Research Article Hybrid Multiphase CFD Solver for...

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2013 Article ID 128936 13 pageshttpdxdoiorg1011552013128936

Research ArticleHybrid Multiphase CFD Solver for CoupledDispersedSegregated Flows in Liquid-Liquid Extraction

Kent E Wardle1 and Henry G Weller2

1 Chemical Sciences and Engineering Division Argonne National Laboratory Argonne IL 60439 USA2OpenCFD Limited Bracknell Berkshire RG12 1BW UK

Correspondence should be addressed to Kent E Wardle kwardleanlgov

Received 26 November 2012 Accepted 19 February 2013

Academic Editor Alırio Rodrigues

Copyright copy 2013 K E Wardle and H G Weller This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The flows in stage-wise liquid-liquid extraction devices include both phase segregated and dispersed flow regimes As a additionallayer of complexity for extraction equipment such as the annular centrifugal contactor free-surface flows also play a critical rolein both the mixing and separation regions of the device and cannot be neglected Traditionally computional fluid dynamics (CFD)of multiphase systems is regime dependentmdashdifferent methods are used for segregated and dispersed flows A hybrid multiphasemethod based on the combination of an Eulerian multifluid solution framework (per-phase momentum equations) and sharpinterface capturing using Volume of Fluid (VOF) on selected phase pairs has been developed using the open-source CFD toolkitOpenFOAM Demonstration of the solver capability is presented through various examples relevant to liquid-liquid extractiondevice flows including three-phase liquid-liquid-air simulations in which a sharp interface is maintained between each liquid andair but dispersed phase modeling is used for the liquid-liquid interactions

1 Introduction

While multiphase flows present unique challenges for com-putational fluid dynamics (CFD) simulation a host of solu-tion methods exist for simulation of well categorized flowsFor ldquodispersedrdquo flows in which one phase is continuousand the other is distributed in fine droplets one can useLagrangian particle tracking for small phase fractions (lessthan sim10) in which each individual fluid particle is followedthrough the fluid in response to local flow conditions Forhigh phase fraction dispersed flows a multi-fluid Eulerian-Eulerian solution method with interphase mass and momen-tum transfer can be applied For stratified flows in whichthe fluid phases have a clearly defined phase interface free-surface capturing methods such as Volume of Fluid (VOF)can be employed Real flows such as those encountered inliquid-liquid extraction devices are not so easily categorizedand can span multiple flow regimes (both spatially andtemporally) In theory interface capturing methods couldbe used for direct simulation of dispersed flows given thata mesh spacing of sim10x smaller than the smallest droplet

can be used however accurate physical capturing of droplet-droplet interactions requires yet finer mesh resolution ordroplet coalescence is severely overpredicted In practice suchmeshingmdashand the small timesteps required (on the order of1119864minus7 s) for stable solutionmdashis not feasible for realistic turbu-lent multiphase flows and will not be in the foreseeable futureeven on large computers unless CFD algorithmdevelopmentsare made which allow significant timestep acceleration ortime parallelization (spatial decomposition is the only optioncurrently for CFD parallelization) This timestep limit is dueto the fact that interface capturing methods require explicitsolution and are limited by Courant number

119862119903 =

Δ119905

Δ119909

asymp 025 (1)

Thus the timestep Δ119905 is directly proportional to the meshspacing Δ119909 (119906 is the flow velocity)mdashthat is if the meshspacing is cut in half the timestep must essentially bedecreased by the same margin Consequently for complexmultiphase flows inwhich both dispersed flow and segregated

2 International Journal of Chemical Engineering

flow regions are present one would like to couple these twomethods into a single solver In such a method interfacecapturing would be used in regions where meshing is suffi-cient to resolve large droplets and bulk fluid-fluid interfacesor for phase pairs where interdispersion can be neglecteddispersed flow models would be used in regions wheredroplet characteristics move into the ldquosubgridrdquo scale As anexample for complexmultiphase flows such as those found inliquid-liquid extraction devices where two immiscible liquidsare mixed and air can also be present one could employsharp interface methods for certain phase pairs (eg liquid-air) and at the same time use dispersed modeling for others(eg liquid-liquid)

The idea of coupling these two methods for solution ofsuch flows was explored by Cerne et al [1] They employeda simplified switching routine based on the gradient of thevolume fraction across neighboring cells to flag cells as eitherVOFor two-fluid and solved the appropriate number of equa-tions in each cellmdashresulting in complicated numerical issuesdue to solving models with different numbers of equationsacross the same domain [2] To avoid such issues this sameresearch team has shifted toward multi-fluid-VOF couplingvia the addition of interface capturing on top of an Eulerianmulti-fluid solver In this way the multi-fluid formulationwith momentum equations for each phase is applied acrossthe entire domain and an interface sharpening algorithm (intheir case a conservative level-set method which is similarin concept to the interface compression method describedlater) is applied for sharp interface regions [2 3] Strubeljand Tiselj [2] give a good overview of methods that havebeen employed for this coupling along with details regardingdifficulties in coupling the phase momentum equations at thesharp interface (where the phase velocities should be equal)Again the simple switching function of Cerne et al [1] hasbeen used by these authors who acknowledge its somewhatarbitrary nature and identify this as a key area ofwork tomakethe coupling more physically based

A recent study by Yan and Che [4] has also attempted tocouple multi-fluid and VOF methods with a ldquoswitchingrdquomechanism based on the identification and transport ofldquolargerdquo (grid-resolved) and ldquosmallrdquo (sub-grid) interface struc-tures with models for exchange between the two While thisprovides a means of phenomenological switching betweenthe grid-resolved and subgrid scales and enables the useof accurate interface reconstruction methods (piece-wiselinear interface construction (PLIC) was used) it requires thesolution of two transport equations for each dispersable phaseand is not easily generalizable to multiple phases beyond two

The current study is part of a broader research effort todeliver computational tools for detailed simulation of liquid-liquid extraction (solvent extraction) unit operationswith theaim of providing a pathway for prediction of key operationalperformancemeasures (eg stage efficiency and other-phasecarryover (backmixing)) for any given set of conditionsSuch predictive capability will help inform process-levelmodeling tools and deliver insight into unit design andoperation To accomplish this methods are required whichcan adequately predict liquid-liquid mixing and interfacialarea generation as well as the formation and transport of

