Research Article Numerical Simulation of Bubble ...

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Research Article Numerical Simulation of Bubble Coalescence and Break-Up in Multinozzle Jet Ejector Dhanesh Patel, 1 Ashvinkumar Chaudhari, 2 Arto Laari, 3 Matti Heiliö, 4 Jari Hämäläinen, 2 and Kishorilal Agrawal 5 1 Centre for Industrial Mathematics and Department of Applied Mathematics, Faculty of Technology and Engineering, e M. S. University of Baroda, Vadodara, Gujarat 390001, India 2 Centre of Computational Engineering and Integrated Design (CEID), Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland 3 Department of Chemistry, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland 4 Department of Mathematics and Physics, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland 5 Department of Chemical Engineering, Faculty of Technology and Engineering, e M. S. University of Baroda, Vadodara, Gujarat 390001, India Correspondence should be addressed to Dhanesh Patel; [email protected] Received 26 October 2015; Accepted 22 December 2015 Academic Editor: Guan H. Yeoh Copyright © 2016 Dhanesh Patel et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Designing the jet ejector optimally is a challenging task and has a great impact on industrial applications. ree different sets of nozzles (namely, 1, 3, and 5) inside the jet ejector are compared in this study by using numerical simulations. More precisely, dynamics of bubble coalescence and breakup in the multinozzle jet ejectors are studied by means of Computational Fluid Dynamics (CFD). e population balance approach is used for the gas phase such that different bubble size groups are included in CFD and the number densities of each of them are predicted in CFD simulations. Here, commercial CFD soſtware ANSYS Fluent 14.0 is used. e realizable - turbulence model is used in CFD code in three-dimensional computational domains. It is clear that Reynolds- Averaged Navier-Stokes (RANS) models have their limitations, but on the other hand, turbulence modeling is not the key issue in this study and we can assume that the RANS models can predict turbulence of the carrying phase accurately enough. In order to validate our numerical predictions, results of one, three, and five nozzles are compared to laboratory experiments data for Cl 2 - NaOH system. Predicted gas volume fractions, bubble size distributions, and resulting number densities of the different bubble size groups as well as the interfacial area concentrations are in good agreement with experimental results. 1. Introduction ere are number of industrial processes in which two- phase flows, that is, gas-liquid mixture, in a jet ejector are encountered. Hence, reactions occurring in gas-liquid systems are of great importance in the chemical as well as in the process industry. Mass transfers in dispersions are directly related to the mass transfer coefficients as well as the interfacial area. e jet ejector is one kind of a venturi scrubber and it is widely used for conducting gas- liquid reactions in practical applications in industry such as pollution control and waste water treatment. Due to their simple construction, low operating cost, high energy efficiency, and good mass transfer characteristics, the jet ejectors have many advantages when used as the gas-liquid contactors. e experimental observations show that dis- persed bubbles towards the bottom of the jet ejector cause highly nonuniform volume distribution in the jet ejector. e gas volume fraction, the interfacial area, and the Sauter mean bubble diameter are the three important parameters that characterize the internal flow structure of gas-liquid flows in the jet ejector [1]. e interfacial transport of mass and Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2016, Article ID 5238737, 19 pages http://dx.doi.org/10.1155/2016/5238737

Transcript of Research Article Numerical Simulation of Bubble ...

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Research ArticleNumerical Simulation of Bubble Coalescence andBreak-Up in Multinozzle Jet Ejector

Dhanesh Patel1 Ashvinkumar Chaudhari2 Arto Laari3 Matti Heiliouml4

Jari Haumlmaumllaumlinen2 and Kishorilal Agrawal5

1Centre for Industrial Mathematics and Department of Applied Mathematics Faculty of Technology and EngineeringThe M S University of Baroda Vadodara Gujarat 390001 India2Centre of Computational Engineering and Integrated Design (CEID) Lappeenranta University of TechnologyPO Box 20 53851 Lappeenranta Finland3Department of Chemistry Lappeenranta University of Technology PO Box 20 53851 Lappeenranta Finland4Department of Mathematics and Physics Lappeenranta University of Technology PO Box 20 53851 Lappeenranta Finland5Department of Chemical Engineering Faculty of Technology and Engineering The M S University of BarodaVadodara Gujarat 390001 India

Correspondence should be addressed to Dhanesh Patel pdhaneshyahoocom

Received 26 October 2015 Accepted 22 December 2015

Academic Editor Guan H Yeoh

Copyright copy 2016 Dhanesh Patel et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Designing the jet ejector optimally is a challenging task and has a great impact on industrial applications Three different setsof nozzles (namely 1 3 and 5) inside the jet ejector are compared in this study by using numerical simulations More preciselydynamics of bubble coalescence and breakup in themultinozzle jet ejectors are studied bymeans of Computational Fluid Dynamics(CFD) The population balance approach is used for the gas phase such that different bubble size groups are included in CFD andthe number densities of each of them are predicted in CFD simulations Here commercial CFD softwareANSYS Fluent 140 is usedThe realizable 119896-120576 turbulence model is used in CFD code in three-dimensional computational domains It is clear that Reynolds-Averaged Navier-Stokes (RANS) models have their limitations but on the other hand turbulence modeling is not the key issuein this study and we can assume that the RANS models can predict turbulence of the carrying phase accurately enough In orderto validate our numerical predictions results of one three and five nozzles are compared to laboratory experiments data for Cl

2-

NaOH system Predicted gas volume fractions bubble size distributions and resulting number densities of the different bubble sizegroups as well as the interfacial area concentrations are in good agreement with experimental results

1 Introduction

There are number of industrial processes in which two-phase flows that is gas-liquid mixture in a jet ejectorare encountered Hence reactions occurring in gas-liquidsystems are of great importance in the chemical as wellas in the process industry Mass transfers in dispersionsare directly related to the mass transfer coefficients as wellas the interfacial area The jet ejector is one kind of aventuri scrubber and it is widely used for conducting gas-liquid reactions in practical applications in industry such

as pollution control and waste water treatment Due totheir simple construction low operating cost high energyefficiency and good mass transfer characteristics the jetejectors have many advantages when used as the gas-liquidcontactors The experimental observations show that dis-persed bubbles towards the bottom of the jet ejector causehighly nonuniform volume distribution in the jet ejectorThegas volume fraction the interfacial area and the Sauter meanbubble diameter are the three important parameters thatcharacterize the internal flow structure of gas-liquid flowsin the jet ejector [1] The interfacial transport of mass and

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 5238737 19 pageshttpdxdoiorg10115520165238737

2 Journal of Applied Mathematics

momentum are proportional to the interfacial area and thedriving forces This is an important parameter required for atwo-fluid model formulation [1] The mean bubble diameterserves as a link between the gas volume fraction and theinterfacial area concentration [1] An accurate knowledgeof local distributions of these three parameters is of greatimportance to eventual understanding and modeling ofthe interfacial transfer processes [1 2] Depending on thegas flow rate two main flow regimes are observed in thejet ejector namely the homogeneous bubbly flow regimeand the heterogeneous (churn-turbulent flow) regime [3]The homogeneous regime is encountered at relatively lowgas velocities and characterized by a narrow bubble sizedistribution and radially uniform gas holdup and it is themost desirable one for practical applications because it offersa large contact area [2ndash4]

The bubble size distribution and gas holdup in gas-liquiddispersions depend extensively on the jet ejector geometryoperating conditions and the physicochemical properties ofthe two phases The design of the jet ejector has primarilybeen carried out by means of empirical or semiempirical cor-relations based mainly on experimental data The scale-up ofthe jet ejector is still poorly understood due to the complexityof flow patterns and their unknown behavior under differentsets of design parameters such as area ratio projectionratio nozzle diameter length of free jet throat diffuserconvergence angle divergence angle and physical proper-ties of the liquid As a whole the phenomenon dependsstrongly on the jet ejector geometry and fluid dynamicsinvolved It is important to note that the similar kind ofstudy has been developed for bubble column reactor by[3]

The method to gain more knowledge and detailedphysical understanding of the hydrodynamics in the jetejector is Computational Fluid Dynamics (CFD) CFD canbe regarded as an effective tool to clarify the importanceof physical effects (eg gravity surface tension) on flowby adding or removing them An increasing number ofpapers deal with CFD applications of bubble columns [3 5ndash7] To analyze the flow pattern of the jet ejector both insteady and transient state conditions employing CFD themajority of researchers use either two- or three-dimensionalmodels when numerical simulations are usually compared toexperimental data As most of the early CFD studies considermonodispersed bubble size distributions ignoring break-up and coalescence mechanisms their validity is limited[3]

During the last two decades significant developmentshave been done in the modeling of two-phase flow processesbecause of the introduction of the two-fluid models [1]In the two-fluid models the interfacial transfer terms arerelated to the interfacial area concentration and the degree ofturbulence near the interfaces [1] Since the interfacial areaconcentration represents the key parameter that links theinteraction of the phases significant attention has been paidtowards developing a better understanding of the coalescenceand breakage effects due to interactions among bubbles andbetween bubbles and turbulent eddies for gas-liquid bubbly

flows [1 2 7ndash11] The population balance method is a well-known method for tracking the size distribution of thedispersed phase and accounting for the breakage and coales-cence dynamics in bubbly flows [1 12ndash22] The populationbalance method was also used by Hamalainen et al [23]and Hamalainen [24] in paper industry for papermakingsuspension flow

In gas-liquid two-phase systems bubble break-up andcoalescence can greatly influence their overall performanceby altering the interfacial area available for the mass transferbetween the phases [3]Therefore in order to develop reliablepredictive tools for designing the jet ejectors it is essen-tial to obtain some insight into the prevailing phenomenathrough simulation models based on the bubble formationand distraction mechanisms that is bubble coalescenceand break-up [3 12 15 25ndash28] incorporated into CFDsimulation making it possible to calculate hydrodynamicvariables such as liquid velocity gas holdup and bubble sizedistributions

In this work an attempt has been made to demonstratethe possibility of combining the population balance modelswith Computational Fluid Dynamics (CFD) for the case of agas-liquid bubbly flow in the jet ejector In all these processesgas holdup 120576

119892 and bubble size distribution are important

design parameters since they define the gas-liquid interfacialarea available for interfacial area mass transfer (119886) which isgiven by

119886 =6120576119892

11988932

(1)

where 11988932

is the mean Sauter diameter of the bubble sizedistribution [3] The MUSIG model implemented in ANSYSFluent 140 which accounts for the nonuniform bubble sizedistribution in a gas-liquid flow [1 2 6 16 29] is used inthis paper Gas volume fraction bubble size distributionnumber density of bubbles gas and liquid pressure variationinterfacial area concentration and gas and liquid velocityvariation in jet ejector are predicated The flow patterndevelopment has been studied at the free jet end and at thethroat end and at the end of the ejector in detail In additionnumerical predictions are compared with experiments andthe predicated gas interfacial area is in a good agreement withexperimental results

2 Jet Ejector

A large choice of gas-liquid contactors for example thefalling film column spray column packed column plate col-umn bubble columnmechanically agitated contactors spraytowers and venturi scrubbers are available for understandingthe mass transfer process Among these the venturi scrubberis a wet type design for gas-liquid contactors In venturi typeof scrubber

(i) liquid is a medium to absorb objectionable gases andparticulates from industrial gaseous waste streams

Journal of Applied Mathematics 3

(ii) a high velocity section of fluid jet is utilized to bringthe liquid and gas into intimate contact with eachother

Venturi scrubbers fall into two categories [30 31]In the first category it uses mechanical blower to draw a

high velocity gas stream through the system The liquid wasoriginally at rest but once the gas accelerates it splits intodroplets Particulates and gases are then confined into thecomparatively slower moving droplets This type is called aldquohigh energy venturi scrubberrdquo (HEVS)The scrubbing liquidcan be introduced in two ways

(i) If the liquid is introduced through nozzles which isusually at the throat it is known as Pearce-Anthonyventuri scrubber

(ii) In this case liquid is introduced as a film which isusually known as the wetted approach type

Secondly a mechanical pump or compressor is used to gen-erate a high velocity to the liquidfluid jet This liquidfluidjet creates suction and gas is entrained into it by transfer ofmomentum This type of the venturi scrubber is called anejector venturi scrubber or ldquojet ejectorrdquo

The jet ejectors have some advantages over other types ofcontactors which are mentioned as follows [31 32]

(i) Lower initial capital cost(ii) Simple construction and being compact(iii) Easy installation and operation(iv) No moving parts so little chances of mechanical

failure and hence highly reliable(v) Being able to deal with wet hot and corrosive gases

and thick aggressive and inflammable particles(vi) Ability to separate gaseous pollutant and fine partic-

ulate matter simultaneously(vii) Being able to handle large gas flow rates(viii) High heat andmass transfer rates and interfacial area

However it is not energy efficient equipment as a fluidmoving device But it has been reported that it has highefficiency as a gas-liquid contacting device [33ndash36]

The jet ejectors are especially effective in chemical andbiochemical industries for gas purification and for col-laborating gas-liquid reactions like chlorination oxidationhydrogenation and hydroformulation processes Jet ejectorsuse high kinetic energy of the operating fluid jet to promotebreak-up and distribution of the suction fluid into smalldropletsbubbles and to pull the gas through the system andpush through the connected outlet

A typical gas-liquid jet ejector is displayed in Figure 1It consists of a converging section a throat section and adiffuserdivergent section The gas is accelerated to atomizethe scrubbing liquid in the convergent section to reach ahigher velocity in the throat Throat is used for interactionof liquid and gases In the diffuserdivergent section thegas is slowing down allowing some recovery of pressure[37 38]

Nozzle

Gas suctionchamber

Free liquid jet

Liquid flow

Gas

Convergent

Throat

Divergent

Contactor

Figure 1 Typical gas-liquid jet ejector [31 113]

Jet ejectors have favorable mixing and mass transfercharacteristics so they are more used among other gas-liquidcontactors A jet ejector is a device in which suction mixingand dispersal of secondary fluid take place and are basedon the principle of utilizing the kinetic energy of a highvelocity motive (primary) fluid jet to entrain the secondaryphase to create fine dispersion of two phases (Utomo et al[39]) The secondary fluid may be dispersed by the shearingaction of the high velocity motive fluid or motive fluidmay get dispersed when it is captured by a secondary fluid[40]

Figure 1 shows the typical ejector system in which the jetof primary fluid typically liquid is pumped into the systemthrough high velocity through a nozzle flowing out of a nozzlewhich creates a low pressure region in the suction chamberinto which secondary fluid typically gas moves according toBernoullirsquos principleThe driving force for entrainment of thesecondary fluid is developed due to the pressure differencebetween the entry point of the secondary fluid and the nozzletipThere is amixing of gas and liquid phases and a gas-liquiddispersion takes place in the mixing tube In the diffuserthe pressure is recovered Coaxial-flow and froth-flow arethe two principal flow regimes in jet ejectors In the annularregion formed between the jet of primary fluid and ejectorwall a central core of primary fluid with secondary fluidflowing was observedThis regime forms coaxial-flow Froth-flow consists of liquid in which gas phase is completelydispersed in the form of bubble [41] The phenomenonof change from coaxial-flow to froth-flow is termed asmixing shock [42] The small bubbles are generated due tomixing shock which turns into creation of large interfacialarea (sim2000m2m3) Therefore greater rates of reaction andsuperior gas-liquid mass transfer rates are obtained in ejec-tors in comparison to other common gas-liquid contactors[31]

Depending on application there may be different objec-tives for design of an ejector which are as follows [43]

4 Journal of Applied Mathematics

(a) To achieve greater entrainment of the secondary fluid(b) To yield deep mixing between the two fluids(c) To inject fluids from a region of low pressure to a

region of higher pressure

Jet ejector may be used as a vacuum producing device as wellas jet pump With the rapid growth of the chemical processindustry their use as entraining and pumping corrosiveliquids slurries fumes and dust-laden gases has increasedTheir use asmass transfer equipment for liquid-liquid extrac-tion gas absorption gas stripping and slurry reaction likehydrogenation oxidation chlorination fermentation and soforth has increased [31 44ndash53]

The numbers of researchers have attempted to optimizeperformance of jet ejector [40 51 54ndash71] due to its increasinggrowth rate in usage Many researchers have studied themass transfer characteristics and performance of the jetejectors followed by contactors draft tube packed columnor bubble column and they have similar conclusion that thereis less mass transfer coefficient in the extended portion incomparison to the ejector [31 34 39 56 69ndash87] Dutta etal [74] and Gamisans et al [60] studied the jet ejectorsand observe that jet ejector without diffuser or throat isless effective in comparison to ejector with them Das andBiswas [40] observe that efficiency of ejector depends on thedesign of the suction chamber the throat the diffuser and thenozzle Apart from the dimensions of the various sections ofthe ejector the factors such as shape of the entrance to theparallel throat shape of entry of convergent section throataspect ratio angle of divergence and convergence and theprojection ratio are also important factors to be reflectedVarious researchers have studied the effect of geometry ofjet ejector in the optimal design of jet ejector Also manyresearchers have worked on flow patterns in jet ejector (see[79 88ndash92])

