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. The SIP – project (strategic institute project). Joint project between UiO and IFE, financed by The Research Council of Norway. 4-yrs, start 2005

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The sip project strategic institute project l.jpg
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)

Droplet distribution and entrainment l.jpg
Droplet distribution and entrainment

  • Simulation by Jie Li from Stephane Zaleski’s web-site

Droplets in turbulence two phase l.jpg
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

Droplet transport three phase l.jpg
Droplet transport (three-phase):

Concentration profiles

Mean velocity profile

Droplet mass fluxes =

Concentration profiles x Velocity profile




  • Additional liquid transport

Droplet concentration profiles depend on l.jpg
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)



Modelling l.jpg

  • 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

The ensemble averaged boltzmann equation l.jpg
The ensemble averaged ”Boltzmann equation”

Conservation equation for the ensemble averaged PDF <W> (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)

Conservation equations for particle gas in 1d stratified turbulent stationary flow l.jpg
Conservation equations for particle gas, in 1D stratified turbulent stationary flow


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.

Rewrite momentum balance for stationary flow vertical mass flux balance l.jpg
Rewrite momentum balance for stationary flow -> 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


Gravitational flux

Note: Must solve for kinetic stress, before particle density is solved for

Test against particle water data l.jpg
Test against mass flux balanceparticle – 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

Vertical profiles re 43000 no added mass effect l.jpg
Vertical profiles, Re=43000, mass flux balanceno added mass effect

Vertical profiles re 43000 added mass in diffusivity l.jpg
Vertical profiles, Re=43000, mass flux balanceadded mass in diffusivity

Vertical profiles re 43000 also calculated normal stress l.jpg
Vertical profiles, Re=43000 mass flux balancealso calculated normal stress

Vertical profiles re 43000 added mass not accounted for l.jpg
Vertical profiles, Re=43000 mass flux balanceadded mass not accounted for

Conclusions l.jpg
Conclusions mass flux balance

  • 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