Upper weir Splash plateSlinger ring

Uppercollector

Lowercollector

Housing

Separatingzone

Rotor

Mixingzone

Rotor inlet Bottom vanes

More-dense-phase inlet

Less-dense-phaseexist

More-dense-phaseexist

Less-dense-phaseinlet

Lowerweir

Figure 1 Sketch of an annular centrifugal contactor

small droplets Liquid-liquid contacting equipment used insolvent extraction processes has the dual purpose of mixingand separating two immiscible fluids Consequently suchdevices inherently encompass a wide variety of multiphaseflow regimes from segregated to dispersed flow types Froma simulation perspective one might perform separate sim-ulations for the mixing [5] and separation [6] functions ofstage-wise extraction devices such as centrifugal contactorsor mixer settlers but in this case assumptions must be maderegarding the coupling between the two regions This isproblematic especially in the case of the annular centrifugalcontactor (Figure 1) In this device mixing occurs in anarrow annulus between the stationary outer housing and theinner rotating cylinder (which itself is the centrifuge) Theconnection between the two regions occurs at the bottomof the rotor where the dispersion of the two fluids entersthe separating region in the hollow rotor These two regionsare not necessarily independent and the coupling is poorlyunderstood These goals and simulation challenges have ledus towards the development of a hybrid method buildingupon the work of the others mentioned above and extendedfor application to the multiphase flow and operation of thesedevices

Of the equipment types generally used for solvent extrac-tion processing of used nuclear fuel centrifugal contactorshave the largest relative knowledge gap and at the sametime the greatest opportunity for significant benefits to afuture fuel cycle facility Thus the present focus has beenon these devices for which this multi-fluid-VOF couplingmethodology is of particular importance to capture both theliquid-liquid dispersion flow as well as the complex dynamicfluid-rotor interaction A thorough review of CFD modelingefforts for annular centrifugal contactors (also called annular

International Journal of Chemical Engineering 3

centrifugal extractors) has recently been published by Vedan-tam et al [7] Three-phase liquid-liquid-air simulation of theflow in an annular centrifugal contactor using a VOF-basedmethod with single momentum equation for the mixture hasbeen performed previously byWardle [8] While simulationsof flow in centrifugal contactor-related geometries fromotherauthors are limited there is one available study in which amulti-fluid solutionmethod is used to simulate the separationof two liquids in a simplified rotor geometry [9] Recent workby Li et al [10] reports three-phase simulations in a coupledmixerseparator centrifugal contactor model using a puremulti-fluid approach Curiously the authors found very littlemixing between the two liquids This appears to have beendue to the assumption of a large dispersed phase diametermdashhowever it may also have been related to smearing of thefluid-rotor contact resulting fromdiffuse liquid-air interfacesCoupling of the multi-fluid method with interface capturingas developed in this current effort allows simulation ofannular centrifugal contactor mixing zone and rotor flows inwhich a liquid-air free surface is captured

2 Computational Methodologyand Implementation

The implementation of the methodology described here hasbeen done using the OpenFOAM CFD package (version21) which provides a collection of libraries and utilitiesfor constructing custom CFD solvers for a wide variety ofapplications A working version of the solver named multi-phaseEulerFoam has been included as part of the release ofversion 21 of the toolkit and is available for download athttpwwwopenfoamorg

3 Multifluid Momentum Equations andImplementation of Interphase Drag Models

Themulti-fluid model equations for incompressible isother-mal flow are given by sets of mass and momentum equationsfor each phase 119896

120597120572119896

120597119905

+ 119896 sdot nabla120572119896 = 0(2)

120597 (120588119896120572119896119896)

120597119905

+ (120588119896120572119896119896 sdot nabla) 119896

= minus120572119896nabla119901 + nabla sdot (120583119896120572119896nabla119896) + 120588119896120572119896 +

119865119863119896 +

119865119904119896

(3)

where the density phase fraction and velocity for phase 119896

are given by 120588119896 120572119896 and 119896 respectively and is gravity Thetwo interfacial forces are the inter-phase momentum transferor drag force

119865119863119896 and the surface tension force

119865119904119896 For theexamples shown here only the drag force is accounted formdashthough surface tension capability (based on the continuumsurface force model of Brackbill [12]) and surface contact

angle effects have also been subsequently added to the solverThe drag term

119865119863119896 is given by

119865119863119896 =

3

4

120588119888120572119888120572119889119862119863

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

(119889 minus 119888)

119889119889

= 120572119888120572119889119870(119889 minus 119888)

(4)

where the subscripts 119888 and 119889 denote the continuous anddispersed phase values and where119870 is

119870 =

3

4

120588119888119862119863

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

119889119889

(5)

As implemented in the solver the drag calculation is genericsuch that the model must simply return the value of 119870 Avariety of correlations could be used for calculation of thedrag coefficient 119862119889 (in (5)) and several common models areavailable in OpenFOAM As an example the commonly usedmodel of Schiller and Naumann [13] was used for the testcases here In this model the drag coefficient is a functionof the Reynolds number Re according to

119862119863 =

24 (1 + 015Re0683)Re

Re le 1000

044 Re gt 1000

(6)

where

Re =

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

119889119889

]119888(7)

in which ]119888 is the continuous phase kinematic viscosity Inthis solver calculation of the drag coefficient can be done byspecifying a dispersed phase or by independent calculationwith each phase as the ldquodispersed phaserdquo and the overall dragcoefficient applied to the momentum equations taken as thevolume fraction weighted average of the two values This is aso-called blended schemewhich is a useful approximation forflows with regions in which either phase is the primary phase

A constant droplet diameter size (independently definedfor each phase) is used in the work reported here but modelsfor variable droplet size are compatible with this flexibleframework Indeed models based on a reduced populationbalance method have been developed and will be reportedseparately

31 Interface Capturing The interface sharpening method-ology employed here is the interface compression methodof Weller [14] Gopala and van Wachem [15] give a usefulcomparison of this type of method with several other inter-face sharpening and reconstruction algorithms employed ina variety of CFD codes The interface compression scheme ofWeller [14] adds an additional ldquoartificialrdquo compression termto the LHS of the volume fraction transport equation (2) as

120597120572119896

120597119905

+ 119896 sdot nabla120572119896 + nabla sdot (119888120572119896 (1 minus 120572119896)) = 0(8)

where the velocity 119888 is applied normally to the interfaceto compress the volume fraction field and maintain a sharp


interface The 120572119896(1 minus 120572119896) term ensures the term is only activein the interface region In addition a bounded differencingscheme is employed for discretization of (8) The value forthe artificial interface ldquocompression velocityrdquo 119888 is given by

119888 = min (119862120572 || max (||)) nabla120572

|nabla120572|

(9)

The nabla120572|nabla120572| term gives the interface unit normal vectorfor the direction of the applied compression velocity Themagnitude of the velocity || is used since dispersion of theinterface (which is being counteracted by the compressionvelocity) can only occur as fast as the magnitude of the localvelocity in the worst case The coefficient 119862120572 is the primarymeans of controlling the interfacial compression While 119862120572

can mathematically be any value ge0 if we restrict 119862120572 le 1 (9)reduces to

