ2θ of the residence time distribution increase

with increasing superficial gas velocity, which is reflected by the wide spreadof the residence time distribution. Furthermore, the dispersion coefficientincreases with increasing superficial gas velocity. This implies that the extentof gas phase backmixing becomes more pronounced at higher superficial gasvelocity.


The simulated dispersion coefficients of the gas phase will subsequently becompared with a number of correlations from the literature. Note that for

169


Table 7.2: Mean and variance of the simulated residence time distributions.

ug [m/s] 0.005 0.01 0.015 0.02 0.025

tm [s] 2.88 2.94 2.90 2.99 3.01σ2 [s2] 0.81 1.35 2.06 2.77 3.38σ2θ [-] 0.10 0.16 0.25 0.31 0.37

Pe [-] 19.50 11.74 7.00 5.25 4.05Dg [m2/s] 0.007 0.011 0.019 0.025 0.033

the earlier introduced correlations, the correlation of Joshi (1982) involves therelation between the superficial gas velocity and the overall gas holdup inthe bubble column. Due to the fact that the overall gas holdups from thesimulations agree with the correlations in the literature quite well, we usedthe overall gas holdup data obtained from the simulations to calculate the gasphase dispersion coefficient from the correlation proposed by Joshi (1982). Acomparison is shown in Figure 7.6.

Diboun (1967)

Men'shchikov (1967)

Towell (1972)

Joshi (1982)

Heijnen (1984)

Wachi (1990)

DBM

Dg [m

2/s]

0

0.02

0.04

0.06

0.08

0.1

0.12

ug [m/s]0 0.005 0.01 0.015 0.02 0.025 0.03

Figure 7.6: Comparison of the simulated gas dispersion coefficient with liter-ature correlations.

The comparison suggests that the simulation results agree very well withthe correlation of Wachi and Nojima (1990) which is derived based on the recir-

170


culation theory. The gas phase dispersion coefficients from the simulations,however, are far below the correlation of Men’shchikov and Aerov (1967).Men’shchikov and Aerov (1967) proposed the correlation according to the ex-perimental data from a bubble column with the diameter of 0.3 m and did notconsider the effect of the column diameter in the correlation. The dependenceof the gas phase dispersion on the column diameter, however, is used in mostof the correlations. Moreover, the effect of the column diameter on the gasphase dispersion coefficient has been proven by Mangartz and Pilhofer (1981).In addition, the simulation results are close to the correlations of Diboun andSchugerl (1967), Towell and Ackermann (1972), Joshi (1982) and Heijnen andVan’t Riet (1984) at low superficial gas velocities (i.e. ug ≤ 0.01 m/s), whilethe difference becomes larger when the superficial gas velocity increases.

7.7.3 Liquid phase dispersion

Comparison of the interpolation methods

The two methods for interpolating the Eulerian velocity to the Lagrangianvelocity, tricubic interpolation and trilinear interpolation are compared witheach other. For instance, the mean-square displacement of the tracer particlesand the autocorrelation of vertical component of the tracer particle velocitiesat superficial gas velocity ug = 0.005 m/s are shown in Figure 7.7.

(a) Meas-square displacement (b) Autocorrelation

Figure 7.7: Comparison of interpolation methods (ug = 0.005 m/s).

It can be seen that there are some minor differences between the two

171


methods with respect to both the mean-square displacement and the autocor-relation, which suggests that the interpolation scheme only slightly influencesthe resulting Lagrangian time series. In the following discussions, only resultsobtained with the tricubic interpolation scheme are considered.

Turbulent diffusion coefficient

The turbulent diffusion coefficient of the liquid phase in the bubble columnis determined according to Equation 7.40. The Lagrangian time scale TL isdetermined as the time for the autocorrelation to decrease to 1/e of its initialvalue (Squires and Eaton, 1991). The results obtained from both sets of tracerparticles are listed in Table 7.3. The turbulent diffusion coefficient determinedfrom the tracer particles “Tracer0” is denoted as Dl0 and that determined fromthe tracer particles “Tracer1” is represented as Dl1. Moreover, for each set oftracer particles, the turbulent diffusion coefficient is determined four timesand the average of these four samples is presented in Table 7.3 along with thestandard deviation which are denoted as σ0 and σ1.

Table 7.3: Turbulent diffusion coefficient of the liquid phase.

ug [m/s] 0.005 0.01 0.015 0.02 0.025

Dl0 [m2/s] 0.006 0.010 0.019 0.016 0.021σ0 [m2/s] 0.002 0.004 0.002 0.002 0.003Dl1 [m2/s] 0.008 0.011 0.020 0.018 0.023σ1 [m2/s] 0.003 0.004 0.003 0.003 0.001

The table shows that the turbulent diffusion coefficients obtained from thetwo set of tracer particles are quite close to each other. However, the standarddeviation of the turbulent diffusion coefficient is relatively large, particularlyat low superficial gas velocity (i.e. ug = 0.005 m/s). Moreover, it can be foundthat the turbulent diffusion coefficient of the liquid phase nearly increaseswith the superficial gas velocity.

172



The turbulent diffusion coefficient of the liquid phase obtained from the sim-ulations are compared with the correlations in the literature and shown inFigure 7.8.

Ohki (1970) Towell (1972)

Deckwer (1974) Hikita (1974)

Zehner (1986) Heijnen (1984)

DBM (Tracer0) DBM (Tracer1)

Dl [m

2/s

]

0

0.01

0.02

0.03

0.04

ug [m/s]0 0.005 0.01 0.015 0.02 0.025 0.03

Figure 7.8: Comparison of the simulated liquid phase dispersion coefficientwith literature correlations.