Performance of jet ejector is also depending on the factorslike bubble size distributions correlation for entrainmentbubble diameter drag force and gas holdup and factorsaffecting the bubble size Measurement of bubble size iscritical issue The researchers in these directions are Lefebvre[93] Liu [94] Schick [95] Xianguo and Tankin [96] DennisGary [97] Azad and Syeda [98] Frank [88] Silva et al [99]Kudzo [100] Ciborowski and Bin [101] Ohkawa et al [102]Sheng and Irons [103] Kitscha andOcamustafaogullari [104]Bailer [105] Evans et al [106] Ceylan et al [107] Havelka etal [76] Mandal et al [81] Zahradnık and Fialova [90] Bailer[105] Pawelczyk and Pindur [108] Dutta and Raghavan [74]Bin [109] and Zheng et al [110]

Figure 2 shows the major parts like primary fluid inletsecondary fluid inlet suction chamber converging sectionthroatmixing zone and diverging sectiondiffuser of anejector The symbols which we have used for describingdifferent components are as follows

(i) Length of throat (119871119879) length of diffuser (119871

119863) and dis-

tance between nozzle and commencement of throat(119871119879119873)

(ii) Diameter of nozzle (119863119873) diameter of throat (119863

119879)

and diameter of suction chamber (119863119878)

Primary fluid inletNozzle

Suction chamber

Convergent

Throat(mixing chamber)

Projection ratioPR = LTNDT

Divergent (diffuser)

Suction chamber

Pd(Hd) Doutlet

LD

DT

LT

LTN

DinletG PS(HS)

G

DN

DS

DinletL

Pj(Hj)

(12)120579divergent

(12)120579convergent

Area ratio AR = ATAN

= D2TD

2N

Area ratio = (D2S minus D2

N)D2N

L (m3s)

(m3s)

Figure 2 Schematic diagram showing geometry of an ejector [31111 112]

(iii) Area of throat (119860119879) area of nozzle (119860

119873) and area of

suction (119860119878)

(iv) Angle of converging sections (120579convergent) and angle ofdiverging sections (120579divergent)

Performance of the ejectors has been studied in terms of

(a) area of throatarea of nozzle that is area ratio (119860119877=

119860119879119860119873)

(b) length of throatdiameter of throat that is throataspect ratio (119871

119879119863119879)

(c) distance between nozzle tip and the commencementof throatdiameter of throat that is projection ratio(119875119877= 119871119879119873119863119879)

(d) suction chamber area ratio (119860119878119860119873= (1198632

119878minus 1198632

119873)

1198632

119873)

3 Mathematical Model

The poly dispersed multiphase flow is observed in jet ejectorIt means that the dispersed phase covers a wide range ofsize groups One of the characteristics of this flow is thatthe different sizes of the dispersed phase interact with eachother through the processes of break-up and coalescence Apopulation balance equation is framed to deal with this kindof a flow It is well known that the population balance modelis best fitted method to calculate the size distribution of adispersed phase which includes break-up and coalescenceeffect [3] Our model is in parallel to the model developed by

Journal of Applied Mathematics 5

Mouza et al [3] The general form of the population balanceequation is

120597119899119894

120597119905+ nabla sdot (

119892119899119894) = 119887119861minus 119889119861+ 119887119862minus 119889119862 (2)

where 119899119894 119887119861 119889119861 119887119862 119889119862 and

119892represent the number density

of size group 119894 the birth rate due to break-up the death ratedue to break-up the birth rate due to coalescence the deathrate due to coalescence and the gas velocity respectivelyTherelation between number density 119899

119894and the volume fraction

120572119892is given by

120572119892119891119894= 119899119894119881119894 (3)

where 119891119894and 119881

119894represent the volume fraction and the

corresponding volume of a bubble of group 119894 respectivelyThe coalescence of two bubbles occurs in three steps[3 19]

(i) First the bubbles collide trapping a slight amount ofliquid between them

(ii) Secondly this liquid film drains until it reaches acritical thickness

(iii) At the end the film breaks and the bubbles jointogether

The process of coalescence depends on the collision rate ofthe two bubbles and the collision efficiency Coalescence isa function of 119905

119894119895and 120591

119894119895 where 119905

119894119895is the time required for

coalescence and 120591119894119895is the contact time Collision is as a result

of turbulence (120579Tu119894119895) laminar shear (120579Ls

119894119895) and buoyancy (120579By

119894119895)

The total coalescence rate is

119876119894119895= (120579

Tu119894119895+ 120579

Bu119894119895+ 120579

Ls119894119895) 119890minus119905119894119895120591119894119895 (4)

Collision is also developed due to the difference in risevelocities of bubbles with different sizes The two conditionsobserved in jet ejector in parallel to the work of Mouza et al[3] for bubble column are as follows

(i) Jet ejectors operate at the homogeneous regime Dueto the narrow bubble size distribution in homoge-nous regime the relative effect of buoyancy can beneglected which leads to the assumption that bubblesrise with the same velocity regardless of their size

(ii) Collision occurs as a result of strong circulationpattern in jet ejector However this happens at gasrates higher than those encountered in the homoge-neous regime and hence laminar shear term may beneglected

Hence there is only the turbulent contribution in the modelwhich is given by

120579119894119895

120591119906= 119899119894119899119895119878119894119895(119906119905119894

2+ 119906119905119895

2) (5)

where 119899119894 119899119895 and119906

119894are the concentrations of bubbles of radius

119903119887119894 concentrations of bubbles of radius 119903

119887119895 and the aver-

age turbulent fluctuating velocity respectively The collision

cross-sectional area (119878119894119895) of the bubble the time required for

coalescence and the contact time are given by (6) (7) and(8) respectively

119878119894119895=120587

4(119877119887119894+ 119877119887119895)2

(6)

119905119894119895= (

119877119894119895

3120588119897

16120590)

05

lnℎ0

ℎ119891

(7)

120591119894119895=119877119894119895

23

12057613 (8)

where 119877119894119895 120588119897 120590 120576 ℎ

0 and ℎ

119891are the equivalent radius the

density of the liquid phase the surface tension the turbulentenergy dissipation the film thickness when collision beginsand the film thickness when ruptures of the film occurrespectively The birth rate of group 119894 due to coalescence ofgroup 119896 and group 119895 bubbles and the death rate of group 119894due to coalescence with other bubbles are given by (9) and(10) respectively

119887119862=1

2

119894

sum

119895=1

119894

sum

119896=1

119876119895119896119899119895119899119896 (9)

119889119862= 119899119894

119873

sum

119895=1

119876119894119895119899119895 (10)

The break-up of bubbles in turbulent dispersions employs themodel developed by Luo and Svendsen [28] The break-uprate of bubbles of size 119894 into bubbles of size 119895 is given by

119892 (V119895 V119894)

= 0923 (1 minus 120572119892)3radic(

120576

1198892119895

)int

1

120583min

(1 + 120583)2

120583113119890minus120591119888119889120583

(11)

where 120576 119889119895 120583 120591119888 and 119873 represent the turbulent energy

dissipation rate the bubble diameter the dimensionless sizeof eddies in the inertial subrange of isotropic turbulencethe critical dimensionless energy for break-up and the totalnumber of groups respectively

The birth rate of group 119894 bubbles due to break-up of largerbubbles and the death rate of group 119894 bubbles due to break-upinto smaller bubbles are given by (12) and (13) respectively

119887119861=

119873

sum

119895=119894+1

119892 (V119895 V119894) 119899119895 (12)

119889119861= 119892119894119899119894 (13)

Themass conservation equation for the liquid phase themassconservation equation for the gas phase and the momentum

6 Journal of Applied Mathematics

conservation equation in the Eulerian framework are givenby (14) (15) and (16) respectively

120597 (120572119897120588119897)

120597119905+ nabla sdot (120572

119897120588119897119897) = 0 (14)

120597 (120572119892119891119894120588119892)

120597119905+ nabla sdot (120572

119892119891119894120588119892119892) = 119878119894 (15)

120597 (120572119898120588119898119898)

120597119905+ nabla

sdot (120572119898120588119898119898119898minus 120583119898120572119898(nabla119898+ (nabla

119898)119879

))

= minus120572119898nabla119901 +119872

119898119899+ 120588119898119892

(16)

where 120588119898 119898 120572119898 120583119898 119901119872

119898119899 and 119892 represent the density

the velocity the volume fraction and the viscosity of the119898th phase the pressure the interphasemomentum exchangebetween phase 119898 and phase 119899 and the gravitational forcerespectively

The momentum exchange between gas and liquid phaseis given by

119872119897119892=3

4120588119898

120572119892

11988932

119862119863(119892minus 119897)10038161003816100381610038161003816119892minus 119897

10038161003816100381610038161003816 (17)

where 119862119863is the interfacial drag coefficient

4 Simulation Parameter forJet Ejector Geometry

Figure 3 shows the details of the geometry of ejector Agrawal[31 111 112] has done the experiments based on the exper-imental setup as shown in Figure 4 It is important to notethat these experimentswere conducted on the industrial stageejector Data for the geometry of the ejector is shown inTable 1

5 The CFD Simulation

Accurate modeling of the jet ejector requires a PopulationBalance Modeling (PBM) to be solved simultaneously witha CFD solver because of the presence of bubble-bubblecontacts or bubble-liquid contacts The PBM-CFD modelthat is MUSIG model (ANSYS Fluent 140) is implementedin this study which combines the population balance methodwith the break-up [28] and coalescence [19] models inorder to predict the bubble size distribution of the gasphase This model uses the Eulerian-Eulerian model TheMUSIGmodel has been extensively used for different systems[1ndash3 11 16 20 27 29]

The equations of continuity momentum and turbulencefor the continuous and dispersed phases give a standardtwo-phase flow calculation It can be extended to includenumber density of bubbles within several size groups usingthe MUSIG model

The size range of the bubbles is split into several groupswith for example groups of equal diameter Equations are

05 Nos holeDIA 36mmPitch 72mm

01 No holeDIA 82mm

03 Nos holeDIA 47mmPitch 94mm

40

80

77

25

25

25112

150

S1

S2

425

75 77

S3

Figure 3 Detail of the ejector used in the experimental setup [31111 112]

then solved for the number density in each group These sizefractions provide a more accurate measure of the interfacialarea density

In this study the simulation is done in three dimensionsand three geometries of the ejector namely with one nozzlethree nozzles and five nozzles as described in Figure 3In the present study bubbles ranging from 14472692 times

10minus5m (bin-20) to 97005842 times 10minus4m (bin-0) in diameter

are equally divided into 21 classes (see Table 2) as theexperimental observation of the gas volume fraction is 100and therefore we consider 0005 0010 0015 0018 00200025 0032 0030 0035 0040 0045 0050 0055 00600065 0070 0075 0080 0085 0090 0095 and 0100whichrepresent the volume fraction of the bubbles of group 119894(119894 = 0 1 2 3 4 5 20) In view of computational timersquosresources we consider the subdivision of the bubble sizesinto 21 size groups All computational results are basedon the discretization of the bubble sizes into 21 groupsANSYS Fluent 140 gives solution to the coupled sets ofgoverning equations for the balances ofmass andmomentumof each phase The conservation equations were discretizedusing the control volume technique Similar kind of study

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Numerical Simulation of Bubble ...

2 Journal of Applied Mathematics

momentum are proportional to the interfacial area and thedriving forces This is an important parameter required for atwo-fluid model formulation [1] The mean bubble diameterserves as a link between the gas volume fraction and theinterfacial area concentration [1] An accurate knowledgeof local distributions of these three parameters is of greatimportance to eventual understanding and modeling ofthe interfacial transfer processes [1 2] Depending on thegas flow rate two main flow regimes are observed in thejet ejector namely the homogeneous bubbly flow regimeand the heterogeneous (churn-turbulent flow) regime [3]The homogeneous regime is encountered at relatively lowgas velocities and characterized by a narrow bubble sizedistribution and radially uniform gas holdup and it is themost desirable one for practical applications because it offersa large contact area [2ndash4]

The bubble size distribution and gas holdup in gas-liquiddispersions depend extensively on the jet ejector geometryoperating conditions and the physicochemical properties ofthe two phases The design of the jet ejector has primarilybeen carried out by means of empirical or semiempirical cor-relations based mainly on experimental data The scale-up ofthe jet ejector is still poorly understood due to the complexityof flow patterns and their unknown behavior under differentsets of design parameters such as area ratio projectionratio nozzle diameter length of free jet throat diffuserconvergence angle divergence angle and physical proper-ties of the liquid As a whole the phenomenon dependsstrongly on the jet ejector geometry and fluid dynamicsinvolved It is important to note that the similar kind ofstudy has been developed for bubble column reactor by[3]

The method to gain more knowledge and detailedphysical understanding of the hydrodynamics in the jetejector is Computational Fluid Dynamics (CFD) CFD canbe regarded as an effective tool to clarify the importanceof physical effects (eg gravity surface tension) on flowby adding or removing them An increasing number ofpapers deal with CFD applications of bubble columns [3 5ndash7] To analyze the flow pattern of the jet ejector both insteady and transient state conditions employing CFD themajority of researchers use either two- or three-dimensionalmodels when numerical simulations are usually compared toexperimental data As most of the early CFD studies considermonodispersed bubble size distributions ignoring break-up and coalescence mechanisms their validity is limited[3]

During the last two decades significant developmentshave been done in the modeling of two-phase flow processesbecause of the introduction of the two-fluid models [1]In the two-fluid models the interfacial transfer terms arerelated to the interfacial area concentration and the degree ofturbulence near the interfaces [1] Since the interfacial areaconcentration represents the key parameter that links theinteraction of the phases significant attention has been paidtowards developing a better understanding of the coalescenceand breakage effects due to interactions among bubbles andbetween bubbles and turbulent eddies for gas-liquid bubbly

flows [1 2 7ndash11] The population balance method is a well-known method for tracking the size distribution of thedispersed phase and accounting for the breakage and coales-cence dynamics in bubbly flows [1 12ndash22] The populationbalance method was also used by Hamalainen et al [23]and Hamalainen [24] in paper industry for papermakingsuspension flow

In gas-liquid two-phase systems bubble break-up andcoalescence can greatly influence their overall performanceby altering the interfacial area available for the mass transferbetween the phases [3]Therefore in order to develop reliablepredictive tools for designing the jet ejectors it is essen-tial to obtain some insight into the prevailing phenomenathrough simulation models based on the bubble formationand distraction mechanisms that is bubble coalescenceand break-up [3 12 15 25ndash28] incorporated into CFDsimulation making it possible to calculate hydrodynamicvariables such as liquid velocity gas holdup and bubble sizedistributions

In this work an attempt has been made to demonstratethe possibility of combining the population balance modelswith Computational Fluid Dynamics (CFD) for the case of agas-liquid bubbly flow in the jet ejector In all these processesgas holdup 120576

119892 and bubble size distribution are important

design parameters since they define the gas-liquid interfacialarea available for interfacial area mass transfer (119886) which isgiven by

119886 =6120576119892

11988932

(1)

where 11988932

is the mean Sauter diameter of the bubble sizedistribution [3] The MUSIG model implemented in ANSYSFluent 140 which accounts for the nonuniform bubble sizedistribution in a gas-liquid flow [1 2 6 16 29] is used inthis paper Gas volume fraction bubble size distributionnumber density of bubbles gas and liquid pressure variationinterfacial area concentration and gas and liquid velocityvariation in jet ejector are predicated The flow patterndevelopment has been studied at the free jet end and at thethroat end and at the end of the ejector in detail In additionnumerical predictions are compared with experiments andthe predicated gas interfacial area is in a good agreement withexperimental results

2 Jet Ejector

A large choice of gas-liquid contactors for example thefalling film column spray column packed column plate col-umn bubble columnmechanically agitated contactors spraytowers and venturi scrubbers are available for understandingthe mass transfer process Among these the venturi scrubberis a wet type design for gas-liquid contactors In venturi typeof scrubber

(i) liquid is a medium to absorb objectionable gases andparticulates from industrial gaseous waste streams

Journal of Applied Mathematics 3

(ii) a high velocity section of fluid jet is utilized to bringthe liquid and gas into intimate contact with eachother

Venturi scrubbers fall into two categories [30 31]In the first category it uses mechanical blower to draw a

high velocity gas stream through the system The liquid wasoriginally at rest but once the gas accelerates it splits intodroplets Particulates and gases are then confined into thecomparatively slower moving droplets This type is called aldquohigh energy venturi scrubberrdquo (HEVS)The scrubbing liquidcan be introduced in two ways

(i) If the liquid is introduced through nozzles which isusually at the throat it is known as Pearce-Anthonyventuri scrubber

