1. Modelling of the particle suspension in turbulent pipe flow
Ui0 23/08/07
Roar Skartlien, IFE

2. 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)

3. Droplet distribution and entrainment
Simulation by Jie Li et.al. from Stephane Zaleski’s web-site

4. 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

5. Droplet transport (three-phase):
Concentration profiles
Mean velocity profile
Gas
Oil
Water
Droplet mass fluxes =
Concentration profiles x Velocity profile
Additional liquid transport

6. 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

7. 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

8. 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)

9. 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.

10. 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
Turbophoresis
Gravitational flux
Note: Must solve for kinetic stress, before particle density is solved for

11. 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

12. Vertical profiles, Re=43000, no added mass effect

13. Vertical profiles, Re=43000, added mass in diffusivity

14. Vertical profiles, Re=43000 also calculated normal stress

15. Vertical profiles, Re=43000 added mass not accounted for

16. 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