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

• 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

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

Droplets in turbulence (two-phase):

Turbulent fluid

Turbulent gasEntrainment and deposition of droplets

Wall film with capillary waves

•Mean shear •Inhomogeneous turbulence•Interfacial waves

Turb. gas/fluid + waves

Droplet transport (three-phase):

Droplet mass fluxes = Concentration profiles x Velocity profile

Water

Oil

Gas

Mean velocity profileConcentration profiles

•Additional liquid transport

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)

t

h

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

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

Dispersion tensor components, depend only on correlations functions of the particle force (set up by the fluid).Here: Explicit forms in homog. approx.

Stress tensor component

Friction Turbulent source

Particle diffusivity

Kinetic wall-normal stress

Rewrite momentum balance for stationary flow -> Vertical mass flux balance

Turbulent diffusionTurbophoresis

Gravitational flux

Particle density

Particle diffusivity

Particle relaxation time Gravity corrected for buoyancy and added mass

Particle kinetic stress

Diffusion due to fluid

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

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

Vertical profiles, Re=43000,no added mass effect

Vertical profiles, Re=43000, added mass in diffusivity

Vertical profiles, Re=43000also calculated normal stress

Vertical profiles, Re=43000added mass not accounted for

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