(ii) In this case liquid is introduced as a film which isusually known as the wetted approach type

Secondly a mechanical pump or compressor is used to gen-erate a high velocity to the liquidfluid jet This liquidfluidjet creates suction and gas is entrained into it by transfer ofmomentum This type of the venturi scrubber is called anejector venturi scrubber or ldquojet ejectorrdquo

The jet ejectors have some advantages over other types ofcontactors which are mentioned as follows [31 32]

(i) Lower initial capital cost(ii) Simple construction and being compact(iii) Easy installation and operation(iv) No moving parts so little chances of mechanical

failure and hence highly reliable(v) Being able to deal with wet hot and corrosive gases

and thick aggressive and inflammable particles(vi) Ability to separate gaseous pollutant and fine partic-

ulate matter simultaneously(vii) Being able to handle large gas flow rates(viii) High heat andmass transfer rates and interfacial area

However it is not energy efficient equipment as a fluidmoving device But it has been reported that it has highefficiency as a gas-liquid contacting device [33ndash36]

The jet ejectors are especially effective in chemical andbiochemical industries for gas purification and for col-laborating gas-liquid reactions like chlorination oxidationhydrogenation and hydroformulation processes Jet ejectorsuse high kinetic energy of the operating fluid jet to promotebreak-up and distribution of the suction fluid into smalldropletsbubbles and to pull the gas through the system andpush through the connected outlet

A typical gas-liquid jet ejector is displayed in Figure 1It consists of a converging section a throat section and adiffuserdivergent section The gas is accelerated to atomizethe scrubbing liquid in the convergent section to reach ahigher velocity in the throat Throat is used for interactionof liquid and gases In the diffuserdivergent section thegas is slowing down allowing some recovery of pressure[37 38]

Nozzle

Gas suctionchamber

Free liquid jet

Liquid flow

Gas

Convergent

Throat

Divergent

Contactor

Figure 1 Typical gas-liquid jet ejector [31 113]

Jet ejectors have favorable mixing and mass transfercharacteristics so they are more used among other gas-liquidcontactors A jet ejector is a device in which suction mixingand dispersal of secondary fluid take place and are basedon the principle of utilizing the kinetic energy of a highvelocity motive (primary) fluid jet to entrain the secondaryphase to create fine dispersion of two phases (Utomo et al[39]) The secondary fluid may be dispersed by the shearingaction of the high velocity motive fluid or motive fluidmay get dispersed when it is captured by a secondary fluid[40]

Figure 1 shows the typical ejector system in which the jetof primary fluid typically liquid is pumped into the systemthrough high velocity through a nozzle flowing out of a nozzlewhich creates a low pressure region in the suction chamberinto which secondary fluid typically gas moves according toBernoullirsquos principleThe driving force for entrainment of thesecondary fluid is developed due to the pressure differencebetween the entry point of the secondary fluid and the nozzletipThere is amixing of gas and liquid phases and a gas-liquiddispersion takes place in the mixing tube In the diffuserthe pressure is recovered Coaxial-flow and froth-flow arethe two principal flow regimes in jet ejectors In the annularregion formed between the jet of primary fluid and ejectorwall a central core of primary fluid with secondary fluidflowing was observedThis regime forms coaxial-flow Froth-flow consists of liquid in which gas phase is completelydispersed in the form of bubble [41] The phenomenonof change from coaxial-flow to froth-flow is termed asmixing shock [42] The small bubbles are generated due tomixing shock which turns into creation of large interfacialarea (sim2000m2m3) Therefore greater rates of reaction andsuperior gas-liquid mass transfer rates are obtained in ejec-tors in comparison to other common gas-liquid contactors[31]

Depending on application there may be different objec-tives for design of an ejector which are as follows [43]

4 Journal of Applied Mathematics

(a) To achieve greater entrainment of the secondary fluid(b) To yield deep mixing between the two fluids(c) To inject fluids from a region of low pressure to a

region of higher pressure

Jet ejector may be used as a vacuum producing device as wellas jet pump With the rapid growth of the chemical processindustry their use as entraining and pumping corrosiveliquids slurries fumes and dust-laden gases has increasedTheir use asmass transfer equipment for liquid-liquid extrac-tion gas absorption gas stripping and slurry reaction likehydrogenation oxidation chlorination fermentation and soforth has increased [31 44ndash53]

The numbers of researchers have attempted to optimizeperformance of jet ejector [40 51 54ndash71] due to its increasinggrowth rate in usage Many researchers have studied themass transfer characteristics and performance of the jetejectors followed by contactors draft tube packed columnor bubble column and they have similar conclusion that thereis less mass transfer coefficient in the extended portion incomparison to the ejector [31 34 39 56 69ndash87] Dutta etal [74] and Gamisans et al [60] studied the jet ejectorsand observe that jet ejector without diffuser or throat isless effective in comparison to ejector with them Das andBiswas [40] observe that efficiency of ejector depends on thedesign of the suction chamber the throat the diffuser and thenozzle Apart from the dimensions of the various sections ofthe ejector the factors such as shape of the entrance to theparallel throat shape of entry of convergent section throataspect ratio angle of divergence and convergence and theprojection ratio are also important factors to be reflectedVarious researchers have studied the effect of geometry ofjet ejector in the optimal design of jet ejector Also manyresearchers have worked on flow patterns in jet ejector (see[79 88ndash92])

Performance of jet ejector is also depending on the factorslike bubble size distributions correlation for entrainmentbubble diameter drag force and gas holdup and factorsaffecting the bubble size Measurement of bubble size iscritical issue The researchers in these directions are Lefebvre[93] Liu [94] Schick [95] Xianguo and Tankin [96] DennisGary [97] Azad and Syeda [98] Frank [88] Silva et al [99]Kudzo [100] Ciborowski and Bin [101] Ohkawa et al [102]Sheng and Irons [103] Kitscha andOcamustafaogullari [104]Bailer [105] Evans et al [106] Ceylan et al [107] Havelka etal [76] Mandal et al [81] Zahradnık and Fialova [90] Bailer[105] Pawelczyk and Pindur [108] Dutta and Raghavan [74]Bin [109] and Zheng et al [110]

Figure 2 shows the major parts like primary fluid inletsecondary fluid inlet suction chamber converging sectionthroatmixing zone and diverging sectiondiffuser of anejector The symbols which we have used for describingdifferent components are as follows

(i) Length of throat (119871119879) length of diffuser (119871

119863) and dis-

tance between nozzle and commencement of throat(119871119879119873)

(ii) Diameter of nozzle (119863119873) diameter of throat (119863

119879)

and diameter of suction chamber (119863119878)

Primary fluid inletNozzle

Suction chamber

Convergent

Throat(mixing chamber)

Projection ratioPR = LTNDT

Divergent (diffuser)

Suction chamber

Pd(Hd) Doutlet

LD

DT

LT

LTN

DinletG PS(HS)

G

DN

DS

DinletL

Pj(Hj)

(12)120579divergent

(12)120579convergent

Area ratio AR = ATAN

= D2TD

2N

Area ratio = (D2S minus D2

N)D2N

L (m3s)

(m3s)

Figure 2 Schematic diagram showing geometry of an ejector [31111 112]

(iii) Area of throat (119860119879) area of nozzle (119860

119873) and area of

suction (119860119878)

(iv) Angle of converging sections (120579convergent) and angle ofdiverging sections (120579divergent)

Performance of the ejectors has been studied in terms of

(a) area of throatarea of nozzle that is area ratio (119860119877=

119860119879119860119873)

(b) length of throatdiameter of throat that is throataspect ratio (119871

119879119863119879)

(c) distance between nozzle tip and the commencementof throatdiameter of throat that is projection ratio(119875119877= 119871119879119873119863119879)

(d) suction chamber area ratio (119860119878119860119873= (1198632

119878minus 1198632

119873)

1198632

119873)

3 Mathematical Model

The poly dispersed multiphase flow is observed in jet ejectorIt means that the dispersed phase covers a wide range ofsize groups One of the characteristics of this flow is thatthe different sizes of the dispersed phase interact with eachother through the processes of break-up and coalescence Apopulation balance equation is framed to deal with this kindof a flow It is well known that the population balance modelis best fitted method to calculate the size distribution of adispersed phase which includes break-up and coalescenceeffect [3] Our model is in parallel to the model developed by

Journal of Applied Mathematics 5

Mouza et al [3] The general form of the population balanceequation is

120597119899119894

120597119905+ nabla sdot (

119892119899119894) = 119887119861minus 119889119861+ 119887119862minus 119889119862 (2)

where 119899119894 119887119861 119889119861 119887119862 119889119862 and

119892represent the number density

of size group 119894 the birth rate due to break-up the death ratedue to break-up the birth rate due to coalescence the deathrate due to coalescence and the gas velocity respectivelyTherelation between number density 119899

119894and the volume fraction

120572119892is given by

120572119892119891119894= 119899119894119881119894 (3)

where 119891119894and 119881

119894represent the volume fraction and the

corresponding volume of a bubble of group 119894 respectivelyThe coalescence of two bubbles occurs in three steps[3 19]

(i) First the bubbles collide trapping a slight amount ofliquid between them

(ii) Secondly this liquid film drains until it reaches acritical thickness

(iii) At the end the film breaks and the bubbles jointogether

The process of coalescence depends on the collision rate ofthe two bubbles and the collision efficiency Coalescence isa function of 119905

119894119895and 120591

119894119895 where 119905

119894119895is the time required for

coalescence and 120591119894119895is the contact time Collision is as a result

of turbulence (120579Tu119894119895) laminar shear (120579Ls

119894119895) and buoyancy (120579By

119894119895)

The total coalescence rate is

119876119894119895= (120579

Tu119894119895+ 120579

Bu119894119895+ 120579

Ls119894119895) 119890minus119905119894119895120591119894119895 (4)

Collision is also developed due to the difference in risevelocities of bubbles with different sizes The two conditionsobserved in jet ejector in parallel to the work of Mouza et al[3] for bubble column are as follows

(i) Jet ejectors operate at the homogeneous regime Dueto the narrow bubble size distribution in homoge-nous regime the relative effect of buoyancy can beneglected which leads to the assumption that bubblesrise with the same velocity regardless of their size

(ii) Collision occurs as a result of strong circulationpattern in jet ejector However this happens at gasrates higher than those encountered in the homoge-neous regime and hence laminar shear term may beneglected

Hence there is only the turbulent contribution in the modelwhich is given by

120579119894119895

120591119906= 119899119894119899119895119878119894119895(119906119905119894

2+ 119906119905119895

2) (5)

where 119899119894 119899119895 and119906

119894are the concentrations of bubbles of radius

119903119887119894 concentrations of bubbles of radius 119903

119887119895 and the aver-

age turbulent fluctuating velocity respectively The collision

cross-sectional area (119878119894119895) of the bubble the time required for

coalescence and the contact time are given by (6) (7) and(8) respectively

119878119894119895=120587

4(119877119887119894+ 119877119887119895)2

(6)

119905119894119895= (

119877119894119895

3120588119897

16120590)

05

lnℎ0

ℎ119891

(7)

120591119894119895=119877119894119895

23

12057613 (8)

where 119877119894119895 120588119897 120590 120576 ℎ

0 and ℎ

119891are the equivalent radius the

density of the liquid phase the surface tension the turbulentenergy dissipation the film thickness when collision beginsand the film thickness when ruptures of the film occurrespectively The birth rate of group 119894 due to coalescence ofgroup 119896 and group 119895 bubbles and the death rate of group 119894due to coalescence with other bubbles are given by (9) and(10) respectively

119887119862=1

2

119894

sum

119895=1

119894

sum

119896=1

119876119895119896119899119895119899119896 (9)

119889119862= 119899119894

119873

sum

119895=1

119876119894119895119899119895 (10)

The break-up of bubbles in turbulent dispersions employs themodel developed by Luo and Svendsen [28] The break-uprate of bubbles of size 119894 into bubbles of size 119895 is given by

119892 (V119895 V119894)

= 0923 (1 minus 120572119892)3radic(

120576

1198892119895

)int

1

120583min

(1 + 120583)2

120583113119890minus120591119888119889120583

(11)

where 120576 119889119895 120583 120591119888 and 119873 represent the turbulent energy

dissipation rate the bubble diameter the dimensionless sizeof eddies in the inertial subrange of isotropic turbulencethe critical dimensionless energy for break-up and the totalnumber of groups respectively

The birth rate of group 119894 bubbles due to break-up of largerbubbles and the death rate of group 119894 bubbles due to break-upinto smaller bubbles are given by (12) and (13) respectively

119887119861=

119873

sum

119895=119894+1

119892 (V119895 V119894) 119899119895 (12)

119889119861= 119892119894119899119894 (13)

Themass conservation equation for the liquid phase themassconservation equation for the gas phase and the momentum

6 Journal of Applied Mathematics

conservation equation in the Eulerian framework are givenby (14) (15) and (16) respectively

120597 (120572119897120588119897)

120597119905+ nabla sdot (120572

119897120588119897119897) = 0 (14)

120597 (120572119892119891119894120588119892)

120597119905+ nabla sdot (120572

119892119891119894120588119892119892) = 119878119894 (15)

120597 (120572119898120588119898119898)

120597119905+ nabla

sdot (120572119898120588119898119898119898minus 120583119898120572119898(nabla119898+ (nabla

119898)119879

))

= minus120572119898nabla119901 +119872

119898119899+ 120588119898119892

(16)

where 120588119898 119898 120572119898 120583119898 119901119872

119898119899 and 119892 represent the density

the velocity the volume fraction and the viscosity of the119898th phase the pressure the interphasemomentum exchangebetween phase 119898 and phase 119899 and the gravitational forcerespectively

The momentum exchange between gas and liquid phaseis given by

119872119897119892=3

4120588119898

120572119892

11988932

119862119863(119892minus 119897)10038161003816100381610038161003816119892minus 119897

10038161003816100381610038161003816 (17)

where 119862119863is the interfacial drag coefficient

4 Simulation Parameter forJet Ejector Geometry

Figure 3 shows the details of the geometry of ejector Agrawal[31 111 112] has done the experiments based on the exper-imental setup as shown in Figure 4 It is important to notethat these experimentswere conducted on the industrial stageejector Data for the geometry of the ejector is shown inTable 1

5 The CFD Simulation

Accurate modeling of the jet ejector requires a PopulationBalance Modeling (PBM) to be solved simultaneously witha CFD solver because of the presence of bubble-bubblecontacts or bubble-liquid contacts The PBM-CFD modelthat is MUSIG model (ANSYS Fluent 140) is implementedin this study which combines the population balance methodwith the break-up [28] and coalescence [19] models inorder to predict the bubble size distribution of the gasphase This model uses the Eulerian-Eulerian model TheMUSIGmodel has been extensively used for different systems[1ndash3 11 16 20 27 29]

The equations of continuity momentum and turbulencefor the continuous and dispersed phases give a standardtwo-phase flow calculation It can be extended to includenumber density of bubbles within several size groups usingthe MUSIG model

The size range of the bubbles is split into several groupswith for example groups of equal diameter Equations are

05 Nos holeDIA 36mmPitch 72mm

01 No holeDIA 82mm

03 Nos holeDIA 47mmPitch 94mm

40

80

77

25

25

25112

150

S1

S2

425

75 77

S3

Figure 3 Detail of the ejector used in the experimental setup [31111 112]

then solved for the number density in each group These sizefractions provide a more accurate measure of the interfacialarea density

In this study the simulation is done in three dimensionsand three geometries of the ejector namely with one nozzlethree nozzles and five nozzles as described in Figure 3In the present study bubbles ranging from 14472692 times

10minus5m (bin-20) to 97005842 times 10minus4m (bin-0) in diameter

are equally divided into 21 classes (see Table 2) as theexperimental observation of the gas volume fraction is 100and therefore we consider 0005 0010 0015 0018 00200025 0032 0030 0035 0040 0045 0050 0055 00600065 0070 0075 0080 0085 0090 0095 and 0100whichrepresent the volume fraction of the bubbles of group 119894(119894 = 0 1 2 3 4 5 20) In view of computational timersquosresources we consider the subdivision of the bubble sizesinto 21 size groups All computational results are basedon the discretization of the bubble sizes into 21 groupsANSYS Fluent 140 gives solution to the coupled sets ofgoverning equations for the balances ofmass andmomentumof each phase The conservation equations were discretizedusing the control volume technique Similar kind of study

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 3: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 3

(ii) a high velocity section of fluid jet is utilized to bringthe liquid and gas into intimate contact with eachother

Venturi scrubbers fall into two categories [30 31]In the first category it uses mechanical blower to draw a

high velocity gas stream through the system The liquid wasoriginally at rest but once the gas accelerates it splits intodroplets Particulates and gases are then confined into thecomparatively slower moving droplets This type is called aldquohigh energy venturi scrubberrdquo (HEVS)The scrubbing liquidcan be introduced in two ways