According to the dispersion coefficient correlations of the liquid phaseintroduced earlier, the correlations of Deckwer et al. (1974), Baird and Rice(1975), Heijnen and Van’t Riet (1984), Zehner (1986) and Kawase and Moo-Young (1986) are very close to each other. All of them exhibit similar de-pendence on the column diameter and the superficial gas velocity, and theconstants of the correlations are also quite similar. Hence, only two of themare used for the comparison in the plot (i.e. Heijnen and Van’t Riet (1984)’scorrelation with the constant equal to 0.33 and Zehner (1986)’s correlation).Moreover, Towell and Ackermann (1972)’s correlation possesses a similartrend with the above correlations because they use a very similar form of theexpression with only slightly different powers of the column diameter and thesuperficial gas velocity. Hikita and Kikukawa (1974)’s correlation reveals aweaker dependence on the superficial gas velocity than the others. Moreover,

173


the dispersion coefficients calculated from Hikita and Kikukawa (1974)’s cor-relation are higher than those obtained from the others at low superficial gasvelocity (i.e. ug < 0.02 m/s). Ohki and Inoue (1970)’s correlation increaseswith the superficial gas velocity faster than the others and is larger than theothers when ug > 0.02 m/s. It is clearly found that the turbulent diffusioncoefficients obtained from the previous section are very close to the disper-sion coefficients calculated from the correlations in the literature within theapplied range of the superficial gas velocity.

7.8 Conclusions

The present chapter utilized the discrete bubble model (DBM) to investigatethe overall gas holdup and the mixing of the both phases in bubble columns.One of the advantages of the Eulerian-Lagrangian approach is that the dis-persed phase is treated as individual elements, i.e. bubbles that are trackedindividually. The total number and the residence times of bubbles in thebubble column reactors can be determined conveniently during simulationsand thus, the overall gas holdup and the residence time distribution of thegas phase can be obtained accordingly. Hence, the dispersion coefficient de-scribing the extent of the gas phase mixing can be calculated. Furthermore,by introducing liquid phase tracer particles, it is possible to study the liquidphase mixing as well.

In the first part of this chapter, the overall gas holdups from the dis-crete bubble model are obtained within the range from ug = 0.005 m/s toug = 0.025 m/s. The results are then compared with correlations in the litera-ture. The comparison reveals that the simulation results agree well with thecorrelations of Kumar et al. (1976), Heijnen and Van’t Riet (1984) and Ruzickaet al. (2001). The results suggest that the overall gas holdup increases nearlylinearly with the superficial gas velocity with a scaling constant of 4 in theadopted range of the superficial gas velocity.

Secondly, the residence time distributions (RTD) of the gas phase are de-termined in the residence times of bubbles from the discrete bubble model. Asharp peak in the residence time distribution is obtained, which means that

174

Nomenclature

most of the bubbles in the bubble column exit the column at the correspondingtime. Moreover, the long tail of the curve suggests that some bubbles remainin the column for longer residence times. As the superficial gas velocity in-creases, one can find that the peak of the residence time distribution shifts tosmaller residence time, which is attributed to the stronger liquid circulation.In addition, the distribution becomes wider when the superficial gas velocityis higher, which is reflected by the increase of the variance of the residencetime distribution.

The comparison with the gas dispersion correlations in the literature showsthat the simulation results agree very well with Wachi and Nojima (1990)’scorrelation which is derived on basis of the recirculation theory.

Finally, the liquid phase mixing is studied through the introduction ofliquid phase tracer particles. Two methods for the interpolation scheme ofthe Lagrangian velocity from the Eulerian velocity of the liquid phase arecompared with each other. Only small differences have been found betweenthe different methods. The turbulent diffusion coefficient is calculated fromthe Lagrangian velocities and compared with correlations in literature. Thecomparison shows that the obtained turbulent diffusion coefficients are veryclose to those from the literature within the applied range of the superficialgas velocity.

Nomenclature

C model coefficient, [-]; concentration of a species, [kg m−3]Bo Bodenstein number, [-]d bubble diameter, [m]D column depth, [m]; dispersion coefficient, [m2 s−1]f function, [-]


σ, [-]

E(t) residence time distribution, [s−1]F force vector, [N]g gravity acceleration, [m s−2]

175


H column height, [m]I unit tensor, [-]n unit normal vector, [-]p pressure, [N m−2]Pe Peclet number, [-]r position vector, [m]R bubble radius, [m]


, [-]

S characteristic filtered strain rate, [s−1]t time, [s]tm mean residence time, [s]u liquid velocity vector, [m s−1]U mean bubble velocity, [m s−1]v bubble velocity vector, [m s−1]V volume, [m3]W column width,[ m]

Greek letters

α local volume fraction, [-]∆ subgrid length scale, [m]δ hole diameter of perforated plate, [m]ε gas holdup, [-]µ viscosity, [kg m−1 s−1]φ column diameter, [m]Φ volume averaged momentum transfer due to interphase forces,

[kg m−2 s−2]ρ density, [kg m−3]σ surface tension, [N m−1]; standard deviationσ2 variance, [s2]σ2θ dimensionless variance, [-]τ stress tensor, [N m−2]

176

Nomenclature

Indices

b bubblecell computational cellD drageff effectiveg gas phaseG gravityi i directionj j directionl liquidL lift; molecular viscosityP pressureS subgridT turbulentVM virtual massW wall

177


7.A Runge-Kutta-Fehlberg method

The Runge-Kutta-Fehlberg method (denoted RKF45) is a numerical techniqueused to solve ordinary differential equations of the form:

drdt= v(r(t), t) (A.I)

Formal integration of A.I yields:

r(t + ∆t) = r(t) +∫ t+∆t

tv(r(t), t) dt (A.II)

The Runge-Kutta-Fehlberg method has a procedure to determine if theproper step size h is being used. At each step, two different approximationsfrom both O(h4) and O(h5) methods for the solution are made and compared.If the two results are in close agreement, the approximation is accepted. Ifthe two results do not agree to a specified accuracy, the step size is reduced.If the results agree to more significant digits than required, the step size isincreased. A brief description is given below.

