Handbook of Multiphase Flow Science and Technology

444 Pages·17.476 MB·English

by  Guan Heng Yeoh (eds.)

Basic Theory and Conceptual Framework
of Multiphase Flows
Guan Heng Yeoh and Jiyuan Tu
Abstract
The fundamentals of computational multiphase fluid dynamics are presented
usingdiscreteandcontinuumframeworks. Dependingonthenumber,type, and
size of phases and their interaction between phases within the flow system, a
multiscaleconsiderationofthemultiphaseflowphysicsallowstheadoptionofa
number of possible approaches. The Lagrangian formulation can be utilized to
track the motion of discrete constituents of identifiable portion of particular
phases occupying the flow system. This represents the most comprehensive
investigation that can be performed to analyze the multiphase flow physics.
Becauseofthecomplexityofthemicroscopicmotionsandthermalcharacteristics
ofeachdiscreteconstituentwhichcanbeprohibitiveatthe(macro)devicescale,
theEulerianformulationwhichcharacterizestheflowofdiscreteconstituentsasa
fluidcanbeadoptedforpracticalanalysisoftheflowsystem.Thisresultsinthe
development of a multifluid approach which solves for the conservation equa-
tions of continuous and dispersed phases. In order to better resolve the micro-
physics of the discrete constituents at the mesoscale, population balance allows
thesynthesizationofthebehavioranddynamicevolutionofthepopulationofthe
discrete constituents occupying the flow system. Such an approach allows the
considerationofthespatialandtemporalevolutionofthegeometricalstructures
as a result of formation and destruction of agglomerates or clusters through
G.H.Yeoh(*)
SchoolofMechanicalandManufacturingEngineering,UniversityofNewSouthWales,Sydney,
NSW,Australia
AustralianNuclearScienceandTechnologyOrganisation(ANSTO),KirraweeDC,NSW,Australia
e-mail:[email protected]
J.Tu
SchoolofAerospace,MechanicalandManufacturingEngineering,PlatformTechnologies
ResearchInstitute(PTRI),RMITUniversity,Melbourne,VIC,Australia
e-mail:[email protected]
#SpringerNatureSingaporePteLtd.2016 1
G.H.Yeoh(ed.),HandbookofMultiphaseFlowScienceandTechnology,
DOI10.1007/978-981-4585-86-6_1-1
2 G.H.YeohandJ.Tu
interactions between the discrete constituents and, more importantly, the colli-
sionswithturbulenteddies.
Keywords
Multiphase flows (cid:129) Lagrangian framework (cid:129) Eulerian framework (cid:129) Population
balance
Contents
Introduction....................................................................................... 2
Gas-SolidFlows............................................................................... 2
Liquid-SolidFlows............................................................................ 3
Gas-LiquidFlows............................................................................. 3
Three-PhaseFlows............................................................................ 4
DefinitionsofBasicTerms....................................................................... 4
Multi-ScaleConsiderationofMultiphaseFlows................................................. 6
LagrangianFormulation.......................................................................... 6
ModelsforParticulate-ParticulateInteraction................................................ 8
ClassificationofParticulate-ParticulateForces............................................... 13
ClassificationofFluid-ParticulateForces.................................................... 18
EulerianFormulation............................................................................. 21
InterpenetratingMediaFramework........................................................... 22
DerivationofGoverningEquations.......................................................... 23
PopulationBalanceApproach................................................................ 35
Summary.......................................................................................... 42
References........................................................................................ 43
Introduction
Multiphaseflowsinthecontextoffluidmechanicscanbeperceivedasaflowsystem
thatconsistsoftwoormoredistinctphasesflowinginafluidmixturewherethelevel
of separation between the phases is at a scale well above the molecular level.
Principally, multiphase flows can be categorized by the number, type, and size of
phases and their respective interaction between each phase in the fluid mixture. A
multiphaseflow system can thus be classified according to different types offlows
depending on the combination of different phases within the fluid mixture. The
physicalunderstandingofmultiphaseflowsespeciallywhendealingwithmorethan
onephaseinthefluidmixtureoffersproblemsofcomplexitythatareimmeasurably
fargreaterthaninsingle-phaseflows.Thesephasesgenerallydonotuniformlymix,
andtheprevalenceofsmall-scaleinteractionsoccurringbetweenthephasescanhave
profound effects on the macroscopic properties of the flow system. This clearly
reflectstheubiquitouschallengesthatpersistwhenmodelingtheinherentlycomplex
natureofthemultiphaseflowphysics.