119888 = 119862120572 ||

nabla120572

|nabla120572|

(10)

and 119862120572 is then simply a binary coefficient which switchesinterface sharpening on (1) or off (0) With 119862120572 set to0 for a given phase pair there is no imposed interfacecompression resulting in phase dispersion according to themulti-fluid model Conversely when it is set to 1 sharpinterface capturing is applied and VOF-style phase fractioncapturing occurs (forcing interface resolution on the mesh)The implementation of the solver is configured such that theinterface compression coefficient 119862120572 is defined and appliedindependently for all phase pairs Thus a sharp interface canbe maintained at all interfaces between specific phases (egair-water and air-oil) and dispersed phase modeling with nointerface compression can be used for other phase pairs (egwater-oil)

In general the interface compression method for inter-face capturing is not as physically accurate as interfacereconstruction methods such as Piecewise Linear InterfaceConstruction (PLIC) However it is much easier to imple-ment and performs faster andmost importantly unlike PLICit is mass conservative [15] One problem that this methodcan suffer from is parasitic wavy currents at the interfacewhich are particularly problematic for small-scale surfacetension driven flows (eg capillary rise) but become lessimportant for advective flows such as is the current target ofthis workThe false interfacial currents can also beminimizedby maintaining a low 119862119903 number through subtimesteppingand restricting 119862120572 = 1 as was done here For a discussion ofother limitations common to VOF methods see [16]

311 Dynamic 119862120572 Switching With the computational frame-work as outlined above it is possible to imagine a unifiedmethod which allows simulation of complex flows with anycombination of regimes ranging from fully dispersed to fullysegregated However as noted by the various researcherswho have attempted coupling between multi-fluid and VOFmethods [1 2 4] the key area of uncertainty is the method bywhich switching occurs between the dispersed and segregatedmodels As outlined above the current implementationcould be made to allow for dynamic switching of the

interface sharpening only in regions where the flow is seg-regated through implementation of a spatially nonuniform119862120572 field(s) Others have suggested that one could chooseto switch according to some predetermined flow regimemap [2] Such an approach however would likely be usefulonly for simple geometry flows (eg pipe flow) and a moregeneral physics-based approach is needed for application toa broader class of flows such as those under investigationhere From a more general physical perspective one wouldlike to use the sharp interface method in regions of theflow From a more general physical perspective one wouldlike to use the sharp interface method in regions of theflow where the actual droplet size and corresponding localmesh resolution is sufficient for accurate capture of dropletcurvature Conversely where the droplet size falls belowthe mesh-resolvable scale inter-phase dispersion modelingwould be employed (sharpening deactivated) Such a schemewould require prediction of local droplet size based on apopulation balance or similar method and comparison withlocal mesh spacing using some cutoff criteria

For the present implementation limited testing was donefor a two-phase version of the solver using a switchingfunction based on the work by Cerne et al [1] and laterused by others [3] This technique applies a switch accordingto the magnitude of the gradient of the volume fractionitselfmdashwith the assumption being that when the gradientof the volume fraction becomes smaller than some cutoffvalue it is an indication of actual phase dispersion andinterface sharpening is turned off In this case the normalizedmagnitude of the gradient of the volume fraction (120574) is usedas defined by

120574 =

|nabla120572|

max (|nabla120572|) (11)

Thus when 120574 ge 120574

⋆ the value of 119862120572 is set to 1 A cutoffvalue of 04 is recommended by Cerne et al [1] howeverthe formulation used here 120574 has been normalized to 1 sothe corresponding value may be somewhat different Onedrawback of this mechanism for switching is that it issomewhat of a ldquoself-fulfilling prophecyrdquo Where the interfaceis already sharp (steep gradient in the volume fraction 120572)you apply interface compression to keep it sharp and whereit is dispersed you let it stay dispersed Even so it gives areasonable preliminary model for the coupling and has beenimplemented and tested for a two-phase version of the solveronly An example case is described in Section 41

For the majority of examples reported here which arethree-phase liquid-liquid-air cases it was assumed that dis-persion of air into the liquids can be neglected and thus the119862120572 parameter was fixed for each phase pair A value of 1(interface capturing on) was used for any air-liquid interfacesand interface capturing was turned off (119862120572 = 0) for the liquid-liquid interfaces In this case no entrained airwill be capturedand the air-liquid interface will be everywhere sharp whilethe liquid phaseswill be allowed to interdisperse and no sharpinterfaces will be imposedThis assumes that the entrainmentof air has a negligible effect on the flow andmixing of the twoliquids


32 Solution Procedure for Multifluid-VOF Coupling Thegeneral solution procedure for the hybrid solver using theequations above is as follows

(1) update timestep according to Courant number limit(ratio of timestep to interface transit time in cell)

(2) solve coupled set of volume fraction equations withinterface sharpening for selected phase pairs ((8) withmultiple subtimesteps)

(3) compute drag coefficients(4) construct equation set for phase velocities and solve

for preliminary values(5) solve pressure-velocity coupling according to Pres-

sure Implicit with Splitting of Operators (PISO) algo-rithm

(a) compute mass fluxes at cell faces(b) define and solve pressure equation (repeat mul-

tiple times for non-orthogonal mesh correctorsteps)

(c) correct fluxes(d) correct velocities and apply BCs(e) repeat for number of PISO corrector steps

(6) compute turbulence and correct velocities(7) repeat from 1 for next timestep

321 Numerical Considerations for Stability of MomentumCoupling and Phase Conservation In the limit of a sharpinterface the velocities on either side of the interface mustbe equal to meet the so-called no-slip interface conditionThis is an inherent feature of traditional VOF simulations asall phases share a single momentum equation and thus thephase velocities are the same everywhere Imposition of asharp interface through the addition of interface compressionldquoon toprdquo of a multi-fluid formulation in which each phasehas its own momentum equation requires that an additionalartificial drag is imposed to equalize the velocities at theinterface In thework of Strubelj andTiselj [2] inwhichmulti-fluid-VOF coupling was performed an arbitrary functionproportional to the inverse of the time step divided by 100(resulting in a large value) was imposed to force large inter-phase drag coefficients at the interface In this case ratherthan devise some arbitrary formulation small ldquoresidual dragrdquoand ldquoresidual phase fractionrdquo constants were added for eachphase pair (typically both equal to 1119864 minus 3) to stabilize thephasemomentumcouplingThese added residual valueswereonly used in calculation of the drag for momentum couplingstability and did not affect actual phase fractions or overallphase conservation

In order to ensure phase conservation for the coupledphase fractions with added interface sharpening it wasnecessary to incorporate limiters on the phase fraction aswell as on the sum of the phase fractions prior to the explicitsolution of the phase fraction equation system These addi-tional limiters have been incorporated in a new multiphaseimplementation of the Multidimensional Universal Limiter

with Explicit Solution (MULES) solver framework withinOpenFOAM This multiphase MULES implementation usedin multiphaseEulerFoam is also leveraged for enhancing thephase conservation performance of the n-phase VOF-onlysolver multiphaseInterFoam As in the case of the standardVOF solver solution of the volume fraction transport equa-tions was done using subtimestepping over several subinter-vals of the overall time step to maintain solution stabilityaccording to the Courant number limit (1) while maximizingoverall timestep tominimize time to solution for the transientsolver It was found that an overall Cr number limit of 15(based on velocities near sharp interface) with 5 subtimestepscould deliver stable results