When integrating Equation A.II with the step size h using the Runge-Kutta-Fehlberg method, let rk be the current position of a certain particle. Each steprequires the use of the following six values:

k1 = hv (rk, tk)

k2 = hv(rk +

14

k1, tk +14

h)

k3 = hv(rk +

332

k1 +932

k2, tk +38

h)

k4 = hv(rk +

19322197

k1 −72002197

k2 +72962197

k3, tk +1213

h)

k5 = hv(rk +

439216

k1 − 8k2 +3680513

k3 −8454104

k4, tk + h)

k6 = hv(rk −

827

k1 + 2k2 −35442565

k3 +18594104

k4 −1110

k5, tk +12

h)

(A.III)

178

7.B. Interpolation methods

Then an approximation to the solution is made using a Runge-Kuttamethod of order 4:

rk+1 = rk +25

216k1 +

14082565

k3 +21974101

k4 −15

k5 (A.IV)

In addition, a better value for the solution is determined using a Runge-Kutta method of order 5:

r′k+1 = rk +16135

k1 +665612825

k3 +2856156430

k4 −950

k5 +255

k6 (A.V)

The two approximations are then compared with each other. If the errorof approximation falls within the appropriate error bound, the approximationwill be accepted. Otherwise, an optimal step size will be computed based onthe two approximations.

7.B Interpolation methods

Trilinear interpolation is a fast and simple interpolation scheme for a threedimensional Cartesian grid. It approximates the value of a point within therectangular grid cell linearly, using data on the corners of the cell. If point p isin the cell shown in Figure B.1 and has the fractional offsets (dx, dy, dz) fromthe grid point p1, then:

f (dx, dy, dz) = f (p1)(1 − dx)(1 − dy)(1 − dz) +

f (p2)dx(1 − dy)(1 − dz) +

f (p3)(1 − dx)dy(1 − dz) +

f (p4)dxdy(1 − dz) +

f (p5)(1 − dx)(1 − dy)dz +

f (p6)dx(1 − dy)dz +

f (p7)(1 − dx)dydz +

f (p8)dxdydz (B.I)

In addition, Lekien and Marsden (2005) introduced a local tricubic interpo-lation scheme in three dimensions that is both C1 and isotropic. The algorithm

179


is based on a specific 64 × 64 matrix that gives the relationship between thederivatives at the corners of the elements and the coefficients of the tricubicinterpolant for this element. A brief description of this method is as follows.

Figure B.1: Element for interpolation in three dimensions

As shown in the Figure B.1, the function f at the position p in the regulargrid can be represented by:

f (dx, dy, dz) =N∑

i=0

N∑j=0

N∑k=0

ai jkdxidy jdzk (B.II)

The coefficients ai jk are determined in such a way that the function f is C1,that is, f is continuous and its 3 first derivatives are also continuous.

The 64 coefficients ai jk are stacked in a vector α by defining:

α1+i+4 j+16k = ai jk ∀i, j, k ∈ 0, 1, 2, 3 (B.III)

Similarly, the constraints on f and its derivatives are stacked in a vector b

180

References

by defining:

bi =

f (pi) if 1 ≤ i ≤ 8

∂∂x

f (pi−8) if 9 ≤ i ≤ 16

∂∂y

f (pi−16) if 17 ≤ i ≤ 24

∂

∂zf (pi−24) if 25 ≤ i ≤ 32

∂2

∂x∂yf (pi−32) if 33 ≤ i ≤ 40

∂2

∂x∂zf (pi−40) if 41 ≤ i ≤ 48

∂2

∂y∂zf (pi−48) if 49 ≤ i ≤ 56

∂3

∂x∂y∂zf (pi−56) if 57 ≤ i ≤ 64

(B.IV)

The derivatives of f can be computed and evaluated for the 8 points pi.This gives a linear system in the 64 unknown coefficients αi:

Bα = b (B.V)

The matrix B is 64 × 64 and has only integer entries. The determinant of Bis 1. As a result, B is invertible and the coefficients αi can be computed usingthe linear relationship:

α = B−1b (B.VI)

The matrix B−1 is the core of the tricubic interpolator and can be found inLekien and Marsden (2005).

References


181


M. H. I. Baird and R. G. Rice. Axial dispersion in large unbaffled columns.The Chemical Engineering Journal, 9(2):171–174, 1975.

D. Bauer, R. Peikert, M. Sato, and M. Sick. A case study in selective visu-alization of unsteady 3d flow. In VIS ’02: Proceedings of the conference onVisualization ’02, pages 525–528, Washington, DC, USA, 2002. IEEE Com-puter Society.

R. S. Brodkey. Mixing: Theory and Practice, volume 1. Academic Press Inc.,1972.

P. V. Danckwerts. Continuous flow systems : Distribution of residence times.Chemical Engineering Science, 2(1):1–13, February 1953.

D. Darmana. On the multiscale modelling of hydrodynamics, mass transfer andchemical reactions in bubble columns. PhD thesis, Enschede, 2006.