(i) If the liquid is introduced through nozzles which isusually at the throat it is known as Pearce-Anthonyventuri scrubber

(ii) In this case liquid is introduced as a film which isusually known as the wetted approach type

Secondly a mechanical pump or compressor is used to gen-erate a high velocity to the liquidfluid jet This liquidfluidjet creates suction and gas is entrained into it by transfer ofmomentum This type of the venturi scrubber is called anejector venturi scrubber or ldquojet ejectorrdquo

The jet ejectors have some advantages over other types ofcontactors which are mentioned as follows [31 32]

(i) Lower initial capital cost(ii) Simple construction and being compact(iii) Easy installation and operation(iv) No moving parts so little chances of mechanical

failure and hence highly reliable(v) Being able to deal with wet hot and corrosive gases

and thick aggressive and inflammable particles(vi) Ability to separate gaseous pollutant and fine partic-

ulate matter simultaneously(vii) Being able to handle large gas flow rates(viii) High heat andmass transfer rates and interfacial area

However it is not energy efficient equipment as a fluidmoving device But it has been reported that it has highefficiency as a gas-liquid contacting device [33ndash36]

The jet ejectors are especially effective in chemical andbiochemical industries for gas purification and for col-laborating gas-liquid reactions like chlorination oxidationhydrogenation and hydroformulation processes Jet ejectorsuse high kinetic energy of the operating fluid jet to promotebreak-up and distribution of the suction fluid into smalldropletsbubbles and to pull the gas through the system andpush through the connected outlet

A typical gas-liquid jet ejector is displayed in Figure 1It consists of a converging section a throat section and adiffuserdivergent section The gas is accelerated to atomizethe scrubbing liquid in the convergent section to reach ahigher velocity in the throat Throat is used for interactionof liquid and gases In the diffuserdivergent section thegas is slowing down allowing some recovery of pressure[37 38]

Nozzle

Gas suctionchamber

Free liquid jet

Liquid flow

Gas

Convergent

Throat

Divergent

Contactor

Figure 1 Typical gas-liquid jet ejector [31 113]

Jet ejectors have favorable mixing and mass transfercharacteristics so they are more used among other gas-liquidcontactors A jet ejector is a device in which suction mixingand dispersal of secondary fluid take place and are basedon the principle of utilizing the kinetic energy of a highvelocity motive (primary) fluid jet to entrain the secondaryphase to create fine dispersion of two phases (Utomo et al[39]) The secondary fluid may be dispersed by the shearingaction of the high velocity motive fluid or motive fluidmay get dispersed when it is captured by a secondary fluid[40]

Figure 1 shows the typical ejector system in which the jetof primary fluid typically liquid is pumped into the systemthrough high velocity through a nozzle flowing out of a nozzlewhich creates a low pressure region in the suction chamberinto which secondary fluid typically gas moves according toBernoullirsquos principleThe driving force for entrainment of thesecondary fluid is developed due to the pressure differencebetween the entry point of the secondary fluid and the nozzletipThere is amixing of gas and liquid phases and a gas-liquiddispersion takes place in the mixing tube In the diffuserthe pressure is recovered Coaxial-flow and froth-flow arethe two principal flow regimes in jet ejectors In the annularregion formed between the jet of primary fluid and ejectorwall a central core of primary fluid with secondary fluidflowing was observedThis regime forms coaxial-flow Froth-flow consists of liquid in which gas phase is completelydispersed in the form of bubble [41] The phenomenonof change from coaxial-flow to froth-flow is termed asmixing shock [42] The small bubbles are generated due tomixing shock which turns into creation of large interfacialarea (sim2000m2m3) Therefore greater rates of reaction andsuperior gas-liquid mass transfer rates are obtained in ejec-tors in comparison to other common gas-liquid contactors[31]

Depending on application there may be different objec-tives for design of an ejector which are as follows [43]

4 Journal of Applied Mathematics

(a) To achieve greater entrainment of the secondary fluid(b) To yield deep mixing between the two fluids(c) To inject fluids from a region of low pressure to a

region of higher pressure

Jet ejector may be used as a vacuum producing device as wellas jet pump With the rapid growth of the chemical processindustry their use as entraining and pumping corrosiveliquids slurries fumes and dust-laden gases has increasedTheir use asmass transfer equipment for liquid-liquid extrac-tion gas absorption gas stripping and slurry reaction likehydrogenation oxidation chlorination fermentation and soforth has increased [31 44ndash53]

The numbers of researchers have attempted to optimizeperformance of jet ejector [40 51 54ndash71] due to its increasinggrowth rate in usage Many researchers have studied themass transfer characteristics and performance of the jetejectors followed by contactors draft tube packed columnor bubble column and they have similar conclusion that thereis less mass transfer coefficient in the extended portion incomparison to the ejector [31 34 39 56 69ndash87] Dutta etal [74] and Gamisans et al [60] studied the jet ejectorsand observe that jet ejector without diffuser or throat isless effective in comparison to ejector with them Das andBiswas [40] observe that efficiency of ejector depends on thedesign of the suction chamber the throat the diffuser and thenozzle Apart from the dimensions of the various sections ofthe ejector the factors such as shape of the entrance to theparallel throat shape of entry of convergent section throataspect ratio angle of divergence and convergence and theprojection ratio are also important factors to be reflectedVarious researchers have studied the effect of geometry ofjet ejector in the optimal design of jet ejector Also manyresearchers have worked on flow patterns in jet ejector (see[79 88ndash92])

Performance of jet ejector is also depending on the factorslike bubble size distributions correlation for entrainmentbubble diameter drag force and gas holdup and factorsaffecting the bubble size Measurement of bubble size iscritical issue The researchers in these directions are Lefebvre[93] Liu [94] Schick [95] Xianguo and Tankin [96] DennisGary [97] Azad and Syeda [98] Frank [88] Silva et al [99]Kudzo [100] Ciborowski and Bin [101] Ohkawa et al [102]Sheng and Irons [103] Kitscha andOcamustafaogullari [104]Bailer [105] Evans et al [106] Ceylan et al [107] Havelka etal [76] Mandal et al [81] Zahradnık and Fialova [90] Bailer[105] Pawelczyk and Pindur [108] Dutta and Raghavan [74]Bin [109] and Zheng et al [110]

Figure 2 shows the major parts like primary fluid inletsecondary fluid inlet suction chamber converging sectionthroatmixing zone and diverging sectiondiffuser of anejector The symbols which we have used for describingdifferent components are as follows

(i) Length of throat (119871119879) length of diffuser (119871

119863) and dis-

tance between nozzle and commencement of throat(119871119879119873)

(ii) Diameter of nozzle (119863119873) diameter of throat (119863

119879)

and diameter of suction chamber (119863119878)

Primary fluid inletNozzle

Suction chamber

Convergent

Throat(mixing chamber)

Projection ratioPR = LTNDT

Divergent (diffuser)

Suction chamber

Pd(Hd) Doutlet

LD

DT

LT

LTN

DinletG PS(HS)

G

DN

DS

DinletL

Pj(Hj)

(12)120579divergent

(12)120579convergent

Area ratio AR = ATAN

= D2TD

2N

Area ratio = (D2S minus D2

N)D2N

L (m3s)

(m3s)

Figure 2 Schematic diagram showing geometry of an ejector [31111 112]

(iii) Area of throat (119860119879) area of nozzle (119860

119873) and area of

suction (119860119878)

(iv) Angle of converging sections (120579convergent) and angle ofdiverging sections (120579divergent)

Performance of the ejectors has been studied in terms of

(a) area of throatarea of nozzle that is area ratio (119860119877=

119860119879119860119873)

(b) length of throatdiameter of throat that is throataspect ratio (119871

119879119863119879)

(c) distance between nozzle tip and the commencementof throatdiameter of throat that is projection ratio(119875119877= 119871119879119873119863119879)

(d) suction chamber area ratio (119860119878119860119873= (1198632

119878minus 1198632

119873)

1198632

119873)

3 Mathematical Model

The poly dispersed multiphase flow is observed in jet ejectorIt means that the dispersed phase covers a wide range ofsize groups One of the characteristics of this flow is thatthe different sizes of the dispersed phase interact with eachother through the processes of break-up and coalescence Apopulation balance equation is framed to deal with this kindof a flow It is well known that the population balance modelis best fitted method to calculate the size distribution of adispersed phase which includes break-up and coalescenceeffect [3] Our model is in parallel to the model developed by

Journal of Applied Mathematics 5

Mouza et al [3] The general form of the population balanceequation is

120597119899119894

120597119905+ nabla sdot (

119892119899119894) = 119887119861minus 119889119861+ 119887119862minus 119889119862 (2)

where 119899119894 119887119861 119889119861 119887119862 119889119862 and

119892represent the number density

of size group 119894 the birth rate due to break-up the death ratedue to break-up the birth rate due to coalescence the deathrate due to coalescence and the gas velocity respectivelyTherelation between number density 119899

119894and the volume fraction

120572119892is given by

120572119892119891119894= 119899119894119881119894 (3)

where 119891119894and 119881

119894represent the volume fraction and the

corresponding volume of a bubble of group 119894 respectivelyThe coalescence of two bubbles occurs in three steps[3 19]

(i) First the bubbles collide trapping a slight amount ofliquid between them

(ii) Secondly this liquid film drains until it reaches acritical thickness

(iii) At the end the film breaks and the bubbles jointogether

The process of coalescence depends on the collision rate ofthe two bubbles and the collision efficiency Coalescence isa function of 119905

119894119895and 120591

119894119895 where 119905

119894119895is the time required for

coalescence and 120591119894119895is the contact time Collision is as a result

of turbulence (120579Tu119894119895) laminar shear (120579Ls

119894119895) and buoyancy (120579By

119894119895)

The total coalescence rate is

119876119894119895= (120579

Tu119894119895+ 120579

Bu119894119895+ 120579

Ls119894119895) 119890minus119905119894119895120591119894119895 (4)

Collision is also developed due to the difference in risevelocities of bubbles with different sizes The two conditionsobserved in jet ejector in parallel to the work of Mouza et al[3] for bubble column are as follows

(i) Jet ejectors operate at the homogeneous regime Dueto the narrow bubble size distribution in homoge-nous regime the relative effect of buoyancy can beneglected which leads to the assumption that bubblesrise with the same velocity regardless of their size

(ii) Collision occurs as a result of strong circulationpattern in jet ejector However this happens at gasrates higher than those encountered in the homoge-neous regime and hence laminar shear term may beneglected

Hence there is only the turbulent contribution in the modelwhich is given by

120579119894119895

120591119906= 119899119894119899119895119878119894119895(119906119905119894

2+ 119906119905119895

2) (5)

where 119899119894 119899119895 and119906

119894are the concentrations of bubbles of radius

119903119887119894 concentrations of bubbles of radius 119903

119887119895 and the aver-

age turbulent fluctuating velocity respectively The collision

cross-sectional area (119878119894119895) of the bubble the time required for

coalescence and the contact time are given by (6) (7) and(8) respectively

119878119894119895=120587

4(119877119887119894+ 119877119887119895)2

(6)

119905119894119895= (

119877119894119895

3120588119897

16120590)

05

lnℎ0

ℎ119891

(7)

120591119894119895=119877119894119895

23

12057613 (8)

where 119877119894119895 120588119897 120590 120576 ℎ

0 and ℎ

119891are the equivalent radius the

density of the liquid phase the surface tension the turbulentenergy dissipation the film thickness when collision beginsand the film thickness when ruptures of the film occurrespectively The birth rate of group 119894 due to coalescence ofgroup 119896 and group 119895 bubbles and the death rate of group 119894due to coalescence with other bubbles are given by (9) and(10) respectively

119887119862=1

2

119894

sum

119895=1

119894

sum

119896=1

119876119895119896119899119895119899119896 (9)

119889119862= 119899119894

119873

sum

119895=1

119876119894119895119899119895 (10)

The break-up of bubbles in turbulent dispersions employs themodel developed by Luo and Svendsen [28] The break-uprate of bubbles of size 119894 into bubbles of size 119895 is given by

119892 (V119895 V119894)

= 0923 (1 minus 120572119892)3radic(

120576

1198892119895

)int

1

120583min

(1 + 120583)2

120583113119890minus120591119888119889120583

(11)

where 120576 119889119895 120583 120591119888 and 119873 represent the turbulent energy

dissipation rate the bubble diameter the dimensionless sizeof eddies in the inertial subrange of isotropic turbulencethe critical dimensionless energy for break-up and the totalnumber of groups respectively

The birth rate of group 119894 bubbles due to break-up of largerbubbles and the death rate of group 119894 bubbles due to break-upinto smaller bubbles are given by (12) and (13) respectively

119887119861=

119873

sum

119895=119894+1

119892 (V119895 V119894) 119899119895 (12)

119889119861= 119892119894119899119894 (13)

Themass conservation equation for the liquid phase themassconservation equation for the gas phase and the momentum

6 Journal of Applied Mathematics

conservation equation in the Eulerian framework are givenby (14) (15) and (16) respectively

120597 (120572119897120588119897)

120597119905+ nabla sdot (120572

119897120588119897119897) = 0 (14)

120597 (120572119892119891119894120588119892)

120597119905+ nabla sdot (120572

119892119891119894120588119892119892) = 119878119894 (15)

120597 (120572119898120588119898119898)

120597119905+ nabla

sdot (120572119898120588119898119898119898minus 120583119898120572119898(nabla119898+ (nabla

119898)119879

))

= minus120572119898nabla119901 +119872

119898119899+ 120588119898119892

(16)

where 120588119898 119898 120572119898 120583119898 119901119872

119898119899 and 119892 represent the density

the velocity the volume fraction and the viscosity of the119898th phase the pressure the interphasemomentum exchangebetween phase 119898 and phase 119899 and the gravitational forcerespectively

The momentum exchange between gas and liquid phaseis given by

119872119897119892=3

4120588119898

120572119892

11988932

119862119863(119892minus 119897)10038161003816100381610038161003816119892minus 119897

10038161003816100381610038161003816 (17)

where 119862119863is the interfacial drag coefficient

4 Simulation Parameter forJet Ejector Geometry

Figure 3 shows the details of the geometry of ejector Agrawal[31 111 112] has done the experiments based on the exper-imental setup as shown in Figure 4 It is important to notethat these experimentswere conducted on the industrial stageejector Data for the geometry of the ejector is shown inTable 1

5 The CFD Simulation

Accurate modeling of the jet ejector requires a PopulationBalance Modeling (PBM) to be solved simultaneously witha CFD solver because of the presence of bubble-bubblecontacts or bubble-liquid contacts The PBM-CFD modelthat is MUSIG model (ANSYS Fluent 140) is implementedin this study which combines the population balance methodwith the break-up [28] and coalescence [19] models inorder to predict the bubble size distribution of the gasphase This model uses the Eulerian-Eulerian model TheMUSIGmodel has been extensively used for different systems[1ndash3 11 16 20 27 29]

The equations of continuity momentum and turbulencefor the continuous and dispersed phases give a standardtwo-phase flow calculation It can be extended to includenumber density of bubbles within several size groups usingthe MUSIG model

The size range of the bubbles is split into several groupswith for example groups of equal diameter Equations are

05 Nos holeDIA 36mmPitch 72mm

01 No holeDIA 82mm

03 Nos holeDIA 47mmPitch 94mm

40

80

77

25

25

25112

150

S1

S2

425

75 77

S3

Figure 3 Detail of the ejector used in the experimental setup [31111 112]

then solved for the number density in each group These sizefractions provide a more accurate measure of the interfacialarea density

In this study the simulation is done in three dimensionsand three geometries of the ejector namely with one nozzlethree nozzles and five nozzles as described in Figure 3In the present study bubbles ranging from 14472692 times

10minus5m (bin-20) to 97005842 times 10minus4m (bin-0) in diameter

are equally divided into 21 classes (see Table 2) as theexperimental observation of the gas volume fraction is 100and therefore we consider 0005 0010 0015 0018 00200025 0032 0030 0035 0040 0045 0050 0055 00600065 0070 0075 0080 0085 0090 0095 and 0100whichrepresent the volume fraction of the bubbles of group 119894(119894 = 0 1 2 3 4 5 20) In view of computational timersquosresources we consider the subdivision of the bubble sizesinto 21 size groups All computational results are basedon the discretization of the bubble sizes into 21 groupsANSYS Fluent 140 gives solution to the coupled sets ofgoverning equations for the balances ofmass andmomentumof each phase The conservation equations were discretizedusing the control volume technique Similar kind of study

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Numerical Simulation of Bubble ...