4 Results Example Cases

The following example cases demonstrate the capability ofthe simulation methods to capture on a per-phase-pairbasis both dispersed and segregated flows Only the firstcase on the breaking of a dam considers treatment of thesharpening coefficient 119862120572 as a volumetric field with localdynamic switching based on the gradient of the volumefraction asmentioned above In all other cases the value of119862120572is generally set to 1 (imposed interface sharpening) for liquid-air interfaces and to 0 for liquid-liquid phase interactionsThe properties used are representative of water oil and airat room temperature conditions As done in the work ofPadial-Collins et al [9] a constant droplet size of 150micronswas assumed for the liquids A value of 1mm was usedfor air Interphase drag was treated via the blended methodmentioned earlier Visualizations were done using ParaViewversion 312

41 Liquid-Liquid ldquoColumnrdquo Collapse This example is amodification to the classic collapsing liquid column 2D testcase in this case for a liquid-liquid system with two initialregions of dispersed phase volume fraction as shown inFigure 2The domain is 584 cm square with the short barrieron the bottom surface having a width of 24 cm and a heightof 48 cm Figure 3 shows a comparison of successive timesnapshots of simulations having set the interface sharpeningcoefficient to 0 (VOF behavior) and 1 (multi-fluid behavior)The behavior is as expected for the two casesmdashthat iswith 119862120572 = 1 immediately upon startup droplets with acharacteristic size similar to the mesh size are formed anddespite this overall segregation of the phases is relativelyslow When interface sharpening is not imposed (119862120572 = 0)and the multi-fluid behavior is governed by the interphasedrag correlation separation of the phases appears to be morephysical and occurs on a faster time scale

A variation of the above case was performed startingfrom the same initial state but in which the value of 119862120572

was allowed to vary locally (as 0 or 1 only) according to(11) Successive time snapshots of phase fraction and thesharpening coefficient field are shown in Figure 4 Note thatthe interphase drag has been increased in this case (througha larger assumed droplet size) resulting in slightly slowerseparation dynamics as compared to the earlier case shownin Figure 3


Gravity

075 05

119883water = 0

Figure 2 Initial condition of collapsing liquid-liquid dispersion testcase

It was observed that around 119905 = 6 s the first region ofactive interface sharpening appears but left-right interfacemotion is sufficient that the sharpened region is not main-tained Between 10 and 12 s a stable sharpened interfaceappears and grows until it covers the length of the phaseinterface around 16-17 s

This example demonstrates the functionality of dynamicinterface sharpening switching based on the volume fractiongradient for a simple test case As noted earlier a switchingfunction of this type is somewhat arbitrary Ideally onewould like a physical basis for governing switching accordingto a comparison of the local predicted droplet size andthe local mesh spacingmdashwhere mesh resolution is sufficientto resolve droplets adequately sharpening is activated andwhere droplet size falls into the subgrid scale sharpening isdeactivated One could imagine a very flexible model whichcould simulate multiple flow regimes in this way in a multi-scale manner with the multifluid method capturing sub-gridphase particle transport analogous in principle to the idea ofthe sub-grid scale modeling done in Large Eddy Simulation(LES)

42 Horizontal Settler The next test case simulates theseparation of a 50 50 liquid-liquid dispersion entering intoa 2D horizontal settler as shown in Figure 5 where the initialstate is stratified layers of oil (red) and water (blue) Gravityacts in the vertical direction Flows of each phase exit from thecorresponding surfaces in the upper right of the domainThelength of themain body of the domain is 10 cm and the overallheight is 225 cm The development of a ldquodispersion bandrdquobetween the regions of separated phases was observed Thedispersion band was not static but was found to be disturbedby longitudinal waves generated by a periodic vortex at theback of the so-called dispersion disk (wall just upstream ofthe inlet) The dispersion disk is placed just upstream ofthe inlet to direct the entering dispersion toward the central

119905 = 025 s

119905 = 10 s

119905 = 20 s

119862120572 = 0119862120572 = 1

(a)

119905 = 30 s

119905 = 40 s

119905 = 50 s

119862120572 = 1 119862120572 = 0

(b)

Figure 3 Comparison of simulation using the solver with 119862

120572= 1

(left columns) and 119862120572 = 0 (right columns) showing the behavior ofVOF versus Euler-Euler Red is oil and blue is water

vertical position of the separating region and try to minimizedownstream disturbances Simulations were done with nodispersion disk in order to verify the effectiveness of thisfeature It was indeed observed that the overall width of thedispersion band is greater without the dispersion disk as inletdisturbances propagate farther downstream and disrupt theseparation of the two phases leading to more entrainment inthe exit streams

43 Annular Mixer As noted above a principle applicationrequiring the capability of this hybrid solution method isthe liquid-liquid-air flow in an annular centrifugal contactorThis device (Figure 1) consists of an annular region with


119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)

Figure 4 Time sequence of the volume fraction field (left of each column) and the 119862120572 field (right of each column) showing the region ofactive interface sharpening showing the evolution of the separation process and appearance of a sharp interface

Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)

Figure 5 Time evolution of phase fraction field for a separating liquid-liquid dispersion in a horizontal settler

a rotating inner cylinder and stationary outer cylinder inwhich the two immiscible liquids are mixed in the presenceof a free surface This complicates the physics significantlyrequiring sharp interface capturing to accurately predict theintermittent liquid-rotor contact [17] At the same timedispersed phasemodeling is needed to predict themixing andflow of the liquid-liquid dispersion

To demonstrate the capability of the solver to capture suchflow dynamics simulations were conducted in an idealizedannular mixer both for a 2D axisymmetric case and a fully3D case In both cases the inner radius is 254 cm and theouter 317 cm (annular gap of 063 cm) and the height of theannulus is 7 cm The top surface is open to air at constant

atmospheric pressure and the bottom surface is treated asa wall Unless otherwise stated the rotation rate of theinner cylinder is 3600 RPM (377 rads) resulting in a surfacevelocity of 956ms Turbulence was treated using Large EddySimulation (LES) with the Smagorinsky sub-grid model Auniform quadrilateral mesh was used for the 2D model withspacing of 02mm (32 cells across the annular gap) In orderto explore mesh dependency of the new solver additionalsimulationswere donewithmesh spacings of 04mm(coarse)and 01mm (fine) The relative mesh spacings for the threesizes can be seen in Figure 6 For the 3D model the basecase simulation was done with a mesh spacing sim04mm (15hexahedral cells across the annular gap 675K cells total)


C

(a)

M

(b)

F

(c)

Figure 6 Relative mesh spacings for the 04mm (C) 02mm (M) and 01mm (F) meshes of the 2D annular geometry Only a short verticalsection showing the initial liquid-liquid interface is shown