BasicTheoryandConceptualFrameworkofMultiphaseFlows 3
Gas-Solid Flows
Gas-solidorgas-particleflowsconcernthemotionofsuspendedsolidparticlesinthe
gas phase. When the particle number density is relatively small, the gas flow
influences the flow of solid particles. Such flows are governed predominantly by
the surface and body forces acting on the solid particles. These types of flows are
generally known as dilute gas-particle flows. When the particle number density is
very small, these solid particles are mere tracers in the gas phase. These types of
flows are commonly known as very dilute gas-particle flows. When the particle
number density is sufficiently large, particle-particle interactions now govern the
motion of solid particles. Collisions that exist between solid particles will signifi-
cantlyalterthemigrationoftheseparticleswithinthegasphase.Thesetypesofflow
are referred to as dense gas-particle or granular flows. For gas-particle flows in
conduits, the motion of solid particles following impact on the boundary walls is
affected by the surface characteristics and material properties, which is different
when compared to the free flight of solid particles in the gas phase. In gas-particle
flows,thesolidparticlesconstitutethedispersedordiscretephaseandthegasisthe
continuousphase.
Liquid-Solid Flows
Liquid-solidflowscomprisethetransportofsolidparticlesintheliquidphase.Such
flowsaredrivenlargelybythepresenceofpressuregradientssincethedensityratio
betweenthetwophasesisgenerallylowandthedragbetweenthephasesconstitutes
the dominant effect in such flows. These types of flows are generally known as
liquid-particle flows or slurry flows. In liquid-particle flows, the solid particles
constitute the dispersed phase and the liquid represents the continuous phase. The
majorconcerninsuchflowsisthecharacterizationofthesedimentationbehaviorin
theliquid-particlemixturewhichisgovernedbytherangeofsizeofsolidparticles
travelingintheliquidphase.
Gas-Liquid Flows
Gas-liquidflowscanexistinanumberofdifferentforms–themotionofgasbubbles
intheliquidphaseorthemotionofliquiddropletsinthegasphase.Fortheformer,
the liquid is taken as the continuous phase and the gas bubbles are considered as
discreteconstituentsofthegasphaseorthedispersedphase.Forthelatter,thegasis
regarded as the continuous phase and the droplets are taken as the finite fluid
particles of the liquid phase or the dispersed phase. Since gas bubbles or liquid
droplets are allowed to deform within the continuous phase, several different
geometrical shapes are possible which include spherical, elliptical, distorted, toroi-
dal,cap,andothercomplexconfigurations.Gas-liquidflowsundergoaspectrumof
flowtransitionregimes.Changeofinterfacialstructuresbetweenthephasesaredue
4 G.H.YeohandJ.Tu
tobubble-bubbleinteractionsviacoalescenceandbreakupofbubblesandanyphase
changeprocessfromgas-to-liquidorliquid-to-gas.Forthespecialcaseofseparated
flowssuchasliquidfilmingasphaseorgasfilminliquidphaseandliquidcoreand
gas film or gas core and liquid film in conduits, such flows possess well-defined
interfaces, and they belong to the specific consideration of immiscible flows. Each
phaseistreatedasacontinuousfluidco-flowingsimultaneouslywitheachother.
Three-Phase Flows
Three-phase gas-liquid-solid flows are encountered in a number of engineering
applications of technical relevance. Principally, this particular class of multiphase
flowsconsidersthesolidparticlesandgasbubblesasbeingthediscreteconstituents
ofthedispersedphaseco-flowingwiththecontinuousliquidphase.Thecoexistence
ofthethreephasesconsiderablycomplicatesthecomputationalmodelingduetothe
requirementofunderstandingthephenomenaassociatedwithparticle-particle,bub-
ble-bubble, particle-bubble, particle-fluid, and bubble-fluid interactions modifying
themultiphaseflowphysics.
Definitions of Basic Terms
Some basic definitions that are fundamental to the description of multiphase flows
are introduced herein. For convenience, the notation that will be adopted corre-
spondstotheCartesiantensorformat.Thelowercasesubscripts(ijk)areemployedin
theconventional mannertodenote vector ortensorcomponents. Thesingleupper-
case subscript (n) refers to the property of a specific phase. In general, the generic
subscripts n = c (continuous liquid), n = d (dispersed or discrete phase), n = l
(liquid), n = g (gas), and n = s (solid) are employed for clarity in depicting the
differentclassesofmultiphaseflows.