4 Journal of Applied Mathematics

(a) To achieve greater entrainment of the secondary fluid(b) To yield deep mixing between the two fluids(c) To inject fluids from a region of low pressure to a

region of higher pressure

Jet ejector may be used as a vacuum producing device as wellas jet pump With the rapid growth of the chemical processindustry their use as entraining and pumping corrosiveliquids slurries fumes and dust-laden gases has increasedTheir use asmass transfer equipment for liquid-liquid extrac-tion gas absorption gas stripping and slurry reaction likehydrogenation oxidation chlorination fermentation and soforth has increased [31 44ndash53]

The numbers of researchers have attempted to optimizeperformance of jet ejector [40 51 54ndash71] due to its increasinggrowth rate in usage Many researchers have studied themass transfer characteristics and performance of the jetejectors followed by contactors draft tube packed columnor bubble column and they have similar conclusion that thereis less mass transfer coefficient in the extended portion incomparison to the ejector [31 34 39 56 69ndash87] Dutta etal [74] and Gamisans et al [60] studied the jet ejectorsand observe that jet ejector without diffuser or throat isless effective in comparison to ejector with them Das andBiswas [40] observe that efficiency of ejector depends on thedesign of the suction chamber the throat the diffuser and thenozzle Apart from the dimensions of the various sections ofthe ejector the factors such as shape of the entrance to theparallel throat shape of entry of convergent section throataspect ratio angle of divergence and convergence and theprojection ratio are also important factors to be reflectedVarious researchers have studied the effect of geometry ofjet ejector in the optimal design of jet ejector Also manyresearchers have worked on flow patterns in jet ejector (see[79 88ndash92])

Performance of jet ejector is also depending on the factorslike bubble size distributions correlation for entrainmentbubble diameter drag force and gas holdup and factorsaffecting the bubble size Measurement of bubble size iscritical issue The researchers in these directions are Lefebvre[93] Liu [94] Schick [95] Xianguo and Tankin [96] DennisGary [97] Azad and Syeda [98] Frank [88] Silva et al [99]Kudzo [100] Ciborowski and Bin [101] Ohkawa et al [102]Sheng and Irons [103] Kitscha andOcamustafaogullari [104]Bailer [105] Evans et al [106] Ceylan et al [107] Havelka etal [76] Mandal et al [81] Zahradnık and Fialova [90] Bailer[105] Pawelczyk and Pindur [108] Dutta and Raghavan [74]Bin [109] and Zheng et al [110]

Figure 2 shows the major parts like primary fluid inletsecondary fluid inlet suction chamber converging sectionthroatmixing zone and diverging sectiondiffuser of anejector The symbols which we have used for describingdifferent components are as follows

(i) Length of throat (119871119879) length of diffuser (119871

119863) and dis-

tance between nozzle and commencement of throat(119871119879119873)

(ii) Diameter of nozzle (119863119873) diameter of throat (119863

119879)

and diameter of suction chamber (119863119878)

Primary fluid inletNozzle

Suction chamber

Convergent

Throat(mixing chamber)

Projection ratioPR = LTNDT

Divergent (diffuser)

Suction chamber

Pd(Hd) Doutlet

LD

DT

LT

LTN

DinletG PS(HS)

G

DN

DS

DinletL

Pj(Hj)

(12)120579divergent

(12)120579convergent

Area ratio AR = ATAN

= D2TD

2N

Area ratio = (D2S minus D2

N)D2N

L (m3s)

(m3s)

Figure 2 Schematic diagram showing geometry of an ejector [31111 112]

(iii) Area of throat (119860119879) area of nozzle (119860

119873) and area of

suction (119860119878)

(iv) Angle of converging sections (120579convergent) and angle ofdiverging sections (120579divergent)

Performance of the ejectors has been studied in terms of

(a) area of throatarea of nozzle that is area ratio (119860119877=

119860119879119860119873)

(b) length of throatdiameter of throat that is throataspect ratio (119871

119879119863119879)

(c) distance between nozzle tip and the commencementof throatdiameter of throat that is projection ratio(119875119877= 119871119879119873119863119879)

(d) suction chamber area ratio (119860119878119860119873= (1198632

119878minus 1198632

119873)

1198632

119873)

3 Mathematical Model

The poly dispersed multiphase flow is observed in jet ejectorIt means that the dispersed phase covers a wide range ofsize groups One of the characteristics of this flow is thatthe different sizes of the dispersed phase interact with eachother through the processes of break-up and coalescence Apopulation balance equation is framed to deal with this kindof a flow It is well known that the population balance modelis best fitted method to calculate the size distribution of adispersed phase which includes break-up and coalescenceeffect [3] Our model is in parallel to the model developed by

Journal of Applied Mathematics 5

Mouza et al [3] The general form of the population balanceequation is

120597119899119894

120597119905+ nabla sdot (

119892119899119894) = 119887119861minus 119889119861+ 119887119862minus 119889119862 (2)

where 119899119894 119887119861 119889119861 119887119862 119889119862 and

119892represent the number density

of size group 119894 the birth rate due to break-up the death ratedue to break-up the birth rate due to coalescence the deathrate due to coalescence and the gas velocity respectivelyTherelation between number density 119899

119894and the volume fraction

120572119892is given by

120572119892119891119894= 119899119894119881119894 (3)

where 119891119894and 119881

119894represent the volume fraction and the

corresponding volume of a bubble of group 119894 respectivelyThe coalescence of two bubbles occurs in three steps[3 19]

(i) First the bubbles collide trapping a slight amount ofliquid between them

(ii) Secondly this liquid film drains until it reaches acritical thickness

(iii) At the end the film breaks and the bubbles jointogether

The process of coalescence depends on the collision rate ofthe two bubbles and the collision efficiency Coalescence isa function of 119905

119894119895and 120591

119894119895 where 119905

119894119895is the time required for

coalescence and 120591119894119895is the contact time Collision is as a result

of turbulence (120579Tu119894119895) laminar shear (120579Ls

119894119895) and buoyancy (120579By

119894119895)

The total coalescence rate is

119876119894119895= (120579

Tu119894119895+ 120579

Bu119894119895+ 120579

Ls119894119895) 119890minus119905119894119895120591119894119895 (4)

Collision is also developed due to the difference in risevelocities of bubbles with different sizes The two conditionsobserved in jet ejector in parallel to the work of Mouza et al[3] for bubble column are as follows

(i) Jet ejectors operate at the homogeneous regime Dueto the narrow bubble size distribution in homoge-nous regime the relative effect of buoyancy can beneglected which leads to the assumption that bubblesrise with the same velocity regardless of their size

(ii) Collision occurs as a result of strong circulationpattern in jet ejector However this happens at gasrates higher than those encountered in the homoge-neous regime and hence laminar shear term may beneglected

Hence there is only the turbulent contribution in the modelwhich is given by

120579119894119895

120591119906= 119899119894119899119895119878119894119895(119906119905119894

2+ 119906119905119895

2) (5)

where 119899119894 119899119895 and119906

119894are the concentrations of bubbles of radius

119903119887119894 concentrations of bubbles of radius 119903

119887119895 and the aver-

age turbulent fluctuating velocity respectively The collision

cross-sectional area (119878119894119895) of the bubble the time required for

coalescence and the contact time are given by (6) (7) and(8) respectively

119878119894119895=120587

4(119877119887119894+ 119877119887119895)2

(6)

119905119894119895= (

119877119894119895

3120588119897

16120590)

05

lnℎ0

ℎ119891

(7)

120591119894119895=119877119894119895

23

12057613 (8)

where 119877119894119895 120588119897 120590 120576 ℎ

0 and ℎ

119891are the equivalent radius the

density of the liquid phase the surface tension the turbulentenergy dissipation the film thickness when collision beginsand the film thickness when ruptures of the film occurrespectively The birth rate of group 119894 due to coalescence ofgroup 119896 and group 119895 bubbles and the death rate of group 119894due to coalescence with other bubbles are given by (9) and(10) respectively

119887119862=1

2

119894

sum

119895=1

119894

sum

119896=1

119876119895119896119899119895119899119896 (9)

119889119862= 119899119894

119873

sum

119895=1

119876119894119895119899119895 (10)

The break-up of bubbles in turbulent dispersions employs themodel developed by Luo and Svendsen [28] The break-uprate of bubbles of size 119894 into bubbles of size 119895 is given by

119892 (V119895 V119894)

= 0923 (1 minus 120572119892)3radic(

120576

1198892119895

)int

1

120583min

(1 + 120583)2

120583113119890minus120591119888119889120583

(11)

where 120576 119889119895 120583 120591119888 and 119873 represent the turbulent energy

dissipation rate the bubble diameter the dimensionless sizeof eddies in the inertial subrange of isotropic turbulencethe critical dimensionless energy for break-up and the totalnumber of groups respectively

The birth rate of group 119894 bubbles due to break-up of largerbubbles and the death rate of group 119894 bubbles due to break-upinto smaller bubbles are given by (12) and (13) respectively

119887119861=

119873

sum

119895=119894+1

119892 (V119895 V119894) 119899119895 (12)

119889119861= 119892119894119899119894 (13)

Themass conservation equation for the liquid phase themassconservation equation for the gas phase and the momentum

6 Journal of Applied Mathematics

conservation equation in the Eulerian framework are givenby (14) (15) and (16) respectively

120597 (120572119897120588119897)

120597119905+ nabla sdot (120572

119897120588119897119897) = 0 (14)

120597 (120572119892119891119894120588119892)

120597119905+ nabla sdot (120572

119892119891119894120588119892119892) = 119878119894 (15)

120597 (120572119898120588119898119898)

120597119905+ nabla

sdot (120572119898120588119898119898119898minus 120583119898120572119898(nabla119898+ (nabla

119898)119879

))

= minus120572119898nabla119901 +119872

119898119899+ 120588119898119892

(16)

where 120588119898 119898 120572119898 120583119898 119901119872

119898119899 and 119892 represent the density

the velocity the volume fraction and the viscosity of the119898th phase the pressure the interphasemomentum exchangebetween phase 119898 and phase 119899 and the gravitational forcerespectively

The momentum exchange between gas and liquid phaseis given by

119872119897119892=3

4120588119898

120572119892

11988932

119862119863(119892minus 119897)10038161003816100381610038161003816119892minus 119897

10038161003816100381610038161003816 (17)

where 119862119863is the interfacial drag coefficient

4 Simulation Parameter forJet Ejector Geometry

Figure 3 shows the details of the geometry of ejector Agrawal[31 111 112] has done the experiments based on the exper-imental setup as shown in Figure 4 It is important to notethat these experimentswere conducted on the industrial stageejector Data for the geometry of the ejector is shown inTable 1

5 The CFD Simulation

Accurate modeling of the jet ejector requires a PopulationBalance Modeling (PBM) to be solved simultaneously witha CFD solver because of the presence of bubble-bubblecontacts or bubble-liquid contacts The PBM-CFD modelthat is MUSIG model (ANSYS Fluent 140) is implementedin this study which combines the population balance methodwith the break-up [28] and coalescence [19] models inorder to predict the bubble size distribution of the gasphase This model uses the Eulerian-Eulerian model TheMUSIGmodel has been extensively used for different systems[1ndash3 11 16 20 27 29]

The equations of continuity momentum and turbulencefor the continuous and dispersed phases give a standardtwo-phase flow calculation It can be extended to includenumber density of bubbles within several size groups usingthe MUSIG model

The size range of the bubbles is split into several groupswith for example groups of equal diameter Equations are

05 Nos holeDIA 36mmPitch 72mm

01 No holeDIA 82mm

03 Nos holeDIA 47mmPitch 94mm

40

80

77

25

25

25112

150

S1

S2

425

75 77

S3

Figure 3 Detail of the ejector used in the experimental setup [31111 112]

then solved for the number density in each group These sizefractions provide a more accurate measure of the interfacialarea density

In this study the simulation is done in three dimensionsand three geometries of the ejector namely with one nozzlethree nozzles and five nozzles as described in Figure 3In the present study bubbles ranging from 14472692 times

10minus5m (bin-20) to 97005842 times 10minus4m (bin-0) in diameter

are equally divided into 21 classes (see Table 2) as theexperimental observation of the gas volume fraction is 100and therefore we consider 0005 0010 0015 0018 00200025 0032 0030 0035 0040 0045 0050 0055 00600065 0070 0075 0080 0085 0090 0095 and 0100whichrepresent the volume fraction of the bubbles of group 119894(119894 = 0 1 2 3 4 5 20) In view of computational timersquosresources we consider the subdivision of the bubble sizesinto 21 size groups All computational results are basedon the discretization of the bubble sizes into 21 groupsANSYS Fluent 140 gives solution to the coupled sets ofgoverning equations for the balances ofmass andmomentumof each phase The conservation equations were discretizedusing the control volume technique Similar kind of study

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Page 5: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 5

Mouza et al [3] The general form of the population balanceequation is

120597119899119894

120597119905+ nabla sdot (

119892119899119894) = 119887119861minus 119889119861+ 119887119862minus 119889119862 (2)

where 119899119894 119887119861 119889119861 119887119862 119889119862 and

119892represent the number density

of size group 119894 the birth rate due to break-up the death ratedue to break-up the birth rate due to coalescence the deathrate due to coalescence and the gas velocity respectivelyTherelation between number density 119899

119894and the volume fraction

120572119892is given by

120572119892119891119894= 119899119894119881119894 (3)

where 119891119894and 119881

119894represent the volume fraction and the

corresponding volume of a bubble of group 119894 respectivelyThe coalescence of two bubbles occurs in three steps[3 19]

(i) First the bubbles collide trapping a slight amount ofliquid between them

(ii) Secondly this liquid film drains until it reaches acritical thickness

(iii) At the end the film breaks and the bubbles jointogether

The process of coalescence depends on the collision rate ofthe two bubbles and the collision efficiency Coalescence isa function of 119905

119894119895and 120591

119894119895 where 119905

119894119895is the time required for

coalescence and 120591119894119895is the contact time Collision is as a result

of turbulence (120579Tu119894119895) laminar shear (120579Ls

119894119895) and buoyancy (120579By

119894119895)

The total coalescence rate is

119876119894119895= (120579

Tu119894119895+ 120579

Bu119894119895+ 120579

Ls119894119895) 119890minus119905119894119895120591119894119895 (4)

Collision is also developed due to the difference in risevelocities of bubbles with different sizes The two conditionsobserved in jet ejector in parallel to the work of Mouza et al[3] for bubble column are as follows

(i) Jet ejectors operate at the homogeneous regime Dueto the narrow bubble size distribution in homoge-nous regime the relative effect of buoyancy can beneglected which leads to the assumption that bubblesrise with the same velocity regardless of their size

(ii) Collision occurs as a result of strong circulationpattern in jet ejector However this happens at gasrates higher than those encountered in the homoge-neous regime and hence laminar shear term may beneglected

Hence there is only the turbulent contribution in the modelwhich is given by

120579119894119895

120591119906= 119899119894119899119895119878119894119895(119906119905119894

2+ 119906119905119895

2) (5)

where 119899119894 119899119895 and119906

119894are the concentrations of bubbles of radius

119903119887119894 concentrations of bubbles of radius 119903

119887119895 and the aver-

age turbulent fluctuating velocity respectively The collision

cross-sectional area (119878119894119895) of the bubble the time required for

coalescence and the contact time are given by (6) (7) and(8) respectively

119878119894119895=120587

4(119877119887119894+ 119877119887119895)2

(6)

119905119894119895= (

119877119894119895

3120588119897

16120590)

05

lnℎ0

ℎ119891

(7)

120591119894119895=119877119894119895

23

12057613 (8)

where 119877119894119895 120588119897 120590 120576 ℎ

0 and ℎ

119891are the equivalent radius the

density of the liquid phase the surface tension the turbulentenergy dissipation the film thickness when collision beginsand the film thickness when ruptures of the film occurrespectively The birth rate of group 119894 due to coalescence ofgroup 119896 and group 119895 bubbles and the death rate of group 119894due to coalescence with other bubbles are given by (9) and(10) respectively

119887119862=1

2

119894

sum

119895=1

119894

sum

119896=1

119876119895119896119899119895119899119896 (9)

119889119862= 119899119894

119873

sum

119895=1

119876119894119895119899119895 (10)

The break-up of bubbles in turbulent dispersions employs themodel developed by Luo and Svendsen [28] The break-uprate of bubbles of size 119894 into bubbles of size 119895 is given by

119892 (V119895 V119894)

= 0923 (1 minus 120572119892)3radic(

120576

1198892119895

)int

1

120583min

(1 + 120583)2

120583113119890minus120591119888119889120583

(11)

where 120576 119889119895 120583 120591119888 and 119873 represent the turbulent energy

dissipation rate the bubble diameter the dimensionless sizeof eddies in the inertial subrange of isotropic turbulencethe critical dimensionless energy for break-up and the totalnumber of groups respectively

The birth rate of group 119894 bubbles due to break-up of largerbubbles and the death rate of group 119894 bubbles due to break-upinto smaller bubbles are given by (12) and (13) respectively