119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s

Figure 7 Sequence of snapshots of phase fraction for water (blue) oil (red) and air (cyan) in the 2D axisymmetric annular mixer geometryat 3600 RPM

though for comparison an additional run was also done fora finer mesh (sim025mm 24M cells)

431 2119863 Axisymmetric Model Figure 7 gives a time seriesof snapshots showing volume fractions for water (blue) oil(red) and air (cyan) from startup through 119905 = 30 s forsimulation on the medium mesh refinement (Figure 6(M))It is clear that even for this very turbulent flow the hybridsolver is able to maintain a sharp interface for the liquid-air free surface while at the same time allowing phase inter-dispersion for the two liquidsUnlike the simplifications oftenused by other CFD studies of this type of flow (eg [18])the presence of air and the existence of the free surfacehas a significant impact on the characteristics of the flow

and breaks down any Taylor-Couette vortices that wouldbe present in the liquid-liquid only case It was observedthat there was one relatively stable vortex at the bottomwhich was characterized by a light-phase rich region atthe center and rotation in the clockwise direction (flowinward along the bottom surface) The companion vortex(counterclockwise rotation) just above this lower one wasfound to be periodically formed and then break away andtravel upward as the liquids are spun off the rotor and movetoward a maximum height on the outer wall

Sharp interface capturing methods such as the VOFmethod used here are inherently mesh dependent the finerthemesh the finer the interface features that can be capturedIn order to explore the mesh dependency that the hybrid


C M F

(a)

C M F

(b)

Figure 8 Snapshots of the phase fractions (a) at 119905 = 0355 s afterstartup and (b) for a time average over the period from 119905 = 2 s to119905 = 5 s for the three meshes

solver inherits from the VOF method simulations for the2D axisymmetric annular mixer model were performed ontwo additional meshesmone twice as coarse and one twiceas fine as the base case Snapshots of the phase fractionsat 119905 = 0355 s after startup (a) and for a time averageover the period from 119905 = 2 s to 119905 = 5 s (b) are shownin Figure 8 Even for the relatively short time after start-upshown in Figure 8(a) the flows observed for the simulationson the three meshes have clear differences Additionallythe differences are apparent not only in the shape of theliquid-air free surface which is to be expected but also inthe multi-fluid dispersion behavior between the two liquidsComparison of the time-averaged behavior (Figure 8(b))however shows better general consistency in the overallliquid height The Taylor-Couette vortex near the bottommentioned previously was found to bemost prominent in themedium mesh case

Though it is not readily apparent from the snapshots inFigure 7 as has been observed previously for flow in thisconfiguration at such conditions [17] the overall height ofthe liquid on the outer wall exhibited an oscillatory behaviorFigure 9 shows a plot of the liquid height on the outer wallversus time for the 2D axisymmetric simulation for the threedifferent meshes As noted previously there is significantvariation in the temporal evolution of the flow on the threemesh densities while the general behavior is similar As willbe shown later the oscillations in liquid height for the 3Dsimulations exhibited a much more clear periodicity for the2D simulations the oscillation magnitude in liquid height

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t

Figure 9 Plot of liquid height on outer wall (stationary) for the 2Dannular mixer simulations at 3600 RPM on the three mesh densities(Figure 6)

07

06

05

04

03

02

01

0

0 02 04 06 08 1

Mean liquid volume fraction

CoarseMedFine

0102

0095

0079

Integral under curve

Nor

mal

ized

adia

l hei

ght

Figure 10 Plot of time-averaged liquid fraction on the rotor side(as a function of normalized height) Integrated values for the totalliquid ldquocoveragerdquo are shown in the legend

was largest for the most coarse mesh The minimum heightof the oscillation of the liquid corresponds with a maximumcontact area between the liquid and the rotor after whichthe liquid is accelerated and spun out and up the housingwall leading to a maximum liquid height corresponding toa minimum in fluid-rotor contact The amount of overallcontact between the liquid and the rotor has an impacton the level of mixing that occurs between the two liquidphases Figure 10 shows a plot of the time-averaged liquidcontact on the rotor side and integrated values correspondingto the fractional liquid ldquocoveragerdquo in each case While not


(a)

(b)

Figure 11 Snapshots of the liquid phase fractions in the 3D annular mixer model at (a) an early time (sim025 s) and sim3 s after startup (b)3600 RPM with the left images showing a side view and the right a cross-section

unexpected there are complex mesh dependencies for thehybrid solver which require additional investigation

In order to provide a point of reference for the addi-tional computational cost of the multi-fluid hybrid schemeversus an all-VOF simulation (single shared momentumequation and sharp interfaces everywhere) a simulation wasdone using OpenFOAMrsquos multiphase-capable VOF solvermultiphaseInterFoam for the fine mesh case (44800 hexahe-dral cells) of the 2D annular mixer problem described hereThe simulation was done using the exact same solver settings(discretization schemes number of subtimesteps on volumefraction solutions etc) and the same number of parallelprocessors (12 cores were used in this case) The simulationswere compared out to 1 second of flow and it was found

that the hybrid multi-fluidVOF solver is only 39 morecostly per timestep than the comparable case with the all-VOFsolver (869CPU secondsstep versus 623 CPU secondsstepfor all-VOF solver) As the bulk of the computational timeis spent in the solution of the volume fractions and more soin the pressure-velocity coupling (Items 2 and 5 resp inthe solution procedure given in Section 32) only a modestcomputational cost is incurred due to the additional phasemomentum equations andmomentum coupling in themulti-fluid formulation that is not required in the VOF-only solver

432 3119863 Model Figure 11 shows snapshots of the liquidphase fractions in the 3D annular mixer model soon afterstart-up (sim025 s) and sim3 s after startup at 3600 RPM In


Table 1 Liquid height oscillation frequency (cycless) at 3600 RPM

Blended drag AQ disp drag Experiment [11]Base mesh Fine mesh 119889aq = 150 120583m 119889aq = 50 120583m 4V 500mLmin CV 1000 Lmin456 plusmn 016 465 plusmn 020 462 plusmn 013 399 plusmn 017 491 plusmn 021 529 plusmn 029

Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)

2400 RPM3600 RPM3600 RPM (fine)

Figure 12 Plot of liquid height for the base case mesh at 2400 RPMand 3600RPM and the fine mesh at 3600 RPM

contrast with the 2D axisymmetric approximation there issignificant azimuthal variation in the flow and liquid-rotorcontact The liquid-liquid dispersion also exhibited a steadyheight oscillation due to being periodically thrown off therotor and up the outer wall A plot of the liquid height on theouter wall (azimuthal average) for two rotor speeds (2400 and3600RPM) are shown in Figure 12 along with the 3600 RPMvalue for the fine mesh for comparison At the lower rotorspeed the liquid exhibits only minimal height variation withsome periodicity while at the high rotor speed a steady oscil-lation develops sim1 s after startup Table 1 gives a comparisonof the liquid height oscillation frequency (calculated fromtrough-trough times for the sim6 cycles between 15ndash3 s) forthe 3Dmodel with different variations the base case blendeddrag model with the droplet diameters for both liquid phasesat 150 microns aqueous dispersed with 150 micron dropletsize and aqueous dispersedwith a 50micron droplet sizeTheresults for the fine mesh (blended case only) are also shownfor comparison There is not a very strong dependence onoscillation frequency for blended versus aqueous dispersed atthe same droplet diameter For the smaller droplet diameterthe oscillation frequency was found to decrease slightlyIn terms of mesh dependency despite oscillation ldquophaserdquooffset due to differences at early times the fully developedoscillation frequency is very comparable for the two meshdensities