Specificpropertiesfrequentlyencounteredareasfollows.Thevolumefractionof
thecontinuousphasecanbedefinedas
δV
α ¼ lim c (1)
c δV!δVo δV
whereδV isthevolumeofthecontinuous phase inthevolumeofδV.Thevolume
c
δVo represents the limiting volume whereby a stationary averaging is performed.
Equivalently,thevolumefractionofthedispersedordiscretephasecanbewrittenas
δV
α ¼ lim d (2)
d δV!δVo δV
whereδV isthevolumeofthedispersedordiscretephaseinthevolumeofδV.This
d
volumefractionisalsoreferredtoasthevoidfractionwhichdescribestheportionof
BasicTheoryandConceptualFrameworkofMultiphaseFlows 5
thechannelorpipeoccupiedbythedispersedgasphaseatanyinstantinspaceand
time.Inthechemicalengineeringterminology,itisknownasholdup.Forthecaseof
two-phaseflow,itfollowsthatα = 1 (cid:2) α .Hence,thesumofthevolumefractions
d c
of different constituents in a multiphase mixture must be equal to unity, i.e., P α
n n
¼1.
Themixturedensitycanbeevaluatedinaccordancewith
X
ρ¼ α ρ (3)
n n n
Thebulkdensityofthedispersedphaseisrelatedtothematerialdensityby
ρ ¼α ρ (4)
d d d
ofwhichthematerialdensity,intermsofalimit,isdefinedas
δM
ρd ¼limδV!δVo δVd (5)
withδM beingthemassofthedispersedphase.Thebulkandmaterialdensitiesof
d
thecontinuousphaseareanalogouslydefinedinaccordancewiththedefinitionofthe
bulkandmaterialdensitiesofthedispersedphase.Conversely,thespecificenthalpy,
h, and specific entropy, s, in terms of per unit mass are weighted similar to the
mixturedensitytothefollowing:
X
ρh¼ α ρ h (6)
n n n n
X
ρs¼ α ρ s (7)
n n n n
It should be noted that properties such as mixture viscosity or thermal conduc-
tivitymaynotbeobtainedthroughsuchsimpleweightedaveraging.Othermeansof
evaluatingthesepropertiesarerequired.
The true velocities (or actual local velocities) of the different phases are the
velocities the phases actually travel, which is the local instantaneous velocities of
the fluids. Defining u and u to be the local instantaneous or phase velocities of
c d
continuousand dispersed phases, thesuperficial velocities and thephasevelocities
arerelatedbythevolumefractionaccordingto
U ¼α u (8)
c c c
U ¼α u (9)
d d d
For the case of two-phase flow, the total superficial velocity is U = U + U .
c d
Hence, the total superficial velocity for a multiphase mixture can be analogously
writtenasU ¼P α u .
n n n
6 G.H.YeohandJ.Tu
Ofmorespecificdefinitions,thequalityofamixturecomprisingliquidandvapor
isdefinedby
ρ
X ¼ d (10)
ρ
whilethedispersedphasemassconcentration,whichistheratioofthemassofthe
dispersedphasetothatofthecontinuousphaseinamixture,isgivenas
ρ
C¼ d (11)
ρ
c
Multi-Scale Consideration of Multiphase Flows
Within the conceptual framework of computational multiphase fluid dynamics, the
Eulerian or Lagrangian formulation of multiphase flows requires the multiscale
consideration of themultiphase flow physics. Thedifferent physical characteristics
at different length scales are illustrated in Fig. 1. For the microscale physics, it is
paramountthatinteractionofthegasbubbles,liquiddrops,andsolidparticleswith
the continuum phase is properly understood through tracking the motion of the
individual discrete constituents in space and time. For the mesoscale physics,
significantinteractionbetweenthediscreteconstituentsmayresultinlocalstructural
changes due to agglomeration/coalescence and breakage/breakup processes of gas
bubbles,liquiddrops,andsolidparticles.Forthemacroscalephysics,thehydrody-
namicbehaviorofthebackgroundfluidontheclustersofgasbubbles,liquiddrops,
and solid particles may yield large scale flow structures influencing the different
individualphasesco-flowingwiththecontinuousphasewithintheflowsystem.
