multiphase.pdf
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Multiphase Flow Modelling Dr. Dr. Gavin Tabor
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What is multiphase flow
Multiphase flow is the flow of 2 (or more) immiscible fluids, or a fluid and a solid component. Examples include : •
Solid particles in air particulate polutants coal dust combustion particle separation fluidized beds •
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Liquid droplets in water/air emulsions, food (eg. mayonaise) diesel combustion •
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Gas bubbles in liquid bioreactors food (eg. ice cream) •
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Solid particles in water slurry flow, hydrotransport, sedimentation •
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Free surface flow marine applications (ship design) sloshing (tanks) free surface channel flow •
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Physics questions
Two phase flow consists of a dispersed phase (droplets/particles/bubbles) intermingled in a continuous phase (gas/liquid). dispersed phase
C o n t i n u o u s P h a s e
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. . . or a macroscopic interface
Physics questions
Several questions arise : How big are the dispersed phase particles? (Also – shape, variations in size etc) How dense are they? (express as a phase fraction – ratio of volumes occupied to total volume of cell) How do they interact with each other? How do they interact with the continuous phase? ....
Modelling techniques
We will look at 3 different approaches – suitable for different flow regimes 1. Lagrangian particle tracking – follow individual particles (or groups) 2. Eulerian 2 phase flow – treat both phases as fluids. 3. Free surface modelling – VOF
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Particle tracking
Applicable for low phase fractions – usually solid particles in air (eg. coal particle combustion) or fluid droplets in air (eg. diesel spray combustion). We know how to solve the NSE to find the motion of the continuous phase. Calculate the hydrodynamic force on that particle, and apply NII to find its tragectory. dvi = f (ρf , ρ p , d , µf , g , ui , u˙i , vi ) dt ....
Equation of Motion
the Basset-Boussinesq-Oseen (BBO) model. We can integrate this numerically to provide the trajectory for each particle. Note : •
Only valid for d local turbulence length scale
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Can be extended to include lift forces
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In some regimes (eg. ρf /ρ p can be neglected
∼
10−3 ) several terms
Stokes number Important scaling parameter in fluid/particle flows : ratio of particle response time τ R to characteristic fluid motion time τ F : τ R St = τ F
For Stokes drag this can be evaluated : ρ p d2 U St = 18µL ....
for characteristic fluid length/velocity scales L, U .
Fluid/particle coupling
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St 1 – particles follow flow exactly St 1 – particles unaffected by continuous phase flow
If τ F is a turbulent scale, then St 1 implies the particles will move with the turbulent motion. However the turbulence will often be modelled (k − model). Need to introduce (often stochastic) model to account for effect of turbulence on particles. ....
Particle/fluid coupling
Particles will also affect the turbulence – either enhance it or dampen it. Distinguish between 1-way and 2-way coupling. •
1-way : fluid affects particles
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2-way : particles also affect fluid
As phase fraction increases, particles can also affect the large-scale mean motion – Einstein correction to visocsity. ....
Higher phase fractions
Individual particle tracking becomes impossible for high phase fractions of small particles – too many particles to track. Track groups of particles – statistical approach. Alternative : Eulerian two-phase flow modelling. A fluid is composed of particles (molecules) but we can treat it as a continuum. Why not model the dispersed phase as a second fluid? ....
Apply this to : liquid/liquid, gas/liquid and solid/liquid flows.
Conditional averaging
Start by defining an indicator function γ , which takes the values γ (x) = 1 if x is in phase a, and
0 if not
We include this in our standard averaging operation. For a quantity φ : αφa = ....
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∆t
γφ dt
Averaged NS Equations
Apply this averaging to the NSE, eg. momentum equation : ∂αua + ∇.αua ua + ∇.αua u a ∂t 1 = − ∇ pa + ν ∇2 αua + interface terms ρa
We also find αa + αa = 1
a set of equations for the continuous phase, and for the dispersed phase. →
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Eulerian 2-phase modelling
We can solve the continuous phase equations, plus the equation αa + αa = 1 plus some model for the dispersed phase. Dispersed phase model can be algebraic, or solve the NSE for this phase. Cond. averaging generates interface terms – represent effect of one phase on the other! ....
Points
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Modelling more complex, but provides mathematical framework to fit into. Eg. need near-wall model Turbulence modelling required. Continous phase – create phase-weighted k − model Dispersed phase – turbulence is some fraction C t of cont. phase . . . but what is dispersed phase turbulence? •
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Numerical problems, particularly with αa + αa = 1
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Much wider range of applicability – can account for : high phase fraction, phase inversion, droplet breakup/coalescence •
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Free surface flow Free surface flow ≡ 2 immiscible fluids separated by an interface. Of importance in : •
Investigating bubble/droplet behaviour
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Slug flow (very large bubbles)
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Large-scale interfaces – ship wakes, ink jets, channel flows etc.
4 basic methods used. We will look at one – Volume Of Fluid (VOF). ....
Modelling overview
If the density ratio is large (air/water) : 1. Particle-based methods (SPH, cellular automata, lattice boltzmann) All other methods solve NSE in both phases, and explicitly follow the position of the interface : 3. Explicit parameterization of the surface 4. Create an indicator function – VOF methods 5. Create a general function G for which G = 0 represents the interface – level set methods ....
VOF Method In the VOF method the indicator function α takes value 1 in one phase and 0 in the other. We have a continuity equation : ∂α + ∇.αu = S α ∂t
and of course αa + αa = 1
The velocity u comes from solving the NSE for the mixture (not individual components). ....
Indicator function The indicator function represents the position of the interface and is advected by the flow. 0
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Computationally the interface smeared over 3-4 cells ....
Reconstruction Reconstructing the interface not trivial : 0
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4 schemes in Fluent – geometric, donor/acceptor, Euler explicit and implicit. ....
Mixture Given the phase fraction α, construct mixture properties as µm = αµa + (1 − α)µb
Solve NSE for velocity, pressure ∂u 1 + ∇.uu = − ∇ p + ν t ∇2 u ∂t ρ
and allocate this u to each component in each cell ....
Additional problems
Note that if there are large differences between properties of the two phases, accuracy may be limited. May need to include surface tension. Important groups Capillary number Weber number ....
µU Ca = σ ρLU 2 We = σ
for Re 1 for Re 1
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