A companion experimental effort has also been initiatedto provide means of validating the advanced simulationcapability presented by this new solver for flows in actualannular centrifugal contactor equipment While the simula-tions presented here are for a simple annular mixer it was

found that certain characteristics of the flow showed goodagreement with preliminary experimental observations anda brief comparison will be made here Experiments wereconducted using a CINC-V2 centrifugal contactor modifiedwith a quartz outer cylinder as reported by Wardle et al[17] The tests included here were done using a liquid-liquidsystem consisting of 1M nitric acid with 1M aluminumnitrate as the aqueous phase (120588 = 117) and 40 (byvolume) tributyl phosphate in dodecane (120588 = 085) Theaqueous phase was dyed with methylene blue to aid invisual phase discrimination While tests were done for avariety of housing vane types and inlet flow rates conditionswere selected for comparison here which gave a comparableannular liquid height in the mixing zone For additionaldetails regarding the experiments please see Wardle 2012[11]

It was observed that while the 3D annularmixer geometryhas a fixed volume with no inletoutlet flow the generalfeatures of the flow compared favorably with observations inexperiment Figure 13 shows a comparison snapshot of theliquid-liquid flow as viewed from the housing side from thesimulation (a) and a high-speed image from experiment (b)Both images are near a minimum liquid height and showsimilar banding and tendrils of the dispersed heavy phase(blue) In both simulation and experiment it was observedthat as the dispersion approached a minimum liquid heightaqueous phase striations such as these appearedAdditionallycontact with the rotor for the collapsing liquid resulted ina sharp ldquospurtingrdquo of air across a line near the top of themain body of liquid followed by the rise of the liquid upthe outer wall In addition to such qualitative comparisonsthe oscillation frequency of the liquid surface was also foundto be quite comparable quantitatively under conditions ofsimilar liquid height As reported previously in Table 1 themean oscillation frequency from simulation was found tobe 46Hz From high-speed video analysis the frequencyobserved in experiments was slightly higher at 49ndash53HzValues from two different cases (4 straight mixing vanes (4V)and 8 curved mixing vanes (CV)) with similar liquid heightare given in Table 1 Note that this is also quite comparableto what was found previously for similar conditions undersingle-phase flow [19] which ranged from 45 to 54Hzat 3600 RPM depending on the total feed flow ratemdashwithhigh frequency being observed at lower flow rates (loweroverall liquid height) This seems to demonstrate that thephenomenon of liquid height oscillation (and the corre-sponding periodic liquidndashrotor contact) is more a functionof the annular geometry and rotor speed than it is of thefluid properties flow configuration (whether net axial flowis present or not) or geometry beneath the rotormdashprovidedthat the conditions and configuration result in comparableliquid volume (height) in the annular region


(a)

(b)

Figure 13 Comparison of snapshots from CFD simulation (a) of3D annular mixer (fine mesh) and experiment [11] (b) in an actualCINC-V2 centrifugal contactor at a comparable overall liquid heightshowing similar aqueous phase (blue) striations

5 Conclusions

A hybrid multiphase CFD solver has been developed whichcombines the Euler-Euler multi-fluid methodology withVOF-type sharp interface capturing on selected phase-pairinterfaces A variety of examples of cases have been pre-sented here which are relevant to liquid-liquid extractionand demonstrate the functionality and flexibility of the newsolver The multiphase flow simulation capability describedhere has application to a variety of complex flows which spanmultiple regimes from fully segregated to fully dispersedWhile the target application is flow in liquid-liquid extractiondevices this methodology could be useful to the simulation

of other multiphase flows which are currently restricted toa single flow regime For example coupling these methodswith heat transfer and phase change could provide a tool forsimulations of gas-liquid channel flows such as those seen innuclear reactors spanning bubbly slug churn-turbulent andannular flows

The primary goal of the overall research effort of whichthis work is a part is the prediction ofmass-transfer efficiencyin stage-wise liquid-liquid extraction devices This requiresprediction of liquid-liquid interfacial area and consequentlycapturing of the dispersed phase droplet size distribution insome manner The solver presented here provides a flexiblefoundation for building in the necessary models from any ofthe availablemethods In the solver as part of theOpenFOAMrelease the droplet diameter of each phase is a field variableand has been implemented as a callable library such that addi-tional droplet diameter models can be easily implementedand selected at runtime Extension of the solver to includevariable droplet size using the reduced population balancemethod of Attarakih et al [20] as implemented in [21] hasrecently been performed and exploration of droplet breakupand coalescence models is underway with very promisingresults

Disclosure

The submitted paper has been created byUChicago ArgonneLLC Operator of Argonne National Laboratory (ldquoArgonnerdquo)Argonne a US Department of Energy Office of ScienceLaboratory is operated under Contract no DE-AC02-06CH11357 The US Government retains for itself and othersacting on its behalf a paid-up nonexclusive irrevocableworldwide license in the said paper to reproduce preparederivative works distribute copies to the public and performpublicly and display publicly by or on behalf of the govern-ment

Acknowledgments

The authors gratefully acknowledge the use of the FusionLinux cluster at ArgonneNational Laboratory and the FissionLinux cluster at IdahoNational Laboratory for computationalresources K E Wardle was supported by the US DOE Officeof Nuclear Energyrsquos Fuel Cycle RampD Program in the areaof Separations K E Wardle would also like to thank JingGao of the University of Illinois at Chicago for experimentalcontributions as a summer research aide

References

[1] G Cerne S Petelin and I Tiselj ldquoCoupling of the interfacetracking and the two-fluid models for the simulation of incom-pressible two-phase flowrdquo Journal of Computational Physics vol171 no 2 pp 776ndash804 2001

[2] L Strubelj and I Tiselj ldquoTwo-fluid model with interface sharp-eningrdquo International Journal forNumericalMethods in Engineer-ing vol 85 pp 575ndash590 2011

[3] L Strubelj I Tiselj and B Mavko ldquoSimulations of free surfaceflows with implementation of surface tension and interface


sharpening in the two-fluid modelrdquo International Journal ofHeat and Fluid Flow vol 30 no 4 pp 741ndash750 2009

[4] K Yan and D Che ldquoA coupled model for simulation of thegas-liquid two-phase flow with complex flow patternsrdquo Inter-national Journal of Multiphase Flow vol 36 no 4 pp 333ndash3482010