W. -D. Deckwer and A. Schumpe. Improved tools for bubble column reactordesign and scale-up. Chemical Engineering Science, 48(5):889–911, 1993.

W. -D. Deckwer, R. Burckhart, and G. Zoll. Mixing and mass transfer in tallbubble columns. Chemical Engineering Science, 29(11):2177–2188, 1974.



J. R. Fair, A. J. Lambright, and J. W. Andersen. Heat transfer and gas holdupin a sparged contactor. IandEC Process Design and Development, 1(1):33–36,1962.

J. J. Heijnen and K. Van’t Riet. Mass transfer, mixing and heat transfer phe-nomena in low viscosity bubble column reactors. The Chemical EngineeringJournal, 28(2):B21–B42, 1984.

182

References

H. Hikita and H. Kikukawa. Liquid-phase mixing in bubble columns: Effectof liquid properties. The Chemical Engineering Journal, 8(3):191–197, 1974.

Andrea J. S. Hin and Frits H. Post. Visualization of turbulent flow withparticles. In VIS ’93: Proceedings of the 4th conference on Visualization ’93,pages 46–53, Washington, DC, USA, 1993. IEEE Computer Society.

B. P. B. Hoomans. Granular dynamics of gas-solid two-phase flows. PhD thesis,Enschede, January 2000.



J. B. Joshi. Axial mixing in multiphase contactors - A unified correlation.Transactions of the Institution of Chemical Engineers, 58(3):155–165, 1980.

J. B. Joshi. Gas phase dispersion in bubble columns. The Chemical EngineeringJournal, 24(2):213–216, 1982.

J. B. Joshi, A. B. Pandit, K. L. Kataria, R. P. Kulkarni, A. N. Sawarkar, D. Tandon,Y. Ram, and M. M. Kumar. Petroleum residue upgradation via visbreaking:A review. Industrial and Engineering Chemistry Research, 47(23):8960–8988,2008.

N. Kantarci, F. Borak, and K. O. Ulgen. Bubble column reactors. ProcessBiochemistry, 40(7):2263–2283, 2005.

Y. Kawase and M. Moo-Young. Liquid phase mixing in bubble columns withnewtonian and non-newtonian fluids. Chemical Engineering Science, 41(8):1969–1977, 1986.

Y. Kawase and M. Moo-Young. Theoretical prediction of gas hold-up inbubble columns with newtonian and non-newtonian fluids. Industrial andEngineering Chemistry Research, 26(5):933–937, 1987.

183


R. Krishna, P.M. Wilkinson, and L.L. Van Dierendonck. A model for gasholdup in bubble columns incorporating the influence of gas density on flowregime transitions. Chemical Engineering Science, 46(10):2491–2496, 1991.

A. Kumar, T. E. Degaleesan, G. S. Laddha, and H. E. Hoelscher. Bubble swarmcharacteristics in bubble columns. Canadian Journal of Chemical Engineering,54(6):503–508, 1976.

D. A. Lane. Scientific visualization of large-scale unsteady fluid flows. InScientific Visualization, Overviews, Methodologies, and Techniques, pages 125–145, Washington, DC, USA, 1997. IEEE Computer Society.

F. Lekien and J. Marsden. Tricubic interpolation in three dimensions. Interna-tional Journal for Numerical Methods in Engineering, 63(3):455–471, 2005.

O. Levenspiel. Chemical Reaction Engineering. John Wiley & Sons, 3rd edition,1999.

S. K. Majumder. Analysis of dispersion coefficient of bubble motion andvelocity characteristic factor in down and upflow bubble column reactor.Chemical Engineering Science, 63(12):3160–3170, 2008.

K. -H. Mangartz and Th. Pilhofer. Interpretation of mass transfer measure-ments in bubble columns considering dispersion of both phases. ChemicalEngineering Science, 36(6):1069–1077, 1981.

Y. Ohki and H. Inoue. Longitudinal mixing of the liquid phase in bubblecolumns. Chemical Engineering Science, 25(1):1–16, 1970.

C. P. Ribeiro Jr. and P. L. C. Lage. Gas-liquid direct-contact evaporation: Areview. Chemical Engineering and Technology, 28(10):1081–1107, 2005.


M. C. Ruzicka, J. Zahradnık, J. Drahos, and N. H. Thomas. Homogeneous-heterogeneous regime transition in bubble columns. Chemical EngineeringScience, 56(15):4609–4626, 2001.

184

References

H. Scott Fogler. Elements of Chemical Reaction Engineering. Prentice Hall PTR,4th edition, August 2005.


A. Shaikh and M. H. Al-Dahhan. A review on flow regime transition in bubblecolumns. International Journal of Chemical Reactor Engineering, 5, 2007.

K. D. Squires and J. K. Eaton. Lagrangian and eulerian statistics obtained fromdirect numerical simulations of homogeneous turbulence. Physics of FluidsA, 3(1):130–143, 1991.





S. Wachi and Y. Nojima. Gas-phase dispersion in bubble columns. ChemicalEngineering Science, 45(4):901–905, 1990.


K. R. Westerterp, W. P. M. van Swaaij, and A. A. C. M. Beenackers. ChemicalReactor Design and Operation. John Wiley & Sons Ltd., 2nd edition, 1987.

185


P. K. Yeung and S. B. Pope. An algorithm for tracking fluid particles innumerical simulations of homogeneous turbulence. Journal of ComputationalPhysics, 79(2):373–416, 1988.