119887119861=

119873

sum

119895=119894+1

119892 (V119895 V119894) 119899119895 (12)

119889119861= 119892119894119899119894 (13)

Themass conservation equation for the liquid phase themassconservation equation for the gas phase and the momentum

6 Journal of Applied Mathematics

conservation equation in the Eulerian framework are givenby (14) (15) and (16) respectively

120597 (120572119897120588119897)

120597119905+ nabla sdot (120572

119897120588119897119897) = 0 (14)

120597 (120572119892119891119894120588119892)

120597119905+ nabla sdot (120572

119892119891119894120588119892119892) = 119878119894 (15)

120597 (120572119898120588119898119898)

120597119905+ nabla

sdot (120572119898120588119898119898119898minus 120583119898120572119898(nabla119898+ (nabla

119898)119879

))

= minus120572119898nabla119901 +119872

119898119899+ 120588119898119892

(16)

where 120588119898 119898 120572119898 120583119898 119901119872

119898119899 and 119892 represent the density

the velocity the volume fraction and the viscosity of the119898th phase the pressure the interphasemomentum exchangebetween phase 119898 and phase 119899 and the gravitational forcerespectively

The momentum exchange between gas and liquid phaseis given by

119872119897119892=3

4120588119898

120572119892

11988932

119862119863(119892minus 119897)10038161003816100381610038161003816119892minus 119897

10038161003816100381610038161003816 (17)

where 119862119863is the interfacial drag coefficient

4 Simulation Parameter forJet Ejector Geometry

Figure 3 shows the details of the geometry of ejector Agrawal[31 111 112] has done the experiments based on the exper-imental setup as shown in Figure 4 It is important to notethat these experimentswere conducted on the industrial stageejector Data for the geometry of the ejector is shown inTable 1

5 The CFD Simulation

Accurate modeling of the jet ejector requires a PopulationBalance Modeling (PBM) to be solved simultaneously witha CFD solver because of the presence of bubble-bubblecontacts or bubble-liquid contacts The PBM-CFD modelthat is MUSIG model (ANSYS Fluent 140) is implementedin this study which combines the population balance methodwith the break-up [28] and coalescence [19] models inorder to predict the bubble size distribution of the gasphase This model uses the Eulerian-Eulerian model TheMUSIGmodel has been extensively used for different systems[1ndash3 11 16 20 27 29]

The equations of continuity momentum and turbulencefor the continuous and dispersed phases give a standardtwo-phase flow calculation It can be extended to includenumber density of bubbles within several size groups usingthe MUSIG model

The size range of the bubbles is split into several groupswith for example groups of equal diameter Equations are

05 Nos holeDIA 36mmPitch 72mm

01 No holeDIA 82mm

03 Nos holeDIA 47mmPitch 94mm

40

80

77

25

25

25112

150

S1

S2

425

75 77

S3

Figure 3 Detail of the ejector used in the experimental setup [31111 112]

then solved for the number density in each group These sizefractions provide a more accurate measure of the interfacialarea density

In this study the simulation is done in three dimensionsand three geometries of the ejector namely with one nozzlethree nozzles and five nozzles as described in Figure 3In the present study bubbles ranging from 14472692 times

10minus5m (bin-20) to 97005842 times 10minus4m (bin-0) in diameter

are equally divided into 21 classes (see Table 2) as theexperimental observation of the gas volume fraction is 100and therefore we consider 0005 0010 0015 0018 00200025 0032 0030 0035 0040 0045 0050 0055 00600065 0070 0075 0080 0085 0090 0095 and 0100whichrepresent the volume fraction of the bubbles of group 119894(119894 = 0 1 2 3 4 5 20) In view of computational timersquosresources we consider the subdivision of the bubble sizesinto 21 size groups All computational results are basedon the discretization of the bubble sizes into 21 groupsANSYS Fluent 140 gives solution to the coupled sets ofgoverning equations for the balances ofmass andmomentumof each phase The conservation equations were discretizedusing the control volume technique Similar kind of study

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Numerical Simulation of Bubble ...

6 Journal of Applied Mathematics

conservation equation in the Eulerian framework are givenby (14) (15) and (16) respectively

120597 (120572119897120588119897)

120597119905+ nabla sdot (120572

119897120588119897119897) = 0 (14)

120597 (120572119892119891119894120588119892)

120597119905+ nabla sdot (120572

119892119891119894120588119892119892) = 119878119894 (15)

120597 (120572119898120588119898119898)

120597119905+ nabla

sdot (120572119898120588119898119898119898minus 120583119898120572119898(nabla119898+ (nabla

119898)119879

))

= minus120572119898nabla119901 +119872

119898119899+ 120588119898119892

(16)

where 120588119898 119898 120572119898 120583119898 119901119872

119898119899 and 119892 represent the density

the velocity the volume fraction and the viscosity of the119898th phase the pressure the interphasemomentum exchangebetween phase 119898 and phase 119899 and the gravitational forcerespectively

The momentum exchange between gas and liquid phaseis given by

119872119897119892=3

4120588119898

120572119892

11988932

119862119863(119892minus 119897)10038161003816100381610038161003816119892minus 119897

10038161003816100381610038161003816 (17)

where 119862119863is the interfacial drag coefficient

4 Simulation Parameter forJet Ejector Geometry

Figure 3 shows the details of the geometry of ejector Agrawal[31 111 112] has done the experiments based on the exper-imental setup as shown in Figure 4 It is important to notethat these experimentswere conducted on the industrial stageejector Data for the geometry of the ejector is shown inTable 1

5 The CFD Simulation

Accurate modeling of the jet ejector requires a PopulationBalance Modeling (PBM) to be solved simultaneously witha CFD solver because of the presence of bubble-bubblecontacts or bubble-liquid contacts The PBM-CFD modelthat is MUSIG model (ANSYS Fluent 140) is implementedin this study which combines the population balance methodwith the break-up [28] and coalescence [19] models inorder to predict the bubble size distribution of the gasphase This model uses the Eulerian-Eulerian model TheMUSIGmodel has been extensively used for different systems[1ndash3 11 16 20 27 29]

The equations of continuity momentum and turbulencefor the continuous and dispersed phases give a standardtwo-phase flow calculation It can be extended to includenumber density of bubbles within several size groups usingthe MUSIG model

The size range of the bubbles is split into several groupswith for example groups of equal diameter Equations are

05 Nos holeDIA 36mmPitch 72mm

01 No holeDIA 82mm

03 Nos holeDIA 47mmPitch 94mm

40

80

77

25

25

25112

150

S1

S2

425

75 77

S3

Figure 3 Detail of the ejector used in the experimental setup [31111 112]

then solved for the number density in each group These sizefractions provide a more accurate measure of the interfacialarea density

In this study the simulation is done in three dimensionsand three geometries of the ejector namely with one nozzlethree nozzles and five nozzles as described in Figure 3In the present study bubbles ranging from 14472692 times

10minus5m (bin-20) to 97005842 times 10minus4m (bin-0) in diameter

are equally divided into 21 classes (see Table 2) as theexperimental observation of the gas volume fraction is 100and therefore we consider 0005 0010 0015 0018 00200025 0032 0030 0035 0040 0045 0050 0055 00600065 0070 0075 0080 0085 0090 0095 and 0100whichrepresent the volume fraction of the bubbles of group 119894(119894 = 0 1 2 3 4 5 20) In view of computational timersquosresources we consider the subdivision of the bubble sizesinto 21 size groups All computational results are basedon the discretization of the bubble sizes into 21 groupsANSYS Fluent 140 gives solution to the coupled sets ofgoverning equations for the balances ofmass andmomentumof each phase The conservation equations were discretizedusing the control volume technique Similar kind of study

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 7

Table 1 Dimensions of ejector [31 111 112]

Setup-IIINozzle diameter 119863

11987382mm 47mm 37mm

Number of nozzles 119899 1 3 5Nozzle number 5 6 7Pitchlowast 2119863

119873

Area ratio (appx)lowastlowast 119860119877

93

Diameter of throatmixing tube 119863119879

25mmLength of throatmixing tubelowastlowastlowast 119871

119879150mm

Projection ratio 119875119877

45

Angle of convergence 120579con Well roundedAngle of divergence of conical diffuser 120579div 7

Length of the conical diffuser 119871119889

425

Diameter of the diffuser exit 119863119862

77

Diameter of extended contactor 119863119862

(mdash)Length of extended contactor 119871

119862(mdash)

Diameter of the suction chamber 119863119878

77mmLength of the suction chamber 119871

119878122mm

Distance between nozzle amp commencement of throat 119871TN 112mmDiameter of secondary gas inlet 119863

119866In 25mmVolume of free jet 119881

1198622632 times 10

minus6m3

Volume of throat 119881119879

736 times 10minus6m3

Volume of divergence 119881119863

115678 times 10minus6m3

Total ejector volume 119881119869

12567 times 10minus6m3

lowastPanchal et al [114] lowastlowastAcharjee et al [54] lowastlowastlowastBiswas et al [57] Yadav and Patwardhan [43] and Mukherjee et al [65]

Pl

P1

V5

V4

V3V2

V1

T1Tank

T2Tank

EjectorS1

S2

S3

Vent

Vent

Air velocitymeter

Cl2rotameter

Soap filmmeter

Circulating pump

Figure 4 Schematic diagram of the experimental setup [31 111 112]

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 8: Research Article Numerical Simulation of Bubble ...

8 Journal of Applied Mathematics

Table 2 Diameter of each bubble class tracked in the simulation

Bubble diameter 119889119894(m)

(bin size)Class index(bin number)

97005842 times 10minus4 Bin-0-fraction

78611387 times 10minus4 Bin-1-fraction

63704928 times 10minus4 Bin-2-fraction

51625064 times 10minus4 Bin-3-fraction

41835809 times 10minus4 Bin-4-fraction

33902814 times 10minus4 Bin-5-fraction

2747409 times 10minus4 Bin-6-fraction

22264395 times 10minus4 Bin-7-fraction

1802573 times 10minus4 Bin-8-fraction

14621301 times 10minus4 Bin-9-fraction

11848779 times 10minus4 Bin-10-fraction

96019883 times 10minus5 Bin-11-fraction

77812388 times 10minus5 Bin-12-fraction

63057437 times 10minus5 Bin-13-fraction

51100352 times 10minus5 Bin-14-fraction

41410594 times 10minus5 Bin-15-fraction

33558229 times 10minus5 Bin-16-fraction

27194846 times 10minus5 Bin-17-fraction

22038101 times 10minus5 Bin-18-fraction

17859189 times 10minus5 Bin-19-fraction

14472692 times 10minus5 Bin-20-fraction

Table 3 Operating conditions

Gas phase Air plus chlorine at 25∘CLiquid phase Water plus sodium hydroxide at 25∘CAverage gas volume fraction 10

can be found from Mouza et al [3] and Ekambara et al[1]

The simulations have been carried out for nozzle 1 byusing 722021 tetrahedral cells 1373357 triangular interiorfaces 140779 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 156799nodes The simulations have been carried out for nozzle 3by using 723519 tetrahedral cells 1376107 triangular interiorfaces 141271 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 157190nodes The simulations have been carried out for nozzle 5by using 715031 tetrahedral cells 1359401 triangular interiorfaces 140731 triangular wall faces 430 triangular pressure-outlet faces 161 triangular velocity-inlet faces and 155599nodes

A second-order discretization scheme was used for theconvective terms At the inlet gas liquid and the averagevolume fraction have been specified At the outlet a relativeaverage static pressure of zero was specified The operatingconditions are summarized in Table 3 The fluid data aretaken at room temperature (25∘C)

Nozzle 1Nozzle 3Nozzle 5

120601G

minus08 minus06 minus04 minus02 0minus1

y (m)

0

001

002

003

004

005

Figure 5 Plot of the volume fraction of gas

6 Results and Discussion

In this work we have made an effort to study Chlorine-Aqueous Sodium Hydroxide system As stated earlier thebubble sizes were distributed into twenty groups (bins) withdiameters between 14472692 times 10

minus5m and 97005842 times

10minus4m

61 Gas Volume Fraction and Liquid Volume Fraction Fig-ures 5 and 6 show the plot of the predicted gas volumefraction and predicted liquid volume fraction for liquidvelocities of 46ms and gas velocity of 02866ms for nozzles1 3 and 5 respectively It can be observed that most of thegas volume fraction tends to migrate towards the bottom ofthe ejector It is also observed that the gas volume increasesin the area of the throat as well as near the gas inlet As thenumber of nozzles increases (from nozzle 1 to nozzle 3 tonozzle 5) the gas volume fraction decreases with the sametrend in all the nozzles As we proceed from nozzle to endof jet ejector we are not adding any gas or liquid so it shouldbe parallel to axis There are three clear cut zones (a) fromdischarge of liquid nozzle to beginning of the throat (b) fromcommencement to the end of the throat and (c) from end ofthe throat to end of the diffuser In zone (a) there is a negativepressure so same quantity of gas has more volume at constanttemperature which is clear by peak zone In the throat thereis almost constant pressure which is shown as almost parallelline in this zone The volume fraction of gas should reducesince the pressure is recovered and higher in zone (c) Thisfact is evident from Figure 5 as the volume fraction of gas isslanting downward

62 Liquid Velocity and Gas Velocity Figures 7 and 8 repre-sent the plots of the velocity magnitude of the gas and liquidphases respectively These results show that the axial liquid

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 9

Nozzle 1Nozzle 3Nozzle 5

120601L

095

096

097

098

099

1

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 6 Plot of the volume fraction of liquid

Nozzle 1Nozzle 3Nozzle 5

UG

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

0

50

100

150

Figure 7 Plot of the velocity magnitude of gas

velocity profile has a slight degree of asymmetry due to thepresence of the gas flow The liquid velocity in the upperregion of the jet ejector is higher than in the lower regionThisis due to the increase in cross-sectional area at the bottom ofthe ejector

63 Bubble Size Distribution Bubble coalescence and break-up determine the bubble size distribution Gas volumefraction and the kinetic energy dissipation rate influence thebubble coalescence and break-up in jet ejector The bubble

0

50

100

150

Nozzle 1Nozzle 3Nozzle 5

UL

(ms

)

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 8 Plot of the velocity magnitude of liquid

size distribution varies with the position because of thenonuniform profiles of the gas volume fraction entrapped byliquid stream and dissipation rate (see [1 3]) The numberdensities of the bubbles along the vertical direction which ismiddle line of the jet ejector for bin-0 bin-10 and bin-20are shown in Figures 10 12 and 14 respectively for liquidvelocities of 46ms and gas velocity of 02866ms Similarlythe number densities of the bubbles in vertical middle planeof the jet ejector for bin-0 bin-10 and bin-20 are shown inFigures 9 11 and 13 respectively It is observed that the bubblesize having more diameters is at the end In zones (a) and

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Numerical Simulation of Bubble ...

10 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

846e + 08

804e + 08

762e + 08

719e + 08

677e + 08

635e + 08

592e + 08

550e + 08

508e + 08

465e + 08

423e + 08

381e + 08

339e + 08

296e + 08

254e + 08

212e + 08

169e + 08

127e + 08

846e + 07

423e + 07

000e + 00

852e + 08

810e + 08

767e + 08

725e + 08

682e + 08

639e + 08

597e + 08

554e + 08

511e + 08

469e + 08

426e + 08

384e + 08

341e + 08

298e + 08

256e + 08

213e + 08

170e + 08

128e + 08

852e + 07

426e + 07

000e + 00

737e + 08

700e + 08

663e + 08

626e + 08

590e + 08

553e + 08

516e + 08

479e + 08

442e + 08

405e + 08

368e + 08

332e + 08

295e + 08

258e + 08

221e + 08

184e + 08

147e + 08

111e + 08

737e + 07

368e + 07

000e + 00

Figure 9 Contour of the number density of bin fraction 0 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times106

minus08 minus06 minus04 minus02 0minus1

y (m)

0

2

4

6

8

10

12

Num

ber d

ensit

y (1

m3)

Figure 10 Plot of the number density of bin fraction 0

(b) the bubble size reduces due to the intimate mixing It canbe seen from these figures that the bubble size distributionfunction reaches an independent state as determined by thebalance between birth and death processes that depend onthe local flow conditions

Figures 15ndash17 represent the discrete number density ofthe bubbles for nozzles 1 3 and 5 respectively for the entirevolume Figures 18ndash20 represent the discrete number densityof the bubbles for nozzles 1 3 and 5 respectively for theplane near to gas inlet It is evident from the figure that as the

diameter increases the number density decreases for entirevolume as well as the plane near to throat

64 Interfacial Area The present simulation results show theinterfacial area variation along the vertical direction whichis middle line of the jet ejector as shown in Figure 22 Thevariation of interfacial area in vertical middle plane is alsoshown in Figure 21 From Figures 5 and 22 it can be seenthat the appearance of interfacial area concentration looks

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Page 11: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 11

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

304e + 10

289e + 10

273e + 10

258e + 10

243e + 10

228e + 10

213e + 10

197e + 10

182e + 10

167e + 10

152e + 10

137e + 10

122e + 10

106e + 10

911e + 09

759e + 09

608e + 09

456e + 09

304e + 09

152e + 09

000e + 00

251e + 10

239e + 10

226e + 10

214e + 10

201e + 10

188e + 10

176e + 10

163e + 10

151e + 10

138e + 10

126e + 10

113e + 10

101e + 10

879e + 09

754e + 09

628e + 09

503e + 09

377e + 09

251e + 09

126e + 09

000e + 00

266e + 10

252e + 10

239e + 10

226e + 10

212e + 10

199e + 10

186e + 10

173e + 10

159e + 10

146e + 10

133e + 10

120e + 10

106e + 10

930e + 09

797e + 09

664e + 09

531e + 09

398e + 09

266e + 09

133e + 09

000e + 00

Figure 11 Contour of the number density of bin fraction 10 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times109

minus08 minus06 minus04 minus02 0minus1

y (m)

0

05

1

15

2

25

3

Num

ber d

ensit

y (1

m3)

Figure 12 Plot of the number density of bin fraction 10

similar to the gas volume fraction But the interfacial areais depending not only on the volume fraction of the gasbut also equally on the distribution of the bubble size Sincethe measurements of the volume fraction and the interfacialarea are independent of each other the data required forcalculating an interfacial area frompopulation balancemodelprovides a valuable test for model prediction [1 3] Henceour results show that the birth and the death processesmodeled in the population balance model are appropriate todescribe the dynamics of bubbles It can be also seen thatthe interfacial area concentration reaches themaximumvalueof 7000m2m3 5600m2m3 and 4500m2m3 in the mixingzone of the jet ejector for the nozzles 1 3 and 5 respectively

It is also observed that the interfacial area is decreasing whenthe number of nozzles is increased The interfacial area is ingood agreement with experimental result [31]

The range of measured values of the interfacial areain the jet ejector is about 3000 to 13000m2m3 [31] Theexperimental values of the interfacial area for nozzle 1 nozzle3 and nozzle 5 are nearly in the range of 4000ndash5500m2m33800ndash4500m2m3 and 2500ndash3500m2m3 respectively fordifferent concentrations of gas and liquid [31] The simulatedvalues of the interfacial area concentration are in the sameorder of magnitude with the experimental results [31] Moredetail on the experimental results can be found in Agrawal[31]

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Numerical Simulation of Bubble ...

12 Journal of Applied Mathematics

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

831e + 12

790e + 12

748e + 12

707e + 12

665e + 12

623e + 12

582e + 12

540e + 12

499e + 12

457e + 12

416e + 12

374e + 12

332e + 12

291e + 12

249e + 12

208e + 12

166e + 12

125e + 12

831e + 11

416e + 11

000e + 00

642e + 12

610e + 12

577e + 12

545e + 12

513e + 12

481e + 12

449e + 12

417e + 12

385e + 12

353e + 12

321e + 12

289e + 12

257e + 12

225e + 12

192e + 12

160e + 12

128e + 12

962e + 11

642e + 11

321e + 11

000e + 00

110e + 13

105e + 13

990e + 12

935e + 12

880e + 12

825e + 12

770e + 12

715e + 12

660e + 12

605e + 12

550e + 12

495e + 12

440e + 12

385e + 12

330e + 12

275e + 12

220e + 12

165e + 12

110e + 12

550e + 11

000e + 00

Figure 13 Contour of the number density of bin fraction 20 for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

times1012

0

1

2

3

4

Num

ber d

ensit

y (1

m3)

minus08 minus06 minus04 minus02 0minus1y (m)

Figure 14 Plot of the number density of bin fraction 20

As shown in Figure 23 the correlation shows that diam-eter is less in zone (a) but increases in zone (b) and is almostconstant in zone (c) In zone (c) there is sudden rise ofthe diameter in the case of nozzle 1 whereas in the casesof nozzles 3 and 5 the diameter does not show appreciablechange The reason is that in nozzle 1 there is only single jetIts singularity nature is not disturbed while fluids are passingthrough the jet ejector Hence there is fast collision of bubblesand it shows sudden rise in the bubble diameter while inthe cases of nozzles 3 and 5 the water jet is divided into

three and five jets of the same velocity respectively so thesestreams maintain their diversity While fluid stream entersinto the diffuser the bubbles entrapped in streams are noteasily disengaged to form higher size of bubbles and for thisreason the two nozzles do not show appreciable increase intheir average diameter

65 Path Lines Figure 24 represents the path lines of liquidfor nozzles 1 3 and 5 respectively It is evident from the figure

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 13N

umbe

r den

sity

(1m

3)

times1010

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 15 Plot of the number density of bubble in the entire volumefor nozzle 1

Num

ber d

ensit

y (1

m3)

times1010

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 16 Plot of the number density of bubble in the entire volumefor nozzle 3

02 04 06 08 10Diameter (m)

0

1

2

3

4

5

6

Num

ber d

ensit

y (1

m3)

times1010

times10minus3

Figure 17 Plot of the number density of bubble in the entire volumefor nozzle 5

Num

ber d

ensit

y (1

m3)

times1011

times10minus3

0

1

2

3

4

5

6

02 04 06 08 10Diameter (m)

Figure 18 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 1

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 19 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 3

Num

ber d

ensit

y (1

m3)

times1011

times10minus302 04 06 08 10

Diameter (m)

0

1

2

3

4

5

6

Figure 20 Plot of the number density of bubble in a plane near tothe gas inlet for nozzle 5

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article Numerical Simulation of Bubble ...

14 Journal of Applied Mathematics

467e + 04

444e + 04

421e + 04

397e + 04

374e + 04

351e + 04

327e + 04

304e + 04

280e + 04

257e + 04

234e + 04

210e + 04

187e + 04

164e + 04

140e + 04

117e + 04

935e + 03

701e + 03

467e + 03

234e + 03

000e + 00

464e + 04

441e + 04

417e + 04

394e + 04

371e + 04

348e + 04

325e + 04

301e + 04

278e + 04

255e + 04

232e + 04

209e + 04

186e + 04

162e + 04

139e + 04

116e + 04

928e + 03

696e + 03

464e + 03

232e + 03

000e + 00

471e + 04

447e + 04

424e + 04

400e + 04

376e + 04

353e + 04

329e + 04

306e + 04

282e + 04

259e + 04

235e + 04

212e + 04

188e + 04

165e + 04

141e + 04

118e + 04

941e + 03

706e + 03

471e + 03

235e + 03

000e + 00

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

Figure 21 Contour of the interfacial area on middle plane for nozzles 1 3 and 5

Nozzle 1Nozzle 3Nozzle 5

a(m

2m

3)

0

2000

4000

6000

8000

minus08 minus06 minus04 minus02 0minus1

y (m)

Figure 22 Plot of the interfacial area

that larger recirculation region occurs in the mixing as thenumber of nozzles increases

7 Conclusion

Population balance approach combinedwith the coalescenceand break-up models is presented to simulate the operationof the jet ejector using CFD software The populationbalance approach was demonstrated by using the equationof bubble number density for gas-liquid flows using ANSYSFluent 140 to explain the temporal and spatial changesof the gas bubble size distribution The CFD results havebeen analyzed to determine bubble size distribution liquidvelocity gas velocity and volume fraction in the jet ejec-tor with the Cl

2-NaOH system Computational results are

compared with the experimental results [31] The simulatedinterfacial area is in good agreement with the experimentalvalues

In order to see the validity of the CFD model we requiremore experimental data on liquid-to-gas ratio geometry ofthe jet ejector properties of gas and liquid reactivity of fluidsand so forthThe coupling of all these properties gives a goodplatform to predict the best model and as a consequence wemay predict the performance of the jet ejector

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 15

Nozzle 1Nozzle 3Nozzle 5

times10minus4

minus08 minus06 minus04 minus02 0minus1

y (m)

0

1

2

3

4

5

Dia

met

er (m

)

Figure 23 Plot of the diameter of bubble

Nozzle 1 Nozzle 3 Nozzle 5

X

Y

ZX

Y

ZX

Y

Z

1586

1507

1427

1348

1269

1190

1110

1031

952

872

793

714

634

555

476

397

317

238

159

79

00

1469

1396

1322

1249

1175

1102

1029

955

882

808

735

661

588

514

441

367

294

220

147

73

00

1371

1302

1234

1165

1097

1028

960

891

823

754

685

617

548

480

411

343

274

206

137

69

00

Figure 24 Path lines of liquid for nozzles 1 3 and 5

Acknowledgment

Thefirst author gratefully acknowledges the financial supportreceived from Department of Mathematics and PhysicsLappeenranta University of Technology Lappeenranta Fin-land for international mobility support making this jointresearch work possible

References

[1] K Ekambara R Sean Sanders K Nandakumar and J HMasliyah ldquoCFD modeling of gas-liquid bubbly flow in hor-izontal pipes influence of bubble coalescence and breakuprdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 620463 20 pages 2012

[2] E Olmos C Gentric C Vial G Wild and N MidouxldquoNumerical simulation of multiphase flow in bubble columnreactors Influence of bubble coalescence and break-uprdquo Chem-ical Engineering Science vol 56 no 21-22 pp 6359ndash6365 2001

[3] K A Mouza N A Kazakis and S V Paras ldquoBubble columnreactor design using a CFD coderdquo in Proceedings of the 1stInternational Conference ldquoFrom Scientific Computing to Compu-tational Engineeringrdquo (IC-SCCE rsquo04) AthensGreece September2004

[4] J B Joshi V S Vitankar A A Kulkarni M T Dhotre andK Ekambara ldquoCoherent flow structures in bubble columnreactorsrdquo Chemical Engineering Science vol 57 no 16 pp 3157ndash3183 2002

[5] G Kocamustafaogullari and W D Huang ldquoInternal structureand interfacial velocity development for bubbly two-phase

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 16: Research Article Numerical Simulation of Bubble ...

16 Journal of Applied Mathematics

flowrdquoNuclear Engineering and Design vol 151 no 1 pp 79ndash1011994

[6] A Kitagawa K Sugiyama and Y Murai ldquoExperimental detec-tion of bubble-bubble interactions in a wall-sliding bubbleswarmrdquo International Journal of Multiphase Flow vol 30 no10 pp 1213ndash1234 2004

[7] G Wild S Poncin H Li and E Olmos ldquoSome aspects ofthe hydrodynamics of bubble columnsrdquo International Journal ofChemical Reactor Engineering vol 1 no 1 pp 1ndash36 2003

[8] W-D Deckwer Bubble Column Reactors John Wiley amp SonsChichester UK 1992

[9] P Spicka M M Dias and J C B Lopes ldquoGas-liquid flow in a2D column comparison between experimental data and CFDmodellingrdquo Chemical Engineering Science vol 56 no 21-22 pp6367ndash6383 2001

[10] J Cao and R N Christensen ldquoAnalysis of moving boundaryproblem for bubble collapse in binary solutionsrdquo NumericalHeat Transfer Part A Applications vol 38 no 7 pp 681ndash6992000

[11] R D S Cavalcanti S R De Farias Neto and E O Vilar ldquoAcomputational fluid dynamics study of hydrogen bubbles in anelectrochemical reactorrdquo Brazilian Archives of Biology andTechnology vol 48 pp 219ndash229 2005

[12] R Krishna and J M Van Baten ldquoScaling up bubble columnreactors with the aid of CFDrdquo Chemical Engineering Researchand Design vol 79 no 3 pp 283ndash309 2001

[13] J M Van Baten and R Krishna ldquoScale up studies on partitionedbubble column reactors with the aid of CFD simulationsrdquoCatalysis Today vol 79-80 pp 219ndash227 2003

[14] K Shimizu S Takada K Minekawa and Y Kawase ldquoPhe-nomenological model for bubble column reactors predictionof gas hold-ups and volumetric mass transfer coefficientsrdquoChemical Engineering Journal vol 78 no 1 pp 21ndash28 2000

[15] V V Buwa and V V Ranade ldquoDynamics of gas-liquid flow ina rectangular bubble column experiments and singlemulti-group CFD simulationsrdquo Chemical Engineering Science vol 57no 22-23 pp 4715ndash4736 2002

[16] S Lo ldquoApplication of population balance to CFD modelling ofgas-liquid reactorsrdquo in Proceedings of the Conference on Trendsin Numerical and Physical Modelling for Industrial MultiphaseFlows Cargese France September 2000

[17] M T Dhotre K Ekambara and J B Joshi ldquoCFD simulationof sparger design and height to diameter ratio on gas hold-upprofiles in bubble column reactorsrdquo Experimental Thermal andFluid Science vol 28 no 5 pp 407ndash421 2004

[18] J-M Delhaye and J B McLaughlin ldquoAppendix 4 report ofstudy group on microphysicsrdquo International Journal of Multi-phase Flow vol 29 no 7 pp 1101ndash1116 2003

[19] M J Prince and H W Blanch ldquoBubble coalescence and break-up in air-sparged bubble columnsrdquo AIChE Journal vol 36 no10 pp 1485ndash1499 1990

[20] J Y Tu G H Yeoh G-C Park and M-O Kim ldquoOn popula-tion balance approach for subcooled boiling flow predictionrdquoJournal of Heat Transfer vol 127 no 3 pp 253ndash264 2005

[21] TWang JWang and Y Jin ldquoA novel theoretical breakup kernelfunction for bubblesdroplets in a turbulent flowrdquo ChemicalEngineering Science vol 58 no 20 pp 4629ndash4637 2003

[22] T Wang J Wang and Y Jin ldquoA CFD-PBM coupled modelfor gas-liquid flowsrdquo AIChE Journal vol 52 no 1 pp 125ndash1402006

[23] J Hamalainen S B Lindstrom T Hamalainen and HNiskanen ldquoPapermaking fibre-suspension flow simulations atmultiple scalesrdquo Journal of Engineering Mathematics vol 71 no1 pp 55ndash79 2011

[24] T HamalainenModelling of fibre orientation and fibre floccula-tion phenomena in paper sheet forming [PhD thesis] TampereUniversity of Technology Publication 768 Tampere Finland2008

[25] F Lehr M Millies and D Mewes ldquoBubble-size distributionsand flow fields in bubble columnsrdquo AIChE Journal vol 48 no11 pp 2426ndash2443 2002

[26] F B Campos and P L C Lage ldquoA numerical method for solvingthe transient multidimensional population balance equationusing an Euler-Lagrange formulationrdquo Chemical EngineeringScience vol 58 no 12 pp 2725ndash2744 2003

[27] P Chen J Sanyal and M P Dudukovic ldquoCFD modeling ofbubble columns flows implementation of population balancerdquoChemical Engineering Science vol 59 no 22-23 pp 5201ndash52072004

[28] H Luo and H F Svendsen ldquoTheoretical model for drop andbubble breakup in turbulent dispersionsrdquo AIChE Journal vol42 no 5 pp 1225ndash1233 1996

[29] G H Yeoh and J Y Tu ldquoPopulation balance modelling forbubbly flowswith heat andmass transferrdquoChemical EngineeringScience vol 59 no 15 pp 3125ndash3139 2004

[30] I Atay Fluid flow and gas absorption in an ejector venturiscrub-ber [Dissertation for theDegree of Doctor of Engineering Science]New Jersey Institute of Technology Newark NJ USA 1986

[31] K S AgrawalModeling of multi nozzle jet ejector for absorptionwith chemical reaction [Doctor of Philosophy Thesis in ChemicalEngineering] Maharaja Sayajirao University of Baroda Vado-dara India 2012

[32] K S Agrawal ldquoJet ejector and its importance in context ofIndian industryrdquo International Journal of Applied EnvironmentalSciences vol 8 no 3 pp 267ndash269 2013

[33] A Laurent and J C Charpentier ldquoAires interfaciales et coeffi-cients de transfert de matiere dans les divers types drsquoabsorbeurset de reacteurs gaz-liquiderdquo The Chemical Engineering Journalvol 8 no 2 pp 85ndash101 1974

[34] M Zlokarnik ldquoEignung und leistungsfahigkeit von beluftungs-vorrichtungen fur die biologische abwasserreinigungrdquo ChemieIngenieur Technik vol 52 no 4 pp 330ndash331 1980

[35] S Ogawa H Yamaguchi S Tone and T Otake ldquoGas-liquidmass transfer in the jet reactor with liquid jet reactorrdquo Journal ofChemical Engineering of Japan vol 16 no 5 pp 419ndash425 1983

[36] J C Charpentier ldquoWhatrsquos new in absorption with chemicalreactionrdquo Transactions of the Institution of Chemical Engineersvol 60 pp 131ndash156 1982

[37] M A M Costa A P R A Ribeiro E R Tognetti M L AguiarJ A S Goncalves and J R Coury ldquoPerformance of a Venturiscrubber in the removal of fine powder from a confined gasstreamrdquoMaterials Research vol 8 no 2 pp 177ndash179 2005

[38] A Majid Y C Qi and K Mehboob ldquoA review of performanceof a venturi scrubberrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 19 pp 3811ndash3818 2012

[39] T Utomo Z JinM Rahman H Jeong andH Chung ldquoInvesti-gation on hydrodynamics and mass transfer characteristics of agas-liquid ejector using three-dimensional CFD modelingrdquoJournal of Mechanical Science and Technology vol 22 no 9 pp1821ndash1829 2008

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 17: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 17

[40] S K Das and M N Biswas ldquoStudies on ejector-venturi fumescrubberrdquo Chemical Engineering Journal vol 119 no 2-3 pp153ndash160 2006

[41] D Li Investigation of an ejector-expansion device in a transcrit-ical carbon dioxide cycle for military ECU applications [PhDthesis] Purdue University West Lafayette Ind USA 2006

[42] J H Witte ldquoMixing shocks in two-phase flowrdquo Journal of FluidMechanics vol 36 no 4 pp 639ndash655 1969

[43] R L Yadav and A W Patwardhan ldquoDesign aspects of ejectorseffects of suction chamber geometryrdquo Chemical EngineeringScience vol 63 no 15 pp 3886ndash3897 2008

[44] H Blenke K Bohner and E Vollmerhaus ldquoUntersuchungenzur berechnung des betriebsverhaltens von treibstrahlforder-ernrdquo Chemie Ingenieur Technik vol 35 no 3 pp 201ndash208 1963

[45] S T Bonnington ldquoWater jet ejectorsrdquo BHRA PublicationRR540 1956

[46] S T Bonnington ldquoWater driven air ejectorsrdquo Tech Rep SP664BHRA Publication 1960

[47] S T Bonnington ldquoA guide to jet pump designrdquo British ChemicalEngineering vol 9 no 3 pp 150ndash154 1964

[48] S T Bonnington and A L King Jet Pumps and Ejectors A Stateof the Art Review and Bibliography BHRA Fluid EngineeringBedford UK 1972

[49] RG Cunningham ldquoGas compressionwith the liquid jet pumprdquoJournal of Fluids Engineering vol 96 no 3 pp 203ndash215 1974

[50] R G Cunningham and R J Dopkin ldquoJet breakup and mixingthroat lengths for the liquid jet gas pumprdquo Transactions of theASME Journal of Fluids Engineering vol 96 no 3 pp 216ndash2261974

[51] X Gamisans M Sarra and F J Lafuente ldquoThe role of the liquidfilm on the mass transfer in venturi-based scrubbersrdquo ChemicalEngineering Research and Design vol 82 no 3 pp 372ndash3802004

[52] G C Govatos ldquoThe slurry pumprdquo Journal of Pipelines vol 1 pp145ndash157 1981

[53] A E Kroll ldquoThe design of jet pumprdquo Chemical EngineeringProgress vol 43 no 2 pp 21ndash24 1947

[54] D K Acharjee P A Bhat A K Mitra and N A Roy ldquoStudieson momentum transfer in vertical liquid jet ejectorsrdquo IndianJournal of Technology vol 13 pp 205ndash210 1975

[55] P A Bhat A KMitra and A N Roy ldquoMomentum transfer in ahorizontal liquid-jet ejectorrdquoTheCanadian Journal of ChemicalEngineering vol 50 no 3 pp 313ndash317 1972

[56] M N Biswas A K Mitra and A N Roy ldquoEffective interfacialarea in a liquid-jet induced horizontal gas-liquid pipelinecontactorrdquo Indian Chemical Engineer vol 19 no 2 pp 15ndash211977

[57] M N Biswas A K Mitra and A N Roy ldquoStudies on gasdispersion in a horizontal liquid jet ejectorrdquo in Proceedings ofthe 2nd Symposium on Jet Pumps and Ejectors and Gas LiftTechniques E3-27-42 BHRA Cambridge UK March 1975

[58] W J Davis ldquoThe effect of the Froude number in estimatingvertical 2-phase gas-liquid friction lossesrdquo British ChemicalEngineering vol 8 pp 462ndash465 1963

[59] G S Davles A K Mitra and A N Roy ldquoMomentum transferstudies in ejectors Correlalations for single-phase and two-phase systemsrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 6 no 3 pp 293ndash299 1967

[60] X Gamisans M Sarra and F J Lafuente ldquoGas pollutantsremoval in a single- and two-stage ejector-venturi scrubberrdquoJournal of HazardousMaterials vol 90 no 3 pp 251ndash266 2002

[61] S Dasappa P J Paul and H J Mukunda ldquoFluid dynamicstudies on ejectors for thermal applications of gasifiersrdquo inProceedings of the 4th National Meet on Recent Advances inBiomass Gasification Technology Mysore India 1993

[62] A K Mitra and A N Roy ldquoStudies on the performanceof ejector correlation for air-air systemrdquo Indian Journal ofTechnology vol 2 no 9 pp 315ndash316 1964

[63] A K Mitra D K Guha and A N Roy ldquoStudies on theperformance of ejector Part-I air-air systemrdquo Indian ChemicalEngineering Transactions vol 5 article 59 1963

[64] D Mukherjee M N Biswas and A K Mitra Holdup Studiesin Liquid Liquid Ejector System vol 1 Institute of ChemicalEngineers 1981

[65] D Mukherjee M N Biswas and A K Mitra ldquoHydrodynamicsof liquid-liquid dispersion in ejectors and vertical two phaseflowrdquo Canadian Journal of Chemical Engineering vol 66 no 6pp 896ndash907 1988

[66] V R Radhakrishnan and A K Mitra ldquoPressure drop holdupand interfacial area in vertical two-phase flow of multi-jetejector induced dispersionsrdquo Canadian Journal of ChemicalEngineering vol 62 no 2 pp 170ndash178 1984

[67] S S Pal A K Mitra and A N Roy ldquoPressure drop and holdupin a vertical two-phase counter current flow with improvedgas mixing liquidrdquo Industrial amp Engineering Chemistry ProcessDesign and Development vol 19 no 1 pp 67ndash75 1979

[68] K P Singh N K Purohit and A K Mitra ldquoPerformancecharacteristics of a horizontal ejector water-water systemrdquoIndian Chemical Engineer vol 16 no 3 pp 1ndash6 1974

[69] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 7 no 11pp 53ndash69 2013

[70] K S Agrawal ldquoInterfacial area andmass transfer characteristicsin multi nozzle jet ejectorrdquo International Journal of EmergingTechnologies in Computational and Applied Sciences vol 3 no5 pp 270ndash280 2013

[71] S K Agrawal ldquoBubble dynamics and interface phenomenonrdquoJournal of Engineering and Technology Research vol 5 no 3 pp42ndash50 2013

[72] S BalamuruganMD Lad VGGaikar andAW PatwardhanldquoEffect of geometry onmass transfer characteristics of ejectorsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8505ndash8517 2007

[73] S Balamurugan V G Gaikar and A W Patwardhan ldquoEffect ofejector configuration on hydrodynamic characteristics of gas-liquid ejectorsrdquo Chemical Engineering Science vol 63 no 3 pp721ndash731 2008

[74] N N Dutta and K V Raghavan ldquoMass transfer and hydrody-namic characteristics of loop reactors with downflow liquid jetejectorrdquoTheChemical Engineering Journal vol 36 no 2 pp 111ndash121 1987

[75] P Havelka V Linek J Sinkule J Zahradnık and M FialovaldquoHydrodynamic and mass transfer characteristics of ejectorloop reactorsrdquo Chemical Engineering Science vol 55 no 3 pp535ndash549 2000

[76] P Havelka V Linek J Sinkule J Zahradnik and M FialovaldquoEffect of the ejector configuration on the gas suction rateand gas hold-up in ejector loop reactorsrdquo Chemical EngineeringScience vol 52 no 11 pp 1701ndash1713 1997

[77] C Li and Y Z Li ldquoInvestigation of entrainment behavior andcharacteristics of gas-liquid ejectors based on CFD simulationrdquoChemical Engineering Science vol 66 no 3 pp 405ndash416 2011

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 18: Research Article Numerical Simulation of Bubble ...

18 Journal of Applied Mathematics

[78] A Mandal ldquoCharacterization of gas-liquid parameters in adown-flow jet loop bubble columnrdquo Brazilian Journal of Chem-ical Engineering vol 27 no 2 pp 253ndash264 2010

[79] A Mandal G Kundu and D Mukherjee ldquoGas-holdup distri-bution and energy dissipation in an ejector-induced downflowbubble column the case of non-Newtonian liquidrdquo ChemicalEngineering Science vol 59 no 13 pp 2705ndash2713 2004

[80] A Mandal G Kundu and D Mukherjee ldquoInterfacial area andliquid-side volumetric mass transfer coefficient in a downflowbubble columnrdquoCanadian Journal of Chemical Engineering vol81 no 2 pp 212ndash219 2003

[81] A Mandal G Kundu and D Mukherjee ldquoGas holdup andentrainment characteristics in a modified downflow bubblecolumn with Newtonian and non-Newtonian liquidrdquo ChemicalEngineering and Processing Process Intensification vol 42 no10 pp 777ndash787 2003

[82] A Mandal G Kundu and D Mukherjee ldquoA comparative studyof gas holdup bubble size distribution and interfacial area in adownflow bubble columnrdquo Chemical Engineering Research andDesign vol 83 no 4 pp 423ndash428 2005

[83] A Mandal G Kundu and D Mukherjee ldquoEnergy analysis andair entrainment in an ejector induced downflow bubble columnwith non-Newtonian motive fluidrdquo Chemical Engineering ampTechnology vol 28 no 2 pp 210ndash218 2005

[84] D A Mitchell ldquoImproving the efficiency of free-jet scrubbersrdquoEnvironment International vol 6 no 1ndash6 pp 21ndash24 1981

[85] F Rahman D B Umesh D Subbarao and M RamasamyldquoEnhancement of entrainment rates in liquid-gas ejectorsrdquoChemical Engineering andProcessing Process Intensification vol49 no 10 pp 1128ndash1135 2010

[86] W Yaici A Laurent N Midoux and J-C Charpentier ldquoDeter-mination of gas-side mass transfer coefficients in trickle-bedreactors in the presence of an aqueous or an organic liquidphaserdquo International Chemical Engineering vol 28 no 2 pp299ndash305 1988

[87] K Yamagiwa D Kusabiraki and A Ohkawa ldquoGas holdup andgas entrainment rate in downflow bubble column with gasentrainment by a liquid jet operating at high liquid throughputrdquoJournal of Chemical Engineering of Japan vol 23 no 3 pp 343ndash348 1990

[88] T Frank ldquoAdvances in computational fluid dynamics (CFD)of 3-dimensional gas-liquid multiphase flowsrdquo in Proceedingsof the NAFEMS Seminar Simulation of Complex Flows (CFD)NiedernhausenWiesbaden Germany April 2005

[89] A A Kendoush and S A W Al-khatab ldquoFlow regimes charac-terization in vertical downward two phase flowrdquo in Proceedingsof the 2nd International Symposium on Multiphase Flow andHeat Transfer X Z Chen T N Veziroglu and C L Tien Edspp 215ndash220 1989

[90] J Zahradnık and M Fialova ldquoThe effect of bubbling regimeon gas and liquid phase mixing in bubble column reactorsrdquoChemical Engineering Science vol 51 no 10 pp 2491ndash25001996

[91] R G Rice and M A Littlefield ldquoDispersion coefficients forideal bubbly flow in truly vertical bubble columnsrdquo ChemicalEngineering Science vol 42 no 8 pp 2045ndash2053 1987

[92] B R Bakshi P Jiang and L S Fan ldquoAnalysis of flow ingas-liquid bubble columns using multi-resolution methodsrdquo inProceedings of the 2nd International Conference on Gas-Liquid-Solid Reactor Engineering pp A1ndashA8 Cambridge Mass USAMarch 1995

[93] H Lefebvre Atomization and Sprays Hemisphere PublishingCorporation New York NY USA 1989

[94] H Liu Science and Engineering of Droplets Fundamentals andApplications Noyes Publications New York NY USA 2000

[95] R J Schick ldquoSpray technology reference guide understandingdrop sizerdquo Bulletne 4598 Spraying Systems Wheaton Ill USA2006

[96] L Xianguo and R S Tankin ldquoDroplet size distribution aderivative of a NukiyamamdashTanasawa type distribution func-tionrdquo Combustion Science and Technology vol 56 no 1 pp 65ndash76 1987

[97] A Dennis Gary ldquoA study of injector spray characteristics insimulated rocket combustion chamber including longitudinalmode pressure oscillationrdquo Tech Rep 730 NASA 1966

[98] M Azad and S R Syeda ldquoA numerical model for bubblesize distribution in turbulent gas-liquid dispersionrdquo Journal ofChemical Engineering vol 24 no 1 pp 25ndash34 2006

[99] M K Silva M A D Avila and M Mori ldquoCFD modelling ofa bubble column with an external loop in the heterogeneousregimerdquoThe Canadian Journal of Chemical Engineering vol 89no 4 pp 671ndash681 2011

[100] A N Kudzo Visualization and characterization of ultrasoniccavitating atomizer and other automotive paint sprayers usinginfrared thermography [A Dissertation Submitted in PartialFulfillment of the Requirements for the Degree of Doctor ofPhilosophy] College of Engineering University of KentuckyLexington Ky USA 2009

[101] J Ciborowski and A Bin ldquoInvestigation of the aeration effect ofplunging liquid jetsrdquo Inzynieria Chemiczna vol 2 pp 557ndash5771972

[102] A Ohkawa D Kusabiraki Y Kawai N Sakai and K EndohldquoSome flow characteristics of a vertical liquid jet system havingdowncomersrdquo Chemical Engineering Science vol 41 no 9 pp2347ndash2361 1986

[103] Y Y Sheng and G A Irons ldquoThe impact of bubble dynamicson the flow in plumes of ladle water modelsrdquoMetallurgical andMaterials Transactions B vol 26 no 3 pp 625ndash635 1995

[104] J Kitscha and G Kocamustafaogullari ldquoBreakup criteria forfluid particlesrdquo International Journal of Multiphase Flow vol 15no 4 pp 573ndash588 1989

[105] FO BailerMass transfer characteristics of a novel gas-liquid con-tactor the advanced buss loop reactor [A Dissertation Submittedto the Swiss Federal Institute of Technology for Degree of Doctorof Technical Sciences] Swiss Federal Institute of TechnologyZurich Switzerland 2001

[106] G M Evans G J Jameson and B W Atkinson ldquoPredictionof the bubble size generated by a plunging liquid jet bubblecolumnrdquo Chemical Engineering Science vol 47 no 13-14 pp3265ndash3272 1992

[107] K Ceylan A Altunbas and G Kelbaliyev ldquoA new model forestimation of drag force in the flow of Newtonian fluids aroundrigid or deformable particlesrdquo Powder Technology vol 119 no2-3 pp 250ndash256 2001

[108] R Pawelczyk and K Pindur ldquoA dynamic method for dispersinggases in liquidsrdquo Chemical Engineering and Processing ProcessIntensification vol 38 no 2 pp 95ndash107 1999

[109] A K Bin ldquoGas entrainment by plunging liquid jetsrdquo ChemicalEngineering Science vol 48 no 21 pp 3585ndash3630 1993

[110] S-Q Zheng Y Yao F-F Guo R-S Bi and J-Y Li ldquoLocalbubble size distribution gas-liquid interfacial areas and gasholdups in an up-flow ejectorrdquo Chemical Engineering Sciencevol 65 no 18 pp 5264ndash5271 2010

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 19: Research Article Numerical Simulation of Bubble ...

Journal of Applied Mathematics 19

[111] K S Agrawal ldquoRemoval efficiency in industrial scale liquid jetejector for chlorine-aqueous caustic soda systemrdquo InternationalJournal of Engineering Trends and Technology vol 4 no 7 pp2931ndash2940 2013

[112] K S Agrawal ldquoExperimental observation for liquid jet ejectorfor chlorine-aqueous caustic soda system at laboratory scalerdquoAmerican International Journal of Research in Science Technol-ogy Engineering andMathematics vol 2 no 2 pp 177ndash181 2013

[113] K S Agrawal ldquoPerformance of venturi scrubberrdquo InternationalJournal of Engineering Research and Development vol 17 no 11pp 53ndash69 2013

[114] N A Panchal S R Bhutada and V G Pangarkar ldquoGas induc-tion and hold up characteristics of liquid jet loop reactor usingmulti orifice nozzlesrdquo Chemical Engineering Communicationsvol 102 no 1 pp 59ndash68 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 20: Research Article Numerical Simulation of Bubble ...

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of