[5] K E Wardle T R Allen M H Anderson and R E SwaneyldquoAnalysis of the effect of mixing vane geometry on the flow inan annular centrifugal contactorrdquo AIChE Journal vol 55 no 9pp 2244ndash2259 2009

[6] K EWardle T R Allen and R Swaney ldquoCFD simulation of theseparation zone of an annular centrifugal contactorrdquo SeparationScience and Technology vol 44 no 3 pp 517ndash542 2009

[7] S Vedantam K E Wardle T V Tamhane V V Ranade andJ B Joshi ldquoCFD simulation of annular centrifugal extractorsrdquoInternational Journal of Chemical Engineering vol 2012 ArticleID 759397 31 pages 2012

[8] K E Wardle ldquoOpen-source CFD simulations of liquid-liquidflow in the annular centrifugal contactorrdquo Separation Scienceand Technology vol 46 no 15 pp 2409ndash2417 2011

[9] N T Padial-Collins D Z Zhang Q Zou X Ma and W BVanderHeyden ldquoCentrifugal contactors separation of an aque-ous and an organic stream in the rotor zone (LA-UR-05-7800)rdquoSeparation Science and Technology vol 41 no 6 pp 1001ndash10232006

[10] S Li W Duan J Chen and J Wang ldquoCFD simulation of gas-liquid-liquid three-phase flow in an annular centrifugal contac-torrdquo Industrial amp Engineering Chemistry Research vol 51 pp11245ndash11253 2012

[11] K E Wardle ldquoFY12 summary report on liquid-liquid contactorexperiments for CFD model validationrdquo Tech Rep ArgonneNational Laboratory 2012

[12] J Brackbill D Kothe and C Zemach ldquoA continuum methodformodeling surface tensionrdquo Journal of Computational Physicsvol 100 no 2 pp 335ndash354 1992

[13] L Schiller and Z Naumann ldquoA drag coefficient corellationrdquoZeitschrift des Vereins Deutscher Ingenieure vol 77 p 318 1935

[14] H G Weller ldquoA new approach to VOF-based interface captur-ing methods for incompressible and compressible flowrdquo TechRep OpenCFD 2008

[15] V R Gopala and B G M van Wachem ldquoVolume of fluidmethods for immiscible-fluid and free-surface flowsrdquo ChemicalEngineering Journal vol 141 no 1ndash3 pp 204ndash221 2008

[16] G Erne S Petelin and I Tiselj ldquoNumerical errors of the vol-ume-of-fluid interface tracking algorithmrdquo International Jour-nal for Numerical Methods in Fluids vol 38 no 4 pp 329ndash3502002

[17] K E Wardle T R Allen M H Anderson and R E SwaneyldquoFree surface flow in the mixing zone of an annular centrifugalcontactorrdquo AIChE Journal vol 54 no 1 pp 74ndash85 2008

[18] M J Sathe S S Deshmukh J B Joshi and S B Koganti ldquoCom-putational fluid dynamics simulation and experimental inves-tigation study of two-phase liquid-liquid flow in a verticalTaylor-couette contactorrdquo Industrial and Engineering ChemistryResearch vol 49 no 1 pp 14ndash28 2010

[19] K E Wardle T R Allen M H Anderson and R E SwaneyldquoExperimental study of the hydraulic operation of an annularcentrifugal contactor with various mixing vane geometriesrdquoAIChE Journal vol 56 no 8 pp 1960ndash1974 2010

[20] M Attarakih M Jaradat C Drumm et al ldquoSolution of thepopulation balance equation using the one primary and one

secondary particle methodrdquo in Proceedings of the 19th EuropeanSymposium on Computer Aided Process Engineering (ESCAPErsquo09) 2009

[21] C Drumm M Attarakih M W Hlawitschka and H J BartldquoOne-group reduced population balance model for CFD sim-ulation of a pilot-plant extraction columnrdquo Industrial andEngineering Chemistry Research vol 49 no 7 pp 3442ndash34522010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



flow regions are present one would like to couple these twomethods into a single solver In such a method interfacecapturing would be used in regions where meshing is suffi-cient to resolve large droplets and bulk fluid-fluid interfacesor for phase pairs where interdispersion can be neglecteddispersed flow models would be used in regions wheredroplet characteristics move into the ldquosubgridrdquo scale As anexample for complexmultiphase flows such as those found inliquid-liquid extraction devices where two immiscible liquidsare mixed and air can also be present one could employsharp interface methods for certain phase pairs (eg liquid-air) and at the same time use dispersed modeling for others(eg liquid-liquid)

The idea of coupling these two methods for solution ofsuch flows was explored by Cerne et al [1] They employeda simplified switching routine based on the gradient of thevolume fraction across neighboring cells to flag cells as eitherVOFor two-fluid and solved the appropriate number of equa-tions in each cellmdashresulting in complicated numerical issuesdue to solving models with different numbers of equationsacross the same domain [2] To avoid such issues this sameresearch team has shifted toward multi-fluid-VOF couplingvia the addition of interface capturing on top of an Eulerianmulti-fluid solver In this way the multi-fluid formulationwith momentum equations for each phase is applied acrossthe entire domain and an interface sharpening algorithm (intheir case a conservative level-set method which is similarin concept to the interface compression method describedlater) is applied for sharp interface regions [2 3] Strubeljand Tiselj [2] give a good overview of methods that havebeen employed for this coupling along with details regardingdifficulties in coupling the phase momentum equations at thesharp interface (where the phase velocities should be equal)Again the simple switching function of Cerne et al [1] hasbeen used by these authors who acknowledge its somewhatarbitrary nature and identify this as a key area ofwork tomakethe coupling more physically based

A recent study by Yan and Che [4] has also attempted tocouple multi-fluid and VOF methods with a ldquoswitchingrdquomechanism based on the identification and transport ofldquolargerdquo (grid-resolved) and ldquosmallrdquo (sub-grid) interface struc-tures with models for exchange between the two While thisprovides a means of phenomenological switching betweenthe grid-resolved and subgrid scales and enables the useof accurate interface reconstruction methods (piece-wiselinear interface construction (PLIC) was used) it requires thesolution of two transport equations for each dispersable phaseand is not easily generalizable to multiple phases beyond two

The current study is part of a broader research effort todeliver computational tools for detailed simulation of liquid-liquid extraction (solvent extraction) unit operationswith theaim of providing a pathway for prediction of key operationalperformancemeasures (eg stage efficiency and other-phasecarryover (backmixing)) for any given set of conditionsSuch predictive capability will help inform process-levelmodeling tools and deliver insight into unit design andoperation To accomplish this methods are required whichcan adequately predict liquid-liquid mixing and interfacialarea generation as well as the formation and transport of