P. Zehner. Momentum, mass and heat transfer in bubble columns. part 2. axialblending and heat transfer. International chemical engineering, 26(1):29–35,1986.

186

Summary

Experimental and Numerical Investigation of Bubble ColumnReactors

Due to various advantages, such as simple geometry, ease of operation, lowoperating and maintenance costs, excellent heat and mass transfer character-istics, bubble column reactors are frequently used in chemical, petrochemical,biochemical, pharmaceutical, metallurgical industries for a variety of pro-cesses, i.e. hydrogenation, oxidation, chlorination, alkylation, chemical gascleaning, various bio-technological applications, etc. However, complex hy-drodynamics and its influence on transport phenomena (i.e. heat and masstransport) make it difficult to achieve reliable design and scale-up of bubblecolumn reactors. Many factors influence the performance of this type of re-actors significantly, such as column dimensions, column internals design, gasdistributor design, operating conditions, i.e. pressure and temperature, su-perficial gas velocity, physical and chemical properties of the involved phases.A large variety of scientific studies on bubble column reactors utilizing bothexperimental and numerical techniques has been carried out during the pastdecades. In this study a bubble column with a square cross-sectional areahas been studied in detail using a combined experimental and computationalapproach.

Chapter 1 introduces bubble column reactors and their variants accordingto practical requirements. Both advantages and disadvantages of these typesof reactors are presented. Key parameters related to the performance of bubblecolumn reactors are also presented. In addition, in Chapter 1 a brief literaturereview is presented on both experimental and numerical techniques utilizedin investigations on the performance of bubble column reactors during thepast decades.

187

Summary

In Chapter 2, accuracy of a four-point optical fibre probe for measuringbubble properties is investigated in a flat bubble column. Photography is usedto validate results obtained from the four-point optical fibre probe. Accordingto the comparison, it is found that the liquid properties have a profoundinfluence on bubble velocity measured by the optical probe. Finally, it isfound that the extent of inaccuracy in the determination of bubble velocitycan be characterized with the Morton number.

The accuracy of the four-point optical fibre probe and its intrusive effect arefurther studied in a square bubble column operating at higher superficial gasvelocity in Chapter 3. Besides bubble velocity, other bubble properties, such aslocal void fraction, chord length and specific interfacial area, are obtained frommeasurements with the four-point optical fibre probe. Furthermore, bubblesize is determined in different ways. Possible reasons for the discrepancy inthe bubble size determination are discussed. The effect of the initial liquidheight in the bubble column on the bubble properties is also investigated.

Chapter 4 studies the effect of the gas sparger properties on the hydrody-namics in a square bubble column with an Eulerian-Lagrangian model. Theperformance of the model is first evaluated by comparison with experimentaldata. Subsequently, the effects of different sparged areas and the sparger loca-tion on hydrodynamics, i.e. liquid velocity, turbulent kinetic energy and voidfraction are investigated. Furthermore, the residence time distribution of thegas phase is extracted from the numerical simulations. These distributions areused to characterize the gas phase mixing in the bubble column by employinga standard axial dispersion model. The results reveal that the extent of mixingincreases when the sparged area decreases. The axial dispersion coefficientincreases as the sparged area is shifted towards the side wall.

For numerical simulation of bubbly flows, reliable closures are required torepresent the interfacial momentum transfer rate (i.e. the effective drag actingon bubbles). Furthermore, the presence of neighboring bubbles in a bubbleswarm may result in deviation of the drag force acting on isolated bubbles.Chapter 5 investigates the performance of several drag correlations reportedin literature for bubble swarms with the aid of a discrete bubble model. Bycomparing with experimental data, it is found that Lima Neto’s drag model

188

and Wen & Yu’s model have a better performance at low superficial gasvelocity and Rusche’s model can predict the hydrodynamics of the bubblyflows better compared to the other models at high superficial gas velocity.

In Chapter 6, breakup models developed in literature are implementedinto the Eulerian-Lagrangian model. Moreover, the critical Weber numberfor bubble breakup studied by many authors in turbulent flows is also in-corporated in the model. The performance of different breakup models andthe critical Weber number for predicting hydrodynamics and the bubble sizedistribution are compared with experimental data.

Finally, the Eulerian-Lagrangian model is further extended to study theperformance of bubble column reactors, i.e. predicting overall gas holdupand phase mixing in Chapter 7. The residence time distribution of the gasphase and tracer particles introduced in the liquid phase are used to study themixing of both the gas and liquid phase. It is found that the applied modelshows very good agreement with empirical correlations reported in literature.

189

Samenvatting

Experimentele en Numerieke Studie van BellenkolomReactoren

Vanwege de vele voordelen, zoals eenvoudige geometrie, eenvoudige oper-atie, lage operatie- en onderhoudskosten, uitstekende warmte- en stofover-dracht karakteristieken worden bellenkolom reactoren veelvuldig gebruiktin de chemische, petrochemische, biochemische en metallurgische industrievoor diverse processen, zoals hydrogenering, oxidatie, chlorering, alkyler-ing, chemische gas zuivering, diverse biochemische toepassingen, etc. Echter,de complexe hydrodynamica en de invloed hiervan op de transport verschi-jnselen (d.w.z. warmte- en stofoverdracht) bemoeilijken een betrouwbaarontwerp en opschaling van bellenkolommen.

Er zijn veel factoren die grote invloed hebben op de prestaties van dittype reactoren, zoals kolom afmetingen, het ontwerp van kolom internals,gasverdeler ontwerp, operatiecondities, d.w.z. druk en temperatuur, gassnel-heid, fysische en chemische eigenschappen van de verschillende fasen. Deafgelopen decennia is een grote verscheidenheid aan wetenschappelijke stud-ies over bellenkolom reactoren uitgevoerd, waarbij gebruik werd gemaakt vanzowel experimentele als numerieke technieken. In dit werk wordt een bel-lenkolom met een vierkante dwarsdoorsnede in detail bestudeerd met behulpvan een gecombineerde experimentele en numerieke aanpak.

In hoofdstuk 1 worden bellenkolom reactoren en hun varianten geıntro-duceerd aan de hand van praktische randvoorwaarden. Zowel voor- ennadelen van dit soort reactoren worden gepresenteerd en belangrijke param-eters worden gerelateerd aan het functioneren van bellenkolom reactoren.Tenslotte bevat hoofdstuk 1 een beknopte literatuurstudie van experimenteleen numerieke technieken die de afgelopen decennia gebruikt zijn in het on-

191

Samenvatting

derzoek naar het functioneren van bellenkolom reactoren.In hoofdstuk 2 wordt de nauwkeurigheid van een vier-punts glasvezel

probe onderzocht voor het meten van beleigenschappen in een platte bel-lenkolom. Digitale fotografie is gebruikt om de resultaten verkregen uitde probe te valideren. Uit deze vergelijking is geconstateerd dat devloeistofeigenschappen een grote invloed hebben op de door de probe geme-ten belsnelheid. Ten slotte is geconstateerd dat de onnauwkeurigheid in debepaling van de belsnelheid kan worden gekarakteriseerd door het Mortonkental.

De nauwkeurigheid van de glasvezel probe en de mate van verstoring vande vloeistof bij een hogere gassnelheid zijn verder onderzocht in een vierkantebellenkolom in hoofdstuk 3. Naast de belsnelheid zijn andere beleigenschap-pen, zoals de lokale gasfractie, de koordelengte en het specifieke oppervlakverkregen uit metingen met de glasvezel probe. Bovendien is de belgroottebepaald op verschillende manieren. Mogelijke redenen voor de discrepantiein de belgrootte bepaling worden besproken. De invloed van de initielevloeistofhoogte in de kolom op de beleigenschappen is ook onderzocht.

In hoofdstuk 4 wordt een Euler-Lagrange model gebruikt om het effect vande eigenschappen van de gasverdeler op de hydrodynamica in een vierkantebellenkolom te bestuderen. De prestaties van het model worden eerst geeval-ueerd aan de hand van vergelijking met experimentele data. Vervolgensworden de effecten van verschillende afmetingen en positionering van degasverdeler op de hydrodynamica, d.w.z. de vloeistofsnelheid, turbulentekinetische energie en de gasfractie onderzocht. Bovendien zijn de verblijfti-jdspreidingen van de gasfase uit de numerieke simulaties verkregen. Dezeworden gebruikt om de menging van de gasfase in de bellenkolom te karak-teriseren aan de hand van een standaard axiaal dispersie model. De resultatentonen aan dat de mate van menging toeneemt wanneer het oppervlak van degasverdeler afneemt. De axiale dispersie coefficient neemt toe naarmate degasverdeler verder naar de zijwand wordt verschoven.

Voor numerieke simulatie van bellenstromen, zijn betrouwbare sluit-ingsrelaties nodig voor de impulsuitwisseling tussen de fasen (d.w.z. deeffectieve wrijvingskracht op bellen). Bovendien kan de aanwezigheid van

192

naburige bellen in een bellenzwerm resulteren in een afwijking op de wrijv-ingskracht die geldt op individuele, geısoleerde bellen. In hoofdstuk 5 wordende prestaties van verschillende wrijvingskrachtcorrelaties voor bellenzwer-men uit de literatuur onderzocht met behulp van een discrete bellenmodel.Aan de hand van een vergelijking met experimentele data is vastgesteld datbij lage gassnelheden de wrijvingskrachtcorrelaties van Lima Neto en Wen& Yu de beste voorspellingen geven, terwijl bij hoge gassnelheden Rusche’scorrelatie de beste resultaten geeft.

In hoofdstuk 6 zijn literatuurmodellen voor het opbreken van bellengeımplementeerd in het Euler-Lagrange model. Bovendien zijn verschillendeliteratuurmodellen voor het kritische Weber kental bij opbreken van bellenin turbulente stromingen verdisconteerd in het model. De prestaties van deverschillende modellen voor het opbreken van bellen en het kritische Weberkental in termen van het voorspellen van hydrodynamica en de belgroot-teverdeling zijn vergeleken met experimentele data.

Ten slotte is in hoofdstuk 7 het Euler-Lagrange model verder uitgebreidom de prestaties van de bellenkolom reactoren te bestuderen, in het bijzondervoor het voorspellen van de totale gasfractie en het mengen van beide fasen.De verblijftijdverdeling van de gasfase en tracer deeltjes geıntroduceerd in devloeistoffase worden gebruikt om het mengen van zowel de gas- als vloeistof-fase te bestuderen. Het is gebleken dat de door het model voorspelde resul-taten zeer goed overeenkomen met empirische correlaties gerapporteerd inde literatuur.