Computational approaches can be utilized to reveal details of particular multi-
phase flow physics that otherwise could not be visualized through experiments or
clarify specific accentuating mechanisms that are consistently being manifested.
Techniques that are being adopted based on the utilization of advanced numerical
methodsandmodelsusuallycontainverydetailedinformation,producinganaccu-
raterealizationofthefluidflow.
Lagrangian Formulation
The concept of Lagrangian tracking entails following the motion of individual
identitiesoftheidentifiableportionofaparticularphaseoccupyingtheflowsystem.
Suchconsiderationthereforeincludesmoleculardynamics,Browniandynamics,and
discreteelementmethod,whichisrepresentedbyanillustrationofaplotdepicting
thecharacteristictimescaleversuslengthscaleinFig.2.Ingeneral,theseidentities
being considered represent a wide range of discrete elements including atoms,
molecules,nuclei,cells,aerosolorcolloidalparticles,andgranules.
BasicTheoryandConceptualFrameworkofMultiphaseFlows 7
Micro-Scale Meso-Scale Macro-Scale
Interactionsof Formation of Hydrodynamic
Discrete Elements Clusters of Bubbles, Behaviour of Phases
with Continuum Fluid Drops and Particles at Device Scale
Motion of Local Structural Large Scale
Discrete Changes Flow Structures
Elements
Length Scale
Fig.1 Multiscaleconsiderationofmultiphaseflows(Yeohetal.2014)
Fig. 2 Illustration of time and length scales for Lagrangian simulations: molecular dynamics,
Browniandynamics,anddiscreteelementmethod(Yeohetal.2014)
Moleculardynamics,Browniandynamics,anddiscreteelementmethodsharethe
commoncharacteristics wherebythe discrete elements are allowed tointeract for a
periodoftimeunderprescribedinteractionlaws,andthemotionofeachindividual
element is resolved by solving the linear momentum and angular momentum
equations, subject to forces and torques arising both from particle interaction with
each other and those imposed on the particles by the surrounding fluid (Li et al.
2011). Consideration of which Lagrangian tracking approach should be adopted
8 G.H.YeohandJ.Tu
stems primarily through the formulation of appropriate particle interaction laws as
wellasbytheimpositionofrandomforcingtomimiccollisionsorinteractionwith
moleculesofthesurroundingfluid.Bycarefullyidentifyingtheseparticleinteraction
laws, the physical behavior of the discrete element at a certain range of time and
lengthscalescanbeefficientlycaptured.
Consideringthediscreteelementswhichcouldbesolidparticles,liquiddroplets,
or gas bubbles, the instantaneous velocity Ud and rotation rate Ωd for the discrete
particulate (particle, droplet, or bubble) through the time-driven discrete element
method can be obtained through solution of the linear and angular momentum
equations(Newton’ssecondlaw):
m DUd ¼XF (12)
d d
Dt
I DΩd ¼XM (13)
d d
Dt
wherem isthemassofparticulate,andI isthemomentofinertiaofparticulate.On
d d
therighthandside,thetimederivativesareessentiallythematerialderivativesofthe
particulate velocity and particulate rotation rate. On the left hand side, the source
terms represent the sum or cumulative of forces and moments acting on the
particulate.
For multiphase flow associated with heat exchange and phase change between
particulateandsurroundingfluid,theseheatandmasstransferprocessesareconcur-
rentlytrackedalongthediscreteparticulatetrajectoriesandsolvedbytheparticulate
conservationequationsofmassandenergy:
Dm
d ¼S (14)
Dt md
DT
d ¼S (15)
Dt Td
where T is the temperature of the particulate. The source term S represents the
d md
interphase mass transfer between the particulate and surrounding fluid, while the
source term S is governed primarily by the interphase convection heat transfer,
Td
latentheattransferassociatedwithmasstransfer,netradiativepowerabsorbedbythe
particulate, and particulate-particulate interaction due to conduction heat transfer.
Notethattheproductoftheparticulatemass,specificheatofconstantpressure,and
materialderivativeoftheparticulatetemperaturedenotethesensibleheatingtermof
theparticulateenergyequation.