Upper weir Splash plateSlinger ring

Uppercollector

Lowercollector

Housing

Separatingzone

Rotor

Mixingzone

Rotor inlet Bottom vanes

More-dense-phase inlet

Less-dense-phaseexist

More-dense-phaseexist

Less-dense-phaseinlet

Lowerweir

Figure 1 Sketch of an annular centrifugal contactor

small droplets Liquid-liquid contacting equipment used insolvent extraction processes has the dual purpose of mixingand separating two immiscible fluids Consequently suchdevices inherently encompass a wide variety of multiphaseflow regimes from segregated to dispersed flow types Froma simulation perspective one might perform separate sim-ulations for the mixing [5] and separation [6] functions ofstage-wise extraction devices such as centrifugal contactorsor mixer settlers but in this case assumptions must be maderegarding the coupling between the two regions This isproblematic especially in the case of the annular centrifugalcontactor (Figure 1) In this device mixing occurs in anarrow annulus between the stationary outer housing and theinner rotating cylinder (which itself is the centrifuge) Theconnection between the two regions occurs at the bottomof the rotor where the dispersion of the two fluids entersthe separating region in the hollow rotor These two regionsare not necessarily independent and the coupling is poorlyunderstood These goals and simulation challenges have ledus towards the development of a hybrid method buildingupon the work of the others mentioned above and extendedfor application to the multiphase flow and operation of thesedevices

Of the equipment types generally used for solvent extrac-tion processing of used nuclear fuel centrifugal contactorshave the largest relative knowledge gap and at the sametime the greatest opportunity for significant benefits to afuture fuel cycle facility Thus the present focus has beenon these devices for which this multi-fluid-VOF couplingmethodology is of particular importance to capture both theliquid-liquid dispersion flow as well as the complex dynamicfluid-rotor interaction A thorough review of CFD modelingefforts for annular centrifugal contactors (also called annular







120597120572119896

120597119905

+ 119896 sdot nabla120572119896 = 0(2)

120597 (120588119896120572119896119896)

120597119905

+ (120588119896120572119896119896 sdot nabla) 119896


119865119863119896 +

119865119904119896

(3)






119865119863119896 is given by

119865119863119896 =

3

4

120588119888120572119888120572119889119862119863

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

(119889 minus 119888)

119889119889

= 120572119888120572119889119870(119889 minus 119888)

(4)


119870 =

3

4

120588119888119862119863

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

119889119889

(5)


119862119863 =

24 (1 + 015Re0683)Re

Re le 1000

044 Re gt 1000

(6)

where

Re =

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

119889119889

]119888(7)




120597120572119896

120597119905





119888 = min (119862120572 || max (||)) nabla120572

|nabla120572|

(9)



119888 = 119862120572 ||

nabla120572

|nabla120572|

(10)






120574 =

|nabla120572|

max (|nabla120572|) (11)

Thus when 120574 ge 120574























Gravity

075 05

119883water = 0





119905 = 025 s

119905 = 10 s

119905 = 20 s

119862120572 = 0119862120572 = 1

(a)

119905 = 30 s

119905 = 40 s

119905 = 50 s

119862120572 = 1 119862120572 = 0

(b)


120572= 1





119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)


Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)






C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















120597120572119896

120597119905

+ 119896 sdot nabla120572119896 = 0(2)

120597 (120588119896120572119896119896)

120597119905

+ (120588119896120572119896119896 sdot nabla) 119896


119865119863119896 +

119865119904119896

(3)






119865119863119896 is given by

119865119863119896 =

3

4

120588119888120572119888120572119889119862119863

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

(119889 minus 119888)

119889119889

= 120572119888120572119889119870(119889 minus 119888)

(4)


119870 =

3

4

120588119888119862119863

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

119889119889

(5)


119862119863 =

24 (1 + 015Re0683)Re

Re le 1000

044 Re gt 1000

(6)

where

Re =

1003816

1003816

1003816

1003816

119889 minus 119888

1003816

1003816

1003816

1003816

119889119889

]119888(7)




120597120572119896

120597119905





119888 = min (119862120572 || max (||)) nabla120572

|nabla120572|

(9)



119888 = 119862120572 ||

nabla120572

|nabla120572|

(10)






120574 =

|nabla120572|

max (|nabla120572|) (11)

Thus when 120574 ge 120574























Gravity

075 05

119883water = 0





119905 = 025 s

119905 = 10 s

119905 = 20 s

119862120572 = 0119862120572 = 1

(a)

119905 = 30 s

119905 = 40 s

119905 = 50 s

119862120572 = 1 119862120572 = 0

(b)


120572= 1





119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)


Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)






C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119888 = min (119862120572 || max (||)) nabla120572

|nabla120572|

(9)



119888 = 119862120572 ||

nabla120572

|nabla120572|

(10)






120574 =

|nabla120572|

max (|nabla120572|) (11)

Thus when 120574 ge 120574























Gravity

075 05

119883water = 0





119905 = 025 s

119905 = 10 s

119905 = 20 s

119862120572 = 0119862120572 = 1

(a)

119905 = 30 s

119905 = 40 s

119905 = 50 s

119862120572 = 1 119862120572 = 0

(b)


120572= 1





119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)


Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)






C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





























Gravity

075 05

119883water = 0





119905 = 025 s

119905 = 10 s

119905 = 20 s

119862120572 = 0119862120572 = 1

(a)

119905 = 30 s

119905 = 40 s

119905 = 50 s

119862120572 = 1 119862120572 = 0

(b)


120572= 1





119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)


Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)






C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Gravity

075 05

119883water = 0





119905 = 025 s

119905 = 10 s

119905 = 20 s

119862120572 = 0119862120572 = 1

(a)

119905 = 30 s

119905 = 40 s

119905 = 50 s

119862120572 = 1 119862120572 = 0

(b)


120572= 1





119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)


Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)






C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119905 = 20 s

119905 = 40 s

119905 = 60 s

119905 = 80 s

(a)

119905 = 100 s

119905 = 120 s

119905 = 140 s

119905 = 160 s

(b)


Time 0000 s

(a)

Time 5000 s

(b)

Time 10000 s(c)

Time 15000 s

(d)

Time 20000 s

(e)






C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C

(a)

M

(b)

F

(c)


119905 = 000 s 119905 = 005 s 119905 = 010 s 119905 = 015 s 119905 = 020 s 119905 = 025 s 119905 = 050 s 119905 = 075 s 119905 = 100 s 119905 = 200 s 119905 = 300 s







C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C M F

(a)

C M F

(b)




1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

CoarseMedFine

Time (s)

Nor

mal

ized

liqu

id h

eigh

t


07

06

05

04

03

02

01

0

0 02 04 06 08 1


CoarseMedFine

0102

0095

0079


Nor

mal

ized

adia

l hei

ght




(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

(b)









Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Nor

mal

ized

liqu

id h

eigh

t

1

095

09

085

08

075

07

065

06

0550 05 1 15 2 25 3

Time (s)








(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

(b)


5 Conclusions




Disclosure


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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