193

总总总结结结

鼓鼓鼓泡泡泡塔塔塔反反反应应应器器器的的的实实实验验验与与与数数数值值值研研研究究究

由于具有结构简单、易操作、低运行与维护成本、高传热传质性质等优点，

鼓泡塔反应器被广泛地应用在化学、石油化学、生物化学、制药、冶金等工

业中的各种过程中，例如制氢、氧化、烷基化、化学气体清洗以及各种生物

技术应用等。然而，复杂的流体力学特征及其对传输性质（例如传热传质）

的影响，使得很难获得可靠设计和按比例放大的鼓泡塔反应器。很多因素，

极大地影响这类反应器的性能，例如反应器尺寸、反应器内部设计、气体分

布器设计、运行条件（如压力、温度）、表观气速以及各相的物理与化学性

质。在过去几十年里，人们做了大量关于鼓泡塔反应器的科学研究，运用了

实验与模拟技术。本研究利用实验与模拟技术详细地研究了一个具有方形截

面的鼓泡塔。

第一章介绍了鼓泡塔反应器及其根据不同实际需求的变体，以及此类反应

器的优缺点。并且介绍了与鼓泡塔反应器性能相关的关键参数。此外，第一

章中，文献综述简要地介绍了在过去几十年里，研究鼓泡塔反应器性能的实

验以及模拟技术。

在第二章里，研究了用于测量气泡性质的四针光纤探针的准确度。高速照

相技术所获得的气泡性质用于比较四针光纤探针的测量结果。比较发现，液

体物性显著地影响着探针所测量的气泡速度。最后，通过Morton数，描述了利用探针测量气泡速度的不准确程度。

第三章通过一个方形鼓泡塔，进一步地研究了在较高的表观气速下，光

纤探针测量的准确度及其侵入性的影响。除了气泡速度，四针光纤探针还测

量了其他气泡性质，例如局部气含率、气泡弦长和比界面积。此外，通过不

同的方法，确定了气泡的尺寸。并且讨论了造成确定气泡尺寸差异的可能原

因。最后，研究了鼓泡塔内初始液体高度对气泡性质的影响。

第四章利用欧拉-拉格朗日模型研究了气体分布器对鼓泡塔内流动性质的影响。首先，通过比较实验数据，验证了模型。随后，利用模型研究了不同的

195

总结

鼓泡区域对流体力学性质，如液体速度，湍动能及气含率的影响。此外，利

用模拟结果，获得了气相驻留时间分布。通过轴向分布模型，利用气相驻留

时间分布，描述了鼓泡塔内的气相混合。结果显示，混合程度随着鼓泡区域

的减小而变大。鼓泡区域越靠近侧壁，轴向分布系数越大。

泡状流的数值模拟需要可靠的封闭模型来描述界面动量传输率，例如作用

在气泡上的有效曳力。此外在大量气泡里，由于周围气泡的影响，作用在气

泡上的曳力可能会不同于作用在单个气泡上的曳力。第五章借助离散气泡模

型，研究了文献中的一些关于气泡群中的曳力关联式。通过比较实验数据发

现，在低表观气速下，Lima Neto以及Wen和Yu的模型优于其他模型。而在高表观气速下，同其他模型相比，Rusche的模型能够较好地预测泡状流的流动。

第六章将文献中发展的破碎模型应用到欧拉-拉格朗日模型。此外，湍流里关于气泡破碎的临界Weber数也被应用到了此模型中。通过比较实验数据，研究了这些不同破碎模型以及临界Weber数预测流动和气泡尺寸分布的性能。最后，在第七章中，欧拉-拉格朗日模型被进一步扩展到用于研究鼓泡塔反

应器的性能，例如预测整体气含率与相混合。气相驻留时间分布和液相中的

示踪粒子被用于研究气相和液相的混合。结果表明，模型所预测的结果同文

献中的实验关联式吻合得很好。

196

List of publications

Publications

• W. Bai, N. G. Deen, J.A.M. Kuipers. Numerical analysis of the effect ofgas sparger on the performance of bubble column, Ind.Eng.Chem.Res.,submitted.

• W. Bai, Y. M. Lau, N. G. Deen, J. A. M. Kuipers. Discrete bubble modelingof bubbly flows: Swarm effects, Chem.Eng.J., submitted.

• W. Bai, Y. M. Lau, N. G. Deen, J. A. M. Kuipers. Discrete bubble modelingof bubbly flows: Implementation of breakup models, Ind.Eng.Chem.Res.,in preparation.

• W. Bai, N. G. Deen, J. A. M. Kuipers. Numerical investigation of gasholdup and phase mixing in bubble column reactors, Ind.Eng.Chem.Res.,in preparation.

Conference papers

• W. Bai, N. G. Deen, and J. A. M. Kuipers. Bubble properties of heteroge-neous bubbly flows in a square bubble column, International Symposiumof Multiphase Flows, July 11-15, 2009, Xi’an, China.

• W. Bai, N. G. Deen, R. F. Mudde and J. A. M. Kuipers. Accuracy of four-point optical probe for determination of bubble velocity, 6th InternationalConference on Computational Fluid Dynamics in the Oil & Gas, Metallurgicaland Process Industries, Trondheim, Norway 10-12 June 2008.

197

Acknowledgement

In this thesis the results of a four-year research study are presented, whichwas conducted at the research group Fundamentals of Chemical ReactionEngineering, Faculty of Science and Technology, University of Twente, theNetherlands. I’d like to thank the Institute of Mechanics, Processes and Con-trol - Twente (IMPACT) for the financial support to the project. Furthermore,I would like to express my sincere thanks to those who have contributed theirhelp, support and care to both my work and my life in Enschede during thepast four years.

First of all, I would like to convey my thanks to my promotor, Prof. HansKuipers. Your careful attitude on research and profound knowledge of processengineering have made a strong impression on me. Your patient guidanceduring monthly meetings and every group activity organized by your familywill be part of my beautiful memories for the rest of my life.