BasicTheoryandConceptualFrameworkofMultiphaseFlows 9
Models for Particulate-Particulate Interaction
Throughspecificconsiderationofusingdiscreteelementmethodforlargeparticulate
assemblies, the hard-sphere or soft-sphere model can be effectively applied. Soft
contact forces are subsequently derived from a point on the bodies for the hard-
spheremodelortheoverlapofbodiesforsoft-spheremodel.
Hard-SphereModel
Mainassumptionsthatareconcernedwiththeparticulateshape,deformationhistory
duringcollision,andnatureofcollisionsare:
(cid:129) Particulatesaregenerallytakentobesphericalandquasirigid
(cid:129) Shapeofparticulatesisretainedafterimpact
(cid:129) Dynamicsofidealizedbinarycollision
(cid:129) Collisionsbetweenparticulatesareinstantaneous
(cid:129) Contactofparticulatesduringcollisionoccursatapoint
(cid:129) Interaction forces are taken to be impulsive and all other finite forces are negli-
gibleduringcollisions
These assumptions are believed to be sufficiently realistic for collisions of
relatively coarse particulates (>100 μm). One characteristic feature of this model
is the ability to process a sequence of collisions one at a time. Also, Lagrangian
trackingofparticulatescanbereadilyperformedwithrealisticvaluesoftherestitu-
tionandfrictioncoefficients.Duringtheimpactoftwoparticlessuchasdescribedin
Fig.3,themotionisgovernedbythelinearandangularimpulsemomentumlawsfor
abinarycollisionoftwospheres:
(cid:2) (cid:3)
md,k Ud,k(cid:2)U0d,k ¼J (16)
(cid:2) (cid:3)
md,l Ud,l(cid:2)U0d,l ¼(cid:2)J (17)
Ird,k(cid:2)Ωd,k(cid:2)Ω0d,k(cid:3)¼J(cid:3)n (18)
k
Ird,l(cid:2)Ωd,l(cid:2)Ω0d,l(cid:3)¼J(cid:3)n (19)
l
where superscript 0 denotes conditions just before collision, m and m are the
d,k d,l
mass,r andr aretheradii,Ω andΩ aretheangularvelocities,I andI are
k l d,k d,l d,k d,l
themomentofinertiaofparticulatekandparticulatel.Velocitiespriortocollisions
arethevelocitiesdeterminedatthelasttimestepjustbeforecollision.Notethatthe
correspondingtimedifferenceshouldnotbelargerthan10(cid:2)4s.
InFig.3,thenormalandtangentialunitvectorsthatdefinethecollisioncoordi-
natesystemare:
10 G.H.YeohandJ.Tu
y
r
l U
r d,l
k n
U
l d,k Wd,l
k
R
Gd,kl R l
W k l
d,k
x
k t J
z
Fig.3 Contactbetweentwoparticlesforhard-spheremodel
x (cid:2)x
n¼ k l (20)
jx (cid:2)xj
k l
(cid:2) (cid:3)
t¼(cid:4)Ud,kl(cid:2)(cid:2)G0d,kl∙n(cid:3)n(cid:4) (21)
(cid:4)(cid:4)Ud,kl(cid:2) G0d,kl∙n n(cid:4)(cid:4)
By definition, I ¼mr2 where r is the radius of gyration of the
gyration gyration
particulateandJistheimpulsevector.Adopting(J (cid:3) n) (cid:3) n = J (cid:2) (J ∙ n)n,the
relative velocity at the contact point between two particulates with velocities U
d,k
andU isderivedas
d,l
Ud,kl(cid:2)U0d,kl ¼Bd,kJ(cid:2)(cid:5)Bd,k(cid:2)Bd,l(cid:6)ðJ∙nÞn (22)
where
Ud,kl ¼Gd,kl(cid:2)(cid:5)rkΩd,k(cid:2)rlΩd,l(cid:6)(cid:3)n (23)
TherelativevelocityofparticulatecentroidsG isgivenby
d,kl
Gd,kl ¼Ud,k(cid:2)Ud,l (24)
Some parameters are established to relate the velocities before and after colli-
sions.Thefirstcollisionparameteristhecoefficientofnormalrestitution,e :
n
(cid:2) (cid:3)
Ud,kl∙n¼(cid:2)en U0d,kl∙n (25)
Thenormalcomponentoftheimpulsevectorcanthusbewrittenas: