Modelling of the particle suspension in turbulent pipe flow

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Modelling of the particle suspension in turbulent
pipe flow

• Ui0 23/08/07
• Roar Skartlien, IFE

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The SIP project (strategic institute project)

• Joint project between UiO and IFE, financed by
  The Research Council of Norway. 4-yrs, start 2005
• Main goal Develop models for droplet transport
  in hydrocarbon pipelines, accounting for
  inhomogeneous turbulence
• UiO Experimental work with particle image
  velocimetry (David Drazen, Atle Jensen)
• IFE Modelling (Roar Skartlien, Sven Nuland)

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Droplet distribution and entrainment

• Simulation by Jie Li et.al. from Stephane
  Zaleskis web-site

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Droplets in turbulence (two-phase)
Wall film with capillary waves
Entrainment and deposition of droplets
Turbulent gas

• Mean shear
• Inhomogeneous turbulence
• Interfacial waves

Turbulent fluid
Turb. gas/fluid waves
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Droplet transport (three-phase)
Mean velocity profile
Concentration profiles
Gas
Oil
Water

• Droplet mass fluxes
• Concentration profiles x Velocity profile

• Additional liquid transport

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Droplet concentration profiles depend on

• Particle diffusivity (turbulence intensity,
  particle inertia and kinetic energy)
• Entrainment rate
• (pressure fluctuation vs. surface tension)
• Droplet size distribution
• (splitting/merging controlled by turbulence)

h
t
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Modelling

• Treat droplets as inertial particles
• Inhomogeneous turbulence
• Splitting and coalescence neglected so far
• Entrainment is a boundary condition
• Use concepts from kinetic theory -- treat the
  particles as a gas use a Boltzmann equation
  approach (Reeks 1992)
• The velocity moments of the pdf yield coupled
  conservation equations for particle density,
  momentum, and kinetic stress

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The ensemble averaged Boltzmann equation
Conservation equation for the ensemble averaged
PDF ltWgt (Reeks 1992, 1993, Hyland et. al. 1999)
Strong property of Reeks theory There is an
exact closure for the diffusion current, if the
fluctuating force obeys Gaussian statistics

• Reduces to the Fokker-Planck equation for heavy
  particles,
• which experience Brownian motion.
• In general, the motion may be considered as a
• Generalized Brownian motion (the force is
  colored noise)

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Conservation equations for particle gas, in 1D
stratified turbulent stationary flow
Friction
Turbulent source
Stress tensor component
Kinetic wall-normal stress
Particle diffusivity
Dispersion tensor components, depend only on
correlations functions of the particle force
(set up by the fluid). Here Explicit forms in
homog. approx.
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Rewrite momentum balance for stationary flow -gt
Vertical mass flux balance
Particle diffusivity
Diffusion due to fluid
Particle kinetic stress
Particle relaxation time
Gravity corrected for buoyancy and added mass
Particle density
Turbulent diffusion
Turbophoresis
Gravitational flux
Note Must solve for kinetic stress, before
particle density is solved for
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Test against particle water data

• Experiments conducted by David Drazen and Atle
  Jensen. Water and polystyrene in horizontal pipe
  flow, 5 cm diameter
• Use Reeks kinetic theory
• Input profiles for fluid wall-normal stress and
  fluid velocity correlation time
• Output particle concentration profile and
  particle wall-normal stress

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Vertical profiles, Re43000,no added mass effect
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Vertical profiles, Re43000, added mass in
diffusivity
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Vertical profiles, Re43000also calculated
normal stress
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Vertical profiles, Re43000added mass not
accounted for
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Conclusions

• The study of turbulent transport of droplets in
  (inhomogeneous) turbulence is experimentally (and
  theoretically) difficult, so
• The PIV-experiments are initiated for water laden
  with polystyrene particles, to test and develop
  theory and experimental method
• Modelling need to include added mass effect for
  current experiments. May need to consider
  particle collisions in dense regions (near pipe
  floor)
• Droplets in gas no added mass effect kinetic
  model less complicated. Next step use glass
  particles in water
• Droplets in gas gas turbulence model (Reynolds
  stress) accounting for gas-fluid interface is
  needed