A special thank to my co-promotor, Dr. Niels Deen, who acted as a dailysupervisor. I couldn’t forget our fruitful weekly discussions, your quick-witted mind and the encouragement you gave. Thank you for being there tohelp me always. I would also like to thank you for all the grammar checkingof the thesis, the Dutch translation of the summary and all other support youoffered. I wish both Hans and you great success at Eindhoven University ofTechnology.

I would like to thank the secretary of the group, Nicole Haitjema, forpreparing all the paperwork for my appointment as a PhD student and ar-ranging housing for me before I came to Enschede. Everything you have donefor me during the past four years is highly appreciated.

Sincere thanks to Gerrit Schorfhaar. I’d like to thank you for constructingthe experimental setup for me. I also want to thank you for the fruitfulconversations and the sharing of your travelling experience and your life.

199

Acknowledgement

I’m impressed that you can still run 5 kilometers twice a week as a seniorperson. I’d like to thank the other technicians in the group, Wim Leppink,Johan Agterhorst and Erik Analbers for their help and kindness. I would alsolike to thank Robert Meijer for his help on PC installing, email settings andeverything related to electronics.

I’d like to send my gratitude to the former group members for their sup-port and kindness: Mao Ye, Christiaan Zeilstra, Dadan Darmana, DongshengZhang, Sabita Sarkar, Wouter Dijkhuizen, Tymen Tiemersma, Sander Noor-man, Willem Godlieb, Jan Albert Laverman and Micheline Abbas. Manythanks to warm-hearted Mao and Chris for their enthusiastic help when I firstcame here. Special thanks to Dongsheng for his helping hand and invitingme for dinners and outings with his family. I would also like to thank all thecurrent members of the group: Prof. Martin van Sint Annaland, Dr. Martinvan der Hoef, Maureen van Buijtenen, Nhi Dang, Jelle de Jong, Tom Kolk-man, Sebastian Kriebitzsch, Yuk Man Lau, Ivo Roghair, Lianghui Tan, MartinTuinier, Olasaju Olaofe and Carles Mesado Melia for bringing a friendly andcreative atmosphere, nice X’mas parties, a memorable sailing event, borrelsand outings. Many thanks to Ivo and Tom for helping me on solving prob-lems with clusters and other things. Thank Yuk Man for discussing the DBMCode with me and offering other help. I’d also like to thank Martin (小马) fortalking to me in Chinese and offering his help.

I would like to extend my sincere thanks to Fausto Gallucci and his family.The delicious Italian food and the hospitality of his family have been embed-ded into my memory. Many thanks to Junwu Wang for the comments on mywork and the nice food at his place.

Many thanks to Wei Zhao & Wei Zhou(F), Yang Zhang & Rui Ge, WeiZhou(M), Xinhui Wang & Xuehui He, Xin Wang, Pengxiang Xu & Xia Bian, YanSong, Rui Guo and Hongmei Zhang for those happy times of dinner gathering,table tennis and dizhu fighting. I would like to thank my former roommatesXuexin Duan, Songyue Chen and Lixian Xu for the warm atmosphere at home,nice conversation and pleasant dinner. I also thank Songyue’s husband YifanHe and Lixian’s family, Hongping Luo and their son, Beibei for unforgettablememories. I would like to thank Huaping Xu for enjoying chats and beer with

200

me. Thank Xiaoying Shao and Shu-Han Hsu for inviting me for dinners andthe trip to Greece. I also thank all the other friends: Rongmei Li, Yixuan Li,Weihua Zhou, Xia Shang, Peng Zhang, Chunlin Song, Jinping Han, Yu Song,Yu Zeng and Xinyan Wen.

Special thanks to Ying Zhang and her parents for taking care of me andtreating me as part of their family. I am grateful to Jie Fan for designing thecover of my thesis.最后，特别感谢我的家人多年来对我的关心和支持：父母业已年迈，尚未

回报丁点养育之恩，深感内疚，而父母仍理解并支持儿子的求学之情。感谢

我的哥哥、嫂子和姐姐、姐夫，他们给我了很大的关心，并赡养父母，使得

我不论在何处学习或工作，均能够安心于学业与工作。感谢我的小外甥和小

侄女，给我带来的开心和快乐。诚挚感谢周彦婷小姐所给予的关心以及在英

文写作上的帮助，使得我能够充满信心地完成论文。

柏巍Wei BaiSeptember 2010, Ames

201

About the author

Wei Bai was born on March 14th 1977 in Hanzhong, Shaanxi, China. Aftercompleting his secondary education in 1996 at Mian county’s 1st high schoolin Shaanxi, China, he continued his study at Xi’an Jiaotong University, Chinaand obtained his bachelor’s degree in Heating, Ventilation and Air Condition-ing Engineering on the subject of “Heating system design of high buildings”in 2000. In 2001, he started his master study under the supervision of Prof.Qiuwang Wang at Xi’an Jiaotong University. He received the master degree inEngineering Thermophysics with a thesis titled ”Three-dimensional numeri-cal simulation of periodical flow and heat transfer in primary surface channel”in 2004. He then joined the Division of Building Science and Technology as aresearch assistant at City University of Hong Kong, China. From 2006 to 2010,he started his PhD research and was supervised by prof.dr.ir. J.A.M. Kuipersand dr.ir. Niels Deen in the Fundamentals of Chemical Reaction Engineeringgroup at University of Twente, the Netherlands. The results of this researchare presented in this dissertation. In June 2010, he joined Prof. Rodney O.Fox’s group as a post doc at Iowa State University in the U.S.A.

203

experimental and numerical investigation of bubble column ... · experimental and numerical...

Documents