A Survey of Turbulence in Pipe Flows with Suspensions, Slurries, and Fluidized Beds

with Applications to Tubular Metal-Air Fuel Cells

Denis VasilescuIllinois Institute of Technology

May 5th 2011

● Suspensions are mixtures of a base fluid with very fine particles, indistinguishable to the naked eye, and low volume fractions. The laminar flow profile is not significantly affected.

● Slurries are like suspensions, but the particles may be coarser and/or have a higher volume fraction. Slurry flows blunt the velocity profile and can be non-Newtonian.

● Fluidized beds are like slurries, but the particles do not inhabit the entire flow domain, rather they collect at the bottom and slide along the wall. Very complicated.

What am I talking about?

● Grains, concrete mix, shale oil, and chemical powders are transported via pipes and ducts throughout product life. Minimizing hydraulic losses can lead to substantial energy savings

● Unwanted particles in a flow can damage pipes through various erosion mechanisms. Maintenance and repair can be reduced by mitigating particle erosion.

● For electric vehicles, metal-air fuel cells with tube geometries may be able to use turbulence in order to enhance power density, and thus car performance, increasing competitiveness

Why does this matter?

● Laminar Flow -Particle Attrition

● Transition to Turbulence

-Matas Experiment

-Effect of Size and Volume Fraction

-Interpretation

● Turbulent Flow

- Kolmogorov scale dependencies

- Eskin numerical experiment for dissipation

- Mixing length as verification

Outline of Topics

Some typical concerns and their principle physics: ● Hydraulic loss

- Deposition: porous media or moving beds

● Wall erosion

- Fracture mechanics: surface free energy

● Particle attrition & chemical reaction rate

- Molecular diffusion: shrinking core model

- Convection: corrosion-erosion model

- Solid dynamics: Particle and wall collisions

Laminar Flow

Some typical concerns and their principle physics: ● Hydraulic loss

- Deposition: porous media or moving beds

● Wall erosion

- Fracture mechanics: surface free energy

● Particle attrition & chemical reaction rate

- Molecular diffusion: shrinking core model

- Convection: corrosion-erosion model

- Solid dynamics: Particle and wall collisions

Laminar Flow

● The onset of intermittent turbulent behavior is demarcated by a critical Reynolds number, which is ~2,100 for pure pipe flows.

● The effect of particles on the transition is not as well studied as their effect on fully turbulent flows. Principle contributions from Matas et al @ IUSTI France and Vlasak et al @ ASCR

Transition

● [Matas 2003] experimental set up:

Transition

● Identify onset of transition from pressure fluctuations

● Measure critical Reynolds number based on flow rate

● Measured p' using electronic manometers:

Transition

● Determine when p' spectra gains energy in nonzero frequencies

● Critical Re taken as midpoint between laminar Re and 'strongly intermittent' Re

● Same method in [Vlasak 2004]

Transition

● Critical Re as function of size and vol. fraction:

Transition

● Smallest two diameters collapse on same curve

● Largest two diameters only collapse for very high volume fractions

● Critical Re as function of size and vol. fraction:

Transition

● Smallest two diameters monotonically increase, delaying transition

● Largest two diameters dip for very small concentrations, hastening transition

● Intuitive reasoning for delaying onset of turbulence: particles increase the effective viscosity of the fluid

μeffective = μ (1 – φ / φm)-1.82

where φm is the maximum spherical packing factor, 0.68.

● Is this effective viscosity hypothesis true?

Scale the curve by viscosities to find out

Transition

■ Critical Re plot scaled by μ /μeffective

Transition

● Smallest two diameters now return critical Re back to ~2,100

● Largest two diameters still hasten transition, effective viscosity does not tell the whole story

● High volume fraction behavior expected; asymptotically, the pipe becomes clogged with particles, becoming a porous media flow which increases friction forces.

● Low volume fraction behavior for particles of large enough size may be tripping the transition by providing just the right amplitude and wave number disturbance.

● What differentiates small versus big diameter behavior either related to D/d ratio or to d/lkol

Transition

● The unifying idea in turbulence is the Kolmogorov energy cascade. For particle flows, the key to understanding their impact is to understand what scales they interact with.

● As was seen in [Matas 2003], slurry behavior changed when particle sizes approached the order of Kolmogorov length scale. Was this a coincidence or no?

● It was also seen that volume fraction changed the effective viscosity of the slurry; the Kolmogorov length scale is based on the effective viscosity

Turbulent Flow

● [Eskin 2004] investigated the matter. First they used an alternative model for effective viscosity:

νeffective = νliquid(ρliquid/ρslurry)(1+2.5 φ + 10 φ2 + 0.0019 e20φ)

● This model was then substituted into

lkol = 4√νeffective3/ε

● Note that [Eskin 2004] believed the effective viscosity model to be applicable for all slurry concentrations based on a Bagnold number estimate.

Turbulent Flow

■ Analytical Kolmogorov scale according to volume fraction (sand, ε = 1)

Turbulent Flow

● Kol. length scale increases with volume fraction, and exponentially for high values.

● This is independent of size, thus one can envision situations where the particles are significantly smaller than the Kol. length scale

● [Eskin 2004] uses [Schook and Roco 1991] model for one-dimensional steady-state turbulent slurry flow:

where τvisc = μslurrydu/dr and τturb = d/dr [αslurry u2]

● Although this model has an analytic solution, [Eskin 2004] solves it numerically to get other quantities and relations, with the promise that the largest disagreement between this numeric model's results and experiments cited in [Schook and Roco 1991] was 15% among the pressure gradients.

Turbulent Flow

■ Energy Dissipation Contribution due to Turbulence According to Slurry Volume Fraction

Turbulent Flow

● Increased concentration reduces turbulence's contribution to energy dissipation

● It was seen earlier that increased volume fraction enlarges the Kol. length scale; smallest eddies transfer energy as kinetic motion to particles rather than as heat!

■ Mixing Length (Prandtl hypothesis for Newtonian fluid) versus radial distance

Turbulent Flow

● Not shown: mixing length was found to be independent of volume fraction

● Independence from vol. fraction and distribution of mixing lengths validate that the model is capturing the physics and the previous graph can be more or less accepted

● Particle concentration delays the onset of transition in general, but for large sizes may hasten it by tripping the flow with Goldilocks perturbations

● When particles are at or below the Kolmogorov length scale, they cannot produce meaningful perturbations and only influence the effective viscosity

● When particles are at or below the Kolmogorov length scale, the smallest eddies no longer occupy the space between particles and dissipate heat, but the particles are swept in their motions and transfer the dissipation into kinetic motion

Summary

Conclusion

Questions?

● Matas, J.P. et al. “Influence of Particles on the Transition to Turbulence in Pipe Flow” Philosophical Transactions from the Royal Society 361. 2003. pp 911-919.

● Vlasak, P. et al. “Laminar and Tyrbulent Transition of Fine-Grained Slurries” Particulate Science and technology 22: 189-200. 2004.

● Eskin, D. et al. “On a Turbulence Model for Slurry Flow in Pipelines” Chemical Engineering Science 59 (2004) pp 557-565.

References